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0704.0002 | Sparsity-certifying Graph Decompositions | Sparsity-certifying Graph Decompositions
Ileana Streinu1∗, Louis Theran2
1 Department of Computer Science, Smith College, Northampton, MA. e-mail: [email protected]
2 Department of Computer Science, University of Massachusetts Amherst. e-mail: [email protected]
Abstract. We describe a new algorithm, the (k, `)-pebble game with colors, and use it to obtain a charac-
terization of the family of (k, `)-sparse graphs and algorithmic solutions to a family of problems concern-
ing tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have
received increased attention in recent years. In particular, our colored pebbles generalize and strengthen
the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characteri-
zation of arboricity. We also present a new decomposition that certifies sparsity based on the (k, `)-pebble
game with colors. Our work also exposes connections between pebble game algorithms and previous
sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9].
1. Introduction and preliminaries
The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs
that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a
graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a
(k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper
range of sparse graphs and 0≤ `≤ k the lower range.
In this paper, we present efficient algorithms for finding decompositions that certify sparsity
in the upper range of `. Our algorithms also apply in the lower range, which was already ad-
dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs
and graphs admitting the decomposition coincide.
Our algorithms are based on a new characterization of sparse graphs, which we call the
pebble game with colors. The pebble game with colors is a simple graph construction rule that
produces a sparse graph along with a sparsity-certifying decomposition.
We define and study a canonical class of pebble game constructions, which correspond to
previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide
a unifying framework for all the previously known special cases, including Nash-Williams-
Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the
properties of the augmenting paths used in matroid union and intersection algorithms[5, 6].
Since the sparse graphs in the upper range are not known to be unions or intersections of the
matroids for which there are efficient augmenting path algorithms, these do not easily apply in
∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO
CCR-0310661 to the first author.
2 Ileana Streinu, Louis Theran
Term Meaning
Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges
Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− `
Block H in G G is sparse, and H is a tight subgraph
Component H of G G is sparse and H is a maximal block
Map-graph Graph that admits an out-degree-exactly-one orientation
(k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs
`Tk Union of ` trees, each vertex is in exactly k of them
Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′)
Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk
Table 1. Sparse graph and decomposition terminology used in this paper.
the upper range. Pebble game with colors constructions may thus be considered a strengthening
of augmenting paths to the upper range of matroidal sparse graphs.
1.1. Sparse graphs
A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤
kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this
paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges
is called tight.
For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges
in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail
in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge.
There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of
a sparse graph. A component is a maximal block.
Table 1 summarizes the sparse graph terminology used in this paper.
1.2. Sparsity-certifying decompositions
A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees.
Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described
by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight
graphs.
A map-graph is a graph that admits an orientation such that the out-degree of each vertex is
exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map-
graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible
configuration certifying that each color forms a map-graph. Map-graphs may be equivalently
defined (see, e.g., [18]) as having exactly one cycle per connected component.1
A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint
map-graphs and ` spanning trees.
Another characterization of map-graphs, which we will use extensively in this paper, is as
the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that
the converse holds as well.
1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases
of the bicycle matroid or spanning pseudoforests.
Sparsity-certifying Graph Decompositions 3
Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a
(2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is
shown with a certifying orientation.
A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each
vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2.
Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the
components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute
multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come
from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition
of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of
tree-pieces in any subgraph G′ has size at least `.
Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an
isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree-
pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges.
These count as three tree-pieces, even though they come from the same back tree when the
whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three
gray tree-pieces and one black one.
Table 1 contains the decomposition terminology used in this paper.
The decomposition problem. We define the decomposition problem for sparse graphs as tak-
ing a graph as its input and producing as output, a decomposition that can be used to certify spar-
sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions;
and the pebble-game-with-colors decomposition, which is defined in the next section.
2. Historical background
The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to
the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint,
4 Ileana Streinu, Louis Theran
Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right
corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray
tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a
single vertex) and one black tree-piece.
Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps-
and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and
matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19].
In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman)
graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay
[21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight
graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the
equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a
direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1.
Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and
proper `Tk graphs for the general upper range. We observe that aside from our new pebble-
game-with-colors decomposition, all the combinatorial characterizations of the upper range of
sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24].
A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick-
son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the
pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and
Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and
left as an open problem using the pebble game to find sparsity certifying decompositions.
3. The pebble game with colors
Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative
integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm
for the decomposition problem later in this paper. Since the phrase “with colors” is necessary
only for comparison to [12], we will omit it in the rest of the paper when the context is clear.
Sparsity-certifying Graph Decompositions 5
We now present the pebble game with colors. The game is played by a single player on a
fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the
addition and/or orientation of an edge. At any moment of time, the state of the game is captured
by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored
by the pebbles on them. While playing the pebble game all edges are directed, and we use the
notation vw to indicate a directed edge from v to w.
We describe the pebble game with colors in terms of its initial configuration and the allowed
moves.
Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices
are shown as black or gray dots. Edges are colored with the color of the pebble on them.
Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start
by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k.
Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume
(w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw
to E(H) and put the pebble picked up from v on the new edge.
Figure 3(a) shows examples of the add-edge move.
Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace
vw with wv in E(H); put the pebble that was on vw on v; and put p on wv.
Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows
examples. The convention in these figures, and throughout this paper, is that pebbles on vertices
are represented as colored dots, and that edges are shown in the color of the pebble on them.
From the definition of the pebble-slide move, it is easy to see that a particular pebble is
always either on the vertex where it started or on an edge that has this vertex as the tail. However,
when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is
sometimes convenient to think of this path reversal sequence as bringing a pebble from the end
of the path to the beginning.
The output of playing the pebble game is its complete configuration.
Output: At the end of the game, we obtain the directed graph H, along with the location
and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble
game configuration colors the edges.
We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game
or that H is a pebble-game graph.
Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions
the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble-
game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a
pebble-game decomposition.
Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges,
and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con-
6 Ileana Streinu, Louis Theran
(a) (b) (c)
Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to
show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an
empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two
black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph.
(c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges
contain a cycle and do not contribute a piece of tree to the subgraph.
Notation Meaning
span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)|
peb(V ′) Number of pebbles on V ′ ⊂V
out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′
pebi(v) Number of pebbles of color ci on v ∈V
outi(v) Number of edges vw colored ci for v ∈V
Table 2. Pebble game notation used in this paper.
nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic
subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′
otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with
the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub-
graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned
by G′.
The properties of pebble-game decompositions are studied in Section 6, and Theorem 2
shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows
this.
For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom-
position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated
vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black
tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not
contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees.
In the following discussion, we use the notation peb(v) for the number of pebbles on v and
pebi(v) to indicate the number of pebbles of colors i on v.
Table 2 lists the pebble game notation used in this paper.
4. Our Results
We describe our results in this section. The rest of the paper provides the proofs.
Sparsity-certifying Graph Decompositions 7
Our first result is a strengthening of the pebble games of [12] to include colors. It says
that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games
discussed in this paper are our pebble game with colors unless noted explicitly.
Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse
with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph.
Next we consider pebble-game decompositions, showing that they are a generalization of
proper `Tk decompositions that extend to the entire matroidal range of sparse graphs.
Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game
graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each
is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse
graphs in the decomposition.
The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus
Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained
by playing the pebble game defined in the previous section. Notice the similarity between the
requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a
proper `Tk .
Our next results show that for any pebble-game graph, we can specialize its pebble game
construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these
specialized pebble game constructions canonical, and using canonical pebble game construc-
tions, we obtain new direct proofs of existing arboricity results.
We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo-
sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning
trees contributes at least one piece of tree to every subgraph.
The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition
is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper
`Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is
sufficient to certify sparsity.
We now state the main theorem for the upper and lower range.
Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game
graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)-
maps-and-trees.
Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game
graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper
`Tk with kn− ` edges.
As corollaries, we obtain the existing decomposition results for sparse graphs.
Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph
G is tight if and only if has a (k, `)-maps-and-trees decomposition.
Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a
proper `Tk .
Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo-
rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem.
Our last result improves on this, showing that a canonical pebble game construction, and thus
8 Ileana Streinu, Louis Theran
a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in
O(n2) time and space.
These time and space bounds mean that our algorithm can be combined with those of [12]
without any change in complexity.
5. Pebble game graphs
In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game
with colors. Since many of the relevant properties of the pebble game with colors carry over
directly from the pebble games of [12], we refer the reader there for the proofs.
We begin by establishing some invariants that hold during the execution of the pebble game.
Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following
invariants are maintained in H:
(I1) There are at least ` pebbles on V . [12]
(I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12]
(I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12]
(I4) For every vertex v ∈V , outi(v)+pebi(v) = 1.
(I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with
a pebble of color ci or a cycle.
Proof. (I1), (I2), and (I3) come directly from [12].
(I4) This invariant clearly holds at the initialization phase of the pebble game with colors.
That add-edge and pebble-slide moves preserve (I4) is clear from inspection.
(I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of
the same color on it. If there is no pebble of that color reachable, then the path must eventually
visit some vertex twice.
From these invariants, we can show that the pebble game constructible graphs are sparse.
Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the
pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight.
The main step in proving that every sparse graph is a pebble-game graph is the following.
Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce
the out degree of v by one.
Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If
peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w.
It follows that any sparse graph has a pebble game construction.
Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse
with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph.
6. The pebble-game-with-colors decomposition
In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We
start with the following lemmas about the structure of monochromatic connected components
in H, the directed graph maintained during the pebble game.
Sparsity-certifying Graph Decompositions 9
Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub-
graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for
i = 1, . . . ,k.
Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex.
Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H
in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these
is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge.
Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′.
Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality
that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and
out-edge tail is the root of a tree-piece.
Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed
path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle.
Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the
monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′.
Since this argument works for any vertex in any color, for each color there is a partitioning
of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each
pebble and out-edge tail is the root of a monochromatic tree, as desired.
Applied to the whole graph Lemma 11 gives us the following.
Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of
color ci is the root of a (possibly empty) monochromatic tree-piece of color ci.
Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′
edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it
to the lower range and giving a construction that finds the tree-pieces, showing the connection
between the `+1 pebble condition and the hereditary condition on proper `Tk .
We conclude our investigation of arbitrary pebble game constructions with a description of
the decomposition induced by the pebble game with colors.
Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game
graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each
is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse
graphs in the decomposition.
Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub-
graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs.
For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can
span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game
decomposition must be sparse. Apply Theorem 1 to complete the proof.
Remark: We observe that a pebble-game decomposition for a Laman graph may be read out
of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed,
pebble game orientations have a natural correspondence with the bipartite matchings used in
10 Ileana Streinu, Louis Theran
Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there
are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be
spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range
pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de-
composition is a proper `Tk .
In the next section, we show that the pebble game can be specialized to correspond to maps-
and-trees and proper `Tk decompositions.
7. Canonical Pebble Game Constructions
In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves-
tigation of decompositions induced by pebble game constructions by studying the case where a
minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15
and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that
this is always possible, implying that monochromatic map-graphs are created only when we
add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that
every color is a forest. Every decomposition characterization of tight graphs discussed above
follows immediately from the main theorem, giving new proofs of the previous results in a
unified framework.
In the proof, we will use two specializations of the pebble game moves. The first is a modi-
fication of the add-edge move.
Canonical add-edge: When performing an add-edge move, cover the new edge with a color
that is on both vertices if possible. If not, then take the highest numbered color present.
The second is a restriction on which pebble-slide moves we allow.
Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a
monochromatic cycle.
We call a pebble game construction that uses only these moves canonical. In this section
we will show that every pebble-game graph has a canonical pebble game construction (Lemma
14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and
maps-and-trees decompositions (Theorem 3 and Theorem 4).
We begin with a technical lemma that motivates the definition of canonical pebble game
constructions. It shows that the situations disallowed by the canonical moves are all the ways
for cycles to form in the lowest ` colors.
Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and
let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created
in exactly one of the following ways:
(M1) The edge vw is added with an add-edge move.
(M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse
edge vw.
Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7.
By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a
connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble
game construction, since the color of an edge only changes when it is inserted the first time or
a new pebble is put on it by a pebble-slide move.
Sparsity-certifying Graph Decompositions 11
vw vw
Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by
adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are
labeled according to their role in the definition of the moves.
Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves,
respectively, in a (2,0)-pebble game construction.
We next show that if a graph has a pebble game construction, then it has a canonical peb-
ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa-
rately. The proof gives two constructions that implement the canonical add-edge and canonical
pebble-slide moves.
Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc-
tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where
`′ = min{k, `}.
Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If
this is not possible, then there are `+1 distinct colors present. Use the highest numbered color
to cover the new edge.
Remark: We note that in the upper range, there is always a repeated color, so no canonical
add-edge moves create cycles in the upper range.
The canonical pebble-slide move is defined by a global condition. To prove that we obtain
the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma
9 to only canonical moves. The main step is to show that if there is any sequence of moves that
reorients a path from v to w, then there is a sequence of canonical moves that does the same
thing.
Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading
to an add-edge move can be replaced with one that has no (M2) steps and allows the same
add-edge move.
In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added,
then it is possible to do this without creating any monochromatic cycles.
12 Ileana Streinu, Louis Theran
Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this
the shortcut construction by analogy to matroid union and intersection augmenting paths used
in previous work on the lower range.
Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step
at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one
application of the shortcut construction reorients a simple path from a vertex w′ to w, and a
path from v to w′ is preserved, the shortcut construction can be applied inductively to find the
sequence of moves we want.
Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines
indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle,
shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part
of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray
tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is
simple, and the shortcut construction can be applied inductively to it.
Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple
path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble
of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v
and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe
that it is rooted at w.
Now consider the edges reversed in our sequence of moves. As noted above, before we make
any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on
this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every
move before the one that reverses some edge yz; prepend onto what is left a sequence of moves
that moves the pebble on w to z in H ′i .
Sparsity-certifying Graph Decompositions 13
Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path.
The path where the pebbles move is indicated by doubled lines.
Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is
(M2); (b) avoiding the (M2) and simplifying the path.
Since no edges change color in the beginning of the new sequence, we have eliminated
the (M2) move. Because our construction does not change any of the edges involved in the
remaining tail of the original sequence, the part of the original path that is left in the new
sequence will still be a simple path in H, meeting our initial hypothesis.
The rest of the lemma follows by induction.
Together Lemma 14 and Lemma 15 prove the following.
Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction.
Using canonical pebble game constructions, we can identify the tight pebble-game graphs
with maps-and-trees and `Tk graphs.
14 Ileana Streinu, Louis Theran
Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game
graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)-
maps-and-trees.
Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game
decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game
graph.
For the reverse direction, consider a canonical pebble game construction of a tight graph.
From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The
definition of the canonical add-edge move implies that there must be at least one pebble of
each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma
12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding
the required ` edge-disjoint spanning trees.
Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph
G is tight if and only if has a (k, `)-maps-and-trees decomposition.
We next consider the decompositions induced by canonical pebble game constructions when
`≥ k +1.
Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb-
ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it
is a proper `Tk with kn− ` edges.
Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show
is that a canonical pebble game construction of a tight graph produces a proper `Tk .
By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom-
position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must
in in exactly k of the trees, as required.
Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a
proper `Tk .
8. Pebble game algorithms for finding decompositions
A naı̈ve implementation of the constructions in the previous section leads to an algorithm re-
quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n)
applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running
time of Θ(n3) for the decomposition problem.
In this section, we describe algorithms for the decomposition problem that run in time
O(n2). We begin with the overall structure of the algorithm.
Algorithm 17 (The canonical pebble game with colors).
Input: A graph G.
Output: A pebble-game graph H.
Method:
– Set V (H) = V (G) and place one pebble of each color on the vertices of H.
– For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide
moves as described by Lemma 15.
Sparsity-certifying Graph Decompositions 15
– If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma
14, otherwise discard vw.
– Finally, return H, and the locations of the pebbles.
Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent
sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction
found is canonical, the main theorem shows that the coloring of the edges in H gives a maps-
and-trees or proper `Tk decomposition.
Complexity. We start by observing that the running time of Algorithm 17 is the time taken to
process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an
edge of G that is added to H.
Each of the pebble game moves can be implemented in constant time. What remains is to
describe an efficient way to find and move the pebbles. We use the following algorithm as a
subroutine of Algorithm 17 to do this.
Algorithm 18 (Finding a canonical path to a pebble.).
Input: Vertices v and w, and a pebble game configuration on a directed graph H.
Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated.
Method:
– Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and
return ‘no.’
– Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi
are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use
the array c[] to keep track of the colors of pebbles on vertices and edges after we move them
and the array s[] to sketch out a canonical path from v to u by finding a successor for each
edge.
– Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in
reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to
the next i. Otherwise, check to see if c[vi+1] = c[ei].
– If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge.
– Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If
a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x
that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i =
1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro-
matic c[x] path from xq−1 to x using this same procedure.
– Finally, slide pebbles along the path from the original endpoints v to u specified by the
successor array s[v], s[s[v]], . . .
The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut
construction. Efficiency comes from the fact that instead of potentially moving the pebble back
and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three
times: once in the initial depth-first search, and twice while converting the initial path to a
canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time
spent processing edges in H.
Although we have not discussed this explicity, for the algorithm to be efficient we need to
maintain components as in [12]. After each accepted edge, the components of H can be updated
in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1)
time each.
16 Ileana Streinu, Louis Theran
Summarizing, we have shown that the canonical pebble game with colors solves the decom-
position problem in time O(n2).
9. An important special case: Rigidity in dimension 2 and slider-pinning
In this short section we present a new application for the special case of practical importance,
k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally
rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the
current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com-
binatorially, we model the bar-slider frameworks as simple graphs together with some loops
placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each
color.
We characterize the minimally rigid bar-slider graphs [20] as graphs that are:
1. (2,3)-sparse for subgraphs containing no loops.
2. (2,0)-tight when loops are included.
We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse
graphs studied in our paper [14].
The connection with the pebble games in this paper is the following.
Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we
replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph.
Proof. Follows from invariant (I3) of Lemma 7.
In [15], we study a special case of slider pinning where every slider is either vertical or
horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction.
For this axis parallel slider case, the minimally rigid graphs are characterized by:
1. (2,3)-sparse for subgraphs containing no loops.
2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each
monochromatic tree spans exactly one loop of its color.
This also has an interpretation in terms of colored pebble games.
Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)-
pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the
graph of a minimally pinned axis-parallel bar-slider framework.
Proof. Follows from Theorem 4, and Lemma 12.
10. Conclusions and open problems
We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and
used it to give an efficient algorithm for finding decompositions of sparse graphs into edge-
disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range
and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the
upper range from [12].
We also used the pebble game with colors to describe a new sparsity-certifying decomposi-
tion that applies to the entire matroidal range of sparse graphs.
Sparsity-certifying Graph Decompositions 17
We defined and studied a class of canonical pebble game constructions that correspond to
either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash-
Williams arboricity theorem and a unified proof of the previously studied decomposition cer-
tificates of sparsity. Canonical pebble game constructions also show the relationship between
the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting
paths, which do not apply in the upper range.
Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2)
algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from
dense ones. Their technique is based on efficiently finding matroid union augmenting paths,
which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to
find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch
scanning, which finds groups of disjoint augmenting paths.
