Datasets:

Kyudan
/

TeX_htmls

Dataset card Files Files and versions Community

Kyudan commited on Mar 27, 2024

Commit

f139f10

verified ·

1 Parent(s): 5e28378

Delete htmls

Browse files

This view is limited to 50 files because it contains too many changes. See raw diff

Files changed (50) hide show

htmls/output_mathjax_example_1.html +0 -124
htmls/output_mathjax_example_10.html +0 -137
htmls/output_mathjax_example_100.html +0 -157
htmls/output_mathjax_example_1000.html +0 -132
htmls/output_mathjax_example_10000.html +0 -123
htmls/output_mathjax_example_10001.html +0 -131
htmls/output_mathjax_example_10002.html +0 -155
htmls/output_mathjax_example_10003.html +0 -169
htmls/output_mathjax_example_10004.html +0 -175
htmls/output_mathjax_example_10005.html +0 -139
htmls/output_mathjax_example_10006.html +0 -156
htmls/output_mathjax_example_10007.html +0 -167
htmls/output_mathjax_example_10008.html +0 -172
htmls/output_mathjax_example_10009.html +0 -169
htmls/output_mathjax_example_1001.html +0 -127
htmls/output_mathjax_example_10010.html +0 -187
htmls/output_mathjax_example_10011.html +0 -163
htmls/output_mathjax_example_10012.html +0 -142
htmls/output_mathjax_example_10013.html +0 -137
htmls/output_mathjax_example_10014.html +0 -150
htmls/output_mathjax_example_10015.html +0 -132
htmls/output_mathjax_example_10016.html +0 -146
htmls/output_mathjax_example_10017.html +0 -134
htmls/output_mathjax_example_10018.html +0 -156
htmls/output_mathjax_example_10019.html +0 -134
htmls/output_mathjax_example_1002.html +0 -124
htmls/output_mathjax_example_10020.html +0 -159
htmls/output_mathjax_example_10021.html +0 -192
htmls/output_mathjax_example_10022.html +0 -195
htmls/output_mathjax_example_10023.html +0 -184
htmls/output_mathjax_example_10024.html +0 -180
htmls/output_mathjax_example_10025.html +0 -204
htmls/output_mathjax_example_10026.html +0 -174
htmls/output_mathjax_example_10027.html +0 -167
htmls/output_mathjax_example_10028.html +0 -131
htmls/output_mathjax_example_10029.html +0 -130
htmls/output_mathjax_example_1003.html +0 -122
htmls/output_mathjax_example_10030.html +0 -126
htmls/output_mathjax_example_10031.html +0 -185
htmls/output_mathjax_example_10032.html +0 -168
htmls/output_mathjax_example_10033.html +0 -162
htmls/output_mathjax_example_10034.html +0 -142
htmls/output_mathjax_example_10035.html +0 -140
htmls/output_mathjax_example_10036.html +0 -123
htmls/output_mathjax_example_10037.html +0 -125
htmls/output_mathjax_example_10038.html +0 -148
htmls/output_mathjax_example_10039.html +0 -196
htmls/output_mathjax_example_1004.html +0 -121
htmls/output_mathjax_example_10040.html +0 -143
htmls/output_mathjax_example_10041.html +0 -139

htmls/output_mathjax_example_1.html DELETED Viewed

@@ -1,124 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $O(n^{2})$ </p>
-    <p> $f$ </p>
-    <p> $n$ </p>
-    <p> $G(v)$ </p>
-    <p> $s_{o}\oplus s_{a}\in\mathbb{V}^{n+m}$ </p>
-    <p> $Z\in\mathbb{R}^{m\times d_{\text{token}}}$ </p>
-    <p> $E_{\psi}(s)$ </p>
-    <p> $\displaystyle=F^{i}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z)).$ </p>
-    <p> $\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text})$ </p>
-    <p> $\cos(\psi_{i},\psi_{j})$ </p>
-    <p> ${}^{4}$ </p>
-    <p> $v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))$ </p>
-    <p> ${}^{*}$ </p>
-    <p> $\displaystyle\text{argmax}_{Z}$ </p>
-    <p> $\rightarrow$ </p>
-    <p> $\mathcal{A}(x,t,s_{o})$ </p>
-    <p> $\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus s_{a}))$ </p>
-    <p> ${}^{1}$ </p>
-    <p> $\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})$ </p>
-    <p> $x_{t}$ </p>
-    <p> $500\times 20=10000$ </p>
-    <p> $w_{i},w_{j}$ </p>
-    <p> $v_{t}^{image}\leftarrow F^{i}(x_{t})$ </p>
-    <p> $m=4$ </p>
-    <p> $s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))$ </p>
-    <p> ${}^{5}$ </p>
-    <p> $Z_{i}$ </p>
-    <p> ${}^{1,*}$ </p>
-    <p> $\text{Proj}_{\psi}(Z)$ </p>
-    <p> $s$ </p>
-    <p> $\displaystyle\text{argmax}_{s_{a}}$ </p>
-    <p> $t$ </p>
-    <p> $s^{\prime}\leftarrow$ </p>
-    <p> $v_{t}^{image}$ </p>
-    <p> $5\times 4\times 100=2000$ </p>
-    <p> ${}^{1,2}$ </p>
-    <p> $\psi\in\mathbb{R}^{|\mathbb{V}|\times d_{\text{token}}}$ </p>
-    <p> $bestloss\leftarrow\mathcal{L},bestZ\leftarrow Z$ </p>
-    <p> $G$ </p>
-    <p> $\lambda=0$ </p>
-    <p> $\text{Proj}_{\psi}:\mathbb{R}^{m\times d_{\text{token}}}\rightarrow\mathbb{R}^%
-{m\times d_{\text{token}}}$ </p>
-    <p> $i\leftarrow 1$ </p>
-    <p> $s\in\mathbb{V}^{*}$ </p>
-    <p> $\displaystyle\text{argmax}_{s_{a}}\mathbb{E}_{x\sim G(F^{t}(E_{\psi}(s_{o}%
-\oplus s_{a})))}\mathcal{A}(x,t,s_{o})~{},$ </p>
-    <p> $\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text}),$ </p>
-    <p> $\cos(a,b)=\frac{a^{T}b}{\|a\|\|b\|}$ </p>
-    <p> $\eta$ </p>
-    <p> $512\times 512$ </p>
-    <p> $x$ </p>
-    <p> $E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})$ </p>
-    <p> $N$ </p>
-    <p> $bestloss>\mathcal{L}$ </p>
-    <p> $v_{t}^{image}=F^{i}(x_{t})$ </p>
-    <p> $d_{\text{emb}}$ </p>
-    <p> $\displaystyle\text{argmax}_{s_{a}}\cos(F^{i}(E_{\psi}(s_{o}\oplus s_{a})),v_{t%
-}).$ </p>
-    <p> $s^{\prime}=$ </p>
-    <p> ${}^{3,*}$ </p>
-    <p> $Z\leftarrow Z-\eta\nabla_{Z}\mathcal{L}$ </p>
-    <p> $100$ </p>
-    <p> $s_{a}$ </p>
-    <p> $s_{o}\oplus s_{a}$ </p>
-    <p> $m$ </p>
-    <p> $v$ </p>
-    <p> $\displaystyle\text{s.t.}\quad v=F^{i}(E_{\psi}(s_{o}\oplus s_{a})),$ </p>
-    <p> $\mathbb{V}=\{w_{1},w_{2},\cdots,w_{|\mathbb{V}|}\}$ </p>
-    <p> $F^{i}$ </p>
-    <p> $\psi$ </p>
-    <p> $\displaystyle\text{s.t.}\quad v$ </p>
-    <p> $s_{o}$ </p>
-    <p> $F^{t}$ </p>
-    <p> ${}^{2}$ </p>
-    <p> $\oplus$ </p>
-    <p> $E_{\psi}(s)_{i}=\psi_{j}$ </p>
-    <p> $5\times 4=20$ </p>
-    <p> $3\times 100$ </p>
-    <p> ${}^{3}$ </p>
-    <p> $v\leftarrow F^{t}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z))$ </p>
-    <p> $\mathcal{L}=-\cos(v,v_{t}^{image})-\lambda\cos(v,v_{t}^{text})$ </p>
-    <p> $s_{o}\in\mathbb{V}^{n}$ </p>
-    <p> $s_{a}\leftarrow E_{\psi}^{-1}(\text{Proj}_{\psi}(bestZ))$ </p>
-    <p> $bestloss\leftarrow\infty,bestZ\leftarrow Z$ </p>
-    <p> $\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))))$ </p>
-    <p> $t\in\mathbb{V}$ </p>
-    <p> $Z$ </p>
-    <p> $(\cdot)$ </p>
-    <p> $x\sim G(v)$ </p>
-    <p> $d_{\text{token}}$ </p>
-    <p> $s_{a}\in\mathbb{V}^{m}$ </p>
-    <p> $v_{t}$ </p>
-    <p> $\lambda$ </p>
-    <p> $\mathbb{V}$ </p>
-    <p> $w_{j}=s_{i}$ </p>
-    <p> $t\in\mathcal{V}$ </p>
-    <p> $x\sim G(F^{t}(E_{\psi}(s)))$ </p>
-    <p> $E_{\psi}$ </p>
-    <p> $j=\text{argmin}_{j^{\prime}}\|\psi_{j^{\prime}}-Z_{i}\|_{2}^{2}$ </p>
-    <p> $|s|\times d_{\text{token}}$ </p>
-    <p> $\displaystyle\text{argmax}_{v_{t}}\mathbb{E}_{x\sim G(v_{t})}\mathcal{A}(x,t,s%
-_{o})~{}.$ </p>
-    <p> $E_{L}\cup E_{R}$ </p>
-    <p> $E_{L}=\{(u,w)|(u,w)\in E,w\neq v\}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10.html DELETED Viewed

@@ -1,137 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $0.01$ </p>
-    <p> $\underset{\pm 0.10}{2.15}$ </p>
-    <p> $(\operatorname{\bm{\theta}}_{\text{client}}^{(t)}=\operatorname{\bm{\theta}}_{%
-\text{client}}^{(0)}$ </p>
-    <p> $r=64$ </p>
-    <p> $297.78$ </p>
-    <p> $\mathbf{0.43}$ </p>
-    <p> $\operatorname{\mathbf{v}}_{i}\in\mathcal{V}$ </p>
-    <p> $0.76$ </p>
-    <p> $\underset{\pm 0.66}{45.89}$ </p>
-    <p> $0$ </p>
-    <p> $\rho$ </p>
-    <p> $0.26$ </p>
-    <p> $0.95$ </p>
-    <p> $p\approx 8.69\times 10^{-8}$ </p>
-    <p> $0.69$ </p>
-    <p> $47.32$ </p>
-    <p> $\mathbf{0.69}$ </p>
-    <p> $2.30$ </p>
-    <p> $\mathbf{A}\in\mathbb{R}^{d\times r}$ </p>
-    <p> $\operatorname{\bm{\theta}}_{\text{client}}^{(t)}$ </p>
-    <p> $500$ </p>
-    <p> $\operatorname{\mathbf{v}}_{i}$ </p>
-    <p> $0.78$ </p>
-    <p> $308$ </p>
-    <p> $\mathbf{0.36}$ </p>
-    <p> $-\frac{1}{|\mathcal{D}^{(t)}_{\bigtriangledown}|}\sum_{\operatorname{\mathbf{d%
-}}^{(t)}\in\mathcal{D}^{(t)}_{\bigtriangledown}}\log p_{(\cdot)}\big{(}%
-\operatorname{\mathbf{d}}^{(t)}\big{|}\operatorname{\bm{\theta}}_{(\cdot)}^{(t%
-)}),$ </p>
-    <p> $\operatorname{\mathbf{pr}}_{\text{client}}$ </p>
-    <p> $p$ </p>
-    <p> $0.66$ </p>
-    <p> $2.65$ </p>
-    <p> $\operatorname{\bm{\theta}}_{\text{client}}^{(0)}$ </p>
-    <p> $25$ </p>
-    <p> $277.25$ </p>
-    <p> $\%$ </p>
-    <p> $57.16$ </p>
-    <p> $0.74$ </p>
-    <p> $0.77$ </p>
-    <p> $\tau=0.5$ </p>
-    <p> $256$ </p>
-    <p> $4.3$ </p>
-    <p> $\operatorname{\bm{\theta}}_{\text{agent}}^{(0)}$ </p>
-    <p> $0.90$ </p>
-    <p> $0.80$ </p>
-    <p> $4$ </p>
-    <p> $0.8$ </p>
-    <p> $9000$ </p>
-    <p> $\operatorname{\bm{\theta}}_{\text{client}}$ </p>
-    <p> $F_{1}$ </p>
-    <p> $\mathbf{55.25}$ </p>
-    <p> $0.60$ </p>
-    <p> $\operatorname{\mathbf{v}}_{\text{next}}\in\text{Children}(\operatorname{%
-\mathbf{v}}_{i})$ </p>
-    <p> $\displaystyle\geq bx_{j}^{\prime}+\sum_{i>j}x_{i}^{\prime}+(b-1)\sum_{i>j}x_{i}$ </p>
-    <p> $x_{i}^{\prime}\geq x_{i}$ </p>
-    <p> $A_{1}$ </p>
-    <p> $\displaystyle(b^{k}-b^{k-1}-(b-1)^{k}+b^{k-1})x_{1}$ </p>
-    <p> $(u,w)$ </p>
-    <p> $C_{1},C_{2}$ </p>
-    <p> $P:(u,v)\cup P^{\prime}$ </p>
-    <p> $e_{>i}$ </p>
-    <p> $(u_{1},u_{2},\dots,u_{k},v)$ </p>
-    <p> $\displaystyle=\frac{(b-1)^{k-1}}{b^{k}-(b-1)^{k}}x.$ </p>
-    <p> $(v,z)\in N(u)\times N(u)$ </p>
-    <p> $1\leq i\leq k$ </p>
-    <p> $x_{\tau+1},\dots,x_{k}\mapsto\textsf{Chunk-Shortest-Edge}(k-\tau,x-\delta)$ </p>
-    <p> $\sum_{i}x_{i}<x$ </p>
-    <p> $\displaystyle=bx+c(v\to t)-c(u,w)-c(w\to t)=\delta+(b-1)x$ </p>
-    <p> $p(e_{k}^{O})<\beta$ </p>
-    <p> $i\geq\tau$ </p>
-    <p> $\displaystyle=\begin{cases*}c(u,y)+cost[y,y,i]&if $(y,z)\in\mathcal{P}^{\prime%
-}(u,y)$\\
-\infty&otherwise\end{cases*}$ </p>
-    <p> $y^{*}\leq\frac{x}{k-1}$ </p>
-    <p> $x-\delta$ </p>
-    <p> $p(e;b_{i})$ </p>
-    <p> $\displaystyle=\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}\left(\frac{b^{k}-(b-1)^{k-1}%
-(b-1+1)}{b^{k}-(b-1)^{k}}\right)x+c(v\to t)$ </p>
-    <p> $(s,s_{1})$ </p>
-    <p> $i+j\leq k$ </p>
-    <p> $\displaystyle=\left(\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}-1\right)x$ </p>
-    <p> $\begin{array}[]{ll}p(e_{\tau})&=bx_{\tau}+c(u_{\tau+1}\to t)\\
-&=bx_{\tau}+\sum_{i=\tau+1}^{k}x_{i}+c(v\to t)\hfill\mbox{(shortest path from
-$u_{\tau+1}$ follows the chunking)}\\
-&=bx_{\tau}+x-\sum_{i=1}^{\tau}x_{i}+c(v\to t)\\
-&=b\cdot\frac{\delta}{\tau}-\tau\cdot\frac{\delta}{\tau}+x+c(v\to t)\hfill%
-\mbox{(substituing $x_{i}=\delta/\tau$ for $i\leq\tau$)}\\
-&=\frac{b\delta}{\tau}+c(u,w)+c(w\to t)\hfill\mbox{(since $\delta=x+c(v\to t)-%
-c(u,w)-c(w\to t)$).}\end{array}$ </p>
-    <p> $(u_{3},z)$ </p>
-    <p> $\displaystyle\frac{(b-1)(b^{k-1}-(b-1)^{k-1})+b^{k-1}}{b^{k-1}-(b-1)^{k-1}}x_{1}$ </p>
-    <p> $\sum_{l>i}x_{l}\leq x$ </p>
-    <p> $\displaystyle(b-1)x_{1}+x$ </p>
-    <p> $c^{n}$ </p>
-    <p> $\delta/k$ </p>
-    <p> $p(e_{i})=bx_{i}+c(u,w)+c(w\to t)$ </p>
-    <p> $\alpha=\beta$ </p>
-    <p> $p(e_{i};b_{1})\leq\alpha_{u}^{(j)}$ </p>
-    <p> $p(e_{i}^{C})=bx_{i}^{C}+c(u_{i}^{C}\to t)$ </p>
-    <p> $x_{i}-x_{i}^{\prime}\geq 0$ </p>
-    <p> $p(e_{i})=\beta^{\prime}$ </p>
-    <p> $\displaystyle\min_{(v,z)\in\mathcal{P}(u,y)}\min(C_{1}(u,v,y),C_{2}(u,y,z),C_{%
-3}(u,v,y,z)).$ </p>
-    <p> $c(u,v)$ </p>
-    <p> $\beta^{*}=p(e_{i})$ </p>
-    <p> $p(e_{j};b_{1})\leq\alpha_{u}^{(1)}$ </p>
-    <p> $\displaystyle=\frac{x}{1-\left(\frac{b-1}{b}\right)^{k-\tau+1}}+c(v\to t)-c(w%
-\to t)-c(u,w).$ </p>
-    <p> $x_{1}$ </p>
-    <p> $O(|E|^{2}k^{3}\log k+|V|)$ </p>
-    <p> $\beta\leftarrow\frac{x-\delta}{z_{\tau}}+c(v\to t)$ </p>
-    <p> $min\_bottleneck\leftarrow\min(\alpha,\beta)$ </p>
-    <p> $\delta/\tau$ </p>
-    <p> $(y,z)$ </p>
-</body>
-</html>

htmls/output_mathjax_example_100.html DELETED Viewed

@@ -1,157 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $x_{t+1},\ldots x_{t+\ell}$ </p>
-    <p> $O(|{\cal X}|^{2L})$ </p>
-    <p> $\forall i=1,\ldots,L$ </p>
-    <p> $1000$ </p>
-    <p> ${\cal M}_{b}$ </p>
-    <p> $(X^{\tau},Z^{L-\tau})\sim{\cal M}_{b}^{L}.$ </p>
-    <p> $(x^{i},y^{j})$ </p>
-    <p> $x^{L}$ </p>
-    <p> $\forall x\in{\cal X}$ </p>
-    <p> $\forall x^{i_{0}}\in{\cal X}^{i_{0}}$ </p>
-    <p> $X^{L}\sim{\cal M}_{s}^{L}$ </p>
-    <p> $\mathbb{E}_{X^{L}\sim{\cal M}_{s}^{L}}[\tau]$ </p>
-    <p> $(X^{L},Y^{L})$ </p>
-    <p> $\ell=L$ </p>
-    <p> $Z^{L-\tau}$ </p>
-    <p> $x^{i}\in{\cal X}^{i}$ </p>
-    <p> ${\cal M}_{b}(\cdot|x^{n})=\text{Ber}(q),$ </p>
-    <p> $\forall\ell\leq L$ </p>
-    <p> $\tau,{\cal M}_{b,{\rm next}}$ </p>
-    <p> $L-1$ </p>
-    <p> $\beta(X^{L},Y^{L})=\tau$ </p>
-    <p> $\displaystyle=\sum_{\ell=0}^{L}\ell\cdot\Pr{\left({\tau=\ell}\right)}=\sum_{%
-\ell=1}^{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\prod_{i=1}^{\ell}\min\{{\cal M}_{%
-b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\},$ </p>
-    <p> $\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{\ell\leq L}\sum_{x^{%
-\ell}}\min\left\{\text{Pr}\left(X^{1:\ell}=x^{\ell}\right),\text{Pr}\left(Y^{1%
-:\ell}=x^{\ell}\right)\right\}$ </p>
-    <p> $\displaystyle=\Pr{\left({X^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr%
-{\left({Y^{i_{0}}=x^{i_{0}},\tau\leq i_{0}}\right)}\cdot p_{\rm res}^{x^{i_{0}%
-}}(x_{i_{0}+1})$ </p>
-    <p> $Y_{\tau+1}\neq X_{\tau+1}$ </p>
-    <p> $\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\prod_{i=1}^{\ell}\min\{{%
-\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}({\cal M}_{s}(%
-x_{\ell+1}\mid x^{\ell})-\min\{{\cal M}_{b}(x_{\ell+1}\mid x^{\ell}),{\cal M}_%
-{s}(x_{\ell+1}\mid x^{\ell})\})$ </p>
-    <p> ${\cal M}_{s}(\cdot\mid X^{i-1}),{\cal M}_{b}(\cdot\mid X^{i-1}),\forall i=0,%
-\ldots,L$ </p>
-    <p> $L=8$ </p>
-    <p> $\displaystyle\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}=\min\{\prod%
-_{i=1}^{\ell}{\cal M}_{s}(x_{i}\mid x^{i-1}),{\cal M}_{b}(x_{i}\mid x^{i-1})\},$ </p>
-    <p> ${\cal M}^{*}(\cdot\mid x^{t})$ </p>
-    <p> $L-\beta(X^{L},Y^{L})$ </p>
-    <p> $X_{1},\ldots,X_{L}\sim{\cal M}_{s}^{L}(\cdot)$ </p>
-    <p> $\forall y^{L-\tau}\in{\cal X}^{{L-\tau}}$ </p>
-    <p> $\Pr{\left({\tau=L}\right)}=\sum_{x^{L}\in{\cal X}^{L}}\prod_{i=1}^{L}\min\{{%
-\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}.$ </p>
-    <p> $\eta_{i}\sim U(0,1)$ </p>
-    <p> $\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})$ </p>
-    <p> $Y\sim{\cal M}_{b}$ </p>
-    <p> $L=4$ </p>
-    <p> $\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\geq\ell_{0}}\right%
-)}+\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau=\ell_{0}-1}\right)}$ </p>
-    <p> $\displaystyle=\Pr{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr%
-{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\leq i_{0}}\right)}$ </p>
-    <p> $L=1$ </p>
-    <p> $\ell=\ell_{0}-1$ </p>
-    <p> $\displaystyle p_{\rm rej}(x^{i})$ </p>
-    <p> $\displaystyle{:=}\sum_{x}{\left({{\cal M}_{b}(x^{i},x)-{\cal M}_{s}(x^{i},x)}%
-\right)}_{+},$ </p>
-    <p> $\displaystyle\;\;\;\Pr{\left({\tau=\ell}\right)}$ </p>
-    <p> $\displaystyle=\sum_{\ell=1}^{L}\Pr{\left({\tau\geq\ell}\right)}=\sum_{\ell=1}^%
-{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}%
-\right)}$ </p>
-    <p> $\displaystyle=\min\{\sum_{x\in{\cal X}}\min\{{\cal M}_{b}(x^{\ell_{0}-1},x),{%
-\cal M}_{s}(x^{\ell_{0}-1},x)\}+p_{\rm remain}(x^{\ell_{0}-1}),$ </p>
-    <p> $\ell>1$ </p>
-    <p> $\forall\ell\leq L-1,x^{\ell}\in{\cal X}^{\ell}$ </p>
-    <p> $x^{t}\in{\cal X}^{t}$ </p>
-    <p> $\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}\right)}=\min\{{\cal M}_{b}(x^{\ell}%
-),{\cal M}_{s}(x^{\ell})\}.$ </p>
-    <p> $\displaystyle{:=}\sum_{x}{\left({{\cal M}_{s}(x^{i},x)-{\cal M}_{b}(x^{i},x)}%
-\right)}_{+}.$ </p>
-    <p> $Y^{L}=(X^{\tau},Z^{L-\tau})$ </p>
-    <p> $\displaystyle\sum_{y^{L}}\pi(x^{L},y^{L})$ </p>
-    <p> $(x)_{+}=\max\{x,0\}.$ </p>
-    <p> ${\cal M}_{b}(\cdot\mid X^{\tau},Y)$ </p>
-    <p> $\displaystyle={\cal M}_{s}(x^{L})\cdot\min\left\{1,\frac{{\cal M}_{b}(x^{L})}{%
-{\cal M}_{s}(x^{L})}\right\}$ </p>
-    <p> ${\cal M}^{L}_{s}(x^{L})$ </p>
-    <p> $(x_{i},\ldots,x_{j})$ </p>
-    <p> $0\leq\tau\leq L$ </p>
-    <p> $\pi\in\Pi({\cal M}^{L}_{s},{\cal M}^{L}_{b})$ </p>
-    <p> $\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\text{Pr}_{X^{L},Y^{L}}\left(X^{%
-1:\ell}=Y^{1:\ell}=x^{\ell},\beta(X^{L},Y^{L})\geq\ell\right)$ </p>
-    <p> $X^{L}$ </p>
-    <p> $Y_{\tau+1}$ </p>
-    <p> $x^{L}\in{\cal X}^{L}$ </p>
-    <p> $\textsc{Verify}_{\pi}$ </p>
-    <p> ${\cal M}_{b,{\rm next}}^{L-\tau}$ </p>
-    <p> $\tau\leq L$ </p>
-    <p> $\min\{{\cal M}_{b}(x^{L}),{\cal M}_{s}(x^{L})\}$ </p>
-    <p> $\displaystyle=\prod_{i=1}^{\ell}\min\{1,\frac{{\cal M}_{b}(x_{i}\mid x^{i-1})}%
-{{\cal M}_{s}(x_{i}\mid x^{i-1})}\}{\left({1-\min\{1,\frac{{\cal M}_{b}(x_{%
-\ell+1}\mid x^{\ell})}{{\cal M}_{s}(x_{\ell+1}\mid x^{\ell})}\}}\right)}.$ </p>
-    <p> $x^{t}$ </p>
-    <p> $X_{i}\sim{\cal M}_{s}(\cdot\mid X^{i-1}).$ </p>
-    <p> $\tau=L-i$ </p>
-    <p> ${\rm E.O.S}\in X^{\tau}$ </p>
-    <p> $\displaystyle=\sum_{x^{L}\in{\cal X}^{L}}\Pr{\left({\tau=\ell,X^{L}=x^{L}}%
-\right)}$ </p>
-    <p> $\pi_{\rm token}$ </p>
-    <p> $\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\Pr{\left({\tau=\ell,X^{%
-\ell+1}=x^{\ell+1}}\right)}$ </p>
-    <p> $Z\sim{\cal M}_{b,{\rm next}}(\cdot)$ </p>
-    <p> $X^{\tau}$ </p>
-    <p> $(a)$ </p>
-    <p> $\ell<L$ </p>
-    <p> $\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\leq\ell_{0}-1}%
-\right)}\cdot\Pr{\left({X^{\ell_{0}-1}\text{ is accepted.}}\right)}$ </p>
-    <p> $Y_{1}=X_{1}$ </p>
-    <p> $\displaystyle=\min\left\{{\cal M}_{b}(x^{\ell_{0}-1}),{\cal M}_{s}(x^{\ell_{0}%
--1})\right\},$ </p>
-    <p> $\displaystyle=\text{Pr}_{\pi}{\left({\beta{{\color[rgb]{0,0,0}=\ell}}\mid X^{L%
-}}\right)}{:=}{{\color[rgb]{0,0,0}\frac{\sum_{\ell:\beta(X^{L},Y^{L})=\ell}\pi%
-(X^{L},Y^{L})}{\pi(X^{L})}}},$ </p>
-    <p> $Y_{2}$ </p>
-    <p> $\displaystyle\max_{\pi\in\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})}\mathbb{E}_{X^%
-{L},Y^{L}\sim\pi}\left[\beta(X^{L},Y^{L})\right]\leq\sum_{\tau=1}^{L}\sum_{x^{%
-\tau}\in{\cal X}^{\tau}}\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}$ </p>
-    <p> $\forall x^{n}$ </p>
-    <p> $\eta_{0}\leq\min\left\{\frac{{\cal M}_{b}(X^{L})}{{\cal M}_{s}(X^{L})},1\right\}$ </p>
-    <p> $O(|{\cal X}|^{L})$ </p>
-    <p> $Z^{L-\tau}\sim{\cal M}_{b,{\rm next}}$ </p>
-    <p> $L=6$ </p>
-    <p> ${\cal M}_{b}(Y_{3}\mid Y_{1},y)$ </p>
-    <p> $Y^{L}=(X^{\tau},Z^{i-\tau})$ </p>
-    <p> $\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\min\{{\cal M}_{s}^{\ell}(x^{%
-\ell}),{\cal M}_{b}^{\ell}(x^{\ell})\}.$ </p>
-    <p> $\displaystyle=\Pr{\left({X^{L}=x^{L}}\right)}\Pr{\left({\tau=L\mid X^{L}=x^{L}%
-}\right)}$ </p>
-    <p> $p_{\rm res}(x)=\frac{{\left({{\cal M}_{b}(x\mid X^{\tau})-{\cal M}_{s}(x\mid X%
-^{\tau})}\right)}_{+}}{\sum_{x^{\prime}}{\left({{\cal M}_{b}(x^{\prime}\mid X^%
-{\tau})-{\cal M}_{s}(x^{\prime}\mid X^{\tau})}\right)}_{+}},$ </p>
-    <p> ${\cal M}_{s}=\text{Ber}(1)$ </p>
-    <p> $\frac{{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}{\sum_{x^{L%
-}}{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}.$ </p>
-    <p> $i=0,1,\ldots,L$ </p>
-</body>
-</html>

htmls/output_mathjax_example_1000.html DELETED Viewed

@@ -1,132 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $M(T)_{\mu\nu}$ </p>
-    <p> $\gamma_{1}:1\succ 0$ </p>
-    <p> $(V_{j},V_{i})$ </p>
-    <p> $\Delta(N,N^{k,n}_{s})=2^{n-k-1}$ </p>
-    <p> $|Pa(N_{s},V_{n})|=k$ </p>
-    <p> $Pa(N_{s},V_{n})\cap Pa(N_{s^{\prime}},V_{n})=\emptyset$ </p>
-    <p> $\binom{2c+1}{c+1}=\binom{2c+1}{c}=\frac{2c+1}{c+1}\binom{2c}{c}=\frac{4c+2}{c+%
-1}\binom{2c-1}{c}$ </p>
-    <p> $2^{k-k^{\prime}+1}-2$ </p>
-    <p> $k^{\prime}=0$ </p>
-    <p> $(2^{n}-2^{n-k})\sum_{k^{\prime}=0}^{k}\binom{k}{k^{\prime}}\binom{n-k-1}{k-k^{%
-\prime}}\geq 3\cdot 2^{n-2}\binom{n-1}{k}\,.$ </p>
-    <p> $V\in Pa(N_{s},V_{n})$ </p>
-    <p> $N_{1},\ldots,N_{t}$ </p>
-    <p> $N^{a}$ </p>
-    <p> $(o,o^{\prime})$ </p>
-    <p> $(T^{n-1,n})_{n\geq 3}$ </p>
-    <p> $T^{n-1,n}$ </p>
-    <p> $o^{\prime}$ </p>
-    <p> $\binom{2d}{d}\leq 2^{2d-1}$ </p>
-    <p> $\mathcal{V^{\prime}}\subseteq\mathcal{V}\setminus{V_{n}}$ </p>
-    <p> $o\succ o^{\prime}$ </p>
-    <p> $\sum_{1\leq s\leq t}|\operatorname{CPT}(N_{s},V_{n})|$ </p>
-    <p> $2^{2k-k^{\prime}}$ </p>
-    <p> $o^{\prime}\succ o$ </p>
-    <p> $s\in\{1,\ldots,\binom{n-1}{k}2^{k}$ </p>
-    <p> $n^{\prime}=\max_{1\leq s\leq t}|Pa(N_{s},V_{n})|$ </p>
-    <p> $T_{\varepsilon}$ </p>
-    <p> $N_{\varepsilon}$ </p>
-    <p> $\binom{2d+2}{d+1}\leq 2^{2d+1}$ </p>
-    <p> $o[V_{j}]=o^{\prime}[V_{j}]=\gamma[V_{j}]$ </p>
-    <p> $CPT(N_{s},V_{n})$ </p>
-    <p> $\operatorname{Inst}(P)$ </p>
-    <p> $|\tau|$ </p>
-    <p> $V_{4}$ </p>
-    <p> $Pa(N_{1},V_{3})$ </p>
-    <p> $000$ </p>
-    <p> $\Leftarrow c\binom{2c-1}{c}\leq(2c+3)\cdot 2^{2c-4}$ </p>
-    <p> $(c+1)\binom{2c+1}{c+1}\leq(c+1)\cdot 2^{2c-1}+\binom{2c}{c}$ </p>
-    <p> $2^{k}-2^{k-k^{\prime}}$ </p>
-    <p> $\gamma:0\succ 1$ </p>
-    <p> $\Leftarrow c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}$ </p>
-    <p> $4c\binom{2c-1}{c}+2\cdot\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-1}+2\cdot\binom{2c%
--1}{c}$ </p>
-    <p> $f_{T^{k,n}}(N^{k,n}_{s})=(2^{n}-2^{n-k})\sum_{k^{\prime}=0}^{k}\binom{k}{k^{%
-\prime}}\binom{n-k-1}{k-k^{\prime}}$ </p>
-    <p> $\mathcal{F}_{bad}=(T_{n})_{n\in\mathbb{N}}$ </p>
-    <p> $o[\{V_{1},V_{2}\}]$ </p>
-    <p> $2\cdot\binom{2c-1}{c}$ </p>
-    <p> $Pa(N^{k,n}_{s},V_{n})\cup P$ </p>
-    <p> $T_{n}=(N^{n}_{1},\ldots,N^{n}_{n-1})$ </p>
-    <p> $s\in\{1,\ldots,t\}$ </p>
-    <p> $N^{k,n}_{2}$ </p>
-    <p> $\Leftarrow 2^{n-2c}\cdot c\binom{2c-1}{c}\leq(2c+2)\cdot 2^{n-4}$ </p>
-    <p> $2^{|\mathcal{U}|}\prod_{s=1}^{t}2^{p_{s}-1}$ </p>
-    <p> $2^{n-k-1}$ </p>
-    <p> $\displaystyle\ \ +2^{k}-2^{k-k^{\prime}}\binom{k}{k^{\prime}}\binom{n-k-1}{k^{%
-\prime}}2^{n-k}\big{]}$ </p>
-    <p> $|\mathcal{V}^{\prime}|=k$ </p>
-    <p> $f_{T}(N)$ </p>
-    <p> $f_{T}(N^{a})>f_{T}(N)$ </p>
-    <p> $p_{s}=|Pa(N_{s},V_{n})|$ </p>
-    <p> $\operatorname{Inst}(\mathcal{V}^{\prime})$ </p>
-    <p> $\gamma:1\succ 0$ </p>
-    <p> $P\setminus Pa(N^{k,n}_{s},V_{n})$ </p>
-    <p> $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})=2^{n-k}-2^{n-2k}$ </p>
-    <p> $\kappa\leq t-\kappa$ </p>
-    <p> $2^{k-k^{\prime}}$ </p>
-    <p> $\operatorname{CPT}(N^{a},V_{n})$ </p>
-    <p> $O(\sum_{1\leq s\leq t}|CPT(N_{s},V_{n})|)$ </p>
-    <p> $f_{T}(N^{a})\leq f_{T}(N)$ </p>
-    <p> $\Delta(N,N_{s})$ </p>
-    <p> $O(2^{|P|}\cdot\sum_{1\leq s\leq t}|CPT(N_{s},V_{n})|)$ </p>
-    <p> $V=\{V_{1},V_{2},V_{3}\}$ </p>
-    <p> $T^{2,3}$ </p>
-    <p> $N_{i}\in T$ </p>
-    <p> $\sum_{\kappa=0}^{\lfloor\frac{t}{2}\rfloor}\kappa\cdot\binom{t}{\kappa}$ </p>
-    <p> $\mathcal{V}^{\prime}$ </p>
-    <p> $freq_{M}(0\succ 1)=t\cdot 2^{n-1}-freq_{M}(1\succ 0)$ </p>
-    <p> $Pa(N^{k,n}_{s},V_{n})\cap P$ </p>
-    <p> $s\neq s^{\prime}$ </p>
-    <p> $2^{n-t-1}$ </p>
-    <p> $\frac{2\cdot(2d+1)}{d+1}$ </p>
-    <p> $2^{n-k}$ </p>
-    <p> $f_{T^{\prime}}(N)=\begin{cases}2\cdot\sum_{\kappa=0}^{c-1}\kappa\binom{2c-1}{%
-\kappa}&\text{if }t=2c-1\\
-2\cdot\sum_{\kappa=0}^{c-1}\kappa\binom{2c}{\kappa}+c\binom{2c}{c}&\text{if }t%
-=2c\\
-\end{cases}$ </p>
-    <p> $N_{4}$ </p>
-    <p> $\Leftarrow c\binom{2c}{c}\leq(2c+3)\cdot 2^{2c-3}$ </p>
-    <p> $freq_{M}(1\succ 0)=\sum\nolimits_{0\leq\mu<2^{n-1}}\sum\nolimits_{1\leq\nu\leq
-t%
-}M_{\mu\nu}$ </p>
-    <p> $\mathcal{F}_{bad}$ </p>
-    <p> $f_{T}(N_{s})=\min\{f_{T}(N_{s^{\prime}})\mid 1\leq s^{\prime}\leq t\}\leq\frac%
-{4}{3}f_{T}(N)$ </p>
-    <p> $N^{k,n}_{s}$ </p>
-    <p> $2^{k}\binom{n-k-1}{k}$ </p>
-    <p> $2^{n-2}$ </p>
-    <p> $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})$ </p>
-    <p> $freq_{M}(1\succ 0)$ </p>
-    <p> $o[V_{i}]=0$ </p>
-    <p> $f_{T^{k,n}}(N)=2^{n-1}\binom{n-1}{k}$ </p>
-    <p> $f_{T^{k,n}}(N^{k,n}_{s})$ </p>
-    <p> $\{V_{1},V_{2}\}$ </p>
-    <p> $(T^{k,n})_{k,n}$ </p>
-    <p> $1\leq s\leq t=2c$ </p>
-    <p> $\Leftarrow 2^{n-2c-1}\cdot c\binom{2c}{c}\leq(2c+3)\cdot 2^{n-4}$ </p>
-    <p> $1\leq k^{\prime}\leq k-1$ </p>
-    <p> $P:=\bigcup_{1\leq s\leq t}Pa(N_{s},V_{n})$ </p>
-    <p> $t=n-1$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10000.html DELETED Viewed

@@ -1,123 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $0.058\scriptscriptstyle\pm\scriptstyle.010$ </p>
-    <p> $0.027\scriptscriptstyle\pm\scriptstyle.004$ </p>
-    <p> $\mu_{c}^{\mathcal{D}_{\text{test}}}$ </p>
-    <p> $0.146\scriptscriptstyle\pm\scriptstyle.002$ </p>
-    <p> $0.124\scriptscriptstyle\pm\scriptstyle.006$ </p>
-    <p> $0.038\scriptscriptstyle\pm\scriptstyle.003$ </p>
-    <p> $0.243\scriptscriptstyle\pm\scriptstyle.002$ </p>
-    <p> $0.338\scriptscriptstyle\pm\scriptstyle.005$ </p>
-    <p> $\mathbf{0.043\scriptscriptstyle\pm\scriptstyle.000}$ </p>
-    <p> $\mathbf{0.010\scriptscriptstyle\pm\scriptstyle.002}$ </p>
-    <p> $1769$ </p>
-    <p> $\mathbf{0.077\scriptscriptstyle\pm\scriptstyle.005}$ </p>
-    <p> $0.045\scriptscriptstyle\pm\scriptstyle.004$ </p>
-    <p> $0.152\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $\mathbf{0.042\scriptscriptstyle\pm\scriptstyle.003}$ </p>
-    <p> $0.025\scriptscriptstyle\pm\scriptstyle.000$ </p>
-    <p> $0.081\scriptscriptstyle\pm\scriptstyle.007$ </p>
-    <p> $0.066\scriptscriptstyle\pm\scriptstyle.005$ </p>
-    <p> $0.013\scriptscriptstyle\pm\scriptstyle.006$ </p>
-    <p> $0.781\scriptscriptstyle\pm\scriptstyle.003$ </p>
-    <p> $0.021\scriptscriptstyle\pm\scriptstyle.003$ </p>
-    <p> $0.012\scriptscriptstyle\pm\scriptstyle.000$ </p>
-    <p> $0.135\scriptscriptstyle\pm\scriptstyle.014$ </p>
-    <p> $0.228\scriptscriptstyle\pm\scriptstyle.007$ </p>
-    <p> $0.069\scriptscriptstyle\pm\scriptstyle.035$ </p>
-    <p> $0.228\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $0.065\scriptscriptstyle\pm\scriptstyle.012$ </p>
-    <p> $0.078\scriptscriptstyle\pm\scriptstyle.007$ </p>
-    <p> $\mathcal{D}_{\text{Test}}$ </p>
-    <p> $0.153\scriptscriptstyle\pm\scriptstyle.002$ </p>
-    <p> $0.077\scriptscriptstyle\pm\scriptstyle.010$ </p>
-    <p> $0.332\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $0.360\scriptscriptstyle\pm\scriptstyle.004$ </p>
-    <p> $0.083\scriptscriptstyle\pm\scriptstyle.002$ </p>
-    <p> $\mathbf{0.033\scriptscriptstyle\pm\scriptstyle.003}$ </p>
-    <p> $0.081\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $0.045\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $0.046\scriptscriptstyle\pm\scriptstyle.002$ </p>
-    <p> $0.018\scriptscriptstyle\pm\scriptstyle.007$ </p>
-    <p> $\mathbf{0.051\scriptscriptstyle\pm\scriptstyle.024}$ </p>
-    <p> $\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.003}$ </p>
-    <p> $0.028\scriptscriptstyle\pm\scriptstyle.006$ </p>
-    <p> $0.042\scriptscriptstyle\pm\scriptstyle.000$ </p>
-    <p> $0.149\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $0.409\scriptscriptstyle\pm\scriptstyle.005$ </p>
-    <p> $0.065\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $0.108\scriptscriptstyle\pm\scriptstyle.003$ </p>
-    <p> $0.028\scriptscriptstyle\pm\scriptstyle.009$ </p>
-    <p> $0.133\scriptscriptstyle\pm\scriptstyle.007$ </p>
-    <p> $0.166\scriptscriptstyle\pm\scriptstyle.003$ </p>
-    <p> $\mathbf{0.034\scriptscriptstyle\pm\scriptstyle.001}$ </p>
-    <p> $\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.001}$ </p>
-    <p> $0.033\scriptscriptstyle\pm\scriptstyle.006$ </p>
-    <p> $0.117\scriptscriptstyle\pm\scriptstyle.002$ </p>
-    <p> $0.083\scriptscriptstyle\pm\scriptstyle.024$ </p>
-    <p> $0.052\scriptscriptstyle\pm\scriptstyle.001$ </p>
-    <p> $0.321\scriptscriptstyle\pm\scriptstyle.002$ </p>
-    <p> $0.058\scriptscriptstyle\pm\scriptstyle.004$ </p>
-    <p> $\mathcal{SRE}$ </p>
-    <p> $\mathcal{SR}$ </p>
-    <p> $\mathbf{PP}(p_{i})=\frac{1}{Length(\overline{p^{e}_{i}\Phi(p^{e}_{i})})+dist^{%
-M(X)}(center,\Phi(p^{e}_{i})},$ </p>
-    <p> $Crosswise$ </p>
-    <p> $Sb_{i}$ </p>
-    <p> $\mathbf{V}_{1},\mathbf{V}_{2},\mathbf{V}_{3},\mathbf{V}_{4}$ </p>
-    <p> $\mathbf{X}\subset\mathbb{R}^{2}$ </p>
-    <p> $p_{1},p_{2}\leftarrow$ </p>
-    <p> $M(\mathbf{X})$ </p>
-    <p> $E_{area}$ </p>
-    <p> $\mathbf{I}=\{I_{i}\},i=0,\dots,N_{I}$ </p>
-    <p> $\mathbf{A}^{c}=\{ac_{k}\}$ </p>
-    <p> $center=\operatorname*{arg\,max}_{m\in M(X)}\;CR(v_{i},v_{j})\;\mid v_{i},v_{j}%
-\in\pi(m),$ </p>
-    <p> $\pi(z)=\{z\in\partial\mathbf{X}:\|z-x\|=D(\mathbf{X})(z)\}$ </p>
-    <p> $p^{\prime}(x^{\prime},y^{\prime})=\zeta(p(x,y)),$ </p>
-    <p> $\mathbf{K_{i}}$ </p>
-    <p> $p(x,y)\in a_{k}$ </p>
-    <p> $p_{triangle}$ </p>
-    <p> $D(\mathbf{X})(z)=\inf_{x\in\partial\mathbf{X}}\|z-x\|,$ </p>
-    <p> $4^{S-1}$ </p>
-    <p> $Sb_{i}^{*}$ </p>
-    <p> $M(\mathbf{X})=\{z\in\mathbf{X}:\lvert\pi(z)\rvert>1\}.$ </p>
-    <p> $P_{\mathbf{X}}$ </p>
-    <p> $x\in\pi(z)$ </p>
-    <p> $D(\mathbf{X}):\mathbb{R}^{2}\mapsto\mathbb{R}$ </p>
-    <p> $\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{H}_{3},\mathbf{H}_{4}$ </p>
-    <p> $z\in\mathbb{R}^{2}$ </p>
-    <p> $\zeta(.)$ </p>
-    <p> $4^{1}\cdot 8/2$ </p>
-    <p> $Crosswise(z)$ </p>
-    <p> $\mathbf{D_{\mathbf{T}}}$ </p>
-    <p> $E_{area}=\sum_{i=1}^{S_{T}}Area(Sb_{i}^{*})$ </p>
-    <p> $\mathbf{\mathcal{O}}^{*}=\operatorname*{arg\,max}_{\mathbf{D_{i}},\mathbf{K_{i%
-}}}\;E_{area},$ </p>
-    <p> $[\;\;]$ </p>
-    <p> $\mathbf{A}^{i}=\{a_{k}\},k=1,\dots,8$ </p>
-    <p> $\Phi\big{(}z\big{)}$ </p>
-    <p> $M_{o}=\dfrac{P_{o}}{P_{\mathbf{X}}},$ </p>
-    <p> $\tau_{p}=0.75$ </p>
-    <p> $Axial$ </p>
-    <p> $M_{n}=\dfrac{1}{N}\sum_{i}\|(L_{i}-L_{ci})\|,$ </p>
-    <p> $Sb^{*}$ </p>
-    <p> $Sb_{i}=[bx_{1},by_{1},bx_{2},by_{2}]$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10001.html DELETED Viewed

@@ -1,131 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\mathfrak{L}_{T}$ </p>
-    <p> $\tau_{e}=2$ </p>
-    <p> $ct\leftarrow G.centroid$ </p>
-    <p> $O(4^{n})$ </p>
-    <p> $dividing\_line\leftarrow$ </p>
-    <p> $\tau_{e}=3$ </p>
-    <p> $4^{3}\cdot n/4$ </p>
-    <p> $\mathfrak{L}_{\mathbf{T}}$ </p>
-    <p> $E_{area}=\sum_{i=1}^{S_{T}}(Area(Sb_{i}^{*})\cdot p_{triangle}(\mathbf{G})),$ </p>
-    <p> $N_{I}>N_{p}$ </p>
-    <p> $N_{I}>>N_{p}$ </p>
-    <p> $\mathfrak{R}_{\mathbf{T}}$ </p>
-    <p> $\mathbf{G_{\mathbf{T}}}$ </p>
-    <p> $\mathbf{S}_{i}=\big{[}N_{I}\cdot\dfrac{Area(p_{i})}{Area(\mathbf{X})}\big{]},$ </p>
-    <p> $slope\leftarrow Axial(ct)$ </p>
-    <p> $Sb$ </p>
-    <p> $N_{c}=N_{I}$ </p>
-    <p> $CR(v_{i},v_{j})=dist^{B}(v_{i},v_{j})-Length(\overline{v_{i}v_{j}}),$ </p>
-    <p> $\Phi(p^{e}_{i})$ </p>
-    <p> $\mathbf{\mathcal{O}}^{*}$ </p>
-    <p> $Axial(z)$ </p>
-    <p> $\|\;\;\|$ </p>
-    <p> $\mathbf{G}_{T}$ </p>
-    <p> $dist^{B}$ </p>
-    <p> $\mathbf{G}\leftarrow p_{2}$ </p>
-    <p> $\bigcup_{i}S_{i}$ </p>
-    <p> $\mathbb{R}^{2}\setminus\mathbf{X}$ </p>
-    <p> $\mathbf{\mathcal{O}}$ </p>
-    <p> $z\in\mathbf{X}$ </p>
-    <p> $M_{a}=\dfrac{|\bigcup_{i}S_{i}|}{P_{\mathbf{X}}},$ </p>
-    <p> $\tau_{e}=1$ </p>
-    <p> $slope$ </p>
-    <p> $dividing\_line$ </p>
-    <p> $p_{triangle}(polygon)=\begin{cases}0.8&polygon\text{ is a triangle}\\
-1.0&\text{otherwise.}\end{cases}$ </p>
-    <p> $8/2$ </p>
-    <p> $R^{m}_{i}$ </p>
-    <p> $S_{i}-1$ </p>
-    <p> $p_{1},p_{2}\leftarrow G$ </p>
-    <p> $dist^{M(X)}$ </p>
-    <p> $S_{T}-1$ </p>
-    <p> $4^{1}\cdot n/2$ </p>
-    <p> $M_{c}=\dfrac{P_{w}}{P_{\mathbf{X}}},$ </p>
-    <p> $\gamma^{u}$ </p>
-    <p> $4^{2^{\tau_{e}}-1}\cdot n/2^{\tau_{e}}$ </p>
-    <p> $\Phi(z,M(\mathbf{X}))=\operatorname*{arg\,min}_{m\in M(\mathbf{X})}\|z-m\|$ </p>
-    <p> $ac_{k}$ </p>
-    <p> $L_{ci}$ </p>
-    <p> $M_{s}=1-\dfrac{|\bigcup_{i}S_{i}|}{\sum_{i}|S_{i}|}.$ </p>
-    <p> $\mathfrak{R}_{T}$ </p>
-    <p> $slope\leftarrow Crosswise(ct)$ </p>
-    <p> $p^{\prime}\in ac_{k}$ </p>
-    <p> $\mathbf{G}\leftarrow p_{1}$ </p>
-    <p> $\mathbf{P}=\{p_{1},\dots,p_{N_{p}}\}$ </p>
-    <p> $p^{e}_{i}$ </p>
-    <p> $Sb=[bx_{1},by_{1},bx_{2},by_{2}]$ </p>
-    <p> $\mathcal{T}_{t}^{(d)}(m)$ </p>
-    <p> $\mathcal{T}_{r}^{(d)}(m)=\lVert(\mathcal{T}^{(d)}(s,m))_{s\in\mathcal{M},s\neq
-t%
-}\rVert_{2}$ </p>
-    <p> $\mathcal{T}_{t}^{(d)}(m)=\lVert(\mathcal{T}^{(d)}(m,t))_{t\in\mathcal{M},t\neq
-s%
-}\rVert_{2}$ </p>
-    <p> $\alpha=\frac{2}{255}$ </p>
-    <p> $\displaystyle x^{t+1}=\prod_{x+\mathcal{B}}(x^{t}+\alpha\text{sgn}(\nabla_{x}L%
-(\theta,x,y)))$ </p>
-    <p> $\mathcal{T}_{r}^{(d)}(m)$ </p>
-    <p> $\mathcal{T}^{(d)}(s,t)$ </p>
-    <p> $\mathcal{A}_{s}^{(d)}=\{(x,y)\}$ </p>
-    <p> $\mathcal{A}_{s}^{(d)}$ </p>
-    <p> $(s,t)\in\mathcal{M}^{2}$ </p>
-    <p> $(7,308+4,542+7,517)*6=116,202$ </p>
-    <p> $\mathcal{T}^{(d)}(s,t)=\frac{|\{(x,y)\in\mathcal{A}_{s}^{(d)};\hat{y}_{t}\neq y%
-\}|}{|\mathcal{A}_{s}^{(d)}|}$ </p>
-    <p> $N_{e}=100$ </p>
-    <p> $l^{\eta}_{i,j\in B}$ </p>
-    <p> $BNM$ </p>
-    <p> $L^{\textit{AttnMask}}_{DM}=L_{DM}(\mathcal{M}\odot x,\mathcal{M}\odot\tilde{x}),$ </p>
-    <p> $[v^{*},\ldots,v^{\&}]:=\operatorname*{arg\,min}_{\mathcal{V}}E_{x_{0},{%
-\epsilon}\sim N(0,I)}\\
-\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,[c_{\theta}(y),v^{*},\ldots,v^{\&}]\|%
-^{2}$ </p>
-    <p> $V=f_{V}(v)$ </p>
-    <p> $v^{\eta}_{i}$ </p>
-    <p> $[y,p^{*}]$ </p>
-    <p> $v^{\eta}_{j}$ </p>
-    <p> $L_{PromptCL}$ </p>
-    <p> $\overline{M}^{p}=1/T\sum_{t=1}^{T}M_{t}^{p}$ </p>
-    <p> $\{v^{*},\ldots,v^{\&}\}$ </p>
-    <p> $v^{*}=c_{\theta}(p^{*})$ </p>
-    <p> $L_{DM}=L_{DM}(x,\tilde{x}):=E_{x_{0},{\epsilon}\sim N(0,I),t\sim\text{Uniform}%
-(1,T)}\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,c_{\phi}(p))\|^{2},$ </p>
-    <p> $6100$ </p>
-    <p> $L=L^{\textit{AttnMask}}_{DM}+\gamma L_{PromptCL}^{adj},$ </p>
-    <p> $M=\text{Softmax}(QK^{T}/\sqrt{d})$ </p>
-    <p> $step=1,\ldots,S$ </p>
-    <p> $\{v^{\&}\}$ </p>
-    <p> $(0.2,0.0005)$ </p>
-    <p> $\mathcal{V}=[v^{*},\ldots,v^{\&}]$ </p>
-    <p> $(p^{*},v^{*})$ </p>
-    <p> $(\mathcal{P},\mathcal{V})$ </p>
-    <p> ${\bm{\epsilon}}\sim\mathcal{N}(\mathbf{0},\textbf{I})$ </p>
-    <p> $L_{PromptCL}^{adj}$ </p>
-    <p> $v=c_{\phi}(p)$ </p>
-    <p> $\eta\in MN$ </p>
-    <p> $L^{\textit{AttnMask}}_{DM}$ </p>
-    <p> $(0.3,0.00075)$ </p>
-    <p> $Q=f_{Q}(z)$ </p>
-    <p> $\mathcal{P}=[p^{*},\ldots,{p}^{\&}]$ </p>
-    <p> $\mathcal{M}=\bigcup_{p\in\mathcal{P}}B(M^{p})$ </p>
-    <p> $\{v^{*}\}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10002.html DELETED Viewed

@@ -1,155 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $K=f_{K}(v)$ </p>
-    <p> $sim(v_{i},v_{j})=v_{i}^{T}.v_{j}/||v_{i}||||v_{j}||$ </p>
-    <p> $B(M^{p}):=\{1\text{ if }M^{p}>k,0\text{ otherwise}\}$ </p>
-    <p> $v^{*}:=\operatorname*{arg\,min}_{v}E_{x_{0},{\epsilon}\sim N(0,I)}\\
-\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,[c_{\theta}(y),v^{*}]\|^{2}$ </p>
-    <p> $\{v_{b}^{n}\}_{b=1}^{B},_{n=1}^{N}$ </p>
-    <p> $\begin{gathered}L_{PromptCL}=\frac{1}{N}\frac{1}{B}\sum_{\eta=1}^{N}\sum_{i=1}%
-^{B}{l^{\eta}_{i,j\in B}},\qquad L_{PromptCL}^{adj}=\frac{1}{NM}\frac{1}{B}%
-\sum_{\eta=1}^{NM}\sum_{i=1}^{B}{l^{\eta}_{i,j\in B}}\end{gathered}$ </p>
-    <p> $(\tau,\gamma)$ </p>
-    <p> $[v^{*},\ldots,v^{\&}]=[c_{\theta}(p^{*}),\ldots,c_{\theta}(p^{\&})]$ </p>
-    <p> $[y,p^{*},\ldots,p^{\&}]$ </p>
-    <p> $c_{\phi}$ </p>
-    <p> $l^{\eta}_{i,j\in B}=-log(\frac{exp(sim(v^{\eta}_{i},v^{\eta}_{j}))/\tau}{\sum_%
-{\eta=1}^{N}\sum_{j=1,j\neq{i}}^{B}exp(sim(v^{\eta}_{i},v^{\eta}_{j})/\tau)})$ </p>
-    <p> $\eta\in N$ </p>
-    <p> $\tau=10^{-4}$ </p>
-    <p> $J^{\tau}(\mathbf{\chi}^{k+1},T)$ </p>
-    <p> $q_{1}=1,\ q_{2}=100$ </p>
-    <p> $d=0.1l$ </p>
-    <p> $\tilde{\chi}_{s}$ </p>
-    <p> $\Phi_{h}=(q_{1}-q_{2})G_{\tau}\ast(T_{h}-T^{\ast}_{h})+\gamma\sqrt{\frac{\pi}{%
-\tau}}G_{\tau}\ast(1-2\chi_{h})+(\kappa_{1}-\kappa_{2})G_{\tau}\ast(\frac{\xi}%
-{2}\nabla T_{h}\cdot\nabla T_{h}+\nabla T_{h}\cdot\nabla T_{h}^{\ast}).$ </p>
-    <p> $\kappa_{1}=5,\ 10,\ 15$ </p>
-    <p> $\tau=3\times 10^{-5}$ </p>
-    <p> $\Omega\in\mathbb{R}^{d}\ (d=2,3)$ </p>
-    <p> $\mathbf{\chi}_{h}$ </p>
-    <p> $\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$ </p>
-    <p> $y=1/2$ </p>
-    <p> $\Omega_{1}\subset\Omega$ </p>
-    <p> $(\ref{ad})$ </p>
-    <p> $\frac{q_{1}}{q_{2}}$ </p>
-    <p> $z=0,\ y=1/2$ </p>
-    <p> $\Phi_{h}$ </p>
-    <p> $T^{\ast}_{h}$ </p>
-    <p> $J^{\tau}(\chi,T)=\int_{\Omega}q(\chi)T\ d\textbf{x}+\frac{\xi}{2}\int_{\Omega}%
-\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}+\gamma\sqrt{\frac{\pi}{\tau}}%
-\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x},$ </p>
-    <p> $\displaystyle\ \ \ \ +\int_{\Omega}\kappa(\chi)\nabla T^{k}\cdot\nabla T^{*k}%
-\ d\textbf{x}-\int_{\Omega}q(\chi)T^{*k}\ d\textbf{x}.$ </p>
-    <p> $\displaystyle\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}=$ </p>
-    <p> $J^{\tau}(\tilde{\chi}_{s},\tilde{T}_{s})>J^{\tau}(\chi^{k},T^{k})$ </p>
-    <p> $\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100,\ \tau=1\times 10^{-4}$ </p>
-    <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\leq\sigma_{1},\chi^{k}(p)=0%
-\big{\}},$ </p>
-    <p> $\tilde{T}_{s}$ </p>
-    <p> $\kappa_{1}=40,\ \kappa_{2}=1$ </p>
-    <p> $\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)\in\arg\min_{\chi\in%
-\mathcal{H}}\tilde{J}^{\tau,k}(\chi).$ </p>
-    <p> $\tilde{B}\subset B$ </p>
-    <p> $A_{1}=\{p\in A:\phi_{A}(p)\leq\sigma_{1}\}\ \ \ \ B_{1}=\{p\in B:\phi_{B}(p)%
-\leq\sigma_{2}\}$ </p>
-    <p> $H^{1}_{\Gamma_{D}}(\Omega)=\{v\in H^{1}(\Omega)\ |\ v|_{\Gamma_{D}}=0\}$ </p>
-    <p> $(\ref{and})$ </p>
-    <p> $\displaystyle\left\{\begin{aligned} -\nabla\cdot(\kappa(\chi^{k})\nabla T)-q(%
-\chi^{k})&=0,\ \ &&\rm in\ \ \Omega,\\
-T&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
-\kappa(\chi^{k})\nabla T\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N},\end{%
-aligned}\right.$ </p>
-    <p> $\displaystyle\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)$ </p>
-    <p> $\sigma_{1}=\tilde{\phi}_{A}^{k}(n_{s})$ </p>
-    <p> $tol>0$ </p>
-    <p> $\displaystyle\int_{\Omega}(q(\chi^{k})T^{k})\ d\textbf{x}+\frac{\xi}{2}\int_{%
-\Omega}\kappa(\chi^{k})\nabla T^{k}\cdot\nabla T^{k}\ d\textbf{x}+\gamma\sqrt{%
-\frac{\pi}{\tau}}\int_{\Omega}\chi^{k}G_{\tau}\ast(1-\chi^{k})\ d\textbf{x}.$ </p>
-    <p> $T\in Q(\chi)$ </p>
-    <p> $\frac{\xi}{2}\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$ </p>
-    <p> $G_{\tau}=\frac{1}{(4\pi\tau)^{d/2}}\exp\left(-\frac{|\textbf{x}|^{2}}{4\tau}\right)$ </p>
-    <p> $\int_{\Omega}\chi^{k+1}\ d\textbf{x}=V_{0}$ </p>
-    <p> $\chi^{k+1}=\tilde{\chi}_{s}$ </p>
-    <p> $\mathbb{R}^{d}\setminus\overline{\Omega}$ </p>
-    <p> $T^{*k}$ </p>
-    <p> $\displaystyle\tilde{B}_{1}$ </p>
-    <p> $\Omega_{1}\in\Omega$ </p>
-    <p> $\Gamma\colon=\Gamma_{D}\cup\Gamma_{N}$ </p>
-    <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma,\chi^{k}(p)=1\big{\}}.$ </p>
-    <p> $\displaystyle\chi^{k+1}=\begin{cases}1&\ \textrm{if}\ \Phi^{k}\leq\sigma,\\
-0&\ \textrm{otherwise}.\end{cases}$ </p>
-    <p> $\frac{\delta\tilde{J}^{\tau}}{\delta T}=0,\ \ \ \frac{\delta\tilde{J}^{\tau}}{%
-\delta T^{*}}=0.$ </p>
-    <p> $\chi^{k+1}=\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)$ </p>
-    <p> $\tilde{\chi}_{s}=\chi_{A_{1}}+\chi^{k}-\chi_{B_{1}}$ </p>
-    <p> $\kappa_{1},\kappa_{2},q_{1},q_{2}$ </p>
-    <p> $T^{\ast}_{h}\in V_{h}^{0}$ </p>
-    <p> $\Phi^{k}=(q_{1}-q_{2})G_{\tau}\ast(T^{k}-T^{*k})+\gamma\sqrt{\frac{\pi}{\tau}}%
-G_{\tau}\ast(1-2\chi^{k})+(k_{1}-k_{2})G_{\tau}\ast(\frac{\xi}{2}\nabla T^{k}%
-\cdot\nabla T^{k}+\nabla T^{k}\cdot\nabla T^{\ast k})$ </p>
-    <p> $\displaystyle\tilde{J}^{\tau,k}(\chi)$ </p>
-    <p> $\displaystyle\int_{\Omega}\left[G_{\tau/2}\ast\chi\right]\left[G_{\tau/2}\ast(%
-1-\chi)\right]\ d\textbf{x}$ </p>
-    <p> $\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100$ </p>
-    <p> $\kappa_{1}=40,\kappa_{2}=1$ </p>
-    <p> $\int_{\Omega}\chi\ d\textbf{x}=V_{0}$ </p>
-    <p> $T^{k+1}$ </p>
-    <p> $\chi^{0}$ </p>
-    <p> $\tilde{\phi}_{A}^{k}$ </p>
-    <p> $\displaystyle\tilde{A}_{2}$ </p>
-    <p> $\chi^{k+1}=Proj_{[0,1]}\left(\chi^{k}-s\left.\frac{\delta J^{\tau}}{\delta\chi%
-}\right|_{\chi^{k}}\right),$ </p>
-    <p> $\kappa(\chi)$ </p>
-    <p> $0.1l$ </p>
-    <p> $\displaystyle\chi^{k+1}=\chi_{A}+\chi^{k}-\chi_{B}.$ </p>
-    <p> $J^{\tau}(\chi^{k+1},T^{k+1})\leq J^{\tau}(\chi^{k+1},T^{k}),$ </p>
-    <p> $\kappa_{2}=1$ </p>
-    <p> $\displaystyle\kappa(\chi)=\kappa_{1}G_{\tau}\ast\chi+\kappa_{2}G_{\tau}\ast(1-%
-\chi),$ </p>
-    <p> $\tilde{J}^{\tau}(\chi,T,T^{*})=J^{\tau}(\chi,T)+\int_{\Omega}(-\nabla\cdot(%
-\kappa(\chi)\nabla T)-q(\chi))\cdot T^{*}d\textbf{x}.$ </p>
-    <p> $q_{2}=100$ </p>
-    <p> $\chi^{1},T^{1},T^{*1},\chi^{2},T^{2},T^{*2},\cdots,\chi^{k},T^{k},T^{*k},\cdots$ </p>
-    <p> $\tilde{J}^{\tau}(\mathbf{\chi}^{k+1},T^{k})\leq\tilde{J}^{\tau}(\mathbf{\chi}^%
-{k},T^{k}),$ </p>
-    <p> $J^{\tau}(\chi,T)$ </p>
-    <p> $\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa(\chi^{k})\nabla T^{*})&%
-=q(\chi^{k})-\xi(\nabla\cdot(\kappa(\chi^{k})\nabla T)),\ \ &&\rm in\ \ \Omega%
-,\\
-T^{*}&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
-\kappa(\chi^{k})\nabla T^{*}\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N}\end{%
-aligned}\right.$ </p>
-    <p> $J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi}^{k},T^{k})$ </p>
-    <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma_{2},\chi^{k}(p)=1\big{%
-\}},$ </p>
-    <p> $\chi^{k+1}\in\mathcal{B}$ </p>
-    <p> $\|\chi^{k+1}-\chi^{k}\|_{2}>tol$ </p>
-    <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\in(\sigma_{1},\sigma),\chi^{k}%
-(p)=0\big{\}},$ </p>
-    <p> $\kappa=\kappa_{1}\chi_{\Omega_{1}}+\kappa_{2}\chi_{\Omega_{2}}$ </p>
-    <p> $\displaystyle\ J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi%
-}^{k},T^{k}).$ </p>
-    <p> $\xi=1\times 10^{-5}$ </p>
-    <p> $600\times 600$ </p>
-    <p> $P_{1}(K)$ </p>
-    <p> $T^{k},T^{*k}$ </p>
-    <p> $\int_{\Omega}\chi^{k+1}(\textbf{x})d\textbf{x}=V_{0}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10003.html DELETED Viewed

