diff --git a/htmls/output_mathjax_example_1.html b/htmls/output_mathjax_example_1.html deleted file mode 100644 index c274364b2071e45aaaea603073f66e46c6d4ecf4..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1.html +++ /dev/null @@ -1,124 +0,0 @@ - - - - MathJax Example - - - - -

$O(n^{2})$

-

$f$

-

$n$

-

$G(v)$

-

$s_{o}\oplus s_{a}\in\mathbb{V}^{n+m}$

-

$Z\in\mathbb{R}^{m\times d_{\text{token}}}$

-

$E_{\psi}(s)$

-

$\displaystyle=F^{i}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z)).$

-

$\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text})$

-

$\cos(\psi_{i},\psi_{j})$

-

${}^{4}$

-

$v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))$

-

${}^{*}$

-

$\displaystyle\text{argmax}_{Z}$

-

$\rightarrow$

-

$\mathcal{A}(x,t,s_{o})$

-

$\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus s_{a}))$

-

${}^{1}$

-

$\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})$

-

$x_{t}$

-

$500\times 20=10000$

-

$w_{i},w_{j}$

-

$v_{t}^{image}\leftarrow F^{i}(x_{t})$

-

$m=4$

-

$s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))$

-

${}^{5}$

-

$Z_{i}$

-

${}^{1,*}$

-

$\text{Proj}_{\psi}(Z)$

-

$s$

-

$\displaystyle\text{argmax}_{s_{a}}$

-

$t$

-

$s^{\prime}\leftarrow$

-

$v_{t}^{image}$

-

$5\times 4\times 100=2000$

-

${}^{1,2}$

-

$\psi\in\mathbb{R}^{|\mathbb{V}|\times d_{\text{token}}}$

-

$bestloss\leftarrow\mathcal{L},bestZ\leftarrow Z$

-

$G$

-

$\lambda=0$

-

$\text{Proj}_{\psi}:\mathbb{R}^{m\times d_{\text{token}}}\rightarrow\mathbb{R}^% -{m\times d_{\text{token}}}$

-

$i\leftarrow 1$

-

$s\in\mathbb{V}^{*}$

-

$\displaystyle\text{argmax}_{s_{a}}\mathbb{E}_{x\sim G(F^{t}(E_{\psi}(s_{o}% -\oplus s_{a})))}\mathcal{A}(x,t,s_{o})~{},$

-

$\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text}),$

-

$\cos(a,b)=\frac{a^{T}b}{\|a\|\|b\|}$

-

$\eta$

-

$512\times 512$

-

$x$

-

$E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})$

-

$N$

-

$bestloss>\mathcal{L}$

-

$v_{t}^{image}=F^{i}(x_{t})$

-

$d_{\text{emb}}$

-

$\displaystyle\text{argmax}_{s_{a}}\cos(F^{i}(E_{\psi}(s_{o}\oplus s_{a})),v_{t% -}).$

-

$s^{\prime}=$

-

${}^{3,*}$

-

$Z\leftarrow Z-\eta\nabla_{Z}\mathcal{L}$

-

$100$

-

$s_{a}$

-

$s_{o}\oplus s_{a}$

-

$m$

-

$v$

-

$\displaystyle\text{s.t.}\quad v=F^{i}(E_{\psi}(s_{o}\oplus s_{a})),$

-

$\mathbb{V}=\{w_{1},w_{2},\cdots,w_{|\mathbb{V}|}\}$

-

$F^{i}$

-

$\psi$

-

$\displaystyle\text{s.t.}\quad v$

-

$s_{o}$

-

$F^{t}$

-

${}^{2}$

-

$\oplus$

-

$E_{\psi}(s)_{i}=\psi_{j}$

-

$5\times 4=20$

-

$3\times 100$

-

${}^{3}$

-

$v\leftarrow F^{t}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z))$

-

$\mathcal{L}=-\cos(v,v_{t}^{image})-\lambda\cos(v,v_{t}^{text})$

-

$s_{o}\in\mathbb{V}^{n}$

-

$s_{a}\leftarrow E_{\psi}^{-1}(\text{Proj}_{\psi}(bestZ))$

-

$bestloss\leftarrow\infty,bestZ\leftarrow Z$

-

$\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))))$

-

$t\in\mathbb{V}$

-

$Z$

-

$(\cdot)$

-

$x\sim G(v)$

-

$d_{\text{token}}$

-

$s_{a}\in\mathbb{V}^{m}$

-

$v_{t}$

-

$\lambda$

-

$\mathbb{V}$

-

$w_{j}=s_{i}$

-

$t\in\mathcal{V}$

-

$x\sim G(F^{t}(E_{\psi}(s)))$

-

$E_{\psi}$

-

$j=\text{argmin}_{j^{\prime}}\|\psi_{j^{\prime}}-Z_{i}\|_{2}^{2}$

-

$|s|\times d_{\text{token}}$

-

$\displaystyle\text{argmax}_{v_{t}}\mathbb{E}_{x\sim G(v_{t})}\mathcal{A}(x,t,s% -_{o})~{}.$

-

$E_{L}\cup E_{R}$

-

$E_{L}=\{(u,w)|(u,w)\in E,w\neq v\}$

- - - diff --git a/htmls/output_mathjax_example_10.html b/htmls/output_mathjax_example_10.html deleted file mode 100644 index b6906530e1c93fd43e4562b311c75d277a044804..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10.html +++ /dev/null @@ -1,137 +0,0 @@ - - - - MathJax Example - - - - -

$0.01$

-

$\underset{\pm 0.10}{2.15}$

-

$(\operatorname{\bm{\theta}}_{\text{client}}^{(t)}=\operatorname{\bm{\theta}}_{% -\text{client}}^{(0)}$

-

$r=64$

-

$297.78$

-

$\mathbf{0.43}$

-

$\operatorname{\mathbf{v}}_{i}\in\mathcal{V}$

-

$0.76$

-

$\underset{\pm 0.66}{45.89}$

-

$0$

-

$\rho$

-

$0.26$

-

$0.95$

-

$p\approx 8.69\times 10^{-8}$

-

$0.69$

-

$47.32$

-

$\mathbf{0.69}$

-

$2.30$

-

$\mathbf{A}\in\mathbb{R}^{d\times r}$

-

$\operatorname{\bm{\theta}}_{\text{client}}^{(t)}$

-

$500$

-

$\operatorname{\mathbf{v}}_{i}$

-

$0.78$

-

$308$

-

$\mathbf{0.36}$

-

$-\frac{1}{|\mathcal{D}^{(t)}_{\bigtriangledown}|}\sum_{\operatorname{\mathbf{d% -}}^{(t)}\in\mathcal{D}^{(t)}_{\bigtriangledown}}\log p_{(\cdot)}\big{(}% -\operatorname{\mathbf{d}}^{(t)}\big{|}\operatorname{\bm{\theta}}_{(\cdot)}^{(t% -)}),$

-

$\operatorname{\mathbf{pr}}_{\text{client}}$

-

$p$

-

$0.66$

-

$2.65$

-

$\operatorname{\bm{\theta}}_{\text{client}}^{(0)}$

-

$25$

-

$277.25$

-

$\%$

-

$57.16$

-

$0.74$

-

$0.77$

-

$\tau=0.5$

-

$256$

-

$4.3$

-

$\operatorname{\bm{\theta}}_{\text{agent}}^{(0)}$

-

$0.90$

-

$0.80$

-

$4$

-

$0.8$

-

$9000$

-

$\operatorname{\bm{\theta}}_{\text{client}}$

-

$F_{1}$

-

$\mathbf{55.25}$

-

$0.60$

-

$\operatorname{\mathbf{v}}_{\text{next}}\in\text{Children}(\operatorname{% -\mathbf{v}}_{i})$

-

$\displaystyle\geq bx_{j}^{\prime}+\sum_{i>j}x_{i}^{\prime}+(b-1)\sum_{i>j}x_{i}$

-

$x_{i}^{\prime}\geq x_{i}$

-

$A_{1}$

-

$\displaystyle(b^{k}-b^{k-1}-(b-1)^{k}+b^{k-1})x_{1}$

-

$(u,w)$

-

$C_{1},C_{2}$

-

$P:(u,v)\cup P^{\prime}$

-

$e_{>i}$

-

$(u_{1},u_{2},\dots,u_{k},v)$

-

$\displaystyle=\frac{(b-1)^{k-1}}{b^{k}-(b-1)^{k}}x.$

-

$(v,z)\in N(u)\times N(u)$

-

$1\leq i\leq k$

-

$x_{\tau+1},\dots,x_{k}\mapsto\textsf{Chunk-Shortest-Edge}(k-\tau,x-\delta)$

-

$\sum_{i}x_{i} -

$\displaystyle=bx+c(v\to t)-c(u,w)-c(w\to t)=\delta+(b-1)x$

-

$p(e_{k}^{O})<\beta$

-

$i\geq\tau$

-

$\displaystyle=\begin{cases*}c(u,y)+cost[y,y,i]&if $(y,z)\in\mathcal{P}^{\prime% -}(u,y)$\\ -\infty&otherwise\end{cases*}$

-

$y^{*}\leq\frac{x}{k-1}$

-

$x-\delta$

-

$p(e;b_{i})$

-

$\displaystyle=\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}\left(\frac{b^{k}-(b-1)^{k-1}% -(b-1+1)}{b^{k}-(b-1)^{k}}\right)x+c(v\to t)$

-

$(s,s_{1})$

-

$i+j\leq k$

-

$\displaystyle=\left(\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}-1\right)x$

-

$\begin{array}[]{ll}p(e_{\tau})&=bx_{\tau}+c(u_{\tau+1}\to t)\\ -&=bx_{\tau}+\sum_{i=\tau+1}^{k}x_{i}+c(v\to t)\hfill\mbox{(shortest path from -$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}$

-

$(u_{3},z)$

-

$\displaystyle\frac{(b-1)(b^{k-1}-(b-1)^{k-1})+b^{k-1}}{b^{k-1}-(b-1)^{k-1}}x_{1}$

-

$\sum_{l>i}x_{l}\leq x$

-

$\displaystyle(b-1)x_{1}+x$

-

$c^{n}$

-

$\delta/k$

-

$p(e_{i})=bx_{i}+c(u,w)+c(w\to t)$

-

$\alpha=\beta$

-

$p(e_{i};b_{1})\leq\alpha_{u}^{(j)}$

-

$p(e_{i}^{C})=bx_{i}^{C}+c(u_{i}^{C}\to t)$

-

$x_{i}-x_{i}^{\prime}\geq 0$

-

$p(e_{i})=\beta^{\prime}$

-

$\displaystyle\min_{(v,z)\in\mathcal{P}(u,y)}\min(C_{1}(u,v,y),C_{2}(u,y,z),C_{% -3}(u,v,y,z)).$

-

$c(u,v)$

-

$\beta^{*}=p(e_{i})$

-

$p(e_{j};b_{1})\leq\alpha_{u}^{(1)}$

-

$\displaystyle=\frac{x}{1-\left(\frac{b-1}{b}\right)^{k-\tau+1}}+c(v\to t)-c(w% -\to t)-c(u,w).$

-

$x_{1}$

-

$O(|E|^{2}k^{3}\log k+|V|)$

-

$\beta\leftarrow\frac{x-\delta}{z_{\tau}}+c(v\to t)$

-

$min\_bottleneck\leftarrow\min(\alpha,\beta)$

-

$\delta/\tau$

-

$(y,z)$

- - - diff --git a/htmls/output_mathjax_example_100.html b/htmls/output_mathjax_example_100.html deleted file mode 100644 index df6c0a0f40e7d5804d11b49aee2f63d498b55af8..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_100.html +++ /dev/null @@ -1,157 +0,0 @@ - - - - MathJax Example - - - - -

$x_{t+1},\ldots x_{t+\ell}$

-

$O(|{\cal X}|^{2L})$

-

$\forall i=1,\ldots,L$

-

$1000$

-

${\cal M}_{b}$

-

$(X^{\tau},Z^{L-\tau})\sim{\cal M}_{b}^{L}.$

-

$(x^{i},y^{j})$

-

$x^{L}$

-

$\forall x\in{\cal X}$

-

$\forall x^{i_{0}}\in{\cal X}^{i_{0}}$

-

$X^{L}\sim{\cal M}_{s}^{L}$

-

$\mathbb{E}_{X^{L}\sim{\cal M}_{s}^{L}}[\tau]$

-

$(X^{L},Y^{L})$

-

$\ell=L$

-

$Z^{L-\tau}$

-

$x^{i}\in{\cal X}^{i}$

-

${\cal M}_{b}(\cdot|x^{n})=\text{Ber}(q),$

-

$\forall\ell\leq L$

-

$\tau,{\cal M}_{b,{\rm next}}$

-

$L-1$

-

$\beta(X^{L},Y^{L})=\tau$

-

$\displaystyle=\sum_{\ell=0}^{L}\ell\cdot\Pr{\left({\tau=\ell}\right)}=\sum_{% -\ell=1}^{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\prod_{i=1}^{\ell}\min\{{\cal M}_{% -b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\},$

-

$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{\ell\leq L}\sum_{x^{% -\ell}}\min\left\{\text{Pr}\left(X^{1:\ell}=x^{\ell}\right),\text{Pr}\left(Y^{1% -:\ell}=x^{\ell}\right)\right\}$

-

$\displaystyle=\Pr{\left({X^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr% -{\left({Y^{i_{0}}=x^{i_{0}},\tau\leq i_{0}}\right)}\cdot p_{\rm res}^{x^{i_{0}% -}}(x_{i_{0}+1})$

-

$Y_{\tau+1}\neq X_{\tau+1}$

-

$\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\prod_{i=1}^{\ell}\min\{{% -\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}({\cal M}_{s}(% -x_{\ell+1}\mid x^{\ell})-\min\{{\cal M}_{b}(x_{\ell+1}\mid x^{\ell}),{\cal M}_% -{s}(x_{\ell+1}\mid x^{\ell})\})$

-

${\cal M}_{s}(\cdot\mid X^{i-1}),{\cal M}_{b}(\cdot\mid X^{i-1}),\forall i=0,% -\ldots,L$

-

$L=8$

-

$\displaystyle\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}=\min\{\prod% -_{i=1}^{\ell}{\cal M}_{s}(x_{i}\mid x^{i-1}),{\cal M}_{b}(x_{i}\mid x^{i-1})\},$

-

${\cal M}^{*}(\cdot\mid x^{t})$

-

$L-\beta(X^{L},Y^{L})$

-

$X_{1},\ldots,X_{L}\sim{\cal M}_{s}^{L}(\cdot)$

-

$\forall y^{L-\tau}\in{\cal X}^{{L-\tau}}$

-

$\Pr{\left({\tau=L}\right)}=\sum_{x^{L}\in{\cal X}^{L}}\prod_{i=1}^{L}\min\{{% -\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}.$

-

$\eta_{i}\sim U(0,1)$

-

$\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})$

-

$Y\sim{\cal M}_{b}$

-

$L=4$

-

$\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\geq\ell_{0}}\right% -)}+\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau=\ell_{0}-1}\right)}$

-

$\displaystyle=\Pr{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr% -{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\leq i_{0}}\right)}$

-

$L=1$

-

$\ell=\ell_{0}-1$

-

$\displaystyle p_{\rm rej}(x^{i})$

-

$\displaystyle{:=}\sum_{x}{\left({{\cal M}_{b}(x^{i},x)-{\cal M}_{s}(x^{i},x)}% -\right)}_{+},$

-

$\displaystyle\;\;\;\Pr{\left({\tau=\ell}\right)}$

-

$\displaystyle=\sum_{\ell=1}^{L}\Pr{\left({\tau\geq\ell}\right)}=\sum_{\ell=1}^% -{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}% -\right)}$

-

$\displaystyle=\min\{\sum_{x\in{\cal X}}\min\{{\cal M}_{b}(x^{\ell_{0}-1},x),{% -\cal M}_{s}(x^{\ell_{0}-1},x)\}+p_{\rm remain}(x^{\ell_{0}-1}),$

-

$\ell>1$

-

$\forall\ell\leq L-1,x^{\ell}\in{\cal X}^{\ell}$

-

$x^{t}\in{\cal X}^{t}$

-

$\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}\right)}=\min\{{\cal M}_{b}(x^{\ell}% -),{\cal M}_{s}(x^{\ell})\}.$

-

$\displaystyle{:=}\sum_{x}{\left({{\cal M}_{s}(x^{i},x)-{\cal M}_{b}(x^{i},x)}% -\right)}_{+}.$

-

$Y^{L}=(X^{\tau},Z^{L-\tau})$

-

$\displaystyle\sum_{y^{L}}\pi(x^{L},y^{L})$

-

$(x)_{+}=\max\{x,0\}.$

-

${\cal M}_{b}(\cdot\mid X^{\tau},Y)$

-

$\displaystyle={\cal M}_{s}(x^{L})\cdot\min\left\{1,\frac{{\cal M}_{b}(x^{L})}{% -{\cal M}_{s}(x^{L})}\right\}$

-

${\cal M}^{L}_{s}(x^{L})$

-

$(x_{i},\ldots,x_{j})$

-

$0\leq\tau\leq L$

-

$\pi\in\Pi({\cal M}^{L}_{s},{\cal M}^{L}_{b})$

-

$\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\text{Pr}_{X^{L},Y^{L}}\left(X^{% -1:\ell}=Y^{1:\ell}=x^{\ell},\beta(X^{L},Y^{L})\geq\ell\right)$

-

$X^{L}$

-

$Y_{\tau+1}$

-

$x^{L}\in{\cal X}^{L}$

-

$\textsc{Verify}_{\pi}$

-

${\cal M}_{b,{\rm next}}^{L-\tau}$

-

$\tau\leq L$

-

$\min\{{\cal M}_{b}(x^{L}),{\cal M}_{s}(x^{L})\}$

-

$\displaystyle=\prod_{i=1}^{\ell}\min\{1,\frac{{\cal M}_{b}(x_{i}\mid x^{i-1})}% -{{\cal M}_{s}(x_{i}\mid x^{i-1})}\}{\left({1-\min\{1,\frac{{\cal M}_{b}(x_{% -\ell+1}\mid x^{\ell})}{{\cal M}_{s}(x_{\ell+1}\mid x^{\ell})}\}}\right)}.$

-

$x^{t}$

-

$X_{i}\sim{\cal M}_{s}(\cdot\mid X^{i-1}).$

-

$\tau=L-i$

-

${\rm E.O.S}\in X^{\tau}$

-

$\displaystyle=\sum_{x^{L}\in{\cal X}^{L}}\Pr{\left({\tau=\ell,X^{L}=x^{L}}% -\right)}$

-

$\pi_{\rm token}$

-

$\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\Pr{\left({\tau=\ell,X^{% -\ell+1}=x^{\ell+1}}\right)}$

-

$Z\sim{\cal M}_{b,{\rm next}}(\cdot)$

-

$X^{\tau}$

-

$(a)$

-

$\ell -

$\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\leq\ell_{0}-1}% -\right)}\cdot\Pr{\left({X^{\ell_{0}-1}\text{ is accepted.}}\right)}$

-

$Y_{1}=X_{1}$

-

$\displaystyle=\min\left\{{\cal M}_{b}(x^{\ell_{0}-1}),{\cal M}_{s}(x^{\ell_{0}% --1})\right\},$

-

$\displaystyle=\text{Pr}_{\pi}{\left({\beta{{\color[rgb]{0,0,0}=\ell}}\mid X^{L% -}}\right)}{:=}{{\color[rgb]{0,0,0}\frac{\sum_{\ell:\beta(X^{L},Y^{L})=\ell}\pi% -(X^{L},Y^{L})}{\pi(X^{L})}}},$

-

$Y_{2}$

-

$\displaystyle\max_{\pi\in\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})}\mathbb{E}_{X^% -{L},Y^{L}\sim\pi}\left[\beta(X^{L},Y^{L})\right]\leq\sum_{\tau=1}^{L}\sum_{x^{% -\tau}\in{\cal X}^{\tau}}\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}$

-

$\forall x^{n}$

-

$\eta_{0}\leq\min\left\{\frac{{\cal M}_{b}(X^{L})}{{\cal M}_{s}(X^{L})},1\right\}$

-

$O(|{\cal X}|^{L})$

-

$Z^{L-\tau}\sim{\cal M}_{b,{\rm next}}$

-

$L=6$

-

${\cal M}_{b}(Y_{3}\mid Y_{1},y)$

-

$Y^{L}=(X^{\tau},Z^{i-\tau})$

-

$\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\min\{{\cal M}_{s}^{\ell}(x^{% -\ell}),{\cal M}_{b}^{\ell}(x^{\ell})\}.$

-

$\displaystyle=\Pr{\left({X^{L}=x^{L}}\right)}\Pr{\left({\tau=L\mid X^{L}=x^{L}% -}\right)}$

-

$p_{\rm res}(x)=\frac{{\left({{\cal M}_{b}(x\mid X^{\tau})-{\cal M}_{s}(x\mid X% -^{\tau})}\right)}_{+}}{\sum_{x^{\prime}}{\left({{\cal M}_{b}(x^{\prime}\mid X^% -{\tau})-{\cal M}_{s}(x^{\prime}\mid X^{\tau})}\right)}_{+}},$

-

${\cal M}_{s}=\text{Ber}(1)$

-

$\frac{{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}{\sum_{x^{L% -}}{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}.$

-

$i=0,1,\ldots,L$

- - - diff --git a/htmls/output_mathjax_example_1000.html b/htmls/output_mathjax_example_1000.html deleted file mode 100644 index d26aedf9f0fc68f188b96833b253b35a6418687c..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1000.html +++ /dev/null @@ -1,132 +0,0 @@ - - - - MathJax Example - - - - -

$M(T)_{\mu\nu}$

-

$\gamma_{1}:1\succ 0$

-

$(V_{j},V_{i})$

-

$\Delta(N,N^{k,n}_{s})=2^{n-k-1}$

-

$|Pa(N_{s},V_{n})|=k$

-

$Pa(N_{s},V_{n})\cap Pa(N_{s^{\prime}},V_{n})=\emptyset$

-

$\binom{2c+1}{c+1}=\binom{2c+1}{c}=\frac{2c+1}{c+1}\binom{2c}{c}=\frac{4c+2}{c+% -1}\binom{2c-1}{c}$

-

$2^{k-k^{\prime}+1}-2$

-

$k^{\prime}=0$

-

$(2^{n}-2^{n-k})\sum_{k^{\prime}=0}^{k}\binom{k}{k^{\prime}}\binom{n-k-1}{k-k^{% -\prime}}\geq 3\cdot 2^{n-2}\binom{n-1}{k}\,.$

-

$V\in Pa(N_{s},V_{n})$

-

$N_{1},\ldots,N_{t}$

-

$N^{a}$

-

$(o,o^{\prime})$

-

$(T^{n-1,n})_{n\geq 3}$

-

$T^{n-1,n}$

-

$o^{\prime}$

-

$\binom{2d}{d}\leq 2^{2d-1}$

-

$\mathcal{V^{\prime}}\subseteq\mathcal{V}\setminus{V_{n}}$

-

$o\succ o^{\prime}$

-

$\sum_{1\leq s\leq t}|\operatorname{CPT}(N_{s},V_{n})|$

-

$2^{2k-k^{\prime}}$

-

$o^{\prime}\succ o$

-

$s\in\{1,\ldots,\binom{n-1}{k}2^{k}$

-

$n^{\prime}=\max_{1\leq s\leq t}|Pa(N_{s},V_{n})|$

-

$T_{\varepsilon}$

-

$N_{\varepsilon}$

-

$\binom{2d+2}{d+1}\leq 2^{2d+1}$

-

$o[V_{j}]=o^{\prime}[V_{j}]=\gamma[V_{j}]$

-

$CPT(N_{s},V_{n})$

-

$\operatorname{Inst}(P)$

-

$|\tau|$

-

$V_{4}$

-

$Pa(N_{1},V_{3})$

-

$000$

-

$\Leftarrow c\binom{2c-1}{c}\leq(2c+3)\cdot 2^{2c-4}$

-

$(c+1)\binom{2c+1}{c+1}\leq(c+1)\cdot 2^{2c-1}+\binom{2c}{c}$

-

$2^{k}-2^{k-k^{\prime}}$

-

$\gamma:0\succ 1$

-

$\Leftarrow c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}$

-

$4c\binom{2c-1}{c}+2\cdot\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-1}+2\cdot\binom{2c% --1}{c}$

-

$f_{T^{k,n}}(N^{k,n}_{s})=(2^{n}-2^{n-k})\sum_{k^{\prime}=0}^{k}\binom{k}{k^{% -\prime}}\binom{n-k-1}{k-k^{\prime}}$

-

$\mathcal{F}_{bad}=(T_{n})_{n\in\mathbb{N}}$

-

$o[\{V_{1},V_{2}\}]$

-

$2\cdot\binom{2c-1}{c}$

-

$Pa(N^{k,n}_{s},V_{n})\cup P$

-

$T_{n}=(N^{n}_{1},\ldots,N^{n}_{n-1})$

-

$s\in\{1,\ldots,t\}$

-

$N^{k,n}_{2}$

-

$\Leftarrow 2^{n-2c}\cdot c\binom{2c-1}{c}\leq(2c+2)\cdot 2^{n-4}$

-

$2^{|\mathcal{U}|}\prod_{s=1}^{t}2^{p_{s}-1}$

-

$2^{n-k-1}$

-

$\displaystyle\ \ +2^{k}-2^{k-k^{\prime}}\binom{k}{k^{\prime}}\binom{n-k-1}{k^{% -\prime}}2^{n-k}\big{]}$

-

$|\mathcal{V}^{\prime}|=k$

-

$f_{T}(N)$

-

$f_{T}(N^{a})>f_{T}(N)$

-

$p_{s}=|Pa(N_{s},V_{n})|$

-

$\operatorname{Inst}(\mathcal{V}^{\prime})$

-

$\gamma:1\succ 0$

-

$P\setminus Pa(N^{k,n}_{s},V_{n})$

-

$\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})=2^{n-k}-2^{n-2k}$

-

$\kappa\leq t-\kappa$

-

$2^{k-k^{\prime}}$

-

$\operatorname{CPT}(N^{a},V_{n})$

-

$O(\sum_{1\leq s\leq t}|CPT(N_{s},V_{n})|)$

-

$f_{T}(N^{a})\leq f_{T}(N)$

-

$\Delta(N,N_{s})$

-

$O(2^{|P|}\cdot\sum_{1\leq s\leq t}|CPT(N_{s},V_{n})|)$

-

$V=\{V_{1},V_{2},V_{3}\}$

-

$T^{2,3}$

-

$N_{i}\in T$

-

$\sum_{\kappa=0}^{\lfloor\frac{t}{2}\rfloor}\kappa\cdot\binom{t}{\kappa}$

-

$\mathcal{V}^{\prime}$

-

$freq_{M}(0\succ 1)=t\cdot 2^{n-1}-freq_{M}(1\succ 0)$

-

$Pa(N^{k,n}_{s},V_{n})\cap P$

-

$s\neq s^{\prime}$

-

$2^{n-t-1}$

-

$\frac{2\cdot(2d+1)}{d+1}$

-

$2^{n-k}$

-

$f_{T^{\prime}}(N)=\begin{cases}2\cdot\sum_{\kappa=0}^{c-1}\kappa\binom{2c-1}{% -\kappa}&\text{if }t=2c-1\\ -2\cdot\sum_{\kappa=0}^{c-1}\kappa\binom{2c}{\kappa}+c\binom{2c}{c}&\text{if }t% -=2c\\ -\end{cases}$

-

$N_{4}$

-

$\Leftarrow c\binom{2c}{c}\leq(2c+3)\cdot 2^{2c-3}$

-

$freq_{M}(1\succ 0)=\sum\nolimits_{0\leq\mu<2^{n-1}}\sum\nolimits_{1\leq\nu\leq -t% -}M_{\mu\nu}$

-

$\mathcal{F}_{bad}$

-

$f_{T}(N_{s})=\min\{f_{T}(N_{s^{\prime}})\mid 1\leq s^{\prime}\leq t\}\leq\frac% -{4}{3}f_{T}(N)$

-

$N^{k,n}_{s}$

-

$2^{k}\binom{n-k-1}{k}$

-

$2^{n-2}$

-

$\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})$

-

$freq_{M}(1\succ 0)$

-

$o[V_{i}]=0$

-

$f_{T^{k,n}}(N)=2^{n-1}\binom{n-1}{k}$

-

$f_{T^{k,n}}(N^{k,n}_{s})$

-

$\{V_{1},V_{2}\}$

-

$(T^{k,n})_{k,n}$

-

$1\leq s\leq t=2c$

-

$\Leftarrow 2^{n-2c-1}\cdot c\binom{2c}{c}\leq(2c+3)\cdot 2^{n-4}$

-

$1\leq k^{\prime}\leq k-1$

-

$P:=\bigcup_{1\leq s\leq t}Pa(N_{s},V_{n})$

-

$t=n-1$

- - - diff --git a/htmls/output_mathjax_example_10000.html b/htmls/output_mathjax_example_10000.html deleted file mode 100644 index 16930b2bf7184f41830644cac59bcd511ed1997a..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10000.html +++ /dev/null @@ -1,123 +0,0 @@ - - - - MathJax Example - - - - -

$0.058\scriptscriptstyle\pm\scriptstyle.010$

-

$0.027\scriptscriptstyle\pm\scriptstyle.004$

-

$\mu_{c}^{\mathcal{D}_{\text{test}}}$

-

$0.146\scriptscriptstyle\pm\scriptstyle.002$

-

$0.124\scriptscriptstyle\pm\scriptstyle.006$

-

$0.038\scriptscriptstyle\pm\scriptstyle.003$

-

$0.243\scriptscriptstyle\pm\scriptstyle.002$

-

$0.338\scriptscriptstyle\pm\scriptstyle.005$

-

$\mathbf{0.043\scriptscriptstyle\pm\scriptstyle.000}$

-

$\mathbf{0.010\scriptscriptstyle\pm\scriptstyle.002}$

-

$1769$

-

$\mathbf{0.077\scriptscriptstyle\pm\scriptstyle.005}$

-

$0.045\scriptscriptstyle\pm\scriptstyle.004$

-

$0.152\scriptscriptstyle\pm\scriptstyle.001$

-

$\mathbf{0.042\scriptscriptstyle\pm\scriptstyle.003}$

-

$0.025\scriptscriptstyle\pm\scriptstyle.000$

-

$0.081\scriptscriptstyle\pm\scriptstyle.007$

-

$0.066\scriptscriptstyle\pm\scriptstyle.005$

-

$0.013\scriptscriptstyle\pm\scriptstyle.006$

-

$0.781\scriptscriptstyle\pm\scriptstyle.003$

-

$0.021\scriptscriptstyle\pm\scriptstyle.003$

-

$0.012\scriptscriptstyle\pm\scriptstyle.000$

-

$0.135\scriptscriptstyle\pm\scriptstyle.014$

-

$0.228\scriptscriptstyle\pm\scriptstyle.007$

-

$0.069\scriptscriptstyle\pm\scriptstyle.035$

-

$0.228\scriptscriptstyle\pm\scriptstyle.001$

-

$0.065\scriptscriptstyle\pm\scriptstyle.012$

-

$0.078\scriptscriptstyle\pm\scriptstyle.007$

-

$\mathcal{D}_{\text{Test}}$

-

$0.153\scriptscriptstyle\pm\scriptstyle.002$

-

$0.077\scriptscriptstyle\pm\scriptstyle.010$

-

$0.332\scriptscriptstyle\pm\scriptstyle.001$

-

$0.360\scriptscriptstyle\pm\scriptstyle.004$

-

$0.083\scriptscriptstyle\pm\scriptstyle.002$

-

$\mathbf{0.033\scriptscriptstyle\pm\scriptstyle.003}$

-

$0.081\scriptscriptstyle\pm\scriptstyle.001$

-

$0.045\scriptscriptstyle\pm\scriptstyle.001$

-

$0.046\scriptscriptstyle\pm\scriptstyle.002$

-

$0.018\scriptscriptstyle\pm\scriptstyle.007$

-

$\mathbf{0.051\scriptscriptstyle\pm\scriptstyle.024}$

-

$\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.003}$

-

$0.028\scriptscriptstyle\pm\scriptstyle.006$

-

$0.042\scriptscriptstyle\pm\scriptstyle.000$

-

$0.149\scriptscriptstyle\pm\scriptstyle.001$

-

$0.409\scriptscriptstyle\pm\scriptstyle.005$

-

$0.065\scriptscriptstyle\pm\scriptstyle.001$

-

$0.108\scriptscriptstyle\pm\scriptstyle.003$

-

$0.028\scriptscriptstyle\pm\scriptstyle.009$

-

$0.133\scriptscriptstyle\pm\scriptstyle.007$

-

$0.166\scriptscriptstyle\pm\scriptstyle.003$

-

$\mathbf{0.034\scriptscriptstyle\pm\scriptstyle.001}$

-

$\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.001}$

-

$0.033\scriptscriptstyle\pm\scriptstyle.006$

-

$0.117\scriptscriptstyle\pm\scriptstyle.002$

-

$0.083\scriptscriptstyle\pm\scriptstyle.024$

-

$0.052\scriptscriptstyle\pm\scriptstyle.001$

-

$0.321\scriptscriptstyle\pm\scriptstyle.002$

-

$0.058\scriptscriptstyle\pm\scriptstyle.004$

-

$\mathcal{SRE}$

-

$\mathcal{SR}$

-

$\mathbf{PP}(p_{i})=\frac{1}{Length(\overline{p^{e}_{i}\Phi(p^{e}_{i})})+dist^{% -M(X)}(center,\Phi(p^{e}_{i})},$

-

$Crosswise$

-

$Sb_{i}$

-

$\mathbf{V}_{1},\mathbf{V}_{2},\mathbf{V}_{3},\mathbf{V}_{4}$

-

$\mathbf{X}\subset\mathbb{R}^{2}$

-

$p_{1},p_{2}\leftarrow$

-

$M(\mathbf{X})$

-

$E_{area}$

-

$\mathbf{I}=\{I_{i}\},i=0,\dots,N_{I}$

-

$\mathbf{A}^{c}=\{ac_{k}\}$

-

$center=\operatorname*{arg\,max}_{m\in M(X)}\;CR(v_{i},v_{j})\;\mid v_{i},v_{j}% -\in\pi(m),$

-

$\pi(z)=\{z\in\partial\mathbf{X}:\|z-x\|=D(\mathbf{X})(z)\}$

-

$p^{\prime}(x^{\prime},y^{\prime})=\zeta(p(x,y)),$

-

$\mathbf{K_{i}}$

-

$p(x,y)\in a_{k}$

-

$p_{triangle}$

-

$D(\mathbf{X})(z)=\inf_{x\in\partial\mathbf{X}}\|z-x\|,$

-

$4^{S-1}$

-

$Sb_{i}^{*}$

-

$M(\mathbf{X})=\{z\in\mathbf{X}:\lvert\pi(z)\rvert>1\}.$

-

$P_{\mathbf{X}}$

-

$x\in\pi(z)$

-

$D(\mathbf{X}):\mathbb{R}^{2}\mapsto\mathbb{R}$

-

$\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{H}_{3},\mathbf{H}_{4}$

-

$z\in\mathbb{R}^{2}$

-

$\zeta(.)$

-

$4^{1}\cdot 8/2$

-

$Crosswise(z)$

-

$\mathbf{D_{\mathbf{T}}}$

-

$E_{area}=\sum_{i=1}^{S_{T}}Area(Sb_{i}^{*})$

-

$\mathbf{\mathcal{O}}^{*}=\operatorname*{arg\,max}_{\mathbf{D_{i}},\mathbf{K_{i% -}}}\;E_{area},$

-

$[\;\;]$

-

$\mathbf{A}^{i}=\{a_{k}\},k=1,\dots,8$

-

$\Phi\big{(}z\big{)}$

-

$M_{o}=\dfrac{P_{o}}{P_{\mathbf{X}}},$

-

$\tau_{p}=0.75$

-

$Axial$

-

$M_{n}=\dfrac{1}{N}\sum_{i}\|(L_{i}-L_{ci})\|,$

-

$Sb^{*}$

-

$Sb_{i}=[bx_{1},by_{1},bx_{2},by_{2}]$

- - - diff --git a/htmls/output_mathjax_example_10001.html b/htmls/output_mathjax_example_10001.html deleted file mode 100644 index 859475e6b0fc70151a936cda7663d86c5dcc2552..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10001.html +++ /dev/null @@ -1,131 +0,0 @@ - - - - MathJax Example - - - - -

$\mathfrak{L}_{T}$

-

$\tau_{e}=2$

-

$ct\leftarrow G.centroid$

-

$O(4^{n})$

-

$dividing\_line\leftarrow$

-

$\tau_{e}=3$

-

$4^{3}\cdot n/4$

-

$\mathfrak{L}_{\mathbf{T}}$

-

$E_{area}=\sum_{i=1}^{S_{T}}(Area(Sb_{i}^{*})\cdot p_{triangle}(\mathbf{G})),$

-

$N_{I}>N_{p}$

-

$N_{I}>>N_{p}$

-

$\mathfrak{R}_{\mathbf{T}}$

-

$\mathbf{G_{\mathbf{T}}}$

-

$\mathbf{S}_{i}=\big{[}N_{I}\cdot\dfrac{Area(p_{i})}{Area(\mathbf{X})}\big{]},$

-

$slope\leftarrow Axial(ct)$

-

$Sb$

-

$N_{c}=N_{I}$

-

$CR(v_{i},v_{j})=dist^{B}(v_{i},v_{j})-Length(\overline{v_{i}v_{j}}),$

-

$\Phi(p^{e}_{i})$

-

$\mathbf{\mathcal{O}}^{*}$

-

$Axial(z)$

-

$\|\;\;\|$

-

$\mathbf{G}_{T}$

-

$dist^{B}$

-

$\mathbf{G}\leftarrow p_{2}$

-

$\bigcup_{i}S_{i}$

-

$\mathbb{R}^{2}\setminus\mathbf{X}$

-

$\mathbf{\mathcal{O}}$

-

$z\in\mathbf{X}$

-

$M_{a}=\dfrac{|\bigcup_{i}S_{i}|}{P_{\mathbf{X}}},$

-

$\tau_{e}=1$

-

$slope$

-

$dividing\_line$

-

$p_{triangle}(polygon)=\begin{cases}0.8&polygon\text{ is a triangle}\\ -1.0&\text{otherwise.}\end{cases}$

-

$8/2$

-

$R^{m}_{i}$

-

$S_{i}-1$

-

$p_{1},p_{2}\leftarrow G$

-

$dist^{M(X)}$

-

$S_{T}-1$

-

$4^{1}\cdot n/2$

-

$M_{c}=\dfrac{P_{w}}{P_{\mathbf{X}}},$

-

$\gamma^{u}$

-

$4^{2^{\tau_{e}}-1}\cdot n/2^{\tau_{e}}$

-

$\Phi(z,M(\mathbf{X}))=\operatorname*{arg\,min}_{m\in M(\mathbf{X})}\|z-m\|$

-

$ac_{k}$

-

$L_{ci}$

-

$M_{s}=1-\dfrac{|\bigcup_{i}S_{i}|}{\sum_{i}|S_{i}|}.$

-

$\mathfrak{R}_{T}$

-

$slope\leftarrow Crosswise(ct)$

-

$p^{\prime}\in ac_{k}$

-

$\mathbf{G}\leftarrow p_{1}$

-

$\mathbf{P}=\{p_{1},\dots,p_{N_{p}}\}$

-

$p^{e}_{i}$

-

$Sb=[bx_{1},by_{1},bx_{2},by_{2}]$

-

$\mathcal{T}_{t}^{(d)}(m)$

-

$\mathcal{T}_{r}^{(d)}(m)=\lVert(\mathcal{T}^{(d)}(s,m))_{s\in\mathcal{M},s\neq -t% -}\rVert_{2}$

-

$\mathcal{T}_{t}^{(d)}(m)=\lVert(\mathcal{T}^{(d)}(m,t))_{t\in\mathcal{M},t\neq -s% -}\rVert_{2}$

-

$\alpha=\frac{2}{255}$

-

$\displaystyle x^{t+1}=\prod_{x+\mathcal{B}}(x^{t}+\alpha\text{sgn}(\nabla_{x}L% -(\theta,x,y)))$

-

$\mathcal{T}_{r}^{(d)}(m)$

-

$\mathcal{T}^{(d)}(s,t)$

-

$\mathcal{A}_{s}^{(d)}=\{(x,y)\}$

-

$\mathcal{A}_{s}^{(d)}$

-

$(s,t)\in\mathcal{M}^{2}$

-

$(7,308+4,542+7,517)*6=116,202$

-

$\mathcal{T}^{(d)}(s,t)=\frac{|\{(x,y)\in\mathcal{A}_{s}^{(d)};\hat{y}_{t}\neq y% -\}|}{|\mathcal{A}_{s}^{(d)}|}$

-

$N_{e}=100$

-

$l^{\eta}_{i,j\in B}$

-

$BNM$

-

$L^{\textit{AttnMask}}_{DM}=L_{DM}(\mathcal{M}\odot x,\mathcal{M}\odot\tilde{x}),$

-

$[v^{*},\ldots,v^{\&}]:=\operatorname*{arg\,min}_{\mathcal{V}}E_{x_{0},{% -\epsilon}\sim N(0,I)}\\ -\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,[c_{\theta}(y),v^{*},\ldots,v^{\&}]\|% -^{2}$

-

$V=f_{V}(v)$

-

$v^{\eta}_{i}$

-

$[y,p^{*}]$

-

$v^{\eta}_{j}$

-

$L_{PromptCL}$

-

$\overline{M}^{p}=1/T\sum_{t=1}^{T}M_{t}^{p}$

-

$\{v^{*},\ldots,v^{\&}\}$

-

$v^{*}=c_{\theta}(p^{*})$

-

$L_{DM}=L_{DM}(x,\tilde{x}):=E_{x_{0},{\epsilon}\sim N(0,I),t\sim\text{Uniform}% -(1,T)}\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,c_{\phi}(p))\|^{2},$

-

$6100$

-

$L=L^{\textit{AttnMask}}_{DM}+\gamma L_{PromptCL}^{adj},$

-

$M=\text{Softmax}(QK^{T}/\sqrt{d})$

-

$step=1,\ldots,S$

-

$\{v^{\&}\}$

-

$(0.2,0.0005)$

-

$\mathcal{V}=[v^{*},\ldots,v^{\&}]$

-

$(p^{*},v^{*})$

-

$(\mathcal{P},\mathcal{V})$

-

${\bm{\epsilon}}\sim\mathcal{N}(\mathbf{0},\textbf{I})$

-

$L_{PromptCL}^{adj}$

-

$v=c_{\phi}(p)$

-

$\eta\in MN$

-

$L^{\textit{AttnMask}}_{DM}$

-

$(0.3,0.00075)$

-

$Q=f_{Q}(z)$

-

$\mathcal{P}=[p^{*},\ldots,{p}^{\&}]$

-

$\mathcal{M}=\bigcup_{p\in\mathcal{P}}B(M^{p})$

-

$\{v^{*}\}$

- - - diff --git a/htmls/output_mathjax_example_10002.html b/htmls/output_mathjax_example_10002.html deleted file mode 100644 index 28739200bc4dba627ecd5956c5c476e1fd1acebc..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10002.html +++ /dev/null @@ -1,155 +0,0 @@ - - - - MathJax Example - - - - -

$K=f_{K}(v)$

-

$sim(v_{i},v_{j})=v_{i}^{T}.v_{j}/||v_{i}||||v_{j}||$

-

$B(M^{p}):=\{1\text{ if }M^{p}>k,0\text{ otherwise}\}$

-

$v^{*}:=\operatorname*{arg\,min}_{v}E_{x_{0},{\epsilon}\sim N(0,I)}\\ -\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,[c_{\theta}(y),v^{*}]\|^{2}$

-

$\{v_{b}^{n}\}_{b=1}^{B},_{n=1}^{N}$

-

$\begin{gathered}L_{PromptCL}=\frac{1}{N}\frac{1}{B}\sum_{\eta=1}^{N}\sum_{i=1}% -^{B}{l^{\eta}_{i,j\in B}},\qquad L_{PromptCL}^{adj}=\frac{1}{NM}\frac{1}{B}% -\sum_{\eta=1}^{NM}\sum_{i=1}^{B}{l^{\eta}_{i,j\in B}}\end{gathered}$

-

$(\tau,\gamma)$

-

$[v^{*},\ldots,v^{\&}]=[c_{\theta}(p^{*}),\ldots,c_{\theta}(p^{\&})]$

-

$[y,p^{*},\ldots,p^{\&}]$

-

$c_{\phi}$

-

$l^{\eta}_{i,j\in B}=-log(\frac{exp(sim(v^{\eta}_{i},v^{\eta}_{j}))/\tau}{\sum_% -{\eta=1}^{N}\sum_{j=1,j\neq{i}}^{B}exp(sim(v^{\eta}_{i},v^{\eta}_{j})/\tau)})$

-

$\eta\in N$

-

$\tau=10^{-4}$

-

$J^{\tau}(\mathbf{\chi}^{k+1},T)$

-

$q_{1}=1,\ q_{2}=100$

-

$d=0.1l$

-

$\tilde{\chi}_{s}$

-

$\Phi_{h}=(q_{1}-q_{2})G_{\tau}\ast(T_{h}-T^{\ast}_{h})+\gamma\sqrt{\frac{\pi}{% -\tau}}G_{\tau}\ast(1-2\chi_{h})+(\kappa_{1}-\kappa_{2})G_{\tau}\ast(\frac{\xi}% -{2}\nabla T_{h}\cdot\nabla T_{h}+\nabla T_{h}\cdot\nabla T_{h}^{\ast}).$

-

$\kappa_{1}=5,\ 10,\ 15$

-

$\tau=3\times 10^{-5}$

-

$\Omega\in\mathbb{R}^{d}\ (d=2,3)$

-

$\mathbf{\chi}_{h}$

-

$\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$

-

$y=1/2$

-

$\Omega_{1}\subset\Omega$

-

$(\ref{ad})$

-

$\frac{q_{1}}{q_{2}}$

-

$z=0,\ y=1/2$

-

$\Phi_{h}$

-

$T^{\ast}_{h}$

-

$J^{\tau}(\chi,T)=\int_{\Omega}q(\chi)T\ d\textbf{x}+\frac{\xi}{2}\int_{\Omega}% -\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}+\gamma\sqrt{\frac{\pi}{\tau}}% -\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x},$

-

$\displaystyle\ \ \ \ +\int_{\Omega}\kappa(\chi)\nabla T^{k}\cdot\nabla T^{*k}% -\ d\textbf{x}-\int_{\Omega}q(\chi)T^{*k}\ d\textbf{x}.$

-

$\displaystyle\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}=$

-

$J^{\tau}(\tilde{\chi}_{s},\tilde{T}_{s})>J^{\tau}(\chi^{k},T^{k})$

-

$\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100,\ \tau=1\times 10^{-4}$

-

$\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\leq\sigma_{1},\chi^{k}(p)=0% -\big{\}},$

-

$\tilde{T}_{s}$

-

$\kappa_{1}=40,\ \kappa_{2}=1$

-

$\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)\in\arg\min_{\chi\in% -\mathcal{H}}\tilde{J}^{\tau,k}(\chi).$

-

$\tilde{B}\subset B$

-

$A_{1}=\{p\in A:\phi_{A}(p)\leq\sigma_{1}\}\ \ \ \ B_{1}=\{p\in B:\phi_{B}(p)% -\leq\sigma_{2}\}$

-

$H^{1}_{\Gamma_{D}}(\Omega)=\{v\in H^{1}(\Omega)\ |\ v|_{\Gamma_{D}}=0\}$

-

$(\ref{and})$

-

$\displaystyle\left\{\begin{aligned} -\nabla\cdot(\kappa(\chi^{k})\nabla T)-q(% -\chi^{k})&=0,\ \ &&\rm in\ \ \Omega,\\ -T&=0,\ \ &&\rm on\ \ \Gamma_{D},\\ -\kappa(\chi^{k})\nabla T\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N},\end{% -aligned}\right.$

-

$\displaystyle\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)$

-

$\sigma_{1}=\tilde{\phi}_{A}^{k}(n_{s})$

-

$tol>0$

-

$\displaystyle\int_{\Omega}(q(\chi^{k})T^{k})\ d\textbf{x}+\frac{\xi}{2}\int_{% -\Omega}\kappa(\chi^{k})\nabla T^{k}\cdot\nabla T^{k}\ d\textbf{x}+\gamma\sqrt{% -\frac{\pi}{\tau}}\int_{\Omega}\chi^{k}G_{\tau}\ast(1-\chi^{k})\ d\textbf{x}.$

-

$T\in Q(\chi)$

-

$\frac{\xi}{2}\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$

-

$G_{\tau}=\frac{1}{(4\pi\tau)^{d/2}}\exp\left(-\frac{|\textbf{x}|^{2}}{4\tau}\right)$

-

$\int_{\Omega}\chi^{k+1}\ d\textbf{x}=V_{0}$

-

$\chi^{k+1}=\tilde{\chi}_{s}$

-

$\mathbb{R}^{d}\setminus\overline{\Omega}$

-

$T^{*k}$

-

$\displaystyle\tilde{B}_{1}$

-

$\Omega_{1}\in\Omega$

-

$\Gamma\colon=\Gamma_{D}\cup\Gamma_{N}$

-

$\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma,\chi^{k}(p)=1\big{\}}.$

-

$\displaystyle\chi^{k+1}=\begin{cases}1&\ \textrm{if}\ \Phi^{k}\leq\sigma,\\ -0&\ \textrm{otherwise}.\end{cases}$

-

$\frac{\delta\tilde{J}^{\tau}}{\delta T}=0,\ \ \ \frac{\delta\tilde{J}^{\tau}}{% -\delta T^{*}}=0.$

-

$\chi^{k+1}=\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)$

-

$\tilde{\chi}_{s}=\chi_{A_{1}}+\chi^{k}-\chi_{B_{1}}$

-

$\kappa_{1},\kappa_{2},q_{1},q_{2}$

-

$T^{\ast}_{h}\in V_{h}^{0}$

-

$\Phi^{k}=(q_{1}-q_{2})G_{\tau}\ast(T^{k}-T^{*k})+\gamma\sqrt{\frac{\pi}{\tau}}% -G_{\tau}\ast(1-2\chi^{k})+(k_{1}-k_{2})G_{\tau}\ast(\frac{\xi}{2}\nabla T^{k}% -\cdot\nabla T^{k}+\nabla T^{k}\cdot\nabla T^{\ast k})$

-

$\displaystyle\tilde{J}^{\tau,k}(\chi)$

-

$\displaystyle\int_{\Omega}\left[G_{\tau/2}\ast\chi\right]\left[G_{\tau/2}\ast(% -1-\chi)\right]\ d\textbf{x}$

-

$\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100$

-

$\kappa_{1}=40,\kappa_{2}=1$

-

$\int_{\Omega}\chi\ d\textbf{x}=V_{0}$

-

$T^{k+1}$

-

$\chi^{0}$

-

$\tilde{\phi}_{A}^{k}$

-

$\displaystyle\tilde{A}_{2}$

-

$\chi^{k+1}=Proj_{[0,1]}\left(\chi^{k}-s\left.\frac{\delta J^{\tau}}{\delta\chi% -}\right|_{\chi^{k}}\right),$

-

$\kappa(\chi)$

-

$0.1l$

-

$\displaystyle\chi^{k+1}=\chi_{A}+\chi^{k}-\chi_{B}.$

-

$J^{\tau}(\chi^{k+1},T^{k+1})\leq J^{\tau}(\chi^{k+1},T^{k}),$

-

$\kappa_{2}=1$

-

$\displaystyle\kappa(\chi)=\kappa_{1}G_{\tau}\ast\chi+\kappa_{2}G_{\tau}\ast(1-% -\chi),$

-

$\tilde{J}^{\tau}(\chi,T,T^{*})=J^{\tau}(\chi,T)+\int_{\Omega}(-\nabla\cdot(% -\kappa(\chi)\nabla T)-q(\chi))\cdot T^{*}d\textbf{x}.$

-

$q_{2}=100$

-

$\chi^{1},T^{1},T^{*1},\chi^{2},T^{2},T^{*2},\cdots,\chi^{k},T^{k},T^{*k},\cdots$

-

$\tilde{J}^{\tau}(\mathbf{\chi}^{k+1},T^{k})\leq\tilde{J}^{\tau}(\mathbf{\chi}^% -{k},T^{k}),$

-

$J^{\tau}(\chi,T)$

-

$\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa(\chi^{k})\nabla T^{*})&% -=q(\chi^{k})-\xi(\nabla\cdot(\kappa(\chi^{k})\nabla T)),\ \ &&\rm in\ \ \Omega% -,\\ -T^{*}&=0,\ \ &&\rm on\ \ \Gamma_{D},\\ -\kappa(\chi^{k})\nabla T^{*}\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N}\end{% -aligned}\right.$

-

$J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi}^{k},T^{k})$

-

$\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma_{2},\chi^{k}(p)=1\big{% -\}},$

-

$\chi^{k+1}\in\mathcal{B}$

-

$\|\chi^{k+1}-\chi^{k}\|_{2}>tol$

-

$\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\in(\sigma_{1},\sigma),\chi^{k}% -(p)=0\big{\}},$

-

$\kappa=\kappa_{1}\chi_{\Omega_{1}}+\kappa_{2}\chi_{\Omega_{2}}$

-

$\displaystyle\ J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi% -}^{k},T^{k}).$

-

$\xi=1\times 10^{-5}$

-

$600\times 600$

-

$P_{1}(K)$

-

$T^{k},T^{*k}$

-

$\int_{\Omega}\chi^{k+1}(\textbf{x})d\textbf{x}=V_{0}$

- - - diff --git a/htmls/output_mathjax_example_10003.html b/htmls/output_mathjax_example_10003.html deleted file mode 100644 index 565944ce039df79fbff0bee143016404b5fac517..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10003.html +++ /dev/null @@ -1,169 +0,0 @@ - - - - MathJax Example - - - - -

$J(\chi,T)$

-

$\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}.$

-

$n_{s}=N-\lfloor N*\theta^{s}\rfloor$

-

$\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\leq\sigma,\chi^{k}(p)=0\big{\}},$

-

$\kappa_{1}=5,\ 10,\ 20$

-

$k_{1}/k_{2}$

-

$\kappa_{1}=20,\ \kappa_{2}=1$

-

$\chi^{k+1}=\arg\min_{\chi\in B}\tilde{J}^{\tau,k}(\chi).$

-

$\chi^{k}$

-

$\chi^{\ast}\in\mathcal{B}$

-

$\mathcal{H}:=\big{\{}\chi\in BV(\Omega)\ |\ \chi(\textbf{x})\in[0,1],\int_{% -\Omega}\chi\ d\textbf{x}=V_{0}\big{\}}.$

-

$\displaystyle=\int_{\Omega}q(\chi)T^{k}\ d\textbf{x}+\frac{\xi}{2}\int_{\Omega% -}\kappa(\chi)\nabla T^{k}\cdot\nabla T^{k}\ d\textbf{x}+\gamma\sqrt{\frac{\pi}% -{\tau}}\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}$

-

$\xi=1\times 10^{-7}$

-

$\tilde{B}_{2}$

-

$\displaystyle=(q(\chi_{h}),\varphi_{h}),\ \ \forall\varphi_{h}\in V_{h}^{0},$

-

$J^{\tau}(\mathbf{\chi}^{k+1},T^{k})\leq J^{\tau}(\mathbf{\chi}^{k},T^{k})$

-

$q_{1}=1,\ 20,\ 80$

-

$q(\chi)$

-

$\displaystyle\chi(\textbf{x}):=\left\{\begin{aligned} &1,\ \ \ \ \textrm{if}\ % -\textbf{x}\in\ \Omega_{1},\\ -&0,\ \ \ \ \textrm{otherwise}.\end{aligned}\right.$

