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https://www1.essex.ac.uk/maths/people/fremlin/materials/ma203notes.tex | essex.ac.uk | CC-MAIN-2018-43 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-43/segments/1539583512434.71/warc/CC-MAIN-20181019191802-20181019213302-00243.warc.gz | 1,149,616,277 | 25,749 | \input fremtex
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\filename{ma203.tex}
\versiondate{14.5.02}
\def\lectureend#1{\discrversionA{\hfill{\twelverm #1}}{}}
\def\omitted#1{\discrversionA{\hfill{\twelverm omitted #1}}{}}
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\Centerline{\bf MA203 Real Analysis}
\Centerline{\smc D.H.Fremlin}
\oldfootnote{}{These notes are made available to students on the
understanding that they have not been fully checked for errors. They
are not intended to provide a substitute for attendance at lectures.
If you notice a mistake, please tell
the lecturer!}
\medskip
This course is intended as a first introduction to the methods required
to prove the basic theorems of real analysis, particularly those
involving continuous and differentiable functions. The theorems
themselves will mostly have been presented in MA206, and should be more
or less familiar. The techniques of proof, however, are likely to be
completely new. They are difficult, and you must expect to have to
work hard.
`Analysis' is one of the traditional subjects taught in university
mathematics degrees -- necessarily so, as it underlies not only most of
modern pure mathematics but also large parts of modern applied
mathematics -- and nearly everywhere gives rise to special problems.
Many students who are quite good at other kinds of mathematics find that
analysis seems to be beyond them. I think that this is for two
separate reasons. The first, and less important, one is that analysis
deals with {\it inequalities}. It may well be that an actual majority
of the formulae in the notes below will have at least one of the symbols
$\le$, $\ge$, $<$ or $>$. These are not really difficult to handle,
but they do require technical skills which you may not yet have
practised enough. So you are going to have to spend some time working
on these. But there is a much more essential difficulty to come to
terms with. All the characteristic sentences of analysis have multiple
quantifiers. `Quantifiers' are the expressions $\Forall$, `for every',
and $\Exists$, `there is'. Of course these appear everywhere in
mathematics. But in analysis we get sentences like
\inset{`for every $\epsilon>0$ there is a $\delta>0$ such that
$|f(x)-f(x_0)|\le\epsilon$ for every $x\in[x_0-\delta,x_0+\delta]$'}
\noindent with three quantifiers
(`$\Forall\epsilon\ldots\Exists\delta\ldots\Forall x\ldots$') one after
the other; and I think some of the sentences later on may have as many
as five. These demand some special ways of thinking, which is what
this course is mostly about.
\bigskip
\noindent{\bf The real number system}
\medskip
The whole of this course will be concerned with the set $\Bbb R$ of
`real numbers'. I will therefore run as quickly as possible over the
key properties of this system. All the facts here are supposed to be
familiar, and I'm not going to prove any of them; I write them out just
so that you can make sure you really do know the things I am going to
assume from now on.
\medskip
\noindent{\bf Addition} For any two real numbers $x$ and $y$, we have a
real number $x+y$, and
\inset{$x+(y+z)=(x+y)+z$ for all real numbers $x$, $y$ and $z$;
there is a real number $0$ such that $x+0=0+x=x$ for every $x\in\Bbb R$;
for every $x\in\Bbb R$ there is a number $-x\in\Bbb R$ such that
$x+(-x)=(-x)+x=0$;
$x+y=y+x$ for all $x$, $y\in\Bbb R$.}
\medskip
\noindent{\bf Multiplication}
For any two real numbers $x$ and $y$, we have a real number $xy$, and
\inset{$x(yz)=(xy)z$ for all real numbers $x$, $y$ and $z$;
there is a real number $1$ such that $x1=1x=x$ for every $x\in\Bbb R$;
for every $x\in\Bbb R\setminus\{0\}$ there is a number $\Bover1x\in\Bbb
R$ such that $x\cdot\Bover1x=\Bover1x\cdot x=1$;
$xy=yx$ for all $x$, $y\in\Bbb R$;
$1\ne 0$.}
\medskip
\noindent{\bf The distributive laws}
\inset{$x(y+z)=xy+xz$ for all $x$, $y$, $z\in\Bbb R$;
$(x+y)z=xz+yz$ for all $x$, $y$, $z\in\Bbb R$.}
\medskip
\noindent{\bf The ordering of $\Bbb R$} We have a relation $\le$ on
$\Bbb R$ such that
\inset{$x\le x$ for every $x\in\Bbb R$,
if $x\le y$ and $y\le z$ then $x\le z$,
if $x\le y$ and $y\le x$ then $x=y$,
for every $x$, $y\in\Bbb R$, either $x\le y$ or $y\le x$.}
\medskip
\noindent{\bf $\le$, $+$ and $\times$} If $x$, $y$, $z\in\Bbb R$, then
\inset{if $x\le y$ then $x+z\le y+z$ and $x-z\le y-z$;
if $x\le y$ and $0\le z$ then $xz\le yz$;
if $x\le y$ and $z\le 0$ then $yz\le xz$.}
\medskip
This list is not quite complete, and there is one further essential fact
about real numbers (`Dedekind completeness') which will play a large
part in this course. But I will deal with it later when we can give it
the time it deserves.
\medskip
\noindent{\bf Chains of inequalities} It will often happen that we have
a string of inequalities
\Centerline{$a\le b\le c\le d\le\ldots$,}
\noindent meaning `$a\le b$ and $b\le c$ and $c\le d$ and $\ldots$'.
In this case, because $\le$ is transitive, we can say
\inset{because $a\le b$ and $b\le c$, $a\le c$,
because $a\le c$ and $c\le d$, $a\le d$,}
\noindent and so on. It sometimes happens that the chain returns to
its starting point, as in
\Centerline{$a\le b\le c\le d\le e\le a$.}
\noindent In this case, at the end, we get $a\le e$ and $e\le a$; so we
must have $a=e$. But this means that we have $a\le d$ and $d\le a$, so
$a=d$, and so on; in the end, we conclude that $a=b=c=d=e$.
\medskip
\noindent{\bf Moduli} For any $x\in\Bbb R$, we write
$$\eqalign{|x|&=x\text{ if }0\le x,\cr
&=-x\text{ if }x\le 0;\cr}$$
\noindent alternatively, we can define $|x|$ as $\max(x,-x)$, that is,
$$\eqalign{|x|&=x\text{ if }-x\le x,\cr
&=-x\text{ if }x\le -x.\cr}$$
\noindent Now the most important fact of all about moduli is the {\bf
triangle inequality}:
\Centerline{$|x+y|\le|x|+|y|$ for all $x$, $y\in\Bbb R$.}
\noindent This is not the kind of thing I am going to spend the course
teaching you how to prove, but the argument is instructive and fairly
easy, so here it is: if $x$, $y\in\Bbb R$, then
\inset{$x\le|x|$ because $|x|=\max(x,-x)$,
so $x+y\le|x|+y$, because you can add anything to an inequality,
also $y\le|y|$, so $|x|+y\le|x|+|y|$,
since $x+y\le|x|+y\le|x|+|y$, $x+y\le|x|+|y|$.
Now $|-x|=\max(-x,-(-x))=\max(-x,x)=|x|$, and similarly $|-y|=|y|$,
so $-(x+y)=(-x)+(-y)\le|-x|+|-y|=|x|+|y|$,
and since $|x+y|$ must be either $x+y$ or $-(x+y)$, we must have
$|x+y|\le|x|+|y|$.}
\noindent (The {\it proof} here is non-examinable. But the {\it fact}
$|x+y|\le|x|+|y|$ is part of the survival kit for the course.)
Two more fundamental facts about moduli are
\Centerline{$|xy|=|x||y|$ for all $x$, $y\in\Bbb R$,}
\Centerline{$|x|\ge 0$ and $|x|=0\iff x=0$.}
\medskip
\noindent{\bf Manipulating moduli} Apart from the three basic facts
above, we are going to need quite a few others. Among these are
\Centerline{$|x|-|y|\le|x-y|$ for all $x$, $y\in\Bbb R$.}
\noindent (To see this, note that $|x|=|(x-y)+y|\le|x-y|+|y|$ and
subtract $|y|$ from both sides.) In the same way
\Centerline{$|y|-|x|\le|y-x|=|x-y|$.}
\noindent But putting these together we have
\Centerline{$\bigl||x|-|y|\bigr|=\max(|x|-|y|,|y|-|x|)\le|x-y|$.}
\noindent Another on the list is
\Centerline{$|x-y|=|x+(-y)|\le|x|+|-y|=|x|+|y|$.}
A particularly important fact is that
\Centerline{if $0<x\le y$ then $\Bover1y\le\Bover1x$;}
\noindent this is because $xy$ and $\Bover1{xy}$ are both greater than
$0$, and if we multiply the inequality $x\le y$ by the positive number
$\Bover1{xy}$ we get $\Bover1y\le\Bover1x$.
We may well need some others; I'll try to point them out as they arise.
\lectureend{02/1}
\bigskip
\noindent{\bf What is a limit?}
\medskip
You should be familiar with the formula
\Centerline{$\lim_{n\to\infty}x_n=a$.}
\noindent In elementary courses in calculus this is generally
interpreted as
\Centerline{`for large enough $n$, $x_n\bumpeq a$.'}
\noindent The trouble with this way of putting it is that it doesn't
tell us how close $x_n$ has to be to $a$ to count as `approximately
equal' to $a$. In any particular application to a practical problem,
we shall probably have a definite idea of how close we need to be. If
you are measuring wallpaper, a millimeter is probably good enough; if
you are designing a computer chip, it certainly isn't. If we are
trying to devise a framework which we shall be able to use in {\it any}
circumstances, we have to leave room for someone to tell us how close
counts as close. That is, we need a parameter $\epsilon$, set by
somebody else, to specify that `$x_n\bumpeq a$' means
`$|x_n-a|\le\epsilon$'. Of course $\epsilon$ must be strictly greater
than $0$; `$|x_n-a|\le 0$' just means `$x_n=a$', and the whole idea of
the limit $\lim_{n\to\infty}x_n$ is that we don't suppose that any
particular value of $x_n$ will be exactly equal to the limit, except by
accident.
So saying that `$\lim_{n\to\infty}x_n=a$' means that
\Centerline{for any acceptable error $\epsilon>0$, $|x_n-a|\le\epsilon$
for all $n$ large enough.}
\noindent How large will `large enough' be? Of course this will depend
on $\epsilon$. But we are claiming that there is {\it some} number
$M$ such that $|x_n-a|\le\epsilon$ whenever $n\ge M$.
Let me write this out again in a more concentrated form.
\Centerline{$\lim_{n\to\infty}x_n=a$}
\noindent means
\Centerline{$\Forall\epsilon>0\Exists n_0\in\Bbb N\Forall n\ge n_0$,
$|x_n-a|\le\epsilon$.}
\noindent This is a typical definition from elementary analysis. It is
really quite difficult to get hold of. We have three quantifiers
$\Forall$, $\Exists$, $\Forall$; these have to be put in the right
order; and any variation of any symbol in the whole formula is liable
to make it wrong.
\medskip
{\bf The Analysis Game} I believe that one way, and a useful way, of
coping with these sentences is to treat them as specifying a game. The
formula
\Centerline{$\lim_{n\to\infty}x_n=a$,
\quad$\Forall\epsilon>0\Exists n_0\in\Bbb N\Forall n\ge n_0$,
$|x_n-a|\le\epsilon$}
\noindent means
\Centerline{for every $\epsilon>0$,
$\Exists n_0\in\Bbb N\Forall n\ge n_0$, $|x_n-a|\le\epsilon$.}
\noindent So if I claim that $\lim_{n\to\infty}x_n=a$, what I am saying
is that for any $\epsilon>0$ which {\it you} (not I) choose, there will
be some $n_0$ such that $|x_n-a|\le\epsilon$ for every $n\ge n_0$. Now
after you have chosen the $\epsilon$, I only have to find one $n_0$
which will work; and I (not you) have the right to point to a number
which I believe is good enough. But I am then claiming that
`$|x_n-a|\le\epsilon$ for every $n\ge n_0$', so {\it you} have the right
to challenge me by naming a particular $n\ge n_0$ and demanding that we
check that $|x_n-a|$ is indeed at most $\epsilon$.
Thus we have the idea of a {\it game} with four moves, in which
\inset{Player I says `$\lim_{n\to\infty}x_n=a$',
Player II chooses some $\epsilon$, and s/he must choose $\epsilon>0$,
Player I chooses some $n_0\in\Bbb N$,
Player II chooses some $n\in\Bbb N$, and s/he must choose $n\ge n_0$,
and at the end of these four moves, the players look at $|x_n-a|$, and
if $|x_n-a|\le\epsilon$ then Player I wins (because that's what s/he
said would happen), while if $|x_n-a|\not\le\epsilon$ then Player II
wins.}
\noindent The actual statement `$\lim_{n\to\infty}x_n=a$' is {\it true}
if, and only if, Player I can always win the game, whatever Player II
does. Of course Player I will have to take care to play the right move
when s/he comes to pick $n_0$; even if the original move
`$\lim_{n\to\infty}x_n=a$' left him/her in a winning position, s/he can
still bungle it by being careless with the second move.
\medskip
{\bf Example} Suppose Player I starts with
\Centerline{$\lim_{n\to\infty}\Bover1n=0$.}
\noindent Will s/he win? Player II can choose any $\epsilon$ s/he
likes, and it's generally good tactics to pick something small; suppose
s/he tries $\epsilon=\Bover1{1000}$. Player I does a little quick
thinking: `at the end, I shall need
$|\Bover1n-0|\le\epsilon=\Bover1{1000}$; now $|\Bover1n-0|$ is just
$\Bover1n$; how can I be sure (remembering that it's Player II who gets
to choose $n$) that s/he will choose $n$ so that
$\Bover1n\le\Bover1{1000}$? well, I'd better make sure that
$n\ge 1000$ -- can I do this? of course! I'll say that $n_0=1000$'.
So s/he does just that: $n_0=1000$. Now Player II has another move;
but whatever s/he does, s/he has to pick $n\ge 1000$, so that
$|\Bover1n-0|=\Bover1n\le\Bover1{1000}=\epsilon$, and Player I will win.
This is what happens if Player II plays $\epsilon=\Bover1{1000}$.
Could s/he have done any better? The trouble is that Player I is
liable to answer with $n_0=\Bover1{\epsilon}$. And if s/he does this,
then for any $n\ge n_0$ we shall have
$|\Bover1n-0|=\Bover1n\le\Bover1{n_0}=\epsilon$. Is there any way of
stopping him/her from playing $n_0=\Bover1{\epsilon}$? Actually, there
is, because if you look at the rules you see that $n_0$ has got to be an
integer. So if we take $\epsilon=\Bover1{1000\pi}$, Player I can't
answer with $n_0=\Bover1{\epsilon}=1000\pi$, because that's not an
integer (it's somewhere between 3141 and 3142). But this does us no
good, because Player I doesn't have to hit any particular mark exactly.
If s/he just chooses {\it some} integer $n_0\ge\Bover1{\epsilon}$ (e.g.,
$n_0=4000$ if $\epsilon=\Bover1{1000\pi}$), then for any $n\ge n_0$ we
shall still have $\Bover1n\le\Bover1{n_0}\le\epsilon$, and Player I will
still win in the end.
Note that Player I must absolutely not tell Player II what his/her
second move is going to be before Player II has committed to a
particular $\epsilon$. Because if Player I gets lazy, and just says
`$n_0=10^{100}$' without looking at what $\epsilon$ was, then Player II
can say `$\epsilon=\exp(-2^{10})$' for his/her first move and then
`$n=n_0$' for his/her second, and when they come to do the sums they
will get
\Centerline{$|\Bover1n-0|=\Bover1n=10^{-100}>e^{-300}>e^{-1024}$}
\noindent and Player II will win.
\medskip
{\bf Example} Suppose that Player I starts by saying that
\Centerline{`$\lim_{n\to\infty}(-1)^n=1$'.}
\noindent Will s/he win? Suppose that Player II again tries a
moderately small $\epsilon>0$ -- e.g., $\epsilon=\Bover1{1000}$, and
Player I answers with his favourite $n_0=10^{100}$. Now Player II has
to pick $n\ge n_0$. S/he does a bit of rough working:
$$\eqalign{|(-1)^n-1|&=|1-1|=0\le\Bover1{1000}\text{ if }n\text{ is
even},\cr
&=|(-1)^n-1|=2\not\le\Bover1{1000}\text{ if }n\text{ is odd}.\cr}$$
\noindent So Player II wants to choose an {\it odd} number $n$. Can
s/he do it? Yes: $n=10^{100}+1$ will do nicely, because the only rules
she has to look out for are $n\in\Bbb N$, $n\ge n_0$. Is there
anything Player I could have done to stop this? No, because whatever
$n_0$ s/he picks, Player II just has to choose $n=n_0$ (if $n_0$ is odd)
or $n=n_0+1$ (if $n_0$ is even) and win. So Player I scuppered his/her
chances right at the beginning by saying that $\sequencen{(-1)^n}$
converges to $1$; this is just not true.
Of course in this case, Player II didn't have to take $\epsilon$ nearly
as small as $\Bover1{1000}$. Any number strictly less than $2$ would
have done; so s/he could have started with $\epsilon=1$, for instance,
showing that Player I wasn't just wrong, but had completely
misunderstood the sequence.
\lectureend{02/2}
\bigskip
\noindent{\bf The first theorem} I will now try to show how these ideas
can be assembled into a rigorous proof of a familiar fact.
\medskip
{\bf Theorem} If $\lim_{n\to\infty}x_n=b$ and $\lim_{n\to\infty}y_n=c$
then $\lim_{n\to\infty}x_n+y_n=b+c$.
\medskip
\noindent{\bf proof} I do not think the ideas of this proof can really
make sense without going through them at least twice, once in the order
in which one remembers them, and once in the order in which one writes
them out.
We are in the position of a Player I who has announced, as her first
move,
\Centerline{$\lim_{n\to\infty}x_n+y_n=b+c$.}
\noindent The theorem says that she can win from this position. If
Player II believes that the theorem is true, I suppose he will resign.
But if he is sceptical, he will choose an $\epsilon>0$ and challenge
Player I to find a successful move in reply. If we are to be sure that
Player I really can win whatever Player II does, we are going to have to
describe a {\it strategy} for
Player I; a set of rules to tell Player I what to do in any of the
positions which Player II can manoeuvre her into. Since we have no
idea what Player II is going to say for his first move (except that it
must be to choose some $\epsilon>0$), the proof more or less has to
begin with
\inset{Let $\epsilon>0$.}
\noindent At this point, Player I is going to have to think ahead.
Because Player II has another move; and Player I has to find an $n_0$
so large that Player II's subsequent choice of $n$ is certain to lose.
So the proof is going to have to look like this:
\inset{Let $\epsilon>0$.\hfill[Player II's first move.]
$\ldots$
Take $n_0$ such that $\ldots$\hfill[Player I's second move.]}
\noindent (Maybe we shall be able to specify a formula for $n_0$; but
we can't count on this, and maybe it will have to be chosen by some
seriously complicated process.) But we know what will come next,
because we know that Player II will choose some $n\ge n_0$; and since
we have no control over this at all, we shall just have to write it out
like that:
\inset{Let $\epsilon>0$.\hfill[Player II's first move.]
$\ldots$
Take $n_0$ such that $\ldots$\hfill[Player I's second move.]
Let $n\ge n_0$.\hfill[Player II's second move.]}
\noindent At this point, the players have finished their moves, and
proceed to see who has won. Now the question they have to decide is
\inset{is $|(x_n+y_n)-(b+c)|\le\epsilon$?}
\noindent because Player I will win if the answer is `yes', and lose if
the answer is `no'.
Everything I have written so far is just a matter of understanding the
structure of the game. It's got nothing to do with the actual
sequences involved. But we have come to the crunch. In order to be
sure that
$|(x_n+y_n)-(b+c)|\le\epsilon$, we need to know something about the
numbers we're looking at. And the {\bf key fact} is that
\Centerline{$|(x_n+y_n)-(b+c)|\le|x_n-b|+|y_n-c|$;}
\noindent this is just a simple application of the triangle inequality,
because $|(x_n+y_n)-(b+c)|=|(x_n-b)+(y_n-c)|\le|x_n-b|+|y_n-c|$; the
modulus of the sum is less than or equal to the sum of the moduli. So
it will be good enough if we can find some reason to be sure that $|x_n-
b|+|y_n-c|\le\epsilon$.
Why should this be so? It's helpful to try to think about what is
supposed to be happening here. We are supposing that
$\lim_{n\to\infty}x_n=b$ and that $\lim_{n\to\infty}y_n=c$ and that
$n\ge n_0$, and presumably Player I chose $n_0$ to be large; in which
case, surely, we shall have $x_n\bumpeq b$ and $y_n\bumpeq c$ in some
sense. So maybe we can actually arrange that
$|x_n-b|\le$ (something) and $|y_n-c|\le$ (something). What
`something' should we try, if we want the sum to be at most $\epsilon$?
There are two pieces, so if we take each of them to be
$\Bover{\epsilon}2$ that ought to work.
Thus our proof (growing slowly) might have a final line
\inset{$|(x_n+y_n)-(b+c)|\le|x_n-b|+|y_n-c|
\le\Bover{\epsilon}2+\Bover{\epsilon}2=\epsilon$\hfill[Player I wins.]}
\noindent But of course we are going to have to work at this a bit.
How can we justify this idea that
\Centerline{if $n\ge n_0$, then $|x_n-b|\le\Bover{\epsilon}2$ and
$|y_n-c|\le\Bover{\epsilon}2$?}
\noindent Well, remember that we know that $\lim_{n\to\infty}x_n=b$, and
also that $\Bover{\epsilon}2>0$. So we know that there is some integer
-- call it $n_1$ -- such that $|x_n-b|\le\Bover{\epsilon}2$ whenever
$n\ge n_1$. (It might happen that the $n_1$ here will itself serve for
the $n_0$ we're looking for. But we mustn't count on it, so we had
better give it a new name.) Similarly, there is some $n_2\in\Bbb N$
such that $|y_n-c|\le\Bover{\epsilon}2$ for every $n\ge n_2$. Putting
these into the framework of the proof we are building, it looks like
this:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $n_1\in\Bbb N$ such that $|x_n-b|\le\Bover{\epsilon}2$ for
every
$n\ge n_1$.
There is an $n_2\in\Bbb N$ such that $|y_n-c|\le\Bover{\epsilon}2$ for
every
$n\ge n_2$.
Take $n_0$ such that $\ldots$.
Let $n\ge n_0$.
$\ldots$
So $|(x_n+y_n)-(b+c)|\le|x_n-b|+|y_n-c|
\le\Bover{\epsilon}2+\Bover{\epsilon}2=\epsilon$.}
What do we need to do to fill this in? Well, we are going to need some
good reason why $|x_n-b|\le\Bover{\epsilon}2$ whenever $n\ge n_0$.
However, we picked $n_1$ so that $|x_n-b|\le\Bover{\epsilon}2$ whenever
$n\ge n_1$. So if $n_0\ge n_1$, that will do fine. Next, we also
need to know that
$|y_n-c|\le\Bover{\epsilon}2$ whenever $n\ge n_0$. Since we picked
$n_2$ so that $|y_n-c|\le\Bover{\epsilon}2$ whenever $n\ge n_2$, we
shall be successful if $n_0\ge n_2$. Thus what Player I needs to do is
to choose some $n_0$ such that $n_0\ge n_1$ and $n_0\ge n_2$. The most
straightforward way of doing this is to make $n_0$ actually the greater
of the two numbers. So the proof becomes
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $n_1\in\Bbb N$ such that $|x_n-b|\le\Bover{\epsilon}2$ for
every
$n\ge n_1$.
There is an $n_2\in\Bbb N$ such that $|y_n-c|\le\Bover{\epsilon}2$ for
every
$n\ge n_2$.
Take $n_0=\max(n_1,n_2)$.
Let $n\ge n_0$.
Then $n\ge n_1$ and $n\ge n_2$, so $|x_n-b|\le\Bover{\epsilon}2$ and
$|y_n-c|\le\Bover{\epsilon}2$ and
\Centerline{$|(x_n+y_n)-(b+c)|\le|x_n-b|+|y_n-c|
\le\Bover{\epsilon}2+\Bover{\epsilon}2=\epsilon$.}
Since this works whatever $\epsilon>0$ and $n\ge n_0$ Player II chooses,
Player I is sure to win and the theorem is true.}
\medskip
{\bf Remarks} I took a great deal of time over this. The points I am
trying to make are
\inset{you don't think of proofs in the order in which you finally write
them out;
you aim to set up the {\it structure} of a proof before filling in very
much of the detail.}
\noindent It won't be clear to you yet, but the proof here has a number
of features which are so common that they're worth putting in your bag
of tricks:
\inset{It starts with `Let $\epsilon>0$'}
\noindent (because the statement of the theorem is Player I's first
move, and we have to give Player II a free choice). It does
occasionally happen that it's worth taking a bit of space to clear the
ground before we ask Player II what he wants to do, but it can never be
actually wrong to get $\epsilon$ into view before we start thinking
about Player I's strategy.
\inset{$n_0=\max(n_1,n_2)$.}
\noindent This is by no means a universal rule; but it is exceedingly
common that when we come to choose $n_0$, we just make it the biggest
integer which has yet appeared.
\inset{$|x_n+y_n-b-c|\le|x_n-b|+|y_n-c|$.}
\noindent When you want to prove that a {\it sum} of any kind is less
than or equal to something, then on most (not all) occasions it's worth
breaking it up into pieces and using the triangle inequality. Of
course it's not always obvious which pieces to use. For instance, it's
perfectly true that
\Centerline{$|(x_n+y_n)-(b+c)|\le|x_n-c|+|y_n-b|$,}
\noindent and this fact is no use at all. However, the original
hypotheses
`$\lim_{n\to\infty}x_n=b$, $\lim_{n\to\infty}y_n=c$' should be a hint
that we want to keep the $x_n$ with $b$ and the $y_n$ with $c$.
\medskip
{\bf The second theorem} I expect you feel that learning forty proofs
like the one above will make this a pretty hard term. So it would, if
you had to learn them all independently. But in fact they run so close
together that for many of them we only have to remember odd clauses in
which they differ from another one. I will try to show this by giving
a result which looks very different from the one here, but uses almost
exactly the same ideas, if we look at it in the right way.
This theorem will be the same result, but for real functions instead of
for sequences. Now there is an extra complication here, so I pause for
a pair of definitions.
\medskip
{\bf Definitions (a)} A {\bf real function} is a function $f$ such that
$\dom f\subseteq\Bbb R$ and $f(x)\in\Bbb R$ for every $x\in\dom f$.
(The point here is that a very large proportion of the important
functions of mathematics aren't defined everywhere, starting with
$\Bover1x$, undefined at $0$. We have to have some way of coping with
this, and in the last fifty years it's become generally accepted that we
should specify the {\it domain} of every function at the moment when we
introduce it.)
\medskip
{\bf (b)} If $f$ is a real function and $a\in\Bbb R$, then
`$\lim_{x\to\infty}f(x)=a$' means
\Centerline{$\Forall\epsilon>0\Exists M\in\Bbb R\Forall x\ge M,\,
x\in\dom f$ and $|f(x)-a|\le\epsilon$.}
\noindent Compare this with the definition of
`$\lim_{n\to\infty}x_n=a$':
\Centerline{$\Forall\epsilon>0\Exists n_0\in\Bbb N\Forall n\ge n_0$,
$|x_n-a|\le\epsilon$.}
\noindent The $n_0\in\Bbb N$ has turned into $M\in\Bbb R$, the $n\ge
n_0$ has turned into $x\ge M$, and $x_n$ has turned into $f(x)$. And
there is a new complication, because when I was dealing with sequences I
took it for granted that $x_n$ would be defined for every relevant $n$.
But when dealing with general real functions, one can't take that sort
of thing without checking carefully; and before saying `$|f(x)-
a|\le\epsilon$' we have to say `$f(x)$ is defined'.
\medskip
{\bf (c)} If $f$ and $g$ are real functions, then $f+g$ is the real
function defined by saying
\Centerline{$\dom(f+g)=\dom f\cap\dom g$,
\quad$(f+g)(x)=f(x)+g(x)$ for every $x\in\dom(f+g)$.}
\noindent (Of course the {\it idea} is in the formula
$(f+g)(x)=f(x)+g(x)$.
The domain $\dom(f+g)$ specified is just the set on which we can
calculate both $f(x)$ and $g(x)$ and use the formula.)
Now the theorem is this:
\medskip
{\bf Theorem} If $f$ and $g$ are real functions and
$\lim_{x\to\infty}f(x)=b$, $\lim_{x\to\infty}g(x)=c$ then
$\lim_{x\to\infty}(f+g)(x)=b+c$.
\medskip
\noindent{\bf proof} What I am going to do is to go through the proof
which worked for sequences, make as few changes as possible, and see if
it still works. Here goes:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.}
\noindent No problems so far.
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $n_1\in\Bbb N$ such that $|x_n-b|\le\Bover{\epsilon}2$ for
every
$n\ge n_1$.}
\noindent Change `$n_1\in\Bbb N$' into `$M_1\in\Bbb R$', `$x_n$' into
`$f(x)$',
`$n\ge n_1$' into `$x\ge M_1$':
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M\in\Bbb R$ such that $|f(x)-b|\le\Bover{\epsilon}2$ for
every
$x\ge M_1$.}
\noindent This won't quite do, because we are talking about $f(x)$
before we've said whether there is such a number; we had better change
to
\inset{There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.}
\noindent Carry on with the next line of the original proof:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.
There is an $n_2\in\Bbb N$ such that $|y_n-c|\le\Bover{\epsilon}2$ for
every
$n\ge n_2$.}
\noindent Translate this line in the same way as the one above:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.
There is an $M_2\in\Bbb R$ such that $x\in\dom g$ and
$|g(x)-c|\le\Bover{\epsilon}2$ for every $x\ge M_2$.}
\noindent Look at the next line:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.
There is an $M_2\in\Bbb R$ such that $x\in\dom g$ and
$|g(x)-c|\le\Bover{\epsilon}2$ for every $x\ge M_2$.
Take $n_0=\max(n_1,n_2)$.}
\noindent We want $M$'s not $n$'s this time:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.
There is an $M_2\in\Bbb R$ such that $x\in\dom g$ and
$|g(x)-c|\le\Bover{\epsilon}2$ for every $x\ge M_2$.
Take $M=\max(M_1,M_2)$.}
\noindent Carry on:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.
There is an $M_2\in\Bbb R$ such that $x\in\dom g$ and
$|g(x)-c|\le\Bover{\epsilon}2$ for every
$x\ge M_2$.}
Take $M=\max(M_1,M_2)$.
Let $n\ge n_0$.
Then $n\ge n_1$ and $n\ge n_2$, so $|x_n-b|\le\Bover{\epsilon}2$ and
$|y_n-c|\le\Bover{\epsilon}2$ and
\Centerline{$|(x_n+y_n)-(b+c)|\le|x_n-b|+|y_n-c|
\le\Bover{\epsilon}2+\Bover{\epsilon}2=\epsilon$.}
\noindent This becomes
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.
There is an $M_2\in\Bbb R$ such that $x\in\dom g$ and
$|g(x)-c|\le\Bover{\epsilon}2$ for every $x\ge M_2$.}
Take $M=\max(M_1,M_2)$.
Let $x\ge M$.
Then $x\ge M_1$ and $x\ge M_2$, so $x\in\dom f$ and $x\in\dom g$ and
$|f(x)-b|\le\Bover{\epsilon}2$ and
$|g(x)-c|\le\Bover{\epsilon}2$ and
\Centerline{$|(f(x)+g(x))-(b+c)|\le|f(x)-b|+|g(y)-c|
\le\Bover{\epsilon}2+\Bover{\epsilon}2=\epsilon$.}
\noindent There is only one thing missing: we want `$(f+g)(x)$' in the
last line instead of `$f(x)+g(x)$'. They are nearly the same thing,
except that before talking about `$(f+g)(x)$' we ought to explain why
$x\in\dom(f+g)$. But this is easy:
\inset{Let $\epsilon>0$. Then $\Bover{\epsilon}2>0$.
There is an $M_1\in\Bbb R$ such that $x\in\dom f$ and
$|f(x)-b|\le\Bover{\epsilon}2$ for every $x\ge M_1$.
There is an $M_2\in\Bbb R$ such that $x\in\dom g$ and
$|g(x)-c|\le\Bover{\epsilon}2$ for every
$x\ge M_2$.}
Take $M=\max(M_1,M_2)$.
Let $x\ge M$.
Then $x\ge M_1$ and $x\ge M_2$, so $x\in\dom f$ and $x\in\dom g$ and
$x\in\dom(f+g)$. Also
$|f(x)-b|\le\Bover{\epsilon}2$ and
$|g(x)-c|\le\Bover{\epsilon}2$, so
\Centerline{$|(f+g)(x)-(b+c)|=|(f(x)+g(x))-(b+c)|\le|f(x)-b|+|f(y)-c|
\le\Bover{\epsilon}2+\Bover{\epsilon}2=\epsilon$.}
\noindent And there is the proof, complete.
\bigskip
\noindent{\bf The next theorem} For a couple of lectures now I shall be
producing variations on the arguments above. I am going to ask you to
learn them. The way to learn them is by comparing and contrasting.
You will find that each proof has one or two points in which it differs
from the others, and many more points in which it is almost the same.
For my first example of this, let me give another result for which you
know the fact but not the proof.
\medskip
{\bf Theorem} If $\lim_{n\to\infty}x_n=b$ and $\lim_{n\to\infty}y_n=c$,
$\lim_{n\to\infty}x_ny_n=bc$.
\medskip
\noindent{\bf proof} I shall try to keep as close as possible to
the line used in the proof for $\lim_{n\to\infty}x_n+y_n=b+c$. Try
this:
\inset{Let $\epsilon>0$.
There is an $n_1\in\Bbb N$ such that $|x_n-b|\le ??$ whenever
$n\ge n_1$.
There is an $n_2\in\Bbb N$ such that $|y_n-c|\le ??$ whenever
$n\ge n_2$.
Set $n_0=\max(n_1,n_2)$.
If $n\ge n_0$, then $n\ge n_1$ and $n\ge n_2$ so $|x_n-b|\le??$ and
$|y_n-c|\le??$ and
\Centerline{$|x_ny_n-bc|\le ?? \le\epsilon$.}}
\noindent Now you see that I've put ?? in place of $\Bover{\epsilon}2$,
because it's hardly credible that the same formula will work. And in
the last line we have a completely new problem to solve, because at this
point we can expect to know something about $x_n-b$ and $y_n-c$, but
it's not at all clear how to turn this into something about $x_ny_n-bc$.
So here we need a new fact, which is
\Centerline{$x_ny_n-bc=(x_n-b)(y_n-c)+b(y_n-c)+c(x_n-b)$.}
\noindent This means that if $x_n-b$ and $y_n-c$ are both practically
zero, the right-hand-side will be a sum of three small pieces and the
left-hand-side will also be nearly zero. But just how small do $x_n-b$
and $y_n-c$ have to be? At this point it makes things much easier to
use a technical trick which will be very useful elsewhere. Since we
don't know what to put in the line
\inset{There is an $n_1\in\Bbb N$ such that $|x_n-b|\le ??$ whenever
$n\ge n_1$,}
\noindent invent a name for it; $\eta$, say. Then those two lines
will read
\inset{There is an $n_1\in\Bbb N$ such that $|x_n-b|\le\eta$ whenever
$n\ge n_1$.
There is an $n_2\in\Bbb N$ such that $|y_n-c|\le\eta$ whenever
$n\ge n_2$,}
\noindent and later on we shall have
\inset{If $n\ge n_0$, then $n\ge n_1$ and $n\ge n_2$ so $|x_n-b|\le\eta$
and $|y_n-c|\le\eta$.}
\noindent Now the point of this method is to be absolutely clear where
$\eta$ came from. This section is part of Player I's scheming after
Player II has chosen $\epsilon$. So is it {\it Player I} who chooses
$\eta$, probably in a way depending on $\epsilon$, as a preliminary to
looking for $n_1$ and $n_2$ and putting these together to produce $n_0$.
Suppose he does this. Then, in the last line, we shall have
\Centerline{$|x_ny_n-bc|=|(x_n-b)(y_n-c)+b(y_n-c)+c(x_n-b)|$.}
\noindent Now there is an absolutely standard way of treating this.
Whenever you have the modulus of a sum, and you want to show that it is
less than or equal to something, the first thing to try is the sum of
the moduli:
$$\eqalign{|x_ny_n-bc|&=|(x_n-b)(y_n-c)+b(y_n-c)+c(x_n-b)|\cr
&\le|(x_n-b)(y_n-c)|+|b(y_n-c)|+|c(x_n-b)|.\cr}$$
\noindent Next, given the modulus of a product, it is usually right to
rewrite it as the product of the moduli:
$$\eqalign{|x_ny_n-bc|&=|(x_n-b)(y_n-c)+b(y_n-c)+c(x_n-b)|\cr
&\le|(x_n-b)(y_n-c)|+|b(y_n-c)|+|c(x_n-b)|\cr
&=|x_n-b||y_n-c|+|b||y_n-c|+|c||x_n-b|.\cr}$$
\noindent Remember that by the time we've got this far, we have
$|x_n-b|\le\eta$ (whatever $\eta$ may be) and $|y_n-c|\le\eta$. So
we get
$$\eqalign{|x_ny_n-bc|&=|(x_n-b)(y_n-c)+b(y_n-c)+c(x_n-b)|\cr
&\le|(x_n-b)(y_n-c)|+|b(y_n-c)|+|c(x_n-b)|\cr
&=|x_n-b||y_n-c|+|b||y_n-c|+|c||x_n-b|\cr
&\le\eta\cdot\eta+|b|\eta+|c|\eta\cr
&=\eta(\eta+|b|+|c|).\cr}$$
We shall therefore be home, with a win for Player I, if she can arrange
that $\eta(\eta+|b|+|c|)\le\epsilon$. Well, if she can solve
quadratics, she will actually be able to get
$\eta(\eta+|b|+|c|)=\epsilon$. But, first, this is a bother, and,
second, there are difficulties with proving that numbers have square
roots; so I much prefer to use a more primitive argument. Remember
that Player I chooses $\eta$ immediately after seeing the $\epsilon$
chosen by Player II. Player I is allowed to choose any $\eta>0$.
($\eta$ really must be strictly greater than $0$ or it may not be true
that $|x_n-b|$ and $|y_n-c|$ are less than or equal to $\eta$ for enough
$n$.) In particular, Player I is certainly allowed to insist that
$\eta\le 1$. Now if $\eta\le 1$, we just need
$\eta(1+|b|+|c|)\le\epsilon$, that is,
$\eta\le\Bover{\epsilon}{1+|b|+|c|}$. So if Player I takes $\eta$ to
be $\min(1,\Bover{\epsilon}{1+|b|+|c|})$, she should win.
Let's write the whole proof out on this basis.
\inset{Let $\epsilon>0$.\hfill[Player I's first move.]
Set $\eta=\min(1,\Bover{\epsilon}{1+|b|+|c|})>0$.
\hfill[Player II is getting into position.]
There is an $n_1\in\Bbb N$ such that $|x_n-b|\le\eta$ whenever
$n\ge n_1$.
There is an $n_2\in\Bbb N$ such that $|y_n-c|\le\eta$ whenever
$n\ge n_2$.
Set $n_0=\max(n_1,n_2)$.\hfill[Player I's second move.]
If $n\ge n_0$, \hfill[Player II's second move]
then $n\ge n_1$ and $n\ge n_2$ so $|x_n-b|\le\eta$ and
$|y_n-c|\le\eta$ and
$$\eqalignno{|x_ny_n-bc|&=|(x_n-b)(y_n-c)+b(y_n-c)+c(x_n-b)|\cr
&\le|(x_n-b)(y_n-c)|+|b(y_n-c)|+|c(x_n-b)|\cr
&=|x_n-b||y_n-c|+|b||y_n-c|+|c||x_n-b|\cr
&\le\eta\cdot\eta+|b|\eta+|c|\eta\cr
&=\eta(\eta+|b|+|c|)\cr
&\le\eta(1+|b|+|c|)&\text{ because }\eta\le 1\cr
&\le\epsilon
&\text{ because }\eta\le\Bover{\epsilon}{1+|b|+|c|}.\cr}$$
So Player I wins and $\lim_{n\to\infty}x_ny_n=bc$.}
\medskip
{\bf More real functions: Definition} Suppose that $f$ is a real
function and that $a$, $b\in\Bbb R$. Then `$\lim_{x\to a}f(x)=b$'
means
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that $f(x)$ is
defined and $|f(x)-b|\le\epsilon$ whenever $0<|x-a|\le\delta$.}
\noindent What is the idea here? The claim is that $f(x)\bumpeq b$
whenever $x\bumpeq a$ and $x\ne a$. (Remember that when we consider
$\lim_{x\to a}f(x)$, one value of $f$ which we {\it never} look at is
$f(a)$, even if it's defined.) As before, it's Player II who decides
what `$f(x)\bumpeq b$' means, by choosing $\epsilon>0$, and following
this Player I decides what `$x\bumpeq a$' means, by choosing a
$\delta>0$; finally Player II gets to choose $x$. As in the
definition of $\lim_{x\to\infty}$, we have to provide for the
possibility that $f$ is not defined everywhere, and I do this by
insisting that Player I must choose a $\delta$ such that $f(x)$ is
defined whenever $|x-a|\le\delta$ (and $x\ne a$); if she can't (for
instance, if $a=0$ and $f(x)=\sqrt{x}$ for $x\ge 0$, but undefined for
$x<0$), then Player I is bound to lose, and I don't allow myself to say
`$\lim_{x\to 0}\sqrt{x}=0$'. (Of course I do allow
`$\lim_{x\downarrow 0}\sqrt{x}=0$', but that's a different game.)
\medskip
{\bf Definition} If $f$ and $g$ are real functions, then $f\times g$ is
the real function defined by saying
\Centerline{$\dom(f\times g)=\dom f\cap\dom g$,
\quad$(f\times g)(x)=f(x)g(x)$ for $x\in\dom(f\times g)$.}
\noindent I give these definitions so that we can have a theorem about
the limit of a product of real functions, as follows.
\medskip
{\bf Theorem} If $f$ and $g$ are real functions, and
$\lim_{x\to a}f(x)=b$ and $\lim_{x\to a}g(x)=c$, then
$\lim_{x\to a}(f\times g)(x)=bc$.
\medskip
\noindent{\bf proof} This is a translation of the theorem on the product
of sequences, just as the theorem
$\lim_{x\to\infty}(f+g)(x)=\lim_{x\to\infty}f(x)+\lim_{x\to\infty}g(x)$
is a translation of the theorem on the sum of sequences. Let me write
it out directly.
Let $\epsilon>0$. Set $\eta=\min(1,\Bover{\epsilon}{1+|b|+|c|})>0$.
There is a $\delta_1>0$ such that $x\in\dom f$ and $|f(x)-b|\le\eta$
whenever $0<|x-a|\le\delta_1$.
There is a $\delta_2>0$ such that $x\in\dom g$ and $|g(x)-c|\le\eta$
whenever $0<|x-a|\le\delta_2$.
Set $\delta=\min(\delta_1,\delta_2)$. If $0<|x-a|\le\delta$, then
$0<|x-a|\le\delta_1$ and $0<|x-a|\le\delta_2$, so $x\in\dom f$ and
$x\in\dom g$ and $x\in\dom(f\times g)$; also $|f(x)-b|\le\eta$ and
$|g(x)-c|\le\eta$, so
$$\eqalignno{|f(x)g(x)-bc|&=|(f(x)-b)(g(x)-c)+b(g(x)-c)+c(f(x)-b)|\cr
&\le|(f(x)-b)(g(x)-c)|+|b(g(x)-c)|+|c(f(x)-b)|\cr
&=|f(x)-b||g(x)-c|+|b||g(x)-c|+|c||f(x)-b|\cr
&\le\eta^2+|b|\eta+|c|\eta
=\eta(\eta+|b|+|c|)
\le\eta(1+|b|+|c|)
\le\epsilon.\cr}$$
\noindent As this works for any $\epsilon>0$, Player I can win whatever
Player II does at his first move, and $\lim_{x\to a}(f\times g)(x)=bc$
is true.
\medskip
\noindent{\bf Remark} Look again at the translation. $\epsilon$ and
$\eta$ do exactly the same things as before. $n_1$ and $n_2$ turn into
$\delta_1$ and $\delta_2$, and $n_0$ turns into $\delta$. But observe
that $\delta$ is the {\it minimum} of $\delta_1$ and $\delta_2$, while
$n_0$ was the {\it maximum} of $n_1$ and $n_2$. This is because of the
rule change concerning the next move. In the case of sequences, Player
II has to choose $n\ge n_0$. So in order to make him choose an $n$
simultaneously greater than or equal to $n_1$ and $n_2$, Player II
chooses $n_0$ so that $n_0\ge n_1$ and $n_0\ge n_2$, and the easiest
such choice is $n_0=\max(n_1,n_2)$. But in the case of
$\lim_{x\to a}$, Player II has to choose $x$ such that $|x-a|\le\delta$,
and Player I wants to be sure that $|x-a|\le\delta_1$ and that
$|x-a|\le\delta_2$; so she takes $\delta=\min(\delta_1,\delta_2)$.
As a general rule, when choosing moves for Player I, you aim to make
things difficult for Player II, by reducing his choices as much as you
can. In this case, it's done by making $\delta$ close to $0$ (but
remembering that $\delta=0$ is cheating).
After this, the difference is mostly that every $x_n$ turns into $f(x)$
and every $y_n$ turns into $g(x)$; but, just as in the theorem on
$\lim_{x\to\infty}(f+g)(x)$, we have to preface every statement about
$f(x)$, $g(x)$ or $(f\times g)(x)$ with an explanation of why $x$
belongs to the domain of the function. In the lines
\inset{there is a $\delta_1>0$ such that $x\in\dom f$ and
$|f(x)-b|\le\eta$ whenever $0<|x-a|\le\delta_1$,
there is a $\delta_2>0$ such that $x\in\dom g$ and $|g(x)-c|\le\eta$
whenever $0<|x-a|\le\delta_2$}
\noindent this came from the definitions of $\lim_{x\to a}f(x)=b$,
$\lim_{x\to a}g(x)=c$; in the line
\inset{$x\in\dom f$ and $x\in\dom g$ and $x\in\dom(f\times g)$}
\noindent it came from the definition of $f\times g$.
\medskip
{\bf Three more definitions} As well as $\lim_{x\to\infty}$ and
$\lim_{x\to a}$, we sometimes want to look at
$\lim_{x\to-\infty}f(x)$, $\lim_{x\uparrow a}f(x)$ and
$\lim_{x\downarrow a}f(x)$. The definitions are as follows.
$\lim_{x\to-\infty}f(x)=b$ means
\inset{for every $\epsilon>0$ there is an $M\in\Bbb R$ such that
$x\in\dom f$ and $|f(x)-b|\le\epsilon$ whenever $x\le M$.}
\noindent (Note that the only difference between this and the definition
of `$\lim_{x\to\infty}f(x)=b$' is that we have `$x\le M$' instead of
`$x\ge M$'. But of course this makes a big difference to Player I's
tactics. When playing from the initial position
`$\lim_{x\to\infty}f(x)=b$', Player I will generally take an $M$ far,
far to the right, so that Player II will be seriously constrained by the
rule `$x\ge M$'. While if
playing from initial position `$\lim_{x\to\-infty}f(x)=b$', Player I
will generally take an $M$ correspondingly far to the left, so that
Player II will have to go to the edge of the universe to satisfy the
requirement `$x\le M$'.
\medskip
$\lim_{x\downarrow a}f(x)=b$ means
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that
$x\in\dom f$ and $|f(x)-b|\le\epsilon$ whenever $a<x\le a+\delta$.}
\noindent The condition `$a<x\le a+\delta$' is an exact way of saying
`$x\bumpeq a$ and $x>a$', or `$x$ is just to the right of $a$'.
\medskip
Finally, $\lim_{x\uparrow a}f(x)=b$ means
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that
$x\in\dom f$ and $|f(x)-b|\le\epsilon$ whenever
$a-\delta\le x<a$.}
\medskip
{\bf Theorem} Suppose that $\lim_{n\to\infty}x_n=b\ne 0$. Then
$\lim_{n\to\infty}\Bover1{x_n}\to\Bover1b$.
\medskip
\noindent{\bf proof} Let $\epsilon>0$. Set
$\eta=\min(\Bover{|b|}2,\bover12b^2\epsilon)>0$. Then there is an
$n_0\in\Bbb N$ such that $|x_n-b|\le\eta$ for every $n\ge n_0$. If
$n\ge n_0$, then
$$\eqalignno{|x_n|
&\ge|b|-|x_n-b|\cr
\displaycause{because $|x_n|+|x_n-b|=|x_n|+|b-x_n|\ge|x_n+b-x_n|$}
&\ge|b|-\eta\ge|b|-\Bover12|b|=\Bover12|b|>0,\cr}$$
\noindent so that $x_n\ne 0$ and $\Bover1{x_n}$ is defined and
$$\eqalignno{|\Bover1{x_n}-\Bover1b|
&=|\Bover{b-x_n}{x_nb}|
=\Bover{|b-x_n|}{|x_n||b|}\cr
&\le\Bover{\eta}{|x_n||b|}\cr
\displaycause{because $|b-x_n|=|x_n-b|\le\eta$}
&\le\bover{\eta}{\bover12|b||b|}\cr
\displaycause{because $|x_n|\ge\Bover12|b|$}
&=\Bover{2\eta}{b^2}\le\Bover2{b^2}\cdot\bover12b^2\epsilon=\epsilon.
\cr}$$
\noindent As $\epsilon$ is arbitrary,
$\lim_{n\to\infty}\Bover1{x_n}=\Bover1b$.
\medskip
\noindent{\bf Remarks} Some of the ideas here are new, some are taken
from earlier proofs. The idea of taking $n_0$ such that $|x_n-
b|\le\eta$ for
$n\ge n_0$, where $\eta$ is some more or less complicated function of
$\epsilon$, is taken from the theorems on products of sequences or
functions.
This time we have only one sequence so we don't have the step
`$n_0=\max(n_1,n_2)$'. What we have instead is a more complicated
string of inequalities at the end. The steps
\Centerline{$|\Bover1{x_n}-\Bover1b|=\ldots=\Bover{|b-x_n|}{|x_n||b|}$}
\noindent are standard; nearly always, in these expressions, if we have
the modulus of a product or quotient we express it as the product or
quotient of the moduli and see what happens. Now in the expression
\Centerline{$\Bover{|b-x_n|}{|x_n||b|}$}
\noindent the $|b-x_n|$ on top is no problem at all; we know that by
the time we've reached this stage, we shall have $|b-x_n|\le\eta$, so we
just replace it by $\eta$. The new difficulty is in the $|x_n|$ on the
bottom. If we are to be sure that that $\Bover{\eta}{|x_n||b|}$ is
{\it less} than or equal to $\epsilon$, we shall need to know that
$|x_n|$ is {\it greater} than or equal to something. And that's where
the line
\Centerline{$|x_n|\ge|b|-|x_n-b|\ldots$}
\noindent comes in. Provided Player I takes $\eta<|b|$, she can be
sure that
$|x_n|\ge|b|-\eta$ so that $\Bover1{|x_n|}\le\Bover1{|b|-\eta}$ and
$\Bover{\eta}{|x_n||b|}\le\Bover{\eta}{(|b|-\eta)|b|}$, and by making
$\eta$ small enough she can ensure that this will be at most $\epsilon$.
The actual
formula
\Centerline{$\eta=\min(\Bover{|b|}2,\Bover12b^2\epsilon)$}
\noindent is just a trick for guaranteeing the three facts
\Centerline{$\eta>0$,\quad $\eta<|b|$,\quad
$\Bover{\eta}{(|b|-\eta)|b|}\le\epsilon$}
\noindent without solving any quadratic equations.
\medskip
{\bf Definition} Let $f$ be a real function. Then the real function
$\Bover1f$ is defined by saying
\Centerline{$\dom\Bover1f=\{x:x\in\dom f,\,f(x)\ne 0\}$,}
\Centerline{$\Bover1f(x)=\Bover1{f(x)}$ for every $x\in\dom\Bover1f$.}
\medskip
{\bf Theorem} Let $f$ be a real function, and suppose that
$\lim_{x\to-\infty}f(x)=b\ne 0$. Then
$\lim_{x\to-\infty}\Bover1f(x)=\Bover1b$.
\medskip
\noindent{\bf proof} Let $\epsilon>0$. Set
$\eta=\min(\Bover{|b|}2,\Bover12b^2\epsilon)>0$. Then there is an
$M\in\Bbb R$ such that $x\in\dom f$ and $|f(x)-b|\le\eta$ for every
$x\le M$. If $x\le M$, then $f(x)$ is defined and
$$\eqalignno{|f(x)|
&\ge|b|-|f(x)-b|\cr
&\ge|b|-\eta\ge\Bover12|b|>0,\cr}$$
\noindent so that $f(x)\ne 0$ and $x\in\dom\Bover1f$ is defined and
$$\eqalignno{|\Bover1{f(x)}-\Bover1b|
&=\Bover{|b-f(x)|}{|f(x)||b|}
\le\bover{\eta}{\bover12|b||b|}\cr
&=\Bover{2\eta}{b^2}\le\epsilon.\cr}$$
\noindent As $\epsilon$ is arbitrary,
$\lim_{x\to-\infty}\Bover1{f(x)}=\Bover1b$.
\bigskip
\noindent{\bf Continuous Functions: Definition} Let $f$ be a real
function. We say that $f$ {\bf is continuous at} $x_0$ if $x_0\in\dom
f$ and
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that
\inset{$|f(x)-f(x_0)|\le\epsilon$ whenever $x\in\dom f$ and
$|x-x_0|\le\delta$.}}
\medskip
\noindent{\bf Remark} Note a very important rule change compared with
the formula $\lim_{x\to x_0}f(x)=f(x_0)$. If Player I says `$f$ is
continuous at $x_0$', then Player II chooses $\epsilon$, Player I
chooses $\delta$ and Player II chooses $x$, just as before. But this
time it is Player II's responsibility to ensure that $x\in\dom f$. For
`$\lim_{x\to x_0}f(x)=b$', Player II was allowed any $x$ such that
$0<|x-x_0|\le\delta$, and if he could find one outside the domain of $f$
he would win. But for `$f$ is continuous at $x_0$', Player II has to
pick $x\in\dom f$. He is now allowed to pick $x=x_0$, but of course
that does him no good at all (which is why it's allowed). So Player II
has a lot less freedom at his second move, and it's easier for Player I
to win.
The reason for this change is that it's useful to be able to say that
$\sqrt{}$ is continuous at $0$. But there is no way of making sense of
the formula
$\lim_{x\uparrow 0}\sqrt x$ (if we want to stick to real numbers), and
if we want to keep the rule
\Centerline{$\lim_{x\to a}f(x)=b$ iff
$\lim_{x\downarrow a}f(x)=\lim_{x\uparrow a}f(x)=b$,}
\noindent then we are going to have to abandon any attempt to interpret
the formula $\lim_{x\to 0}\sqrt x=0$, at least for the `real' function
$\sqrt{}$.
\medskip
{\bf Theorem} If $f$ and $g$ are continuous real functions, so are
$f+g$, $f\times g$ and $\Bover1f$.
\medskip
\noindent{\bf proof (a)} Take $x_0\in\dom(f+g)$ and $\epsilon>0$. Let
$\delta_1$, $\delta_2>0$ be such that
\Centerline{$|f(x)-f(x_0)|\le\Bover12\epsilon$ whenever $x\in\dom f$ and
$|x-x_0|\le\delta_1$,}
\Centerline{$|g(x)-g(x_0)|\le\Bover12\epsilon$ whenever $x\in\dom g$ and
$|x-x_0|\le\delta_2$.}
\noindent Set $\delta=\min(\delta_1,\delta_2)>0$. Then if
$x\in\dom(f+g)$ and $|x-x_0|\le\delta$,
\Centerline{$|(f+g)(x)-(f+g)(x_0)|= |f(x)+g(x)-f(x_0)-g(x_0)|
\le|f(x)-f(x_0)|+|g(x)-
g(x_0)|\le\Bover12\epsilon+\Bover12\epsilon=\epsilon$.}
\noindent As $x_0$ and $\epsilon$ are arbitrary, $f+g$ is continuous.
\medskip
{\bf (b)} Take $x_0\in\dom(f\times g)$ and $\epsilon>0$. Set
$\eta=\min(1,\Bover{\epsilon}{1+|f(x_0)|+|g(x_0)|})>0$. Let
$\delta_1$, $\delta_2>0$ be such that
\Centerline{$|f(x)-f(x_0)|\le\eta$ whenever $x\in\dom f$ and
$|x-x_0|\le\delta_1$,}
\Centerline{$|g(x)-g(x_0)|\le\eta$ whenever $x\in\dom g$ and
$|x-x_0|\le\delta_2$.}
\noindent Set $\delta=\min(\delta_1,\delta_2)>0$. Then if
$x\in\dom(f\times g)$ and $|x-x_0|\le\delta$,
$$\eqalign{|(f\times g)(x)-(f\times g)(x_0)|
&= |f(x)g(x)-f(x_0)g(x_0)|\cr
&\le|f(x)-f(x_0)||g(x)-g(x_0)|+|f(x_0)||g(x)-g(x_0)|+|f(x)-
f(x_0)||g(x_0)|\cr
&\le\eta(\eta+|f(x_0)|+|g(x_0)|
\le\eta(1+|f(x_0)|+|g(x_0)|
\le\epsilon.\cr}$$
\noindent As $x_0$ and $\epsilon$ are arbitrary, $f+g$ is continuous.
\medskip
{\bf (c)} Take $x_0\in\dom(\Bover1f)$ and $\epsilon>0$. Set
$\eta=\min(\Bover12|f(x_0)|,\Bover12\epsilon|f(x_0)|^2)>0$. Let
$\delta>0$ be such that
\Centerline{$|f(x)-f(x_0)|\le\eta$ whenever $x\in\dom f$ and
$|x-x_0|\le\delta$.}
\noindent Then if
$x\in\dom(\Bover1f)$ and $|x-x_0|\le\delta$,
\Centerline{$|f(x)|\ge|f(x_0)|-|f(x)-f(x_0)|\ge|f(x_0)|-\eta
\ge\Bover12|f(x_0)|$,}
\noindent so
$$\eqalign{|(\bover1f)(x)-(\bover1f)(x_0)|
&=\bigl|\bover1{f(x)}-\bover1{f(x_0)}\bigr|
=\bover{|f(x_0)-f(x)|}{|f(x)||f(x_0)|}\cr
&\le\bover{\eta}{\bover12|f(x_0)||f(x_0)|}
\le\epsilon.\cr}$$
\noindent As $x_0$ and $\epsilon$ are arbitrary, $\Bover1f$ is
continuous.
\medskip
{\bf Definition} If $f$ and $g$ are real functions, their {\bf
composition} $f\smallcirc g$ is defined by saying
\Centerline{$\dom(f\smallcirc g)=\{x:x\in\dom g,\,g(x)\in\dom f\}$,
\quad$(f\smallcirc g)(x)=f(g(x))$ for $x\in\dom(f\smallcirc g)$.}
\medskip
{\bf Theorem} If $f$ and $g$ are continuous real functions, then
$f\smallcirc g$ is continuous.
\medskip
\noindent{\bf proof} Take $x_0\in\dom(f\smallcirc g)$ and $\epsilon>0$.
Then $g(x_0)\in\dom f$ so there is an $\eta>0$ such that
$|f(y)-f(g(x_0))|\le\epsilon$ whenever $y\in\dom f$ and $|y-
g(x_0)|\le\eta$. Next, $x_0\in\dom g$ so there is a $\delta>0$ such
that $|g(x)-g(x_0)|\le\eta$ whenever $x\in\dom g$ and $|x-
x_0|\le\delta$.
If $x\in\dom(f\smallcirc g)$ and $|x-x_0|\le\delta$, then $g(x)\in\dom
f$ and $|g(x)-g(x_0)|\le\eta$, so
\Centerline{$|(f\smallcirc g)(x)-(f\smallcirc g)(x_0)|
=|f(g(x))-f(g(x_0))|\le\epsilon$.}
\noindent As $x_0$ and $\epsilon$ are arbitrary, $f\smallcirc g$ is
continuous.
\medskip
{\bf Examples (a)} Constant functions are continuous. \Prf\ Suppose
that $f(x)=c$ for $x\in\dom f$. Take $x_0\in\dom f$ and $\epsilon>0$.
Set $\delta=1$. Then if $x\in\dom f$ and $|x-x_0|\le\delta$,
\Centerline{$|f(x)-f(x_0)|=|c-c|=0\le\epsilon$.}
\noindent As $x_0$ and $\epsilon$ are arbitrary, $f$ is continuous.\
\Qed
\noindent{\bf Remark} Observe that in this (quite exceptional) case,
Player I can announce her second move $\delta=1$ {\it before} Player II
has played his first move; her position is so strong that he has no way
of using the advance knowledge.
\medskip
\quad{\bf (b)} Identity functions are continuous. \Prf\ Suppose that
$f(x)=x$ for $x\in\dom f$.
Take $x_0\in\dom f$ and $\epsilon>0$. Set $\delta=\epsilon$. Then
if $x\in\dom f$ and $|x-x_0|\le\delta$,
\Centerline{$|f(x)-f(x_0)|=|x-x_0|\le\delta=\epsilon$.}
\noindent As $x_0$ and $\epsilon$ are arbitrary, $f$ is continuous.\
\Qed
\noindent{\bf Remark} In {\it this} case, Player I has to know part of
Player II's move; but if she doesn't pay attention properly and misses
the announcement of $x_0$, she can still win by playing
$\delta=\epsilon$, and ask what $x_0$ is afterwards, when they come to
check the calculation of $f(x)-f(x_0)$.
\medskip
\quad{\bf (c)} The function $x\mapsto|x|:\Bbb R\to\Bbb R$ is continuous.
\Prf\ Take $x_0\in\Bbb R$ and $\epsilon>0$. Set $\delta=\epsilon$.
Then if $x\in\dom f$ and $|x-x_0|\le\delta$,
\Centerline{$||x|-|x_0||\le|x-x_0|\le\delta=\epsilon$.}
\noindent As $x_0$ and $\epsilon$ are arbitrary, $|\,\,|$ is
continuous.\ \Qed
\medskip
{\bf Corollary} For any continuous real function $f$,
$x\mapsto|f(x)|:\dom f\to\Bbb R$ is continuous. \Prf\ This is just the
composition $|\,\|\smallcirc f$ of two continuous functions.\ \Qed
\medskip
{\bf Remark} We now know that $f+g$, $f\times g$, $\Bover1f$ and
$f\smallcirc g$ are continuous whenever $f$ and $g$ are, and that
$x\mapsto c$, $x\mapsto x$ and $x\mapsto|x|$ are continuous functions
from $\Bbb R$ to itself for any $c\in\Bbb R$. Putting these together,
we see (for instance) that
\inset{$x\mapsto x^2:\Bbb R\to\Bbb R$ is continuous}
\noindent (because this is just $f\times f$, where $f(x)=x$ for every
$x\in\Bbb R$),
\inset{$x\mapsto 2x^2:\Bbb R\to\Bbb R$ is continuous}
\noindent (being the product of the continuous functions $x\mapsto x^2$
and $x\mapsto 2$),
\inset{$x\mapsto 2x^3:\Bbb R\to\Bbb R$ is continuous}
\noindent (being the product of the continuous functions $x\mapsto 2x^2$
and $x\mapsto x$),
\inset{$x\mapsto\Bover1{x^2}:\Bbb R\setminus\{0\}\to\Bbb R$ is
continuous}
\noindent (being the reciprocal of a continuous function),
\inset{$x\mapsto x+\Bover1{x^2}:\Bbb R\setminus\{0\}\to\Bbb R$ is
continuous}
\noindent (being the sum of two continuous functions),
\inset{$x\mapsto 2(x+\Bover1{x^2})^3:\Bbb R\setminus\{0\}\to\Bbb R$ is
continuous}
\noindent (being the composition of the functions $x\mapsto 2x^3$ and
$x\mapsto x+\Bover1{x^2}$). Generally, most of the functions we have
names for are continuous; and we can prove that a function with a
formula like
\Centerline{$h(x)=\Bover{\sin(3x+1)}{1-\ln\cos x}$}
\noindent is continuous as soon as we know that $\sin$, $\cos$, $\ln$
are continuous (which I am afraid will be just outside the scope of this
course).
\bigskip
\noindent{\bf Dedekind completeness} I come now to the final basic
property of the real number system which I left out of the initial list
of properties of addition, multiplication and the order relation. For
this we need some terminology. If $A\subseteq\Bbb R$ is any set, an
{\bf upper bound} of $A$ is an $x\in\Bbb R$ such that $a\le x$ for every
$a\in A$, and a {\bf lower bound} of $A$ is an $x\in\Bbb R$ such that
$x\le a$ for every $a\in A$.
\medskip
\inset{{\bf Examples} If $A=[0,1]$ then $2$, $\pi$, $1$ are upper bounds
of $A$ and $-1$, $0$ are lower bounds.
If $A=\Bbb N$ then $-1$, $-\bover14$, $0$ are lower bounds of $A$, but
$A$ has no upper bounds.
If $A=\Bbb Z$ then $A$ has no upper bounds and no lower bounds.
If $A=\ooint{0,1}$ then $1$ is an upper bound of $A$ but
$0{\cdot}999999$ is not, because $0{\cdot}9999991\in A$.}
If $A=\emptyset$ then every real number is both an upper bound and a
lower bound for $A$.
\medskip
If $A$ has a least upper bound, I will call it the {\bf supremum} of
$A$, $\sup A$; if it has a greatest lower bound, I call it the {\bf
infimum} of $A$, $\inf A$.
\medskip
\inset{{\bf Examples} If $A=[0,1]$ then $\sup A=1$ and $\inf A=0$.
If $A=\Bbb N$ then $\inf A=0$ but $A$ has no supremum (because it has no
upper bounds at all).
If $A=\Bbb Z$ then $A$ has no supremum and no infimum.
If $A=\ooint{0,1}$ then $\sup A=1$ and $\inf A=0$.
If $A=\emptyset$ then $A$ has no supremum and no infimum.}
\medskip
Note that $\ooint{0,1}$ and $[0,1]$ have exactly the same upper bounds;
$x\in\Bbb R$ is an upper bound for either iff $x\ge 1$, and $1$ is the
least upper bound of both. This shows that $\sup A$ (when defined) may
or may not belong to the set $A$.
\medskip
{\bf FUNDAMENTAL FACT} If $A\subseteq\Bbb R$ is non-empty and has an
upper bound, it has a least upper bound.
Similarly, if $A\subseteq\Bbb R$ is non-empty and has a lower bound, it
has a greatest lower bound.
\medskip
\noindent{\bf Remarks} What these principles are saying is that a subset
of $\Bbb R$ has a supremum and an infimum unless it plainly can't,
either because it doesn't have any bounds on the appropriate side, or
because it is empty and has altogether too many upper and lower bounds.
The idea goes back to classical times; in a geometric form it was
proposed by Eudoxus. It re-surfaced in the nineteenth century as part
of the general programme of putting calculus on a sound logical footing,
and is now generally called `Dedekind's axiom' or `the principle of
Dedekind completeness of $\Bbb R$'.
\bigskip
\noindent{\bf Convergent sequences} A large part of the rest of the
course will involve Dedekind completeness in one way or another. For
the first application I will give two of the most important theorems on
convergence of sequences. We need some definitions.
\medskip
{\bf Definitions} A subset $A$ of $\Bbb R$ is {\bf bounded} if it has
both upper and lower bounds. A real sequence $\sequencen{x_n}$ is {\bf
bounded} if $\{x_n:n\in\Bbb N\}$ is bounded, that is, there are $a$,
$b\in\Bbb R$ such that $a\le x_n\le b$ for every $n\in\Bbb N$.
A real sequence $\sequencen{x_n}$ is {\bf non-decreasing} if
$x_n\le x_{n+1}$ for every $n\in\Bbb N$, and {\bf non-increasing} if
$x_{n+1}\le x_n$ for every $n$. Finally, a sequence is {\bf monotonic}
if it is either non-decreasing or non-increasing (or both).
\medskip
{\bf Examples} (i) If $x_n=2^n$ for $n\in\Bbb N$, then $\sequencen{x_n}$
is unbounded, monotonic (non-decreasing), not convergent.
(ii) If $x_n=\Bover1n$ for $n\ge 1$, then $\langle x_n\rangle_{n\ge 1}$
is bounded, monotonic (non-increasing), convergent (to $0$).
(iii) If $x_n=(-1)^n$ for $n\in\Bbb N$, then $\sequencen{x_n}$ is
bounded, not monotonic, not convergent.
(iv) If $x_n=(-2)^n$ for $n\in\Bbb N$, then $\sequencen{x_n}$ is
unbounded, not monotonic, not convergent.
(v) If $x_n=3$ for $n\in\Bbb N$, then $\sequencen{x_n}$ is bounded,
monotonic (simultaneously non-decreasing and non-increasing), convergent
(to $3$).
(vi) If $x_n=\Bover{(-1)^n}{n}$ for $n\ge 1$, then
$\langle x_n\rangle_{n\ge 1}$ is bounded, not monotonic, convergent (to
$0$).
\medskip
{\bf Theorem} A bounded monotonic sequence is convergent.
\medskip
\noindent{\bf proof} Let $\sequencen{x_n}$ be a bounded monotonic
sequence.
\medskip
\quad{\bf case 1} Suppose that $\sequencen{x_n}$ is non-decreasing.
Set $A=\{x_n:n\in\Bbb N\}$. Because $\sequencen{x_n}$ is bounded, $A$
has an upper bound; because $x_0\in A$, $A$ is not empty; so $\sup A$
is defined in $\Bbb R$; call it $b$.
Let $\epsilon>0$. Then $b-\epsilon<b$, so $b-\epsilon$ is not an upper
bound of $A$; let $n_0\in\Bbb N$ be such that $x_{n_0}>b-\epsilon$.
If $n\ge n_0$, then
$$\eqalignno{b-\epsilon&\le x_{n_0}\le x_n\cr
\displaycause{because $\sequence{i}{x_i}$ is non-decreasing and
$n_0\le n$}
\le b\cr
\displaycause{because $x_n\in A$ and $b$ is an upper bound of $A$}
\le b+\epsilon.\cr}$$
\noindent So $x_n\in[b-\epsilon,b+\epsilon]$ and $|x_n-b|\le\epsilon$.
As $\epsilon$ is arbitrary, $\lim_{n\to\infty}x_n=b$ and
$\sequencen{x_n}$ is convergent.
\medskip
\quad{\bf case 2} Suppose that $\sequencen{x_n}$ is non-increasing.
\medskip
\noindent{\bf first method} Set $A=\{x_n:n\in\Bbb N\}$. Because
$\sequencen{x_n}$ is bounded, $A$ has a lower bound; because
$x_0\in A$, $A$ is not empty; so $\inf A$ is defined in $\Bbb R$;
call it $b$.
Let $\epsilon>0$. Then $b+\epsilon>b$, so $b+\epsilon$ is not a lower
bound of $A$; let $n_0\in\Bbb N$ be such that $x_{n_0}<b+\epsilon$.
If $n\ge n_0$, then
$$\eqalignno{b+\epsilon&\ge x_{n_0}\ge x_n\cr
\displaycause{because $\sequence{i}{x_i}$ is non-increasing and
$n_0\le n$}
\ge b\cr
\displaycause{because $x_n\in A$ and $b$ is a lower bound of $A$}
\ge b-\epsilon.\cr}$$
\noindent So $x_n\in[b-\epsilon,b+\epsilon]$ and $|x_n-b|\le\epsilon$.
As $\epsilon$ is arbitrary, $\lim_{n\to\infty}x_n=b$ and
$\sequencen{x_n}$ is convergent.
\medskip
\noindent{\bf second method} $\sequencen{-x_n}$ is non-decreasing and
bounded, so by case 1 there is a $b$ such that
$\lim_{n\to\infty}(-x_n)=b$. Now $\lim_{n\to\infty}x_n=-b$ so
$\sequencen{x_n}$ is convergent.
\medskip
{\bf Remarks} If we think of this theorem in terms of the Analysis Game,
the moves are as follows:
\inset{Player I says `bounded monotonic sequences are convergent'.}
\noindent This is of the form `every bounded monotonic sequence
$\ldots$', so
\inset{Player II chooses a bounded monotonic sequence
$\sequencen{x_n}$.}
\noindent Now Player I is claiming, of this particular bounded monotonic
sequence, that it has a limit. So she has to tell us what that limit
is; that is,
\inset{Player I chooses $b$.}
\noindent At this point, Player I is claiming that
$\lim_{n\to\infty}x_n=b$, so they turn to the very first game:
\inset{Player II chooses $\epsilon>0$
Player I chooses $n_0\in\Bbb N$
Player II chooses $n\ge n_0$
and they look to see whether $|x_n-b|\le\epsilon$, or not.}
\noindent The proof of the theorem consists, as usual, of a description
of a strategy for Player I. Because there are two separate moves (the
choice of $b$ and the choice of $n_0$) to manage, we either have to be
very good at looking ahead or remember at least one of them. The rule
for $b$ is
$$\eqalign{b
&=\sup\{x_n:n\in\Bbb N\}\text{ if }\sequencen{x_n}\text{ is
non-decreasing},\cr
&=\inf\{x_n:n\in\Bbb N\}\text{ if }\sequencen{x_n}\text{ is
non-increasing}.\cr}$$
\noindent The rule for $n_0$ is
$$\eqalign{\text{choose }n_0\text{ such that }x_{n_0}
\ge b-\epsilon\text{ if }\sequencen{x_n}\text{ is
non-decreasing},\cr
\le b+\epsilon\text{ if }\sequencen{x_n}\text{ is
non-increasing}.\cr}$$
\noindent Most of the proof amounts, in fact, to making sure that these
rules can be applied; if we are going to set
$b=\sup\{x_n:n\in\Bbb N\}$, we must first check that
$\{x_n:n\in\Bbb N\}$ is non-empty and bounded above.
Note that the theorem depends on two hypotheses: the sequence must be
simultaneously bounded and monotonic; if it's not bounded (like
$x_n=2^n$), or not monotonic (like $x_n=(-1)^n$) it may fail to be
convergent. In the proof, therefore, we must use both these facts.
The assumption that $\sequencen{x_n}$ is bounded is used when Player I
chooses $b$, and the assumption that it is monotonic is used at the
checking stage, to see that $x_n$ is on the correct side of $x_{n_0}$
and is therefore at least as close to $b$ as $x_{n_0}$ is.
There is a similar exact economy in our use of the properties of $b$.
Looking at case 1 in the proof above, in which $b=\sup A$, we use the
fact that $b$ is an upper bound of $A$ at the end, where we say that
$x_n\le b\le b+\epsilon$; we have already used the fact that $A$ has no
upper bound less than $b$, when we said that $b-\epsilon$ is not an
upper bound of $A$, so there is an $n_0\in\Bbb N$ such that
$x_{n_0}>b-\epsilon$. When you are writing out a proof of this kind,
you should look for these things; they are a check that you aren't
leaving anything out.
\medskip
{\bf Cauchy sequences: Definition} A real sequence $\sequencen{x_n}$ is
a {\bf Cauchy sequence} if for every $\epsilon>0$ there is an
$n_0\in\Bbb N$ such that $|x_m-x_n|\le\epsilon$ for all $m$, $n\ge n_0$.
\medskip
{\bf Proposition} Every Cauchy sequence is bounded.
\medskip
\noindent{\bf proof} Let $\sequencen{x_n}$ be a Cauchy sequence. Then
there is an $n_0\in\Bbb N$ such that $|x_m-x_n|\le 1$ for every $m$,
$n\ge n_0$; in particular, $|x_n-x_{n_0}|\le 1$ for every $n\ge n_0$.
Set
\Centerline{$M=\max(|x_0|,|x_1|,\ldots,|x_{n_0}|,|x_{n_0}|+1)$;}
\noindent then $M$ is the maximum of a finite string of real numbers, so
is finite. If $n\le n_0$, then $|x_n|\le M$ because $|x_n|$ is listed
in the string; if $n\ge n_0$, then
\Centerline{$|x_n|\le|x_{n_0}|+|x_n-x_{n_0}|\le|x_{n_0}|+1\le M$.}
\noindent So $|x_n|\le M$, that is, $-M\le x_n\le M$, for every
$n\in\Bbb N$, and $\{x_n:n\in\Bbb N\}$ is bounded.
\medskip
{\bf Theorem} Every Cauchy sequence is convergent.
\medskip
\noindent{\bf proof} Let $\sequencen{x_n}$ be a Cauchy sequence. Then
it is bounded, by the proposition just above; say $a$, $b\in\Bbb R$ are
such that $a\le x_n\le b$ for every $n\in\Bbb N$. For each $n\in\Bbb
N$ set $A_n=\{x_i:i\ge n\}$. Then $a$ is a lower bound for $A_n$ and
$x_n\in A_n$, so $A_n$ is a non-empty set with a lower bound and has an
infimum $y_n=\inf A_n$, with $a\le y_n\le x_n$.
Next, $A_{n+1}\subseteq A_n$ (in fact, $A_n=A_{n+1}\cup\{x_n\}$), so
$y_n$ is also a lower bound for $A_{n+1}$, and $y_n\le y_{n+1}$.
This is true for every $n\in\Bbb N$, so $\sequencen{y_n}$ is a
non-decreasing sequence. Moreover, $a\le y_n\le x_n\le b$ for every
$n$, so $\sequencen{y_n}$ is bounded. It is therefore convergent, by
the last theorem; let $c$ be its limit.
Now $\lim_{n\to\infty}x_n-y_n=0$. \Prf\ Let $\epsilon>0$. Then there
is an $n_0\in\Bbb N$ such that $|x_i-x_n|\le\epsilon$ whenever $n$,
$i\ge n_0$. Let $n\ge n_0$. Then, for any $i\ge n$,
$|x_i-x_n|\le\epsilon$, so $x_i\ge x_n-\epsilon$. This means that
$x_n-\epsilon$ is a lower bound for $A_n$; since $y_n$ is the greatest
lower bound, $x_n-\epsilon\le y_n$. But we already know that
$y_n\le x_n$, so $|x_n-y_n|\le\epsilon$, and this is true for every
$n\ge n_0$. As $\epsilon$ is arbitrary, $\lim_{n\to\infty}x_n-y_n=0$.\
\Qed
Since $\lim_{n\to\infty}x_n-y_n=0$ and $\lim_{n\to\infty}y_n=c$,
\Centerline{$\lim_{n\to\infty}x_n
=\lim_{n\to\infty}x_n-y_n+y_n=0+c=c$.}
\noindent Thus $\sequencen{x_n}$ is convergent; as $\sequencen{x_n}$
was arbitrary, the theorem is proved.
\medskip
{\bf *Remark} Observe that we have a formula for the limit of
$\sequencen{x_n}$: it is
$$\eqalignno{c&=\lim_{n\to\infty}y_n=\sup\{y_n:n\in\Bbb N\}\cr
\displaycause{see the proof of the theorem that bounded monotonic
sequences are convergent}
&=\sup\{\inf\{x_i:i\ge n\}:n\in\Bbb N\}.\cr}$$
\noindent This last formula makes sense for any {\it bounded} sequence
$\sequencen{x_n}$ (note that in the proof above we used the fact that
$\sequencen{x_n}$ is bounded right at the beginning, but didn't need to
know any more until we came to look at $\lim_{n\to\infty}x_n-y_n$); and
it has a name; it's called `$\liminf_{n\to\infty}x_n$'.
\medskip
{\bf Theorem} Every convergent sequence is Cauchy.
\medskip
\noindent{\bf proof} Let $\sequencen{x_n}$ be a convergent sequence, with limit $b$. Let $\epsilon>0$. Then there is an $n_0\in\Bbb N$ such that
$|x_n-b|\le\Bover{\epsilon}2$ for every $n\ge n_0$. If $m$, $n\ge n_0$,
\Centerline{$|x_m-x_n|\le|x_m-b|+|x_n-b|
\le\Bover{\epsilon}2+\Bover{\epsilon}2=\epsilon$.}
\noindent As $\epsilon$ is arbitrary, $\sequencen{x_n}$ is Cauchy.
\medskip
{\bf Cauchy's General Principle of Convergence} Putting the last two theorems together, we have the following fundamental principle:
\inset{\inset{A real sequence is convergent iff it is Cauchy.}}
\noindent This is one of the most basic properties of the real numbers.
\bigskip
\noindent{Summation: Definitions}
A `series' is a sequence we mean to try to add up.
If $\sequence{k}{x_k}$ is a series, its {\bf sequence of partial sums} is the sequence $\sequencen{s_n}$ defined by
\Centerline{$s_n=\sum_{k=0}^nx_k$ for every $n\in\Bbb N$.}
\noindent The {\bf sum} of the series is
\Centerline{$\sum_{k=0}^{\infty}x_k=\lim_{n\to\infty}s_n
=\lim_{n\to\infty}\sum_{k=0}^nx_k$}
\noindent if this is defined.
\inset{\inset{The sum of the series is the limit of the sequence of partial sums.}}
\noindent A series $\sequence{k}{x_k}$ is {\bf summable} if its sequence of partial sums is convergent, that is, $\sum_{k=0}^{\infty}x_k$ is defined as a real number (not allowing $\pm\infty$).
A series $\sequence{k}{x_k}$ is {\bf absolutely summable} if the series $\sequence{k}{|x_k|}$ of its absolute values is summable, that is,
$\sum_{k=0}^{\infty}|x_k|=\lim_{n\to\infty}\sum_{k=0}^n|x_k|$ is defined as a real number (not allowing $\infty$).
\medskip
{\bf Remark} Some of our favourite series don't begin at $x_0$; e.g., $x_k=\Bover1k$. For such a series $\langle x_k\rangle_{k\ge 1}$, the sequence of partial sums has to start at the same point, and we have
$\langle s_n\rangle_{n\ge 1}$, where $s_n=\sum_{k=1}^nx_k$, and
\Centerline{$\sum_{k=1}^{\infty}x_k=\lim_{n\to\infty}\sum_{k=1}^nx_k$}
\noindent if this is defined.
\medskip
{\bf Theorem} An absolutely summable series is summable.
\medskip
\noindent{\bf proof} Let $\sequence{k}{x_k}$ be an absolutely summable series.
For each $n\in\Bbb N$, set
\Centerline{$s_n=\sum_{k=0}^nx_k$,
\quad$t_n=\sum_{k=0}^n|x_k|$.}
\noindent Then $|s_m-s_n|\le|t_m-t_n|$ for all $m$, $n\in\Bbb N$. \Prf\ (i) If $n<m$, then
\Centerline{$|s_m-s_n|=|x_{n+1}+x_{n+2}+\ldots+x_m|
\le|x_{n+1}|+|x_{n+2}|+\ldots+|x_m|=t_m-t_n=|t_m-t_n|$.}
\noindent (ii) If $m=n$ then $|s_m-s_n|=0=|t_m-t_n|$. (iii) If $n<m$ then
$|s_m-s_n|=|s_n-s_m|\le t_n-t_m=|t_m-t_n|$.\ \Qed
Now we are supposing that $\sequence{k}{x_k}$ is absolutely summable, that is, that $\sequencen{t_n}$ is convergent. We know that convergent sequences are Cauchy, so $\sequencen{t_n}$ is Cauchy. It follows that $\sequencen{s_n}$ is Cauchy. \Prf\ For any $\epsilon>0$, there is an $n_0\in\Bbb N$ such that $|t_m-t_n|\le\epsilon$ for all $m$, $n\ge n_0$. So
$|s_m-s_n|\le|t_m-t_n|\le\epsilon$ for all $m$, $n\ge n_0$. As $\epsilon$ is arbitrary, $\sequencen{s_n}$ is Cauchy.\ \Qed
But now remember that all Cauchy sequences are convergent. So $\sequencen{s_n}$ is convergent, that is, $\sequence{k}{x_k}$ is summable.
\medskip
The same ideas can be used to prove the Comparison Test. I give a simple form.
\medskip
{\bf Theorem} Suppose that $0\le x_k\le y_k$ for every $k\in\Bbb N$ and that $\sequence{k}{y_k}$ is summable. Then $\sequence{k}{x_k}$ is summable.
\medskip
\noindent{\bf proof} For each $n\in\Bbb N$, set
\Centerline{$s_n=\sum_{k=0}^nx_k$,
\quad$t_n=\sum_{k=0}^ny_k$.}
\noindent Then $|s_m-s_n|\le|t_m-t_n|$ for all $m$, $n\in\Bbb N$. \Prf\ (i) If $n<m$, then
\Centerline{$|s_m-s_n|=x_{n+1}+x_{n+2}+\ldots+x_m
\le y_{n+1}+y_{n+2}+\ldots+y_m=|t_m-t_n|$.}
\noindent (ii) If $m=n$ then $|s_m-s_n|=0=|t_m-t_n|$. (iii) If $n<m$ then
$|s_m-s_n|=s_n-s_m\le t_n-t_m=|t_m-t_n|$.\ \Qed
Now we are supposing that $\sequence{k}{y_k}$ is summable, that is, that $\sequencen{t_n}$ is convergent. We know that convergent sequences are Cauchy, so $\sequencen{t_n}$ is Cauchy. It follows that $\sequencen{s_n}$ is Cauchy, as in the last theorem. So $\sequencen{s_n}$ is convergent, that is, $\sequence{k}{x_k}$ is summable.
\medskip
{\bf Series of positive terms} Note that if $\sequence{k}{x_k}$ is a series with
every $x_k\ge 0$, and $s_n=\sum_{k=0}^nx_k$ for each $n$, then $s_{n+1}=s_n+x_{n+1}\ge s_n$ for each $n$, and $\sequencen{s_n}$ is
non-decreasing. It follows that $\sequence{k}{x_k}$ is summable iff $\sequencen{s_n}$ is bounded. (I gave as a theorem the fact that a bounded monotonic sequence is convergent. But note also that every convergent sequence is bounded, because it is Cauchy, or otherwise.)
\bigskip
\noindent{\bf Continuous functions on closed bounded intervals} Continuous functions on closed bounded intervals have several special properties of the greatest importance. One of the ones which we use most often is the following.
\medskip
{\bf Theorem} Let $f$ be a real function which is defined and continuous at every point of a closed interval $[a,b]$, where $a\le b$. Then $f$ is bounded on $[a,b]$, that is, $\{f(x):x\in[a,b]\}$ is a bounded set.
\medskip
\noindent{\bf proof} Set $A=\{x:x\in[a,b]$, $f$ is bounded on $[a,x]\}$. Then $A$ is bounded above by $b$ and $a\in A$ (because $[a,a]=\{a\}$, so $\{f(y):y\in[a,a]\}=\{f(a)\}$ is bounded). So $A$ has a least upper bound $c$ say, and $a\le c\le b$. Now $f$ is continuous at $c$, so there is a $\delta>0$ such that $|f(x)-f(c)|\le 1$ whenever $x\in[c-\delta,c+\delta]\cap\dom f$. Set $z=\min(b,c+\delta)$. Then $a\le c\le z\le b$ so $z\in[a,b]$.
Now $c-\delta<c$, so there is an $x\in A$ such that $c-\delta\le x$. We know that $\{f(y):y\in[a,x]\}$ is bounded; let $M_0$, $M_1\in\Bbb R$ be such that $M_0\le f(y)\le M_1$ whenever $y\in[a,x]$. Set $M_0'=\min(M_0,f(x)-1)$, $M'_1=\max(M_1,f(c)+1)$. If $y\in[a,z]$, then either $y\le x$ and $y\in[a,x]$ and
\Centerline{$M'_0\le M_0\le f(y)\le M_1\le M'_1$,}
\noindent or $x\le y$ and $c-\delta\le x\le y\le z\le c+\delta$ and
$y\in[c-\delta,c+\delta]\cap[a,b]\subseteq[c-\delta,c+\delta]\cap\dom f$, so
$|f(y)-c|\le 1$ and
\Centerline{$M'_0\le f(c)-1\le f(y)\le f(c)+1\le M'_1$.}
\noindent This shows that $M'_0$, $M'_1$ are bounds for $\{f(y):y\in[a,z]\}$, so
$z\in A$. But this means that $z\le c$; since $z=\min(b,c+\delta)$, we must have $z=b$ so $b\in A$. But this means that $f$ is bounded on $[a,b]$, which is what we set out to prove.
\medskip
In fact rather more is true.
\medskip
{\bf Theorem} Let $f$ be a real function defined, and continuous, at each point of a closed interval $[a,b]$, where $a\le b$. Then $f$ attains its bounds on $[a,b]$, that is, there are points $z_1$, $z_2\in[a,b]$ such that $f(z_1)\le f(x)\le f(z_2)$ for every $x\in[a,b]$.
\medskip
\noindent{\bf proof} Set $B=\{f(x):x\in[a,b]$. Then $B$ is non-empty (since it contains $f(a)$) and bounded (by the last theorem); so $c_1=\inf B$ and $c_2=\sup B$ are defined in $\Bbb R$.
\Quer\ Suppose, if possible, that $f(x)\ne c_1$ for every $x\in[a,b]$, that is, that $c_1\notin B$. Then $f(x)>c_1$ for every $x\in[a,b]$. Set $g(x)=\Bover1{f(x)-c_1}$ whenever this is defined, that is, whenever $x\in\dom f$ and $f(x)\ne c_1$. Then $g$ is continuous at any point where $f$ is continuous and $f(x)\ne c_1$ (because the function $y\mapsto\Bover1{y-c_1}$ is continuous); in particular, $g$ is continuous at every point of $[a,b]$. There is therefore some $K$ such that $g(x)\le K$ for every $x\in[a,b]$, because continuous functions on closed bounded intervals are bounded. But this means that
$f(x)-c_1\ge\Bover1K$ for every $x\in[a,b]$, that is,
$f(x)\ge c_1+\Bover1K$ for every $x\in[a,b]$, that is, $c_1+\Bover1K$ is a lower bound for $B$, and $c_1$ is not the greatest lower bound of $B$.\ \Bang
So we have to conclude that there is some $z_1\in[a,b]$ such that $f(z_1)=c_1$, and now we have $f(z_1)\le f(x)$ for every $x\in[a,b]$.
I have still to find $z_2$. To do this, {\it either} repeat the argument just above, upside down, as follows:
\inset{\Quer\ Suppose, if possible, that $f(x)\ne c_2$ for every $x\in[a,b]$, that is, that $c_2\notin B$. Then $f(x)<c_2$ for every $x\in[a,b]$. Set $g(x)=\Bover1{c_2-f(x)}$ whenever this is defined, that is, whenever $x\in\dom f$ and $f(x)\ne c_2$. Then $g$ is continuous at any point where $f$ is continuous and $f(x)\ne c_2$ (because the function $y\mapsto\Bover1{c_2-y}$ is continuous); in particular, $g$ is continuous at every point of $[a,b]$. There is therefore some $K$ such that $g(x)\le K$ for every $x\in[a,b]$, because continuous functions on closed bounded intervals are bounded. But this means that
$c_2-f(x)\ge\Bover1K$ for every $x\in[a,b]$, that is,
$f(x)\le c_2-\Bover1K$ for every $x\in[a,b]$, that is, $c_2-\Bover1K$ is an upper bound for $B$, and $c_2$ is not the least upper bound of $B$.\ \Bang
So we have to conclude that there is some $z_2\in[a,b]$ such that $f(z_2)=c_2$, and now we have $f(x)\le f(z_2)$ for every $x\in[a,b]$}
\noindent {\it or} do the same inversion by introducing a new function $h$ say, defined by saying that $h(x)=-f(x)$ for every $x\in\dom f$. Then $h$ is defined and continuous at any point where $f$ is (because the function $y\mapsto -y$ is continuous), in particular, at every point of $[a,b]$. By the previous argument, there is a point $z_2\in[a,b]$ such that $-f(z_2)=h(z_2)\le h(x)=-f(x)$ for every $x\in[a,b]$, that is, $f(x)\le f(z_2)$ for every $x\in[a,b]$.
\medskip
{\bf Theorem} Let $f$ be a real function defined, and continuous, at each point of a closed interval $[a,b]$, where $a\le b$. If $c$ lies between $f(a)$ and $f(b)$, then there is a $z\in[a,b]$ such that $f(z)=c$.
\medskip
\noindent{\bf proof (a)} Suppose that $f(a)\le c\le f(b)$. Set
\Centerline{$A=\{x:x\in[a,b],\,f(x)\le c\}$.}
\noindent Then $A$ is bounded above by $b$, and $a\in A$, so $z=\sup A$ is defined and $a\le z\le b$. As $z\in[a,b]$, $f$ is continuous at $z$.
\Quer\ If $f(z)<c$, then $c-f(z)>0$, so there is a $\delta>0$ such that
$|f(x)-f(c)|\le c-f(z)$ whenever $x\in\dom f$ and $|x-z|\le\delta$. Consider $x=\min(b,z+\delta)$. Then $a\le z\le x\le b$ so $f(x)$ is defined, and $z\le x\le z+\delta$ so $f(x)\le f(z)+(c-f(z))=c$. But this means that $x\in A$. On the other hand, since $f(z)<c\le f(b)$, $z$ cannot be equal to $b$, so $z<b$ and $z<x$ and $z$ is not an upper bound for $A$.\ \Bang
\Quer\ If $f(z)>c$, then $\Bover12(f(z)-c)>0$, so there is a $\delta>0$ such that $| f(x)-f(c)|\le \bover12(f(z)-c)$ whenever $x\in\dom f$ and
$|x-z|\le\delta$. Now there must be an $x\in A$ such that
$z-\delta\le x\le z$, in which case $f(x)\le c$ and $|x-z|\le\delta$, so
\Centerline{$f(z)-c\le f(z)-f(x)\le|f(x)-f(z)|\le\Bover12(f(z)-c)$,}
\noindent which is impossible.\ \Bang
We are forced to conclude that $f(z)=c$; which is what we were looking for.
\medskip
{\bf (b)} We still have to deal with the case in which
$f(b)\le c\le f(a)$. Just as in the last theorem, we have a choice: {\it either} repeat the argument above with half the signs exchanged, as follows:
\inset{Set
\Centerline{$A=\{x:x\in[a,b],\,f(x)\ge c\}$.}
\noindent Then $A$ is bounded above by $b$, and $a\in A$, so $z=\sup A$ is defined and $a\le z\le b$. As $z\in[a,b]$, $f$ is continuous at $z$.
\Quer\ If $f(z)>c$, then $f(z)-c>0$, so there is a $\delta>0$ such that
$|f(x)-f(c)|\le f(z)-c$ whenever $x\in\dom f$ and $|x-z|\le\delta$. Consider $x=\min(b,z+\delta)$. Then $a\le z\le x\le b$ so $f(x)$ is defined, and $z\le x\le z+\delta$ so $f(x)\ge f(z)-(f(z)-c)=c$. But this means that $x\in A$. On the other hand, since $f(z)>c\ge f(b)$, $z$ cannot be equal to $b$, so $z<b$ and $z<x$ and $z$ is not an upper bound for $A$.\ \Bang
\Quer\ If $f(z)<c$, then $\Bover12(c-f(z))>0$, so there is a $\delta>0$ such that $|f(x)-f(c)|\le\bover12(c-f(z))$ whenever $x\in\dom f$ and
$|x-z|\le\delta$. Now there must be an $x\in A$ such that
$z-\delta\le x\le z$, in which case $f(x)\ge c$ and $|x-z|\le\delta$, so
\Centerline{$c-f(z)\le f(x)-f(z)\le|f(x)-f(z)|\le\Bover12(c-f(z))$,}
\noindent which is impossible.\ \Bang
We are forced to conclude that $f(z)=c$}
\noindent {\it or} define a new function $h$ by saying that
\Centerline{$h(x)=-f(x)$ for every $x\in\dom f$,}
\noindent so that $h$ is defined and continuous wherever $f$ is, in particular, at every point of $[a,b]$. Now
\Centerline{$h(a)=-f(a)\le -c\le -f(b)=h(b)$,}
\noindent so by part (a) we know that there is a $z\in[a,b]$ such that
$h(z)=-c$, that is, $f(z)=c$.
So the theorem is true in this case also.
\bigskip
\noindent{\bf Differentiable Functions}
\medskip
{\bf Definition} Let $f$ be a real function. We say that $f$ is {\bf differentiable} at $a\in\Bbb R$, with {\bf derivative} $b=f'(a)$, if
$a\in\dom f$ and $\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}=b$.
\medskip
\noindent{\bf Remark} Note that if the limit is to be defined, then there must be some $\delta>0$ such that $f(x)$ is defined whenever $0<|x-a|\le\delta$; since we must also be able to calculate $f(a)$, we see that the whole interval $[x-\delta,x+\delta]$ must be included in $\dom f$.
\medskip
{\bf Lemma} If $f$ is a real function, $a\in\dom f$ and
$\lim_{x\to a}f(x)=f(a)$, then $f$ is continuous at $a$.
\medskip
\noindent{\bf proof} For every $\epsilon>0$ there is a $\delta>0$ such that $x\in\dom f$ and $|f(x)-f(a)|\le\epsilon$ whenever $0<|x-a|\le\delta$. Of course $|f(a)-f(a)|\le\epsilon$, so we see that if $|x-a|\le\delta$ and $x\in\dom f$ then $|f(x)-f(a)|\le\epsilon$. As $\epsilon$ is arbitrary, $f$ is continuous at $a$.
\medskip
{\bf Theorem} If a real function $f$ is differentiable at $a\in\Bbb R$, then $f$ is continuous at $a$.
\medskip
\noindent{\bf proof} Of course $a\in\dom f$. I seek to show that $f(a)=\lim_{x\to a}f(x)$. We know that
$\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}=f'(a)$ and that $\lim_{x\to a}x-a=0$. Since the limit of a product is the product of the limits whenever the latter is defined, we have
$$\eqalign{\lim_{x\to a}f(x)-f(a)
&=\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}\cdot(x-a)\cr
&=\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}\cdot\lim_{x\to a}x-a
=f'(a)\cdot 0=0.\cr}$$
\noindent Next, the limit of a sum is the sum of the limits whenever the latter is defined, so
$$\eqalign{\lim_{x\to a}f(x)
&=\lim_{x\to a}f(x)-f(a)+f(a)\cr
&=\lim_{x\to a}f(x)-f(a)+\lim_{x\to a}f(a)
=0+f(a)=f(a).\cr}$$
\noindent By the last lemma, $f$ is continuous at $a$.
\medskip
{\bf Proposition} Let $f$ and $g$ be real functions, both differentiable at $a\in\Bbb R$. Then $f+g$ is differentiable at $a$ and $(f+g)'(a)=f'(a)+g'(a)$.
\medskip
\noindent{\bf proof} We know that
\Centerline{$\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}=f'(a)$,
\quad$\lim_{x\to a}\Bover{g(x)-g(a)}{x-a}=g'(a)$.}
\noindent Now the limit of a sum is the sum of the limits whenever the latter is defined, so
$$\eqalign{\lim_{x\to a}\Bover{(f+g)(x)-(f+g)(a)}{x-a}
&=\lim_{x\to a}\Bover{f(x)+g(x)-f(a)-g(a)}{x-a}\cr
&=\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}+\Bover{g(x)-g(a)}{x-a}\cr
&=\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}
+\lim_{x\to a}\Bover{g(x)-g(a)}{x-a}
=f'(a)+g'(a),\cr}$$
\noindent as required.
\medskip
{\bf Proposition} Let $f$ and $g$ be real functions, both differentiable at $a\in\Bbb R$. Then $f\times g$ is differentiable at $a$ and $(f\times g)'(a)=f'(a)g(a)+f(a)g'(a)$.
\medskip
\noindent{\bf proof} We know that
\Centerline{$\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}=f'(a)$,
$\lim_{x\to a}g(a)=g(a)$;}
\noindent because the limit of a product is the product of the limits when the latter exists,
\Centerline{$\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}g(a)=f'(a)g(a)$.}
\noindent Similarly,
\Centerline{$\lim_{x\to a}\Bover{g(x)-g(a)}{x-a}=g'(a)$,
$\lim_{x\to a}f(a)=f(a)$,}
\noindent so
$\lim_{x\to a}f(a)\Bover{g(x)-g(a)}{x-a}=f(a)g'(a)$. Moreover,
$\lim_{x\to a}x-a=0$, so
\Centerline{$\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}
\cdot\Bover{g(x)-g(a)}{x-a}\cdot(x-a)=f'(a)g'(a)\cdot 0=0$.}
\noindent Now we know also that the limit of a sum is the sum of the limits when the latter exists, so
$$\eqalign{\lim_{x\to a}\Bover{(f\times g)(x)-(f\times g)(a)}{x-a}
&=\lim_{x\to a}\Bover{f(x)g(x)-f(a)g(a)}{x-a}\cr
&=\lim_{x\to a}\Bover{(f(x)-f(a))(g(x)-g(a))+(f(x)-f(a))g(a)
+f(a)(g(x)-g(a))}{x-a}\cr
&=\lim_{x\to a}\Bover{(f(x)-f(a))(g(x)-g(a))}{x-a}
+\Bover{f(x)-f(a)}{x-a}g(a)
+f(a)\Bover{g(x)-g(a)}{x-a}\cr
&=\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}\Bover{g(x)-g(a)}{x-a}(x-a)
+\Bover{f(x)-f(a)}{x-a}g(a)
+f(a)\Bover{g(x)-g(a)}{x-a}\cr
&=0+f'(a)g(a)+f(a)g'(a)
=f'(a)g(a)+f(a)g'(a).\cr}$$
\noindent But this is just what it means to say that $(f\times g)'(a)$ is defined and equal to $f'(a)g(a)+f(a)g'(a)$.
\medskip
{\bf Three basic functions} (a) If $f(x)=c$ for $x$ near $a$, then
\Centerline{$f'(a)=\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}
=\lim_{x\to a}\Bover{c-c}{x-a}
=\lim_{x\to a}0=0$.}
\noindent (Constant functions are differentiable, with derivative zero.)
\medskip
(b) If $f(x)=x$ for $x$ near $a$, then
\Centerline{$f'(a)=\lim_{x\to a}\Bover{f(x)-f(a)}{x-a}
=\lim_{x\to a}\Bover{x-a}{x-a}
=\lim_{x\to a}1=1$.}
\noindent (The identity function is differentiable, with derivative 1.)
\medskip
(c) If $f(x)=\Bover1x$ for $x$ near $a$, then
$$\eqalignno{f'(a)
&=\lim_{x\to a}\bover{f(x)-f(a)}{x-a}
=\lim_{x\to a}\bover{\bover1x-\bover1a}{x-a}\cr
&=\lim_{x\to a}-\bover1{xa}
=-\bover1a\lim_{x\to a}1x
=-\bover1a\cdot\bover1a\cr
\displaycause{because the function $x\mapsto\Bover1x$ is continuous and is defined near $a$}
&=-\bover1{a^2}.\cr}$$
\noindent (The reciprocal function is differentiable.)
\medskip
{\bf Derivatives without division} It is very useful to know the following fact.
\medskip
\noindent{\bf Lemma} Let $f$ be a real function, and $a\in\dom f$. Then $f'(a)$ is defined and equal to $b$ iff
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that $|f(x)-f(a)-b(x-a)|$ is defined and less than or equal to $\epsilon|x-a|$ whenever $|x-a|\le\delta$.}
\medskip
\noindent{\bf proof} We have
$$\eqalignno{f'(a)=b
&\iff\Forall\epsilon>0\Exists\delta>0,\,|\Bover{f(x)-f(a)}{x-a}-b|\text{ exists }\le\epsilon\text{ whenever }0<|x-a|\le\delta\cr
&\iff\Forall\epsilon>0\Exists\delta>0,\,|f(x)-f(a)-b(x-a)|
\text{ exists }\le\epsilon|x-a|\text{ whenever }0<|x-a|\le\delta\cr
\displaycause{multiplying or dividing both sides of the inequality by the strictly positive number $|x-a|$, and remembering that $|y||z|=|yz|$ for all $y$, $z\in\Bbb R$}
&\iff\Forall\epsilon>0\Exists\delta>0,\,|f(x)-f(a)-b(x-a)|
\text{ exists }\le\epsilon|x-a|\text{ whenever }|x-a|\le\delta\cr
}$$
\noindent because if $x=a$ then $|f(x)-f(a)-b(x-a)|=0=\epsilon|x-a|$.
\medskip
{\bf Theorem} (Chain Rule for differentiable functions) Let $f$ and $g$ be real functions, and suppose that $g$ is differentiable at $a$ and that $f$ is differentiable at $g(a)$. Then $f\circ g$ is differentiable at $a$, with $(f\circ g)'(a)=f'(g(a))\cdot g'(a)$.
\medskip
\noindent{\bf proof} Write $c=f'(g(a))$, $b=g'(a)$. Let $\epsilon>0$. Set $\eta=\min(1,\Bover1{1+|b|+|c|})$. Let $\delta_1>0$ be such that $g(x)$ is defined and $|g(x)-g(a)-b(x-a)|\le\eta|x-a|$ whenever $|x-a|\le\delta_1$.
Then
\inset{$|g(x)-g(a)|\le|b(x-a)|+\eta|x-a|=(|b|+\eta)|x-a|$
\hfill[key step}
\noindent whenever $|x-a|\le\delta_1$. Let $\delta_2>0$ be such that $f(y)$ is defined and $|f(y)-f(g(a))-c(y-g(a))|\le\eta|y-g(a)|$ whenever
$|y-g(a)|\le\delta_2$.
Set $\delta=\min(\delta_1,\Bover{\delta_2}{|b|+\eta})>0$. If $|x-a|\le\delta$, then $|x-a|\le\delta_1$ so $g(x)$ is defined and
$|g(x)-g(a)-b(x-a)|\le\eta|x-a|$ and
\Centerline{$|g(x)-g(a)|\le(|b|+\eta)|x-a|\le(|b|+\eta)\delta\le\delta_2$.}
\noindent This means that $f(g(x))$ is defined and
$$\eqalignno{|(f\circ g)(x)-(f\circ g)(a)-cb(x-a)|
&=|f(g(x))-f(g(a))-cb(x-a)|\cr
&\le|f(g(x))-f(g(a))-c(g(x)-g(a))|+|c(g(x)-g(a))-cb(x-a)|
&\text{[key step}\cr
&\le\eta|g(x)-g(a)|+|c||g(x)-g(a)-b(x-a)|\cr
\displaycause{because $|g(x)-g(a)|\le\delta_2$}
&\le\eta(|b|+\eta)|x-a|+|c|\eta|x-a|\cr
&=\eta(|b|+\eta+|c|)|x-a|
\le\eta(|b|+1+|c|)|x-a|
\le\epsilon|x-a|.\cr}$$
\noindent As $\epsilon$ is arbitrary, $(f\circ g)'(a)$ is defined and equal to $cb$, as claimed.
\bigskip
{\bf Rolle's Theorem and the Mean Value Theorem}
The last section of the course introduces one of the fundamental theorems of analysis. Before embarking on the main results we need to tidy things up a little.
\medskip
{\bf Proposition} (a) Suppose that $f$ is a real function, that $d<a$ in
$\Bbb R$, and that $f(x)\ge c$ whenever $d\le x<a$. If
$\lim_{x\uparrow a}f(x)$ is defined, then $\lim_{x\uparrow a}f(x)\ge c$.
(b) Suppose that $f$ is a real function, that $a<d$ in
$\Bbb R$, and that $f(x)\ge c$ whenever $a<x\le d$. If
$\lim_{x\downarrow a}f(x)$ is defined, then $\lim_{x\downarrow a}f(x)\ge c$.
\medskip
\noindent{\bf proof (a)} Set $b=\lim_{x\uparrow a}f(x)$. \Quer\ Suppose, if possible, that $b<c$. Then $c-b>0$; set $\epsilon=\Bover12(c-b)>0$. There is a $\delta>0$ such that $f(x)$ is defined and $|f(x)-b|\le\epsilon$ whenever $a-\delta\le x<a$. Set $x=\max(a-\delta,d)$; then $f(x)\ge c$, and also
$|f(x)-b|\le\epsilon$, so
\Centerline{$b\ge f(x)-|f(x)-b|\ge c-\epsilon
=b+2\epsilon-\epsilon=b+\epsilon>b$,}
\noindent which is impossible.\ \Bang
So $b$ must be greater than or equal to $c$, as claimed.
\medskip
{\bf (b)} Set $b=\lim_{x\downarrow a}f(x)$. \Quer\ Suppose, if possible, that $b<c$. Then $c-b>0$; set $\epsilon=\Bover12(c-b)>0$. There is a $\delta>0$ such that $f(x)$ is defined and $|f(x)-b|\le\epsilon$ whenever
$a<x\le a+\delta$. Set $x=\min(a+\delta,d)$; then $f(x)\ge c$, and also
$|f(x)-b|\le\epsilon$, so
\Centerline{$b\ge f(x)-|f(x)-b|\ge c-\epsilon
=b+2\epsilon-\epsilon=b+\epsilon>b$,}
\noindent which is impossible.\ \Bang
So $b$ must be greater than or equal to $c$, as claimed.
\medskip
{\bf Proposition} Let $f$ be a real function and $a$, $b\in\Bbb R$. Then $\lim_{x\to a}f(x)=b$ iff $\lim_{x\uparrow a}f(x)=\lim_{x\downarrow a}f(x)=b$.
\medskip
\noindent{\bf proof (a)} Suppose that $\lim_{x\to a}f(x)=b$.
\medskip
\quad{\bf (i)} Let $\epsilon>0$. Then there is a $\delta>0$ such that $f(x)$ is defined and $|f(x)-b|\le\epsilon$ whenever $0<|x-a|\le\delta$. If now
$a-\delta\le x<a$, $|x-a|=a-x\in\ocint{0,\delta}$ so $f(x)$ is defined and $|f(x)-b|\le\epsilon$. As $\epsilon$ is arbitrary, $\lim_{x\uparrow a}f(x)=b$.
\medskip
\quad{\bf (ii)} Let $\epsilon>0$. Then there is a $\delta>0$ such that $f(x)$ is defined and $|f(x)-b|\le\epsilon$ whenever $0<|x-a|\le\delta$. If now
$a<x\le a+\delta$, $|x-a|=x-a\in\ocint{0,\delta}$ so $f(x)$ is defined and $|f(x)-b|\le\epsilon$. As $\epsilon$ is arbitrary,
$\lim_{x\downarrow a}f(x)=b$.
\medskip
{\bf (b)} Now suppose that
$\lim_{x\downarrow a}f(x)=\lim_{x\uparrow a}f(x)=b$. Let $\epsilon>0$. Then
\inset{there is a $\delta_1>0$ such that $x\in\dom f$ and
$|f(x)-b|\le\epsilon$ whenever $a-\delta_1\le x<a$,
there is a $\delta_2>0$ such that $x\in\dom f$ and
$|f(x)-b|\le\epsilon$ whenever $a<x\le a+\delta_2$.}
\noindent Set $\delta=\min(\delta_1,\delta_2)>0$. If
$0<|x-a|\le\delta$, then
\inset{{\it either} $x<a$, in which case
$a-\delta_1\le a-\delta\le x<a$, so $x\in\dom f$ and
$|f(x)-b|\le\epsilon$,
{\it or} $x>a$, in which case
$a<x\le a+\delta\le a+\delta_1$, so $x\in\dom f$ and
$|f(x)-b|\le\epsilon$.}
\noindent Thus in either case $f(x)$ is defined and
$|f(x)-b|\le\epsilon$; as $\epsilon$ is arbitrary,
$\lim_{x\to a}f(x)=b$.
\medskip
{\bf Rolle's Theorem} Let $f$ be a real function, defined and continuous at each point of a closed interval $[a,b]$, where $a<b$ in $\Bbb R$. Suppose that $f$ is differentiable at each point of the open interval $\ooint{a,b}$, and that $f(a)=f(b)$. Then there is a $z\in\ooint{a,b}$ such that $f'(z)=0$.
\medskip
\noindent{\bf proof} Recall that a continuous function on a closed bounded interval is bounded and attains its bounds. So there are $z_1$, $z_2\in[a,b]$ such that $f(z_1)\le f(x)\le f(z_2)$ for every $x\in[a,b]$.
\medskip
{\bf case 1} Suppose that $z_1\in\ooint{a,b}$. Then $f'(z_1)$ is defined, and
\Centerline{$f'(z_1)=\lim_{x\to z_1}\Bover{f(x)-f(z_1)}{x-z_1}
=\lim_{x\downarrow z_1}\Bover{f(x)-f(z_1)}{x-z_1}
=\lim_{x\uparrow z_1}\Bover{f(x)-f(z_1)}{x-z_1}$}
\noindent by the last proposition.
If $z_1<x\le b$, then $f(x)-f(z_1)\ge 0$ so
$\Bover{f(x)-f(z_1)}{x-z_1}\ge 0$; but this means that
$f'(z_1)=\lim_{x\downarrow z_1}\Bover{f(x)-f(z_1)}{z-z_1}\ge 0$, because the limit of a non-negative function (if defined) must be non-negative.
If $a\le x<z_1$, then $f(x)-f(z_1)\ge 0$ so
$\Bover{f(x)-f(z_1)}{x-z_1}\le 0$; but this means that
$f'(z_1)=\lim_{x\downarrow z_1}\Bover{f(x)-f(z_1)}{z-z_1}\le 0$.
Putting these together, we see that $f'(z_1)=0$ and we can take $z=z_1$.
\medskip
{\bf case 2} Suppose that $z_2\in\ooint{a,b}$. Then $f'(z_2)$ is defined, and
\Centerline{$f'(z_2)=\lim_{x\to z_2}\Bover{f(x)-f(z_2)}{x-z_2}
=\lim_{x\downarrow z_2}\Bover{f(x)-f(z_2)}{x-z_2}
=\lim_{x\uparrow z_2}\Bover{f(x)-f(z_2)}{x-z_2}$}
\noindent by the last proposition.
If $z_2<x\le b$, then $f(x)-f(z_2)\le 0$ so
$\Bover{f(x)-f(z_2)}{x-z_2}\le 0$; but this means that
$f'(z_2)=\lim_{x\downarrow z_2}\Bover{f(x)-f(z_2)}{z-z_2}\le 0$.
If $a\le x<z_2$, then $f(x)-f(z_2)\le 0$ so
$\Bover{f(x)-f(z_2)}{x-z_2}\ge 0$; but this means that
$f'(z_2)=\lim_{x\downarrow z_2}\Bover{f(x)-f(z_2)}{z-z_2}\ge 0$.
Putting these together, we see that $f'(z_2)=0$ and we can take $z=z_2$.
\medskip
{\bf case 3} Suppose that neither $z_1$ nor $z_2$ belong to $\ooint{a,b}$; that is, that both are either $a$ or $b$. Then $f(z_1)$ must be either $f(a)$ or $f(b)$; but these are equal, so $f(z_1)=f(a)$. Similarly $f(z_2)=f(a)$. This means that
$f(a)\le f(x)\le f(a)$, that is, $f(x)=f(a)$, for every $x\in\ooint{a,b}$; and if we take any $z\in\ooint{a,b}$ (e.g., $z=\Bover12(a+b)$), we shall have $f'(z)=0$.
(Note that it is only in case 3 that we need to know that $f(a)=f(b)$.)
\medskip
{\bf Mean Value Theorem} Let $f$ be a real function, defined and continuous at each point of a closed interval $[a,b]$, where $a<b$ in $\Bbb R$. Suppose that $f$ is differentiable at each point of the open interval $\ooint{a,b}$. Then there is a $z\in\ooint{a,b}$ such that $f'(z)=\Bover{f(b)-f(a)}{b-a}$.
\medskip
\noindent{\bf proof} Set $h(x)=f(x)-\Bover{f(b)-f(a)}{b-a}x$ for $x\in\dom f$. Observe that because the constant function
$x\mapsto-\Bover{f(b)-f(a)}{b-a}$ and the identity function $x\mapsto x$ are differentiable everywhere, so is their product
$x\mapsto-\Bover{f(b)-f(a)}{b-a}x$, and $h$ will be differentiable at every point at which $f$ is differentiable and continuous at every point at which $f$ is continuous; in particular, $h$ is continuous everywhere in $[a,b]$ and differentiable everywhere in $\ooint{a,b}$. Also
\Centerline{$h(b)-h(a)=f(b)-f(a)-\Bover{f(b)-f(a)}{b-a}(b-a)=0$.}
\noindent By Rolle's theorem, there is a $z\in\ooint{a,b}$ such that
\Centerline{$0=h'(z)=f'(z)-\Bover{f(b)-f(a)}{b-a}$,}
\noindent and $f'(z)=\Bover{f(b)-f(a)}{b-a}$, as required.
\medskip
{\bf Cauchy's Mean Value Theorem} Let $f$ and $g$ be real functions, both defined and continuous at each point of a closed interval $[a,b]$, where $a<b$ in $\Bbb R$. Suppose that both $f$ and $g$ are differentiable at each point of the open interval $\ooint{a,b}$, and that $g'(x)\ne 0$ for every $x\in\ooint{a,b}$. Then there is a $z\in\ooint{a,b}$ such that
$\Bover{f'(z)}{g'(z)}=\Bover{f(b)-f(a)}{g(b)-g(a)}$.
\medskip
\noindent{\bf proof} We need to know that $g(a)\ne g(b)$. But Rolle's theorem tells us that if $g(a)$ and $g(b)$ were equal, there would be a point $x\in\ooint{a,b}$ such that $g'(x)=0$; which isn't so.
Set $h(x)=f(x)(g(b)-g(a))-g(x)(f(b)-f(a))$ for $x\in\dom f$. Then $h$ will be differentiable at every point at which $f$ and $g$ are differentiable and continuous at every point at which $f$ and $g$ are continuous; in particular, $h$ is continuous everywhere in $[a,b]$ and differentiable everywhere in $\ooint{a,b}$. Also
\Centerline{$h(b)-h(a)
=(f(b)-f(a))(g(b)-g(a))-(g(b)-g(a))(f(b)-f(a))=0$.}
\noindent By Rolle's theorem, there is a $z\in\ooint{a,b}$ such that
\Centerline{$0=h'(z)=f'(z)(g(b)-g(a))-g'(z)(f(b)-f(a))$.}
\noindent Since $g(b)-g(a)$ and $g'(z)$ are both non-zero,
$\Bover{f'(z)}{g'(z)}=\Bover{f(b)-f(a)}{g(b)-g(a)}$, as required.
\end
|
http://dlmf.nist.gov/10.20.E3.tex | nist.gov | CC-MAIN-2017-22 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2017-22/segments/1495463608659.43/warc/CC-MAIN-20170526105726-20170526125726-00319.warc.gz | 133,631,122 | 672 | \[\frac{2}{3}(-\zeta)^{\frac{3}{2}}=\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\mathrm{%
d}t=\sqrt{z^{2}-1}-\mathop{\mathrm{arcsec}\/}\nolimits z,\] |
http://ofap.ulstu.ru/resources/1443/two_certificate.tex?association=resources&parent_scaffold=fremantle%2Fguest%2Fpeople&person_id=203 | ulstu.ru | CC-MAIN-2022-05 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-05/segments/1642320304471.99/warc/CC-MAIN-20220124023407-20220124053407-00548.warc.gz | 44,568,740 | 2,640 | %% LaTeX2e-файл
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{\Large ОФАП УОЦ НИТ}
\bigskip\bigskip
\begin{tabu}{ b{5cm} >{\centering}b{19cm} }
\includegraphics [height=4.5cm]{logo.png} &
Ульяновский государственный технический университет
Ульяновский областной центр новых информационных технологий
Областной фонд алгоритмов и программ
\bigskip
{\Huge \textit{Свидетельство №\,1443}}
\bigskip
о регистрации программно-информационного продукта
\end{tabu}
\bigskip
\begin{tabu}{ b{5cm} >{\raggedright \itshape}p{19cm} }\hline \\
Наименование: & \textbf{Истоки понятия коммуникации в социологической теории Ю. Хабермаса} \\
Тип: & Научная публикация \\
URI: & \textbf{http://ofap.ulstu.ru/1443} \\
Авторы: & Алхасов Алхас Ярахмедович --
Кафедра философии, социологии и политологии (УлГУ/ФГНиСТ) \\
Дата регистрации: & 31 мая 2016 г. \\
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& Директор Ульяновского областного центра НИТ & & \textit{К.В.\,Святов} \\
\psbarcode{2016-05-31 http://ofap.ulstu.ru/1443}{format=compact layers=4}{azteccode}
\rowfont{\small} & Администратор базы данных ОФАП & & \textit{Ю.А.\,Лапшов} \\
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https://projects.mako.cc/source/state_of_wikimedia_research_2015/blob_plain/cf79c51c1cffe95eed4fe82aa967fb08f992cf5a:/20150717-wikimania_research.tex | mako.cc | CC-MAIN-2021-31 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-31/segments/1627046153391.5/warc/CC-MAIN-20210727103626-20210727133626-00112.warc.gz | 482,067,611 | 6,180 | \documentclass[xcolor=dvipsnames]{beamer}
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\newcommand\dropshadow[3]{%
\node[black!30!white] at (#1+0.1,#2-0.1) {
\scalebox{2}{\Huge \textbf{#3}}
};
\node at (#1,#2) {
\scalebox{2}{\Huge \e{#3}}
};
}
% create an empty quotetxt so we can reuse it
\newcommand{\quotetxt}{}
% more flexible non-tikz alternative with no dropshadow
\newlength{\centertxtlen}
\makeatletter
\newcommand\centertext[2]{%
\setlength{\centertxtlen}{#1}%
\setlength{\centertxtlen}{0.48\centertxtlen}%
{\centering
\fontsize{#1}{2\centertxtlen}\selectfont
\e{#2}
}
}
% add function to stop numbering appendix slides
\newcommand{\backupbegin}{
\newcounter{framenumberappendix}
\setcounter{framenumberappendix}{\value{framenumber}}
}
\newcommand{\backupend}{
\addtocounter{framenumberappendix}{-\value{framenumber}}
\addtocounter{framenumber}{\value{framenumberappendix}}
}
% packages i use in essentially every document
\usepackage{graphicx}
\usepackage{url}
% \usepackage{dcolumn}
% \usepackage{booktabs}
% replace footnotes with symbols instead of numbers
\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
\usepackage{perpage}
\MakePerPage{footnote}
%% BEAMER THEME STUFF
\usetheme[pageofpages=/,% String used between the current page and the
% total page count.
bullet=default,% Use circles instead of squares for bullets.
titleline=false,% Show a line below the frame title.
alternativetitlepage=true,% Use the fancy title page.
titlepagelogo=figures/logo.pdf,% Logo for the first page.
%watermark=watermark-polito,% Watermark used in every page.
watermarkheight=100px,% Height of the watermark.
watermarkheightmult=4,% The watermark image is 4 times bigger
% than watermarkheight.
]{Torino}
\usecolortheme{mako}
\useinnertheme{rectangles}
%\setbeamertemplate{blocks}[rounded][]
\setbeamercolor{block title}{bg=makopurple3, fg=White}
\setbeamertemplate{items}[default]
\setbeamertemplate{blocks}[shadow=true]
\usepackage{tcolorbox}
% These options will be applied to all `tcolorboxes`
\tcbset{%
noparskip,
colback=makopurple5, %background color of the box
colframe=makopurple1, %color of frame and title background
coltext=black, %color of body text
coltitle=white, %color of title text
arc=0em,
left=0.1em,
right=0.1em,
fonttitle=\bfseries,
alerted/.style={coltitle=red,
colframe=gray!40},
example/.style={coltitle=black,
colframe=green!20,
colback=green!5},
}
%\useoutertheme{infolines}
%\usepackage[breaklinks]{hyperref}
\hypersetup{colorlinks=true, linkcolor=Black, citecolor=Black, filecolor=makopurple1,
urlcolor=Plum, unicode=true}
% create a boldface version of the header
\setbeamerfont{frametitle}{series=\bfseries}
\setbeamerfont{title}{series=\bfseries}
% tweak the beamer font to make it a bit lists a bit smaller
\setbeamerfont*{itemize/enumerate body}{size=\small}
\setbeamerfont*{itemize/enumerate subbody}{size=\footnotesize}
\setbeamerfont*{itemize/enumerate subsubbody}{size=\footnotesize}
% indent the margins of the itemize lists a little bit
\setlength{\leftmargin}{0pt}
\setlength{\leftmargini}{0.7cm}
\setlength{\leftmarginii}{0.7cm}
% create a new \e{} command to make things purple and bold
\newcommand{\e}[1]{\textcolor{makopurple1}{\textbf{#1}}}
% remove the nagivation symbols
\setbeamertemplate{navigation symbols}{}
\title{Presentation Title}
% \subtitle{Presentation Subtitle}
\author[Benj. Mako Hill]{\textbf{Benjamin Mako Hill}\\ [email protected]}
\institute[UW/Harvard]{\textbf{University of Washington}\\
Department of Communication\\
\emph{Assistant Professor}\\
\hspace{1pt}\\
\textbf{Harvard University}\\
Berkman Center for Internet and Society\\
\emph{Resident Fellow}}
\date{December 2, 1980}
\newcommand{\credit}[1]{%
\tikz[overlay]{\node at (current page.south east)
[anchor=south east,yshift=1.1em,xshift=0.35em]
{\smaller \smaller {[}#1{]}};}}
\begin{document}
% remove some of the space in the itemize to make it quite compact
\let\olditemize\itemize
\renewcommand\itemize{\olditemize\itemsep-1pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% SLIDE: Title Slide
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[plain]
\begin{tikzpicture}
\node at (current page.center) [xshift=-3.5cm, yshift=0.5cm, opacity=0.4]
{\includegraphics[height=\paperheight]{figures/wikimedia_projects.png}};
\node at (current page.south east)
[anchor=south east,text width=1.8\paperwidth,align=right,color=black]
{
{\spaceskip 0.3em%
\fontsize{2.5em}{2.5em}
\selectfont {\bf \color{makopurple4} The State of Wikimedia\\
Research: 2014-2015} \par}
\vspace{1em}
{\spaceskip 0.3em%
\fontsize{2.0em}{2.1em}
\selectfont {\bf \color{black} Benjamin Mako Hill\\
Tilman Bayer\\
Aaron Shaw\\
Wikimania 2015, Mexico City\\
July 17, 2015} \par}
};
\end{tikzpicture}
% Removed to accommodate Tilman's setup
% \input{vc}
%
% \tikz[overlay,shift=(current page.south west)]{\node [xshift=5.6em,yshift=0.5em]{\colorbox{makopurple1}{\color{white} \tt \smaller \smaller \smaller revision:\ \VCRevision\ (\VCDateTEX)}};}
\note{I've been doing this for many years. I started in 2008 and
have done this almost every single year since.
This began as an excuse for me to make sure I was up to date on
Wikimedia Research.}
\end{frame}
%% SLIDE: Anecdote from Wikimania 2008
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\quotetxt}{``This talk will try to [provide] a quick
tour – a literature review in the scholarly parlance – of the last
year's academic landscape around Wikimedia and its projects geared
at non-academic editors and readers. It will try to categorize,
distill, and describe, from a birds eye view, the academic landscape
as it is shaping up around
our project.''\\
\hfill – \e{From my Wikimania 2008 Submission}}
\begin{frame}
{\smaller \quotetxt}
\pause
\includegraphics[width=\textwidth]{figures/google_scholar_result.png}
\pause
\tikz{\draw (current page.center) [xshift=-2.1cm, yshift=0.9cm, color=red]
ellipse (1.5cm and 0.5cm);}
\note<1>{Back in Wikimania 2008, I set out to run a session at
Wikimania that would provide a comprehensive literature review of
articles in Wikipedia published in the last year.
\begin{quote}
\quotetxt
\end{quote}
Then, about two weeks before Wikimania, I did the scholar search
so I could build the literature.}
\note<2->{I tried to import the whole list into Zotero and managed
to get banned for abusing the Google Scholar because they thought
that no human being could realistically consume the amount of
material published on Wikipedia that year.
So anyway, I had a 45 minute talk so it worked out to 3.45 seconds
to per paper...
And believe it or not, this year is even bigger.
And my talk is even shorter.}
\end{frame}
%% SLIDE: Citations Per Year
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\includegraphics[width=\textwidth]{figures/citations_by_year.pdf}
\centering
{\smaller \emph{Number of citation, per year, with the term
“wikipedia” in the title.\\
(Source: Google scholar results. Accessed: 2013-08-06)}}
\note{Academics have written \e{a lot} of papers about
Wikipedia. There are more than 500 papers published about
Wikipedia each year and although we've reached and moved past a
peak it seems, it's not slowing by much.}
\end{frame}
\begin{frame}
\begin{itemize}
\larger \larger
\item \e{2968} Wikipedia-related publications in the Scopus database
as of November 2013
\item \e{160} recent publications reviewed or mentioned in the 12 issues
of the Wikimedia Research Newsletter August 2013-July 2014.
\end{itemize}
\end{frame}
%% SLIDE: My Scope Conditions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\includegraphics[width=\textwidth]{figures/multiple_issues.png}
\larger \larger
In selecting papers for this session, the goal is always to choose
examples of work that:
\begin{itemize}
\larger \larger
\item Represent \e{important themes} from Wikipedia in the last year.
\item Research that is likely to be of \e{interest} to Wikimedians.
\item Research by people who are \e{not at Wikimania}.
\item \ldots with a bias towards \e{peer-reviewed} publications
\end{itemize}
\note{This is my disclaimer slide...
Within these goals, the selections are \e{incomplete}, and \e{wrong}.}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Paper Summaries}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Event Prediction}
% \begin{frame}
% \centertext{6em}{Event Prediction}
% \note{Mako
% This was the year that studies of readership of Wikipedia really
% blossomed. People figured out how to use the view data. Much of
% what they used it for was prediction.}
% \end{frame}
% \begin{frame}
% \frametitle{Wikipedia Viewership and Flu Prediction}
% \larger \larger McIver, David J., and John
% S. Brownstein. ``\e{Wikipedia Usage Estimates Prevalence of
% Influenza-Like Illness in the United States in Near Real-Time}.''
% PLoS Comput Biol 10, no. 4 (April 17, 2014):
% e1003581. \href{http://dx.doi.org/10.1371/journal.pcbi.1003581}{doi:10.1371/journal.pcbi.1003581}.
% \end{frame}
% \begin{frame}
% \frametitle{Wikipedia Viewership and Flu Prediction: Motivation}
% \begin{itemize}
% \larger \larger
% \item \e{Google Flu Trends} uses search engine queries to try to
% predict influenza epidemics more quickly than traditional methods.
% \item ..but it has been criticized as being biased (e.g., by media coverage).
% \item WP is freely available and viewership data is free, unlike
% Google which is proprietary.
% \end{itemize}
% \note{2009 H1N1 Swine Flu broke GFT.}
% \end{frame}
% \begin{frame}
% \frametitle{Wikipedia Viewership and Flu Prediction: Methods}
% \begin{itemize}
% \larger \larger \larger
% \item Measure traffic to flu related articles on Wikipedia
% \item Compare to the ``gold standard'' data from the Center for
% Disease Control (CDC)
% \end{itemize}
% \end{frame}
% \begin{frame}
% \frametitle{Wikipedia Viewership and Flu Prediction: Results}
% \centering
% \includegraphics[width=\textwidth]{figures/flu.png}
% \note{\begin{itemize}
% \larger \larger
% \item Wikipedia better than Google at predicting peak flu weeks.
% \item Wikipedia better at predicting relative influenza rates.
% \end{itemize}}
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% SLIDE: Other Resources
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{More Resources}
\begin{itemize}
\larger \larger
\item \e{Wikimedia Research Newsletter} [[:meta:Research:Newsletter]] / @WikiResearch
\item \e{WikiSym/OpenSym} (This August in San Francisco!)
\item \e{WikiPapers Repository} [http://wikipapers.referata.com]
\item \e{Much More}
\end{itemize}
{\centering
\includegraphics[width=0.25\textwidth]{figures/Wikimedia_Research_Newsletter_Logo.png}
}
\note{Those are my six exemplary studies from the past year.
There has been just tons and tons of work in this area. Trying to
talk about this in 20 minutes strikes me as increasingly crazy
every year I try to do it.
The most important source, now going for a couple years, is the
Wikimedia Research Newsletter which is published monthly in the (English)
Signpost and syndicated on the Wikimedia Research.
But there are other resources as well. And I encourage you to get
involved.}
\end{frame}
\subsection{Meta-Analyses}
\begin{frame}
\frametitle{Meta-Analyses}
\begin{itemize}
\larger \larger
\item Okoli et al.,
\href{https://spectrum.library.concordia.ca/978618/}{``The sum of
all human knowledge'': a systematic review of scholarly research
on the content of Wikipedia}.
\item Bar-Ilan and Aharony,
\href{http://dl.acm.org/citation.cfm?doid=2615569.2615643}{Twelve
years of Wikipedia research}.
\item Taraborelli. \href{https://meta.wikimedia.org/wiki/Research:Newsletter/2013/August\#Keynote\_on\_applicable\_Wikipedia\_research}{Keynote
on Wikipedia Research}. OpenSym 2013. Hong Kong.
\item Benkler, Shaw, and Hill,
\href{http://mako.cc/academic/benkler\_shaw\_hill-peer\_production\_ci.pdf}{Peer
Production: A Modality of Collective Intelligence}.
\end{itemize}
\end{frame}
\end{document}
|
http://porocila.imfm.si/2010/mat/clani/kandic.tex | imfm.si | CC-MAIN-2023-14 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296949097.61/warc/CC-MAIN-20230330035241-20230330065241-00096.warc.gz | 36,782,232 | 1,440 | \clan {Marko Kandi}
%--------------------------------------------------------
% A. objavljene znanstvene monografije
%--------------------------------------------------------
%\begin{skupina}{A}
%\disertacija
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\magisterij
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\monografija
% {AVTORJI}
% {NASLOV}
% {ZALOZBA}
% {KRAJ} {DRZAVA} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% B. raziskovalni clanki sprejeti v objavo v znanstvenih
% revijah in v zbornikih konferenc
%--------------------------------------------------------
\begin{skupina}{B}
\sprejetoRevija
{}
{Ideal-triangularizability of nil-algebras generated by positive operators}
{Proc.\ Amer.\ Math.\ Soc.}
\sprejetoRevija
{R.~Drnov\v{s}ek, \crta}
{More on positive commutators}
{J. Math.\ Anal.\ Appl.}
%\sprejetoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% C. raziskovalni clanki objavljeni v znanstvenih revijah
% in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{C}
%\objavljenoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\objavljenoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% D. urednistvo v znanstvenih revijah in zbornikih
% znanstvenih konferenc
%--------------------------------------------------------
%\begin{skupina}{D}
%\urednikRevija
% {OPIS}
% {REVIJA}
%\urednikZbornik
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% E. organizacija mednarodnih in domacih znanstvenih
% srecanj
%--------------------------------------------------------
%\begin{skupina}{E}
%\organizacija
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% F. vabljena predavanja na tujih ustanovah in
% mednarodnih konferencah
%--------------------------------------------------------
%\begin{skupina}{F}
%\predavanjeUstanova
% {NASLOV}
% {OPIS}
% {USTANOVA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\predavanjeKonferenca
% {NASLOV}
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% G. aktivne udelezbe na mednarodnih in domacih
% konferencah
%--------------------------------------------------------
%\begin{skupina}{G}
%\konferenca
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% H. strokovni clanki
%--------------------------------------------------------
%\begin{skupina}{H}
%\clanekRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\clanekZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% I. razno
%--------------------------------------------------------
%\begin{skupina}{I}
%\razno
% {OPIS}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% tuji gosti
%--------------------------------------------------------
%\begin{seznam}
%\gost {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO} {POVABILO}
%\end{seznam}
%--------------------------------------------------------
% gostovanja
%--------------------------------------------------------
%\begin{seznam}
%\gostovanje {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{seznam}
|
http://www-verimag.imag.fr/~monniaux/download/ifenoughspace.sty | imag.fr | CC-MAIN-2019-13 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2019-13/segments/1552912202530.49/warc/CC-MAIN-20190321172751-20190321194751-00359.warc.gz | 223,306,245 | 762 | \ProvidesPackage{ifenoughspace}[2011/05/12 David Monniaux]
\newcommand{\ifenoughspace}[3]{%
\@tempdimc\pagegoal \advance\@tempdimc-\pagetotal%
\ifdim #1>\@tempdimc #3 \else #2\fi}
|
https://www.authorea.com/users/330376/articles/457162/download_latex | authorea.com | CC-MAIN-2021-39 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-39/segments/1631780057973.90/warc/CC-MAIN-20210926205414-20210926235414-00473.warc.gz | 676,641,126 | 6,930 | \documentclass[10pt]{article}
\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}
\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
\errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother
\usepackage{natbib}
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}
\usepackage{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[ngerman,english]{babel}
\usepackage{float}
\begin{document}
\title{Salvage chemotherapy after failure of targeted therapy in a child with
BRAF V600E low grade glioma}
\author[1]{Musthafa Raswoli}%
\author[1]{Liana Nobre}%
\author[2]{Cynthia Hawkins}%
\author[2]{Ute Bartels}%
\author[3]{Uri Tabori}%
\author[1]{Eric Bouffet}%
\affil[1]{Hospital for Sick Children}%
\affil[2]{The Hospital for Sick Children}%
\affil[3]{The Hospital for Sick Children , toronto}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
Targeted therapies are increasingly used in the management of pediatric
low grade glioma. How-ever for patients who show resistance to these
treatments, limited options are available. We pre-sent the case of a
patient with BRAFV600 mutated low grade glioma who showed progression on
a combination of trametinib and dabrafenib. Discontinuation of treatment
was associated with a life-threatening deterioration and reintroduction
of targeted therapy had no effect. The patient eventually showed a
dramatic response to TPCV (thiguanine, procarbazine, CCNU and
vincris-tine) , which suggests a role of chemotherapy in these
situations.%
\end{abstract}%
\sloppy
\textbf{Introduction}
The last decades have witnessed a progressive paradigm shift in the
non-surgical management of pediatric low-grade gliomas (PLGG). While
radiation was historically the standard treatment, its use has
progressively decreased with the development of chemotherapy strategies
that have shown the possibility to delay or avoid radiotherapy in most
patients\textsuperscript{1}. A new shift is currently happening with
evidence that alterations within the mitogen-activated protein kinase
(MAPK) pathway affect most PLGGs and represent potential therapeutic
targets\textsuperscript{2}. Phase I and II trials have shown the
efficacy of BRAF and MEK inhibitors in recurrent and/or refractory
PLGG\textsuperscript{34}. Response rates to these agents are promising
and appear to be superior to those observed with chemotherapy, and
clinical trials are ongoing to compare the efficacy of the new agents
with standard chemotherapy in treatment na\selectlanguage{ngerman}ïve patients
\textsuperscript{5}. However, a number of questions remain unanswered
regarding these new compounds, and in particular with regard to the
management of patients who fail to respond or progress while treated
with MEK or BRAF inhibitors. Herein we describe a patient with BRAFV600E
mutated PLGG who progressed on a combination of dabrafenib and
tramatenib. This patient eventually responded to a chemotherapeutic
regimen consisting of thioguanine, procarbazine, lomustine and
vincristine (TPCV) resulting in reversal of life-threatening symptoms.
\textbf{Case description}
A 2-year old male with a 1.5-year history of slight developmental delay
and progressive visual disturbance associated nystagmus was referred for
MRI by his ophthalmologist. Family history was potentially contributory
due to consanguineous marriage. The child was showing marked nystagmus,
and poor vision. No stigmata of neurofibromatosis type I (NF1) were
present. MRI revealed a suprasellar mass, measuring 4.9cm x 5.6cm x
3.8cm extending to the hypothalamus, basal ganglia, thalami, posterior
limb of internal capsules, optic tract and lateral geniculate nuclei, as
well as the cerebral peduncles (Fig 1). The patient underwent a biopsy
and histological examination was consistent with the diagnosis of PLGG
with piloid features. Immunostains showed positivity for GFAP and
BRAFV600 mutation. Tumor cells were immunopositive for MLH1, MSH2, MSH6
and PMS2. The MIB-1 proliferation index was up to 2\%. Further testing
revealed the presence of a FGFR1 N546K mutation.
Postoperatively, the patient started weekly vinblastine for 70 weeks. He
showed a mixed response to this treatment, with regression of the
chiasmatic component, while the temporal component experienced mild
progression. Due to the significant size of the tumor and poor vision,
the child was then placed on dabrafenib. His tumor remained stable,
although the vision continued to deteriorate. In this context,
tramatenib was added 15 months later. No clear benefit of this addition
was observed clinically and radiologically, and vision continued to
progressively deteriorate to complete blindness. After two years of dual
treatment, the patient presented with dysphagia, ataxia, dizziness,
right-sided weakness and slurred speech and the decision was made switch
to a trial of immune checkpoint inhibitor, as the tumor tested positive
for PD-L1. One week after termination of the combination therapy, the
patient experienced acute deterioration, with poor responsiveness,
unsteady gait and worsening speech. His MRI showed substantial increase
in tumor size. Re-introduction of dabrafenib did not improve his
symptoms, and the GSC of the child was fluctuating between 7 and 11,
despite high-doses of dexamethasone (4 mg//m2 QID). Parents declined the
option of whole brain radiotherapy and it was decided to initiate TPCV.
His condition remained severely compromised for 2 months. Due to severe
side effects of high-dose dexamethasone, bevacizumab was initiated at
10mg/kg via IV biweekly for 4 cycles. His clinical condition started to
improve after two cycles of TPCV, and after 4 cycles, the MRI scan show
marked improvement (Fig 1). TPCV was continued for 6 cycles, after which
it was discontinued due to thrombocytopenia. During treatment, he was
offered intensive rehabilitation and was able to resume school 4 months
following initiation of treatment. Two years after completion of TPCV,
the patient is clinically well, with a Lansky score of 100\% and stable
MRI scan.
\textbf{Discussion}
PLGG encompass several entities characterized by different
histopahtological features. During the last decade, molecular
characterization of PLGG has identified a number of recurrent
alterations, with the majority involving the MAPK
pathway\textsuperscript{6}. In children without NF1, the most common
alterations in this pathway include the\emph{KIAA1549-BRAF} fusion and
the BRAFV600E mutation\textsuperscript{7}. Both alterations can be
targeted, and the use of MEK inhibitors in patients with PLGG harboring
BRAF fusion or BRAF inhibitors in patients with PLGG harboring BRAFV600E
mutation have shown promising response rate\textsuperscript{43}.
Although PLGGs associated with BRAF mutation appear to have a more
aggressive behavior \textsuperscript{8}, they show an excellent response
to BRAF inhibitors. Hargrave et al conducted a trial of dabrafenib in
children with BRAF mutated PLGG and reported a partial response in 19 of
27 evaluable patients, and a 1-year event free survival of
85\%\textsuperscript{3}. More recently, Nobre et al compared 2 cohorts
of patients with BRAF mutated PLGG treated with chemotherapy or BRAF
inhibitors (vemurafenib or dabrafenib) and demonstrated a clear
advantage for targeted treatments, with an overall objective response
rate of 28\% and 71\%, respectively\textsuperscript{5}. Report on the
combination of BRAF and MEK inhibitors in BRAF mutated LGG are pending.
The rationale for this combination is based on superior outcome observed
in patients with BRAF mutated melanomas randomized to either dabrafenib
or dabrafenib and trametinib\textsuperscript{9}.
However, the risk of developing resistance to BRAF inhibition exists and
has been well documented in melanoma. In PLGG, there has been limited
focus on this issue and the management of patients who show progression
during treatment with BRAF inhibitors remains challenging. Mulcahy Levy
et al recently described a patient with BRAFV600E mutated~ganglioglioma
who developed resistance to vemurafenib. The addition of chloroquine to
vemurafenib was associated with durable clinical improvement as well
radiographic response\textsuperscript{10}. A clinical trial is ongoing
to confirm these early data.
As the response rate of BRAFV600E PLGG is extremely high for BRAF
inhibition, the fact that our patient had limited benefit of BRAF
inhibition is intriguing. A plausible mechanism can be the additional
FGFR1 N546K mutation. In contrast to FGFR gene fusions, point mutations
in FGFR1 tend to be associated with other RAS/MAPK mutations and are
showing less favorable outcome\textsuperscript{7}.
Interestingly, the acute clinical deterioration of our patient within a
week of discontinuation of the BRAF/MEK inhibition is not uncommon in
BRAFV600E tumors. This precluded his inclusion in any clinical trial.
Re-challenging with dabrafenib did not show any evidence of efficacy and
the decision was made to proceed with chemotherapy, using the TPCV
regimen. This regimen has been compared to the combination of
vincristine and carboplatin in a randomized trial and has shown better
event free survival at 5 years\textsuperscript{11}. However, this trial
did not include any molecular study and whether chemotherapy regimens
have a better activity in specific molecular subgroups is unknown.
Our experience is intriguing and provides some evidence that salvage
chemotherapy is still an option when targeted therapy fail. As most
BRAFV600E PLGG recur rapidly after cessation of targeted therapies, the
sustained tumor control, 2 years after completion of chemotherapy is
encouraging, suggesting a different and potentially synergistic role for
chemotherapy is such situations. Further studies will determine whether
a combination of targeted and chemotherapy regimens are superior to each
of these as a single modality.
\textbf{Disclosures:}
Eric Bouffet is a member of an advisory board of Novartis. Other authors
do not report any conflict of interest.
\textbf{Figure 1:}
MRI scan (FLAIR Sequence) at the time of diagnosis (A); at the time of
progression, before starting TPCV (B); 6 months after initiation of TPCV
(C); 18 months after completion of TPCV (D)
References
1. Reddy AT, Packer RJ. Chemotherapy for low-grade gliomas.
\emph{Child's nervous system : ChNS : official journal of the
International Society for Pediatric Neurosurgery} 1999;\textbf{15} (10):
506-13.
2. Packer RJ, Pfister S, Bouffet E, et al. Pediatric low-grade gliomas:
implications of the biologic era.\emph{Neuro-oncology} 2016.
3. Hargrave DR, Bouffet E, Tabori U, et al. Efficacy and Safety of
Dabrafenib in Pediatric Patients with BRAF V600 Mutation-Positive
Relapsed or Refractory Low-Grade Glioma: Results from a Phase I/IIa
Study. \emph{Clinical cancer research : an official journal of the
American Association for Cancer Research} 2019;\textbf{25} (24):
7303-11.
4. Fangusaro J, Onar-Thomas A, Young Poussaint T, et al. Selumetinib in
paediatric patients with BRAF-aberrant or neurofibromatosis type
1-associated recurrent, refractory, or progressive low-grade glioma: a
multicentre, phase 2 trial. \emph{The Lancet Oncology} 2019; \textbf{20}
(7): 1011-22.
5. Nobre L, Zapotocy M, Ramaswamy V, et al. Outcomes of BRAF V600E
Pediatric Gliomas Treated With Targeted BRAF Inhibition. \emph{JCO
Precision Oncology} 2020; \textbf{4} : 561-71.
6. Zhang J, Wu G, Miller CP, et al. Whole-genome sequencing identifies
genetic alterations in pediatric low-grade gliomas. \emph{Nature
genetics} 2013; \textbf{45} (6): 602-12.
7. Ryall S, Zapotocky M, Fukuoka K, et al. Integrated Molecular and
Clinical Analysis of 1,000 Pediatric Low-Grade Gliomas. \emph{Cancer
Cell} 2020; \textbf{37} (4): 569-83 e5.
8. Lassaletta A, Zapotocky M, Mistry M, et al. Therapeutic and
Prognostic Implications of BRAF V600E in Pediatric Low-Grade Gliomas.
\emph{Journal of clinical oncology : official journal of the American
Society of Clinical Oncology} 2017;\textbf{35} (25): 2934-41.
9. Long GV, Flaherty KT, Stroyakovskiy D, et al. Dabrafenib plus
trametinib versus dabrafenib monotherapy in patients with metastatic
BRAF V600E/K-mutant melanoma: long-term survival and safety analysis of
a phase 3 study. \emph{Annals of oncology : official journal of the
European Society for Medical Oncology} 2017; \textbf{28} (7): 1631-9.
10. Mulcahy Levy JM, Zahedi S, Griesinger AM, et al. Autophagy
inhibition overcomes multiple mechanisms of resistance to BRAF
inhibition in brain tumors. \emph{eLife} 2017;\textbf{6} .
11. Ater JL, Zhou T, Holmes E, et al. Randomized study of two
chemotherapy regimens for treatment of low-grade glioma in young
children: a report from the Children's Oncology Group.\emph{Journal of
clinical oncology : official journal of the American Society of Clinical
Oncology} 2012; \textbf{30} (21): 2641-7.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Figure-1/Figure-1}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\end{document}
|
https://ctan.math.washington.edu/tex-archive/info/examples/Math/03-05-4.ltx | washington.edu | CC-MAIN-2022-27 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-27/segments/1656103324665.17/warc/CC-MAIN-20220627012807-20220627042807-00186.warc.gz | 245,115,648 | 1,283 | %%
%% Ein Beispiel der DANTE-Edition
%% Mathematiksatz mit LaTeX
%% 2. Auflage
%%
%% Beispiel 03-05-4 auf Seite 31.
%%
%% Copyright (C) 2012 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
%%
%% ====
% Show page(s) 1
%%
%%
\documentclass[]{exaarticle}
\pagestyle{empty}
\setlength\textwidth{352.81416pt}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\setlength\parindent{0pt}
\setlength\parskip{1ex plus 0.2ex}
\StartShownPreambleCommands
\usepackage{varwidth}
\StopShownPreambleCommands
\begin{document}
Ähnlich zum Zeilenmodus kann man auch abgesetzte Formeln einrahmen, allerdings
müssen \ldots\ (dieser Text dient nur der Demonstration der Absatzbreite und hat
sonst keinen Sinn)
\begin{center}
\fbox{\begin{varwidth}{\dimexpr\linewidth-2\fboxsep-2\fboxrule}
\[ f(x)=\int\limits_1^{\infty}\frac{1}{x^2}\,\mathrm{d}x=1 \]
\end{varwidth}}
\fbox{\begin{varwidth}{\dimexpr\linewidth-2\fboxsep-2\fboxrule}
\[ f(x)=\int\limits_1^{\infty}\frac{1}{x^2}\,\mathrm{d}x=1 \]
\vspace{\belowdisplayshortskip}
\end{varwidth}}
\end{center}
\end{document}
|
https://cheatography.com/deleted-2754/cheat-sheets/self-measures-for-self-esteem/latex/ | cheatography.com | CC-MAIN-2021-39 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-39/segments/1631780057416.67/warc/CC-MAIN-20210923013955-20210923043955-00686.warc.gz | 221,979,062 | 3,795 | \documentclass[10pt,a4paper]{article}
% Packages
\usepackage{fancyhdr} % For header and footer
\usepackage{multicol} % Allows multicols in tables
\usepackage{tabularx} % Intelligent column widths
\usepackage{tabulary} % Used in header and footer
\usepackage{hhline} % Border under tables
\usepackage{graphicx} % For images
\usepackage{xcolor} % For hex colours
%\usepackage[utf8x]{inputenc} % For unicode character support
\usepackage[T1]{fontenc} % Without this we get weird character replacements
\usepackage{colortbl} % For coloured tables
\usepackage{setspace} % For line height
\usepackage{lastpage} % Needed for total page number
\usepackage{seqsplit} % Splits long words.
%\usepackage{opensans} % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem} % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath} % Symbols
\usepackage{MnSymbol} % Symbols
\usepackage{wasysym} % Symbols
%\usepackage[english,german,french,spanish,italian]{babel} % Languages
% Document Info
\author{{[}deleted{]}}
\pdfinfo{
/Title (self-measures-for-self-esteem.pdf)
/Creator (Cheatography)
/Author ({[}deleted{]})
/Subject (Self Measures for Self Esteem Cheat Sheet)
}
% Lengths and widths
\addtolength{\textwidth}{6cm}
\addtolength{\textheight}{-1cm}
\addtolength{\hoffset}{-3cm}
\addtolength{\voffset}{-2cm}
\setlength{\tabcolsep}{0.2cm} % Space between columns
\setlength{\headsep}{-12pt} % Reduce space between header and content
\setlength{\headheight}{85pt} % If less, LaTeX automatically increases it
\renewcommand{\footrulewidth}{0pt} % Remove footer line
\renewcommand{\headrulewidth}{0pt} % Remove header line
\renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit
% This two commands together give roughly
% the right line height in the tables
\renewcommand{\arraystretch}{1.3}
\onehalfspacing
% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
\newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols
\newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns
\newcommand{\tn}{\tabularnewline} % Required as custom column type in use
% Font and Colours
\definecolor{HeadBackground}{HTML}{333333}
\definecolor{FootBackground}{HTML}{666666}
\definecolor{TextColor}{HTML}{333333}
\definecolor{DarkBackground}{HTML}{0DA3A3}
\definecolor{LightBackground}{HTML}{EFF9F9}
\renewcommand{\familydefault}{\sfdefault}
\color{TextColor}
% Header and Footer
\pagestyle{fancy}
\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
\SetRowColor{DarkBackground}
\vspace{-7pt}
{\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
\hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
}
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
\vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Self Measures for Self Esteem Cheat Sheet}}}} \\
\normalsize{by \textcolor{DarkBackground}{{[}deleted{]}} via \textcolor{DarkBackground}{\uline{cheatography.com/2754/cs/14564/}}}
\end{tabulary}
\end{multicols}}
\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
\SetRowColor{FootBackground}
\mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}} \\
\vspace{-2pt}{[}deleted{]} \\
\uline{cheatography.com/deleted-2754} \\
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
\SetRowColor{FootBackground}
\mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}} \\
\vspace{-2pt}Published 4th February, 2018.\\
Updated 4th February, 2018.\\
Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
\SetRowColor{FootBackground}
\mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}} \\
\SetRowColor{white}
\vspace{-5pt}
%\includegraphics[width=48px,height=48px]{dave.jpeg}
Measure your website readability!\\
www.readability-score.com
\end{tabulary}
\end{multicols}}
\begin{document}
\raggedright
\raggedcolumns
% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.
\begin{multicols*}{2}
\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Self Measures for Self Esteem}} \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Below is a list of statements dealing with your general feelings about yourself. Please indicate how strongly you agree or disagree with each statement. \newline % Row Count 4 (+ 4)
Response \newline % Row Count 5 (+ 1)
SA -Strongly Agree \newline % Row Count 6 (+ 1)
A - Agree \newline % Row Count 7 (+ 1)
D -Disagree \newline % Row Count 8 (+ 1)
SD -Strongly Disagree% Row Count 9 (+ 1)
} \tn
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{Source: \seqsplit{http://fetzer.org/sites/default/files/images/stories/pdf/selfmeasures/Self\_Measures\_for\_Self-Esteem\_ROSENBERG\_SELF-ESTEEM.pdf}} \tn
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}
\begin{tabularx}{8.4cm}{x{4.332 cm} x{2.28 cm} x{0.988 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Assessment}} \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Question}} & {\bf{Response}} & {\bf{Score}} \tn
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
1.On the whole, I am satisfied with myself. & {[}\_\_\_{]} & {[}\_\_\_{]} \tn
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
2.At times I think I am no good at all. & {[}\_\_\_{]} & {[}\_\_\_{]}-R \tn
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
3. I feel that I have a number of good qualities. & {[}\_\_\_{]} & {[}\_\_\_{]} \tn
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
4. I am able to do things as well as most other people. & {[}\_\_\_{]} & {[}\_\_\_{]} \tn
% Row Count 12 (+ 3)
% Row 5
\SetRowColor{white}
5. I feel I do not have much to be proud of. & {[}\_\_\_{]} & {[}\_\_\_{]}-R \tn
% Row Count 14 (+ 2)
% Row 6
\SetRowColor{LightBackground}
6. I certainly feel useless at times. & {[}\_\_\_{]} & {[}\_\_\_{]}-R \tn
% Row Count 16 (+ 2)
% Row 7
\SetRowColor{white}
7. I feel that I'm a person of worth, at least on an equal plane with others. & {[}\_\_\_{]} & {[}\_\_\_{]} \tn
% Row Count 20 (+ 4)
% Row 8
\SetRowColor{LightBackground}
8. I wish I could have more respect for myself. & {[}\_\_\_{]} & {[}\_\_\_{]}-R \tn
% Row Count 23 (+ 3)
% Row 9
\SetRowColor{white}
9. All in all, I am inclined to feel that I am a failure. & {[}\_\_\_{]} & {[}\_\_\_{]}-R \tn
% Row Count 26 (+ 3)
% Row 10
\SetRowColor{LightBackground}
10. I take a positive attitude toward myself. & {[}\_\_\_{]} & {[}\_\_\_{]} \tn
% Row Count 29 (+ 3)
% Row 11
\SetRowColor{white}
& {\bf{Total Score}}\{\{ar\}\} & {[}\_\_\_{]} \tn
% Row Count 31 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{Scoring:}} \newline Items 2, 5, 6, 8, 9 are reverse scored. \newline "Strongly Disagree" 1 point, "Disagree" 2 points, \newline "Agree" 3 points, and "Strongly Agree" 4 points. \newline \newline Sum scores for all ten items. Keep scores on a continuous scale. Higher scores indicate higher self-esteem.} \tn
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}
\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Rosenberg}} \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/davidpol_1517769457_rses-tmp.png}}} \tn
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}
% That's all folks
\end{multicols*}
\end{document}
|
https://mirror.las.iastate.edu/tex-archive/macros/latex/contrib/exsheets/exsheets_en.tex | iastate.edu | CC-MAIN-2023-06 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2023-06/segments/1674764494976.72/warc/CC-MAIN-20230127101040-20230127131040-00413.warc.gz | 410,571,934 | 27,906 | % !arara: pdflatex
% !arara: biber
% arara: pdflatex
% arara: pdflatex
% arara: pdflatex
% --------------------------------------------------------------------------
% the EXSHEETS package
%
% Yet another package for the creation of exercise sheets
%
% --------------------------------------------------------------------------
% Clemens Niederberger
% Web: https://bitbucket.org/cgnieder/exsheets/
% E-Mail: [email protected]
% --------------------------------------------------------------------------
% Copyright 2011-2019 Clemens Niederberger
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3
% of this license or (at your option) any later version.
% The latest version of this license is in
% http://www.latex-project.org/lppl.txt
% and version 1.3 or later is part of all distributions of LaTeX
% version 2005/12/01 or later.
%
% This work has the LPPL maintenance status `maintained'.
%
% The Current Maintainer of this work is Clemens Niederberger.
% --------------------------------------------------------------------------
% If you have any ideas, questions, suggestions or bugs to report, please
% feel free to contact me.
% --------------------------------------------------------------------------
\documentclass[load-preamble+]{cnltx-doc}
\usepackage{exsheets}
\usepackage{bookmark}
\setcnltx{
package = {exsheets} ,
authors = Clemens Niederberger ,
email = [email protected] ,
url = {http://www.mychemistry.eu/forums/forum/exsheets/} ,
title = the \ExSheets\ bundle ,
info = {%
the packages \ExSheets{} and \ExSheetslistings \\
\emph{or}\\
Yet another package for the creation of exercise sheets and exams.%
} ,
module-sep = {\,>>\,} ,
index-setup = { othercode=\footnotesize,level=\section} ,
abstract = {%
\ExSheets\ provides means to create exercises or questions and their
corresponding solutions. The questions can be divided into classes and
can be printed selectively. Meta-data to questions can be added and
recovered.
\par
The solutions may be printed where they are, can be collected and printed
at a later point in the document alltogether or section-wise or
selectively by \acs{id}.\par
\ExSheets\ provides a comprehensive interface for styling the headings of
questions and solutions.%
} ,
add-cmds = {
% exsheets:
addpoints,
blank,
C,
checkedchoicebox,choice,choicebox,
correct,
CurrentQuestionID,
DebugExSheets,
DeclareExSheetsHeadingContainer,
DeclareQuestionClass,DeclareQuestionProperty,
examspace,
exlabel,
ForEachQuestion,
GetQuestionClass,
GetQuestionProperty,
grade,
includequestions,iflastquestion,
IfQuestionPropertyF,IfQuestionPropertyT,IfQuestionPropertyTF,
IfQuestionSubtitleF,IfQuestionSubtitleT,IfQuestionSubtitleTF,
lastvariant,
NewQuSolPair,NewLstQuSolPair,
numberofquestions,
points,pointssum,
PrintIfIncludeActiveF,PrintIfIncludeActiveT,PrintIfIncludeActiveTF,
PrintQuestionClassF,PrintQuestionClassT,PrintQuestionClassTF,
printsolutions,
PrintSolutionsF,PrintSolutionsTF,PrintSolutionsT,
questionsincludedlast,QuestionNumber,RenewQuSolPair,
S,
SetQuestionProperties,
SetupExSheets,
SetVariations,
startnewitemline,
sumpoints,
totalpoints,
variant,vary,
% cntformats:
@cntfmts@parsed@pattern,
AddCounterPattern,
eSaveCounterPattern,
NewCounterPattern,
ReadCounterFrom,
ReadCounterPattern,ReadCounterPatternFrom,
SaveCounterPattern,
% tasks:
NewTasks,
settasks,
task
} ,
add-envs = {
question,
solution,
tasks
} ,
add-silent-cmds = {
acs,
bigstar,bottomrule,
citetitle,cs,color,
DeclareInstance,DeclareTemplateInterface,
endmdframed,endspacing,
keyis,
leftthumbsup,
mdframed,midrule,
rightarrow,rlap,
s,sample,setlength,
spacing,square,
tabcolsep,
textcite,
textcolor,
tmpa,tmpb,
toprule
}
}
\usepackage{exsheets-listings}
\microtypesetup{tracking=scshape}
\defbibheading{bibliography}[\bibname]{\section{#1}}
\usepackage[biblatex]{embrac}[2012/06/29]
\ChangeEmph{[}[,.02em]{]}[.055em,-.08em]
\ChangeEmph{(}[-.01em,.04em]{)}[.04em,-.05em]
\usepackage{booktabs,array,ragged2e}
\newpackagename\ExSheets{ExSheets}
\newpackagename\ExSheetslistings{ExSheets-listings}
\newpackagename\cntformats{cntformats}
\newpackagename\Tasks{tasks}
% ----------------------------------------------------------------------------
% other packages, bibliography, index
\usepackage{xcoffins,wasysym,enumitem,siunitx}
\usepackage[accsupp]{acro}
\DeclareAcronym{faq}{
short = faq ,
long = Frequently Asked Questions ,
format = \scshape ,
pdfstring = FAQ ,
accsupp = FAQ
}
\DeclareAcronym{id}{
short = id ,
long = Identifier ,
format = \scshape ,
pdfstring = ID ,
accsupp = ID
}
\DeclareAcronym{ctan}{
short = ctan ,
long = Comprehensive \TeX\ Archive Network ,
format = \scshape ,
pdfstring = CTAN ,
accsupp = CTAN
}
\usepackage{filecontents}
\usepackage{csquotes}
\addbibresource{biblatex-examples.bib}
\addbibresource{\jobname.bib}
\begin{filecontents*}{\jobname.bib}
@online{tex.sx:131546,
title = {Fixing lstlisting inside \ExSheets\ \code{question} and
\code{solution} environments} ,
author = {Stefano Bragaglia} ,
url = {http://tex.stackexchange.com/q/131546/5049} ,
date = {2013-09-04} ,
urldate = {2013-09-22}
}
@online{tex.sx:133907,
title = {How do I extract repeated functionality (\ExSheets\ workaround) to
make it reusable?} ,
author = {Forkrul Assail} ,
url = {http://tex.stackexchange.com/q/133907/5049} ,
date = {2013-09-18} ,
urldate = {2013-09-22}
}
\end{filecontents*}
% ----------------------------------------------------------------------------
% example definitions that have to be done in the preamble:
\DeclareQuestionClass{difficulty}{difficulties}
\DeclareQuestionProperty{notes}
\DeclareQuestionProperty{reference}
\DeclareQuestionProperty{topic}
\DeclareRelGrades{
1 = 1 ,
{1,5} = .9167 ,
2 = .8333 ,
{2,5} = .75 ,
3 = .6667 ,
{3,5} = .5833 ,
4 = .5
}
\usepackage{amssymb}
\let\checkmark\relax
\usepackage{dingbat}
\DeclareRobustCommand*\questionstar{\texorpdfstring{\bonusquestionsign}{* }}
\DeclareRobustCommand*\bonusquestionsign{\llap{$\bigstar$\space}}
\NewQuSolPair
{question*}[name=\questionstar Bonus Question]
{solution*}[name=\questionstar Solution]
\NewTasks[style=multiplechoice]{multiplechoice}[\choice](3)
\newcommand*\correct{\PrintSolutionsTF{\checkedchoicebox}{\choicebox}}
\usepackage{alphalph}
\NewPatternFormat{aa}{\alphalph}
\NewCounterPattern{testa}{ta}
% \AfterPackage!{hyperref}{%
% \pdfstringdefDisableCommands{\def\questionstar{* }}%
% }
\begin{document}
\part{Preliminaries}
\section{Licence and Requirements}
\license
\ExSheets\ loads and needs the following packages:
\needpackage{l3kernel}~\cite{bnd:l3kernel}, \pkg{xparse}, \pkg{xtemplate},
\pkg{l3keys2e}\footnote{all three \CTANurl{l3packages}}~\cite{bnd:l3packages},
\pkg{l3sort}\footnote{\CTANurl{l3experimental}}~\cite{bnd:l3experimental},
\needpackage{xcolor}~\cite{pkg:xcolor}, \needpackage{ulem}~\cite{pkg:ulem},
\needpackage{etoolbox}~\cite{pkg:etoolbox},
\needpackage{environ}~\cite{pkg:environ}, and
\pkg{pgfcore}\footnote{\CTANurl[graphics]{pgf}}~\cite{pkg:pgf}. \ExSheets\
calls \cs*{normalem} (from the \pkg{ulem} package).
\section{Motivation}
There are already quite a number of packages that allow the creation of
exercise sheets or written exams. Just to name the most common ones:
\pkg{eqexam}~\cite{pkg:eqexam}, \pkg{exam}~\cite{cls:exam},
\pkg{examdesign}~\cite{pkg:examdesign}, \pkg{exercise}~\cite{pkg:exercise},
\pkg{probsoln}~\cite{pkg:probsoln}, \pkg{answers}~\cite{pkg:answers},
\pkg{esami}~\cite{pkg:esami}, \pkg{exsol}~\cite{pkg:exsol} (and many
more \ldots).
One thing I missed in all packages that I've tried out\footnote{Well, probably
I didn't try hard enough\ldots} was a high flexibility in choosing which
questions and solutions should be printed, where which solutions should be
printed and so on, combined with the possibility to assign questions to
different classes so one could for example create two versions of an exam out
of the box. And --~I can't get enough~-- I also want to be able to use/design
different layouts for questions additional to a standard section-like format.
All these points are realized in \ExSheets.
Additionally one should be able to assign some sort of meta-data to questions
that of course should be easily reusable. How this can be done is explained
in section~\ref{sec:additional_info}.
Then there is --~at least in Germany~-- the habit of having lists of exercises
aligned in columns but counting from the left to the right instead from up to
down. That's why the \pkg{tasks} package was developed as part of \ExSheets{}
and was distributed as part of the bundle\changedversion{0.15}. Now it is a
package of its own but is loaded by \ExSheets{} automatically with the
necessary setup to make them work together nicely.
\ExSheets{} has no native support for multiple choice tests but that doesn't
mean that you can't create them with \ExSheets. It just means that they may
be a bit more work with \ExSheets{} than with other packages.
I had the idea for this package in 2008. Back then my \TeX{} skills were by
far not good enough to write it. Actually, even today I wouldn't have been
able to realize it without all the l3 packages like \pkg{l3kernel} and
\pkg{l3packages}. I actively began to develop \ExSheets\ in spring~2011 but
it wasn't until now (September~2012) that I consider it stable enough for
wider usage. At the time of writing (\today) there still are probably lots of
rough edges let alone bugs so I am very interested in all kinds of feedback.
\section{Additional Packages}
\ExSheets\ actually bundles two packages: \ExSheets, \ExSheetslistings.
\ExSheetslistings\ is an add-on to \ExSheets\ that offers some functionality
to use \pkg{listings} with \ExSheets. It is presented in
part~\ref{part:listings}.
\ExSheets\ used to bundle the \pkg{translations} package,
too\changedversion{0.9i}, but doesn't any more. You can find the
\pkg{translations} package as a package of it's own on the \ac{ctan}. It also
used to bundle the packages \pkg{tasks} and
\pkg{cntformats}\changedversion{0.15}. They're available now as packages of
their own as well.
% \section{Installation and Documentation}
% If you install \ExSheets\ manually beware to put the files
% \begin{itemize}
% \item[]\verb+exsheets_headings.def+
% \item[]\verb+exsheets_headings.cfg+
% \end{itemize}
% in the same directory as the \code{exsheets.sty} file\footnote{That is, a
% directory like \code{texmf-local/tex/latex/exsheets}, probably}.
% As with every manual package installation you need to make sure to put the
% files in a directory where \TeX\ can find them and afterwards update the
% database.
% \subsection{The \pkg*{tasks} Package}
% The \pkg{tasks} package~\cite{pkg:tasks} used to be part of the \ExSheets\
% bundle but is a package of its own now\changedversion{0.15} and released
% independently. You can find it as every other package on \ctan\ and in a full
% \TeX~Live or \hologo{MiKTeX} installation.
% \subsection{The \pkg*{cntformats} Package}
% The \pkg{cntformats} package~\cite{pkg:cntformats} used to be part of the
% \ExSheets\ bundle but is a package of its own now\changedversion{0.15} and
% released independently. You can find it as every other package on \ctan\ and
% in a full \TeX~Live or \hologo{MiKTeX} installation.
% \subsection{The \pkg*{translations} Package}
% The \pkg{translations} package~\cite{pkg:translations} used to be part of the
% \ExSheets\ bundle but is a package of its own now\changedversion{0.9i} and
% released independently. You can find it as every other package on \ctan\ and
% in a full \TeX~Live or \hologo{MiKTeX} installation.
% \section{News}
% \begin{description}
% \item[Version 0.7]
% With version~0.7 there has been a potentially breaking change: the
% \code{tasks} environment previously provided by \ExSheets\ has been
% extracted into a package of its own. This does not change any syntax
% \emph{per se}. However, if you used custom settings then you'll probably run
% into some problems. The options for the environment are no longer set with
% \cs{SetupExSheets} but with \cs{settasks}. Also the object that is used for
% the list template and its instances has been renamed from
% \code{exsheets-tasks} into \code{tasks}.
% What's probably even more of a breaking change is a syntax difference of the
% \code{tasks} environment: the optional argument for the number of columns is
% \emph{no longer set in braces but parentheses}. This is deliberate as it
% reflects the optional nature of the argument better and is consistent with
% the syntax of \cs{NewTasks}, too.
% Additionally the labels of the list got an additional offset of \code{1ex}
% from the items which will lead to slightly different output. In some cases
% this might actually lead to the most annoying changes. In this case say
% \cs{settasks}\Marg{label-offset=0pt} which should cure things again.
% I am very sorry for any inconvenience! I am trying to keep such changes as
% minimal and rare as possibly. However, it is not always avoidable when a
% package is new and still a child. It will grow up eventually.
% \ExSheets' other packages -- \href{tasks_en.pdf}{\Tasks} and
% \href{cntformats_en.pdf}{\cntformats} -- have gotten their own documentation
% which are essentially extracted from this very document you're reading now.
% This manual contains links to the respective manuals.
% \item[Version v0.9i]
% The \pkg{translations} package~\cite{pkg:translations} is no longer part of
% the \ExSheets\ bundle. From now on (July~17.\@ 2013) it is provided as a
% package of its own.
% \item[Version 0.10]
% The \ExSheets\ family has got a new member: \ExSheetslistings. This package
% proposes a solution for the problem of using verbatim material in \ExSheets'
% \env{question} and \env{solution} environments. It is presented in
% part~\ref{part:listings}.
% Question now can get subtitles that are printed if the heading instance
% supports it, see section~\ref{sec:subtitles-questions}.
% \item[Version 0.11]
% The commands \cs{GetQuestionClass} and \cs{PrintQuestionClassTF} have been
% added. They're explained in section~\ref{sec:retr-class-value}.
% \item[Version 0.12]
% The \option{auto-label} is now more flexible to allow the use together with
% packages \pkg{cleveref}.
% Question properties can now be retrieved before the question is printed (by
% writing the properties to the \code{aux} file).
% \item[Version 0.13]
% New options:
% \begin{itemize}
% \item \option{chapter-hook} allows to add code to the list of solutions
% when the solutions of a new chapter are printed, see
% section~\ref{sec:solutions-print-all}.
% \item \option{section-hook} allows to add code to the list of solutions
% when the solutions of a new section are printed, see
% section~\ref{sec:solutions-print-all}.
% \end{itemize}
% \item[Version 0.14]
% New options:
% \begin{itemize}
% \item New option \option{pre-hook} to the \env{question} environment that
% allows to add code directly before the question body, see
% section~\ref{sec:opti-ques-envir}.
% \item New option \option{post-hook} to the \env{question} environment that
% allows to add code directly after the question body, see
% section~\ref{sec:opti-ques-envir}.
% \item New command \cs{ExSheetsHeading}, see
% section~\ref{sec:using-an-exsheets}.
% \item New pre-defined question properties \code{question-body},
% \code{bonus-points} and \code{counter}, see
% section~\ref{sec:additional_info}.
% \item New option \option{save-to-aux}, see
% section~\ref{sec:additional_info}.
% \end{itemize}
% \item[Version 0.15]
% \begin{itemize}
% \item The packages \pkg{tasks} and \pkg{cntformats} have been removed from
% the bundle and are now distributed as packages of their own.
% \item The options \option*{load-headings} and \option*{load-tasks} have
% been dropped. The optional functionality they provided is now provided
% all the time.
% \item New command \cs{IfQuestionPropertyTF}, see
% section~\ref{sec:additional_info}.
% \end{itemize}
% \item[Version 0.16]
% New options/changes:
% \begin{itemize}
% \item The option \option{pre-hook} to the \env{question} environment now
% places its contents before the question heading, see
% section~\ref{sec:opti-ques-envir}.
% \item New option \option{pre-body-hook} to the \env{question} environment
% which adds its contents before the question body, see
% section~\ref{sec:opti-ques-envir}.
% \item New option \option{post-body-hook} to the \env{question} environment
% which adds its contents after the question body, see
% section~\ref{sec:opti-ques-envir}.
% \item New option \option{pre-hook} to the \env{solution} environment which
% adds code before a solution, see section~\ref{sec:opti-soli-envir}.
% \item New option \option{post-hook} to the \env{solution} environment which
% adds code after a solution, see section~\ref{sec:opti-soli-envir}.
% \item New option \option{pre-body-hook} to the \env{solution} environment
% which adds its contents before the solution body, see
% section~\ref{sec:opti-soli-envir}.
% \item New option \option{post-body-hook} to the \env{solution} environment
% which adds its contents after the solution body, see
% section~\ref{sec:opti-soli-envir}.
% \end{itemize}
% \item[Version 0.17]
% New option:
% \begin{itemize}
% \item The option \option{use-saved-counter-format} has been introduced. It
% is described in section~\ref{sec:solutions} on
% page~\pageref{option:use-saved-counter-format}.
% \end{itemize}
% \item[Version 0.18]
% The package now provides the correct Danish translations, thanks to Jonas
% Nyrup.
% The macro \cs{exsheetsprintsolution} is introduced, see
% page~\pageref{exsheetsprintsolution} for a little bit of an explanation.
% The option \option{no-skip-below} is introduced which disables the insertion
% of vertical space after the question and solution environments.
% \item[Version 0.20]
% New command \cs{DeclareExSheetsHeadingContainer}.
% \item[Version 0.21] Changes:
% \begin{itemize}
% \item \cs{includequestions} issues an error if it can't find the file to
% include.
% \item question properties are now also accessable when the corresponding
% question isn't printed.
% \item The variables \verbcode+\l_exsheets_counter_qu_int+ and \\
% \verbcode+\g_exsheets_question_identification_prop+ are now public.
% \end{itemize}
% \end{description}
\section{Thanks}
I need to thank the many users who gave me feedback so far! For one thing
this shows me that \ExSheets\ is useful to people. It also led to many
improvements like new features and countless bug fixes.
\part{The \ExSheets\ package}\label{part:exsheets}
\section{Setup}
The \ExSheets\ package has three different types of options, kind of. The
first type are the classic package options which are used when you load
\ExSheets:
\begin{sourcecode}
\usepackage[<options>]{exsheets}
\end{sourcecode}
All general options can be used this way and most of them are described in
section~\ref{sec:options}. All of those options also can be set via the setup
command:
\begin{commands}
\command{SetupExSheets}[\oarg{module}\marg{options}]
\end{commands}
The second type are options that belong to a specific environment or command.
These options are either used directly with the environment/command
\begin{sourcecode}
\begin{env}[<options>]
...
\end{env}
\end{sourcecode}
or can also be set with the setup command. In the first case they only act
upon the environment or command where they're used. In the second case they
are set for all following uses of the corresponding environment or command.
The options of the second type all belong to \module*{modules}. Let's say you
want to specify some options of the \env{question} environment. You can then
say the following:
\begin{sourcecode}
\SetupExSheets[question]{option1,option2=value2}
% or:
\SetupExSheets{question/option1,question/option2=value2}
\end{sourcecode}
The \module*{module} an option belongs to is written in the left margin next
to the when the option is described.
The third type aren't options at all, actually. However, thanks to the great
\pkg{xtemplate} package you are able to define your own instances of some of
the objects used by \ExSheets. This is explained in a little more detail in
part~\ref{part:style} on page~\pageref{part:style}\,ff. This third type,
however, brings in a possible instability: the \pkg{xtemplate} package is in
an experimental and developing state. This means that the sytax of the
package may and possibly will change sometime in the future. I cannot foresee
what any consequences of that will be for \ExSheets.
\section{General Options}\label{sec:options}
The package \ExSheets\ has some options, namely the following ones:
\begin{options}
%% counter-format
\keyval{counter-format}{counter-format}\Default{qu.}
Formatting of the counter of the questions. This option takes a special
kind of string that is described in section~\ref{ssec:counter}.
\keyval{counter-within}{counter}\Default
Resets the \code{question} counter with every step of \meta{counter}.
%% auto-label
\keybool{auto-label}\Default{false}
If set to \code{true} \ExSheets\ will automatically place a
\cs*{label}\Marg{qu:\meta{id}} for each question. See
section~\ref{sec:auto-label-opti} for ways to customize this. It will
also create the question properties \code{ref} and \code{pageref}, see
section~\ref{sec:additional_info} for more on this.
%% headings
\keyval{headings}{instance}\Default{block}
Choose the style of the questions' and solutions' headings. There are two
predefined styles: \code{block} and \code{runin}.
%% headings-format
\keyval{headings-format}{code}\Default{\cs*{normalsize}\cs*{bfseries}}
This code is placed immediately before the headings of the questions and
solutions.
\keyval{subtitle-format}{code}\Default{\cs*{normalsize}\cs*{itshape}}
This code is placed immediately before the subtitle of the questions and
solutions. It only has an effect with a title instance that uses the
subtitle coffin, see section~\ref{sec:exsheets-headings}.
% skip-below
\keyval{skip-below}{dim}\Default{.5\cs*{baselineskip}}
\sinceversion{0.18}Sets the vertical space that is inserted after the
question and solution environments.
% no-skip-below
\keybool{no-skip-below}\Default{false}
\sinceversion{0.18}Disables the insertion of vertical space after the
question and solution environments.
%% totoc
\keybool{totoc}\Default{false}
This option adds the questions and solutions with their names and numbers
to the table of contents.
%% questions-totoc
\keybool{questions-totoc}\Default{false}
This option adds the questions with their names and numbers to the table
of contents.
%% solutions-totoc
\keybool{solutions-totoc}\Default{false}
This option adds the solutions with their names and numbers to the table
of contents.
%% toc-level
\keyval{toc-level}{toc level}\Default{subsection}
This option sets the level in which questions and solutions should appear
in the table of contents.
%% questions-toc-level
\keyval{questions-toc-level}{toc level}\Default{subsection}
This option sets the level in which questions should appear in the table
of contents.
%% solutions-toc-level
\keyval{solutions-toc-level}{toc level}\Default{subsection}
This option sets the level in which solutions should appear in the table
of contents.
%% use-ref
\keybool{use-ref}\Default{false}
enable referencing to sections and chapters in a way that the references
can be used with \cs{printsolutions}, see
section~\ref{sssec:print_specific_section} for details.
\end{options}
The \code{toc} options are demonstrated with section~\ref{sec:solutions:list}
and the solutions printed there being listed in the table of contents.
\section{Create Questions/Exercises and their Solutions}
Now, let's start with the most important part: the questions and (possibly)
their respective solutions.
\subsection{The \env*{question} Environment}\label{ssec:questions}
Questions are written inside the \env{question} environment:
\begin{environments}
\environment{question}[\oarg{options}\marg{points}]
The main environment: creates a new exercise/question. Both arguments are
optional!
\end{environments}
\begin{example}
\begin{question}
This is our very first very difficult to solve question!
\end{question}
\end{example}
As you can see a heading is automatically created and the question is
numbered. You can of course change both the numbering and the naming, but
more on that later.
The \env{question} environment takes an optional argument \marg{points} that
can be used to assign points to the question (as is common in written exams):
\begin{example}
\begin{question}{3}
This is our first difficult question that is worth 3 points!
\end{question}
\end{example}
These points are saved internally (see section~\ref{sec:points} for reasons
why) and are written to the right margin next to the question heading in the
default setting.
You can also assign bonus points by inserting \code{\meta{point}+\meta{bonus
points}} as argument.
\begin{example}
\begin{question}{1+1}
This question is worth 1 point and 1 bonus point.
\end{question}
\begin{question}{+3}
This question is a bonus question. It is worth 3 bonus points.
\end{question}
\end{example}
The points are counted and added to the total sum of points, see
section~\ref{sec:points} for details on this. \sinceversion{0.12}Should you
want that the points of a specific question \emph{should not be added} to the
total sum then precede it with a bang \code{!}:
\begin{example}
\begin{question}{!3}
This question's points won't be added to the total sum.
\end{question}
\end{example}
Beware that this also prevents bonus points. The points simply will be
written where the heading instance puts them.
\sinceversion{0.3}One additional thing: you might want to define custom
commands that should behave differently if they're inside or outside of the
\env{question} environment. In this case you can use these commands:
\begin{commands}
\expandable\command{IfInsideQuestionTF}[\marg{true code}\marg{false code}]
Check if inside of a question and either leave \meta{true code} or
\meta{false code} in the input stream.
\expandable\command{IfInsideQuestionT}[\marg{true code}]
Check if inside of a question and either leave \meta{true code} in the
input stream if true.
\expandable\command{IfInsideQuestionF}[\marg{false code}]
Check if inside of a question and either leave \meta{false code} in the
input stream if not.
\end{commands}
\subsection{Options to the \env*{question} Environment}\label{sec:opti-ques-envir}
The \env{question} environment takes one or more of the following options:
\begin{options}
\keychoice{type}{exam,exercise}\Module{question}\Default{exercise}
Determines the type of question and changes the default name of a question
from ``Exercise'' to ``Question''. These default names are language
dependent.\par
If you use \cs*{usepackage}\oarg{ngerman}\marg{babel}, for example, then
the names are ``Übung'' and ``Aufgabe''.
\keyval{name}{name}\Module{question}\Default
Sets a custom name. All predefined names are discarded.
\keyval{subtitle}{subtitle}\Module{question}\Default
Adds a subtitle \meta{subtitle} for the question that is used by headings
instances that make use of the subtitle coffin, see
section~\ref{sec:exsheets-headings}.
\keyval{skip-below}{dim}\Module{question}\Default{.5\cs*{baselineskip}}
\sinceversion{0.18}Sets the vertical space that is inserted after the
question environment.
\keybool{no-skip-below}\Module{question}\Default{false}
\sinceversion{0.18}Disables the insertion of vertical space after the
question environment.
\keybool{print}\Module{question}\Default{true}
Prints or hides the question.
\keyval{ID}{id}\Module{question}\Default
Assigns a custom \acs{id} to the question. See section~\ref{ssec:ids} for
further information.
\keyval{label}{label}\Module{question}\Default
Places a \cs*{label}\marg{label} for the question. This will overwrite
any label that is placed by the \option{auto-label} option.
\keyval{class}{class}\Module{question}\Default
Assigns a class \meta{class} to the question. See
section~\ref{sec:classes} for further information.
\keyval{topic}{topic}\Module{question}\Default
Assigns a topic \meta{topic} to the question. See
section~\ref{sec:topics} for further information.
\keybool{use}\Module{question}\Default{true}
Discards the question. Or not.
\keyval{pre-hook}{code}\Module{question}\Default
\changedversion{0.16}Adds \meta{code} directly before the question title.
\keyval{post-hook}{code}\Module{question}\Default
\changedversion{0.16}Adds \meta{code} directly after the question.
\keyval{pre-body-hook}{code}\Module{question}\Default
\sinceversion{0.16}Adds \meta{code} directly before the question body.
\keyval{post-body-hook}{code}\Module{question}\Default
\sinceversion{0.16}Adds \meta{code} directly after the question body.
\end{options}
\begin{example}
\begin{question}[type=exam]
This question has the type \keyis{type}{exam}. The default name has changed
from ``Exercise'' to ``Question''.
\end{question}
\begin{question}[name=Fancy name]
This question has a custom name.
\end{question}
\begin{question}[print=false]
This question is not printed.