@@ -1,169 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $J(\chi,T)$ </p>
-    <p> $\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}.$ </p>
-    <p> $n_{s}=N-\lfloor N*\theta^{s}\rfloor$ </p>
-    <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\leq\sigma,\chi^{k}(p)=0\big{\}},$ </p>
-    <p> $\kappa_{1}=5,\ 10,\ 20$ </p>
-    <p> $k_{1}/k_{2}$ </p>
-    <p> $\kappa_{1}=20,\ \kappa_{2}=1$ </p>
-    <p> $\chi^{k+1}=\arg\min_{\chi\in B}\tilde{J}^{\tau,k}(\chi).$ </p>
-    <p> $\chi^{k}$ </p>
-    <p> $\chi^{\ast}\in\mathcal{B}$ </p>
-    <p> $\mathcal{H}:=\big{\{}\chi\in BV(\Omega)\ |\ \chi(\textbf{x})\in[0,1],\int_{%
-\Omega}\chi\ d\textbf{x}=V_{0}\big{\}}.$ </p>
-    <p> $\displaystyle=\int_{\Omega}q(\chi)T^{k}\ d\textbf{x}+\frac{\xi}{2}\int_{\Omega%
-}\kappa(\chi)\nabla T^{k}\cdot\nabla T^{k}\ d\textbf{x}+\gamma\sqrt{\frac{\pi}%
-{\tau}}\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}$ </p>
-    <p> $\xi=1\times 10^{-7}$ </p>
-    <p> $\tilde{B}_{2}$ </p>
-    <p> $\displaystyle=(q(\chi_{h}),\varphi_{h}),\ \ \forall\varphi_{h}\in V_{h}^{0},$ </p>
-    <p> $J^{\tau}(\mathbf{\chi}^{k+1},T^{k})\leq J^{\tau}(\mathbf{\chi}^{k},T^{k})$ </p>
-    <p> $q_{1}=1,\ 20,\ 80$ </p>
-    <p> $q(\chi)$ </p>
-    <p> $\displaystyle\chi(\textbf{x}):=\left\{\begin{aligned} &1,\ \ \ \ \textrm{if}\ %
-\textbf{x}\in\ \Omega_{1},\\
-&0,\ \ \ \ \textrm{otherwise}.\end{aligned}\right.$ </p>
-    <p> $\begin{cases}&\min\limits_{\mathbf{\chi}\in\mathcal{B}}J^{\tau}(\mathbf{\chi},%
-T),\\
-&\textrm{s.t.}\ T\in Q(\chi),\end{cases}$ </p>
-    <p> $\tau=1\times 10^{-4}$ </p>
-    <p> $\ \tau>0$ </p>
-    <p> $\chi^{k+1}=\arg\min_{\chi\in\mathcal{H}}\mathcal{L}^{\tau,k}_{r^{k}}(\chi)=%
-\arg\min_{\chi\in\mathcal{H}}\int_{\Omega}\chi\Phi^{k}\ d\textbf{x},$ </p>
-    <p> $\chi_{1}.$ </p>
-    <p> $\displaystyle\chi^{k+1}=\chi_{\tilde{A}_{1}}+\chi^{k}-\chi_{\tilde{B}_{1}}.$ </p>
-    <p> $Q(\chi)$ </p>
-    <p> $\displaystyle\chi^{k+1}=$ </p>
-    <p> $\tilde{J}^{\tau,k}(\chi)\approx\tilde{J}^{\tau,k}(\chi^{k})+\mathcal{L}^{\tau,%
-k}_{\chi^{k}}(\chi-\chi^{k}),$ </p>
-    <p> $\chi^{k+1}$ </p>
-    <p> $\xi=1e-7$ </p>
-    <p> $\displaystyle\tilde{A}_{1}$ </p>
-    <p> $\Phi^{k}=(q_{1}-q_{2})G_{\tau}\ast(T^{k}-T^{*k})+\gamma\sqrt{\frac{\pi}{\tau}}%
-G_{\tau}\ast(1-2\chi^{k})+(k_{1}-k_{2})G_{\tau}\ast(\frac{\xi}{2}\nabla T^{k}%
-\cdot\nabla T^{k}+\nabla T^{k}\cdot\nabla T^{\ast k}).$ </p>
-    <p> $\chi^{k+1}(x)=\begin{cases}1\ \ \textrm{if}\ \Phi^{k}(x)\leq\sigma,\\
-0\ \ \textrm{otherwise,}\end{cases}$ </p>
-    <p> $\begin{cases}\dfrac{\delta(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{%
-\Omega}\chi\ d\textbf{x}-V_{0}))}{\delta\chi}(x)=\Phi^{k}-\sigma\leq 0\ \ \ %
-\textrm{if}\ \ \chi(x)=1,\\
-\dfrac{\delta(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{\Omega}\chi\ %
-d\textbf{x}-V_{0}))}{\delta\chi}(x)=\Phi^{k}-\sigma\geq 0\ \ \ \textrm{if}\ \ %
-\chi(x)=0,\\
-\dfrac{d(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{\Omega}\chi\ d%
-\textbf{x}-V_{0}))}{d\sigma}=\int_{\Omega}\chi\ d\textbf{x}-V_{0}=0.\end{cases}$ </p>
-    <p> $T_{h}\in V_{h}^{0}$ </p>
-    <p> $\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa\nabla T)+q&=0,\ \ &&\rm
-in%
-\ \Omega,\\
-(\kappa\nabla T)\cdot\mathbf{n}&=0,\ \ &&\rm on\ \Gamma_{N},\\
-T&=0,\ \ &&\rm on\ \Gamma_{D},\\
-|\Omega_{1}|&=\beta|\Omega|,\ \ &&\rm with\ a\ fixed\ parameter\ \beta\in(0,1)%
-,\end{aligned}\right.$ </p>
-    <p> $\beta=0.1,\ 0.2,\ 0.3$ </p>
-    <p> $\int_{\Omega}\chi\ d\textbf{x}$ </p>
-    <p> $\tilde{\phi}_{B}^{k}$ </p>
-    <p> $\displaystyle\int_{\Omega}\chi G_{\tau/2}\ast\left[G_{\tau/2}\ast(1-\chi)%
-\right]\ d\textbf{x}$ </p>
-    <p> $\chi_{\Omega_{i}}$ </p>
-    <p> $\kappa_{1}=40,\ \kappa_{2}=1,\gamma=20$ </p>
-    <p> $\displaystyle\left\{\begin{aligned} -\nabla\cdot(\kappa(\chi)\nabla T)-q(\chi)%
-&=0,\ \ &&\rm in\ \ \Omega,\\
-T&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
-\kappa(\chi)\nabla T\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N},\end{aligned%
-}\right.$ </p>
-    <p> $\chi_{h}\in\mathcal{B}_{h}$ </p>
-    <p> $\displaystyle J^{\tau}(\mathbf{\chi}^{k},T^{k})=$ </p>
-    <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\in(\sigma,\sigma_{2}),\chi^{k}%
-(p)=1\big{\}},$ </p>
-    <p> $\gamma=12,\ 15,\ 50$ </p>
-    <p> $|\Gamma|\approx\sqrt{\frac{\pi}{\tau}}\int_{\Omega}\chi G_{\tau}*(1-\chi)\ d%
-\textbf{x},$ </p>
-    <p> $J^{\tau}(\mathbf{\chi}^{k},T^{k})$ </p>
-    <p> $\tilde{A}\subset A$ </p>
-    <p> $q=q_{1}\chi_{\Omega_{1}}+q_{2}\chi_{\Omega_{2}}$ </p>
-    <p> $J^{\tau}(\chi^{k+1},T^{k+1})\leq J^{\tau}(\chi^{k},T^{k}).$ </p>
-    <p> $J^{\tau}(\tilde{\chi}_{s},\tilde{T}_{s})$ </p>
-    <p> $\displaystyle\tilde{B}_{2}$ </p>
-    <p> $q_{1}/q_{2}$ </p>
-    <p> $\chi^{k+1}=\arg\min_{\chi\in\mathcal{\mathcal{H}}}\tilde{J}^{\tau,k}(\chi),$ </p>
-    <p> $\tilde{J}^{\tau}(\chi,T^{k},T^{*k})$ </p>
-    <p> $BV(\Omega)$ </p>
-    <p> $G_{\tau}\ast$ </p>
-    <p> $\gamma=15$ </p>
-    <p> $\kappa_{1}=10,\ \kappa_{2}=1$ </p>
-    <p> $60\times 60\times 60$ </p>
-    <p> $\kappa_{1},\ \kappa_{2},\ q_{1},\ q_{2},\ \tau,\ \gamma,\ \xi$ </p>
-    <p> $\frac{\kappa_{1}}{\kappa_{2}}$ </p>
-    <p> $\displaystyle q(\chi)=q_{1}G_{\tau}\ast\chi+q_{2}G_{\tau}\ast(1-\chi).$ </p>
-    <p> ${0}\times[0.45,0.55]$ </p>
-    <p> $\tilde{J}^{\tau,k}(\chi)$ </p>
-    <p> $\displaystyle=\int_{\Omega}\chi\Phi^{k}\ d\textbf{x},$ </p>
-    <p> $\mathcal{B}:=\big{\{}\chi\in BV(\Omega)\ |\ \chi(\textbf{x})=\{0,1\},\int_{%
-\Omega}\chi\ d\textbf{x}=V_{0}\big{\}},$ </p>
-    <p> $\displaystyle=(q(\chi_{h}),\varphi_{h})+\xi(\kappa(\chi_{h})\nabla T_{h},%
-\nabla\varphi_{h}),\ \ \forall\varphi_{h}\in V_{h}^{0}.$ </p>
-    <p> $\mathbf{\chi}$ </p>
-    <p> $\displaystyle V_{h}=\{v\in H^{1}(\Omega)\ |\ v\in P_{1}(K),\forall K\in%
-\mathcal{T}_{h}\},\quad V_{h}^{0}=V_{h}\cap H^{1}_{\Gamma_{D}}(\Omega),$ </p>
-    <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma,\chi^{k}(p)=1\big{\}},$ </p>
-    <p> $\displaystyle(\kappa(\chi_{h})\nabla T_{h},\nabla\varphi_{h})$ </p>
-    <p> $\tau=0.35\times 10^{-4}$ </p>
-    <p> $\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa(\chi)\nabla T^{*})&=q(%
-\chi)-\xi(\nabla\cdot(\kappa(\chi)\nabla T)),\ \ &&\rm in\ \ \Omega,\\
-T^{*}&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
-\kappa(\chi)\nabla T^{*}\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N}.\end{%
-aligned}\right.$ </p>
-    <p> $\displaystyle-(\kappa(\chi_{h})\nabla T^{\ast}_{h},\nabla\varphi_{h})$ </p>
-    <p> $\sigma_{2}=\tilde{\phi}_{B}^{k}(n_{s})$ </p>
-    <p> $\chi^{1},\chi^{2},\cdots,\chi^{k+1},\cdots$ </p>
-    <p> $\chi^{k},~{}T^{k},~{}T^{*k}$ </p>
-    <p> $q_{1}=1,\ 40,\ 80$ </p>
-    <p> $\displaystyle\int_{\Omega}\left[G_{\tau/2}\ast\chi\right]\left[1-G_{\tau/2}%
-\ast\chi\right]\ d\textbf{x},$ </p>
-    <p> $G_{\tau/2}\ast$ </p>
-    <p> $Proj_{[0,1]}(v)=\begin{cases}v&{\rm if}\ v\in[0,1],\\
-0&{\rm if}\ v<0,\\
-1&{\rm if}\ v>1.\end{cases}$ </p>
-    <p> $\displaystyle\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)$ </p>
-    <p> $\kappa_{1}=40$ </p>
-    <p> $\Omega_{2}=\Omega\setminus\Omega_{1}\in\Omega$ </p>
-    <p> $\kappa_{1}/\kappa_{2}$ </p>
-    <p> $V_{0}=\beta|\Omega|$ </p>
-    <p> $J^{\tau}(\mathbf{\chi}^{k},T)$ </p>
-    <p> $\min_{(\Omega_{1},T)}J(\Omega_{1},T)=\int_{\Omega}qTd\textbf{x}+\gamma|\Gamma|,$ </p>
-    <p> $\displaystyle=\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{0})}{p%
-_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}%
-\mathbb{E}_{\hat{Z}_{i/N}\sim\int p_{v_{\theta}}(Z_{i/N}|Z_{1})q(Z_{1}|Z_{0})%
-dZ_{1}}$ </p>
-    <p> $Z_{7/8}$ </p>
-    <p> $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
-||g_{v_{\theta}}(Z_{t},t)-(\alpha_{t}[\log(b)-\log(a)]X^{S})||_{2}^{2}.$ </p>
-    <p> $\alpha_{1}X^{S}$ </p>
-    <p> $\forall Z_{1-i/K}$ </p>
-    <p> $q(\cdot|\cdot)$ </p>
-    <p> $\displaystyle\mathcal{L}_{\textrm{OFM-KT}}=\mathbb{E}_{(X^{S},Y)}[\frac{1}{N}%
-\sum_{i=0}^{N-1}$ </p>
-    <p> $\min L(g^{S}(X^{S}),g^{T}(X^{T}))$ </p>
-    <p> $\displaystyle\mathcal{L}_{\textrm{FM-KT}}=\mathbb{E}_{(X^{S},X^{T})}\frac{1}{N%
-}\sum_{i=1}^{N}||\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{\theta}}(Z_{1-i/%
-N},1-i/N))/-\nabla_{t}\sigma_{t})-X^{T}||_{2}^{2},$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10004.html DELETED Viewed

@@ -1,175 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $p_{v_{\theta}}(\cdot|\cdot)$ </p>
-    <p> $\displaystyle=\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)%
-}||g_{v_{\theta}}(Z_{t},t)-(\nabla_{t}\alpha_{t}X^{S}+\nabla_{t}\sigma_{t}X^{T%
-})||_{2}^{2}.$ </p>
-    <p> $\left\lfloor{\beta_{d}}B\right\rfloor$ </p>
-    <p> $\displaystyle Z_{t}=\alpha_{t}X^{S}+\sigma_{t}X^{T},\ s.t.\ \lim_{t\rightarrow
-0%
-}\alpha_{t}=0,\lim_{t\rightarrow 0}\sigma_{t}=1,\lim_{t\rightarrow 1}\sigma_{t%
-}=0.$ </p>
-    <p> $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
-||g_{v_{\theta}}(Z_{t},t)-((\frac{1}{2}a(1-t)+\frac{1}{2}b)\alpha_{t}X^{S}-%
-\frac{\alpha_{t}}{\sqrt{1-\alpha_{t}^{2}}}\alpha_{t}(\frac{1}{2}a(1-t)+\frac{1%
-}{2}b)X^{T})||_{2}^{2}.$ </p>
-    <p> $g^{S}(\cdot)$ </p>
-    <p> $\displaystyle+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{%
-\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the %
-ground truth label}}].$ </p>
-    <p> $\{X_{t}\}_{t}$ </p>
-    <p> $\displaystyle=\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)\mathcal{K}(1-i/%
-K).$ </p>
-    <p> $Z_{1-i/N}=Z_{1-(i-1)/N}-g_{v_{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N$ </p>
-    <p> $\mathcal{E}(Z_{1-i/K})$ </p>
-    <p> $\hat{Z}_{1}\sim\pi_{1}$ </p>
-    <p> $\displaystyle\mathcal{L}_{\textrm{FM-KT}^{\Theta}}=$ </p>
-    <p> $Z_{1}=\alpha_{1}X^{S}$ </p>
-    <p> $\frac{\sigma_{t}-\sigma_{t-\Delta t}}{t-\Delta t}$ </p>
-    <p> $\displaystyle-\log p_{v_{\theta}}(Z_{0})\leq\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}%
-\left[\log\frac{q(Z_{1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1%
-/N})}+\sum_{i=1}^{N}\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{%
-(i-1)/N}|Z_{i/N})}\right].$ </p>
-    <p> $\sigma_{t}\in\mathbb{R}^{+}$ </p>
-    <p> ${}_{\textrm{L}}$ </p>
-    <p> $\displaystyle\left[||q(Z_{(i-1)/N}|Z_{i/N},Z_{0})-p_{v_{\theta}}(Z_{(i-1)/N}|%
-\hat{Z}_{i/N})||_{2}^{2}\right],\quad s.t.\quad\textrm{Law}(Z_{i/N})\stackrel{%
-{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i/N}).$ </p>
-    <p> $\mathcal{L}_{\textrm{FM-KT}}$ </p>
-    <p> $X_{t}=tX^{S}+(1-t)X^{T}$ </p>
-    <p> $\displaystyle Z_{1-(i+1)/K}$ </p>
-    <p> $(X^{S},X^{T},Y)$ </p>
-    <p> $\mathit{a=0.02}$ </p>
-    <p> $\displaystyle\quad\quad-g_{v_{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N,\quad s.t.%
-\quad i\geq 1.$ </p>
-    <p> $\displaystyle\mathcal{E}(Z_{1-2/K})=\mathcal{E}(Z_{1-1/K})(1-(1/K)\psi(1-1/K))%
-+(1/K)\mathcal{K}(1-1/K)$ </p>
-    <p> $\{Z_{1-i/K}\}_{i=1}^{K}$ </p>
-    <p> $Z_{i/N}$ </p>
-    <p> $\displaystyle,X^{T})+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v%
-_{\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the %
-ground truth label (optional)}}],$ </p>
-    <p> $\frac{d\hat{Z}_{t}}{dt}=-g^{*}_{v_{\theta}}(\hat{Z}_{t},t)$ </p>
-    <p> $Z_{1-(i-1)/K}$ </p>
-    <p> $\sigma_{t}=1,\ s.t.\quad a=0.02,b=100$ </p>
-    <p> $\{dZ_{t-\Delta_{t}},dZ_{t-2\Delta_{t}},\cdots,dZ_{s}\}$ </p>
-    <p> $\mathit{b=0.1}$ </p>
-    <p> $\textrm{Law}(Z_{i/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i%
-/N})$ </p>
-    <p> $g_{v_{\theta}}(Z_{t},t)$ </p>
-    <p> $\displaystyle\mathcal{E}(Z_{0})=(1/K)\left[\sum_{i=0}^{K-1}\mathcal{K}(1-i/K)%
-\right]+(1/K^{2})\left[\sum_{j=1}^{K-1}\left[\psi(1-j/K)\left(\sum_{i=0}^{j-1}%
-\mathcal{K}(1-i/K)\right)\right]\right]+\mathcal{O}(1/K^{3}).$ </p>
-    <p> $\displaystyle\textrm{the sampling process:}\quad Z_{1-i/N}=Z_{1-(i-1)/N}-g_{v_%
-{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N,\quad s.t.\quad i\geq 1,$ </p>
-    <p> $\mathbb{E}_{\hat{Z}_{i/N},Z_{1},Z_{0}}D_{\mathrm{KL}}(q(Z_{(i-1)/N}|Z_{i/N},Z_%
-{0})||p_{v_{\theta}}(Z_{(i-1)/N}|\hat{Z}_{i/N}))$ </p>
-    <p> $Z_{1-i/K}=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})$ </p>
-    <p> $Z_{1-i/K}$ </p>
-    <p> $Z_{1-i/N}\!=\!Z_{1\!-\!(i\!-\!1)/N}\!-\!g_{v_{\theta}}(Z_{1\!-\!(i\!-\!1)/N},1%
-\!-\!(i\!-\!1)/N)/N$ </p>
-    <p> $\displaystyle=Z_{1-i/K}-(1/K)g_{v_{\theta}}(X_{1-i/K}+\mathcal{E}(Z_{1-i/K}),1%
--i/K)$ </p>
-    <p> ${}_{\textrm{M}}$ </p>
-    <p> $(X^{S},X^{T})\in(\mathbb{R}^{d},\mathbb{R}^{d})$ </p>
-    <p> $\displaystyle\approx X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)\left[g_{v_{\theta}%
-}(X_{1-i/K},1-i/K)+\mathcal{E}(Z_{1-i/K})\nabla_{X_{t}}g_{v_{\theta}}(X_{1-i/K%
-},1-i/K)\right],$ </p>
-    <p> $X^{S}$ </p>
-    <p> $\log p_{v_{\theta}}(Z_{0})\geq-\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}[\log\frac{q(Z_{%
-1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}+\sum_{i=1}^{N}%
-\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{(i-1)/N}|Z_{i/N})}]$ </p>
-    <p> $\displaystyle\mathbb{E}[L(\mathcal{T}_{\textrm{vanilla}}(X^{S}),\mathcal{T}(Z_%
-{0}))+\alpha^{\Theta}L(\mathcal{T}_{\textrm{vanilla}}(X^{S}),Y)]+\mathcal{L}_{%
-\textrm{FM-KT}},$ </p>
-    <p> $\mathcal{L}_{\textrm{OFM-KT}}$ </p>
-    <p> $B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor$ </p>
-    <p> $\lim_{t\rightarrow 1}\nabla_{t}\alpha_{t}=+\infty$ </p>
-    <p> $p_{v_{\theta}}(\cdot|Z_{i/N})$ </p>
-    <p> $X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]$ </p>
-    <p> $X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]=\textbf{{%
-shuffle}}\left(X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]%
-\right).$ </p>
-    <p> $\mathcal{E}(Z_{1-(i+1)/K})=\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)%
-\mathcal{K}(1-i/K)$ </p>
-    <p> $\left[\frac{dX_{t}}{dt}-g_{v_{\theta}}(\mathcal{H}(t),t)\right]$ </p>
-    <p> $Z_{1}=\alpha_{1}X_{S}$ </p>
-    <p> $\mathit{b=100}$ </p>
-    <p> $Z_{t}+\int_{t}^{s}g_{v_{\theta}}(Z_{\tau},\tau)d\tau$ </p>
-    <p> $\displaystyle-\log p_{v_{\theta}}(Z_{0})\leq\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}%
-\left[\log\frac{q(Z_{1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1%
-/N})}+\sum_{i=1}^{N}\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{%
-(i-1)/N}|Z_{i/N})}\right]$ </p>
-    <p> $\{\hat{Z}_{i/N}\}_{i}$ </p>
-    <p> $\mathcal{K}(t)$ </p>
-    <p> $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
-||g_{v_{\theta}}(Z_{t},t)-\nabla_{t}Z_{t}||_{2}^{2}$ </p>
-    <p> $\nabla_{t}\sigma_{t}$ </p>
-    <p> $\nabla_{t}\sigma_{t}\equiv 0$ </p>
-    <p> $\displaystyle=Z_{1-i/K}-(1/K)g_{v_{\theta}}(Z_{1-i/K},1-i/K)$ </p>
-    <p> $g^{T}(\cdot)$ </p>
-    <p> ${}_{\textrm{50}}$ </p>
-    <p> $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
-||g_{v_{\theta}}(Z_{t},t)-(X^{S}-X^{T})||_{2}^{2}.$ </p>
-    <p> $\rho_{t}(Z):\mathbb{R}^{d}\times[0,1]\rightarrow\mathbb{R}^{d}$ </p>
-    <p> $\psi(t)=\nabla_{X_{t}}g_{v_{\theta}}(X_{t},t)$ </p>
-    <p> $\textrm{Law}(Z_{1})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{1})$ </p>
-    <p> $\hat{Z}_{0}\sim\pi_{0}$ </p>
-    <p> $B-\left\lfloor{\beta_{d}}B\right\rfloor$ </p>
-    <p> $\alpha_{t}=a(\frac{b}{a})^{t}$ </p>
-    <p> $\displaystyle\approx\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{%
-0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}%
-\mathbb{E}_{\hat{Z}_{i/N}\sim\int p_{v_{\theta}}(Z_{i/N}|Z_{1})q(Z_{1}|Z_{0})%
-dZ_{1}}$ </p>
-    <p> $\displaystyle\mathcal{L}_{\textrm{FM-KT++}}=\mathbb{E}_{(X^{S},X^{T},Y)}\frac{%
-1}{N}\sum_{i=0}^{N-1}L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{\theta}}(Z%
-_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),X^{T})$ </p>
-    <p> $\{Z_{1-i/N}\}_{i}$ </p>
-    <p> $\displaystyle=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)g_{v_{\theta}}(X_{1-i/K}+%
-\mathcal{E}(Z_{1-i/K}),1-i/K)$ </p>
-    <p> $\mathcal{K}(t)\geq 0$ </p>
-    <p> $\mathcal{L}_{\textrm{guided}}$ </p>
-    <p> $\displaystyle\left[D_{\mathrm{KL}}(q(Z_{(i-1)/N}|Z_{i/N},Z_{0})||p_{v_{\theta}%
-}(Z_{(i-1)/N}|\hat{Z}_{i/N}))\right],\quad s.t.\quad\textrm{Law}(Z_{i/N})%
-\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i/N})$ </p>
-    <p> $\displaystyle\quad\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{%
-\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Z_{0})}_{\textrm{the Online %
-KD loss}}$ </p>
-    <p> $\displaystyle\approx X_{1-(i+1)/K}+\mathcal{E}(Z_{1-i/K})+(1/K)\mathcal{K}(1-i%
-/K)-(1/K)\mathcal{E}(Z_{1-i/K})\psi(1-i/K)$ </p>
-    <p> ${}_{\textrm{S}}$ </p>
-    <p> $\displaystyle Z_{1-(i+1)/K}-X_{1-(i+1)/K}$ </p>
-    <p> $Z_{1-(i+1)/K}\!=\!Z_{1-i/K}-g_{v_{\theta}}(Z_{1-{i/K}},1-i/K)dt$ </p>
-    <p> $\frac{\alpha_{t}-\alpha_{t-\Delta t}}{t-\Delta t}$ </p>
-    <p> $\nabla_{t}\alpha_{t}$ </p>
-    <p> $\mathcal{E}(Z_{1-i/K})\geq 0$ </p>
-    <p> $X^{T}\in\mathbb{R}^{B\times C\times H\times W}$ </p>
-    <p> $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}$ </p>
-    <p> $g^{*}_{v_{\theta}}(\cdot)$ </p>
-    <p> $dZ_{t}$ </p>
-    <p> $i\!\geq\!1$ </p>
-    <p> $\{\textrm{Law}(Z_{i/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_%
-{i/N})\}_{i=0}^{N-1}$ </p>
-    <p> $\displaystyle=X_{1-(i+1)/K}+\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)%
-\mathcal{K}(1-i/K)$ </p>
-    <p> $\sigma(t)=1-0.1t$ </p>
-    <p> $\rho_{0}(X^{T})=\rho_{1}(X^{S})+\int_{\rho_{1}(Z)}^{\rho_{0}(Z)}\partial\rho_{%
-t}(Z)$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10005.html DELETED Viewed

@@ -1,139 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $L(\cdot,\cdot)$ </p>
-    <p> $\displaystyle\int_{0}^{1}\mathbb{E}[||\partial\rho_{t}(Z)/\partial t-g_{v_{%
-\theta}}(Z_{t},t)||]dt.$ </p>
-    <p> $g_{v_{\theta}}(\cdot)$ </p>
-    <p> ${}_{\textrm{75}}$ </p>
-    <p> $\hat{Z}_{0}$ </p>
-    <p> ${Z}_{i/N}$ </p>
-    <p> $\displaystyle=\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{0})}{p%
-_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}%
-\mathbb{E}_{q(Z_{i/N}|Z_{0})}\mathbb{E}_{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}\left[%
-\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{(i-1)/N}|Z_{i/N})}\right]$ </p>
-    <p> $\displaystyle\mathcal{L}_{\textrm{FM-KT}}\!=\!\mathbb{E}[\frac{1}{N}\sum_{i=0}%
-^{N-1}\!L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}\!\!-\!\!g_{v_{\theta}}(Z_{1\!%
--\!i/N},1-i/N))/\!-\!\nabla_{t}\sigma_{t})$ </p>
-    <p> $\sigma_{t}=\sqrt{1-\alpha_{t}^{2}},\ s..t.\quad a=19.9,b=0.1$ </p>
-    <p> $\displaystyle=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)\left[g_{v_{\theta}}(X_{1-%
-i/K},1-i/K)+\mathcal{E}(Z_{1-i/K})\psi(1-i/K)\right],$ </p>
-    <p> $\hat{Z}_{i/N}$ </p>
-    <p> $\alpha^{\Theta}$ </p>
-    <p> $\textrm{Law}(Z_{(i-1)/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z%
-}_{(i-1)/N})$ </p>
-    <p> $\displaystyle+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{%
-\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the %
-ground truth label (optional)}},$ </p>
-    <p> $\alpha_{t}=\textrm{exp}(-\frac{1}{4}a(1-t)^{2}-\frac{1}{2}b(1-t))$ </p>
-    <p> $\lim_{t\rightarrow 1}\alpha_{t}=1$ </p>
-    <p> $\{Z_{t}\}_{t}$ </p>
-    <p> $\sigma(t)=1$ </p>
-    <p> $X^{S}-X^{T}=\frac{dX_{t}}{dt}$ </p>
-    <p> $\displaystyle\mathcal{E}(Z_{1-1/K})=(1/K)\mathcal{K}(1)$ </p>
-    <p> $\sigma_{t}=\sqrt{1-\alpha_{t}^{2}}$ </p>
-    <p> $\mathit{a=19.9}$ </p>
-    <p> $\displaystyle\textrm{where}\quad Z_{1-i/N}=Z_{1-(i-1)/N}$ </p>
-    <p> $\mathcal{H}(t)=\operatorname*{arg\,sup}_{X_{t}}\{||\frac{dX_{t}}{dt}-g_{v_{%
-\theta}}(X_{t},t)||_{2}^{2}\}$ </p>
-    <p> $||\frac{dX_{t}}{dt}-g_{v_{\theta}}(X_{t},t)||_{2}^{2}$ </p>
-    <p> $t\sim\mathcal{U}[0,1]$ </p>
-    <p> $\mathcal{T}_{\textrm{vanilla}}(\cdot)$ </p>
-    <p> $\sim 10^{6}\times 10^{4}=10^{10}$ </p>
-    <p> $\sim 1\,\mathrm{KB}$ </p>
-    <p> $\displaystyle-~{}NCC(\mathcal{X}_{\mathrm{fx}},\mathcal{X}_{\mathrm{wp,n}})+%
-\lambda\sum_{p\in\Omega}||\nabla\varphi(p)||^{2}$ </p>
-    <p> $\varphi=\sum_{i=0}^{n}\varphi_{i}$ </p>
-    <p> $\mathcal{X}_{\mathrm{wp}}$ </p>
-    <p> $w_{k,i}=|m(i\in\omega)|^{g}$ </p>
-    <p> $\mathcal{X}_{\mathrm{mv}}$ </p>
-    <p> $0.7\times 0.7\times 3.0$ </p>
-    <p> $\mathcal{X}_{\mathrm{fx}}$ </p>
-    <p> $0.837\pm 0.021$ </p>
-    <p> $1.2\times 1.2\times 3.0$ </p>
-    <p> $\displaystyle\begin{split}\mathcal{X}_{\mathrm{wp,n}}=\mathcal{X}_{\mathrm{mv}%
-}\circ\sum_{i=0}^{n}\varphi_{i}.\end{split}$ </p>
-    <p> $\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ </p>
-    <p> $\mathcal{X}_{\mathrm{wp,1}}$ </p>
-    <p> $\mathcal{X}_{\mathrm{wp,n}}$ </p>
-    <p> $0.926\pm 0.012$ </p>
-    <p> $0.847\pm 0.008$ </p>
-    <p> $\centering\mathcal{X}_{\mathrm{wp}}=\mathcal{X}_{\mathrm{mv}}\circ\varphi%
-\approx\mathcal{X}_{\mathrm{fx}}.\@add@centering$ </p>
-    <p> $\mathcal{L}_{\mathrm{sim}}$ </p>
-    <p> $0.866\pm 0.020$ </p>
-    <p> $0.915\pm 0.012$ </p>
-    <p> $230\times 230$ </p>
-    <p> $\mathcal{X}_{\mathrm{wp,0}}$ </p>
-    <p> $0.811\pm 0.023$ </p>
-    <p> $0.807\pm 0.23$ </p>
-    <p> $290\times 250$ </p>
-    <p> $lr_{\mathrm{epoch}}=3\cdot 10^{-4}\cdot e^{-3\mathrm{epoch}/500}$ </p>
-    <p> $H\times W\times L$ </p>
-    <p> $0.920\pm 0.014$ </p>
-    <p> $\%|J_{\phi}|\leq 0$ </p>
-    <p> $\%|J_{\varphi}|<0$ </p>
-    <p> $\mathcal{L}_{\mathrm{smooth}}$ </p>
-    <p> $\displaystyle~{}\mathcal{L}_{\mathrm{sim}}+\lambda\mathcal{L}_{\mathrm{smooth}}$ </p>
-    <p> $0.866\pm 0.013$ </p>
-    <p> $94.65\%$ </p>
-    <p> $\langle search\rangle$ </p>
-    <p> $P(\hat{a}=\langle search\rangle|\theta^{\prime},q)$ </p>
-    <p> $LM_{\theta^{\prime}}$ </p>
-    <p> $LM_{\theta^{\prime}}:Q\mapsto\Omega\cup\{\langle search\rangle\}$ </p>
-    <p> $LM_{\theta}:Q\mapsto\Omega$ </p>
-    <p> $\psi\circ LM_{\theta}:Q\mapsto\Omega\cup\{\langle search\rangle\}$ </p>
-    <p> $alpha=32$ </p>
-    <p> $\underset{\theta}{\text{argmin}}\prod_{q\in Q}\left[P(\hat{a}=\langle search%
-\rangle|\theta^{\prime},q)+\lambda P(\hat{a}\notin A|\theta^{\prime},q)\right]$ </p>
-    <p> $(\langle search\rangle)$ </p>
-    <p> $7e-5$ </p>
-    <p> $LM_{\theta}$ </p>
-    <p> $\psi(LM_{\theta}(q))=\begin{cases}\mathds{1}(\hat{a}),&\text{if }\hat{a}\in A%
-\\
-\langle search\rangle,&\text{otherwise}\end{cases}$ </p>
-    <p> $P(\hat{a}\notin A|\theta^{\prime},q)$ </p>
-    <p> $733.5$ </p>
-    <p> $TR^{(1)},TR^{(2)},\cdots,TR^{(m)}$ </p>
-    <p> $\mathcal{V}_{\mathrm{target}}$ </p>
-    <p> $\hat{P}_{v}$ </p>
-    <p> $\mathcal{V}_{target}$ </p>
-    <p> $\operatorname*{arg\,min}_{\theta}\sum_{G}\mathcal{L}(G,\theta).$ </p>
-    <p> $p^{(1)},p^{(2)},\cdots,p^{(m)}$ </p>
-    <p> $q^{(1)},q^{(2)},\cdots,q^{(n)}$ </p>
-    <p> $\hat{\mathcal{V}}_{target}$ </p>
-    <p> $\in TR^{i}$ </p>
-    <p> $\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\notin\mathcal{V}_{\textrm{target}%
-}}\log P(\hat{P}_{v}=0|\mathcal{G},\theta)$ </p>
-    <p> $\displaystyle\mathcal{L}(G,\theta)=$ </p>
-    <p> $\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\in\mathcal{V}_{\textrm{target}}}%
-\log P(\hat{P}_{v}=1|\mathcal{G},\theta)$ </p>
-    <p> $1,2,\cdots,m$ </p>
-    <p> $TR^{i}$ </p>
-    <p> $N=1923$ </p>
-    <p> $p^{(1)}_{ref},p^{(2)}_{ref},\cdots,p^{(m)}_{ref}$ </p>
-    <p> $\mathcal{V}_{\mathrm{target}}\subset\mathcal{V}$ </p>
-    <p> $[0.1,0.01,0.001,0.0001]$ </p>
-    <p> $\mathrm{EE}_{y}$ </p>
-    <p> $\left[\mathcal{D}_{t},\mathcal{D}_{r}\right]$ </p>
-    <p> $\left[\mathcal{D}_{r},\mathcal{H}_{r}\right]$ </p>
-    <p> $W=F\cdot S$ </p>
-    <p> $\mathcal{L}_{GPs}=\mathcal{L}_{G}+w_{P}\mathcal{L}_{P}+w_{S}\mathcal{L}_{s}$ </p>
-    <p> $\eta_{t}>6$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10006.html DELETED Viewed

@@ -1,156 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\left[\mathcal{D}_{t},\mathcal{H}_{t}\right]$ </p>
-    <p> $\eta_{min}<6$ </p>
-    <p> $X_{syn}$ </p>
-    <p> $\displaystyle l_{3}+l_{2}<l_{1}+l_{4}$ </p>
-    <p> $\delta S_{out}$ </p>
-    <p> $F\cdot S=\tau\cdot\theta$ </p>
-    <p> $\mathrm{EE}_{x}$ </p>
-    <p> $\left[\mathcal{D}_{t},\mathcal{H}_{t}\right]=\left[\{d_{t,1},d_{t,2},...,d_{t,%
-i}\},\{\eta_{t,1},\eta_{t,2},...,\eta_{t,i}\}\right]$ </p>
-    <p> $\mathcal{L}_{P}=\frac{1}{N}\sum^{N}_{i=1}\left[(d_{r,i}-d_{t,i})^{2}+(\eta_{r,%
-i}-\eta_{t,i})^{2}\right]$ </p>
-    <p> $\left[\mathcal{H}_{t},\mathcal{H}_{r}\right]$ </p>
-    <p> $|F_{in}\cdot\delta S_{in}|=|F_{out}\cdot\delta S_{out}|$ </p>
-    <p> $\eta=\left|\frac{\delta S_{out}}{\delta\theta_{in}}\right|$ </p>
-    <p> $(\mathrm{x}_{i},\mathrm{x}_{j})$ </p>
-    <p> $i=100$ </p>
-    <p> $d_{t}=1.0,\eta_{t}=2.0$ </p>
-    <p> $d_{r},\eta_{r}$ </p>
-    <p> $\delta\theta_{in}$ </p>
-    <p> $|\tau_{in}\cdot\delta\theta_{in}|=|F_{out}\cdot\delta S_{out}|$ </p>
-    <p> $\mathcal{L}_{s}=-\frac{1}{N}\sum^{N}_{i=1}\min{D\left(\mathrm{x}_{i},\mathrm{x%
-}_{j}\right)}$ </p>
-    <p> $|W_{in}|=|W_{out}|$ </p>
-    <p> $\left[\mathcal{D}_{r},\mathcal{H}_{r}\right]=\left[\{d_{r,1},d_{r,2},...,d_{r,%
-i}\},\{\eta_{r,1},\eta_{r,2},...,\eta_{r,i}\}\right]$ </p>
-    <p> $y\left[i,j\right]=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}h\left[m,n%
-\right]\cdot x\left[i-m,j-n\right]$ </p>
-    <p> $0-\infty$ </p>
-    <p> $\mathcal{M}=\{M_{i}|0\leq i<h\}$ </p>
-    <p> $\mathcal{M}=\{M_{i|0\leq i<h}\}$ </p>
-    <p> $\{1e^{-5},2e^{-5}\}$ </p>
-    <p> $M=10,50,70,100$ </p>
-    <p> $R_{t+1,A_{t},B_{t}}$ </p>
-    <p> $\displaystyle\mathbb{E}_{0}\left[\sqrt{\sum_{a}\left(\tilde{C}^{+}_{T,i}\right%
-)^{2}}\right]\leqslant\mathbb{E}_{0}\left[\sqrt{\sum_{t=1}^{T}\sum_{a}\left(%
-\tilde{\Re}_{t,a}\right)^{2}}\right]\leqslant\sqrt{\sum_{t=1}^{T}\frac{2(1+%
-\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma_{t}}}$ </p>
-    <p> $\displaystyle w_{t}(a,b)^{2}=\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{%
-t}=a,B_{t}=b)=\frac{\beta^{\prime}_{t}\log(1+\sigma_{w}^{-2}\sigma^{2}_{t}(a,b%
-))}{\log(1+\sigma_{w}^{-2})}\geqslant\beta^{\prime}_{t}\sigma^{2}_{t}(a,b).$ </p>
-    <p> $\displaystyle+C\sum_{t=0}^{T}\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{%
-\prime},A_{t},B_{t})\cup\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))$ </p>
-    <p> $\mathbb{P}(\max_{i}y_{ti}\leqslant\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{%
-\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+\sigma_{n}^{2}}})\geqslant 1-\delta%
-_{1}$ </p>
-    <p> $\displaystyle\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{R})$ </p>
-    <p> $\displaystyle\leqslant\sum_{a}\left({\tilde{C}_{T-1,a}^{+}}\right)^{2}+\sum_{i%
-}(\tilde{\Re}_{T,a})^{2}$ </p>
-    <p> $\displaystyle{\tilde{C}_{T,a}}={\tilde{C}_{T-1,a}}+\tilde{\Re}_{T,a},$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}%
-\left[\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]$ </p>
-    <p> $\pi=\pi^{\operatorname{adv-est}}$ </p>
-    <p> $\phi(a,b)=P_{a}(\theta)/(P_{n_{0}}+P_{b}\mathds{1}(a=b))$ </p>
-    <p> $\Re_{full}(T,\operatorname{adv},\tilde{R})$ </p>
-    <p> $\displaystyle\mathbf{k}_{t}(a,b)$ </p>
-    <p> $\Re_{t1}$ </p>
-    <p> $\displaystyle\mathbb{P}(\tilde{R}_{t1}\geqslant\tilde{R}_{t2})$ </p>
-    <p> $\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}},(r_{t})_{t})=%
-\max_{a\in\mathcal{A}}\mathbb{E}\left[\sum_{t=0}^{T-1}r_{t}(a)-r_{t}(A_{t})\right]$ </p>
-    <p> $c(\Delta,\sigma_{n})>0$ </p>
-    <p> $GP(0,k((a,b),(a^{\prime},b^{\prime})))$ </p>
-    <p> $X_{2}=X_{1}$ </p>
-    <p> $\mathcal{N}(0.5,2.0)$ </p>
-    <p> $r_{t}(a)=f_{\theta}(a,B_{t})$ </p>
-    <p> $\tilde{f}_{t+1}(a,b)$ </p>
-    <p> $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}).$ </p>
-    <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\mathbb{P}(N_{t}\leqslant 0)$ </p>
-    <p> $\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ </p>
-    <p> $\displaystyle\Re_{t}(a)-\mathbb{E}_{t}\left[\tilde{\Re}_{t,a}\right]$ </p>
-    <p> $\displaystyle\mathbf{R}_{t}$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{t=0}^{T-1}\tilde{R}_{t+1}(a)-\tilde{R}_{t+%
-1}(A_{t})\right].$ </p>
-    <p> $f_{1},f_{2},f_{3}\in\mathcal{F}$ </p>
-    <p> $\displaystyle\mathbb{P}(\omega_{t}|\Omega_{t-1})$ </p>
-    <p> $\frac{x}{\log(1+x)}$ </p>
-    <p> $X_{2}=[0,1]$ </p>
-    <p> $\displaystyle\sum_{t=0}^{T-1}\mathbb{E}\left[f_{\theta}(a,B_{t})-f_{\theta}(A_%
-{t},B_{t})\>|\>\theta\right]$ </p>
-    <p> $\pi=\pi^{\operatorname{adv-OTS}}$ </p>
-    <p> $r^{i}(a^{i},a^{-i})$ </p>
-    <p> $\displaystyle\geqslant(1-\delta_{1})\mathbb{P}(\max_{i}x_{ti}\geqslant\frac{t}%
-{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+%
-\sigma_{n}^{2}}}\mid\epsilon)$ </p>
-    <p> $\begin{cases}x_{ti}&\sim\mathcal{N}_{i}(0,1),\quad i=1,\ldots,M\\
-y_{ti}&\sim\mathcal{N}_{i}\left(\frac{t}{t+\sigma_{n}^{2}}(1-\Delta),\frac{%
-\sigma_{n}^{2}}{\sigma_{n}^{2}+t}\right),\quad i=1,\ldots,M\end{cases}$ </p>
-    <p> $(10^{-1})$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\tilde{R}_{t+1}(a)-\tilde{R}_{t+1}(A_{t})\mid%
-\theta\right],$ </p>
-    <p> $\neg{\mathcal{E}}$ </p>
-    <p> $N(\mu_{t}(a,b),\sigma_{t}(a,b))$ </p>
-    <p> $f_{\theta}(a,b)$ </p>
-    <p> $\tilde{R}=(\tilde{R}_{1},\ldots,\tilde{R}_{t+1},\ldots)$ </p>
-    <p> $U=(U_{t}\>|\>t\in\mathbb{N})$ </p>
-    <p> $A^{*}=\operatorname*{arg\,max}_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}%
-\left[R_{t+1,a,B_{t}}\>|\>\theta\right]$ </p>
-    <p> $reg_{1}=\tilde{R}_{1}-\tilde{R}_{1}^{T}X_{1}\cdot\mathbf{1},$ </p>
-    <p> $\mathcal{O}(d\log T)$ </p>
-    <p> $(a^{i},a^{-i})$ </p>
-    <p> $[a^{i}]_{e}=U^{i}$ </p>
-    <p> $0.15(\sum_{i=1}^{N}[a^{i}]_{e}/C_{e})^{4}$ </p>
-    <p> $\displaystyle=\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})\left(%
-\mathbb{P}(\Omega_{t-1})+\mathbb{P}(\bar{\Omega}_{t-1})\right)$ </p>
-    <p> $(A_{t},B_{t})$ </p>
-    <p> $[a^{i}]_{e}$ </p>
-    <p> $\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}})$ </p>
-    <p> $\displaystyle\left\langle\tilde{C}_{t-1}^{+},\tilde{\Re}_{t}\right\rangle$ </p>
-    <p> $W_{t+1}=Y_{t+1,A_{t},B_{t}}-f_{\theta}(A_{t},B_{t})$ </p>
-    <p> $\displaystyle\Re(T,\pi^{\operatorname{alg}},\pi^{B})=\mathbb{E}\left[\Re(T,\pi%
-^{\operatorname{alg}},\pi^{B},\theta)\right]$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{t}I_{t}(\theta;Z_{t})\right]=\sum_{t=0}^{T%
--1}I(\theta;Z_{t}\>|\>Z_{0},\ldots,Z_{t-1})$ </p>
-    <p> $[0,1]^{\mathcal{A}}$ </p>
-    <p> $\displaystyle I_{t}\left(\theta;R_{t+1,A_{t},B_{t}}\mid A_{t}=a,B_{t}=b\right)%
-=\frac{1}{2}\log\left(1+\frac{\sigma^{2}_{t}(a,b)}{\sigma_{w}^{2}}\right)$ </p>
-    <p> $W_{t}=Y_{t+1,A_{t},B_{t}}-f_{\theta}(A_{t},B_{t})$ </p>
-    <p> $\displaystyle\quad+C\sum_{t=0}^{T-1}\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{%
-R},U^{\prime},A_{t},B_{t}))+2\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{%
-t}))+\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}\right)$ </p>
-    <p> $t_{e}(u)=c_{e}(1+0.5(\frac{u}{C_{e}})^{4}),$ </p>
-    <p> $k(x,x)\leqslant 1$ </p>
-    <p> $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\left(1-\frac{f}{\sqrt{2\pi}%
-(f^{2}+1)e^{f^{2}/2}}\right)^{M}$ </p>
-    <p> $\displaystyle\leqslant\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{%
-t},B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}(U^{\prime}_{t}(A_{t},B_{t}%
-)-L_{t}(A_{t},B_{t}))$ </p>
-    <p> $\mathcal{O}((\log T)^{d+1})$ </p>
-    <p> $\displaystyle=\underbrace{\mathbb{E}\left[\tilde{R}_{t+1}(a)-\tilde{R}_{t+1}(A%
-_{t})\>|\>\theta\right]}_{(I)}+\underbrace{\mathbb{E}\left[f_{\theta}({a,B_{t}%
-})-\tilde{R}_{t+1}(a)\>|\>\theta\right]}_{(II)}+\underbrace{\mathbb{E}\left[%
-\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})\>|\>\theta\right]}_{(III)}$ </p>
-    <p> $\Re^{*}(T,\text{IWE-Hedge})=\mathcal{O}(\sqrt{T\mathcal{A}\log\mathcal{A}}).$ </p>
-    <p> $x_{t}^{-i}$ </p>
-    <p> $\Omega_{t-1}$ </p>
-    <p> $\beta_{t}=\frac{2\beta^{\prime}_{t}}{\log(1+\sigma_{w}^{-2})}.$ </p>
-    <p> $\Re_{t}=[0.5m_{1}+\sum_{k=2}^{t}m_{k},-0.5m_{1}]$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10007.html DELETED Viewed

@@ -1,167 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\{z^{j}_{t+1}\}_{j\in[M]}$ </p>
-    <p> $\mu(a,b)=\mathbb{E}\left[f_{\theta}(a,b)\right]$ </p>
-    <p> $\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2\beta_{T}=\mathcal{O}(\log\mathcal{A}%
-T+\log T+\log\log\mathcal{A}T).$ </p>
-    <p> $(M=10,20,30)$ </p>
-    <p> $\Re_{\operatorname{adv}}(a;T,\text{Hedge},\tilde{R})=\mathcal{O}(2c\sqrt{T\log%
-\mathcal{A}})$ </p>
-    <p> $R_{t+1,A_{t},B_{t}}=\theta_{A_{t},B_{t}}$ </p>
-    <p> $a^{+}=\max\{a,0\}$ </p>
-    <p> $o=f_{\theta}(a)+w$ </p>
-    <p> $\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}\right)$ </p>
-    <p> $B_{0:T}$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}I_{t}(\theta;A_{t},B_{t},R_{t+1,A%
-_{t},B_{t}})\right]$ </p>
-    <p> $\displaystyle\gamma_{T}:=\max_{A_{0:T},B_{0:T}}I(\theta;A_{0},B_{0},\ldots,A_{%
-T-1},B_{T-1})$ </p>
-    <p> $f_{\theta}(a,b)=\phi(a,b)^{\top}\theta$ </p>
-    <p> $\Re_{t}\in\mathbb{R}^{\mathcal{A}}$ </p>
-    <p> $\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A%
-_{t}}}-\frac{\hat{X}_{t,A_{t}}}{{X}_{t,A_{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)$ </p>
-    <p> $\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t+1},t=0,1,\ldots)$ </p>
-    <p> $\tilde{R}_{t}=\left[\tilde{R}_{t1},\tilde{R}_{t2}\right]$ </p>
-    <p> $L=(L_{t}\>|\>t\in\mathbb{N})$ </p>
-    <p> $\mathbb{E}_{t}\left[R_{t,a,b}^{2}\right]:=\mathbb{E}_{t}\left[(f_{\theta]}(a,b%
-)+W_{t+1})^{2}\right]=\mathbb{E}_{t}\left[f_{\theta]}(a,b)^{2}+W_{t+1}^{2}%
-\right]\leqslant 1+\sigma_{w}^{2}$ </p>
-    <p> $\epsilon_{t}\sim\mathcal{N}(0,0.1)$ </p>
-    <p> $\tilde{R}^{\operatorname{est}}$ </p>
-    <p> $N_{t}=\sum\limits_{k=2}^{t}m_{k}\sim\mathcal{N}\left(-\sum\limits_{k=2}^{t}%
-\frac{k}{k+\sigma_{n}^{2}}(1-\Delta),\ \sum\limits_{k=2}^{t}(1+\frac{\sigma_{n%
-}^{2}}{k+\sigma_{n}^{2}})\right)\triangleq\mathcal{N}(\mu_{t},\sigma_{t}^{2}).$ </p>
-    <p> $\displaystyle=\log\prod_{t=1}^{\infty}\mathbb{P}(\omega_{t}|\Omega_{t-1})=\sum%
-_{t=1}^{\infty}\log\mathbb{P}(\omega_{t}|\Omega_{t-1})$ </p>
-    <p> $\displaystyle=\sum_{t=1}^{\infty}\log\Phi\left(\frac{\sum_{k=2}^{t}\frac{k}{k+%
-\sigma_{n}^{2}}(1-\Delta)}{\sqrt{\sum_{k=2}^{t}(1+\frac{\sigma_{n}^{2}}{k+%
-\sigma_{n}^{2}})}}\right)$ </p>
-    <p> $\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{R})$ </p>
-    <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}\left(\tilde{\Re}_{t,a}%
-\right)^{2}\right]$ </p>
-    <p> $\tilde{R}_{t}(2nd)$ </p>
-    <p> $X_{1}=[0.5,0.5]$ </p>
-    <p> $\mathcal{E}(c)$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}%
-\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta\right]\right]$ </p>
-    <p> $\textbf{r}_{t}=(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}}$ </p>
-    <p> $m_{1}<0$ </p>
-    <p> ${N}(\mu_{p},\Sigma_{p})$ </p>
-    <p> $\mathcal{O}\big{(}\sqrt{T\mathcal{A}}+\sqrt{\gamma_{T}\beta T}\big{)}$ </p>
-    <p> $c^{\prime}=-0.62$ </p>
-    <p> $(b^{i},b^{-i})$ </p>
-    <p> $U^{\prime}_{t}\geqslant L_{t}$ </p>
-    <p> $r^{i}(a^{i},a^{-i})=-\ell^{i}(a^{i},a^{-i})$ </p>
-    <p> $H_{t+1}=(H_{t},A_{t},B_{t},R_{t+1,A_{t},B_{t}})$ </p>
-    <p> $\mathcal{O}\left(\left(\sqrt{\log\mathcal{A}}+\sqrt{\log(\mathcal{A}T)\log(T)^%
-{d+1}}\right)\sqrt{T}\right)$ </p>
-    <p> $\displaystyle\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t}=a,B_{t}=b)%
-\geqslant\beta^{\prime}_{t}\sigma^{2}_{t}(a,b).$ </p>
-    <p> $\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t}):=\{\tilde{R}_{t+1}(A_{t}%
-)\leqslant U^{\prime}_{t}(A_{t},B_{t})\}.$ </p>
-    <p> $X_{1}=\hat{X}_{1}$ </p>
-    <p> $k_{\rm L}(\cdot,\cdot)$ </p>
-    <p> $(a+b)^{+}\leqslant(a^{+}+b)^{+}\leqslant\left|a^{+}+b\right|.$ </p>
-    <p> $\tilde{R}_{t+1}$ </p>
-    <p> $\displaystyle\Re^{*}(T,\pi,\theta)$ </p>
-    <p> $\displaystyle=\sum_{a}(\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/\mathcal{A})\sum_{b}%
-Y_{t,b}f_{\theta}(a,b)$ </p>
-    <p> $\sqrt{\gamma_{T}}$ </p>
-    <p> $\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant%
-\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b),\forall a\in\mathcal{A}\mid H_{t})%
-\leqslant 2\mathcal{A}\exp(-\beta^{\prime}_{t}/2),$ </p>
-    <p> $\displaystyle=\sum_{a}\frac{\mathbb{E}_{t}\left[R_{t+1,a,B_{t}}^{2}\right]}{X_%
-{t,a}}+\sum_{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\mathbb{E}_{t}\left[R_{t+1,A_{t}%
-,B_{t}}^{2}\right]\left(\left|\mathcal{A}\right|\hat{X}_{t,a}-2\right)$ </p>
-    <p> $\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t},t\in\mathbb{Z}_{++})$ </p>
-    <p> $\displaystyle\leqslant\mathbb{E}_{0}\left[\tilde{C}_{T,a}^{+}\right]=\mathbb{E%
-}_{0}\left[\sqrt{(\tilde{C}_{T,a}^{+})^{2}}\right]\leqslant\mathbb{E}_{0}\left%
-[\sqrt{\sum_{a}\left(\tilde{C}_{T,a}^{+}\right)^{2}}\right],$ </p>
-    <p> $M_{1}=M_{2}=\ldots=M_{T}=M=\mathcal{O}(\log\mathcal{A}T)$ </p>
-    <p> $\sigma(H_{t},A_{t},R_{t+1,A_{t},B_{t}})$ </p>
-    <p> $\mathcal{R}:\mathbb{R}\mapsto[0,1]$ </p>
-    <p> $\displaystyle\mathbb{E}_{0}\left[\sqrt{\sum_{a}\left(\tilde{C}^{+}_{T,i}\right%
-)^{2}}\right]\leqslant\mathbb{E}_{0}\left[\sqrt{\sum_{t=1}^{T}\sum_{a}\left(%
-\tilde{\Re}_{t,a}\right)^{2}}\right]$ </p>
-    <p> $H_{t},B_{t}$ </p>
-    <p> $\tilde{R}_{t2}$ </p>
-    <p> $\displaystyle\leqslant\min_{t>0}\frac{\exp(\sigma^{2}t^{2}/2)}{\exp(tc)}=\exp(%
--c^{2}/\sigma^{2})$ </p>
-    <p> $\displaystyle\mu_{t+1}=\Sigma_{t+1}\left(\Sigma_{t}^{-1}\mu_{t}+\frac{R_{t+1,A%
-_{t},B_{t}}}{\sigma_{w}^{2}}\phi(A_{t},B_{t})\right)$ </p>
-    <p> $\displaystyle\mathbb{P}(X-\mu\geqslant c)$ </p>
-    <p> $\pi_{t}(H_{t})$ </p>
-    <p> $v\in\mathbb{R}^{\mathcal{A}}$ </p>
-    <p> $\sigma_{p}\leqslant 1$ </p>
-    <p> $\hat{X}_{t+1}$ </p>
-    <p> $\tilde{R}_{t+1}(a)\in[0,C]$ </p>
-    <p> $\tilde{R}_{t}=\left[z_{t},\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{%
-\sigma_{n}^{2}}{\sigma_{n}^{2}+t}}z^{\prime}_{t}\right],$ </p>
-    <p> $H_{t},A_{t},B_{t}$ </p>
-    <p> $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{{X}_{t,a}}+\sum%
-_{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|\hat{X}_{t,a}%
--2\right)\right)$ </p>
-    <p> $f(x)=\theta^{\top}x,\theta\sim N(0,\sigma_{0}I)$ </p>
-    <p> $\mathbb{E}_{t}\left[\cdot\right]=\mathbb{E}\left[\cdot\mid H_{t},\theta\right]$ </p>
-    <p> $\bm{a}=(a^{i},a^{-i})$ </p>
-    <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\left(-\frac{1}{\sqrt{2\pi}f_{t}(%
-\Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$ </p>
-    <p> $\displaystyle V^{*}=\max_{P\in\mathcal{D}(\mathcal{A})}\min_{Q\in\mathcal{D}(%
-\mathcal{B})}\mathbb{E}_{A\sim P,B\sim Q}\left[f_{\theta}(A,B)\right],$ </p>
-    <p> $\mathbb{P}(f_{\theta}\in\mathcal{F}_{t})\geqslant 1-2\mathcal{A}\exp(-\beta^{%
-\prime}_{t}/2).$ </p>
-    <p> $\operatorname{clip}_{[-c,c]}(x)\geqslant\min(x,c)$ </p>
-    <p> $0.01\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ </p>
-    <p> $\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A_{t},B_{t}}\>|\>\theta\right]$ </p>
-    <p> $\text{SINR}(a,b;\theta)=\phi(a,b)^{T}\theta$ </p>
-    <p> $\delta=1/\sqrt{t}$ </p>
-    <p> $\displaystyle\lim_{t\to\infty}\log\mathbb{P}(\Omega_{t})$ </p>
-    <p> $\text{Regret}^{i}(T)=\frac{1}{T}\max_{a\in\Delta^{\mathcal{D}(\mathcal{A}^{i})%
-}}\mathbb{E}\left[\sum_{t=1}^{T}\phi\left(a,x_{t}^{-i}\right)-\phi\left(x_{t}^%
-{i},x_{t}^{-i}\right)\right],$ </p>
-    <p> $\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv})+\sqrt{%
-\beta I(\theta;H_{T})T},$ </p>
-    <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant 2^{4%
-/3}(1+\sigma_{w}^{2})^{1/3}\left|\mathcal{A}\right|^{2/3}T^{2/3}.$ </p>
-    <p> $g_{t}(\cdot):\mathbb{R}_{+}^{\mathcal{A}}\times\mathbb{R}_{+}^{\mathcal{A}}%
-\mapsto\mathbb{R}_{+}^{\mathcal{A}}$ </p>
-    <p> $m_{t}=z_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)-\sqrt{\frac{\sigma_{n}^{2}}{t%
-+\sigma_{n}^{2}}}z^{\prime}_{t}=z_{t}-\sqrt{\frac{\sigma_{n}^{2}}{t+\sigma_{n}%
-^{2}}}z^{\prime}_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)$ </p>
-    <p> $f_{t}(\Delta,\sigma_{n})=\frac{(1-\Delta)\left(t-\sigma_{n}^{2}\ln{(t+\sigma_{%
-n}^{2})}+2\sigma_{n}^{2}\ln{\sigma_{n}}-1/(\sigma_{n}^{2}+1)\right)}{\sqrt{t+%
-\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}-2\sigma_{n}^{2}\ln{\sigma_{n}}-(\sigma_{%
-n}^{2}+2)/(\sigma_{n}^{2}+1)}},$ </p>
-    <p> $\displaystyle\leqslant\sum_{a}\left|\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/%
-\mathcal{A}\right|$ </p>
-    <p> $\tilde{R}_{t1}\geqslant\tilde{R}_{t2}$ </p>
-    <p> $(\gamma_{t})_{t\geqslant 0}$ </p>
-    <p> $\displaystyle\mathbb{P}(\mathcal{E}_{t}(\tilde{R},U,B_{t})\mid H_{t},B_{t})$ </p>
-    <p> $\phi(a,b)\in\mathbb{R}^{d}$ </p>
-    <p> $\sigma^{2}_{t}(a,b)=k((a,b),(a,b))-\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}%
-+\sigma^{2}{\bm{I}}_{t})\mathbf{k}_{t}(a,b)$ </p>
-    <p> $\tilde{C}^{+}_{t-1,a}=0$ </p>
-    <p> $\mathcal{A}_{J}=\mathcal{F}$ </p>
-    <p> $(r_{t})_{t}$ </p>
-    <p> $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\exp{\left(\frac{-Mf}{\sqrt{%
-2\pi}(f^{2}+1)e^{f^{2}/2}}\right)}$ </p>
-    <p> $\displaystyle(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{t}))$ </p>
-    <p> $Y_{t}=\max_{y}X_{t}^{\top}\theta y$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10008.html DELETED Viewed

@@ -1,172 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $X_{t,a}$ </p>
-    <p> $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad
-U%
-^{\prime}=(\mu_{t}(a,b)+\sqrt{2\log(M\sqrt{t})}\sigma_{t}(a,b):t\in\mathbb{N}).$ </p>
-    <p> $(f_{\theta}(a,b):(a,b)\in\mathcal{A}\times\mathcal{B})$ </p>
-    <p> $\mathcal{O}\big{(}\sqrt{T\mathcal{A}}\big{)}$ </p>
-    <p> $a^{i},b^{i}\in\mathcal{A}^{i}$ </p>
-    <p> $\displaystyle=\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime}_{t}}}+%
-1\right)\sqrt{\beta_{t}I_{t}(\theta;A_{t},B_{t},R_{t+1,A_{t},B_{t}})}$ </p>
-    <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\left(1-\frac{1}{\sqrt{2\pi}f_{t}%
-(\Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$ </p>
-    <p> $\Delta,\sigma_{n}$ </p>
-    <p> $\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime%
-}_{t}}}+1\right)\sqrt{\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t},B_{t%
-})}$ </p>
-    <p> $\left|\mathcal{A}\right|$ </p>
-    <p> $k(x,x^{\prime})=\exp(-(2l^{2})^{-1}\left\|x-x^{\prime}\right\|^{2}s)$ </p>
-    <p> $\Phi(\beta^{\prime}_{t})^{M}=1/\sqrt{t}$ </p>
-    <p> $\tilde{R}_{t}\in[0,1]^{\mathcal{A}}$ </p>
-    <p> $m_{t}=\tilde{R}_{t1}-\tilde{R}_{t2}$ </p>
-    <p> $\overline{x}_{T}=\frac{1}{T}\sum_{t=1}^{T}x_{t},\quad\overline{y}_{T}=\frac{1}%
-{T}\sum_{t=1}^{T}y_{t}.$ </p>
-    <p> $k((a,b),(a^{\prime},b^{\prime}))$ </p>
-    <p> $\left\langle\tilde{C}_{t-1}^{+},\tilde{\Re}_{t}\right\rangle\leqslant 0$ </p>
-    <p> $N(0,\sigma_{w}^{2})$ </p>
-    <p> $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}R_{t+1,A_{t},B_{t}}^{2}\left(%
-\frac{1}{X_{t,A_{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_{t}}^{2}}+\left|%
-\mathcal{A}\right|\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}}\right)\right]$ </p>
-    <p> $\displaystyle\tilde{\Re}_{t,a}=\frac{\mathbb{I}_{A_{t}=a}R_{t+1,A_{t},B_{t}}}{%
-X_{t,a}}-R_{t+1,A_{t},B_{t}}\frac{\hat{X}_{t,A_{t}}}{X_{t,A_{t}}}$ </p>
-    <p> $(X_{a})_{a\in\mathcal{A}}$ </p>
-    <p> $Y_{t}=\min_{y}X_{t}^{\top}\theta y$ </p>
-    <p> $\log(\text{average regret})\propto(1/2)\log(M+N)$ </p>
-    <p> $H_{t}=\left(A_{0},B_{0},Y_{1,A_{0},B_{0}},\ldots,A_{t-1},B_{t-1},Y_{t,A_{t-1},%
-B_{t-1}}\right)$ </p>
-    <p> $U^{\prime}\geqslant U\geqslant L$ </p>
-    <p> $\mathcal{O}(T^{d(d+1)/(2\nu+d(d+1))}(\log T))$ </p>
-    <p> $\mathcal{O}\big{(}\sqrt{T\mathcal{A}\log\mathcal{A}}\big{)}$ </p>
-    <p> $\displaystyle:=\operatorname{clip}_{[0,1]}\left(\tilde{f}_{t+1}^{\operatorname%
-{OTS}}(a,B_{t})\right).$ </p>
-    <p> $n_{t}(a,b)$ </p>
-    <p> $\tilde{\Re}_{t}$ </p>
-    <p> $\Phi(x)\leqslant 1-\frac{x}{\sqrt{2\pi}(x^{2}+1)e^{x^{2}/2}}$ </p>
-    <p> $\displaystyle=\sum_{t=1}^{\infty}\log\Phi(-\frac{\mu_{t}}{\sigma_{t}})$ </p>
-    <p> $g_{t}:\Delta^{\mathcal{A}}\times[0,1]^{\mathcal{A}}\mapsto\Delta^{\mathcal{A}}$ </p>
-    <p> $c=0.54$ </p>
-    <p> $\tilde{\mathcal{O}}(\sqrt{MN/T})$ </p>
-    <p> $X_{t+1}=g_{t}(X_{t},(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}})$ </p>
-    <p> $\displaystyle=\sum_{a}(\hat{X}_{t,a}-X_{t,a})\sum_{b}Y_{t,b}f_{\theta}(a,b)$ </p>
-    <p> $\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))$ </p>
-    <p> $Z_{t+1}$ </p>
-    <p> $\displaystyle\geqslant(1-\delta_{1})\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y%
-_{i}\mid\epsilon)$ </p>
-    <p> $X_{2}=[1,0]$ </p>
-    <p> $\mathcal{O}\big{(}\sqrt{T\log\mathcal{A}}\big{)}$ </p>
-    <p> $\displaystyle\mathcal{F}_{t}:=\left\{f_{\theta}:\left|f_{\theta}(a,B_{t})-\mu_%
-{t}(a,B_{t})\right|\leqslant\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t}),%
-\forall a\in\mathcal{A}\right\}$ </p>
-    <p> $w\sim N\left(0,\sigma_{w}^{2}\right)$ </p>
-    <p> $\frac{\sqrt{2\log M\sqrt{t}}}{\sqrt{\beta^{\prime}_{t}}}=\sqrt{\frac{2\log M%
-\sqrt{t}}{\log\mathcal{A}\sqrt{t}}}$ </p>
-    <p> $G_{J}$ </p>
-    <p> $\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})\leqslant(\mathds{1%
-}_{\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})})(U^{\prime}_{t}(A_{t%
-},B_{t})-f_{\theta}(A_{t},B_{t}))+C(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R%
-},U^{\prime},A_{t},B_{t})}).$ </p>
-    <p> $\displaystyle=\sum_{t=1}^{\infty}\log\mathbb{P}(\mu_{t}+\sigma_{t}Z\leqslant 0%
-),\quad Z\sim\mathcal{N}(0,1)$ </p>
-    <p> $\text{regret-matching}^{+}(\text{RM}^{+})$ </p>
-    <p> $\displaystyle f_{\theta}(a,B_{t})-f_{\theta}(A_{t},B_{t})=\underbrace{\tilde{R%
-}_{t+1}(a)-\tilde{R}_{t+1}(A_{t})}_{(I)\leavevmode\nobreak\ \operatorname{adv}%
-_{t+1}(a)}+\underbrace{f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)}_{(II)%
-\leavevmode\nobreak\ \operatorname{pess}_{t+1}(a)}+\underbrace{\tilde{R}_{t+1}%
-(A_{t})-f_{\theta}(A_{t},B_{t})}_{(III)\operatorname{est}_{t+1}}$ </p>
-    <p> $\sum_{a}\tilde{C}_{t-1,a}^{+}\leqslant 0$ </p>
-    <p> $\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant C(1-\mathds{1}_{%
-\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t}%
-)}),\quad\forall a\in\mathcal{A}.$ </p>
-    <p> $\displaystyle\leqslant\sqrt{T\left(8\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2%
-\beta_{T}\right)I(\theta;H_{T})}$ </p>
-    <p> $\mathbb{P}(\tilde{R}_{t1}\geqslant\tilde{R}_{t2})$ </p>
-    <p> $\displaystyle\Re(T,\pi^{\operatorname{alg}},\pi^{B})$ </p>
-    <p> $\displaystyle=\mathbb{P}(0.5m_{1}+\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$ </p>
-    <p> $(H_{t},A_{t},B_{t},\theta)$ </p>
-    <p> $\beta=\mathcal{O}(\log\mathcal{A}T)$ </p>
-    <p> $\sigma_{n}=0.1$ </p>
-    <p> $[-c,c]^{\mathcal{A}}$ </p>
-    <p> $\displaystyle\geqslant\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$ </p>
-    <p> $\lim_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant c>0,$ </p>
-    <p> $\displaystyle=\mathbb{P}\left(\max_{j\in[M]}z^{j}_{t+1}\geqslant\sqrt{\beta^{%
-\prime}_{t}}\right)$ </p>
-    <p> $\sigma_{t}(a,b)=\left\|\phi(a,b)\right\|_{\Sigma_{t}}$ </p>
-    <p> $\mathbb{P}(\Omega_{t})\geqslant\lim\limits_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant
-c$ </p>
-    <p> $\displaystyle\operatorname{NashRegret}_{t}=\mathbb{E}\left[V^{*}-R_{t+1,A_{t},%
-B_{t}}\right]$ </p>
-    <p> $\tilde{R}_{t}\in\mathbb{R}^{\mathcal{A}}$ </p>
-    <p> $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\Phi^{M}\left(\frac{t}{t+%
-\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+%
-\sigma_{n}^{2}}}\right)$ </p>
-    <p> $I(\theta;H_{T})=I(\theta;A_{0},B_{0},\ldots,A_{T-1},B_{T-1})\leqslant\gamma_{T}.$ </p>
-    <p> $\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant%
-\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b)\mid H_{t})$ </p>
-    <p> $r=(\sqrt{2\nu}/l)\left\|x-x^{\prime}\right\|$ </p>
-    <p> $(a,b),(a^{\prime},b^{\prime})\in\mathcal{A}\times\mathcal{B}$ </p>
-    <p> $\mu_{t}(a,b)$ </p>
-    <p> $\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}\leqslant\sum_{t=1}^{T}\sum_{a}%
-\left(\tilde{\Re}_{t,a}\right)^{2}$ </p>
-    <p> $\tilde{R}_{t}$ </p>
-    <p> $\displaystyle\tilde{f}^{\operatorname{OTS}}_{t+1}(a,B_{t}):=(\max_{j\in[M]}z_{%
-t+1}^{j})\cdot\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t})+\mu_{t}(a,B_{t})%
-\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \tilde{R}_{t+1}(a)=%
-\operatorname{clip}_{[-c,c]}\left(\tilde{f}_{t+1}^{\operatorname{OTS}}(a,B_{t}%
-)\right),\forall a\in\mathcal{A}.$ </p>
-    <p> $\displaystyle\tilde{R}^{\operatorname{OTS}}_{t+1}(a)$ </p>
-    <p> $\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A%
-_{t}}}-\frac{\tilde{C}_{t-1,A_{t}}^{+}/\sum_{a}\tilde{C}_{t-1,a}^{+}}{{X}_{t,A%
-_{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)=0$ </p>
-    <p> $\gamma=\sqrt[3]{((1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2})/{2T}}$ </p>
-    <p> $\tilde{R}_{t+1}=E(H_{t+1},Z_{t+1})\in\mathbb{R}^{\mathcal{A}}.$ </p>
-    <p> $\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv},\tilde{R}%
-)+\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{%
-t}))\right]$ </p>
-    <p> $\displaystyle\leqslant\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_%
-{t})-L_{t}(A_{t},B_{t}))\right]$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[f_{\theta}({a,B_{t}})-\tilde{R}_{t+1}(a)\mid%
-\theta\right],$ </p>
-    <p> $(III)$ </p>
-    <p> $\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant\mathds{1}_{%
-\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}(f_{\theta}(a,B_{t})-U_{t}(a,B_{t}))+C(%
-1-\mathds{1}_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})}).$ </p>
-    <p> $\Re(T)\geqslant 2\mathbb{P}(\Omega_{T})\Delta\cdot T.$ </p>
-    <p> $\displaystyle\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\>|\>\theta%
-\right]=(I)+(II)+(III)$ </p>
-    <p> $reg_{t}$ </p>
-    <p> $\displaystyle=I(\theta;Z_{0},\ldots,Z_{T-1})=I(\theta;H_{T})$ </p>
-    <p> $w\in\mathbb{R}_{+}$ </p>
-    <p> $\mathbb{P}(\Omega_{t})$ </p>
-    <p> $\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime%
-}_{t}}}+1\right)\sqrt{\beta^{\prime}_{t}}\sigma_{t}(A_{t},B_{t})$ </p>
-    <p> $\Re_{\operatorname{full}}(a;T,\text{RM},\tilde{R})=\mathcal{O}(2c\sqrt{T%
-\mathcal{A}})$ </p>
-    <p> $\sigma^{2}_{t}(a,b)\leqslant k((a,b),(a,b))\leqslant 1$ </p>
-    <p> $\displaystyle\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{%
-t}))\leqslant\frac{1}{\sqrt{t}}.$ </p>
-    <p> $\displaystyle\leqslant 2\exp\left(-\frac{\beta^{\prime}_{t}}{2}\right)$ </p>
-    <p> $\mathbb{P}(f_{\theta}(a,B_{t})\geqslant\tilde{R}_{t+1}(a)\mid\theta)\leqslant%
-\mathcal{O}(1/\sqrt{T}).$ </p>
-    <p> $\displaystyle=\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y_{i})$ </p>
-    <p> $\sqrt{2\log\mathcal{A}\sqrt{T}},0.2\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}},%
-0.05\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ </p>
-    <p> $\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10009.html DELETED Viewed