-

$\begin{cases}&\min\limits_{\mathbf{\chi}\in\mathcal{B}}J^{\tau}(\mathbf{\chi},% -T),\\ -&\textrm{s.t.}\ T\in Q(\chi),\end{cases}$

-

$\tau=1\times 10^{-4}$

-

$\ \tau>0$

-

$\chi^{k+1}=\arg\min_{\chi\in\mathcal{H}}\mathcal{L}^{\tau,k}_{r^{k}}(\chi)=% -\arg\min_{\chi\in\mathcal{H}}\int_{\Omega}\chi\Phi^{k}\ d\textbf{x},$

-

$\chi_{1}.$

-

$\displaystyle\chi^{k+1}=\chi_{\tilde{A}_{1}}+\chi^{k}-\chi_{\tilde{B}_{1}}.$

-

$Q(\chi)$

-

$\displaystyle\chi^{k+1}=$

-

$\tilde{J}^{\tau,k}(\chi)\approx\tilde{J}^{\tau,k}(\chi^{k})+\mathcal{L}^{\tau,% -k}_{\chi^{k}}(\chi-\chi^{k}),$

-

$\chi^{k+1}$

-

$\xi=1e-7$

-

$\displaystyle\tilde{A}_{1}$

-

$\Phi^{k}=(q_{1}-q_{2})G_{\tau}\ast(T^{k}-T^{*k})+\gamma\sqrt{\frac{\pi}{\tau}}% -G_{\tau}\ast(1-2\chi^{k})+(k_{1}-k_{2})G_{\tau}\ast(\frac{\xi}{2}\nabla T^{k}% -\cdot\nabla T^{k}+\nabla T^{k}\cdot\nabla T^{\ast k}).$

-

$\chi^{k+1}(x)=\begin{cases}1\ \ \textrm{if}\ \Phi^{k}(x)\leq\sigma,\\ -0\ \ \textrm{otherwise,}\end{cases}$

-

$\begin{cases}\dfrac{\delta(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{% -\Omega}\chi\ d\textbf{x}-V_{0}))}{\delta\chi}(x)=\Phi^{k}-\sigma\leq 0\ \ \ % -\textrm{if}\ \ \chi(x)=1,\\ -\dfrac{\delta(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{\Omega}\chi\ % -d\textbf{x}-V_{0}))}{\delta\chi}(x)=\Phi^{k}-\sigma\geq 0\ \ \ \textrm{if}\ \ % -\chi(x)=0,\\ -\dfrac{d(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{\Omega}\chi\ d% -\textbf{x}-V_{0}))}{d\sigma}=\int_{\Omega}\chi\ d\textbf{x}-V_{0}=0.\end{cases}$

-

$T_{h}\in V_{h}^{0}$

-

$\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa\nabla T)+q&=0,\ \ &&\rm -in% -\ \Omega,\\ -(\kappa\nabla T)\cdot\mathbf{n}&=0,\ \ &&\rm on\ \Gamma_{N},\\ -T&=0,\ \ &&\rm on\ \Gamma_{D},\\ -|\Omega_{1}|&=\beta|\Omega|,\ \ &&\rm with\ a\ fixed\ parameter\ \beta\in(0,1)% -,\end{aligned}\right.$

-

$\beta=0.1,\ 0.2,\ 0.3$

-

$\int_{\Omega}\chi\ d\textbf{x}$

-

$\tilde{\phi}_{B}^{k}$

-

$\displaystyle\int_{\Omega}\chi G_{\tau/2}\ast\left[G_{\tau/2}\ast(1-\chi)% -\right]\ d\textbf{x}$

-

$\chi_{\Omega_{i}}$

-

$\kappa_{1}=40,\ \kappa_{2}=1,\gamma=20$

-

$\displaystyle\left\{\begin{aligned} -\nabla\cdot(\kappa(\chi)\nabla T)-q(\chi)% -&=0,\ \ &&\rm in\ \ \Omega,\\ -T&=0,\ \ &&\rm on\ \ \Gamma_{D},\\ -\kappa(\chi)\nabla T\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N},\end{aligned% -}\right.$

-

$\chi_{h}\in\mathcal{B}_{h}$

-

$\displaystyle J^{\tau}(\mathbf{\chi}^{k},T^{k})=$

-

$\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\in(\sigma,\sigma_{2}),\chi^{k}% -(p)=1\big{\}},$

-

$\gamma=12,\ 15,\ 50$

-

$|\Gamma|\approx\sqrt{\frac{\pi}{\tau}}\int_{\Omega}\chi G_{\tau}*(1-\chi)\ d% -\textbf{x},$

-

$J^{\tau}(\mathbf{\chi}^{k},T^{k})$

-

$\tilde{A}\subset A$

-

$q=q_{1}\chi_{\Omega_{1}}+q_{2}\chi_{\Omega_{2}}$

-

$J^{\tau}(\chi^{k+1},T^{k+1})\leq J^{\tau}(\chi^{k},T^{k}).$

-

$J^{\tau}(\tilde{\chi}_{s},\tilde{T}_{s})$

-

$\displaystyle\tilde{B}_{2}$

-

$q_{1}/q_{2}$

-

$\chi^{k+1}=\arg\min_{\chi\in\mathcal{\mathcal{H}}}\tilde{J}^{\tau,k}(\chi),$

-

$\tilde{J}^{\tau}(\chi,T^{k},T^{*k})$

-

$BV(\Omega)$

-

$G_{\tau}\ast$

-

$\gamma=15$

-

$\kappa_{1}=10,\ \kappa_{2}=1$

-

$60\times 60\times 60$

-

$\kappa_{1},\ \kappa_{2},\ q_{1},\ q_{2},\ \tau,\ \gamma,\ \xi$

-

$\frac{\kappa_{1}}{\kappa_{2}}$

-

$\displaystyle q(\chi)=q_{1}G_{\tau}\ast\chi+q_{2}G_{\tau}\ast(1-\chi).$

-

${0}\times[0.45,0.55]$

-

$\tilde{J}^{\tau,k}(\chi)$

-

$\displaystyle=\int_{\Omega}\chi\Phi^{k}\ d\textbf{x},$

-

$\mathcal{B}:=\big{\{}\chi\in BV(\Omega)\ |\ \chi(\textbf{x})=\{0,1\},\int_{% -\Omega}\chi\ d\textbf{x}=V_{0}\big{\}},$

-

$\displaystyle=(q(\chi_{h}),\varphi_{h})+\xi(\kappa(\chi_{h})\nabla T_{h},% -\nabla\varphi_{h}),\ \ \forall\varphi_{h}\in V_{h}^{0}.$

-

$\mathbf{\chi}$

-

$\displaystyle V_{h}=\{v\in H^{1}(\Omega)\ |\ v\in P_{1}(K),\forall K\in% -\mathcal{T}_{h}\},\quad V_{h}^{0}=V_{h}\cap H^{1}_{\Gamma_{D}}(\Omega),$

-

$\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma,\chi^{k}(p)=1\big{\}},$

-

$\displaystyle(\kappa(\chi_{h})\nabla T_{h},\nabla\varphi_{h})$

-

$\tau=0.35\times 10^{-4}$

-

$\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa(\chi)\nabla T^{*})&=q(% -\chi)-\xi(\nabla\cdot(\kappa(\chi)\nabla T)),\ \ &&\rm in\ \ \Omega,\\ -T^{*}&=0,\ \ &&\rm on\ \ \Gamma_{D},\\ -\kappa(\chi)\nabla T^{*}\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N}.\end{% -aligned}\right.$

-

$\displaystyle-(\kappa(\chi_{h})\nabla T^{\ast}_{h},\nabla\varphi_{h})$

-

$\sigma_{2}=\tilde{\phi}_{B}^{k}(n_{s})$

-

$\chi^{1},\chi^{2},\cdots,\chi^{k+1},\cdots$

-

$\chi^{k},~{}T^{k},~{}T^{*k}$

-

$q_{1}=1,\ 40,\ 80$

-

$\displaystyle\int_{\Omega}\left[G_{\tau/2}\ast\chi\right]\left[1-G_{\tau/2}% -\ast\chi\right]\ d\textbf{x},$

-

$G_{\tau/2}\ast$

-

$Proj_{[0,1]}(v)=\begin{cases}v&{\rm if}\ v\in[0,1],\\ -0&{\rm if}\ v<0,\\ -1&{\rm if}\ v>1.\end{cases}$

-

$\displaystyle\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)$

-

$\kappa_{1}=40$

-

$\Omega_{2}=\Omega\setminus\Omega_{1}\in\Omega$

-

$\kappa_{1}/\kappa_{2}$

-

$V_{0}=\beta|\Omega|$

-

$J^{\tau}(\mathbf{\chi}^{k},T)$

-

$\min_{(\Omega_{1},T)}J(\Omega_{1},T)=\int_{\Omega}qTd\textbf{x}+\gamma|\Gamma|,$

-

$\displaystyle=\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{0})}{p% -_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}% -\mathbb{E}_{\hat{Z}_{i/N}\sim\int p_{v_{\theta}}(Z_{i/N}|Z_{1})q(Z_{1}|Z_{0})% -dZ_{1}}$

-

$Z_{7/8}$

-

$\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}% -||g_{v_{\theta}}(Z_{t},t)-(\alpha_{t}[\log(b)-\log(a)]X^{S})||_{2}^{2}.$

-

$\alpha_{1}X^{S}$

-

$\forall Z_{1-i/K}$

-

$q(\cdot|\cdot)$

-

$\displaystyle\mathcal{L}_{\textrm{OFM-KT}}=\mathbb{E}_{(X^{S},Y)}[\frac{1}{N}% -\sum_{i=0}^{N-1}$

-

$\min L(g^{S}(X^{S}),g^{T}(X^{T}))$

-

$\displaystyle\mathcal{L}_{\textrm{FM-KT}}=\mathbb{E}_{(X^{S},X^{T})}\frac{1}{N% -}\sum_{i=1}^{N}||\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{\theta}}(Z_{1-i/% -N},1-i/N))/-\nabla_{t}\sigma_{t})-X^{T}||_{2}^{2},$

- - - diff --git a/htmls/output_mathjax_example_10004.html b/htmls/output_mathjax_example_10004.html deleted file mode 100644 index 6a3116182f28efe4ce278b6d89f1379393e0d3ef..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10004.html +++ /dev/null @@ -1,175 +0,0 @@ - - - - MathJax Example - - - - -

$p_{v_{\theta}}(\cdot|\cdot)$

-

$\displaystyle=\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)% -}||g_{v_{\theta}}(Z_{t},t)-(\nabla_{t}\alpha_{t}X^{S}+\nabla_{t}\sigma_{t}X^{T% -})||_{2}^{2}.$

-

$\left\lfloor{\beta_{d}}B\right\rfloor$

-

$\displaystyle Z_{t}=\alpha_{t}X^{S}+\sigma_{t}X^{T},\ s.t.\ \lim_{t\rightarrow -0% -}\alpha_{t}=0,\lim_{t\rightarrow 0}\sigma_{t}=1,\lim_{t\rightarrow 1}\sigma_{t% -}=0.$

-

$\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}% -||g_{v_{\theta}}(Z_{t},t)-((\frac{1}{2}a(1-t)+\frac{1}{2}b)\alpha_{t}X^{S}-% -\frac{\alpha_{t}}{\sqrt{1-\alpha_{t}^{2}}}\alpha_{t}(\frac{1}{2}a(1-t)+\frac{1% -}{2}b)X^{T})||_{2}^{2}.$

-

$g^{S}(\cdot)$

-

$\displaystyle+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{% -\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the % -ground truth label}}].$

-

$\{X_{t}\}_{t}$

-

$\displaystyle=\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)\mathcal{K}(1-i/% -K).$

-

$Z_{1-i/N}=Z_{1-(i-1)/N}-g_{v_{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N$

-

$\mathcal{E}(Z_{1-i/K})$

-

$\hat{Z}_{1}\sim\pi_{1}$

-

$\displaystyle\mathcal{L}_{\textrm{FM-KT}^{\Theta}}=$

-

$Z_{1}=\alpha_{1}X^{S}$

-

$\frac{\sigma_{t}-\sigma_{t-\Delta t}}{t-\Delta t}$

-

$\displaystyle-\log p_{v_{\theta}}(Z_{0})\leq\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}% -\left[\log\frac{q(Z_{1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1% -/N})}+\sum_{i=1}^{N}\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{% -(i-1)/N}|Z_{i/N})}\right].$

-

$\sigma_{t}\in\mathbb{R}^{+}$

-

${}_{\textrm{L}}$

-

$\displaystyle\left[||q(Z_{(i-1)/N}|Z_{i/N},Z_{0})-p_{v_{\theta}}(Z_{(i-1)/N}|% -\hat{Z}_{i/N})||_{2}^{2}\right],\quad s.t.\quad\textrm{Law}(Z_{i/N})\stackrel{% -{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i/N}).$

-

$\mathcal{L}_{\textrm{FM-KT}}$

-

$X_{t}=tX^{S}+(1-t)X^{T}$

-

$\displaystyle Z_{1-(i+1)/K}$

-

$(X^{S},X^{T},Y)$

-

$\mathit{a=0.02}$

-

$\displaystyle\quad\quad-g_{v_{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N,\quad s.t.% -\quad i\geq 1.$

-

$\displaystyle\mathcal{E}(Z_{1-2/K})=\mathcal{E}(Z_{1-1/K})(1-(1/K)\psi(1-1/K))% -+(1/K)\mathcal{K}(1-1/K)$

-

$\{Z_{1-i/K}\}_{i=1}^{K}$

-

$Z_{i/N}$

-

$\displaystyle,X^{T})+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v% -_{\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the % -ground truth label (optional)}}],$

-

$\frac{d\hat{Z}_{t}}{dt}=-g^{*}_{v_{\theta}}(\hat{Z}_{t},t)$

-

$Z_{1-(i-1)/K}$

-

$\sigma_{t}=1,\ s.t.\quad a=0.02,b=100$

-

$\{dZ_{t-\Delta_{t}},dZ_{t-2\Delta_{t}},\cdots,dZ_{s}\}$

-

$\mathit{b=0.1}$

-

$\textrm{Law}(Z_{i/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i% -/N})$

-

$g_{v_{\theta}}(Z_{t},t)$

-

$\displaystyle\mathcal{E}(Z_{0})=(1/K)\left[\sum_{i=0}^{K-1}\mathcal{K}(1-i/K)% -\right]+(1/K^{2})\left[\sum_{j=1}^{K-1}\left[\psi(1-j/K)\left(\sum_{i=0}^{j-1}% -\mathcal{K}(1-i/K)\right)\right]\right]+\mathcal{O}(1/K^{3}).$

-

$\displaystyle\textrm{the sampling process:}\quad Z_{1-i/N}=Z_{1-(i-1)/N}-g_{v_% -{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N,\quad s.t.\quad i\geq 1,$

-

$\mathbb{E}_{\hat{Z}_{i/N},Z_{1},Z_{0}}D_{\mathrm{KL}}(q(Z_{(i-1)/N}|Z_{i/N},Z_% -{0})||p_{v_{\theta}}(Z_{(i-1)/N}|\hat{Z}_{i/N}))$

-

$Z_{1-i/K}=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})$

-

$Z_{1-i/K}$

-

$Z_{1-i/N}\!=\!Z_{1\!-\!(i\!-\!1)/N}\!-\!g_{v_{\theta}}(Z_{1\!-\!(i\!-\!1)/N},1% -\!-\!(i\!-\!1)/N)/N$

-

$\displaystyle=Z_{1-i/K}-(1/K)g_{v_{\theta}}(X_{1-i/K}+\mathcal{E}(Z_{1-i/K}),1% --i/K)$

-

${}_{\textrm{M}}$

-

$(X^{S},X^{T})\in(\mathbb{R}^{d},\mathbb{R}^{d})$

-

$\displaystyle\approx X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)\left[g_{v_{\theta}% -}(X_{1-i/K},1-i/K)+\mathcal{E}(Z_{1-i/K})\nabla_{X_{t}}g_{v_{\theta}}(X_{1-i/K% -},1-i/K)\right],$

-

$X^{S}$

-

$\log p_{v_{\theta}}(Z_{0})\geq-\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}[\log\frac{q(Z_{% -1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}+\sum_{i=1}^{N}% -\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{(i-1)/N}|Z_{i/N})}]$

-

$\displaystyle\mathbb{E}[L(\mathcal{T}_{\textrm{vanilla}}(X^{S}),\mathcal{T}(Z_% -{0}))+\alpha^{\Theta}L(\mathcal{T}_{\textrm{vanilla}}(X^{S}),Y)]+\mathcal{L}_{% -\textrm{FM-KT}},$

-

$\mathcal{L}_{\textrm{OFM-KT}}$

-

$B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor$

-

$\lim_{t\rightarrow 1}\nabla_{t}\alpha_{t}=+\infty$

-

$p_{v_{\theta}}(\cdot|Z_{i/N})$

-

$X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]$

-

$X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]=\textbf{{% -shuffle}}\left(X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]% -\right).$

-

$\mathcal{E}(Z_{1-(i+1)/K})=\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)% -\mathcal{K}(1-i/K)$

-

$\left[\frac{dX_{t}}{dt}-g_{v_{\theta}}(\mathcal{H}(t),t)\right]$

-

$Z_{1}=\alpha_{1}X_{S}$

-

$\mathit{b=100}$

-

$Z_{t}+\int_{t}^{s}g_{v_{\theta}}(Z_{\tau},\tau)d\tau$

-

$\displaystyle-\log p_{v_{\theta}}(Z_{0})\leq\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}% -\left[\log\frac{q(Z_{1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1% -/N})}+\sum_{i=1}^{N}\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{% -(i-1)/N}|Z_{i/N})}\right]$

-

$\{\hat{Z}_{i/N}\}_{i}$

-

$\mathcal{K}(t)$

-

$\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}% -||g_{v_{\theta}}(Z_{t},t)-\nabla_{t}Z_{t}||_{2}^{2}$

-

$\nabla_{t}\sigma_{t}$

-

$\nabla_{t}\sigma_{t}\equiv 0$

-

$\displaystyle=Z_{1-i/K}-(1/K)g_{v_{\theta}}(Z_{1-i/K},1-i/K)$

-

$g^{T}(\cdot)$

-

${}_{\textrm{50}}$

-

$\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}% -||g_{v_{\theta}}(Z_{t},t)-(X^{S}-X^{T})||_{2}^{2}.$

-

$\rho_{t}(Z):\mathbb{R}^{d}\times[0,1]\rightarrow\mathbb{R}^{d}$

-

$\psi(t)=\nabla_{X_{t}}g_{v_{\theta}}(X_{t},t)$

-

$\textrm{Law}(Z_{1})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{1})$

-

$\hat{Z}_{0}\sim\pi_{0}$

-

$B-\left\lfloor{\beta_{d}}B\right\rfloor$

-

$\alpha_{t}=a(\frac{b}{a})^{t}$

-

$\displaystyle\approx\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{% -0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}% -\mathbb{E}_{\hat{Z}_{i/N}\sim\int p_{v_{\theta}}(Z_{i/N}|Z_{1})q(Z_{1}|Z_{0})% -dZ_{1}}$

-

$\displaystyle\mathcal{L}_{\textrm{FM-KT++}}=\mathbb{E}_{(X^{S},X^{T},Y)}\frac{% -1}{N}\sum_{i=0}^{N-1}L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{\theta}}(Z% -_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),X^{T})$

-

$\{Z_{1-i/N}\}_{i}$

-

$\displaystyle=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)g_{v_{\theta}}(X_{1-i/K}+% -\mathcal{E}(Z_{1-i/K}),1-i/K)$

-

$\mathcal{K}(t)\geq 0$

-

$\mathcal{L}_{\textrm{guided}}$

-

$\displaystyle\left[D_{\mathrm{KL}}(q(Z_{(i-1)/N}|Z_{i/N},Z_{0})||p_{v_{\theta}% -}(Z_{(i-1)/N}|\hat{Z}_{i/N}))\right],\quad s.t.\quad\textrm{Law}(Z_{i/N})% -\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i/N})$

-

$\displaystyle\quad\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{% -\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Z_{0})}_{\textrm{the Online % -KD loss}}$

-

$\displaystyle\approx X_{1-(i+1)/K}+\mathcal{E}(Z_{1-i/K})+(1/K)\mathcal{K}(1-i% -/K)-(1/K)\mathcal{E}(Z_{1-i/K})\psi(1-i/K)$

-

${}_{\textrm{S}}$

-

$\displaystyle Z_{1-(i+1)/K}-X_{1-(i+1)/K}$

-

$Z_{1-(i+1)/K}\!=\!Z_{1-i/K}-g_{v_{\theta}}(Z_{1-{i/K}},1-i/K)dt$

-

$\frac{\alpha_{t}-\alpha_{t-\Delta t}}{t-\Delta t}$

-

$\nabla_{t}\alpha_{t}$

-

$\mathcal{E}(Z_{1-i/K})\geq 0$

-

$X^{T}\in\mathbb{R}^{B\times C\times H\times W}$

-

$\displaystyle\operatorname*{arg\,min}_{v_{\theta}}$

-

$g^{*}_{v_{\theta}}(\cdot)$

-

$dZ_{t}$

-

$i\!\geq\!1$

-

$\{\textrm{Law}(Z_{i/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_% -{i/N})\}_{i=0}^{N-1}$

-

$\displaystyle=X_{1-(i+1)/K}+\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)% -\mathcal{K}(1-i/K)$

-

$\sigma(t)=1-0.1t$

-

$\rho_{0}(X^{T})=\rho_{1}(X^{S})+\int_{\rho_{1}(Z)}^{\rho_{0}(Z)}\partial\rho_{% -t}(Z)$

- - - diff --git a/htmls/output_mathjax_example_10005.html b/htmls/output_mathjax_example_10005.html deleted file mode 100644 index 2432b2ae8eec0d4bfb387f791050a4e0905f78dc..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10005.html +++ /dev/null @@ -1,139 +0,0 @@ - - - - MathJax Example - - - - -

$L(\cdot,\cdot)$

-

$\displaystyle\int_{0}^{1}\mathbb{E}[||\partial\rho_{t}(Z)/\partial t-g_{v_{% -\theta}}(Z_{t},t)||]dt.$

-

$g_{v_{\theta}}(\cdot)$

-

${}_{\textrm{75}}$

-

$\hat{Z}_{0}$

-

${Z}_{i/N}$

-

$\displaystyle=\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{0})}{p% -_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}% -\mathbb{E}_{q(Z_{i/N}|Z_{0})}\mathbb{E}_{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}\left[% -\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{(i-1)/N}|Z_{i/N})}\right]$

-

$\displaystyle\mathcal{L}_{\textrm{FM-KT}}\!=\!\mathbb{E}[\frac{1}{N}\sum_{i=0}% -^{N-1}\!L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}\!\!-\!\!g_{v_{\theta}}(Z_{1\!% --\!i/N},1-i/N))/\!-\!\nabla_{t}\sigma_{t})$

-

$\sigma_{t}=\sqrt{1-\alpha_{t}^{2}},\ s..t.\quad a=19.9,b=0.1$

-

$\displaystyle=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)\left[g_{v_{\theta}}(X_{1-% -i/K},1-i/K)+\mathcal{E}(Z_{1-i/K})\psi(1-i/K)\right],$

-

$\hat{Z}_{i/N}$

-

$\alpha^{\Theta}$

-

$\textrm{Law}(Z_{(i-1)/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z% -}_{(i-1)/N})$

-

$\displaystyle+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{% -\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the % -ground truth label (optional)}},$

-

$\alpha_{t}=\textrm{exp}(-\frac{1}{4}a(1-t)^{2}-\frac{1}{2}b(1-t))$

-

$\lim_{t\rightarrow 1}\alpha_{t}=1$

-

$\{Z_{t}\}_{t}$

-

$\sigma(t)=1$

-

$X^{S}-X^{T}=\frac{dX_{t}}{dt}$

-

$\displaystyle\mathcal{E}(Z_{1-1/K})=(1/K)\mathcal{K}(1)$

-

$\sigma_{t}=\sqrt{1-\alpha_{t}^{2}}$

-

$\mathit{a=19.9}$

-

$\displaystyle\textrm{where}\quad Z_{1-i/N}=Z_{1-(i-1)/N}$

-

$\mathcal{H}(t)=\operatorname*{arg\,sup}_{X_{t}}\{||\frac{dX_{t}}{dt}-g_{v_{% -\theta}}(X_{t},t)||_{2}^{2}\}$

-

$||\frac{dX_{t}}{dt}-g_{v_{\theta}}(X_{t},t)||_{2}^{2}$

-

$t\sim\mathcal{U}[0,1]$

-

$\mathcal{T}_{\textrm{vanilla}}(\cdot)$

-

$\sim 10^{6}\times 10^{4}=10^{10}$

-

$\sim 1\,\mathrm{KB}$

-

$\displaystyle-~{}NCC(\mathcal{X}_{\mathrm{fx}},\mathcal{X}_{\mathrm{wp,n}})+% -\lambda\sum_{p\in\Omega}||\nabla\varphi(p)||^{2}$

-

$\varphi=\sum_{i=0}^{n}\varphi_{i}$

-

$\mathcal{X}_{\mathrm{wp}}$

-

$w_{k,i}=|m(i\in\omega)|^{g}$

-

$\mathcal{X}_{\mathrm{mv}}$

-

$0.7\times 0.7\times 3.0$

-

$\mathcal{X}_{\mathrm{fx}}$

-

$0.837\pm 0.021$

-

$1.2\times 1.2\times 3.0$

-

$\displaystyle\begin{split}\mathcal{X}_{\mathrm{wp,n}}=\mathcal{X}_{\mathrm{mv}% -}\circ\sum_{i=0}^{n}\varphi_{i}.\end{split}$

-

$\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$

-

$\mathcal{X}_{\mathrm{wp,1}}$

-

$\mathcal{X}_{\mathrm{wp,n}}$

-

$0.926\pm 0.012$

-

$0.847\pm 0.008$

-

$\centering\mathcal{X}_{\mathrm{wp}}=\mathcal{X}_{\mathrm{mv}}\circ\varphi% -\approx\mathcal{X}_{\mathrm{fx}}.\@add@centering$

-

$\mathcal{L}_{\mathrm{sim}}$

-

$0.866\pm 0.020$

-

$0.915\pm 0.012$

-

$230\times 230$

-

$\mathcal{X}_{\mathrm{wp,0}}$

-

$0.811\pm 0.023$

-

$0.807\pm 0.23$

-

$290\times 250$

-

$lr_{\mathrm{epoch}}=3\cdot 10^{-4}\cdot e^{-3\mathrm{epoch}/500}$

-

$H\times W\times L$

-

$0.920\pm 0.014$

-

$\%|J_{\phi}|\leq 0$

-

$\%|J_{\varphi}|<0$

-

$\mathcal{L}_{\mathrm{smooth}}$

-

$\displaystyle~{}\mathcal{L}_{\mathrm{sim}}+\lambda\mathcal{L}_{\mathrm{smooth}}$

-

$0.866\pm 0.013$

-

$94.65\%$

-

$\langle search\rangle$

-

$P(\hat{a}=\langle search\rangle|\theta^{\prime},q)$

-

$LM_{\theta^{\prime}}$

-

$LM_{\theta^{\prime}}:Q\mapsto\Omega\cup\{\langle search\rangle\}$

-

$LM_{\theta}:Q\mapsto\Omega$

-

$\psi\circ LM_{\theta}:Q\mapsto\Omega\cup\{\langle search\rangle\}$

-

$alpha=32$

-

$\underset{\theta}{\text{argmin}}\prod_{q\in Q}\left[P(\hat{a}=\langle search% -\rangle|\theta^{\prime},q)+\lambda P(\hat{a}\notin A|\theta^{\prime},q)\right]$

-

$(\langle search\rangle)$

-

$7e-5$

-

$LM_{\theta}$

-

$\psi(LM_{\theta}(q))=\begin{cases}\mathds{1}(\hat{a}),&\text{if }\hat{a}\in A% -\\ -\langle search\rangle,&\text{otherwise}\end{cases}$

-

$P(\hat{a}\notin A|\theta^{\prime},q)$

-

$733.5$

-

$TR^{(1)},TR^{(2)},\cdots,TR^{(m)}$

-

$\mathcal{V}_{\mathrm{target}}$

-

$\hat{P}_{v}$

-

$\mathcal{V}_{target}$

-

$\operatorname*{arg\,min}_{\theta}\sum_{G}\mathcal{L}(G,\theta).$

-

$p^{(1)},p^{(2)},\cdots,p^{(m)}$

-

$q^{(1)},q^{(2)},\cdots,q^{(n)}$

-

$\hat{\mathcal{V}}_{target}$

-

$\in TR^{i}$

-

$\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\notin\mathcal{V}_{\textrm{target}% -}}\log P(\hat{P}_{v}=0|\mathcal{G},\theta)$

-

$\displaystyle\mathcal{L}(G,\theta)=$

-

$\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\in\mathcal{V}_{\textrm{target}}}% -\log P(\hat{P}_{v}=1|\mathcal{G},\theta)$

-

$1,2,\cdots,m$

-

$TR^{i}$

-

$N=1923$

-

$p^{(1)}_{ref},p^{(2)}_{ref},\cdots,p^{(m)}_{ref}$

-

$\mathcal{V}_{\mathrm{target}}\subset\mathcal{V}$

-

$[0.1,0.01,0.001,0.0001]$

-

$\mathrm{EE}_{y}$

-

$\left[\mathcal{D}_{t},\mathcal{D}_{r}\right]$

-

$\left[\mathcal{D}_{r},\mathcal{H}_{r}\right]$

-

$W=F\cdot S$

-

$\mathcal{L}_{GPs}=\mathcal{L}_{G}+w_{P}\mathcal{L}_{P}+w_{S}\mathcal{L}_{s}$

-

$\eta_{t}>6$

- - - diff --git a/htmls/output_mathjax_example_10006.html b/htmls/output_mathjax_example_10006.html deleted file mode 100644 index d71f8ddb0a67572fffe2620cbeb10c9c12e85546..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10006.html +++ /dev/null @@ -1,156 +0,0 @@ - - - - MathJax Example - - - - -

$\left[\mathcal{D}_{t},\mathcal{H}_{t}\right]$

-

$\eta_{min}<6$

-

$X_{syn}$

-

$\displaystyle l_{3}+l_{2} -

$\delta S_{out}$

-

$F\cdot S=\tau\cdot\theta$

-

$\mathrm{EE}_{x}$

-

$\left[\mathcal{D}_{t},\mathcal{H}_{t}\right]=\left[\{d_{t,1},d_{t,2},...,d_{t,% -i}\},\{\eta_{t,1},\eta_{t,2},...,\eta_{t,i}\}\right]$

-

$\mathcal{L}_{P}=\frac{1}{N}\sum^{N}_{i=1}\left[(d_{r,i}-d_{t,i})^{2}+(\eta_{r,% -i}-\eta_{t,i})^{2}\right]$

-

$\left[\mathcal{H}_{t},\mathcal{H}_{r}\right]$

-

$|F_{in}\cdot\delta S_{in}|=|F_{out}\cdot\delta S_{out}|$

-

$\eta=\left|\frac{\delta S_{out}}{\delta\theta_{in}}\right|$

-

$(\mathrm{x}_{i},\mathrm{x}_{j})$

-

$i=100$

-

$d_{t}=1.0,\eta_{t}=2.0$

-

$d_{r},\eta_{r}$

-

$\delta\theta_{in}$

-

$|\tau_{in}\cdot\delta\theta_{in}|=|F_{out}\cdot\delta S_{out}|$

-

$\mathcal{L}_{s}=-\frac{1}{N}\sum^{N}_{i=1}\min{D\left(\mathrm{x}_{i},\mathrm{x% -}_{j}\right)}$

-

$|W_{in}|=|W_{out}|$

-

$\left[\mathcal{D}_{r},\mathcal{H}_{r}\right]=\left[\{d_{r,1},d_{r,2},...,d_{r,% -i}\},\{\eta_{r,1},\eta_{r,2},...,\eta_{r,i}\}\right]$

-

$y\left[i,j\right]=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}h\left[m,n% -\right]\cdot x\left[i-m,j-n\right]$

-

$0-\infty$

-

$\mathcal{M}=\{M_{i}|0\leq i -

$\mathcal{M}=\{M_{i|0\leq i -

$\{1e^{-5},2e^{-5}\}$

-

$M=10,50,70,100$

-

$R_{t+1,A_{t},B_{t}}$

-

$\displaystyle\mathbb{E}_{0}\left[\sqrt{\sum_{a}\left(\tilde{C}^{+}_{T,i}\right% -)^{2}}\right]\leqslant\mathbb{E}_{0}\left[\sqrt{\sum_{t=1}^{T}\sum_{a}\left(% -\tilde{\Re}_{t,a}\right)^{2}}\right]\leqslant\sqrt{\sum_{t=1}^{T}\frac{2(1+% -\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma_{t}}}$

-

$\displaystyle w_{t}(a,b)^{2}=\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{% -t}=a,B_{t}=b)=\frac{\beta^{\prime}_{t}\log(1+\sigma_{w}^{-2}\sigma^{2}_{t}(a,b% -))}{\log(1+\sigma_{w}^{-2})}\geqslant\beta^{\prime}_{t}\sigma^{2}_{t}(a,b).$

-

$\displaystyle+C\sum_{t=0}^{T}\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{% -\prime},A_{t},B_{t})\cup\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))$

-

$\mathbb{P}(\max_{i}y_{ti}\leqslant\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{% -\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+\sigma_{n}^{2}}})\geqslant 1-\delta% -_{1}$

-

$\displaystyle\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{R})$

-

$\displaystyle\leqslant\sum_{a}\left({\tilde{C}_{T-1,a}^{+}}\right)^{2}+\sum_{i% -}(\tilde{\Re}_{T,a})^{2}$

-

$\displaystyle{\tilde{C}_{T,a}}={\tilde{C}_{T-1,a}}+\tilde{\Re}_{T,a},$

-

$\displaystyle\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}% -\left[\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]$

-

$\pi=\pi^{\operatorname{adv-est}}$

-

$\phi(a,b)=P_{a}(\theta)/(P_{n_{0}}+P_{b}\mathds{1}(a=b))$

-

$\Re_{full}(T,\operatorname{adv},\tilde{R})$

-

$\displaystyle\mathbf{k}_{t}(a,b)$

-

$\Re_{t1}$

-

$\displaystyle\mathbb{P}(\tilde{R}_{t1}\geqslant\tilde{R}_{t2})$

-

$\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}},(r_{t})_{t})=% -\max_{a\in\mathcal{A}}\mathbb{E}\left[\sum_{t=0}^{T-1}r_{t}(a)-r_{t}(A_{t})\right]$

-

$c(\Delta,\sigma_{n})>0$

-

$GP(0,k((a,b),(a^{\prime},b^{\prime})))$

-

$X_{2}=X_{1}$

-

$\mathcal{N}(0.5,2.0)$

-

$r_{t}(a)=f_{\theta}(a,B_{t})$

-

$\tilde{f}_{t+1}(a,b)$

-

$U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}).$

-

$\displaystyle\geqslant\sum_{t=1}^{\infty}\log\mathbb{P}(N_{t}\leqslant 0)$

-

$\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$

-

$\displaystyle\Re_{t}(a)-\mathbb{E}_{t}\left[\tilde{\Re}_{t,a}\right]$

-

$\displaystyle\mathbf{R}_{t}$

-

$\displaystyle=\mathbb{E}\left[\sum_{t=0}^{T-1}\tilde{R}_{t+1}(a)-\tilde{R}_{t+% -1}(A_{t})\right].$

-

$f_{1},f_{2},f_{3}\in\mathcal{F}$

-

$\displaystyle\mathbb{P}(\omega_{t}|\Omega_{t-1})$

-

$\frac{x}{\log(1+x)}$

-

$X_{2}=[0,1]$

-

$\displaystyle\sum_{t=0}^{T-1}\mathbb{E}\left[f_{\theta}(a,B_{t})-f_{\theta}(A_% -{t},B_{t})\>|\>\theta\right]$

-

$\pi=\pi^{\operatorname{adv-OTS}}$

-

$r^{i}(a^{i},a^{-i})$

-

$\displaystyle\geqslant(1-\delta_{1})\mathbb{P}(\max_{i}x_{ti}\geqslant\frac{t}% -{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+% -\sigma_{n}^{2}}}\mid\epsilon)$

-

$\begin{cases}x_{ti}&\sim\mathcal{N}_{i}(0,1),\quad i=1,\ldots,M\\ -y_{ti}&\sim\mathcal{N}_{i}\left(\frac{t}{t+\sigma_{n}^{2}}(1-\Delta),\frac{% -\sigma_{n}^{2}}{\sigma_{n}^{2}+t}\right),\quad i=1,\ldots,M\end{cases}$

-

$(10^{-1})$

-

$\displaystyle=\mathbb{E}\left[\tilde{R}_{t+1}(a)-\tilde{R}_{t+1}(A_{t})\mid% -\theta\right],$

-

$\neg{\mathcal{E}}$

-

$N(\mu_{t}(a,b),\sigma_{t}(a,b))$

-

$f_{\theta}(a,b)$

-

$\tilde{R}=(\tilde{R}_{1},\ldots,\tilde{R}_{t+1},\ldots)$

-

$U=(U_{t}\>|\>t\in\mathbb{N})$

-

$A^{*}=\operatorname*{arg\,max}_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}% -\left[R_{t+1,a,B_{t}}\>|\>\theta\right]$

-

$reg_{1}=\tilde{R}_{1}-\tilde{R}_{1}^{T}X_{1}\cdot\mathbf{1},$

-

$\mathcal{O}(d\log T)$

-

$(a^{i},a^{-i})$

-

$[a^{i}]_{e}=U^{i}$

-

$0.15(\sum_{i=1}^{N}[a^{i}]_{e}/C_{e})^{4}$

-

$\displaystyle=\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})\left(% -\mathbb{P}(\Omega_{t-1})+\mathbb{P}(\bar{\Omega}_{t-1})\right)$

-

$(A_{t},B_{t})$

-

$[a^{i}]_{e}$

-

$\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}})$

-

$\displaystyle\left\langle\tilde{C}_{t-1}^{+},\tilde{\Re}_{t}\right\rangle$

-

$W_{t+1}=Y_{t+1,A_{t},B_{t}}-f_{\theta}(A_{t},B_{t})$

-

$\displaystyle\Re(T,\pi^{\operatorname{alg}},\pi^{B})=\mathbb{E}\left[\Re(T,\pi% -^{\operatorname{alg}},\pi^{B},\theta)\right]$

-

$\displaystyle=\mathbb{E}\left[\sum_{t}I_{t}(\theta;Z_{t})\right]=\sum_{t=0}^{T% --1}I(\theta;Z_{t}\>|\>Z_{0},\ldots,Z_{t-1})$

-

$[0,1]^{\mathcal{A}}$

-

$\displaystyle I_{t}\left(\theta;R_{t+1,A_{t},B_{t}}\mid A_{t}=a,B_{t}=b\right)% -=\frac{1}{2}\log\left(1+\frac{\sigma^{2}_{t}(a,b)}{\sigma_{w}^{2}}\right)$

-

$W_{t}=Y_{t+1,A_{t},B_{t}}-f_{\theta}(A_{t},B_{t})$

-

$\displaystyle\quad+C\sum_{t=0}^{T-1}\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{% -R},U^{\prime},A_{t},B_{t}))+2\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{% -t}))+\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}\right)$

-

$t_{e}(u)=c_{e}(1+0.5(\frac{u}{C_{e}})^{4}),$

-

$k(x,x)\leqslant 1$

-

$\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\left(1-\frac{f}{\sqrt{2\pi}% -(f^{2}+1)e^{f^{2}/2}}\right)^{M}$

-

$\displaystyle\leqslant\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{% -t},B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}(U^{\prime}_{t}(A_{t},B_{t}% -)-L_{t}(A_{t},B_{t}))$

-

$\mathcal{O}((\log T)^{d+1})$

-

$\displaystyle=\underbrace{\mathbb{E}\left[\tilde{R}_{t+1}(a)-\tilde{R}_{t+1}(A% -_{t})\>|\>\theta\right]}_{(I)}+\underbrace{\mathbb{E}\left[f_{\theta}({a,B_{t}% -})-\tilde{R}_{t+1}(a)\>|\>\theta\right]}_{(II)}+\underbrace{\mathbb{E}\left[% -\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})\>|\>\theta\right]}_{(III)}$

-

$\Re^{*}(T,\text{IWE-Hedge})=\mathcal{O}(\sqrt{T\mathcal{A}\log\mathcal{A}}).$

-

$x_{t}^{-i}$

-

$\Omega_{t-1}$

-

$\beta_{t}=\frac{2\beta^{\prime}_{t}}{\log(1+\sigma_{w}^{-2})}.$

-

$\Re_{t}=[0.5m_{1}+\sum_{k=2}^{t}m_{k},-0.5m_{1}]$

- - - diff --git a/htmls/output_mathjax_example_10007.html b/htmls/output_mathjax_example_10007.html deleted file mode 100644 index 110017e4faebf2feb502642076d58e80c592f813..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10007.html +++ /dev/null @@ -1,167 +0,0 @@ - - - - MathJax Example - - - - -

$\{z^{j}_{t+1}\}_{j\in[M]}$

-

$\mu(a,b)=\mathbb{E}\left[f_{\theta}(a,b)\right]$

-

$\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2\beta_{T}=\mathcal{O}(\log\mathcal{A}% -T+\log T+\log\log\mathcal{A}T).$

-

$(M=10,20,30)$

-

$\Re_{\operatorname{adv}}(a;T,\text{Hedge},\tilde{R})=\mathcal{O}(2c\sqrt{T\log% -\mathcal{A}})$

-

$R_{t+1,A_{t},B_{t}}=\theta_{A_{t},B_{t}}$

-

$a^{+}=\max\{a,0\}$

-

$o=f_{\theta}(a)+w$

-

$\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}\right)$

-

$B_{0:T}$

-

$\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}I_{t}(\theta;A_{t},B_{t},R_{t+1,A% -_{t},B_{t}})\right]$

-

$\displaystyle\gamma_{T}:=\max_{A_{0:T},B_{0:T}}I(\theta;A_{0},B_{0},\ldots,A_{% -T-1},B_{T-1})$

-

$f_{\theta}(a,b)=\phi(a,b)^{\top}\theta$

-

$\Re_{t}\in\mathbb{R}^{\mathcal{A}}$

-

$\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A% -_{t}}}-\frac{\hat{X}_{t,A_{t}}}{{X}_{t,A_{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)$

-

$\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t+1},t=0,1,\ldots)$

-

$\tilde{R}_{t}=\left[\tilde{R}_{t1},\tilde{R}_{t2}\right]$

-

$L=(L_{t}\>|\>t\in\mathbb{N})$

-

$\mathbb{E}_{t}\left[R_{t,a,b}^{2}\right]:=\mathbb{E}_{t}\left[(f_{\theta]}(a,b% -)+W_{t+1})^{2}\right]=\mathbb{E}_{t}\left[f_{\theta]}(a,b)^{2}+W_{t+1}^{2}% -\right]\leqslant 1+\sigma_{w}^{2}$

-

$\epsilon_{t}\sim\mathcal{N}(0,0.1)$

-

$\tilde{R}^{\operatorname{est}}$

-

$N_{t}=\sum\limits_{k=2}^{t}m_{k}\sim\mathcal{N}\left(-\sum\limits_{k=2}^{t}% -\frac{k}{k+\sigma_{n}^{2}}(1-\Delta),\ \sum\limits_{k=2}^{t}(1+\frac{\sigma_{n% -}^{2}}{k+\sigma_{n}^{2}})\right)\triangleq\mathcal{N}(\mu_{t},\sigma_{t}^{2}).$

-

$\displaystyle=\log\prod_{t=1}^{\infty}\mathbb{P}(\omega_{t}|\Omega_{t-1})=\sum% -_{t=1}^{\infty}\log\mathbb{P}(\omega_{t}|\Omega_{t-1})$

-

$\displaystyle=\sum_{t=1}^{\infty}\log\Phi\left(\frac{\sum_{k=2}^{t}\frac{k}{k+% -\sigma_{n}^{2}}(1-\Delta)}{\sqrt{\sum_{k=2}^{t}(1+\frac{\sigma_{n}^{2}}{k+% -\sigma_{n}^{2}})}}\right)$

-

$\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{R})$

-

$\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}\left(\tilde{\Re}_{t,a}% -\right)^{2}\right]$

-

$\tilde{R}_{t}(2nd)$

-

$X_{1}=[0.5,0.5]$

-

$\mathcal{E}(c)$

-

$\displaystyle=\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}% -\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta\right]\right]$

-

$\textbf{r}_{t}=(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}}$

-

$m_{1}<0$

-

${N}(\mu_{p},\Sigma_{p})$

-

$\mathcal{O}\big{(}\sqrt{T\mathcal{A}}+\sqrt{\gamma_{T}\beta T}\big{)}$

-

$c^{\prime}=-0.62$

-

$(b^{i},b^{-i})$

-

$U^{\prime}_{t}\geqslant L_{t}$

-

$r^{i}(a^{i},a^{-i})=-\ell^{i}(a^{i},a^{-i})$

-

$H_{t+1}=(H_{t},A_{t},B_{t},R_{t+1,A_{t},B_{t}})$

-

$\mathcal{O}\left(\left(\sqrt{\log\mathcal{A}}+\sqrt{\log(\mathcal{A}T)\log(T)^% -{d+1}}\right)\sqrt{T}\right)$

-

$\displaystyle\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t}=a,B_{t}=b)% -\geqslant\beta^{\prime}_{t}\sigma^{2}_{t}(a,b).$

-

$\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t}):=\{\tilde{R}_{t+1}(A_{t}% -)\leqslant U^{\prime}_{t}(A_{t},B_{t})\}.$

-

$X_{1}=\hat{X}_{1}$

-

$k_{\rm L}(\cdot,\cdot)$

-

$(a+b)^{+}\leqslant(a^{+}+b)^{+}\leqslant\left|a^{+}+b\right|.$

-

$\tilde{R}_{t+1}$

-

$\displaystyle\Re^{*}(T,\pi,\theta)$

-

$\displaystyle=\sum_{a}(\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/\mathcal{A})\sum_{b}% -Y_{t,b}f_{\theta}(a,b)$

-

$\sqrt{\gamma_{T}}$

-

$\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant% -\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b),\forall a\in\mathcal{A}\mid H_{t})% -\leqslant 2\mathcal{A}\exp(-\beta^{\prime}_{t}/2),$

-

$\displaystyle=\sum_{a}\frac{\mathbb{E}_{t}\left[R_{t+1,a,B_{t}}^{2}\right]}{X_% -{t,a}}+\sum_{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\mathbb{E}_{t}\left[R_{t+1,A_{t}% -,B_{t}}^{2}\right]\left(\left|\mathcal{A}\right|\hat{X}_{t,a}-2\right)$

-

$\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t},t\in\mathbb{Z}_{++})$

-

$\displaystyle\leqslant\mathbb{E}_{0}\left[\tilde{C}_{T,a}^{+}\right]=\mathbb{E% -}_{0}\left[\sqrt{(\tilde{C}_{T,a}^{+})^{2}}\right]\leqslant\mathbb{E}_{0}\left% -[\sqrt{\sum_{a}\left(\tilde{C}_{T,a}^{+}\right)^{2}}\right],$

-

$M_{1}=M_{2}=\ldots=M_{T}=M=\mathcal{O}(\log\mathcal{A}T)$

-

$\sigma(H_{t},A_{t},R_{t+1,A_{t},B_{t}})$

-

$\mathcal{R}:\mathbb{R}\mapsto[0,1]$

-

$\displaystyle\mathbb{E}_{0}\left[\sqrt{\sum_{a}\left(\tilde{C}^{+}_{T,i}\right% -)^{2}}\right]\leqslant\mathbb{E}_{0}\left[\sqrt{\sum_{t=1}^{T}\sum_{a}\left(% -\tilde{\Re}_{t,a}\right)^{2}}\right]$

-

$H_{t},B_{t}$

-

$\tilde{R}_{t2}$

-

$\displaystyle\leqslant\min_{t>0}\frac{\exp(\sigma^{2}t^{2}/2)}{\exp(tc)}=\exp(% --c^{2}/\sigma^{2})$

-

$\displaystyle\mu_{t+1}=\Sigma_{t+1}\left(\Sigma_{t}^{-1}\mu_{t}+\frac{R_{t+1,A% -_{t},B_{t}}}{\sigma_{w}^{2}}\phi(A_{t},B_{t})\right)$

-

$\displaystyle\mathbb{P}(X-\mu\geqslant c)$

-

$\pi_{t}(H_{t})$

-

$v\in\mathbb{R}^{\mathcal{A}}$

-

$\sigma_{p}\leqslant 1$

-

$\hat{X}_{t+1}$

-

$\tilde{R}_{t+1}(a)\in[0,C]$

-

$\tilde{R}_{t}=\left[z_{t},\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{% -\sigma_{n}^{2}}{\sigma_{n}^{2}+t}}z^{\prime}_{t}\right],$

-

$H_{t},A_{t},B_{t}$

-

$\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{{X}_{t,a}}+\sum% -_{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|\hat{X}_{t,a}% --2\right)\right)$

-

$f(x)=\theta^{\top}x,\theta\sim N(0,\sigma_{0}I)$

-

$\mathbb{E}_{t}\left[\cdot\right]=\mathbb{E}\left[\cdot\mid H_{t},\theta\right]$

-

$\bm{a}=(a^{i},a^{-i})$

-

$\displaystyle\geqslant\sum_{t=1}^{\infty}\left(-\frac{1}{\sqrt{2\pi}f_{t}(% -\Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$

-

$\displaystyle V^{*}=\max_{P\in\mathcal{D}(\mathcal{A})}\min_{Q\in\mathcal{D}(% -\mathcal{B})}\mathbb{E}_{A\sim P,B\sim Q}\left[f_{\theta}(A,B)\right],$

-

$\mathbb{P}(f_{\theta}\in\mathcal{F}_{t})\geqslant 1-2\mathcal{A}\exp(-\beta^{% -\prime}_{t}/2).$

-

$\operatorname{clip}_{[-c,c]}(x)\geqslant\min(x,c)$

-

$0.01\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$

-

$\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A_{t},B_{t}}\>|\>\theta\right]$

-

$\text{SINR}(a,b;\theta)=\phi(a,b)^{T}\theta$

-

$\delta=1/\sqrt{t}$

-

$\displaystyle\lim_{t\to\infty}\log\mathbb{P}(\Omega_{t})$

-

$\text{Regret}^{i}(T)=\frac{1}{T}\max_{a\in\Delta^{\mathcal{D}(\mathcal{A}^{i})% -}}\mathbb{E}\left[\sum_{t=1}^{T}\phi\left(a,x_{t}^{-i}\right)-\phi\left(x_{t}^% -{i},x_{t}^{-i}\right)\right],$

-

$\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv})+\sqrt{% -\beta I(\theta;H_{T})T},$

-

$\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant 2^{4% -/3}(1+\sigma_{w}^{2})^{1/3}\left|\mathcal{A}\right|^{2/3}T^{2/3}.$

-

$g_{t}(\cdot):\mathbb{R}_{+}^{\mathcal{A}}\times\mathbb{R}_{+}^{\mathcal{A}}% -\mapsto\mathbb{R}_{+}^{\mathcal{A}}$

-

$m_{t}=z_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)-\sqrt{\frac{\sigma_{n}^{2}}{t% -+\sigma_{n}^{2}}}z^{\prime}_{t}=z_{t}-\sqrt{\frac{\sigma_{n}^{2}}{t+\sigma_{n}% -^{2}}}z^{\prime}_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)$

-

$f_{t}(\Delta,\sigma_{n})=\frac{(1-\Delta)\left(t-\sigma_{n}^{2}\ln{(t+\sigma_{% -n}^{2})}+2\sigma_{n}^{2}\ln{\sigma_{n}}-1/(\sigma_{n}^{2}+1)\right)}{\sqrt{t+% -\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}-2\sigma_{n}^{2}\ln{\sigma_{n}}-(\sigma_{% -n}^{2}+2)/(\sigma_{n}^{2}+1)}},$

-

$\displaystyle\leqslant\sum_{a}\left|\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/% -\mathcal{A}\right|$

-

$\tilde{R}_{t1}\geqslant\tilde{R}_{t2}$

-

$(\gamma_{t})_{t\geqslant 0}$

-

$\displaystyle\mathbb{P}(\mathcal{E}_{t}(\tilde{R},U,B_{t})\mid H_{t},B_{t})$

-

$\phi(a,b)\in\mathbb{R}^{d}$

-

$\sigma^{2}_{t}(a,b)=k((a,b),(a,b))-\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}% -+\sigma^{2}{\bm{I}}_{t})\mathbf{k}_{t}(a,b)$

-

$\tilde{C}^{+}_{t-1,a}=0$

-

$\mathcal{A}_{J}=\mathcal{F}$

-

$(r_{t})_{t}$

-

$\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\exp{\left(\frac{-Mf}{\sqrt{% -2\pi}(f^{2}+1)e^{f^{2}/2}}\right)}$

-

$\displaystyle(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{t}))$

-

$Y_{t}=\max_{y}X_{t}^{\top}\theta y$

- - - diff --git a/htmls/output_mathjax_example_10008.html b/htmls/output_mathjax_example_10008.html deleted file mode 100644 index 83fb408bf05e2028410a76e1841c2d9e83cbec1c..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10008.html +++ /dev/null @@ -1,172 +0,0 @@ - - - - MathJax Example - - - - -

$X_{t,a}$

-

$U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad -U% -^{\prime}=(\mu_{t}(a,b)+\sqrt{2\log(M\sqrt{t})}\sigma_{t}(a,b):t\in\mathbb{N}).$

-

$(f_{\theta}(a,b):(a,b)\in\mathcal{A}\times\mathcal{B})$

-

$\mathcal{O}\big{(}\sqrt{T\mathcal{A}}\big{)}$

-

$a^{i},b^{i}\in\mathcal{A}^{i}$

-

$\displaystyle=\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime}_{t}}}+% -1\right)\sqrt{\beta_{t}I_{t}(\theta;A_{t},B_{t},R_{t+1,A_{t},B_{t}})}$

-

$\displaystyle\geqslant\sum_{t=1}^{\infty}\log\left(1-\frac{1}{\sqrt{2\pi}f_{t}% -(\Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$

-

$\Delta,\sigma_{n}$

-

$\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime% -}_{t}}}+1\right)\sqrt{\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t},B_{t% -})}$

-

$\left|\mathcal{A}\right|$

-

$k(x,x^{\prime})=\exp(-(2l^{2})^{-1}\left\|x-x^{\prime}\right\|^{2}s)$

-

$\Phi(\beta^{\prime}_{t})^{M}=1/\sqrt{t}$

-

$\tilde{R}_{t}\in[0,1]^{\mathcal{A}}$

-

$m_{t}=\tilde{R}_{t1}-\tilde{R}_{t2}$

-

$\overline{x}_{T}=\frac{1}{T}\sum_{t=1}^{T}x_{t},\quad\overline{y}_{T}=\frac{1}% -{T}\sum_{t=1}^{T}y_{t}.$