\end{question}
\end{example}
The difference between \option{print} and \option{use} lies behind the scenes:
with \keyis{print}{false} the question is not printed, but it still gets an
individual \ac{id}, is numbered, and a possible solution is saved. This is
for example useful when you want to print a sample solution for an exam. With
\keyis{use}{false} it is fully discarded which means it is not accessible
through an \acs{id} and a possible solution will not be saved.
\subsection{Subtitles to Questions}\label{sec:subtitles-questions}
The \option{subtitle} option mentioned in section~\ref{sec:opti-ques-envir}
can be used to add a subtitle to a question. However, unless you choose a
suitable heading (see section~\ref{sec:exsheets-headings}) it won't be
printed. Currently there is \emph{one} heading instance that uses the
subtitles but it should be easy to create a custom heading using one of the
existing ones as a starter example. When creating such a heading you may want
to distinguish between the cases when a subtitle has been given and when no
subtitle is present. This can be done with the following commands:
\begin{commands}
\expandable\command{IfQuestionSubtitleTF}[\marg{true code}\marg{false code}]
Tests if the current question has a subtitle. Leaves either \meta{true
code} or \meta{false code} in the input stream.
\expandable\command{IfQuestionSubtitleT}[\marg{true code}]
Tests if the current question has a subtitle. Leaves \meta{true code} in
the input stream if it has.
\expandable\command{IfQuestionSubtitleF}[\marg{false code}]
Tests if the current question has a subtitle. Leaves \meta{false code} in
the input stream if it hasn't.
\end{commands}
A subtitle is also a property of a question in the sense of
section~\ref{sec:additional_info}. That means if a subtitle is given it can
be retrieved with \cs{GetQuestionProperty}.
As an example you could define your own heading instance that prints the
\acs{id} of a question and (if given) the subtitle:
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{QE}{default}{
join = {
title[r,B]number[l,B](.333em,0pt) ;
title[r,B]subtitle[l,B](1em,0pt)
} ,
attach = {
main[l,vc]title[l,vc](0pt,0pt) ;
main[r,vc]points[l,vc](\marginparsep,0pt)
} ,
subtitle-post-code = {ID: \CurrentQuestionID} ,
number-post-code = {\IfQuestionSubtitleF{ID: \CurrentQuestionID}}
}
\end{sourcecode}
Please see section~\ref{sec:exsheets-headings} for more details on heading
instances.
\subsection{The \env*{solution} Environment}
If you want to save/print (more on the exact usage in
section~\ref{sec:solutions}) a solution you have to use the \env{solution}
environment \emph{after} the question it belongs to and \emph{before} the next
question.
\begin{environments}
\environment{solution}[\oarg{options}]
The main environment for adding solutions to exercises/questions.
\end{environments}
\begin{example}
\begin{question}[ID=first]\label{qu:question_with_solution}
This is our first question that gets a solution!
\end{question}
\begin{solution}
This is the solution to exercise~\ref{qu:question_with_solution}!
\end{solution}
\end{example}
You can see that in the default settings the solution is \emph{not} written to
the document. It has been saved, though, for possible later usage. We will
see the solution later!
\subsection{Options to the \env*{solution} Environment}\label{sec:opti-soli-envir}
The \env{solutions} environment also has options, namely these:
\begin{options}
\keyval{name}{name}\Module{solution}\Default
Sets a custom name.
\keybool{print}\Module{solution}\Default{false}
Prints or hides the solution.
\keyval{skip-below}{dim}\Module{solution}\Default{.5\cs*{baselineskip}}
\sinceversion{0.18}Sets the vertical space that is inserted after the
solution environment.
\keybool{no-skip-below}\Module{solution}\Default{false}
\sinceversion{0.18}Disables the insertion of vertical space after the
solution environment.
\keyval{pre-hook}{code}\Module{solution}\Default
\sinceversion{0.16}Adds \meta{code} directly before the solution title.
\keyval{post-hook}{code}\Module{solution}\Default
\sinceversion{0.16}Adds \meta{code} directly after the solution.
\keyval{pre-body-hook}{code}\Module{solution}\Default
\sinceversion{0.16}Adds \meta{code} directly before the solution body.
\keyval{post-body-hook}{code}\Module{solution}\Default
\sinceversion{0.16}Adds \meta{code} directly after the solution body.
\end{options}
Their meaning is the same as those for the \code{question} environment.
\begin{example}
\begin{question}{5}
The solution to this questions gets printed where it is.
\end{question}
\begin{solution}[print]
See? This solution gets printed where you have put it in the code of
your document.
\end{solution}
\begin{question}{2.5}
The solution to this questions gets printed where it is \emph{and}
has a fancy name. Have you noticed that you can assign partial
points?
\end{question}
\begin{solution}[print,name=Fancy name]
See? This solution gets printed where you have put it and has a fancy
name!
\end{solution}
\end{example}
\subsection{Setting the Counter}\label{ssec:counter}
The package option \option{counter-format} allows you to specify how the question
counter (a counter unsurprisingly name \code{question}) is formatted.
The input is an arbitrary string which means you can have anything as counter
number. However, the letter combinations \code{ch}, \code{se}, \code{qu} and
\code{tsk} are replaced with the counters for the chapter, section, question
or tasks (see the \Tasks\ package), respectively. While the last one is not
really useful in this case the others allow for a combined numbering. Each of
these letter combinations can have an optional argument that specifies the
format of the respective counter. \code{1}: \cs*{arabic}, \code{a}:
\cs*{alph}, \code{A}: \cs*{Alph}, \code{r}: \cs*{roman} and \code{R}:
\cs*{Roman}.
\begin{example}
\SetupExSheets{counter-format=Nr~se~(qu[a])}
\begin{question}
A question with a differently formatted number.
\end{question}
\end{example}
Since the strings associated with the counters are replaced one has to hide
them if they are actually wanted in the counter format. The easiest way would
to hide them in braces.
\begin{example}
\SetupExSheets{counter-format={section}\,se~{question}\,(qu[a])}
\begin{question}
A question with a yet differently formatted number.
\end{question}
\end{example}
\subsection{Language Settings}
The names of the questions and solutions are language dependent. If you use
\pkg{babel} or \pkg{polyglossia} \ExSheets\ will adapt to the document language.
\ExSheets\ has a number of translations but surely not all! If you miss a
language please drop me a line in an
email\footnote{\href{mailto:[email protected]}{[email protected]}}
containing the \pkg{babel} language name and the correct translations for
questions (possibly distinguishing between exercises and exam questions) and
solutions.
Until I implement it you can add something like this to your preamble (example
for Danish) and try if it works:
\begin{sourcecode}
\DeclareTranslation{Danish}{exsheets-exercise-name}{\O{}velse}
\DeclareTranslation{Danish}{exsheets-question-name}{Opgave}
\DeclareTranslation{Danish}{exsheets-solution-name}{Opl\o{}sning}
\end{sourcecode}
If this isn't working it means that the language you're using is unknown to
the \pkg{translations} package. In this case please notify me, too. You then
can still use the \option{name} options.
\section{Counting Points}\label{sec:points}
\subsection{The Commands}
You have seen in section~\ref{ssec:questions} that you can assign points to a
question. If you do so these points are printed into the
margin\footnote{Well, not necessarily. It depends on the heading style you
have chosen.} and are counted internally. But there are additional commands
to assign points or bonus points and a number of commands to retrieve the sum
of points and/or bonus points.
\begin{commands}
\command{addpoints}[\sarg\marg{num}]
This command can be used to add points assigned to subquestions.
\cs{addpoints} will print the points (with ``unit'') \emph{and} add them
to the sum of all points, \cs{addpoints}\sarg\ will only add them but print
nothing.
\command{points}[\sarg\marg{num}]
This command will only print the points (with ``unit'') but won't add them
to the sum of points.
\command{addbonus}[\sarg\marg{num}]
This command can be used to add bonus points assigned to subquestions.
\cs{addbonus} will print the points (with ``unit'') \emph{and} add them
to the sum of all bonus points, \cs{addbonus}\sarg\ will only add them but
print nothing.
\command{bonus}[\sarg\marg{num}]
This command will only print the bonus points (with ``unit'') but won't
add them to the sum of bonus points.
\command{pointssum}[\sarg]
Prints the sum of all points with or without (starred version) ``unit'':
\pointssum
\command{currentpointssum}[\sarg]
Prints the current sum of points with or without (starred version)
``unit'': \currentpointssum
\command{bonussum}[\sarg]
Prints the sum of all bonus points with or without (starred version)
``unit'': \bonussum
\command{currentbonussum}[\sarg]
Prints the current sum of bonus points with or without (starred version)
``unit'': \currentbonussum
\command{totalpoints}[\sarg]
prints the sum of the points \emph{and} the sum of the bonus points with
``unit'': \totalpoints\space The starred version prints the sum of the
points without ``unit'': \totalpoints*.
\end{commands}
The commands \cs{pointssum}, \cs{bonussum} and \cs{totalpoints} need at
least \emph{two} \LaTeX\ runs to get the sum right.
Suppose you have an exercise worth \points{4} which consists of four questions
listed with an \env{enumerate} environment that are all worth \points{1}
each. You have two possibilities to display and count them:
\begin{example}
% uses package `enumitem'
\begin{question}{4}
\begin{enumerate}[label=\alph*)]
\item blah (\points{1})
\item blah (\points{1})
\item blah (\points{1})
\item blah (\points{1})
\end{enumerate}
\end{question}
\begin{question}
\begin{enumerate}[label=\alph*)]
\item blah (\addpoints{1})
\item blah (\addpoints{1})
\item blah (\addpoints{1})
\item blah (\addpoints{1})
\end{enumerate}
\end{question}
\end{example}
\subsection{Options}
\begin{options}
\keyval{name}{name}\Module{points}\Default{P.}
Choose the ``unit'' for the points. If you like to differentiate between
a single point and more than one point you can give a plural ending
separated with a slash: \keyis{name}{point/s}. This sets also the name of
the bonus points.
\keyval{name-plural}{plural form of name}\Module{points}\Default
Instead of forming the plural form with an ending to the singular form
this option allows to set an extra word for it. This sets also the plural
form for the bonus points.
\keyval{bonus-name}{name}\Module{points}\Default{P.}
Choose the ``unit'' for the bonus points. If you like to differentiate
between a single point and more than one point you can give a plural
ending separated with a slash: \key{bonus-name}{point/s}.
\keyval{bonus-plural}{plural form of name}\Module{points}\Default
Instead of forming the plural form with an ending to the singular form
this option allows to set an extra word for it.
\keybool{use-name}\Module{points}\Default{true}
Don't display the name at all. Or do.
\keyval{format}{code}\Module{points}\Default{\cs*{@firtsofone}}
\sinceversion{0.9d}Format number plus name as a whole. Ideally
\meta{code} would end with a command that takes an argument. Else
number plus name will be braced.
\keyval{number-format}{any code}\Module{points}\Default
This option allows formatting of the number, \eg, italics:
\keyis{number-format}{\cs*{textit}}.
\keyval{bonus-format}{any code}\Module{points}\Default
This option allows formatting of the number of the bonus points, \eg,
italics: \keyis{bonus-format}{\cs*{textit}}.
\keybool{parse}\Module{points}\Default{true}
If set to \code{false} the points are not counted and the
\cs{totalpoints}, \cs{pointssum} and \cs{bonussum} commands won't know
their value.
\keybool{separate-bonus}\Module{points}\Default{false}
This option determines whether points and bonus points each get their own
unit when they appear together (in the margin or with \cs{totalpoints}).
\keyval{pre-bonus}{tokens}\Module{points}\Default{\cs*{space}(+}
Code to be inserted before the bonus points when they follow normal
points.
\keyval{post-bonus}{tokens}\Module{points}\Default{)}
Code to be inserted after the bonus points when they follow normal
points.
\end{options}
\begin{example}[add-sourcecode-options={literate=}]
\SetupExSheets[points]{name=point/s,number-format=\color{red}}
\begin{question}{1}
This one's easy so only 1 point can be earned.
\end{question}
\begin{question}{7.5}
But this one's hard! 7.5 points are in there for you!
\end{question}
\end{example}
\section{Printing Solutions}\label{sec:solutions}
You have already seen that you can print solutions where they are using the
\option{print} option. But \ExSheets\ offers you quite more
possibilities.
In the next subsections the usage of the following command is discussed.
\begin{commands}
\command{printsolutions}[\oarg{setting}]
Print solutions of questions/exercises.
\end{commands}
Before we do that a hint: remember that you can set the option \option{print}
globally:
\begin{sourcecode}
% in the preamble
\SetupExSheets{solution/print=true}
\end{sourcecode}
Now if you want to typeset some text depending on the option being true or not
you can use the following commands:
\begin{commands}
\expandable\command{PrintSolutionsTF}[\marg{true code}\marg{false code}]
Either leaves \meta{true code} or \meta{false code} in the input stream
depending on wether solutions are printed or not, \ie, on the value of the
solution's option \option{print}. Inside a \env{solution} environment
this always prints \meta{true code}.
\expandable\command{PrintSolutionsT}[\marg{true code}]
Either leaves \meta{true code} or nothing in the input stream depending on
wether solutions are printed or not, \ie, on the value of the solution's
option \option{print}. Inside a \env{solution} environment this always
prints \meta{true code}.
\expandable\command{PrintSolutionsF}[\marg{false code}]
Either leaves nothing or \meta{false code} in the input stream depending
on wether solutions are printed or not, \ie, on the value of the
solution's option \option{print}. Inside a \env{solution}environment this
always prints nothing.
\end{commands}
They might come in handy if you want two versions of an exercise sheet, one
with the exercises and one with the solutions, and you want to add different
titles to these versions, for instance.
% When solutions are saved a lot of information is saved. One of them is the
% current counter format. The following option determines wether the saved
% counter format or the currently active one is used when \cs{printsolutions} is
% called:
% \begin{options}
% \keybool{use-saved-counter-format}\Default{true}
% \changedversion{0.21}When set to true the counter format of solutions
% printed by \cs{printsolutions}\label{option:use-saved-counter-format} are
% independent from the setting of \option{counter-format}. The saved format
% is used instead.
% \end{options}
\subsection{Print all}\label{sec:solutions-print-all}
The first and easiest usage of \cs{printsolutions} is the following:
\begin{sourcecode}
\printsolutions
\end{sourcecode}
There is nothing more to say, really. It prints all solutions you have
specified except those belonging to a question with option \keyis{use}{false}.
Yes, there's one more point: \cs{printsolutions} only knows the solutions
that have been set \emph{before} its usage! This is also true for every usage
explained in the next sections.
\begin{example}
\printsolutions
\end{example}
Two options allow to add code to the list of solutions when used with
\cs{printsolutions}\Oarg{all} (which is the same as using it without option):
\begin{options}
\keyval{chapter-hook}{code}
\sinceversion{0.13}Adds \meta{code} to the list of solutions every time
solutions from a new chapter are printed (before the solutions of the
corresponding chapter are printed).
\keyval{section-hook}{code}
\sinceversion{0.13}Adds \meta{code} to the list of solutions every time
solutions from a new section are printed (before the solutions of the
corresponding section are printed).
\end{options}
\subsection{Print per chapter/section}
\minisec{Current chapter/section}
If you are not creating an exercise sheet or an exam but are writing a
textbook you maybe want a section at the end of each chapter showing the
solution to the exercises presented in that chapter. In this case use the
command as follows:
\begin{sourcecode}
\printsolutions[section]
% or
\printsolutions[chapter]
\end{sourcecode}
Again, this is pretty much self-explaining. The solutions to the questions of
the current chapter\footnote{Only if the document class you're using
\emph{has} chapters, of course!} or section are printed.
\begin{example}
\begin{question}
This is the first and only question in this section.
\end{question}
\begin{solution}
This will be one of a few solutions printed by the following call of
\cs{printsolutions}.
\end{solution}
And now:
\printsolutions[section]
\end{example}
\minisec{Specific chapter/section}\label{sssec:print_specific_section}
You can also print only the solutions from chapters or sections other than the
current ones. The syntax is fairly easy:
\begin{example}
\printsolutions[section={1-7,10}]
% the same for chapters:
% \printsolutions[chapter={1-7,10}]
\end{example}
Don't forget that \cs{printsolutions} cannot know the solutions from
section~10 yet. It is just used to demonstrate the syntax. You can also use
an open range, \eg, something like
\begin{sourcecode}
\printsolutions[section={-4,10-}]
\end{sourcecode}
This would print the solutions from sections~1--4 and from all sections with
number 10\footnote{Or rather where \cs*{value}\Marg{section} is 10 or greater --
the actual counter formatting is irrelevant.} and greater.
There is an obvious disadvantage: you have to know the section numbers! But
there is a solution: use the package option \keyis{use-ref}{true}. Then you
can do something like
\begin{sourcecode}
% in the preamble:
\usepackage[use-ref]{exsheets}
% somewhere in your code after \section{A really cool section title}:
\label{sec:ReallyCool}
% somewhere later in your code:
\printsolutions[section={-\S{sec:ReallyCool}}]
% which will print all solutions from questions up to and
% including the really cool section
\end{sourcecode}
With the package option \keyis{use-ref}{true} each usage of \cs*{label} will
create additional labels (one preceded with \code{exse:} and another one with
\code{exch:}) which store the section number and the chapter number,
respectively. These are used internally by two commands \cs{S} and \cs{C}
which refer to the section number and the chapter number the label was created
in. \emph{These commands are only available as arguments of}
\cs{printsolutions}.
Since some packages like the well known \pkg{hyperref} for example redefine
\cs*{label} \option{use-ref} won't work in together with it. In this case
don't use \option{use-ref} and set \cs{exlabel}\marg{label} instead to
remember the section/the chapter number. Its usage is just like \cs*{label}.
So the safest way is as follows:
\begin{sourcecode}
% in the preamble:
\usepackage{exsheets}
% somewhere in your code after \section{A really cool section title}:
\exlabel{sec:ReallyCool}
% somewhere later in your code:
\printsolutions[section={-\S{sec:ReallyCool}}]
% which will print all solutions from questions up to and
% including the really cool section
\end{sourcecode}
Please be aware that the labels must be processed in a previous \LaTeX\ run
before \cs{S} and \cs{C} can pass them on to \cs{printsolutions}.
\subsection{Print by \acs{id}}\label{ssec:ids}
Now comes the best part: you can also print selected solutions! Every
question has an \acs{id}. To see which \acs{id} a question has you can call
the following command:
\begin{commands}
% \command{DebugExSheets}[\Marg{\choices{true,false}}]
% Enable or disable visual \ExSheets' debugging.
\expandable\command{CurrentQuestionID}
\sinceversion{0.4a}Expands to the current question \acs{id}.
\end{commands}
\begin{options}
\keybool{debug}
Enable or disable visual \ExSheets' debugging.
\end{options}
Let's create some more questions and take a look what this command does:
\begin{example}
\SetupExSheets{debug=true}
\begin{question}[ID=nice!]
A question with a nice \acs{id}!
\end{question}
\begin{solution}
The solution to the question with the nice \acs{id}.
\end{solution}
\begin{question}{3.75}
Yet another question. But this time with quarter points!
\end{question}
\begin{solution}
Yet another solution.
\end{solution}
\end{example}
So now we can call some specific solutions:
\begin{example}
\printsolutions[byID={first,nice!,10,14}]
\end{example}
This makes use of the \pkg{l3sort} package which at the time of writing is
still considered experimental. In case you wonder where solution~14 is:
question~14 has no solution given.
If you don't want that the solutions are sorted automatically but appear in
the order given you can use the option
\begin{options}
\keybool{sorted}\Module{solution}\Default{true}
Sort solutions given by \acs{id} or don't.
\end{options}
\section{Conditional Printing of Questions}\label{sec:cond-print-quest}
\subsection{Using Classes}\label{sec:classes}
For creating different variants of a written exam or different difficulty
levels of an exercise sheet it comes in handy if one can assign certain
classes to questions and then tell \ExSheets\ only to use one ore more
specific classes.
\begin{options}
\keyval{use-classes}{list of classes}\Default
When this option is used only the questions belonging to the specified
classes are printed and have their solutions saved.
\end{options}
\begin{example}
\SetupExSheets{use-classes={A,C}}
\begin{question}[class=A]
Belonging to class A.
\end{question}
\begin{question}[class=B]
Belonging to class B.
\end{question}
\begin{question}[class=C]
Belonging to class C!
\end{question}
\end{example}
Questions of classes that are not used are fully discarded. \emph{This also
means that questions that don't have a class assigned are discarded.}
\ExplSyntaxOn
\bool_set_false:N \g__exsheets_use_classes_bool
\ExplSyntaxOff
\subsection{Using Topics}\label{sec:topics}
Similarly to classes one can assign topics to questions. The usage is
practically identical, the semantic meaning is different.
\begin{options}
\keyval{use-topics}{list of topics}\Default
When this option is used only the questions belonging to the specified
topics are printed and have their solutions saved.
\end{options}
\begin{example}
\SetupExSheets{use-topics={trigonometry}}
\begin{question}[topic=trigonometry]
A trigonometry question.
\end{question}
\begin{question}[topic=arithmetics]
A arithmetics question
\end{question}
\end{example}
Questions of topics that are not used are fully discarded. \emph{This also
means that questions that don't have a topic assigned are discarded.}
If you set both \option{use-classes} and \option{use-topics} then only
questions will be used that \emph{match both categories}.
Ideally one could assign more than one topic to a question but this is
\emph{not} supported yet.
\ExplSyntaxOn
\bool_set_false:N \g__exsheets_use_topics_bool
\ExplSyntaxOff
\subsection{Own Dividing Concepts}
Actually\sinceversion{0.8} both classes and topics are introduced into
\ExSheets\ internally this way:
\begin{sourcecode}
\DeclareQuestionClass{class}{classes}
\DeclareQuestionClass{topic}{topics}
\end{sourcecode}
which means you can do the same introducing your own dividing concepts.
\begin{commands}
\command{DeclareQuestionClass}[\marg{singular name}\marg{plural name}]
Introduces a new dividing concept and defines both new options for the
\env{question} environment and new global options.
\end{commands}
For example you could decide you want to group your questions according to
their difficulty. You could place the following line in your preamble:
\begin{sourcecode}
\DeclareQuestionClass{difficulty}{difficulties}
\end{sourcecode}
This would define an option \option*{use-difficulties} analogous to
\option{use-classes} and \option{use-topics}. It would also define an option
\option{difficulty} for the \env{question} environment. This means you could
now do something like the following:
\begin{example}
\SetupExSheets{use-difficulties={easy,hard}}
\begin{question}[difficulty=easy]
An easy question.
\end{question}
\begin{question}[difficulty=medium]
This one's a bit harder.
\end{question}
\begin{question}[difficulty=hard]
Now let's see if you can solve this one.
\end{question}
\end{example}
\subsection{Retrieving the Class Value in a Question}\label{sec:retr-class-value}
Sometimes it may be desirable to retrieve the value of a class defined by
\cs{DeclareQuestionClass} that a question has in order to be able to print,
say. This is possible with the following commands:
\begin{commands}
\expandable\command{GetQuestionClass}[\marg{class}]
Prints the value of \meta{class} a question has. The command is
expandable. If the class does not exist or the value is empty the command
expands to nothing.
\command{PrintQuestionClassTF}[\marg{class}\marg{true}\marg{false}]
Test if a question has a non-empty value for class \meta{class} and either
leaves \meta{true} or \meta{false} in the input stream. In the
\meta{true} argument you can refer to the value with \code{\#1} where you
want it printed.
\command{PrintQuestionClassT}[\marg{class}\marg{true}]
Like \cs{PrintQuestionClassTF} but only has the \meta{true} branch.
\command{PrintQuestionClassF}[\marg{class}\marg{false}]
Like \cs{PrintQuestionClassTF} but only has the \meta{false} branch.
\end{commands}
\begin{example}
\begin{question}[difficulty=hard]
This question has the difficulty level
``\PrintQuestionClassTF{difficulty}{#1}{??}''.
\end{question}
\end{example}
\ExplSyntaxOn
\bool_set_false:N \g__exsheets_use_difficulties_bool
\ExplSyntaxOff
\subsection{Tagging Questions}
There\sinceversion{0.20} is another way of dividing questions: you can assign
tags to questions:
\begin{sourcecode}
\begin{question}[tags={foo,bar,baz}]
...
\end{question}
\end{sourcecode}
You can then decide to print only questions with certain tags by using the
following option:
\begin{options}
\keyval{use-tags}{csv list of tags to include}
Select tags. When used only questions being tagged with at least one of
the tags in \meta{csv list of tags to include} are printed.
\end{options}
\section{Adding and Using Additional Information to
Questions}\label{sec:additional_info}
\subsection{Question Properties -- the Basics}
For managing lots of questions and corresponding solutions it can be very
useful to be able to save and recover additional information to the questions.
This is possible with the following commands. First the ones for saving:
\begin{commands}
\command{DeclareQuestionProperty}[\marg{name}]
This command defines a question property \meta{name}. It can only be
used in the document preamble.
\command{SetQuestionProperties}[\Marg{\meta{name}=\meta{value},...}]
Set the properties for a specific question. this command can only be used
inside the \env{question} environment.
\end{commands}
Now the commands for recovering the properties:
\begin{commands}
\command{QuestionNumber}[\marg{id}]
Recover the number of the question with the \acs{id} \meta{id}. The
number is displayed according to the format set with
\option{counter-format}.
\expandable\command{GetQuestionProperty}[\marg{name}\marg{id}]
Recover the property \meta{name} of the question with the \acs{id}
\meta{id}. Of course the property must have been declared before. The
command is expandable. Since\changedversion{0.12} the properties of a
question are written to the \code{aux} file it is possible to retrieve
them before the corresponding \env{question} environment has been used.
\expandable\command{IfQuestionPropertyTF}[\marg{name}\marg{id}\marg{true}\marg{false}]
A command\sinceversion{0.15} that returns \meta{true} if the question with
the \acs{id} \meta{id} has the property \meta{name} and \meta{false}
otherwise. The variants \cs{IfQuestionPropertyT} and
\cs{IfQuestionPropertyF} also exist which only have the \meta{true} or the
\meta{false} branch.
\end{commands}
Let's say we have declared the properties \code{notes}, \code{reference} and
\code{topic}. By default the property \code{points} is available and gets the
value of the optional argument of the \code{question} environment.
We can now do the following:
\begin{example}
% uses `biblatex'
\begin{question}[ID=center,topic=LaTeX]{3}
Explain how you could center text in a \LaTeX\ document.
\SetQuestionProperties{
topic = \TeX/\LaTeX ,
notes = {How to center text.},
reference = {\textcite{companion}}}
\end{question}
\begin{solution}
To center a short part of the text body one can use the \env*{center}
environment (\points{1}). Inside an environment like \env*{table} one
should use \cs*{centering} (\points{1}). For single lines there is also
the \cs*{centerline} command (\points{1}).
\end{solution}
\begin{question}[ID=knuthbooks,topic=LaTeX]{2}
Name two books by D.\,E.\,Knuth.
\SetQuestionProperties{
topic = \TeX/\LaTeX ,
notes = {Books by Knuth.},
reference = {\textcite{knuth:ct:a,knuth:ct:b,knuth:ct:c,knuth:ct:d,knuth:ct:e}}}
\end{question}
\begin{solution}
For example two volumes from \citetitle{knuth:ct}:
\citetitle{knuth:ct:a,knuth:ct:b,knuth:ct:c,knuth:ct:d,knuth:ct:e}. Each
valid answer is worth \points{1}
\end{solution}
\end{example}
It is now possible to recover these values later:
\begin{example}
% uses `booktabs'
\begin{center}
\begin{tabular}{lll}
\toprule
Question & Property & \\
\midrule
\QuestionNumber{center}
& Points & \GetQuestionProperty{points}{center} \\
& Topic & \GetQuestionProperty{topic}{center} \\
& References & \GetQuestionProperty{reference}{center} \\
& Note & \GetQuestionProperty{notes}{center} \\
\midrule
\QuestionNumber{knuthbooks}
& Points & \GetQuestionProperty{points}{knuthbooks} \\
& Topic & \GetQuestionProperty{topic}{knuthbooks} \\
& References & \GetQuestionProperty{reference}{knuthbooks} \\
& Note & \GetQuestionProperty{notes}{knuthbooks} \\
\bottomrule
\end{tabular}
\end{center}
\end{example}
Please note that properties \emph{are not the same} as the dividing concepts
explained in section~\ref{sec:cond-print-quest} although they may seem
similar in meaning or even have the same name.
When properties are set they are also written to the \code{aux} file which
means they can be retrieved \emph{before} the corresponding question. Of
course this means that two compilation runs are necessary.
\subsection{Pre-defined Properties}
A few properties are already defined by \ExSheets:
\begin{itemize}
\item \code{counter}:\sinceversion{0.14} this property holds the actual
question number formatted according to the formatting set with option
\option{counter-format}.
\item \code{subtitle}:\sinceversion{0.12} this property holds the subtitle
of the question if given.
\item \code{question-body}:\sinceversion{0.14} this property holds the body
of the corresponding \env{question} environment. Unlike the other
properties it is per default \emph{not} written to the \code{aux} file.
\item \code{points}: this property holds the sum of points given to a
question.
\item \code{bonus-points}:\sinceversion{0.14} this property holds the sum of
bonus points given to a question.
\item \code{ref}:\sinceversion{0.7f} when the option \option{auto-label} is
used this property is defined and expands to the corresponding \cs*{ref}.
Also see section~\ref{sec:auto-label-opti}.
\item \code{page-ref}:\sinceversion{0.7f} when the option
\option{auto-label} is used this property is defined and expands to the
corresponding \cs*{pageref}. Also see section~\ref{sec:auto-label-opti}.
\end{itemize}
There is one option affecting the property \code{question-body}:
\begin{options}
\keybool{save-to-aux}\Module{question}\Default{false}
When set to \code{true} the property \code{question-body} is also written
to the \code{aux} file.
\end{options}
\subsection{Advanced Usage}
There are additional commands\sinceversion{0.3} that might prove useful. They
allow advanced usage of defined properties. Below an example is shown how
they can be used to generate a grading table.
\begin{commands}
\command{ForEachQuestion}[\marg{code to be executed for each used question}]
\changedversion{0.14}Inside the argument one can refer to the \ac{id} of a
question with \code{\#1}. You can also refer to the number of the
question with \code{\#2}. \emph{Number} means that if you \emph{use}
seven questions then those questions have numbers~1 to~7.
\expandable\command{numberofquestions}
\changedversion{0.14} returns the complete number of used questions.
\expandable\command{iflastquestion}[\marg{true code}\marg{false code}]
Although this command is available in the whole document it is only useful
inside \cs{ForEachQuestion}. It tells you if the end of the loop is
reached or not.
\end{commands}
One could use these commands to create a grading table, for instance:
\begin{sourcecode}
\begin{tabular}{|l|*{\numberofquestions}{c|}c|}\hline
Question &
\ForEachQuestion{\QuestionNumber{#1}\iflastquestion{}{&}} &
Total \\ \hline
Points &
\ForEachQuestion{\GetQuestionProperty{points}{#1}\iflastquestion{}{&}} &
\pointssum* \\ \hline
Reached &
\ForEachQuestion{\iflastquestion{}{&}} & \\ \hline
\end{tabular}
\end{sourcecode}
For four questions the table now would look similar to
figure~\ref{fig:grading-table}.
\begin{figure}[ht]
\centering
\begin{tabular}{|l|*{4}{c|}c|}\hline
Question & 1. & 2. & 3. & 4. & Total \\ \hline
Points & 3 & 5 & 10 & 8 & 26 \\ \hline
Reached & & & & & \\ \hline
\end{tabular}
\caption{An example for a grading table. (Actually this is a fake. See the
\code{grading-table.tex} file shipped with exsheets for the real use case.)}
\label{fig:grading-table}
\end{figure}
\section{Variations of an Exam}
It is a quite common task\sinceversion{0.6} to design an exam in two different
variants. This is of course possible with \ExSheets' classes (see
section~\ref{sec:classes}). However, often not the whole question is to be
different but only small details, the numbers in a maths exam, say. For this
purpose \ExSheets\ provides the following commands:
\begin{commands}
\command{SetVariations}[\marg{num}]
Set the number of different variants. This will determine how many
arguments the command \cs{vary} will get. \meta{num} must at least be
\code{2} and is initially set to \code{2}.
\command{variant}[\marg{num}]
Choose the active variant. The argument must be a number between \code{1}
and the number set with \cs{SetVariations}. Initially set to \code{1}.
\command{vary}[\marg{variant 1}\marg{variant 2}]
This command is the one actually used in the document. It has a number of
required arguments equal to the number set with \cs{SetVariations}. All
of its arguments are discarded except the one specified with
\cs{variant}.
\command{lastvariant}
\sinceversion{0.7b}Each time \cs{vary} is called it stores the value it
chose in \cs{lastversion}. This might be convenient to use if one
otherwise would have to repeatedly write the same \cs{vary}.
\end{commands}
\begin{example}
\SetVariations{6}%
\variant{6}\vary{A}{B}{C}{D}{E}{F}
(last variant: \lastvariant)
\variant{1}\vary{A}{B}{C}{D}{E}{F}
(last variant: \lastvariant)
\variant{5}\vary{A}{B}{C}{D}{E}{F}
(last variant: \lastvariant)
\variant{2}\vary{A}{B}{C}{D}{E}{F}
(last variant: \lastvariant)
\variant{4}\vary{A}{B}{C}{D}{E}{F}
(last variant: \lastvariant)
\variant{3}\vary{A}{B}{C}{D}{E}{F}
(last variant: \lastvariant)
\end{example}
\section{A Grade Distribution}
Probably this is a rather esoteric feature but it could proof useful in some
cases. Suppose you are a German math teacher and want to grade exactly
corresponding to the number of points relative to the sum of total points,
regardless of how big that might be. You could do something like this to
present your grading decisions for the exam:
\begin{example}
% preamble:
% \DeclareRelGrades{
% 1 = 1 ,
% {1,5} = .9167 ,
% 2 = .8333 ,
% {2,5} = .75 ,
% 3 = .6667 ,
% {3,5} = .5833 ,
% 4 = .5
% }
\small\setlength\tabcolsep{2pt}
\begin{tabular}{r|*8c}
Punkte
& $\grade*{1}$ & $\le\grade*{1}$ & $\le\grade*{1,5}$ & $\le\grade*{2}$
& $\le\grade*{2,5}$ & $\le\grade*{3}$ & $\le\grade*{3,5}$ & $<\grade*{4}$ \\
Note
& 1 & 1--2 & 2 & 2--3 & 3 & 3--4 & 4 & 5
\end{tabular}
\end{example}
These are the available commands and options:
\begin{commands}
\command{DeclareRelGrades}[\Marg{\meta{grade}=\meta{num},...}]
This command is used to define grades and assign the percentage of total
points to them.
\command{grade}[\sarg\marg{grade}]
Gives the number of points corresponding to a grade depending on the value
of \cs{pointssum} with or without (starred version) ``unit''.
\end{commands}
\begin{options}
\keyval{round}{num}\Module{grades}\Default{0}
The number of decimals the points of a grade are rounded to. This doesn't
apply to the maximum number of points if the rounded number would be
bigger than the actual sum.
\keybool{half}\Module{grades}\Default{false}
If set to \code{true} points are rounded either to full or to half
points.
\end{options}
\section{Selectively Include Questions from External Files}\label{sec:include}
\subsection{Caveat}
I need to say some words of caution: the \cs{includequestions} that will be
presented shortly is probably \ExSheets' most experimental one at the time of
writing (\today). Thanks to feedback of users it is constantly improved and
bugs are fixed. It is not a very efficient way to insert question regarding
performance and you shouldn't wonder if compilation slows down when you use
it. It probably needs to be re-written all over but on the one hand that
would introduce new bugs and on the other hand for the time being I don't have
the capacities, anyway, so you'll have to live it, I'm afraid.
\subsection{How it works}
Suppose you have one or more files with questions prepared to use them as a
kind of database. One for class A, say, one for class B, one for class C and
so one, something like this:
\begin{sourcecode}
% this is file classA.tex
\begin{question}[class=A,ID=A1,topic=X]
First question of class A, topic X.
\end{question}
\begin{solution}
First solution of class A.
\end{solution}
\begin{question}[class=A,ID=A2,topic=Y]
Second question of class A, topic Y.
\end{question}
\begin{solution}
Second solution of class A.
\end{solution}
...
% end of file classA.tex
\endinput
\end{sourcecode}
You can of course just \cs*{input} or \cs*{include} it but that would of
course include the whole file into your document. But would't it be nice to
just include selected questions? Or maybe a five random questions from the
file? That is possible with the following command:
\begin{commands}
\command{includequestions}[\oarg{options}\marg{list of filenames}]
Include questions from external files.
\end{commands}
If you use it without options it will have the same effect as \cs*{input}.
There are however the following options:
\begin{options}
\keybool{all}\Module{include}
\keyval{IDs}{list of IDs}\Module{include}\Default
Includes only the specified questions.
\keyval{random}{num}\Module{include}\Default
Includes \meta{num} randomly selected questions. This option uses the
\pkg{pgfcore} package to create the pseudo-random numbers.
\keyval{exclude}{list of IDs}\Module{include}\Default
Questions who's \acp{id} are specified here are \emph{not} included. This
option can be combined with the \option{random} option.
\end{options}
The usage should be self-explainable:
\begin{sourcecode}
% include questions A1, A3 and A4:
\includequestions[IDs={A1,A3,A4}]{classA.tex}
% or include 3 random questions:
\includequestions[random=3]{classA}
\end{sourcecode}
In order to be able to select the questions \ExSheets\ needs to \cs*{input}
the file twice. The first time the available questions are determined, the
second time the selected questions are used. This unfortunately means that
anything that is \emph{not} part of a question or solution is also input
twice. Either don't put anything else into the file or use one of the
following commands for control:
\begin{commands}
\command{PrintIfIncludeActiveTF}[\marg{true code}\marg{false code}]
Checks if the questions are actively included or not and puts \meta{true
code} or \meta{false code} in the input stream depending on the answer.
\command{PrintIfIncludeActiveT}[\marg{true code}]
Checks if the questions are actively included or not and puts \meta{true
code} in the input stream if the answer is yes.
\command{PrintIfIncludeActiveF}[\marg{false code}]
Checks if the questions are actively included or not and puts \meta{false
code} in the input stream if the answer is no.
\end{commands}
The selection can be refined further by selecting questions belonging to a
specific class of questions (see section~\ref{sec:cond-print-quest}) before
using \cs{includequestions}.
\sinceversion{0.8}After you've used \cs{includequestions} the \acp{id} of the
included questions is available as an unordered comma separated list in the
following macro:
\begin{commands}
\command{questionsincludedlast}
Unordered comma separated list of question \acp{id} included with the last
usage of \cs{includequestions}.
\end{commands}
\section{The \option*{auto-label} Option}\label{sec:auto-label-opti}
The\sinceversion{0.12} package option \option{auto-label} sets a
\cs*{label}\Marg{qu:\meta{id}} every time the question environment is used.
Both the used command and the automated label can be customized using the
following options:
\begin{options}
\keyval{label-format}{code}\Default{qu:\#1}
The pattern for generating the automatic label. \code{\#1} gets replaced
by the \ac{id} of the corresponding question.
\keyval{label-cmd}{macro}\Default{\cs*{label}}
The command used for generating the label. A command that should take one
mandatory argument.
\keyval{ref-cmd}{macro}\Default{\cs*{ref}}
The command used in the \code{ref} property created by the
\option{auto-label} option, also see section \ref{sec:additional_info}.
The command should take one mandatory argument.
\keyval{pageref-cmd}{macro}\Default{\cs*{pageref}}
The command used in the \code{pageref} property created by the
\option{auto-label} option, also see section \ref{sec:additional_info}.
The command should take one mandatory argument.
\end{options}
\section{Own Question/Solution Pairs}
\ExSheets\changedversion{0.9} provides the possibility to create new
environments that behave like the \env{question} and \env{solution}
environments. This would allow, for example, to define a
\env*{question*}/\env*{solution*} environment pair for bonus questions. The
following commands may be used in the document preamble:
\begin{commands}
\command{NewQuSolPair}[\marg{question}\oarg{question options}\oarg{general
options}\marg{solution}\oarg{solution options}\oarg{general options}]
Define a new pair of question and solution environments.
\command{RenewQuSolPair}[\marg{question}\oarg{question options}\oarg{general
options}\marg{solution}\oarg{solution options}\oarg{general options}]
Redefine an existing pair of question and solution environments.
\end{commands}
The standard environments are defined as follows:
\begin{sourcecode}
\NewQuSolPair{question}{solution}
\end{sourcecode}
Let's say we want the possibility to add bonus questions. A simple way would
be to define starred variants that add a star in the margin left to the title:
\begin{example}
% preamble:
% - \texorpdfstring is provided by `hyperref'
% - \bigstar is provided by `amssymb'
% \DeclareRobustCommand*\questionstar{\texorpdfstring{\bonusquestionsign}{* }}
% \DeclareRobustCommand*\bonusquestionsign{\llap{$\bigstar$\space}}
%
% \NewQuSolPair
% {question*}[name=\questionstar Bonus Question]
% {solution*}[name=\questionstar Solution]
\begin{question*}
This is a bonus question.
\end{question*}
\begin{solution*}[print]
This is what the solution looks like.
\end{solution*}
\end{example}
As you can see the environments take the same options as are described for the
standard \env{question} and \env{solution} environments.
\section{Filling in the Blanks}
\subsection{Cloze}
Both\changedversion{0.4} in exercise sheets and in exams it is sometimes
desirable to be able to create \blank{blanks} that have to be filled in. Or
maybe some more lines: \blank[width=5\linewidth]{}
\begin{commands}
\command{blank}[\sarg\oarg{options}\marg{text to be filled in}]
creates a blank in normal text or in a question but fills the text of its
argument if inside a solution. If used at the \emph{begin of a paragraph}
\cs{blank} will do two things: it will set the linespread according to an
option explained below and will insert \cs*{par} after the lines. If you
don't want that use the starred version.
\end{commands}
The options are these:
\begin{options}
\keychoice{style}{line,wave,dline,dotted,dashed}\Module{blank}\Default{line}
The style of the line. This uses the corresponding command from the
\pkg{ulem} package and is the whole reason why \ExSheets\ loads it in the
first place.
\keyval{scale}{num}\Module{blank}\Default{1}
Scales the width of the blank by factor \meta{num} unless the width is
explicitly set.
\keyval{width}{dim}\Module{blank}\Default
The width of the line. If it is not used the width of the filled in text
is used.
\keyval{linespread}{num}\Module{blank}\Default{1}
Set the linespread for the blank lines. This only has an effect if
\cs{blank} is used at the begin of a paragraph.
\keyval{line-increment}{dim}\Module{blank}\Default{1pt}
\sinceversion{0.21h}When the blank line ist built it is built in multiples
of this value. If the value is too large you may end up with uneven
lines. If the value is too small you may end up with a non-ending
compilation.
\keyval{line-minimum-length}{dim}\Module{blank}\Default{2em}
\sinceversion{0.21h}The minimal length a line must have before it is built
step by step.
\end{options}
\begin{example}
\begin{question}
Try to fill in \blank[width=4cm]{these} blanks. All of them
\blank[style=dotted]{are created} by using the \cs{blank}
\blank[style=dashed]{command}.
\end{question}
\begin{solution}[print]
Try to fill in \blank[width=4cm]{these} blanks. All of them
\blank[style=dotted]{are created} by using the \cs{blank}
\blank[style=dashed]{command}.
\end{solution}
\end{example}
A number of empty lines are easily created by setting the width option:
\begin{example}
\blank[width=4.8\linewidth,linespread=1.5]{}
\end{example}
\subsection{Vertical Space for answers}
When\sinceversion{0.3} you're creating an exam you might want to add some
vertical space where the students can write down their answers. While you can
always use \cs*{vspace} this is not always handy when the space left on the
page is less than you want. In this case it would be nice if a) there would
be no warning and b) the rest of the space would be added at the top of the
next page. This is what the following command is for:
\begin{commands}
\command{examspace}[\sarg\marg{dim}]
Add space as specified in \meta{dim}. If the space available on the
current page is not enough the rest of the space will be added at the top
of the next page. The starred version will silently drop any leftover
space instead of adding it to the next page.
\end{commands}
\begin{example}[side-by-side]
\begin{question}
What do you think of this feature?
\examspace{3cm}
\end{question}
This line comes after the space.
\end{example}
\section{Styling your Exercise/Exam Sheets}\label{part:style}
\subsection{Background}
The \ExSheets\ package makes extensive use of \LaTeX3's coffins\footnote{See
the documentation to the \pkg{xcoffins} package for more information on
that.} as well as its templates concept\footnote{Have a look into the
documentation to the \pkg{xtemplate} package.}. The latter allows a
rather easy extension and customization of some of \ExSheets' environments.
To be more precise: you can define your own instances for the headings used
for questions and solutions.
What this package doesn't provide is changing the background of questions or
framing them. But this is easily possible using the \pkg{mdframed} package
and its \cs*{surroundwithmdframed} command.
\ExSheets{} also provides the options \option{pre-hook}, \option{post-hook},
\option{pre-body-hook} and \option{post-body-hook} to both the question and
the solution environment. With them it is rather straightforward to add a
\pkg{mdframed} frame for instance:
\begin{sourcecode}
\SetupExSheets{
solution/pre-hook = \mdframed ,
solution/post-hook = \endmdframed
}
\end{sourcecode}
Then\sinceversion{0.18} there is the macro
\cs{exsheetsprintsolution}\marg{heading}\marg{body}\label{exsheetsprintsolution}
which may be redefined to suit your needs. The default definition is
equivalent to
\begin{sourcecode}
\newcommand\exsheetsprintsolution[2]{#1#2}
\end{sourcecode}
\subsection{The \code{exsheets-headings} Object}\label{sec:exsheets-headings}
\ExSheets\ defines the object \code{exsheets-headings} and one template for it,
the `default' template. The package also defines two instances of this
template, the `block' instance and the `runin' instance.
\begin{example}
\SetupExSheets{headings=block}
\begin{question}{1}
a `block' heading
\end{question}
\SetupExSheets{headings=runin}
\begin{question}{1}
a `runin' heading
\end{question}
\end{example}
\subsubsection{Available Options}
This section only lists the options that can be used when defining an instance
of the `default' template. The following subsections will give loads of
examples of their usage. The options are listed in the definition for the
template interface:
\begin{sourcecode}
\DeclareTemplateInterface{exsheets-heading}{default}{3}{
% option : type = default
inline : boolean = false ,
runin : boolean = false ,
indent-first : boolean = false ,
toc-reversed : boolean = false ,
vscale : real = 1 ,
above : length = 2pt ,
below : length = 2pt ,
main : tokenlist = ,
pre-code : tokenlist = ,
post-code : tokenlist = ,
title-format : tokenlist = ,
title-pre-code : tokenlist = ,
title-post-code : tokenlist = ,
number-format : tokenlist = ,
number-pre-code : tokenlist = ,
number-post-code : tokenlist = ,
subtitle-format : tokenlist = ,
subtitle-pre-code : tokenlist = ,
subtitle-post-code : tokenlist = ,
points-format : tokenlist = ,
points-pre-code : tokenlist = ,
points-post-code : tokenlist = ,
join : tokenlist = ,
attach : tokenlist =
}
\end{sourcecode}
Each heading is built with at most five coffins available with the names
`main', `title', `subtitle', `number' and `points'. Those coffins place
possibly the whole heading, the title, the subtitle, the question number and
the assigned points. The only coffin that's always typeset is the `main'
coffin, which is empty per default.
Coffins can be joined (two become one, the first extends its bounding box to
contain the second) using the following syntax:
\begin{sourcecode}
join = coffin1[handle11,handle12]coffin2[handle21,handle22](x-offset,y-offset)
\end{sourcecode}
The syntax for attaching (two become one, the first does \emph{not} extend its
bounding box around the second) is the same.
More on coffin handles is described in the documentation for the
\pkg{xcoffins}. Figure~\ref{fig:handles} briefly demonstrates the available
handle pairs.
\begin{figure}[ht]
\centering
\parbox{4.5cm}{%
\NewCoffin\ExampleCoffin
\SetHorizontalCoffin\ExampleCoffin{\color{gray!30}\rule{4cm}{4cm}}%
\DisplayCoffinHandles\ExampleCoffin{blue}%
}
\caption{Available handles for a horizontal coffin.}\label{fig:handles}
\end{figure}
It is possible\sinceversion{0.20} to add own static coffins:
\begin{commands}
\command{DeclareExSheetsHeadingContainer}[\marg{name}\marg{code}]
Defines a new coffin \meta{name} containing \meta{code}. You can refer to
the current question's \ac{id} with \cs{CurrentQuestionID}.
\end{commands}
The following subsections will show all definitions of the instances available
and how they look. This will hopefully give you enough ideas to create your
own instance if you want to have another heading style than the ones
available. Each of the following instances is available through the option
\key{headings}{instance}.
The following examples use a sample text defined as follows:
\begin{sourcecode}
\def\s{This is some sample text we will use to create a somewhat
longer text spanning a few lines.}
\def\sample{\s\ \s\par\s}
\end{sourcecode}
\def\s{This is some sample text we will use to create a somewhat longer text
spanning a few lines.}
\def\sample{\s\ \s\par\s}
All of the following examples use the same question call:
\begin{sourcecode}
\SetupExSheets{headings=<name>}
\begin{question}[subtitle=The subtitle of the question]{1}
A `<name>' heading. \sample
\end{question}
\end{sourcecode}
\subsubsection{The `block' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{block}{default}{
join = { title[r,B]number[l,B](.333em,0pt) } ,
attach =
{
main[l,vc]title[l,vc](0pt,0pt) ;
main[r,vc]points[l,vc](\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=block}
\begin{question}[subtitle=The subtitle of the question]{1}
A `block' heading. \sample
\end{question}
\subsubsection{The `runin' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{runin}{default}{
runin = true ,
number-post-code = \space ,
attach =
{ main[l,vc]points[l,vc](\linewidth+\marginparsep,0pt) } ,
join =
{
main[r,vc]title[r,vc](0pt,0pt) ;
main[r,vc]number[l,vc](.333em,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=runin}
\begin{question}[subtitle=The subtitle of the question]{1}
A `runin' heading. \sample
\end{question}
\subsubsection{The `simple' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{simple}{default}{
title-format = \normalsize ,
points-pre-code = ( ,
points-post-code = ) ,
attach = { main[l,t]number[l,t](0pt,0pt) } ,
join =
{
number[r,b]title[l,b](.333em,0pt) ;
main[l,b]points[l,t](1em,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=simple}
\begin{question}[subtitle=The subtitle of the question]{1}
A `simple' heading. \sample
\end{question}
\subsubsection{The `empty' Instance}
\sinceversion{0.9a}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{empty}{default}{
runin = true ,
above = \parskip ,
below = \parskip ,
attach = { main[l,vc]points[l,vc](\linewidth+\marginparsep,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=empty}
\begin{question}[subtitle=The subtitle of the question]{1}
An `empty' heading. \sample
\end{question}
\subsubsection{The `block-rev' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{block-rev}{default}{
toc-reversed = true ,
join = { number[r,B]title[l,B](.333em,0pt) } ,
attach =
{
main[l,vc]number[l,vc](0pt,0pt) ;
main[r,vc]points[l,vc](\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=block-rev}
\begin{question}[subtitle=The subtitle of the question]{1}
A `block-rev' heading. \sample
\end{question}
\subsubsection{The `block-subtitle' Instance}
\sinceversion{0.10}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{block-subtitle}{default}{
join = {
title[r,B]number[l,B](.333em,0pt) ;
title[r,B]subtitle[l,B](1em,0pt)
} ,
attach = {
main[l,vc]title[l,vc](0pt,0pt) ;
main[r,vc]points[l,vc](\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=block-subtitle}
\begin{question}[subtitle=The subtitle of the question]{1}
A `block-subtitle' heading. \sample
\end{question}
\subsubsection{The `block-wp' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{block-wp}{default}{
points-pre-code = ( ,
points-post-code = ) ,
join =
{
title[r,B]number[l,B](.333em,0pt) ;
title[r,B]points[l,B](.333em,0pt)
} ,
attach = { main[l,vc]title[l,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=block-wp}
\begin{question}[subtitle=The subtitle of the question]{1}
A `block-wp' heading. \sample
\end{question}
\subsubsection{The `block-wp-rev' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{block-wp-rev}{default}{
toc-reversed = true ,
points-pre-code = ( ,
points-post-code = ) ,
join =
{
number[r,B]title[l,B](.333em,0pt) ;
number[r,B]points[l,B](.333em,0pt)
} ,
attach = { main[l,vc]number[l,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=block-wp-rev}
\begin{question}[subtitle=The subtitle of the question]{1}
A `block-wp-rev' heading. \sample
\end{question}
\subsubsection{The `block-nr' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{block-nr}{default}{
attach =
{
main[l,vc]number[l,vc](0pt,0pt) ;
main[r,vc]points[l,vc](\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=block-nr}
\begin{question}[subtitle=The subtitle of the question]{1}
A `block-nr' heading. \sample
\end{question}
\subsubsection{The `block-nr-wp' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{block-nr-wp}{default}{
points-pre-code = ( ,
points-post-code = ) ,
join = { number[r,vc]points[l,vc](.333em,0pt) } ,
attach = { main[l,vc]number[l,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=block-nr-wp}
\begin{question}[subtitle=The subtitle of the question]{1}
A `block-nr-wp' heading. \sample
\end{question}
\subsubsection{The `runin-rev' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{runin-rev}{default}{
toc-reversed = true ,
runin = true ,
title-post-code = \space ,
attach =
{ main[l,vc]points[l,vc](\linewidth+\marginparsep,0pt) } ,
join =
{
main[r,vc]number[r,vc](0pt,0pt) ;
main[r,vc]title[l,vc](.333em,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=runin-rev}
\begin{question}[subtitle=The subtitle of the question]{1}
A `runin-rev' heading. \sample
\end{question}
\subsubsection{The `runin-wp' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{runin-wp}{default}{
runin = true ,
points-pre-code = ( ,
points-post-code = )\space ,
join =
{
main[r,vc]title[r,vc](0pt,0pt) ;
main[r,vc]number[l,vc](.333em,0pt) ;
main[r,vc]points[l,vc](.333em,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=runin-wp}
\begin{question}[subtitle=The subtitle of the question]{1}
A `runin-wp' heading. \sample
\end{question}
\subsubsection{The `runin-wp-rev' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{runin-wp-rev}{default}{
toc-reversed = true ,
runin = true ,
points-pre-code = ( ,
points-post-code = )\space ,
join =
{
main[r,vc]number[r,vc](0pt,0pt) ;
main[r,vc]title[l,vc](.333em,0pt) ;
main[r,vc]points[l,vc](.333em,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=runin-wp-rev}
\begin{question}[subtitle=The subtitle of the question]{1}
A `runin-wp-rev' heading. \sample
\end{question}
\subsubsection{The `runin-nr' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{runin-nr}{default}{
runin = true ,
number-post-code = \space ,
attach =
{ main[l,vc]points[l,vc](\linewidth+\marginparsep,0pt) } ,
join = { main[r,vc]number[l,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=runin-nr}
\begin{question}[subtitle=The subtitle of the question]{1}
A `runin-nr' heading. \sample
\end{question}
\subsubsection{The `runin-fixed-nr' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{runin-fixed-nr}{default}{
runin = true ,
number-pre-code = \hbox to 2em \bgroup ,
number-post-code = \hfil\egroup ,
attach =
{ main[l,vc]points[l,vc](\linewidth+\marginparsep,0pt) } ,
join = { main[r,vc]number[l,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=runin-fixed-nr}
\begin{question}[subtitle=The subtitle of the question]{1}
A `runin-fixed-nr' heading. \sample
\end{question}
\subsubsection{The `runin-nr-wp' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{runin-nr-wp}{default}{
runin = true ,
points-pre-code = ( ,
points-post-code = )\space ,
join =
{
main[r,vc]number[l,vc](0pt,0pt) ;
main[r,vc]points[l,vc](.333em,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=runin-nr-wp}
\begin{question}[subtitle=The subtitle of the question]{1}
A `runin-nr-wp' heading. \sample
\end{question}
\subsubsection{The `inline' Instance}
\sinceversion{0.5}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{inline}{default}{
inline = true ,
number-pre-code = \space ,
number-post-code = \space ,
join =
{
main[r,vc]title[r,vc](0pt,0pt) ;
main[r,vc]number[l,vc](0pt,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=inline}
Text before
\begin{question}[subtitle=The subtitle of the question]{1}
An `inline' heading. \sample
\end{question}
Text after
\subsubsection{The `inline-wp' Instance}
\sinceversion{0.5}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{inline-wp}{default}{
inline = true ,
number-pre-code = \space ,
number-post-code = \space ,
points-pre-code = ( ,
points-post-code = )\space ,
join =
{
main[r,vc]title[r,vc](0pt,0pt) ;
main[r,vc]number[l,vc](0pt,0pt) ;
main[r,vc]points[l,vc](0pt,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=inline-wp}
Text before
\begin{question}[subtitle=The subtitle of the question]{1}
An `inline-wp' heading. \sample
\end{question}
Text after
\subsubsection{The `inline-nr' Instance}
\sinceversion{0.5}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{inline-nr}{default}{
inline = true ,
number-post-code = \space ,
join = { main[r,vc]number[l,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=inline-nr}
Text before
\begin{question}[subtitle=The subtitle of the question]{1}
An `inline-nr' heading. \sample
\end{question}
Text after
\subsubsection{The `centered' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{centered}{default}{
join = { title[r,B]number[l,B](.333em,0pt) } ,
attach =
{
main[hc,vc]title[hc,vc](0pt,0pt) ;
main[r,vc]points[l,vc](\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=centered}
\begin{question}[subtitle=The subtitle of the question]{1}
A `centered' heading. \sample
\end{question}
\subsubsection{The `centered-wp' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{centered-wp}{default}{
points-pre-code = ( ,
points-post-code = ) ,
join =
{
title[r,B]number[l,B](.333em,0pt) ;
title[r,B]points[l,B](.333em,0pt)
} ,
attach = { main[hc,vc]title[hc,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=centered-wp}
\begin{question}[subtitle=The subtitle of the question]{1}
A `centered-wp' heading. \sample
\end{question}
\subsubsection{The `margin' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{margin}{default}{
runin = true ,
number-post-code = \space ,
points-pre-code = ( ,
points-post-code = )\space ,
join = { title[r,b]number[l,b](.333em,0pt) } ,
attach =
{
main[l,vc]title[r,vc](0pt,0pt) ;
main[l,b]points[r,t](0pt,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=margin}
\begin{question}[subtitle=The subtitle of the question]{1}
A `margin' heading. \sample
\end{question}
\subsubsection{The `margin-nr' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{margin-nr}{default}{
runin = true ,
attach =
{
main[l,vc]number[r,vc](-.333em,0pt) ;
main[r,vc]points[l,vc](\linewidth+\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=margin-nr}
\begin{question}[subtitle=The subtitle of the question]{1}
A `margin-nr' heading. \sample
\end{question}
\subsubsection{The `raggedleft' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{raggedleft}{default}{
join = { title[r,B]number[l,B](.333em,0pt) } ,
attach =
{
main[r,vc]title[r,vc](0pt,0pt) ;
main[r,vc]points[l,vc](\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=raggedleft}
\begin{question}[subtitle=The subtitle of the question]{1}
A `raggedleft' heading. \sample
\end{question}
\subsubsection{The `fancy' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{fancy}{default}{
toc-reversed = true ,
indent-first = true ,
vscale = 2 ,
pre-code = \rule{\linewidth}{1pt} ,
post-code = \rule{\linewidth}{1pt} ,
title-format = \large\scshape\color{rgb:red,0.65;green,0.04;blue,0.07} ,
number-format = \large\bfseries\color{rgb:red,0.02;green,0.04;blue,0.48} ,
points-format = \itshape ,
join = { number[r,B]title[l,B](.333em,0pt) } ,
attach =
{
main[hc,vc]number[hc,vc](0pt,0pt) ;
main[l,vc]points[r,vc](-\marginparsep,0pt)
}
}
\end{sourcecode}
\SetupExSheets{headings=fancy}
\begin{question}[subtitle=The subtitle of the question]{1}
A `fancy' heading. \sample
\end{question}
\subsubsection{The `fancy-wp' Instance}
\begin{sourcecode}
\DeclareInstance{exsheets-heading}{fancy-wp}{default}{
toc-reversed = true ,
indent-first = true ,
vscale = 2 ,
pre-code = \rule{\linewidth}{1pt} ,
post-code = \rule{\linewidth}{1pt} ,
title-format = \large\scshape\color{rgb:red,0.65;green,0.04;blue,0.07} ,
number-format = \large\bfseries\color{rgb:red,0.02;green,0.04;blue,0.48} ,
points-format = \itshape ,
points-pre-code = ( ,
points-post-code = ) ,
join =
{
number[r,B]title[l,B](.333em,0pt) ;
number[r,B]points[l,B](.333em,0pt)
} ,
attach = { main[hc,vc]number[hc,vc](0pt,0pt) }
}
\end{sourcecode}
\SetupExSheets{headings=fancy-wp}
\begin{question}[subtitle=The subtitle of the question]{1}
A `fancy-wp' heading. \sample
\end{question}
\subsection{Using an \ExSheets{} Heading in Custom Code}\label{sec:using-an-exsheets}
It can be useful to have access to \ExSheets{} headings in custom code. This
is possible with the following command\sinceversion{0.14}:
\begin{commands}
\command{ExSheetsHeading}[\marg{instance}\marg{title}\marg{number}%
\marg{points}\marg{bonus}\marg{id}]
The meaning of the arguments is as follows:
\begin{itemize}
\item \meta{instance}: the name of the headings instance to be used.
\item \meta{title}: the contents of the \code{title} coffin.
\item \meta{number}: the contents of the \code{number} coffin.
\item \meta{points}: The number of points given to the question. If
non-zero this will cause the points to be printed in the \code{points}
coffin.
\item \meta{bonus}: the same as \meta{points} but for bonus points.
\item \meta{id}: the \acs{id} of the question this heading belongs to.
\end{itemize}
\end{commands}
In combination with \cs{ForEachQuestion} the command can be used to build a
custom list of questions. An example of its usage can be seen at the German
Q\&A~site \TeX welt: \url{http://texwelt.de/wissen/fragen/6698#6738}.
\subsection{Load Custom Configurations}
If you have custom configurations you want to be loaded automatically then save
them in a file \code{exsheets\_configurations.cfg}. If this file is present it
will be loaded \cs*{AtBeginDocument}.
\SetupExSheets{headings=block}
\part{The \ExSheetslistings\ Package}\label{part:listings}
\section{The Problem}
I knew the day would come when people would ask how to include verbatim
material in the \env{question} and \env{solution} environments. Since they're
defined with the \pkg{environ} package they're reading their environment
bodies like macros read their arguments. This makes it impossible to use
verbatim material inside them\footnote{See the \TeX\ \acs{faq}
\url{http://www.tex.ac.uk/cgi-bin/texfaq2html?label=verbwithin} for reasons
why.}. Now the day has come~\cite{tex.sx:131546}. Soon after the first
question appeared I wrote the first draft for \ExSheetslistings\ for a question
on \TeX.sx~\cite{tex.sx:133907}.
\section{The Proposed Solution}
The \ExSheetslistings\ package defines \pkg{listings} environments that place
their contents inside \env{question} and \env{solution} environments. They do
this by writing the listing to a unique auxiliary file --
questions to \code{\cs*{jobname}-ex\meta{num}.lst} and solutions to
\code{\cs*{jobname}-sol\meta{num}.lst} where \meta{num} is an increasing
integer that ensures that each listing gets a unique file name. Those files
are then included with \cs{lstinputlisting} if and when the question or
solution is printed.
\begin{environments}
\environment{lstquestion}[\oarg{options}]
A \pkg{listings} environment placed in a \env{question}.
\environment{lstsolution}[\oarg{options}]
A \pkg{listings} environment placed in a \env{solution}.
\end{environments}
All you have to do to use the package is loading it the usual way:
\begin{sourcecode}
\usepackage{exsheets-listings}
\end{sourcecode}
This will also load the packages \ExSheets\ and \pkg{listings} if they're not
loaded already.
\begin{example}
% this uses my listings style used in this documentation for all pieces of
% code:
\begin{lstquestion}[%
pre=Explain what this piece of \TeX\ code does:,
listings={style=cnltx}]
\begingroup\expandafter\expandafter\expandafter\endgroup
\expandafter\ifx\csname foo\endcsname\relax
...
\else
...
\fi
\end{lstquestion}
\end{example}
The example already shows two options of these environments. Here is the
complete list:
\begin{options}
\keyval{pre}{text}
\meta{text} is placed before the code in the \env{question} or
\env{solution} environment.
\keyval{post}{text}
\meta{text} is placed after the code in the \env{question} or
\env{solution} environment.
\keyval{options}{options}
Options passed to underlying the \env{question} or \env{solution}
environment.
\keyval{points}{points}
The points assigned to the underlying \env{question} environment.
\keyval{listings}{options}
Options passed to the underlying \pkg{listings} environment.
\end{options}
There are also two new options for \ExSheets\ that can be set with
\cs{SetupExSheets}:
\begin{options}
\keyval{listings}{options}\Module{question}
Options passed to the underlying \pkg{listings} environment of
\env{lstquestion}.
\keyval{listings}{options}\Module{solution}
Options passed to the underlying \pkg{listings} environment of
\env{lstsolution}.
\end{options}
\section{Own Environments}
\begin{commands}
\command{NewLstQuSolPair}[\oarg{options for both environments}\marg{lst question
env}\marg{question env}\oarg{options for lst question env}\marg{lst
solution env}\marg{solution env}\oarg{options for lst solution env}]
Defines two new \pkg{listings} environments that place the listing in a
question environment \meta{question env} or a solution environment
\meta{solution env}. Those underlying environments should be
environments as defined by \cs{NewQuSolPair}. The different options
allow to preset options for the newly defined environments.
\end{commands}
The existing environments have been defined like this:
\begin{sourcecode}
\NewLstQuSolPair{lstquestion}{question}{lstsolution}{solution}
\end{sourcecode}
\appendix
\part{Appendix}
\section{A List of all Solutions used in this Manual}\label{sec:solutions:list}
\SetupExSheets{headings=block-wp,solutions-totoc}
\printsolutions
\end{document}
|
https://repo.iut.ac.ir/tex-archive/macros/generic/misc/mandel.tex | iut.ac.ir | CC-MAIN-2023-14 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2023-14/segments/1679296943589.10/warc/CC-MAIN-20230321002050-20230321032050-00524.warc.gz | 579,000,845 | 2,045 | % TeX Mandelbrot Program
%
% My entry for the contest ``The most useless TeX program ever written''.
% Can anyone top that? (Don't mention MusicTeX in this context. That's not
% meant to be a joke.)
% Also welcome: Improvements of this saving on TeX's resources. This takes
% more than five minutes on my NeXT 486/66.
%
% Seb, 6.7.94
%
% ([email protected])
%
%
% parameters that you can modify:
%
% \x, \y, \xmax, \ymax in units of 10000
\newcount\x
\x=-20000 % means -2.000 for real part of start value
\newcount\y
\y=-20000 % imaginary part
\newcount\xmax
\xmax=20000 % end value, real
\newcount\ymax
\ymax=20000 % end value, imaginary
%
% calculation depth
\newcount\mandeldepth
\mandeldepth=100
%
%
% recursation depth = number of pixels.
\newcount\xsize % (Don't go wild, tail recursion
\xsize=100 % uses loads of TeX memory!)
\newcount\ysize
\ysize=100
%
% picture size
\newdimen\height
\newdimen\width
\height=10cm
\width=10cm
%
% that was all the parameters!
%
%
% pixel size:
\newdimen\xgrid
\xgrid=\height
\divide\xgrid by\xsize
\newdimen\ygrid
\ygrid=\width
\divide\ygrid by\ysize
%
\nopagenumbers
%
\newcount\stepX % work out step sizes
\stepX=\xmax
\advance\stepX by -\x
\divide\stepX by\xsize
\newcount\stepY
\stepY=\ymax
\advance\stepY by -\y
\divide\stepY by\ysize
%
\def\cout#1{\immediate\write16{#1}} % type out debug message
%
\def\yes{Y} % some stupid logical macros
\def\no{N}
\def\notyet{x}
%
\newcount\u % variables for calculation
\newcount\v
\newcount\newV
\newcount\newU
\newcount\d
\newcount\h
\def\runmandel{% % recursive Mandelbrot macro
\newU=\u%
\multiply\newU by \u% % newU := u^2
\h=\v%
\multiply\h by \v% % h := v^2
\advance\newU by -\h% % newU := u^2 - v^2
\h=\x%
\multiply\h by 10000% % h := x * 10000 (normalize)
\advance\newU by \h% % newU := newU + h
\divide\newU by 10000% % renormalize u
\newV=\u%
\multiply\newV by \v%
\multiply\newV by 2% % newV := 2 *u * v
\h=\y%
\multiply\h by 10000% % h := y * 10000
\advance\newV by \h% % newV := newV + h
\divide\newV by 10000% % renormalize v
\u=\newU% % newU = u^2 - v^2 + x
\v=\newV% % newV = 2 * u * v + y
\advance\d by 1%
% \cout{d: \the\d, u: \the\u, v: \the\v}%
\ifnum\d>\mandeldepth% % member ofthe Mandelbrot set
\def\result{\yes}%
\fi%
\ifnum\u<-20000%
\def\result{\no}% % real part exceeded -2
\fi%
\ifnum\u>20000%
\def\result{\no}% % real part exceeded 2
\fi%
\ifnum\v<-20000%
\def\result{\no}% % imaginary part exceeded -2
\fi%
\ifnum\v>20000%
\def\result{\no}% % real part exceeded 2
\fi%
\if\notyet\result\runmandel\fi% % play it again...
}%
\def\result{\yes}%
\def\makeline{\advance\x by \stepX%
%\cout{testing \the\x = i * \the\y}%
\u=0%
\v=0%
\d=0%
\def\result{\notyet}%
\runmandel% result expands to either yes or no now
\if\result\yes%
\vrule height \xgrid depth 0pt width \ygrid%
\else%
\vrule height \xgrid depth 0pt width 0pt\hskip\xgrid%
\fi%
\ifnum\x<\xmax%
\makeline%
\else\fi}%
\def\makepicture{\advance\y by \stepY%
\hbox{\makeline}%
%\cout{y in makepicture: \the\y}%
\ifnum\y<\ymax%
\makepicture%
\else\fi}%
\vbox{\offinterlineskip\makepicture} % now, let's go tail
\bye % recursive!
|
http://porocila.imfm.si/2004/mat/clani/lipovec.tex | imfm.si | CC-MAIN-2023-14 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296949958.54/warc/CC-MAIN-20230401094611-20230401124611-00693.warc.gz | 41,603,036 | 879 | \clan {Alenka Lipovec}
\begin{skupina}{C}
\objavljenoRevija
{B.~Bre\v{s}ar, S.~Klav\v{z}ar, \crta, B.~Mohar}
{Cubic inflation, mirror graphs, regular maps, and partial cubes}
{European J.~Combin.} {25} {2004} {1} {5{5}--64}
\objavljenoRevija
{S.~Klav\v{z}ar, \crta}
{Edge-critical isometric subgraphs of hypercubes}
{Ars Combin.} {70} {2004} {1} {139--147}
\end{skupina}
\begin{skupina}{I}
\razno
{Ena recenzija u\v cbenika.}
\razno
{Mentorica pri 2 diplomskih delih.}
\end{skupina}
|
https://authorea.com/users/225395/articles/281858/download_latex | authorea.com | CC-MAIN-2021-10 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-10/segments/1614178373241.51/warc/CC-MAIN-20210305183324-20210305213324-00529.warc.gz | 211,899,771 | 2,764 | \documentclass[10pt]{article}
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\begin{document}
\title{Title}
\author[1]{Jaquelinne Jáquez-Amador}%
\author[2]{MAGALY CEBALLOS-MORALES}%
\affil[1]{Instituto Tecnológico Superior Zacatecas Occidente}%
\affil[2]{Affiliation not available}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{spanish}
\begin{abstract}
En este primer trabajo analizamos la herramienta
\href{http://www.authorea.com}{Authorea} la cual sirve para la
elaboraci\selectlanguage{ngerman}ón de textos académicos. Desglosamos cada una de sus
funcionalidades.%
\end{abstract}\selectlanguage{ngerman}%
\sloppy
\selectlanguage{ngerman}\section*{Introducción (Fer)}
{\label{891068}}
Hola
\section*{Desarrollo (Luis)}
{\label{404653}}
Una de las capacidades de esta plataforma es la inclusión de las
imágenes como podemos apreciar a continuación:\selectlanguage{spanish}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.28\columnwidth]{figures/presa-de-jales/presa-de-jales}
\caption{\selectlanguage{ngerman}{Representación esquemática de una presa de jales obtenida de
este\href{http://www.loscardones.com.mx/presa_de_jales.php}{sitio
web}~\protect\cite{cardones2015}
{\label{490708}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}
\subsection*{Además de la inclusión de figuras también podemos incorporar
tablas, como los ejemplificaremos a
continuación:}
{\label{812809}}\par\null\selectlanguage{spanish}
\begin{table}[H]
\centering
\normalsize\begin{tabulary}{1.0\textwidth}{CCC}
& Nombre del alumno & Calificaci\selectlanguage{ngerman}ón \\
& Juanito & 90 \\
& Rub\selectlanguage{ngerman}én & 100 \\
\end{tabulary}
\caption{\selectlanguage{ngerman}{Resultados de la evaluación de la unidad 1
{\label{601338}}%
}}
\end{table}\selectlanguage{ngerman}\par\null
Los problemas ambientales asociados a la minería ha causado diversas
dificultades a lo largo del país. Entre ellas se encuentra las
relacionadas con la salud debido a los metales pesados que están
contenidos en las presas de jales que desafortunadamente pueden entrar
en contacto con la población debido a su volatilidad.\citep{puga2006contaminacion}
\par\null
\subsection*{Los menús en Authorea}
{\label{812809}}
\subsubsection*{formatos de
exportación}
{\label{792061}}
\section*{conclusiones (Fer y Luis)}
{\label{434783}}
\section*{}
{\label{958360}}
\selectlanguage{spanish}
\FloatBarrier
\bibliographystyle{plainnat}
\bibliography{bibliography/converted_to_latex.bib%
}
\end{document}
|
https://source.contextgarden.net/tex/context/base/mkiv/context-todo.tex?dl=1 | contextgarden.net | CC-MAIN-2023-06 | text/plain | application/x-tex | crawl-data/CC-MAIN-2023-06/segments/1674764500384.17/warc/CC-MAIN-20230207035749-20230207065749-00481.warc.gz | 535,779,143 | 2,754 | \setuplayout
[width=middle,
height=middle,
topspace=2cm,
header=0pt,
footer=1cm]
\setupbodyfont
[bookman]
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[punk,abr-02]
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[section]
[color=ColorThree,
style=\bfb]
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style=bold]
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% \def\aterm{\sym{?}}
% \def\rterm{\sym{--}}
% \def\dterm{\sym{+}}
% \def\pterm{\sym{!}}
%
% \startitemize[packed]
% \aterm on the agenda (update, extension, rewrite)
% \rterm no longer on the agenda, rejected
% \dterm no longer on the agenda, done
% \pterm work in progress (so keep an eye on the betas)
% \stopitemize
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endfor ;
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picture l ; l := image ( draw for i=1 upto n : if i > 1 : -- fi p[i] endfor ) ;
picture t ; t := textext("\framed[frame=off,align={middle,lohi},foregroundcolor=ColorThree,foregroundstyle=\PunkFont]{\ConTeXt\endgraf MkIV\endgraf\kern-\strutdepth RoadMap}") ;
fill Page enlarged 10 withcolor "ColorOne" ;
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draw thelabel.ulft((t xsized .5PaperWidth),lrcorner Page shifted - (PaperWidth/20,-PaperWidth/40)) ;
StopPage ;
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\startsubject[title={Introduction}]
There is not really a long term roadmap for development. One reason is that there
is already a lot available. When we started with \LUATEX, the \CONTEXT\ code was
mostly rewritten from \MKII\ to \MKIV, and that process is more of less finished.
Of course there is always work left. We have moved on to \LUAMETATEX\ and \LMTX\
as follow up on \MKIV. Deep down some things might (and will) change but the user
interface will stay the same (as usual).
This file is not a complete overview of our plans but users can at least get an
idea of what we're working on and what is coming. Feel free to submit
suggestions.
\startlines
Hans Hagen
Hasselt NL
\currentdate
\stoplines
\stopsubject
\startsubject[title={On the agenda for \LUAMETATEX}]
The agenda for \LUATEX\ is basically empty as this program is (supposed to be)
frozen, but for \LUAMETATEX\ we can still consider improvements. In fact, there
are quite some changes between the engines already (internally). Items that
concern \LUATEX\ have been removed from the agenda but some improvements in
\LUAMETATEX\ might trickle back.
\startitemize
\startitem
cleanup passive nodes in the par builder
\stopitem
\startitem
optimize some callback resolution (more direct) but there is not that
much to gain there
\stopitem
\startitem
remove local par in head of line when done with linebreak and maybe also
ensure leftskip and rightskip
\stopitem
\startitem
only return nil in lua helpers when we expect multiple calls in in one
line
\stopitem
\startitem
experiment a bit more with the new protrusion code
\stopitem
\stopitemize
\stopsubject
\startsubject[title={On the agenda for \CONTEXT\ \MKIV}]
\startitemize
\startitem
play with par callback and properties
\stopitem
\startitem
optimize positions for columnareas and parpos (sequential)
\stopitem
\startitem
add flag to font for math engine
\stopitem
\startitem
get rid of components in glyph nodes
\stopitem
\startitem
play with box attributes
\stopitem
\startitem
check consistency between footnotes and running text (main color,
styles, properties)
\stopitem
\startitem
freeze actions and tasks (by name or function), maybe a register
function that does that so no one can overload built-in features
\startitem
redo some of the spacing (adapt to improvements in engine)
\stopitem
\startitem
more node and code injections
\stopitem
\startitem
maybe reorganize position data (more subtables)
\stopitem
\startitem
use \type {\matheqnogapstep}, \type {\Ustack}, \type {\mathscriptsmode},
\type {\mathdisplayskipmode} and other new math primitives
\stopitem
\startitem
cleanup some lua helpers (io.exists vs lfs.isfile)
\stopitem
\stopitemize
% should be in lpdf namespace:
%
% ./grph-pat.lua 69: local l = new_literal(lpdf.patternstream(p,width,height))
\stopsubject
\stoptext
|
https://www.zentralblatt-math.org/matheduc/en/?id=69978&type=tex | zentralblatt-math.org | CC-MAIN-2019-22 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-22/segments/1558232256764.75/warc/CC-MAIN-20190522063112-20190522085112-00303.warc.gz | 1,022,244,519 | 1,132 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2002b.01665}
\itemau{Dahn, Ingo}
\itemti{Automatic textbook construction and web delivery in the 21st century.}
\itemso{J. Struct. Learn. Intell. Syst. 14, No. 4, 401-413 (2001).}
\itemab
We explain the basic concepts of Slicing Book Technology, which is a new method to augment textbooks with online personalization services. This technology is then compared with techniques used for computer-based training. Finally, some related technological and economic trends are discussed. (Author's abstract)
\itemrv{~}
\itemcc{U50}
\itemut{computer-based training; personalization; slicing book technology}
\itemli{}
\end |
http://milde.users.sourceforge.net/Matheschriften/kerkis-math-mathdesign-test.tex | sourceforge.net | CC-MAIN-2018-05 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2018-05/segments/1516084891539.71/warc/CC-MAIN-20180122193259-20180122213259-00543.warc.gz | 215,663,599 | 881 | % Kerkis
% ------
% `Kerkis fonts <http://iris.math.aegean.gr/kerkis/>`_ sind neoklassizistische
% griechische Schriften als Ergänzung zu URW Bookman.
% kerkis-math: nutzt Kerkis fonts als Mathealphabete
% kombinieren mit anderen Mathepaketen für speziele Mathematik-Symbole
\newcommand{\schriftart}{Kerkis + Mathdesign}
\newcommand{\fontsetup}{
\usepackage[charter,expert]{mathdesign}
\usepackage{kerkis-math,kerkis}
}
\input{mathfonttest.tex}
|
https://floridaclimateinstitute.org/refbase/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%2C%20area%20FROM%20refs%20WHERE%20serial%20%3D%201982%20ORDER%20BY%20created_date%20DESC%2C%20created_time%20DESC%2C%20modified_date%20DESC%2C%20modified_time%20DESC%2C%20serial%20DESC&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=20&rowOffset=&wrapResults=1&citeOrder=creation-date&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg= | floridaclimateinstitute.org | CC-MAIN-2018-34 | application/x-latex | application/x-latex | crawl-data/CC-MAIN-2018-34/segments/1534221211126.22/warc/CC-MAIN-20180816152341-20180816172341-00518.warc.gz | 671,109,164 | 1,371 |
%&LaTeX
\documentclass{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Bertassello_etal2018} Bertassello, L. E., Rao, P. ~S. C., Park, J., Jawitz, J. W., \& Botter, G. (2018). Stochastic modeling of wetland-groundwater systems. \textit{Advances in Water Resources}, \textit{112}, 214--223.
\end{thebibliography}
\end{document}
|
https://lib.anarhija.net/library/michael-william-the-ecology-montreal-party-a-libertarian-frankenstein.tex | anarhija.net | CC-MAIN-2022-49 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-49/segments/1669446710237.57/warc/CC-MAIN-20221127105736-20221127135736-00211.warc.gz | 399,900,734 | 13,073 | \documentclass[DIV=12,%
BCOR=0mm,%
headinclude=false,%
footinclude=false,open=any,%
fontsize=10pt,%
oneside,%
paper=a5]%
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\usepackage{fontspec}
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% global style
\pagestyle{plain}
\usepackage{microtype} % you need an *updated* texlive 2012, but harmless
\usepackage{graphicx}
\usepackage{alltt}
\usepackage{verbatim}
% http://tex.stackexchange.com/questions/3033/forcing-linebreaks-in-url
\PassOptionsToPackage{hyphens}{url}\usepackage[hyperfootnotes=false,hidelinks,breaklinks=true]{hyperref}
\usepackage{bookmark}
\usepackage[shortlabels]{enumitem}
\usepackage{tabularx}
\usepackage[normalem]{ulem}
\def\hsout{\bgroup \ULdepth=-.55ex \ULset}
% https://tex.stackexchange.com/questions/22410/strikethrough-in-section-title
% Unclear if \protect \hsout is needed. Doesn't looks so
\DeclareRobustCommand{\sout}[1]{\texorpdfstring{\hsout{#1}}{#1}}
\usepackage{wrapfig}
\usepackage{indentfirst}
% remove the numbering
\setcounter{secnumdepth}{-2}
% remove labels from the captions
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\renewcommand*{\tableformat}{}
\KOMAoption{captions}{belowfigure,nooneline}
\addtokomafont{caption}{\centering}
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% footnote handling
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\usepackage{perpage}
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% avoid breakage on multiple <br><br> and avoid the next [] to be eaten
\newcommand*{\forcelinebreak}{\strut\\*{}}
\newcommand*{\hairline}{%
\bigskip%
\noindent \hrulefill%
\bigskip%
}
% reverse indentation for biblio and play
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\leftskip=\parindent
\parindent=-\parindent
\smallskip
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}{\smallskip}
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\newcommand*{\Slash}{\slash\hspace{0pt}}
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% forbid widows/orphans
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% http://tex.stackexchange.com/questions/304802/how-not-to-hyphenate-the-last-word-of-a-paragraph
\finalhyphendemerits=10000
% given that we said footinclude=false, this should be safe
\setlength{\footskip}{2\baselineskip}
\title{The Ecology Montreal Party: A “Libertarian” Frankenstein}
\date{1994}
\author{Michael William}
\subtitle{}
% https://groups.google.com/d/topic/comp.text.tex/6fYmcVMbSbQ/discussion
\hypersetup{%
pdfencoding=auto,
pdftitle={Ecology Montreal Party: A “Libertarian” Frankenstein},%
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\begin{document}
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\begin{center}
{\usekomafont{title}{\huge The Ecology Montreal Party: A “Libertarian” Frankenstein\par}}%
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\vskip 2em
{\usekomafont{author}{Michael William\par}}%
\vskip 1.5em
\vfill
{\usekomafont{date}{1994\par}}%
\end{center}
\end{titlepage}
\cleardoublepage
\tableofcontents
% start a new right-handed page
\cleardoublepage
Shortly before the last Canadian elections, the head of the ruling Conservatives, Brian Mulroney, resigned. Enormously unpopular, his approval rating approaching 10\%, Mulroney was visibly damaging the party’s already slim chances of winning the upcoming elections. Replacing Mulroney at the party helm was Kim Campbell, a one-time member of the Social Credit Party, a right-wing populist party which is now defunct except in one province.
During the elections any mention of Mulroney by the Conservatives was predictably avoided. Their campaign, though, went further. The party, incredibly, attempted to present itself as outsider, as anti-establishment. It was almost as if the party in power was running against itself.
This desperate reality-bending was ultimately more amusing than effective. The Conservatives were virtually wiped out, going from a comfortable majority to two seats. Such events, however, eloquently reflected a climate in which politicians and parties are despised as never before.
The response of the parties to what negates them — their attempt to integrate and neutralize it — is populism. Significantly, when the Conservatives were elbowed out, they were displaced on the right by the populist Reform Party, which went from three seats to fifty-two. The Party is run almost single-handedly by Preston Manning, an evangelical Christian who presents himself as an anti-politician, ostentatiously refusing a few of the perks of office, but is in fact the son of a former premier and a consummate politician.
Ross Perot, a paranoid, unvarnished authoritarian, evokes electronic town halls while running essentially a one-man show. Demonstrated by Perot is populism’s ability to transcend traditional political categories and draw support from both the left and right.
In Russia, a potent nationalist-populist brew allowed a ranting buffoon, Zhirinovsky, to gobble a quarter of the parliamentary vote.
Today populism is ubiquitous, seemingly obligatory. Above all, it is a sure-fire indicator of demagogy.
\begin{center}
* * *
\end{center}
One of the newest kids on the populist block is the libertarian municipalism-inspired Ecology Montreal Party. If “Vote for me, and the people will be in power” constitutes populism’s usual refrain, libertarian municipalism’s spin might be phrased: “Vote for me, and the state will eventually wither away.”
Uh huh.
Montreal is a major centre of libertarian municipalism. Ecology Montreal in effect was initiated primarily by one person, Dimitri Roussopoulos, a self-described anarchist who was a candidate in the last elections, in which more than one “anarchist” ran. Ecology Montreal’s members take “their inspiration from the social ecology and urban theories of Murray Bookchin,” according to Phillip Chee, a party militant, and many libertarian municipalist books, including Bookchin’s, emanate from Roussopoulos’ Black Rose Books\Slash{}\emph{Our Generation} magazine operation. Bookchin himself was brought in to address an Ecology Montreal policy conference. An international social ecology conference with libertarian municipalism as the featured topic will take place in Montreal in 1994\dots{}
Until recently, libertarian municipalism has been primarily confined to institutes and academia. Now that it is generating actual political parties and is acquiring a history, it is useful to look at how that history is being represented by the ideology’s adherents.
In its “Living in the City” special issue (Fall 93), the Murray Bookchin-influenced Toronto journal \emph{Kick It Over} published an excerpt from a text on libertarian municipalism by Bookchin and an article on Ecology Montreal by Phillip Chee. A one-two, the Bookchin reprint theoretically softens us up for Chee’s Ecology Montreal sucker punch.
In his piece Bookchin encourages anarchists to become politicians and to run for office, and drools over “cybernetic devices,” making clear his desire to mediate experience through them.
Central to libertarian municipalism is drawing a dubious distinction between the nation-state and the municipal state. Libertarian municipalism legitimizes the city-state but turns up its nose at the nation-state (although Ecology Montreal is clearly willing to coexist with it). Differences between these states, however, are far outweighed by what they have in common: the omnipresence of the money\Slash{}commodity economy, the existence of politicians, the laws they impose and the cops and courts that back them up, and the reign of the technocrats necessary to run modern industrial capital. We deal with municipal cops, not the army, on a daily basis.
Chee’s article about Ecology Montreal is a classic illustration of Party Thinking eclipsing autonomous thought — of the political organization imposing its own logic and imperatives. Once set in motion, a party rapidly takes on a life of its own. For the party militant people are either inside or outside the party and those on the inside, having internalized the party’s imperatives, view those on the outside in a reified, manipulative way (ultimately principally as vote fodder). Thus Chee reels off the banal facts of party life, seemingly blissfully unaware of how it sounds to the unconverted, that Ecology Montreal, rather than a radical departure, is actually more akin to partyism-as-usual: choosing candidates, counting voters, setting up party structures, putting out position papers, making deals with other parties etc. \emph{ad nauseam} — these staples of party “life” provide a structure, a bureaucratic playpen to keep the militants’ hands occupied. Psychologically the militant needs to assign the party and his or her activities a key role — to be convinced, in Chee’s words, that Ecology Montreal “has the potential to ignite a movement.” Also key is the moral superiority which justifies the militant’s leading role. For Chee, the party becomes the model of the future society; it is the very purity of the militants’ lives which justifies handing them state power.
For the militant the organization becomes the beacon. Thus Ecology Montreal presents itself as an “educational organization,” and puts on “educational events.” Having come up empty-handed in the most recent elections, libertarian municipalists in Syracuse are presently producing and distributing a journal in order to “educate the public.” This vanguard aspect is crucial to Bookchinism. In a recent issue of \emph{Green Perspectives}, for example, editors Murray Bookchin and Janet Biehl defend an “educational” approach, and specifically advocate vanguardism, attempting to put an innocuous, cultural spin on the concept: “The word \emph{vanguard}, we should add, does not throw us into a panic. An avant-garde teacher (or artist) is still a teacher (or artist), and there is no point in pretending otherwise.” Present-day anarchists who question vanguardism are referred to with the word anarchist in quotation marks, implying that being an anarchist and questioning vanguardism is incompatible, as the duo pines after the good old days, \emph{i.e}. the “nineteenth and early twentieth centuries,” when anarchists and “their organizations” adopted a vanguardist posture. Displayed here is how much Bookchin and Biehl have in common with the foibles of the nineteenth century anarchists — their Enlightenment-based religious belief in techno-rationalism and the ideology of progress, which finds its natural culmination in Bookchin’s “cybernetic devices.” Also key here are specialization and division of labour: the student\Slash{}teacher dichotomy and its institutionalization become the initial hierarchy on which all the others are built.
If academia-drenched, this is not simply an academic question. In the early ’80s an attempt took place to put this outlook into practice with the creation of the briefly very active and now moribund Anarchos Institute. Initiated in large part by Bookchin and Roussopoulos, the Institute epitomized their vision of a coterie of academics implementing a top-down relationship vis-a-vis non-academic anarchists initially, and, presumably, eventually a broader milieu. In Bookchinist discourse this is theorized as the “indispensable radical intelligentsia” without which “a libertarian movement” will be unable to “emerge.” This, however, was not the approach of everyone involved in the Institute. Rapidly a crisis took place, triggered off by Roussopoulos’ authoritarianism and unilateral decision-making. When the non-academics in the local Montreal group objected, they were purged by the profs in a clear instance of academic class solidarity. (If they don’t support Roussopoulos, where are they going to publish?) At a key meeting Bookchin was parachuted in to lend his authority to the purge exercise. In the resulting scandal the Institute rapidly became a ghost of its former self, as the academic rump group implemented classical sleaze techniques like refusing access to the mailing list to the non-academics so they couldn’t inform the membership about what was going on. This is just one in a long string of similar incidents involving Roussopoulos, including firing two anarchists at Black Rose Books when they attempted to collectivize the project.\footnote{I was not a member of the Anarchos Institute, but followed events closely. Documents about the Institute and the Black Rose firings are available by writing to: Michael, C.P. 1554 Succ. B, Montreal, Quebec, Canada H3B 3L2.}
\begin{center}
* * *
\end{center}
Despite abundant talk about triggering off “participatory, face to face” activity, no examples are provided by Chee of Ecology Montreal causing anyone to do anything. On the contrary, as he acknowledges, “By far the most publicly visible activity Ecology Montreal has engaged in has been its electoral efforts.” Chee’s account is a classic case of electoralism imposing its logic and priorities. “During the election campaign,” he recounts, “the fundamental disagreements about the movement’s structure were pushed below the surface. The crux of the matter was what type of leadership the party should adopt.” And, Chee informs us, presently Ecology Montreal is “putting considerable effort into creating an electoral strategy for the 1994 elections.”
Ghee goes to considerable lengths to distance Ecology Montreal from other parties, especially the social democrats. Evoked by Roussopoulos in Chee’s piece is the term “anti-party party,” using the German Greens as a model(!). But Ecology Montreal’s main concern is clearly grabbing parliamentary power (entirely understandable from an electoral viewpoint, seeing that no Ecology Montreal candidates won in the last election). Thus the party is currently hammering out a “common platform” with “independent city councillors” and other “progressives.” This is only more of the tired leftism that has been discredited worldwide, notably, in Canada, with the arrival in power for the first time in the province of Ontario of the New Democratic Party (social democrats). Within a year the popularity of the party plummeted; few retain any illusions about “really-existing” NDPism. Ecology Montreal’s desperate attempt to elect a candidate or two also involves an infusion of traditional political horse-trading, as “Alliance 94” proceeds to “divide up the electoral map so as not to run alliance candidates against each other.”
Another example of opportunist tinkering with the system is the party’s reaction to a proposal to reduce the number of politicians from the current 51. Instead, Ecology Montreal proposed that “Montreal adopt a partial system of proportional representation. Thirty-one seats would remain single member constituencies with election by direct majority, and 20 seats to be distributed among representatives of the municipalities proportionally to the percentage of the popular vote gained by each party to the city as a whole.” Demonstrated here is that despite obligatory complaints about “impersonal bureaucracies and professional politicians,” Chee \emph{really believes} in representational democracy — that politicians are legitimate, that parties represent people, that people can be represented by politicians. Thus Ecology Montreal’s pathetic solution becomes sprinkling in a few councillors from presently marginalized parties, or otherwise slightly shifting the final party tallies. These token councillors of course would probably be powerless. Disappeared here is that its totalitarian nature is what most defines representative democracy: even when most people don’t vote (often the case), politicians get in, backed by the entire state\Slash{}police apparatus.
Another bureaucratic horror story, to go by Chee’s account, has been Ecology Montreal’s internal functioning, including factions exiting the party, periods where people weren’t talking to each other, and a tendency for power to accumulate in a coordinating committee. At one point, for example, a coordinating committee had to ’“clean up’ the movement” (what movement? Ecology Montreal is a groupuscule, not a movement). In an another example of centralization of power, it is also the coordinating committee which is discussing the agreement with other opposition groups not to run candidates against each other. In fact, Ecology Montreal is presently dysfunctional with respect to the structure it has set up, which invests some power in “local associations.” However “Ecology Montreal currently does not have any local associations in existence,” Chee informs us, so the ubiquitous coordinating committee is presently acting as the “principal coordinating council.” Which is hardly surprising: these municipal parties are basically empty shells which only come “alive” at election time.
An Ecology Montreal program was produced by the coordinating committee and adopted by the membership in 1992. Dense fog and rhetoric render navigating this document a perilous undertaking. Much is clarified though when we learn that the ruling MCM party “can no longer be considered an instrument for progressive change.” In other words the MCM \emph{once was}, to use Ecology Montreal’s Old Left terminology, “progressive.” Ecology Montreal is in large part a back-to-the-roots MCM (a party in which Roussopoulos was once a militant).
Instead of abolishing money, Ecology Montreal intends to preserve the law of value, wage labour and the commodity economy, ensuring that people will continue to buy and sell each other as before. The party’s call for full employment makes it clear that they wish to retain high levels of production, and talk of “hiring and promotion practices” underlines that bosses and hierarchy will endure.
Ecology Montreal’s call for “the application of a user-pay system on all highways” typifies the Band-aid solutions to be expected on an ecological level. Thus the party is reduced to grumbling about the “excessive use of the automobile,” and vaguely wants to “reduce pollution from industrial sources.” These people obviously intend to keep the techno-grid fundamentally intact.
Also of note is a section on non-violence. Here we learn that Ecology Montreal is “simply opposed to the use of force.” They certainly don’t want non-pacifist hordes of uncontrollables dislodging \emph{their} politicians. The document explicitly rules out going on the offensive against the cops (e.g. riots), and advocates a “weapons-free zone,” disarming people against fascists and Stalinists, who are hardly in the habit of beating swords into plowshares.
Concerning elected candidates, the Party’s approach is democratic centralism. Once arrived at, in other words, the party line must be toed. “Defending and promoting the programme and strategy” is obligatory, the party statutes outline, and “the final decision of Ecology Montreal on any matter must be accepted.” Mindless obedience is of course the very definition of the party hack.
Lumping libertarian municipalism in with other strains of populism will elicit objections from some, no doubt. After all, Bookchin and Chee often \emph{sound} anti-authoritarian, even anti-statist. However, implementing change top-down through the state is clearly not anti-statist: it’s leftism. Roussopoulos’ idiotic position papers which hope “to unite the left” demonstrate that, despite the anarcho-rhetoric, he’s just a leftist. Libertarian municipalism is a form of left populism because instead of locating all legitimacy in autonomous activity, it posits political parties which claim to \emph{represent} widespread disgust with “impersonal bureaucracies and professional politicians” (in Chee’s words). People, however, can only represent themselves; the party has no role to play. The role of the party in other words is to immediately abolish itself.
Ecology Montreal wishes to recuperate our disgust and to channel it towards electoralism, the reformist Ecology Montreal racket, and leftism — “\dots{} so unpopular is the MCM that the 1994 election may reflect enormous political ferment, according to Phillip Chee,” we learn for example in \emph{Green Perspectives}. At the same time Chee fears that Ecology Montreal “will fall on the deaf ears of a people fed up and increasingly cynical of the current political system” — in other words that his gang will get the boot along with the rest. Cynicism is corrosive and a double-edged sword to be sure but it is also an antidote to false hopes. Unfortunately there are always new parties popping up, propping up a more and more discredited system. With enough negativity, however, there might just be a qualitative leap\dots{}
\begin{center}
* * *
\end{center}
Ecology Montreal, Chee, and Bookchin also exalt “the citizen,” a term which, like “the proletariat” of yore, becomes the defining role — \emph{the} role we are all expected to play. Max Stirner notes this term’s relation to the (anti-monarchist) bourgeois revolution, whereby everyone is “raised or lowered to the dignity of the \emph{citizen}: (\dots{}) the \emph{third} estate becomes the sole estate, namely, the estate of — \emph{citizens of the state}.” Or, in Ecology Montreal’s words, citizens “must be aware of their duties and rights as citizens.” As Stirner notes, “\dots{} few qualms are felt about changing existing laws. But who would dare sin against the \emph{idea} of the State, or refuse to submit to the \emph{idea} of law. So people remain ‘law-respecting’ loyal ‘citizens.’” Libertarian municipalism proposes to decentralize the state, to create a profusion of mini-states. Thus “neighbourhood councils should be empowered to enact laws,” according to Ecology Montreal. With laws of course come the cops to back them up (green-uniformed, no doubt). Hardly surprisingly the police question propels Ecology Montreal to new heights of Orwellian obfuscation: in Ecology Montreal-speak, the police become yet another brand of \emph{coordinator} — they “coordinate \dots{} efforts to enhance and protect public safety.” How sweet.
Instead of using the ever-changing desires of unique individuals as a starting point, Chee imposes a pre-fab, abstract, all-purpose councilism. “Mandated and recallable delegates” become the theoretical antidote to bureaucratiza-tion. But as John Zerzan notes, “delegates and recall have always been, in practice, direct routes to bureaucratization and the rule of experts (consult all trade union history).” In an industrial economy these so-called mandated and recallable delegates become mouthpieces of the desires of the megamachine, which are relayed back to the base \emph{as necessities}.
Ecology Montreal’s role is to legitimize the present municipal state through their participation and to legitimize the cybernetic state to come. Ecology Montreal wants us to internalize — to self-manage — the state. With our resistance to it weakened, authority will circulate more freely through the pyramid of power. As opposed to a Japanese-style implanted technobureaucracy, Ecology Montreal proposes a more participatory self-alienation where we choose our technocrats more directly (if we vote for them, they must be ours). Integral to this approach are the “cybernetic devices,” “mass technology” and “sophisticated technology” marketed in Bookchinism. I have already discussed this aspect in a previous article in \emph{Anarchy} in a passage which began with a quote from Bookchin:
\begin{quote}
“I believe that science and technology should be used in the service of refurbishing and rehabilitating a new balance with nature.”
But Bookchin’s vision of a high-tech apparatus passively “in the service” of humanity — a discourse he shares with all the technocrats — denies the qualitative leap, the autonomization of technology which occurs with the implementation of mass tech-niques in the metropolis. Later, Bookchin backhandedly ac-knowledges this autonomization, when the underlying technor determinism of his discourse makes “sophisticated technology” a universal given: “\dots{}the very things we are using presuppose a great deal of sophisticated technology. Let’s face the fact that we need these technologies.” Rather than presupposing a great deal of sophisticated technology, isn’t it more appropriate to question “the very things we are using”? When Bookchin says “we need” these technologies, he is speaking only for himself.
— \emph{Anarchy} \#33
\end{quote}
Not surprisingly, anti-civilizationists are the object of particular scorn in the Bookchin organ \emph{Green Perspectives}, where “anarcho-primitivism” is termed a “pathology.” That civilization thinks it needs to cure us is par for the course. It is more and more obvious, though, that it is civilization which is the problem.
\begin{center}
* * *
\end{center}
Once parties and the municipal state are swallowed, accepting the nation-state is only a short theoretical step away, as demonstrated by anarcho-nationalist Serge Roy’s call for Quebec separatism in the Bookchin-oriented Quebec City journal \emph{Hors d’Ordre}.
Meanwhile, Bookchinism continues to spread. The most recent issue of \emph{Green Perspectives} lists works by Bookchin translated into Norwegian, Dutch, German, Greek, Italian, Japanese, Portuguese, Spanish and Turkish. This interest in effect is hardly surprising. Apart from its academic appeal, Bookchinism can be very attractive to a wide variety of middle-of-the-road anarchists who are searching for simplistic, seemingly squeaky-clean solutions.
This essay is not intended as an over-all critique of Bookchinism, which hopefully someone will undertake. In the meantime, John Zerzan’s brief but pointed review of Bookchin’s \emph{The Rise of Urbanization and the Decline of Citizenship} remains the most incisive critique to date.\footnote{Appeared in \emph{Anarchy}, \emph{Demolition Derby} and \emph{Interrogations Pour La Communaute Humaine}.}
\section{Update}
On February 24, Alliance ’94 made its first public appearance in the form of a forum on the role of the opposition at City Hall. The event was a complete flop; as many journalists showed up as members of the public. Four Alliance hacks gave pep talks, followed by a discussion\Slash{}question period. It quickly became apparent that yet another coordinating committee was running the show; people could offer comments but had no real input in decision-making. One person called for a debate about what is apparently a major feature of the Alliance — running a candidate for mayor. Roussopoulos immediately squelched the idea of a debate. Running a mayoral candidate was the “center,” the “heart” of the Alliance, he enthused, waxing lyrical, a necessary “symbol of unity.” Besides, the question had already been dealt with by the coordinating committee.
Much hand-wringing took place over the fact that there was no chance that anything approaching 50\% of the electorate would vote. Figures were tossed around as to what would be a reasonable Alliance tally. Marcel Sevigny, a leftist councillor, said that winning six or seven seats could be counted a success.
The evening was co-chaired by Bernard Bourbonnais, who also gave a talk as the Ecology Montreal rep. At one point he excused himself after making a clumsy statement, joking that he “wasn’t enough of a politician yet.” Not to worry, chump, you’re learning fast. Also at the presiding table were three people from the \emph{Our Generation,} crowd. In effect the Alliance apparently consists of Ecology Montreal, two leftist councillors and a handful of academics and hangers on. The few people who showed up to check out the event seemed primarily wary. One man who had been sent an invitation complained bitterly about being confronted with a “\emph{fait accompli}” concerning process and decision-making. “The community isn’t here,” another man noted, injecting a refreshing breath of reality into this stale, tedious non-event.
\section{April 19 Update}
Alliance ’94 has now collapsed. Ecology Montreal and the DCM (a small leftist party) are presently courting each other with an eye to stitching together a “federation” for the election campaign. “Our hope is to form a federation, meaning there would be a single party, but membership in the party would be limited to associations [Ecology Montreal and the DCM],” Ecology Montreal spokesperson Andrea Levy is quoted as saying in \emph{Hour}, a local cultural\Slash{}news-weekly.
“There is considerable interest and enthusiasm on both sides at this point,” chirped DCM leader Sam Boskey. O the mating rituals of marginalized leftist groupuscles!
Meanwhile, the international social ecology conference on libertarian municipalism will take place on May 7 and 8. Bookchin will be the predictable featured speaker and Andrea Levy will give a talk as the Ecology Montreal rep. Some local anti-authoritarians are contemplating showing up to protest the libertarian municipalism racket and to distribute this text.
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Library.Anarhija.Net
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\begin{center}
Michael William
The Ecology Montreal Party: A “Libertarian” Frankenstein
1994
\bigskip
Originally published in “Anarchy: A Journal of Desire Armed” \#40 Spring\Slash{}Summer ’94. Vol. 14, No. 2.
\bigskip
\textbf{lib.anarhija.net}
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\begin{document}
\title{{\bf Percolation and number of phases\\ in the 2D Ising
model}}
\author{
Hans-Otto Georgii\\ {\small\sl Mathematisches Institut der
Universit\"at M\"unchen}\\ {\small\sl Theresienstr.\ 39, D-80333
M\"unchen, Germany.}
\and Yasunari Higuchi\\ {\small\sl Department of Mathematics, Faculty of
Science}\\ {\small\sl Kobe University, Rokko, Kobe 657-8501, Japan.}
}
\date{}
\maketitle
\begin{quote}
We reconsider the percolation approach of Russo, Aizenman and Higuchi
for showing that there exist only two phases in the Ising model on the
square lattice.
We give a fairly short alternative proof which is only based on FKG
monotonicity and avoids the use of GKS-type inequalities
originally needed for some background results. Our proof extends to
the Ising model on other planar lattices such as the triangular
and honeycomb lattice.
We can also treat the Ising antiferromagnet in an external field and
the hard-core lattice gas model on $\Z^2$.
\end{quote}
\section{Introduction}
\medskip\noindent
One of the fundamental results on the two-dimensional ferromagnetic
Ising model is the following theorem obtained independently
in the late 1970s by Aizenman \cite{Aiz} and Higuchi \cite{Hig}
on the basis of the seminal work of Russo \cite{Rus}.
\medskip\noindent
{\bf Theorem}.
{\em For the ferromagnetic Ising model on $\Z^2$ with no external
field and inverse temperature $\b>\b_c$,
there exist precisely two
extremal Gibbs measures $\mu^+$ and $\mu^-$.}
\medskip\noindent
The basic technique initiated by Russo
consists of an interplay of three features of the Ising model:
\bit
\item[--] the strong Markov property for random sets defined by
geometric conditions involving clusters of constant spin,
\item[--] the symmetry of the interaction under spin-flip and lattice
automorphisms, and
\item[--] the ferromagnetic character of the interaction which
manifests itself in FKG order and positive correlations.
\eit
These ingredients led to a detailed understanding of the geometric
features of typical configurations as described by the concepts of
percolation theory.
In addition to these tools, the authors of \cite{Aiz,Hig,Rus} also
needed the result that the
limiting Gibbs measure with $\pm$ boundary condition is a mixture of
the two
pure phases. This result was obtained by quite different means,
namely some symmetry
correlation inequalities of GKS and Lebowitz type \cite{MM}. While
these symmetry inequalities are a beautiful and powerful tool, they
are quite different in character from the FKG inequality and have
their own restrictions. It is therefore natural to ask whether
Russo's random cluster method is flexible enough to prove the theorem
without recourse to symmetry inequalities. On the one hand, this
would allow to extend the theorem to models with less symmetries,
while on the other hand one might gain a deeper understanding of
possible geometric features of typical configurations.
In this paper we propose such a purely geometric reasoning which is
only based on the three features above and avoids the use of the
symmetry inequalities of Messager and Miracle-Sole \cite{MM}.
Despite this reduction of tools we could simplify the proof by
an efficient combination of known geometric arguments. These include
\bit
\item [--] the Burton-Keane uniqueness theorem for infinite clusters
\cite{BK},
\item [--] a version of Zhang's argument for the impossibility of
simultaneous plus- and minus-percolation in $\Z^2$ (cf.\ Theorem 5.18 of
\cite{GHM}),
\item [--] Russo's symmetry trick for simultaneous flipping of spins and
reflection of the lattice \cite{Rus}, and
\item [--] Aizenman's idea of looking at contour intersections in a
duplicated system \cite{Aiz}.
\eit
We have tried to keep the paper reasonably
self-contained, so that the reader will find a complete proof of the
theorem.
As a payoff of the method we also obtain
some generalizations. On the one hand, the arguments
carry over to the Ising model on other planar lattices such as the
triangular or the hexagonal lattice. On the other hand, in the case
of the square lattice they cover
also the antiferromagnetic Ising model in an external field as well
as the hard-core lattice gas model.
\section{Set-up and basic facts}
\label{facts}
Although we assume that the reader is familiar with the
definition of the Ising model, let us start recalling a number of
fundamental facts and introducing some notations. We assume
throughout that the inverse temperature $\b$ exceeds the Onsager
threshold $\b_{c}$, and that there is no external field, $h=0$.
The main ingredients we need are the following:
\smallskip
$\bullet$
the {\em configuration space\/} $\O=\{-1,1\}^{\Z^2}$, which is
equipped
with the Borel $\s$-algebra $\F$ and the local $\s$-algebras $\F_{\L}$
of events depending only on the spins in $\L\subset\Z^2$.
$\bullet$
the {\em Gibbs distributions} $\mu_{\L}^\o$
in finite regions $\L\subset\Z^2$ with boundary condition $\o\in\O$;
these enjoy the {\em Markov property} which says that
$\mu_{\L}^\o(A)$ for
$A\in\F_{\L}$ depends only on the restriction of $\o$
to the boundary $\partial\L=\{x\not\in\L:|x-y|=1
\mbox{ for some }y\in\L\}$ of $\L$, and the {\em finite-energy
property}, which states that $\mu_{\L}^\o(A)>0$.
$\bullet$
the {\em Gibbs measures} $\mu$ on $(\O,\F)$ which, by
definition,
satisfy $\mu(\cdot\,|\F_{\L^c})(\o) =\mu_{\L}^\o$ for $\mu$-almost
all $\o$ and any finite $\L$; we write $\G$ for the set
of all Gibbs measures and $\Gex$
for the set of all extremal Gibbs measures.
$\bullet$
the {\em strong Markov property} of Gibbs measures,
stating that
$\mu(\cdot\,|\F_{\Gamma^c})(\o) =\mu_{\Gamma(\o)}^\o$ for
$\mu$-almost
all $\o$ whenever $\Gamma$ is
a {\em random\/} finite subset of $\Z^2$ satisfying
$\{\Gamma=\L\}\in\F_{\L^c}$ for all finite $\L$;
$\F_{\Gamma^c}$ is the set of all events $A$ satisfying
$A\cap \{\Gamma=\L\}\in\F_{\L^c}$ for all finite $\L$.
$\bullet$
the {\em stochastic monotonicity} (or FKG order) of Gibbs
distributions; writing $\mu\preceq\nu$ when
$\mu(f)\leq\nu(f)$ for all increasing local (or, equivalently,
all increasing bounded measurable) real functions $f$ on $\O$,
we have $\mu_{\L}^\o\preceq\mu_{\L}^{\o'}$ when $\o\leq\o'$, and
$\mu_{\L}^\o\preceq\mu_{\D}^\o$ when $\D\subset\L$ and $\o\equiv
+1$ on $\L\setminus\D$ (the opposite relation holds when $\o\equiv
-1$ on $\L\setminus\D$).
$\bullet$
the {\em pure phases}
$\mu^+,\,\mu^-\in\G$ obtained as limits for $\L\uparrow\Z^2$ of
$\mu_{\L}^\o$ with $\o\equiv+1$ resp.\ $-1$, their invariance
under all graph automorphisms of $\Z^2$,
the sandwich relation
$\mu^-\preceq\mu\preceq\mu^+$ for any other $\mu\in\G$,
and the resulting extremality of $\mu^+$ and $\mu^-$.
$\bullet$
the characterization of extremal Gibbs measures by their
{\em triviality on the tail $\s$-algebra}
$\T=\bigcap\{\F_{\L^c}:\L\subset\Z^2\mbox{ finite}\}$;
the fact that extremal Gibbs measures have {\em positive correlations};
and the {\em extremal
decomposition} representing any Gibbs measure as the barycenter of a
mass distribution on $\Gex$.
\smallskip\noindent
A general account of Gibbs measures can be found in \cite{Gii}, and
\cite{GHM} contains an exposition of the
Ising model and its properties related to stochastic monotonicity.
We will also use a class of transformations of $\O$ which preserve
the Ising Hamiltonian, and thereby the class $\G$ of Gibbs measures.
These transformations are
\smallskip
$\bullet$
the {\em spin-flip transformation}
$T:\o=(\o(x))_{x\in\Z^2}\to (-\o(x))_{x\in\Z^2}\,;$
$\bullet$ the {\em translations} $\th_{x}$, $x\in\Z^2$, which are
defined by $\th_{x}\o(y)=\o(y-x)$ for $y\in \Z^2$, and in particular
the horizontal and vertical shifts
$\th_{\h}=\th_{(1,0)}$ resp.\ $\th_{\v}=\th_{(0,1)}$; and
$\bullet$ the {\em reflections\/} in lines $\ell$ through lattice
sites: for any $k\in\Z$ we write
\[
R_{k,\h}: \Z^2\ni x=(x_1,x_2) \to (x_1,2k-x_2)\in\Z^2
\]
for the reflection in the horizontal line $\{x_2=k\}$,
and similarly $R_{k,\v}$ for the reflection in the vertical line
$\{x_1=k\}$. For $k=0$ we simply write $R_\h=R_{0,\h}$ and
$R_\v=R_{0,\v}$. All these reflections act canonically on $\O$.
\medskip
We will investigate the geometric behavior of typical configurations
in {\em half-planes\/} of $\Z^2$. These are sets of the form
\[
\pi=\{ x=(x_1, x_2) \in \Z^2 :\ x_i \ge k\}
\]
with $k\in\Z$, $i\in\{1,2\}$, or with `$\ge$' replaced by `$\leq$'.
The line $\ell =\{ x\in \Z^2: \ x_i= k \}$ is called the associated
{\em boundary line}. In particular, we will consider
\bit
\item the upper half-plane
$ \pi_\up = \{ x=(x_1, x_2) \in \Z^2: \ x_2 \geq 0 \} $,
\item the downwards half-plane
$ \pi_\lo = \{ x = (x_1, x_2) \in \Z^2: \ x_2 \leq 0 \} $,
\eit
and the analogously defined right half-plane $ \pi_\ri$
and left half-plane $\pi_\le$. We will also work with
\bit
\item the left horizontal semiaxis
$\ell_\le = \{x=(x_1,x_2) \in \Z^2: \ x_1 \leq 0,\, x_2=0\}$, and
\item the right semiaxis
$ \ell_\ri = \{x = (x_1, x_2) \in \Z^2: \ x_1 \geq 0,\, x_2 = 0 \}$.
\eit
\medskip
In the rest of this section we state three fundamental results on
percolation in the Ising model. By the symmetry between
the spin values $+1$ and $-1$,
these results also hold when `$-$' and `$+$' are
interchanged. Similarly, all notations introduced with one
sign will be used accordingly for the opposite sign.
We assume that the reader is familiar with the basic
concepts of percolation theory such as
paths and $*$paths, $+$paths and $+*$paths (consisting of plus
spins), circuits and $*$circuits, semicircuits in half-planes,
clusters, $+$ and $+*$clusters, and so on. These can be found in the
paper of Russo \cite{Rus}.
The starting point is the following result of \cite{CNPR,Rus}.
Let $E^+$ denote the
event that there exists an infinite $+$cluster in
$\Z^2$, and define $E^-$, $E^{+*}$, $E^{-*}$ analogously.
(Throughout this paper we will use the letter $E$ to denote events
concerning existence of infinite clusters.)
%
\begin{lem}{\bf (Existence of infinite clusters)}\label{lem:CNPR}
If $\mu\in\G$ is different from $\mu^-$, there exists with
positive probability an infinite $+$cluster.
That is, $\mu(E^+)>0$ when $\mu\ne\mu^-$.
\end{lem}
%
{\sl Sketch Proof}.
Suppose that $\mu(E^+)=0$. Then any given square
$\D$ is almost surely surrounded by a $-*$circuit, and with
probability close to $1$ such a circuit can already be found within a
square $\L\supset\D$ provided $\L$ is large enough. Define a
random set $\Gamma$ as the largest subset of $\L$ which is the
interior of such a $-*$circuit. (A largest such set exists because
the union of such sets is again the interior of a $-*$circuit.) By
maximality, $\Gamma$ satisfies the conditions of the strong Markov
property. This together with the stochastic monotonicity
$\mu^-_\Gamma\preceq \mu^-$ implies (in the limit $\L\uparrow\Z^2$)
that $\mu\preceq\mu^-$ on $\F_\D$.
Since $\D$ was arbitrary and $\mu^-$ is minimal we find that
$\mu=\mu^-$, and the lemma is proved. $\Box$
\medskip\noindent
The next lemma is a variant of another result of Russo \cite{Rus}.
%
\begin{lem}{\bf (Flip-reflection domination)}\label{TR}
Let $\mu \in \G$ and $R$ any reflection, and suppose that for
$\mu$-almost all $\o$ each finite $\D\subset \Z^2$ is surrounded by
an $R$-invariant
$*$circuit $c$ such that $\o \ge R\circ T(\o)$ on $c$.
Then $\mu \succeq \mu \circ R \circ T$.
\end{lem}
%
{\sl Proof}. Another way of stating the assumption
is that for
any finite $R$-invariant $\D$ and $\mu$-almost all $\o$
there exists a finite $R$-invariant random set $\Ga(\o)\supset\D$
such that $\o \ge R\circ T(\o) \mbox{ on }\partial\Ga(\o)$.
Given any $\e>0$, we can thus find an $R$-invariant $\L$ so large that
with probability at least $1-\e$ such
an $R$-invariant $\Ga(\o)$ exists within $\L$.
Since the union of any two such $\Ga(\o)$'s enjoys the same
properties,
we can assume that $\Ga(\o)$ is chosen maximal in $\L$; in the case
when no
such $\Ga(\o)$ exists we set $\Ga(\o)=\emptyset$.
The maximality of $\Ga$ implies that the events $\{\Ga = G\}$ are
measurable with respect to $\F_{\L\setminus G}$.
For any increasing ${\cal F}_{\D}$-measurable function $f\geq 0$
we thus get from the strong Markov property
$$
\mu(f) \geq \mu\bigg( \mu_\Ga^\cdot( f)\;;\;\Ga \ne\emptyset\bigg)\;.
$$
However, if $\Ga(\o)\ne\emptyset$ then
$\o \ge R\circ T(\o) \mbox{ on }\partial \Ga(\o)$. By stochastic
monotonicity, for such $\o$ we have
$$
\mu_{\Ga(\o)}^\o( f )\ge
\mu_{\Ga(\o)}^{R\circ T(\o)}( f )=\mu_{\Ga(\o)}^\o( f\circ R\circ T
)\;,
$$
where the identity follows from the
$R$-invariance of $\Ga$ and the $R\circ T$ -invariance of the
interaction. Hence
$$
\mu(f)\geq \mu\bigg( f\circ R\circ T \,;\, \Ga \ne \emptyset\bigg)
\geq \mu ( f\circ R\circ T ) - \e\,\| f \|_\infty \; .
$$
The lemma thus follows by letting $ \e \to 0 $ and $\D\uparrow\Z^2$.
$\Box$
\medskip\noindent
A third useful result of Russo \cite{Rus} is the following. To state
it we need to introduce two notations. First,
let
\[
\theta=\mu^+(0 \in I^{+*})
\]
be the $\mu^+$-probability that the origin belongs to
an infinite $+*$cluster. Lemma \ref{lem:CNPR} implies that $\theta>0$.
Secondly, for a half-plane $\pi$ with boundary line $\ell$ and a
$*$semicircuit $\s$ in $\pi$ we write $\mbox{Int\,}\s$ for the unique
subset of $\Z^2$ which is invariant under the reflection $R$ in
$\ell$
and satisfies $\pi\cap\partial(\mbox{Int\,}\s)=\s$; we call
$\mbox{Int\,}\s$ the interior of $\s$.
%
\begin{lem} {\bf(Point-to-semicircuit lemma)}
\label{point-to-semicircuit}
Let $\pi$ be some half-plane with boundary line $\ell$, $x\in\ell$,
and $\s$ a $*$semicircuit in $\pi$ with interior
$\L=\mbox{\rm Int\,}\s\ni x$. Let $\o\in\O$ be such that
$\o\equiv +1$ on $\s$. Then
\[
\mu_\L^\o\bigg( x \mbox{ \rm is in $\L$ $+*$connected to $\s$} \bigg) \geq
\theta/2\;.
\]
\end{lem}
%
{\sl Proof}. By hypothesis we have $\o\geq R\circ T(\o)$ on $\partial\L$,
and therefore
$\mu_\L^\o\succeq\mu_\L^\o\circ R\circ T$. To exploit this relation
we let $B_{x,\s}$ be the event that there exists a $+*$paths in $\L$
from $x$ to $\s$, $C_{x,\s}$ the event that $x$ is surrounded by a
$+*$circuit in $\L$ which is $+*$connected to $\s$, and
$D_{x,\s}=B_{x,\s}\cup C_{x,\s}$.
Then $\mu_\L^\o(D_{x,\s}\cup R\circ T(D_{x,\s}))=1$, and
therefore $\mu_\L^\o(D_{x,\s})\geq1/2$.
Hence
$$\mu_\L^\o(B_{x,\s})
\ge\mu_\L^\o(B_{x,\s}|C_{x,\s})\,\mu(D_{x,\s})
\ge \mu_\L^\o(B_{x,\s}|C_{x,\s})/2\,.
$$
But if $C_{x,\s}$ occurs then there exists a largest random set
$\Ga\subset\L$ containing $x$ such that $\partial\Ga$ forms a
$+*$circuit and is $+*$connected to $\s$. Writing $B_{x,\partial\Ga}$
for the event that $x$ is $+*$connected to $\partial\Ga$ and
using the strong Markov property we thus find that
$$
\mu_\L^\o(B_{x,\s}|C_{x,\s})=
\mu_\L^\o\,(\,\mu^+_{\Gamma}(B_{x,\partial\Ga})\,|\,C_{x,\s}\,)
\geq\theta
$$
because $\mu^+_{\Gamma}(B_{x,\partial\Ga})\geq\theta$ by
stochastic monotonicity. Together with the previous inequality this
gives the result. $\Box$
\section{Percolation in half-planes}
In this section we will prove that there exist plenty of infinite
clusters of constant spin in the half-planes of $\Z^2$.
In particular, this
will show that all translation invariant $\mu\in{\G}$
are mixtures of $\mu^+$ and $\mu^-$.
We will use two pearls of percolation theory,
the Burton-Keane uniqueness theorem \cite{BK}, and Zhang's argument
for
the non-existence of two infinite clusters of opposite sign in $\Z^2$.
For a given half-plane $\pi$ we let $E^+_\pi$ denote the event that
there exists an infinite $+$cluster in $\pi$. When this occurs, we
will write $I^+_\pi$ for such an infinite $+$cluster in $\pi$. (As we
will see, such clusters are unique, so that this notation does not
lead into conflicts.) In case of the standard half-planes,
we will only keep the
directional index and omit the $\pi$; for example, we write
$E^+_{\up}$ for $E^+_{\pi_{\mbox{\tiny up}}}$.
Similar notations will be used for $+*$clusters
and for the sign $-$ instead of $+$.
Let us say that $(\pi,\,\pi')$ is a pair of {\em conjugate
half-planes\/} if $\pi$ and $\pi'$ share only a common boundary line.
An
associated
pair $(I^+_{\pi},I^+_{\pi'})$ or $(I^-_{\pi},I^-_{\pi'})$ of infinite
clusters of the same sign in $\pi$ and $\pi'$ will be called an
{\em infinite butterfly}. (This name alludes to the assumption that
the two infinite `wings' have the same `color', but is not meant to
suggest that they are symmetric and connected to each other,
although the latter will turn out to be true.)
%
\begin{lem}{\bf(Butterfly lemma)}\label{butterfly}
$\G$-almost surely there exists at least one infinite butterfly.
\end{lem}
%
{\bf Proof: } Suppose the contrary. By the extremal decomposition
theorem
and the fact that the existence of infinite butterflies is a tail
measurable event, there is then some $\mu\in{\Gex}$ for which there
exists no infinite butterfly $\mu$-almost surely.
We will show that this is impossible.
{\em Step 1. }First we observe that $\mu$ is $R\circ T$-invariant
for all reflections $R=R_{k,\h}$ or $R_{k,\v}$,
and in particular is periodic under translations.
Indeed, let $(\pi,\,\pi')$ be conjugate half-planes with common
boundary line $\ell$ and $R$ the
reflection in $\ell$ mapping $\pi$ onto $\pi'$. By the absence of
infinite butterflies, at least
one of the half-planes $\pi$ and $\pi'$ contains no infinite
$-$cluster, and this or the other half-plane contains no infinite
$+$cluster. In view of the tail triviality of $\mu$, we can assume
that $\mu(E^-_{\pi})=0$. This means that for $\mu$-almost all $\o$
every finite $\D\subset\pi$ is surrounded
by some $+*$semicircuit $\g$ in $\pi$. For such a $\g$, $c=\g\cup
R(\g)$
is an $R$-invariant $*$circuit that surrounds $\D\cup R(\D)$
and satisfies $\o \ge R\circ T(\o)$ on $c$. By Lemma \ref{TR},
this gives the flip-reflection domination
$\mu \succeq \mu \circ R \circ T$. Since also $\mu(E^+_{\pi})=0$ or
$\mu(E^+_{\pi'})=0$, we conclude in the same way that
$\mu \preceq \mu \circ R \circ T$, so that $\mu=\mu\circ R\circ T$.
Since both $\th_\h^2$ and $\th_\v^2$ are compositions of two
reflections, the invariance under the translation group
$(\th_x)_{x\in 2\,\Z^2}$ follows.
{\em Step 2. }We now take advantage of
the Burton--Keane uniqueness theorem \cite{BK}, stating that for
every periodic $\mu$ with finite energy there exists at most
one infinite $+$ (resp.\ $-$) cluster, and Zhang's symmetry argument
(cf.\ \cite{GHM}, Theorem 5.18)
deducing from this uniqueness the absence of simultaneous $+$ and
$-$percolation.
We start noting that, by the flip-reflection symmetry of $\mu$,
$\mu$ is different
from $\mu^+$ and $\mu^-$, so that by Lemma \ref{lem:CNPR} and
the Burton--Keane uniqueness theorem there exist both a unique
infinite
$+$cluster $I^+$ and a unique infinite $-$cluster $I^-$
in the whole plane $\Z^2$ $\mu$-almost surely.
We now choose a square $\L=[-n,n]^2\cap\Z^2$ so large that
$\mu(\L\cap I^+\ne\emptyset)>1-2^{-12}$.
Let $\partial_k\L$ be the intersection of
$\partial\L$ with the $k$'th quadrant, and let $A^+_{k}$
be the increasing event that there exists an
infinite $+$path in $\L^c$ starting from some site
in $\partial_k\L$. Define $A^-_{k}$ analogously. Since
\[
\{\L\cap I^+\ne\emptyset\}\subset \bigcup_{k=1}^4 A^+_{k}
\]
and $\mu$ (as an extremal Gibbs measure) has positive correlations,
it follows that
\[
\prod_{k=1}^4 \mu(\O\setminus A^+_{k})\leq \mu(\bigcap_{k=1}^4
\O\setminus{A^+_{k}})
\leq\mu(\L\cap I^+=\emptyset) < 2^{-12}\;,
\]
whence there exists some $k\in\{1,\ldots,4\}$ such that
$\mu(\O\setminus{A^+_{k}})<2^{-3}$. For notational convenience
we assume that $k=1$.
By the flip-reflection symmetry shown above, we find that
\[
\mu(A^+_{1}\cap A^-_{2} \cap A^+_{3} \cap A^-_{4})>1-4\cdot
2^{-3}=1/2\;,
\]
which is impossible because on this intersection the infinite clusters
$I^+$ and $I^-$ cannot be both unique. This contradiction concludes
the proof of the lemma. $\Box$
\medskip\noindent
The butterfly lemma leads immediately to the following result first
obtained by Messager and Miracle-Sole \cite{MM} by means of
correlation inequalities of symmetry type; the following proof
appeared first in \cite{GHM}.
%
\begin{cor}\label{cor:MM}
{\bf (Periodic Gibbs measures) }
Any periodic $\mu\in\G$ is a mixture of $\mu^+$ and $\mu^-$.
\end{cor}
%
{\sl Proof}. Suppose $\mu\in\G$ is invariant under
$(\th_x)_{x\in p\Z^2}$ for some period $p\geq 1$. Conditioning $\mu$
on any periodic tail event $E$ we obtain again a periodic Gibbs measure.
It is therefore sufficient to show that $\mu(E^+\cap E^-)=0$. Indeed,
the butterfly lemma then shows that $\mu(E^+) +\mu(E^-)=1$, and Lemma
\ref{lem:CNPR} implies that $\mu(\,\cdot\,|E^+)=\mu^+$ and
$\mu(\,\cdot\,|E^-)=\mu^-$ whenever these conditional probabilities
are defined. Hence $\mu=\mu(E^+)\,\mu^+ +\mu(E^-)\,\mu^-$.
Suppose by contraposition that $\mu(E^+\cap E^-)>0$. Since
$E^+\cap E^-$ is invariant and tail measurable, we can in fact
assume that $\mu(E^+\cap E^-)=1$; otherwise we replace $\mu$ by
$\mu(\,\cdot\,|E^+\cap E^-)$. By the butterfly lemma,
there exists a pair $(\pi,\pi')$ of conjugate halfplanes, say
$\pi_\up$ and $\pi_\lo$, and a sign, say $+$, such that both
half-planes contain an infinite clusters of this sign with
positive probability. Since $\mu(E^-)=1$ by assumption,
we can find a large square $\D$ such that with
positive probability $\D$ meets infinite $+$clusters in $\pi_\up$ and
$\pi_\lo$ and also an infinite $-$cluster. This $-$cluster leaves
$\D$ either on the left or on the right between the two infinite
$+$clusters. We can assume that the latter occurs with positive
probability. By the
finite energy property, it then follows
that also $\mu(A_0)>0$, where for $k\in p\Z$ we write
$A_k$ for the event that the point $(k,0)$ belongs to a two-sided
infinite $+$path with its two halves staying in $\pi_\up$ resp.\
$\pi_\lo$, and $(k+1,0)$ belongs to an infinite $-$cluster.
Let $A$ be the event that $A_k$ occurs for infinitely many $k<0$ and
infinitely many $k>0$. The horizontal periodicity and Poincar\'e's
recurrence theorem (cf.\ Lemma (18.15) of \cite{Gii}) then show that
$\mu(A_0\setminus A)=0$, and therefore $\mu(A)>0$. But on $A$ there
exist infinitely many $-$clusters which are separated from each other
by the infinitely many `vertical' $+$paths. This contradicts the
Burton--Keane theorem.
$\Box$
\medskip\noindent
The preceding argument actually shows that $\mu(E^{-*}\cap
E^{+*})=0$ whenever $\mu\in\G$ is periodic. Since $\mu^+(E^+)=1$ by
Lemma \ref{lem:CNPR} and tail triviality, this shows that in the
$+$phase the $+$spins
form an infinite sea with only finite islands.
%
\begin{cor}\label{ocean}
{\bf (Plus-sea in the plus-phase)}
$\mu^+(E^{-*})=0$. Hence,
$\mu^+$-almost surely there exists a unique infinite
$+$cluster $I^+$ in $\Z^2$ which surrounds each finite set.
\end{cor}
%
We note that in contrast to Zhang's argument
(cf.\ Theorem 5.18 of \cite{GHM})
our proof of the preceding corollary does not rely on
the reflection invariance of $\mu^+$ but only on its periodicity, and
thus can be extended to the setting of Section \ref{extensions} below.
We conclude this section with the observation that percolation in
half-planes is not affected by spatial shifts.
%
\begin{lem}\label{shift-lemma}
{\bf (Shift lemma)}
Let $\pi$ and $\tilde\pi$ be two half-planes such that
$\pi\supset\tilde\pi$,
i.e., $\pi$ and $\tilde\pi$ are translates of each other. Then
$E^+_{\pi}= E^+_{\tilde\pi}$ $\G$-almost surely, and similarly with
$-$ instead of $+$.
\end{lem}
%
{\sl Proof}. Since trivially $E^+_{\pi}\supset E^+_{\tilde\pi}$,
we only need to show that $E^+_{\pi}\subset E^+_{\tilde\pi}$
$\G$-almost surely. For definiteness we consider the case when
$\pi=\pi_\up=\{x_2\ge0 \}$ and
$\tilde\pi =\{x_2\ge 1\}$. Take any $\mu\in\Gex$, and
suppose that $\mu( E^+_{\tilde\pi})=0$. Then for almost all $\o$ and
any $n\geq1$
there exists a smallest $-*$semicircuit $\s_n(\o)$ in $\tilde\pi$
containing $\D_n\cup\s_{n-1}(\o)$ in its interior;
here $\D_n= [-n,n]\times[1,n]$ and $\s_0=\emptyset$.
Let $x_n(\o)\in\ell_\le$ and $y_n(\o)\in\ell_\ri$ be the two points
facing the two endpoints
of $\s_n(\o)$; these are $\F_{\tilde\pi}$-measurable functions of
$\o$, and the random sets $\{x_n,y_n\}$ are pairwise disjoint. Let
$A_n$
be the event that the spins at $x_n$ and $y_n$ take value $-1$.
We claim that $A_n$ occurs for infinitely many $n$ with probability
1. Indeed, for each $N\ge1$ and any $x\in\ell_\le$, $y\in\ell_\ri$
we have
\bea
&&\mu\bigg(A_{N}\,\bigg|\,x_N=x,y_N=y,\bigcap_{n>N}A_n^c\bigg)\\
&&=\mu\bigg(\mu_{\{x,y\}}^\cdot(\o(x)=\o(y)=-1)\,\bigg|\,
x_N=x,y_N=y,\bigcap_{n>N}A_n^c\bigg)\\
&&\ge \d^2>0
\eea
because the event in the condition is measurable with respect to
$\{x,y\}^c$
and the one-point conditional probabilities of $\mu$
are bounded from below by $\d=[ 1 + e^{8\beta } ]^{-1}$.
Hence $\mu(A_{N}^c|\bigcap_{n>N}A_n^c)\leq 1-\d^2$, and by iteration
we get
$\mu(\bigcap_{n\ge N}A_n^c)=0$, proving the claim.
We thus conclude that
with probability $1$ each box $[-n,n]\times[0,n]$ is surrounded
by a $-*$semicircuit in $\pi_\up$, which means
that $\mu( E^+_{\up})=0$. As $\mu(E^+_{\tilde\pi})$ is either 0 or 1,
the lemma follows. $\Box$
\section{Uniqueness of semi-infinite clusters}
Our next subject is the uniqueness of infinite
clusters in half-planes, together with the stronger property that
such
clusters touch the boundary line infinitely often.
This result has already been obtained by Russo \cite{Rus}
on the basis of the subsequent Lemma \ref{semi-unique} which we
derive here differently from the preceding Corollary
\ref{cor:MM}.
%
\begin{lem} {\bf (Line touching lemma)}\label{line_touching}
For any half-plane $\pi$, there exists $\G$-almost surely at most
one infinite $+$ (resp.\ $+*$)
cluster $I^+_\pi$ (resp.\ $I^{+*}_\pi$) in $\pi$.
When it exists, this infinite cluster $\G$-almost surely
intersects the boundary line $\ell$ of $\pi$ infinitely often, in the
sense that outside any finite $\D$ one can find an infinite path in
this cluster starting from $\ell$.
\end{lem}
%
{\sl Proof}. For definiteness we assume that
$\pi=\pi_\up$; other half-planes merely correspond to a change of
coordinates. We consider only infinite
$+$clusters in $\pi_\up$; the case of $+*$clusters
is similar. It is also clear that any result proved for the
$+$sign is also valid with the $-$sign.
{\em Uniqueness: }The uniqueness of infinite $+$clusters in
$\pi_\up$ is a consequence of the second statement, the line-touching
property for infinite $-*$clusters. Indeed,
suppose there exists no infinite $-*$cluster in $\pi_\up$; then each
finite set in $\pi_\up$ is surrounded by a $+$semicircuit, so that
any
two infinite $+$paths are necessarily $+$connected to each other. In
the alternative case when an infinite $-*$cluster $I^{-*}_\up$ in
$\pi_\up$ exists,
this $I^{-*}_\up$ meets $\ell_\le$ or $\ell_\ri$ infinitely often, so
that each infinite $+$cluster must meet the other half-line
infinitely often. Hence, two such $+$clusters must cross each
other, and are thus identical.
{\em Line touching: } Let $\mu\in\Gex$ and $x\in\pi_\up$ and
consider the event $A^+_x$ that $x$
belongs to an infinite $+$cluster in $\pi_\up$ which does not touch
the
horizontal axis $\ell_\h$. We will show that $\mu(A^+_x)=0$. Once
this is
established, we can take the union over all $x$ and use the finite
energy property to see that for each finite $\D$ the event
``an infinite $+$cluster in $\pi_\up$ is not connected to $\ell_\h$
outside
$\D$''
also has probability zero, which means that almost surely any
infinite
$+$cluster in $\pi_\up$ must meet $\ell_\h$ infinitely often.
Intuitively, if $A^+_x$ occurs then the infinite $+$cluster
containing $x$ is separated from $\ell_\h$ by an infinite $-*$path;
but
the spins `above' this path feel only the $-$boundary condition and
thus believe to be in the $-$phase $\mu^-$, so that they will not
form an
infinite $+$cluster.
To make this intuition precise we fix some integer $k\geq1$ and
consider the event $A^+_{x,k}$ that $x$ belongs to a
$+$cluster of size at least $k$ which does not meet $\ell_\h$. Take a
large box $\D\subset\pi_\up$ containing $x$.
For $\o\in A^+_{x,k}$ we consider the largest set $\Ga(\o)\subset
\D$ containing $x$ such that $\o=-1$ on
$\partial\Ga(\o)\setminus\partial_\up\D$, where
$\partial_\up\D=\partial\D\cap \pi_\up$. By
the strong Markov property and the stochastic monotonicity of Gibbs
distributions, we find that
\[
\mu(A^+_{x,k})\leq \mu\bigg(\mu^\cdot_\Ga(E^+_{x,k}) \bigg)
\leq \mu_\D^{\pm}(E^+_{x,k})\;,
\]
where $E^+_{x,k}$ is the event that $x$ belongs to a $+$cluster in
$\pi_\up$ of size
at least $k$ and $\pm$ stands for the configuration which is $+1$
on $\pi_\up$ and $-1$ on $\pi_\up^c$.
Now, again by stochastic monotonicity the semi-infinite limit
$\mu^\pm_{\up}=\lim_{\D\uparrow\pi_\up}\mu_\D^{\pm}$ exists.
Letting first $\D\uparrow\Z^2$ and then $k\to\infty$ we thus find
that $\mu(A^+_{x})\leq \mu^\pm_{\up}(E^+_\up)$. The lemma thus follows
from the subsequent lemma. $\Box$
\medskip\noindent
By the argument of Lemma \ref{lem:CNPR}, the following result
implies the uniqueness of the Gibbs measure on $\pi_\up$
with $-$boundary condition in $\pi_\lo$.
%
\begin{lem}\label{semi-unique}
{\bf (No percolation on a bordered half-plane)}
$\mu^\pm_{\up}(E^{+*}_\up)=0$.
\end{lem}
%
{\sl Proof}. To begin we note that $\mu^\pm_{\up}$ is invariant
under horizontal translations and stochastically maximal
in the set of all Gibbs measures on $\pi_\up$ with $-$boundary
condition in $\pi_\lo$. This follows just as in the case of
the plus-phase $\mu^+$ on the whole lattice. In particular,
$\mu^\pm_{\up}$ is trivial on the $\pi_\up$-tail $\T_\up=
\bigcap \{\F_{\pi_\up\setminus\L}:
{\L\subset\pi_\up\mbox{ finite}}\}$.
We think of $\mu^\pm_{\up}$ as
a probability measure on $\O$ for which almost all configurations are
identically equal to $-1$ on $\pi_\up^c$.
Next we consider the downwards translates
$\mu^+_{n,-}=\mu^\pm_{\up}\circ \th_\v^{-n}$, $n\geq 0$.
Evidently, $\mu^+_{n,-}$ is obtained by an analogous infinite-volume
limit in the half-plane $\{x_2\geq-n\}$. This shows that
$\mu^+_{n,-}\preceq \mu^+_{n+1,-}$ by stochastic monotonicity,
so that the stochastically increasing limit
$\mu^+_{-}=\lim_{n\to\infty}\mu^+_{n,-}$ exists.
Clearly $\mu^+_{-}\in\G$. Also, $\mu^+_{-}$
inherits the horizontal invariance of the $\mu^+_{n,-}$ and is in
addition vertically invariant. Corollary \ref{cor:MM} therefore
implies that $\mu^+_{-}=a\,\mu^- +(1-a)\mu^+$ for some coefficient
$a\in[0,1]$.
We claim that $a>0$. For $n\ge1$ let $B_n$ denote the event that the
origin is $-*$connected to the horizontal line $\{x_2=-n\}$. By the
horizontal ergodicity of $\mu^+_{n,-}$, there exist for
$\mu^+_{n,-}$-almost all $\o$ some random integers
$m_\le(\o)<0<m_\ri(\o)$
such that $\o\equiv-1$ on
$$
\s(\o)=\bigg\{x\in\Z^2: x_1\in\{m_\le(\o),m_\ri(\o)\},\;
-n\leq x_2\leq 0\bigg\}\;.
$$
Together with a segment of the line $\{x_2=-n-1\}$ on which $\o=-1$
$\mu^+_{n,-}$-almost surely, $\s(\o)$ forms a $-$semicircuit in
$\pi_\lo$ surrounding the origin. An immediate application of the
strong Markov property and the point-to-semicircuit lemma thus
implies that $\mu^+_{n,-}(B_n)\geq
\theta/2$. Therefore, writing $E^{-*}_{0,m}$ for the event that the
origin belongs to some $-*$ cluster of size at least $m$ we find
$\mu^+_{n,-}(E^{-*}_{0,m})\geq \theta/2$ when $n\geq m$. Letting
first
$n\to\infty$ and then $m\to\infty$ we see that $\mu^+_{-}(E^{-*})
\geq \theta/2$.
Since $\mu^+(E^{-*})=0$ by Corollary \ref{ocean}, it follows that
$a\geq\theta/2$, and the claim is proved.
To conclude the proof we observe that
\[
\mu^\pm_{\up}(E^{+*}_\up) \leq \mu^+_{-}(E^{+*}) = 1-a <1\;,
\]
again by Corollary \ref{ocean}.
Since $\mu^\pm_{\up}$ is trivial on $\T_\up$, the lemma follows.
$\Box$
\medskip\noindent
The butterfly lemma and shift lemma together still leave the
possibility that all infinite butterflies have the same orientation,
either horizontal or vertical. As a consequence of the line touching
lemma, we can now show that both orientations must occur.
%
\begin{lem}\label{orthogonal-butterflies}
{\bf(Orthogonal butterflies)}
$\G$-almost surely there exist both a horizontal infinite butterfly
in $\pi_\up$ and $\pi_\lo$ as well as a vertical infinite butterfly
in $\pi_\le$ and $\pi_\ri$.
\end{lem}
%
{\sl Proof}. Suppose there exists some $\mu \in \Gex$ having almost
surely no vertical infinite butterfly.
By the first step in the proof of the butterfly lemma, it then follows
that $\mu=\mu\circ R_{k,\v}\circ T$ for all $k\in\Z$, and thus
$\mu=\mu\circ\th_{\h}^{-2}$.
By the tail triviality, $\mu$ is in fact ergodic under $\th_{\h}^2$.
By the butterfly lemma, horizontal infinite
butterflies do exist, say of color $+$.
We now use an argument similar to that in Corollary \ref{cor:MM},
with the line touching lemma in place of the Burton-Keane theorem.
Fix any $n\ge1$. For $k\in\Z$ let $A_k$ denote the event that the
point $(k,0)$ is connected by
straight $+$paths to infinite $+$clusters in both
$\pi_{n,\up}=\{x\in\Z^2:x_2\ge n\}$ and $\pi_{n,\lo}=\{x_2\leq -n\}$.
Let $A$ be the event that $A_k$ occurs for infinitely many $k<0$
and infinitely many $k>0$. The finite energy property then shows
that $\mu(A_0)>0$, and the horizontal ergodicity and Poincar\'e's
recurrence theorem (or the ergodic theorem) imply that
$\mu(A)=1$. But the line
touching lemma guarantees that the infinitely many doubly-infinite
`vertical' $+$paths passing through the horizontal axis are connected
to each other in $\pi_{n,\up}$ and $\pi_{n,\lo}$. As
$n$ was arbitrary, it follows that
almost surely each finite set is surrounded by a
$+$circuit, and an infinite $-$cluster cannot exist.
In view of Lemma \ref{lem:CNPR}, this implies that
$\mu=\mu^+$. But $\mu^+$ is not invariant under $R_\v\circ T$,
in contradiction to what we derived for $\mu$. $\Box$
\medskip\noindent
The preceding argument can be used to derive the result of
Russo \cite{Rus} that $\mu^+$ and $\mu^-$ are the only phases which
are periodic in one direction. We will not need this intermediate
result.
\section{Non-coexistence of phases}
\label{inv}
In this section we will prove the following proposition.
%
\begin{prop} \label{invariance}
{\bf (Absence of non-periodic phases) }
Any Gibbs measure $\mu\in\G$ is invariant under translations, i.e.,
$\mu=\mu\circ\th_\h^{-1}$ and $\mu=\mu\circ\th_\v^{-1}$.
\end{prop}
%
Together with Corollary \ref{cor:MM} this will immediately imply the
main theorem that each Gibbs measure is a mixture of the two phases
$\mu^+$ and $\mu^-$. Our starting point is the following lemma
estimating the probability that a semi-infinite cluster can be pinned
at a prescribed point.
%
\begin{lem}
{\bf (Pinning lemma) } \label{pinning}
Let $\mu\in\G$, and suppose that there exists an infinite $+*$cluster
$I^{+*}_\up$ in $\pi_\up$ which meets the right semiaxis $\ell_\ri$
infinitely often. Then for each finite
square $\D=[-n,n]^2$ and $x\in\ell\ri$ we have
\[
\mu\bigg(\mbox{\rm $x$ is $+*$connected in
$(\D\cup\ell_\le)^c$ to $I^{+*}_\up$}\bigg)
\ge \theta/4
\]
provided $x$ lies sufficiently far to the right. The same
holds when `left' and `right' or `up' and `down' are interchanged.
\end{lem}
%
{\sl Proof}. By hypothesis, the infinite component of
$I^{+*}_\up\setminus\D$ almost surely contains infinitely many points
of $\ell_\ri$.
Thus, if $x\in\ell_\ri$ is located far enough to the right
then, with probability exceeding $1/2$, at least one such
point can be found left from $x$, and another such point can be found
right from $x$. This means that $x$ is surrounded by a
$+*$semicircuit $\s$ in $\pi_\up$
which belongs to $I^{+*}_\up$ and satisfies
$\D\cap\mbox{Int\,}\s=\emptyset$.
Let $\L$ be a large square box containing $x$. If $\L$ is large
enough, a semicircuit $\s$ as above can be found within $\L$ with
probability still larger than $1/2$. We then can assume that $\s$
has the largest interior among all such semicircuits in $\L$. Using
the strong Markov property and the point-to-semicircuit lemma we get
the result.
$\Box$
\medskip\noindent
Our main task in the following is to analyze the situation when a
half-plane contains both an infinite $+$cluster and an infinite
$-$cluster.
(The line-touching lemma still allows this possibility.)
In this situation it is useful to consider contours.
%
\begin{rem}\label{contours}
{\bf (Contours in half-planes) }
{\rm As is usually done in the Ising model, we draw lines of unit
length
between adjacent spins of
opposite sign. We then obtain a system of polygonal curves running
through the sites of
the dual lattice $\Z^2+(\frac12,\frac12)$. A {\em contour\/} in a
half-plane $\pi$ is a part of
these polygonal curves which separates a $-$cluster in $\pi$ from a
$+*$cluster in $\pi$.
This corresponds to the convention that at crossing points the
contours
are supposed to bend around the $-$spins. (The artificial asymmetry
between $+$ and $-$ does not matter, and we could clearly make the
opposite convention.)
Suppose now that $\pi$ contains both an infinite $+*$cluster $I^{+*}_\pi$
and an infinite $-$cluster $I^-_\pi$. Since these are unique by the
line touching lemma, it follows that $\pi$ contains a unique
semi-infinite contour $\g_\pi$ which starts between two points of the
boundary line $\ell$. On its two sides, $\g_\pi$ is accompanied by an
infinite $+*$path $f^{+}_\pi$
and an infinite $-$path $f^{-}_\pi$. We call $f^{+}_\pi$ and
$f^-_\pi$ the $+$ resp.\ $-$face of $\g_\pi$.
By the shift lemma, $\g_\pi$ intersects each line $\ell'\subset\pi$
parallel to $\ell$ only finitely often. Indeed, the shifted
half-plane $\pi'$ with boundary line $\ell'$ also contains a unique
semi-infinite contour $\g_{\pi'}$. Since the faces of $\g_{\pi'}$ are
connected to $\ell$ by the line touching lemma, $\g_{\pi'}$
necessarily coincides with $\g_\pi$ up to finitely many steps.
}
\end{rem}
%
From now on we consider a fixed extremal Gibbs measure $\mu\in\Gex$.
We want to prove that $\mu$ is horizontally invariant. (The proof of
vertical invariance is similar.) To this end we consider its
horizontal translate
$\hat\mu=\mu\circ\th_\h^{-1}$, as well as the product measure
$\hat\nu=\mu\times\hat\mu$ on $\O\times\O$. It is convenient two
think of the latter as a duplicated system consisting of two
independent layers.
The following lemma is a slight variation of a result of Aizenman
\cite{Aiz} in his
proof of the main theorem.
%
\begin{lem}\label{intersections}
{\bf (Fluctuations of the infinite contour) }
Suppose $\pi_\up$ contains a semi-infinite contour $\g_\up$
$\mu$-almost surely. Then
for $\hat\nu$-almost all
$(\o,\hat\o)\in\O^2$, $\g_\up(\o)$ and $\g_\up(\hat\o)$ intersect
each
other infinitely often.
\end{lem}
%
{\sl Proof}. By the line touching lemma and tail triviality, we can
assume that $I^-_\up$ intersects $\ell_\ri$ infinitely often and
$I^{+*}_\up$ intersects $\ell_\le$ infinitely often. (The alternative
case is analogous.) For
$n\geq 1$ let $\pi_{n,\up}=\th_\v^n\pi_\up$ be the half-plane
above the horizontal line through $(0,n)$, $I^{-}_{n,\up}$
the infinite $-$cluster in this half-plane and
\[
a_n = \min\{k\in\Z: (k,n)\in I^{-}_{n,\up}\}
\]
the abscissa of the point at which $\g_\up$ enters definitely into
$\pi_{n,\up}$.
Since $\g_\up$ is a continuous curve, it is sufficient to show
that for $\hat\nu$-almost all $(\o,\hat\o)$ we have
$a_n(\o)\geq a_n(\hat\o)$ and $a_{n+1}(\o)<
a_{n+1}(\hat\o)$ infinitely often. Using the definition
$\hat\mu=\mu\circ\th_\h^{-1}$ and setting $\nu=\mu\times\mu$ and
$d_n(\o,\o')=a_n(\o)-a_n(\o')$ for $\o,\o'\in\O$, we need to
show that $d_n\geq 1>d_{n+1}$ infinitely often $\nu$-almost
surely.
We observe first that $d_n=0$ infinitely often $\nu$-almost surely.
Indeed, the complementary event consists of the two parts
$A=\{d_n\geq1 \mbox{ eventually}\}$ and
$B=\{d_n\leq-1 \mbox{ eventually}\}$. Symmetry implies that
$\nu(A)=\nu(B)$. On the other hand, the tail-triviality of $\mu$
implies that $\nu(A)=0$ or $1$. This follows from Fubini's theorem
because $A$ is measurable with respect
to the `product-tail' $\T^{(2)}=\bigcap\{\F_{\L^c}^2:\L\subset\Z^2
\mbox{ finite}\}$ in $\O^2$.
(One should not be mistaken to
believe that $A$ was measurable with respect to the smaller
`tail-product' $\T^2$. It is only the case that the
$\o$-section $A_\o$ of $A$ belongs to $\T$ for any $\o$, and the
function $\o\to\mu(A_\o)$ is $\T$-measurable.) The disjointness of
$A$ and $B$ thus implies that $\nu(A)=\nu(B)=0$.
Next we claim that there exists some constant $\d>0$ such that
$$
\nu(d_{n-1}\geq1|d_{n},d_{n+1},\ldots)\geq \d\quad\mbox{ on
}\,\{d_n=0\}\;.
$$
To see this let $A_{n,k}=\{(\o,\o'):a_n(\o)=a_n(\o')=k \}$,
$\D_{n,k}$ the two-point set consisting of the points $(k,n-1)$ and
$(k+1,n-1)$, and $B_{n,k}$ the event that $\o=(-1,-1)$ on $\D_{n,k}$
and $\o'=(-1,+1)$ on $\D_{n,k}$. Then
\[
\nu(B_{n,k}|\F_{\D_{n,k}^c}^2)(\o,\o')=
\mu^\o_{\D_{n,k}}\times \mu^{\o'}_{\D_{n,k}}(B_{n,k})\geq
[1+e^{8\b}]^{-4}\equiv\d
\]
and thus
\[
\nu(d_{n-1}\geq1|\F_{n,\up}^2)
\geq \nu(B_{n,k}|\F_{n,\up}^2)
\geq\d \quad \mbox{ on $A_{n,k}\,$ $\nu$-almost surely}
\]
because $A_{n,k}\cap B_{n,k}\subset\{d_{n-1}\geq1\}$ and
$\pi_{n,\up}\subset \D_{n,k}^c$. Summing over $k$ and
conditioning on $\s(d_{n},d_{n+1},\ldots)$ we get the claim.
Now, given any integers $1\leq N<L$ let $\t=\t_{N,L}$ be the largest
$n\in\{N,\ldots,L\}$ such that $d_n=0$, provided such an $n$
exists,
and $\t=0$ otherwise. Then
$\{\t=n\}\in\s(d_{n},d_{n+1},\ldots)$ for
all $n$, and therefore
\bea
&&\nu\bigg(\exists\, n\geq N: d_{n-1}\geq 1\ >\ d_n\bigg)\geq
\sum_{n=N}^L \nu(d_{n-1}\geq 1,\,\t=n)\\
&&=\sum_{n=N}^L \nu\bigg(\nu(d_{n-1}\geq
1|d_{n},d_{n+1},\ldots)\,1_{\{\t=n\}}\bigg)
\ \geq\ \d\;\nu\bigg(\exists\, N\leq n\leq L: d_n=0\bigg)\;.
\eea
Letting first $L\to\infty$ and then $N\to\infty$ we see that
$d_{n-1}\geq 1> d_n$
infinitely often with probability at least $\d$. This gives the
result because $\nu$ is trivial on the product-tail $\T^{(2)}$.
$\Box$
\medskip\noindent
Our key observation is the following percolation result for the
duplicated system with distribution $\hat\nu$.
We will say that a path in $\Z^2$ is a $\ls$path
for a pair $(\o,\hat\o)\in\O^2$ if $\o(x)\leq\hat\o(x)$ for all its
sites $x$. In the same way we define $\lss$paths, and we can speak
of $\lss$circuits and $\lss$clusters.
%
\begin{lem}\label{lss-circuits}
{\bf(No $(+,-)$percolation in the duplicated system) }
$\hat\nu$-almost surely each finite square $\D=[-n,n]^2$ is
surrounded by
a $\lss$circuit in $\Z^2$.
\end{lem}
%
{\sl Proof}. Consider any two points $x\in\ell_\le$ and
$y\in\ell_\ri$.
We claim that with $\hat\nu$-probability at least $(\theta/4)^2$
there exists a $\lss$path from $x$ to $y$ `above' $\D$, provided $x$
and $y$ are located sufficiently far to the left resp.\ to the right.
We distinguish three cases.
{\em Case 1: $\mu(E^+_\up)=0$}. By Lemma
\ref{orthogonal-butterflies}, $\pi_\up$ then almost surely contains an
infinite $-$cluster $I^-_\up$, and each finite subset of $\pi_\up$ is
surrounded by a $-*$semicircuit in $\pi_\up$. In other words, an
infinite $-*$cluster $I^{-*}_\up$ in $\pi_\up$ exists and touches
both $\ell_\le$ and $\ell_\ri$ infinitely often. By the pinning lemma
and the positive correlations of $\mu$,
with $\mu$-probability at least $(\theta/4)^2$ both $x$ and $y$ are
$-*$connected to $I^{-*}_\up$ outside $\D$, and therefore also
$-*$connected to each other by a $-*$path $p$ above $\D$. However,
this $-*$path $p$ on the first layer is certainly also a $\lss$path
for the duplicated system, and the claim follows.
{\em Case 2: $\mu(E^-_\up)=0$}. In this case we also have
$\hat\mu(E^-_\up)=0$. Interchanging $+$ and $-$ and replacing $\mu$
by $\hat\mu$ in Case 1, we find that with $\hat\mu$-probability at
least $(\theta/4)^2$, there exists a $+*$path $\hat p$ in the second
layer above $\D$ from $x$ to $y$. Since $\hat p$
is again a $\lss$path
for the duplicated system, the claim follows as in the first case.
{\em Case 3: $\mu(E^+_\up)=\mu(E^-_\up)=1$ }.
Then $\mu$-almost surely there exists a unique semi-infinite contour
$\g_\up$, and by tail triviality we can assume (for definiteness)
that $\g_\up$ has its $+$face on the left-hand side $\mu$-almost
surely, and thus also $\hat\mu$-almost surely. By the pinning lemma
and the independence of the two layers, the following event has
$\hat\nu$-probability at least $(\theta/4)^2$:
\bit
\item[--]in the first layer, $y$ is $-*$connected off $\D$ to
$I^{-}_\up(\o)$, and thus to the $-$face $f^-_\up(\o)$ of
$\g_\up(\o)$; that is, there exists an infinite $-*$path $p_y^-(\o)$
from $y$ outside $\D$ eventually running along $\g_\up(\o)$;
\item[--]in the second layer, $x$ is $+*$connected off $\D$ to
$I^{+*}_\up(\hat\o)$, and thus to the $+$face $f^+_\up(\hat\o)$ of
$\g_\up(\hat\o)$; that is, there exists an infinite $+*$path
$p_x^+(\hat\o)$ from $x$ outside $\D$ eventually running along
$\g_\up(\hat\o)$.
\eit
Since $\g_\up(\o)$ and $\g_\up(\hat\o)$ intersect each other
infinitely often by Lemma \ref{intersections}, the union of
$p_y^-(\o)$ and $p_x^+(\hat\o)$ contains a
$*$path from $x$ to $y$ which by construction is a $\lss$path for the
duplicated system. This proves the claim in the final case.
To conclude the proof of the lemma, we let $A_{x,y}$ denote the event
that there exist a $\lss$path from $x$ to $y$ above $\D$, and
$B_{x,y}$ the event that such a path exists below $\D$. The indicator
functions of these events can be written as increasing functions $f$
resp.\ $g$ of the difference configuration $\hat\o-\o$. Using the
positive correlations of $\mu$ and $\hat\mu$ we thus obtain
\bea
\hat\nu(A_{x,y}\cap B_{x,y}) &=& \int\mu(d\o)\int\hat\mu(d\hat\o)\;
f(\hat\o-\o)\, g(\hat\o-\o)\\
&\geq& \int\mu(d\o)\; \hat\mu(f(\cdot-\o))\,\hat\mu(g(\cdot-\o))\\
&\geq& \hat\nu(A_{x,y})\,\hat\nu(B_{x,y})
\ \geq\ (\theta/4)^4\;.
\eea
The last inequality follows from the claim and its analogue for the
lower half-plane. However, if $A_{x,y}\cap B_{x,y}$ occurs then $\D$
is surrounded by a $\lss$circuit for the duplicated system. Letting
$\D\uparrow\Z^2$ we see that with probability at least $(\theta/4)^4$
each finite set is surrounded by a $\lss$circuit. Since this event is
measurable with respect to the product-tail $\T^{(2)}$ on which
$\hat\nu$ is trivial, the lemma follows.
$\Box$
\medskip\noindent
It is now easy to complete the proof of Proposition \ref{invariance}.
\medskip\noindent
{\sl Proof of Proposition \ref{invariance}}. Consider any square
$\D=[-n,n]^2$, and let $\e>0$. By Lemma \ref{lss-circuits}, $\D$ is
$\hat\nu$-almost surely surrounded by a $\lss$circuit, and with
probability at least $1-\e$ such a $\lss$circuit can be
found in a sufficiently large square $\L$. Let $\Ga$ be the interior
of the largest such $\lss$circuit; if no such $\lss$circuit exists
let $\Ga=\emptyset$. Then we find for any increasing
$\F_\D$-measurable function $0\leq f\leq1$, using the strong
Markov property of $\hat\nu$ and
the fact that $\mu_\Ga^\o\preceq \mu_\Ga^{\hat\o}$ when
$\Ga(\o,\o')\ne\emptyset$,
\bea
\mu(f)&=&\hat\nu(f\otimes 1)\ \leq\
\int_{\{\Ga\ne\emptyset\}} d\hat\nu(\o,\o')\;
\mu^{\o}_{\Ga(\o,\o')}(f) +\e\\
&\leq&\int d\hat\nu(\o,\o')\;\mu^{\o'}_{\Ga(\o,\o')}(f) +\e
\ =\ \hat\nu(1\otimes f)+\e\ = \ \hat\mu(f)+\e\;.
\eea
Letting $\e\to0$ and $\D\uparrow\Z^2$ we find that
$\mu\preceq\hat\mu$. Interchanging $\mu$ and $\hat\mu$
(i.e., the roles of the layers) we get the reverse relation.
Hence $\mu=\hat\mu$, so that $\mu$ is horizontally invariant.
The vertical invariance follows similarly by an interchange of
coordinates. $\Box$
\section{Extensions}
\label{extensions}
Which properties of the square lattice $\Z^2$ entered into
the preceding arguments? The only essential feature was its
invariance
under the reflections in all horizontal and vertical lines with
integer coordinates.
We claim that the theorem remains true for the Ising model
on any graph $\LL$ with these properties. (The Ising model on
triangular and honeycomb lattices has already been treated in
\cite{Fuk}.)
To be more precise, let $\R=\{R_{k,\h},R_{k,\v}:k\in\Z\}$ denote the
set of all reflections of the Euclidean
plane ${\bf R}^2$ in horizontal or vertical lines with integer
coordinates, and suppose $\LL$ is a countable subset of ${\bf R}^2$
which (after suitable scaling and rotation) is $R$-invariant
for all $R\in\R$.
Such an $\LL$ is uniquely determined by its finite intersection
with the unit cube $[0,1]^2$, and it is periodic with period $2$.
Suppose further that $\LL$ is equipped with a symmetric
neighbor relation `$\sim$' satisfying
\bit
\item[(L1)] each $x\in\LL$ has only finitely many `neighbors'
$y\in\LL$
satisfying $x\sim y$;
\item[(L2)] $x\sim y$ if and only if $Rx\sim Ry$ for all $R\in\R$.
\eit
If $x\sim y$ we say that $x$ and $y$ are connected by an edge, which
is
visualized by a straight line segment between $x$ and $y$.
The preceding assumptions simply mean that $(\LL,\sim)$ is a locally
finite graph admitting the reflections $R\in\R$ (and thereby the
translations $\th_x$, $x\in 2\Z^2$) as graph automorphisms.
The fundamental further assumption is
\bit
\item[(L3)] $(\LL,\sim)$ is planar, i.e.,
the edges between different pairs of neighboring points do not cross
each other except possibly at some lattice points.
\eit
The complement (in ${\bf R}^2$) of
the union of all edges then splits into connected components called
the faces of $(\LL,\sim)$.
One basic consequence of planarity is that $(\LL,\sim)$ admits a
conjugate matching graph $(\LL,\stackrel{*}{\sim})$. As indicated by
the
notation, this conjugate graph has the same set of vertices, but the
relation $x\stackrel{*}{\sim} y$ holds if either $x\sim y$ or $x$ and
$y$ are distinct points (on the border) of the same face of $(\LL,\sim)$.
(Note that this matching dual is in general not planar.)
The edges of $(\LL,\stackrel{*}{\sim})$ are then used to define the
concept of
$*$connectedness. The construction implies that
every path in $(\LL,\sim)$ is also a $*$path (i.e., a path in
$(\LL,\stackrel{*}{\sim})$), and that the outer boundary of any
cluster
is a $*$path, and vice versa.
(The latter property holds for arbitrary matching pairs of graphs
as defined in Kesten \cite{Kes}, e.g. However, we also
used repeatedly the former property which does not extend to general
matching pairs. In particular, this means that our results do not
apply to the Ising model on the matching conjugate of $\Z^2$ having
nearest-neighbor interactions {\em and\/} diagonal interactions.)
Another consequence of planarity is that we can draw contours
separating clusters from $*$clusters. Such contours can either be
visualized by broken lines passing through the edges of $(\LL,\sim)$,
or simply as a pair consisting of a path and an adjacent $*$path,
namely the two faces of the contour.
In order to see how to work with a graph $(\LL,\sim)$ as above we
will discuss the proper definition of half-planes and their boundary
lines. A half-plane $\pi$ is
still the intersection of $\LL$ with a set of the form $\{x\in {\bf
R}^2:x_i\geq k\}$, $k\in\Z$, $i\in\{1,2\}$, or with $\leq$
instead of $\geq$. However, the `boundary line' $\ell$ is now in
general not a straight line but rather the set $\ell=\{x\in\pi:x\sim
y
\mbox{ for some } y\notin\pi\}$. In particular, $\ell$ is
not necessarily a line of fixed points for the reflection $R\in\R$
mapping $\pi$ onto its conjugate halfplane $\pi'$. However,
we can simply replace a point $x\in\ell$ by a pair $(x,x')$
consisting
of $x$ and its $R$-image $x'$ (which
either coincides with $x$ or is a neighbor of $x$).
Similarly, the semiaxes $\ell_\up$ etc.\ should be considered as sets
of such pairs. With these
modifications, all geometric arguments still work in the obvious way.
So, as a consequence of the preceding discussion we see that the
square lattice $\Z^2$ can be replaced, for example, by
\medskip
$\bullet$ the {\em triangular lattice\/} {\bf T}. This is the
$\R$-invariant lattice satisfying
${\bf T}\cap[0,1]^2=\{(1,0),(0,1)\}$ and
$(-1,0)\sim(1,0)\sim(0,1)\sim(2,1)$; the
remaining edges result from (L2). Since all
faces are triangles, {\bf T} is self-matching. While $\pi_\up$ and
$\pi_\lo$ have a common straight boundary line, the boundaries of
$\pi_\ri$ and $\pi_\le$ are not straight; besides a common part on
the vertical axis they also contain the adjacent points $(1,k)$
resp.\ $(-1,k)$, $k\in 2\Z$.
$\bullet$ the hexagonal or {\em honeycomb lattice\/} {\bf H}. Here,
for example, ${\bf H}\cap[0,1]^2=\{(\frac13,1),(\frac23,0) \}$
and $(-\frac13,1)\sim(\frac13,1)\sim(\frac23,0)\sim
(\frac43,0)$; all other edges are
again determined by (L2). As in the triangular
lattice, $\pi_\up$ and
$\pi_\lo$ have a common straight boundary line, but $\pi_\ri$ and
$\pi_\le$
have no common points.
$\bullet$ the {\em diced lattice}. This is obtained from the
honeycomb lattice by placing points in the centers of the hexagonal
faces and connecting them to the three points in the west, north-east
and south-east of these faces; to obtain reflection symmetry an
additional shift by $(-\frac13,0)$ is necessary.
See p.\ 16 of \cite{Kes} for more details.
$\bullet$ the covering lattice of the honeycomb lattice, the
{\em Kagom\'e lattice}, cf.\ p.\ 37 of \cite{Kes}.
\bigskip\noindent
As for the interaction,
it is neither necessary that the interaction along all bonds is the
same,
nor that it is invariant under the spin flip. Except for attractivity,
we need only the invariance under simultaneous flip-reflections
(which in particular
implies periodicity with period 2). For example, we did not need
that
$\mu^+$ and $\mu^-$ are invariant under all $R\in\R$ and related to
each other by the spin flip $T$ (cf.\ the comments after Corollary
\ref{ocean}). We rather needed that $\mu^+=\mu^-\circ
R\circ T$ for all $R\in\R$ (implying that $\mu^+$ and $\mu^-$ are
periodic, and that any flip-reflection invariant $\mu$ is different
from these phases; the latter was used in Lemmas \ref{butterfly} and
\ref{orthogonal-butterflies}).
This, however, holds whenever the interaction is invariant
under flip-reflections. The same invariance is sufficient for
spin-reflection domination and the point-to-semicircuit lemma. So,
the only thing to observe is that whenever we work with translations
(as in the proof of Lemma \ref{semi-unique} and in Section \ref{inv}
\ref{intersections} and below) we have to confine ourselves to {\em
even\/} translations, which does not raise any problems.
As a result,
we can consider any system of spins $\o(x)=\pm1$ with formal
Hamiltonian
of the form
\be{Ham}
H(\o)=\sum_{x\sim y} U_{x,y}(\o(x),\o(y)) +\sum_{x\in\LL}
V_x(\o(x))\;,
\ee
where for all $a,b\in\{-1,1\}$ we have $U_{x,y}(a,b)=U_{y,x}(b,a)$ and
\bit
\item[(H1)] $U_{x,y}(1,\cdot)-U_{x,y}(-1,\cdot)$ is decreasing on
$\{-1,1\}$;
\item[(H2)] $U_{x,y}(a,b)=U_{Rx,Ry}(-a,-b)$ and $V_x(a)=V_{Rx}(-a)$
for all $R\in\R$.
\eit
Assumption (H1) implies that the FKG inequality is applicable.
We thus obtain the following general result.
%
\begin{thm}\label{general}
Consider a planar graph $(\LL,\sim)$ as above and an
interaction of the form \rf{Ham} satisfying (H1) and (H2).
Then there exist no more than two extremal Gibbs measures.
\end{thm}
%
The standard case, of course, is the ferromagnetic Ising model
without external field; this corresponds to the choice
$U_{x,y}(a,b)=-\b ab$ and $V_x\equiv 0$. But there is also
another case of particular interest. Consider
$\LL=\Z^2+(\frac12,\frac12)$, the shifted square lattice with its
usual
graph structure. $\LL$ is bipartite, in the sense that $\LL$ splits
into to disjoint sublattices, $\LL_{even}$ and $\LL_{odd}$, such that
all
edges run from one sublattice to the other. If we set
$U_{x,y}(a,b)=-\b ab$
and define a staggered external field
\[
V_x(a)=\left\{\ba{rl}-h a&\mbox{if }x\in \LL_{even}\\
h a& \mbox{if }x\in \LL_{odd}\ea \right.
\]
with $h\in{\bf R}$ then the conditions (H1) and (H2) hold;
here we take advantage of the fact that the
reflections $R\in\R$ map
$\LL_{even}$ into $\LL_{odd}$ and vice versa.
But it is well-known that this model is isomorphic to the
{\em anti\/}ferromagnetic Ising model on $\Z^2$ with external field
$h$; the
isomorphism consists in flipping all spins in $\LL_{odd}$. This
gives us the following result.
%
\begin{cor}
For the Ising antiferromagnet on $\Z^2$ for any inverse temperature
and
arbitrary external field there exist at most two extremal
Gibbs measures.
\end{cor}
%
This corollary does not extend to non-bipartite lattices such as the
triangular lattice. In fact, for the Ising antiferromagnet on {\bf T}
one expects the existence of three different phases for suitable $h$.
However, there is a similar repulsive model to which our theorem
applies, namely the hard-core lattice gas on $\Z^2$. In this model,
the values
$-1$ and $1$ are interpreted as the absence resp.\ presence of a
particle, and no particles are allowed to sit on adjacent places.
Interchanging the values $\pm1$ on $\LL_{odd}$ we obtain an
isomorphic
model which is defined by setting
$U_{x,y}(a,b)=\infty$ if $x\in\LL_{even}$ and $a=-b=1$,
or $x\in\LL_{odd}$ and $a=-b=-1$, and
$U_{x,y}(a,b)=0$ otherwise;
and
$V_x(a)=-\log\lambda$ if $x\in\LL_{even}$ and $a=1$,
or $x\in\LL_{odd}$ and $a=-1$, and
$V_x(a)=0$ otherwise.
The parameter $\lambda>0$ is called the activity.
This model satisfies all conditions of Theorem \ref{general}, except
that the interaction takes the value $+\infty$. This implies that the
finite energy condition does not hold as it stands. However, it is
easily seen that there are still enough admissible configurations to
satisfy all needs of the Burton-Keane theorem and our other
applications of
the finite energy property. This leads us to the following corollary.
%
\begin{cor}
For the hard-core lattice gas on $\Z^2$ at any activity $\lambda>0$
there exist at most two extremal Gibbs measures.
\end{cor}
%
\small
\begin{thebibliography}{19}
\bibitem{Aiz} Aizenman, M. (1980) Translation invariance and
instability of
phase coexistence in the two-dimensional Ising system, {\sl Commun.
Math. Phys.} {\bf 73}, 83--94.
\bibitem{BK} Burton, R. and Keane, M. (1989) Density and uniqueness in
percolation, {\sl Commun. Math. Phys.} {\bf 121}, 501--505.
\bibitem{CNPR} Coniglio, A., Nappi, C.R., Peruggi, F. and Russo, L.
(1976)
Percolation and phase transitions in the Ising model,
{\sl Commun. Math. Phys}{\bf 51}, 315--323.
\bibitem{Fuk} Fukuda, H. (1995) The two dimensional Ising model and
Gibbs states, an approach with percolation method, Master thesis,
Osaka University (in Japanese).
\bibitem{Gii} Georgii, H.-O. (1988) {\sl Gibbs Measures and Phase
Transitions}, de Gruyter, Berlin New York.
\bibitem{GHM} Georgii, H.-O., H\"aggstr\"om, O. and Maes, C. (1999)
The random geometry of equilibrium phases, in: Domb and J.L.
Lebowitz (eds.), {\sl Critical phenomena}, Academic Press.
\bibitem{Grimmett} Grimmett, G.R. (1999) {\sl Percolation}, 2.\
edition, Springer, New York.
\bibitem{Hig} Higuchi, Y. (1981) On the absence of non-translation
invariant
Gibbs states for the two-dimensional Ising model, in: J. Fritz, J.L.
Lebowitz and D. Sz\'asz (eds.), {\sl Random fields}, Esztergom
(Hungary) 1979. Amsterdam: North-Holland, Vol. I, 517--534.
\bibitem{Kes} Kesten, H. (1982) {\sl Percolation theory for
mathematicians}. Boston MA, Birkh\"auser.
\bibitem{MM} Messager, A. and Miracle-Sole, S. (1975) Equilibrium
states of the two-dimensional Ising model in the two-phase region,
{\sl Commun. Math. Phys.} {\bf 40}, 187--196.
\bibitem{Rus} Russo, L. (1979)
The infinite cluster method in the two-dimen\-si\-o\-nal
Ising model, {\sl Commun. Math. Phys.} {\bf 67}, 251--266.
\end{thebibliography}
\end{document}
---------------9907280857952--
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\input zb-matheduc
\iteman{ZMATH 2015d.00960}
\itemau{Kern Florian; Burgeth, Bernhard; Eichhorn, Dieter}
\itemti{Mathworld. Algorithms for image processing. (MatheWelt. Algorithmen zur Bildbearbeitung.)}
\itemso{Math. Lehren 32, No. 188, 16 p, pull-out section (2015).}
\itemab
Die Smartphone-Kamera bietet einige M\"oglichkeiten, Schnappsch\"usse zu bearbeiten: Kontraste erh\"ohen, Konturen sch\"arfen, Bilder \"uberlagern oder Effekte anwenden. In dem ab Klasse 9 geeigneten Arbeitsheft besch\"aftigen sich die Sch\"ulerinnen und Sch\"uler u.a. mit den Fragen, was mit Grauwert-Bildern passiert, wenn man Funktionen bzw. Effekte auf sie anwendet, und welche Aussagen das Histogramm dabei \"uber die Grauwertanteile des Bildes macht.
\itemrv{Renate St\"urmer (Zweibr\"ucken)}
\itemcc{R40 U60}
\itemut{educational media; workbooks; algorithms; digital image processing; functions; inverse of a function; grade 9; grade 10; upper secondary}
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Anarhija.info lancia il concorso per l’immagine più \href{https://anarhija.info/library/anarhija-info-3-vizio-capitale-la-lussuria-it}{politically correct}.\forcelinebreak L’immagine vincitrice indicherà la nuova linea grafica del sito.
\begin{figure}[p]
\centering
\includegraphics[keepaspectratio=true,height=0.75\textheight,width=\textwidth]{c-a-concorso-anarhija-info-it-2.jpg}
\end{figure}
\clearpage
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\emph{(si ringraziano compagn! per l’invio delle immagini)}
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Anarhija.info
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Concorso anarhija.info
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%
% Are Canadian academics attributing too much influence on policy to globalization?
%
% Created by Milan on 2013-03-11.
% Copyright (c) 2013 Milan Ilnyckyj. All rights reserved.
%
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\title{Are Canadian academics attributing too much influence on policy to globalization?}
\author{Milan Ilnyckyj}
\date{2013-03-19}
% The preamble is ending now
\begin{document}
\maketitle
\section*{Introduction}
To a large degree, globalization now establishes the context in which public policy is made.
While policy-makers are rarely rigidly bound to particular choices as a consequence of global economic, political, technological, and social integration, these phenomena have all influenced their thinking and planning in the post-WWII era.
Economic policy is made in the context of global competitiveness, and within global institutional structures like the World Trade Organization and its associated agreements.
There is a general pattern of the opening of the Canadian economy, as well as integration with markets elsewhere, particularly in the United States.\footcite{Skogstad2008}
This has an impact in policy areas as diverse as taxation, agriculture, and the scope and functioning of Canada's social safety net.
Security is also considered from a global perspective.
In the short-to-medium term, this includes factors like the activities of transnational terrorist groups and the intersections between state failure and global security; in the long-term, it includes strategic considerations about how a changing balance of economic power internationally affects security priorities.
Less directly, other manifestations of globalization make themselves felt in the establishment of public policy --- for instance, in terms of the changing characteristics of immigration, wherein those who immigrate have much more capability to remain in contact with the people, culture, and current events in their country of origin.
All that being acknowledged, it can be difficult to identify precisely how much influence globalization had on the establishment of a particular policy, and this has been an area of contention for Canadian scholars of political science.
There is even debate about whether the phenomenon of globalization should be treated as a single thing, or whether it would be better divided analytically into `globalization' defined by ``structural economic factors'' and `internationalization' defined by ``when policies within domestic jurisdictions face increased scrutiny, participation, or influence from transnational actors and international institutions''.\footcite[][p. 407]{Haddow2004}
While the two ideas are conceptually separable, they may nonetheless be practically intertwined to a degree that makes their consideration in isolation from one another infeasible.
Economic integration establishes the landscape in which policy decisions are made about taxation, industrial policy, and environmental standards.
Even in the area of social policy, performance is generally evaluated comparatively by intergovernmental organizations, non-governmental organizations, and by government departments and ministers themselves.
When the question of `how Canada is doing' in any particular field arises, the answer is usually formulated in comparison with a group of peer countries.
As such, Canadian academics have rightly accorded considerable importance to globalization in policy-making, though identifying the precise degree of influence upon any particular policy decision is challenging.
\section*{A constraining international environment}
Canada's recent federal budget documents are laden with international comparisons in areas including research and development, the global competitiveness of firms, trade, taxation rates, and immigration.
\emph{Budget 2012} is peppered with references to how Canada's policies and performance compare with those of other G-7 states.\footcite[][http://www.budget.gc.ca/2012/plan/toc-tdm-eng.html]{Budget2012}
The budget also speaks directly to the ways in which international competitiveness concerns drive policy-making.
Under the heading of `Improving Conditions for Business Investment', for instance, it describes ``streamlining'' the regulatory system, expanding trade, ``keeping taxes low for job-creating businesses'', and ``further developing our financial sector advantage''.
Generally speaking, the hypothesis that ``globalization is causing nations to converge towards neoliberal and market-oriented options'' is convincing.\footcite[][p. 403]{Haddow2004}
Richard Simeon identifies some of the dimensions of this constraint, including a more limited scope of policy instruments to be employed, ``capital mobility and the impact of global problems [that] exceed the regulatory grasp of the state'', and ``debts and deficits [that] also constrain state innovation''. \footcite[][p.379]{Simeon1996}
This international focus is replicated in Canadian political science scholarship, including in terms of the growth of comparative studies, and participation in international collaborations.\footcite[][p.376]{Simeon1996}\footcite{White2008}
Nonetheless, scholars have identified gaps in the literature.
Grace Skogstad, for instance, argues that more work needs to be done on the function of policy networks ``in the context of both multilevel governance and internationalization''.\footcite[][p.219]{SkogstadComp}
Rodney Haddow identifies how ``little is known about the actual work of policy analysts in contemporary Canadian governments'', arguing that what little is known suggests a ``very lumpy or uneven distribution of policy analytical capacity''.\footcite[][p.165]{Haddow2004}
Decades ago, Richard Simeon pointed out how despite the growth in interest in the study of policy-making, the work that had been done largely comprised ``a proliferation of isolated studies, and of different methods and approaches'' in which political scientists have not even decided what the dependent variables to be studied should be.\footcite[][p.548]{Simeon1976}
He goes on to criticize work in the field as having ``apolitical, atheoretical, non-cumulative, and non- comparative characteristics''.\footcite[][p.580]{Simeon1976}
It is not entirely clear whether subsequent progress has addressed these concerns.
While a great deal of academic work has certainly been done, it may not employ the methods or serve the purposes that Simeon would endorse.
At the same time, it isn't clear that Simeon's preferred approach, which focuses on description and explanation rather than prescription, is really the most appropriate option for academics involved in the study of public policy.
Given some of the analytical shortcomings identified in government and policy networks by numerous scholars, it may be that political scientists can play a valuable role in pointing out the substantive shortcomings in current public policy and in making suggestions about how it might be improved.
Political scientists have examined and described many of the mechanisms through which the forces of globalization feed through into domestic policy formulation.
Often, this has taken the form of a narrowing of the range of possible policies under consideration and the exclusion of specific policy options that are expected to reduce Canadian competitiveness.
For example, Grace Skogstad describes how:
\begin{quote}
``as the liberalization of capital and goods markets contributes to consolidation of investment and financial capital in a small number of transnational corporations, governments find themselves forced to compete with one another to have value added within their territory but not elsewhere.''\footcite[][p. 19]{Skogstad2008}
\end{quote}
Some policy consequences of this include:
\begin{quote}
``a limited and non-interventionist role for the state in the market, freedom of trade and capital mobility, removal of welfare benefits that create disincentives to market participation, and in general a smaller public sphere.''\footcite[][p. 19]{Skogstad2008}
\end{quote}
This analysis is echoed by other scholars who highlight how globalization empowers individuals and firms to relocate capital and production to jurisdictions that promulgate policies that conform to their preferences.
Societal consequences arising from this may include increasing inequality (both economically and in terms of the ability of different individuals and groups to shape the policy agenda), a weakened social safety net, and an increased ability for multinational firms to set the terms under which they operate.
The perception of international constraint can also be exploited by sophisticated actors.
Agents like wealthy individuals and multinational companies can highlight their ability to relocate when trying to encourage the promulgation of policies that favour their interests.
Similarly, governments can use arguments about international constraint to legitimize policies that may otherwise be controversial or unpopular, arguing that their hands are effectively tied.
The most roundabout form of such public relations work may be cases where a government privately wishes to push forward an unpopular policy, fears the political backlash, and therefore encourages an international institution like the European Union or International Monetary Fund to call for it, allowing the government to accede reluctantly to implementing its own policy preference, under cover of a story about outside imposition.
\section*{Policy independence maintained}
Ultimately, states retain their sovereignty.
Resolutions of the United Nations Security Council are frequently ignored, and this is an international institution empowered to ``take such action by air, sea, or land forces as may be necessary to maintain or restore international peace and security''.
Less dramatically, states often fail to live up to their international obligations, such as when Canada failed to ever develop a credible plan to meet its greenhouse gas pollution reduction targets under the United Nations Framework Convention on Climate Change and Kyoto Protocol, prior to withdrawing from the latter agreement entirely.
If a government has sufficient will, it can make the kind of choices that are sometimes alleged to be barred by globalization --- raising taxes on individuals or businesses that could opt to relocate, raising labour and environmental standards above those of competing jurisdictions, and so on.