@@ -1,169 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\mathcal{O}\big{(}\sqrt{T\log\mathcal{A}}+\sqrt{\gamma_{T}\beta T}\big{)}$ </p>
-    <p> $\displaystyle\sim N(\mu_{t}(a,B_{t}),\sigma_{t}(a,B_{t})),j\in[M_{t+1}]$ </p>
-    <p> $\ell^{i}(a^{i},a^{-i})=\sum_{e\in\mathcal{E}}[a^{i}]_{e}t_{e}([a^{i}]_{e}+[g(a%
-^{-i})]_{e}).$ </p>
-    <p> $a^{i}\in\mathcal{A}^{i}\subset\mathbb{R}^{|\mathcal{E}|}$ </p>
-    <p> $\beta=\log(\mathcal{A}T)$ </p>
-    <p> $K^{i}((a^{i},a^{-i}),(b^{i},b^{-i}))=k_{\rm L}(a^{i},b^{i})k_{\rm P}(a^{i}+g(a%
-^{-i}),b^{i}+g(b^{-i})),$ </p>
-    <p> $\tilde{R}_{t2}=\max\limits_{i}y_{ti}$ </p>
-    <p> $\beta_{t}=\sqrt{2\log{\mathcal{A}\sqrt{t}}}$ </p>
-    <p> $\tilde{R}^{OTS}_{t+1}(1st)$ </p>
-    <p> $\log(\text{average regret})\propto\frac{1}{2}\log(MN)$ </p>
-    <p> $\tilde{C}_{t,a}=\sum_{s=0}^{t}\tilde{\Re}_{s,a}$ </p>
-    <p> $\mathcal{O}(\sqrt{T\mathcal{A}\log\mathcal{A}})$ </p>
-    <p> $X_{t+1}=g_{t}(X_{t},r_{t})$ </p>
-    <p> $min(x,c)$ </p>
-    <p> $\pi^{\operatorname{alg}}=(\pi_{t})_{t\in\mathbb{N}}$ </p>
-    <p> $a^{-i},b^{-i}\in\mathcal{A}^{-i}$ </p>
-    <p> $\displaystyle\begin{cases}\textrm{Hedge:}\ &g_{t,a}(X_{t},r_{t})=X_{t,a}\exp%
-\left(\eta_{t}r_{t}(a)\right),\\
-\textrm{RM:}\ &g_{t,a}(X_{t},r_{t})=\max\left(0,\sum\limits_{s=0}^{t}r_{t}(a)-%
-r_{t}(A_{s})\right),\end{cases}$ </p>
-    <p> $\displaystyle\hat{X}_{t+1,a}=\begin{cases}{\tilde{C}^{+}_{t,a}}/{\sum_{a\in%
-\mathcal{A}}\tilde{C}_{t,a}^{+}},&\text{if }\sum_{a\in\mathcal{A}}\tilde{C}_{t%
-,a}^{+}>0,\\
-\text{arbitrary vector on simplex, e.g. }1/\mathcal{A},&\text{otherwise}\end{cases}$ </p>
-    <p> $\sigma_{t}^{2}(a,b)\leqslant\frac{1}{\log(1+\sigma_{w}^{-2})}\log(1+\sigma_{w}%
-^{-2}\sigma^{2}_{t}(a,b)).$ </p>
-    <p> $\displaystyle\left((a+b)^{+}\right)^{2}\leqslant(a^{+})^{2}+2(a^{+})b+b^{2}$ </p>
-    <p> $[a^{i}]_{e}+[g(a^{-i})]_{e}$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}U^{\prime}_{t}(A_{t},B_{t})-L_{t}%
-(A_{t},B_{t})\right]$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})%
-\mid\theta\right].$ </p>
-    <p> $\Sigma_{p}=\sigma_{p}I$ </p>
-    <p> $\mathbb{P}(\Omega_{t})=\mathbb{P}(\omega_{1})\mathbb{P}(\omega_{2}|\Omega_{1})%
-\ldots\mathbb{P}(\omega_{t}|\Omega_{t-1})$ </p>
-    <p> $\displaystyle=[R_{1,A_{0},B_{0}},\ldots,R_{t,A_{t-1},B_{t-1}}]^{\top}$ </p>
-    <p> $\displaystyle\geqslant\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})%
-\mathbb{P}(\Omega_{t-1})+\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\bar{\Omega%
-}_{t-1})\mathbb{P}(\bar{\Omega}_{t-1})$ </p>
-    <p> $\displaystyle=\Re_{\operatorname{full}}(T,\operatorname{adv},\tilde{R})+%
-\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}\left[%
-\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]+\sum_{t=0}^{T-1}\mathbb{E%
-}\left[\operatorname{est}_{t+1}\right]$ </p>
-    <p> $\operatorname{NashRegret}_{t}$ </p>
-    <p> $\theta=\begin{bmatrix}1&1-\Delta\\
-1-\Delta&1\end{bmatrix},$ </p>
-    <p> $\displaystyle\operatorname{NashRegret}(T)=\sum_{t=1}^{T}\operatorname{%
-NashRegret}_{t}$ </p>
-    <p> $I(\theta;R_{t+1,A_{t},B_{t}}|H_{t})$ </p>
-    <p> $f_{\theta}(a,b)=\theta(a,b)$ </p>
-    <p> $P_{X}(i)={X_{i}}/\sum_{i\in\mathcal{A}}X_{i}$ </p>
-    <p> $\displaystyle\sigma_{t}(a,b)=\sqrt{\frac{\sigma_{w}^{2}}{\sigma_{w}^{2}/\sigma%
-^{2}_{p}(a,b)+n_{t}(a,b)}}$ </p>
-    <p> $KL(\overline{x}_{T},x^{)}$ </p>
-    <p> $\Re_{\operatorname{full}}$ </p>
-    <p> $k((a,b),(a,b))=\phi(a,b)^{\top}\Sigma_{p}\phi(a,b)$ </p>
-    <p> $\{\tilde{R}_{t}:t\in\mathbb{Z}_{++}\}$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\tilde{\Re}_{t,a}\mid\theta,H_{t}\right]=\mathbb{%
-E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]-\sum_{a}\hat{X}_{t,a}%
-\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]$ </p>
-    <p> $\displaystyle\tilde{f}^{\operatorname{OTS}}_{t+1}(a,B_{t})$ </p>
-    <p> $I(\theta;H_{T})$ </p>
-    <p> $\mathbb{P}\left(\eta_{j}\leqslant w\right)=\mathbb{P}({\eta_{j}}/{\sigma}%
-\leqslant{w}/{\sigma})=\Phi({w}/{\sigma}).$ </p>
-    <p> $\displaystyle\Re(T)\geqslant 2\mathbb{P}(\Omega_{T})\Delta\cdot T$ </p>
-    <p> $\phi(a,b)$ </p>
-    <p> $\mathcal{E}_{t}(\tilde{R},U,B_{t}):=\{\tilde{R}_{t+1}(a)\geqslant U_{t}(a,B_{t%
-}),\forall a\in\mathcal{A}\}.$ </p>
-    <p> $\displaystyle\stackrel{{\scriptstyle(ii)}}{{\geqslant}}\mathbb{P}\left(\left(%
-\max_{j\in[M]}z^{j}_{t+1}\right)\sigma_{t}(a,B_{t})\geqslant\sqrt{\beta^{%
-\prime}_{t}}\sigma_{t}(a,B_{t}),\forall a\in\mathcal{A}\>|\>H_{t},B_{t}\right)$ </p>
-    <p> $\phi(a,b)=e_{a,b}$ </p>
-    <p> $\displaystyle\mathbb{E}_{0}\left[\tilde{C}_{T,a}\right]$ </p>
-    <p> $f(t,\Delta,\sigma_{n})=\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2%
-\sigma_{n}^{2}\log(M/\delta_{1})}{t+\sigma_{n}^{2}}}$ </p>
-    <p> $\mathcal{A}_{R}=\mathcal{F}\times\mathcal{F}\times\mathcal{F}$ </p>
-    <p> $B_{t}\sim P_{Y_{t}}$ </p>
-    <p> $\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t}))$ </p>
-    <p> $\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}},(r_{t})_{t})=%
-\max_{a\in\mathcal{A}}\mathbb{E}\left[\sum_{t=0}^{T-1}r_{t}(a)-r_{t}(A_{t})%
-\right].$ </p>
-    <p> $\tilde{R}_{t}(1nd)$ </p>
-    <p> $\lim_{t\to\infty}\log\mathbb{P}(\Omega_{t})\geqslant\log c^{\prime}>-\infty$ </p>
-    <p> $0\leqslant L_{t}\leqslant U_{t}\leqslant C$ </p>
-    <p> $\sigma_{n},M$ </p>
-    <p> $0\leqslant{\hat{X}_{t,a}}/{X}_{t,a}\leqslant 1/(1-\gamma_{t})$ </p>
-    <p> $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\Phi^{M}(f)$ </p>
-    <p> $X_{t+1,a}\propto X_{t,a}\exp(\eta_{t}\tilde{R}_{t+1}(a))$ </p>
-    <p> $\displaystyle\mathbb{P}(\left|f_{\theta}(a,B_{t})-{\mu_{t}(a,B_{t})}\right|%
-\geqslant\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b),\forall a\in\mathcal{A}\mid H%
-_{t})\leqslant 2\mathcal{A}\exp(-\beta^{\prime}_{t}/2).$ </p>
-    <p> $X_{2}\propto\max(reg_{1},0)$ </p>
-    <p> $f_{\theta}(a,b)=\theta_{a,b}$ </p>
-    <p> $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{\left|\mathcal{A}%
-\right|}{\gamma_{t}}+\min(\frac{\left|\mathcal{A}\right|}{\gamma_{t}},\frac{%
-\left|\mathcal{A}\right|}{1-\gamma_{t}})(\left|\mathcal{A}\right|-2)\right)%
-\leqslant\frac{2(1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma_{t}}$ </p>
-    <p> $\displaystyle\tilde{f}^{\operatorname{TS},j}_{t+1}(a,B_{t})$ </p>
-    <p> $B_{0:T}=(B_{0}=b_{0},B_{1}=b_{1},\ldots,B_{T-1}=b_{T-1})$ </p>
-    <p> $\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})\geqslant\mathbb{P}(%
-\sum_{k=2}^{t}m_{k}\leqslant 0|\bar{\Omega}_{t-1})$ </p>
-    <p> $(W_{t}:t\in\mathbb{Z}_{++})$ </p>
-    <p> $\displaystyle\tilde{h}(a)=\mathbb{I}_{A=a}\frac{h(A)}{X_{a}},\forall a\in%
-\mathcal{A}$ </p>
-    <p> $\displaystyle\mathbb{E}_{t}\left[R_{t+1,A_{t},B_{t}}^{2}\left(\frac{1}{X_{t,A_%
-{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_{t}}^{2}}+\left|\mathcal{A}\right|%
-\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}}\right)\right]$ </p>
-    <p> $(10$ </p>
-    <p> $f_{\theta}(a,B_{t})$ </p>
-    <p> $\displaystyle\Re^{*}(T,\pi,\theta)\leqslant\Re_{\operatorname{full}}(T,%
-\operatorname{adv},\tilde{R}^{\operatorname{est}})+\sqrt{\log(\mathcal{A}T)I(%
-\theta;H_{T})T}$ </p>
-    <p> $\displaystyle I(\theta;o)=H(o)-H(o\mid\theta)=\frac{1}{2}\log 2\pi e(\sigma(a)%
-^{2}+\sigma_{w}^{2})-\frac{1}{2}\log 2\pi e\sigma_{w}^{2}=\frac{1}{2}\log(1+%
-\sigma_{w}^{-2}\sigma(a))$ </p>
-    <p> $\tilde{r}t=A{ij}+\epsilon_{t}$ </p>
-    <p> $\operatorname{adv}$ </p>
-    <p> $M=\frac{\log(\mathcal{A}\sqrt{T})}{\log\frac{1}{\Phi(\beta^{\prime}_{t})}},%
-\beta^{\prime}_{t}=2\log\mathcal{A}\sqrt{T}$ </p>
-    <p> $\tilde{R}_{t+1,A_{t},B_{t}}$ </p>
-    <p> $X_{t}=[0,1],\forall t\geqslant 2$ </p>
-    <p> $\mathcal{O}\big{(}T^{2/3}\mathcal{A}^{2/3}\big{)}$ </p>
-    <p> $\phi:\mathcal{A}\times\mathcal{B}\mapsto\mathbb{R}^{d}$ </p>
-    <p> $\mathbb{P}\left(\max_{j\in[M]}\eta_{j}\leqslant\sqrt{2\sigma^{2}\log(M/\delta)%
-}\right)\geqslant 1-\delta.$ </p>
-    <p> $(II)$ </p>
-    <p> $k(x,x^{\prime})=(2^{1-\nu}/\Gamma(\nu))r^{\nu}B_{\nu}(r)$ </p>
-    <p> $\mathcal{F}=\{f_{1},f_{2},f_{3}\}$ </p>
-    <p> $\displaystyle=C\sum_{t=0}^{T-1}\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde%
-{R},U,B_{t})}\cup\neg{\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}\right)$ </p>
-    <p> $|\phi(a,b)|\leqslant 1$ </p>
-    <p> $N=528$ </p>
-    <p> $reg_{t}=\tilde{R}_{t}-X_{t}^{T}\tilde{R}_{t}\cdot\mathbf{1}=[m_{t},0],\quad%
-\forall t\geqslant 2$ </p>
-    <p> $R_{t+1,A_{t},B_{t}}=\mathcal{R}(Y_{t+1,A_{t},B_{t}})$ </p>
-    <p> $A_{t}\sim P_{X_{t}}$ </p>
-    <p> $X_{t+1}=g_{t}(X_{t},\tilde{R}_{t+1}).$ </p>
-    <p> $a^{-i}\in\mathcal{A}^{-i}$ </p>
-    <p> $w_{t}(a,b)$ </p>
-    <p> $\Re^{*}(T,\text{IWE-RM})=\mathcal{O}(T^{2/3}\mathcal{A}^{2/3}).$ </p>
-    <p> $(H_{t},\theta,A_{t},B_{t})$ </p>
-    <p> $\mathbb{E}\left[\operatorname{pess}_{t+1}\>|\>\theta\right]=\mathbb{E}\left[%
-\operatorname{est}_{t+1}\>|\>\theta\right]=0$ </p>
-    <p> $\displaystyle\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\>|\>\theta%
-\right]=\mathbb{E}\left[f_{\theta}(a,B_{t})-f_{\theta}(A_{t},B_{t})\>|\>\theta\right]$ </p>
-    <p> $\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))\leqslant 2\mathcal{A}%
-\exp(-\beta_{t}^{\prime}/2)=2/\sqrt{t}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_1001.html DELETED Viewed

@@ -1,127 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\min\{f_{T}(N_{s})\mid 1\leq s\leq t\}<2f_{T}(N)$ </p>
-    <p> $\gamma^{\prime}[V]=\gamma^{\prime\prime}[V]$ </p>
-    <p> $N^{k,n}_{s^{\prime}}$ </p>
-    <p> $\mathcal{V}\setminus(\mathcal{V}^{\prime}\cup\{V_{n}\})$ </p>
-    <p> $V_{n-1}$ </p>
-    <p> $Pa(N,V_{i})$ </p>
-    <p> $Pa(N^{k,n}_{s},V_{n})$ </p>
-    <p> $\Leftarrow(2c-1)\cdot 2^{n-2}-2^{n-2c}\cdot c\binom{2c-1}{c}\geq(6c-6)\cdot 2^%
-{n-4}$ </p>
-    <p> $1\leq s\leq t=2c-1$ </p>
-    <p> $(2^{k}-1)\cdot 1=2^{k}-1$ </p>
-    <p> $1\succ 0$ </p>
-    <p> $0\leq k^{\prime}\leq k$ </p>
-    <p> $\mathcal{V}^{\prime}\subseteq\mathcal{V}\setminus\{V_{n}\}$ </p>
-    <p> $\Leftarrow(8c-4)\cdot 2^{n-4}-2^{n-2c}\cdot c\binom{2c-1}{c}\geq(6c-6)\cdot 2^%
-{n-4}$ </p>
-    <p> $\mathcal{V}^{\prime}\subseteq\mathcal{V}$ </p>
-    <p> $0\leq\kappa\leq\lfloor\frac{t}{2}\rfloor$ </p>
-    <p> $\mathcal{V}=\{V_{1},\ldots,V_{n}\}$ </p>
-    <p> $M(T^{\prime})$ </p>
-    <p> $2c\cdot 2^{n-2}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq\frac{3}{4}(2c-2)\cdot 2^{n-2}$ </p>
-    <p> $o[V_{i}]=b$ </p>
-    <p> $\mathcal{V}^{\prime},\mathcal{V}^{\prime\prime}\subseteq\mathcal{V}$ </p>
-    <p> $Pa(N,V_{n})=P_{N}\subseteq P$ </p>
-    <p> $k\in\{2,\ldots,n-1\}$ </p>
-    <p> $1\leq s^{\prime}\leq\binom{n-1}{k}2^{k}$ </p>
-    <p> $\mathcal{V}\setminus{V_{n}}$ </p>
-    <p> $001$ </p>
-    <p> $P\subseteq\{V_{1},\ldots,V_{n-1}\}$ </p>
-    <p> $o,o^{\prime}$ </p>
-    <p> $2^{k-k^{\prime}}-1$ </p>
-    <p> $I=\{1,\ldots,m\}$ </p>
-    <p> $2^{k+1}-2$ </p>
-    <p> $\{V_{1},\ldots,V_{n-1}\}$ </p>
-    <p> $T_{\varepsilon}=(N^{\varepsilon}_{1},\ldots,N^{\varepsilon}_{t_{\varepsilon}})$ </p>
-    <p> $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})=2^{n-k}$ </p>
-    <p> $4c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-1}\,.$ </p>
-    <p> $(s,o,o^{\prime})$ </p>
-    <p> $1\leq s\leq t$ </p>
-    <p> $1\leq s\leq\binom{n-1}{k}2^{k}$ </p>
-    <p> $o[V]=o^{\prime}[V]=\gamma[V]$ </p>
-    <p> $(c+1)\binom{2c+1}{c+1}\leq(c+2)\cdot 2^{2c-1}$ </p>
-    <p> $f_{T}(N_{s})$ </p>
-    <p> $\operatorname{CPT}(N_{s},V_{n})$ </p>
-    <p> $|Pa(N^{k,n}_{s},V_{n})\cap Pa(N^{k,n}_{s^{\prime}},V_{n})|=:k^{\prime}$ </p>
-    <p> $s^{\prime}\neq s$ </p>
-    <p> $2^{k-k^{\prime}+1}$ </p>
-    <p> $T^{k,n}$ </p>
-    <p> $\mathcal{F}_{t\in O(1)}$ </p>
-    <p> $f_{T}(N)=$ </p>
-    <p> $f_{T}(N)=\begin{cases}t\cdot 2^{n-2}-2^{n-t-1}\cdot c\binom{2c-1}{c}&\text{if %
-}t=2c-1\\
-t\cdot 2^{n-2}-2^{n-t-1}\cdot c\binom{2c}{c}&\text{if }t=2c\\
-\end{cases}$ </p>
-    <p> $2^{k}-1$ </p>
-    <p> $\gamma:b\succ b^{\prime}$ </p>
-    <p> $f_{T^{k,n}}(N^{k,n}_{s})=\sum_{s^{\prime}\neq s}\Delta(N^{k,n}_{s},N^{k,n}_{s^%
-{\prime}})$ </p>
-    <p> $Pa(N^{a},V_{n})\subseteq P$ </p>
-    <p> $\displaystyle\sum_{k^{\prime}=1}^{k-1}\big{[}2^{k-k^{\prime}}\binom{k}{k^{%
-\prime}}\binom{n-k-1}{k^{\prime}}(2^{n-k}-2^{n-2k+k^{\prime}})$ </p>
-    <p> $c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}=(2c+2)\cdot 2^{2c-4}$ </p>
-    <p> $|\mathcal{V}^{\prime}|$ </p>
-    <p> $001,101$ </p>
-    <p> $\Leftarrow 8c\cdot 2^{n-4}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq(6c-3)\cdot 2^{n-4}$ </p>
-    <p> $Pa(N^{k,n}_{s},V_{n})\cap P=k^{\prime}$ </p>
-    <p> $c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}\,,$ </p>
-    <p> $Pa(N^{a},V_{n})\subseteq Pa(N_{s},V_{n})$ </p>
-    <p> $\displaystyle(2^{k}-1)\cdot 2^{n-k}+2^{k}\binom{n-k-1}{k}(2^{n-k}-2^{n-2k})$ </p>
-    <p> $f_{T^{k,n}}(N^{k,n}_{s})\geq(3/2)f_{T^{k,n}}(N)$ </p>
-    <p> $Pa(N,V_{n})\subseteq P$ </p>
-    <p> $t-\kappa$ </p>
-    <p> $\mathcal{V}=\{V_{1},\ldots,V_{n-1}\}$ </p>
-    <p> $P\in\{Pa(N_{s},V_{n})\mid 1\leq s\leq t\}$ </p>
-    <p> $\gamma_{2}:1\succ 0$ </p>
-    <p> $\Leftarrow c\binom{2c-1}{c}\leq(2c+2)\cdot 2^{2c-4}$ </p>
-    <p> $V_{i}\in\mathcal{V}^{\prime}$ </p>
-    <p> $o^{\prime}[V_{i}]=1$ </p>
-    <p> $\gamma\in\operatorname{Inst}(\mathcal{V}^{\prime})$ </p>
-    <p> $2^{p_{s}-1}$ </p>
-    <p> $\frac{2\cdot(2d+1)}{d+1}<4$ </p>
-    <p> $Pa(N,V_{i})=\emptyset$ </p>
-    <p> $f_{T_{n}}(N^{*})$ </p>
-    <p> $\gamma_{2}\in\operatorname{Inst}(P)$ </p>
-    <p> $2\leq k\leq n-1$ </p>
-    <p> $Pa(N_{s},V_{n})$ </p>
-    <p> $\gamma[V_{i}]$ </p>
-    <p> $freq_{M^{\prime}}(0\succ 1)\leq freq_{M^{\prime}}(1\succ 0)$ </p>
-    <p> $CPT(N_{i},V_{3})$ </p>
-    <p> $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}}=2^{n-k}$ </p>
-    <p> $\gamma^{\prime}\in\operatorname{Inst}(P)\setminus\{\gamma\}$ </p>
-    <p> $T^{k,n}=(N^{k,n}_{1},\ldots,N^{k,n}_{t})$ </p>
-    <p> $V\in\mathcal{V}^{\prime}\cap\mathcal{V}^{\prime\prime}$ </p>
-    <p> $\gamma\in\operatorname{Inst}(Pa(N_{s},V_{n}))$ </p>
-    <p> $(o^{\prime},o)$ </p>
-    <p> $V\in\mathcal{V}^{\prime}$ </p>
-    <p> $\gamma_{1}=00$ </p>
-    <p> $Pa(N^{k,n}_{s^{\prime}},V_{n})$ </p>
-    <p> $f_{T^{n-1,n}}(N^{n-1,n}_{s})>(2-\varepsilon)f_{T^{n-1,n}}(N)$ </p>
-    <p> $Inst(\mathcal{V}^{\prime})$ </p>
-    <p> $\gamma_{1}\in\operatorname{Inst}(Pa(N^{k,n}_{s},V_{n}))$ </p>
-    <p> $|P|\leq\max\{|Pa(N_{s},V_{n})\mid 1\leq i\leq t\}$ </p>
-    <p> $1\leq s\leq 2^{n-1}$ </p>
-    <p> $T=(N_{1},\ldots,N_{t})$ </p>
-    <p> $\Leftarrow 2c\cdot 2^{n-2}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq(6c-3)\cdot 2^{n-4}$ </p>
-    <p> $\{b,b^{\prime}\}=\{0,1\}$ </p>
-    <p> $\gamma^{\prime}\in\operatorname{Inst}(\mathcal{V}^{\prime})$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10010.html DELETED Viewed

@@ -1,187 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\tilde{\mathcal{O}}(\sqrt{(M+N)/T})$ </p>
-    <p> $\text{SINR}(a,b;\theta)$ </p>
-    <p> $\sqrt{2\log\mathcal{A}\sqrt{T}}$ </p>
-    <p> $\operatorname{NashRegret}(T)$ </p>
-    <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\Phi(f_{t}(\Delta,\sigma_{n}))$ </p>
-    <p> $(\Delta,\sigma_{n}^{2})$ </p>
-    <p> $\displaystyle\tilde{R}_{t+1}(a)=1-\frac{\mathbb{I}_{A_{t}=a}(1-R_{t+1,A_{t},B_%
-{t}})}{X_{t,a}}$ </p>
-    <p> $w_{t}(a,b)=\sqrt{\beta_{t}I_{t}\left(\theta;R_{t+1,A_{t},B_{t}}\mid A_{t}=a,B_%
-{t}=b\right)}\quad\text{ with }\quad\beta_{t}=\frac{2\beta^{\prime}_{t}}{\log(%
-1+\sigma_{w}^{-2})}.$ </p>
-    <p> $\mu_{t}(a,b)=\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}+\sigma^{2}{\bm{I}}_{t%
-})^{-1}\mathbf{R}_{t}$ </p>
-    <p> $\eta_{1},\eta_{2},\ldots,\eta_{M}$ </p>
-    <p> $z_{t},z^{\prime}_{t}\sim\mathcal{N}(0,1)$ </p>
-    <p> $\theta\in\mathbb{R}^{\mathcal{A}\times\mathcal{B}}$ </p>
-    <p> $\beta_{t}=2\beta^{\prime}_{t}/\log(1+\sigma_{w}^{-2})$ </p>
-    <p> $\Re_{\operatorname{full}}(T,{\operatorname{Hedge}})=\mathcal{O}(\sqrt{T\log%
-\mathcal{A}})$ </p>
-    <p> $\displaystyle:=\max_{j\in[M_{t+1}]}\tilde{f}^{\operatorname{TS},j}_{t+1}(a,B_{%
-t}),$ </p>
-    <p> $\displaystyle\leqslant\max_{a\in\mathcal{A}}\Re_{\operatorname{full}}(a;T,%
-\operatorname{adv},\tilde{R})+\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}%
-^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]+%
-\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\right]$ </p>
-    <p> $\tilde{r}_{t}=A_{ij}+\epsilon_{t}$ </p>
-    <p> $N(\mu_{p},\Sigma_{p})$ </p>
-    <p> $M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}$ </p>
-    <p> $\displaystyle=\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0)$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}\operatorname{est}_{t+1}\right]$ </p>
-    <p> $\displaystyle\mathbb{E}_{t}\left[\tilde{R}_{t+1}(a)\right]=1-\mathbb{E}_{t}%
-\left[\mathbb{I}_{A_{t}=a}\frac{1-R_{t+1,a,B_{t}}}{X_{t,a}}\right]=1-\mathbb{E%
-}_{t}\left[\mathbb{I}_{A_{t}=a}\right]\frac{1-\mathbb{E}_{t}\left[f_{\theta}(a%
-,B_{t})\right]}{X_{t,a}}=\mathbb{E}_{t}\left[f_{\theta}(a,B_{t})\right].$ </p>
-    <p> $\sigma_{t}(a,b)$ </p>
-    <p> $\displaystyle=k((A_{i},B_{i}),(A_{j},B_{j}))$ </p>
-    <p> $\Re(T,\pi^{\operatorname{alg}},B_{0:T},\theta)$ </p>
-    <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]=\mathbb{E}_{0%
-}\left[\sum_{t=1}^{T}\mathbb{E}_{t}\left[\tilde{\Re}_{t,a}\right]+\sum_{t=1}^{%
-T}2\gamma_{t}\right]=\mathbb{E}_{0}\left[\tilde{C}_{T,a}+\sum_{t=1}^{T}2\gamma%
-_{t}\right]\leqslant\sqrt{\sum_{t=1}^{T}\frac{2(1+\sigma_{w}^{2})\left|%
-\mathcal{A}\right|^{2}}{\gamma_{t}}}+2\gamma_{t}$ </p>
-    <p> $\displaystyle=\mathbb{P}(\tilde{R}_{t+1}(a)\geqslant U_{t}(a,B_{t}),\forall a%
-\in\mathcal{A}\mid H_{t},B_{t})$ </p>
-    <p> $\Re_{t}(a):=\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta,H_{t%
-}\right]=\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]-\sum_{a}{X%
-}_{t,a}\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]$ </p>
-    <p> $\displaystyle\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A^{*},B_{t}}-R_{t+1,A_{t},%
-B_{t}}\>|\>\theta\right],$ </p>
-    <p> $\displaystyle\leqslant\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{%
-R})+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta%
-\right]+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\>|\>\theta\right]$ </p>
-    <p> $\mathbb{E}\left[\tilde{R}_{t+1}(a)\>|\>H_{t},\theta\right]=\mathbb{E}\left[f_{%
-\theta}(a,B_{t})\>|\>H_{t},\theta\right]=\mathbb{E}\left[g_{\theta}(e_{a},Y_{t%
-})\>|\>H_{t},\theta\right]$ </p>
-    <p> $\displaystyle\leqslant\mathbb{E}\left[\sum_{t=0}^{T-1}\sqrt{\left(\sqrt{2\log(%
-M\sqrt{t})/\beta^{\prime}_{t}}+1\right)^{2}\beta_{t}I_{t}(\theta;A_{t},B_{t},R%
-_{t+1,A_{t},B_{t}})}\right]$ </p>
-    <p> $\Re_{\operatorname{full}}(T,\text{RM},\tilde{R}^{\operatorname{est}})=\mathcal%
-{O}(\sqrt{T\mathcal{A}})$ </p>
-    <p> $(a=b)$ </p>
-    <p> $\mathbb{P}(\Omega_{t})\geqslant c$ </p>
-    <p> $\beta^{\prime}_{t}=2\log\mathcal{A}\sqrt{t}.$ </p>
-    <p> $\displaystyle\Re^{*}(T,\pi^{\operatorname{alg}})=\sup_{B_{0:T}\in\mathcal{B}^{%
-T}}\Re(T,\pi^{\operatorname{alg}},B_{0:T})=o(T),$ </p>
-    <p> $\mathbf{K}_{t}$ </p>
-    <p> $\tilde{R}^{OTS}_{t+1}(1st)>\tilde{R}^{OTS}_{t+1}(2nd)$ </p>
-    <p> $\displaystyle\leqslant\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}%
-\mathbb{E}\left[C(1-\mathds{1}_{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap%
-\mathcal{E}^{c}_{t}(f_{\theta},B_{t})})\mid\theta\right]\right]$ </p>
-    <p> $\nu\rightarrow\infty$ </p>
-    <p> $\displaystyle\quad+C\left(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R},U,A_{t},%
-B_{t})}\mathds{1}_{\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}\right)$ </p>
-    <p> $\displaystyle X_{t,a}=(1-\gamma_{t})\hat{X}_{t,a}+\gamma_{t}(1/\mathcal{A}),%
-\forall a\in\mathcal{A}$ </p>
-    <p> $\textrm{Hedge:}\ g_{t,a}(X_{t},r_{t})=X_{t,a}\exp(\eta_{t}r_{t}(a)),\quad%
-\textrm{RM:}\ g_{t,a}(X_{t},r_{t})=\max\left(0,\sum_{s=0}^{t}r_{t}(a)-r_{t}(A_%
-{s})\right).$ </p>
-    <p> $\displaystyle\mathcal{E}_{t}(f_{\theta},B_{t}):=\{\forall a\in\mathcal{A},f_{%
-\theta}(a,B_{t})\in[L_{t}(a,B_{t}),U_{t}(a,B_{t})]\}.$ </p>
-    <p> $reg_{1}$ </p>
-    <p> $f_{\theta}(a,b)\mid H_{t}$ </p>
-    <p> $[a^{i}]_{e}=0$ </p>
-    <p> $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}R_{t+1,A_{t},B_{t}}^{2%
-}\left(\left(\frac{\mathbb{I}_{A_{t}=a}}{X_{t,A_{t}}}\right)^{2}-\frac{2\hat{X%
-}_{t,A_{t}}}{X_{t,A_{t}}^{2}}\mathbb{I}_{A_{t}=a}+\frac{\hat{X}_{t,A_{t}}^{2}}%
-{X_{t,A_{t}}^{2}}\right)\right]$ </p>
-    <p> $\tilde{R}_{t1}=\max\limits_{i}x_{ti}$ </p>
-    <p> $\displaystyle\mathbb{P}(\neg\mathcal{E}_{t}(\tilde{R},U,B_{t}))\leqslant\frac{%
-1}{\sqrt{t}}.$ </p>
-    <p> $\displaystyle\mathbf{K}_{t}(i,j)$ </p>
-    <p> $\nu>1$ </p>
-    <p> $\tilde{R}_{t+1}(a)$ </p>
-    <p> $\tilde{R}:=\{\tilde{R}_{t}:t\in\mathbb{Z}_{++}\}$ </p>
-    <p> $\mathbb{P}(A_{t}\in\cdot\>|\>\pi_{t})=\mathbb{P}(A_{t}\in\cdot\>|\>H_{t})=\pi_%
-{t}(\cdot)$ </p>
-    <p> $\displaystyle\begin{cases}\textrm{UCB:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}=%
-\mu_{t}(a,B_{t})+\beta_{t}\sigma_{t}(a,B_{t}),\\
-&\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.%
-\\
-\textrm{TS:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}\sim N(\mu_{t}(a,B_{t}),%
-\sigma_{t}(a,B_{t})),\\
-&\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.%
-\end{cases}$ </p>
-    <p> $\Sigma_{0}=\Sigma_{p}$ </p>
-    <p> $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{X_{t,a}}+\sum_{%
-a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|-2\right)\right)$ </p>
-    <p> $\mathds{1}(a=b)$ </p>
-    <p> $\operatorname{est}$ </p>
-    <p> $\mathbf{k}_{t}((a,b))$ </p>
-    <p> $\mathcal{E}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})$ </p>
-    <p> $\displaystyle\leqslant\sum_{a}\left(\left|\gamma_{t}\hat{X}_{t,a}\right|+\left%
-|\gamma_{t}/\mathcal{A}\right|\right)=2\gamma_{t}$ </p>
-    <p> $\mathcal{B}={1,\ldots,\left|\mathcal{B}\right|}$ </p>
-    <p> $M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}.$ </p>
-    <p> $\tilde{R}_{t+1}(a)=\min(\tilde{f}_{t+1}(a,B_{t}),1)$ </p>
-    <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant\sqrt%
-{T}\sqrt{\frac{2(1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma}}+2%
-\gamma T,$ </p>
-    <p> $\displaystyle\mathbb{P}\left(\max_{j\in[M]}\eta_{j}\geqslant w\right)=1-\left[%
-\Phi\left(\frac{w}{\sigma}\right)\right]^{M}$ </p>
-    <p> $Z_{t}=(A_{t},B_{t},R_{t+1,A_{t},B_{t}}))$ </p>
-    <p> $(r_{t})_{t\in[T]}\in[0,1]^{\mathcal{A}\times T}$ </p>
-    <p> $\displaystyle\stackrel{{\scriptstyle(iii)}}{{=}}1-\Phi\left(\sqrt{\beta_{t}^{%
-\prime}}\right)^{M}.$ </p>
-    <p> $\displaystyle>-\infty,$ </p>
-    <p> $KL(\overline{y}_{T},y^{)}$ </p>
-    <p> $\displaystyle\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t})=\mathds{1}_{\mathcal{E}^{o%
-}_{t}(\tilde{R},U,B_{t})}(\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t}))+(1-\mathds{1%
-}_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})})(\tilde{R}_{t+1}(a)-f_{\theta}(a,%
-B_{t}))$ </p>
-    <p> $\mathcal{A}={1,\ldots,\left|\mathcal{A}\right|}$ </p>
-    <p> $\Re^{*}(T,\text{UCB-Hedge})=\mathcal{O}(\sqrt{T\log\mathcal{A}}+\sqrt{\gamma_{%
-T}\beta T}),\ \Re^{*}(T,\text{UCB-RM})=\mathcal{O}(\sqrt{T\mathcal{A}}+\sqrt{%
-\gamma_{T}\beta T}).$ </p>
-    <p> $\Re_{\operatorname{full}}(T,\text{Hedge},\tilde{R}^{\operatorname{est}})=%
-\mathcal{O}(\sqrt{T\log\mathcal{A}})$ </p>
-    <p> $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad
-L%
-=(\mu_{t}(a,b)-\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}).$ </p>
-    <p> $c(\Delta,\sigma_{n})\approx 0.54$ </p>
-    <p> $k_{\rm P}(\cdot,\cdot)$ </p>
-    <p> $\sigma_{n}>0$ </p>
-    <p> $k((a,b),(a^{\prime},b^{\prime}))=\mathbb{E}\left[(f_{\theta}(a,b)-\mu(a,b))(f_%
-{\theta}(a^{\prime},b^{\prime})-\mu(a^{\prime},b^{\prime}))\right]$ </p>
-    <p> $\displaystyle=[k((A_{0},B_{0}),(a,b)),\ldots,k((A_{t-1},B_{t-1}),(a,b))]^{\top}$ </p>
-    <p> $\sigma_{t}^{2}(a,b)$ </p>
-    <p> $\displaystyle\leqslant\sum_{a}\left(\left({\tilde{C}_{T-1,a}^{+}}\right)^{2}+2%
-{\tilde{C}_{T-1,a}^{+}}{\tilde{\Re}_{T,a}}+\left(\tilde{\Re}_{T,a}\right)^{2}\right)$ </p>
-    <p> $A\in\mathbb{R}^{10\times 5}$ </p>
-    <p> $\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})$ </p>
-    <p> $g(a^{-i})=\sum_{j\neq i}a^{j}$ </p>
-    <p> $\gamma_{t}=\gamma$ </p>
-    <p> $P_{a}(\theta)$ </p>
-    <p> $\displaystyle\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}$ </p>
-    <p> $\sum\limits_{k=1}^{t}\frac{1}{k+a}\leqslant\int_{0}^{t}\frac{1}{k+a}dk$ </p>
-    <p> $y^{*}=\operatorname*{arg\,min}\limits_{y\in\Delta}y^{T}(Ax),$ </p>
-    <p> $\displaystyle\mathbb{P}(\max_{j\in[M]}\eta_{j}\geqslant w)=1-\mathbb{P}(\max_{%
-j\in[M]}\eta_{j}\leqslant w)=1-\mathbb{P}(\forall j\in[M],\eta_{j}\leqslant w)%
-=1-\left[\Phi\left(\frac{w}{\sigma}\right)\right]^{M}.$ </p>
-    <p> $m_{1}\leqslant 0$ </p>
-    <p> $\tilde{R}_{t+1}\in[0,C]$ </p>
-    <p> $\mathcal{D}(\mathcal{A}^{i})$ </p>
-    <p> $\displaystyle\sum_{t=1}^{T}\frac{1}{\sqrt{t}}=2\sum_{t=1}^{T}\frac{t-(t-1)}{%
-\sqrt{t}+\sqrt{t}}\leqslant 2\sum_{t=1}^{T}\frac{t-(t-1)}{\sqrt{t}+\sqrt{t-1}}%
-=2\sum_{t=1}^{T}(\sqrt{t}-\sqrt{t-1})=2\sqrt{T}$ </p>
-    <p> $50)$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10011.html DELETED Viewed

@@ -1,163 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\mu_{0}=\mu_{p}$ </p>
-    <p> $\text{Gap}(x,y)=\max_{\left(x^{\prime},y^{\prime}\right)\in\Delta}\mathbb{E}%
-\left[\phi\left(x^{\prime},y\right)-\phi\left(x,y^{\prime}\right)\right]$ </p>
-    <p> $Y_{t+1,A_{t},B_{t}}=f_{\theta}(A_{t},B_{t})+\eta_{t},$ </p>
-    <p> $(1-x)^{M}\leqslant e^{-Mx}$ </p>
-    <p> $\theta(a,b)\in\mathbb{R}$ </p>
-    <p> $\Re_{\operatorname{full}}(T,{\operatorname{Hedge}},(r_{t})_{t})=\mathcal{O}(%
-\sqrt{T\log\mathcal{A}}),\ \Re_{\operatorname{full}}(T,{\operatorname{RM}},(r_%
-{t})_{t})=\mathcal{O}(\sqrt{T\mathcal{A}}).$ </p>
-    <p> $f_{t}(\Delta,\sigma_{n}^{2})$ </p>
-    <p> $\displaystyle=\max_{(r_{t})_{t}}\Re_{\operatorname{full}}(T,{\operatorname{adv%
-}},(r_{t})_{t}).$ </p>
-    <p> $I(\theta;o)=\frac{1}{2}\log\left(1+{\sigma_{w}^{-2}\sigma(a)}\right)$ </p>
-    <p> $(\overline{x}_{T},\overline{y}_{T})$ </p>
-    <p> $N\left(0,\sigma^{2}\right)$ </p>
-    <p> $v^{+}=(v_{a}^{+})_{a\in\mathcal{A}}$ </p>
-    <p> $\displaystyle\leqslant\mathcal{O}\left(\sqrt{T\log(\mathcal{A}T)I(\theta;H_{T}%
-)}\right).$ </p>
-    <p> $\pi=(\pi_{t})_{t\in\mathbb{N}}$ </p>
-    <p> $\mathbb{I}_{A_{t}=a}R_{t+1,A_{t},B_{t}}=\mathbb{I}_{A_{t}=a}R_{t+1,a,B_{t}}$ </p>
-    <p> $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\mathbb{E}_{t}\left[R_{t+1,A_{%
-t},B_{t}}^{2}\left(\frac{1}{X_{t,A_{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_%
-{t}}^{2}}+\left|\mathcal{A}\right|\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}%
-}\right)\right]\right],$ </p>
-    <p> $M=5,10,20,50,70,100$ </p>
-    <p> $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}\left(\frac{\mathbb{I}%
-_{A_{t}=a}R_{t+1,A_{t},B_{t}}}{X_{t,a}}-R_{t+1,A_{t},B_{t}}\frac{\hat{X}_{t,A_%
-{t}}}{X_{t,A_{t}}}\right)^{2}\right]$ </p>
-    <p> $c\geqslant 0$ </p>
-    <p> $k(x,x^{\prime})=x^{\top}x^{\prime}$ </p>
-    <p> $U^{\prime},U,L$ </p>
-    <p> $\Re_{\operatorname{full}}(T,{\operatorname{RM}})=\mathcal{O}(\sqrt{T\mathcal{A%
-}})$ </p>
-    <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\Phi\left(\frac{(1-\Delta)\left(t%
--\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}+2\sigma_{n}^{2}\ln{\sigma_{n}}-1/(%
-\sigma_{n}^{2}+1)\right)}{\sqrt{t+\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}-2%
-\sigma_{n}^{2}\ln{\sigma_{n}}-(\sigma_{n}^{2}+2)/(\sigma_{n}^{2}+1)}}\right)$ </p>
-    <p> $A\in\mathcal{R}^{10\times 5}$ </p>
-    <p> $U^{\prime}=(\mu_{t}(a,b)+\sqrt{2\log(M\sqrt{t})}\sigma_{t}(a,b):t\in\mathbb{N}).$ </p>
-    <p> $\tilde{R}^{OTS}_{t+1}(2nd)$ </p>
-    <p> $\displaystyle\leqslant\sqrt{\mathbb{E}\left[\sum_{t=0}^{T-1}\left(\sqrt{2\log(%
-M\sqrt{t})/\beta^{\prime}_{t}}+1\right)^{2}\beta_{t}\right]}\sqrt{\mathbb{E}%
-\left[\sum_{t=0}^{T-1}I_{t}(\theta;A_{t},B_{t},R_{t+1,A_{t},B_{t}})\right]}$ </p>
-    <p> $(P^{*},Q^{*})$ </p>
-    <p> $\pi_{\operatorname{alg}}$ </p>
-    <p> $\pm\tilde{r}_{t}$ </p>
-    <p> $\displaystyle\Sigma_{t+1}=\left(\Sigma_{t}^{-1}+\frac{1}{\sigma_{w}^{2}}\phi(A%
-_{t},B_{t})\phi(A_{t},B_{t})^{\top}\right)^{-1}$ </p>
-    <p> $\mathbb{P}(\Omega_{t})\geqslant c(\Delta,\sigma_{n})$ </p>
-    <p> $\mathcal{E}^{c}_{t}(f_{\theta},B_{t})$ </p>
-    <p> $\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{t}%
-))\right]$ </p>
-    <p> $\mu_{t}(a,b)=\phi(a,b)^{\top}\mu_{t}$ </p>
-    <p> $c=e^{c^{\prime}}$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\tilde{h}(a)\right]=\mathbb{E}\left[\mathbb{I}_{A%
-=a}\right]h(a)/X_{a}=h(a).$ </p>
-    <p> $\displaystyle\stackrel{{\scriptstyle(i)}}{{\geqslant}}\mathbb{P}(\tilde{R}_{t+%
-1}(a)\geqslant\min\{\mu_{t}(a,B_{t})+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{%
-t}),c\},\forall a\in\mathcal{A}\>|\>H_{t},B_{t})$ </p>
-    <p> $\displaystyle=\sum_{t=1}^{\infty}\log\mathbb{P}(Z\leqslant-\frac{\mu_{t}}{%
-\sigma_{t}})$ </p>
-    <p> $f_{\theta}(a)\sim N(\mu(a),\sigma(a))$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{t=0}^{T-1}\mathbb{E}\left[C(1-\mathds{1}_{%
-\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t}%
-)})\right]\right]$ </p>
-    <p> $31.9\pm 1.6$ </p>
-    <p> $46.4\pm 0.8$ </p>
-    <p> $\text{Uniform}[0.05,0.1]$ </p>
-    <p> $71.6\%\rightarrow 78.9\%$ </p>
-    <p> $65.2\pm 1.1$ </p>
-    <p> $\phi:\mathcal{X}\rightarrow\mathbb{R}^{k}$ </p>
-    <p> $93.8\pm 0.3$ </p>
-    <p> $b\in{\bm{b}}$ </p>
-    <p> ${\rho^{\prime}},{\alpha^{\prime}},{\beta^{\prime}}$ </p>
-    <p> $\mathcal{A}_{\text{ft}}(x^{\prime}\mid x)=1$ </p>
-    <p> $\frac{{\beta}}{{\gamma}}$ </p>
-    <p> $\displaystyle\mathcal{L}_{\text{pretrain}}(\phi)=-2\cdot$ </p>
-    <p> ${\alpha^{\prime}}$ </p>
-    <p> $67.84\pm 0.70$ </p>
-    <p> $32.1\pm 0.8$ </p>
-    <p> $31.2\%\rightarrow 31.9\%$ </p>
-    <p> $50.6\pm 1.3$ </p>
-    <p> $91.4\pm 0.9$ </p>
-    <p> $\displaystyle\mathcal{A}_{\text{prop}}({x^{\prime}}\mid x)=\begin{cases}{\rho^%
-{\prime}}&x={x^{\prime}}\\
-{\alpha^{\prime}}&\{x^{\prime},x\}\in\{\{1,3\},\{3,5\},\{5,7\},\{2,4\},\{4,6\}%
-,\{6,8\},\{1,7\},\{2,8\}\\
-{\beta^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}\\
-{\gamma^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,4\},\{2,3\},\{3,6\},\{4,5\},\{5,8%
-\},\{6,7\},\{1,8\},\{2,7\}\}\\
-\end{cases}.$ </p>
-    <p> $30.4\%\rightarrow 34.5\%$ </p>
-    <p> $\text{21M}\rightarrow\text{69M}$ </p>
-    <p> $27.9\pm 0.5$ </p>
-    <p> $y_{x}\in\mathbb{R}^{k}$ </p>
-    <p> $p^{*}(\cdot\mid x)$ </p>
-    <p> $\text{loguniform}(0.95z,\;\text{min}(1.5(1+z)-1,\;5z))$ </p>
-    <p> ${\bm{X}}^{\prime}=\{\tilde{{\bm{X}}^{\prime}}_{:,j}+\epsilon_{j}\}_{j=1}^{W},{%
-\bm{X}}^{\prime}_{\text{err}}=\left\{\sqrt{\tilde{{\bm{X}}^{\prime}}_{\text{%
-err},:,j}^{2}+\epsilon_{j}^{2}}\right\}_{j=1}^{W}.$ </p>
-    <p> $z=z^{\prime}$ </p>
-    <p> $46.4\%\rightarrow 50.6\%$ </p>
-    <p> $\{F(t_{i},w_{j})\}_{i=1,j=1}^{T,W}$ </p>
-    <p> ${\rho^{\prime}}>\max\{{\alpha^{\prime}},{\beta^{\prime}}\}$ </p>
-    <p> $\mathbf{51.7\pm 0.8}$ </p>
-    <p> $65.2\%\rightarrow 91.4\%$ </p>
-    <p> $64.5\pm 1.2$ </p>
-    <p> $i\in\{1,...,T\},j\in\{1,...,W\}$ </p>
-    <p> $0.320\pm 0.009$ </p>
-    <p> $92.3\pm 0.7$ </p>
-    <p> $\mathcal{T}=\{3,4,5,6,7,8\}$ </p>
-    <p> $0.304\pm 0.010$ </p>
-    <p> $\mathcal{A}_{\text{ft}}(x^{\prime}\mid x)=\sum_{z^{\prime}}T(x^{\prime}\mid x,%
-z^{\prime})\hat{p_{T}}(z^{\prime}\mid z)$ </p>
-    <p> $67.54\pm 0.32$ </p>
-    <p> $\mathbf{79.90\pm 0.60}$ </p>
-    <p> $\{F^{\prime}_{\text{err}}(t_{\text{new},i},w_{\text{new},i})\}_{i=1,j=1}^{T,W}$ </p>
-    <p> $\widehat{f}_{\text{erm}}\in\operatorname*{arg\,min}_{f}\mathcal{L}_{\text{ERM}%
-}(f)$ </p>
-    <p> $\{F(t_{i},b_{j})\}_{i=1,j=1}^{T,W},\{F_{\text{err}}(t_{i},b_{j})\}_{i=1,j=1}^{%
-T,W}$ </p>
-    <p> $\mathbf{0.247\pm 0.005}$ </p>
-    <p> ${\bm{X}}^{\prime}_{\text{err}}$ </p>
-    <p> $\mathcal{L}_{0-1}(\widehat{f})=0$ </p>
-    <p> $T(x^{\prime}|x,z^{\prime})$ </p>
-    <p> $((y_{1},d_{1}),(y_{2},d_{2}))$ </p>
-    <p> $x\in\{1,3,5,7\}$ </p>
-    <p> $46.4\%\rightarrow 48.5\%$ </p>
-    <p> $\displaystyle\mathcal{A}_{\text{pre}}({x^{\prime}}\mid x)=\begin{cases}{\rho^{%
-\prime}}&x={x^{\prime}}\\
-{\alpha^{\prime}}&\{x^{\prime},x\}\in\{\{1,4\},\{3,5\},\{5,7\},\{2,5\},\{4,6\}%
-,\{6,8\},\{1,8\},\{2,7\}\\
-{\beta^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}\\
-{\gamma^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,3\},\{2,4\},\{3,6\},\{4,5\},\{5,8%
-\},\{6,7\},\{1,7\},\{2,8\}\}\\
-\end{cases}.$ </p>
-    <p> $y_{x}=1$ </p>
-    <p> $0.27\rightarrow 0.25$ </p>
-    <p> $96.7\pm 0.0$ </p>
-    <p> $48.5\pm 3.2$ </p>
-    <p> $\widehat{\phi}=\operatorname*{arg\,min}_{\phi}\mathcal{L}_{\text{pretrain}}(\phi)$ </p>
-    <p> $46.4\%\rightarrow 47.5\%$ </p>
-    <p> $d_{-}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10012.html DELETED Viewed

@@ -1,142 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $0.277\pm 0.004$ </p>
-    <p> $77.72\pm 0.59$ </p>
-    <p> $\mathcal{T}=\{x\in\mathcal{T}:d_{x}=2\}$ </p>
-    <p> $0.274\pm 0.016$ </p>
-    <p> $\displaystyle\mathcal{L}_{\text{ERM}}(f)=\mathbb{E}_{x\sim P_{S},x^{\prime}%
-\sim\mathcal{A}_{\text{ft}}(\cdot\mid x)}[\ell(f(x^{\prime}),y_{x})].$ </p>
-    <p> $\mathbf{36.9\pm 0.7}$ </p>
-    <p> $\mathcal{A}_{\text{ft}}$ </p>
-    <p> $34.5\pm 1.4$ </p>
-    <p> $71.6\%\rightarrow 68.8\%$ </p>
-    <p> ${\beta}>{\gamma}$ </p>
-    <p> $\mathcal{L}_{0-1}(\widehat{f}_{\text{erm}})=1/3$ </p>
-    <p> $\mathbf{98.5\pm 0.0}$ </p>
-    <p> $\text{SNR}(x,x_{\text{err}})=\frac{|x|}{x_{\text{err}}}$ </p>
-    <p> $30.4\%\rightarrow 32.1\%$ </p>
-    <p> $x\in\{2,4,6,8\}$ </p>
-    <p> $\text{SNR}({\bm{X}}^{\prime}_{i,j},{\bm{X}}^{\prime}_{\text{err},i,j})\geq 5$ </p>
-    <p> $\displaystyle\mathcal{A}_{\text{ft}}({x^{\prime}}\mid x)=\begin{cases}1&\{{x^{%
-\prime}},x\}\in\{1,4\},\{2,3\}\\
-1&x={x^{\prime}}\text{ and }x\notin\{1,2\}\\
-0&\text{otherwise}\end{cases}$ </p>
-    <p> $61.26\pm 1.10$ </p>
-    <p> $\widehat{f}_{\text{erm}}$ </p>
-    <p> $x,{x^{\prime}}$ </p>
-    <p> $\text{loss}_{\text{ft}}:\mathbb{R}^{n}\times\mathcal{Y}\rightarrow\mathbb{R}$ </p>
-    <p> $86.1\pm 1.3$ </p>
-    <p> $\tilde{x^{\prime}}$ </p>
-    <p> $96.7\%\rightarrow 98.5\%$ </p>
-    <p> $P_{U}=\beta P_{S}+(1-\beta)P_{T}$ </p>
-    <p> $\min\{{\alpha^{\prime}},{\beta^{\prime}}\}>{\gamma^{\prime}}$ </p>
-    <p> $F^{\prime},F^{\prime}_{\text{err}}$ </p>
-    <p> $z^{\prime}\sim\text{loguniform}(0.95z_{\text{orig}},\;\text{min}(1.5(1+z_{%
-\text{orig}})-1,\;5z_{\text{orig}}))$ </p>
-    <p> $90.5\pm 0.4$ </p>
-    <p> $\widehat{f}(x)=\operatorname*{arg\,max}_{i\in[r]}(\widehat{B}\widehat{\phi}(x)%
-)_{i}$ </p>
-    <p> $92.3\pm 0.2$ </p>
-    <p> ${x^{\prime}}\in\mathcal{X}$ </p>
-    <p> $65.2\rightarrow 91.4$ </p>
-    <p> $\tilde{{\bm{X}}^{\prime}}_{\text{err}}=10^{0.4(d(z^{\prime})-d(z_{\text{orig}}%
-))}\{F^{\prime}_{\text{err}}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T%
-,W},$ </p>
-    <p> $\alpha>\gamma+\beta$ </p>
-    <p> $\displaystyle\mathcal{L}_{\text{ft}}(h)=\mathbb{E}_{x\sim P_{S},y\sim p^{*}(%
-\cdot\mid x),{x^{\prime}}\sim\mathcal{A}_{\text{ft}}(\cdot|x)}[\text{loss}_{%
-\text{ft}}(h(\widehat{\phi}({x^{\prime}})),\;y;\;\theta)]$ </p>
-    <p> $\displaystyle\mathcal{L}_{\text{pretrain}}(\phi)=\mathbb{E}_{(x,x^{+})\sim S_{%
-+}}[d_{+}(\phi(x),\phi(x^{+}))]-\mathbb{E}_{x,{x^{\prime}}\sim P_{U}}[d_{-}(%
-\phi(x),\phi({x^{\prime}}))].$ </p>
-    <p> $93.8\%\rightarrow 94.9\%$ </p>
-    <p> $61.3\%\rightarrow 67.8\%$ </p>
-    <p> $d(z)$ </p>
-    <p> $0.32\rightarrow 0.28$ </p>
-    <p> $\{F_{\text{err}}(t_{i},w_{j})\}_{i=1,j=1}^{T,W}$ </p>
-    <p> $\mathcal{A}_{\text{pre}}(\cdot\mid x)$ </p>
-    <p> ${\bm{t}}_{\text{new}}=\frac{1+z^{\prime}}{1+z_{\text{orig}}}{\bm{t}}$ </p>
-    <p> $\hat{p_{T}}(z^{\prime}|z)$ </p>
-    <p> $78.84\pm 0.97$ </p>
-    <p> $40.5\pm 1.6$ </p>
-    <p> $\mathcal{A}(\cdot|x)$ </p>
-    <p> $\mathcal{A}_{\text{prop}}$ </p>
-    <p> $78.9\%\rightarrow 68.8\%$ </p>
-    <p> $\gamma>\beta$ </p>
-    <p> $\hat{p_{T}}(z^{\prime}\mid z)$ </p>
-    <p> $46.4\pm 0.5$ </p>
-    <p> $0.286\pm 0.007$ </p>
-    <p> $30.4\rightarrow 31.2$ </p>
-    <p> $30.4\%\rightarrow 37.2\%$ </p>
-    <p> $10^{\text{Uniform}[-3,-2]}$ </p>
-    <p> $36.1\pm 0.7$ </p>
-    <p> $89.3\%\rightarrow 92.3\%$ </p>
-    <p> $30.4\pm 0.6$ </p>
-    <p> $\displaystyle\begin{cases}{\rho}&y_{1}=y_{2},d_{1}=d_{2}~{}\text{~{}~{}(same %
-class, same domain)}\\
-{\alpha}&y_{1}=y_{2},d_{1}\neq d_{1}\text{~{}~{}(same class, different domain)%
-}\\
-{\beta}&y_{1}\neq y_{2},d_{1}=d_{2}\text{~{}~{}(different class, same domain)}%
-\\
-{\gamma}&y_{1}\neq y_{2},d_{1}\neq d_{2}\text{~{}~{}(different class and %
-domain)}\\
-\end{cases},$ </p>
-    <p> $68.75\pm 0.95$ </p>
-    <p> $\mathbf{94.9\pm 0.4}$ </p>
-    <p> $10^{\text{Uniform}[-5,-2]}$ </p>
-    <p> $\mathcal{S}=\{1,2\}$ </p>
-    <p> $F,F_{\text{err}}:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ </p>
-    <p> $h:\mathbb{R}^{k}\rightarrow\mathbb{R}^{n}$ </p>
-    <p> $77.40$ </p>
-    <p> $\mathcal{Y}=\{1,\dots,k\}$ </p>
-    <p> $\mathbf{80.54\pm 1.20}$ </p>
-    <p> $S_{+}(x,{x^{\prime}})$ </p>
-    <p> $71.59\pm 1.10$ </p>
-    <p> $z_{\text{orig}}$ </p>
-    <p> $\displaystyle\mathbb{E}_{(x,x^{+})\sim S_{+}}\left[\phi(x)^{\top}\phi(x^{+})%
-\right]+\mathbb{E}_{x,x^{\prime}\sim P_{U}}\left[\left(\phi(x)^{\top}\phi(x^{%
-\prime})\right)^{2}\right].$ </p>
-    <p> $L_{T}(f)=\mathbb{E}_{x\sim P_{T},y\sim p^{*}(\cdot\mid x)}[\ell(f(x),y)]$ </p>
-    <p> $89.3\pm 0.9$ </p>
-    <p> $\mathbf{51.4\pm 0.6}$ </p>
-    <p> $\mathcal{A}_{\text{pre}}$ </p>
-    <p> ${\bm{X}}^{\prime},{\bm{X}}^{\prime}_{\text{err}}$ </p>
-    <p> $89.3\rightarrow 92.3$ </p>
-    <p> $65.15\pm 0.67$ </p>
-    <p> ${\bm{w}}_{\text{new}}=\frac{1+z^{\prime}}{1+z_{\text{orig}}}{\bm{w}}$ </p>
-    <p> $62.3\pm 1.9$ </p>
-    <p> $36.3\%\rightarrow 37.2\%$ </p>
-    <p> $47.5\pm 1.0$ </p>
-    <p> $y_{x}=-1$ </p>
-    <p> ${\rho},{\alpha},{\beta},{\gamma}$ </p>
-    <p> $46.4\rightarrow 46.4$ </p>
-    <p> $\mathbf{0.256\pm 0.005}$ </p>
-    <p> ${\bm{X}}_{\text{err}}\in\mathbb{R}^{T\times W}$ </p>
-    <p> $\epsilon\in\mathbb{R}^{W}$ </p>
-    <p> $\widehat{\phi}$ </p>
-    <p> $\widehat{\phi}:\mathcal{X}\to\mathbb{R}^{k}$ </p>
-    <p> $d_{+}$ </p>
-    <p> ${\bm{X}}\in\mathbb{R}^{T\times W}$ </p>
-    <p> ${\alpha}>{\gamma}$ </p>
-    <p> $({\bm{t}}_{\text{new}},{\bm{w}}_{\text{new}})$ </p>
-    <p> $31.2\pm 0.6$ </p>
-    <p> $0.289\pm 0.003$ </p>
-    <p> $0.310\pm 0.006$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10013.html DELETED Viewed

@@ -1,137 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\displaystyle\mathcal{L}_{\text{MAE}}(\phi)=\mathbb{E}_{x\sim P_{U},{x^{\prime%
-}}\sim\mathcal{A}_{\text{pre}}(\cdot\mid x)}[(\phi({x^{\prime}})-x)^{2}]$ </p>
-    <p> $\mathbf{0.246\pm 0.015}$ </p>
-    <p> $\frac{{\alpha}}{{\gamma}}$ </p>
-    <p> $\tilde{{\bm{X}}^{\prime}}=10^{0.4(d(z^{\prime})-d(z_{\text{orig}}))}\{F^{%
-\prime}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T,W},$ </p>
-    <p> $\mathcal{S}=\{x\in\mathcal{X}:d_{x}=1\}$ </p>
-    <p> $92.3\%\rightarrow 96.7\%$ </p>
-    <p> $\displaystyle\mathcal{L}(B)=\mathbb{E}_{x\sim P_{S}}\left[\ell(B\widehat{\phi}%
-(x),y_{x})\right]+\eta\|B\|_{F}^{2},$ </p>
-    <p> $x\sim P_{S}$ </p>
-    <p> $({\bm{t}},{\bm{w}})$ </p>
-    <p> $\text{Uniform}[0.5,0.9]$ </p>
-    <p> $S_{+}(x,x^{+})=\mathbb{E}_{\bar{x}\sim P_{U}}[\mathcal{A}_{\text{pre}}(x\mid%
-\bar{x})\mathcal{A}_{\text{pre}}(x^{+}\mid\bar{x})]$ </p>
-    <p> $\{F^{\prime}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T,W}$ </p>
-    <p> $\widehat{h}=\operatorname*{arg\,min}_{h}\mathcal{L}_{\text{ft}}(h)$ </p>
-    <p> $5Ã$ </p>
-    <p> $0.8\AA$ </p>
-    <p> $\left(\,\overline{\text{Ch.}},\text{Ch.}\,\right)$ </p>
-    <p> $match(Ch.,\cdot{})$ </p>
-    <p> $\left(\,\text{Ch.},\overline{\text{Ch.}}\,\right)$ </p>
-    <p> $\\
-A$ </p>
-    <p> $\min(match)$ </p>
-    <p> $\left(\,\text{H.2},\overline{\text{H.2}}\,\right)$ </p>
-    <p> $10\AA$ </p>
-    <p> $\max(d_{\rho})$ </p>
-    <p> $match(\cdot{},Ch.)$ </p>
-    <p> $B(\mathbf{p},\rho_{\mathbf{p}})$ </p>
-    <p> $\gamma:M\rightarrow M^{\prime}$ </p>
-    <p> $d_{\rho}(C,C^{\prime}):=\sqrt{\sum_{\mathbf{p}\in M}|\,\rho_{\mathbf{p}}-\rho_%
-{\gamma(\mathbf{p})}\,|^{2}},$ </p>
-    <p> $\overline{\text{H.1}}$ </p>
-    <p> $\AA$ </p>
-    <p> $\mathbf{p}^{\prime}\in C^{\prime}$ </p>
-    <p> $d_{\rho}(C,C^{\prime})$ </p>
-    <p> $d_{\rho}$ </p>
-    <p> $\overline{\text{Ch.}}$ </p>
-    <p> $1.4\AA$ </p>
-    <p> $\mathbf{p}_{N}$ </p>
-    <p> $match(C,C^{\prime})$ </p>
-    <p> $\left(\,\overline{\text{H.1}},\text{H.1}\,\right)$ </p>
-    <p> $\alpha-\pi$ </p>
-    <p> $V(C,\rho):=\iiint_{\bigcup_{\mathbf{p}\in{}C}B(\mathbf{p},\rho_{\mathbf{p}})}1dxdydz.$ </p>
-    <p> $\overline{\text{H.2}}$ </p>
-    <p> $\left(\,\overline{\text{H.2}},\text{H.2}\,\right)$ </p>
-    <p> $L(C):=\sum_{j=1}^{N}\|\mathbf{p}_{j}-\mathbf{p}_{j-1}\|_{2}.$ </p>
-    <p> $s(C):=\dfrac{1}{tortuousness(C)},$ </p>
-    <p> $\left(\,\text{H.1},\overline{\text{H.1}}\,\right)$ </p>
-    <p> $-\mathbf{v}$ </p>
-    <p> $\alpha_{t}x_{t}+\beta_{t}\epsilon_{t}$ </p>
-    <p> $\max_{x^{adv}}\ \ J(x^{adv},y)\ \ \ \ s.t.\left\|x-x^{adv}\right\|_{\infty}<\epsilon.$ </p>
-    <p> $\max_{x^{adv}}\ \ \left\|f^{m}(x,p)-f^{m}(x^{adv},p)\right\|_{2}\ \ \ \ s.t.%
-\left\|x-x^{adv}\right\|_{\infty}<\epsilon.$ </p>
-    <p> $f^{m}(\cdot)$ </p>
-    <p> $\epsilon^{\prime}=0.01$ </p>
-    <p> $\left\|x-x^{adv}\right\|_{p}<\epsilon$ </p>
-    <p> $\displaystyle\mathbb{E}[\sum_{t=0}^{m}\gamma^{t}r(y_{{\mathrm{LM}},t})],{\rm s%
-.t.,}y_{{\mathrm{LM}},t}\sim M_{\mathrm{LM}}(\cdot|\hat{s_{t}},x),$ </p>
-    <p> $c_{i}=Z_{1}(f_{i})$ </p>
-    <p> ${}^{\clubsuit,\heartsuit}$ </p>
-    <p> $M_{\mathrm{LM}}$ </p>
-    <p> $\rm answer$ </p>
-    <p> $s_{0}=[-1]$ </p>
-    <p> $P^{\prime\prime}=\{c_{i}\}_{i=0}^{n-1}$ </p>
-    <p> $S_{\mathrm{semantic}}$ </p>
-    <p> $V=\{v_{1},...,v_{m}\}$ </p>
-    <p> $P^{\prime}=\{f_{i}\}_{i=0}^{n-1}$ </p>
-    <p> $\rm question$ </p>
-    <p> $k=a_{t}$ </p>
-    <p> $(\rm question,\rm context,\rm answer)$ </p>
-    <p> $q=Z_{2}(l)$ </p>
-    <p> $\zeta(y,\hat{y})=\lambda\cdot S_{\mathrm{textual}}(y,\hat{y})+(1-\lambda)\cdot
-S%
-_{\mathrm{semantic}}(y,\hat{y}),$ </p>
-    <p> $\displaystyle\underset{\theta}{\mathrm{max}}$ </p>
-    <p> $r=\alpha\cdot\zeta(y,\hat{y})$ </p>
-    <p> $l=\mathrm{concat}(g,h)$ </p>
-    <p> $P=\{p_{i}\}_{i=0}^{n-1}$ </p>
-    <p> $\rm context$ </p>
-    <p> $\hat{s_{t}}\sim\prod_{i=0}^{t}\pi_{\theta}(a_{i}|s_{<i},x)$ </p>
-    <p> $S_{\mathrm{textual}}$ </p>
-    <p> $\{v_{i}\}_{i=0}^{m}$ </p>
-    <p> $s_{t}=\mathrm{append}(s_{t-1},a_{t})$ </p>
-    <p> $\pi_{\theta}(a_{t}|s_{<t},x)$ </p>
-    <p> $v_{t+1}=p_{k}$ </p>
-    <p> $y_{\mathrm{LM}}$ </p>
-    <p> ${}^{*~{}\heartsuit}$ </p>
-    <p> $M_{\mathrm{LM}}(\cdot|v_{0},v_{1},...,v_{m},x)$ </p>
-    <p> $(v_{0},...,v_{t})\times a_{t}\rightarrow(v_{0},...,v_{t},v_{t+1})$ </p>
-    <p> $\underset{V\subset P}{\mathrm{max}}R(y_{\mathrm{LM}}\sim M_{\mathrm{LM}}(\cdot%
-|v_{0},v_{1},...,v_{m},x)),$ </p>
-    <p> $s_{0}=(v_{0},x)$ </p>
-    <p> $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)>1,\\
-x_{(i,j)}&\text{ if }S\left({V_{\left(i,j\right)}}\right)=1,\\
-0&\text{otherwise}\end{cases}$ </p>
-    <p> $x_{\left(i,j\right)}$ </p>
-    <p> $Q:\{0,1\}$ </p>
-    <p> $u{\left({x}\right)}^{t}$ </p>
-    <p> $MFA=\left\langle{\mathbb{Z}^{2},Q,V,F,w}\right\rangle$ </p>
-    <p> $Q:{Wb}_{i}=0$ </p>
-    <p> $f(0,x_{i+1},x_{i+2})=f(1,x_{i+1},x_{i+2})$ </p>
-    <p> $\left[0,51,204,255\right]$ </p>
-    <p> $\left({x_{i-1},x_{i},x_{i+1},x_{i+2}}\right)$ </p>
-    <p> $\Delta m_{\text{init}-t}=\sum_{i=0}^{n}{x_{i}^{t_{0}}}-\sum_{i=0}^{n}{x_{i}^{t}}$ </p>
-    <p> $f(x_{i},x_{i+1},x_{i+2})$ </p>
-    <p> $\displaystyle x^{t+1}_{i}=\begin{cases}x^{t}&\text{if }w^{t}=1\\
-f\left({u\left({x_{i}}\right)}^{t}\right)&\text{otherwise}\end{cases}$ </p>
-    <p> $Q:{Wb}_{i}=1$ </p>
-    <p> $f(x_{i-1},x_{i},x_{i+1})$ </p>
-    <p> $\begin{split}&\delta_{t}=\frac{\Delta(X_{t-1},X_{t})}{n}\quad\text{with}\quad t%
-\geq 1,n\in\mathbb{N}\quad\text{and}\\
-&\Delta(X_{t-1},X_{t})=X_{t-1}\oplus X_{t}\end{split}$ </p>
-    <p> $m_{t}=\sum_{i=0}^{n}{x_{i}^{t}}$ </p>
-    <p> $V\left(i_{0},j_{0}\right)=\left\{{\left({i,j}\right):\mid{i-i_{0}}\mid+\mid{j-%
-j_{0}}\mid\leq 1}\right\}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10014.html DELETED Viewed