-

$k((a,b),(a^{\prime},b^{\prime}))$

-

$\left\langle\tilde{C}_{t-1}^{+},\tilde{\Re}_{t}\right\rangle\leqslant 0$

-

$N(0,\sigma_{w}^{2})$

-

$\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}R_{t+1,A_{t},B_{t}}^{2}\left(% -\frac{1}{X_{t,A_{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_{t}}^{2}}+\left|% -\mathcal{A}\right|\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}}\right)\right]$

-

$\displaystyle\tilde{\Re}_{t,a}=\frac{\mathbb{I}_{A_{t}=a}R_{t+1,A_{t},B_{t}}}{% -X_{t,a}}-R_{t+1,A_{t},B_{t}}\frac{\hat{X}_{t,A_{t}}}{X_{t,A_{t}}}$

-

$(X_{a})_{a\in\mathcal{A}}$

-

$Y_{t}=\min_{y}X_{t}^{\top}\theta y$

-

$\log(\text{average regret})\propto(1/2)\log(M+N)$

-

$H_{t}=\left(A_{0},B_{0},Y_{1,A_{0},B_{0}},\ldots,A_{t-1},B_{t-1},Y_{t,A_{t-1},% -B_{t-1}}\right)$

-

$U^{\prime}\geqslant U\geqslant L$

-

$\mathcal{O}(T^{d(d+1)/(2\nu+d(d+1))}(\log T))$

-

$\mathcal{O}\big{(}\sqrt{T\mathcal{A}\log\mathcal{A}}\big{)}$

-

$\displaystyle:=\operatorname{clip}_{[0,1]}\left(\tilde{f}_{t+1}^{\operatorname% -{OTS}}(a,B_{t})\right).$

-

$n_{t}(a,b)$

-

$\tilde{\Re}_{t}$

-

$\Phi(x)\leqslant 1-\frac{x}{\sqrt{2\pi}(x^{2}+1)e^{x^{2}/2}}$

-

$\displaystyle=\sum_{t=1}^{\infty}\log\Phi(-\frac{\mu_{t}}{\sigma_{t}})$

-

$g_{t}:\Delta^{\mathcal{A}}\times[0,1]^{\mathcal{A}}\mapsto\Delta^{\mathcal{A}}$

-

$c=0.54$

-

$\tilde{\mathcal{O}}(\sqrt{MN/T})$

-

$X_{t+1}=g_{t}(X_{t},(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}})$

-

$\displaystyle=\sum_{a}(\hat{X}_{t,a}-X_{t,a})\sum_{b}Y_{t,b}f_{\theta}(a,b)$

-

$\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))$

-

$Z_{t+1}$

-

$\displaystyle\geqslant(1-\delta_{1})\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y% -_{i}\mid\epsilon)$

-

$X_{2}=[1,0]$

-

$\mathcal{O}\big{(}\sqrt{T\log\mathcal{A}}\big{)}$

-

$\displaystyle\mathcal{F}_{t}:=\left\{f_{\theta}:\left|f_{\theta}(a,B_{t})-\mu_% -{t}(a,B_{t})\right|\leqslant\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t}),% -\forall a\in\mathcal{A}\right\}$

-

$w\sim N\left(0,\sigma_{w}^{2}\right)$

-

$\frac{\sqrt{2\log M\sqrt{t}}}{\sqrt{\beta^{\prime}_{t}}}=\sqrt{\frac{2\log M% -\sqrt{t}}{\log\mathcal{A}\sqrt{t}}}$

-

$G_{J}$

-

$\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})\leqslant(\mathds{1% -}_{\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})})(U^{\prime}_{t}(A_{t% -},B_{t})-f_{\theta}(A_{t},B_{t}))+C(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R% -},U^{\prime},A_{t},B_{t})}).$

-

$\displaystyle=\sum_{t=1}^{\infty}\log\mathbb{P}(\mu_{t}+\sigma_{t}Z\leqslant 0% -),\quad Z\sim\mathcal{N}(0,1)$

-

$\text{regret-matching}^{+}(\text{RM}^{+})$

-

$\displaystyle f_{\theta}(a,B_{t})-f_{\theta}(A_{t},B_{t})=\underbrace{\tilde{R% -}_{t+1}(a)-\tilde{R}_{t+1}(A_{t})}_{(I)\leavevmode\nobreak\ \operatorname{adv}% -_{t+1}(a)}+\underbrace{f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)}_{(II)% -\leavevmode\nobreak\ \operatorname{pess}_{t+1}(a)}+\underbrace{\tilde{R}_{t+1}% -(A_{t})-f_{\theta}(A_{t},B_{t})}_{(III)\operatorname{est}_{t+1}}$

-

$\sum_{a}\tilde{C}_{t-1,a}^{+}\leqslant 0$

-

$\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant C(1-\mathds{1}_{% -\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t}% -)}),\quad\forall a\in\mathcal{A}.$

-

$\displaystyle\leqslant\sqrt{T\left(8\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2% -\beta_{T}\right)I(\theta;H_{T})}$

-

$\mathbb{P}(\tilde{R}_{t1}\geqslant\tilde{R}_{t2})$

-

$\displaystyle\Re(T,\pi^{\operatorname{alg}},\pi^{B})$

-

$\displaystyle=\mathbb{P}(0.5m_{1}+\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$

-

$(H_{t},A_{t},B_{t},\theta)$

-

$\beta=\mathcal{O}(\log\mathcal{A}T)$

-

$\sigma_{n}=0.1$

-

$[-c,c]^{\mathcal{A}}$

-

$\displaystyle\geqslant\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$

-

$\lim_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant c>0,$

-

$\displaystyle=\mathbb{P}\left(\max_{j\in[M]}z^{j}_{t+1}\geqslant\sqrt{\beta^{% -\prime}_{t}}\right)$

-

$\sigma_{t}(a,b)=\left\|\phi(a,b)\right\|_{\Sigma_{t}}$

-

$\mathbb{P}(\Omega_{t})\geqslant\lim\limits_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant -c$

-

$\displaystyle\operatorname{NashRegret}_{t}=\mathbb{E}\left[V^{*}-R_{t+1,A_{t},% -B_{t}}\right]$

-

$\tilde{R}_{t}\in\mathbb{R}^{\mathcal{A}}$

-

$\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\Phi^{M}\left(\frac{t}{t+% -\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+% -\sigma_{n}^{2}}}\right)$

-

$I(\theta;H_{T})=I(\theta;A_{0},B_{0},\ldots,A_{T-1},B_{T-1})\leqslant\gamma_{T}.$

-

$\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant% -\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b)\mid H_{t})$

-

$r=(\sqrt{2\nu}/l)\left\|x-x^{\prime}\right\|$

-

$(a,b),(a^{\prime},b^{\prime})\in\mathcal{A}\times\mathcal{B}$

-

$\mu_{t}(a,b)$

-

$\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}\leqslant\sum_{t=1}^{T}\sum_{a}% -\left(\tilde{\Re}_{t,a}\right)^{2}$

-

$\tilde{R}_{t}$

-

$\displaystyle\tilde{f}^{\operatorname{OTS}}_{t+1}(a,B_{t}):=(\max_{j\in[M]}z_{% -t+1}^{j})\cdot\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t})+\mu_{t}(a,B_{t})% -\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \tilde{R}_{t+1}(a)=% -\operatorname{clip}_{[-c,c]}\left(\tilde{f}_{t+1}^{\operatorname{OTS}}(a,B_{t}% -)\right),\forall a\in\mathcal{A}.$

-

$\displaystyle\tilde{R}^{\operatorname{OTS}}_{t+1}(a)$

-

$\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A% -_{t}}}-\frac{\tilde{C}_{t-1,A_{t}}^{+}/\sum_{a}\tilde{C}_{t-1,a}^{+}}{{X}_{t,A% -_{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)=0$

-

$\gamma=\sqrt[3]{((1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2})/{2T}}$

-

$\tilde{R}_{t+1}=E(H_{t+1},Z_{t+1})\in\mathbb{R}^{\mathcal{A}}.$

-

$\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv},\tilde{R}% -)+\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{% -t}))\right]$

-

$\displaystyle\leqslant\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_% -{t})-L_{t}(A_{t},B_{t}))\right]$

-

$\displaystyle=\mathbb{E}\left[f_{\theta}({a,B_{t}})-\tilde{R}_{t+1}(a)\mid% -\theta\right],$

-

$(III)$

-

$\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant\mathds{1}_{% -\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}(f_{\theta}(a,B_{t})-U_{t}(a,B_{t}))+C(% -1-\mathds{1}_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})}).$

-

$\Re(T)\geqslant 2\mathbb{P}(\Omega_{T})\Delta\cdot T.$

-

$\displaystyle\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\>|\>\theta% -\right]=(I)+(II)+(III)$

-

$reg_{t}$

-

$\displaystyle=I(\theta;Z_{0},\ldots,Z_{T-1})=I(\theta;H_{T})$

-

$w\in\mathbb{R}_{+}$

-

$\mathbb{P}(\Omega_{t})$

-

$\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime% -}_{t}}}+1\right)\sqrt{\beta^{\prime}_{t}}\sigma_{t}(A_{t},B_{t})$

-

$\Re_{\operatorname{full}}(a;T,\text{RM},\tilde{R})=\mathcal{O}(2c\sqrt{T% -\mathcal{A}})$

-

$\sigma^{2}_{t}(a,b)\leqslant k((a,b),(a,b))\leqslant 1$

-

$\displaystyle\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{% -t}))\leqslant\frac{1}{\sqrt{t}}.$

-

$\displaystyle\leqslant 2\exp\left(-\frac{\beta^{\prime}_{t}}{2}\right)$

-

$\mathbb{P}(f_{\theta}(a,B_{t})\geqslant\tilde{R}_{t+1}(a)\mid\theta)\leqslant% -\mathcal{O}(1/\sqrt{T}).$

-

$\displaystyle=\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y_{i})$

-

$\sqrt{2\log\mathcal{A}\sqrt{T}},0.2\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}},% -0.05\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$

-

$\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})$

- - - diff --git a/htmls/output_mathjax_example_10009.html b/htmls/output_mathjax_example_10009.html deleted file mode 100644 index 0f2c1eedafd13b5674cdff751357b552ddd4a22e..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10009.html +++ /dev/null @@ -1,169 +0,0 @@ - - - - MathJax Example - - - - -

$\mathcal{O}\big{(}\sqrt{T\log\mathcal{A}}+\sqrt{\gamma_{T}\beta T}\big{)}$

-

$\displaystyle\sim N(\mu_{t}(a,B_{t}),\sigma_{t}(a,B_{t})),j\in[M_{t+1}]$

-

$\ell^{i}(a^{i},a^{-i})=\sum_{e\in\mathcal{E}}[a^{i}]_{e}t_{e}([a^{i}]_{e}+[g(a% -^{-i})]_{e}).$

-

$a^{i}\in\mathcal{A}^{i}\subset\mathbb{R}^{|\mathcal{E}|}$

-

$\beta=\log(\mathcal{A}T)$

-

$K^{i}((a^{i},a^{-i}),(b^{i},b^{-i}))=k_{\rm L}(a^{i},b^{i})k_{\rm P}(a^{i}+g(a% -^{-i}),b^{i}+g(b^{-i})),$

-

$\tilde{R}_{t2}=\max\limits_{i}y_{ti}$

-

$\beta_{t}=\sqrt{2\log{\mathcal{A}\sqrt{t}}}$

-

$\tilde{R}^{OTS}_{t+1}(1st)$

-

$\log(\text{average regret})\propto\frac{1}{2}\log(MN)$

-

$\tilde{C}_{t,a}=\sum_{s=0}^{t}\tilde{\Re}_{s,a}$

-

$\mathcal{O}(\sqrt{T\mathcal{A}\log\mathcal{A}})$

-

$X_{t+1}=g_{t}(X_{t},r_{t})$

-

$min(x,c)$

-

$\pi^{\operatorname{alg}}=(\pi_{t})_{t\in\mathbb{N}}$

-

$a^{-i},b^{-i}\in\mathcal{A}^{-i}$

-

$\displaystyle\begin{cases}\textrm{Hedge:}\ &g_{t,a}(X_{t},r_{t})=X_{t,a}\exp% -\left(\eta_{t}r_{t}(a)\right),\\ -\textrm{RM:}\ &g_{t,a}(X_{t},r_{t})=\max\left(0,\sum\limits_{s=0}^{t}r_{t}(a)-% -r_{t}(A_{s})\right),\end{cases}$

-

$\displaystyle\hat{X}_{t+1,a}=\begin{cases}{\tilde{C}^{+}_{t,a}}/{\sum_{a\in% -\mathcal{A}}\tilde{C}_{t,a}^{+}},&\text{if }\sum_{a\in\mathcal{A}}\tilde{C}_{t% -,a}^{+}>0,\\ -\text{arbitrary vector on simplex, e.g. }1/\mathcal{A},&\text{otherwise}\end{cases}$

-

$\sigma_{t}^{2}(a,b)\leqslant\frac{1}{\log(1+\sigma_{w}^{-2})}\log(1+\sigma_{w}% -^{-2}\sigma^{2}_{t}(a,b)).$

-

$\displaystyle\left((a+b)^{+}\right)^{2}\leqslant(a^{+})^{2}+2(a^{+})b+b^{2}$

-

$[a^{i}]_{e}+[g(a^{-i})]_{e}$

-

$\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}U^{\prime}_{t}(A_{t},B_{t})-L_{t}% -(A_{t},B_{t})\right]$

-

$\displaystyle=\mathbb{E}\left[\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})% -\mid\theta\right].$

-

$\Sigma_{p}=\sigma_{p}I$

-

$\mathbb{P}(\Omega_{t})=\mathbb{P}(\omega_{1})\mathbb{P}(\omega_{2}|\Omega_{1})% -\ldots\mathbb{P}(\omega_{t}|\Omega_{t-1})$

-

$\displaystyle=[R_{1,A_{0},B_{0}},\ldots,R_{t,A_{t-1},B_{t-1}}]^{\top}$

-

$\displaystyle\geqslant\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})% -\mathbb{P}(\Omega_{t-1})+\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\bar{\Omega% -}_{t-1})\mathbb{P}(\bar{\Omega}_{t-1})$

-

$\displaystyle=\Re_{\operatorname{full}}(T,\operatorname{adv},\tilde{R})+% -\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}\left[% -\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]+\sum_{t=0}^{T-1}\mathbb{E% -}\left[\operatorname{est}_{t+1}\right]$

-

$\operatorname{NashRegret}_{t}$

-

$\theta=\begin{bmatrix}1&1-\Delta\\ -1-\Delta&1\end{bmatrix},$

-

$\displaystyle\operatorname{NashRegret}(T)=\sum_{t=1}^{T}\operatorname{% -NashRegret}_{t}$

-

$I(\theta;R_{t+1,A_{t},B_{t}}|H_{t})$

-

$f_{\theta}(a,b)=\theta(a,b)$

-

$P_{X}(i)={X_{i}}/\sum_{i\in\mathcal{A}}X_{i}$

-

$\displaystyle\sigma_{t}(a,b)=\sqrt{\frac{\sigma_{w}^{2}}{\sigma_{w}^{2}/\sigma% -^{2}_{p}(a,b)+n_{t}(a,b)}}$

-

$KL(\overline{x}_{T},x^{)}$

-

$\Re_{\operatorname{full}}$

-

$k((a,b),(a,b))=\phi(a,b)^{\top}\Sigma_{p}\phi(a,b)$

-

$\{\tilde{R}_{t}:t\in\mathbb{Z}_{++}\}$

-

$\displaystyle\mathbb{E}\left[\tilde{\Re}_{t,a}\mid\theta,H_{t}\right]=\mathbb{% -E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]-\sum_{a}\hat{X}_{t,a}% -\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]$

-

$\displaystyle\tilde{f}^{\operatorname{OTS}}_{t+1}(a,B_{t})$

-

$I(\theta;H_{T})$

-

$\mathbb{P}\left(\eta_{j}\leqslant w\right)=\mathbb{P}({\eta_{j}}/{\sigma}% -\leqslant{w}/{\sigma})=\Phi({w}/{\sigma}).$

-

$\displaystyle\Re(T)\geqslant 2\mathbb{P}(\Omega_{T})\Delta\cdot T$

-

$\phi(a,b)$

-

$\mathcal{E}_{t}(\tilde{R},U,B_{t}):=\{\tilde{R}_{t+1}(a)\geqslant U_{t}(a,B_{t% -}),\forall a\in\mathcal{A}\}.$

-

$\displaystyle\stackrel{{\scriptstyle(ii)}}{{\geqslant}}\mathbb{P}\left(\left(% -\max_{j\in[M]}z^{j}_{t+1}\right)\sigma_{t}(a,B_{t})\geqslant\sqrt{\beta^{% -\prime}_{t}}\sigma_{t}(a,B_{t}),\forall a\in\mathcal{A}\>|\>H_{t},B_{t}\right)$

-

$\phi(a,b)=e_{a,b}$

-

$\displaystyle\mathbb{E}_{0}\left[\tilde{C}_{T,a}\right]$

-

$f(t,\Delta,\sigma_{n})=\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2% -\sigma_{n}^{2}\log(M/\delta_{1})}{t+\sigma_{n}^{2}}}$

-

$\mathcal{A}_{R}=\mathcal{F}\times\mathcal{F}\times\mathcal{F}$

-

$B_{t}\sim P_{Y_{t}}$

-

$\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t}))$

-

$\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}},(r_{t})_{t})=% -\max_{a\in\mathcal{A}}\mathbb{E}\left[\sum_{t=0}^{T-1}r_{t}(a)-r_{t}(A_{t})% -\right].$

-

$\tilde{R}_{t}(1nd)$

-

$\lim_{t\to\infty}\log\mathbb{P}(\Omega_{t})\geqslant\log c^{\prime}>-\infty$

-

$0\leqslant L_{t}\leqslant U_{t}\leqslant C$

-

$\sigma_{n},M$

-

$0\leqslant{\hat{X}_{t,a}}/{X}_{t,a}\leqslant 1/(1-\gamma_{t})$

-

$\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\Phi^{M}(f)$

-

$X_{t+1,a}\propto X_{t,a}\exp(\eta_{t}\tilde{R}_{t+1}(a))$

-

$\displaystyle\mathbb{P}(\left|f_{\theta}(a,B_{t})-{\mu_{t}(a,B_{t})}\right|% -\geqslant\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b),\forall a\in\mathcal{A}\mid H% -_{t})\leqslant 2\mathcal{A}\exp(-\beta^{\prime}_{t}/2).$

-

$X_{2}\propto\max(reg_{1},0)$

-

$f_{\theta}(a,b)=\theta_{a,b}$

-

$\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{\left|\mathcal{A}% -\right|}{\gamma_{t}}+\min(\frac{\left|\mathcal{A}\right|}{\gamma_{t}},\frac{% -\left|\mathcal{A}\right|}{1-\gamma_{t}})(\left|\mathcal{A}\right|-2)\right)% -\leqslant\frac{2(1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma_{t}}$

-

$\displaystyle\tilde{f}^{\operatorname{TS},j}_{t+1}(a,B_{t})$

-

$B_{0:T}=(B_{0}=b_{0},B_{1}=b_{1},\ldots,B_{T-1}=b_{T-1})$

-

$\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})\geqslant\mathbb{P}(% -\sum_{k=2}^{t}m_{k}\leqslant 0|\bar{\Omega}_{t-1})$

-

$(W_{t}:t\in\mathbb{Z}_{++})$

-

$\displaystyle\tilde{h}(a)=\mathbb{I}_{A=a}\frac{h(A)}{X_{a}},\forall a\in% -\mathcal{A}$

-

$\displaystyle\mathbb{E}_{t}\left[R_{t+1,A_{t},B_{t}}^{2}\left(\frac{1}{X_{t,A_% -{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_{t}}^{2}}+\left|\mathcal{A}\right|% -\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}}\right)\right]$

-

$(10$

-

$f_{\theta}(a,B_{t})$

-

$\displaystyle\Re^{*}(T,\pi,\theta)\leqslant\Re_{\operatorname{full}}(T,% -\operatorname{adv},\tilde{R}^{\operatorname{est}})+\sqrt{\log(\mathcal{A}T)I(% -\theta;H_{T})T}$

-

$\displaystyle I(\theta;o)=H(o)-H(o\mid\theta)=\frac{1}{2}\log 2\pi e(\sigma(a)% -^{2}+\sigma_{w}^{2})-\frac{1}{2}\log 2\pi e\sigma_{w}^{2}=\frac{1}{2}\log(1+% -\sigma_{w}^{-2}\sigma(a))$

-

$\tilde{r}t=A{ij}+\epsilon_{t}$

-

$\operatorname{adv}$

-

$M=\frac{\log(\mathcal{A}\sqrt{T})}{\log\frac{1}{\Phi(\beta^{\prime}_{t})}},% -\beta^{\prime}_{t}=2\log\mathcal{A}\sqrt{T}$

-

$\tilde{R}_{t+1,A_{t},B_{t}}$

-

$X_{t}=[0,1],\forall t\geqslant 2$

-

$\mathcal{O}\big{(}T^{2/3}\mathcal{A}^{2/3}\big{)}$

-

$\phi:\mathcal{A}\times\mathcal{B}\mapsto\mathbb{R}^{d}$

-

$\mathbb{P}\left(\max_{j\in[M]}\eta_{j}\leqslant\sqrt{2\sigma^{2}\log(M/\delta)% -}\right)\geqslant 1-\delta.$

-

$(II)$

-

$k(x,x^{\prime})=(2^{1-\nu}/\Gamma(\nu))r^{\nu}B_{\nu}(r)$

-

$\mathcal{F}=\{f_{1},f_{2},f_{3}\}$

-

$\displaystyle=C\sum_{t=0}^{T-1}\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde% -{R},U,B_{t})}\cup\neg{\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}\right)$

-

$|\phi(a,b)|\leqslant 1$

-

$N=528$

-

$reg_{t}=\tilde{R}_{t}-X_{t}^{T}\tilde{R}_{t}\cdot\mathbf{1}=[m_{t},0],\quad% -\forall t\geqslant 2$

-

$R_{t+1,A_{t},B_{t}}=\mathcal{R}(Y_{t+1,A_{t},B_{t}})$

-

$A_{t}\sim P_{X_{t}}$

-

$X_{t+1}=g_{t}(X_{t},\tilde{R}_{t+1}).$

-

$a^{-i}\in\mathcal{A}^{-i}$

-

$w_{t}(a,b)$

-

$\Re^{*}(T,\text{IWE-RM})=\mathcal{O}(T^{2/3}\mathcal{A}^{2/3}).$

-

$(H_{t},\theta,A_{t},B_{t})$

-

$\mathbb{E}\left[\operatorname{pess}_{t+1}\>|\>\theta\right]=\mathbb{E}\left[% -\operatorname{est}_{t+1}\>|\>\theta\right]=0$

-

$\displaystyle\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\>|\>\theta% -\right]=\mathbb{E}\left[f_{\theta}(a,B_{t})-f_{\theta}(A_{t},B_{t})\>|\>\theta\right]$

-

$\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))\leqslant 2\mathcal{A}% -\exp(-\beta_{t}^{\prime}/2)=2/\sqrt{t}$

- - - diff --git a/htmls/output_mathjax_example_1001.html b/htmls/output_mathjax_example_1001.html deleted file mode 100644 index f8aa0f6faac52629e96013442c4b83b93380d1a5..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1001.html +++ /dev/null @@ -1,127 +0,0 @@ - - - - MathJax Example - - - - -

$\min\{f_{T}(N_{s})\mid 1\leq s\leq t\}<2f_{T}(N)$

-

$\gamma^{\prime}[V]=\gamma^{\prime\prime}[V]$

-

$N^{k,n}_{s^{\prime}}$

-

$\mathcal{V}\setminus(\mathcal{V}^{\prime}\cup\{V_{n}\})$

-

$V_{n-1}$

-

$Pa(N,V_{i})$

-

$Pa(N^{k,n}_{s},V_{n})$

-

$\Leftarrow(2c-1)\cdot 2^{n-2}-2^{n-2c}\cdot c\binom{2c-1}{c}\geq(6c-6)\cdot 2^% -{n-4}$

-

$1\leq s\leq t=2c-1$

-

$(2^{k}-1)\cdot 1=2^{k}-1$

-

$1\succ 0$

-

$0\leq k^{\prime}\leq k$

-

$\mathcal{V}^{\prime}\subseteq\mathcal{V}\setminus\{V_{n}\}$

-

$\Leftarrow(8c-4)\cdot 2^{n-4}-2^{n-2c}\cdot c\binom{2c-1}{c}\geq(6c-6)\cdot 2^% -{n-4}$

-

$\mathcal{V}^{\prime}\subseteq\mathcal{V}$

-

$0\leq\kappa\leq\lfloor\frac{t}{2}\rfloor$

-

$\mathcal{V}=\{V_{1},\ldots,V_{n}\}$

-

$M(T^{\prime})$

-

$2c\cdot 2^{n-2}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq\frac{3}{4}(2c-2)\cdot 2^{n-2}$

-

$o[V_{i}]=b$

-

$\mathcal{V}^{\prime},\mathcal{V}^{\prime\prime}\subseteq\mathcal{V}$

-

$Pa(N,V_{n})=P_{N}\subseteq P$

-

$k\in\{2,\ldots,n-1\}$

-

$1\leq s^{\prime}\leq\binom{n-1}{k}2^{k}$

-

$\mathcal{V}\setminus{V_{n}}$

-

$001$

-

$P\subseteq\{V_{1},\ldots,V_{n-1}\}$

-

$o,o^{\prime}$

-

$2^{k-k^{\prime}}-1$

-

$I=\{1,\ldots,m\}$

-

$2^{k+1}-2$

-

$\{V_{1},\ldots,V_{n-1}\}$

-

$T_{\varepsilon}=(N^{\varepsilon}_{1},\ldots,N^{\varepsilon}_{t_{\varepsilon}})$

-

$\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})=2^{n-k}$

-

$4c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-1}\,.$

-

$(s,o,o^{\prime})$

-

$1\leq s\leq t$

-

$1\leq s\leq\binom{n-1}{k}2^{k}$

-

$o[V]=o^{\prime}[V]=\gamma[V]$

-

$(c+1)\binom{2c+1}{c+1}\leq(c+2)\cdot 2^{2c-1}$

-

$f_{T}(N_{s})$

-

$\operatorname{CPT}(N_{s},V_{n})$

-

$|Pa(N^{k,n}_{s},V_{n})\cap Pa(N^{k,n}_{s^{\prime}},V_{n})|=:k^{\prime}$

-

$s^{\prime}\neq s$

-

$2^{k-k^{\prime}+1}$

-

$T^{k,n}$

-

$\mathcal{F}_{t\in O(1)}$

-

$f_{T}(N)=$

-

$f_{T}(N)=\begin{cases}t\cdot 2^{n-2}-2^{n-t-1}\cdot c\binom{2c-1}{c}&\text{if % -}t=2c-1\\ -t\cdot 2^{n-2}-2^{n-t-1}\cdot c\binom{2c}{c}&\text{if }t=2c\\ -\end{cases}$

-

$2^{k}-1$

-

$\gamma:b\succ b^{\prime}$

-

$f_{T^{k,n}}(N^{k,n}_{s})=\sum_{s^{\prime}\neq s}\Delta(N^{k,n}_{s},N^{k,n}_{s^% -{\prime}})$

-

$Pa(N^{a},V_{n})\subseteq P$

-

$\displaystyle\sum_{k^{\prime}=1}^{k-1}\big{[}2^{k-k^{\prime}}\binom{k}{k^{% -\prime}}\binom{n-k-1}{k^{\prime}}(2^{n-k}-2^{n-2k+k^{\prime}})$

-

$c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}=(2c+2)\cdot 2^{2c-4}$

-

$|\mathcal{V}^{\prime}|$

-

$001,101$

-

$\Leftarrow 8c\cdot 2^{n-4}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq(6c-3)\cdot 2^{n-4}$

-

$Pa(N^{k,n}_{s},V_{n})\cap P=k^{\prime}$

-

$c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}\,,$

-

$Pa(N^{a},V_{n})\subseteq Pa(N_{s},V_{n})$

-

$\displaystyle(2^{k}-1)\cdot 2^{n-k}+2^{k}\binom{n-k-1}{k}(2^{n-k}-2^{n-2k})$

-

$f_{T^{k,n}}(N^{k,n}_{s})\geq(3/2)f_{T^{k,n}}(N)$

-

$Pa(N,V_{n})\subseteq P$

-

$t-\kappa$

-

$\mathcal{V}=\{V_{1},\ldots,V_{n-1}\}$

-

$P\in\{Pa(N_{s},V_{n})\mid 1\leq s\leq t\}$

-

$\gamma_{2}:1\succ 0$

-

$\Leftarrow c\binom{2c-1}{c}\leq(2c+2)\cdot 2^{2c-4}$

-

$V_{i}\in\mathcal{V}^{\prime}$

-

$o^{\prime}[V_{i}]=1$

-

$\gamma\in\operatorname{Inst}(\mathcal{V}^{\prime})$

-

$2^{p_{s}-1}$

-

$\frac{2\cdot(2d+1)}{d+1}<4$

-

$Pa(N,V_{i})=\emptyset$

-

$f_{T_{n}}(N^{*})$

-

$\gamma_{2}\in\operatorname{Inst}(P)$

-

$2\leq k\leq n-1$

-

$Pa(N_{s},V_{n})$

-

$\gamma[V_{i}]$

-

$freq_{M^{\prime}}(0\succ 1)\leq freq_{M^{\prime}}(1\succ 0)$

-

$CPT(N_{i},V_{3})$

-

$\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}}=2^{n-k}$

-

$\gamma^{\prime}\in\operatorname{Inst}(P)\setminus\{\gamma\}$

-

$T^{k,n}=(N^{k,n}_{1},\ldots,N^{k,n}_{t})$

-

$V\in\mathcal{V}^{\prime}\cap\mathcal{V}^{\prime\prime}$

-

$\gamma\in\operatorname{Inst}(Pa(N_{s},V_{n}))$

-

$(o^{\prime},o)$

-

$V\in\mathcal{V}^{\prime}$

-

$\gamma_{1}=00$

-

$Pa(N^{k,n}_{s^{\prime}},V_{n})$

-

$f_{T^{n-1,n}}(N^{n-1,n}_{s})>(2-\varepsilon)f_{T^{n-1,n}}(N)$

-

$Inst(\mathcal{V}^{\prime})$

-

$\gamma_{1}\in\operatorname{Inst}(Pa(N^{k,n}_{s},V_{n}))$

-

$|P|\leq\max\{|Pa(N_{s},V_{n})\mid 1\leq i\leq t\}$

-

$1\leq s\leq 2^{n-1}$

-

$T=(N_{1},\ldots,N_{t})$

-

$\Leftarrow 2c\cdot 2^{n-2}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq(6c-3)\cdot 2^{n-4}$

-

$\{b,b^{\prime}\}=\{0,1\}$

-

$\gamma^{\prime}\in\operatorname{Inst}(\mathcal{V}^{\prime})$

- - - diff --git a/htmls/output_mathjax_example_10010.html b/htmls/output_mathjax_example_10010.html deleted file mode 100644 index 7cd86bea727afe670fc15428990e12e8b476fe8c..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10010.html +++ /dev/null @@ -1,187 +0,0 @@ - - - - MathJax Example - - - - -

$\tilde{\mathcal{O}}(\sqrt{(M+N)/T})$

-

$\text{SINR}(a,b;\theta)$

-

$\sqrt{2\log\mathcal{A}\sqrt{T}}$

-

$\operatorname{NashRegret}(T)$

-

$\displaystyle\geqslant\sum_{t=1}^{\infty}\log\Phi(f_{t}(\Delta,\sigma_{n}))$

-

$(\Delta,\sigma_{n}^{2})$

-

$\displaystyle\tilde{R}_{t+1}(a)=1-\frac{\mathbb{I}_{A_{t}=a}(1-R_{t+1,A_{t},B_% -{t}})}{X_{t,a}}$

-

$w_{t}(a,b)=\sqrt{\beta_{t}I_{t}\left(\theta;R_{t+1,A_{t},B_{t}}\mid A_{t}=a,B_% -{t}=b\right)}\quad\text{ with }\quad\beta_{t}=\frac{2\beta^{\prime}_{t}}{\log(% -1+\sigma_{w}^{-2})}.$

-

$\mu_{t}(a,b)=\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}+\sigma^{2}{\bm{I}}_{t% -})^{-1}\mathbf{R}_{t}$

-

$\eta_{1},\eta_{2},\ldots,\eta_{M}$

-

$z_{t},z^{\prime}_{t}\sim\mathcal{N}(0,1)$

-

$\theta\in\mathbb{R}^{\mathcal{A}\times\mathcal{B}}$

-

$\beta_{t}=2\beta^{\prime}_{t}/\log(1+\sigma_{w}^{-2})$

-

$\Re_{\operatorname{full}}(T,{\operatorname{Hedge}})=\mathcal{O}(\sqrt{T\log% -\mathcal{A}})$

-

$\displaystyle:=\max_{j\in[M_{t+1}]}\tilde{f}^{\operatorname{TS},j}_{t+1}(a,B_{% -t}),$

-

$\displaystyle\leqslant\max_{a\in\mathcal{A}}\Re_{\operatorname{full}}(a;T,% -\operatorname{adv},\tilde{R})+\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}% -^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]+% -\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\right]$

-

$\tilde{r}_{t}=A_{ij}+\epsilon_{t}$

-

$N(\mu_{p},\Sigma_{p})$

-

$M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}$

-

$\displaystyle=\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0)$

-

$\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}\operatorname{est}_{t+1}\right]$

-

$\displaystyle\mathbb{E}_{t}\left[\tilde{R}_{t+1}(a)\right]=1-\mathbb{E}_{t}% -\left[\mathbb{I}_{A_{t}=a}\frac{1-R_{t+1,a,B_{t}}}{X_{t,a}}\right]=1-\mathbb{E% -}_{t}\left[\mathbb{I}_{A_{t}=a}\right]\frac{1-\mathbb{E}_{t}\left[f_{\theta}(a% -,B_{t})\right]}{X_{t,a}}=\mathbb{E}_{t}\left[f_{\theta}(a,B_{t})\right].$

-

$\sigma_{t}(a,b)$

-

$\displaystyle=k((A_{i},B_{i}),(A_{j},B_{j}))$

-

$\Re(T,\pi^{\operatorname{alg}},B_{0:T},\theta)$

-

$\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]=\mathbb{E}_{0% -}\left[\sum_{t=1}^{T}\mathbb{E}_{t}\left[\tilde{\Re}_{t,a}\right]+\sum_{t=1}^{% -T}2\gamma_{t}\right]=\mathbb{E}_{0}\left[\tilde{C}_{T,a}+\sum_{t=1}^{T}2\gamma% -_{t}\right]\leqslant\sqrt{\sum_{t=1}^{T}\frac{2(1+\sigma_{w}^{2})\left|% -\mathcal{A}\right|^{2}}{\gamma_{t}}}+2\gamma_{t}$

-

$\displaystyle=\mathbb{P}(\tilde{R}_{t+1}(a)\geqslant U_{t}(a,B_{t}),\forall a% -\in\mathcal{A}\mid H_{t},B_{t})$

-

$\Re_{t}(a):=\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta,H_{t% -}\right]=\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]-\sum_{a}{X% -}_{t,a}\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]$

-

$\displaystyle\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A^{*},B_{t}}-R_{t+1,A_{t},% -B_{t}}\>|\>\theta\right],$

-

$\displaystyle\leqslant\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{% -R})+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta% -\right]+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\>|\>\theta\right]$

-

$\mathbb{E}\left[\tilde{R}_{t+1}(a)\>|\>H_{t},\theta\right]=\mathbb{E}\left[f_{% -\theta}(a,B_{t})\>|\>H_{t},\theta\right]=\mathbb{E}\left[g_{\theta}(e_{a},Y_{t% -})\>|\>H_{t},\theta\right]$

-

$\displaystyle\leqslant\mathbb{E}\left[\sum_{t=0}^{T-1}\sqrt{\left(\sqrt{2\log(% -M\sqrt{t})/\beta^{\prime}_{t}}+1\right)^{2}\beta_{t}I_{t}(\theta;A_{t},B_{t},R% -_{t+1,A_{t},B_{t}})}\right]$

-

$\Re_{\operatorname{full}}(T,\text{RM},\tilde{R}^{\operatorname{est}})=\mathcal% -{O}(\sqrt{T\mathcal{A}})$

-

$(a=b)$

-

$\mathbb{P}(\Omega_{t})\geqslant c$

-

$\beta^{\prime}_{t}=2\log\mathcal{A}\sqrt{t}.$

-

$\displaystyle\Re^{*}(T,\pi^{\operatorname{alg}})=\sup_{B_{0:T}\in\mathcal{B}^{% -T}}\Re(T,\pi^{\operatorname{alg}},B_{0:T})=o(T),$

-

$\mathbf{K}_{t}$

-

$\tilde{R}^{OTS}_{t+1}(1st)>\tilde{R}^{OTS}_{t+1}(2nd)$

-

$\displaystyle\leqslant\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}% -\mathbb{E}\left[C(1-\mathds{1}_{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap% -\mathcal{E}^{c}_{t}(f_{\theta},B_{t})})\mid\theta\right]\right]$

-

$\nu\rightarrow\infty$

-

$\displaystyle\quad+C\left(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R},U,A_{t},% -B_{t})}\mathds{1}_{\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}\right)$

-

$\displaystyle X_{t,a}=(1-\gamma_{t})\hat{X}_{t,a}+\gamma_{t}(1/\mathcal{A}),% -\forall a\in\mathcal{A}$

-

$\textrm{Hedge:}\ g_{t,a}(X_{t},r_{t})=X_{t,a}\exp(\eta_{t}r_{t}(a)),\quad% -\textrm{RM:}\ g_{t,a}(X_{t},r_{t})=\max\left(0,\sum_{s=0}^{t}r_{t}(a)-r_{t}(A_% -{s})\right).$

-

$\displaystyle\mathcal{E}_{t}(f_{\theta},B_{t}):=\{\forall a\in\mathcal{A},f_{% -\theta}(a,B_{t})\in[L_{t}(a,B_{t}),U_{t}(a,B_{t})]\}.$

-

$reg_{1}$

-

$f_{\theta}(a,b)\mid H_{t}$

-

$[a^{i}]_{e}=0$

-

$\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}R_{t+1,A_{t},B_{t}}^{2% -}\left(\left(\frac{\mathbb{I}_{A_{t}=a}}{X_{t,A_{t}}}\right)^{2}-\frac{2\hat{X% -}_{t,A_{t}}}{X_{t,A_{t}}^{2}}\mathbb{I}_{A_{t}=a}+\frac{\hat{X}_{t,A_{t}}^{2}}% -{X_{t,A_{t}}^{2}}\right)\right]$

-

$\tilde{R}_{t1}=\max\limits_{i}x_{ti}$

-

$\displaystyle\mathbb{P}(\neg\mathcal{E}_{t}(\tilde{R},U,B_{t}))\leqslant\frac{% -1}{\sqrt{t}}.$

-

$\displaystyle\mathbf{K}_{t}(i,j)$

-

$\nu>1$

-

$\tilde{R}_{t+1}(a)$

-

$\tilde{R}:=\{\tilde{R}_{t}:t\in\mathbb{Z}_{++}\}$

-

$\mathbb{P}(A_{t}\in\cdot\>|\>\pi_{t})=\mathbb{P}(A_{t}\in\cdot\>|\>H_{t})=\pi_% -{t}(\cdot)$

-

$\displaystyle\begin{cases}\textrm{UCB:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}=% -\mu_{t}(a,B_{t})+\beta_{t}\sigma_{t}(a,B_{t}),\\ -&\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.% -\\ -\textrm{TS:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}\sim N(\mu_{t}(a,B_{t}),% -\sigma_{t}(a,B_{t})),\\ -&\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.% -\end{cases}$

-

$\Sigma_{0}=\Sigma_{p}$

-

$\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{X_{t,a}}+\sum_{% -a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|-2\right)\right)$

-

$\mathds{1}(a=b)$

-

$\operatorname{est}$

-

$\mathbf{k}_{t}((a,b))$

-

$\mathcal{E}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})$

-

$\displaystyle\leqslant\sum_{a}\left(\left|\gamma_{t}\hat{X}_{t,a}\right|+\left% -|\gamma_{t}/\mathcal{A}\right|\right)=2\gamma_{t}$

-

$\mathcal{B}={1,\ldots,\left|\mathcal{B}\right|}$

-

$M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}.$

-

$\tilde{R}_{t+1}(a)=\min(\tilde{f}_{t+1}(a,B_{t}),1)$

-

$\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant\sqrt% -{T}\sqrt{\frac{2(1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma}}+2% -\gamma T,$

-

$\displaystyle\mathbb{P}\left(\max_{j\in[M]}\eta_{j}\geqslant w\right)=1-\left[% -\Phi\left(\frac{w}{\sigma}\right)\right]^{M}$

-

$Z_{t}=(A_{t},B_{t},R_{t+1,A_{t},B_{t}}))$

-

$(r_{t})_{t\in[T]}\in[0,1]^{\mathcal{A}\times T}$

-

$\displaystyle\stackrel{{\scriptstyle(iii)}}{{=}}1-\Phi\left(\sqrt{\beta_{t}^{% -\prime}}\right)^{M}.$

-

$\displaystyle>-\infty,$

-

$KL(\overline{y}_{T},y^{)}$

-

$\displaystyle\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t})=\mathds{1}_{\mathcal{E}^{o% -}_{t}(\tilde{R},U,B_{t})}(\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t}))+(1-\mathds{1% -}_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})})(\tilde{R}_{t+1}(a)-f_{\theta}(a,% -B_{t}))$

-

$\mathcal{A}={1,\ldots,\left|\mathcal{A}\right|}$

-

$\Re^{*}(T,\text{UCB-Hedge})=\mathcal{O}(\sqrt{T\log\mathcal{A}}+\sqrt{\gamma_{% -T}\beta T}),\ \Re^{*}(T,\text{UCB-RM})=\mathcal{O}(\sqrt{T\mathcal{A}}+\sqrt{% -\gamma_{T}\beta T}).$

-

$\Re_{\operatorname{full}}(T,\text{Hedge},\tilde{R}^{\operatorname{est}})=% -\mathcal{O}(\sqrt{T\log\mathcal{A}})$

-

$U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad -L% -=(\mu_{t}(a,b)-\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}).$

-

$c(\Delta,\sigma_{n})\approx 0.54$

-

$k_{\rm P}(\cdot,\cdot)$

-

$\sigma_{n}>0$

-

$k((a,b),(a^{\prime},b^{\prime}))=\mathbb{E}\left[(f_{\theta}(a,b)-\mu(a,b))(f_% -{\theta}(a^{\prime},b^{\prime})-\mu(a^{\prime},b^{\prime}))\right]$

-

$\displaystyle=[k((A_{0},B_{0}),(a,b)),\ldots,k((A_{t-1},B_{t-1}),(a,b))]^{\top}$

-

$\sigma_{t}^{2}(a,b)$

-

$\displaystyle\leqslant\sum_{a}\left(\left({\tilde{C}_{T-1,a}^{+}}\right)^{2}+2% -{\tilde{C}_{T-1,a}^{+}}{\tilde{\Re}_{T,a}}+\left(\tilde{\Re}_{T,a}\right)^{2}\right)$

-

$A\in\mathbb{R}^{10\times 5}$

-

$\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})$

-

$g(a^{-i})=\sum_{j\neq i}a^{j}$

-

$\gamma_{t}=\gamma$

-

$P_{a}(\theta)$

-

$\displaystyle\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}$

-

$\sum\limits_{k=1}^{t}\frac{1}{k+a}\leqslant\int_{0}^{t}\frac{1}{k+a}dk$

-

$y^{*}=\operatorname*{arg\,min}\limits_{y\in\Delta}y^{T}(Ax),$

-

$\displaystyle\mathbb{P}(\max_{j\in[M]}\eta_{j}\geqslant w)=1-\mathbb{P}(\max_{% -j\in[M]}\eta_{j}\leqslant w)=1-\mathbb{P}(\forall j\in[M],\eta_{j}\leqslant w)% -=1-\left[\Phi\left(\frac{w}{\sigma}\right)\right]^{M}.$

-

$m_{1}\leqslant 0$

-

$\tilde{R}_{t+1}\in[0,C]$

-

$\mathcal{D}(\mathcal{A}^{i})$

-

$\displaystyle\sum_{t=1}^{T}\frac{1}{\sqrt{t}}=2\sum_{t=1}^{T}\frac{t-(t-1)}{% -\sqrt{t}+\sqrt{t}}\leqslant 2\sum_{t=1}^{T}\frac{t-(t-1)}{\sqrt{t}+\sqrt{t-1}}% -=2\sum_{t=1}^{T}(\sqrt{t}-\sqrt{t-1})=2\sqrt{T}$

-

$50)$

- - - diff --git a/htmls/output_mathjax_example_10011.html b/htmls/output_mathjax_example_10011.html deleted file mode 100644 index 3d913153c47a9841ffa3c8d2058ab576b094e2d9..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10011.html +++ /dev/null @@ -1,163 +0,0 @@ - - - - MathJax Example - - - - -

$\mu_{0}=\mu_{p}$

-

$\text{Gap}(x,y)=\max_{\left(x^{\prime},y^{\prime}\right)\in\Delta}\mathbb{E}% -\left[\phi\left(x^{\prime},y\right)-\phi\left(x,y^{\prime}\right)\right]$

-

$Y_{t+1,A_{t},B_{t}}=f_{\theta}(A_{t},B_{t})+\eta_{t},$

-

$(1-x)^{M}\leqslant e^{-Mx}$

-

$\theta(a,b)\in\mathbb{R}$

-

$\Re_{\operatorname{full}}(T,{\operatorname{Hedge}},(r_{t})_{t})=\mathcal{O}(% -\sqrt{T\log\mathcal{A}}),\ \Re_{\operatorname{full}}(T,{\operatorname{RM}},(r_% -{t})_{t})=\mathcal{O}(\sqrt{T\mathcal{A}}).$

-

$f_{t}(\Delta,\sigma_{n}^{2})$

-

$\displaystyle=\max_{(r_{t})_{t}}\Re_{\operatorname{full}}(T,{\operatorname{adv% -}},(r_{t})_{t}).$

-

$I(\theta;o)=\frac{1}{2}\log\left(1+{\sigma_{w}^{-2}\sigma(a)}\right)$

-

$(\overline{x}_{T},\overline{y}_{T})$

-

$N\left(0,\sigma^{2}\right)$

-

$v^{+}=(v_{a}^{+})_{a\in\mathcal{A}}$

-

$\displaystyle\leqslant\mathcal{O}\left(\sqrt{T\log(\mathcal{A}T)I(\theta;H_{T}% -)}\right).$

-

$\pi=(\pi_{t})_{t\in\mathbb{N}}$

-

$\mathbb{I}_{A_{t}=a}R_{t+1,A_{t},B_{t}}=\mathbb{I}_{A_{t}=a}R_{t+1,a,B_{t}}$

-

$\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\mathbb{E}_{t}\left[R_{t+1,A_{% -t},B_{t}}^{2}\left(\frac{1}{X_{t,A_{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_% -{t}}^{2}}+\left|\mathcal{A}\right|\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}% -}\right)\right]\right],$

-

$M=5,10,20,50,70,100$

-

$\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}\left(\frac{\mathbb{I}% -_{A_{t}=a}R_{t+1,A_{t},B_{t}}}{X_{t,a}}-R_{t+1,A_{t},B_{t}}\frac{\hat{X}_{t,A_% -{t}}}{X_{t,A_{t}}}\right)^{2}\right]$

-

$c\geqslant 0$

-

$k(x,x^{\prime})=x^{\top}x^{\prime}$

-

$U^{\prime},U,L$

-

$\Re_{\operatorname{full}}(T,{\operatorname{RM}})=\mathcal{O}(\sqrt{T\mathcal{A% -}})$

-

$\displaystyle\geqslant\sum_{t=1}^{\infty}\log\Phi\left(\frac{(1-\Delta)\left(t% --\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}+2\sigma_{n}^{2}\ln{\sigma_{n}}-1/(% -\sigma_{n}^{2}+1)\right)}{\sqrt{t+\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}-2% -\sigma_{n}^{2}\ln{\sigma_{n}}-(\sigma_{n}^{2}+2)/(\sigma_{n}^{2}+1)}}\right)$

-

$A\in\mathcal{R}^{10\times 5}$

-

$U^{\prime}=(\mu_{t}(a,b)+\sqrt{2\log(M\sqrt{t})}\sigma_{t}(a,b):t\in\mathbb{N}).$

-

$\tilde{R}^{OTS}_{t+1}(2nd)$

-

$\displaystyle\leqslant\sqrt{\mathbb{E}\left[\sum_{t=0}^{T-1}\left(\sqrt{2\log(% -M\sqrt{t})/\beta^{\prime}_{t}}+1\right)^{2}\beta_{t}\right]}\sqrt{\mathbb{E}% -\left[\sum_{t=0}^{T-1}I_{t}(\theta;A_{t},B_{t},R_{t+1,A_{t},B_{t}})\right]}$

-

$(P^{*},Q^{*})$

-

$\pi_{\operatorname{alg}}$

-

$\pm\tilde{r}_{t}$

-

$\displaystyle\Sigma_{t+1}=\left(\Sigma_{t}^{-1}+\frac{1}{\sigma_{w}^{2}}\phi(A% -_{t},B_{t})\phi(A_{t},B_{t})^{\top}\right)^{-1}$

-

$\mathbb{P}(\Omega_{t})\geqslant c(\Delta,\sigma_{n})$

-

$\mathcal{E}^{c}_{t}(f_{\theta},B_{t})$

-

$\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{t}% -))\right]$

-

$\mu_{t}(a,b)=\phi(a,b)^{\top}\mu_{t}$

-

$c=e^{c^{\prime}}$

-

$\displaystyle\mathbb{E}\left[\tilde{h}(a)\right]=\mathbb{E}\left[\mathbb{I}_{A% -=a}\right]h(a)/X_{a}=h(a).$

-

$\displaystyle\stackrel{{\scriptstyle(i)}}{{\geqslant}}\mathbb{P}(\tilde{R}_{t+% -1}(a)\geqslant\min\{\mu_{t}(a,B_{t})+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{% -t}),c\},\forall a\in\mathcal{A}\>|\>H_{t},B_{t})$

-

$\displaystyle=\sum_{t=1}^{\infty}\log\mathbb{P}(Z\leqslant-\frac{\mu_{t}}{% -\sigma_{t}})$

-

$f_{\theta}(a)\sim N(\mu(a),\sigma(a))$

-

$\displaystyle=\mathbb{E}\left[\sum_{t=0}^{T-1}\mathbb{E}\left[C(1-\mathds{1}_{% -\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t}% -)})\right]\right]$

-

$31.9\pm 1.6$

-

$46.4\pm 0.8$

-

$\text{Uniform}[0.05,0.1]$

-

$71.6\%\rightarrow 78.9\%$

-

$65.2\pm 1.1$

-

$\phi:\mathcal{X}\rightarrow\mathbb{R}^{k}$

-

$93.8\pm 0.3$

-

$b\in{\bm{b}}$

-

${\rho^{\prime}},{\alpha^{\prime}},{\beta^{\prime}}$

-

$\mathcal{A}_{\text{ft}}(x^{\prime}\mid x)=1$

-

$\frac{{\beta}}{{\gamma}}$

-

$\displaystyle\mathcal{L}_{\text{pretrain}}(\phi)=-2\cdot$

-

${\alpha^{\prime}}$

-

$67.84\pm 0.70$

-

$32.1\pm 0.8$

-

$31.2\%\rightarrow 31.9\%$

-

$50.6\pm 1.3$

-

$91.4\pm 0.9$

-

$\displaystyle\mathcal{A}_{\text{prop}}({x^{\prime}}\mid x)=\begin{cases}{\rho^% -{\prime}}&x={x^{\prime}}\\ -{\alpha^{\prime}}&\{x^{\prime},x\}\in\{\{1,3\},\{3,5\},\{5,7\},\{2,4\},\{4,6\}% -,\{6,8\},\{1,7\},\{2,8\}\\ -{\beta^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}\\ -{\gamma^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,4\},\{2,3\},\{3,6\},\{4,5\},\{5,8% -\},\{6,7\},\{1,8\},\{2,7\}\}\\ -\end{cases}.$

-

$30.4\%\rightarrow 34.5\%$

-

$\text{21M}\rightarrow\text{69M}$

-

$27.9\pm 0.5$

-

$y_{x}\in\mathbb{R}^{k}$

-

$p^{*}(\cdot\mid x)$

-

$\text{loguniform}(0.95z,\;\text{min}(1.5(1+z)-1,\;5z))$

-

${\bm{X}}^{\prime}=\{\tilde{{\bm{X}}^{\prime}}_{:,j}+\epsilon_{j}\}_{j=1}^{W},{% -\bm{X}}^{\prime}_{\text{err}}=\left\{\sqrt{\tilde{{\bm{X}}^{\prime}}_{\text{% -err},:,j}^{2}+\epsilon_{j}^{2}}\right\}_{j=1}^{W}.$

-

$z=z^{\prime}$

-

$46.4\%\rightarrow 50.6\%$

-

$\{F(t_{i},w_{j})\}_{i=1,j=1}^{T,W}$

-

${\rho^{\prime}}>\max\{{\alpha^{\prime}},{\beta^{\prime}}\}$

-

$\mathbf{51.7\pm 0.8}$

-

$65.2\%\rightarrow 91.4\%$

-

$64.5\pm 1.2$

-

$i\in\{1,...,T\},j\in\{1,...,W\}$

-

$0.320\pm 0.009$

-

$92.3\pm 0.7$

-

$\mathcal{T}=\{3,4,5,6,7,8\}$

-

$0.304\pm 0.010$

-

$\mathcal{A}_{\text{ft}}(x^{\prime}\mid x)=\sum_{z^{\prime}}T(x^{\prime}\mid x,% -z^{\prime})\hat{p_{T}}(z^{\prime}\mid z)$

-

$67.54\pm 0.32$

-

$\mathbf{79.90\pm 0.60}$

-

$\{F^{\prime}_{\text{err}}(t_{\text{new},i},w_{\text{new},i})\}_{i=1,j=1}^{T,W}$

-

$\widehat{f}_{\text{erm}}\in\operatorname*{arg\,min}_{f}\mathcal{L}_{\text{ERM}% -}(f)$

-

$\{F(t_{i},b_{j})\}_{i=1,j=1}^{T,W},\{F_{\text{err}}(t_{i},b_{j})\}_{i=1,j=1}^{% -T,W}$

-

$\mathbf{0.247\pm 0.005}$

-

${\bm{X}}^{\prime}_{\text{err}}$

-

$\mathcal{L}_{0-1}(\widehat{f})=0$

-

$T(x^{\prime}|x,z^{\prime})$

-

$((y_{1},d_{1}),(y_{2},d_{2}))$

-

$x\in\{1,3,5,7\}$

-

$46.4\%\rightarrow 48.5\%$

-

$\displaystyle\mathcal{A}_{\text{pre}}({x^{\prime}}\mid x)=\begin{cases}{\rho^{% -\prime}}&x={x^{\prime}}\\ -{\alpha^{\prime}}&\{x^{\prime},x\}\in\{\{1,4\},\{3,5\},\{5,7\},\{2,5\},\{4,6\}% -,\{6,8\},\{1,8\},\{2,7\}\\ -{\beta^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}\\ -{\gamma^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,3\},\{2,4\},\{3,6\},\{4,5\},\{5,8% -\},\{6,7\},\{1,7\},\{2,8\}\}\\ -\end{cases}.$