We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester-
mann’s algorithm without changing the running time. The data structures used in the implemen-
tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those
used to support cyclic scanning.
The two major open algorithmic problems related to the pebble game are then:
Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain
an implementable O(n3/2) algorithm for the lower range.
Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble
game algorithm for the upper range.
In particular, it would be of practical importance to find an implementable O(n3/2) algorithm
for decompositions into edge-disjoint spanning trees.
References
1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th
European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003)
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and applications. Algorithmica 7(1), 465–497 (1992)
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sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11
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9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing
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18 Ileana Streinu, Louis Theran
10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the
pebble game. Journal of Computational Physics 137, 346–365 (1997)
11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering
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308(8), 1425–1437 (2008)
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dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg.
cs.uwindsor.ca/papers/72.pdf
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Computer Science 13(10) (2007)
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Canadian Conference on Computational Geometry (CCCG’07) (2007)
16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland,
Amsterdam (1979)
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Mathematical Society 39, 12 (1964)
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(1992)
19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees.
Mathematics of Operations Research 10(4), 701–708 (1985)
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http://cccg.cs.uwindsor.ca/papers/72.pdf
http://cccg.cs.uwindsor.ca/papers/72.pdf
Introduction and preliminaries
Historical background
The pebble game with colors
Our Results
Pebble game graphs
The pebble-game-with-colors decomposition
Canonical Pebble Game Constructions
Pebble game algorithms for finding decompositions
An important special case: Rigidity in dimension 2 and slider-pinning
Conclusions and open problems
|
0704.0003 | The evolution of the Earth-Moon system based on the dark matter field
fluid model | The evolution of the Earth-Moon system based on the dark fluid model
The evolution of the Earth-Moon system based on
the dark matter field fluid model
Hongjun Pan
Department of Chemistry
University of North Texas, Denton, Texas 76203, U. S. A.
Abstract
The evolution of Earth-Moon system is described by the dark matter field fluid
model with a non-Newtonian approach proposed in the Meeting of Division of Particle
and Field 2004, American Physical Society. The current behavior of the Earth-Moon
system agrees with this model very well and the general pattern of the evolution of the
Moon-Earth system described by this model agrees with geological and fossil evidence.
The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago,
which is far beyond the Roche’s limit. The result suggests that the tidal friction may not
be the primary cause for the evolution of the Earth-Moon system. The average dark
matter field fluid constant derived from Earth-Moon system data is 4.39 × 10-22 s-1m-1.
This model predicts that the Mars’s rotation is also slowing with the angular acceleration
rate about -4.38 × 10-22 rad s-2.
Key Words. dark matter, fluid, evolution, Earth, Moon, Mars
1. Introduction
The popularly accepted theory for the formation of the Earth-Moon system is that
the Moon was formed from debris of a strong impact by a giant planetesimal with the
Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the
formation of the Earth-Moon system, it has been evolving at all time scale. It is well
known that the Moon is receding from us and both the Earth’s rotation and Moon’s
rotation are slowing. The popular theory is that the tidal friction causes all those changes
based on the conservation of the angular momentum of the Earth-Moon system. The
situation becomes complicated in describing the past evolution of the Earth-Moon
system. Because the Moon is moving away from us and the Earth rotation is slowing, this
means that the Moon was closer and the Earth rotation was faster in the past. Creationists
argue that based on the tidal friction theory, the tidal friction should be stronger and the
recessional rate of the Moon should be greater in the past, the distance of the Moon
would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon
would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence
indicates that the recession of the Moon in the past was slower than the present rate, i. e.,
the recession has been accelerating with time. Therefore, it must be concluded that tidal
friction was very much less in the remote past than we would deduce on the basis of
present-day observations (Stacey 1977). This was called “geological time scale
difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the
tidal friction theory (Brush 1983).
But we have to consider the case carefully in various aspects. One possible
scenario is that the Earth has been undergoing dynamic evolution at all time scale since
its inception, the geological and physical conditions (such as the continent positions and
drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at
remote past could be substantially different from currently, in which the tidal friction
could be much less; therefore, the receding rate of the Moon could be slower. Various
tidal friction models were proposed in the past to describe the evolution of the Earth-
Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable
distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray
et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present
rate of tidal dissipation is anomalously high because the tidal force is close to a resonance
in the response function of ocean (Brush 1983). Kagan gave a detailed review about those
tidal friction models (Kagan 1997). Those models are based on many assumptions about
geological (continental position and drifting) and physical conditions in the past, and
many parameters (such as phase lag angle, multi-mode approximation with time
dependent frequencies of the resonance modes, etc.) have to be introduced and carefully
adjusted to make their predictions close to the geological evidence. However, those
assumptions and parameters are still challenged, to certain extent, as concoction.
The second possible scenario is that another mechanism could dominate the
evolution of the Earth-Moon system and the role of the tidal friction is not significant. In
the Meeting of Division of Particle and Field 2004, American Physical Society,
University of California at Riverside, the author proposed a dark matter field fluid model
(Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with
this model very well. This paper will demonstrate that the past evolution of Moon-Earth
system can be described by the dark matter field fluid model without any assumptions
about past geological and physical conditions. Although the subject of the evolution of
the Earth-Moon system has been extensively studied analytically or numerically, to the
author’s knowledge, there are no theories similar or equivalent to this model.
2. Invisible matter
In modern cosmology, it was proposed that the visible matter in the universe is
about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently
invisible which is called dark matter and dark energy, such invisible matter has an anti-
gravity property to make the universe expanding faster and faster.
If the ratio of the matter components of the universe is close to this hypothesis,
then, the evolution of the universe should be dominated by the physical mechanism of
such invisible matter, such physical mechanism could be far beyond the current
Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian
physics could reflect only a corner of the iceberg of the greater physics.
If the ratio of the matter components of the universe is close to this hypothesis,
then, it should be more reasonable to think that such dominant invisible matter spreads in
everywhere of the universe (the density of the invisible matter may vary from place to
place); in other words, all visible matter objects should be surrounded by such invisible
matter and the motion of the visible matter objects should be affected by the invisible
matter if there are interactions between the visible matter and the invisible matter.
If the ratio of the matter components of the universe is close to this hypothesis,
then, the size of the particles of the invisible matter should be very small and below the
detection limit of the current technology; otherwise, it would be detected long time ago
with such dominant amount.
With such invisible matter in mind, we move to the next section to develop the
dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible
matter (dark matter, dark energy and possible other terms) is called dark matter here.
3. The dark matter field fluid model
In this proposed model, it is assumed that:
1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly
filled with the dark matter which is in quiescent state relative to the motion of the
celestial body. The dark matter possesses a field property and a fluid property; it can
interact with the celestial body with its fluid and field properties; therefore, it can have
energy exchange with the celestial body, and affect the motion of the celestial body.
2. The fluid property follows the general principle of fluid mechanics. The dark matter
field fluid particles may be so small that they can easily permeate into ordinary
“baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter
field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the
manner of a sponge moving thru water. The nature of the field property of the dark matter
field fluid is unknown. It is here assumed that the interaction of the field associated with
the dark matter field fluid with the celestial body is proportional to the mass of the
celestial body. The dark matter field fluid is assumed to have a repulsive force against the
gravitational force towards baryonic matter. The nature and mechanism of such repulsive
force is unknown.
With the assumptions above, one can study how the dark matter field fluid may
influence the motion of a celestial body and compare the results with observations. The
common shape of celestial bodies is spherical. According to Stokes's law, a rigid non-
permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds
number experiences a resistance force F
rvF πμ6−= (1)
where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity
constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the
velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual
properties of the dark matter field fluid and its permeation into the sphere, the force F
may not be proportional to the radius of the sphere. Also, F may be proportional to the
mass of the sphere due to the field interaction. Therefore, with the combined effects of
both fluid and field, the force exerted on the sphere by the dark matter field fluid is
assumed to be of the scaled form
(2) mvrF n−−= 16πη
where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be
viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η
is the dark matter field fluid constant, which is equivalent to μ. The direction of the
resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force
described by Eq. 2 is velocity-dependent and causes negative acceleration. According to
Newton's second law of motion, the equation of motion for the sphere is
mvr
m n−−= 16πη (3)
Then
(4) )6exp( 10 vtrvv
n−−= πη
where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a
massive gravitational center, there are three forces in the line between the sphere and the
gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and
(3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the
orbital velocity and causes the sphere to move inward to the gravitational center.
However, if the sum of the centripetal acceleration force and the repulsive force is
stronger than the gravitational force, then, the sphere will move outward and recede from
the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is
sufficiently slow and the repulsive force is small compared to the gravitational force and
centripetal acceleration force, then the rate of receding will be accordingly relatively
slow. Therefore, the gravitational force and the centripetal acceleration force can be
approximately treated in equilibrium at any time. The pseudo equilibrium equation is
GMm 2
2 = (5)
where G is the gravitational constant, M is the mass of the gravitational center, and R is
the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields
)12exp( 1
R n−= πη (6)
(7) )12exp( 10 trRR
n−= πη
where
R = (8)
R0 is the initial distance to the gravitational center. Note that R exponentially increases
with time. The increase of orbital energy with the receding comes from the repulsive
force of dark matter field fluid. The recessional rate of the sphere is
dR n−= 112πη (9)
The acceleration of the recession is
( Rr
Rd n 21
12 −= πη ) . (10)
The recessional acceleration is positive and proportional to its distance to the
gravitational center, so the recession is faster and faster.