Recognition of this reality is important for being able to ``account for continued variety and for equally broad similarities in patterns of change across many settings''.\footcite[][p. 412]{Haddow2004}
Globalization, then, operates more as a series of soft barriers that adjust the relative costs and benefits associated with different policy options, rather than an array of rigid barriers that constrain policy choices to just one option or a small handful thereof.
As Grace Skogstad argues convincingly, globalization and the top-down policy constraints that accompany it has not been the only significant change impacting the state in recent decades --- there is also new bottom-up pressure from policy networks and communities.\footcite{SkogstadComp}\footcite[][p.31]{Skogstad2008}\footcite[See also:][p.163]{Howlett2009}
These sub-national entities now play a role in both policy formulation and implementation, and function within a dynamic of ``regularized patterns of interaction between state actors and representatives of societal interests''.\footcite[][p.207-8]{SkogstadComp}
Skogstad highlights how the behaviour of these groups must be interpreted with an appreciation for the importance of ``internationalized policy environments, multilevel governance, and social actors who transcend national borders''.\footcite[][p.206]{SkogstadComp}
Thus, even the sub-national groups that help to drive policy are subject to some of the manifestations of globalization, and their influence must be interpreted within that context.
Skogstad argues that the increased complexity of the policy-making environment which has arisen from globalization increases the dependence of the state on expert policy networks and suggests that these networks may be ``the crucial linchpin in the capacity of governments to adjust their economies and public policies to the constrains and opportunities posed by globalization''.\footcite[][p.2167]{SkogstadComp}
One example of a nation bucking global trends for internal ideological reasons can be seen in the current Canadian government's hostility toward scientific research, and arguably toward expert advice generally.
Nobody focused on international competitiveness or evidence-driven policy would have expected the elimination of the long-form census (a move noisily protested by many businesses) or the general marginalization of Statistics Canada.
Similarly, it is hard to see why Canada competing in a global market would choose to make disproportionate cuts to scientific research and development, while also implementing policies to hamper the ability of government-funded scientists to collaborate and communicate their findings.
These decisions highlight how a government's priorities can push aside the influence of globalization.
In seeking to shore up support among a core base of supporters --- as well as in endeavouring to shield itself from credible criticism --- a government can make choices that weaken the analysis underlying its policy-making and which set the country at a disadvantage relative to its international peers.
Generally speaking, Canada's current government has shown especially little regard for approaches that ``prioritize evidentiary or data-based decision-making'', as discussed by Michael Howlett.\footcite{Howlett2009}
One area in which this is especially evident is in the government's ongoing conflicts with the Parliamentary Budget Officer (PBO) about the accurate costing of its policy promises.
Despite numerous occasions in which PBO estimates have proven justified after the fact, the government attacks each new one with the same vehemence as those which preceded it.
Another example, in the area of health policy, is the government's hostility to Vancouver's `InSite' safe injection facility for intravenous drug users.
Widely endorsed as a harm reduction mechanism by health researchers, the facility's exemption from the Controlled Drugs and Substances Act was only maintained by means of a Supreme Court decision in 2011 which held that the government's decision was ``arbitrary, undermining the very purposes of the CDSA, which include public health and safety'' as well as ``grossly disproportionate''.\footcite[][http://scc.lexum.org/decisia-scc-csc/scc-csc/scc-csc/en/item/7960/index.do]{CourtInsite}
\section*{Conclusions}
Globalization now sets the stage for many policy decisions, but staging is not destiny and both states and other actors retain the ability to make choices other than those encouraged by globalization and international economic integration.
Continuing to chronicle and analyze these behaviours will remain an important role for political scientists, requiring the development of new methodologies that can tease apart the causal inputs of decisions and seek to identify those that have been dominant in particular instances.
Canadian political scientists have largely accepted the argument that globalization is an important development affecting public policy formulation, and a great deal of work has been done with the intention of better understanding this dynamic.
The more complex a decision-making process becomes --- and the more actors and inputs are involved --- the harder it is to definitively attribute particular outcomes to specific inputs and identify the overall degree of influence of one phenomenon upon another.
As such, even as globalization can be plausibly interpreted to have a growing role on policy development and implementation, it is also part of a pattern of increasing complexity that makes it challenging (and sometimes even impossible) to isolate and measure the degree of influence associated with any one component in that mixture.
Scholars attempting to understand a globalized world must therefore maintain a measure of humility in making causal assertions, including about the degree of influence globalization itself had on any particular policy outcome.
\nocite{White2008}
\printbibliography
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{\large{\bf IMSc-93/21}}
\end{flushright}
\vspace{.2cm}
\begin{center}
{\large{\bf Generalized Fock spaces, new forms of quantum }}
\vspace{.2cm}
{\large{\bf statistics and their algebras}}
\vspace{1cm}
{\bf A.K.Mishra and G.Rajasekaran}
\medskip
{\it Institute of Mathematical Sciences, } \\
{\it Taramani, Madras-600 113} \\
{\it INDIA.} \\
{\it e-mail:~mishra~,[email protected]}
\end{center}
\vspace{.3cm}
\baselineskip=18pt
{\centerline{\bf Abstract}}
We formulate a theory of generalized Fock spaces which underlies the
different forms of quantum statistics such as ``infinite'', Bose-Einstein
and Fermi-Dirac statistics. Single-indexed systems as well as
multi-indexed systems that cannot be mapped into single-indexed systems
are studied. Our theory is based on a three-tiered structure consisting
of Fock space, statistics and algebra. This general formalism not only
unifies the
various forms of statistics and algebras, but also allows us to
construct many new forms of quantum statistics as well as many algebras
of creation and destruction operators. Some of these are : new algebras
for infinite statistics, q-statistics and its many avatars, a consistent
algebra for fractional statistics, null statistics or statistics of
frozen order, ``doubly-infinite'' statistics, many representations of
orthostatistics, Hubbard statistics and its variations.
\vspace{.3cm}
\noindent {\bf Key Words :} Fock space -- multioscillator algebras --
$q$-deformation -- \linebreak quantum groups -- generalized statistics
-- infinite statistics -- fractional \linebreak statistics -- Hubbard model.
\vspace{.3cm}
\noindent {\bf PACS Nos: 03.70 ; 05.30 ; 02.20 ; 71.27}
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\noindent {\bf{2. Generalized Fock Spaces}}
Given a set of oscillators with indices g,h,i ${\ldots}$ m, we
construct the state vector
$$
\vert n_g, n_h \ldots n_m ; 1 \rangle \,=\, \vert \underbrace{1_g \cdots
1_g}_{n_g} \, \underbrace{1_h \cdots 1_h}_{n_h} \, \cdots \underbrace
{1_m \cdots 1_m}_{n_m} \, \rangle \eqno(2.1)
$$
\noindent On the right-hand-side of this equation, $1_g, 1_h \ldots 1_m$
appear $n_g, n_h \ldots n_m$ times respectively and $n_g \ldots n_m$
denote the number of quanta with indices g ... m respectively. Together
with the state (2.1) we consider the set of all states obtained through
all distinct permutations of the entries on the right-hand-side of (2.1).
Thus, we have
$$
\vert n_g, n_h \ldots n_m ; 2 \rangle \,=\, \vert \underbrace{1_g \cdots
1_g}_{n_g-1} \, 1_h 1_g \underbrace{1_h \cdots 1_h}_{n_h-1} \cdots
\underbrace{1_m \cdots 1_m}_{n_m} \rangle \eqno(2.2)
$$
$$
\vdots
$$
$$
\vert n_g, n_h \cdots n_m ; 35 \rangle \, = \, \vert 1_g 1_g 1_m 1_m \,
\underbrace{1_h \cdots 1_h}_{n_h} \ldots \underbrace{1_m \cdots
1_m}_{n_m-2} \rangle \eqno(2.3)
$$
$$
\vdots
$$
$$
\vert n_g, n_h \cdots n_m ; s \rangle \, = \, \vert \underbrace{1_m
\cdots 1_m}_{n_m} \, \ldots \underbrace{1_h \cdots 1_h}_{n_h} \,
\underbrace{1_g \cdots 1_g}_{n_g} \, \rangle \eqno(2.4)
$$
\noindent Here we have given examples of a few permutations. In (2.2),
the positions of one g-quantum and one h-quantum has been
interchanged and in (2.3), we have indicated a few more interchanges.
Collectively, we shall denote the set of all these states as
$$
\vert n_g, n_h \ldots n_m ; \mu \rangle \quad , \quad \mu = {
1,2} \ldots s \eqno(2.5)
$$
\noindent where $s$ is the total number of distinct permutations and
$\mu$ labels each of these states. It is easy to see that
$$
s = \frac{(n_g + n_h + \ldots n_m)!}{n_g!n_h! \ldots n_m!} \eqno(2.6)
$$
\noindent (In eq.(2.3), $\mu$ has been put 35 rather arbitrarily). We
assume the existence of a unique vacuum state corresponding to zero
occupation number for all the oscillators :
$$
\vert 0 \rangle \equiv \vert 0, 0, 0 \ldots 0 \rangle \eqno(2.7)
$$
We shall first consider the set of all the s states given in (2.5) as
linearly independent. Although they are linearly independent, they may
not be orthogonal to each other in general, nor are they normalized.
However, a state in one sector characterized by the occupation
numbers ${(n_g,n_h \ldots n_m)}$ is orthogonal to any state in
another sector characterized by another set of occupation numbers
${(n^{'}_g, n^{'}_h \ldots n^{'}_m)}$. We can summarize these
statements by the equation :
$$
\langle n^{'}_g, n^{'}_h \ldots n^{'}_m ; \alpha \vert n_g,n_h \ldots n_m ;
\beta \rangle \, = \, \delta_{n^{'}_g n_g} \, \delta_{n^{'}_h n_h}
\ldots \delta_{n^{'}_m n_m} \, M_{\alpha \beta} \eqno(2.8)
$$
\noindent Note in particular that the inner product vanishes even if
a single occupation number does not match. Within the same sector,
the inner product is given by
$$
\langle n_g \ldots n_m ; \alpha \vert n_g \ldots n_m ; \beta \rangle
\,=\, M_{\alpha \beta} \eqno(2.8a)
$$
where $M$ is a $s \times s$ hermitian matrix. In fact, there is an infinite
set of matrices of varying dimensions, one corresponding to each sector
$\{n_g \ldots n_m\}$. We choose all these matrices to be positive
definite. This set of inner-product matrices M plays an important role
in the general formalism.
>From the set of linearly independent vectors given in (2.5), it is
possible to construct an orthonormal set of vectors which we shall
denote by a double-barred ket:
$$
\parallel n_g \ldots n_m; \mu \gg ; \quad \mu = 1 . . . . . s
\eqno(2.9)
$$
These satisfy the orthonormality relation
$$
\ll n^{'}_g \ldots n^{'}_m ; \alpha \parallel n_g \ldots n_m ; \beta \gg \,
= \, \delta_{n^{'}_g n_g} \ldots \delta_{n^{'}_m n_m} \,
\delta_{\alpha \beta} \eqno(2.10)
$$
\noindent There is no unique way of doing this and the resulting
orthonormal set is
not unique. One may use Gram - Schmidt orthogonalization procedure or
calculate the eigenstates of the inner-product matrix $M$ or follow any
other method. Whatever may be the method, one can write the relation
connecting the two sets of kets :
$$
\parallel n_g \ldots n_m ; \mu \gg \, =\, \sum_\nu X_{\nu \mu} \vert n_g
\ldots n_m ; \nu \rangle \eqno(2.11)
$$
\noindent and the inverse relation :
$$
\vert n_g \ldots n_m ; \alpha \rangle \, = \, \sum_{\beta}
(X^{-1})_{\beta \alpha} \, \parallel n_g \ldots n_m ; \beta \gg
\eqno(2.12)
$$
\noindent where $X$ is a nonsingular matrix. Although $X$ is not unique
(since it depends on the particular orthogonalization procedure used),
it is possible to show, using (2.8), (2.10), (2.11) and (2.12),
that $XX^{\dagger}$ is the inverse of the innerproduct
matrix\footnote{Eq(2.13) is analogous to the relation
between the metric tensor and the vierbien in general relativity.}:
$$
M^{-1} \,=\, XX^{\dagger} \eqno(2.13)
$$
\noindent Thus, one simple way of ensuring positivity of the
inner-product matrix is to choose a non-singular matrix $X$ and then
determine $M$ using (2.13). Again it must be kept in mind that we are
dealing with an infinite set of matrices, $X$, one for each sector $\{ n_g
\ldots n_m \}$. Also the orthonormality relation holds between vectors
from two different sectors and we have already used this in writing
(2.10).
The completeness relation for the orthonormal set of states
{\hbox{$\parallel
n_g \ldots n_m ; \mu \gg $}} can be written in the form:
$$
I \,=\, \sum_{n_g \ldots n_m} \, \sum_{\mu} \, \parallel n_g \ldots n_m
; \mu \gg \, \ll n_g \ldots n_m ; \mu \parallel \eqno(2.14)
$$
\noindent where I is the identity operator. Substituting from (2.11) into
(2.14) and using (2.13), we get the resolution of the identity operator in
terms of the non-orthonormal set of states :
$$
I \,=\, \sum_{n_g \ldots n_m} \, \sum_{\lambda, \nu} \, \vert n_g \ldots
n_m ; \nu \rangle (M^{-1})_{\nu \lambda} \, \langle n_g \ldots n_m ;
\lambda \vert \eqno(2.14a)
$$
\noindent It is convenient to define the projection operator
$$
P(n_g \ldots n_m) \,=\, \sum_{\mu} \parallel n_g \ldots n_m ; \mu \gg \,
\ll n_g \ldots n_m ; \mu \parallel \eqno(2.15)
$$
$$
= \sum_{\lambda, \nu} \, \vert n_g \ldots n_m ; \nu \rangle
(M^{-1})_{\nu \lambda} \, \langle n_g \ldots n_m ; \lambda \vert
\eqno(2.16)
$$
\noindent so that we have
$$
I = \sum_{n_g \ldots n_m} \, P (n_g \ldots n_m) \eqno(2.17)
$$
\noindent One can easily verify the following properties of the
projection operators :
$$
P (n_g \ldots n_m) \parallel n^{'}_g \ldots n^{'}_m ; \mu \gg \, =\,
\delta_{n_g n^{'}_g} \ldots \delta_{n_m n^{'}_m } \, \parallel n_g
\ldots n_m ; \mu \gg \eqno(2.18)
$$
$$
P (n_g \ldots n_m) \vert n^{'}_g \ldots n^{'}_m ;\mu \rangle \, =\,
\delta_{n_g n^{'}_g} \ldots \delta_{n_m n^{'}_m} \, \vert n_g \ldots n_m
; \mu \rangle \eqno(2.19)
$$
\noindent It is worth noting that ${P (n_g \ldots n_m )}$
projects out any single state not only from the orthonormal set
$\parallel n_g \ldots n_m ; \mu \gg $ but also from the non-orthonormal
set $\vert n_g \ldots n_m ; \mu \rangle$.
In terms of the above projection operators it is very easy to construct
the number operators :
$$
N_k \, =\, \sum_{n_g \ldots n_k \ldots n_m} \quad n_k P(n_g \ldots n_k
\ldots n_m) \eqno(2.20)
$$
\noindent which satisfy the following properties
$$
N_k \parallel n_g \ldots n_k \ldots n_m ; \mu \gg \,=\, n_k \parallel
n_g \ldots n_k \ldots n_m ; \mu \gg \eqno(2.21)
$$
$$
N_k \vert n_g \ldots n_k \ldots n_m ; \mu \rangle \, =\, n_k \vert n_g
\ldots n_k \ldots n_m ; \mu \rangle \eqno(2.22)
$$
$$
[N_k, N_j] \, = \, 0 \quad {\rm for \ any} \ k \ {\rm and} \ j
\eqno(2.23)
$$
We now introduce the transition operators which connect states, lying in
different sectors. Obviously it is enough to define the so called
annihilation and creation operators $c_j$ and $c^{\dagger}_j$. We
define
$$
c^{\dagger}_j \,=\, \sum_{n_g \ldots n_j \ldots n_m} \, \sum_{\mu^{'}
\nu} \, A_{\mu^{'} \nu} \, \vert n_g \ldots (n_{j}+1) \ldots n_m ; \mu^{'}
\rangle \, \langle n_g \ldots n_j \dots n_m ; \nu \vert \eqno(2.24)
$$
\noindent and $c_j$ as the hermitian conjugate of c$^{\dagger}_j$,
where $A_{\mu^{'} \nu}$ are a set of arbitrary (complex) numbers. Note
that the span of $\mu^{'}$ is larger than that of $\nu$ and this is the
reason for the prime on $\mu$.
\noindent Specifically,
$$
\mu^{'} \, = \, 1 \ldots s^{'} \, ; \, s^{'} \,=\, \frac{(n_g + \ldots
(n_{j}+1) + \ldots n_m) !}{n_g ! \ldots (n_j +1) ! \ldots n_m !} \eqno(2.25a)
$$
$$
\nu \, = 1 \, \ldots s \, ; \, s = \frac{(n_g + \ldots n_j + \ldots n_m)!}{n_g!
\ldots n_j ! \ldots n_m!} \eqno(2.25b)
$$
\noindent Hence A is a rectangular matrix. Since A is arbitrary in
general, the relation(2.24) provides the most general definition of the creation
operator. Even in this general case, it is possible to verify the
following commutation relation between the number operator defined in
(2.20) and the creation operator defined in (2.24)
$$
[c^{\dagger}_j, N_k ] \, =\, - c^{\dagger}_j \, \delta_{jk} \eqno(2.26)
$$
\noindent The projection property given in (2.19) plays a crucial role in
the proof of (2.26).
So far we did not specify how the ordered state vectors $\vert n_g
\ldots n_m ; \mu \rangle $ are constructed. In fact, in general there
is no need to specify any procedure for their explicit construction.
The formalism given so far holds whatever may be the explicit form of
their construction. However, once annihilation and creation operators
c and $c^{\dagger}$ are introduced, there exists a natural procedure to
construct the set of ordered states using $c^{\dagger}$ and the vacuum
state $\vert 0 \rangle$. This procedure has the advantage that the
arbitrariness of the matrix A introduced in (2.24)
disappears and A in fact gets determined in terms of M. Hence let us do
it.
In terms of creation operators, the ordered state (2.2) for instance is
constructed as follows :
$$
\vert \underbrace{1_g \cdots 1_g}_{n_g-1} \, 1_h 1_g \underbrace{1_h
\cdots 1_h}_{n_h-1} \ldots \underbrace{1_m \cdots 1_m}_{n_m} \rangle \,
= \, (c^{\dagger}_g)^{n_g-1} c^{\dagger}_h c^{\dagger}_g
(c^{\dagger}_h)^{n_h-1} \cdots (c^{\dagger}_m)^{n_m} \, \vert 0 \rangle
\eqno(2.27)
$$
\noindent and other states are constructed in a similar fashion. We may
introduce the notation
$$
|n_g \ldots n_m ; \mu > \ = \ (c^{\dagger^{n_g}}_g \ldots
c^{\dagger^{n_m}}_m ; \mu ) | 0 > \eqno(2.28)
$$
$$
<n_g \ldots n_m ; \mu | \ = \ < 0 | (c^{n_m}_m \ldots
c^{n_g}_g ; \mu ) \eqno(2.29)
$$
\noindent where $(c^{\dagger^{n_g}}_g \ldots c^{\dagger^{n_m}}_m ; \mu)$
is a permutation of the creation operators similar to the permutations
defined in eqn(2.1) - (2.5) and $(c^{n_m}_m \, \cdots c^{n_g}_g ; \mu)$
is defined
as the hermitian conjugate of $(c^{\dagger^{n_g}}_g
\ldots c^{\dagger^{n_m}}_m ; \mu)$. For
states constructed in this manner, there exists a simple formula
connecting states in ``adjacent'' sectors. For instance,
$$
\vert 1_j, \underbrace{1_g \cdots 1_g}_{n_g} \ldots \underbrace{
1_m \cdots 1_m}_{n_m} \rangle \, = \, c^{\dagger}_j \vert \underbrace{1_g \cdots 1_g}_{n_g}
\ldots \underbrace{1_m \cdots 1_m}_{n_m} \, \rangle \eqno(2.30)
$$
\noindent More generally, we may write
$$
c^{\dagger}_j \vert n_g \ldots n_j \ldots ; \lambda \rangle \,=\, \vert
1_j, n_g \ldots n_j \ldots ; \lambda \rangle \eqno(2.31)
$$
\noindent where $\vert 1_j, n_g \ldots n_j \ldots ; \lambda \rangle$ on
the right is a subset of states in which one `j'
quantum appears on the extreme left. Although the total number
of states in the set $\vert n_g
\ldots (n_j + 1) \ldots ; \lambda^{'} \rangle$ is $s^{'}$ given by
(2.25a), the total number of states in the subset $\vert 1_j, n_g \ldots
n_j \ldots ; \lambda \rangle$ is s given by (2.25b). Further the states
in the subset $\vert 1_j, n_g \ldots n_j ...; \lambda \rangle$ are given the
same ordinal number $\lambda$ as in the set
$\vert n_g \ldots n_j \dots ; \lambda \rangle$. This is possible
since the quanta $\{ n_g \ldots n_j \ldots \}$ are permuted among
themselves without disturbing the extra j-quantum
sitting on the extreme left.
We now substitute the expression for $c^{\dagger}_j$ given by
(2.24) into the left hand side of (2.31). We have
$$
c^{\dagger}_j \vert n_g \ldots n_j \ldots ; \lambda \rangle
$$
$$
= \, \sum_{n^{'}_g \ldots n^{'}_j\ldots} \, \sum_{\mu^{'} \nu} \, A_{\mu^{'}
\nu} \, \vert n^{'}_g \ldots (n^{'}_{j}+1) \ldots ; \mu^{'} \rangle \,
\langle n^{'}_g \ldots n^{'}_j \ldots ; \nu \vert n_g \ldots n_j
\ldots ; \lambda \rangle
$$
$$
= \sum_{\mu^{'}\nu} \, A_{\mu^{'} \nu} M_{\nu \lambda} \, \vert
n_g \ldots (n_{j}+1) \ldots ; \mu^{'} \rangle \eqno(2.32)
$$
\noindent where we have used (2.8). On comparing with (2.31), we get
$$
\sum_{\nu} \, A_{\mu^{'} \nu} \, M_{\nu \lambda} \,=\, \delta_{\mu^{'}
\lambda} \eqno(2.33)
$$
\noindent From (2.33) we see that $\mu^{'} \,=\, \lambda$.
This means that in (2.24),
only the subset $\vert 1_j, n_g \ldots n_j \ldots ; \lambda \rangle $
contributes and hence A is in fact a square matrix and is equal to the
inverse of M$^{-1}$:
$$A \,=\, M^{-1} \eqno(2.34)
$$
So, we may rewrite (2.24):
$$
c^{\dagger}_j \,=\, \sum_{n_g \ldots n_j \ldots} \, \sum_{\lambda, \nu}
\, (M^{-1})_{\lambda \nu} \, \vert 1_j, n_g \ldots n_j \ldots ; \lambda
\rangle \, \langle n_g \ldots n_j \ldots ; \nu \vert \eqno(2.35)
$$
\noindent Thus, A and hence c and c$^{\dagger}$ are completely determined in
terms of the set of M matrices.
Next, consider the expression for the number operator (eqs.(2.20),(2.15)
and (2.16)) :
$$
N_k = \sum_{n_g \ldots n_k \ldots n_m} n_k \sum_\mu \parallel n_g \ldots
n_k \ldots n_m ; \mu \gg \ll n_g \ldots n_k \ldots n_m ; \mu \parallel
\eqno(2.36)
$$
$$
= \sum_{n_g \ldots n_k \ldots n_m} n_k \sum_{\lambda \nu}
\vert n_g \ldots n_k \ldots n_m ; \lambda > (M^{-1})_{\lambda\nu}
< n_g \ldots n_k \ldots n_m ; \nu \vert \eqno(2.37)
$$
\noindent One may try to express this in terms of $c$ and $c^\dagger$.
Using eqs.(2.28) and (2.29),
$$
N_k = \sum_{n_g \ldots n_k \ldots n_m} n_k \sum_{\lambda \nu}
(c^{\dagger^{n_g}}_g \ldots c^{\dagger^{n_k}}_k \ldots
c^{\dagger^{n_m}}_m ; \lambda) |0><0| (c^{n_m}_m \ldots c^{n_k}_k \ldots
c^{n_g}_g ; \nu ) (M^{-1})_{\lambda \nu} \eqno(2.38)
$$
\noindent Apart from $c^\dagger$ and $c$, the above expression contains
the vacuum projector $|0><0|$. Later we shall give examples where $|0><0|$ is
determined as products of $c^\dagger$ and $c$ so that $N_k$ can be
expressed entirely in terms of $c^\dagger$ and $c$. However, in general,
this does not lead to simple results, whereas eq.(2.36) provides us with a
universal representation of number operators which is valid in all
cases.
So far, we regarded the set of state vectors $|n_g, n_h \ldots , \mu > ;
\mu = 1 \ldots s$ (where $s$ is given by (2.6)) to be linearly
independent and the resulting
generalized Fock space is the complete Fock space, which we shall call
the {\it super Fock space}.
We shall now show how to construct reduced Fock spaces. The motivation
for this is that many Fock spaces of physical interest such as the
bosonic Fock space or fermionic Fock space are reduced Fock spaces.
There are various ways of doing this. One may postulate relationships
between states connected by permutations, or one may disallow certain
permutations by equating them to null vectors. Yet another way to
achieve this is to use the permutation group $S_n$ acting on the
$n$-particle state. The super Fock space we have constructed consists of
all the representations of the permutation group. If we allow only
certain representations of $S_n$, we get a reduced Fock space.
All these possibilities are contained in the statement that in the space
of vectors $|n_g, n_h \ldots ;\mu>$, there are $r$ null vectors $(r <s)$
$$
\sum_{\mu} B^p_\mu |n_g,n_h \ldots ; \mu > = 0 \ ; \ p = 1,2, \ldots r ; r
<s. \eqno(2.39)
$$
\noindent where $B^p_{\mu}$ are constants. So, the dimension of the vector space in the sector $\{n_g, n_h \ldots
\}$ is reduced to $d$ given by
$$
d \ = \ s \ - \ r\,. \eqno(2.40)
$$
An important class of reduced Fock spaces are those for which $d = 1$. Here, all
the states connected by permutation of indices will be taken to be
related to each other through equations of the type(2.39). In
otherwords,the number of relations $r$ in eq. (2.39) is
$s-1$. We shall call this space as the {\it bosonic Fock space}. If we impose
the additional restriction : $n_g = 0$ or $1$ only, the resulting space
will be called {\it fermionic Fock space}.
This restriction can also be stated in the form of eq.(2.39) :
$$
|n_g, n_h \ldots ; \mu > \ = \ 0 \quad {\rm for} \quad n_g, n_h \ldots
\ge 2\,. \eqno(2.41)
$$
We can define a new reduced Fock space, also of dimension $d=1$ in each
sector, by taking the set of all the permuted states as null states
except a single state (of a chosen order) which is taken to be the
allowed state. We shall call this as the Fock space of frozen order.
Another important class of reduced Fock spaces are those associated with
parastatistics [35-37], which we shall call parabosonic and parafermionic
Fock spaces. For these, the number of relations $r$ is smaller than for
bosonic or fermionic Fock spaces, so that the dimension $d$ satisfies
$$
1 < d < s \,. \eqno(2.42)
$$
{\it The formalism constructed in the present section is valid for all these
reduced Fock spaces also}, with the modification that all the summations
over $\mu,\nu$ etc. will now go over the range $1 \ldots d$ and
correspondingly, $X,M,$ and $A$ become $d\times d$ matrices. There is
an arbitrariness in the choice of the d states.
Any choice of $d$ states will do, as long as they are non-null
states.
All these Fock spaces, the super Fock space as well as the reduced ones
will be collectively called generalized Fock spaces. To sum up this
discussion we may note that a generalized Fock space is completely
defined by stating what are the allowed states of the system.
We shall define statistics by the precise relationship linking states
obtained by permutation. In general, many relationships can be envisaged
and hence many different forms of statistics can reside within a particular reduced Fock space. However,
in super Fock space, all the states obtained by permutation are
independent and so there is a unique statistics associated with this
Fock space, namely ``infinite statistics'', which is the same as
Boltzmann statistics, but implemented quantum mechanically. Similarly,
in the Fock space of frozen order too, there exists only a single
statistics, named ``null statistics''.
In this section, we started with the generalized Fock space consisting
of the set of allowed states of the system and constructed the creation,
annihilation and number oeprators in terms of the outer products of state
vectors. Do these $c$ and $c^\dagger$ form an operator algebra?
In general, $c$ and
$c^\dagger$ constructed in this way may not form a simple algebra, or
even a closed algebra. Historically, it is the reverse route that has
been followed ; one postulates an algebra of $c$ and $c^\dagger$ and
then deduces the states allowed by the algebra. In this sense, a given
relation involving $c$ and $c^\dagger$ implicitly defines an inner
product and through it specifies the allowed and the null states of the
system. In practice, starting with an algebra is an easier procedure
and we shall use it in the later sections. Actually, it is
complementary to the approach described in this section.
Therefore we have two equivalent ways of dealing with the generalized Fock
spaces. In the first approach, which we have formulated in this section
we construct $c$ and $c^\dagger$ in terms of the allowed states of the
system and the algebra of $c$ and $c^\dagger$ is then a derived
consequence. This is the more fundamental approach and is of universal
validity. In the second approach, which also we shall use in the latter
sections, we start with an algebra of $c$ and $c^\dagger$
and then determine the states of the system allowed by the $cc^\dagger$
algebra. Since the restrictions on the allowed states of the system can
generally be stated in the form of $cc$ relations, the first
approach can be characterized as $cc \rightarrow cc^\dagger$ while the
second is $cc^\dagger \rightarrow cc$. Within the second approach,
we shall describe an elegant method to derive $cc$ relations from
$cc^\dagger$ algebras.
For infinite statistics, there is no restriction on the allowed
states and $cc$ relations do not exist. Hence, for this statistics,
the first approach should be interpreted as ``no $cc$
relation'' $\rightarrow cc^\dagger$ algebra while in the second approach,
starting with any particular $cc^\dagger$ algebra describing infinite
statistics one shows that there does not exist any $cc$ relation. In
those cases where the $cc^\dagger$ algebra depends on a continuous
parameter $q$, one can determine the values of $q$ where $cc$ relation
exists. Generally, these values of $q$ correspond to the boundary
of the region in the parameter space where infinite statistics
with positive definite $M$
exists. On this boundary, one or more eigenvalues of the $M$ matrices
become zero, thus leading to the emergence of the same number of null
vectors in the Fock space which can equivalently be interpreted as the
emergence of $cc$ relations. Thus the formalism unifies infinite
statistics residing on the super Fock space with the various forms of
statistics residing on reduced Fock spaces.
It must be noted that the inner product matrices $M$ occuring in
eq.(2.8a) are quite arbitrary. Consequently, more than one realization or
representation of creation and destruction operators is possible. In
fact, it is this freedom to select arbitrary $M$ which enables one to
construct different algebras involving $c$ and $c^\dagger$, all
operating over the same Fock space.
To sum up, we construct a three-tiered structure consisting of Fock
space, statistics and algebra. Fock space is specified by the set of the
allowed states of the system. Statistics is defined by the nature of the
symmetry of the allowed states under permutation. Algebra of the
creation and destruction operators is determined by the choice of the
inner product matrices $M$.
We shall construct different representations of
infinite statistics in Sec.3. In the bosonic and fermionic Fock spaces,
many forms of quantum statistics which include Bose and Fermi statistics
are possible. These and the null statistics in the Fock space of frozen
order will be taken up in Sec.4.
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\noindent {\bf 3. Super Fock Space and Infinite Statistics}
\noindent {\bf 3.1 The standard representation $(M = 1)$}
Having taken all the s states to be independent, the simplest choice of
the matrix X is the unit matrix which implies M and A also to be unit
matrices (of appropriate dimensions) for all the sectors
$\{ n_g, n_h \ldots\}$ :
$$
X = M = A = 1 \eqno(3.1)
$$
\noindent For this choice, which we shall call the standard
representation of infinite statistics, the ordered states (2.5) themselves
form an orthonormal set :
$$
\langle n_g, n_h \ldots ; \mu \vert n_g, n_h \ldots ; \nu \rangle \,=\,
\delta_{\mu \nu} \eqno(3.2)
$$
\noindent Further, the creation and annihilation operators are given by
$$
c^{\dagger}_j \,=\, \sum_{n_g, n_h\ldots} \, \sum_{\mu} \,
\vert 1_{j} , n_g, n_h \ldots ; \mu
\rangle \, \langle n_g, n_h \ldots ; \mu \vert
\eqno(3.3)
$$
$$
c_j \,=\, \sum_{n_g,n_h \ldots} \, \sum_{\mu} \vert n_g, n_h \ldots ;
\mu \rangle \, \langle 1_{j} , n_g, n_h \ldots ; \mu \vert \eqno(3.4)
$$
\noindent From (3.3) and (3.4), using the orthogonality and completeness
relations, one can derive the cc$^\dagger$ algebra :
$$
c_i c^\dagger_j \, =\, \delta_{ij} \eqno(3.5)
$$
\noindent This algebra was first proposed by Greenberg [13], in an
important paper on infinite statistics.
In the standard representation, we also have the useful identity :
$$
\sum_i c^\dagger_i c_i \ = 1 - |0><0| \eqno(3.6)
$$
\noindent Although this identity can be obtained from the eqs(3.3)
and (3.4) that define $c^\dagger$ and $c$, it is important to
note that within Fock space it follows
from the $cc^\dagger$ algebra (eq.(3.5)). We shall prove it by showing
that eq.(3.6) is valid, when applied on any state in the generalized
Fock space. First applying on $|0>$, we find that both sides are zero.
Next we apply on the other states $c^\dagger_k c^\dagger_\ell \ldots |0> $:
$$
\left\{ \sum_i c^\dagger_i c_i - 1 + |0><0| \right\} c^\dagger_j
c^\dagger_k \ldots |0> = 0 \eqno(3.7)
$$
\noindent Using eq.(3.5) and noting $<0|c^\dagger_j \ = \ 0$
we see that the left side of eq(3.7) infact vanishes, thus
completing the proof of eq(3.6).
In a different context, Cuntz [38] had studied the algebra defined by
eq(3.5) and by the relation :
$$
\sum_i c^\dagger_i c_i \ = \ 1\,. \eqno(3.8)
$$
\noindent Cuntz algebra is inconsistent with Fock
space, as can be seen by applying both sides of eq (3.8) on $|0>$.
In contrast, our eq(3.6), because of the inclusion of the vacuum
projector term in it, is consistent with Fock space and is infact a
consequence of the algebra of $c$ and
$c^\dagger$ in the standard representation of infinite statistics.
Putting $M=1$, the number operator $N_k$ given in eq(2.38) becomes
$$
N_k \ = \ \sum_{n_g..n_k..n_m} n_k \sum_\mu \left( c^{\dagger^{n_g}}_g
\ldots c^{\dagger^{n_k}}_k \ldots c^{\dagger^{n_m}}_m ; \mu\right)
|0><0| \left(c^{n_m}_m \ldots c^{n_k}_k \ldots c^{n_g}_g ; \mu
\right)\eqno(3.9)
$$
\noindent Substitution of $|0><0|$ from (3.6) into this equation and a
straightforward but tedius calcualtion finally leads to
$$
N_k \ = \ \sum_{n_g..n_k..n_m} \sum_\mu \left( c^{\dagger^{n_g}}_g
\ldots c^{\dagger^{n_k}}_k \ldots c^{\dagger^{n_m}}_m ; \mu\right)
c^\dagger_k c_k \left(c^{n_m}_m \ldots c^{n_k}_k \ldots c^{n_g}_g ; \mu
\right) \eqno(3.9a)
$$
\noindent This expression is identical to Greenberg's formula for $N_k$
[13], though it is written in a different form. This illustrates the
derivation of the representation of $N_k$ in terms of $c$ and
$c^\dagger$. As already mentioned, in general neither the procedure nor
the result is simple and in any case one does not need it. The universal
representation of $N_k$ given in eq(2.36) is sufficient.
\vspace{.5cm}
\noindent {\bf 3.2 $q$-mutators with real $q$}
Many other choices of M are possible. A particularly interesting choice is
the one in which M is given as a function of a real parameter q. Consider the inner product between the n-particle state
vectors with all occupation numbers unity : $\vert 1_g, 1_h \ldots ; \mu
\rangle $. The inner product matrix M in this sector has dimension n !
$\times$ n! and its matrix element is taken to be
$$
\langle 1_g, 1_h \ldots ; \mu \vert \,
1_g, 1_h \ldots ; \nu \rangle \, = q^J \eqno(3.10)
$$
\noindent where q is a real number lying in the range $-1 < q < + 1$
and $J$ is the number of inversions required to transform the
state $\vert 1_g, 1_h \ldots ; \nu \rangle $ into the state $\vert 1_g,
1_h \ldots ; \mu \rangle$. Number of inversions is the minimum number of
successive interchanges between adjacent quanta that will take the state
$\vert 1_g, 1_h \ldots ; \nu \rangle$ to $\vert 1_g, 1_h \ldots ; \mu
\rangle $. For example,
$$
\langle 1_h 1_g 1_k \vert 1_g 1_h 1_k \rangle \,=\, q \eqno(3.11)
$$
$$
\langle 1_h 1_k 1_g \vert 1_g 1_h 1_k \rangle \, =\, q^2 \eqno(3.12)
$$
The positivity of the q-dependent M matrices defined above has been
proved by Fivel [15] and Zagier [16]. In particular, Zagier has given the
explicit form of the determinant of the (n ! $\times$ n!) dimensional M
matrix for arbitrary n :
$$
det M \,=\, \prod^{n-1}_{k=1} \, [1-q^{k(k+1)}]^{(n-k)n!/k(k+1)}
\eqno(3.13)
$$
This determinant is positive in the range $-1$ $<$ q $<$ +1.
The inner product for states with occupation numbers larger than unity
(which are the same as states with repeated indices) is obtained from
the above inner product for states with distinct indices by symmetrizing
with respect to the repeated indices. For example, consider
$$
\langle 2_g 1_m \vert 1_g 1_m 1_g \rangle \,=\, \langle 1_g 1_g 1_m
\vert 1_g 1_m 1_g \rangle \eqno(3.14)
$$
\noindent Replace one of the g's by h in both the initial and final
states and thus get a matrix element with distinct indices which can be
calculated using(3.10). This replacement can be done in (2!)$^2$ ways.
The sum of these (2!)$^2$ matrix elements divided by 2! is the required
answer. Thus,
$$
\langle 1_g 1_g 1_m \vert 1_g 1_m 1_g \rangle \, =\, \frac{1}{2} \left[
\langle 1_g 1_h 1_m \vert 1_g 1_m 1_h \rangle +
\langle 1_g 1_h 1_m \vert 1_h 1_m 1_g \rangle + \right.
$$
$$
\left. \langle 1_h 1_g 1_m \vert 1_g 1_m 1_h \rangle +
\langle 1_h 1_g 1_m \vert 1_h 1_m 1_g \rangle \right] \eqno(3.15)
$$
$$
= q + q^2 \eqno(3.16)
$$
\noindent It is clear that the matrix M for repeated indices is obtained
from the higher dimensional matrix for distinct indices by a process of
collapse or reduction. It can be shown that this process retains the
positivity of the matrix.
It is worth noting that the norm for $n$-particle state with all
the indices repeated is
$$
<n_g|n_g> = [n_g!]_q = [n_g]_q [n_g-1]_q \ldots [2]_q [1]_q \eqno(3.17)
$$
\noindent where the ``$q$-number'' $[n]_q$ is defined by
$$
[n]_q \ \equiv \ \frac{1-q^n}{1-q} = 1 + q + q^2 + \ldots q^{n-1}.
\eqno(3.18)
$$
Inverting these $M$ matrices and defining the creation operator through
(2.35), one can derive the algebra of $c$ and $c^\dagger$. This
is more difficult than the reverse procedure which is the way this
subject really developed. Greenberg [14] proposed the $q$-mutator
algebra :
$$
c_i c^\dagger_j - q c^\dagger_j c_i \ = \ \delta_{ij} \eqno(3.19)
$$
\noindent The $q$-dependent $M$ matrices defined through (3.10), (3.15)
and (3.16) follow from this algebra. Fivel and Zagier then proved the
positivity of these $M$ matrices for $-1 < q < + 1$. Inspite of the fact
that we have not derived eq(3.19) from the $M$ matrices defined here, we
can assert its validity because the $M$ matrices completely determine
$c$ and $c^\dagger$. It is in order to emphasize this point that we have
presented the $M$ matrices first and the algebra of $c$ and $c^\dagger$
as a derived consequence.
\vspace{.5cm}
\noindent {\bf 3.3. New representations of infinite statistics}
First we briefly consider the $2$-parameter algebra
$$
c_ic^\dagger_j - q_1 c^\dagger_j c_i - q_2 \delta_{ij} \sum_k
c^\dagger_k c_k \ = \ \delta_{ij} \eqno(3.20)
$$
\noindent where $q_1$ and $q_2$ are real parameters. This may be
regarded as another representation of infinite statistics. Although the
determination of the full region in the $\{q_1,q_2\}$ parameter space
for which $M$ is positive definite is still an unsolved problem, one can
show [19] positivity of $M$ on the straight line defined by $q_1=0 ;
-1<q_2 < \infty$. Infact it is possible to map this whole line on to the
point $q_1=0 ; q_2=0$ so that we just get back the standard
representation defined by eq (3.5).
We next present two new algebras :
$$
c_i c^\dagger_j - c^\dagger_j c_i \ = \ \delta_{ij} p^{2\sum_{k<i}N_k}
p^{N_i} \eqno(3.21)
$$
\noindent and
$$ \left. \begin{array}{c}
c_i c^\dagger_j - p^{-1} c^\dagger_j c_i \ = 0 \quad {\rm for} \quad i \ne
j \\
\\
c_i c^\dagger_i - c^\dagger_i c_i \ = \ p^{N_i} \end{array} \right\}
\eqno(3.22)
$$
\noindent where $p$ is a real parameter and $N_i$ are number operators
already written down in Sec.2. Of these two algebras, the first one
(eq.3.21) is based on ordered indices, that
is, given any two indices $i$ and $j$, one must be considered larger or
smaller than the other. So, we may take the indices to be the natural
numbers $1,2,3\ldots$.
Eqs (3.21) and (3.22) are again representations of the same infinite
statistics, for it is possible to map both these algebras on to
Greenberg's $q$-mutator algebra of eq (3.19) with the following
identification of the parameters: $p=q^{-1}$. Temporarilly renaming the
$(c,c^\dagger)$ of eqs (3.21) and (3.22) by $(b,b^\dagger)$
and $(d,d^\dagger)$ respectively, the mapping or transformation
equations are
$$
b_i \ = \ p^{\sum_{k<i} N_k} p^{\frac{1}{2}N_i} c_i \eqno(3.23)
$$
$$
d_i \ = \ p^{\frac{1}{2}N_i} c_i\,. \eqno(3.24)
$$
\noindent Hence, it is clear that both the algebras of eqs (3.21) and
(3.22) lead to positive definite $M$ matrices for $p >+1$ and $p <-1$
(corresponding to $-1 < q < + 1$).
An interesting feature of the algebra (3.21) is the validity of ordinary
commutation relation between $c_i$ and $c^\dagger_j$ for $i \ne j$ as in
Bose statistics ; nevertheless the full algebra (3.21) describes
infinite statistics!
In contrast to the situation for the $q$-mutator algebra of eq (3.19), where
the number operators can be expressed in terms of $c$ and $c^\dagger$
only after considerable algebraic manouvers [16,17] and the resulting
expressions are quite complicated, the corresponding expressions for the
algebras (3.21) and (3.22) are simple. For (3.21), we get
$({\hbox{taking}} \ p>0)$
$$ \left. \begin{array}{lcl}
N_1 & = & \frac{1}{log \ p} \ log \ (c_1 c_1^\dagger - c^\dagger_1 c_1)
\\
& & \qquad \quad \vdots \\
N_i & = & \frac{1}{log \ p} \ log \ (c_i c_i^\dagger - c^\dagger_i c_i) - 2
\sum_{k<i} N_k \end{array} \right\} \eqno(3.25)
$$
\noindent For (3.22), the number operator is even simpler :
$$
N_i \ = \ \frac{1}{log \ p} \ log \ (c_i c_i^\dagger - c^\dagger_i c_i)
\quad {\rm for \ all} \quad i\,. \eqno(3.26)
$$
\noindent Inspite of our ability to write down such formal expressions for
$N_i$ in terms of $c$'s and $c^\dagger $s, we must also point out that
they are not of much use. All that one ever needs of the number
operators are the properties contained in the eqs (2.21) - (2.23) and
(2.26) and as for explicit representation, eqs (2.36) and (2.37) will
do.
\vspace{.5cm}
\noindent {\bf 3.4. $q$-mutators with complex $q$}
We now generalize Greenberg's q-mutator algebra to complex q. This
generalized algebra is based on ordered indices and is defined by the
following equations :
$$
c_i c^\dagger_j - q c^\dagger_j c_i = 0 \quad {\rm for} \quad i < j
\eqno(3.27)
$$
$$
c_i c^\dagger_i - p c^\dagger_i c_i = 1 \eqno(3.28)
$$
where q and p are complex and real parameters respectively and the
indices i,j etc. refer to any of the natural numbers 1,2,3, $\ldots$ .
The relation for the opposite order $i > j$ is derivable from (3.27) by
hermitian conjugation :
$$
c_i c^\dagger_j - q^\ast c^\dagger_j c_i = 0 \quad {\hbox{for}}
\quad i > j \eqno(3.27')
$$
\noindent and so it is not an independent relation.
Let us now calculate the inner product matrix for this algebra. For
distinct indices, we find
$$
\langle 1_g 1_h 1_k \ldots ; \mu \vert 1_g 1_h 1_k \ldots ; \nu \rangle
\,=\, (q^\ast)^{{J}_+} \, q^{J_ -} \eqno(3.29)
$$
\noindent where J$_+$ and J$_ {-}$ are the number of positive and
negative inversions in the permutation $\nu \rightarrow \mu$. The total
number of inversions (the sum of positive and negative inversions)
is the same as the number of inversions already defined below eq.(3.10).
We further define an inversion as positive if it is a
transposition of indices from the ascending order to the descending
order and as negative if it is the reverse transposition. For example,
(1,2) $\rightarrow$ (2,1) is a positive inversion while (2,1) $\rightarrow$
(1,2) is a negative inversion. Thus we have
$$
\langle 1_3 1_1 1_2 \vert 1_2 1_1 1_3 \rangle \,=\, (q^\ast)^2 q
\eqno(3.29')
$$
\noindent since the permutation (213) $\rightarrow$ (312) contains two
positive and one negative inversions as shown in Fig.1.
The relationship between the algebra defined by eqs(3.27) and (3.28)
with complex $q$ (and $p = |q|$) and Greenberg's algebra defined by
eq.(3.19) with real $q$ can be given. Calling the creation operators for
the former and latter algebras as $c^\dagger(q)$ and $c^\dagger(|q|)$
respectively, the relationship is
$$
c^\dagger_i(q) \ = \ e^{i\theta \sum_{k<i} N_k} c^\dagger_i(|q|)
\eqno(3.30)
$$
\noindent where $\theta$ is the phase of $q$ :
$$
q \ = \ |q| e^{i\theta} \eqno(3.31)
$$
\noindent and $N_k$ are the number operators defined in eq.(2.37).
As a consequence, the inner product matrices for the two algebras are
related by $\theta$-dependent unitary matrices:
$$
M(q) \ = \ T^\dagger(\theta) M (|q|) T(\theta) \eqno(3.32)
$$
\noindent where
$$
T^\dagger T \ = \ TT^\dagger \ = \ 1\,. \eqno(3.33)
$$
\noindent We have already given in eq.(3.13), Zagier's result
for the determinant of $M$ for real $q$. For our algebra with complex
$q$, because of eq.(3.32), the determinant of $M$ for $n$ particles with
distinct indices is
$$
{\rm det} M_n (q) \,=\, \prod^{n-1}_{k=1} (1-\mid q
\mid^{k(k+1)})^{\frac{(n-k)n!}{k(k+1)}} \eqno(3.34)
$$
\noindent Hence, $M_n(q)$ for complex $q$ is positive for $|q| <1$, and
we have thus extended the Fivel-Zagier result for positivity to complex
$q$.
In contrast to the situation for Greenberg's algebra, the inner product
matrix M for states with repeated indices cannot be derived from that
for states with distinct indices, for the present algebra. For
n-particle states with all the indices repeated, the norm is the same as
in eqn(3.17) and (3.18), with $q$ replaced by $p$ :
$$
\langle n_g \mid n_g \rangle \,=\, (1+p) (1+p+p^2) \ldots
(1+p+p^2+\ldots p^{n_g -1}) \eqno(3.35)
$$
\noindent which is positive for p $> -1$. For states with only some
indices repeated, M is a function of both q and p and the problem is
more complicated ; but we have verified positivity of the M's upto
3-particle states for $-1 < p < \mid q \mid^{-2}$.
If we choose $p = -1$, all states with repeated indices are forbidden.
This is just Pauli's exclusion principle. Thus, the algebra defined by
eqs.(3.27) and (3.28) with $p = -1$ leads to {\it infinite or Boltzmann
statistics with exclusion principle}.
Infact, we can restrict both eqs(3.5) and (3.19) to $i \ne j$ and for
$i = j$ replace them by
$$
c_ic_i^\dagger + c^\dagger_i c_i = 1 \eqno(3.36)
$$
\noindent In all these cases, we have Boltzmann statistics with
exclusion principle.
We give a compilation of the various algebras all representing infinite
statistics in Table I.
\noindent {\bf 3.5. Unitary transformations}
We must point out an important difference between the algebras
described by eqs(3.5), (3.19) and (3.20) on the one hand and the algebras
of eqs(3.21), (3.22), (3.27) and (3.28) on the other hand. It is easy to
see that the former are covariant under the unitary transformations on
the indices:
$$
c_k \ \rightarrow \ \sum_m \ U_{km} \ c_m \ ; \ U^\dagger U \
= \ UU^\dagger \ = \ 1\,. \eqno(3.37)
$$
\noindent In fact, one can show [19] that under certain
conditions eq.(3.20) is the most general bilinear algebra of $c_i$ and
$c^\dagger_j$ invariant under this unitary transformation.
This property is violated for the complex $q$-algebra of eqs(3.27) and
(3.28) as well as the algebras of eqs (3.21) and (3.22). Covariance
under the unitary
transformation is desirable in a general context since it is closely
connected to the superposition principle in quantum mechanics [3].
Nevertheless, algebras violating this requirement have been proposed in
the recent literature, either because of the possibility of applications
to specific systems in condensed matter physics or because of
mathematical motivation. Hence, we include such algebras in our
investigation.
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\noindent {\bf{4. Statistics and Algebras in Bosonic,
Fermionic and Frozen Fock Spaces}}
We introduce two types of reduced Fock spaces in sub section 4.1. In one,
we take any two states obtained by permutation to be related to each
other and we allow all occupation numbers : $n_i = 0,1,2, \ldots$ for
all $i$. In the other, any two states obtained by permutation are again
related to each other but occupation numbers are restricted by exclusion
principle : $ n_i = 0$ and 1 only. We shall call the former as bosonic
Fock space and the latter as fermionic Fock space.
In subsection 4.2, we introduce the Fock
space of frozen order, in which permutations are forbidden and
this also comes in two varieties, bosonic and fermionic.
\vspace{.5cm}
\noindent {\bf 4.1 Bosonic and Fermionic Fock Spaces}
A consistent way of defining a general relationship among permuted
states for both the bosonic and fermionic Fock spaces is to take
$$
\vert n_1, n_2 \ldots ; \mu \rangle \,=\, q^J \vert n_1, n_2 \ldots ;
1 \rangle \eqno(4.1)
$$
\noindent where $q$ is a complex number and $J$ is the number of
inversions in the permutation : $1 \rightarrow \mu$. The number of
inversions was already defined in Sec.3. Although $J$ was defined there
for the case in which the occupation numbers were restricted to unity,
the same definition can be extended to larger occupation numbers.
We assume $q \ne 0$. The case $q=0$ will be considered separately.
Many different forms of statistics as well as various algebras of
$c$ and $c^\dagger$ can be shown to be contained as particular cases of
eq.(4.1).
We first make some general remarks amplifying the meaning of eq(4.1).
This equation can be obtained by repeated application of the elementary
relation :
$$
| \ldots 1_i \ 1_j \ldots > \ = \ q | \ldots 1_j 1_i \ldots >\,, \quad
{\rm for} \quad i > j \eqno(4.2)
$$
\noindent where, except for the two adjacent quanta of indices $i$ and
$j$ which are interchanged, all other quanta are left unchanged. In
eq.(4.2), we have used the same notation for the state vectors as on the
right hand side of eqs(2.1)-(2.4). It is clear that, for any $q$ other
than $\pm 1$, eq.(4.2) makes sense only if the indices $i$ and $j$, are
ordered through an inequality (to be specific, $i > j$). Hence we have
taken the indices to be the natural numbers $1,2,3\ldots$in writing
eq.(4.1).
The relationship among the state vectors given in eq(4.1) or (4.2) can
be equivalently expressed as a quadratic relation between two creation
operators or two annihilation operators. For any state $|\ldots >$,
using eq.(2.30) or (2.31) we have
$$
c^\dagger_i c^\dagger_j | \ldots > \ = \ | 1_i 1_j \ldots >
$$
$$
c^\dagger_j c^\dagger_i | \ldots > \ = \ | 1_j 1_i \ldots >
$$
\noindent Comparing with eq.(4.2), we get
$$
c^\dagger_i c_j^\dagger | \ldots > \ = \ q c_j^\dagger c^\dagger_i |
\ldots >\,, \quad {\rm for } \quad i > j
$$
\noindent Since this is valid for any state $| \ldots >$, we can write
$$
c_i^\dagger c_j^\dagger - q c_j^\dagger c_i^\dagger \, = 0 \quad
{\rm for} \quad i > j
$$
\noindent or,
$$
c_j c_i - q^\star c_i c_j \ = \ 0 \quad
{\rm for} \quad i > j. \eqno(4.3)
$$
The above $cc$ relation (eq 4.3) is common to both bosonic and
fermionic spaces. But for the fermionic space there exists the additonal
restriction :
$$
c^\dagger_i c^\dagger_i \ = \ 0\,, \quad {\rm or} \quad c_i c_i \ = \ 0
\eqno(4.4)
$$
All the states $|n_1,n_2 \ldots ; \mu>$ for fixed occupation numbers
$(n_1,n_2 \ldots)$ but different values of $\mu$ being related to each
other through eq.(4.1), the dimension of the vector space in any sector
$(n_1, n_2 \ldots)$ is reduced to unity. So, for each sector we choose
one standard vector $|n_1,n_2 \ldots ; 1>$ which we rewrite as $|n_1,n_2
\ldots>$, dispensing with $\mu$ completely.
The matrices $X$ and $M$ then become numbers related by
$$
M^{-1} \ = \ X X^\star\,. \eqno(4.5)
$$
\noindent Eq.(2.11) becomes
$$
\parallel n_1, n_2 \ldots \gg \,=\, X(n_1, n_2 \ldots) \vert n_1, n_2
\ldots \rangle \eqno(4.6)
$$
\noindent and we have
$$
\ll n_1, n_2 \ldots \parallel n^{'}_1, n^{'}_2 \ldots \gg \,=\,
\delta_{n_1 n^{'}_1} \, \delta_{n_2 n^{'}_2} \ldots \eqno(4.7)
$$
\noindent From eq.(2.35), we get
$$
\begin{array}{lll}
c^\dagger_j & = & {\displaystyle{\sum_{n_1 \ldots n_j\ldots}}} \,
\vert 1_j, n_1, n_2 \ldots
n_j .. \rangle \, \langle n_1, n_2 \ldots n_j .. \vert \, M^{-1} (n_1,
n_2 ..) \\
& \\
&=& {\displaystyle{\sum_{n_1 \ldots n_j\ldots}}} \, q^{\sum_{i<j} n_i} \,
\vert n_1, n_2 \ldots
(n_{j}+1) .. \rangle \, \langle n_1, n_2 \ldots n_j .. \vert \,
M^{-1} (n_1, n_2 \ldots) \end{array}
$$
\noindent where we have pushed the $j$ quantum to the right of all $i$
quanta for $i < j$ using eq.(4.2). Also using eqs.(4.5) and (4.6),
$$
\begin{array}{lll}
c^\dagger_j = & {\displaystyle{\sum_{n_1 \ldots n_j}}} \,
q^{\sum_{i<j} n_i} \,{\displaystyle{ \frac{X(n_1 .. n_j
..)}{X(n_1 .. (n_{j}+1)..)}}} \, \parallel n_1, n_2 \ldots (n_{j}+1) .. \gg \,
\ll n_1, n_2 \ldots n_j .. \parallel
\end{array} \eqno(4.8)
$$
\noindent Eq.(4.8) and its hermitian conjugate define the most
general form of creation and
destruction operators for the bosonic and fermionic Fock spaces.
If we assume factorization of the norm M as well as that of X :
$$
X(n_1, n_2 \ldots) \,=\, \phi (n_1) \phi (n_2) \ldots, \eqno(4.9)
$$
\noindent then we get
$$
c^{\dagger}_j = \sum_{n_1 \ldots n_j \ldots} \, q^{\sum_{i < j}n_i} \,
\frac{\phi(n_j)}{\phi(n_j+1)} \, \parallel n_1 .. n_j +1 .. \gg \, \ll
n_1 .. n_j .. \parallel \eqno(4.10)
$$
$$
c_j = \sum_{n_1 \ldots n_j \ldots} \, (q^{\ast})^{\sum_{i < j}n_i} \,
\frac{\phi^\ast(n_j)}{\phi^\ast(n_j+1)} \, \parallel n_1 .. n_j .. \gg \, \ll
n_1 .. (n_j+1) .. \parallel \eqno(4.11)
$$
We now construct the operator algebras for c$^{\dagger}$ and c given by
eqs.(4.10) and (4.11). First of all, {\it for any form of $\phi(n_i)$}, we
get
$$
\left. \begin{array}{llr}
c_i c^{\dagger}_j - q c^{\dagger}_j c_i & = & 0 \quad for \quad i < j \\
& \\
c_i c^{\dagger}_j - q^\ast c^{\dagger}_j c_i & = & 0 \quad for \quad i > j
\end{array} \right\} \eqno(4.12)
$$
\noindent For i = j, the algebra depends on the choice of q and
$\phi(n_i)$. We find
$$
c_j c_j^\dagger \quad = \quad |q|^{2 \sum_{i<j} N_i} \left\vert
\frac{\phi(N_j)}{\phi(N_j+1)} \right\vert^2 \eqno(4.13)
$$
$$
c_j^\dagger c_j \quad = \quad |q|^{2 \sum_{i<j} N_i} \left\vert
\frac{\phi(N_j-1)}{\phi(N_j)} \right\vert^2 \eqno(4.14)
$$
\noindent where $N_i, N_j \ldots$ are number operators and $\phi(N_i)$
is the function introduced in
eq.(4.9). So, we can write the generalized commutation relation
$$ \left. \begin{array}{c}
c_j c_j^\dagger - p c^\dagger_j c_j \ = \ |q|^{2 \sum_{i<j} N_i} f(N_j)
\\
\\
f(N_j) \ = \ \left \vert \frac{\phi (N_j)}{\phi(N_j+1)} \right \vert^2
-p \left \vert \frac{\phi(N_j-1)}{\phi(N_j)} \right \vert^2 \end{array}
\right\} \eqno(4.15)
$$
\noindent where $p$ is any (complex) number. Eqs.(4.12) and (4.15) constitute
the $cc^\dagger$ algebra for $q$-statistics defined by the $cc$ algebra
of eq.(4.3).
What we have derived above can be regarded as the most general algebra
of creation and annihilation operators for $q$-statistics, subject to
the assumption of the factorization of the normalization factor implied
by eq.(4.9). In particular, since the function $\phi(N_j)$ occuring in
eq.(4.15) is arbitrary, we may regard our equations as constituting an
infinite-parameter deformation of the algebra of $c$ and $c^\dagger$.
Exploiting the arbitrariness of $\phi(N_j)$ we can get many simpler
forms of the algebra which we now describe.
\noindent {\bf (i)} Choosing $\phi(n_j)$ to be a constant independent of $n_j$
but noting that $\phi(-1)$ must vanish\footnote{On applying both sides
of eq.(4.14) on the vacuum state, the left hand side vanishes and so for
consistency, $\phi(-1)=0$}, eq.(4.15) becomes
$$
c_j c^\dagger_j - p c^\dagger_j c_j \ = \ |q|^{2 \sum_{i<j} N_i} \left\{
1-p(1-\delta_{N_j,0}) \right\} \eqno(4.16)
$$
\noindent where $\delta_{N_j,0}$ is the Kronecker delta, which can also be
represented as $sin \ \pi N_j / \pi N_j$.
\vspace{.5cm}
\noindent{\bf (ii)} Putting $p=0$ in eq.(4.16) we get the simple algebra :
$$
c_j c^\dagger_j \ = \ |q|^{2 \sum_{i<j} N_i} \eqno(4.17)
$$
\vspace{.5cm}
\noindent{(iii)} As a third possibility, we may choose $f(N_j)$ in eq.(4.15)
to be unity and $p$ real so that we have the algebra :
$$
c_j c^{\dagger}_j - p c^{\dagger}_j c_j \, = \, |q|^{2 \sum_{i<j} N_i}
\eqno(4.18)
$$
\noindent This choice requires that the function $\phi(n)$ must satisfy
the equation
$$
\left\vert \frac{\phi(n_j)}{\phi(n_j+1)} \right \vert^2 \ - \ p
\left\vert \frac{\phi(n_j-1)}{\phi(n_j)} \right\vert^2 \ = \ 1 \eqno(4.19)
$$
\noindent The solution of this functional equation (with the constraints
that $\phi(0)$ is finite and $\phi(-1)$ vanishes) is
$$
|\phi(n)|^{-2} \ = \ [(n)!]_p \equiv [n]_p [n-1]_p \ldots [1]_p
\eqno(4.20)
$$
\noindent where
$$
[n]_p \ = \ \frac{p^n-1}{p-1} \ = \ 1+p+p^2+\ldots p^{n-1} \eqno(4.21)
$$
\noindent For $p=1$, the ``p-numbers'' in
eqs(4.20) and (4.21) become ordinary numbers.
\vspace{.5cm}
\noindent {\bf (iv)} As a final choice we put $p=|q|^2$ in eqs (4.18) - (4.21).
As a consequence of this choice, the right hand side of eq (4.18)
becomes a bilinear function of $c$ and $c^\dagger$. For, from eq.(4.14),
we can prove an identity :
$$
(|q|^2-1) \sum_{k<j} c^\dagger_k c_k = \sum_{k<j} |q|^{2 \sum_{i<k}
N_i} \left \vert \frac{\phi(N_k-1)}{\phi(N_k)}\right \vert^2 (|q|^2-1)
$$
$$
= \sum_{k<j} |q|^{2 \sum_{i<k} N_i} (|q|^{2N_k}-1)
$$
$$
= |q|^{2 \sum_{i<j} N_i} -1 \eqno(4.22)
$$
\noindent where we have used eqs(4.20) and (4.21) with $p=|q|^2$. Hence
eq.(4.18) can be replaced by
$$
c_jc^\dagger_j - |q|^2 c^\dagger_j c_j \ = \ 1 + (|q|^2-1) \sum_{k<j}
c^\dagger_k c_k \eqno(4.23)
$$
Any one of the eqs(4.16), (4.17), (4.18) or (4.23) together with
eq.(4.12) constitute a $cc^\dagger$ algebra and one can construct many
more examples of $cc^\dagger$ algebras, all of which correspond to the
same $q$-statistics specified by eq.(4.3). We have so far assumed that
the occupation numbers are unrestricted : $n_j \ge 0$ and so these
algebras are valid for the bosonic $q$-statistics, namely $q$-statistics
in the bosonic space.
The fermionic $q$-statistics is defined by eqs(4.3) and (4.4). In this
case, $n_j$ in eqs(4.10) and (4.11) can take the value $0$ only, while
all the other occupation numbers will range over $0$ and $1$. So, the
only arbitrary parameter that enters the definition of $c$ and
$c^\dagger$ is $r = \phi(0)/\phi(1)$. The $cc^\dagger$ relation for $i
\ne j$ is still the same as in eq(4.12) while the $c_j c^\dagger_j$
relation of eq(4.15) can be simplified to
$$
c_jc^\dagger_j - p c^\dagger_j c_j \ = \ |q|^{2 \sum_{i<j} N_i} |r|^2
\left\{ \delta_{N_j,0} - p \delta_{N_j,1} \right \} \eqno(4.24)
$$
\noindent Further we can generally prove the fermionic analogue of
the identity which was earlier proved in eq.(4.22) for the bososnic
case only for a special choice of $\phi(n)$:
$$ \begin{array}{lcl}
(|q|^2-1) \sum_{k<j} c^\dagger_k c_k & = & (|q|^2-1) \sum_{k<j}
|q|^{2 \sum_{i<k} N_i} |r|^2 \delta_{N_k,1} \\
\\
& = & \sum_{k<j} |q|^{2 \sum_{i<k} N_i} (|q|^{2N_k}-1) |r|^2 \\
\\
& = & (|q|^{2 \sum_{i<j} N_i} -1)|r|^2 \end{array} \eqno(4.25)
$$
\noindent Hence, eq (4.24) can be rewritten as
$$
c_jc^\dagger_j - p c^\dagger_j c_j \ = \ \{ |r|^2 + (|q|^2 - 1)
\sum_{k<j} c^\dagger_k c_k\} (\delta_{N_j,0} - p \delta_{N_j,1})
\eqno(4.26)
$$
\noindent Eqs (4.12) and (4.26) constitute the general algebra for the
fermionic $q$-statistics. In contrast to the eq(4.15) of the bosonic
case, the fermionic case does not have the freedom of infinite - parameter
deformation.
On the righthand side of eq.(4.26), apart from the curly bracket \{~~\}
which is bilinear in $c$ and $c^\dagger$, there are kronecker deltas
which depend on the operator $N_j$. A simpler relation is obtained for
the choice $p = -1$, since $\delta_{N_j,0} + \delta_{N_j,1} \ = \ 1$. We
then have
$$
c_jc^\dagger_j + c^\dagger_j c_j \ = \ |r|^2 + (|q|^2 -1)
\sum_{k<j} c^\dagger_k c_k \eqno(4.27)
$$
\noindent A further choice $|r|^2 = 1$ gives
$$
c_jc^\dagger_j + c^\dagger_j c_j \ = \ 1 + (|q|^2 -1)
\sum_{k<j} c^\dagger_k c_k \eqno(4.28)
$$
\noindent which is the fermionic analogue of eq.(4.23).
The algebra given by eqs(4.3), (4.12) and (4.23) (for real $q$) are
covariant under {\it the quantum group} $SU_q(n)$ where $n$ is the total
number of indices. This is true of the fermionic algebra of eqs(4.3),
(4.12) and (4.28) also. See ref [20-31] for this relationship to the
theory of quantum groups. However, our approach based on states related
by $q$-statistics (eq(4.1)) does not involve any notion of quantum group
{\it per se} and so we shall not describe it. The formalism of the
generalized Fock space appears to be
capable of incorporating many general algebraic structures. In
particular, the $c_jc_j^\dagger$ algebra of eq(4.15) constitutes a
general infinite-parameter deformation while the quantum-group related
algebra of eq(4.23) is only a particular case of this.
We have constructed the general algebras for arbitrary complex $q$.
The choice of $q$ defines the symmetry of the state under permutation.
Hence we call the different choices of $q$ as different forms of
statistics. Particular values of $q$ such as $q = \pm 1$ or
$q = e^{i\theta}$ are of
special interest, although these cases are all contained in the general
formulae already given. We shall call the statistics obtained for $q =
+1, -1$ and $e^{i\theta}$ as Bose statistics, Fermi statistics or
``fractional'' statistics respectively. Either the bosonic or fermionic
Fock space can be used to construct any one of the various forms of
statistics. Thus, within our
terminology, it is perfectly possible for instance to have Fermi
statistics residing in bosonic Fock space or vice versa.
Within a particular statistics, different choices of $M$ or $\phi$
correspond to different representations of $c_j$ and $c^\dagger_j$,
which in turn lead to different $cc^\dagger$ algebras. If we assume the
factorization given in eq(4.9), for a given $q$, only the $i=j$ part of
the $cc^\dagger$ algebra depends on the representation.
For the sake of clarity, all these forms of statistics and algebras are
exhibited in Tables II and III for the bosonic and fermionic spaces
respectively.
Several comments are in order, concerning the contents of these tables.
The algebra
$$
c_ic_j - c_j c_i = 0, \quad {\rm for} \quad i \ne j \eqno(4.29)
$$
$$
c_ic^\dagger_j - c^\dagger_j c_i = 0, \quad {\rm for} \quad i \ne j
\eqno(4.30)
$$
$$
c_j c^\dagger_j \ = \ 1 \eqno(4.31)
$$
\noindent provide the simplest representation for Bose statistics based on
the simple choice of $M$ or $\phi$ as unity. This is analogous to the
standard representation (eq(3.5)) for infinite statistics but very
different from it. In fact they live in entirely different Fock spaces.
In recent literature, eq(4.31) has been sometimes confused with
infinite statistics. Eqs.(4.31) coupled with eqs(4.29) and (4.30) infact
describes Bose statistics only, although in a noncanonical
representation.
Attention may be drawn to the ``anticommuting
bosons'' and the ``commuting fermions'' shown in Tables II and III
respectively. The ``commuting fermions'' have been called hard-core
bosons in condensed-matter-physics literature. Our terminology seems
more appropriate since they are really fermions in disguise, living in
the fermionic Fock space.
The precise connection of our ``fractional'' statistics to the exchange
property of the wavefunctions proposed [39,40] for one and
two-dimensional systems needs further study. In particular, a suitable
mapping of the ordered indices to coordinates in one and two dimensions
is necessary.
Among all the different algebras given in Tables II and III, only two of
them, namely the canonical representations for the bosons and fermions
have the distinction of being covariant under the unitary
transformations that mix the indices (see eq.(3.37)).
All the other algebras violate this requirement, although, as we
have already mentioned, two of them, one in the bosonic and the
other in the fermionic Fock spaces, are covariant under the quantum
groups SU$_q$(n).
The examples contained in these tables can be called deformed-oscillator
algebras on which many papers [1-12,20-27,32,41-42] have been written
recently. Inspite of the multiplicity of these deformed algebras, we
must not ignore the basic fact that {\it they are all avatars of just two
primary constructions which may be taken to be the canonical bosonic
algebra for the bosonic Fock space and the canonical fermionic algebra for
the fermionic Fock space}. All the different forms of statistics
belonging to the bosonic Fock space as well as their various algebraic
representations are related to each other and to the canonical bosonic
algebra and similar is the situation
for the fermionic Fock space. Given two different forms of statistics
within the bosonic Fock space characterized by the statistics parameters
$q_1$ and $q_2$ respectively, or/and two different representations
characterized by the functions $\phi_1(n)$ and $\phi_2(n)$ respectively,
it is easy to get from (4.10) the relationship
$$
c^\dagger_j(2) \ = \ \left( \frac{q_2}{q_1}\right)^{\sum_{i<j} N_i}
\frac{\phi_2(N_j-1)}{\phi_2(N_j)} \frac{\phi_1(N_j)}{\phi_1(N_j-1)}
c^\dagger_j(1) \eqno(4.32)
$$
\noindent The corresponding equation for the fermionic Fock space is
$$
c^\dagger_j(2) \ = \ \left( \frac{q_2}{q_1}\right)^{\sum_{i<j} N_i} \left(
\frac{r_2}{r_1}\right) c^\dagger_j(1) \eqno(4.33)
$$
\noindent Hence, from the point of view of Fock space, there is nothing
new in all these deformed oscillators.
In the above equations, the number operator $N_i$ are given by
$$
N_i \ = \ \sum_{n_1 \ldots n_i \ldots} n_i \vert\vert n_1 \ldots n_i
\ldots \gg \ll n_1 \ldots n_i \ldots \vert\vert \eqno(4.34)
$$
\noindent and, written in this fashion in terms of the normalized states
$\vert\vert \gg$, {\it $N_i$ is the same in all the different forms of
statistics (within the bosonic or fermonic Fock spaces) and in all
the different representations}.
Using eq.(4.14) it is also possible to express the number
operators in terms of c and c$^{\dagger}$ in any statistics or any
representation within the bosonic or fermionic Fock space.
Noting that for the canonical bosonic algebra
$$
q = 1 \ ; \ \phi(n) = \frac{1}{(n!)^{1/2}} \quad , \eqno(4.35)
$$
\noindent we get from eq.(4.14)
$$
N_i = c^{\dagger}_i c_i. \eqno(4.36)
$$
\noindent For other statistics and representations also, the set of
equations obtained from eq.(4.14) starting with j $=$ 1 and
successively increasing j, can be seen to be implicit recursion formulae
for all $N_j$. Thus we have
$$ \left. \begin{array}{lcl}
{\displaystyle{c_1^\dagger c_1}} & = & \left\vert
{\displaystyle{\frac{\phi(N_1-1)}{\phi(N_1)}}} \right \vert^2 \\
{\displaystyle{c_2^\dagger c_2}} & = & |q|^{2N_1} \left\vert
{\displaystyle{\frac{\phi(N_2-1)}{\phi(N_2)}}} \right \vert^2 \\
\vdots & & \end{array}\right\}
\eqno(4.37)
$$
\noindent By choosing $\phi(N_j)$ one then gets the desired
explicit expressions for N$_j$. However, as already emphasized, such
expressions are not of much use; the universal representation of N$_j$
given in eq.(4.34) is sufficient for all purposes.
The formalism of Fock space developed here has sufficient flexibility so
as to allow many strightforward extensions or generalizations. For
instance, the relationship between permuted states defined by eqs(4.1)
or (4.2) can be generalized further. Instead of eq(4.3), we may take
$$
c_j c_i - q^\ast_{ij} c_i c_j \ = \ 0 \quad {\rm for} \quad i > j\,,
\eqno(4.38)
$$
\noindent where $q_{ij}$ are complex parameters. This may be called {\it
multiparameter $q$-statistics}. The algebras corresponding to such
extensions as well as further generalizations can all be constructed
essentially by the same procedure as given in this section. Details will
be presented elsewhere. Suffices it to say that, once the relationship
among the permuted states is specified through the $cc$ relation, the
rest of the story follows.
Special cases of such algebras have been reported in the literature on
quantum groups [20-33]. We have already referred to the generality of our
approach as compared to quantum groups. As further points of comparison
between the two approaches we must mention the following. From the point
of view of quantum groups, the whole set of relations among $c$ and
$c^\dagger$ are taken for granted. In contrast, our analysis based on the
underlying Fock space reveals the $cc$ relation as the key to the whole
algebraic structure, although the normalization function $\phi(n)$ also
plays a role in determining the actual representation of the operators.
Hence, depending on the mode of expressing the $cc$ relation and the
choice of $\phi(n)$, one can generate any number of algebras of $c$ and
$c^\dagger$. Thus, our approach helps to demystify the quantum - group
related algebras by reducing them to their basic essentials which are
identified to be simple properties of state vectors in the Fock space.
Reversing the procedure, one can probably reconstruct the whole edifice
of the quantum group itself starting from the more elementary notions
relating to states in the Fock space. This of course lies outside the
purview of the present paper.
\vspace{.5cm}
\noindent {\bf 4.2 Fock space of frozen order and null statistics}
In subsection 4.1 we had taken any two states obtained by
permutation to be related to each other. We now consider a limiting
situation in which the particles are frozen in a particular order, with
no permutation allowed. This is the {\it Fock space of frozen
states } and the asosciated statistics is {\it the statistics of frozen
states}. Whereas in infinite statistics each permutation leads to a new
state, in the statistics of frozen states no permutation is allowed. To
emphasize this contrast, the latter may be called {\it null statistics}.
Although this may be obtained as the limit $q \rightarrow 0$ of
the $q$-statistics, more properly, it must be regarded as an independent
statistics. For, whereas the statistics for two (non-zero) values of $q$
are related to each other through eqs(4.32) or (4.33), the statistics
for $q=0$ cannot be related to statistics for $q\ne 0$. Hence, we shall
construct the algebra of $c$ and $c^\dagger$ for this system of frozen
states directly from the definition. Again, depending on whether the
particles obey exclusion principle or not, we have two different
versions of the system which we shall call the fermionic and the bosonic
versions respectively.
Let us first consider the bosonic version. Referring to eq(4.1) the
Fock space of frozen states is obtained by taking $|n_1 n_2 \ldots ; 1>$
as the allowed state and requiring
$$
|n_1 n_2 \ldots ; \mu> \ = \ 0 \quad {\rm for} \quad \mu \ne 1
\eqno(4.39)
$$
\noindent Or, equivalently
$$
| \ldots 1_j 1_i \ldots > \ = \ 0 \quad {\rm for} \quad i < j
\eqno(4.40)
$$
\noindent and hence
$$
c_i c_j \ = \ 0 \quad {\rm for} \quad i < j \eqno(4.41)
$$
Assuming factorization of the norm as in eq(4.9), we can represent the
creation operator in the form :
$$
c^\dagger_j \ = \ \sum_{n_j, n_{j+1} \ldots}
\frac{\phi(n_j)}{\phi(n_j+1)} \vert \vert 0_1, \ldots 0_{j-1},
(n_j +1), n_{j+1} \ldots \gg \ll 0_1, \ldots 0_{j-1}, n_j, n_{j+1}
\ldots \vert \vert \eqno(4.42)
$$
\noindent The zeroes in the state vectors arise from the fact that
$c^\dagger_j$ can create a quantum with index $j$ only if indices $i<j$
are unoccupied. From eq(4.41) and the orthogonality of the states, we
get
$$
c_k c^\dagger_j \ = \ 0 \quad {\rm for} \quad k \ne j \eqno(4.43)
$$
\noindent The $c_jc^\dagger_j$ algebra will depend on the choice of
$\phi(n_j)/\phi(n_j+1)$. We shall assume $\phi(n_j) = \phi(n_j+1)$ for
simplicity. Then, by using the completeness relation for the states, one
can verify
$$
c_j c^\dagger_j \ = \ 1 - \sum_{k < j} c^\dagger_k c_k \eqno(4.44)
$$
\noindent Eqs (4.41), (4.43) and (4.44) together define the algebra for
the bosonic version of the statistics of frozen order.
For the fermionic version, eq(4.41) is replaced by
$$
c_i c_j \ = \ 0 \quad {\rm for} \quad i \le j \eqno(4.45)
$$
\noindent and one again gets
$$
c_k c^\dagger_j \ = \ 0 \quad {\rm for} \quad k \ne j \eqno(4.46)
$$
\noindent but, instead of eq (4.44), one finds
$$
c_j c^\dagger_j \ = \ 1 - \sum_{k \le j} c^\dagger_k c_k
\eqno(4.47)
$$
\noindent Thus, the fermionic version of the statistics of frozen states
is described by the algebra of eqs(4.45)-(4.47). It is interesting to
note that the replacement of the sign $<$ in the bosonic algebra by the
sign $\le$ yields the fermionic algebra.
Finally we note that the expression for the number operators in terms of
$c$ and $c^\dagger$ for the above algebra of null statistics can be
shown to be essentially the same as that for the standard representation
of infinite statistics (eq 3.9a), but because of eqs (4.41) or (4.45),
it can be simplified to read as
$$
N_k = \sum_{n_1,n_2\dots n_k} c_1^{\dagger n_1}
c_2^{\dagger n_2} \ldots c_k^{\dagger n_k} c_k^{\dagger} c_k
c_k^{n_k} \ldots c_2^{n_2} c_1^{n_1}
\eqno(4.48)
$$
\noindent where the sum over $n_1,n_2 \ldots n_k$ is unrestricted $(\ge
0)$ for the bosonic version, but is restricted for the fermionic
version : $n_1,n_2 \ldots n_{k-1}$ go over 0 and 1 while $n_k = 0$.
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\noindent {\bf {5. Derivation of cc relations from $cc^\dagger$
algebras}}
In the last section we showed how to construct the bilinear algebra of c
and c$^{\dagger}$ starting from states related by eq.(4.1). In other
words, cc$^{\dagger}$ algebra has been derived from the cc algebra given
by eq.(4.3). The converse is also true ; the cc algebra can be derived
from the cc$^{\dagger}$ algebra, as will be shown in this section.
Thus, within the
framework of Fock space it is unnecessary to give both cc and
cc$^{\dagger}$ relations. Either the $cc$ relation or the $cc^\dagger$
relation can be used to define the Fock space and the other can be
derived. But there are some caveats:
\noindent (i) Although the $cc$ relation does define the Fock space, the
$cc^\dagger$ algebra does not follow uniquely ; as already pointed out,
the operators $c,c^\dagger$ and their algebras depend on the choice of
$M$.
\noindent (ii) In order to define the Fock space completely, the
$cc^\dagger$ algebra must fulfill certain conditions. The $cc^\dagger$
algebra must be such that an arbitrary inner product $<0|c_ic_j \ldots
c^\dagger_g c^\dagger_h |0>$ or infact the vacuum matrix element of any
polynomial in the $c$'s and $c^{\dagger}$ 's arbitrarily ordered, can be
calculated using the $cc^\dagger$ algebra and the definition of the
vacuum state $|0>$ :
$$
c_k |0> \ = \ 0 \quad {\rm for\ all } \quad k\,. \eqno(5.1)
$$
\noindent A general form of the $cc^\dagger$ algebra that satisfies this
requirement is :
$$
c_i c^\dagger_j \ = \ A_{ij} + \sum_{k,m} B_{ijkm} c^\dagger_k c_m
\eqno(5.2)
$$
\noindent where $A_{ij}$ and $B_{ijkm}$ can be constants, but more
generally they can be functions of the number operators.
All the $cc^\dagger$ algebras considered in this
paper are of this form.
There exists an elegant method [19] to derive cc relations from
the $cc^{\dagger}$ algebras. Restricting ourselves to relations quadratic in
$c$, we define
$$
Q_{ij} \, = \, c_i c_j - q^{'} c_j c_i \eqno(5.3)
$$
\noindent where q$^{'}$ may be an arbitrary complex parameter. Suppose
that, by using the given cc$^{\dagger}$ algebra, we are able to prove
$$
Q_{ij} c^{\dagger}_k \,=\, \sum_{i'j'k'} \, F_{ijk; i'j'k'} \,
c^{\dagger}_{k'} \, Q_{i'j'} \eqno(5.4)
$$
\noindent for all i,j and k where F$_{ijk;i'j'k'}$ may be a c-number or
operator. Then, by applying this equation successively, we get
$$
Q_{ij} c^{\dagger}_k c^{\dagger}_m \ldots \,=\, \sum_{i'j'k'} \,
\sum_{i''j''m'} \, F_{ijk;i'j'k'} \, F_{i'j'm; i''j''m'} \, c^{\dagger}_{k'}
\, c^{\dagger}_{m'}\ldots Q_{i''j''} \eqno(5.5)
$$
\noindent Allowing both sides of this equation to act on $\vert 0 >$, the
right side vanishes and so we
see that $Q_{ij}$ acting on any Fock state $c^{\dagger}_k c^{\dagger}_m
\ldots \vert 0 >$ gives zero. Hence we may write the operator identity
:
$$
Q_{ij} \,=\, 0 \eqno(5.6)
$$
\noindent which is the cc relation sought after. It may also be pointed
out that often one finds
$$
F_{ijk;i'j'k'} \, =\, f_{ijk} \delta_{ii'} \, \delta_{jj'} \,
\delta_{kk'} \eqno(5.7)
$$
\noindent so that eq.(5.4) is simplified to
$$
Q_{ij} \, c^{\dagger}_k \,=\, f_{ij k} \, c^{\dagger}_k Q_{ij}
\eqno(5.8)
$$
\noindent Thus, the form-invariance of $Q_{ij}$ on being pushed to the
right of $c^{\dagger}_k$, as explicitly given in eqs.(5.4) or (5.8) is the
necessary and sufficient condition for the existence of the $cc$
relation. We may now
apply this method on various $cc^\dagger$ algebras discussed in the
previous sections.
\noindent {\bf{5.1 $q$-mutator algebra with real $q$}}:
The $cc^\dagger$ algebra is
$$
c_i c^{\dagger}_j - q c^{\dagger}_j c_i \, = \, \delta_{ij} \quad ,
\quad \forall i,j. \eqno(5.9)
$$
\noindent We define
$$
Q_{ij} \,=\, c_i c_j - q' c_j c_i \quad , \quad \forall i,j \eqno(5.10)
$$
\noindent From eq.(5.9),
$$
Q_{ij} c^{\dagger}_k \,=\, (1-qq') c_i \delta_{kj} + (q-q') c_j
\delta_{ki} + q^2 c^{\dagger}_k Q_{ij} \eqno(5.11)
$$
\noindent The form-invariance of $Q_{ij}$ requires
$$
q' = q = \pm 1. \eqno(5.12)
$$
\noindent So, we get the cc relations
$$
c_i c_j \pm c_j c_i = 0 \eqno(5.13)
$$
\noindent corresponding to Bose and Fermi statistics ${\rm
at} \, q = \pm 1$. For $-1 < q < 1$, there are no $cc$ relations and
we have infinite statistics. The inner product matrix $M$ which remains
positive definite for $-1 < q < 1$, develops zero eigenvalues at $q=\pm
1$. (see eq.(3.13)) corresponding to the null states such as
$(c^\dagger_i c^\dagger_j \pm c^\dagger_j c^\dagger_i)|0>$ arising from
eq.(5.13).
\noindent {\bf{5.2 $q$-mutator algebra with complex $q$}}
The $cc^\dagger$ relations are
$$
c_i c^{\dagger}_j - q c^{\dagger}_j c_i = 0 \quad {\rm for} \, i <
j \eqno(5.14)
$$
$$
c_i c^{\dagger}_i - p c^{\dagger}_i c_i = 1 \eqno(5.15)
$$
\noindent We define
$$
Q_{ij} \, = \, c_i c_j - q^{'} c_jc_i \quad {\rm for} \quad i < j
\eqno(5.16)
$$
$$
\, = \, c_i c_i \quad {\rm for} \quad i = j \eqno(5.17)
$$
\noindent Using eqs.(5.14) and (5.15), we find
$$
\left. \begin{array}{lcl}
Q_{ij} c^{\dagger}_k & = & q^2 c^{\dagger}_k Q_{ij} \, \quad for \,
i \le j < k \\
&\\
& = & q^{\ast^2} c^{\dagger}_k Q_{ij} \, \quad for \, k < i \le j \\
&\\
& = & \vert q \vert^2 c^{\dagger}_k Q_{ij} \, \quad for \, i < k < j \\
&\\
& = & (q^{\ast} - q^{'}) c_j + pq^{\ast} c^{\dagger}_i Q_{ij} \, \quad for
\, k = i < j \\
&\\
& = & (1-qq^{'}) c_i + pq c^{\dagger}_j Q_{ij} \, \quad for \, i < j = k \\
&\\
& = & (1+p) c_i + p^2 c^{\dagger}_i Q_{ii} \, \quad for \, i = j = k.
\end{array} \right\} \eqno(5.18)
$$
\noindent The form-invariance of $Q_{ij}$ for $i < j$ requires
$$
q^{'} \, = \, q^{\ast} = e^{-i\theta} \eqno(5.19)
$$
\noindent where $\theta$ is real. We thus get the cc relation
corresponding to fractional statistics at the boundary $\vert q \vert =$
1 of the disc $\vert q \vert < 1$ in the complex plane.
We already know that the disc $|q|<1$ corresponds to the region of
positive-definite inner product matrix $M$ for the complex $q$-mutator
algebra of infinite statistics. At the boundary of this positivity
region, we have fractional statistics living in a reduced Fock space
(see Fig.2). It is the states of vanishing norm of the form
$$
( \ldots c^{\dagger}_i c^{\dagger}_j \ldots \vert 0 > ) \quad - e^{i
\theta} (\ldots c^{\dagger}_j c^{\dagger}_i \ldots | 0 > ) \quad {\rm
for} \quad i < j \eqno(5.20)
$$
\noindent which contribute to the zeroes of det M$_n$ in eq.(3.34) at
$\vert q \vert = $ 1. Once we remove these states, we get the reduced
Fock space of positive-definite norm.
The form-invariance of $Q_{ij}$ \, for i = j requires
$$
p = -1 \eqno(5.21)
$$
\noindent with the corresponding cc relation
$$
c_i c_i \, = \, 0 \eqno(5.22)
$$
\noindent This leads to the exclusion principle namely $n_i = 0$ or 1 only ;
equivalently, the norms of states with repeated indices are zero as seen
by putting $p = -1$ in eq.(3.35).
To sum up, we may note four possibilities all contained in the algebra
of eqs (5.14) and (5.15) :
\noindent (a) Infinite statistics with multiple occupation
$$
\vert q \vert \neq 1 \quad ; \quad p \neq - 1. \eqno(5.23)
$$
\noindent (b) Infinite statistics with exclusion principle
$$
\vert q \vert \neq 1 \quad ; \quad p = -1 \eqno(5.24)
$$
\noindent (c) Fractional statistics with multiple occupation
$$
q = e^{i\theta} \quad ; \quad p \neq -1 \eqno(5.25)
$$
\noindent (d) Fractional statistics with exclusion principle
$$
q = e^{i \theta} ; p = - 1. \eqno(5.26)
$$
\noindent {\bf{5.3 $q$-statistics }}
The cc$^{\dagger}$ relations are
$$
c_i c^{\dagger}_j - q c^{\dagger}_j c_i \, = \, 0 \quad {\rm for} \quad
i < j \eqno(5.27)
$$
$$
c_j c^{\dagger}_j - p c^{\dagger}_j c_j \, = \, |q|^{2
\sum_{i<j} N_i} f(N_j) \eqno(5.28)
$$
\noindent We define
$$
Q_{ij} \, = \, c_i c_j - q' c_j c_i \quad {\rm for} \quad i > j
\eqno(5.29)
$$
\noindent Using eqs.(5.27), (5.28) we get
$$
\left. \begin{array}{lcl}
Q_{ij} c^{\dagger}_k & = & q^2 c^{\dagger}_k Q_{ij} \quad
{\rm for} \quad i < j < k \\
& \\
& = & q^{\ast^2} c^{\dagger}_k Q_{ij} \, \quad {\rm for} \quad k < i < j \\
& \\
& = & \vert q \vert^2 c^{\dagger}_k Q_{ij} \, \quad {\rm for} \quad
i < k < j \\
& \\
& = & p q^\ast c^{\dagger}_j Q_{ij} + c_i |q|^{2\sum_{m < j}
N_m} f(N_j) (1-q'q^\ast) \quad {\rm for} \quad i > j = k \\
& \\
& = & pq c^{\dagger}_i Q_{ij} + \vert q\vert^{2\sum_{m < i}N_m} f(N_i)
c_j(q-q'|q|^2) \quad {\rm for} \quad k = i > j
\end{array} \right\} \eqno(5.30)
$$
\noindent Repeating the argument of form-invariance of $Q_{ij}$,
we conclude
$$
c_i c_j - q^{\star^{-1}} c_j c_i = 0 \quad {\rm for} \quad i > j
\eqno(5.31)
$$
\noindent In this proof we have not used any particular form of $f(N_j)$,
but have used only the standard commutation relations among $N_k$ and
$c_i$ (eqs(2.23) and (2.26).
This derivation of the $c_ic_j$ relation for $i \ne j$ is valid for the
bosonic as well as the fermionic fock spaces. However, for the fermionic
space, there exists the additional relation :
$$
c_ic_i \ = \ 0 \eqno(5.32)
$$
To derive this, we now commute $c^\dagger_k$ through $c_ic_i$
using the same eqs(5.27 - 5.28), we get
$$ \left. \begin{array}{lcl}
c_ic_ic^\dagger_k & = & q^{\ast 2} c^\dagger_k c_i c_i \quad {\rm for}
\quad i > k \\
& = & q^2 c^\dagger_k c_i c_i \quad {\rm for} \quad k > i \\
\\
& = & p^2 c^\dagger_i c_i c_i + |q|^{2 \sum_{m<i} N_m} c_i \{ pf(N_i-1) +
f(N_i)\} \ {\rm for} \ i=k \end{array} \right\} \eqno(5.33)
$$
\noindent So, for the validity of eq(5.32), we require
$$
c_i\{pf(N_i-1) + f(N_i)\} \ = \ 0 \eqno(5.34)
$$
\noindent Substituting the form of $f(N_i)$ for the fermionic Fock space
(from eq.(4.24))
$$
f(N_i) \ = \ |r|^2 (\delta_{N_i,0} - p \delta _{N_i,1})\,,
\eqno(5.35)
$$
\noindent we see that eq.(5.34) is satisfied, since in the fermionic
Fock space, $\delta_{N_i,2} = 0$ and $c_i\delta_{N_i,0} = 0.$
One may also note that eq.(5.34) is also satisfied for $p=-1$ and
$f(N_i)$ = constant (which may be chosen to be unity). So, for bosonic
Fock space, i.e. $c_ic_i \ne 0$, we must avoid $p=-1$ in eq.(5.28) and
(4.15).
The above derivations of $cc$ relations have used the general
$cc^\dagger$ algebra and hence includes the cases of
quantum-group-covariant algebras, independent deformed oscillators,
commuting fermions, anticommuting bosons etc. Further, it must be noted
that the presence of the term with the factor
$|q|^{2 \sum_{i<j} N_i}$ in eq.(5.28) is crucial for
the validity of the $cc$ relations ; without this factor, there will be
no $cc$ relation and we will get infinite statistics.
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\noindent {\bf 6. Two-indexed Systems}
\vspace{.5cm}
We have so far considered generalized Fock spaces consisting of states $|n_g,
n_h, \ldots ; \mu >$ where the indices $g,h,\ldots$ may refer either to
a single quantum number, or to a collection of quantum numbers,
specifying the space, spin and other internal degrees of freedom. In the
latter case, it may be supposed that one has mapped a collection of indices
to a single index. What we are envisaging in this
section are situations where such mapping is not possible. This can
happen in various ways. To be specific, let us consider oscillators with
a pair of indices, a latin index $(g,h,\ldots)$ and a greek index
$(\alpha, \beta \ldots)$. There exists a class of systems in which the
occupation numbers with a single index $n_g$ or $n_\alpha$ are defined,
but occupation numbers with both the indices $n_{g\alpha}$ are not
defined. Such systems cannot be mapped into single-indexed systems. In
another class of systems, $n_{g\alpha}$ do exist, but the subsidiary
conditions that define the reduced Fock space depend on the two indices
$g$ and $\alpha$ in such a way that prevents mapping of $(g,\alpha)$
into a single index. We consider these two classes of systems in
subsections 6.1 and 6.2 respectively.
It may also be mentioned that we first encountered such systems in the
study of the Hubbard model in the limit of infinite Coulomb repulsion
[43-46]. Although this was our original motivation, this has now opened the
door to a more general framework encompassing novel forms of statistics
and algebras.
\vspace{1cm}
\noindent {\bf 6.1 \ Systems in which $n_{g\alpha}$ do not exist}
We specify the states as $|n_g,n_h\ldots ; n_\alpha, n_\beta \ldots ;
\mu>$, where $n_g, n_h$ $\ldots$ are the numbers of quanta with indices $g,h$
$\ldots$ respectively while $n_\alpha, n_\beta, \ldots$ are the numbers of
quanta with indices $\alpha, \beta\ldots$ respectively and we have the
constraint :
$$
n_g + n_h + \ldots \ = \ n_\alpha + n_\beta + \ldots \eqno(6.1)
$$
\noindent We may regard $n_g$ as the total number of quanta with index
$g$ whatever may be their greek index and similarly for $n_\alpha$.
In such a state, the occupancy number with both indices such as
$n_{k\alpha}$ is not defined. The latin indices and the greek indices
are independently permutated and this leads to a much enlarged space in
each sector. Now $\mu$ goes over $1 \ldots s'$ where
$$
s' \ = \ \frac{(n_g + n_h + \ldots)!}{n_g ! n_h! \ldots } \times
\frac{(n_\alpha + n_\beta + \ldots)!}{n_\alpha ! n_\beta! \ldots } \eqno(6.2)
$$
\noindent This is to be compared to the size of the space of the states
$|n_{g\alpha}, n_{h\beta} \ldots ; \mu>$ for which the range of $\mu$ is
$$
s \ = \ \frac{(n_{g\alpha} + n_{h\beta} + \ldots)!}{n_{g\alpha} !
n_{h\beta }! \ldots } \eqno(6.3)
$$
\noindent In general, $s'$ is larger than $s$.
Let us consider the two-particle sector as an example. If we specify
occupation numbers with both indices, we have the two states in the
sector $(1_{k\alpha}, 1_{m\beta}) \ : \ | 1_{k\alpha}, 1_{m\beta} ; \mu>$
with $\mu = 1 $ or $2$ which correspond to $|1_{k\alpha}, 1_{m\beta}>$
and $|1_{m\beta}, 1_{k\alpha}>$ and another two states in the sector
$(1_{m\alpha}, 1_{k\beta}) : | 1_{m\alpha}, 1_{k\beta} ; \mu>$ with $\mu
= 1$ or $2$ corresponding to $|1_{m\alpha}, 1_{k\beta}>$ and
$|1_{k\beta}, 1_{m\alpha}>$. (Here we are not using $k\alpha, m\beta$ etc
in their generic sense ; they are used to denote specific values of the
indices.) On the other hand, with the new type of states with decoupled
indices for which
occupation numbers with both indices do not exist, we have the
two-particle states $|1_k, 1_m ; 1_\alpha, 1_\beta ; \mu>$ and now $\mu$
goes over 1 to 4 corresponding to the four state vectors $|1_{k\alpha},
1_{m\beta}>, \linebreak |1_{m\beta}, 1_{k\alpha}>, |1_{m\alpha}, 1_{k\beta}>,
|1_{k\beta}, 1_{m\alpha}>$. Thus, both the sectors considered earlier
are combined to form a single enlarged sector. Such a regrouping of
sectors with consequent enlargement occurs throughout the Fock space,
in the case of the decoupled indices.
The construction of the orthonormal set as well as the other properties
of the generalized Fock spaces given in Sec.2 goes through for the present
case of decoupled indices also except that the matrices $X,M$ etc will
be of higher dimensions. The creation and destruction operators
$c^\dagger_{k\alpha}, c_{k\alpha}$ also can be constructed in the same
way as in Sec.2. The relevant equations and formulae with the
appropriate changes incorporated are given in Appendix A. They are
self-explanatory.
Just as in the case of single-indexed systems (cf.Sec.2), we can again
have a super Fock space in which all the states connected by independent
permutation of the latin and greek indices are taken to be independent.
We shall call the associated statistics as ``doubly-infinite''
statistics since it is infinite statistics in latin and greek indices
separately. By imposing relations among the permutated states, one can
get many kinds of reduced Fock spaces. Because of the larger number of
available states in each sector, many new types of statistics become
possible. These can be discussed and the associated algebras can be
constructed by the same procedure as in Sec.3, 4 and 5. However, we
shall be brief and restrict ourselves to presenting some of the
important results only. Some of these algebras and the new kinds of
statistics implied by them have been discussed by us in greater detail
in the earlier papers [18,19,46].
\noindent {\bf Doubly-infinite statistics}
Consider the $cc^\dagger$ algebra described by
$$
c_{k\alpha} c^\dagger_{m\beta} - q \delta_{\alpha \beta} \sum_\gamma
c^\dagger_{m\gamma} c_{k\gamma} \ = \ \delta_{km} \delta_{\alpha\beta}
\eqno(6.4)
$$
\noindent where $q$ is a real parameter lying in the range
$$
-1 < q < 1\,. \eqno(6.5)
$$
\noindent It is the second term on the left of eq(6.4) in which $\alpha$
and $\beta$ have been dissociated from $k$ and $m$ respectively that
leads to the decoupling of the latin and greek indices and prevents the
mapping of lattin and greek indices to a single index. One can show
[19] that the inner-product matrices following from this algebra are
all positive definite for $-1<q<1$. Further, one can show using
arguments similar to those in Sec.5 that there is no $cc$ relation for
this algebra in the same parameter range and so all the states connected
by {\it independent} permutation of the latin and greek indices are
independent. The underlying Fock space is the full super Fock space of
dimension given by eq.(6.2) and we have infinite statistics in latin and
greek indices separately, or doubly-infinite statistics.
For $q=0$ in eq(6.4), the two indices can be mapped into a single index
and the algebra reduces to the standard representation of single-indexed
infinite statistics described in Sec.3.1.
The algebra of eq.(6.4) is covariant under the unitary
transformations on the latin indices :
$$
d_{k\alpha} = \sum_m U_{km} c_{m\alpha} \ ; \ U^\dagger U \ = \
UU^\dagger \ = \ 1 \eqno(6.6)
$$
\noindent as well as under separate unitary transformations on the greek
indices :
$$
e_{k\alpha} \ = \ \sum_{\lambda} V_{\alpha\lambda} c_{k\lambda} \ ; \
V^\dagger V \ = \ VV^{\dagger} = 1. \eqno(6.7)
$$
\noindent But the algebra is not covariant under the enlarged unitary
transformations involving both the latin and greek indices (for $q \ne
0$). A special case of the unitary transformations of eqs(6.6) and (6.7)
is the phase transformation. Eq.(6.4) is covariant under either of the
following phase transformations :
$$
d_{k\alpha} \ = \ e^{i \phi_k} c_{k\alpha} \eqno(6.8)
$$
$$
e_{k\alpha} \ = \ e^{i \phi_\alpha} c_{k\alpha} \eqno(6.9)
$$
\noindent As a consequence, the number operators $N_k$ and $N_\alpha$
exist. However, eq(6.4) is not covariant under the transformation :
$$
f_{k\alpha} \ = \ e^{i \phi_{k\alpha}} c_{k\alpha} \eqno(6.10)
$$
\noindent and correspondingly, $N_{k\alpha}$ does not exist.
We can make $q$ in (6.4) complex provided we order the latin indices and
thus we get an algebra which is the analogue of the $q$-mutator algebra
with complex $q$ for the single-indexed systems (Sec.3) :
$$
c_{i\alpha} c^\dagger_{j\beta} - q \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{j\gamma} c_{i\gamma} \ = 0 \quad {\rm for} \quad i < j
\eqno(6.11)
$$
$$
c_{j\alpha} c^\dagger_{j\beta} - p \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{j\gamma} c_{j\gamma} \ = \ \delta_{\alpha\beta} \eqno(6.12)
$$
\noindent where $q$ is complex, but $p$ is real. This again describes
the same doubly-infinite statistics ;
only the representation and algebra are different. This algebra is no
longer covariant under the unitary transformations of eqs (6.6) and
(6.7), but is still covariant under the phase transformations of eqs
(6.8) and (6.9). Positivity of the inner-product matrices $M$ requires
$|q|<1$. As for $p$, similar statements as in Sec.3 can be made.
Coming back to eq(6.4), at the boundary of the range of $q$ given in
eq(6.5), namely at $q = \pm 1$, we get two new forms of statistics
called orthobose and orthofermi statistics [18,19,46] which reside in
reduced Fock spaces defined by $cc$ relations. The algebras for these
are given below :
\noindent {\bf Orthobose Statistics}
$$
c_{k\alpha} c^\dagger_{m\beta} - \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{m\gamma} c_{k\gamma} \ = \ \delta_{km} \delta_{\alpha\beta} \eqno(6.13)
$$
$$
c_{k\alpha} c_{m\beta} - c_{m\alpha} c_{k\beta} \ = \ 0\,.
\eqno(6.14)
$$
\noindent {\bf Orthofermi Statistics}
$$
c_{k\alpha} c^\dagger_{m\beta} + \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{m\gamma} c_{k\gamma} \ = \ \delta_{km}
\delta_{\alpha\beta} \eqno(6.15)
$$
$$
c_{k\alpha} c_{m\beta} + c_{m\alpha} c_{k\beta} \ = \ 0\,.
\eqno(6.16)
$$
The inner product matrices $M$ for the reduced Fock spaces generated by
these algebras (eqs(6.13) - (6.16)) can be shown to be positive. In both
these statistics, any two states obtained by permuting the greek indices
are independent. On the other hand, states obtained by permuting the
latin indices are either equal to each other, or related by $(-1)^J$
where $J$ is the number of inversions in the permutation of the latin
indices, for the orthobose or orthofermi statistics respectively. Thus,
we have infinite statistics in the greek indices and bose or fermi
statistics in the latin indices. For orthofermi statistics, we have the
further condition :
$$
n_k \ = \ 0 \ {\rm or} \ 1 \ {\rm only}\,. \eqno(6.17)
$$
\noindent It must be noted that the exclusion implied by eq(6.17) is
stronger than the usual Pauli exclusion principle ; eq(6.17) requires
that there cannot be more than one particle with index $k$ whatever may
be its greek index.
Orthostatistics can be generalised to $q$-orthostatistics. We keep the
infinite statistics in the greek indices but have $q$-bose or $q$-fermi
statistics in the latin indices. More precisely, we must regard these as
$q$-orthostatistics lying in orthobosonic and orthofermionic Fock
spaces. We can call them $q$-orthobose or $q$-orthofermi statistics.
\noindent {\bf $q$ - Orthobose Statistics}
$$
c_{i\alpha} c^\dagger_{j\beta} + q \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{j\gamma} c_{i\gamma} \ = \ 0 \quad {\rm for} \quad i < j \eqno(6.18)
$$
$$
c_{j\alpha} c^\dagger_{j\beta} - |q|^2 \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{j\gamma} c_{j\gamma} \ = \ \delta_{\alpha\beta} + \delta_{\alpha\beta}
(|q|^2-1) \sum_{k<j} \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma} \eqno(6.19)
$$
$$
c_{i\alpha} c_{j\beta} + q^\star c_{j\alpha } c_{i\beta} \ = \ 0 \quad
{\rm for } \quad i < j \eqno(6.20)
$$
\noindent {\bf $q$-Orthofermi Statistics}
$$
c_{i\alpha} c^\dagger_{j\beta} + q \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{j\gamma} c_{i\gamma} \ = \ 0 \quad {\rm for} \quad i < j \eqno(6.21)
$$
$$
c_{j\alpha} c^\dagger_{j\beta} + \delta_{\alpha\beta} \sum_\gamma
c^\dagger_{j\gamma} c_{j\gamma} \ = \ \delta_{\alpha\beta} +
\delta_{\alpha\beta}
(|q|^2-1) \sum_{k<j} \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma} \eqno(6.22)
$$
$$
c_{i\alpha} c_{j\beta} + q^\star c_{j\alpha} c_{i\beta} \ = \ 0 \quad
{\rm for } \quad i < j \eqno(6.23)
$$
$$
c_{j\alpha} c_{j\beta} \ = \ 0 \eqno(6.24)
$$
In eqs(6.18) - (6.24), $q$ is an arbitrary complex parameter. For
$q=e^{i\theta}$, and $q=0$ we shall have fractional statistics and
statistics of frozen order respectively, but in the latin indices only.
Of course one can construct many other algebras corresponding to the
same statistics, just as in the case of the single-indexed systems.
Further, these equations are analogous to eqs(4.3), (4.12),(4.23) and
(4.28) which are covariant under quantum groups. So, another direction
is indicated here for the further generalization of
quantum-group-theoretic structures to two-indexed systems.
\newpage
\noindent {\bf 6.2 Systems in which $n_{g\alpha}$ exist}
We shall now consider a different kind of double-indexed systems. Here,
the occupation numbers $n_{g\alpha}$ exist, nevertheless mapping to
single-indexed systems is not possible because of the subsidiary
conditions that define the reduced Fock space. We give below
three examples of such double indexed systems, described by the algebras
:
\begin{description}
\item[(a)]
$$
c_{k\alpha} c^\dagger_{m\beta} + (1-\delta_{km})
c^\dagger_{m\beta} c_{k\alpha} \ = \ \delta_{km} \delta_{\alpha\beta}
\left( 1 - \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma}\right) \eqno(6.25)
$$
\item[]
$$
c_{k\alpha} c_{m\beta} + (1-\delta_{km}) c_{m\beta} c_{k\alpha}
\ = \ 0\,. \eqno(6.26)
$$
\item[(b)]
$$
c_{k\alpha} c^\dagger_{m\beta} - (1-\delta_{km})
c^\dagger_{m\beta} c_{k\alpha} \ = \ \delta_{km} \delta_{\alpha\beta}
\left( 1 + \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma}\right) \eqno(6.27)
$$
\item[]
$$
(1-\delta_{km}) (c_{k\alpha} c_{m\beta} - c_{m\beta} c_{k\alpha})
\ = \ 0 \,. \eqno(6.28)
$$
\item[(c)]
$$
c_{k\alpha} c^\dagger_{m\beta} - (1-\delta_{km})
c^\dagger_{m\beta} c_{k\alpha} \ = \ \delta_{km} \delta_{\alpha\beta}
\left( 1 - \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma}\right) \eqno(6.29)
$$
\item[]
$$
c_{k\alpha} c_{m\beta} - (1-\delta_{km}) c_{m\beta} c_{k\alpha}
\ = \ 0 \,. \eqno(6.30)
$$
\end{description}
Each of these algebras is covariant under the unitary transformation on
the greek indices defined by eq.(6.7), but not covariant under the
unitary transformation on the latin indices defined by eq.(6.6).
However, all of them are covariant under not only the phase
transformations in eqs(6.8) and (6.9) but also the $k\alpha$-dependent
phase transformation of eq.(6.10). hence all the occupation numbers
$n_k, n_\alpha$ as well as $n_{k\alpha}$ exist.
Algebra (a) leads to states which are antisymmetric for simultaneous
interchange of latin and greek indices as in Fermi-Dirac statistics, but
the usual Pauli exclusion principle is replaced by the stronger or more
exclusive exclusion principle, as in orthofermi statistics (see eq.6.17)
:
$$
n_k \ = \ 0 \quad {\rm or} \quad 1 \quad {\rm only} \,, \eqno(6.31)
$$
\noindent where
$$
n_k \ \equiv \ \sum_\alpha n_{k\alpha} \eqno(6.32)
$$
\noindent These are precisely the states that are allowed in the Hubbard
model of strongly correlated electrons in the limit of infinite
intrasite Coulomb repulsion if we interpret the latin index as the site
and the greek index as the spin. Hence, this algebra and the statistics can
be called Hubbard algebra and Hubbard statistics respectively. For more
details, the reader is referred to [46]. Since $n_{k\alpha}$ exists, the
states of the system can be characterized as $|n_{k\alpha}, n_{m\beta}
\ldots ; \mu>$. However, because of the above constraint (eq(6.31))
which can be rewritten in the form :
$$
\sum_\alpha n_{k\alpha} \ = \ 0 \quad {\rm or} \quad 1 \quad {\rm
only} \eqno(6.33)
$$
\noindent this sytem cannot be mapped into a single-indexed system.
Algebra (b) can be regarded as the bosonic ``counterpoint'' of algebra
(a). States are symmetric under simultanueous exchange of latin and
greek indices for quanta with different latin indices $(k \ne m)$. But
for quanta with identical latin indices $(k = m)$, there is no
restriction on the symmetry with respect to the greek indices. In other
words, there is infinite statistics in greek indices if the
corresponding latin indices are identical. Whereas algebra (a) leads to
more exclusive states than allowed by Pauli, algebra (b) leads to ``more
inclusive'' states than allowed by Bose. For this reason we
may call algebra (b) as the ``inclusive counterpoint'' to algebra (a).
Such a restriction on the allowed states of a two-indexed system which
distinguishes $k=m$ from $k\ne m$ cannot be mapped into a condition for
a single-indexed system.
Finally, algebra (c) leads to states that are symmetric for simultaneous
exchange of latin and greek indices, but the stronger exclusion
principle of eq (6.33) is also valid as in algebra (a). Hence, algebras
(a) and (c) represent two forms of statistics which may be called
antisymmetric and symmetric Hubbard statistics respectively
both residing within the same reduced Fock space.
On the other hand, algebra (b) and its statistics lie in a
different reduced Fock space which is a Fock space with the new
``inclusion'' principle.
A compilation of the algebras and statistics for two-indexed systems is
given in Table IV. The $cc$ relations are not included since they can be
shown to follow from the $cc^\dagger$ algebra, whenever they exist.
\newpage
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\noindent{\bf 7. Summary and Discussion}
We have formulated a theory of generalized Fock spaces which is
sufficiently general so as to encompass the well known Fock spaces and
many newer ones. We have shown that such a theory can be constructed
without introducing creation and annihilation operators. The only
requirements for constructing a generalized Fock space are to specify the
set of allowed states, and to make it an inner product space. By
freeing the notion of the underlying state space from c and
c$^{\dagger}$, we are able to define different forms of quantum
statistics in a representation independent manner. Subsequently, one
can construct c and c$^{\dagger}$ and their algebras in any desired
representation.
Our general formalism not only unifies the various forms of statistics
and algebras proposed so far but also allows one to construct many new
forms of quantum statistics as well as algebras of c and c$^{\dagger}$
in a systematic manner. Some of these are the following :
\noindent (a) Many new algebras for infinite statistics \\
\noindent (b) Complex q-statistics and a number of cc$^{\dagger}$
algebras representing them \\
\noindent (c) A consistent algebra of c and c$^{\dagger}$ for
``fractional" statistics \\
\noindent (d) Null statistics or statistics of frozen order \\
\noindent (e) ``Doubly-infinite" statistics and its representations \\
\noindent (f) q-orthobose and q-orthofermi statistics \\
\noindent (g) A statistics for two-indexed systems with a new
``inclusion principle''. \\
\noindent (h) A symmetric version of Hubbard statistics.
Our primary concept is that of generalized Fock space, of which many
categories have been introduced in this paper. Next comes the notion
of statistics which is defined by the type of symmetry or relationship
among the state vectors residing in the particular type of Fock space.
In a given Fock space, more than one type of symmetry can be postulated,
the prime example of this being the symmetry, antisymmetry or q-symmetry
in the bosonic and fermionic Fock spaces. For a given statistics,
there can exist different representations of c and c$^{\dagger}$,
leading to different $cc^{\dagger}$ relations. To summarize, a
particular Fock space can admit different statistics, and a particular
statistics can be represented by more than one $cc^{\dagger}$
algebra. But {\it {the important point is that various
statistics and algebras residing in a given Fock space are all
inter-related}}. These interconnections are given by generalized
versions of the well-known Jordan-Wigner-Klein transformations.
No such interconnections exist among statistics and
algebras belonging to distinct Fock spaces.
For the sake of clarity, the above-described logical order of concepts
as well as their logical interconnections are presented in the form of
flow charts or block diagrams in Figs.3 and 4. The single-indexed
systems are considered in Fig.3. The Fock spaces of higher dimension
are shown to the right of those of lower dimension . The Fock space of
frozen order as well as the bosonic and fermionic Fock spaces have the
lowest dimension $d = 1$ in any sector $\{n_g, n_h \ldots\}$. Next come the
parafermionic and parabosonic Fock spaces which have d $>$ 1. At the
extreme right, we have the super Fock space which has the largest
dimension $d = s$ in each sector with s given by eq.(2.6). Null
statistics and infinite statistics can be regarded as the opposite
limiting cases of generalized statistics and hence these two forms of
statistics along with their Fock spaces occupy the opposite ends of the
diagram. Although not shown separately in Fig.3 because of lack of
space, the bosonic and fermionic Fock spaces are distinct and each must
be separately associated with the complete set of statistics and
algebras shown. Same is true of the parabosonic and parafermionic Fock
spaces. Further, there are two Fock spaces of frozen order, the bosonic
and fermionic type. And finally, there exists another super Fock space
with exclusion principle, which is not shown separately.
Within the parafermionic and parabosonic Fock spaces many
``deformations" of parastatistics and many other representations and
algebras apart from Green's trilinear algebra [35] are possible.
These are indicated by the hanging arrows in Fig.3. Further, as shown
by the dotted lines, there is enough room for many new varieties of Fock
spaces and associated statistics and algebras. These possibilities may
be pursued in the future.
Coming to Fig.4 depicting the systems with two indices, here again Fock
spaces of higher dimension generally lie to the right. Although shown
together, the orthobosonic and orthofermionic Fock spaces must be regarded
distinct. Here, one can envisage a richer harvest of new Fock spaces,
statistics and algebras because of the two indices and this again is for
the future.
We now conclude with some general comments :
\noindent 1. We must once again repeat and emphasize the point that most of the
$q$-deformations on oscillators discussed in the literature amount to
only a change of variable and hence must be regarded as different
avatars of bosonic or fermionic systems. However, one must clearly
distinguish those deformations such as the $q$-mutator algebra of
Greenberg that require the construction of new types of Fock spaces.
Obviously, Greenberg-type of deformations can never be reduced to change
of variables living within the bosonic or fermionic Fock space. Some
degree of confusion prevails in recent literature since this distinction
is not kept in mind. (See for instance [7,32,41,42,47]).
\vspace{.5cm}
\noindent 2. In Sec.4, we have shown that algebras that are covariant
under quantum groups are only a particular case of the more general
class of algebras that can be derived from the formalism of generalized
Fock spaces. This formalism is based on linear vector space and linear
operators acting on this space; mathematically, no more sophistication
is required. And yet it is capable of handling quantum-group related
structures in a self-contained manner. It would seem that the basic
concepts of quantum groups are contained in the theory of generalized
Fock spaces and it must be possible to construct quantum group itself
starting from this theory.
\vspace{.5cm}
\noindent 3. We have already referred to the desirability of covariance under
unitary transformations that mix the indices as a requirement for the
algebras of creation and annihilation operators. We shall call the
algebras that satisfy this requirement as covariant algebras. This
property stems from the superposition principle in quantum mechanics.
Since the indices describe quantum states of a single particle, if we
demand that, for any orthonormal set of quantum states obtained by
superposition of the original set of quantum states, the algebra should
retain the same form, then covariance under unitary transformations
follows. Many of the algebras presented in this paper violate this
requirement. Such noncovariant algebras probably cannot be used in a general
context, as for instance, in constructing a quantum field theory that
respects many of the known invariance principles such as translational
or rotational invariance. Nevertheless, these algebras may be useful to
describe specific systems in specific states such as those encountered
in condensed matter physics.
Some of these noncovariant algebras do have other nice properties,
although these are motivated mainly from a mathematical point of view.
This is the case of those algebras that are covariant under quantum groups.
Among the algebras for single-indexed systems that have been
discussed, Greenberg's $q$-mutator
algebra is the only $q$-deformation that is covariant under unitary
transformations, but then one has to pay the price of the enlarged Fock
space. Every other known $q$-deformation leads to a noncovariant
algebra.
Greenberg's $q$-mutator algebra (including the case $q=0$ which is the
standard representation), the canonical bosonic and fermionic algebras
and Green's trilinear algebras for parabosons and parafermions are the
covariant representatives living respectively in the super Fock space,
bosonic and fermionic Fock spaces and the parabosonic and parafermionic
Fock spaces. All the other algebras living in these three catagories of
Fock spaces, although noncovariant, can be transformed to these
covariant algebras through equations such as eq(4.32). This is not the
case for the algebra of null statisitcs or the algebra of Boltzmann
statistics with Pauli principle living respectively in the Fock space of
frozen order and the super Fock space with Pauli principle. In these Fock
spaces, covariant algebras do not exist.
\vspace{.5cm}
\noindent 4. Quantum mechanics is sometimes viewed as a deformation of
classical mechanics since the commutator bracket of quantum mechanics
can be related to the deformation of the classical Poisson bracket, the
Planck's constant playing the role of the deformation parameter.
Relying on similar reasoning it has been proposed that a deformation of
canonical commutation relations will lead to fundamentally new
mathematical or physical structures [48,49,50]. The analysis presented in
this paper shows that nothing of this sort happens, if viewed within the
framework of Fock space. The transition from
classical to quantum mechanics requires the replacement of
the notion of the phase space by that of the Hilbert space or Fock
space. In contrast, we have seen that all the deformations of
commutation relations can be formulated within the framework of Fock
space. In fact most of the deformed structures proposed in the
literature exist within the time-honoured bosonic and fermionic Fock
spaces only. Even Greenberg's infinite statistics lives within a Fock
space, although an enlarged one.
\vspace{.5cm}
\noindent 5. While remaining within the framework of quantum mechanics,
the general theory of Fock spaces presented here throws light on the
enlarged framework within which the familiar quantum field theory and
statistical mechanics reside and hence may lead to newer forms of
quantum field theory and statistical mechanics. This is infact the
main motivation behind
our work. Apart from earlier work on parastatistics [37], we may
mention as examples of new forms of quantum field theories, Greenberg's
construction [13,14,51] of a nonrelativistic quantum field theory based
on infinite statistics and our construction [52] of a local
relativistic quantum field theory based on orthostatistics.
Many other forms of quantum field
theories based on the generalized Fock spaces may be possible. Their
formulation and study is an agenda for the future.
\vspace{.5cm}
\noindent 6. Although one may not be able to construct local relativistic
quantum field theories corresponding to many of the newer forms of
statistics and algebras, nonrelativistic quantum field theories based on
these are still possible. Condensed matter physics is a rich field where
applications of such theories may be relevant. In fact there is no
reason why the quasiparticles encountered in condensed matter systems
should be bosons or fermions only. We have shown that any of the
generalized Fock spaces provides a perfectly valid quantum-mechanical
framework for many-particle systems. Hence, quasiparticles living in a
generalized Fock space offer an important field of study.
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{\centerline{\bf Appendix A : Generalised Fock Spaces for Two-indexed
Systems}}
\bigskip
Here we consider only those two-indexed systems in which $n_{g\alpha}$ do
not exist. See Sec.6.1.
\medskip
\noindent{\underline {The state vectors, inner products and orthonormal
sets}}
$$
\langle n^{'}_g, n^{'}_h \ldots; n^{'}_{\alpha}, n^{'}_{\beta} \ldots ;
\mu \vert n_g, n_h \ldots ; n_{\alpha}, n_{\beta} \ldots; \nu \rangle
$$
$$
\quad = \quad \delta_{n^{'}_g n_g}
\delta_{n'_h n_h} \, \ldots \delta_{n^{'}_{\alpha} n_{\alpha}}
\delta_{n^{'}_{\beta} n_{\beta}} \ldots M_{\mu \nu} \eqno(A.1)
$$
$$
\parallel n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots ; \mu \gg =
\sum_{\nu} \, X_{\nu \mu} \,
\vert n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots ; \nu \rangle
\eqno(A.2)
$$
$$
\ll n^{'}_g, n^{'}_h \ldots; n^{'}_{\alpha}, n^{'}_{\beta} \ldots ;
\mu \parallel n_g, n_h \ldots ; n_{\alpha}, n_{\beta} \ldots; \nu \gg
$$
$$
\quad = \quad \delta_{n^{'}_g n_g}
\delta_{n^{'}_h n_h} \, \ldots \delta_{n^{'}_{\alpha} n_{\alpha}}
\delta_{n^{'}_{\beta} n_{\beta}} \ldots \delta_{\mu \nu} \eqno(A.3)
$$
$$
M^{-1} \, = \, XX^{\dagger} \eqno(A.4)
$$
$$
I \, = \, \sum_{{\stackrel{n_g,n_h ..}{n_{\alpha}, n_{\beta}..}}}
\, \sum_{\mu} \parallel n_g, n_h ..; n_{\alpha}, n_{\beta} .. ;
\mu \gg \ \ll n_g, n_h .. ; n_{\alpha}, n_{\beta} \ldots; \mu
\parallel \eqno(A.5)
$$
$$
= \sum_{{\stackrel{n_g,n_h \ldots}{n_{\alpha}, n_{\beta}\ldots}}} \,
\sum_{\lambda, \nu} \vert n_g, n_h \ldots ; n_{\alpha}, {n_\beta} \ldots ;
\nu \rangle \, (M^{-1})_{\nu \lambda} \, \langle n_g \ldots ; n_{\alpha}
\ldots ; \lambda \vert \eqno(A.6)
$$
\vspace{0.2cm}
\noindent {\underline {Projection Operators :}}
$$
P (n_g,n_h \ldots; n_{\alpha}, n_{\beta} \ldots) \,
$$
$$
= \, \sum_{\mu} \parallel n_g, n_h \ldots; n_{\alpha},
n_{\beta} \ldots ; \mu \gg \, \ll n_g, n_h \ldots ; n_{\alpha},
n_{\beta} \ldots; \mu \parallel \eqno(A.7)
$$
$$
= \sum_{\lambda, \nu} \vert n_g, n_h \ldots ; n_{\alpha}, n_{\beta} \ldots ;
\nu \rangle \, (M^{-1})_{\nu \lambda} \, \langle n_g \ldots ; n_{\alpha}
\ldots ; \lambda \vert \eqno(A.8)
$$
$$
I \, = \, \sum_{{\stackrel{n_g,n_h \ldots}{n_{\alpha}, n_{\beta}\ldots}}}
\, P(n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots) \eqno(A.9)
$$
$$
P(n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots) \,
\parallel n^{'}_g, n^{'}_h \ldots; n^{'}_{\alpha}, n^{'}_{\beta} \ldots ;
\mu \gg \,
$$
$$
\quad = \quad \delta_{n_g n^{'}_g} \ldots \delta_{n_{\alpha}
n^{'}_{\alpha}} \ldots \parallel n_g \ldots ; n_\alpha \ldots ;
\mu \gg \eqno(A.10)
$$
$$
P (n_g \ldots ; n_{\alpha} \ldots ) \, \vert n^{'}_g \ldots ;
n^{'}_{\alpha} \ldots ; \mu \rangle \,
$$
$$
\qquad = \quad \delta_{n_g n^{'}_g} \ldots
\delta_{n_{\alpha}n^{'}_{\alpha}} \ldots \vert n_g \ldots ; n_{\alpha}
\ldots ; \mu \rangle \eqno(A.11)
$$
\noindent {\underline {Number operators}}
$$
N_k \, = \, \sum_{{\stackrel{n_g \ldots n_k \ldots}{n_{\alpha} \ldots}}}
\, n_k \, P (n_g \ldots n_k \ldots ; n_{\alpha} \ldots) \eqno(A.12)
$$
$$
N_{\beta} \, = \, \sum_{{\stackrel{n_g \ldots}{n_{\alpha}, n_{\beta} \ldots}}}
\, n_{\beta} \, P (n_g \ldots ; n_{\alpha}, n_{\beta} \ldots)
\eqno(A.13)
$$
$$
N_k \parallel n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu
\gg
$$
$$
\qquad = \quad n_k \parallel n_g \ldots n_k \ldots ;
n_{\alpha}, n_{\beta} \ldots ; \mu \gg \eqno(A.14)
$$
$$
N_k \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu
> =
n_k \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu
\rangle \eqno(A.15)
$$
$$
N_{\beta} \parallel n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu
\gg
$$
$$
\qquad = \quad n_{\beta} \parallel n_g \ldots n_k \ldots ;
n_{\alpha}, n_{\beta} \ldots ; \mu \gg \eqno(A.16)
$$
$$
N_{\beta} \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu
\rangle =
n_{\beta} \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu
\rangle \eqno(A.17)
$$
$$
[N_k, N_j] \, = \, [N_{\alpha}, N_{\beta}] \, = \, [N_k, N_{\alpha}] \,
$$
$$
\qquad = \quad 0 \quad {\rm for \, any } \ k, j
\ {\rm and \, any} \ \alpha, \beta \eqno(A.18)
$$
\noindent Total number operator is
$$
N = \sum_k N_k = \sum_\alpha N_\alpha \eqno(A.19)
$$
\noindent {\underline {Creation and destruction operators}} :
$$
c^{\dagger}_{j \beta} \, = \, \sum_{{\stackrel{n_g,n_j
\ldots}{n_{\alpha}, n_{\beta} \ldots}}} \, \sum_{\mu^{'} \nu} \,
A_{\mu^{'} \nu} \, \vert n_g \ldots (n_{j}+1) \ldots ; n_{\alpha},
n_{\beta} + 1 \ldots ; \mu^{'} \rangle \,
$$
$$
\qquad \otimes \langle n_g \ldots n_j \ldots ; n_{\alpha},n_{\beta}
\ldots \nu \vert \eqno(A.20)
$$
$$
{[c^{\dagger}_{j \beta}, N_k]} = - c^{\dagger}_{j \beta}
\delta_{jk} \eqno(A.21)
$$
$$
{[c^{\dagger}_{j \beta}, N_{\alpha}]} = - c^{\dagger}_{j \beta}
\delta_{\alpha \beta} \eqno(A.22)
$$
\noindent For some particular $\mu$,
$$
\vert 3_g, 2_h; 4_{\alpha}, 1_{\beta} ; \mu \rangle
= \vert 1_{g \alpha} 1_{g \alpha} 1_{g \beta} 1_{h \alpha} 1_{h \alpha}
\rangle \, = \, (c^{\dagger}_{g\alpha})^2 \, c^{\dagger}_{g \beta} \,
(c^{\dagger}_{h\alpha})^2 \, \vert 0 \rangle \eqno(A.23)
$$
$$
c^{\dagger}_{j\beta} \, \vert n_g \ldots \, n_{j \ldots } ; n_{\alpha},
n_{\beta} \ldots ; \lambda \rangle \, = \, \vert \underbrace{1_{j \beta,}
; n_g \cdots n_j \cdots ; n_{\alpha}, n_{\beta} \cdots ;}_
{n_g \cdots n_j +1 \cdots ; n_{\alpha}, n_{\beta} + 1} \lambda \rangle
\eqno(A.24)
$$
$$
c^{\dagger}_{j \sigma} \, \vert n_g \ldots n_j \ldots ; n_{\alpha}
\ldots n_{\sigma} \ldots ; \lambda \rangle
$$
$$
= \sum_{{\stackrel{n^{'}_g \ldots n^{'}_j \ldots}{n^{'}_{\alpha} \ldots
n^{'}_{\sigma}\ldots}}} \, \sum_{\mu^{'} \nu} \, A_{\mu^{'} \nu} \, \vert
n^{'}_g \ldots (n^{'}_{j}+1) \ldots ; n^{'}_{\alpha} \ldots n^{'}_{\sigma}
+1 \ldots ; \mu^{'} \rangle \eqno(A.25)
$$
$$
\otimes \langle n^{'}_g \ldots n^{'}_j \ldots ; n^{'}_{\alpha} \ldots
n^{'}_{\sigma} \ldots ; \nu \vert n_g \ldots n_j \ldots
; n_{\alpha} \ldots n_{\sigma} \ldots ; \lambda \rangle
$$
$$
= \sum_{\mu^{'} \nu} \, A_{\mu^{'} \nu} M_{\nu \lambda} \, \vert n_g
\ldots (n_{j}+1) \ldots ; n_{\alpha} \ldots n_{\sigma} +1 \ldots ; \mu^{'}
\rangle \eqno(A.26)
$$
$$
\sum_{\nu} \, A_{\mu^{'}\nu} M_{\nu \lambda} \, = \, \delta_{\mu^{'}
\lambda} \eqno(A.27)
$$
$$
A \, = \, M^{-1} \eqno(A.28)
$$
$$
c^{\dagger}_{j \sigma} \, = \, \sum_{{\stackrel{n_g \ldots
n_j..}{n_{\alpha} .. n_{\sigma}..}}} \, \sum_{\lambda \nu} \,
(M^{-1})_{\lambda \nu} \vert 1_{j\sigma} ; n_g \ldots n_j \ldots ;
n_{\alpha} \ldots n_{\sigma} \ldots ; \lambda \rangle \,
$$
$$
\qquad \otimes \langle n_g \ldots n_j \ldots ; n_{\alpha} \ldots
n_{\sigma} \ldots ; \nu \vert \eqno(A.29)
$$
\newpage
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\noindent {\bf References}
\bigskip
\begin{enumerate}
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L411.
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149.
\item{} S.L.Woronowicz, Publ. RIMS Kyoto University {\bf 23} (1987) 117.
\item S.L.Woronowicz, Commun. Math. Phys. {\bf 111} (1987) 613.
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(1991) 43.
\item M.Chaichian and P.Kulish, in Nonperturbative methods in
low-dimensional quantum field theories (Proc. 14th John Hopkins Workshop
on Current Problems in Particle Theory, Debrecen, Hungary, 1990) (World
Sci. Pub. Co., River Edge, NJ, 1991) p.213.
\item{} R.Jagannathan, R.Sridhar, R.Vasudevan, S.Chaturvedi,
M.Krishnakumari, P.Santha and V.Srinivasan, J. Phys. A {\bf
25} (1992) 6429.
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{\bf 50} (1991) 321.
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49} (1991) 317.
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(1982) 957.
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2407.
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537 ; ibid {\bf 37} (1991) 455 (E).
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3240.
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\end{enumerate}
\end{document}
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\begin{tabular}{|c|l|l|} \hline
& & \\
\multicolumn{1}{|c|}{\bf Statistics} & \multicolumn{1}{|c|}{\bf Representation}
& \multicolumn{1}{|c|}{\bf Algebra} \\
& & \\ \hline
& & \\
Boltzmann & Standard representation & $c_i c^{\dagger}_j = \delta_{ij}$ \\
& & \\
'' & q-mutator (with real q) & $c_i c^{\dagger}_j
-qc^{\dagger}_j c_i = \delta_{ij}$ \\
& & \\
'' & Two-parameter algebra & \{ \begin{tabular}{l}
$c_i c^{\dagger}_j - q_1 c^{\dagger}_jc_i$ \\ $-q_2 \delta_{ij}
\Sigma_k c^{\dagger}_k c_k = \delta_{ij}$ \end{tabular} \\
& & \\
'' & q-mutator, transformed & $c_i c^{\dagger}_j -
c^{\dagger}_j c_i = \delta_{ij} p^{2 \Sigma_{k<i}{N_k}} p^{N_i}$ \\
& & \\
'' & $q$-mutator, transformed & \{ \begin{tabular}{l}
$c_ic^{\dagger}_j - p^{-1} c^{\dagger}_j c_i = 0, \quad {\rm for} \quad
i \neq j$ \\
$ c_i c^{\dagger}_i - c^{\dagger}_i c_i = p^{N_i}$
\end{tabular} \\
& & \\
'' & q-mutator (with complex q) & \{ \begin{tabular}{l}
$c_ic^{\dagger}_j-qc^{\dagger}_jc_i = 0, \quad {\rm for} \quad i < j$ \\
$ c_i c^{\dagger}_i -pc^{\dagger}_i c_i = 1$ \end{tabular} \\
& & \\
\begin{tabular}{c}Boltzmann \\ with \\ Pauli principle \end{tabular}
& Standard representation & \{ \begin{tabular}{l} $c_ic^\dagger_j = 0,
\quad {\rm for} \quad i \neq j$ \\
$ c_i c^{\dagger}_i + c^{\dagger}_i c_i = 1$ \end{tabular} \\
& & \\
''
& q-mutator (with real $q$) & \{ \begin{tabular}{l}
$c_ic^{\dagger}_j-qc^{\dagger}_jc_i = 0, \quad {\rm for} \quad i \ne j$ \\
$ c_i c^{\dagger}_i + c^{\dagger}_i c_i = 1$. \end{tabular} \\
& & \\
''
& q-mutator (with complex $q$) & \{ \begin{tabular}{l}
$c_ic^{\dagger}_j-qc^{\dagger}_jc_i = 0, \quad {\rm for} \quad i < j$ \\
$ c_i c^{\dagger}_i + c^{\dagger}_i c_i = 1$ \end{tabular} \\
& & \\ \hline
\end{tabular}
\vspace{.8cm}
{\bf Table I. Representations of infinite statistics.}
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{\small {
\begin{tabular}{|c|c|c|l|l|} \hline
& & & & \\
{\bf Statistics} & {\bf $cc$ algebra} & {\bf $c_i
c_j^\dagger$ algebra} & \multicolumn{1}{|c|}{\bf $c_j c_j^\dagger$ algebra} &
\multicolumn{1}{|c|}{\bf Remarks} \\
& & {\bf for $i\ne j$} & & \\
& & & & \\ \hline
& & & & \\
$q$-statistics & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ & $c_j c^\dagger_j - p c^\dagger_j c_j = |q|^{2
\sum_{i<j} N_i} f(N_j)$ & General \\
& for $i < j$ & for $ i < j$ & & representation \\
& & & & \\
'' & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ &
$c_j c^\dagger_j - p c^\dagger_j c_j = |q|^{2 \sum_{i<j} N_i}
\left\{ 1-p (1-\delta_{N_j,0})\right\}$ & \\
& for $i < j$ & for $ i < j$ & & \\
& & & & \\
'' & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ &
$c_j c^\dagger_j = |q|^{2 \sum_{i<j} N_i}$ & \\
& for $i < j$ & for $ i < j$ & & \\
& & & & \\
'' & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ & $c_j c^\dagger_j- pc^\dagger_j c_j =
|q|^{2 \sum_{i<j} N_i}$ & \\
& for $i < j$ & for $ i < j$ & & \\
& & & & \\
'' & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ & $c_j c^\dagger_j- |q|^2 c^\dagger_j c_j = 1 +
(|q|^2-1) \sum_{k<j} c^\dagger_k c_k$ & Covariant \\
& for $i < j$ & for $ i < j$ & & under $SU_q(n)$ \\
& & & & \\
bose statistics & $c_i c_j = c_j c_i$ & $c_i c^\dagger_j
= c^\dagger_j c_i$ & $c_j c^\dagger_j- pc^\dagger_j c_j = f(N_j)$
& General \\
& & for $ i \ne j$ & & representation \\
& & & & \\
'' & $c_i c_j = c_j c_i$ & $c_i c^\dagger_j
= c^\dagger_j c_i$ &
$c_j c^\dagger_j - p c^\dagger_j c_j = \left\{ 1-p (1-\delta_{N_j,0})\right\}$
& \\
& & for $ i \ne j$ & & \\
& & & & \\
'' & $c_i c_j = c_j c_i$ & $c_i c^\dagger_j
= c^\dagger_j c_i$ & $c_j c^\dagger_j = 1$ & simplest \\
& & for $ i \ne j$ & & representation \\
& & & & \\ \hline
\end{tabular}}}
\vspace{1cm}
{\bf Table II. \ Statistics and algebras in the bosonic Fock space.}
\end{center}
\newpage
\noindent {\bf Table II (Continued)}
\vspace{.5cm}
\begin{center}
{\small {
\begin{tabular}{|c|c|c|l|l|} \hline
& & & & \\
bose statistics & $c_i c_j = c_j c_i$ & $c_i c^\dagger_j
= c^\dagger_j c_i$ & $c_j c^\dagger_j - pc^\dagger_j c_j = 1 ; p \ne -1 $
& Commuting \\
& & for $ i \ne j$ & & deformed oscillators \\
& & & & \\
'' & $c_i c_j = c_j c_i$ & $c_i c^\dagger_j
= c^\dagger_j c_i$ & $c_j c^\dagger_j - c^\dagger_j c_j = 1$
& Canonical rep. \\
& & for $ i \ne j$ & & of bosons \\
& & & & \\
fermi statistics & $c_i c_j = -c_j c_i$ & $c_i c^\dagger_j
= -c^\dagger_j c_i$ & $c_j c^\dagger_j- pc^\dagger_j c_j = f(N_j)$
& General \\
& for $i \ne j$ & for $ i \ne j$ & & representation \\
& & & & \\
'' & $c_i c_j = -c_j c_i$ & $c_i c^\dagger_j
= -c^\dagger_j c_i$ & $c_j c^\dagger_j - p c^\dagger_j c_j =
\left\{ 1-p (1-\delta_{N_j,0})\right\}$ & \\
& for $i \ne j$ & for $ i \ne j$ & & \\
& & & & \\
'' & $c_i c_j = -c_j c_i$ & $c_i c^\dagger_j
= -c^\dagger_j c_i$ & $c_j c^\dagger_j = 1$
& \\
& for $i \ne j$ & for $ i \ne j$ & & \\
& & & & \\
'' & $c_i c_j = -c_j c_i$ & $c_i c^\dagger_j
= -c^\dagger_j c_i$ & $c_j c^\dagger_j - pc^\dagger_j c_j = 1 ; p \ne
-1$ & Anticommuting \\
& for $i \ne j$ & for $ i \ne j$ & & deformed oscillators \\
& & & & \\
`` & $c_i c_j = -c_j c_i$ & $c_i c^\dagger_j
= -c^\dagger_j c_i$ & $c_j c^\dagger_j - c^\dagger_j c_j = 1$
& Anticommuting \\
& for $i \ne j$ & for $ i \ne j$ & & bosons \\
& & & & \\
fractional & $c_i c_j = e^{-i\theta} c_j c_i$ &
$c_i c^\dagger_j = e^{i\theta} c^\dagger_j c_i$ &
Any one of the same $c_j c^\dagger_j$ & \\
statistics & for $i < j$ & for $ i < j$ &
relations given in bose or & \\
& & & fermi statistics above & \\
& & & & \\ \hline
\end{tabular}}}
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{\small {
\begin{tabular}{|c|c|c|l|l|} \hline
& & & & \\
{\bf Statistics} & {\bf $cc$ algebra} & {\bf $c_i
c_j^\dagger$ algebra} & \multicolumn{1}{|c|}{\bf $c_j c_j^\dagger$ algebra} &
\multicolumn{1}{|c|}{\bf Remarks} \\
& & {\bf for $i\ne j$} & & \\
& & & & \\ \hline
& & & & \\
$q$-statistics & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ &
$c_j c^\dagger_j - p c^\dagger_j c_j = (\delta_{N_j,0} - p
\delta_{N_j,1})$ & General \\
& for $i < j$ & for $ i < j$ & $ \times \{ |r|^2+(|q|^2-1) \sum_{k<j}
c^\dagger_k c_k\}$ & representation \\
& $c_ic_i = 0$ & & & \\
& & & & \\
'' & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ & $c_jc_j^\dagger + c^\dagger_jc_j=|r|^2+(|q|^2-1)
\sum_{k<j} c_k^\dagger c_k$ & \\
& for $i < j$ & for $ i < j$ & & \\
& $c_ic_i = 0$ & & & \\
& & & & \\
'' & $c_i c_j = q^\ast c_j c_i$ & $c_i c^\dagger_j
= q c^\dagger_j c_i$ & $c_j c^\dagger_j + c^\dagger_j c_j =
1 + (|q|^2-1) \sum_{k<j} c^\dagger_k c_k$ & Covariant \\
& for $i < j$ & for $ i < j$ & & under $SU_q(n)$ \\
& $c_ic_i = 0$ & & & \\
& & & & \\
bose-statistics & $c_i c_j = c_j c_i$ & $c_i c^\dagger_j
= c^\dagger_j c_i$ & $c_j c^\dagger_j - p c^\dagger_j c_j =
(\delta_{N_j,0} - p \delta_{N_j,1})|r|^2$ & \\
& $c_i c_i = 0$ & for $ i \ne j$ & & \\
& & & & \\
'' & $c_i c_j = c_j c_i$ & $c_i c^\dagger_j
= c^\dagger_j c_i$ & $c_j c^\dagger_j + c^\dagger_j c_j = 1$
& Commuting \\
& $c_i c_i = 0$ & for $ i \ne j$ & & fermions \\
& & & & \\
fermi-statistics & $c_i c_j = -c_j c_i$ & $c_i c^\dagger_j
= -c^\dagger_j c_i$ & $c_j c^\dagger_j - p c^\dagger_j c_j =
(\delta_{N_j,0} - p \delta_{N_j,1})|r|^2$ & \\
& for all $i,j$ & for $i \ne j$ & & \\
& & & & \\
'' & $c_i c_j = -c_j c_i$ & $c_i c^\dagger_j
= -c^\dagger_j c_i$ & $c_j c^\dagger_j + c^\dagger_j c_j = 1$
& Canonical rep. \\
& for all $i,j$ & for $i \ne j$ & & of fermions. \\
& & & & \\
fractional & $c_i c_j = e^{-i\theta} c_j c_i$ &
$c_i c^\dagger_j = e^{i\theta} c^\dagger_j c_i$ &
$c_j c^\dagger_j - p c^\dagger_j c_j =
(\delta_{N_j,0} - p \delta_{N_j,1})|r|^2$ & \\
statistics & for $i < j$ & for $ i < j$ & & \\
& $c_i c_i = 0$ & & & \\
& & & & \\
'' & $c_i c_j = e^{-i\theta} c_j c_i$ &
$c_i c^\dagger_j = e^{i\theta} c^\dagger_j c_i$ &
$c_j c^\dagger_j + c^\dagger_j c_j = 1$ & \\
& for $i < j$ & for $ i < j$ & & \\
& $c_i c_i = 0$ & & & \\
& & & & \\ \hline
\end{tabular}}}
\vspace{.5cm}
{\bf Table III. \ Statistics and algebras in the fermionic Fock space.}
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\begin{tabular}{|c|c|l|} \hline
& & \\
{\bf $N_{k\alpha}$} &
\parbox[c]{2cm}{\begin{tabular}{c}
{\bf Statistics in} \\ {\bf latin indices}
\end{tabular}} \hspace{1cm}
\begin{tabular}{c}
\parbox[c]{3cm}{{\bf Statistics in} \\ {\bf greek indices} }
\end{tabular} &
\multicolumn{1}{|c|}{\bf $cc^\dagger$ algebra} \\
& & \\ \hline
& & \\
Do not exist &
\parbox[c]{2cm}{~~~~~~~~infinite} \hspace{2cm}
\parbox[c]{3cm}{infinite} &
$c_{k\alpha} c^\dagger_{m\beta} - q \delta_{\alpha\beta}
\sum_\gamma c^\dagger_{m\gamma} c_{k\gamma} =
\delta_{km} \delta_{\alpha\beta} $ (real $q$) \\
& & \\
'' &
\parbox[c]{2cm}{~~~~~~~~''} \hspace{2cm}
\parbox[c]{3cm}{~~~~''} &
$\left\{ \begin{tabular}{ll}
$c_{k\alpha} c^\dagger_{m\beta} - q
\delta_{\alpha\beta} \sum_\gamma c^\dagger_{m\gamma} c_{k\gamma} = 0$ \,,
& for \ $k < m$ \\
\\ $c_{k\alpha} c^\dagger_{k\beta} - p
\delta_{\alpha\beta} \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma} =
\delta_{\alpha\beta}$ ( & complex $q$ \\ & and real $p$)
\end{tabular} \right.$ \\
& & \\
'' & \begin{tabular}{c}
\parbox[c]{2cm}{~~~~~~~Fermi/} \\
\parbox[c]{2cm}{~~~~~~~Bose} \end{tabular} \hspace{2cm}
\parbox[c]{3cm}{~~~''} &
$c_{k\alpha} c^\dagger_{m\beta} \pm
\delta_{\alpha\beta} \sum_\gamma c^\dagger_{m\gamma} c_{k\gamma} =
\delta_{km} \delta_{\alpha\beta} $ \\
& & \\
'' & \begin{tabular}{c}
\parbox[c]{2cm}{~~~~~~~$q$-fermi/} \\
\parbox[c]{2cm}{~~~~~~$q$-bose} \end{tabular} \hspace{2cm}
\parbox[c]{3cm}{~~~''} &
$\left\{ \begin{tabular}{l}
$c_{i\alpha} c^\dagger_{j\beta} - q
\delta_{\alpha\beta} \sum_\gamma c^\dagger_{j\gamma} c_{i\gamma} = 0
\quad$ for $\quad i < j$ \\ \\
$c_{j\alpha} c^\dagger_{j\beta} - x
\delta_{\alpha\beta} \sum_\gamma c^\dagger_{j\gamma} c_{j\gamma}$ \\ \\
$\quad \quad = \delta_{\alpha\beta} + \delta_{\alpha\beta} (|q|^2-1)
\sum_{k<j} \sum_\gamma c^\dagger_{k\gamma} c^\dagger_{k\gamma}$ \\ \\
$x = |q|^2$ \ for \ $q$-bose ; $x=-1$ \ for \ $q$-fermi
\end{tabular} \right.$ \\
& & \\
Exist & \begin{tabular}{c} Antisymmetric for total exchange, \\
but $n_k \le 1$ \end{tabular} &
\begin{tabular}{l}
$c_{k\alpha} c^\dagger_{m\beta} + (1-\delta_{km}) c^\dagger_{m\beta}
c_{k\alpha}$ \\ \\ $\quad \quad = \delta_{km} \delta_{\alpha\beta}
\left( 1 - \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma} \right)$
\end{tabular} \\
& & \\
'' & \begin{tabular}{c} Symmetric for total exchange for \\
$k \ne m$, but infinite statistics \\ in greek indices for $k = m$ \\
\end{tabular} &
\begin{tabular}{l}
$c_{k\alpha} c^\dagger_{m\beta} - (1-\delta_{km}) c^\dagger_{m\beta}
c_{k\alpha}$ \\ \\ $\quad \quad = \delta_{km} \delta_{\alpha\beta}
\left( 1 + \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma} \right)$
\end{tabular} \\
& & \\
'' & \begin{tabular}{c} Symmetric for total exchange but \\
$n_k \ge 1$ \end{tabular} &
\begin{tabular}{l}
$c_{k\alpha} c^\dagger_{m\beta} - (1-\delta_{km}) c^\dagger_{m\beta}
c_{k\alpha}$ \\ \\ $\quad \quad = \delta_{km} \delta_{\alpha\beta}
\left( 1 - \sum_\gamma c^\dagger_{k\gamma} c_{k\gamma} \right)$
\end{tabular} \\
& & \\ \hline
\end{tabular}
\vspace{1cm}
{\bf Table IV. Statistics and algebras for 2-indexed
systems.}
\end{center}
\newpage
%\end{document}
%\documentstyle[12pt]{article}
%\begin{document}
\baselineskip=24pt
\pagestyle{empty}
\noindent {\bf Figure Captions }
\begin{description}
\item[{Fig.1 }] Inversion diagram for the permutation (213) $\rightarrow $
(312). \\
Positive inversions : (1,3) $\rightarrow$ (3,1) and (2,3) $\rightarrow$
(3,2). \\
Negative inversion : (2,1) $\rightarrow$ (1,2).
\item[{Fig.2}] The complex $q$-plane of the $q$-mutator algebra. The disc
$|q|<1$ corresponds to infinite statistics and the circle $|q|=1$
corresponds to fractional statistics. $F$ and $B$ are the Fermi-Dirac
and Bose-Einstein points.
\item[{Fig.3}] Generalized Fock spaces, quantum statistics, algebras and
their interconnections.
\item[{Fig.4}] Same as Fig.3 for systems with two indices that cannot be
mapped into a single index.
\end{description}
\end{document}
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The Project Gutenberg EBook of Letters to Dead Authors, by Andrew Lang
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org
Title: Letters to Dead Authors
Author: Andrew Lang
Posting Date: January 26, 2009 [EBook \#3319]
Release Date: July, 2002
Language: English
Character set encoding: ASCII
*** START OF THIS PROJECT GUTENBERG EBOOK LETTERS TO DEAD AUTHORS ***
Produced by A. Elizabeth Warren
LETTERS TO DEAD AUTHORS
By Andrew Lang
Contents.
I. To W. M. Thackeray
II. To Charles Dickens
III. To Pierre De Ronsard
IV. To Herodotus
V. Epistle to Mr. Alexander Pope
VI. To Lucian of Samosata
VII. To Maitre Francoys Rabelais
VIII. To Jane Austen
IX. To Master Isaak Walton
X. To M. Chapelain
XI. To Sir John Manndeville, Kt
XII. To Alexandre Dumas
XIII. To Theocritus
XIV. To Edgar Allan Poe
XV. To Sir Walter Scott, Bart.
XVI. To Eusebius of Caesarea
XVII. To Percy Bysshe Shelley
XVIII. To Monsieur De Moliere, Valet De Chambre du Roi
XIX. To Robert Burns
XX. To Lord Byron
XXI. To Omar Khayya'm
XXII. To Q. Horatius Flaccus
Preface.
Sixteen of these Letters, which were written at the suggestion of the
editor of the `St. James's Gazette,' appeared in that journal, from
which they are now reprinted, by the editor's kind permission. They have
been somewhat emended, and a few additions have been made. The Letters
to Horace, Byron, Isaak Walton, Chapelain, Ronsard, and Theocritus have
not been published before.
The gem published for the first time on the title-page is a red
cornelian in the British Museum, probably Graeco-Roman, and treated in
an archaistic style. It represents Hermes Psychogogos, with a Soul, and
has some likeness to the Baptism of Our Lord, as usually shown in art.
Perhaps it may be post-Christian. The gem was selected by Mr. A. S.
Murray.
It is, perhaps, superfluous to add that some of the Letters are written
rather to suit the Correspondent than to express the writer's own taste
or opinions. The Epistle to Lord Byron, especially, is 'writ in a manner
which is my aversion.'
LETTERS TO DEAD AUTHORS
This text was converted to LaTeX by means of \textbf{GutenMark} software (version Jul 12 2014).
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\gutchapter{I. To W. M. Thackeray.}
Sir,---There are many things that stand in the way of the critic when he has a mind to praise the living. He may dread the charge of writing rather to vex a rival than to exalt the subject of his applause. He shuns the appearance of seeking the favour of the famous, and would not willingly be regarded as one of the many parasites who now advertise each movement and action of contemporary genius. 'Such and such men of letters are passing their summer holidays in the Val d'Aosta,' or the Mountains of the Moon, or the Suliman Range, as it may happen. So reports our literary `Court Circular,' and all our \textit{Precieuses} read the tidings with enthusiasm. Lastly, if the critic be quite new to the world of letters, he may superfluously fear to vex a poet or a novelist by the abundance of his eulogy. No such doubts perplex us when, with all our hearts, we would commend the departed; for they have passed almost beyond the reach even of envy; and to those pale cheeks of theirs no commendation can bring the red.
You, above all others, were and remain without a rival in your many-sided excellence, and praise of you strikes at none of those who have survived your day. The increase of time only mellows your renown, and each year that passes and brings you no successor does but sharpen the keenness of our sense of loss. In what other novelist, since Scott was worn down by the burden of a forlorn endeavour, and died for honour's sake, has the world found so many of the fairest gifts combined? If we may not call you a poet (for the first of English writers of light verse did not seek that crown), who that was less than a poet ever saw life with a glance so keen as yours, so steady, and so sane? Your pathos was never cheap, your laughter never forced; your sigh was never the pulpit trick of the preacher. Your funny people---your Costigans and Fokers---were not mere characters of trick and catch-word, were not empty comic masks. Behind each the human heart was beating; and ever and again we were allowed to see the features of the man.
Thus fiction in your hands was not simply a profession, like another, but a constant reflection of the whole surface of life: a repeated echo of its laughter and its complaint. Others have written, and not written badly, with the stolid professional regularity of the clerk at his desk; you, like the Scholar Gipsy, might have said that 'it needs heaven-sent moments for this skill.' There are, it will not surprise you, some honourable women and a few men who call you a cynic; who speak of 'the withered world of Thackerayan satire;' who think your eyes were ever turned to the sordid aspects of life---to the mother-in-law who threatens to `take away her silver bread-basket;' to the intriguer, the sneak, the termagant; to the Beckys, and Barnes Newcomes, and Mrs. Mackenzies of this world. The quarrel of these sentimentalists is really with life, not with you; they might as wisely blame Monsieur Buffon because there are snakes in his Natural History. Had you not impaled certain noxious human insects, you would have better pleased Mr. Ruskin; had you confined yourself to such performances, you would have been more dear to the Neo-Balzacian school in fiction.
You are accused of never having drawn a good woman who was not a doll, but the ladies that bring this charge seldom remind us either of Lady Castlewood or of Theo or Hetty Lambert. The best women can pardon you Becky Sharp and Blanche Amory; they find it harder to forgive you Emmy Sedley and Helen Pendennis. Yet what man does not know in his heart that the best women---God bless them---lean, in their characters, either to the sweet passiveness of Emmy or to the sensitive and jealous affections of Helen? 'Tis Heaven, not you, that made them so; and they are easily pardoned, both for being a very little lower than the angels and for their gentle ambition to be painted, as by Guido or Guercino, with wings and harps and haloes. So ladies have occasionally seen their own faces in the glass of fancy, and, thus inspired, have drawn Romola and Consuelo. Yet when these fair idealists, \textit{Mdme}. Sand and George Eliot, designed Rosamund Vincy and Horace, was there not a spice of malice in the portraits which we miss in your least favourable studies?
That the creator of Colonel Newcome and of Henry Esmond was a snarling cynic; that he who designed Rachel Esmond could not draw a good woman: these are the chief charges (all indifferent now to you, who were once so sensitive) that your admirers have to contend against. A French critic, M. Taine, also protests that you do preach too much. Did any author but yourself so frequently break the thread (seldom a strong thread) of his plot to converse with his reader and moralise his tale, we also might be offended. But who that loves Montaigne and Pascal, who that likes the wise trifling of the one and can bear with the melancholy of the other, but prefers your preaching to another's playing!
Your thoughts come in, like the intervention of the Greek Chorus, as an ornament and source of fresh delight. Like the songs of the Chorus, they bid us pause a moment over the wider laws and actions of human fate and human life, and we turn from your persons to yourself, and again from yourself to your persons, as from the odes of Sophocles or Aristophanes to the action of their characters on the stage. Nor, to my taste, does the mere music and melancholy dignity of your style in these passages of meditation fall far below the highest efforts of poetry. I remember that scene where Clive, at Barnes Newcome's Lecture on the Poetry of the Affections, sees Ethel who is lost to him. 'And the past and its dear histories, and youth and its hopes and passions, and tones and looks for ever echoing in the heart and present in the memory---these, no doubt, poor Clive saw and heard as he looked across the great gulf of time, and parting and grief, and beheld the wonmn he had loved for many years.'
\textit{For ever echoing in the heart and present in the memory:} who has not heard these tones, who does not hear them as he turns over your books that, for so many years, have been his companions and comforters? We have been young and old, we have been sad and merry with you, we have listened to the mid-night chimes with Pen and Warrington, have stood with you beside the death-bed, have mourned at that yet more awful funeral of lost love, and with you have prayed in the inmost chapel sacred to our old and immortal affections, \textit{a' leal souvenir!} And whenever you speak for yourself, and speak in earnest, how magical, how rare, how lonely in our literature is the beauty of your sentences! 'I can't express the charm of them' (so you write of George Sand; so we may write of you): 'they seem to me like the sound of country bells, provoking I don't know what vein of music and meditation, and falling sweetly and sadly on the ear.' Surely that style, so fresh, so rich, so full of surprises---that style which stamps as classical your fragments of slang, and perpetually astonishes and delights---would alone give immortality to an author, even had he little to say. But you, with your whole wide world of fops and fools, of good women and brave men, of honest absurdities and cheery adventurers: you who created the Steynes and Newcomes, the Beckys and Blanches, Captain Costigan and F. B., and the Chevalier Strong---all that host of friends imperishable---you must survive with Shakespeare and Cervantes in the memory and affection of men.
\gutchapter{II. To Charles Dickens.}
Sir,---It has been said that every man is born a Platonist or an Aristotelian, though the enormous majority of us, to be sure, live and die without being conscious of any invidious philosophic partiality whatever. With more truth (though that does not imply very much) every Englishman who reads may be said to be a partisan of yourself or of Mr. Thackeray. Why should there be any partisanship in the matter; and why, having two such good things as your novels and those of your contemporary, should we not be silently happy in the possession? Well, men are made so, and must needs fight and argue over their tastes in enjoyment. For myself, I may say that in this matter I am what the Americans do not call a `Mugwump,' what English politicians dub a 'superior person'---that is, I take no side, and attempt to enjoy the best of both.
It must be owned that this attitude is sometimes made a little difficult by the vigour of your special devotees. They have ceased, indeed, thank Heaven! to imitate you; and even in `descriptive articles' the touch of Mr. Gigadibs, of him whom `we almost took for the true Dickens,' has disappeared. The young lions of the Press no longer mimic your less admirable mannerisms---do not strain so much after fantastic comparisons, do not (in your manner and Mr. Carlyle's) give people nick-names derived from their teeth, or their complexion; and, generally, we are spared second-hand copies of all that in your style was least to be commended. But, though improved by lapse of time in this respect, your devotees still put on little conscious airs of virtue, robust manliness, and so forth, which would have irritated you very much, and there survive some press men who seem to have read you a little (especially your later works), and never to have read anything else. Now familiarity with the pages of 'Our Mutual Friend'and `Dombey and Son' does not precisely constitute a liberal education, and the assumption that it does is apt (quite unreasonably) to prejudice people against the greatest comic genius of modern times.
On the other hand, Time is at last beginning to sift the true admirers of Dickens from the false. Yours, Sir, in the best sense of the word, is a popular success, a popular reputation. For example, I know that, in a remote and even Pictish part of this kingdom, a rural household, humble and under the shadow of a sorrow inevitably approaching, has found in `David Copperfield' oblivion of winter, of sorrow, and of sickness. On the other hand, people are now picking up heart to say that 'they cannot read Dickens,' and that they particularly detest `Pickwick.' I believe it was young ladies who first had the courage of their convictions in this respect. `Tout sied aux belles,' and the fair, in the confidence of youth, often venture on remarkable confessions. In your 'Natural History of Young Ladies' I do not remember that you describe the Humorous Young Lady (1). She is a very rare bird indeed, and humour generally is at a deplorably low level in England.
(1) I am informed that the \textit{Natural History of Young}
\textit{ Ladies} is attributed, by some writers, to another
philosopher, the author of \textit{The Art of Pluck}.
Hence come all sorts of mischief, arisen since you left us; and, it may be said, that inordinate philanthropy, genteel sympathy with Irish murder and arson, Societies for Badgering the Poor, Esoteric Buddhism, and a score of other plagues, including what was once called Aestheticism, are all, primarily, due to want of humour. People discuss, with the gravest faces, matters which properly should only be stated as the wildest paradoxes. It naturally follows that, in a period almost destitute of humour, many respectable persons `cannot read Dickens,' and are not ashamed to glory in their shame. We ought not to be angry with others for their misfortunes; and yet when one meets the \textit{cretins} who boast that they cannot read Dickens, one certainly does feel much as Mr. Samuel Weller felt when he encountered Mr. Job Trotter.
How very singular has been the history of the decline of humour. Is there any profound psychological truth to be gathered from consideration of the fact that humour has gone out with cruelty? A hundred years ago, eighty years ago---nay, fifty years ago---we were a cruel but also a humorous people. We had bull-baitings, and badger-drawings, and hustings, and prize-fights, and cock-fights; we went to see men hanged; the pillory and the stocks were no empty `terrors unto evil-doers,' for there was commonly a malefactor occupying each of these institutions. With all this we had a broad blown comic sense. We had Ho-garth, and Bunbury, and George Cruikshank, and Gilray; we had Leech and Surtees, and the creator of Tittlebat Titmouse; we had the Shepherd of the `Noctes,' and, above all, we had \textit{you}.
From the old giants of English fun---burly persons delighting in broad caricature, in decided colours, in cockney jokes, in swashing blows at the more prominent and obvious human follies---from these you derived the splendid high spirits and unhesitating mirth of your earlier works. Mr. Squeers, and Sam Weller, and Mrs. Gamp, and all the Pickwickians, and Mr. Dowler, and John Browdie---these and their immortal companions were reared, so to speak, on the beef and beer of that naughty, fox-hunting, badger-baiting old England, which we have improved out of existence. And these characters, assuredly, are your best; by them, though stupid people cannot read about them, you will live while there is a laugh left among us. Perhaps that does not assure you a very prolonged existence, but only the future can show.
The dismal seriousness of the time cannot, let us hope, last for ever and a day. Honest old Laughter, the true \textit{lutin} of your inspiration, must have life left in him yet, and cannot die; though it is true that the taste for your pathos, and your melodrama, and plots constructed after your favourite fashion ('Great Expectations' and the 'Tale of Two Cities' are exceptions) may go by and never be regretted. Were people simpler, or only less clear-sighted, as far as your pathos is concerned, a generation ago? Jeffrey, the hard-headed shallow critic, who declared that Wordsworth `would never do,' cried, `wept like anything,' over your Little Nell. One still laughs as heartily as ever with Dick Swiveller; but who can cry over Little Nell?
Ah, Sir, how could you---who knew so intimately, who remembered so strangely well the fancies, the dreams, the sufferings of childhood---how could you `wallow naked in the pathetic,' and massacre holocausts of the Innocents? To draw tears by gloating over a child's death-bed, was it worthy of you? Was it the kind of work over which our hearts should melt? I confess that Little Nell might die a dozen times, and be welcomed by whole legions of Angels, and I (like the bereaved fowl mentioned by Pet Marjory) would remain unmoved.
She was more than usual calm,
She did not give a single dam,
wrote the astonishing child who diverted the leisure of Scott. Over your Little Nell and your Little Dombey I remain more than usual calm; and probably so do thousands of your most sincere admirers. But about matter of this kind, and the unsealing of the fountains of tears, who can argue? Where is taste? where is truth? What tears are 'manly, Sir, manly,' as Fred Bayham has it; and of what lamentations ought we rather to be ashamed? \textit{Sunt lacrymae rerum}; one has been moved in the cell where Socrates tasted the hemlock; or by the river-banks where Syracusan arrows slew the parched Athenians among the mire and blood; or, in fiction, when Colonel Newcome said \textit{Adsum}, or over the diary of Clare Doria Forey, or where Aramis laments, with strange tears, the death of Porthos. But over Dombey (the Son), or Little Nell, one declines to snivel.
When an author deliberately sits down and says, 'Now, let us have a good cry,' he poisons the wells of sensibility and chokes, at least in many breasts, the fountain of tears. Out of `Dombey and Son' there is little we care to remember except the deathless Mr. Toots; just as we forget the melodramatics of `Martin Chuzzlewit.' I have read in that book a score of times; I never see it but I revel in it---in Pecksniff, and Mrs. Gamp, and the Americans. But what the plot is all about, what Jonas did, what Montagu Tigg had to make in the matter, what all the pictures with plenty of shading illustrate, I have never been able to comprehend. In the same way, one of your most thorough-going admirers has allowed (in the licence of private conversation) that 'Ralph Nickleby and Monk are too steep;' and probably a cultivated taste will always find them a little precipitous.
'Too steep:'---the slang expresses that defect of an ardent genius, carried above itself, and out of the air we breathe, both in its grotesque and in its gloomy imaginations. To force the note, to press fantasy too hard, to deepen the gloom with black over the indigo, that was the failing which proved you mortal. To take an instance in little: when Pip went to Mr. Pumblechook's, the boy thought the seedsman 'a very happy man to have so many little drawers in his shop.' The reflection is thoroughly boyish; but then you add, 'I wondered whether the flower-seeds and bulbs ever wanted of a fine day to break out of those jails and bloom.' That is not boyish at all; that is the hard-driven, jaded literary fancy at work.
`So we arraign her; but she,' the Genius of Charles Dickens, how brilliant, how kindly, how beneficent she is! dwelling by a fountain of laughter imperishable; though there is something of an alien salt in the neighbouring fountain of tears. How poor the world of fancy would be, how `dispeopled of her dreams,' if, in some ruin of the social system, the books of Dickens were lost; and if The Dodger, and Charley Bates, and Mr. Crinkle, and Miss Squeers, and Sam Weller, and Mrs. Gamp, and Dick Swiveller were to perish, or to vanish with Menander's men and women! We cannot think of our world without them; and, children of dreams as they are, they seem more essential than great statesmen, artists, soldiers, who have actually worn flesh and blood, ribbons and orders, gowns and uniforms. May we not almost welcome `Free Education'? for every Englishman who can read, unless he be an Ass, is a reader the more for you.
\gutchapter{III. To Pierre de Ronsard (Prince of Poets.)}
Master and Prince of Poets,---As we know what choice thou madest of a sepulchre (a choice how ill fulfilled by the jealousy of Fate), so we know well the manner of thy chosen immortality. In the Plains Elysian, among the heroes and the ladies of old song, there was thy Love with thee to enjoy her paradise in an eternal spring.
La' du plaisant Avril la saison imortelle
Sans eschange le suit,
La terre sans labeur, de sa grasse mamelle,
Tout chose y produit;
D'enbas la troupe sainte autrefois amoureuse,
Nous honorant sur tous,
Viendra nous saluer, s'estimant bien-heureuse
De s'accointer de nous.
There thou dwellest, with the learned lovers of old days, with Belleau, and Du Bellay, and Bai'f, and the flower of the maidens of Anjou. Surely no rumour reaches thee, in that happy place of reconciled affections, no rumour of the rudeness of Time, the despite of men, and the change which stole from thy locks, so early grey, the crown of laurels and of thine own roses. How different from thy choice of a sepulchre have been the fortunes of thy tomb!
I will that none should break
The marble for my sake,
Wishful to make more fair
My sepulchre.
So didst thou sing, or so thy sweet numbers run in my rude English. Wearied of Courts and of priories, thou didst desire a grave beside thine own Loire, not remote from
The caves, the founts that fall
From the high mountain wall,
That fall and flash and fleet,
Wilh silver fret.
Only a laurel tree
Shall guard the grave of me;
Only Apollo's bough
Shall shade me now!
Far other has been thy sepulchre: not in the free air, among the field flowers, but in thy priory of Saint Cosme, with marble for a monument, and no green grass to cover thee. Restless wert thou in thy life; thy dust was not to be restful in thy death. The Huguenots, \textit{ces nouveaux Chretiens qui la France ont pillee}, destroyed thy tomb, and the warning of the later monument,
ABI, NEFASTE, QUAM CALCAS HUMUM SACRA EST,
has not scared away malicious men. The storm that passed over France a hundred years ago, more terrible than the religious wars that thou didst weep for, has swept the column from the tomb. The marble was broken by violent hands, and the shattered sepulchre of the Prince of Poets gained a dusty hospitality from the museum of a country town. Better had been the laurel of thy desire, the creeping vine, and the ivy tree.
Scarce more fortunate, for long, than thy monument was thy memory. Thou hast not encountered, Master, in the Paradise of Poets, Messieurs Malherbe, De Balzac, and Boileau---Boileau who spoke of thee as \textit{Ce poete orgueilleux trebuche de si haut!}
These gallant gentlemen, I make no doubt, are happy after their own fashion, backbiting each other and thee in the Paradise of Critics. In their time they wrought thee much evil, grumbling that thou wrotest in Greek and Latin (of which tongues certain of them had but little skill), and blaming thy many lyric melodies and the free flow of thy lines. What said M. de Balzac to M. Chapelain? 'M. de Malherbe, M. de Grasse, and yourself must be very little poets, if Ronsard be a great one.' Time has brought in his revenges, and Messieurs Chapelain and De Grasse are as well forgotten as thou art well remembered. Men could not always be deaf to thy sweet old songs, nor blind to the beauty of thy roses and thy loves. When they took the wax out of their ears that M. Boileau had given them lest they should hear the singing of thy Sirens, then they were deaf no longer, then they heard the old deaf poet singing and made answer to his lays. Hast thou not heard these sounds? have they not reached thee, the voices and the lyres of Theophile Gautier and Alfred de Musset? Methinks thou hast marked them, and been glad that the old notes were ringing again and the old French lyric measures tripping to thine ancient harmonies, echoing and replying to the Muses of Horace and Catullus. Returning to Nature, poets returned to thee. Thy monument has perished, but not thy music, and the Prince of Poets has returned to his own again in a glorious Restoration.
Through the dust and smoke of ages, and through the centuries of wars we strain our eyes and try to gain a glimpse of thee, Master, in thy good days, when the Muses walked with thee. We seem to mark thee wandering silent through some little village, or dreaming in the woods, or loitering among thy lonely places, or in gardens where the roses blossom among wilder flowers, or on river banks where the whispering poplars and sighing reeds make answer to the murmur of the waters. Such a picture hast thou drawn of thyself in the summer afternoons.
Je m'en vais pourmener tantost parmy la plaine,
Tantost en un village, et tantost en un bois,
Et tantost par les lieux solitaires et cois.
J'aime fort les jardins qui sentent le sauvage,
J'aime le flot de l'eau qui gazou'ille au rivage.
Still, methinks, there was a book in the hand of the grave and learned poet; still thou wouldst carry thy Horace, thy Catullus, thy Theocritus, through the gem-like weather of the \textit{Renouveau}, when the woods were enamelled with flowers, and the young Spring was lodged, like a wandering prince, in his great palaces hung with green:
Orgueilleux de ses fleurs, enfle de sa jeunesse,
Loge comme un grand Prince en ses vertes maisons!
Thou sawest, in these woods by Loire side, the fair shapes of old religion, Fauns, Nymphs, and Satyrs, and heard'st in the nightingale's music the plaint of Philomel. The ancient poets came back in the train of thyself and of the Spring, and learning was scarce less dear to thee than love; and thy ladies seemed fairer for the names they borrowed from the beauties of forgotten days, Helen and Cassandra. How sweetly didst thou sing to them thine old morality, and how gravely didst thou teach the lesson of the Roses! Well didst thou know it, well didst thou love the Rose, since thy nurse, carrying thee, an infant, to the holy font, let fall on thee the sacred water brimmed with floating blossoms of the Rose!
Mignonne, allons voir si la Rose,
Qui ce matin avoit desclose
Sa robe de pourpre au soleil,
A point perdu ceste vespree
Les plis de sa robe pourpree,
Et son teint au votre pareil.
And again,
La belle Rose du Printemps,
Aubert, admoneste les hommes
Passer joyeusement le temps,
Et pendant que jeunes nous sommes,
Esbattre la fleur de nos ans.
In the same mood, looking far down the future, thou sangest of thy lady's age, the most sad, the most beautiful of thy sad and beautiful lays; for if thy bees gathered much honey 't was somewhat bitter to taste, as that of the Sardinian yews. How clearly we see the great hall, the grey lady spinning and humming among her drowsy maids, and how they waken at the word, and she sees her spring in their eyes, and they forecast their winter in her face, when she murmurs ''Twas Ronsard sang of me.'
Winter, and summer, and spring, how swiftly they pass, and how early time brought thee his sorrows, and grief cast her dust upon thy head.
Adieu ma Lyre, adieu fillettes,
Jadis mes douces amourettes,
Adieu, je sens venir ma fin,
Nul passetemps de ma jeunesse
Ne m'accompagne en la vieillesse,
Que le feu, le lict et le vin.
Wine, and a soft bed, and a bright fire: to this trinity of poor pleasures we come soon, if, indeed, wine be left to us. Poetry herself deserts us; is it not said that Bacchus never forgives a renegade? and most of us turn recreants to Bacchus. Even the bright fire, I fear, was not always there to warm thine old blood, Master, or, if fire there were, the wood was not bought with thy book-seller's money. When autumn was drawing in during thine early old age, in 1584, didst thou not write that thou hadst never received a sou at the hands of all the publishers who vended thy books? And as thou wert about putting forth the folio edition of 1584, thou didst pray Buon, the bookseller, to give thee sixty crowns to buy wood withal, and make thee a bright fire in winter weather, and comfort thine old age with thy friend Gallandius. And if Buon will not pay, then to try the other book-sellers, 'that wish to take everything and give nothing.'
Was it knowledge of this passage, Master, or ignorance of everything else, that made certain of the common steadfast dunces of our days speak of thee as if thou hadst been a starveling, neglected poetaster, jealous forsooth, of Maitre Francoys Rabelais? See how ignorantly M. Fleury writes, who teaches French literature withal to them of Muscovy, and hath indited a Life of Rabelais. 'Rabelais etait revetu d'un emploi honorable; Ronsard etait traite en subalterne,' quoth this wondrous professor. What! Pierre de Ronsard, a gentleman of a noble house, holding the revenue of many abbeys, the friend of Mary Stuart, of the Duc d'Orleans, of Charles IX., \textit{he} is \textit{traite en subalterne}, and is jealous of a frocked or unfrocked \textit{manant} like Maitre Francoys! And then this amazing Fleury falls foul of thine epitaph on Mai'tre Francoys and cries, 'Ronsard a voulu faire des vers mechants; il n'a fait que de mechants vers.' More truly saith M. Sainte-Beuve, 'If the good Rabelais had returned to Meudon on the day when this epitaph was made over the wine, he would, methinks, have laughed heartily.' But what shall be said of a Professor like the egregious M. Fleury, who holds that Ronsard was despised at Court? Was there a party at tennis when the king would not fain have had thee on his side, declaring that he ever won when Ronsard was his partner? Did he not give thee benefices, and many priories, and call thee his father in Apollo, and even, so they say, bid thee sit down beside him on his throne? Away, ye scandalous folk, who tell us that there was strife between the Prince of Poets and the King of Mirth. Naught have ye by way of proof of your slander but the talk of Jean Bernier, a scurrilous, starveling apothecary, who put forth his fables in 1697, a century and a half after Mai'tre Francoys died. Bayle quoted this fellow in a note, and ye all steal the tattle one from another in your dull manner, and know not whence it comes, nor even that Bayle would none of it and mocked its author. With so little knowledge is history written, and thus doth each chattering brook of a 'Life swell with its tribute, that great Mississippi of falsehood,' Biography.
\gutchapter{IV. To Herodotus.}
To Herodotus of Halicarnassus, greeting.---Concerning the matters set forth in your histories, and the tales you tell about both Greeks and barbarians, whether they be true, or whether they be false, men dispute not little but a great deal. Wherefore I, being concerned to know the verity, did set forth to make search in every manner, and came in my quest even unto the ends of the earth. For there is an island of the Cimmerians beyond the Straits of Heracles, some three days' voyage to a ship that hath a fair following wind in her sails; and there it is said that men know many things from of old: thither, then, I came in my inquiry. Now, the island is not small, but large, greater than the whole of Hellas; and they call it Britain. In that island the east wind blows for ten parts of the year, and the people know not how to cover themselves from the cold. But for the other two months of the year the sun shines fiercely, so that some of them die thereof, and others die of the frozen mixed drinks; for they have ice even in the summer, and this ice they put to their liquor. Through the whole of this island, from the west even to the east, there flows a river called Thames: a great river and a laborious, but not to be likened to the River of Egypt.
The mouth of this river, where I stepped out from my ship, is exceedingly foul and of an evil savour by reason of the city on the banks. Now this city is several hundred parasangs in circumference. Yet a man that needed not to breathe the air might go round it in one hour, in chariots that run under the earth; and these chariots are drawn by creatures that breathe smoke and sulphur, such as Orpheus mentions in his `Argonautica,' if it be by Orpheus. The people of the town, when I inquired of them concerning Herodotus of Halicarnassus, looked on me with amazement, and went straightway about their business,---namely, to seek out whatsoever new thing is coming to pass all over the whole inhabited world, and as for things old, they take no keep of them.
Nevertheless, by diligence I learned that he who in this land knew most concerning Herodotus was a priest, and dwelt in the priests' city on the river which is called the City of the Ford of the Ox. But whether Io, when she wore a cow's shape, had passed by that way in her wanderings, and thence comes the name of that city, I could not (though I asked all men I met) learn aught with certainty. But to me, considering this, it seemed that Io must have come thither. And now farewell to Io.
To the City of the Priests there are two roads: one by land; and one by water, following the river. To a well-girdled man, the land journey is but one day's travel; by the river it is longer but more pleasant. Now that river flows, as I said, from the west to the east. And there is in it a fish called chub, which they catch; but they do not eat it, for a certain sacred reason. Also there is a fish called trout, and this is the manner of his catching. They build far this purpose great dams of wood, which they call weirs. Having built the weir they sit upon it with rods in their hands, and a line on the rod, and at the end of the line a little fish. There then they `sit and spin in the sun,' as one of their poets says, not for a short time but for many days, having rods in their hands and eating and drinking. In this wise they angle for the fish called trout; but whether they ever catch him or not, not having seen it, I cannot say; for it is not pleasant to me to speak things concerning which I know not the truth.
Now, after sailing and rowing against the stream for certain days, I came to the City of the Ford of the Ox. Here the river changes his name, and is called Isis, after the name of the goddess of the Egyptians. But whether the Britons brought the name from Egypt or whether the Egyptians took it from the Britons, not knowing I prefer not to say. But to me it seems that the Britons are a colony of the Egyptians, or the Egyptians a colony of the Britons. Moreover, when I was in Egypt I saw certain soldiers in white helmets, who were certainly British. But what they did there (as Egypt neither belongs to Britain nor Britain to Egypt) I know not, neither could they tell me. But one of them replied to me in that line of Homer (if the Odyssey be Homer's), 'We have come to a sorry Cyprus, and a sad Egypt.' Others told me that they once marched against the Ethiopians, and having defeated them several times, then came back again, leaving their property to the Ethiopians. But as to the truth of this I leave it to every man to form his own opinion.
Having come into the City of the Priests, I went forth into the street, and found a priest of the baser sort, who for a piece of silver led me hither and thither among the temples, discoursing of many things.
Now it seemed to me a strange thing that the city was empty, and no man dwelling therein, save a few priests only, and their wives, and their children, who are drawn to and fro in little carriages dragged by women, but the priest told me that during half the year the city was desolate, for that there came somewhat called `The Long,' or `The Vac,' and drave out the young priests. And he said that these did no other thing but row boats, and throw balls from one to the other, and this they were made to do, he said, that the young priests might learn to be humble, for they are the proudest of men. But whether he spoke truth or not I know not, only I set down what he told me. But to anyone considering it, this appears rather to jump with his story---namely, that the young priests have houses on the river, painted of divers colours, all of them empty.
Then the priest, at my desire, brought me to one of the temples, that I might seek out all things concerning Herodotus the Halicarnassian, from one who knew. Now this temple is not the fairest in the city, but less fair and goodly than the old temples, yet goodlier and more fair than the new temples; and over the roof there is the image of an eagle made of stone---no small marvel, but a great one, how men came to fashion him; and that temple is called the House of Queens. Here they sacrifice a boar once every year; and concerning this they tell a certain sacred story which I know but will not utter.
Then I was brought to the priest who had a name for knowing most about Egypt, and the Egyptians, and the Assyrians, and the Cappadocians, and all the kingdoms of the Great King. He came out to me, being attired in a black robe, and wearing on his head a square cap. But why the priests have square caps I know, and he who has been initiated into the mysteries which they call `Matric' knows, but I prefer not to tell. Concerning the square cap, then, let this be sufficient. Now, the priest received me courteously, and when I asked him, concerning Herodotus, whether he were a true man or not, he smiled, and answered `Abu Goosh,' which, in the tongue of the Arabians, means `The Father of Liars.' Then he went on to speak concerning Herodotus, and he said in his discourse that Herodotus not only told the thing which was not, but that he did so wilfully, as one knowing the truth but concealing it. For example, quoth he, 'Solon never went to see Croesus, as Herodotus avers; nor did those about Xerxes ever dream dreams; but Herodotus, out of his abundant wickedness, invented these things.
`Now behold,' he went on, 'how the curse of the Gods falls upon Herodotus. For he pretends that he saw Cadmeian inscriptions at Thebes. Now I do not believe there were any Cadmeian inscriptions there: therefore Herodotus is most manifestly lying. Moreover, this Herodotus never speaks of Sophocles the Athenian, and why not? Because he, being a child at school, did not learn Sophocles by heart: for the tragedies of Sophocles could not have been learned at school before they were written, nor can any man quote a poet whom he never learned at school. Moreover, as all those about Herodotus knew Sophocles well, he could not appear to them to be learned by showing that he knew what they knew also.' Then I thought the priest was making game and sport, saying first that Herodotus could know no poet whom he had not learned at school, and then saying that all the men of his time well knew this poet, 'about whom everyone was talking'. But the priest seemed not to know that Herodotus and Sophocles were friends, which is proved by this, that Sophocles wrote an ode in praise of Herodotus.
Then he went on, and though I were to write with a hundred hands (like Briareus, of whom Homer makes mention) I could not tell you all the things that the priest said against Herodotus, speaking truly, or not truly, or sometimes correctly and sometimes not, as often befalls mortal men. For Herodotus, he said, was chiefly concerned to steal the lore of those who came before him, such as Hecataeus, and then to escape notice as having stolen it. Also he said that, being himself cunning and deceitful, Herodotus was easily beguiled by the cunning of others, and believed in things manifestly false, such as the story of the Phoenix-bird.
Then I spoke, and said that Herodotus himself declared that he could not believe that story; but the priest regarded me not. And he said that Herodotus had never caught a crocodile with cold pig, nor did he ever visit Assyria, nor Babylon, nor Elephantine; but, saying that he had been in these lands, said that which was not true. He also declared that Herodotus, when he travelled, knew none of the Fat Ones of the Egyptians, but only those of the baser sort. And he called Herodotus a thief and a beguiler, and `the same with intent to deceive,' as one of their own poets writes, and, to be short, Herodotus, I could not tell you in one day all the charges which are now brought against you; but concerning the truth of these things, \textit{you} know, not least, but most, as to yourself being guilty or innocent. Wherefore, if you have anything to show or set forth whereby you may be relieved from the burden of these accusations, now is the time. Be no more silent; but, whether through the Oracle of the Dead, or the Oracle of Branchidae, or that in Delphi, or Dodona, or of Amphiaraus at Oropus, speak to your friends and lovers (whereof I am one from of old) and let men know the very truth.
Now, concerning the priests in the City of the Ford of the Ox, it is to be said that of all men whom we know they receive strangers most gladly, feasting them all day. Moreover, they have many drinks, cunningly mixed, and of these the best is that they call Archdeacon, naming it from one of the priests' offices. Truly, as Homer says (if the Odyssey be Homer's), 'when that draught is poured into the bowl then it is no pleasure to refrain.'
Drinking of this wine, or nectar, Herodotus, I pledge you, and pour forth some deal on the ground, to Herodotus of Halicarnassus, in the House of Hades.