-

$y_{x}=1$

-

$0.27\rightarrow 0.25$

-

$96.7\pm 0.0$

-

$48.5\pm 3.2$

-

$\widehat{\phi}=\operatorname*{arg\,min}_{\phi}\mathcal{L}_{\text{pretrain}}(\phi)$

-

$46.4\%\rightarrow 47.5\%$

-

$d_{-}$

- - - diff --git a/htmls/output_mathjax_example_10012.html b/htmls/output_mathjax_example_10012.html deleted file mode 100644 index f77c3b18cb06ccc34c51dd6ec4f683fb5916f279..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10012.html +++ /dev/null @@ -1,142 +0,0 @@ - - - - MathJax Example - - - - -

$0.277\pm 0.004$

-

$77.72\pm 0.59$

-

$\mathcal{T}=\{x\in\mathcal{T}:d_{x}=2\}$

-

$0.274\pm 0.016$

-

$\displaystyle\mathcal{L}_{\text{ERM}}(f)=\mathbb{E}_{x\sim P_{S},x^{\prime}% -\sim\mathcal{A}_{\text{ft}}(\cdot\mid x)}[\ell(f(x^{\prime}),y_{x})].$

-

$\mathbf{36.9\pm 0.7}$

-

$\mathcal{A}_{\text{ft}}$

-

$34.5\pm 1.4$

-

$71.6\%\rightarrow 68.8\%$

-

${\beta}>{\gamma}$

-

$\mathcal{L}_{0-1}(\widehat{f}_{\text{erm}})=1/3$

-

$\mathbf{98.5\pm 0.0}$

-

$\text{SNR}(x,x_{\text{err}})=\frac{|x|}{x_{\text{err}}}$

-

$30.4\%\rightarrow 32.1\%$

-

$x\in\{2,4,6,8\}$

-

$\text{SNR}({\bm{X}}^{\prime}_{i,j},{\bm{X}}^{\prime}_{\text{err},i,j})\geq 5$

-

$\displaystyle\mathcal{A}_{\text{ft}}({x^{\prime}}\mid x)=\begin{cases}1&\{{x^{% -\prime}},x\}\in\{1,4\},\{2,3\}\\ -1&x={x^{\prime}}\text{ and }x\notin\{1,2\}\\ -0&\text{otherwise}\end{cases}$

-

$61.26\pm 1.10$

-

$\widehat{f}_{\text{erm}}$

-

$x,{x^{\prime}}$

-

$\text{loss}_{\text{ft}}:\mathbb{R}^{n}\times\mathcal{Y}\rightarrow\mathbb{R}$

-

$86.1\pm 1.3$

-

$\tilde{x^{\prime}}$

-

$96.7\%\rightarrow 98.5\%$

-

$P_{U}=\beta P_{S}+(1-\beta)P_{T}$

-

$\min\{{\alpha^{\prime}},{\beta^{\prime}}\}>{\gamma^{\prime}}$

-

$F^{\prime},F^{\prime}_{\text{err}}$

-

$z^{\prime}\sim\text{loguniform}(0.95z_{\text{orig}},\;\text{min}(1.5(1+z_{% -\text{orig}})-1,\;5z_{\text{orig}}))$

-

$90.5\pm 0.4$

-

$\widehat{f}(x)=\operatorname*{arg\,max}_{i\in[r]}(\widehat{B}\widehat{\phi}(x)% -)_{i}$

-

$92.3\pm 0.2$

-

${x^{\prime}}\in\mathcal{X}$

-

$65.2\rightarrow 91.4$

-

$\tilde{{\bm{X}}^{\prime}}_{\text{err}}=10^{0.4(d(z^{\prime})-d(z_{\text{orig}}% -))}\{F^{\prime}_{\text{err}}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T% -,W},$

-

$\alpha>\gamma+\beta$

-

$\displaystyle\mathcal{L}_{\text{ft}}(h)=\mathbb{E}_{x\sim P_{S},y\sim p^{*}(% -\cdot\mid x),{x^{\prime}}\sim\mathcal{A}_{\text{ft}}(\cdot|x)}[\text{loss}_{% -\text{ft}}(h(\widehat{\phi}({x^{\prime}})),\;y;\;\theta)]$

-

$\displaystyle\mathcal{L}_{\text{pretrain}}(\phi)=\mathbb{E}_{(x,x^{+})\sim S_{% -+}}[d_{+}(\phi(x),\phi(x^{+}))]-\mathbb{E}_{x,{x^{\prime}}\sim P_{U}}[d_{-}(% -\phi(x),\phi({x^{\prime}}))].$

-

$93.8\%\rightarrow 94.9\%$

-

$61.3\%\rightarrow 67.8\%$

-

$d(z)$

-

$0.32\rightarrow 0.28$

-

$\{F_{\text{err}}(t_{i},w_{j})\}_{i=1,j=1}^{T,W}$

-

$\mathcal{A}_{\text{pre}}(\cdot\mid x)$

-

${\bm{t}}_{\text{new}}=\frac{1+z^{\prime}}{1+z_{\text{orig}}}{\bm{t}}$

-

$\hat{p_{T}}(z^{\prime}|z)$

-

$78.84\pm 0.97$

-

$40.5\pm 1.6$

-

$\mathcal{A}(\cdot|x)$

-

$\mathcal{A}_{\text{prop}}$

-

$78.9\%\rightarrow 68.8\%$

-

$\gamma>\beta$

-

$\hat{p_{T}}(z^{\prime}\mid z)$

-

$46.4\pm 0.5$

-

$0.286\pm 0.007$

-

$30.4\rightarrow 31.2$

-

$30.4\%\rightarrow 37.2\%$

-

$10^{\text{Uniform}[-3,-2]}$

-

$36.1\pm 0.7$

-

$89.3\%\rightarrow 92.3\%$

-

$30.4\pm 0.6$

-

$\displaystyle\begin{cases}{\rho}&y_{1}=y_{2},d_{1}=d_{2}~{}\text{~{}~{}(same % -class, same domain)}\\ -{\alpha}&y_{1}=y_{2},d_{1}\neq d_{1}\text{~{}~{}(same class, different domain)% -}\\ -{\beta}&y_{1}\neq y_{2},d_{1}=d_{2}\text{~{}~{}(different class, same domain)}% -\\ -{\gamma}&y_{1}\neq y_{2},d_{1}\neq d_{2}\text{~{}~{}(different class and % -domain)}\\ -\end{cases},$

-

$68.75\pm 0.95$

-

$\mathbf{94.9\pm 0.4}$

-

$10^{\text{Uniform}[-5,-2]}$

-

$\mathcal{S}=\{1,2\}$

-

$F,F_{\text{err}}:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$

-

$h:\mathbb{R}^{k}\rightarrow\mathbb{R}^{n}$

-

$77.40$

-

$\mathcal{Y}=\{1,\dots,k\}$

-

$\mathbf{80.54\pm 1.20}$

-

$S_{+}(x,{x^{\prime}})$

-

$71.59\pm 1.10$

-

$z_{\text{orig}}$

-

$\displaystyle\mathbb{E}_{(x,x^{+})\sim S_{+}}\left[\phi(x)^{\top}\phi(x^{+})% -\right]+\mathbb{E}_{x,x^{\prime}\sim P_{U}}\left[\left(\phi(x)^{\top}\phi(x^{% -\prime})\right)^{2}\right].$

-

$L_{T}(f)=\mathbb{E}_{x\sim P_{T},y\sim p^{*}(\cdot\mid x)}[\ell(f(x),y)]$

-

$89.3\pm 0.9$

-

$\mathbf{51.4\pm 0.6}$

-

$\mathcal{A}_{\text{pre}}$

-

${\bm{X}}^{\prime},{\bm{X}}^{\prime}_{\text{err}}$

-

$89.3\rightarrow 92.3$

-

$65.15\pm 0.67$

-

${\bm{w}}_{\text{new}}=\frac{1+z^{\prime}}{1+z_{\text{orig}}}{\bm{w}}$

-

$62.3\pm 1.9$

-

$36.3\%\rightarrow 37.2\%$

-

$47.5\pm 1.0$

-

$y_{x}=-1$

-

${\rho},{\alpha},{\beta},{\gamma}$

-

$46.4\rightarrow 46.4$

-

$\mathbf{0.256\pm 0.005}$

-

${\bm{X}}_{\text{err}}\in\mathbb{R}^{T\times W}$

-

$\epsilon\in\mathbb{R}^{W}$

-

$\widehat{\phi}$

-

$\widehat{\phi}:\mathcal{X}\to\mathbb{R}^{k}$

-

$d_{+}$

-

${\bm{X}}\in\mathbb{R}^{T\times W}$

-

${\alpha}>{\gamma}$

-

$({\bm{t}}_{\text{new}},{\bm{w}}_{\text{new}})$

-

$31.2\pm 0.6$

-

$0.289\pm 0.003$

-

$0.310\pm 0.006$

- - - diff --git a/htmls/output_mathjax_example_10013.html b/htmls/output_mathjax_example_10013.html deleted file mode 100644 index 798c672a86a4966a71073afb64e4da4bbc9979c2..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10013.html +++ /dev/null @@ -1,137 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\mathcal{L}_{\text{MAE}}(\phi)=\mathbb{E}_{x\sim P_{U},{x^{\prime% -}}\sim\mathcal{A}_{\text{pre}}(\cdot\mid x)}[(\phi({x^{\prime}})-x)^{2}]$

-

$\mathbf{0.246\pm 0.015}$

-

$\frac{{\alpha}}{{\gamma}}$

-

$\tilde{{\bm{X}}^{\prime}}=10^{0.4(d(z^{\prime})-d(z_{\text{orig}}))}\{F^{% -\prime}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T,W},$

-

$\mathcal{S}=\{x\in\mathcal{X}:d_{x}=1\}$

-

$92.3\%\rightarrow 96.7\%$

-

$\displaystyle\mathcal{L}(B)=\mathbb{E}_{x\sim P_{S}}\left[\ell(B\widehat{\phi}% -(x),y_{x})\right]+\eta\|B\|_{F}^{2},$

-

$x\sim P_{S}$

-

$({\bm{t}},{\bm{w}})$

-

$\text{Uniform}[0.5,0.9]$

-

$S_{+}(x,x^{+})=\mathbb{E}_{\bar{x}\sim P_{U}}[\mathcal{A}_{\text{pre}}(x\mid% -\bar{x})\mathcal{A}_{\text{pre}}(x^{+}\mid\bar{x})]$

-

$\{F^{\prime}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T,W}$

-

$\widehat{h}=\operatorname*{arg\,min}_{h}\mathcal{L}_{\text{ft}}(h)$

-

$5Ã$

-

$0.8\AA$

-

$\left(\,\overline{\text{Ch.}},\text{Ch.}\,\right)$

-

$match(Ch.,\cdot{})$

-

$\left(\,\text{Ch.},\overline{\text{Ch.}}\,\right)$

-

$\\ -A$

-

$\min(match)$

-

$\left(\,\text{H.2},\overline{\text{H.2}}\,\right)$

-

$10\AA$

-

$\max(d_{\rho})$

-

$match(\cdot{},Ch.)$

-

$B(\mathbf{p},\rho_{\mathbf{p}})$

-

$\gamma:M\rightarrow M^{\prime}$

-

$d_{\rho}(C,C^{\prime}):=\sqrt{\sum_{\mathbf{p}\in M}|\,\rho_{\mathbf{p}}-\rho_% -{\gamma(\mathbf{p})}\,|^{2}},$

-

$\overline{\text{H.1}}$

-

$\AA$

-

$\mathbf{p}^{\prime}\in C^{\prime}$

-

$d_{\rho}(C,C^{\prime})$

-

$d_{\rho}$

-

$\overline{\text{Ch.}}$

-

$1.4\AA$

-

$\mathbf{p}_{N}$

-

$match(C,C^{\prime})$

-

$\left(\,\overline{\text{H.1}},\text{H.1}\,\right)$

-

$\alpha-\pi$

-

$V(C,\rho):=\iiint_{\bigcup_{\mathbf{p}\in{}C}B(\mathbf{p},\rho_{\mathbf{p}})}1dxdydz.$

-

$\overline{\text{H.2}}$

-

$\left(\,\overline{\text{H.2}},\text{H.2}\,\right)$

-

$L(C):=\sum_{j=1}^{N}\|\mathbf{p}_{j}-\mathbf{p}_{j-1}\|_{2}.$

-

$s(C):=\dfrac{1}{tortuousness(C)},$

-

$\left(\,\text{H.1},\overline{\text{H.1}}\,\right)$

-

$-\mathbf{v}$

-

$\alpha_{t}x_{t}+\beta_{t}\epsilon_{t}$

-

$\max_{x^{adv}}\ \ J(x^{adv},y)\ \ \ \ s.t.\left\|x-x^{adv}\right\|_{\infty}<\epsilon.$

-

$\max_{x^{adv}}\ \ \left\|f^{m}(x,p)-f^{m}(x^{adv},p)\right\|_{2}\ \ \ \ s.t.% -\left\|x-x^{adv}\right\|_{\infty}<\epsilon.$

-

$f^{m}(\cdot)$

-

$\epsilon^{\prime}=0.01$

-

$\left\|x-x^{adv}\right\|_{p}<\epsilon$

-

$\displaystyle\mathbb{E}[\sum_{t=0}^{m}\gamma^{t}r(y_{{\mathrm{LM}},t})],{\rm s% -.t.,}y_{{\mathrm{LM}},t}\sim M_{\mathrm{LM}}(\cdot|\hat{s_{t}},x),$

-

$c_{i}=Z_{1}(f_{i})$

-

${}^{\clubsuit,\heartsuit}$

-

$M_{\mathrm{LM}}$

-

$\rm answer$

-

$s_{0}=[-1]$

-

$P^{\prime\prime}=\{c_{i}\}_{i=0}^{n-1}$

-

$S_{\mathrm{semantic}}$

-

$V=\{v_{1},...,v_{m}\}$

-

$P^{\prime}=\{f_{i}\}_{i=0}^{n-1}$

-

$\rm question$

-

$k=a_{t}$

-

$(\rm question,\rm context,\rm answer)$

-

$q=Z_{2}(l)$

-

$\zeta(y,\hat{y})=\lambda\cdot S_{\mathrm{textual}}(y,\hat{y})+(1-\lambda)\cdot -S% -_{\mathrm{semantic}}(y,\hat{y}),$

-

$\displaystyle\underset{\theta}{\mathrm{max}}$

-

$r=\alpha\cdot\zeta(y,\hat{y})$

-

$l=\mathrm{concat}(g,h)$

-

$P=\{p_{i}\}_{i=0}^{n-1}$

-

$\rm context$

-

$\hat{s_{t}}\sim\prod_{i=0}^{t}\pi_{\theta}(a_{i}|s_{ -

$S_{\mathrm{textual}}$

-

$\{v_{i}\}_{i=0}^{m}$

-

$s_{t}=\mathrm{append}(s_{t-1},a_{t})$

-

$\pi_{\theta}(a_{t}|s_{ -

$v_{t+1}=p_{k}$

-

$y_{\mathrm{LM}}$

-

${}^{*~{}\heartsuit}$

-

$M_{\mathrm{LM}}(\cdot|v_{0},v_{1},...,v_{m},x)$

-

$(v_{0},...,v_{t})\times a_{t}\rightarrow(v_{0},...,v_{t},v_{t+1})$

-

$\underset{V\subset P}{\mathrm{max}}R(y_{\mathrm{LM}}\sim M_{\mathrm{LM}}(\cdot% -|v_{0},v_{1},...,v_{m},x)),$

-

$s_{0}=(v_{0},x)$

-

$\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)>1,\\ -x_{(i,j)}&\text{ if }S\left({V_{\left(i,j\right)}}\right)=1,\\ -0&\text{otherwise}\end{cases}$

-

$x_{\left(i,j\right)}$

-

$Q:\{0,1\}$

-

$u{\left({x}\right)}^{t}$

-

$MFA=\left\langle{\mathbb{Z}^{2},Q,V,F,w}\right\rangle$

-

$Q:{Wb}_{i}=0$

-

$f(0,x_{i+1},x_{i+2})=f(1,x_{i+1},x_{i+2})$

-

$\left[0,51,204,255\right]$

-

$\left({x_{i-1},x_{i},x_{i+1},x_{i+2}}\right)$

-

$\Delta m_{\text{init}-t}=\sum_{i=0}^{n}{x_{i}^{t_{0}}}-\sum_{i=0}^{n}{x_{i}^{t}}$

-

$f(x_{i},x_{i+1},x_{i+2})$

-

$\displaystyle x^{t+1}_{i}=\begin{cases}x^{t}&\text{if }w^{t}=1\\ -f\left({u\left({x_{i}}\right)}^{t}\right)&\text{otherwise}\end{cases}$

-

$Q:{Wb}_{i}=1$

-

$f(x_{i-1},x_{i},x_{i+1})$

-

$\begin{split}&\delta_{t}=\frac{\Delta(X_{t-1},X_{t})}{n}\quad\text{with}\quad t% -\geq 1,n\in\mathbb{N}\quad\text{and}\\ -&\Delta(X_{t-1},X_{t})=X_{t-1}\oplus X_{t}\end{split}$

-

$m_{t}=\sum_{i=0}^{n}{x_{i}^{t}}$

-

$V\left(i_{0},j_{0}\right)=\left\{{\left({i,j}\right):\mid{i-i_{0}}\mid+\mid{j-% -j_{0}}\mid\leq 1}\right\}$

- - - diff --git a/htmls/output_mathjax_example_10014.html b/htmls/output_mathjax_example_10014.html deleted file mode 100644 index 7071a3b4c772802c7557dc68b13246496fb99060..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10014.html +++ /dev/null @@ -1,150 +0,0 @@ - - - - MathJax Example - - - - -

$m_{\text{inter}}=\sum_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{n}}^{F,w_{0}}(i,j)}}-\sum% -_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{n}}^{F,w_{k}}(i,j)}}$

-

$w=\left\{{H1V1},{H2V2},{H4V4},\text{cut\&rel}\right\}$

-

$\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)=2,\\ -0&\text{otherwise}\end{cases}$

-

$x^{t_{0}}$

-

$\left({HV}\right)^{20}$

-

${Wb}_{i}$

-

$f(x_{i-1},x_{i},0)=f(x_{i-1},x_{i},1)$

-

$\left[1,10,30,60\right]$

-

$\delta_{\text{inter}}=\frac{{X_{t_{n}}^{F,w_{0}}}\oplus{X_{t_{n}}^{F,w_{k}}}}{% -N^{2}}$

-

$w^{t+1}={g\left({u\left({x}\right)}^{t}\right)}$

-

$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}$

-

$\left({HV}\right)^{*}$

-

${Wb}_{\left(i,j\right)}$

-

$\left({HHVV}\right)^{*}$

-

$t_{0},t_{1},\ldots,t_{n}$

-

${H,V}$

-

$\begin{split}&\delta_{\text{init state}}=\frac{\Delta(X_{t_{0}},X_{t})}{n}% -\quad\text{with}\quad t\geq 1,n\in\mathbb{N}\quad\text{and}\\ -&\Delta(X_{t_{0}},X_{t})=X_{t_{0}}\oplus X_{t}\end{split}$

-

$\displaystyle x^{t+1}_{i}=\begin{cases}0&\text{if }w^{t}=1\\ -f\left({u\left({x_{i}}\right)}^{t}\right)&\text{otherwise}\end{cases}$

-

$x(i,j)$

-

$x_{t_{n}}^{F,w}(i,j)[0\mapsto-1]=\begin{cases}x_{t_{n}}^{F,w}(i,j)&\text{ if }% -x_{t_{n}}^{F,w}(i,j)=1\\ --1&\text{ otherwise. }\end{cases}$

-

$\left({HHHHVVVV}\right)^{*}$

-

$Q=\{0,1\}$

-

$\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)\geq 2,\\ -0&\text{otherwise}\end{cases}$

-

$w_{0}=\emptyset$

-

$x^{t+1}_{i}=f(x^{t}_{i-1},x^{t}_{i},x^{t}_{i+1})$

-

$V_{\left(i,j\right)}=\left[x_{\left(i-1,j\right)},x_{\left(i+1,j\right)},x_{% -\left(i,j-1\right)},x_{\left(i,j+1\right)}\right]$

-

$\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)>2,\\ -x_{(i,j)}&\text{ if }S\left({V_{\left(i,j\right)}}\right)=2,\\ -0&\text{otherwise}\end{cases}$

-

$m_{\text{intra}}=\sum_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{0}}^{F,w}(i,j)}}-\sum_{i=% -0}^{m}{\sum_{j=0}^{m}}{x_{t_{n}}^{F,w}(i,j)}$

-

$\delta_{\text{intra}}=\frac{X_{t_{0}}^{F,w}\oplus X_{t_{n}}^{F,w}}{N^{2}}$

-

$\left[30,32,90,110,150\right]$

-

$\displaystyle\begin{split}{\text{Cid}(x_{t})}=\frac{\sum_{i=0}^{m}{\sum_{j=0}^% -{m}{x_{t_{n}}^{F,w}(i,j)[0\mapsto-1]}}}{N^{2}}\end{split}$

-

$\displaystyle f_{ij}=MLP(FC_{q}(F_{i})\oplus FC_{k}(F_{j})),$

-

$(x_{i2},y_{i2})$

-

$FC_{q}$

-

$\{s_{kj},k=1,2,...,n\}$

-

$b_{i}=(x_{i1},y_{i1},x_{i2},y_{i2})$

-

$T=\{T_{i}|i=1,2,3,\ldots,K\}$

-

$(x_{i1},y_{i1})$

-

$FC_{k}$

-

$C\in\mathbb{Z}^{N\times N}$

-

$\displaystyle s_{ij}=\frac{\exp(f_{ij})}{\sum_{i=1}^{N}\exp(f_{ij})},$

-

$R\in\mathbb{R}^{N\times N}$

-

$\{T_{i}|i=1,2,3,\ldots,K\}$

-

$\displaystyle=\min\limits_{1\leq i\leq R}{\Delta_{l2_{i}}},$

-

$\displaystyle=\{[t_{1},t_{2},...,t_{m}]|t_{i}\in T_{j}\},\forall j\in[1..R],$

-

$\displaystyle L_{b_{j}}$

-

$\displaystyle=\{l_{2ij}|i\in D_{b_{j}}\},L_{n_{j}}=\{l_{2ij}|i\in D_{n}\},$

-

$>0.17$

-

$\text{\text{EmbMarker}}+\text{{CSE}}$

-

$>0.98$

-

${\bm{e}}_{p2}$

-

$\lambda(S)$

-

$>0.10$

-

$j\in[1..R]$

-

$\displaystyle=\{\cos_{ij}|i\in D_{b_{j}}\},C_{n_{j}}=\{\cos_{ij}|i\in D_{n}\},$

-

${\bm{e}}_{p1}$

-

$\displaystyle C_{b_{j}}$

-

${\bm{e}}_{o}$

-

$\Delta_{cos}(\%)\downarrow$

-

$>0.20$

-

${\bm{u}}^{\langle k+1\rangle}$

-

${\bm{e}}_{s}$

-

$\displaystyle\Delta_{\cos_{k}}$

-

$\displaystyle=\frac{1}{|L_{b_{k}}|}\sum_{i\in L_{b_{k}}}i-\frac{1}{|L_{n_{k}}|% -}\sum_{j\in L_{n_{k}}}j.$

-

$\displaystyle=\frac{{\bm{e}}_{i}\cdot{\bm{w}}_{j}}{||{\bm{e}}_{i}||\cdot||{\bm% -{w}}_{j}||},\quad l_{2ij}=\biggl{|}\biggl{|}\frac{{\bm{e}}_{i}}{||{\bm{e}}_{i}% -||}-\frac{{\bm{w}}_{j}}{||{\bm{w}}_{j}||}\biggl{|}\biggl{|}^{2},$

-

$\Delta_{l2}(\%)\uparrow$

-

$>0.47$

-

$\lambda(S)=\lambda_{1}(S)+\lambda_{2}(S)+…+\lambda_{R}(S)$

-

$\displaystyle\Delta_{l2_{k}}$

-

$\displaystyle D_{p}=Rank(D_{v})-Rank(D_{s}),$

-

$>0.57$

-

$>0.02$

-

${\bm{u}}^{\langle k+1\rangle}={\bm{e}}^{\langle k\rangle}-\text{Proj}({\bm{e}}% -^{\langle k\rangle},{\bm{c}}^{\langle k\rangle}).$

-

${\bm{W}}=\{{\bm{w}}_{1},{\bm{w}}_{2},...,{\bm{w}}_{R}\}$

-

${\bm{e}}_{p}$

-

$\displaystyle D_{b_{j}}$

-

${\bm{e}}^{\langle k+1\rangle}$

-

$>0.26$

-

$\displaystyle D_{n}$

-

$\displaystyle\min_{\boldsymbol{\alpha}}\biggl{\|}{\bm{w}}-\sum_{k=1}^{K}{% -\alpha}_{k}\cdot{\bm{c}}^{\langle k\rangle}\biggl{\|}^{2}.$

-

$>0.04$

-

$\text{p-value}_{j}$

-

$\displaystyle=\frac{1}{|C_{b_{k}}|}\sum_{i\in C_{b_{k}}}i-\frac{1}{|C_{n_{k}}|% -}\sum_{j\in C_{n_{k}}}j,$

-

$>0.22$

-

$Rank$

-

$\displaystyle=\min\limits_{1\leq i\leq R}{\text{p-value}_{i}}.$

-

$>0.56$

-

$\text{Norm}\left({\bm{u}}^{\langle k+1\rangle}\right)=\frac{{\bm{u}}^{\langle k% -+1\rangle}}{||{\bm{u}}^{\langle k+1\rangle}||}$

-

$\displaystyle\Delta_{l2}$

-

$\text{Proj}({\bm{e}}^{\langle k\rangle},{\bm{c}}^{\langle k\rangle})=\frac{{% -\bm{c}}^{\langle k\rangle}\cdot{\bm{e}}^{\langle k\rangle}}{||{\bm{c}}^{% -\langle k\rangle}||}\cdot{\bm{c}}^{\langle k\rangle}.$

-

$\displaystyle{\bm{e}}_{p}=\text{Norm}\left((1-\sum_{r=1}^{R}\lambda_{r}(S))% -\cdot{\bm{e}}_{o}+\sum_{r=1}^{R}\lambda_{r}(S)\cdot{\bm{w}}_{r}\right).$

-

${\bm{e}}_{s2}$

-

$>10^{-3}$

-

$\displaystyle=\{[t_{1},t_{2},...,t_{m}]|t_{i}\notin T\}.$

-

${\bm{c}}^{\langle k\rangle}$

-

$\Delta_{l2}(\%)$

-

$>0.36$

-

${\bm{e}}_{p}=f({\bm{e}}_{o},t)$

-

$>0.62$

-

$>0.55$

- - - diff --git a/htmls/output_mathjax_example_10015.html b/htmls/output_mathjax_example_10015.html deleted file mode 100644 index 92d5eab5c1f5e00d14c68139b05af5e040b2555f..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10015.html +++ /dev/null @@ -1,132 +0,0 @@ - - - - MathJax Example - - - - -

$\Delta_{cos}(\%)$

-

${\bm{e}}^{\langle 0\rangle}$

-

${\bm{e}}^{\langle k\rangle}$

-

$>0.08$

-

$>0.83$

-

${\bm{e}}_{s1}$

-

$\displaystyle=\max\limits_{1\leq i\leq R}{\Delta_{\cos_{i}}},$

-

$>10^{-4}$

-

$m=4,n=20,\text{and frequency interval}=[0.5\%,1\%]$

-

$\displaystyle\Delta_{\cos}$

-

$\displaystyle\cos_{ij}$

-

$C_{b_{j}}$

-

$<10^{-3}$

-

$\Theta_{v}$

-

$1,801,350$

-

$C_{n_{j}}$

-

$>0.21$

-

$Pops(\pi_{a})=\\ -\{\{1,7\},\{5\},\{11,37\},\{13,19\},\{15\},\{22\},\{24\}\}$

-

$\delta_{sgo}$

-

$\delta_{flex}=\frac{4}{9}=0.4$

-

$Pops(\pi_{b})=\\ -\{\{1,7\},\{5,11\},\{13,19\},\{15\},\{22\},\{24,37\}\}$

-

$sg\neq X$

-

$SubGoals(\pi_{b})=``XXXCBXXXXA"$

-

$D(\pi_{a},\pi_{b})=1-\delta(\pi_{a},\pi_{b})$

-

$\delta_{\text{sgo}}(\pi_{a},\pi_{b})=1-\frac{HDist(SubGoals(\pi_{a}),SubGoals(% -\pi_{b}))}{max(SubGoals(\pi_{a}),SubGoals(\pi_{b}))}$

-

$HDist(SubGoals(\pi_{a}),SubGoals(\pi_{b}))=5$

-

$x\in\{a,s,c\}$

-

$SubGoals(\pi_{b})$

-

$\delta(\pi_{a},\pi_{b})\rightarrow{[0,1]}$

-

$state\leftarrow PerformAction(state,a)$

-

$subgoalLetter\leftarrow GetEncodedSubgoals(PI)$

-

$CBA$

-

$\delta_{flex}$

-

$\pi_{a}=\{1,7,5,11,37,13,19,15,22,24\}$

-

$\pi_{b}=\{1,7,5,11,13,19,15,22,24,37\}$

-

$sg\leftarrow GetSubGoal(state,PI)$

-

$state\leftarrow GetInitialState(PI)$

-

$\delta_{x}(\pi_{a},\pi_{b})=|A(\pi_{a})\cap A(\pi_{b})|/|A(\pi_{b})\cup A(\pi_% -{a})|$

-

$seq\leftarrow AppendTo(seq,subgoalLetter[sg])$

-

$SubGoals$

-

$HDist(s_{\pi_{a}},s_{\pi_{b}})\rightarrow\mathbb{R}$

-

$max(a,b)\rightarrow\mathbb{N}$

-

$SubGoals(\pi_{a})=``XXBXXXXAXC"$

-

$\delta_{u}(\pi_{a},\pi_{b})={\begin{cases}1,\ if\ \pi_{a}\setminus\pi_{b}=% -\emptyset\\ -1,\ if\ \pi_{a}\subset\pi_{b}\\ -0,\ otherwise\end{cases}}$

-

$\delta_{\text{flex}}(\pi_{a},\pi_{b})=\frac{|Pop(\pi_{a})\cap Pop(\pi_{b})|}{|% -Pop(\pi_{a})\cup Pop(\pi_{b})|}$

-

$seq\leftarrow AppendTo(seq,``X")$

-

$seq\leftarrow``"$

-

$Pop(\pi)$

-

$\delta_{a}(\pi_{a},\pi_{b})=\frac{|1,7,5,11,37,13,19,15,22,24|}{|1,7,5,11,13,1% -9,15,22,24,37|}=1$

-

$HDist$

-

$SubGoals(\pi_{a})$

-

$BAC$

-

$A(\pi)$

-

$\xi=0.031$

-

$\mathcal{D^{\prime}}(\mathcal{D}(x)+\eta)$

-

$||\delta||_{\infty}\leq\xi$

-

$Z_{l}(\cdot)$

-

$u\leftarrow\max(|\delta_{init}|)$

-

$s\in\{-1,1\}^{d}$

-

$\mathrm{Cos}$

-

$y\in\mathbb{R}^{2}=\{0,1\}$

-

$s\sim\mathrm{\textbf{Bernoulli}}(p),\quad\delta_{s}\leftarrow\delta_{p}\odot s.$

-

$\delta_{0}\leftarrow 0$

-

$f(x)_{i}$

-

$x_{adv}\leftarrow x_{K}$

-

$\delta\leftarrow x^{p}-x$

-

$\epsilon=0.062$

-

$\mathcal{D^{\prime}}(\cdot)$

-

$x_{adv}\leftarrow\mathrm{Clip}_{x,\ \epsilon^{\prime}-\kappa}(x_{idct}),$

-

$[0,\max(\delta_{init})]$

-

$m\leftarrow(l+u)/2$

-

$c(x_{t})=0$

-

$c(x_{adv})=0$

-

$x_{0}\leftarrow x_{r}$

-

$x_{idct}^{p}\leftarrow\mathcal{D’}(x_{dct}^{p}+\eta)$

-

$f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$

-

$\delta\leftarrow\mathrm{Clip}_{0,\ u}(\delta_{init})$

-

$x_{dct}^{p}\leftarrow\mathcal{D}(x+\delta)$

-

$\mathop{\mathrm{argmin}}\limits_{\delta}\ \mathrm{Cos}(Z_{l}(x),\ Z_{l}(x_{r}+% -\delta)),\quad s.t.\quad||\delta||_{\infty}\leq\xi,$

-

$p=0.999$

-

$x_{idct}\leftarrow\mathcal{D^{\prime}}(\mathcal{D}(x_{adv})+\eta),\quad\eta% -\sim\{-\gamma,\gamma\}^{d},$

-

$\delta\leftarrow\mathrm{\textbf{BinarySearch}}(x,\delta_{init})$

-

$i\in[1,\ k]$

-

$\gamma=1.75$

-

$||\delta||_{\infty}\leq\epsilon^{\prime},\epsilon^{\prime}\in(0,\max|\delta_{% -init}|]$

-

$c(x)=\mathrm{argmax}_{i\in\{0,1\}}\ f(x)_{i}$

-

$Z_{l}(x)$

-

$\mathrm{Clip}_{x_{r},\ \xi}(\cdot)$

-

$\eta\sim\{-\gamma,\gamma\}^{d}$

-

$c(x^{p})=0$

-

$\mathrm{\#\ Perturbation\ Random\ Flip}$

-

$i\in[1,\ K]$

-

$50.34$

-

$\delta_{init}\leftarrow\mathrm{\textbf{CrossPerturbInit}}(x,x_{r},Z,f)$

-

$c(x+\delta)=0$

-

$Z_{l}(x_{r}+\delta)$

-

$\delta^{p}\leftarrow\delta\odot s$

-

$\mathrm{\#\ Frequency\ Noise\ Projection}$

-

$x^{p}\leftarrow\mathrm{Clip}_{x,\ \epsilon^{\prime}-\kappa}(x_{dct}^{p})$

-

$\delta\leftarrow\delta^{p}$

- - - diff --git a/htmls/output_mathjax_example_10016.html b/htmls/output_mathjax_example_10016.html deleted file mode 100644 index e521fcc9823314947a797cd93630ac3a76c1d926..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10016.html +++ /dev/null @@ -1,146 +0,0 @@ - - - - MathJax Example - - - - -

$x_{adv}\leftarrow x+\delta$

-

$J(x_{i-1},x)\leftarrow\mathrm{-Cos}(Z_{l}(x_{i-1}),\ Z_{l}(x))$

-

$\mathcal{D^{\prime}}(\mathcal{D}(x))$

-

$l\leftarrow m$

-

$s\sim\mathrm{\textbf{Bernoulli}}(p)$

-

$\min_{\delta}||\delta||_{\infty}\quad\mathrm{s.t.}\quad c(x+\delta)=0.$

-

$\delta_{i}\leftarrow\delta_{i-1}+\frac{\xi}{K}\cdot\mathrm{sign}(\nabla_{x_{i-% -1}}J(x_{i-1},x))$

-

$c(x+\delta^{p})==0$

-

$x_{r}+\delta$

-

$p\in(0,1)^{d}$

-

$x_{i}\leftarrow\mathrm{Clip}_{x_{r},\ \xi}(x_{i-1}+\delta_{i})$

-

$x_{t}\leftarrow\mathrm{Clip}_{x_{r},\ m}(x+\delta_{init})$

-

$\mathrm{\#\ Perturbation\ Initialization}$

-

$\delta_{init}$

-

$\epsilon^{\prime}\leftarrow||\delta||_{\infty}$

-

$i,j,k\in[1,n]$

-

$\varepsilon_{2}=0.9\varepsilon$

-

$\langle\gamma\rangle_{1}=\sum_{i=1}^{n}\langle\gamma_{i}\rangle_{1}$

-

$d_{max}^{\prime}\leftarrow\mathsf{max}(d_{1}^{\prime},...,d_{n}^{\prime})$

-

$f=\langle f\rangle_{1}+\langle f\rangle_{2}$

-

$\langle v_{4},v_{5}\rangle$

-

$Lap(\frac{\triangle}{\varepsilon_{2}})$

-

$\displaystyle\frac{e^{\frac{-\varepsilon_{2}.|\widetilde{T}-T(G)|}{\triangle}}% -}{e^{\frac{-\varepsilon_{2}.|\widetilde{T}-T(G^{\prime})|}{\triangle}}}=e^{% -\frac{\varepsilon_{2}.(|\widetilde{T}-T(G^{\prime})|-|\widetilde{T}-T(G)|)}{% -\triangle}}$

-

$Pr[\mathcal{M}_{i}(A_{i})\in S]\leq e^{\epsilon}Pr[\mathcal{M}_{i}(A_{i}^{% -\prime})\in S]$

-

$\hat{T}(G,d_{max}^{\prime})$

-

$T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime})$

-

$\displaystyle\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime% -}),\hat{T}(G,d_{max}^{\prime}))]=\mathbb{V}[T^{\prime}(G,\varepsilon_{2},d_{% -max}^{\prime})]$

-

$ds=[0]*n,\hat{A_{i}}=\varnothing$

-

$e=a-x,f=b-y,g=c-z$

-

$\langle u\rangle_{i}$

-

$\langle T^{\prime}\rangle_{2}=\langle T\rangle_{2}+\langle\gamma\rangle_{2}$

-

$\mathsf{Local2Rounds_{\triangle}}$

-

$\langle x\rangle_{2},\langle y\rangle_{2},\langle z\rangle_{2},\langle w% -\rangle_{2},\langle o\rangle_{2},\langle p\rangle_{2},\langle q\rangle_{2}% -\rightarrow S_{2}$

-

$\varepsilon_{1}=0.1\varepsilon$

-

$\Gamma(\beta)=\int_{0}^{\infty}x^{\beta-1}e^{-x}dx$

-

$O\binom{n}{2}$

-

$\gamma_{i}=(Gam_{1}-Gam_{2})$

-

$\langle\gamma\rangle_{2}=\sum_{i=1}^{n}\langle\gamma_{i}\rangle_{2}$

-

$\mathsf{Count}$

-

$\displaystyle e^{\frac{\varepsilon_{2}|T(G)-T(G^{\prime})|}{\triangle}}=e^{% -\varepsilon_{2}}$

-

$\hat{A_{i}}$

-

$d_{max}^{\prime}$

-

$\hat{A}=\{\hat{A_{1}},...,\hat{A_{n}}\}$

-

$\langle a_{ij}\rangle_{1}$

-

$l_{2}^{2}(d_{max}^{\prime},d_{max})<0.009d_{max}$

-

$\langle v_{2},v_{1}\rangle$

-

$\mathbb{E}[l_{2}^{2}(T(G),\hat{T}(G,d_{max}^{\prime}))]=(T(G)-\hat{T}(G,d_{max% -}^{\prime}))^{2}$

-

$\langle v_{j},v_{i}\rangle$

-

$v_{i},i\in[1,n]$

-

$Gam_{2}=\mathsf{Gamma}(n,\frac{d_{max}^{\prime}}{\varepsilon_{2}})$

-

$\langle x\rangle_{2}=(x-r)$

-

$r\in\mathbb{Z}_{2^{l}}$

-

$T(G^{\prime})$

-

$D^{\prime},d_{max}^{\prime}$

-

$\displaystyle\frac{Pr[d^{\prime}=d_{i}+x]}{Pr[d^{\prime}=d_{i}^{\prime}+x^{% -\prime}]}=\frac{Pr[x=d^{\prime}-d_{i}]}{Pr[x^{\prime}=d^{\prime}-d_{i}^{\prime% -}]}$

-

$\langle T\rangle_{1}=\langle T\rangle_{2}=0$

-

$\langle a_{ij}\rangle_{2}$

-

$\displaystyle\mathbb{V}[Lap(\frac{d_{max}^{\prime}}{\varepsilon_{2}})]=O(\frac% -{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$

-

$\langle x\rangle_{1}=r$

-

$v_{k},k\in[n]$

-

$\langle g\rangle_{i}=\langle c\rangle_{i}-\langle z\rangle_{i}$

-

$\displaystyle\frac{Pr[\widetilde{T}=T+(r_{1}+...+r_{n})]}{Pr[\widetilde{T}=T^{% -\prime}+(r_{1}^{\prime}+...+r_{n}^{\prime})]}$

-

$\hat{A_{i}}\leftarrow A_{i}$

-

$a_{ij}=1,j\in[n]$

-

$|d_{i}-d_{i}^{\prime}|=1$

-

$T^{\prime}=\langle T^{\prime}\rangle_{1}+\langle T^{\prime}\rangle_{2}$

-

$D^{\prime}\leftarrow D^{\prime}\cup\{d_{i}^{\prime}\}$

-

$O(\frac{d_{max}^{2}}{\varepsilon^{2}})$

-

$r=\{r_{1},...,r_{n}\}$

-

$O(\theta)$

-

$+\langle x\rangle_{2}fg+\langle y\rangle_{2}eg+\langle z\rangle_{2}ef+efg$

-

$e=\langle e\rangle_{1}+\langle e\rangle_{2}$

-

$O(n^{2}+nd_{max}^{2})$

-

$d_{i}>d_{max}^{\prime}$

-

$(d_{i}-d_{max}^{\prime})$

-

$(i\in\{1,2\})$

-

$\displaystyle\frac{Pr[\widetilde{T}=T(G)+x]}{Pr[\widetilde{T}=T(G^{\prime})+x^% -{\prime}]}=\frac{Pr[x=\widetilde{T}-T(G)]}{Pr[x^{\prime}=\widetilde{T}-T(G^{% -\prime})]}$

-

$\langle x\rangle_{2},\langle y\rangle_{2},\langle z\rangle_{2},\langle w% -\rangle_{2},\langle o\rangle_{2},\langle p\rangle_{2},\langle q\rangle_{2}$

-

$O(d_{max}^{\prime})$

-

$\langle e\rangle_{1}=\langle a_{ij}\rangle_{1}-\langle x\rangle_{1}$

-

$\langle\gamma_{i}\rangle_{1}\rightarrow S_{1}$

-

$\hat{ds}\leftarrow ds[1:d_{max}^{\prime}]$

-

$\mathsf{view}_{S_{i}}^{\Pi}$

-

$x=r_{1}+...+r_{n}$

-

$d_{max}^{\prime}\approx d_{max}$

-

$\langle e\rangle_{i}=\langle a\rangle_{i}-\langle x\rangle_{i},$

-

$\displaystyle\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime% -}),\hat{T}(G,d_{max}^{\prime}))]$

-

$\times 10^{10}$

-

$k=j+1$

-

$e,f,$

-

$\displaystyle\frac{Pr[\widetilde{T}=\langle T\rangle_{1}+\langle T\rangle_{2}+% -(r_{1}+...+r_{n})]}{Pr[\widetilde{T}=\langle T^{\prime}\rangle_{1}+\langle T^{% -\prime}\rangle_{2}+(r_{1}^{\prime}+...+r_{n}^{\prime})]}$

-

$\mathsf{Sim}_{S_{i}}$

-

$\langle\gamma_{i}\rangle_{1},\langle\gamma_{i}\rangle_{2}$

-

$\langle u\rangle_{1}$

-

$r^{\prime}=\{r_{1}^{\prime},...,r_{n}^{\prime}\}$

-

$Pr[\mathcal{M}(D)\in S]\leq e^{\epsilon}Pr[\mathcal{M}(D^{\prime})\in S]$

-

$O\binom{d_{max}^{\prime}}{2}$

-

$d=\langle d\rangle_{1}+\langle d\rangle_{2}=\langle w\rangle_{1}+\langle xy% -\rangle_{1}g+\langle xz\rangle_{1}f+\langle yz\rangle_{1}e+\langle x\rangle_{1% -}fg+\langle y\rangle_{1}eg+\langle z\rangle_{1}ef+\langle w\rangle_{2}+\langle -xy% -\rangle_{2}g+\langle xz\rangle_{2}f+\langle yz\rangle_{2}e+\langle x\rangle_{2% -}fg+\langle y\rangle_{2}eg+\langle z\rangle_{2}ef+efg$

-

$\langle x\rangle_{2}+\langle y\rangle_{1}+\langle y\rangle_{2}=x+y$

-

$\langle e\rangle_{2}=\langle a_{ij}\rangle_{2}-\langle x\rangle_{2}$

- - - diff --git a/htmls/output_mathjax_example_10017.html b/htmls/output_mathjax_example_10017.html deleted file mode 100644 index b848d81f1497f17a959faa9407877f2049576cd5..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10017.html +++ /dev/null @@ -1,134 +0,0 @@ - - - - MathJax Example - - - - -

$D=\{d_{1},...,d_{n}\}$

-

$\hat{ds}$

-

$\langle T\rangle=\{\langle T\rangle_{1},\langle T\rangle_{2}\}$

-

$T\neq$

-

$\mathsf{CentralLap_{\triangle}}$

-

$\langle g\rangle_{2}=\langle a_{jk}\rangle_{2}-\langle z\rangle_{2}$

-

$d_{i}^{\prime}\leftarrow d_{i}+\mathsf{Lap}(\frac{1}{\varepsilon_{1}})$

-

$Lap(\frac{1}{\varepsilon_{1}})$

-

$\langle d\rangle_{i}=\langle w\rangle_{i}+\langle xy\rangle_{i}g+\langle xz% -\rangle_{i}f+\langle yz\rangle_{i}e+\langle x\rangle_{i}fg+\langle y\rangle_{i% -}eg+\langle z\rangle_{i}ef+(i-1)efg$

-

$u_{1}=\langle w\rangle_{1}+\langle xy\rangle_{1}g+\langle xz\rangle_{1}f+% -\langle yz\rangle_{1}e$

-

$re(T,T^{\prime})=\frac{|T-T^{\prime}|}{T}$

-

$\langle f\rangle_{i}=\langle b\rangle_{i}-\langle y\rangle_{i},$

-

$Gam_{1}(n,\frac{\triangle}{\varepsilon_{2}})-Gam_{2}(n,\frac{\triangle}{% -\varepsilon_{2}})$

-

$D,D^{\prime}\in\mathcal{X}^{n}$

-

$A_{ij}==1$

-

$u=a_{ij}\times a_{ik}\times a_{jk}(i -

$A_{i}=\{a_{i1},...,a_{in}\}$

-

$\varepsilon,n,d_{max},d_{max}^{\prime}$

-

$y\in range(\mathcal{M})$

-

$\mathsf{Project}$

-

$\displaystyle(\mathbb{E}[T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime})]-\hat{% -T}(G,d_{max}^{\prime}))^{2}+\mathbb{V}[T^{\prime}(G,\varepsilon_{2},d_{max}^{% -\prime})]$

-

$\mathsf{Gamma}$

-

$\theta=1000$

-

$\mathsf{Perturb}$

-

$Gam_{2}$

-

$\mathsf{Perturb}()$

-

$O(nd_{max})$

-

$\mathsf{GraphProjection}$

-

$\langle x\rangle_{1},\langle y\rangle_{1},\langle z\rangle_{1},\langle w% -\rangle_{1},\langle o\rangle_{1},\langle p\rangle_{1},\langle q\rangle_{1}$

-

$S\subseteq Range(\mathcal{M}_{i})$

-

$\mathbb{E}[l_{2}^{2}(T(G),\hat{T}(G,d_{max}^{\prime}))]=0$

-

$\varepsilon=\varepsilon_{1}+\varepsilon_{2}$

-

$\langle T^{\prime}\rangle_{1}=\langle T\rangle_{1}+\langle\gamma\rangle_{1}$

-

$\langle v_{i},v_{j}\rangle\in E$

-

$\langle y\rangle_{i}$

-

$ds[j]$

-

$\langle f\rangle_{1}=\langle a_{ik}\rangle_{1}-\langle y\rangle_{1}$

-

$Gam_{1}(n,\lambda)$

-

$\mathsf{project}$

-

$\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{0\}$

-

$O(\frac{e^{\varepsilon}}{(e^{\varepsilon}-1)^{2}}(d_{max}^{3}n+\frac{e^{% -\varepsilon}}{\varepsilon^{2}}d_{max}^{2}n))$

-

$u=\langle u\rangle_{1}+\langle u\rangle_{2}=\langle x\rangle_{1}$

-

$\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{1\}$

-

$\langle T\rangle\leftarrow\{\langle T\rangle_{1},\langle T\rangle_{2}\}$

-

$Laplace$

-

$Pr[\mathcal{M}(G)=y]\leq e^{\varepsilon}Pr[\mathcal{M}(G^{\prime})=y]$

-

$\langle x\rangle_{i}$

-

$\langle T\rangle_{1}$

-

$1\times\sim 2\times$

-

$S_{i\in\{1,2\}}$

-

$A=\{A_{1},...,A_{n}\}$

-

$\langle T\rangle_{2}\leftarrow\langle T\rangle_{2}+u_{2}$

-

$G,G^{\prime}\in\mathcal{G}$

-

$\langle\gamma_{i}\rangle_{2}\rightarrow S_{2}$

-

$\mathsf{Lap(\frac{1}{\varepsilon_{1}})}$

-

$O(\frac{d_{max}^{\prime 2}}{\varepsilon^{2}})$

-

$u_{2}=\langle w\rangle_{2}+\langle xy\rangle_{2}g+\langle xz\rangle_{2}f+% -\langle yz\rangle_{2}e$

-

$\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime}),\hat{T}(G,% -d_{max}^{\prime}))]=O(\frac{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$

-

$\langle v_{i},v_{j}\rangle$

-

$Gam_{1}$

-

$l_{2}(T,T^{\prime})=(T-T^{\prime})^{2}$

-

$=w+xyg+xzf+yze+xfg+yeg+zef+efg$

-

$\langle u\rangle_{2}$

-

$D^{\prime}=\{d_{1}^{\prime},...,d_{n}^{\prime}\}$

-

$\mathsf{Laplace}$

-

$w=xyz,o=xy,p=xz,q=yz$

-

$ds[j]\leftarrow\mathsf{DS}(d_{i},d_{j}^{\prime})$

-

$d_{max}^{\prime}\geq 0$

-

$\langle f\rangle_{2}=\langle a_{ik}\rangle_{2}-\langle y\rangle_{2}$

-

$S\subseteq Range(\mathcal{M})$

-

$\{d_{1}^{\prime},...,d_{n}^{\prime}\}$

-

$Gam_{2}(n,\lambda)$

-

$\displaystyle\frac{e^{-\varepsilon_{1}.|d^{\prime}-d_{i}|}}{e^{-\varepsilon_{1% -}.|d^{\prime}-d_{i}^{\prime}|}}=e^{\varepsilon_{1}.(|d^{\prime}-d_{i}^{\prime}% -|-|d^{\prime}-d_{i}|)}\leq e^{\varepsilon_{1}|d_{i}-d_{i}^{\prime}|}=e^{% -\varepsilon_{1}},$

-

$(r_{1},...,r_{n})$

-

$\mathsf{view}_{S_{i}}^{\Pi}\approx\mathsf{Sim}_{S_{i}}$

-

$T(G)=\langle T\rangle_{1}+\langle T\rangle_{2}$

-

$\mathsf{Project}()$

-

$Gam_{1}=\mathsf{Gamma}(n,\frac{d_{max}^{\prime}}{\varepsilon_{2}})$

-

$Gamma(x,n,\lambda)=\frac{(1/\lambda)^{1/n}}{\Gamma(1/n)}x^{\frac{1}{n}-1}e^{-% -\frac{x}{\lambda}},$

-

$g=\langle g\rangle_{1}+\langle g\rangle_{2}$

-

$d=a\times b\times c$

-

$Lap(.)$

-

$ds[k]$

-

$a_{ij}\times a_{ik}\times a_{jk}=1$

-

$(D^{\prime},d_{max}^{\prime})\leftarrow\mathsf{Max}(D,\varepsilon_{1})$

-

$\langle x\rangle=\langle x\rangle_{1}+\langle x\rangle_{2}$

-

$Lap(\lambda)=\sum_{i=1}^{n}[Gam_{1}(n,\lambda)-Gam_{2}(n,\lambda)],$

-

$\mathsf{Count}()$

-

$A=\{A_{1},A_{2},...,A_{n}\}$

-

$\langle v_{2},v_{5}\rangle$

-

$DS(d_{1},d_{2})$

-

$\mathsf{Max}(.)$

-

$u=a_{ij}\times a_{ik}\times a_{jk}$

-

$\langle T\rangle\leftarrow\mathsf{Count}(\hat{A})$

-

$|T(G)-T(G^{\prime})|=\triangle$

-

$+\langle x\rangle_{1}fg+\langle y\rangle_{1}eg+\langle z\rangle_{1}ef$

-

$d_{max}^{\prime}\geq d_{max}$

-

$\gamma_{i}=\langle\gamma_{i}\rangle_{1}+\langle\gamma_{i}\rangle_{2}$

-

$w=x\times y\times z,o=x\times y,p=x\times z,q=y\times z$

-

$Pr[\mathcal{M}(G)\in S]\leq e^{\epsilon}Pr[\mathcal{M}(G^{\prime})\in S]$

- - - diff --git a/htmls/output_mathjax_example_10018.html b/htmls/output_mathjax_example_10018.html deleted file mode 100644 index 721ad5e1bbcdb17f82825a152670a3905b8b6c78..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10018.html +++ /dev/null @@ -1,156 +0,0 @@ - - - - MathJax Example - - - - -

$d_{i}\leq d_{max}^{\prime}$

-

$\hat{A}\leftarrow\mathsf{Project}(A,D,D^{\prime},d_{max}^{\prime})$

-

$\langle g\rangle_{1}=\langle a_{jk}\rangle_{1}-\langle z\rangle_{1}$

-

$\langle x\rangle_{1},\langle y\rangle_{1},\langle z\rangle_{1},\langle w% -\rangle_{1},\langle o\rangle_{1},\langle p\rangle_{1},\langle q\rangle_{1}% -\rightarrow S_{1}$

-

$\langle T\rangle_{1}\leftarrow\langle T\rangle_{1}+u_{1}$

-

$x,y,z,w,o,p,q$

-

$O(\frac{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$

-

$T^{\prime}\leftarrow\mathsf{Perturb}(\langle T\rangle,d_{max}^{\prime},% -\varepsilon_{2})$

-

$T(G^{\prime})=\langle T^{\prime}\rangle_{1}+\langle T^{\prime}\rangle_{2}$

-

$x^{\prime}=r_{1}^{\prime}+...+r_{n}^{\prime}$

-

$a_{ij},a_{ik},a_{jk}$

-

$\varepsilon_{2}\geq 0$

-

$f(x,\lambda)=\frac{1}{2\lambda}e^{\frac{|x|}{\lambda}}$

-

$DS(d_{1},d_{2})=\frac{|d_{1}-d_{2}|}{d_{1}}$

-

$\langle T\rangle_{2}$

-

${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}% -]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{% -\mathcal{E}}}\cdot\pi_{RT}$

-

$\pi_{FiT}$

-

${\bigl{\{}Dev_{C}^{Tot}\bigr{\}}_{\mathcal{E}}}$

-

${\bigl{\{}V^{P2P}\bigr{\}}_{\mathcal{E}}}$

-

$N_{U},{\bigl{\{}V^{P2P}\bigr{\}}_{\mathcal{E}}},{\bigl{\{}V^{Real}\bigr{\}}_{% -\mathcal{E}}},Dev_{C}^{Tot},Dev_{P}^{Tot},\pi_{P2P},\pi_{RT}$

-

${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}% -]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{% -\mathcal{E}}}\cdot\pi_{P2P}$

-

$\pi_{P2P}$

-

$KGen_{pe}(k)$

-

$each~{}m_{l}\pcin~{}M(u_{k})$

-

${\bigl{\{}inDev\bigr{\}}_{\mathcal{E}}},{\bigl{\{}inDev_{M}\bigr{\}}_{\mathcal% -{E}}}$