@@ -1,150 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $m_{\text{inter}}=\sum_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{n}}^{F,w_{0}}(i,j)}}-\sum%
-_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{n}}^{F,w_{k}}(i,j)}}$ </p>
-    <p> $w=\left\{{H1V1},{H2V2},{H4V4},\text{cut\&rel}\right\}$ </p>
-    <p> $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)=2,\\
-0&\text{otherwise}\end{cases}$ </p>
-    <p> $x^{t_{0}}$ </p>
-    <p> $\left({HV}\right)^{20}$ </p>
-    <p> ${Wb}_{i}$ </p>
-    <p> $f(x_{i-1},x_{i},0)=f(x_{i-1},x_{i},1)$ </p>
-    <p> $\left[1,10,30,60\right]$ </p>
-    <p> $\delta_{\text{inter}}=\frac{{X_{t_{n}}^{F,w_{0}}}\oplus{X_{t_{n}}^{F,w_{k}}}}{%
-N^{2}}$ </p>
-    <p> $w^{t+1}={g\left({u\left({x}\right)}^{t}\right)}$ </p>
-    <p> $x^{t+1}_{i}=\begin{cases}&\text{if $w^{t}_{i}=1$ and $f\left(x^{t}_{i-1},x^{t}%
-_{i},1\right)\neq f\left(x^{t}_{i-1},x^{t}_{i},0\right)$}\\
-x^{t}_{i}&\qquad\qquad\qquad\text{ or}\\
-&\text{if $w^{t}_{i}=1$ and $f\left(1,x^{t}_{i},x^{t}_{i+1}\right)\neq f\left(%
-0,x^{t}_{i},x^{t}_{i+1}\right)$}\\
-\\
-f(x^{t}_{i-1},x^{t}_{i},x^{t}_{i+1})&\text{otherwise}\end{cases}$ </p>
-    <p> $\left({HV}\right)^{*}$ </p>
-    <p> ${Wb}_{\left(i,j\right)}$ </p>
-    <p> $\left({HHVV}\right)^{*}$ </p>
-    <p> $t_{0},t_{1},\ldots,t_{n}$ </p>
-    <p> ${H,V}$ </p>
-    <p> $\begin{split}&\delta_{\text{init state}}=\frac{\Delta(X_{t_{0}},X_{t})}{n}%
-\quad\text{with}\quad t\geq 1,n\in\mathbb{N}\quad\text{and}\\
-&\Delta(X_{t_{0}},X_{t})=X_{t_{0}}\oplus X_{t}\end{split}$ </p>
-    <p> $\displaystyle x^{t+1}_{i}=\begin{cases}0&\text{if }w^{t}=1\\
-f\left({u\left({x_{i}}\right)}^{t}\right)&\text{otherwise}\end{cases}$ </p>
-    <p> $x(i,j)$ </p>
-    <p> $x_{t_{n}}^{F,w}(i,j)[0\mapsto-1]=\begin{cases}x_{t_{n}}^{F,w}(i,j)&\text{ if }%
-x_{t_{n}}^{F,w}(i,j)=1\\
--1&\text{ otherwise. }\end{cases}$ </p>
-    <p> $\left({HHHHVVVV}\right)^{*}$ </p>
-    <p> $Q=\{0,1\}$ </p>
-    <p> $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)\geq 2,\\
-0&\text{otherwise}\end{cases}$ </p>
-    <p> $w_{0}=\emptyset$ </p>
-    <p> $x^{t+1}_{i}=f(x^{t}_{i-1},x^{t}_{i},x^{t}_{i+1})$ </p>
-    <p> $V_{\left(i,j\right)}=\left[x_{\left(i-1,j\right)},x_{\left(i+1,j\right)},x_{%
-\left(i,j-1\right)},x_{\left(i,j+1\right)}\right]$ </p>
-    <p> $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)>2,\\
-x_{(i,j)}&\text{ if }S\left({V_{\left(i,j\right)}}\right)=2,\\
-0&\text{otherwise}\end{cases}$ </p>
-    <p> $m_{\text{intra}}=\sum_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{0}}^{F,w}(i,j)}}-\sum_{i=%
-0}^{m}{\sum_{j=0}^{m}}{x_{t_{n}}^{F,w}(i,j)}$ </p>
-    <p> $\delta_{\text{intra}}=\frac{X_{t_{0}}^{F,w}\oplus X_{t_{n}}^{F,w}}{N^{2}}$ </p>
-    <p> $\left[30,32,90,110,150\right]$ </p>
-    <p> $\displaystyle\begin{split}{\text{Cid}(x_{t})}=\frac{\sum_{i=0}^{m}{\sum_{j=0}^%
-{m}{x_{t_{n}}^{F,w}(i,j)[0\mapsto-1]}}}{N^{2}}\end{split}$ </p>
-    <p> $\displaystyle f_{ij}=MLP(FC_{q}(F_{i})\oplus FC_{k}(F_{j})),$ </p>
-    <p> $(x_{i2},y_{i2})$ </p>
-    <p> $FC_{q}$ </p>
-    <p> $\{s_{kj},k=1,2,...,n\}$ </p>
-    <p> $b_{i}=(x_{i1},y_{i1},x_{i2},y_{i2})$ </p>
-    <p> $T=\{T_{i}|i=1,2,3,\ldots,K\}$ </p>
-    <p> $(x_{i1},y_{i1})$ </p>
-    <p> $FC_{k}$ </p>
-    <p> $C\in\mathbb{Z}^{N\times N}$ </p>
-    <p> $\displaystyle s_{ij}=\frac{\exp(f_{ij})}{\sum_{i=1}^{N}\exp(f_{ij})},$ </p>
-    <p> $R\in\mathbb{R}^{N\times N}$ </p>
-    <p> $\{T_{i}|i=1,2,3,\ldots,K\}$ </p>
-    <p> $\displaystyle=\min\limits_{1\leq i\leq R}{\Delta_{l2_{i}}},$ </p>
-    <p> $\displaystyle=\{[t_{1},t_{2},...,t_{m}]|t_{i}\in T_{j}\},\forall j\in[1..R],$ </p>
-    <p> $\displaystyle L_{b_{j}}$ </p>
-    <p> $\displaystyle=\{l_{2ij}|i\in D_{b_{j}}\},L_{n_{j}}=\{l_{2ij}|i\in D_{n}\},$ </p>
-    <p> $>0.17$ </p>
-    <p> $\text{\text{EmbMarker}}+\text{{CSE}}$ </p>
-    <p> $>0.98$ </p>
-    <p> ${\bm{e}}_{p2}$ </p>
-    <p> $\lambda(S)$ </p>
-    <p> $>0.10$ </p>
-    <p> $j\in[1..R]$ </p>
-    <p> $\displaystyle=\{\cos_{ij}|i\in D_{b_{j}}\},C_{n_{j}}=\{\cos_{ij}|i\in D_{n}\},$ </p>
-    <p> ${\bm{e}}_{p1}$ </p>
-    <p> $\displaystyle C_{b_{j}}$ </p>
-    <p> ${\bm{e}}_{o}$ </p>
-    <p> $\Delta_{cos}(\%)\downarrow$ </p>
-    <p> $>0.20$ </p>
-    <p> ${\bm{u}}^{\langle k+1\rangle}$ </p>
-    <p> ${\bm{e}}_{s}$ </p>
-    <p> $\displaystyle\Delta_{\cos_{k}}$ </p>
-    <p> $\displaystyle=\frac{1}{|L_{b_{k}}|}\sum_{i\in L_{b_{k}}}i-\frac{1}{|L_{n_{k}}|%
-}\sum_{j\in L_{n_{k}}}j.$ </p>
-    <p> $\displaystyle=\frac{{\bm{e}}_{i}\cdot{\bm{w}}_{j}}{||{\bm{e}}_{i}||\cdot||{\bm%
-{w}}_{j}||},\quad l_{2ij}=\biggl{|}\biggl{|}\frac{{\bm{e}}_{i}}{||{\bm{e}}_{i}%
-||}-\frac{{\bm{w}}_{j}}{||{\bm{w}}_{j}||}\biggl{|}\biggl{|}^{2},$ </p>
-    <p> $\Delta_{l2}(\%)\uparrow$ </p>
-    <p> $>0.47$ </p>
-    <p> $\lambda(S)=\lambda_{1}(S)+\lambda_{2}(S)+â¦+\lambda_{R}(S)$ </p>
-    <p> $\displaystyle\Delta_{l2_{k}}$ </p>
-    <p> $\displaystyle D_{p}=Rank(D_{v})-Rank(D_{s}),$ </p>
-    <p> $>0.57$ </p>
-    <p> $>0.02$ </p>
-    <p> ${\bm{u}}^{\langle k+1\rangle}={\bm{e}}^{\langle k\rangle}-\text{Proj}({\bm{e}}%
-^{\langle k\rangle},{\bm{c}}^{\langle k\rangle}).$ </p>
-    <p> ${\bm{W}}=\{{\bm{w}}_{1},{\bm{w}}_{2},...,{\bm{w}}_{R}\}$ </p>
-    <p> ${\bm{e}}_{p}$ </p>
-    <p> $\displaystyle D_{b_{j}}$ </p>
-    <p> ${\bm{e}}^{\langle k+1\rangle}$ </p>
-    <p> $>0.26$ </p>
-    <p> $\displaystyle D_{n}$ </p>
-    <p> $\displaystyle\min_{\boldsymbol{\alpha}}\biggl{\|}{\bm{w}}-\sum_{k=1}^{K}{%
-\alpha}_{k}\cdot{\bm{c}}^{\langle k\rangle}\biggl{\|}^{2}.$ </p>
-    <p> $>0.04$ </p>
-    <p> $\text{p-value}_{j}$ </p>
-    <p> $\displaystyle=\frac{1}{|C_{b_{k}}|}\sum_{i\in C_{b_{k}}}i-\frac{1}{|C_{n_{k}}|%
-}\sum_{j\in C_{n_{k}}}j,$ </p>
-    <p> $>0.22$ </p>
-    <p> $Rank$ </p>
-    <p> $\displaystyle=\min\limits_{1\leq i\leq R}{\text{p-value}_{i}}.$ </p>
-    <p> $>0.56$ </p>
-    <p> $\text{Norm}\left({\bm{u}}^{\langle k+1\rangle}\right)=\frac{{\bm{u}}^{\langle k%
-+1\rangle}}{||{\bm{u}}^{\langle k+1\rangle}||}$ </p>
-    <p> $\displaystyle\Delta_{l2}$ </p>
-    <p> $\text{Proj}({\bm{e}}^{\langle k\rangle},{\bm{c}}^{\langle k\rangle})=\frac{{%
-\bm{c}}^{\langle k\rangle}\cdot{\bm{e}}^{\langle k\rangle}}{||{\bm{c}}^{%
-\langle k\rangle}||}\cdot{\bm{c}}^{\langle k\rangle}.$ </p>
-    <p> $\displaystyle{\bm{e}}_{p}=\text{Norm}\left((1-\sum_{r=1}^{R}\lambda_{r}(S))%
-\cdot{\bm{e}}_{o}+\sum_{r=1}^{R}\lambda_{r}(S)\cdot{\bm{w}}_{r}\right).$ </p>
-    <p> ${\bm{e}}_{s2}$ </p>
-    <p> $>10^{-3}$ </p>
-    <p> $\displaystyle=\{[t_{1},t_{2},...,t_{m}]|t_{i}\notin T\}.$ </p>
-    <p> ${\bm{c}}^{\langle k\rangle}$ </p>
-    <p> $\Delta_{l2}(\%)$ </p>
-    <p> $>0.36$ </p>
-    <p> ${\bm{e}}_{p}=f({\bm{e}}_{o},t)$ </p>
-    <p> $>0.62$ </p>
-    <p> $>0.55$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10015.html DELETED Viewed

@@ -1,132 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\Delta_{cos}(\%)$ </p>
-    <p> ${\bm{e}}^{\langle 0\rangle}$ </p>
-    <p> ${\bm{e}}^{\langle k\rangle}$ </p>
-    <p> $>0.08$ </p>
-    <p> $>0.83$ </p>
-    <p> ${\bm{e}}_{s1}$ </p>
-    <p> $\displaystyle=\max\limits_{1\leq i\leq R}{\Delta_{\cos_{i}}},$ </p>
-    <p> $>10^{-4}$ </p>
-    <p> $m=4,n=20,\text{and frequency interval}=[0.5\%,1\%]$ </p>
-    <p> $\displaystyle\Delta_{\cos}$ </p>
-    <p> $\displaystyle\cos_{ij}$ </p>
-    <p> $C_{b_{j}}$ </p>
-    <p> $<10^{-3}$ </p>
-    <p> $\Theta_{v}$ </p>
-    <p> $1,801,350$ </p>
-    <p> $C_{n_{j}}$ </p>
-    <p> $>0.21$ </p>
-    <p> $Pops(\pi_{a})=\\
-\{\{1,7\},\{5\},\{11,37\},\{13,19\},\{15\},\{22\},\{24\}\}$ </p>
-    <p> $\delta_{sgo}$ </p>
-    <p> $\delta_{flex}=\frac{4}{9}=0.4$ </p>
-    <p> $Pops(\pi_{b})=\\
-\{\{1,7\},\{5,11\},\{13,19\},\{15\},\{22\},\{24,37\}\}$ </p>
-    <p> $sg\neq X$ </p>
-    <p> $SubGoals(\pi_{b})=``XXXCBXXXXA"$ </p>
-    <p> $D(\pi_{a},\pi_{b})=1-\delta(\pi_{a},\pi_{b})$ </p>
-    <p> $\delta_{\text{sgo}}(\pi_{a},\pi_{b})=1-\frac{HDist(SubGoals(\pi_{a}),SubGoals(%
-\pi_{b}))}{max(SubGoals(\pi_{a}),SubGoals(\pi_{b}))}$ </p>
-    <p> $HDist(SubGoals(\pi_{a}),SubGoals(\pi_{b}))=5$ </p>
-    <p> $x\in\{a,s,c\}$ </p>
-    <p> $SubGoals(\pi_{b})$ </p>
-    <p> $\delta(\pi_{a},\pi_{b})\rightarrow{[0,1]}$ </p>
-    <p> $state\leftarrow PerformAction(state,a)$ </p>
-    <p> $subgoalLetter\leftarrow GetEncodedSubgoals(PI)$ </p>
-    <p> $CBA$ </p>
-    <p> $\delta_{flex}$ </p>
-    <p> $\pi_{a}=\{1,7,5,11,37,13,19,15,22,24\}$ </p>
-    <p> $\pi_{b}=\{1,7,5,11,13,19,15,22,24,37\}$ </p>
-    <p> $sg\leftarrow GetSubGoal(state,PI)$ </p>
-    <p> $state\leftarrow GetInitialState(PI)$ </p>
-    <p> $\delta_{x}(\pi_{a},\pi_{b})=|A(\pi_{a})\cap A(\pi_{b})|/|A(\pi_{b})\cup A(\pi_%
-{a})|$ </p>
-    <p> $seq\leftarrow AppendTo(seq,subgoalLetter[sg])$ </p>
-    <p> $SubGoals$ </p>
-    <p> $HDist(s_{\pi_{a}},s_{\pi_{b}})\rightarrow\mathbb{R}$ </p>
-    <p> $max(a,b)\rightarrow\mathbb{N}$ </p>
-    <p> $SubGoals(\pi_{a})=``XXBXXXXAXC"$ </p>
-    <p> $\delta_{u}(\pi_{a},\pi_{b})={\begin{cases}1,\ if\ \pi_{a}\setminus\pi_{b}=%
-\emptyset\\
-1,\ if\ \pi_{a}\subset\pi_{b}\\
-0,\ otherwise\end{cases}}$ </p>
-    <p> $\delta_{\text{flex}}(\pi_{a},\pi_{b})=\frac{|Pop(\pi_{a})\cap Pop(\pi_{b})|}{|%
-Pop(\pi_{a})\cup Pop(\pi_{b})|}$ </p>
-    <p> $seq\leftarrow AppendTo(seq,``X")$ </p>
-    <p> $seq\leftarrow``"$ </p>
-    <p> $Pop(\pi)$ </p>
-    <p> $\delta_{a}(\pi_{a},\pi_{b})=\frac{|1,7,5,11,37,13,19,15,22,24|}{|1,7,5,11,13,1%
-9,15,22,24,37|}=1$ </p>
-    <p> $HDist$ </p>
-    <p> $SubGoals(\pi_{a})$ </p>
-    <p> $BAC$ </p>
-    <p> $A(\pi)$ </p>
-    <p> $\xi=0.031$ </p>
-    <p> $\mathcal{D^{\prime}}(\mathcal{D}(x)+\eta)$ </p>
-    <p> $||\delta||_{\infty}\leq\xi$ </p>
-    <p> $Z_{l}(\cdot)$ </p>
-    <p> $u\leftarrow\max(|\delta_{init}|)$ </p>
-    <p> $s\in\{-1,1\}^{d}$ </p>
-    <p> $\mathrm{Cos}$ </p>
-    <p> $y\in\mathbb{R}^{2}=\{0,1\}$ </p>
-    <p> $s\sim\mathrm{\textbf{Bernoulli}}(p),\quad\delta_{s}\leftarrow\delta_{p}\odot s.$ </p>
-    <p> $\delta_{0}\leftarrow 0$ </p>
-    <p> $f(x)_{i}$ </p>
-    <p> $x_{adv}\leftarrow x_{K}$ </p>
-    <p> $\delta\leftarrow x^{p}-x$ </p>
-    <p> $\epsilon=0.062$ </p>
-    <p> $\mathcal{D^{\prime}}(\cdot)$ </p>
-    <p> $x_{adv}\leftarrow\mathrm{Clip}_{x,\ \epsilon^{\prime}-\kappa}(x_{idct}),$ </p>
-    <p> $[0,\max(\delta_{init})]$ </p>
-    <p> $m\leftarrow(l+u)/2$ </p>
-    <p> $c(x_{t})=0$ </p>
-    <p> $c(x_{adv})=0$ </p>
-    <p> $x_{0}\leftarrow x_{r}$ </p>
-    <p> $x_{idct}^{p}\leftarrow\mathcal{Dâ}(x_{dct}^{p}+\eta)$ </p>
-    <p> $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ </p>
-    <p> $\delta\leftarrow\mathrm{Clip}_{0,\ u}(\delta_{init})$ </p>
-    <p> $x_{dct}^{p}\leftarrow\mathcal{D}(x+\delta)$ </p>
-    <p> $\mathop{\mathrm{argmin}}\limits_{\delta}\ \mathrm{Cos}(Z_{l}(x),\ Z_{l}(x_{r}+%
-\delta)),\quad s.t.\quad||\delta||_{\infty}\leq\xi,$ </p>
-    <p> $p=0.999$ </p>
-    <p> $x_{idct}\leftarrow\mathcal{D^{\prime}}(\mathcal{D}(x_{adv})+\eta),\quad\eta%
-\sim\{-\gamma,\gamma\}^{d},$ </p>
-    <p> $\delta\leftarrow\mathrm{\textbf{BinarySearch}}(x,\delta_{init})$ </p>
-    <p> $i\in[1,\ k]$ </p>
-    <p> $\gamma=1.75$ </p>
-    <p> $||\delta||_{\infty}\leq\epsilon^{\prime},\epsilon^{\prime}\in(0,\max|\delta_{%
-init}|]$ </p>
-    <p> $c(x)=\mathrm{argmax}_{i\in\{0,1\}}\ f(x)_{i}$ </p>
-    <p> $Z_{l}(x)$ </p>
-    <p> $\mathrm{Clip}_{x_{r},\ \xi}(\cdot)$ </p>
-    <p> $\eta\sim\{-\gamma,\gamma\}^{d}$ </p>
-    <p> $c(x^{p})=0$ </p>
-    <p> $\mathrm{\#\ Perturbation\ Random\ Flip}$ </p>
-    <p> $i\in[1,\ K]$ </p>
-    <p> $50.34$ </p>
-    <p> $\delta_{init}\leftarrow\mathrm{\textbf{CrossPerturbInit}}(x,x_{r},Z,f)$ </p>
-    <p> $c(x+\delta)=0$ </p>
-    <p> $Z_{l}(x_{r}+\delta)$ </p>
-    <p> $\delta^{p}\leftarrow\delta\odot s$ </p>
-    <p> $\mathrm{\#\ Frequency\ Noise\ Projection}$ </p>
-    <p> $x^{p}\leftarrow\mathrm{Clip}_{x,\ \epsilon^{\prime}-\kappa}(x_{dct}^{p})$ </p>
-    <p> $\delta\leftarrow\delta^{p}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10016.html DELETED Viewed

@@ -1,146 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $x_{adv}\leftarrow x+\delta$ </p>
-    <p> $J(x_{i-1},x)\leftarrow\mathrm{-Cos}(Z_{l}(x_{i-1}),\ Z_{l}(x))$ </p>
-    <p> $\mathcal{D^{\prime}}(\mathcal{D}(x))$ </p>
-    <p> $l\leftarrow m$ </p>
-    <p> $s\sim\mathrm{\textbf{Bernoulli}}(p)$ </p>
-    <p> $\min_{\delta}||\delta||_{\infty}\quad\mathrm{s.t.}\quad c(x+\delta)=0.$ </p>
-    <p> $\delta_{i}\leftarrow\delta_{i-1}+\frac{\xi}{K}\cdot\mathrm{sign}(\nabla_{x_{i-%
-1}}J(x_{i-1},x))$ </p>
-    <p> $c(x+\delta^{p})==0$ </p>
-    <p> $x_{r}+\delta$ </p>
-    <p> $p\in(0,1)^{d}$ </p>
-    <p> $x_{i}\leftarrow\mathrm{Clip}_{x_{r},\ \xi}(x_{i-1}+\delta_{i})$ </p>
-    <p> $x_{t}\leftarrow\mathrm{Clip}_{x_{r},\ m}(x+\delta_{init})$ </p>
-    <p> $\mathrm{\#\ Perturbation\ Initialization}$ </p>
-    <p> $\delta_{init}$ </p>
-    <p> $\epsilon^{\prime}\leftarrow||\delta||_{\infty}$ </p>
-    <p> $i,j,k\in[1,n]$ </p>
-    <p> $\varepsilon_{2}=0.9\varepsilon$ </p>
-    <p> $\langle\gamma\rangle_{1}=\sum_{i=1}^{n}\langle\gamma_{i}\rangle_{1}$ </p>
-    <p> $d_{max}^{\prime}\leftarrow\mathsf{max}(d_{1}^{\prime},...,d_{n}^{\prime})$ </p>
-    <p> $f=\langle f\rangle_{1}+\langle f\rangle_{2}$ </p>
-    <p> $\langle v_{4},v_{5}\rangle$ </p>
-    <p> $Lap(\frac{\triangle}{\varepsilon_{2}})$ </p>
-    <p> $\displaystyle\frac{e^{\frac{-\varepsilon_{2}.|\widetilde{T}-T(G)|}{\triangle}}%
-}{e^{\frac{-\varepsilon_{2}.|\widetilde{T}-T(G^{\prime})|}{\triangle}}}=e^{%
-\frac{\varepsilon_{2}.(|\widetilde{T}-T(G^{\prime})|-|\widetilde{T}-T(G)|)}{%
-\triangle}}$ </p>
-    <p> $Pr[\mathcal{M}_{i}(A_{i})\in S]\leq e^{\epsilon}Pr[\mathcal{M}_{i}(A_{i}^{%
-\prime})\in S]$ </p>
-    <p> $\hat{T}(G,d_{max}^{\prime})$ </p>
-    <p> $T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime})$ </p>
-    <p> $\displaystyle\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime%
-}),\hat{T}(G,d_{max}^{\prime}))]=\mathbb{V}[T^{\prime}(G,\varepsilon_{2},d_{%
-max}^{\prime})]$ </p>
-    <p> $ds=[0]*n,\hat{A_{i}}=\varnothing$ </p>
-    <p> $e=a-x,f=b-y,g=c-z$ </p>
-    <p> $\langle u\rangle_{i}$ </p>
-    <p> $\langle T^{\prime}\rangle_{2}=\langle T\rangle_{2}+\langle\gamma\rangle_{2}$ </p>
-    <p> $\mathsf{Local2Rounds_{\triangle}}$ </p>
-    <p> $\langle x\rangle_{2},\langle y\rangle_{2},\langle z\rangle_{2},\langle w%
-\rangle_{2},\langle o\rangle_{2},\langle p\rangle_{2},\langle q\rangle_{2}%
-\rightarrow S_{2}$ </p>
-    <p> $\varepsilon_{1}=0.1\varepsilon$ </p>
-    <p> $\Gamma(\beta)=\int_{0}^{\infty}x^{\beta-1}e^{-x}dx$ </p>
-    <p> $O\binom{n}{2}$ </p>
-    <p> $\gamma_{i}=(Gam_{1}-Gam_{2})$ </p>
-    <p> $\langle\gamma\rangle_{2}=\sum_{i=1}^{n}\langle\gamma_{i}\rangle_{2}$ </p>
-    <p> $\mathsf{Count}$ </p>
-    <p> $\displaystyle e^{\frac{\varepsilon_{2}|T(G)-T(G^{\prime})|}{\triangle}}=e^{%
-\varepsilon_{2}}$ </p>
-    <p> $\hat{A_{i}}$ </p>
-    <p> $d_{max}^{\prime}$ </p>
-    <p> $\hat{A}=\{\hat{A_{1}},...,\hat{A_{n}}\}$ </p>
-    <p> $\langle a_{ij}\rangle_{1}$ </p>
-    <p> $l_{2}^{2}(d_{max}^{\prime},d_{max})<0.009d_{max}$ </p>
-    <p> $\langle v_{2},v_{1}\rangle$ </p>
-    <p> $\mathbb{E}[l_{2}^{2}(T(G),\hat{T}(G,d_{max}^{\prime}))]=(T(G)-\hat{T}(G,d_{max%
-}^{\prime}))^{2}$ </p>
-    <p> $\langle v_{j},v_{i}\rangle$ </p>
-    <p> $v_{i},i\in[1,n]$ </p>
-    <p> $Gam_{2}=\mathsf{Gamma}(n,\frac{d_{max}^{\prime}}{\varepsilon_{2}})$ </p>
-    <p> $\langle x\rangle_{2}=(x-r)$ </p>
-    <p> $r\in\mathbb{Z}_{2^{l}}$ </p>
-    <p> $T(G^{\prime})$ </p>
-    <p> $D^{\prime},d_{max}^{\prime}$ </p>
-    <p> $\displaystyle\frac{Pr[d^{\prime}=d_{i}+x]}{Pr[d^{\prime}=d_{i}^{\prime}+x^{%
-\prime}]}=\frac{Pr[x=d^{\prime}-d_{i}]}{Pr[x^{\prime}=d^{\prime}-d_{i}^{\prime%
-}]}$ </p>
-    <p> $\langle T\rangle_{1}=\langle T\rangle_{2}=0$ </p>
-    <p> $\langle a_{ij}\rangle_{2}$ </p>
-    <p> $\displaystyle\mathbb{V}[Lap(\frac{d_{max}^{\prime}}{\varepsilon_{2}})]=O(\frac%
-{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$ </p>
-    <p> $\langle x\rangle_{1}=r$ </p>
-    <p> $v_{k},k\in[n]$ </p>
-    <p> $\langle g\rangle_{i}=\langle c\rangle_{i}-\langle z\rangle_{i}$ </p>
-    <p> $\displaystyle\frac{Pr[\widetilde{T}=T+(r_{1}+...+r_{n})]}{Pr[\widetilde{T}=T^{%
-\prime}+(r_{1}^{\prime}+...+r_{n}^{\prime})]}$ </p>
-    <p> $\hat{A_{i}}\leftarrow A_{i}$ </p>
-    <p> $a_{ij}=1,j\in[n]$ </p>
-    <p> $|d_{i}-d_{i}^{\prime}|=1$ </p>
-    <p> $T^{\prime}=\langle T^{\prime}\rangle_{1}+\langle T^{\prime}\rangle_{2}$ </p>
-    <p> $D^{\prime}\leftarrow D^{\prime}\cup\{d_{i}^{\prime}\}$ </p>
-    <p> $O(\frac{d_{max}^{2}}{\varepsilon^{2}})$ </p>
-    <p> $r=\{r_{1},...,r_{n}\}$ </p>
-    <p> $O(\theta)$ </p>
-    <p> $+\langle x\rangle_{2}fg+\langle y\rangle_{2}eg+\langle z\rangle_{2}ef+efg$ </p>
-    <p> $e=\langle e\rangle_{1}+\langle e\rangle_{2}$ </p>
-    <p> $O(n^{2}+nd_{max}^{2})$ </p>
-    <p> $d_{i}>d_{max}^{\prime}$ </p>
-    <p> $(d_{i}-d_{max}^{\prime})$ </p>
-    <p> $(i\in\{1,2\})$ </p>
-    <p> $\displaystyle\frac{Pr[\widetilde{T}=T(G)+x]}{Pr[\widetilde{T}=T(G^{\prime})+x^%
-{\prime}]}=\frac{Pr[x=\widetilde{T}-T(G)]}{Pr[x^{\prime}=\widetilde{T}-T(G^{%
-\prime})]}$ </p>
-    <p> $\langle x\rangle_{2},\langle y\rangle_{2},\langle z\rangle_{2},\langle w%
-\rangle_{2},\langle o\rangle_{2},\langle p\rangle_{2},\langle q\rangle_{2}$ </p>
-    <p> $O(d_{max}^{\prime})$ </p>
-    <p> $\langle e\rangle_{1}=\langle a_{ij}\rangle_{1}-\langle x\rangle_{1}$ </p>
-    <p> $\langle\gamma_{i}\rangle_{1}\rightarrow S_{1}$ </p>
-    <p> $\hat{ds}\leftarrow ds[1:d_{max}^{\prime}]$ </p>
-    <p> $\mathsf{view}_{S_{i}}^{\Pi}$ </p>
-    <p> $x=r_{1}+...+r_{n}$ </p>
-    <p> $d_{max}^{\prime}\approx d_{max}$ </p>
-    <p> $\langle e\rangle_{i}=\langle a\rangle_{i}-\langle x\rangle_{i},$ </p>
-    <p> $\displaystyle\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime%
-}),\hat{T}(G,d_{max}^{\prime}))]$ </p>
-    <p> $\times 10^{10}$ </p>
-    <p> $k=j+1$ </p>
-    <p> $e,f,$ </p>
-    <p> $\displaystyle\frac{Pr[\widetilde{T}=\langle T\rangle_{1}+\langle T\rangle_{2}+%
-(r_{1}+...+r_{n})]}{Pr[\widetilde{T}=\langle T^{\prime}\rangle_{1}+\langle T^{%
-\prime}\rangle_{2}+(r_{1}^{\prime}+...+r_{n}^{\prime})]}$ </p>
-    <p> $\mathsf{Sim}_{S_{i}}$ </p>
-    <p> $\langle\gamma_{i}\rangle_{1},\langle\gamma_{i}\rangle_{2}$ </p>
-    <p> $\langle u\rangle_{1}$ </p>
-    <p> $r^{\prime}=\{r_{1}^{\prime},...,r_{n}^{\prime}\}$ </p>
-    <p> $Pr[\mathcal{M}(D)\in S]\leq e^{\epsilon}Pr[\mathcal{M}(D^{\prime})\in S]$ </p>
-    <p> $O\binom{d_{max}^{\prime}}{2}$ </p>
-    <p> $d=\langle d\rangle_{1}+\langle d\rangle_{2}=\langle w\rangle_{1}+\langle xy%
-\rangle_{1}g+\langle xz\rangle_{1}f+\langle yz\rangle_{1}e+\langle x\rangle_{1%
-}fg+\langle y\rangle_{1}eg+\langle z\rangle_{1}ef+\langle w\rangle_{2}+\langle
-xy%
-\rangle_{2}g+\langle xz\rangle_{2}f+\langle yz\rangle_{2}e+\langle x\rangle_{2%
-}fg+\langle y\rangle_{2}eg+\langle z\rangle_{2}ef+efg$ </p>
-    <p> $\langle x\rangle_{2}+\langle y\rangle_{1}+\langle y\rangle_{2}=x+y$ </p>
-    <p> $\langle e\rangle_{2}=\langle a_{ij}\rangle_{2}-\langle x\rangle_{2}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10017.html DELETED Viewed

@@ -1,134 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $D=\{d_{1},...,d_{n}\}$ </p>
-    <p> $\hat{ds}$ </p>
-    <p> $\langle T\rangle=\{\langle T\rangle_{1},\langle T\rangle_{2}\}$ </p>
-    <p> $T\neq$ </p>
-    <p> $\mathsf{CentralLap_{\triangle}}$ </p>
-    <p> $\langle g\rangle_{2}=\langle a_{jk}\rangle_{2}-\langle z\rangle_{2}$ </p>
-    <p> $d_{i}^{\prime}\leftarrow d_{i}+\mathsf{Lap}(\frac{1}{\varepsilon_{1}})$ </p>
-    <p> $Lap(\frac{1}{\varepsilon_{1}})$ </p>
-    <p> $\langle d\rangle_{i}=\langle w\rangle_{i}+\langle xy\rangle_{i}g+\langle xz%
-\rangle_{i}f+\langle yz\rangle_{i}e+\langle x\rangle_{i}fg+\langle y\rangle_{i%
-}eg+\langle z\rangle_{i}ef+(i-1)efg$ </p>
-    <p> $u_{1}=\langle w\rangle_{1}+\langle xy\rangle_{1}g+\langle xz\rangle_{1}f+%
-\langle yz\rangle_{1}e$ </p>
-    <p> $re(T,T^{\prime})=\frac{|T-T^{\prime}|}{T}$ </p>
-    <p> $\langle f\rangle_{i}=\langle b\rangle_{i}-\langle y\rangle_{i},$ </p>
-    <p> $Gam_{1}(n,\frac{\triangle}{\varepsilon_{2}})-Gam_{2}(n,\frac{\triangle}{%
-\varepsilon_{2}})$ </p>
-    <p> $D,D^{\prime}\in\mathcal{X}^{n}$ </p>
-    <p> $A_{ij}==1$ </p>
-    <p> $u=a_{ij}\times a_{ik}\times a_{jk}(i<j<k)$ </p>
-    <p> $A_{i}=\{a_{i1},...,a_{in}\}$ </p>
-    <p> $\varepsilon,n,d_{max},d_{max}^{\prime}$ </p>
-    <p> $y\in range(\mathcal{M})$ </p>
-    <p> $\mathsf{Project}$ </p>
-    <p> $\displaystyle(\mathbb{E}[T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime})]-\hat{%
-T}(G,d_{max}^{\prime}))^{2}+\mathbb{V}[T^{\prime}(G,\varepsilon_{2},d_{max}^{%
-\prime})]$ </p>
-    <p> $\mathsf{Gamma}$ </p>
-    <p> $\theta=1000$ </p>
-    <p> $\mathsf{Perturb}$ </p>
-    <p> $Gam_{2}$ </p>
-    <p> $\mathsf{Perturb}()$ </p>
-    <p> $O(nd_{max})$ </p>
-    <p> $\mathsf{GraphProjection}$ </p>
-    <p> $\langle x\rangle_{1},\langle y\rangle_{1},\langle z\rangle_{1},\langle w%
-\rangle_{1},\langle o\rangle_{1},\langle p\rangle_{1},\langle q\rangle_{1}$ </p>
-    <p> $S\subseteq Range(\mathcal{M}_{i})$ </p>
-    <p> $\mathbb{E}[l_{2}^{2}(T(G),\hat{T}(G,d_{max}^{\prime}))]=0$ </p>
-    <p> $\varepsilon=\varepsilon_{1}+\varepsilon_{2}$ </p>
-    <p> $\langle T^{\prime}\rangle_{1}=\langle T\rangle_{1}+\langle\gamma\rangle_{1}$ </p>
-    <p> $\langle v_{i},v_{j}\rangle\in E$ </p>
-    <p> $\langle y\rangle_{i}$ </p>
-    <p> $ds[j]$ </p>
-    <p> $\langle f\rangle_{1}=\langle a_{ik}\rangle_{1}-\langle y\rangle_{1}$ </p>
-    <p> $Gam_{1}(n,\lambda)$ </p>
-    <p> $\mathsf{project}$ </p>
-    <p> $\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{0\}$ </p>
-    <p> $O(\frac{e^{\varepsilon}}{(e^{\varepsilon}-1)^{2}}(d_{max}^{3}n+\frac{e^{%
-\varepsilon}}{\varepsilon^{2}}d_{max}^{2}n))$ </p>
-    <p> $u=\langle u\rangle_{1}+\langle u\rangle_{2}=\langle x\rangle_{1}$ </p>
-    <p> $\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{1\}$ </p>
-    <p> $\langle T\rangle\leftarrow\{\langle T\rangle_{1},\langle T\rangle_{2}\}$ </p>
-    <p> $Laplace$ </p>
-    <p> $Pr[\mathcal{M}(G)=y]\leq e^{\varepsilon}Pr[\mathcal{M}(G^{\prime})=y]$ </p>
-    <p> $\langle x\rangle_{i}$ </p>
-    <p> $\langle T\rangle_{1}$ </p>
-    <p> $1\times\sim 2\times$ </p>
-    <p> $S_{i\in\{1,2\}}$ </p>
-    <p> $A=\{A_{1},...,A_{n}\}$ </p>
-    <p> $\langle T\rangle_{2}\leftarrow\langle T\rangle_{2}+u_{2}$ </p>
-    <p> $G,G^{\prime}\in\mathcal{G}$ </p>
-    <p> $\langle\gamma_{i}\rangle_{2}\rightarrow S_{2}$ </p>
-    <p> $\mathsf{Lap(\frac{1}{\varepsilon_{1}})}$ </p>
-    <p> $O(\frac{d_{max}^{\prime 2}}{\varepsilon^{2}})$ </p>
-    <p> $u_{2}=\langle w\rangle_{2}+\langle xy\rangle_{2}g+\langle xz\rangle_{2}f+%
-\langle yz\rangle_{2}e$ </p>
-    <p> $\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime}),\hat{T}(G,%
-d_{max}^{\prime}))]=O(\frac{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$ </p>
-    <p> $\langle v_{i},v_{j}\rangle$ </p>
-    <p> $Gam_{1}$ </p>
-    <p> $l_{2}(T,T^{\prime})=(T-T^{\prime})^{2}$ </p>
-    <p> $=w+xyg+xzf+yze+xfg+yeg+zef+efg$ </p>
-    <p> $\langle u\rangle_{2}$ </p>
-    <p> $D^{\prime}=\{d_{1}^{\prime},...,d_{n}^{\prime}\}$ </p>
-    <p> $\mathsf{Laplace}$ </p>
-    <p> $w=xyz,o=xy,p=xz,q=yz$ </p>
-    <p> $ds[j]\leftarrow\mathsf{DS}(d_{i},d_{j}^{\prime})$ </p>
-    <p> $d_{max}^{\prime}\geq 0$ </p>
-    <p> $\langle f\rangle_{2}=\langle a_{ik}\rangle_{2}-\langle y\rangle_{2}$ </p>
-    <p> $S\subseteq Range(\mathcal{M})$ </p>
-    <p> $\{d_{1}^{\prime},...,d_{n}^{\prime}\}$ </p>
-    <p> $Gam_{2}(n,\lambda)$ </p>
-    <p> $\displaystyle\frac{e^{-\varepsilon_{1}.|d^{\prime}-d_{i}|}}{e^{-\varepsilon_{1%
-}.|d^{\prime}-d_{i}^{\prime}|}}=e^{\varepsilon_{1}.(|d^{\prime}-d_{i}^{\prime}%
-|-|d^{\prime}-d_{i}|)}\leq e^{\varepsilon_{1}|d_{i}-d_{i}^{\prime}|}=e^{%
-\varepsilon_{1}},$ </p>
-    <p> $(r_{1},...,r_{n})$ </p>
-    <p> $\mathsf{view}_{S_{i}}^{\Pi}\approx\mathsf{Sim}_{S_{i}}$ </p>
-    <p> $T(G)=\langle T\rangle_{1}+\langle T\rangle_{2}$ </p>
-    <p> $\mathsf{Project}()$ </p>
-    <p> $Gam_{1}=\mathsf{Gamma}(n,\frac{d_{max}^{\prime}}{\varepsilon_{2}})$ </p>
-    <p> $Gamma(x,n,\lambda)=\frac{(1/\lambda)^{1/n}}{\Gamma(1/n)}x^{\frac{1}{n}-1}e^{-%
-\frac{x}{\lambda}},$ </p>
-    <p> $g=\langle g\rangle_{1}+\langle g\rangle_{2}$ </p>
-    <p> $d=a\times b\times c$ </p>
-    <p> $Lap(.)$ </p>
-    <p> $ds[k]$ </p>
-    <p> $a_{ij}\times a_{ik}\times a_{jk}=1$ </p>
-    <p> $(D^{\prime},d_{max}^{\prime})\leftarrow\mathsf{Max}(D,\varepsilon_{1})$ </p>
-    <p> $\langle x\rangle=\langle x\rangle_{1}+\langle x\rangle_{2}$ </p>
-    <p> $Lap(\lambda)=\sum_{i=1}^{n}[Gam_{1}(n,\lambda)-Gam_{2}(n,\lambda)],$ </p>
-    <p> $\mathsf{Count}()$ </p>
-    <p> $A=\{A_{1},A_{2},...,A_{n}\}$ </p>
-    <p> $\langle v_{2},v_{5}\rangle$ </p>
-    <p> $DS(d_{1},d_{2})$ </p>
-    <p> $\mathsf{Max}(.)$ </p>
-    <p> $u=a_{ij}\times a_{ik}\times a_{jk}$ </p>
-    <p> $\langle T\rangle\leftarrow\mathsf{Count}(\hat{A})$ </p>
-    <p> $|T(G)-T(G^{\prime})|=\triangle$ </p>
-    <p> $+\langle x\rangle_{1}fg+\langle y\rangle_{1}eg+\langle z\rangle_{1}ef$ </p>
-    <p> $d_{max}^{\prime}\geq d_{max}$ </p>
-    <p> $\gamma_{i}=\langle\gamma_{i}\rangle_{1}+\langle\gamma_{i}\rangle_{2}$ </p>
-    <p> $w=x\times y\times z,o=x\times y,p=x\times z,q=y\times z$ </p>
-    <p> $Pr[\mathcal{M}(G)\in S]\leq e^{\epsilon}Pr[\mathcal{M}(G^{\prime})\in S]$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10018.html DELETED Viewed

@@ -1,156 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $d_{i}\leq d_{max}^{\prime}$ </p>
-    <p> $\hat{A}\leftarrow\mathsf{Project}(A,D,D^{\prime},d_{max}^{\prime})$ </p>
-    <p> $\langle g\rangle_{1}=\langle a_{jk}\rangle_{1}-\langle z\rangle_{1}$ </p>
-    <p> $\langle x\rangle_{1},\langle y\rangle_{1},\langle z\rangle_{1},\langle w%
-\rangle_{1},\langle o\rangle_{1},\langle p\rangle_{1},\langle q\rangle_{1}%
-\rightarrow S_{1}$ </p>
-    <p> $\langle T\rangle_{1}\leftarrow\langle T\rangle_{1}+u_{1}$ </p>
-    <p> $x,y,z,w,o,p,q$ </p>
-    <p> $O(\frac{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$ </p>
-    <p> $T^{\prime}\leftarrow\mathsf{Perturb}(\langle T\rangle,d_{max}^{\prime},%
-\varepsilon_{2})$ </p>
-    <p> $T(G^{\prime})=\langle T^{\prime}\rangle_{1}+\langle T^{\prime}\rangle_{2}$ </p>
-    <p> $x^{\prime}=r_{1}^{\prime}+...+r_{n}^{\prime}$ </p>
-    <p> $a_{ij},a_{ik},a_{jk}$ </p>
-    <p> $\varepsilon_{2}\geq 0$ </p>
-    <p> $f(x,\lambda)=\frac{1}{2\lambda}e^{\frac{|x|}{\lambda}}$ </p>
-    <p> $DS(d_{1},d_{2})=\frac{|d_{1}-d_{2}|}{d_{1}}$ </p>
-    <p> $\langle T\rangle_{2}$ </p>
-    <p> ${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}%
-]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{%
-\mathcal{E}}}\cdot\pi_{RT}$ </p>
-    <p> $\pi_{FiT}$ </p>
-    <p> ${\bigl{\{}Dev_{C}^{Tot}\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> ${\bigl{\{}V^{P2P}\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> $N_{U},{\bigl{\{}V^{P2P}\bigr{\}}_{\mathcal{E}}},{\bigl{\{}V^{Real}\bigr{\}}_{%
-\mathcal{E}}},Dev_{C}^{Tot},Dev_{P}^{Tot},\pi_{P2P},\pi_{RT}$ </p>
-    <p> ${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}%
-]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{%
-\mathcal{E}}}\cdot\pi_{P2P}$ </p>
-    <p> $\pi_{P2P}$ </p>
-    <p> $KGen_{pe}(k)$ </p>
-    <p> $each~{}m_{l}\pcin~{}M(u_{k})$ </p>
-    <p> ${\bigl{\{}inDev\bigr{\}}_{\mathcal{E}}},{\bigl{\{}inDev_{M}\bigr{\}}_{\mathcal%
-{E}}}$ </p>
-    <p> $\{.\}_{\mathcal{E}}$ </p>
-    <p> $Bal^{Tot}_{sup}$ </p>
-    <p> ${\bigl{\{}V^{Real}\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> $V^{Real}$ </p>
-    <p> ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
-]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
-\mathcal{E}}}\cdot{\pi_{RT}}$ </p>
-    <p> $Dev_{P}^{Tot}>Dev_{C}^{Tot}c$ </p>
-    <p> ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
-]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
-\mathcal{E}}}/Dev_{P}^{Tot}\cdot TotRev_{P}$ </p>
-    <p> $Stat_{M}[u_{k}]$ </p>
-    <p> $KGen_{pe}$ </p>
-    <p> ${\bigl{\{}V^{Real}[u_{k}]\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> ${\bigl{\{}inDev\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> ${\bigl{\{}Stat\bigr{\}}_{\mathcal{E}}},{\bigl{\{}Stat_{M}\bigr{\}}_{\mathcal{E%
-}}}$ </p>
-    <p> ${\bigl{\{}inDev_{M}\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> ${{\bigl{\{}inDev[u_{k}]\bigr{\}}_{\mathcal{E}}}}$ </p>
-    <p> $inDev$ </p>
-    <p> ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
-]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
-\mathcal{E}}}\cdot\pi_{P2P}$ </p>
-    <p> $V^{P2P}$ </p>
-    <p> $V^{Real}[u_{k}]$ </p>
-    <p> ${\bigl{\{}inDev[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{Real}[u_{%
-k}]\bigr{\}}_{\mathcal{E}}}-{\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}};$ </p>
-    <p> $M(u_{k})$ </p>
-    <p> $H{({\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}})}$ </p>
-    <p> $PK_{sup}$ </p>
-    <p> $\xrightarrow{}PK_{sup},SK_{sup}$ </p>
-    <p> ${\bigl{\{}inDev_{M}[M(m_{l})]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{%
-Real}[M(m_{l})]\bigr{\}}_{\mathcal{E}}}-{\bigl{\{}V^{P2P}[M(m_{l})]\bigr{\}}_{%
-\mathcal{E}}};$ </p>
-    <p> $Dev_{P}^{Tot}=Dev_{C}^{Tot}$ </p>
-    <p> $Dev_{P}^{Tot}>Dev_{C}^{Tot}$ </p>
-    <p> $TotRev_{P}=(Dev_{C}^{Tot}\cdot\pi_{P2P}+(Dev_{P}^{Tot}-Dev_{C}^{Tot})\cdot\pi_%
-{FiT})$ </p>
-    <p> $\footnotesize{\bigl{\{}Dev_{C}^{Tot}\bigr{\}}_{\mathcal{E}}}\leftarrow\sum_{i=%
-0}^{N_{C}-1}{\bigl{\{}inDev_{C}[c_{i}]\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> ${inDev_{P}[u_{k}]}/Dev_{P}^{Tot}$ </p>
-    <p> ${\bigl{\{}stat^{Tot}[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}stat^{%
-Tot}[u_{k}]\bigr{\}}_{\mathcal{E}}}+{\bigl{\{}stat[u_{k}]\bigr{\}}_{\mathcal{E%
-}}}$ </p>
-    <p> $Dev^{Tot}$ </p>
-    <p> $\pi_{P2P},\pi_{FiT},\pi_{RT}$ </p>
-    <p> $N_{C}=N_{P}$ </p>
-    <p> ${\bigl{\{}inDev[c_{i}]\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> $each~{}u_{k}$ </p>
-    <p> ${\bigl{\{}Dev_{P}^{Tot}\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> ${Dev_{C}^{Tot}}$ </p>
-    <p> $Bal_{sup}\leftarrow 0$ </p>
-    <p> $\pi_{RT}$ </p>
-    <p> $Stat$ </p>
-    <p> ${Dev_{P}^{Tot}}$ </p>
-    <p> $N_{C}<N_{P}$ </p>
-    <p> ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
-]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
-\mathcal{E}}}\cdot{\pi_{P2P}}$ </p>
-    <p> $TotRev_{P}$ </p>
-    <p> $Bal_{sup}\leftarrow-(Dev_{P}^{Tot}-Dev_{C}^{Tot})\cdot\pi_{FiT}$ </p>
-    <p> $Stat[u_{k}]$ </p>
-    <p> $\footnotesize{\bigl{\{}Dev_{P}^{Tot}\bigr{\}}_{\mathcal{E}}}\leftarrow\sum_{j=%
-0}^{N_{C}-1}{\bigl{\{}inDev_{P}[p_{j}]\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> $H{({\bigl{\{}V^{Real}[u_{k}]\bigr{\}}_{\mathcal{E}}})}$ </p>
-    <p> ${\bigl{\{}stat_{M}^{Tot}[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}%
-stat^{Tot}[m_{l}]\bigr{\}}_{\mathcal{E}}}+{\bigl{\{}stat[m_{l}]\bigr{\}}_{%
-\mathcal{E}}}$ </p>
-    <p> $stat^{Tot}$ </p>
-    <p> $N_{C}>N_{P}$ </p>
-    <p> $Bal_{sup}$ </p>
-    <p> $(Dev_{P}^{Tot}-Dev_{C}^{Tot})$ </p>
-    <p> $Dev_{P}^{Tot}<Dev_{C}^{Tot}$ </p>
-    <p> ${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}%
-]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{%
-\mathcal{E}}}/Dev_{P}^{Tot}\cdot TotRev_{P}$ </p>
-    <p> $SK_{sup}$ </p>
-    <p> ${\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}}$ </p>
-    <p> $Bal_{sup}\leftarrow(Dev_{C}^{Tot}-Dev_{P}^{Tot})\cdot\pi_{RT}.$ </p>
-    <p> $k(\mu,\mu^{\prime})=\left(1+\left\|(\mu-\mu^{\prime})/\sigma\right\|_{2}^{2}%
-\right)^{-1}$ </p>
-    <p> $\displaystyle=-\log(\mathcal{N}(h(\mathbf{x});\mathbf{0},\mathbf{I})))-\log%
-\left|\det\mathbf{J}_{h}(\mathbf{x})\right|$ </p>
-    <p> $K=D-d$ </p>
-    <p> $\mathbf{x}=h^{-1}(\mathbf{z})$ </p>
-    <p> $\mathcal{L}_{\mathrm{MMD}}(p(\mathbf{x}_{D}),p(h^{-1}([\mathbf{y}_{d},\mathbf{%
-z}_{K}])))$ </p>
-    <p> $D>d$ </p>
-    <p> $(x_{\text{soma}},y_{\text{soma}})$ </p>
-    <p> $\displaystyle\mathcal{L}_{\mathrm{NLL}}=-\log(p(\mathbf{x}))$ </p>
-    <p> $I(x,y)=\exp\left(-\frac{(x-x_{\text{stim}})^{2}+(y-y_{\text{stim}})^{2}}{2\rho%
-^{2}}-\frac{(x-x_{\text{soma}})^{2}+(y-y_{\text{soma}})^{2}}{2\lambda^{2}}%
-\right),$ </p>
-    <p> $\{\mu\},\{\mu^{\prime}\}$ </p>
-    <p> $g_{\theta}:\tilde{\mathbf{x}}\mapsto\mathbf{x}_{s}$ </p>
-    <p> $\mathbf{z}\neq\mathbf{0}$ </p>
-    <p> $\mathbf{J}_{h}(\mathbf{x})=\partial h(\mathbf{x})/\partial\mathbf{x}^{T}$ </p>
-    <p> $h(\mathbf{x}_{D})=[h_{\mathbf{y}_{d}}(\mathbf{x}_{D}),h_{\mathbf{z}_{K}}(%
-\mathbf{x}_{D})]$ </p>
-    <p> $P_{M^{\prime}}$ </p>
-    <p> $h:\mathbf{x}_{D}\mapsto[\mathbf{y}_{d},\mathbf{z}_{K}]$ </p>
-    <p> $h_{\mathbf{y}_{d}}(\mathbf{x}_{D})\approx s(\mathbf{x}_{D})$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10019.html DELETED Viewed

@@ -1,134 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\mathcal{L}_{\mathrm{MMD}}(p_{M},p_{M^{\prime}})=\left(\mathbb{E}_{i,j}[k(\mu_%
-{i},\mu_{j})-2k(\mu_{i},\mu_{j}^{\prime})+k(\mu_{i}^{\prime},\mu_{j}^{\prime})%
-]\right)^{\frac{1}{2}},$ </p>
-    <p> $\rho=400\,\mu m$ </p>
-    <p> $\lambda=1550\,\mu m$ </p>
-    <p> $\mathbf{y}_{d}$ </p>
-    <p> $\mathbf{z}\sim\pi(\mathbf{z})=\mathcal{N}(\mathbf{z};\mathbf{0},\mathbf{I}))$ </p>
-    <p> $\mathcal{L}_{\mathrm{NLL}}\simeq\frac{1}{2}\|h(\mathbf{x};\mathbf{c})\|_{2}^{2%
-}-\log\left|\det\mathbf{J}_{h}(\mathbf{x})\right|$ </p>
-    <p> $\min_{\theta}\mathcal{L}_{\mathrm{MSE}}\left(\tilde{\mathbf{x}},f_{\phi}\left(%
-g_{\theta}\left(\tilde{\mathbf{x}}\right)\right)\right),$ </p>
-    <p> $\mathbf{z}_{K}\sim\pi(\mathbf{z}_{K})=\mathcal{N}(\mathbf{z}_{K};\mathbf{0},%
-\mathbf{I}_{K}))$ </p>
-    <p> ${\mathbf{x}_{s}}$ </p>
-    <p> $\mathcal{L}_{\mathrm{MSE}}=\mathbb{E}[(\mathbf{y}_{d}-h_{\mathbf{y}_{d}}(%
-\mathbf{x}_{D}))^{2}]$ </p>
-    <p> $\mu^{\prime}\sim p_{M^{\prime}}$ </p>
-    <p> $p(\mathbf{x})=\pi(\mathbf{z}=h(\mathbf{x}))\left|\det\frac{\partial h(\mathbf{%
-x})}{\partial\mathbf{x}^{T}}\right|,$ </p>
-    <p> $\mathbf{x}=h^{-1}(\mathbf{z};\mathbf{c})$ </p>
-    <p> $s:\mathbf{x}_{D}\in\mathbb{R}^{D}\mapsto\mathbf{y}_{d}\in\mathbb{R}^{d}$ </p>
-    <p> $\displaystyle\simeq\frac{1}{2}\|h(\mathbf{x})\|_{2}^{2}-\log\left|\det\mathbf{%
-J}_{h}(\mathbf{x})\right|.$ </p>
-    <p> $(x_{\text{stim}},y_{\text{stim}})$ </p>
-    <p> $\mathbf{z}=h(\mathbf{x};\mathbf{c})$ </p>
-    <p> $\mathbf{z}=h(\mathbf{x})$ </p>
-    <p> $\mathbf{z}_{K}$ </p>
-    <p> $\mathcal{L}_{\mathrm{MMD}}(p(h(\mathbf{x}_{D})),p(\mathbf{y}_{d})p(\mathbf{z}_%
-{K}))$ </p>
-    <p> $\mathbf{x}_{D}$ </p>
-    <p> $\mathbf{y}_{p}$ </p>
-    <p> $\mu\sim p_{M}$ </p>
-    <p> $\Psi\in\mathbb{R}^{z\times h}$ </p>
-    <p> $\bm{0.982}$ </p>
-    <p> $24.90$ </p>
-    <p> $\bm{0.861}$ </p>
-    <p> $\bm{0.044}$ </p>
-    <p> $z=8,192$ </p>
-    <p> $r:\Re\times\Re^{3}\rightarrow\Re^{3}$ </p>
-    <p> $24.88$ </p>
-    <p> $\bm{0.047}$ </p>
-    <p> $24.13$ </p>
-    <p> $\bm{0.018}$ </p>
-    <p> $\bm{32.70}$ </p>
-    <p> $\bm{W}_{\Psi}$ </p>
-    <p> $M^{\prime}_{\Delta}=(\bm{W}^{\prime}_{\Delta},\bm{b}^{\prime}_{\Delta})$ </p>
-    <p> $\bm{W}_{\gamma}$ </p>
-    <p> $27.89$ </p>
-    <p> $\bm{0.714}$ </p>
-    <p> $\bm{0.070}$ </p>
-    <p> $\bm{0.025}$ </p>
-    <p> $\bm{0.911}$ </p>
-    <p> $\bm{h}^{\prime}_{\Delta}$ </p>
-    <p> $\bm{28.14}$ </p>
-    <p> $32.53$ </p>
-    <p> $\bm{b}=\{\bm{b}_{\Delta},\bm{b}_{c},\bm{b}_{\psi}$ </p>
-    <p> $29.57$ </p>
-    <p> $\bm{h}_{c}$ </p>
-    <p> $M^{\prime}_{\Delta}$ </p>
-    <p> $\bm{W}=\{\bm{W}_{\Delta},\bm{W}_{c},\bm{W}_{\psi}$ </p>
-    <p> $\bm{0.991}$ </p>
-    <p> $\bm{0.093}$ </p>
-    <p> $\bm{W}_{\mu},\bm{W}_{\gamma},\bm{W}_{\Psi},\bm{W}^{\prime}_{\Delta},\bm{W}^{%
-\prime}_{c}\}$ </p>
-    <p> $\bm{37.72}$ </p>
-    <p> $\bm{b}_{\Psi}$ </p>
-    <p> $\bm{0.988}$ </p>
-    <p> $\bm{35.41}$ </p>
-    <p> $26.11$ </p>
-    <p> $\bm{f}_{\Delta}=\sigma\left(\bm{h}_{\Delta}\right),$ </p>
-    <p> $\bm{0.029}$ </p>
-    <p> $\bm{h}_{c}^{\prime}=[\bm{\Psi}\otimes\bm{f}_{c},\mathbf{d}].$ </p>
-    <p> $\bm{0.956}$ </p>
-    <p> $\bm{32.34}$ </p>
-    <p> $\bm{31.13}$ </p>
-    <p> $\bm{26.51}$ </p>
-    <p> $26,29$ </p>
-    <p> $27.31$ </p>
-    <p> $\bm{20.80}$ </p>
-    <p> $\bm{\mu}=\bm{f}_{\mu}\otimes\bm{\gamma}.$ </p>
-    <p> $\bm{36.76}$ </p>
-    <p> $\bm{W}_{\psi}$ </p>
-    <p> $\bm{\gamma}=tanh\left({\bm{W}_{\gamma}[\bm{h}_{\Delta},\bm{h}_{c}]+\bm{b}_{%
-\gamma}}\right),$ </p>
-    <p> $\bm{27.56}$ </p>
-    <p> $\bm{b}_{\mu},\bm{b}_{\gamma},\bm{b}_{\Psi},\bm{b}^{\prime}_{\Delta},\bm{b}^{%
-\prime}_{c}\}$ </p>
-    <p> $\bm{W}_{\mu}$ </p>
-    <p> $23.49$ </p>
-    <p> $\bm{0.007}$ </p>
-    <p> $\bm{h}_{\Delta}$ </p>
-    <p> $\bm{0.103}$ </p>
-    <p> $\bm{f}_{c}=\sigma\left({\bm{h}_{c}}\right),$ </p>
-    <p> $\bm{0.068}$ </p>
-    <p> $MSE(\cdot)$ </p>
-    <p> ${\cal L}=MSE(r(\Delta,\bm{c}),\bm{g}),$ </p>
-    <p> $32.45$ </p>
-    <p> $\bm{0.009}$ </p>
-    <p> $\bm{f}_{\psi}=\sigma\left({\bm{W}_{\psi}\bm{[h}_{\Delta},\bm{h}_{c}]+\bm{b}_{%
-\psi}}\right),$ </p>
-    <p> $\bm{0.026}$ </p>
-    <p> $\mathbf{l}=(x,y,z)$ </p>
-    <p> $\bm{f}_{\mu}=\sigma\left({\bm{W}_{\mu}[\bm{h}_{\Delta},\bm{h}_{c}]+\bm{b}_{\mu%
-}}\right),$ </p>
-    <p> $\bm{0.882}$ </p>
-    <p> $\bm{0.837}$ </p>
-    <p> $[\bm{h}_{\Delta},\bm{h}_{c}]$ </p>
-    <p> $\bm{f}_{\Delta}$ </p>
-    <p> $\bm{26.06}$ </p>
-    <p> $\bm{b_{\mu}}$ </p>
-    <p> $29.56$ </p>
-    <p> $\bm{27.01}$ </p>
-    <p> $26.05$ </p>
-    <p> $F_{\Theta}:(\mathbf{l},\mathbf{d})\rightarrow(\mathbf{c},\Delta)$ </p>
-</body>
-</html>

htmls/output_mathjax_example_1002.html DELETED Viewed

@@ -1,124 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\operatorname{Inst}(\emptyset)$ </p>
-    <p> $\prod_{s=1}^{t}2^{p_{s}-1}$ </p>
-    <p> ${\rm D}_{\rm NP}\leq_{0\hbox{-}T}{\rm D}_{\rm P}$ </p>
-    <p> $\p^{A}=\np^{A}$ </p>
-    <p> $\p^{L}=\p$ </p>
-    <p> $\langle M^{\prime},\epsilon\rangle$ </p>
-    <p> $\np^{\mathcal{O}}$ </p>
-    <p> $\langle M,1^{n},1^{t}\rangle\in{\rm U}_{\rm NP}$ </p>
-    <p> $\np^{L}=\np$ </p>
-    <p> $x\notin{\rm HP}$ </p>
-    <p> $\np^{\p}=\np$ </p>
-    <p> $\p$ </p>
-    <p> $A\leq_{f(n)\hbox{-}T}B$ </p>
-    <p> ${\rm D}_{\rm NP}\not\in\p^{{\rm D}_{\rm P}}$ </p>
-    <p> $x\in\mathcal{O}$ </p>
-    <p> $L\in{\rm\Sigma_{1}^{0}}$ </p>
-    <p> $n,t\in{\mathbb{N}^{+}}$ </p>
-    <p> $t^{\prime}\in{\mathbb{N}}$ </p>
-    <p> $\Theta(2^{n})$ </p>
-    <p> ${\rm D}_{\rm NP}=\{\langle M,1^{n}\rangle\mid n\in{\mathbb{N}}\land(\exists x%
-\in\{0,1\}^{n})[M$ </p>
-    <p> $\langle M,1^{n}\rangle\in{\rm D}_{\rm NP}$ </p>
-    <p> ${\rm U}_{\rm P}=\{\langle M,x,1^{t}\rangle\mid t\in{\mathbb{N}^{+}}\land M$ </p>
-    <p> $M^{\mathcal{O}}$ </p>
-    <p> ${\rm HP}=\{\langle M,w\rangle\mid M\text{ halts on input }w\}.$ </p>
-    <p> $(\forall x)[x\in A\iff f(x)\in B]$ </p>
-    <p> $z\notin{\rm D}_{\rm NP}$ </p>
-    <p> $(\forall L\in{\rm\Sigma_{1}^{0}})[L\leq_{m}A]$ </p>
-    <p> $\p^{\p}=\p$ </p>
-    <p> $A\leq_{m}B$ </p>
-    <p> ${\mathbb{N}^{+}}=\{1,2,3,\ldots\}$ </p>
-    <p> ${\rm D}_{\rm NP}\in{\rm\Sigma_{1}^{0}}$ </p>
-    <p> $\Omega(2^{n})$ </p>
-    <p> $\p^{X}\neq\np^{X}$ </p>
-    <p> $\p^{A}\neq\np^{A}\iff\p\neq\np$ </p>
-    <p> $\p\neq\np$ </p>
-    <p> $]\}$ </p>
-    <p> ${\rm U}_{\rm NP}$ </p>
-    <p> $L\leq_{m}{\rm D}_{\rm NP}$ </p>
-    <p> $x]\}$ </p>
-    <p> $\p^{\mathcal{O}}$ </p>
-    <p> $\min(c,2^{n})$ </p>
-    <p> ${\rm D}_{\rm NP}\leq_{m}{\rm D}_{\rm P}$ </p>
-    <p> $z\not\in{\rm D}_{\rm NP}$ </p>
-    <p> $y=\epsilon$ </p>
-    <p> $\p^{\p}$ </p>
-    <p> ${\rm U}_{\rm P}$ </p>
-    <p> $\{0,1\}^{n}\cap L(M)$ </p>
-    <p> $A\leq_{T}B$ </p>
-    <p> ${\rm D}_{\rm P}$ </p>
-    <p> $U_{\text{\bf P}}$ </p>
-    <p> ${\rm PSPACE}$ </p>
-    <p> $x\in L(M)$ </p>
-    <p> $\p^{B}\neq\np^{B}$ </p>
-    <p> $A\in\p$ </p>
-    <p> $\langle M,x\rangle\in{\rm D}_{\rm P}\iff(\exists t>0)[\langle M,x,1^{t}\rangle%
-\in{\rm U}_{\rm P}]$ </p>
-    <p> ${\mathbb{N}}=\{0,1,2,\ldots\}$ </p>
-    <p> $A=L(M^{B})$ </p>
-    <p> $\mathcal{C}^{\mathcal{D}}=\bigcup_{A\in\mathcal{D}}\mathcal{C}^{A}$ </p>
-    <p> $\np$ </p>
-    <p> ${\rm U}_{\rm NP}\not\in\p^{{\rm U}_{\rm P}}$ </p>
-    <p> $\np^{\p}$ </p>
-    <p> $f:{\mathbb{N}}\rightarrow{\mathbb{N}}$ </p>
-    <p> $\p=\np$ </p>
-    <p> $f(x)\in{\rm D}_{\rm NP}$ </p>
-    <p> ${\rm\Sigma_{0}^{0}}$ </p>
-    <p> ${\rm\Sigma_{0}^{0}}^{\mathcal{O}}$ </p>
-    <p> $A\in{\rm\Sigma_{1}^{0}}$ </p>
-    <p> $x\not\in\mathcal{O}$ </p>
-    <p> $\p^{A}=\p$ </p>
-    <p> ${\rm U}_{\rm NP}=\{\langle M,1^{n},1^{t}\rangle\mid n\in{\mathbb{N}}\land t\in%
-{\mathbb{N}^{+}}\land(\exists x\in\{0,1\}^{n})[M$ </p>
-    <p> $\langle M,1^{n}\rangle$ </p>
-    <p> $x\in{\rm HP}$ </p>
-    <p> $\langle M,1^{n}\rangle\in{\rm D}_{\rm NP}\iff(\exists t>0)[\langle M,1^{n},1^{%
-t}\rangle\in{\rm U}_{\rm NP}]$ </p>
-    <p> $x\}$ </p>
-    <p> $w\in\{0,1\}^{n}$ </p>
-    <p> $\epsilon\notin L(M^{\prime})$ </p>
-    <p> $A\leq_{1\hbox{-}T}B$ </p>
-    <p> ${\rm HP}\leq_{m}{\rm D}_{\rm NP}$ </p>
-    <p> $\np^{A}=\np$ </p>
-    <p> $(\forall x)[x\notin{\rm HP}\implies f(x)\notin{\rm D}_{\rm NP}]$ </p>
-    <p> ${\rm\Sigma_{1}^{0}}$ </p>
-    <p> ${\rm HP}$ </p>
-    <p> $\langle M,w\rangle$ </p>
-    <p> $(\forall x)[x\in{\rm HP}\iff f(x)\in{\rm D}_{\rm NP}]$ </p>
-    <p> $\mathcal{C},\mathcal{D}$ </p>
-    <p> $\epsilon\in L(M^{\prime})$ </p>
-    <p> ${\rm D}_{\rm NP}$ </p>
-    <p> ${\rm D}_{\rm NP}\leq_{h\hbox{-}T}{\rm D}_{\rm P}$ </p>
-    <p> $L\in\p$ </p>
-    <p> ${\rm D}_{\rm P}=\{\langle M,x\rangle\mid M$ </p>
-    <p> ${\rm D}_{\rm NP}\leq_{1\hbox{-}T}{\rm D}_{\rm P}$ </p>
-    <p> $c\in{\mathbb{N}^{+}}$ </p>
-    <p> $(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}$ </p>
-    <p> $G\setminus X:=G[V(G)\setminus X]$ </p>
-    <p> $V(C_{2})$ </p>
-    <p> $p_{j}\in B$ </p>
-    <p> $H\setminus u$ </p>
-    <p> $\chi(G)\leq k$ </p>
-    <p> $u\in V(H)$ </p>
-    <p> $\{\overline{K_{2}\cup C_{2k+1}}\mid k\in\mathbb{N}\}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10020.html DELETED Viewed