And I wish you farewell, and good be with you. Whether the priest spoke truly, or not truly, even so may such good things betide you as befall dead men.
\gutchapter{V. Epistle to Mr. Alexander Pope.}
From mortal Gratitude, decide, my Pope,
Have Wits Immortal more to fear or hope?
Wits toil and travail round the Plant of Fame,
Their Works its Garden, and its Growth their Aim,
Then Commentators, in unwieldy Dance,
Break down the Barriers of the trim Pleasance,
Pursue the Poet, like Actaeon's Hounds,
Beyond the fences of his Garden Grounds,
Rend from the singing Robes each borrowed gem,
Rend from the laurel'd Brows the Diadem,
And, if one Rag of Character they spare,
Comes the Biographer, and strips it bare!
Such, Pope, has been thy Fortune, such thy Doom.
Swift the Ghouls gathered at the Poet's Tomb,
With Dust of Notes to clog each lordly Line,
Warburton, Warton, Croker, Bowles, combine!
Collecting Cackle, Johnson condescends
To \textit{interview} the Drudges of your Friends.
Though still your Courthope holds your merits high,
And still proclaims your Poems poetry,
Biographers, un-Boswell-like, have sneered,
And Dunces edit him whom Dunces feared!
They say; what say they? Not in vain You ask.
To tell you what they say, behold my Task!
`Methinks already I your Tears survey'
As I repeat `the horrid Things they say.' (1)
(1) \textit{Rape of the Lock}.
Comes El---n first: I fancy you'll agree
Not frenzied Dennis smote so fell as he;
For El---n's Introduction, crabbed and dry,
Like Churchill's Cudgel's (2) marked with Lie, and Lie!
(2) In Mr Hogarth's Caricatura.
'Too dull to know what his own System meant,
Pope yet was skilled new Treasons to invent;
A Snake that puffed himself and stung his Friends,
Few Lied so frequent, for such little Ends;
His mind, like Flesh inflamed, (3) was raw and sore,
And still, the more he writhed, he stung the more!
Oft in a Quarrel, never in the Right,
His Spirit sank when he was called to fight.
Pope, in the Darkness mining like a Mole,
Forged on Himself, as from Himself he stole,
And what for Caryll once he feigned to feel,
Transferred, in Letters never sent, to Steele!
Still he denied the Letters he had writ,
And still mistook Indecency for Wit.
His very Grammar, so De Quincey cries,
``Detains the Reader, and at times defies!"'
(3) Elwyn's Pope, ii. 15.
Fierce El---n thus: no Line escapes his Rage,
And furious Foot-notes growl 'neath every Page:
See St-ph-n next take up the woful Tale,
Prolong the Preaching, and protract the Wail!
'Some forage Falsehoods from the North and South,
But Pope, poor D----l, lied from Hand to Mouth; (1)
Affected, hypocritical, and vain,
A Book in Breeches, and a Fop in Grain;
A Fox that found not the high Clusters sour,
The Fanfaron of Vice beyond his power,
Pope yet possessed'---(the Praise will make you start)---
'Mean, morbid, vain, he yet possessed a Heart!
And still we marvel at the Man, and still
Admire his Finish, and applaud his Skill:
Though, as that fabled Barque, a phantom Form,
Eternal strains, nor rounds the Cape of Storm,
Even so Pope strove, nor ever crossed the Line
That from the Noble separates the Fine!'
(1) `Poor Pope was always a hand-to-mouth liar.'
---\textit{Pope}, by Leslie Stephen, 139.
The Learned thus, and who can quite reply,
Reverse the Judgment, and Retort the Lie?
You reap, in armed Hates that haunt Your name,
Reap what you sowed, the Dragon's Teeth of Fame:
You could not write, and from unenvious Time
Expect the Wreath that crowns the lofty Rhyme,
You still must fight, retreat, attack, defend,
And oft, to snatch a Laurel, lose a Friend!
The Pity of it! And the changing Taste
Of changing Time leaves half your Work a Waste!
My Childhood fled your couplet's clarion tone,
And sought for Homer in the Prose of Bohn.
Still through the Dust of that dim Prose appears
The Flight of Arrows and the Sheen of Spears;
Still we may trace what Hearts heroic feel,
And hear the Bronze that hurtles on the Steel!
But, ah, your Iliad seems a half-pretence,
Where Wits, not Heroes, prove their Skill in Fence,
And great Achilles' Eloquence doth show
As if no Centaur trained him, but Boileau!
Again, your Verse is orderly,---and more,---
`The Waves behind impel the Waves before;'
Monotonously musical they glide,
Till Couplet unto Couplet hath replied.
But turn to Homer! How his Verses sweep!
Surge answers Surge and Deep doth call on Deep;
This Line in Foam and Thunder issues forth,
Spurred by the West or smitten by the North,
Sombre in all its sullen Deeps, and all
Clear at the Crest, and foaming to the Fall,
The next with silver Murmur dies away,
Like Tides that falter to Calypso's Bay!
Thus Time, with sordid Alchemy and dread,
Turns half the Glory of your Gold to Lead;
Thus Time,---at Ronsard's wreath that vainly bit,---
Has marred the Poet to preserve the Wit,
Who almost left on Addison a stain,
Whose knife cut cleanest with a poisoned pain,---
Yet Thou (strange Fate that clings to all of Thine!)
When most a Wit dost most a Poet shine.
In Poetry thy Dunciad expires,
When Wit has shot `her momentary Fires.'
'T is Tragedy that watches by the Bed
`Where tawdry Yellow strove with dirty Red,'
And men, remembering all, can scarce deny
To lay the Laurel where thine Ashes lie!
\section*{\raggedright VI. To Lucian of Samosata.}
In what bower, oh Lucian, of your rediscovered Islands Fortunate are you
now reclining; the delight of the fair, the learned, the witty, and the
brave? In that clear and tranquil climate, whose air breathes of 'violet
and lily, myrtle, and the flower of the vine,'
Where the daisies are rose-scented,
And the Rose herself has got
Perfume which on earth is not,
among the music of all birds, and the wind-blown notes of flutes hanging
on the trees, methinks that your laughter sounds most silvery sweet,
and that Helen and fair Charmides are still of your company. Master of
mirth, and Soul the best contented of all that have seen the world's
ways clearly, most clear-sighted of all that have made tranquillity
their bride, what other laughers dwell with you, where the crystal and
fragrant waters wander round the shining palaces and the temples of
amethyst?
Heine surely is with you; if, indeed, it was not one Syrian soul that
dwelt among alien men, Germans and Romans, in the bodily tabernacles of
Heine and of Lucian. But he was fallen on evil times and evil tongues;
while Lucian, as witty as he, as bitter in mockery, as happily dowered
with the magic of words, lived long and happily and honoured, imprisoned
in no `mattress-grave.' Without Rabelais, without Voltaire, without
Heine, you would find, methinks, even the joys of your Happy Islands
lacking in zest; and, unless Plato came by your way, none of the
ancients could meet you in the lists of sportive dialogue.
There, among the vines that bear twelve times in the year, more
excellent than all the vineyards of Touraine, while the song-birds bring
you flowers from vales enchanted, and the shapes of the Blessed come
and go, beautiful in wind-woven raiment of sunset hues; there, in a land
that knows not age nor winter, midnight, nor autumn, nor noon, where the
silver twilight of summer-dawn is perennial, where youth does not wax
spectre-pale and die; there, my Lucian, you are crowned the Prince of
the Paradise of Mirth.
Who would bring you, if he had the power, from the banquet where Homer
sings: Homer, who, in mockery of commentators, past and to come, German
and Greek, informed you that he was by birth a Babylonian? Yet, if
you, who first wrote Dialogues of the Dead, could hear the prayer of
an epistle wafted to `lands indiscoverable in the unheard-of West,' you
might visit once more a world so worthy of such a mocker, so like the
world you knew so well of old.
Ah, Lucian, we have need of you, of your sense and of your mockery!
Here, where faith is sick and superstition is waking afresh; where gods
come rarely, and spectres appear at five shillings an interview; where
science is popular, and philosophy cries aloud in the market-place,
and clamour does duty for government, and Thais and Lais are names of
power---here, Lucian, is room and scope for you. Can I not imagine a
new `Auction of Philosophers,' and what wealth might be made by him who
bought these popular sages and lecturers at his estimate, and vended
them at their own?
HERMES: Whom shall we put first up to auction?
ZEUS: That German in spectacles; he seems a highly respectable man.
HERMES: Ho, pessimist, come down and let the public view you.
ZEUS: Go on, put him up and have done with him.
HERMES: Who bids for the Life Miserable, for extreme, complete, perfect,
unredeemable perdition? What offers for the universal extinction of the
species, and the collapse of the Conscious?
A PURCHASER: He does not look at all a bad lot. May one put him through
his paces?
HERMES: Certainly; try your luck.
PURCHASER: What is your name?
PESSIMIST: Hartmann.
PURCHASER: What can you teach me?
PESSIMIST: That Life is not worth Living.
PURCHASER: Wonderful! Most edifying! How much for this lot?
HERMES: Two hundred pounds.
PURCHASER: I will write you a cheque for the money. Come home,
Pessimist, and begin your lessons without more ado.
HERMES: Attention! Here is a magnificent article---the Positive Life, the
Scientific Life, the Enthusiastic Life. Who bids for a possible place in
the Calendar of the Future?
PURCHASER: What does he call himself? he has a very French air.
HERMES: Put your own questions.
PURCHASER: What's your pedigree, my Philosopher, and previous
performances?
POSITIVIST: I am by Rousseau out of Catholicism, with a strain of the
Evolution blood.
PURCHASER: What do you believe in?
POSITIVIST: In Man, with a large M.
PURCHASER: Not in individual Man?
POSITIVIST: By no means; not even always in Mr. Gladstone. All men, all
Churches, all parties, all philosophies, and even the other sect of our
own Church, are perpetually in the wrong. Buy me, and listen to me, and
you will ahvays be in the right.
PURCHASER: And, after this life, what have you to offer me?
POSITIVIST: A distinguished position in the Choir Invisible: but not, of
course, conscious immortality.
PURCHASER: Take him away, and put up another lot.
Then the Hegelian, with his Notion, and the Darwinian, with his
notions, and the Lotzian, with his Broad Church mixture of Religion and
Evolution, and the Spencerian, with that Absolute which is a sort of
a something, might all be offered with their divers wares; and cheaply
enough, Lucian, you would value them in this auction of Sects. 'There is
but one way to Corinth,' as of old; but which that way may be, oh master
of Hermotimus, we know no more than he did of old; and still we find, of
all philosophies, that the Stoic route is most to be recommended. But
we have our Cyrenaics too, though they are no longer 'clothed in
purple, and crowned with flowers, and fond of drink and of female
flute-players.' Ah, here too, you might laugh, and fail to see where the
Pleasure lies, when the Cyrenaics are no `judges of cakes' (nor of
ale, for that matter), and are strangers in the Courts of Princes. 'To
despise all things, to make use of all things, in all things to follow
pleasure only:' that is not the manner of the new, if it were the secret
of the older Hedonism.
Then, turning from the philosophers to the seekers after a sign, what
change, Lucian, would you find in them and their ways? None; they are
quite unaltered. Still our Perigrinus, and our Perigrina too, come to
us from the East, or, if from the West, they take India on their
way---India, that secular home of drivelling creeds, and of religion in
its sacerdotage. Still they prattle of Brahmins and Buddhism; though,
unlike Peregrinus, they do not publicly burn themselves on pyres, at
Epsom Downs, after the Derby. We are not so fortunate in the demise of
our Theosophists; and our police, less wise than the Hellenodicae, would
probably not permit the Immolation of the Quack. Like your Alexander,
they deal in marvels and miracles, oracles and warnings. All such bogy
stories as those of your `Philopseudes,' and the ghost of the lady who
took to table-rapping because one of her best slippers had not been
burned with her body, are gravely investigated by the Psychical Society.
Even your ignorant Bibliophile is still with us---the man without a tinge
of letters, who buys up old manuscripts 'because they are stained and
gnawed, and who goes, for proof of valued antiquity, to the testimony
of the book-worms.' And the rich Bibliophile now, as in your satire,
clothes his volumes in purple morocco and gay \textit{dorures}, while their
contents are sealed to him.
As to the topics of satire and gay curiosity which occupy the lady known
as `Gyp,' and M. Halevy in his `Les Petites Cardinal,' if you had not
exhausted the matter in your `Dialogues of Hetairai,' you would be
amused to find the same old traits surviving without a touch of change.
One reads, in Halevy's French, of Madame Cardinal, and, in your Greek,
of the mother of Philinna, and marvels that eighteen hundred years
have not in one single trifle altered the mould. Still the old shabby
light-loves, the old greed, the old luxury and squalor. Still the
unconquerable superstition that now seeks to tell fortunes by the
cards, and, in your time, resorted to the sorceress with her magical
`bull-roarer' or `\textit{turndun}.' (1)
(1)The Greek \textit{rombos} [transliterated], mentioned by Lucian
and Theocritus, was the magical weapon of the Australians---
the \textit{turndun}.
Yes, Lucian, we are the same vain creatures of doubt and dread, of
unbelief and credulity, of avarice and pretence, that you knew, and at
whom you smiled. Nay, our very `social question' is not altered. Do
you not write, in `The Runaways,' 'The artisans will abandon their
workshops, and leave their trades, when they see that, with all the
labour that bows their bodies from dawn to dark, they make a petty and
starveling pittance, while men that toil not nor spin are floating in
Pactolus'?
They begin to see this again as of yore; but whether the end of their
vision will be a laughing matter, you, fortunate Lucian, do not need to
care. Hail to you, and farewell!
\gutchapter{VII. To Maitre Francoys Rabelais.}
Of the Coming of the Coqcigrues.
Master,---In the Boreal and Septentrional lands, turned aside from the noonday and the sun, there dwelt of old (as thou knowest, and as Olaus voucheth) a race of men, brave, strong, nimble, and adventurous, who had no other care but to fight and drink. There, by reason of the cold (as Virgil witnesseth), men break wine with axes. To their minds, when once they were dead and gotten to Valhalla, or the place of their Gods, there would be no other pleasure but to swig, tipple, drink, and boose till the coming of that last darkness and Twilight, wherein they, with their deities, should do battle against the enemies of all mankind; which day they rather desired than dreaded.
So chanced it also with Pantagruel and Brother John and their company, after they had once partaken of the secret of the \textit{Dive Bouteille}. Thereafter they searched no longer; but, abiding at their ease, were merry, frolic, jolly, gay, glad, and wise; only that they always and ever did expect the awful Coming of the Coqcigrues. Now concerning the day of that coming, and the nature of them that should come, they knew nothing; and for his part Panurge was all the more adread, as Aristotle testifieth that men (and Panurge above others) most fear that which they know least. Now it chanced one day, as they sat at meat, with viands rare, dainty, and precious as ever Apicius dreamed of, that there fluttered on the air a faint sound as of sermons, speeches, orations, addresses, discourses, lectures, and the like; whereat Panurge, pricking up his ears, cried, `Methinks this wind bloweth from Midlothian,' and so fell a trembling.
Next, to their aural orifices, and the avenues audient of the brain, was borne a very melancholy sound as of harmoniums, hymns, organ-pianos, psalteries, and the like, all playing different airs, in a kind most hateful to the Muses. Then said Panurge, as well as he might for the chattering of his teeth: 'May I never drink if here come not the Coqcigrues!' and this saying and prophecy of his was true and inspired. But thereon the others began to mock, flout, and gird at Panurge for his cowardice. `Here am I!' cried Brother John, 'well-armed and ready to stand a siege; being entrenched, fortified, hemmed-in and surrounded with great pasties, huge pieces of salted beef, salads, fricassees, hams, tongues, pies, and a wilderness of pleasant little tarts, jellies, pastries, trifles, and fruits of all kinds, and I shall not thirst while I have good wells, founts, springs, and sources of Bordeaux wine, Burgundy, wine of the Champagne country, sack and Canary. A fig for thy Coqcigrues!'
But even as he spoke there ran up suddenly a whole legion, or rather army, of physicians, each armed with laryngoscopes, stethoscopes, horoscopes, microscopes, weighing machines, and such other tools, engines, and arms as they had who, after thy time, persecuted Monsieur de Pourceaugnac! And they all, rushing on Brother John, cried out to him, `Abstain! Abstain!' And one said, 'I have well diagnosed thee, and thou art in a fair way to have the gout.' 'I never did better in my days,' said Brother John. `Away with thy meats and drinks!' they cried. And one said, `He must to Royat;' and another, `Hence with him to Aix;' and a third, `Banish him to Wiesbaden;' and a fourth, 'Hale him to Gastein;' and yet another, `To Barbouille with him in chains!'
And while others felt his pulse and looked at his tongue, they all wrote prescriptions for him like men mad. `For thy eating,' cried he that seemed to be their leader, `No soup!' `No soup!' quoth Brother John; and those cheeks of his, whereat you might have warmed your two hands in the winter solstice, grew white as lilies. 'Nay! and no salmon nor any beef nor mutton! A little chicken by times, but \textit{periculo tuo}! Nor any game, such as grouse, partridge, pheasant, capercailzie, wild duck; nor any cheese, nor fruit, nor pastry, nor coffee, nor eau de vie; and avoid all sweets. No veal, pork, nor made dishes of any kind.' 'Then what may I eat?' quoth the good Brother, whose valour had oozed out of the soles of his sandals. `A little cold bacon at breakfast---no eggs,' quoth the leader of the strange folk, `and a slice of toast without butter.' 'And for thy drink'---('What?' gasped Brother John)---'one dessert-spoonful of whisky, with a pint of the water of Apollinaris at luncheon and dinner. No more!' At this Brother John fainted, falling like a great buttress of a hill, such as Taygetus or Erymanthus.
While they were busy with him, others of the frantic folk had built great platforms of wood, whereon they all stood and spoke at once, both men and women. And of these some wore red crosses on their garments, which meaneth `Salvation;' and others wore white crosses, with a little black button of crape, to signify `Purity;' and others bits of blue to mean `Abstinence.' While some of these pursued Panurge others did beset Pantagruel; asking him very long questions, whereunto he gave but short answers. Thus they asked:
Have ye Local Option here?---Pan.: What?
May one man drink if his neighbour be not athirst?---Pan.: Yea!
Have ye Free Education?---Pan.: What?
Must they that have, pay to school them that have not?---Pan.: Nay
Have ye free land?---Pan.: What?
Have ye taken the land from the farmer, and given it to the tailor out of work and the candlemaker masterless?---Pan.: Nay!
Have your women folk votes?---Pan.: Bosh!
Have ye got religion?---Pan.: How?
Do you go about the streets at night, brawling, blowing a trumpet before you, and making long prayers?---Pan.: Nay
Have you manhood suffrage?---Pan.: Eh?
Is Jack as good as his master? Pan.: Nay!
Have you joined the Arbitration Society?---Pan.: \textit{Quoy?}?
Will you let another kick you, and will you ask his neighbour if you deserve the same?---Pan.: Nay?
Do you cat what you list?---Pan.: Ay!
Do you drink when you are athirst? Pan.: Ay!
Are you governed by the free expression of the popular will?---Pan.:
How?
Are you servants of priests, pulpits, and penny papers?---Pan.: No!
Now, when they heard these answers of Pantagruel they all fell, some a weeping, some a praying, some a swearing, some an arbitrating, some a lecturing, some a caucussing, some a preaching, some a faith-healing, some a miracle-working, some a hypnotising, some a writing to the daily press; and while they were thus busy, like folk distraught, 'reforming the island,' Pantagruel burst out a laughing; whereat they were greatly dismayed; for laughter killeth the whole race of Coqcigrues, and they may not endure it.
Then Pantagruel and his company stole aboard a barque that Panurge had ready in the harbour. And having provisioned her well with store of meat and good drink, they set sail for the kingdom of Entelechy, where, having landed, they were kindly entreated; and there abide to this day; drinking of the sweet and eating of the fat, under the protection of that intellectual sphere which hath in all places its centre and nowhere its circumference.
Such was their destiny; there was their end appointed, and thither the Coqcigrues can never come. For all the air of that land is full of laughter, which killeth Coqcigrues; and there aboundeth the herb Pantagruelion. But for thee, Master Francoys, thou art not well liked in this island of ours, where the Coqcigrues are abundant, very fierce, cruel, and tyrannical. Yet thou hast thy friends, that meet and drink to thee and wish thee well wheresoever thou hast found thy \textit{grand peut-etre}.
\gutchapter{VIII. To Jane Austen.}
Madame,---If to the enjoyments of your present state be lacking a view of the minor infirmities or foibles of men, I cannot but think (were the thought permitted) that your pleasures are yet incomplete. Moreover, it is certain that a woman of parts who has once meddled with literature will never wholly lose her love for the discussion of that delicious topic, nor cease to relish what (in the cant of our new age) is styled `literary shop.' For these reasons I attempt to convey to you some inkling of the present state of that agreeable art which you, madam, raised to its highest pitch of perfection.
As to your own works (immortal, as I believe), I have but little that is wholly cheering to tell one who, among women of letters, was almost alone in her freedom from a lettered vanity. You are not a very popular author: your volumes are not found in gaudy covers on every bookstall; or, if found, are not perused with avidity by the Emmas and Catherines of our generation. 'Tis not long since a blow was dealt (in the estimation of the unreasoning) at your character as an author by the publication of your familiar letters. The editor of these epistles, unfortunately, did not always take your witticisms, and he added others which were too unmistakably his own. While the injudicious were disappointed by the absence of your exquisite style and humour, the wiser sort were the more convinced of your wisdom. In your letters (knowing your correspondents) you gave but the small personal talk of the hour, for them sufficient; for your books you reserved matter and expression which are imperishable. Your admirers, if not very numerous, include all persons of taste, who, in your favour, are apt somewhat to abate the rule, or shake off the habit, which commonly confines them to but temperate laudation.
'T is the fault of all art to seem antiquated and faded in the eyes of the succeeding generation. The manners of your age were not the manners of to-day, and young gentlemen and ladies who think Scott `slow,' think Miss Austen `prim' and `dreary.' Yet, even could you return among us, I scarcely believe that, speaking the language of the hour, as you might, and versed in its habits, you would win the general admiration. For how tame, madam, are your characters, especially your favourite heroines! how limited the life which you knew and described! how narrow the range of your incidents! how correct your grammar!
As heroines, for example, you chose ladies like Emma, and Elizabeth, and Catherine: women remarkable neither for the brilliance nor for the degradation of their birth; women wrapped up in their own and the parish's concerns, ignorant of evil, as it seems, and unacquainted with vain yearnings and interesting doubts. Who can engage his fancy with their match-makings and the conduct of their affections, when so many daring and dazzling heroines approach and solicit his regard?
Here are princesses dressed in white velvet stamped with golden fleurs-de-lys---ladies with hearts of ice and lips of fire, who count their roubles by the million, their lovers by the score, and even their husbands, very often, in figures of some arithmetical importance. With these are the immaculate daughters of itinerant Italian musicians, maids whose souls are unsoiled amidst the contaminations of our streets, and whose acquaintance with the art of Phidias and Praxiteles, of Daedalus and Scopas, is the more admirable, because entirely derived from loving study of the inexpensive collections vended by the plaster-of-Paris man round the corner. When such heroines are wooed by the nephews of Dukes, where are your Emmas and Elizabeths? Your volumes neither excite nor satisfy the curiosities provoked by that modern and scientific fiction, which is greatly admired, I learn, in the United States, as well as in France and at home.
You erred, it cannot be denied, with your eyes open. Knowing Lydia and Kitty so intimately as you did, why did you make of them almost insignificant characters? With Lydia for a heroine you might have gone far; and, had you devoted three volumes, and the chief of your time, to the passions of Kitty, you might have held your own, even now, in the circulating library. How Lyddy, perched on a corner of the roof, first beheld her Wickham; how, on her challenge, he climbed up by a ladder to her side; how they kissed, caressed, swung on gates together, met at odd seasons, in strange places, and finally eloped: all this might have been put in the mouth of a jealous elder sister, say Elizabeth, and you would not have been less popular than several favourites of our time. Had you cast the whole narrative into the present tense, and lingered lovingly over the thickness of Mary's legs and the softness of Kitty's cheeks, and the blonde fluffiness of Wickham's whiskers, you would have left a romance still dear to young ladies.
Or again, you might entrance your students still, had you concentrated your attention on Mrs. Rushworth, who eloped with Henrv Crawford. These should have been the chief figures of `Mansfield Park.' But you timidly decline to tackle Passion. `Let other pens,' you write, 'dwell on guilt and misery. I quit such odious subjects as soon as I can.' Ah, \textit{there} is the secret of your failure! Need I add that the vulgarity and narrowness of the social circles you describe impair your popularity? I scarce remember more than one lady of title, and but very few lords (and these unessential) in all your tales. Now, when we all wish to be in society, we demand plenty of titles in our novels, at any rate, and we get lords (and very queer lords) even from Republican authors, born in a country which in your time was not renowned for its literature. I have heard a critic remark, with a decided air of fashion, on the brevity of the notice which your characters give each other when they offer invitations to dinner. 'An invitation to dinner next day was despatched,' and this demonstrates that your acquaintance `went out' very little, and had but few engagements. How vulgar, too, is one of your heroines, who bids Mr. Darcy 'keep his breath to cool his porridge.' I blush for Elizabeth! It were superfluous to add that your characters are debased by being invariably mere members of the Church of England as by law established. The Dissenting enthusiast, the open soul that glides from Esoteric Buddhism to the Salvation Army, and from the Higher Pantheism to the Higher Paganism, we look for in vain among your studies of character. Nay, the very words I employ are of unknown sound to you; so how can you help us in the stress of the soul's travailings?
You may say that the soul's travailings are no affair of yours; proving thereby that you have indeed but a lowly conception of the duty of the novelist. I only remember one reference, in all your works, to that controversy which occupies the chief of our attention---the great controversy on Creation or Evolution. Your Jane Bennet cries: 'I have no idea of there being so much Design in the world as some persons imagine.' Nor do you touch on our mighty social question, the Land Laws, save when Mrs. Bennet appears as a Land Reformer, and rails bitterly against the cruelty 'of settling an estate away from a family of five daughters, in favour of a man whom nobody cared anything about.' There, madam, in that cruelly unjust performance, what a text you had for a \textit{Tendenz-Roman}. Nay, you can allow Kitty to report that a Private had been flogged, without introducing a chapter on Flogging in the Army. But you formally declined to stretch your matter out, here and there, 'with solemn specious nonsense about something unconnected with the story.' No `padding' for Miss Austen! In fact, madam, as you were born before Analysis came in, or Passion, or Realism, or Naturalism, or Irreverence, or Religious Open-mindedness, you really cannot hope to rival your literary sisters in the minds of a perplexed generation. Your heroines are not passionate, we do not see their red wet cheeks, and tresses dishevelled in the manner of our frank young Maenads. What says your best successor, a lady who adds fresh lustre to a name that in fiction equals yours? She says of Miss Austen: 'Her heroines have a stamp of their own. They have a \textit{certain gentle self-respect and humour and hardness of heart}... Love with them does not mean a passion as much as an interest, deep and silent.' I think one prefers them so, and that Englishwomen should be more like Anne Elliot than Maggie Tulliver. 'All the privilege I claim for my own sex is that of loving longest when existence or when hope is gone,' said Anne; perhaps she insisted on a monopoly that neither sex has all to itself. Ah, madam, what a relief it is to come back to your witty volumes, and forget the follies of to-day in those of Mr. Collins and of Mrs. Bennet! How fine, nay, how noble is your art in its delicate reserve, never insisting, never forcing the note, never pushing the sketch into the caricature! You worked without thinking of it, in the spirit of Greece, on a labour happily limited, and exquisitely organised. `Dear books,' we say, with Miss Thackeray---'dear books, bright, sparkling with wit and animation, in which the homely heroines charm, the dull hours fly, and the very bores are enchanting.'
\gutchapter{IX. To Master Isaak Walton.}
Father Isaak,---When I would be quiet and go angling it is my custom to carry in my wallet thy pretty book, `The Compleat Angler.' Here, methinks, if I find not trout I shall find content, and good company, and sweet songs, fair milkmaids, and country mirth. For you are to know that trout be now scarce, and whereas he was ever a fearful fish, he hath of late become so wary that none but the cunningest anglers may be even with him.
It is not as it was in your time, Father, when a man might leave his shop in Fleet Street, of a holiday, and, when he had stretched his legs up Tottenham Hill, come lightly to meadows chequered with waterlilies and lady-smocks, and so fall to his sport. Nay, now have the houses so much increased, like a spreading sore (through the breaking of that excellent law of the Conscientious King and blessed Martyr, whereby building beyond the walls was forbidden), that the meadows are all swallowed up in streets. And as to the River Lea, wherein you took many a good trout, I read in the news sheets that 'its bed is many inches thick in horrible filth, and the air for more than half a mile on each side of it is polluted with a horrible, sickening stench,' so that we stand in dread of a new Plague, called the Cholera. And so it is all about London for many miles, and if a man, at heavy charges, betake himself to the fields, lo you, folk are grown so greedy that none will suffer a stranger to fish in his water.
So poor anglers are in sore straits. Unless a man be rich and can pay great rents, he may not fish, in England, and hence spring the discontents of the times, for the angler is full of content, if he do but take trout, but if he be driven from the waterside, he falls, perchance, into evil company, and cries out to divide the property of the gentle folk. As many now do, even among Parliament, men, whom you loved not, Father Isaak, neither do I love them more than Reason and Scripture bid each of us be kindly to his neighbour. But, behold, the causes of the ill content are not yet all expressed, for even where a man hath licence to fish, he will hardly take trout in our age, unless he be all the more cunning. For the fish, harried this way and that by so many of your disciples, is exceeding shy and artful, nor will he bite at a fly unless it falleth lightly, just above his mouth, and floateth dry over him, for all the world like the natural \textit{ephemeris}. And we may no longer angle with worm for him, nor with penk or minnow, nor with the natural fly, as was your manner, but only with the artificial, for the more difficulty the more diversion. For my part I may cry, like Viator in your book, 'Master, I can neither catch with the first nor second Angle: I have no fortune.'
So we fare in England, but somewhat better north of the Tweed, where trout are less wary, but for the most part small, except in the extreme rough north, among horrid hills and lakes. Thither, Master, as methinks you may remember, went Richard Franck, that called himself \textit{Philanthropus}, and was, as it were, the Columbus of anglers, discovering for them a new Hyperborean world. But Franck, doubtless, is now an angler in the Lake of Darkness, with Nero and other tyrants, for he followed after Cromwell, the man of blood, in the old riding days. How wickedly doth Franck boast of that leader of the giddy multitude, 'when they raged, and became restless to find out misery for themselves and others, and the rabble would herd themselves together,' as you said, `and endeavour to govern and act in spite of authority.' So you wrote; and what said Franck, that recreant angler? Doth he not praise 'Ireton, Vane, Nevill, and Martin, and the most renowned, valorous, and victorious conqueror, Oliver Cromwell.' Natheless, with all his sins on his head, this Franck discovered Scotland for anglers, and my heart turns to him when he praises 'the glittering and resolute streams of Tweed.'
In those wilds of Assynt and Loch Rannoch, Father, we, thy followers, may yet take trout, and forget the evils of the times. But, to be done with Franck, how harshly he speaks of thee and thy book. 'For you may dedicate your opinion to what scribbling putationer you please; the \textit{Compleat Angler} if you will, who tells you of a tedious fly story, extravagantly collected from antiquated authors, such as Gesner and Dubravius.' Again, he speaks of 'Isaac Walton, whose authority to me seems alike authentick, as is the general opinion of the vulgar prophet,' \&c.
Certain I am that Franck, if a better angler than thou, was a worse man, who, writing his `Dialogues Piscatorial' or `Northern Memoirs' five years after the world welcomed thy `Compleat Angler,' was jealous of thy favour with the people, and, may be, hated thee for thy loyalty and sound faith. But, Master, like a peaceful man avoiding contention, thou didst never answer this blustering Franck, but wentest quietly about thy quiet Lea, and left him his roaring Brora and windy Assynt. How could this noisy man know thee---and know thee he did, having argued with thee in Stafford---and not love Isaak Walton? A pedant angler, I call him, a plaguy angler, so let him huff away, and turn we to thee and to thy sweet charm in fishing for men.
How often, studying in thy book, have I hummed to myself that of Horace---
Laudis amore tumes? Sunt certa piacula quae te
Ter pure lecto poterunt recreare libello.
So healing a book for the frenzy of fame is thy discourse on meadows, and pure streams, and the country life. How peaceful, men say, and blessed must have been the life of this old man, how lapped in content, and hedged about by his own humility from the world! They forget, who speak thus, that thy years, which were many, were also evil, or would have seemed evil to divers that had tasted of thy fortunes. Thou wert poor, but that, to thee, was no sorrow, for greed of money was thy detestation. Thou wert of lowly rank, in an age when gentle blood was alone held in regard; yet tiny virtues made thee hosts of friends, and chiefly among religious men, bishops, and doctors of the Church. Thy private life was not unacquainted with sorrow; thy first wife and all her fair children were taken from thee like flowers in spring, though, in thine age, new love and new offspring comforted thee like 'the primrose of the later year.' Thy private griefs might have made thee bitter, or melancholy, so might the sorrows of the State and of the Church, which were deprived of their heads by cruel men, despoiled of their wealth, the pious driven, like thee, from their homes; fear everywhere, everywhere robbery and confusion: all this ruin might have angered another temper. But thou, Father, didst bear all with so much sweetness as perhaps neither natural temperament, nor a firm faith, nor the love of angling could alone have displayed. For we see many anglers (as witness Richard Franck aforesaid) who are angry men, and myself, when I get my hooks entangled at every cast in a tree, have come nigh to swear prophane.
Also we see religious men that are sour and fanatical, no rare thing in the party that professes godliness. But neither private sorrow nor public grief could abate thy natural kindliness, nor shake a religion which was not untried, but had, indeed, passed through the furnace like fine gold. For if we find not Faith at all times easy, because of the oppositions of Science, and the searching curiosity of men's minds, neither was Faith a matter of course in thy day. For the learned and pious were greatly tossed about, like worthy Mr. Chillingworth, by doubts wavering between the Church of Rome and the Reformed Church of England. The humbler folk, also, were invited, now here, now there, by the clamours of fanatical Nonconformists, who gave themselves out to be somebody, while Atheism itself was not without many to witness to it. Therefore, such a religion as thine was not, so to say, a mere innocence of evil in the things of our Belief, but a reasonable and grounded faith, strong in despite of oppositions. Happy was the man in whom temper, and religion, and the love of the sweet country and an angler's pastime so conveniently combined; happy the long life which held in its hand that threefold clue through the labyrinth of human fortunes! Around thee Church and State might fall in ruins, and might be rebuilded, and thy tears would not be bitter, nor thy triumph cruel.
Thus, by God's blessing, it befell thee
Nec turpem senectam
Degere, nec cithara carentem.
I would, Father, that I could get at the verity about thy poems. Those recommendatory verses with which thou didst grace the Lives of Dr. Donne and others of thy friends, redound more to the praise of thy kind heart than thy fancy. But what or whose was the pastoral poem of 'Thealma and Clearchus,' which thou didst set about printing in 1678, and gavest to the world in 1683? Thou gavest John Chalkhill for the author's name, and a John Chalkhill of thy kindred died at Winchester, being eighty years of his age, in 1679. Now thou speakest of John Chalkhill as 'a friend of Edmund Spenser's,' and how could this be?
Are they right who hold that John Chalkhill was but a name of a friend, borrowed by thee out of modesty, and used as a cloak to cover poetry of thine own inditing? When Mr. Flatman writes of Chalkhill, 't is in words well fitted to thine own merit:
Happy old man, whose worth all mankind knows
Except himself, who charitably shows
The ready road to virtue and to praise,
The road to many long and happy days.
However it be, in that road, by quiet streams and through green pastures, thou didst walk all thine almost century of years, and we, who stray into thy path out of the highway of life, we seem to hold thy hand, and listen to thy cheerful voice. If our sport be worse, may our content be equal, and our praise, therefore, none the less. Father, if Master Stoddard, the great fisher of Tweed-side, be with thee, greet him for me, and thank him for those songs of his, and perchance he will troll thee a catch of our dear River.
Tweed! winding and wild! where the heart is unbound,
They know not, they dream not, who linger around,
How the saddened will smile, and the wasted rewin
From thee---the bliss withered within.
Or perhaps thou wilt better love,
The lanesome Tala and the Lyne,
And Mahon wi' its mountain rills,
An' Etterick, whose waters twine
Wi' Yarrow frae the forest hills;
An' Gala, too, and Teviot bright,
An' mony a stream o' playfu' speed,
Their kindred valleys a' unite
Amang the braes o' bonnie Tweed!
So, Master, may you sing against each other, you two good old anglers, like Peter and Corydon, that sang in your golden age.
\gutchapter{X. To M. Chapelain.}
Monsieur,---You were a popular writer, and an honourable, over-educated, upright gentleman. Of the latter character you can never be deprived, and I doubt not it stands you in better stead where you are, than the laurels which flourished so gaily, and faded so soon.
Laurel is green for a season, and Love is fair for a day,
But Love grows bitter with treason, and laurel out-lives not May.
I know not if Mr. Swinburne is correct in his botany, but \textit{your} laurel certainly outlived not May, nor can we hope that you dwell where Orpheus and where Homer are. Some other crown, some other Paradise, we cannot doubt it, awaited \textit{un si bon homme}. But the moral excellence that even Boileau admitted, \textit{ladfoi, l'honneur, la probiite,} do not in Parnassus avail the popular poet, and some luckless Musset or Theophile, Regnier or Villars attains a kind of immortality denied to the man of many contemporary editions, and of a great commercial success.
If ever, for the confusion of Horace, any Poet was Made, you, Sir, should have been that fortunately manufactured article. You were, in matters of the Muses, the child of many prayers. Never, since Adam's day, have any parents but yours prayed for a poet-child. Then Destiny, that mocks the desires of men in general, and fathers in particular, heard the appeal, and presented M. Chapelain and Jeanne Corbiere his wife with the future author of `La Pucelle.' Oh futile hopes of men, \textit{O pectora caeca!} All was done that education could do for a genius which, among other qualities, `especially lacked fire and imagination,' and an ear for verse---sad defects these in a child of the Muses. Your training in all the mechanics and metaphysics of criticism might have made you exclaim, like Rasselas, 'Enough! Thou hast convinced me that no human being can ever be a Poet.' Unhappily, you succeeded in convincing Cardinal Richelieu that to be a Poet was well within your powers, you received a pension of one thousand crowns, and were made Captain of the Cardinal's minstrels, as M. de Treville was Captain of the King's Musketeers.
Ah, pleasant age to live in, when good intentions in poetry were more richly endowed than ever is Research, even Research in Prehistoric English, among us niggard moderns! How I wish I knew a Cardinal, or, even as you did, a Prime Minister, who would praise and pension me; but Envy be still! Your existence was more happy indeed; you constructed odes, corrected sonnets, presided at the Ho'tel Rambouillet, while the learned ladies were still young and fair, and you enjoyed a prodigious celebrity on the score of your yet unpublished Epic. `Who, indeed,' says a sympathetic author, M. Theophile Gautier, 'who could expect less than a miracle from a man so deeply learned in the laws of art---a perfect Turk in the science of poetry, a person so well pensioned, and so favoured by the great?' Bishops and politicians combined in perfect good faith to advertise your merits. Hard must have been the heart that could resist the testimonials of your skill as a poet offered by the Duc de Montausier, and the learned Huet, Bishop of Avranches, and Monseigneur Godeau, Bishop of Vence, or M. Colbert, who had such a genius for finance.
If bishops and politicians and prime ministers skilled in finance, and some critics, Menage and Sarrazin and Vaugetas, if ladies of birth and taste, if all the world in fact, combined to tell you that you were a great poet, how can we blame you for taking yourself seriously, and appraising yourself at the public estimate?
It was not in human nature to resist the evidence of the bishops especially, and when every minor poet believes in himself on the testimony of his own conceit, you may be acquitted of vanity if you listened to the plaudits of your friends. Nay, you ventured to pronounce judgment on contemporaries whom Posterity has preferred to your perfections. `Moliere,' said you, 'understands the nature of comedy, and presents it in a natural style. The plot of his best pieces is borrowed, but not without judgment; his \textit{morale} is fair, and he has only to avoid scurrility.'
Excellent, unconscious, popular Chapelain!
Of yourself you observed, in a Report on contemporary literature, that your 'courage and sincerity never allowed you to tolerate work not absolutely good.' And yet you regarded 'La Pucelle with some complacency.
On the 'Pucelle you were occupied during a generation of mortal men. I marvel not at the length of your labours, as you received a yearly pension till the Epic was finished, but your Muse was no Alcmena, and no Hercules was the result of that prolonged night of creations. First you gravely wrote out (it was the task of five years) all the compositions in prose. Ah, why did you not leave it in that commonplace but appropriate medium? What says the Precieuse about you in Boileau's satire?
In Chapelain, for all his foes have said,
She finds but one defect, he can't be read;
Yet thinks the world might taste his maiden's woes,
If only he would turn his verse to prose!
The verse had been prose, and prose, perhaps, it should have remained. Yet for this precious `Pucelle,' in the age when `Paradise Lost' was sold for five pounds, you are believed to have received about four thousand. Horace was wrong, mediocre poets may exist (now and then), and he was a wise man who first spoke of \textit{aurea mediocritas}. At length the great work was achieved, a work thrice blessed in its theme, that divine Maiden to whom France owes all, and whom you and Voltaire have recompensed so strangely. In folio, in italics, with a score of portraits and engravings, and \textit{culs de lampe}, the great work was given to the world, and had a success. Six editions in eighteen months are figures which fill the poetic heart with envy and admiration. And then, alas! the bubble burst. A great lady, Madame de Longveille, hearing the 'Pucelle read aloud, murmured that it was 'perfect indeed, but perfectly wearisome.' Then the satires began, and the satirists never left you till your poetic reputation was a rag, till the mildest Abbe at Menages had his cheap sneer for Chapelain.
I make no doubt, Sir, that envy and jealousy had much to do with the onslaught on your `Pucelle.' These qualities, alas! are not strange to literary minds; does not even Hesiod tell us 'potter hates potter, and poet hates poet'? But contemporary spites do not harm true genius. Who suffered more than Moliere from cabals? Yet neither the court nor the town ever deserted him, and he is still the joy of the world. I admit that his adversaries were weaker than yours. What were Boursault and Le Boulanger, and Thomas Corneille and De Vise, what were they all compared to your enemy, Boileau? Brossette tells a story which really makes a man pity you. There was a M. de Puimorin who, to be in the fashion, laughed at your once popular Epic. 'It is all very well for a man to laugh who cannot even read.' Whereon M. de Puimorin replied: 'Qu'il n'avoit que trop su' lire, depuis que Chapelain s'etoit avise de faire imprimer.' A new horror had been added to the accomplishment of reading since Chapelain had published. This repartee was applauded, and M. de Puimorin tried to turn it into an epigram. He did complete the last couplet,
Helas! pour mes peches, je n'ai su' que trop lire
Depuis que tu fais imprimer.
But by no labour would M. de Puimorin achieve the first two lines of his epigram. Then you remember what great allies came to his assistance. I almost blush to think that M. Despreaux, M. Racine, and M. de Moliere, the three most renowned wits of the time, conspired to complete the poor jest, and madden you. Well, bubble as your poetry was, you may be proud that it needed all these sharpest of pens to prick the bubble. Other poets, as popular as you, have been annihilated by an article. Macaulay puts forth his hand, and `Satan Montgomery' was no more. It did not need a Macaulay, the laughter of a mob of little critics was enough to blow into space; but you probably have met Montgomery, and of contemporary failures or successes I do not speak.
I wonder, sometimes, whether the consensus of criticism ever made you doubt for a moment whether, after all, you were not a false child of Apollo? Was your complacency tortured, as the complacency of true poets has occasionally been, by doubts? Did you expect posterity to reverse the verdict of the satirists, and to do you justice? You answered your earliest assailant, Liniere, and, by a few changes of words, turned his epigrams into flattery. But I fancy, on the whole, you remained calm, unmoved, wrapped up in admiration of yourself. According to M. de Marivaux, who reviewed, as I am doing, the spirits of the mighty dead, you 'conceived, on the strength of your reputation, a great and serious veneration for yourself and your genius.' Probably you were protected by this invulnerable armour of an honest vanity, probably you declared that mere jealousy dictates the lines of Boileau, and that Chapelain's real fault was his popularity, and his pecuniary success, Qu'il soit le mieux rente de tous les beaux-esprits.
This, you would avow, was your offence, and perhaps you were not altogether mistaken. Yet posterity declines to read a line of yours, and, as we think of you, we are again set face to face with that eternal problem, how far is popularity a test of poetry? Burns was a poet, and popular. Byron was a popular poet, and the world agrees in the verdict of their own generation. But Montgomery, though he sold so well, was no poet, nor, Sir, I fear, was your verse made of the stuff of immortality. Criticism cannot hurt what is truly great; the Cardinal and the Academy left Chimene as fair as ever, and as adorable. It is only pinchbeck that perishes under the acids of satire: gold defies them. Yet I sometimes ask myself, does the existence of popularity like yours justify the malignity of satire, which blesses neither him who gives, nor him who takes? Are poisoned arrows fair against a bad poet? I doubt it, Sir, holding that, even unpricked, a poetic bubble must soon burst by its own nature. Yet satire will assuredly be written so long as bad poets are successful, and bad poets will assuredly reflect that their assailants are merely envious, and, while their vogue lasts, that Prime Ministers and the purchasing public are the only judges.
Monsieur,
Votre tres humble serviteur,
Andrew Lang.
\gutchapter{XI. To Sir John Manndeville, Kt.}
(Of the Ways Into Ynde.)
Sir John,---wit you well that men holden you but light, and some clepen you a Liar. And they say that you never were born in Englond, in the town of Seynt Albones, nor have seen and gone through manye diverse Londes. And there goeth an old knight at arms, and one that connes Latyn, and hath been beyond the sea, and hath seen Prester John's country. And he hath been in an Yle that men clepen Burmah, and there bin women bearded. Now men call him Colonel Henry Yule, and he hath writ of thee in his great booke, Sir John, and he holds thee but lightly. For he saith that ye did pill your tales out of Odoric his book, and that ye never saw snails with shells as big as houses, nor never met no Devyls, but part of that ye say, ye took it out of William of Boldensele his book, yet ye took not his wisdom, withal, but put in thine own foolishness. Nevertheless, Sir John, for the frailty of Mankynde, ye are held a good fellow, and a merry; so now, come, I shall tell you of the new ways into Ynde.
In that Lond they have a Queen that governeth all the Lond, and all they ben obeyssant to her. And she is the Queen of Englond; for Englishmen have taken all the Lond of Ynde. For they were right good werryoures of old, and wyse, noble, and worthy. But of late hath risen a new sort of Englishman very puny and fearful, and these men clepen Radicals. And they go ever in fear, and they scream on high for dread in the streets and the houses, and they fain would flee away from all that their fathers gat them with the sword. And this sort men call Scuttleres, but the mean folk and certain of the womenkind hear them gladly, and they say ever that Englishmen should flee out of Ynde. Fro England men gon to Ynde by many dyverse Contreyes. For Englishmen ben very stirring and nymble. For they ben in the seventh climate, that is of the Moon. And the Moon (ye have said it yourself, Sir John, natheless, is it true) is of lightly moving, for to go diverse ways, and see strange things, and other diversities of the Worlde. Wherefore Englishmen be lightly moving, and far wandering. And they gon to Ynde by the great Sea Ocean. First come they to Gibraltar, that was the point of Spain, and builded upon a rock; and there ben apes, and it is so strong that no man may take it. Natheless did Englishmen take it fro the Spanyard, and all to hold the way to Ynde. For ye may sail all about Africa, and past the Cape men clepen of Good Hope, but that way unto Ynde is long and the sea is weary. Wherefore men rather go by the Midland sea, and Englishmen have taken many Yles in that sea.
For first they have taken an Yle that is clept Malta; and therein built they great castles, to hold it against them of Fraunce, and Italy, and of Spain. And from this Ile of Malta Men gon to Cipre. And Cipre is right a good Yle, and a fair, and a great, and it hath 4 principal Cytees within him. And at Famagost is one of the principal Havens of the sea that is in the world, and Englishmen have but a lytel while gone won that Yle from the Sarazynes. Yet say that sort of Englishmen where of I told you, that is puny and sore adread, that the Lond is poisonous and barren and of no avail, for that Lond is much more hotter than it is here. Yet the Englishmen that ben werryoures dwell there in tents, and the skill is that they may ben the more fresh.
From Cypre, Men gon to the Lond of Egypte, and in a Day and a Night he that hath a good wind may come to the Haven of Alessandrie. Now the Lond of Egypt longeth to the Soudan, yet the Soudan longeth not to the Lond of Egypt. And when I say this, I do jape with words, and may hap ye understond me not. Now Englishmen went in shippes to Alessandrie, and brent it, and over ran the Lond, and their soudyours warred agen the Bedoynes, and all to hold the way to Ynde. For it is not long past since Frenchmen let dig a dyke, through the narrow spit of lond, from the Midland sea to the Red sea, wherein was Pharaoh drowned. So this is the shortest way to Ynde there may be, to sail through that dyke, if men gon by sea.
But all the Lond of Egypt is clepen the Vale enchaunted; for no man may do his business well that goes thither, but always fares he evil, and therefore clepen they Egypt the Vale perilous, and the sepulchre of reputations. And men say there that is one of the entrees of Helle. In that Vale is plentiful lack of Gold and Silver, for many misbelieving men, and many Christian men also, have gone often time for to take of the Thresoure that there was of old, and have pilled the Thresoure, wherefore there is none left. And Englishmen have let carry thither great store of our Thresoure, 9,000,000 of Pounds sterling, and whether they will see it agen I misdoubt me. For that Vale is alle fulle of Develes and Fiendes that men clepen Bondholderes, for that Egypt from of olde is the Lond of Bondage. And whatsoever Thresoure cometh into the Lond, these Devyls of Bondholders grabben the same. Natheless by that Vale do Englishmen go unto Ynde, and they gon by Aden, even to Kurrachee, at the mouth of the Flood of Ynde. Thereby they send their souldyours, when they are adread of them of Muscovy.
For, look you, there is another way into Ynde, and thereby the men of Muscovy are fain to come, if the Englishmen let them not. That way cometh by Desert and Wildernesse, from the sea that is clept Caspian, even to Khiva, and so to Merv; and then come ye to Zulfikar and Penjdeh, and anon to Herat, that is called the Key of the Gates of Ynde. Then ye win the lond of the Emir of the Afghauns, a great prince and a rich, and he hath in his Thresoure more crosses, and stars, and coats that captains wearen, than any other man on earth.
For all they of Muscovy, and all Englishmen maken him gifts, and he keepeth the gifts, and he keepeth his own counsel. For his lond lieth between Ynde and the folk of Muscovy, wherefore both Englishmen and men of Muscovy would fain have him friendly, yea, and independent. Wherefore they of both parties give him clocks, and watches, and stars, and crosses, and culverins, and now and again they let cut the throats of his men some deal, and pill his country. Thereby they both set up their rest that the Emir will be independent, yea, and friendly. But his men love him not, neither love they the English nor the Muscovy folk, for they are worshippers of Mahound, and endure not Christian men. And they love not them that cut their throats, and burn their country.
Now they of Muscovy ben Devyls, und they ben subtle for to make a thing seme otherwise than it is, for to deceive mankind. Wherefore Englishmen putten no trust in them of Muscovy, save only the Englishmen ciept Radicals, for they make as if they loved these Develes, out of the fear and dread of war wherein they go, and would be slaves sooner than fight. But the folk of Ynde know not what shall befall, nor whether they of Muscovy will take the Lond, or Englishmen shall keep it, so that their hearts may not enduren for drede. And methinks that soon shall Englishmen and Muscovy folk put their bodies in adventure, and war one with another, and all for the way to Ynde.
But St. George for Englond, I say, and so enough; and may the Seyntes hele thee, Sir John, of thy Gowtes Artetykes, that thee tormenten. But to thy Boke I list not to give no credence.
\gutchapter{XII. To Alexandre Dumas.}
Sir,---There are moments when the wheels of life, even of such a life as yours, run slow, and when mistrust and doubt overshadow even the most intrepid disposition. In such a moment, towards the ending of your days, you said to your son, M. Alexandre Dumas, 'I seem to see myself set on a pedestal which trembles as if it were founded on the sands.' These sands, your uncounted volumes, are all of gold, and make a foundation more solid than the rock. As well might the singer of Odysseus, or the authors of the `Arabian Nights', or the first inventors of the stories of Boccaccio, believe that their works were perishable (their names, indeed, have perished), as the creator of `Les Trois Mousquetaires' alarm himself with the thought that the world could ever forget Alexandre Dumas.
Than yours there has been no greater nor more kindly and beneficent force in modern letters. To Scott, indeed, you owed the first impulse of your genius; but, once set in motion, what miracles could it not accomplish? Our dear Porthos was overcome, at last, by a superhuman burden; but your imaginative strength never found a task too great for it. What an extraordinary vigour, what health, what an overflow of force was yours! It is good, in a day of small and laborious ingenuities, to breathe the free air of your books, and dwell in the company of Dumas's men---so gallant, so frank, so indomitable, such swordsmen, and such trenchermen. Like M. de Rochefort in `Vingt Ans Apres,' like that prisoner of the Bastille, your genius 'n'est que d'un parti, c'est du parti du grand air.'
There seems to radiate from you a still persistent energy and enjoyment; in that current of strength not only your characters live, frolic, kindly, and sane, but even your very collaborators were animated by the virtue which went out of you. How else can we explain it, the dreary charge which feeble and envious tongues have brought against you, in England and at home? They say you employed in your novels and dramas that vicarious aid which, in the slang of the studio, the 'sculptor's ghost' is fabled to afford.
Well, let it be so; these ghosts, when uninspired by you, were faint and impotent as `the strengthless tribes of the dead' in Homer's Hades, before Odysseus had poured forth the blood that gave them a momentary valour. It was from you and your inexhaustible vitality that these collaborating spectres drew what life they possessed; and when they parted from you they shuddered back into their nothingness. Where are the plays, where the romances which Maquet and the rest wrote in their own strength? They are forgotten with last year's snows; they have passed into the wide waste-paper basket of the world. You say of D'Artagnan, when severed from his three friends---from Porthos, Athos, and Aramis---'he felt that he could do nothing, save on the condition that each of these companions yielded to him, if one may so speak, a share of that electric fluid which was his gift from heaven.'
No man of letters ever had so great a measure of that gift as you; none gave of it more freely to all who came---to the chance associate of the hour, as to the characters, all so burly and full-blooded, who flocked from your brain. Thus it was that you failed when you approached the supernatural. Your ghosts had too much flesh and blood, more than the living persons of feebler fancies. A writer so fertile, so rapid, so masterly in the ease with which he worked, could not escape the reproaches of barren envy. Because you overflowed with wit, you could not be `serious;' Because you created with a word, you were said to scamp your work; because you were never dull, never pedantic, incapable of greed, you were to be censured as desultory, inaccurate, and prodigal.
A generation suffering from mental and physical anaemia---a generation devoted to the `chiselled phrase,' to accumulated `documents,' to microscopic porings over human baseness, to minute and disgustful records of what in humanity is least human---may readily bring these unregarded and railing accusations. Like one of the great and good-humoured Giants of Rabelais, you may hear the murmurs from afar, and smile with disdain. To you, who can amuse the world---to you who offer it the fresh air of the highway, the battle-field, and the sea---the world must always return, escaping gladly from the boudoirs and the \textit{bouges}, from the surgeries and hospitals, and dead rooms, of M. Daudet and M. Zola and of the wearisome De Goncourt.
With all your frankness, and with that queer morality of the Camp which, if it swallows a camel now and again, never strains at a gnat, how healthy and wholesome, and even pure, are your romances! You never gloat over sin, nor dabble with an ugly curiosity in the corruptions of sense. The passions in your tales are honourable and brave, the motives are clearly human. Honour, Love, Friendship make the threefold cord, the clue your knights and dames follow through how delightful a labyrinth of adventures! Your greatest books, I take the liberty to maintain, are the Cycle of the Valois ('La Reine Margot, `La Dame de Montsoreau,' 'Les Quarante-cinq'), and the Cycle of Louis Treize and Louis Quatorze ('Les Trois Mousquetaires,' `Vingt Ans Apres,' 'Le Vicomte de Bragelonne); and, beside these two trilogies---a lonely monument, like the sphinx hard by the three pyramids---'Monte Cristo.'
In these romances how easy it would have been for you to burn incense to that great goddess, Lubricity, whom our critic says your people worship. You had Branto'me, you had Tallemant, you had Retif, and a dozen others, to furnish materials for scenes of voluptuousness and of blood that would have outdone even the present \textit{naturalistes}. From these alcoves of `Les Dames Galantes,' and from the torture chambers (M. Zola would not have spared us one starting sinew of brave La Mole on the rack) you turned, as Scott would have turned, without a thought of their profitable literary uses. You had other metal to work on: you gave us that superstitious and tragical true love of La Moles, that devotion---how tender and how pure!---of Bussy for the Dame de Montsoreau. You gave us the valour of D'Artagnan, the strength of Porthos, the melancholy nobility of Athos: Honour, Chivalry, and Friendship. I declare your characters are real people to me and old friends. I cannot bear to read the end of `Bragelonne,' and to part with them for ever. 'Suppose Perthos, Athos, and Aramis should enter with a noiseless swagger, curling their moustaches.' How we would welcome them, forgiving D'Artagnan even his hateful \textit{fourberie} in the case of Milady. The brilliance of your dialogue has never been approached: there is wit everywhere; repartees glitter and ring like the flash and clink of small-swords. Then what duels are yours! and what inimitable battle-pieces! I know four good fights of one against a multitude, in literature. These are the Death of Gretir the Strong, the Death of Gunnar of Lithend, the Death of Hereward the Wake, the Death of Bussy d'Amboise. We can compare the strokes of the heroic fighting-times with those described in later days; and, upon my word, I do not know that the short sword of Gretir, or the bill of Skarphedin, or the bow of Gunnar was better wielded than the rapier of your Bussy or the sword and shield of Kingsley's Hereward.
They say your fencing is unhistorical; no doubt it is so, and you knew it. La Mole could not have lunged on Coconnas `after deceiving circle;' for the parry was not invented except by your immortal Chicot, a genius in advance of his time. Even so Hamlet and Laertes would have fought with shields and axes, not with small swords. But what matters this pedantry? In your works we hear the Homeric Muse again, rejoicing in the clash of steel; and even, at times, your very phrases are unconsciously Homeric.
Look at these men of murder, on the Eve of St. Bartholomew, who flee in terror from the Queen's chamber, and 'find the door too narrow for their flight:' the very words were anticipated in a line of the `Odyssey' concerning the massacre of the Wooers. And the picture of Catherine de Medicis, prowling `like a wolf among the bodies and the blood,' in a passage of the Louvre---the picture is taken unwittingly from the `Iliad.' There was in you that reserve of primitive force, that epic grandeur and simplicity of diction. This is the force that animates `Monte Cristo,' the earlier chapters, the prison, and the escape. In later volumes of that romance, methinks, you stoop your wing. Of your dramas I have little room, and less skill, to speak. `Antony,' they tell me, was `the greatest literary event of its time,' was a restoration of the stage. 'While Victor Hugo needs the cast-off clothes of history, the wardrobe and costume, the sepulchre of Charlemagne, the ghost of Barbarossa, the coffins of Lucretia Borgia, Alexandre Dumas requires no more than a room in an inn, where people meet in riding cloaks, to move the soul with the last degree of terror and of pity.'
The reproach of being amusing has somewhat dimmed your fame---for a moment. The shadow of this tyranny will soon be overpast; and when 'La Curee and 'Pot-Bouille are more forgotten than `Le Grand Cyrus,' men and women---and, above all, boys---will laugh and weep over the page of Alexandre Dumas. Like Scott himself, you take us captive in our childhood. I remember a very idle little boy who was busy with the `Three Musketeers' when he should have been occupied with 'Wilkins's Latin Prose.' `Twenty years after' (alas and more) he is still constant to that gallant company; and, at this very moment, is breathlessly wondering whether Grimand will steal M. de Beaufort out of the Cardinal's prison.
\gutchapter{XIII. To Theocritus}
`Sweet, methinks, is the whispering sound of yonder pine-tree,' so, Theocritus, with that sweet word \textit{ade}*, didst thou begin and strike the keynote of thy songs. `Sweet,' and didst thou find aught of sweet, when thou, like thy Daphnis, didst 'go down the stream, when the whirling wave closed over the man the Muses loved, the man not hated of the Nymphs?' Perchance below those waters of death thou didst find, like thine own Hylas, the lovely Nereids waiting thee, Eunice, and Malis, and Nycheia with her April eyes. In the House of Hades, Theocritus, doth there dwell aught that is fair, and can the low light on the fields of asphodel make thee forget thy Sicily? Nay, methinks thou hast not forgotten, and perchance for poets dead there is prepared a place more beautiful than their dreams. It was well for the later minstrels of another day, it was well for Ronsard and Du Bellay to desire a dim Elysium of their own, where the sunlight comes faintly through the shadow of the earth, where the poplars are duskier, and the waters more pale than in the meadows of Anjou.
* Transliterated.
There, in that restful twilight, far remote from war and plot, from sword and fire, and from religions that sharpened the steel and lit the torch, there these learned singers would fain have wandered with their learned ladies, satiated with life and in love with an unearthly quiet. But to thee, Theocritus, no twilight of the Hollow Land was dear, but the high suns of Sicily and the brown cheeks of the country maidens were happiness enough. For thee, therefore, methinks, surely is reserved an Elysium beneath the summer of a far-off system, with stars not ours and alien seasons. There, as Bion prayed, shall Spring, the thrice desirable, be with thee the whole year through, where there is neither frost, nor is the heat so heavy on men, but all is fruitful, and all sweet things blossom, and evenly meted are darkness and dawn. Space is wide, and there be many worlds, and suns enow, and the Sun-god surely has had a care of his own. Little didst thou need, in thy native land, the isle of the three capes, little didst thou need but sunlight on land and sea. Death can have shown thee naught dearer than the fragrant shadow of the pines, where the dry needles of the fir are strewn, or glades where feathered ferns make `a couch more soft than Sleep.' The short grass of the cliffs, too, thou didst love, where thou wouldst lie, and watch, with the tunny watcher till the deep blue sea was broken by the burnished sides of the tunny shoal, and afoam with their gambols in the brine. There the Muses met thee, and the Nymphs, and there Apollo, remembering his old thraldom with Admetus, would lead once more a mortal's flocks, and listen and learn, Theocritus, while thou, like thine own Comatas, `didst sweetly sing.'
There, methinks, I see thee as in thy happy days, 'reclined on deep beds of fragrant lentisk, lowly strewn, and rejoicing in new stript leaves of the vine, while far above thy head waved many a poplar, many an elm-tree, and close at hand the sacred waters sang from the mouth of the cavern of the nymphs.' And when night came, methinks thou wouldst flee from the merry company and the dancing girls, from the fading crowds of roses or white violets, from the cottabos, and the minstrelsy, and the Bibline wine, from these thou wouldst slip away into the summer night. Then the beauty of life and of the summer would keep thee from thy couch, and wandering away from Syracuse by the sandhills and the sea, thou wouldst watch the low cabin, roofed with grass, where the fishing-rods of reed were leaning against the door, while the Mediterranean floated up her waves, and filled the waste with sound. There didst thou see thine ancient fishermen rising ere the dawn from their bed of dry sea-weed, and heardst them stirring, drowsy, among their fishing gear, and heardst them tell their dreams.
Or again thou wouldst wander with dusty feet through the ways that the dust makes silent, while the breath of the kine, as they were driven forth with the morning, came fresh to thee, and the trailing dewy branch of honeysuckle struck sudden on thy cheek. Thou wouldst see the Dawn awake in rose and saffron across the waters, and Etna, grey and pale against the sky, and the setting crescent would dip strangely in the glow, on her way to the sea. Then, methinks, thou wouldst murmur, like thine own Simaetha, the love-lorn witch, 'Farewell, Selene, bright and fair; farewell, ye other stars, that follow the wheels of the quiet Night.' Nay, surely it was in such an hour that thou didst behold the girl as she burned the laurel leaves and the barley grain, and melted the waxen image, and called on Selene to bring her lover home. Even so, even now, in the islands of Greece, the setting Moon may listen to the prayers of maidens. 'Bright golden Moon, that now art near the waters, go thou and salute my lover, he that stole my love, and that kissed me, saying ``Never will I leave thee.'' And lo, he hath left me as men leave a field reaped and gleaned, like a church where none cometh to pray, like a city desolate.'
So the girls still sing in Greece, for though the Temples have fallen, and the wandering shepherds sleep beneath the broken columns of the god's house in Selinus, yet these ancient fires burn still to the old divinities in the shrines of the hearths of the peasants. It is none of the new creeds that cry, in the dirge of the Sicilian shepherds of our time, 'Ah, light of mine eyes, what gift shall I send thee, what offering to the other world? The apple fadeth, the quince decayeth, and one by one they perish, the petals of the rose. I will send thee my tears shed on a napkin, and what though it burneth in the flame, if my tears reach thee at the last.'
Yes, little is altered, Theocritus, on these shores beneath the sun, where thou didst wear a tawny skin stripped from the roughest of he-goats, and about thy breast an old cloak buckled with a plaited belt. Thou wert happier there, in Sicily, methinks, and among vines and shadowy lime-trees of Cos, than in the dust, and heat, and noise of Alexandria. What love of fame, what lust of gold tempted thee away from the red cliffs, and grey olives, and wells of black water wreathed with maidenhair?
The music of the rustic flute
Kept not for long its happy country tone;
Lost it too soon, and learned a stormy note
Of men contention tost, of men who groan,
Which tasked thy pipe too sore, and tired thy throat---
It failed, and thou wast mute!
What hadst \textit{thou} to make in cities, and what could Ptolemies and Princes give thee better than the goat-milk cheese and the Ptelean wine? Thy Muses were meant to be the delight of peaceful men, not of tyrants and wealthy merchants, to whom they vainly went on a begging errand. 'Who will open his door and gladly receive our Muses within his house, who is there that will not send them back again without a gift? And they with naked feet and looks askance come homewards, and sorely they upbraid me when they have gone on a vain journey, and listless again in the bottom of their empty coffer they dwell with heads bowed over their chilly knees, where is their drear abode, when portionless they return.' How far happier was the prisoned goat-herd, Comatas, in the fragrant cedar chest where the blunt-faced bees from the meadow fed him with food of tender flowers, because still the Muse dropped sweet nectar on his lips!
Thou didst leave the neat-herds and the kine, and the oaks of Himera, the galingale hummed over by the bees, and the pine that dropped her cones, and Amaryllis in her cave, and Bombyca with her feet of carven ivory. Thou soughtest the City, and strife with other singers, and the learned write still on thy quarrels with Apollonius and Callimachus, and Antagoras of Rhodes. So ancient are the hatreds of poets, envy, jealousy, and all unkindness.
Not to the wits of Courts couldst thou teach thy rural song, though all these centuries, more than two thousand years, they have laboured to vie with thee. There has come no new pastoral poet, though Virgil copied thee, and Pope, and Phillips, and all the buckram band of the teacup time; and all the modish swains of France have sung against thee, as the \textit{son challenged Athene}. They never knew the shepherd's life, the long' winter nights on dried heather by the fire, the long summer days, when over the dry grass all is quiet, and only the insects hum, and the shrunken burn whispers a silver tune. Swains in high-heeled shoon, and lace, shepherdesses in rouge and diamonds, the world is weary of all concerning them, save their images in porcelain, effigies how unlike the golden figures, dedicate to Aphrodite, of Bombyca and Battus. Somewhat, Theocritus, thou hast to answer for, thou that first of men brought the shepherd to Court, and made courtiers wild to go a Maying with the shepherds.
\gutchapter{XIV. To Edgar Allan Poe.}
Sir,---Your English readers, better acquainted with your poems and romances than with your criticisms, have long wondered at the indefatigable hatred which pursues your memory. You, who knew the men, will not marvel that certain microbes of letters, the survivors of your own generation, still harass your name with their malevolence, while old women twitter out their incredible and heeded slanders in the literary papers of New York. But their persistent animosity does not quite suffice to explain the dislike with which many American critics regard the greatest poet, perhaps the greatest literary genius, of their country. With a commendable patriotism, they are not apt to rate native merit too low; and you, I think, are the only example of an American prophet almost without honour in his own country.
The recent publication of a cold, careful, and in many respects admirable study of your career ('Edgar Allan Poe,' by George Woodberry: Houghton, Mifflin and Co., Boston) reminds English readers who have forgotten it, and teaches those who never knew it, that you were, unfortunately, a Reviewer. How unhappy were the necessities, how deplorable the vein, that compelled or seduced a man of your eminence into the dusty and stony ways of contemporary criticism! About the writers of his own generation a leader of that generation should hold his peace, he should neither praise nor blame nor defend his equals; he should not strike one blow at the buzzing ephemerae of letters. The breath of their life is in the columns of `Literary Gossip;' and they should be allowed to perish with the weekly advertisements on which they pasture. Reviewing, of course, there must needs be; but great minds should only criticise the great who have passed beyond the reach of eulogy or fault-finding.
Unhappily, taste and circumstances combined to make you a censor; you vexed a continent, and you are still unforgiven. What 'irritation of a sensitive nature, chafed by some indefinite sense of wrong,' drove you (in Mr. Longfellow's own words) to attack his pure and beneficent Muse we may never ascertain. But Mr. Longfellow forgave you easily; for pardon comes easily to the great. It was the smaller men, the Daweses, Griswolds, and the like, that knew not how to forget. 'The New Yorkers never forgave him,' says your latest biographer; and one scarcely marvels at the inveteracy of their malice. It was not individual vanity alone, but the whole literary class that you assailed. 'As a literary people,' you wrote, `we are one vast perambulating humbug.' After that declaration of war you died, and left your reputation to the vanities yet writhing beneath your scorn. They are writhing and writing still. He who knows them need not linger over the attacks and defences of your personal character; he will not waste time on calumnies, tale-bearing, private letters, and all the noisome dust which takes so long in settling above your tomb.
For us it is enough to know that you were compelled to live by your pen, and that in an age when the author of `To Helen' and' The Cask of Amontillado' was paid at the rate of a dollar a column. When such poverty was the mate of such pride as yours, a misery more deep than that of Burns, an agony longer than Chatterton's, were inevitable and assured. No man was less fortunate than you in the moment of his birth---\textit{infelix opportunitate vitae}. Had you lived a generation later, honour, wealth, applause, success in Europe and at home, would all have been yours. Within thirty years so great a change has passed over the profession of letters in America; and it is impossible to estimate the rewards which would have fallen to Edgar Poe, had chance made him the contemporary of Mark Twain and of `Called Back.' It may be that your criticisms helped to bring in the new era, and to lift letters out of the reach of quite unlettered scribblers. Though not a scholar, at least you had a respect for scholarship. You might still marvel over such words as `objectional' in the new biography of yourself, and might ask what is meant by such a sentence as 'his connection with it had inured to his own benefit by the frequent puffs of himself,' and so forth.
Best known in your own day as a critic, it is as a poet and a writer of short tales that you must live. But to discuss your few and elaborate poems is a waste of time, so completely does your own brief definition of poetry, `the rhythmic creation of the beautiful,' exhaust your theory, and so perfectly is the theory illustrated by the poems. Natural bent, and reaction against the example of Mr. Longfellow, combined to make you too intolerant of what you call the `didactic' element in verse. Even if morality be not seven-eighths of our life (the exact proportion as at present estimated), there was a place even on the Hellenic Parnassus for gnomic bards, and theirs in the nature of the case must always be the largest public.
`Music is the perfection of the soul or the idea of poetry,' so you wrote; 'the vagueness of exaltation aroused by a sweet air (which should be indefinite and never too strongly suggestive), is precisely what we should aim at in poetry.' You aimed at that mark, and struck it again and again, notably in `Helen, thy beauty is to me,' in 'The Haunted Palace,' `The Valley of Unrest,' and `The City in the Sea.' But by some Nemesis which might, perhaps, have been foreseen, you are, to the world, the poet of one poem---'The Raven:' a piece in which the music is highly artificial, and the `exaltation' (what there is of it) by no means particularly `vague.' So a portion of the public know little of Shelley but the `Skylark,' and those two incongruous birds, the lark and the raven, bear each of them a poet's name \textit{vivu' per ora virum}. Your theory of poetry, if accepted, would make you (after the author of `Kubla Khan') the foremost of the poets of the world; at no long distance would come Mr. William Morris as he was when he wrote 'Golden Wings,' `The Blue Closet,' and `The Sailing of the Sword;' and, close up, Mr. Lear, the author of `The Yongi Bongi Bo,' and the lay of the `Jumblies.'
On the other hand Homer would sink into the limbo to which you consigned Moliere. If we may judge a theory by its results, when compared with the deliberate verdict of the world, your aesthetic does not seem to hold water. The `Odyssey' is not really inferior to `Ulalume,' as it ought to be if your doctrine of poetry were correct, nor 'Le Festin de Pierre to `Undine.' Yet you deserve the praise of having been constant, in your poetic practice, to your poetic principles---principles commonly deserted by poets who, like Wordsworth, have published their aesthetic system. Your pieces are few; and Dr. Johnson would have called you, like Fielding, `a barren rascal.' But how can a writer's verses be numerous if with him, as with you, 'poetry is not a pursuit but a passion... which cannot at will be excited with an eye to the paltry compensations or the more paltry commendations of mankind!' Of you it may be said, more truly than Shelley said it of himself, that 'to ask you for anything human, is like asking at a gin-shop for a leg of mutton.'
Humanity must always be, to the majority of men, the true stuff of poetry; and only a minority will thank you for that rare music which (like the strains of the fiddler in the story) is touched on a single string, and on an instrument fashioned from the spoils of the grave. You chose, or you were destined
To vary from the kindly race of men;
and the consequences, which wasted your life, pursue your reputation. For your stories has been reserved a boundless popularity, and that highest success---the success of a perfectly sympathetic translation. By this time, of course, you have made the acquaintance of your translator, M. Charles Baudelaire, who so strenuously shared your views about Mr. Emerson and the Transcendentalists, and who so energetically resisted all those ideas of `progress' which `came from Hell or Boston.' On this point, however, the world continues to differ from you and M. Baudelaire, and perhaps there is only the choice between our optimism and universal suicide or universal opium-eating. But to discuss your ultimate ideas is perhaps a profitless digression from the topic of your prose romances.
An English critic (probably a Northerner at heart) has described them as `Hawthorne and delirium tremens.' I am not aware that extreme orderliness, masterly elaboration, and unchecked progress towards a predetermined effect are characteristics of the visions of delirium. If they be, then there is a deal of truth in the criticism, and a good deal of delirium tremens in your style. But your ingenuity, your completeness, your occasional luxuriance of fancy and wealth of jewel-like words, are not, perhaps, gifts which Mr. Hawthorne had at his command. He was a great writer---the greatest writer in prose fiction whom America has produced. But you and he have not much in common, except a certain mortuary turn of mind and a taste for gloomy allegories about the workings of conscience.
I forbear to anticipate your verdict about the latest essays of American fiction. These by no means follow in the lines which you laid down about brevity and the steady working to one single effect. Probably you would not be very tolerant (tolerance was not your leading virtue) of Mr. Roe, now your countrymen's favourite novelist. He is long, he is didactic, he is eminently uninspired. In the works of one who is, what you were called yourself, a Bostonian, you would admire, at least, the acute observation, the subtlety, and the unfailing distinction. But, destitute of humour as you unhappily but undeniably were, you would miss, I fear, the charm of `Daisy Miller.' You would admit the unity of effect secured in `Washington Square,' though that effect is as remote as possible from the terror of `The House of Usher' or the vindictive triumph of 'The Cask of Amontillado.'
Farewell, farewell, thou sombre and solitary spirit: a genius tethered to the hack-work of the press, a gentleman among \textit{canaille}, a poet among poetasters, dowered with a scholar's taste without a scholar's training, embittered by his sensitive scorn, and all unsupported by his consolations.
\gutchapter{XV. To Sir Walter Scott, Bart.}
Rodono, St. Mary's Loch:
Sept. 5, 1885.
Sir,---In your biography it is recorded that you not only won the favour of all men and women; but that a domestic fowl conceived an affection for you, and that a pig, by his will, had never been severed from your company. If some Circe had repeated in my case her favourite miracle of turning mortals into swine, and had given me a choice, into that fortunate pig, blessed among his race, would I have been converted! You, almost alone among men of letters, still, like a living friend, win and charm us out of the past; and if one might call up a poet, as the scholiast tried to call Homer, from the shades, who would not, out of all the rest, demand some hours of your society? Who that ever meddled with letters, what child of the irritable race, possessed even a tithe of your simple manliness, of the heart that never knew a touch of jealousy, that envied no man his laurels, that took honour and wealth as they came, but never would have deplored them had you missed both and remained but the Border sportsman and the Border antiquary?
Were the word `genial' not so much profaned, were it not misused in easy good-nature, to extenuate lettered and sensual indolence, that worn old term might be applied, above all men, to `the Shirra.' But perhaps we scarcely need a word (it would be seldom in use) for a character so rare, or rather so lonely, in its nobility and charm as that of Walter Scott. Here, in the heart of your own country, among your own grey round-shouldered hills (each so like the other that the shadow of one falling on its neighbour exactly outlines that neighbour's shape), it is of you and of your works that a native of the Forest is most frequently brought in mind. All the spirits of the river and the hill, all the dying refrains of ballad and the fading echoes of story, all the memory of the wild past, each legend of burn and loch, seem to have combined to inform your spirit, and to secure themselves an immortal life in your song. It is through you that we remember them; and in recalling them, as in treading each hillside in this land, we again remember you and bless you.
It is not `Sixty Years Since' the echo of Tweed among his pebbles fell for the last time on your ear; not sixty years since, and how much is altered! But two generations have passed; the lad who used to ride from Edinburgh to Abbotsford, carrying new books for you, and old, is still vending, in George Street, old books and new. Of politics I have not the heart to speak. Little joy would you have had in most that has befallen since the Reform Bill was passed, to the chivalrous cry of 'burke Sir Walter.' We are still very Radical in the Forest, and you were taken away from many evils to come. How would the cheek of Walter Scott, or of Leyden, have blushed at the names of Majuba, The Soudan, Maiwand, and many others that recall political cowardice or military incapacity! On the other hand, who but you could have sung the dirge of Gordon, or wedded with immortal verse the names of Hamilton (who fell with Cavagnari), of the two Stewarts, of many another clansman, brave among the bravest! Only he who told how
The stubborn spearmen still made good
Their dark impenetrable wood
could have fitly rhymed a score of feats of arms in which, as at M'Neill's Zareeba and at Abu Klea,
Groom fought like noble, squire like knight,
As fearlessly and well.
Ah, Sir, the hearts of the rulers may wax faint, and the voting classes may forget that they are Britons; but when it comes to blows our fighting men might cry, with Leyden,
My name is little Jock Elliot,
And wha daur meddle wi' me!
Much is changed, in the country-side as well as in the country; but much remains. The little towns of your time are populous and excessively black with the smoke of factories---not, I fear, at present very flourishing. In Galashiels you still see the little change-house and the cluster of cottages round the Laird's lodge, like the clachan of Tully Veolan. But these plain remnants of the old Scotch towns are almost buried in a multitude of 'smoky dwarf houses'---a living poet, Mr. Matthew Arnold, has found the fitting phrase for these dwellings, once for all. All over the Forest he waters are dirty and poisoned: I think they are filthiest below Hawick; but this may be mere local prejudice in a Selkirk man. To keep them clean costs money; and, though improvements are often promised, I cannot see much change for the better. Abbotsford, luckily, is above Galashiels, and only receives the dirt and dyes of Selkirk, Peebles, Walkerburn, and Innerlethen. On the other hand, your ill-omened later dwelling, `the unhappy palace of your race,' is overlooked by villas that prick a cockney ear among their larches, hotels of the future. Ah, Sir, Scotland is a strange place. Whisky is exiled from some of our caravanserais, and they have banished Sir John Barleycorn. It seems as if the views of the excellent critic (who wrote your life lately, and said you had left no descendants, \textit{le pauvre homme}) were beginning to prevail. This pious biographer was greatly shocked by that capital story about the keg of whisky that arrived at the Liddesdale farmer's during family prayers. Your Toryism also was an offence to him.
Among these vicissitudes of things and the overthrow of customs, let us be thankful that, beyond the reach of the manufacturers, the Border country remains as kind and homely as ever. I looked at Ashiestiel some days ago: the house seemed just as it may have been when you left it for Abbotsford, only there was a lawn-tennis net on the lawn, the hill on the opposite bank of the Tweed was covered to the crest with turnips, and the burn did not sing below the little bridge, for in this arid summer the burn was dry. But there was still a grilse that rose to a big March brown in the shrunken stream below Elibank. This may not interest you, who styled yourself
No fisher,
But a well-wisher
To the game!
Still, as when you were thinking over Marmion, a man might have 'grand gallops among the hills'---those grave wastes of heather and bent that sever all the watercourses and roll their sheep-covered pastures from Dollar Law to White Combe, and from White Combe to the Three Brethren Cairn and the Windburg and Skelf-hill Pen. Yes, Teviotdale is pleasant still, and there is not a drop of dye in the water, \textit{purior electro}, of Yarrow. St. Mary's Loch lies beneath me, smitten with wind and rain---the St. Mary's of North and of the Shepherd. Only the trout, that see a myriad of artificial flies, are shyer than of yore. The Shepherd could no longer fill a cart up Meggat with trout so much of a size that the country people took them for herrings.
The grave of Piers Cockburn is still not desecrated: hard by it lies, within a little wood; and beneath that slab of old sandstone, and the graven letters, and the sword and shield, sleep 'Piers Cockburn and Marjory his wife.' Not a hundred yards off was the castle door where they hanged him; this is the tomb of the ballad, and the lady that buried him rests now with her wild lord.
Oh, wat ye no my heart was sair,
When I happit the mouls on his yellow hair;
Oh, wat ye no my heart was wae,
When I turned about and went my way! (1)
Here too hearts have broken, and there is a sacredness in the shadow and beneath these clustering berries of the rowan-trees. That sacredness, that reverent memory of our old land, it is always and inextricably blended with our memories, with our thoughts, with our love of you. Scotchmen, methinks, who owe so much to you, owe you most for the example you gave of the beauty of a life of honour, showing them what, by Heaven's blessing, a Scotchman still might be.
(1) Lord Napier and Ettrick points out to me that, unluckily, the tradition is erroneous. Piers was not executed at all. William Cockburn suffered in Edinburgh. But the \textit{Border Minstrelsy} overrides history.
\textit{Criminal Trials in Scotland} by Robert Pitcairn, Esq. Vol. i. part I. p. 144, A. D. 1530. 17 Jac. V.
May 16. William Cokburne of Henderland, convicted (in presence of the King) of high treason committed by him in bringing Alexander Forestare and his son, Englishmen, to the plundering of Archibald Somervile; and for treasunably bringing certain Englishmen to the lands of Glenquhome; and for common theft, common reset of theft, out-putting and in-putting thereof. Sentence. For which causes and crimes he has forfeited his life, lands, and goods, movable and immovable; which shall be escheated to the King. Beheaded.
Words, empty and unavailing---for what words of ours can speak our thoughts or interpret our affections! From you first, as we followed the deer with King James, or rode with William of Deloraine on his midnight errand, did we learn what Poetry means and ali the happiness that is in the gift of song. This and more than may be told you gave us, that are not forgetful, not ungrateful, though our praise be unequal to our gratitude. \textit{Fungor inani munere!}
\gutchapter{XVI. To Eusebius of Caesarea.}
(Concerning the Gods of the Heathen.)
Touching the Gods of the Heathen, most reverend Father, thou art not ignorant that even now, as in the time of thy probation on earth, there is great dissension. That these feigned Deities and idols, the work of men's hands, are no longer worshipped thou knowest; neither do men eat meat offered to idols. Even as spoke that last Oracle which murmured forth, the latest and the only true voice from Delphi, even so 'the fair-wrought court divine hath fallen; no more hath Phoebus his home, no more his laurel-bough, nor the singing well of water; nay, the sweet-voiced water is silent.' The fane is ruinous, and the images of men's idolatry are dust.
Nevertheless, most worshipful, men do still dispute about the beginnings of those sinful Gods: such as Zeus, Athene, and Dionysus: and marvel how first they won their dominion over the souls of the foolish peoples. Now, concerning these things there is not one belief, but many; howbeit, there are two main kinds of opinion. One sect of philosophers believes---as thyself, with heavenly learning, didst not vainly persuade---that the Gods were the inventions of wild and bestial folk, who, long before cities were builded or life was honourably ordained, fashioned forth evil spirits in their own savage likeness; ay, or in the likeness of the very beasts that perish. To this judgment, as it is set forth in thy Book of the Preparation for the Gospel, I, humble as I am, do give my consent. But on the other side are many and learned men, chiefly of the tribes of the Alemanni, who have almost conquered the whole inhabited world. These, being unwilling to suppose that the Hellenes were in bondage to superstitions handed down from times of utter darkness and a bestial life, do chiefly hold with the heathen philosophers, even with the writers whom thou, most venerable, didst confound with thy wisdom and chasten with the scourge of small cords of thy wit.
Thus, like the heathen, our doctors and teachers maintain that the Gods of the nations were, in the beginning, such pure natural creatures as the blue sky, the sun, the air, the bright dawn, and the fire; but, as time went on, men, forgetting the meaning of their own speech and no longer understanding the tongue of their own fathers, were misled and beguiled into fashioning all those lamentable tales: as that Zeus, for love of mortal women, took the shape of a bull, a ram, a serpent, an ant, an eagle, and sinned in such wise as it is a shame even to speak of.
Behold, then, most worshipful, how these doctors and learned men argue, even like the philosophers of the heathen whom thou didst confound. For they declare the Gods to have been natural elements, sun and sky and storm, even as did thy opponents; and, like them, as thou saidst, 'they are nowise at one with each other in their explanations.' For of old some boasted that Hera was the Air; and some that she signified the love of woman and man; and some that she was the waters above the Earth; and others that she was the Earth beneath the waters; and yet others that she was the Night, for that Night is the shadow of Earth: as if, forsooth, the men who first worshipped Hera had understanding of these things! And when Hera and Zeus quarrel unseemly (as Homer declareth), this meant (said the learned in thy days) no more than the strife and confusion of the elements, and was not in the beginning an idle slanderous tale.
To all which, most worshipful, thou didst answer wisely: saying that Hera could not be both night, and earth, and water, and air, and the love of sexes, and the confusion of the elements; but that all these opinions were vain dreams, and the guesses of the learned. And why---thou saidst---even if the Gods were pure natural creatures, are such foul things told of them in the Mysteries as it is not fitting for me to declare. 'These wanderings, and drinkings, and loves, and corruptions, that would be shameful in men, why,' thou saidst, 'were they attributed to the natural elements; and wherefore did the Gods constantly show themselves, like the sorcerers called were-wolves, in the shape of the perishable beasts?' But, mainly, thou didst argue that, till the philosophers of the heathen were agreed among themselves, not all contradicting each the other, they had no semblance of a sure foundation for their doctrine.
To all this and more, most worshipful Father, I know not what the heathen answered thee. But, in our time, the learned men who stand to it that the heathen Gods were in the beginning the pure elements, and that the nations, forgetting their first love and the significance of their own speech, became confused and were betrayed into foul stories about the pure Gods---these learned men, I say, agree no whit among themselves. Nay, they differ one from another, not less than did Plutarch and Porphyry and Theagenes, and the rest whom thou didst laugh to scorn. Bear with me, Father, while I tell thee how the new Plutarchs and Porphyrys do contend among themselves; and yet these differences of theirs they call `Science'!
Consider the goddess Athene, who sprang armed from the head of Zeus, even as---among the fables of the poor heathen folk of seas thou never knewest---goddesses are fabled to leap out from the armpits or feet of their fathers. Thou must know that what Plato, in the `Cratylus,' made Socrates say in jest, the learned among us practise in sad earnest. For, when they wish to explain the nature of any God, they first examine his name, and torment the letters thereof, arranging and altering them according to their will, and flying off to the speech of the Indians and Medes and Chaldeans, and other Barbarians, if Greek will not serve their turn. How saith Socrates? 'I bethink me of a very new and ingenious idea that occurs to me; and, if I do not mind, I shall be wiser than I should be by to-morrow's dawn. My notion is that we may put in and pull out letters at pleasure and alter the accents.' Even so do our learned---not at pleasure, maybe, but according to certain fixed laws (so they declare); yet none the more do they agree among themselves. And I deny not that they discover many things true and good to be known; but, as touching the names of the Gods, their learning, as it standeth, is confusion. Look, then, at the goddess Athene: taking one example out of hundreds. We have dwelling in our coasts Muellerus, the most erudite of the doctors of the Alemanni, and the most golden-mouthed. Concerning Athene, he saith that her name is none other than, in the ancient tongue of the Brachmanae, \textit{Ahana'}, which, being interpreted, means the Dawn. `And that the morning light,' saith he, 'offers the best starting-point; for the later growth of Athene has been proved, I believe, beyond the reach of doubt or even cavil.' (1)
(1) 'The Lesson of Jupiter.'---\textit{Nineteenth Century}, October, 1885.
Yet this same doctor candidly lets us know that another of his nation, the witty Benfeius, hath devised another sense and origin of Athene, taken from the speech of the old Medes. But Muellerus declares to us that whosoever shall examine the contention of Benfeius 'will be bound, in common honesty, to confess that it is untenable.' This, Father, is one for Benfeius, as the saying goes. And as Muellerus holds that these matters `admit of almost mathematical precision,' it would seem that Benfeius is but a \textit{Dummkopf}, as the Alemanni say, in their own language, when they would be pleasant among themselves.
Now, wouldst thou credit it? despite the mathematical plainness of the facts, other Alemanni agree neither with Muellerus, nor yet with Benfeius, and will neither hear that Athene was the Dawn, nor yet that she is `the feminine of the Zend \textit{Thra'eta'na athwya'na}.' Lo, you! how Prellerus goes about to show that her name is drawn not from \textit{Ahana'} and the old Brachmanae, nor \textit{athwya'na} and the old Medes, but from 'the root \textit{aith}*, whence \textit{aither}*, the air, or \textit{ath}*, whence \textit{anthos}*, a flower.' Yea, and Prellerus will have it that no man knows the verity of this matter. None the less he is very bold, and will none of the Dawn; but holds to it that Athene was, from the first, 'the clear pure height of the Air, which is exceeding pure in Attica.'
Now, Father, as if all this were not enough, comes one Roscherus in, with a mighty great volume on the Gods, and Furtwaenglerus, among others, for his ally. And these doctors will neither with Rueckertus and Hermannus, take Athene for `wisdom in person;' nor with Welckerus and Prellerus, for `the goddess of air;' nor even, with Muellerus and mathematical certainty, for `the Morning-Red:' but they say that Athene is the `black thunder-cloud, and the lightning that leapeth therefrom'! I make no doubt that other Alemanni are of other minds: \textit{quot Alemanni tot sententiae}.
Yea, as thou saidst of the learned heathen, \textit{Oude gar allelois symphona} \textit{physiologousis}. Yet these disputes of theirs they call `Science'! But if any man says to the learned: 'Best of men, you are erudite, and laborious and witty; but, till you are more of the same mind, your opinions cannot be styled knowledge. Nay, they are at present of no avail whereon to found any doctrine concerning the Gods'---that man is railed at for his `mean' and `weak' arguments.
* Transliterated from Greek.
Was it thus, Father, that the heathen railed against thee? But I must still believe, with thee, that these evil tales of the Gods were invented `when man's life was yet brutish and wandering' (as is the life of many tribes that even now tell like tales), and were maintained in honour of the later Greeks 'because none dared alter the ancient beliefs of his ancestors.' Farewell, Father; and all good be with thee, wishes thy well-wisher and thy disciple.
\gutchapter{XVII. To Percy Bysshe Shelley.}
Sir,---In your lifetime on earth you were not more than commonly curious as to what was said by `the herd of mankind,' if I may quote your own phrase. It was that of one who loved his fellow-men, but did not in his less enthusiastic moments overestimate their virtues and their discretion. Removed so far away from our hubbub, and that world where, as you say, we `pursue our serious folly as of old,' you are, one may guess, but moderately concerned about the fate of your writings and your reputation. As to the first, you have somewhere said, in one of your letters, that the final judgment on your merits as a poet is in the hands of posterity, and that you fear the verdict will be `Guilty,' and the sentence `Death.' Such apprehensions cannot have been fixed or frequent in the mind of one whose genius burned always with a clearer and steadier flame to the last. The jury of which you spoke has met: a mixed jury and a merciful. The verdict is `Well done,' and the sentence Immortality of Fame. There have been, there are, dissenters; yet probably they will be less and less heard as the years go on.
One judge, or juryman, has made up his mind that prose was your true province, and that your letters will outlive your lays. I know not whether it was the same or an equally well-inspired critic, who spoke of your most perfect lyrics (so Beau Brummell spoke of his ill-tied cravats) as `a gallery of your failures.' But the general voice does not echo these utterances of a too subtle intellect. At a famous University (not your own) once existed a band of men known as 'The Trinity Sniffers.' Perhaps the spirit of the sniffer may still inspire some of the jurors who from time to time make themselves heard in your case. The `Quarterly Review', I fear, is still unreconciled. It regards your attempts as tainted by the spirit of 'The Liberal Movement in English Literature;' and it is impossible, alas! to maintain with any success that you were a Throne and Altar Tory. At Oxford you are forgiven; and the old rooms where you let the oysters burn (was not your founder, King Alfred, once guilty of similar negligence?) are now shown to pious pilgrims.
But Conservatives, 't is rumoured, are still averse to your opinions, and are believed to prefer to yours the works of the Reverend Mr. Keble, and, indeed, of the clergy in general. But, in spite of all this, your poems, like the affections of the true lovers in Theocritus, are still `in the mouths of all, and chiefly on the lips of the young.' It is in your lyrics that you live, and I do not mean that every one could pass an examination in the plot of ``Prometheus Unbound'' Talking of this piece, by the way, a Cambridge critic finds that it reveals in you a hankering after life in a cave---doubtless an unconsciously inherited memory from cave-man. Speaking of cave-man reminds me that you once spoke of deserting song for prose, and of producing a history of the moral, intellectual, and political elements in human society, which, we now agree, began, as Asia would fain have ended, in a cave.
Fortunately you gave us 'Adonai, and `Hellas' instead of this treatise, and we have now successfully written the natural history of Man for ourselves. Science tells us that before becoming cave-dweller he was a brute; Experience daily proclaims that he constantly reverts to his original condition. \textit{L'homme est un mechant animal}, in spite of your boyish efforts to add pretty girls 'to the list of the good, the disinterested, and the free.'
Ah, not in the wastes of Speculation, nor the sterile din of Politics, were `the haunts meet for thee.' Watching the yellow bees in the ivy bloom, and the reflected pine forest in the water-pools, watching the sunset as it faded, and the dawn as it fired, and weaving all fair and fleeting things into a tissue where light and music were at one, that was the task of Shelley! `To ask you for anything human,' you said, 'was like asking for a leg of mutton at a gin-shop.' Nay, rather, like asking Apollo and Hebe, in the Olympian abodes, to give us beef for ambrosia, and port for nectar. Each poet gives what he has, and what he can offer; you spread before us fairy bread, and enchanted wine, and shall we turn away, with a sneer, because, out of all the multitudes of singers, one is spiritual and strange, one has seen Artemis unveiled? One, like Anchises, has been beloved of the Goddess, and his eyes, when he looks on the common works of common men, are, like the eyes of Anchises, blind with excess of light. Let Shelley sing of what he saw, what none saw but Shelley!
Notwithstanding the popularity of your poems (the most romantic of things didactic), our world is no better than the world you knew. This will disappoint you, who had `a passion for reforming it.' Kings and priests are very much where you left them. True, we have a poet who assails them, at large, frequently and fearlessly; yet Mr. Swinburne has never, like `kind Hunt,' been in prison, nor do we fear for him a charge of treason. Moreover, chemical science has discovered new and ingenious ways of destroying principalities and powers. You would be interested in the methods, but your peaceful Revolutionism, which disdained physical force, would regret their application.
Our foreign affairs are not in a state which even you would consider satisfactory; for we have just had to contend with a Revolt of Islam, and we still find in Russia exactly the qualities which you recognised and described. We have a great statesman whose methods and eloquence somewhat resemble those you attribute to Laon and Prince Athanase. Alas! he is a youth of more than seventy summers; and not in his time will Prometheus retire to a cavern and pass a peaceful millennium in twining buds and beams.
In domestic affairs most of the Reforms you desired to see have been carried. Ireland has received Emancipation, and almost everything else she can ask for. I regret to say that she is still unhappy; her wounds unstanched, her wrongs unforgiven. At home we have enfranchised the paupers, and expect the most happy results. Paupers (as Mr. Gladstone says) are `our own flesh and blood,' and, as we compel them to be vaccinated, so we should permit them to vote. Is it a dream that Mr. Jesse Collings (how you would have loved that man!) has a Bill for extending the priceless boon of the vote to inmates of Pauper Lunatic Asylums? This may prove that last element in the Elixir of political happiness which we have sought in vain. Atheists, you will regret to hear, are still unpopular; but the new Parliament has done something for Mr. Bradlaugh. You should have known our Charles while you were in the `Queen Mab' stage. I fear you wandered, later, from his robust condition of intellectual development.
As to your private life, many biographers contrive to make public as much of it as possible. Your name, even in life, was, alas! a kind of \textit{ducdame} to bring people of no very great sense into your circle. This curious fascination has attracted round your memory a feeble folk of commentators, biographers, anecdotists, and others of the tribe. They swarm round you like carrion-flies round a sensitive plant, like night-birds bewildered by the sun. Men of sense and taste have written on you, indeed; but your weaker admirers are now disputing as to whether it was your heart, or a less dignified and most troublesome organ, which escaped the flames of the funeral pyre. These biographers fight terribly among themselves, and vainly prolong the memory of 'old unhappy far-off things, and \textit{sorrows} long ago.' Let us leave them and their squabbles over what is unessential, their raking up of old letters and old stories.
The town has lately yawned a weary laugh over an enemy of yours, who has produced two heavy volumes, styled by him `The Real Shelley.' The real Shelley, it appears, was Shelley as conceived of by a worthy gentleman so prejudiced and so skilled in taking up things by the wrong handle that I wonder he has not made a name in the exact science of Comparative Mythology. He criticises you in the spirit of that Christian Apologist, the Englishman who called you `a damned Atheist' in the post-office at Pisa. He finds that you had `a little turned-up nose,' a feature no less important in his system than was the nose of Cleopatra (according to Pascal) in the history of the world. To be in harmony with your nose, you were a `phenomenal' liar, an ill-bred, ill-born, profligate, partly insane, an evil-tempered monster, a self-righteous person, full of self-approbation---in fact you were the Beast of this pious Apocalypse. Your friend Dr. Lind was an embittered and scurrilous apothecary, 'a bad old man.' But enough of this inopportune brawler. For Humanity, of which you hoped such great things, Science predicts extinction in a night of Frost. The sun will grow cold, slowly---as slowly as doom came on Jupiter in your `Prometheus,' but as surely. If this nightmare be fulfilled, perhaps the Last Man, in some fetid hut on the ice-bound Equator, will read. by a fading lamp charged with the dregs of the oil in his cruse, the poetry of Shelley. So reading, he, the latest of his race, will not wholly be deprived of those sights which alone (says the nameless Greek) make life worth enduring. In your verse he will have sight of sky, and sea, and cloud, the gold of dawn and the gloom of earthquake and eclipse, he will be face to face, in fancy, with the great powers that are dead, sun, and ocean, and the illimitable azure of the heavens. In Shelley's poetry, while Man endures, all those will survive; for your `voice is as the voice of winds and tides,' and perhaps more deathless than all of these, and only perishable with the perishing of the human spirit.
\gutchapter{XVIII. To Monsieur de Moliere, Valet de Chambre du Roi.}
Monsieur,---With what awe does a writer venture into the presence of the great Moliere! As a courtier in your time would scratch humbly (with his comb!) at the door of the Grand Monarch, so I presume to draw near your dwelling among the Immortals. You, like the king who, among all his titles, has now none so proud as that of the friend of Moliere---you found your dominions small, humble, and distracted; you raised them to the dignity of an empire: what Louis XIV. did for France you achieved for French comedy; and the ba'ton of Scapin still wields its sway though the sword of Louis was broken at Blenheim. For the King the Pyrenees, or so he fancied, ceased to exist; by a more magnificent conquest you overcame the Channel. If England vanquished your country's arms, it was through you that France \textit{ferum victorem cepit}, and restored the dynasty of Comedy to the land whence she had been driven. Ever since Dryden borrowed `L'Etourdi,' our tardy apish nation has lived (in matters theatrical) on the spoils of the wits of France.
In one respect, to be sure, times and manners have altered. While you lived, taste kept the French drama pure; and it was the congenial business of English playwrights to foist their rustic grossness and their large Fescennine jests into the urban page of Moliere. Now they are diversely occupied; and it is their affair to lend modesty where they borrow wit, and to spare a blush to the cheek of the Lord Chamberlain. But still, as has ever been our wont since Etherege saw, and envied, and imitated your successes---still we pilfer the plays of France, and take our \textit{bien}, as you said in your lordly manner, wherever we can find it. We are the privateers of the stage; and it is rarely, to be sure, that a comedy pleases the town which has not first been `cut out' from the countrymen of Moliere. Why this should be, and what `tenebriferous star' (as Paracelsus, your companion in the 'Dialogues des Morts,' would have believed) thus darkens the sun of English humour, we know not; but certainly our dependence on France is the sincerest tribute to you. Without you, neither Rotrou, nor Corneille, nor 'a wilderness of monkeys' like Scarron, could ever have given Comedy to France and restored her to Europe.
While we owe to you, Monsieur, the beautiful advent of Comedy, fair and beneficent as Peace in the play of Aristophanes, it is still to you that we must turn when of comedies we desire the best. If you studied with daily and nightly care the works of Plautus and Terence, if you 'let no musty \textit{bouquin} escape you' (so your enemies declared), it was to some purpose that you laboured. Shakespeare excepted, you eclipsed all who came before you; and from those that follow, however fresh, we turn: we turn from Regnard and Beaumarchais, from Sheridan: and Goldsmith, from Musset and Pailleron and Labiche, to that crowded world of your creations. `Creations' one may well say, for you anticipated Nature herself: you gave us, before she did, in Alceste a Rousseau who was a gentleman not a lacquey; in a \textit{mot} of Don Juan's, the secret of the new Religion and the watchword of Comte, \textit{l'amour de l'humanite}.
Before you where can we find, save in Rabelais, a Frenchman with humour; and where, unless it be in Montaigne, the wise philosophy of a secular civilisalion? With a heart the most tender, delicate, loving, and generous, a heart often in agony and torment, you had to make life endurable (we cannot doubt it) without any whisper of promise, or hope, or warning from Religion. Yes, in an age when the greatest mind of all, the mind of Pascal, proclaimed that the only help was in voluntary blindness, that the only chance was to hazard all on a bet at evens, you, Monsieur, refused to be blinded, or to pretend to see what you found invisible.
In Religion you beheld no promise of help. When the Jesuits and Jansenists of your time saw, each of them, in Tartufe the portrait of their rivals (as each of the laughable Marquises in your play conceived that you were girding at his neighbour), you all the while were mocking every credulous excess of Faith. In the sermons preached to Agnes we surely hear your private laughter; in the arguments for credulity which are presented to Don Juan by his valet we listen to the eternal self-defence of superstition. Thus, desolate of belief, you sought for the permanent element of life---precisely where Pascal recognised all that was most fleeting and unsubstantial---in \textit{divertissement}; in the pleasure of looking on, a spectator of the accidents of existence, an observer of the follies of mankind. Like the Gods of the Epicurean, you seem to regard our life as a play that is played, as a comedy; yet how often the tragic note comes in! What pity, and in the laughter what an accent of tears, as of rain in the wind! No comedian has been so kindly and human as you; none has had a heart, like you, to feel for his butts, and to leave them sometimes, in a sense, superior to their tormentors. Sganarelle, M. de Pourceaugnac, George Dandin, and the rest---our sympathy, somehow, is with them, after all; and M. de Pourceaugnac is a gentleman, despite his misadventures.
Though triumphant Youth and malicious Love in your plays may batter and defeat Jealousy and Old Age, yet they have not all the victory, or you did not mean that they should win it. They go off with laughter, and their victim with a grimace; but in him we, that are past our youth, behold an actor in an unending tragedy, the defeat of a generation. Your sympathy is not wholly with the dogs that are having their day; you can throw a bone or a crust to the dog that has had his, and has been taught that it is over and ended. Yourself not unlearned in shame, in jealousy, in endurance of the wanton pride of men (how could the poor player and the husband of Celimene be untaught in that experience?), you never sided quite heartily, as other comedians have done, with young prosperity and rank and power.
I am not the first who has dared to approach you in the Shades; for just after your own death the author of `Les Dialogues des Morts' gave you Paracelsus as a companion, and the author of `Le Jugement de Pluton' made the `mighty warder' decide that 'Moliere should not talk philosophy.' These writers, like most of us, feel that, after all, the comedies of the \textit{Contemplateur}, of the translator of Lucretius, are a philosophy of life in themselves, and that in them we read the lessons of human experience writ small and clear.
What comedian but Moliere has combined with such depths---with the indignation of Alceste, the self-deception of Tartufe, the blasphemy of Don Juan---such wildness of irresponsible mirth, such humour, such wit! Even now, when more than two hundred years have sped by, when so much water has flowed under the bridges and has borne away so many trifles of contemporary mirth (\textit{cetera} \textit{fluminis ritu feruntur}), even now we never laugh so well as when Mascarille and Vadius and M. Jourdain tread the boards in the Maison de Moliere. Since those mobile dark brows of yours ceased to make men laugh, since your voice denounced the `demoniac' manner of contemporary tragedians, I take leave to think that no player has been more worthy to wear the \textit{canons} of Mascarille or the gown of Vadius than M. Coquelin of the Comedie Francaise. In him you have a successor to your Mascarille so perfect, that the ghosts of play-goers of your date might cry, could they see him, that Moliere had come again. But, with all respect to the efforts of the fair, I doubt if Mdlle. Barthet, or \textit{Mdme}. Croizette herself, would reconcile the town to the loss of the fair De Brie, and Madeleine, and the first, the true Celimene, Armande. Yet had you ever so merry a \textit{soubrette} as \textit{Mdme}. Samary, so exquisite a Nicole?
Denounced, persecuted, and buried hugger-mugger two hundred years ago, you are now not over-praised, but more worshipped, with more servility and ostentation, studied with more prying curiosity than you may approve. Are not the Molieristes a body who carry adoration to fanaticism? Any scrap of your handwriting (so few are these), any anecdote even remotely touching on your life, any fact that may prove your house was numbered 15 not 22, is eagerly seized and discussed by your too minute historians. Concerning your private life, these men often write more like malicious enemies than friends; repeating the fabulous scandals of Le Boulanger, and trying vainly to support them by grubbing in dusty parish registers. It is most necessary to defend you from your friends---from such friends as the veteran and inveterate M. Arsene Houssaye, or the industrious but puzzle-headed M. Loiseleur. Truly they seek the living among the dead, and the immortal Moliere among the sweepings of attorneys' offices. As I regard them (for I have tarried in their tents) and as I behold their trivialities---the exercises of men who neglect Molieres works to write about Moliere's great-grandmother's second-best bed---I sometimes wish that Moliere were here to write on his devotees a new comedy, `Les Molieristes.' How fortunate were they, Monsieur, who lived and worked with you, who saw you day by day, who were attached, as Lagrange tells us, by the kindest loyalty to the best and most honourable of men, the most open-handed in friendship, in charity the most delicate, of the heartiest sympathy! Ah, that for one day I could behold you, writing in the study, rehearsing on the stage, musing in the lace-seller's shop, strolling through the Palais, turning over the new books at Billaine's, dusting your ruffles among the old volumes on the sunny stalls. Would that, through the ages, we could hear you after supper, merry with Boileau, and with Racine,---not yet a traitor,---laughing over Chapelain, combining to gird at him in an epigram, or mocking at Cotin, or talking your favourite philosophy, mindful of Descartes. Surely of all the wits none was ever so good a man, none ever made life so rich with humour and friendship.
\gutchapter{XIX. To Robert Burns.}
Sir,---Among men of Genius, and especially among Poets, there are some to whom we turn with a peculiar and unfeigned affection; there are others whom we admire rather than love. By some we are won with our will, by others conquered against our desire. It has been your peculiar fortune to capture the hearts of a whole people---a people not usually prone to praise, but devoted with a personal and patriotic loyalty to you and to your reputation. In you every Scot who \textit{is} a Scot sees, admires, and compliments Himself, his ideal self---independent, fond of whisky, fonder of the lassies; you are the true representative of him and of his nation. Next year will be the hundredth since the press of Kilmarnock brought to light its solitary masterpiece, your Poems; and next year, therefore, methinks, the revenue will receive a welcome accession from the abundance of whisky drunk in your honour. It is a cruel thing for any of your countrymen to feel that, where all the rest love, he can only admire; where all the rest are idolators, he may not bend the knee; but stands apart and beats upon his breast, observing, not adoring---a critic. Yet to some of us---petty souls, perhaps, and envious---that loud indiscriminating praise of `Robbie Burns' (for so they style you in their Change-house familiarity) has long been ungrateful; and, among the treasures of your songs, we venture to select and even to reject. So it must be! We cannot all love Haggis, nor `painch, tripe, and thairm,' and all those rural dainties which you celebrate as `warm-reekin, rich!' `Rather too rich,' as the Young Lady said on an occasion recorded by Sam Weller.
Auld Scotland wants nae skinking ware
That jaups in luggies;
But, if ye wish her gratefu' prayer,
Gie her a Haggis!
You \textit{have} given her a Haggis, with a vengeance, and her `gratefu' prayer' is yours for ever. But if even an eternity of partridge may pall on the epicure, so of Haggis too, as of all earthly delights, cometh satiety at last. And yet what a glorious Haggis it is---the more emphatically rustic and even Fescennine part of your verse! We have had many a rural bard since Theocritus `watched the visionary flocks,' but you are the only one of them all who has spoken the sincere Doric. Yours is the talk of the byre and the plough-tail; yours is that large utterance of the early hinds. Even Theocritus minces matters, save where Lacon and Comatas quite outdo the swains of Ayrshire. `But thee, Theocritus, wha matches?' you ask, and yourself out-match him in this wide rude region, trodden only by the rural Muse.
`\textit{Thy} rural loves are nature's sel';' and the wooer of Jean Armour speaks more like a true shepherd than the elegant Daphnis of the `Oaristys.'
Indeed it is with this that moral critics of your life reproach you, forgetting, perhaps, that in your amours you were but as other Scotch ploughmen and shepherds of the past and present. Ettrick may still, with Afghanistan, offer matter for idylls, as Mr. Carlyle (your antithesis, and the complement of the Scotch character) supposed; but the morals of Ettrick are those of rural Sicily in old days, or of Mossgiel in your days. Over these matters the Kirk, with all her power, and the Free Kirk too, have had absolutely no influence whatever. To leave so delicate a topic, you were but as other swains, or, as `that Birkie ca'd a lord,' Lord Byron; only you combined (in certain of your letters) a libertine theory with your practice; you poured out in song your audacious raptures, your half-hearted repentance, your shame and your scorn. You spoke the truth about rural lives and loves. We may like it or dislike it; but we cannot deny the verity.
Was it not as unhappy a thing, Sir, for you, as it was fortunate for Letters and for Scotland, that you were born at the meeting of two ages and of two worlds---precisely in the moment when bookish literature was beginning to reach the people, and when Society was first learning to admit the low-born to her Minor Mysteries? Before you how many singers not less truly poets than yourself---though less versatile not less passionate, though less sensuous not less simple---had been born and had died in poor men's cottages! There abides not even the shadow of a name of the old Scotch song-smiths, of the old ballad-makers. The authors of `Clerk Saunders,' of `The Wife of Usher's Well,' of `Fair Annie,' and `Sir Patrick Spens,' and `The Bonny Hind,' are as unknown to us as Homer, whom in their directness and force they resemble. They never, perhaps, gave their poems to writing; certainly they never gave them to the press. On the lips and in the hearts of the people they have their lives; and the singers, after a life obscure and untroubled by society or by fame, are forgotten. 'The Iniquity of Oblivion blindly scattereth his Poppy.'
Had you been born some years earlier you would have been even as these unnamed Immortals, leaving great verses to a little clan---verses retained only by Memory. You would have been but the minstrel of your native valley: the wider world would not have known you, nor you the world. Great thoughts of independence and revolt would never have burned in you; indignation would not have vexed you. Society would not have given and denied her caresses. You would have been happy. Your songs would have lingered in all `the circle of the summer hills;' and your scorn, your satire, your narrative verse, would have been unwritten or unknown. To the world what a loss! and what a gain to you! We should have possessed but a few of your lyrics, as
When o'er the hill the eastern star
Tells bughtin-time is near, my jo;
And owsen frae the furrowed field,
Return sae dowf and wearie O!
How noble that is, how natural, how unconsciously Greek! You found, oddly, in good Mrs. Barbauld, the merits of the Tenth Muse:
In thy sweet sang, Barbauld, survives
Even Sappho's flame!
But how unconsciously you remind us both of Sappho and of Homer in these strains about the Evening Star and the hour when the Day \textit{metenisseto boulytoide}?* Had you lived and died the pastoral poet of some silent glen, such lyrics could not but have survived; free, too, of all that in your songs reminds us of the Poet's Corner in the 'Kirkcudbright Advertiser.' We should not have read how
Phoebus, gilding the brow o' morning,
Banishes ilk darksome shade!
Still we might keep a love-poem unexcelled by Catullus,
Had we never loved sae kindly,
Had we never loved sae blindly,
Never met---or never parted,
We had ne'er been broken-hearted.
But the letters to Clarinda would have been unwritten, and the thrush would have been untaught in `the style of the Bird of Paradise.'
\textit{Transliterated from Greek.}
A quiet life of song, \textit{fallentis semita vitae}', was not to be yours. Fate otherwise decreed it. The touch of a lettered society, the strife with the Kirk, discontent with the State, poverty and pride, neglect and success, were needed to make your Genius what it was, and to endow the world with `Tam o' Shanter,' the `Jolly Beggars,' and 'Holy Willie's Prayer.' Who can praise them too highly---who admire in them too much the humour, the scorn, the wisdom, the unsurpassed energy and courage? So powerful, so commanding, is the movement of that Beggars' Chorus, that, methinks, it unconsciously echoed in the brain of our greatest living poet when he conceived the Vision of Sin. You shall judge for yourself. Recall:
Here's to budgets, bags, and wallets!
Here's to all the wandering train!
Here's our ragged bairns and callers!
One and all cry out, Amen!
A fig for those by law protected!
Liberty's a glorious feast!
Courts for cowards were erected!
Churches built to please the priest!
Then read this: Drink to lofty hopes that cool
Visions of a perfect state:
Drink we, last, the public fool,
Frantic love and frantic hate.
......... Drink to Fortune, drink to Chance,
While we keep a little breath!
Drink to heavy Ignorance
Hob and nob with brother Death!
Is not the movement the same, though the modern speaks a wilder
recklessness?
So in the best company we leave you, who were the life and soul of so much company, good and bad. No poet, since the Psalmist of Israel, ever gave the world more assurance of a man; none lived a life more strenuous, engaged in an eternal conflict of the passions, and by them overcome---'mighty and mightily fallen.' When we think of you, Byron seems, as Plato would have said, remote by one degree from actual truth, and Musset by a degree more remote than Byron.
\gutchapter{XX. To Lord Byron.}
My Lord, (Do you remember how Leigh Hunt
Enraged you once by writing \textit{My dear Byron}?)
Books have their fates,---as mortals have who punt,
And \textit{yours} have entered on an age of iron.
Critics there be who think your satin blunt,
Your pathos, fudge; such perils must environ
Poets who in their time were quite the rage,
Though now there's not a soul to turn their page.
Yes, there is much dispute about your worth,
And much is said which you might like to know
By modern poets here upon the earth,
Where poets live, and love each other so;
And, in Elysium, it may move your mirth
To hear of bards that pitch your praises low,
Though there be some that for your credit stickle,
As---Glorious Mat,---and not inglorious Nichol.
This kind of writing is my pet aversion,
I hate the slang, I hate the personalities,
I loathe the aimless, reckless, loose dispersion,
Of every rhyme that in the singer's wallet is,
I hate it as you hated the \textit{Excursion},
But, while no man a hero to his valet is,
The hero's still the model; I indite
The kind of rhymes that Byron oft would write.
There's a Swiss critic whom I cannot rhyme to,
One Scherer, dry as sawdust, grim and prim.
Of him there's much to say, if I had time to
Concern myself in any wise with him.
He seems to hate the heights he cannot climb to,
He thinks your poetry a coxcomb's whim,
A good deal of his sawdust he has spilt on
Shakspeare, and Moliere, and you, and Milton.
Ay, much his temper is like Vivien's mood,
Which found not Galahad pure, nor Lancelot brave;
Cold as a hailstorm on an April wood,
He buries poets in an icy grave,
His Essays---he of the Genevan hood!
Nothing so good, but better doth he crave.
So stupid and so solemn in his spite
He dares to print that Moliere could not write!
Enough of these excursions; I was saying
That half our English Bards are turned Reviewers,
And Arnold was discussing and assaying
The weight and value of that work of yours,
Examining and testing it and weighing,
And proved, the gems are pure, the gold endures.
While Swinburne cries with an exceeding joy,
the stones are paste, and half the gold, alloy.
In Byron, Arnold finds the greatest force,
Poetic, in this later age of ours
His song, a torrent from a mountain source,
Clear as the crystal, singing with the showers,
Sweeps to the sea in unrestricted course
Through banks o'erhung with rocks and sweet with flowers;
None of your brooks that modestly meander,
But swift as Awe along the Pass of Brander.
And when our century has clomb its crest,
And backward gazes o'er the plains of Time,
And counts its harvest, yours is still the best,
The richest garner in the field of rhyme
(The metaphoric mixture, 't is confest,
Is all my own, and is not quite sublime).
But fame's not yours alone; you must divide all
The plums and pudding with the Bard of Rydal!
WORDSWORTH and BYRON, these the lordly names
And these the gods to whom most incense burns.
`Absurd!' cries Swinburne, and in anger flames,
And in an AEschylean fury spurns
With impious foot your altar, and exclaims
And wreathes his laurels on the golden urns
Where Coleridge's and Shelley's ashes lie,
Deaf to the din and heedless of the cry.
For Byron (Swinburne shouts) has never woven
One honest thread of life within his song;
As Offenbach is to divine Beethoven
So Byron is to Shelley (\textit{This} is strong!),
And on Parnassus' peak, divinely cloven,
He may not stand, or stands by cruel wrong;
For Byron's rank (the Examiner has reckoned)
Is in the third class or a feeble second.
`A Bernesque poet' at the very most,
And never earnest save in politics---
The Pegasus that he was wont to boast
A blundering, floundering hackney, full of tricks,
A beast that must be driven to the post
By whips and spurs and oaths and kicks and sticks,
A gasping, ranting, broken-winded brute,
That any judge of Pegasi would shoot;
In sooth, a half-bred Pegasus, and far gone
In spavin, curb, and half a hundred woes.
And Byron's style is `jolter-headed jargon;'
His verse is `only bearable in prose.'
So living poets write of those that are gone,
And o'er the Eagle thus the Bantam crows;
And Swinburne ends where Verisopht began,
By owning you `a very clever man.'
Or rather does not end: he still must utter
A quantity of the unkindest things.
Ah! were you here, I marvel, would you flutter
O'er such a foe the tempest of your wings?
'T is `rant and cant and glare and splash and splutter'
That rend the modest air when Byron sings.
There Swinburne stops: a critic rather fiery.
\textit{Animis caelestibus tantaene irae}?
But whether he or Arnold in the right is,
Long is the argument, the quarrel long;
\textit{Non nobis est to settle tantas lites};
No poet I, to judge of right or wrong:
But of all things I always think a fight is
The most unpleasant in the lists of song;
When Marsyas of old was flayed, Apollo
Set an example which we need not follow.
The fashion changes! Maidens do not wear,
As once they wore, in necklaces and lockets
A curl ambrosial of Lord Byron's hair;
`Don Juan' is not always in our pockets
Nay, a NEW WRITER's readers do not care
Much for your verse, but are inclined to mock its
Manners and morals. Ay, and most young ladies
To yours prefer the `Epic' called `of Hades'!
I do not blame them; I'm inclined to think
That with the reigning taste 't is vain to quarrel,
And Burns might teach his votaries to drink,
And Byron never meant to make them moral.
You yet have lovers true, who will not shrink
From lauding you and giving you the laurel;
The Germans too, those men of blood and iron,
Of all our poets chiefly swear by Byron.
Farewell, thou Titan fairer than the gods!
Farewell, farewell, thou swift and lovely spirit,
Thou splendid warrior with the world at odds,
Unpraised, unpraisable, beyond thy merit;
Chased, like Oresres, by the furies' rods,
Like him at length thy peace dost thou inherit;
Beholding whom, men think how fairer far
Than all the steadfast stars the wandering star!
\textit{Note} Mr. Swlnburne's and Mr. Arnold's diverse views of Byron will be found in the \textit{Selections} by Mr. Arnold and in the \textit{Nineteenth Century}.
\gutchapter{XXI. To Omar Kha'yya'm.}
Wise Omar, do the Southern Breezes fling
Above your Grave, at ending of the Spring,
The Snowdrift of the petals of the Rose,
The wild white Roses you were wont to sing?
Far in the South I know a Land divine, (1)
And there is many a Saint and many a Shrine,
And over all the shrines the Blossom blows
Of Roses that were dear to you as wine.
(1) The hills above San Remo, where rose-bushes are planted
by the shrines. Omar desired that his grave might be where
the wind would scatter rose-leaves over it.
You were a Saint of unbelieving days,
Liking your Life and happy in men's Praise;
Enough for you the Shade beneath the Bough,
Enough to watch the wild World go its Ways.
Dreadless and hopeless thou of Heaven or Hell,
Careless of Words thou hadst not Skill to spell,
Content to know not all thou knowest now,
What's Death? Doth any Pitcher dread the Well?
The Pitchers we, whose Maker makes them ill,
Shall He torment them if they chance to spill?
Nay, like the broken potsherds are we cast
Forth and forgotten,---and what will be will!
So still were we, before the Months began
That rounded us and shaped us into Man.
So still we shall be, surely, at the last,
Dreamless, untouched of Blessing or of Ban!
Ah, strange it seems that this thy common thought
How all things have been, ay, and shall be nought
Was ancient Wisdom in thine ancient East,
In those old Days when Senlac fight was fought,
Which gave our England for a captive Land
To pious Chiefs of a believing Band,
A gift to the Believer from the Priest,
Tossed from the holy to the blood-red Hand! (1)
(1) Omar was contemporary with the battle of Hastings.
Yea, thou wert singing when that Arrow clave
Through helm and brain of him who could not save
His England, even of Harold Godwin's son;
The high tide murmurs by the Hero's grave! (1)
(1) Per mandata Ducis, Rex hic, Heralde, quiescis,
Ut custos maneas littoris et pelagi.
And \textit{thou} wert wreathing Roses---who can tell?---
Or chanting for some girl that pleased thee well,
Or satst at wine in Nasha'pu'r, when dun
The twilight veiled the field where Harold fell!
The salt Sea-waves above him rage and roam!
Along the white Walls of his guarded Home
No Zephyr stirs the Rose, but o'er the wave
The wild Wind beats the Breakers into Foam!
And dear to him, as Roses were to thee,
Rings long the Roar of Onset of the Sea;
The \textit{Swan's Path} of his Fathers is his grave:
His sleep, methinks, is sound as thine can be.
His was the Age of Faith, when all the West
Looked to the Priest for torment or for rest;
And thou wert living then, and didst not heed
The Saint who banned thee or the Saint who blessed!
Ages of Progress! These eight hundred years
Hath Europe shuddered with her hopes or fears,
And now!---she listens in the wilderness
To thee, and half believeth what she hears!
Hadst \textit{thou} THE SECRET? Ah, and who may tell?
`An hour we have,' thou saidst. `Ah, waste it well!'
An hour we have, and yet Eternity
Looms o'er us, and the thought of Heaven or Hell!
Nay, we can never be as wise as thou,
O idle singer 'neath the blossomed bough.
Nay, and we cannot be content to die.
\textit{We} cannot shirk the questions `Where?' and `How?'
Ah, not from learned Peace and gay Content
Shall we of England go the way he went
The Singer of the Red Wine and the Rose
Nay, otherwise than his our Day is spent!
Serene he dwelt in fragrant Nasha'pu'r,
But we must wander while the Stars endure.
\textit{He} knew THE SECRET: we have none that knows,
No Man so sure as Omar once was sure!
\section*{\raggedright XXII. To Q. Horatius Flaccus.}
In what manner of Paradise are we to conceive that you, Horace, are
dwelling, or what region of immortality can give you such pleasures as
this life afforded? The country and the town, nature and men, who knew
them so well as you, or who ever so wisely made the best of those two
worlds? Truly here you had good things, nor do you ever, in all your
poems, look for more delight in the life beyond; you never expect
consolation for present sorrow, and when you once have shaken hands with
a friend the parting seems to you eternal.
Quis desiderio sit pudor aut modus
Tam cari capitis?
So you sing, for the dear head you mourn has sunk for ever beneath the
wave. Virgil might wander forth bearing the golden branch 'the Sibyl
doth to singing men allow,' and might visit, as one not wholly without
hope, the dim dwellings of the dead and the unborn. To him was it
permitted to see and sing 'mothers and men, and the bodies outworn
of mighty heroes, boys and unwedded maids, and young men borne to the
funeral fire before their parents' eyes.' The endless caravan swept past
him---'many as fluttering leaves that drop and fall in autumn woods when
the first frost begins; many as birds that flock landward from the great
sea when now the chill year drives them o'er the deep and leads them to
sunnier lands.' Such things was it given to the sacred poet to behold,
and the happy seats and sweet pleasances of fortunate souls, where the
larger light clothes all the plains and dips them in a rosier gleam,
plains with their own new sun and stars before unknown. Ah, not \textit{frustra
pius} was Virgil, as you say, Horace, in your melancholy song. In him,
we fancy, there was a happier mood than your melancholy patience. 'Not,
though thou wert sweeter of song than Thracian Orpheus, with that lyre
whose lay led the dancing trees, not so would the blood return to the
empty shade of him whom once with dread wand the inexorable god hath
folded with his shadowy flocks; but patience lighteneth what heaven
forbids us to undo.'
\textit{Durum, sed levius fit patientia}?
It was all your philosophy in that last sad resort to which we are
pushed so often---
'With close-lipped Patience for our only friend,
Sad Patience, too near neighbour of Despair.'
The Epicurean is at one with the Stoic at last, and Horace with Marcus
Aurelius. 'To go away from among men, if there are gods, is not a thing
to be afraid of; but if indeed they do not exist, or if they have no
concern about human affairs, what is it to me to live in a universe
devoid of gods or devoid of providence?'
An excellent philosophy, but easier to those for whom no Hope had dawn
or seemed to set. Yet it is harder than common, Horace, for us to think
of you, still glad somewhere, among rivers like Liris and plains and
vine-clad hills, that
Solemque suum, sua sidera borunt.
It is hard, for you looked for no such thing.
\textit{Omnes una manet nox}
\textit{
Et calcanda semel via leti}.
You could not tell Maecenas that you would meet him again; you could
only promise to tread the dark path with him.
\textit{Ibimus, ibimus,}
\textit{
Utcunque praecedes, supremum}
\textit{
Carpere iter comites parati}.
Enough, Horace, of these mortuary musings. You loved the lesson of the
roses, and now and again would speak somewhat like a death's head over
thy temperate cups of Sabine \textit{ordinaire}. Your melancholy moral was
but meant to heighten the joy of thy pleasant life, when wearied Italy,
after all her wars and civic bloodshed, had won a peaceful haven. The
harbour might be treacherous; the prince might turn to the tyrant; far
away on the wide Roman marches might be heard, as it were, the endless,
ceaseless monotone of beating horses' hoofs and marching feet of men.
They were coming, they were nearing, like footsteps heard on wool; there
was a sound of multitudes and millions of barbarians, all the North,
\textit{officina gentium}, mustering and marshalling her peoples. But their
coming was not to be to-day, nor to-morrow; nor to-day was the budding
princely sway to blossom into the blood-red flower of Nero. In the hall
between the two tempests of Republic and Empire your odes sound 'like
linnets in the pauses of the wind.'
What joy there is in these songs! what delight of life, what an
exquisite Hellenic grace of art, what a manly nature to endure, what
tenderness and constancy of friendship, what a sense of all that is fair
in the glittering stream, the music of the waterfall, the hum of bees,
the silvery grey of the olive woods on the hillside! How human are all
your verses, Horace! what a pleasure is yours in the straining poplars,
swaying in the wind! what gladness you gain from the white crest of
Soracte, beheld through the fluttering snowflakes while the logs are
being piled higher on the hearth. You sing of women and wine---not all
whole-hearted in your praise of them, perhaps, for passion frightens
you, and 't is pleasure more than love that you commend to the young.
Lydia and Glycera, and the others, are but passing guests of a heart at
ease in itself, and happy enough when their facile reign is ended. You
seem to me like a man who welcomes middle age, and is more glad than
Sophocles was to `flee from these hard masters' the passions. In the
`fallow leisure of life' you glance round contented, and find all very
good save the need to leave all behind. Even that you take with an
Italian good-humour, as the folk of your sunny country bear poverty and
hunger.
\textit{Durum, sed levius fit patientia}!
To them, to you, the loveliness of your land is, and was, a thing to
live for. None of the Latin poets your fellows, or none but Virgil, seem
to me to have known so well as you, Horace, how happy and fortunate a
thing it was to be born in Italy. You do not say so, like your Virgil,
in one splendid passage, numbering the glories of the land as a lover
might count the perfections of his mistress. But the sentiment is ever
in your heart and often on your lips.
Me nec tam patiens Lacedaemon,
Nec tam Larissae percussit campus opimae,
Quam domus Albuneae resonantis
Et praeceps Anio, ac Tiburni lucus, et uda
Mobilibus pomaria rivis. (1)
\begin{quotation}
(1) 'Me neither resolute Sparta nor the rich Larissaean
plain so enraptures as the fane of echoing Albunea, the
headlong Anio, the grove of Tibur, the orchards watered by
the wandering rills.\end{quotation}
So a poet should speak, and to every singer his own land should be
dearest. Beautiful is Italy with the grave and delicate outlines of her
sacred hills, her dark groves, her little cities perched like eyries on
the crags, her rivers gliding under ancient walls; beautiful is Italy,
her seas, and her suns: but dearer to me the long grey wave that bites
the rock below the minster in the north; dearer is the barren moor and
black peat-water swirling in tanny foam, and the scent of bog myrtle
and the bloom of heather, and, watching over the lochs, the green
round-shouldered hills.
In affection for your native land, Horace, certainly the pride in great
Romans dead and gone made part, and you were, in all senses, a lover
of your country, your country's heroes, your country's gods. None but a
patriot could have sung that ode on Regulus, who died, as our own hero
died, on an evil day for the honour of Rome, as Gordon for the honour of
England.
Fertur pudicae conjujis osculum,
Parvosque natos, ut capitis minor,
Ab se removisse, et virilem
Torvus humi pusuisse voltum:
Donec labantes consilio patres
Firmaret auctor nunquam alias dato,
Interque maerentes amicos
Egregius properaret exul.
Atqui sciebat, quae sibi barbarus
Tortor pararet: non aliter tamen
Dimovit obstantes propinquos,
Et populum reditus morantem,
Quam si clientum longa negotia
Dijudicata lite relinqueret,
Tendens Venafranos in agros
Aut Lacedaemonium Tarentum. (1)
(1) 'They say he put aside from him the pure lips of his wife and
his little children, like a man unfree, and with his brave face bowed
earthward sternly he waited till with such counsel as never mortal gave
he might strengthen the hearts of the Fathers, and through his mourning
friends go forth, a hero, into exile. Yet well he knew what things were
being prepared for him at the hands of the tormenters, who, none the
less, put aside the kinsmen that barred his path and the people that
would fain have held him back, passing through their midst as he
might have done, if, his retainers' weary business ended and the suits
adjudged, he were faring to his Venafran lands or to Dorian Tarentum.'
We talk of the Greeks as your teachers. Your teachers they were, but
that poem could only have been written by a Roman! The strength, the
tenderness, the noble and monumental resolution and resignation---these
are the gift of the lords of human things, the masters of the world.
Your country's heroes are dear to you, Horace, but you did not sing them
better than your country's Gods, the pious protecting spirits of the
hearth, the farm, the field, kindly ghosts, it may be, of Latin fathers
dead or Gods framed in the image of these. What you actually believed
we know not, \textit{you} knew not. Who knows what he believes? \textit{Parcus Deorum
cultor} you bowed not often, it may be, in the temples of the state
religion and before the statues of the great Olympians; but the pure and
pious worship of rustic tradition, the faith handed down by the homely
elders, with that you never broke. Clean hands and a pure heart, these,
with a sacred cake and shining grains of salt, you could offer to the
Lares. It was a benignant religion, uniting old times and new, men
living and men long dead and gone, in a kind of service and sacrifice
solemn yet familiar.
Te nihil attinet
Tentare multa caede bidentium
Parvos coronantem marino
Rore deos fragilique myrto.
Immunis aram si tetigit manus,
Non sumptuosa blandior hostia
Mollivit aversos Penates
Farre pio et salienta mica. (1)
\begin{quotation}
(1) Thou, Phidyle, hast no need to besiege the gods with
slaughter so great of sheep, thou who crownest thy tiny
deities with myrtle rare and rosemary. If but the hand be
clean that touches the altar, then richest sacrifice will
not more appease the angered Penates than the duteous cake
and salt that crackles in the blaze.'\end{quotation}
Farewell, dear Horace; farewell, thou wise and kindly heathen; of
mortals the most human, the friend of my friends and of so many
generations of men.
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http://www.numerical.rl.ac.uk/people/nimg/oumsc/lectures/pf3.8.tex | rl.ac.uk | CC-MAIN-2018-05 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-05/segments/1516084890314.60/warc/CC-MAIN-20180121060717-20180121080717-00059.warc.gz | 534,613,290 | 2,028 | \noindent
%\subsection{Proof of Theorem \thelecture.8}
\textcolor{red}{{\bf PROOF OF THEOREM \thelecture.8}}
\noindent
Suppose otherwise that $f_k$ is bounded from below,
and that there is a subsequence $\{t_i\} \subseteq {\cal S}$, such that
\eqn{uct-w1}{\|g_{t_i}\| \geq 2 \epsilon > 0}
for some $\epsilon > 0 $ and for all $i$.
Theorem~ \thelecture.7 \implies
$\exists \{\ell_i\} \subseteq \calS$ such that
\eqn{uct-w2}{\|g_k \| \geq \epsilon \tim {for} t_i \leq k < \ell_i
\tim{and} \|g_{\ell_i}\| < \epsilon.}
Now restrict attention to indices in
\disp{ {\cal K} \eqdef \{ k \in {\cal S} \mid t_i \leq k < \ell_i \}.}
\newpage
\setlength{\unitlength}{1,25mm}
\begin{figure}[h]
\begin{center}
\begin{picture}(170,95)(-18,-5)
\put(0,0){\vector(0,1){80}}
\put(2,80){$\|g_k\|$}
\put(0,40){\dashbox{1}(150,0){}}
\put(-5,39){$2\epsilon$}
\put(0,20){\dashbox{1}(150,0){}}
\put(-5,19){$\epsilon$}
\put(0,0){\vector(1,0){150}}
\put(152,0){$k$}
\put(-5,-1){$\cal S$}
\multiput(0,0)(5,0){30}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(0,-4){\vector(1,0){150}}
\put(-7,-5){$\scriptstyle \{t_i\}$}
\multiput(0,-4)(5,0){5}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(35,-4){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(50,-4)(5,0){3}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(85,-4)(5,0){3}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(125,-4)(5,0){2}{\makebox(0,0){\rule{0.75mm}{2mm}}}
%\put(0,-8){\vector(1,0){150}}
\put(-7,-9){$\scriptstyle \{\ell_i\}$}
\multiput(0,-8)(5,0){5}{\line(0,1){2}}
\put(35,-8){\line(0,1){2}}
\put(0,-8){\vector(1,0){39}}
\put(40,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(50,-8)(5,0){3}{\line(0,1){2}}
\put(50,-8){\vector(1,0){24}}
\put(75,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(85,-8)(5,0){3}{\line(0,1){2}}
\put(85,-8){\vector(1,0){19}}
\put(105,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(125,-8)(5,0){2}{\line(0,1){2}}
\put(125,-8){\vector(1,0){14}}
\put(140,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(0,-12){\vector(1,0){150}}
\put(-5,-13){$\cal K$}
\multiput(0,-12)(5,0){8}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(50,-12)(5,0){5}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(85,-12)(5,0){4}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(125,-12)(5,0){3}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(0,75){\redcircle}
\put(5,60){\redcircle}
\put(10,65){\redcircle}
\put(15,50){\redcircle}
\put(20,52){\redcircle}
\put(25,37){\yellowcircle}
\put(30,34){\yellowcircle}
\put(35,43){\redcircle}
\put(40,18){\bluestar}
\put(45,30){\yellowcircle}
\put(50,41){\redcircle}
\put(55,46){\redcircle}
\put(60,42){\redcircle}
\put(65,24){\yellowcircle}
\put(70,31){\yellowcircle}
\put(75,15){\bluestar}
\put(80,27){\yellowcircle}
\put(85,51){\redcircle}
\put(90,47){\redcircle}
\put(95,41){\redcircle}
\put(100,29){\yellowcircle}
\put(105,13){\bluestar}
\put(110,17){\bluestar}
\put(115,23){\yellowcircle}
\put(120,22){\yellowcircle}
\put(125,46){\redcircle}
\put(130,41){\redcircle}
\put(135,21){\yellowcircle}
\put(140,16){\bluestar}
\put(145,8){\bluestar}
\end{picture}
\end{center}
\vspace*{5mm}
\caption{\label{uct-subseq_fig}The subsequences of the proof of
Theorem~\thelecture.8}
\end{figure}
\setlength{\unitlength}{1mm}
\newpage
\noindent
As in proof of Theorem~\thelecture.7, \req{uct-w2} \implies
\eqn{uct-w4}{f_k - f_{k+1}
\geq \eta_s [f_k - m_k(s_k)]
\geq \half \eta_s \epsilon
\min\left[ \frac{\epsilon}{1 + \kappa_b}, \Delta_k \right]}
% \min\left[ \frac{\epsilon}{1 + \kappa_b}, \kappa_s \Delta_k \right]}
for all $k \in {\cal K}$ \implies LHS of
\req{uct-w4} $\longrightarrow 0$ as $k \longrightarrow \infty$ \implies
\disp{ \lim_{\stackrel{k \rightarrow \infty}{k \in {\cal K}}} \Delta_k = 0}
\implies
\disp{
\Delta_k \leq \frac{2}{\epsilon \eta_s }[ f_k - f_{k+1} ].}
%\Delta_k \leq \frac{2}{\epsilon \eta_s \kappa_s}[ f_k - f_{k+1} ].}
for $k \in {\cal K}$ sufficiently large
\implies
\eqn{uct-ww4}{\| x_{t_i} - x_{\ell_i} \|
\leq \sum_{\stackrel{j=t_i}{j \in {\cal K}}}^{\ell_i-1} \|x_j - x_{j+1}\|
\leq \sum_{\stackrel{j=t_i}{j \in {\cal K}}}^{\ell_i-1} \Delta_j
\leq \frac{2}{\epsilon \eta_s } [ f_{t_i} - f_{\ell_i} ].}
%\leq \frac{2}{\epsilon \eta_s \kappa_s} [ f_{t_i} - f_{\ell_i} ].}
for $i$ sufficiently large.
\noindent
But RHS of \req{uct-ww4} $\longrightarrow 0$
%\noindent
\implies $\| x_{t_i} - x_{\ell_i} \| \longrightarrow 0$ as $i$ tends to infinity
\noindent
+ continuity \implies $\|g_{t_i}-g_{\ell_i}\|$ $\longrightarrow 0$.
\noindent
Impossible as $\|g_{t_i} - g_{\ell_i}\| \geq \epsilon$
by definition of $\{t_i\}$ and $\{\ell_i\}$ \implies
no subsequence satisfying \req{uct-w1} can exist.
|
http://uubu.fr/tex/docker-network-ls.tex | uubu.fr | CC-MAIN-2018-47 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-47/segments/1542039741628.8/warc/CC-MAIN-20181114041344-20181114063344-00091.warc.gz | 360,914,985 | 1,674 | \documentclass{report}
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\lfoot{\Large \textit{Uubu.fr}}
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Title={docker-network-ls},
Author={Sylvain Girod},
Creator={Bash script from uubu.fr's format version 1},
Producer={Bash script and PDFLaTeX},
Subject={Lister les réseaux},
CreationDate={D:20160313171404},
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Keywords={docker;containers;lxc}
}}
\begin{document}
\begin{center}
{\Huge docker-network-ls }
\end{center}
\vspace{1cm}
\begin{flushleft}
{\large Lister les réseaux}
\vspace{10mm}
\hspace{1mm} Liste tous les réseaux que le service connait. Inclus les réseaux connectés sur plusieurs hôte dans un cluster. L'option --no-trunc affiche l'id complet des réseaux. -f ou --filter filtre la sortie au format key=value. S'il y a plus d'un filtre, agit comme un filtre OR.
\vspace{5mm} \\
\vspace{1cm}{\huge type }
\
\hspace{1mm} Le fitre type support 2 valeurs: builtin qui affiche les réseaux prédéfinis (bridge, none, host), et custom qui affiche les réseaux utilisateur. Le filtre suivant affiche les réseaux utilisateurs: \\
\hspace{1mm} \textbf{docker network ls --filter type=custom} \\
\hspace{1mm} Ce filtre permet de supprimer tous les réseaux utilisateurs: \\
\hspace{1mm} \textbf{docker network rm `docker network ls --filter type=custom -q`} \\
\vspace{1cm}{\huge Name }
\
\hspace{1mm} Le filtre name correspond à tout ou partie du nom d'un réseau. Le filtre suivant matche tous les réseaux avec un nom contenant la chaîne foobar: \\
\hspace{1mm} \textbf{docker network ls --filter name=foobar} \\
\vspace{1cm}{\huge ID }
\
\hspace{1mm} Le filtre id matche tout ou partie de l'ID d'un réseaux: \\
\hspace{1mm} \textbf{docker network ls --filter id=63d1ff1f77b07ca51070a8c227e962238358bd310bde1529cf62e6c307ade161} \\
\vspace{1cm}{\huge OPTIONS }
\
\begin{description} \normalsize
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|
http://essca2018.servicelaboratory.ch/cfp/essca-example.tex | servicelaboratory.ch | CC-MAIN-2019-04 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2019-04/segments/1547583657510.42/warc/CC-MAIN-20190116134421-20190116160421-00586.warc.gz | 75,020,644 | 965 | % ESSCA template: Please do not change the next five lines.
\documentclass[11pt,twocolumn]{article}
\usepackage[utf8]{inputenc}
\usepackage{url}
\usepackage[cm]{fullpage}
\pagenumbering{gobble}
\title{Cloud Functions in Space}
\author{In Cognito\\
Extraterrestrial University\\
\url{[email protected]}}
\begin{document}
\maketitle
\section{Introduction}
We argue about the usefulness of deploying cloud functions in space and show a viable realisation.
\section{Conclusion}
We have demonstrated how to use cloud functions in space for lowering cost and increasing productivity.
Our raw data and further material is available online.
\end{document}
|
https://theanarchistlibrary.org/library/lola-ridge-the-ghetto-and-other-poems.tex | theanarchistlibrary.org | CC-MAIN-2021-39 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-39/segments/1631780056297.61/warc/CC-MAIN-20210918032926-20210918062926-00347.warc.gz | 611,092,901 | 29,545 | \documentclass[DIV=12,%
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\title{The Ghetto and Other Poems}
\date{}
\author{Lola Ridge}
\subtitle{}
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\cleardoublepage
\tableofcontents
% start a new right-handed page
\cleardoublepage
\chapter{To the American People}
\begin{verse}
Will you feast with me, American People? \\{}
But what have I that shall seem good to you!
On my board are bitter apples \\{}
And honey served on thorns, \\{}
And in my flagons fluid iron, \\{}
Hot from the crucibles.
How should such fare entice you!
\end{verse}
\chapter{The Ghetto}
\section{I}
\begin{verse}
Cool, inaccessible air \\{}
Is floating in velvety blackness shot with steel-blue lights, \\{}
But no breath stirs the heat \\{}
Leaning its ponderous bulk upon the Ghetto \\{}
And most on Hester street\dots{}
The heat\dots{} \\{}
Nosing in the body’s overflow, \\{}
Like a beast pressing its great steaming belly close, \\{}
Covering all avenues of air\dots{}
The heat in Hester street, \\{}
Heaped like a dray \\{}
With the garbage of the world.
Bodies dangle from the fire escapes \\{}
Or sprawl over the stoops\dots{} \\{}
Upturned faces glimmer pallidly— \\{}
Herring-yellow faces, spotted as with a mold, \\{}
And moist faces of girls \\{}
Like dank white lilies, \\{}
And infants’ faces with open parched mouths that suck at the air \\{}
~~~~~as at empty teats.
Young women pass in groups, \\{}
Converging to the forums and meeting halls, \\{}
Surging indomitable, slow \\{}
Through the gross underbrush of heat. \\{}
Their heads are uncovered to the stars, \\{}
And they call to the young men and to one another \\{}
With a free camaraderie. \\{}
Only their eyes are ancient and alone\dots{}
The street crawls undulant, \\{}
Like a river addled \\{}
With its hot tide of flesh \\{}
That ever thickens. \\{}
Heavy surges of flesh \\{}
Break over the pavements, \\{}
Clavering like a surf— \\{}
Flesh of this abiding \\{}
Brood of those ancient mothers who saw the dawn break over Egypt\dots{} \\{}
And turned their cakes upon the dry hot stones \\{}
And went on \\{}
Till the gold of the Egyptians fell down off their arms\dots{} \\{}
Fasting and athirst\dots{} \\{}
And yet on\dots{}
Did they vision—with those eyes darkly clear, \\{}
That looked the sun in the face and were not blinded— \\{}
Across the centuries \\{}
The march of their enduring flesh? \\{}
Did they hear— \\{}
Under the molten silence \\{}
Of the desert like a stopped wheel— \\{}
(And the scorpions tick-ticking on the sand\dots{}) \\{}
The infinite procession of those feet?
\end{verse}
\section{II}
\begin{verse}
I room at Sodos’—in the little green room that was Bennie’s— \\{}
With Sadie \\{}
And her old father and her mother, \\{}
Who is not so old and wears her own hair.
Old Sodos no longer makes saddles. \\{}
He has forgotten how. \\{}
He has forgotten most things—even Bennie who stays away \\{}
~~~~~and sends wine on holidays— \\{}
And he does not like Sadie’s mother \\{}
Who hides God’s candles, \\{}
Nor Sadie \\{}
Whose young pagan breath puts out the light— \\{}
That should burn always, \\{}
Like Aaron’s before the Lord.
Time spins like a crazy dial in his brain, \\{}
And night by night \\{}
I see the love-gesture of his arm \\{}
In its green-greasy coat-sleeve \\{}
Circling the Book, \\{}
And the candles gleaming starkly \\{}
On the blotched-paper whiteness of his face, \\{}
Like a miswritten psalm\dots{} \\{}
Night by night \\{}
I hear his lifted praise, \\{}
Like a broken whinnying \\{}
Before the Lord’s shut gate.
Sadie dresses in black. \\{}
She has black-wet hair full of cold lights \\{}
And a fine-drawn face, too white. \\{}
All day the power machines \\{}
Drone in her ears\dots{} \\{}
All day the fine dust flies \\{}
Till throats are parched and itch \\{}
And the heat—like a kept corpse— \\{}
Fouls to the last corner.
Then—when needles move more slowly on the cloth \\{}
And sweaty fingers slacken \\{}
And hair falls in damp wisps over the eyes— \\{}
Sped by some power within, \\{}
Sadie quivers like a rod\dots{} \\{}
A thin black piston flying, \\{}
One with her machine.
She—who stabs the piece-work with her bitter eye \\{}
And bids the girls: “Slow down— \\{}
You’ll have him cutting us again!” \\{}
She—fiery static atom, \\{}
Held in place by the fierce pressure all about— \\{}
Speeds up the driven wheels \\{}
And biting steel—that twice \\{}
Has nipped her to the bone.
Nights, she reads \\{}
Those books that have most unset thought, \\{}
New-poured and malleable, \\{}
To which her thought \\{}
Leaps fusing at white heat, \\{}
Or spits her fire out in some dim manger of a hall, \\{}
Or at a protest meeting on the Square, \\{}
Her lit eyes kindling the mob\dots{} \\{}
Or dances madly at a festival. \\{}
Each dawn finds her a little whiter, \\{}
Though up and keyed to the long day, \\{}
Alert, yet weary\dots{} like a bird \\{}
That all night long has beat about a light.
The Gentile lover, that she charms and shrews, \\{}
Is one more pebble in the pack \\{}
For Sadie’s mother, \\{}
Who greets him with her narrowed eyes \\{}
That hold some welcome back. \\{}
“What’s to be done?” she’ll say, \\{}
“When Sadie wants she takes\dots{} \\{}
Better than Bennie with his Christian woman\dots{} \\{}
A man is not so like, \\{}
If they should fight, \\{}
To call her Jew\dots{}”
Yet when she lies in bed \\{}
And the soft babble of their talk comes to her \\{}
And the silences\dots{} \\{}
I know she never sleeps \\{}
Till the keen draught blowing up the empty hall \\{}
Edges through her transom \\{}
And she hears his foot on the first stairs.
Sarah and Anna live on the floor above. \\{}
Sarah is swarthy and ill-dressed. \\{}
Life for her has no ritual. \\{}
She would break an ideal like an egg for the winged thing at the core. \\{}
Her mind is hard and brilliant and cutting like an acetylene torch. \\{}
If any impurities drift there, they must be burnt up as in a clear flame. \\{}
It is droll that she should work in a pants factory. \\{}
—Yet where else\dots{} tousled and collar awry at her olive throat. \\{}
Besides her hands are unkempt. \\{}
With English\dots{} and everything\dots{} there is so little time. \\{}
She reads without bias— \\{}
Doubting clamorously— \\{}
Psychology, plays, science, philosophies— \\{}
Those giant flowers that have bloomed and withered, scattering their seed\dots{} \\{}
—And out of this young forcing soil what growth may come— \\{}
~~~~~what amazing blossomings.
Anna is different. \\{}
One is always aware of Anna, and the young men turn their heads \\{}
~~~~~to look at her. \\{}
She has the appeal of a folk-song \\{}
And her cheap clothes are always in rhythm. \\{}
When the strike was on she gave half her pay. \\{}
She would give anything—save the praise that is hers \\{}
And the love of her lyric body.
But Sarah’s desire covets nothing apart. \\{}
She would share all things\dots{} \\{}
Even her lover.
\end{verse}
\section{III}
\begin{verse}
The sturdy Ghetto children \\{}
March by the parade, \\{}
Waving their toy flags, \\{}
Prancing to the bugles— \\{}
Lusty, unafraid\dots{} \\{}
Shaking little fire sticks \\{}
At the night— \\{}
The old blinking night— \\{}
Swerving out of the way, \\{}
Wrapped in her darkness like a shawl.
But a small girl \\{}
Cowers apart. \\{}
Her braided head, \\{}
Shiny as a black-bird’s \\{}
In the gleam of the torch-light, \\{}
Is poised as for flight. \\{}
Her eyes have the glow \\{}
Of darkened lights.
She stammers in Yiddish, \\{}
But I do not understand, \\{}
And there flits across her face \\{}
A shadow \\{}
As of a drawn blind. \\{}
I give her an orange, \\{}
Large and golden, \\{}
And she looks at it blankly. \\{}
I take her little cold hand and try to draw her to me, \\{}
But she is stiff\dots{} \\{}
Like a doll\dots{}
Suddenly she darts through the crowd \\{}
Like a little white panic \\{}
Blown along the night— \\{}
Away from the terror of oncoming feet\dots{} \\{}
And drums rattling like curses in red roaring mouths\dots{} \\{}
And torches spluttering silver fire \\{}
And lights that nose out hiding-places\dots{} \\{}
To the night— \\{}
Squatting like a hunchback \\{}
Under the curved stoop— \\{}
The old mammy-night \\{}
That has outlived beauty and knows the ways of fear— \\{}
The night—wide-opening crooked and comforting arms, \\{}
Hiding her as in a voluminous skirt.
The sturdy Ghetto children \\{}
March by the parade, \\{}
Waving their toy flags, \\{}
Prancing to the bugles, \\{}
Lusty, unafraid. \\{}
But I see a white frock \\{}
And eyes like hooded lights \\{}
Out of the shadow of pogroms \\{}
Watching\dots{} watching\dots{}
\end{verse}
\section{IV}
\begin{verse}
Calicoes and furs, \\{}
Pocket-books and scarfs, \\{}
Razor strops and knives \\{}
(Patterns in check\dots{})
Olive hands and russet head, \\{}
Pickles red and coppery, \\{}
Green pickles, brown pickles, \\{}
(Patterns in tapestry\dots{})
Coral beads, blue beads, \\{}
Beads of pearl and amber, \\{}
Gewgaws, beauty pins— \\{}
Bijoutry for chits— \\{}
Darting rays of violet, \\{}
Amethyst and jade\dots{} \\{}
All the colors out to play, \\{}
Jumbled iridescently\dots{} \\{}
(Patterns in stained glass \\{}
Shivered into bits!)
Nooses of gay ribbon \\{}
Tugging at one’s sleeve, \\{}
Dainty little garters \\{}
Hanging out their sign\dots{} \\{}
Here a pout of frilly things— \\{}
There a sonsy feather\dots{} \\{}
(White beards, black beards \\{}
Like knots in the weave\dots{})
And ah, the little babies— \\{}
Shiny black-eyed babies— \\{}
(Half a million pink toes \\{}
Wriggling altogether.) \\{}
Baskets full of babies \\{}
Like grapes on a vine.
Mothers waddling in and out, \\{}
Making all things right— \\{}
Picking up the slipped threads \\{}
In Grand street at night— \\{}
Grand street like a great bazaar, \\{}
Crowded like a float, \\{}
Bulging like a crazy quilt \\{}
Stretched on a line.
But nearer seen \\{}
This litter of the East \\{}
Takes on a garbled majesty.
The herded stalls \\{}
In dissolute array\dots{} \\{}
The glitter and the jumbled finery \\{}
And strangely juxtaposed \\{}
Cans, paper, rags \\{}
And colors decomposing, \\{}
Faded like old hair, \\{}
With flashes of barbaric hues \\{}
And eyes of mystery\dots{} \\{}
Flung \\{}
Like an ancient tapestry of motley weave \\{}
Upon the open wall of this new land.
Here, a tawny-headed girl\dots{} \\{}
Lemons in a greenish broth \\{}
And a huge earthen bowl \\{}
By a bronzed merchant \\{}
With a tall black lamb’s wool cap upon his head\dots{} \\{}
He has no glance for her. \\{}
His thrifty eyes \\{}
Bend—glittering, intent \\{}
Their hoarded looks \\{}
Upon his merchandise, \\{}
As though it were some splendid cloth \\{}
Or sumptuous raiment \\{}
Stitched in gold and red\dots{}
He seldom talks \\{}
Save of the goods he spreads— \\{}
The meager cotton with its dismal flower— \\{}
But with his skinny hands \\{}
That hover like two hawks \\{}
Above some luscious meat, \\{}
He fingers lovingly each calico, \\{}
As though it were a gorgeous shawl, \\{}
Or costly vesture \\{}
Wrought in silken thread, \\{}
Or strange bright carpet \\{}
Made for sandaled feet\dots{}
Here an old grey scholar stands. \\{}
His brooding eyes— \\{}
That hold long vistas without end \\{}
Of caravans and trees and roads, \\{}
And cities dwindling in remembrance— \\{}
Bend mostly on his tapes and thread.
What if they tweak his beard— \\{}
These raw young seed of Israel \\{}
Who have no backward vision in their eyes— \\{}
And mock him as he sways \\{}
Above the sunken arches of his feet— \\{}
They find no peg to hang their taunts upon. \\{}
His soul is like a rock \\{}
That bears a front worn smooth \\{}
By the coarse friction of the sea, \\{}
And, unperturbed, he keeps his bitter peace.
What if a rigid arm and stuffed blue shape, \\{}
Backed by a nickel star \\{}
Does prod him on, \\{}
Taking his proud patience for humility\dots{} \\{}
All gutters are as one \\{}
To that old race that has been thrust \\{}
From off the curbstones of the world\dots{} \\{}
And he smiles with the pale irony \\{}
Of one who holds \\{}
The wisdom of the Talmud stored away \\{}
In his mind’s lavender.
But this young trader, \\{}
Born to trade as to a caul, \\{}
Peddles the notions of the hour. \\{}
The gestures of the craft are his \\{}
And all the lore \\{}
As when to hold, withdraw, persuade, advance\dots{} \\{}
And be it gum or flags, \\{}
Or clean-all or the newest thing in tags, \\{}
Demand goes to him as the bee to flower. \\{}
And he—appraising \\{}
All who come and go \\{}
With his amazing \\{}
Slight-of-mind and glance \\{}
And nimble thought \\{}
And nature balanced like the scales at nought— \\{}
Looks Westward where the trade-lights glow, \\{}
And sees his vision rise— \\{}
A tape-ruled vision, \\{}
Circumscribed in stone— \\{}
Some fifty stories to the skies.
\end{verse}
\section{V}
\begin{verse}
As I sit in my little fifth-floor room— \\{}
Bare, \\{}
Save for bed and chair, \\{}
And coppery stains \\{}
Left by seeping rains \\{}
On the low ceiling \\{}
And green plaster walls, \\{}
Where when night falls \\{}
Golden lady-bugs \\{}
Come out of their holes, \\{}
And roaches, sepia-brown, consort\dots{} \\{}
I hear bells pealing \\{}
Out of the gray church at Rutgers street, \\{}
Holding its high-flung cross above the Ghetto, \\{}
And, one floor down across the court, \\{}
The parrot screaming: \\{}
Vorwärts\dots{} Vorwärts\dots{}
The parrot frowsy-white, \\{}
Everlastingly swinging \\{}
On its iron bar.
A little old woman, \\{}
With a wig of smooth black hair \\{}
Gummed about her shrunken brows, \\{}
Comes sometimes on the fire escape. \\{}
An old stooped mother, \\{}
The left shoulder low \\{}
With that uneven droopiness that women know \\{}
Who have suckled many young\dots{} \\{}
Yet I have seen no other than the parrot there.
I watch her mornings as she shakes her rugs \\{}
Feebly, with futile reach \\{}
And fingers without clutch. \\{}
Her thews are slack \\{}
And curved the ruined back \\{}
And flesh empurpled like old meat, \\{}
Yet each conspires \\{}
To feed those guttering fires \\{}
With which her eyes are quick.
On Friday nights \\{}
Her candles signal \\{}
Infinite fine rays \\{}
To other windows, \\{}
Coupling other lights, \\{}
Linking the tenements \\{}
Like an endless prayer.
She seems less lonely than the bird \\{}
That day by day about the dismal house \\{}
Screams out his frenzied word\dots{} \\{}
That night by night— \\{}
If a dog yelps \\{}
Or a cat yawls \\{}
Or a sick child whines, \\{}
Or a door screaks on its hinges, \\{}
Or a man and woman fight— \\{}
Sends his cry above the huddled roofs: \\{}
Vorwärts\dots{} Vorwärts\dots{}
\end{verse}
\section{VI}
\begin{verse}
In this dingy cafe \\{}
The old men sit muffled in woollens. \\{}
Everything is faded, shabby, colorless, old\dots{} \\{}
The chairs, loose-jointed, \\{}
Creaking like old bones— \\{}
The tables, the waiters, the walls, \\{}
Whose mottled plaster \\{}
Blends in one tone with the old flesh.
Young life and young thought are alike barred, \\{}
And no unheralded noises jolt old nerves, \\{}
And old wheezy breaths \\{}
Pass around old thoughts, dry as snuff, \\{}
And there is no divergence and no friction \\{}
Because life is flattened and ground as by many mills.
And it is here the Committee— \\{}
Sweet-breathed and smooth of skin \\{}
And supple of spine and knee, \\{}
With shining unpouched eyes \\{}
And the blood, high-powered, \\{}
Leaping in flexible arteries— \\{}
The insolent, young, enthusiastic, undiscriminating Committee, \\{}
Who would placard tombstones \\{}
And scatter leaflets even in graves, \\{}
Comes trampling with sacrilegious feet!
The old men turn stiffly, \\{}
Mumbling to each other. \\{}
They are gentle and torpid and busy with eating. \\{}
But one lifts a face of clayish pallor, \\{}
There is a dull fury in his eyes, like little rusty grates. \\{}
He rises slowly, \\{}
Trembling in his many swathings like an awakened mummy, \\{}
Ridiculous yet terrible. \\{}
—And the Committee flings him a waste glance, \\{}
Dropping a leaflet by his plate.
A lone fire flickers in the dusty eyes. \\{}
The lips chant inaudibly. \\{}
The warped shrunken body straightens like a tree. \\{}
And he curses\dots{} \\{}
With uplifted arms and perished fingers, \\{}
Claw-like, clutching\dots{} \\{}
So centuries ago \\{}
The old men cursed Acosta, \\{}
When they, prophetic, heard upon their sepulchres \\{}
Those feet that may not halt nor turn aside for ancient things.
\end{verse}
\section{VII}
\begin{verse}
Here in this room, bare like a barn, \\{}
Egos gesture one to the other— \\{}
Naked, unformed, unwinged \\{}
Egos out of the shell, \\{}
Examining, searching, devouring— \\{}
Avid alike for the flower or the dung\dots{} \\{}
(Having no dainty antennae for the touch and withdrawal— \\{}
Only the open maw\dots{})
Egos cawing, \\{}
Expanding in the mean egg\dots{} \\{}
Little squat tailors with unkempt faces, \\{}
Pale as lard, \\{}
Fur-makers, factory-hands, shop-workers, \\{}
News-boys with battling eyes \\{}
And bodies yet vibrant with the momentum of long runs, \\{}
Here and there a woman\dots{}
Words, words, words, \\{}
Pattering like hail, \\{}
Like hail falling without aim\dots{} \\{}
Egos rampant, \\{}
Screaming each other down. \\{}
One motions perpetually, \\{}
Waving arms like overgrowths. \\{}
He has burning eyes and a cough \\{}
And a thin voice piping \\{}
Like a flute among trombones.
One, red-bearded, rearing \\{}
A welter of maimed face bashed in from some old wound, \\{}
Garbles Max Stirner. \\{}
His words knock each other like little wooden blocks. \\{}
No one heeds him, \\{}
And a lank boy with hair over his eyes \\{}
Pounds upon the table. \\{}
—He is chairman.
Egos yet in the primer, \\{}
Hearing world-voices \\{}
Chanting grand arias\dots{} \\{}
Majors resonant, \\{}
Stunning with sound\dots{} \\{}
Baffling minors \\{}
Half-heard like rain on pools\dots{} \\{}
Majestic discordances \\{}
Greater than harmonies\dots{} \\{}
—Gleaning out of it all \\{}
Passion, bewilderment, pain\dots{}
Egos yearning with the world-old want in their eyes— \\{}
Hurt hot eyes that do not sleep enough\dots{} \\{}
Striving with infinite effort, \\{}
Frustrate yet ever pursuing \\{}
The great white Liberty, \\{}
Trailing her dissolving glory over each hard-won barricade— \\{}
Only to fade anew\dots{}
Egos crying out of unkempt deeps \\{}
And waving their dreams like flags— \\{}
Multi-colored dreams, \\{}
Winged and glorious\dots{}
A gas jet throws a stunted flame, \\{}
Vaguely illumining the groping faces. \\{}
And through the uncurtained window \\{}
Falls the waste light of stars, \\{}
As cold as wise men’s eyes\dots{} \\{}
Indifferent great stars, \\{}
Fortuitously glancing \\{}
At the secret meeting in this shut-in room, \\{}
Bare as a manger.
\end{verse}
\section{VIII}
\begin{verse}
Lights go out \\{}
And the stark trunks of the factories \\{}
Melt into the drawn darkness, \\{}
Sheathing like a seamless garment.
And mothers take home their babies, \\{}
Waxen and delicately curled, \\{}
Like little potted flowers closed under the stars.
Lights go out \\{}
And the young men shut their eyes, \\{}
But life turns in them\dots{}
Life in the cramped ova \\{}
Tearing and rending asunder its living cells\dots{} \\{}
Wars, arts, discoveries, rebellions, travails, immolations, \\{}
~~~~~cataclysms, hates\dots{} \\{}
Pent in the shut flesh. \\{}
And the young men twist on their beds in languor and dizziness \\{}
~~~~~unsupportable\dots{} \\{}
Their eyes—heavy and dimmed \\{}
With dust of long oblivions in the gray pulp behind— \\{}
Staring as through a choked glass. \\{}
And they gaze at the moon—throwing off a faint heat— \\{}
The moon, blond and burning, creeping to their cots \\{}
Softly, as on naked feet\dots{} \\{}
Lolling on the coverlet\dots{} like a woman offering her white body.
Nude glory of the moon! \\{}
That leaps like an athlete on the bosoms of the young girls stripped \\{}
~~~~~of their linens; \\{}
Stroking their breasts that are smooth and cool as mother-of-pearl \\{}
Till the nipples tingle and burn as though little lips plucked at them. \\{}
They shudder and grow faint. \\{}
And their ears are filled as with a delirious rhapsody, \\{}
That Life, like a drunken player, \\{}
Strikes out of their clear white bodies \\{}
As out of ivory keys.
Lights go out\dots{} \\{}
And the great lovers linger in little groups, still passionately debating, \\{}
Or one may walk in silence, listening only to the still summons of Life— \\{}
Life making the great Demand\dots{} \\{}
Calling its new Christs\dots{} \\{}
Till tears come, blurring the stars \\{}
That grow tender and comforting like the eyes of comrades; \\{}
And the moon rolls behind the Battery \\{}
Like a word molten out of the mouth of God.
Lights go out\dots{} \\{}
And colors rush together, \\{}
Fusing and floating away\dots{} \\{}
Pale worn gold like the settings of old jewels\dots{} \\{}
Mauves, exquisite, tremulous, and luminous purples \\{}
And burning spires in aureoles of light \\{}
Like shimmering auras.
They are covering up the pushcarts\dots{} \\{}
Now all have gone save an old man with mirrors— \\{}
Little oval mirrors like tiny pools. \\{}
He shuffles up a darkened street \\{}
And the moon burnishes his mirrors till they shine like phosphorus\dots{} \\{}
The moon like a skull, \\{}
Staring out of eyeless sockets at the old men trundling home the pushcarts.
\end{verse}
\section{IX}
\begin{verse}
A sallow dawn is in the sky \\{}
As I enter my little green room. \\{}
Sadie’s light is still burning\dots{} \\{}
Without, the frail moon \\{}
Worn to a silvery tissue, \\{}
Throws a faint glamour on the roofs, \\{}
And down the shadowy spires \\{}
Lights tip-toe out\dots{} \\{}
Softly as when lovers close street doors.
Out of the Battery \\{}
A little wind \\{}
Stirs idly—as an arm \\{}
Trails over a boat’s side in dalliance— \\{}
Rippling the smooth dead surface of the heat, \\{}
And Hester street, \\{}
Like a forlorn woman over-born \\{}
By many babies at her teats, \\{}
Turns on her trampled bed to meet the day.
LIFE! \\{}
Startling, vigorous life, \\{}
That squirms under my touch, \\{}
And baffles me when I try to examine it, \\{}
Or hurls me back without apology. \\{}
Leaving my ego ruffled and preening itself.
Life, \\{}
Articulate, shrill, \\{}
Screaming in provocative assertion, \\{}
Or out of the black and clotted gutters, \\{}
Piping in silvery thin \\{}
Sweet staccato \\{}
Of children’s laughter,
Or clinging over the pushcarts \\{}
Like a litter of tiny bells \\{}
Or the jingle of silver coins, \\{}
Perpetually changing hands, \\{}
Or like the Jordan somberly \\{}
Swirling in tumultuous uncharted tides, \\{}
Surface-calm.
Electric currents of life, \\{}
Throwing off thoughts like sparks, \\{}
Glittering, disappearing, \\{}
Making unknown circuits, \\{}
Or out of spent particles stirring \\{}
Feeble contortions in old faiths \\{}
Passing before the new.
Long nights argued away \\{}
In meeting halls \\{}
Back of interminable stairways— \\{}
In Roumanian wine-shops \\{}
And little Russian tea-rooms\dots{}
Feet echoing through deserted streets \\{}
In the soft darkness before dawn\dots{} \\{}
Brows aching, throbbing, burning— \\{}
Life leaping in the shaken flesh \\{}
Like flame at an asbestos curtain.
Life— \\{}
Pent, overflowing \\{}
Stoops and façades, \\{}
Jostling, pushing, contriving, \\{}
Seething as in a great vat\dots{}
Bartering, changing, extorting, \\{}
Dreaming, debating, aspiring, \\{}
Astounding, indestructible \\{}
Life of the Ghetto\dots{}
Strong flux of life, \\{}
Like a bitter wine \\{}
Out of the bloody stills of the world\dots{} \\{}
Out of the Passion eternal.
\end{verse}
\part{Manhattan Lights}
\chapter{Manhattan}
\begin{verse}
Out of the night you burn, Manhattan, \\{}
In a vesture of gold— \\{}
Span of innumerable arcs, \\{}
Flaring and multiplying— \\{}
Gold at the uttermost circles fading \\{}
Into the tenderest hint of jade, \\{}
Or fusing in tremulous twilight blues, \\{}
Robing the far-flung offices, \\{}
Scintillant-storied, forking flame, \\{}
Or soaring to luminous amethyst \\{}
Over the steeples aureoled—
Diaphanous gold, \\{}
Veiling the Woolworth, argently \\{}
Rising slender and stark \\{}
Mellifluous-shrill as a vender’s cry, \\{}
And towers squatting graven and cold \\{}
On the velvet bales of the dark, \\{}
And the Singer’s appraising \\{}
Indolent idol’s eye, \\{}
And night like a purple cloth unrolled—
Nebulous gold \\{}
Throwing an ephemeral glory about life’s vanishing points, \\{}
Wherein you burn\dots{} \\{}
You of unknown voltage \\{}
Whirling on your axis\dots{} \\{}
Scrawling vermillion signatures \\{}
Over the night’s velvet hoarding\dots{} \\{}
Insolent, towering spherical \\{}
To apices ever shifting.
\end{verse}
\chapter{Broadway}
\begin{verse}
Light! \\{}
Innumerable ions of light, \\{}
Kindling, irradiating, \\{}
All to their foci tending\dots{}
Light that jingles like anklet chains \\{}
On bevies of little lithe twinkling feet, \\{}
Or clingles in myriad vibrations \\{}
Like trillions of porcelain \\{}
Vases shattering\dots{}
Light over the laminae of roofs, \\{}
Diffusing in shimmering nebulae \\{}
About the night’s boundaries, \\{}
Or billowing in pearly foam \\{}
Submerging the low-lying stars\dots{}
Light for the feast prolonged— \\{}
Captive light in the goblets quivering\dots{} \\{}
Sparks evanescent \\{}
Struck of meeting looks— \\{}
Fringed eyelids leashing \\{}
Sheathed and leaping lights\dots{} \\{}
Infinite bubbles of light \\{}
Bursting, reforming\dots{} \\{}
Silvery filings of light \\{}
Incessantly falling\dots{} \\{}
Scintillant, sided dust of light \\{}
Out of the white flares of Broadway— \\{}
Like a great spurious diamond \\{}
In the night’s corsage faceted\dots{}
Broadway, \\{}
In ambuscades of light, \\{}
Drawing the charmed multitudes \\{}
With the slow suction of her breath— \\{}
Dangling her naked soul \\{}
Behind the blinding gold of eunuch lights \\{}
That wind about her like a bodyguard.
Or like a huge serpent, iridescent-scaled, \\{}
Trailing her coruscating length \\{}
Over the night prostrate— \\{}
Triumphant poised, \\{}
Her hydra heads above the avenues, \\{}
Values appraising \\{}
And her avid eyes \\{}
Glistening with eternal watchfulness\dots{}
Broadway— \\{}
Out of her towers rampant, \\{}
Like an unsubtle courtezan \\{}
Reserving nought for some adventurous night.
\end{verse}
\chapter{Flotsam}
\begin{verse}
Crass rays streaming from the vestibules; \\{}
Cafes glittering like jeweled teeth; \\{}
High-flung signs \\{}
Blinking yellow phosphorescent eyes; \\{}
Girls in black \\{}
Circling monotonously \\{}
About the orange lights\dots{}
Nothing to guess at\dots{} \\{}
Save the darkness above \\{}
Crouching like a great cat.
In the dim-lit square, \\{}
Where dishevelled trees \\{}
Tustle with the wind—the wind like a scythe \\{}
Mowing their last leaves— \\{}
Arcs shimmering through a greenish haze— \\{}
Pale oval arcs \\{}
Like ailing virgins, \\{}
Each out of a halo circumscribed, \\{}
Pallidly staring\dots{}
Figures drift upon the benches \\{}
With no more rustle than a dropped leaf settling— \\{}
Slovenly figures like untied parcels, \\{}
And papers wrapped about their knees \\{}
Huddled one to the other, \\{}
Cringing to the wind— \\{}
The sided wind, \\{}
Leaving no breach untried\dots{}
So many and all so still\dots{} \\{}
The fountain slobbering its stone basin \\{}
Is louder than They— \\{}
Flotsam of the five oceans \\{}
Here on this raft of the world.
This old man’s head \\{}
Has found a woman’s shoulder. \\{}
The wind juggles with her shawl \\{}
That flaps about them like a sail, \\{}
And splashes her red faded hair \\{}
Over the salt stubble of his chin. \\{}
A light foam is on his lips, \\{}
As though dreams surged in him \\{}
Breaking and ebbing away\dots{} \\{}
And the bare boughs shuffle above him \\{}
And the twigs rattle like dice\dots{}
She—diffused like a broken beetle— \\{}
Sprawls without grace, \\{}
Her face gray as asphalt, \\{}
Her jaws sagging as on loosened hinges\dots{} \\{}
Shadows ply about her mouth— \\{}
Nimble shadows out of the jigging tree, \\{}
That dances above her its dance of dry bones.
\end{verse}
\section{II}
\begin{verse}
A uniformed front, \\{}
Paunched; \\{}
A glance like a blow, \\{}
The swing of an arm, \\{}
Verved, vigorous; \\{}
Boot-heels clanking \\{}
In metallic rhythm; \\{}
The blows of a baton, \\{}
Quick, staccato\dots{}
—There is a rustling along the benches \\{}
As of dried leaves raked over\dots{} \\{}
And the old man lifts a shaking palsied hand, \\{}
Tucking the displaced paper about his knees.
Colder\dots{} \\{}
And a frost under foot, \\{}
Acid, corroding, \\{}
Eating through worn bootsoles.
Drab forms blur into greenish vapor. \\{}
Through boughs like cross-bones, \\{}
Pale arcs flare and shiver \\{}
Like lilies in a wind.
High over Broadway \\{}
A far-flung sign \\{}
Glitters in indigo darkness \\{}
And spurts again rhythmically, \\{}
Spraying great drops \\{}
Red as a hemorrhage.
\end{verse}
\chapter{Spring}
\begin{verse}
A spring wind on the Bowery, \\{}
Blowing the fluff of night shelters \\{}
Off bedraggled garments, \\{}
And agitating the gutters, that eject little spirals of vapor \\{}
Like lewd growths.
Bare-legged children stamp in the puddles, splashing each other, \\{}
One—with a choir-boy’s face \\{}
Twits me as I pass\dots{} \\{}
The word, like a muddied drop, \\{}
Seems to roll over and not out of \\{}
The bowed lips, \\{}
Yet dewy red \\{}
And sweetly immature.
People sniff the air with an upward look— \\{}
Even the mite of a girl \\{}
Who never plays\dots{} \\{}
Her mother smiles at her \\{}
With eyes like vacant lots \\{}
Rimming vistas of mean streets \\{}
And endless washing days\dots{} \\{}
Yet with sun on the lines \\{}
And a drying breeze.
The old candy woman \\{}
Shivers in the young wind. \\{}
Her eyes—littered with memories \\{}
Like ancient garrets, \\{}
Or dusty unaired rooms where someone died— \\{}
Ask nothing of the spring.
But a pale pink dream \\{}
Trembles about this young girl’s body, \\{}
Draping it like a glowing aura.
She gloats in a mirror \\{}
Over her gaudy hat, \\{}
With its flower God never thought of\dots{}
And the dream, unrestrained, \\{}
Floats about the loins of a soldier, \\{}
Where it quivers a moment, \\{}
Warming to a crimson \\{}
Like the scarf of a toreador\dots{}
But the delicate gossamer breaks at his contact \\{}
And recoils to her in strands of shattered rose.
\end{verse}
\chapter{Bowery Afternoon}
\begin{verse}
Drab discoloration \\{}
Of faces, façades, pawn-shops, \\{}
Second-hand clothing, \\{}
Smoky and fly-blown glass of lunch-rooms, \\{}
Odors of rancid life\dots{}
Deadly uniformity \\{}
Of eyes and windows \\{}
Alike devoid of light\dots{} \\{}
Holes wherein life scratches— \\{}
Mangy life \\{}
Nosing to the gutter’s end\dots{}
Show-rooms and mimic pillars \\{}
Flaunting out of their gaudy vestibules \\{}
Bosoms and posturing thighs\dots{}
Over all the Elevated \\{}
Droning like a bloated fly.
\end{verse}
\chapter{Promenade}
\begin{verse}
~~~~~Undulant rustlings, \\{}
~~~~~Of oncoming silk, \\{}
~~~~~Rhythmic, incessant, \\{}
~~~~~Like the motion of leaves\dots{} \\{}
~~~~~Fragments of color \\{}
~~~~~In glowing surprises\dots{} \\{}
~~~~~Pink inuendoes \\{}
~~~~~Hooded in gray \\{}
~~~~~Like buds in a cobweb \\{}
~~~~~Pearled at dawn\dots{} \\{}
~~~~~Glimpses of green \\{}
~~~~~And blurs of gold \\{}
~~~~~And delicate mauves \\{}
~~~~~That snatch at youth\dots{} \\{}
~~~~~And bodies all rosily \\{}
~~~~~Fleshed for the airing, \\{}
~~~~~In warm velvety surges \\{}
~~~~~Passing imperious, slow\dots{}
Women drift into the limousines \\{}
That shut like silken caskets \\{}
On gems half weary of their glittering\dots{} \\{}
Lamps open like pale moon flowers\dots{} \\{}
Arcs are radiant opals \\{}
Strewn along the dusk\dots{} \\{}
No common lights invade. \\{}
And spires rise like litanies— \\{}
Magnificats of stone \\{}
Over the white silence of the arcs, \\{}
Burning in perpetual adoration.
\end{verse}
\chapter{The Fog}
\begin{verse}
Out of the lamp-bestarred and clouded dusk— \\{}
Snaring, illuding, concealing, \\{}
Magically conjuring— \\{}
Turning to fairy-coaches \\{}
Beetle-backed limousines \\{}
Scampering under the great Arch— \\{}
Making a decoy of blue overalls \\{}
And mystery of a scarlet shawl— \\{}
Indolently— \\{}
Knowing no impediment of its sure advance— \\{}
Descends the fog.
\end{verse}
\chapter{Faces}
\begin{verse}
A late snow beats \\{}
With cold white fists upon the tenements— \\{}
Hurriedly drawing blinds and shutters, \\{}
Like tall old slatterns \\{}
Pulling aprons about their heads.
Lights slanting out of Mott Street \\{}
Gibber out, \\{}
Or dribble through bar-room slits, \\{}
Anonymous shapes \\{}
Conniving behind shuttered panes \\{}
Caper and disappear\dots{} \\{}
Where the Bowery \\{}
Is throbbing like a fistula \\{}
Back of her ice-scabbed fronts.
Livid faces \\{}
Glimmer in furtive doorways, \\{}
Or spill out of the black pockets of alleys, \\{}
Smears of faces like muddied beads, \\{}
Making a ghastly rosary \\{}
The night mumbles over \\{}
And the snow with its devilish and silken whisper\dots{} \\{}
Patrolling arcs \\{}
Blowing shrill blasts over the Bread Line \\{}
Stalk them as they pass, \\{}
Silent as though accouched of the darkness, \\{}
And the wind noses among them, \\{}
~~~~~Like a skunk \\{}
That roots about the heart\dots{}
Colder: \\{}
And the Elevated slams upon the silence \\{}
Like a ponderous door. \\{}
Then all is still again, \\{}
Save for the wind fumbling over \\{}
The emptily swaying faces— \\{}
The wind rummaging \\{}
Like an old Jew\dots{}
Faces in glimmering rows\dots{} \\{}
(No sign of the abject life— \\{}
Not even a blasphemy\dots{}) \\{}
But the spindle legs keep time \\{}
To a limping rhythm, \\{}
And the shadows twitch upon the snow \\{}
~~~~~Convulsively— \\{}
As though death played \\{}
With some ungainly dolls.
\end{verse}
\part{Labor}
\chapter{Debris}
\begin{verse}
I love those spirits \\{}
That men stand off and point at, \\{}
Or shudder and hood up their souls— \\{}
Those ruined ones, \\{}
Where Liberty has lodged an hour \\{}
And passed like flame, \\{}
Bursting asunder the too small house.
\end{verse}
\chapter{Dedication}
\begin{verse}
I would be a torch unto your hand, \\{}
A lamp upon your forehead, Labor, \\{}
In the wild darkness before the Dawn \\{}
That I shall never see\dots{}
We shall advance together, my Beloved, \\{}
Awaiting the mighty ushering\dots{} \\{}
Together we shall make the last grand charge \\{}
And ride with gorgeous Death \\{}
With all her spangles on \\{}
And cymbals clashing\dots{} \\{}
And you shall rush on exultant as I fall— \\{}
Scattering a brief fire about your feet\dots{}
Let it be so\dots{} \\{}
Better—while life is quick \\{}
And every pain immense and joy supreme, \\{}
And all I have and am \\{}
Flames upward to the dream\dots{} \\{}
Than like a taper forgotten in the dawn, \\{}
Burning out the wick.
\end{verse}
\chapter{The Song of Iron}
\section{I}
\begin{verse}
Not yet hast Thou sounded \\{}
Thy clangorous music, \\{}
Whose strings are under the mountains\dots{} \\{}
Not yet hast Thou spoken \\{}
The blooded, implacable Word\dots{}
But I hear in the Iron singing— \\{}
In the triumphant roaring of the steam and pistons pounding— \\{}
Thy barbaric exhortation\dots{} \\{}
And the blood leaps in my arteries, unreproved, \\{}
Answering Thy call\dots{} \\{}
All my spirit is inundated with the tumultuous passion of Thy Voice, \\{}
And sings exultant with the Iron, \\{}
For now I know I too am of Thy Chosen\dots{}
Oh fashioned in fire— \\{}
Needing flame for Thy ultimate word— \\{}
Behold me, a cupola \\{}
Poured to Thy use!
Heed not my tremulous body \\{}
That faints in the grip of Thy gauntlet. \\{}
Break it\dots{} and cast it aside\dots{} \\{}
But make of my spirit \\{}
That dares and endures \\{}
Thy crucible\dots{} \\{}
Pour through my soul \\{}
Thy molten, world-whelming song.
\dots{} Here at Thy uttermost gate \\{}
Like a new Mary, I wait\dots{}
\end{verse}
\section{II}
\begin{verse}
Charge the blast furnace, workman\dots{} \\{}
Open the valves— \\{}
Drive the fires high\dots{} \\{}
(Night is above the gates).
How golden-hot the ore is \\{}
From the cupola spurting, \\{}
Tossing the flaming petals \\{}
Over the silt and furnace ash— \\{}
Blown leaves, devastating, \\{}
Falling about the world\dots{}
Out of the furnace mouth— \\{}
Out of the giant mouth— \\{}
The raging, turgid, mouth— \\{}
Fall fiery blossoms \\{}
Gold with the gold of buttercups \\{}
In a field at sunset, \\{}
Or huskier gold of dandelions, \\{}
Warmed in sun-leavings, \\{}
Or changing to the paler hue \\{}
At the creamy hearts of primroses.
Charge the converter, workman— \\{}
Tired from the long night? \\{}
But the earth shall suck up darkness— \\{}
The earth that holds so much\dots{} \\{}
And out of these molten flowers, \\{}
Shall shape the heavy fruit\dots{}
Then open the valves— \\{}
Drive the fires high, \\{}
Your blossoms nurturing. \\{}
(Day is at the gates \\{}
And a young wind\dots{})
Put by your rod, comrade, \\{}
And look with me, shading your eyes\dots{} \\{}
Do you not see— \\{}
Through the lucent haze \\{}
Out of the converter rising— \\{}
In the spirals of fire \\{}
Smiting and blinding, \\{}
A shadowy shape \\{}
White as a flame of sacrifice, \\{}
Like a lily swaying?