-

$\{.\}_{\mathcal{E}}$

-

$Bal^{Tot}_{sup}$

-

${\bigl{\{}V^{Real}\bigr{\}}_{\mathcal{E}}}$

-

$V^{Real}$

-

${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}% -]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{% -\mathcal{E}}}\cdot{\pi_{RT}}$

-

$Dev_{P}^{Tot}>Dev_{C}^{Tot}c$

-

${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}% -]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{% -\mathcal{E}}}/Dev_{P}^{Tot}\cdot TotRev_{P}$

-

$Stat_{M}[u_{k}]$

-

$KGen_{pe}$

-

${\bigl{\{}V^{Real}[u_{k}]\bigr{\}}_{\mathcal{E}}}$

-

${\bigl{\{}inDev\bigr{\}}_{\mathcal{E}}}$

-

${\bigl{\{}Stat\bigr{\}}_{\mathcal{E}}},{\bigl{\{}Stat_{M}\bigr{\}}_{\mathcal{E% -}}}$

-

${\bigl{\{}inDev_{M}\bigr{\}}_{\mathcal{E}}}$

-

${{\bigl{\{}inDev[u_{k}]\bigr{\}}_{\mathcal{E}}}}$

-

$inDev$

-

${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}% -]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{% -\mathcal{E}}}\cdot\pi_{P2P}$

-

$V^{P2P}$

-

$V^{Real}[u_{k}]$

-

${\bigl{\{}inDev[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{Real}[u_{% -k}]\bigr{\}}_{\mathcal{E}}}-{\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}};$

-

$M(u_{k})$

-

$H{({\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}})}$

-

$PK_{sup}$

-

$\xrightarrow{}PK_{sup},SK_{sup}$

-

${\bigl{\{}inDev_{M}[M(m_{l})]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{% -Real}[M(m_{l})]\bigr{\}}_{\mathcal{E}}}-{\bigl{\{}V^{P2P}[M(m_{l})]\bigr{\}}_{% -\mathcal{E}}};$

-

$Dev_{P}^{Tot}=Dev_{C}^{Tot}$

-

$Dev_{P}^{Tot}>Dev_{C}^{Tot}$

-

$TotRev_{P}=(Dev_{C}^{Tot}\cdot\pi_{P2P}+(Dev_{P}^{Tot}-Dev_{C}^{Tot})\cdot\pi_% -{FiT})$

-

$\footnotesize{\bigl{\{}Dev_{C}^{Tot}\bigr{\}}_{\mathcal{E}}}\leftarrow\sum_{i=% -0}^{N_{C}-1}{\bigl{\{}inDev_{C}[c_{i}]\bigr{\}}_{\mathcal{E}}}$

-

${inDev_{P}[u_{k}]}/Dev_{P}^{Tot}$

-

${\bigl{\{}stat^{Tot}[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}stat^{% -Tot}[u_{k}]\bigr{\}}_{\mathcal{E}}}+{\bigl{\{}stat[u_{k}]\bigr{\}}_{\mathcal{E% -}}}$

-

$Dev^{Tot}$

-

$\pi_{P2P},\pi_{FiT},\pi_{RT}$

-

$N_{C}=N_{P}$

-

${\bigl{\{}inDev[c_{i}]\bigr{\}}_{\mathcal{E}}}$

-

$each~{}u_{k}$

-

${\bigl{\{}Dev_{P}^{Tot}\bigr{\}}_{\mathcal{E}}}$

-

${Dev_{C}^{Tot}}$

-

$Bal_{sup}\leftarrow 0$

-

$\pi_{RT}$

-

$Stat$

-

${Dev_{P}^{Tot}}$

-

$N_{C} -

${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}% -]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{% -\mathcal{E}}}\cdot{\pi_{P2P}}$

-

$TotRev_{P}$

-

$Bal_{sup}\leftarrow-(Dev_{P}^{Tot}-Dev_{C}^{Tot})\cdot\pi_{FiT}$

-

$Stat[u_{k}]$

-

$\footnotesize{\bigl{\{}Dev_{P}^{Tot}\bigr{\}}_{\mathcal{E}}}\leftarrow\sum_{j=% -0}^{N_{C}-1}{\bigl{\{}inDev_{P}[p_{j}]\bigr{\}}_{\mathcal{E}}}$

-

$H{({\bigl{\{}V^{Real}[u_{k}]\bigr{\}}_{\mathcal{E}}})}$

-

${\bigl{\{}stat_{M}^{Tot}[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}% -stat^{Tot}[m_{l}]\bigr{\}}_{\mathcal{E}}}+{\bigl{\{}stat[m_{l}]\bigr{\}}_{% -\mathcal{E}}}$

-

$stat^{Tot}$

-

$N_{C}>N_{P}$

-

$Bal_{sup}$

-

$(Dev_{P}^{Tot}-Dev_{C}^{Tot})$

-

$Dev_{P}^{Tot} -

${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}% -]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{% -\mathcal{E}}}/Dev_{P}^{Tot}\cdot TotRev_{P}$

-

$SK_{sup}$

-

${\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}}$

-

$Bal_{sup}\leftarrow(Dev_{C}^{Tot}-Dev_{P}^{Tot})\cdot\pi_{RT}.$

-

$k(\mu,\mu^{\prime})=\left(1+\left\|(\mu-\mu^{\prime})/\sigma\right\|_{2}^{2}% -\right)^{-1}$

-

$\displaystyle=-\log(\mathcal{N}(h(\mathbf{x});\mathbf{0},\mathbf{I})))-\log% -\left|\det\mathbf{J}_{h}(\mathbf{x})\right|$

-

$K=D-d$

-

$\mathbf{x}=h^{-1}(\mathbf{z})$

-

$\mathcal{L}_{\mathrm{MMD}}(p(\mathbf{x}_{D}),p(h^{-1}([\mathbf{y}_{d},\mathbf{% -z}_{K}])))$

-

$D>d$

-

$(x_{\text{soma}},y_{\text{soma}})$

-

$\displaystyle\mathcal{L}_{\mathrm{NLL}}=-\log(p(\mathbf{x}))$

-

$I(x,y)=\exp\left(-\frac{(x-x_{\text{stim}})^{2}+(y-y_{\text{stim}})^{2}}{2\rho% -^{2}}-\frac{(x-x_{\text{soma}})^{2}+(y-y_{\text{soma}})^{2}}{2\lambda^{2}}% -\right),$

-

$\{\mu\},\{\mu^{\prime}\}$

-

$g_{\theta}:\tilde{\mathbf{x}}\mapsto\mathbf{x}_{s}$

-

$\mathbf{z}\neq\mathbf{0}$

-

$\mathbf{J}_{h}(\mathbf{x})=\partial h(\mathbf{x})/\partial\mathbf{x}^{T}$

-

$h(\mathbf{x}_{D})=[h_{\mathbf{y}_{d}}(\mathbf{x}_{D}),h_{\mathbf{z}_{K}}(% -\mathbf{x}_{D})]$

-

$P_{M^{\prime}}$

-

$h:\mathbf{x}_{D}\mapsto[\mathbf{y}_{d},\mathbf{z}_{K}]$

-

$h_{\mathbf{y}_{d}}(\mathbf{x}_{D})\approx s(\mathbf{x}_{D})$

- - - diff --git a/htmls/output_mathjax_example_10019.html b/htmls/output_mathjax_example_10019.html deleted file mode 100644 index a7683d3cdd52f2d26ef3ab934812277fb6548e5e..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10019.html +++ /dev/null @@ -1,134 +0,0 @@ - - - - MathJax Example - - - - -

$\mathcal{L}_{\mathrm{MMD}}(p_{M},p_{M^{\prime}})=\left(\mathbb{E}_{i,j}[k(\mu_% -{i},\mu_{j})-2k(\mu_{i},\mu_{j}^{\prime})+k(\mu_{i}^{\prime},\mu_{j}^{\prime})% -]\right)^{\frac{1}{2}},$

-

$\rho=400\,\mu m$

-

$\lambda=1550\,\mu m$

-

$\mathbf{y}_{d}$

-

$\mathbf{z}\sim\pi(\mathbf{z})=\mathcal{N}(\mathbf{z};\mathbf{0},\mathbf{I}))$

-

$\mathcal{L}_{\mathrm{NLL}}\simeq\frac{1}{2}\|h(\mathbf{x};\mathbf{c})\|_{2}^{2% -}-\log\left|\det\mathbf{J}_{h}(\mathbf{x})\right|$

-

$\min_{\theta}\mathcal{L}_{\mathrm{MSE}}\left(\tilde{\mathbf{x}},f_{\phi}\left(% -g_{\theta}\left(\tilde{\mathbf{x}}\right)\right)\right),$

-

$\mathbf{z}_{K}\sim\pi(\mathbf{z}_{K})=\mathcal{N}(\mathbf{z}_{K};\mathbf{0},% -\mathbf{I}_{K}))$

-

${\mathbf{x}_{s}}$

-

$\mathcal{L}_{\mathrm{MSE}}=\mathbb{E}[(\mathbf{y}_{d}-h_{\mathbf{y}_{d}}(% -\mathbf{x}_{D}))^{2}]$

-

$\mu^{\prime}\sim p_{M^{\prime}}$

-

$p(\mathbf{x})=\pi(\mathbf{z}=h(\mathbf{x}))\left|\det\frac{\partial h(\mathbf{% -x})}{\partial\mathbf{x}^{T}}\right|,$

-

$\mathbf{x}=h^{-1}(\mathbf{z};\mathbf{c})$

-

$s:\mathbf{x}_{D}\in\mathbb{R}^{D}\mapsto\mathbf{y}_{d}\in\mathbb{R}^{d}$

-

$\displaystyle\simeq\frac{1}{2}\|h(\mathbf{x})\|_{2}^{2}-\log\left|\det\mathbf{% -J}_{h}(\mathbf{x})\right|.$

-

$(x_{\text{stim}},y_{\text{stim}})$

-

$\mathbf{z}=h(\mathbf{x};\mathbf{c})$

-

$\mathbf{z}=h(\mathbf{x})$

-

$\mathbf{z}_{K}$

-

$\mathcal{L}_{\mathrm{MMD}}(p(h(\mathbf{x}_{D})),p(\mathbf{y}_{d})p(\mathbf{z}_% -{K}))$

-

$\mathbf{x}_{D}$

-

$\mathbf{y}_{p}$

-

$\mu\sim p_{M}$

-

$\Psi\in\mathbb{R}^{z\times h}$

-

$\bm{0.982}$

-

$24.90$

-

$\bm{0.861}$

-

$\bm{0.044}$

-

$z=8,192$

-

$r:\Re\times\Re^{3}\rightarrow\Re^{3}$

-

$24.88$

-

$\bm{0.047}$

-

$24.13$

-

$\bm{0.018}$

-

$\bm{32.70}$

-

$\bm{W}_{\Psi}$

-

$M^{\prime}_{\Delta}=(\bm{W}^{\prime}_{\Delta},\bm{b}^{\prime}_{\Delta})$

-

$\bm{W}_{\gamma}$

-

$27.89$

-

$\bm{0.714}$

-

$\bm{0.070}$

-

$\bm{0.025}$

-

$\bm{0.911}$

-

$\bm{h}^{\prime}_{\Delta}$

-

$\bm{28.14}$

-

$32.53$

-

$\bm{b}=\{\bm{b}_{\Delta},\bm{b}_{c},\bm{b}_{\psi}$

-

$29.57$

-

$\bm{h}_{c}$

-

$M^{\prime}_{\Delta}$

-

$\bm{W}=\{\bm{W}_{\Delta},\bm{W}_{c},\bm{W}_{\psi}$

-

$\bm{0.991}$

-

$\bm{0.093}$

-

$\bm{W}_{\mu},\bm{W}_{\gamma},\bm{W}_{\Psi},\bm{W}^{\prime}_{\Delta},\bm{W}^{% -\prime}_{c}\}$

-

$\bm{37.72}$

-

$\bm{b}_{\Psi}$

-

$\bm{0.988}$

-

$\bm{35.41}$

-

$26.11$

-

$\bm{f}_{\Delta}=\sigma\left(\bm{h}_{\Delta}\right),$

-

$\bm{0.029}$

-

$\bm{h}_{c}^{\prime}=[\bm{\Psi}\otimes\bm{f}_{c},\mathbf{d}].$

-

$\bm{0.956}$

-

$\bm{32.34}$

-

$\bm{31.13}$

-

$\bm{26.51}$

-

$26,29$

-

$27.31$

-

$\bm{20.80}$

-

$\bm{\mu}=\bm{f}_{\mu}\otimes\bm{\gamma}.$

-

$\bm{36.76}$

-

$\bm{W}_{\psi}$

-

$\bm{\gamma}=tanh\left({\bm{W}_{\gamma}[\bm{h}_{\Delta},\bm{h}_{c}]+\bm{b}_{% -\gamma}}\right),$

-

$\bm{27.56}$

-

$\bm{b}_{\mu},\bm{b}_{\gamma},\bm{b}_{\Psi},\bm{b}^{\prime}_{\Delta},\bm{b}^{% -\prime}_{c}\}$

-

$\bm{W}_{\mu}$

-

$23.49$

-

$\bm{0.007}$

-

$\bm{h}_{\Delta}$

-

$\bm{0.103}$

-

$\bm{f}_{c}=\sigma\left({\bm{h}_{c}}\right),$

-

$\bm{0.068}$

-

$MSE(\cdot)$

-

${\cal L}=MSE(r(\Delta,\bm{c}),\bm{g}),$

-

$32.45$

-

$\bm{0.009}$

-

$\bm{f}_{\psi}=\sigma\left({\bm{W}_{\psi}\bm{[h}_{\Delta},\bm{h}_{c}]+\bm{b}_{% -\psi}}\right),$

-

$\bm{0.026}$

-

$\mathbf{l}=(x,y,z)$

-

$\bm{f}_{\mu}=\sigma\left({\bm{W}_{\mu}[\bm{h}_{\Delta},\bm{h}_{c}]+\bm{b}_{\mu% -}}\right),$

-

$\bm{0.882}$

-

$\bm{0.837}$

-

$[\bm{h}_{\Delta},\bm{h}_{c}]$

-

$\bm{f}_{\Delta}$

-

$\bm{26.06}$

-

$\bm{b_{\mu}}$

-

$29.56$

-

$\bm{27.01}$

-

$26.05$

-

$F_{\Theta}:(\mathbf{l},\mathbf{d})\rightarrow(\mathbf{c},\Delta)$

- - - diff --git a/htmls/output_mathjax_example_1002.html b/htmls/output_mathjax_example_1002.html deleted file mode 100644 index 3b13300946f143ac221de083487dcb56f50870e9..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1002.html +++ /dev/null @@ -1,124 +0,0 @@ - - - - MathJax Example - - - - -

$\operatorname{Inst}(\emptyset)$

-

$\prod_{s=1}^{t}2^{p_{s}-1}$

-

${\rm D}_{\rm NP}\leq_{0\hbox{-}T}{\rm D}_{\rm P}$

-

$\p^{A}=\np^{A}$

-

$\p^{L}=\p$

-

$\langle M^{\prime},\epsilon\rangle$

-

$\np^{\mathcal{O}}$

-

$\langle M,1^{n},1^{t}\rangle\in{\rm U}_{\rm NP}$

-

$\np^{L}=\np$

-

$x\notin{\rm HP}$

-

$\np^{\p}=\np$

-

$\p$

-

$A\leq_{f(n)\hbox{-}T}B$

-

${\rm D}_{\rm NP}\not\in\p^{{\rm D}_{\rm P}}$

-

$x\in\mathcal{O}$

-

$L\in{\rm\Sigma_{1}^{0}}$

-

$n,t\in{\mathbb{N}^{+}}$

-

$t^{\prime}\in{\mathbb{N}}$

-

$\Theta(2^{n})$

-

${\rm D}_{\rm NP}=\{\langle M,1^{n}\rangle\mid n\in{\mathbb{N}}\land(\exists x% -\in\{0,1\}^{n})[M$

-

$\langle M,1^{n}\rangle\in{\rm D}_{\rm NP}$

-

${\rm U}_{\rm P}=\{\langle M,x,1^{t}\rangle\mid t\in{\mathbb{N}^{+}}\land M$

-

$M^{\mathcal{O}}$

-

${\rm HP}=\{\langle M,w\rangle\mid M\text{ halts on input }w\}.$

-

$(\forall x)[x\in A\iff f(x)\in B]$

-

$z\notin{\rm D}_{\rm NP}$

-

$(\forall L\in{\rm\Sigma_{1}^{0}})[L\leq_{m}A]$

-

$\p^{\p}=\p$

-

$A\leq_{m}B$

-

${\mathbb{N}^{+}}=\{1,2,3,\ldots\}$

-

${\rm D}_{\rm NP}\in{\rm\Sigma_{1}^{0}}$

-

$\Omega(2^{n})$

-

$\p^{X}\neq\np^{X}$

-

$\p^{A}\neq\np^{A}\iff\p\neq\np$

-

$\p\neq\np$

-

$]\}$

-

${\rm U}_{\rm NP}$

-

$L\leq_{m}{\rm D}_{\rm NP}$

-

$x]\}$

-

$\p^{\mathcal{O}}$

-

$\min(c,2^{n})$

-

${\rm D}_{\rm NP}\leq_{m}{\rm D}_{\rm P}$

-

$z\not\in{\rm D}_{\rm NP}$

-

$y=\epsilon$

-

$\p^{\p}$

-

${\rm U}_{\rm P}$

-

$\{0,1\}^{n}\cap L(M)$

-

$A\leq_{T}B$

-

${\rm D}_{\rm P}$

-

$U_{\text{\bf P}}$

-

${\rm PSPACE}$

-

$x\in L(M)$

-

$\p^{B}\neq\np^{B}$

-

$A\in\p$

-

$\langle M,x\rangle\in{\rm D}_{\rm P}\iff(\exists t>0)[\langle M,x,1^{t}\rangle% -\in{\rm U}_{\rm P}]$

-

${\mathbb{N}}=\{0,1,2,\ldots\}$

-

$A=L(M^{B})$

-

$\mathcal{C}^{\mathcal{D}}=\bigcup_{A\in\mathcal{D}}\mathcal{C}^{A}$

-

$\np$

-

${\rm U}_{\rm NP}\not\in\p^{{\rm U}_{\rm P}}$

-

$\np^{\p}$

-

$f:{\mathbb{N}}\rightarrow{\mathbb{N}}$

-

$\p=\np$

-

$f(x)\in{\rm D}_{\rm NP}$

-

${\rm\Sigma_{0}^{0}}$

-

${\rm\Sigma_{0}^{0}}^{\mathcal{O}}$

-

$A\in{\rm\Sigma_{1}^{0}}$

-

$x\not\in\mathcal{O}$

-

$\p^{A}=\p$

-

${\rm U}_{\rm NP}=\{\langle M,1^{n},1^{t}\rangle\mid n\in{\mathbb{N}}\land t\in% -{\mathbb{N}^{+}}\land(\exists x\in\{0,1\}^{n})[M$

-

$\langle M,1^{n}\rangle$

-

$x\in{\rm HP}$

-

$\langle M,1^{n}\rangle\in{\rm D}_{\rm NP}\iff(\exists t>0)[\langle M,1^{n},1^{% -t}\rangle\in{\rm U}_{\rm NP}]$

-

$x\}$

-

$w\in\{0,1\}^{n}$

-

$\epsilon\notin L(M^{\prime})$

-

$A\leq_{1\hbox{-}T}B$

-

${\rm HP}\leq_{m}{\rm D}_{\rm NP}$

-

$\np^{A}=\np$

-

$(\forall x)[x\notin{\rm HP}\implies f(x)\notin{\rm D}_{\rm NP}]$

-

${\rm\Sigma_{1}^{0}}$

-

${\rm HP}$

-

$\langle M,w\rangle$

-

$(\forall x)[x\in{\rm HP}\iff f(x)\in{\rm D}_{\rm NP}]$

-

$\mathcal{C},\mathcal{D}$

-

$\epsilon\in L(M^{\prime})$

-

${\rm D}_{\rm NP}$

-

${\rm D}_{\rm NP}\leq_{h\hbox{-}T}{\rm D}_{\rm P}$

-

$L\in\p$

-

${\rm D}_{\rm P}=\{\langle M,x\rangle\mid M$

-

${\rm D}_{\rm NP}\leq_{1\hbox{-}T}{\rm D}_{\rm P}$

-

$c\in{\mathbb{N}^{+}}$

-

$(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}$

-

$G\setminus X:=G[V(G)\setminus X]$

-

$V(C_{2})$

-

$p_{j}\in B$

-

$H\setminus u$

-

$\chi(G)\leq k$

-

$u\in V(H)$

-

$\{\overline{K_{2}\cup C_{2k+1}}\mid k\in\mathbb{N}\}$

- - - diff --git a/htmls/output_mathjax_example_10020.html b/htmls/output_mathjax_example_10020.html deleted file mode 100644 index 15713f347e44c713faa7bf1264f4ce75463f6ff6..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10020.html +++ /dev/null @@ -1,159 +0,0 @@ - - - - MathJax Example - - - - -

$\Theta_{\Delta}=(\bm{W}_{\Delta},\bm{b}_{\Delta})$

-

$\bm{0.954}$

-

$(\Delta,\bm{c})$

-

$\bm{0.873}$

-

$\bm{0.796}$

-

$M_{\Delta}$

-

$\bm{0.055}$

-

$\bm{33.99}$

-

$\bm{b}_{\gamma}$

-

$\bm{0.122}$

-

$\bm{0.914}$

-

$24.54$

-

$\bm{36.46}$

-

$M^{\prime}_{c}=(\bm{W}^{\prime}_{c},\bm{W}^{\prime}_{c})$

-

$\bm{h}^{\prime}_{c}$

-

$20-30$

-

$\bm{f}_{\psi}$

-

$\bm{36.13}$

-

$30.91$

-

$\bm{f}_{\mu}$

-

$\bm{28.60}$

-

$\bm{h}_{\Delta}^{\prime}=[\bm{\Psi}\otimes\bm{f}_{\Delta},\mathbf{l}],$

-

$\bm{22.23}$

-

$\bm{0.010}$

-

$33.91$

-

$33.09$

-

$31.75$

-

$26,73$

-

$\bm{\Psi}=tanh\left({\bm{W}_{\Psi}\left(\bm{\mu}+\left(\bm{f}_{\psi}\otimes% -\Psi\right)\right)+\bm{b}_{\Psi}}\right),$

-

$34.56$

-

$\bm{35.83}$

-

$\bm{b_{\psi}}$

-

$\Theta_{c}=(\bm{W}_{c},\bm{b}_{c})$

-

$\lfloor p^{2}\cdot r\rfloor$

-

$E_{SP}$

-

$E_{SE}$

-

$\displaystyle E_{SP}(sp,2i)=\sin(\frac{sp}{\Omega^{\frac{2i}{d}}})\quad E_{SP}% -(sp,2i+1)=\cos(\frac{sp}{\Omega^{\frac{2i}{d}}})$

-

$E_{SE}(s,2i)$

-

$\displaystyle E_{P}(pos,2i)=\sin(\frac{pos}{\Omega^{\frac{2i}{d}}}),\text{ }E_% -{P}(pos,2i+1)=\cos(\frac{pos}{\Omega^{\frac{2i}{d}}})$

-

$j\in\{i,...,n\}$

-

$p_{i}c_{j}$

-

$(p,M)$

-

$\displaystyle\leq\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},% -\tilde{y})\geq A\right]+\mathbb{P}\left[\frac{1}{(\mathcal{T}+1)}\geq\frac{1}{% -B}\right]$

-

$\hat{x}^{1,k},\hat{y}^{1,k}$

-

$s_{x}^{t}$

-

$F_{1},...,F_{M}:\mathbb{R}^{d}\to\mathbb{R}$

-

$\nabla F_{\mathcal{D}}(w^{t})$

-

$\displaystyle=\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w^{t};B^{t}),x^{t}% --x\rangle$

-

$\displaystyle\leq 2\tau q\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};% -B^{t})\|_{\infty}^{2}+2\tau\sum_{k=t_{0}}^{t-1}\Big{(}\|\nabla f(\hat{w}^{t_{0% -}};B^{k})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\Big{)}$

-

$L_{2}^{F}$

-

$L_{2}^{G}\leq L_{2}^{F}D^{3}$

-

$\mathbb{E}\big{[}\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})-\min_{w\in% -\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})\big{]}\leq\frac{2L_{1}}{T}+\frac{2L_{1% -}^{2}}{\lambda T}+\lambda\log(d_{x}).$

-

$\displaystyle\qquad+\frac{L_{0}\sqrt{(TK/q+qK)\log(1/\delta)}\log(d_{x})}{n% -\varepsilon}+\frac{L_{2}}{K}+\frac{L_{1}q\log(T/q)}{T}.$

-

$F(w^{T})-F(x)\leq\frac{1}{\sum_{j\in[T]}\beta_{j}}\bigg{(}\operatorname{Regret% -}_{T}(x)+\sum_{t\in[T]}\langle\beta_{t}\nabla F(w^{t})-g_{t},x_{t}-x\rangle% -\bigg{)}.$

-

$2\Delta_{s}-$

-

$\Lambda^{T}$

-

$\beta_{t}\in\mathbb{R}^{+},g^{t}\in\mathbb{R}^{d_{x}}$

-

$\color[rgb]{1,0,0}\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n% -\varepsilon}\right)^{1/2}$

-

$\ell_{t}(x)=\langle g^{t},x\rangle$

-

$\mathbb{R}^{d_{x}}$

-

$\mathcal{Z}=\{z_{1},...,z_{|\mathcal{Z}|}\}$

-

$\displaystyle\leq\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|},x\in\Delta_{|% -\mathcal{Z}|}}(F_{\mathcal{D}}(\tilde{x},y)-F_{\mathcal{D}}(x,\bar{y})).\right]$

-

$(g_{x},g_{y})=\text{BiasReducedGradient}(x,y,N,B)$

-

$\displaystyle\leq 2\tau\sum_{k=t_{0}}^{t-1}\Big{(}\|\nabla f(w^{t_{0}};B^{t})-% -\nabla f(\hat{w}^{t_{0}};B^{t})\|_{\infty}^{2}+\|\nabla f(\hat{w}^{t_{0}};B^{k% -})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\Big{)}$

-

$a(x)\leq Cb(x)$

-

$(\tilde{x}^{t},\tilde{y}^{t})_{t\in[T]}$

-

$\max\{L_{0},B\}$

-

$\displaystyle=\max_{j\in[d]}\left|\nabla_{j}F\left(\bar{a}\right)-\nabla_{j}F% -\left(\bar{x}\right)\right|^{2}=\max_{j\in[d]}\left|\langle\nabla F(\bar{a})-% -\nabla F(\bar{x}),e_{j}\rangle\right|^{2}$

-

$\mathbb{E}[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})]\lesssim\sqrt{[(% -\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2% -}]U}+L_{2}\sqrt{U}.$

-

$\max\{L_{1},L_{2}\}$

-

$\tilde{x}^{t+1},\tilde{y}^{t+1}$

-

$\displaystyle\leq 5\sqrt{\frac{\log(|{\cal Q}|)}{n}},$

-

$\mathbb{P}\left[F(\bar{a}^{T})-F(\bar{x}^{T})\geq\frac{L_{1}D^{2}}{2}\sum_{t=1% -}^{T}\lambda_{t}^{2}+\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{% -t}^{2}}\right]\leq\exp(-\beta^{2}).$

-

$(\tilde{x},\tilde{y})$

-

$\min_{x\in\Delta_{x}}\max_{y\in\Delta_{y}}F_{\mathcal{D}}(x,y),$

-

$g_{y}=C_{M}2^{N}(\nabla_{y}f(\bar{x}_{+},\bar{y}_{+};B)-\nabla_{y}f(\bar{x}_{-% -},\bar{y}_{-};B))+\nabla_{y}f(x_{0},y_{0};B)$

-

$\mathbb{E}\left[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})\right]$

-

$i\in\{K_{x}+1,...,K_{x}+K_{y}\}$

-

$\textstyle s^{t}_{y}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{y,j}\right)$

-

$\alpha_{t}=\lambda_{t}\left\langle\nabla F\left(\sum_{k=t}^{T}\lambda_{k}x^{k}% -+\sum_{k=1}^{t-1}\lambda_{k}a^{k}\right),a^{t}-x^{t}\right\rangle$

-

$(\alpha_{t})_{t\in[T]}$

-

$\displaystyle\|\mathbb{E}[\Phi_{\mathcal{D}}(x,y)-(g_{x},-g_{y})]\|_{\infty}$

-

$f:\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}\to\mathbb{R}$

-

$\Delta(s_{y}^{t})$

-

$\tilde{y}^{1}$

-

$\{\mathcal{T}\geq t-1\}=\Big{\{}\sum_{k\in[t-1]}2^{N_{k}}\leq U\Big{\}}$

-

$\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}% -^{2}+(\log(d_{x})+\log(d_{y}))ML_{1}^{2}]\frac{U}{M}+\frac{L_{2}U}{M2^{M}}$

-

$\displaystyle\leq\|\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t})\|_{\infty}% -\|x^{t}-x\|_{1}$

-

$\displaystyle\langle\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle$

-

$\displaystyle\geq\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\right]=% -\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\mathbb{E}[2^{N_{t}}\mathbbm{1}_{(% -\mathcal{T}+1\geq t)}|N_{t-1},...,N_{1}]\right]$

-

$(\mathcal{A}_{n})_{n\geq 1}$

-

$\displaystyle=|F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-F(\Lambda(\lambda% -_{2},\mu_{2})+(x_{1},y_{1}))|$

-

$\displaystyle=\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},% -\tilde{y})\cdot\frac{1}{\mathcal{T}+1}\geq\frac{A}{B}\right]$

-

$\tau\eqsim\min\left\{\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+(L_{0}^{2}+L_{1}^{2})% -q\sqrt{\log(d_{x})/K}+L_{1}^{2}q/[\sqrt{\log(d_{x})}K^{3/2}])T}},\frac{1}{L_{0% -}q},\frac{n\varepsilon}{TL_{0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}.$

-

$\hat{x}^{t+1,k}$

-

$\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\lesssim L_{1}\frac{\sqrt{\log(% -d_{x})+\log(d_{y})}}{\sqrt{T}}+\frac{(\log(d_{x})+\log(d_{y}))}{\tau T}+\tau L% -_{0}^{2}+\frac{L_{2}}{K}.$

-

$\displaystyle=\tau\Big{|}2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{% -+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})\Big{)}+\nabla% -_{x}f(x_{0}^{r},y_{0}^{r};B^{r})$

-

$\mathbb{E}\Big{\|}\nabla_{x}f(\bar{x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(x,y;B% -_{k})\Big{\|}_{\infty}^{2}\leq\frac{20L_{2}^{2}}{2^{2(k+1)}}+\frac{(12+2\log(d% -_{x}))L_{1}^{2}}{2^{k+1}},$

-

$\bar{x}=\frac{1}{T}\sum_{t=1}^{T}x^{t}$

-

$\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\leq U$

- - - diff --git a/htmls/output_mathjax_example_10021.html b/htmls/output_mathjax_example_10021.html deleted file mode 100644 index 9ded2c4b31919aa2965858d9f7596f80dd17fb6a..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10021.html +++ /dev/null @@ -1,192 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\leq\tau\Big{|}\frac{2^{N_{r}}\alpha}{2^{N_{r}}}\Big{[}\Big{(}% -\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r};z^{*})-\nabla_{x}f(\bar{x}_{+}^{r}% -,\bar{y}_{+}^{r};z^{\prime*})\Big{)}+\Big{(}\nabla_{x}f(\bar{x}_{-}^{r},\bar{y% -}_{-}^{r};z^{*})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};z^{\prime*})\Big{% -)}\Big{]}$

-

$\displaystyle\lesssim\begin{cases}q\tau\big{[}\frac{L_{1}^{2}}{\sqrt{\log(d_{x% -})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}}{\sqrt{K}}\big{]}&% -\text{without second order smoothnes}\\ -q\tau\big{[}\frac{L_{2}^{2}}{K^{2}}+\frac{L_{1}^{2}\log(d_{x})}{K}\big{]}&% -\text{with second order smoothness}\end{cases},$

-

$(w^{t},v^{t})$

-

$M=d$

-

$\mbox{TG}(p,M)$

-

$\displaystyle\quad+\mathbb{E}\big{[}\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(% -\hat{w}^{t_{0}};B^{t}),x^{t}-x\rangle\big{]}$

-

$i\in[K_{x}]$

-

$\langle\nabla F(\bar{a})-\nabla F(\bar{x}),e_{j}\rangle\leq\frac{|F(\bar{a}+re% -_{j})-F(\bar{x}+re_{j})|+|F(\bar{x})-F(\bar{a})|}{r}+L_{1}r.$

-

$q=\sqrt{T/\log(d_{x})},K=T/q=\sqrt{T\log(d_{x})}$

-

$J:=\{e_{1},...,e_{d}\}$

-

$s_{y}^{t}$

-

$L_{2}\lesssim(L_{0}+L_{1})\left\{\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n% -}}+\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n% -\varepsilon}\right)^{1/2}\right\}$

-

$y^{1}=(1/d_{y},...,1/d_{y})$

-

$\displaystyle\lesssim\frac{2(\log(d_{x})+\log(d_{y}))}{\tau T}+5\tau L_{0}^{2}% -+\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t% -})-g^{t}\mid\mathcal{F}_{t}]\|_{\infty}.$

-

$2^{N_{t}}\leq 2^{M}$

-

$\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n\varepsilon}\right)^{2/5}$

-

$\displaystyle\leq\frac{2\log(|\mathcal{Z}|)}{\tau_{x}T}+\frac{\log(|{\cal Q}|)% -}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+\frac{1}{T}\sum_{t=1}^{T}\langle\nabla_{x}F_% -{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t}-w^{t}\rangle+2\|-\nabla_{y}F_{% -\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty}.$

-

$\operatorname{Regret}_{T}(x)\leq\frac{\log(d)}{\tau}+\frac{\tau}{2}\sum_{t\in[% -T]}\|g^{t}\|_{\infty}^{2},$

-

$a^{t}\sim P_{x^{t}}$

-

$w^{t}=\frac{(t-1)w^{t-1}+x^{t}}{t}$

-

$\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},\tilde{y})-% -\operatorname{Gap}(\bar{x},\bar{y}))\right]$

-

$F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)\leq\frac{1}{T}\left[\operatorname{% -Regret}_{T}(x)+\sum_{t\in[T]}\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-% -x\rangle\right],$

-

$f:\mathbb{R}^{d}\mapsto\mathbb{R}$

-

$X\sim{\cal N}(0,1/2)$

-

$\displaystyle=\max_{k\in[K_{x}+K_{y}]}\big{|}\nabla_{k}\langle\Lambda^{T}_{j},% -\nabla F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_% -{2},\mu_{2})+(x_{1},y_{1}))\rangle\big{|}$

-

$\displaystyle=3\left(\frac{|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}+L_{1}^{2}r^{2}+% -\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{2}}{r^{2}}\right)$

-

$\displaystyle\leq 4\tau\sum_{k=t_{0}}^{t-1}\left\|\nabla f(\hat{w}^{t_{0}};B^{% -k})-\nabla f(w^{t_{0}};B^{k})\right\|_{\infty}.$

-

$\displaystyle\lesssim\left(L_{0}^{2}+L_{2}^{2}+\log(d_{x})ML_{1}^{2}\right)% -\mathbb{E}\left[\sum_{t=1}^{U}\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$

-

$\tau\leq\frac{B\varepsilon}{8L_{0}\sqrt{2(TK/q+qK)\log(1/\delta)}}$

-

$\displaystyle\lesssim\sqrt{\frac{(L_{0}^{2}+L_{1}^{2}q\log(d_{x})/K+L_{2}^{2}q% -/K^{2})\log(d_{x})}{T}}+\frac{L_{0}q\log(d_{x})}{T}$

-

$\displaystyle+\langle\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle$

-

$f_{i}:\mathcal{X}\times\mathcal{Z}\to[-B,B]$

-

$\hat{x}\sim P_{x}$

-

$\mathcal{Y}=\Delta_{y}$

-

$\tilde{x}_{j}^{t+1}\propto\tilde{x}_{j}^{t}\exp\big{(}-\tau\nabla_{j}f(w^{t_{0% -}};B^{k})\big{)}$

-

$N(S)=N((\varepsilon_{n}(S))_{n\geq 1},(\delta_{n}(S))_{n\geq 1})$

-

$\displaystyle\leq\tau\alpha L_{0}(4+1/2^{N_{r}})\leq 4.5\tau\alpha L_{0}.$

-

$A\eqsim\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(% -d_{y}))L_{1}^{2}]U}\quad,\quad B\eqsim U/M$

-

$\displaystyle\leq\frac{\log(|\mathcal{Z}|)}{\tau_{x}}+\frac{\tau_{x}}{2}\sum_{% -t\in[T]}\|g^{t}_{x}\|_{\infty}^{2}+\frac{\log(|{\cal Q}|)}{\tau_{y}}+\frac{% -\tau_{y}}{2}\sum_{t\in[T]}\|g^{t}_{y}\|_{\infty}^{2}+\sum_{t=1}^{T}\langle\Phi% -_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w,v)\rangle.$

-

$\lambda_{i}=\frac{1}{T}$

-

$(\lambda_{1},\ldots,\lambda_{T})\in\Delta_{T}$

-

$\mathbb{P}\left[\sum_{t=1}^{T}\alpha_{t}\geq\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{% -\sum_{t=1}^{T}\lambda_{t}^{2}}\right]\leq\exp(-\beta^{2}),$

-

$F_{\mathcal{D}}(x_{\mathcal{D}},\bar{y})=0$

-

$\displaystyle T\operatorname{Gap}(\bar{x},\bar{y})$

-

$x^{1},...,x^{T}\in\Delta_{d}$

-

$\|\nabla F(x^{1})-\nabla F(x^{2})\|_{\infty}\leq L_{1}\|x^{1}-x^{2}\|_{1}$

-

$\frac{4\tau L_{0}}{B}\leq\frac{\varepsilon}{2\sqrt{2(TK/q+(q+1)K)\log(1/\delta% -)}}.$

-

$T\eqsim\min\left\{n,\frac{n\varepsilon}{\sqrt{(\log(d_{x})+\log(d_{y}))\log(1/% -\delta)}}\right\},$

-

$T\eqsim\min\left\{n,\frac{(n\varepsilon)^{2/3}}{\log(1/\delta)^{1/3}}\right\}$

-

$|(x^{t}-\tilde{x}^{t})_{j}|\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum% -_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}\underbrace{\max\left\{2\max% -_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|,\exp\left(2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_% -{j}|\right)-1\right\}}_{:=\psi},$

-

$\frac{4L_{0}\tau}{B}$

-

$\displaystyle\leq L_{0}^{F}\sum_{i=1}^{K_{x}}|\lambda_{1,i}-\lambda_{2,i}|\|x_% -{i}-x_{1}\|_{1}+L_{0}\sum_{j=1}^{K_{y}}|\mu_{1,j}-\mu_{2,j}|\|y_{j}-y_{1}\|_{1}$

-

$U=\min\left\{\frac{n\varepsilon\sqrt{L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{% -y}))L_{1}^{2}}}{\sqrt{4\cdot 48\cdot 81(\log(d_{x})+\log(d_{y}))\log(1/\delta)% -}L_{0}},\frac{n}{2}\right\}$

-

$T=\min\left\{n,\frac{n\varepsilon}{\log(d_{x})\sqrt{\log(1/\delta)}}\right\}$

-

$\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\langle\Phi_{\mathcal{D}}(x^{t},y^{t% -})-(g_{x}^{t},g_{y}^{t}),(x^{t},y^{t})-(w^{t},v^{t})\rangle\right]\lesssim% -\frac{L_{2}\mathbb{E}[\mathcal{T}+1]}{2^{M}}.$

-

$\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})$

-

$\mathbb{E}\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F\left(% -\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\|^{2}_{\infty}\lesssim\frac{L_{2}^{2}}{T% -^{2}}+\frac{L_{1}^{2}(1+\log(d))}{T}$

-

$x_{i}-x_{1}$

-

$2^{N_{t}}$

-

$\|x\|_{1}=\sum_{j\in[d]}|x_{j}|$

-

$0 -

$\mathbb{E}[\nabla_{j}F(\bar{x})-\nabla_{j}F(\bar{a})]=\mathbb{E}[\langle\nabla -F% -(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle]\leq\frac{4L_{1}}{rT}+L_{1}r.$

-

$\{x_{1},\ldots,x_{K_{x}}\}$

-

$M=\log_{2}(\sqrt{U})$

-

$\displaystyle\leq 6\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\leq 6\sum_{t=1}^{% -\mathcal{T}}2^{N_{t}}+6\cdot 2^{M}\leq 6(U-2^{M})+6\cdot 2^{M}=6U,$

-

$\displaystyle\leq 2L_{1}\|w^{t}-w^{t_{0}}\|_{1}\leq 2L_{1}\sum_{k=t_{0}}^{t-1}% -\|w^{k+1}-w^{k}\|_{1}$

-

$\frac{2\tau L_{0}}{B}$

-

$|s_{x}^{t}-s_{x}^{\prime t}|$

-

$\displaystyle\leq\tau\|\nabla\operatorname{LSE}(-\tau G(S))\|_{1}\|G_{j}(S)-G_% -{j}(S^{\prime})\|_{\infty}$

-

$\displaystyle 4\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}+2(\mathcal{T}+1)$

-

$\displaystyle\leq 0+\frac{4L_{1}q}{q\lfloor t/q\rfloor+1}+2\|\mathbb{E}_{\hat{% -w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})]\|_{\infty}$

-

$K=T/\log(d_{x})$

-

$\displaystyle\leq\frac{2L_{1}}{T}+\frac{2L_{1}^{2}}{\lambda T}+\lambda\log(d_{% -y}).$

-

$\tilde{x}^{t+1}$

-

$x^{1}=(1/d_{x},...,1/d_{x})$

-

$N:\mathbb{R}^{\infty}_{\geq 0}\times\mathbb{R}^{\infty}_{\geq 0}\to\mathbb{N}$

-

$\{y_{1},\ldots,y_{K_{y}}\}$

-

$\mathbb{P}(j=e_{i})\propto\exp\left(\frac{\varepsilon s(S,i)}{2\Delta_{s}}\right)$

-

$N^{t+1}\sim\mbox{TG}(0.5,M)$

-

$\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-% -\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}% -\leq\frac{4L_{1}}{\sqrt{T}}.$

-

$\displaystyle\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|% -^{2}\right]$

-

$\displaystyle\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^% -{t_{0}};B^{t}),x^{t}-x\rangle]$

-

$\displaystyle\leq\sum_{t\in[T]}[F_{\mathcal{D}}(x^{t},v)-F_{\mathcal{D}}(w,y^{% -t})]\leq\sum_{t\in[T]}\langle\Phi_{\mathcal{D}}(x^{t},y^{t}),(x^{t},y^{t})-(w,% -v)\rangle$

-

$\bar{x}^{T}=\sum_{t=1}^{T}\lambda_{t}x^{t}$

-

$T,K,q,\tau$

-

$\|g^{t}\|_{\infty}\leq L_{0}$

-

$\displaystyle\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t}% -,y^{t})-(w,v)\rangle=\sum_{t=1}^{T}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t% -})-g^{t}_{x},x^{t}-w\rangle+\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{% -t}_{y},y^{t}-v\rangle$

-

$\displaystyle\operatorname{Gap}(\bar{x},\bar{y})\leq\frac{2(\log(d_{x})+\log(d% -_{y}))}{\tau T}+5\tau L_{0}^{2}+\frac{1}{T}\sum_{t=1}^{T}\langle\Phi_{\mathcal% -{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w^{t},v^{t})\rangle.$

-

$\displaystyle\leq\frac{2L_{1}}{T}+\frac{2L_{1}^{2}}{\lambda T}+\lambda\log(d_{% -y})+\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]+\frac{2L_{1}}{T}+\frac{2L_% -{1}^{2}}{\lambda T}+\lambda\log(d_{x})$

-

$\displaystyle\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^% -{t};B^{t}),x^{t}-x\rangle]$

-

$s^{t}_{x}$

-

$f(\cdot,\cdot;z)$

-

$G_{j}(S)=\left(\sum_{k\in[i-1]}g^{k}_{j}\right)_{i\in[t]}$

-

$\hat{G}_{j}=-\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(\hat{w}^{t_{0}};B^{k}),G_{j}=% --\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(w^{t_{0}};B^{k})$

-

$\xi^{1},\xi^{2},...$

-

$\|g^{t}\|_{\infty}^{2}+\|\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\|_{\infty}^{2}% -\leq 5L_{0}^{2}$

-

$T\eqsim\min\left\{n,\frac{(n\varepsilon)^{2/3}}{\log(1/\delta)^{1/3}}\right\},$

-

$\mathcal{T}+1$

-

$x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau g^{t}_{j}\right),\quad\forall j\in% -[d_{x}]$

-

$\mathbb{E}[A_{1}]=\mathbb{E}[\langle\tilde{\Delta}^{t_{0}},\tilde{x}^{t}-x% -\rangle]\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second order % -smoothnes}\\ -\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$

- - - diff --git a/htmls/output_mathjax_example_10022.html b/htmls/output_mathjax_example_10022.html deleted file mode 100644 index 20bb13cc4ca3b37241aaa29f4c267dbb1285b65f..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10022.html +++ /dev/null @@ -1,195 +0,0 @@ - - - - MathJax Example - - - - -

$\mathbb{E}[\|x^{t}\|_{\infty}]<\infty$

-

$x^{1}=(1/|\mathcal{Z}|,...,1/|\mathcal{Z}|),y^{1}=(1/|\mathcal{Q}|,...,1/|% -\mathcal{Q}|)$

-

$\langle-\nabla F(\bar{a}),e_{j}\rangle\leq\frac{F(\bar{a}+re_{j})-F(\bar{a})}{% -r}+\frac{L_{1}r}{2}.$

-

$F_{1}(\cdot)=F(\cdot)$

-

$2^{N_{t}+1}$

-

$\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},\tilde{y})\geq A% -\right]\lesssim\frac{\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(% -\log(d_{x})+\log(d_{y}))L_{1}^{2}]U}}{A}.$

-

$\lambda=L_{1}\frac{2}{\sqrt{T(\log(d_{x})+\log(d_{y}))}}$

-

$\displaystyle\lesssim\frac{\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2% -}+(\log(d_{x})+\log(d_{y}))L_{1}^{2}]U}}{A}$

-

$(x_{i}-x_{1},\mathbf{0}_{d_{y}})$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1}+L_{2})% -{\small\bigg{[}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{n}}+\left(\frac{(\log(d_{x% -})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{2/5}\bigg{]}.}$

-

$\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(\hat{G}_{j}\right)}{\sum_{i\in[d_{x% -}]}x^{t_{0}}_{i}\exp\left(\hat{G}_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{% -j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}$

-

$\lambda=\sqrt{8}/\sqrt{\log(|{\cal Q}|)T}$

-

$\mathcal{A}_{1:N(S)}$

-

$L_{0}D$

-

$\bar{a}^{T}=\sum_{t\in[T]}\lambda_{t}a^{t}$

-

$L/\mu=\Omega(d)$

-

$\ell_{x}=\log(d_{x})$

-

$x^{1},\ldots,x^{T}\in\mathbb{R}^{d}$

-

$\displaystyle\leq\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|+|F(\bar{x})-F(% -\bar{a})|}{r}+L_{1}r$

-

$\displaystyle\leq\frac{2(\log(d_{x})+\log(d_{y}))}{\tau}+\tau\sum_{t=1}^{T}(\|% -g^{t}\|_{\infty}^{2}+\|\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\|_{\infty}^{2})$

-

$\displaystyle=\mathbb{E}\left[\nabla_{x}F_{\mathcal{D}}(\bar{x}_{+,M},\bar{y}_% -{+,M})-\nabla_{x}F_{\mathcal{D}}(x_{-,0},y_{-,0})+\nabla_{x}F_{\mathcal{D}}(x_% -{0},y_{0})\right]$

-

$g^{t}_{x,j}$

-

$\operatorname{Gap}(x,y)=\max_{v\in\mathcal{X},w\in\mathcal{Y}}(F_{\mathcal{D}}% -(x,w)-F_{\mathcal{D}}(v,y))$

-

$\|x\|_{\infty}=\max_{j\in[d]}|x_{j}|$

-

$B=n/T$

-

$\displaystyle\leq\max_{i\in[K_{x}+K_{y}]}\|\Lambda^{T}_{i}\|_{1}\|\nabla F(% -\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{% -2})+(x_{1},y_{1}))\|_{\infty}$

-

$\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-% -\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}% -\leq\frac{2L_{2}}{T}.$

-

$w^{t}_{j}\propto\exp(\operatorname{LSE}(\tau G_{j}(S))).$

-

$\displaystyle\left(\sum_{i=1}^{K_{x}}\lambda_{i}(x_{i}-x_{1}),\sum_{j=1}^{K_{y% -}}\mu_{j}(y_{j}-y_{1})\right).$

-

$\displaystyle\lesssim\sqrt{[(\log(d_{x})+\log(d_{y})]L_{1}^{2}\frac{U}{M}}$

-

$\tilde{x}^{t+1},\hat{x}^{t+1,k}$

-

$\displaystyle\leq A^{2}+C^{2}\log(M)+2C^{2}+2AC$

-

$\displaystyle\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^{t_{0}};B^{% -t}),x^{t}-x\rangle$

-

$\mathbb{P}[F(\bar{a}^{T})-F\left(\bar{x}^{T}\right)\geq\alpha]\leq\mathbb{P}% -\left[\sum_{t=1}^{T}\alpha_{t}+\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{% -2}\geq\alpha\right]=\mathbb{P}\left[\sum_{t\in[T]}\alpha_{t}\geq\alpha^{\prime% -}\right]$

-

$s_{x}^{t}=s_{x}^{\prime t}$

-

$s^{t}_{x}=-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r% -},\bar{y}_{+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})% -\Big{)}+\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}.$

-

$\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|^{2}\right]% -\leq\frac{5L_{1}^{2}D^{4}}{4}\left(\sum_{t=1}^{T}\lambda_{t}^{2}\right)^{2}+% -\frac{L_{0}^{2}D^{2}(6+\log(M))}{2}\sum_{t=1}^{T}\lambda_{t}^{2}.$

-

$\mathbb{E}\big{[}\sum_{t\in[T]}\alpha_{t}\big{]}=0$

-

$g^{t}\in\partial f_{t}(x)$

-

$\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-% -\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}% -\lesssim\frac{L_{2}}{T}$

-

$\displaystyle\leq 2L_{0}^{2}+4C_{M}\sum_{k=0}^{M}2^{k}\mathbb{E}\Big{\|}\nabla% -_{x}f(\bar{x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(x,y;B_{k})\Big{\|}_{\infty}^{2}$

-

$\sum_{t\in[\mathcal{T}+1]}\max\{1,2^{N_{t}}/\alpha\}\leq n$

-

$\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{% -1}^{2}q\log(d_{x})}{K}+\frac{L_{2}^{2}q}{K^{2}}\big{]}+\frac{L_{2}}{K}+\frac{L% -_{1}q\log(T/q)}{T},$

-

$\operatorname{Regret}_{T}(x)\leq\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}\sum_{t% -\in[T]}\|g^{t}\|_{\infty}^{2}$

-

$\displaystyle\mathbb{E}\left[\max_{q\in{\cal Q}}|\mathbb{E}_{z}[q(z)]-q(\tilde% -{S})|\right]$

-

$M=\log(\sqrt{U})$

-

$\mathbb{R}^{d_{x}},\mathbb{R}^{d_{y}}$

-

$\mathbb{P}\left[\frac{1}{\mathcal{T}+1}\geq\frac{1}{B}\right]=\mathbb{P}\left[% -\mathcal{T}\leq B-1\right]\leq\mathbb{P}\left[\sum_{t\in[B]}2^{N_{t}}>U-2^{M}% -\right]\leq\frac{B\mathbb{E}[2^{N_{1}}]}{U-2^{M}},$

-

$F_{\mathcal{D}}(\cdot)$

-

$\displaystyle\leq L_{0}^{F}D(\|\lambda_{1}-\lambda_{2}\|_{1}+\|\mu_{1}-\mu_{2}% -\|_{1})$

-

$\displaystyle\lesssim(L_{0}+L_{1})\sqrt{\frac{\log(d_{x})}{T}}+\frac{L_{2}}{% -\sqrt{T}\log(d_{x})^{1/4}}$

-

$(\hat{x}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{x^{t}}$

-

$\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},% -\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\}$

-

$g^{t}=\nabla f(\hat{w}^{t};B^{t})$

-

$L_{1}^{G}\leq L_{1}^{F}D^{2}$

-

$g^{t}_{x}=\nabla_{x}f(x^{t},\hat{y}^{t};S)$

-

$\displaystyle=\mathbb{E}[\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v)-% -\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\bar{x},v)]+\mathbb{E}[\max_{v\in\Delta_{% -y}}F_{\mathcal{D}}(\bar{x},v)-\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})]$

-

$\frac{\varepsilon}{2\sqrt{2T(K+1)\log(1/\delta)}}$

-

$\frac{4TL_{0}\tau_{y}}{n}\leq\frac{\varepsilon}{2\sqrt{2T\log(1/\delta)}}.$

-

$\displaystyle=\frac{(A+B)^{2}}{2}+2MC\left[C\exp\left(-(B/C)^{2}\right)+A% -\mathbb{P}[X\geq B/C]\right]$

-

$\displaystyle+\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})% -,x^{t}-x\rangle.$

-

$\mathbb{E}[x_{t+1}|x_{1:t}]=0$

-

$M(\log(d_{x})+\log(d_{y}))$

-

$\mathbb{P}[X\geq B/C]\leq\exp\left(-(B/C)^{2})\right)$

-

$T=\frac{6n\varepsilon}{16\sqrt{2\log(1/\delta)}\log(|{\cal Q}|)},\tau_{x}=% -\sqrt{\frac{\log(|\mathcal{Z}|)}{9T}},\tau_{y}=\frac{\log(|{\cal Q}|)}{6\sqrt{% -\log(|\mathcal{Z}|)T}}$

-

$(\varepsilon^{\prime},T\delta+\delta^{\prime})$

-

$\tilde{x}^{t}\sim P_{x^{t}}$

-

$\sum_{t\in[T]}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t}-w% -\rangle\\ -\leq\sum_{t\in[T]}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t% -}-w^{t}\rangle+\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|\nabla_{% -x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}.$

-

$\displaystyle\leq\mathbb{E}\left[(\mathcal{T}+1)\left(\frac{4L_{1}}{\mathcal{T% -}+1}+\frac{2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{\mathcal{T}+1}}\right)\right]$

-

$\displaystyle\leq\max_{k\in[K_{x}+K_{y}]}\sum_{i\in[K_{x}+K_{y}]}|\Lambda_{j,i% -}|\|\Lambda^{T}_{k}\|_{1}\|\nabla\nabla_{i}F(\Lambda(\lambda_{1},\mu_{1})+(x_{% -1},y_{1}))-\nabla\nabla_{i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\|_{\infty}$

-

$(f_{i})_{i\in[d_{y}]}$

-

$\sqrt{\log(d)/n}+(\log(d)^{3/2}/[n\varepsilon])^{2/5}$

-

$f:\mathcal{X}\times\mathcal{Z}\to\mathbb{R}$

-

$(g^{t}_{x},g^{t}_{y})=\operatorname{BiasReducedGradient}(x^{t},y^{t},N^{t},B^{% -t})$