@@ -1,159 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\Theta_{\Delta}=(\bm{W}_{\Delta},\bm{b}_{\Delta})$ </p>
-    <p> $\bm{0.954}$ </p>
-    <p> $(\Delta,\bm{c})$ </p>
-    <p> $\bm{0.873}$ </p>
-    <p> $\bm{0.796}$ </p>
-    <p> $M_{\Delta}$ </p>
-    <p> $\bm{0.055}$ </p>
-    <p> $\bm{33.99}$ </p>
-    <p> $\bm{b}_{\gamma}$ </p>
-    <p> $\bm{0.122}$ </p>
-    <p> $\bm{0.914}$ </p>
-    <p> $24.54$ </p>
-    <p> $\bm{36.46}$ </p>
-    <p> $M^{\prime}_{c}=(\bm{W}^{\prime}_{c},\bm{W}^{\prime}_{c})$ </p>
-    <p> $\bm{h}^{\prime}_{c}$ </p>
-    <p> $20-30$ </p>
-    <p> $\bm{f}_{\psi}$ </p>
-    <p> $\bm{36.13}$ </p>
-    <p> $30.91$ </p>
-    <p> $\bm{f}_{\mu}$ </p>
-    <p> $\bm{28.60}$ </p>
-    <p> $\bm{h}_{\Delta}^{\prime}=[\bm{\Psi}\otimes\bm{f}_{\Delta},\mathbf{l}],$ </p>
-    <p> $\bm{22.23}$ </p>
-    <p> $\bm{0.010}$ </p>
-    <p> $33.91$ </p>
-    <p> $33.09$ </p>
-    <p> $31.75$ </p>
-    <p> $26,73$ </p>
-    <p> $\bm{\Psi}=tanh\left({\bm{W}_{\Psi}\left(\bm{\mu}+\left(\bm{f}_{\psi}\otimes%
-\Psi\right)\right)+\bm{b}_{\Psi}}\right),$ </p>
-    <p> $34.56$ </p>
-    <p> $\bm{35.83}$ </p>
-    <p> $\bm{b_{\psi}}$ </p>
-    <p> $\Theta_{c}=(\bm{W}_{c},\bm{b}_{c})$ </p>
-    <p> $\lfloor p^{2}\cdot r\rfloor$ </p>
-    <p> $E_{SP}$ </p>
-    <p> $E_{SE}$ </p>
-    <p> $\displaystyle E_{SP}(sp,2i)=\sin(\frac{sp}{\Omega^{\frac{2i}{d}}})\quad E_{SP}%
-(sp,2i+1)=\cos(\frac{sp}{\Omega^{\frac{2i}{d}}})$ </p>
-    <p> $E_{SE}(s,2i)$ </p>
-    <p> $\displaystyle E_{P}(pos,2i)=\sin(\frac{pos}{\Omega^{\frac{2i}{d}}}),\text{ }E_%
-{P}(pos,2i+1)=\cos(\frac{pos}{\Omega^{\frac{2i}{d}}})$ </p>
-    <p> $j\in\{i,...,n\}$ </p>
-    <p> $p_{i}c_{j}$ </p>
-    <p> $(p,M)$ </p>
-    <p> $\displaystyle\leq\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},%
-\tilde{y})\geq A\right]+\mathbb{P}\left[\frac{1}{(\mathcal{T}+1)}\geq\frac{1}{%
-B}\right]$ </p>
-    <p> $\hat{x}^{1,k},\hat{y}^{1,k}$ </p>
-    <p> $s_{x}^{t}$ </p>
-    <p> $F_{1},...,F_{M}:\mathbb{R}^{d}\to\mathbb{R}$ </p>
-    <p> $\nabla F_{\mathcal{D}}(w^{t})$ </p>
-    <p> $\displaystyle=\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w^{t};B^{t}),x^{t}%
--x\rangle$ </p>
-    <p> $\displaystyle\leq 2\tau q\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};%
-B^{t})\|_{\infty}^{2}+2\tau\sum_{k=t_{0}}^{t-1}\Big{(}\|\nabla f(\hat{w}^{t_{0%
-}};B^{k})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\Big{)}$ </p>
-    <p> $L_{2}^{F}$ </p>
-    <p> $L_{2}^{G}\leq L_{2}^{F}D^{3}$ </p>
-    <p> $\mathbb{E}\big{[}\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})-\min_{w\in%
-\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})\big{]}\leq\frac{2L_{1}}{T}+\frac{2L_{1%
-}^{2}}{\lambda T}+\lambda\log(d_{x}).$ </p>
-    <p> $\displaystyle\qquad+\frac{L_{0}\sqrt{(TK/q+qK)\log(1/\delta)}\log(d_{x})}{n%
-\varepsilon}+\frac{L_{2}}{K}+\frac{L_{1}q\log(T/q)}{T}.$ </p>
-    <p> $F(w^{T})-F(x)\leq\frac{1}{\sum_{j\in[T]}\beta_{j}}\bigg{(}\operatorname{Regret%
-}_{T}(x)+\sum_{t\in[T]}\langle\beta_{t}\nabla F(w^{t})-g_{t},x_{t}-x\rangle%
-\bigg{)}.$ </p>
-    <p> $2\Delta_{s}-$ </p>
-    <p> $\Lambda^{T}$ </p>
-    <p> $\beta_{t}\in\mathbb{R}^{+},g^{t}\in\mathbb{R}^{d_{x}}$ </p>
-    <p> $\color[rgb]{1,0,0}\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n%
-\varepsilon}\right)^{1/2}$ </p>
-    <p> $\ell_{t}(x)=\langle g^{t},x\rangle$ </p>
-    <p> $\mathbb{R}^{d_{x}}$ </p>
-    <p> $\mathcal{Z}=\{z_{1},...,z_{|\mathcal{Z}|}\}$ </p>
-    <p> $\displaystyle\leq\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|},x\in\Delta_{|%
-\mathcal{Z}|}}(F_{\mathcal{D}}(\tilde{x},y)-F_{\mathcal{D}}(x,\bar{y})).\right]$ </p>
-    <p> $(g_{x},g_{y})=\text{BiasReducedGradient}(x,y,N,B)$ </p>
-    <p> $\displaystyle\leq 2\tau\sum_{k=t_{0}}^{t-1}\Big{(}\|\nabla f(w^{t_{0}};B^{t})-%
-\nabla f(\hat{w}^{t_{0}};B^{t})\|_{\infty}^{2}+\|\nabla f(\hat{w}^{t_{0}};B^{k%
-})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\Big{)}$ </p>
-    <p> $a(x)\leq Cb(x)$ </p>
-    <p> $(\tilde{x}^{t},\tilde{y}^{t})_{t\in[T]}$ </p>
-    <p> $\max\{L_{0},B\}$ </p>
-    <p> $\displaystyle=\max_{j\in[d]}\left|\nabla_{j}F\left(\bar{a}\right)-\nabla_{j}F%
-\left(\bar{x}\right)\right|^{2}=\max_{j\in[d]}\left|\langle\nabla F(\bar{a})-%
-\nabla F(\bar{x}),e_{j}\rangle\right|^{2}$ </p>
-    <p> $\mathbb{E}[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})]\lesssim\sqrt{[(%
-\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2%
-}]U}+L_{2}\sqrt{U}.$ </p>
-    <p> $\max\{L_{1},L_{2}\}$ </p>
-    <p> $\tilde{x}^{t+1},\tilde{y}^{t+1}$ </p>
-    <p> $\displaystyle\leq 5\sqrt{\frac{\log(|{\cal Q}|)}{n}},$ </p>
-    <p> $\mathbb{P}\left[F(\bar{a}^{T})-F(\bar{x}^{T})\geq\frac{L_{1}D^{2}}{2}\sum_{t=1%
-}^{T}\lambda_{t}^{2}+\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{%
-t}^{2}}\right]\leq\exp(-\beta^{2}).$ </p>
-    <p> $(\tilde{x},\tilde{y})$ </p>
-    <p> $\min_{x\in\Delta_{x}}\max_{y\in\Delta_{y}}F_{\mathcal{D}}(x,y),$ </p>
-    <p> $g_{y}=C_{M}2^{N}(\nabla_{y}f(\bar{x}_{+},\bar{y}_{+};B)-\nabla_{y}f(\bar{x}_{-%
-},\bar{y}_{-};B))+\nabla_{y}f(x_{0},y_{0};B)$ </p>
-    <p> $\mathbb{E}\left[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})\right]$ </p>
-    <p> $i\in\{K_{x}+1,...,K_{x}+K_{y}\}$ </p>
-    <p> $\textstyle s^{t}_{y}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{y,j}\right)$ </p>
-    <p> $\alpha_{t}=\lambda_{t}\left\langle\nabla F\left(\sum_{k=t}^{T}\lambda_{k}x^{k}%
-+\sum_{k=1}^{t-1}\lambda_{k}a^{k}\right),a^{t}-x^{t}\right\rangle$ </p>
-    <p> $(\alpha_{t})_{t\in[T]}$ </p>
-    <p> $\displaystyle\|\mathbb{E}[\Phi_{\mathcal{D}}(x,y)-(g_{x},-g_{y})]\|_{\infty}$ </p>
-    <p> $f:\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}\to\mathbb{R}$ </p>
-    <p> $\Delta(s_{y}^{t})$ </p>
-    <p> $\tilde{y}^{1}$ </p>
-    <p> $\{\mathcal{T}\geq t-1\}=\Big{\{}\sum_{k\in[t-1]}2^{N_{k}}\leq U\Big{\}}$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}%
-^{2}+(\log(d_{x})+\log(d_{y}))ML_{1}^{2}]\frac{U}{M}+\frac{L_{2}U}{M2^{M}}$ </p>
-    <p> $\displaystyle\leq\|\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t})\|_{\infty}%
-\|x^{t}-x\|_{1}$ </p>
-    <p> $\displaystyle\langle\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle$ </p>
-    <p> $\displaystyle\geq\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\right]=%
-\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\mathbb{E}[2^{N_{t}}\mathbbm{1}_{(%
-\mathcal{T}+1\geq t)}|N_{t-1},...,N_{1}]\right]$ </p>
-    <p> $(\mathcal{A}_{n})_{n\geq 1}$ </p>
-    <p> $\displaystyle=|F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-F(\Lambda(\lambda%
-_{2},\mu_{2})+(x_{1},y_{1}))|$ </p>
-    <p> $\displaystyle=\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},%
-\tilde{y})\cdot\frac{1}{\mathcal{T}+1}\geq\frac{A}{B}\right]$ </p>
-    <p> $\tau\eqsim\min\left\{\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+(L_{0}^{2}+L_{1}^{2})%
-q\sqrt{\log(d_{x})/K}+L_{1}^{2}q/[\sqrt{\log(d_{x})}K^{3/2}])T}},\frac{1}{L_{0%
-}q},\frac{n\varepsilon}{TL_{0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}.$ </p>
-    <p> $\hat{x}^{t+1,k}$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\lesssim L_{1}\frac{\sqrt{\log(%
-d_{x})+\log(d_{y})}}{\sqrt{T}}+\frac{(\log(d_{x})+\log(d_{y}))}{\tau T}+\tau L%
-_{0}^{2}+\frac{L_{2}}{K}.$ </p>
-    <p> $\displaystyle=\tau\Big{|}2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{%
-+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})\Big{)}+\nabla%
-_{x}f(x_{0}^{r},y_{0}^{r};B^{r})$ </p>
-    <p> $\mathbb{E}\Big{\|}\nabla_{x}f(\bar{x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(x,y;B%
-_{k})\Big{\|}_{\infty}^{2}\leq\frac{20L_{2}^{2}}{2^{2(k+1)}}+\frac{(12+2\log(d%
-_{x}))L_{1}^{2}}{2^{k+1}},$ </p>
-    <p> $\bar{x}=\frac{1}{T}\sum_{t=1}^{T}x^{t}$ </p>
-    <p> $\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\leq U$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10021.html DELETED Viewed

@@ -1,192 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\displaystyle\leq\tau\Big{|}\frac{2^{N_{r}}\alpha}{2^{N_{r}}}\Big{[}\Big{(}%
-\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r};z^{*})-\nabla_{x}f(\bar{x}_{+}^{r}%
-,\bar{y}_{+}^{r};z^{\prime*})\Big{)}+\Big{(}\nabla_{x}f(\bar{x}_{-}^{r},\bar{y%
-}_{-}^{r};z^{*})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};z^{\prime*})\Big{%
-)}\Big{]}$ </p>
-    <p> $\displaystyle\lesssim\begin{cases}q\tau\big{[}\frac{L_{1}^{2}}{\sqrt{\log(d_{x%
-})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}}{\sqrt{K}}\big{]}&%
-\text{without second order smoothnes}\\
-q\tau\big{[}\frac{L_{2}^{2}}{K^{2}}+\frac{L_{1}^{2}\log(d_{x})}{K}\big{]}&%
-\text{with second order smoothness}\end{cases},$ </p>
-    <p> $(w^{t},v^{t})$ </p>
-    <p> $M=d$ </p>
-    <p> $\mbox{TG}(p,M)$ </p>
-    <p> $\displaystyle\quad+\mathbb{E}\big{[}\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(%
-\hat{w}^{t_{0}};B^{t}),x^{t}-x\rangle\big{]}$ </p>
-    <p> $i\in[K_{x}]$ </p>
-    <p> $\langle\nabla F(\bar{a})-\nabla F(\bar{x}),e_{j}\rangle\leq\frac{|F(\bar{a}+re%
-_{j})-F(\bar{x}+re_{j})|+|F(\bar{x})-F(\bar{a})|}{r}+L_{1}r.$ </p>
-    <p> $q=\sqrt{T/\log(d_{x})},K=T/q=\sqrt{T\log(d_{x})}$ </p>
-    <p> $J:=\{e_{1},...,e_{d}\}$ </p>
-    <p> $s_{y}^{t}$ </p>
-    <p> $L_{2}\lesssim(L_{0}+L_{1})\left\{\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n%
-}}+\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n%
-\varepsilon}\right)^{1/2}\right\}$ </p>
-    <p> $y^{1}=(1/d_{y},...,1/d_{y})$ </p>
-    <p> $\displaystyle\lesssim\frac{2(\log(d_{x})+\log(d_{y}))}{\tau T}+5\tau L_{0}^{2}%
-+\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t%
-})-g^{t}\mid\mathcal{F}_{t}]\|_{\infty}.$ </p>
-    <p> $2^{N_{t}}\leq 2^{M}$ </p>
-    <p> $\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n\varepsilon}\right)^{2/5}$ </p>
-    <p> $\displaystyle\leq\frac{2\log(|\mathcal{Z}|)}{\tau_{x}T}+\frac{\log(|{\cal Q}|)%
-}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+\frac{1}{T}\sum_{t=1}^{T}\langle\nabla_{x}F_%
-{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t}-w^{t}\rangle+2\|-\nabla_{y}F_{%
-\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty}.$ </p>
-    <p> $\operatorname{Regret}_{T}(x)\leq\frac{\log(d)}{\tau}+\frac{\tau}{2}\sum_{t\in[%
-T]}\|g^{t}\|_{\infty}^{2},$ </p>
-    <p> $a^{t}\sim P_{x^{t}}$ </p>
-    <p> $w^{t}=\frac{(t-1)w^{t-1}+x^{t}}{t}$ </p>
-    <p> $\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},\tilde{y})-%
-\operatorname{Gap}(\bar{x},\bar{y}))\right]$ </p>
-    <p> $F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)\leq\frac{1}{T}\left[\operatorname{%
-Regret}_{T}(x)+\sum_{t\in[T]}\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-%
-x\rangle\right],$ </p>
-    <p> $f:\mathbb{R}^{d}\mapsto\mathbb{R}$ </p>
-    <p> $X\sim{\cal N}(0,1/2)$ </p>
-    <p> $\displaystyle=\max_{k\in[K_{x}+K_{y}]}\big{|}\nabla_{k}\langle\Lambda^{T}_{j},%
-\nabla F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_%
-{2},\mu_{2})+(x_{1},y_{1}))\rangle\big{|}$ </p>
-    <p> $\displaystyle=3\left(\frac{|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}+L_{1}^{2}r^{2}+%
-\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{2}}{r^{2}}\right)$ </p>
-    <p> $\displaystyle\leq 4\tau\sum_{k=t_{0}}^{t-1}\left\|\nabla f(\hat{w}^{t_{0}};B^{%
-k})-\nabla f(w^{t_{0}};B^{k})\right\|_{\infty}.$ </p>
-    <p> $\displaystyle\lesssim\left(L_{0}^{2}+L_{2}^{2}+\log(d_{x})ML_{1}^{2}\right)%
-\mathbb{E}\left[\sum_{t=1}^{U}\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$ </p>
-    <p> $\tau\leq\frac{B\varepsilon}{8L_{0}\sqrt{2(TK/q+qK)\log(1/\delta)}}$ </p>
-    <p> $\displaystyle\lesssim\sqrt{\frac{(L_{0}^{2}+L_{1}^{2}q\log(d_{x})/K+L_{2}^{2}q%
-/K^{2})\log(d_{x})}{T}}+\frac{L_{0}q\log(d_{x})}{T}$ </p>
-    <p> $\displaystyle+\langle\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle$ </p>
-    <p> $f_{i}:\mathcal{X}\times\mathcal{Z}\to[-B,B]$ </p>
-    <p> $\hat{x}\sim P_{x}$ </p>
-    <p> $\mathcal{Y}=\Delta_{y}$ </p>
-    <p> $\tilde{x}_{j}^{t+1}\propto\tilde{x}_{j}^{t}\exp\big{(}-\tau\nabla_{j}f(w^{t_{0%
-}};B^{k})\big{)}$ </p>
-    <p> $N(S)=N((\varepsilon_{n}(S))_{n\geq 1},(\delta_{n}(S))_{n\geq 1})$ </p>
-    <p> $\displaystyle\leq\tau\alpha L_{0}(4+1/2^{N_{r}})\leq 4.5\tau\alpha L_{0}.$ </p>
-    <p> $A\eqsim\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(%
-d_{y}))L_{1}^{2}]U}\quad,\quad B\eqsim U/M$ </p>
-    <p> $\displaystyle\leq\frac{\log(|\mathcal{Z}|)}{\tau_{x}}+\frac{\tau_{x}}{2}\sum_{%
-t\in[T]}\|g^{t}_{x}\|_{\infty}^{2}+\frac{\log(|{\cal Q}|)}{\tau_{y}}+\frac{%
-\tau_{y}}{2}\sum_{t\in[T]}\|g^{t}_{y}\|_{\infty}^{2}+\sum_{t=1}^{T}\langle\Phi%
-_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w,v)\rangle.$ </p>
-    <p> $\lambda_{i}=\frac{1}{T}$ </p>
-    <p> $(\lambda_{1},\ldots,\lambda_{T})\in\Delta_{T}$ </p>
-    <p> $\mathbb{P}\left[\sum_{t=1}^{T}\alpha_{t}\geq\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{%
-\sum_{t=1}^{T}\lambda_{t}^{2}}\right]\leq\exp(-\beta^{2}),$ </p>
-    <p> $F_{\mathcal{D}}(x_{\mathcal{D}},\bar{y})=0$ </p>
-    <p> $\displaystyle T\operatorname{Gap}(\bar{x},\bar{y})$ </p>
-    <p> $x^{1},...,x^{T}\in\Delta_{d}$ </p>
-    <p> $\|\nabla F(x^{1})-\nabla F(x^{2})\|_{\infty}\leq L_{1}\|x^{1}-x^{2}\|_{1}$ </p>
-    <p> $\frac{4\tau L_{0}}{B}\leq\frac{\varepsilon}{2\sqrt{2(TK/q+(q+1)K)\log(1/\delta%
-)}}.$ </p>
-    <p> $T\eqsim\min\left\{n,\frac{n\varepsilon}{\sqrt{(\log(d_{x})+\log(d_{y}))\log(1/%
-\delta)}}\right\},$ </p>
-    <p> $T\eqsim\min\left\{n,\frac{(n\varepsilon)^{2/3}}{\log(1/\delta)^{1/3}}\right\}$ </p>
-    <p> $|(x^{t}-\tilde{x}^{t})_{j}|\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum%
-_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}\underbrace{\max\left\{2\max%
-_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|,\exp\left(2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_%
-{j}|\right)-1\right\}}_{:=\psi},$ </p>
-    <p> $\frac{4L_{0}\tau}{B}$ </p>
-    <p> $\displaystyle\leq L_{0}^{F}\sum_{i=1}^{K_{x}}|\lambda_{1,i}-\lambda_{2,i}|\|x_%
-{i}-x_{1}\|_{1}+L_{0}\sum_{j=1}^{K_{y}}|\mu_{1,j}-\mu_{2,j}|\|y_{j}-y_{1}\|_{1}$ </p>
-    <p> $U=\min\left\{\frac{n\varepsilon\sqrt{L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{%
-y}))L_{1}^{2}}}{\sqrt{4\cdot 48\cdot 81(\log(d_{x})+\log(d_{y}))\log(1/\delta)%
-}L_{0}},\frac{n}{2}\right\}$ </p>
-    <p> $T=\min\left\{n,\frac{n\varepsilon}{\log(d_{x})\sqrt{\log(1/\delta)}}\right\}$ </p>
-    <p> $\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\langle\Phi_{\mathcal{D}}(x^{t},y^{t%
-})-(g_{x}^{t},g_{y}^{t}),(x^{t},y^{t})-(w^{t},v^{t})\rangle\right]\lesssim%
-\frac{L_{2}\mathbb{E}[\mathcal{T}+1]}{2^{M}}.$ </p>
-    <p> $\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})$ </p>
-    <p> $\mathbb{E}\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F\left(%
-\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\|^{2}_{\infty}\lesssim\frac{L_{2}^{2}}{T%
-^{2}}+\frac{L_{1}^{2}(1+\log(d))}{T}$ </p>
-    <p> $x_{i}-x_{1}$ </p>
-    <p> $2^{N_{t}}$ </p>
-    <p> $\|x\|_{1}=\sum_{j\in[d]}|x_{j}|$ </p>
-    <p> $0<c\leq 2e$ </p>
-    <p> $\mathbb{E}[\nabla_{j}F(\bar{x})-\nabla_{j}F(\bar{a})]=\mathbb{E}[\langle\nabla
-F%
-(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle]\leq\frac{4L_{1}}{rT}+L_{1}r.$ </p>
-    <p> $\{x_{1},\ldots,x_{K_{x}}\}$ </p>
-    <p> $M=\log_{2}(\sqrt{U})$ </p>
-    <p> $\displaystyle\leq 6\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\leq 6\sum_{t=1}^{%
-\mathcal{T}}2^{N_{t}}+6\cdot 2^{M}\leq 6(U-2^{M})+6\cdot 2^{M}=6U,$ </p>
-    <p> $\displaystyle\leq 2L_{1}\|w^{t}-w^{t_{0}}\|_{1}\leq 2L_{1}\sum_{k=t_{0}}^{t-1}%
-\|w^{k+1}-w^{k}\|_{1}$ </p>
-    <p> $\frac{2\tau L_{0}}{B}$ </p>
-    <p> $|s_{x}^{t}-s_{x}^{\prime t}|$ </p>
-    <p> $\displaystyle\leq\tau\|\nabla\operatorname{LSE}(-\tau G(S))\|_{1}\|G_{j}(S)-G_%
-{j}(S^{\prime})\|_{\infty}$ </p>
-    <p> $\displaystyle 4\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}+2(\mathcal{T}+1)$ </p>
-    <p> $\displaystyle\leq 0+\frac{4L_{1}q}{q\lfloor t/q\rfloor+1}+2\|\mathbb{E}_{\hat{%
-w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})]\|_{\infty}$ </p>
-    <p> $K=T/\log(d_{x})$ </p>
-    <p> $\displaystyle\leq\frac{2L_{1}}{T}+\frac{2L_{1}^{2}}{\lambda T}+\lambda\log(d_{%
-y}).$ </p>
-    <p> $\tilde{x}^{t+1}$ </p>
-    <p> $x^{1}=(1/d_{x},...,1/d_{x})$ </p>
-    <p> $N:\mathbb{R}^{\infty}_{\geq 0}\times\mathbb{R}^{\infty}_{\geq 0}\to\mathbb{N}$ </p>
-    <p> $\{y_{1},\ldots,y_{K_{y}}\}$ </p>
-    <p> $\mathbb{P}(j=e_{i})\propto\exp\left(\frac{\varepsilon s(S,i)}{2\Delta_{s}}\right)$ </p>
-    <p> $N^{t+1}\sim\mbox{TG}(0.5,M)$ </p>
-    <p> $\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
-\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}%
-\leq\frac{4L_{1}}{\sqrt{T}}.$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|%
-^{2}\right]$ </p>
-    <p> $\displaystyle\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^%
-{t_{0}};B^{t}),x^{t}-x\rangle]$ </p>
-    <p> $\displaystyle\leq\sum_{t\in[T]}[F_{\mathcal{D}}(x^{t},v)-F_{\mathcal{D}}(w,y^{%
-t})]\leq\sum_{t\in[T]}\langle\Phi_{\mathcal{D}}(x^{t},y^{t}),(x^{t},y^{t})-(w,%
-v)\rangle$ </p>
-    <p> $\bar{x}^{T}=\sum_{t=1}^{T}\lambda_{t}x^{t}$ </p>
-    <p> $T,K,q,\tau$ </p>
-    <p> $\|g^{t}\|_{\infty}\leq L_{0}$ </p>
-    <p> $\displaystyle\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t}%
-,y^{t})-(w,v)\rangle=\sum_{t=1}^{T}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t%
-})-g^{t}_{x},x^{t}-w\rangle+\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{%
-t}_{y},y^{t}-v\rangle$ </p>
-    <p> $\displaystyle\operatorname{Gap}(\bar{x},\bar{y})\leq\frac{2(\log(d_{x})+\log(d%
-_{y}))}{\tau T}+5\tau L_{0}^{2}+\frac{1}{T}\sum_{t=1}^{T}\langle\Phi_{\mathcal%
-{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w^{t},v^{t})\rangle.$ </p>
-    <p> $\displaystyle\leq\frac{2L_{1}}{T}+\frac{2L_{1}^{2}}{\lambda T}+\lambda\log(d_{%
-y})+\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]+\frac{2L_{1}}{T}+\frac{2L_%
-{1}^{2}}{\lambda T}+\lambda\log(d_{x})$ </p>
-    <p> $\displaystyle\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^%
-{t};B^{t}),x^{t}-x\rangle]$ </p>
-    <p> $s^{t}_{x}$ </p>
-    <p> $f(\cdot,\cdot;z)$ </p>
-    <p> $G_{j}(S)=\left(\sum_{k\in[i-1]}g^{k}_{j}\right)_{i\in[t]}$ </p>
-    <p> $\hat{G}_{j}=-\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(\hat{w}^{t_{0}};B^{k}),G_{j}=%
--\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(w^{t_{0}};B^{k})$ </p>
-    <p> $\xi^{1},\xi^{2},...$ </p>
-    <p> $\|g^{t}\|_{\infty}^{2}+\|\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\|_{\infty}^{2}%
-\leq 5L_{0}^{2}$ </p>
-    <p> $T\eqsim\min\left\{n,\frac{(n\varepsilon)^{2/3}}{\log(1/\delta)^{1/3}}\right\},$ </p>
-    <p> $\mathcal{T}+1$ </p>
-    <p> $x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau g^{t}_{j}\right),\quad\forall j\in%
-[d_{x}]$ </p>
-    <p> $\mathbb{E}[A_{1}]=\mathbb{E}[\langle\tilde{\Delta}^{t_{0}},\tilde{x}^{t}-x%
-\rangle]\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second order %
-smoothnes}\\
-\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10022.html DELETED Viewed

@@ -1,195 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\mathbb{E}[\|x^{t}\|_{\infty}]<\infty$ </p>
-    <p> $x^{1}=(1/|\mathcal{Z}|,...,1/|\mathcal{Z}|),y^{1}=(1/|\mathcal{Q}|,...,1/|%
-\mathcal{Q}|)$ </p>
-    <p> $\langle-\nabla F(\bar{a}),e_{j}\rangle\leq\frac{F(\bar{a}+re_{j})-F(\bar{a})}{%
-r}+\frac{L_{1}r}{2}.$ </p>
-    <p> $F_{1}(\cdot)=F(\cdot)$ </p>
-    <p> $2^{N_{t}+1}$ </p>
-    <p> $\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},\tilde{y})\geq A%
-\right]\lesssim\frac{\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(%
-\log(d_{x})+\log(d_{y}))L_{1}^{2}]U}}{A}.$ </p>
-    <p> $\lambda=L_{1}\frac{2}{\sqrt{T(\log(d_{x})+\log(d_{y}))}}$ </p>
-    <p> $\displaystyle\lesssim\frac{\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2%
-}+(\log(d_{x})+\log(d_{y}))L_{1}^{2}]U}}{A}$ </p>
-    <p> $(x_{i}-x_{1},\mathbf{0}_{d_{y}})$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1}+L_{2})%
-{\small\bigg{[}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{n}}+\left(\frac{(\log(d_{x%
-})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{2/5}\bigg{]}.}$ </p>
-    <p> $\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(\hat{G}_{j}\right)}{\sum_{i\in[d_{x%
-}]}x^{t_{0}}_{i}\exp\left(\hat{G}_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{%
-j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}$ </p>
-    <p> $\lambda=\sqrt{8}/\sqrt{\log(|{\cal Q}|)T}$ </p>
-    <p> $\mathcal{A}_{1:N(S)}$ </p>
-    <p> $L_{0}D$ </p>
-    <p> $\bar{a}^{T}=\sum_{t\in[T]}\lambda_{t}a^{t}$ </p>
-    <p> $L/\mu=\Omega(d)$ </p>
-    <p> $\ell_{x}=\log(d_{x})$ </p>
-    <p> $x^{1},\ldots,x^{T}\in\mathbb{R}^{d}$ </p>
-    <p> $\displaystyle\leq\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|+|F(\bar{x})-F(%
-\bar{a})|}{r}+L_{1}r$ </p>
-    <p> $\displaystyle\leq\frac{2(\log(d_{x})+\log(d_{y}))}{\tau}+\tau\sum_{t=1}^{T}(\|%
-g^{t}\|_{\infty}^{2}+\|\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\|_{\infty}^{2})$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\nabla_{x}F_{\mathcal{D}}(\bar{x}_{+,M},\bar{y}_%
-{+,M})-\nabla_{x}F_{\mathcal{D}}(x_{-,0},y_{-,0})+\nabla_{x}F_{\mathcal{D}}(x_%
-{0},y_{0})\right]$ </p>
-    <p> $g^{t}_{x,j}$ </p>
-    <p> $\operatorname{Gap}(x,y)=\max_{v\in\mathcal{X},w\in\mathcal{Y}}(F_{\mathcal{D}}%
-(x,w)-F_{\mathcal{D}}(v,y))$ </p>
-    <p> $\|x\|_{\infty}=\max_{j\in[d]}|x_{j}|$ </p>
-    <p> $B=n/T$ </p>
-    <p> $\displaystyle\leq\max_{i\in[K_{x}+K_{y}]}\|\Lambda^{T}_{i}\|_{1}\|\nabla F(%
-\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{%
-2})+(x_{1},y_{1}))\|_{\infty}$ </p>
-    <p> $\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
-\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}%
-\leq\frac{2L_{2}}{T}.$ </p>
-    <p> $w^{t}_{j}\propto\exp(\operatorname{LSE}(\tau G_{j}(S))).$ </p>
-    <p> $\displaystyle\left(\sum_{i=1}^{K_{x}}\lambda_{i}(x_{i}-x_{1}),\sum_{j=1}^{K_{y%
-}}\mu_{j}(y_{j}-y_{1})\right).$ </p>
-    <p> $\displaystyle\lesssim\sqrt{[(\log(d_{x})+\log(d_{y})]L_{1}^{2}\frac{U}{M}}$ </p>
-    <p> $\tilde{x}^{t+1},\hat{x}^{t+1,k}$ </p>
-    <p> $\displaystyle\leq A^{2}+C^{2}\log(M)+2C^{2}+2AC$ </p>
-    <p> $\displaystyle\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^{t_{0}};B^{%
-t}),x^{t}-x\rangle$ </p>
-    <p> $\mathbb{P}[F(\bar{a}^{T})-F\left(\bar{x}^{T}\right)\geq\alpha]\leq\mathbb{P}%
-\left[\sum_{t=1}^{T}\alpha_{t}+\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{%
-2}\geq\alpha\right]=\mathbb{P}\left[\sum_{t\in[T]}\alpha_{t}\geq\alpha^{\prime%
-}\right]$ </p>
-    <p> $s_{x}^{t}=s_{x}^{\prime t}$ </p>
-    <p> $s^{t}_{x}=-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r%
-},\bar{y}_{+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})%
-\Big{)}+\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}.$ </p>
-    <p> $\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|^{2}\right]%
-\leq\frac{5L_{1}^{2}D^{4}}{4}\left(\sum_{t=1}^{T}\lambda_{t}^{2}\right)^{2}+%
-\frac{L_{0}^{2}D^{2}(6+\log(M))}{2}\sum_{t=1}^{T}\lambda_{t}^{2}.$ </p>
-    <p> $\mathbb{E}\big{[}\sum_{t\in[T]}\alpha_{t}\big{]}=0$ </p>
-    <p> $g^{t}\in\partial f_{t}(x)$ </p>
-    <p> $\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
-\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}%
-\lesssim\frac{L_{2}}{T}$ </p>
-    <p> $\displaystyle\leq 2L_{0}^{2}+4C_{M}\sum_{k=0}^{M}2^{k}\mathbb{E}\Big{\|}\nabla%
-_{x}f(\bar{x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(x,y;B_{k})\Big{\|}_{\infty}^{2}$ </p>
-    <p> $\sum_{t\in[\mathcal{T}+1]}\max\{1,2^{N_{t}}/\alpha\}\leq n$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
-1}^{2}q\log(d_{x})}{K}+\frac{L_{2}^{2}q}{K^{2}}\big{]}+\frac{L_{2}}{K}+\frac{L%
-_{1}q\log(T/q)}{T},$ </p>
-    <p> $\operatorname{Regret}_{T}(x)\leq\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}\sum_{t%
-\in[T]}\|g^{t}\|_{\infty}^{2}$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\max_{q\in{\cal Q}}|\mathbb{E}_{z}[q(z)]-q(\tilde%
-{S})|\right]$ </p>
-    <p> $M=\log(\sqrt{U})$ </p>
-    <p> $\mathbb{R}^{d_{x}},\mathbb{R}^{d_{y}}$ </p>
-    <p> $\mathbb{P}\left[\frac{1}{\mathcal{T}+1}\geq\frac{1}{B}\right]=\mathbb{P}\left[%
-\mathcal{T}\leq B-1\right]\leq\mathbb{P}\left[\sum_{t\in[B]}2^{N_{t}}>U-2^{M}%
-\right]\leq\frac{B\mathbb{E}[2^{N_{1}}]}{U-2^{M}},$ </p>
-    <p> $F_{\mathcal{D}}(\cdot)$ </p>
-    <p> $\displaystyle\leq L_{0}^{F}D(\|\lambda_{1}-\lambda_{2}\|_{1}+\|\mu_{1}-\mu_{2}%
-\|_{1})$ </p>
-    <p> $\displaystyle\lesssim(L_{0}+L_{1})\sqrt{\frac{\log(d_{x})}{T}}+\frac{L_{2}}{%
-\sqrt{T}\log(d_{x})^{1/4}}$ </p>
-    <p> $(\hat{x}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{x^{t}}$ </p>
-    <p> $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
-\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\}$ </p>
-    <p> $g^{t}=\nabla f(\hat{w}^{t};B^{t})$ </p>
-    <p> $L_{1}^{G}\leq L_{1}^{F}D^{2}$ </p>
-    <p> $g^{t}_{x}=\nabla_{x}f(x^{t},\hat{y}^{t};S)$ </p>
-    <p> $\displaystyle=\mathbb{E}[\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v)-%
-\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\bar{x},v)]+\mathbb{E}[\max_{v\in\Delta_{%
-y}}F_{\mathcal{D}}(\bar{x},v)-\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})]$ </p>
-    <p> $\frac{\varepsilon}{2\sqrt{2T(K+1)\log(1/\delta)}}$ </p>
-    <p> $\frac{4TL_{0}\tau_{y}}{n}\leq\frac{\varepsilon}{2\sqrt{2T\log(1/\delta)}}.$ </p>
-    <p> $\displaystyle=\frac{(A+B)^{2}}{2}+2MC\left[C\exp\left(-(B/C)^{2}\right)+A%
-\mathbb{P}[X\geq B/C]\right]$ </p>
-    <p> $\displaystyle+\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})%
-,x^{t}-x\rangle.$ </p>
-    <p> $\mathbb{E}[x_{t+1}|x_{1:t}]=0$ </p>
-    <p> $M(\log(d_{x})+\log(d_{y}))$ </p>
-    <p> $\mathbb{P}[X\geq B/C]\leq\exp\left(-(B/C)^{2})\right)$ </p>
-    <p> $T=\frac{6n\varepsilon}{16\sqrt{2\log(1/\delta)}\log(|{\cal Q}|)},\tau_{x}=%
-\sqrt{\frac{\log(|\mathcal{Z}|)}{9T}},\tau_{y}=\frac{\log(|{\cal Q}|)}{6\sqrt{%
-\log(|\mathcal{Z}|)T}}$ </p>
-    <p> $(\varepsilon^{\prime},T\delta+\delta^{\prime})$ </p>
-    <p> $\tilde{x}^{t}\sim P_{x^{t}}$ </p>
-    <p> $\sum_{t\in[T]}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t}-w%
-\rangle\\
-\leq\sum_{t\in[T]}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t%
-}-w^{t}\rangle+\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|\nabla_{%
-x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}.$ </p>
-    <p> $\displaystyle\leq\mathbb{E}\left[(\mathcal{T}+1)\left(\frac{4L_{1}}{\mathcal{T%
-}+1}+\frac{2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{\mathcal{T}+1}}\right)\right]$ </p>
-    <p> $\displaystyle\leq\max_{k\in[K_{x}+K_{y}]}\sum_{i\in[K_{x}+K_{y}]}|\Lambda_{j,i%
-}|\|\Lambda^{T}_{k}\|_{1}\|\nabla\nabla_{i}F(\Lambda(\lambda_{1},\mu_{1})+(x_{%
-1},y_{1}))-\nabla\nabla_{i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\|_{\infty}$ </p>
-    <p> $(f_{i})_{i\in[d_{y}]}$ </p>
-    <p> $\sqrt{\log(d)/n}+(\log(d)^{3/2}/[n\varepsilon])^{2/5}$ </p>
-    <p> $f:\mathcal{X}\times\mathcal{Z}\to\mathbb{R}$ </p>
-    <p> $(g^{t}_{x},g^{t}_{y})=\operatorname{BiasReducedGradient}(x^{t},y^{t},N^{t},B^{%
-t})$ </p>
-    <p> $w\in\Delta_{|\mathcal{Z}|},v\in\Delta_{|{\cal Q}|}$ </p>
-    <p> $2^{N_{1}},...,2^{N_{t-1}}$ </p>
-    <p> $\frac{4\tau L_{0}}{B}$ </p>
-    <p> $\sum_{t\in[B]}2^{N_{i}}>U$ </p>
-    <p> $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}\{F_{\mathcal{D}}(x,y)=\mathbb{E}_%
-{z\sim\mathcal{D}}[f(x,y;z)]$ </p>
-    <p> $\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{%
-\infty}\leq 2L_{2}/K$ </p>
-    <p> $\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\lesssim\sqrt{\frac{[(%
-\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2%
-}]\log(U)^{2}}{U}}\right]\geq 0.99.$ </p>
-    <p> $\phi(y)=\sum_{j\in[d_{y}]}y_{j}\log(1/y_{j})$ </p>
-    <p> $\bar{x}^{T}=\sum_{t\in[T]}\lambda_{t}x^{t}$ </p>
-    <p> $F(\cdot+re_{j})$ </p>
-    <p> $\displaystyle\|\nabla\nabla_{j}G(\lambda_{1},\mu_{1})-\nabla\nabla_{j}G(%
-\lambda_{2},\mu_{2})\|_{\infty}=\max_{k\in[K_{x}+K_{y}]}\big{|}\nabla_{k,j}G(%
-\lambda_{1},\mu_{1})-\nabla_{k,j}G(\lambda_{2},\mu_{2})\big{|}$ </p>
-    <p> $x^{t+1}_{j}\propto\exp\left\{-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_%
-{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y%
-}_{-}^{r};B^{r})\Big{)}+\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}\right\},$ </p>
-    <p> $w\in\mathcal{X}$ </p>
-    <p> $\displaystyle+\mathbb{E}[\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})-\min_%
-{w\in\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})]$ </p>
-    <p> $\displaystyle=\frac{5L_{1}^{2}D^{4}}{4}\left(\sum_{t=1}^{T}\lambda_{t}^{2}%
-\right)^{2}+\frac{L_{0}^{2}D^{2}(6+\log(M))}{2}\sum_{t=1}^{T}\lambda_{t}^{2}.$ </p>
-    <p> $\displaystyle|G(\lambda_{1},\mu_{1})-G(\lambda_{2},\mu_{2})|$ </p>
-    <p> $B^{r}=B^{\prime r}$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{k=0}^{M}\big{(}\nabla_{x}F_{\mathcal{D}}(%
-\bar{x}_{+,k},\bar{y}_{+,k})-\nabla_{x}F_{\mathcal{D}}(\bar{x}_{-,k},\bar{y}_{%
--,k})\big{)}+\nabla_{x}F_{\mathcal{D}}(x_{0},y_{0})\right]$ </p>
-    <p> $\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w^{t};B^{t}),x^{t}-x%
-\rangle]=0$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})-\operatorname{Gap}(\bar{x},%
-\bar{y})]\leq\frac{4}{n}+6\sqrt{\frac{\log(|{\cal Q}|)}{n}}$ </p>
-    <p> $g_{y}^{1}$ </p>
-    <p> $\hat{G}_{j}$ </p>
-    <p> $\displaystyle\leq L_{2}^{F}D^{3}\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})%
-\|_{1}.$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau L_{0}^{2}+\frac{1}{T}\sum%
-_{t\in[T]}\left(\frac{L_{2}}{K}+\mathbbm{1}_{(t>q)}\left[\frac{4L_{1}q}{q%
-\lfloor t/q\rfloor+1}+q\frac{L_{2}^{2}\tau}{K^{2}}+q\frac{L_{1}^{2}\log(d_{x})%
-\tau}{K}\right]\right)$ </p>
-    <p> $\tau\eqsim\min\left\{\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+(L_{0}^{2}+L_{1}^{2})%
-q\sqrt{\log(d_{x})/K}+L_{1}^{2}q/[\sqrt{\log(d_{x})}K^{3/2}])T}},\frac{1}{L_{0%
-}q},\frac{n\varepsilon}{TL_{0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}$ </p>
-    <p> $\mathbb{E}\Big{\|}\nabla_{x}f(x,y;B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B%
-_{k})\Big{\|}_{\infty}^{2}\leq\frac{20L_{2}^{2}}{2^{2k}}+\frac{(12+2\log(d_{x}%
-))L_{1}^{2}}{2^{k}}.$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10023.html DELETED Viewed

@@ -1,184 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}%
-x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j%
-}|\right].$ </p>
-    <p> $(\hat{w}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{w^{t}}$ </p>
-    <p> $\min_{x\in\mathcal{X}}\{\max_{i\in[d_{y}]}F_{i}(x)\}=\min_{x\in\mathcal{X}}%
-\max_{y\in\Delta_{y}}\mathbb{E}_{z\sim\mathcal{D}}\left[\sum_{i\in[d_{y}]}y_{i%
-}f_{i}(x;z)\right].$ </p>
-    <p> $(\bar{x}_{-,k},\bar{y}_{-,k})$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
-1}^{2}q}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x%
-})}q}{\sqrt{K}}\Big{]}+\frac{L_{1}}{\sqrt{K}}+\frac{1}{T}\sum_{t=q+1}^{T}\left%
-[\frac{L_{1}q}{q\lfloor t/q\rfloor+1}\right]$ </p>
-    <p> $\max_{j\in[d]}\left|s^{t}_{x}(S,j)-s^{t}_{x}(S^{\prime},j)\right|\leq\frac{2L_%
-{0}\tau}{B}$ </p>
-    <p> $Q(S)=\frac{1}{n}\sum_{j\in[n]}Q(z_{i_{j}})$ </p>
-    <p> $x^{1},x^{2}\in\mathcal{X}$ </p>
-    <p> $a^{1},...,a^{T}$ </p>
-    <p> $y^{t+1}_{i}\propto y^{t}_{i}\exp\left(\tau g^{t}_{y,i}\right),\quad\forall i%
-\in[d_{y}]$ </p>
-    <p> $r=2/\sqrt{T}$ </p>
-    <p> $TK/q+qK+K$ </p>
-    <p> $s_{x}^{\prime t}$ </p>
-    <p> $\sqrt{\log(d)/n}+\log(d)/\sqrt{n\varepsilon}$ </p>
-    <p> $\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}x^{%
-t_{0}}_{i}\exp\left(G_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)%
-\exp\left(\hat{G}_{j}-G_{j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G%
-_{i}\right)\exp\left(\hat{G}_{i}-G_{i}\right)}$ </p>
-    <p> $s^{t}_{x}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{x,j}\right).$ </p>
-    <p> $\varepsilon=\frac{\varepsilon^{\prime}}{2\sqrt{2T\log(1/\delta^{\prime})}}$ </p>
-    <p> $\langle\nabla F(\bar{x}),e_{j}\rangle\leq\frac{F(\bar{x})-F(\bar{x}+re_{j})}{r%
-}+\frac{L_{1}r}{2},$ </p>
-    <p> $\hat{w}^{t}=\hat{w}^{t-1}$ </p>
-    <p> $\displaystyle\leq\sqrt{\mathbb{E}[\|Q(S)-\mathbb{E}_{z}[Q(z)]\|_{\infty}^{2}]}$ </p>
-    <p> $\tau=\sqrt{\frac{(\log(d_{x})+\log(d_{y}))}{(L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+%
-\log(d_{y}))L_{1}^{2})U}}$ </p>
-    <p> $q(\mathcal{D})=\mathbb{E}_{z\sim\mathcal{D}}[q(z)]$ </p>
-    <p> $g^{t}_{x}$ </p>
-    <p> $\mathbb{E}[F(\bar{a}^{T})-F(\bar{x}^{T})]\leq\frac{L_{1}}{2}\sum_{t=1}^{T}%
-\lambda_{t}^{2}\mathbb{E}\left[\left\|a^{t}-x^{t}\right\|^{2}\right]$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|}}\sum_{j\in[|{\cal Q%
-}|]}y_{j}(\mathbb{E}_{z}[q(z)]-q(\tilde{S}))\right]$ </p>
-    <p> $a,b:\mathbb{R}^{p}\mapsto\mathbb{R}$ </p>
-    <p> $(w{t+1},v^{t+1})$ </p>
-    <p> $\|x^{t}-a^{t}\|\leq D$ </p>
-    <p> $(x_{0},y_{0})=(\hat{x}^{1},\hat{y}^{1})$ </p>
-    <p> $U\geq 4$ </p>
-    <p> $\displaystyle\leq 5A^{2}+(6+\log(M))C^{2}$ </p>
-    <p> $\displaystyle\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\geq\frac{%
-A}{B}\right]$ </p>
-    <p> $\displaystyle\leq 4L_{1}+2L_{1}\sqrt{[\log(d_{x})+\log(d_{y})]\mathbb{E}[(%
-\mathcal{T}+1)]}$ </p>
-    <p> $\max\{L_{0},L_{1}\}$ </p>
-    <p> $\Delta(s_{x}^{t})$ </p>
-    <p> $\displaystyle\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)%
-\right\|^{2}_{\infty}$ </p>
-    <p> $2\Delta(s^{t}_{y})$ </p>
-    <p> $L_{2}^{G}$ </p>
-    <p> $(\mathbb{R}^{d},\|\cdot\|_{1})$ </p>
-    <p> $\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n\varepsilon}\right)^{1/3}$ </p>
-    <p> $q=\sqrt{T}/\log(d_{x})$ </p>
-    <p> $f(\cdot;z)$ </p>
-    <p> $\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq n\alpha/2$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\mathbb{E}[\|g^{t}_{x}\|_{\infty}^%
-{2}\mathbbm{1}_{(\mathcal{T}\geq t-1)}\mid\mathcal{F}_{t}]\right]$ </p>
-    <p> $\displaystyle\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},%
-\tilde{y})-\operatorname{Gap}(\bar{x},\bar{y}))\right]$ </p>
-    <p> $0<\varepsilon<8\log(1/\delta)$ </p>
-    <p> $\operatorname{Regret}_{T}^{x}(w)\leq\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}%
-\sum_{t\in[T]}\|g^{t}_{x}\|_{\infty}^{2}\text{ and }\operatorname{Regret}_{T}^%
-{y}(v)\leq\frac{\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|g^{t}_{y}\|_{%
-\infty}^{2}$ </p>
-    <p> $F_{\mathcal{D}}(x,y)$ </p>
-    <p> $\mathbb{E}[a^{t}|a^{t-1},..,a^{1}]=x^{t}$ </p>
-    <p> $\mathbb{E}[F_{\lambda}(\tilde{x})-F_{\lambda}(\bar{x})]\leq\frac{2L_{1}}{T}+%
-\frac{2L_{1}^{2}}{\lambda T}.$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[(\mathcal{T}+1)\mathbb{E}_{\tilde{x},\tilde{y}}%
-\left[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},\bar{%
-y})\mid\mathcal{T}\right]\right]$ </p>
-    <p> $f:{\cal X}\times{\cal Y}\mapsto\mathbb{R}$ </p>
-    <p> $(\bar{x}_{+},\bar{y}_{+})=\frac{1}{2^{N+1}}\sum_{i\in[2^{N+1}]}(\hat{x}^{i},%
-\hat{y}^{i})$ </p>
-    <p> $\log(d_{x})\simeq\log(d_{y})$ </p>
-    <p> $\displaystyle\lesssim\left(L_{0}^{2}+L_{2}^{2}+\log(d_{x})ML_{1}^{2}\right)%
-\mathbb{E}[\mathcal{T}+1],$ </p>
-    <p> $\Delta_{d}=\{x\in\mathbb{R}^{d}:\|x\|_{1}=1,x_{j}\geq 0\text{ for all }j\in[d]\}$ </p>
-    <p> $F(\bar{x}+re_{j})\leq F(\bar{x})+\langle\nabla F(\bar{x}),re_{j}\rangle+\frac{%
-L_{1}r^{2}}{2}\|e_{j}\|_{1}^{2},$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}%
-^{2}+(\log(d_{x})+\log(d_{y}))ML_{1}^{2}]\mathbb{E}[\mathcal{T}+1]+\frac{L_{2}%
-\mathbb{E}[\mathcal{T}+1]}{2^{M}}.$ </p>
-    <p> $\displaystyle=\frac{4L_{1}}{T}+\frac{4L_{1}^{2}}{\lambda T}+\lambda(\log(d_{x}%
-)+\log(d_{y}))+\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})].$ </p>
-    <p> $\displaystyle\quad\mathbb{E}[g_{x}]=\mathbb{E}\left[\sum_{k=0}^{M}(\nabla_{x}f%
-(\bar{x}_{+,k},\bar{y}_{+,k};B_{k})-\nabla_{x}f(\bar{x}_{-,k},\bar{y}_{-,k};B_%
-{k}))+\frac{2^{-k}}{C_{M}}\nabla_{x}f(x_{0},y_{0};B_{k})\right]$ </p>
-    <p> $U-\sqrt{U}\geq U/2$ </p>
-    <p> $g^{t}=\nabla f(\hat{w}^{t_{0}};B^{t})$ </p>
-    <p> $\displaystyle F(\bar{a}^{T})$ </p>
-    <p> $\displaystyle\mathbb{E}\big{[}\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v%
-)-\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\bar{x},v)\big{]}$ </p>
-    <p> $\displaystyle\|\mathbb{E}[(g_{x},g_{y})]-\nabla F_{\mathcal{D}}(x,y)\|_{\infty}$ </p>
-    <p> $\displaystyle=\mathbb{E}_{x^{t}}[\langle\mathbb{E}_{\hat{w}^{t}}[\nabla f(w^{t%
-};B^{t})-\nabla f(\hat{w}^{t};B^{t})\mid x^{t}],x^{t}-x\rangle]$ </p>
-    <p> $\displaystyle\lesssim\begin{cases}\frac{L_{1}}{\sqrt{K}}+q\tau\big{[}\frac{L_{%
-1}^{2}}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x}%
-)}}{\sqrt{K}}\big{]}&\text{without second order smoothnes}\\
-\frac{L_{2}}{K}+q\tau\big{[}\frac{L_{2}^{2}}{K^{2}}+\frac{L_{1}^{2}\log(d_{x})%
-}{K}\big{]}&\text{with second order smoothness}\end{cases}$ </p>
-    <p> $\displaystyle\mathbb{E}\|g_{x}\|_{\infty}^{2}$ </p>
-    <p> $\tau\leq 1/(8qL_{0})$ </p>
-    <p> $\mathbb{E}[F_{\cal D}(\hat{w}^{T})-F_{\cal D}(x)]\lesssim(L_{0}+L_{1})\left[%
-\sqrt{\frac{\log(d_{x})}{n}}+L_{0}\frac{\log(d_{x})^{7/10}\log(1/\delta)^{1/5}%
-}{(n\varepsilon)^{2/5}}\right]\\
-+L_{1}\log(\sqrt{n}\log(d_{x}))\left[\frac{1}{\log(d_{x})\sqrt{n}}+\frac{\log(%
-1/\delta)^{1/5}}{\log(d_{x})^{4/5}(n\varepsilon)^{2/5}}\right].$ </p>
-    <p> $\frac{L_{1}}{\sqrt{T}}$ </p>
-    <p> $\displaystyle\qquad+\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}%
-,(x^{t},y^{t})-(w^{t},v^{t})\rangle.$ </p>
-    <p> $\alpha_{t}=\Big{\langle}\nabla F\Big{(}\sum_{k=t}^{T}\lambda_{k}x^{k}+\sum_{k=%
-1}^{t-1}\lambda_{k}a^{k}\Big{)},\lambda_{t}(a^{t}-x^{t})\Big{\rangle}$ </p>
-    <p> $(\mathbf{0}_{d_{x}},y_{i}-y_{1})$ </p>
-    <p> $\tilde{x}^{1},\tilde{y}^{1}$ </p>
-    <p> $x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau g^{t}_{x,j}\right),\quad\forall j%
-\in[d_{x}]$ </p>
-    <p> $U\geq 1$ </p>
-    <p> $U\leq\min\left\{\frac{\varepsilon^{2}}{48\log(1/\delta)(9\tau\alpha L_{0})^{2}%
-},n\alpha/2,n/2\right\}$ </p>
-    <p> $a^{1},...,a^{T}\in\Delta_{d}$ </p>
-    <p> $\displaystyle=\langle\tilde{\Delta}^{t_{0}},x^{t}-\tilde{x}^{t}\rangle$ </p>
-    <p> $\mathbb{E}[\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)%
-\right\|^{2}_{\infty}]\leq\frac{43L_{1}^{2}}{\sqrt{12+\log(d)}T^{3/2}}+\frac{1%
-7(L_{0}^{2}+L_{1}^{2})\sqrt{(12+\log(d))}}{\sqrt{T}}.$ </p>
-    <p> $\displaystyle\leq L_{0}^{F}\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},%
-\mu_{2})\|_{1}$ </p>
-    <p> $x\in\Delta_{x},y\in\Delta_{y}$ </p>
-    <p> $w^{t}=\frac{1}{t}\sum_{i\in[t]}x^{i}$ </p>
-    <p> $T=\min\left\{n,\frac{(n\varepsilon)^{4/5}}{(\log(d_{x})\log(1/\delta))^{2/5}}\right\}$ </p>
-    <p> $(v^{t})_{t\in[T]}$ </p>
-    <p> $\tilde{y}^{t}\sim P_{y^{t}}$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau L_{0}^{2}+\frac{1}{T}\sum%
-_{t\in[T]}\left(\frac{L_{1}}{\sqrt{K}}+\mathbbm{1}_{(t>q)}\left[\frac{4L_{1}q}%
-{q\lfloor t/q\rfloor+1}+\frac{L_{1}^{2}q\tau}{\sqrt{\log(d_{x})}K^{3/2}}+\frac%
-{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}q\tau}{\sqrt{K}}\right]\right)$ </p>
-    <p> $(x_{-,0},y_{-,0})$ </p>
-    <p> $r^{2}=\sqrt{\frac{8(12+\log(d))}{T}}$ </p>
-    <p> $(\bar{x}_{+,k},\bar{y}_{+,k})$ </p>
-    <p> $\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\|g^{t}_{y}\|_{\infty}^{2}\right]%
-\lesssim L_{0}^{2}+L_{2}^{2}+\log(d_{y})ML_{1}^{2}\mathbb{E}[\mathcal{T}+1]$ </p>
-    <p> $(\mathbf{E},\|\cdot\|)$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2\log(|\mathcal{Z}|)}%
-{\tau_{x}T}+\frac{\log(|{\cal Q}|)}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+\frac{2}{T%
-}\sum_{t=1}^{T}\mathbb{E}[\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}%
-\|_{\infty}].$ </p>
-    <p> $\|a^{t}-x^{t}\|_{1}\leq D$ </p>
-    <p> $\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^{t};B^{t}),x^%
-{t}-x\rangle]\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second %
-order smoothnes}\\
-\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$ </p>
-    <p> $\displaystyle\mathbb{E}[F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)]$ </p>
-    <p> $|\mathbb{E}[-\nabla_{y,i}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y,i}\mid\mathcal{%
-F}_{t}]|\leq 2L_{2}/K$ </p>
-    <p> $\displaystyle+\frac{L_{1}\log(\sqrt{n\log(d_{x})})}{\sqrt{T\log(d_{x})}}+\frac%
-{L_{0}\sqrt{T\log(1/\delta)}\log^{3/2}(d_{x})}{n\varepsilon}.$ </p>
-    <p> $T,\tau$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10024.html DELETED Viewed

@@ -1,180 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $w\in\Delta_{x},v\in\Delta_{y}$ </p>
-    <p> $T,\tau,L$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{%
-t=1}^{\mathcal{T}+1}(L_{0}^{2}+\|g^{t}_{x}\|_{\infty}^{2}+\|g^{t}_{y}\|_{%
-\infty}^{2})$ </p>
-    <p> $F_{\mathcal{D}}(x)=\mathbb{E}_{z\sim\mathcal{D}}[f(x;z)]$ </p>
-    <p> $\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t}),x^{t}-x\rangle$ </p>
-    <p> $t_{0}=q\lfloor t/q\rfloor$ </p>
-    <p> $C=\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{t}^{2}}$ </p>
-    <p> $s^{t}_{y}=-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_{y}f(\bar{x}_{+}^{r%
-},\bar{y}_{+}^{r};B^{r})-\nabla_{y}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})%
-\Big{)}+\nabla_{y}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}.$ </p>
-    <p> $\Delta(s_{y}^{t})\leq 4.5\tau\alpha L_{0}$ </p>
-    <p> $\hat{w}^{t_{0}}$ </p>
-    <p> $\displaystyle\longmapsto$ </p>
-    <p> $(\hat{x}^{i})_{i\in[2^{N+1}]}\overset{\text{iid}}{\sim}P_{x}$ </p>
-    <p> $(\hat{y}^{i})_{i\in[2^{N+1}]}\overset{\text{iid}}{\sim}P_{y}$ </p>
-    <p> $\{a^{i}\}_{i\in[T]}$ </p>
-    <p> $\displaystyle=\frac{(A+B)^{2}}{2}+\sum_{j\in[M]}\int_{B/C}^{\infty}\mathbb{P}%
-\left[|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|\geq A+C\beta\right](A+C\beta)Cd\beta$ </p>
-    <p> $\frac{2TL_{0}\tau}{n}$ </p>
-    <p> $L_{1}=2$ </p>
-    <p> $\delta^{\prime},\delta^{\prime\prime}>0$ </p>
-    <p> $N((\varepsilon_{n})_{n\geq 1},(\delta_{n})_{n\geq 1})=\inf\left\{n:\varepsilon%
-<\sqrt{2\log\left(\frac{1}{\delta^{\prime}}\right)\sum_{m\leq n+1}\varepsilon_%
-{m}^{2}}+\frac{1}{2}\sum_{\sum_{m\leq n+1}}\varepsilon_{m}^{2}\text{ or }%
-\delta^{\prime\prime}<\sum_{m\leq n+1}\delta_{m}\right\}.$ </p>
-    <p> $\hat{G}_{j},G_{j}$ </p>
-    <p> $(x^{t},y^{t})_{t\in[T]}$ </p>
-    <p> $\mathbb{P}\left[F\left(\sum_{t=1}^{T}\lambda_{t}a^{t}\right)-F\left(\sum_{t=1}%
-^{T}\lambda_{t}x^{t}\right)\geq\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{%
-2}+\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{t}^{2}}\,\right]%
-\leq\exp(-\beta^{2}).$ </p>
-    <p> $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
-\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K\eqsim\frac{T}{%
-\log(d_{x})+\log(d_{y})}.$ </p>
-    <p> $\displaystyle=\int_{0}^{\infty}\mathbb{P}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F%
-_{j}(\bar{a}^{T})|\geq\beta\right]\beta d\beta$ </p>
-    <p> $\mathbb{E}\left[F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a})%
-\right]\leq 4L_{1}/T$ </p>
-    <p> $g^{t}_{x}=\nabla_{x}F_{\mathcal{D}}(\hat{x}^{t},\hat{y}^{t};B^{t})$ </p>
-    <p> $(\tilde{x},\tilde{y})=\frac{1}{T}\sum_{t=1}^{T}(\tilde{x}^{t},\tilde{y}^{t})$ </p>
-    <p> $\hat{y}^{2}$ </p>
-    <p> $\displaystyle=\sqrt{\mathbb{E}\left[\left\|\sum_{j\in[n]}\frac{Q(z_{i_{j}})-%
-\mathbb{E}_{z}[Q(z)]}{n}\right\|_{\infty}^{2}\right]}$ </p>
-    <p> $L_{2}=0$ </p>
-    <p> $\displaystyle\leq\sqrt{2e\log(|{\cal Q}|)\sum_{j\in[n]}\left\|\frac{Q(z_{i_{j}%
-})-\mathbb{E}_{z}[Q(z)]}{n}\right\|_{\infty}^{2}}$ </p>
-    <p> $\displaystyle\text{and}\quad\mathbb{E}[\|(g_{x},-g_{y})\|_{\infty}^{2}]$ </p>
-    <p> $\displaystyle=\mathbb{E}[\|Q(S)-\mathbb{E}_{z}[Q(z)]\|_{\infty}]$ </p>
-    <p> $N\sim\mbox{TG}(0.5,M)$ </p>
-    <p> ${}^{(\ast)}$ </p>
-    <p> $\hat{x}^{t+1,k},\hat{y}^{t+1,k}$ </p>
-    <p> $\|\nabla F(x)\|_{\infty}\leq L_{0}$ </p>
-    <p> $\operatorname{LSE}$ </p>
-    <p> $\alpha=\left(\frac{2\varepsilon^{2}}{48\cdot 81\log(1/\delta)(\tau L_{0})^{2}n%
-}\right)^{1/3}$ </p>
-    <p> $(x^{t}-\tilde{x}^{t})_{j}=\frac{x^{t_{0}}_{j}\exp\left(-\tau\sum_{k=t_{0}}^{t-%
-1}\nabla_{j}f(\hat{w}^{t_{0}};B^{k})\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}%
-\exp\left(-\tau\sum_{k=t_{0}}^{t-1}\nabla_{i}f(\hat{w}^{t_{0}};B^{k})\right)}-%
-\frac{x^{t_{0}}_{j}\exp\left(-\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(w^{t_{0}};B^%
-{k})\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(-\tau\sum_{k=t_{0}}^{t-1%
-}\nabla_{i}f(w^{t_{0}};B^{k})\right)}.$ </p>
-    <p> $\hat{x}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{x}^{t,k}$ </p>
-    <p> $(x^{t},y^{t})$ </p>
-    <p> $g^{t}_{y}$ </p>
-    <p> $K\eqsim\frac{T}{\log(d_{x})+\log(d_{y})}$ </p>
-    <p> $A=\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2}$ </p>
-    <p> $w^{1}\in\mathbb{R}^{d}$ </p>
-    <p> $\mathbb{E}\left[\max_{q\in{\cal Q}}|\mathbb{E}_{z}[q(z)]-q(\tilde{S})|\right]%
-\leq\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})]\leq 20\sqrt{\frac{\log(|%
-{\cal Q}|)}{n}}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{%
-\log(|{\cal Q}|)}}{\sqrt{n\varepsilon}}.$ </p>
-    <p> ${\cal X}\subseteq\mathbb{R}^{d_{x}}$ </p>
-    <p> $\operatorname{Gap}(\tilde{x},\tilde{y})\leq A/B$ </p>
-    <p> $\sqrt{\log(d)/n}+(\log(d)^{3/2}/[n\varepsilon])^{1/3}$ </p>
-    <p> $\mathcal{A}:\mathcal{Z}^{n}\mapsto\mathcal{X}$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})-\operatorname{Gap}(\bar{x},%
-\bar{y})]\leq\frac{4}{n}+\frac{8}{\lambda n}+\lambda\log(|{\cal Q}|).$ </p>
-    <p> $\sum_{t=q+1}^{T}\left[\frac{1}{q\lfloor t/q\rfloor+1}\right]\leq q\left(\frac{%
-1}{q}+\frac{1}{2q}+...+\frac{1}{(T/q)q}\right)\lesssim\log\Big{(}\frac{T}{q}%
-\Big{)}$ </p>
-    <p> $\mathbb{E}\|g_{x}\|_{\infty}^{2}\lesssim L_{0}^{2}+L_{2}^{2}+M\log(d_{x})L_{1}%
-^{2}$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
-1}^{2}q\log(d_{x})}{K}+\frac{L_{2}^{2}q}{K^{2}}\big{]}+\frac{L_{2}}{K}+\frac{1%
-}{T}\sum_{t=q+1}^{T}\left[\frac{L_{1}q}{q\lfloor t/q\rfloor+1}\right]$ </p>
-    <p> $(\hat{y}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{y^{t}}$ </p>
-    <p> $\displaystyle=\max\{\|\mathbb{E}[g_{x}]-\nabla_{x}F_{\mathcal{D}}(x,y)\|_{%
-\infty},\|\mathbb{E}[g_{y}]-\nabla_{y}F_{\mathcal{D}}(x,y)\|_{\infty}\}.$ </p>
-    <p> $\mathcal{T}+1\leq n/2$ </p>
-    <p> $\displaystyle=\big{\|}\Lambda^{T}\big{[}\nabla F(\Lambda(\lambda_{1},\mu_{1})+%
-(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\big{]}\big%
-{\|}_{\infty}$ </p>
-    <p> $\mathbb{E}\left[F\left(\sum_{t=1}^{T}\lambda_{t}a^{t}\right)-F\left(\sum_{t=1}%
-^{T}\lambda_{t}x^{t}\right)\right]\leq\frac{L_{1}}{2}\sum_{t=1}^{T}\lambda_{t}%
-^{2}\mathbb{E}\left[\left\|a^{t}-x^{t}\right\|^{2}\right].$ </p>
-    <p> $|{\cal Z}|$ </p>
-    <p> $\displaystyle\sqrt{2\log(1/\delta)\sum_{t=1}^{\mathcal{T}+1}(4\cdot 2^{N_{t}}+%
-2)(9\tau\alpha L_{0})^{2}}+\frac{1}{2}\sum_{t=1}^{\mathcal{T}+1}(4\cdot 2^{N_{%
-t}}+2)(9\tau\alpha L_{0})^{2}\leq\varepsilon.$ </p>
-    <p> $0<\varepsilon^{\prime}<1$ </p>
-    <p> $t>q$ </p>
-    <p> $\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}%
-x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[\exp\left(2\max_{j\in[d_{x}]}|\hat{G%
-}_{j}-G_{j}|\right)-1\right],$ </p>
-    <p> $\min_{x\in\Delta_{x}}F_{\mathcal{D}}(x)$ </p>
-    <p> $\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}%
-x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[1-\exp\left(-2\max_{j\in[d_{x}]}|%
-\hat{G}_{j}-G_{j}|\right)\right]$ </p>
-    <p> $\displaystyle(x^{t}-\tilde{x}^{t})_{j}$ </p>
-    <p> $\displaystyle\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second %
-order smoothnes}\\
-\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$ </p>
-    <p> $S=\{z_{i_{1}},...,z_{i_{n}}\}\overset{\text{iid}}{\sim}\mathcal{D}$ </p>
-    <p> $\max\{1,2^{N^{t}}/\alpha\}$ </p>
-    <p> $(\hat{w}^{T,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{w^{T}}$ </p>
-    <p> $\tau\eqsim\min\left\{\left(\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+L_{1}^{2}q\log(%
-d_{x})/K+L_{2}^{2}q/K^{2})T}}\right),\frac{1}{L_{0}q},\frac{n\varepsilon}{TL_{%
-0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\},$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|}}F_{\mathcal{D}}(%
-\tilde{x},y)\right]$ </p>
-    <p> $\ell=\log(d_{x})+\log(d_{y})$ </p>
-    <p> $\displaystyle=\mathbb{E}[2^{N_{1}}]\mathbb{E}\left[\mathcal{T}+1\right]\eqsim M%
-\mathbb{E}\left[\mathcal{T}+1\right].$ </p>
-    <p> $\ell_{1}/\ell_{q}$ </p>
-    <p> $\displaystyle\qquad+\frac{\alpha}{2^{N_{r}}}\Big{(}\nabla_{x}f(x_{0}^{r},y_{0}%
-^{r};z^{*})-\nabla_{x}f(x_{0}^{r},y_{0}^{r};z^{\prime*})\Big{)}\Big{|}$ </p>
-    <p> $\|\nabla\operatorname{LSE}(x)\|_{1}=1$ </p>
-    <p> $\displaystyle=\mathbb{E}[\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v)-%
-\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})]$ </p>
-    <p> $a^{1},...,a^{t-1}$ </p>
-    <p> $\displaystyle\leq 2\|\mathbb{E}_{\hat{w}^{t}}[\nabla f(w^{t};B^{t})-\nabla f(%
-\hat{w}^{t};B^{t})]\|_{\infty}$ </p>
-    <p> $\mathcal{T}\leq B-1$ </p>
-    <p> $(\bar{x},\bar{y})=\frac{1}{T}\sum_{t=1}^{T}(x^{t},y^{t})$ </p>
-    <p> $\displaystyle\quad+4C_{M}\sum_{k=0}^{M}\mathbb{E}2^{k}\Big{\|}\nabla_{x}f(x,y;%
-B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})\Big{\|}_{\infty}^{2}.$ </p>
-    <p> $y^{1}=(1/d_{y},...,1/d_{y}),t=1,N^{1}\sim\mbox{TG}(0.5,M)$ </p>
-    <p> $\tau\leq 1/(4L_{0}q)$ </p>
-    <p> $\log(n)\left(L_{0}+L_{2}+L_{1}\sqrt{(\log(d_{x})+\log(d_{y}))}\right)\sqrt{%
-\frac{\log(d_{x})+\log(d_{y})}{n}}\\
-+\log(n)\sqrt{L_{0}\sqrt{L_{0}^{2}+L_{2}^{2}+L_{1}^{2}\log(n)(\log(d_{x})+\log%
-(d_{y}))}}\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n%
-\varepsilon}\right)^{1/2}.$ </p>
-    <p> $\left|\mathbb{E}\left[F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-F\left(%
-\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right|\leq\frac{2L_{1}}{T}.$ </p>
-    <p> $\displaystyle=\tau\|G_{j}(S)-G_{j}(S^{\prime})\|_{\infty}.$ </p>
-    <p> $\color[rgb]{1,0,0}\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}^{2/3}}{(n%
-\varepsilon)^{2/3}}$ </p>
-    <p> $\displaystyle+\frac{BM}{U-\sqrt{U}}.$ </p>
-    <p> $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}F(x,y)$ </p>
-    <p> $\bar{x}=\frac{1}{T}\sum_{t\in[T]}x^{t},\bar{a}=\frac{1}{T}\sum_{t\in[T]}a^{t}$ </p>
-    <p> $L_{1}^{F}$ </p>
-    <p> $\mathcal{Q}=\{q:\mathcal{Z}\to[-1,1]\}$ </p>
-    <p> $\displaystyle=\mathbb{E}[\langle\nabla f(w^{t};B^{t})-\nabla f(\hat{w}^{t};B^{%
-t}),x^{t}-x\rangle]$ </p>
-    <p> $L_{1}^{G}$ </p>
-    <p> $\max_{q\in{\cal Q}}|q(\tilde{S})-q(\mathcal{D})|$ </p>
-    <p> $g^{t}=(g^{t}_{x},g^{t}_{y})$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10025.html DELETED Viewed