According to the mechanics of fluids, for a rigid non-permeable sphere rotating
about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere
ωπμ 38 rT −= (11)
where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is
opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent
dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted
on the sphere is
( ) ωπη mrT n 318 −−= (12)
The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The
torque causes the negative angular acceleration
= (13)
where I is the moment of inertia of the sphere in the dark matter field fluid
( )21
2 nrmI −= (14)
Therefore, the equation of rotation for the sphere in the dark matter field fluid is
ωπη
d −−= 120 (15)
Solving this equation yields
(16) )20exp( 10 tr
n−−= πηωω
where ω0 is the initial angular velocity. One can see that the angular velocity of the
sphere exponentially decreases with time and the angular deceleration is proportional to
its angular velocity.
For the same celestial sphere, combining Eq. 9 and Eq. 15 yields
(17)
The significance of Eq. 17 is that it contains only observed data without assumptions and
undetermined parameters; therefore, it is a critical test for this model.
For two different celestial spheres in the same system, combining Eq. 9 and Eq.
15 yields
67.1
1 −=−=⎟⎟
(18)
This is another critical test for this model.
4. The current behavior of the Earth-Moon system agrees with the model
The Moon-Earth system is the simplest gravitational system. The solar system is
complex, the Earth and the Moon experience not only the interaction of the Sun but also
interactions of other planets. Let us consider the local Earth-Moon gravitational system as
an isolated local gravitational system, i.e., the influence from the Sun and other planets
on the rotation and orbital motion of the Moon and on the rotation of the Earth is
assumed negligible compared to the forces exerted by the moon and earth on each other.
In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data
about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are
listed below for the readers' convenience to verify the calculation because the data may
vary slightly with different data sources.
Moon:
Mean radius: r = 1738.0 km
Mass: m = 7.3483 × 1025 gram
Rotation period = 27.321661 days
Angular velocity of Moon = 2.6617 × 10-6 rad s-1
Mean distance to Earth Rm= 384400 km
Mean orbital velocity v = 1.023 km s-1
Orbit eccentricity e = 0.0549
Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2
= (-1.255 ± 0.024) × 10-4 rad century-2
= (-1.260 ± 0.024) × 10-23 rad s-2
Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1
Earth:
Mean radius: r = 6371.0 km
Mass: m = 5.9742 × 1027 gram
Rotation period = 23 h 56m 04.098904s = 86164.098904s
Angular velocity of rotation = 7.292115 × 10-5 rad s-1
Mean distance to the Sun Rm= 149,597,870.61 km
Mean orbital velocity v = 29.78 km s-1
Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2
The Moon's angular rotation acceleration rate and increase in mean distance to the Earth
(receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey
et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the
result is
039.054.1
10662.21021.1
1092509.31026.1
(19)
The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number
384400 km is assumed to be the distance from the Moon's surface to the Earth's surface.
Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord
with the model used here. The difference (about 7.8%) between the values of -1.54 and -
1.67 may come from several sources:
1. Moon's orbital is not a perfect circle
2. Moon is not a perfect rigid sphere.
3. The effect from Sun and other planets.
4. Errors in data.
5. Possible other unknown reasons.
The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data
sets. The third data set can be used to further test the model. If this model correctly
describes the situation at hand, it should give consistent results for different motions. The
values of n and η calculated from three different data sets are listed below (Note, the
mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are
used in the calculation).
The value of n: n = 0.64
From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1
From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1
From the Moon's recession: η = 4.64 × 10-22 s-1 m-1
One can see that the three values of η are consistent within the range of error in the data.
The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1
By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into
Eq. 18, the result is
14.053.1
6371000
1738000
1021.11029.7
1092509.3105.5
)64.01(
(20)
This is also in accord with the model used here.
The dragging force exerted on the Moon's orbital motion by the dark matter field
fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between
the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field
fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm,
respectively.
5. The evolution of Earth-Moon system
Sonett et al. found that the length of the terrestrial day 900 million years ago was
about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al.,
1996). According to the model presented here, back in that time, the length of the day
was about 19.2 hours, this agrees very well with Sonett et al.'s result.
Another critical aspect of modeling the evolution of the Earth-Moon system is to
give a reasonable estimate of the closest distance of the Moon to the Earth when the
system was established at 4.5 billion years ago. Based on the dark matter field fluid
model, and the above result, the closest distance of the Moon to the Earth was about
259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago,
this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and
McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion
years ago was 250000 km, this is probably the most reasonable number that most
astronomers believe and it agrees excellently with the result of this model. The closest
distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or
242000 km (Hansen, 1982).
According to this model, the length of day of the Earth was about 8 hours at 4.5
billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the
length of day of the Earth with the age of the Earth-Moon system described by this model
along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One
can see that those data fit this model very well in their time range.
Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from
Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model
for past 900 million years. One can see that the model agrees with the geological and
fossil data beautifully.
The important difference of this model with early models in describing the early
evolution of the Earth-Moon system is that this model is based only on current data of the
Moon-Earth system and there are no assumptions about the conditions of earlier Earth
rotation and continental drifting. Based on this model, the Earth-Moon system has been
smoothly evolving to the current position since it was established and the recessional rate
of the Moon has been gradually increasing, however, this description does not take it into
account that there might be special events happened in the past to cause the suddenly
significant changes in the motions of the Earth and the Moon, such as strong impacts by
giant asteroids and comets, etc, because those impacts are very common in the universe.
The general pattern of the evolution of the Moon-Earth system described by this model
agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially
increases with time. One may then imagine that the recessional rate will quickly become
very large. The increase is in fact extremely slow. The Moon's recessional rate will be
3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years.
However, whether the Moon's recession will continue or at some time later another
mechanism will take over is not known. It should be understood that the tidal friction
does affect the evolution of the Earth itself such as the surface crust structure, continental
drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s
rotation, however, such role is not a dominant mechanism.
Unfortunately, there is no data available for the changes of the Earth's orbital
motion and all other members of solar system. According to this model and above results,
the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km
century-1, the length of a year increases about 6.8 ms and the change of the temperature is
-1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on
the Earth. The length of a year at 1 billion years ago would be 80% of the current length
of the year. However, much evidence (growth-bands of corals and shellfish as well as
some other evidences) suggest that there has been no apparent change in the length of the
year over the billion years and the Earth's orbital motion is more stable than its rotation.
This suggests that dark matter field fluid is circulating around Sun with the same
direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the
Earth's orbital motion experiences very little or no dragging force from the dark matter
field fluid. However, this is a conjecture, extensive research has to be conducted to verify
if this is the case.
6. Skeptical description of the evolution of the Mars
The Moon does not have liquid fluid on its surface, even there is no air, therefore,
there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the
Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees
with the model very well. Based on this, one may reasonably think that the Mars’s
rotation should be slowing also.
The Mars is our nearest neighbor which has attracted human’s great attention
since ancient time. The exploration of the Mars has been heating up in recent decades.
NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect
data and study this mysterious planet. So far there is still not enough data about the
history of this planet to describe its evolution. Same as the Earth, the Mars rotates about
its central axis and revolves around the Sun, however, the Mars does not have a massive
moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no
liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to
slow its rotation by tidal friction theories. Based on the above result and current the
Mars's data, this model predicts that the angular acceleration of the Mars should be about
-4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the
solar days/Mars year, the vertical dash line marks the current age of the Mars with
assumption that the Mars was formed in a similar time period of the Earth formation.
Such description was not given before according to the author’s knowledge and is
completely skeptical due to lack of reliable data. However, with further expansion of the
research and exploration on the Mars, we shall feel confident that the reliable data about
the angular rotation acceleration of the Mars will be available in the near future which
will provide a vital test for the prediction of this model. There are also other factors
which may affect the Mars’s rotation rate such as mass redistribution due to season
change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be
carefully analyzed.
7. Discussion about the model
From the above results, one can see that the current Earth-Moon data and the
geological and fossil data agree with the model very well and the past evolution of the
Earth-Moon system can be described by the model without introducing any additional
parameters; this model reveals the interesting relationship between the rotation and
receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in
the same gravitational system, such relationship is not known before. Such success can
not be explained by “coincidence” or “luck” because of so many data involved (current
Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a
“ad hoc” or a wrong model, although the chance for the natural happening of such
“coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s
data will clarify this; otherwise, a new theory from different approach can be developed
to give the same or better description as this model does. It is certain that this model is
not perfect and may have defects, further development may be conducted.
James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar
regions will no longer be regarded as waste places in the universe, which the Creator has
not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find
them to be already full of this wonderful medium; so full, that no human power can
remove it from the smallest portion of space, or produce the slightest flaw in its infinite
continuity. It extends unbroken from star to star ….” The medium that Maxwell talked
about is the aether which was proposed as the carrier of light wave propagation. The
Michelson-Morley experiment only proved that the light wave propagation does not
depend on such medium and did not reject the existence of the medium in the interstellar
space. In fact, the concept of the interstellar medium has been developed dramatically
recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field
fluid is just a part of such wonderful medium and “precisely” described by Maxwell.
7. Conclusion
The evolution of the Earth-Moon system can be described by the dark matter field
fluid model with non-Newtonian approach and the current data of the Earth and the Moon
fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the
Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of
day was about 8 hours. The general pattern of the evolution of the Moon-Earth system
described by this model agrees with geological and fossil evidence. The tidal friction may
not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation
is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2.
References
S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century:
Creationist Arguments for a young Earth. Scientists confront creationism. W. W.
Norton & Company, New York, London, pp 49.
E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood
Cliffs, NJ 07632.
J. O. Dickey, et al., 1994. Science, 265, 482.
D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109.
K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457.
W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504.
B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173.
B. A. Kagan, 1997. Prog. Oceanog. 40, 109.
E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J.
Sediment. Res. 69(6), 1154.
K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York.
H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135.
R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653.
C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the
Earth’s Rotation. Springer-Verlag, Berlin, pp. 154.