-

$w\in\Delta_{|\mathcal{Z}|},v\in\Delta_{|{\cal Q}|}$

-

$2^{N_{1}},...,2^{N_{t-1}}$

-

$\frac{4\tau L_{0}}{B}$

-

$\sum_{t\in[B]}2^{N_{i}}>U$

-

$\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}\{F_{\mathcal{D}}(x,y)=\mathbb{E}_% -{z\sim\mathcal{D}}[f(x,y;z)]$

-

$\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{% -\infty}\leq 2L_{2}/K$

-

$\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\lesssim\sqrt{\frac{[(% -\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2% -}]\log(U)^{2}}{U}}\right]\geq 0.99.$

-

$\phi(y)=\sum_{j\in[d_{y}]}y_{j}\log(1/y_{j})$

-

$\bar{x}^{T}=\sum_{t\in[T]}\lambda_{t}x^{t}$

-

$F(\cdot+re_{j})$

-

$\displaystyle\|\nabla\nabla_{j}G(\lambda_{1},\mu_{1})-\nabla\nabla_{j}G(% -\lambda_{2},\mu_{2})\|_{\infty}=\max_{k\in[K_{x}+K_{y}]}\big{|}\nabla_{k,j}G(% -\lambda_{1},\mu_{1})-\nabla_{k,j}G(\lambda_{2},\mu_{2})\big{|}$

-

$x^{t+1}_{j}\propto\exp\left\{-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_% -{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y% -}_{-}^{r};B^{r})\Big{)}+\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}\right\},$

-

$w\in\mathcal{X}$

-

$\displaystyle+\mathbb{E}[\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})-\min_% -{w\in\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})]$

-

$\displaystyle=\frac{5L_{1}^{2}D^{4}}{4}\left(\sum_{t=1}^{T}\lambda_{t}^{2}% -\right)^{2}+\frac{L_{0}^{2}D^{2}(6+\log(M))}{2}\sum_{t=1}^{T}\lambda_{t}^{2}.$

-

$\displaystyle|G(\lambda_{1},\mu_{1})-G(\lambda_{2},\mu_{2})|$

-

$B^{r}=B^{\prime r}$

-

$\displaystyle=\mathbb{E}\left[\sum_{k=0}^{M}\big{(}\nabla_{x}F_{\mathcal{D}}(% -\bar{x}_{+,k},\bar{y}_{+,k})-\nabla_{x}F_{\mathcal{D}}(\bar{x}_{-,k},\bar{y}_{% --,k})\big{)}+\nabla_{x}F_{\mathcal{D}}(x_{0},y_{0})\right]$

-

$\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w^{t};B^{t}),x^{t}-x% -\rangle]=0$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})-\operatorname{Gap}(\bar{x},% -\bar{y})]\leq\frac{4}{n}+6\sqrt{\frac{\log(|{\cal Q}|)}{n}}$

-

$g_{y}^{1}$

-

$\hat{G}_{j}$

-

$\displaystyle\leq L_{2}^{F}D^{3}\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})% -\|_{1}.$

-

$\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau L_{0}^{2}+\frac{1}{T}\sum% -_{t\in[T]}\left(\frac{L_{2}}{K}+\mathbbm{1}_{(t>q)}\left[\frac{4L_{1}q}{q% -\lfloor t/q\rfloor+1}+q\frac{L_{2}^{2}\tau}{K^{2}}+q\frac{L_{1}^{2}\log(d_{x})% -\tau}{K}\right]\right)$

-

$\tau\eqsim\min\left\{\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+(L_{0}^{2}+L_{1}^{2})% -q\sqrt{\log(d_{x})/K}+L_{1}^{2}q/[\sqrt{\log(d_{x})}K^{3/2}])T}},\frac{1}{L_{0% -}q},\frac{n\varepsilon}{TL_{0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}$

-

$\mathbb{E}\Big{\|}\nabla_{x}f(x,y;B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B% -_{k})\Big{\|}_{\infty}^{2}\leq\frac{20L_{2}^{2}}{2^{2k}}+\frac{(12+2\log(d_{x}% -))L_{1}^{2}}{2^{k}}.$

- - - diff --git a/htmls/output_mathjax_example_10023.html b/htmls/output_mathjax_example_10023.html deleted file mode 100644 index bd4a24d468cf4f531fc5223d63f198d528a6258e..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10023.html +++ /dev/null @@ -1,184 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}% -x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j% -}|\right].$

-

$(\hat{w}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{w^{t}}$

-

$\min_{x\in\mathcal{X}}\{\max_{i\in[d_{y}]}F_{i}(x)\}=\min_{x\in\mathcal{X}}% -\max_{y\in\Delta_{y}}\mathbb{E}_{z\sim\mathcal{D}}\left[\sum_{i\in[d_{y}]}y_{i% -}f_{i}(x;z)\right].$

-

$(\bar{x}_{-,k},\bar{y}_{-,k})$

-

$\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{% -1}^{2}q}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x% -})}q}{\sqrt{K}}\Big{]}+\frac{L_{1}}{\sqrt{K}}+\frac{1}{T}\sum_{t=q+1}^{T}\left% -[\frac{L_{1}q}{q\lfloor t/q\rfloor+1}\right]$

-

$\max_{j\in[d]}\left|s^{t}_{x}(S,j)-s^{t}_{x}(S^{\prime},j)\right|\leq\frac{2L_% -{0}\tau}{B}$

-

$Q(S)=\frac{1}{n}\sum_{j\in[n]}Q(z_{i_{j}})$

-

$x^{1},x^{2}\in\mathcal{X}$

-

$a^{1},...,a^{T}$

-

$y^{t+1}_{i}\propto y^{t}_{i}\exp\left(\tau g^{t}_{y,i}\right),\quad\forall i% -\in[d_{y}]$

-

$r=2/\sqrt{T}$

-

$TK/q+qK+K$

-

$s_{x}^{\prime t}$

-

$\sqrt{\log(d)/n}+\log(d)/\sqrt{n\varepsilon}$

-

$\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}x^{% -t_{0}}_{i}\exp\left(G_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)% -\exp\left(\hat{G}_{j}-G_{j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G% -_{i}\right)\exp\left(\hat{G}_{i}-G_{i}\right)}$

-

$s^{t}_{x}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{x,j}\right).$

-

$\varepsilon=\frac{\varepsilon^{\prime}}{2\sqrt{2T\log(1/\delta^{\prime})}}$

-

$\langle\nabla F(\bar{x}),e_{j}\rangle\leq\frac{F(\bar{x})-F(\bar{x}+re_{j})}{r% -}+\frac{L_{1}r}{2},$

-

$\hat{w}^{t}=\hat{w}^{t-1}$

-

$\displaystyle\leq\sqrt{\mathbb{E}[\|Q(S)-\mathbb{E}_{z}[Q(z)]\|_{\infty}^{2}]}$

-

$\tau=\sqrt{\frac{(\log(d_{x})+\log(d_{y}))}{(L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+% -\log(d_{y}))L_{1}^{2})U}}$

-

$q(\mathcal{D})=\mathbb{E}_{z\sim\mathcal{D}}[q(z)]$

-

$g^{t}_{x}$

-

$\mathbb{E}[F(\bar{a}^{T})-F(\bar{x}^{T})]\leq\frac{L_{1}}{2}\sum_{t=1}^{T}% -\lambda_{t}^{2}\mathbb{E}\left[\left\|a^{t}-x^{t}\right\|^{2}\right]$

-

$\displaystyle=\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|}}\sum_{j\in[|{\cal Q% -}|]}y_{j}(\mathbb{E}_{z}[q(z)]-q(\tilde{S}))\right]$

-

$a,b:\mathbb{R}^{p}\mapsto\mathbb{R}$

-

$(w{t+1},v^{t+1})$

-

$\|x^{t}-a^{t}\|\leq D$

-

$(x_{0},y_{0})=(\hat{x}^{1},\hat{y}^{1})$

-

$U\geq 4$

-

$\displaystyle\leq 5A^{2}+(6+\log(M))C^{2}$

-

$\displaystyle\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\geq\frac{% -A}{B}\right]$

-

$\displaystyle\leq 4L_{1}+2L_{1}\sqrt{[\log(d_{x})+\log(d_{y})]\mathbb{E}[(% -\mathcal{T}+1)]}$

-

$\max\{L_{0},L_{1}\}$

-

$\Delta(s_{x}^{t})$

-

$\displaystyle\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)% -\right\|^{2}_{\infty}$

-

$2\Delta(s^{t}_{y})$

-

$L_{2}^{G}$

-

$(\mathbb{R}^{d},\|\cdot\|_{1})$

-

$\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n\varepsilon}\right)^{1/3}$

-

$q=\sqrt{T}/\log(d_{x})$

-

$f(\cdot;z)$

-

$\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq n\alpha/2$

-

$\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\mathbb{E}[\|g^{t}_{x}\|_{\infty}^% -{2}\mathbbm{1}_{(\mathcal{T}\geq t-1)}\mid\mathcal{F}_{t}]\right]$

-

$\displaystyle\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},% -\tilde{y})-\operatorname{Gap}(\bar{x},\bar{y}))\right]$

-

$0<\varepsilon<8\log(1/\delta)$

-

$\operatorname{Regret}_{T}^{x}(w)\leq\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}% -\sum_{t\in[T]}\|g^{t}_{x}\|_{\infty}^{2}\text{ and }\operatorname{Regret}_{T}^% -{y}(v)\leq\frac{\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|g^{t}_{y}\|_{% -\infty}^{2}$

-

$F_{\mathcal{D}}(x,y)$

-

$\mathbb{E}[a^{t}|a^{t-1},..,a^{1}]=x^{t}$

-

$\mathbb{E}[F_{\lambda}(\tilde{x})-F_{\lambda}(\bar{x})]\leq\frac{2L_{1}}{T}+% -\frac{2L_{1}^{2}}{\lambda T}.$

-

$\displaystyle=\mathbb{E}\left[(\mathcal{T}+1)\mathbb{E}_{\tilde{x},\tilde{y}}% -\left[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},\bar{% -y})\mid\mathcal{T}\right]\right]$

-

$f:{\cal X}\times{\cal Y}\mapsto\mathbb{R}$

-

$(\bar{x}_{+},\bar{y}_{+})=\frac{1}{2^{N+1}}\sum_{i\in[2^{N+1}]}(\hat{x}^{i},% -\hat{y}^{i})$

-

$\log(d_{x})\simeq\log(d_{y})$

-

$\displaystyle\lesssim\left(L_{0}^{2}+L_{2}^{2}+\log(d_{x})ML_{1}^{2}\right)% -\mathbb{E}[\mathcal{T}+1],$

-

$\Delta_{d}=\{x\in\mathbb{R}^{d}:\|x\|_{1}=1,x_{j}\geq 0\text{ for all }j\in[d]\}$

-

$F(\bar{x}+re_{j})\leq F(\bar{x})+\langle\nabla F(\bar{x}),re_{j}\rangle+\frac{% -L_{1}r^{2}}{2}\|e_{j}\|_{1}^{2},$

-

$\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}% -^{2}+(\log(d_{x})+\log(d_{y}))ML_{1}^{2}]\mathbb{E}[\mathcal{T}+1]+\frac{L_{2}% -\mathbb{E}[\mathcal{T}+1]}{2^{M}}.$

-

$\displaystyle=\frac{4L_{1}}{T}+\frac{4L_{1}^{2}}{\lambda T}+\lambda(\log(d_{x}% -)+\log(d_{y}))+\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})].$

-

$\displaystyle\quad\mathbb{E}[g_{x}]=\mathbb{E}\left[\sum_{k=0}^{M}(\nabla_{x}f% -(\bar{x}_{+,k},\bar{y}_{+,k};B_{k})-\nabla_{x}f(\bar{x}_{-,k},\bar{y}_{-,k};B_% -{k}))+\frac{2^{-k}}{C_{M}}\nabla_{x}f(x_{0},y_{0};B_{k})\right]$

-

$U-\sqrt{U}\geq U/2$

-

$g^{t}=\nabla f(\hat{w}^{t_{0}};B^{t})$

-

$\displaystyle F(\bar{a}^{T})$

-

$\displaystyle\mathbb{E}\big{[}\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v% -)-\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\bar{x},v)\big{]}$

-

$\displaystyle\|\mathbb{E}[(g_{x},g_{y})]-\nabla F_{\mathcal{D}}(x,y)\|_{\infty}$

-

$\displaystyle=\mathbb{E}_{x^{t}}[\langle\mathbb{E}_{\hat{w}^{t}}[\nabla f(w^{t% -};B^{t})-\nabla f(\hat{w}^{t};B^{t})\mid x^{t}],x^{t}-x\rangle]$

-

$\displaystyle\lesssim\begin{cases}\frac{L_{1}}{\sqrt{K}}+q\tau\big{[}\frac{L_{% -1}^{2}}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x}% -)}}{\sqrt{K}}\big{]}&\text{without second order smoothnes}\\ -\frac{L_{2}}{K}+q\tau\big{[}\frac{L_{2}^{2}}{K^{2}}+\frac{L_{1}^{2}\log(d_{x})% -}{K}\big{]}&\text{with second order smoothness}\end{cases}$

-

$\displaystyle\mathbb{E}\|g_{x}\|_{\infty}^{2}$

-

$\tau\leq 1/(8qL_{0})$

-

$\mathbb{E}[F_{\cal D}(\hat{w}^{T})-F_{\cal D}(x)]\lesssim(L_{0}+L_{1})\left[% -\sqrt{\frac{\log(d_{x})}{n}}+L_{0}\frac{\log(d_{x})^{7/10}\log(1/\delta)^{1/5}% -}{(n\varepsilon)^{2/5}}\right]\\ -+L_{1}\log(\sqrt{n}\log(d_{x}))\left[\frac{1}{\log(d_{x})\sqrt{n}}+\frac{\log(% -1/\delta)^{1/5}}{\log(d_{x})^{4/5}(n\varepsilon)^{2/5}}\right].$

-

$\frac{L_{1}}{\sqrt{T}}$

-

$\displaystyle\qquad+\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}% -,(x^{t},y^{t})-(w^{t},v^{t})\rangle.$

-

$\alpha_{t}=\Big{\langle}\nabla F\Big{(}\sum_{k=t}^{T}\lambda_{k}x^{k}+\sum_{k=% -1}^{t-1}\lambda_{k}a^{k}\Big{)},\lambda_{t}(a^{t}-x^{t})\Big{\rangle}$

-

$(\mathbf{0}_{d_{x}},y_{i}-y_{1})$

-

$\tilde{x}^{1},\tilde{y}^{1}$

-

$x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau g^{t}_{x,j}\right),\quad\forall j% -\in[d_{x}]$

-

$U\geq 1$

-

$U\leq\min\left\{\frac{\varepsilon^{2}}{48\log(1/\delta)(9\tau\alpha L_{0})^{2}% -},n\alpha/2,n/2\right\}$

-

$a^{1},...,a^{T}\in\Delta_{d}$

-

$\displaystyle=\langle\tilde{\Delta}^{t_{0}},x^{t}-\tilde{x}^{t}\rangle$

-

$\mathbb{E}[\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)% -\right\|^{2}_{\infty}]\leq\frac{43L_{1}^{2}}{\sqrt{12+\log(d)}T^{3/2}}+\frac{1% -7(L_{0}^{2}+L_{1}^{2})\sqrt{(12+\log(d))}}{\sqrt{T}}.$

-

$\displaystyle\leq L_{0}^{F}\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},% -\mu_{2})\|_{1}$

-

$x\in\Delta_{x},y\in\Delta_{y}$

-

$w^{t}=\frac{1}{t}\sum_{i\in[t]}x^{i}$

-

$T=\min\left\{n,\frac{(n\varepsilon)^{4/5}}{(\log(d_{x})\log(1/\delta))^{2/5}}\right\}$

-

$(v^{t})_{t\in[T]}$

-

$\tilde{y}^{t}\sim P_{y^{t}}$

-

$\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau L_{0}^{2}+\frac{1}{T}\sum% -_{t\in[T]}\left(\frac{L_{1}}{\sqrt{K}}+\mathbbm{1}_{(t>q)}\left[\frac{4L_{1}q}% -{q\lfloor t/q\rfloor+1}+\frac{L_{1}^{2}q\tau}{\sqrt{\log(d_{x})}K^{3/2}}+\frac% -{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}q\tau}{\sqrt{K}}\right]\right)$

-

$(x_{-,0},y_{-,0})$

-

$r^{2}=\sqrt{\frac{8(12+\log(d))}{T}}$

-

$(\bar{x}_{+,k},\bar{y}_{+,k})$

-

$\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\|g^{t}_{y}\|_{\infty}^{2}\right]% -\lesssim L_{0}^{2}+L_{2}^{2}+\log(d_{y})ML_{1}^{2}\mathbb{E}[\mathcal{T}+1]$

-

$(\mathbf{E},\|\cdot\|)$

-

$\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2\log(|\mathcal{Z}|)}% -{\tau_{x}T}+\frac{\log(|{\cal Q}|)}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+\frac{2}{T% -}\sum_{t=1}^{T}\mathbb{E}[\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}% -\|_{\infty}].$

-

$\|a^{t}-x^{t}\|_{1}\leq D$

-

$\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^{t};B^{t}),x^% -{t}-x\rangle]\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second % -order smoothnes}\\ -\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$

-

$\displaystyle\mathbb{E}[F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)]$

-

$|\mathbb{E}[-\nabla_{y,i}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y,i}\mid\mathcal{% -F}_{t}]|\leq 2L_{2}/K$

-

$\displaystyle+\frac{L_{1}\log(\sqrt{n\log(d_{x})})}{\sqrt{T\log(d_{x})}}+\frac% -{L_{0}\sqrt{T\log(1/\delta)}\log^{3/2}(d_{x})}{n\varepsilon}.$

-

$T,\tau$

- - - diff --git a/htmls/output_mathjax_example_10024.html b/htmls/output_mathjax_example_10024.html deleted file mode 100644 index f649204959c179f9ea9606205d6cedf3a2a75863..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10024.html +++ /dev/null @@ -1,180 +0,0 @@ - - - - MathJax Example - - - - -

$w\in\Delta_{x},v\in\Delta_{y}$

-

$T,\tau,L$

-

$\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{% -t=1}^{\mathcal{T}+1}(L_{0}^{2}+\|g^{t}_{x}\|_{\infty}^{2}+\|g^{t}_{y}\|_{% -\infty}^{2})$

-

$F_{\mathcal{D}}(x)=\mathbb{E}_{z\sim\mathcal{D}}[f(x;z)]$

-

$\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t}),x^{t}-x\rangle$

-

$t_{0}=q\lfloor t/q\rfloor$

-

$C=\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{t}^{2}}$

-

$s^{t}_{y}=-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_{y}f(\bar{x}_{+}^{r% -},\bar{y}_{+}^{r};B^{r})-\nabla_{y}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})% -\Big{)}+\nabla_{y}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}.$

-

$\Delta(s_{y}^{t})\leq 4.5\tau\alpha L_{0}$

-

$\hat{w}^{t_{0}}$

-

$\displaystyle\longmapsto$

-

$(\hat{x}^{i})_{i\in[2^{N+1}]}\overset{\text{iid}}{\sim}P_{x}$

-

$(\hat{y}^{i})_{i\in[2^{N+1}]}\overset{\text{iid}}{\sim}P_{y}$

-

$\{a^{i}\}_{i\in[T]}$

-

$\displaystyle=\frac{(A+B)^{2}}{2}+\sum_{j\in[M]}\int_{B/C}^{\infty}\mathbb{P}% -\left[|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|\geq A+C\beta\right](A+C\beta)Cd\beta$

-

$\frac{2TL_{0}\tau}{n}$

-

$L_{1}=2$

-

$\delta^{\prime},\delta^{\prime\prime}>0$

-

$N((\varepsilon_{n})_{n\geq 1},(\delta_{n})_{n\geq 1})=\inf\left\{n:\varepsilon% -<\sqrt{2\log\left(\frac{1}{\delta^{\prime}}\right)\sum_{m\leq n+1}\varepsilon_% -{m}^{2}}+\frac{1}{2}\sum_{\sum_{m\leq n+1}}\varepsilon_{m}^{2}\text{ or }% -\delta^{\prime\prime}<\sum_{m\leq n+1}\delta_{m}\right\}.$

-

$\hat{G}_{j},G_{j}$

-

$(x^{t},y^{t})_{t\in[T]}$

-

$\mathbb{P}\left[F\left(\sum_{t=1}^{T}\lambda_{t}a^{t}\right)-F\left(\sum_{t=1}% -^{T}\lambda_{t}x^{t}\right)\geq\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{% -2}+\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{t}^{2}}\,\right]% -\leq\exp(-\beta^{2}).$

-

$\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},% -\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K\eqsim\frac{T}{% -\log(d_{x})+\log(d_{y})}.$

-

$\displaystyle=\int_{0}^{\infty}\mathbb{P}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F% -_{j}(\bar{a}^{T})|\geq\beta\right]\beta d\beta$

-

$\mathbb{E}\left[F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a})% -\right]\leq 4L_{1}/T$

-

$g^{t}_{x}=\nabla_{x}F_{\mathcal{D}}(\hat{x}^{t},\hat{y}^{t};B^{t})$

-

$(\tilde{x},\tilde{y})=\frac{1}{T}\sum_{t=1}^{T}(\tilde{x}^{t},\tilde{y}^{t})$

-

$\hat{y}^{2}$

-

$\displaystyle=\sqrt{\mathbb{E}\left[\left\|\sum_{j\in[n]}\frac{Q(z_{i_{j}})-% -\mathbb{E}_{z}[Q(z)]}{n}\right\|_{\infty}^{2}\right]}$

-

$L_{2}=0$

-

$\displaystyle\leq\sqrt{2e\log(|{\cal Q}|)\sum_{j\in[n]}\left\|\frac{Q(z_{i_{j}% -})-\mathbb{E}_{z}[Q(z)]}{n}\right\|_{\infty}^{2}}$

-

$\displaystyle\text{and}\quad\mathbb{E}[\|(g_{x},-g_{y})\|_{\infty}^{2}]$

-

$\displaystyle=\mathbb{E}[\|Q(S)-\mathbb{E}_{z}[Q(z)]\|_{\infty}]$

-

$N\sim\mbox{TG}(0.5,M)$

-

${}^{(\ast)}$

-

$\hat{x}^{t+1,k},\hat{y}^{t+1,k}$

-

$\|\nabla F(x)\|_{\infty}\leq L_{0}$

-

$\operatorname{LSE}$

-

$\alpha=\left(\frac{2\varepsilon^{2}}{48\cdot 81\log(1/\delta)(\tau L_{0})^{2}n% -}\right)^{1/3}$

-

$(x^{t}-\tilde{x}^{t})_{j}=\frac{x^{t_{0}}_{j}\exp\left(-\tau\sum_{k=t_{0}}^{t-% -1}\nabla_{j}f(\hat{w}^{t_{0}};B^{k})\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}% -\exp\left(-\tau\sum_{k=t_{0}}^{t-1}\nabla_{i}f(\hat{w}^{t_{0}};B^{k})\right)}-% -\frac{x^{t_{0}}_{j}\exp\left(-\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(w^{t_{0}};B^% -{k})\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(-\tau\sum_{k=t_{0}}^{t-1% -}\nabla_{i}f(w^{t_{0}};B^{k})\right)}.$

-

$\hat{x}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{x}^{t,k}$

-

$(x^{t},y^{t})$

-

$g^{t}_{y}$

-

$K\eqsim\frac{T}{\log(d_{x})+\log(d_{y})}$

-

$A=\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2}$

-

$w^{1}\in\mathbb{R}^{d}$

-

$\mathbb{E}\left[\max_{q\in{\cal Q}}|\mathbb{E}_{z}[q(z)]-q(\tilde{S})|\right]% -\leq\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})]\leq 20\sqrt{\frac{\log(|% -{\cal Q}|)}{n}}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{% -\log(|{\cal Q}|)}}{\sqrt{n\varepsilon}}.$

-

${\cal X}\subseteq\mathbb{R}^{d_{x}}$

-

$\operatorname{Gap}(\tilde{x},\tilde{y})\leq A/B$

-

$\sqrt{\log(d)/n}+(\log(d)^{3/2}/[n\varepsilon])^{1/3}$

-

$\mathcal{A}:\mathcal{Z}^{n}\mapsto\mathcal{X}$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})-\operatorname{Gap}(\bar{x},% -\bar{y})]\leq\frac{4}{n}+\frac{8}{\lambda n}+\lambda\log(|{\cal Q}|).$

-

$\sum_{t=q+1}^{T}\left[\frac{1}{q\lfloor t/q\rfloor+1}\right]\leq q\left(\frac{% -1}{q}+\frac{1}{2q}+...+\frac{1}{(T/q)q}\right)\lesssim\log\Big{(}\frac{T}{q}% -\Big{)}$

-

$\mathbb{E}\|g_{x}\|_{\infty}^{2}\lesssim L_{0}^{2}+L_{2}^{2}+M\log(d_{x})L_{1}% -^{2}$

-

$\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{% -1}^{2}q\log(d_{x})}{K}+\frac{L_{2}^{2}q}{K^{2}}\big{]}+\frac{L_{2}}{K}+\frac{1% -}{T}\sum_{t=q+1}^{T}\left[\frac{L_{1}q}{q\lfloor t/q\rfloor+1}\right]$

-

$(\hat{y}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{y^{t}}$

-

$\displaystyle=\max\{\|\mathbb{E}[g_{x}]-\nabla_{x}F_{\mathcal{D}}(x,y)\|_{% -\infty},\|\mathbb{E}[g_{y}]-\nabla_{y}F_{\mathcal{D}}(x,y)\|_{\infty}\}.$

-

$\mathcal{T}+1\leq n/2$

-

$\displaystyle=\big{\|}\Lambda^{T}\big{[}\nabla F(\Lambda(\lambda_{1},\mu_{1})+% -(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\big{]}\big% -{\|}_{\infty}$

-

$\mathbb{E}\left[F\left(\sum_{t=1}^{T}\lambda_{t}a^{t}\right)-F\left(\sum_{t=1}% -^{T}\lambda_{t}x^{t}\right)\right]\leq\frac{L_{1}}{2}\sum_{t=1}^{T}\lambda_{t}% -^{2}\mathbb{E}\left[\left\|a^{t}-x^{t}\right\|^{2}\right].$

-

$|{\cal Z}|$

-

$\displaystyle\sqrt{2\log(1/\delta)\sum_{t=1}^{\mathcal{T}+1}(4\cdot 2^{N_{t}}+% -2)(9\tau\alpha L_{0})^{2}}+\frac{1}{2}\sum_{t=1}^{\mathcal{T}+1}(4\cdot 2^{N_{% -t}}+2)(9\tau\alpha L_{0})^{2}\leq\varepsilon.$

-

$0<\varepsilon^{\prime}<1$

-

$t>q$

-

$\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}% -x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[\exp\left(2\max_{j\in[d_{x}]}|\hat{G% -}_{j}-G_{j}|\right)-1\right],$

-

$\min_{x\in\Delta_{x}}F_{\mathcal{D}}(x)$

-

$\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}% -x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[1-\exp\left(-2\max_{j\in[d_{x}]}|% -\hat{G}_{j}-G_{j}|\right)\right]$

-

$\displaystyle(x^{t}-\tilde{x}^{t})_{j}$

-

$\displaystyle\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second % -order smoothnes}\\ -\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$

-

$S=\{z_{i_{1}},...,z_{i_{n}}\}\overset{\text{iid}}{\sim}\mathcal{D}$

-

$\max\{1,2^{N^{t}}/\alpha\}$

-

$(\hat{w}^{T,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{w^{T}}$

-

$\tau\eqsim\min\left\{\left(\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+L_{1}^{2}q\log(% -d_{x})/K+L_{2}^{2}q/K^{2})T}}\right),\frac{1}{L_{0}q},\frac{n\varepsilon}{TL_{% -0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\},$

-

$\displaystyle=\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|}}F_{\mathcal{D}}(% -\tilde{x},y)\right]$

-

$\ell=\log(d_{x})+\log(d_{y})$

-

$\displaystyle=\mathbb{E}[2^{N_{1}}]\mathbb{E}\left[\mathcal{T}+1\right]\eqsim M% -\mathbb{E}\left[\mathcal{T}+1\right].$

-

$\ell_{1}/\ell_{q}$

-

$\displaystyle\qquad+\frac{\alpha}{2^{N_{r}}}\Big{(}\nabla_{x}f(x_{0}^{r},y_{0}% -^{r};z^{*})-\nabla_{x}f(x_{0}^{r},y_{0}^{r};z^{\prime*})\Big{)}\Big{|}$

-

$\|\nabla\operatorname{LSE}(x)\|_{1}=1$

-

$\displaystyle=\mathbb{E}[\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v)-% -\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})]$

-

$a^{1},...,a^{t-1}$

-

$\displaystyle\leq 2\|\mathbb{E}_{\hat{w}^{t}}[\nabla f(w^{t};B^{t})-\nabla f(% -\hat{w}^{t};B^{t})]\|_{\infty}$

-

$\mathcal{T}\leq B-1$

-

$(\bar{x},\bar{y})=\frac{1}{T}\sum_{t=1}^{T}(x^{t},y^{t})$

-

$\displaystyle\quad+4C_{M}\sum_{k=0}^{M}\mathbb{E}2^{k}\Big{\|}\nabla_{x}f(x,y;% -B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})\Big{\|}_{\infty}^{2}.$

-

$y^{1}=(1/d_{y},...,1/d_{y}),t=1,N^{1}\sim\mbox{TG}(0.5,M)$

-

$\tau\leq 1/(4L_{0}q)$

-

$\log(n)\left(L_{0}+L_{2}+L_{1}\sqrt{(\log(d_{x})+\log(d_{y}))}\right)\sqrt{% -\frac{\log(d_{x})+\log(d_{y})}{n}}\\ -+\log(n)\sqrt{L_{0}\sqrt{L_{0}^{2}+L_{2}^{2}+L_{1}^{2}\log(n)(\log(d_{x})+\log% -(d_{y}))}}\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n% -\varepsilon}\right)^{1/2}.$

-

$\left|\mathbb{E}\left[F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-F\left(% -\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right|\leq\frac{2L_{1}}{T}.$

-

$\displaystyle=\tau\|G_{j}(S)-G_{j}(S^{\prime})\|_{\infty}.$

-

$\color[rgb]{1,0,0}\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}^{2/3}}{(n% -\varepsilon)^{2/3}}$

-

$\displaystyle+\frac{BM}{U-\sqrt{U}}.$

-

$\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}F(x,y)$

-

$\bar{x}=\frac{1}{T}\sum_{t\in[T]}x^{t},\bar{a}=\frac{1}{T}\sum_{t\in[T]}a^{t}$

-

$L_{1}^{F}$

-

$\mathcal{Q}=\{q:\mathcal{Z}\to[-1,1]\}$

-

$\displaystyle=\mathbb{E}[\langle\nabla f(w^{t};B^{t})-\nabla f(\hat{w}^{t};B^{% -t}),x^{t}-x\rangle]$

-

$L_{1}^{G}$

-

$\max_{q\in{\cal Q}}|q(\tilde{S})-q(\mathcal{D})|$

-

$g^{t}=(g^{t}_{x},g^{t}_{y})$

- - - diff --git a/htmls/output_mathjax_example_10025.html b/htmls/output_mathjax_example_10025.html deleted file mode 100644 index 28615fbea3d7aafa671f77a18a63cd92d7111fec..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10025.html +++ /dev/null @@ -1,204 +0,0 @@ - - - - MathJax Example - - - - -

$0\leq\phi(y)\leq\log(d_{y})$

-

$y_{i}-y_{1}$

-

$f(x,y;z)=\sum_{j\in[|\mathcal{Q}|]}y_{j}(q_{j}(z)-\langle q_{j},x\rangle)$

-

$S=\{z^{1},...,z^{n}\}$

-

$\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\|g^{t}_{x}\|_{\infty}^{2}\mathbbm% -{1}_{(\mathcal{T}+1\geq t)}\right]$

-

$\displaystyle\leq 4\tau\sum_{k=t_{0}}^{t-1}\|\nabla f(w^{t_{0}};B^{t})-\nabla f% -(\hat{w}^{t_{0}};B^{t})\|_{\infty}\|\nabla f(\hat{w}^{t_{0}};B^{k})-\nabla f(w% -^{t_{0}};B^{k})\|_{\infty}$

-

$\displaystyle\operatorname{Gap}(\bar{x},\bar{y})$

-

$\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq 10\sqrt{\frac{\log(|{\cal Q% -}|)}{n}}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{\log(|{% -\cal Q}|)}}{\sqrt{n\varepsilon}}.$

-

$\displaystyle\qquad-2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r% -};B^{\prime r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{\prime r})\Big{% -)}-\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{\prime r})\Big{|}$

-

$\displaystyle\leq\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})\|% -_{\infty}\|x^{t}-\tilde{x}^{t}\|_{1}$

-

$\mathbb{E}_{\mathcal{A},S}\Big{[}\max_{q\in\mathcal{Q}}|\mathbb{E}_{z\sim% -\mathcal{D}}[q(z)]-q(\tilde{S})|\Big{]}\leq 20\sqrt{\frac{\log(|{\cal Q}|)}{n}% -}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{\log(|{\cal Q}% -|)}}{\sqrt{n\varepsilon}}.$

-

$y^{t+1}_{j}\propto y^{t}_{j}\exp\left(\tau_{y}g^{t}_{y,j}\right),\quad\forall j% -\in[|\mathcal{Q}|]$

-

$\displaystyle\Lambda:\Delta_{K_{x}}\times\Delta_{K_{y}}$

-

$\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(w^{T})]\lesssim\frac{L% -_{1}}{K}$

-

$\displaystyle\|x^{t}-\tilde{x}^{t}\|_{1}\leq\psi$

-

$\displaystyle\leq\frac{\log(d_{x})}{\tau T}+\frac{\tau L_{0}^{2}}{2}+\frac{1}{% -T}\sum_{t\in[T]}\mathbb{E}\left[\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{% -t}-x\rangle\right]$

-

$\mathbb{E}\left[\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{2}}% -{r^{2}}\right]\leq\frac{20L_{1}^{2}}{T^{2}r^{2}}+\frac{8L_{0}^{2}(6+\log(d))}{% -Tr^{2}}$

-

$\tilde{\Delta}^{t_{0}}$

-

$\displaystyle=\mathbb{E}\big{[}\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},% -v)-F_{\lambda}(\tilde{x})\big{]}+\mathbb{E}[F_{\lambda}(\tilde{x})-F_{\lambda}% -(\bar{x})]+\mathbb{E}\big{[}F_{\lambda}(\bar{x})-\max_{v\in\Delta_{y}}F_{% -\mathcal{D}}(\bar{x},v)\big{]}$

-

$\psi\leq 8\tau qL_{0}$

-

$\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{% -1}^{2}q}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x% -})}q}{\sqrt{K}}\Big{]}+\frac{L_{1}}{\sqrt{K}}+\frac{L_{1}q\log(T/q)}{T}.$

-

$\mathbb{E}\left[\operatorname{Gap}(\bar{x},\bar{y})\right]$

-

$\displaystyle\leq\sum_{t=1}^{T}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g% -^{t}_{x},x^{t}-w\rangle+2\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|% -_{\infty}$

-

$\displaystyle\|x^{t}-\tilde{x}^{t}\|_{1}=\sum_{j\in[d_{x}]}|(x^{t}-\tilde{x}^{% -t})_{j}|\leq\sum_{j\in[d_{x}]}\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_% -{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}\psi=\psi.$

-

$F(\bar{a}^{T})=F(\bar{x}^{T}+\bar{a}^{T}-\bar{x}^{T})$

-

$\mathcal{X}=\Delta_{x}$

-

$\displaystyle=\mathbb{E}\left[\nabla_{x}F_{\mathcal{D}}(\bar{x}_{+,M},\bar{y}_% -{+,M})\right],$

-

$P_{w^{t-1}}$

-

$\mathbb{E}\left[\frac{|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}\right]\leq\frac{20L_% -{1}^{2}}{T^{2}r^{2}}+\frac{48L_{0}^{2}}{Tr^{2}}.$

-

$T=\min\Big{\{}n,\frac{n\varepsilon}{\log(d_{x})\sqrt{\log(1/\delta)}}\Big{\}},% -K=\sqrt{T\log(d_{x})},q=\sqrt{T/\log(d_{x})},$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},% -\bar{y})]\leq\frac{4L_{1}}{T}+\frac{2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}}{% -\sqrt{T}}.$

-

$f(x,y;z)=\sum_{i\in[d_{y}]}y_{i}f_{i}(x;z)$

-

$\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)\exp\left(\hat{G}_{j}-G% -_{j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)\exp\left(% -\hat{G}_{i}-G_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i% -\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}$

-

$\displaystyle\leq\frac{F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a% -})}{r}+L_{1}r$

-

$\max_{v\in\Delta_{y}}F_{\mathcal{D}}(x,v)\leq F_{\lambda}(x)\leq\lambda\log(d_% -{y})+\max_{v\in\Delta_{y}}F_{\mathcal{D}}(x,v)$

-

$2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|\leq 1$

-

$i\in[d_{y}]$

-

$w^{t+1}=\operatorname{argmin}_{x\in\mathcal{X}}D_{\psi}(w^{t},x)+\langle\xi^{t% -},x\rangle$

-

$\displaystyle\quad+2\tau q\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}}% -;B^{t})\|_{\infty}^{2}+2\tau\sum_{k=t_{0}}^{t-1}\mathbb{E}\left[\|\nabla f(% -\hat{w}^{t_{0}};B^{k})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\right]$

-

$\mathbb{E}[\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)% -\right\|^{2}_{\infty}]\leq\frac{120L_{1}^{2}}{T^{2}r^{2}}+\frac{24L_{0}^{2}(12% -+\log(d))}{Tr^{2}}+3L_{1}^{2}r^{2}.$

-

$\mathbb{E}\|g_{y}\|_{\infty}^{2}\lesssim L_{0}^{2}+L_{2}^{2}+M\log(d_{y})L_{1}% -^{2}$

-

$\operatorname{LSE}((x_{1},...,x_{k}))=\log(\exp(x_{1})+...+\exp(x_{k}))$

-

$\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|^{2}\right]% -\leq\frac{(A+B)^{2}}{2}+2MC(C+A)\exp\left(-(B/C)^{2}\right).$

-

$|\alpha_{i}|\leq L_{0}D\lambda_{i}$

-

$\displaystyle\lesssim(L_{0}+L_{1})\sqrt{\frac{\log(d_{x})}{T}}+\frac{L_{2}}{T^% -{3/4}\log(d_{x})^{1/4}}+\frac{L_{1}+L_{2}}{\sqrt{T\log(d_{x})}}$

-

$z^{\prime*}$

-

$x,y,N,B$

-

$p\leq C_{M}\leq 1$

-

$v^{1}=(1/d_{y},...,1/d_{y})\in\mathbb{R}^{d_{y}},v^{t+1}:=\operatorname{argmin% -}_{y\in\Delta_{y}}\left(\tau\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g_{% -y}^{t},y\rangle+\sum_{i\in[d_{y}]}v^{t}_{i}\log(v^{t}_{i}/y_{i})\right).$

-

$\mathbb{E}\left[\operatorname{Gap}(\tilde{x},\bar{y})\right]$

-

$\displaystyle\leq F(\bar{x}^{T}+\bar{a}^{T-1}-\bar{x}^{T-1})+\langle\nabla F(% -\bar{x}^{T}+\bar{a}^{T-1}-\bar{x}^{T-1}),\lambda_{T}(a^{T}-x^{T})\rangle+\frac% -{L_{1}\lambda_{T}^{2}\left\|a^{T}-x^{T}\right\|^{2}}{2}$

-

$F_{\lambda}(x):=\max_{y\in\Delta_{y}}[F_{\mathcal{D}}(x,y)+\lambda\phi(y)]$

-

$\hat{y}^{t}\sim P_{y^{t}}$

-

$\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\mathbb{E}[\|g^{t}_{x}\|_{\infty}^% -{2}]\mathbbm{1}_{(\mathcal{T}\geq t-1)}\right]$

-

$Q(z)=(q_{1}(z),...,q_{|{\cal Q}|}(z))$

-

$\mathbb{P}[\operatorname{Gap}(\tilde{x},\tilde{y})\geq A/B]\leq 0.01$

-

$\sum_{t\in[T]}\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y},y^{t}-v% -\rangle\\ -\leq\sum_{t\in[T]}\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y},y^{% -t}-v^{t}\rangle+\frac{\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|-\nabla% -_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty}^{2}.$

-

$\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq U$

-

$x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau_{x}g^{t}_{x,j}\right),\quad\forall -j% -\in[|\mathcal{Z}|]$

-

$\tilde{y}^{i}\sim P_{y^{i}}$

-

$\displaystyle\leq 4\tau\max_{j\in[d_{x}]}\sum_{k=t_{0}}^{t-1}\left|\nabla_{j}f% -(\hat{w}^{t_{0}};B^{k})-\nabla_{j}f(w^{t_{0}};B^{k})\right|$

-

$\displaystyle\leq\langle\nabla\operatorname{LSE}(-\tau G_{j}(S)),\tau G_{j}(S^% -{\prime})-\tau G_{j}(S)\rangle$

-

$\hat{x}^{1},\tilde{x}^{1},\hat{y}^{1},\tilde{y}^{1},...,\hat{x}^{t},\tilde{x}^% -{t},\hat{y}^{t},\tilde{y}^{t}$

-

$\displaystyle\leq 4L_{0}^{2}+64L_{2}^{2}+3ML_{1}^{2}[\log(2d_{x})+\log(2d_{y})% -]+8ML_{2},$

-

$\displaystyle\qquad+\sum_{t=1}^{\mathcal{T}+1}\langle\Phi_{\mathcal{D}}(x^{t},% -y^{t})-(g_{x}^{t},g_{y}^{t}),(x^{t},y^{t})-(w^{t},v^{t})\rangle$

-

$\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},% -\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K\eqsim\sqrt{\frac{% -T}{\log(d_{x})+\log(d_{y})}}.$

-

$\max_{j\in[d_{x}]}\tau\|G_{j}(S)-G_{j}(S^{\prime})\|_{\infty}=\tau\max_{j\in[d% -_{x}],i\in[t]}\left|\sum_{k\in[i-1]}(g^{k}_{j}-g^{\prime k}_{j})\right|\leq% -\frac{2L_{0}}{B}.$

-

$\displaystyle\leq 3\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{% -2}+|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}+L_{1}^{2}r^{2}$

-

$w^{t}_{j}=\frac{1}{t}\sum_{i\in[t]}x^{i}_{j}\propto\sum_{i\in[t]}\exp\left(-% -\tau\sum_{k\in[i-1]}g^{k}_{j}\right)=\exp\left(\log\left(\sum_{i\in[t]}\exp% -\left(-\tau\sum_{k\in[i]}g^{k}_{j}\right)\right)\right).$

-

$\displaystyle\leq\operatorname{Regret}_{T}^{x}(w)+\operatorname{Regret}_{T}^{y% -}(v)+\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-% -(w,v)\rangle.$

-

$t\in[T],k\in[K]$

-

$\displaystyle=4L_{1}+2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}\mathbb{E}\left[\sqrt% -{\mathcal{T}+1}\right]$

-

$\displaystyle\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]$

-

$\hat{y}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{y}^{t,k}$

-

$(\tilde{x},\tilde{y})=\frac{1}{t}\sum_{i=1}^{t}(\tilde{x}^{i},\tilde{y}^{i})$

-

$\displaystyle\leq F\left(\bar{x}^{T}\right)+\sum_{t\in[T]}\underbrace{\Big{% -\langle}\nabla F\Big{(}\bar{x}^{T}+\bar{a}^{t-1}-\bar{x}^{t-1}\Big{)},\lambda_% -{t}(a^{t}-x^{t})\Big{\rangle}}_{\alpha_{t}}+\frac{L_{1}}{2}\sum_{t\in[T]}% -\lambda_{t}^{2}\|a^{t}-x^{t}\|^{2}$

-

$2\Delta(s^{t}_{x})$

-

$(x^{t}-\tilde{x}^{t})_{j}$

-

$T\eqsim\min\left\{n,\left[\frac{n\varepsilon}{(\log(d_{x})+\log(d_{y}))^{1/4}% -\sqrt{\log(1/\delta)}}\right]^{4/5}\right\},$

-

$\mathcal{T}+1\leq\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq U$

-

$S=\{z^{1},...,z^{n}\}\overset{\text{iid}}{\sim}\mathcal{D}$

-

$\|w^{t}-w^{t-1}\|_{1}=\|x^{t}-w^{t-1}\|_{1}/t\leq 2/t$

-

$\bar{x}=\frac{1}{T}\sum_{t\in[T]}x^{t}$

-

$\{\alpha_{i}\}_{i\in[T]}$

-

$\displaystyle=L_{0}^{F}\left\|\left(\sum_{i=1}^{K_{x}}(\lambda_{1,i}-\lambda_{% -2,i})(x_{i}-x_{1}),\sum_{j=1}^{K_{y}}(\mu_{1,j}-\mu_{2,j})(y_{j}-y_{1})\right)% -\right\|_{1}$

-

$\alpha=\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2}+\alpha^{\prime}$

-

$\frac{4TL_{0}\tau}{n}$

-

$\mathbb{E}[a^{t}|a^{t-1},\ldots,a^{1}]=x^{t}$

-

$\displaystyle=\mathbb{E}[2^{N_{1}}]\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}% -\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$

-

$\hat{w}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{w}^{t,k}$

-

$\|\mathbb{E}[g_{x}]-\nabla_{x}F_{\mathcal{D}}(x,y)\|_{\infty}=\max_{j\in[d]}|% -\mathbb{E}[g_{x,j}]-\nabla_{x,j}F_{\mathcal{D}}(x,y)|\leq\frac{2L_{2}}{2^{M}}$

-

$\mathcal{A}_{1:n-1}$

-

$\displaystyle\mathbb{E}[\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_% -{\infty}]$

-

$\tau\eqsim\min\left\{\left(\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+L_{1}^{2}q\log(% -d_{x})/K+L_{2}^{2}q/K^{2})T}}\right),\frac{1}{L_{0}q},\frac{n\varepsilon}{TL_{% -0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}.$

-

$\mathcal{X}\subseteq\mathbb{R}^{d_{x}}$

-

$g_{x},g_{y}$

-

$\displaystyle\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]$

-

$T=\min\left\{n,\frac{(n\varepsilon)^{4/5}}{(\log(d_{x})\log(1/\delta))^{2/5}}% -\right\},q=\sqrt{T}/\log(d_{x}),K=T/\log(d_{x})$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\left[% -\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n}}+\bigg{(}\frac{(\log(d_{x})+% -\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{1/3}\bigg{]}.$

-

$\displaystyle|s_{x}^{t}-s_{x}^{\prime t}|$

- - - diff --git a/htmls/output_mathjax_example_10026.html b/htmls/output_mathjax_example_10026.html deleted file mode 100644 index 09cb73fa813b86f2a1918e12828550872b2c1527..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10026.html +++ /dev/null @@ -1,174 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle=\langle\mathbb{E}_{\hat{w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-% -\nabla f(\hat{w}^{t_{0}};B^{t})],\tilde{x}^{t}-x\rangle$

-

$\mathbb{E}[\hat{x}]=x$

-

$x^{1},...,x^{T}$

-

$\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})\|_{1}\leq D\|(% -\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})\|_{1}$

-

$\displaystyle\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(x)]$

-

$\log(|{\cal Q}|)\leq C\log(|\mathcal{Z}|)$

-

$w^{t}=\sum_{i\in[t]}\frac{\beta_{i}x^{i}}{\sum_{j\in[t]}\beta_{j}}$

-

$\textstyle x^{t+1}_{j}\propto\exp\left\{-\tau\left(\sum_{i=1}^{t}g^{i}_{x,j}% -\right)\right\},$

-

$|\mathcal{Q}|\leq|\mathcal{Z}|^{C}$

-

$8\tau qL_{0}\leq 2$

-

$\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}^{7/10}}{(n\varepsilon)^{2/5}}$

-

$q(S)=\frac{1}{|S|}\sum_{z\in S}q(z)$

-

$\Phi(x,y)=\big{(}\nabla_{x}f(x,y),-\nabla_{y}f(x,y)\big{)}$

-

$\varepsilon\leq 8\log(1/\delta)$

-

$g_{x}=C_{M}2^{N}(\nabla_{x}f(\bar{x}_{+},\bar{y}_{+};B)-\nabla_{x}f(\bar{x}_{-% -},\bar{y}_{-};B))+\nabla_{x}f(x_{0},y_{0};B)$

-

$\displaystyle\operatorname{LSE}(-\tau G_{j}(S))-\operatorname{LSE}(-\tau G_{j}% -(S^{\prime}))$

-

$\mathbb{E}\left[\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-x\rangle\right]$

-

$\displaystyle\leq L_{1}^{F}D^{2}\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})% -\|_{1}.$

-

$\frac{4L_{0}\tau}{B}\leq\frac{\varepsilon}{2\sqrt{2T(K+1)\log(1/\delta)}}$

-

$B^{\prime r}$

-

$\frac{L_{1}^{2}}{T^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d)}}{\sqrt{T}}$

-

$n,d_{x},d_{y}$

-

$F_{j}=\nabla_{j}F$

-

$\tau_{x},\tau_{y}$

-

$L_{0}^{G}\leq L_{0}^{F}D$

-

$x^{t}\in\Delta_{d}$

-

$\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{% -\infty}\leq\frac{4L_{1}}{\sqrt{K}}$

-

$\mathbb{P}[\mathcal{A}(S)\in\mathcal{E}]\leq e^{\varepsilon}\mathbb{P}[% -\mathcal{A}(S^{\prime})\in\mathcal{E}]+\delta$

-

$(\varepsilon_{n},\delta_{n})$

-

$\sum_{t=1}^{\mathcal{T}}2^{N_{t}}\leq U-2^{M}$

-

$\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq\min\{n,n\alpha/2\}$

-

$\color[rgb]{1,0,0}\frac{\ell}{\sqrt{n}}+\frac{\ell}{\sqrt{n\varepsilon}}{}^{(% -\ast)}$

-

$\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\lesssim\sqrt{\frac{[(% -\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2% -}]M^{2}}{U}}\,\right]\geq 0.99.$

-

$x\in\mathcal{X},$

-

$\displaystyle\leq 2\|\mathbb{E}_{\hat{w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-% -\nabla f(\hat{w}^{t_{0}};B^{t})]\|_{\infty}$

-

$\displaystyle(\tilde{x}^{t}-x^{t})_{j}$

-

$\lambda\phi(y)$

-

$\nabla f(w^{t};B^{t})$

-

$\displaystyle\langle\nabla F(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle$

-

$\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}% -^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2}]U+\frac{L_{2}U}{2^{M}}.$

-

$j\in[d_{x}]$

-

$\log(d)/\sqrt{n}+\log(d)/[n\varepsilon]^{1/2}$

-

$\displaystyle\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\|g^{t}_{x}\|_{\infty}^% -{2}\right]$

-

$\displaystyle\leq\sum_{t=1}^{T}\langle(\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-% -g^{t}_{x},x^{t}-w^{t}\rangle)+\frac{\log(|\mathcal{Z}|)}{\tau_{x}}+\sum_{t=1}^% -{T}\tau_{x}\|\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}+2% -\|\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty},$

-

$B=C\sqrt{\log(M)}$

-

$\displaystyle\leq 2\sum_{k=0}^{M}C_{M}2^{k}\mathbb{E}\Big{\|}\nabla_{x}f(\bar{% -x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})\Big{\|}_{% -\infty}^{2}+\frac{2^{-k}}{C_{M}}\mathbb{E}\|\nabla_{x}f(x_{0},y_{0};B_{k})\|_{% -\infty}^{2}$

-

$\displaystyle=\max_{i\in[K_{x}+K_{y}]}|\langle\Lambda^{T}_{i},\nabla F(\Lambda% -(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{2})+(x_% -{1},y_{1}))\rangle|$

-

$\displaystyle\sqrt{2\log(1/\delta)\sum_{t=1}^{\mathcal{T}+1}(2^{N_{t}+1}+1)(4% -\Delta(s_{x}^{t})^{2}+4\Delta(s_{y}^{t})^{2})}+\frac{1}{2}\sum_{t=1}^{\mathcal% -{T}+1}(2^{N_{t}+1}+1)(4\Delta(s_{x}^{t})^{2}+4\Delta(s_{y}^{t})^{2})\leq\varepsilon.$

-

$\min_{x\in\Delta_{|\mathcal{Z}|}}F_{\mathcal{D}}(x,\bar{y})\leq 0$

-

$U\leq\min\{n,n\alpha/2\}$

-

$n,1/\delta$

-

$g^{t}_{y}=\nabla_{y}f(x^{t},\hat{y}^{t};S)$

-

$\hat{w}^{T}=\frac{1}{K}\sum_{k\in[K]}\hat{w}^{T,k}$

-

$\tilde{y}^{t+1}$

-

$\min_{\lambda\in\Delta_{K_{x}}}\max_{\mu\in\Delta_{K_{y}}}[G(\lambda,\mu):=F(% -\Lambda(\lambda,\mu)+(x_{1},y_{1}))].$

-

$\displaystyle(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})$

-

$\Delta(s^{t}_{x})$

-

$\displaystyle\leq\max_{k\in[K_{x}+K_{y}]}\|\Lambda^{T}_{k}\|_{1}\|\Lambda^{T}_% -{j}\|_{1}L_{2}^{F}\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})% -\|_{1}$

-

$w^{t},v^{t}$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\left[% -\sqrt{\frac{\log(d_{x})+\log(d_{y})}{n}}+\left(\frac{\sqrt{\log(1/\delta)}(% -\log(d_{x})+\log(d_{y}))^{3/2}}{n\varepsilon}\right)^{1/3}\right].$

-

$\mathcal{E}\subseteq\mathcal{X}$

-

$\tilde{x}=\frac{1}{n}\sum_{k=1}^{n}\tilde{x}^{k}$

-

$\tau\leq B\varepsilon/[8L_{0}\sqrt{2T(K+1)\log(1/\delta)}].$

-

$\displaystyle=\max_{k\in[K_{x}+K_{y}]}\left|\sum_{i\in[K_{x}+K_{y}]}\Lambda^{T% -}_{j,i}\langle\Lambda^{T}_{k},\nabla\nabla_{i}F(\Lambda(\lambda_{1},\mu_{1})+(% -x_{1},y_{1}))-\nabla\nabla_{i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\right|$

-

$\displaystyle\|\nabla G(\lambda_{1},\mu_{1})-\nabla G(\lambda_{2},\mu_{2})\|_{\infty}$

-

$L_{2}\lesssim(L_{0}+L_{1})\left\{\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n% -}}+\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n% -\varepsilon}\right)^{1/2}\right\},$

-

$\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(x)]\lesssim\frac{\log(% -d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{1}^{2}q}{\sqrt{\log(d_{x})}K^{3/% -2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}q}{\sqrt{K}}\Big{]}+\frac{L_{% -1}}{\sqrt{K}}+\frac{L_{1}q\log(T/q)}{T}.$

-

$\big{(}L_{1}+\frac{L_{1}^{2}}{\lambda}\big{)}$

-

$U=\min\left\{\frac{(n\varepsilon)^{2/3}}{(4\cdot 48\cdot 81\log(1/\delta)^{1/3% -}(\tau L_{0})^{2/3}},\frac{n}{2}\right\}$