@@ -1,204 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $0\leq\phi(y)\leq\log(d_{y})$ </p>
-    <p> $y_{i}-y_{1}$ </p>
-    <p> $f(x,y;z)=\sum_{j\in[|\mathcal{Q}|]}y_{j}(q_{j}(z)-\langle q_{j},x\rangle)$ </p>
-    <p> $S=\{z^{1},...,z^{n}\}$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\|g^{t}_{x}\|_{\infty}^{2}\mathbbm%
-{1}_{(\mathcal{T}+1\geq t)}\right]$ </p>
-    <p> $\displaystyle\leq 4\tau\sum_{k=t_{0}}^{t-1}\|\nabla f(w^{t_{0}};B^{t})-\nabla f%
-(\hat{w}^{t_{0}};B^{t})\|_{\infty}\|\nabla f(\hat{w}^{t_{0}};B^{k})-\nabla f(w%
-^{t_{0}};B^{k})\|_{\infty}$ </p>
-    <p> $\displaystyle\operatorname{Gap}(\bar{x},\bar{y})$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq 10\sqrt{\frac{\log(|{\cal Q%
-}|)}{n}}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{\log(|{%
-\cal Q}|)}}{\sqrt{n\varepsilon}}.$ </p>
-    <p> $\displaystyle\qquad-2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r%
-};B^{\prime r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{\prime r})\Big{%
-)}-\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{\prime r})\Big{|}$ </p>
-    <p> $\displaystyle\leq\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})\|%
-_{\infty}\|x^{t}-\tilde{x}^{t}\|_{1}$ </p>
-    <p> $\mathbb{E}_{\mathcal{A},S}\Big{[}\max_{q\in\mathcal{Q}}|\mathbb{E}_{z\sim%
-\mathcal{D}}[q(z)]-q(\tilde{S})|\Big{]}\leq 20\sqrt{\frac{\log(|{\cal Q}|)}{n}%
-}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{\log(|{\cal Q}%
-|)}}{\sqrt{n\varepsilon}}.$ </p>
-    <p> $y^{t+1}_{j}\propto y^{t}_{j}\exp\left(\tau_{y}g^{t}_{y,j}\right),\quad\forall j%
-\in[|\mathcal{Q}|]$ </p>
-    <p> $\displaystyle\Lambda:\Delta_{K_{x}}\times\Delta_{K_{y}}$ </p>
-    <p> $\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(w^{T})]\lesssim\frac{L%
-_{1}}{K}$ </p>
-    <p> $\displaystyle\|x^{t}-\tilde{x}^{t}\|_{1}\leq\psi$ </p>
-    <p> $\displaystyle\leq\frac{\log(d_{x})}{\tau T}+\frac{\tau L_{0}^{2}}{2}+\frac{1}{%
-T}\sum_{t\in[T]}\mathbb{E}\left[\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{%
-t}-x\rangle\right]$ </p>
-    <p> $\mathbb{E}\left[\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{2}}%
-{r^{2}}\right]\leq\frac{20L_{1}^{2}}{T^{2}r^{2}}+\frac{8L_{0}^{2}(6+\log(d))}{%
-Tr^{2}}$ </p>
-    <p> $\tilde{\Delta}^{t_{0}}$ </p>
-    <p> $\displaystyle=\mathbb{E}\big{[}\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},%
-v)-F_{\lambda}(\tilde{x})\big{]}+\mathbb{E}[F_{\lambda}(\tilde{x})-F_{\lambda}%
-(\bar{x})]+\mathbb{E}\big{[}F_{\lambda}(\bar{x})-\max_{v\in\Delta_{y}}F_{%
-\mathcal{D}}(\bar{x},v)\big{]}$ </p>
-    <p> $\psi\leq 8\tau qL_{0}$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
-1}^{2}q}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x%
-})}q}{\sqrt{K}}\Big{]}+\frac{L_{1}}{\sqrt{K}}+\frac{L_{1}q\log(T/q)}{T}.$ </p>
-    <p> $\mathbb{E}\left[\operatorname{Gap}(\bar{x},\bar{y})\right]$ </p>
-    <p> $\displaystyle\leq\sum_{t=1}^{T}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g%
-^{t}_{x},x^{t}-w\rangle+2\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|%
-_{\infty}$ </p>
-    <p> $\displaystyle\|x^{t}-\tilde{x}^{t}\|_{1}=\sum_{j\in[d_{x}]}|(x^{t}-\tilde{x}^{%
-t})_{j}|\leq\sum_{j\in[d_{x}]}\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_%
-{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}\psi=\psi.$ </p>
-    <p> $F(\bar{a}^{T})=F(\bar{x}^{T}+\bar{a}^{T}-\bar{x}^{T})$ </p>
-    <p> $\mathcal{X}=\Delta_{x}$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\nabla_{x}F_{\mathcal{D}}(\bar{x}_{+,M},\bar{y}_%
-{+,M})\right],$ </p>
-    <p> $P_{w^{t-1}}$ </p>
-    <p> $\mathbb{E}\left[\frac{|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}\right]\leq\frac{20L_%
-{1}^{2}}{T^{2}r^{2}}+\frac{48L_{0}^{2}}{Tr^{2}}.$ </p>
-    <p> $T=\min\Big{\{}n,\frac{n\varepsilon}{\log(d_{x})\sqrt{\log(1/\delta)}}\Big{\}},%
-K=\sqrt{T\log(d_{x})},q=\sqrt{T/\log(d_{x})},$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},%
-\bar{y})]\leq\frac{4L_{1}}{T}+\frac{2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}}{%
-\sqrt{T}}.$ </p>
-    <p> $f(x,y;z)=\sum_{i\in[d_{y}]}y_{i}f_{i}(x;z)$ </p>
-    <p> $\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)\exp\left(\hat{G}_{j}-G%
-_{j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)\exp\left(%
-\hat{G}_{i}-G_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i%
-\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}$ </p>
-    <p> $\displaystyle\leq\frac{F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a%
-})}{r}+L_{1}r$ </p>
-    <p> $\max_{v\in\Delta_{y}}F_{\mathcal{D}}(x,v)\leq F_{\lambda}(x)\leq\lambda\log(d_%
-{y})+\max_{v\in\Delta_{y}}F_{\mathcal{D}}(x,v)$ </p>
-    <p> $2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|\leq 1$ </p>
-    <p> $i\in[d_{y}]$ </p>
-    <p> $w^{t+1}=\operatorname{argmin}_{x\in\mathcal{X}}D_{\psi}(w^{t},x)+\langle\xi^{t%
-},x\rangle$ </p>
-    <p> $\displaystyle\quad+2\tau q\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}}%
-;B^{t})\|_{\infty}^{2}+2\tau\sum_{k=t_{0}}^{t-1}\mathbb{E}\left[\|\nabla f(%
-\hat{w}^{t_{0}};B^{k})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\right]$ </p>
-    <p> $\mathbb{E}[\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)%
-\right\|^{2}_{\infty}]\leq\frac{120L_{1}^{2}}{T^{2}r^{2}}+\frac{24L_{0}^{2}(12%
-+\log(d))}{Tr^{2}}+3L_{1}^{2}r^{2}.$ </p>
-    <p> $\mathbb{E}\|g_{y}\|_{\infty}^{2}\lesssim L_{0}^{2}+L_{2}^{2}+M\log(d_{y})L_{1}%
-^{2}$ </p>
-    <p> $\operatorname{LSE}((x_{1},...,x_{k}))=\log(\exp(x_{1})+...+\exp(x_{k}))$ </p>
-    <p> $\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|^{2}\right]%
-\leq\frac{(A+B)^{2}}{2}+2MC(C+A)\exp\left(-(B/C)^{2}\right).$ </p>
-    <p> $|\alpha_{i}|\leq L_{0}D\lambda_{i}$ </p>
-    <p> $\displaystyle\lesssim(L_{0}+L_{1})\sqrt{\frac{\log(d_{x})}{T}}+\frac{L_{2}}{T^%
-{3/4}\log(d_{x})^{1/4}}+\frac{L_{1}+L_{2}}{\sqrt{T\log(d_{x})}}$ </p>
-    <p> $z^{\prime*}$ </p>
-    <p> $x,y,N,B$ </p>
-    <p> $p\leq C_{M}\leq 1$ </p>
-    <p> $v^{1}=(1/d_{y},...,1/d_{y})\in\mathbb{R}^{d_{y}},v^{t+1}:=\operatorname{argmin%
-}_{y\in\Delta_{y}}\left(\tau\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g_{%
-y}^{t},y\rangle+\sum_{i\in[d_{y}]}v^{t}_{i}\log(v^{t}_{i}/y_{i})\right).$ </p>
-    <p> $\mathbb{E}\left[\operatorname{Gap}(\tilde{x},\bar{y})\right]$ </p>
-    <p> $\displaystyle\leq F(\bar{x}^{T}+\bar{a}^{T-1}-\bar{x}^{T-1})+\langle\nabla F(%
-\bar{x}^{T}+\bar{a}^{T-1}-\bar{x}^{T-1}),\lambda_{T}(a^{T}-x^{T})\rangle+\frac%
-{L_{1}\lambda_{T}^{2}\left\|a^{T}-x^{T}\right\|^{2}}{2}$ </p>
-    <p> $F_{\lambda}(x):=\max_{y\in\Delta_{y}}[F_{\mathcal{D}}(x,y)+\lambda\phi(y)]$ </p>
-    <p> $\hat{y}^{t}\sim P_{y^{t}}$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\mathbb{E}[\|g^{t}_{x}\|_{\infty}^%
-{2}]\mathbbm{1}_{(\mathcal{T}\geq t-1)}\right]$ </p>
-    <p> $Q(z)=(q_{1}(z),...,q_{|{\cal Q}|}(z))$ </p>
-    <p> $\mathbb{P}[\operatorname{Gap}(\tilde{x},\tilde{y})\geq A/B]\leq 0.01$ </p>
-    <p> $\sum_{t\in[T]}\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y},y^{t}-v%
-\rangle\\
-\leq\sum_{t\in[T]}\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y},y^{%
-t}-v^{t}\rangle+\frac{\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|-\nabla%
-_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty}^{2}.$ </p>
-    <p> $\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq U$ </p>
-    <p> $x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau_{x}g^{t}_{x,j}\right),\quad\forall
-j%
-\in[|\mathcal{Z}|]$ </p>
-    <p> $\tilde{y}^{i}\sim P_{y^{i}}$ </p>
-    <p> $\displaystyle\leq 4\tau\max_{j\in[d_{x}]}\sum_{k=t_{0}}^{t-1}\left|\nabla_{j}f%
-(\hat{w}^{t_{0}};B^{k})-\nabla_{j}f(w^{t_{0}};B^{k})\right|$ </p>
-    <p> $\displaystyle\leq\langle\nabla\operatorname{LSE}(-\tau G_{j}(S)),\tau G_{j}(S^%
-{\prime})-\tau G_{j}(S)\rangle$ </p>
-    <p> $\hat{x}^{1},\tilde{x}^{1},\hat{y}^{1},\tilde{y}^{1},...,\hat{x}^{t},\tilde{x}^%
-{t},\hat{y}^{t},\tilde{y}^{t}$ </p>
-    <p> $\displaystyle\leq 4L_{0}^{2}+64L_{2}^{2}+3ML_{1}^{2}[\log(2d_{x})+\log(2d_{y})%
-]+8ML_{2},$ </p>
-    <p> $\displaystyle\qquad+\sum_{t=1}^{\mathcal{T}+1}\langle\Phi_{\mathcal{D}}(x^{t},%
-y^{t})-(g_{x}^{t},g_{y}^{t}),(x^{t},y^{t})-(w^{t},v^{t})\rangle$ </p>
-    <p> $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
-\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K\eqsim\sqrt{\frac{%
-T}{\log(d_{x})+\log(d_{y})}}.$ </p>
-    <p> $\max_{j\in[d_{x}]}\tau\|G_{j}(S)-G_{j}(S^{\prime})\|_{\infty}=\tau\max_{j\in[d%
-_{x}],i\in[t]}\left|\sum_{k\in[i-1]}(g^{k}_{j}-g^{\prime k}_{j})\right|\leq%
-\frac{2L_{0}}{B}.$ </p>
-    <p> $\displaystyle\leq 3\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{%
-2}+|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}+L_{1}^{2}r^{2}$ </p>
-    <p> $w^{t}_{j}=\frac{1}{t}\sum_{i\in[t]}x^{i}_{j}\propto\sum_{i\in[t]}\exp\left(-%
-\tau\sum_{k\in[i-1]}g^{k}_{j}\right)=\exp\left(\log\left(\sum_{i\in[t]}\exp%
-\left(-\tau\sum_{k\in[i]}g^{k}_{j}\right)\right)\right).$ </p>
-    <p> $\displaystyle\leq\operatorname{Regret}_{T}^{x}(w)+\operatorname{Regret}_{T}^{y%
-}(v)+\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-%
-(w,v)\rangle.$ </p>
-    <p> $t\in[T],k\in[K]$ </p>
-    <p> $\displaystyle=4L_{1}+2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}\mathbb{E}\left[\sqrt%
-{\mathcal{T}+1}\right]$ </p>
-    <p> $\displaystyle\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]$ </p>
-    <p> $\hat{y}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{y}^{t,k}$ </p>
-    <p> $(\tilde{x},\tilde{y})=\frac{1}{t}\sum_{i=1}^{t}(\tilde{x}^{i},\tilde{y}^{i})$ </p>
-    <p> $\displaystyle\leq F\left(\bar{x}^{T}\right)+\sum_{t\in[T]}\underbrace{\Big{%
-\langle}\nabla F\Big{(}\bar{x}^{T}+\bar{a}^{t-1}-\bar{x}^{t-1}\Big{)},\lambda_%
-{t}(a^{t}-x^{t})\Big{\rangle}}_{\alpha_{t}}+\frac{L_{1}}{2}\sum_{t\in[T]}%
-\lambda_{t}^{2}\|a^{t}-x^{t}\|^{2}$ </p>
-    <p> $2\Delta(s^{t}_{x})$ </p>
-    <p> $(x^{t}-\tilde{x}^{t})_{j}$ </p>
-    <p> $T\eqsim\min\left\{n,\left[\frac{n\varepsilon}{(\log(d_{x})+\log(d_{y}))^{1/4}%
-\sqrt{\log(1/\delta)}}\right]^{4/5}\right\},$ </p>
-    <p> $\mathcal{T}+1\leq\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq U$ </p>
-    <p> $S=\{z^{1},...,z^{n}\}\overset{\text{iid}}{\sim}\mathcal{D}$ </p>
-    <p> $\|w^{t}-w^{t-1}\|_{1}=\|x^{t}-w^{t-1}\|_{1}/t\leq 2/t$ </p>
-    <p> $\bar{x}=\frac{1}{T}\sum_{t\in[T]}x^{t}$ </p>
-    <p> $\{\alpha_{i}\}_{i\in[T]}$ </p>
-    <p> $\displaystyle=L_{0}^{F}\left\|\left(\sum_{i=1}^{K_{x}}(\lambda_{1,i}-\lambda_{%
-2,i})(x_{i}-x_{1}),\sum_{j=1}^{K_{y}}(\mu_{1,j}-\mu_{2,j})(y_{j}-y_{1})\right)%
-\right\|_{1}$ </p>
-    <p> $\alpha=\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2}+\alpha^{\prime}$ </p>
-    <p> $\frac{4TL_{0}\tau}{n}$ </p>
-    <p> $\mathbb{E}[a^{t}|a^{t-1},\ldots,a^{1}]=x^{t}$ </p>
-    <p> $\displaystyle=\mathbb{E}[2^{N_{1}}]\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}%
-\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$ </p>
-    <p> $\hat{w}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{w}^{t,k}$ </p>
-    <p> $\|\mathbb{E}[g_{x}]-\nabla_{x}F_{\mathcal{D}}(x,y)\|_{\infty}=\max_{j\in[d]}|%
-\mathbb{E}[g_{x,j}]-\nabla_{x,j}F_{\mathcal{D}}(x,y)|\leq\frac{2L_{2}}{2^{M}}$ </p>
-    <p> $\mathcal{A}_{1:n-1}$ </p>
-    <p> $\displaystyle\mathbb{E}[\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_%
-{\infty}]$ </p>
-    <p> $\tau\eqsim\min\left\{\left(\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+L_{1}^{2}q\log(%
-d_{x})/K+L_{2}^{2}q/K^{2})T}}\right),\frac{1}{L_{0}q},\frac{n\varepsilon}{TL_{%
-0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}.$ </p>
-    <p> $\mathcal{X}\subseteq\mathbb{R}^{d_{x}}$ </p>
-    <p> $g_{x},g_{y}$ </p>
-    <p> $\displaystyle\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]$ </p>
-    <p> $T=\min\left\{n,\frac{(n\varepsilon)^{4/5}}{(\log(d_{x})\log(1/\delta))^{2/5}}%
-\right\},q=\sqrt{T}/\log(d_{x}),K=T/\log(d_{x})$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\left[%
-\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n}}+\bigg{(}\frac{(\log(d_{x})+%
-\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{1/3}\bigg{]}.$ </p>
-    <p> $\displaystyle|s_{x}^{t}-s_{x}^{\prime t}|$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10026.html DELETED Viewed

@@ -1,174 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\displaystyle=\langle\mathbb{E}_{\hat{w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-%
-\nabla f(\hat{w}^{t_{0}};B^{t})],\tilde{x}^{t}-x\rangle$ </p>
-    <p> $\mathbb{E}[\hat{x}]=x$ </p>
-    <p> $x^{1},...,x^{T}$ </p>
-    <p> $\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})\|_{1}\leq D\|(%
-\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})\|_{1}$ </p>
-    <p> $\displaystyle\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(x)]$ </p>
-    <p> $\log(|{\cal Q}|)\leq C\log(|\mathcal{Z}|)$ </p>
-    <p> $w^{t}=\sum_{i\in[t]}\frac{\beta_{i}x^{i}}{\sum_{j\in[t]}\beta_{j}}$ </p>
-    <p> $\textstyle x^{t+1}_{j}\propto\exp\left\{-\tau\left(\sum_{i=1}^{t}g^{i}_{x,j}%
-\right)\right\},$ </p>
-    <p> $|\mathcal{Q}|\leq|\mathcal{Z}|^{C}$ </p>
-    <p> $8\tau qL_{0}\leq 2$ </p>
-    <p> $\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}^{7/10}}{(n\varepsilon)^{2/5}}$ </p>
-    <p> $q(S)=\frac{1}{|S|}\sum_{z\in S}q(z)$ </p>
-    <p> $\Phi(x,y)=\big{(}\nabla_{x}f(x,y),-\nabla_{y}f(x,y)\big{)}$ </p>
-    <p> $\varepsilon\leq 8\log(1/\delta)$ </p>
-    <p> $g_{x}=C_{M}2^{N}(\nabla_{x}f(\bar{x}_{+},\bar{y}_{+};B)-\nabla_{x}f(\bar{x}_{-%
-},\bar{y}_{-};B))+\nabla_{x}f(x_{0},y_{0};B)$ </p>
-    <p> $\displaystyle\operatorname{LSE}(-\tau G_{j}(S))-\operatorname{LSE}(-\tau G_{j}%
-(S^{\prime}))$ </p>
-    <p> $\mathbb{E}\left[\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-x\rangle\right]$ </p>
-    <p> $\displaystyle\leq L_{1}^{F}D^{2}\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})%
-\|_{1}.$ </p>
-    <p> $\frac{4L_{0}\tau}{B}\leq\frac{\varepsilon}{2\sqrt{2T(K+1)\log(1/\delta)}}$ </p>
-    <p> $B^{\prime r}$ </p>
-    <p> $\frac{L_{1}^{2}}{T^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d)}}{\sqrt{T}}$ </p>
-    <p> $n,d_{x},d_{y}$ </p>
-    <p> $F_{j}=\nabla_{j}F$ </p>
-    <p> $\tau_{x},\tau_{y}$ </p>
-    <p> $L_{0}^{G}\leq L_{0}^{F}D$ </p>
-    <p> $x^{t}\in\Delta_{d}$ </p>
-    <p> $\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{%
-\infty}\leq\frac{4L_{1}}{\sqrt{K}}$ </p>
-    <p> $\mathbb{P}[\mathcal{A}(S)\in\mathcal{E}]\leq e^{\varepsilon}\mathbb{P}[%
-\mathcal{A}(S^{\prime})\in\mathcal{E}]+\delta$ </p>
-    <p> $(\varepsilon_{n},\delta_{n})$ </p>
-    <p> $\sum_{t=1}^{\mathcal{T}}2^{N_{t}}\leq U-2^{M}$ </p>
-    <p> $\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq\min\{n,n\alpha/2\}$ </p>
-    <p> $\color[rgb]{1,0,0}\frac{\ell}{\sqrt{n}}+\frac{\ell}{\sqrt{n\varepsilon}}{}^{(%
-\ast)}$ </p>
-    <p> $\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\lesssim\sqrt{\frac{[(%
-\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2%
-}]M^{2}}{U}}\,\right]\geq 0.99.$ </p>
-    <p> $x\in\mathcal{X},$ </p>
-    <p> $\displaystyle\leq 2\|\mathbb{E}_{\hat{w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-%
-\nabla f(\hat{w}^{t_{0}};B^{t})]\|_{\infty}$ </p>
-    <p> $\displaystyle(\tilde{x}^{t}-x^{t})_{j}$ </p>
-    <p> $\lambda\phi(y)$ </p>
-    <p> $\nabla f(w^{t};B^{t})$ </p>
-    <p> $\displaystyle\langle\nabla F(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle$ </p>
-    <p> $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}%
-^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2}]U+\frac{L_{2}U}{2^{M}}.$ </p>
-    <p> $j\in[d_{x}]$ </p>
-    <p> $\log(d)/\sqrt{n}+\log(d)/[n\varepsilon]^{1/2}$ </p>
-    <p> $\displaystyle\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\|g^{t}_{x}\|_{\infty}^%
-{2}\right]$ </p>
-    <p> $\displaystyle\leq\sum_{t=1}^{T}\langle(\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-%
-g^{t}_{x},x^{t}-w^{t}\rangle)+\frac{\log(|\mathcal{Z}|)}{\tau_{x}}+\sum_{t=1}^%
-{T}\tau_{x}\|\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}+2%
-\|\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty},$ </p>
-    <p> $B=C\sqrt{\log(M)}$ </p>
-    <p> $\displaystyle\leq 2\sum_{k=0}^{M}C_{M}2^{k}\mathbb{E}\Big{\|}\nabla_{x}f(\bar{%
-x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})\Big{\|}_{%
-\infty}^{2}+\frac{2^{-k}}{C_{M}}\mathbb{E}\|\nabla_{x}f(x_{0},y_{0};B_{k})\|_{%
-\infty}^{2}$ </p>
-    <p> $\displaystyle=\max_{i\in[K_{x}+K_{y}]}|\langle\Lambda^{T}_{i},\nabla F(\Lambda%
-(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{2})+(x_%
-{1},y_{1}))\rangle|$ </p>
-    <p> $\displaystyle\sqrt{2\log(1/\delta)\sum_{t=1}^{\mathcal{T}+1}(2^{N_{t}+1}+1)(4%
-\Delta(s_{x}^{t})^{2}+4\Delta(s_{y}^{t})^{2})}+\frac{1}{2}\sum_{t=1}^{\mathcal%
-{T}+1}(2^{N_{t}+1}+1)(4\Delta(s_{x}^{t})^{2}+4\Delta(s_{y}^{t})^{2})\leq\varepsilon.$ </p>
-    <p> $\min_{x\in\Delta_{|\mathcal{Z}|}}F_{\mathcal{D}}(x,\bar{y})\leq 0$ </p>
-    <p> $U\leq\min\{n,n\alpha/2\}$ </p>
-    <p> $n,1/\delta$ </p>
-    <p> $g^{t}_{y}=\nabla_{y}f(x^{t},\hat{y}^{t};S)$ </p>
-    <p> $\hat{w}^{T}=\frac{1}{K}\sum_{k\in[K]}\hat{w}^{T,k}$ </p>
-    <p> $\tilde{y}^{t+1}$ </p>
-    <p> $\min_{\lambda\in\Delta_{K_{x}}}\max_{\mu\in\Delta_{K_{y}}}[G(\lambda,\mu):=F(%
-\Lambda(\lambda,\mu)+(x_{1},y_{1}))].$ </p>
-    <p> $\displaystyle(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})$ </p>
-    <p> $\Delta(s^{t}_{x})$ </p>
-    <p> $\displaystyle\leq\max_{k\in[K_{x}+K_{y}]}\|\Lambda^{T}_{k}\|_{1}\|\Lambda^{T}_%
-{j}\|_{1}L_{2}^{F}\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})%
-\|_{1}$ </p>
-    <p> $w^{t},v^{t}$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\left[%
-\sqrt{\frac{\log(d_{x})+\log(d_{y})}{n}}+\left(\frac{\sqrt{\log(1/\delta)}(%
-\log(d_{x})+\log(d_{y}))^{3/2}}{n\varepsilon}\right)^{1/3}\right].$ </p>
-    <p> $\mathcal{E}\subseteq\mathcal{X}$ </p>
-    <p> $\tilde{x}=\frac{1}{n}\sum_{k=1}^{n}\tilde{x}^{k}$ </p>
-    <p> $\tau\leq B\varepsilon/[8L_{0}\sqrt{2T(K+1)\log(1/\delta)}].$ </p>
-    <p> $\displaystyle=\max_{k\in[K_{x}+K_{y}]}\left|\sum_{i\in[K_{x}+K_{y}]}\Lambda^{T%
-}_{j,i}\langle\Lambda^{T}_{k},\nabla\nabla_{i}F(\Lambda(\lambda_{1},\mu_{1})+(%
-x_{1},y_{1}))-\nabla\nabla_{i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\right|$ </p>
-    <p> $\displaystyle\|\nabla G(\lambda_{1},\mu_{1})-\nabla G(\lambda_{2},\mu_{2})\|_{\infty}$ </p>
-    <p> $L_{2}\lesssim(L_{0}+L_{1})\left\{\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n%
-}}+\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n%
-\varepsilon}\right)^{1/2}\right\},$ </p>
-    <p> $\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(x)]\lesssim\frac{\log(%
-d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{1}^{2}q}{\sqrt{\log(d_{x})}K^{3/%
-2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}q}{\sqrt{K}}\Big{]}+\frac{L_{%
-1}}{\sqrt{K}}+\frac{L_{1}q\log(T/q)}{T}.$ </p>
-    <p> $\big{(}L_{1}+\frac{L_{1}^{2}}{\lambda}\big{)}$ </p>
-    <p> $U=\min\left\{\frac{(n\varepsilon)^{2/3}}{(4\cdot 48\cdot 81\log(1/\delta)^{1/3%
-}(\tau L_{0})^{2/3}},\frac{n}{2}\right\}$ </p>
-    <p> $F_{j}(\cdot)=F(\cdot+re_{j})$ </p>
-    <p> $\mathbb{P}(j=e_{i})\propto\exp\left(s(S,i)\right)$ </p>
-    <p> $\tilde{x}^{i}\sim P_{x^{i}}$ </p>
-    <p> $P_{x^{t}}$ </p>
-    <p> $\min_{x\in\Delta_{|\mathcal{Z}|}}\max_{y\in\Delta_{|\mathcal{Q}|}}\mathbb{E}_{%
-z\sim\mathcal{D}}\Big{[}\sum_{j\in[|\mathcal{Q}|]}y_{j}(q_{j}(z)-\langle q_{j}%
-,x\rangle)\Big{]}.$ </p>
-    <p> $\displaystyle T[F_{\mathcal{D}}(\bar{x},v)-F_{\mathcal{D}}(w,\bar{y})]$ </p>
-    <p> $s(S,j)=\operatorname{LSE}(\tau G_{j}(S))$ </p>
-    <p> $\mathbb{E}\left\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F%
-\left(\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\right\|^{2}_{\infty}\leq\frac{20L_%
-{2}^{2}}{T^{2}}+\frac{8L_{1}^{2}(6+\log(d))}{T}.$ </p>
-    <p> $\displaystyle\leq\int_{0}^{A+B}\beta d\beta+\sum_{j\in[M]}\int_{A+B}^{\infty}%
-\mathbb{P}\left[|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|\geq\beta\right]\beta d\beta$ </p>
-    <p> $L_{0}^{G}$ </p>
-    <p> $\mathcal{T}+1\geq t$ </p>
-    <p> $\displaystyle\mathbb{E}[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})]$ </p>
-    <p> $s^{t}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{y,j}\right)$ </p>
-    <p> $\displaystyle T[F_{\mathcal{D}}(\bar{x},v)-F_{\mathcal{D}}(w,\bar{y})]\leq%
-\operatorname{Regret}_{T}^{x}(w)+\operatorname{Regret}_{T}^{y}(v)+\sum_{t=1}^{%
-T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w,v)\rangle$ </p>
-    <p> $\displaystyle\leq\max_{i\in[K_{x}+K_{y}]}\|\Lambda^{T}_{i}\|_{1}L_{1}^{F}\|%
-\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})\|_{1}$ </p>
-    <p> $\langle\tilde{\Delta}^{t_{0}},x^{t}-x\rangle=\underbrace{\langle\tilde{\Delta}%
-^{t_{0}},\tilde{x}^{t}-x\rangle}_{=:A_{1}}+\underbrace{\langle\tilde{\Delta}^{%
-t_{0}},x^{t}-\tilde{x}^{t}\rangle}_{=:A_{2}},$ </p>
-    <p> $\tilde{x}^{1},...,\tilde{x}^{n}\overset{\text{iid}}{\sim}P_{\bar{x}}$ </p>
-    <p> $\mathbb{E}\left[\mathcal{T}+1\right]\lesssim U/\mathbb{E}[2^{N_{1}}]\eqsim U/M$ </p>
-    <p> $\langle\tilde{\Delta}^{t_{0}},x^{t}-\tilde{x}^{t}\rangle\leq\|\tilde{\Delta}^{%
-t_{0}}\|_{\infty}\|x^{t}-\tilde{x}^{t}\|_{1}$ </p>
-    <p> $\displaystyle\mathbb{E}_{\hat{w}^{t_{0}}}[\langle\tilde{\Delta}^{t_{0}},\tilde%
-{x}^{t}-x\rangle]$ </p>
-    <p> $\max_{i\in[d_{y}]}\{F_{i}(x):=\mathbb{E}_{z\sim\mathcal{D}}[f_{i}(x;z)]\}$ </p>
-    <p> $\sum_{i\in[t-1]}2^{N^{i}}\leq U-2^{M}$ </p>
-    <p> $\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},\tilde{y})\geq A%
-\right]\leq\frac{\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},%
-\tilde{y})-\operatorname{Gap}(\bar{x},\bar{y}))\right]+\mathbb{E}\left[(%
-\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})\right]}{A}.$ </p>
-    <p> ${\cal Y}\subseteq\mathbb{R}^{d_{y}}$ </p>
-    <p> $a\eqsim b$ </p>
-    <p> $\mathbb{P}[N=k]=p^{k}\mathbbm{1}_{(k\in\{0,1,...,M\})}/C_{M}$ </p>
-    <p> $(w^{t})_{t\in[T]}$ </p>
-    <p> $\displaystyle\leq\frac{(A+B)^{2}}{2}+2MC\left[C\int_{B/C}^{\infty}\exp\left(-%
-\beta^{2}\right)\beta d\beta+A\int_{B/C}^{\infty}\exp\left(-\beta^{2}\right)d%
-\beta\right]$ </p>
-    <p> $\bar{a}^{T}=\sum_{t=1}^{T}\lambda_{t}a^{t}$ </p>
-    <p> $g^{t}_{y}=-\nabla_{y}F_{\mathcal{D}}(\hat{x}^{t},\hat{y}^{t};B^{t})$ </p>
-    <p> $U\leq\frac{\varepsilon^{2}}{48\log(1/\delta)(9\tau\alpha L_{0})^{2}}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10027.html DELETED Viewed

@@ -1,167 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $4\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}+2(\mathcal{T}+1)\leq 6\sum_{t=1}^{%
-\mathcal{T}+1}2^{N_{t}}\leq\frac{\varepsilon^{2}}{8\log(1/\delta)(9\tau\alpha L%
-_{0})^{2}}.$ </p>
-    <p> $\displaystyle=L_{0}^{F}D\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})\|_{1}.$ </p>
-    <p> $\psi=4\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|$ </p>
-    <p> $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
-\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K=1.$ </p>
-    <p> $\displaystyle\leq 2L_{1}\sum_{k=t_{0}}^{t-1}\frac{2}{k+1}\leq\frac{4L_{1}q}{q%
-\lfloor t/q\rfloor+1},$ </p>
-    <p> $\displaystyle=\mathbb{E}\big{[}\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w%
-^{t};B^{t}),x^{t}-x\rangle\big{]}+\mathbb{E}\big{[}\langle\nabla f(w^{t};B^{t}%
-)-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle\big{]}$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\bigg{%
-[}\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n}}+\left(\frac{(\log(d_{x})+%
-\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{1/2}\bigg{]}.$ </p>
-    <p> $\textstyle\mathbb{E}\Big{[}\big{\|}\sum_{t=1}^{T}x^{t}\Big{\|}_{\infty}^{2}%
-\Big{]}\leq c\log(d)\sum_{t=1}^{T}\mathbb{E}[\|x^{t}\|_{\infty}^{2}].$ </p>
-    <p> $b\lesssim a$ </p>
-    <p> $\hat{y}^{t+1,k}$ </p>
-    <p> $P_{w^{t}}$ </p>
-    <p> $F(\bar{a}^{T})-F\left(\bar{x}^{T}\right)\leq\sum_{t=1}^{T}\alpha_{t}+\frac{L_{%
-1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2},$ </p>
-    <p> $\displaystyle\mathcal{X}\times\mathcal{Y}-(x_{1},y_{1})$ </p>
-    <p> $\mathbb{E}\left\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F%
-\left(\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\right\|^{2}_{\infty}\leq\frac{43L_%
-{1}^{2}}{\sqrt{12+\log(d)}T^{3/2}}+\frac{17(L_{0}^{2}+L_{1}^{2})\sqrt{(12+\log%
-(d))}}{\sqrt{T}}.$ </p>
-    <p> $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\mathbb{E}[2^{N_{t}}]%
-\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$ </p>
-    <p> $9\tau\alpha L_{0}$ </p>
-    <p> $\mathbb{E}[F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)]\leq\frac{\log(d_{x})}{%
-\tau T}+\frac{\tau L_{0}^{2}}{2}+\frac{1}{T}\sum_{t\in[T]}\mathbb{E}\left[%
-\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-x\rangle\right].$ </p>
-    <p> $\Delta(s^{t}_{y})$ </p>
-    <p> $\mathbb{E}[F_{\cal D}(\hat{w}^{T})-F_{\cal D}(x)]\lesssim(L_{0}+L_{1})\left[%
-\sqrt{\frac{\log(d_{x})}{n}}+\frac{\log(d_{x})\log(1/\delta)^{1/4}}{\sqrt{n%
-\varepsilon}}\right]+\frac{L_{2}}{\sqrt{n}\log(d_{x})^{1/4}}\\
-+\frac{L_{2}\log(d_{x})^{1/4}\log(1/\delta)^{1/4}}{\sqrt{n\varepsilon}}+L_{1}%
-\log(\sqrt{n\log(d_{x})})\Big{[}\frac{1}{\sqrt{n}}+\frac{\log(1/\delta)^{1/4}}%
-{\sqrt{n\varepsilon}}\Big{]}.$ </p>
-    <p> $\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}}{\sqrt{n\varepsilon}}$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},%
-\bar{y})]\leq 8L_{1}\sqrt{\log(d_{x})+\log(d_{y})}/\sqrt{T}$ </p>
-    <p> $|\mathbb{E}[\nabla_{x,j}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x,j}\mid\mathcal{F%
-}_{t}]|\leq\frac{2L_{2}}{K}$ </p>
-    <p> $\Delta_{s}:=\max_{j\in J}\max_{S\sim S^{\prime}}|s(S,j)-s(S^{\prime},j)|$ </p>
-    <p> $\displaystyle=\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})%
-,x^{t}-\tilde{x}^{t}\rangle$ </p>
-    <p> $L_{0}^{F}$ </p>
-    <p> $\displaystyle=\max_{k\in[K_{x}+K_{y}]}\left|\sum_{i\in[K_{x}+K_{y}]}\Lambda^{T%
-}_{j,i}\Big{[}\nabla_{k,i}F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla%
-_{k,i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\Big{]}\right|$ </p>
-    <p> $\displaystyle\leq\frac{2L_{2}}{2^{M}},$ </p>
-    <p> $x\in\Delta_{x}$ </p>
-    <p> $(\bar{x}_{-},\bar{y}_{-})=\frac{1}{2^{N}}\sum_{i\in[2^{N}]}(\hat{x}^{i},\hat{y%
-}^{i})$ </p>
-    <p> $s:{\cal Z}^{n}\times J\mapsto\mathbb{R}$ </p>
-    <p> $\mathbb{E}[\langle\nabla F(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle]\leq\frac{%
-\mathbb{E}\left[F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a})%
-\right]}{r}+L_{1}r.$ </p>
-    <p> $\hat{w}^{T}$ </p>
-    <p> $\displaystyle(\lambda,\mu)$ </p>
-    <p> $\operatorname{LSE}(x)$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2(\log(d_{x})+\log(d_%
-{y}))}{\tau T}+5\tau L_{0}^{2}+\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\mathbb{E}%
-[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{\infty}.$ </p>
-    <p> $x_{\mathcal{D}}$ </p>
-    <p> $\sum_{t\in[T]}\langle\tau(\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}),w^%
-{t}-w\rangle\leq\log(d_{x})+\frac{1}{2}\sum_{t\in[T]}\tau^{2}\|\nabla_{x}F_{%
-\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}.$ </p>
-    <p> $\mathbb{E}[\mathcal{T}+1]$ </p>
-    <p> $\sqrt{\log(d)/n}+\log(d)^{7/10}/[n\varepsilon]^{2/5}$ </p>
-    <p> $\displaystyle=4\max_{j\in[d_{x}]}\tau\left|\sum_{k=t_{0}}^{t-1}[\nabla_{j}f(%
-\hat{w}^{t_{0}};B^{k})-\nabla_{j}f(w^{t_{0}};B^{k})]\right|$ </p>
-    <p> $\mathcal{T}=\sup\left\{T\in\mathbb{N}:\sum_{t=1}^{T}2^{N_{t}}\leq U-2^{M}%
-\right\}.$ </p>
-    <p> $\bar{y}=\sum_{t\in[T]}y^{t}$ </p>
-    <p> $\displaystyle=\sum_{k=0}^{M}\mathbb{E}\Big{\|}C_{M}2^{k}[\nabla_{x}f(\bar{x}_{%
-+},\bar{y}_{+};B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})]+\nabla_{x}f(%
-x_{0},y_{0};B_{k})\Big{\|}_{\infty}^{2}\cdot\frac{2^{-k}}{C_{M}}$ </p>
-    <p> $\left|\mathbb{E}\left[\nabla_{j}F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
-\nabla_{j}F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right|\leq\frac{%
-2L_{2}}{T}.$ </p>
-    <p> $w^{1}=(1/d_{x},...,1/d_{x})\in\mathbb{R}^{d_{x}},w^{t+1}:=\operatorname{argmin%
-}_{x\in\Delta_{x}}\left(\tau\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g_{x%
-}^{t},x\rangle+\sum_{i\in[d_{x}]}w^{t}_{i}\log(w^{t}_{i}/x_{i})\right)$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\leq\frac{8L_{1}\sqrt{\log(%
-d_{x})+\log(d_{y})}}{\sqrt{T}}+\frac{2(\log(d_{x})+\log(d_{y}))}{\tau T}+5\tau
-L%
-_{0}^{2}+\frac{2\sqrt{2}L_{1}}{\sqrt{K}}.$ </p>
-    <p> $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2\log(|\mathcal{Z}|)}%
-{\tau_{x}T}+\frac{\log(|{\cal Q}|)}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+10\sqrt{%
-\frac{\log(|{\cal Q}|)}{n}}.$ </p>
-    <p> $\sum_{t\in[T]}\langle\xi^{t},w^{t}-w\rangle\leq D_{\psi}(w^{1},w)+\frac{1}{2}%
-\sum_{t\in[T]}\|\xi^{t}\|^{2}_{*}$ </p>
-    <p> $20.28\pm 1.09$ </p>
-    <p> $\textbf{20.60}\pm 0.89$ </p>
-    <p> $20.84\pm 0.83$ </p>
-    <p> $19.82\pm 1.05$ </p>
-    <p> $20.46\pm 1.16$ </p>
-    <p> $20.05\pm 1.10$ </p>
-    <p> $20.30\pm 0.74$ </p>
-    <p> $20.96\pm 0.86$ </p>
-    <p> $19.74\pm 1.00$ </p>
-    <p> $\textbf{20.41}\pm 1.19$ </p>
-    <p> $20.80\pm 0.88$ </p>
-    <p> $20.25\pm 1.10$ </p>
-    <p> $20.94\pm 0.85$ </p>
-    <p> $\textbf{21.44}\pm 0.80$ </p>
-    <p> $20.19\pm 1.04$ </p>
-    <p> $\{(o_{t},a_{t}),\dots,(o_{t+n},a_{t+n})\}$ </p>
-    <p> $20.73\pm 0.84$ </p>
-    <p> $20.09\pm 1.02$ </p>
-    <p> $21.22\pm 0.87$ </p>
-    <p> $20.95\pm 0.87$ </p>
-    <p> $\textbf{20.54}\pm 1.21$ </p>
-    <p> $20.24\pm 0.98$ </p>
-    <p> $20.83\pm 0.94$ </p>
-    <p> $97.89\%$ </p>
-    <p> $\textbf{20.95}\pm 0.92$ </p>
-    <p> $\textbf{21.58}\pm 1.10$ </p>
-    <p> $\textbf{20.47}\pm 1.11$ </p>
-    <p> $\textbf{20.41}\pm 0.90$ </p>
-    <p> $20.12\pm 1.12$ </p>
-    <p> $20.26\pm 1.05$ </p>
-    <p> $21.15\pm 0.91$ </p>
-    <p> $20.51\pm 1.02$ </p>
-    <p> $20.20\pm 0.24$ </p>
-    <p> $20.81\pm 0.90$ </p>
-    <p> $20.31\pm 1.07$ </p>
-    <p> $\textbf{20.97}\pm 0.92$ </p>
-    <p> $21.01\pm 0.85$ </p>
-    <p> $20.91\pm 0.79$ </p>
-    <p> $20.96\pm 0.84$ </p>
-    <p> $20.83\pm 0.82$ </p>
-    <p> $256\times 160$ </p>
-    <p> $20.33\pm 1.20$ </p>
-    <p> $20.31\pm 1.12$ </p>
-    <p> $w_{p_{xy}}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(x-m)^{2}+(y-n)^{2}}{2\sigma^{2}}},$ </p>
-    <p> $p(I_{pred})$ </p>
-    <p> $\displaystyle L_{VQVAE}=||x-D(e)||^{2}_{2}+||sg[E(x)]-e||^{2}_{2}+$ </p>
-    <p> $T_{style}$ </p>
-    <p> $I_{x_{ij}}=\mathop{\arg\min}_{\theta\in\Theta}Dist(Z_{q_{ij}},Z_{\theta}),$ </p>
-    <p> $L_{VQVAE}$ </p>
-    <p> $\displaystyle\beta||sg[e]-E(x)||^{2}_{2},$ </p>
-    <p> $I_{GT}$ </p>
-    <p> $i=1,{\ldots}\,,T$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10028.html DELETED Viewed

@@ -1,131 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $T_{text}$ </p>
-    <p> $p_{xy}$ </p>
-    <p> $S_{text}$ </p>
-    <p> $P(I_{0},I_{1},...,I_{F}|I_{0})=\prod_{i=1}^{F}P(I_{i}|I_{0}),$ </p>
-    <p> $w_{p_{xy}}$ </p>
-    <p> $L_{Transformer}=-\sum p(I_{pred})\log q(I_{GT}),$ </p>
-    <p> $Dist(Z_{q_{ij}},Z_{\theta})=\left\|Z_{q_{ij}}-Z_{\theta}\right\|_{2},$ </p>
-    <p> $S_{style}$ </p>
-    <p> $q(I_{GT})$ </p>
-    <p> $P_{i_{xy}}=\frac{\sum_{p_{xy}>0}w_{p_{xy}}p_{xy}}{\sum_{p_{xy}>0}w_{p_{xy}}},$ </p>
-    <p> $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(G)\leq c\cdot\mathrm{RAC%
-}(\mathcal{P}_{n},G)$ </p>
-    <p> $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f)\leq c\cdot\mathrm{RAC%
-}(\mathcal{P}_{n},f)$ </p>
-    <p> $\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2}\log n)$ </p>
-    <p> $H=S_{k}$ </p>
-    <p> $v_{\sqrt{n}}$ </p>
-    <p> $H=([h],E)$ </p>
-    <p> $O(1/n^{c})$ </p>
-    <p> $n/{2.2^{i}}$ </p>
-    <p> $\Pr(E_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge\neg E_{j-1})=\Theta%
-\left(\frac{2^{t}}{1.1^{t}}\cdot\frac{1}{n^{1/10}}\right)$ </p>
-    <p> $\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2})$ </p>
-    <p> $\displaystyle\mathrm{RAC}(\mathcal{P},G)=\min_{A\in\mathcal{A}_{\mathcal{P}}}%
-\max_{\pi\in\Gamma}\text{Queries}^{\mathcal{P}}_{A}(\pi(G)),$ </p>
-    <p> $|H|/(|B|-i)=\Theta(1/n)$ </p>
-    <p> $x\neq y\in[n]$ </p>
-    <p> $n^{1-c}/{1.1^{t}}$ </p>
-    <p> $|C|=\Theta(n/\log n)$ </p>
-    <p> $1\leq i\leq\sqrt{n}-1$ </p>
-    <p> $I\in\mathcal{I}_{i}\setminus\mathcal{L}_{i}$ </p>
-    <p> $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n^{1/10}%
-\log(n))$ </p>
-    <p> $n^{0.9}$ </p>
-    <p> $\Pr(E)=1-1/\Theta(n^{1/4})$ </p>
-    <p> $C\sqrt{n}$ </p>
-    <p> $(1+o(1))p$ </p>
-    <p> $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n)$ </p>
-    <p> $|V_{t}|=n-O(t)\geq n/2$ </p>
-    <p> $\alpha n^{1/4}$ </p>
-    <p> $I\in\mathcal{L}_{i}$ </p>
-    <p> $O(n^{0.1})$ </p>
-    <p> $\mathop{\mathbb{E}}[I_{i}]\leq\alpha q_{i}/n^{0.1}$ </p>
-    <p> $x_{1},\ldots x_{h}\in[n]$ </p>
-    <p> $\{x_{1},\ldots,x_{h}\}$ </p>
-    <p> $\Pr(E)=1-O(1/\sqrt{n})$ </p>
-    <p> $b_{t}=n^{9/10}/1.1^{t}$ </p>
-    <p> $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\mathrm{RAC}(\mathcal{P},G)=O(n^{1/10})$ </p>
-    <p> $|\mathcal{L}_{i}|\leq\frac{\kappa n^{0.1}}{2^{i-2}}.$ </p>
-    <p> $I\in\mathcal{R}_{i}$ </p>
-    <p> $O(1/p)=O(1/n^{1/4})$ </p>
-    <p> $H_{a,b,c}$ </p>
-    <p> $y_{i},y_{j},y_{l}$ </p>
-    <p> $2^{t}\cdot n^{9/10}/1.1^{t}$ </p>
-    <p> $\text{deg}(v)=1$ </p>
-    <p> $H=P_{k}$ </p>
-    <p> $\mathcal{A}=\{A_{n}\}_{n\in\mathbb{N}}$ </p>
-    <p> $f\circ\pi$ </p>
-    <p> $V_{i}=\bigcup_{I\in\mathcal{I}_{i}}V(I)$ </p>
-    <p> $Cn^{0.1}\log n$ </p>
-    <p> $Q\subset I$ </p>
-    <p> $\mathcal{A}_{\mathcal{P}}$ </p>
-    <p> $\{G_{n}\}_{n\in\mathbb{N}}$ </p>
-    <p> $p\in\{p_{i_{1}},\ldots,p_{i_{k}}\}$ </p>
-    <p> $f(x_{u})=x_{v}$ </p>
-    <p> $2^{i}<2^{t}$ </p>
-    <p> $B(t)/|V_{t}|\leq 1/n^{0.1+\Omega(1)}$ </p>
-    <p> $\mathcal{P}_{S_{k}}$ </p>
-    <p> $\alpha\kappa$ </p>
-    <p> $n^{1-c}/1.1^{i}$ </p>
-    <p> $B(n)\leq n^{9/10}/1.1^{\log(n)/1000}=n^{9/10-\Omega(1)}$ </p>
-    <p> $G^{\pi}=(V,E^{\pi})$ </p>
-    <p> $2^{i-2}$ </p>
-    <p> $f(x_{1})=\ldots=f(x_{k})$ </p>
-    <p> $2/2^{i-1}=4/2^{i}$ </p>
-    <p> $O(n^{3/4}\cdot\log^{4}n/n^{1/4})=o(n^{3/4})$ </p>
-    <p> $B=[n]\setminus C$ </p>
-    <p> $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f_{n})\geq\omega(n)\cdot%
-\mathrm{RAC}(\mathcal{P}_{n},f_{n}).$ </p>
-    <p> $b_{i}=n^{9/10}/1.1^{i}$ </p>
-    <p> $f\in\Delta$ </p>
-    <p> $\frac{2^{t}}{1.1^{t}}\cdot\frac{1.1^{t}\cdot n}{n^{1-c}\cdot 2^{t}}=n^{c}.$ </p>
-    <p> $\mathop{\mathbb{E}}_{f\leftarrow\Delta}RAC(\mathcal{P}_{H},f)=O(n^{3/4})$ </p>
-    <p> $\frac{n^{1-c}\cdot 2^{t}}{n\cdot 1.1^{t}}$ </p>
-    <p> $\Omega(n^{0.1}\log n)$ </p>
-    <p> $2^{t}+4$ </p>
-    <p> $\mathop{\mathbb{E}}\left[X_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge%
-\neg E_{j-1}\right]=O\left(\frac{2^{t}}{1.1^{t}}\right)$ </p>
-    <p> $c=1/10$ </p>
-    <p> $\Pr({E_{\text{long}}})=O(\kappa)$ </p>
-    <p> $\frac{a_{i}\cdot 2^{i}}{n}=\frac{1}{1.1^{i}}$ </p>
-    <p> $\omega\colon\mathbb{N}\to\mathbb{N}$ </p>
-    <p> $C_{i_{1}},\ldots,C_{i_{T}}$ </p>
-    <p> $C=\bigcup_{i=1}^{N}C_{i}$ </p>
-    <p> $\mathcal{P}_{H}$ </p>
-    <p> $\mathop{\mathbb{E}}[\text{\# red vertices encountered}]\leq\sum_{\frac{1}{1000%
-}\log n\leq i\leq\frac{1}{100}\log n}\alpha\frac{q_{i}}{n^{0.1}}\leq\frac{%
-\alpha}{n^{0.1}}cn^{0.1}\log n\leq\alpha c\log n,$ </p>
-    <p> $N=\alpha n^{1/4}/\log(n)$ </p>
-    <p> $\pi\colon[n]\to[n]$ </p>
-    <p> $\bigcup_{i=1}^{N}C_{i}$ </p>
-    <p> $\Omega(n^{c}\log n)$ </p>
-    <p> $\Pr({E_{\text{long}}})$ </p>
-    <p> $\Pr({E_{\text{long}}})\leq\sum_{I\in\mathcal{L}_{i}}\Pr(I\in\mathcal{R}_{i})%
-\leq|\mathcal{L}_{i}|\cdot(1+o(1))p\leq(1+o(1))\cdot\frac{\kappa n^{0.1}}{2^{i%
--2}}\cdot\frac{2^{i}}{n^{0.1}}=O(\kappa).$ </p>
-    <p> $\pi\colon V\to V$ </p>
-    <p> $G=(V,E)\in\mathcal{P}$ </p>
-    <p> $i=1,...,\Theta(\sqrt{n}\log n)$ </p>
-    <p> ${E_{\text{short}}}$ </p>
-    <p> $n^{9/10}/1.1^{t}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10029.html DELETED Viewed

@@ -1,130 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\text{Queries}_{A}(\cdot)$ </p>
-    <p> $\mathcal{B}_{i}=\mathcal{I}_{i}\setminus\mathcal{R}_{i}$ </p>
-    <p> $\mathrm{RAC}(\mathcal{P}_{S_{k}},\Delta)=O(n^{1/2})$ </p>
-    <p> $\mathcal{P}_{S_{3}}$ </p>
-    <p> $\mathcal{RAC}(\mathcal{P}_{H},f)=O(n^{3/4})$ </p>
-    <p> $p=\frac{n^{9/10}/1.1^{i}}{n/2.2^{i}}=\frac{2^{i}}{n^{0.1}}$ </p>
-    <p> $\tilde{\theta}(n^{1/4})$ </p>
-    <p> $O(2^{t}/1.1^{t})$ </p>
-    <p> $\frac{1}{1000}\log n\leq t\leq\frac{1}{100}\log n$ </p>
-    <p> $\omega(n)=n^{\Omega(1)}$ </p>
-    <p> $\mathcal{R}_{i}\subseteq\mathcal{I}_{i}$ </p>
-    <p> $\alpha\frac{n}{\log n}$ </p>
-    <p> $Cn^{1/10}\log n$ </p>
-    <p> $p_{i_{1}},\ldots,p_{i_{T}}$ </p>
-    <p> $1/|I|$ </p>
-    <p> $\text{Queries}^{\mathcal{P}}_{A}(f,r)$ </p>
-    <p> $\mathrm{RAC}(\mathcal{P},G)\in O(n^{1/10})$ </p>
-    <p> $H=H_{a,b,c}$ </p>
-    <p> $\displaystyle\text{Queries}_{A}(f)\leq\alpha\cdot\max_{\pi}\text{Queries}_{A^{%
-\prime}}(f\circ\pi)$ </p>
-    <p> $(\pi(u),\pi(v))\in E^{\pi}$ </p>
-    <p> $([3],\{1\to 3,2\to 3\})$ </p>
-    <p> $a_{i}=n/2.2^{i}$ </p>
-    <p> $\mathop{\mathbb{E}}_{f\leftarrow\Delta}\text{Queries}_{A}(f)=\Omega(n/\log n)$ </p>
-    <p> $\frac{1}{1000}\log n\leq i\leq\frac{1}{100}\log n$ </p>
-    <p> $I\notin\mathcal{R}_{i}$ </p>
-    <p> $n^{1/10}$ </p>
-    <p> $n^{1/10+\Omega(1)}$ </p>
-    <p> ${E_{\text{short}}}=E\setminus{E_{\text{long}}}$ </p>
-    <p> $\Pr(E)=1-o(1)$ </p>
-    <p> $\text{Queries}_{A}(f)$ </p>
-    <p> $\Omega(n^{1/10}\log n)$ </p>
-    <p> $f\circ\pi\in\mathcal{P}$ </p>
-    <p> $\Pr(E)=O(\kappa)$ </p>
-    <p> $Cn^{1/2}$ </p>
-    <p> $\sum_{i=0}^{t}2^{i}\cdot\frac{1}{1.1^{i}}+\sum_{i=t}^{\log n}2^{t}\cdot\frac{1%
-}{1.1^{i}}=O\left(\frac{2^{t}}{1.1^{t}}\right)$ </p>
-    <p> $n^{\frac{3}{4}}$ </p>
-    <p> $P\in\mathcal{P}_{t}$ </p>
-    <p> $\displaystyle\text{Queries}_{A}(f)\geq\omega(n)\cdot\max_{\pi}\text{Queries}_{%
-A^{\prime}}(f\circ\pi).$ </p>
-    <p> $t=o(n)$ </p>
-    <p> $\Pr({E_{\text{short}}})=O(\kappa)$ </p>
-    <p> $[\frac{\sqrt{n}}{4},\frac{3\sqrt{n}}{2}]$ </p>
-    <p> $v_{0},v_{1},\ldots,v_{\sqrt{n}}$ </p>
-    <p> $n^{\frac{1}{10}+\varepsilon}$ </p>
-    <p> $n\geq\mathbb{N}$ </p>
-    <p> $O(n^{1/10})$ </p>
-    <p> $P_{1},...,P_{\sqrt{n}}$ </p>
-    <p> $\{\mathcal{A}_{n}\}_{n\in\mathbb{N}}$ </p>
-    <p> $O(\log^{4}n/\sqrt{n})$ </p>
-    <p> $\frac{4}{2^{i}}\cdot(1+o(1))\cdot\frac{2^{i}}{n^{0.1}}=O\left(\frac{1}{n^{0.1}%
-}\right).$ </p>
-    <p> $2^{i}/n^{c}$ </p>
-    <p> $P\setminus\{u,v\}$ </p>
-    <p> $f\colon[n]\to[n]$ </p>
-    <p> $\{\mathcal{P}_{n}\}$ </p>
-    <p> $\Theta(\sqrt{n}\log n)$ </p>
-    <p> $\mathcal{P}=\{\mathcal{P}_{n}\}_{n\in\mathbb{N}}$ </p>
-    <p> $\{A_{n}\}_{n\geq N}$ </p>
-    <p> $Cn^{1/4}$ </p>
-    <p> $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(G_{n})\geq\omega(n)\cdot%
-\mathrm{RAC}(\mathcal{P}_{n},G_{n}).$ </p>
-    <p> $(n^{1/4}/4,{n^{1/4}}/2)$ </p>
-    <p> $f(y)=x$ </p>
-    <p> $\text{Queries}^{\mathcal{P}}_{A}(f)=\mathop{\mathbb{E}}_{r}\text{Queries}^{%
-\mathcal{P}}_{A}(f,r)$ </p>
-    <p> $\min(2^{i},2^{t})$ </p>
-    <p> $2\leq i\leq\sqrt{n}-1$ </p>
-    <p> $\frac{4}{2^{i}}$ </p>
-    <p> $\Pr({E_{\text{short}}})$ </p>
-    <p> $n^{1/4}/\log n$ </p>
-    <p> $p_{i_{k}}$ </p>
-    <p> $v_{0},\ldots,v_{\sqrt{n}}$ </p>
-    <p> $W_{1},\ldots,W_{m}$ </p>
-    <p> $I=\{\frac{1}{1000}\log n\leq i\leq\frac{1}{100}\log n:\text{there exists a %
-path of length $2^{i}$ with a red end}\}$ </p>
-    <p> $i_{1},...,i_{T}$ </p>
-    <p> $f_{n}\colon[n]\to[n]$ </p>
-    <p> $O(\log^{1.1}n)$ </p>
-    <p> $G\in\Delta$ </p>
-    <p> $|Q|<2^{i-2}$ </p>
-    <p> ${E_{\text{long}}}$ </p>
-    <p> $E={E_{\text{long}}}\cup{E_{\text{short}}}$ </p>
-    <p> $G^{\pi}\in\mathcal{P}$ </p>
-    <p> $\displaystyle\mathrm{RAC}(\mathcal{P},f)=\min_{A\in\mathcal{A}_{\mathcal{P}}}%
-\max_{\pi}\text{Queries}_{A}(f\circ\pi),$ </p>
-    <p> $i_{1},\ldots,i_{T}$ </p>
-    <p> $I\setminus Q$ </p>
-    <p> $n/2.2^{i}$ </p>
-    <p> $cn^{0.1}\log n$ </p>
-    <p> $\frac{1}{1000}\log n$ </p>
-    <p> $\kappa n^{0.1}$ </p>
-    <p> $\sqrt{n}-2$ </p>
-    <p> $Path\mbox{ }of\mbox{ }length\mbox{ }\sqrt{n}$ </p>
-    <p> $s\in B\Rightarrow s_{i}\in M_{B}\;\forall_{s_{i}\in s}\land((s_{0}\in E_{B}^{0%
-}\land s_{N}\in E_{B}^{1})\lor(s_{N}\in E_{B}^{0}\land s_{0}\in E_{B}^{1})).$ </p>
-    <p> $3\times N$ </p>
-    <p> $\mathrm{LM}_{k}(s):=\left\{||l_{k}-s_{i}||\forall s_{i}\in s\right\},$ </p>
-    <p> $b\in\{1000,3000\}$ </p>
-    <p> $5\,mm$ </p>
-    <p> $s\equiv(s_{0},...,s_{N})$ </p>
-    <p> $1\times N$ </p>
-    <p> $56\times N$ </p>
-    <p> $\mathrm{LM}[s]$ </p>
-    <p> $W_{ij}=\left\{\begin{array}[]{ll}1&s_{j}\in R_{i}\\
-0&otherwise\end{array}\right.,$ </p>
-    <p> $M_{B},E_{B}^{0},E_{B}^{1}$ </p>
-    <p> $3\cdot\mathrm{10}^{-5}$ </p>
-    <p> $J_{mm}(A,B)$ </p>
-</body>
-</html>

htmls/output_mathjax_example_1003.html DELETED Viewed

@@ -1,122 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $x=a$ </p>
-    <p> $C_{1},\ldots,C_{k}\cup(C\setminus\{v\})$ </p>
-    <p> $z^{v}$ </p>
-    <p> $\{2,\dots,s-1\}$ </p>
-    <p> $S\cap(A\cup C)$ </p>
-    <p> $H:=G[A\cup B\cup S]$ </p>
-    <p> $z\in V(C)$ </p>
-    <p> $N_{G}(p_{s})\cap V(H)=\{b_{2}\}$ </p>
-    <p> $H\setminus C$ </p>
-    <p> $k\geq\chi(\overline{G})\geq 2$ </p>
-    <p> $H_{1},H_{2}\in\mathcal{M}_{\mathcal{C}}$ </p>
-    <p> $\{p_{0},\dots,p_{s-1}\}$ </p>
-    <p> $(x,c)$ </p>
-    <p> ${\cal G}_{1}$ </p>
-    <p> $q_{t}=y$ </p>
-    <p> $q_{2},\dots,q_{t}\in B)$ </p>
-    <p> $G\in\mathcal{G}_{k}$ </p>
-    <p> $S=(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}$ </p>
-    <p> $X_{v}$ </p>
-    <p> $\mathcal{M}_{\mathcal{G}_{2}}=\{\overline{K_{2}\cup C_{2k+1}}\mid k\in\mathbb{%
-N}\}$ </p>
-    <p> $\{a_{i},b_{j}\}$ </p>
-    <p> $\mathcal{O}(n^{\omega}\log n)$ </p>
-    <p> $|V(H)|\geq 2$ </p>
-    <p> $V(C)$ </p>
-    <p> $q_{i},\ldots,q_{t},b_{1},b_{2},a_{2},q_{i}$ </p>
-    <p> $N_{G}(v)\cap V(H)\subseteq A$ </p>
-    <p> $G[N_{G}(v)]$ </p>
-    <p> $G_{A}\setminus S^{\prime}$ </p>
-    <p> $\mathcal{O}(n^{2k})$ </p>
-    <p> $\mathbb{Z}_{k}$ </p>
-    <p> $\{X_{v}\}_{v\in V(H)}$ </p>
-    <p> $2K_{1}\vee K_{2}$ </p>
-    <p> $G[\{a_{1},a_{2},b_{1},b_{2}\}\cup V(Q)]$ </p>
-    <p> $v^{xy}\in S$ </p>
-    <p> $S\cap A$ </p>
-    <p> $(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}\subseteq S$ </p>
-    <p> $X_{w}\subseteq V(D)$ </p>
-    <p> $\mathcal{G}_{k}=\mathcal{G}_{\mathcal{C}_{k}}$ </p>
-    <p> $N_{G}[u]=V(G)$ </p>
-    <p> $\mathcal{O}(n^{2.373}(n+m))$ </p>
-    <p> $F\in{\cal F}$ </p>
-    <p> $r_{i}\in C$ </p>
-    <p> $G\setminus x$ </p>
-    <p> $G\setminus N_{G}[v]$ </p>
-    <p> $S^{*}:=(S\setminus\{v^{xy}\})\cup\{x,y\}$ </p>
-    <p> $U:=\{v\in V(H)\mid X_{v}\cap V(C)\neq\emptyset\}$ </p>
-    <p> $N_{A}\cap N_{B}=\emptyset$ </p>
-    <p> $G\setminus(A\cup B)$ </p>
-    <p> $N_{G}(q_{t})\cap V(H)=\{a_{1},b_{1}\}$ </p>
-    <p> $S\subseteq S^{*}$ </p>
-    <p> $K_{\ell+1}\in\mathcal{G}_{\mathcal{C}}$ </p>
-    <p> ${\mathcal{C}_{k}}$ </p>
-    <p> $V(H)=A\cup B$ </p>
-    <p> $K_{\ell}$ </p>
-    <p> $N_{G}(v)\cap V(H)=B$ </p>
-    <p> $X_{w}$ </p>
-    <p> $H\setminus a$ </p>
-    <p> $H[A]$ </p>
-    <p> $G^{\prime}[S]$ </p>
-    <p> $G_{B}:=G[B\cup C]$ </p>
-    <p> $S^{*}\subseteq S\cup(V(G)\setminus V(H))$ </p>
-    <p> $G^{\prime}:=G/xy$ </p>
-    <p> $p_{i},\dots,p_{s},b_{1},b_{3},a_{3},a_{2},q_{t},\dots,q_{0},p_{i}$ </p>
-    <p> $q_{2},\dots,q_{t}\in Y$ </p>
-    <p> $a_{1},q_{i}$ </p>
-    <p> $\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}}\}$ </p>
-    <p> $p_{1},\dots,p_{i-1}\in A$ </p>
-    <p> $v^{xy}$ </p>
-    <p> $\mathcal{C}\subseteq\mathcal{G}_{\mathcal{C}}$ </p>
-    <p> $S^{\prime}\setminus\{x,y\}$ </p>
-    <p> $p_{1}=x$ </p>
-    <p> $N_{G}(q_{t})\cap V(H)=\{a_{2}\}$ </p>
-    <p> $G\setminus N_{G}[x]$ </p>
-    <p> $\overline{\overline{G}}=G$ </p>
-    <p> $H[S]\in\mathcal{C}$ </p>
-    <p> $S^{\prime}=S$ </p>
-    <p> $\bigcup_{v\in V(H)}X_{v}\not\subseteq V(H_{2})$ </p>
-    <p> $c\in C\setminus S$ </p>
-    <p> $N_{G}(p_{s})\cap V(H)=\{b_{1}\}$ </p>
-    <p> $\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{0}}\}=\{2K_{1}\vee K_{1}\}=\{%
-P_{3}\}$ </p>
-    <p> $K_{\ell+1}$ </p>
-    <p> $H\in\mathcal{H}$ </p>
-    <p> $A=C\cup\{v\}$ </p>
-    <p> $U=\{v\in V(H)\mid X_{v}\subseteq V(C)\}$ </p>
-    <p> $S_{A}:=S\setminus B$ </p>
-    <p> $2K_{1}\vee 3K_{1}$ </p>
-    <p> $q_{j}\in C$ </p>
-    <p> $p_{i-1},p_{j+1}\in C$ </p>
-    <p> $\{a_{1},b_{1}\}$ </p>
-    <p> $Y\subseteq V(G)\setminus\{x\}$ </p>
-    <p> $\{a,b\}=\{x,y\}$ </p>
-    <p> $\{X_{v}\}_{v\in\{a,b\}\cup V(H)}$ </p>
-    <p> $|V(H)|=2k$ </p>
-    <p> $(x,b)$ </p>
-    <p> $C\cap S$ </p>
-    <p> $\mathcal{O}(|V(G)|^{2})$ </p>
-    <p> $G[a_{j},a_{k},b_{j},b_{k},q_{i},q_{i+1},\dots,q_{t}]$ </p>
-    <p> $2K_{1}$ </p>
-    <p> $a_{i},a_{j}$ </p>
-    <p> $p_{i}\in B$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10030.html DELETED Viewed