L. B. Slichter, 1963. J. Geophys. Res. 68, 14.
C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100.
F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons.
J. W. Wells, 1963. Nature, 197, 948.
Caption
Figure 1, the evolution of Moon’s distance and the length of day of the earth with
the age of the Earth-Moon system. Solid lines are calculated according to the dark matter
field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the
length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al.
The dash line marks the current age of the Earth-Moon system.
Figure 2, the evolution of Solar days of year with the age of the Earth-Moon
system. The solid line is calculated according to dark matter field fluid model. The data
are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current
age (4.5 billion years).
Figure 3, the skeptical description of the evolution of Mars’s length of day and the
solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5
billion years). The vertical dash line marks the current age of Mars.
Figure 1, Moon's distance and the length of day of Earth
change with the age of Earth-Moon system
The age of Earth-Moon system (109 years)
0 1 2 3 4 5
Distance
Length of day
Roche's limit
Hansen's result
Figure 2, the solar days / year vs. the age of the Earth
The age of the Earth (109 years)
3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6
|
0704.0004 | A determinant of Stirling cycle numbers counts unlabeled acyclic
single-source automata | A Determinant of Stirling Cycle Numbers Counts Unlabeled
Acyclic Single-Source Automata
DAVID CALLAN
Department of Statistics
University of Wisconsin-Madison
1300 University Ave
Madison, WI 53706-1532
[email protected]
March 30, 2007
Abstract
We show that a determinant of Stirling cycle numbers counts unlabeled acyclic
single-source automata. The proof involves a bijection from these automata to
certain marked lattice paths and a sign-reversing involution to evaluate the deter-
minant.
1 Introduction The chief purpose of this paper is to show bijectively that
a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata.
Specifically, let Ak(n) denote the kn × kn matrix with (i, j) entry
[ ⌊ i−1
⌊ i−1
⌋+1+i−j
, where
is the Stirling cycle number, the number of permutations on [i] with j cycles. For
example,
A2(5) =
1 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0
0 1 3 2 0 0 0 0 0 0
0 0 1 3 2 0 0 0 0 0
0 0 0 1 6 11 6 0 0 0
0 0 0 0 1 6 11 6 0 0
0 0 0 0 0 1 10 35 50 24
0 0 0 0 0 0 1 10 35 50
0 0 0 0 0 0 0 1 15 85
0 0 0 0 0 0 0 0 1 15
http://arxiv.org/abs/0704.0004v1
As evident in the example, Ak(n) is formed from k copies of each of rows 2 through n+1
of the Stirling cycle triangle, arranged so that the first nonzero entry in each row is a 1
and, after the first row, this 1 occurs just before the main diagonal; in other words, Ak(n)
is a Hessenberg matrix with 1s on the infra-diagonal. We will show
Main Theorem. The determinant of Ak(n) is the number of unlabeled acyclic single-
source automata with n transient states on a (k + 1)-letter input alphabet.
Section 2 reviews basic terminology for automata and recurrence relations to count
finite acyclic automata. Section 3 introduces column-marked subdiagonal paths, which
play an intermediate role, and a way to code them. Section 4 presents a bijection from
these column-marked subdiagonal paths to unlabeled acyclic single-source automata. Fi-
nally, Section 5 evaluates detAk(n) using a sign-reversing involution and shows that the
determinant counts the codes for column-marked subdiagonal paths.
2 Automata
A (complete, deterministic) automaton consists of a set of states and an input alphabet
whose letters transform the states among themselves: a letter and a state produce another
state (possibly the same one). A finite automaton (finite set of states, finite input alphabet
of, say, k letters) can be represented as a k-regular directed multigraph with ordered edges:
the vertices represent the states and the first, second, . . . edge from a vertex give the effect
of the first, second, . . . alphabet letter on that state. A finite automaton cannot be acyclic
in the usual sense of no cycles: pick a vertex and follow any path from it. This path must
ultimately hit a previously encountered vertex, thereby creating a cycle. So the term
acyclic is used in the looser sense that only one vertex, called the sink, is involved in
cycles. This means that all edges from the sink loop back to itself (and may safely be
omitted) and all other paths feed into the sink.
A non-sink state is called transient. The size of an acyclic automaton is the number of
transient states. An acyclic automaton of size n thus has transient states which we label
1, 2, . . . , n and a sink, labeled n + 1. Liskovets [1] uses the inclusion-exclusion principle
(more about this below) to obtain the following recurrence relation for the number ak(n)
of acyclic automata of size n on a k-letter input alphabet (k ≥ 1):
ak(0) = 1; ak(n) =
(−1)n−j−1
(j + 1)k(n−j)ak(j), n ≥ 1.
A source is a vertex with no incoming edges. A finite acyclic automaton has at least
one source because a path traversed backward v1 ← v2 ← v3 ← . . . must have distinct
vertices and so cannot continue indefinitely. An automaton is single-source (or initially
connected) if it has only one source. Let Bk(n) denote the set of single-source acyclic
finite (SAF) automata on a k-letter input alphabet with vertices 1, 2, . . . , n + 1 where 1
is the source and n + 1 is the sink, and set bk(n) = | Bk(n) |. The two-line representation
of an automaton in Bk(n) is the 2× kn matrix whose columns list the edges in order. For
example,
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5
2 4 6 6 6 6 6 6 6 3 5 3 2 2 6
is in B3(5) and the source-to-sink paths in B include 1
→ 6, 1
→ 6, 1
→ 6, where the alphabet is {a, b, c}.
Proposition 1. The number bk(n) of SAF automata of size n on a k-letter input alphabet
(n, k ≥ 1) is given by
bk(n) =
(−1)n−i
(i+ 1)k(n−i)ak(i)
Remark This formula is a bit more succinct than the the recurrence in [1, Theorem
3.2].
Proof Consider the setA of acyclic automata with transient vertices [n] = {1, 2, . . . , n}
in which 1 is a source. Call 2, 3, . . . , n the interior vertices. For X ⊆ [2, n], let
f(X) = # automata in A whose set of interior vertices includes X,
g(X) = # automata in A whose set of interior vertices is precisely X.
Then f(X) =
Y :X⊆Y⊆[2,n] g(Y ) and by Möbius inversion [2] on the lattice of subsets of
[2, n], g(X) =
Y :X⊆Y⊆[2,n] µ(X, Y )f(Y ) where µ(X, Y ) is the Möbius function for this
lattice. Since µ(X, Y ) = (−1)|Y |−|X| if X ⊆ Y , we have in particular that
g(∅) =
Y⊆[2,n]
(−1)| Y |f(Y ). (1)
Let | Y | = n − i so that 1 ≤ i ≤ n. When Y consists entirely of sources, the vertices
in [n+ 1]\Y and their incident edges form a subautomaton with i transient states; there
are ak(i) such. Also, all edges from the n − i vertices comprising Y go directly into
[n + 1]\Y : (i + 1)k(n−i) choices. Thus f(Y ) = (i + 1)k(n−i)ak(i). By definition, g(∅) is
the number of automata in A for which 1 is the only source, that is, g(∅) = bk(n) and the
Proposition now follows from (1).
An unlabeled SAF automaton is an equivalence class of SAF automata under relabeling
of the interior vertices. Liskovets notes [1] (and we prove below) that Bk(n) has no
nontrivial automorphisms, that is, each of the (n− 1)! relabelings of the interior vertices
of B ∈ Bk(n) produces a different automaton. So unlabeled SAF automata of size n on
a k-letter alphabet are counted by 1
(n−1)!
bk(n). The next result establishes a canonical
representative in each relabeling class.
Proposition 2. Each equivalence class in Bk(n) under relabeling of interior vertices has
size (n− 1)! and contains exactly one SAF automaton with the “last occurrences increas-
ing” property: the last occurrences of the interior vertices—2, 3, . . . , n—in the bottom row
of its two-line representation occur in that order.
Proof The first assertion follows from the fact that the interior vertices of an au-
tomatonB ∈ bk(n) can be distinguished intrinsically, that is, independent of their labeling.
To see this, first mark the source, namely 1, with a mark (new label) v1 and observe that
there exists at least one interior vertex whose only incoming edge(s) are from the source
(the only currently marked vertex) for otherwise a cycle would be present. For each such
interior vertex v, choose the last edge from the marked vertex to v using the built-in
ordering of these edges. This determines an order on these vertices; mark them in order
v2, v3, . . . , vj (j ≥ 2). If there still remain unmarked interior vertices, at least one of them
has incoming edges only from a marked vertex or again a cycle would be present. For
each such vertex, use the last incoming edge from a marked vertex, where now edges are
arranged in order of initial vertex vi with the built-in order breaking ties, to order and
mark these vertices vj+1, vj+2, . . .. Proceed similarly until all interior vertices are marked.
For example, for
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5
2 4 6 6 6 6 6 6 6 3 5 3 2 2 6
v1 = 1 and there is just one interior vertex, namely 4, whose only incoming edge is from
the source, and so v2 = 4 and 4 becomes a marked vertex. Now all incoming edges to
both 3 and 5 are from marked vertices and the last such edges (built-in order comes into
play) are 4
→ 5 and 4
→ 3 putting vertices 3, 5 in the order 5, 3. So v3 = 5 and v4 = 3.
Finally, v5 = 2. This proves the first assertion. By construction of the vs, relabeling each
interior vertex i with the subscript of its corresponding v produces an automaton in Bk(n)
with the “last occurrences increasing” property and is the only relabeling that does so.
The example yields
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5
5 2 6 4 3 4 5 5 6 6 6 6 6 6 6
Now let Ck(n) denote the set of canonical SAF automata in Bk(n) representing un-
labeled automata; thus | Ck(n) | =
(n−1)!
bk(n). Henceforth, we identify an unlabeled au-
tomaton with its canonical representative.