\end{verse}
\section{III}
\begin{verse}
The ore leaping in the crucibles, \\{}
The ore communicant, \\{}
Sending faint thrills along the leads\dots{} \\{}
Fire is running along the roots of the mountains\dots{} \\{}
I feel the long recoil of earth \\{}
As under a mighty quickening\dots{} \\{}
(Dawn is aglow in the light of the Iron\dots{}) \\{}
All palpitant, I wait\dots{}
\end{verse}
\section{IV}
\begin{verse}
Here ye, Dictators—late Lords of the Iron, \\{}
Shut in your council rooms, palsied, depowered— \\{}
The blooded, implacable Word? \\{}
Not whispered in cloture, one to the other, \\{}
(Brother in fear of the fear of his brother\dots{}) \\{}
But chanted and thundered \\{}
On the brazen, articulate tongues of the Iron \\{}
Babbling in flame\dots{}
Sung to the rhythm of prisons dismantled, \\{}
Manacles riven and ramparts defaced\dots{} \\{}
(Hearts death-anointed yet hearing life calling\dots{}) \\{}
Ankle chains bursting and gallows unbraced\dots{}
Sung to the rhythm of arsenals burning\dots{} \\{}
Clangor of iron smashing on iron, \\{}
Turmoil of metal and dissonant baying \\{}
Of mail-sided monsters shattered asunder\dots{}
Hulks of black turbines all mangled and roaring, \\{}
Battering egress through ramparted walls\dots{} \\{}
Mouthing of engines, made rabid with power, \\{}
Into the holocaust snorting and plunging\dots{}
Mighty converters torn from their axis, \\{}
Flung to the furnaces, vomiting fire, \\{}
Jumbled in white-heaten masses disshapen\dots{} \\{}
Writhing in flame-tortured levers of iron\dots{}
Gnashing of steel serpents twisting and dying\dots{} \\{}
Screeching of steam-glutted cauldrons rending\dots{} \\{}
Shock of leviathans prone on each other\dots{} \\{}
Scaled flanks touching, ore entering ore\dots{} \\{}
Steel haunches closing and grappling and swaying \\{}
In the waltz of the mating locked mammoths of iron, \\{}
Tasting the turbulent fury of living, \\{}
Mad with a moment’s exuberant living! \\{}
Crash of devastating hammers despoiling.. \\{}
Hands inexorable, marring \\{}
What hands had so cunningly moulded\dots{}
Structures of steel welded, subtily tempered, \\{}
Marvelous wrought of the wizards of ore, \\{}
Torn into octaves discordantly clashing, \\{}
Chords never final but onward progressing \\{}
In monstrous fusion of sound ever smiting on sound \\{}
~~~~~in mad vortices whirling\dots{}
Till the ear, tortured, shrieks for cessation \\{}
Of the raving inharmonies hatefully mingling\dots{} \\{}
The fierce obligato the steel pipes are screaming\dots{} \\{}
The blare of the rude molten music of Iron\dots{}
\end{verse}
\chapter{Frank Little at Calvary}
\section{I}
\begin{verse}
He walked under the shadow of the Hill \\{}
Where men are fed into the fires \\{}
And walled apart\dots{} \\{}
Unarmed and alone, \\{}
He summoned his mates from the pit’s mouth \\{}
Where tools rested on the floors \\{}
And great cranes swung \\{}
Unemptied, on the iron girders. \\{}
And they, who were the Lords of the Hill, \\{}
Were seized with a great fear, \\{}
When they heard out of the silence of wheels \\{}
The answer ringing \\{}
In endless reverberations \\{}
Under the mountain\dots{}
So they covered up their faces \\{}
And crept upon him as he slept\dots{} \\{}
Out of eye-holes in black cloth \\{}
They looked upon him who had flung \\{}
Between them and their ancient prey \\{}
The frail barricade of his life\dots{} \\{}
And when night—that has connived at so much— \\{}
Was heavy with the unborn day, \\{}
They haled him from his bed\dots{}
Who might know of that wild ride? \\{}
Only the bleak Hill— \\{}
The red Hill, vigilant, \\{}
Like a blood-shot eye \\{}
In the black mask of night— \\{}
Dared watch them as they raced \\{}
By each blind-folded street \\{}
Godiva might have ridden down\dots{} \\{}
But when they stopped beside the Place, \\{}
I know he turned his face \\{}
Wistfully to the accessory night\dots{}
And when he saw—against the sky, \\{}
Sagged like a silken net \\{}
Under its load of stars— \\{}
The black bridge poised \\{}
Like a gigantic spider motionless\dots{} \\{}
I know there was a silence in his heart, \\{}
As of a frozen sea, \\{}
Where some half lifted arm, mid-way \\{}
Wavers, and drops heavily\dots{}
I know he waved to life, \\{}
And that life signaled back, transcending space, \\{}
To each high-powered sense, \\{}
So that he missed no gesture of the wind \\{}
Drawing the shut leaves close\dots{} \\{}
So that he saw the light on comrades’ faces \\{}
Of camp fires out of sight\dots{} \\{}
And the savor of meat and bread \\{}
Blew in his nostrils\dots{} and the breath \\{}
Of unrailed spaces \\{}
Where shut wild clover smelled as sweet \\{}
As a virgin in her bed.
I know he looked once at America, \\{}
Quiescent, with her great flanks on the globe, \\{}
And once at the skies whirling above him\dots{} \\{}
Then all that he had spoken against \\{}
And struck against and thrust against \\{}
Over the frail barricade of his life \\{}
Rushed between him and the stars\dots{}
\end{verse}
\section{II}
\begin{verse}
Life thunders on\dots{} \\{}
Over the black bridge \\{}
The line of lighted cars \\{}
Creeps like a monstrous serpent \\{}
Spooring gold\dots{}
Watchman, what of the track?
Night\dots{} silence\dots{} stars\dots{} \\{}
All’s Well!
\end{verse}
\section{III}
\begin{verse}
Light\dots{} \\{}
(Breaking mists\dots{} \\{}
Hills gliding like hands out of a slipping hold\dots{}) \\{}
Light over the pit mouths, \\{}
Streaming in tenuous rays down the black gullets of the Hill\dots{} \\{}
(The copper, insensate, sleeping in the buried lode.) \\{}
Light\dots{} \\{}
Forcing the clogged windows of arsenals\dots{} \\{}
Probing with long sentient fingers in the copper chips\dots{} \\{}
Gleaming metallic and cold \\{}
In numberless slivers of steel\dots{} \\{}
Light over the trestles and the iron clips \\{}
Of the black bridge—poised like a gigantic spider motionless— \\{}
Sweet inquisition of light, like a child’s wonder\dots{} \\{}
Intrusive, innocently staring light \\{}
That nothing appals\dots{}
Light in the slow fumbling summer leaves, \\{}
Cooing and calling \\{}
All winged and avid things \\{}
Waking the early flies, keen to the scent\dots{} \\{}
Green-jeweled iridescent flies \\{}
Unerringly steering— \\{}
Swarming over the blackened lips, \\{}
The young day sprays with indiscriminate gold\dots{}
Watchman, what of the Hill?
Wheels turn; \\{}
The laden cars \\{}
Go rumbling to the mill, \\{}
And Labor walks beside the mules\dots{} \\{}
All’s Well with the Hill!
\end{verse}
\chapter{Spires}
\begin{verse}
Spires of Grace Church, \\{}
For you the workers of the world \\{}
Travailed with the mountains\dots{} \\{}
Aborting their own dreams \\{}
Till the dream of you arose— \\{}
Beautiful, swaddled in stone— \\{}
Scorning their hands.
\end{verse}
\chapter{The Legion of Iron}
\begin{verse}
They pass through the great iron gates— \\{}
Men with eyes gravely discerning, \\{}
Skilled to appraise the tunnage of cranes \\{}
Or split an inch into thousandths— \\{}
Men tempered by fire as the ore is \\{}
And planned to resistance \\{}
Like steel that has cooled in the trough; \\{}
Silent of purpose, inflexible, set to fulfilment— \\{}
To conquer, withstand, overthrow\dots{} \\{}
Men mannered to large undertakings, \\{}
Knowing force as a brother \\{}
And power as something to play with, \\{}
Seeing blood as a slip of the iron, \\{}
To be wiped from the tools \\{}
Lest they rust.
But what if they stood aside, \\{}
Who hold the earth so careless in the crook of their arms?
What of the flamboyant cities \\{}
And the lights guttering out like candles in a wind\dots{} \\{}
And the armies halted\dots{} \\{}
And the train mid-way on the mountain \\{}
And idle men chaffing across the trenches\dots{} \\{}
And the cursing and lamentation \\{}
And the clamor for grain shut in the mills of the world? \\{}
What if they stayed apart, \\{}
Inscrutably smiling, \\{}
Leaving the ground encumbered with dead wire \\{}
And the sea to row-boats \\{}
And the lands marooned— \\{}
Till Time should like a paralytic sit, \\{}
A mildewed hulk above the nations squatting?
\end{verse}
\chapter{Fuel}
\begin{verse}
What of the silence of the keys \\{}
And silvery hands? The iron sings\dots{} \\{}
Though bows lie broken on the strings, \\{}
The fly-wheels turn eternally\dots{}
Bring fuel—drive the fires high\dots{} \\{}
Throw all this artist-lumber in \\{}
And foolish dreams of making things\dots{} \\{}
(Ten million men are called to die.)
As for the common men apart, \\{}
Who sweat to keep their common breath, \\{}
And have no hour for books or art— \\{}
What dreams have these to hide from death!
\end{verse}
\chapter{A Toast}
\begin{verse}
Not your martyrs anointed of heaven— \\{}
~~~~~The ages are red where they trod— \\{}
But the Hunted—the world’s bitter leaven— \\{}
~~~~~Who smote at your imbecile God—
A being to pander and fawn to, \\{}
~~~~~To propitiate, flatter and dread \\{}
As a thing that your souls are in pawn to, \\{}
~~~~~A Dealer who traffics the dead;
A Trader with greed never sated, \\{}
~~~~~Who barters the souls in his snares, \\{}
That were trapped in the lusts he created, \\{}
~~~~~For incense and masses and prayers—
They are crushed in the coils of your halters; \\{}
~~~~~‘Twere well—by the creeds ye have nursed— \\{}
That ye send up a cry from your altars, \\{}
~~~~~A mass for the Martyrs Accursed;
A passionate prayer from reprieval \\{}
~~~~~For the Brotherhood not understood— \\{}
For the Heroes who died for the evil, \\{}
~~~~~Believing the evil was good.
To the Breakers, the Bold, the Despoilers, \\{}
~~~~~Who dreamed of a world over-thrown\dots{} \\{}
They who died for the millions of toilers— \\{}
~~~~~Few—fronting the nations alone!
—To the Outlawed of men and the Branded, \\{}
~~~~~Whether hated or hating they fell— \\{}
I pledge the devoted, red-handed, \\{}
~~~~~Unfaltering Heroes of Hell!
\end{verse}
\part{Accidentals}
\chapter{“The Everlasting Return”}
\begin{verse}
It is dark\dots{} so dark, I remember the sun on Chios\dots{} \\{}
It is still\dots{} so still, I hear the beat of our paddles on the Aegean\dots{}
Ten times we had watched the moon \\{}
Rise like a thin white virgin out of the waters \\{}
And round into a full maternity\dots{} \\{}
For thrice ten moons we had touched no flesh \\{}
Save the man flesh on either hand \\{}
That was black and bitter and salt and scaled by the sea.
The Athenian boy sat on my left\dots{} \\{}
His hair was yellow as corn steeped in wine\dots{} \\{}
And on my right was Phildar the Carthaginian, \\{}
Grinning Phildar \\{}
With his mouth pulled taut as by reins from his black gapped teeth. \\{}
Many a whip had coiled about him \\{}
And his shoulders were rutted deep as wet ground under chariot wheels, \\{}
And his skin was red and tough as a bull’s hide cured in the sun. \\{}
He did not sing like the other slaves, \\{}
But when a big wind came up he screamed with it. \\{}
And always he looked out to sea, \\{}
Save when he tore at his fish ends \\{}
Or spat across me at the Greek boy, whose mouth was red and apart \\{}
~~~~~like an opened fruit.
We had rowed from dawn and the green galley hard at our stern. \\{}
She was green and squat and skulked close to the sea. \\{}
All day the tish of their paddles had tickled our ears, \\{}
And when night came on \\{}
And little naked stars dabbled in the water \\{}
And half the crouching moon \\{}
Slid over the silver belly of the sea thick-scaled with light, \\{}
We heard them singing at their oars\dots{} \\{}
We who had no breath for song.
There was no sound in our boat \\{}
Save the clingle of wrist chains \\{}
And the sobbing of the young Greek. \\{}
I cursed him that his hair blew in my mouth, tasting salt of the sea\dots{} \\{}
I cursed him that his oar kept ill time\dots{} \\{}
When he looked at me I cursed him again, \\{}
That his eyes were soft as a woman’s.
How long\dots{} since their last shell gouged our batteries? \\{}
How long\dots{} since we rose at aim with a sleuth moon astern? \\{}
(It was the damned green moon that nosed us out\dots{} \\{}
The moon that flushed our periscope till it shone like a silver flame\dots{})
They loosed each man’s right hand \\{}
As the galley spent on our decks\dots{} \\{}
And amazed and bloodied we reared half up \\{}
And fought askew with the left hand shackled\dots{} \\{}
But a zigzag fire leapt in our sockets \\{}
And knotted our thews like string\dots{} \\{}
Our thews grown stiff as a crooked spine that would not straighten\dots{}
How long\dots{} since our gauges fell \\{}
And the sea shoved us under? \\{}
It is dark\dots{} so dark\dots{} \\{}
Darkness presses hairy-hot \\{}
Where three make crowded company\dots{} \\{}
And the rank steel smells\dots{} \\{}
It is still\dots{} so still\dots{} \\{}
I seem to hear the wind \\{}
On the dimpled face of the water fathoms above\dots{}
It was still\dots{} so still\dots{} we three that were left alive \\{}
Stared in each other’s faces\dots{} \\{}
But three make bitter company at one man’s bread\dots{} \\{}
And our hate grew sharp and bright as the moon’s edge in the water.
One grinned with his mouth awry from the long gapped teeth\dots{} \\{}
And one shivered and whined like a gull as the waves pawed us over\dots{} \\{}
But one struck with his hate in his hand\dots{}
After that I remember \\{}
Only the dead men’s oars that flapped in the sea\dots{} \\{}
The dead men’s oars that rattled and clicked like idiots’ tongues.
It is still\dots{} so still, with the jargon of engines quiet. \\{}
We three awaiting the crunch of the sea \\{}
Reach our hands in the dark and touch each other’s faces\dots{} \\{}
We three sheathing hate in our hearts\dots{} \\{}
But when hate shall have made its circuit, \\{}
Our bones will be loving company \\{}
Here in the sea’s den\dots{} \\{}
And one whimpers and cries on his God \\{}
And one sits sullenly \\{}
But both draw away from me\dots{} \\{}
For I am the pyre their memories burn on\dots{} \\{}
Like black flames leaping \\{}
Our fiery gestures light the walled-in darkness of the sea\dots{} \\{}
The sea that kneels above us\dots{} \\{}
And makes no sign.
\end{verse}
\chapter{Palestine}
\begin{verse}
Old plant of Asia— \\{}
Mutilated vine \\{}
Holding earth’s leaping sap \\{}
In every stem and shoot \\{}
That lopped off, sprouts again— \\{}
Why should you seek a plateau walled about, \\{}
Whose garden is the world?
\end{verse}
\chapter{The Song}
\begin{verse}
That day, in the slipping of torsos and straining flanks \\{}
~~~~~on the bloodied ooze of fields plowed by the iron, \\{}
And the smoke bluish near earth and bronze in the sunshine \\{}
~~~~~floating like cotton-down, \\{}
And the harsh and terrible screaming, \\{}
And that strange vibration at the roots of us\dots{} \\{}
Desire, fierce, like a song\dots{} \\{}
And we heard \\{}
(Do you remember?) \\{}
All the Red Cross bands on Fifth avenue \\{}
And bugles in little home towns \\{}
And children’s harmonicas bleating
~~~~~America!
And after\dots{} \\{}
(Do you remember?) \\{}
The drollery of the wind on our faces, \\{}
And horizons reeling, \\{}
And the terror of the plain \\{}
Heaving like a gaunt pelvis to the sun\dots{} \\{}
Under us—threshing and twanging \\{}
Torn-up roots of the Song\dots{}
\end{verse}
\chapter{To The Others}
\begin{verse}
I see you, refulgent ones, \\{}
Burning so steadily \\{}
Like big white arc lights\dots{} \\{}
There are so many of you. \\{}
I like to watch you weaving— \\{}
Altogether and with precision \\{}
Each his ray— \\{}
Your tracery of light, \\{}
Making a shining way about America.
I note your infinite reactions— \\{}
In glassware \\{}
And sequin \\{}
And puddles \\{}
And bits of jet— \\{}
And here and there a diamond\dots{}
But you do not yet see me, \\{}
Who am a torch blown along the wind, \\{}
Flickering to a spark \\{}
But never out.
\end{verse}
\chapter{Babel}
\begin{verse}
Oh, God did cunningly, there at Babel— \\{}
Not mere tongues dividing, but soul from soul, \\{}
So that never again should men be able \\{}
To fashion one infinite, towering whole.
\end{verse}
\chapter{The Fiddler}
\begin{verse}
In a little Hungarian cafe \\{}
Men and women are drinking \\{}
Yellow wine in tall goblets.
Through the milky haze of the smoke, \\{}
The fiddler, under-sized, blond, \\{}
Leans to his violin \\{}
As to the breast of a woman. \\{}
Red hair kindles to fire \\{}
On the black of his coat-sleeve, \\{}
Where his white thin hand \\{}
Trembles and dives, \\{}
Like a sliver of moonlight, \\{}
When wind has broken the water.
\end{verse}
\chapter{Dawn Wind}
\begin{verse}
Wind, just arisen— \\{}
(Off what cool mattress of marsh-moss \\{}
In tented boughs leaf-drawn before the stars, \\{}
Or niche of cliff under the eagles?) \\{}
You of living things, \\{}
So gay and tender and full of play— \\{}
Why do you blow on my thoughts—like cut flowers \\{}
Gathered and laid to dry on this paper, rolled out of dead wood?
I see you \\{}
Shaking that flower at me with soft invitation \\{}
And frisking away, \\{}
Deliciously rumpling the grass\dots{}
So you fluttered the curtains about my cradle, \\{}
Prattling of fields \\{}
Before I had had my milk\dots{} \\{}
Did I stir on my pillow, making to follow you, Fleet One? \\{}
I—swaddled, unwinged, like a bird in the egg.
Let be \\{}
My dreams that crackle under your breath\dots{} \\{}
You have the dust of the world to blow on\dots{} \\{}
Do not tag me and dance away, looking back\dots{} \\{}
I am too old to play with you, \\{}
Eternal Child.
\end{verse}
\chapter{North Wind}
\begin{verse}
I love you, malcontent \\{}
Male wind— \\{}
Shaking the pollen from a flower \\{}
Or hurling the sea backward from the grinning sand.
Blow on and over my dreams\dots{} \\{}
Scatter my sick dreams\dots{} \\{}
Throw your lusty arms about me\dots{} \\{}
Envelop all my hot body\dots{} \\{}
Carry me to pine forests— \\{}
Great, rough-bearded forests\dots{} \\{}
Bring me to stark plains and steppes\dots{} \\{}
I would have the North to-night— \\{}
The cold, enduring North.
And if we should meet the Snow, \\{}
Whirling in spirals, \\{}
And he should blind my eyes\dots{} \\{}
Ally, you will defend me— \\{}
You will hold me close, \\{}
Blowing on my eyelids.
\end{verse}
\chapter{The Destroyer}
\begin{verse}
I am of the wind\dots{} \\{}
A wisp of the battering wind\dots{}
I trail my fingers along the Alps \\{}
And an avalanche falls in my wake\dots{} \\{}
I feel in my quivering length \\{}
When it buries the hamlet beneath\dots{}
I hurriedly sweep aside \\{}
The cities that clutter our path\dots{} \\{}
As we whirl about the circle of the globe\dots{} \\{}
As we tear at the pillars of the world\dots{} \\{}
Open to the wind, \\{}
The Destroyer! \\{}
The wind that is battering at your gates.
\end{verse}
\chapter{Lullaby}
\begin{verse}
Rock-a-by baby, woolly and brown\dots{} \\{}
(There’s a shout at the door an’ a big red light\dots{}) \\{}
Lil’ coon baby, mammy is down\dots{} \\{}
Han’s that hold yuh are steady an’ white\dots{}
Look piccaninny—such a gran’ blaze \\{}
Lickin’ up the roof an’ the sticks of home— \\{}
Ever see the like in all yo’ days! \\{}
—Cain’t yuh sleep, mah bit-of-honey-comb?
Rock-a-by baby, up to the sky! \\{}
Look at the cherries driftin’ by— \\{}
Bright red cherries spilled on the groun’— \\{}
Piping-hot cherries at nuthin’ a poun’!
Hush, mah lil’ black-bug—doan yuh weep. \\{}
Daddy’s run away an’ mammy’s in a heap \\{}
By her own fron’ door in the blazin’ heat \\{}
Outah the shacks like warts on the street\dots{}
An’ the singin’ flame an’ the gleeful crowd \\{}
Circlin’ aroun’\dots{} won’t mammy be proud! \\{}
With a stone at her hade an’ a stone on her heart, \\{}
An’ her mouth like a red plum, broken apart\dots{}
See where the blue an’ khaki prance, \\{}
Adding brave colors to the dance \\{}
About the big bonfire white folks make— \\{}
Such gran’ doin’s fo’ a lil’ coon’s sake!
Hear all the eagah feet runnin’ in town— \\{}
See all the willin’ han’s reach outah night— \\{}
Han’s that are wonderful, steady an’ white! \\{}
To toss up a lil’ babe, blinkin’ an’ brown\dots{}
Rock-a-by baby—higher an’ higher! \\{}
Mammy is sleepin’ an’ daddy’s run lame\dots{} \\{}
(Soun’ may yuh sleep in yo’ cradle o’ fire!) \\{}
Rock-a-by baby, hushed in the flame\dots{}
(An incident of the East St. Louis Race Riots, when some white women \\{}
flung a living colored baby into the heart of a blazing fire.)
\end{verse}
\chapter{The Foundling}
\begin{verse}
Snow wraiths circle us \\{}
Like washers of the dead, \\{}
Flapping their white wet cloths \\{}
Impatiently \\{}
About the grizzled head, \\{}
Where the coarse hair mats like grass, \\{}
And the efficient wind \\{}
With cold professional baste \\{}
Probes like a lancet \\{}
Through the cotton shirt\dots{}
About us are white cliffs and space. \\{}
No façades show, \\{}
Nor roof nor any spire\dots{} \\{}
All sheathed in snow\dots{} \\{}
The parasitic snow \\{}
That clings about them like a blight.
Only detached lights \\{}
Float hazily like greenish moons, \\{}
And endlessly \\{}
Down the whore-street, \\{}
Accouched and comforted and sleeping warm, \\{}
The blizzard waltzes with the night.
\end{verse}
\chapter{The Woman With Jewels}
\begin{verse}
The woman with jewels sits in the cafe, \\{}
Spraying light like a fountain. \\{}
Diamonds glitter on her bulbous fingers \\{}
And on her arms, great as thighs, \\{}
Diamonds gush from her ear-lobes over the goitrous throat. \\{}
She is obesely beautiful. \\{}
Her eyes are full of bleared lights, \\{}
Like little pools of tar, spilled by a sailor in mad haste for shore\dots{} \\{}
And her mouth is scarlet and full—only a little crumpled— \\{}
~~~~~like a flower that has been pressed apart\dots{}
Why does she come alone to this obscure basement— \\{}
She who should have a litter and hand-maidens to support her \\{}
~~~~~on either side?
She ascends the stairway, and the waiters turn to look at her, \\{}
~~~~~spilling the soup. \\{}
The black satin dress is a little lifted, showing the dropsical legs \\{}
~~~~~in their silken fleshings\dots{} \\{}
The mountainous breasts tremble\dots{} \\{}
There is an agitation in her gems, \\{}
That quiver incessantly, emitting trillions of fiery rays\dots{} \\{}
She erupts explosive breaths\dots{} \\{}
Every step is an adventure \\{}
From this\dots{} \\{}
The serpent’s tooth \\{}
Saved Cleopatra.
\end{verse}
\chapter{Submerged}
\begin{verse}
I have known only my own shallows— \\{}
Safe, plumbed places, \\{}
Where I was wont to preen myself.
But for the abyss \\{}
I wanted a plank beneath \\{}
And horizons\dots{}
I was afraid of the silence \\{}
And the slipping toe-hold\dots{}
Oh, could I now dive \\{}
Into the unexplored deeps of me— \\{}
Delve and bring up and give \\{}
All that is submerged, encased, unfolded, \\{}
That is yet the best.
\end{verse}
\chapter{Art and Life}
\begin{verse}
When Art goes bounding, lean, \\{}
Up hill-tops fired green \\{}
To pluck a rose for life.
Life like a broody hen \\{}
Cluck-clucks him back again.
But when Art, imbecile, \\{}
Sits old and chill \\{}
On sidings shaven clean, \\{}
And counts his clustering \\{}
Dead daisies on a string \\{}
With witless laughter\dots{}
Then like a new Jill \\{}
Toiling up a hill \\{}
Life scrambles after.
\end{verse}
\chapter{Brooklyn Bridge}
\begin{verse}
Pythoness body—arching \\{}
Over the night like an ecstasy— \\{}
I feel your coils tightening\dots{} \\{}
And the world’s lessening breath.
\end{verse}
\chapter{Dreams}
\begin{verse}
Men die\dots{} \\{}
Dreams only change their houses. \\{}
They cannot be lined up against a wall \\{}
And quietly buried under ground, \\{}
And no more heard of\dots{} \\{}
However deep the pit and heaped the clay— \\{}
Like seedlings of old time \\{}
Hooding a sacred rose under the ice cap of the world— \\{}
Dreams will to light.
\end{verse}
\chapter{The Fire}
\begin{verse}
The old men of the world have made a fire \\{}
To warm their trembling hands. \\{}
They poke the young men in. \\{}
The young men burn like withes.
If one run a little way, \\{}
The old men are wrath. \\{}
They catch him and bind him and throw him again to the flames. \\{}
Green withes burn slow\dots{} \\{}
And the smoke of the young men’s torment \\{}
Rises round and sheer as the trunk of a pillared oak, \\{}
And the darkness thereof spreads over the sky\dots{}
Green withes burn slow\dots{} \\{}
And the old men of the world sit round the fire \\{}
And rub their hands\dots{} \\{}
But the smoke of the young men’s torment \\{}
Ascends up for ever and ever.
\end{verse}
\chapter{A Memory}
\begin{verse}
I remember \\{}
The crackle of the palm trees \\{}
Over the mooned white roofs of the town\dots{} \\{}
The shining town\dots{} \\{}
And the tender fumbling of the surf \\{}
On the sulphur-yellow beaches \\{}
As we sat\dots{} a little apart\dots{} in the close-pressing night.
The moon hung above us like a golden mango, \\{}
And the moist air clung to our faces, \\{}
Warm and fragrant as the open mouth of a child \\{}
And we watched the out-flung sea \\{}
Rolling to the purple edge of the world, \\{}
Yet ever back upon itself\dots{} \\{}
As we\dots{}
Inadequate night\dots{} \\{}
And mooned white memory \\{}
Of a tropic sea\dots{} \\{}
How softly it comes up \\{}
Like an ungathered lily.
\end{verse}
\chapter{The Edge}
\begin{verse}
I thought to die that night in the solitude where they would never find me\dots{} \\{}
But there was time\dots{} \\{}
And I lay quietly on the drawn knees of the mountain, \\{}
~~~~~staring into the abyss\dots{} \\{}
I do not know how long\dots{} \\{}
I could not count the hours, they ran so fast \\{}
Like little bare-foot urchins—shaking my hands away\dots{} \\{}
But I remember \\{}
Somewhere water trickled like a thin severed vein\dots{} \\{}
And a wind came out of the grass, \\{}
Touching me gently, tentatively, like a paw.
As the night grew \\{}
The gray cloud that had covered the sky like sackcloth \\{}
Fell in ashen folds about the hills, \\{}
Like hooded virgins, pulling their cloaks about them\dots{} \\{}
There must have been a spent moon, \\{}
For the Tall One’s veil held a shimmer of silver\dots{}
That too I remember\dots{} \\{}
And the tenderly rocking mountain \\{}
Silence \\{}
And beating stars\dots{}
Dawn \\{}
Lay like a waxen hand upon the world, \\{}
And folded hills \\{}
Broke into a sudden wonder of peaks, stemming clear and cold, \\{}
Till the Tall One bloomed like a lily, \\{}
Flecked with sun, \\{}
Fine as a golden pollen— \\{}
It seemed a wind might blow it from the snow.
I smelled the raw sweet essences of things, \\{}
And heard spiders in the leaves \\{}
And ticking of little feet, \\{}
As tiny creatures came out of their doors \\{}
To see God pouring light into his star\dots{}
\dots{} It seemed life held \\{}
No future and no past but this\dots{}
And I too got up stiffly from the earth, \\{}
And held my heart up like a cup\dots{}
\end{verse}
\chapter{The Garden}
\begin{verse}
Bountiful Givers, \\{}
I look along the years \\{}
And see the flowers you threw\dots{} \\{}
Anemones \\{}
And sprigs of gray \\{}
Sparse heather of the rocks, \\{}
Or a wild violet \\{}
Or daisy of a daisied field\dots{} \\{}
But each your best.
I might have worn them on my breast \\{}
To wilt in the long day\dots{} \\{}
I might have stemmed them in a narrow vase \\{}
And watched each petal sallowing\dots{} \\{}
I might have held them so—mechanically— \\{}
Till the wind winnowed all the leaves \\{}
And left upon my hands \\{}
A little smear of dust.
Instead \\{}
I hid them in the soft warm loam \\{}
Of a dim shadowed place\dots{} \\{}
Deep \\{}
In a still cool grotto, \\{}
Lit only by the memories of stars \\{}
And the wide and luminous eyes \\{}
Of dead poets \\{}
That love me and that I love\dots{} \\{}
Deep\dots{} deep\dots{} \\{}
Where none may see—not even ye who gave— \\{}
About my soul your garden beautiful.
\end{verse}
\chapter{Under-Song}
\begin{verse}
There is music in the strong \\{}
~~~~~Deep-throated bush, \\{}
Whisperings of song \\{}
~~~~~Heard in the leaves’ hush— \\{}
Ballads of the trees \\{}
~~~~~In tongues unknown— \\{}
A reminiscent tone \\{}
~~~~~On minor keys\dots{}
Boughs swaying to and fro \\{}
~~~~~Though no winds pass\dots{} \\{}
Faint odors in the grass \\{}
~~~~~Where no flowers grow, \\{}
And flutterings of wings \\{}
~~~~~And faint first notes, \\{}
Once babbled on the boughs \\{}
~~~~~Of faded springs.
Is it music from the graves \\{}
~~~~~Of all things fair \\{}
Trembling on the staves \\{}
~~~~~Of spacious air— \\{}
Fluted by the winds \\{}
~~~~~Songs with no words— \\{}
Sonatas from the throats \\{}
~~~~~Of master birds?
One peering through the husk \\{}
~~~~~Of darkness thrown \\{}
May hear it in the dusk— \\{}
~~~~~That ancient tone, \\{}
Silvery as the light \\{}
~~~~~Of long dead stars \\{}
Yet falling through the night \\{}
~~~~~In trembling bars.
\end{verse}
\chapter{A Worn Rose}
\begin{verse}
Where to-day would a dainty buyer \\{}
Imbibe your scented juice, \\{}
Pale ruin with a heart of fire; \\{}
Drain your succulence with her lips, \\{}
Grown sapless from much use\dots{} \\{}
Make minister of her desire \\{}
A chalice cup where no bee sips— \\{}
~~~~~Where no wasp wanders in?
Close to her white flesh housed an hour, \\{}
~~~~~One held you\dots{} her spent form \\{}
Drew on yours for its wasted dower— \\{}
What favour could she do you more? \\{}
~~~~~Yet, of all who drink therein, \\{}
~~~~~None know it is the warm \\{}
Odorous heart of a ravished flower \\{}
Tingles so in her mouth’s red core\dots{}
\end{verse}
\chapter{Iron Wine}
\begin{verse}
The ore in the crucible is pungent, smelling like acrid wine, \\{}
It is dusky red, like the ebb of poppies, \\{}
And purple, like the blood of elderberries. \\{}
Surely it is a strong wine—juice distilled of the fierce iron. \\{}
I am drunk of its fumes. \\{}
I feel its fiery flux \\{}
Diffusing, permeating, \\{}
Working some strange alchemy\dots{} \\{}
So that I turn aside from the goodly board, \\{}
So that I look askance upon the common cup, \\{}
And from the mouths of crucibles \\{}
Suck forth the acrid sap.
\end{verse}
\chapter{Dispossessed}
\begin{verse}
Tender and tremulous green of leaves \\{}
Turned up by the wind, \\{}
Twanging among the vines— \\{}
Wind in the grass \\{}
Blowing a clear path \\{}
For the new-stripped soul to pass\dots{}
The naked soul in the sunlight\dots{} \\{}
Like a wisp of smoke in the sunlight \\{}
On the hill-side shimmering.
Dance light on the wind, little soul, \\{}
Like a thistle-down floating \\{}
Over the butterflies \\{}
And the lumbering bees\dots{}
Come away from that tree \\{}
And its shadow grey as a stone\dots{}
Bathe in the pools of light \\{}
On the hillside shimmering— \\{}
Shining and wetted and warm in the sun-spray falling like golden rain—
But do not linger and look \\{}
At that bleak thing under the tree.
\end{verse}
\chapter{The Star}
\begin{verse}
Last night \\{}
I watched a star fall like a great pearl into the sea, \\{}
Till my ego expanding encompassed sea and star, \\{}
Containing both as in a trembling cup.
\end{verse}
\chapter{The Tidings}
(Easter 1916)
\begin{verse}
Censored lies that mimic truth\dots{} \\{}
~~~~~Censored truth as pale as fear\dots{} \\{}
My heart is like a rousing bell— \\{}
~~~~~And but the dead to hear\dots{}
My heart is like a mother bird, \\{}
~~~~~Circling ever higher, \\{}
And the nest-tree rimmed about \\{}
~~~~~By a forest fire\dots{}
My heart is like a lover foiled \\{}
~~~~~By a broken stair— \\{}
They are fighting to-night in Sackville Street, \\{}
~~~~~And I am not there!
\end{verse}
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\begin{center}
The Anarchist Library
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Lola Ridge
The Ghetto and Other Poems
\bigskip
Retrieved on May 23, 2012 from http:\Slash{}\Slash{}www.gutenberg.org\Slash{}cache\Slash{}epub\Slash{}4332\Slash{}pg4332.txt
Produced by Catherine Daly\forcelinebreak The larger part of the poem entitled “The Ghetto” appeared originally in \emph{The New Republic} and some of poems were printed in \emph{The International}, \emph{Others}, \emph{Poetry}, etc. To the editors who first published the poems the author makes due acknowledgment.
\bigskip
\textbf{theanarchistlibrary.org}
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https://www.emis.de/journals/EJC/Volume_13/Abstracts/v13i1r72.abs.tex | emis.de | CC-MAIN-2020-29 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-29/segments/1593655879738.16/warc/CC-MAIN-20200702174127-20200702204127-00115.warc.gz | 821,124,183 | 1,474 | \magnification=1200
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%\def\frac#1 #2 {{#1\over #2}}
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\noindent
%
%
{\bf David J. Galvin}
%
%
\medskip
\noindent
%
%
{\bf Bounding the Partition Function of Spin-Systems}
%
%
\vskip 5mm
\noindent
%
%
%
%
With a graph $G=(V,E)$ we associate a collection of non-negative real
weights $\bigcup_{v\in V}\{\lambda_{i,v}:1\leq i \leq m\} \cup
\bigcup_{uv \in E} \{\lambda_{ij,uv}:1\leq i \leq j \leq m\}.$ We
consider the probability distribution on
$\{f:V\rightarrow\{1,\ldots,m\}\}$ in which each $f$ occurs with
probability proportional to $\prod_{v \in V}\lambda_{f(v),v}\prod_{uv
\in E}\lambda_{f(u)f(v),uv}$. Many well-known statistical physics
models, including the Ising model with an external field and the
hard-core model with non-uniform activities, can be framed as such a
distribution. We obtain an upper bound, independent of $G$, for the
partition function (the normalizing constant which turns the
assignment of weights on $\{f:V\rightarrow\{1,\ldots,m\}\}$ into a
probability distribution) in the case when $G$ is a regular bipartite
graph. This generalizes a bound obtained by Galvin and Tetali who
considered the simpler weight collection $\{\lambda_i:1 \leq i \leq
m\} \cup \{\lambda_{ij}:1 \leq i \leq j \leq m\}$ with each
$\lambda_{ij}$ either $0$ or $1$ and with each $f$ chosen with
probability proportional to $\prod_{v \in V}\lambda_{f(v)}\prod_{uv
\in E}\lambda_{f(u)f(v)}$. Our main tools are a generalization to list
homomorphisms of a result of Galvin and Tetali on graph homomorphisms
and a straightforward second-moment computation.
\bye
|
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https://cs.uwaterloo.ca/journals/JIS/VOL10/Holdener/holdener7.tex | uwaterloo.ca | CC-MAIN-2022-33 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-33/segments/1659882571153.86/warc/CC-MAIN-20220810100712-20220810130712-00653.warc.gz | 193,410,372 | 22,653 | \documentclass[12pt,reqno]{article}
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\vskip 1cm{\LARGE\bf
Abundancy ``Outlaws" of the Form $\frac{\sigma(N) + t}{N}$}
%\vskip .1in of the Form $\frac{\sigma(N) + t}{N}$}
\vskip 1cm \large
William G. Stanton and Judy A. Holdener\\
Department of Mathematics\\
Kenyon College\\
Gambier, Ohio 43022 \\
USA \\
\href{mailto:[email protected]}{\tt [email protected]}\\
\href{mailto:[email protected]}{\tt [email protected]} \\
\end{center}
\vskip .2 in
\begin{abstract}
The abundancy index of a positive integer $n$ is defined to be the
rational number $I(n)=\sigma(n)/n$, where $\sigma$ is the sum of
divisors function $\sigma(n)=\sum_{d|n}d$. An abundancy outlaw is
a rational number greater than 1 that fails to be in the image of
of the map $I$. In this paper, we consider rational numbers of the
form $(\sigma(N)+t)/N$ and prove that under certain conditions
such rationals are abundancy outlaws.
\end{abstract}
%\newtheorem{theorem}{Theorem}
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\section{Introduction}
The abundancy index of a positive integer $n$ is defined to be the
rational number $I(n)=\sigma(n)/n$, where $\sigma$ is the sum of
divisors function, $\sigma(n)=\sum_{d|n}d$. Positive integers
having integer-valued abundancy indices are said to be
\textit{multiperfect} numbers, and if $I(n)=2$ in particular, then
$n$ is \emph{perfect}. More generally, the abundancy index of a
number $n$ can be thought of as a measure of its perfection; if
$I(n)<2$ then $n$ is said to be \textit{deficient}, and if
$I(n)>2$ then $n$ is \textit{abundant}. In this way, the
abundancy index is a useful tool in gaining a better understanding
of perfect numbers. In fact, the following theorem provides
conditions equivalent to the existence of an odd perfect number
\cite{HoldJ2006}.
\begin{theorem} \label{thm:oddperfequiv}
There exists an odd perfect number if and only if there exist
positive integers $p, n,$ and $\alpha$ such that $p\equiv \alpha
\equiv 1\mod 4$, where $p$ is a prime not dividing $n$, and
\begin{displaymath}
I(n)=\frac{2p^\alpha(p-1)}{p^{\alpha+1}-1}.
\end{displaymath}
\end{theorem}
So, for example, if one could find an integer $n$ having abundancy
index equal to $5/3$, then one would be able to produce an odd
perfect number. Hence, it is useful to try to characterize those
rational numbers in $(1, \infty)$ that do not appear as the
abundancy index of some positive integer. We will call such
numbers \emph{abundancy outlaws}.
\begin{definition}
A rational number $r/s$ greater than 1 is said to be an
\textit{abundancy outlaw} if $I(x)=r/s$ has no solution among the
positive integers.
\end{definition}
In this paper, we consider the sequence of rational numbers in
$(1,\infty)$. For each numerator $a > 1$, list the rationals
$a/b$, with gcd$(a,b)=1$, so that denominators $1\leq b < a$
appear in ascending order:
\begin{displaymath}
\frac{2}{1}, \frac{3}{1}, \frac{3}{2}, \frac{4}{1}, \frac{4}{3},
\frac{5}{1}, \frac{5}{2}, \frac{5}{3}, \frac{5}{4}, \frac{6}{1},
\frac{6}{5}, \frac{7}{1}, \frac{7}{2}, \frac{7}{3}, \frac{7}{4},
\frac{7}{5}, \frac{7}{6}, ...
\end{displaymath}
While each term in the sequence is either an abundancy index or an
abundancy outlaw, it is generally difficult to determine the
status of a given rational. The sequence can be partitioned into
three sets: those rationals that are known to be abundancy
indices, those that are known to be outlaws, and those with
abundancy index/outlaw status unknown. Our goal is to capture
outlaws from the third category, increasing the size of the second
category. Since rationals of the form $(\sigma(N)-t)/N$, with $t
\geq 1$, are known to be outlaws (see Property 2.3 below), we
consider rational numbers of the form $(\sigma(N)+t)/N$. We prove
that under certain conditions $(\sigma(N)+t)/N$ is an abundancy
outlaw. It is worth noting that our original interest in such
rationals stemmed from our interest in the fraction
$5/3=(\sigma(3)+1)/3$. Unfortunately, our results do not allow us
to say anything about rationals of the form $(\sigma(p)+1)/p$
where $p$ is a prime. Such elusive rationals remain in category
three. Nonetheless, we do prove that $(\sigma(2p)+1)/(2p)$ is an
abundancy outlaw for all primes $p>3$. (Since $I(6)=
(\sigma(2^2)+1)/2^2$ and $I(18)= (\sigma(6)+1)/6$, this provides a
complete characterization of fractions of this form.)
\section{Preliminaries}
It is useful to think of the abundancy index $I$ as a function
mapping the natural numbers $n\geq 2$ into the set of rational
numbers in $(1,\infty)$. Defining $D$ to be the image of $I$:
\begin{displaymath}
D=\{I(n):n\in \mathbb{N},n\geq 2\},
\end{displaymath}
we can ask many questions about $D$. For instance, how are the
abundancy indices distributed among the set $(1,\infty)$?
Certainly, we can find elements of $D$ arbitrarily close to 1
because $I(p)=(p+1)/p$ for all primes $p$. Moreover, it is not
hard to show that $I(n!)\geq \sum_i^n 1/i$, and therefore $D$ is
unbounded. In fact, $D$ is dense in $(1,\infty)$
\cite{Laatsch1986}. Even more interesting, P. Weiner proved that
the set of abundancy outlaws is \emph{also} dense in $(1,\infty)$
\cite{Weiner2000}! Hence it seems that the situation is both
complex and interesting. For our purposes, the following
properties will be helpful.
\smallskip
\begin{description}
\item[Property 2.1] $I(kN) \geq I(N)$ for all natural numbers
$k$ and $N \geq 2$. (See \cite{Laatsch1986}, page 84.)
\medskip
\item[Property 2.2] If $I(n) = k/m$ with
$\gcd(k,m)=1$, then $m|n$. This follows directly from setting
$\sigma(n)/n = k/m$. Clearly, $m|(nk)$ and since $k$ and $m$ are
relatively prime, it must be that $m|n$.
\medskip
\item[Property 2.3] If $m < k < \sigma(m)$ and $k$
is relatively prime to $m$, then $k/m$ is an abundancy outlaw.
Hence if $r/s$ is an abundancy index with $\gcd(r,s)=1$, then $r
\geq \sigma(s)$. (See \cite{Weiner2000}, page 309. The property
also appears in \cite{Anderson1974}.)
\end{description}
\medskip
Property 2.3 reveals a class of abundancy outlaws. Indeed, it was
using this property that Weiner was able to prove that the set of
outlaws is dense in $(1,\infty)$. It also worth noting that
Property 2.3 implies that $(k+1)/k$ is an abundancy index if and
only if $k$ is prime. Similarly, $(k+2)/k$ is an abundancy outlaw
whenever $k$ is an odd composite number. If $p$ is a prime greater
than 2, then it is unknown whether $(p+2)/p$ is an outlaw. (See
\cite{Ryan2002}, pages 512-513 for more discussion about this.)
Finally, recent progress has been made by R. Ryan in finding
abundancy outlaws. In 2002, Ryan produced an example of an
abundancy outlaw not captured by Property 2.3. (See Theorem B.6 in
\cite{Ryan2002}.) Because the conditions describing Ryan's outlaw
are quite technical, we will not restate them here. Nonetheless,
we want to mention that the search for outlaws employed in the
next section was inspired by Ryan's work.
\section{A search for abundancy outlaws}
As Property 2.1 implies, multiplying any number $N = \prod_{i=1}^n
p_i^{k_i}$ by one of its prime divisors, $p_j$, will serve to
increase its abundancy. The following lemma measures this
increase.
\begin{lemma}
Let $N = \prod_{i=1}^n p_i^{k_i}$ for primes $p_1,p_2,...,p_n$.
Then
\begin{displaymath}
\frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j + 1}) - 1} =
\frac{\sigma(p_j N)}{p_j \sigma(N)}
\end{displaymath}
for all $1 \leq j \leq n$.
\end{lemma}
\begin{proof}
The result follows from the fact that
\begin{displaymath}
\begin{array}{lll}
p_j \sigma(N) &=& p_j \sigma(p_j^{k_j}) \sigma(N/p_j^{k_j})\\
&=& p_j (\sum_{i=1}^{k_j} p_j^i) \sigma(N/p_j^{k_j}) \\
&=& (\sigma(p_j^{k_j + 1}) - 1) \sigma(N/p_j^{k_j}).
\end{array}
\end{displaymath}
Therefore,
\begin{eqnarray}
\frac{\sigma\left(p_j N\right)}{p_j \sigma(N)} &=&
\frac{\sigma(p_j^{k_j + 1})
\sigma(N/p_j^{k_j})}{(\sigma(p_j^{k_j +
1}) - 1) \sigma(N/p_j^{k_j})}
\\
&=& \frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j +
1})-1}.
\end{eqnarray}
\end{proof}
Next we present criteria that can be used in the search for
abundancy outlaws. As the assumptions given in Theorem 3.2 below
indicate, our search focuses on those fractions $r/s$ (in reduced
form) that satisfy $I(N)<r/s<I(p_iN)$ for some prime divisor $p_i$
of a positive integer $N$. To be more specific, keeping
Properties 2.1 and 2.2 in mind, we look for those values of $s$
having divisors that lead to abundancy values exceeding $I(p_iN)$.
\begin{theorem}
Let $r/s>1$ be a fraction in lowest terms such that there exists a
divisor $N = \prod_{i=1}^n p_i^{k_i}$ of $s$ satisfying the
following two conditions:
\begin{enumerate}
\item $r/s < I(p_i N)$ for all $i \leq n$
\item The product \, $\sigma(N) (s/N)$ has a divisor $M$ such that $(M,r) =
1$ and $I(M) \geq \frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j + 1}) -
1}$ for some positive integer $j \leq n$.
\end{enumerate}
Then $r/s$ is an abundancy outlaw.
\end{theorem}
\begin{proof}
Let $r/s>1$ be a fraction in lowest terms satisfying the above
hypotheses. Suppose that $I(x) = r/s$ for some natural number
$x$. Since $r/s$ is in lowest terms, Property 2.2 ensures that $s
| x$. Then, because $N | s$, $N | x$, and therefore, $x = d N$ for
some positive integer $d$. However, since $r/s < I(p_i N)$ for $1
\leq i \leq n$, the first assumption requires that $p_i^{k_i + 1}
\nmid x$ for all $1 \leq i \leq n$, and thus, $d$ is relatively
prime to $N$.
\par
Consequently, we can write
\begin{displaymath}
I(x) = I(d N) = I(d) I(N) = \frac{r}{s}.
\end{displaymath}
Hence,
\begin{displaymath}
\sigma(d)\sigma(N)(s/N) = r d.
\end{displaymath}
Now, by the second assumption, we know that there exists a
positive integer $M$ such that
\begin{displaymath}
M | \sigma(N) (s/N)
\end{displaymath}
and $M$ is relatively prime to $r$. Therefore, $M | d$ and we can
say that
\begin{displaymath}
I(x) = I(d)I(N) \geq I(M) I(N)
\end{displaymath}
by Property 2.1. Then, by assumption 2) and Lemma 3.1,
\begin{displaymath}
I(M) \geq \frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j + 1}) -1} =
\frac{\sigma(p_j N)}{p_j \sigma(N)}
\end{displaymath}
for some $1 \leq j \leq n$, so we know that
\begin{displaymath}
I(x) \geq I(M) I(N) \geq \frac{\sigma(p_j N)}{p_j \sigma(N)}
\frac{\sigma(N)}{N} = \frac{\sigma(p_j N)}{p_j N} = I(p_j N)
\end{displaymath}
Therefore, $I(x) \geq I(p_j N)$, which contradicts our assumption
that $I(x) = r/s < I(p_i N)$ for all $1 \leq i \leq n$. We
conclude, then, that $r/s$ is an abundancy outlaw.
\end{proof}
\begin{example}
Theorem 3.2 can be used to show that $37/22$ is an abundancy
outlaw. Certainly $37/22 < I(2^2) = 7/4$. Thus, assumption 1) is
satisfied for $N = 2$. Next, note that $M = 3$ divides
$\sigma(2)\cdot (22/2)$, and because $gcd(3,37) = 1$, with $I(3)
> \frac{\sigma(4)}{\sigma(4) - 1} = 7/6$, assumption 2) is
satisfied as well. A computer search reveals many more examples.
(See Table 3.4)
\end{example}
\begin{center}
$
\begin{array}{ccc}
\begin{tabular}{|l|c|l|}
\hline
Abundancy & & \\
outlaw $r/s$ & $s$ & $\sigma(s)$ \\
\hline
$ 29/12 $ & $ 2^2\cdot 3 $ & $ 28 $ \\
$ 37/22 $ & $ 2\cdot 11 $ & $ 36 $ \\
$ 43/20 $ & $ 2^2\cdot 5 $ & $ 42 $ \\
$ 43/26 $ & $ 2\cdot 13 $ & $ 42 $ \\
$ 55/34 $ & $ 2\cdot 17 $ & $ 54 $ \\
$ 59/34 $ & $ 2\cdot 17 $ & $ 54 $ \\
$ 61/24 $ & $ 2^3\cdot 3 $ & $ 60 $ \\
$ 61/38 $ & $ 2\cdot 19 $ & $ 60 $ \\
$ 65/38 $ & $ 2\cdot 19 $ & $ 60 $ \\
$ 73/30 $ & $ 2\cdot 3\cdot 5 $ & $ 72 $ \\
$ 73/46 $ & $ 2\cdot 23 $ & $ 72 $ \\
$ 73/51 $ & $ 3\cdot 17 $ & $ 72 $ \\
$ 77/46 $ & $ 2\cdot 23 $ & $ 72 $ \\
$ 79/45 $ & $ 3^2\cdot 5 $ & $ 78 $ \\
$ 79/46 $ & $ 2\cdot 23 $ & $ 72 $ \\
$ 91/40 $ & $ 2^3\cdot 5 $ & $ 90 $ \\
$ 91/58 $ & $ 2\cdot 29 $ & $ 90 $ \\
$ 95/58 $ & $ 2\cdot 29 $ & $ 90 $ \\
$ 97/42 $ & $ 2\cdot 3\cdot 7 $ & $ 96 $ \\
$ 97/58 $ & $ 2\cdot 29 $ & $ 90 $ \\
$ 97/62 $ & $ 2\cdot 31 $ & $ 96 $ \\
$ 97/69 $ & $ 3\cdot 23 $ & $ 96 $ \\
$ 101/58 $ & $ 2\cdot 29 $ & $ 90 $ \\
$ 101/62 $ & $ 2\cdot 31 $ & $ 96 $ \\
$ 103/62 $ & $ 2\cdot 31 $ & $ 96 $ \\
$ 107/62 $ & $ 2\cdot 31 $ & $ 96 $ \\
$ 115/74 $ & $ 2\cdot 37 $ & $ 114 $ \\
$ 119/74 $ & $ 2\cdot 37 $ & $ 114 $ \\
$ 121/56 $ & $ 2^3\cdot 7 $ & $ 120 $ \\
$ 121/74 $ & $ 2\cdot 37 $ & $ 114 $ \\
$ 121/87 $ & $ 3\cdot 29 $ & $ 120 $ \\
$ 125/48 $ & $ 2^4\cdot 3 $ & $ 124 $ \\
\hline
\end{tabular}
& \hspace{1cm} &
\begin{tabular}{|l|c|l|}
\hline
Abundancy & & \\
outlaw $r/s$ & $s$ & $\sigma(s)$ \\
\hline
$ 125/74 $ & $ 2\cdot 37 $ & $ 114 $ \\
$ 125/87 $ & $ 3\cdot 29 $ & $ 120 $ \\
$ 127/68 $ & $ 2^2\cdot 17 $ & $ 126 $ \\
$ 127/74 $ & $ 2\cdot 37 $ & $ 114 $ \\
$ 127/82 $ & $ 2\cdot 41 $ & $ 126 $ \\
$ 131/82 $ & $ 2\cdot 41 $ & $ 126 $ \\
$ 131/93 $ & $ 3\cdot 31 $ & $ 128 $ \\
$ 133/82 $ & $ 2\cdot 41 $ & $ 126 $ \\
$ 133/86 $ & $ 2\cdot 43 $ & $ 132 $ \\
$ 133/93 $ & $ 3\cdot 31 $ & $ 128 $ \\
$ 137/82 $ & $ 2\cdot 41 $ & $ 126 $ \\
$ 137/86 $ & $ 2\cdot 43 $ & $ 132 $ \\
$ 139/82 $ & $ 2\cdot 41 $ & $ 126 $ \\
$ 139/86 $ & $ 2\cdot 43 $ & $ 132 $ \\
$ 141/76 $ & $ 2^2\cdot 19 $ & $ 140 $ \\
$ 143/82 $ & $ 2\cdot 41 $ & $ 126 $ \\
$ 143/86 $ & $ 2\cdot 43 $ & $ 132 $ \\
$ 145/66 $ & $ 2\cdot 3\cdot 11 $ & $ 144 $ \\
$ 145/86 $ & $ 2\cdot 43 $ & $ 132 $ \\
$ 145/94 $ & $ 2\cdot 47 $ & $ 144 $ \\
$ 149/86 $ & $ 2\cdot 43 $ & $ 132 $ \\
$ 149/94 $ & $ 2\cdot 47 $ & $ 144 $ \\
$ 151/94 $ & $ 2\cdot 47 $ & $ 144 $ \\
$ 155/94 $ & $ 2\cdot 47 $ & $ 144 $ \\
$ 155/111 $ & $ 3\cdot 37 $ & $ 152 $ \\
$ 157/94 $ & $ 2\cdot 47 $ & $ 144 $ \\
$ 157/99 $ & $ 3^2\cdot 11 $ & $ 156 $ \\
$ 157/111 $ & $ 3\cdot 37 $ & $ 152 $ \\
$ 161/94 $ & $ 2\cdot 47 $ & $ 144 $ \\
$ 163/94 $ & $ 2\cdot 47 $ & $ 144 $ \\
$ 163/106 $ & $ 2\cdot 53 $ & $ 162 $ \\
$ 167/106 $ & $ 2\cdot 53 $ & $ 162 $ \\
\hline
\end{tabular}
\end{array}
$
\vspace{0.3cm} \textbf{Table 3.4}: A list of abundancy outlaws
found using Theorem 3.2. These outlaws are captured by Theorem 3.2,
but not by Property 2.3.
\end{center}
\section{The main results}
Table 3.4 reveals some recognizable patterns. Most obvious is the
indication that, for small odd values of $t$ (and $p$ prime),
reduced rational numbers of the form
\begin{displaymath}
\frac{r}{s}=\frac{\sigma(2^mp^n)+t}{2^mp^n}
\end{displaymath}
often lead to abundancy outlaws. In fact, the results of this
section will show that $\frac{\sigma(2^mp^{2n+1})+1}{2^mp^{2n+1}}$
is an abundancy outlaw whenever gcd$(p, \sigma(2^m))=1$. In
particular, if $p>3$, then $(\sigma(2p)+1)/(2p)$ is
\textit{always} an abundancy outlaw. Our main result, however,
will address the more general situation where $r/s$ is a reduced
rational number of the form $(\sigma(N)+t)/N$. First we will need
a lemma.
\begin{lemma}
Let $N = \prod_{i=1}^n p_i^{k_i}$, where $p_i$ is a prime for all
$1 \leq i \leq n$. Then, for a given $1 \leq j \leq n$ and a
positive integer $t$,
\begin{displaymath}
p_j < \frac{1}{t}\sigma\left(\frac{N}{p_j^{k_j}}\right)
\end{displaymath}
if and only if
\begin{displaymath}
\frac{\sigma(N) + t}{N} < I(p_j N).
\end{displaymath}
\end{lemma}
\begin{proof}
Assume that
\begin{displaymath}
\frac{\sigma(N) + t}{N} < I(p_j N)
\end{displaymath}
for a given natural number $j \leq n$. This is equivalent to
\begin{displaymath}
p_j \sigma(N) + p_j t < \sigma(p_j N).
\end{displaymath}
Then, since $p_j \sigma(p_j^{k_j}) = \sigma(p_j^{k_j + 1}) - 1$,
we find that this inequality is equivalent to
\begin{displaymath}
\begin{array}{lll}
(\sigma(p_j^{k_j + 1}) - 1)\sigma\left(\frac{N}{p_j^{k_j}}\right)+
p_j t & < & \sigma(p_j N),
\end{array}
\end{displaymath}
or
\begin{displaymath}
\begin{array}{lll}
\sigma(p_j N) - \sigma\left(\frac{N}{p_j^{k_j}}\right) + p_j t & <
& \sigma(p_j N).
\end{array}
\end{displaymath}
Therefore,
\begin{displaymath}
p_j < \frac{1}{t}\sigma\left(\frac{N}{p_j^{k_j}}\right).
\end{displaymath}
\end{proof}
We are now ready to present the main result.
\begin{theorem}
For a positive integer $t$, let $\frac{\sigma(N) + t}{N}$ be a
fraction in lowest terms, and let $N = \prod_{i=1}^n p_i^{k_i}$
for primes $p_1,p_2,...,p_n$. If there exists a positive integer
$j \leq n$ such that $p_j < \frac{1}{t} \sigma(N/p_j^{k_j})$ and
$\sigma(p_j^{k_j})$ has a divisor $D>1$ such that at least one of
the following is true:
\begin{enumerate}
\item $I(p_j^{k_j}) I(D) > \frac{\sigma(N)
+ t}{N}$ and $gcd(D,t) = 1$
\item $gcd(D,Nt) = 1$
\end{enumerate}
\noindent then $\frac{\sigma(N) + t}{N}$ is an abundancy outlaw.
\end{theorem}
\begin{proof}
Case 1: Let $N$ and $t$ be natural numbers such that
$\frac{\sigma(N) + t}{N}$ is a fraction in lowest terms, and let
$j \leq n$ be a natural number satisfying $p_j < \frac{1}{t}
\sigma(N/p_j^{k_j})$ and $D$ a divisor of $\sigma(p_j^{k_j})$
satisfying hypothesis (1). Suppose further that $I(x) =
\frac{\sigma(N)+t}{N}$ for some natural number $x$. Since
$(\sigma(N) + t,N) = 1$, $N | x$ by Property 2.2. Say $x = d N$,
where $d$ is a positive integer. Since $p_j$ satisfies $p_j <
\frac{1}{t}\sigma(N/p_j^{k_j})$, Lemma 4.1 implies that $I(x) =
\frac{\sigma(N) +t}{N} < I(p_j N)$. Hence, $gcd(p_j^{k_j}, d
N/p_j^{k_j}) = 1$ and
\begin{displaymath}
I(x) = I(p_j^{k_j}) I(d N/p_j^{k_j}) = \frac{\sigma(N) + t}{N}.
\end{displaymath}
Equivalently,
\begin{displaymath}
\sigma(p_j^{k_j}) \sigma(d N/p_j^{k_j}) = (\sigma(N) + t) d.
\end{displaymath}
Since $\sigma(p_j^{k_j}) | \sigma(N)$ and $gcd(D,t) = 1$, the
divisor $D$ of $\sigma(p_j^{k_j})$ satisfying hypothesis (1) also
divides $d$. Hence $D$ divides $d N/p_j^{k_j}$, and by Property
2.1, $I(D) \leq I(d N/p_j^{k_j})$. Thus
\begin{displaymath}
I(p_j^{k_j}) I(D) \leq I(p_j^{k_j}) I(dN/p_j^{k_j}) = I(x).
\end{displaymath}
Given $I(x) = \frac{\sigma(N) + t}{N}$, we conclude then that
\begin{displaymath}
I(p_j^{k_j})I(D) \leq \frac{\sigma(N) + t}{N}.
\end{displaymath}
This contradicts the assumption that $I(p_j^{k_j}) I(D) >
\frac{\sigma(N) + t}{N}$. Therefore, $\frac{\sigma(N) + t}{N}$ is
an
abundancy outlaw.\\
Case 2: Now let $N=\prod_{i=1}^n p_i^{k_i}$ and $t$ be natural
numbers such that $\sigma(p_j)^{k_j}$ has a divisor $D$ satisfying
hypothesis (2) for some $1\leq j\leq n$, and assume that $I(x) =
\frac{\sigma(N) + t}{N}$ for some natural number $x$. Since
$\frac{\sigma(N) + t}{N}$ is in lowest terms, $N | x$, so that $x
= s N$ for some natural number $s$. By assumption, there exists a
natural number $j \leq n$ so that $p_j < \frac{1}{t}
\sigma(N/p_j^{k_j})$. By Lemma 4.1, $I(x) = \frac{\sigma(N) +
t}{N} < I(p_j N)$. Hence, $p_j \nmid s$, so that
\begin{displaymath}
\begin{array}{lll}
I(x) &=& I\left(p_j^{k_j} s \frac{N}{p_j^{k_j}}\right) \\
&=& I(p_j^{k_j}) I\left(s \frac{N}{p_j^{k_j}}\right).
\end{array}
\end{displaymath}
Next, we factor out the part of $s$, $\prod_{i=1}^n
p_i^{\gamma_i}$, that has divisors in common with $N/p_j^{k_j}$,
so that
\begin{displaymath}
\overline{s} = \frac{s}{\prod_{i=1}^{n} p_i^{\gamma_i}},
\end{displaymath}
where $\gamma_i$ is a non-negative integer for all $i \leq n$.\par
Then, we can rewrite $I(x)$ once more in the following form:
\begin{displaymath}
I(x) = I(p_j^{k_j}) I(\overline{s}) I\left(\frac{N}{p_j^{k_j}}
\prod_{i=1}^n p_i^{\gamma_i}\right)
\end{displaymath}
Because $I(x) = \frac{\sigma(N) + t}{N}$, we can see, then, that
\begin{displaymath}
\begin{array}{lll}
I(p_r^{k_r})I(\overline{s}) I\left(\frac{N}{p_r^{k_r}}
\prod_{i=1}^n p_i^{\gamma_i} \right) &=& \frac{\sigma(N) + t}{N},
\end{array}
\end{displaymath}
or equivalently,
\begin{displaymath}
\begin{array}{lll} \sigma(p_r^{k_r})
\sigma(\overline{s}) \sigma\left(\frac{N}{p_r^{k_r}} \prod_{i=1}^n
p_i^{\gamma_i} \right) &=& (\sigma(N) + t) \overline{s}
\prod_{i=1}^n p_i^{\gamma_i}.
\end{array}
\end{displaymath}
Now consider the divisor $D$ of $\sigma(p_j^{k_j})$ satisfying
hypothesis (2). Since $\sigma(p_j^{k_j}) | \sigma(N)$ and
$gcd(D,Nt) = 1$, $D$ divides $\overline{s}$ (so $\overline{s}>1)$.
This, then, means that $I(\overline{s}) \geq I(D)$. Then, since
$p_j < \frac{1}{t} \sigma(N/p_j^{k_j})$,
\begin{equation*}
p_j \sigma(p_j^{k_j}) < \frac{1}{t} \sigma(N),
\end{equation*}
which implies that the following is true:
\begin{equation*}
\frac{1}{p_j \sigma(p_j^{k_j})} > \frac{t}{\sigma(N)}.
\end{equation*}
Thus, since $D | \sigma(p_j^{k_j})$, $D < p_j \sigma(p_j^{k_j})$,
so that $1/D > 1/p_j \sigma(p_j^{k_j})$.
Hence, the following are true:
\begin{eqnarray*}
I(\overline{s}) &\geq & I(D) \\
&\geq & 1 + \frac{1}{D} \\
&>& 1 + \frac{1}{p_j \sigma(p_j^{k_j})} \\
&>& 1 + \frac{t}{\sigma(N)} \\
&=& \frac{\sigma(N) + t}{\sigma(N)} \\
&=& \frac{\sigma(N) + t}{I(p_j^{k_j}) I(N/p_j^{k_j})
N}\\
&\geq& \frac{\sigma(N) + t}{I(p_j^{k_j}) I((N/p_j^{k_j})\prod_{i=1}^n p_i^{\gamma_i}) N}.
\end{eqnarray*}
Hence
\begin{displaymath}
I(p_j^{k_j})I(\overline{s}) I\left(\frac{N}{p_j^{k_j}}
\prod_{i=1}^n p_i^{\gamma_i} \right) = I(x) > \frac{\sigma(N) +
t}{N},
\end{displaymath}
which contradicts our original assumption that
$I(x)=\frac{\sigma(N)+t}{N}$. Thus, $\frac{\sigma(N) + t}{N}$ is
an abundancy outlaw.
\end{proof}
\vspace{1cm}
If $t = 1$ we get the following corollary.
\begin{corollary}
Let $\frac{\sigma(N) + 1}{N}$ be a fraction in lowest terms, and
let $N = \prod_{i=1}^n p_i^{k_i}$ for primes $p_1,p_2,...,p_n$. If
there exists a natural number $j \leq n$ such that $p_j <
\sigma(N/p_j^{k_j})$ and $\sigma(p_j^{k_j})$ has a divisor $D$
such that at least one of the following is true:
\begin{enumerate}
\item $I(p_j^{k_j}) I(D) > \frac{\sigma(N)
+ 1}{N}$
\item $gcd(D,N) = 1$
\end{enumerate}
then $\frac{\sigma(N) + 1}{N}$ is an abundancy outlaw.
\end{corollary}
\section{Constructing Sequences of Abundancy Outlaws}
Using the results in the previous section, we can find and
construct sequences of abundancy outlaws. The following lemma
will be helpful.
\begin{lemma}
Let $N = \prod_{i=1}^n p_i^{k_i}$ for primes $p_1, p_2,...,p_n$.
Then $N$ is relatively prime to $\sigma(N) + 1$ if and only if
$p_i$ is relatively prime to $\sigma(N/p_i^{k_i}) + 1$ for all $1
\leq i \leq n$.
\end{lemma}
\begin{proof}
Since $\sigma(p_i^{k_i}) \equiv 1 \; (mod \; p_i)$,
\begin{displaymath}
\begin{array}{lll}
\sigma(N) + 1 &=& \sigma(p_i^{k_i})\sigma(N/p_i^{k_i}) + 1 \\
&\equiv& \sigma(N/p_i^{k_i}) + 1 \; (mod \; p_i).
\end{array}
\end{displaymath}
Thus, any prime divisor $p_i$ of $N$ and $\sigma(N) + 1$ must also
be a prime divisor of $\sigma(N/p_i^{k_i}) + 1$, and conversely,
if $p_i | (\sigma(N/p_i^{k_i}) + 1)$, then $p_i$ divides both $N$
and $\sigma(N) + 1$.
\end{proof}
\subsection*{Outlaws with even denominators}
\begin{corollary}
For all natural numbers $m$ and nonnegative integers $n$, and for
all odd primes $p$ such that $gcd(p,\sigma(2^m))=1$, the rational
number
\begin{equation*}
\frac{\sigma(2^m p^{2n + 1}) + 1}{2^m p^{2n + 1}}
\end{equation*}
is an abundancy outlaw.
\end{corollary}
\begin{proof}
To see that $(\sigma(2^m p^{2n+1})+1)/2^m p^{2n+1}$ is in lowest
terms, apply Lemma 5.1. Since $p$ and $2n+1$ are odd,
$\sigma(p^{2n+1}) + 1$ is odd, so $gcd(2^m, \sigma(p^{2n+1}) + 1)
= 1$. Also, $gcd(p^{2n+1}, \sigma(2^m) + 1) = 1$, because
$\sigma(2^m) + 1 = 2^{m+1}$. Next, we apply Corollary 4.3. Since
$2 < \sigma(p^{2n+1})$, we consider divisors $D$ of $\sigma(2^m)$.
Clearly, $gcd(\sigma(2^m), 2^m) = 1$, and because $p$ does not
divide $\sigma(2^m)$, $gcd(\sigma(2^m), 2^m p^{2n+1}) = 1$. Hence,
$D = \sigma(2^m)$ is the divisor required to apply Corollary 4.3.
Thus, $(\sigma(2^m p^{2n+1}))/2^m p^{2n+1}$
is an abundancy outlaw.\\
\end{proof}
\begin{corollary}
For all primes $p > 3$,
\begin{displaymath}
\frac{\sigma(2p) + 1}{2p}
\end{displaymath}
is an abundancy outlaw. If $p=2$ or $p=3$ then $\frac{\sigma(2p)
+ 1}{2p}$ is an abundancy index.
\end{corollary}
\begin{proof}
This result follows directly from Corollary 5.2. To prove that
$\frac{\sigma(2p) + 1}{2p}$ is an abundancy index when $p=2$ and
$p=3$, note that $I(6)=\frac{\sigma(2^2) + 1}{2^2}$ and
$I(18)=\frac{\sigma(6) + 1}{6}$.
\end{proof}
\begin{remark}
Corollary 5.3 actually has a very simple proof (and in fact, the
results presented in section 4 represent our attempt to push this
simple proof as far as possible). Clearly, $(\sigma(2p) + 1)/(2p)
= (3p + 4)/(2p)$ is in lowest terms. Thus, if $I(N) = (\sigma(2p)
+ 1)/2p$, $2p | N$. Because $p > 3$, it can be shown that $I(4p) >
(\sigma(2p) + 1)/2p$, so $4 \nmid N$. Therefore, $\sigma(2) |
\sigma(N)$, and since $\sigma(2) = 3$ does not divide $\sigma(2p)
+ 1$, $3$ divides $N$. Hence, $I(N) > I(6 p) > 2 > (\sigma(2p) +
1)/2p$. This is a contradiction, so $(\sigma(2p) + 1)/2p$ is an
abundancy outlaw, and we have captured the following sequence of
outlaws:
\begin{displaymath}
\frac{19}{10}, \frac{25}{14}, \frac{37}{22}, \frac{43}{26},
\frac{55}{34}, \frac{61}{38}, \frac{73}{46}, \frac{91}{58},
\frac{97}{62}, \frac{115}{74}, \frac{127}{82}, \frac{133}{86},
\frac{145}{94}, \frac{163}{106}, \frac{181}{118},
\frac{187}{122},...
\end{displaymath}
\end{remark}
Corollary 5.3 also captures another (potentially infinite) set of
outlaws having even denominators...
\begin{corollary}
If $N$ is an even perfect number,
\begin{equation*}
\frac{\sigma(2N) + 1}{2N}
\end{equation*}
is an abundancy outlaw.
\end{corollary}
\begin{proof}
Since $N$ is an even perfect number, $N = 2^{p-1} (2^p - 1)$,
where $p$ and $2^p - 1$ are both prime. Hence,
\begin{equation*}
2N = 2^p (2^p - 1).
\end{equation*}
Applying Corollary 5.2, we need only show that $gcd(2^p - 1,
\sigma(2^p)) = 1$. This follows from the fact that
\begin{equation*}
2(2^p - 1) = (2^{p+1} - 1) - 1,
\end{equation*}
which is clearly relatively prime to $\sigma(2^p) = 2^{p+1} - 1.$
Therefore, $(\sigma(2N) + 1)/2N$ is an abundancy outlaw.
\end{proof}
\subsection*{Outlaws with odd denominators}
\begin{corollary}
Let $M$ be an odd natural number, and let $p$, $\alpha$, and $t$
be odd natural numbers such that $p \nmid M$ and $p < \frac{1}{t}
\sigma(M)$. Then, if $(\sigma(p^\alpha M) + t)/p^\alpha M$ is in
lowest terms,
\begin{equation*}
\frac{\sigma(p^\alpha M) + t}{p^\alpha M}
\end{equation*}
is an abundancy outlaw.
\end{corollary}
\begin{proof}
Since $(\sigma(p^\alpha M) + t)/p^\alpha M$ is in lowest terms, we
apply Theorem 4.2 to show that it is an abundancy outlaw. Since
$p$ and $\alpha$ are both odd, $\sigma(p^{\alpha})$ is even.
Hence, $D = 2$ divides $\sigma(p^{\alpha})$, and since $p$, $M$,
and $t$ are all odd, $gcd(D, p M t) = 1$. Thus, $(\sigma(p^\alpha
M) + t)/p^\alpha M$ is an abundancy outlaw.
\end{proof}
\begin{corollary}
For primes $p$ and $q$, with $3 < q$, $p < q$, and $gcd(p,q+2) =
gcd(q,p+2) = 1$,
\begin{displaymath}
\frac{\sigma(pq) + 1}{pq}
\end{displaymath}
is an abundancy outlaw.
\end{corollary}
\begin{proof}
The case for $p = 2$ follows from Corollary 5.3. Now suppose that
$q > p > 2$. In this case, neither $p$ nor $q$ divides
$\sigma(pq) + 1 = pq + p + q + 2$, because $gcd(p,q+2) =
gcd(q,p+2) = 1$. Thus, $\frac{\sigma(pq) + 1}{pq}$ is in lowest
terms. The result now follows from Corollary 5.6.
\end{proof}
Corollary 5.7 produces outlaws with ease. To illustrate, let $p$
and $q$ be odd primes with $3 < p < q$, and assume $q \equiv 1 \;
(mod \; p)$. Then $p \nmid q + 2$ and $q \nmid p + 2$. Since
Dirichlet's theorem on arithmetic progressions of primes ensures
the existence of an infinite sequence of primes $q$ satisfying $q
\equiv 1 \; (mod \; p)$, Corollary 5.7 reveals an infinite class
of outlaws corresponding to each odd prime $p>3$. The sequences
corresponding to the primes 5, 7, and 11 follow.
\begin{description}
\item[$p=5$ : ] $\frac{73}{55}, \frac{193}{155}, \frac{253}{205}, \frac{373}{305}, \frac{433}{355}, \frac{613}{505}, \frac{793}{655},
\frac{913}{755}, \frac{1093}{905}, \frac{1153}{955}, \frac{1273}{1055}, \frac{1513}{1255},\frac{1633}{1355},\frac{1693}{1405},
\frac{1873}{1555},\frac{1993}{1655},\frac{2413}{2005}, \frac{2533}{2105},...$
\item[$p=7$ : ] $\frac{241}{203}, \frac{353}{301}, \frac{577}{497}, \frac{913}{791}, \frac{1025}{889}, \frac{1585}{1379}, \frac{1697}{1477}, \frac{1921}{1673}, \frac{2257}{1967}, \frac{2705}{2359}, \frac{3041}{2653},
\frac{3377}{2947}, \frac{3601}{3143}, \frac{3713}{3241}, \frac{3937}{3437},\frac{4385}{3829},\frac{4945}{4319},...$
\item[$p=11$ :] $\frac{289}{253}, \frac{817}{737}, \frac{1081}{979}, \frac{2401}{2189}, \frac{3985}{3641}, \frac{4249}{3883}, \frac{4777}{4367},
\frac{5041}{4609}, \frac{5569}{5093}, \frac{7417}{6787},\frac{7945}{7271},\frac{8209}{7513},\frac{8737}{7997},
\frac{10321}{9449},\frac{10585}{9691},\frac{11377}{10417},...$
\end{description}
\begin{remark}
It is interesting to note that if $p$ and $q = p+2$ are twin
primes then Corollary 5.7 does not apply, and in this case
$$\frac{\sigma(p(p+2))+1}{p(p+2)}=\frac{\sigma(p)+1}{p}=\frac{p+2}{p}.$$
Once again, we are faced with that elusive ratio $\frac{p+2}{p}$!
\end{remark}
\section{Further explorations}
R. Ryan considered the equation
\begin{equation}
I(x) = \frac{p+2}{p} = \frac{\sigma(p) + 1}{p},
\end{equation}
where $p$ is an odd prime \cite{Ryan2002}. Whether or not a
solution $x$ exists is still an open question. This problem is
interesting for two reasons. First, $\frac{p+2}{p}$ is just barely
out of reach of Weiner's 2000 result, since $p + 2 = \sigma(p) +
1$ for all primes $p$. Second, as we have already mentioned, if
$\frac{5}{3} = \frac{3+2}{3}$ is an abundancy index then there
exists an odd perfect number.
It seems to be just as difficult to find a solution to the above
equation as it is to show that no such $x$ exists. R. Ryan
reports that $I(x) = \frac{p+2}{p}$ has no solution for $x <
10^{16}$ \cite{Ryan2002}. The investigations that led us to
Theorems 3.2 and 4.2 were motivated in part by a desire to show
that equation 5.1 has no solutions. However, $\frac{p+2}{p}$ has
proven to be an elusive fraction. The techniques we have employed
in this paper cannot be applied to it. The difficulty lies in the
fact that
\begin{displaymath}
\frac{p+2}{p} > \frac{p}{p-1} > I(p^{\alpha})
\end{displaymath}
for all primes $p$ and natural numbers $\alpha$. Thus, one can
never ``trap" $\frac{p+2}{p}$ between two numbers:
\begin{displaymath}
I(p^{\alpha}N)<\frac{p+2}{p} < I(p^{\alpha+1}N).
\end{displaymath}
Hence, Theorems 3.2 and 4.2 fail to capture this potential outlaw.
In fact, proving that $\frac{p+2}{p}$ is an abundancy outlaw seems
to require the discovery of an entirely new category of abundancy outlaws.\\
There are also many interesting questions to consider relating to
the size of these sets of abundancy outlaws, in the sense of
asymptotic density. Based on some preliminary computer
experiments, the proportion of outlaws captured by Theorem 4.2
seems to approach $1.7$ percent. See the appendix for empirical
data. Our future work will involve a deeper exploration of the
sizes of the sets of outlaws, indices, and rationals of unknown
status.
\color{black}{
\section{Appendix}
The following is a table containing empirical data relating to the
asymptotic density of the set of outlaws captured by Theorem 3.4.
The second and third columns give the number of outlaws with
numerators less than or equal to $n$ captured by Theorem 3.4 and
Proposition 2.3, respectively. The fourth column gives the number
of rationals in the image of the abundancy index of the first one
million natural numbers with numerators less than or equal to $n$.
The fifth column gives the total number of rationals with
numerators less than or equal to $n$, and the sixth column gives
the value of column 2 divided by column 5. In other words, the
sixth column gives the proportion of outlaws captured by Theorem
3.4 in the set of rationals with numerator less than or equal to
$n$.
\begin{tabular}{|l|l|l|l|l|l|}
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
\hline
$n$ & Thm. 3.4 & Prop. 2.3 & Abundancy Index & Total rationals & Proportion \\
\hline
10 & 0 & 4 & 20 & 32 & 0 \\
100 & 44 & 720 & 553 & 3044 & 0.0145 \\
1000 & 5170 & 74927 & 7803 & 304192 & 0.01700 \\
10000 & 518193 & 750174 & 62064 & 30397486 & 0.01705 \\
\hline
\end{tabular}
\vspace{0.3cm} \textbf{Table 7.1}: A table of empirical data on
the asymptotic densities of the abundancy outlaws and the
abundancy indices.\\ \\
As a way to visualize the distribution of abundancy outlaws, we
include a list of the rationals with numerators less than or equal
to 100 (under the ordering described in the introduction). Each
rational number $q$ is colored according to its abundancy
index/outlaw status:}
\color{black}
\begin{itemize}
\color{black}
\item \color{blue} {Blue}: \color{black}$q$ is in the image of the abundancy index of the first one million natural numbers, or a natural number from $2$ to $11$ (the abundancy index of a known multiperfect number)
\item \color{green} {Green}: \color{black} $q$ is an outlaw captured by Property 2.3.
\item \color{red} {Red}: \color{black} $q$ is an outlaw captured by Theorem 4.2.
\item \color{black} {Black}: \color{black} $q$ is not in any of the other categories (the outlaw status of $q$ is ``unknown")
\end{itemize}
\noindent In the following list, $450$ rationals are blue, $720$ are green, $44$ are red, and $1961$ are black.\\ \\
\noindent \color{blue}{2}, \color{blue}{3}, \color{blue}{3/2},
\color{blue}{4}, \color{blue}{4/3}, \color{blue}{5},
\color{blue}{5/2}, \color{black}{5/3}, \color{green}{5/4},
\color{blue}{6}, \color{blue}{6/5}, \color{blue}{7},
\color{blue}{7/2}, \color{blue}{7/3}, \color{blue}{7/4},
\color{black}{7/5}, \color{green}{7/6}, \color{blue}{8},
\color{blue}{8/3}, \color{blue}{8/5}, \color{blue}{8/7},
\linebreak
\color{blue}{9}, \color{black}{9/2}, \color{blue}{9/4}, \color{blue}{9/5}, \color{black}{9/7}, \color{green}{9/8}, \color{blue}{10}, \color{blue}{10/3}, \color{black}{10/7}, \color{green}{10/9}, \color{blue}{11}, \color{black}{11/2}, \color{blue}{11/3}, \color{blue}{11/4}, \color{black}{11/5}, \color{green}{11/6}, \color{black}{11/7}, \linebreak
\color{green}{11/8}, \color{green}{11/9}, \color{green}{11/10}, \color{black}{12}, \color{blue}{12/5}, \color{blue}{12/7}, \color{blue}{12/11}, \color{black}{13}, \color{black}{13/2}, \color{black}{13/3}, \color{blue}{13/4}, \color{blue}{13/5}, \color{blue}{13/6}, \color{black}{13/7}, \color{green}{13/8}, \linebreak
\color{blue}{13/9}, \color{green}{13/10}, \color{black}{13/11}, \color{green}{13/12}, \color{black}{14}, \color{black}{14/3}, \color{blue}{14/5}, \color{blue}{14/9}, \color{black}{14/11}, \color{blue}{14/13}, \color{black}{15}, \color{black}{15/2}, \color{blue}{15/4}, \color{blue}{15/7}, \color{blue}{15/8}, \linebreak
\color{black}{15/11}, \color{black}{15/13}, \color{green}{15/14},
\color{black}{16}, \color{black}{16/3}, \color{blue}{16/5},
\color{blue}{16/7}, \color{blue}{16/9}, \color{blue}{16/11},
\color{blue}{16/13}, \color{green}{16/15}, \color{black}{17},
\color{black}{17/2}, \color{black}{17/3}, \color{blue}{17/4},
\linebreak
\color{blue}{17/5}, \color{black}{17/6}, \color{black}{17/7}, \color{black}{17/8}, \color{black}{17/9}, \color{green}{17/10}, \color{black}{17/11}, \color{green}{17/12}, \color{black}{17/13}, \color{green}{17/14}, \color{green}{17/15}, \color{green}{17/16}, \color{black}{18}, \color{blue}{18/5}, \linebreak
\color{blue}{18/7}, \color{blue}{18/11}, \color{black}{18/13},
\color{blue}{18/17}, \color{black}{19}, \color{black}{19/2},
\color{black}{19/3}, \color{black}{19/4}, \color{blue}{19/5},
\color{blue}{19/6}, \color{blue}{19/7}, \color{black}{19/8},
\color{black}{19/9}, \color{red}{19/10}, \color{black}{19/11},
\linebreak \color{green}{19/12}, \color{black}{19/13},
\color{green}{19/14}, \color{green}{19/15}, \color{green}{19/16},
\color{black}{19/17}, \color{green}{19/18}, \color{black}{20},
\color{black}{20/3}, \color{blue}{20/7}, \color{blue}{20/9},
\color{black}{20/11}, \color{black}{20/13}, \color{black}{20/17},
\linebreak \color{blue}{20/19}, \color{black}{21},
\color{black}{21/2}, \color{black}{21/4}, \color{blue}{21/5},
\color{blue}{21/8}, \color{blue}{21/10}, \color{blue}{21/11},
\color{blue}{21/13}, \color{green}{21/16}, \color{black}{21/17},
\color{black}{21/19}, \color{green}{21/20}, \color{black}{22},
\linebreak
\color{black}{22/3}, \color{blue}{22/5}, \color{blue}{22/7}, \color{blue}{22/9}, \color{black}{22/13}, \color{green}{22/15}, \color{black}{22/17}, \color{black}{22/19}, \color{green}{22/21}, \color{black}{23}, \color{black}{23/2}, \color{black}{23/3}, \color{black}{23/4}, \color{black}{23/5}, \linebreak
\color{black}{23/6}, \color{black}{23/7}, \color{black}{23/8}, \color{black}{23/9}, \color{black}{23/10}, \color{black}{23/11}, \color{green}{23/12}, \color{black}{23/13}, \color{green}{23/14}, \color{green}{23/15}, \color{green}{23/16}, \color{black}{23/17}, \color{green}{23/18}, \linebreak
\color{black}{23/19}, \color{green}{23/20}, \color{green}{23/21},
\color{green}{23/22}, \color{black}{24}, \color{black}{24/5},
\color{blue}{24/7}, \color{blue}{24/11}, \color{blue}{24/13},
\color{blue}{24/17}, \color{blue}{24/19}, \color{blue}{24/23},
\color{black}{25}, \color{black}{25/2}, \linebreak
\color{black}{25/3}, \color{black}{25/4}, \color{black}{25/6}, \color{blue}{25/7}, \color{blue}{25/8}, \color{blue}{25/9}, \color{black}{25/11}, \color{green}{25/12}, \color{black}{25/13}, \color{red}{25/14}, \color{green}{25/16}, \color{black}{25/17}, \color{green}{25/18}, \color{black}{25/19}, \linebreak
\color{green}{25/21}, \color{green}{25/22}, \color{black}{25/23}, \color{green}{25/24}, \color{black}{26}, \color{black}{26/3}, \color{black}{26/5}, \color{blue}{26/7}, \color{blue}{26/9}, \color{blue}{26/11}, \color{blue}{26/15}, \color{blue}{26/17}, \color{black}{26/19}, \color{green}{26/21}, \linebreak
\color{black}{26/23}, \color{green}{26/25}, \color{black}{27}, \color{black}{27/2}, \color{black}{27/4}, \color{black}{27/5}, \color{blue}{27/7}, \color{blue}{27/8}, \color{blue}{27/10}, \color{blue}{27/11}, \color{black}{27/13}, \color{black}{27/14}, \color{green}{27/16}, \color{blue}{27/17}, \linebreak
\color{black}{27/19}, \color{green}{27/20}, \color{green}{27/22}, \color{black}{27/23}, \color{green}{27/25}, \color{green}{27/26}, \color{black}{28}, \color{black}{28/3}, \color{black}{28/5}, \color{blue}{28/9}, \color{blue}{28/11}, \color{blue}{28/13}, \color{blue}{28/15}, \color{blue}{28/17}, \linebreak
\color{black}{28/19}, \color{black}{28/23}, \color{green}{28/25},
\color{green}{28/27}, \color{black}{29}, \color{black}{29/2},
\color{black}{29/3}, \color{black}{29/4}, \color{black}{29/5},
\color{black}{29/6}, \color{black}{29/7}, \color{black}{29/8},
\color{black}{29/9}, \color{black}{29/10}, \linebreak
\color{black}{29/11}, \color{red}{29/12}, \color{black}{29/13}, \color{black}{29/14}, \color{black}{29/15}, \color{green}{29/16}, \color{black}{29/17}, \color{green}{29/18}, \color{black}{29/19}, \color{green}{29/20}, \color{green}{29/21}, \color{green}{29/22}, \color{black}{29/23}, \linebreak
\color{green}{29/24}, \color{green}{29/25}, \color{green}{29/26}, \color{green}{29/27}, \color{green}{29/28}, \color{black}{30}, \color{black}{30/7}, \color{blue}{30/11}, \color{blue}{30/13}, \color{black}{30/17}, \color{blue}{30/19}, \color{black}{30/23}, \color{blue}{30/29}, \color{black}{31}, \linebreak
\color{black}{31/2}, \color{black}{31/3}, \color{black}{31/4},
\color{black}{31/5}, \color{black}{31/6}, \color{black}{31/7},
\color{blue}{31/8}, \color{blue}{31/9}, \color{blue}{31/10},
\color{blue}{31/11}, \color{blue}{31/12}, \color{blue}{31/13},
\color{blue}{31/14}, \color{black}{31/15}, \linebreak
\color{blue}{31/16}, \color{black}{31/17}, \color{green}{31/18}, \color{black}{31/19}, \color{green}{31/20}, \color{green}{31/21}, \color{green}{31/22}, \color{black}{31/23}, \color{green}{31/24}, \color{blue}{31/25}, \color{green}{31/26}, \color{green}{31/27}, \color{green}{31/28}, \linebreak
\color{black}{31/29}, \color{green}{31/30}, \color{black}{32}, \color{black}{32/3}, \color{black}{32/5}, \color{black}{32/7}, \color{blue}{32/9}, \color{blue}{32/11}, \color{blue}{32/13}, \color{blue}{32/15}, \color{blue}{32/17}, \color{blue}{32/19}, \color{blue}{32/21}, \color{blue}{32/23}, \linebreak
\color{blue}{32/25}, \color{green}{32/27}, \color{black}{32/29}, \color{blue}{32/31}, \color{black}{33}, \color{black}{33/2}, \color{black}{33/4}, \color{black}{33/5}, \color{black}{33/7}, \color{black}{33/8}, \color{blue}{33/10}, \color{black}{33/13}, \color{blue}{33/14}, \color{blue}{33/16}, \linebreak
\color{black}{33/17}, \color{black}{33/19}, \color{green}{33/20},
\color{black}{33/23}, \color{black}{33/25}, \color{green}{33/26},
\color{green}{33/28}, \color{black}{33/29}, \color{black}{33/31},
\color{green}{33/32}, \color{black}{34}, \color{black}{34/3},
\color{black}{34/5}, \color{black}{34/7}, \linebreak
\color{black}{34/9}, \color{black}{34/11}, \color{black}{34/13},
\color{black}{34/15}, \color{black}{34/19}, \color{black}{34/21},
\color{black}{34/23}, \color{black}{34/25}, \color{green}{34/27},
\color{black}{34/29}, \color{black}{34/31}, \color{green}{34/33},
\color{black}{35}, \color{black}{35/2}, \linebreak
\color{black}{35/3}, \color{black}{35/4}, \color{black}{35/6},
\color{black}{35/8}, \color{black}{35/9}, \color{blue}{35/11},
\color{blue}{35/12}, \color{blue}{35/13}, \color{black}{35/16},
\color{black}{35/17}, \color{green}{35/18}, \color{blue}{35/19},
\color{green}{35/22}, \color{black}{35/23}, \linebreak
\color{green}{35/24}, \color{green}{35/26}, \color{green}{35/27},
\color{black}{35/29}, \color{black}{35/31}, \color{green}{35/32},
\color{green}{35/33}, \color{green}{35/34}, \color{black}{36},
\color{black}{36/5}, \color{black}{36/7}, \color{blue}{36/11},
\color{blue}{36/13}, \color{blue}{36/17}, \linebreak
\color{blue}{36/19}, \color{blue}{36/23}, \color{black}{36/25},
\color{blue}{36/29}, \color{black}{36/31}, \color{green}{36/35},
\color{black}{37}, \color{black}{37/2}, \color{black}{37/3},
\color{black}{37/4}, \color{black}{37/5}, \color{black}{37/6},
\color{black}{37/7}, \color{black}{37/8}, \color{black}{37/9},
\linebreak \color{black}{37/10}, \color{black}{37/11},
\color{blue}{37/12}, \color{black}{37/13}, \color{black}{37/14},
\color{black}{37/15}, \color{blue}{37/16}, \color{black}{37/17},
\color{green}{37/18}, \color{black}{37/19}, \color{green}{37/20},
\color{black}{37/21}, \color{red}{37/22}, \linebreak
\color{black}{37/23}, \color{green}{37/24}, \color{black}{37/25},
\color{green}{37/26}, \color{green}{37/27}, \color{green}{37/28},
\color{black}{37/29}, \color{green}{37/30}, \color{black}{37/31},
\color{green}{37/32}, \color{green}{37/33}, \color{green}{37/34},
\color{green}{37/35}, \linebreak \color{green}{37/36},
\color{black}{38}, \color{black}{38/3}, \color{black}{38/5},
\color{black}{38/7}, \color{black}{38/9}, \color{blue}{38/11},
\color{blue}{38/13}, \color{black}{38/15}, \color{black}{38/17},
\color{blue}{38/21}, \color{black}{38/23}, \color{black}{38/25},
\color{green}{38/27}, \linebreak
\color{black}{38/29}, \color{black}{38/31}, \color{green}{38/33}, \color{green}{38/35}, \color{blue}{38/37}, \color{black}{39}, \color{black}{39/2}, \color{black}{39/4}, \color{black}{39/5}, \color{black}{39/7}, \color{black}{39/8}, \color{blue}{39/10}, \color{blue}{39/11}, \color{black}{39/14}, \linebreak
\color{black}{39/16}, \color{blue}{39/17}, \color{black}{39/19}, \color{green}{39/20}, \color{black}{39/22}, \color{black}{39/23}, \color{black}{39/25}, \color{green}{39/28}, \color{black}{39/29}, \color{black}{39/31}, \color{green}{39/32}, \color{green}{39/34}, \color{green}{39/35}, \linebreak
\color{black}{39/37}, \color{green}{39/38}, \color{black}{40}, \color{black}{40/3}, \color{black}{40/7}, \color{black}{40/9}, \color{blue}{40/11}, \color{blue}{40/13}, \color{blue}{40/17}, \color{blue}{40/19}, \color{blue}{40/21}, \color{black}{40/23}, \color{blue}{40/27}, \color{blue}{40/29}, \linebreak
\color{black}{40/31}, \color{green}{40/33}, \color{blue}{40/37}, \color{green}{40/39}, \color{black}{41}, \color{black}{41/2}, \color{black}{41/3}, \color{black}{41/4}, \color{black}{41/5}, \color{black}{41/6}, \color{black}{41/7}, \color{black}{41/8}, \color{black}{41/9}, \color{black}{41/10}, \color{black}{41/11}, \linebreak
\color{black}{41/12}, \color{black}{41/13}, \color{black}{41/14}, \color{black}{41/15}, \color{black}{41/16}, \color{black}{41/17}, \color{black}{41/18}, \color{black}{41/19}, \color{green}{41/20}, \color{black}{41/21}, \color{red}{41/22}, \color{black}{41/23}, \color{green}{41/24}, \linebreak
\color{black}{41/25}, \color{green}{41/26}, \color{black}{41/27}, \color{green}{41/28}, \color{black}{41/29}, \color{green}{41/30}, \color{black}{41/31}, \color{green}{41/32}, \color{green}{41/33}, \color{green}{41/34}, \color{green}{41/35}, \color{green}{41/36}, \color{black}{41/37}, \linebreak
\color{green}{41/38}, \color{green}{41/39}, \color{green}{41/40}, \color{black}{42}, \color{black}{42/5}, \color{blue}{42/11}, \color{blue}{42/13}, \color{blue}{42/17}, \color{blue}{42/19}, \color{blue}{42/23}, \color{black}{42/25}, \color{black}{42/29}, \color{black}{42/31}, \linebreak
\color{black}{42/37}, \color{blue}{42/41}, \color{black}{43}, \color{black}{43/2}, \color{black}{43/3}, \color{black}{43/4}, \color{black}{43/5}, \color{black}{43/6}, \color{black}{43/7}, \color{black}{43/8}, \color{black}{43/9}, \color{black}{43/10}, \color{black}{43/11}, \color{black}{43/12}, \linebreak
\color{black}{43/13}, \color{black}{43/14}, \color{black}{43/15}, \color{black}{43/16}, \color{black}{43/17}, \color{black}{43/18}, \color{black}{43/19}, \color{red}{43/20}, \color{black}{43/21}, \color{black}{43/22}, \color{black}{43/23}, \color{green}{43/24}, \color{black}{43/25}, \linebreak
\color{red}{43/26}, \color{black}{43/27}, \color{green}{43/28}, \color{black}{43/29}, \color{green}{43/30}, \color{black}{43/31}, \color{green}{43/32}, \color{green}{43/33}, \color{green}{43/34}, \color{green}{43/35}, \color{green}{43/36}, \color{black}{43/37}, \color{green}{43/38}, \linebreak
\color{green}{43/39}, \color{green}{43/40}, \color{black}{43/41}, \color{green}{43/42}, \color{black}{44}, \color{black}{44/3}, \color{black}{44/5}, \color{black}{44/7}, \color{black}{44/9}, \color{black}{44/13}, \color{blue}{44/15}, \color{blue}{44/17}, \color{black}{44/19}, \color{black}{44/21}, \linebreak
\color{black}{44/23}, \color{black}{44/25}, \color{blue}{44/27}, \color{black}{44/29}, \color{black}{44/31}, \color{green}{44/35}, \color{black}{44/37}, \color{green}{44/39}, \color{black}{44/41}, \color{blue}{44/43}, \color{black}{45}, \color{black}{45/2}, \color{black}{45/4}, \color{black}{45/7}, \linebreak
\color{black}{45/8}, \color{black}{45/11}, \color{blue}{45/13}, \color{blue}{45/14}, \color{blue}{45/16}, \color{blue}{45/17}, \color{blue}{45/19}, \color{blue}{45/22}, \color{blue}{45/23}, \color{black}{45/26}, \color{green}{45/28}, \color{blue}{45/29}, \color{black}{45/31}, \linebreak
\color{green}{45/32}, \color{green}{45/34}, \color{black}{45/37}, \color{green}{45/38}, \color{black}{45/41}, \color{black}{45/43}, \color{green}{45/44}, \color{black}{46}, \color{black}{46/3}, \color{black}{46/5}, \color{black}{46/7}, \color{black}{46/9}, \color{black}{46/11}, \color{black}{46/13}, \linebreak
\color{black}{46/15}, \color{black}{46/17}, \color{black}{46/19}, \color{black}{46/21}, \color{black}{46/25}, \color{black}{46/27}, \color{black}{46/29}, \color{black}{46/31}, \color{green}{46/33}, \color{green}{46/35}, \color{black}{46/37}, \color{green}{46/39}, \color{black}{46/41}, \linebreak
\color{black}{46/43}, \color{green}{46/45}, \color{black}{47}, \color{black}{47/2}, \color{black}{47/3}, \color{black}{47/4}, \color{black}{47/5}, \color{black}{47/6}, \color{black}{47/7}, \color{black}{47/8}, \color{black}{47/9}, \color{black}{47/10}, \color{black}{47/11}, \color{black}{47/12}, \color{black}{47/13}, \linebreak
\color{black}{47/14}, \color{black}{47/15}, \color{black}{47/16}, \color{black}{47/17}, \color{black}{47/18}, \color{black}{47/19}, \color{black}{47/20}, \color{black}{47/21}, \color{black}{47/22}, \color{black}{47/23}, \color{green}{47/24}, \color{black}{47/25}, \color{red}{47/26}, \linebreak
\color{black}{47/27}, \color{green}{47/28}, \color{black}{47/29}, \color{green}{47/30}, \color{black}{47/31}, \color{green}{47/32}, \color{green}{47/33}, \color{green}{47/34}, \color{green}{47/35}, \color{green}{47/36}, \color{black}{47/37}, \color{green}{47/38}, \color{green}{47/39}, \linebreak
\color{green}{47/40}, \color{black}{47/41}, \color{green}{47/42}, \color{black}{47/43}, \color{green}{47/44}, \color{green}{47/45}, \color{green}{47/46}, \color{black}{48}, \color{black}{48/5}, \color{black}{48/7}, \color{blue}{48/11}, \color{blue}{48/13}, \color{blue}{48/17}, \color{blue}{48/19}, \linebreak
\color{blue}{48/23}, \color{blue}{48/25}, \color{blue}{48/29},
\color{blue}{48/31}, \color{blue}{48/35}, \color{blue}{48/37},
\color{blue}{48/41}, \color{blue}{48/43}, \color{blue}{48/47},
\color{black}{49}, \color{black}{49/2}, \color{black}{49/3},
\color{black}{49/4}, \color{black}{49/5}, \linebreak
\color{black}{49/6}, \color{black}{49/8}, \color{black}{49/9}, \color{black}{49/10}, \color{black}{49/11}, \color{black}{49/12}, \color{blue}{49/13}, \color{blue}{49/15}, \color{blue}{49/16}, \color{blue}{49/17}, \color{blue}{49/18}, \color{black}{49/19}, \color{black}{49/20}, \linebreak
\color{black}{49/22}, \color{black}{49/23}, \color{green}{49/24}, \color{black}{49/25}, \color{blue}{49/26}, \color{black}{49/27}, \color{black}{49/29}, \color{green}{49/30}, \color{black}{49/31}, \color{green}{49/32}, \color{red}{49/33}, \color{green}{49/34}, \color{green}{49/36}, \linebreak
\color{black}{49/37}, \color{green}{49/38}, \color{green}{49/39}, \color{green}{49/40}, \color{black}{49/41}, \color{black}{49/43}, \color{green}{49/44}, \color{green}{49/45}, \color{green}{49/46}, \color{black}{49/47}, \color{green}{49/48}, \color{black}{50}, \color{black}{50/3}, \color{black}{50/7}, \linebreak
\color{black}{50/9}, \color{black}{50/11}, \color{black}{50/13}, \color{blue}{50/17}, \color{blue}{50/19}, \color{black}{50/21}, \color{black}{50/23}, \color{black}{50/27}, \color{black}{50/29}, \color{black}{50/31}, \color{black}{50/33}, \color{black}{50/37}, \color{green}{50/39}, \linebreak
\color{black}{50/41}, \color{black}{50/43}, \color{black}{50/47}, \color{green}{50/49}, \color{black}{51}, \color{black}{51/2}, \color{black}{51/4}, \color{black}{51/5}, \color{black}{51/7}, \color{black}{51/8}, \color{black}{51/10}, \color{black}{51/11}, \color{blue}{51/13}, \color{blue}{51/14}, \linebreak
\color{blue}{51/16}, \color{black}{51/19}, \color{blue}{51/20}, \color{black}{51/22}, \color{black}{51/23}, \color{black}{51/25}, \color{black}{51/26}, \color{green}{51/28}, \color{black}{51/29}, \color{black}{51/31}, \color{green}{51/32}, \color{black}{51/35}, \color{black}{51/37}, \linebreak
\color{green}{51/38}, \color{green}{51/40}, \color{black}{51/41}, \color{black}{51/43}, \color{green}{51/44}, \color{green}{51/46}, \color{black}{51/47}, \color{green}{51/49}, \color{green}{51/50}, \color{black}{52}, \color{black}{52/3}, \color{black}{52/5}, \color{black}{52/7}, \color{black}{52/9}, \linebreak
\color{black}{52/11}, \color{blue}{52/15}, \color{blue}{52/17},
\color{blue}{52/19}, \color{blue}{52/21}, \color{blue}{52/23},
\color{black}{52/25}, \color{black}{52/27}, \color{blue}{52/29},
\color{black}{52/31}, \color{blue}{52/33}, \color{black}{52/35},
\color{black}{52/37}, \linebreak \color{black}{52/41},
\color{black}{52/43}, \color{green}{52/45}, \color{black}{52/47},
\color{green}{52/49}, \color{green}{52/51}, \color{black}{53},
\color{black}{53/2}, \color{black}{53/3}, \color{black}{53/4},
\color{black}{53/5}, \color{black}{53/6}, \color{black}{53/7},
\color{black}{53/8}, \linebreak
\color{black}{53/9}, \color{black}{53/10}, \color{black}{53/11}, \color{black}{53/12}, \color{black}{53/13}, \color{black}{53/14}, \color{black}{53/15}, \color{black}{53/16}, \color{black}{53/17}, \color{black}{53/18}, \color{black}{53/19}, \color{black}{53/20}, \color{black}{53/21}, \linebreak
\color{black}{53/22}, \color{black}{53/23}, \color{green}{53/24},
\color{black}{53/25}, \color{black}{53/26}, \color{black}{53/27},
\color{green}{53/28}, \color{black}{53/29}, \color{green}{53/30},
\color{black}{53/31}, \color{green}{53/32}, \color{black}{53/33},
\color{green}{53/34}, \linebreak
\color{black}{53/35}, \color{green}{53/36}, \color{black}{53/37}, \color{green}{53/38}, \color{green}{53/39}, \color{green}{53/40}, \color{black}{53/41}, \color{green}{53/42}, \color{black}{53/43}, \color{green}{53/44}, \color{green}{53/45}, \color{green}{53/46}, \color{black}{53/47}, \linebreak
\color{green}{53/48}, \color{green}{53/49}, \color{green}{53/50}, \color{green}{53/51}, \color{green}{53/52}, \color{black}{54}, \color{black}{54/5}, \color{black}{54/7}, \color{black}{54/11}, \color{black}{54/13}, \color{blue}{54/17}, \color{blue}{54/19}, \color{blue}{54/23}, \color{black}{54/25}, \linebreak
\color{blue}{54/29}, \color{black}{54/31}, \color{black}{54/35}, \color{black}{54/37}, \color{black}{54/41}, \color{black}{54/43}, \color{black}{54/47}, \color{green}{54/49}, \color{blue}{54/53}, \color{black}{55}, \color{black}{55/2}, \color{black}{55/3}, \color{black}{55/4}, \color{black}{55/6}, \linebreak
\color{black}{55/7}, \color{black}{55/8}, \color{black}{55/9}, \color{black}{55/12}, \color{black}{55/13}, \color{black}{55/14}, \color{black}{55/16}, \color{blue}{55/17}, \color{blue}{55/18}, \color{blue}{55/19}, \color{black}{55/21}, \color{black}{55/23}, \color{green}{55/24}, \linebreak
\color{black}{55/26}, \color{black}{55/27}, \color{green}{55/28},
\color{black}{55/29}, \color{black}{55/31}, \color{green}{55/32},
\color{red}{55/34}, \color{green}{55/36}, \color{black}{55/37},
\color{green}{55/38}, \color{green}{55/39}, \color{black}{55/41},
\color{green}{55/42}, \linebreak \color{black}{55/43},
\color{green}{55/46}, \color{black}{55/47}, \color{green}{55/48},
\color{green}{55/49}, \color{green}{55/51}, \color{green}{55/52},
\color{black}{55/53}, \color{green}{55/54}, \color{black}{56},
\color{black}{56/3}, \color{black}{56/5}, \color{black}{56/9},
\color{black}{56/11}, \linebreak
\color{blue}{56/13}, \color{blue}{56/15}, \color{blue}{56/17}, \color{blue}{56/19}, \color{blue}{56/23}, \color{blue}{56/25}, \color{blue}{56/27}, \color{blue}{56/29}, \color{blue}{56/31}, \color{blue}{56/33}, \color{black}{56/37}, \color{blue}{56/39}, \color{blue}{56/41}, \linebreak
\color{black}{56/43}, \color{green}{56/45}, \color{black}{56/47}, \color{green}{56/51}, \color{black}{56/53}, \color{green}{56/55}, \color{black}{57}, \color{black}{57/2}, \color{black}{57/4}, \color{black}{57/5}, \color{black}{57/7}, \color{black}{57/8}, \color{black}{57/10}, \color{black}{57/11}, \linebreak
\color{black}{57/13}, \color{blue}{57/14}, \color{blue}{57/16}, \color{blue}{57/17}, \color{black}{57/20}, \color{black}{57/22}, \color{black}{57/23}, \color{black}{57/25}, \color{blue}{57/26}, \color{blue}{57/28}, \color{black}{57/29}, \color{black}{57/31}, \color{green}{57/32}, \linebreak
\color{black}{57/34}, \color{black}{57/35}, \color{blue}{57/37}, \color{green}{57/40}, \color{black}{57/41}, \color{black}{57/43}, \color{green}{57/44}, \color{green}{57/46}, \color{black}{57/47}, \color{blue}{57/49}, \color{green}{57/50}, \color{green}{57/52}, \color{black}{57/53}, \linebreak
\color{green}{57/55}, \color{green}{57/56}, \color{black}{58}, \color{black}{58/3}, \color{black}{58/5}, \color{black}{58/7}, \color{black}{58/9}, \color{black}{58/11}, \color{black}{58/13}, \color{black}{58/15}, \color{black}{58/17}, \color{black}{58/19}, \color{black}{58/21}, \color{black}{58/23}, \linebreak
\color{black}{58/25}, \color{black}{58/27}, \color{black}{58/31}, \color{black}{58/33}, \color{black}{58/35}, \color{black}{58/37}, \color{black}{58/39}, \color{black}{58/41}, \color{black}{58/43}, \color{green}{58/45}, \color{black}{58/47}, \color{black}{58/49}, \color{green}{58/51}, \linebreak
\color{black}{58/53}, \color{green}{58/55}, \color{green}{58/57}, \color{black}{59}, \color{black}{59/2}, \color{black}{59/3}, \color{black}{59/4}, \color{black}{59/5}, \color{black}{59/6}, \color{black}{59/7}, \color{black}{59/8}, \color{black}{59/9}, \color{black}{59/10}, \color{black}{59/11}, \color{black}{59/12}, \linebreak
\color{black}{59/13}, \color{black}{59/14}, \color{black}{59/15}, \color{black}{59/16}, \color{black}{59/17}, \color{black}{59/18}, \color{black}{59/19}, \color{black}{59/20}, \color{black}{59/21}, \color{black}{59/22}, \color{black}{59/23}, \color{green}{59/24}, \color{black}{59/25}, \linebreak
\color{black}{59/26}, \color{black}{59/27}, \color{black}{59/28}, \color{black}{59/29}, \color{green}{59/30}, \color{black}{59/31}, \color{green}{59/32}, \color{black}{59/33}, \color{red}{59/34}, \color{black}{59/35}, \color{green}{59/36}, \color{black}{59/37}, \color{green}{59/38}, \linebreak
\color{red}{59/39}, \color{green}{59/40}, \color{black}{59/41}, \color{green}{59/42}, \color{black}{59/43}, \color{green}{59/44}, \color{green}{59/45}, \color{green}{59/46}, \color{black}{59/47}, \color{green}{59/48}, \color{black}{59/49}, \color{green}{59/50}, \color{green}{59/51}, \linebreak
\color{green}{59/52}, \color{black}{59/53}, \color{green}{59/54},
\color{green}{59/55}, \color{green}{59/56}, \color{green}{59/57},
\color{green}{59/58}, \color{black}{60}, \color{black}{60/7},
\color{black}{60/11}, \color{black}{60/13}, \color{blue}{60/17},
\color{blue}{60/19}, \linebreak
\color{blue}{60/23}, \color{blue}{60/29}, \color{blue}{60/31}, \color{blue}{60/37}, \color{black}{60/41}, \color{black}{60/43}, \color{black}{60/47}, \color{blue}{60/49}, \color{black}{60/53}, \color{blue}{60/59}, \color{black}{61}, \color{black}{61/2}, \color{black}{61/3}, \color{black}{61/4}, \linebreak
\color{black}{61/5}, \color{black}{61/6}, \color{black}{61/7}, \color{black}{61/8}, \color{black}{61/9}, \color{black}{61/10}, \color{black}{61/11}, \color{black}{61/12}, \color{black}{61/13}, \color{black}{61/14}, \color{black}{61/15}, \color{black}{61/16}, \color{black}{61/17}, \color{black}{61/18}, \linebreak
\color{black}{61/19}, \color{black}{61/20}, \color{black}{61/21}, \color{black}{61/22}, \color{black}{61/23}, \color{red}{61/24}, \color{black}{61/25}, \color{blue}{61/26}, \color{black}{61/27}, \color{black}{61/28}, \color{black}{61/29}, \color{green}{61/30}, \color{black}{61/31}, \linebreak
\color{green}{61/32}, \color{black}{61/33}, \color{red}{61/34}, \color{black}{61/35}, \color{green}{61/36}, \color{black}{61/37}, \color{red}{61/38}, \color{blue}{61/39}, \color{green}{61/40}, \color{black}{61/41}, \color{green}{61/42}, \color{black}{61/43}, \color{green}{61/44}, \linebreak
\color{green}{61/45}, \color{green}{61/46}, \color{black}{61/47}, \color{green}{61/48}, \color{black}{61/49}, \color{green}{61/50}, \color{green}{61/51}, \color{green}{61/52}, \color{black}{61/53}, \color{green}{61/54}, \color{green}{61/55}, \color{green}{61/56}, \color{green}{61/57}, \linebreak
\color{green}{61/58}, \color{black}{61/59}, \color{green}{61/60},
\color{black}{62}, \color{black}{62/3}, \color{black}{62/5},
\color{black}{62/7}, \color{black}{62/9}, \color{black}{62/11},
\color{black}{62/13}, \color{blue}{62/15}, \color{blue}{62/17},
\color{blue}{62/19}, \color{blue}{62/21}, \linebreak
\color{blue}{62/23}, \color{blue}{62/25}, \color{black}{62/27}, \color{black}{62/29}, \color{black}{62/33}, \color{black}{62/35}, \color{black}{62/37}, \color{blue}{62/39}, \color{black}{62/41}, \color{black}{62/43}, \color{green}{62/45}, \color{black}{62/47}, \color{black}{62/49}, \linebreak
\color{green}{62/51}, \color{black}{62/53}, \color{green}{62/55}, \color{green}{62/57}, \color{black}{62/59}, \color{blue}{62/61}, \color{black}{63}, \color{black}{63/2}, \color{black}{63/4}, \color{black}{63/5}, \color{black}{63/8}, \color{black}{63/10}, \color{black}{63/11}, \color{black}{63/13}, \linebreak
\color{black}{63/16}, \color{blue}{63/17}, \color{blue}{63/19}, \color{blue}{63/20}, \color{blue}{63/22}, \color{blue}{63/23}, \color{blue}{63/25}, \color{blue}{63/26}, \color{blue}{63/29}, \color{blue}{63/31}, \color{blue}{63/32}, \color{blue}{63/34}, \color{black}{63/37}, \linebreak
\color{black}{63/38}, \color{green}{63/40}, \color{blue}{63/41}, \color{black}{63/43}, \color{green}{63/44}, \color{green}{63/46}, \color{black}{63/47}, \color{green}{63/50}, \color{green}{63/52}, \color{black}{63/53}, \color{green}{63/55}, \color{green}{63/58}, \color{black}{63/59}, \linebreak
\color{black}{63/61}, \color{green}{63/62}, \color{black}{64},
\color{black}{64/3}, \color{black}{64/5}, \color{black}{64/7},
\color{black}{64/9}, \color{black}{64/11}, \color{black}{64/13},
\color{black}{64/15}, \color{blue}{64/17}, \color{blue}{64/19},
\color{blue}{64/21}, \color{blue}{64/23}, \linebreak
\color{blue}{64/25}, \color{black}{64/27}, \color{blue}{64/29}, \color{blue}{64/31}, \color{blue}{64/33}, \color{blue}{64/35}, \color{blue}{64/37}, \color{blue}{64/39}, \color{blue}{64/41}, \color{blue}{64/43}, \color{green}{64/45}, \color{blue}{64/47}, \color{black}{64/49}, \linebreak
\color{green}{64/51}, \color{black}{64/53}, \color{green}{64/55}, \color{green}{64/57}, \color{black}{64/59}, \color{blue}{64/61}, \color{green}{64/63}, \color{black}{65}, \color{black}{65/2}, \color{black}{65/3}, \color{black}{65/4}, \color{black}{65/6}, \color{black}{65/7}, \color{black}{65/8}, \linebreak
\color{black}{65/9}, \color{black}{65/11}, \color{black}{65/12},
\color{black}{65/14}, \color{black}{65/16}, \color{black}{65/17},
\color{black}{65/18}, \color{blue}{65/19}, \color{blue}{65/21},
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\color{black}{65/27}, \linebreak
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\linebreak
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\color{black}
\section{Acknowledgment}The authors would like to thank Carl
Pomerance and an anonymous referee for their feedback, which
helped to improve both the content and the presentation of this
article.
\bibliographystyle{amsplain}
\begin{thebibliography}{10}
\bibitem {Anderson1974} C. W. Anderson,
The solution of $\Sigma(n)=\sigma(n)/n=a/b,$ $\Phi(n)=\phi(n)/n=a/b$
and some related considerations, unpublished manuscript, 1974.
\bibitem {HoldJ2006} J. Holdener, Conditions equivalent to the existence of odd perfect numbers,
\textit{Math. Mag.} \textbf{79} (2006), 389--391.
\bibitem {Laatsch1986} R. Laatsch, Measuring the abundancy of
integers, \textit{Math. Mag.} \textbf{59} (1986), 84--92.
\bibitem {Ryan2002} R. Ryan,
Results concerning uniqueness for $\sigma(x)/x= \sigma(p^nq^m)/(p^nq^m)$ and related topics,
\textit{Int. Math. J.} \textbf{2} (2002),
497--514.
\bibitem {Weiner2000} P. A. Weiner,
The abundancy ratio, a measure of perfection,
\textit{Math. Mag.} \textbf{73} (2000), 307--310.
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2000 {\it Mathematics Subject Classification}: Primary
11A25;
Secondary 11Y55, 11Y70.\\
\noindent {\it Keywords}: abundancy index, abundancy outlaw, sum
of divisors function, perfect numbers.
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received October 25 2006;
revised version received August 31 2007; September 25 2007.
Published in {\it Journal of Integer Sequences}, September 25 2007.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
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\title{Ein Nekrolog auf Identitätspolitik}
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\emph{Ich begann mit dem Schreiben dieses Textes einige Monate vor den Krawallen nach George Floyds Tod. Der Aufstand, der mittlerweile zu einem globalen Ereignis geworden ist, hat mich motiviert, meine Perspektive in diesem Text darzulegen. Meine Erfahrungen in Minneapolis vom 26. bis zum 30. Mai haben meine Verachtung für Identitätspolitik gestärkt, weshalb ich diese um zusätzliche Kritiken daran, die aufgrund dieser Erfahrungen entstanden sind, ergänzt habe.}
Gehen wir zurück zu einer Zeit und einem Ort, an dem die Menschen Pager und Münztelefone nutzten. Als Verandas und öffentliche Parks die Orte waren, an denen man herumlungerte. Eine Zeit, in der Konflikte von Angesicht zu Angesicht geklärt wurden und Scheißelabern direkte Konsequenzen im realen Leben hatte. Das waren die Zeiten vor „Callout-Culture“, „Troll-baiting“ und anderen internetdominierten Aktivitäten. Manche sagen das Internet und die Ausbreitung von Technologie hätten den Kampf gegen Unterdrückung vorangebracht. Meine Meinung? Das Internet ist der Ort, an dem alle Potenziale für soziale Revolte absterben. Zusätzlich zu unsinnigen Petitionen und endlosen Memes kann die [Selbst-] Wahrnehmung als Rebell*in durch Selbstmitleidsorgien und akademische Loyalität erzielt werden, anstatt durch praktische Direkte Aktion. Während das Internet die ideale Brutstätte für Tastatur-Krieger*innen und überhebliche Akademiker*innen ist, bewirkt es auch die Rückentwicklung der sozialen Fähigkeit von Angesicht zu Angesicht zu kommunizieren. Konfliktlösung nimmt die Formen eines undefinierten Internet-Schauspiels und meist auch die einer widerlichen, dem realen Leben nachempfundenen Nachbildung von Richter, Geschworenen und Henker an. Interaktion von Angesicht zu Angesicht ist beinnahe unnötig in der Techno-Gesellschaft, in der ein Smartphone zu einem persönlichen Bedarfsartikel geworden ist, der scheinbar mit den Händen seines Besitzers verwachsen ist. Von einem individuell verdunkelbaren Bildschirm kann nun das ganze Spektrum emotionalen Ausdrucks aus einem Pufferspeicher von Emotionen digital repräsentiert werden.
Das Internet ist auch ein Ort, an dem die Lynchmob-Mentalität der „Callout-Culture“ die Menschen ermutigt, sich gegenseitig als eindimensionale Wesen zu betrachten, die sich ausschließlich durch Fehler und Unvollkommenheiten definieren. Im Namen von „sozialer Gerechtigkeit“ und dem „Outing von Tätern“ entsteht ein neuer Etatismus, der Angst und Schuld dazu nutzt, die Konformität von Allies zu erzwingen. Und ähnlich dem Bestraftwerden durch den Staat, kann ein Individuum, wenn es einmal im Internet verurteilt wurde, niemals wieder diesem Stigma entfliehen. Stattdessen bleibt jegliche persönliche Entwicklung unbedeutend im Vergleich zu der statischen Natur vergangener Fehler. Unabhängig von persönlicher Entwicklung ist ein verurteiltes Individuum auf ewig dazu verdammt, der Aussage seiner Online-Darstellung unterworfen zu sein.
In meiner Erfahrung als „marginalisierte Stimme“ habe ich gesehen, wie Identitätspolitik von Aktivist*innen als Werkzeug der sozialen Kontrolle instrumentalisiert wurde, das gegen alle genutzt wurde, die in das identitäre Schema des „Unterdrückers“ passten. Der ursprüngliche Machtkampf um Gleichheit hat sich in eine olympische Sportart für sozialen Einfluss verwandelt, die die gleiche soziale Hierarchie umkehrt, die ursprünglich hätte zerstört werden sollen. Viele Identitätspolitiker*innen, die ich getroffen habe sind mehr daran interessiert „weiße Schuld“ für ihren persönlichen (und sogar finanziellen) Vorteil zu nutzen, als die Organisationsstrukturen des Rassismus [white supremacy] physisch anzugreifen. Ich habe erlebt, wie die Opferrolle dazu genutzt wurde, haarsträubende Lüg\#LISTtitle Nekrolog auf Identitätspolitiken und Schikanen zu rechtfertigen, die durch persönliche Rache motiviert waren. Viel zu häufig habe ich gesehen, wie Identitätspolitik eine Kultur schafft, in der persönliche Erfahrungen auf passives Schweigen reduziert werden. Aber all das ist nichts Neues. Jede*r, die*der sich selbst (seit einiger Zeit) als Anarchist*in definiert, wird schon einmal die eine oder andere Form von „call-out“ oder „silencing“ gesehen oder auch selbst erlebt haben. Warum erwähne ich das also? Weil diese Scheiße \emph{immer noch} passiert und ich \emph{immer noch} so viele Menschen sehe, denen der Mut fehlt, sie offen zu konfrontieren.
Ich erwarte nicht, dass dieser Text der Identitätspolitik zu einem plötzlichen Ende verhilft. Vielmehr drücke ich meine Feindschaft ihr gegenüber und ihrem autoritären, anti-individualistischen Charakter aus. Ich sehe \emph{noch immer} selbstbezeichnende Anarchist*innen über „weiße“ Dreads lästern (ebenso wie ich Menschen sehe, die ihre Dreads unter sozialem Druck abschneiden). Ich sehe \emph{noch immer} Menschen, die das Wählengehen rechtfertigen, wie sie es bei Obama getan haben (dieses Mal für Bernie [Sanders]\footnote{Ein demokratisch-sozialistischer US-Politiker, der wiederholt als Präsidentschaftskandidat antrat und dabei Unterstützung von vielen (radikalen) Linken der USA bekam (Anm. d. Übers.).}). Und ich sehe \emph{noch immer} „Allies“, die ihre Frustration in sich hinein fressen, weil sie zu verängstigt sind, den Autoritarismus, den sie vor sich sehen, zu konfrontieren.
Wie viele „weiße“ Anarchist*innen wurden Rassist*innen (oder Privilegierte) genannt und für ihre Weigerung an der vergangenen Wahl 2020 teilzunehmen, angedisst?
Stell dir vor, wie Anarchie aussähe, wenn die Menschen sich weigern würden, den herablassenden Forderungen von Identitätspolitiker*innen zu gehorchen. Würden sich die Menschen freier fühlen, ihre Leben jenseits der engstirnigen Grenzen vorgegebener Identitäten zu erforschen? Würden sie sich furchtlos ermächtigen, sich ihre eigene Meinung zu bilden? Gibt es irgendeine Freude zu erleben innerhalb der angespannten Farce des akademischen Elitarismus?
Wäre dieser Text weniger stichhaltig, wenn er nicht von einer queeren Person of Color geschrieben worden wäre? Was, wenn er von einem „weißen“ „cis“ „Mann“ stammen würde? Warum wäre das relevant?
Im Großen und Ganzen wäre es das nicht. Denn schließlich geht es hier nicht nur um Identität, sondern um anti-autoritäre Anarchie. Wenn es eine Sache gibt, die ich in den letzten paar Jahren zur Genüge beobachten konnte, dann ist es Identitätspolitik, die sich wie eine Plage ausbreitet und jeden sozialen Raum verschlingt – ironischerweise inklusive den anarchistischer Kreise. Für mich bedeutet Anarchie die Zerstörung sämtlicher sozial konstruierter Identitäten und aller Einschränkungen, die diese der Vorstellungskraft setzen. Anarchie ist eine individualistische Erfahrung, die sich selbst im Gefängnis zugewiesener Identitäten gefangen sieht. Statt dieses Gefängnis gemeinsam mit der Gesellschaft, die es errichtet, zu zerstören, ist der heutige Anarchismus zu einem Friedhof erstorbenen Potenzials, internalisierter Opferrollen und eines ideologischen Wettkampfes darum, wer „am meisten unterdrückt“ ist, geworden.
Anstatt Identität selbst anzugreifen und den Apparat, der dieses Paradigma aufrechterhält, vergeudet mensch seine Energie damit, sich gegenseitig fertig zu machen, indem mensch die Komplexität individueller Einzigartigkeit ausblendet und die staatliche Rolle einnimmt, einander aufgrund von Zugehörigkeit zu identitären Kategorien zu definieren. Eine bestimmte Identität anzunehmen bestätigt nur die Existenz dieser Identität als eine universelle Wahrheit und damit – aufgrund der kolonialen Intention zugewiesener Identitäten – auch die Knechtschaft und Versklavung einiger durch andere als eine ebenso universelle Wahrheit.
Ich weigere mich daran teilzunehmen, Versklavung als Voraussetzung meiner Existenz hochzuhalten und deshalb sind diese „Wahrheiten“ nichts weiter als politische Märchenerzählungen. Sie sind Produkt einer perfektionierten, sozial entwickelten Gottesvorstellung, die wie ein parasitärer Cordyceps\footnote{Eine parasitäre Pilzgattung (Anm. d. Übers.).} in den Verstand eindringt und unbedingten Gehorsam einfordert. Bestandteil dieser mentalen Manipulation ist ein durch die Einkerkerung der industriellen Gesellschaft institutionalisierter Verstand. Identitätspolitik ist die verstaubte Kette der Kolonisation, poliert von denen, die ihr persönlichen Wert beimessen. Diese „Wahrheiten“ sind die sozialen Konstrukte der Kontrolle, die ein Leben der Rebellion im tiefen und finsteren Brunnen der Reform in Ketten legen. Und während sich viele dort eingerichtet haben, bin ich ausgebrochen, um das unendliche unbekannte Terrain des Hedonismus und der anti-politischen Anarchie zu erforschen. „Schwarze“, „Braune“ oder „Weiße“ Macht [Power] ist die Antithese der Freiheit; sie ist die ideologische Wohltätigkeitsarbeit einer zivilisierten, humanistischen Form der Rebellion. Identitätspolitik ist die Sterilisation der Individualität, die sie sowohl gehorsam gegenüber der kollektivistischen Autorität, als auch gutgläubig gegenüber dem nationalistischen Mythos der Überlegenheit macht.
Letztlich ist der „Mensch“ ein Tier, das innerhalb sozial konstruierter Kategorien in eine Hierarchie des ökonomischen Status domestiziert wurde. Und auch wenn sich diese Hierarchie über die Jahre verändert hat, wird sie dennoch beständig durch die Beziehung derer, die Befehle erteilen und derer, die gehorchen, aufrecht erhalten. Unabhängig davon, wie die Kategorien innerhalb dieser Hierarchie positioniert sind, bleibt die Hierarchie autoritär; die Gruppe dominiert das Individuum. Was einen „Menschen“ definiert, ist der Grad seines Gehorsams und seiner Unterwürfigkeit gegenüber zivilisierten Rollen und Verhaltensweisen, die die industrielle Gesellschaft erfordert. Je weniger kooperativ ein „Mensch“ ist, desto wahrscheinlicher wird dieser „Mensch“ mit einem Tier verglichen werden. Das Tier ist das unerwünschte Wesen – selbst für die Identitätspolitiker*innen, die es vorziehen, den ideologischen Anthropozentrismus der Kolonisateur*innen zu übernehmen. Vielleicht erklärt das, warum es so wenig Diskussion um Tierbefreiung im links-anarchistischen Diskurs gibt. Die marginalisierte Stimme soll lieber als gleichwertig zu den zivilisierten Kolonisator*innen porträtiert werden, als als verlorengegangene Beziehung zwischen ihrer Animalität und der Erde. Im Zentrum linker Politik steht das humanistische Ziel sozialer Gleichheit innerhalb des industriellen Fortschritts – während die Erde zugleich weiterhin in Nationalstaaten zerteilt bleibt und für die anthropozentrische Ausbeutung und Ausbreitung verwüstet wird.
Meine Meinung ist, dass solange eine*r eine persönliche Beziehung zur „menschlichen“ Identität – ähnlich wie der zu „weißen“ oder „männlichen“ Identitäten – pflegt, das Individuum das koloniale Paradigma von zivilisiert vs. wild bekräftigt. Und solange diese Bekräftigung andauert, bleibt das Individuum auch anfällig dafür in anderen identitären Konstrukten gefangen zu werden, die das ungezähmte Potential weiter unterdrücken.
Ich frage mich, wann bzw. ob Anarchist*innen im Allgemeinen jemals über die Gruppen-Mentalität des Linksradikalismus hinaus hin zum individualistischen Aufstand gelangen werden, in dem sie die Konfrontation von Identität als einen Akt der persönlichen Emanzipation begreifen. Werden Anarchist*innen eines Tages begreifen, dass alles über dem Individuum eine Autorität darstellt, egal ob es sich dabei um „die Kommune“, die „Bewegung“ oder die kulturelle Herrschaft von Identität handelt? Vielleicht einige, aber ich bin mir sicher, nicht alle.
\section{Die heilige Opferrolle}
\emph{Nach 45-minütiger Fahrt sind wir endlich da. Wir haben einen langen Tag des Ladendiebstahls hinter uns und das ist unser letzter Halt. Ich bin an der Reihe und ich nehme mir vor, den Laden mit Waren im Wert von mindestens 500 Dollar zu verlassen, um diese später online zu verkaufen. Aber ich habe ein schlechtes Gefühl bei diesem Laden. Anders als die Läden davor ist dieser Laden wesentlich kleiner, was bedeutet, dass die Ladendetektive die Türen besser im Blick behalten können. Größere Läden haben meist ausgedehntere Ein- und Ausgangsbereiche. Außerdem ist es schwieriger, alle Einkäufer*innen durch die Kameras im Auge zu behalten, je größer ein Geschäft ist. Ich entscheide mich, trotzdem zu gehen. Sei dir nie sicher über irgendetwas, wenn du es nicht bereits versucht hast.}
\emph{Ich gehe hinein, nehme mir einen Wagen und schaue mich nach den Waren um, die ich klauen will. Ich checke auch die Schlangen an den Kassen und den Kundenserviceschalter. Zwei der Mitarbeiter*innen am Kundenserviceschalter sind damit beschäftigt, sich zu unterhalten, die Kassen sind alle, bis auf eine am Eingang und zwei am Ausgang, geschlossen. Am Eingang steht ein*e Mitarbeiter*in, die*der die Einkaufswägen abwischt. An einer der Kassen sitzt ein*e Kassierer*in, während die andere unbesetzt ist. Ich speichere die Situation als „zu einfach“ aussehend ab, aber ich entscheide mich, mich zunächst darauf zu fokussieren, wo die Dinge sind, die ich brauche. Nachdem ich meinen Einkaufswagen beladen habe, beginne ich meine Reise in Richtung Ausgang. Jede*r, die*der Ladendiebstahl zum Lebensunterhalt begeht, weiß, dass das der aufregende Teil ist. Die ganze Zeit bis zu diesem Moment war ich nur ein*e gewöhnliche*r Einkäufer*in. Aber nun, da ich auf den Ausgang zugehe, werfe ich mein Kostüm der*des „Einkäufers*in“ ab und bereite mich auf die kriminelle Erfahrung der*des „Ladendiebs*in“ vor. Als mein Herz zu pochen beginnt, kann ich spüren wie meine Nerven ein Wohlgefühl auslösen – eine beruhigende Antwort als eine temporäre Ablenkung von der Panik, um meine Sinne scharf und fokussiert zu halten. Ich muss auf alles vorbereitet sein. Und noch immer muss ich mein(e) „gewöhnliche*r Einkäufer*in“-Gesicht und -Körpersprache bewahren. Als ich die „zu leichte“ Bahn zum Ausgang durchquere, sieht alles gut aus.}
\emph{Die Menschen am Kundenservice unterhalten sich immer noch und widmen mir keine Aufmerksamkeit, der*die eine Kassierer*in ist zu beschäftigt, irgendwen anzurufen, um mich zu bemerken. Ich hole meinen Fake-Kassenzettel heraus und gehe ganz gewöhnlich durch die ersten Schleusentüren des Ausgangs. Wenn ich gesehen oder erwischt worden wäre, wäre das der Moment in dem sich mir von hinten jemand nähern würde oder ich jemanden nach meiner Schulter greifen spüren würde. Als ich die zweiten Schleusentüren passiere, ist noch immer alles gut. Zeit, mich auf den Weg zur Rückseite des Parkplatzes zu machen – und dann passiert es \dots{}}
\emph{Jede*r, die*der lange genug geklaut hat, kennt diese gefürchteten Worte: „Entschuldigen Sie \dots{} Entschuldigen Sie!“, höre ich jemanden hinter mir rufen. Ich tue so, als hätte ich nichts gehört. Dann höre ich schnelle Schritte von hinten auf mich zukommen. „Entschuldigung, ich muss Ihren Kassenzettel sehen“, sagt er, als er mir seinen Ladendetektiv-Ausweis vorhält. Fuck. Wo hat mich dieser geleckte Hipster gesehen? Muss in der Kleidungsabteilung hinter mir gewesen sein \dots{} Vielleicht war diese Bahn eine verdammte Falle. Egal. Lass den Einkaufswagen stehen und geh weg. Ich beginne wegzulaufen und höre „Nein, nein \dots{} Wir müssen wieder reingehen und Papierkram ausfüllen. Keine Sorge, Sie werden nicht angezeigt.“ Ja klar, Papierkram mit all meinen Daten ausfüllen und für ihre Akten fotografiert werden – vergisss es. Ich laufe weiter davon. Ein anderer Ladendetektiv kommt rausgerannt und ist am Telefon. Er ruft die Polizei. Ich erkenne sofort, dass der erste Typ mich heimlich hinhalten wollte, bis die Polizei eintrifft! Ich sprinte los. Ich höre, wie die beiden mir dicht hinterherrennen. Ich überquere die Sraße und stürze in eine Wohnwagensiedlung. Im Zickzack renne ich zwischen den Wohnwagen weiter und verstecke mich schließlich in einem Blechschuppen. Ich zwinge mich, ruhig und tief durchzuatmen. Ich beruhige mich und lausche, wie die beiden in der Nähe nach mir suchen.}
\emph{Als ich sie schließlich nicht mehr höre, texte ich meinen Kompliz*innen meinen ungefähren Aufenthaltsort. Ich verlasse den Schuppen, versuche einige Dinge aufzuräumen, die heruntergefallen sind, als ich dort hineingestürmt bin. Die Bullen müssen jede Sekunde hier sein. Ich sehe, wie das Auto meiner Kompliz*innen langsam vorüberfährt und winke sie zu mir. Ich springe hinein und lege mich auf den Boden, als wir wegfahren.}
\emph{Ich hätte meinem Instinkt vertrauen sollen. Das war Pech. Aber es hätte schlimmer kommen können. Statt die Nacht in einer Zelle zu verbringen, sitze ich gemütlich hier und schreibe diesen Text. Aber das ist die Realität des Ladendiebstahls – oder jedes Verbrechens, was das angeht. Egal wie viele Male du damit durchkommst, es ist wichtig darauf vorbereitet zu sein, eines Tages gefasst zu werden. Sei bereit dafür. Und wenn es passiert, untersuche die Panik, die Emotionen und die physischen Reaktionen deines Körpers \dots{} Lerne sie kennen. So dass du das nächste Mal, wenn du ein Verbrechen begehst, ein besseres Verständnis vom worst-case-Szenarion besitzt. Für mich ist das elementar und es gibt keinen Platz für die Opferrolle oder einen Ausruf der Unschuldigkeit.}
Während Covid-19 die Bedingungen für staatliche Repression in Form von „Ausgangssperren“ schuf, hat es ironischerweise zugleich meine Möglichkeiten für illegalistisches Vergnügen erweitert! Viele Unternehmen blieben wochenlang unbeaufsichtigt, was bedeutet, dass Sachbeschädigungen länger unbemerkt blieben. Inmitten der Panik waren die Supermarkt-Detektiv*innen und das Sicherheitspersonal damit beschäftigt, die Anzahl der Gegenstände, die die Menschen bezahlten zu überprüfen, ohne die Wagenladungen an Lebensmitteln zu bemerken, die heimlich zur anderen Tür hinausgeschoben wurden.
Bevor sie ihre Läden schlossen, deaktivierten viele Geschäfte wie REI, L.L Bean und andere ihre Sicherheitsschranken. Ich vermute, dass dies aufgrund der großen Menge an Menschen geschah, die diese mit bezahlten Waren passierten, auf denen ein versteckter Diebstahlschutz noch immer aufgeklebt war. Vielleicht wurden die Schleusen abgeschaltet, um einen sekündlich losgehenden Alarm zu vermeiden. Das eröffnete die gute Gelegenheit, gesicherte Waren stressfrei hinauszutragen.
Die letzten Wochen ließen in mir alte Erinnerungen an die Zeit aufsteigen, zu der ich Anarchie noch als eine Aktivität verstand, die nur solange andauerte, wie eine Erste-Mai-Demo, eine Demonstration oder ein nächtliches Vergnügen. Ich erinnere mich daran, dass ich das Gefühl hatte, Anarchie sei der Moment, in dem ich schwarze Hosen, Schuhe und Handschuhe trug und mit einem T-Shirt vermummt war. Nach diesen Aktivitäten galt es, in die „wirkliche Welt“ zurückzukehren. Zurück zur Lohnsklaverei, zurück zur täglichen Routine des Mietezahlens und Abzählens meiner Lebensmittelgutscheine für den Supermarkt. Sicher, neben den Büchertischen bei Punkkonzerten und radikalen Veranstaltungen gab es die gelegentlichen klandestinen Aktivitäten. Aber es gab diesen Zwiespalt, der immer eine Trennung kreierte, die Anarchie zu einer Art extrakurrikularen Aktivität machte. Sicher widmete ich mein Leben der Rebellion; Das ganze Konzept eines Zine-Distros, das ich später „Warzone Distro“ nennen würde, entwickelte ich, während ich meine Arbeitszeit auf dem Scheißhaus verstreichen ließ. Trotz der Lohnsklaverei war ich immer damit beschäftigt, den einfachsten Weg einzuschlagen und für den größtmöglichen Lohn so wenig wie nur möglich zu arbeiten. Ich war der*die Arbeiter*in, die*der seine*ihre Überstunden an andere weitergab. Einen halben Tag Arbeit für eine leichte LKW-Ladung? Scheiße Mann, ich bin raus!
Mit der Zeit war mir Anarchie als bloße extrakurrikulare Aktivität nicht mehr genug. Und was ich damit meine, ist, dass ich zunehmend intoleranter gegenüber Bossen, Lohnsklaverei, Weckern, Mietezahlen und Münzen abzählen wurde. Ich erinnerte mich daran, wie es war Kind zu sein und keine solche Verpflichtungen zu haben. Ich erinnerte mich an meine ganztägigen Abenteuer draußen, vom frühen Morgen bis spät in die Nacht. Jeder Tag war ein neues Abenteuer und jeden Tag lernte ich etwas Neues über mich selbst. Dann, als verantwortungsbewusste*r Erwachsene*r lernte ich etwas Neues über mich. Ich hasste das Erwachsensein, mich erwachsen zu verhalten und die performative Rolle und Identität des „Erwachsenwerdens“. Aber ich versuchte nicht wieder zum Kind zu werden. Diese Tage waren vorbei. Ich fragte mich, wie ein anarchistisches Leben, das die Erwachsenen\Slash{}Kind Binarität überwinden würde, aussehen könnte.
Schnell vergangene Jahre später stehe ich hier, arbeitslos, aber nicht länger Münzenzählend, älter, aber jugendlicher als ich jemals war. Einige sagen, ich verkörpere die schlimmste aller Welten; hedonistisch, gewaltsam und kindisch. Selbstverständlich obliegt es der subjektiven Interpretation, was diese Worte bedeuten und wie diese auf mich angewandt werden, aber eines ist sicher: Ich fühle mich freier als ich mich je gefühlt habe. Und ich habe eine Liebesaffäre mit dem Verbrechen. Es ist eine intime Erfahrung – Verbrechen in wütender Verachtung für die Gesellschaft und das Gesetz zu begehen. Brüche zu erzeugen und damit durchzukommen vervollkommnet mein Verlangen nach Anarchie mit jedem Moment. Heutzutage erlebe ich den ganzen Tag draußen, vom frühen Morgen bis spät in die Nacht. Und mit jeder kriminellen Aktivität lerne ich mich besser kennen. Aufgrund der Tatsache, dass meine lustvollen Tage des Lebengenießens entweder im Gefängnis oder mit meinem plötzlichen Tod enden werden, lerne ich, die Gegenwart der Vergangenheit und der Zukunft vorzuziehen.
Eine Sache, die ich über Verbrechen gelernt habe, ist das einzigartige Gefühl, das mit dem Gesetzebrechen einhergeht, ein Sinn für individuelle Handlungsmöglich- und Unmöglichkeiten, ein Sinn für Stärken und Schwächen. All das kann durch die Erfahrung des Gesetzebrechens entdeckt werden. Und es ist \emph{diese} Erfahrung, die ich ausweiten will, um mehr über mich selbst zu lernen, um unkontrollierbar im anti-sozialen Sinn zu werden.
Ich blicke auf meine Vergangenheit zurück, in der ich vom Kult der Identitätspolitik eingesperrt war. Ich erinnere mich daran, dass ein Grund dafür, die Opferrolle einzunehmen, war, soziale Aufmerksamkeit zu generieren und die (marginalisierten) Identitäten, die mir zugewiesen worden waren, in einem positiven Licht zu zeichnen. „Sieh mich an! Eine verantwortungsvolle queere Person of Color, die einer Arbeit nachgeht und ein*e gesetzestreue*r Bürger*in ist!“ Aber warum? Um zu beweisen, wie ähnlich ich all diesen „weißen“, hart arbeitenden proletarischen Helden war, die Amerika braucht, um sein koloniales Establishment aufrechtzuerhalten? Ein*e andere*r Lohnsklav*in, die*der passiv und bereitwillig die eigene Versklavung akzeptiert? Um ein*e weitere*r Christ*in of Color zu werden, die*der vorgibt, dass es ein imaginäres Königreich dort oben gibt für all diejenigen von uns Strolchen, die niemals eine faire Chance im Leben hatten? Scheiß auf all das.
Der Grund aus dem all die Rassist*innen, Homofeind*innen, Patriarchen und Patriot*innen Menschen wie mich fürchten, liegt jenseits von Identitätspolitik; Ich bin ein*e erbitterte Feind*in ihrer Kontrolle und Ordnung. Das gesellschaftliche Schloss, das sie zu errichten und aufrechtzuerhalten trachten, wird immer Ziel meiner Sabotage sein!
Ich denke die meisten Menschen können nachvollziehen und verstehen, dass es \emph{nicht} nötig ist, sozial zugewiesene Identitäten anzunehmen, um zu verstehen \emph{wie} die Gesellschaft diese als Werkzeuge der sozialen Kontrolle einsetzt. Ich denke es ist ebenso leicht zu erkennen, inwiefern Identität als Werkzeug der Revolution beschränkt ist und vielmehr zu internen Konflikten in vielen revolutionären Projekten geführt hat. Aber was mich umhaut, ist die Tatsache, dass so viele diese Identitäten nicht sofort als vorrangige persönliche Form der Rebellion zurückweisen. Ich denke mensch kann sagen, dass diese Identitäten die Hierarchien, die sie aufrechterhalten, gerade deshalb aufrechterhalten, weil sie so häufig von linken Organisationen für moralische Überzeugung genutzt werden. Durch Opferrolle und Unschuld wird Identitätspolitik als eine alle ansprechende Methode zur Schaffung einer kollektiven Gesinnung genutzt, die das Individuum letztlich ermutigt, unabhängiges Denken zugunsten eines Gottkomplexes von Moralität und Kollektivismus aufzugeben. Ich denke das spielt auch eine große Rolle im Etatismus und der Zurückweisung von illegalistischer Revolte.
Ich lehne die statische, ziviliserte Binarität von Schuld und Unschuld ab und damit auch die Internalisierung der Opferrolle. Ich habe keine Verwendung für eine „Call-out-Culture“ oder einen Internet-Lynchmob gegen meine Feind*innen. Im Internet werden alle meine Versuche öffentliche Unterstützung gegen eine*n Feind*in zu erlangen, nur einen anderen Feind (den Staat) informieren und ihn ermächtigen, mir meine Verantwortlichkeit zu stehlen. Schuld und Unschuld sind auch eine legalistische Binarität, die nur dazu dient, nach einem moralischen Determinismus zu urteilen. Ich verachte den Staat und alle seine Manifestationen, sowie seine Repression gegen das Chaos. Deshalb bin ich kein Opfer; Ich bin (selbst)erklärte*r Feind*in in einem Krieg gegen ihn [den Staat]. Ich erwarte kein Mitleid, keine Begnadigung oder Wohltätigkeit, weder von ihm, noch von seinen Befürworter*innen.
\emph{Es war der Tag, an dem Chicago sein Ausgangssperren-Dekret verhängte. Mein*e Kompliz*in und ich waren in meiner Heimatstadt bei meiner Mutter zu Besuch. Als wir, nachdem wir meiner Mutter ein paar Lebensmittel besorgt hatten, heimfuhren, bemerkte ich jemanden alleine auf einer Parkbank sitzen. Ihr Name ist „Big Momma“. Ich war überrascht, sie draußen in der Kälte zu sehen und nicht in einer der lokalen Zufluchtsstätten. Wir fanden heraus, dass die Unterkünfte ihre Türen geschlossen hatten, vermutlich wegen Covid-19. Ich fragte mich, wie viele andere draußen in der Kälte waren \dots{}}
\emph{Mein*e Kompliz*in und ich gingen zu einem Park, in dem ich früher Food Not Bombs [eine Art Volksküche; Anm. d. Übers.] veranstaltet hatte und zu meiner Überraschung hatten dort um die 20 Personen ein Camp vor der Belüftungsanlage eines Gebäudes errichtet, aus der warme Luft strömte. Wir gingen hinüber und fragten die Menschen, wie es ihnen gehe. Einige Menschen, die mich noch von aktivistischen Projekten Jahre zuvor kannten, kamen aufgeregt herbeigelaufen, um mich zu begrüßen. Sie alle gehörten zu den Unglücklichen, die zumindest für dieses Wochenende aus den Unterkünften ausgesperrt worden waren. Mein*e Kompliz*in und ich gingen zurück zum Wagen und heckten einen Plan aus.}
\emph{Eine halbe Stunde später stehen wir in einem anderen Lebensmittelgeschäft. Anders als sonst ist es nicht ganz leicht, Lebensmittel aus diesem Geschäft zu tragen ohne sie zu bezahlen. Aufgrund der erhöhten Sicherheit an der Tür wegen Covid-19 und der Angst vor Plünderungen hat sich das Szenario verändert. Aber es ist noch immer möglich, einen vollen Einkaufswagen aus dem Laden zu schieben. Wir befüllen den unteren Teil des Wagens mit Wasserflaschen, zahlreichen Brotlaiben, Erdnusbutter, Marmelade und über 20 Packungen gemischter Nüsse, frischen Äpfeln und Bananen. Fertig. Ich voraus bahnen wir uns den Weg zur Tür. Meine Aufgabe ist es, um die Ecke nach zwei Angestellten an den Selbstbedienungskassen zu spähen, um sicherzustellen, dass sie uns nicht beobachten. Wenn sie gerade schauen, würde ich mein Handy herausholen, als würde ich einen Anruf tätigen. Falls nicht, gehe ich einfach weiter. Mein*e Kompliz*in ist mit dem Wagen dicht hinter mir. Die Luft ist rein. Die ersten Schleusentüren \dots{} die zweiten Schleusentüren \dots{} Alles ist gut gelaufen. Schließlich erreichen wir den Wagen und beladen den Kofferraum. Geschafft! Nächster Halt ist ein anderer Lebensmittelladen, aber wir werden dort keine Lebensmittel besorgen: Wir überfallen die Männer- und Frauentoiletten, um große Rollen Toilettenpapier abzustauben. Manchmal ist es ein wenig laut, die Spender zu öffnen, aber es geht relativ einfach mit jeder Art von Hausschlüssel. Wir füllen zwei Rucksäcke mit je drei großen Rollen und schon sind wir fertig.}
\emph{Zurück bei meiner Mutter waschen wir unsere Hände sorgfältig, bevor wir Beutel um Beutel Peanutbutter-Jelly-Sandwiches zubereiten. Nachdem wir damit fertig sind, kehren wir zurück zum Camp der Obdachlosen. Jede Person bekommt zwei Sandwiches, zwei Äpfel, Zwei Bananen, getrocknete Früchte und eine Flasche Wasser. Zusätzlich hüllen wir die Toilettenpapier-Rollen in Plastiktüten vom Supermarkt, um sie trocken zu halten und übergeben sie. Wir bleiben noch eine Weile, lachen gemeinsam und reden Scheiße über die Cops. Es tat gut, neue Bekanntschaften zu schließen und alten Freund*innen zu begegnen. Es tat gut zu sehen, dass sie alle zurechtkommen und trotz der Umstände des Wetters und der geschlossenen Unterkünfte guter Dinge waren. Wir gingen und beschlossen auch in anderen Parks nach den Menschen zu sehen. Wir fanden ein paar einsame Wölfe, die freudig nahmen, was wir an Wasser und Sandwiches übrig hatten. Dann kehrten wir zurück zum Haus meiner Mutter, wo wir die Nacht verbrachten. Ich öffnete den Kühlschrank und musste kichern, als ich all das gestohlene vegane Essen sah und überlegte, was ich zum Abendessen wollte.}
\section{Der Allyship-Feigling}
Meiner Meinung nach begann das Konzept des „Allyship“ mit guten Absichten, aber ebenso wie andere Aspekte der Identitätspolitik wurde es schlecht und hätte dringend entsorgt werden müssen. Ich empfinde „Allyship“ folgendermaßen: Wenn du ein politisches Buzzword und Konzept \emph{brauchst}, um dich zu motivieren, dich mit Menschen jenseits gegenderter oder rassifizierter Kategorien zu verbünden, dann ist deine „Solidarität“ unaufrichtig. Wenn deine Art zu kommunizieren voller vorab von einem „Woke Ally 1×1“-Workshop genehmigter Themen ist, bist du zu einer freilaufenden Marionette geworden. Aufrichtige gegenseitige Hilfe oder Solidarität benötigt keine trendigen Twitter-Phrasen, um dich zu motivieren, Beziehungen zu knüpfen. In anderen Worten: Gib dich nicht mit mir ab, nur weil du gelesen hast, dass das das Richtige ist, oder weil dein*e progressive*r College-Professor*in dir das gesagt hat. Kriech mir nicht in den Arsch und laufe mir hinterher, weil ich eine \emph{viktimisierte}, „marginalisierte“ oder „PoC Stimme“ bin. Oder weil deine Freund*innen und Kamarad*innen dich dazu drängen. Lass nicht zu, dass etwas so Künstliches wie sozial konstruierte Kategorien unsere Beziehung definieren. Gib dich mit mir nur ab, wenn du persönlich Interesse an unserer Interaktion und meiner Persönlichkeit hast und du – das ist das wichtigste – das aus individuellem Verlangen willst. Ich glaube nicht an erzwungene gegenseitige Hilfe: Damit machst du zwei Menschen zugleich zur*zum Idiot*in.
Es gibt auch diejenigen, die aufgrund von rassifizierten oder vergeschlechtlichten Zuschreibungen annehmen, sie wüssten, wie andere Menschen denken. Das sind die Identitätspolitiker*innen, die sowohl als Polizei als auch als Repräsentant*innen anderer auftreten und versuchen, Allyship durch Schuldgefühle und Anprangerungskampagnen zu erzwingen. Sie nutzen ihre Identität, um sich selbst als unverantwortlich zu erklären, während sie zugleich eine passiv-agressive Form der Kommunikation zur Einschüchterung nutzen. Aber meiner Ansicht nach ist niemand \emph{verpflichtet} ihnen oder irgendjemand anderem, besonders nicht aufgrund von etwas so flachem wie Identität, zuzuhören oder sie zu unterstützen. Ich bin stets derer überdrüssig, die sprechen, als würden sie die Interessen von Menschen vertreten, die sie nie getroffen haben. Es ist idiotisch zu glauben, dass nur weil Menschen sozial ähnliche Identitäten zugewiesen bekommen, jedes Individuum den Stereotypen dieser Identitäten entspricht.
Identitätspolitik war erfolgreich darin, ein Verständnis dafür zu entwickeln, wie die zivilisierte Gesellschaft funktioniert, aber als Lösung, um diese Gesellschaft einzureißen, führt sie nur dazu, Identitäten zu beschränken, zu Nationalismus, internalisierten Opferrollen und weiteren Stereotypen für Menschen, die dagegen ankämpfen.
Du willst etwas über die Erfahrungen einer Person erfahren? Sprich doch mit ihr direkt. Triff keine Annahmen aufgrund von sozialen Konstruktionen. Du willst deine Solidarität zu Menschen zeigen? Behandle sie als Individuen mit einzigartigen Erfahrungen und Geschichten, nicht als bloße Schwarmmitglieder irgendwelcher homogenisierter Gruppen. Und für all diejenigen, die noch immer gehorchen ohne zu hinterfragen gilt \emph{noch immer}: Ein anderes Wort für \emph{Weißer Ally} ist Feigling!
\section{Die Woke Führung}
Persönlich mag ich das Wort „bilden“ nicht, um den Austausch von Ideen zwischen zwei Individuen zu beschreiben. „Bilden“ impliziert die Etablierung universeller „Wahrheiten“ anstelle des horizontalen Austauschs persönlicher Perspektiven. Der Kontext, in dem ich das Wort „Bilden“ am häufigsten wahrnehme, kreiert eine soziale Hierarchie zwischen denjenigen, die „woke“ [dt. etwa \textasciitilde{}politisch achtsam\textasciitilde{}, bzw. etwas polemisch ausgedrückt \textasciitilde{}politisch konform\textasciitilde{}] sind und denjenigen, die das nicht sind. Lernen Menschen überhaupt irgendetwas, wenn die Kommunikation von Ideen von oben herab stattfindet? Vielleicht. Aber ich bevorzuge es, diese Hierarchie nicht zu befördern.
Individuelle Menschen sind mehr als nur „weiß“, „braun“ oder „schwarz“, „Männer“ oder „Frauen“ oder welche soziale Konstruktion ihnen auch immer bei ihrer Geburt zugewiesen wurde. Deshalb kommt die Kommunikation aufgrund identitätsbasierter Annahmen so gut wie immer herablassend rüber. Ich sehe Scheiße wie „Bilde deine Freunde“ oder „lass dich bilden“, als würde auf eine Kirche der sozialen Gerechtigkeit verwiesen, in der mensch „erweckt“ würde. Und offensichtlich ist die kapitalistische Mentalität, Informationen zu verkaufen, ohne Frage gerechtfertigt. Einige glauben die „Arbeit“, Fragen zu beantworten, sei einen Lohn wert und zitieren etwas vom Umfang einer Google-Suche, wenn eine*r nicht in der Lage dazu ist, sie zu bezahlen. Ironischerweise kommen viele Fragen in guter Absicht und stammen von gutmeinenden Aktivist*innen, die es ertragen, zunächst von oben herab behandelt zu werden. Meiner Meinung nach entmutigt diese elitäre Art des Antwortens gegenüber wohlmeinenden Menschen diese, indem sie deren persönliche Geschichten trivialisiert und ihnen einredet, andere als an der Spitze stehend zu akzeptieren. Dieser Methode des „Bildens“ liegt ein Kollektivismus zugrunde, der die Grundlage für ein weiteres soziales System des Zwangs legt. Ich habe kein Interesse daran, mich daran zu beteiligen. Ich kann eine kritische Perspektive einnehmen oder einem Punkt widersprechen ohne einen Austausch zu hierarchisieren.
Ich betrachte jeden individuellen Verstand als rauschenden, wilden Strom von Ideen, die über die Ufer treten, wenn der Damm der sozialen Unterwerfung zusammenbricht. Die Gesellschaft verhindert kollektiv jegliche Wildheit, indem sie das Individuum domestiziert und letzlich ein im Käfig eingesperrtes Tier aus dem Verstand macht. Unter all der sozialen Konditionierung gibt es ein einzigartiges Individuum, das sich selbst in chaotischem Widerspruch gegen die Gesellschaft entdeckt.
Gleichförmigkeit ist die Feindin des freien Ausdrucks. Es gibt keine „Bildung“, nur eine verbreitete Meinung, die von denen, die beabsichtigen für andere zu denken, durchgesetzt wird. Ich denke, Ideen und Perspektiven können auf eine Art und Weise ausgetauscht werden, die keinem autoritären Modell der Kommunikation von oben herab gleicht. Ich bin kein*e Lehrer*in und ich strebe nicht danach, andere zu bilden. Stattdessen teile ich meine persönlichen Erfahrungen und Ideen, sowie sie entstehen und sich entwickeln, mit der Welt in dem Verständnis, dass andere uneinverstanden sind und einzigartige eigene Erfahrungen haben.
Beispielsweise habe ich gelernt, dass das illegalistische Leben nicht jeder*jedem taugt. Ich habe einige Menschen, die eine Weile so gelebt haben, unter dem Gewicht des realen Stresses krimineller Aktivitäten zusammenbrechen sehen. Wenn ich also diese Worte über Kriminalität schreibe – ebenso wie meine Verachtung gegenüber Identitätspolitik – spreche ich nur für mich selbst. Als ich anfing „Descending into Madness“ zu schreiben, war ich in der selben Nacht mit zwei Taschen im Wert von je über 300 Dollar aus einem REI in Seattle gelaufen. Die Sicherheitstürme schlugen keinen Alarm, als ich mit zwei Diebstahlsicherungen durch sie durchging. Ehe ich hinauslief dachte ich bei mir selbst, dass meine kriminellen Aktivitäten nahelegen, dass ich dem Wahnsinn verfallen [descending into madness] sei, weil das zu versuchen verdammt noch mal verrückt war. Aber ich war erfolgreich. Und auf der Autofahrt nach Hause bemerkte ich, dass wenn ich solche mutigen Verrücktheiten nicht befördern würde, ich möglicherweise niemals bemerkt hätte, dass die Sicherheitstürme in diesen Geschäften nicht funktionierten.
Meiner Meinung nach führt die „woke Führung“ des Linksradikalismus den Anarchismus über eine Klippe in einen zunehmenden Zerfall\footnote{Nun, ich denke wenn dem so wäre, so wäre \emph{der} Anarchismus ja auch irgendwie selbst schuld, wenn \emph{er} sich von irgendetwas oder irgendwem „führen“ lässt. Aber vielleicht ist es – bei allem Verständnis für Polemik – auch ein wenig über die Strenge geschlagen, Anarchist*innen (in den USA und sonstwo) so sehr über einen Kamm zu scheren (Anm. d. Übers.).}. Vor Angst und Scham, die von einer neuen Ordnung erzwungen wird, werden einige Anarchist*innen es niemals zu Selbstemanzipation oder unabhängigem Denken als Zurückweisung der Autorität eines Gruppendenkens bringen. Viele Menschen bezeichnen sich trotz einer engstirnigen, liberalen Definition von Anti-Diskriminierung selbst als Anarchist*innen – einer Definition, die Anti-Diskriminierung auf die moralistischen, humanistischen Grenzen der zivilisierten Gesellschaft beschränkt. Es ist kein Zufall, dass die meiste anti-diskriminatorische Praxis einen staatlichen Apparat erfordert, um Gesetze durchzusetzen, die gleiche Rechte schaffen. Und während es an gleichen Rechten für alle Menschen im Kapitalismus nichts auszusetzen gibt, wird mit diesem Sieg die staatliche Reform statt des antiautoritären Angriffs gefeiert. Und an der Spitze dieser staatlichen Macht stehen die „Community-Anführer*innen“ oder diejenigen, die kein Interesse daran haben, Herrschaft zu kritisieren. Stattdessen haben sie ihre soziopolitischen Karrieren mit belanglosen Reformen im Namen „der Gemeinschaft“ gemacht und Radikale diffamiert – indem sie sie als Aufrührer*innen bezeichnen. Und hinter diesen Anführer*innen stehen „weiße“ anarchistische Allies, verwirrt und frustriert, die sich zwischen den Alternativen entscheiden müssen, ein*e Rassist*in genannt zu werden, weil sie die Scheiße in Brand stecken oder eine*n gute*n Ally genannt zu werden, weil sie den Arsch eines*einer „schwarzen“ Predigers*in küssen.
\begin{quote}
Was du oder ich als „taktisch“ oder nicht bezeichnen, ist nicht wirklich relevant. Dies ist weniger ein Krieg im herkömmlichen Sinne als ein Sturm – unkontrollierbar und chaotisch. Das ist eines der Probleme mit der linken Charakterisierung „der Bewegung“ als etwas Einheitliches, Monolithisches und ideologisch Konsistentes. Das ist sie nicht. Das wird sie nie sein. „Die Bewegung“ besteht aus einer Million Individuen mit ihren eigenen individuellen Ansichten, Meinungen und Handlungen und es hilft niemandem irgendetwas, wenn du jede*n verspottest, die*der die Dinge nicht genauso macht, wie du es gerne hättest. – Baba Yaga
\end{quote}
\section{Ein anderes Wort für „Schwarze*r Anführer*in“ [Black Leadership] ist Autoritarismus}
\emph{Am Ende unserer Demo kommen wir am Dritten Revier an der Ecke East Lake St.\Slash{}Minnehaha Ave an. Organisator*innen von Black Lives Matter beginnen etwas über Forderungen, gemischt mit einigen Gebeten und dumpfen Anfeuerungsrufen in ihr Megafon zu heulen. Ich bemerke, wie sich jemand hinter mir langsam heranschleicht und anfängt mit seiner Faust gegen das Fenster zu schlagen. Weil sie fürchten, es könne zerbrechen, beginnen drei in der Nähe Stehende leise, ihn zu ermahnen: „Das ist nicht der richtige Ort dafür, bleib freidlich!“ Die Person antwortet leise, aber mit wütender Anspannung in seiner Stimme, „Das ist das verdammte Problem, Ihr Motherfucker wollt nie etwas tun, außer zu marschieren und Sprechchöre zu rufen.“ Entmutigt läuft er weg. „Ich bin deiner Meinung, wirklich“, sage ich zu ihm. „Das ist es, was läuft – Scheiß auf den anderen Scheiß,“ antwortet er und geht weg. Ungefähr eine Minute später verliere ich die Geduld, der BLM Rede übers Friedlichsein zuzuhören und entscheide, nach dieser Person zu suchen. Ich gehe um die Ecke zur Hinterseite der Polizeiwache und bemerke einen Aufruhr. Eine Gruppe von 5-7 „schwarzen“ Typen blockieren die hinteren Glastüren der Polizeiwache und diskutieren mit rund 20 „schwarzen“ und „braunen“ wütenden Jugendlichen – inklusive dem von vorhin. Unfähig meine eigene Frustration an mich zu halten, lasse ich mich ebenfalls auf eine Diskussion mit den Polizei-Verteidiger*innen ein. Schließlich, inmitten des Gebrülls beginnen einige „schwarze“ und „braune“ Jugendliche damit „fuck 12“\footnote{„Fuck 12“ bedeutet soviel wie „Fuck the Police“ [Fick die Polizei]. Die 12 bezieht sich dabei meines Wissens nach ursprünglich auf die Cop-interne – moglicherweise lokale – Einheitsnummerierung für das Drogendezernat (12). Die Bezeichnung wird allerdings allgemein für alle Bullen verwendet (Anm. d. Übers.).} neben die Blockade zu sprühen. Von der Menschenmenge hinter mir, die sich mittlerweile verdreifacht hat, ertönt Jubel. An den Türen bricht ein Gerangel los und dann zerschmettert ein einzelner Stein ein Fenster der Polizeiwache, gefolgt von einem Hagel aus Steinen, Gehwegplatten, Flaschen und allem anderen in Reichweite. Die Gruppe von 5-7 „schwarzen“ Pazifist*innen schreien vor Verzweiflung, um die Zerstörung zu stoppen, gehen sogar soweit, Menschen physisch festzuhalten, aber werden schließlich überwältigt. Sie versuchen die bereits geworfenen Steine aufzusammeln und finden sich dabei in zahlreichen physischen Konfrontationen wieder. Menschen von vor dem Gebäude rennen herbei und stimmen in den Vandalismus mit ein. Nachdem alle Fenster eingeworfen sind, bewegt sich die Meute auf den Parkplatz der Polizei zu und beginnt, die Polizeifahrzeuge zu zerstören. Ich schnappe gerade nach Luft, als ich eine Blendgranate explodieren höre. Die Polizei kommt aus einer anderen Tür gerannt und beginnt, Gummischrot und Tränengas zu verschießen. Die Meute wird versprengt, aber unter hysterischem Lachen der Freude und Erfüllung. Das Dritte Revier liegt in Trümmern – und ich ahnte nicht, dass das erst der Anfang sein sollte.}
Am nächsten Tag tauchte eine größere Menge vorrangig „schwarzer“ und „brauner“ Jugendlicher auf und setzte den Krieg gegen das Dritte Revier fort. Bis zur Nacht wurde von den Menschen auf diesen Straßen ein Radius von drei Meilen von der Kontrolle der Polizei befreit. Das Dritte Revier wurde aufgebrochen und gestürmt. Die Polizei verließ das gesamte Areal. Ihre Wache wurde in Brand gesteckt und Polizeifahrzeuge wurden auf die Straße gefahren und abgefackelt. In ein Gebäude, das an den Parkplatz angrenzte, wurde eingebrochen und dieses wurde mit anderen Läden nebenan in Brand gesteckt. Die Menschen feierten ihren Sieg, indem sie ihre Waffen in die Luft abfeuerten. Fremde sangen und tanzten um ausgebrannte Polizeifahrzeuge, gaben sich im Vorbeilaufen High-Fives und teilten geplünderte Lebensmittel untereinander. Vor brennenden Gebäuden standen angeregt plaudernde Menschen, während andere Steine auf die Überreste von Ladenfenstern warfen, um Zielen zu üben.
Auch wenn das Ganze wie eine perfekte Utopie ausgesehen haben mag, so war es doch nicht von der Realität entkoppelt. Zwischen kleinen Fraktionen von Menschen brachen Kämpfe aus und lange währende persönliche Konflikte wurden in den nun Cop-freien Straßen geklärt. Ladeninhaber*innen schossen auf Plünderer*innen und töteten sie und Sozialwohnungen wurden niedergebrannt. Aber das ist der Unterschied zwischen den zuckerüberzogenen Lehrbuch-Ideologien der Politik und roher, unvermittelter Wut. Die Revolution folgte keinen Lehren von Mao oder religiösen Botschaften eines Gottes. Die Feuer, Plünderungen und Angriffe auf die Polizei brauchten keinen Marxismus, kein Transkript \emph{des kommenden Aufstands} oder einen akademischen Kurs zur Geschichte des Anarchismus. Alles Benötigte war der chaotische Ausdruck von Wut gegen die Repräsentationsformen von Herrschaft.
Wie es zu erwarten war, gaben viele Menschen im Internet – darunter auch viele selbstbezeichnende Anarchist*innen – ihr Urteil zu der Situation ab – die meisten davon aus einer ideologischen Position, die Gleichförmigkeit einen Wert einräumt und „akzeptablen“ Formen der Revolte engstirnige Grenzen setzt. Meiner Erfahrung nach entwickeln sich Aufstände wie dieser am Besten, wenn sie nicht kontrolliert oder organisiert werden. Je mehr der Ausdruck der Wut kontrolliert und organisiert wird, desto weniger anarchistisch wird er – und wird schließlich befriedet, um einer bestimmten politischen Vision in den Sattel zu verhelfen. Für mich ist das nicht wünschenswert und außerdem unrealistisch. Zerstörung ist Zerstörung, Gewalt wird immer Gewalt sein und von einem Aufstand irgendetwas Geringeres zu erwarten ist bestenfalls naiv. Während einige am Rand sitzen und spezifische Taktiken und Formen des emotionalen Ausdrucks moralisieren, missachten sie die Realität, dass ein ausgewachsener Krieg keine inhärente Moral kennt. Geschäfte, die vernagelt und als „Schwarzen gehörend“ ausgewiesen wurden, wurden nicht durch irgendeine moralische Überlegung ausgespart; auch in sie wurde eingebrochen, sie wurden geplündert und anschließend niedergebrannt.
Außerdem hat die Polizei meiner Meinung nach umso weniger Möglichkeiten, sich den Protesten anzupassen und sie zu dominieren, desto unkontrollierbarer und unverwaltbarer ein Aufstand bleibt. Die Polizei hatte nicht die geringste Kontrolle über hunderte Individuen, die so chaotisch rebellierten, dass sie sie überwältigten und in die Flucht schlugen.
Während der nächsten Tage fanden Angriffe gegen das 5. Polizeirevier statt, während Liberale, Pazifist*innen und Identitätspolitiker*innen sich still zurückzogen, um sich für ihre Unfähigkeit, den ersten Riot zu kontrollieren, zu rächen. Das Internet wurde zu ihrem Ausgangspunkt für eine der schlimmsten Lügenkampagnen und Panikmache, die ich je gesehen habe.
Als die Siege brennender Bullenwägen und Polizeiwachen online kursierten, traten Liberale von überall aus den Staaten auf die Bühne, in einem verzweifelten autoritären Versuch, ihre ideologische Moral und ihr politisches Programm durchzusetzen. Sie verbreiteten ein Narrativ, in dem jede*r, die*der an den Sabotageakten teilnimmt, als „Nazi“ [white supremacist] oder „Undercover Bulle“ gebrandmarkt wurde, die*der den Aufstand „infiltrieren“ würde.
Bei einigen dieser Liberalen handelt es sich um die gleichen „schwarzen“ Personen, die gescheitert waren, „schwarze“ und „braune“ Rebell*innen an Plünderungen und der Zerstörung von Eigentum zu hindern. Sie waren daran gescheitert, alle „weißen“ Menschen davon zu überzeugen, die Riots zu verlassen (weil sogar einige „weiße“ Menschen wussten, dass nicht alle „schwarzen“ und „braunen“ Menschen ein Problem mit ihrer Anwesenheit hätten, sondern sie als Kompliz*innen schätzten). Und in dem Versuch, kapitalistische, reformistische Werte zu wahren, strebten Liberale aller Hautfarben danach, die Plünderungen und den Vandalismus zu stoppen, indem sie die sozialen Medien mit offensichtlich falschen Informationen fluteten. Diese falschen Informationen sind gespickt mit Schlagwörtern wie „Agent Provocateur“ und „Nazis“, um die Leser*innen emotional dazu zu bringen innerhalb einer falschen Dichotomie Seite zu beziehen. Und diejenigen, die nicht physisch auf den Straßen sind oder zusammen mit der Polizei die Rebellen bekämpfen, sind die Zielgruppe dieser beschränkten, ungenauen Darstellungen der Realität.
Unterschiedliche ideologische Motive führen zu unterschiedlichen Interpretationen der Ereignisse. Und da Liberale und Pazifist*innen dazu tendieren, die sozialen Medien zu dominieren, während andere auf den Straßen beschäftigt sind, haben sie [dort] einen Vorteil. Da Liberale \emph{alle} Personen of Color moralisch als gehorsame und aufopferungsvolle Held*innen framen, haben die meisten Menschen Schwierigkeiten, sich klarzumachen, dass Personen of Color verantwortlich \emph{für} die Zerstörung von Eigentum und die Teilnahme an gewaltsamen Formen des Protests sind. Das spielt ebenfalls mit, wenn „weiße“ Personen für als moralisch verwerflich betrachtete Formen der Revolte verantwortlich gemacht werden. Riots\Slash{}Aufstände sind niemals vollkommen utopisch und angenehm. Sie sind die gefährlichen Elemente der Befreiung, die auftreten, wenn alle anderen Möglichkeiten gescheitert sind. Egal ob Menschen Angst vor Gewalt haben oder nicht, wird das nichts an der Tatsache ändern, dass die Polizei tötet und auch weiterhin töten wird, solange das Konzept der Vollstreckung von Gesetzen existiert. Meiner Meinung nach gibt es keine „Verbesserung“ der Polizei und es gibt keine „Gerechtigkeit“, wenn sich jemand bereits sechs Fuß tief unter der Erde befindet.
Und Polizist*innen sind nicht alle „weiß“. Auch „schwarze“ Cops töten „schwarze“ Menschen.
Das Schlimmste an der Online-Deutung der Ereignisse ist, dass die Menschen, die diese Falschinformationen verbreiten, der Online-Welt nicht auch die Freude, das Lächeln, das Singen und Tanzen der Rebell*innen diverser Hautfarbe vermittelten, als diese die Zerstörung des 3. Polizeireviers feierten.
Scheiße, stell dir vor du wärst eine Person of Color, die ihr Leben lang von der Polizei schikaniert wird und dann kommt der Tag und die Nacht, in der du tatsächlich eine Polizeiwache brennen siehst und die Polizei sich vollkommen aus diesem Gebiet zurückzieht. All das wird aus der Geschichte getilgt, wenn Liberale das einer Gruppe von Menschen – Rassist*innen – zuschreiben, die gar nicht Teil dieser Kämpfe waren.
Bis heute, verbreiten noch immer Menschen diese Verschwörungstheorien im Internet, etwa das berühmte „Ziegelstein-Köder“-Video, in dem Cops (hinter ihrem eigenen Gebäude, nicht in einer Allee, wie ursprünglich behauptet) Ziegelsteine entladen. Auch wenn ich nicht mit absoluter Sicherheit sagen kann, dass \emph{gar} keine Nazis anwesend waren (ich meine, ich konnte einige in Pickups vorüberfahren sehen, die white power-Scheiße riefen und auch einen „braunen“ Typen, der in einem Truck vorüberfuhr und Pro-Polizei-Slogans rief und eine Konföderierten-Flagge schwenkte), aber ich habe sicherlich keine von ihnen während der Kämpfe gesehen. Ich habe „schwarze“ Menschen einander unterhaken sehen, um die Riot-Polizei zu beschützen, ich habe weiße Allies gesehen, die andere „weiße“ Menschen der Polizei übergaben, im Namen der Unterstützung „Schwarzer“, und schließlich wie die Polizei die Kontrolle zurückgewann und diese befriedenden Bemühungen nutzte, um \emph{friedliche} Protestierende niederzuknüppeln.
\section{Ungezähmte Delinquenz}
Meiner Meinung nach offenbaren die letzten Monate die Schwächen der Zivilisation ziemlich offensichtlich. Als panische Reaktion auf soziale Spannungen und spontane Ausbrüche illegaler Aktivitäten hat die Kontrolle der Regierung zugenommen. Covid-19 durchbrach die Ordnung der täglichen Produktivität und der zivilisierten Sklaverei und verschaffte den Menschen mehr Zeit, um über ihre Leben nachzudenken und den Wert ihrer freien Zeit außerhalb der Arbeit schätzen zu lernen. Die Aufstände in Reaktion auf die Ermordung George Floyds offenbarten die Schwächen der polizeilichen Macht und Kontrolle – selbst in ihrem Hoheitsgebiet. An diesem Punkt habe ich keine Vorstellung, was als Nächstes kommt.
Ich gebe zu, es faszinierend zu finden, nichtmenschliche Lebewesen und die Natur inmitten der industriellen Verzweiflung gedeihen zu sehen. Klarere Himmel, verschiedene Tiere auf den Straßen, Überschwemmungen, die die Grundmauern dieser Betonwüste lockern. Ich kann nicht umhin, beides, die Pandemie und diese andauernden Brüche mit der Autorität besser als eine Rückkehr zur Normalität zu empfinden, einer Normalität, in der der Tod durch die industrielle Zivilisation und den Staat ebenso Routine ist wie in einem Schlachthaus.
Ich frage mich, welche Art von Unterhaltungen die Menschen miteinander oder mit sich selbst während dieser erblühenden Destabilisierung der domestizierten Ordnung führen. Werden mehr und mehr Menschen diese Gelegenheit ergreifen, um ihrem Ärger und ihrer Frustration durch zufällige Akte der Gewalt und Sabotage gegeneinander Luft zu machen? Gegen die Vollstreckung der Gesetze? Gegen die Institutionen, die aufgrund der finanziellen Einbußen geschwächt sind und nun anfälliger sind als jemals zuvor? Ich kann nur hoffen, dass die Aufstände in irgendeiner Form weitergehen – offen oder im Untergrund, was für mich persönlich wünschenswerter wäre.
Werden die Menschen auf die Rückkehr der alltäglichen Misere der Monotomie hoffen oder werden sie die Tiefen permanenter Unsicherheit erforschen? Zur Arbeit zurückkehren oder zur Ungezähmtheit? Ich vermute nur die Zeit wird darüber Aufschluss geben.
Aber hier kann ich nur für mich selbst sprechen. Meine Anarchie ist die meine, ebenso wie es meine Gedanken und Worte in diesem Text sind. Ich schreibe nicht, um irgendeinen Club von Internet-Anarchist*innen zu beeindrucken, die mit intellektuellen Texten jonglieren, um in Selbstlob zu verfallen. Ich mache mein Tagebuch öffentlich in dem antagonistischen Versuch, das Opfer- und anti-individualistische Narrativ des Linksradikalismus, das den gegenwärtigen Anarchismus dominiert, zu verhöhnen.
Ich sehne mich nicht nach einer Rückkehr zur Normalität und der alltäglichen Misere industrieller Produktion. Ich habe kein Verlangen danach, lächerliche „Siege“, wie das zur Verantwortungziehen der Polizei, Entlassungen oder Gefängnisstrafen zu feiern, die nur von der Wiedererrichtung ihres zerstörten Reviers oder möglicherweise eines ebenso autoritären „community-basierten“ Ersatzes gefolgt sein werden. Ich sehne mich nach nichts anderem als nach der totalen Abschaffung von \emph{jeder} Regierung und Normierung. Und vermutlich werden diejenigen, die irgendeine Form elitistischer Macht besitzen mich unbequem finden und eine Diffamierungskampagne gegen mich starten, um meine Schriften und mich aus ihrer Bewegung zu verbannen. Aber sie haben keine Ahnung, dass die Tage und Nächte zwischen weiten Feldern und den Sternen und zwischen Baumwipfeln und dem Boden das Terrain meines Abenteuers ist! Und damit einher geht eine Wonne, die Anarchie als eine pulsierende Lebenserfahrung ausmacht, anstatt einem Maß von digitalem Sozialkapital oder einer eingefrorenen Theorie aus einem akademischen Journal.
Das Internet hat eine Kultur geschaffen, die verzweifelt nach sozialer Kontinuität und digitaler Bestätigung sucht. Es ist die Brutstätte für „neue“ Konzepte des Anarchismus, die nichts weiter sind als kommunistische Leichen mit Hipster-Ästhetik. Antizivilisatorische Anarchie durchsetzt vom Linksradikalismus zeigt nun die Ausdehnung ihrer Macht durch endlose Twitter-Debatten über „Ökofaschismus“. Twitter – ein Ort wo die Wiederaneignung des eigenen Lebens und Körpers durch die Jünger*innen der Privilegien-Politik beschämt wird – ist ein Friedhof an Stimmen, die ihren eigenen Tod-durch-Internet verherrlichen.
Mein Animalismus strebt nicht danach, das Aussehen und die Verhaltensweisen existenter Tiere zu adaptieren. Stattdessen ist er die Silhouette einer illegalistischen, ungezähmten Bedrohung, die um die brennenden Gefängnisse der Domestizierung tanzt. Meine Ablehnung der Opferrolle ist eine Absage sowohl an die Mitleidspolitik moralitätsbasierter Organisierung, als auch an das Heiligtum der Unschuld. Meine Anarchie ist ein Nekrolog auf die Identitätspolitik. Sie ist ein persönlicher Aufstand ohne Zukunft, ein Traum ohne die Anästhesie der Hoffnung, ein Ausdruck der Wonne mit der Lebensdauer einer explodierenden Bombe.
\emph{Dieser Text ist all den Rebell*innen gewidmet, deren einzige Interaktion mit Autorität in Feuer und Zerstörung besteht \dots{} Ich bin für immer von eurem mutigen Zorn jenseits rassifizierter und vergeschlechtlichter Grenzen inspiriert \dots{} Gewidmet der Jugend, die am 26. Mai Geschichte schrieb, den Rebell*innen, die umkamen und denen, die derzeit für ihre Beteiligung an diesem Krieg gegen den Staat gefangen gehalten werden. RIP George Floyd}
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Anarchistische Bibliothek
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Anticopyright
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Flower Bomb
Ein Nekrolog auf Identitätspolitik
2020
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Zündlumpen \#66; Entnommen am 11.03.20121 von \href{https://zuendlumpen.noblogs.org/post/2020/06/27/ein-nekrolog-auf-identitaetspolitik/}{https:\Slash{}\Slash{}zuendlumpen.noblogs.org\Slash{}post\Slash{}2020\Slash{}06\Slash{}27\Slash{}ein-nekrolog-auf-identitaetspolitik\Slash{}}
\emph{Übersetzung des englischen Originaltextes „An Obituary for Identity Politics“ von Flower Bomb, ursprünglich erschienen beim Warzone Distro.}
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https://latex-cookbook.net/tex/minimal-plot.tex | latex-cookbook.net | CC-MAIN-2021-31 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2021-31/segments/1627046153709.26/warc/CC-MAIN-20210728092200-20210728122200-00514.warc.gz | 365,830,537 | 1,280 | % Plotting a function with minimal axes
% Author: Stefan Kottwitz
% https://www.packtpub.com/hardware-and-creative/latex-cookbook
\documentclass[border=10pt]{standalone}
%%%<
\usepackage{verbatim}
%%%>
\begin{comment}
:Title: Plotting a function with minimal axes
:Tags: Mathematics;Functions;Plots;Graphics;PGFPlots;Chapter10
:Author: Stefan Kottwitz
:Slug: minimal-plot
We plot a sine function with PGFplots, similar to:
http://latex-cookbook.net/cookbook/examples/grid-plot/
While a grid and a lot of ticks can be useful to inspect
specific values of a function, an overall view onto
a function can be nicer with reduced axes, maybe even
shifted away a bit.
There's a style which you can use. If you download the file at
http://pgfplots.net/media/tikzlibrarypgfplots.shift.code.tex,
and put it into your document folder, you can load this
style in your preamble, as written below.
Then, just change the axis option grid to shift.
The code, and more, is explained in the LaTeX Cookbook,
Chapter 10, Advanced Mathematics, Plotting functions
in two dimensions.
\end{comment}
\usepackage{pgfplots}
\usepgfplotslibrary{shift}
\begin{document}
\begin{tikzpicture}
\begin{axis} [shift, xtick = {-360,-270,...,360}]
\addplot [domain=-360:360, samples=100, thick] { sin(x) };
\end{axis}
\end{tikzpicture}
\end{document}
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%% The LaTeX Graphics Companion, 2ed (first printing May 2007)
%%
%% Example 5-9-13 on page 259.