-

$F_{j}(\cdot)=F(\cdot+re_{j})$

-

$\mathbb{P}(j=e_{i})\propto\exp\left(s(S,i)\right)$

-

$\tilde{x}^{i}\sim P_{x^{i}}$

-

$P_{x^{t}}$

-

$\min_{x\in\Delta_{|\mathcal{Z}|}}\max_{y\in\Delta_{|\mathcal{Q}|}}\mathbb{E}_{% -z\sim\mathcal{D}}\Big{[}\sum_{j\in[|\mathcal{Q}|]}y_{j}(q_{j}(z)-\langle q_{j}% -,x\rangle)\Big{]}.$

-

$\displaystyle T[F_{\mathcal{D}}(\bar{x},v)-F_{\mathcal{D}}(w,\bar{y})]$

-

$s(S,j)=\operatorname{LSE}(\tau G_{j}(S))$

-

$\mathbb{E}\left\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F% -\left(\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\right\|^{2}_{\infty}\leq\frac{20L_% -{2}^{2}}{T^{2}}+\frac{8L_{1}^{2}(6+\log(d))}{T}.$

-

$\displaystyle\leq\int_{0}^{A+B}\beta d\beta+\sum_{j\in[M]}\int_{A+B}^{\infty}% -\mathbb{P}\left[|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|\geq\beta\right]\beta d\beta$

-

$L_{0}^{G}$

-

$\mathcal{T}+1\geq t$

-

$\displaystyle\mathbb{E}[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})]$

-

$s^{t}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{y,j}\right)$

-

$\displaystyle T[F_{\mathcal{D}}(\bar{x},v)-F_{\mathcal{D}}(w,\bar{y})]\leq% -\operatorname{Regret}_{T}^{x}(w)+\operatorname{Regret}_{T}^{y}(v)+\sum_{t=1}^{% -T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w,v)\rangle$

-

$\displaystyle\leq\max_{i\in[K_{x}+K_{y}]}\|\Lambda^{T}_{i}\|_{1}L_{1}^{F}\|% -\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})\|_{1}$

-

$\langle\tilde{\Delta}^{t_{0}},x^{t}-x\rangle=\underbrace{\langle\tilde{\Delta}% -^{t_{0}},\tilde{x}^{t}-x\rangle}_{=:A_{1}}+\underbrace{\langle\tilde{\Delta}^{% -t_{0}},x^{t}-\tilde{x}^{t}\rangle}_{=:A_{2}},$

-

$\tilde{x}^{1},...,\tilde{x}^{n}\overset{\text{iid}}{\sim}P_{\bar{x}}$

-

$\mathbb{E}\left[\mathcal{T}+1\right]\lesssim U/\mathbb{E}[2^{N_{1}}]\eqsim U/M$

-

$\langle\tilde{\Delta}^{t_{0}},x^{t}-\tilde{x}^{t}\rangle\leq\|\tilde{\Delta}^{% -t_{0}}\|_{\infty}\|x^{t}-\tilde{x}^{t}\|_{1}$

-

$\displaystyle\mathbb{E}_{\hat{w}^{t_{0}}}[\langle\tilde{\Delta}^{t_{0}},\tilde% -{x}^{t}-x\rangle]$

-

$\max_{i\in[d_{y}]}\{F_{i}(x):=\mathbb{E}_{z\sim\mathcal{D}}[f_{i}(x;z)]\}$

-

$\sum_{i\in[t-1]}2^{N^{i}}\leq U-2^{M}$

-

$\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},\tilde{y})\geq A% -\right]\leq\frac{\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},% -\tilde{y})-\operatorname{Gap}(\bar{x},\bar{y}))\right]+\mathbb{E}\left[(% -\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})\right]}{A}.$

-

${\cal Y}\subseteq\mathbb{R}^{d_{y}}$

-

$a\eqsim b$

-

$\mathbb{P}[N=k]=p^{k}\mathbbm{1}_{(k\in\{0,1,...,M\})}/C_{M}$

-

$(w^{t})_{t\in[T]}$

-

$\displaystyle\leq\frac{(A+B)^{2}}{2}+2MC\left[C\int_{B/C}^{\infty}\exp\left(-% -\beta^{2}\right)\beta d\beta+A\int_{B/C}^{\infty}\exp\left(-\beta^{2}\right)d% -\beta\right]$

-

$\bar{a}^{T}=\sum_{t=1}^{T}\lambda_{t}a^{t}$

-

$g^{t}_{y}=-\nabla_{y}F_{\mathcal{D}}(\hat{x}^{t},\hat{y}^{t};B^{t})$

-

$U\leq\frac{\varepsilon^{2}}{48\log(1/\delta)(9\tau\alpha L_{0})^{2}}$

- - - diff --git a/htmls/output_mathjax_example_10027.html b/htmls/output_mathjax_example_10027.html deleted file mode 100644 index d2a2313386dc87b65c490d17db14ff59147a91ae..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10027.html +++ /dev/null @@ -1,167 +0,0 @@ - - - - MathJax Example - - - - -

$4\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}+2(\mathcal{T}+1)\leq 6\sum_{t=1}^{% -\mathcal{T}+1}2^{N_{t}}\leq\frac{\varepsilon^{2}}{8\log(1/\delta)(9\tau\alpha L% -_{0})^{2}}.$

-

$\displaystyle=L_{0}^{F}D\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})\|_{1}.$

-

$\psi=4\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|$

-

$\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},% -\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K=1.$

-

$\displaystyle\leq 2L_{1}\sum_{k=t_{0}}^{t-1}\frac{2}{k+1}\leq\frac{4L_{1}q}{q% -\lfloor t/q\rfloor+1},$

-

$\displaystyle=\mathbb{E}\big{[}\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w% -^{t};B^{t}),x^{t}-x\rangle\big{]}+\mathbb{E}\big{[}\langle\nabla f(w^{t};B^{t}% -)-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle\big{]}$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\bigg{% -[}\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n}}+\left(\frac{(\log(d_{x})+% -\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{1/2}\bigg{]}.$

-

$\textstyle\mathbb{E}\Big{[}\big{\|}\sum_{t=1}^{T}x^{t}\Big{\|}_{\infty}^{2}% -\Big{]}\leq c\log(d)\sum_{t=1}^{T}\mathbb{E}[\|x^{t}\|_{\infty}^{2}].$

-

$b\lesssim a$

-

$\hat{y}^{t+1,k}$

-

$P_{w^{t}}$

-

$F(\bar{a}^{T})-F\left(\bar{x}^{T}\right)\leq\sum_{t=1}^{T}\alpha_{t}+\frac{L_{% -1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2},$

-

$\displaystyle\mathcal{X}\times\mathcal{Y}-(x_{1},y_{1})$

-

$\mathbb{E}\left\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F% -\left(\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\right\|^{2}_{\infty}\leq\frac{43L_% -{1}^{2}}{\sqrt{12+\log(d)}T^{3/2}}+\frac{17(L_{0}^{2}+L_{1}^{2})\sqrt{(12+\log% -(d))}}{\sqrt{T}}.$

-

$\displaystyle=\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\mathbb{E}[2^{N_{t}}]% -\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$

-

$9\tau\alpha L_{0}$

-

$\mathbb{E}[F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)]\leq\frac{\log(d_{x})}{% -\tau T}+\frac{\tau L_{0}^{2}}{2}+\frac{1}{T}\sum_{t\in[T]}\mathbb{E}\left[% -\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-x\rangle\right].$

-

$\Delta(s^{t}_{y})$

-

$\mathbb{E}[F_{\cal D}(\hat{w}^{T})-F_{\cal D}(x)]\lesssim(L_{0}+L_{1})\left[% -\sqrt{\frac{\log(d_{x})}{n}}+\frac{\log(d_{x})\log(1/\delta)^{1/4}}{\sqrt{n% -\varepsilon}}\right]+\frac{L_{2}}{\sqrt{n}\log(d_{x})^{1/4}}\\ -+\frac{L_{2}\log(d_{x})^{1/4}\log(1/\delta)^{1/4}}{\sqrt{n\varepsilon}}+L_{1}% -\log(\sqrt{n\log(d_{x})})\Big{[}\frac{1}{\sqrt{n}}+\frac{\log(1/\delta)^{1/4}}% -{\sqrt{n\varepsilon}}\Big{]}.$

-

$\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}}{\sqrt{n\varepsilon}}$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},% -\bar{y})]\leq 8L_{1}\sqrt{\log(d_{x})+\log(d_{y})}/\sqrt{T}$

-

$|\mathbb{E}[\nabla_{x,j}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x,j}\mid\mathcal{F% -}_{t}]|\leq\frac{2L_{2}}{K}$

-

$\Delta_{s}:=\max_{j\in J}\max_{S\sim S^{\prime}}|s(S,j)-s(S^{\prime},j)|$

-

$\displaystyle=\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})% -,x^{t}-\tilde{x}^{t}\rangle$

-

$L_{0}^{F}$

-

$\displaystyle=\max_{k\in[K_{x}+K_{y}]}\left|\sum_{i\in[K_{x}+K_{y}]}\Lambda^{T% -}_{j,i}\Big{[}\nabla_{k,i}F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla% -_{k,i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\Big{]}\right|$

-

$\displaystyle\leq\frac{2L_{2}}{2^{M}},$

-

$x\in\Delta_{x}$

-

$(\bar{x}_{-},\bar{y}_{-})=\frac{1}{2^{N}}\sum_{i\in[2^{N}]}(\hat{x}^{i},\hat{y% -}^{i})$

-

$s:{\cal Z}^{n}\times J\mapsto\mathbb{R}$

-

$\mathbb{E}[\langle\nabla F(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle]\leq\frac{% -\mathbb{E}\left[F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a})% -\right]}{r}+L_{1}r.$

-

$\hat{w}^{T}$

-

$\displaystyle(\lambda,\mu)$

-

$\operatorname{LSE}(x)$

-

$\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2(\log(d_{x})+\log(d_% -{y}))}{\tau T}+5\tau L_{0}^{2}+\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\mathbb{E}% -[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{\infty}.$

-

$x_{\mathcal{D}}$

-

$\sum_{t\in[T]}\langle\tau(\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}),w^% -{t}-w\rangle\leq\log(d_{x})+\frac{1}{2}\sum_{t\in[T]}\tau^{2}\|\nabla_{x}F_{% -\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}.$

-

$\mathbb{E}[\mathcal{T}+1]$

-

$\sqrt{\log(d)/n}+\log(d)^{7/10}/[n\varepsilon]^{2/5}$

-

$\displaystyle=4\max_{j\in[d_{x}]}\tau\left|\sum_{k=t_{0}}^{t-1}[\nabla_{j}f(% -\hat{w}^{t_{0}};B^{k})-\nabla_{j}f(w^{t_{0}};B^{k})]\right|$

-

$\mathcal{T}=\sup\left\{T\in\mathbb{N}:\sum_{t=1}^{T}2^{N_{t}}\leq U-2^{M}% -\right\}.$

-

$\bar{y}=\sum_{t\in[T]}y^{t}$

-

$\displaystyle=\sum_{k=0}^{M}\mathbb{E}\Big{\|}C_{M}2^{k}[\nabla_{x}f(\bar{x}_{% -+},\bar{y}_{+};B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})]+\nabla_{x}f(% -x_{0},y_{0};B_{k})\Big{\|}_{\infty}^{2}\cdot\frac{2^{-k}}{C_{M}}$

-

$\left|\mathbb{E}\left[\nabla_{j}F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-% -\nabla_{j}F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right|\leq\frac{% -2L_{2}}{T}.$

-

$w^{1}=(1/d_{x},...,1/d_{x})\in\mathbb{R}^{d_{x}},w^{t+1}:=\operatorname{argmin% -}_{x\in\Delta_{x}}\left(\tau\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g_{x% -}^{t},x\rangle+\sum_{i\in[d_{x}]}w^{t}_{i}\log(w^{t}_{i}/x_{i})\right)$

-

$\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\leq\frac{8L_{1}\sqrt{\log(% -d_{x})+\log(d_{y})}}{\sqrt{T}}+\frac{2(\log(d_{x})+\log(d_{y}))}{\tau T}+5\tau -L% -_{0}^{2}+\frac{2\sqrt{2}L_{1}}{\sqrt{K}}.$

-

$\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2\log(|\mathcal{Z}|)}% -{\tau_{x}T}+\frac{\log(|{\cal Q}|)}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+10\sqrt{% -\frac{\log(|{\cal Q}|)}{n}}.$

-

$\sum_{t\in[T]}\langle\xi^{t},w^{t}-w\rangle\leq D_{\psi}(w^{1},w)+\frac{1}{2}% -\sum_{t\in[T]}\|\xi^{t}\|^{2}_{*}$

-

$20.28\pm 1.09$

-

$\textbf{20.60}\pm 0.89$

-

$20.84\pm 0.83$

-

$19.82\pm 1.05$

-

$20.46\pm 1.16$

-

$20.05\pm 1.10$

-

$20.30\pm 0.74$

-

$20.96\pm 0.86$

-

$19.74\pm 1.00$

-

$\textbf{20.41}\pm 1.19$

-

$20.80\pm 0.88$

-

$20.25\pm 1.10$

-

$20.94\pm 0.85$

-

$\textbf{21.44}\pm 0.80$

-

$20.19\pm 1.04$

-

$\{(o_{t},a_{t}),\dots,(o_{t+n},a_{t+n})\}$

-

$20.73\pm 0.84$

-

$20.09\pm 1.02$

-

$21.22\pm 0.87$

-

$20.95\pm 0.87$

-

$\textbf{20.54}\pm 1.21$

-

$20.24\pm 0.98$

-

$20.83\pm 0.94$

-

$97.89\%$

-

$\textbf{20.95}\pm 0.92$

-

$\textbf{21.58}\pm 1.10$

-

$\textbf{20.47}\pm 1.11$

-

$\textbf{20.41}\pm 0.90$

-

$20.12\pm 1.12$

-

$20.26\pm 1.05$

-

$21.15\pm 0.91$

-

$20.51\pm 1.02$

-

$20.20\pm 0.24$

-

$20.81\pm 0.90$

-

$20.31\pm 1.07$

-

$\textbf{20.97}\pm 0.92$

-

$21.01\pm 0.85$

-

$20.91\pm 0.79$

-

$20.96\pm 0.84$

-

$20.83\pm 0.82$

-

$256\times 160$

-

$20.33\pm 1.20$

-

$20.31\pm 1.12$

-

$w_{p_{xy}}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(x-m)^{2}+(y-n)^{2}}{2\sigma^{2}}},$

-

$p(I_{pred})$

-

$\displaystyle L_{VQVAE}=||x-D(e)||^{2}_{2}+||sg[E(x)]-e||^{2}_{2}+$

-

$T_{style}$

-

$I_{x_{ij}}=\mathop{\arg\min}_{\theta\in\Theta}Dist(Z_{q_{ij}},Z_{\theta}),$

-

$L_{VQVAE}$

-

$\displaystyle\beta||sg[e]-E(x)||^{2}_{2},$

-

$I_{GT}$

-

$i=1,{\ldots}\,,T$

- - - diff --git a/htmls/output_mathjax_example_10028.html b/htmls/output_mathjax_example_10028.html deleted file mode 100644 index 2002ad2157bb31d15a12106a3703ec267731f1f6..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10028.html +++ /dev/null @@ -1,131 +0,0 @@ - - - - MathJax Example - - - - -

$T_{text}$

-

$p_{xy}$

-

$S_{text}$

-

$P(I_{0},I_{1},...,I_{F}|I_{0})=\prod_{i=1}^{F}P(I_{i}|I_{0}),$

-

$w_{p_{xy}}$

-

$L_{Transformer}=-\sum p(I_{pred})\log q(I_{GT}),$

-

$Dist(Z_{q_{ij}},Z_{\theta})=\left\|Z_{q_{ij}}-Z_{\theta}\right\|_{2},$

-

$S_{style}$

-

$q(I_{GT})$

-

$P_{i_{xy}}=\frac{\sum_{p_{xy}>0}w_{p_{xy}}p_{xy}}{\sum_{p_{xy}>0}w_{p_{xy}}},$

-

$\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(G)\leq c\cdot\mathrm{RAC% -}(\mathcal{P}_{n},G)$

-

$\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f)\leq c\cdot\mathrm{RAC% -}(\mathcal{P}_{n},f)$

-

$\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2}\log n)$

-

$H=S_{k}$

-

$v_{\sqrt{n}}$

-

$H=([h],E)$

-

$O(1/n^{c})$

-

$n/{2.2^{i}}$

-

$\Pr(E_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge\neg E_{j-1})=\Theta% -\left(\frac{2^{t}}{1.1^{t}}\cdot\frac{1}{n^{1/10}}\right)$

-

$\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2})$

-

$\displaystyle\mathrm{RAC}(\mathcal{P},G)=\min_{A\in\mathcal{A}_{\mathcal{P}}}% -\max_{\pi\in\Gamma}\text{Queries}^{\mathcal{P}}_{A}(\pi(G)),$

-

$|H|/(|B|-i)=\Theta(1/n)$

-

$x\neq y\in[n]$

-

$n^{1-c}/{1.1^{t}}$

-

$|C|=\Theta(n/\log n)$

-

$1\leq i\leq\sqrt{n}-1$

-

$I\in\mathcal{I}_{i}\setminus\mathcal{L}_{i}$

-

$\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n^{1/10}% -\log(n))$

-

$n^{0.9}$

-

$\Pr(E)=1-1/\Theta(n^{1/4})$

-

$C\sqrt{n}$

-

$(1+o(1))p$

-

$\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n)$

-

$|V_{t}|=n-O(t)\geq n/2$

-

$\alpha n^{1/4}$

-

$I\in\mathcal{L}_{i}$

-

$O(n^{0.1})$

-

$\mathop{\mathbb{E}}[I_{i}]\leq\alpha q_{i}/n^{0.1}$

-

$x_{1},\ldots x_{h}\in[n]$

-

$\{x_{1},\ldots,x_{h}\}$

-

$\Pr(E)=1-O(1/\sqrt{n})$

-

$b_{t}=n^{9/10}/1.1^{t}$

-

$\mathop{\mathbb{E}}_{G\leftarrow\Delta}\mathrm{RAC}(\mathcal{P},G)=O(n^{1/10})$

-

$|\mathcal{L}_{i}|\leq\frac{\kappa n^{0.1}}{2^{i-2}}.$

-

$I\in\mathcal{R}_{i}$

-

$O(1/p)=O(1/n^{1/4})$

-

$H_{a,b,c}$

-

$y_{i},y_{j},y_{l}$

-

$2^{t}\cdot n^{9/10}/1.1^{t}$

-

$\text{deg}(v)=1$

-

$H=P_{k}$

-

$\mathcal{A}=\{A_{n}\}_{n\in\mathbb{N}}$

-

$f\circ\pi$

-

$V_{i}=\bigcup_{I\in\mathcal{I}_{i}}V(I)$

-

$Cn^{0.1}\log n$

-

$Q\subset I$

-

$\mathcal{A}_{\mathcal{P}}$

-

$\{G_{n}\}_{n\in\mathbb{N}}$

-

$p\in\{p_{i_{1}},\ldots,p_{i_{k}}\}$

-

$f(x_{u})=x_{v}$

-

$2^{i}<2^{t}$

-

$B(t)/|V_{t}|\leq 1/n^{0.1+\Omega(1)}$

-

$\mathcal{P}_{S_{k}}$

-

$\alpha\kappa$

-

$n^{1-c}/1.1^{i}$

-

$B(n)\leq n^{9/10}/1.1^{\log(n)/1000}=n^{9/10-\Omega(1)}$

-

$G^{\pi}=(V,E^{\pi})$

-

$2^{i-2}$

-

$f(x_{1})=\ldots=f(x_{k})$

-

$2/2^{i-1}=4/2^{i}$

-

$O(n^{3/4}\cdot\log^{4}n/n^{1/4})=o(n^{3/4})$

-

$B=[n]\setminus C$

-

$\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f_{n})\geq\omega(n)\cdot% -\mathrm{RAC}(\mathcal{P}_{n},f_{n}).$

-

$b_{i}=n^{9/10}/1.1^{i}$

-

$f\in\Delta$

-

$\frac{2^{t}}{1.1^{t}}\cdot\frac{1.1^{t}\cdot n}{n^{1-c}\cdot 2^{t}}=n^{c}.$

-

$\mathop{\mathbb{E}}_{f\leftarrow\Delta}RAC(\mathcal{P}_{H},f)=O(n^{3/4})$

-

$\frac{n^{1-c}\cdot 2^{t}}{n\cdot 1.1^{t}}$

-

$\Omega(n^{0.1}\log n)$

-

$2^{t}+4$

-

$\mathop{\mathbb{E}}\left[X_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge% -\neg E_{j-1}\right]=O\left(\frac{2^{t}}{1.1^{t}}\right)$

-

$c=1/10$

-

$\Pr({E_{\text{long}}})=O(\kappa)$

-

$\frac{a_{i}\cdot 2^{i}}{n}=\frac{1}{1.1^{i}}$

-

$\omega\colon\mathbb{N}\to\mathbb{N}$

-

$C_{i_{1}},\ldots,C_{i_{T}}$

-

$C=\bigcup_{i=1}^{N}C_{i}$

-

$\mathcal{P}_{H}$

-

$\mathop{\mathbb{E}}[\text{\# red vertices encountered}]\leq\sum_{\frac{1}{1000% -}\log n\leq i\leq\frac{1}{100}\log n}\alpha\frac{q_{i}}{n^{0.1}}\leq\frac{% -\alpha}{n^{0.1}}cn^{0.1}\log n\leq\alpha c\log n,$

-

$N=\alpha n^{1/4}/\log(n)$

-

$\pi\colon[n]\to[n]$

-

$\bigcup_{i=1}^{N}C_{i}$

-

$\Omega(n^{c}\log n)$

-

$\Pr({E_{\text{long}}})$

-

$\Pr({E_{\text{long}}})\leq\sum_{I\in\mathcal{L}_{i}}\Pr(I\in\mathcal{R}_{i})% -\leq|\mathcal{L}_{i}|\cdot(1+o(1))p\leq(1+o(1))\cdot\frac{\kappa n^{0.1}}{2^{i% --2}}\cdot\frac{2^{i}}{n^{0.1}}=O(\kappa).$

-

$\pi\colon V\to V$

-

$G=(V,E)\in\mathcal{P}$

-

$i=1,...,\Theta(\sqrt{n}\log n)$

-

${E_{\text{short}}}$

-

$n^{9/10}/1.1^{t}$

- - - diff --git a/htmls/output_mathjax_example_10029.html b/htmls/output_mathjax_example_10029.html deleted file mode 100644 index 75d0462f58ec977cc684dc3331c322b5fa12005d..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10029.html +++ /dev/null @@ -1,130 +0,0 @@ - - - - MathJax Example - - - - -

$\text{Queries}_{A}(\cdot)$

-

$\mathcal{B}_{i}=\mathcal{I}_{i}\setminus\mathcal{R}_{i}$

-

$\mathrm{RAC}(\mathcal{P}_{S_{k}},\Delta)=O(n^{1/2})$

-

$\mathcal{P}_{S_{3}}$

-

$\mathcal{RAC}(\mathcal{P}_{H},f)=O(n^{3/4})$

-

$p=\frac{n^{9/10}/1.1^{i}}{n/2.2^{i}}=\frac{2^{i}}{n^{0.1}}$

-

$\tilde{\theta}(n^{1/4})$

-

$O(2^{t}/1.1^{t})$

-

$\frac{1}{1000}\log n\leq t\leq\frac{1}{100}\log n$

-

$\omega(n)=n^{\Omega(1)}$

-

$\mathcal{R}_{i}\subseteq\mathcal{I}_{i}$

-

$\alpha\frac{n}{\log n}$

-

$Cn^{1/10}\log n$

-

$p_{i_{1}},\ldots,p_{i_{T}}$

-

$1/|I|$

-

$\text{Queries}^{\mathcal{P}}_{A}(f,r)$

-

$\mathrm{RAC}(\mathcal{P},G)\in O(n^{1/10})$

-

$H=H_{a,b,c}$

-

$\displaystyle\text{Queries}_{A}(f)\leq\alpha\cdot\max_{\pi}\text{Queries}_{A^{% -\prime}}(f\circ\pi)$

-

$(\pi(u),\pi(v))\in E^{\pi}$

-

$([3],\{1\to 3,2\to 3\})$

-

$a_{i}=n/2.2^{i}$

-

$\mathop{\mathbb{E}}_{f\leftarrow\Delta}\text{Queries}_{A}(f)=\Omega(n/\log n)$

-

$\frac{1}{1000}\log n\leq i\leq\frac{1}{100}\log n$

-

$I\notin\mathcal{R}_{i}$

-

$n^{1/10}$

-

$n^{1/10+\Omega(1)}$

-

${E_{\text{short}}}=E\setminus{E_{\text{long}}}$

-

$\Pr(E)=1-o(1)$

-

$\text{Queries}_{A}(f)$

-

$\Omega(n^{1/10}\log n)$

-

$f\circ\pi\in\mathcal{P}$

-

$\Pr(E)=O(\kappa)$

-

$Cn^{1/2}$

-

$\sum_{i=0}^{t}2^{i}\cdot\frac{1}{1.1^{i}}+\sum_{i=t}^{\log n}2^{t}\cdot\frac{1% -}{1.1^{i}}=O\left(\frac{2^{t}}{1.1^{t}}\right)$

-

$n^{\frac{3}{4}}$

-

$P\in\mathcal{P}_{t}$

-

$\displaystyle\text{Queries}_{A}(f)\geq\omega(n)\cdot\max_{\pi}\text{Queries}_{% -A^{\prime}}(f\circ\pi).$

-

$t=o(n)$

-

$\Pr({E_{\text{short}}})=O(\kappa)$

-

$[\frac{\sqrt{n}}{4},\frac{3\sqrt{n}}{2}]$

-

$v_{0},v_{1},\ldots,v_{\sqrt{n}}$

-

$n^{\frac{1}{10}+\varepsilon}$

-

$n\geq\mathbb{N}$

-

$O(n^{1/10})$

-

$P_{1},...,P_{\sqrt{n}}$

-

$\{\mathcal{A}_{n}\}_{n\in\mathbb{N}}$

-

$O(\log^{4}n/\sqrt{n})$

-

$\frac{4}{2^{i}}\cdot(1+o(1))\cdot\frac{2^{i}}{n^{0.1}}=O\left(\frac{1}{n^{0.1}% -}\right).$

-

$2^{i}/n^{c}$

-

$P\setminus\{u,v\}$

-

$f\colon[n]\to[n]$

-

$\{\mathcal{P}_{n}\}$

-

$\Theta(\sqrt{n}\log n)$

-

$\mathcal{P}=\{\mathcal{P}_{n}\}_{n\in\mathbb{N}}$

-

$\{A_{n}\}_{n\geq N}$

-

$Cn^{1/4}$

-

$\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(G_{n})\geq\omega(n)\cdot% -\mathrm{RAC}(\mathcal{P}_{n},G_{n}).$

-

$(n^{1/4}/4,{n^{1/4}}/2)$

-

$f(y)=x$

-

$\text{Queries}^{\mathcal{P}}_{A}(f)=\mathop{\mathbb{E}}_{r}\text{Queries}^{% -\mathcal{P}}_{A}(f,r)$

-

$\min(2^{i},2^{t})$

-

$2\leq i\leq\sqrt{n}-1$

-

$\frac{4}{2^{i}}$

-

$\Pr({E_{\text{short}}})$

-

$n^{1/4}/\log n$

-

$p_{i_{k}}$

-

$v_{0},\ldots,v_{\sqrt{n}}$

-

$W_{1},\ldots,W_{m}$

-

$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}\}$

-

$i_{1},...,i_{T}$

-

$f_{n}\colon[n]\to[n]$

-

$O(\log^{1.1}n)$

-

$G\in\Delta$

-

$|Q|<2^{i-2}$

-

${E_{\text{long}}}$

-

$E={E_{\text{long}}}\cup{E_{\text{short}}}$

-

$G^{\pi}\in\mathcal{P}$

-

$\displaystyle\mathrm{RAC}(\mathcal{P},f)=\min_{A\in\mathcal{A}_{\mathcal{P}}}% -\max_{\pi}\text{Queries}_{A}(f\circ\pi),$

-

$i_{1},\ldots,i_{T}$

-

$I\setminus Q$

-

$n/2.2^{i}$

-

$cn^{0.1}\log n$

-

$\frac{1}{1000}\log n$

-

$\kappa n^{0.1}$

-

$\sqrt{n}-2$

-

$Path\mbox{ }of\mbox{ }length\mbox{ }\sqrt{n}$

-

$s\in B\Rightarrow s_{i}\in M_{B}\;\forall_{s_{i}\in s}\land((s_{0}\in E_{B}^{0% -}\land s_{N}\in E_{B}^{1})\lor(s_{N}\in E_{B}^{0}\land s_{0}\in E_{B}^{1})).$

-

$3\times N$

-

$\mathrm{LM}_{k}(s):=\left\{||l_{k}-s_{i}||\forall s_{i}\in s\right\},$

-

$b\in\{1000,3000\}$

-

$5\,mm$

-

$s\equiv(s_{0},...,s_{N})$

-

$1\times N$

-

$56\times N$

-

$\mathrm{LM}[s]$

-

$W_{ij}=\left\{\begin{array}[]{ll}1&s_{j}\in R_{i}\\ -0&otherwise\end{array}\right.,$

-

$M_{B},E_{B}^{0},E_{B}^{1}$

-

$3\cdot\mathrm{10}^{-5}$

-

$J_{mm}(A,B)$

- - - diff --git a/htmls/output_mathjax_example_1003.html b/htmls/output_mathjax_example_1003.html deleted file mode 100644 index a0cbe670afa81a862b25a3cf7754585bd49c79c2..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1003.html +++ /dev/null @@ -1,122 +0,0 @@ - - - - MathJax Example - - - - -

$x=a$

-

$C_{1},\ldots,C_{k}\cup(C\setminus\{v\})$

-

$z^{v}$

-

$\{2,\dots,s-1\}$

-

$S\cap(A\cup C)$

-

$H:=G[A\cup B\cup S]$

-

$z\in V(C)$

-

$N_{G}(p_{s})\cap V(H)=\{b_{2}\}$

-

$H\setminus C$

-

$k\geq\chi(\overline{G})\geq 2$

-

$H_{1},H_{2}\in\mathcal{M}_{\mathcal{C}}$

-

$\{p_{0},\dots,p_{s-1}\}$

-

$(x,c)$

-

${\cal G}_{1}$

-

$q_{t}=y$

-

$q_{2},\dots,q_{t}\in B)$

-

$G\in\mathcal{G}_{k}$

-

$S=(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}$

-

$X_{v}$

-

$\mathcal{M}_{\mathcal{G}_{2}}=\{\overline{K_{2}\cup C_{2k+1}}\mid k\in\mathbb{% -N}\}$

-

$\{a_{i},b_{j}\}$

-

$\mathcal{O}(n^{\omega}\log n)$

-

$|V(H)|\geq 2$

-

$V(C)$

-

$q_{i},\ldots,q_{t},b_{1},b_{2},a_{2},q_{i}$

-

$N_{G}(v)\cap V(H)\subseteq A$

-

$G[N_{G}(v)]$

-

$G_{A}\setminus S^{\prime}$

-

$\mathcal{O}(n^{2k})$

-

$\mathbb{Z}_{k}$

-

$\{X_{v}\}_{v\in V(H)}$

-

$2K_{1}\vee K_{2}$

-

$G[\{a_{1},a_{2},b_{1},b_{2}\}\cup V(Q)]$

-

$v^{xy}\in S$

-

$S\cap A$

-

$(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}\subseteq S$

-

$X_{w}\subseteq V(D)$

-

$\mathcal{G}_{k}=\mathcal{G}_{\mathcal{C}_{k}}$

-

$N_{G}[u]=V(G)$

-

$\mathcal{O}(n^{2.373}(n+m))$

-

$F\in{\cal F}$

-

$r_{i}\in C$

-

$G\setminus x$

-

$G\setminus N_{G}[v]$

-

$S^{*}:=(S\setminus\{v^{xy}\})\cup\{x,y\}$

-

$U:=\{v\in V(H)\mid X_{v}\cap V(C)\neq\emptyset\}$

-

$N_{A}\cap N_{B}=\emptyset$

-

$G\setminus(A\cup B)$

-

$N_{G}(q_{t})\cap V(H)=\{a_{1},b_{1}\}$

-

$S\subseteq S^{*}$

-

$K_{\ell+1}\in\mathcal{G}_{\mathcal{C}}$

-

${\mathcal{C}_{k}}$

-

$V(H)=A\cup B$

-

$K_{\ell}$

-

$N_{G}(v)\cap V(H)=B$

-

$X_{w}$

-

$H\setminus a$

-

$H[A]$

-

$G^{\prime}[S]$

-

$G_{B}:=G[B\cup C]$

-

$S^{*}\subseteq S\cup(V(G)\setminus V(H))$

-

$G^{\prime}:=G/xy$

-

$p_{i},\dots,p_{s},b_{1},b_{3},a_{3},a_{2},q_{t},\dots,q_{0},p_{i}$

-

$q_{2},\dots,q_{t}\in Y$

-

$a_{1},q_{i}$

-

$\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}}\}$

-

$p_{1},\dots,p_{i-1}\in A$

-

$v^{xy}$

-

$\mathcal{C}\subseteq\mathcal{G}_{\mathcal{C}}$

-

$S^{\prime}\setminus\{x,y\}$

-

$p_{1}=x$

-

$N_{G}(q_{t})\cap V(H)=\{a_{2}\}$

-

$G\setminus N_{G}[x]$

-

$\overline{\overline{G}}=G$

-

$H[S]\in\mathcal{C}$

-

$S^{\prime}=S$

-

$\bigcup_{v\in V(H)}X_{v}\not\subseteq V(H_{2})$

-

$c\in C\setminus S$

-

$N_{G}(p_{s})\cap V(H)=\{b_{1}\}$

-

$\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{0}}\}=\{2K_{1}\vee K_{1}\}=\{% -P_{3}\}$

-

$K_{\ell+1}$

-

$H\in\mathcal{H}$

-

$A=C\cup\{v\}$

-

$U=\{v\in V(H)\mid X_{v}\subseteq V(C)\}$

-

$S_{A}:=S\setminus B$

-

$2K_{1}\vee 3K_{1}$

-

$q_{j}\in C$

-

$p_{i-1},p_{j+1}\in C$

-

$\{a_{1},b_{1}\}$

-

$Y\subseteq V(G)\setminus\{x\}$

-

$\{a,b\}=\{x,y\}$

-

$\{X_{v}\}_{v\in\{a,b\}\cup V(H)}$

-

$|V(H)|=2k$

-

$(x,b)$

-

$C\cap S$

-

$\mathcal{O}(|V(G)|^{2})$

-

$G[a_{j},a_{k},b_{j},b_{k},q_{i},q_{i+1},\dots,q_{t}]$

-

$2K_{1}$

-

$a_{i},a_{j}$

-

$p_{i}\in B$

- - - diff --git a/htmls/output_mathjax_example_10030.html b/htmls/output_mathjax_example_10030.html deleted file mode 100644 index 28940319caddf5faa7aaba8228f27ce1e3e4d751..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10030.html +++ /dev/null @@ -1,126 +0,0 @@ - - - - MathJax Example - - - - -

${}_{mm\_morph}$

-

$c=\frac{max(|A|,|B|)}{min(|A|,|B|)}$

-

$J\_{mm\_syn}$

-

$C_{WALS}$

-

$m:\mathbb{L}\mapsto\mathbb{R}$

-

$\rho=0.69$

-

$J_{mm}$

-

$1\dots|Z|$

-

$J_{mm\_syn}$

-

$J_{mm}(\mathbf{a},\mathbf{b})=\frac{\sum_{j=1}^{|Z|}min(a_{j},b_{j})}{\sum_{j=% -1}^{|Z|}max(a_{j},b_{j})}$

-

$\{Z=t(y):y\in Y\}=\{(y_{i},z_{j})\}$

-

$\{Y=m(x):x\in X\}=\{(x_{i},y_{i})\}$

-

$J_{mm\_morph}$

-

${}_{mm}$

-

$1\dots|X|$

-

$J\_{mm\_morph}$

-

$ch\_ttr_{500}$

-

$s_{l}=r\cdot ch\_ttr_{l\_500}$

-

${}_{morph}$

-

${}_{mm\_syn}$

-

${}_{syn}$

-

$S=\textbf{C}\circ\textbf{U}(X)$

-

$val_{1}$

-

$\sigma(G;D)$

-

$\log P(G|D,\lambda)\propto\log P(D|G)+\log P(G|\lambda)$

-

$\sigma(G;D)=\sum_{i=1}^{n}\sigma\left(v_{i},\operatorname{pa}\left(v_{i}\right% -);D\right)$

-

$X=\{x_{1},x_{2},\cdots,x_{n}\}$

-

$sym_{1}$

-

$P\left(v_{i}\mid\text{pa}\left(v_{i}\right)\right)$

-

$val_{n}$

-

$x_{i_{1}}\rightarrow x_{j_{1}}$

-

$S{\prime}$

-

$\mathcal{A}=\{x_{i}\leadsto x_{j}\mid(x_{i},x_{j})\in S^{\prime}\}$

-

$P(G\mid D,\lambda)=\frac{P(D\mid G,\lambda)P(G\mid\lambda)}{P(D\mid\lambda)}$

-

$Domain$

-

$sym_{i}$

-

$val_{i}$

-

$\text{pa}(v_{i})$

-

$x\leadsto y$

-

$P(D|\lambda)$

-

$S=\{(x_{i},x_{j})\}$

-

$\sigma(G;D,\lambda)=\sigma(G;D)+\sigma(G;\lambda)$

-

$PromtU$

-

$sym_{n}$

-

$x_{i}\leadsto x_{j}\Rightarrow x_{j}\not\in\text{pa}(x_{i}),x_{i} -

$x\not\rightarrow y$

-

$S^{\prime}={(x_{i},x_{j})}$

-

$S=\{(x_{i_{1}},x_{j_{1}}),\cdots,(x_{i_{m}},x_{j_{m}})\mid x_{i_{k}},x_{j_{k}}% -\in X\}$

-

$\frac{2\cdot\text{precision}\cdot\text{recall}}{\text{precision}+\text{recall}}$

-

$x_{i_{m}}\rightarrow x_{j_{m}}$

-

$S^{\prime}=\textbf{R}\circ\textbf{C}\circ\textbf{U}(X)$

-

$P(D\mid G,\lambda)=P(D\mid G)$

-

$T=\textbf{U}(X)$

-

$\sigma(G;\lambda)$

-

$x_{i}\leadsto x_{j}$

-

$T=\{t_{1},t_{2},\cdots,t_{n}\}$

-

$g_{i}(\textbf{x})$

-

$h_{j}(\textbf{x})$

-

$\displaystyle\mathrm{s.t.}:$

-

$\displaystyle f({\bf{x}}),x\in D$

-

$\displaystyle\mathrm{Min}:$

-

$\displaystyle g_{i}({\bf{x}})\leq 0,{i}=1,\dots,{p}$

-

$\displaystyle h_{j}({\bf{x}})\leq 0,{j}=1,\dots,{q}$

-

$S\neq\phi$

-

$W=\phi$

-

$W=W\cup(S\cap T)$

-

$E^{\prime}[i]=E^{\prime}[j]$

-

$\lvert E^{\prime}\rvert$

-

$\rm{LayoutLMv3_{LARGE}}$

-

$V[j]=V[j]+1$

-

$T=P[j*2-1]$

-

$batch\_size=8$

-

$S=S[j+1:\lvert S\rvert]$

-

$L=L[1:\lvert L\rvert]$

-

$E=E\cup ParseEntityValue(D,J^{\prime})$

-

$~{}XX|YY_{segment}$

-

$T.subtypes=\phi$

-

$G=\phi$

-

$\lvert E\rvert$

-

$M=\{``s.x|s.y"\mapsto s|s\in D.segments\}$

-

$G^{\prime}.value=\bigcup_{w\in W}w.text\_value$

-

$P[j*2]\notin M$

-

$\bigcup_{T^{\prime}\in T}MajorityVoting(\bigcup_{S^{\prime}\in S}DecodeForType% -(ParseJson(S^{\prime}),T^{\prime},D))$

-

$800train/100dev/100test$

-

$E^{\prime}.subtypes=\bigcup_{T^{\prime}\in T.subtypes}DecodeForType(J^{\prime}% -,T^{\prime},D)$

-

$R.split(E[i])$

-

$(segment~{}text,segment~{}identifier)$

-

$S=D.pages[i].segments$

-

$G^{\prime}.bounding\_box=\{\min(b.x),\min(b.y),\max(b.x),\max(b.y)\}_{w\in W,b% -=w.bounding\_box}$

-

$E=\phi$

-

$F(S[1:j])\leq L$

-

$V=[0,0,...,0]\in\mathbb{R}\textsuperscript{$\lvert E\rvert$}$

-

$L=\{T\}$

-

$\lvert P\rvert/2$

-

$G=G\cup\{G^{\prime}\}$

-

$learning\_rate=2\cdot 10^{-5}$

-

$E=E\cup\{E^{\prime}\}$

-

$\mathbf{LayoutLMv3_{LARGE}}$

-

$T^{\prime}=L[0]$

-

$C=C\cup\{S[1:j]\}$

- - - diff --git a/htmls/output_mathjax_example_10031.html b/htmls/output_mathjax_example_10031.html deleted file mode 100644 index 8eb726714ad955ea12847ebd9a6c3df1fdf5db84..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10031.html +++ /dev/null @@ -1,185 +0,0 @@ - - - - MathJax Example - - - - -

$\lvert D.pages\rvert$

-

$[x\textsubscript{center},y\textsubscript{center}]$

-

$E^{\prime}=\phi$

-

$E^{\prime}[j]$

-

$J^{\prime}=J[j][T.type]$

-

$j=\lvert S\rvert$

-

$C=\phi$

-

$S=M[P[j*2]]$

-

$E^{\prime}=E^{\prime}\cup\{e|e\in E[j],e.type=T^{\prime}\}$

-

$R=Regex(``(\char 92d\char 92d\char 92|\char 92d\char 92d)")$

-

$\mathbf{Donut}$

-

$L\neq\phi$

-

$L=L\cup T^{\prime}.subtypes$

-

$Donut$

-

$E[argmax(V)]$

-

$[x\textsubscript{min},y\textsubscript{min},x\textsubscript{max},y% -\textsubscript{max}]$

-

$D=10/50/100/200$

-

$V[i]=V[i]+1$

-

$~{}XX|YY_{segment_{1}}\char 92n~{}XX|YY_{segment_{2}}\char 92n~{}...$

-

$\displaystyle\bm{q}(t):=\begin{bmatrix}q_{0}^{(1)}(t)&q_{0}^{(2)}(t)&q_{1}^{(1% -)}(t)&q_{1}^{(2)}(t)&\dots&q_{N-2}^{(1)}(t)&q_{N-2}^{(2)}(t)\end{bmatrix}^{T},$

-

$\displaystyle q_{3}:$

-

$40\text{\,}\mathrm{ms}$

-

$\displaystyle q_{0}:\quad$

-

$\displaystyle\;a_{i+1,i+1}=-\frac{1}{\psi_{\frac{i-1}{2}}}\left(\frac{\zeta_{% -\frac{i-3}{2}}+\zeta_{\frac{i-1}{2}}}{2\zeta_{\frac{i-3}{2}}\zeta_{\frac{i-1}{% -2}}}\right),$

-

$\displaystyle\lim_{t\to 0}\bm{q}(0,t)$

-

$M_{\textrm{left}}$

-

$(\xi_{n})_{n=0,\ldots,N-1}$

-

$n=1,2,...,N-1$

-

$\displaystyle\dot{\bm{q}}(t)$

-

$\bm{y}(t_{0})={(0.02,0.01)^{T}}$

-

$\displaystyle D_{x}\bm{q}(t)$

-

$\displaystyle=\bm{y}_{0}.$

-

$\dot{\bm{y}}(t)$

-

$\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)\in\mathbb{R}^{N}$

-

$\displaystyle\frac{243d^{5}}{120}(\partial^{5}_{\xi}q)_{0}.$

-

$\nicefrac{{\partial q(\xi(x),t)}}{{\partial\xi}}\rvert_{x_{n}}$

-

$\displaystyle\frac{d^{3}}{24}b_{1}$

-

$\displaystyle t\in[0,T],$

-

$\displaystyle c\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}+% -\frac{Nc}{N-1}\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{1}}\approx$

-

$\displaystyle\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$

-

$\displaystyle\frac{81d^{3}}{24}b_{3}$

-

$\displaystyle\frac{d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$

-

$\displaystyle\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t):=\begin{bmatrix}\frac{2}{2+% -\gamma\psi_{0}}\bm{f}^{(1)}(\bm{q}_{0}(t),\bm{y}(t),t)\\ -\frac{2}{2+\gamma\psi_{0}}\bm{f}^{(2)}(\bm{q}_{0}(t),\bm{y}(t),t)\\ -0\\ -\vdots\\ -0\end{bmatrix}.$

-

$\displaystyle M_{2}\;D_{x}^{2}\bm{q}(t)$

-

$\displaystyle\approx B_{2}\;D_{x}\bm{q}(t).$

-

$R=\nicefrac{{7}}{{3}}$

-

$-\nicefrac{{1}}{{2}}$

-

$\displaystyle\left(\frac{4Nc}{N-n}\right)\frac{\partial q(\xi(x),t)}{\partial x% -}\biggr{\rvert}_{n}+\left(\frac{Nc}{N-n-1}\right)\frac{\partial q(\xi(x),t)}{% -\partial x}\biggr{\rvert}_{n+1}$

-

$u^{(1)}=0$

-

$\displaystyle B_{1}:=\frac{1}{d}\begin{bmatrix}\frac{12\alpha dc-17\gamma}{18% -\gamma}&0&\nicefrac{{1}}{{2}}&0&\nicefrac{{1}}{{2}}&0&-\nicefrac{{1}}{{18}}&0&% -\dots&0\\ -0&\frac{12\alpha dc-17\gamma}{18\gamma}&0&\nicefrac{{1}}{{2}}&0&\nicefrac{{1}}% -{{2}}&0&-\nicefrac{{1}}{{18}}&\ddots&0\\ --3&0&0&0&3&0&0&0&\dots&0\\ -0&-3&0&0&0&3&0&0&\ddots&0\\ -\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ -\vdots&\ddots&\ddots&\ddots&\ddots&0&-3&0&0&0\\ -0&\dots&\dots&\dots&\dots&\dots&0&-3&0&0\end{bmatrix},$

-

$\displaystyle+\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{4}% -\gamma\sin(\lambda t)}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})% -\lambda}dk+$

-

$\displaystyle\;y_{0}^{(2)}+\frac{1-\cos(\lambda t)}{\lambda}+\frac{2}{\pi}\int% -_{0}^{\infty}\;v_{0}^{(2)}\frac{\gamma(1-e^{-k^{2}t})}{k^{2}\gamma^{2}+(k^{2}-% -\alpha)^{2}}\;dk+$

-

$\displaystyle d\;(\partial^{2}_{\xi}q)_{0}$

-

$\displaystyle\underbrace{\begin{bmatrix}\dot{\bm{q}}(t)\\ -\dot{\bm{y}}(t)\end{bmatrix}}_{=:\dot{\bm{\eta}}(t)}=\underbrace{\left[\begin{% -array}[]{c | c}A_{s}&\begin{array}[]{c c}0&0\\ -\vdots&\vdots\\ -0&0\\ -\end{array}\\ -\hline\cr\\ -\begin{array}[]{c c c c c}1&0&0&\dots&0\\ -0&1&0&\dots&0\end{array}&\begin{array}[]{c c}0&0\\ -0&0\end{array}\end{array}\right]}_{=:A}\underbrace{\begin{bmatrix}\bm{q}(t)\\ -\bm{y}(t)\end{bmatrix}}_{=:\bm{\eta}(t)}+\underbrace{\begin{bmatrix}\bm{v}(\bm% -{q}_{0}(t),\bm{y}(t),t)\\ -\bm{u}(\bm{y}(t),t)\end{bmatrix}}_{=:\bm{\omega}(\bm{q}_{0}(t),\bm{y}(t),t)}$

-

$R=\nicefrac{{1}}{{3}}$

-

$\displaystyle\frac{d^{2}}{2}a_{1}$

-

$\displaystyle\underbrace{\left(\mathbb{I}-\frac{\Delta t}{2}A\right)}_{=:M_{% -\textrm{left}}}\bm{\eta}^{k+1}=\left(\mathbb{I}+\frac{\Delta t}{2}A\right)\bm{% -\eta}^{k}+\frac{\Delta t}{2}\left(\bm{\omega}^{k}+\bm{\omega}^{k+1}\right),$

-

$\displaystyle\frac{\partial^{2}q(\xi(x),t)}{\partial x^{2}}\biggr{\rvert}_{x_{% -n}}\approx\;\frac{1}{\psi_{n}}\left[\frac{\partial q(\xi(x),t)}{\partial x}% -\biggr{\rvert}_{x_{n+\nicefrac{{1}}{{2}}}}-\frac{\partial q(\xi(x),t)}{% -\partial x}\biggr{\rvert}_{n-\nicefrac{{1}}{{2}}}\right]$

-

$\displaystyle:=M_{2}-\frac{2c}{3\gamma}B_{2}M_{1}^{-1}\mathbb{P}.$

-

$\displaystyle-\frac{2c}{3\gamma}\left(\alpha q_{0}-\gamma(\partial_{x}q)_{0}\right)$

-

$\bm{v}(t_{0})$

-

$\displaystyle\frac{16d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$

-

$R=\nicefrac{{7}}{{9}}$

-

$\bm{q}(0,t_{0})={(0,0.1)}^{T}$

-

$\left(\nicefrac{{d\xi(x)}}{{dx}}\right)_{x_{n}}$

-

$\displaystyle\approx M_{1}^{-1}\;B_{1}\bm{q}(t)+M_{1}^{-1}\;K_{1}(\bm{q}_{0}(t% -),\bm{y}(t),t)+M_{1}^{-1}\;V_{1}(\dot{\bm{q}}_{0}(t)),$

-

$\displaystyle\approx B_{2}\;D_{x}\bm{q}(t),$

-

$\displaystyle\beta=\frac{\rho_{p}}{\rho_{f}},\quad R=\frac{1+2\beta}{3},\quad S% -=\frac{a^{2}}{3T\nu},$

-

$a_{0},a_{1},b_{0},\hat{b}_{0},b_{1},\hat{b}_{1},b_{2},b_{3}\in\mathbb{R}$

-

$\displaystyle(\partial_{\xi}q)_{0}:\quad$

-

$\displaystyle M_{2}:=\begin{bmatrix}c&0&3\frac{Nc}{N-1}&0&0&0&0&0&\dots&0\\ -0&c&0&3\frac{Nc}{N-1}&0&\ddots&\ddots&\ddots&\ddots&0\\ -c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&\ddots&0\\ -0&c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&0\\ -0&0&\frac{Nc}{N-1}&0&\frac{4Nc}{N-2}&0&\frac{Nc}{N-3}&0&\ddots&0\\ -\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ -\vdots&\ddots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}&0\\ -\vdots&\ddots&\ddots&\ddots&0&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}\\ -0&\ddots&\ddots&\ddots&\ddots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}&0\\ -0&0&\dots&\dots&\dots&\dots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}\end{bmatrix},$

-

$\displaystyle+\frac{1}{2d}q_{2}(t)-\frac{1}{18d}q_{3}(t)+$

-

$\bm{q}(0,t_{0})={(0.5414,0)}^{T}$

-

$t\in[0,5]$

-

$\displaystyle a_{11}=$

-

$\displaystyle\frac{d^{3}}{6}a_{1}$

-

$m\in\{1,2,\dots,N-2\}$

-

$\displaystyle(\partial^{4}_{\xi}q)_{0}:\quad$

-

$\bm{u}(\bm{y}(t),t)$

-

$\displaystyle\left(\frac{Nc}{N-n+1}\right)\frac{\partial q(\xi(x),t)}{\partial -x% -}\biggr{\rvert}_{n-1}+$

-

$\displaystyle x>0,$

-

$n\in\{1,2,\dots,N-1\}$

-

$\displaystyle\frac{d^{2}}{2}(\partial^{3}_{\xi}q)_{0}$

-

$f^{(1)}\equiv 0$

-

$S=0.3$

-

$\displaystyle 3b_{3}$

-

$\bm{f}(\bm{q}_{0}(t),\bm{y}(t),t)$

-

$\frac{\partial}{\partial x}\bm{q}(\xi(x),t)$

-

$\Psi(x,y,t)=-U_{0}L\text{tanh}(y/L)+\sum_{i=1}^{3}A_{i}U_{0}L\text{sech}^{2}(y% -/L)\cos(k_{i}x-\sigma_{i}t)$

-

$\displaystyle\frac{1}{d}b_{3}$

-

$\displaystyle\frac{d^{4}}{120}b_{1}$

-

$\displaystyle=\frac{D\bm{u}}{Dt}-\frac{1}{S}(\bm{v}-\bm{u})-\sqrt{\frac{3}{\pi -S% -}}\left\{\frac{1}{\sqrt{t}}\left(\bm{v}(0)-\bm{u}(0)\right)+\int_{0}^{t}\frac{% -(\dot{\bm{v}}(s)-\dot{\bm{u}}(s))}{\sqrt{t-s}}ds\right\},$

-

$\displaystyle\alpha:=\frac{1}{RS},\quad\gamma:=\frac{1}{R}\sqrt{\frac{3}{S}}$

-

$d:=\xi_{n+1}-\xi_{n}=\frac{1}{N}$

-

$\bm{q}(0,t_{0})={(0.1,0)}^{T}$

-

$\displaystyle\frac{32d^{4}}{120}b_{2}$

-

$\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}% -\approx\frac{1}{2}\left[\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_% -{x_{\nicefrac{{1}}{{2}}}}+\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert% -}_{x_{-\nicefrac{{1}}{{2}}}}\right].$

-

$\displaystyle\frac{16d^{3}}{24}b_{2}$

-

$\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n}}=% -\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}\cdot\frac{1}{c}% -\left(1-\frac{n}{N}\right),$

-

$\displaystyle M_{1}\;D_{x}\bm{q}(t)$

- - - diff --git a/htmls/output_mathjax_example_10032.html b/htmls/output_mathjax_example_10032.html deleted file mode 100644 index 029ad45da135862a4e4323f62a2287baf064422c..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10032.html +++ /dev/null @@ -1,168 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle-\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{2}% -\gamma\cos(\lambda t)}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})% -}dk.$

-

$\displaystyle q_{0}:$

-

$\bm{q}(x,t)$

-

$\displaystyle\frac{4d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$

-

$\displaystyle\dot{\bm{y}}(t)$

-

$\displaystyle\frac{243d^{4}}{120}b_{3}.$

-

$M_{1}^{-1}$

-

$A_{s}\in\mathbb{R}^{N\times N}$

-

$(x_{n})_{n=0,\ldots,N-1}$

-

$\displaystyle\frac{d}{2}b_{1}$

-

$\bm{y}={(0.02,0.01)}^{T}$

-

$\displaystyle V_{1}(\dot{\bm{q}}_{0}(t))=\frac{2c}{3\gamma}\begin{bmatrix}\dot% -{q}^{(1)}_{0}(t)\\ -\dot{q}^{(2)}_{0}(t)\\ -0\\ -0\\ -\vdots\\ -0\end{bmatrix}=\frac{2c}{3\gamma}\underbrace{\begin{bmatrix}1&0&0&\dots\\ -0&1&0&\ddots\\ -0&0&0&\ddots\\ -\vdots&\ddots&\ddots&\ddots\end{bmatrix}}_{=:\mathbb{P}}\dot{\bm{q}}(t)=\frac{% -2c}{3\gamma}\;\mathbb{P}\;\dot{\bm{q}}(t).$