3 Column-Marked Subdiagonal Paths
A subdiagonal (k, n, p)-path is a lattice path of steps E = (1, 0) and N = (0, 1), E for
east and N for north, from (0, 0) to (kn, p) that never rise above the line y = 1
x. Let
Ck(n, p) denote the set of such paths.For k ≥ 1, it is clear that Ck(n, p) is nonempty only
for 0 ≤ p ≤ n and it is known (generalized ballot theorem) that
|Ck(n, p) | =
kn− kp+ 1
kn+ p+ 1
kn+ p + 1
A path P in Ck(n, n) can be coded by the heights of its E steps above the line y = −1;
this gives a a sequence (bi)
i=1 subject to the restrictions 1 ≤ b1 ≤ b2 ≤ . . . ≤ bkn and
bi ≤ ⌈i/k⌉ for all i.
A column-marked subdiagonal (k, n, p)-path is one in which, for each i ∈ [1, kn], one of
the lattice squares below the ith E step and above the horizontal line y = −1 is marked,
say with a ‘ ∗ ’. Let C
k(n, p) denote the set of such marked paths.
b b b
b b b b
b b b b
∗ ∗ ∗
(0,0)
(8,4)
y = −1
y = 1
A path in C
2(4, 3)
A marked path P ∗ in C
k(n, n) can be coded by a sequence of pairs
(ai, bi)
where
i=1 is the code for the underlying path P and ai ∈ [1, bi] gives the position of the ∗ in the
ith column. The example is coded by (1, 1), (1, 1), (1, 2), (2, 2), (1, 2), (3, 3), (1, 3), (2, 3).
An explicit sum for |C
k(n, n) | is
k(n, n) | =
1≤b1≤b2≤...≤bkn,
bi ≤ ⌈i/k⌉ for all i
b1b2 . . . bkn,
because the summand b1b2 . . . bkn is the number of ways to insert the ‘ ∗ ’s in the underlying
path coded by (bi)
It is also possible to obtain a recurrence for |C
k(n, p) |, and then, using Prop. 1, to
show analytically that |C
k(n, n) | = | Ck+1(n) |. However, it is much more pleasant to
give a bijection and in the next section we will do so. In particular, the number of SAF
automata on a 2-letter alphabet is
| C2(n) | = |C
1(n, n) | =
1≤b1≤b2≤...≤bn
bi ≤ i for all i
b1b2 . . . bn = (1, 3, 16, 127, 1363, . . .)n≥1,
sequence A082161 in [3].
4 Bijection from Paths to Automata
In this section we exhibit a bijection from C
k(n, n) to Ck+1(n). Using the illustrated
path as a working example with k = 2 and n = 4,
b b b
b b b b
b b b b
∗ ∗ ∗
(0,0)
(8,4)
y = −1
y = 1
first construct the top row of a two-line representation consisting of k + 1 each 1s, 2s,
. . . ,n s and number them left to right:
The last step in the path is necessarily anN step. For the second last, third last,. . .N steps
in the path, count the number of steps following it. This gives a sequence i1, i2, . . . , in−1
satisfying 1 ≤ i1 < i2 < . . . < in−1 and ij ≤ (k + 1)j for all j. Circle the positions
i1, i2, . . . , in−1 in the two-line representation and then insert (in boldface) 2, 3, . . . , n in
the second row in the circled positions:
2 3 4
These will be the last occurrences of 2, 3, . . . , n in the second row. Working from the last
column in the path back to the first, fill in the blanks in the second row left to right as
follows. Count the number of squares from the ∗ up to the path (including the ∗ square)
http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082161
and add this number to the nearest boldface number to the left of the current blank entry
(if there are no boldface numbers to the left, add this number to 1) and insert the result
in the current blank square. In the example the numbers of squares are 2,3,1,2,1,2,1,1
yielding
2 4 5 3 3 5 4 5 4 5 5
This will fill all blank entries except the last. Note that ∗ s in the bottom row correspond
to sink (that is, n+1) labels in the second row. Finally, insert n+1 into the last remaining
blank space to give the image automaton:
1 1 1 2 2 2 3 3 3 4 4 4
2 4 5 3 3 5 4 5 4 5 5 5
This process is fully reversible and the map is a bijection.
5 Evaluation of detAk(n)
For simplicity, we treat the case k = 1, leaving the generalization to arbitrary k
as a not-too-difficult exercise for the interested reader. Write A(n) for A1(n). Thus
A(n) =
1≤i,j≤n
. From the definition of detA(n) as a sum of signed products, we
show that detA(n) is the total weight of certain lists of permutations, each list carrying
weight ±1. Then a weight-reversing involution cancels all −1 weights and reduces the
problem to counting the surviving lists. These surviving lists are essentially the codes for
paths in C
1(n, p), and the Main Theorem follows from §4.
To describe the permutations giving a nonzero contribution to detA(n) =
σ sgn σ×
i=1 ai,σ(i), define the code of a permutation σ on [n] to be the list c = (ci)
i=1 with
ci = σ(i)−(i−1). Since the (i, j) entry of A(n),
, is 0 unless j ≥ i−1, we must have
σ(i) ≥ i−1 for all i. It is well known that there are 2n−1 such permutations, corresponding
to compositions of n, with codes characterized by the following four conditions: (i) ci ≥ 0
for all i, (ii) c1 ≥ 1, (iii) each ci ≥ 1 is immediately followed by ci − 1 zeros in the list,
i=1 ci = n. Let us call such a list a padded composition of n: deleting the zeros
is a bijection to ordinary compositions of n. For example, (3, 0, 0, 1, 2, 0) is a padded
composition of 6. For a permutation σ with padded composition code c, the nonzero
entries in c give the cycle lengths of σ. Hence sgnσ, which is the parity of “n−#cycles
in σ”, is given by (−1)#0s in c.
We have detA(n) =
σ sgn σ
i=1 ai,σ(i) =
σ sgn σ
2i−σ(i)
, and so
detA(n) =
(−1)#0s in c
i+ 1− ci
where the sum is restricted to padded compositions c of n with ci ≤ i for all i (A002083)
because
i+1−ci
= 0 unless ci ≤ i.
Henceforth, let us write all permutations in standard cycle form whereby the smallest
entry occurs first in each cycle and these smallest entries increase left to right. Thus,
with dashes separating cycles, 154-2-36 is the standard cycle form of the permutation
( 1 2 3 4 5 65 2 6 1 4 3 ). We define a nonfirst entry to be one that does not start a cycle. Thus the
preceding permutation has 3 nonfirst entries: 5,4,6. Note that the number of nonfirst
entries is 0 only for the identity permutation. We denote an identity permutation (of any
size) by ǫ.
By definition of Stirling cycle number, the product in (2) counts lists (πi)
i=1 of permu-
tations where πi is a permutation on [i+1] with i+1− ci cycles, equivalently, with ci ≤ i
nonfirst entries. So define Ln to be the set all lists of permutations π = (πi)
i=1 where πi
is a permutation on [i + 1], #nonfirst entries in πi is ≤ i, π1 is the transposition (1,2),
each nonidentity permutation πi is immediately followed by ci − 1 ǫ’s where ci ≥ 1 is the
number of nonfirst entries in πi (so the total number of nonfirst entries is n). Assign a
weight to π ∈ Ln by wt(π) = (−1)
# ǫ’s in π. Then
detA(n) =
wt(π).
We now define a weight-reversing involution on (most of) Ln. Given π ∈ Ln, scan the
list of its component permutations π1 = (1, 2), π2, π3, . . . left to right. Stop at the first
one that either (i) has more than one nonfirst entry, or (ii) has only one nonfirst entry, b
say, and b > maximum nonfirst entry m of the next permutation in the list. Say πk is the
permutation where we stop.
http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002083
In case (i) decrement (i.e. decrease by 1) the number of ǫ’s in the list by splitting πk
into two nonidentity permutations as follows. Let m be the largest nonfirst entry of πk
and let ℓ be its predecessor. Replace πk and its successor in the list (necessarily an ǫ) by
the following two permutations: first the transposition (ℓ,m) and second the permutation
obtained from πk by erasing m from its cycle and turning it into a singleton. Here are
two examples of this case (recall permutations are in standard cycle form and, for clarity,
singleton cycles are not shown).
i 1 2 3 4 5 6
πi 12 13 23 14-253 ǫ ǫ
i 1 2 3 4 5 6
πi 12 13 23 25 14-23 ǫ
i 1 2 3 4 5 6
πi 12 23 14 13-24 ǫ 23
i 1 2 3 4 5 6
πi 12 23 14 24 13 23
The reader may readily check that this sends case (i) to case (ii).
In case (ii), πk is a transposition (a, b) with b > maximum nonfirst entry m of πk+1. In
this case, increment the number of ǫ’s in the list by combining πk and πk+1 into a single
permutation followed by an ǫ: in πk+1, b is a singleton; delete this singleton and insert b
immediately after a in πk+1 (in the same cycle). The reader may check that this reverses
the result in the two examples above and, in general, sends case (ii) to case (i). Since the
map alters the number of ǫ’s in the list by 1, it is clearly weight-reversing. The map fails
only for lists that both consist entirely of transpositions and have the form
(a1, b1), (a2, b2), . . . , (an, bn) with b1 ≤ b2 ≤ . . . ≤ bn.
Such lists have weight 1. Hence detA(n) is the number of lists
(ai, bi)
satisfying
1 ≤ ai < bi ≤ i+ 1 for 1 ≤ i ≤ n, and b1 ≤ b2 ≤ . . . ≤ bn. After subtracting 1 from each
bi, these lists code the paths in C
1(n, n) and, using §4, detA(n) = |C
1(n, n) | = | C2(n) |.