%%
%% Copyright (C) 2007 Michel Goossens, Frank Mittelbach, Denis Roegel, Sebastian Rahtz, Herbert Vo\ss
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http://www.public.iastate.edu/~vardeman/stat305/webchap1forms.tex | iastate.edu | CC-MAIN-2018-26 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2018-26/segments/1529267867095.70/warc/CC-MAIN-20180624215228-20180624235228-00010.warc.gz | 478,594,530 | 1,386 | %% This document created by Scientific Word (R) Version 3.5
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\begin{document}
{\Large Chapter 1 Formula Sheet}
\bigskip\bigskip
engineering statistics
\bigskip
observational study
\bigskip
experimental study
\bigskip
enumerative study
\bigskip
analytical study
\bigskip
population
\bigskip
sample
\bigskip
categorical data
\bigskip
quantitative data
\bigskip
univariate data
\bigskip
multivariate data
\bigskip
paired data
\bigskip
factors
\bigskip
levels
\bigskip
combinations
\bigskip
complete factorial study
\bigskip
fractional factorial study
\bigskip
validity
\bigskip
precision
\bigskip
accuracy
\bigskip
mathematical model
\bigskip
probability/stochastic model
\end{document} |
https://hallaweb.jlab.org/experiment/E08-027/meetings/030308.tex | jlab.org | CC-MAIN-2022-05 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-05/segments/1642320301309.22/warc/CC-MAIN-20220119094810-20220119124810-00518.warc.gz | 353,641,500 | 5,102 | \documentclass[12pt,epsfig]{article}
\usepackage{graphicx}
\begin{document}
\pagestyle{empty}
\section{March 03, 2008}
{\bf Present:} Paul Brindza, Alexandre Camsonne, J.P. Chen, Don Crabb, Al Gavalya,
Kees De Jager, John LeRose, Mike Seely, Karl Slifer.
%\begin{itemize}
%\item
This was the first Hall A planning meeting for the $g_2^p$ experiment.
Kees explained the purpose of this and subsequent meetings:
to define the resources needed
to implement the experiment (manpower/money) and to find a way in which this
experiment can be made compatible with QWeak.
Because of Qweak, Kees points out that running with the cryo septa may
be difficult. Asks for spokesmen to provide justification for why the
warm septa can not be used. Kees also requests that we provide a solution
in which the beam can be run `straight-thru' the chicane for ease of
transition to subsequent experiments which do not use the chicane.
%\item
We presented a short overview of the experiment and the
necessary equipment. During this presentation, several issues were discussed
by the group:
\begin{enumerate}
\item {\bf Target: } The target roots pumps will need to be serviced prior to the run ( balance the rotors, service the motors). There may be money in the budget next year to do this.
Kees requests details of target operation. For example, how often do we have to exchange
target material?
Hall C used three platforms. The only platform that is really essential
is the platform that contains the target support ring.
Other equipment could probably be located on the floor.
The buffer dewar must be close to target. Roots pumps can be on the floor.
We should ensure that any new platforms are
compatibile
with Hall A space constraints (ie. produce schematics and drawings).
We will need a well shielded location for the EIO power supply.
\item {\bf Beamline: }
Alexandre is working with Jay on the beamline design.
He points out that
all planned changes are downstream of the Moller quad. The Moller detector is unmoved,
but the ``French Bench'' has to be pulled. The question is raised as to whether the
beamline schematic accounts for the upstream target shift.
Paul suggests to use the Moller dipole as the first chicane magnet, by
changing the polarity. This would require a straight-thru path for standard
Moller operation. This will be investigated further.
The planned chicane supports will obstruct use of the man-lift on the far side of
the hall.
Hall C has requested that we locate a replacement magnet for HKS magnet if
we want to use it for our chicane. There appears to be many workable replacements.
We plan to move the target back 140 cm from the pivot center
in order to allow some space between the two septa for passage of the beam.
This is 60 cm more than was done previously. The question is raised as
to whether this coincides with the location of the distribution
can support beam.
The overhead clearance above the target is complicated by moving the target upstream.
The polarized target is not much different in size
from the standard cryotarget, but shifting up stream means
its very tight overhead.
We need to produce diagrams/drawings to ensure that the target can be located
where we have planned.
({\bf Note:} After the meeting we asked Al Gavalya to check
using his Hall A model and a model of the SANE target can. There
appears to be sufficient clearance infront of the post to put the target
at -140cm. This must be verified in the hall.
The beam spotsize depends on energy and materials traversed.
With rastering and multiple scattering we expect atleast
a 3 cm beamspot after the target.
Retracting the target to -140 cm allows the cold septa to be
separated by 12 cm ( 2*60 cm * tan(6) = 12 cm ).
If we can only retract the target to -120 cm we will still have 8 cm space
between the cold septa.
The yoke of the warm septa would need to be modified to pass the beam. Also, the
warm septa coils are only separated by 3 cm in the present design.
We'll need to perform a full simulation to verify the beam can pass for either
choice of septa.
%Paul says that when the septa are at the lowest possible angle, the
%Q1's are touching. Have to back off to get the beam through. The lowest momentum
%beams make it through the Septa gap, but crash into the Q1s downstream.
%Paul : might be ok to hit downstream of $Q^2$, but before that would be bad.
%Ed doesn't think there is enough space to get the rastered beam and halo
%through to a dump. Size of opening is really small.
%John points out that we have to calculate the multiple scattering for the
%configuration we have and whether we have sufficient horizontal clearance.
%Paul : Cryogenic Septa touch at six degrees, so the minimal angle possible would be
%something like 7 degrees. %33:53.
%J.P. points out that we have moved back an additional 60 cm for just this purpose.
%However, Ed believes that this coincides with the location of the distribution
%can support beam. We will check with Al.
We should remove the whole downstream beampipe section instead of modifying hot pipe.
There is a blue beam which supports the
downstream pipe and carries utilities. It would interfere with the beamdump
if the dump
is below the beamline.
Ed notes that the beamline supports have to come out along with the shielding
at the dump.
The instrumentation and controls that go back to support the dump will have
to be rerouted.
\item {\bf Septa :} We discussed 4 possible Septa options:
\begin{enumerate}
\item Two Cold Septa: This is what we assumed in the proposal and is the ideal situation from physics standpoint, but requires repairs
to the right septum. It may also be difficult to run both septa during QWEAK.
\item One Cold Septa: Allows us to obtain all physics goals, and reduces cryo needs.
Would require some additional
beamtime. Minimizes repair overhead. Lose valuable cross check of two arms.
Paul points out that if we are using only one septa, we should warm up the right
HRS to reduce the load.
%could turn off the right HRS, which has a huge load (15-20 g/s).
\item Two Warm Septa: In this scenario, we lose a significant portion of high $Q^2$ portion of the proposal.
We need to follow up with Paul on the possibility of using the warm Septa
at angles greater than 8 degrees, by moving the target.
The warm septa will not pass the primary (vertically deflected) beam
as they are now configured since the vertical
deflection angle can be as great as 11 degrees.
The design would need to be modified to accomodate this.
%In order to use the warm septa with the
%chicaned beam, we'll have to
%remove a slice out of the yoke.
John points out we will lose acceptance
and probably have worse systematics.
%47:45
\item Two Warm Septa plus a HRS only run.
Good for physics, but requires significant overhead to remove the septa and
relocate the target. In the displayed scenario, there is a gap introduced
in the $Q^2$ coverage, but it could easily be filled with an additional
energy.
% Kees and Ed point out that once you move
% the target, the chicane beampipe won't be aligned anymore.
Kees and Ed point out that if you move the target, the beam pipe
(girder) between the second chican magnet and the target chamber
will no longer be aligned.
\end{enumerate}
\end{enumerate}
\section{Conclusions/Open Issues}
\begin{enumerate}
\item Ed estimates 5-6 month installation time. Kees says this fits with the planned
6 month down for 12 GeV upgrade.
\item Regarding cryo consumption: The buffer dewar requires something like 1 g/s. The
target needs something like 5L/hr (5\% impact on hall).
Qweak has a two kW target. A rough estimate is that they will require half of the ESR
capacity plus all of the
available CHL capacity.
This leaves 1/2 of the ESR capacity for Hall A. Hall B has no impact on this.
%With 1/2 of the ESR, Hall A can run the HRS with 10 uA cryotarg, nothing more.
\item Cryo Septa: Ed estimates 1 man-year to repair the right Septum.
Ed will test the flow needed for left septum this summer (time permitting before
Transversity which runs mid October to mid January).
He believes it is best to do this in the Hall. Some of the Septa controls have
been scavenged and have to be located.
Ed does
not have time to train the left septum this summer. This can possibly be
done during transversity, but only if right Q1 can be warmed up.
\item Open Questions:
\begin{itemize}
\item We need a description of target operations.
Specifically what happens if we loose the ESR? Does the traget get
warmed up and do we then have to replace the target? We might not
have the overhead room to do that in place, etc.
\item We need to get realistic drawings of the target layout with supports in the Hall.
\item What modifications are needed to the chicane if the target is at the standard position?
\item We need to discuss with Jay to get details of the movable beam pipe.
\item Passage of beam after the target should be verified via simulation.
\item How far upstream can we move the target?
\end{itemize}
\item We will meet again in April or May.
\end{enumerate}
%\end{itemize}
\section{Note}
We later met with Al Gavalya, Joyce Miller and Bert Metzger. Bert provided
CAD models of the SANE target can to Al who then placed them in the
Hall A model. In the model it appears that there is sufficient space to retract the
target to 140 cm upstream as planned without interfering with the support
beam. Al and Joyce will work (as their schedule permits) to create full target
models and verify this completely. We will also perform visual inspection
in the hall.
Also Don Crabb informed us that the target `donut' which takes much of
the overhead clearance can actually be rotated to horizontal ( pointing to the left HRS side).
%\begin{figure}
%\begin{center}
%\includegraphics[angle=0,width=1.0\textwidth]{}
%\caption{\label{}}
%\end{center}
%\end{figure}
%
\end{document}
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http://build.openvpn.net/doxygen/latex/dir_a0910e16ec470873ec69e2574eb80ec7.tex | openvpn.net | CC-MAIN-2018-13 | application/octet-stream | application/x-tex | crawl-data/CC-MAIN-2018-13/segments/1521257647251.74/warc/CC-MAIN-20180320013620-20180320033620-00160.warc.gz | 53,118,138 | 766 | \hypertarget{dir_a0910e16ec470873ec69e2574eb80ec7}{}\subsection{/root/openvpn/src/plugins Directory Reference}
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\subsubsection*{Directories}
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directory \hyperlink{dir_a27999ce7ae039ffcae98618682a8d0f}{auth-\/pam}
\item
directory \hyperlink{dir_c88459c12ea28212540b37ae6e8f707a}{down-\/root}
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\title{Contrôle commun}
\date{~-01-12-10-}
\author{Terminales ES, 2010-2011}
\begin{document}
\Large
\renewcommand{\labelitemi}{$\star$}
\maketitle
\section*{Problème commun- (11 points)}
\noindent Soit $f:\R-\{0\}\rightarrow\R,~x\mapsto\frac{x^3+7x-3}{x^2}$.\\
\noindent Soit $\mathcal C_f$ la courbe représentative de $f$ dans un repère orthonormé d'unité $0.5$cm. \\
\begin{enumerate}
\item Déterminer la limite de $f$ en $0$. Interpréter géométriquement le résultat.
\item
\begin{enumerate}
\item Calculer la limite de $f$ en $+\infty$ puis en $-\infty$.
\item Démontrer que pour tout $x\neq 0$, $f(x)=x+\frac 7x-\frac 3{x^2}$.
\item Démontrer que la droite $\Delta$ d'équation $y=x$ est asymptote à $\mathcal C_f$ en $+\infty$ et $-\infty$.
\item Étudier la position relative de $\mathcal C_f$ et $\Delta$.
\end{enumerate}
\item
\begin{enumerate}
\item Montrer que pour tout $x\neq 0$, $f'(x)=\frac {(x+3)(x-1)(x-2)}{x^3}$.
\item Dresser le tableau de variations de $f$.
\end{enumerate}
\item
\begin{enumerate}
\item Montrer que l'équation $f(x)=0$ admet une solution unique $\alpha$ sur l'intervalle $]0.1;1[$.
\item Montrer que $f(x)=0$ admet $\alpha$ pour seule solution sur $\R$.
\item Donner une valeur approchée de $\alpha$ à $0,1$ près
\end{enumerate}
\item
\begin{enumerate}
\item Déterminer l'équation de la tangente $T$ à $\mathcal C_f$ au point d'abscisse $-1$.
\item Montrer que $\mathcal C_f$ possède une tangente $T'$ à $\mathcal C_f$ parallèle à $\Delta$ et déterminer son équation.
\end{enumerate}
\item Représenter $\Delta$, $T$, $T'$ et $\mathcal C_f$ dans un repère orthonormé d'unité $1$cm.
\end{enumerate}
\newpage
\section*{Exercice commun. (4 points)}
\medskip
\emph{Cet exercice est un questionnaire à choix multiples (QCM). Pour chacune des questions, une seule réponse est exacte. Le candidat notera à chaque fois sur sa copie le numéro de la question suivi de la proposition qui lui semble correcte. Aucune justification n'est demandée.}
\medskip
\emph{Le barème sera établi comme suit :
\setlength\parindent{10mm}
\begin{itemize}
\item[] pour une réponse exacte aux questions 1, 2, 3 et 4 : 0,5 point,
\item[] pour une réponse exacte aux questions 5 et 6 : 1 point,
\item[] pour une réponse fausse ou l'absence de réponse: 0 point.
\end{itemize}
\setlength\parindent{0mm}}
\medskip
Pour toutes les questions, on considère la fonction $f$ définie sur l'intervalle $]-1; + \infty[$ par : $f(x) = 2 - \dfrac{1}{x + 1}.$
On appelle $\mathcal{C}$ sa courbe représentative dans un repère donné du plan.
\medskip
\begin{enumerate}
\item On a :\\
\begin{tabularx}{\linewidth}{*{3}{@{$\bullet~~$}X}}
$\displaystyle\lim_{x \to -1} f(x) = - 1$&$\displaystyle\lim_{x \to -1} f(x) = 2$&$\displaystyle\lim_{x \to -1} f(x) = - \infty$\\
\end{tabularx}
\medskip
\item La courbe $\mathcal{C}$ admet une asymptote d'équation:\\
\begin{tabularx}{\linewidth}{*{3}{@{$\bullet~~$}X}}
$y = 2$&$y = -1$&$x = 2$\\
\end{tabularx}
\medskip
\item Pour tout réel $x$ de l'intervalle $]-1~;~+ \infty[,~f(x)$ peut s'écrire:\\
\begin{tabularx}{\linewidth}{*{3}{@{$\bullet~~$}X}}
$f(x) = \dfrac{2x}{x + 1}$&$f(x) = \dfrac{2x + 1}{x + 1}$&$f(x) = \dfrac{1}{x + 1}$\\
\end{tabularx}
\medskip
\item Le signe de $f(x)$ sur l'intervalle $]-1~;~+ \infty[$ est donné par le tableau :
\medskip
\begin{tabularx}{\linewidth}{*{3}{X}}
\psset{xunit=0.9cm,yunit=0.7cm}\begin{pspicture}(3.9,2)
\psframe(3.9,2) \psline(0,1)(3.9,1)\psline(0.9,0)(0.9,2)
\psline(1.05,0)(1.05,1)\psline(1.15,0)(1.15,1)\uput[u](0.45,1){$x$} \uput[u](1.1,1){$-1$} \uput[u](2.4,1){$0$}
\uput[u](3.6,1){$+ \infty$}
\uput[u](0.45,0){$f(x)$}\uput[u](1.7,0){$-$} \uput[u](2.4,0){$0$}
\uput[u](3.15,0){$+$}\uput[u](0.1,2){$\bullet$}
\end{pspicture}&\psset{xunit=0.9cm,yunit=0.7cm}\begin{pspicture}(3.9,2)
\psframe(3.9,2) \psline(0,1)(3.9,1)\psline(1.05,0)(1.05,1)\psline(1.15,0)(1.15,1)\psline(0.9,0)(0.9,2)
\uput[u](0.45,1){$x$}\uput[u](1.1,1){$-1$}
\uput[u](3.6,1){$+ \infty$}
\uput[u](0.45,0){$f(x)$} \uput[u](2.4,0){$+$}\uput[u](0.1,2){$\bullet$}
\end{pspicture}&\psset{xunit=0.9cm,yunit=0.7cm}\begin{pspicture}(3.9,2)
\psframe(3.9,2) \psline(0,1)(3.9,1)\psline(1.05,0)(1.05,1)\psline(1.15,0)(1.15,1)\psline(0.9,0)(0.9,2)
\uput[u](0.45,1){$x$}\uput[u](1.1,1){$-1$}\uput[u](2.4,1){$-\frac{1}{2}$}
\uput[u](3.6,1){$+ \infty$} \uput[u](0.45,0){$f(x)$}\uput[u](1.7,0){$-$} \uput[u](2.4,0){$0$} \uput[u](3.1,0){$+$}\uput[u](0.1,2){$\bullet$}
\end{pspicture}\\
\end{tabularx}
\medskip
\item Le coefficient directeur de la tangente à la courbe $\mathcal{C}$ au point d'abscisse $1$ est:\\
\begin{tabularx}{\linewidth}{*{3}{@{$\bullet~~$}X}}
$\dfrac{3}{2}$&$\dfrac{1}{4}$&$-\dfrac{1}{2}$\\
\end{tabularx}
\medskip
\item En $+\infty$, $\mathcal C$ admet une asymptote d'équation
\medskip
\begin{tabularx}{\linewidth}{*{3}{@{$\bullet~~$}X}}
$y=2$ &$y=2x$ &$x=2$
\end{tabularx}
\medskip
\end{enumerate}
\newpage
\section*{Exercice (obligatoire - pas d'option math) : 5 points }
Le coût total de fabrication d'un produit est donnée par $C(q)=\frac{q^3}3-6q^2+40q$ pour $q\in[0;12]$ où $q$ représente le nombre de milliers d’unités fabriquées et $C(q)$ le coût de fabrication en centaines d’euros. \\
\begin{enumerate}
\item On rappelle que le coût unitaire moyen est donné par $C_M(q)=\frac{C(q)}q$ pour tout $q>0$.
\begin{enumerate}
\item Exprimer en fonction de $q$ le coût unitaire moyen.
\item Calculer le nombre $q0$ d’unités à fabriquer pour que le coût moyen soit minimal.
\end{enumerate}
\item On appelle coût marginal la dépense occasionnée par la production d’un objet supplémentaire. On modélise ce coût marginal par $C_m(q)=C'(q)$ où $C'$ est la dérivée de $C$.
\begin{enumerate}
\item Exprimer en fonction de $q$ le coût marginal.
\item Vérifier que pour $q0$, le coût marginal est égal au coût moyen.
\end{enumerate}
\item
On suppose que l'entreprise vend toute sa production. Pour $q\in]0;12]$ le bénéfice en centaines d’euros, pour la production et la vente de $q$ milliers d’unités est $B(q)=-\frac{q^3}3+2q^2+21q$.
\begin{enumerate}
\item Calculer le nombre d’unités à produire pour que l’entreprise soit rentable.
\item Déterminer le nombre d'unités à fabriquer pour obtenir le bénéfice maximum. Que vaut ce bénéfice maximal ?
\end{enumerate}
\end{enumerate}
\end{document}
% Découper suivant les pointillés
\hspace{-3em}\raisebox{-7pt}[0pt][\height]{\ScissorRight} \hrulefill~\raisebox{-7pt}[0pt][\height]{\ScissorLeft}
|
https://www.zentralblatt-math.org/matheduc/en/?id=15368&type=tex | zentralblatt-math.org | CC-MAIN-2019-30 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-30/segments/1563195525483.62/warc/CC-MAIN-20190718001934-20190718023934-00558.warc.gz | 860,279,520 | 1,563 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2013d.00198}
\itemau{Joubert, Marie}
\itemti{Using digital technologies in mathematics teaching: developing an understanding of the landscape using three ``grand challenge'' themes.}
\itemso{Educ. Stud. Math. 82, No. 3, 341-359 (2013).}
\itemab
Summary: This paper develops an understanding of the issues, interests and concerns within the mathematics education community related to the use of computers and other digital technologies in the teaching and learning of mathematics. It begins by arguing for the importance of understanding this landscape of interests and concerns, and then turns to the theoretical and methodological choices made in this study, explaining how it has drawn on the approach developed by the STELLAR European Network of Excellence. Analysing the titles and abstracts of a conference chosen to represent the mathematics education community, it maps out the landscape framed by three ``grand challenges'', finding that an understanding of orchestrating learning is at the heart of the interests of the community, and that the community is interested in exploring new and different contexts for the teaching and learning of mathematics. However, there is currently less interest in investigating and exploiting the increasing connectedness of learners within this community. Further, while the ``grand challenges'' framing is useful in mapping the landscape, it fails to take into account both the personal concerns of teachers and students, such as attitude and confidence, and issues related to doing research and understanding research concerns.
\itemrv{~}
\itemcc{D20}
\itemut{research; digital technology; computers; landscape; grand challenge; education community}
\itemli{doi:10.1007/s10649-012-9430-x}
\end |
https://ctan.math.washington.edu/tex-archive/macros/latex/contrib/sides/sides-sample.tex | washington.edu | CC-MAIN-2022-21 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-21/segments/1652662531762.30/warc/CC-MAIN-20220520061824-20220520091824-00446.warc.gz | 235,187,212 | 1,487 | \documentclass[12pt]{sides}
\title{A Sample Play}
\author{Wing L. Mui
\\ \\
17 Imaginary Road \\
Real Town, MA 01002
\\ \\
http://www.wingie.org
}
\date{5/9/2005}
\begin{document}
\maketitle
\castpage
\cast{Eleanor} A girl of age 18. She likes pies and $\pi$.
\cast{Cat} A large creature that meows.
\notes
This is a sample of what the $sides$ package for \LaTeX\ can do.
\newact
\newscene
\stagedir{2003, New York City. The \chara{Cat} and \chara{Eleanor} are engaging in conversation upon a meadow.}
\repl{Cat} Meow. I am a kitten. Kittens like to meow.
\repl{Eleanor} Yes, indeed. \pause\ But the question is, do cats like to typeset in \LaTeX ?
\repl{Cat} I do not know. Let us find out!
\stagedir{They exit.}
\newscene
\stagedir{Five hours later. \chara{Eleanor} enters and rushes across the stage.}
\repl{Ellie} \paren{Angrily.} My document class does not work! The cat has destroyed my code!
\stagedir{\chara{Eleanor} exits.}
\newactnamed{Interlude}
\stagedir{The \chara{Cat} is eating a biscuit on top of a roof.}
\repl{Cat} Cats are not known to give monologues. But they are also not known to talk. So I am an exception.
You see, all is relative---including class paths. Eleanor failed in that she did not take this into account.
This is a mighty tasty biscuit.
\stagedir{The \chara{Cat} exits.}
\newact
\newscenenamed{The Final Scene}
\stagedir{Three days later. \chara{Eleanor} and the \chara{Cat} speak to each other upon a roof on a stormy night. \chara{Eleanor} is about to jump off the edge.}
\repl{Cat} Do not jump, dear girl!
\repl{Eleanor} But I must! For I cannot write my documents in \LaTeX ! \paren{She jumps.}
\repl{Cat} No!
\stagedir{Blackout. We hear the sound of a ``splat''. End of play.}
\end{document} |
https://www.emis.de/journals/EJC/Volume_13/Abstracts/v13i1n3.abs.tex | emis.de | CC-MAIN-2020-29 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-29/segments/1593655879738.16/warc/CC-MAIN-20200702174127-20200702204127-00465.warc.gz | 836,080,993 | 1,021 | \magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Arthur H. Busch}
%
%
\medskip
\noindent
%
%
{\bf A Note on the Number of Hamiltonian Paths in Strong Tournaments}
%
%
\vskip 5mm
\noindent
%
%
%
%
We prove that the minimum number of distinct hamiltonian paths in a
strong tournament of order $n$ is $5^{{n-1}\over{3}}$. A known
construction shows this number is best possible when $n \equiv 1
\hbox{ \rm mod } 3$ and gives similar minimal values for $n$ congruent
to $0$ and $2$ modulo $3$.
\bye
|
http://math.huji.ac.il/~ijmath/index_tex/index137.tex | huji.ac.il | CC-MAIN-2017-39 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2017-39/segments/1505818690591.29/warc/CC-MAIN-20170925092813-20170925112813-00358.warc.gz | 215,693,840 | 2,116 | %File content.tex, 9 September, 1992
%\magnification=\magstep1
\input ij.sty
\input mssymb
\def\ {\thinspace}
\lineskiplimit=100pt
\lineskip=7pt %This controls the spacing between indexitems of
%both kinds
\tenpoint
\null\vskip 3cm
\def\index\indexvolume#1\indexyear#2\endindextitle{
\rm\parindent=0pt%\overfullrule=0pt
\centerline{ISRAEL JOURNAL OF MATHEMATICS}
\vskip0.4cm
\centerline{TABLE OF CONTENTS, VOLUME #1\unskip, #2\unskip}
\vskip1.2cm
\frenchspacing}
\def\endindex{}
\def\indexitem\indexauthor#1\indextitle#2\indexpage#3\endindexitem
{\halign {\normalbaselines%
\vbox{\hsize=11cm\hangindent=2em\hangafter=1 ##}%
&\hbox to 1.3cm{\hfill ##}\cr%
{\smc #1\unskip}\quad{\sl #2} & #3 \cr}\par}
\def\indexauthor{}
\def\endindexitem{}
\def\indexpage{}
\def\indextitle{}
\def\indexsee\indexauthor#1\partner#2\endindexitem
{{\smc #1\unskip\enspace}{\sl See }{\smc #2\unskip}\par}
\index
\indexvolume 137
\indexyear 2003
\endindextitle
\indexitem
\indexauthor An, J. and E.~A.~O'Brien
\indextitle Conjectures on the character degrees of the
Harada--Norton simple group HN
\indexpage 157
\endindexitem
\indexitem
\indexauthor Anantharaman-Delaroche, C.
\indextitle On spectral characterizations of\break amenability
\indexpage 1
\endindexitem
\indexitem
\indexauthor Ball, K.
\indextitle Entropy and $\sigma$-algebra equivalence of certain
random walks on random sceneries
\indexpage 35
\endindexitem
\indexitem
\indexauthor Baracco, L.
\indextitle Analytic discs in conormal bundles to real submanifolds
of ${\mathbb C}^n$
\indexpage 149
\endindexitem
\indexitem
\indexauthor Bhattacharya, S.
\indextitle Higher order mixing and rigidity of algebraic\break actions on
compact abelian groups
\indexpage 211
\endindexitem
\indexitem
\indexauthor Bhattacharya, S. and K.~Schmidt
\indextitle Homoclinic points and isomorphism rigidity of algebraic
${\mathbb Z}^d$-actions on zero-dimensional compact abelian groups
\indexpage 189
\endindexitem
\indexitem
\indexauthor Bouhjar, K. and J.~J.~Dijkstra
\indextitle On the structure of $n$-point sets
\indexpage 321
\endindexitem
\indexitem
\indexauthor Bruin, H. and S.~van Strien
\indextitle Expansion of derivatives in one-\break dimensional dynamics
\indexpage 223
\endindexitem
\indexitem
\indexauthor Bruin, H. and S.~Troubetzkoy
\indextitle The Gauss map on a class of interval translation
mappings
\indexpage 125
\endindexitem
\indexitem
\indexauthor Cooper, S.~B., A.~Li, A.~Sorbi and Y.~Yang
\indextitle There exists a maximal 3-c.e. enumeration degree
\indexpage 285
\endindexitem
\indexsee
\indexauthor Dijkstra, J.~J.
\partner K.~Bouhjar
\endindexitem
\indexitem
\indexauthor Herzog, M., G.~Kaplan and A.~Lucchini
\indextitle On subgroups containing non-trivial normal subgroups
\indexpage 183
\endindexitem
\indexitem
\indexauthor Ivanov, A.~A.
\indextitle Cayley graphs having nice enumerations
\indexpage 61
\endindexitem
\indexsee
\indexauthor Kaplan, G.
\partner M.~Herzog
\endindexitem
\indexitem
\indexauthor Klopsch, B.
\indextitle Enumerating finite groups without abelian composition
factors
\indexpage 265
\endindexitem
\indexitem
\indexauthor Lee, S.~T. and H.~Y.~Loke
\indextitle Degenerate principal series representations of
Sp($p,q$)
\indexpage 355
\endindexitem
\indexsee
\indexauthor Li, A.
\partner S.~B.~Cooper
\endindexitem
\indexsee
\indexauthor Loke, H.~Y.
\partner S.~T.~Lee
\endindexitem
\indexsee
\indexauthor Lucchini, A.
\partner M.~Herzog
\endindexitem
\indexsee
\indexauthor O'Brien, E.~A.
\partner J.~An
\endindexitem
\indexsee
\indexauthor Schmidt, K.
\partner S.~Bhattacharya
\endindexitem
\indexsee
\indexauthor Sorbi, A.
\partner S.~B.~Cooper
\endindexitem
\indexsee
\indexauthor Strien, S.~van
\partner H.~Bruin
\endindexitem
\indexitem
\indexauthor Su, Y., K.~Zhao and L.~Zhu
\indextitle Simple Lie color algebras of Weyl type
\indexpage 109
\endindexitem
\indexsee
\indexauthor Troubetzkoy, S.
\partner H.~Bruin
\endindexitem
\indexsee
\indexauthor Yang, Y.
\partner S.~B.~Cooper
\endindexitem
\indexsee
\indexauthor Zhao, K.
\partner Y.~Su
\endindexitem
\indexsee
\indexauthor Zhu, L.
\partner Y.~Su
\endindexitem
\endindex
%End of file, content.tex
|
http://ida.darksky.org/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%2C%20area%20FROM%20refs%20WHERE%20serial%20%3D%20510%20ORDER%20BY%20first_author%2C%20author_count%2C%20author%2C%20year%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=5&rowOffset=&wrapResults=1&citeOrder=&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg= | darksky.org | CC-MAIN-2020-45 | application/x-latex | application/x-latex | crawl-data/CC-MAIN-2020-45/segments/1603107881551.11/warc/CC-MAIN-20201023234043-20201024024043-00212.warc.gz | 55,309,763 | 1,286 | %&LaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Bray+Young2012} Bray, M. S., \& Young, M. E. (2012). Chronobiological Effects on Obesity. \textit{Curr Obes Rep}, \textit{1}(1), 9--15.
\end{thebibliography}
\end{document}
|
https://ctan.math.utah.edu/tex-archive/support/easylatex/testFiles/transpose.tex.correct | utah.edu | CC-MAIN-2022-40 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030336978.73/warc/CC-MAIN-20221001230322-20221002020322-00375.warc.gz | 222,474,095 | 796 | \documentclass{article}
\usepackage{amsmath} % pkg required by EasyLatex
\usepackage{amsfonts} % pkg required by EasyLatex
\usepackage{amssymb} % pkg required by EasyLatex
\usepackage{psfrag} % pkg required by EasyLatex
\usepackage[dvips]{graphicx} % pkg required by EasyLatex
\begin{document}
The transpose of
\begin{align*}
\left[ \begin{array}{lll}
0 & 1 & 0
\\ 0 & 1 & 0
\\ 0 & 1 & 0
\end{array} \right]
\end{align*}
is
\begin{align*}
\left[ \begin{array}{lll}
0 & 0 & 0
\\ 1 & 1 & 1
\\ 0 & 0 & 0
\end{array} \right]
\end{align*}
.
\end{document}
|
http://dlmf.nist.gov/12.14.E33.tex | nist.gov | CC-MAIN-2017-17 | application/x-tex | null | crawl-data/CC-MAIN-2017-17/segments/1492917121665.69/warc/CC-MAIN-20170423031201-00097-ip-10-145-167-34.ec2.internal.warc.gz | 99,511,241 | 772 | \[\mathop{W\/}\nolimits\!\left(\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)\sim%
\frac{\pi^{\frac{1}{2}}\mu^{\frac{1}{3}}l(\mu)}{2^{-\frac{1}{2}}e^{-\frac{1}{4%
}\pi\mu^{2}}}\phi(\zeta)\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(-\mu^{%
\frac{4}{3}}\zeta\right)\sum_{s=0}^{\infty}(-1)^{s}\frac{A_{s}(\zeta)}{\mu^{4s%
}}+\frac{\mathop{\mathrm{Ai}\/}\nolimits'\!\left(-\mu^{\frac{4}{3}}\zeta\right%
)}{\mu^{\frac{8}{3}}}\sum_{s=0}^{\infty}(-1)^{s}\frac{B_{s}(\zeta)}{\mu^{4s}}%
\right),\] |
http://edshare.soton.ac.uk/2343/23/MA222qu26.tex | soton.ac.uk | CC-MAIN-2019-43 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2019-43/segments/1570986676227.57/warc/CC-MAIN-20191017200101-20191017223601-00525.warc.gz | 61,023,383 | 1,330 | \documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}
{\bf Question}
Sudden dips in a supply of electrical current in occur in
accordance with the assumptions of a Poisson process with a rate
of 3 dips per hour. What is the probability that the first dip in
the afternoon occurs later than 1.00 p.m.?
If a device fails when it has experienced the cumulative effect of
27 dips in the current, what is the probability density function
of the life of the device? Find approximately the probability that
such a device will last for longer than 12 hours.
\vspace{.25in}
{\bf Answer}
Dips $\sim$ Poisson(3)
$P$(1st afternoon dip occurs after 1p.m.) $= P(W_1 > 1 {\rm hour})
= e^{-3} = 0.0498$ (=$P$(no events))
The life of a device = waiting time until 27th dip, and so has
p.d.f.
$\ds \frac{3e^{-3t}(3t)^{26}}{26!} \hspace{.3in} t \geq 0$ $\ \ \
(\Gamma(3,27))$
Now $\ds \mu = \frac{27}{3} = 9 \, {\rm and} \, \sigma^2 =
\frac{27}{9} = 3$
So the lifetime $\sim N(9,3)$
$\ds P(L>12) = P\left(Z > \frac{12-9}{\sqrt 3} \right) = P(Z>\sqrt
3) \approx P(Z > 1.73) \approx 0.4182$ from tables.
\end{document}
|
https://www.zentralblatt-math.org/matheduc/en/?id=15234&type=tex | zentralblatt-math.org | CC-MAIN-2019-39 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-39/segments/1568514573439.64/warc/CC-MAIN-20190919040032-20190919062032-00272.warc.gz | 1,078,960,279 | 1,386 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2013d.00901}
\itemau{B\"uchter, Andreas; Haug, Reinhold}
\itemti{Learning with materials. Supporting in the building of basic mathematical ideas. (Lernen mit Material. Anker setzen beim Aufbau mathematischer Grundvorstellungen.)}
\itemso{Math. Lehren 30, No. 176, 2-7 (2013).}
\itemab
Mathematik betreiben hei\ss t insbesondere, Begirffe zu entwickeln und mit ihnen gedanklich zu arbeiten. Dazu werden tragf\"ahige Vorstellungen ben\"otigt, deren Ausbildung auch in den Sekundarstufen durch didaktisch reflektierten Einsatz von Material unterst\"utzt werden kann. Der Beitrag zeigt lerntheoretische und psychologische Gr\"unde f\"ur praktisch-gegenst\"andliche Handlungserfahrungen auf und gibt Anregungen zu Planung, Gestaltung und Durchf\"uhrung von materialbasiertem Unterricht.
\itemrv{Renate St\"urmer (Zweibr\"ucken)}
\itemcc{U63 D43}
\itemut{lesson planning; teaching methods; edudcational media; object teaching; educational analysis; concept formation; student activities; survey articles}
\itemli{}
\end |
http://theanarchistlibrary.org/library/anonymous-reclaim-your-mind-manifesto.tex | theanarchistlibrary.org | CC-MAIN-2020-29 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-29/segments/1593657151197.83/warc/CC-MAIN-20200714181325-20200714211325-00139.warc.gz | 98,241,507 | 15,962 | \documentclass[DIV=12,%
BCOR=10mm,%
headinclude=false,%
footinclude=false,open=any,%
fontsize=11pt,%
twoside,%
paper=210mm:11in]%
{scrbook}
\usepackage{fontspec}
\usepackage{polyglossia}
\setmainfont{Linux Libertine O}
% these are not used but prevents XeTeX to barf
\setsansfont[Scale=MatchLowercase]{CMU Sans Serif}
\setmonofont[Scale=MatchLowercase]{CMU Typewriter Text}
\setmainlanguage{english}
% global style
\pagestyle{plain}
\usepackage{microtype} % you need an *updated* texlive 2012, but harmless
\usepackage{graphicx}
\usepackage{alltt}
\usepackage{verbatim}
% http://tex.stackexchange.com/questions/3033/forcing-linebreaks-in-url
\PassOptionsToPackage{hyphens}{url}\usepackage[hyperfootnotes=false,hidelinks,breaklinks=true]{hyperref}
\usepackage{bookmark}
% footnote handling
\usepackage{bigfoot}
\usepackage{perpage}
\DeclareNewFootnote{default}
\DeclareNewFootnote{B}
\MakeSorted{footnoteB}
\renewcommand*\thefootnoteB{(\arabic{footnoteB})}
% continuous numbering across the document. Defaults to resetting at chapter. Unclear
% \usepackage{chngcntr}
% \counterwithout{footnote}{chapter}
\usepackage[shortlabels]{enumitem}
\usepackage{tabularx}
\usepackage[normalem]{ulem}
\def\hsout{\bgroup \ULdepth=-.55ex \ULset}
% https://tex.stackexchange.com/questions/22410/strikethrough-in-section-title
% Unclear if \protect \hsout is needed. Doesn't looks so
\DeclareRobustCommand{\sout}[1]{\texorpdfstring{\hsout{#1}}{#1}}
\usepackage{wrapfig}
\usepackage{indentfirst}
% remove the numbering
\setcounter{secnumdepth}{-2}
% remove labels from the captions
\renewcommand*{\captionformat}{}
\renewcommand*{\figureformat}{}
\renewcommand*{\tableformat}{}
\KOMAoption{captions}{belowfigure,nooneline}
\addtokomafont{caption}{\centering}
% avoid breakage on multiple <br><br> and avoid the next [] to be eaten
\newcommand*{\forcelinebreak}{\strut\\*{}}
\newcommand*{\hairline}{%
\bigskip%
\noindent \hrulefill%
\bigskip%
}
% reverse indentation for biblio and play
\newenvironment*{amusebiblio}{
\leftskip=\parindent
\parindent=-\parindent
\smallskip
\indent
}{\smallskip}
\newenvironment*{amuseplay}{
\leftskip=\parindent
\parindent=-\parindent
\smallskip
\indent
}{\smallskip}
\newcommand*{\Slash}{\slash\hspace{0pt}}
\addtokomafont{disposition}{\rmfamily}
\addtokomafont{descriptionlabel}{\rmfamily}
% forbid widows/orphans
\frenchspacing
\sloppy
\clubpenalty=10000
\widowpenalty=10000
% http://tex.stackexchange.com/questions/304802/how-not-to-hyphenate-the-last-word-of-a-paragraph
\finalhyphendemerits=10000
% given that we said footinclude=false, this should be safe
\setlength{\footskip}{2\baselineskip}
\title{Reclaim Your Mind : Manifesto}
\date{April 13, 2011}
\author{Anonymous}
\subtitle{An Urgent Message for all those who have or are in danger of being labelled mentally ill}
% https://groups.google.com/d/topic/comp.text.tex/6fYmcVMbSbQ/discussion
\hypersetup{%
pdfencoding=auto,
pdftitle={Reclaim Your Mind : Manifesto},%
pdfauthor={Anonymous},%
pdfsubject={An Urgent Message for all those who have or are in danger of being labelled mentally ill},%
pdfkeywords={alienation; psychology; social control}%
}
\begin{document}
\begin{titlepage}
\strut\vskip 2em
\begin{center}
{\usekomafont{title}{\huge Reclaim Your Mind : Manifesto\par}}%
\vskip 1em
{\usekomafont{subtitle}{An Urgent Message for all those who have or are in danger of being labelled mentally ill\par}}%
\vskip 2em
{\usekomafont{author}{Anonymous\par}}%
\vskip 1.5em
\vfill
{\usekomafont{date}{April 13, 2011\par}}%
\end{center}
\end{titlepage}
\cleardoublepage
\tableofcontents
% start a new right-handed page
\cleardoublepage
\chapter{\textbf{Introduction to “Reclaim Your Mind”}}
We are pleased to publish this anonymous text which appeared on the old website of the 325 Collective in 2003 and which was re-edited and published online in 2011. We don’t agree with this text entirely and are pessimistic about the prospect of a wide-spread ‘healing’ of Earth or the masses of society – as has always been the case, minorities will throw off the chains of social obligations, will seize their own life trajectory and wilfully define themselves, finding each other and creating unique moments of beauty, life and freedom. Maybe in the ruins of techno-industrial mass society a widespread healing will occur. For the time being, we aim to liberate ourselves from the cage we’ve been born into, alongside as many others as possible. This means overcoming the limiting constraints of our mental patterns and freeing ourselves from the self-fulfilling diagnoses of mental illness, to glory in our inscrutable uniqueness and our dysfunctionality – our refusal to be working, integrated components of the megamachine.
The zine ‘Beyond Amnesty’ (downloadable from 325.nostate.net) is also well worth reading, a piercing and personal attack on psychiatry and this prison society that drives us to self-destruction. Anti-civilisation thinkers such as Chellis Glendinning, John Zerzan and Derrick Jensen have traced the pathology of modern society to its root in domestication, and the growing patterns of control, repression, abuse and self-destruction. It will take the deliberate violation of the control patterns instilled by the tranquilising institutions of society to reclaim our wilful self-empowered autonomy.
We remember the suicides and drug abuse and say the deaths are murders by this dominator system. We are in an existential struggle and have declared revolutionary war to the end.
\begin{flushright}
Dark Matter Publications, Spring 2012
\end{flushright}
\chapter{\textbf{Selling cures for the problems they created}}
It is well known that depression has been on a steady rise in the past few decades. This increase apparently isn’t about to stop since the World Health Organization (WHO) recently predicted that, by 2020, depression would be the second most prevalent health problem in the world, just below heart disease, and offered as an explanation that this was due to a previous underestimation of the number of people suffering from this “illness”.
Couldn’t the increasing feeling of emptiness and worthlessness characteristic of depression be related to the society we live in, at a time when people lose themselves in consumption and mass entertainment to avoid thinking about their miserable life, their economic survival or the ongoing destruction of the planet? While the “experts” paid by pharmacology corporations will invariably answer that depression is a brain disorder due to a “chemical imbalance”, the result of some faulty genes they have yet to identity, we cannot help to wonder how this could not be environmental considering there was no such thing as depression in Africa before colonization?
\begin{quote}
\emph{Depression has widely been touted as endemic to the 20-something generation. Severe depression is 10 times more prevalent today than it was 50 years ago, and it strikes a full decade earlier in life on average than it did a generation ago. Such feelings and behaviours testify to frustration and despair that have nowhere to go when the social landscape is so frozen. Disaffection or even opposition are quickly marketed into sellable style images; alienation as fashion. Meanwhile suicide, perhaps the ultimate regression, has been on a steady rise for several decades.}
\end{quote}
\begin{quote}
\textbf{- John Zerzan}
\end{quote}
At best depression is considered as a necessary side-effect of “progress” — just like civilian murders are a “collateral damage” of war — but “experts” always affirm that more scientific research is the only solution to the problem. Until these mythical genes have been found and we can all live in the promised techno-virtual “paradise” as genetically-modified human beings, the pharmacology industry is of course delighted to sell drugs to help humanity deal with its “chemical imbalances”, just like we are sold bottled water to “solve” the problem of water pollution.
\chapter{\textbf{A brave new world?}}
Unfortunately if they discover a perfect drug to remove all the symptoms of depression and stress, they won’t stop here. As it is becoming increasingly clear, their interest is not happiness and well-being of the whole of humanity, as they promise, but “progress” and the on-going technological race. Such a drug would in reality be an opportunity to increase the demands of society and the stress- and depression-inducing effects of our environment. Our dependency on drugs will force in turn others to use them to be able to compete and survive.
\begin{quote}
\emph{Imagine a society that subjects people to conditions that make them terribly unhappy, then gives them the drugs to take away their unhappiness. Science fiction? It is already happening to some extent in our own society. Instead of removing the conditions that make people depressed, modern society gives them antidepressant drugs. In effect, antidepressants are a means of modifying an individual’s internal state in such a way as to enable him to tolerate social conditions that he would otherwise find intolerable.}
\end{quote}
\begin{quote}
\textbf{- Theodore Kaczynski}
\end{quote}
While until recently society had to adapt to the limits of human beings, the situation has been reversed and it’s now human beings who have to adapt to society. Is that their idea of a “perfect world”? And more importantly, is that yours?
Since this essential question never generates any debate in mainstream media, it seems that pharmacology corporations have already answered for us. Using the false promises of consumerism — “our product will solve all your problems and make you feel fulfilled just like these idealized images you see on our ads” — the sales of anti-depressants has increased by 800\% in the 90′s alone. They now claim that one american out of five, a market of over 50 millions people, “needs a cure urgently”. Is it urgent because people are finally waking up and they need to be plugged back into the Matrix of illusion before they see the desert of the real world they live in?
\chapter{\textbf{The myth of permanent happiness}}
One effect of the excessive marketing from the pharmacology industry, and the many other industries promoting “health” as a value, is the creation of an unique ideal that everyone is supposed to cling to. These “fullfilled” consumers we see on ads have become, consciously or unconsciously, some kind of role models for most of us. They make us believe that permanent happiness is possible, the biggest myth of them all.
\begin{quote}
\emph{Suffering is a misunderstanding. It exists. It’s real. I can call it a misunderstanding, but I can’t pretend that it doesn’t exist, or will ever cease to exist\dots{} There are times I — I am very frightened. Any happiness seems trivial. And yet, I wonder if it isn’t all a misunderstanding — this grasping after happiness, this fear of pain\dots{} If instead of fearing it and running from it, one could get through it, go beyond it. I don’t know how to say it. But I believe that the reality of pain is not pain. If you can endure it all the way.}
\end{quote}
\begin{quote}
\textbf{- Ursula Le Guin}
\end{quote}
Human beings, like any conscious organisms, are dualistic by nature. We can’t know the sweet without knowing the sour. We can’t experience happiness without experiencing sadness. With our attempt to eliminate sadness, with our obsession for positivism, we have only found emotional death. This is the way someone suffering from depression feels. In fact, it could be argued that a depressive person is different from a “normal” person only in her awareness of the poverty of her emotional life. Too afraid of terror, we have become unable to feel joy. To be able to go beyond that, pain should be embraced like any other feelings.
\begin{quote}
\emph{Chagrin, shame, fear, terror, anger are transient madness.}
\end{quote}
\begin{quote}
\textbf{- Benjamin Rush, father of American psychiatry}
\end{quote}
Individuals labelled with manic-depression (bipolar disorder), on the other hand, experience both states intensely. While mania is accepted and even promoted by our society (think about shopping sprees), the depressive episodes are frowned upon. Take a minute and ask yourself: is this intensity necessarily bad for the person or is it a problem only because it doesn’t fit in the myth of permanent happiness promoted by modern society? Yes, manic-depression cause suffering and makes it difficult to live a “normal” life, but isn’t it also a way to experience life more deeply?
\chapter{\textbf{Normality vs. diversity}}
What we observe is that, while some problems such as depression and stress seem to be the result of our decadent mental landscape, other states of mind have become a problem only because they don’t fit within the grand schemes of civilization, where normality makes it easier to enforce “order”. They use cultural differences for marketing purpose but in the end everyone is supposed to buy the same products and have the same desires: a stable income, an happy family, a nice house, a perfectly-sized body, a great confidence. Those who can afford it have “choices”, but they are mostly limited to different tastes, such as the color of their car.
Yet diversity is essential for the survival of any eco-system. If nature has survived until our “conquest”, it’s because of this biological diversity. If the climate or environment changed, some species would die but others would survive, making it possible for evolution to continue. When civilization will have reduced cows to one “perfect” race, it will only require one virus to kill them all.
What interests us most though is diversity amongst human beings. We see that society doesn’t support this simply by the way public schools work: all children have to follow the same subjects, regardless of their interests, and even if at the end of compulsory school they can “choose” between different careers, this is because of the needs of the job marketplace only. As a result a lot of teenagers finish school alienated from their initial aspirations and struggle to find a job which interests them.
\chapter{\textbf{Disorders as differences}}
Let’s now look at some other widespread “mental illnesses” from the point of view that they are more a difference, part of the diversity of any ecosystem, than a disorder:
Attention-Deficit Hyperactivity Disorder (ADHD) is a particularly sad example of the world we live in: many children are unable to stay all day long in these prisons called “schools” because they have too much energy and creativity. “Concerned” about their future in this society, influenced by psychiatrists, their parents feed them with Ritalin, sometimes since the age of 4, to numb them down and kill their flame. They fit again in the illusion of normality, but does that makes them happier? Schizophrenia is often mentioned when talking about mental illnesses for it can be deeply disturbing and very long. To understand this “illness” we should take a look at “primitive” cultures: in all of them, we can find shamans who had the gift to travel on the “other world” and heal people. The initiation was involuntary (although young shamans could be identified early by their tribes) and required several deeply disturbing years until the shaman was able to master his or her skills. What’s interesting is that the effects of this initiation are extremely similar to the “symptoms” of schizophrenia. Indeed some primitive tribes were fooled by western psychiatrists that their future shaman was “schizophrenic” and had to be medicated. Unfortunately it appears that anti-psychotic drugs prevent the process to be finished, in such a way that the individual gets lost in the void between the two worlds.
People are increasingly put under the label of Asperger’s syndrome or highly-functioning autism. Individuals diagnosed with this “disorder” usually have a high IQ and no impairment other than a difficulty to interact and communicate with others. We suggest that their isolation and obsessive thinking may set them apart and make communication more difficult (or less meaningful) as they are not on the same wavelength as others. That doesn’t have to be a disadvantage though: some psychiatrists have in fact recently “back-diagnosed” Newton and Einstein with Asperger’s syndrome. The question is: would these two geniuses have delivered their wisdom if they had been labelled as autistic and medicated in their youth?
Social anxiety is also on the rise. Apart from the fact it’s easy to become self-conscious about our behavior and look when we live in a society which judge everyone on their appearance, it’s worth mentionning that amongst all animals a percentage of them are naturally shy. Shy animals have greater chances of survival since their fears put them less at risk. For human beings, shyness may make it more difficult to be part of society but it’s also a great opportunity to develop inner capabilities that others, too busy socializing, don’t have the time to care about.
What we observe is that most of the pain felt by “mentally ill” individuals is caused more by a rejection of society than by the “illness” itself. Alienation, loneliness, homelessness, low self-esteem are all the destructive results of a society which doesn’t tolerate differences. Furthmore, the belief that there is something “wrong” that has to be “corrected” (or at least repressed) can only alienate people from themselves and make them feel miserable and worthless. Indeed almost all of these “illnesses” are usually coupled with depression.
\begin{quote}
\emph{Our society tends to regard as a “sickness” any mode of thought or behavior that is inconvenient for the system, and this is plausible because when an individual doesn’t fit into the system it causes pain to the individual as well as problems for the system. Thus the manipulation of an individual to adjust him to the system is seen as a “cure” for a “sickness” and therefore as good.}
\end{quote}
\begin{quote}
\textbf{- Theodore Kaczynski}
\end{quote}
\chapter{\textbf{Healing or repression?}}
All this makes us wonder if psychiatrists and psychologists are really interested in healing or if their role is to keep the illusion of “order”, “normality” and “sanity” within society? Indeed the primary goals of asylums has always been to keep the “insanes” outside of society because they were considered “dangerous”. But how are they dangerous considering that there is statistically the same amount of criminals amongst the “sanes” than amongst the “insanes”? Is it perhaps because they do not fit in society and their mere existence exposes the lies of this system?
It’s interesting to note that, before the appearance of asylums, heretics, witches, prostitutes, madmen and basically anyone “socially deviant” were being “treated” (tortured, exorcised, burnt) by the Inquisition and, when the Church started to lose its power, some of the witch-hunters “converted” to psychiatry and kept on doing basically the same job, using pseudoscience instead of religion to put themselves above the possessed\Slash{}mentally ill and try to adjust them to society’s standards. These standards change enormously through time and space. For example, homosexuality was considered to be a disorder by the bible of psychiatry, the DSM, until the 70′s.
Curiously, many psychiatrists today believe that witches were “misdiagnosed”, that they were in fact “suffering” of “mental illness”, not “demonic possession”. They are the only ones who don’t believe in the theory of the scapegoat (a figure whom the fears — or repressed desires — of society are projected on), agreed by all historians. Could it be because this theory also applies perfectly for the “mentally ills” of today?
The ones suffering the most from psychiatry are perhaps the children, who do not have choices regarding their medication because they are not supposed to be “responsible” enough. Apart from ADHD and Asperger’s syndrome mentionned previously, the label of “Oppositional Defiant Disorder” offers a convenient explanation for parents who do not want to understand why their children are rebelling against their oppressive ideology and pointless consumer lifestyle. For these parents, medication appears to be the only “solution”, especially if they are themselves victims of the psychiatric industry.
\chapter{\textbf{Psychiatry as political repression}}
More striking cases of how the myth of “mental illness” has been used by the system for repression include the Soviet Union where political dissenters were regularly “diagnosed” as “mentally ill” and confined in asylums. Similar repression was done in the USA where socially deviants were locked — such as Timothy Leary for advocating the use of “illegal” drugs. And as recently as April 2003 someone who was reading and talking about conspiracy theories within the US government was diagnosed as “paranoid” and held in an asylum for 9 days!
\begin{quote}
\emph{The excess of the passion for liberty produced, in many people, opinions and conducts which could not be removed by reason not retrained by government\dots{} The extensive influence which these opinions had upon the understandings, passions, and morals of many of the citizens of the United States, constituted a form of insanity, which I shall take the liberty of distinguishing by the name of anarchy.}
\end{quote}
\begin{quote}
\textbf{- Benjamin Rush, father of American psychiatry}
\end{quote}
Just like new laws are constantly added to create new classes of criminals and force people into an ever-narrowing range of legality, new mental disorders are “discovered” all the time to create new classes of “insanes”, open new markets for the pharmacology industry and force people into an ever-narrowing range of “sanity”. Actually, the “symptoms” of mental disorders — the only things on which the existence of these “disorders” are based on — are so broad and common that anyone could be “diagnosed” with 2–3 of them by just visiting a psychiatrist!
\begin{quote}
\emph{We need a program of psychosurgery and political control of our society. The purpose is physical control of the mind. Everyone who deviates from the given norm can be surgically mutilated. The individual may think that the most important reality is his own existence, but this is only his personal point of view. This lacks historical perspective. Man does not have the right to develop his own mind. This kind of liberal orientation has great appeal. We must electrically control the brain. Some day armies and generals will be controlled by electrical stimulation of the brain.}
\end{quote}
\begin{quote}
\textbf{- Dr. Jose Delgado}
\end{quote}
This quote, coming from a psychiatrist who was recruited by the CIA for the MKULTRA program of mind control after having served the fascist regime in Spain, could not be more explicit. This obsession for control is nothing new for the male-driven civilization we live in, and controlling the human mind is no doubt their biggest challenge. They dream to kill the animal (the life-force) in us, to finally transform us into perfect machines working exclusively in the name of “progress”. The founding father of American psychiatry, Benjamin Rush, even considered insanes as “untamed animals whom it is the psychiatrist’s duty to discipline”, a comparison which remind us of the way non-white people were treated during the colonization.
\chapter{\textbf{The reality of medication}}
Electroshock “therapy” and lobotomy are not as common as a few decades ago, although the fact these barbaric practices still exist is deeply revealing of the society we live in. Forced medication has been mostly replacing them, often with a threat of forced confinment if the drugs are not taken.
This change is not caused by a new sense of humanity amongst psychiatrists but by pressures from insurance companies who find it cheaper to send patients back home with drugs prescriptions, as well as from the needs of the pharmacology industry to increase its revenues. Another cause is that people are generally more willing to get “fixed” but we shouldn’t necessarily see voluntary treatment as a progress over forced one. Indeed, it may just prove that we have been so effectively brainwashed that we do not resist anymore.
\begin{quote}
\emph{Of all the tyrannies a tyranny sincerely exercised for the good of its victims may be the most oppressive.}
\end{quote}
\begin{quote}
\textbf{- C. S. Lewis}
\end{quote}
Let’s have a look at the way these prescription drugs “work”. While ads and psychiatrists explicitly or implicitly claim that they help to heal, the reality appears different: they work by merely hiding the “symptoms” and keeping the brain quiet. Once the drugs have finished their effects, the individual is on the same situation as before or even worse, since all antipsychotic drugs can damage the brain after several months or years of “treatment”, a phenomenon known as tardive dyskinesia. All of them are strongly addictive as well.
So we see that drugs do nothing other than keeping an artificial state in the brain which makes things more bearable for the individual. There is nothing wrong with that, considering that some mental disorders are deeply disturbing and can push to suicide, but it’s not fair to tell people that their medication is going to heal them. Prescription drugs, like any other drugs, should be used as cautiously as possible and along with a real treatment, assisted or not by a professional. Otherwise the person will stay a consumer\Slash{}victim of the pharmacology industry all her life. Sadly, this may be what they are hoping for.
\chapter{\textbf{ Natural healing}}
The most problematic effect of these drugs, however, is that they often prevent individuals to go through the natural healing process which requires a dynamic chaotic void before a healthy restoration is possible. Real healing is not a slow gradual process, as psychiatrists would like to believe, but is cyclic, with its lows and its highs, until the brain has been “purged” from old conditioned neurons and a new freedom can be found. Unfortunately the people working on the mental health field don’t like anything chaotic so they do all they can to suppress these “symptoms”, preventing at the same time patients to reach the end of the tunnel.
If a person is lucky enough to be “allowed” to complete the healing process, it results in a new outlook on her image, life, reality and society, allowing her to adopt a more healthy lifestyle, perhaps away from mindless consumerism and weapons of mass distraction like television. Challenges and difficulties are necessary elements of any spiritual growth, and what could be more challenging than a mental “illness” which forces us to understand how our brain works and to evolve our consciousness in order to be able to keep on living? Our “disorders” truly are dangerous gifts which should be cultivated and respected rather than repressed and hated.
\chapter{\textbf{Reclaiming our minds}}
Make no mistake: this lengthy analysis on civilization and psychiatry is not used to push off our responsibility, to blame all our problems on others. We are all too eager to finally reclaim our minds! We don’t need “professionals” to tell us how to live, people have found this by themselves for millions of years, and those “experts” prevent people thinking for themselves, providing instead ready-made explanations for any difficulty they may encounter in their development.
\begin{quote}
\emph{Either you think — or else others have to think for you and take power from you, pervert and discipline your natural tastes, civilize and sterilize you.}
\end{quote}
\begin{quote}
\textbf{- F. Scott Fitzgerald}
\end{quote}
\textbf{Victimism} leads people to feel that they must have some kind of professional to help them — mental health agent, religious leader, educator, fashion adviser — because they are incapable of independently making their own decisions or carrying out their own activities. This is not the case! We are all capable of finding our way to healing, it’s only the belief that we are not which makes us stuck!
Who can know better than ourselves what’s going on in our minds? Our deep fears, motivations, desires are usually beyond words and they reach such a deep layer of our reality that few psychiatrist could find about them, especially not under the pressure of insurance companies to be more “efficient”. The way we think and see the world is entirely dependent on our past experiences. No one can truly understand us without re-experiencing our whole life!
The idea is not necessarily to reject psychiatry as a whole but to let people choose what they feel is best for them, by showing them the different alternatives available and educating them about the lies of the mental health industry. Most importantly, we are not looking for an unique “Truth”, we want each individual to understand how their mind works, seek their own solutions and have the freedom to enact whatever course of action they feel is best. Self-exploration allows any of us to evolve from the status of helpless victims to the one of healers. This is the spirit of do-it-yourself applied to the brain!
Alternatives to traditional “treatments” exist and most have existed long before a professional class of psychiatrists was created. Examples are : Meditation, yoga, magick, self-hypnosis, herbal \& nutritional treatment, cognitive therapy, neuro-linguistic programming (NLP). All are useful pieces in our toolboxes. Most of them also have a holistic point of view, which emphasizes the importance of the whole and the interdependence of the parts, an idea most psychiatrists completely reject!
The solution will not come from above or outside, but from below and within. Our unconscious wants to help our conscious mind to heal, if only we listen to it. Changes inevitably happen when we finally take responsibility for who we are, for our life, for our community, for our planet, for our future. We aren’t fooled anymore by society’s doubletalk which tells us to be “responsible citizens” while asking us to follow orders from above without questioning. We want real responsibility and real freedom!
\chapter{\textbf{More than healing}}
At this point something should be clarified: this is not just about healing, because this idea would suppose there is a plateau to be reached, a sense of eternal well-being to be found. There isn’t any, except on fairy tales and advertisements. Remember: permanent happiness is their myth! Personal development (or whatever name you prefer to give it) is in fact an on-going cyclic process where the travel matters more than the destination.
To be able to travel more freely, you’ll want to unload from your shoulders all the burden of psychiatric conditioning and especially the idea that there is something wrong with you. As it should be said more often: you are perfect as you are! You have done your best given your situation and, even if the path you took until now has been more difficult than others, this doesn’t mean you’re a failure! “Normal” people will have to go through this one day too, or else they will never have the chance to grow up.
You are what you believe you are. If you insist on believing that you are a helpless victim of a terrible illness whose salvation lies on the hands of a few mega-corporations, this is what you’ll eventually experience all your life. What do you want?
\begin{quote}
\emph{They lied to you, sold you ideas of good \& evil, gave you distrust of your body \& shame for your prophethood of chaos, invented words of disgust for your molecular love, mesmerized you with inattention, bored you with civilization \& all its usurious emotions. There is no becoming, no revolution, no struggle, no path; already you’re the monarch of your own skin — your inviolable freedom waits to be completed only by the love of other monarchs: a politics of dream, urgent as the blueness of sky.}
\end{quote}
\begin{quote}
\textbf{- Hakim Bey}
\end{quote}
Depending on how many years have been spent on the mental health industry (or under the judgment of society), changing this belief can be difficult but it’s possible! Pay attentions to your beliefs in your day-by-day life, play with different paradigms (belief-systems) for a day, for a week, for a month. Hack into your reality-tunnel and accept the idea that you are not inferior but simply different, that we are in fact all different. If you need some inspiration or if you aren’t quite convinced that beliefs have a role to play, read some books on cognitive liberty, neuro-linguistic programming or chaos. It might forever change the way you view “reality”.
\chapter{\textbf{Following our path}}
Our mind is self-created, it has developed over the years as we were bouncing on “reality” and other human beings. There is no rigid structure that every brain follows, even if some models are useful to understand how we think. As a consequence, we all have completely different potentials and shortcomings. Instead of focusing on our “problems”, wouldn’t it be more sensible to promote our gifts, skills, desires, sensitivities, so that each one of us make the most of their potential during their limited lifetime, no matter how different they are from the current “norms”?
The norms are and have always been illusory anyway. Quantum physics tell us we are the co-creators of the universe, that the only fact of observing an object changes its nature and that our “subjective” mind has a much more important role in reality than most materialists think. The way we see “reality” and ourselves has more to do with our mental environment (“the Matrix”) than with any kind of materialist reality or genetic predispositions. If we are able to somehow go beyond the mental structures of civilization, if we are able to transcend them, then everything becomes possible!
\begin{quote}
\emph{Man is ignorant of the nature of his own being and powers. Even his idea of his limitations is based on experience of the past, and every step in his progress extends his empire. There is therefore no reason to assign theoretical limits to what he may be, or what he may do.}
\end{quote}
\begin{quote}
\textbf{- Aleister Crowley}
\end{quote}
To follow our path, we must first know what we want. We must learn to listen to our inner voice, our unconscious, our true will, that something inside of each one of us which intuitively know what’s the best direction to take. This may mean turning off, at least for a while, all the background noise of civilization, like television, radio, newspapers and eventually friends. All our conditioned fears, desires and ideas of our limitations will not go away at the second we isolate ourselves but we may find meditation valuable to help us in this process.
\begin{quote}
\emph{Two roads diverged in the woods. I took the one less travelled by, and that has made all the difference.}
\end{quote}
\begin{quote}
\textbf{- Robert Frost}
\end{quote}
As far as we know, we only have one life on this planet. Why should we waste it trying to adapt ourselves to the always more demanding expectations of this insane society when there is so much to live, explore, experience and discover?
Changes always come from below and the old structures of oppression will inexorably fall when we stop relying on them. We will then finally be able to create a new culture of diversity and solidarity where everyone is accepted (and loved!) for who they are!
The good news is that any change we make in society through our actions is likely to have much more impact on our lives than any treatment done alone at home! This doesn’t mean we have to create a new class of professionals who will tell others how to live, that’s the very thing we are opposing! Just like the position of a teacher over his students automatically prevents any teaching to occur, the authority of a healer rules out any real healing. Effective healing is non-hierarchical, with all individuals healing and being healed at the same time.
We have been staying alone for too long! After too many years of isolation and alienation, some have given up the hope that someone will one day truly understand them (as opposed to just being “compassionate” like their psychiatrist — if they are lucky). Enough despair, enough division! We are here, the “insane”, the “angry”, “the unstable”, the “chaotic”, the “depressed” \dots{}
The saying “none of us is free as long as some are not free” seems more true than ever in these oppressive times. We can’t expect to find joy and wholeness without changing our environment, without changing the very structures of reality. Thus any real, profound healing will necessarily involve a healing of the planet and society as a whole.
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The Anarchist Library
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Anti-Copyright
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Anonymous
Reclaim Your Mind : Manifesto
An Urgent Message for all those who have or are in danger of being labelled mentally ill
April 13, 2011
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Retrieved on June 7, 2012 from \href{http://325.nostate.net/?p=2162}{325.nostate.net}
Dark Matter Publications
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\textbf{theanarchistlibrary.org}
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|
https://people.eecs.berkeley.edu/~fateman/papers/levin.tex | berkeley.edu | CC-MAIN-2022-05 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-05/segments/1642320306181.43/warc/CC-MAIN-20220129122405-20220129152405-00625.warc.gz | 511,703,116 | 16,732 | \documentclass{article}
\usepackage{fullpage}
\title{DRAFT:
Integration of oscillatory integrals, a computer-algebra approach}
\author {Richard Fateman\\
Computer Science\\
University of California\\
Berkeley, CA, USA}
\begin{document}
\maketitle
\begin{abstract}
The numerical integration of oscillatory integrals is an important and
well-studied area of mathematical inquiry. See \cite{HOP} for a
series of recent conferences on the topic, including
applications. Examining some of the proffered methods from the
position of a computer algebra system (CAS) provides several
opportunities not available with purely numerical approaches. These
include the computing of symbolic derivatives as part of the
processing, the provision of a framework for error control based on
exact or arbitrarily-high-precision arithmetic, and a relatively
simple exposition upon which further elaborations may be built. By
executing code in a symbolic system it is also possible to compute
approximate answers as expressions in terms of symbolic parameters
such as frequency, displaying the nature of dependencies as well as
re-useable formulas for different values of parameters.
\end{abstract}
\section{Introduction}
Assume that we have a computer algebra system at our disposal, and
we need numerical or semi-numerical approximations to certain integrals
that we can write in this form:
$$I~=~\int_{a}^b e^{i \omega f(x)}g(x)dx.$$
As the real function $f(x)$ varies for $x$ in $[a,b]$, the complex
exponential (alternatively the literature may use sines or cosines
since $\exp(i x)~=~\cos(x)+i \sin(x)$) produces oscillations in
the integrand. As is done commonly in this context, we multiply $f(x)$
by a real parameter $\omega$ to allow us to demonstrate how the
situation changes with frequency. That is, given a particular $f(x)$,
an exponent where
$\omega=100$ will have a higher frequency, and be more oscillatory
than $\omega=1$. In our subsequent considerations, the situation is
that $f(x)$ and $g(x)$ are analytic and vary relatively modestly in
the interval $[a,b]$. If they are not, the integral may be broken
up into pieces where these assumptions hold.
Sometimes it is computationally convenient to deal
with the real and imaginary parts of the integral separately, but
notationally, the complex exponential is more often used in the literature.
This type of integral is problematical for typical quadrature programs
since almost any procedure which is based on sampling of the integrand
will be subject to the unreliable numerical consequences resulting
from computing values of the rapidly varying integrand. Other
approaches, targeted to oscillatory integrands, can be quite
successful on these. A readable survey of various approaches from the
theoretical and numerical analysis perspective has been composed by
Iserles \cite{iserles05}.
There are apparently significant applications of this type of integral
in investigations of optics, \cite{HOP}. There are also many papers
and computer programs ranging back to work by Filon \cite{filon28},
addressing this application. One recent computer algebra package by
Andrew Moylan \cite{LevinIntegrate}, has been incorporated into
Mathematica version 8.0's {\tt NIntegrate} command. {\tt NIntegrate}
is an impressive collection of methods; unfortunately as apparently is
the case with ``automatic'' quadrature packages, particular problems
may not fit well with the default choices made by the system. In such
cases the user may be required to deal with the mundane aspects of
trying to track many options, estimating errors, adjusting precision,
etc. Especially in these cases, understanding some of the consequences
of the user-available choices may be helpful.
In subsequent sections we review a few of the approaches specifically for oscillatory integrals that have
occurred in the literature and view each from a simple tutorial
perspective to see how these ideas fit in a computer algebra
framework. Understanding simple programs may allow the reader to
adjust them to particular situations. Rather than dealing with the
full complexity of packages intended to account for all eventualities
for all inputs, we hope to provide a readable exposition.
\section{Filon's method, more or less}
The idea from Filon \cite{filon28} is to approximate $g(x)$ piecewise
as a polynomial, each section looking like $g_0+g_1x+ g_2x^2$.
Instead of integrating $g(x)$ using Simpson's Rule,
consider that you can integrate $x^k\exp(i \omega f(x))$ for
$k=0,1,2$. This gives you alternatives for the Simpson's Rule
weights, and the integral can be written as the (piecewise)
sum of the modified Simpson's Rule. If we were actually integrating
a quadratic polynomial times an exponential, we could write it out:
$${{e^{i\,\omega\,x}}\over{\omega^3}}\left(
\left(-{ g_2}\,\omega^2\,x^2-{ g_1}\,\omega^2\,x-{ g_0}\,
\omega^2+2\,{ g_2}\right)\,i+2\,{ g_2}\,\omega\,x+{ g_1}\,
\omega\right)$$
Of course you may not start with the coefficients in that polynomial
but only its value at several points. Three points determine a
quadratic, and thus the polynomial can be computed by interpolation.
As an example, the interpolation algebra works out simply if you have
values of $g$ at the points $-1,~0,~+1$ in which case
$$ { g_0}=g\left(0\right) , { g_1}={{2\,g\left(1\right)-2
\,g\left(-1\right)}\over{4}} , { g_2}=-{{-g\left(1\right)+2\,g
\left(0\right)-g\left(-1\right)}\over{2}} $$
There are advantages to alternative techniques in accuracy and speed
if (more) points are not equally spaced, but chosen instead to
facilitate Gaussian
quadrature.
In any case, details are then encoded in numerical routines. The
result is to entirely eliminate the problematic oscillatory term from
the quadrature. Programs for Filon integration for $f(x)=x$ can be
found in scientific subroutine libraries. One precaution of note
included in the ACM library implementation is that it evaluates
certain subexpressions by an alternative series calculation less
subject to cancellation error than the most obvious formula. Another
technique to avoid accumulation of this error, available in a CAS, is
to simply increase the floating-point precision temporarily. There
are numerous tweaks possible on this idea.
The QUADPACK library routine QAWO\footnote{Available
at http://www.netlib.org, and included in the CAS Maxima.} and
related programs can also be used. These use a ``modified
Clenshaw-Curtis'' quadrature scheme. The adaptation provides that the
function $g(x)$ is approximated by a single Chebyshev series
approximated over the whole range.
%The weights used are those tabulated by the integrals of ($T_n(x) \exp
%(i \omega x)$). We display $n=0 \cdots 4$ here:
%
%$$ -{{i\,e^{i\,\omega\,x}}\over{\omega}} $$
% $$ -{{\left(i\,\omega\,
% x-1\right)\,e^{i\,\omega\,x}}\over{\omega^2}} $$
% $$ -{{\left(2\,i\,
% \omega^2\,x^2-4\,\omega\,x-i\,\omega^2-4\,i\right)\,e^{i\,\omega\,x}
% }\over{\omega^3}} $$
% $$ -{{\left(4\,i\,\omega^3\,x^3-12\,\omega^2\,x^2+
% \left(-3\,i\,\omega^3-24\,i\,\omega\right)\,x+3\,\omega^2+24\right)
% \,e^{i\,\omega\,x}}\over{\omega^4}} $$
%$$ -{{\left(8\,i\,\omega^4\,x^4-
% 32\,\omega^3\,x^3+\left(-8\,i\,\omega^4-96\,i\,\omega^2\right)\,x^2+
% \left(16\,\omega^3+192\,\omega\right)\,x+i\,\omega^4+16\,i\,\omega^2
% +192\,i\right)\,e^{i\,\omega\,x}}\over{\omega^5}} $$
We provide a version of Filon's procedure using the Maxima computer
algebra system to illustrate its brevity. It is based on a FORTRAN program
by Chase and Fosdick\cite{chase}\footnote {We do not display lengthier
``optimized'' code which computes arrays of sines/cosines by
differencing, and/or uses faster machine floats rather than
higher-precision software floats, deals with cosines rather than
sines, uses general endpoints, attempts to estimate error,
etc.}. This is the code for the imaginary part $\sin(\omega x)$;
similar code for $\cos (\omega x)$ computes the real part; combined
they compute the code for $\exp(i \omega x)$.
%% note
%% for all n,
%% Integrate[ChebyshevT[n_, x] Sin[x]] :=
%% -Cos[x]*ChebyshevT[n, x] +
%% Integrate[n* ChebyshevU[n-1, x]* Cos[x], x]
%%
%% II[ChebyshevU[n_, x]*Cos[x]] :=
%% ChebyshevU[n, x]*Sin[x] -
%% Integrate[(((-1 - n) ChebyshevU[-1 + n, x] +
%% n *x* ChebyshevU[n, x])* Sin[x])/(-1 + x^2), x]
%%
%%, um, express ChebyU.. U[n]=2*x*U[n-1]-U[n-2]; U0=1, u1=2*x
%% we could always express stuff in terms of 1,x,x^2 and convert to
%% cheby
\begin{verbatim}
filons0(f,m,p):= /*integrate f(x)*sin(m x) from 0 to 1, p points.
for notation, see CACM Algorithms: Chase & Fosdick Alg 353 */
block([h:bfloat(1/(2*p)),
k:bfloat(m), theta,s2p,s2pm1,alf,bet,gam],
theta:bfloat(k*h),
s2p:
block([sum:0],
for i:0 thru p do sum:sum+f(2*h*i)*sin(2*k*h*i),
sum-1/2*(f(1)*sin(k))),
s2pm1:
block([sum:0],
for i:1 thru p do sum:sum+f(h*(2*i-1))*sin (k*h*(2*i-1)), sum),
fpprec:2*fpprec, /*double the floating precision, precaution */
alf: 1/theta + sin(2*theta)/2/theta^2-2*sin(theta)^2/theta^3,
bet: 2*((1+cos(theta)^2)/theta^2 -sin(2*theta)/theta^3),
gam: 4*(sin(theta)/theta^3-cos(theta)/theta^2),
fpprec:fpprec/2,
h*(alf*(f(0)-f(1))*cos(k) +bet*s2p+ gam*s2pm1))
\end{verbatim}
\section{What if the argument of sin/cos/exp is complicated?}
The case $f(x)=mx$, simply a constant times $x$, is solved by a direct
application of Filon's method or modified Clenshaw-Curtis.
If the function $f(x)$ is more
complicated than this, we can try to make a substitution (change of
variables) \cite{evans} to convert $f(x)$ to this simple form. Let
$y=f(x)$. Solve for $x=f^{(-1)}(y)$, compute $dy$ and integrate
$g(f^{(-1)}(y)) e^{i \omega y} dy$ between transformed bounds.
Finding a suitable inverse function can be tricky\footnote{ Sufficient
conditions, mathematically speaking: $f$ is differentiable and
$f^(-1)$ is continuous. Solution methods in computer algebra systems
may not necessarily assure this second condition, or for that
matter, may not find an explicit inverse.}. Some computer algebra
systems provide a command (e.g. Maxima's {\tt changevar}).
Example:
$$\int_{a}^ {b} {e^{i\,w\,\sinh x}}\,\cos x\;dx$$
becomes suitable for Filon quadrature after it is changed to
$$\int_{\sinh a}^{\sinh b}
{{{ e^{i\,\omega\,y}\,\cos {\rm arcsinh}\;
y}\over{\sqrt{y^2+1}}}\;dy}.$$
%integrate(exp(%i*w*y)*cos(asinh (y))/sqrt(y^2+1),y,sinh(a),sinh(b))
%integrate(exp(%i*w*y)*cos(asinh(y))/sqrt(y^2+1),y,sinh(-1),sinh(1))
%quad_qawo(cos(asinh(y))/sqrt(y^2+1),y,sinh(-1),sinh(1),1000,cos);
It is interesting to see how this change can be used
to pre-process out the difficulty and just use an ordinary
(non-oscillatory) adaptive quadrature program. Given a simple test of
$a=-1,~b=1,~\omega=1000$, Mathematica 7's default numerical
integration program ran on the original problem for 0.8 seconds before
signalling failure. On the transformed result the same program
produced a result in 0.05 seconds. In each case a result of
approximately 1.69206e-4 was obtained, so the failure message was
actually produced erroneously. Quadpack's {\tt QAWO}, called from Maxima,
returns in about 0.001 seconds\footnote{QAWO returns in 0.0002 sec if the
function $\cos({\rm asinh} (y))/\sqrt{y^2+1}$ is compiled.}
with an answer 1.69206436906{\em 66968}e-4, where we know from more
accurate evaluation that the digits in italics are in error.
A Mathematica program oriented toward oscillatory integrands, {\tt
LevinIntegrate} by Andrew Moylan \cite{LevinIntegrate} obtained the same result in 0.05
seconds.
(More about this method later.)
Note that the inverse function could in fact be represented by
an approximation, and computed in a separate step, rather than
seeing it substituted literally into the transformed integrand.
In passing we also note that there is sometimes a requirement to integrate
$\sin(x)^ng(x)$ for some integer $n$. Since the power of
$\sin(x)$ can be written as a sum of multiple angles, (e.g. $\sin mx
~=~ 1/4 (3 \sin mx - \sin 3mx)$) the task can be reduced to the previous
case.
\section{Integration by Parts}
In this section we follow the approach suggested by work by Iserles and Olver on using
derivatives \cite{iserles05}; computer algebra systems provide an efficient
route to setting up this kind of method.
Here's the idea. We can use integration by parts as follows:
$$\int e^{i \omega f(x)}g(x)dx~ = ~\int u dv~=~ uv-\int v~ du.$$
Let $u=g/(i \omega f')$, $dv=e^{i \omega f}\times i \omega f'$. Then $v=e^{i \omega f}$, and
$du=(f'g'-gf'')/(- i \omega(f')^2)$
There are several points to note. We consider the $uv$ part as
``solved'' and that it requires no further attention other than
ordinary evaluation. We can factor out $-1/(i \omega)$ from the
remaining integral, $K=\int v~ du$. Significantly, $K$ is itself
again of the same form as the original problem, so integration by
parts can be used again. Because the remaining integral will have
that coefficient out front of $1/\omega ^2$, we can consider this as a
``correction'' to the $uv$, and that this term will drop even faster
if the oscillation parameter $\omega$ is larger. (That is, the approximation
gets better for {\em more oscillatory} integrands.) Indeed, by
repeating this operation (recursively, in our program) we can generate
a series in $1/ \omega$, although this asymptotic series does not
actually converge for fixed $\omega$ as we add more terms.
This expansion in series requires the
computation of (perhaps many) derivatives of $g$, although ultimately
these will be evaluated only at the endpoints of the definite
integral. The slightly odd display of $du$ with the $i \omega$ in the
denominator makes for a slightly better appearance of the
formula/program below.
\begin{verbatim}
/*integrate by parts oscillatory integral, ibpoi */
ibpoi(f,g,x,w,L):= ibpoi1(f,g,x,w,L,0); /* set count to 0 */
ibpoi1(f,g,x,w,L,c):= if c>=L then 0 else
block([df:diff(f,x)],
exp(%i*w*f)*g/(%i*w*df)
- 1/(%i*w)*ibpoi1(f, (df*diff(g,x)-g*diff(df,x))/df^2,x,w,L,c+1))
/*our example */
define(v(x),ibpoi(sinh(x),cos(x),x,1000,1));
expand(v(1.0)-v(-1.0));
\end{verbatim}
In this case, just one term yields 1.{\em 702}e-4, which is correct to
3 decimal places. Carrying 3 terms provides 1.69206436{\em 7290}e-4,
good for 9 decimal places. Continuing for 6 terms, the result 1.69206436906{\em
6987}e-4, is good for about 13 places. Ultimately, increasing the
number of terms ceases to increase the accuracy in this asymptotic
series for fixed $\omega$. The final, otherwise omitted, error term
can be evaluated by some other means. Increasing $\omega$,
predictably, will make the series converge faster. Because {\tt
ipboi} is a symbolic program, we can leave the parameter $\omega$
symbolic instead of using the number 1000, and compute the resulting
formula. For $v(x)$ with one step in the expansion:
$$-{{i\,e^{i\,\omega\,\sinh x}\,\cos x}\over{\omega\,\cosh x}}.$$
Clearly this is not expensive to evaluate.
For further analysis, see Iserles, (Lemma 2.1) \cite{iserles05}. While
additional intuition may be gleaned from examining these formulas, in
practice just running the four-line program may be quite useful. An
elaboration of the program in case many terms are deemed useful would
take a different approach to computing derivatives: in fact, one need
not compute $f'$, $f''$, etc. One only needs $f'(a)$, $f'(b)$
$f''(a)$, $f''(b)$ etc. These are likely more rapidly calculated by computing
two Taylor series for $f$, one centered at $a$ and one centered at
$b$, and similarly for $g$. Ordinary arithmetic on Taylor series (so
long as they are expanded at the same point) is also provided, in most
computer algebra systems.
The Maxima program then looks like this:
\begin{verbatim}
(ibpoilims(f,g,x,w,L,a,b):= /* integrate from a to b */
block([keepfloat:true, ratprint:false],
at(ibpoilims1(taylor(f,x,b,L),taylor(g,x,b,L),x,w,L,0),x=b)
-at(ibpoilims1(taylor(f,x,a,L),taylor(g,x,a,L),x,w,L,0),x=a)),
ibpoilims1(f,g,x,w,L,c):= if c>=L then 0 else
block([df:diff(f,x,1)],
exp(%i*w*f)*g/(%i*w*df)
- 1/(%i*w)*ibpoilims1(f,(df*diff(g,x,1)-g*diff(df,x,1))/df^2,x,w,L,c+1)))
/*our example */
ibpoilims(sinh(x),cos(x),x,1000.0,3,-1.0,1.0); /* integrate from -1 to 1 */
\end{verbatim}
Another tack, similar in concept to the program above but
quite different in execution would be to use ``automatic
differentiation'' \footnote{http://www.autodiff.org}.
While the example given runs quite fast as given, note if we used the
command {\tt ibpoilims(sinh(x),cos(x), x,1000,3,-1,1)} using integer
values rather than floats, the program produces a large symbolic
expression including exact terms like $\sinh(1)$, which can later be
evaluated to any desired precision. Furthermore, if we change the
first line of {\tt ibpoilims1} to return a non-zero function of $x$,
say {\tt ErrorTerm(x)} we can see how its value enters into the result.
Note that this idea as well as variable substitution (change of
variables) essentially approaches the definite integration
problem by modifying the related {\em
indefinite} integral. Change of variables removes the oscillation,
and integration by parts attenuates the contribution of the
oscillation.
\section{Elaborations and improvements: Levin}
When the integrand is recognized as a case of a smooth function
multiplied by a simple oscillation, Filon's method or modified
Clenshaw-Curtis can be used directly, and oscillation presents no
problem. If we intend for the computer system to use this procedure
automatically as part of a general integration routine
then we need to programmatically recognize that the integral is in fact
appropriately oscillatory, and the routine must separate the pieces.
A major alternative can be based on an idea due to Levin
\cite{levin-82}, which works as follows: Again we compute the {\em
indefinite} integral, a function of $x$
$$H(x)~=~ \int e^{ i \omega f(x)}g(x)dx.$$ Given $H$ one can
simply computes $H(b)-H(a)$.
The problem is neatly solved if any of the infinite
number of formulas for $H$ can be found. $H$, in this
case is a formula for an indefinite
integral which is valid up to an arbitrary constant! All we
need is some formula which can be evaluated (perhaps only approximately)
at two points. In the course of the subtraction the arbitrary constant
cancels.
Finding some $H$ may be time-consuming though it is something that
computer algebra systems can sometimes do.
{If we can anticipated that it is too time-consuming,
perhaps we won't bother to look! In the case of solving many
problems with the same integrand but different endpoints or internal
parameters, it may be well worth looking.}
% (cite something re Risch \cite{}).
A more
likely case is that it is {\em not especially convenient or possible to find a
closed form}. It is still possible to find a method to approximate
$H$ as closely as desired.
%%%% not ... and in a form that gets better when $\omega$ increases.
The Levin technique \cite{levin-82} proceeds by finding an
approximation for $H$ by observing that we are looking for (or
approximating) a function $p(x)$ such that
$$ e^{f(x)} p(x)~\approx
~\int e^{ f(x)}g(x)dx.$$
If we compute the derivative with respect to $x$ of both sides and divide
out by $e^{f(x)}$ we have a differential equation $$p'(x)+f'(x)p(x)=g(x)$$.
All we need to do is find $p$, and then $H(x)=e^{f(x)} p(x)$, leading to
the approximate value of the desired integral, $H(b)-H(a)$.
Levin observes that under the assumptions that $g$ and $f'$ are at
most slowly oscillatory, then among the solutions for $p$
there is one that is not rapidly oscillating. Furthermore,
we will find it by a collocation
method. Using a computer algebra system instead of a
purely numerical approach to finding $p(x)$ can provide additional
information -- like variation of values (and error) with respect to
$\omega$.
%%%
In reviewing options for solving the associated differential equation,
one thought in using computer algebra was to try direct generation of
a Taylor series for $p$, using one of several symbolic-iterative
methods. In fact this is not a good idea because such a solution
requires choosing an expansion point (we could select the midpoint of
the interval) and producing a series whose coefficients are computed
by matching derivatives at that point. Since the integrand is highly
oscillatory, this would not provide a high quality result at any
distance from the point of expansion, and as $\omega$ increased, the
approximation would worsen. Basically, we would find the wrong
solution to the ODE.
Instead, as demonstrated by Levin, we can find an approximation to
a less oscillatory solution by collocation.
{This transforms the computation of finding $p$ into a set of linear
equations for finding a polynomial fit to the solution. Numerous
variations on this technique have emerged to try to overcome
problems with this numerical approach, since it appears that the
generated equations may be an unstable system.}
Let us return to a simple form where we have, for the moment absorbed
$i \omega$ into $f$. We wish to compute or approximate $p$ in:
$$p'+f'p=g$$
We proceed by assuming some expansion of $p$ in the same (or other)
basis functions but with coefficients to be found by collocation.
Given the basis, $\{x^j\}$, and therefore
assuming $p_k=\alpha_0 + \alpha_1 x + ... +\alpha_{k-1}x^{k-1}$, we have a
corresponding $p'= \alpha_1 + ...+ (k-1) \alpha_{k-1}x^{k-2}$.
Choose some set of collocation points, which might be chosen for
simplicity of exposition (but not optimality) as equally-spaced from
$a$ to $b$. $a=r_0, r_1, ...., r_{k-1}=b$. We set up $k$ linear equations
by which one can determine the $\{\alpha_j\}, j=0,...,k-1$.
This requires computing $f'(x_j)$ and $g(x_j)$ at those points, as part
of the setup.
Once one has the values of $\{\alpha_j\}, j=0,...,k-1$, and hence an
approximate functional form for $p$, we can use that to compute $p(a)$
and $p(b)$.
Here's a sample program in Maxima's language:
\begin{verbatim}
/* integrate (exp(i*w*f)*g,a,b) by Levin method, k points*/
LevinInt(f,g,a,b,omega,k):= block([varlist,p,pointlist,sol,ratprint:false],
varlist: makelist(alpha[i],i,0,k-1),
p: varlist . makelist(x^i,i,0,k-1),
pointlist : makelist(a+ (b-a)*i/(k-1),i,0,k-1),
define(eq(x), diff(p,x)+%i*omega*diff(f,x)*p-g),
sol:linsolve(eqs:map(eq,pointlist),varlist),
define(H(x), exp(%i*omega*f)*ev(p,sol)),
rectform(H(b)-H(a)))
\end{verbatim}
Using our standard example, {\tt
LevinInt(sinh(x),cos(x),-1.0,1.0,1000,8);} in which we have chosen
collocation at only 8 points, produces a result of 1.69{\em 178...}e-4.
Using 23 subdivisions we get 1.6920643{\em 3981}e-4.
%0.000169206436906715960940227786164 + 0.*10^-43
This program is only a single example, and building a more-or-less
automatic program would require attempting some measurement of likely
error, measuring effort, increasing precision, and for that matter,
checking that $f$ and $g$ are sufficiently smooth, and the formulation makes sense.
We have experimented with increasing the precision (above machine
double-floats), with somewhat inconclusive results. In particular an
interesting alternative to the numerical operation of the above
program is to allow for the complete answer to be generated
(symbolically including expressions like $\sinh(1/30)$) and then
proceed to simplify that [large] expression, and/or evaluate it using
ever-higher numerical precision to see if that affects the
result\footnote{Methods for the exact solution of a linear system with
symbolic coefficients are quite different from numerical methods,
and can result in huge solution expressions.}. Other variation
include using points and a set of orthogonal functions that provide
better fit with fewer terms, for example Gauss-Lobatto points and
Chebyshev polynomials.
%%% i think, from mathematica, the answer may
Incidentally, we believe an accurate value is about 1.6920643690671596e-4.
\section{Extension to other functions}
Let us sketch the basic idea of extension to other functions as shown by Levin \cite{levin-96} where the exponential can be replaced by an element of a vector of oscillators
which has the required form ${\bf W}'={\bf A} \cdot {\bf W}$.
This paper solves for
(composed) functions for collocation\footnote{ On the computer algebra
front, Levin's 1996 paper expands on this and provides an example
computed by Mathematica, but other than setting up the collocation
matrix more or less conveniently, no symbolic features are used: the
example is primarily using Mathematica to access a library routine for
numerical solution of a set of linear equations. Moylan makes all this
work more automatically \cite{LevinIntegrate}.}. Extensions to this
idea include matching derivatives as well as values at grid
points\cite{iserles04}.
As a specific example, take the Bessel functions of the first kind, ($J$) which satisfy the differential equation
$$\left(
\begin{array}{c}
J_0(x)' \\
J_1(x)'
\end{array}
\right)
=
{\bf A}
\left(
\begin{array}{c}
J_0(x) \\
J_1(x)
\end{array}
\right)
$$
where
$${\bf A} = {\left(
\begin{array}{cc}
0 & -1 \\
1 & -\frac{1}{x}
\end{array}
\right)}.
$$
The product of any two oscillators satisfying the criterion with an invertible
matrix also satisfies the criterion. For example, to add products of sin, cos, $J_0$ and $J_1$ to the possible oscillators, consider the matrix
\left(
\begin{array}{cccccccc}
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & -\frac{1}{x} & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 & -\frac{1}{x}
\end{array}
\right)
corresponding to the vector ${\bf W}^T =(\sin(x),cos(x), J_0(x), J_1(x), \sin(x) J_0(x), \cos(x) J_1(x), \cos(x) J_0(x), \sin(x) J_1(x)) $
We reduce the integration problem to finding the (now vector) solutions for $p$ of the
differential equation $${\bf p}'+ {\bf A} {\bf p} = {\bf g}.$$
We do this via collocation: by choosing a set of points, say $\{c_j\}$ and a proposed
expansion of ${\bf p}$ in some orthogonal set (e.g. powers of $x$, or Chebyshev polynomials of degree $r$ or less), evaluating and solving
$${\bf p)'(c_j)+{\bf A}(r,c_J) {\bf p}(c_j)={\bf g}(c_j).$$
Again, finding ${\bf p}$ allows us to integrate from $a$ to $b$ by computing
${\bf p}(b){\bf W}(b)-{\bf p}(a){\bf W}(a)$.
How best to arrange this computation demands attention to many details; for example subdivision of the range from $a$ to $b$ may be advisable: fitting a sequence of low-degree curves may be superior to trying to make a single polynomial fit. Moylan's program \cite{levin-integrate} addresses these issues.
%***
\section{CAS quadrature and error estimation}
It is easily demonstrated that any deterministic mechanism for
quadrature that is based on sampling can be stymied by an adversarial
integrand-evaluation procedure to produce an arbitrarily incorrect
result, and an under-estimated error.
A CAS may have a different prospect in this regard, since it is plausible
that the integrand is presented as a mathematical formula which can be
analyzed for various characteristics including continuity,
differentiability, and presence of poles and singularities. An obvious upper and
lower bound can be obtained by even such a simple mechanism as the
rectangular rule. If $M=\max_{a \le x \le b}f(x)$, and
$m=\min_{a \le x \le b}f(x)$,
then $|b-a|m < \int_a^b f(x) dx < |b-a|M$. For some of
the techniques here, $|b-a|$ can be made small by subdivision, or
$f(x)$ can be made slowly-varying.
It is also possible, in many cases, to bound the truncation error, that
is, the error incurred in replacing a mathematical function by an
approximation: say, replacing a function by its Taylor series, or
Chebyshev series.
Finally, there will usually be a contribution to the error in a reported
result caused by the finite-precision arithmetic used for evaluation.
This can be alleviated in a CAS in several ways.
\begin{itemize}
\item The expressions being evaluated can be mechanically rearranged
to minimize cancellation.
\item The expression, or parts of it, can sometimes be evaluated
{\em exactly} using rational arithmetic.
\item The expression can be evaluated in arbitrary-precision
arithmetic. That is, the same expression can be used but with a
working precision of any specified number of bits, and thus some
computational deficiency at precision $n$ can be overwhelmed by
recomputation at precision $2n$ or higher.
\end{itemize}
Overall,
we have found that in studying particular formulations of quadrature,
we can more easily experiment in this framework:
given an integral, how best can we approximate it with one of these
techniques? Which improves the value, a higher order procedure (more
terms, smaller subdivisions, etc.) or more precise arithmetic? While
a standard numerical language or system can generally allow for more
terms, a CAS provides easy access for other arithmetics, in
particular, arbitrary precision experiments, or perhaps interval
arithmetic. These techniques can be honed on chosen examples
whose exact values are known.
\section{Conclusion}
We have added a few nuances to previous discussions of numerical
integration in a symbolic context (\cite{Fateman81, Geddes86}) which
were primarily concerned with dealing with singularities in the
integrand, stationary points in the oscillatory component,
or with infinite integrals. We have by no means completely
surveyed the literature on the numerical treatment of highly
oscillatory problems, but have simply reviewed the kinds of
contributions that rather simple computer algebra programs can make
toward advancing some of the methods. We hope these demonstrations are
rather simple to understand. We have not, in this brief paper,
attempted to provide a ``one-stop'' complete automated solution, nor
have we included necessary code in our examples to make estimates of
error. We have not covered a number of other approaches or variants
(in particular, variations on the method of stationary phase) that
seem either less accurate, more expensive, or more difficult to
program than the methods here.
The recently released version 8.0 of Mathematica, which was released
after the first draft of this paper, has an expanded facility,
enlarging its already ambitious numerical integration suite of
programs. It includes the Levin-like methods contributed by Andrew
Moylan \cite{LevinIntegrate}. The resulting system is quite complex,
although in our experimentation, capable of good results with many
``automatic'' settings. Even so,a better understandinf of how these
ideas work may be fruitful in finding efficient methods for particular
kinds of documented or new oscillatory integrals.
for exploration of techniques for oscillatory integrals to increase
efficiency and accuracy.
{\begin{thebibliography}{99}
\bibitem{chase}
S.M. Chase and L.D. Fosdick,
``An algorithm for Filon Quadrature,''
{\em CACM 12 no 8} August 1969 453--457.
\bibitem{evans} G.A. Evans, An alternative method for irregular oscillatory integrals over a finite range, {\em Internat. J. Comput. Math. 53} (1994) 185-193.
\bibitem{Fateman81} R.J.~Fateman, ``Computer Algebra and Numerical Integration,
Proc. SYMSAC'81 (ISSAC), August, 1981, 228--232.
\bibitem{filon28}
L.N.G. Filon,
``On a quadrature formula for trigonometric integrals,''
{\em Proc. Roy Soc. Edinburgh 49} 1928-29 38.
\bibitem{Geddes86} K.O.~Geddes,
``Numerical Integration in a Symbolic Context''
Proc. SYMSAC-86, (ISSAC) July, 1986, 185--191.
\bibitem{iserles05}
Arieh Iserles and
Syvert P Norsett.
``Efficient quadrature of highly oscillatory integrals using derivatives,''
{\em Proc. R. Soc. A 8} May 2005 vol. 461 no. 2057 1383---1399 .
\bibitem{iserles04}
A. Iserles and S. P. Norsett,
``On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation''
BIT Numerical Mathematics
Volume 44, Number 4, 755---772, DOI: 10.1007/s10543-004-5243-3
\bibitem{levin-82} David Levin. ``Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations,''
{\em Math. Comput. 38 (1982)} 531---538.
\bibitem{levin-96} David Levin,
``Fast integration of rapidly oscillatory functions,''
{\em J. Computational and Applied Math. 67 (1996)}, 95---101.
\bibitem{LevinIntegrate} David Levin,{\tt http://sites.google.com/site/andrewjmoylan/levinintegrate}. Also see arxiv 0710.3140v1.pdf.
\bibitem{HOP} Newton Institute Seminars on Highly Oscillatory Problems,
{\tt http://www.newton.ac.uk/webseminars/pg+ws/2007/hop/}
\end{thebibliography}
}
\end{document}
some examples
picardfode(y,y,x,1,10,0); exponential
picardfode(sqrt(1/(x^2+1))*y,y,x,1,10,0);
\end{verbatim}
OLD
We need to choose a starting point, but it does not matter, so long as it is
not a zero of $f'$, which by hypothesis allows us to choose $(b+a)/2$.
As an example, consider the differential equation
$r'(x) = \sqrt{r(x)^2+x^2+4}, ~~~r(0)=0.$
Solving it as a Taylor series to 8th order by \verb|newtonfode(sqrt(x^2+y^2+4),y,x,0,8);|
yields
$$ y~=~2\,x+{{5\,x^3}\over{12}}+{{x^5}\over{192}}+{{149\,x^7}\over{32256}}$$
Substitution into
$${{d}\over{dx}}\,y-\sqrt{y^2+x^2+4}$$
yields $1683 x^8/114688$, making us wonder if it is right:
back-substitution shows the result is zero to order 7, which seems pretty close.
(Indeed, it is easily shown by computing more terms that the displayed value for $r$ is
actually correct to order 8).
Given that we can compute a value for $r(x)$, we can compute $H(x)=\exp(f(x))\times r(x)$ and are done.
We can even give a formula for $H(x,\omega)$ showing how the error varies, as a function of $\omega$,
according to the last term in the series. The formula for $H$ may be used, subject
to sufficient accuracy, anywhere within the radius of convergence of the series. The
functional dependency on $\omega$ is explicit, and
(arguably?) we do not have to write out graphs or tables
to illustrate its effectiveness.
There are alternative formulations for solving an ODE semi-symbolically, at
least one of which, using Chebyshev series, seems particularly appropriate. In
this case, a series solution in a specific range, say $[-1, 1]$ would not suffer from
increasing error away from the point of expansion, (zero) but would instead present
nearly optimal approximations for a given degree within a pre-determined range or
at its edges (cite chebfun site).
\subsection{Example}
Consider
$$ I_1 ~=~ \int_{-1}^1 \frac{1}{1+x^2} \exp(i \omega (x^3+4 x )) dx.$$
For this example $f(x) = (i \omega (x^3+4x))$ and hence $f'(x) = (i \omega (3 x^2+ 4))$.
Also, $g(x) = \frac{1}{1+x^2}.$
The differential equation is $r'(x)= g(x)- f'(x)*r(x)$ and can be solved by calling
{\tt newtonfode} as
\verb|newtonfode(1/(x^2+1)-(%i*w*(3*x^2+4))*r, r,x,1,15)| where we have chosen an
expansion to power 10 in $x$. Whether this is an appropriate size may be judged by
dropping off one or more terms to see if the answer changes significantly.
\begin{verbatim}
Notes:
MUST account for Newton seminar activity on this topic, see
http://www.comlab.ox.ac.uk/sheehan.olver/papers/storysofar.pdf
http://www.newton.ac.uk/programmes/HOP/seminars/070509001.pdf
and the detailed Mathematica program from Andrew Moylan's 2008 thesis
Bibliography etc.
\end{verbatim}
iserles/norsett asymptotic series.
sig[0](f,g,x):=f$
sig[k](f,g,x):= diff(sig[k-1](f,g,x)/diff(g,x),x)
kill(sig,dg,h,IS,f,g);
sig[0](f,x):=f;
f:1+cosh(x)$
g:x^3+x^2+4*x+1$ /* g' must be non-zero at 0 and 1 */
dg:diff(g,x);
sig[k](f,x):=diff( sig[k-1](f,x)/dg,x);
h(a,x):=subst(a,x,exp(%i*w*g)/dg*sig[m-1](f,x))$
S[m]:=-(1/(-%i*w)^m*(h(1,x)-h(0,x)))$
IS[L]:=sum(S[m],m,1,L)$
picardfode(F,y,x,a0,n,pt):=
block([s:a0,deg:0],
while deg <= n do
s:a0+integrate( taylor(subst(y=s,F),x,pt,deg:deg+1),x),
return(taylor(s,x,pt,n)))$
%..................
show w out front ..
r' + i*w*f'*r = g
assume r(x) = a1(x)*exp(-i*w*f(x)).
then r'(x)= (a1'(x) - a1(x)*i*w*f'(x)) *exp(-i*w*f(x)).
r' + i*w*f'*r =G is then
(a1'(x) - a1(x)*i*w*f'(x)) *exp(i*w*f(x))
+ i*w*f'(x)*a1(x)*exp(-i*w*f(x)) =
(a1'(x) *exp(-i*w*f(x)) =G
so a1'(x) = G*exp(i*w*f(x)).
which tells us nothing since that is the problem we started with..
of course, not surprising.
what if we let s= exp(-i*w*f(x)), and express r(x) as
a0 +a1*s +a2*s^2 + ... which is a Fourier expansion.
r' is then a1*s' + 2*a2*s*s' + ... = s'*(a1+2*a2*s+3*a3*s^2+ ...)
s' is -i*w*f'*s .
r'+i*w*f'*r =G is then
s'*(a1+2*a2*s+ ...)- s'*(a0+a1*s+a2*s^2 + ...)= G
s'*(a0 + a1(1-s)+a2(2s-s^2) +a3 (3s^2-s^3) + ....) =G
(a0+a1) + (-a1-2*a2)*s + (-a2+3*a3)*s^2 + ...) =G/s'
if we can expand G/s' in powers of s, do we have anything useful?
it would look sort of like a series in sin(k*w*x), k=1,2,.../
presumably the coefficients would show the spectrum of G.
Not helpful? Dunno.
...................
consider integrating factor in general..
y'+P(x)*y=Q(x)
M(x) is an integrating factor ..
M*y' + MPy =MQ
we want lhs to be
(My)' = MQ
to integrate as ..
My= int(MQ,x)
so
y= (1/M)*int(MQ,x)
what is M?
M' = PM or
M'/M =P
log(M)=int(P) or
M= exp(int(P))
put iw in there..
M=exp(int(iwP),x))
look at int(MQ,x)...
integrate by parts.
du=Q/(iwP')dx
v=M=exp(int(iwP)*(iwP')
u =1/(iw)* int(Q/P',x)
dv=iwP'*exp(int... yeech.
uv-int(u dv) ...
uv is int(Q,x)*M- i*w* ... no. try the other way
du=Mdx
u= (1/(i w))... no, no good there..
go back to iserles
r'+i*w*f'*r=g, asymptotically large w
r= int(g-i*w*f'*r,x)= int(g,x) - int (i*w*f'*r,c).
assume r=exp(i*w*h).
r'=i*w*h
$$\int e^{i \omega f(x)}g(x)dx~ = ~\int u dv~=~ uv-\int v~ du.$$
Let $u=g/(i \omega f')$, $dv=e^{i \omega f}\times f'$. Then $v=e^{i \omega f}$, and
$du=(f'g'-gf'')/(- i \omega(f')^2)$
int(exp(iwf)*g,x) = int(u dv) = uv-int(v du).
let u = g/(iwf'),
dv=exp(iwf)*f'*iw
v =exp(iwf)
du =(- f'g'-gf'')/ (-i w f'^2). So
u*v = (1/iw) * exp(iwf)*(g/f'). f' better not be zero...
int(v du) is 1/(-iw)* int(exp(iwf)* (-f'g'-gf'')/f'^2,x).
looks smaller as w increases.
..............
OI(f,g,a,b,omega,k):= block([varlist,p,pointlist,sol,ratprint:false, keepfloat:true],
varlist: makelist(alpha[i],i,0,k), /*start at 0 or 1*/
p: varlist . makelist(x^i,i,0,k),
pointlist : makelist(float(a+ (b-a)*i/k),i,0,k),
define(eq(x), diff(p,x)+%i*omega*diff(f,x)*p-g), ldisplay(yy),
eqs:ev(map(eq,pointlist),numer), ldisplay(sol),
sol:linsolve(eqs,varlist), ldisplay(sol),
define(H(x), exp(%i*omega*f)*ev(p,sol)),
rectform(H(b)-H(a)))
OI(cos(x),x,0,1,50,9)
transformations
consider integrate(f(x),x,a,b). assume -inf <a<b<inf
= (b-a)/2 * integrate(f( ((b-a)*y+(b+a)) /2),x,-1,1)....
if f = exp(iw)*h(x)...
(b-a)/2 * integrate( exp( i*w* ((b-a)*y+(b+a)) /2)* h( ((b-a)*y+(b+a)) /2) dy)
=(b-a)/2 * exp(i*w*(b+a)/2 * integrate( exp( i*w* ((b-a)*y)/2)* h( ((b-a)*y+(b+a)) /2) dy)
=1/2* (b-a)* exp(i*w*(b+a)/2) * integrate( exp( i*w* (b-a)/2*y)* h( ((b-a)*y+(b+a)) /2), y,-1,1).
Longer program for integration by parts...
(ibpoilims(f,g,x,w,L,a,b):= /* integrate from a to b */
block([keepfloat:true, ratprint:false],
at(ibpoilims1(taylor(f,x,b,L),taylor(g,x,b,L),x,w,L,0),x=b)
-at(ibpoilims1(taylor(f,x,a,L),taylor(g,x,a,L),x,w,L,0),x=a)),
ibpoilims1(f,g,x,w,L,c):= if c>=L then 0 else
block([df:diff(f,x,1)],
exp(%i*w*f)*g/(%i*w*df)
- 1/(%i*w)*ibpoilims1(f,(df*diff(g,x,1)-g*diff(df,x,1))/df^2,x,w,L,c+1)))
/*our example */
ibpoilims(sinh(x),cos(x),x,1000.0,3.0,-1.0,1.0); /* integrate from -1 to 1 */
(ibpoilims(f,g,x,w,L,a,b):= /* integrate from a to b */
block([keepfloat:true, ratprint:false],
at(ibpoilims1(taylor(f,x,b,L),taylor(g,x,b,L),x,w,L,0),x=b)
-at(ibpoilims1(taylor(f,x,a,L),taylor(g,x,a,L),x,w,L,0),x=a))
ibpoilims1(f,g,x,w,L,c):= if c>=L then 0 else
block([df:diff(f,x,1)],
exp(%i*w*f)*g/(%i*w*df)
- 1/(%i*w)*ibpoilims1(f,(df*diff(g,x,1)-g*diff(df,x,1))/df^2,x,w,L,c+1)))
/*our example */
ibpoilims(sinh(x),cos(x),x,1000.0,3.0,-1.0,1.0); /* integrate from -1 to 1 */
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
let's try Levin for Bessel functions.
integrate(bessel_j(1,200*x) *(x^2+1),x,0,1)
which is 1/200*(1-bessel_j(0,200)+bessel_j(2,200))
exactly, or about
0.005151659172396532004783694962224718046978102435
consider ibpoi,
consider a version of LevinInt
Let bessel_j[1,x] == J1(x).
diff(J1(x),x) is 1/2*(J0(x)-J2(x)), by the way.
H= integrate(J1(200*x)*(x^2+1),x,0,1)
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https://ctan.math.washington.edu/tex-archive/info/examples/Einfuehrung2/17-03-1.ltx | washington.edu | CC-MAIN-2022-27 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-27/segments/1656103915196.47/warc/CC-MAIN-20220630213820-20220701003820-00292.warc.gz | 241,913,076 | 1,396 | %%
%% Ein Beispiel der DANTE-Edition
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%% Beispiel 17-03-1 auf Seite 844.
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%% Copyright (C) 2016 Herbert Voss
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Das momentane Datum ist \PrintDatum. \par \Test[Datum]{Neues Datum: } \par
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http://dlmf.nist.gov/4.21.E35.tex | nist.gov | CC-MAIN-2017-17 | application/x-tex | null | crawl-data/CC-MAIN-2017-17/segments/1492917118831.16/warc/CC-MAIN-20170423031158-00108-ip-10-145-167-34.ec2.internal.warc.gz | 107,901,051 | 646 | \[\mathop{\sin\/}\nolimits\!\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\mathop{\sin%
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https://cheatography.com/bernattorras/cheat-sheets/laravel/latex/ | cheatography.com | CC-MAIN-2022-40 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030334974.57/warc/CC-MAIN-20220927002241-20220927032241-00521.warc.gz | 209,529,366 | 4,997 | \documentclass[10pt,a4paper]{article}
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'password' =\textgreater{} '123456' \newline % Row Count 21 (+ 1)
); \newline % Row Count 22 (+ 1)
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'email' =\textgreater{} '[email protected]', \newline % Row Count 26 (+ 1)
'password' =\textgreater{} 'abc' \newline % Row Count 27 (+ 1)
); \newline % Row Count 28 (+ 1)
DB::table('users')-\textgreater{}where('email', \$data{[}'email'{]})-\textgreater{}update(\$data); \newline % Row Count 30 (+ 2)
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\$post-\textgreater{}title('updated title'); \newline % Row Count 21 (+ 1)
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{\bf{ // DELETE }} \newline % Row Count 23 (+ 1)
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|
http://www.tcl.tk/cgi-bin/tct/tip/28.tex | tcl.tk | CC-MAIN-2013-20 | application/x-latex | null | crawl-data/CC-MAIN-2013-20/segments/1368706009988/warc/CC-MAIN-20130516120649-00028-ip-10-60-113-184.ec2.internal.warc.gz | 742,372,010 | 10,577 | \documentclass[]{article}
\usepackage{amsmath,graphicx,supertabular,hyperref,tabularx,ifthen}
\title{TIP \#28: How to be a good maintainer for Tcl/Tk}
\date{February 23, 2001}
\author{Don Porter}
\urlstyle{sf}
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\begin{document}\maketitle
\begin{center}\begin{tabularx}{\linewidth}{|r@{: }X|}\hline
\textbf{TIP \#28}&\textbf{How to be a good maintainer for Tcl/Tk}\\\hline
Author&
Don Porter \tipmail{dgp}{users.sourceforge.net}
\\
Created&Friday, \(\text{23}^{\text{rd}}\) February 2001\\
Type&Informative\\
State&Draft\\
Vote&Pending\\
Version&\tipversion$Revision: 1.21 $\\
Post History&\\
\hline\end{tabularx}\end{center}
\thispagestyle{empty}\pagestyle{empty}
\begin{tipabstract}
This document presents information and advice to maintainers in the form of a Frequently Asked Questions (FAQ) list.
\end{tipabstract}
\tableofcontents\setcounter{page}{0}\clearpage\pagestyle{plain}
\section{Preface}
Notice in the header above that this is a Draft document. It won't be the \textit{official} word of the TCT unless/until it is accepted by the TCT. Meanwhile, it should still be a helpful guide to those serving or considering service as maintainers. At the very least it's a useful straw man to revise into something better. Help us make it even more useful by using the [Edit] link at the bottom of this page (if any) to add/revise the questions and answers, or add your comments.
\section{Background}
TCT procedures (see \cite{tip0}) calls for one or more \textit{maintainers} to take responsibility for each functional area of the Tcl (\cite{tip16}) or Tk (\cite{tip23}) source code. Every source code patch to Tcl or Tk will be committed to the official branches of the appropriate CVS repository only after approval by an appropriate set of maintainers.
\section{Can I be a Tcl/Tk maintainer?}
Most likely. To be a maintainer, you should have...
\begin{itemize}
\item{}
...an interest in Tcl/Tk.
\item{}
...access to the Internet (Web and e-mail).
\item{}
...some volunteer time to contribute.
\item{}
...the ability and the support software to code in C and/or Tcl, use CVS, use SourceForge facilities, and familiarity with a portion of the Tcl/Tk source code to be maintained, or the willingness to acquire these things.
\end{itemize}
For the most part, if you are reading this document, you probably have what it takes to be a Tcl/Tk maintainer.
\section{What can I maintain?}
The Tcl Core Team (TCT) has divided up the Tcl/Tk source code into functional areas as described in \cite{tip16} and \cite{tip23}. You can volunteer to help maintain as many areas as you think you can handle. Select those you have experience with or an interest in.
\section{What does a maintainer do?}
Maintainers are the people who make changes to the files that make up the source code distribution of Tcl or Tk -{}- code, documentation, and tests. That's what a maintainer does: check in changes to the official source code in the area he/she maintains.
The source code can be changed for several reasons: to correct a bug, to add a new feature, or to re-implement an existing feature in a new way. The reason for a change controls how much oversight the maintainer must have while making the change. More on this below.
\section{How do I prepare to be a maintainer?}
The official repositories of Tcl and Tk source code are kept at SourceForge, so you need to register for a SourceForge account [\url{https://sourceforge.net/account/register.php}]. As part of the registration, you will select a login name. When you volunteer as a maintainer, the administrators of the Tcl or Tk projects will need that name to give you write access to the appropriate repository.
Once you have a SourceForge account, get familiar with the tools it provides. Most important is that you get set up to use CVS over SSH to access the repository. This can be difficult. There are some notes [\url{http://tcltk.org/sourceforge}] on how other Developers on the Tcl and Tk projects have been able to successfully get this done.
This document does not include instructions on how to use CVS. See the following references for assistance with learning CVS.
\begin{itemize}
\item{}
\url{http://cvsbook.red-bean.com/cvsbook.html}
\end{itemize}
\textit{Add more references here please.}
\section{How do I volunteer to be a maintainer?}
Send a message to \tipmail{tcl-core}{lists.sourceforge.net} telling the TCT your SourceForge login name and what area(s) you want to help maintain. Someone will add you to the list of \textit{Developers} on the Tcl or Tk projects and enable your access to SourceForge features like the Bug Tracker and Patch Manager. As a Developer, you will have write access to the appropriate repository of official source code.
\section{Write access! So I can just start changing Tcl/Tk?!}
For some purposes, yes. For others, you'll need to get approval from the TCT first. Read on...
\section{What Internet resources does a maintainer use?}
A maintainer uses the SourceForge Bug Tracker for Tcl or Tk to learn what bugs are reported in his area (browse by Category).
\begin{itemize}
\item{}
\url{http://sourceforge.net/bugs/?group_id=10894}
\item{}
\url{http://sourceforge.net/bugs/?group_id=12997}
\end{itemize}
A maintainer uses the SourceForge Patch Manager for Tcl or Tk to learn what patches make changes in his area (browse by Category).
\begin{itemize}
\item{}
\url{http://sourceforge.net/patch/?group_id=10894}
\item{}
\url{http://sourceforge.net/patch/?group_id=12997}
\end{itemize}
A maintainer uses CVS via SSH to access, track, and modify the various branches of development in the repository of official Tcl or Tk source code.
\begingroup\small\begin{verbatim}
cvs -d :ext:[email protected]:/cvsroot/tcl \
checkout -r $BRANCH_TAG -d $LOCAL_DIR tcl
cvs -d :ext:[email protected]:/cvsroot/tktoolkit \
checkout -r $BRANCH_TAG -d $LOCAL_DIR tk
\end{verbatim}
\endgroup
A maintainer examines the state of Tcl Improvement Proposals (TIPs) and adds his comments to them at the TIP Document Collection.
\begin{itemize}
\item{}
\url{http://purl.org/tcl/home/cgi-bin/tct/tip/}
\end{itemize}
A maintainer may follow and participate in TCT discussions about TIPs and other matters concerning Tcl/Tk development on the TCLCORE mailing list.
\begin{itemize}
\item{}
\url{http://lists.sourceforge.net/lists/listinfo/tcl-core}
\end{itemize}
A maintainer may receive e-mail notification every time any change is made to any entry in Tcl's or Tk's Bug Tracker or Patch Manager by subscribing to the TCLBUGS mailing list.
\begin{itemize}
\item{}
\url{http://lists.sourceforge.net/lists/listinfo/tcl-bugs}
\end{itemize}
\section{There are multiple maintainers in my area. What do I do?}
The maintainer tasks are the same; you just have more hands to get the job done. It is up to the maintainers of an area to decide among themselves how they will divide the tasks. They might each take on a particular subset of files. Or they might let some maintainers fix bugs while others review new features. Or they might appoint one maintainer as the \textit{lead} and let him assign tasks to the others. Whatever works for you, and gets the work done.
\section{I found a bug in my area. What do I do?}
Bug finding and reporting is a job for the whole community, so when you find a bug, take off your maintainer hat. Report it to the Bug Tracker just like anyone would. If you recognize that the bug is in your area, go ahead and assign it to the Category for your area and to yourself or one of the other maintainers who share responsibility for that area.
\section{Why do I report the bug to myself?}
So that the bug appears in the database. Someone else may find it too, and when they go to report it to the Bug Tracker, they should discover that it's an already reported problem. A registered bug report is also the place where progress on fixing the bug can be recorded for all to see.
\section{There's a bug reported in the Category for the area I maintain. What do I do?}
First, understand the bug report. The best bug reports are clear and come with a demonstration script, but not all reports are so well crafted. You may need to exchange messages with the person who reported the bug. If the reporter logged in to SourceForge as \textit{username} before submitting a report, then you can write back to \textit{[email protected]}. If the bug was reported by \textit{nobody}, the best you can do is post a followup comment to the bug asking for more information, and hope the reporter comes back to check.
Next, confirm that the bug report is valid, original, and that it belongs in your area. Does it correctly assert that some public interface provided by your area behaves differently from its documented behavior? If not, then you should take the appropriate action:
\begin{enumerate}
\item{}
If the bug report notes a problem in another project, assign it to a Developer who is an Admin of the other project. Add a comment asking them to reassign to the correct project. Assigned To: \textit{an Admin of the other project}.
If no Developer is an Admin of the other project, or the other project isn't hosted by SourceForge, note the error in a comment, and mark the report invalid. Resolution: Invalid; Status: Closed; Assigned To: \textit{yourself}.
\item{}
If the bug report notes a problem due to a bug in another area, reassign it to the appropriate Category. Category: \textit{correct category}
\item{}
If the reporter's expectations are incorrect, point them to the documentation. You may also want to revise the documentation if it is not clear. Resolution: Invalid; Status: Closed; Assigned To: \textit{yourself}.
\item{}
If the bug report notes a problem already noted by another bug report, note the duplication. Resolution: Duplicate; Status: Closed; Assigned To: \textit{yourself}.
\item{}
If the bug report acknowledges that the code is behaving as documented, but argues that the documented behavior should be revised, then the report is a feature request rather than a bug report. More on handling feature requests below. Group: Feature Request.
\end{enumerate}
Valid, original bug reports in your area should be assigned to a maintainer of your area. If you are the only maintainer of your area, assign the bug to yourself. If there are multiple maintainers, you should decide among yourselves how to divide up the bug report assignments.
\section{There's a bug assigned to me. What do I do?}
Now we get the the heart of what a maintainer does. This is where you unleash the energies and talents you bring to the table. So, the best answer is ``Do what works best for you.'' The rest of this answer should be read as additional guidelines and tips that have worked well for others and might help you, but not as a mandatory checklist you must follow. If some advice below seems more burdensome than helpful, fall back to ``Do what works best for you.'' The goal is to register a patch that fixes the bug with the SourceForge Patch Manager. Do whatever helps you accomplish that goal.
Try to enlist the assistance of the person who reported the bug. This is especially important if the problem is platform-specific on a platform you do not have access to. Gaining the participation of the person who reported the bug can have many other benefits too. They see that progress is being made. They can offer additional insights they have, but left out of their original report. They can see how better bug reports lead to faster, better solutions, so their next reports may be of higher quality. They may even gain enough experience that their next report may come with the correction already attached. Eventually, they may even become maintainers themselves.
First, try to develop a test that demonstrates the bug and add it to the section of the test suite for your area. If the original bug report contained a demonstration script, perhaps you can adapt that. The new test will help you verify when you have fixed the bug.
If a fix for the bug is offered with the report, give it a try. Otherwise develop a fix yourself. Take care that while fixing the bug, you do not create new bugs by changing the correct behavior of other parts of the code in your section. The test suite for your area is very helpful. Use it.
It may become apparent that the best fix for your bug can only be accomplished after another bug is fixed first, or perhaps after a new feature is added to Tcl/Tk. In those cases, add a comment to the original bug report so those interested will know what is causing the delay. SourceForge may offer a way to denote these dependencies as well.
If you have trouble fixing the bug, ask for help. Try the other maintainers of your area first. Then try posting comments attached to the original bug report. Using \textit{cvs log}, you can get a list of developers who've recently made changes to the files you maintain. They might be able to offer advice, or explanations about why the code is the way it is. If none of these focused searches for help bears fruit, then try broader requests to the TCLCORE mailing lists, or the \url{news:comp.lang.tcl} newsgroup.
At any time, you may have several bugs assigned to you. It will help guide the expectations of the Tcl community if you can assign priority values to the bugs indicating the importance you assign to them. Try to work on fixing higher priority bugs before lower priority bugs. Some reasons you might give a bug a higher priority include:
\begin{enumerate}
\item{}
The bug causes a panic or core dump.
\item{}
Documentation is missing or incorrect.
\item{}
Other bug fixes are waiting on this bug fix.
\item{}
Several duplicate reports or ``me too'' comments about the bug are coming in from the community.
\end{enumerate}
Some reasons you might give a bug a lower priority include:
\begin{enumerate}
\item{}
A workaround is identified (add it as a comment attached to the bug report).
\item{}
Feature requests tend to get lower priority since they should be handled through the TIP process.
\end{enumerate}
Once you have crafted a fix for the bug, create a patch to the official source code (including the new tests that test for the fixed bug) and register it with the SourceForge Patch Manager. Note the number of the bug report fixed by the patch somewhere in the summary or comments associated with the patch. Assign the patch to yourself. Assign the Category to the area you maintain.
\section{There's a patch registered under the Category I maintain. What do I do?}
The SourceForge Patch Manager is used to review and revise patches before they are committed to the official source code. Your actions depend on what the patch does to your area, and who the patch is assigned to. The patch may change the public interface provided by your area (feature change); or the change may be completely internal (bug fix, or re-implementation) within your area. The patch may be assigned to you, to someone else, or to nobody. The person the patch is assigned to is the person who is leading the effort to integrate the patch into the official source code.
\section{What if the patch is assigned to nobody?}
The patch has probably been contributed by someone not on the list of Developers. It may be a contributed bug fix, or a contributed implementation of a TIP. Assign contributed bug fixes to the same maintainer who is assigned the corresponding bug report. If there is no corresponding bug report, add one. Assign TIP implementations to the Developer identified in the TIP as the one responsible for implementation of that TIP, or the TCT member who sponsored the TIP.
If the patch changes only your area (and shared or generated files), then leave the Category in your area. If the patch changes other areas as well as yours, change the category to None.
\section{What if the patch is assigned to me?}
Presumably you've assigned it to yourself to indicate that you're taking charge of integrating that patch into the official sources. If that's a mistake, treat the patch as if it were assigned to nobody. If you are the one leading the integration effort, see below (How do I integrate a patch into the official sources?).
\section{What if the patch is assigned to someone else?}
If the patch is assigned to another maintainer in your area, let him handle it. Leave it alone.
If the patch makes no changes in your area, change the Category of the patch to None.
If the patch makes changes in your area, and is assigned to a Developer who is not a maintainer of your area, that Developer is asking for review of the patch's changes to your area. You or one of the other maintainers of your area should review the patch and accept or reject it. Read on...
\section{What special review does a "feature change" patch require?}
Changes to the public interface of your section must be proposed to and accepted by the TCT through the TIP process before they can be added to the official Tcl source code. If the patch changes the public interface of your section, then there should be an associated TIP describing the new feature(s) that patch implements. Until there is such a TIP, and that TIP has been accepted by the TCT (check the value of the State header), you should not approve the patch.
Once there is an approved TIP corresponding to the patch, you should confirm that the patch correctly implements the accepted feature as described by the TIP. If not, you should not approve the patch.
After confirming that the patch correctly implements the feature change described in an accepted TIP, you should still review the technical merit of the patch's changes to your area before approving it.
\section{How do I review the technical merits of a patch?}
Apply the patch and run the test suites that cover your area. Check that the patch does not add any new test failures. If the patch is a bug fix, check that it actually fixes the bug. Think five times before approving a patch that causes new test failures or incompletely fixes a bug or incompletely implements an approved TIP.
Keep in mind that once the patch is integrated into the official sources, you'll be expected to maintain it. It is not in your interest to approve patches that make your job harder. Think four times before approving a patch that you do not understand.
Check that the patch keeps the features offered on different platforms consistent. If not, be certain that the documentation properly notes the platform-specific behavior. Think three times before approving a patch that causes the capabilities of Tcl/Tk to further diverge on different platforms.
Check that the patch follows Tcl's established coding conventions. See the Tcl/Tk Engineering Manual [\url{http://purl.org/tcl/home/doc/engManual.pdf}] and the Tcl Style Guide [\url{http://purl.org/tcl/home/doc/styleGuide.pdf}] for details. This is especially important when accepting contributed patches. Think twice before approving a patch that doesn't conform to these conventions.
Check the effect of the patch on the performance of Tcl/Tk. Use the tclbench set of benchmarks.
\begingroup\small\begin{verbatim}
cvs -d :pserver:[email protected]:/cvsroot/tcllib \
checkout tclbench
\end{verbatim}
\endgroup
Think carefully before approving a patch that significantly degrades the performance of important operations.
Finally, while examining the patch, you may see a better way to accomplish the effect of the changes in your area. If you can provide that alternative implementation reasonably quickly, then propose it as a revision to the patch. However, be careful not to let the perfect be the enemy of the good. If a patch works, do not reject just because you can imagine a better way it could be done. Provide the better way, or accept the less good way in the patch, and leave migration to the better way for later when you have the time.
To approve the patch's changes to your area, simply note your approval in a followup comment on the patch. Indicate in your comment the Category of the area for which you approve the changes. If the patch changes multiple areas, set the Category of the patch back to None.
To reject the patch, you also indicate your rejection in a followup comment. You should explain the reasons for your rejection so that the patch can be revised with the goal of gaining your approval. If you can supply the needed revisions with reasonable effort, do so. If the patch changes multiple areas, set the Category of the patch back to None.
Unless the patch is assigned to you, do not change the Status of the patch. Leave that to the Developer assigned to the patch.
\section{How do I integrate a patch into the official sources?}
First you need the approval of at least one maintainer of each section changed by the patch.
\section{How do I get approval for integration?}
First, assign the patch to yourself to indicate that you are leading the integration effort. Next, determine the list of categories corresponding to the areas changed by the patch. It may help if you list them in a comment attached to the patch.
For each category in the list, assign the Category of the patch to that category. Then wait for a maintainer for that area to review the patch. If one approves it, then assign the next Category in the list. If maintainers for all areas on the list approve the same patch, you may integrate the patch into the official sources.
If a maintainer rejects the patch, revise the patch to address his concerns. Then start the review again. Start with the maintainer who rejected the first patch to be sure his concerns are addressed first.
Note that if the patch changes only the area you maintain, then you may immediately integrate the patch into the official sources once you are satisfied with it and it is registered in the Patch Manager.
\section{The patch is approved. How should it be integrated?}
Get a CVS working directory that is up to date with the HEAD branch of the official source repository. Apply the patch to your working directory, and then 'cvs commit' the changes to the HEAD branch.
At the same time you commit the patch, be sure to add an entry to the ChangeLog file describing the change. Follow the established format, which is derived from the GNU coding conventions. The description should be brief, but should describe the change reasonably completely. Include the SourceForge Bug and Patch ID numbers in the ChangeLog entry, but do not assume that the reader will have access to the Bug Tracker and Patch Manager to be able to understand the entry. You may assume the reader has access to the documentation.
Finally, with the patch integrated, change the Status of the patch in the Patch Manager to Accepted. If any bugs were fixed by the patch, change their Resolution to Fixed, and their Status to Closed.
\section{I want a patch review even though the patch changes only my area.}
Keep in mind that integrating a patch into the official sources is not an irreversible act. Commits to the HEAD branch will be checked out and tested by members of the Tcl community who are tracking Tcl/Tk development. Alpha and beta releases of Tcl/Tk that include your patch will also get your changes reviewed in practical settings.
That said, if you really want a pre-commit review of your patch, you can add a comment to the patch asking for review. Someone will probably respond. It's up to your judgment how long to wait, keeping in mind that you are the maintainer, so your judgment on the quality of patches in your area is implicitly trusted.
\section{What about CVS branches?}
When you integrate a patch into the official source code, you will usually 'cvs commit' the patch onto the HEAD branch. If the patch includes a feature change, it must (except in unusual circumstances approved by the TCT) be committed to the HEAD branch. The HEAD branch is the development branch from which alpha releases of Tcl/Tk are generated.
At any time, there is also one or more \textit{stable} branches of development. As of February, 2001, the branch 'core-8-3-1-branch' indicates the sequence of revisions from which the 8.3.x releases of Tcl/Tk are generated.
Since the Tcl Core Team took over development of Tcl/Tk, no changes have been committed to a stable branch, so we really have not established procedures on how we will decide what bug fixes should and should not be applied to the stable branch. It is possible that maintainers will be involved, though. It is also possible that a special team will be appointed to update the stable branch in preparation for the next stable release. In the case that you as a maintainer are asked to commit to the stable branch, be aware that the only patches that should be committed to a stable branch are those that fix bugs. No new features should be committed here.
The other kind of branch is a \textit{feature} branch. This is a development branch on which a sequence of several revisions may be committed as work in progress on a new feature, or re-implementation of existing features. Typically a feature branch will be created if the effort...
\begin{itemize}
\item{}
...touches on several functional areas;
\item{}
...is worked on jointly by several Developers;
\item{}
...is complex enough to require several revisions;
\item{}
...needs prototyping to determine the best TIP proposal to make; or
\item{}
...makes an incompatible change to Tcl/Tk that properly belongs on the next major version of Tcl/Tk before the HEAD branch has been designated for work toward the next major version.
\end{itemize}
As a Developer, feel free to create a feature branch if you have a reason to use one. Make a note of your branch tags in \cite{tip31}. Avoid the use of a branch tag matching core-* . Save the core-* branch tags for the tags of official stable branches and releases. To avoid conflict with other Developers, consider using your SourceForge login name as a prefix on the feature branch tags you create. Try to also make the branch tag descriptive of the purpose of the branch.
One big advantage of a feature branch is that any Developer may commit changes to a feature branch without all the publication, review, and approval overhead required when committing patches to the HEAD or stable branches. On the feature branches you can go through multiple revisions reasonably quickly and spend the administrative overhead only at the end when it is time to apply the finished product to the official branches.
\section{What other things does a maintainer do?}
The tasks of fixing bugs and approving and committing patches to the official source code of Tcl and Tk are the core tasks that maintainers perform. That's all the job actually requires.
You will probably want to keep an eye on the TCT's plans for Tcl/Tk development as well. If a TIP proposes a new feature in your area, it is in your interest to know about it, and propose revisions and improvements to it. Ultimately you will be asked to approve the patch that implements the new feature, and then you will be expected to maintain it, so if you have concerns about a proposal, it's best to make them known early. TCT members will probably ask your opinion on TIPs that propose changes to your area for this reason.
\section{Comments}
Please add your comments here.
\begin{quote}
Well, since I drafted this SourceForge has replaced the Bug Tracker and Patch Manager with a \textit{Tracker}. This TIP \textit{really} needs revision now.
\end{quote}
\section{Copyright}
This document has been placed in the public domain.
\section{Colophon}
\textit{TIP AutoGenerator --- written by Donal K. Fellows}
\begin{thebibliography}{TIP \#4}
\addcontentsline{toc}{section}{References}
\bibitem[TIP \#0]{tip0}
John Ousterhout,
\emph{Tcl Core Team Basic Rules},
on-line at \url{http://purl.org/tcl/tip/0.html}
\bibitem[TIP \#16]{tip16}
Don Porter,
Daniel Steffen,
\emph{Tcl Functional Areas for Maintainer Assignments},
on-line at \url{http://purl.org/tcl/tip/16.html}
\bibitem[TIP \#23]{tip23}
Kevin Kenny,
Jim Ingham,
Don Porter,
Daniel A. Steffen,
Donal K. Fellows,
\emph{Tk Toolkit Functional Areas for Maintainer Assignments},
on-line at \url{http://purl.org/tcl/tip/23.html}
\bibitem[TIP \#31]{tip31}
Don Porter,
miguel sofer,
Jeff Hobbs,
Kevin Kenny,
David Gravereaux,
Donal K. Fellows,
Andreas Kupries,
Donal K. Fellows,
,
Kevin Kenny,
\emph{CVS tags in the Tcl and Tk repositories},
on-line at \url{http://purl.org/tcl/tip/31.html}
\end{thebibliography}
\end{document}
|
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\title{Teorija i kritika rascjepa vrijednosti}
\date{2018}
\author{Roswitha Scholz}
\subtitle{Teze o kapitalizmu i hijerarhijskim rodnim odnosima}
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U tekstu koji slijedi predstavit ću neke od teza o kritici rascjepa vrijednosti (Wertabspaltung) koju sam razvila i iscrpno razložila u knjizi „Rod kapitalizma" (prvo izdanje objavljeno 2000. godine). U 1990-im godinama su u polju istraživanja o ženama, nakon kraha Istočnog bloka, visoko na cijeni bili kulturološki pristupi i pristupi teorije razlike, koji su zatim mutirali u teorije roda. S druge strane, do krize 2007\Slash{}2008, Marx već dugo ni u feminizmu ne predstavlja tabu. Teorija rascjepa vrijednosti moj je odgovor na te dvije tendencije. Moje je teorijsko polazište je, uz Adornovu kritičku teoriju, jedna nova fundamentalno kritična teorija „vrijednosti" i teorija „apstraktnog rada" kao daljnje razvijanje marksijanske kritike političke ekonomije, čiji su najprominentniji predstavnici u posljednjih nekoliko desetljeća bili Robert Kurz i djelomično Moishe Postone (usp. Kurz 1991, Kurz 1999, Postone 1988, Postone 2003). Namjera mi je pridodati novu kvalitetu njihovim pristupima.
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\textbf{1.}
U skladu s tim novim shvaćanjem kritike vrijednosti, u središtu kritike ne stoji takozvani višak vrijednosti za sebe, odnosno eksploatacija koja je izvana definirana kroz kapital preko pravno-vlasničkih odnosa, već društveni karakter robnoproizvodnog sistema, a time i karakter apstraktnog rada u širem kapitalističkom kontekstu. Prema tom pristupu, „rad " nastaje tek u kapitalizmu, zajedno s poopćenjem robne proizvodnje, te ga stoga nije moguće ontologizirati.
Kao robe, proizvodi predstavljaju „minuli apstraktni rad" te utoliko i vrijednost, to jest predstavljaju jednu određenu (prepoznatu na tržištu kao društveno vrijednu) količinu uložene ljudske energije. To „predstavljanje" se opet očituje u novcu kao univerzalnom posredniku i istovremenoj samosvrhovitosti oblika kapitala. Proizvodne aktivnosti su u moderno doba posredovane odnosom apstraktnog rada. Razvojem kapitalizma cjelokupni je život na Zemlji obilježen slobodnim kretanjem kapitala, a zajedno s time, u kapitalizmu po prvi put nastaje apstraktni rad, koji se danas doima ahistorijskim - kao ontološki princip.
\textbf{2.}
Osnovni oblik kapitalizma, po mojem mišljenju, iz perpektive forme „vrijednosti", odnosno apstraktnog rada još uvijek nije u dovoljnoj mjeri definiran kao fetišistički odnos. Valja uzeti u obzir da se uz formu vrijednosti i apstraktni rad, u kapitalizmu pojavljuju i reproduktivne aktivnosti koje ponajprije obavljaju žene. U skladu s time, rascjep vrijednosti podrazumijeva da su reproduktivne aktivnosti u srži određene kao ženske aktivnosti tako što su pomoću pripisivanja određenih orođenih osjećaja, osobina, držanja (emocionalnost, senzualnost, brižnost i dr.), jednostavno odvojene od vrijednostiZapstraktnog rada. Ženski životni kontekst, ženske reproduktivne aktivnosti u kapitalizmu imaju stoga drugačiji karakter od apstraktnog rada, zbog čega se ne mogu bez poteškoća supsumirati pod pojam „rada“. Radi se o jednoj strani kapitalističkog društva koju se ne može obuhvatiti marksijanskim pojmovnim instrumentarijem. Međutim, u jednu ruku, reproduktivni rad je povezan s vrijednošću, njezina je pretpostavka te joj nužno pripada; u drugu, on se ipak nalazi izvan nje.
Vrijednost i njen rascjep stoje, dakle, u dijalektičkom odnosu jedno prema drugome. Jedno se ne može izvesti iz drugoga, već se jedan od drugoga razilaze istovremeno međusobno se određujući. Utoliko se rascjep vrijednosti može razumjeti kao nadređenu logiku koja premašuje unutarnje kategorije zasnovane na robnom obliku. Upravo se u tom smislu može doći do razumijevanja (fetišističkog) podruštvljenja, a ne samo preko forme „vrijednosti“. Trebalo bi također naglasiti kako je ono što nam je naizgled osjetilno neposredno dato u području reprodukcije, potrošnje i aktivnosti koje ih okružuju, postalo historijskim u kontekstu rascjepa vrijednosti kao sveukupnog procesa.
Kategorije kritike političke ekonomije ipak nisu dovoljne i iz još jedne perspektive. Rascjep vrijednosti podrazumijeva specifičan sociopsihološki odnos. Određene manje vrijedne osobine (senzualnost, emocionalnost, slabosti u rezoniranju i karakteru itd.) pripisane su „ženi“, te su odvojene od muško-modernoga subjekta. Takvo orođenje osobina znatno obilježava simbolički poredak robnoproizvodnog patrijarhata. Posebice se na sociopsihološkoj i kulturalnosimboličkoj razini rascijepa vrijednosti to orođivanje ispostavlja kao formalni princip robnoproizvodnog patrijarhata.
\textbf{3.}
Asimetričan rodni odnos se po mojem mišljenju u teorijskom smislu može istraživati samo ograničimo li se na modernu\Slash{}postmodernu. To ne treba značiti da taj odnos nema svoju predmodernu historiju. Međutim, s podruštvljenjem robnog oblika on je zadobio sasvim novu kvalitetu. U takvom bi sustavu žene trebale biti prvenstveno zadužene za manje vrednovano područje reprodukcije koje se ne može predstaviti u novcu, a muškarci za kapitalističku proizvodnu sferu i javnost. Tome proturječe tumačenja koja na rodne odnose gledaju kao na „ostatke“ predkapitalističkih razdoblja. Primjerice, nuklearna obitelj, kakvu mi poznajemo, pojavljuje se tek u 18. stoljeću, kao što su općenito javna i privatna sfera onako kako ih mi razumijemo nastale u moderno doba. Istodobno tvrdim da kapitalistička robna proizvodnja u tom razdoblju nije naprosto išla svojim tijekom, nego da se naprotiv tada pokrenula društvena dinamika čiji je glavni princip odnos rascjepa vrijednosti. Odnos između privatne i javne sfere razjašnjava i postojanje muških saveza koji su nastali na osnovi afekata usmjerenih protiv ženskoga. Tako su se u muške saveze ustrojile cjelokupna država i politika. Ipak, efekt rascijepa vrijednosti kao društveno-formalnog principa prožima sve razine i sfere, dakle i različite javne sfere. Npr. žene su zapravo cijelo to vrijeme također bile zaposlene, no unatoč tome, u prosjeku zarađuju manje od muškaraca te do danas imaju poteškoća u tome da dospiju na više pozicije.
\textbf{4.}
Kao i Frigga Haug, polazim od ideje da se robnoproizvodni patrijarhat mora pojmiti kao civilizacijski model (usp. Haug 1996, str. 229 nadalje), ali ću njena promišljanja prilagoditi teoriji rascjepa vrijednosti (usp. Haug 1996, od str. 229 nadalje). Na „muškarca" se tako gleda kao na čovjeka\Slash{}duhovno biće\Slash{}onoga koji je nadišao tijelo, a na ženu pak kao na nečovjeka, kao na tijelo. Muškarci moraju stremiti slavi, odvažnosti, besmrtnim djelima, te ih se drži radnima. Žene za zadaću imaju brigu o pojedincima i o ljudskom rodu. Pri tome se priroda mora proizvodno podčiniti, njome se mora ovladati. Muškarac se pak neprestano nalazi u nadmetanju s drugima.
Takve predodžbe također određuju sliku poretka modernog društva u cjelini. Proizvodna sposobnost i voljnost, racionalno, ekonomsko, efektivno trošenje vremena podjednako određuju civilizacijski model u njegovim objektivnim strukturama, njegovim mehanizmima i njegovoj historiji, kao i principe djelovanja pojedinaca.
Istodobno s napuštanjem društvenoga i prirode, podčinjavanje i marginaliziranje žena stoga su pretpostavka robnoproizvodnog civilizacijskog modela (usp. Haug 1996, od str. 229 nadalje).
Empirijski se pojedinci ne mogu otrgnuti društveno-kulturnim uzorcima, čak i kad se više s njima ne poistovjećuju. Pri tome, predodžbe o rodu podliježu historijskoj mijeni, što ćemo još kasnije vidjeti.
\textbf{5.}
strukturi, a da se istovremeno različitost supsumira te se štoviše prenose mehanizmi, strukture, obilježja robnoproizvodnog patrijarhata na ne-robnoproizvodna društva, što podrazumijeva stavljanje različitih razina, sfera, područja u samom robnoproizvodnom patrijarhatu u istu kutiju, uz iznimku kvalitativnih razlika. Za razliku od polaženja od forme vrijednosti, kod odnosa rascjepa vrijednosti polazi se od jedne temeljne društvene strukture kojoj odgovara androcentrično-univerzalističko i identitetsko-logičko razmišljanje. Jer od presudne važnosti nije samo to da postoje - uz iznimku različitih kvaliteta - trećina vremena provedena zajedno, prosječno radno vrijeme, apstraktni rad, što djelomično stoji iza ekvivalentnog oblika novca. Prije će biti da je društvenoj strukturi vrijednost nužna kako bi mogla gledati na kućanski rad, životnosvjetno, senzualno, emocionalno, nepojmljivo, nejednoznačno kao manje vrijedno te sve to staviti po stranu.
Dakle, rascjep je pretpostavka za to da kontingentno, ne-regularno, ne-analitično, znanstvenim sredstvima nespoznatljivo u muški dominiranim područjima znanosti, politike i ekonomije ostane značajno neistraženo. Osim toga, najvažniju ulogu igra klasifikacijsko razmišljanje koje ne može kritički sagledati specifičnu kvalitetu, samu stvar i ne može opaziti i poduprijeti popratne razlike, lomove, ambivalencije, neistovremenosti itd.
Drugim riječima, za „podruštvljeno društvo" kapitalizma, da uporabimo Adornovu formulaciju, to zapravo znači da spomenute razine i područja nisu naprosto neraskidivo, kao „realna", povezana jedna s drugima, nego da ih se u podjednakoj mjeri mora promatrati u njihovoj objektivnoj „unutarnjoj" povezanosti prema bazalnoj razini rascjepa vrijednosti kao principa forme društvenog totaliteta koje tvori društvo uopće i na razini biti i na razini pojave.
U isti mah teorija rascijepa vrijednosti je uvijek svjesna i svojih granica kao teorija.
Prema tome, ono što se podudara s pojmom rascijepa vrijednosti kao formalnog principa se ne može podići na razinu „glavnog proturječja". Jer prema do sada iznesenom, teorija rascijepa vrijednosti se kao i teorija vrijednosti može donekle nazvati i „logikom jednoga". Ona u svojoj kritici identitetske logike ostaje vjerna samoj sebi te može postojati ako samu sebe ne relativizira ili niječe u slučaju nužde. A to također znači da teorija rascijepa vrijednosti mora dati prostora drugim oblicima društvene diskriminacije (ekonomske nejednakosti, rasizma i antisemitizma) teoretski ravnopravno mjesto (u što ne mogu ovdje detaljnije ulaziti).
\textbf{6.}
U teoriji se robnoproizvodni-patrijarhalni razvoj u različitim regijama svijeta odvijao na različite načine, osim u slučaju (nekada) rodnosimetričnih društava koja do danas nisu ili nisu u potpunosti preuzela moderne rodne odnose (usp. primjerice Weiss 1995). No treba razmotriti i „drugačije ispletene" patrijarhalne odnose koji su zakriveni od strane robnoproizvodnog patrijarhata, poostvarenog u moderno-zapadnom društvu, razvojem svjetskog tržišta, pri čemu nisu sasvim izgubili svoju specifičnost.
U tom kontekstu moramo nadalje uzeti u obzir da se rodni odnosi i predodžbe o muškosti i ženskosti također nisu uvijek na jednak način ispoljavali unutar zapadnjačko-moderne historije. Valja reći kako su moderni pojmovi rada i rodne binarnosti, proizvod specifičnog razvoja kapitalizma, te da idu ruku pod ruku s njime. Tek je u 18. stoljeću nastao moderni „sistem dualne rodnosti" (Carol Hagemann-White) te je došlo do „polarizacije rodnih karaktera" (Karin Hausen). Premda su žene već ranije slovile kao manje vrijedne, imale su mnogo mogućnosti da vrše utjecaj na neformalne načine, barem dok se u značajnijoj mjeri nije pojavila moderna javnost. Muškarac je u predmodernim i ranomodernim društvima radije imao simboličnu poziciju prvenstva. Žene još nisu bile definirane isključivo kao kućanice i majke, kao što je to bio slučaj od 18. stoljeća. Ženski doprinos materijalnoj reprodukciji u agrarnim je društvima bio smatran podjednako važnim kao onaj muškaraca (usp. Heintz\Slash{}Honegger 1981). Ako su moderni rodni odnosi s odgovarajućim polarnim rodnim etiketama isprva privremeno bili ograničeni na buržoaziju, s poopćenjem uže obitelji su se postupno proširili na sve slojeve i klase s posljednjim zamahom u fordističkom razvoju u 1950-im godinama.
\textbf{7}.
Rascjep vrijednosti stoga nije nikakva kruta struktura, onako kao što je možemo susresti kod nekih socioloških strukturnih modela. Radi se o procesu koji nije statična, pa ga se ne može uvijek poimati na jednak način. U postmoderno doba je ponovno pokazao novo lice. Žene se sada smatra „dvostruko podruštvljenima“ (Regina Becker-Schmidt), hoćereći, u istoj su mjeri odgovorne za obitelj i profesiju. Štoviše, ta činjenica nije ništa novo - velik je dio žena bio i prije već na neki način zaposlen. Novo je to što je ta činjenica tijekom promjena u zadnjih nekoliko desetljeća i za vrijeme strukturnih proturječja koji su ih pratili sada napokon prodrla u opću svijest. Na žene se već dugo više ne gleda samo kao na kućanice i majke. Zbog toga uostalom nije samo besmisleno, nego krajnje upitno, kad kvir pokreti, čija je klasična teoretska začetnica Judith Butler, smatraju da moraju dekonstruirati modernu binarnost rodova. Problem je to što oni nastoje učiniti neuvjerljivom karikaturom ono što je u kapitalističkom smislu već samo po sebi suvišno. Već se dugo vremena odvijaju „realne dekonstrukcije“ koje se mogu iščitati iz primjerice „dvostrukog podruštvljenja“ žena, ali i odjeće, habitusa muškaraca i žena itd., a da rodna hijerarhija u suštini nije nestala uslijed toga.
\textbf{8.}
Pri analizi rodnih odnosa u postmoderni od presudne je važnosti ustrajati na dijalektici između bića i pojave. Drugim riječima, promjene rodnih odnosa moraju biti protumačene počevši od mehanizama i struktura rascjepa vrijednosti koji kao formalni princip određuje sve razine društva. Nadalje, razvoj proizvodnih snaga i tržišna dinamika, koji i sami u tom smislu počivaju na rascjepu vrijednosti, potkopavaju svoju vlastitu pretpostavku utoliko što dovode do udaljavanja dobrog dijela žena od njihovih tradicionalnih uloga. Tako se od 1950-ih godina sve više žena uključuje u sferu apstraktnog rada što je između ostalog uvjetovano procesima racionalizacije u kućanstvu, mogućnošću korištenja kontracepcijskih sredstava, izjednačavanjem obrazovnih mogućnosti s muškarcima, lakšim ostvarivanjem roditeljstva uz zaposlenje i sl., kao što je pokazao Ulrich Beck (Beck 1986, str. 174 nadalje). U tom je pogledu „dvostruka podruštvljenost“ žena zadobila novu kvalitetu.
Iako su žene dobrim dijelom integrirane u „“javnu“ sferu, još uvijek su usprkos svemu odgovorne za kućanstvo i djecu, moraju se boriti više od muškaraca kako bi dospjele na više pozicije, u prosjeku zarađuju daleko manje od muškaraca itd. Struktura rascjepa vrijednosti se stoga preobrazila, ali je zapravo još uvijek itekako prisutna.
Stari buržoaski odnosi među rodovima nisu skrojeni po mjeri „turbokapitalizma“ s obzirom na njegove rigorozne zahtjeve za fleksibilnošću uslijed kojih dolazi do izgradnje prinudnih fleksi- identiteta. Međutim, ti identiteti ipak ostaju specifično orođeni (usp. primjerice Schultz 1994, Wichterich 1998). Stara predodžba o ženama je suvišna te sada nastupa ona „dvostruko podruštvljena“. Usto, novije analize „globalizacije i rodnih odnosa“ navode na zaključak da su globalizacijske tendencije dovele do propadanja patrijarhata uslijed toga što su se žene izborile za sve više sloboda, a što je navodno imanentno sistemu.
Međutim, i ovdje se moraju uzeti u obzir različiti društveno-kulturni konteksti u različitim regijama svijeta. Unatoč tim različitim položajima žena koji se moraju uzeti u obzir, kad u društvu dominira logika pobjednika i gubitnika, ona prijeti da će propašću srednje klase progutati i pobjednike (usp. Kurz 2005). Dobro situirane žene (s karijerom) mogle su si priuštiti mahom loše plaćene migrantkinje iz Istočnog bloka kao „sluškinje" i njegovateljice. Na taj se način odvila preraspodjela njegovateljskog i brižnog rada unutar ženskih životnih svjetova.
Za velik dio populacije, „barbarizacija patrijarhata" u Europi podrazumijeva da su sve više vidljive sljedeće tendencije, kako ih mi barem poznajemo jednim dijelom iz („crnih") geta SAD-a ili slamova u takozvanim zemljama Trećeg svijeta: žene su u istoj mjeri odgovorne za novac i za preživljavanje; sve su više integrirane u svjetsko tržište, a da nemaju nikakve šanse dogurati do toga da si same mogu osigurati egzistenciju; odgajaju djecu uz pomoć svoje ženske rodbine i susjeda (i tu se odvija preraspodjela brižnog rada unutar ženskog roda); muškarci dolaze i odlaze, lutajući od posla do posla i od žene do žene koje ih možda čak i djelomično uzdržavaju. Kroz prekarizaciju radnih odnosa, što je povezano s erozijom tradicionalne obiteljske strukture, muškarac više ne nosi ulogu hranitelja obitelji (Schultz 1994). Društvena atomizacija i individualizacija napreduju sve više u pozadini nesigurnih oblika egzistencije, zbog sve lošijeg općeg ekonomskom stanja, povlačenja socijalne države i nametanja prisilnih mjera upravljanja krizom, a da rodna hijerarhija u principu ne nestaje. Rascjep vrijednosti se kao temeljni društveni princip u određenoj mjeri odvaja od rigidnih institucionalnih nositelja modernosti (prije svega obitelji i rada). Robnoproizvodni patrijarhat samo blijedi, dok odnos između vrijednosti, odnosno apstraktnog rada i od njega odvojene reprodukcije, nije napušten. Potrebno je i konstatirati da su razmjeri muškog nasilja porasli na više razina.
Turbokapitalizam, kao što je već rečeno, iziskuje rodno specifične fleksi-prisilne identitete. Međutim, postmoderni rodni model „dvostruko podruštvljene žene" u suvremenom kriznom kapitalizmu ne može trajno stabilizirati društvenu reprodukciju, jer se i sam sve više okreće na mjestu i dokazuje racionalnost u iracionalnosti „kolapsa modernizacije" (Kurz 1991). „Dvostruka podruštvljenost" individualnih žena se paradoksalno nalazi u procesu raspadanja po pitanju njihove funkcionalnosti za robnoproizvodni patrijarhat. Primjerice, voditeljice grupa za samopomoć koje bi trebale pomoći u prebrođivanju krize u takozvanom Trećem svijetu su ponajprije žene. Pri tome valja reći da reproduktivne aktivnosti u doba Just-in-Time tendencija imaju daleko manje šanse za uspjeh. Te su aktivnosti kao neželjeno smeće ponajviše dodijeljene dvostruko opterećenim ženama, čime one preuzimaju ulogu kriznih upraviteljica. Sad, kad smo duboko zaglibili u blatu, žene bi trebale služiti kao „sredstva za čišćenje i dezinfekciju" (Christina Turmer-Rohr).
Apele za ženske kvote na vodećim pozicijama (koji su postali naročito prominentni nakon 2008. godine) također treba sagledati u ovom kontekstu.
\textbf{9.}
Kao druga strana postmodernizma, od druge polovice 1980-ih godina identitarni i desni pokreti u određenoj su mjeri dobili na zamahu. Etnički konflikti i etnograđanski ratovi diljem svijeta na tužan način tome svjedoče. Moglo bi se stoga govoriti o dijalektici dekonstruktivističkog identitarnog poricanja i, s druge strane, rigidnog identitetskog izražavanja u oronulom kapitalističkom patrijarhatu. Primjerice, u ratnim pothvatima žene nisu samo žrtve sistematskih silovanja i nasilja u privatnim odnosima, već su istovremeno i krizne upraviteljice u jednoj autonomnoj kriznoj dinamici. Međutim, muška nadmoć se ponovno otvoreno uspostavlja bez pokušaja ideološkog maskiranja. U posljednjih nekoliko godina nakon financijskog sloma 2008. došlo je do ogromnog zaokreta na desno čak i u zapadnim centrima. Ključne riječi: Trump, AfD (Alternativa za Njemačku) u Njemačkoj, sudjelovanje FPO-a (Slobodarske stranke Austrije) u austrijskoj vladi itd. Odgovarajuće su stranke i pokreti, kao što je poznato, zauzeli maskulinističke stavove. Građanska obitelj i tradicionalni rodni odnosi trebaju ponovno biti uspostavljeni zajedno s homofobnim, rasističkim, antisemitskim i antiromskim pozicijama. No to nije u skladu sa stvarnošću: hranitelj obitelji i kućanica su u postfordističko doba globalizacije u propadajućem kapitalističkom patrijarhatu definitivno zastarjeli. Žene su danas nolens volens prisiljene biti radno aktivne kako bi preživjele. Tu se otkriva nova proturječna struktura u procesu propadanja kapitalističkog patrijarhata. Protiv desnih i restauracijskih tendencija se unatoč tome podiže otpor koji se naposljetku manifestira i u ponovnom jačanju ženskog pokreta.
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\begin{center}
Anarhistička biblioteka
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-yu}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Roswitha Scholz
Teorija i kritika rascjepa vrijednosti
Teze o kapitalizmu i hijerarhijskim rodnim odnosima
2018
\bigskip
\href{https://mi2.hr/2019/06/roswitha-scholz-teorija-i-kritika-rascjepa-vrijednosti-teze-o-kapitalizmu-i-hijerarhijskim-rodnim-odnosima-1/}{https:\Slash{}\Slash{}mi2.hr\Slash{}2019\Slash{}06\Slash{}roswitha-scholz-teorija-i-kritika-rascjepa-vrijednosti-teze-o-kapitalizmu-i-hijerarhijskim-rodnim-odnosima-1\Slash{}}
Prevela: Maria Ćaćić. Uredile: Mia i Lina Gonan. Tekst predavanja koje autorica održala u okviru \emph{Škole neophodnog znanja,} 03. 12. 2018, u net klubu MaMa u Zagrebu. Iznesene teze su iscrpnije izložene u tekstu \href{https://anarhisticka-biblioteka.net/library/roswitha-scholz-vrednost-je-muskarac}{Vrijednost je muškarac}.
\bigskip
\textbf{anarhisticka-biblioteka.net}
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\noindent Compiled on \today~ for DTK 2022-03
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\begin{document}
\title{Splenic vein enlargement, a rare cause of nutcracker syndrome}
\author[1]{Mehdi Karami}%
\author[2]{hossein kouhi}%
\author[1]{Seyedeh Fatemeh Sadatmadani}%
\author[1]{Bahar Sadeghi}%
\author[1]{Zeynab Rostamiyan}%
\author[1]{Mozhdeh Hashemzadeh}%
\affil[1]{Isfahan University of Medical Sciences}%
\affil[2]{Affiliation not available}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
Nutcracker syndrome refers to symptomatic compression of left renal
vein. There are many reasons for that. We report a case that the
enlargement of splenic vein has caused to nutcracker syndrome. Results
implies that not firm venous structures can be cause of nutcracker
syndrome.%
\end{abstract}%
\sloppy
\textbf{Key clinical message}
Abnormal enlargement of the splenic vein is one of the etiologies of
Natcracker syndrome that should be considered when examining the causes
of this syndrome. Because knowing rare etiologies can help correctly
diagnose Natcracker syndrome and reduce its mortality.
\textbf{Introduction}
Nutcracker syndrome (NCS) is the symptomatic clinical condition in which
of left renal vein (LRV) becomes compressed usually between abdominal
aorta and superior mesenteric artery (SMA). NCS is important because
secondary chronic LRV hypertension may lead to chronic renal disease or
renal vein thrombosis (1). The most common cause of NCS is the short
distance between SMA and abdominal aorta (1-4). Other reported causes
are retroperitoneal pathologies as tumors or lymphadenopathies. Venous
causes are very rare but splenic vein enlargement has not been reported.
(1,3,5). NCS can happen at any age from childhood to seventh decade
(6-11) with peaking spread in middle age adults (10,11). In this case,
NCS is secondary to compression effect of enlarged splenic vein on LRV.
\textbf{Case history}
A 55 y/o woman, known case of hairy cell leukemia complained from
intermittent abdominal pain (mainly in left flank), nausea, weight loss
and loss of appetite from 3 months ago. The patient`s family history was
unremarkable and has undergone chemotherapy treatment from four months
ago. Physical examination positive findings~were huge splenomegaly \&
cachexia. Her height was 140 centimeter and her weight was 35 kilograms
(BMI=17.85). Vital signs were normal. Laboratory positive finding was:
HB= 8.1 g/dl. Urine analysis was normal.
In abdominal ultrasound, Heterogeneity and increase in splenic
parenchymal echogenicity, huge splenomegaly as spleen span of 210 mm was
detected.
Computerized tomography scan (CT scan) confirmed huge splenomegaly and
heterogeneity of splenic parenchymal density that indicated splenic
infarction. Splenic vein was markedly enlarged in diameter of 19 mm
which had compression effect on LRV. Marked prominency of both gonadal
veins and congestion of pelvic veins bilaterally was detected which
represents that left gonadal vein and bilateral pelvic veins are
drainaged via right gonadal vein and this is secondary to LRV
compression. These findings lead us that NCS had been occurred.
CT scan showed the normal angle and distance between the Abdominal Aorta
and SMA. No retroperitoneal~pathology was shown. Both kidneys were
normal. Mild ascites, Mild hepatomegaly and evidence of secondary portal
hypertension were also noticed. The patient was referred to surgery
department for splenectomy to reveal her symptoms.
\textbf{Discussion}
NCS should be considered in differential diagnosis of patients with
intermittent flank pain. According to this case, abnormally enlarged
venous structures such as splenic vein which don't have a firm
consistency can be mentioned as unusual causes of NCS.
The causes of NCS are divided to three main groups:
1. Arterial causes: the most common cause of NCS is proximity of SMA to
abdominal aorta (1-4). Also other arterial causes are abdominal aortic
aneurysm, overarching testicular artery and ectopic ventral right renal
artery
2. Retroperitoneal tumors or pathologies such as pancreatic neoplasm,
chronic pancreatitis, para-aortic lymphadenopathy and decreased
retro-peritoneal mesenteric fat tissue
3. Venous causes (that are very rare): LRV duplication, left renal
ptosis, Left-sided IVC, hemi-azygos continuation and persistent left
superior vena cava combination (as high pressure veins) (1,3,5). Splenic
vein enlargement has not been reported as a cause of NCS and this is the
first time that an enlarged organic vein is reported as a cause of NCS.
All of the mechanisms involved in LRV compression lead to LRV outflow
obstruction (1).
In our case SMA syndrome was one of the differential diagnoses due to
rapid weight loss and nausea but imaging data ruled it out (12). CT scan
findings showed huge splenomegaly, splenic infarction, and enlarged
splenic vein with compression effect on LRV (NCS) (Figure 1). Secondary
prominency of both gonadal veins and pelvic congestion were also noticed
that represented drainage of these veins through right gonadal vein in
the setting of compression of LRV in NCS (Figure 2). The interesting and
unique point of our case is a compression effect on LRV that has been
made by an enlarged venous structure (enlarged splenic vein) without
having a firm or muscular consistency, instead of the arteries or other
solid pathologies. In a similar case report, the massively dilated
common bile duct which hasn't a firm structure was reported as an
unusual etiology for NCS (13). The most common symptoms and signs of NCS
are abdominal pain, left flank pain and hematuria (14).
As the another interesting point, abdominal and left flank pain existed
in our case while hematuria as a main clinical manifestation of NCS
wasn't obtained (15).
NCS is an important diagnosis due to the significant morbidity
associated with it, including the risk of chronic renal disease from
long-term LRV hypertension and thrombosis (1).
Knowing about the rare etiologies can help in accurate diagnosis of NCS,
which will lead to a reduction of its morbidity. To our knowledge,
abnormal enlargement of a venous structure has not been reported as an
etiology for NCS.
\textbf{Declaration of patient consent:} The authors certify that they
have obtained appropriate patient consent forms.
\textbf{Financial support and sponsorship}
Nil.
\textbf{Acknowledgment:}
Research Team Appreciates clinical informationsist research center
dependent on Deputy of research and technology of Isfahan University of
Medical Sciences
\textbf{Authorship}
Author 1: Mehdi Karami: Responsible for monitoring the accuracy of
medical content
Author 2: Hossein Kouhi: Responsible for manuscript preparation,
editing, review and its guarantee
Email: [email protected]
Author 3: Seyedeh Fatemeh Sadatmadani: Responsible for literature
search, data acquisition and help in manuscript preparation
Author 4: Bahar Sadeghi: Responsible for literature search, data
acquisition and help in manuscript preparation
Author 5: Narges Rostamiyan: Responsible for literature search, data
acquisition and help in manuscript preparation
Author 6: Mozhdeh Hashemzadeh: Responsible for reviewing content in
terms of writing principles and help in literature search
\textbf{References}
1. Said SM, Gloviczki P, Kalra M, Oderich GS, Duncan AA, D Fleming M, et
al. 2013. Renal nutcracker syndrome: surgical options. Semin Vasc Surg;
26:35--42.
2. Scultetus AH, Villavicencio JL, Gillespie DL.2001. The nutcracker
syndrome: its role in the pelvic venous disorders. J Vasc Surg;
34:812-9.
3. Mahmood SK, Oliveira GR, Rosovsky RP. An easily missed diagnosis:
flank pain and nutcracker syndrome. BMJ Case Rep 2013.
http://dx.doi.org/10.1136/bcr-2013-009447.
4. Shin JI, Park JM, Lee JS, Kim MJ. 2007. Effect of renal Doppler
ultrasound on the detection of nutcracker syndrome in children with
hematuria. Eur J Pediatr; 166:399-404.
5. Okada M, Tsuzuki K, Ito S. 1998. Diagnosis of the nutcracker
phenomenon using two-dimensional ultrasonography. Clin Nephrol;
49:35--40.
6. Shin Jl, Lee JS, Kim MJ. 2006. The prevalence, physical
characteristics and diagnosis of nutcracker syndrome. Eur J Vasc
Endovasc Surg; 32:335-6.
7. Venkatachalam S, Bumpus K, Kapadia SR, Gray B, Lyden S, Shishehbor
MH. 2011. The nutcracker syndrome. Ann Vasc Surg; 25:1154--64.
8. Buschi AJ, Harrison RB, Norman A, Brenbridge AG, Williamson BR,
Gentry RR, et al. 1980. Distended left renal vein: CT/sonographic normal
variant. AJR Am J Roentgenol; 135:339--42.
9. Ahmed K, Sampath R, Khan MS. 2006. Current trends in the diagnosis
and management of renal nutcracker syndrome: a review. Eur J Vasc
Endovasc Surg; 31:410-6.
10. He Y, Wu Z, Chen S, Tian L, Li D, Li M, et al. 2014.Nutcracker
syndrome-how well do we know it? Urology; 83:12--7.
11. Kurklinsky AK, Rooke TW. 2010. Nutcracker phenomenon and nutcracker
syndrome. Mayo Clin Proc; 85:552--9.
12. Ananthan K, Onida S, Davies AH. 2017. Nutcracker Syndrome: An Update
on Current Diagnostic Criteria and Management Guidelines. Eur J Vasc
Endovasc Surg; 53(6):886--94.
13. Orczyk K, Wysiadecki G, Majos A, Stefa\selectlanguage{polish}ń\selectlanguage{english}czyk L, Topol M, Polguj M.
2017. What Each Clinical Anatomist Has to Know about Left Renal Vein
Entrapment Syndrome (Nutcracker Syndrome): A Review of the Most
Important Findings. Biomed Res Int; 2017:1--7.
14. Mina E, El Sadr AR. 1974. Proceedings: Anatomical and surgical
aspects in the operative management of varicocele. West Afr J Pharmacol
Drug Res; 2 (1):100-101.
15. Do\selectlanguage{polish}ł\selectlanguage{english}owy J, Stoinska A, Ku\selectlanguage{polish}ś\selectlanguage{english}mierska M, Kuniej T, Pluci\selectlanguage{polish}ń\selectlanguage{english}ska I, Ja\selectlanguage{polish}ź\selectlanguage{english}wiec
P. 2011. A case of a spontaneous splenorenal shunt associated with the
nutcracker syndrome. Polish J Radiol ; 76(4):49--51.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/photo1/photo1}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/photo2/photo2}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\end{document}
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https://tug.ctan.org/fonts/eiad/doc/recreat.tex | ctan.org | CC-MAIN-2022-33 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-33/segments/1659882570827.41/warc/CC-MAIN-20220808122331-20220808152331-00542.warc.gz | 534,625,495 | 2,036 | % This file is public domain.
% Originally written 1998, Ivan A Derzhanski.
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\newcommand\english{\fontfamily{cmr}\selectfont}
\newcommand\irish{\fontfamily{eiad}\selectfont}
\irish Amhra/n macaro/nach a chuir {\english Marion Gunn}
chuig \textsc{\english gaelic-l}.
Le/iri/onn an te/acs seo aicme na gclo/nna Gaelach rialta
chomh maith le foirinn chlo/scri/ofa an leithid luainigh,
\english Dunhill {\irish7`} Fibonacci.
(This text demonstrates the Regular Irish family of typefaces
as well as the variable-width typewriter fount, Dunhill and Fibonacci.)
\begin{verse}
{\fontseries{b}\english One day for recreation \quad
\irish Is gan e/inne beo im chuideachta, \\
\english I spied a charming fair maid, \quad
\irish Ina haonar is i/ i siopa istigh.} \\
{\bfseries\english She was singing like an angel, \quad
\irish Is me/ ag e/isteacht lena binne-ghuth; \\
\english I whispered soft and easy--- \quad
\irish Is e/ du/irt si/: Stad ded radaireacht!}
\qquad \irish Curfa//\english Chorus:
\irish \qquad\qquad Anonn is anall, a Mha/iri/n, \quad
\quad Do mha/lai/ is do bheilteanna; \\
\qquad\qquad Is a bhean na stocai/ mba/na, \quad
\quad Ba bhrea/ liom bheith ag iomaidh leat.
\slshape
\english Her amber locks most neatly \quad
\irish Go dre/imreach si/os ag titim le/i, \\
\english Adown her back and waist, \quad
\irish Is gur phreab mo chroi/ le taitneamh di. \\
{\bfseries\english I asked her was she the fair one, \quad
\irish An bhandia u/d bhi/ ag Iu/patar \\
\english Or the brightsome vestal deity \quad
\irish A chaith tre/imhse seal in Ifreann.}
\qquad Curfa//\english Chorus.
\itshape
\english She answered me most daintily: \quad
\irish Ni/ he/inne da/r thugais me/. \\
\english I fear you are a \irish re/ice, \quad
Is na/ taobhaigh a thuilleadh me/! \\
{\bfseries\english Indeed I am no \irish re/ice, \quad
Na/ straeire a bhre/agfadh bruinneall seal. \\
\english I'm a pupil of Jack Lahey's; \quad
\irish Is is i/ an a/it a gco/nai/m Mucros.}
\qquad Curfa//\english Chorus.
\upshape
{\scshape\english I asked her who her father was, \quad
\irish Is du/irt si/ liom an ministir. \\
\english I knew I stood in danger, \quad
\irish Is gur bhaolach da/ bhfeicfi/ sinn!} \\
\let\english\cmvtt \let\irish\eiadvtt
\english If I had you in a neat grove, \quad
\irish Idir Claonach is Mucros, \\
\english Your sparkling eyes do tease me, \quad
\irish Tri/ la/r mo chroi/ ta/ taitneamh duit.
\qquad Curfa//\english Chorus.
\let\english\cmdh \let\irish\eiaddh
\english Her syllables most charming, \quad
\irish Is gur bhrea/ liom bheith ina cuideachta. \\
Is gur bhinne liom na/ an chla/irseach \quad
Gach ardphort da/ seinneadh si/. \\
\let\english\cmfib \let\irish\eiadfib
\english You'll get my stock and farm, \quad
\irish Ma/ the/ann tu/ liom go Mucros. \\
\english And then she sang most charming: \quad
\irish A ghra/ geal, \english I'm fond of you!
\qquad \irish Curfa//\english Chorus.
\end{verse}
\end{document}
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http://www2.informatik.uni-freiburg.de/~frank/latex-kurs/latex-kurs-3/Handout-better.tex | uni-freiburg.de | CC-MAIN-2022-33 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-33/segments/1659882572163.61/warc/CC-MAIN-20220815085006-20220815115006-00745.warc.gz | 104,547,127 | 1,039 | % Dieser Text ist urheberrechtlich geschützt
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https://www.zentralblatt-math.org/matheduc/en/?id=8404&type=tex | zentralblatt-math.org | CC-MAIN-2019-35 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-35/segments/1566027319915.98/warc/CC-MAIN-20190824063359-20190824085359-00421.warc.gz | 1,048,256,715 | 1,501 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2015c.00076}
\itemau{Mousoulides, Nicholas G.}
\itemti{Using modeling-based learning as a facilitator of parentel engagement in mathematics: the role of parents' beliefs.}
\itemso{Liljedahl, Peter (ed.) et al., Proceedings of the 38th conference of the International Group for the Psychology of Mathematics Education ``Mathematics education at the edge", PME 38 held jointly with the 36th conference of PME-NA, Vancouver, Canada, July 15--20, 2014, Vol. 4. [s. l.]: International Group for the Psychology of Mathematics Education (ISBN 978-0-86491-360-9/set; 978-0-86491-364-7/v.4). 265-272 (2014).}
\itemab
Summary: Being part of a larger research project aimed at connecting mathematics and science to the world of work by promoting mathematical modeling as an inquiry based approach, the present study aimed to: (a) describe parents' beliefs about inquiry-based mathematical modeling and parental engagement, and (b) explore the impact of a modeling-based learning environment on enhancing parental engagement. Results from semi-structured interviews with 19 parents from one elementary school classroom revealed strong positive beliefs on their engagement in their children learning, an appreciation of the modeling approach for bridging school mathematics and home, and their willingness to collaborate with teachers. Implications for parental engagement in mathematics learning are discussed.
\itemrv{~}
\itemcc{C20 D40 D30 M10}
\itemut{modeling; parental engagement; parents' beliefs; modeling-based learning}
\itemli{}
\end |
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\begin{tabular}{|l|l|}%
\hline
name & amount in \$ \\
\hline
Achterberg & 100 \\
Bosman & 150\notea \\
Evers & 125\noteb \\
Gerritsen & 145 \\
Hooier & 170\notec \\
Jansen & 165\notea \\
\hline
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\Set{setlab_contact_phone}{0384000000}
% For new Set lab style
\Set{irtesset_contact_name}{{}~{Thomas PIOTROWSKI}}
\Set{irtesset_contact_email}{[email protected]}
\Set{irtesset_contact_phone}{0384000000}
% For Multiagent group style
\Set{mafr_contact_name}{{}~{Thomas PIOTROWSKI}}
\Set{mafr_contact_email}{[email protected]}
\Set{mafr_contact_phone}{0384000000}
\gdef\MYBIO{}
\setdocabstract{\MYBIO}
\begin{document}
\begin{center}
\includegraphics[width=.25\linewidth]{anonymous_avatar.png}
\end{center}
\section{Identification}
\begin{tabularx}{\linewidth}{lX}
{\bf Name, Firstname:} & PIOTROWSKI Thomas \\
{\bf Professional Position:} & PhD Student \\
{\bf French National Section:} & 27 - Informatique \\
{\bf Teaching Institution:}& Universit\'e de Technologie de Belfort-Montb\'eliard, Rue Thierry Mieg 90010 \textsc{Belfort} cedex, France\\
{\bf Research Laboratory:} & IRTES-SET (Laboratoire Syst\`emes et Transports, Institut de Recherche Transport \'Energie Soci\'et\'e, Belfort, France) \\
{\bf Email:} & [email protected] \\
{\bf Web site:} & http://multiagent.fr/People:Piotrowski\_thomas \\
\end{tabularx}
\vfill
\textcolor{red}{\large This curriculum vit{\ae} is generated by a bot automatically, from data published on the website above. To obtain an official curriculum vit{\ae}, please contact Thomas PIOTROWSKI directly.}
\end{document} |
https://matthewdaws.github.io/files/bsa/corrections.tex | github.io | CC-MAIN-2021-31 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-31/segments/1627046153971.20/warc/CC-MAIN-20210730154005-20210730184005-00138.warc.gz | 405,448,035 | 6,261 | \documentclass[twoside,12pt,a4paper]{article}
\usepackage[margin=2cm]{geometry}
%\usepackage{times}
\usepackage{amsmath,amssymb,amsthm,latexsym}
%\usepackage{amsfonts}
\theoremstyle{plain}
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}[proposition]{Theorem}
\newtheorem{corollary}[proposition]{Corollary}
\newtheorem{lemma}[proposition]{Lemma}
\newtheorem{claim}[proposition]{Claim}
\newtheorem{definition}[proposition]{Definition}
\newtheorem{conjecture}[proposition]{Conjecture}
\theoremstyle{definition}
\newtheorem{example}[proposition]{Example}
\newtheorem{remark}[proposition]{Remark}
\newcommand{\mc}{\mathcal}
\newcommand{\Sp}{\operatorname{Sp}}
\begin{document}
\noindent{MATH5002: Ongoing corrections and comments}
\section{Theorem 3.26}
At the top of page~124, why is it ``elementary to see'' that $f(A)$ is open?
By a ``topological vector space'' I mean a vector space which has a topology making
the vector space operations continuous. Any normed space, or locally convex space,
is a topological vector space.
\begin{lemma}
Let $E$ be a topological vector space, and let $f:E\rightarrow\mathbb R$ be a
continuous linear functional. Then either $f=0$ or $f$ is an open mapping.
\end{lemma}
\begin{proof}
Suppose $f\not=0$, so there is $x_0\in E$ with $f(x_0)\not=0$.
Let $A\subseteq E$ be open, and let $a\in A$. As $f(x_0)\not=0$ we can
find some scalar $\lambda$ such that $f(\lambda x_0) = \lambda f(x_0)$ does
not equal $f(a)$. Set $x_1=\lambda x_0$. Consider the function
$\varphi:\mathbb R\rightarrow E; t\mapsto tx_1 +(1-t)a$. This is continuous
(as it only involves the vector space operations) and $\varphi(0)=a\in A$.
As $A$ is open, there is $\epsilon>0$ such that $\varphi(t)\in A$ if
$-\epsilon < t <\epsilon$. Then consider the set $\{ f(\varphi(t)) :
-\epsilon < t <\epsilon\}$, which is the image of a line segment under a linear
map, and is hence the open interval between $-\epsilon f(x_1) + (1+\epsilon)
f(a)$ and $\epsilon f(x_1) + (1-\epsilon) f(a)$. The actual values are
irrelevant-- the point is that $f(x_1)\not=f(a)$, so this is a proper open
interval containing $f(a)$. Thus every point in $f(A)$ has an open
neighbourhood, and we conclude that $f(A)$ is open.
\end{proof}
In the proof of part (iii) (also on page~124) it's written ``since $A$
is compact, there is a convex open neighbourhood $V$ of $0_E$ with $(A+V)
\cap B=\emptyset$''. Why is this? We first need a lemma.
\begin{lemma}
Let $E$ be a real locally convex space. For each $x\in E$ and each
open set $U$ containing $x$, there is a convex open neighbourhood $W$ of
$0_E$ with $x+W+W \subseteq U$, where $x+W+W=\{ x+w+v : w,v\in W \}$.
\end{lemma}
\begin{proof}
As the topology is translation invariant, $U=x+V$ for some open set $V$
containing $0$. As addition is continuous and $0_E+0_E=0_E$, there are
neighbourhoods $V_1,V_2$ of $0_E$ with $V_1+V_2 \subseteq V$. As the
topology is locally convex, we can find a convex open neighbourhood $W$ of
$0_E$ with $W\subseteq V_1\cap V_2$. Thus $W+W\subseteq V$, or equivalently,
$x+W+W \subseteq U$.
\end{proof}
We now show the claimed result. For each $a\in A$, as $B$ is closed
and $a\not\in B$, by the lemma, there is a convex open neighbourhood $W_a$
of $0_E$ with $a+W_a+W_a \cap B = \emptyset$. Then the family
$\{ a+W_a : a\in A \}$ is an open cover for $A$, so as $A$ is
compact, there is a finite subcover, say $\{ a_i+W_{a_i} : 1\leq i\leq n\}$.
Set $V=W_{a_1} \cap\cdots\cap W_{a_n}$, which is a convex open neighbourhood
of $0_E$. Then
\[ A+V \subseteq \bigcup_{i=1}^n a_i + W_{a_i} + V
\subseteq \bigcup_{i=1}^n a_i + W_{a_i} + W_{a_i}, \]
which is disjoint from $B$, as required.
\section{Lemma~3.38}
Let $T:E\rightarrow F$ be a (bounded) linear map.
The first part of this lemma suggests that it is obvious to see that if
\[ \text{for all } y\in F \text{ there is }x\in E \text{ with }
T(x)=y, \|x\|\leq K\|y\|, \]
then $T$ is open. Why is this?
Well, let $U\subseteq E$ be open. Then let $y_0\in T(U)$, so $y_0=T(x_0)$ for some
$x_0\in U$. As $U$ is open, $B(x_0,\epsilon)\subseteq U$ for some $\epsilon>0$.
Let $z\in F$ with $\|z\| < \epsilon/K$, so by assumption, there is $x'\in E$ with
$T(x')=z$ and $\|x'\|\leq K\|z\|<\epsilon$. Then $x_0+x' \in B(x_0,\epsilon)\subseteq U$
and $T(x_0+x') = y_0+z$. As $z$ was arbitrary, we've shown that $B(y_0,\epsilon/K)
\subseteq T(U)$. As $y_0$ are arbitrary, we've shown that $T(U)$ is open.
If you think about it for a moment, we have actually just proved the following lemma:
\begin{lemma}
Let $E,F$ be normed space and $T:E\rightarrow F$ be linear. Let $B=\{x\in E:
\|x\|<1\}$ be the open unit ball of $E$, and suppose that $T(B)$ contains an open
neighbourhood of $0$. Then $T$ is open.
\end{lemma}
\section{Theorem~3.40}
What is the ``elementary argument'' in the proof. I think it is the following.
We know that $\overline{T(B_N)}$ has non-empty interior, which means we can find
$y\in \overline{T(B_N)}$ and $\epsilon>0$ with $B(y,\epsilon) \subseteq
\overline{T(B_N)}$. Thus we can find $(x_n)\subseteq E$ with $\|x_n\|< N$ for all
$n$, and with
$T(x_n)\rightarrow y$. Let $w\in F$ with $\|w\| < \epsilon/2$. Then
$y+2w \in B(y,\epsilon) \subseteq \overline{T(B_N)}$ and so we can find
$(x_n')\subseteq E$ with $\|x_n'\|<N$ for all $n$,
and with $T(x_n')\rightarrow y+2w$. Then
\[ \Big\| \frac{1}{2}(x_n'-x_n) \Big\| < N \text{ for all }n, \quad
\text{and}\quad T\Big( \frac{1}{2}(x_n'-x_n) \Big)
\rightarrow \frac12(y+2w - y) = w. \]
That is, $w\in \overline{T(B_N)}$. We've hence shown that $B(0,\epsilon/2)
\subseteq \overline{T(B_N)}$, which is what we need.
\section{Section~5.3: The Jacobson Radical}
Let $A$ be a unital algebra. For a left ideal $L$ of $A$, define
$L:A=\{a\in A:ab\in L \ (b\in A) \}$ which agrees with the kernel of the natural
representation of $A$ on $A/L$. An ideal is \emph{primitive} if it equals $L:A$
for some maximal left ideal $L$; equivalently, the primitive ideals are the
kernels of irreducible representations.
If $A$ is a Banach algebra, then a maximal (left) ideal must be closed.
If $L$ is closed, then so is $L:A$.
Thus primitive ideals are automatically closed.
If $A$ is commutative and Banach, then the maximal ideals correspond to the
kernels of characters. It's claimed in the book (page~231) that it's ``obvious''
that the primitive ideals are simply the maximal ideals.
\begin{lemma}
In a commutative unital algebra $A$, the primitive ideals are the maximal ideals.
\end{lemma}
\begin{proof}
Let $M\subseteq A$ be a maximal ideal. Then by a standard Zorn's lemma
argument, there is a maximal left ideal $L$ containing $M$. Set $P=L:A$,
so $P$ is primitive. If $m\in M$, then for $b\in A$, also $mb\in M \subseteq
L$; it follows that $m\in L:A=P$ so $M\subseteq P$. As $M$ is maximal, we
conclude that $M=P$ is primitive (and we didn't use that $A$ is commutative).
Conversely, let $P$ be a primitive ideal, so $P=L:A$ for some maximal left ideal
$L$. As $A$ is commutative, $L$ is a maximal (two-sided) ideal and so if
$a\in L$ and $b\in A$, then $ab\in L$; this shows that $L\subseteq L:A=P$.
By maximality, $L=P$ and so $P$ is a maximal ideal.
\end{proof}
\{With hindsight that was easy! But the book starts talking about characters,
which seems misleading to me.\}
\section{Comments on various exercises}
\subsection{Exercise 2.9}
To show that $C(\mathbb I)$ is isomorphic to $C(\mathbb I)\oplus C(\mathbb I)$
is pretty hard-- this is proved in Banach's book, for example! I cannot
see how to give a nice ``hint''.
\subsection{Exercise 4.4}
This is very hard. Here are some hints which make it (a bit) easier:
\begin{itemize}
\item Suppose that the exercise is true for Banach algebras of the
form $\mathcal B(E)$. Let $A\rightarrow\mathcal B(A); a\mapsto L_a$ where
$L_a(b)=ab$, the left-regular representation. Use this to show the result.
\item Suppose that the exercise is true for Banach algebras of the
form $\mathcal B(E^*)$. Show that it's true for $\mathcal B(E)$.
\item Use the Krein-Milman theorem applied to the unit ball of $E^*$,
to show that the exercise is true for Banach algebras of the
form $\mathcal B(E^*)$.
\end{itemize}
\subsection{Exercise 4.5}
This appears to be false under any reasonable interpretation.
\subsection{Exercise 4.9}
Of course, this should ask you to show that $LR=I_H$ but $RL\not=I_H$.
\subsection{Exercise 4.10}
This is false, as stated. We have that $\|T\|\leq 1$ and that
$\bigcap_n T^n(E) = \{0\}$ (notice the typo here in the question).
It is true that $0\in\Sp(T)$ and that $T$ has no eigenvalues. You can
follow the construction with the operators $U_\zeta$ to show correctly
that $\Sp(T)$ is rotationally invariant. So $\Sp(T)$ is a union of
circles, all inside the closed unit disc in $\mathbb C$. However, this
does not mean that $\Sp(T)$ is itself a disc (or $\{0\}$).