@@ -1,126 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> ${}_{mm\_morph}$ </p>
-    <p> $c=\frac{max(|A|,|B|)}{min(|A|,|B|)}$ </p>
-    <p> $J\_{mm\_syn}$ </p>
-    <p> $C_{WALS}$ </p>
-    <p> $m:\mathbb{L}\mapsto\mathbb{R}$ </p>
-    <p> $\rho=0.69$ </p>
-    <p> $J_{mm}$ </p>
-    <p> $1\dots|Z|$ </p>
-    <p> $J_{mm\_syn}$ </p>
-    <p> $J_{mm}(\mathbf{a},\mathbf{b})=\frac{\sum_{j=1}^{|Z|}min(a_{j},b_{j})}{\sum_{j=%
-1}^{|Z|}max(a_{j},b_{j})}$ </p>
-    <p> $\{Z=t(y):y\in Y\}=\{(y_{i},z_{j})\}$ </p>
-    <p> $\{Y=m(x):x\in X\}=\{(x_{i},y_{i})\}$ </p>
-    <p> $J_{mm\_morph}$ </p>
-    <p> ${}_{mm}$ </p>
-    <p> $1\dots|X|$ </p>
-    <p> $J\_{mm\_morph}$ </p>
-    <p> $ch\_ttr_{500}$ </p>
-    <p> $s_{l}=r\cdot ch\_ttr_{l\_500}$ </p>
-    <p> ${}_{morph}$ </p>
-    <p> ${}_{mm\_syn}$ </p>
-    <p> ${}_{syn}$ </p>
-    <p> $S=\textbf{C}\circ\textbf{U}(X)$ </p>
-    <p> $val_{1}$ </p>
-    <p> $\sigma(G;D)$ </p>
-    <p> $\log P(G|D,\lambda)\propto\log P(D|G)+\log P(G|\lambda)$ </p>
-    <p> $\sigma(G;D)=\sum_{i=1}^{n}\sigma\left(v_{i},\operatorname{pa}\left(v_{i}\right%
-);D\right)$ </p>
-    <p> $X=\{x_{1},x_{2},\cdots,x_{n}\}$ </p>
-    <p> $sym_{1}$ </p>
-    <p> $P\left(v_{i}\mid\text{pa}\left(v_{i}\right)\right)$ </p>
-    <p> $val_{n}$ </p>
-    <p> $x_{i_{1}}\rightarrow x_{j_{1}}$ </p>
-    <p> $S{\prime}$ </p>
-    <p> $\mathcal{A}=\{x_{i}\leadsto x_{j}\mid(x_{i},x_{j})\in S^{\prime}\}$ </p>
-    <p> $P(G\mid D,\lambda)=\frac{P(D\mid G,\lambda)P(G\mid\lambda)}{P(D\mid\lambda)}$ </p>
-    <p> $Domain$ </p>
-    <p> $sym_{i}$ </p>
-    <p> $val_{i}$ </p>
-    <p> $\text{pa}(v_{i})$ </p>
-    <p> $x\leadsto y$ </p>
-    <p> $P(D|\lambda)$ </p>
-    <p> $S=\{(x_{i},x_{j})\}$ </p>
-    <p> $\sigma(G;D,\lambda)=\sigma(G;D)+\sigma(G;\lambda)$ </p>
-    <p> $PromtU$ </p>
-    <p> $sym_{n}$ </p>
-    <p> $x_{i}\leadsto x_{j}\Rightarrow x_{j}\not\in\text{pa}(x_{i}),x_{i}<x_{j}$ </p>
-    <p> $x\not\rightarrow y$ </p>
-    <p> $S^{\prime}={(x_{i},x_{j})}$ </p>
-    <p> $S=\{(x_{i_{1}},x_{j_{1}}),\cdots,(x_{i_{m}},x_{j_{m}})\mid x_{i_{k}},x_{j_{k}}%
-\in X\}$ </p>
-    <p> $\frac{2\cdot\text{precision}\cdot\text{recall}}{\text{precision}+\text{recall}}$ </p>
-    <p> $x_{i_{m}}\rightarrow x_{j_{m}}$ </p>
-    <p> $S^{\prime}=\textbf{R}\circ\textbf{C}\circ\textbf{U}(X)$ </p>
-    <p> $P(D\mid G,\lambda)=P(D\mid G)$ </p>
-    <p> $T=\textbf{U}(X)$ </p>
-    <p> $\sigma(G;\lambda)$ </p>
-    <p> $x_{i}\leadsto x_{j}$ </p>
-    <p> $T=\{t_{1},t_{2},\cdots,t_{n}\}$ </p>
-    <p> $g_{i}(\textbf{x})$ </p>
-    <p> $h_{j}(\textbf{x})$ </p>
-    <p> $\displaystyle\mathrm{s.t.}:$ </p>
-    <p> $\displaystyle f({\bf{x}}),x\in D$ </p>
-    <p> $\displaystyle\mathrm{Min}:$ </p>
-    <p> $\displaystyle g_{i}({\bf{x}})\leq 0,{i}=1,\dots,{p}$ </p>
-    <p> $\displaystyle h_{j}({\bf{x}})\leq 0,{j}=1,\dots,{q}$ </p>
-    <p> $S\neq\phi$ </p>
-    <p> $W=\phi$ </p>
-    <p> $W=W\cup(S\cap T)$ </p>
-    <p> $E^{\prime}[i]=E^{\prime}[j]$ </p>
-    <p> $\lvert E^{\prime}\rvert$ </p>
-    <p> $\rm{LayoutLMv3_{LARGE}}$ </p>
-    <p> $V[j]=V[j]+1$ </p>
-    <p> $T=P[j*2-1]$ </p>
-    <p> $batch\_size=8$ </p>
-    <p> $S=S[j+1:\lvert S\rvert]$ </p>
-    <p> $L=L[1:\lvert L\rvert]$ </p>
-    <p> $E=E\cup ParseEntityValue(D,J^{\prime})$ </p>
-    <p> $<segment~{}text>~{}XX|YY_{segment}$ </p>
-    <p> $T.subtypes=\phi$ </p>
-    <p> $G=\phi$ </p>
-    <p> $\lvert E\rvert$ </p>
-    <p> $M=\{``s.x|s.y"\mapsto s|s\in D.segments\}$ </p>
-    <p> $G^{\prime}.value=\bigcup_{w\in W}w.text\_value$ </p>
-    <p> $P[j*2]\notin M$ </p>
-    <p> $\bigcup_{T^{\prime}\in T}MajorityVoting(\bigcup_{S^{\prime}\in S}DecodeForType%
-(ParseJson(S^{\prime}),T^{\prime},D))$ </p>
-    <p> $800train/100dev/100test$ </p>
-    <p> $E^{\prime}.subtypes=\bigcup_{T^{\prime}\in T.subtypes}DecodeForType(J^{\prime}%
-,T^{\prime},D)$ </p>
-    <p> $R.split(E[i])$ </p>
-    <p> $(segment~{}text,segment~{}identifier)$ </p>
-    <p> $S=D.pages[i].segments$ </p>
-    <p> $G^{\prime}.bounding\_box=\{\min(b.x),\min(b.y),\max(b.x),\max(b.y)\}_{w\in W,b%
-=w.bounding\_box}$ </p>
-    <p> $E=\phi$ </p>
-    <p> $F(S[1:j])\leq L$ </p>
-    <p> $V=[0,0,...,0]\in\mathbb{R}\textsuperscript{$\lvert E\rvert$}$ </p>
-    <p> $L=\{T\}$ </p>
-    <p> $\lvert P\rvert/2$ </p>
-    <p> $G=G\cup\{G^{\prime}\}$ </p>
-    <p> $learning\_rate=2\cdot 10^{-5}$ </p>
-    <p> $E=E\cup\{E^{\prime}\}$ </p>
-    <p> $\mathbf{LayoutLMv3_{LARGE}}$ </p>
-    <p> $T^{\prime}=L[0]$ </p>
-    <p> $C=C\cup\{S[1:j]\}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10031.html DELETED Viewed

@@ -1,185 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\lvert D.pages\rvert$ </p>
-    <p> $[x\textsubscript{center},y\textsubscript{center}]$ </p>
-    <p> $E^{\prime}=\phi$ </p>
-    <p> $E^{\prime}[j]$ </p>
-    <p> $J^{\prime}=J[j][T.type]$ </p>
-    <p> $j=\lvert S\rvert$ </p>
-    <p> $C=\phi$ </p>
-    <p> $S=M[P[j*2]]$ </p>
-    <p> $E^{\prime}=E^{\prime}\cup\{e|e\in E[j],e.type=T^{\prime}\}$ </p>
-    <p> $R=Regex(``(\char 92d\char 92d\char 92|\char 92d\char 92d)")$ </p>
-    <p> $\mathbf{Donut}$ </p>
-    <p> $L\neq\phi$ </p>
-    <p> $L=L\cup T^{\prime}.subtypes$ </p>
-    <p> $Donut$ </p>
-    <p> $E[argmax(V)]$ </p>
-    <p> $[x\textsubscript{min},y\textsubscript{min},x\textsubscript{max},y%
-\textsubscript{max}]$ </p>
-    <p> $D=10/50/100/200$ </p>
-    <p> $V[i]=V[i]+1$ </p>
-    <p> $<text~{}on~{}segment_{1}>~{}XX|YY_{segment_{1}}\char 92n<text~{}on~{}segment_{%
-2}>~{}XX|YY_{segment_{2}}\char 92n~{}...$ </p>
-    <p> $\displaystyle\bm{q}(t):=\begin{bmatrix}q_{0}^{(1)}(t)&q_{0}^{(2)}(t)&q_{1}^{(1%
-)}(t)&q_{1}^{(2)}(t)&\dots&q_{N-2}^{(1)}(t)&q_{N-2}^{(2)}(t)\end{bmatrix}^{T},$ </p>
-    <p> $\displaystyle q_{3}:$ </p>
-    <p> $40\text{\,}\mathrm{ms}$ </p>
-    <p> $\displaystyle q_{0}:\quad$ </p>
-    <p> $\displaystyle\;a_{i+1,i+1}=-\frac{1}{\psi_{\frac{i-1}{2}}}\left(\frac{\zeta_{%
-\frac{i-3}{2}}+\zeta_{\frac{i-1}{2}}}{2\zeta_{\frac{i-3}{2}}\zeta_{\frac{i-1}{%
-2}}}\right),$ </p>
-    <p> $\displaystyle\lim_{t\to 0}\bm{q}(0,t)$ </p>
-    <p> $M_{\textrm{left}}$ </p>
-    <p> $(\xi_{n})_{n=0,\ldots,N-1}$ </p>
-    <p> $n=1,2,...,N-1$ </p>
-    <p> $\displaystyle\dot{\bm{q}}(t)$ </p>
-    <p> $\bm{y}(t_{0})={(0.02,0.01)^{T}}$ </p>
-    <p> $\displaystyle D_{x}\bm{q}(t)$ </p>
-    <p> $\displaystyle=\bm{y}_{0}.$ </p>
-    <p> $\dot{\bm{y}}(t)$ </p>
-    <p> $\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)\in\mathbb{R}^{N}$ </p>
-    <p> $\displaystyle\frac{243d^{5}}{120}(\partial^{5}_{\xi}q)_{0}.$ </p>
-    <p> $\nicefrac{{\partial q(\xi(x),t)}}{{\partial\xi}}\rvert_{x_{n}}$ </p>
-    <p> $\displaystyle\frac{d^{3}}{24}b_{1}$ </p>
-    <p> $\displaystyle t\in[0,T],$ </p>
-    <p> $\displaystyle c\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}+%
-\frac{Nc}{N-1}\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{1}}\approx$ </p>
-    <p> $\displaystyle\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$ </p>
-    <p> $\displaystyle\frac{81d^{3}}{24}b_{3}$ </p>
-    <p> $\displaystyle\frac{d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t):=\begin{bmatrix}\frac{2}{2+%
-\gamma\psi_{0}}\bm{f}^{(1)}(\bm{q}_{0}(t),\bm{y}(t),t)\\
-\frac{2}{2+\gamma\psi_{0}}\bm{f}^{(2)}(\bm{q}_{0}(t),\bm{y}(t),t)\\
-0\\
-\vdots\\
-0\end{bmatrix}.$ </p>
-    <p> $\displaystyle M_{2}\;D_{x}^{2}\bm{q}(t)$ </p>
-    <p> $\displaystyle\approx B_{2}\;D_{x}\bm{q}(t).$ </p>
-    <p> $R=\nicefrac{{7}}{{3}}$ </p>
-    <p> $-\nicefrac{{1}}{{2}}$ </p>
-    <p> $\displaystyle\left(\frac{4Nc}{N-n}\right)\frac{\partial q(\xi(x),t)}{\partial x%
-}\biggr{\rvert}_{n}+\left(\frac{Nc}{N-n-1}\right)\frac{\partial q(\xi(x),t)}{%
-\partial x}\biggr{\rvert}_{n+1}$ </p>
-    <p> $u^{(1)}=0$ </p>
-    <p> $\displaystyle B_{1}:=\frac{1}{d}\begin{bmatrix}\frac{12\alpha dc-17\gamma}{18%
-\gamma}&0&\nicefrac{{1}}{{2}}&0&\nicefrac{{1}}{{2}}&0&-\nicefrac{{1}}{{18}}&0&%
-\dots&0\\
-0&\frac{12\alpha dc-17\gamma}{18\gamma}&0&\nicefrac{{1}}{{2}}&0&\nicefrac{{1}}%
-{{2}}&0&-\nicefrac{{1}}{{18}}&\ddots&0\\
--3&0&0&0&3&0&0&0&\dots&0\\
-0&-3&0&0&0&3&0&0&\ddots&0\\
-\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
-\vdots&\ddots&\ddots&\ddots&\ddots&0&-3&0&0&0\\
-0&\dots&\dots&\dots&\dots&\dots&0&-3&0&0\end{bmatrix},$ </p>
-    <p> $\displaystyle+\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{4}%
-\gamma\sin(\lambda t)}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})%
-\lambda}dk+$ </p>
-    <p> $\displaystyle\;y_{0}^{(2)}+\frac{1-\cos(\lambda t)}{\lambda}+\frac{2}{\pi}\int%
-_{0}^{\infty}\;v_{0}^{(2)}\frac{\gamma(1-e^{-k^{2}t})}{k^{2}\gamma^{2}+(k^{2}-%
-\alpha)^{2}}\;dk+$ </p>
-    <p> $\displaystyle d\;(\partial^{2}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle\underbrace{\begin{bmatrix}\dot{\bm{q}}(t)\\
-\dot{\bm{y}}(t)\end{bmatrix}}_{=:\dot{\bm{\eta}}(t)}=\underbrace{\left[\begin{%
-array}[]{c | c}A_{s}&\begin{array}[]{c c}0&0\\
-\vdots&\vdots\\
-0&0\\
-\end{array}\\
-\hline\cr\\
-\begin{array}[]{c c c c c}1&0&0&\dots&0\\
-0&1&0&\dots&0\end{array}&\begin{array}[]{c c}0&0\\
-0&0\end{array}\end{array}\right]}_{=:A}\underbrace{\begin{bmatrix}\bm{q}(t)\\
-\bm{y}(t)\end{bmatrix}}_{=:\bm{\eta}(t)}+\underbrace{\begin{bmatrix}\bm{v}(\bm%
-{q}_{0}(t),\bm{y}(t),t)\\
-\bm{u}(\bm{y}(t),t)\end{bmatrix}}_{=:\bm{\omega}(\bm{q}_{0}(t),\bm{y}(t),t)}$ </p>
-    <p> $R=\nicefrac{{1}}{{3}}$ </p>
-    <p> $\displaystyle\frac{d^{2}}{2}a_{1}$ </p>
-    <p> $\displaystyle\underbrace{\left(\mathbb{I}-\frac{\Delta t}{2}A\right)}_{=:M_{%
-\textrm{left}}}\bm{\eta}^{k+1}=\left(\mathbb{I}+\frac{\Delta t}{2}A\right)\bm{%
-\eta}^{k}+\frac{\Delta t}{2}\left(\bm{\omega}^{k}+\bm{\omega}^{k+1}\right),$ </p>
-    <p> $\displaystyle\frac{\partial^{2}q(\xi(x),t)}{\partial x^{2}}\biggr{\rvert}_{x_{%
-n}}\approx\;\frac{1}{\psi_{n}}\left[\frac{\partial q(\xi(x),t)}{\partial x}%
-\biggr{\rvert}_{x_{n+\nicefrac{{1}}{{2}}}}-\frac{\partial q(\xi(x),t)}{%
-\partial x}\biggr{\rvert}_{n-\nicefrac{{1}}{{2}}}\right]$ </p>
-    <p> $\displaystyle:=M_{2}-\frac{2c}{3\gamma}B_{2}M_{1}^{-1}\mathbb{P}.$ </p>
-    <p> $\displaystyle-\frac{2c}{3\gamma}\left(\alpha q_{0}-\gamma(\partial_{x}q)_{0}\right)$ </p>
-    <p> $\bm{v}(t_{0})$ </p>
-    <p> $\displaystyle\frac{16d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$ </p>
-    <p> $R=\nicefrac{{7}}{{9}}$ </p>
-    <p> $\bm{q}(0,t_{0})={(0,0.1)}^{T}$ </p>
-    <p> $\left(\nicefrac{{d\xi(x)}}{{dx}}\right)_{x_{n}}$ </p>
-    <p> $\displaystyle\approx M_{1}^{-1}\;B_{1}\bm{q}(t)+M_{1}^{-1}\;K_{1}(\bm{q}_{0}(t%
-),\bm{y}(t),t)+M_{1}^{-1}\;V_{1}(\dot{\bm{q}}_{0}(t)),$ </p>
-    <p> $\displaystyle\approx B_{2}\;D_{x}\bm{q}(t),$ </p>
-    <p> $\displaystyle\beta=\frac{\rho_{p}}{\rho_{f}},\quad R=\frac{1+2\beta}{3},\quad S%
-=\frac{a^{2}}{3T\nu},$ </p>
-    <p> $a_{0},a_{1},b_{0},\hat{b}_{0},b_{1},\hat{b}_{1},b_{2},b_{3}\in\mathbb{R}$ </p>
-    <p> $\displaystyle(\partial_{\xi}q)_{0}:\quad$ </p>
-    <p> $\displaystyle M_{2}:=\begin{bmatrix}c&0&3\frac{Nc}{N-1}&0&0&0&0&0&\dots&0\\
-0&c&0&3\frac{Nc}{N-1}&0&\ddots&\ddots&\ddots&\ddots&0\\
-c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&\ddots&0\\
-0&c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&0\\
-0&0&\frac{Nc}{N-1}&0&\frac{4Nc}{N-2}&0&\frac{Nc}{N-3}&0&\ddots&0\\
-\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
-\vdots&\ddots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}&0\\
-\vdots&\ddots&\ddots&\ddots&0&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}\\
-0&\ddots&\ddots&\ddots&\ddots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}&0\\
-0&0&\dots&\dots&\dots&\dots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}\end{bmatrix},$ </p>
-    <p> $\displaystyle+\frac{1}{2d}q_{2}(t)-\frac{1}{18d}q_{3}(t)+$ </p>
-    <p> $\bm{q}(0,t_{0})={(0.5414,0)}^{T}$ </p>
-    <p> $t\in[0,5]$ </p>
-    <p> $\displaystyle a_{11}=$ </p>
-    <p> $\displaystyle\frac{d^{3}}{6}a_{1}$ </p>
-    <p> $m\in\{1,2,\dots,N-2\}$ </p>
-    <p> $\displaystyle(\partial^{4}_{\xi}q)_{0}:\quad$ </p>
-    <p> $\bm{u}(\bm{y}(t),t)$ </p>
-    <p> $\displaystyle\left(\frac{Nc}{N-n+1}\right)\frac{\partial q(\xi(x),t)}{\partial
-x%
-}\biggr{\rvert}_{n-1}+$ </p>
-    <p> $\displaystyle x>0,$ </p>
-    <p> $n\in\{1,2,\dots,N-1\}$ </p>
-    <p> $\displaystyle\frac{d^{2}}{2}(\partial^{3}_{\xi}q)_{0}$ </p>
-    <p> $f^{(1)}\equiv 0$ </p>
-    <p> $S=0.3$ </p>
-    <p> $\displaystyle 3b_{3}$ </p>
-    <p> $\bm{f}(\bm{q}_{0}(t),\bm{y}(t),t)$ </p>
-    <p> $\frac{\partial}{\partial x}\bm{q}(\xi(x),t)$ </p>
-    <p> $\Psi(x,y,t)=-U_{0}L\text{tanh}(y/L)+\sum_{i=1}^{3}A_{i}U_{0}L\text{sech}^{2}(y%
-/L)\cos(k_{i}x-\sigma_{i}t)$ </p>
-    <p> $\displaystyle\frac{1}{d}b_{3}$ </p>
-    <p> $\displaystyle\frac{d^{4}}{120}b_{1}$ </p>
-    <p> $\displaystyle=\frac{D\bm{u}}{Dt}-\frac{1}{S}(\bm{v}-\bm{u})-\sqrt{\frac{3}{\pi
-S%
-}}\left\{\frac{1}{\sqrt{t}}\left(\bm{v}(0)-\bm{u}(0)\right)+\int_{0}^{t}\frac{%
-(\dot{\bm{v}}(s)-\dot{\bm{u}}(s))}{\sqrt{t-s}}ds\right\},$ </p>
-    <p> $\displaystyle\alpha:=\frac{1}{RS},\quad\gamma:=\frac{1}{R}\sqrt{\frac{3}{S}}$ </p>
-    <p> $d:=\xi_{n+1}-\xi_{n}=\frac{1}{N}$ </p>
-    <p> $\bm{q}(0,t_{0})={(0.1,0)}^{T}$ </p>
-    <p> $\displaystyle\frac{32d^{4}}{120}b_{2}$ </p>
-    <p> $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}%
-\approx\frac{1}{2}\left[\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_%
-{x_{\nicefrac{{1}}{{2}}}}+\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert%
-}_{x_{-\nicefrac{{1}}{{2}}}}\right].$ </p>
-    <p> $\displaystyle\frac{16d^{3}}{24}b_{2}$ </p>
-    <p> $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n}}=%
-\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}\cdot\frac{1}{c}%
-\left(1-\frac{n}{N}\right),$ </p>
-    <p> $\displaystyle M_{1}\;D_{x}\bm{q}(t)$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10032.html DELETED Viewed

@@ -1,168 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\displaystyle-\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{2}%
-\gamma\cos(\lambda t)}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})%
-}dk.$ </p>
-    <p> $\displaystyle q_{0}:$ </p>
-    <p> $\bm{q}(x,t)$ </p>
-    <p> $\displaystyle\frac{4d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle\dot{\bm{y}}(t)$ </p>
-    <p> $\displaystyle\frac{243d^{4}}{120}b_{3}.$ </p>
-    <p> $M_{1}^{-1}$ </p>
-    <p> $A_{s}\in\mathbb{R}^{N\times N}$ </p>
-    <p> $(x_{n})_{n=0,\ldots,N-1}$ </p>
-    <p> $\displaystyle\frac{d}{2}b_{1}$ </p>
-    <p> $\bm{y}={(0.02,0.01)}^{T}$ </p>
-    <p> $\displaystyle V_{1}(\dot{\bm{q}}_{0}(t))=\frac{2c}{3\gamma}\begin{bmatrix}\dot%
-{q}^{(1)}_{0}(t)\\
-\dot{q}^{(2)}_{0}(t)\\
-0\\
-0\\
-\vdots\\
-0\end{bmatrix}=\frac{2c}{3\gamma}\underbrace{\begin{bmatrix}1&0&0&\dots\\
-0&1&0&\ddots\\
-0&0&0&\ddots\\
-\vdots&\ddots&\ddots&\ddots\end{bmatrix}}_{=:\mathbb{P}}\dot{\bm{q}}(t)=\frac{%
-2c}{3\gamma}\;\mathbb{P}\;\dot{\bm{q}}(t).$ </p>
-    <p> $\bm{y}(t_{0})={(0.02,0.01)}^{T}$ </p>
-    <p> $\displaystyle\;a_{24}=\frac{\gamma}{\zeta_{0}(2+\gamma\psi_{0})}$ </p>
-    <p> $q(\xi(x),t)$ </p>
-    <p> $\displaystyle a_{i,i-2}=$ </p>
-    <p> $\bm{q}(0,t_{0})={(0,0)}^{T}$ </p>
-    <p> $\displaystyle(\partial_{\xi}q)_{0}+(\partial_{\xi}q)_{1}=-\frac{2c}{3\gamma}%
-\left(\alpha q_{0}-\frac{\gamma}{c}(\partial_{\xi}q)_{0})\right)+\frac{1}{d}%
-\left(\frac{12\alpha dc-17\gamma}{18\gamma}\right)q_{0}+\frac{1}{2d}q_{1}+%
-\frac{1}{2d}q_{2}-\frac{1}{18d}q_{3},$ </p>
-    <p> $\dot{\bm{q}}(t)=D_{x}^{2}\bm{q}(t)$ </p>
-    <p> $\displaystyle\bm{q}_{0}(t):=\begin{bmatrix}q_{0}^{(1)}(t)&q_{0}^{(2)}(t)\end{%
-bmatrix}^{T}.$ </p>
-    <p> $\displaystyle:=\Psi^{-1}B_{2}M_{1}^{-1}K_{1}(\bm{q}_{0}(t),\bm{y}(t),t),$ </p>
-    <p> $q^{(2)}(0,t_{0})>0$ </p>
-    <p> $\xi_{n}:=\frac{n}{N}$ </p>
-    <p> $\bm{y}(t_{0})={(1,0)}^{T}$ </p>
-    <p> $\displaystyle a_{0}(\partial_{\xi}q)_{0}+a_{1}(\partial_{\xi}q)_{1}=\hat{b}_{0%
-}\left(\alpha q_{0}-\gamma(\partial_{x}q)_{0}\right)+\frac{1}{d}\left\{b_{0}q_%
-{0}+b_{1}q_{1}+b_{2}q_{2}+b_{3}q_{3}\right\},$ </p>
-    <p> $\displaystyle\frac{8d^{2}}{6}b_{2}$ </p>
-    <p> $\displaystyle+\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{2}%
-\gamma e^{-k^{2}t}}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})}dk+$ </p>
-    <p> $q^{(3)}_{0}(t)$ </p>
-    <p> $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n+%
-\nicefrac{{1}}{{2}}}}\approx\;\frac{q_{n+1}(t)-q_{n}(t)}{2\zeta_{n}},$ </p>
-    <p> $\bm{\omega}^{k+1}$ </p>
-    <p> $\displaystyle\bm{q}_{t}(0,t)+\alpha\bm{q}(0,t)-\gamma\bm{q}_{x}(0,t)$ </p>
-    <p> $\displaystyle\bm{\tilde{\eta}}^{k+1}=\left(\mathbb{I}+\Delta t\;A\right)\bm{%
-\eta}^{k}+\Delta t\;\bm{\omega}^{k}$ </p>
-    <p> $\displaystyle q_{1}:$ </p>
-    <p> $q^{(3)}_{1}(t)$ </p>
-    <p> $\bm{y}(t_{0})={(0,0)}^{T}$ </p>
-    <p> $\displaystyle 2b_{2}$ </p>
-    <p> $R=7/9$ </p>
-    <p> $\bm{q}(t_{0})={(0,0)}^{T}$ </p>
-    <p> $\displaystyle\approx B_{2}M_{1}^{-1}B_{1}\bm{q}(t)+B_{2}M_{1}^{-1}\;K_{1}(\bm{%
-q}_{0}(t),\bm{y}(t),t).$ </p>
-    <p> $\displaystyle\frac{d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle x_{n}:=x(\xi_{n})=-c\ln(1-\xi_{n}),$ </p>
-    <p> $\rho_{p}\to 0$ </p>
-    <p> $\displaystyle\;a_{i+1,i-1}=\frac{1}{2\psi_{\frac{i-1}{2}}\zeta_{\frac{i-3}{2}}},$ </p>
-    <p> $R=4/3$ </p>
-    <p> $\displaystyle\frac{4d}{2}b_{2}$ </p>
-    <p> $\displaystyle q_{0}$ </p>
-    <p> $V_{1}(\dot{\bm{q}}_{0}(t))$ </p>
-    <p> $\displaystyle\;y^{(1)}_{0}+u^{(1)}t+2\frac{v_{0}^{(1)}}{\pi}\int_{0}^{\infty}%
-\frac{\gamma(1-e^{-k^{2}t})}{(\alpha-k^{2})^{2}+(k\gamma)^{2}}dk,$ </p>
-    <p> $\displaystyle\frac{d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$ </p>
-    <p> $\bm{q}(0,t)$ </p>
-    <p> $\displaystyle\frac{d^{5}}{120}(\partial^{5}_{\xi}q)_{0},$ </p>
-    <p> $\displaystyle=\bm{q}(0,t)+\bm{u}(\bm{y}(t),t),$ </p>
-    <p> $S=0.01$ </p>
-    <p> $\displaystyle\iff\xi(x)=1-e^{-\nicefrac{{x}}{{c}}}$ </p>
-    <p> $S\in\{0.01,0.1,0.5,1,2,4\}$ </p>
-    <p> $\displaystyle 3d(\partial_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle\approx\frac{3}{d}\left(q_{n+1}(t)-q_{n-1}(t)\right)$ </p>
-    <p> $\displaystyle(\partial_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle(\partial^{2}_{\xi}q)_{0}:\quad$ </p>
-    <p> $i=2m+1$ </p>
-    <p> $\displaystyle M_{1}:=\begin{bmatrix}c&0&\frac{Nc}{N-1}&0&0&0&0&\dots&0\\
-0&c&0&\frac{Nc}{N-1}&0&\ddots&\ddots&\ddots&0\\
-c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&0\\
-0&c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&0\\
-\vdots&0&\frac{Nc}{N-1}&0&\frac{4Nc}{N-2}&0&\frac{Nc}{N-3}&\ddots&0\\
-\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
-\vdots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}&0\\
-\vdots&\ddots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}\\
-0&\ddots&\ddots&\ddots&\ddots&\frac{Nc}{3}&0&\frac{4Nc}{2}&0\\
-0&0&\dots&\dots&\dots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}\end{bmatrix},$ </p>
-    <p> $u^{(1)}=0.05$ </p>
-    <p> $\bm{\omega}^{k}:=\bm{\omega}(\bm{q}_{0}(t^{k}),\bm{y}(t^{k}),t^{k})$ </p>
-    <p> $115\times 86$ </p>
-    <p> $\displaystyle\bm{q}(x,0)$ </p>
-    <p> $\displaystyle=A_{s}\bm{q}(t)+\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$ </p>
-    <p> $\displaystyle 2d(\partial_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle-\frac{2c}{3\gamma}\left(f(\bm{q}_{0}(t),\bm{y}(t),t)-\frac{%
-\partial q(\xi(x),t)}{\partial t}\biggr{\rvert}_{x_{0}}\right).$ </p>
-    <p> $\displaystyle\alpha\hat{b}_{0}$ </p>
-    <p> $\displaystyle\bm{u}=\left\lVert\bm{y}(t)\right\lVert\omega\;\bm{e}_{\theta}=%
-\omega\begin{bmatrix}-y^{(2)}\\
-y^{(1)}\end{bmatrix},$ </p>
-    <p> $\displaystyle=\bm{v}_{0}-\bm{u}_{0},$ </p>
-    <p> $\displaystyle B_{2}:=\frac{1}{d}\begin{bmatrix}-\nicefrac{{17}}{{6}}&0&%
-\nicefrac{{3}}{{2}}&0&\nicefrac{{3}}{{2}}&0&-\nicefrac{{1}}{{6}}&0&\dots&0\\
-0&-\nicefrac{{17}}{{6}}&0&\nicefrac{{3}}{{2}}&0&\nicefrac{{3}}{{2}}&0&-%
-\nicefrac{{1}}{{6}}&\dots&0\\
--3&0&0&0&3&0&0&0&\dots&0\\
-0&-3&0&0&0&3&0&\ddots&\ddots&0\\
-0&0&-3&0&0&0&3&0&\ddots&\vdots\\
-\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
-\vdots&\ddots&\ddots&\ddots&\ddots&0&-3&0&0&0\\
-0&\dots&\dots&\dots&\dots&\dots&0&-3&0&0\end{bmatrix},$ </p>
-    <p> $\bm{y}(t_{0})=(1,0)^{T}$ </p>
-    <p> $1056\text{\,}\mathrm{s}$ </p>
-    <p> $\displaystyle M_{2}D_{x}^{2}\bm{q}(t)\approx B_{2}\left(M_{1}^{-1}\;B_{1}\bm{q%
-}(t)+M_{1}^{-1}\;K_{1}(\bm{q}_{0}(t),\bm{y}(t),t)+M_{1}^{-1}\;V_{1}(\dot{\bm{q%
-}}_{0}(t))\right).$ </p>
-    <p> $R=7/3$ </p>
-    <p> $\displaystyle\frac{9d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$ </p>
-    <p> $\bm{f}\left(\bm{q}(0,t),\bm{y}(t),t\right):=\left(\frac{1}{R}-1\right)\frac{D%
-\bm{u}}{Dt}-\bm{q}(0,t)\cdot\nabla_{y}\bm{u}(\bm{y}(t),t)$ </p>
-    <p> $R<1$ </p>
-    <p> $\displaystyle\bm{u}(t)=\begin{bmatrix}u^{(1)}\\
-\sin(\lambda t)\end{bmatrix},$ </p>
-    <p> $\displaystyle\frac{d^{4}}{24}a_{1}$ </p>
-    <p> $\displaystyle\bm{y}(0)$ </p>
-    <p> $\displaystyle\frac{32d^{5}}{120}(\partial^{5}_{\xi}q)_{0},$ </p>
-    <p> $S=0.5$ </p>
-    <p> $\displaystyle\frac{1}{d}b_{0}$ </p>
-    <p> $\displaystyle\frac{\partial\xi(x)}{\partial x}\biggr{\rvert}_{x_{n}}$ </p>
-    <p> $\displaystyle(\partial^{3}_{\xi}q)_{0}:\quad$ </p>
-    <p> $\displaystyle d(\partial_{\xi}q)_{0}$ </p>
-    <p> $\frac{d^{4}}{60}(\partial^{5}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle=\bm{f}(\bm{q}(0,t),\bm{y}(t),t),$ </p>
-    <p> $\displaystyle R\dot{\bm{v}}(t)$ </p>
-    <p> $f^{(2)}(s)=\left(\frac{1}{R}-1\right)\left(\lambda\cos(\lambda s)\right)$ </p>
-    <p> $\displaystyle:=\Psi^{-1}B_{2}M_{1}^{-1}B_{1},$ </p>
-    <p> $\displaystyle\frac{27d^{2}}{6}b_{3}$ </p>
-    <p> $\displaystyle\bm{q}_{t}(x,t)$ </p>
-    <p> $\displaystyle a_{12}=$ </p>
-    <p> $\displaystyle\frac{27d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle y^{(2)}(t)=$ </p>
-    <p> $\bm{q}(0,t_{0})={(0.00052558,-0.00064947)}^{T}$ </p>
-    <p> $\displaystyle x(\xi)=-c\ln(1-\xi)$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10033.html DELETED Viewed

@@ -1,162 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $10\text{\,}\mathrm{ms}$ </p>
-    <p> $\displaystyle a_{0}(\partial_{\xi}q)_{0}+a_{1}(\partial_{\xi}q)_{1}=\hat{b}_{0%
-}\left(\alpha q_{0}-\frac{\gamma}{c}(\partial_{\xi}q)_{0}\right)+\frac{1}{d}%
-\left\{b_{0}q_{0}+b_{1}q_{1}+b_{2}q_{2}+b_{3}q_{3}\right\}.$ </p>
-    <p> $\displaystyle\frac{81d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle(\partial^{5}_{\xi}q)_{0}:\quad$ </p>
-    <p> $\displaystyle da_{1}$ </p>
-    <p> $\displaystyle q_{2}:$ </p>
-    <p> $\displaystyle\psi_{n}:=x_{n+\nicefrac{{1}}{{2}}}-x_{n-\nicefrac{{1}}{{2}}},%
-\quad\zeta_{n}:=x_{n+\nicefrac{{3}}{{4}}}-x_{n+\nicefrac{{1}}{{4}}}$ </p>
-    <p> $R=1/3$ </p>
-    <p> $\beta=2/3$ </p>
-    <p> $\displaystyle(\partial_{\xi}q)_{0}:$ </p>
-    <p> $\displaystyle\frac{9d}{2}b_{3}$ </p>
-    <p> $\displaystyle\frac{1}{d}b_{2}$ </p>
-    <p> $\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$ </p>
-    <p> $\displaystyle a_{i,i+2}=$ </p>
-    <p> $\displaystyle c\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}+%
-\left(\frac{3Nc}{N-1}\right)\frac{\partial q(\xi(x),t)}{\partial x}\biggr{%
-\rvert}_{x_{1}}\approx\frac{1}{d}\left(-\frac{17}{6}q_{0}(t)+\frac{3}{2}q_{1}(%
-t)+\frac{3}{2}q_{2}(t)-\frac{1}{6}q_{3}(t)\right).$ </p>
-    <p> $\displaystyle\approx B_{1}\bm{q}(t)+K_{1}(\bm{q}_{0}(t),\bm{y}(t),t)+V_{1}(%
-\dot{\bm{q}}_{0}(t)),$ </p>
-    <p> $R\neq 1$ </p>
-    <p> $D_{x}\bm{q}(t)$ </p>
-    <p> $\displaystyle\frac{\gamma}{c}\hat{b}_{0}$ </p>
-    <p> $\displaystyle\frac{d^{4}}{24}(\partial^{5}_{\xi}q)_{0},$ </p>
-    <p> $\displaystyle x>0,t\in(0,T],$ </p>
-    <p> $\displaystyle\;a_{i+1,i+3}=\frac{1}{2\psi_{\frac{i-1}{2}}\zeta_{\frac{i-1}{2}}},$ </p>
-    <p> $\displaystyle=\bm{q}_{xx}(x,t),$ </p>
-    <p> $\bm{q}(0,t_{0})=(0,0.1)$ </p>
-    <p> $n\in\{0,1,\dots,N-1\}$ </p>
-    <p> $\bm{q}_{x}(0,t)$ </p>
-    <p> $\displaystyle\;a_{22}=-\frac{\gamma+2\alpha\zeta_{0}}{\zeta_{0}(2+\gamma\psi_{%
-0})},$ </p>
-    <p> $\displaystyle=\frac{1}{c}e^{-\nicefrac{{x_{n}}}{{c}}}=\frac{1}{c}\left(1-\xi_{%
-n}\right)=\frac{1}{c}\left(1-\frac{n}{N}\right).$ </p>
-    <p> $\bm{y}={(0,0)}^{T}$ </p>
-    <p> $\displaystyle(\partial_{\xi}q)_{1}:$ </p>
-    <p> $\displaystyle+\frac{1}{d}\left(\frac{12\alpha dc-17\gamma}{18\gamma}\right)q_{%
-0}+\frac{1}{2d}q_{1}+\frac{1}{2d}q_{2}-\frac{1}{18d}q_{3}.$ </p>
-    <p> $\displaystyle\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n-1}}+%
-\frac{4\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}+\frac{\partial
-q%
-(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n+1}}\approx\frac{3}{d}\left(q_{n+1%
-}(t)-q_{n-1}(t)\right)$ </p>
-    <p> $\displaystyle\left(\frac{12\alpha dc-17\gamma}{18\gamma d}\right)q_{0}(t)+%
-\frac{1}{2d}q_{1}(t)+$ </p>
-    <p> $\displaystyle K_{1}:=\begin{bmatrix}-\frac{2c}{3\gamma}f^{(1)}(\bm{q}_{0}(t),%
-\bm{y}_{0}(t),t)\\
--\frac{2c}{3\gamma}f^{(2)}(\bm{q}_{0}(t),\bm{y}_{0}(t),t)\\
-0\\
-0\\
-\vdots\\
-0\end{bmatrix}\quad\text{and}\quad V_{1}:=\frac{2c}{3\gamma}\begin{bmatrix}%
-\frac{\partial q^{(1)}(\xi(x),t)}{\partial t}\biggr{\rvert}_{0}\\
-\frac{\partial q^{(2)}(\xi(x),t)}{\partial t}\biggr{\rvert}_{0}\\
-0\\
-0\\
-\vdots\\
-0\end{bmatrix}.$ </p>
-    <p> $q^{(2)}(x,t_{0})=0$ </p>
-    <p> $R=\nicefrac{{4}}{{3}}$ </p>
-    <p> $50\text{\,}\mathrm{Hz}$ </p>
-    <p> $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n}}=%
-\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}\cdot\frac{d\xi(%
-x)}{dx}\biggr{\rvert}_{x_{n}}.$ </p>
-    <p> $\displaystyle\hat{b}_{0}=-\frac{2c}{3\gamma}\quad b_{0}=\frac{12\alpha dc-17%
-\gamma}{18\gamma}\quad b_{1}=\frac{1}{2}\quad b_{2}=\frac{1}{2}\quad b_{3}=-%
-\frac{1}{18},$ </p>
-    <p> $\displaystyle a_{i,i}=$ </p>
-    <p> $\displaystyle\frac{d^{2}}{6}b_{1}$ </p>
-    <p> $(a_{i,j})_{1\leq i\leq 2N,\;1\leq j\leq 2N}$ </p>
-    <p> $\displaystyle M_{2}\;D_{x}^{2}\bm{q}(t)-B_{2}M_{1}^{-1}V_{1}(\dot{\bm{q}}_{0}(%
-t))$ </p>
-    <p> $\displaystyle\frac{8d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle=\bm{v}(t),$ </p>
-    <p> $\bm{\eta}^{k}:=\bm{\eta}(t^{k})$ </p>
-    <p> $\displaystyle(\partial_{\xi}q)_{0},$ </p>
-    <p> $2\text{\,}\mathrm{mm}$ </p>
-    <p> $t\in[0,10]$ </p>
-    <p> $\displaystyle y^{(1)}(t)=$ </p>
-    <p> $\displaystyle\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{0}}+3%
-\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{1}}\approx\frac{1}{%
-d}\left(-\frac{17}{6}q_{0}(t)+\frac{3}{2}q_{1}(t)+\frac{3}{2}q_{2}(t)-\frac{1}%
-{6}q_{3}(t)\right).$ </p>
-    <p> $\displaystyle q_{0},$ </p>
-    <p> $\displaystyle\frac{1}{d}b_{1}$ </p>
-    <p> $\displaystyle\frac{d^{3}}{6}(\partial^{4}_{\xi}q)_{0}$ </p>
-    <p> $\displaystyle-\frac{2c}{3\gamma}\left(f(q_{0},x_{0},t)-(\partial_{t}q)_{0}\right)$ </p>
-    <p> $\displaystyle c(\partial_{x}q)_{0}+\frac{Nc}{N-1}(\partial_{x}q)_{1}=$ </p>
-    <p> $(\bar{1}+\bar{3})(\bar{2}+\bar{3})(1+2+3)$ </p>
-    <p> ${{}^{2,1}}$ </p>
-    <p> ${{}^{5,1}}$ </p>
-    <p> ${{}^{3,1}}$ </p>
-    <p> $(i,~{}j)$ </p>
-    <p> $CS_{1}\diamond CS_{2}=\{u\cup v|u\in CS_{1},v\in CS_{2}\}$ </p>
-    <p> $(N,~{}\preccurlyeq)$ </p>
-    <p> ${}^{1,\text{\Letter}}$ </p>
-    <p> ${}^{\text{\Letter}}$ </p>
-    <p> $checking\_logic\_block$ </p>
-    <p> $\Phi(n)$ </p>
-    <p> $(\bar{1}+\bar{4})(\bar{2}+\bar{4})(1+2+4)$ </p>
-    <p> $\Phi(n)=\left\{\begin{aligned} \{\{n\}\}&:n\in PIs\\
-\{\{n\}\}\cup\Phi(n_{1})\diamond\Phi(n_{2})&:otherwise\end{aligned}\right\}.$ </p>
-    <p> $CS_{1}\diamond CS_{2}$ </p>
-    <p> $ge\in N$ </p>
-    <p> $\{|u\cup v|\leq k\}$ </p>
-    <p> $(\bar{1}+4+5)(\bar{1}+\bar{3}+5)(\bar{1}+\bar{2}+5)(2+3+\bar{4}+\bar{5})(1+%
-\bar{5})$ </p>
-    <p> $\mathbf{F_{q}}$ </p>
-    <p> $\mathbf{F_{q}}\in\mathbb{R}^{H/8\times W/8\times C}$ </p>
-    <p> $\mathbf{F_{s,s}}\in\mathbb{R}^{H/8\times W/8\times C}$ </p>
-    <p> $\mathbf{P^{\prime}_{r}}=(1+\alpha\mathbf{W}\otimes\mathbf{P_{r}})$ </p>
-    <p> $\mathbf{P}\in\mathbb{R}^{1\times 1\times C}$ </p>
-    <p> $\mathbf{P^{\prime\prime}}$ </p>
-    <p> $\mathbf{C_{train}}$ </p>
-    <p> $\mathbf{M_{q}}$ </p>
-    <p> $\mathbf{P_{r}}\in\mathbb{R}^{1\times 1\times C}$ </p>
-    <p> $\mathbf{F_{r}}$ </p>
-    <p> $\mathbf{M_{s}^{k}}$ </p>
-    <p> $\mathbf{I_{s}^{k}}$ </p>
-    <p> $\mathbf{M_{s}}$ </p>
-    <p> $\mathbf{Q_{i}}=\{(\mathbf{I_{q}},\mathbf{M_{q}})\}_{i}$ </p>
-    <p> $\mathbf{S_{i}}=\{(\mathbf{I_{s}^{k}},\mathbf{M_{s}^{k}}),k\in\{1,\dots,K\}\}_{i}$ </p>
-    <p> $\mathbf{\hat{y}}=\mathrm{softmax}(\mathrm{cosine}(\mathbf{P^{\prime\prime}},%
-\mathbf{F^{\prime}_{q}}))$ </p>
-    <p> $L=L_{seg}+\lambda_{1}L_{s}+\lambda_{2}L_{q}$ </p>
-    <p> $\mathbf{P^{\prime}}=(1+\alpha\mathbf{W}\otimes\mathbf{P})$ </p>
-    <p> $\mathbf{P^{\prime\prime}}\in\mathbb{R}^{1\times 1\times C}$ </p>
-    <p> $\mathbf{P_{r}}$ </p>
-    <p> $\mathbf{F^{\prime}_{r}}$ </p>
-    <p> $L_{s}=\mathrm{BCE}(\mathrm{cosine}(\mathbf{P^{\prime}},\mathbf{F_{s,s}}),%
-\mathbf{M_{s}})\\
-+\mathrm{BCE}(\mathrm{cosine}(\mathbf{P^{\prime}_{r}},\mathbf{F_{s,r}}),%
-\mathbf{M_{s}})$ </p>
-    <p> $\mathbf{I_{q}}$ </p>
-    <p> $\mathrm{MAP}$ </p>
-    <p> $L_{q}=\mathrm{BCE}(\mathrm{cosine}(\mathrm{MAP}(\mathbf{F^{\prime}_{q}}),%
-\mathbf{F^{\prime}_{q}}),\mathbf{M_{q}})$ </p>
-    <p> $\mathbf{F^{\prime}_{q}}$ </p>
-    <p> $\mathbf{F_{s,r}}\in\mathbb{R}^{H/8\times W/8\times C}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10034.html DELETED Viewed

@@ -1,142 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\mathbf{C_{test}}$ </p>
-    <p> $L_{seg}=\mathrm{BCE}(\mathbf{\hat{y}},\mathbf{M_{q}})$ </p>
-    <p> $\mathbf{X}\in\mathbb{R}^{1\times 1\times 2C}$ </p>
-    <p> $\mathbf{P^{\prime}}\in\mathbb{R}^{1\times 1\times C}$ </p>
-    <p> $\mathbf{W}=\mathrm{sigmoid}(f_{2}(\sigma(f_{1}(\mathbf{X}))))$ </p>
-    <p> $\mathbf{F_{r}}\in\mathbb{R}^{H/8\times W/8\times C}$ </p>
-    <p> $\mathbf{F^{\prime}_{q}}\in\mathbb{R}^{H/4\times W/4\times C}$ </p>
-    <p> $\mathbf{\hat{y}}\in\mathbb{R}^{H\times W\times 1}$ </p>
-    <p> $e_{i}=\{\mathbf{S_{i}},\mathbf{Q_{i}}\}$ </p>
-    <p> ${\sf msg^{*}}$ </p>
-    <p> ${\sf sk_{sanit}}$ </p>
-    <p> ${\sf AD}_{i}({\sf MODIFY})=1\}$ </p>
-    <p> $({\sf msg},\sigma)$ </p>
-    <p> ${\sf Fixed_{AD}(msg^{*})}={\sf Fixed_{AD}}({\sf msg}_{i})$ </p>
-    <p> $\{\mathcal{S}_{1},\mathcal{F},\mathcal{S}_{2}\}$ </p>
-    <p> ${\sf msg^{*},\sigma^{*},pk^{*}_{sig},pk_{sanit}}$ </p>
-    <p> $({\sf sk_{sign},pk_{sign}})$ </p>
-    <p> $\sigma_{2}^{{}^{\prime}}$ </p>
-    <p> $({\sf msg_{1}},{\sf MODIFY}_{1})$ </p>
-    <p> ${\sf sec_{k}}$ </p>
-    <p> $({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}^{{\sf
-Signature%
-(\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$ </p>
-    <p> ${\sf msg_{0}=\mathcal{H}(0||{\sf msg_{fix}}||{AD}||{\sf pk_{sanit}})}$ </p>
-    <p> $\displaystyle\mbox{or }{\sf msg^{*}\notin\{MODIFY(msg_{i})\;|\;MODIFY\mbox{ %
-with }}$ </p>
-    <p> $r_{(1)}\left(\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=\dots=r_{(k)}\left%
-(\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=0.$ </p>
-    <p> $({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})=(0,{\sf msg}^{*}_{{\sf fixed%
-},i},{\sf AD}^{*}_{i},{\sf pk}^{*}_{{\sf sanit},i})$ </p>
-    <p> $({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})$ </p>
-    <p> $\{\mathcal{Q},\mathcal{X},\mathcal{Y}\}$ </p>
-    <p> $\sigma^{*}=(\sigma_{1}^{*},\sigma_{2}^{*},{\sf AD}^{*})$ </p>
-    <p> $({\sf msg^{*}_{fixed}},{\sf AD^{*},pk_{sign},pk^{*}_{sanit},\sigma_{1}^{*}})$ </p>
-    <p> $({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit}^%
-{{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$ </p>
-    <p> ${\sf pk_{sanit}}$ </p>
-    <p> ${\sf x_{0}}=\mathcal{S}_{1}^{-1}({\sf y})\in\mathbb{F}_{q}^{m},{\sf x_{1}}=%
-\mathcal{F}_{1}^{-1}({\sf x_{0}})\in\mathbb{F}_{q}^{n}$ </p>
-    <p> $\mathcal{H}:\{0,1\}^{*}\rightarrow\mathbb{F}^{m}$ </p>
-    <p> $\alpha_{1}=\mathcal{S}^{-1}({\sf msg_{1}}),\beta_{1}=\mathcal{F}^{-1}(\alpha_{%
-1})$ </p>
-    <p> $\displaystyle\mbox{and }{\sf Judge({\sf msg^{*},\sigma^{*},pk_{sign},pk*_{%
-sanit}})=Sig}$ </p>
-    <p> $\forall i=1,2,\ldots,\Delta$ </p>
-    <p> $({\sf pk^{*}_{sign},{\sf msg^{*}},\sigma^{*}})$ </p>
-    <p> ${\sf msg}\in\{0,1\}^{*}$ </p>
-    <p> ${\sf msg_{1}=\mathcal{H}(1||msg||pk_{sanit}||pk_{sign})}$ </p>
-    <p> $(n=160,m=64)$ </p>
-    <p> $(b=0)$ </p>
-    <p> ${\sf msg^{*}_{fixed}}={\sf FIXED_{AD^{*}}(msg^{*})}$ </p>
-    <p> ${\sf Fixed_{AD}}$ </p>
-    <p> $\mathcal{X}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{m}$ </p>
-    <p> $\mathcal{S}_{2}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$ </p>
-    <p> ${\sf msg^{\prime}}$ </p>
-    <p> ${\sf msg_{2}=\mathcal{H}(1||msg^{\prime}||pk_{sanit}||pk_{sign})}$ </p>
-    <p> $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}=1]$ </p>
-    <p> $({\sf msg}_{i},{\sf AD}_{i},{\sf pk_{sign}},{\sf pk_{sanit}}_{i})$ </p>
-    <p> $j=q+1,\dots,r$ </p>
-    <p> $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Sanitizer\mbox{-}Acc{\mathcal{G}}^{%
-SSS}}=1]$ </p>
-    <p> ${\sf y}=\mathcal{P}({\sf x})$ </p>
-    <p> ${\sf(msg_{0},{\sf MODIFY}_{0}),(msg_{1},{\sf MODIFY}_{1})}$ </p>
-    <p> ${\sf msg^{\prime}}\in\{{\sf MODIFY(msg)}\;|\;{\sf MODIFY}\mbox{ with }{\sf AD(%
-MODIFY)}=1\}$ </p>
-    <p> $\displaystyle{\sf msg^{\prime}}\leftarrow{\sf MODIFY(msg)}$ </p>
-    <p> ${\sf(pk_{sanit},sk_{sanit})}\leftarrow{\sf KGen\mbox{-}Sanit(1^{\kappa})}$ </p>
-    <p> $a\leftarrow\mathcal{G}^{{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)},{\sf
-Sanitization%
-(\cdot,\cdot,sk_{sanit},\cdot)},{\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{%
-sanit},b)}}({\sf pk_{sign},pk_{sanit}})$ </p>
-    <p> ${\sf(msg^{\prime}_{j},\sigma^{\prime}_{j})\leftarrow Sanitization(msg_{j,b},%
-MODIFY_{j,b},\sigma_{j,b},pk_{sign},sk_{sanit})}$ </p>
-    <p> $({\sf msg_{j,0}},{\sf Modify_{j,0}}),({\sf msg_{j,1}},{\sf Modify_{j,0}},)$ </p>
-    <p> ${\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{sanit},b)}$ </p>
-    <p> ${\sf msg_{fixed}}\leftarrow{\sf FIXED}_{{\sf AD}_{i}}({\sf msg}_{i})$ </p>
-    <p> $(\cdot,{\sf sk_{sign}},\cdot,\cdot)$ </p>
-    <p> $\mathcal{R}=\left(r_{(1)}(\delta,\dots,\delta_{n}),\dots,r_{(k)}(\delta_{1},%
-\dots,\delta_{n})\right)$ </p>
-    <p> ${\sf msg,{\sf MODIFY},\sigma,pk_{sign},}\\
-{\sf sk_{sanit}}$ </p>
-    <p> $\mathcal{G}_{\sf sanit}$ </p>
-    <p> $\sigma_{j,b}\leftarrow{\sf Signature({\sf msg_{j,b},sk_{sign},pk_{sanit},AD_{j%
-}})}$ </p>
-    <p> ${\sf EXP\mbox{-}Unforgeability_{\mathcal{G}}^{SSS}}$ </p>
-    <p> $\mathcal{R}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg_{2}}$ </p>
-    <p> $(\bar{\delta}_{1},\dots,\bar{\delta}_{n})\in\mathbb{F}_{q}^{n}$ </p>
-    <p> $\mathcal{S}_{1}\circ\mathcal{F}\circ\mathcal{S}_{2}$ </p>
-    <p> $({\sf msg^{\prime}}_{i},{\sf\sigma^{\prime}}_{i},{\sf pk}_{{\sf sanit},i})$ </p>
-    <p> ${\sf msg_{FIX}=FIXED_{AD}(msg)}$ </p>
-    <p> ${\sf(pk_{sign},sk_{sign})}\leftarrow{\sf KGen\mbox{-}Sign(1^{\kappa})}$ </p>
-    <p> ${\sf pk^{*}_{sign}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sign},i},{\sf msg^{%
-\prime}}_{i})$ </p>
-    <p> ${\sf\sigma_{1}}$ </p>
-    <p> ${\sf Mul\mbox{-}SAN}$ </p>
-    <p> $GF(16)$ </p>
-    <p> $1,{\sf msg^{*},\sigma^{*},pk^{*}_{sanit},pk_{sign}}$ </p>
-    <p> ${\sf msg_{fix}}$ </p>
-    <p> ${\sf MODIFY}$ </p>
-    <p> $(b=1)$ </p>
-    <p> $\mathcal{Y}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$ </p>
-    <p> ${\sf msg,sk_{sign},pk_{sanit},AD}$ </p>
-    <p> $\delta(\kappa)$ </p>
-    <p> ${\sf msg}_{i}$ </p>
-    <p> $\mathcal{O}_{\sf LoR}$ </p>
-    <p> ${\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}$ </p>
-    <p> ${\sf pub_{k}}$ </p>
-    <p> ${\sf msg,\sigma,pk_{sign},pk_{sanit}}$ </p>
-    <p> ${\sf AD}_{j}$ </p>
-    <p> $\mathcal{R}=\mathcal{Q}\circ\mathcal{X}\circ\mathcal{Y}:\mathbb{F}_{q}^{n}%
-\rightarrow\mathbb{F}_{q}^{m}$ </p>
-    <p> ${\sf Sig}$ </p>
-    <p> $({\sf pk^{*}_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit%
-}({\sf pk_{sign}})$ </p>
-    <p> $i=1,\dots,\Delta$ </p>
-    <p> ${\sf AD(MODIFY)}\in\{0,1\}$ </p>
-    <p> ${\sf(pk_{sanit},sk_{sanit})}\leftarrow$ </p>
-    <p> ${\sf msg^{*}\notin\{MODIFY(msg)\;|\;MODIFY\mbox{ with }}$ </p>
-    <p> $\displaystyle{\sf Verification({\sf msg^{*},\sigma^{*},pk_{sign},pk^{*}_{sanit%
-}})=true}$ </p>
-    <p> ${\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}$ </p>
-    <p> ${\sf msg_{fix}=Fixed_{AD}({\sf msg})}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10035.html DELETED Viewed

@@ -1,140 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\frac{m(n+2)(n+1)}{2}$ </p>
-    <p> $\displaystyle{\sf AD}_{i}({\sf MODIFY})=1\}$ </p>
-    <p> ${\sf pub_{k},sec_{k}\leftarrow Kg(\kappa}$ </p>
-    <p> ${\sf AD(MODIFY)}=1$ </p>
-    <p> ${\sf msg^{*},\sigma^{*},pk^{*}_{sign},pk_{sanit}}$ </p>
-    <p> $({\sf 1,{msg}^{*},AD^{*},pk^{*}_{sanit},pk_{sign}})$ </p>
-    <p> $(\mathbb{F}_{q})$ </p>
-    <p> $r_{(i)}\in\mathbb{F}_{q}[\delta_{1},\dots,\delta_{n}]$ </p>
-    <p> $\sigma^{\prime}=(\sigma_{1},\sigma^{\prime}_{2},{\sf AD})$ </p>
-    <p> $({\sf msg^{*}},\sigma^{*})\leftarrow\mathcal{G}^{{\sf Signature(\cdot,sk_{sign%
-},\cdot,\cdot)},{\sf Sanitization(\cdot,\cdot,sk_{sanit},\cdot)}}({\sf pk_{%
-sign},pk_{sanit}})$ </p>
-    <p> $\in\{0,1\}^{*}$ </p>
-    <p> ${\sf EXP\mbox{-}Signer\mbox{-}Acc_{\mathcal{G}_{sign}}^{SSS}}$ </p>
-    <p> $\mathcal{P}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg}_{1}$ </p>
-    <p> ${\sf msg^{\prime}\leftarrow MODIFY(msg)}$ </p>
-    <p> $({\sf msg}_{j},{\sf MODIFY}_{j},\sigma_{j},{\sf pk}_{{\sf sign},i})$ </p>
-    <p> $\mathcal{S}_{1}:\mathbb{F}_{q}^{m}\rightarrow\mathbb{F}_{q}^{m}$ </p>
-    <p> $\displaystyle({\sf pk_{sanit}^{*}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i}%
-,{\sf msg^{\prime}}_{j})\forall i=1,2,\dots,q$ </p>
-    <p> ${\sf pk*_{sanit}\neq pk_{{sanit},i}}$ </p>
-    <p> $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}San\mbox{-}Acc_{\mathcal{G}_{sanit}}%
-^{SSS}}]$ </p>
-    <p> $\sigma^{*}_{2}$ </p>
-    <p> ${\sf EXP\mbox{-}San\mbox{-}Acc_{\mathcal{G}_{sanit}}^{SSS}}$ </p>
-    <p> $i=1,2,\ldots,q$ </p>
-    <p> $\mathcal{Q}:\mathbb{F}_{q}^{m}\rightarrow\mathbb{F}_{q}^{m}$ </p>
-    <p> $\displaystyle{\sf Verification({\sf msg^{*},\sigma^{*},pk_{sign},pk*_{sanit}})%
-=true}$ </p>
-    <p> ${\sf(msg_{0},}{\sf MODIFY}_{0})$ </p>
-    <p> ${\sf 0/1\leftarrow Ver(x,pub_{k})}$ </p>
-    <p> $y\in\mathbb{F}_{q}^{m}$ </p>
-    <p> ${\sf sk_{sign}}$ </p>
-    <p> $({\sf msg}_{i},{\sf pk}_{{\sf sanit},i},{\sf pk}_{{\sf sign},i})$ </p>
-    <p> $\displaystyle\mbox{and }\forall i=1,2,\dots q,\;\;{\sf pk^{*}_{sanit}\neq pk_{%
-{sanit},i}}$ </p>
-    <p> ${\sigma}_{1}$ </p>
-    <p> $i=1,2,\dots,q$ </p>
-    <p> ${\sf FIXED_{AD}}$ </p>
-    <p> ${\sf msg_{0}}$ </p>
-    <p> ${\sigma_{i,2}}$ </p>
-    <p> $({\sf pk*_{sign},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sign}^{{%
-\sf Sanitization(\cdot,\cdot,\cdot,sk_{sign})}}({\sf pk_{sanit}})$ </p>
-    <p> ${\sf msg^{*},\sigma^{*},pk_{sign},pk*_{sanit}}$ </p>
-    <p> ${\sf y}$ </p>
-    <p> ${\sf msg^{\prime}},\sigma^{\prime})\leftarrow$ </p>
-    <p> $\sigma^{\prime}_{2}=\mathcal{Y}^{-1}(\beta_{2})$ </p>
-    <p> $\sigma_{1}=\mathcal{T}^{-1}(\beta_{0})$ </p>
-    <p> $({\sf msg}_{j,0},{\sf MODIFY}_{j,0},{\sf AD}_{j})\equiv({\sf msg}_{j,1},{\sf
-MODIFY%
-}_{j,1},{\sf AD}_{j})$ </p>
-    <p> ${\sf pk_{sign}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sign},j},{\sf msg^{\prime}}_%
-{j})$ </p>
-    <p> ${\sf AD(MODIFY)}=\begin{cases}1&\mbox{ if modifications are valid \newline
-with respect }\\
-&\mbox{ to }{\sf AD};\\
-0&\text{ otherwise.}\end{cases}$ </p>
-    <p> $\sigma=(\sigma_{1},\sigma_{2},{\sf AD})$ </p>
-    <p> $(0,{\sf msg_{fix},AD,pk_{sanit}})$ </p>
-    <p> $\alpha_{0}=\mathcal{S}^{-1}({\sf msg_{0}}),\beta_{0}=\mathcal{F}^{-1}(\alpha_{%
-0})$ </p>
-    <p> $\mathcal{B}^{\prime}s$ </p>
-    <p> ${\sf EXP\mbox{-}Privacy_{\mathcal{G}}^{SSS}}$ </p>
-    <p> ${\sf pk_{sanit}^{*}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i},{\sf msg^{%
-\prime}}_{j})$ </p>
-    <p> ${\sf AD}_{i}={\sf AD^{*}}$ </p>
-    <p> ${\sf(pk_{sign},sk_{sign})}\leftarrow$ </p>
-    <p> ${\sf x\leftarrow Sig(y,sec_{k})}$ </p>
-    <p> $({\sf msg}_{i},{\sf AD}_{i},{\sf pk}_{{\sf sanit},i})$ </p>
-    <p> $\mathcal{P}{\sf(\sigma_{1})\stackrel{{\scriptstyle?}}{{=}}msg_{0}}$ </p>
-    <p> $\mathcal{G}_{\sf signer}$ </p>
-    <p> ${\sf AD}$ </p>
-    <p> ${\sf pk_{sign}}$ </p>
-    <p> $\{\mathcal{S},\mathcal{F},\mathcal{T}\}$ </p>
-    <p> ${\sf FIXED_{AD}(msg^{\prime})}\neq{\sf FIXED_{AD}(msg)}$ </p>
-    <p> $\sigma_{2}=\mathcal{T}^{-1}(\beta_{1})$ </p>
-    <p> $n^{2}+m^{2}+C$ </p>
-    <p> ${\sf msg_{1}}$ </p>
-    <p> $({\sf msg^{*},\sigma^{*},pk^{*}_{sanit}})$ </p>
-    <p> $\alpha_{2}=\mathcal{Q}^{-1}({\sf msg_{2}}),\beta_{2}=\mathcal{X}^{-1}(\alpha_{%
-2})$ </p>
-    <p> $({\sf msg^{\prime}_{j}},\sigma^{\prime}_{j})$ </p>
-    <p> ${\sf pk*_{sanit}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i},{\sf msg}_{i})$ </p>
-    <p> $i\in\{1,\ldots,\Delta\}$ </p>
-    <p> $({\sf pk^{*}_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}({\sf pk_{%
-sign}})$ </p>
-    <p> ${\sf MODIFY}({\sf msg}_{i})$ </p>
-    <p> ${\sf San}$ </p>
-    <p> $\mathcal{F}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{m}$ </p>
-    <p> $({\sf sk_{sanit},pk_{sanit}})$ </p>
-    <p> $(\sf x,y)$ </p>
-    <p> ${\sf x}$ </p>
-    <p> ${\sf x}=\mathcal{S}_{2}^{-1}({\sf x_{1}})\in\mathbb{F}_{q}^{n}$ </p>
-    <p> $\mathcal{P}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg_{1}}$ </p>
-    <p> ${\sf msg,{\sf MODIFY},\sigma,pk_{sign},sk_{sanit}}$ </p>
-    <p> $j\in\{\Delta+1,\ldots,r\}$ </p>
-    <p> $\sigma=(\sigma_{i,1},\sigma_{i,2},{\sf AD}_{i})$ </p>
-    <p> $\mathcal{P}=\mathcal{S}\circ\mathcal{F}\circ\mathcal{T}:\mathbb{F}_{q}^{n}%
-\rightarrow\mathbb{F}_{q}^{m}$ </p>
-    <p> $\hat{p}_{i}\in\mathbb{R}^{T}$ </p>
-    <p> $\mathcal{L}=\sum_{i}\ell_{\textrm{FL}}\left(\hat{p}_{i},g_{i}\right)$ </p>
-    <p> $\mathbf{A}\in\mathbb{B}^{L\times T}$ </p>
-    <p> $g_{i}\in\mathbb{B}^{T}$ </p>
-    <p> $\mathcal{C}\setminus c$ </p>
-    <p> $\{t^{\prime}_{j}\}$ </p>
-    <p> $\hat{\textbf{p}}$ </p>
-    <p> $\textbf{p}_{\textbf{cf}}$ </p>
-    <p> $F_{avg}=70.09.$ </p>
-    <p> $`\backslash w+^{\prime}$ </p>
-    <p> $F_{avg}=50.1$ </p>
-    <p> $PP({\tilde{p}},q)=b^{H({\tilde{p}},q)}=b^{\mathbb{E}_{\tilde{p}}[log_{b}q]}$ </p>
-    <p> $\begin{split}\arg\max_{i}\frac{P(y_{i}|x)}{P(y_{i}|x_{cf})}\approx\arg\max_{i}%
-\frac{P(y_{i}|x)}{P(y_{i}|x_{cf})}.\end{split}$ </p>
-    <p> $P(y_{i})\text{ or }p_{cf}$ </p>
-    <p> $(W\hat{p})_{i}=\frac{\hat{p}_{i}}{(p_{cf})_{i}}=\frac{P(y_{i}|x)}{P(y_{i}|x_{%
-cf})}$ </p>
-    <p> $F_{avg}$ </p>
-    <p> $F_{avg}=\frac{F_{\text{favor}}+F_{\text{against}}}{2}$ </p>
-    <p> $\text{argmax}(\textbf{q})$ </p>
-    <p> $\arg\max_{i}\prod_{j=1}^{i}\log P(y_{j}\mid x,y_{1,\cdots,j-1})$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10036.html DELETED Viewed