References
[1] Valery A. Liskovets, Exact enumeration of acyclic deterministic au-
tomata, Disc. Appl. Math., in press, 2006. Earlier version available at
http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html
http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html
[2] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed., Cambridge
University Press, NY, 2001.
[3] Neil J. Sloane (founder and maintainer), The On-Line Encyclopedia of Integer Se-
quences http://www.research.att.com:80/ njas/sequences/index.html?blank=1
http://www.research.att.com:80/~njas/sequences/index.html?blank=1
|
0704.0005 | From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha}$ | FROM DYADIC Λα TO Λα
WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
Abstract. In this paper we show how to compute the Λα norm , α ≥ 0,
using the dyadic grid. This result is a consequence of the description of
the Hardy spaces Hp(RN ) in terms of dyadic and special atoms.
Recently, several novel methods for computing the BMO norm of a function
f in two dimensions were discussed in [9]. Given its importance, it is also of
interest to explore the possibility of computing the norm of a BMO function,
or more generally a function in the Lipschitz class Λα, using the dyadic grid
in RN . It turns out that the BMO question is closely related to that of
approximating functions in the Hardy space H1(RN ) by the Haar system.
The approximation in H1(RN ) by affine systems was proved in [2], but this
result does not apply to the Haar system. Now, if HA(R) denotes the closure
of the Haar system in H1(R), it is not hard to see that the distance d(f,HA)
of f ∈ H1(R) to HA is ∼
f(x) dx
∣, see [1]. Thus, neither dyadic atoms
suffice to describe the Hardy spaces, nor the evaluation of the norm in BMO
can be reduced to a straightforward computation using the dyadic intervals.
In this paper we address both of these issues. First, we give a characterization
of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms, and then,
by a duality argument, we show how to compute the norm in Λα(R
N ), α ≥ 0,
using the dyadic grid.
We begin by introducing some notations. Let J denote a family of cubes
Q in RN , and Pd the collection of polynomials in R
N of degree less than or
equal to d. Given α ≥ 0, Q ∈ J , and a locally integrable function g, let pQ(g)
denote the unique polynomial in P[α] such that [g − pQ(g)]χQ has vanishing
moments up to order [α].
For a locally square-integrable function g, we consider the maximal function
α,J g(x) given by
α,J g(x) = sup
x∈Q,Q∈J
|Q|α/N
|g(y)− pQ(g)(y)|
1991 Mathematics Subject Classification. 42B30,42B35.
http://arxiv.org/abs/0704.0005v1
2 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
The Lipschitz space Λα,J consists of those functions g such that M
α,J g is
in L∞, ‖g‖Λα,J = ‖M
α,J g‖∞; when the family in question contains all cubes
in RN , we simply omit the subscript J . Of course, Λ0 = BMO.
Two other families, of dyadic nature, are of interest to us. Intervals in R of
the form In,k = [ (k−1)2
n, k2n], where k and n are arbitrary integers, positive,
negative or 0, are said to be dyadic. In RN , cubes which are the product of
dyadic intervals of the same length, i.e., of the form Qn,k = In,k1 ×· · ·×In,kN ,
are called dyadic, and the collection of all such cubes is denoted D.
There is also the family D0. Let I
n,k = [(k− 1)2
n, (k+ 1)2n], where k and
n are arbitrary integers. Clearly I ′n,k is dyadic if k is odd, but not if k is even.
Now, the collection {I ′n,k : n, k integers} contains all dyadic intervals as well
as the shifts [(k − 1)2n + 2n−1, k 2n + 2n−1] of the dyadic intervals by their
half length. In RN , put D0 = {Q
n,k : Q
n,k = I
× · · · × I ′n,kN }; Q
n,k is
called a special cube. Note that D0 contains D properly.
Finally, given I ′n,k, let I
n,k = [(k − 1)2
n, k2n], and I
n,k = [k2
n, (k + 1)2n].
The 2N subcubes of Q′n,k = I
× · · · × I ′n,kN of the form I
× · · · × I
Sj = L or R, 1 ≤ j ≤ N , are called the dyadic subcubes of Q
Let Q0 denote the special cube [−1, 1]
N . Given α ≥ 0, we construct a
family Sα of piecewise polynomial splines in L
2(Q0) that will be useful in
characterizing Λα. Let A be the subspace of L
2(Q0) consisting of all functions
with vanishing moments up to order [α] which coincide with a polynomial
in P[α] on each of the 2
N dyadic subcubes of Q0. A is a finite dimensional
subspace of L2(Q0), and, therefore, by the Graham-Schmidt orthogonalization
process, say, A has an orthonormal basis in L2(Q0) consisting of functions
p1, . . . , pM with vanishing moments up to order [α], which coincide with a
polynomial in P[α] on each dyadic subinterval of Q0. Together with each p
we also consider all dyadic dilations and integer translations given by
pLn,k,α(x) = 2
n(N+α)pL(2nx1 + k1, . . . , 2
nxN + kN ) , 1 ≤ L ≤ M ,
and let
Sα = {p
n,k,α : n, k integers, 1 ≤ L ≤ M} .
Our first result shows how the dyadic grid can be used to compute the
norm in Λα.
Theorem A. Let g be a locally square-integrable function and α ≥ 0. Then,
g ∈ Λα if, and only if, g ∈ Λα,D and Aα(g) = supp∈Sα
∣〈g, p〉
∣ < ∞. Moreover,
‖g‖Λα ∼ ‖g‖Λα,D +Aα(g) .
Furthermore, it is also true, and the proof is given in Proposition 2.1 be-
low, that ‖g‖Λα ∼ ‖g‖Λα,D0 . However, in this simpler formulation, the tree
structure of the cubes in D has been lost.
FROM DYADIC Λα TO Λα 3
The proof of Theorem A relies on a close investigation of the predual of
Λα, namely, the Hardy space H
p(RN ) with 0 < p = (α + N)/N ≤ 1. In the
process we characterize Hp in terms of simpler subspaces: H
, or dyadic Hp,
and H
, the space generated by the special atoms in Sα. Specifically, we
Theorem B. Let 0 < p ≤ 1, and α = N(1/p− 1). We then have
Hp = H
where the sum is understood in the sense of quasinormed Banach spaces.
The paper is organized as follows. In Section 1 we show that individual
Hp atoms can be written as a superposition of dyadic and special atoms;
this fact may be thought of as an extension of the one-dimensional result of
Fridli concerning L∞ 1- atoms, see [5] and [1]. Then, we prove Theorem B.
In Section 2 we discuss how to pass from Λα,D, and Λα,D0 , to the Lipschitz
space Λα.
1. Characterization of the Hardy spaces Hp
We adopt the atomic definition of the Hardy spaces Hp, 0 < p ≤ 1, see
[6] and [10]. Recall that a compactly supported function a with [N(1/p− 1)]
vanishing moments is an L2 p -atom with defining cube Q if supp(a) ⊆ Q, and
|Q|1/p
| a(x) |2dx
≤ 1 .
The Hardy space Hp(RN ) = Hp consists of those distributions f that can be
written as f =
λjaj , where the aj ’s are H
p atoms,
|λj |
p < ∞, and the
convergence is in the sense of distributions as well as in Hp. Furthermore,
‖f‖Hp ∼ inf
|λj |
where the infimum is taken over all possible atomic decompositions of f . This
last expression has traditionally been called the atomic Hp norm of f .
Collections of atoms with special properties can be used to gain a better
understanding of the Hardy spaces. Formally, let A be a non-empty subset
of L2 p -atoms in the unit ball of Hp. The atomic space H
spanned by A
consists of those ϕ in Hp of the form
λjaj , aj ∈ A ,
|λj |
p < ∞ .
It is readily seen that, endowed with the atomic norm
‖ϕ‖Hp
= inf
|λj |
: ϕ =
λj aj , aj ∈ A
becomes a complete quasinormed space. Clearly, H
⊆ Hp, and, for
f ∈ H
, ‖f‖Hp ≤ ‖f‖Hp
4 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
Two families are of particular interest to us. When A is the collection
of all L2 p -atoms whose defining cube is dyadic, the resulting space is H
or dyadic Hp. Now, although ‖f‖Hp ≤ ‖f‖Hp
, the two quasinorms are not
equivalent on H
. Indeed, for p = 1 and N = 1, the functions
fn(x) = 2
n[χ[1−2−n,1](x) − χ[1,1+2−n](x)] ,
satisfy ‖fn‖H1 = 1, but ‖fn‖H1
∼ |n| tends to infinity with n.
Next, when Sα is the family of piecewise polynomial splines constructed
above with α = N(1/p − 1), in analogy with the one-dimensional results in
[4] and [1], H
is referred to as the space generated by special atoms.
We are now ready to describe Hp atoms as a superposition of dyadic and
special atoms.
Lemma 1.1. Let a be an L2 p -atom with defining cube Q, 0 < p ≤ 1,
and α = N(1/p − 1). Then a can be written as a linear combination of 2N
dyadic atoms ai, each supported in one of the dyadic subcubes of the smallest
special cube Qn,k containing Q, and a special atom b in Sα. More precisely,
a(x) =
i=1 di ai(x) +
L=1 cL p
−n,−k,α(x), with |di| , |cL| ≤ c.