It is easy to see that
\[ T^n(x_1,x_2,x_3,\cdots) = (0,\cdots,0,w_1w_2\cdots w_nx_1,
w_2w_3\cdots w_{n+1}x_2, \cdots), \]
where there are $n$ zeros. It follows that
\[ \|T^n\| = \sup_{m\geq 1} \| w_m \cdots w_{m+n-1} \|
\implies \rho(T) = \lim_n \sup_{m\geq 1} \| w_m \cdots w_{m+n-1} \|^{1/n}. \]
\subsubsection{Counter-example}
Define
\[ w_n = \begin{cases} 1/k &: n=2^k \text{ for some }k\in\mathbb N, \\
1 &: \text{otherwise}. \end{cases} \]
Because the gaps between successive powers of $2$ increase without bound,
it follows from the above formula that $\rho(T)=1$. By rotational invariance,
$\Sp(T)$ must contain the unit circle.
We'll now show that $0\in\Sp_{\text{ap}}(T)$,
\subsection{Exercise 4.11(i)}
$K$ is a compact Hausdorff space, $F\subseteq K$ is closed, we define
\[ I(F) = \{ f\in C(K) : f|F=0 \}. \]
This is a closed ideal in $C(K)$ (which is not too hard to show).
Why does every closed ideal arise in this way? I think that this is slightly
tricky to answer. Let $J\subseteq C(K)$ be a closed ideal, and then set
\[ F = \{ k\in K : f(k)=0 \ (f\in J) \}. \]
It's not too hard to show that this is a closed subset of $K$, and that
$J \subseteq I(F)$. But why do we have equality?
Form the quotient algebra $A=C(K)/J$, so that $A$ is a commutative
Banach algebra.
\begin{claim}
With notation as above, $A$ is semi-simple, that is, the Gelfand
transform $\mc G:A\rightarrow C(\Phi_A)$ is injective.
\end{claim}
If we believe this, then suppose that $J\not=I(F)$. Thus there is
$g\in I(F)$ with $g\not\in J$, and so $g+J\not=0$ in $A$, and so
$\mc G(g+J)\not=0$. Thus there is a character $\varphi$ on $A$ with
$\varphi(g+J)\not=0$. Then $\phi:C(K)\rightarrow\mathbb C; f\mapsto
\varphi(f+J)$ is a character on $C(K)$, and so there is $k\in K$ with
$\phi(f)=f(k)$ for all $f$. Thus $g(k)\not=0$. As $g\in I(F)$, we must
have that $k\not\in F$. However, for any $f\in J$ we have that $f(k)=
\phi(f) = \varphi(f+J)=0$, and so $k\in F$, contradiction. So $J=I(F)$.
How do we prove the claim? We could use that $C(K)$ is a C$^*$-algebra,
that $J$ is $*$-closed, and that thus $C(K)/J$ is also a C$^*$-algebra.
Then use that the Gelfand transform of a commutative C$^*$-algebra is
always injective (actually, an isomorphism).
\subsubsection{A direct proof}
Again let $g\in I(F)$. For $\epsilon>0$ let $U=\{ k\in K : |g(k)|<\epsilon\}$
so $U$ is an open set containing $F$. For each $x\not\in U$, as $x\not\in K$,
we can find $f_x\in J$ with $f_x(x)=1$ say (by definition of $F$ we can find
$f_x\in J$ with $f_x(x)\not=0$, and then rescale). Then $U_x=\{ k\in K:
|f_x(k)|>1/2 \}$ is open and contains $x$. As $K\setminus U$ is
closed, hence compact, we can find $x_1,\cdots,x_n$ with $U_{x_1}\cup
\cdots\cup U_{x_n} \supseteq K\setminus U$.
Given $f\in J$, notice that $|f|^2 = f \overline{f} \in J$ as $J$ is an ideal.
Thus $h = |f_{x_1}|^2 + \cdots + |f_{x_n}|^2\in J$. Then $h(k)=0$ for each
$k\in F$, while for each $x\not\in U$, there is $i$ with $x\in U_{x_i}$, and so
$h(x)>(1/2)^2 = 1/4$.
Now consider\footnote{Thanks to George Berkley for point this trick out.}
$g_n\in C(K)$ defined by
\[ g_n(x) = g(x) \frac{n h(x)}{1+nh(x)}. \]
Notice that $nh(x)/(1+nh(x)) \in [0,1)$ for all $x$ and $n$.
If $x\not\in U$ then $h(x)>1/4$ and so $nh(x)/(1+nh(x)) \rightarrow 1$ as
$n\rightarrow\infty$, \emph{uniformly} for $x\not\in U$. In particular, if
$n$ is large, then $|g_n(x)-g(x)| < \epsilon$ for all $x\not\in U$.
If $x\in U$ then $|g(x)|<\epsilon$ and so also $|g_n(x)|<\epsilon$, and so
$|g_n(x)-g(x)| < 2\epsilon$. We conclude that for $n$ large, $g_n$ approximates
$g$ in the supremum norm. However, notice that
\[ g_n = \frac{ng}{1+nh} h, \]
and so as $J$ is an ideal, $g_n\in J$.
As $J$ is closed, we conclude that $g\in J$, as required.
\subsection{An example}
Consider $B=C^1([0,1])$ the continuous
differentiable functions on $[0,1]$ with the norm $\|f\|=\|f\|_\infty +
\|f'\|_\infty$. This is a natural Banach function algebra on $[0,1]$
(see Exercise~4.12).
Let $J$ be the collection of functions with $f(1/2)=f'(1/2)=0$.
This is a linear subspace, and an ideal, as for any $g$,
\[ (gf)(1/2) =0, \qquad (gf)'(1/2) = g'(1/2)f(1/2) + g(1/2) f'(1/2) =0. \]
It's easy to see that it's closed (thanks to the norm we used). However,
I claim that the associated $F$ must be $\{1/2\}$, and so $I(F)\not=J$.
Indeed, the function $f(x)=(x-1/2)^2$ is in $J$ but vanishes only at $1/2$.
What goes wrong with the above proof? The problem is that while $\|g_n-g\|$
is small, we have no control over $\| g_n' - g'\|$.
\end{document} |
https://www.aanda.org/articles/aa/full/2006/11/aa4041-05/table5.tex | aanda.org | CC-MAIN-2022-40 | text/plain | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030335276.85/warc/CC-MAIN-20220928180732-20220928210732-00296.warc.gz | 690,594,311 | 1,388 | \begin{table}%t5
\caption{\label{tab:5}Statistical overview of the averages and ranges of the physical properties of the 63 oscillations in coronal loop footpoints found in this study combined with that in \citet{dm2002}. Note that the uncertainty in the parameters is taken to be the standard error in the mean,~$\sigma_{M}$.}
%\centering
\par
\begin{tabular}{c c c}
\hline\hline
Parameter & Average & Range \\
\hline
Footpoint Length, $L$ & $28.1 \pm 1.3$~Mm& $7.0{-}54.6$~Mm \\
Footpoint Width, ${\it w}$ & $8.6 \pm 0.3$~Mm& $3.5{-}14.9$~Mm \\
Footpoint Divergence ${\it w_{\rm d}}$ & $0.24 \pm 0.02$ & $0.05{-}0.71$ \\
Oscillation Period, $P$ & $284.0 \pm 10.4$~s& $145{-}550$~s \\
Propagation Speed, $v$ & $99.7 \pm 3.9$~km~s$^{-1}$ & ${\it O}(45){-}{\it O}(205)$~km~s$^{-1}$ \\
Relative Amplitude, $A$ & $3.7\% \pm 0.2\%$ & $0.7{-}14.6\%$ \\
Detection Length, $L_{\rm d}$ & $8.3 \pm 0.6$~Mm & $2.9{-}23.2$~Mm \\
Energy Flux, $F$ & $313 \pm 26$~erg~cm$^{-2}$~s$^{-1}$ & $68{-}1560$~erg~cm$^{-2}$~s$^{-1}$\\
\hline
\end{tabular}
\end{table} |
https://www.cs.cornell.edu/info/people/raman/aster/latex/xx00.tex | cornell.edu | CC-MAIN-2021-21 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-21/segments/1620243991904.6/warc/CC-MAIN-20210511060441-20210511090441-00378.warc.gz | 708,706,099 | 722 | \input {headers}
\Huge
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$$a+\frac{b}{c} +d$$
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https://fisica.cab.cnea.gov.ar/bt/refbase/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%20FROM%20refs%20WHERE%20serial%20%3D%20458%20ORDER%20BY%20first_author%2C%20author_count%2C%20author%2C%20year%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=10&rowOffset=&wrapResults=1&citeOrder=&citeStyle=Chicago&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg= | cnea.gov.ar | CC-MAIN-2022-21 | application/x-latex | application/x-latex | crawl-data/CC-MAIN-2022-21/segments/1652662560022.71/warc/CC-MAIN-20220523163515-20220523193515-00792.warc.gz | 309,704,214 | 1,310 | %&LaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Dediu_etal1995} Dediu, V. I., Q. D. Jiang, F. C. Matacotta, P. Scardi, M. Lazzarino, G. Nieva, and L. Civale. "Deposition of MBa2Cu3O7-x thin films by channel-spark method." \textit{Superconductor Science and Technology} 8, no. 3 (1995): 160.
\end{thebibliography}
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|
http://web.mit.edu/xavid/Public/resume07.tex | mit.edu | CC-MAIN-2022-05 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-05/segments/1642320303884.44/warc/CC-MAIN-20220122194730-20220122224730-00498.warc.gz | 73,477,404 | 2,314 | \documentclass[left]{res}
\name{Xavid Pretzer}
\address{14 Buckingham St.\\
Somerville, MA 02143\\
\texttt{[email protected]}}
\begin{document}
\begin{resume}
\section{Objective}
To gain teaching experience and provide my unique insight to help others learn
by working as a Teaching Assistant.
\section{Experience}
\begin{description}
\item[Sept. 2004--January 2008] Metacarta, Inc., Cambridge, MA---Software Engineer
Worked alone and in groups full- and part-time to develop a variety of tools for Metacarta in the Office of the Chief Technology Officer. Integrated open-source projects such as Heritrix with Metacarta's geographic search systems. Coordinated and did advanced development work on a dynamic web map interface for Metacarta's product. Contributed to the founding of an open-source project called OpenLayers.
\item[June--August 2006] Software R\&D Group, Ricoh, T\={o}ky\={o}, Japan---Paid Internship
Researched and implemented an automated process for converting existing
parts catalog entries from PDF into an interactive format
that could be easily searched and accessed.
\item[February 2002--June 2004] Shaker Heights High School, Shaker Heights, OH
Assisted in teaching AP Computer Science.
\item[July--August 2002] Villanova Summer Research Institute, Villanova, PA---Paid Internship
Created a computer model of language evolution in a prelinguistic society to evaluate theories on human language evolution. Developed programs to implement this model.
\end{description}
\section{Education}
Bachelors in Computer Science and Engineering at the Massachusetts Institute of Technology
with a minor in Applied International Studies.
%Courses include the Structure and Interpretation of Computer Programs, Circuits and Electronics, and Artificial Intelligence.
%Currently in his third year of Japanese language studies.
%Graduated in 2004 with an honors diploma from Shaker Heights High School with a 4.4 GPA, including 7 semesters of Computer Science.
\section{Honors}
\begin{itemize}
\item Intel Science Talent Search Semifinalist, 2004
\item Eagle Scout Award, Boy Scouts of America, 2003
\item Intel International Science and Engineering Fair Finalist, 2002
\end{itemize}
\section{Skills}
\begin{itemize}
\item Extensive programming experience in C, C++, JavaScript, Python, Scheme, Java, and
\TeX, with some experience in other languages.
\item Experience writing, building, and packaging software
and administering servers and software installations on Debian/Ubuntu Linux and
Athena.
\item Web design experience with CSS, HTML/XHTML, SVG, and JavaScript.
\item Experience with circuit design and programming microcontrollers in C and Assembly.
%\item Familiarity with the design of applications, applets, tools, CGI scripts, online environments, and basic robotic control systems.
\item Conversational proficiency in Japanese and familiarity with Japanese culture and etiquette.
\item Basic Spanish language proficiency.
%\item Currently serving as Chamberlain for the MIT branch of the Society for Creative Anachronism, an international medieval re-creation group.
%\item Additional leadership experience as a Scribe, Patrol Leader, and Guide in the Boy Scouts of America.
\end{itemize}
\section{References}
\begin{itemize}
\item John Frank, CTO, Metacarta, Inc., 857-928-0614, \texttt{[email protected]}
\item M. Frans Kaashoek, Professor, MIT, 617-253-7149, \texttt{[email protected]}
\end{itemize}
\end{resume}
\end{document} |
http://www.emis.de/journals/EJC/Volume_18/Abstracts/v18i1p232.abs.tex | emis.de | CC-MAIN-2018-26 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-26/segments/1529267867050.73/warc/CC-MAIN-20180624180240-20180624200240-00408.warc.gz | 397,465,937 | 915 | \documentclass[12pt]{article}
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\noindent
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{\bf J\'anos Bar\'at, Gwena\"el Joret and David R. Wood}
%
%
\medskip
\noindent
%
%
{\bf Disproof of the List Hadwiger Conjecture}
%
%
\vskip 5mm
\noindent
%
%
%
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The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph
is $t$-choosable. We disprove this conjecture by constructing a
$K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.
\end{document}
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https://smlnj-gforge.cs.uchicago.edu/scm/viewvc.php/*checkout*/trunk/doc/report/intro.tex?revision=2636&root=diderot | uchicago.edu | CC-MAIN-2021-49 | text/x-tex | application/postscript | crawl-data/CC-MAIN-2021-49/segments/1637964363689.56/warc/CC-MAIN-20211209061259-20211209091259-00111.warc.gz | 572,616,697 | 2,515 | %!TEX root = report.tex
%
\chapter{Introduction}
Diderot is a parallel domain-specific language for programming image analysis and
visualization algorithms.
It supports a high-level programming model based on the mathematics of continuous
tensor fields.
These fields are reconstructed from discrete image-data sets (\eg{}, MRI data) using
separable convolution kernels.
We use \emph{tensors} to refer collectively to scalars,
vectors, and matrices, which encompasses the types of values produced
by the imaging modalities mentioned above, as well as values produced
by taking spatial derivatives of images.
Diderot permits programmers to express algorithms directly in terms of
tensors, tensor fields, and tensor field operations, using the same
mathematical notation that would be used in vector and tensor calculus
(such as $\nabla$ for the gradient).
Diderot is indended to be useful for prototyping image
analysis and visualization methods in contexts where a meaningful
evaluation of the methods requires its application to real image data,
but the real data volumes are of a size that requires efficient
parallel computation.
Diderot is also suited for educational contexts where the conceptual
transparency of the implementation is of primary importance.
In addition to providing a high-level mathematical programming model,
Diderot is also designed to be implemented on a range of parallel hardware,
including shared-memory multiprocessors and GPUs.
\section{Basic concepts}
The design of Diderot is based on three core concepts: image data, the continuous fields
that are reconstructed from them, and the strands that represent the computations over
the fields.
We give a high-level overview of these in this section.
\subsection{Images}
\subsection{Fields}
Computing with continuous tensor fields is one of the unique characteristics of Diderot.
Field values are constructed by convolving image data with
kernels (\CD{img${\color{\kwColor}\circledast}$bspln3}), but they can also be defined
by using higher-order operations, such as addition, subtraction, and scaling of fields.
Most importantly, Diderot supports differentiation of fields using the operators $\nabla$
(for scalar fields) and $\nabla\otimes$ (for higher-order tensor fields).
Two operations on fields are testing whether a point \CD{x} lies within the domain
of a field \CD{F} (\CD{inside(x, F)}) and \emph{probing} a field \CD{F} at a point \CD{x}
(\CD{F(x)}).
Probing and differentiating are fundamental to extracting geometric
information from fields.
\subsection{Strands}
The class of applications that Diderot targets are characterized as consisting of many
largely independent subcomputations.
For example, the rays in a volume renderer, the paths from fiber tractography, and
the particles in a particle system.
In Diderot, these mostly independent computations are modeled as \emph{strands}, which
execute in a \emph{bulk synchronous}
fashion~\cite{valiant:bridging-model-for-parallel,q-n-a-about-bsp}.
\section{Diderot by example}
% include VR-lite example here
\begin{figure}[p]
\begin{quote}
\lstset{language=Diderot}
\input{vr-lite}
\end{quote}%
\caption{A simple volume renderer in Diderot}
\label{fig:vr-lite}
\end{figure}%
<script type="text/javascript">//<![CDATA[
function toggle_ffErrors() {
var errorsblock = document.getElementById("ffErrorsBlock");
var errorsgroup = document.getElementById("ffErrors");
if (errorsblock.style.display == "none") {
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errorsgroup.style.right = "10px";
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errorsblock.style.display = "none";
errorsgroup.style.right = "300px";
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//]]></script>
<div id="ffErrors">
<a href="javascript:toggle_ffErrors();">Click to toggle</a>
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<div class="error">does not end with </html> tag</div>
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\end{quote}%
\caption{A simple volume renderer in Diderot}
\label{fig:vr-lite}
\end{figure}%</div>
</div></div>
</body></html>
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https://www.ttp.kit.edu/Preprints/ttp/ttp95/ttp95-28/ttp95-28.tex | kit.edu | CC-MAIN-2019-35 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2019-35/segments/1566027315132.71/warc/CC-MAIN-20190819221806-20190820003806-00282.warc.gz | 974,847,949 | 10,091 | %Title: Virtual Top Effects on Low-Mass Higgs Interactions at
% Next-to-Leading Order in QCD
%Authors: B.K. Kniehl, M. Steinhauser
%Published: <EM>Phys. Lett.</EM> <B>B365</B> (1995) 297-301
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\begin{document}
\title{Virtual Top Effects on Low-Mass Higgs Interactions at Next-to-Leading
Order in QCD}
\author{{\sc Bernd A. Kniehl}\thanks{Permanent address:
Max-Planck-Institut f\"ur Physik, Werner-Heisenberg-Institut,
F\"ohringer Ring 6, 80805 Munich, Germany.}\\
{\normalsize Theoretical Physics Department, Fermi National Accelerator
Laboratory,}\\
{\normalsize P.O. Box 500, Batavia, IL 60510, USA}\\
{\normalsize and}\\
{\normalsize Department of Physics, University of Wisconsin,}\\
{\normalsize 1150 University Avenue, Madison, WI~53706, USA}\\ \\
{\sc Matthias Steinhauser}\\
{\normalsize Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe,}\\
{\normalsize Kaiserstra\ss e 12, 76128 Karlsruhe, Germany}}
\date{}
\maketitle
\begin{abstract}
We present the next-to-leading-order QCD corrections of
${\cal O}(\alpha_s^2G_FM_t^2)$ to the low-$M_H$ effective $\ell^+\ell^-H$,
$W^+W^-H$, and $ZZH$ interaction Lagrangians in the high-$M_t$ limit.
In the on-shell scheme formulated with $G_F$, the
${\cal O}(\alpha_s^2G_FM_t^2)$ corrections support the
${\cal O}(\alpha_sG_FM_t^2)$ ones and further increase the screening of the
${\cal O}(G_FM_t^2)$ terms.
The coefficients of $(\alpha_s/\pi)^2$ range from $-6.847$ to $-16.201$, being
in line with the value $-14.594$ recently found for $\Delta\rho$.
All four QCD expansions converge considerably more rapidly, if they are written
with $\mu_t=m_t(\mu_t)$, where $m_t(\mu)$ is the $\overline{\mbox{MS}}$ mass,
rather than the pole mass, $M_t$.
\medskip
\noindent
PACS numbers: 12.15.-y, 12.38.Bx, 14.65.Ha, 14.80.Bn
\end{abstract}
\newpage
Now that the existence of the top quark has been established \cite{abe}, the
Higgs boson is the last missing link in the Standard Model (SM).
The discovery of this particle and the study of its properties are among the
most urgent goals of present and future high-energy colliding-beam experiments.
The Higgs boson is currently being searched for with the CERN Large
Electron-Positron Collider (LEP1) and the SLAC Linear Collider (SLC) via
Bjorken's process \cite{bjo}, $e^+e^-\to Z\to f\bar fH$.
At the present time, the failure of this search allows one to rule out the
mass range $M_H\le64.3$~GeV at the 95\% confidence level \cite{jan}.
The hunt for the Higgs boson will be continued with LEP2 via the
Higgs-strahlung mechanism \cite{ell}, $e^+e^-\to ZH\to f\bar fH$.
In next-generation $e^+e^-$ linear supercolliders (NLC), also
$e^+e^-\to\bar\nu_e\nu_eH$ via $W^+W^-$ fusion and, to a lesser extent,
$e^+e^-\to e^+e^-H$ via $ZZ$ fusion will provide copious sources of
Higgs bosons.
Once a novel scalar particle is discovered, it will be crucial to decide if it
is the very Higgs boson of the SM or if it lives in some more extended Higgs
sector.
For that purpose, precise knowledge of the SM predictions will be mandatory,
{\it i.e.}, quantum corrections must be taken into account.
The status of the radiative corrections to the production and decay processes
of the SM Higgs boson has recently been reviewed \cite{pr}.
Since the top quark is by far the heaviest known elementary particle,
with a pole mass of $M_t=(180\pm12)$~GeV \cite{abe}, the leading high-$M_t$
terms, of ${\cal O}(G_FM_t^2)$, are particularly important, and it is desirable
to gain control over their QCD corrections.
During the last year, a number of papers have appeared in which the two-loop
${\cal O}(\alpha_sG_FM_t^2)$ corrections to various Higgs-boson production and
decay processes are presented.
The list of these processes includes
$H\to f\bar f$, with $f\ne b$ \cite{hll} and $f=b$ \cite{ks1,kwi},
$Z\to f\bar fH$ and $e^+e^-\to ZH$ \cite{ks2},
$e^+e^-\to\bar\nu_e\nu_eH$ via $W^+W^-$ fusion \cite{ks3},
$gg\to H$ \cite{ks3,gam},
and more \cite{ks3}.
In this paper, we shall proceed one step beyond and tackle with three-loop
${\cal O}(\alpha_s^2G_FM_t^2)$ corrections.
To simplify matters, we shall work in the limit $M_H\ll M_t$ and concentrate on
reactions with colourless external legs.
Such reactions typically involve the $\ell^+\ell^-H$, $W^+W^-H$, and $ZZH$
couplings together with the gauge couplings of the $W$ and $Z$ bosons to the
leptons.
Our primary task is thus to find the next-to-leading QCD corrections to the
low-$M_H$ effective $\ell^+\ell^-H$, $W^+W^-H$, and $ZZH$ interaction
Lagrangians.
Recently, the ${\cal O}(\alpha_s^2G_FM_t^2)$ correction to $\Delta\rho$ has
been calculated and found to be quite sizeable \cite{avd},
being right at the edge of affecting ongoing precision tests of the standard
electroweak theory.
For $N_c=3$ and $n_f=6$, the QCD expansion of $\Delta\rho$ reads \cite{avd,djo}
\begin{equation}
\label{drhoos}
\Delta\rho=3X_t\left[1-2.859\,912\,a(1+a\beta_0L)
-a^2(25.311\,305-1.786\,213\,n_f)\right],
\end{equation}
where $X_t=(G_FM_t^2/8\pi^2\sqrt2)$, $a=\alpha_s(\mu)/\pi$,
$L=\ln(\mu^2/M_t^2)$, $\beta_0=11/4-n_f/6$ is the one-loop coefficient of the
QCD beta function, $G_F$ is Fermi's constant, and $\mu$ is the QCD
renormalization scale.
It is of great theoretical interest to find out whether the occurrence of
significant ${\cal O}(\alpha_s^2G_FM_t^2)$ corrections is specific to
$\Delta\rho$ or whether this is a common feature among the electroweak
parameters with a quadratic $M_t$ dependence at one loop.
In the latter case, there must be some underlying principle which is
responsible for this phenomenon.
Our analysis will put us into a position where we can investigate this issue
for four independent quantities.
We shall work in the electroweak on-shell renormalization scheme, with $G_F$
as a basic parameter \cite{sir}.
We shall take the colour gauge group to be SU(3), so that $N_c=C_A=3$,
$C_F=4/3$, and $T_F=1/2$.
We shall explicitly include five massless quark flavours plus the massive top
quark in our calculation, so that we have $n_f=6$ active quark flavours
altogether.
We shall evaluate the strong coupling constant, $\alpha_s(\mu)$, at
next-to-leading order in the modified minimal-subtraction
($\overline{\mbox{MS}}$) scheme \cite{msb}.
The $W$-, $Z$-, and Higgs-boson self-energies $\Pi_{WW}(q^2)$, $\Pi_{ZZ}(q^2)$,
and $\Pi_{HH}(q^2)$ will be the basic ingredients of our analysis.
In the case of $\Pi_{HH}(q^2)$, we shall actually need the first derivative
$\Pi_{HH}^\prime(q^2)$ for the Higgs-boson wave-function renormalization.
Since we wish to extract the leading high-$M_t$ terms, we may put $q^2=0$.
While the ${\cal O}(\alpha_s^2G_FM_t^2)$ results for $\Pi_{WW}(0)$ and
$\Pi_{ZZ}(0)$ are now well established \cite{avd}, $\Pi_{HH}^\prime(0)$
requires a separate analysis, which we shall carry out here.
Our calculation will proceed along the lines of Ref.~\cite{avd}.
We shall present our main results in this letter.
The technical details and a variety of applications are reported elsewhere
\cite{long}.
The Feynman diagrams pertinent to $\Pi_{HH}(q^2)$ in
${\cal O}(\alpha_s^2G_FM_t^2)$ come in twenty different topologies.
Typical specimen are depicted in Fig.~\ref{one}.
Using dimensional regularization, with $n=4-2\epsilon$ space-time dimensions
and a 't~Hooft mass $\mu$, and adopting from Ref.~\cite{gra} the QCD coupling
and mass counterterms in the $\overline{\mbox{MS}}$ scheme, we find
\begin{eqnarray}
\label{pihh}
\Pi_{HH}^\prime(0)&\n=\n&{3G_Fm_t^2(\mu)\over8\pi^2\sqrt2}
\left\{{2\over\epsilon}+2l-{4\over3}
+a\left(-{2\over\epsilon^2}+{5\over3\epsilon}+2l^2-{10\over3}l-{37\over18}
\right)
\right.\nonumber\\
&\n+\n&
a^2\left[{5\over2\epsilon^3}-{79\over12\epsilon^2}-{1\over3\epsilon}
\left(\zeta(3)-{311\over 36}\right)
+{5\over2}l^3-{7\over2}l^2-l\left(\zeta(3)+{1073\over72}\right)
\right.\nonumber\\
&\n-\n&\left.\left.
{16\over3}\li\left({1\over2}\right)+{11\over3}\zeta(4)+{37\over9}\zeta(3)
+{4\over3}\zeta(2)\ln^22-{2\over9}\ln^42+{17\over54}\right]\right\},
\end{eqnarray}
where $m_t(\mu)$ is the top-quark $\overline{\mbox{MS}}$ mass,
$l=\ln[\mu^2/m_t^2(\mu)]$, $\li{}$ is the quadrilogarithm, and $\zeta$ is
Riemann's zeta function.
In Eq.~(\ref{pihh}), we have omitted terms containing $\gamma_E-\ln(4\pi)$,
where $\gamma_E$ is Euler's constant.
These may be retrieved by substituting $\mu^2\to4\pi e^{-\gamma_E}\mu^2$.
We observe that, up to an overall minus sign, the ultraviolet divergences in
Eq.~(\ref{pihh}) precisely match those of the corresponding expression for
$\Pi_{WW}(0)/M_W^2$ in Ref.~\cite{avd}.
In the following, we shall employ $M_t$ instead of $m_t(\mu)$, since $M_t$
directly corresponds to the parameter which is being extracted from experiment
\cite{abe}.
The two-loop relation between $M_t$ and $m_t(M_t)$ may be found in
Ref.~\cite{gra}, and the $\mu$ evolution of $m_t(\mu)$ is determined by the
respective renormalization-group (RG) equation.
The QCD corrections to the $\ell^+\ell^-H$ Yukawa coupling originate in the
renormalizations of the Higgs-boson wave function and vacuum expectation
value.
For $M_H\ll M_t$, they may be accommodated in the $\ell^+\ell^-H$ interaction
Lagrangian by writing \cite{hff}
\begin{equation}
{\cal L}_{\ell\ell H}=-2^{1/4}G_F^{1/2}m_\ell\bar\ell\ell H(1+\delta_u),
\end{equation}
where
\begin{equation}
\delta_u=-{1\over2}\left[{\Pi_{WW}(0)\over M_W^2}+\Pi_{HH}^\prime(0)\right]
\end{equation}
is manifestly finite, gauge independent, and RG invariant.
Here, the subscript $u$ is to remind us that this term appears as a universal
building block in the radiative corrections to all production and decay
processes of the Higgs boson.
Combining Eq.~(\ref{pihh}) with the corresponding expression for
$\Pi_{WW}(0)/M_W^2$ in Ref.~\cite{avd} and eliminating $m_t(\mu)$ in favour of
$M_t$, we obtain
\begin{equation}
\label{duos}
\delta_u={7\over2}X_t\left[1-1.797\,105\,a(1+a\beta_0L)
-a^2(28.649\,053-2.074\,701\,n_f)\right].
\end{equation}
Equation~(\ref{duos}) reproduces the well-known ${\cal O}(G_FM_t^2)$
\cite{hff,cha} and ${\cal O}(\alpha_sG_FM_t^2)$ \cite{hll} terms.
Next, we shall derive the ${\cal O}(\alpha_s^2G_FM_t^2)$ correction to the
low-$M_H$ effective $W^+W^-H$ interaction Lagrangian.
In contrast to the $\ell^+\ell^-H$ case, we are now faced with the task of
computing genuine three-point amplitudes at three loops, which, at first sight,
appears to be enormously hard.
In fact, we are not aware of any three-loop calculation of a three-point
function in the literature.
Fortunately, in the limit that we are interested in, this problem may be
reduced to one involving just three-loop two-point diagrams by means of a
low-energy theorem, whose lowest-order version has been introduced in
Refs.~\cite{ell,vai}.
Generally speaking, this theorem relates the amplitudes of two processes which
differ by the insertion of an external Higgs-boson line carrying zero
four-momentum.
It allows us to compute a loop amplitude with an external Higgs boson which is
light compared to the virtual particles by differentiating the respective
amplitude without that Higgs boson with respect to the virtual-particle masses.
In Refs.~\cite{ks1,kil}, it has been shown how the applicability of this
theorem may be extended beyond the leading order.
Proceeding along the lines of Refs.~\cite{ks2,ks3}, we obtain
\begin{equation}
\label{lwwh}
{\cal L}_{W^+W^-H}=2^{5/4}G_F^{1/2}M_W^2W_\mu^+W^{-\mu}H(1+\delta_{WWH}),
\end{equation}
with
\begin{equation}
\label{dwwhnu}
\delta_{WWH}=\delta_u+\left[1-{(m_t^0)^2\partial\over\partial(m_t^0)^2}\right]
{\Pi_{WW}(0)\over M_W^2},
\end{equation}
where $m_t^0$ is the bare top-quark mass.
In Ref.~\cite{avd}, $\Pi_{WW}(0)$ is expressed in terms of $m_t(\mu)$.
Thus, we have to undo the top-quark mass renormalization \cite{gra} before we
can apply Eq.~(\ref{dwwhnu}).
Then, after evaluating the right-hand side of Eq.~(\ref{dwwhnu}), we introduce
$M_t$ and so obtain
\begin{equation}
\label{dwwhos}
\delta_{WWH}=-{5\over2}X_t\left[1-2.284\,053\,a(1+a\beta_0L)
-a^2(20.612\,157-1.632\,629\,n_f)\right].
\end{equation}
We recover the well-known ${\cal O}(G_FM_t^2)$ \cite{cha,hww}
and ${\cal O}(\alpha_sG_FM_t^2)$ \cite{ks3} terms.
The derivation of the ${\cal O}(\alpha_s^2G_FM_t^2)$ correction to the
low-$M_H$ effective $ZZH$ interaction Lagrangian proceeds in close analogy to
the $W^+W^-H$ case, and we merely list the result:
\begin{equation}
{\cal L}_{ZZH}=2^{1/4}G_F^{1/2}M_Z^2Z_\mu Z^\mu H(1+\delta_{ZZH}),
\end{equation}
where
\begin{equation}
\label{dzzhos}
\delta_{ZZH}=-{5\over2}X_t\left[1-4.684\,053\,a(1+a\beta_0L)
-a^2(18.410\,658-1.927\,313\,n_f)\right].
\end{equation}
Equation~(\ref{dzzhos}) contains the well-known ${\cal O}(G_FM_t^2)$
\cite{cha,hzz} and ${\cal O}(\alpha_sG_FM_t^2)$ \cite{ks2} terms.
The analytic versions of Eqs.~(\ref{duos}), (\ref{dwwhos}), and (\ref{dzzhos})
for $N_c$ arbitrary and in terms of fundamental functions and one master
integral, which may be solved numerically with high precision \cite{avd}, are
included in Ref.~\cite{long}.
We have presented the three-loop ${\cal O}(\alpha_s^2G_FM_t^2)$ corrections to
the effective Lagrangians for the interactions of light Higgs bosons with pairs
of charged leptons, $W$ bosons, and $Z$ bosons in the SM.
As a corollary, we note that $\Gamma(H\to\ell^+\ell^-)$, $\Gamma(H\to W^+W^-)$,
and $\Gamma(H\to ZZ)$ receive the correction factors $(1+\delta_u)^2$,
$(1+\delta_{WWH})^2$, and $(1+\delta_{ZZH})^2$, respectively.
At first sight, the conditions $M_H>2M_V$, with $V=W,Z$, which are necessary
for the decays $H\to W^+W^-$ and $H\to ZZ$ to happen, seem to conflict with
our assumption that $M_H\ll M_t$.
However, as may be already gleaned from the corresponding one-loop analyses
\cite{hww,hzz}, the actual expansion parameter is $r=(M_H^2/4M_t^2)$, which is
still small against unity if $M_H\gsim2M_V$.
{\it E.g.}, at one loop, the term linear in $r$ to be included within the
square brackets of Eq.~(\ref{dzzhos}) has the coefficient $+4/25$.
Moreover, these results may be used to refine the theoretical predictions for a
variety of four- and five-point production and decay processes of light Higgs
bosons at present and future $e^+e^-$ colliders.
This is relegated to our comprehensive report \cite{long}.
Here, we would like to focus attention on an interesting theoretical point.
In fact, our analysis allows us to recognize a certain universal pattern in the
structure of the QCD perturbation series.
In addition to $\Delta\rho$, we have now three more independent observables
with quadratic $M_t$ dependence at our disposal for which the QCD expansion is
known up to next-to-leading order, namely $\delta_u$, $\delta_{WWH}$, and
$\delta_{ZZH}$.
In the on-shell scheme of electroweak and QCD renormalization, these four
electroweak parameters exhibit striking common properties.
In fact, the leading- and next-to-leading-order QCD corrections act in the
same direction and screen the ${\cal O}(G_FM_t^2)$ terms.
Furthermore, the sets of $\alpha_s/\pi$ and $(\alpha_s/\pi)^2$ coefficients
each lie in the same ball park.
{}For the choice $\mu=M_t$, the coefficients of $\alpha_s/\pi$ range between
$-1.797$ and $-4.684$, and those of $(\alpha_s/\pi)^2$ between $-6.847$ and
$-16.201$.
We would like to point out that the corresponding coefficients of the ratio
$\mu_t^2/M_t^2$, where $\mu_t=m_t(\mu_t)$, are $-2.667$ and $-11.140$
\cite{long}, {\it i.e.}, they lie right in the centres of these ranges.
Therefore, it suggests itself that the use of $M_t$ is the origin of these
similarities.
In fact, if we express the QCD expansions in terms of $\mu_t$ rather than $M_t$
and choose $\mu=\mu_t$, then the coefficients of $\alpha_s/\pi$ and
$(\alpha_s/\pi)^2$ nicely group themselves around zero;
they range from $-2.017$ to 0.870 and from $-3.970$ to $1.344$, respectively
\cite{long}.
This indicates that the perturbation expansions converge more rapidly if we
renormalize the top-quark mass according to the $\overline{\mbox{MS}}$ scheme.
Without going into details, we would like to mention that the study of
renormalons \cite{ren} offers a possible theoretical explanation of this
observation.
Since the on-shell and $\overline{\mbox{MS}}$ results coincide in lowest
order, this does, of course, not imply that the QCD corrections are any
smaller in the $\overline{\mbox{MS}}$ scheme.
It just means that, as a rule, the ${\cal O}(G_FM_t^2)$ terms with $M_t$
replaced by the two-loop expression for $\mu_t$ \cite{long} are likely to
provide fair approximations for the full three-loop results.
In all the cases considered here, the QCD corrections now appear to be well
under control.
Finally, we would like to test Broadhurst's rule concerning the na\"\i ve
non-abelianization of QCD \cite{bro}.
Guided by the observation that the $n_f$-independent term of $\beta_0$ emerges
from the coefficient of $n_f$ by multiplication with $-33/2$, Broadhurst
conjectured that this very relation between the $n_f$-independent term and the
coefficient of $n_f$ approximately holds for any observable at next-to-leading
order in QCD.
This rule is trivially satisfied for the $L$-dependent terms in
Eqs.~(\ref{drhoos}), (\ref{duos}), (\ref{dwwhos}), and (\ref{dzzhos}).
As for the constant terms of ${\cal O}(\alpha_s^2G_FM_t^2)$ in these
expressions, multiplication of the coefficients of $n_f$ with $-33/2$ yields
$-29.5$, $-34.2$, $-26.9$, and $-31.8$, which has to be compared with the
respective $n_f$-independent terms, $-25.3$, $-28.6$, $-20.6$, and $-18.4$.
We observe that, in all these cases, Broadhurst's rule correctly predicts the
sign and the order of magnitude of the $n_f$-independent terms, which are much
harder to compute.
Except for the last case, these predictions are, in fact, very close to the
true values.
\bigskip
We would like to thank Bill Bardeen, David Broadhurst, Kostja Chetyrkin, and
Michael Spira for very useful discussions.
One of us (BAK) is indebted to the FNAL Theory Group for inviting him as a
Guest Scientist.
He is also grateful to the Phenomenology Department of the University of
Wisconsin at Madison for the great hospitality extended to him during a
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\end{thebibliography}
\newpage
\begin{figure}[ht]
\begin{center}
\begin{tabular}{ccc}
\epsfxsize=5.0cm
\leavevmode
\epsffile[130 260 470 530]{hse1.ps}
&
\epsfxsize=5.0cm
\leavevmode
\epsffile[130 260 470 530]{hse11.ps}
&
\epsfxsize=5.0cm
\leavevmode
\epsffile[130 260 470 530]{hse20.ps}
\end{tabular}
\caption{\label{one}Typical Feynman diagrams pertinent to $\Pi_{HH}(q^2)$ in
${\cal O}(\alpha_s^2G_FM_t^2)$.
$f$ stands for any quark.}
\end{center}
\end{figure}
\end{document}
|
http://o.castera.free.fr/tex/Nombres_complexes.tex | free.fr | CC-MAIN-2017-22 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2017-22/segments/1495463612018.97/warc/CC-MAIN-20170529053338-20170529073338-00080.warc.gz | 337,385,815 | 8,193 | \documentclass[francais, a4paper, 12pt, reqno]{amsart}%article ams
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pdftitle={Les nombres complexes}, % title
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\theoremstyle{plain}
\newtheorem{theoreme}{Th\'eor\`eme}[section]
\theoremstyle{definition}
\newtheorem{definition}{D\'efinition}[section]
\newtheorem*{regle}{R\`egles de calcul}
\newtheorem*{propriete}{Propri\'et\'es}
\theoremstyle{remark}
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\newtheorem*{remarque}{Remarque}
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\title{Les Nombres complexes}
\author{Olivier Cast\'era}
\email{[email protected]}
\urladdr{http://o.castera.free.fr/}
\date{\today}
\begin{document}
\begin{abstract}
Le corps des nombres complexes $\mathbbm C$ forme une extension quadratique du corps des nombres r\'eels $\mathbbm R$. Les nombres complexes de la forme $(x,0)$ forment un sous-corps de $\mathbbm C$ qui est isomorphe au corps $\mathbbm R$, par l'application qui \`a $(x,0)$ fait correspondre $x$.
\end{abstract}
\maketitle
\tableofcontents
\section{Groupe}
Un groupe est une structure alg\'ebrique relativement simple puisqu'elle ne contient qu'une seule op\'eration.
Elle est utilis\'ee dans beaucoup d'autres structures alg\'ebrique.
\begin{definition}
Un ensemble non vide $G$ muni de l'op\'eration~$\Box$, est un groupe, not\'e $(G,\Box)$, ssi
\begin{enumerate}
\item l'op\'eration binaire~$\Box$ est une loi de composition interne~: \`a chaque paire d'\'el\'ements de $G$, elle associe un \'el\'ement de $G$
\begin{align*}
\forall (a,b)\in G^{2},\quad a\Box b\in G
\end{align*}
\item l'op\'eration $\Box$ est associative
\begin{align*}
\forall (a,b,c)\in G^{3},\quad a\Box(b\Box c)=(a\Box b)\Box c
\end{align*}
\item Il existe un \'el\'ement neutre (ou identit\'e) $e$ dans $G$, pour l'op\'e\-ra\-tion~$\Box$
\begin{align*}
\exists e \in G\ /\ \forall a\in G,\quad e\Box a=a\Box e=a
\end{align*}
\item Tout \'el\'ement $a$ de $G$ poss\`ede un sym\'etrique $b$ dans $G$
\begin{align*}
\forall a\in G,\ \exists b\in G\ /\quad a\Box b=b\Box a=e
\end{align*}
\end{enumerate}
Pour un groupe multiplicatif, la loi de composition est not\'ee par une juxtaposition des \'el\'ements.
L'\'el\'ement neutre est l'unit\'e. Le sym\'etrique de $a$ est appel\'e inverse de $a$, et not\'e $a^{-1}$.\\
Pour un groupe additif, la loi de composition est not\'ee par $+$.
L'\'el\'ement neutre est l'\'el\'ement nul ou z\'ero. Le sym\'etrique de $a$ est appel\'e l'oppos\'e de $a$, et not\'e $-a$.
\end{definition}
\begin{theoreme}
Quel que soit le groupe $(G,\Box)$, l'\'el\'ement neutre est unique.
\end{theoreme}
\begin{proof}[D\'emonstration]
Supposons que $e$ et $e'$ soient les \'el\'ements neutres du groupe $(G,\Box)$
\begin{align*}
&\forall x\in G,\ x\Box e=e\Box x=x\\
&(x=e')\Rightarrow(e'\Box e=e\Box e'=e')\\
\\
&\forall x\in G,\ x\Box e'=e'\Box x=x\\
&(x=e)\Rightarrow(e\Box e'=e'\Box e=e)\\
\\
&(e'\Box e=e'\ \ \text{et}\ \ e'\Box e=e)\Rightarrow (e'=e)
\end{align*}
\end{proof}
\begin{definition}
Un groupe $(G,\Box)$ est dit ab\'elien ssi l'op\'eration $\Box$ est commutative
\begin{align*}
\forall (a,b)\in G^{2},\quad a\Box b=b\Box a
\end{align*}
\end{definition}
\begin{exemple}
Les ensembles des entiers naturels $\mathbbm Z$, des rationnels $\mathbbm Q$, et des r\'eels $\mathbbm R$, sont des groupes ab\'eliens pour l'addition $+$. Les ensembles des rationnels priv\'es de z\'ero (z\'ero n'a pas d'inverse), $\mathbbm Q^{*}$, et des r\'eels priv\'es de z\'ero, $\mathbbm R^{*}$, sont des groupes ab\'eliens pour la multiplication $\times$.
\end{exemple}
\subsection{Produit direct de groupes}
\begin{definition}
Soient $(G,\star)$ et $(H,\star)$ deux groupes munis de la m\^eme loi de composition interne $\star$.
Consid\'erons le produit cart\'esien $G\times H$ des ensembles $G$ et $H$, c'est \`a dire l'ensemble des paires or\-don\-n\'ees ${\{g\in G,h\in H,(g,h)\}}$.
On munit le produit cart\'esien $G\times H$ de l'op\'eration $\ostar$
\begin{align*}
\forall (g_{1},g_{2})\in G^{2}&,\ \forall (h_{1},h_{2})\in H^{2},\\
(g_{1},h_{1})\ostar(g_{2},h_{2})&=(g_{1}\star g_{2},h_{1}\star h_{2})
\end{align*}
$(G\times H,\ostar)$ est appel\'e produit direct de $G$ et $H$.
\end{definition}
\begin{theoreme}\label{direct}
Le produit direct $(G\times H,\ostar)$ forme un groupe.
\end{theoreme}
\begin{proof}[D\'emonstration]\hfill
\begin{enumerate}
\item l'op\'eration $\ostar$ est une loi de composition interne
\begin{align*}
&(G,\star)\,est\,un\,groupe\ :\ \forall (g_{1},g_{2})\in G^{2},\ (g_{1}\star g_{2})\in G\\
&(H,\star)\,est\,un\,groupe\ :\ \forall (h_{1},h_{2})\in H^{2},\ (h_{1}\star h_{2})\in H\\
&\forall (g_{1},h_{1})\in (G\times H),\ \forall (g_{2},h_{2})\in (G\times H),\\
&\qquad\qquad(g_{1}\star g_{2},h_{1}\star h_{2})\in (G\times H)\\
&\qquad\qquad(g_{1},h_{1})\ostar(g_{2},h_{2})\in (G\times H)
\end{align*}
\item l'op\'eration $\ostar$ est associative
\begin{align*}
\forall (g_{1},h_{1}),(g_{2},h_{2}),(g_{3},h_{3})&\in (G\times H)^{3},\\
(g_{1},h_{1})\ostar[(g_{2},h_{2})\ostar(g_{3},h_{3})]&=(g_{1},h_{1})\ostar[(g_{2}\star g_{3},h_{2}\star h_{3})\\
&=[g_{1}\star (g_{2}\star g_{3}),h_{1}\star (h_{2}\star h_{3})]
\end{align*}
et,
\begin{align*}
[(g_{1},h_{1})\ostar(g_{2},h_{2})]\ostar(g_{3},h_{3})&=(g_{1}\star g_{2},h_{1}\star h_{2})]\ostar(g_{3},h_{3})\\
&=[(g_{1}\star g_{2})\star g_{3},(h_{1}\star h_{2})\star h_{3}]\\
&=[g_{1}\star (g_{2}\star g_{3}),h_{1}\star (h_{2}\star h_{3})]
\end{align*}
o\`u l'on a utilis\'e l'associativit\'e de la loi $\star$ dans $G$ et dans $H$.\\
Par cons\'equent,
\begin{align*}
(g_{1},h_{1})\ostar[(g_{2},h_{2})\ostar(g_{3},h_{3})]=[(g_{1},h_{1})\ostar(g_{2},h_{2})]\ostar(g_{3},h_{3})
\end{align*}
\item l'op\'eration $\ostar$ admet un \'el\'ement neutre dans $G\times H$.\\
Soit $e$ l'\'el\'ement neutre du groupe $G$, et soit $e'$ l'\'el\'ement neutre du groupe $H$
\begin{align*}
\forall (g,h)\in (G\times H),\ (e,e')\ostar(g,h)&=(e\star g,e'\star h)\\
&=(g,h)\\
\\
(g,h)\ostar(e,e')&=(g\star e,h\star e')\\
&=(g,h)
\end{align*}
\end{enumerate}
\end{proof}
\begin{theoreme}\label{t3}
Si $(G,\star)$ et $(H,\star)$ sont des groupes ab\'eliens, alors le produit direct $(G\times H,\ostar)$ forme un groupe ab\'elien.
\end{theoreme}
\begin{proof}[D\'emonstration]
\begin{align*}
(G,\star)\,est\,un\,groupe\,ab\acute elien\ :\ g_{1}\star g_{2}&=g_{2}\star g_{1}\\
(H,\star)\,est\,un\,groupe\,ab\acute elien\ :\ h_{1}\star h_{2}&=h_{2}\star h_{1}\\
\forall (g_{1},h_{1}),(g_{2},h_{2})\in (G\times H)^{2},(g_{1},h_{1})\ostar(g_{2},h_{2})&=(g_{1}\star g_{2},h_{1}\star h_{2})\\
&=(g_{2}\star g_{1},h_{2}\star h_{1})\\
&=(g_{2},h_{2})\ostar(g_{1},h_{1})
\end{align*}
\end{proof}
\subsection{Morphisme de groupes}
\begin{definition}
Soient deux groupes $(G,*)$ et $(G',\star)$. L'application $f$ de $(G,*)$ dans $(G',\star)$ est un morphisme de groupes ssi
\begin{align*}
f : G&\rightarrow G'\\
\forall (x,y)\in G^{2},\ f(x*y)&=f(x)\star f(y)
\end{align*}
$f$ est un isomorphisme de groupes ssi $f$ est un morphisme bijectif. Dans ce cas, $f^{-1}$ est aussi un morphisme de groupes.
\end{definition}
\section{Anneau}
\begin{definition}\label{anneau}
Un ensemble $A$ muni de deux op\'erations, not\'ees $\oplus$ et $\odot$, est un anneau, not\'e $(A,\oplus,\odot)$, ssi
\begin{enumerate}
\item $(A,\oplus)$ est un groupe ab\'elien
\item l'op\'eration $\odot$ est une loi de composition interne
\begin{align*}
\forall (a,b)\in A^{2},\ a\odot b\in A
\end{align*}
\item l'op\'eration $\odot$ est associative
\begin{align*}
\forall (a,b,c)\in A^{3},\ a\odot(b\odot c)=(a\odot b)\odot c
\end{align*}
\item l'op\'eration $\odot$ admet un \'el\'ement neutre $e'$ dans $A$
\begin{align*}
\exists e' \in A\ /\ \forall a\in A,\ e'\odot a=a\odot e'=a
\end{align*}
\item l'op\'eration $\odot$ est distributive \`a gauche et \`a droite par rapport \`a l'op\'eration $\oplus$
\begin{align*}
\forall x,y,z\in A^{3},\ a\odot(b\oplus c)&=(a\odot b)\oplus(a\odot c)\\
\forall x,y,z\in A^{3},\ (b\oplus c)\odot a&=(b\odot a)\oplus(c\odot a)
\end{align*}
\end{enumerate}
\end{definition}
\begin{definition}\label{anneaucommutatif}
Un anneau $(A,\oplus,\odot)$ est dit commutatif ssi l'op\'eration $\odot$ est commutative
\begin{align*}
\forall (a,b)\in A^{2},\ a\odot b=b\odot a
\end{align*}
\end{definition}
\begin{regle}
Quel que soit l'anneau $(A,\oplus,\odot)$
\begin{align*}
&Soit\ e\ l'\acute el\acute ement\ neutre\ de\ la\ loi\ \oplus\ :\ \forall x \in A,\ e\odot x=x \odot e=e\\
&Soit\ \ominus y\ le\ sym\acute etrique\ de\ y\ pour\ la\ loi\ \oplus\ :\\
&\qquad\qquad\forall (x,y)\in A^{2},\ x\odot(\ominus y)=\ominus(x\odot y)\\
&\forall (x,y,z)\in A^{3},\ x\odot(y\ominus z)=(x\odot y)\ominus(x\odot z)\\
&Si\ (A,\oplus,\odot)\ est\ un\ anneau\ commutatif,\ bin\hat ome\ de\ Newton~:\\
&\forall n\in \mathbbm N,\ \forall(x,y)\in A^{2},\ (x\oplus y)^{n}=\sum_{k=0}^{n}C_{n}^{k}x^{k}\odot y^{n\ominus k}
\end{align*}
\end{regle}
\subsection{Anneau int\`egre}
\begin{definition}
Un \'el\'ement non nul $a$ d'un anneau $(A,\oplus,\odot)$ est un diviseur de z\'ero \`a gauche, ssi
\begin{align*}
\exists b\neq e\in A\ /\ a\odot b=e
\end{align*}
\end{definition}
\begin{definition}
Un \'el\'ement non nul $a$ d'un anneau $(A,\oplus,\odot)$ est un diviseur de z\'ero \`a droite, ssi
\begin{align*}
\exists b\neq e\in A\ /\ b\odot a=e
\end{align*}
\end{definition}
\begin{definition}
Un anneau $(A,\oplus,\odot)$ est int\`egre s'il est diff\'erent de l'\'el\'ement nul $\{e\}$, commutatif, et sans diviseur de z\'ero.
\end{definition}
\begin{theoreme}
Pour tout anneau $(A,\oplus,\odot)$ int\`egre
\begin{align*}
\forall (a,b)\in A^{2},\ (a\odot b=e)\Rightarrow(a=e\ \text{ou}\ b=e)
\end{align*}
\end{theoreme}
\subsection{Sous-anneau}
\begin{definition}\label{sousanneau}
Toute partie $A'$ de l'ensemble $A$ est un sous-anneau de l'anneau $(A,\oplus,\odot)$ ssi
\begin{enumerate}
\item $(A',\oplus,\odot)$ est un anneau
\item $A'$ est stable pour la loi $\oplus$
\begin{align*}
\forall (a,b)\in A'^{2},\ a\oplus b\in A'
\end{align*}
\item $A'$ est stable pour la loi $\odot$
\begin{align*}
\forall (a,b)\in A'^{2},\ a\odot b\in A'
\end{align*}
\end{enumerate}
\end{definition}
\subsection{Morphisme d'anneaux}
\begin{definition}\label{morphismed'anneau}
Soient deux anneaux $(A_{1},\oplus,\odot)$ et $(A_{2},\boxplus,\boxdot)$, et soient $e_{1}$ l'\'el\'ement neutre de $\odot$ dans $A_{1}$, et $e_{2}$ l'\'el\'ement neutre de $\boxdot$ dans $A_{2}$. L'application $f$ de $(A_{1},\oplus,\odot)$ dans $(A_{2},\boxplus,\boxdot)$ est un morphisme d'anneaux ssi
\begin{enumerate}
\item $\forall (x,y)\in A_{1}^{2},\ f(x\oplus y)=f(x)\boxplus f(y)$\\
\item $\forall (x,y)\in A_{1}^{2},\ f(x\odot y)=f(x)\boxdot f(y)$\\
\item $f(e_{1})=e_{2}$
\end{enumerate}
$f$ est un isomorphisme d'anneaux ssi $f$ est un morphisme bijectif. Dans ce cas, $f^{-1}$ est aussi un morphisme d'anneaux.
\end{definition}
\section{Corps}
\begin{definition}\label{corps}
Un ensemble $K$ muni de deux op\'erations, not\'ees $\oplus$ et $\odot$, est un corps, not\'e $(K,\oplus,\odot)$, ssi
\begin{enumerate}
\item $(K,\oplus,\odot)$ est un anneau
\item tout \'el\'ement non nul $a$ de $K$ poss\`ede un inverse, not\'e $a^{-1}$, dans $K$ pour l'op\'eration $\odot$
\begin{align*}
\forall a\neq e\in K,\ \exists a^{-1}\in K\ /\ a\odot a^{-1}=a^{-1}\odot a=e'
\end{align*}
\end{enumerate}
\end{definition}
\begin{definition}
Un corps $(K,\oplus,\odot)$ est dit commutatif ssi l'op\'eration $\odot$ est commutative
\begin{align*}
\forall (a,b)\in K^{2},\ a\odot b=b\odot a
\end{align*}
\end{definition}
\begin{theoreme}\label{diviseur}
Si $(K,\oplus,\odot)$ est un corps alors il n'a pas de diviseur de z\'ero.
\end{theoreme}
\begin{proof}[D\'emonstration]
D'apr\`es la d\'efinition \ref{corps}, si $(K,\oplus,\odot)$ est un corps
\begin{align*}
\forall a\neq e\in K,\ \exists a^{-1}\in K\ /\ a\odot a^{-1}&=a^{-1}\odot a=e'
\end{align*}
Pour d\'emontrer que
\begin{align*}
(a\odot b=e)\Rightarrow(a=e\ \text{ou}\ b=e)
\end{align*}
nous allons montrer que si $a\odot b=e$ alors il est impossible d'avoir $a\neq e\ \text{et}\ b\neq e$
\begin{align*}
\forall (a,b)\in K^{2},\ a\odot b&=e\\
\forall a\neq e\in K,\ a^{-1}\odot a\odot b&=a^{-1}\odot e\\
e'\odot b&=e\\
b&=e
\end{align*}
et par sym\'etrie des r\^oles de $a$ et $b$, si $b\neq e$ alors $a=e$.
\end{proof}
\subsection{Sous-corps}
\begin{definition}\label{souscorps}
Toute partie $K'$ de l'ensemble $K$ est un sous-corps du corps $(K,\oplus,\odot)$ ssi
\begin{enumerate}
\item $(K',\oplus,\odot)$ est un corps
\item $K'$ est stable pour la loi $\oplus$
\begin{align*}
\forall (a,b)\in K'^{2},\ a\oplus b\in K'
\end{align*}
\item $K'$ est stable pour la loi $\odot$
\begin{align*}
\forall (a,b)\in K'^{2},\ a\odot b\in K'
\end{align*}
\end{enumerate}
\end{definition}
\section{Racines Carr\'ees}\label{RC}
Soit $(K,\oplus,\odot)$ un anneau commutatif. Nous dirons qu'un \'el\'ement $\alpha$ de $K$ est un carr\'e dans $K$ ssi
\begin{align*}
\exists x\in K\ /\ x^{2}=\alpha
\end{align*}
$x$ est appel\'ee racine carr\'ee de $\alpha$ dans $(K,\oplus,\odot)$. Si $x$ est une racine carr\'ee de $\alpha$ dans $K$ alors il en est de m\^eme de $\ominus x$, car ${(\ominus x)^{2}=\alpha}$. Si l'anneau $(K,\oplus,\odot)$ est int\`egre, $\alpha$ ne peut admettre plus de deux racines carr\'ees dans $K$, car la relation ${x^{2}=y^{2}}$, qui s'\'ecrit dans tous les cas sous la forme ${(x\ominus y)(x\oplus y)=e}$, implique soit ${x=y}$, soit ${y=\ominus x}$.\\
\begin{exemple}
Si $(K,\oplus,\odot)$ est le corps des nombres r\'eels $(\mathbbm R,+,\times)$, $\alpha$ est un carr\'e dans $K$ ssi $\alpha\geqslant0$, par exemple pour $\alpha=2$. Si $(K,\oplus,\odot)$ est le corps des nombres rationnels $(\mathbbm Q,+,\times)$, $2$ n'est pas un carr\'e.
\end{exemple}
On est ainsi conduit \`a examiner le probl\`eme suivant :\\
Soit $(K,\oplus,\odot)$ un anneau commutatif et $\alpha$ un \'el\'ement de $K$ qui n'est pas un carr\'e dans $K$. Est-il possible de construire un anneau com\-mu\-ta\-tif $(L,\boxplus,\boxdot)$ poss\'edant les propri\'et\'es suivantes :
$K$ est un sous-anneau de $L$, et $\alpha$ est un carr\'e dans $L$ ?\\
\\
Soit $\alpha$ un \'el\'ement d'un anneau commutatif $(K,\oplus,\odot)$, qui n'est pas un carr\'e dans $K$. Supposons le probl\`eme r\'esolu et d\'esignons par $(L,\boxplus,\boxdot)$ un anneau commutatif dont $K$ soit un sous-anneau
\begin{align*}
(K,\oplus,\odot)\subseteq (L,\oplus,\odot)
\end{align*}
et par $\omega$ une racine carr\'ee de $\alpha$ dans $L$
\begin{align*}
\omega^{2}=\alpha
\end{align*}
D\'esignons par $L'$ l'ensemble des \'el\'ements $z$ de l'anneau commutatif $L$ poss\'edant la propri\'et\'e
\begin{align*}
\forall z\in L',\ \exists (x,y)\in K^{2}\ /\ z=x\oplus(\omega\odot y)
\end{align*}
\begin{notation}
Nous posons que la loi de composition $\odot$ est prioritaire sur la loi de composition $\oplus$, et nous omettrons souvent le symbole $\odot$ pour faciliter la lecture. Par cons\'equent, nous \'ecrirons
\begin{align*}
z=x\oplus\omega y
\end{align*}
\end{notation}
\begin{theoreme}
$(L',\oplus,\odot)$ est un sous-anneau de $(L,\boxplus,\boxdot)$ contenant $(K,\oplus,\odot)$ et $\omega$.
\end{theoreme}
\begin{proof}
En utilisant les propi\'et\'es des lois de composition internes d'un anneau, donn\'ees en d\'efinition \ref{anneau}, nous avons
\begin{align}
\forall (x'\oplus\omega y')\in L'\ &\text{et}\ \forall (x''\oplus\omega y'')\in L',\notag\\
\notag\\
(x'\oplus\omega y')\oplus(x''\oplus\omega y'')&=x'\oplus\omega y'\oplus x''\oplus\omega y''\notag\\
&=x'\oplus x''\oplus\omega y'\oplus\omega y''\notag\\
&=(x'\oplus x'')\oplus\omega(y'\oplus y'')\in L'\label{a}\\
\notag\\
(x'\oplus\omega y')\odot(x''\oplus\omega y'')&=x'x''\oplus x'\omega y''\oplus\omega y'x''\oplus\omega y'\omega y''\notag\\
&=x'x''\oplus\alpha y'y''\oplus x'\omega y''\oplus\omega y'x''\notag\\
&=(x'x''\oplus\alpha y'y'')\oplus\omega(x'y''\oplus y'x'')\in L'\label{b}
\end{align}
D'apr\`es la d\'efinition~\ref{sousanneau}, $(L',\oplus,\odot)$ est un sous-anneau de $(L,\boxplus,\boxdot)$.
De plus, l'anneau $L'$ contient l'anneau $K$ (poser $y=e$), et il contient aussi $\omega$ (poser $x=e$ et $y=e'$).
\end{proof}
Ce r\'esultat montre que si le probl\`eme admet une solution, alors on peut construire l'anneau commutatif $(L,\boxplus,\boxdot)$ de telle sorte que chacun de ses \'el\'ements s'\'ecrive sous la forme $x\oplus\omega y$ avec $(x,y)\in K^{2}$.\\
Autrement dit, si nous introduisons l'application $f$, telle que
\begin{align*}
f:K\times K&\rightarrow L\\
f(x,y)&=x\oplus\omega y
\end{align*}
alors $f$ est \textit{surjective}
\begin{align*}
\forall u \in L,\ \exists (x,y)\in (K\times K)\ /\ f(x,y)=u
\end{align*}
\begin{remarque}
Si $(K,\oplus,\odot)$ est un corps et si $\alpha$ n'est pas un carr\'e dans $(K,\oplus,\odot)$, l'application $f$ est en outre \textit{injective}
\begin{align*}
\forall (x',y')\!\in\! K^{2},\forall (x'',y'')\!\in\! K^{2},\ f(x',y')=f(x'',y'')\Rightarrow (x'=x'',y'=y'')
\end{align*}
\begin{proof}
En posant $x=x'\ominus x''$ et $y=y'\ominus y''$, nous avons
\begin{align*}
f(x',y')&=f(x'',y'')\\
x'\oplus\omega y'&=x''\oplus\omega y''\\
(x'\ominus x'')\oplus\omega(y'\ominus y'')&=e\\
x\oplus\omega y&=e
\end{align*}
Raisonnons par l'absurde en supposant $y\neq e$.\\
$y$ est inversible dans $K$, puisque par hypoth\`ese $K$ est un corps, et
$y$ est aussi inversible dans $L$ puisque $(K,\oplus,\odot)\subseteq (L,\boxplus,\boxdot)$.\\
Par cons\'equent
\begin{align*}
\omega=\ominus xy^{-1}\in K
\end{align*}
contrairement \`a l'hypoth\`ese que $\alpha$ n'est pas un carr\'e dans $K$.\\
Donc $y=e$. Par cons\'equent $x=e$, $x'=x''$, $y'=y''$ et $f$ est \textit{injective}.
\end{proof}
\end{remarque}
En introduisant l'application $f$, les \'egalit\'es \eqref{a} et \eqref{b} s'\'ecrivent
\begin{align*}
f(x',y')\oplus f(x'',y'')&=f(x'\oplus x'',y'\oplus y'')\\
f(x',y')\odot f(x'',y'')&=f(x'x''\oplus\alpha y'y'',x'y''\oplus y'x'')
\end{align*}
Ces \'egalit\'es, obtenues en supposant le probl\`eme r\'esolu, vont maintenant nous servir de point de d\'epart pour construire une solution au probl\`eme pos\'e.
\section{L'anneau $L$}
Soit $\alpha$ un \'el\'ement d'un anneau commutatif $(K,\oplus,\odot)$, qui n'est pas un carr\'e dans $K$. Nous allons construire un nouvel anneau $(L,\boxplus,\boxdot)$ dans lequel $\alpha$ est un carr\'e. Soit l'ensemble $L$, produit cart\'esien de $K\times K$, tel qu'un \'el\'ement $(x,y)$ de $L$ soit une paire ordonn\'ee de deux \'el\'ements $x$ et $y$ de $K$.
\begin{definition}\label{mul}
Les deux op\'erations $\boxplus$ et $\boxdot$ dans $(L,\boxplus,\boxdot)$ sont d\'e\-fi\-nies comme suit
\begin{align*}
(x',y')\boxplus(x'',y'')&=(x'\oplus x'',y'\oplus y'')\\
(x',y')\boxdot(x'',y'')&=(x'x''\oplus\alpha y'y'',x'y''\oplus y'x'')
\end{align*}
\end{definition}
lesquelles font intervenir \`a la fois l'\'el\'ement $\alpha$ et les lois de composition dans l'anneau~$(K,\oplus,\odot)$.
\begin{theoreme}
L'ensemble $L$ muni des lois de composition internes $\boxplus$ et $\boxdot$ est un anneau.
\end{theoreme}
\begin{proof}
Suivons les cinq points de la d\'efinition \ref{anneau}.
\begin{enumerate}
\item Montrons que $(L,\boxplus)$ est un groupe ab\'elien.\\
L'ensemble $K$ muni de la loi de composition $\oplus$ est un groupe ab\'elien.
D'apr\`es le th\'eor\`eme~\ref{t3}, le produit direct des groupes ab\'eliens $K\times K$ est un groupe ab\'elien, donc $(L,\boxplus)$ est un groupe ab\'elien.
\item Montrons que $\boxdot$ est une loi de composition interne.\\
$\forall x',y',x'',y''$, quatre \'el\'ements de l'anneau $K$.\\
$x'x''\oplus\alpha y'y''\in K$ et $x'y''\oplus\alpha y'x''\in K$\\
Si l'on pose $a=(x',y')$ et $b=(x'',y'')$, on a~:
\begin{align*}
\forall (a,b)\in L^2,\ a\boxdot b\in L
\end{align*}
\item Montrons que la loi $\boxdot$ est associative.\\
En utilisant la d\'efinition~\ref{mul}
\begin{align*}
&(x,y)\boxdot[(x',y')\boxdot(x'',y'')]=(x,y)\boxdot(x'x''\oplus\alpha y'y'', x'y''\oplus y'x'')\\
&=(x(x'x''\!\oplus\alpha y'y'')\!\oplus\alpha y(x'y''\!\oplus y'x''), x(x'y''\!\oplus y'x'')\!\oplus y(x'x''\!\oplus\alpha y'y''))\\
&=(xx'x''\!\oplus x\alpha y'y''\!\oplus\alpha yx'y'\!\oplus\alpha yy'x'',xx'y''\!\oplus xy'x''\!\oplus yx'x''\!\oplus y\alpha y'y'')
\end{align*}
et d'autre part
\begin{align*}
&[(x,y)\boxdot(x',y')]\boxdot(x'',y'')=(xx'\oplus\alpha yy', xy'\oplus yx')\boxdot(x'',y'')\\
&=[(xx'\oplus\alpha yy')x''\oplus\alpha (xy'\oplus yx')y'', (xx'\oplus\alpha yy')y''\oplus (xy'\oplus yx')x'']\\
&=(xx'x''\!\oplus\alpha yy'x''\!\oplus\alpha xy'y''\!\oplus\alpha yx'y'',xx'y''\!\oplus\alpha yy'y''\!\oplus xy'x''\!\oplus x'yx'')
\end{align*}
L'associativit\'e s'obtient en comparant les r\'esultats.
\item Montrons que la loi $\boxdot$ admet $(e',e)$ comme \'el\'ement neutre
\begin{align*}
(e',e)\boxdot(x,y)&=(e'x\oplus\alpha ey,e'y\oplus ex)\\
&=(x\oplus\alpha e,y\oplus e)\\
&=(x,y)
\end{align*}
\item Montrons que la loi $\boxdot$ est distributive \`a gauche par rapport \`a la loi~$\boxplus$
\begin{align*}
&(x,y)\boxdot[(x',y')\boxplus(x'',y'')]=(x,y)\boxdot(x'\oplus x'',y'\oplus y'')\\
&=(x(x'\oplus x'')\oplus\alpha y(y'\oplus y''),x(y'\oplus y'')\oplus y(x'\oplus x''))\\
&=(xx'\oplus xx''\oplus\alpha yy'\oplus\alpha yy'',xy'\oplus xy''\oplus yx'\oplus yx'')\\
&=((xx'\oplus\alpha yy')\oplus(xx''\oplus\alpha yy''),(xy'\oplus yx')\oplus(xy''\oplus yx''))\\
&=(xx'\oplus\alpha yy',xy'\oplus yx')\boxplus(xx''\oplus\alpha yy'',xy''\oplus yx'')\\
&=[(x,y)\boxdot(x',y')]\boxplus[(x,y)\boxdot(x'',y'')]
\end{align*}
De m\^eme pour la distributivit\'e \`a droite.
\end{enumerate}
\end{proof}
\begin{theoreme}
$(L,\boxplus,\boxdot)$ est un anneau commutatif.
\end{theoreme}
\begin{proof}
Montrons que la loi $\boxdot$ est commutative
\begin{align*}
(x,y)\boxdot(x',y')&=(xx'\oplus\alpha yy',xy'\oplus yx')\\
&=(x'x\oplus\alpha y'y,x'y\oplus y'x)\\
&=(x',y')\boxdot(x,y)
\end{align*}
D'apr\`es la d\'efinition~\ref{anneaucommutatif}, l'ensemble $L$ muni des lois de composition $\boxplus$ et $\boxdot$ est donc un anneau commutatif.
\end{proof}
\begin{theoreme}
L'anneau $(L,\boxplus,\boxdot)$ contient un sous-anneau i\-so\-mor\-phe \`a l'anneau $(K,\oplus,\odot)$.
\end{theoreme}
\begin{proof}
Consid\'erons l'application
\begin{align*}
f : K&\rightarrow K\times K\\
f(x)&=(x,e)
\end{align*}
A tout \'el\'ement $(x,e)$ de l'ensemble $K\times K$ correspond un \'el\'ement unique $x$ de l'ensemble $K$ par $f$.
Par cons\'equent $f$ est bijective
\begin{align*}
\forall(x,e)\in (K\times K),\ \exists!\ x\in K\ /\ f(x)=(x,e)
\end{align*}
De plus, nous avons les relations suivantes
\begin{align*}
f(x')\boxplus f(x'')&=(x',e)\boxplus(x'',e)\\
&=(x'\oplus x'',e\oplus e)\\
&=(x'\oplus x'',e)\\
&=f(x'\oplus x'')\\
\\
f(x')\boxdot f(x'')&=(x',e)\boxdot(x'',e)\\
&=(x'x''\oplus\alpha ee,x'e\oplus x''e)\\
&=(x'x'',e)\\
&=f(x'x'')\\
\\
f(e')&=(e',e)
\end{align*}
D'apr\`es la d\'efinition~\ref{morphismed'anneau}, $f$ est un isomorphisme de l'anneau $(K,\oplus,\odot)$ sur un sous-anneau de l'anneau $(L,\boxplus,\boxdot)$.
\begin{notation}
Comme $f$ transforme les lois de compositions de l'anneau $(K,\oplus,\odot)$ en celles du sous-anneau $f(K)$ de l'anneau $(L,\boxplus,\boxdot)$, il n'y a aucun inconv\'enient \`a identifier chaque \'el\'ement $x$ de l'anneau $(K,\oplus,\odot)$ \`a l'\'el\'ement $f(x)$ de l'anneau $(L,\boxplus,\boxdot)$. Nous utiliserons la notation (fausse) suivante
\begin{align*}
(x,e)&=x
\end{align*}
or
\begin{align*}
(x,y)&=(x\oplus e,e\oplus y)\\
&=(x,e)\boxplus(e,y)\\
&=(x,e)\boxplus(ey\oplus\alpha e'e,ee\oplus e'y)\\
&=(x,e)\boxplus[(e,e')\boxdot(y,e)]
\end{align*}
d'o\`u la notation suivante
\begin{align}
(x,y)=x\oplus\omega y\label{notation}
\end{align}
avec en particulier,
\begin{align*}
(e',e)&=e'\\
(e,e')&=e\oplus\omega e'\\
&=\omega
\end{align*}
En utilisant cette notation, les d\'efinitions \ref{mul} s'\'ecrivent
\begin{align}
(x'\oplus\omega y')\oplus(x''\oplus\omega y'')&=(x'\oplus x'')\oplus\omega (y'\oplus y'')\label{c}\\
(x'\oplus\omega y')\odot(x''\oplus\omega y'')&=(x'x''\oplus\alpha y'y'')\oplus\omega(x'y''\oplus y'x'')\label{d}
\end{align}
\end{notation}
Il reste \`a montrer que $\alpha$ est un carr\'e dans l'anneau $(L,\boxplus,\boxdot)$. Con\-si\-d\'e\-rons l'\'el\'ement $\omega=(e,e')$ de l'anneau $(L,\boxplus,\boxdot)$. On a alors
\begin{align*}
\omega ^{2}&=(e,e')(e,e')\\
&=(ee\oplus\alpha e'e',ee'\oplus e'e)\\
&=(\alpha,e)\\
&=\alpha
\end{align*}
puisqu'on a convenu d'identifier chaque \'el\'ement $x$ de l'anneau $(K,\oplus,\odot)$ \`a l'\'el\'ement $f(x)$ de l'anneau $(L,\boxplus,\boxdot)$.
\end{proof}
L'anneau $(L,\boxplus,\boxdot)$ se note $K[\sqrt{\alpha}]$ et s'appelle une \textit{extension quadratique de $K$}. On dit que $K[\sqrt{\alpha}]$ s'obtient par \textit{adjonction \`a $K$ d'une racine carr\'e de $\alpha$}.
\section{El\'ements inversibles d'une extension quadratique}
Soient $K$ un anneau commutatif et $\alpha$ un \'el\'ement de $K$. On consid\`ere l'extension quadratique $L=K[\sqrt{\alpha}]$.
\begin{definition}
Soit $z=(x,y)=x\oplus\omega y\in L$. On appelle conjugu\'e de $z$, l'\'el\'ement $\bar z$ de $L$, tel que
\begin{align*}
\bar z&=(x,\ominus y)\\
&=x\ominus\omega y
\end{align*}
\end{definition}
\begin{definition}\label{norme}
Soit $z=x\oplus\omega y\in L$. On appelle norme de $z$, l'\'el\'ement
\begin{align*}
N(z)&=\bar zz\\
&=(x\ominus\omega y)\odot(x\oplus\omega y)\\
&=x^{2}\ominus\omega^{2}y^{2}\\
&=x^{2}\ominus\alpha y^{2}
\end{align*}
\end{definition}
On remarque que $N(e')=e'$.
\begin{theoreme}
\begin{align*}
\overline{z'\oplus z''}=\bar z'\oplus \bar z''
\end{align*}
\end{theoreme}
\begin{proof}[D\'emonstration]
\begin{align*}
\overline{z'\oplus z''}&=\overline{(x'\oplus\omega y')\oplus(x''\oplus\omega y'')}\\
&=\overline{(x'\oplus x'')\oplus\omega(y'\oplus y'')}\\
&=(x'\oplus x'')\ominus \omega (y'\oplus y'')\\
&=(x'\oplus x'')\oplus[\ominus\omega y'\oplus(\ominus\omega y'')]\\
&=(x'\ominus \omega y')\oplus(x''\ominus \omega y'')\\
&=\bar z'\oplus \bar z''
\end{align*}
\end{proof}
\begin{theoreme}
\begin{align*}
\overline{z'z''}=\bar z'\bar z''
\end{align*}
\end{theoreme}
\begin{proof}[D\'emonstration]
\begin{align*}
\overline{z'z''}&=\overline{(x'\oplus\omega y')\odot(x''\oplus\omega y'')}\\
&=\overline{(x'x''\oplus\alpha y'y'')\oplus\omega(x'y''\oplus y'x'')}\\
&=(x'x''\oplus\alpha y'y'')\ominus\omega(x'y''\oplus y'x'')\\
&=(x'x''\oplus\alpha y'y'')\oplus[\ominus\omega x'y''\oplus(\ominus\omega y'x'')]\\
&=(x'\ominus\omega y')\odot(x''\ominus\omega y'')\\
&=\bar z'\bar z''
\end{align*}
\end{proof}
\begin{theoreme}
\begin{align*}
N(z'z'')=N(z')N(z'')
\end{align*}
\end{theoreme}
\begin{proof}[D\'emonstration]
\begin{align*}
N(z'z'')&=\overline{z'z''}\odot z'z''\\
&=\overline{z'}\odot\overline{z''}\odot z'\odot z''\\
&=\overline{z'}\odot z'\odot\overline{z''}\odot z''\\
&=\overline{z'}z'\odot \overline{z''}z''\\
&=N(z')N(z'')
\end{align*}
\end{proof}
\begin{theoreme}\label{inv}
Soient $K$ un anneau commutatif, $\alpha$ un \'el\'ement de $K$, et $z$ un \'el\'ement de l'anneau $K[\sqrt{\alpha}]$. Pour que $z$ soit inversible dans l'anneau $K[\sqrt{\alpha}]$, il faut et il suffit que $N(z)$ le soit dans $K$. On a alors
\begin{align}
z^{-1}=N(z)^{-1}\,\bar z\label{zinv}
\end{align}
\end{theoreme}
\begin{proof}[D\'emonstration]
Supposons $z$ inversible, alors
\begin{align*}
z^{-1}z&=e'\\
N(z^{-1}z)&=N(e')\\
N(z^{-1})N(z)&=e'
\end{align*}
$N(z)$ est donc bien un \'el\'ement inversible de l'anneau $K$.\\
Inversement, supposons $N(z)$ inversible dans $K$, alors
\begin{align*}
\bar zz&=N(z)\\
N(z)^{-1}\,\bar zz&=e'\\
N(z)^{-1}\,\bar z&=z^{-1}
\end{align*}
donc $z$ est inversible.
\end{proof}
\section{Le corps $\mathbbm R$ des nombres r\'eels}
Prenons le cas o\`u $K=\mathbbm R$, corps des nombres r\'eel, et $\alpha=-1$. D'apr\`es le th\'eor\`eme \ref{inv}, l'anneau $\mathbbm R[\sqrt{-1}]$ est un corps commutatif. Il s'appelle corps des nombres complexes et se note $\mathbbm C$. Un nombre complexe est un couple de nombres r\'eels $(x,y)$. Les calculs sur les nombres complexes se font gr\^ace aux \'egalit\'es \ref{c} et \ref{d}.
\begin{propriete}
Dans la pratique on utilise seulement les propri\'et\'es suivantes des nombres complexes
\begin{enumerate}
\item les nombres complexes forment un corps commutatif $\mathbbm C$
\item le corps $\mathbbm R$ des nombres r\'eels est un sous-corps de $\mathbbm C$
\item il existe un nombre complexe $i$ \textnormal{(cette notation remplace la notation $\omega$ utilis\'ee pour les extensions quadratiques g\'en\'erales)}, tel que
\begin{align*}
i^{2}=-1
\end{align*}
\item $\forall(x,y)\in\mathbbm R^{2}$, tout nombre complexe $z$ s'\'ecrit d'une fa\c con et d'une seule, sous la forme
\begin{align*}
z=x+iy
\end{align*}
$x+iy$ est appel\'ee forme alg\'ebrique du nombre complexe $(x,y)$. $x$ est la partie r\'eelle de $z$, et $y$ est sa partie imaginaire.
\end{enumerate}
\end{propriete}
On utilise les notations suivantes
\begin{align*}
x&=Re(z)\\
y&=Im(z)
\end{align*}
Nous pouvons v\'erifier que tout \'el\'ement non nul de $\mathbbm C$ admet un inverse. Soit $z=x+iy$ un nombre complexe. D'apr\`es la d\'efinition \ref{norme}, sa norme s'\'ecrit
\begin{align*}
N(z)&=x^{2}-\alpha y^{2}\\
&=x^{2}+y^{2}
\end{align*}
et d'apr\`es le th\'eor\`eme \ref{inv}, son inverse s'\'ecrit
\begin{align*}
z^{-1}&=N(z)^{-1}\,\bar z\\
&=\dfrac{x}{x^{2}+y^{2}}-i\dfrac{y}{x^{2}+y^{2}}
\end{align*}
le d\'enominateur ne peut s'annuler que si $x=y=0$, c'est \`a dire si $z=0$.
\end{document} |
https://pages.stat.wisc.edu/~callan/notes/motzkin_schroder/motzkin_schroder_notes.tex | wisc.edu | CC-MAIN-2023-06 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2023-06/segments/1674764500158.5/warc/CC-MAIN-20230205000727-20230205030727-00049.warc.gz | 472,186,178 | 12,454 | %&amstex
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\setbox\strutbox\hbox{\vrule height10.2\p@ depth4.2\p@ width\z@}%
\setbox\strutbox@\hbox{\lower.6\normallineskiplimit\vbox{%
\kern-\normallineskiplimit\copy\strutbox}}%
\setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.4\ht\z@
\fi
\normalbaselines\rm\[email protected]\jot3.6\ex@\the\twelvepoint@}
\font@\fourteenrm=cmr10 scaled\magstep2
\font@\fourteenit=cmti10 scaled\magstep2
\font@\fourteensl=cmsl10 scaled\magstep2
\font@\fourteensmc=cmcsc10 scaled\magstep2
\font@\fourteentt=cmtt10 scaled\magstep2
\font@\fourteenbf=cmbx10 scaled\magstep2
\font@\fourteeni=cmmi10 scaled\magstep2
\font@\fourteensy=cmsy10 scaled\magstep2
\font@\fourteenex=cmex10 scaled\magstep2
\font@\fourteenmsa=msam10 scaled\magstep2
\font@\fourteeneufm=eufm10 scaled\magstep2
\font@\fourteenmsb=msbm10 scaled\magstep2
\newtoks\fourteenpoint@
\def\fourteenpoint{\normalbaselineskip15\p@
\abovedisplayskip18\p@ plus4.3\p@ minus12.9\p@
\belowdisplayskip\abovedisplayskip
\abovedisplayshortskip\z@ plus4.3\p@
\belowdisplayshortskip10.1\p@ plus4.3\p@ minus5.8\p@
\textonlyfont@\rm\fourteenrm \textonlyfont@\it\fourteenit
\textonlyfont@\sl\fourteensl \textonlyfont@\bf\fourteenbf
\textonlyfont@\smc\fourteensmc \textonlyfont@\tt\fourteentt
%Erg„nzung des fetten Small-Capitals-Fonts:
%
\ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}%
\let\Big\big \let\bigg\big \let\Bigg\big
\else
\textfont\z@=\fourteenrm \scriptfont\z@=\twelverm \scriptscriptfont\z@=\tenrm
\textfont\@ne=\fourteeni \scriptfont\@ne=\twelvei \scriptscriptfont\@ne=\teni
\textfont\tw@=\fourteensy \scriptfont\tw@=\twelvesy \scriptscriptfont\tw@=\tensy
\textfont\thr@@=\fourteenex \scriptfont\thr@@=\twelveex
\scriptscriptfont\thr@@=\twelveex
\textfont\itfam=\fourteenit \scriptfont\itfam=\twelveit
\scriptscriptfont\itfam=\twelveit
\textfont\bffam=\fourteenbf \scriptfont\bffam=\twelvebf
\scriptscriptfont\bffam=\tenbf
\setbox\strutbox\hbox{\vrule height12.2\p@ depth5\p@ width\z@}%
\setbox\strutbox@\hbox{\lower.72\normallineskiplimit\vbox{%
\kern-\normallineskiplimit\copy\strutbox}}%
\setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.7\ht\z@
\fi
\normalbaselines\rm\[email protected]\jot4.3\ex@\the\fourteenpoint@}
\font@\seventeenrm=cmr10 scaled\magstep3
\font@\seventeenit=cmti10 scaled\magstep3
\font@\seventeensl=cmsl10 scaled\magstep3
\font@\seventeensmc=cmcsc10 scaled\magstep3
\font@\seventeentt=cmtt10 scaled\magstep3
\font@\seventeenbf=cmbx10 scaled\magstep3
\font@\seventeeni=cmmi10 scaled\magstep3
\font@\seventeensy=cmsy10 scaled\magstep3
\font@\seventeenex=cmex10 scaled\magstep3
\font@\seventeenmsa=msam10 scaled\magstep3
\font@\seventeeneufm=eufm10 scaled\magstep3
\font@\seventeenmsb=msbm10 scaled\magstep3
\newtoks\seventeenpoint@
\def\seventeenpoint{\normalbaselineskip18\p@
\abovedisplayskip21.6\p@ plus5.2\p@ minus15.4\p@
\belowdisplayskip\abovedisplayskip
\abovedisplayshortskip\z@ plus5.2\p@
\belowdisplayshortskip12.1\p@ plus5.2\p@ minus7\p@
\textonlyfont@\rm\seventeenrm \textonlyfont@\it\seventeenit
\textonlyfont@\sl\seventeensl \textonlyfont@\bf\seventeenbf
\textonlyfont@\smc\seventeensmc \textonlyfont@\tt\seventeentt
%Erg„nzung des fetten Small-Capitals-Fonts:
%
\ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}%
\let\Big\big \let\bigg\big \let\Bigg\big
\else
\textfont\z@=\seventeenrm \scriptfont\z@=\fourteenrm \scriptscriptfont\z@=\twelverm
\textfont\@ne=\seventeeni \scriptfont\@ne=\fourteeni \scriptscriptfont\@ne=\twelvei
\textfont\tw@=\seventeensy \scriptfont\tw@=\fourteensy \scriptscriptfont\tw@=\twelvesy
\textfont\thr@@=\seventeenex \scriptfont\thr@@=\fourteenex
\scriptscriptfont\thr@@=\fourteenex
\textfont\itfam=\seventeenit \scriptfont\itfam=\fourteenit
\scriptscriptfont\itfam=\fourteenit
\textfont\bffam=\seventeenbf \scriptfont\bffam=\fourteenbf
\scriptscriptfont\bffam=\twelvebf
\setbox\strutbox\hbox{\vrule height14.6\p@ depth6\p@ width\z@}%
\setbox\strutbox@\hbox{\lower.86\normallineskiplimit\vbox{%
\kern-\normallineskiplimit\copy\strutbox}}%
\setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=2\ht\z@
\fi
\normalbaselines\rm\[email protected]\jot5.2\ex@\the\seventeenpoint@}
\catcode`\@=13
\font\Bf=cmbx12
\font\Rm=cmr12
\def\LL{\leavevmode\setbox0=\hbox{L}\hbox to\wd0{\hss\char'40L}}
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\de{\delta}
\def\ep{\varepsilon}
\def\ze{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\vt{\vartheta}
\def\io{\iota}
\def\ka{\kappa}
\def\la{\lambda}
\def\rh{\rho}
\def\si{\sigma}
\def\ta{\tau}
\def\ph{\varphi}
\def\ch{\chi}
\def\ps{\psi}
\def\om{\omega}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\Th{\Theta}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\Ph{\Phi}
\def\Ps{\Psi}
\def\Om{\Omega}
\def\row#1#2#3{#1_{#2},\ldots,#1_{#3}}
\def\rowup#1#2#3{#1^{#2},\ldots,#1^{#3}}
\def\x{\times}
\def\crf{} %used for crossreferencing, Tex should ignore.
\def\rf{} %used for refencing (section-numbers)
\def\rfnew{} %used for new-section numbers
\def\P{{\Bbb P}}
\def\R{{\Bbb R}}
\def\X{{\Cal X}}
\def\C{{\Bbb C}}
\def\Mf{{\Cal Mf}}
\def\FM{{\Cal F\Cal M}}
\def\F{{\Cal F}}
\def\G{{\Cal G}}
\def\V{{\Cal V}}
\def\T{{\Cal T}}
\def\A{{\Cal A}}
\def\N{{\Bbb N}}
\def\Z{{\Bbb Z}}
\def\Q{{\Bbb Q}}
\def\ddt{\left.\tfrac \partial{\partial t}\right\vert_0}
\def\dd#1{\tfrac \partial{\partial #1}}
\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\space\number\day, \number\year}
\def\nmb#1#2{#2} %zum Nummerieren
\def\dfrac#1#2{{\displaystyle{#1\over#2}}}
\def\tfrac#1#2{{\textstyle{#1\over#2}}}
\def\iprod#1#2{\langle#1,#2\rangle}
\def\pder#1#2{\frac{\partial #1}{\partial #2}}
\def\iint{\int\!\!\int}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\supp{\operatorname{supp}}
\def\Df{\operatorname{Df}}
\def\dom{\operatorname{dom}}
\def\Ker{\operatorname{Ker}}
\def\Tr{\operatorname{Tr}}
\def\Res{\operatorname{Res}}
\def\Aut{\operatorname{Aut}}
\def\kgV{\operatorname{kgV}}
\def\ggT{\operatorname{ggT}}
\def\diam{\operatorname{diam}}
\def\Im{\operatorname{Im}}
\def\Re{\operatorname{Re}}
\def\ord{\operatorname{ord}}
\def\rang{\operatorname{rang}}
\def\rng{\operatorname{rng}}
\def\grd{\operatorname{grd}}
\def\inv{\operatorname{inv}}
\def\maj{\operatorname{maj}}
\def\des{\operatorname{des}}
\def\varmaj{\operatorname{\overline{maj}}}
\def\vardes{\operatorname{\overline{des}}}
\def\pvarmaj{\operatorname{\overline{maj}'}}
\def\pmaj{\operatorname{maj'}}
\def\ln{\operatorname{ln}}
\def\der{\operatorname{der}}
\def\Hom{\operatorname{Hom}}
\def\tr{\operatorname{tr}}
\def\Span{\operatorname{Span}}
\def\grad{\operatorname{grad}}
\def\div{\operatorname{div}}
\def\rot{\operatorname{rot}}
\def\Sp{\operatorname{Sp}}
\def\sgn{\operatorname{sgn}}
\def\liml{\lim\limits}
\def\supl{\sup\limits}
\def\bigcupl{\bigcup\limits}
\def\bigcapl{\bigcap\limits}
\def\limsupl{\limsup\limits}
\def\liminfl{\liminf\limits}
\def\intl{\int\limits}
\def\suml{\sum\limits}
\def\maxl{\max\limits}
\def\minl{\min\limits}
\def\prodl{\prod\limits}
\def\tg{\operatorname{tan}}
\def\ctg{\operatorname{cot}}
\def\arctg{\operatorname{arctan}}
\def\arccot{\operatorname{arccot}}
\def\arcctg{\operatorname{arccot}}
\def\tgh{\operatorname{tanh}}
\def\ctgh{\operatorname{coth}}
\def\arcsinh{\operatorname{arcsinh}}
\def\arccosh{\operatorname{arccosh}}
\def\arctgh{\operatorname{arctanh}}
\def\arcctgh{\operatorname{arccoth}}
\def\3{\ss}
\catcode`\@=11
\def\dddot#1{\vbox{\ialign{##\crcr
.\hskip-.5pt.\hskip-.5pt.\crcr\noalign{\kern1.5\p@\nointerlineskip}
$\hfil\displaystyle{#1}\hfil$\crcr}}}
\newif\iftab@\tab@false
\newif\ifvtab@\vtab@false
\def\tab{\bgroup\tab@true\vtab@false\vst@bfalse\Strich@false%
\def\\{\global\hline@@false%
\ifhline@\global\hline@false\global\hline@@true\fi\cr}
\edef\l@{\the\leftskip}\ialign\bgroup\hskip\l@##\hfil&&##\hfil\cr}
\def\endtab{\cr\egroup\egroup}
\def\vtab{\vtop\bgroup\vst@bfalse\vtab@true\tab@true\Strich@false%
\bgroup\def\\{\cr}\ialign\bgroup&##\hfil\cr}
\def\endvtab{\cr\egroup\egroup\egroup}
\def\stab{\[email protected]\null
\bgroup\tab@true\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@
\normalbaselines\offinterlineskip
\openup\spreadmlines@
\edef\l@{\the\leftskip}\ialign
\bgroup\hskip\l@##\hfil&&##\hfil\crcr}
\def\endstab{\crcr\egroup
\egroup}
\newif\ifvst@b\vst@bfalse
\def\vstab{\[email protected]\null
\vtop\bgroup\tab@true\vtab@false\vst@btrue\Strich@true\bgroup\Let@@\vspace@
\normalbaselines\offinterlineskip
\openup\spreadmlines@\bgroup}
\def\endvstab{\crcr\egroup\egroup
\egroup\tab@false\Strich@false}
\newdimen\htstrut@
\[email protected]\p@
\newdimen\htStrut@
\htStrut@12\p@
\newdimen\dpstrut@
\[email protected]\p@
\newdimen\dpStrut@
\[email protected]\p@
\def\openup{\afterassignment\@penup\dimen@=}
\def\@penup{\advance\lineskip\dimen@
\advance\baselineskip\dimen@
\advance\lineskiplimit\dimen@
\divide\dimen@ by2
\advance\htstrut@\dimen@
\advance\htStrut@\dimen@
\advance\dpstrut@\dimen@
\advance\dpStrut@\dimen@}
\def\Let@@{\relax\iffalse{\fi%
\def\\{\global\hline@@false%
\ifhline@\global\hline@false\global\hline@@true\fi\cr}%
\iffalse}\fi}
\def\matrix{\null\,\vcenter\bgroup
\tab@false\vtab@false\vst@bfalse\Strich@false\Let@@\vspace@
\normalbaselines\openup\spreadmlines@\ialign
\bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr
\Mathstrut@\crcr\noalign{\kern-\baselineskip}}
\def\endmatrix{\crcr\Mathstrut@\crcr\noalign{\kern-\baselineskip}\egroup
\egroup\,}
\def\smatrix{\[email protected]\null\,
\vcenter\bgroup\tab@false\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@
\normalbaselines\offinterlineskip
\openup\spreadmlines@\ialign
\bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr}
\def\endsmatrix{\crcr\egroup
\egroup\,\Strich@false}
\newdimen\D@cke
\def\Dicke#1{\global\D@cke#1}
\newtoks\tabs@\tabs@{&}
\newif\ifStrich@\Strich@false
\newif\iff@rst
\def\Stricherr@{\iftab@\ifvtab@\errmessage{\noexpand\s not allowed
here. Use \noexpand\vstab!}%
\else\errmessage{\noexpand\s not allowed here. Use \noexpand\stab!}%
\fi\else\errmessage{\noexpand\s not allowed
here. Use \noexpand\smatrix!}\fi}
\def\format{\ifvst@b\else\crcr\fi\egroup\iffalse{\fi\ifnum`}=0 \fi\format@}
\def\format@#1\\{\def\preamble@{#1}%
\def\Str@chfehlt##1{\ifx##1\s\Stricherr@\fi\ifx##1\\\let\Next\relax%
\else\let\Next\Str@chfehlt\fi\Next}%
\def\c{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
depth\dpstrut@ width\z@}\noexpand\fi%
\ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
\fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}%
\def\r{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
depth\dpstrut@ width\z@}\noexpand\fi%
\ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
\fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi}%
\def\l{\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
depth\dpstrut@ width\z@}\noexpand\fi%
\ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
\fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}%
\def\s{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@%
\fi\the\tabs@\ }%
\def\sa{\ifStrich@\vrule width\D@cke\the\hashtoks@%
\the\tabs@\ %
\fi}%
\def\se{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@\fi}%
\def\cd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
depth\dpstrut@ width\z@}\noexpand\fi%
\ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
\fi$\dsize\m@th\the\hashtoks@$\hfil}%
\def\rd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
depth\dpstrut@ width\z@}\noexpand\fi%
\ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
\fi$\dsize\m@th\the\hashtoks@$}%
\def\ld{\noexpand\ifhline@@\hbox{\vrule height\htStrut@%
depth\dpstrut@ width\z@}\noexpand\fi%
\ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}%
\fi$\dsize\m@th\the\hashtoks@$\hfil}%
\ifStrich@\else\Str@chfehlt#1\\\fi%
\setbox\z@\hbox{\xdef\Preamble@{\preamble@}}\ifnum`{=0 \fi\iffalse}\fi
\ialign\bgroup\span\Preamble@\crcr}
\newif\ifhline@\hline@false
\newif\ifhline@@\hline@@false
\def\hlinefor#1{\multispan@{\strip@#1 }\leaders\hrule height\D@cke\hfill%
\global\hline@true\ignorespaces}
\def\Item "#1"{\par\noindent\hangindent2\parindent%
\hangafter1\setbox0\hbox{\rm#1\enspace}\ifdim\wd0>2\parindent%
\box0\else\hbox to 2\parindent{\rm#1\hfil}\fi\ignorespaces}
\def\ITEM #1"#2"{\par\noindent\hangafter1\hangindent#1%
\setbox0\hbox{\rm#2\enspace}\ifdim\wd0>#1%
\box0\else\hbox to 0pt{\rm#2\hss}\hskip#1\fi\ignorespaces}
\def\item"#1"{\par\noindent\hang%
\setbox0=\hbox{\rm#1\enspace}\ifdim\wd0>\the\parindent%
\box0\else\hbox to \parindent{\rm#1\hfil}\enspace\fi\ignorespaces}
\let\plainitem@\item
\catcode`\@=13
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Input-Datei zum Erzeugen von Gitterpunktwegen.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%entering a path:
% \Pfad(x-coordinate of starting point,y-coordinate of starting point),path
% as 1-2-3-4-word\endPfad
%(1=step in x-direction, 2=step in y-direction, 3=upward diagonal step, 4=downward diagonal step)
%
%entering a dotted path:
% \SPfad(x-coordinate of starting point,y-coordinate of starting point),path
% as 1-2-3-4-word\endSPfad
%(1=step in x-direction, 2=step in y-direction, 3=upward diagonal step, 4=downward diagonal step)
%
%coordinate-axes:
% \Koordinatenachsen(length of positive x-axes, length of positive y-axes)(length of negative x-axes, length of negative y-axes)
%The length of negative axes are entered as negative numbers.
%
%lattice:
% \Gitter(number of points in positive x-direction, number of points in positive y-direction)(number of points in negative x-direction, number of points in negative y-direction)
%
%diagonal lines:
% \Diagonale(x-coordinate of SW-most point,y-coordinate of SW-most point)length of the projection on the x-axes
%
%antidiagonal lines:
% \AntiDiagonale(x-coordinate of NW-most point,y-coordinate of NW-most point)length of the projection on the x-axes
%
%vectors:
% \Vektor(x-coordinate of incline, y-coordinate of incline)length(x-coordinate of starting point, y-coordinate of starting point)
%
%labelling of points:
% \Label[location?]{[label]}(x-coordinate,y-coordinate)
%where:
% [location?]=\l,\lo,\lu,\r,\ro,\ru,\o,\u
%and l=left, r=right, u=bottom, o=top.
%In addition, if by \Einheit?cm the basic unit is changed, there exist
%\llo,\loo,\llu,\luu,\rro,\roo,\rru,\ruu.
%
%The basic unit can be changed by entering
% \Einheit=?cm
%The default is \Einheit=0.5cm.
%
%The thickness of the paths can be changed by entering
% \PfadDicke{?cm}
%The default is \PfadDicke=1pt.
%
%The following point sizes are available:
%\DuennPunkt, \NormalPunkt, \DickPunkt. Syntax:
% \DickPunkt(x-coordinate,y-coordinate), etc.
%
%Besides, a circle is available by \Kreis. Syntax:
% \Kreis(x-coordinate,y-coordinate)
%
\catcode`\@=11
\font\tenln = line10
\font\tenlnw = linew10
\newskip\Einheit \Einheit=0.5cm
\newcount\xcoord \newcount\ycoord
\newdimen\xdim \newdimen\ydim \newdimen\PfadD@cke \newdimen\Pfadd@cke
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%LaTeX counters, dimensions, variables for lines%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcount\@tempcnta
\newcount\@tempcntb
\newdimen\@tempdima
\newdimen\@tempdimb
\newdimen\@wholewidth
\newdimen\@halfwidth
\newcount\@xarg
\newcount\@yarg
\newcount\@yyarg
\newbox\@linechar
\newbox\@tempboxa
\newdimen\@linelen
\newdimen\@clnwd
\newdimen\@clnht
\newif\if@negarg
\def\@whilenoop#1{}
\def\@whiledim#1\do #2{\ifdim #1\relax#2\@iwhiledim{#1\relax#2}\fi}
\def\@iwhiledim#1{\ifdim #1\let\@nextwhile=\@iwhiledim
\else\let\@nextwhile=\@whilenoop\fi\@nextwhile{#1}}
\def\@whileswnoop#1\fi{}
\def\@whilesw#1\fi#2{#1#2\@iwhilesw{#1#2}\fi\fi}
\def\@iwhilesw#1\fi{#1\let\@nextwhile=\@iwhilesw
\else\let\@nextwhile=\@whileswnoop\fi\@nextwhile{#1}\fi}
\def\thinlines{\let\@linefnt\tenln \let\@circlefnt\tencirc
\@wholewidth\fontdimen8\tenln \@halfwidth .5\@wholewidth}
\def\thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw
\@wholewidth\fontdimen8\tenlnw \@halfwidth .5\@wholewidth}
\thinlines
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\PfadD@cke1pt \[email protected]
\def\PfadDicke#1{\PfadD@cke#1 \divide\PfadD@cke by2 \Pfadd@cke\PfadD@cke \multiply\PfadD@cke by2}
\long\def\LOOP#1\REPEAT{\def\BODY{#1}\ITERATE}
\def\ITERATE{\BODY \let\next\ITERATE \else\let\next\relax\fi \next}
\let\REPEAT=\fi
\def\Punkt{\hbox{\raise-2pt\hbox to0pt{\hss$\ssize\bullet$\hss}}}
\def\DuennPunkt(#1,#2){\unskip
\raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
\raise-2.5pt\hbox to0pt{\hss$\bullet$\hss}\hss}}
\def\NormalPunkt(#1,#2){\unskip
\raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
\raise-3pt\hbox to0pt{\hss\twelvepoint$\bullet$\hss}\hss}}
\def\DickPunkt(#1,#2){\unskip
\raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
\raise-4pt\hbox to0pt{\hss\fourteenpoint$\bullet$\hss}\hss}}
\def\Kreis(#1,#2){\unskip
\raise#2 \Einheit\hbox to0pt{\hskip#1 \Einheit
\raise-4pt\hbox to0pt{\hss\fourteenpoint$\circ$\hss}\hss}}
%%%%%%%%%%%%%%%%%%%%%
%LaTeX line macros%
%%%%%%%%%%%%%%%%%%%%%
\def\Line@(#1,#2)#3{\@xarg #1\relax \@yarg #2\relax
\@linelen=#3\Einheit
\ifnum\@xarg =0 \@vline
\else \ifnum\@yarg =0 \@hline \else \@sline\fi
\fi}
\def\@sline{\ifnum\@xarg< 0 \@negargtrue \@xarg -\@xarg \@yyarg -\@yarg
\else \@negargfalse \@yyarg \@yarg \fi
\ifnum \@yyarg >0 \@tempcnta\@yyarg \else \@tempcnta -\@yyarg \fi
\ifnum\@tempcnta>6 \@badlinearg\@tempcnta0 \fi
\ifnum\@xarg>6 \@badlinearg\@xarg 1 \fi
\setbox\@linechar\hbox{\@linefnt\@getlinechar(\@xarg,\@yyarg)}%
\ifnum \@yarg >0 \let\@upordown\raise \@clnht\z@
\else\let\@upordown\lower \@clnht \ht\@linechar\fi
\@clnwd=\wd\@linechar
\if@negarg \hskip -\wd\@linechar \def\@tempa{\hskip -2\wd\@linechar}\else
\let\@tempa\relax \fi
\@whiledim \@clnwd <\@linelen \do
{\@upordown\@clnht\copy\@linechar
\@tempa
\advance\@clnht \ht\@linechar
\advance\@clnwd \wd\@linechar}%
\advance\@clnht -\ht\@linechar
\advance\@clnwd -\wd\@linechar
\@tempdima\@linelen\advance\@tempdima -\@clnwd
\@tempdimb\@tempdima\advance\@tempdimb -\wd\@linechar
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\catcode`\@=13
\def\aigner{1}
\def\callan{2}
\def\donaghey76{3}
\def\gkp{4}
\def\rogers{5}
\def\sloane{6}
\def\sch{Schr\"{o}der }
\vbox{ }
\vskip-20pt
\topmatter
\title Notes on Motzkin and Schr\"{o}der Numbers
\endtitle
\author David Callan
\endauthor
\affil
Department of Statistics \\
University of Wisconsin-Madison \\
1210 W. Dayton Street \\
Madison, WI \ 53706-1693 \\
{\bf callan\@stat.wisc.edu}
\endaffil
\endtopmatter
\document
\head 1. Motzkin numbers \endhead
%\Section{Introduction}
The Motzkin number $M_{n}$ is the cardinality of the set of
sequences $\Cal{M}_{n}=
\{(x_{i})_{i=1}^{n}:x_{i}\in\{-1,0,1\},$ all partial sums
$\sum_{i=1}^{k}x_{i}$ are nonnegative and $\sum_{i=1}^{n}x_{i}=0\}$.
These sequences have a
pictorial representation as {\it Motzkin paths\/}: lattice paths of
upsteps (corresponding to +1), downsteps ($-1$) and flatsteps (0)
that begin and end at, but never dip below,
``ground level''.
For example, the $M_{3}=4$ Motzkin 3-paths are pictured in Figure 1.
\vskip10pt
\vbox{
$$
\Pfad(-9,0),111\endPfad
\Pfad(-4,0),134\endPfad
\Pfad(1,0),341\endPfad
\Pfad(6,0),314\endPfad
\DuennPunkt(-9,0)
\DuennPunkt(-8,0)
\DuennPunkt(-7,0)
\DuennPunkt(-6,0)
\DuennPunkt(-4,0)
\DuennPunkt(-3,0)
\DuennPunkt(-2,1)
\DuennPunkt(-1,0)
\DuennPunkt(1,0)
\DuennPunkt(2,1)
\DuennPunkt(3,0)
\DuennPunkt(4,0)
\DuennPunkt(6,0)
\DuennPunkt(7,1)
\DuennPunkt(8,1)
\DuennPunkt(9,0)
$$
}
\centerline{\eightpoint Figure 1}
\vskip10pt
\noindent If flatsteps are disallowed, the
resulting paths, known as Dyck paths (``mountain ranges''), are counted by the
Catalan number $C_{n}$ \cite{\callan,\gkp} where $n$ now denotes the number of upsteps (=
number of downsteps).
The sequence of Motzkin numbers
$(M_{n})_{n\geq 0}=\{1,1,2,4,9,21,51,\ldots\}$ is
logarithmically convex, that is,
$M_{n}^{2} \leq M_{n-1}M_{n+1},\ n\geq 1$ \cite{\aigner}. Our first
note is
a combinatorial proof of that fact: an injection
$\Cal{M}_{n} \times \Cal{M}_{n} \rightarrow \Cal{M}_{n-1} \times
\Cal{M}_{n+1}.$ Start with a pair of Motzkin $n$-paths, the second
placed so that it begins one unit to the right of the first.
Scan the paths left to right and locate the first ``close encounter''
defined as a point of intersection---either at a lattice point as in
Figure 2a or at the
center point of crossing diagonal steps as in Figure 2b---or a pair of
flatsteps forming the
top and bottom of a unit square as in Figure 2c.
\vskip10pt
\vbox{
$$
\Pfad(-11,0),33144\endPfad
\SPfad(-10,0),11314\endSPfad
\Pfad(-3,0),33144\endPfad
\SPfad(-2,0),13344\endSPfad
\Pfad(5,0),33144\endPfad
\SPfad(6,0),31141\endSPfad
\DuennPunkt(-11,0)
\DuennPunkt(-10,1)
\DuennPunkt(-9,2)
\DuennPunkt(-8,2)
\DuennPunkt(-7,1)
\DuennPunkt(-10,0)
\DuennPunkt(-9,0)
\DuennPunkt(-8,0)
\DuennPunkt(-5,0)
\DuennPunkt(-6,0)
\DuennPunkt(-6,1)
\DuennPunkt(-3,0)
\DuennPunkt(-2,1)
\DuennPunkt(-1,2)
\DuennPunkt(-2,0)
\DuennPunkt(-1,0)
\DuennPunkt(0,1)
\DuennPunkt(0,2)
\DuennPunkt(1,1)
\DuennPunkt(2,0)
\DuennPunkt(1,2)
\DuennPunkt(2,1)
\DuennPunkt(3,0)
\DuennPunkt(5,0)
\DuennPunkt(6,0)
\DuennPunkt(6,1)
\DuennPunkt(7,2)
\DuennPunkt(7,1)
\DuennPunkt(8,1)
\DuennPunkt(8,2)
\DuennPunkt(9,1)
\DuennPunkt(10,0)
\DuennPunkt(11,0)
$$
}
\centerline{\eightpoint a \qquad \qquad \qquad\qquad\qquad \qquad\ b \qquad
\qquad\qquad\qquad\qquad\qquad \quad c}
\centerline{\eightpoint Figure 2}
\vskip10pt
Certainly at least one such close
encounter exists. In the situation of Figure 2a, reassign the two initial
segments to the other path. In that of Figure 2b, swing the crossing steps
$45^{\circ}$ so they become horizontal. In that of Figure 2c, change the lower
horizontal step to an upstep and the upper one to a downstep. The
pairs in Figure 2 thus yield respectively the pairs in Figure 3.
\vskip10pt
\vbox{
$$
\Pfad(-11,0),331414\endPfad
\SPfad(-10,0),1134\endSPfad
\Pfad(-3,0),331144\endPfad
\SPfad(-2,0),1314\endSPfad
\Pfad(5,0),334141\endPfad
\SPfad(6,0),3344\endSPfad
\DuennPunkt(-11,0)
\DuennPunkt(-10,1)
\DuennPunkt(-9,2)
\DuennPunkt(-8,2)
\DuennPunkt(-7,1)
\DuennPunkt(-10,0)
\DuennPunkt(-9,0)
\DuennPunkt(-8,0)
\DuennPunkt(-5,0)
\DuennPunkt(-6,0)
\DuennPunkt(-6,1)
\DuennPunkt(-3,0)
\DuennPunkt(-2,1)
\DuennPunkt(-1,2)
\DuennPunkt(-2,0)
\DuennPunkt(-1,0)
\DuennPunkt(0,1)
\DuennPunkt(0,2)
\DuennPunkt(1,1)
\DuennPunkt(2,0)
\DuennPunkt(1,2)
\DuennPunkt(2,1)
\DuennPunkt(3,0)
\DuennPunkt(5,0)
\DuennPunkt(6,0)
\DuennPunkt(6,1)
\DuennPunkt(7,2)
\DuennPunkt(7,1)
\DuennPunkt(8,1)
\DuennPunkt(8,2)
\DuennPunkt(9,1)
\DuennPunkt(10,0)
\DuennPunkt(11,0)
$$
}
\centerline{\eightpoint a \qquad \qquad \qquad\qquad\qquad \qquad\ b \qquad
\qquad\qquad\qquad\qquad\qquad \quad c}
\centerline{\eightpoint Figure 3}
\vskip10pt
In all cases, the result will be a pair of paths, one in
$\Cal{M}_{n-1}$, the other in $\Cal{M}_{n+1}$. Furthermore, the
location of the first close encounter will remain invariant; so the
mapping is reversible and hence an injection.
The only elements of $\Cal{M}_{n-1} \times
\Cal{M}_{n+1}$ not hit will be the pairs of nonintersecting paths
having no flatsteps as sides of a unit square. Such pairs exist
except for $n=2$; this proves the desired inequality with equality
only for $n=2.$
\head 2. Bijection from bushes to Motzkin paths \endhead
Just as for the Catalan numbers, there are numerous combinatorial
manifestations of the Motzkin numbers besides lattice paths (see
\cite{\donaghey76} for a comprehensive survey). We'll consider one
related to trees. First we recall that there are several
easily-grasped bijections
from tree-like structures to lattice paths that involve ``walking
around the tree''. For example, rooted ordered trees with
$n$ edges
correspond to Dyck paths of $n$ upsteps (and $n$ downsteps) as follows.
Draw the tree up from the root.
Then walk around the tree clockwise starting at the root, closely
following the edges. Thus each edge gets traversed twice in opposite
directions. Simply record an upstep when you travel up an edge and a downstep
when you travel down an edge.
As another example, full binary trees with $n$ interior vertices
(and hence $2n$ edges and $n+1$ leaf vertices) also correspond to Dyck
paths of $n$ upsteps. Again walk around the tree but this time processing in turn
each edge {\it that has not previously been traversed\/}: a
left-leaning edge becomes an upstep and a right-leaning edge becomes a
downstep. Note that in the former case each edge corresponds to {\it two}
steps in the lattice path---an upstep
{\it and} a downstep. In the latter case, each edge corresponds to
only one step and we rely on the ``full binary'' property of the tree
to ensure equal numbers of upsteps and downsteps.
To return to Motzkin numbers,
a {\it bush} or {\it branch-reduced tree} is a
rooted, ordered tree in which only the root is allowed exactly one
child; in other words, each non-root vertex is either a leaf or has at
least 2 children. The 4 bushes with 4 edges are shown in Figure 4
(drawn downward).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% (Following current practice, we draw trees downward because the %
% analogy with family trees is deemed stronger than that with nature's trees. %
% Still, it's convenient to retain some terminology from the latter.) %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip10pt
\vbox{
$$
\Pfad(-4,0),344\endPfad
\Pfad(-3,-1),3\endPfad
\Pfad(1,-1),34\endPfad
\Pfad(2,0),34\endPfad
\Pfad(7,-1),22\endPfad
\Pfad(6,-1),34\endPfad
\DuennPunkt(-9,0)
\DuennPunkt(-8,0)
\DuennPunkt(-7.5,1)
\DuennPunkt(-7,0)
\DuennPunkt(-6,0)
\DuennPunkt(-4,0)
\DuennPunkt(-3,1)
\DuennPunkt(-3,-1)
\DuennPunkt(-2,0)
\DuennPunkt(-1,-1)
\DuennPunkt(1,-1)
\DuennPunkt(2,0)
\DuennPunkt(3,1)
\DuennPunkt(3,-1)
\DuennPunkt(4,0)
\DuennPunkt(7,0)
\DuennPunkt(7,1)
\DuennPunkt(6,-1)
\DuennPunkt(7,-1)
\DuennPunkt(8,-1)
$$
}
\centerline{\eightpoint Figure 4}
\vskip10pt
\noindent Let $\Cal{B}_{n}$ denote the set of bushes with $n$ edges.
It is shown in \cite{\donaghey76} that $\vert \Cal{B}_{n} \vert =
M_{n-1}.$ For our second note, we will confirm this by exhibiting
another walk-around
bijection, from $\Cal{B}_{n}$ to $\Cal{M}_{n-1}$, that incorporates
features of both the preceding bijections.
Start with a
bush in $\Cal{B}_{n}$. Walk around it counterclockwise as usual.
Process in turn each edge that has not
previously been traversed as follows: an edge incident to the root
becomes a flatstep, otherwise a leftmost edge (from its parent vertex) becomes
an upstep, a rightmost edge becomes a downstep, and an interior edge
becomes a flatstep.
Finally, delete the initial step (necessarily a flatstep).
For example, the bush in Figure 5a (with
edges numbered in the order they are first encountered) corresponds to
the Motzkin path $AB$ in Figure 5b. Also, the 4 bushes in Figure 4
correspond respectively to the 4 Motzkin paths in Figure 1.
%\newpage
\vskip10pt
\vbox{
$$
\Einheit=0.8cm.
\Pfad(-5,-1),334\endPfad
\Pfad(-4,-1),24\endPfad
\SPfad(2,0),1\endSPfad
\Pfad(3,0),3141\endPfad
\Label\u{\eightpoint A}(3,0)
\Label\u{\eightpoint B}(7,0)
\Label\o{\longrightarrow}(0,-0.3)
\Label\lo{\eightpoint 1}(-3.3,0.0)
\Label\lo{\eightpoint 2}(-4.3,-1.0)
\Label\l{\eightpoint 3}(-3.8,-0.6)
\Label\l{\eightpoint 4}(-3.3,-0.6)
\Label\l{\eightpoint 5}(-2.4,0.4)
\DuennPunkt(-5,-1)
\DuennPunkt(-4,-1)
\DuennPunkt(-4,0)
\DuennPunkt(-3,-1)
\DuennPunkt(-3,1)
\DuennPunkt(-2,0)
%
\DuennPunkt(2,0)
\DuennPunkt(3,0)
\DuennPunkt(4,1)
\DuennPunkt(5,1)
\DuennPunkt(6,0)
\DuennPunkt(7,0)
$$
}
\centerline{\eightpoint \qquad \qquad a \qquad\qquad\qquad \qquad\ \qquad
\qquad\qquad\qquad\qquad\qquad \quad b}
\centerline{\eightpoint Figure 5}
\vskip10pt
The inverse mapping is somewhat trickier to describe. A little
terminology for Motzkin paths is helpful. We distinguish between {\it
ground-level} and {\it raised} flatsteps. Each downstep has an {\it
associated} upstep: head straight west from the downstep until you
encounter an upstep. Similarly, each raised flatstep has an associated
upstep: start just beneath the flatstep and head west. Thus in Figure
6, the downsteps 5 and 7 are associated respectively to upsteps 4 and
2 while the raised flatsteps 3 and 6 are both associated to upstep 2.
(Nothing is associated to the ground-level flatsteps 1 and 8.)
\vskip10pt
\vbox{
$$
\Einheit=0.8cm.
\Pfad(-4,-2),13134141\endPfad
\Label\o{\eightpoint 1}(-3.5,-2.1)
\Label\o{\eightpoint 2}(-2.7,-1.8)
\Label\o{\eightpoint 3}(-1.5,-1.1)
\Label\o{\eightpoint 4}(-0.7,-0.8)
\Label\o{\eightpoint 5}(0.3,-0.9)
\Label\o{\eightpoint 6}(1.5,-1.1)
\Label\o{\eightpoint 7}(2.3,-1.9)
\Label\o{\eightpoint 8}(3.5,-2.1)
\DuennPunkt(-4,-2)
\DuennPunkt(-3,-2)
\DuennPunkt(-2,-1)
\DuennPunkt(-1,-1)
\DuennPunkt(0,0)
\DuennPunkt(1,-1)
\DuennPunkt(2,-1)
\DuennPunkt(3,-2)
\DuennPunkt(4,-2)
$$
}
\centerline{\eightpoint Figure 6}
\vskip10pt
Now given a Motzkin path, prepend a (ground-level) flatstep and turn
each step of the path into an edge of a bush, working from left to
right, as follows. Each new
edge will be joined to an existing vertex (parent) thus introducing a
new vertex (child) which will always be placed to the right of any
existing children of the parent. First create the root. Proceeding, a
ground-level flatstep becomes an edge from the root. An upstep becomes an
edge from the vertex introduced by the previous step (and this is the
only way a {\it first} child of a non-root parent arises). Finally, a
raised flatstep or a downstep introduces a (right) sibling to the
vertex introduced by its associated upstep. Since upsteps and
downsteps come in associated pairs, this ensures that each non-root parent
vertex will have at least 2 children. It is not hard to see that
these mappings are indeed inverses of one another.
\head 3. \sch numbers and royal paths \endhead
Counting bushes by number of leaf vertices (rather than by number of
edges) yields the \sch numbers \cite{\rogers}: $S_{n}$ is the number
of bushes with
$n+1$ leafs. Under the bijection of the preceding section, a leaf vertex
in a bush corresponds to a path vertex from which an upstep
does {\it not} emanate. Noting that the terminal point of a path is
always such a vertex, we conclude that $S_{n}$ is the number of Motzkin
paths with a total of $n$ flatsteps and downsteps. Flipping Motzkin
paths across ground level and rotating $45^{\circ}$ counterclockwise,
we get the more common interpretation of $S_{n}$ as the number of lattice
paths from $(0,0)$ to $(n,n)$ consisting of steps $(1,0), (0,1), (1,1) $
that never rise above the line $y=x$ (called {\it royal} paths
in \cite{\sloane}). The $S_{2}=6$ bushes with 3 leaf vertices
are given below their corresponding royal paths in Figure 7.
\vskip5pt
\vbox{
$$
\Einheit=0.4cm.
%\Einheit=0.5cm.
\Pfad(-13,-3),33\endPfad
\Pfad(-9,-3),123\endPfad
\Pfad(-4,-3),312\endPfad
\Pfad(1,-3),132\endPfad
\Pfad(5,-3),1122\endPfad
\Pfad(10,-3),1212\endPfad
\DuennPunkt(-13,-3)
\DuennPunkt(-12,-2)
\DuennPunkt(-11,-1)
\DuennPunkt(-9,-3)
\DuennPunkt(-8,-3)
\DuennPunkt(-8,-2)
\DuennPunkt(-7,-1)
\DuennPunkt(-4,-3)
\DuennPunkt(-3,-2)
\DuennPunkt(-2,-2)
\DuennPunkt(-2,-1)
\DuennPunkt(1,-3)
\DuennPunkt(2,-3)
\DuennPunkt(3,-2)
\DuennPunkt(3,-1)
\DuennPunkt(5,-3)
\DuennPunkt(6,-3)
\DuennPunkt(7,-3)
\DuennPunkt(10,-3)
\DuennPunkt(11,-3)
\DuennPunkt(7,-2)
\DuennPunkt(11,-2)
\DuennPunkt(12,-2)
\DuennPunkt(7,-1)
\DuennPunkt(12,-1)
$$
}
\vskip5pt
\vbox{
$$
\Einheit=0.4cm.
%\Einheit=0.5cm.
\Pfad(-13,-1),34\endPfad
\Pfad(-12,-1),2\endPfad
\Pfad(-9,-2),34\endPfad
\Pfad(-8,-1),34\endPfad
\Pfad(-4,-1),34\endPfad
\Pfad(-3,-2),34\endPfad
\Pfad(1,-2),34\endPfad
\Pfad(2,-2),22\endPfad
\Pfad(5,-3),34\endPfad
\Pfad(6,-2),34\endPfad
\Pfad(7,-1),2\endPfad
\Pfad(11,-3),34\endPfad
\Pfad(10,-2),34\endPfad
\Pfad(11,-1),2\endPfad
\DuennPunkt(-13,-1)
\DuennPunkt(-12,-1)
\DuennPunkt(-12,0)
\DuennPunkt(-11,-1)
\DuennPunkt(-9,-2)
\DuennPunkt(-8,-1)
\DuennPunkt(-7,-2)
\DuennPunkt(-7,0)
\DuennPunkt(-6,-1)
\DuennPunkt(-4,-1)
\DuennPunkt(-3,-2)
\DuennPunkt(-3,0)
\DuennPunkt(-2,-1)
\DuennPunkt(-1,-2)
\DuennPunkt(1,-2)
\DuennPunkt(2,-1)
\DuennPunkt(2,0)
\DuennPunkt(2,-2)
\DuennPunkt(3,-2)
\DuennPunkt(5,-3)
\DuennPunkt(6,-2)
\DuennPunkt(7,0)
\DuennPunkt(7,-1)
\DuennPunkt(7,-3)
\DuennPunkt(8,-2)
\DuennPunkt(10,-2)
\DuennPunkt(11,0)
\DuennPunkt(11,-1)
\DuennPunkt(11,-3)
\DuennPunkt(12,-2)
\DuennPunkt(13,-3)
$$
}
\centerline{\eightpoint Figure 7}
\vskip10pt
The \sch number sequence $(S_{n})_{n \geq 0}$ begins
$1,\,2,\,6,\,22,\,90,\,394,\ldots$ and $S_{n}$ is even for $n\geq 1.$
The ``bush'' interpretation of \sch numbers makes this obvious: there
are as many planted (i.e. root has degree one) bushes
on $n$ leaf vertices as non-planted ones (simply remove
the root edge from the planted bushes). This observation
has the following translation to royal paths:
the $S_{n}$ royal paths from $(0,0)$ to $(n,n)$ split into 2
equal-size classes according as they possess a diagonal step on
the line $y=x$
or not. For example, the first 3 royal paths in Figure 7 do so and the
second 3 don't. A concluding exercise: find a bijection between these equal-size
classes. (Hint. Look for the {\it last} diagonal step on the line $y=x$,
and failing that, look for the {\it first} return to the line $y=x$.)
\Refs
\ref\no \aigner\by M. Aigner\paper Motzkin numbers\jour
Europ. J.~Combinatorics\vol 19\yr 1998\pages 663--675\endref
\ref\no \callan\by D.~Callan\paper Pair them up!: A visual approach to the
Chung-Feller theorem\jour
Coll. Math. J.\vol 26\yr 1995\pages 196--198\endref
\ref\no \donaghey76\by R.~Donaghey and L.~W.~Shapiro\paper Motzkin
numbers\jour J.~Comb. Th.~A\vol 23 \yr 1977\pages 291--301\endref
\ref\no \gkp \by R.~Graham, D.~Knuth, O.~Patashnik \book Concrete
Mathematics \publ Addison-Wesley\publaddr
New York\yr 1994\endref
\ref\no \rogers\by D.G.~Rogers and L.~W.~Shapiro\paper Some
correspondences involving the \sch numbers and relations
\book Combinatorial Mathematics, Lecture Notes in Mathematics
\vol686\publ Springer-Verlag\publaddr
New York \yr 1978\pages 267--274\endref
\ref\no \sloane \by Sloane and Plouffe \book The Encyclopedia of
Integer Sequences \publ Academic Press\publaddr
New York\yr 1995\endref
\endRefs
\enddocument
|
http://porocila.imfm.si/2017/mat/clani/kosir.tex | imfm.si | CC-MAIN-2020-29 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-29/segments/1593655897844.44/warc/CC-MAIN-20200709002952-20200709032952-00591.warc.gz | 107,358,366 | 2,531 | \clan {Tomaž Košir}
%--------------------------------------------------------
% A. objavljene znanstvene monografije
%--------------------------------------------------------
%\begin{skupina}{A}
%\disertacija
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\magisterij
% {NASLOV}
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% {FAKULTETA}
% {ODDELEK}
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%\monografija
% {AVTORJI}
% {NASLOV}
% {ZALOZBA}
% {KRAJ} {DRZAVA} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
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% B. raziskovalni clanki sprejeti v objavo v znanstvenih
% revijah in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{B}
%\sprejetoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA}
%\sprejetoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
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% C. raziskovalni clanki objavljeni v znanstvenih revijah
% in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{C}
%\objavljenoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\objavljenoZbornik
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% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
\begin{skupina}{C}
\objavljenoRevija
% 1.01: {\bf 1}. KO\v{S}IR, Toma\v{z}, OMLADI\v{C}, Matja\v{z}. Normalized tight vs. general frames in sampling problems. {\it Adv.\ Oper.\ Theory}, 2017, vol. 2, iss. 2, str. 114-125. $[$COBISS.SI-ID 17977177$]$\\
{\crta, M.~Omladi\v{c}}
{Normalized tight vs. general frames in sampling problems}
{Adv.\ Oper.\ Theory} {2} {2017} {} {1{1}4--125}
\end{skupina}
%--------------------------------------------------------
% D. urednistvo v znanstvenih revijah in zbornikih
% znanstvenih konferenc
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%\begin{skupina}{D}
%\urednikRevija
% {OPIS}
% {REVIJA}
%\urednikZbornik
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% E. organizacija mednarodnih in domacih znanstvenih
% srecanj
%--------------------------------------------------------
%\begin{skupina}{E}
%\organizacija
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
\begin{skupina}{E}
\organizacija
{Član organizacijskega odbora}
{8\uth Linear Algebra Workshop}
{Ljubljana} {} {junij} {2017}
\end{skupina}
%--------------------------------------------------------
% F. vabljena predavanja na tujih ustanovah in
% mednarodnih konferencah
%--------------------------------------------------------
%\begin{skupina}{F}
%\predavanjeUstanova
% {NASLOV}
% {OPIS}
% {USTANOVA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\predavanjeKonferenca
% {NASLOV}
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% G. aktivne udelezbe na mednarodnih in domacih
% konferencah
%--------------------------------------------------------
%\begin{skupina}{G}
%\konferenca
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% H. strokovni clanki
%--------------------------------------------------------
%\begin{skupina}{H}
%\clanekRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\clanekZbornik
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% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% I. razno
%--------------------------------------------------------
\begin{skupina}{I}
\razno
{Mentorstvo pri 9 magistrskih delih in enem delu diplomskega seminarja}
\end{skupina}
%\begin{skupina}{I}
%POZOR: Bibliografija2017.tex > 2017\mat\clani\kosir.tex 4637/192: Stevilo neopredeljenih zadetkov: 10
%\razno
% Ured: {\bf 2}. {\it Rendiconti dell'Istituto di Matematica dell'Universit\`{a} di Trieste}. Ko\v{s}ir, Toma\v{z} (\v{c}lan uredni\v{s}kega odbora 2008-). Trieste: Istituto di matematica. Universit\`{a} di Trieste, 1969-. ISSN 0049-4704. $[$COBISS.SI-ID 26269952$]$\\
%\razno
% Ment: {\bf 3}. \v{S}EGA, Tadej{\it . Analiza modela stresnih testov : magistrsko delo}. Ljubljana: $[$T. \v{S}ega$]$, 2017. XI, 53 str., ilustr. $[$COBISS.SI-ID 18215513$]$\\
%\razno
% Ment: %list {\bf 4}. ER\v{Z}EN, Miha{\it . Generator ekonomskih scenarijev : magistrsko delo}. Ljubljana: $[$M. Er\v{z}en$]$, 2017. 61 f., ilustr. $[$COBISS.SI-ID 18148441$]$\\
%\razno
% Ment: %list {\bf 5}. MAR\v{S}I\v{C}, Andrej{\it . Heston model derivation, discretisation and calibration in R : magistrsko delo}. Ljubljana: $[$A. Mar\v{s}i\v{c}$]$, 2017. 76 str., ilustr. $[$COBISS.SI-ID 18127961$]$\\
%\razno
% Ment: %list {\bf 6}. LE\v{C}NIK, Jure{\it . Napoved urne porabe elektri\v{c}ne energije za dan vnaprej z metodami strojnega u\v{c}enja : magistrsko delo}. Ljubljana: $[$J. Le\v{c}nik$]$, 2017. 57 str., ilustr. $[$COBISS.SI-ID 18214233$]$\\
%\razno
% Ment: %list {\bf 7}. VODOPIJA, Aljo\v{s}a{\it . Napovedovanje umrljivosti za manj\v{s}e populacije : magistrsko delo}. Ljubljana: $[$A. Vodopija$]$, 2017. IV, 54 f., ilustr. $[$COBISS.SI-ID 18122073$]$\\
%\razno
% Ment: %list {\bf 8}. HRVATI\v{C}, Anteja{\it . Pregled in analiza ob\v{c}utljivosti izbranih metod za izra\v{c}unavanje \v{s}kodnih rezervacij premo\v{z}enjskih zavarovanj : magistrsko delo}. Ljubljana: $[$A. Hrvati\v{c}$]$, 2017. VII, 75 str., ilustr. $[$COBISS.SI-ID 18106457$]$\\
%\razno
% Ment: %list {\bf 9}. DREN\v{S}EK, Irena{\it . Solventnostne kapitalske zahteve in tr\v{z}no tveganje : magistrsko delo}. Ljubljana: $[$I. Dren\v{s}ek$]$, 2017. 48 str., ilustr. $[$COBISS.SI-ID 18107737$]$\\
%\razno
% Ment: %list {\bf 10}. BOH, Samo{\it . Vrednotenje na OTC trgih : magistrsko delo}. Ljubljana: $[$S. Boh$]$, 2017. XIII, 58 str., ilustr. $[$COBISS.SI-ID 18106969$]$\\
%\razno
% Ment: {\bf 11}. MITROVI\'{C}, Sanja{\it . Geometrija kubi\v{c}nih krivulj : delo diplomskega seminarja}. Ljubljana: $[$S. Mitrovi\'{c}$]$, 2017. 28 str., ilustr. $[$COBISS.SI-ID 18188377$]$\\
%\end{skupina}
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% tuji gosti
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%\gost {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO} {POVABILO}
%\end{seznam}
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% gostovanja
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%\begin{seznam}
%\gostovanje {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{seznam}
|
http://deeke.org/stat2008-prova1.tex | deeke.org | CC-MAIN-2017-47 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2017-47/segments/1510934806736.55/warc/CC-MAIN-20171123050243-20171123070243-00217.warc.gz | 73,189,331 | 1,971 | \documentclass[a4paper,10pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[latin1]{inputenc}
\usepackage[portuges]{babel}
\usepackage{graphics}
\usepackage{makeidx}
\usepackage{graphicx}
\usepackage{color}
\font\tenrm=cmr10 \font\tenit=cmti10 \font\elevenbf=cmbx10
scaled\magstep 1 \font\elevenrm=cmr10 scaled\magstep 1
\font\elevenit=cmti10 scaled\magstep 1 \font\ninebf=cmbx9
\font\ninerm=cmr9 \font\nineit=cmti9 \font\eightbf=cmbx8
\font\eightrm=cmr8 \font\eightit=cmti8 \font\sevenrm=cmr7
\newcommand{\bibit}{\nineit}
\newcommand{\bibbf}{\ninebf}
\topmargin-1cm \oddsidemargin 1cm \textwidth14cm \textheight 23cm
\pagestyle{plain}
\newtheorem{exercise}{}
\date{23 de abril de 2008}
\begin{document}
\thispagestyle{empty}
\begin{center}
\Large{\textsc{Probabilidade e Estatística - C}}\\
\normalsize
{\tenrm Prof. Fernando Deeke Sasse} \\
\tenit {Departamento de Matem\'atica, UDESC - Joinville%
}\\\vspace{0.4cm}
\large{Prova 1 - Probabilidade Básica\\ \small{23 de abril de 2008}}\\
\end{center}
\vspace{0.8cm} \textbf{1.} Amostras de uma peça moldada de
plástico são classificadas com base no acabamento da superfície e
acabamento da borda. Os resultados envolvendo 110 partes
são resumidos como segue: \\ \\
\begin{tabular}{l| c c r}
% \hline
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
& & acabamento da borda & \\ \hline
& & excelente & bom \\
acabamento da superfície & excelente & 77 & 4 \\
& bom & 9 & 20 \\
% \hline
\end{tabular}\\ \\
Denotando por A o evento correspondente a um acabamento de
superfície excelente, e B o evento correspondente a um acabamento
de borda excelente. Se uma parte é selecionada aleatoriamente,
determine: $P(A),\, P(A'),\, P(B)$, $P(A \cup B),\,P(A\cap B)$,
$P(A\cap B')$.\\ \\
\textbf{2}
Um lote de 120 chips semicondutores contém 22 que são defeituosos. \\
(a) Dois chips são selecionados aleatoriamente do lote, sem substituição. Qual
a probabilidade de que o segundo seja defeituoso ? \\
(b) Três chips são selecionados aleatoriamente do lote, sem substituição. Qual
a probabilidade de que todos sejam defeituosos ?\\
(c) Três chips são selecionados aleatoriamente do lote, sem substituição. Qual
a probabilidade de que ao menos um seja defeituoso ? \\ \\
\textbf{3} Entre 5 engenheiros e 7 físicos, deve-se formar uma
comissão de 2 engenheiros e 3 físicos. De quantas maneiras isso
pode ser feito se:\\
(a) Qualquer engenheiro e qualquer físico pode ser selecionado.\\
(b) Um determinado físico deve ser incluído \\
(c) Dois determinados engenheiros não devem ser incluídos. \\ \\
\textbf{4} Um inspetor trabalhando para uma companhia de
manufatura tem uma probabilidade de $99\%$ de identificar
corretamente um item com defeito e $0.5\%$ de chance de
classificar incorretamente um produto bom como defeituoso. A
companhia tem evidências de que sua linha produz $0.9\%$ de ítens
defeituosos.\\
(a) Qual a probabilidade de que um item selecionado para inspeção
seja classificado como defeituoso ?\\
(b) Se um item selecionado aleatoriamente é classificado como
não-defeituoso, qual a \linebreak probabilidade de que ele seja
realmente bom.
\end{document}
|
https://hallcweb.jlab.org/DocDB/0009/000990/002/Beam_in_hcana.tex | jlab.org | CC-MAIN-2021-43 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-43/segments/1634323585281.35/warc/CC-MAIN-20211019202148-20211019232148-00647.warc.gz | 408,638,031 | 4,136 | \documentclass[]{article}
\usepackage{graphicx}
%opening
\title{Beam in HCANA}
\begin{document}
\maketitle
\tableofcontents
\section{Introduction}
For most purposes in Hall C data analysis, the beam apparatus that is used is
THcRasteredBeam and the beam detector is THcRaster. A sample in the replay script
is :
\begin{verbatim}
THaApparatus* beam = new THcRasteredBeam("H.rb", "Rastered Beamline")
\end{verbatim}
This is for when the HallCSpectrometer is defined as "H" ( for SHMS it is custom
to be defined as "P") . The beam object is given the name "H.rb" and
can be used later in physics modules:
\begin{verbatim}
THcReactionPoint* hrp = new THcReactionPoint("H.react", "", "H", "H.rb");
THcExtTarCor* hext = new THcExtTarCor("H.extcor", "", "H", "H.react");
THcPrimaryKine* hkin = new THcPrimaryKine("H.kin", "", "H", "H.rb");
\end{verbatim}
Methods for getting the information on the beam positions and direction that are calculated in {\it THcRaster.cxx} .
\section{Raster and EPICS BPMs}
The Hall C raster consists of two sets of X and Y coils which are labeled in the code as "A" and "B". For each event, the raster voltage for all four coils is sampled in a FADC250. This voltage is proportional to the magnetic
field in the raster coils which is directly related to the deflection of the
beam at the target. The two X (Y) coils are in sync and give enough deflection
so, for 11 GeV beam, the maxiumum deflection at the maximum voltage in the supplies is about 5x5mm ( 5mm is the full width, 2.5mm is the half-width). The
raster voltage only gives the relative deflection compared to the average beam position.
In Hall C, there is presently no measure of the beam position on an
event-by-event basis. The EPICS data is used to give the average beam position with the assumption that the average beam position has a small variation over the course of a run.
The EPICS data is in units of mm and the coordinate system has +X pointing beam right
and +Y pointing up.
The {\it THcRaster.cxx} reads in the raster current data. The {\it Init}
method sets {\it THcTHcRasterRawHit} based on the detector map:
\begin{verbatim}
! HRASTER_ID=18 :: ADC
DETECTOR=18
ROC=1
SLOT=18
12, 1, 1, 0 ! FR-YA
13, 1, 2, 0 ! FR-XA
14, 1, 3, 0 ! FR-YB
15, 1, 4, 0 ! FR-XA
\end{verbatim}
with a similar setup for the SHMS detector map.
The {\it Decode} method reads in the raw ADC values for each event.
The internal variable names
and the names in the output tree are given in Table~\ref{tab:rawname}.
\begin{table} [h]
\begin{center}
\begin{tabular}[]{|c|c|c|} \hline\hline
Variable Name & Tree name & Comment \\ \hline
frxaRawAdc & FRXA\_rawadc & \\ \hline
frxbRawAdc & FRXB\_rawadc & \\ \hline
fryaRawAdc & FRYA\_rawadc & \\ \hline
frybRawAdc & FRYB\_rawadc & \\ \hline
\end{tabular}
\caption{Raster Raw ADC variables}
\label{tab:rawname}
\end{center}
\end{table}
If the EPICS handler is initialized in the replay script, then the "RAW" EPICS X-Y position data for the three BPMS along the girder before the target is read to the variables: BPMXA\_raw, BPMXB\_raw, BPMXC\_raw, BPMXA\_raw, BPMXB\_raw and BPMXC\_raw.
The raw variables are not available in the output tree. One could get them from the EPICS tree if the definition file is setup properly.
The {\it Process} method converts the raw raster ADC and EPICS BPM data into calibrated positions.
The variable raster ADC (listed in Table~\ref{tab:adcname}) is calculated by subtracting an offset from the raw raster ADC. The offsets are parameters : gfrxa\_adc\_zero\_offset, gfrxb\_adc\_zero\_offset, gfrya\_adc\_zero\_offset, gfryb\_adc\_zero\_offset.
\begin{table} [h]
\begin{center}
\begin{tabular}[]{|c|c|c|} \hline\hline
Variable Name & Tree name & Comment \\ \hline
frxaRawAdc & FRXA\_rawadc & \\ \hline
frxbRawAdc & FRXB\_rawadc & \\ \hline
fryaRawAdc & FRYA\_rawadc & \\ \hline
frybRawAdc & FRYB\_rawadc & \\ \hline
\end{tabular}
\caption{Raster ADC variables}
\label{tab:adcname}
\end{center}
\end{table}
The relative raster position is calculated as:
\begin{equation}
\mbox{fXA\_pos} = \frac{\mbox{fXA\_ADC}}{\mbox{fFrXA\_ADCperCM}}*\frac{\mbox{fFrCalMom}}{\mbox{fgpbeam}}
\end{equation}
The parameters are explained in Table~\ref{tab:adcparam}.
\begin{table} [h]
\begin{center}
\begin{tabular}[]{|c|c|c|} \hline\hline
Variable Name & Parameter name & Comment \\ \hline
fFrXA\_ADCperCM & gfrxa\_adcpercm & Scale factor Counts per cm\\ \hline
fFrXB\_ADCperCM & gfrxb\_adcpercm & Scale factor Counts per cm\\ \hline
fFrYA\_ADCperCM & gfrya\_adcpercm & Scale factor Counts per cm\\ \hline
fFrYB\_ADCperCM & gfryb\_adcpercm & Scale factor Counts per cm\\ \hline
fFrCalMom & gfr\_cal\_mom & Beam energy that calibration was done\\ \hline
fgpbeam & gpbeam & Beam energy in kinematics file \\ \hline
\end{tabular}
\caption{Raster Parameters}
\label{tab:adcparam}
\end{center}
\end{table}
The desire was to have the raster position is the EPICS coordinate system (+X beam right
and +Y up). To achieve this the positions for the Y variables had to be multiplied by -1.
Calibration of the raster positions is done by looking at the raster raw ADC histograms.
The lower and upper edges of the raster raw ADC can be found for each raster coil.
The offset is the average of the edges and the scale factor is the difference of the
edges divided by the expected full width of the raster. The size can be checked by comparison
to the carbon hole size which has a diameter of 2mm.
The BPMs positions at the target can be set as parameters which are the average over the entire run or determined from the EPICS data
throughout the run. The parameters are shown in Table~\ref{tab:bpmparam}. It is optional that the parameters are read in. If the code reads in the any of parameters, then the BPM target positions and angles will be all set to the parameter ( if it happens that only some of the parameters are read in then the others are set to zero.). If you are
using the parameter option to set the target BPM position, then it is best to do it in the
{\it standard.kinematics} file since it can change run-to-run. A script has to be
written to calculate the average target positions from the EPICS data.
\begin{table} [h]
\begin{center}
\begin{tabular}[]{|c|c|c|} \hline\hline
Parameter name & Variable Name & Comment \\ \hline
gbeam\_x & fgbeam\_x & Average target X position from EPICS BPMs (+X beam right)\\ \hline
gbeam\_y& fgbeam\_y & Average target Y position from EPICS BPMs (+Y beam up)\\ \hline
gbeam\_xp& fgbeam\_xp & Average target X angle from EPICS BPMs \\ \hline
gbeam\_yp& fgbeam\_yp & Average target Y angle from EPICS BPMs\\ \hline
\end{tabular}
\caption{BPM Parameters}
\label{tab:bpmparam}
\end{center}
\end{table}
The BPMs are calibrated relative to the HARPs. A script has been developed to fit a group
of HARP scans to determine the calibration constants. The script and instructions are
in the CALIBRATION/bpm\_calib subdirectory of hallc\_replay git repository. The HARP coordinate system has +X pointing beam left and +Y up. The BPM position in the HARP
coordinate system is given by
\begin{equation}
\mbox{BPMXA\_pos} = 0.1*(\mbox{fgbpmxa\_slope}*\mbox{BPMXA\_raw} + \mbox{fgbpmxa\_off})
\end{equation}
The 0.1 is to convert the calibration to units of cm. One note is that the slopes for all
the BPMX should be negative to convert the EPICS raw value to the HARP coordinate system. The BPM calibration parameters are given in Table~\ref{tab:bpmcalparam}.
\begin{table} [h]
\begin{center}
\begin{tabular}[]{|c|c|c|} \hline\hline
Parameter name & Variable Name & Comment \\ \hline
gbpmxa\_slope & fgbpmxa\_slope & Slope of HARP fit to BPMXA data\\ \hline
gbpmxb\_slope & fgbpmxb\_slope & Slope of HARP fit to BPMXB data\\ \hline
gbpmya\_slope & fgbpmya\_slope & Slope of HARP fit to BPMYA data\\ \hline
gbpmyb\_slope & fgbpmyb\_slope & Slope of HARP fit to BPMYB data\\ \hline
gbpmxa\_off & fgbpmxa\_off & Intercept of HARP fit to BPMXA data\\ \hline
gbpmxb\_off & fgbpmxb\_off & Intercept of HARP fit to BPMXB data\\ \hline
gbpmya\_off & fgbpmya\_off & Intercept of HARP fit to BPMYA data\\ \hline
gbpmyb\_off & fgbpmyb\_off & Intercept of HARP fit to BPMYB data\\ \hline
\end{tabular}
\caption{BPM Calibration Parameters}
\label{tab:bpmcalparam}
\end{center}
\end{table}
The X and Y BPM position and angles at the target (in the HARP coordinate system) are calculated using the BPMA\_pos and BPMC\_pos positions.
In the code ,these are called xbeam and ybeam and fXbpm\_tar = -xbeam and fYbpm\_tar=ybeam to put it in the EPICs coordinate system. If the parameters gbeam\_x and gbeam\_y are read-in then fXbpm\_tar = gbeam\_x and fYbpm\_tar = gbeam\_y.
The tree names of the BPM positions in the EPICS coordinate system are listed in Table~\ref{tab:bpmtree}.
\begin{table} [h]
\begin{center}
\begin{tabular}[]{|c|c|c|} \hline\hline
Variable name & Tree Name & Comment \\ \hline
fXbpm\_A & fr\_xbpmA & \\ \hline
fYbpm\_A & fr\_ybpmA & \\ \hline
fXbpm\_B & fr\_xbpmB & \\ \hline
fYbpm\_B & fr\_ybpmB & \\ \hline
fXbpm\_C & fr\_xbpmC & \\ \hline
fYbpm\_C & fr\_ybpmC & \\ \hline
fXbpm\_tar & fr\_xbpm\_tar & \\ \hline
fYbpm\_tar & fr\_ybpm\_tar & \\ \hline
\end{tabular}
\caption{BPM positions variables and tree names}
\label{tab:bpmtree}
\end{center}
\end{table}
\section{Target positions}
The predicted Y target position ($Y_{pred}$) measured by the spectrometer depends on the target position along the beam line ($Z_H$),
the horizontal beam position ($X_{H}$), the horizontal mispointing of the spectrometer ($Y_{mis}$), the central angle of the
spectrometer ($\theta_c$) and the scattering angle of the particle relative to the spectrometer coordinate system ($Y'$).
\begin{equation}
Y_{pred} = X_H*(\cos\theta_c-Y'*\sin\theta_c)-Z_H*(\sin\theta_c+Y'*\cos\theta_c)-Y_{mis}
\end{equation}
For scattering into the HMS then $\theta_c$ should have a negative sign. The $X_{H}$ is equal to the negative of P.rb.raster.fr\_xbpm\_tar.
$Y_{mis}$ is taken from spectrometer survey. For the SHMS, $Y_{mis} = -0.06$~cm. For the HMS,
\begin{equation}
Y_{mis} = 0.052 - 0.0012*abs(\theta_c) + 0.0002*\theta_c^2
\end{equation}
where $\theta_c$ is in degrees and $Y_{mis}$ in cm. $+Z_H$ is defined to be pointing beam downstream.
\end{document} |
https://wiki.horde.org/Project/Undo?actionID=export&format=tex | horde.org | CC-MAIN-2021-43 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-43/segments/1634323585507.26/warc/CC-MAIN-20211022114748-20211022144748-00561.warc.gz | 751,913,157 | 1,732 | \documentclass{article}
\usepackage{ulem}
\pagestyle{headings}
\begin{document}
\tableofcontents
Undo support in HordeBugsPeopleDescriptionResources
\part{Undo support in Horde}
\section{Bugs}
https://bugs.horde.org/ticket/11187\footnote{https://bugs.horde.org/ticket/11187}
\section{People}
\section{Description}
Instead of deletion, support undo in Horde applications. This might take the form of a Horde-wide (per-user of course) "trash folder", or some other form of undeletion or revision history.
One suggested approach is to embed this into a larger task to implement journaling in Horde. This would look include these tasks:
\begin{itemize}
\item{Horde\_Journal library with SQL and potentially MongoDB backends (5 hours)}
\item{Journaling for apps (10 hours): Use Horde\_Journaling in Kronolith, Nag, Turba, Mnemo to track additions, changes, deletions.}
\item{Collection (calendars, address books, etc) history (10 hours): Application support (backend and UI) to retrieve and display journal summaries for a groupware collection. Maybe on-demand for dynamic frontends?}
\item{Object history (10 hours): Application support (backend and UI) to retrieve and display complete journal for a groupware object. Maybe on-demand for dynamic frontends?}
\item{Undo (25 hours): Implement undo functionality based on the journaling features for end users.}
\end{itemize}
Time estimations are probably too optimistic, especially for the base journaling functionality.
\section{Resources}
http://www.alistapart.com/articles/neveruseawarning\footnote{http://www.alistapart.com/articles/neveruseawarning}\newline
http://bergie.iki.fi/blog/undeletion\_in\_midgard.html\footnote{http://bergie.iki.fi/blog/undeletion\_in\_midgard.html}\newline
http://paulbuchheit.blogspot.com/2007/06/quick-all-actions-should-have-undo.html\footnote{http://paulbuchheit.blogspot.com/2007/06/quick-all-actions-should-have-undo.html}\newline
http://humanized.com/weblog/2007/09/14/undo-made-easy-with-ajax-part-1/\footnote{http://humanized.com/weblog/2007/09/14/undo-made-easy-with-ajax-part-1/}\newline
http://azarask.in/blog/post/undo\_with\_ajax\_2/\footnote{http://azarask.in/blog/post/undo\_with\_ajax\_2/}\newline
http://ajaxian.com/archives/addressbook-an-example-of-the-form-history-pattern\footnote{http://ajaxian.com/archives/addressbook-an-example-of-the-form-history-pattern}
\noindent\rule{\textwidth}{1pt}
Back to the Project List\footnote{http://example.com/index.php?page=Project}
\end{document}
|
https://www.cip.ifi.lmu.de/~grinberg/algebra/cherednik-errata.tex | lmu.de | CC-MAIN-2020-45 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-45/segments/1603107880878.30/warc/CC-MAIN-20201023073305-20201023103305-00535.warc.gz | 673,532,893 | 21,651 | \documentclass[12pt,final,notitlepage,onecolumn]{article}%
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\textbf{Lecture Notes on Cherednik Algebras}
\textit{Pavel Etingof and Xiaoguang Ma}
\href{https://arxiv.org/abs/1001.0432v4}{arXiv:1001.0432v4 (version 4, 19 Apr
2010)}
\textbf{Errata and questions by Darij Grinberg}
\bigskip
\end{center}
These are corrections and comments to the ``Lecture Notes on Cherednik
Algebras'' by Pavel Etingof and Xiaoguang Ma. In their current form, they
cover only the first ca. 10 pages of the notes.
\section*{Section 1}
\begin{itemize}
\item \textbf{Page 4:} Replace ``irrieducible'' by ``irreducible''.
\item \textbf{Page 5:} Replace ``shperical'' by ``spherical''.
\end{itemize}
\section*{Section 2}
\begin{itemize}
\item \textbf{Page 6, Theorem 2.1:} I think the words ``rational
coefficients'', ``lower order terms'' and ``homogeneous'' need some more
explanations. Here is how I understand them; please correct me if I am getting
something wrong:
\textit{``rational coefficients''} means ``coefficients which are rational
functions in the variables $x_{1},x_{2},...,x_{n}$'' (not ``coefficients which
are rational numbers'' or ``coefficients which are polynomials over
$\mathbb{Q}$'').
\textit{``lower order terms''} means the following: Let $\mathbf{D}$ be the
$\mathbb{C}$-algebra of all partial differential operators in the variables
$x_{1},x_{2},...,x_{n}$ whose coefficients are rational functions in the
variables $x_{1},x_{2},...,x_{n}$. Define a $\mathbb{C}$-algebra filtration on
$\mathbf{D}$ by requiring that all rational functions in $x_{1},x_{2}%
,...,x_{n}$ are in filtration degree $0$, and all $\dfrac{\partial}{\partial
x_{j}}$ are in filtration degree $1$. Then,
\[
L_{j}=\sum\limits_{i=1}^{n}\left( \dfrac{\partial}{\partial x_{i}}\right)
^{j}+\text{lower order terms}%
\]
means that%
\[
L_{j}\equiv\sum\limits_{i=1}^{n}\left( \dfrac{\partial}{\partial x_{i}%
}\right) ^{j}\operatorname{mod}\left( \left( j-1\right) \text{-th filtered
part of }\mathbf{D}\right) .
\]
And the \textit{order} of a partial differential operator $E\in\mathbf{D}$
means the smallest $n\in\mathbb{N}$ such that $E$ lies in the $n$-th filtered
part of $\mathbf{D}$. Am I seeing this right?
Note that this $\mathbb{C}$-algebra filtration on $\mathbf{D}$ can be also
characterized differently: Let $\mathbf{D}_{\operatorname*{const}}$ denote the
$\mathbb{C}$-algebra of all partial differential operators in the variables
$x_{1},x_{2},...,x_{n}$ whose coefficients are constant. Let the unadorned
$\otimes$ sign denote $\otimes_{\mathbb{C}}$. Then, $\mathbf{D}=\mathbb{C}%
\left( x_{1},x_{2},...,x_{n}\right) \otimes\mathbf{D}_{\operatorname*{const}%
}$ as vector spaces. Since the algebra $\mathbf{D}_{\operatorname*{const}}$ is
canonically graded (by giving all $\dfrac{\partial}{\partial x_{j}}$ the
degree $1$) and the algebra $\mathbb{C}\left( x_{1},x_{2},...,x_{n}\right) $
is trivially graded (by giving its every element the degree $0$), the tensor
product $\mathbb{C}\left( x_{1},x_{2},...,x_{n}\right) \otimes
\mathbf{D}_{\operatorname*{const}}$ is also graded. Since $\mathbf{D}%
=\mathbb{C}\left( x_{1},x_{2},...,x_{n}\right) \otimes\mathbf{D}%
_{\operatorname*{const}}$ as vector spaces, this yields that the vector space
$\mathbf{D}$ is also graded (albeit this is not a grading of the $\mathbb{C}%
$\textbf{-algebra }$\mathbf{D}$, since generally $\mathbf{D}\neq
\mathbb{C}\left( x_{1},x_{2},...,x_{n}\right) \otimes\mathbf{D}%
_{\operatorname*{const}}$ as algebras), hence filtered. This filtration is
easily seen to be the same filtration on $\mathbf{D}$ as defined above.)
Note that as vector spaces,%
\begin{align*}
& \left( j\text{-th filtered part of }\mathbf{D}\right) \diagup\left(
\left( j-1\right) \text{-th filtered part of }\mathbf{D}\right) \\
& \cong\left( j\text{-th graded part of }\mathbf{D}\right) \\
& \ \ \ \ \ \ \ \ \ \ \left( \text{since the filtration of }\mathbf{D}\text{
comes from a vector space grading on }\mathbf{D}\right) \\
& =\left( j\text{-th graded part of }\mathbb{C}\left( x_{1},x_{2}%
,...,x_{n}\right) \otimes\mathbf{D}_{\operatorname*{const}}\right) \\
& =\mathbb{C}\left( x_{1},x_{2},...,x_{n}\right) \otimes\left( j\text{-th
graded part of }\mathbf{D}_{\operatorname*{const}}\right) \\
& \ \ \ \ \ \ \ \ \ \ \left( \text{since }\mathbb{C}\left( x_{1}%
,x_{2},...,x_{n}\right) \text{ is concentrated in degree }0\right) .
\end{align*}
\textit{``homogeneous''} means the following: Let $\mathbf{D}_{\hom}$ be the
$\mathbb{C}$-subalgebra of the algebra $\mathbf{D}$ (defined above) generated
by all homogeneous rational functions in $x_{1},x_{2},...,x_{n}$ and the
derivations $\dfrac{\partial}{\partial x_{j}}$. This is a graded algebra,
where the degree of a homogeneous rational function is its usual degree, and
the degree of a derivation $\dfrac{\partial}{\partial x_{j}}$ is $-1$. Then,
when we say that a differential operator in $\mathbf{D}$ is
\textit{homogeneous of degree }$k$ (for some integer $k$), we mean that this
operator lies in $\mathbf{D}_{\hom}$ and has degree $k$.
\item \textbf{Page 6, four lines above Definition 2.3:} You speak of an
``inner product''. Maybe point out that it is supposed to be bilinear, not
sesquilinear (some people might be confused).
\item \textbf{Page 6, two lines above Definition 2.3:} You say ``equivalently,
$s$ is conjugate to $\operatorname*{diag}\left( -1,1,...,1\right) $''.
Conjugate where? in $\operatorname*{GL}\left( \mathfrak{h}\right) $ or in
$\operatorname*{O}\left( \mathfrak{h}\right) $ ? In this case, both are true
(as long as we suppose $s$ to lie in $\operatorname*{O}\left( \mathfrak{h}%
\right) $), but it would be better if you would point this out more explicitly.
\item \textbf{Page 6, Theorem 2.4, and many times after:} I think Theorem 2.4
is called the Chevalley-Shep\textbf{h}ard-Todd theorem, with two ``h'''s in
``Shephard'' (cf. \url{http://en.wikipedia.org/wiki/Geoffrey_Colin_Shephard} ).
\item \textbf{Page 6, one line below Theorem 2.4:} Maybe add ``if $G$ is a
complex reflection group'' into the sentence that comes directly after Theorem 2.4.
\item \textbf{Page 6, two lines below Theorem 2.4:} You write: ``The numbers
$d_{i}$ are uniquely determined''. You need to add here that you require
$d_{1}\leq d_{2}\leq...\leq d_{\dim\mathfrak{h}}$ (else, the ``$L_{1}=H$''
part of Theorem 2.9 makes no sense).
\item \textbf{Page 7, Example 2.5:} It is not clear what $p_{i}$ are, and why
you write $P_{i}\left( p_{1},...,p_{n}\right) $ (the $p_{i}$ are definitely
not polynomial variables, since they are algebraically dependent). Let me just
record the answer (which you explained in an email): You want $p_{i}%
=e_{i}-\dfrac{e_{1}+e_{2}+\cdots+e_{n}}{n}\in\mathfrak{h}$ (where $e_{1}%
,e_{2},...,e_{n}$ are the standard basis vectors of $\mathbb{C}^{n}$), and
instead of $P_{i}\left( p_{1},...,p_{n}\right) $ you simply want to write
$P_{i}$.
\item \textbf{Page 7, between Definition 2.6 and Example 2.7:} You write:
``Note that by Chevalley's theorem, a parabolic subgroup of a complex
(respectively, real) reflection group is itself a complex (respectively, real)
reflection group.'' What Chevalley's theorem do you mean? If you are applying
Theorem 2.4, isn't it quite an overkill? (Or is there really no simpler proof?)
\item \textbf{Page 7, between Example 2.7 and \S 2.4:} You write: ``and we can
define the open set $\mathfrak{h}_{\operatorname*{reg}}^{\ast G^{\prime}}$ of
all $\lambda\in\mathfrak{h}^{G^{\prime}}$ for which $G_{\lambda}=G^{\prime}%
$''. I think the ``$\mathfrak{h}^{G^{\prime}}$'' should be ``$\mathfrak{h}%
^{\ast G^{\prime}}$'' here.
\item \textbf{Page 7, first line of \S 2.4:} Replace ``Let $s\subset
\operatorname*{GL}\left( \mathfrak{h}\right) $'' by ``Let $s\in
\operatorname*{GL}\left( \mathfrak{h}\right) $''.
\item \textbf{Page 7, second line of \S 2.4:} You might want to point out that
a ``nontrivial eigenvalue'' of a reflection means an eigenvalue $\neq1$.
(Normally, in linear algebra, I tend to mean $\neq0$ by ``nontrivial''.)
\item \textbf{Page 7, one line above Definition 2.8:} What do you mean by a
``conjugation invariant function''? Invariant under conjugation by elements of
$W$, or by conjugation by any element of $\operatorname*{O}\left(
\mathfrak{h}\right) $ (or even $\operatorname*{GL}\left( \mathfrak{h}%
\right) $ ?) that happens to send an element of $\mathcal{S}$ to another
element of $\mathcal{S}$ ?
\item \textbf{Page 7, Definition 2.8:} This is hardly an error, but maybe it
would improve the exposition if you would define what $\Delta_{\mathfrak{h}}$
means. (It's just that I don't like algebra texts relying on geometry
preknowledge.) I assume we can define it by $\Delta_{\mathfrak{h}}%
=\sum\limits_{i=1}^{r}\partial_{y_{i}}^{2}$ for any orthonormal basis $\left(
y_{1},y_{2},...,y_{r}\right) $ of $\mathfrak{h}$ ?
\item \textbf{Page 7, one line above Theorem 2.9:} When you write
``$P_{1}\left( \mathbf{p}\right) =\mathbf{p}^{2}$'', it wouldn't hurt to
point out that $\mathbf{p}$ is a variable vector in $\mathfrak{h}^{\ast}$ (not
$\mathfrak{h}$), so ``$P_{1}\left( \mathbf{p}\right) =\mathbf{p}^{2}$''
describes $P_{1}$ as a polynomial function on $\mathfrak{h}^{\ast}$ (that is,
an element of $\mathbb{C}\left[ \mathfrak{h}^{\ast}\right] =S\mathfrak{h}$).
\item \textbf{Page 8, two lines above Remark 2.10:} You write: ``This theorem
is obviously a generalization of Theorem 1 about $W=\mathfrak{S}_{n}$.'' Given
that the representation $\mathbb{C}^{n}$ of $\mathfrak{S}_{n}$ is not
irreducible, while lifting the $L_{j}$ from the representation $\mathbb{C}%
^{n-1}$ of $\mathfrak{S}_{n}$ to $\mathbb{C}^{n}$ requires some work (as our
emails showed), I don't think the word ``obviously'' is justified here. See
below for a proposal how to improve this (by getting rid of the standing
assumption that $\mathfrak{h}$ be irreducible).
\item \textbf{Page 8, fourth line of \S 2.5:} You write: ``We normalize them
in such a way that $\left\langle \alpha_{s},\alpha_{s}^{\vee}\right\rangle
=2$.'' At this point, I had to think for a while about why this is possible
(i. e., why we don't have $\left\langle \alpha_{s},\alpha_{s}^{\vee
}\right\rangle =0$). This is quite easy to see by diagonalizing the matrix
$s$, but maybe you should make this an explicit exercise. (Remark 2.13, too,
could be an exercise.)
\item \textbf{Page 8, fifth line of \S 2.5:} Again, you speak of a ``function
invariant with respect to conjugation'', and it is not clear by what you allow
to conjugate. (I will henceforth assume that you allow conjugation by $G$.)
\item \textbf{Page 8, Definition 2.11:} Please say that $\mathbb{C}\left(
\mathfrak{h}\right) $ means the quotient field of $S\left( \mathfrak{h}%
^{\ast}\right) $. (I know that this follows from standard algebraic geometry
notation, but I didn't expect that you are using algebraic geometry notation.)
Also, please add ``Let $a\in\mathfrak{h}$.'' at the beginning of this Definition.
\item \textbf{Page 8, Proposition 2.14:} Beginning with part (i) of this
proposition, you seem to systematically write $\left( \cdot,\cdot\right) $
for the bilinear form on $\mathfrak{h}^{\ast}\times\mathfrak{h}$ that you
formerly denoted by $\left\langle \cdot,\cdot\right\rangle $. I don't like
this notation very much, because $\left( \cdot,\cdot\right) $ already means
two different bilinear forms (one on $\mathfrak{h}\times\mathfrak{h}$ and one
on $\mathfrak{h}^{\ast}\times\mathfrak{h}^{\ast}$) in the case when
$G\subseteq\operatorname*{O}\left( \mathfrak{h}\right) $, but it's okay
since one can always infer types. But you should point out the change in
notation, or else it appears as if you suddenly switched to the case
$G\subseteq\operatorname*{O}\left( \mathfrak{h}\right) $ !
\item \textbf{Page 9, proof of Theorem 2.15:} I think that%
\[
-\sum_{s\in\mathcal{S}}c_{s}\left( a,\alpha_{s}\right) \left( x,\alpha
_{s}^{\vee}\right) \left( b,\alpha_{s}\right) sD_{\alpha_{s}^{\vee}}%
\cdot\dfrac{1-\lambda_{s}^{-1}}{2}%
\]
should be%
\[
-\sum_{s\in\mathcal{S}}c_{s}\left( a,\alpha_{s}\right) \left( x,\alpha
_{s}^{\vee}\right) \left( b,\alpha_{s}\right) sD_{\alpha_{s}^{\vee}}%
\cdot\dfrac{1-\lambda_{s}}{2}%
\]
(leaving aside the fact that you are still using the notation $\left(
\cdot,\cdot\right) $ for what was formerly called $\left\langle \cdot
,\cdot\right\rangle $). To make sure that I haven't done any mistakes, let me
write up the details of this computation. (They are completely straightforward
and I don't think you should explicit them in the paper, but I am doing them
here so you can tell me where I am going wrong.)
It is clearly enough to prove that every $s\in\mathcal{S}$ satisfies%
\begin{equation}
\left[ s,D_{b}\right] =\left\langle b,\alpha_{s}\right\rangle sD_{\alpha
_{s}^{\vee}}\cdot\dfrac{1-\lambda_{s}}{2}. \label{p9.1}%
\end{equation}
First, we show that%
\begin{equation}
\text{every }b\in\mathfrak{h}\text{ satisfies }b-s^{-1}b=\dfrac{1-\lambda_{s}%
}{2}\left\langle b,\alpha_{s}\right\rangle \alpha_{s}^{\vee}. \label{p9.2}%
\end{equation}
(This is similar to Proposition 2.14 (i), but with $\mathfrak{h}$ instead of
$\mathfrak{h}^{\ast}$.)
\textit{Proof of (\ref{p9.2}):} WLOG, assume that $\mathfrak{h}=\mathbb{C}%
^{n}$, $s=\operatorname*{diag}\left( \lambda_{s},1,1,...,1\right) $,
$\alpha_{s}=e_{1}^{\ast}$ and $\alpha_{s}^{\vee}=2e_{1}$, where $\left(
e_{1},e_{2},...,e_{n}\right) $ is the standard basis of $\mathbb{C}^{n}$ and
$\left( e_{1}^{\ast},e_{2}^{\ast},...,e_{n}^{\ast}\right) $ is its dual
basis. (This situation can always be achieved by an appropriate change of
basis in $\mathfrak{h}$.) By linearity, it is enough to prove (\ref{p9.2}) in
the cases when $b=e_{i}$ for $i\in\left\{ 1,2,...,n\right\} $. So consider
this case. If $i>1$, then both sides of (\ref{p9.2}) are $0$, and thus
(\ref{p9.2}) holds. Remains the case $i=1$. In this case, $b=e_{1}=\dfrac
{1}{2}\alpha_{s}^{\vee}$, so that%
\[
b-s^{-1}b=\dfrac{1}{2}\left( \alpha_{s}^{\vee}-\underbrace{s^{-1}\alpha
_{s}^{\vee}}_{=\lambda_{s}\alpha_{s}^{\vee}}\right) =\dfrac{1-\lambda_{s}}%
{2}\alpha_{s}^{\vee}%
\]
and%
\[
\dfrac{1-\lambda_{s}}{2}\left\langle \underbrace{b}_{=e_{1}}%
,\underbrace{\alpha_{s}}_{=e_{1}^{\ast}}\right\rangle \alpha_{s}^{\vee}%
=\dfrac{1-\lambda_{s}}{2}\underbrace{\left\langle e_{1},e_{1}^{\ast
}\right\rangle }_{=1}\alpha_{s}^{\vee}=\dfrac{1-\lambda_{s}}{2}\alpha
_{s}^{\vee}.
\]
Thus, (\ref{p9.2}) holds in the case $i=1$ as well, and thus (\ref{p9.2}) is proven.
\textit{Proof of (\ref{p9.1}):} We have%
\begin{align*}
\left[ s,D_{b}\right] & =sD_{b}-D_{b}s=s\left( D_{b}-\underbrace{s^{-1}%
D_{b}s}_{\substack{=D_{s^{-1}b}\\\text{(by Proposition 2.14 (ii))}}}\right) \\
& =s\underbrace{\left( D_{b}-D_{s^{-1}b}\right) }_{\substack{=D_{b-s^{-1}%
b}=D_{\dfrac{1-\lambda_{s}}{2}\left\langle b,\alpha_{s}\right\rangle
\alpha_{s}^{\vee}}\\\text{(by (\ref{p9.1}))}}}=sD_{\dfrac{1-\lambda_{s}}%
{2}\left\langle b,\alpha_{s}\right\rangle \alpha_{s}^{\vee}}\\
& =\left\langle b,\alpha_{s}\right\rangle sD_{\alpha_{s}^{\vee}}\cdot
\dfrac{1-\lambda_{s}}{2},\ \ \ \ \ \ \ \ \ \ \text{and (\ref{p9.1}) is
proven.}%
\end{align*}
\item \textbf{Page 9, proof of Theorem 2.15:} You write: ``since this algebra
acts faithfully on $\mathbb{C}\left( \mathfrak{h}\right) $'' (where ``this
algebra'' is the semidirect product $\mathbb{C}G\ltimes\mathcal{D}\left(
\mathfrak{h}_{\operatorname*{reg}}\right) $). I am wondering how you prove
this. I have a proof, but it is rather messy: First, the claim that
$\mathbb{C}G\ltimes\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}%
}\right) $ acts faithfully on $\mathbb{C}\left( \mathfrak{h}\right) $ can
be rewritten as follows: If $\left( D_{g}\right) _{g\in G}$ is a family of
differential operators indexed by elements of $G$ such that $\sum\limits_{g\in
G}gD_{g}$ is $0$ as an endomorphism of $\mathbb{C}\left( \mathfrak{h}\right)
$, then each $g\in G$ satisfies $D_{g}=0$. To prove this, we first notice that
we can WLOG assume that every $D_{g}$ has polynomial coefficients (because we
can move denominators to the left, moving them past derivations by means of
the quotient rule and moving them past the $g$'s by using the formula%
\[
g\circ f=\left( gf\right) g\ \ \ \ \ \ \ \ \ \ \text{for any }g\in G\text{
and }f\in\mathbb{C}\left( \mathfrak{h}\right)
\]
). Now, let $v\in\mathfrak{h}$ be a point which is not fixed by any $g\in
G\diagdown\left\{ \operatorname*{id}\right\} $. Recall that $\sum
\limits_{g\in G}gD_{g}$ is $0$ as an endomorphism of $\mathbb{C}\left(
\mathfrak{h}\right) $. In particular, $\sum\limits_{g\in G}gD_{g}$ acts as
$0$ on $\mathbb{C}\left[ \mathfrak{h}\right] $. Thus, for every
$p\in\mathbb{C}\left[ \mathfrak{h}\right] $, a certain $\mathbb{C}\left[
\mathfrak{h}\right] $-linear combination of the partial derivatives of $p$
(of various orders) taken at the points $gv$ for varying $g\in G$ is
identically $0$ (and the coefficients of this combination don't depend on
$p$). But since we can find a polynomial with any given set of finitely many
prescribed values of partial derivatives at finitely many points\footnote{This
follows from the Chinese remainder theorem, applied to the ring $\mathbb{C}%
\left[ \mathfrak{h}\right] $. In fact, by prescribing the values of finitely
many partial derivatives of a polynomial $p\in\mathbb{C}\left[ \mathfrak{h}%
\right] $ at some point $w\in\mathfrak{h}$, we put a condition on the residue
class of $p$ modulo a certain power of the maximal ideal $\mathfrak{m}%
_{w}\subseteq\mathfrak{h}$ (where $\mathfrak{m}_{w}$ is the ideal of all
polynomials that vanish at $w$). Such a condition is always satisfiable. Thus,
if we prescribe the values of finitely many partial derivatives of a
polynomial $p\in\mathbb{C}\left[ \mathfrak{h}\right] $ at finitely many
points $w_{1},w_{2},...,w_{\ell}\in\mathfrak{h}$, we put conditions on the
residue classes of $p$ modulo powers of $\mathfrak{m}_{w_{1}}$, $\mathfrak{m}%
_{w_{2}}$, $...$, $\mathfrak{m}_{w_{\ell}}$. Each of these $\ell$ conditions
alone is satisfiable; thus, the conjunction of these $\ell$ conditions is also
satisfiable (because the Chinese remainder theorem says that $\left(
\mathbb{C}\left[ \mathfrak{h}\right] \right) \diagup\left( \mathfrak{m}%
_{w_{1}}^{\alpha_{1}}\mathfrak{m}_{w_{2}}^{\alpha_{2}}...\mathfrak{m}%
_{w_{\ell}}^{\alpha_{\ell}}\right) =\prod\limits_{i=1}^{\ell}\left(
\mathbb{C}\left[ \mathfrak{h}\right] \right) \diagup\mathfrak{m}_{w_{i}%
}^{\alpha_{i}}$, so that every $\ell$-tuple in $\prod\limits_{i=1}^{\ell
}\left( \mathbb{C}\left[ \mathfrak{h}\right] \right) \diagup
\mathfrak{m}_{w_{i}}^{\alpha_{i}}$ has a common representative in
$\mathbb{C}\left[ \mathfrak{h}\right] $), i. e., we can find a polynomial
with our given set of prescribed values.}, this yields that the $\mathbb{C}%
\left[ \mathfrak{h}\right] $-linear combination must be trivial at $v$; in
other words, $D_{g}$ is identically $0$ at $v$ for every $g\in G$. Since this
holds for every point $v\in\mathfrak{h}$ which is not fixed by any $g\in
G\diagdown\left\{ \operatorname*{id}\right\} $, and since the set of such
points is Zariski-dense in $\mathfrak{h}$, this yields that $D_{g}$ is
identically $0$ everywhere for every $g\in G$. This proves that each $g\in G$
satisfies $D_{g}=0$, qed.
\item \textbf{Page 9, \S 2.6, just before Proposition 2.16:} It would be nice
to explain when an element of $\mathbb{C}W\ltimes\mathcal{D}\left(
\mathfrak{h}_{\operatorname*{reg}}\right) $ or an operator on the space of
regular functions of $\mathfrak{h}_{\operatorname*{reg}}$ is said to be
$W$\textit{-invariant}. (Short answer: When it commutes with every $g\in W$.)
\item \textbf{Page 10, two lines above Corollary 2.17:} You write: ``the
algebra $\left( S\mathfrak{h}\right) ^{W}$ is free''. By ``free'', you mean
``free as a commutative algebra'', not ``free as an algebra''. (I know, this
is some nitpicking.)
\item \textbf{Page 10, Corollary 2.17:} In my opinion, you should explain what
$P_{j}\left( D_{y_{1}},...,D_{y_{r}}\right) $ means, because $P_{j}$ is just
an element of $S\mathfrak{h}$, and not a polynomial. (The meaning of
$P_{j}\left( D_{y_{1}},...,D_{y_{r}}\right) $ is the following: Since
$\left\{ y_{1},y_{2},...,y_{r}\right\} $ is a basis of $\mathfrak{h}$, we
can identify the symmetric algebra $S\mathfrak{h}$ with the ring of
polynomials in the $r$ variables $y_{1}$, $y_{2}$, $...$, $y_{r}$ over
$\mathbb{C}$. Thus, $P_{j}\in S\mathfrak{h}$ becomes a polynomial in the $r$
variables $y_{1}$, $y_{2}$, $...$, $y_{r}$. If we now substitute $D_{y_{1}}$,
$D_{y_{2}}$, $...$, $D_{y_{r}}$ for these variables $y_{1}$, $y_{2}$, $...$,
$y_{r}$ in $P_{j}$ (this is allowed because the Dunkl operators $D_{a}$
commute), we obtain an element of $\mathbb{C}W\ltimes\mathcal{D}\left(
\mathfrak{h}_{\operatorname*{reg}}\right) $. This element is what you denote
by $P_{j}\left( D_{y_{1}},...,D_{y_{r}}\right) $.
\item \textbf{Page 10, proof of Corollary 2.17:} Replace ``$L_{j}$'' by
``$\overline{L}_{j}$'' twice in this proof.
\item \textbf{Page 10, proof of Corollary 2.17:} This proof would be more
readable if you would explain why the $P_{j}\left( D_{y_{1}},...,D_{y_{r}%
}\right) $ is $W$-invariant for all $j$. The \textit{proof} is not too immediate:
First, it is easy to see that the map%
\begin{align*}
\mathfrak{h} & \rightarrow\mathbb{C}W\ltimes\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) ,\\
a & \mapsto D_{a}%
\end{align*}
is $\mathbb{C}$-linear\footnote{This is because the map%
\begin{align*}
\mathfrak{h} & \rightarrow\mathbb{C}W\ltimes\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) ,\\
a & \mapsto\partial_{a}%
\end{align*}
is $\mathbb{C}$-linear, and because $\alpha_{s}$ is $\mathbb{C}$-linear for
every $s\in W$.}. Denote this map by $T$.
Since $\left\{ y_{1},y_{2},...,y_{r}\right\} $ is a basis of $\mathfrak{h}$,
we can identify the symmetric algebra $S\mathfrak{h}$ with the ring of
polynomials in the $r$ variables $y_{1}$, $y_{2}$, $...$, $y_{r}$ over
$\mathbb{C}$. Thus, every $P\in S\mathfrak{h}$ becomes a polynomial in the $r$
variables $y_{1}$, $y_{2}$, $...$, $y_{r}$. As a consequence, for every $P\in
S\mathfrak{h}$, we will denote by $P\left( D_{y_{1}},D_{y_{2}},...,D_{y_{r}%
}\right) $ the result of substituting $D_{y_{1}}$, $D_{y_{2}}$, $...$,
$D_{y_{r}}$ for these variables $y_{1}$, $y_{2}$, $...$, $y_{r}$ in $P$. When
$P\in\mathfrak{h}$, then $P\left( D_{y_{1}},D_{y_{2}},...,D_{y_{r}}\right) $
is a $\mathbb{C}$-linear combination of $D_{y_{1}}$, $D_{y_{2}}$, $...$,
$D_{y_{r}}$ (here, we regard $\mathfrak{h}$ as a subspace of $S\mathfrak{h}$,
so $P\in\mathfrak{h}$ yields $P\in S\mathfrak{h}$).
It is easy to see that
\begin{equation}
\text{every }a\in\mathfrak{h}\text{ satisfies }D_{a}=a\left( D_{y_{1}%
},D_{y_{2}},...,D_{y_{r}}\right) \label{p10.1}%
\end{equation}
(where $a\left( D_{y_{1}},D_{y_{2}},...,D_{y_{r}}\right) $ is to be
understood as just explained, with $a$ being regarded as an element of
$S\mathfrak{h}$). (In fact, the equation (\ref{p10.1}) is $\mathbb{C}$-linear
in $a$ (because of the $\mathbb{C}$-linearity of $T$), and thus in order to
prove it for all $a\in\mathfrak{h}$, it is enough to prove it in the case when
$a\in\left\{ y_{1},y_{2},...,y_{r}\right\} $ (since $\left\{ y_{1}%
,y_{2},...,y_{r}\right\} $ is a basis of $\mathfrak{h}$), but in this case it
is trivial.)
Now, for any $j\in\left\{ 1,2,...,\dim\mathfrak{h}\right\} $ and any $g\in
W$, we have%
\begin{align*}
& gP_{j}\left( D_{y_{1}},...,D_{y_{r}}\right) g^{-1}\\
& =P_{j}\left( gD_{y_{1}}g^{-1},...,gD_{y_{r}}g^{-1}\right)
\ \ \ \ \ \ \ \ \ \ \left( \text{since conjugation by }g\text{ is an algebra
automorphism}\right) \\
& =P_{j}\left( D_{gy_{1}},...,D_{gy_{r}}\right) \ \ \ \ \ \ \ \ \ \ \left(
\text{since }gD_{y_{i}}g^{-1}=D_{gy_{i}}\text{ for every }i\text{ due to
Proposition 2.14 (ii)}\right) \\
& =P_{j}\left( \left( gy_{1}\right) \left( D_{y_{1}},D_{y_{2}%
},...,D_{y_{r}}\right) ,...,\left( gy_{r}\right) \left( D_{y_{1}}%
,D_{y_{2}},...,D_{y_{r}}\right) \right) \\
& \ \ \ \ \ \ \ \ \ \ \left( \text{since (\ref{p10.1}) yields that
}D_{gy_{i}}=\left( gy_{i}\right) \left( D_{y_{1}},D_{y_{2}},...,D_{y_{r}%
}\right) \text{ for every }i\right) \\
& =\underbrace{\left( P_{j}\left( gy_{1},...,gy_{r}\right) \right)
}_{\substack{=gP_{j}=P_{j}\\\text{(since }P_{j}\in\left( S\mathfrak{h}%
\right) ^{W}\text{)}}}\left( D_{y_{1}},...,D_{y_{r}}\right) \\
& \ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{here (as explained above) }\left( P_{j}\left( gy_{1},...,gy_{r}%
\right) \right) \left( D_{y_{1}},...,D_{y_{r}}\right) \text{ means}\\
\text{ ``the polynomial }P_{j}\left( gy_{1},...,gy_{r}\right) \text{ with
}D_{y_{1}}\text{, }D_{y_{2}}\text{, }...\text{, }D_{y_{r}}\text{ substituted
for }y_{1}\text{, }y_{2}\text{, }...\text{, }y_{r}\text{''}%
\end{array}
\right) \\
& =P_{j}\left( D_{y_{1}},...,D_{y_{r}}\right) ,
\end{align*}
so that $P_{j}\left( D_{y_{1}},...,D_{y_{r}}\right) $ is $W$-invariant, qed.
The main idea of this proof, I think, was the $\mathbb{C}$-linearity of $T$.
While trivial, it is (in my opinion) unexpected for such a complicated map.
\item \textbf{Pages 7-10:} Here are various suggested changes to make the
proofs clearer (these changes should be made at the same time, as they depend
one on another):
-- In Theorem 2.9, add the claim that the $L_{j}$ are $W$-invariant.
(Otherwise, Theorem 2.1 doesn't directly follow from Theorem 2.9, because
Theorem 2.1 claims the $\mathfrak{S}_{n}$-invariance of the $L_{j}$.)
-- In Corollary 2.17, add the claim that the $\overline{L}_{j}$ are
$W$-invariant. (Otherwise, Theorem 2.9 with the added claim that the $L_{j}$
are $W$-invariant doesn't directly follow from Theorem 2.9.)
-- On page 9, you write:
``For any element $B\in\mathbb{C}W\ltimes\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $, define $m\left( B\right) $ to be the
differential operator $\mathbb{C}\left( \mathfrak{h}\right) ^{W}%
\rightarrow\mathbb{C}\left( \mathfrak{h}\right) $, defined by $B$. That is,
if $B=\sum_{g\in W}B_{g}g$, $B_{g}\in\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $, then $m\left( B\right) =\sum_{g\in W}%
B_{g}$.''
This is slightly confusing, since you later (in the proof of Corollary 2.17)
want $m\left( B\right) $ to be defined on the whole $\mathbb{C}\left(
\mathfrak{h}\right) $ rather than just on $\mathbb{C}\left( \mathfrak{h}%
\right) ^{W}$. In my opinion, you should replace the text I've just quoted by
the following:
``For any element $B\in\mathbb{C}W\ltimes\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $, define a differential operator $m\left(
B\right) \in\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $
by $m\left( B\right) =\sum\limits_{g\in W}B_{g}$, where $B$ is being written
in the form $B=\sum\limits_{g\in W}B_{g}g$ with $B_{g}\in\mathcal{D}\left(
\mathfrak{h}_{\operatorname*{reg}}\right) $. The differential operator $B$
defined this way satisfies the following properties:
(i) If $f\in\mathbb{C}\left( \mathfrak{h}\right) ^{W}$, then $m\left(
B\right) f=Bf$.
(ii) If $B\in\mathbb{C}W\ltimes\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $ is $W$-invariant, then $m\left( B\right) $
is $W$-invariant as well\footnote{\textit{Proof.} Let $B\in\mathbb{C}%
W\ltimes\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $ be
$W$-invariant. Write $B$ in the form $B=\sum\limits_{g\in W}B_{g}g$ with
$B_{g}\in\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $.