@@ -1,123 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $P(y_{i}|x_{cf})$ </p>
-    <p> $\frac{P(y_{i}|x)}{P(y_{i})}$ </p>
-    <p> $H({\tilde{p}},q)=-\sum_{i=1}^{n}\tilde{p}(x_{i})\log_{b}q(x_{i})$ </p>
-    <p> $F_{avg}=72.6$ </p>
-    <p> $F_{avg}=82.29$ </p>
-    <p> $F_{avg}=84.43$ </p>
-    <p> $\langle tweet\rangle$ </p>
-    <p> $F_{avg}=71.0$ </p>
-    <p> $F_{avg}=84.28$ </p>
-    <p> $F=\frac{2\cdot Precision\cdot Recall}{Precision+Recall}$ </p>
-    <p> $\text{PMI}(x,y_{i})$ </p>
-    <p> $\textbf{W}=\text{diag}(\textbf{p}_{\textbf{cf}})^{-1}$ </p>
-    <p> $77.11$ </p>
-    <p> $78.43$ </p>
-    <p> $\textbf{q}=softmax(\textbf{W}\hat{\textbf{p}}+\textbf{b})$ </p>
-    <p> $t+5\Delta$ </p>
-    <p> $\mathcal{WJ}$ </p>
-    <p> $B^{\prime}_{k}$ </p>
-    <p> $\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})$ </p>
-    <p> $\mathsf{status}$ </p>
-    <p> $v^{\prime}-1\geq v+1$ </p>
-    <p> $=\delta$ </p>
-    <p> $\max(t_{g},t)+2\Delta$ </p>
-    <p> $n-2f^{\prime}$ </p>
-    <p> $\mathcal{C}_{v-1}(B_{k-1})$ </p>
-    <p> $\mathcal{T}_{v-1}$ </p>
-    <p> $\langle\mathsf{fb\text{-}vote},H(B_{h}),v\rangle_{i}$ </p>
-    <p> $v^{*}>v$ </p>
-    <p> $\mathcal{C}_{v}^{\prime}(B_{k^{\prime}})$ </p>
-    <p> $v^{\prime}>v+1$ </p>
-    <p> $55.7\%-69.0\%$ </p>
-    <p> $\mathcal{C}_{v}(B_{k})$ </p>
-    <p> $f^{\prime}=33$ </p>
-    <p> $\mathcal{C}_{v^{\prime}-1}(B_{k^{\prime}-1})$ </p>
-    <p> $\langle\mathsf{status},v,{\sf lock}_{i}\rangle$ </p>
-    <p> $\langle{\sf opt\text{-}propose},B_{k+1},v+1\rangle$ </p>
-    <p> $\mathcal{C}_{v}=\mathcal{C}_{v}(B_{k})$ </p>
-    <p> $B_{k^{\prime}}=B_{k+1}$ </p>
-    <p> $t\geq t_{g}$ </p>
-    <p> $\mathcal{T}_{v^{\prime\prime}-1}$ </p>
-    <p> $\langle\mathsf{propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B_{h}),v%
-^{\prime}\rangle$ </p>
-    <p> $675\%$ </p>
-    <p> $\mathcal{C}^{f}_{v}(B_{l})$ </p>
-    <p> $\mathcal{TC}_{v}$ </p>
-    <p> $\mathcal{C}_{v^{\prime}-1}$ </p>
-    <p> $\Delta=500ms$ </p>
-    <p> $f^{\prime}=0$ </p>
-    <p> $\mathcal{C}_{v^{\prime\prime}-1}$ </p>
-    <p> ${\sf view\text{-}timer}$ </p>
-    <p> $v^{\prime}=v+1$ </p>
-    <p> $=3\Delta$ </p>
-    <p> $\langle\mathsf{commit},H(B_{k}),v\rangle_{i}$ </p>
-    <p> $\langle\mathsf{timeout},v-1,{\sf lock}_{i}\rangle_{i}$ </p>
-    <p> $\mathcal{C}^{n}_{v}(B_{l})$ </p>
-    <p> $\mathcal{C}^{f}_{v}(B_{h})$ </p>
-    <p> $5f-1$ </p>
-    <p> $448\%$ </p>
-    <p> $v^{*}\geq v+1$ </p>
-    <p> $v\geq v^{\prime}$ </p>
-    <p> $\langle{\sf fb\text{-}propose},B_{k},\mathcal{C}_{v^{\prime}}(B_{h}),\mathcal{%
-TC}_{v-1},v\rangle$ </p>
-    <p> $t<t_{g}$ </p>
-    <p> $\mathcal{C}_{v^{\prime\prime}}(B_{h})\geq{\sf lock}_{i}$ </p>
-    <p> $n-3f^{\prime}$ </p>
-    <p> $2f^{\prime}$ </p>
-    <p> ${\sf timeout\_view}_{i}<v$ </p>
-    <p> $\geq 2\delta$ </p>
-    <p> $\langle\mathsf{propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B_{k^{%
-\prime\prime}}),v^{\prime}\rangle$ </p>
-    <p> $174\%$ </p>
-    <p> $\mathsf{fb\text{-}vote}$ </p>
-    <p> $\langle\mathsf{propose},B_{h+1},\mathcal{C}_{v-1}(B_{h}),v\rangle$ </p>
-    <p> $=(f+1)\delta$ </p>
-    <p> $\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})\geq{\sf lock}_{i}$ </p>
-    <p> $=\Lambda+2\rho$ </p>
-    <p> ${\sf lock}_{i}>\mathcal{C}_{v-1}(B_{h})$ </p>
-    <p> $\mathcal{C}_{v+1}(B_{k^{\prime}})$ </p>
-    <p> $\mathcal{WM}$ </p>
-    <p> $\langle\mathsf{vote},H(B_{k}),v\rangle_{i}$ </p>
-    <p> $\langle\mathsf{opt\text{-}vote},H(B_{k}),v\rangle_{i}$ </p>
-    <p> $B^{\prime}_{k^{\prime}}$ </p>
-    <p> $P_{j}\neq P_{i}$ </p>
-    <p> ${\sf view\text{-}timer}_{i}$ </p>
-    <p> $v^{\prime}+1>v$ </p>
-    <p> $B_{l}=B_{k}$ </p>
-    <p> $t+\lambda$ </p>
-    <p> $,P_{i}$ </p>
-    <p> $C_{v}(B_{k})$ </p>
-    <p> ${\sf timeout\_view}_{i}\geq v$ </p>
-    <p> $\mathcal{C}_{v}(B_{h})$ </p>
-    <p> $\mathcal{C}_{v}(B_{l})$ </p>
-    <p> $=2\Lambda+\rho$ </p>
-    <p> $L_{v+1}$ </p>
-    <p> $\langle{\sf opt\text{-}propose},B_{k},v\rangle$ </p>
-    <p> $B_{k^{\prime\prime}}$ </p>
-    <p> $\mathcal{TC}_{v+1}$ </p>
-    <p> $43\%\text{--}54\%$ </p>
-    <p> $214\%-230\%$ </p>
-    <p> $(f+1)\delta$ </p>
-    <p> ${\sf lock}_{i}<\mathcal{C}_{v}(B_{k})$ </p>
-    <p> $\mathsf{opt\text{-}vote}$ </p>
-    <p> $\mathcal{C}_{v+1}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10037.html DELETED Viewed

@@ -1,125 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\mathcal{C}_{v^{\prime}}(B_{l})$ </p>
-    <p> $\lx@sectionsign\ref{sec:commit-moonshot}$ </p>
-    <p> $t_{g}+2\Delta<\max(t_{g},t)+3\Delta$ </p>
-    <p> $\mathcal{T}_{v^{*}}$ </p>
-    <p> $B_{k^{\prime}}$ </p>
-    <p> $7\delta^{*}$ </p>
-    <p> $702\%$ </p>
-    <p> $\langle\mathsf{propose},B_{h},\mathcal{C}_{v^{\prime}}(B_{h-1}),v\rangle$ </p>
-    <p> ${\sf lock}_{i}$ </p>
-    <p> $\mathsf{timeout}_{v}$ </p>
-    <p> $\mathcal{TC}_{v-1}$ </p>
-    <p> $\mathcal{T}_{v^{\prime}}$ </p>
-    <p> $\mathcal{C}_{v}\leq\mathcal{C}_{v^{\prime}}$ </p>
-    <p> $B_{k}\neq B^{\prime}_{k}$ </p>
-    <p> $\max(t_{g},t)+3\Delta$ </p>
-    <p> $\langle\mathsf{propose},B_{l},\mathcal{C}_{v^{\prime}}(B_{h}),v+1\rangle$ </p>
-    <p> $\mathcal{C}_{v+1}(B_{k+1})$ </p>
-    <p> $\mathcal{C}_{v^{\prime}}(B_{h})=\mathcal{C}_{v}(B_{k})$ </p>
-    <p> $\mathsf{vote}$ </p>
-    <p> $t+4\Delta$ </p>
-    <p> $\lceil\frac{n+f+1}{2}\rceil$ </p>
-    <p> $t_{g}+3\Delta=\max(t_{g},t)+3\Delta$ </p>
-    <p> $\langle\mathsf{commit},H(B_{k}),v\rangle_{*}$ </p>
-    <p> $\mathcal{C}_{v-1}(B_{h})$ </p>
-    <p> $\mathcal{T}_{v^{\prime}-1}$ </p>
-    <p> $\delta\leq\Delta$ </p>
-    <p> $\rho<\Lambda$ </p>
-    <p> $v^{\prime\prime}\geq v$ </p>
-    <p> $\mathcal{C}^{n}_{v+1}(B_{k^{\prime}})$ </p>
-    <p> $\mathcal{C}^{o}_{v}(B_{h})$ </p>
-    <p> $B_{k^{\prime}}=B_{k}$ </p>
-    <p> $\mathcal{C}_{v^{*}}(B_{k^{*}})$ </p>
-    <p> $p\leq 1.8$ </p>
-    <p> $\mathcal{T}_{v^{\prime\prime}}$ </p>
-    <p> $\langle{\sf fb\text{-}propose},B_{h},\mathcal{C}_{v^{\prime}}(B_{h-1}),%
-\mathcal{TC}_{v-1},v\rangle$ </p>
-    <p> $\mathcal{C}_{v^{\prime\prime}}$ </p>
-    <p> $\langle\mathsf{timeout},v,{\sf lock}_{i}\rangle_{i}$ </p>
-    <p> $3f^{\prime}$ </p>
-    <p> $v\leq v^{\prime\prime}<v^{\prime}$ </p>
-    <p> ${\sf timeout\_view}<v-1$ </p>
-    <p> $v^{\prime\prime}<v^{\prime}$ </p>
-    <p> $\langle\mathsf{timeout},v\rangle_{i}$ </p>
-    <p> $\lx@sectionsign\ref{sec:moonshot_v2}$ </p>
-    <p> ${\color[rgb]{0.0078125,0.38671875,0.46875}\definecolor[named]{pgfstrokecolor}{%
-rgb}{0.0078125,0.38671875,0.46875}\pgfsys@color@rgb@stroke{0.0078125}{0.386718%
-75}{0.46875}\pgfsys@color@rgb@fill{0.0078125}{0.38671875}{0.46875}\checkmark}$ </p>
-    <p> $\langle{\sf fb\text{-}propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B%
-_{k^{\prime\prime}}),\mathcal{TC}_{v^{\prime}-1},v^{\prime}\rangle$ </p>
-    <p> $\mathcal{C}_{v-1}(B_{h-1})$ </p>
-    <p> $\mathcal{C}_{v^{\prime}}(B_{h})$ </p>
-    <p> $C_{v-1}$ </p>
-    <p> $\lx@sectionsign\ref{sec:moonshot_v1}$ </p>
-    <p> $2f+1th$ </p>
-    <p> $\max(t_{g},t)+\Delta$ </p>
-    <p> $70.3\%-74.4\%$ </p>
-    <p> $\langle\mathsf{propose},B_{k},\mathcal{C}_{v-1}(B_{k-1}),v\rangle$ </p>
-    <p> $\mathcal{TC}_{v^{\prime\prime}-1}$ </p>
-    <p> $v\leq v^{*}<v^{\prime}$ </p>
-    <p> $\mathcal{C}_{v-1}$ </p>
-    <p> ${\sf lock}_{i}\leq\mathcal{C}_{v^{\prime}}(B_{h})$ </p>
-    <p> $\mathcal{C}^{o}_{v}(B_{k})$ </p>
-    <p> $\langle\mathsf{propose},B_{l},\mathcal{C}_{v^{\prime}}(B_{h}),v\rangle$ </p>
-    <p> $t_{g}+\Delta$ </p>
-    <p> $p\leq 9$ </p>
-    <p> $7\Delta$ </p>
-    <p> $=3\delta$ </p>
-    <p> $\geq 4\delta$ </p>
-    <p> $\mathcal{C}^{n}_{v}(B_{h})$ </p>
-    <p> $H(B_{k-1})$ </p>
-    <p> $v^{\prime\prime}>v$ </p>
-    <p> $\mathcal{C}_{v^{\prime}}(B_{h-1})$ </p>
-    <p> $\langle\mathsf{propose},B_{h+1},\mathcal{C}_{v^{\prime}}(B_{h}),v\rangle$ </p>
-    <p> $70.5\%-71.8\%$ </p>
-    <p> $B_{k^{\prime}-1}$ </p>
-    <p> $v^{\prime}+1$ </p>
-    <p> $51\%\text{--}53\%$ </p>
-    <p> $\mathsf{timeout}_{v^{*}}$ </p>
-    <p> $B_{k}=B_{h+1}$ </p>
-    <p> $B_{k}=B_{l}$ </p>
-    <p> $\mathcal{C}^{o}_{v+1}(B_{k^{\prime}})$ </p>
-    <p> $P_{i}\in\mathcal{V}$ </p>
-    <p> $B_{k}\neq B_{l}$ </p>
-    <p> $f^{\prime}\leq f=\lfloor\frac{n-1}{3}\rfloor$ </p>
-    <p> $v^{\prime\prime}+1$ </p>
-    <p> $\mathcal{V}=($ </p>
-    <p> $B_{k}:=(b_{v},H(B_{k-1}))$ </p>
-    <p> $=4\delta$ </p>
-    <p> $v^{\prime}-1$ </p>
-    <p> $5\Delta$ </p>
-    <p> $\mathcal{C}_{v+1}(B_{l})$ </p>
-    <p> $t+3\Delta$ </p>
-    <p> $\mathcal{TC}_{v^{\prime}-1}$ </p>
-    <p> $B_{k}=B^{\prime}_{k}$ </p>
-    <p> $\mathcal{C}_{v^{\prime\prime}}(B_{h})$ </p>
-    <p> $\mathcal{TC}_{v^{\prime}}$ </p>
-    <p> $\mathcal{C}_{v}(B_{h+1})$ </p>
-    <p> $\mathcal{C}^{n}_{v}(B_{k})$ </p>
-    <p> $\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})\geq\mathcal{C}_{v}(B_{k})$ </p>
-    <p> $B_{h+1}$ </p>
-    <p> $\mathcal{C}_{v^{\prime}}(B_{k^{\prime}})$ </p>
-    <p> $=2\delta$ </p>
-    <p> $MaxStepsBacktracking=NumberOfEmptyCells$ </p>
-    <p> $\textit{Success Rate}\boldsymbol{\propto}\textit{max steps}*\textit{$\frac{1}{%
-emptycells}$}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10038.html DELETED Viewed

@@ -1,148 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $s+r\approx o$ </p>
-    <p> $\mathbf{I}_{t}^{s,r}\in\mathbb{Z}^{|\mathcal{E}|\times|\mathcal{R}|\times|%
-\mathcal{E}|}$ </p>
-    <p> $\textbf{W}^{(l)}_{6}$ </p>
-    <p> $\textbf{t}_{i}^{\prime\prime}=\sigma(\sum_{r\in\mathcal{R}_{TG}}\frac{1}{%
-\mathcal{N}_{i}}\textbf{W}_{r}\textbf{t}_{j}^{\prime}+\textbf{W}_{11}\textbf{t%
-}_{i}^{\prime})\textnormal{,}$ </p>
-    <p> $y^{e}_{t}$ </p>
-    <p> $\textbf{W}_{r}\in\mathbb{R}^{d\times d}$ </p>
-    <p> $\textbf{W}_{9}\in\mathbb{R}^{d\times 32}$ </p>
-    <p> $\alpha_{o,s}$ </p>
-    <p> $\textbf{o}^{(l+1)}_{t}=\sigma(\sum_{(s,r,o)\in\mathcal{F}_{t}}\textbf{W}^{(l)}%
-_{1}(\psi(\textbf{s}^{(l)}_{t}||\textbf{r}_{t}))+\textbf{W}^{(l)}_{2}\textbf{o%
-}^{(l)}_{t})\textnormal{,}$ </p>
-    <p> $\textbf{UE}_{t-k:t-1}$ </p>
-    <p> $\textbf{W}_{11}\in\mathbb{R}^{d\times d}$ </p>
-    <p> $f_{q}\in\{G_{t-k:t-1}\}$ </p>
-    <p> $\textbf{R}_{t}=\mathrm{GRU}(\textbf{R}_{t-1},[pooling(\textbf{E}^{\mathcal{R}}%
-_{t-1})||\textbf{R}])\textnormal{,}$ </p>
-    <p> $(s,r,o,t)$ </p>
-    <p> $\mathcal{R}_{TG}$ </p>
-    <p> $\textbf{E}^{\mathcal{R}}$ </p>
-    <p> $t>t_{T}$ </p>
-    <p> $\alpha^{(l)}_{o,s}=\frac{exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}^{(l)}_{4}[%
-\textbf{s}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime\prime}]))}{%
-\sum_{s^{\prime}\in\mathcal{N}_{(o)}}exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}%
-^{(l)}_{4}[\textbf{s'}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime%
-\prime}]))}\textnormal{,}$ </p>
-    <p> $q=(s,r,?,t)$ </p>
-    <p> $p_{R}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},%
-\textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t})\textnormal{.}$ </p>
-    <p> $\textbf{R}_{t}$ </p>
-    <p> $\textbf{t}^{\prime}=\textbf{W}_{8}(\textbf{W}_{9}\textbf{t}||\sigma(\textbf{W}%
-_{10}\textbf{t}))\textnormal{,}$ </p>
-    <p> $\mathbf{W}^{(l)}_{1}$ </p>
-    <p> $\displaystyle=\beta\sum_{(s,r,t)\in\mathcal{Q}^{e}_{t}}y^{e}_{t}\log p(o|s,r,t%
-,G_{t-1})$ </p>
-    <p> $\textbf{W}_{10}\in\mathbb{R}^{d\times 32}$ </p>
-    <p> $\textbf{s},\textbf{o}\in\textbf{UE}_{t-k:t-1}$ </p>
-    <p> $\textbf{W}^{(l)}_{3}\in\mathbb{R}^{4d}$ </p>
-    <p> $p_{H}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},%
-\textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t}\odot\mathbf{I}_{t}^{s,r})%
-\textnormal{,}$ </p>
-    <p> $\mathcal{Q}^{r}_{t}$ </p>
-    <p> $\textbf{UE}_{t}$ </p>
-    <p> $\Theta\in\mathbb{R}^{|\mathcal{E}|\times d}$ </p>
-    <p> $\{\textbf{E}_{t-k+1},\textbf{E}_{t-k+2},...,\textbf{E}_{t}\}$ </p>
-    <p> $\mathcal{N}_{(o)}$ </p>
-    <p> $y^{r}_{t}$ </p>
-    <p> $G=\{\mathcal{E},\mathcal{R},\mathcal{F},\mathcal{T}\}$ </p>
-    <p> $TG=\{\mathcal{E}_{TG},\mathcal{R}_{TG}\}$ </p>
-    <p> $\textbf{W}_{8}\in\mathbb{R}^{d\times 2d}$ </p>
-    <p> $\mathcal{Q}^{e}_{t}$ </p>
-    <p> $\textbf{W}^{(l)}_{4}\in\mathbb{R}^{4d\times 4d}$ </p>
-    <p> $\mathsf{LMS}$ </p>
-    <p> $\textbf{W}_{7}\in\mathbb{R}^{1\times d}$ </p>
-    <p> $\textbf{UE}_{t-k:t-1}=\mathrm{MEAN}(\sum_{i=t-k+1}^{t}\textbf{E}_{i})%
-\textnormal{.}$ </p>
-    <p> $\displaystyle+(1-\beta)\sum_{(s,o,t)\in\mathcal{Q}^{r}_{t}}y^{r}_{t}\log p(r|s%
-,o,t,G_{t-1})\textnormal{,}$ </p>
-    <p> $\textbf{E}_{t}=\mathrm{GRU}(\textbf{E}_{t-1},\textbf{G}_{t-1})\textnormal{.}$ </p>
-    <p> $\textbf{W}^{(l)}_{5}$ </p>
-    <p> $\mathbf{I}_{t}^{s,r}$ </p>
-    <p> $p(o|s,r,t,G_{t-1})=\alpha p_{H}(o|s,r,t,G_{t-1})+(1-\alpha)p_{R}(o|s,r,t,G_{t-%
-1})\textnormal{.}$ </p>
-    <p> $\mathbf{W}^{(l)}_{2}$ </p>
-    <p> $\textbf{o}^{(l+1)}=\sigma(\sum_{(s,r,o)\in UG}\alpha^{(l)}_{o,s}\textbf{W}^{(l%
-)}_{5}\psi(\textbf{s}^{(l)}||\textbf{r})+\textbf{W}^{(l)}_{6}\textbf{o}^{(l)})%
-\textnormal{,}$ </p>
-    <p> $\textbf{GE}_{t}=\sigma(\textbf{W}_{7}\Theta_{e})\textbf{E}_{t}+(1-\sigma(%
-\textbf{W}_{7}\Theta_{e}))\textbf{UE}_{t}\textnormal{,}$ </p>
-    <p> $\mathcal{E}_{TG}$ </p>
-    <p> $G=\{G_{1},G_{2},...,G_{\mathcal{T}}\}$ </p>
-    <p> $p(o|s,r,t,G_{t-1})$ </p>
-    <p> $1-A_{h}$ </p>
-    <p> ${\mathcal{X}_{1}}\in\mathbb{R}^{{1024}\times{{3}}}$ </p>
-    <p> $P=\{{\mathcal{P}_{1}}\in\mathbb{R}^{{2}\times{{N}\times{C}}},{\mathcal{P}_{2}}%
-\in\mathbb{R}^{{2}\times{{N_{1}}}\times{C_{1}}}$ </p>
-    <p> ${\mathcal{X}_{3}}\in\mathbb{R}^{{128}\times{{3}}}$ </p>
-    <p> $\displaystyle+||(\Pi(\mathcal{J}(\mathcal{M}^{h}_{MANO}))-\Pi(\mathcal{J}(\hat%
-{\mathcal{M}}^{h}_{MANO})))||_{2}.$ </p>
-    <p> $X_{h}=\{X_{l}\in\mathbb{R}^{{N}\times{C}},X_{r}\in\mathbb{R}^{{N}\times{C}}\}$ </p>
-    <p> $i<num\_layers$ </p>
-    <p> $\mathcal{\hat{P}}=\mathcal{P}\odot\bm{\alpha}+\bm{\beta},(\bm{\alpha},\bm{%
-\beta})=\psi(\mathcal{F}).$ </p>
-    <p> $\mathcal{L}_{V}=\sum_{h\in\{L,R\}}||\mathcal{M}^{h}_{GCN}-\hat{\mathcal{M}}^{h%
-}_{GCN}||_{1}+||\mathcal{M}^{h}_{MANO}-\hat{\mathcal{M}}^{h}_{MANO}||_{1}.$ </p>
-    <p> ${\mathcal{F}_{3}}\in\mathbb{R}^{{\frac{H}{4}}\times{\frac{W}{4}}\times{256}}\}$ </p>
-    <p> ${G_{out}}=\sum_{k=0}^{K-1}C_{k}(\hat{L})G_{in}W_{k}.$ </p>
-    <p> $M=\{M_{l}\in\mathbb{R}^{{H}\times{{W}}},M_{r}\in\mathbb{R}^{{H}\times{{W}}}\}$ </p>
-    <p> $\hat{P_{i}},NumPoints\_{i},BallRadius\_{i}$ </p>
-    <p> $\mathcal{I}_{c}\in\mathbb{R}^{{H}\times{W}\times{3}}$ </p>
-    <p> $num\_layers$ </p>
-    <p> $\psi_{i}(\hat{F_{i}},P_{i})$ </p>
-    <p> $\displaystyle\mathcal{L}_{rep}$ </p>
-    <p> $cat(P_{i},PointNet(group(S_{i})))$ </p>
-    <p> $BallRadius$ </p>
-    <p> $(G\odot(\alpha+1)+\beta)$ </p>
-    <p> $NumPoints$ </p>
-    <p> $(P_{i}\odot(\alpha+1)+\beta)$ </p>
-    <p> $\hat{P_{i}}$ </p>
-    <p> $A_{h}\in[0,1]$ </p>
-    <p> $F=\{{\mathcal{F}_{1}}\in\mathbb{R}^{{H}\times{{W}\times{3}}},{\mathcal{F}_{2}}%
-\in\mathbb{R}^{{\frac{H}{2}}\times{\frac{W}{2}}\times{64}}$ </p>
-    <p> $P_{ct}=\{P_{l}\in\mathbb{R}^{2},P_{r}\in\mathbb{R}^{2}\}$ </p>
-    <p> $\psi_{i+1}(C,G)$ </p>
-    <p> $\mathcal{G}\in\mathbb{R}^{{2}\times{1024}\times{1}}$ </p>
-    <p> $\mathcal{I}_{d}\in\mathbb{R}^{{H}\times{W}\times{1}}$ </p>
-    <p> $Fetch(F_{i}|u,v)$ </p>
-    <p> $\mathcal{L}_{J}=\sum_{h\in\{L,R\}}||\mathcal{J}(\mathcal{M}^{h}_{MANO})-%
-\mathcal{J}(\hat{\mathcal{M}}^{h}_{MANO})||_{1}.$ </p>
-    <p> $K^{-1}X_{i}$ </p>
-    <p> $\mathcal{L}_{m}=||M-\hat{M}||_{1},$ </p>
-    <p> $[1,num\_layers]$ </p>
-    <p> $W_{k}\in\mathbb{R}^{{C_{in}}\times{C_{out}}}$ </p>
-    <p> $G_{in}\in\mathbb{R}^{{N}\times{C_{in}}}$ </p>
-    <p> $PointNet(\hat{P_{i}})$ </p>
-    <p> $\hat{L}\in\mathbb{R}^{{N}\times{N}}$ </p>
-    <p> $\mathcal{L}_{root}=\sum_{h\in\{L,R\}}||Root^{h}-\hat{Root^{h}}||_{1}.$ </p>
-    <p> $\mathcal{L}_{smooth}=\sum_{i=1}^{3}||e_{i}\cdot\hat{n}||_{1}+||e-\hat{e}||_{1},$ </p>
-    <p> ${\mathcal{P}_{3}}\in\mathbb{R}^{{2}\times{{N_{2}}}\times{C_{2}}}\}$ </p>
-    <p> $\mathcal{G_{V}}\in\mathbb{R}^{{N}\times{C}},(N=63,126,252),(C=512,256,128)$ </p>
-    <p> $\mathcal{L}_{c}=\sum_{h\in\{L,R\}}(1-A_{h})^{\gamma}\log(A_{h}),$ </p>
-    <p> ${\mathcal{X}_{2}}\in\mathbb{R}^{{512}\times{{3}}}$ </p>
-    <p> $G_{out}\in\mathbb{R}^{{N}\times{C_{out}}}$ </p>
-    <p> $\displaystyle=\sum_{h\in\{L,R\}}||(\Pi(\mathcal{M}^{h}_{MANO})-\Pi(\hat{%
-\mathcal{M}}^{h}_{MANO}))||_{2}$ </p>
-    <p> $F(t)=\frac{(1-t)(1-2t)-\sqrt{(1-t)(1-5t)}}{2t(2-t)}.$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10039.html DELETED Viewed

@@ -1,196 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\delta(R)$ </p>
-    <p> $F(t)=F(t,1,1,1,1)=t^{5}C(t)^{4}$ </p>
-    <p> $v_{n}\sim 4^{n+2}/\sqrt{\pi}n^{-3/2}$ </p>
-    <p> $W(t)=\frac{1}{1-P(t)/t}$ </p>
-    <p> $-(5t^{6}-16t^{5}+15t^{4}-28t^{3}+23t^{2}-8t+1)(t-1)^{2}F-(2t^{5}-5t^{4}+4t^{3}%
--10t^{2}+6t-1)(t-1)^{2}=0$ </p>
-    <p> $(2143,3\underline{41}2)$ </p>
-    <p> $F(t,x_{1},x_{2},x_{3},x_{4})$ </p>
-    <p> $F(t)=\frac{t(1-t)(1-7t+16t^{2}-11t^{3}+2t^{4})}{(1-4t+2t^{2})(1-3t+t^{2})^{2}}.$ </p>
-    <p> $\mathrm{mod}\ 4$ </p>
-    <p> $2\underline{14}3,3\underline{41}2$ </p>
-    <p> $P_{1}=$ </p>
-    <p> $(3,2,1,2)$ </p>
-    <p> $\left.{D=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tA}{1-t}}%
--1\right)A}\right\}$ </p>
-    <p> $e_{m}:=[t^{m}]\prod_{i=1}^{4}(1+tx_{i})$ </p>
-    <p> $F(t,x_{1},x_{2},x_{3},x_{4})=t^{5}\frac{1}{2\alpha}\left(\beta-\sqrt{\beta^{2}%
--4\alpha e_{4}^{2}}\right)$ </p>
-    <p> $(n+4)v_{n}-6(n+2)v_{n-1}+4(2n-1)v_{n-2}=0$ </p>
-    <p> $\left.\left(\frac{1}{(1-t)^{2}}-1\right)D,\ \ {D=\frac{t^{2}}{1-t}+\left(\frac%
-{1}{(1-t)^{2}}-1\right)A+\frac{S_{A}^{2}}{(1-t)^{2}}}\right\}$ </p>
-    <p> $\alpha F^{2}-\beta F+e_{2}^{4}$ </p>
-    <p> $\left.\frac{t^{2}}{1-t}+\frac{tS_{A}}{1-t}\right\}$ </p>
-    <p> $\left.{S_{D}=\frac{t^{2}}{1-t}+\frac{tS_{A}}{1-t}}\right\}$ </p>
-    <p> $V(t)=tC^{2}(t)(1+t^{2}C^{4}(t))$ </p>
-    <p> $(1,2,3,1)$ </p>
-    <p> ${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A}$ </p>
-    <p> $(45312,213)$ </p>
-    <p> $F(t)=\frac{1-3t-t^{2}+2t^{3}-\sqrt{1-6t+7t^{2}+2t^{3}+t^{4}}}{2t^{2}(2-t)}.$ </p>
-    <p> $\left\{S_{A}=\frac{t^{2}}{1-t}+\frac{tS_{D}}{1-t},\ \ S_{D}=\frac{(t+S_{A})^{2%
-}}{1-(t+S_{A})}\right\}$ </p>
-    <p> $O(\sqrt{n}\ln n)$ </p>
-    <p> $F(t)=\frac{1-3t+t^{2}-\sqrt{1-6t+7t^{2}-2t^{3}+t^{4}}}{2t}.$ </p>
-    <p> $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D},\right.$ </p>
-    <p> $F(t)=\frac{1-t-\sqrt{1-6t+t^{2}}}{2}$ </p>
-    <p> $\alpha:=\prod_{i=1}^{4}(1-x_{i}+tx_{i}^{2})$ </p>
-    <p> $P_{4}=\leavevmode\hbox to15.51pt{\vbox to15.51pt{\pgfpicture\makeatletter%
-\raise-3.91434pt\hbox{\hskip 0.5pt\lower-0.5pt\hbox to 0.0pt{%
-\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
-\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
-{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
-to%
- 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
-{}{{}}{}
-{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
-\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{7.25558pt}\pgfsys@lineto{14.51118pt}{%
-7.25558pt}\pgfsys@stroke\pgfsys@invoke{ }
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
-{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
-\pgfsys@invoke{ }{}\pgfsys@moveto{10.15784pt}{7.25558pt}\pgfsys@lineto{10.1578%
-4pt}{14.51118pt}\pgfsys@stroke\pgfsys@invoke{ }
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
-{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
-\pgfsys@invoke{ }{}\pgfsys@moveto{4.35333pt}{0.0pt}\pgfsys@lineto{4.35333pt}{7%
-.25558pt}\pgfsys@stroke\pgfsys@invoke{ }
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
-\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%
-\lxSVG@closescope\endpgfpicture}}$ </p>
-    <p> $P_{4}=$ </p>
-    <p> $P_{3}=$ </p>
-    <p> $2143,3\underline{41}2$ </p>
-    <p> $\left\{{S_{A}=\frac{t^{2}}{1-t}+\left(\frac{1}{1-\frac{tS_{D}}{1-t}}\frac{1}{(%
-1-t)^{2}}-1\right)S_{D}},\right.$ </p>
-    <p> $t^{8}(t-2)^{2}F^{4}-t^{3}(t^{2}-3t+2)(t^{5}-7t^{4}+4t^{3}-6t^{2}+5t-1)F^{3}-t(%
-t-1)(4t^{7}-22t^{6}+37t^{5}-42t^{4}+53t^{3}-35t^{2}+10t-1)F^{2}$ </p>
-    <p> $\left\{{A=\frac{(t+D)^{2}}{1-(t+D)}},\ \ {D=\frac{(t+A)^{2}}{1-(t+A)}}\right\}$ </p>
-    <p> $532642x_{1}=2x_{1}\times 11^{2}\times 31\times 71$ </p>
-    <p> $(2143,3412)$ </p>
-    <p> $\beta:=(2e_{4}t^{2}-4t(e_{4}-3e_{3}+2e_{2})+e_{4}-e_{3}+e_{2}-e_{1}+2)e_{4}$ </p>
-    <p> $\left\{{S_{A}=\frac{t^{2}}{1-t}+\frac{S_{D}}{1-t}},\ \ S_{D}=\frac{t^{2}}{1-t}%
-+\right.$ </p>
-    <p> $tF^{3}+2tF^{2}+(2t-1)F+t=0.$ </p>
-    <p> $\!,$ </p>
-    <p> $\left\{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tD}{1-t}}%
--1\right)D,\ \ D=\frac{(t+A)^{2}}{1-(t+A)}\right\}$ </p>
-    <p> $C(t)=\frac{1-\sqrt{1-4t}}{2t}$ </p>
-    <p> $2x_{1}x_{2}^{2}x_{3}x_{4}$ </p>
-    <p> $2413,3142)$ </p>
-    <p> $Z(t)=\frac{t(1-2t)}{1-4t+2t^{2}}$ </p>
-    <p> $2\underline{14}3,45312$ </p>
-    <p> $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D},\ \ {D=\frac{%
-(t+A)^{2}}{1-(t+A)}}\right\}$ </p>
-    <p> $(2413,3142)$ </p>
-    <p> $(2\underline{14}3,231)$ </p>
-    <p> $P_{3}=\leavevmode\hbox to15.51pt{\vbox to15.51pt{\pgfpicture\makeatletter%
-\raise-3.91434pt\hbox{\hskip 0.5pt\lower-0.5pt\hbox to 0.0pt{%
-\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
-\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
-{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
-to%
- 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
-{}{{}}{}
-{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
-\pgfsys@invoke{ }{}\pgfsys@moveto{7.25558pt}{0.0pt}\pgfsys@lineto{7.25558pt}{1%
-4.51118pt}\pgfsys@stroke\pgfsys@invoke{ }
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
-{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
-\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{4.35333pt}\pgfsys@lineto{7.25558pt}{4%
-.35333pt}\pgfsys@stroke\pgfsys@invoke{ }
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
-{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
-\pgfsys@invoke{ }{}\pgfsys@moveto{7.25558pt}{10.15784pt}\pgfsys@lineto{14.5111%
-8pt}{10.15784pt}\pgfsys@stroke\pgfsys@invoke{ }
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
-\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
-\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%
-\lxSVG@closescope\endpgfpicture}}$ </p>
-    <p> $\delta(\mathcal{R})$ </p>
-    <p> $(2143,45312)$ </p>
-    <p> $d(f(t),g(t))=2^{-\operatorname{val}(f(t)-g(t))}$ </p>
-    <p> $P(t)/t$ </p>
-    <p> ${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A+\frac{S_{D}^{2}}{(1-t%
-)^{2}}}$ </p>
-    <p> $F(t)=\frac{t(1-2t)}{1-4t+2t^{2}}.$ </p>
-    <p> $(2143,231)$ </p>
-    <p> $\!\}.$ </p>
-    <p> $[1..b]$ </p>
-    <p> $2143,3412$ </p>
-    <p> $F(t)=t+A(t)+D(t)$ </p>
-    <p> $21354,45312$ </p>
-    <p> $2143$ </p>
-    <p> $\operatorname{val}(f(t))=+\infty$ </p>
-    <p> $V(t)=W(t)Z(t)=\frac{(1-2t)\big{(}\,1-4t+2t^{2}+(1-2t)\sqrt{1-4t}\,\big{)}}{2t^%
-{3}}=tC^{2}(t)(1+t^{2}C^{4}(t)),$ </p>
-    <p> $\left\{{S_{A}=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)S_{D}},\ \ S%
-_{D}=\right.$ </p>
-    <p> $A(t)=D(t)$ </p>
-    <p> $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D+\frac{S_{D}^{2%
-}}{(1-t)^{2}}},\ \ {D=\frac{(t+A)^{2}}{1-(t+A)}}\right\}$ </p>
-    <p> $\{\!$ </p>
-    <p> $f(t)=\sum_{n\geq 0}f_{n}t^{n}$ </p>
-    <p> $(2\underline{14}3,3\underline{41}2)$ </p>
-    <p> $a_{i+1}>1$ </p>
-    <p> $P_{2}=$ </p>
-    <p> $(2\underline{14}3,45312)$ </p>
-    <p> $P(t)=t^{5}C^{4}(t)\left(\frac{2}{1-\left(\frac{1}{(1-t)^{2}}-1\right)}-1\right).$ </p>
-    <p> $2143,45312$ </p>
-    <p> $t^{4}(t-2)^{2}F^{4}+t(t-2)(4t^{3}-7t^{2}+6t-1)F^{3}+(2t^{4}-t^{3}-2t^{2}+5t-1)%
-F^{2}-(4t^{3}-7t^{2}+6t-1)F+t^{2}=0$ </p>
-    <p> $2\underline{14}3$ </p>
-    <p> $F(t)=\frac{1-t-\sqrt{1-6t+t^{2}}}{2}.$ </p>
-    <p> $\left.\left(\frac{1}{(1-t)^{2}}-1\right)S_{A}+\left(\frac{1}{1-t}\frac{t^{2}}{%
-1-2t}\right)^{2}\right\}$ </p>
-    <p> $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tD}{1-t}%
-}-1\right)D},\ \ {D=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A+%
-\frac{S_{A}^{2}}{(1-t)^{2}}}\right\}$ </p>
-    <p> $1,2,\ldots,b$ </p>
-    <p> $F(t,x_{1},11,31,71)$ </p>
-    <p> $(21354,45312)$ </p>
-    <p> ${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tA}{1-t}}-1%
-\right)A}$ </p>
-    <p> $f_{n}\neq 0$ </p>
-    <p> $\left\{A=\frac{t^{2}}{1-t}+\right.$ </p>
-    <p> $(1,b,1,1)$ </p>
-    <p> $A=\frac{(t+A)^{2}}{1-(t+A)}$ </p>
-    <p> $F(t)=\frac{t(1-16t+11t^{2}-434t^{3}+1045t^{4}-1590t^{5}+1508t^{6}-846t^{7}+252%
-t^{8}-30t^{9})}{(1-2t)^{4}(1-3t+t^{2})^{2}(1-4t+2t^{2})}.$ </p>
-    <p> $t^{5}C^{4}(t)$ </p>
-    <p> $\begin{array}[]{lcccl}\text{1.}\ (a,\ b,\ 1,\ 1)\ \longrightarrow\ (1,\ b+1,\ %
-1,\ 1),&&&&\text{3.}\ (1,\ b,\ c,\ d)\ \longrightarrow\ (1,\ b,\ [1..c],\ d+1)%
-,\\
-\text{2.}\ (1,\ b,\ c,\ 1)\ \longrightarrow\ (1,\ [1..b],\ c+1,\ 1),&&&&\text{%
-4.}\ (a,\ b,\ c,\ d)\ \longrightarrow\ (a+1,\ b,\ c,\ [1..d]).\end{array}$ </p>
-    <p> $\displaystyle\begin{split}&F(t,x_{1},x_{2},x_{3},x_{4})=t^{5}x_{1}x_{2}x_{3}x_%
-{4}+tx_{1}x_{2}x_{3}x_{4}[x_{3}x_{4}]F(t,1,x_{2},x_{3},x_{4})\\
-&+tx_{1}x_{3}x_{4}\frac{[x_{1}x_{4}]F(t,x_{1},x_{2},x_{3},x_{4})-x_{2}[x_{1}x_%
-{4}]F(t,x_{1},1,x_{3},x_{4})}{x_{2}-1}\\
-&+tx_{1}x_{4}\frac{[x_{1}]F(t,x_{1},x_{2},x_{3},x_{4})-x_{3}[x_{1}]F(t,x_{1},x%
-_{2},1,x_{4})}{x_{3}-1}\\
-&+tx_{1}\frac{F(t,x_{1},x_{2},x_{3},x_{4})-x_{4}F(t,x_{1},x_{2},x_{3},1)}{x_{4%
-}-1}.\end{split}$ </p>
-    <p> $21354$ </p>
-    <p> $(\mathbf{c},\sigma)=F_{\theta}^{c}(\mathbf{\gamma}(\mathbf{x}),\mathbf{\gamma}%
-(\mathbf{d}))$ </p>
-    <p> $\mathbf{f}_{z}\in\mathbb{R}^{4}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_1004.html DELETED Viewed

@@ -1,121 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $G\setminus\{a_{1},b_{1}\}$ </p>
-    <p> $y\in V(F)$ </p>
-    <p> $q_{1}=c$ </p>
-    <p> $v\in V(G)\setminus V(H)$ </p>
-    <p> $G[q_{j},\dots,,q_{t},b_{1},a_{1},a_{2}]$ </p>
-    <p> $N_{G}(q_{t})\cap V(H)=B$ </p>
-    <p> $G[S]=H[S]$ </p>
-    <p> $\omega<2.3728596$ </p>
-    <p> $x\in A$ </p>
-    <p> $G_{i}-v_{i}$ </p>
-    <p> $H\in\mathcal{C}$ </p>
-    <p> $\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{1}}\}=\{2K_{1}\vee 2K_{1}\}=%
-\{C_{4}\}$ </p>
-    <p> $N_{G}(v)\cap V(H)\subseteq B$ </p>
-    <p> $a_{2},a_{3},b_{1}$ </p>
-    <p> $Q=q_{0},\dots,q_{t}$ </p>
-    <p> $H\cong 2K_{1}$ </p>
-    <p> $\mathcal{G}_{k}$ </p>
-    <p> $V(G)\setminus V(H)$ </p>
-    <p> $E(H_{1})\cup E(H_{2})\cup\{v_{1}v_{2}\mid v_{1}\in V(H_{1}),v_{2}\in V(H_{2})\}$ </p>
-    <p> $G[S^{\prime}]\in\mathcal{C}$ </p>
-    <p> $G_{A}:=G[A\cup C]$ </p>
-    <p> $\bigcup_{v\in V(H)}X_{v}\subseteq V(2K_{1})$ </p>
-    <p> $\mathcal{O}(n^{2}(n+m))$ </p>
-    <p> $K_{2,3}$ </p>
-    <p> $p_{1},\dots,p_{i},q_{j},\dots,q_{t}$ </p>
-    <p> $K_{p,q}$ </p>
-    <p> $\mathcal{F}_{\mathcal{C}}$ </p>
-    <p> $S^{\prime}\subseteq S^{*}$ </p>
-    <p> $G\in\mathcal{C}_{k}$ </p>
-    <p> $\mathcal{O}(n^{4})$ </p>
-    <p> $r_{1},\dots,r_{i},c$ </p>
-    <p> $S=V(G)$ </p>
-    <p> $\mathcal{O}(n^{2k+2})$ </p>
-    <p> $S\subseteq A\cup C$ </p>
-    <p> $S^{\prime}\subseteq S\cap(A\cup C)$ </p>
-    <p> $S\subseteq V(G^{\prime})\setminus\{a,b,v^{xy}\}=V(G)\setminus\{a,b,x,y\}$ </p>
-    <p> $A=\{a_{1},a_{2},a_{3}\}$ </p>
-    <p> $G/e$ </p>
-    <p> $p_{s}=c$ </p>
-    <p> $S\subseteq V(G^{\prime})\setminus\{v^{xy}\}=V(G)\setminus\{x,y\}$ </p>
-    <p> $a_{i},a_{j},a_{k}\in A$ </p>
-    <p> $G[a_{1},a_{2},b_{1},b_{2},p_{0},\dots,p_{s}]$ </p>
-    <p> $x,y\in V(G^{\prime})$ </p>
-    <p> $S\subseteq V(G)\setminus\{a,b\}$ </p>
-    <p> $\sum_{x\in X}w(x)$ </p>
-    <p> $S_{v}=\{v,z^{v}\}$ </p>
-    <p> $\mathcal{O}(n^{2+\epsilon})$ </p>
-    <p> $p_{1},\ldots,p_{k}$ </p>
-    <p> $r_{1}=x$ </p>
-    <p> $K_{2,k+1}\notin\mathcal{G}_{k}$ </p>
-    <p> $p_{1},\dots,p_{i-1},p_{j+1},\dots,p_{s}$ </p>
-    <p> $P^{3}$ </p>
-    <p> $p_{1},\dots,p_{i},q_{j+1},\dots,q_{t}$ </p>
-    <p> $\chi(\overline{G})\leq k$ </p>
-    <p> $r_{i},c\in C$ </p>
-    <p> $G[S^{*}]\in\mathcal{C}$ </p>
-    <p> $F\in{\mathcal{C}}$ </p>
-    <p> $\overline{C_{6}}$ </p>
-    <p> $|V(H)|-1$ </p>
-    <p> $G\setminus S^{\prime}$ </p>
-    <p> $S^{*}\cap V(H)\subseteq S$ </p>
-    <p> $G^{\prime}[S]=G[S]$ </p>
-    <p> $\overline{K_{2}\cup C_{2k+1}}\cong 2K_{1}\vee\overline{C_{2k+1}}$ </p>
-    <p> $S\cap B=\emptyset$ </p>
-    <p> $h_{1},\dots,h_{i}$ </p>
-    <p> $y=b$ </p>
-    <p> $a_{2},q_{j},b_{1}$ </p>
-    <p> $K_{2,k}$ </p>
-    <p> $a,b\in V(G)\setminus S$ </p>
-    <p> $y\in B$ </p>
-    <p> $N_{G}(q_{t})\cap V(H)=\{b_{1}\}$ </p>
-    <p> $\mathcal{M}_{\mathcal{G}_{0}}=\{P_{3}\}$ </p>
-    <p> $\mathcal{O}(n^{3+\epsilon})$ </p>
-    <p> $K_{k-1}$ </p>
-    <p> $\mathcal{O}(n^{2+\epsilon}).$ </p>
-    <p> $p_{i}\in C$ </p>
-    <p> $r_{k}=y$ </p>
-    <p> $K_{2,k+1}$ </p>
-    <p> $V(C_{1})$ </p>
-    <p> $N_{G}[y]\setminus N_{G}[x]$ </p>
-    <p> $p_{s}\in N_{B}$ </p>
-    <p> ${\mathcal{C}}$ </p>
-    <p> $N_{G}[x]:=\{x\}\cup N_{G}(x)$ </p>
-    <p> $N_{B}=\emptyset$ </p>
-    <p> $\alpha(G^{*})=\alpha(G)+|E(G)|$ </p>
-    <p> $G[v,a_{i},a_{j},a_{k}]$ </p>
-    <p> $p_{1}\in A$ </p>
-    <p> $G[x_{1},\dots,x_{t}]$ </p>
-    <p> $G^{\prime}\in\mathcal{G}_{\mathcal{C}}$ </p>
-    <p> $K_{\ell}\notin\mathcal{C}$ </p>
-    <p> $V(H)\setminus U$ </p>
-    <p> $G\in\mathcal{C}$ </p>
-    <p> $uw\in E(H)$ </p>
-    <p> $a_{2}\in N_{G}(p_{s})\cap V(H)\subseteq\{a_{2},b_{2}\}$ </p>
-    <p> $G[A\cup B\cup\{q_{0},q_{1}\}]$ </p>
-    <p> $\{a_{i},b_{i}\}$ </p>
-    <p> $|N_{G}(v)\cap V(H)|\geq 2$ </p>
-    <p> $a,b\in V(G^{\prime})\setminus S$ </p>
-    <p> $X=\{x\}$ </p>
-    <p> $E(H_{1})\cup E(H_{2})$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10040.html DELETED Viewed

@@ -1,143 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $\tilde{z}^{i}_{t},{z}^{i}_{t}$ </p>
-    <p> $\mathcal{L}_{rec}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-\hat%
-{Z}^{i}(\mathbf{r})\right\|^{2}$ </p>
-    <p> $\hat{Z^{i}}(\mathbf{r})$ </p>
-    <p> $\mathcal{L}_{Mrec}$ </p>
-    <p> $\mathcal{L}_{ref}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-%
-\tilde{Z}^{i}(\mathbf{r})\right\|^{2}\hskip 2.84544pt,\textrm{where}\hskip 2.8%
-4544pt\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i})$ </p>
-    <p> $\hat{Z}^{i}(\mathbf{r})=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\gamma(t))%
-\mathbf{f}_{z}(\mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544%
-ptT(t)=\textrm{exp}\left(\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$ </p>
-    <p> $t\mathbf{d}$ </p>
-    <p> $\nabla\mathcal{L}_{MDDS}$ </p>
-    <p> $\nabla_{\theta}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta}\mathcal{L}_{\mathrm{%
-SDS}}(\mathbf{z},y_{src})-\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\hat{%
-\mathbf{z}},y_{trg}),$ </p>
-    <p> $\lambda_{im}$ </p>
-    <p> $\mathcal{L}_{rtot}=\lambda_{rec}\mathcal{L}_{rec}+\lambda_{ref}\mathcal{L}_{ref}$ </p>
-    <p> $\lambda_{ref}$ </p>
-    <p> $y_{src}$ </p>
-    <p> $\mathcal{L}_{MDDS}$ </p>
-    <p> $\tilde{{Z}}^{i}$ </p>
-    <p> ${F_{\phi}(\cdot)}$ </p>
-    <p> $\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta,\phi}\mathcal{L}%
-_{\mathrm{SDS}}(\mathbf{z}^{i},y_{src})-\nabla_{\theta,\phi}\mathcal{L}_{%
-\mathrm{SDS}}(\tilde{\mathbf{z}}^{i},y_{trg}).$ </p>
-    <p> $(\mathbf{f}_{z},\sigma)=F_{\theta}(\mathbf{\gamma}(\mathbf{x}),\mathbf{\gamma}%
-(\mathbf{d}))$ </p>
-    <p> $\hat{C}(r)=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\mathbf{r}(t))\mathbf{c}(%
-\mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544ptT(t)=\textrm{%
-exp}\left(-\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$ </p>
-    <p> $\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\mathbf{z},y_{trg},\epsilon,t)=%
-\omega(t)(\epsilon_{\psi}\left(\mathbf{z}_{\mathbf{t}},y_{trg},t\right)-%
-\epsilon)\frac{\partial\mathbf{z}_{\mathbf{t}}}{\partial\theta}$ </p>
-    <p> $504\times 378$ </p>
-    <p> $\lambda_{rec}$ </p>
-    <p> $\mathcal{L}_{\mathrm{Mrec}}=\lambda_{im}\cdot\mathcal{M}\cdot\mathcal{L}_{%
-\mathrm{rtot}}+\lambda_{om}\cdot(1-\mathcal{M})\cdot\mathcal{L}_{\mathrm{rtot}}.$ </p>
-    <p> $I=\{I^{i}\}_{i=1}^{N}$ </p>
-    <p> ${z}:=\{{z}^{i}\}_{i=1}^{N}$ </p>
-    <p> $\tilde{z}^{i}$ </p>
-    <p> ${z^{i}}=\mathcal{E}({{I^{i}}})\in\mathbb{R}^{64\times 64\times 4}$ </p>
-    <p> $y_{trg}$ </p>
-    <p> $\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{MDDS}}=\mathcal{M}\cdot(\nabla_{%
-\theta,\phi}\mathcal{L}_{\mathrm{DDS}}),$ </p>
-    <p> $[t_{near},t_{far}]$ </p>
-    <p> $\mathcal{L}_{\mathrm{tot}}=\mathcal{L}_{\mathrm{MDDS}}+\mathcal{L}_{\mathrm{%
-Mrec}}$ </p>
-    <p> $\lambda_{om}$ </p>
-    <p> $\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i}).$ </p>
-    <p> $\mathcal{L}_{rtot}$ </p>
-    <p> $\mathbf{r}(t)=\mathbf{o}$ </p>
-    <p> $[T\cdot G,Q]$ </p>
-    <p> $\displaystyle+R_{\text{upright}}+R_{\text{heading}}$ </p>
-    <p> $205\pm 8$ </p>
-    <p> $10P$ </p>
-    <p> $\displaystyle+R_{\text{effort}}+R_{\text{act}}+R_{\text{dof}}$ </p>
-    <p> $P=100,T=50,W=10$ </p>
-    <p> $12+3\mathbb{A}+2\mathbb{F}$ </p>
-    <p> $\mathbb{A},\mathbb{F}$ </p>
-    <p> $\displaystyle+R_{\text{death}}\times\mathbf{1}_{\{\text{head\_height}\leq\text%
-{termination\_height}\}}$ </p>
-    <p> $\displaystyle\ R_{\text{progress}}+R_{\text{alive}}\times\mathbf{1}_{\{\text{%
-head\_height}\geq\text{termination\_height}\}}$ </p>
-    <p> $G=\lfloor(Q-P)/T\rfloor$ </p>
-    <p> $(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\frac{1}{2}\Phi(p^{m-6i}))+1$ </p>
-    <p> $v\in\mathbb{Z}_{p^{m}}$ </p>
-    <p> $\tau^{\prime}\circ\tau_{i}(E)=\tau^{\prime}(E)=E^{\prime},\ \forall\ i\in\{1,2%
-,3,...,t\}$ </p>
-    <p> $j(E_{1})\neq 0,1728$ </p>
-    <p> $\begin{split}\quad y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\end{split}$ </p>
-    <p> $a^{\frac{p-1}{2}}\equiv 1\mod p)$ </p>
-    <p> $N_{R}(\mathbb{Z}_{p})=\Phi(p^{2})$ </p>
-    <p> $N_{R}(\mathcal{R})$ </p>
-    <p> $\ \ \frac{\Phi(n^{4})}{|Aut(E)|}$ </p>
-    <p> $32505856$ </p>
-    <p> $E_{2}/\mathbb{Z}_{p^{m}}:y^{2}=x^{3}+\bar{a}x+\bar{b}$ </p>
-    <p> $\displaystyle b_{6}$ </p>
-    <p> $\Delta^{v}(p^{m})$ </p>
-    <p> $2q-4+6+4=2q+6$ </p>
-    <p> $E_{1}/\mathbb{F}_{q}:y^{2}=x^{3}+ax+b$ </p>
-    <p> $|Aut(E)|=4$ </p>
-    <p> $\{u^{\prime},\alpha u^{\prime},\alpha^{2}u^{\prime},-u^{\prime},-\alpha u^{%
-\prime},-\alpha^{2}u^{\prime}\}$ </p>
-    <p> $\{6,19\}$ </p>
-    <p> $\begin{split}C_{R}(\mathbb{Z}_{n})=\sum_{\mathbb{E}_{k}}C^{(k)}_{R}(\mathbb{Z}%
-_{n})\end{split}$ </p>
-    <p> $34091302912$ </p>
-    <p> $15400$ </p>
-    <p> $(3^{-1})^{3}a^{3}\equiv-(2^{-1})^{2}b^{2}\ \mod p$ </p>
-    <p> $\mathbb{Z}_{2^{m}}$ </p>
-    <p> $k_{2}=4$ </p>
-    <p> $6\nmid char(\mathbb{Z}_{n})$ </p>
-    <p> $\displaystyle(E)=c_{4}^{3}/\Delta$ </p>
-    <p> $\displaystyle b_{8}$ </p>
-    <p> $3.5115653\times 10^{13}$ </p>
-    <p> $\frac{q-1}{|Aut(E)|}$ </p>
-    <p> $\{11,14\}$ </p>
-    <p> $2p^{m}-4$ </p>
-    <p> $28672$ </p>
-    <p> $P(1,1)$ </p>
-    <p> $N_{q}=2q+3+\left(\frac{-4}{q}\right)+2\left(\frac{-3}{q}\right)$ </p>
-    <p> $N_{G}(\mathbb{Z}_{n})=\Phi(n^{5})$ </p>
-    <p> $x=X/Z,y=Y/Z$ </p>
-    <p> $Aut(E^{\prime})$ </p>
-    <p> $y^{2}=x^{3}+3x$ </p>
-    <p> $a=0\;\text{and}\;b\neq 0$ </p>
-    <p> $|Aut(E)|=6$ </p>
-    <p> $\tau^{\prime}\neq\tau_{i}$ </p>
-    <p> $2p^{i-1}$ </p>
-    <p> $|Aut(E_{1})|=6$ </p>
-    <p> $q\equiv 1\mod k$ </p>
-    <p> $E/\mathbb{F}_{q}$ </p>
-    <p> $\{2,23\}$ </p>
-    <p> $|Aut(\mathbb{E}_{k})|\;\text{over}\;\mathbb{Z}_{n}=\prod\limits_{i=1}^{l}|Aut(%
-\mathbb{E}_{k_{i}})|\;\text{over}\;\mathbb{Z}_{p_{i}}$ </p>
-    <p> $\mathbb{F}_{2^{m}},1\leq m\leq 10$ </p>
-    <p> $F(X,Y,Z)=Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}-X^{3}-a_{2}X^{2}Z-a_{4}XZ^{2}-a_{6}Z^{3}=0$ </p>
-    <p> $N^{\prime\prime}_{R}(\mathbb{Z}_{n})$ </p>
-    <p> $y^{2}=x^{3}+4x+2$ </p>
-    <p> $\mathbb{Z}_{p^{m}}^{*}$ </p>
-    <p> $Aut(E)=\{\tau_{1},\tau_{2},...,\tau_{t}\}$ </p>
-    <p> $\mathbb{Z}_{7^{m}}$ </p>
-</body>
-</html>

htmls/output_mathjax_example_10041.html DELETED Viewed

@@ -1,139 +0,0 @@
-<!DOCTYPE html>
-<html>
-<head>
-    <title>MathJax Example</title>
-    <script>
-      MathJax = {
-        tex: {
-          inlineMath: [['$', '$'], ['\(', '\)']]
-        },
-        svg: {
-          fontCache: 'global'
-        }
-      };
-    </script>
-    <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
-</head>
-<body>
-        <p> $y^{2}=x^{3}+x$ </p>
-    <p> $p^{2(i-1)}$ </p>
-    <p> $2q-4$ </p>
-    <p> $\tau:(x,y)\rightarrow(u^{2}x,u^{3}y),\;u\in\;\mathbb{Z}_{n}^{*}$ </p>
-    <p> $2058$ </p>
-    <p> $y^{2}=x^{3}-x+9$ </p>
-    <p> $N_{R}(\mathbb{Z}_{n})$ </p>
-    <p> $|Aut(E_{k})|=2,4$ </p>
-    <p> $a=-3c^{2}$ </p>
-    <p> $\begin{split}s_{1}^{3}+s_{2}^{2}&\equiv 0\mod p\\
-\implies s_{1}^{3}&\equiv-s_{2}^{2}\mod p\end{split}$ </p>
-    <p> $\displaystyle=a_{1}^{2}a_{6}+4a_{2}a_{6}-a_{1}a_{3}a_{4}+a_{2}a_{3}^{2}-a_{4}^%
-{2}$ </p>
-    <p> $\displaystyle=a_{1}^{2}+4a_{2}$ </p>
-    <p> $\displaystyle\implies k_{1}$ </p>
-    <p> $\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),\;u\in\mathbb{K}^{*}$ </p>
-    <p> $\Delta=-16(4a^{3}+27b^{2})$ </p>
-    <p> $\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),u\in\;\mathbb{Z}_{n}^{*},r,s,%
-t\in\mathbb{Z}_{n}$ </p>
-    <p> $E(\mathbb{K})$ </p>
-    <p> $\displaystyle=2a_{4}+a_{1}a_{3}$ </p>
-    <p> $p^{m}=2j+1,\ i,j\in\mathbb{N}$ </p>
-    <p> $\gcd(p,6)=1$ </p>
-    <p> $\begin{split}E:Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}&=X^{3}+a_{2}X^{2}Z+a_{4}XZ^{2}\\
-&\ \ \ +a_{6}Z^{3}\end{split}$ </p>
-    <p> $|A^{3}_{0}|=p^{\lfloor\frac{2m}{3}\rfloor}$ </p>
-    <p> $3\mid\Phi(p^{m})$ </p>
-    <p> $\begin{split}E:y^{2}+a_{1}xy+a_{3}y=&x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\\
-&\text{where}\ a_{i}\in\;\mathbb{Z}_{n}\end{split}$ </p>
-    <p> $C_{G}(\mathbb{Z}_{n})$ </p>
-    <p> $\left(.\right)$ </p>
-    <p> $|A^{2}_{0}|=p^{\lfloor\frac{m}{2}\rfloor}$ </p>
-    <p> $E_{2}/\mathbb{F}_{q}:y^{2}=x^{3}+\bar{a}x+\bar{b}$ </p>
-    <p> $u^{4}\bar{a}=a$ </p>
-    <p> $q^{5}-q^{4}$ </p>
-    <p> $(\sum_{i=0}^{\left\lfloor\frac{m-1}{2}\right\rfloor}\frac{1}{2}\Phi(p^{m-2i}))%
-+1.$ </p>
-    <p> $Aut(E)$ </p>
-    <p> $1\leq i\leq\left\lfloor\frac{m-1}{2}\right\rfloor+1$ </p>
-    <p> $\gcd(p^{m},6)=1$ </p>
-    <p> $\bar{a}=0$ </p>
-    <p> $\displaystyle=c_{4}^{3}/\Delta$ </p>
-    <p> $\frac{\sum\limits_{i=0}^{\left\lfloor\frac{m-1}{3}\right\rfloor}\Phi(p^{m-3i})%
-}{3}+1.$ </p>
-    <p> $N_{R}(\mathbb{Z}_{p^{m}})=\Phi(p^{2m})$ </p>
-    <p> $\tau^{\prime}\circ\tau_{i}=\tau^{\prime}\circ\tau_{j}$ </p>
-    <p> $A^{k}_{u}(p^{m})$ </p>
-    <p> $|\#E(\mathbb{F}_{q})-q-1|\leq 2\sqrt{q}$ </p>
-    <p> $\mathbb{Z}_{p}^{m}$ </p>
-    <p> $y^{2}=x^{3}+1$ </p>
-    <p> $16807$ </p>
-    <p> $p^{m}\equiv 1\mod 4$ </p>
-    <p> $\frac{\partial F}{\partial X},\frac{\partial F}{\partial Y},\frac{\partial F}{%
-\partial Z}$ </p>
-    <p> $n^{3}\Phi(n)=\Phi(n^{4})$ </p>
-    <p> $2\leq l\leq 4$ </p>
-    <p> $\begin{split}C^{(k)}_{R}(\mathbb{Z}_{n})&=\frac{\prod\limits_{i=1}^{l}N^{(k_{i%
-})}_{R}(\mathbb{Z}_{p_{i}}).\prod\limits_{i=1}^{l}|Aut(\mathbb{E}_{k_{i}})|}{%
-\prod\limits_{i=1}^{l}\Phi(p_{i})}\\
-&=\prod\limits_{i=1}^{l}\frac{N^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}}).|Aut(\mathbb%
-{E}_{(k_{i})})|}{\Phi(p_{i})}\\
-&=\prod\limits_{i=1}^{l}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\end{split}$ </p>
-    <p> $\displaystyle=-1728\frac{4a^{3}}{\Delta}$ </p>
-    <p> $\frac{p-1}{6}$ </p>
-    <p> $C^{(k)}_{R}(\mathbb{Z}_{n})$ </p>
-    <p> $\mathbb{Z}_{p_{i}}(1\leq i\leq l)$ </p>
-    <p> $\sum_{E_{k}}\frac{\Phi(n)}{|Aut(E_{k})|}=\Phi(n^{2})$ </p>
-    <p> $y^{2}=x^{3}+3x+2$ </p>
-    <p> $\Phi(n^{5})$ </p>
-    <p> $\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),\ u\in\mathbb{Z}_{n}^{*}$ </p>
-    <p> $u^{\prime}\in\mathbb{F}_{q}^{*}$ </p>
-    <p> $A^{2}_{u}(25)$ </p>
-    <p> $\begin{split}s_{1}^{3}+s_{2}^{2}&\equiv 0\mod(p^{m})\\
-\implies s_{1}^{3}&\equiv-s_{2}^{2}\mod(p^{m})\end{split}$ </p>
-    <p> $p\equiv 3\mod 4$ </p>
-    <p> $N_{G}(\mathcal{R})$ </p>
-    <p> $|Aut(E_{1})|=4$ </p>
-    <p> $n=p_{1}p_{2}\ldots p_{k}$ </p>
-    <p> $\mathbb{F}_{(2g+1)^{n}}$ </p>
-    <p> $y^{2}=x^{3}+x+2$ </p>
-    <p> $\displaystyle=-16(4a^{3}+27b^{2})\in\mathbb{Z}_{n}^{*}$ </p>
-    <p> $Aut(E^{\prime})=t$ </p>
-    <p> $char(\mathbb{Z}_{n})\nmid 2,3$ </p>
-    <p> $\bar{b}=0$ </p>
-    <p> $\gcd(q,6)=1$ </p>
-    <p> $\displaystyle(\frac{\Phi(p^{m})}{6})k_{1}$ </p>
-    <p> $E(\mathbb{Z}_{n})\cong E(\mathbb{Z}_{p_{1}})\oplus E(\mathbb{Z}_{p_{2}})\oplus%
-\ldots\oplus E(\mathbb{Z}_{p_{l}})$ </p>
-    <p> $q\equiv 1\mod 12$ </p>
-    <p> $p=3j+1$ </p>
-    <p> $\displaystyle=b_{2}^{2}-24b_{4}$ </p>
-    <p> $\Phi(n),n,n,n$ </p>
-    <p> $p\equiv 2\mod 3$ </p>
-    <p> $N_{R}(\mathbb{Z}_{p^{m}})$ </p>
-    <p> $A^{k}_{0}(p^{m})=\{x\ |\ x^{k}\equiv 0\mod p^{m}\}$ </p>
-    <p> $|Aut(E)|=2$ </p>
-    <p> $q\equiv 1,5,7,11\mod 12$ </p>
-    <p> $E_{1}/\mathbb{Z}_{p^{m}}:y^{2}=x^{3}+ax+b$ </p>
-    <p> $p^{m}\equiv 1\mod 2$ </p>
-    <p> $\begin{split}C_{R}(\mathbb{Z}_{n})&=\sum_{\mathbb{E}_{k}}C^{(k)}_{R}(\mathbb{Z%
-}_{n})\\
-&=\sum_{\mathbb{E}_{k}}\prod\limits_{i=1}^{l}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}%
-})\end{split}$ </p>
-    <p> $(3^{-1}a,2^{-1}b)=(s_{1},s_{2})$ </p>
-    <p> $r,s,t\in\mathbb{K}$ </p>
-    <p> $\mathbb{Z}_{3125}$ </p>
-    <p> $u^{6}\bar{b}=b$ </p>
-    <p> $\tau^{\prime}(E)=E^{\prime}$ </p>
-    <p> $j(E_{1})=0$ </p>
-    <p> $4\mid\Phi(p^{m})$ </p>
-    <p> $\mathbb{Z}_{p^{m}}$ </p>
-    <p> $p^{m}=3j+1,\ i,j\in\mathbb{N}$ </p>
-    <p> $C^{(k)}_{R}(\mathbb{Z}_{n})=\frac{N^{(k)}_{R}(\mathbb{Z}_{n}).|Aut(\mathbb{E}_%
-{k})|}{\Phi(n)}$ </p>
-    <p> $E:y^{2}=x^{3}+ax+b$ </p>
-    <p> $\frac{q^{4}-q^{3}}{|Aut(E)|}$ </p>
-    <p> $2\mid\Phi(p^{m})$ </p>
-    <p> $\{3,22\}$ </p>
-    <p> $\frac{p-1}{2}\mid(p-1)$ </p>
-</body>
-</html>