Proof. Suppose first that the defining cube of a is Q0, and let Q1, . . . , Q2N
denote the dyadic subcubes of Q0. Furthermore, let {e
i , . . . , e
i } denote an
orthonormal basis of the subspace Ai of L
2(Qi) consisting of polynomials in
P[α], 1 ≤ i ≤ 2
N . Put
αi(x) = a(x)χQi (x)−
〈aχQi , e
j(x) , 1 ≤ i ≤ 2
and observe that 〈αi, e
j〉 = 0 for 1 ≤ j ≤ M . Therefore, αi has [α] vanishing
moments, is supported in Qi, and
‖αi‖2 ≤ ‖aχQi‖2 +
‖aχQi‖2 ≤ (M + 1) ‖aχQi‖2 .
ai(x) =
2N(1/2−1/p)
M + 1
αi(x) , 1 ≤ i ≤ N ,
is an L2 p - dyadic atom. Finally, put
b(x) = a(x) −
M + 1
2N(1/2−1/p)
ai(x) .
FROM DYADIC Λα TO Λα 5
Clearly b has [α] vanishing moments, is supported in Q0, coincides with a
polynomial in P[α] on each dyadic subcube of Q0, and
‖b‖22 ≤
|〈aχQi , e
2 ≤ M ‖a‖22 .
So, b ∈ A, and, consequently, b(x) =
L=1 cL p
L(x), where
|cL| = |〈b, p
L〉| ≤ c , 1 ≤ L ≤ M .
In the general case, let Q be the defining cube of a, side-length Q = ℓ, and
let n and k = (k1, . . . , kN ) be chosen so that 2
n−1 ≤ ℓ < 2n, and
Q ⊂ [(k1 − 1)2
n, (k1 + 1)2
n]× · · · × [(kN − 1)2
n, (kN + 1)2
Then, (1/2)N ≤ |Q|/2nN < 1.
Now, given x ∈ Q0, let a
′ be the translation and dilation of a given by
a′(x) = 2nN/pa(2nx1 − k1, . . . , 2
nxN − kN ) .
Clearly, [α] moments of a′ vanish, and
‖a′‖2 = 2
nN/p 2−nN/2‖a‖2 ≤ c |Q|
1/p|Q|−1/2‖a‖2 ≤ c .
Thus, a′ is a multiple of an atom with defining cube Q0. By the first part of
the proof,
a′(x) =
i(x) +
L(x) , x ∈ Q0 .
The support of each a′i is contained in one of the dyadic subcubes of Q0, and,
consequently, there is a k such that
ai(x) = 2
−nN/pa′i(2
−nx1 − k1, . . . , 2
−nxN − kN )
ai is an L
2p -atom supported in one of the dyadic subcubes of Q. Similarly
for the pL’s. Thus,
a(x) =
di ai(x) +
−n,−k,N(1/p−1)(x) ,
and we have finished. �
Theorem B follows readily from Lemma 1.1. Clearly, H
→֒ Hp.
Conversely, let f =
j λj aj be in H
p. By Lemma 1.1 each aj can be written
as a sum of dyadic and special atoms, and, by distributing the sum, we can
write f = fd + fs, with fd in H
, fs in H
, and
‖fd‖Hp
, ‖fs‖Hp
|λj |
Taking the infimum over the decompositions of f we get ‖f‖Hp
c ‖f‖Hp , and H
p →֒ H
. This completes the proof.
6 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
The meaning of this decomposition is the following. Cubes in D are con-
tained in one of the 2N non-overlapping quadrants of RN . To allow for the
information carried by a dyadic cube to be transmitted to an adjacent dyadic
cube, they must be connected. The pLn,k,α’s channel information across ad-
jacent dyadic cubes which would otherwise remain disconnected. The reader
will have no difficulty in proving the quantitative version of this observation:
Let T be a linear mapping defined on Hp, 0 < p ≤ 1, that assumes values in
a quasinormed Banach space X . Then, T is continuous if, and only if, the
restrictions of T to H
and H
are continuous.
2. Characterizations of Λα
Theorem A describes how to pass from Λα,D to Λα, and we prove it next.
Since (Hp)∗ = Λα and (H
)∗ = Λα,D, from Theorem B it follows readily that
Λα = Λα,D ∩ (H
)∗, so it only remains to show that (H
)∗ is characterized
by the condition Aα(g) < ∞.
First note that if g is a locally square-integrable function with Aα(g) < ∞
and f =
j,L cj,L p
nj ,kj ,α
, since 0 < p ≤ 1,
|〈g, f〉| ≤
|cj,L| |〈g, p
nj ,kj ,α
≤ Aα(g)
|cj,L|
and, consequently, taking the infimum over all atomic decompositions of f in
, we get g ∈ (H
)∗ and ‖g‖(Hp
)∗ ≤ Aα(g).
To prove the converse we proceed as in [3]. Let Qn = [−2
n, 2n]N . We begin
by observing that functions f in L2(Qn) that have vanishing moments up to
order [α] and coincide with polynomials of degree [α] on the dyadic subcubes
of Qn belong to H
‖f‖Hp
≤ |Qn|
1/p−1/2‖f‖2 .
Given ℓ ∈ (H
)∗, for a fixed n let us consider the restriction of ℓ to the space
of L2 functions f with [α] vanishing moments that are supported in Qn. Since
|ℓ(f)| ≤ ‖ℓ‖ ‖f‖Hp
≤ ‖ℓ‖ |Qn|
1/p−1/2‖f‖2 ,
this restriction is continuous with respect to the norm in L2 and, consequently,
it can be extended to a continuous linear functional in L2 and represented as
ℓ(f) =
f(x) gn(x) dx ,
FROM DYADIC Λα TO Λα 7
where gn ∈ L
2(Qn) and satisfies ‖gn‖2 ≤ ‖ℓ‖ |Qn|
1/p−1/2. Clearly, gn is
uniquely determined in Qn up to a polynomial pn in P[α]. Therefore,
gn(x) − pn(x) = gm(x)− pm(x) , a.e. x ∈ Qmin(n,m) .
Consequently, if
g(x) = gn(x)− pn(x) , x ∈ Qn ,
g(x) is well defined a.e. and, if f ∈ L2 has [α] vanishing moments and is
supported in Qn, we have
ℓ(f) =
f(x) gn(x) dx
f(x) [gn(x)− pn(x)] dx
f(x) g(x) dx .
Moreover, since each 2nN/ppL(2n ·+k) is an L2 p-atom, 1 ≤ L ≤ M , it readily
follows that
Aα(g) = sup
1≤L≤M
n,k∈Z
|〈g, 2−n/ppL(2n ·+k)〉|
≤ ‖ℓ‖ sup
‖pL‖Hp ≤ ‖ℓ‖ ,
and, consequently, Aα(g) ≤ ‖ℓ‖ , and (H
)∗ is the desired space. �
The reader will have no difficulty in showing that this result implies the
following: Let T be a bounded linear operator from a quasinormed space X
into Λα,D. Then, T is bounded from X into Λα if, and only if, Aα(Tx) ≤
c ‖x‖X for every x ∈ X .
The process of averaging the translates of dyadic BMO functions leads to
BMO, and is an important tool in obtaining results in BMO once they are
known to be true in its dyadic counterpart, BMOd, see [7]. It is also known
that BMO can be obtained as the intersection of BMOd and one of its shifted
counterparts, see [8]. These results motivate our next proposition, which
essentially says that g ∈ Λα if, and only if, g ∈ Λα,D and g is in the Lipschitz
class obtained from the shifted dyadic grid. Note that the shifts involved in
this class are in all directions parallel to the coordinate axis and depend on
the side-length of the cube.
Proposition 2.1. Λα = Λα,D0 , and ‖g‖Λα ∼ ‖g‖Λα,D0 .
Proof. It is obvious that ‖g‖Λα,D0 ≤ ‖g‖Λα . To show the other inequality we
invoke Theorem A. Since D ⊂ D0, it suffices to estimate Aα(g), or, equiva-
lently, |〈g, p〉| for p ∈ Sα, α = N(1/p − 1). So, pick p = p
n,k,α in Sα. The
defining cube Q of pLn,k,α is in D0, and, since p
n,k,α has [α] vanishing moments,
8 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
〈pLn,k,α, pQ(g)〉 = 0. Therefore,
|〈g, pLn,k,α〉| = |〈g − pQ(g), p
n,k,α〉|
≤ ‖pLn,k,α‖2 ‖g − pQ(g)‖L2(Q)
≤ |Q|α/N |Q|1/2‖pLn,k,α‖2 ‖g‖Λα,D0 .
Now, a simple change of variables gives |Q|α/N |Q|1/2‖pLn,k,α‖2 ≤ 1, and, con-
sequently, also Aα(g) ≤ ‖g‖Λα,D0 . �
References
[1] W. Abu-Shammala, J.-L. Shiu, and A. Torchinsky, Characterizations of the Hardy
space H1 and BMO, preprint.
[2] H.-Q. Bui and R. S. Laugesen, Approximation and spanning in the Hardy space, by
affine systems, Constr. Approx., to appear.
[3] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a
distibution, II, Advances in Math., 24 (1977), 101–171.
[4] G. S. de Souza, Spaces formed by special atoms, I, Rocky Mountain J. Math. 14 (1984),
no. 2, 423–431.
[5] S. Fridli, Transition from the dyadic to the real nonperiodic Hardy space, Acta Math.
Acad. Paedagog. Niházi (N.S.) 16 (2000), 1–8, (electronic).
[6] J. Garćıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related
topics, Notas de Matemática 116, North Holland, Amsterdam, 1985.
[7] J. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2,
351–371.
[8] T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad.
Sci. Paris 336 (2003), no. 12, 1003–1006.
[9] T. M. Le and L. A. Vese, Image decomposition using total variation and div( BMO)∗,
Multiscale Model. Simul. 4, (2005), no. 2, 390–423.
[10] A. Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc.,
Mineola, NY, 2004.
Department of Mathematics, Indiana University, Bloomington IN 47405
E-mail address: [email protected]
Department of Mathematics, Indiana University, Bloomington IN 47405
E-mail address: [email protected]
1. Characterization of the Hardy spaces Hp
2. Characterizations of
References
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