Then, $m\left( B\right) =\sum\limits_{g\in W}B_{g}$. Let $h\in W$. Since $B$
is $W$-invariant, we have $hB=Bh$, so that%
\begin{align*}
\sum\limits_{g\in W}hB_{g}g & =h\underbrace{\sum\limits_{g\in W}B_{g}g}%
_{=B}=hB=\underbrace{B}_{=\sum\limits_{g\in W}B_{g}g}h=\sum\limits_{g\in
W}B_{g}gh=\sum\limits_{g\in W}B_{gh^{-1}}g\underbrace{h^{-1}h}%
_{=\operatorname*{id}}\\
& \ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{here, we substituted }gh^{-1}\text{ for }g\text{ in the sum}\\
\text{(since the map }W\rightarrow W,\ g\mapsto gh^{-1}\text{ is a bijection)}%
\end{array}
\right) \\
& =\sum\limits_{g\in W}B_{gh^{-1}}g.
\end{align*}
Compared to%
\begin{align*}
\sum\limits_{g\in W}hB_{g}\underbrace{g}_{=h^{-1}hg} & =\sum\limits_{g\in
W}hB_{g}h^{-1}hg=\sum\limits_{g\in W}hB_{g}h^{-1}hg=\sum\limits_{g\in
W}hB_{h^{-1}g}h^{-1}\underbrace{hh^{-1}}_{=\operatorname*{id}}g\\
& \ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{here, we substituted }h^{-1}g\text{ for }g\text{ in the sum}\\
\text{(since the map }W\rightarrow W,\ g\mapsto h^{-1}g\text{ is a bijection)}%
\end{array}
\right) \\
& =\sum\limits_{g\in W}hB_{h^{-1}g}h^{-1}g,
\end{align*}
this yields $\sum\limits_{g\in W}B_{gh^{-1}}g=\sum\limits_{g\in W}hB_{h^{-1}%
g}h^{-1}g$. Notice that every $g\in W$ satisfies $B_{gh^{-1}}\in
\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $ and
$hB_{h^{-1}g}h^{-1}\in\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}%
}\right) $.
\par
But any element of $\mathbb{C}W\ltimes\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $ can be \textbf{uniquely} written in the form
$\sum\limits_{g\in W}C_{g}g$ with $C_{g}\in\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $. Hence, if we have $\sum\limits_{g\in W}%
C_{g}g=\sum\limits_{g\in W}D_{g}g$ for some choice of $C_{g}\in\mathcal{D}%
\left( \mathfrak{h}_{\operatorname*{reg}}\right) $ and $D_{g}\in
\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $, then every
$g\in W$ satisfies $C_{g}=D_{g}$. Applied to $C_{g}=B_{gh^{-1}}$ and
$C_{g}=hB_{h^{-1}g}h^{-1}$, this yields that
\par%
\[
\text{every }g\in W\text{ satisfies }B_{gh^{-1}}=hB_{h^{-1}g}h^{-1}%
\]
(because $\sum\limits_{g\in W}B_{gh^{-1}}g=\sum\limits_{g\in W}hB_{h^{-1}%
g}h^{-1}g$). Hence,
\begin{align*}
\sum\limits_{g\in W}B_{gh^{-1}} & =\sum\limits_{g\in W}hB_{h^{-1}g}%
h^{-1}=\sum\limits_{g\in W}hB_{g}h^{-1}\\
& \ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{here, we substituted }g\text{ for }h^{-1}g\text{ in the sum}\\
\text{(since the map }W\rightarrow W,\ g\mapsto h^{-1}g\text{ is a bijection)}%
\end{array}
\right) \\
& =h\underbrace{\left( \sum\limits_{g\in W}B_{g}\right) }_{=m\left(
B\right) }h^{-1}=hm\left( B\right) h^{-1}.
\end{align*}
Compared to%
\begin{align*}
\sum\limits_{g\in W}B_{gh^{-1}} & =\sum\limits_{g\in W}B_{g}%
\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{here, we substituted }g\text{ for }gh^{-1}\text{ in the sum}\\
\text{(since the map }W\rightarrow W,\ g\mapsto gh^{-1}\text{ is a bijection)}%
\end{array}
\right) \\
& =m\left( B\right) ,
\end{align*}
this yields $m\left( B\right) =hm\left( B\right) h^{-1}$, so that
$m\left( B\right) h=hm\left( B\right) $.
\par
Since this holds for every $h\in W$, this yields that $m\left( B\right) $ is
$W$-invariant, qed.}.
(iii) Any $s\in W$ and any $B\in\mathbb{C}W\ltimes\mathcal{D}\left(
\mathfrak{h}_{\operatorname*{reg}}\right) $ satisfy $m\left( B\right)
=m\left( Bs\right) $. (This is used, e. g., in the proof that $m\left(
D_{y}^{2}\right) =m\left( D_{y}\partial_{y}\right) $ in the proof of
Proposition 2.16.)
(iv) Any $A\in\mathbb{C}W\ltimes\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $ and any $W$-invariant $B\in\mathbb{C}%
W\ltimes\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $
satisfy $m\left( AB\right) =m\left( A\right) m\left( B\right) $.''
\item \textbf{Pages 7-10:} In much of Section 2, you work with a standing
assumption requiring that $\mathfrak{h}$ be an irreducible $W$-module. This
makes deducing Theorem 2.1 from Theorem 2.9 unnecessarily hard. There is a
very easy way to get rid of the standing assumption:
-- On page 7, replace ``Let us assume that $\mathfrak{h}$ is an irreducible
representation of $W$ (i. e. $W$ is an irreducible finite Coxeter group, and
$\mathfrak{h}$ is its reflection representation.) In this case, we can take
$P_{1}\left( \mathbf{p}\right) =\mathbf{p}^{2}$'' by ``Note that if
$\mathfrak{h}$ is an irreducible representation of $W$ (i. e. $W$ is an
irreducible finite Coxeter group, and $\mathfrak{h}$ is its reflection
representation), then we can take $P_{1}\left( \mathbf{p}\right)
=\mathbf{p}^{2}$. If $W=\mathfrak{S}_{n}$ and $\mathfrak{h}=\mathbb{C}^{n}$
(with the standard permutation representation of $\mathfrak{S}_{n}$), then we
can take $P_{2}\left( \mathbf{p}\right) =\mathbf{p}^{2}$''.
-- In Theorem 2.9, replace ``$L_{1}=H$'' by ``if $\ell\in\left\{
1,2,...,\dim\mathfrak{h}\right\} $ is such that $P_{\ell}\left(
\mathbf{p}\right) =\mathbf{p}^{2}$, then $L_{\ell}=H$''.
-- One line above Corollary 2.17, replace ``$P_{1}=\mathbf{p}^{2}$'' by
``$P_{1}$''.
-- In Corollary 2.17, replace ``$\overline{L}_{1}=\overline{H}$'' by ``if
$\ell\in\left\{ 1,2,...,\dim\mathfrak{h}\right\} $ is such that $P_{\ell
}\left( \mathbf{p}\right) =\mathbf{p}^{2}$, then $\overline{L}_{\ell
}=\overline{H}$''.{}
\item \textbf{Page 10, proof of Proposition 2.18:} The expression
``$\sum\limits_{i=1}^{r}\partial_{y_{i}}\left( \log\delta_{c}\right)
\partial_{y_{i}}$'' is ambiguous: Does $\partial_{y_{i}}\left( \log\delta
_{c}\right) $ mean the product $\partial_{y_{i}}\cdot\left( \log\delta
_{c}\right) $ in $\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}%
}\right) $ or the $y_{i}$-derivative of $\log\delta_{c}$ ? (I know it means
the latter.)
\item \textbf{Page 10, proof of Proposition 2.18:} As I don't like this proof
(it uses a strange function $\delta_{c}$, which is in general not algebraic
and doesn't have a very obvious interpretation as a power series), let me
reformulate it in a more algebraic way. First, I will show some lemmas:
\textbf{Lemma 2.18a.} Let $A$ and $D$ be $\mathbb{C}$-algebras. Let
$\mathcal{G}$ be a subset of $A$ which generates $A$ as a $\mathbb{C}%
$-algebra. Let $f:A\rightarrow D$ be a $\mathbb{C}$-linear map. Assume that%
\[
f\left( ab\right) =f\left( a\right) f\left( b\right)
\ \ \ \ \ \ \ \ \ \ \text{for every }a\in\mathcal{G}\text{ and }b\in A\text{.}%
\]
Also, assume that $f\left( 1\right) =1$. Then, $f$ is a $\mathbb{C}$-algebra homomorphism.
\textit{Proof of Lemma 2.18a.} Let $\mathcal{H}$ be the subset%
\[
\left\{ x\in A\ \mid\ f\left( xb\right) =f\left( x\right) f\left(
b\right) \text{ for every }b\in A\right\} .
\]
Every $a\in\mathcal{G}$ satisfies $a\in\mathcal{H}$ (because every
$a\in\mathcal{G}$ satisfies $f\left( ab\right) =f\left( a\right) f\left(
b\right) $ for every $b\in A$, and thus $a\in\left\{ x\in A\ \mid\ f\left(
xb\right) =f\left( x\right) f\left( b\right) \text{ for every }b\in
A\right\} =\mathcal{H}$). In other words, $\mathcal{G}\subseteq\mathcal{H}$.
Also, any $\lambda\in\mathbb{C}$, $\mu\in\mathbb{C}$, $a\in\mathcal{H}$ and
$a^{\prime}\in\mathcal{H}$ satisfy $\lambda a+\mu a^{\prime}\in\mathcal{H}%
$\ \ \ \ \footnote{\textit{Proof.} Let $\lambda\in\mathbb{C}$, $\mu
\in\mathbb{C}$, $a\in\mathcal{H}$ and $a^{\prime}\in\mathcal{H}$. Since
$a\in\mathcal{H}=\left\{ x\in A\ \mid\ f\left( xb\right) =f\left(
x\right) f\left( b\right) \text{ for every }b\in A\right\} $, we have
$f\left( ab\right) =f\left( a\right) f\left( b\right) $ for every $b\in
A$. Since $a^{\prime}\in\mathcal{H}=\left\{ x\in A\ \mid\ f\left( xb\right)
=f\left( x\right) f\left( b\right) \text{ for every }b\in A\right\} $, we
have $f\left( a^{\prime}b\right) =f\left( a^{\prime}\right) f\left(
b\right) $ for every $b\in A$. Now,%
\begin{align*}
f\left( \underbrace{\left( \lambda a+\mu a^{\prime}\right) b}_{=\lambda
ab+\mu a^{\prime}b}\right) & =f\left( \lambda ab+\mu a^{\prime}b\right)
=\lambda\underbrace{f\left( ab\right) }_{=f\left( a\right) f\left(
b\right) }+\mu\underbrace{f\left( a^{\prime}b\right) }_{=f\left(
a^{\prime}\right) f\left( b\right) }\ \ \ \ \ \ \ \ \ \ \left( \text{since
}f\text{ is }\mathbb{C}\text{-linear}\right) \\
& =\lambda f\left( a\right) f\left( b\right) +\mu f\left( a^{\prime
}\right) f\left( b\right) =\underbrace{\left( \lambda f\left( a\right)
+\mu f\left( a^{\prime}\right) \right) }_{\substack{=f\left( \lambda a+\mu
a^{\prime}\right) \\\text{(since }f\text{ is }\mathbb{C}\text{-linear)}%
}}f\left( b\right) =f\left( \lambda a+\mu a^{\prime}\right) f\left(
b\right)
\end{align*}
for every $b\in A$. In other words, $\lambda a+\mu a^{\prime}\in\left\{ x\in
A\ \mid\ f\left( xb\right) =f\left( x\right) f\left( b\right) \text{ for
every }b\in A\right\} =\mathcal{H}$, qed.}. Combined with the trivial fact
that $0\in\mathcal{H}$ (this quickly follows from $f\left( 0\right) =0$),
this yields that $\mathcal{H}$ is a $\mathbb{C}$-vector subspace of $A$.
Also, $1\in\mathcal{H}$ (since $f\left( \underbrace{1b}_{=b}\right)
=f\left( b\right) =\underbrace{1}_{=f\left( 1\right) }f\left( b\right)
=f\left( 1\right) f\left( b\right) $ for every $b\in A$, so that
$1\in\left\{ x\in A\ \mid\ f\left( xb\right) =f\left( x\right) f\left(
b\right) \text{ for every }b\in A\right\} =\mathcal{H}$). Besides, any
$a\in\mathcal{H}$ and $a^{\prime}\in\mathcal{H}$ satisfy $aa^{\prime}%
\in\mathcal{H}$\ \ \ \ \footnote{\textit{Proof.} Let $a\in\mathcal{H}$ and
$a^{\prime}\in\mathcal{H}$. Since $a\in\mathcal{H}=\left\{ x\in
A\ \mid\ f\left( xb\right) =f\left( x\right) f\left( b\right) \text{ for
every }b\in A\right\} $, we have%
\begin{equation}
f\left( ab\right) =f\left( a\right) f\left( b\right)
\ \ \ \ \ \ \ \ \ \ \text{for every }b\in A \label{p10.2.18a.0}%
\end{equation}
for every $b\in A$. Since $a^{\prime}\in\mathcal{H}=\left\{ x\in
A\ \mid\ f\left( xb\right) =f\left( x\right) f\left( b\right) \text{ for
every }b\in A\right\} $, we have $f\left( a^{\prime}b\right) =f\left(
a^{\prime}\right) f\left( b\right) $ for every $b\in A$. Now, every $b\in
A$ satisfies%
\begin{align*}
f\left( aa^{\prime}b\right) & =f\left( a\right) \underbrace{f\left(
a^{\prime}b\right) }_{=f\left( a^{\prime}\right) f\left( b\right)
}\ \ \ \ \ \ \ \ \ \ \left( \text{by (\ref{p10.2.18a.0}), applied to
}a^{\prime}b\text{ instead of }b\right) \\
& =f\left( a\right) f\left( a^{\prime}\right) f\left( b\right)
\end{align*}
and%
\[
\underbrace{f\left( aa^{\prime}\right) }_{\substack{=f\left( a\right)
f\left( a^{\prime}\right) \\\text{(by (\ref{p10.2.18a.0}), applied to
}a^{\prime}\\\text{instead of }b\text{)}}}f\left( b\right) =f\left(
a\right) f\left( a^{\prime}\right) f\left( b\right) .
\]
Hence, $f\left( aa^{\prime}b\right) =f\left( a\right) f\left( a^{\prime
}\right) f\left( b\right) =f\left( aa^{\prime}\right) f\left( b\right)
$ for every $b\in A$. Hence, $aa^{\prime}\in\left\{ x\in A\ \mid\ f\left(
xb\right) =f\left( x\right) f\left( b\right) \text{ for every }b\in
A\right\} =\mathcal{H}$, qed.}. Combining this with the fact that
$1\in\mathcal{H}$ and that $\mathcal{H}$ is a $\mathbb{C}$-vector subspace of
$A$, we conclude that $\mathcal{H}$ is a $\mathbb{C}$-subalgebra of $A$.
Combined with $\mathcal{G}\subseteq\mathcal{H}$, this yields that
$\mathcal{H}$ is a $\mathbb{C}$-subalgebra of $A$ containing $\mathcal{G}$ as
a subset. But since every $\mathbb{C}$-subalgebra of $A$ containing
$\mathcal{G}$ as a subset must contain $A$ as a
subset\footnote{\textit{Proof.} We know that $\mathcal{G}$ generates $A$ as a
$\mathbb{C}$-algebra. In other words, $A$ is the smallest $\mathbb{C}%
$-subalgebra of $A$ containing $\mathcal{G}$ as a subset. Hence, every
$\mathbb{C}$-subalgebra of $A$ containing $\mathcal{G}$ as a subset must
contain $A$ as a subset, qed.}, this yields that $\mathcal{H}$ contains $A$ as
a subset. In other words, $A\subseteq\mathcal{H}$. Thus, every $a\in A$
satisfies $a\in A\subseteq\mathcal{H}=\left\{ x\in A\ \mid\ f\left(
xb\right) =f\left( x\right) f\left( b\right) \text{ for every }b\in
A\right\} $, so that $f\left( ab\right) =f\left( a\right) f\left(
b\right) $ for every $b\in A$.
We thus have proven that every $a\in A$ and $b\in A$ satisfy $f\left(
ab\right) =f\left( a\right) f\left( b\right) $. Combined with $f\left(
1\right) =1$ and with the $\mathbb{C}$-linearity of the map $f$, this yields
that $f$ is a $\mathbb{C}$-algebra homomorphism. Lemma 2.18a is proven.
\textbf{Lemma 2.18b.} Let $V$ be a finite-dimensional $\mathbb{C}$-vector
space, and let $U$ be a Zariski-dense open subset of $V$. Let $\mathbb{C}%
\left[ U\right] $ and $\mathbb{C}\left[ V\right] $ be the coordinate rings
of $U$ and $V$, respectively (so that $\mathbb{C}\left[ V\right] =S\left(
V^{\ast}\right) $, and $\mathbb{C}\left[ U\right] $ is a localization of
$\mathbb{C}\left[ V\right] $). Let $\tau:V\rightarrow\mathbb{C}\left[
U\right] $ be a $\mathbb{C}$-linear map. Assume that%
\begin{equation}
\left[ \partial_{a}+\tau\left( a\right) ,\partial_{b}+\tau\left( b\right)
\right] =0\ \ \ \ \ \ \ \ \ \ \text{for any }a\in V\text{ and }b\in V.
\label{p10.2.18b.1}%
\end{equation}
Then, there exists a unique $\mathbb{C}$-algebra homomorphism $\varsigma
:\mathcal{D}\left( U\right) \rightarrow\mathcal{D}\left( U\right) $ which
satisfies the following two conditions:
\textit{Condition 1:} We have $\varsigma\left( f\right) =f$ for every
$f\in\mathbb{C}\left[ U\right] $ (where $\mathbb{C}\left[ U\right] $ is
canonically embedded into $\mathcal{D}\left( U\right) $).
\textit{Condition 2:} We have $\varsigma\left( \partial_{a}\right)
=\partial_{a}+\tau\left( a\right) $ for every $a\in V$.
\textit{Proof of Lemma 2.18b.} Since the $\mathbb{C}$-algebra $\mathcal{D}%
\left( U\right) $ is generated by the elements of $\mathbb{C}\left[
U\right] $ and the elements $\partial_{a}$ for $a\in V$, it is clear that
there exists \textbf{at most one} $\mathbb{C}$-algebra homomorphism
$\varsigma:\mathcal{D}\left( U\right) \rightarrow\mathcal{D}\left(
U\right) $ satisfying Conditions 1 and 2. Hence, in order to prove that there
exists \textbf{exactly one} such homomorphism, we need only check that there
exists \textbf{at least one} such homomorphism. Let us do this now.
Let $\mathcal{D}_{\operatorname*{const}}\left( V\right) $ be the
$\mathbb{C}$-algebra of differential operators on $V$ with constant
coefficients. Recall that $\mathcal{D}\left( U\right) =\mathbb{C}\left[
U\right] \otimes\mathcal{D}_{\operatorname*{const}}\left( V\right) $ as a
vector space (where $\otimes$ means $\otimes_{\mathbb{C}}$). In particular,
for any $f\in\mathbb{C}\left[ U\right] $ and any $D\in\mathcal{D}%
_{\operatorname*{const}}\left( V\right) $, the operator $fD\in
\mathcal{D}\left( U\right) $ is the tensor product $f\otimes D\in
\mathbb{C}\left[ U\right] \otimes\mathcal{D}_{\operatorname*{const}}\left(
V\right) $.
Moreover, we can define a map $\partial:V\rightarrow\mathcal{D}%
_{\operatorname*{const}}\left( V\right) $ by%
\[
\partial\left( v\right) =\partial_{v}\ \ \ \ \ \ \ \ \ \ \text{for every
}v\in V.
\]
Then, $\partial$ is a $\mathbb{C}$-linear injection, and the image
$\partial\left( V\right) $ is the space of all degree-$1$ differential
operators on $V$ with constant coefficients. Denote by $\partial^{-1}%
:\partial\left( V\right) \rightarrow V$ the inverse of $\partial$ on
$\partial\left( V\right) $.
Let $D^{\prime}$ be the $\mathbb{C}$-subalgebra of $\mathcal{D}\left(
U\right) $ generated by $\left\{ \partial_{v}+\tau\left( v\right)
\ \mid\ v\in V\right\} $. Then, the algebra $D^{\prime}$ is commutative
(because (\ref{p10.2.18b.1}) shows that its generators commute). Define a
$k$-linear map $\xi:V\rightarrow D^{\prime}$ by%
\[
\xi\left( v\right) =\partial_{v}+\tau\left( v\right)
\ \ \ \ \ \ \ \ \ \ \text{for every }v\in V.
\]
Then, $\xi\circ\partial^{-1}$ is a $\mathbb{C}$-linear map $\partial\left(
V\right) \rightarrow D^{\prime}$. By the universal property of the symmetric
algebra, the $\mathbb{C}$-linear map $\xi\circ\partial^{-1}:\partial\left(
V\right) \rightarrow D^{\prime}$ can be extended to a $\mathbb{C}$-algebra
homomorphism $\Xi:S\left( \partial\left( V\right) \right) \rightarrow
D^{\prime}$ such that%
\begin{equation}
\Xi\left( z\right) =\left( \xi\circ\partial^{-1}\right) \left( z\right)
\ \ \ \ \ \ \ \ \ \ \text{for every }z\in\partial\left( V\right)
\label{p10.2.18b.2}%
\end{equation}
(because $D^{\prime}$ is commutative). Consider this $\Xi$.
Since $\partial\left( V\right) $ is the space of all degree-$1$ differential
operators on $V$ with constant coefficients, we have $\mathcal{D}%
_{\operatorname*{const}}\left( V\right) \cong S\left( \partial\left(
V\right) \right) $. Hence, we can regard $\Xi:S\left( \partial\left(
V\right) \right) \rightarrow D^{\prime}$ as a $\mathbb{C}$-algebra
homomorphism $\mathcal{D}_{\operatorname*{const}}\left( V\right) \rightarrow
D^{\prime}$. Since $\Xi$ is a $\mathbb{C}$-algebra homomorphism, we have
$\Xi\left( 1\right) =1$.
Now, define a $\mathbb{C}$-linear map $\varsigma:\mathbb{C}\left[ U\right]
\otimes\mathcal{D}_{\operatorname*{const}}\left( V\right) \rightarrow
\mathcal{D}\left( U\right) $ by%
\[
\varsigma\left( f\otimes D\right) =f\Xi\left( D\right)
\ \ \ \ \ \ \ \ \ \ \text{for every }f\in\mathbb{C}\left[ U\right] \text{
and }D\in\mathcal{D}_{\operatorname*{const}}\left( V\right) .
\]
Since $\mathbb{C}\left[ U\right] \otimes\mathcal{D}_{\operatorname*{const}%
}\left( V\right) =\mathcal{D}\left( U\right) $, this map $\varsigma$ is a
$\mathbb{C}$-linear map $\mathcal{D}\left( U\right) \rightarrow
\mathcal{D}\left( U\right) $. We claim that $\varsigma$ is a $\mathbb{C}%
$-algebra homomorphism satisfying Conditions 1 and 2.
In fact, every $f\in\mathbb{C}\left[ U\right] $ satisfies%
\begin{align*}
\varsigma\left( \underbrace{f}_{=f\otimes1}\right) & =\varsigma\left(
f\otimes1\right) =f\underbrace{\Xi\left( 1\right) }_{=1}%
\ \ \ \ \ \ \ \ \ \ \left( \text{by the definition of }\varsigma\right) \\
& =f.
\end{align*}
Thus, $\varsigma$ satisfies Condition 1. Applied to $f=1$, Condition 1 yields
$\varsigma\left( 1\right) =1$.
Every $a\in V$ satisfies%
\begin{align*}
\varsigma\left( \underbrace{\partial_{a}}_{=1\otimes\partial_{a}}\right) &
=\zeta\left( 1\otimes\partial_{a}\right) =1\Xi\left( \partial_{a}\right)
\ \ \ \ \ \ \ \ \ \ \left( \text{by the definition of }\varsigma\right) \\
& =\Xi\left( \partial_{a}\right) =\left( \xi\circ\partial^{-1}\right)
\left( \partial_{a}\right) \ \ \ \ \ \ \ \ \ \ \left( \text{by
(\ref{p10.2.18b.2}), applied to }z=\partial_{a}\right) \\
& =\xi\left( \underbrace{\partial^{-1}\left( \partial_{a}\right)
}_{\substack{=a\\\text{(since }\partial_{a}=\partial\left( a\right)
\text{)}}}\right) =\xi\left( a\right) =\partial_{a}+\tau\left( a\right)
\ \ \ \ \ \ \ \ \ \ \left( \text{by the definition of }\xi\right) .
\end{align*}
Thus, $\varsigma$ satisfies Condition 2.
We now will prove that $\varsigma$ is a $\mathbb{C}$-algebra homomorphism. For
this, define a subset $\mathcal{G}$ of $\mathcal{D}\left( U\right) $ by
$\mathcal{G}=\mathbb{C}\left[ U\right] \cup\partial\left( V\right) $.
Then, $\mathcal{G}$ generates $\mathcal{D}\left( U\right) $ as a
$\mathbb{C}$-algebra. Hence, Lemma 2.18a (applied to $A=\mathcal{D}\left(
U\right) $, $D=\mathcal{D}\left( U\right) $ and $f=\varsigma$), in order to
prove that $\varsigma$ is a $\mathbb{C}$-algebra homomorphism, it will be
enough to prove that%
\begin{equation}
\varsigma\left( ab\right) =\varsigma\left( a\right) \varsigma\left(
b\right) \ \ \ \ \ \ \ \ \ \ \text{for every }a\in\mathcal{G}\text{ and }%
b\in\mathcal{D}\left( U\right) \text{.} \label{p10.2.18b.3}%
\end{equation}
So let us prove this now:
\textit{Proof of (\ref{p10.2.18b.3}):} Let $a\in\mathcal{G}$ and
$b\in\mathcal{D}\left( U\right) $. Since the equality (\ref{p10.2.18b.3}) is
$\mathbb{C}$-linear in $b$, we can WLOG assume that $b$ has the form $gE$ for
some $g\in\mathbb{C}\left[ U\right] $ and $E\in\mathcal{D}\left( U\right)
$ (because every element of $\mathcal{D}\left( U\right) $ is a $\mathbb{C}%
$-linear combination of elements of this form). Assume this. Thus,
$b=gE=g\otimes E$. Hence,%
\begin{equation}
\varsigma\left( b\right) =\varsigma\left( g\otimes E\right) =g\Xi\left(
E\right) \ \ \ \ \ \ \ \ \ \ \left( \text{by the definition of }%
\varsigma\right) . \label{p10.2.18b.5}%
\end{equation}
Since $a\in\mathcal{G}=\mathbb{C}\left[ U\right] \cup\partial\left(
V\right) $, we have $a\in\mathbb{C}\left[ U\right] $ or $a\in
\partial\left( V\right) $. Thus, we must be in one of the following cases:
\textit{Case 1:} We have $a\in\mathbb{C}\left[ U\right] $.
\textit{Case 2:} We have $a\in\partial\left( V\right) $.
Let us consider Case 1 first. In this case, $a\in\mathbb{C}\left[ U\right]
$. Hence, $\varsigma\left( a\right) =a$ (by Condition 1, applied to $f=a$),
and%
\begin{align*}
\varsigma\left( a\underbrace{b}_{=gE}\right) & =\varsigma\left(
\underbrace{agE}_{=ag\otimes E}\right) =\varsigma\left( ag\otimes E\right)
=\underbrace{a}_{=\varsigma\left( a\right) }\underbrace{g\Xi\left(
E\right) }_{\substack{=\varsigma\left( b\right) \\\text{(by
(\ref{p10.2.18b.5}))}}}\ \ \ \ \ \ \ \ \ \ \left( \text{by the definition of
}\varsigma\right) \\
& =\varsigma\left( a\right) \varsigma\left( b\right) .
\end{align*}
Hence, (\ref{p10.2.18b.3}) is proven in Case 1.
Let us now consider Case 2. In this case, $a\in\partial\left( V\right) $.
Thus, there exists some $v\in V$ such that $a=\partial_{v}$. Consider this
$v$. Let $\partial_{v}g$ denote the product of the elements $\partial_{v}$ and
$g$ in the $\mathbb{C}$-algebra $\mathcal{D}\left( V\right) $, whereas
$\partial_{v}\left( g\right) $ denotes the image of $g$ under the
differential operator $\partial_{v}$. Then,%
\[
\partial_{v}g=g\partial_{v}+\partial_{v}\left( g\right) ,
\]
so that%
\[
\underbrace{a}_{=\partial_{v}}\underbrace{b}_{=gE}=\underbrace{\partial_{v}%
g}_{=g\partial_{v}+\partial_{v}\left( g\right) }E=g\partial_{v}%
E+\partial_{v}\left( g\right) E=g\otimes\partial_{v}E+\partial_{v}\left(
g\right) \otimes E,
\]
and thus%
\begin{align*}
\varsigma\left( ab\right) & =\varsigma\left( g\otimes\partial
_{v}E+\partial_{v}\left( g\right) \otimes E\right) =\underbrace{\varsigma
\left( g\otimes\partial_{v}E\right) }_{\substack{=g\Xi\left( \partial
_{v}E\right) \\\text{(by the definition of }\varsigma\text{)}}%
}+\underbrace{\varsigma\left( \partial_{v}\left( g\right) \otimes E\right)
}_{\substack{=\partial_{v}\left( g\right) \Xi\left( E\right) \\\text{(by
the definition of }\varsigma\text{)}}}\\
& =g\underbrace{\Xi\left( \partial_{v}E\right) }_{\substack{=\Xi\left(
\partial_{v}\right) \Xi\left( E\right) \\\text{(since }\Xi\text{ is a
}\mathbb{C}\text{-algebra}\\\text{homomorphism)}}}+\partial_{v}\left(
g\right) \Xi\left( E\right) =g\underbrace{\Xi\left( \partial_{v}\right)
}_{\substack{=\left( \xi\circ\partial^{-1}\right) \left( \partial
_{v}\right) \\\text{(by (\ref{p10.2.18b.2}), applied}\\\text{to }%
z=\partial_{v}\text{)}}}\Xi\left( E\right) +\partial_{v}\left( g\right)
\Xi\left( E\right) \\
& =g\underbrace{\left( \xi\circ\partial^{-1}\right) \left( \partial
_{v}\right) }_{=\xi\left( \partial^{-1}\left( \partial_{v}\right) \right)
}\Xi\left( E\right) +\partial_{v}\left( g\right) \Xi\left( E\right)
=g\xi\left( \underbrace{\partial^{-1}\left( \partial_{v}\right)
}_{\substack{=v\\\text{(since }\partial_{v}=\partial\left( v\right)
\text{)}}}\right) \Xi\left( E\right) +\partial_{v}\left( g\right)
\Xi\left( E\right) \\
& =g\underbrace{\xi\left( v\right) }_{\substack{=\partial_{v}+\tau\left(
v\right) \\\text{(by the definition of }\xi\text{)}}}\Xi\left( E\right)
+\partial_{v}\left( g\right) \Xi\left( E\right) =g\left( \partial
_{v}+\tau\left( v\right) \right) \Xi\left( E\right) +\partial_{v}\left(
g\right) \Xi\left( E\right) \\
& =\left( \underbrace{g\left( \partial_{v}+\tau\left( v\right) \right)
}_{=g\partial_{v}+g\tau\left( v\right) }+\partial_{v}\left( g\right)
\right) \Xi\left( E\right) =\left( g\partial_{v}+g\tau\left( v\right)
+\partial_{v}\left( g\right) \right) \Xi\left( E\right) .
\end{align*}
On the other hand, Condition 2 (applied to $v$ instead of $a$) yields
$\varsigma\left( \partial_{v}\right) =\partial_{v}+\tau\left( v\right) $,
so that%
\begin{align*}
\varsigma\left( \underbrace{a}_{=\partial_{v}}\right) \varsigma\left(
\underbrace{b}_{=gE=g\otimes E}\right) & =\underbrace{\varsigma\left(
\partial_{v}\right) }_{=\partial_{v}+\tau\left( v\right) }%
\underbrace{\varsigma\left( g\otimes E\right) }_{\substack{=g\Xi\left(
E\right) \\\text{(by the definition of }\varsigma\text{)}}}\\
& =\left( \partial_{v}+\tau\left( v\right) \right) g\Xi\left( E\right)
=\underbrace{\partial_{v}g}_{=g\partial_{v}+\partial_{v}\left( g\right) }%
\Xi\left( E\right) +\underbrace{\tau\left( v\right) g}_{=g\tau\left(
v\right) }\Xi\left( E\right) \\
& =g\partial_{v}\Xi\left( E\right) +\partial_{v}\left( g\right)
\Xi\left( E\right) +g\tau\left( v\right) \Xi\left( E\right) \\
& =\left( g\partial_{v}+\partial_{v}\left( g\right) +g\tau\left(
v\right) \right) \Xi\left( E\right) =\left( g\partial_{v}+g\tau\left(
v\right) +\partial_{v}\left( g\right) \right) \Xi\left( E\right) \\
& =\varsigma\left( ab\right) .
\end{align*}
Hence, (\ref{p10.2.18b.3}) is proven in Case 2.
So we have proven (\ref{p10.2.18b.3}) in each of the Cases 1 and 2. Since
Cases 1 and 2 are the only possible cases, this yields that (\ref{p10.2.18b.3}%
) always holds.
Thus, Lemma 2.18a (applied to $A=\mathcal{D}\left( U\right) $,
$D=\mathcal{D}\left( U\right) $ and $f=\varsigma$) yields that $\varsigma$
is a $\mathbb{C}$-algebra homomorphism. Hence, $\varsigma$ is a $\mathbb{C}%
$-algebra homomorphism satisfying Conditions 1 and 2. We thus have verified
the existence of a $\mathbb{C}$-algebra homomorphism $\varsigma:\mathcal{D}%
\left( U\right) \rightarrow\mathcal{D}\left( U\right) $ satisfying
Conditions 1 and 2. This completes the proof of Lemma 2.18b.
\textbf{Corollary 2.18c.} Let $\mathfrak{h}$ be a $\mathbb{C}$-vector space.
Let $\mathcal{S}$ be a finite set. For every $s\in\mathcal{S}$, let $c_{s}$ be
an element of $\mathbb{C}$ and let $\alpha_{s}$ be an element of
$\mathfrak{h}^{\ast}$. Let $\mathfrak{h}_{\operatorname*{reg}}$ be a
Zariski-dense open subset of $\mathfrak{h}$ such that every $a\in
\mathfrak{h}_{\operatorname*{reg}}$ and every $s\in\mathcal{S}$ satisfy
$\alpha_{s}\left( a\right) \neq0$. Then, there exists a unique $\mathbb{C}%
$-algebra homomorphism $\varsigma:\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) \rightarrow\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $ which satisfies the following two conditions:
\textit{Condition 1:} We have $\varsigma\left( f\right) =f$ for every
$f\in\mathbb{C}\left[ \mathfrak{h}_{\operatorname*{reg}}\right] $ (where
$\mathbb{C}\left[ \mathfrak{h}_{\operatorname*{reg}}\right] $ is canonically
embedded into $\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $).
\textit{Condition 2:} We have $\varsigma\left( \partial_{a}\right)
=\partial_{a}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left(
a\right) }{\alpha_{s}}$ for every $a\in\mathfrak{h}$.
\textit{Proof of Corollary 2.18c.} Let $V=\mathfrak{h}$ and $U=\mathfrak{h}%
_{\operatorname*{reg}}$. Define a $\mathbb{C}$-linear map $\tau:\mathfrak{h}%
\rightarrow\mathbb{C}\left[ \mathfrak{h}_{\operatorname*{reg}}\right] $ by
\[
\tau\left( a\right) =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha
_{s}\left( a\right) }{\alpha_{s}}\ \ \ \ \ \ \ \ \ \ \text{for every }%
a\in\mathfrak{h}.
\]
Then, obviously, Conditions 1 and 2 of Corollary 2.18c are equivalent to
Conditions 1 and 2 of Lemma 2.18b, respectively.
Every $a\in\mathfrak{h}$ and $b\in\mathfrak{h}$ satisfy%
\begin{align*}
& \left[ \partial_{a}+\tau\left( a\right) ,\partial_{b}+\tau\left(
b\right) \right] \\
& =\left[ \partial_{a}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha
_{s}\left( a\right) }{\alpha_{s}},\partial_{b}+\sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}\alpha_{s}\left( b\right) }{\alpha_{s}}\right] \\
& \ \ \ \ \ \ \ \ \ \ \left( \text{since }\tau\left( a\right)
=\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( a\right) }%
{\alpha_{s}}\text{ and }\tau\left( b\right) =\sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}\alpha_{s}\left( b\right) }{\alpha_{s}}\right) \\
& =\underbrace{\left[ \partial_{a},\partial_{b}\right] }_{=0}+\left[
\partial_{a},\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left(
b\right) }{\alpha_{s}}\right] +\underbrace{\left[ \sum\limits_{s\in
\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( a\right) }{\alpha_{s}},\partial
_{b}\right] }_{=-\left[ \partial_{b},\sum\limits_{s\in\mathcal{S}}%
\dfrac{c_{s}\alpha_{s}\left( a\right) }{\alpha_{s}}\right] }%
+\underbrace{\left[ \sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha
_{s}\left( a\right) }{\alpha_{s}},\sum\limits_{s\in\mathcal{S}}\dfrac
{c_{s}\alpha_{s}\left( b\right) }{\alpha_{s}}\right] }_{=0}\\
& =\underbrace{\left[ \partial_{a},\sum\limits_{s\in\mathcal{S}}\dfrac
{c_{s}\alpha_{s}\left( b\right) }{\alpha_{s}}\right] }_{=\partial
_{a}\left( \sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left(
b\right) }{\alpha_{s}}\right) =\sum\limits_{s\in\mathcal{S}}c_{s}\alpha
_{s}\left( b\right) \cdot\partial_{a}\left( \dfrac{1}{\alpha_{s}}\right)
}-\underbrace{\left[ \partial_{b},\sum\limits_{s\in\mathcal{S}}\dfrac
{c_{s}\alpha_{s}\left( a\right) }{\alpha_{s}}\right] }_{=\partial
_{b}\left( \sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left(
a\right) }{\alpha_{s}}\right) =\sum\limits_{s\in\mathcal{S}}c_{s}\alpha
_{s}\left( a\right) \cdot\partial_{b}\left( \dfrac{1}{\alpha_{s}}\right)
}\\
& =\sum\limits_{s\in\mathcal{S}}c_{s}\alpha_{s}\left( b\right)
\cdot\underbrace{\partial_{a}\left( \dfrac{1}{\alpha_{s}}\right)
}_{\substack{=-\dfrac{\partial_{a}\left( \alpha_{s}\right) }{\alpha_{s}^{2}%
}=-\dfrac{\alpha_{s}\left( a\right) }{\alpha_{s}^{2}}\\\text{(since
}\partial_{a}\left( \alpha_{s}\right) =\alpha_{s}\left( a\right) \text{
(because }\alpha_{s}\\\text{is linear))}}}-\sum\limits_{s\in\mathcal{S}}%
c_{s}\alpha_{s}\left( a\right) \cdot\underbrace{\partial_{b}\left(
\dfrac{1}{\alpha_{s}}\right) }_{\substack{=-\dfrac{\partial_{b}\left(
\alpha_{s}\right) }{\alpha_{s}^{2}}=.\dfrac{\alpha_{s}\left( b\right)
}{\alpha_{s}^{2}}\\\text{(since }\partial_{b}\left( \alpha_{s}\right)
=\alpha_{s}\left( b\right) \text{ (because }\alpha_{s}\\\text{is linear))}%
}}\\
& =\sum\limits_{s\in\mathcal{S}}\underbrace{c_{s}\alpha_{s}\left( b\right)
\cdot\left( -\dfrac{\alpha_{s}\left( a\right) }{\alpha_{s}^{2}}\right)
}_{=\dfrac{-c_{s}\alpha_{s}\left( a\right) \alpha_{s}\left( b\right)
}{\alpha_{s}^{2}}}-\sum\limits_{s\in\mathcal{S}}\underbrace{c_{s}\alpha
_{s}\left( a\right) \cdot\left( -\dfrac{\alpha_{s}\left( b\right)
}{\alpha_{s}^{2}}\right) }_{=\dfrac{-c_{s}\alpha_{s}\left( a\right)
\alpha_{s}\left( b\right) }{\alpha_{s}^{2}}}\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( a\right)
\alpha_{s}\left( b\right) }{\alpha_{s}^{2}}-\sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}\alpha_{s}\left( a\right) \alpha_{s}\left( b\right) }%
{\alpha_{s}^{2}}=0.
\end{align*}
Hence, Lemma 2.18b yields that there exists a unique $\mathbb{C}$-algebra
homomorphism $\varsigma:\mathcal{D}\left( U\right) \rightarrow
\mathcal{D}\left( U\right) $ which satisfies the Conditions 1 and 2 of Lemma
2.18b. In other words, there exists a unique $\mathbb{C}$-algebra homomorphism
$\varsigma:\mathcal{D}\left( U\right) \rightarrow\mathcal{D}\left(
U\right) $ which satisfies the Conditions 1 and 2 of Corollary 2.18c (because
we know that Conditions 1 and 2 of Corollary 2.18c are equivalent to
Conditions 1 and 2 of Lemma 2.18b, respectively). Corollary 2.18c is thus proven.
\textbf{Definition.} Let $\mathfrak{h}$ be a $\mathbb{C}$-vector space with a
nondegenerate bilinear inner product $\left( \cdot,\cdot\right) $. Let
$W\subseteq\operatorname*{O}\left( \mathfrak{h}\right) $ be a real
reflection group, and $\mathcal{S}\subseteq W$ the set of reflections. Let
$c:\mathcal{S}\rightarrow\mathbb{C}$ be a function invariant under conjugation
(by elements of $W$). For every $s\in\mathcal{S}$, we will write $c_{s}$ for
$c\left( s\right) $. For every $s\in\mathcal{S}$, let $\alpha_{s}%
\in\mathfrak{h}^{\ast}$ be the unique (up to scaling by an element of
$\mathbb{C}^{\times}$) nonzero eigenvector of $s$ (acting on $\mathfrak{h}%
^{\ast}$) with eigenvalue $-1$, and let $\alpha_{s}^{\vee}\in\mathfrak{h}$ be
the unique (up to scaling by an element of $\mathbb{C}^{\times}$) nonzero
eigenvector of $s$ (acting on $\mathfrak{h}$) with eigenvalue $-1$. Define $H$
as in Definition 2.8, and define $\overline{H}$ as in Proposition 2.16. Let
$\mathfrak{h}_{\operatorname*{reg}}$ be the subset $\left\{ x\in
\mathfrak{h}\ \mid\ W_{x}=\left\{ \operatorname*{id}\right\} \right\} $ of
$\mathfrak{h}$. According to Corollary 2.18c, there exists a unique
$\mathbb{C}$-algebra homomorphism $\varsigma:\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) \rightarrow\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $ which satisfies the Conditions 1 and 2 of
Corollary 2.18c\footnote{In fact, every $a\in\mathfrak{h}_{\operatorname*{reg}%
}$ and every $s\in\mathcal{S}$ satisfy $\alpha_{s}\left( a\right) \neq0$
(because otherwise, $a$ would be fixed under $s$, contradicting $W_{a}%
=\left\{ \operatorname*{id}\right\} $).}. This homomorphism $\varsigma$ will
be denoted by $\varsigma_{c}$. Due to Condition 1, it satisfies%
\begin{equation}
\varsigma_{c}\left( f\right) =f\ \ \ \ \ \ \ \ \ \ \text{for every }%
f\in\mathbb{C}\left[ \mathfrak{h}_{\operatorname*{reg}}\right]
\label{p10.2.18c.1}%
\end{equation}
(where $\mathbb{C}\left[ \mathfrak{h}_{\operatorname*{reg}}\right] $ is
canonically embedded into $\mathcal{D}\left( \mathfrak{h}%
_{\operatorname*{reg}}\right) $). Due to Condition 2, it satisfies%
\begin{equation}
\varsigma_{c}\left( \partial_{a}\right) =\partial_{a}+\sum\limits_{s\in
\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( a\right) }{\alpha_{s}%
}\ \ \ \ \ \ \ \ \ \ \text{for every }a\in\mathfrak{h}. \label{p10.2.18c.2}%
\end{equation}
\textbf{Remark.} In terms of your Proposition 2.18, this homomorphism
$\varsigma_{c}$ is the conjugation by $\delta_{c}$ (that is, it is given by
$D\mapsto\delta_{c}^{-1}\circ D\circ\delta_{c}$). However, our definition of
$\varsigma_{c}$ was purely algebraic, while your $\delta_{c}$ is a
transcendental function (in general).
Now, our elementary version of Proposition 2.18 rewrites as follows:
\textbf{Proposition 2.18d.} We have $\varsigma_{c}\left( \overline{H}\right)
=H$.
Before we prove this, another lemma:
\textbf{Lemma 2.18e.} Let $\mathfrak{h}$ be a $\mathbb{C}$-vector space with a
nondegenerate bilinear inner product $\left( \cdot,\cdot\right) $. Let
$W\subseteq\operatorname*{O}\left( \mathfrak{h}\right) $ be a real
reflection group, and $\mathcal{S}\subseteq W$ the set of reflections. Let
$c:\mathcal{S}\rightarrow\mathbb{C}$ be a function invariant under conjugation
(by elements of $W$). For every $s\in\mathcal{S}$, we will write $c_{s}$ for
$c\left( s\right) $. For every $s\in\mathcal{S}$, let $\alpha_{s}%
\in\mathfrak{h}^{\ast}$ be the unique (up to scaling by an element of
$\mathbb{C}^{\times}$) nonzero eigenvector of $s$ (acting on $\mathfrak{h}%
^{\ast}$) with eigenvalue $-1$. Let $\mathfrak{h}_{\operatorname*{reg}}$ be
the subset $\left\{ x\in\mathfrak{h}\ \mid\ G_{x}=\left\{ \operatorname*{id}%
\right\} \right\} $ of $\mathfrak{h}$. Then:
\textbf{(a)} Every $t\in\mathcal{S}$ satisfies%
\[
t\left( \prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) =-\prod
\limits_{s\in\mathcal{S}}\alpha_{s}%
\]
(where $t\left( \prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) $ denotes
the action of $t\in W$ on $\prod\limits_{s\in\mathcal{S}}\alpha_{s}\in
S\left( \mathfrak{h}^{\ast}\right) $).
\textbf{(b)} We have%
\[
\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}%
}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha
_{u}}=0.
\]
\textbf{(c)} We have%
\[
\sum\limits_{s\in\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}\left(
\alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}}=\sum\limits_{s\in
\mathcal{S}}\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha
_{s}^{2}}.
\]
\textit{Proof of Lemma 2.18e.} Let us notice that
\begin{equation}
\text{if two }t\in\mathcal{S}\text{ and }s\in\mathcal{S}\text{ satisfy
}\operatorname*{Ker}\left( \alpha_{t}\right) \subseteq\operatorname*{Ker}%
\left( \alpha_{s}\right) \text{, then }t=s \label{p10.2.18e.1}%
\end{equation}
\footnote{\textit{Proof of (\ref{p10.2.18e.1}):} Let $t\in\mathcal{S}$ and
$s\in\mathcal{S}$ satisfy $\operatorname*{Ker}\left( \alpha_{t}\right)
\subseteq\operatorname*{Ker}\left( \alpha_{s}\right) $. Then,
$\operatorname*{Ker}\left( \alpha_{t}\right) =\operatorname*{Ker}\left(
\alpha_{s}\right) $ (since $\operatorname*{Ker}\left( \alpha_{t}\right) $
and $\operatorname*{Ker}\left( \alpha_{s}\right) $ are hyperplanes in
$\mathfrak{h}$, and thus have the same dimension).
\par
But $s$ is the reflection in the hyperplane $\operatorname*{Ker}\left(
\alpha_{s}\right) $ (because $s$ is a reflection, and $\alpha_{s}%
\in\mathfrak{h}^{\ast}$ is the unique (up to scaling by an element of
$\mathbb{C}^{\times}$) nonzero eigenvector of $s$ (acting on $\mathfrak{h}%
^{\ast}$) with eigenvalue $-1$). Similarly, $t$ is the reflection in the
hyperplane $\operatorname*{Ker}\left( \alpha_{t}\right) $. Thus,%
\[
s=\left( \text{the reflection in the hyperplane }%
\underbrace{\operatorname*{Ker}\left( \alpha_{s}\right) }%
_{=\operatorname*{Ker}\left( \alpha_{t}\right) }\right) =\left( \text{the
reflection in the hyperplane }\operatorname*{Ker}\left( \alpha_{t}\right)
\right) =t,
\]
qed.}. As a consequence, the polynomials $\alpha_{s}\in\mathbb{C}\left[
\mathfrak{h}\right] $ for $s\in\mathcal{S}$ are pairwise
coprime.\footnote{\textit{Proof.} Assume the contrary. Then, there exist two
distinct elements $t\in\mathcal{S}$ and $s\in\mathcal{S}$ such that the
polynomials $\alpha_{s}$ and $\alpha_{t}$ have a nontrivial common divisor.
Consider these $t$ and $s$. The polynomials $\alpha_{s}$ and $\alpha_{t}$ have
a nontrivial common divisor, but are both linear. Therefore, $\alpha_{s}$ and
$\alpha_{t}$ must be proportional to each other, i. e., there exists a
$\lambda\in\mathbb{C}^{\times}$ such that $\alpha_{s}=\lambda\alpha_{t}$.
Therefore, $\operatorname*{Ker}\left( \alpha_{t}\right) =\operatorname*{Ker}%
\left( \alpha_{s}\right) $, so that $t=s$ (by (\ref{p10.2.18e.1})),
contradicting the assumption that $t$ and $s$ be distinct. This contradiction
proves that our assumption was wrong, qed.}
Also, for every $t\in\mathcal{S}$ and every $s\in\mathcal{S}$, we have
$t\alpha_{s}\in\mathbb{C}^{\times}\alpha_{tst^{-1}}$%
\ \ \ \ \footnote{\textit{Proof.} Let $t\in\mathcal{S}$ and $s\in\mathcal{S}$.
Then, $\alpha_{s}$ is a nonzero eigenvector of $s$ (acting on $\mathfrak{h}%
^{\ast}$) with eigenvalue $-1$. Thus, $s\alpha_{s}=-1\alpha_{s}=-\alpha_{s}$,
so that $\left( tst^{-1}\right) \left( t\alpha_{s}\right)
=t\underbrace{s\alpha_{s}}_{=-\alpha_{s}}=-t\alpha_{s}$. In other words,
$t\alpha_{s}$ is an eigenvector of $tst^{-1}$ (acting on $\mathfrak{h}^{\ast}%
$) with eigenvalue $-1$. Also, $t\alpha_{s}\neq0$ (since $\alpha_{s}\neq0$).
Hence, $t\alpha_{s}$ is a nonzero eigenvector of $tst^{-1}$ (acting on
$\mathfrak{h}^{\ast}$) with eigenvalue $-1$. Thus, $t\alpha_{s}\in
\mathbb{C}^{\times}\alpha_{tst^{-1}}$ (because $\alpha_{tst^{-1}}$ is the
unique (up to scaling by an element of $\mathbb{C}^{\times}$) nonzero
eigenvector of $tst^{-1}$ (acting on $\mathfrak{h}^{\ast}$) with eigenvalue
$-1$), qed.}. In other words, for every $t\in\mathcal{S}$ and every
$s\in\mathcal{S}$, there exists a $\mu_{t,s}\in\mathbb{C}^{\times}$ such that%
\begin{equation}
t\alpha_{s}=\mu_{t,s}\alpha_{tst^{-1}}. \label{p10.2.18e.mu}%
\end{equation}
Consider these $\mu_{t,s}$.
\textbf{(a)} Let $t\in\mathcal{S}$. Then,%
\begin{align*}
t\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha
_{s}\right) & =\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\underbrace{\left( t\alpha_{s}\right) }_{=\mu_{t,s}\alpha_{tst^{-1}}}%
=\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}%
\underbrace{\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }%
\alpha_{tst^{-1}}}_{\substack{=\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\alpha_{s}\\\text{(because the map}\\\mathcal{S}\diagdown\left\{
t\right\} \rightarrow\mathcal{S}\diagdown\left\{ t\right\} ,\ s\mapsto
tst^{-1}\\\text{is a bijection)}}}\\
& =\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}%
\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}.
\end{align*}
Hence,%
\begin{align*}
t^{2}\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }%
\alpha_{s}\right) & =t\cdot\underbrace{t\left( \prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}\right) }_{=\prod
\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}\prod
\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}}\\
& =\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}%
\cdot\underbrace{t\left( \prod\limits_{s\in\mathcal{S}}\alpha_{s}\right)
}_{=\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}%
\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}}=\left(
\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}\right)
^{2}\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}.
\end{align*}
Compared with $t^{2}\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\alpha_{s}\right) =\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\alpha_{s}$ (because $t$ is a reflection and thus satisfies
$t^{2}=\operatorname*{id}$), this yields $\left( \prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}\right) ^{2}\prod
\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}%
=\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}$. Since
$\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}$ is
nonzero, this yields $\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\mu_{t,s}\right) ^{2}=1$. Hence, $\prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}=1$ or $\prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}=-1$.
Let us first assume that $\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\mu_{t,s}=-1$. In this case,%
\begin{equation}
t\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha
_{s}\right) =\underbrace{\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\mu_{t,s}}_{=-1}\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\alpha_{s}=-\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\alpha_{s}. \label{p10.2.18e.2}%
\end{equation}
Now, $\operatorname*{Ker}\left( \alpha_{t}\right) \not \subseteq
\bigcup\limits_{s\in\mathcal{S};\ s\neq t}\operatorname*{Ker}\left(
\alpha_{s}\right) $\ \ \ \ \footnote{\textit{Proof.} Assume the contrary.
Then, $\operatorname*{Ker}\left( \alpha_{t}\right) \subseteq\bigcup
\limits_{s\in\mathcal{S};\ s\neq t}\operatorname*{Ker}\left( \alpha
_{s}\right) $. Since $\operatorname*{Ker}\left( \alpha_{t}\right) $ and
$\operatorname*{Ker}\left( \alpha_{s}\right) $ are vector subspaces of
$\mathfrak{h}$, this yields that there exists some $s\in\mathcal{S}$ such that
$s\neq t$ and $\operatorname*{Ker}\left( \alpha_{t}\right) \subseteq
\operatorname*{Ker}\left( \alpha_{s}\right) $ (because there is a well-known
linear-algebraic fact that if a vector subspace $U$ of a finite-dimensional
$\mathbb{C}$-vector space $V$ is a subset of the union $\bigcup\limits_{i\in
I}W_{i}$ of finitely many subspaces $W_{i}$ of $V$, then there exists some
$i\in I$ such that $U\subseteq W_{i}$). Consider this $s$. Then,
$\operatorname*{Ker}\left( \alpha_{t}\right) \subseteq\operatorname*{Ker}%
\left( \alpha_{s}\right) $, so that $t=s$ (by (\ref{p10.2.18e.1})),
contradicting $s\neq t$. This contradiction shows that our assumption was
wrong, qed.}. Hence, there exists a $p\in\operatorname*{Ker}\left( \alpha
_{t}\right) $ such that $p\notin\bigcup\limits_{s\in\mathcal{S};\ s\neq
t}\operatorname*{Ker}\left( \alpha_{s}\right) $. Pick such a $p$.
Now, $t$ is the reflection in the hyperplane $\operatorname*{Ker}\left(
\alpha_{t}\right) $ (because $t$ is a reflection, and $\alpha_{t}%
\in\mathfrak{h}^{\ast}$ is the unique (up to scaling by an element of
$\mathbb{C}^{\times}$) nonzero eigenvector of $t$ (acting on $\mathfrak{h}%
^{\ast}$) with eigenvalue $-1$). Thus, $\operatorname*{Ker}\left( \alpha
_{t}\right) =\left\{ \text{set of fixed points of }t\text{ in }%
\mathfrak{h}\right\} $.
Since $p\in\operatorname*{Ker}\left( \alpha_{t}\right) =\left\{ \text{set
of fixed points of }t\text{ in }\mathfrak{h}\right\} $, the point $p$ is
fixed under $t$, so that $tp=p$ and thus $t^{-1}p=p$. Thus,%
\[
\left( t\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\alpha_{s}\right) \right) \left( p\right) =\left( \prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}\right)
\underbrace{\left( t^{-1}p\right) }_{=p}=\left( \prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}\right) \left( p\right)
.
\]
Compared to%
\[
\underbrace{\left( t\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\alpha_{s}\right) \right) }_{\substack{=-\prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}\\\text{(by
(\ref{p10.2.18e.2}))}}}\left( p\right) =-\left( \prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}\right) \left( p\right)
,
\]
this yields $\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\alpha_{s}\right) \left( p\right) =-\left( \prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}\right) \left( p\right)
$. Thus, $\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\alpha_{s}\right) \left( p\right) =0$. In other words, $\prod
\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}\left(
p\right) =0$. Hence, there exists some $s\in\mathcal{S}\diagdown\left\{
t\right\} $ such that $p\in\operatorname*{Ker}\left( \alpha_{s}\right) $.
In other words, $p\in\bigcup\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\operatorname*{Ker}\left( \alpha_{s}\right) =\bigcup
\limits_{s\in\mathcal{S};\ s\neq t}\operatorname*{Ker}\left( \alpha
_{s}\right) $, contradicting $p\notin\bigcup\limits_{s\in\mathcal{S};\ s\neq
t}\operatorname*{Ker}\left( \alpha_{s}\right) $.
This contradiction shows that our assumption (the assumption that
$\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}=-1$) was
wrong. So we don't have $\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\mu_{t,s}=-1$. Since we know that we have $\prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}=1$ or $\prod\limits_{s\in
\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}=-1$, this yields that
$\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\mu_{t,s}=1$.
But we know that $\alpha_{t}$ is an eigenvector of $t$ (acting on
$\mathfrak{h}^{\ast}$) with eigenvalue $-1$. Thus, $t\alpha_{t}=-1\alpha
_{t}=-\alpha_{t}$.
Now, $\prod\limits_{s\in\mathcal{S}}\alpha_{s}=\alpha_{t}\cdot\prod
\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\} }\alpha_{s}$, so that%
\begin{align*}
t\left( \prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) & =t\left(
\alpha_{t}\cdot\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\alpha_{s}\right) =\underbrace{t\alpha_{t}}_{=-\alpha_{t}}\cdot
\underbrace{t\left( \prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\alpha_{s}\right) }_{=\prod\limits_{s\in\mathcal{S}\diagdown\left\{
t\right\} }\mu_{t,s}\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\alpha_{s}}\\
& =-\underbrace{\prod\limits_{s\in\mathcal{S}\diagdown\left\{ t\right\}
}\mu_{t,s}}_{=1}\cdot\underbrace{\alpha_{t}\prod\limits_{s\in\mathcal{S}%
\diagdown\left\{ t\right\} }\alpha_{s}}_{=\prod\limits_{s\in\mathcal{S}%
}\alpha_{s}}=-\prod\limits_{s\in\mathcal{S}}\alpha_{s}.
\end{align*}
This proves Lemma 2.18e \textbf{(a)}.
\textbf{(b)} Let $P$ be the function%
\[
\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}%
}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha
_{u}}\in\mathbb{C}\left[ \mathfrak{h}_{\operatorname*{reg}}\right] .
\]
Then, every $t\in\mathcal{S}$ satisfies%
\begin{align*}
tP & =t\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq
u}}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}%
\alpha_{u}}=\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq
u}}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\left(
t\alpha_{s}\right) \left( t\alpha_{u}\right) }=\sum\limits_{\substack{s\in
\mathcal{S};\ u\in\mathcal{S};\\s\neq u}}\dfrac{c_{s}c_{u}\left( t\alpha
_{s},t\alpha_{u}\right) }{\left( t\alpha_{s}\right) \left( t\alpha
_{u}\right) }\\
& \ \ \ \ \ \ \ \ \ \ \left( \text{since }t\in\mathcal{S}\subseteq
W\subseteq\operatorname*{O}\left( \mathfrak{h}\right) \text{ and thus
}\left( \alpha_{s},\alpha_{u}\right) =\left( t\alpha_{s},t\alpha
_{u}\right) \right) \\
& =\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}%
}\dfrac{c_{s}c_{u}\left( \mu_{t,s}\alpha_{tst^{-1}},\mu_{t,u}\alpha
_{tut^{-1}}\right) }{\mu_{t,s}\alpha_{tst^{-1}}\cdot\mu_{t,u}\alpha
_{tut^{-1}}}\\
& \ \ \ \ \ \ \ \ \ \ \left( \text{since (\ref{p10.2.18e.mu}) yields
}t\alpha_{s}=\mu_{t,s}\alpha_{tst^{-1}}\text{ and }t\alpha_{u}=\mu_{t,u}%
\alpha_{tut^{-1}}\right) \\
& =\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}%
}\dfrac{c_{s}c_{u}\mu_{t,s}\mu_{t,u}\left( \alpha_{tst^{-1}},\alpha
_{tut^{-1}}\right) }{\mu_{t,s}\alpha_{tst^{-1}}\cdot\mu_{t,u}\alpha
_{tut^{-1}}}=\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq
u}}\dfrac{c_{s}c_{u}\left( \alpha_{tst^{-1}},\alpha_{tut^{-1}}\right)
}{\alpha_{tst^{-1}}\alpha_{tut^{-1}}}\\
& =\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}%
}\dfrac{c_{tst^{-1}}c_{tut^{-1}}\left( \alpha_{tst^{-1}},\alpha_{tut^{-1}%
}\right) }{\alpha_{tst^{-1}}\alpha_{tut^{-1}}}\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{since the function }c\text{ is invariant under}\\
\text{conjugation, and thus }c_{s}=c_{tst^{-1}}\text{ and }c_{u}=c_{tut^{-1}}%
\end{array}
\right) \\
& =\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}%
}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha
_{u}}\\
& \ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{here, we substituted }\left( s,u\right) \text{ for }\left(
tst^{-1},tut^{-1}\right) \text{ in the sum, because the map}\\
\left\{ \left( s,u\right) \in\mathcal{S}\times\mathcal{S}\ \mid\ s\neq
u\right\} \rightarrow\left\{ \left( s,u\right) \in\mathcal{S}%
\times\mathcal{S}\ \mid\ s\neq u\right\} ,\ \left( s,u\right)
\mapsto\left( tst^{-1},tut^{-1}\right) \\
\text{is a bijection}%
\end{array}
\right) \\
& =P.
\end{align*}
Moreover, since $P=\sum\limits_{\substack{s\in\mathcal{S};\ u\in
\mathcal{S};\\s\neq u}}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right)
}{\alpha_{s}\alpha_{u}}$ and $\prod\limits_{s\in\mathcal{S}}\alpha_{s}%
=\prod\limits_{q\in\mathcal{S}}\alpha_{q}$, we have%
\begin{align*}
P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s} & =\sum
\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}}\dfrac
{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}}%
\cdot\prod\limits_{q\in\mathcal{S}}\alpha_{q}=\sum\limits_{\substack{s\in
\mathcal{S};\ u\in\mathcal{S};\\s\neq u}}c_{s}c_{u}\left( \alpha_{s}%
,\alpha_{u}\right) \cdot\underbrace{\dfrac{\prod\limits_{q\in\mathcal{S}%
}\alpha_{q}}{\alpha_{s}\alpha_{u}}}_{=\prod\limits_{q\in\mathcal{S}%
\diagdown\left\{ s,u\right\} }\alpha_{q}}\\
& =\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}%
}c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) \cdot\prod\limits_{q\in
\mathcal{S}\diagdown\left\{ s,u\right\} }\alpha_{q}.
\end{align*}
This yields immediately that $P\cdot\prod\limits_{s\in\mathcal{S}}\alpha
_{s}\in\mathbb{C}\left[ \mathfrak{h}\right] $ and $\deg\left( P\cdot
\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) \leq\left\vert \mathcal{S}%
\right\vert -2$. Also, every $t\in\mathcal{S}$ satisfies%
\begin{equation}
t\left( P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right)
=\underbrace{tP}_{=P}\cdot\underbrace{t\left( \prod\limits_{s\in\mathcal{S}%
}\alpha_{s}\right) }_{\substack{=-\prod\limits_{s\in\mathcal{S}}\alpha
_{s}\\\text{(by Lemma 2.18e \textbf{(a)})}}}=-P\cdot\prod\limits_{s\in
\mathcal{S}}\alpha_{s}. \label{p10.2.18e.6}%
\end{equation}
Thus, for every $t\in\mathcal{S}$, we have $\alpha_{t}\mid P\cdot
\prod\limits_{s\in\mathcal{S}}\alpha_{s}$ in $\mathbb{C}\left[ \mathfrak{h}%
\right] $\ \ \ \ \footnote{\textit{Proof.} Let $t\in\mathcal{S}$. We know
that $t$ is the reflection in the hyperplane $\operatorname*{Ker}\left(
\alpha_{t}\right) $ (because $t$ is a reflection, and $\alpha_{t}%
\in\mathfrak{h}^{\ast}$ is the unique (up to scaling by an element of
$\mathbb{C}^{\times}$) nonzero eigenvector of $t$ (acting on $\mathfrak{h}%
^{\ast}$) with eigenvalue $-1$). Thus, $\operatorname*{Ker}\left( \alpha
_{t}\right) =\left\{ \text{set of fixed points of }t\text{ in }%
\mathfrak{h}\right\} $.
\par
Now, let $x\in\operatorname*{Ker}\left( \alpha_{t}\right) $. Then, $tx=x$
(because $x\in\operatorname*{Ker}\left( \alpha_{t}\right) =\left\{
\text{set of fixed points of }t\text{ in }\mathfrak{h}\right\} $) and thus
$t^{-1}x=x$, so that%
\[
\left( t\left( P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right)
\right) \left( x\right) =\left( P\cdot\prod\limits_{s\in\mathcal{S}}%
\alpha_{s}\right) \left( \underbrace{t^{-1}x}_{=x}\right) =\left(
P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) \left( x\right) .
\]
Compared to%
\[
\underbrace{\left( t\left( P\cdot\prod\limits_{s\in\mathcal{S}}\alpha
_{s}\right) \right) }_{\substack{=-P\cdot\prod\limits_{s\in\mathcal{S}%
}\alpha_{s}\\\text{(by (\ref{p10.2.18e.6}))}}}\left( x\right) =-\left(
P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) \left( x\right) ,
\]
this yields $\left( P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right)
\left( x\right) =-\left( P\cdot\prod\limits_{s\in\mathcal{S}}\alpha
_{s}\right) \left( x\right) $, so that $\left( P\cdot\prod\limits_{s\in
\mathcal{S}}\alpha_{s}\right) \left( x\right) =0$.
\par
Now forget that we fixed $x$. We thus have proven that every $x\in
\operatorname*{Ker}\left( \alpha_{t}\right) $ satisfies $\left( P\cdot
\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) \left( x\right) =0$. In
other words, the polynomial $P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}$
vanishes on the kernel of the linear function $\alpha_{t}$. Thus, $\alpha
_{t}\mid P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}$ in $\mathbb{C}\left[
\mathfrak{h}\right] $ (because a polynomial which vanishes on the kernel of a
linear function must be divisible by that function), qed.}. Since the
polynomials $\alpha_{s}\in\mathbb{C}\left[ \mathfrak{h}\right] $ for
$s\in\mathcal{S}$ are pairwise coprime, this yields that $\prod\limits_{t\in
\mathcal{S}}\alpha_{t}\mid P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}$ in
$\mathbb{C}\left[ \mathfrak{h}\right] $ (because $\mathbb{C}\left[
\mathfrak{h}\right] $ is a unique factorization domain). Since $\deg\left(
P\cdot\prod\limits_{s\in\mathcal{S}}\alpha_{s}\right) \leq\left\vert
\mathcal{S}\right\vert -2<\left\vert \mathcal{S}\right\vert =\deg\left(
\prod\limits_{t\in\mathcal{S}}\alpha_{t}\right) $, this leads to $P\cdot
\prod\limits_{s\in\mathcal{S}}\alpha_{s}=0$ (because if a polynomial is
divisible by a polynomial of greater degree, then the former polynomial must
be $0$). Hence, $P=0$ (since $\mathbb{C}\left[ \mathfrak{h}%
_{\operatorname*{reg}}\right] $ is an integral domain, and $\prod
\limits_{s\in\mathcal{S}}\alpha_{s}\neq0$). Since $P=\sum
\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}}\dfrac
{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}}$,
this rewrites as $\sum\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S}%
;\\s\neq u}}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }%
{\alpha_{s}\alpha_{u}}=0$. Lemma 2.18e \textbf{(b)} is proven.
\textbf{(c)} We have%
\begin{align*}
\sum\limits_{s\in\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}\left(
\alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}} & =\underbrace{\sum
\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s\neq u}}\dfrac
{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}}%
}_{\substack{=0\\\text{(by Lemma 2.18e \textbf{(b)})}}}+\underbrace{\sum
\limits_{\substack{s\in\mathcal{S};\ u\in\mathcal{S};\\s=u}}\dfrac{c_{s}%
c_{u}\left( \alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}}}%
_{=\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}\alpha_{s}}}\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}c_{s}\left( \alpha_{s}%
,\alpha_{s}\right) }{\alpha_{s}\alpha_{s}}=\sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}},
\end{align*}
and thus Lemma 2.18e \textbf{(c)} is proven.
\textit{Proof of Proposition 2.18d.} Let $\left\{ y_{1},y_{2},...,y_{r}%
\right\} $ be an orthonormal basis of $\mathfrak{h}$. Then, by the definition
of the Laplace operator, $\Delta_{\mathfrak{h}}=\sum\limits_{i=1}^{r}%
\partial_{y_{i}}^{2}$.
For every $s\in\mathcal{S}$, we have%
\begin{equation}
\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) y_{i}=\dfrac{1}{2}\left(
\alpha_{s},\alpha_{s}\right) \alpha_{s}^{\vee}. \label{p10.2.18d.1}%
\end{equation}
\footnote{\textit{Proof of (\ref{p10.2.18d.1}):} Let $s\in\mathcal{S}$. Then,
the bilinear form $\left( \cdot,\cdot\right) $ is $W$-invariant (because
$W\subseteq\operatorname*{O}\left( \mathfrak{h}\right) $).
\par
Since the bilinear form $\left( \cdot,\cdot\right) $ is nondegenerate, it
induces an isomorphism $J:\mathfrak{h}^{\ast}\rightarrow\mathfrak{h}$. This
isomorphism $J$ is $W$-linear (since $\left( \cdot,\cdot\right) $ is
$W$-invariant). Also, it satisfies%
\[
J\left( \varphi\right) =\sum\limits_{i=1}^{r}\varphi\left( y_{i}\right)
y_{i}\ \ \ \ \ \ \ \ \ \ \text{for every }\varphi\in\mathfrak{h}^{\ast}%
\]
(since $\left\{ y_{1},y_{2},...,y_{r}\right\} $ is an orthonormal basis of
$\mathfrak{h}$). Applied to $\varphi=\alpha_{s}$, this yields%
\[
J\left( \alpha_{s}\right) =\sum\limits_{i=1}^{r}\alpha_{s}\left(
y_{i}\right) y_{i}.
\]
But since $\alpha_{s}$ is an eigenvector of $s$ (acting on $\mathfrak{h}%
^{\ast}$) with eigenvalue $-1$, we have $s\alpha_{s}=-1\alpha_{s}=-\alpha_{s}%
$. Thus, $J\left( s\alpha_{s}\right) =J\left( -\alpha_{s}\right)
=-J\left( \alpha_{s}\right) $. Compared with $J\left( s\alpha_{s}\right)
=sJ\left( \alpha_{s}\right) $ (since $J$ is $W$-linear), this yields
$sJ\left( \alpha_{s}\right) =-J\left( \alpha_{s}\right) =-1J\left(
\alpha_{s}\right) $. In other words, $J\left( \alpha_{s}\right) $ is an
eigenvector of $s$ (acting on $\mathfrak{h}$) with eigenvalue $-1$. This
yields that $J\left( \alpha_{s}\right) \in\mathbb{C}\alpha_{s}^{\vee}$
(because $\alpha_{s}^{\vee}\in\mathfrak{h}$ is the unique (up to scaling by an
element of $\mathbb{C}^{\times}$) nonzero eigenvector of $s$ (acting on
$\mathfrak{h}$) with eigenvalue $-1$). In other words, there exists a
$\lambda\in\mathbb{C}$ such that $J\left( \alpha_{s}\right) =\lambda
\alpha_{s}^{\vee}$. We now will prove that $\lambda=\dfrac{1}{2}\left(
\alpha_{s},\alpha_{s}\right) $.
\par
In fact, from $J\left( \alpha_{s}\right) =\sum\limits_{i=1}^{r}\alpha
_{s}\left( y_{i}\right) y_{i}$, we deduce that%
\[
\left\langle \alpha_{s},J\left( \alpha_{s}\right) \right\rangle
=\left\langle \alpha_{s},\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right)
y_{i}\right\rangle =\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right)
\underbrace{\left\langle \alpha_{s},y_{i}\right\rangle }_{=\alpha_{s}\left(
y_{i}\right) }=\sum\limits_{i=1}^{r}\left( \alpha_{s}\left( y_{i}\right)
\right) ^{2}=\left( \alpha_{s},\alpha_{s}\right)
\]
(since $\left\{ y_{1},y_{2},...,y_{r}\right\} $ is an orthonormal basis of
$\mathfrak{h}$). Compared with%
\[
\left\langle \alpha_{s},\underbrace{J\left( \alpha_{s}\right) }%
_{=\lambda\alpha_{s}^{\vee}}\right\rangle =\lambda\underbrace{\left\langle
\alpha_{s},\alpha_{s}^{\vee}\right\rangle }_{=2}=2\lambda,
\]
this yields $2\lambda=\left( \alpha_{s},\alpha_{s}\right) $, so that
$\lambda=\dfrac{1}{2}\left( \alpha_{s},\alpha_{s}\right) $. Now,%
\[
\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) y_{i}=J\left( \alpha
_{s}\right) =\underbrace{\lambda}_{=\dfrac{1}{2}\left( \alpha_{s},\alpha
_{s}\right) }\alpha_{s}^{\vee}=\dfrac{1}{2}\left( \alpha_{s},\alpha
_{s}\right) \alpha_{s}^{\vee}.
\]
This proves (\ref{p10.2.18d.1}).} Thus, for every $s\in\mathcal{S}$, we have%
\begin{align}
\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) \partial_{y_{i}} &
=\partial_{\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) y_{i}%
}=\partial_{\dfrac{1}{2}\left( \alpha_{s},\alpha_{s}\right) \alpha_{s}%
^{\vee}}\ \ \ \ \ \ \ \ \ \ \left( \text{by (\ref{p10.2.18d.1})}\right)
\nonumber\\
& =\dfrac{1}{2}\left( \alpha_{s},\alpha_{s}\right) \partial_{\alpha
_{s}^{\vee}}. \label{p10.2.18d.2}%
\end{align}
Also, for every $s\in\mathcal{S}$ and $g\in\mathfrak{h}^{\ast}$, we have%
\begin{equation}
\left( \alpha_{s},\alpha_{s}\right) g\left( \alpha_{s}^{\vee}\right)
=2\left( \alpha_{s},g\right) . \label{p10.2.18d.3}%
\end{equation}
\footnote{\textit{Proof.} Let $s\in\mathcal{S}$ and $g\in\mathfrak{h}^{\ast}$.
Then, (\ref{p10.2.18d.1}) yields $\dfrac{1}{2}\left( \alpha_{s},\alpha
_{s}\right) \alpha_{s}^{\vee}=\sum\limits_{i=1}^{r}\alpha_{s}\left(
y_{i}\right) y_{i}$, so that $\left( \alpha_{s},\alpha_{s}\right)
\alpha_{s}^{\vee}=2\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) y_{i}%
$, and thus%
\[
g\left( \left( \alpha_{s},\alpha_{s}\right) \alpha_{s}^{\vee}\right)
=g\left( 2\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) y_{i}\right)
=2\underbrace{\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) g\left(
y_{i}\right) }_{\substack{=\left( \alpha_{s},g\right) \\\text{(since
}\left\{ y_{1},y_{2},...,y_{r}\right\} \text{ is an}\\\text{orthonormal
basis of }\mathfrak{h}\text{)}}}=2\left( \alpha_{s},g\right) .
\]
Since $g\left( \left( \alpha_{s},\alpha_{s}\right) \alpha_{s}^{\vee
}\right) =\left( \alpha_{s},\alpha_{s}\right) g\left( \alpha_{s}^{\vee
}\right) $, this rewrites as $\left( \alpha_{s},\alpha_{s}\right) g\left(
\alpha_{s}^{\vee}\right) =2\left( \alpha_{s},g\right) $. This proves
(\ref{p10.2.18d.3}).}
On the other hand, from $\Delta_{\mathfrak{h}}=\sum\limits_{i=1}^{r}%
\partial_{y_{i}}^{2}$, we obtain%
\begin{align*}
\varsigma_{c}\left( \Delta_{\mathfrak{h}}\right) & =\varsigma_{c}\left(
\sum\limits_{i=1}^{r}\partial_{y_{i}}^{2}\right) =\sum\limits_{i=1}%
^{r}\left( \underbrace{\varsigma_{c}\left( \partial_{y_{i}}\right)
}_{\substack{=\partial_{y_{i}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}%
\alpha_{s}\left( y_{i}\right) }{\alpha_{s}}\\\text{(by (\ref{p10.2.18c.2}),
applied to }a=y_{i}\text{)}}}\right) ^{2}\\
& \ \ \ \ \ \ \ \ \ \ \left( \text{since }\varsigma_{c}\text{ is a
}\mathbb{C}\text{-algebra homomorphism}\right) \\
& =\sum\limits_{i=1}^{r}\underbrace{\left( \partial_{y_{i}}+\sum
\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right) }%
{\alpha_{s}}\right) ^{2}}_{=\partial_{y_{i}}^{2}+\partial_{y_{i}}%
\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right)
}{\alpha_{s}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left(
y_{i}\right) }{\alpha_{s}}\partial_{y_{i}}+\left( \sum\limits_{s\in
\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right) }{\alpha_{s}}\right)
^{2}}\\
& =\sum\limits_{i=1}^{r}\left( \partial_{y_{i}}^{2}+\partial_{y_{i}}%
\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right)
}{\alpha_{s}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left(
y_{i}\right) }{\alpha_{s}}\partial_{y_{i}}+\left( \sum\limits_{s\in
\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right) }{\alpha_{s}}\right)
^{2}\right) \\
& =\underbrace{\sum\limits_{i=1}^{r}\partial_{y_{i}}^{2}}_{=\Delta
_{\mathfrak{h}}}+\underbrace{\sum\limits_{i=1}^{r}\partial_{y_{i}}%
\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right)
}{\alpha_{s}}}_{=\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}^{r}%
\alpha_{s}\left( y_{i}\right) \partial_{y_{i}}\dfrac{1}{\alpha_{s}}}\\
& \ \ \ \ \ \ \ \ \ \ +\underbrace{\sum\limits_{i=1}^{r}\sum\limits_{s\in
\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right) }{\alpha_{s}}%
\partial_{y_{i}}}_{=\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}%
^{r}\alpha_{s}\left( y_{i}\right) \dfrac{1}{\alpha_{s}}\partial_{y_{i}}%
}+\sum\limits_{i=1}^{r}\underbrace{\left( \sum\limits_{s\in\mathcal{S}}%
\dfrac{c_{s}\alpha_{s}\left( y_{i}\right) }{\alpha_{s}}\right) ^{2}}%
_{=\sum\limits_{s\in\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}\alpha
_{s}\left( y_{i}\right) }{\alpha_{s}}\cdot\dfrac{c_{u}\alpha_{u}\left(
y_{i}\right) }{\alpha_{u}}}\\
& =\Delta_{\mathfrak{h}}+\sum\limits_{s\in\mathcal{S}}c_{s}\sum
\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) \underbrace{\partial_{y_{i}%
}\dfrac{1}{\alpha_{s}}}_{=\dfrac{1}{\alpha_{s}}\partial_{y_{i}}+\partial
_{y_{i}}\left( \dfrac{1}{\alpha_{s}}\right) }\\
& \ \ \ \ \ \ \ \ \ \ +\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}%
^{r}\alpha_{s}\left( y_{i}\right) \dfrac{1}{\alpha_{s}}\partial_{y_{i}%
}+\underbrace{\sum\limits_{i=1}^{r}\sum\limits_{s\in\mathcal{S};\ u\in
\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left( y_{i}\right) }{\alpha_{s}}%
\cdot\dfrac{c_{u}\alpha_{u}\left( y_{i}\right) }{\alpha_{u}}}_{=\sum
\limits_{s\in\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}}{\alpha_{s}%
\alpha_{u}}\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) \alpha
_{u}\left( y_{i}\right) }%
\end{align*}%
\begin{align*}
& =\Delta_{\mathfrak{h}}+\underbrace{\sum\limits_{s\in\mathcal{S}}c_{s}%
\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) \left( \dfrac{1}%
{\alpha_{s}}\partial_{y_{i}}+\partial_{y_{i}}\left( \dfrac{1}{\alpha_{s}%
}\right) \right) }_{=\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}%
^{r}\alpha_{s}\left( y_{i}\right) \dfrac{1}{\alpha_{s}}\partial_{y_{i}}%
+\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}^{r}\alpha_{s}\left(
y_{i}\right) \partial_{y_{i}}\left( \dfrac{1}{\alpha_{s}}\right) }\\
& \ \ \ \ \ \ \ \ \ \ +\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}%
^{r}\alpha_{s}\left( y_{i}\right) \dfrac{1}{\alpha_{s}}\partial_{y_{i}}%
+\sum\limits_{s\in\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}}{\alpha
_{s}\alpha_{u}}\underbrace{\sum\limits_{i=1}^{r}\alpha_{s}\left(
y_{i}\right) \alpha_{u}\left( y_{i}\right) }_{\substack{=\left( \alpha
_{s},\alpha_{u}\right) \\\text{(since }\left\{ y_{1},y_{2},...,y_{r}%
\right\} \text{ is an orthonormal}\\\text{basis of }\mathfrak{h}\text{) }}}\\
& =\Delta_{\mathfrak{h}}+\sum\limits_{s\in\mathcal{S}}c_{s}\sum
\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) \dfrac{1}{\alpha_{s}}%
\partial_{y_{i}}+\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}^{r}%
\alpha_{s}\left( y_{i}\right) \partial_{y_{i}}\left( \dfrac{1}{\alpha_{s}%
}\right) \\
& \ \ \ \ \ \ \ \ \ \ +\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}%
^{r}\alpha_{s}\left( y_{i}\right) \dfrac{1}{\alpha_{s}}\partial_{y_{i}}%
+\sum\limits_{s\in\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}}{\alpha
_{s}\alpha_{u}}\left( \alpha_{s},\alpha_{u}\right)
\end{align*}%
\begin{align*}
& =\Delta_{\mathfrak{h}}+2\sum\limits_{s\in\mathcal{S}}c_{s}\underbrace{\sum
\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) \dfrac{1}{\alpha_{s}}%
\partial_{y_{i}}}_{=\dfrac{1}{\alpha_{s}}\sum\limits_{i=1}^{r}\alpha
_{s}\left( y_{i}\right) \partial_{y_{i}}}+\sum\limits_{s\in\mathcal{S}}%
c_{s}\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right) \underbrace{\partial
_{y_{i}}\left( \dfrac{1}{\alpha_{s}}\right) }_{\substack{=-\dfrac
{\partial_{y_{i}}\left( \alpha_{s}\right) }{\alpha_{s}^{2}}=-\dfrac
{\alpha_{s}\left( y_{i}\right) }{\alpha_{s}^{2}}\\\text{(since }%
\partial_{y_{i}}\left( \alpha_{s}\right) =\alpha_{s}\left( y_{i}\right)
\text{ (because }\alpha_{s}\\\text{is linear))}}}\\
& \ \ \ \ \ \ \ \ \ \ +\underbrace{\sum\limits_{s\in\mathcal{S}%
;\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}}{\alpha_{s}\alpha_{u}}\left( \alpha
_{s},\alpha_{u}\right) }_{\substack{=\sum\limits_{s\in\mathcal{S}%
;\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}\left( \alpha_{s},\alpha_{u}\right)
}{\alpha_{s}\alpha_{u}}=\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}^{2}\left(
\alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}\\\text{(by Lemma 2.18e
\textbf{(c)})}}}\\
& =\Delta_{\mathfrak{h}}+2\sum\limits_{s\in\mathcal{S}}c_{s}\dfrac{1}%
{\alpha_{s}}\underbrace{\sum\limits_{i=1}^{r}\alpha_{s}\left( y_{i}\right)
\partial_{y_{i}}}_{\substack{=\dfrac{1}{2}\left( \alpha_{s},\alpha
_{s}\right) \partial_{\alpha_{s}^{\vee}}\\\text{(by (\ref{p10.2.18d.2}))}%
}}+\underbrace{\sum\limits_{s\in\mathcal{S}}c_{s}\sum\limits_{i=1}^{r}%
\alpha_{s}\left( y_{i}\right) \left( -\dfrac{\alpha_{s}\left(
y_{i}\right) }{\alpha_{s}^{2}}\right) }_{=-\sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}}{\alpha_{s}^{2}}\sum\limits_{i=1}^{r}\left( \alpha_{s}\left(
y_{i}\right) \right) ^{2}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}%
^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}%
\end{align*}%
\begin{align}
& =\Delta_{\mathfrak{h}}+\underbrace{2\sum\limits_{s\in\mathcal{S}}%
c_{s}\dfrac{1}{\alpha_{s}}\cdot\dfrac{1}{2}\left( \alpha_{s},\alpha
_{s}\right) \partial_{\alpha_{s}^{\vee}}}_{=\sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}}%
\partial_{\alpha_{s}^{\vee}}}-\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}%
}{\alpha_{s}^{2}}\underbrace{\sum\limits_{i=1}^{r}\left( \alpha_{s}\left(
y_{i}\right) \right) ^{2}}_{\substack{=\left( \alpha_{s},\alpha_{s}\right)
\\\text{(since }\left\{ y_{1},y_{2},...,y_{r}\right\} \text{ is an}\\\text{
orthonormal basis of }\mathfrak{h}\text{) }}}+\sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}%
}\nonumber\\
& =\Delta_{\mathfrak{h}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left(
\alpha_{s},\alpha_{s}\right) }{\alpha_{s}}\partial_{\alpha_{s}^{\vee}}%
-\sum\limits_{s\in\mathcal{S}}\underbrace{\dfrac{c_{s}}{\alpha_{s}^{2}}\left(
\alpha_{s},\alpha_{s}\right) }_{=\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}^{2}}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}%
^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}\nonumber\\
& =\Delta_{\mathfrak{h}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left(
\alpha_{s},\alpha_{s}\right) }{\alpha_{s}}\partial_{\alpha_{s}^{\vee}}%
-\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}^{2}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}%
^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}.
\label{p10.2.18d.6}%
\end{align}
On the other hand, every $t\in\mathcal{S}$ satisfies%
\begin{align}
\varsigma_{c}\left( \partial_{\alpha_{t}^{\vee}}\right) & =\partial
_{a_{t}^{\vee}}+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\alpha_{s}\left(
\alpha_{t}^{\vee}\right) }{\alpha_{s}}\ \ \ \ \ \ \ \ \ \ \left( \text{by
(\ref{p10.2.18c.2}), applied to }a=\alpha_{t}^{\vee}\right) \nonumber\\
& =\partial_{a_{t}^{\vee}}+\sum\limits_{u\in\mathcal{S}}\dfrac{c_{u}%
\alpha_{u}\left( \alpha_{t}^{\vee}\right) }{\alpha_{u}}%
\ \ \ \ \ \ \ \ \ \ \left( \text{here, we renamed }s\text{ as }u\text{ in the
sum}\right) . \label{p10.2.18d.7}%
\end{align}
Now,%
\begin{align}
& \varsigma_{c}\left( \sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left(
\alpha_{s},\alpha_{s}\right) }{\alpha_{s}}\partial_{\alpha_{s}^{\vee}}\right)
\nonumber\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}}\underbrace{\varsigma_{c}\left( \partial
_{\alpha_{s}^{\vee}}\right) }_{\substack{=\partial_{a_{s}^{\vee}}%
+\sum\limits_{u\in\mathcal{S}}\dfrac{c_{u}\alpha_{u}\left( \alpha_{s}^{\vee
}\right) }{\alpha_{u}}\\\text{(by (\ref{p10.2.18d.7}), applied to
}t=s\text{)}}}=\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha
_{s},\alpha_{s}\right) }{\alpha_{s}}\left( \partial_{a_{s}^{\vee}}%
+\sum\limits_{u\in\mathcal{S}}\dfrac{c_{u}\alpha_{u}\left( \alpha_{s}^{\vee
}\right) }{\alpha_{u}}\right) \nonumber\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}}\partial_{a_{s}^{\vee}}+\underbrace{\sum
\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha_{s}\right)
}{\alpha_{s}}\sum\limits_{u\in\mathcal{S}}\dfrac{c_{u}\alpha_{u}\left(
\alpha_{s}^{\vee}\right) }{\alpha_{u}}}_{=\sum\limits_{s\in\mathcal{S}%
;\ u\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha_{s}\right) }%
{\alpha_{s}}\cdot\dfrac{c_{u}\alpha_{u}\left( \alpha_{s}^{\vee}\right)
}{\alpha_{u}}}\nonumber\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}}\partial_{a_{s}^{\vee}}+\sum\limits_{s\in
\mathcal{S};\ u\in\mathcal{S}}\underbrace{\dfrac{c_{s}\left( \alpha
_{s},\alpha_{s}\right) }{\alpha_{s}}\cdot\dfrac{c_{u}\alpha_{u}\left(
\alpha_{s}^{\vee}\right) }{\alpha_{u}}}_{=\dfrac{c_{s}c_{u}}{\alpha_{s}%
\alpha_{u}}\left( \alpha_{s},\alpha_{s}\right) \alpha_{u}\left( \alpha
_{s}^{\vee}\right) }\nonumber\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}}\partial_{a_{s}^{\vee}}+\sum\limits_{s\in
\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}}{\alpha_{s}\alpha_{u}%
}\underbrace{\left( \alpha_{s},\alpha_{s}\right) \alpha_{u}\left(
\alpha_{s}^{\vee}\right) }_{\substack{=2\left( \alpha_{s},\alpha_{u}\right)
\\\text{(by (\ref{p10.2.18d.3}), applied to }g=\alpha_{u}\text{)}}}\nonumber\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}}\partial_{a_{s}^{\vee}}+2\sum\limits_{s\in
\mathcal{S};\ u\in\mathcal{S}}\underbrace{\dfrac{c_{s}c_{u}}{\alpha_{s}%
\alpha_{u}}\left( \alpha_{s},\alpha_{u}\right) }_{=\dfrac{c_{s}c_{u}\left(
\alpha_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}}}\nonumber\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}}\partial_{a_{s}^{\vee}}+2\underbrace{\sum
\limits_{s\in\mathcal{S};\ u\in\mathcal{S}}\dfrac{c_{s}c_{u}\left( \alpha
_{s},\alpha_{u}\right) }{\alpha_{s}\alpha_{u}}}_{\substack{=\sum
\limits_{s\in\mathcal{S}}\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha_{s}\right)
}{\alpha_{s}^{2}}\\\text{(by Lemma 2.18e \textbf{(c)})}}}\nonumber\\
& =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}}\partial_{a_{s}^{\vee}}+2\sum\limits_{s\in
\mathcal{S}}\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha
_{s}^{2}}. \label{p10.2.18d.8}%
\end{align}
Now, $\overline{H}=\Delta_{\mathfrak{h}}-\sum\limits_{s\in\mathcal{S}}%
\dfrac{c_{s}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}}\partial
_{\alpha_{s}^{\vee}}$, so that
\begin{align*}
\varsigma_{c}\left( \overline{H}\right) & =\varsigma_{c}\left(
\Delta_{\mathfrak{h}}-\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left(
\alpha_{s},\alpha_{s}\right) }{\alpha_{s}}\partial_{\alpha_{s}^{\vee}%
}\right) =\varsigma_{c}\left( \Delta_{\mathfrak{h}}\right) -\varsigma
_{c}\left( \sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha_{s}%
,\alpha_{s}\right) }{\alpha_{s}}\partial_{\alpha_{s}^{\vee}}\right) \\
& =\left( \Delta_{\mathfrak{h}}+\sum\limits_{s\in\mathcal{S}}\dfrac
{c_{s}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}}\partial_{\alpha
_{s}^{\vee}}-\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( \alpha
_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}+\sum\limits_{s\in\mathcal{S}}%
\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}\right)
\\
& \ \ \ \ \ \ \ \ \ \ -\left( \sum\limits_{s\in\mathcal{S}}\dfrac
{c_{s}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}}\partial_{a_{s}%
^{\vee}}+2\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}^{2}\left( \alpha
_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}\right) \\
& \ \ \ \ \ \ \ \ \ \ \left( \text{by (\ref{p10.2.18d.6}) and
(\ref{p10.2.18d.8})}\right) \\
& =\Delta_{\mathfrak{h}}-\underbrace{\left( \sum\limits_{s\in\mathcal{S}%
}\dfrac{c_{s}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}%
+\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha
_{s}\right) }{\alpha_{s}^{2}}\right) }_{=\sum\limits_{s\in\mathcal{S}%
}\left( \dfrac{c_{s}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}%
}+\dfrac{c_{s}^{2}\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}%
}\right) =\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left( c_{s}+1\right)
\left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}}\\
& =\Delta_{\mathfrak{h}}-\sum\limits_{s\in\mathcal{S}}\dfrac{c_{s}\left(
c_{s}+1\right) \left( \alpha_{s},\alpha_{s}\right) }{\alpha_{s}^{2}}=H.
\end{align*}
This proves Proposition 2.18d.
\textit{Proof of Theorem 2.9.} The $\mathbb{C}$-algebra homomorphism
$\varsigma_{c}:\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right)
\rightarrow\mathcal{D}\left( \mathfrak{h}_{\operatorname*{reg}}\right) $
preserves the degree of homogeneous differential operators and their symbols
and commutes with the action of $W$ by conjugation (these facts all are easy
to prove), and maps $\overline{H}$ to $H$ (by Proposition 2.18d). Hence,
applying $\varsigma_{c}$ to Corollary 2.17, we obtain Theorem 2.9 (at least,
if we add to Corollary 2.17 the claim that the $\overline{L}_{j}$ are
$W$-invariant, as I suggested above), with the $L_{j}$ being given by
$L_{j}=\varsigma_{c}\left( \overline{L}_{j}\right) $.
Theorem 2.9 is thus proven, and with it Theorem 2.1.
\item \textbf{Page 10, Remark 2.20:} It would be better to replace ``$L_{i}$''
by ``$\overline{L}_{i}$'' here.
\item[...] [To be continued?]
\item \textbf{Page 12, Example 2.25:} In this example, you regard
$\mathfrak{h}$ as being embedded into $\mathbb{C}^{n}$ as the subspace
consisting of the vectors whose coordinates sum to zero. (This is the same
embedding as in Example 2.5.) The $p_{i}$ are the same as in Example 2.5. The
$x_{i}$ (for each $i\in\left\{ 1,2,\ldots,n\right\} $) is the linear map
sending each element of $\mathfrak{h}$ to its $i$-th coordinate. This all is
worth pointing out explicitly, since it is far from obvious.
\end{itemize}
\protect\begin{noncompile}
\section*{Section 3}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\section*{Section 4}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\section*{Section 5}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\section*{Section 6}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\section*{Section 7}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\section*{Section 8}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\section*{Section 9}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\section*{Section 10}
\begin{itemize}
\item \textbf{.}
\end{itemize}
\end{noncompile}
\end{document} |
https://theanarchistlibrary.org/library/peter-gelderloos-international-solidarity-when-things-are-not-black-and-white.tex | theanarchistlibrary.org | CC-MAIN-2022-49 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00484.warc.gz | 602,393,491 | 9,346 | \documentclass[DIV=12,%
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\title{International solidarity when things are not black and white}
\date{January 2, 2020}
\author{Peter Gelderloos}
\subtitle{}
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{\usekomafont{title}{\huge International solidarity when things are not black and white\par}}%
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As 2020 is off to a rebellious start, a wave of struggles with growing intensity continues to sweep across the globe, from Chile to Hong Kong. People are rising up against government repression and austerity measures, or trying to defend themselves from right-wing coups. None of these uprisings are simple or homogeneous; some include elements anti-capitalists may strongly disagree with, and the necessity of self-defense against the advances of the far-right often puts us in a position of defending left-wing governments we may have well founded criticisms of.
When these complexities and critiques are brought up, the ensuing debate usually devolves into a total polarization in which one side denies any possible criticism and the other side prioritizes their criticism over solidarity. As an end result of this kind of posturing, each side denies legitimacy to the other and claims their own position is above reproach. But criticism is oxygen for the struggle. Revolutionary movements that do not honestly consider their own weaknesses are setting themselves up for failure. And when a movement cannot develop relevant responses to a situation of growing misery and exploitation, when it betrays the dreams that launched it in the first place, it is paving the way for the right to come back with a vengeance.
We can do better than this. In order to extend effective solidarity, we need to identify some principles and patterns that will help us achieve this.
\section{Urgency and Proportionality}
When people are dying in the streets, questions of survival need to take priority. This means understanding the alliances people make in context. A progressive political party making a tacit alliance with a far-right party to stay in power another term — as happened after the last elections in Barcelona — is an entirely different kind of pragmatism than Kurdish fighters accepting US aid in a fight for their very survival against ISIS and Turkey, or anticapitalist protesters in Hong Kong, facing a brutal onslaught by police and an extradition law that promises future repression, fighting alongside those who want a liberal political system in the US sphere of influence.
We should be honest about the complexities of a struggle and openly discuss the authoritarian tendencies of certain allies, while keeping things in perspective and correctly identifying who presents an immediate threat to our survival or freedom, or that of the people we are trying to support from a distance.
People in Bolivia have been fighting in the streets for the future of their country. The mostly Indigenous protesters opposed to the coup that ousted Evo Morales have already suffered massacres and disappearances, while the groups behind the coup are receiving support from the US and right-wing governments in the OAS. Which is the bigger threat right now; specific policies of Morales over the past ten years that damaged Indigenous autonomy and destroyed large swathes of rainforest, or the evangelist, neo-fascist political groups with military and paramilitary support that want to annihilate Indigenous cultures, subjugate all the working class and indigenous people in Bolivia who have been fighting for their dignity, and accelerate the despoliation of the earth? Obviously, the latter.
In a somewhat similar vein, anarchists in Ukraine had to find a coherent, effective position in the 2014 Maidan movement and the subsequent outbreak of war in Donbas. On the one side, there was a heterogeneous alliance of Ukrainian progressives, centrists and fascists, most of whom favored a closer relationship with the EU. On the other side were Russian nationalists, Stalinists and fascists — and the Russian military. Anti-capitalists from the region tend to be critical of both the EU and Moscow, as well as fascists of any stripe.
The debate largely centers around prioritizing different threats. One relevant argument holds that Russia is the dominant imperialist power in that part of the world, and people are safer if they remain outside its orbit. Russia and its client states like Belarus routinely arrest, torture and assassinate anarchists and other dissidents. A large part of the movement has had to flee into exile, and it is pertinent that Ukraine is one of the relatively safe countries they flee to.
Any criticism of the different positions comrades adopt that does not take into account very real questions of survival with regards to Ukrainian and Russian fascists as well as Russian police actions is liable to be ignored by those who personally face those dangers.
Survival must be a priority. But we also have to keep in mind that as long as capitalism controls our survival, it cannot be the chief compass point for revolutionary struggles. There is also an important criticism to be made of survival in a liberal, individualistic sense: the survival of our specific bodies, and not the survival of our dreams and desires, the survival of our communities, cultures, or histories.
While we fight for our survival and that of allies near and far, we also need to create other paths of struggle so that the very way we live increasingly takes on revolutionary implications. Otherwise, we will always be sacrificing long-term considerations for short-term necessities. If survival means continuously supporting the latest government or political party that won’t put us on the chopping block so quickly, we will never be able to contest capitalist designs on our lives.
For survival to be a revolutionary consideration, it must also include the survival of our communities, histories and dreams.
\section{Who is the Protagonist?}
“The people” or another more specific but equally essentialist category almost always make an appearance in debates around how to position ourselves with respect to a complex conflict. There is an authoritarian habit of claiming to speak on their behalf, of justifying our own position as the only one that is in the interests of said people.
It is quite possible that the first time in history the term “the people” was used in a politically effective way, it was already a manipulation: segments of the bourgeoisie, legitimizing their own interests and justifying a most profitable change to the structures of government and society, namely, giving property owning males the vote, privatizing land and enclosing the commons under the guise of abolishing the privileges of Church and aristocracy. They included themselves as part of “the people,” the new source of political legitimacy, even though they had very little in common, and a great deal of antagonism, with most of the other people included in that group.
Nonetheless, many on the left still use this term uncritically, without acknowledging that any iteration of a “people” is a multifaceted, shifting, fluid, heterogeneous group with no consensus, no fixed interests, and with their own voices, their own capacity to constantly redefine their interests.
This essentialist operation smoothes over — or tramples down — the many ever-changing differences between people, because to represent a group you must first deprive it of its own multitude of voices. And you cannot climb onto the backs of a group of people and steal the power they generate without claiming at some level to represent them.
When it is a case of someone making essentialist statements in support of a distant group they do not belong to, it is obvious how this is problematic. But it can also be problematic when people position themselves as representatives of a group they actually belong to.
This is by no means a call for liberal atomization. “Individuals” are probably an even more artificial category than most groups (topic for a whole other article, but if you even just take, say, respiration, immune systems, or knowledge, \emph{no one} functions as an individual; rather, we function as a part of networks that include all living things; nobody breathes without trees or learns without those who came before us). It is instead a call for nuance, a distinction between speaking up for collective experiences, and implying that everyone in a collective agrees with us or can be represented by us.
It is inevitable to use simplifying phrases like, “solidarity with the Bolivian people.” Already, we are leaving out the racist evangelists and capitalists who also make up a part of the Bolivian people, though when one sector of a society attacks and dominates the rest, espousing a racist and classist logic, they are asking to be discounted. Whether or not we are a part of the group in question, we should be clear these are \emph{our} values — values we are happy to explain and defend — that justify delegitimizing a group of people.
But when we go a step further and claim “those who do not support Morales are anti-Indigenous” or “those who do not vote for Obama (or, as will probably be the case in 2020, a white Democratic candidate) are racist,” we are insisting in an underhanded way not only that all Indigenous or all Black people have a similar experience of racism, but that all of them believe in the same strategy for change, and it just so happens to be the strategy we espouse.
This is an authoritarian operation, silencing all the revolutionary Indigenous and Black people, in these two examples, who have different ideas on resistance, and appropriating an extreme degree of unaccountable power as one claims to speak on behalf of so many others — unaccountable because they are obscuring the fact they are expressing their own values and instead imputing those values as the natural, essential belief of hundreds of millions of people.
This would be ridiculously horrible if I, a white person, were to do it, but it is still an essentialist, authoritarian operation that silences difference when someone does it within their own category. And those most likely to be silenced by this operation are those with the least access to institutional power and dominant technologies of communication.
\section{Anti-Imperialist Realities}
I have learned a great deal in conversation with a Venezuelan friend who is a Chavista. While she is more critical of Maduro, she believes that Chávez sincerely and effectively tried to use the state to support popular movements in Venezuela, while also maintaining and expanding petroleum extraction so that the country could acquire the foreign exchange needed for survival. She did not work for the government: her experience and her perspective is from the streets, from the popular movements.
She fully acknowledges that petroleum and coal extraction exacerbated conflicts with multiple Indigenous communities, but also that centuries of colonial and neocolonial economic structuring meant that the country was utterly dependent on global capital flows just to feed itself. This falls in line with Walter Rodney’s analysis of the Soviet Union and the possibility for socialism in Africa: a revolution within a country does not entirely save that country from occupying a colonial or extractivist niche within the global capitalist economy.
This view is not by any means a free pass for authoritarian socialists; rather, it requires us to make distinctions between different degrees and strategies of authoritarianism. In my friend’s experience, Chávez was valuable to popular movements precisely because he gave those movements space to grow and temporarily kept the racist aristocracy off their backs, but it was primarily the movements that were making things better, though government resources played an important role.
On the contrary, the Soviet Union quickly curtailed the autonomy of the social movements and soon crushed those movements altogether. It is worth noting that the state unopposed proved to be the quickest path back to capitalism. But in Venezuela under Chávez, though the government did resort to violence against those who opposed it, as all governments do, it also adopted a wholly different relationship with social movements.
Venezuela is different from Bolivia; Evo focused on institutionalizing the movements rather than empowering them, and did not hesitate to attack them when they protested some policy of his. Again, this approach is not the same as that practiced in socialist Russia, China or Cuba, but the frictions and disappointments it caused can also help explain the early success of the coup against Morales.
I can recognize the reality of my Chavista friend’s experiences without believing states are an appropriate tool for revolutions. I can argue that government, private property and wage labor are themselves intrinsically colonial forms antithetical to liberation. That it is all but inevitable for a competent bureaucrat like Maduro to follow a charismatic visionary like Chávez, and similarly for a Stalin to follow a Lenin.
I can argue that a total rupture with the global capitalist economy — which no socialist state has ever undertaken and which no government is structurally capable of accomplishing — is actually the only hope for a real revolution. I can use the example of the great revolutionary gains when peasants and workers won their autonomy, free from the state, in Ukraine in 1919, in Shinmin Province in Manchuria from 1929 to 1931, in Catalonia and Aragón in 1936, every single time crushed not by the right, but by an authoritarian left that to this day is allergic to criticism.
I can also point to the much more recent example of the \emph{período especial} in Cuba, in the ‘90s, when people subjected to a total blockade organized their own survival at the margins of a government on the edge of collapse, afraid to get in their way.
I can fervently believe that an anti-state strategy is the best one in all of these cases. But that belief is irrelevant if I do not also recognize that it is a question of life and death if a country cuts itself off from global capitalism without transforming its economy. As such, many people will prefer to keep one foot in either world, despite the difficulties and as yet unresolved dilemmas that strategy entails.
\section{Common Enemies}
In closing, I want to offer an image, a proposal, that transcends facile polemics. Despite our disagreements, would we stand on the same side of the barricades? Who among us would not be on the same side if we were suddenly together in Bolivia, at Standing Rock, in Charlottesville, in Ferguson, in Chile, in Lebanon? We would not all go to the same protests, not all the time, nor participate in the same initiatives, but when things got hot, when on the other side of the street it was the cops, the fascists, shooting at us, getting ready to charge, I would like to think we would fight together, looking out for one another’s survival.
Those who do the most to keep these flame wars going are ensconced behind computer screens or in ivory towers and do not have to face situations of actual danger. But the rest of us have long become accustomed to the impoverished forms of solidarity favored by these types. On social media, a post insulting one faction or the other, denying their revolutionary credentials, gets passed around tens of thousands of times. Another one, suggesting we find which evangelical churches or private companies support the far-right in Bolivia, gets ignored.
A suggestion that we identify common enemies, power structures that all of us would oppose, who sow misery from the very poorest to the very richest of countries, would require us to give up our shallow posturing and take risks together, despite our differences.
That is exactly what needs to happen.
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The Anarchist Library
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Peter Gelderloos
International solidarity when things are not black and white
January 2, 2020
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Retrieved on 5\textsuperscript{th} March 2022 from \href{https://roarmag.org/essays/international-solidarity-gelderloos/}{roarmag.org}
The internationalist left often fails to transcend facile polemics, even when effective solidarity can mean the difference between life and death. We can do better than this.
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\title{Anarchism: What it Is and What it Is Not}
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IT was an observation of John Stuart Mill’s that to know a thing it is necessary to realize, not only what it is, but also what it is not. Applying this definition or test to that passage of Mr. Thomas B. Preston’s paper on “Are We Socialists?” (ARENA, December) in which he states and criticises the principles of anarchism, we find ourselves entitled to affirm that Mr. Preston scarcely possesses such familiarity with, and comprehension of, the essential doctrines of anarchism as would justify confident criticism of that school.
What is anarchism, and who are the anarchists? Loosely speaking, there are two schools of anarchism, two species of anarchists. There is the school of communist anarchism. This school rigorously adheres to the economic and ‘political teachings of Michael Bakounin. It insists on the “ expropriation of the expropriators”—capitalists and men who live on rent, interest, or profit—and the total abolition of private property in capital, or the means and instruments of production. It favors the use of physical force, and is‘ openly revolutionary. In short, most of Mr. Preston’s statements concerning anarchists certainly may be accepted as\textasciitilde{} tolerably exact with reference to this school. The school to which Mr. Preston’s predications do not apply is that of individualist anarchism. Strictly speaking, this school is the only one in the field which possesses the right to the term “ anarchist,” since, as will presently be established, it is the only school which logically and consistently follows out the principle of non-interference with personal liberty. Whether it carries out the principle specified “to an exaggerated absurdity,” is, of course, a matter of opinion; but before delivering any judgment, let us ascertain the precise significance of the principle of “ personal liberty” espoused by the individualist anarchists.
Few are aware that the anarchistic principle of “personal liberty” is absolutely coincident with the famous Spencerian “first principle of human happiness,”—the principle of “equal freedom,”—to which precise expression is given in the following formula: Every man is free to do that which he wills, provided he infringes not the equal freedom of any other man. This principle of equal freedom the individualist anarchists accept without reservation or qualification, recognizing no exceptions to scientific ethical laws, any more than to physical laws. By accepting the principle is naturally meant the acceptance of all its corollaries or logical deductions; and these corollaries are: the right to physical integrity, which negatives murder, assault, and minor trespasses; the rights to free motion and locomotion, which imply the freedom to move from place to place without hindrance; the right to the uses of natural media — land, light, air; the right to property, in products as well as in means of production, which negatives any species of robbery and any system of compulsory “nationalism” or communism; the rights of gift and bequests; the rights of free exchange and free contract; the right of free industry; the rights of free belief and worship; and the rights of free speech and publication. These rights are natural social rights, and no society can be stable and harmonious which tolerates their infringement. The test of social progress is observance and respect of these personal rights, and not any form of government. “Crime” can mean nothing else than the violation of one or more of these positive rights; no individual who refrains from aggression or invasion of rights can possibly be criminal. Now, from this point of view let us examine the ethical character of our present political practices. It is admitted without hesitation that no individual can rationally or justly claim the right to trench upon the freedom of any one of his fellows; but it is generally assumed that a government that is, a body representing a majority of the individuals -is entitled to traverse and violate many of the rights of the individual. If the government should attempt to murder a citizen against whom no crime was alleged, it would certainly cause a revolution, it being universally felt that murder does not cease to be a crime when committed by public authorities. Yet when government breaks the law of equal freedom by taxing men against their consent, and thus denying the right to property; or when it imposes a “duty” on imports, and prohibits men from exchanging
freely with people of other lands, and thereby tramples upon the right of free exchange; or when it passes laws in restriction of banking and the issuing of circulating notes, in distinct contradiction of the rights to free industry, free exchange, and free contract; or when it compels the observance of religious holidays in spite of the right to free belief; or when it monopolizes the letter-carrying industry regardless of the prohibition of such actions by the rightful freedom of industry, the great majority of men do not dream of interposing any objection or raising the question of the ethical propriety of such conduct. In other words, the great majority of people act upon the tacit or avowed belief that there are two ethical standards, not one, and that governments are not to be judged in the same manner as individuals. That which is a crime, a punishable act, when committed by a private citizen, may be a legitimate and even praiseworthy act when done in the name of the government. Is this belief rational?
No, answers the individualist anarchist. That which the ethical law interdicts is a crime when proceeding from the government no less than when proceeding from the private citizen. This answer clearly implies more than is embraced in the position of Spencerian individualists. According to these, it is wrong for the government to assume any function save that of protecting the rights of individuals, of enforcing the corollaries of the law of equal freedom. But it is claimed that there is an ethical warrant for compelling men to support a government organized for such a purpose ; that there is an ethical warrant for compulsory taxation and for government not based on individual consent. On the other hand, the individualist anarchists maintain that a government not based on the actual consent of the governed is pure tyranny, and that compulsory taxation is robbery. To interfere with a man who acts within the limits of equal freedom, who invades no one’s proper sphere, is a crime, and hence all governments resting on compulsory taxation are unethical. It is undoubtedly true that men are confronted with the necessity of providing for systematic and organized protection of their rights or freedoms; still, he who declines to accept the protection of government and to contribute toward its support, can only be said to be guilty of folly, and of folly which by no means necessarily involves the injury of his fellows; therefore there is no warrant for any interference with him.
In view of these elucidations, is it correct to assert that individualist anarchists contemplate the utter abolition of “all law and government”? The answer is, yes and no. It is important to distinguish and to bear in mind the anarchistic definitions of the terms used. If by “government” be understood voluntary co-operation for purposes of defence against aggression, then the anarchists are emphatically in favor of it. As long as anti-social feelings and tendencies exist, co-operation against invaders is a necessity. If by. “law” be understood ethical law, the law of social life, then the anarchists strenuously insist on its faithful observance. But if by government he meant the coercion of the non-aggressive individual, then anarchism wages eternal war upon it; if by law be meant the statutes enacted by men both ignorant and reckless of the essential conditions of social happiness, then anarchism posits “no law.” Those who imagine that “the abolition of all law and government” is equivalent, in intention and fact, to the deliberate abandonment of all attempts to enforce justice and punish aggression, are betrayed into error by their definitions of the terms “law” and “government.” In proclaiming the sovereignty of the individual, the anarchist demands for him the full enjoyment of every liberty except the liberty to trespass. In other words, the anarchist contends for equal liberty, and wants every individual to count for one and no more than one. Invasion of rights he would punish, and he would co-operate voluntarily with his fellows for this as for numerous other purposes. But he would not coerce non-invasive citizens into co-operation of any kind. While, if left free, men’s self-interest, as well as their love of fair play, will prompt them to co-operate in the organization of protection against crime, there is no ethical warrant for compelling men to belong to any defensive or insurance associations. The anarchist thus upholds the right of the non—aggressive individual to “ignore the state.”
Two considerations have to be emphasized before proceeding to review and meet Mr. Preston’s criticisms seriatim. In the first place, the anarchists do not expect to obtain golden conduct out of leaden instincts, and to realize the perfect political system under conditions so unfavorable as those of to-day. The fundamental question of voluntary taxation is not with them at present a question of practical politics, but one of scientific politics, or rather of ethical and social science. They believe, with Spencer, that “an ideal, far in advance of practicability though it may be, is always needful for right guidance.” They are not impatient, and are satisfied with slow and gradual progress; but they insist on moving towards the ideal, not away from it. Anarchists gladly work with other reformers whenever the demand is really for an enlargement of liberty and opportunity, and for a restriction of governmental activity, but they do not mistake one plank for the entire platform, a part for the whole. Free trade is a step in advance, and the anarchists would aid in securing it. Free banking and free credit they deem one of the most vital of economic reforms, and they are ready to devote themselves to its furtherance. Land reform they regard as of great importance, and any movement tending to make occupation and use the title to land will command their warm approval. And so on. But they never permit themselves to forget that the goal, the ideal, is the abolition of all forms of compulsory co-operation, and that the progress of society has been from the principle of militarism to that of industrialism, from status to contract, “from a condition in which agreement results from authority (to use the words of G. H. Lewes) to a condition where authority results from agreement.”
The second fact requiring explicit and emphatic asseveration is that the individualist anarchists are not revolutionists, and do not rely on physical force. They do, however, favor passive resistance to despotism and governmental invasion.
A refusal of the Irish tenants to pay rent would be applauded by them, as would also an attempt to disregard any law not sanctioned by equity and reason. Disregard of tariff laws or banking laws or Sunday laws meets with their indorsement, but the methods of the so-called “anarchist communists” they reject as suicidal. As far as possible they would go with Carlyle in endeavoring “to do justice justly.” Dissemination of true conceptions of economic and political justice is their chief task and method.
And now descending to the specific and particular, let us deal with Mr. Preston. Anarchism, he avers. “would abolish all government, and leave individuals subject only to natural laws.” This is true, though not in the sense intended. Anarchism would insist on obedience to all natural social laws, and would abolish all laws and all government not in harmony with the real laws of social life. “In a perfect state of society, the anarchists claim, men would do right without any laws. Education and self-control would rule the individual,” etc. Yes, anarchists do claim all this, but their claim is not original. Philosophical Christians and evolutionists are in accord with them in this matter. But an anarchist society may be far from perfect, and hence stand in need of penal institutions and defensive organizations; and these are wholly compatible with anarchist principles. Anarchism does “not tolerate crime; it merely insists on the right of the non-criminal to ignore the defensive bodies, as we are allowed to-day to ignore insurance companies. Crime would be punished by anarchism, since courts and juries and prisons would remain. “Communities would be formed of individuals attracted to each other by a similarity of tastes and desires. If a member of one of these groups became dissatisfied, he would leave it, and join some other group, more congenial to his tastes.” Communist anarchists will recognize in these descriptions a more or less faithful outline of their system; but to individualist anarchists they have a queer, unfamiliar, and unpleasant sound. Individualist anarchists scout the notion that to work for wages 1S degrading, and that the wage system necessarily involves exploitation of labor. Under a system of equality of opportunity, the laborer would receive the full product of his labor in the form of wages, and the capitalist would receive nothing but proper compensation for his services as organizer and captain of industry. Really free competition (which does not exist to-day) would bring about this condition of things. The trouble with us is not that workmen are forced to work for others for wages, but that monopoly and law-created privilege place capital in a position to dictate terms to the laborer. The supply of labor exceeds the demand for it, and therefore wages are below their natural level — the total product of the laborer. Under a system of free land—or occupying ownership—and free credit, the demand for labor would exceed the supply, and wages would rise. Still, the individualist anarchists believe, with Mill and Cairnes, that association is to be the watchword of the future, and that future industrial relations will be prevailingly based on the co-operative principle. The talk about “communities” and “similarity of tastes,” however, is as irrelevant to the industrial ideal of the individualists as it is to that of the economists named. “Theoretical anarchy may thus be defined as a state of society in which every one does as he pleases without doing wrong.” No; theoretical anarchy is to be defined as a state of society in which every one is allowed to do as he pleases so long as he does not please to break the law of equal freedom. “ As long as men are subject to the physical necessities of the body, \dots{} there will be a clash of material interests which requires regulation; and such regulation requires government.” Defining “government” as the coercion of now invasive, “ such regulation ” does not require government in the opinion of the anarchists. To assert that it does, is to beg the very question at issue. Institutions to protect rights and restrain aggression are not to be confounded with government. If the institutions are formed on the voluntary principle, they are not “government.” Is a fire insurance company “government”? That which is based on actual consent is not government. “ The trouble with many anarchists is that they wish to bring about their system by violence,” etc. This is true of the so-called communistic anarchists, who are not really entitled to the name they usurp, since they believe in compulsory communism and violate the law of equal freedom; but it is not true of the real anarchists,—the individualist anarchists, who abjure violent methods. “ In theory they simply carry out to an exaggerated absurdity the doctrine of non-interference with personal liberty.” It is manifest that this was written on the assumption that anarchists would not resist crime and would not undertake to enforce the law of justice or equal freedom. Since, however, as has been explained, only the inoffensive are to be allowed to ignore the defensive organizations, while aggressors are to be punished and coerced, the charge of exaggerated absurdity falls to the ground.
But perhaps Mr. Preston holds that it is absurd to favor voluntary taxation, “government by actual consent,” and that the attempt to carry out the law of equal freedom would be fatal to society. If so, I can only say that anarchists differ with him.
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\title{Not Falling for It: How the Uprising in Chile Has Outlasted State Repression}
\date{November 8, 2019}
\author{CrimethInc.}
\subtitle{And the Questions for Movements to Come}
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\par
As of today—Friday, November 8, 2019—the government of Chile has spent three full weeks switching back and forth between strategies of brutality, division, and deceit without yet succeeding in stemming the tide of resistance. The events of these weeks offer a useful primer in strategies of state repression and how to outmaneuver, outsmart, and outlast them.
On October 6, the Chilean government headed by rapacious billionaire Sebastián Piñera announced a new austerity package that would further impoverish struggling Chileans. Unfortunately for the authorities, it was an inopportune moment to squeeze an already restless population. The next day, in Ecuador, thousands of indigenous people arrived in the capital city to protest an austerity package, occupying the Parliament building and clashing with police forces. On October 14, the Ecuadorian government backed down, repealing the austerity bill.
That same day, students swung into action in Chile, organizing a series of mass fare-dodging protests against the hike in public transit costs. These culminated on October 18 in clashes, vandalism, and arsons that damaged 16 buses and 78 metro stations, as well as various banks and several other major buildings, including the headquarters of the Italian energy company Enel. In retaliation, Piñera announced a state of emergency and curfew, hoping to bludgeon the population back into submission.
Speculation has circulated about the arsons to the effect that they were orchestrated or permitted by the security forces. Certainly, police in Chile are well-known for engaging in undercover operations—and US intelligence agencies have engaged in all manner of disruptive activities to delegitimize social movements in Chile and elsewhere. At the same time, all around the world, whenever ordinary people manage to get the better of the authorities, those who take it for granted that the state is the only protagonist of history always attribute this to the work of agents provocateurs. Is it really possible that \emph{all} the arsons of October 18 were the work of police agents? What would the government stand to gain by arranging for the destruction of its own public transit infrastructure? It might seem strategic to attribute the arsons to police agents in order to delegitimize the police in the eyes of the general public, but this could have the unintended effect of mobilizing popular rage against the most radical or confrontational participants in the movement—on the absurd grounds that they must be police infiltrators, no less. Rather than legitimizing the sort of confrontational tactics that a powerful movement must sometimes employ, this approach implies that what is needed is \emph{better policing.}
Although we should not underestimate the extent to which state forces can act irrationally against their own interests, it is disempowering to assume that popular movements are \emph{not} capable of confrontational tactics. Conspiracy theories about the arsons obscure what was \emph{strategic} about them. State false-flag operations would be aimed at discrediting the movement, not deepening the crisis itself. In this regard, it seems more likely that the reports of suspicious people attacking working-class markets represent genuine undercover police or far-right activity, or that, as some have alleged, police have concealed some of the murders they have carried out by dragging the victims in burning buildings; in those cases, at least, their motivations would be clear. But the authorities stood to gain nothing by dramatically escalating the conflict on October 18 by burning their own metro stations. Whether by smashing the turnstiles or burning entire stations, it was precisely by making business as usual \emph{impossible} that demonstrators made the desperate circumstances of their daily lives a problem for their rulers and set the stage for the movement to expand. Without the vandalism, the movement would never have become the force that it is.
In any case, the next day, October 19, Piñera suspended the metro price increase. The speed with which he did this shows that he knew he had pushed people too far. If he could have waited to suspend the fare increase, he might have been able to announce it later, in order to give demonstrators a feeling of accomplishment and get them out of the streets; instead, having already pushed his luck, he had to suspend it immediately in hopes of discharging popular resentment before the crisis deepened. It didn’t work.
For a government, the goal of making concessions is only to trick enough people into leaving the streets that it will be possible to isolate and defeat those who remain. On October 20, Piñera expanded the state of emergency to most of the country, announcing from the headquarters of the army that his government was “at war against a powerful and implacable enemy.” This gesture, and above all the place from which he spoke, was a not-so-coded declaration that he intended to return Chile to the murderous state violence of the Pinochet dictatorship.
Yet once again, the people in the streets did not back down. They continued to demonstrate, even as the military injured and killed people, and they refused to permit the authorities to sow divisions, sticking together with the same cohesion that has given the movement in Hong Kong its long life.
This is why, on October 23, Piñera was forced to announce the suspension of the whole austerity package and the introduction of some minor reforms—what Chileans have been calling “table scraps.”
Again, Chileans knew better than to settle for this. That same day, Chile’s trade unions declared a general strike. On October 25, the largest demonstration in Chilean history took place, bringing 1.2 million people into the streets of Santiago to show that they supported this movement that had originated in massive public criminal activity and continued in defiance of the express orders of the government.
This was a massive defeat for Piñera—it showed that he could neither resolve the situation by brute force nor by petty bribery. This is why, on October 26, he promised to lift the State of Emergency and to swap out some of the ministers in his government—though not to relinquish power himself. He also changed his rhetoric, congratulating Chileans on a “peaceful” demonstration and suggesting a distinction between law-abiding families and criminal hooligans.
Let’s review: when Piñera couldn’t suppress the movement by police violence, he played for time by suspending the fare increase—while declaring martial law and mobilizing the army. When didn’t work, he shifted to a new strategy of divide and conquer, flattering the majority of Chileans by suggesting that their concerns were legitimate while demonizing the brave demonstrators who launched the movement.
Now that things seem to have plateaued—not to say calmed down—Piñera is trying, yet again, to return to his original strategy of brute force. On November 7, he introduced an array of bills to increase the penalties for militant protest tactics including self-defense against police and concealing one’s identity against state surveillance. Congratulate the movement on its victories, but crack down on the means by which it won them.
Over 7000 people have been arrested and many thousands injured; despite their obvious loyalty to the uniformed mercenaries of the state, prosecutors admit to over 800 allegations of police abuse, torture, rape, and battery. Piñera has expressed his “total support” for the conduct of the police and military throughout this sequence of events, but he is saying that all this brutality is not enough—in addition to arresting, beating, shooting, and killing people, he wants the police and military to be able to imprison additional massive numbers of people for long periods of time.
\bigskip
Make no mistake, the movement in Chile would not have gotten off the ground if not for the students organizing mass illegal activity. It would not have spread countrywide if not for the vandalism, arson, and acts of self-defense against police attacks. It would not have created a crisis that demanded a response if not for looting and disruption. To make a distinction between the “law-abiding” participants and the “criminals” in the movement is to say that it would be better if the movement had never taken place—it is an attempt to ensure that no such movement will ever take place again.
We have seen this many times before. The movement against police and white supremacy that burst into the public consciousness with the riots in Ferguson only got off the ground because the original participants openly attacked police officers, burned down buildings, and refused to divide into “violent” and “nonviolent” factions. Democracy itself, the system via which Chile, the United States, and so many other nations are governed, began in blazing crime; if not for criminal revolutionaries, we would still be living under the heel of hereditary monarchs.
Once again, the movement in Chile faces a crucial juncture. If the majority of the participants accept Piñera’s flattery and congratulate themselves on being “peaceful” and “honest” in contrast to those who are “criminals,” this will enable him to push through draconian measures to ensure that it will never be possible for Chileans to defend themselves against austerity measures again. On the contrary, what is needed is for the tactics of the “criminals” to spread to every honest citizen, to every person who sincerely wants peace. Neither Piñera nor anyone else who aims to rule by force will ever create peace; it can only arise when their totalitarian aspirations are thwarted.
To understand what Piñera wants, we need only look at what has happened in Egypt. Since regaining control of the country with the military coup in 2013 and introducing new measures like the ones Piñera is proposing, military strongman al-Sisi has crushed protests of all kinds. He now aspires to rule until at least the year 2034. Those who make only half a revolution dig their own graves, as the saying goes.
So the stakes are high. Demonstrators in Chile must permanently delegitimize the instruments of state power such as the police, the courts, and the army, making it impossible for them to maintain order by any combination of brutality, concessions, and prosecution. This is the only way out of the nightmare of neoliberal austerity.
This is how movements win against oppressive governments: by a winning combination of confrontational direct action, solidarity across different demographics and tactics, persistence, and strategic innovation. The movement in Chile has demonstrated this already.
To support our comrades in Chile, we have arranged the translation and design of our texts The Illegitimacy of Violence, the Violence of Legitimacy and What They Mean When They Say Peace, both of which treat these issues. We wish them strength in the struggle ahead. May every Piñera fall.
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CrimethInc.
Not Falling for It: How the Uprising in Chile Has Outlasted State Repression
And the Questions for Movements to Come
November 8, 2019
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Retrieved on 17\textsuperscript{th} June 2021 from \href{https://crimethinc.com/2019/11/08/not-falling-for-it-how-the-uprising-in-chile-has-outlasted-state-repression-and-the-questions-for-movements-to-come}{crimethinc.com}
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\item For example, $T_n = \overline{X}_n = \frac{1}{n}\sum_{i=1}^nX_i$. \pause
\item $n$ is the sample size. \pause
\item $n \rightarrow \infty$ is an approximation of what happens for large samples. \pause
\item Good things should happen when estimates are based on more information.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Convergence}
%\framesubtitle{}
\begin{itemize}
\item Convergence of $T_n$ as $n \rightarrow \infty$ is not an ordinary limit, because probability is involved. \pause
\item There are several different types of convergence. \pause
\item We will work with \emph{convergence in probability} and \emph{convergence in distribution}.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Convergence in Probability to a random variable}
%\framesubtitle{}
Definition: The sequence of random variables $X_1, X_2, \ldots$ is said to converge in probability to the random variable $Y$ if for all $\epsilon > 0$,
$\displaystyle \lim_{n \rightarrow \infty}P\{|X_n-Y|\geq\epsilon\} = 0$, and we write
$X_n \stackrel{p}{\rightarrow} Y$.\pause
\begin{columns}
\column{0.4\textwidth}
{\small
\begin{eqnarray*}
|X_n-Y| < \epsilon \pause & \Leftrightarrow & -\epsilon < X_n-Y < \epsilon \\ \pause
& \Leftrightarrow & Y-\epsilon < X_n < Y+\epsilon \\
\end{eqnarray*} \pause
} % End size
\vspace{30mm} ~
\column{0.6\textwidth}
\includegraphics[width=2.75in]{strip}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Convergence in Probability to a constant}
\framesubtitle{More immediate applications in statistics: We will focus on this.} \pause
Definition: The sequence of random variables $T_1, T_2, \ldots$ is said to converge in probability to the constant $c$ if for all $\epsilon > 0$,
{\LARGE
\begin{displaymath}
\lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0
\end{displaymath}
} % End size
% Or equivalently, \pause
% \begin{displaymath}
% \lim_{n \rightarrow \infty}P\{|T_n-c|\leq\epsilon\} = 1
% \end{displaymath} \pause
and we write $T_n \stackrel{p}{\rightarrow} c$. \pause
\begin{eqnarray*}
|T_n-c| < \epsilon \pause & \Leftrightarrow & -\epsilon < T_n-c < \epsilon \\ \pause
& \Leftrightarrow & c-\epsilon < T_n < c+\epsilon \\ \pause
\end{eqnarray*}
\begin{picture}(10,10)
% Line, direction (1,0), horizontal extent 200, starting point (50,0)
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(148,-15){$c$}
\put(100,-2){(} % Left parenthesis
\put(200,-2){)} % Right parenthesis
\put(90,-15){$c-\epsilon$}
\put(190,-15){$c+\epsilon$}
\end{picture}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example: $T_n \sim U(-\frac{1}{n}, \frac{1}{n})$}
\framesubtitle{Convergence in probability means $\lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0$}
\begin{picture}(10,10)(25,-25)
% Line, direction (1,0), horizontal extent 200, starting point (50,0)
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(148,-15){$c$}
\put(100,-2){(} % Left parenthesis
\put(200,-2){)} % Right parenthesis
\put(90,-15){$c-\epsilon$}
\put(190,-15){$c+\epsilon$}
\end{picture} \pause
\begin{itemize}
\item $T_1$ is uniform on $(-1,1)$. \pause Height of the density is $\frac{1}{2}$. \pause
\item $T_2$ is uniform on $(-\frac{1}{2},\frac{1}{2})$. \pause Height of the density is 1. \pause
\item $T_3$ is uniform on $(-\frac{1}{3},\frac{1}{3})$. \pause Height of the density is $\frac{3}{2}$. \pause
\item Eventually, $\frac{1}{n} < \epsilon$ \pause and $P\{|T_n-0|\geq\epsilon\} = 0$\pause, forever. \pause
\item Eventually means for all $n>\frac{1}{\epsilon}$.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example: $X_1, \ldots, X_n$ are independent $U(0,\theta)$}
\framesubtitle{Convergence in probability means $\lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0$} \pause
For $0 < x < \theta$, \pause
\begin{itemize}
\item[] $F_{_{X_i}}(x) = \int_0^x \frac{1}{\theta} \, dt \pause = \frac{x}{\theta}$. \pause
\item[] $Y_n = \max_i (X_i)$. \pause
\item[] $F_{_{Y_n}}(y) = \left(\frac{y}{\theta}\right)^n$ \pause
\end{itemize} \vspace{2mm}
\begin{picture}(10,10) % (25,-25)
% Line, direction (1,0), horizontal extent 200, starting point (50,0)
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(148,-15){$\theta$}
\put(100,-2){(} % Left parenthesis
\put(200,-2){)} % Right parenthesis
\put(90,-15){$\theta-\epsilon$}
\put(190,-15){$\theta+\epsilon$}
\end{picture} \pause \vspace{5mm}
\begin{eqnarray*}
P\{|Y_n-\theta|\geq\epsilon\} & = & F_{_{Y_n}}(\theta-\epsilon) \\ \pause
& = & \left(\frac{\theta-\epsilon}{\theta}\right)^n \\ \pause
& \rightarrow & 0 \mbox{ ~~~because } \frac{\theta-\epsilon}{\theta}<1. \pause
\end{eqnarray*}
So the observed maximum data value goes in probability to $\theta$, the theoretical maximum data value.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Markov's inequality: Theorem 3.6.1}
\framesubtitle{A stepping stone} \pause
Let $Y$ be a random variable with $P(Y \geq 0)=1$. \pause Then for any $a>0$,
$E(Y) \geq a \, P(Y \geq a)$. \pause \\
{ \small \vspace{3mm}
Proof (for continuous random variables): \pause
\begin{eqnarray*}
E(Y) & = & \int_0^\infty y f(y) \, dy \\ \pause
& = & \int_0^a y f(y) \, dy + \int_a^\infty y f(y) \, dy \\ \pause
& \geq & \int_a^\infty {\color{red}y} f(y) \, dy \\ \pause
& \geq & \int_a^\infty {\color{red}a} f(y) \, dy \\ \pause
& = & {\color{red}a} \int_a^\infty f(y) \, dy \\ \pause
& = & a \, P(Y \geq a) ~~~~~ \blacksquare
\end{eqnarray*}
} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Variance Rule}
\framesubtitle{Not in the text, I believe}
{\large
Let $T_1, T_2, \ldots$ be a sequence of random variables, and let $c$ be a constant. If
\begin{itemize}
\item $\displaystyle \lim_{n \rightarrow \infty}E(X_n) = c$ and
\item $\displaystyle \lim_{n \rightarrow \infty}Var(X_n) = 0$
\end{itemize}
Then $T_n \stackrel{p}{\rightarrow} c$.
} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Proof of the Variance Rule}
\framesubtitle{Using Markov's inequality: $E(Y) \geq a \, P(Y \geq a)$} \pause
{\small
Seek to show $\forall \epsilon > 0$, $\displaystyle \lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0$. \pause
Denote $E(T_n)$ by $\mu_n$. \pause
In Markov's inequality, let $Y=(T_n-c)^2$, and $a = \epsilon^2$. \pause
\begin{eqnarray*}
E[(T_n-c)^2] & \geq & \epsilon^2 P\{ (T_n-c)^2 \geq \epsilon^2 \} \\ \pause
& = & \epsilon^2 P\{ |T_n-c| \geq \epsilon \}, \mbox{ so} \\ \pause
%\end{eqnarray*}
%\begin{eqnarray*}
0 & \leq & P\{ |T_n-c| \geq \epsilon \}
\leq \frac{1}{\epsilon^2} E[(T_n-c)^2] \\ \pause
& = & \frac{1}{\epsilon^2} E[(T_n-\mu_n + \mu_n - c)^2] \\ \pause
& = & \frac{1}{\epsilon^2} E[(T_n-\mu_n)^2 +2(T_n-\mu_n)(\mu_n-c) + (\mu_n-c)^2] \\ \pause
& = & \frac{1}{\epsilon^2} \left(
E(T_n-\mu_n)^2 + 2(\mu_n-c)
{ \color{red}E(T_n-\mu_n) }
+ E(\mu_n-c)^2
\right) \\ \pause
& = & \frac{1}{\epsilon^2} \left(
E(T_n-\mu_n)^2 + 2(\mu_n-c)
({ \color{red}E(T_n)}-{\color{red}\mu_n})
+ (\mu_n-c)^2
\right) \\ \pause
& = & \frac{1}{\epsilon^2} \left(
E(T_n-\mu_n)^2 + 0 + (\mu_n-c)^2 \right)
\end{eqnarray*}
} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Continuing the proof}
%\framesubtitle{}
{\small
Have
\begin{eqnarray*}
0 &\leq& P\{ |T_n-c| \geq \epsilon \} \\
&\leq& \frac{1}{\epsilon^2} \left(
E(T_n-\mu_n)^2 + (\mu_n-c)^2 \right) \\ \pause
&=& \frac{1}{\epsilon^2} \left(
Var(T_n) + (\mu_n-c)^2 \right) \pause \mbox{, so that} \\ \pause
{\color{red} 0 }
&{\color{red}\leq}& {\color{red} \lim_{n \rightarrow\infty} P\{ |T_n-c| \geq \epsilon \} } \\
&{\color{red}\leq}& \lim_{n \rightarrow\infty} \frac{1}{\epsilon^2}
\left( Var(T_n) + (\mu_n-c)^2 \right) \\ \pause
&=& \frac{1}{\epsilon^2} \left(
\lim_{n \rightarrow\infty} Var(T_n) +
\lim_{n \rightarrow\infty} (\mu_n-c)^2
\right) \\ \pause
&=& \frac{1}{\epsilon^2} \left(
\lim_{n \rightarrow\infty} Var(T_n) +
\left( \lim_{n \rightarrow\infty}\mu_n - \lim_{n \rightarrow\infty}c \right)^2
\right) \\ \pause
&=& \frac{1}{\epsilon^2} \left(
0 + (c-c)^2
\right) \pause = {\color{red} 0 } \pause
\end{eqnarray*}
Squeeze. \hspace{5mm} $\blacksquare$
} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Law of Large Numbers}
\framesubtitle{That is, the ``Weak" Law of Large Numbers} \pause
Theorem: Let $X_1, \ldots, X_n$ be independent random variables with expected value $\mu$ and variance $\sigma^2$. \pause Then the sample mean
{\Large
\begin{displaymath}
\overline{X}_n = \frac{1}{n}\sum_{i=1}^nX_i \stackrel{p}{\rightarrow} \mu.
\end{displaymath} \pause
} % End size
Proof: $E(\overline{X}_n) = \mu$ and $Var(\overline{X}_n) = \frac{\sigma^2}{n}$.\pause
As $n \rightarrow \infty$, $E(\overline{X}_n) \rightarrow \mu$ and
$Var(\overline{X}_n)\rightarrow 0$. \pause
So by the Variance Rule,
$\overline{X}_n \stackrel{p}{\rightarrow} \mu$.\hspace{5mm} $\blacksquare$
\vspace{5mm} \pause
The implications are huge.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Probability is long-run relative frequency}
\framesubtitle{Sometimes offered as a \emph{definition} of probability!} \pause
This follows from the Law of Large Numbers. \pause
Repeat some process over and over a lot of times, and count how many times the event $A$ occurs. \pause Independently for $i=1, \ldots, n$,
\begin{itemize}
\item Let $X_i(s)=1$ if $s \in A$, and $X_i(s)=0$ if $s \notin A$. \pause
\item So $X_i$ is an \emph{indicator} for the event $A$. \pause
\item $X_i$ is Bernoulli, with $P(X_i=1) = \theta = P(A)$. \pause
\item $E(X_i) = \sum_{x=0}^1 x \, p(x) \pause = 0\cdot(1-\theta) + 1\cdot \theta
= \theta$. \pause
\item $\overline{X}_n$ is the proportion of times the event occurs in $n$ independent trials. \pause
\item The proportion of successes converges in probability to $P(A)$. % \pause
% \item So while $\overline{X}_n$ is a random quantity with its own probability distribution, \pause
% \item That distribution shrinks to fit in a tiny interval around $P(A)$, no matter how small the interval.
\end{itemize} \vspace{3mm}
\begin{picture}(10,10)
% Line, direction (1,0), horizontal extent 200, starting point (50,0)
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(148,-15){$\theta$}
\put(100,-2){(} % Left parenthesis
\put(200,-2){)} % Right parenthesis
\put(90,-15){$\theta-\epsilon$}
\put(190,-15){$\theta+\epsilon$}
\end{picture}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{More comments}
%\framesubtitle{}
\begin{itemize}
\item Law of Large Numbers is the basis of using \emph{simulation} to estimate probabilities. \pause
\item Have things like $\frac{1}{n}\sum_{i=1}^nX_i^2 \stackrel{p}{\rightarrow} E(X^2)$ \pause
\item In fact, $\frac{1}{n}\sum_{i=1}^ng(X_i) \stackrel{p}{\rightarrow} E[g(X)]$ \pause
\item Convergence in probability also applies to \emph{vectors} of random variables, like $(X_n,Y_n) \stackrel{p}{\rightarrow} (c_1,c_2)$.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Theorem}
\framesubtitle{Continuous Mapping Theorem for convergence in probability} \pause
Let $g(x)$ be a function that is continuous at $x=c$. If $T_n \stackrel{p}{\rightarrow} c$, then $g(T_n) \stackrel{p}{\rightarrow} g(c)$. \pause \vspace{5mm}
%
Examples:
\begin{itemize}
\item A Geometric distribution has expected value $\frac{1-\theta}{\theta}$. \pause $g(\overline{X}_n) = 1/(1+\overline{X}_n)$ converges in probability to \pause
\begin{eqnarray*}
\frac{1}{1+E(X_i)} \pause & = & \frac{1}{1+\frac{1-\theta}{\theta}} \\
& = & \theta
\end{eqnarray*}
\pause
\item A Uniform($0,\theta$) distribution has expected value $\theta/2$. So \\ \pause $2\overline{X}_n \stackrel{p}{\rightarrow} 2E(X_i) \pause = 2\frac{\theta}{2}=\theta$
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Background } \pause
\framesubtitle{For the proof of the continuous mapping theorem}
\begin{itemize}
\item $T_n \stackrel{p}{\rightarrow} c$ means that for all $\epsilon > 0$,
\begin{eqnarray*}
& & \lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0 \\
& \Leftrightarrow &
\lim_{n \rightarrow \infty}P\{|T_n-c|< \epsilon\} = 1
\end{eqnarray*}
\vspace{7mm}
\begin{picture}(10,10)(25,-25)
% Line, direction (1,0), horizontal extent 200, starting point (50,0)
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(148,-15){$c$}
\put(100,-2){(} % Left parenthesis
\put(200,-2){)} % Right parenthesis
\put(90,-15){$c-\epsilon$}
\put(190,-15){$c+\epsilon$}
\end{picture} \pause
% \vspace{5mm}
\item $g(x)$ continuous at $c$ means that for all $\epsilon > 0$, there exists $\delta>0$ such that if $|x-c|<\delta$, then $|g(x)-g(c)| < \epsilon$.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Proof of the Continuous Mapping Theorem}
\framesubtitle{For convergence in probability}
\begin{columns}
\column{1.1\textwidth} % To use more margin
Have $T_n \stackrel{p}{\rightarrow} c$ and $g(x)$ continuous at $c$. Seek to show that for all $\epsilon > 0$, \pause $ \lim_{n \rightarrow \infty}P\{|g(T_n)-g(c)|< \epsilon\} = 1$. \pause
Let $\epsilon > 0$ be given. \pause
$g(x)$ continuous at $c$ means there exists $\delta>0$ such that for $s\in S$, if $|X_n(s)-c|<\delta$, then $|g(X_n(s))-g(c)| < \epsilon$. \pause That is,
\vspace{3mm}
If $s_0 \in \{s: |X_n(s)-c|<\delta\}$, then $s_0 \in \{s: |g(X_n(s))-g(c)| < \epsilon\}$. \pause
This is the definition of containment\pause:
\begin{eqnarray*}
&& \{s: |X_n(s)-c|<\delta\} \subseteq \{s: |g(X_n(s))-g(c)| < \epsilon\} \\ \pause
& \Rightarrow & P(|X_n-c|<\delta) \leq P(|g(X_n)-g(c)| < \epsilon) \pause
\leq 1 \\ \pause
& \Rightarrow & \lim_{n \rightarrow \infty} P(|X_n-c|<\delta) \leq \lim_{n \rightarrow \infty}P(|g(X_n)-g(c)| < \epsilon) \leq 1 \\ \pause
&& \hspace{20mm} \equalto{}{\mbox{1}}
\end{eqnarray*}
\hspace{10mm} Squeeze $\blacksquare$
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Central Limit Theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Convergence in distribution}
\framesubtitle{Another mode of convergence} \pause
Definition: Let the random variables $X_1, X_2 \ldots$ have cumulative distribution functions $F_{_{X_1}}(x), F_{_{X_2}}(x) \ldots$\pause, and let the random variable $X$ have cumulative distribution function $F_{_X}(x)$. \pause The (sequence of) random variable(s) $X_n$ is said to \emph{converge in distribution} to $X$ if \pause
{\LARGE
\begin{displaymath}
\lim_{n \rightarrow \infty}F_{_{X_n}}(x) = F_{_X}(x)
\end{displaymath} \pause \vspace{4mm}
} % End size
at every point where $F_{_X}(x)$ is continuous\pause, and we write $X_n \stackrel{d}{\rightarrow} X$.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example: Convergence to a Bernoulli with $p=\frac{1}{2}$}
\framesubtitle{$\lim_{n \rightarrow \infty}F_{_{X_n}}(x) = F_{_X}(x)$ at all continuity points of $F_{_X}(x)$} \pause
\begin{displaymath}
p_{_{X_n}}(x) =
\left\{ \begin{array}{cl} % ll means left left
1/2 & \mbox{for } x=\frac{1}{n} \\
1/2 & \mbox{for } x=1+\frac{1}{n} \\
0 & \mbox{Otherwise}
\end{array} \right.
\end{displaymath} \vspace{3mm} \pause
\begin{picture}(10,10)(0,-10)
\put(15,-2){$n=1$}
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(100,5){\line(0,-1){10} }
\put(200,5){\line(0,-1){10} }
\put(98,-15){0}
\put(148,-15){1}
\put(198,-15){2}
\put(197.5,-2){$\bullet$}
\put(147.5,-2){$\bullet$}
\end{picture} \pause
\begin{picture}(10,10)(0,10)
\put(15,-2){$n=2$}
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(100,5){\line(0,-1){10} }
\put(200,5){\line(0,-1){10} }
\put(98,-15){0}
\put(148,-15){1}
\put(198,-15){2}
\put(172.5,-2){$\bullet$}
\put(122.5,-2){$\bullet$}
\end{picture} \pause
\begin{picture}(10,10)(0,30)
\put(15,-2){$n=3$}
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(100,5){\line(0,-1){10} }
\put(200,5){\line(0,-1){10} }
\put(98,-15){0}
\put(148,-15){1}
\put(198,-15){2}
\put(164.7,-2){$\bullet$}
\put(114.7,-2){$\bullet$}
\end{picture} \pause \vspace{15mm}
\begin{itemize}
\item For $x<0$, $\lim_{n \rightarrow \infty}F_{_{X_n}}(x)=$ \pause $0$ \pause
\item For $0<x<1$, $\lim_{n \rightarrow \infty}F_{_{X_n}}(x)=$ \pause $\frac{1}{2}$ \pause
\item For $x>1$, $\lim_{n \rightarrow \infty}F_{_{X_n}}(x)=$ \pause $1$ \pause
\item What happens at $x=0$ and $x=1$ does not matter.
\end{itemize}
\end{frame} % A picture of the cdf would be really good.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Convergence to a constant} \pause
%\framesubtitle{}
{\small
Consider a ``degenerate" random variable $X$ with $P(X=c)=1$. \pause \vspace{2mm}
\begin{picture}(10,10) % (25,-25)
% Line, direction (1,0), horizontal extent 200, starting point (50,0)
\put(50,0){\line(1,0){200} }
\put(150,5){\line(0,-1){10} }
\put(148,-15){$c$}
\put(100,-2){(} % Left parenthesis
\put(200,-2){)} % Right parenthesis
\put(90,-15){$c-\epsilon$}
\put(190,-15){$c+\epsilon$}
\end{picture} \pause \vspace{5mm}
Suppose $X_n$ converges in probability to $c$. \pause
\begin{itemize}
\item Then for any $x>c$, $F_{_{X_n}}(x) \rightarrow 1$ for $\epsilon$ small enough. \pause
\item And for any $x<c$, $F_{_{X_n}}(x) \rightarrow 0$ for $\epsilon$ small enough. \pause
\item So $X_n$ converges in distribution to $c$.
\end{itemize} \pause
Suppose $X_n$ converges in distribution to $c$, so that $F_{_{X_n}}(x) \rightarrow 1$ for all $x>c$ and $F_{_{X_n}}(x) \rightarrow 0$ for all $x<c$. \pause Let $\epsilon>0$ be given.
% If necessary make it smaller.
\pause
\begin{eqnarray*}
P\{|X_n-c|<\epsilon\} & = & P\{ c-\epsilon < X_n < c+\epsilon \} \\ \pause
& = & F_{_{X_n}}(c+\epsilon)-F_{_{X_n}}(c-\epsilon) \pause \mbox{ so} \\ \pause
\lim_{n \rightarrow \infty}P\{|X_n-c|<\epsilon\}
& = & \lim_{n \rightarrow \infty}F_{_{X_n}}(c+\epsilon) -
\lim_{n \rightarrow \infty}F_{_{X_n}}(c-\epsilon) \\ \pause
& = & 1-0 = 1
\end{eqnarray*} \pause
And $X_n$ converges in probability to $c$.
} % End size of whole slide.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Comment}
%\framesubtitle{}
\begin{itemize}
\item Convergence in probability might seem redundant, because it's just convergence in distribution to a constant. \pause
\item But that's only true when the convergence is to a constant. \pause
\item Convergence in probability to a non-degenerate random variable \pause implies convergence in distribution. \pause
\item But convergence in distribution
does not imply convergence in probability when the convergence is to a non-degenerate variable.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Big Theorem about convergence in distribution}
\framesubtitle{Theorem 4.4.2 in the text} \pause
Let the random variables $X_1, X_2 \ldots$ have cumulative distribution functions $F_{_{X_1}}(x), F_{_{X_2}}(x) \ldots$ and moment-generating functions $M_{_{X_1}}(t), M_{_{X_2}}(t) \ldots$. \pause
Let the random variable $X$ have cumulative distribution function $F_{_X}(x)$ and moment-generating function $M_{_X}(t)$. \pause
If
\begin{displaymath}
\lim_{n \rightarrow \infty} M_{_{X_n}}(t) = M_{_X}(t)
\end{displaymath}
for all $t$ in an open interval containing $t=0$, \pause then $X_n$ converges in distribution to $X$. \pause \vspace{5mm}
The idea is that convergence of moment-generating functions implies convergence of distribution functions. This makes sense because moment-generating functions and distribution functions are one-to-one.
\end{frame}
% _{_{X_1}} _{_{X_n}} _{_X}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example: Poisson approximation to the binomial}
\framesubtitle{We did this before with probability mass functions and it was a challenge.} \pause
Let $X_n$ be a binomial ($n,p_n$) random variable with $p_n=\frac{\lambda}{n}$, so that $n \rightarrow \infty$ and $p \rightarrow 0$ in such a way that the value of $n \, p_n=\lambda$ remains fixed. Find the limiting distribution of $X_n$. \pause \vspace{1mm}
Recalling that the MGF of a Poisson is $e^{\lambda(e^t-1)}$ and $\left(1 + \frac{x}{n}\right)^n \rightarrow e^x$, \pause
\begin{eqnarray*}
M_{_{X_n}}(t) & = & (\theta e^t+1-\theta )^n \\ \pause
& = & \left(\frac{\lambda}{n}e^t+1-\frac{\lambda}{n} \right)^n \\ \pause
& = & \left(1+\frac{\lambda(e^t-1)}{n} \right)^n \\ \pause
& \rightarrow & e^{\lambda(e^t-1)} \\ \pause
\end{eqnarray*}
MGF of Poisson($\lambda$).
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Central Limit Theorem}
\framesubtitle{Proved using limiting moment-generating functions} \pause
Let $X_1, \ldots, X_n$ be independent random variables from a distribution with expected value $\mu$ and variance $\sigma^2$. \pause Then
\begin{displaymath}
Z_n = \frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma} \stackrel{d}{\rightarrow} Z \sim N(0,1)
\end{displaymath} \pause
In practice, $Z_n$ is often treated as standard normal for $n>25$\pause, although the $n$ required for an accurate approximation really depends on the distribution.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Sometimes we say the distribution of the sample mean is approximately normal, or ``asymptotically" normal.} \pause
%\framesubtitle{}
\begin{itemize}
\item This is justified by the Central Limit Theorem. \pause
\item But it does \emph{not} mean that $\overline{X}_n$ converges in distribution to a normal random variable. \pause
\item The Law of Large Numbers says that $\overline{X}_n$ converges in probability to a constant, $\mu$. \pause
\item So $\overline{X}_n$ converges to $\mu$ in distribution as well. \pause
\item That is, $\overline{X}_n$ converges in distribution to a degenerate random variable with all its probability at $\mu$.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Why would we say that for large $n$, the sample mean is approximately $N(\mu,\frac{\sigma^2}{n})$?}
\pause
\vspace{5mm}
Have
$Z_n = \frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma}$ \pause converging to
$ Z \sim N(0,1)$.
\pause
{\footnotesize
\begin{eqnarray*}
Pr\{\overline{X}_n \leq x\} \pause
& = &
Pr\left\{ \frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma} \leq
\frac{\sqrt{n}(x-\mu)}{\sigma}\right\} \\ \pause
& = &
Pr\left\{ Z_n \leq \frac{\sqrt{n}(x-\mu)}{\sigma}\right\} \pause
\approx \Phi\left( \frac{\sqrt{n}(x-\mu)}{\sigma} \right)
\end{eqnarray*} }
\pause
Suppose $Y$ is \emph{exactly} $N(\mu,\frac{\sigma^2}{n})$:
\pause
{\footnotesize
\begin{eqnarray*}
Pr\{Y \leq x\} \pause
& = &
Pr\left\{ \frac{\sqrt{n}(Y-\mu)}{\sigma} \leq
\frac{x-\mu}{\sigma/\sqrt{n}}\right\} \\ \pause
& = &
Pr\left\{ Z_n \leq \frac{\sqrt{n}(x-\mu)}{\sigma}\right\} \pause
= \Phi\left( \frac{\sqrt{n}(x-\mu)}{\sigma} \right)
\end{eqnarray*}
} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Copyright Information}
This slide show was prepared by \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
{Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\vspace{5mm}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
and $Var(Y) = \sigma^2$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{} \pause
%\framesubtitle{}
\begin{itemize}
\item \pause
\item \pause
\item
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\includegraphics[width=2in]{BivariateNormal}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
%\framesubtitle{}
{\LARGE
\begin{eqnarray*}
m_1 & = & a + b \\
m_2 & = & c + d
\end{eqnarray*}
} % End size
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{} \pause
%\framesubtitle{}
\begin{itemize}
\item \pause
\item \pause
\item
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# R code for the convergence in probability plot
rm(list=ls())
epsilon = 1
Xn = seq(from=-10,to=10,by=0.1)
Y = Xn
y1 = Xn+epsilon
y2 = Xn-epsilon
plot(Xn,Y,pch=' ',axes=F,xlab='',ylab='')
lines(Xn,Y,lty=2)
lines(Xn,y1,lty=1); lines(Xn,y2,lty=1)
# Draw my own axes
xx = c(-9,9); yy = c(0,0); lines(xx,yy,lty=1)
xx = c(0,0); yy = c(-9,9); lines(xx,yy,lty=1)
text(10,0,'x'); text(0,10,'y')
text(-0.3,epsilon,expression(paste(epsilon)))
text(-0.5,-epsilon,expression(paste(-epsilon)))
# R code for plots of normal MGFs
tt = seq(from=-1,to=1,by=0.05)
mu = 0; sigsq = 1
zero = exp(mu*tt + 0.5*sigsq*tt^2)
mu = 1; one = exp(mu*tt + 0.5*sigsq*tt^2)
mu = -1; minusone = exp(mu*tt + 0.5*sigsq*tt^2)
x = c(tt,tt,tt); y = c(zero,one,minusone)
plot(x,y,pch=' ',xlab='t',ylab = 'M(t)')
lines(tt,zero,lty=1)
lines(tt,one,lty=2)
lines(tt,minusone,lty=3)
title("Fingerprints of the normal distribution")
# Legend
x1 <- c(-0.4,0) ; y1 <- c(4,4) ; lines(x1,y1,lty=1)
text(0.25,4,expression(paste(mu," = 0, ",sigma^2," = 1")))
x2 <- c(-0.4,0) ; y2 <- c(3.75,3.75) ; lines(x2,y2,lty=2)
text(0.25,3.75,expression(paste(mu," = 1, ",sigma^2," = 1")))
x3 <- c(-0.4,0) ; y3 <- c(3.5,3.5) ; lines(x3,y3,lty=3)
text(0.25,3.5,expression(paste(mu," = -1, ",sigma^2," = 1")))
# R code for plots of chi-squared MGFs
tt = seq(from=-0.25,to=0.25,by=0.005)
nu = 1; one = (1-2*tt)^(-nu/2)
nu = 2; two = (1-2*tt)^(-nu/2)
nu = 3; three = (1-2*tt)^(-nu/2)
x = c(tt,tt,tt); y = c(one,two,three)
plot(x,y,pch=' ',xlab='t',ylab = 'M(t)')
lines(tt,one,lty=1)
lines(tt,two,lty=2)
lines(tt,three,lty=3)
title("Fingerprints of the chi-squared distribution")
# Legend
x1 <- c(-0.2,-0.1) ; y1 <- c(2.5,2.5) ; lines(x1,y1,lty=1)
text(-0.05,2.5,expression(paste(nu," = 1")))
x2 <- c(-0.2,-0.1) ; y2 <- c(2.3,2.3) ; lines(x2,y2,lty=2)
text(-0.05,2.3,expression(paste(nu," = 2")))
x3 <- c(-0.2,-0.1) ; y3 <- c(2.1,2.1) ; lines(x3,y3,lty=3)
text(-0.05,2.1,expression(paste(nu," = 3")))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% I cut this out because it hurt the continuity of LLN.
\begin{frame}
\frametitle{Calculation of $Var(\overline{X}_n)$}
\framesubtitle{An aside}
\begin{eqnarray*}
Var\left( \frac{1}{n}\sum_{i=1}^nX_i \right) \pause
& = & \frac{1}{n^2} Var\left(\sum_{i=1}^nX_i \right) \\ \pause
& \stackrel{ind}{=} & \frac{1}{n^2} \sum_{i=1}^nVar(X_i) \\ \pause
& = & \frac{1}{n^2} \sum_{i=1}^n \sigma^2 \pause = \frac{1}{n^2} n\sigma^2 \pause
= \frac{\sigma^2}{n}
\end{eqnarray*}
\end{frame}
|
http://dlmf.nist.gov/7.5.E1.tex | nist.gov | CC-MAIN-2014-10 | application/x-tex | null | crawl-data/CC-MAIN-2014-10/segments/1393999661726/warc/CC-MAIN-20140305060741-00040-ip-10-183-142-35.ec2.internal.warc.gz | 52,725,334 | 628 | \[\mathop{F\/}\nolimits\!\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{{-z^{2}}%
}-\mathop{w\/}\nolimits\!\left(z\right)\right)=-\tfrac{1}{2}i\sqrt{\pi}e^{{-z^%
{2}}}\mathop{\mathrm{erf}\/}\nolimits\!\left(iz\right).\] |
https://lhcbproject.web.cern.ch/lhcbproject/Publications/p/Directory_LHCb-PAPER-2014-001/Table_2.tex | cern.ch | CC-MAIN-2020-10 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-10/segments/1581875145818.81/warc/CC-MAIN-20200223154628-20200223184628-00271.warc.gz | 445,590,744 | 1,146 |
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https://anarhija.info/library/germania-azioni-di-solidarieta-con-i-prigionieri-di-g20-07-08-2017-it.tex | anarhija.info | CC-MAIN-2021-10 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-10/segments/1614178362899.14/warc/CC-MAIN-20210301182445-20210301212445-00318.warc.gz | 194,491,034 | 3,149 | \documentclass[DIV=12,%
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\title{Germania: Azioni di solidarietà con i prigionieri di G20 (07-08\Slash{}2017)}
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\textbf{Amburgo}: Nella notte di 3 agosto 2017 una macchina della POWER (Personen-Objekt-Werkschutz GmbH) è stata data alle fiamme. Solidarietà con i prigioneri di G20!
\textbf{Flensburgo}: Nelle notti tra il 1° e il 2 agosto, e l’8 eil 9 agosto, i commissariati di Flensburg e di Tarp (nei dintorni di Flensburg) sono stati presi di mira in solidarietà con tutti i prigionieri di G20. In questi due attacchi è stata gettata della pittura sulla facciata e lasciata la scritta “Welcome to Hell”. Infine, nella notte tra il 14 e il 15 agosto un commissariato di Handewittt è anche stato preso di mira (lasciata pittura e scritte sulla facciata in relazione al vertice G20).
\textbf{Münster (Monaco di Vestfalia)}: Alle prime ore di 6 agosto, nei dintorni di via Weseler, sono state lasciate delle scritte su sette veicoli di polizia: “All Cops are targets” [\emph{tutti gli sbirri sono bersagli, ndt}] o “Aprite gli occhi sulla scelta della vostra professione!”. Nella notte di 13 luglio scorso, 15 veicoli di polizia sono stati danneggiati poche ore dopo il vertice di G20, necessitando un ricambio di tutti i vetri (70.000 euro di danni). Gli inquirenti suppongono che ci sia un legame tra questi due attacchi avvenuti quasi a un mese di distanza l’uno dall’altro.
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Anarhija.info
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Germania: Azioni di solidarietà con i prigionieri di G20 (07-08\Slash{}2017)
\bigskip
\href{https://sansattendre.noblogs.org/post/2017/08/15/allemagne-attaques-solidaires-avec-les-prisonniers-du-g20-aout-2017/}{sansattendre.noblogs.org}
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\textbf{anarhija.info}
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https://mirror.anarhija.net/sv.theanarchistlibrary.org/mirror/a/aa/anonym-appendix-till-en-avbruten-debatt-om-anonymitet-och-attack.tex | anarhija.net | CC-MAIN-2021-25 | application/octet-stream | application/x-tex | crawl-data/CC-MAIN-2021-25/segments/1623488519735.70/warc/CC-MAIN-20210622190124-20210622220124-00339.warc.gz | 339,059,205 | 11,999 | \documentclass[DIV=12,%
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\title{Appendix till en avbruten debatt om anonymitet och attack}
\date{Mars 2014}
\author{Anonym}
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En debatt är ett djuptågende utforskande av en speciell fråga genom konfrontation mellan två eller fler sidor, var och en sida med dess egna position. Till skillnad från de som anser att debatter bör undvikas för att inte provocera fram splittringar, anser vi att de behöver förses med näring. Målet med debatten inte är att utse en vinnare, som vi alla måste buga inför, utan istället att berika de olika sidornas medvetanden. Debatter tydliggör idéer.
Uttalandet av och konfrontationen mellan olika idéer – detta är precis vad en debatt är! - lyser upp de skumma delarna och indikerar idéernas svaga punkter. Detta hjälper alla, undantagslöst. Det hjälper alla deltagande sidor att finslipa, korrigera och förstärka sina egna idéer. Det hjälper samtidigt alla som ämnar delta i debatten, att välja vilken sida att stå på (må det vara den ena sidan, den andra sidan eller ingen av de diskuterande sidorna).
Den anarkistiska rörelsens historia är full av debatter. Alla har varit användbara även om de ibland har varit smärtsamma. Dess historia är tillika full med avsaknade debatter, olika idéer som aldrig konfronterades, som i slutändan lämnade alla åt sina egna övertygelser (eller tvivel). Var detta till det bättre, då det på så vis undivikits steril polemik? Nej, enligt oss var det till det sämre då fruktsamma diskussioner inte fick komma till skott.
En av dessa avsaknade debatter är den om användandet av akronymer, att representera riktiga organisationer och att ta ansvar för direkta aktioner mot den härskande ordningen. För oss verkar denna debatt, om än viktig, ha avbrutits samma ögonblick som den föddes.
På en internationell nivå gjordes ett försök att öppna upp en sådan debatt genom texten Brev till den Anarkistiska Galaxen som dök upp i slutet av 2011. Denna text var en presentation av idéerna som förespråkar anonymitet och tar avstånd ifrån användandet av organisatoriska och ansvarstagande akronymer. Den talade även om de insurrektionella perspektiven, föreställningen om informalitet och mångfalden av attacker.
Exakt ett år senare i november 2012, vid tillfället för det anarkistiska mötet i Zürich, spred anarkisterna i Conspiracy Cells of Fire en text där de presenterade idéerna som förespråkar användandet av organisatoriska akronymer och de som tar avstånd från anonymitet. Denna text presenterade även några mer allmänna idéer kring anarkistisk intervention, som relationen till ”medlande kamper” eller formandet av urbana gerillagrupper. Gott. Med utgångspunkt i olika idéer har de båda sidorna gjort sina egna presentationer. Det enda som nu saknades för att starta en debatt var konfrontationen av dessa olika idéer. Detta är vad till exempel anarkisterna som i augusti 2013 spred texten Anonymity försökte göra, med sin explicita utgångspunkt i CCF:s texter för kritik och svar på tal.
Vid tillfället för det Internationella Anarkistiska Symposiet i Mexiko i december 2013, spred CCF texten (Let’s become dangerous\dots{} for the spreading of the Black International), i vilken kapitlet ”FAI, akronymer och den 'anarkistiska galaxens' anonymitet” inleds med följande antydan: ”Vi är medvetna om den tillplattade polemik som släppts lös mot FAI av kamrater och ”kamrater””. En talande premiss, då den reducerar det som kunde ha varit en debatt för alla till polemik mot någon. Därtill görs här även en åtskillnad mellan de som har försökt att skapa en sådan debatt, mellan kamrater och ”kamrater”(?). Detta bidrag refererar specifikt till ett par texter såsom Brev till den Anarkistiska Galaxen och Anonymity, avfärdandes det sistnämnda bidraget som en text ”skriven av anarkister från den politiska anonymitetens spänning [\dots{}] utan någon kamratlig inställning.” En debatt vore möjlig och önskvärd för att fördjupa idéer, för att specifikt unvika att blockera och låsa allt utrymme med enkla ”pro” och ”contra” men det verkar för oss som att anklagelser i stil med ”teoretiker som inte gör något” snarare sätter punkt för diskussionen. Så vi kunde ha håll käften eller bara låtit det vara. Och förvisso skulle vi gladeligen ha besparat oss själva från att nära en debatt som – i motsats till vad författarna till Anonymity tänkte – uppenbarligen är oönskad.
Så om vi fortfarande ska föra vår talan, så är det bara för att vi inte vill att en eventuell tystnad ska ses som ett förslag, ett fel som i dessa mörka och ledsamma dagar skulle kunna ske. Det är därför, trots den uppenbara meningslösheten, vi ansåg att det fortfarande är viktigt att skriva ett appendix till en debatt som nu har avbrutits. Det kommer att vara ett slutgiltigt appendix, ett som kommer att ha svårt med uppföljning, ett appendix författat i en splittrad motvilja, bara för att undvika att verka inställsamma.
Vad sade texten Anonymity? I princip två saker. Först och främst, och detta är i den ordning det skrevs, inte utifrån relevans, så sade texten att anonymitet är att föredra framför så kallad ”taktiskt” perspektiv. Att hålla fast vid en identitet ger mer utrymme åt juridiken att låta associativa anklagelser hagla över kamrater, då snarare än att lämna uppgiften att uppfinna någon ”organisation” (så som repression ofta har gjort i anarkismens historia) åt polisen och domarna i en skrattspegel av deras repressiva spektakel, erbjuder anarkister med en fascination för organisations-identitet detta direkt till dem. Repression kommer alltid att försöka reducera upproret till en enda organisation (existerande eller påhittad), en enda grupp eller om så bara några få individer, för att försöka skapa en klyfta mellan de påstådda ”gärningspersonerna” och ”åskådarna” och att stämpla detta på det anarkistiska och revolutionära upprorets träsk, på de enstaka sociala spänningarna och individuella handlingar, på affiniteter och omgrupperingar, på informaliteten och mångfalden av attacker och metoder. Ett diagram som reflekterar dess egna auktoritära struktur (för domare känner inte till något annat och kan inte föreställa sig att det existerar ett diffust och onkontrollerbart uppror), med en juridisk översättning av roller (ledare, skattmästare, strateger, bombexperter, gevärsmän, sympatisörer, sabotörer,\dots{}) i total motsats till de anarkistiska och antiauktoritära idéerna. För att dessa idéer utgår från individen – från den individuella kapaciteten att tänka, handla och förena sig med andra i kampen mot makten – och stöter bort anslutandet eller uppslukandet av individen i strukturer som stympar dess vilja och idéer. Vi är förstås väl medvetna om att repression också slår mot anarkister även när de inte använder sig av akronymer och frågan är över huvud taget inte om en skäms över sina handlingar och idér eller ej. I detta fallet handlar det helt enkelt om att komplicera domarnas uppgift för att förlänga motsättningarna, att få dem att hålla ut och konstant öppna upp mer utrymme för anarkister och rebeller att kasta sig in i striden. Anonyma aktioner – och med anonyma menar vi aktioner som följs upp av den mest absoluta tystnad, aktioner som följs upp av minimala kommunikéer utan akronymer eller i vart fall utan återkommande akronymer – underlättar inte fiendens repressiva uppgift, eftersom att, vid sidan av handlingen i sig, så måste fienden hitta på allting själva. Ingen säger till dem att ”det var jag som gjorde det,” ingen ger några ytterligare ledtrådar (som till exempel ligvistiska koder i kommunikéer, en organisationsakronym,\dots{}) så att de kan finna gärningspersonerna.
På dessa anmärkningar, föreslagna i Anonymity, varken svarar eller besvarar anarkisterna i CCF i sitt citat från Odysséen. Istället begränsar de sig själva till att konstatera att ”ytlig kunskap är värre än okunskap” och till att erinra sig om att ”Odysseus, när han lämnade ön Polyphemus, skrek från sitt skepp ”Jag, Odysseus, förblindade er\dots{}”.” Det är hemskt att se någon som krypandes krampaktigt greppar efter grässtrån. Odysseus tog ansvar för sin handling först efter att han hade lämnat fiendens ö, när han trodde sig vara säker i sin båt (och för övrigt, gjorde han detta mot sina kamraters varningar). Med andra ord så tog han på sig ansvaret först när han trodde att kriget med cykloperna var över. Under tiden kriget rasade förblev han tyst.
Låt oss nu lämna de litterära myterna för tillfället. Den andra poängen i Anonymity var att endast med fråvaron av identiteter som reser sig över andra, även genom massmedias exploatering, är jämlikhet möjlig. Där det inte finns några ledare, finns det inte heller några anhängare. Där det inte finns några celebriteter, finns det heller inga beundrare. Där det inte finns någon som kan sätta sig över andra, finns det inte heller någon som hamnar under. I anonymitetens mörker är alla jämlikar. Vad är det för mening med att ta detta steg längre än de andra mörka rebeller som attackerar makten?
I ett bidrag till Symposiumet i Mexiko kan vi läsa att ”FAI är helt enkelt det osynliga community (sic!) där begären efter att attackera vår era möts.” Men så varför ska begären att attackera vår era bara mötas i det begränsade utrymme som ryms i tre bokstäver och inte i hela det upproriska alfabetet? Ett argument som lyftes fram av anarkisterna i CCF, är att de vill göra skillnad mellan dem själva och de anarkister som följer i vänsterns släptåg. Men varför skulle ett namn skilja oss från de taktlösa syndikalisterna och de luriga medborgarrättsmilitanterna, mer än användandet av direkt aktion som ett uttryck för en permanent och inte bara en snitsigt omväxlande konfliktualitet? Vi läste också att ”Handlingar talar för sig själva genom kommunikéer, då de håller sin distans till den 'anarkistiska' oppositionen, som ibland bränner ner banker i 'de fattigas namn och mot plutokratins kapital', för att bevisa att den i vart fall gör någonting. Nej, hetlevrade celler. Ni kommer inte att sälja oss sådan förvirring. Handlingar talar för sig själva eller så talar de genom kommunikéer. Det är inte samma sak; det har aldrig varit samma sak. Enligt er så talar handlingar genom kommunikéer. Enligt oss så talar de för sig själva. Och detta är hela denna frågas kärna.
En behöver inte leta så länge för att hitta suggestiva exempel. I Aten den första november förra året, öppnade någon eld mot några medlemmar i Gyllene Gryning. Två fascister döda. En handling som talar för sig själv. Med fascister bör en inte diskutera, inte förhandla, en bör inte be den demokratiska Staten att dra tillbaka sina stormtrupper. Nej, vi bekämpar dem direkt, utan medling, med alla metoder av attack en finner lämpligt. Den dagen, när denna aktion var anonym, hyllades den av anarkister över hela världen. Mycket vanligt folk, i Grekland och i resten av världen, hyllade aktionen. Vad mer behövdes egentligen? På vilket sätt berikade Kämpande Folkliga Revolutionära Krafter aktionen med kommunikéen som publicerades den sextonde november? Inte på något sätt. Nej, kommunikéen snarare försvagade aktionen genom att den kopplade handlingen till identitet och till ideologin hos en av den revolutionära rörelsens alla utbrytargrupper. Hade det varit någon skillnad om det istället för FPRF hade varit GRA, eller FLG, eller BPC, eller BPRKJ, eller XJT, eller ZZPPHQWX som hade tagit på sig handlingen? Självklart inte. Förra året visade några kamrater att kärnkraftsetablissemanget är sårbart. Denna aktion klargjorde att det fortfarande finns de som är ansvariga och att det är möjligt att attackera dem. På vilket sätt berikade den efterföljande kommunikéen aktionen? Var inte denna aktion tydlig, precis och riktig?
Ja, handling talar för sig själv. De behöver inte bombastiska kommunikéer. Det är de kämpande organisationerna som behöver kommunikéer för inrätta sin hegemoni i rörelsen, för att få sitt ljus att lysa starkare än resten av den anarkistiska galaxen, för att bli stjärnor att referera till omgivna av satelitter.
En skulle kunna svara att om aktionerna förblev anonyma, så skulle de också kunna genomföras i andra syften som en inte delar eller med olämpliga motiv. Det skulle till och med kunna vara ondskefulla krafters verk, av maffia eller gangsters, av fascister eller till och med Staten själv. Och därför, för att undvika all förvirring och för att våld verkligen inte är blott privilegerat anarkister och antiauktoritära, så bör en ta på sig ansvaret för sina handlingar. Men i spegelbilden av den demokratiska förvaltningen av den sociala freden, i detta kadavers spektakel, så förlorar alltid orden sin mening: de anarkistiska idéerna kan inte spridas på andra sätt än på de anarkistiska, genom själva kampen, långt utom räckhåll för Statens klor; om inte, så stympas de i olika grad beroende på maktens behov av kontroll och konsensusproduktion. Den organiserade förvirringen är en aspekt av repression, till och med en stöttepelare, men det går inte att krossa den med kommunikéer utan den går endast att krossa i de kamputrymmen där ord och betydelser sammansmids av rebellerna själva, att använda i dialoger mellan varandra, utan mediering, utan representation.
Om attackerna som anarkister föreslår och gör verklighet av syftar till att förstöra maktens strukturer och individer, så är den viktiga aspekten förstörandet i sig. Vi vill ha frihet och för att detta ska bli verklighet, måste det som kväver oss förstöras. Bra. Från frihet eller från kaos om du föredrar det, om bara så tillfälligt eller kortvarigt, kan många tendenser till anarki framväxa men också tendenser till mycket mindre vackra saker. En kan inte lura sig själv till att tro att detta står och faller med ansvarstagande kommunikéer: det hänger på idéerna som vi är kapabla till att utveckla och sprida, på den förståelse och bedömning som anarkister lyckas skapa av den verklighet som förändras och störtas genom attackerna och revolterna. Här kommer vi åter till samma grundläggande problem: tanke och dynamit, så som en en anarkist i slutet av artonhundratalet konstaterade. Dynamiten kan inte ersätta idéerna; idéerna kan inte ersätta dynamiten. De är två intimt sammanbundna aspekter av anarkism, aspekter som fräter på det auktoritära samhället: på dess ideologier såväl som dess strukturer, på dess invånare såväl som deras värderingar, på dess sociala relationer såväl som på dess snutar. Relationen mellan dessa två aspekter är perspektivet och det är faktiskt detta som debatten borde handla om. Problemet med perspektivet kan inte lösas genom att skicka högtravande kommunikéer eller genom att förstärka identitets-organisations-loggan, inte heller genom att hela tiden upprepa samma tio anarkismens grundläggande banaliteter eller vad som utgör individualismens trossats.
CCF tycker inte om ”de som gömmer sig bakom anonymitet.” De väljer ett namn och ”dess namn är FAI och det är vårt ”vi”. Ett kollektivt ”vi”.” Det får oss att tänka på de trista militanta anarkisterna från förr som anklagade Emile Henry för att inte låta sig själv arresteras som en viss August Vaillant gjorde, för att inte ha velat ta ansvar för sin handling på platsen (för att han ville fortsätta sina attacker!). CCF föreslår att ”vi ska lämna den ”anarkistiska” galaxens teoretiker, som predikar politisk anonymitet utan att göra något, bakom oss. För att, om vi ska tala sanning, en del av den politiska anonymitetens spänningar i hög grad gömmer sin rädsla för repression bakom sina teorier.” Att de anonyma kamraterna håller sig ”bakom” CCF, det råder det ingen tvekan om. Det vill säga, om du anser att CCF:s hysteri är att gå framåt, att göra sig sedda, att föra sin talan\dots{} men de kamrater som beslutat sig för att inte bara ställa sina aktioner under massmedians nåd, som vill fortsätta att vara ”mörka individer bland andra mörka individer,” skulle ju bara göra detta för att dölja sin egen inaktivitet eller sin rädsla för repression, detta är verkligen en uppvisning av en ond cirkel. Ett perfekt argument för att avbryta en debatt: de som kritiserar, gör bara det för att de inte handlar och för att de är rädda.
Men lusten att förbli anonym uttrycker på samma gång vägran av all avantgardism och ett försök att hålla sig utom räckhåll för repressionens klor, för att kunna förlänga fientligheterna och inte skammen över våra handlingar. Förresten, hysterin kring att ta ansvar för aktioner har inte alltid funnits. Eller var Ravachol, Henry, Di Giovanni\dots{} kanske ”gömda” bakom anonymitet? Nej, de bara handlade. De handlade utan behov av att beundra sig själva i mediaspegeln, som fortsätter att reflektera ens egna identitetslogga. Och om dessa anarkisters handlingar inte var tydliga eller förståeliga, så försökte hela den anarkistiska rörelsen, genom debatter, tidningar, affischer, pamfletter och så vidare, att göra dem det. I slutändan tillgörde dessa aktioner alla som såg sig själva i den anarkistiska kampen. På så sätt försökte tanken och dynamiten att gå hand i hand. Båda anarkismens aspekter, i kampperspektivets utrymme. Men ja, detta var den Gamla Anarkin.
Idag hör vi alltmer prat om den ”Nya Anarkin.” Hur löjlig denna pretention är framgår redan av själva namnet. Sedan det förra milleniet har anarkister från Spanien och Italien, från Frankrike och Argentina, från här och där, vuxit upp med samma gamla syndikalistiskt militanta refräng i sina öron, föreställandes att de enda äkta anarkisterna är de som är med i FAI (Federación Anarquista Ibérica, Federazione Anarchica Italiana, Fédération Anarchiste Française, FORA i Argentina\dots{}). Utanför FAI finns det ingen frälsning, bara tvetydigheter. Utanför anarkismens representativa organisationer, finns det ingenting. Ja och idag, då kommer det anarkister från hela världen för att påminna om att de sanna anarkisterna, praktikens anarkister, bara är de som tillhör FAI (Informal Anarchist Federation). Möjligen kan de tolerera de som accepterar att följa den Svarta Internationalen eller de som ”av estetiska skäl,” så som CCF har uttryckt det, väljer att agera anonymt. Den Nya Anarkin verkar inte så nytt för oss, den bara reproducerar den gamla: federationer, program, pakter, kommunikéer, akronymer och uppsvällda slogans.
Flera texter och bidrag har försökt och försöker fortfarande att öppna upp debatten om informalitet och även Brev till den anarkistiska galaxen fokuserade på detta. Vi är ”bewildered” av hur någon seriöst kan komma på tanken att försöka sälja en stabil revolutionär organisation, en permanent och formell akronym, en rigid handlingsmetod som alltid är densamma och fördefinierad (gör en aktion, skriv en kommuniké och skicka runt den), som informalitet till oss. Även i de mest simpla av ordet ”informells” betydelser, som var och en trots allt pekar på frånvaron av alla formaliteter, verkar det svårt att förneka att en akronym är en formalitet. Alltså är den Informella AnarkistFederationen, den Internationella Revolutionära Fronten eller vad de än vill kalla sig, inga informella organisationer. Problemet är inte att strida om ”the paternity” i ordet ”informell” ( vi är inte intresserade av att starta ett parti med sina dogmer, sina förutbestämda definitioner, som alltid är frånkopplade kampen i sig och därmed bara parasitisk) – problemet är förvirringen som omöjliggör en riktig debatt. Om en står för uppbyggandet av en permanent anarkistisk kamporganisation, så borde den bara säga det och den kan bli förstådd av alla anarkister. Om en står för ett syndikalistiskt föhållningssätt till kampen och accepterar ”steg för steg”-logiken och ”the revendicative” kampen för att förbättra det existerande och på så sätt få det berömda ”proletära medvetandet” att växa, så är det inte till någon hjälp (mer än till att sprida förvirring) att presentera detta förhållningssätt som ett insurrektionellt. Informalitet, i alla fall så som vi alltid har förstått det, är vägran av alla låsta strukturer, alla program, alla företablerade metoder, alla stämplar, all representation. Informalitet och informell organisering existerar därför bara i det kontinuerliga experimenterandet mellan kamrater som fördjupar sina affiniteter och ömsesidigt föreslår projekt av attack och kamp. Informalitet har ingen grundande text och inte heller några representanter. Det existerar endast som ett stöd i den anarkistiska kampen, för anarkister i kampen, för att möjliggöra för oss att uppnå det vi vill göra. I sina bidrag säger anarkisterna i CCF att ”På ett naturligt sätt saknar FAI exklusivitet. Det är därför vi inte föreslår FAI:s kvantitativa tillväxt [\dots{}] Vårt förslag är att organisera väpnade celler och affinitetsgrupper, att forma ett internationellt nätverk av praktikens anarkister.” Då frågar vi oss om förslaget är multipliceringen av affinitetsgrupper (vi vill inte gå in på detaljer i bruket av ord som ”celler”, då vi återkopplar till - i alla fall historiskt men trots allt var det kanske den Gamla Anarkin – hierarkier och partiorganisationer), varför FAI? Som ett stöd för detta förslag? Men en affinitetsgrupp är just precis ett möte mellan individer och den verkliga autonomin att handla, det är inte det grundläggande elementet i en stor superstruktur och ännu mindre i en superstruktur som etablerades för flera år sedan. Kopplingen mellan affinitetsgrupper skulle kunna vara informaliteten, det vill säga utbytet av idéer och perspektiv, utvecklandet av gemensamma projekt, en utveckling som aldrig når sitt slut, som alltid utvecklas, alltid utan någon formalitet. FAI:s förslag sätter bara upp ett staket runt informalitetens vidsträckta terräng.
Staten, partierna, församlingarna, organisationerna\dots{} alla dessa entiteter grundar sig i ett ”kollektivt vi”: medborgare eller militanta eller aktivister. Och individen, det vet de inte ens vad det är för något. Vi å andra sidan, vi älskar individen, med dess tankar och dess unika och ”singulära” handlingar. Såväl när den är ensam som i plural, när deras stigar korsats av andra individer. Därför hatar vi Staten och partierna (som alltid är auktoritära) och misstror församlingar och organisationer (som ibland kan vara frihetliga). Till skillnad från CCF så tänker inte vi att ”Rebell-jaget” kan göra sig hemmastadd i det ”kollektiva vi:et.” Till skillnad från flera FAI-kommunikéer, så är vi inte intresserade av dela ut certfikat för gott eller dåligt uppförande, till anarkister som försöker att kämpa, som definierar en som en ”praktikens anarkist” och en annan som ”en teoretiker som ingenting gör.” Det är en uppenbar lögn som stänger ner allt utrymme för debatt och fördjupning, för att sedan få det att verka som att de enda anarkister som attackerar makten skulle vara de, som stödjer FAI:s förslag och de som håller käft även om de inte håller med den ideologiska hegemonin som FAI försöker inrätta (genom tvång eller annat) över informell anarkism och användandet av attack och sabotage. Debatter och diskussioner lyser smärtsamt med sin frånvaro i den internationella anarkistiska rörelsen och förslagen som kommer prêt-à-porter, stänger fler dörrar och utrymmen för uppror än vad de öppnar. Detta problem fick oss att delta i denna avbrutna debatt och detta samma problem kommer att fortsätta driva oss framåt.
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Det Anarkistiska Biblioteket
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Anonym
Appendix till en avbruten debatt om anonymitet och attack
Mars 2014
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Översatt från engelska av UpprorsBladet juni 2014.
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