-

$\bm{y}(t_{0})={(0.02,0.01)}^{T}$

-

$\displaystyle\;a_{24}=\frac{\gamma}{\zeta_{0}(2+\gamma\psi_{0})}$

-

$q(\xi(x),t)$

-

$\displaystyle a_{i,i-2}=$

-

$\bm{q}(0,t_{0})={(0,0)}^{T}$

-

$\displaystyle(\partial_{\xi}q)_{0}+(\partial_{\xi}q)_{1}=-\frac{2c}{3\gamma}% -\left(\alpha q_{0}-\frac{\gamma}{c}(\partial_{\xi}q)_{0})\right)+\frac{1}{d}% -\left(\frac{12\alpha dc-17\gamma}{18\gamma}\right)q_{0}+\frac{1}{2d}q_{1}+% -\frac{1}{2d}q_{2}-\frac{1}{18d}q_{3},$

-

$\dot{\bm{q}}(t)=D_{x}^{2}\bm{q}(t)$

-

$\displaystyle\bm{q}_{0}(t):=\begin{bmatrix}q_{0}^{(1)}(t)&q_{0}^{(2)}(t)\end{% -bmatrix}^{T}.$

-

$\displaystyle:=\Psi^{-1}B_{2}M_{1}^{-1}K_{1}(\bm{q}_{0}(t),\bm{y}(t),t),$

-

$q^{(2)}(0,t_{0})>0$

-

$\xi_{n}:=\frac{n}{N}$

-

$\bm{y}(t_{0})={(1,0)}^{T}$

-

$\displaystyle a_{0}(\partial_{\xi}q)_{0}+a_{1}(\partial_{\xi}q)_{1}=\hat{b}_{0% -}\left(\alpha q_{0}-\gamma(\partial_{x}q)_{0}\right)+\frac{1}{d}\left\{b_{0}q_% -{0}+b_{1}q_{1}+b_{2}q_{2}+b_{3}q_{3}\right\},$

-

$\displaystyle\frac{8d^{2}}{6}b_{2}$

-

$\displaystyle+\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{2}% -\gamma e^{-k^{2}t}}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})}dk+$

-

$q^{(3)}_{0}(t)$

-

$\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n+% -\nicefrac{{1}}{{2}}}}\approx\;\frac{q_{n+1}(t)-q_{n}(t)}{2\zeta_{n}},$

-

$\bm{\omega}^{k+1}$

-

$\displaystyle\bm{q}_{t}(0,t)+\alpha\bm{q}(0,t)-\gamma\bm{q}_{x}(0,t)$

-

$\displaystyle\bm{\tilde{\eta}}^{k+1}=\left(\mathbb{I}+\Delta t\;A\right)\bm{% -\eta}^{k}+\Delta t\;\bm{\omega}^{k}$

-

$\displaystyle q_{1}:$

-

$q^{(3)}_{1}(t)$

-

$\bm{y}(t_{0})={(0,0)}^{T}$

-

$\displaystyle 2b_{2}$

-

$R=7/9$

-

$\bm{q}(t_{0})={(0,0)}^{T}$

-

$\displaystyle\approx B_{2}M_{1}^{-1}B_{1}\bm{q}(t)+B_{2}M_{1}^{-1}\;K_{1}(\bm{% -q}_{0}(t),\bm{y}(t),t).$

-

$\displaystyle\frac{d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$

-

$\displaystyle x_{n}:=x(\xi_{n})=-c\ln(1-\xi_{n}),$

-

$\rho_{p}\to 0$

-

$\displaystyle\;a_{i+1,i-1}=\frac{1}{2\psi_{\frac{i-1}{2}}\zeta_{\frac{i-3}{2}}},$

-

$R=4/3$

-

$\displaystyle\frac{4d}{2}b_{2}$

-

$\displaystyle q_{0}$

-

$V_{1}(\dot{\bm{q}}_{0}(t))$

-

$\displaystyle\;y^{(1)}_{0}+u^{(1)}t+2\frac{v_{0}^{(1)}}{\pi}\int_{0}^{\infty}% -\frac{\gamma(1-e^{-k^{2}t})}{(\alpha-k^{2})^{2}+(k\gamma)^{2}}dk,$

-

$\displaystyle\frac{d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$

-

$\bm{q}(0,t)$

-

$\displaystyle\frac{d^{5}}{120}(\partial^{5}_{\xi}q)_{0},$

-

$\displaystyle=\bm{q}(0,t)+\bm{u}(\bm{y}(t),t),$

-

$S=0.01$

-

$\displaystyle\iff\xi(x)=1-e^{-\nicefrac{{x}}{{c}}}$

-

$S\in\{0.01,0.1,0.5,1,2,4\}$

-

$\displaystyle 3d(\partial_{\xi}q)_{0}$

-

$\displaystyle\approx\frac{3}{d}\left(q_{n+1}(t)-q_{n-1}(t)\right)$

-

$\displaystyle(\partial_{\xi}q)_{0}$

-

$\displaystyle(\partial^{2}_{\xi}q)_{0}:\quad$

-

$i=2m+1$

-

$\displaystyle M_{1}:=\begin{bmatrix}c&0&\frac{Nc}{N-1}&0&0&0&0&\dots&0\\ -0&c&0&\frac{Nc}{N-1}&0&\ddots&\ddots&\ddots&0\\ -c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&0\\ -0&c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&0\\ -\vdots&0&\frac{Nc}{N-1}&0&\frac{4Nc}{N-2}&0&\frac{Nc}{N-3}&\ddots&0\\ -\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ -\vdots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}&0\\ -\vdots&\ddots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}\\ -0&\ddots&\ddots&\ddots&\ddots&\frac{Nc}{3}&0&\frac{4Nc}{2}&0\\ -0&0&\dots&\dots&\dots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}\end{bmatrix},$

-

$u^{(1)}=0.05$

-

$\bm{\omega}^{k}:=\bm{\omega}(\bm{q}_{0}(t^{k}),\bm{y}(t^{k}),t^{k})$

-

$115\times 86$

-

$\displaystyle\bm{q}(x,0)$

-

$\displaystyle=A_{s}\bm{q}(t)+\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$

-

$\displaystyle 2d(\partial_{\xi}q)_{0}$

-

$\displaystyle-\frac{2c}{3\gamma}\left(f(\bm{q}_{0}(t),\bm{y}(t),t)-\frac{% -\partial q(\xi(x),t)}{\partial t}\biggr{\rvert}_{x_{0}}\right).$

-

$\displaystyle\alpha\hat{b}_{0}$

-

$\displaystyle\bm{u}=\left\lVert\bm{y}(t)\right\lVert\omega\;\bm{e}_{\theta}=% -\omega\begin{bmatrix}-y^{(2)}\\ -y^{(1)}\end{bmatrix},$

-

$\displaystyle=\bm{v}_{0}-\bm{u}_{0},$

-

$\displaystyle B_{2}:=\frac{1}{d}\begin{bmatrix}-\nicefrac{{17}}{{6}}&0&% -\nicefrac{{3}}{{2}}&0&\nicefrac{{3}}{{2}}&0&-\nicefrac{{1}}{{6}}&0&\dots&0\\ -0&-\nicefrac{{17}}{{6}}&0&\nicefrac{{3}}{{2}}&0&\nicefrac{{3}}{{2}}&0&-% -\nicefrac{{1}}{{6}}&\dots&0\\ --3&0&0&0&3&0&0&0&\dots&0\\ -0&-3&0&0&0&3&0&\ddots&\ddots&0\\ -0&0&-3&0&0&0&3&0&\ddots&\vdots\\ -\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ -\vdots&\ddots&\ddots&\ddots&\ddots&0&-3&0&0&0\\ -0&\dots&\dots&\dots&\dots&\dots&0&-3&0&0\end{bmatrix},$

-

$\bm{y}(t_{0})=(1,0)^{T}$

-

$1056\text{\,}\mathrm{s}$

-

$\displaystyle M_{2}D_{x}^{2}\bm{q}(t)\approx B_{2}\left(M_{1}^{-1}\;B_{1}\bm{q% -}(t)+M_{1}^{-1}\;K_{1}(\bm{q}_{0}(t),\bm{y}(t),t)+M_{1}^{-1}\;V_{1}(\dot{\bm{q% -}}_{0}(t))\right).$

-

$R=7/3$

-

$\displaystyle\frac{9d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$

-

$\bm{f}\left(\bm{q}(0,t),\bm{y}(t),t\right):=\left(\frac{1}{R}-1\right)\frac{D% -\bm{u}}{Dt}-\bm{q}(0,t)\cdot\nabla_{y}\bm{u}(\bm{y}(t),t)$

-

$R<1$

-

$\displaystyle\bm{u}(t)=\begin{bmatrix}u^{(1)}\\ -\sin(\lambda t)\end{bmatrix},$

-

$\displaystyle\frac{d^{4}}{24}a_{1}$

-

$\displaystyle\bm{y}(0)$

-

$\displaystyle\frac{32d^{5}}{120}(\partial^{5}_{\xi}q)_{0},$

-

$S=0.5$

-

$\displaystyle\frac{1}{d}b_{0}$

-

$\displaystyle\frac{\partial\xi(x)}{\partial x}\biggr{\rvert}_{x_{n}}$

-

$\displaystyle(\partial^{3}_{\xi}q)_{0}:\quad$

-

$\displaystyle d(\partial_{\xi}q)_{0}$

-

$\frac{d^{4}}{60}(\partial^{5}_{\xi}q)_{0}$

-

$\displaystyle=\bm{f}(\bm{q}(0,t),\bm{y}(t),t),$

-

$\displaystyle R\dot{\bm{v}}(t)$

-

$f^{(2)}(s)=\left(\frac{1}{R}-1\right)\left(\lambda\cos(\lambda s)\right)$

-

$\displaystyle:=\Psi^{-1}B_{2}M_{1}^{-1}B_{1},$

-

$\displaystyle\frac{27d^{2}}{6}b_{3}$

-

$\displaystyle\bm{q}_{t}(x,t)$

-

$\displaystyle a_{12}=$

-

$\displaystyle\frac{27d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$

-

$\displaystyle y^{(2)}(t)=$

-

$\bm{q}(0,t_{0})={(0.00052558,-0.00064947)}^{T}$

-

$\displaystyle x(\xi)=-c\ln(1-\xi)$

- - - diff --git a/htmls/output_mathjax_example_10033.html b/htmls/output_mathjax_example_10033.html deleted file mode 100644 index 084df4ddebff96abc64bbcca54597d5dcd2724ac..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10033.html +++ /dev/null @@ -1,162 +0,0 @@ - - - - MathJax Example - - - - -

$10\text{\,}\mathrm{ms}$

-

$\displaystyle a_{0}(\partial_{\xi}q)_{0}+a_{1}(\partial_{\xi}q)_{1}=\hat{b}_{0% -}\left(\alpha q_{0}-\frac{\gamma}{c}(\partial_{\xi}q)_{0}\right)+\frac{1}{d}% -\left\{b_{0}q_{0}+b_{1}q_{1}+b_{2}q_{2}+b_{3}q_{3}\right\}.$

-

$\displaystyle\frac{81d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$

-

$\displaystyle(\partial^{5}_{\xi}q)_{0}:\quad$

-

$\displaystyle da_{1}$

-

$\displaystyle q_{2}:$

-

$\displaystyle\psi_{n}:=x_{n+\nicefrac{{1}}{{2}}}-x_{n-\nicefrac{{1}}{{2}}},% -\quad\zeta_{n}:=x_{n+\nicefrac{{3}}{{4}}}-x_{n+\nicefrac{{1}}{{4}}}$

-

$R=1/3$

-

$\beta=2/3$

-

$\displaystyle(\partial_{\xi}q)_{0}:$

-

$\displaystyle\frac{9d}{2}b_{3}$

-

$\displaystyle\frac{1}{d}b_{2}$

-

$\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$

-

$\displaystyle a_{i,i+2}=$

-

$\displaystyle c\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}+% -\left(\frac{3Nc}{N-1}\right)\frac{\partial q(\xi(x),t)}{\partial x}\biggr{% -\rvert}_{x_{1}}\approx\frac{1}{d}\left(-\frac{17}{6}q_{0}(t)+\frac{3}{2}q_{1}(% -t)+\frac{3}{2}q_{2}(t)-\frac{1}{6}q_{3}(t)\right).$

-

$\displaystyle\approx B_{1}\bm{q}(t)+K_{1}(\bm{q}_{0}(t),\bm{y}(t),t)+V_{1}(% -\dot{\bm{q}}_{0}(t)),$

-

$R\neq 1$

-

$D_{x}\bm{q}(t)$

-

$\displaystyle\frac{\gamma}{c}\hat{b}_{0}$

-

$\displaystyle\frac{d^{4}}{24}(\partial^{5}_{\xi}q)_{0},$

-

$\displaystyle x>0,t\in(0,T],$

-

$\displaystyle\;a_{i+1,i+3}=\frac{1}{2\psi_{\frac{i-1}{2}}\zeta_{\frac{i-1}{2}}},$

-

$\displaystyle=\bm{q}_{xx}(x,t),$

-

$\bm{q}(0,t_{0})=(0,0.1)$

-

$n\in\{0,1,\dots,N-1\}$

-

$\bm{q}_{x}(0,t)$

-

$\displaystyle\;a_{22}=-\frac{\gamma+2\alpha\zeta_{0}}{\zeta_{0}(2+\gamma\psi_{% -0})},$

-

$\displaystyle=\frac{1}{c}e^{-\nicefrac{{x_{n}}}{{c}}}=\frac{1}{c}\left(1-\xi_{% -n}\right)=\frac{1}{c}\left(1-\frac{n}{N}\right).$

-

$\bm{y}={(0,0)}^{T}$

-

$\displaystyle(\partial_{\xi}q)_{1}:$

-

$\displaystyle+\frac{1}{d}\left(\frac{12\alpha dc-17\gamma}{18\gamma}\right)q_{% -0}+\frac{1}{2d}q_{1}+\frac{1}{2d}q_{2}-\frac{1}{18d}q_{3}.$

-

$\displaystyle\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n-1}}+% -\frac{4\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}+\frac{\partial -q% -(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n+1}}\approx\frac{3}{d}\left(q_{n+1% -}(t)-q_{n-1}(t)\right)$

-

$\displaystyle\left(\frac{12\alpha dc-17\gamma}{18\gamma d}\right)q_{0}(t)+% -\frac{1}{2d}q_{1}(t)+$

-

$\displaystyle K_{1}:=\begin{bmatrix}-\frac{2c}{3\gamma}f^{(1)}(\bm{q}_{0}(t),% -\bm{y}_{0}(t),t)\\ --\frac{2c}{3\gamma}f^{(2)}(\bm{q}_{0}(t),\bm{y}_{0}(t),t)\\ -0\\ -0\\ -\vdots\\ -0\end{bmatrix}\quad\text{and}\quad V_{1}:=\frac{2c}{3\gamma}\begin{bmatrix}% -\frac{\partial q^{(1)}(\xi(x),t)}{\partial t}\biggr{\rvert}_{0}\\ -\frac{\partial q^{(2)}(\xi(x),t)}{\partial t}\biggr{\rvert}_{0}\\ -0\\ -0\\ -\vdots\\ -0\end{bmatrix}.$

-

$q^{(2)}(x,t_{0})=0$

-

$R=\nicefrac{{4}}{{3}}$

-

$50\text{\,}\mathrm{Hz}$

-

$\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n}}=% -\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}\cdot\frac{d\xi(% -x)}{dx}\biggr{\rvert}_{x_{n}}.$

-

$\displaystyle\hat{b}_{0}=-\frac{2c}{3\gamma}\quad b_{0}=\frac{12\alpha dc-17% -\gamma}{18\gamma}\quad b_{1}=\frac{1}{2}\quad b_{2}=\frac{1}{2}\quad b_{3}=-% -\frac{1}{18},$

-

$\displaystyle a_{i,i}=$

-

$\displaystyle\frac{d^{2}}{6}b_{1}$

-

$(a_{i,j})_{1\leq i\leq 2N,\;1\leq j\leq 2N}$

-

$\displaystyle M_{2}\;D_{x}^{2}\bm{q}(t)-B_{2}M_{1}^{-1}V_{1}(\dot{\bm{q}}_{0}(% -t))$

-

$\displaystyle\frac{8d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$

-

$\displaystyle=\bm{v}(t),$

-

$\bm{\eta}^{k}:=\bm{\eta}(t^{k})$

-

$\displaystyle(\partial_{\xi}q)_{0},$

-

$2\text{\,}\mathrm{mm}$

-

$t\in[0,10]$

-

$\displaystyle y^{(1)}(t)=$

-

$\displaystyle\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{0}}+3% -\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{1}}\approx\frac{1}{% -d}\left(-\frac{17}{6}q_{0}(t)+\frac{3}{2}q_{1}(t)+\frac{3}{2}q_{2}(t)-\frac{1}% -{6}q_{3}(t)\right).$

-

$\displaystyle q_{0},$

-

$\displaystyle\frac{1}{d}b_{1}$

-

$\displaystyle\frac{d^{3}}{6}(\partial^{4}_{\xi}q)_{0}$

-

$\displaystyle-\frac{2c}{3\gamma}\left(f(q_{0},x_{0},t)-(\partial_{t}q)_{0}\right)$

-

$\displaystyle c(\partial_{x}q)_{0}+\frac{Nc}{N-1}(\partial_{x}q)_{1}=$

-

$(\bar{1}+\bar{3})(\bar{2}+\bar{3})(1+2+3)$

-

${{}^{2,1}}$

-

${{}^{5,1}}$

-

${{}^{3,1}}$

-

$(i,~{}j)$

-

$CS_{1}\diamond CS_{2}=\{u\cup v|u\in CS_{1},v\in CS_{2}\}$

-

$(N,~{}\preccurlyeq)$

-

${}^{1,\text{\Letter}}$

-

${}^{\text{\Letter}}$

-

$checking\_logic\_block$

-

$\Phi(n)$

-

$(\bar{1}+\bar{4})(\bar{2}+\bar{4})(1+2+4)$

-

$\Phi(n)=\left\{\begin{aligned} \{\{n\}\}&:n\in PIs\\ -\{\{n\}\}\cup\Phi(n_{1})\diamond\Phi(n_{2})&:otherwise\end{aligned}\right\}.$

-

$CS_{1}\diamond CS_{2}$

-

$ge\in N$

-

$\{|u\cup v|\leq k\}$

-

$(\bar{1}+4+5)(\bar{1}+\bar{3}+5)(\bar{1}+\bar{2}+5)(2+3+\bar{4}+\bar{5})(1+% -\bar{5})$

-

$\mathbf{F_{q}}$

-

$\mathbf{F_{q}}\in\mathbb{R}^{H/8\times W/8\times C}$

-

$\mathbf{F_{s,s}}\in\mathbb{R}^{H/8\times W/8\times C}$

-

$\mathbf{P^{\prime}_{r}}=(1+\alpha\mathbf{W}\otimes\mathbf{P_{r}})$

-

$\mathbf{P}\in\mathbb{R}^{1\times 1\times C}$

-

$\mathbf{P^{\prime\prime}}$

-

$\mathbf{C_{train}}$

-

$\mathbf{M_{q}}$

-

$\mathbf{P_{r}}\in\mathbb{R}^{1\times 1\times C}$

-

$\mathbf{F_{r}}$

-

$\mathbf{M_{s}^{k}}$

-

$\mathbf{I_{s}^{k}}$

-

$\mathbf{M_{s}}$

-

$\mathbf{Q_{i}}=\{(\mathbf{I_{q}},\mathbf{M_{q}})\}_{i}$

-

$\mathbf{S_{i}}=\{(\mathbf{I_{s}^{k}},\mathbf{M_{s}^{k}}),k\in\{1,\dots,K\}\}_{i}$

-

$\mathbf{\hat{y}}=\mathrm{softmax}(\mathrm{cosine}(\mathbf{P^{\prime\prime}},% -\mathbf{F^{\prime}_{q}}))$

-

$L=L_{seg}+\lambda_{1}L_{s}+\lambda_{2}L_{q}$

-

$\mathbf{P^{\prime}}=(1+\alpha\mathbf{W}\otimes\mathbf{P})$

-

$\mathbf{P^{\prime\prime}}\in\mathbb{R}^{1\times 1\times C}$

-

$\mathbf{P_{r}}$

-

$\mathbf{F^{\prime}_{r}}$

-

$L_{s}=\mathrm{BCE}(\mathrm{cosine}(\mathbf{P^{\prime}},\mathbf{F_{s,s}}),% -\mathbf{M_{s}})\\ -+\mathrm{BCE}(\mathrm{cosine}(\mathbf{P^{\prime}_{r}},\mathbf{F_{s,r}}),% -\mathbf{M_{s}})$

-

$\mathbf{I_{q}}$

-

$\mathrm{MAP}$

-

$L_{q}=\mathrm{BCE}(\mathrm{cosine}(\mathrm{MAP}(\mathbf{F^{\prime}_{q}}),% -\mathbf{F^{\prime}_{q}}),\mathbf{M_{q}})$

-

$\mathbf{F^{\prime}_{q}}$

-

$\mathbf{F_{s,r}}\in\mathbb{R}^{H/8\times W/8\times C}$

- - - diff --git a/htmls/output_mathjax_example_10034.html b/htmls/output_mathjax_example_10034.html deleted file mode 100644 index 0098fa8acb67202c154fe817751012a0cf84224f..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10034.html +++ /dev/null @@ -1,142 +0,0 @@ - - - - MathJax Example - - - - -

$\mathbf{C_{test}}$

-

$L_{seg}=\mathrm{BCE}(\mathbf{\hat{y}},\mathbf{M_{q}})$

-

$\mathbf{X}\in\mathbb{R}^{1\times 1\times 2C}$

-

$\mathbf{P^{\prime}}\in\mathbb{R}^{1\times 1\times C}$

-

$\mathbf{W}=\mathrm{sigmoid}(f_{2}(\sigma(f_{1}(\mathbf{X}))))$

-

$\mathbf{F_{r}}\in\mathbb{R}^{H/8\times W/8\times C}$

-

$\mathbf{F^{\prime}_{q}}\in\mathbb{R}^{H/4\times W/4\times C}$

-

$\mathbf{\hat{y}}\in\mathbb{R}^{H\times W\times 1}$

-

$e_{i}=\{\mathbf{S_{i}},\mathbf{Q_{i}}\}$

-

${\sf msg^{*}}$

-

${\sf sk_{sanit}}$

-

${\sf AD}_{i}({\sf MODIFY})=1\}$

-

$({\sf msg},\sigma)$

-

${\sf Fixed_{AD}(msg^{*})}={\sf Fixed_{AD}}({\sf msg}_{i})$

-

$\{\mathcal{S}_{1},\mathcal{F},\mathcal{S}_{2}\}$

-

${\sf msg^{*},\sigma^{*},pk^{*}_{sig},pk_{sanit}}$

-

$({\sf sk_{sign},pk_{sign}})$

-

$\sigma_{2}^{{}^{\prime}}$

-

$({\sf msg_{1}},{\sf MODIFY}_{1})$

-

${\sf sec_{k}}$

-

$({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}^{{\sf -Signature% -(\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$

-

${\sf msg_{0}=\mathcal{H}(0||{\sf msg_{fix}}||{AD}||{\sf pk_{sanit}})}$

-

$\displaystyle\mbox{or }{\sf msg^{*}\notin\{MODIFY(msg_{i})\;|\;MODIFY\mbox{ % -with }}$

-

$r_{(1)}\left(\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=\dots=r_{(k)}\left% -(\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=0.$

-

$({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})=(0,{\sf msg}^{*}_{{\sf fixed% -},i},{\sf AD}^{*}_{i},{\sf pk}^{*}_{{\sf sanit},i})$

-

$({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})$

-

$\{\mathcal{Q},\mathcal{X},\mathcal{Y}\}$

-

$\sigma^{*}=(\sigma_{1}^{*},\sigma_{2}^{*},{\sf AD}^{*})$

-

$({\sf msg^{*}_{fixed}},{\sf AD^{*},pk_{sign},pk^{*}_{sanit},\sigma_{1}^{*}})$

-

$({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit}^% -{{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$

-

${\sf pk_{sanit}}$

-

${\sf x_{0}}=\mathcal{S}_{1}^{-1}({\sf y})\in\mathbb{F}_{q}^{m},{\sf x_{1}}=% -\mathcal{F}_{1}^{-1}({\sf x_{0}})\in\mathbb{F}_{q}^{n}$

-

$\mathcal{H}:\{0,1\}^{*}\rightarrow\mathbb{F}^{m}$

-

$\alpha_{1}=\mathcal{S}^{-1}({\sf msg_{1}}),\beta_{1}=\mathcal{F}^{-1}(\alpha_{% -1})$

-

$\displaystyle\mbox{and }{\sf Judge({\sf msg^{*},\sigma^{*},pk_{sign},pk*_{% -sanit}})=Sig}$

-

$\forall i=1,2,\ldots,\Delta$

-

$({\sf pk^{*}_{sign},{\sf msg^{*}},\sigma^{*}})$

-

${\sf msg}\in\{0,1\}^{*}$

-

${\sf msg_{1}=\mathcal{H}(1||msg||pk_{sanit}||pk_{sign})}$

-

$(n=160,m=64)$

-

$(b=0)$

-

${\sf msg^{*}_{fixed}}={\sf FIXED_{AD^{*}}(msg^{*})}$

-

${\sf Fixed_{AD}}$

-

$\mathcal{X}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{m}$

-

$\mathcal{S}_{2}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$

-

${\sf msg^{\prime}}$

-

${\sf msg_{2}=\mathcal{H}(1||msg^{\prime}||pk_{sanit}||pk_{sign})}$

-

$\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}=1]$

-

$({\sf msg}_{i},{\sf AD}_{i},{\sf pk_{sign}},{\sf pk_{sanit}}_{i})$

-

$j=q+1,\dots,r$

-

$\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Sanitizer\mbox{-}Acc{\mathcal{G}}^{% -SSS}}=1]$

-

${\sf y}=\mathcal{P}({\sf x})$

-

${\sf(msg_{0},{\sf MODIFY}_{0}),(msg_{1},{\sf MODIFY}_{1})}$

-

${\sf msg^{\prime}}\in\{{\sf MODIFY(msg)}\;|\;{\sf MODIFY}\mbox{ with }{\sf AD(% -MODIFY)}=1\}$

-

$\displaystyle{\sf msg^{\prime}}\leftarrow{\sf MODIFY(msg)}$

-

${\sf(pk_{sanit},sk_{sanit})}\leftarrow{\sf KGen\mbox{-}Sanit(1^{\kappa})}$

-

$a\leftarrow\mathcal{G}^{{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)},{\sf -Sanitization% -(\cdot,\cdot,sk_{sanit},\cdot)},{\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{% -sanit},b)}}({\sf pk_{sign},pk_{sanit}})$

-

${\sf(msg^{\prime}_{j},\sigma^{\prime}_{j})\leftarrow Sanitization(msg_{j,b},% -MODIFY_{j,b},\sigma_{j,b},pk_{sign},sk_{sanit})}$

-

$({\sf msg_{j,0}},{\sf Modify_{j,0}}),({\sf msg_{j,1}},{\sf Modify_{j,0}},)$

-

${\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{sanit},b)}$

-

${\sf msg_{fixed}}\leftarrow{\sf FIXED}_{{\sf AD}_{i}}({\sf msg}_{i})$

-

$(\cdot,{\sf sk_{sign}},\cdot,\cdot)$

-

$\mathcal{R}=\left(r_{(1)}(\delta,\dots,\delta_{n}),\dots,r_{(k)}(\delta_{1},% -\dots,\delta_{n})\right)$

-

${\sf msg,{\sf MODIFY},\sigma,pk_{sign},}\\ -{\sf sk_{sanit}}$

-

$\mathcal{G}_{\sf sanit}$

-

$\sigma_{j,b}\leftarrow{\sf Signature({\sf msg_{j,b},sk_{sign},pk_{sanit},AD_{j% -}})}$

-

${\sf EXP\mbox{-}Unforgeability_{\mathcal{G}}^{SSS}}$

-

$\mathcal{R}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg_{2}}$

-

$(\bar{\delta}_{1},\dots,\bar{\delta}_{n})\in\mathbb{F}_{q}^{n}$

-

$\mathcal{S}_{1}\circ\mathcal{F}\circ\mathcal{S}_{2}$

-

$({\sf msg^{\prime}}_{i},{\sf\sigma^{\prime}}_{i},{\sf pk}_{{\sf sanit},i})$

-

${\sf msg_{FIX}=FIXED_{AD}(msg)}$

-

${\sf(pk_{sign},sk_{sign})}\leftarrow{\sf KGen\mbox{-}Sign(1^{\kappa})}$

-

${\sf pk^{*}_{sign}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sign},i},{\sf msg^{% -\prime}}_{i})$

-

${\sf\sigma_{1}}$

-

${\sf Mul\mbox{-}SAN}$

-

$GF(16)$

-

$1,{\sf msg^{*},\sigma^{*},pk^{*}_{sanit},pk_{sign}}$

-

${\sf msg_{fix}}$

-

${\sf MODIFY}$

-

$(b=1)$

-

$\mathcal{Y}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$

-

${\sf msg,sk_{sign},pk_{sanit},AD}$

-

$\delta(\kappa)$

-

${\sf msg}_{i}$

-

$\mathcal{O}_{\sf LoR}$

-

${\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}$

-

${\sf pub_{k}}$

-

${\sf msg,\sigma,pk_{sign},pk_{sanit}}$

-

${\sf AD}_{j}$

-

$\mathcal{R}=\mathcal{Q}\circ\mathcal{X}\circ\mathcal{Y}:\mathbb{F}_{q}^{n}% -\rightarrow\mathbb{F}_{q}^{m}$

-

${\sf Sig}$

-

$({\sf pk^{*}_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit% -}({\sf pk_{sign}})$

-

$i=1,\dots,\Delta$

-

${\sf AD(MODIFY)}\in\{0,1\}$

-

${\sf(pk_{sanit},sk_{sanit})}\leftarrow$

-

${\sf msg^{*}\notin\{MODIFY(msg)\;|\;MODIFY\mbox{ with }}$

-

$\displaystyle{\sf Verification({\sf msg^{*},\sigma^{*},pk_{sign},pk^{*}_{sanit% -}})=true}$

-

${\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}$

-

${\sf msg_{fix}=Fixed_{AD}({\sf msg})}$

- - - diff --git a/htmls/output_mathjax_example_10035.html b/htmls/output_mathjax_example_10035.html deleted file mode 100644 index f8d2ec355a81ec5f3640d51332c50ee4eab208a2..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10035.html +++ /dev/null @@ -1,140 +0,0 @@ - - - - MathJax Example - - - - -

$\frac{m(n+2)(n+1)}{2}$

-

$\displaystyle{\sf AD}_{i}({\sf MODIFY})=1\}$

-

${\sf pub_{k},sec_{k}\leftarrow Kg(\kappa}$

-

${\sf AD(MODIFY)}=1$

-

${\sf msg^{*},\sigma^{*},pk^{*}_{sign},pk_{sanit}}$

-

$({\sf 1,{msg}^{*},AD^{*},pk^{*}_{sanit},pk_{sign}})$

-

$(\mathbb{F}_{q})$

-

$r_{(i)}\in\mathbb{F}_{q}[\delta_{1},\dots,\delta_{n}]$

-

$\sigma^{\prime}=(\sigma_{1},\sigma^{\prime}_{2},{\sf AD})$

-

$({\sf msg^{*}},\sigma^{*})\leftarrow\mathcal{G}^{{\sf Signature(\cdot,sk_{sign% -},\cdot,\cdot)},{\sf Sanitization(\cdot,\cdot,sk_{sanit},\cdot)}}({\sf pk_{% -sign},pk_{sanit}})$

-

$\in\{0,1\}^{*}$

-

${\sf EXP\mbox{-}Signer\mbox{-}Acc_{\mathcal{G}_{sign}}^{SSS}}$

-

$\mathcal{P}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg}_{1}$

-

${\sf msg^{\prime}\leftarrow MODIFY(msg)}$

-

$({\sf msg}_{j},{\sf MODIFY}_{j},\sigma_{j},{\sf pk}_{{\sf sign},i})$

-

$\mathcal{S}_{1}:\mathbb{F}_{q}^{m}\rightarrow\mathbb{F}_{q}^{m}$

-

$\displaystyle({\sf pk_{sanit}^{*}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i}% -,{\sf msg^{\prime}}_{j})\forall i=1,2,\dots,q$

-

${\sf pk*_{sanit}\neq pk_{{sanit},i}}$

-

$\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}San\mbox{-}Acc_{\mathcal{G}_{sanit}}% -^{SSS}}]$

-

$\sigma^{*}_{2}$

-

${\sf EXP\mbox{-}San\mbox{-}Acc_{\mathcal{G}_{sanit}}^{SSS}}$

-

$i=1,2,\ldots,q$

-

$\mathcal{Q}:\mathbb{F}_{q}^{m}\rightarrow\mathbb{F}_{q}^{m}$

-

$\displaystyle{\sf Verification({\sf msg^{*},\sigma^{*},pk_{sign},pk*_{sanit}})% -=true}$

-

${\sf(msg_{0},}{\sf MODIFY}_{0})$

-

${\sf 0/1\leftarrow Ver(x,pub_{k})}$

-

$y\in\mathbb{F}_{q}^{m}$

-

${\sf sk_{sign}}$

-

$({\sf msg}_{i},{\sf pk}_{{\sf sanit},i},{\sf pk}_{{\sf sign},i})$

-

$\displaystyle\mbox{and }\forall i=1,2,\dots q,\;\;{\sf pk^{*}_{sanit}\neq pk_{% -{sanit},i}}$

-

${\sigma}_{1}$

-

$i=1,2,\dots,q$

-

${\sf FIXED_{AD}}$

-

${\sf msg_{0}}$

-

${\sigma_{i,2}}$

-

$({\sf pk*_{sign},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sign}^{{% -\sf Sanitization(\cdot,\cdot,\cdot,sk_{sign})}}({\sf pk_{sanit}})$

-

${\sf msg^{*},\sigma^{*},pk_{sign},pk*_{sanit}}$

-

${\sf y}$

-

${\sf msg^{\prime}},\sigma^{\prime})\leftarrow$

-

$\sigma^{\prime}_{2}=\mathcal{Y}^{-1}(\beta_{2})$

-

$\sigma_{1}=\mathcal{T}^{-1}(\beta_{0})$

-

$({\sf msg}_{j,0},{\sf MODIFY}_{j,0},{\sf AD}_{j})\equiv({\sf msg}_{j,1},{\sf -MODIFY% -}_{j,1},{\sf AD}_{j})$

-

${\sf pk_{sign}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sign},j},{\sf msg^{\prime}}_% -{j})$

-

${\sf AD(MODIFY)}=\begin{cases}1&\mbox{ if modifications are valid \newline -with respect }\\ -&\mbox{ to }{\sf AD};\\ -0&\text{ otherwise.}\end{cases}$

-

$\sigma=(\sigma_{1},\sigma_{2},{\sf AD})$

-

$(0,{\sf msg_{fix},AD,pk_{sanit}})$

-

$\alpha_{0}=\mathcal{S}^{-1}({\sf msg_{0}}),\beta_{0}=\mathcal{F}^{-1}(\alpha_{% -0})$

-

$\mathcal{B}^{\prime}s$

-

${\sf EXP\mbox{-}Privacy_{\mathcal{G}}^{SSS}}$

-

${\sf pk_{sanit}^{*}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i},{\sf msg^{% -\prime}}_{j})$

-

${\sf AD}_{i}={\sf AD^{*}}$

-

${\sf(pk_{sign},sk_{sign})}\leftarrow$

-

${\sf x\leftarrow Sig(y,sec_{k})}$

-

$({\sf msg}_{i},{\sf AD}_{i},{\sf pk}_{{\sf sanit},i})$

-

$\mathcal{P}{\sf(\sigma_{1})\stackrel{{\scriptstyle?}}{{=}}msg_{0}}$

-

$\mathcal{G}_{\sf signer}$

-

${\sf AD}$

-

${\sf pk_{sign}}$

-

$\{\mathcal{S},\mathcal{F},\mathcal{T}\}$

-

${\sf FIXED_{AD}(msg^{\prime})}\neq{\sf FIXED_{AD}(msg)}$

-

$\sigma_{2}=\mathcal{T}^{-1}(\beta_{1})$

-

$n^{2}+m^{2}+C$

-

${\sf msg_{1}}$

-

$({\sf msg^{*},\sigma^{*},pk^{*}_{sanit}})$

-

$\alpha_{2}=\mathcal{Q}^{-1}({\sf msg_{2}}),\beta_{2}=\mathcal{X}^{-1}(\alpha_{% -2})$

-

$({\sf msg^{\prime}_{j}},\sigma^{\prime}_{j})$

-

${\sf pk*_{sanit}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i},{\sf msg}_{i})$

-

$i\in\{1,\ldots,\Delta\}$

-

$({\sf pk^{*}_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}({\sf pk_{% -sign}})$

-

${\sf MODIFY}({\sf msg}_{i})$

-

${\sf San}$

-

$\mathcal{F}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{m}$

-

$({\sf sk_{sanit},pk_{sanit}})$

-

$(\sf x,y)$

-

${\sf x}$

-

${\sf x}=\mathcal{S}_{2}^{-1}({\sf x_{1}})\in\mathbb{F}_{q}^{n}$

-

$\mathcal{P}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg_{1}}$

-

${\sf msg,{\sf MODIFY},\sigma,pk_{sign},sk_{sanit}}$

-

$j\in\{\Delta+1,\ldots,r\}$

-

$\sigma=(\sigma_{i,1},\sigma_{i,2},{\sf AD}_{i})$

-

$\mathcal{P}=\mathcal{S}\circ\mathcal{F}\circ\mathcal{T}:\mathbb{F}_{q}^{n}% -\rightarrow\mathbb{F}_{q}^{m}$

-

$\hat{p}_{i}\in\mathbb{R}^{T}$

-

$\mathcal{L}=\sum_{i}\ell_{\textrm{FL}}\left(\hat{p}_{i},g_{i}\right)$

-

$\mathbf{A}\in\mathbb{B}^{L\times T}$

-

$g_{i}\in\mathbb{B}^{T}$

-

$\mathcal{C}\setminus c$

-

$\{t^{\prime}_{j}\}$

-

$\hat{\textbf{p}}$

-

$\textbf{p}_{\textbf{cf}}$

-

$F_{avg}=70.09.$

-

$`\backslash w+^{\prime}$

-

$F_{avg}=50.1$

-

$PP({\tilde{p}},q)=b^{H({\tilde{p}},q)}=b^{\mathbb{E}_{\tilde{p}}[log_{b}q]}$

-

$\begin{split}\arg\max_{i}\frac{P(y_{i}|x)}{P(y_{i}|x_{cf})}\approx\arg\max_{i}% -\frac{P(y_{i}|x)}{P(y_{i}|x_{cf})}.\end{split}$

-

$P(y_{i})\text{ or }p_{cf}$

-

$(W\hat{p})_{i}=\frac{\hat{p}_{i}}{(p_{cf})_{i}}=\frac{P(y_{i}|x)}{P(y_{i}|x_{% -cf})}$

-

$F_{avg}$

-

$F_{avg}=\frac{F_{\text{favor}}+F_{\text{against}}}{2}$

-

$\text{argmax}(\textbf{q})$

-

$\arg\max_{i}\prod_{j=1}^{i}\log P(y_{j}\mid x,y_{1,\cdots,j-1})$

- - - diff --git a/htmls/output_mathjax_example_10036.html b/htmls/output_mathjax_example_10036.html deleted file mode 100644 index 8d813908e9ea28169fedd078f68c71b6bc8c581c..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10036.html +++ /dev/null @@ -1,123 +0,0 @@ - - - - MathJax Example - - - - -

$P(y_{i}|x_{cf})$

-

$\frac{P(y_{i}|x)}{P(y_{i})}$

-

$H({\tilde{p}},q)=-\sum_{i=1}^{n}\tilde{p}(x_{i})\log_{b}q(x_{i})$

-

$F_{avg}=72.6$

-

$F_{avg}=82.29$

-

$F_{avg}=84.43$

-

$\langle tweet\rangle$

-

$F_{avg}=71.0$

-

$F_{avg}=84.28$

-

$F=\frac{2\cdot Precision\cdot Recall}{Precision+Recall}$

-

$\text{PMI}(x,y_{i})$

-

$\textbf{W}=\text{diag}(\textbf{p}_{\textbf{cf}})^{-1}$

-

$77.11$

-

$78.43$

-

$\textbf{q}=softmax(\textbf{W}\hat{\textbf{p}}+\textbf{b})$

-

$t+5\Delta$

-

$\mathcal{WJ}$

-

$B^{\prime}_{k}$

-

$\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})$

-

$\mathsf{status}$

-

$v^{\prime}-1\geq v+1$

-

$=\delta$

-

$\max(t_{g},t)+2\Delta$

-

$n-2f^{\prime}$

-

$\mathcal{C}_{v-1}(B_{k-1})$

-

$\mathcal{T}_{v-1}$

-

$\langle\mathsf{fb\text{-}vote},H(B_{h}),v\rangle_{i}$

-

$v^{*}>v$

-

$\mathcal{C}_{v}^{\prime}(B_{k^{\prime}})$

-

$v^{\prime}>v+1$

-

$55.7\%-69.0\%$

-

$\mathcal{C}_{v}(B_{k})$

-

$f^{\prime}=33$

-

$\mathcal{C}_{v^{\prime}-1}(B_{k^{\prime}-1})$

-

$\langle\mathsf{status},v,{\sf lock}_{i}\rangle$

-

$\langle{\sf opt\text{-}propose},B_{k+1},v+1\rangle$

-

$\mathcal{C}_{v}=\mathcal{C}_{v}(B_{k})$

-

$B_{k^{\prime}}=B_{k+1}$

-

$t\geq t_{g}$

-

$\mathcal{T}_{v^{\prime\prime}-1}$

-

$\langle\mathsf{propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B_{h}),v% -^{\prime}\rangle$

-

$675\%$

-

$\mathcal{C}^{f}_{v}(B_{l})$

-

$\mathcal{TC}_{v}$

-

$\mathcal{C}_{v^{\prime}-1}$

-

$\Delta=500ms$

-

$f^{\prime}=0$

-

$\mathcal{C}_{v^{\prime\prime}-1}$

-

${\sf view\text{-}timer}$

-

$v^{\prime}=v+1$

-

$=3\Delta$

-

$\langle\mathsf{commit},H(B_{k}),v\rangle_{i}$

-

$\langle\mathsf{timeout},v-1,{\sf lock}_{i}\rangle_{i}$

-

$\mathcal{C}^{n}_{v}(B_{l})$

-

$\mathcal{C}^{f}_{v}(B_{h})$

-

$5f-1$

-

$448\%$

-

$v^{*}\geq v+1$

-

$v\geq v^{\prime}$

-

$\langle{\sf fb\text{-}propose},B_{k},\mathcal{C}_{v^{\prime}}(B_{h}),\mathcal{% -TC}_{v-1},v\rangle$

-

$t -

$\mathcal{C}_{v^{\prime\prime}}(B_{h})\geq{\sf lock}_{i}$

-

$n-3f^{\prime}$

-

$2f^{\prime}$

-

${\sf timeout\_view}_{i} -

$\geq 2\delta$

-

$\langle\mathsf{propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B_{k^{% -\prime\prime}}),v^{\prime}\rangle$

-

$174\%$

-

$\mathsf{fb\text{-}vote}$

-

$\langle\mathsf{propose},B_{h+1},\mathcal{C}_{v-1}(B_{h}),v\rangle$

-

$=(f+1)\delta$

-

$\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})\geq{\sf lock}_{i}$

-

$=\Lambda+2\rho$

-

${\sf lock}_{i}>\mathcal{C}_{v-1}(B_{h})$

-

$\mathcal{C}_{v+1}(B_{k^{\prime}})$

-

$\mathcal{WM}$

-

$\langle\mathsf{vote},H(B_{k}),v\rangle_{i}$

-

$\langle\mathsf{opt\text{-}vote},H(B_{k}),v\rangle_{i}$

-

$B^{\prime}_{k^{\prime}}$

-

$P_{j}\neq P_{i}$

-

${\sf view\text{-}timer}_{i}$

-

$v^{\prime}+1>v$

-

$B_{l}=B_{k}$

-

$t+\lambda$

-

$,P_{i}$

-

$C_{v}(B_{k})$

-

${\sf timeout\_view}_{i}\geq v$

-

$\mathcal{C}_{v}(B_{h})$

-

$\mathcal{C}_{v}(B_{l})$

-

$=2\Lambda+\rho$

-

$L_{v+1}$

-

$\langle{\sf opt\text{-}propose},B_{k},v\rangle$

-

$B_{k^{\prime\prime}}$

-

$\mathcal{TC}_{v+1}$

-

$43\%\text{--}54\%$

-

$214\%-230\%$

-

$(f+1)\delta$

-

${\sf lock}_{i}<\mathcal{C}_{v}(B_{k})$

-

$\mathsf{opt\text{-}vote}$

-

$\mathcal{C}_{v+1}$

- - - diff --git a/htmls/output_mathjax_example_10037.html b/htmls/output_mathjax_example_10037.html deleted file mode 100644 index 7ad1a913cde9eee4e807ca08c21832b4d75c9e08..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10037.html +++ /dev/null @@ -1,125 +0,0 @@ - - - - MathJax Example - - - - -

$\mathcal{C}_{v^{\prime}}(B_{l})$

-

$\lx@sectionsign\ref{sec:commit-moonshot}$

-

$t_{g}+2\Delta<\max(t_{g},t)+3\Delta$

-

$\mathcal{T}_{v^{*}}$

-

$B_{k^{\prime}}$

-

$7\delta^{*}$

-

$702\%$

-

$\langle\mathsf{propose},B_{h},\mathcal{C}_{v^{\prime}}(B_{h-1}),v\rangle$

-

${\sf lock}_{i}$

-

$\mathsf{timeout}_{v}$

-

$\mathcal{TC}_{v-1}$

-

$\mathcal{T}_{v^{\prime}}$

-

$\mathcal{C}_{v}\leq\mathcal{C}_{v^{\prime}}$

-

$B_{k}\neq B^{\prime}_{k}$

-

$\max(t_{g},t)+3\Delta$

-

$\langle\mathsf{propose},B_{l},\mathcal{C}_{v^{\prime}}(B_{h}),v+1\rangle$

-

$\mathcal{C}_{v+1}(B_{k+1})$

-

$\mathcal{C}_{v^{\prime}}(B_{h})=\mathcal{C}_{v}(B_{k})$

-

$\mathsf{vote}$

-

$t+4\Delta$

-

$\lceil\frac{n+f+1}{2}\rceil$

-

$t_{g}+3\Delta=\max(t_{g},t)+3\Delta$

-

$\langle\mathsf{commit},H(B_{k}),v\rangle_{*}$

-

$\mathcal{C}_{v-1}(B_{h})$

-

$\mathcal{T}_{v^{\prime}-1}$

-

$\delta\leq\Delta$

-

$\rho<\Lambda$

-

$v^{\prime\prime}\geq v$

-

$\mathcal{C}^{n}_{v+1}(B_{k^{\prime}})$

-

$\mathcal{C}^{o}_{v}(B_{h})$

-

$B_{k^{\prime}}=B_{k}$

-

$\mathcal{C}_{v^{*}}(B_{k^{*}})$

-

$p\leq 1.8$

-

$\mathcal{T}_{v^{\prime\prime}}$

-

$\langle{\sf fb\text{-}propose},B_{h},\mathcal{C}_{v^{\prime}}(B_{h-1}),% -\mathcal{TC}_{v-1},v\rangle$

-

$\mathcal{C}_{v^{\prime\prime}}$

-

$\langle\mathsf{timeout},v,{\sf lock}_{i}\rangle_{i}$

-

$3f^{\prime}$

-

$v\leq v^{\prime\prime} -

${\sf timeout\_view} -

$v^{\prime\prime} -

$\langle\mathsf{timeout},v\rangle_{i}$

-

$\lx@sectionsign\ref{sec:moonshot_v2}$

-

${\color[rgb]{0.0078125,0.38671875,0.46875}\definecolor[named]{pgfstrokecolor}{% -rgb}{0.0078125,0.38671875,0.46875}\pgfsys@color@rgb@stroke{0.0078125}{0.386718% -75}{0.46875}\pgfsys@color@rgb@fill{0.0078125}{0.38671875}{0.46875}\checkmark}$

-

$\langle{\sf fb\text{-}propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B% -_{k^{\prime\prime}}),\mathcal{TC}_{v^{\prime}-1},v^{\prime}\rangle$

-

$\mathcal{C}_{v-1}(B_{h-1})$

-

$\mathcal{C}_{v^{\prime}}(B_{h})$

-

$C_{v-1}$

-

$\lx@sectionsign\ref{sec:moonshot_v1}$

-

$2f+1th$

-

$\max(t_{g},t)+\Delta$

-

$70.3\%-74.4\%$

-

$\langle\mathsf{propose},B_{k},\mathcal{C}_{v-1}(B_{k-1}),v\rangle$

-

$\mathcal{TC}_{v^{\prime\prime}-1}$

-

$v\leq v^{*} -

$\mathcal{C}_{v-1}$

-

${\sf lock}_{i}\leq\mathcal{C}_{v^{\prime}}(B_{h})$

-

$\mathcal{C}^{o}_{v}(B_{k})$

-

$\langle\mathsf{propose},B_{l},\mathcal{C}_{v^{\prime}}(B_{h}),v\rangle$

-

$t_{g}+\Delta$

-

$p\leq 9$

-

$7\Delta$

-

$=3\delta$

-

$\geq 4\delta$

-

$\mathcal{C}^{n}_{v}(B_{h})$

-

$H(B_{k-1})$

-

$v^{\prime\prime}>v$

-

$\mathcal{C}_{v^{\prime}}(B_{h-1})$

-

$\langle\mathsf{propose},B_{h+1},\mathcal{C}_{v^{\prime}}(B_{h}),v\rangle$

-

$70.5\%-71.8\%$

-

$B_{k^{\prime}-1}$

-

$v^{\prime}+1$

-

$51\%\text{--}53\%$

-

$\mathsf{timeout}_{v^{*}}$

-

$B_{k}=B_{h+1}$

-

$B_{k}=B_{l}$

-

$\mathcal{C}^{o}_{v+1}(B_{k^{\prime}})$

-

$P_{i}\in\mathcal{V}$

-

$B_{k}\neq B_{l}$

-

$f^{\prime}\leq f=\lfloor\frac{n-1}{3}\rfloor$

-

$v^{\prime\prime}+1$

-

$\mathcal{V}=($

-

$B_{k}:=(b_{v},H(B_{k-1}))$

-

$=4\delta$

-

$v^{\prime}-1$

-

$5\Delta$

-

$\mathcal{C}_{v+1}(B_{l})$

-

$t+3\Delta$

-

$\mathcal{TC}_{v^{\prime}-1}$

-

$B_{k}=B^{\prime}_{k}$

-

$\mathcal{C}_{v^{\prime\prime}}(B_{h})$

-

$\mathcal{TC}_{v^{\prime}}$

-

$\mathcal{C}_{v}(B_{h+1})$

-

$\mathcal{C}^{n}_{v}(B_{k})$

-

$\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})\geq\mathcal{C}_{v}(B_{k})$

-

$B_{h+1}$

-

$\mathcal{C}_{v^{\prime}}(B_{k^{\prime}})$

-

$=2\delta$

-

$MaxStepsBacktracking=NumberOfEmptyCells$

-

$\textit{Success Rate}\boldsymbol{\propto}\textit{max steps}*\textit{$\frac{1}{% -emptycells}$}$

- - - diff --git a/htmls/output_mathjax_example_10038.html b/htmls/output_mathjax_example_10038.html deleted file mode 100644 index dca87a73b3a1fd68740313b3004715b0791428c2..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10038.html +++ /dev/null @@ -1,148 +0,0 @@ - - - - MathJax Example - - - - -

$s+r\approx o$

-

$\mathbf{I}_{t}^{s,r}\in\mathbb{Z}^{|\mathcal{E}|\times|\mathcal{R}|\times|% -\mathcal{E}|}$

-

$\textbf{W}^{(l)}_{6}$

-

$\textbf{t}_{i}^{\prime\prime}=\sigma(\sum_{r\in\mathcal{R}_{TG}}\frac{1}{% -\mathcal{N}_{i}}\textbf{W}_{r}\textbf{t}_{j}^{\prime}+\textbf{W}_{11}\textbf{t% -}_{i}^{\prime})\textnormal{,}$

-

$y^{e}_{t}$

-

$\textbf{W}_{r}\in\mathbb{R}^{d\times d}$

-

$\textbf{W}_{9}\in\mathbb{R}^{d\times 32}$

-

$\alpha_{o,s}$

-

$\textbf{o}^{(l+1)}_{t}=\sigma(\sum_{(s,r,o)\in\mathcal{F}_{t}}\textbf{W}^{(l)}% -_{1}(\psi(\textbf{s}^{(l)}_{t}||\textbf{r}_{t}))+\textbf{W}^{(l)}_{2}\textbf{o% -}^{(l)}_{t})\textnormal{,}$

-

$\textbf{UE}_{t-k:t-1}$

-

$\textbf{W}_{11}\in\mathbb{R}^{d\times d}$

-

$f_{q}\in\{G_{t-k:t-1}\}$

-

$\textbf{R}_{t}=\mathrm{GRU}(\textbf{R}_{t-1},[pooling(\textbf{E}^{\mathcal{R}}% -_{t-1})||\textbf{R}])\textnormal{,}$

-

$(s,r,o,t)$

-

$\mathcal{R}_{TG}$

-

$\textbf{E}^{\mathcal{R}}$

-

$t>t_{T}$

-

$\alpha^{(l)}_{o,s}=\frac{exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}^{(l)}_{4}[% -\textbf{s}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime\prime}]))}{% -\sum_{s^{\prime}\in\mathcal{N}_{(o)}}exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}% -^{(l)}_{4}[\textbf{s'}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime% -\prime}]))}\textnormal{,}$

-

$q=(s,r,?,t)$

-

$p_{R}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},% -\textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t})\textnormal{.}$

-

$\textbf{R}_{t}$

-

$\textbf{t}^{\prime}=\textbf{W}_{8}(\textbf{W}_{9}\textbf{t}||\sigma(\textbf{W}% -_{10}\textbf{t}))\textnormal{,}$

-

$\mathbf{W}^{(l)}_{1}$

-

$\displaystyle=\beta\sum_{(s,r,t)\in\mathcal{Q}^{e}_{t}}y^{e}_{t}\log p(o|s,r,t% -,G_{t-1})$

-

$\textbf{W}_{10}\in\mathbb{R}^{d\times 32}$

-

$\textbf{s},\textbf{o}\in\textbf{UE}_{t-k:t-1}$

-

$\textbf{W}^{(l)}_{3}\in\mathbb{R}^{4d}$

-

$p_{H}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},% -\textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t}\odot\mathbf{I}_{t}^{s,r})% -\textnormal{,}$

-

$\mathcal{Q}^{r}_{t}$

-

$\textbf{UE}_{t}$

-

$\Theta\in\mathbb{R}^{|\mathcal{E}|\times d}$

-

$\{\textbf{E}_{t-k+1},\textbf{E}_{t-k+2},...,\textbf{E}_{t}\}$

-

$\mathcal{N}_{(o)}$

-

$y^{r}_{t}$

-

$G=\{\mathcal{E},\mathcal{R},\mathcal{F},\mathcal{T}\}$

-

$TG=\{\mathcal{E}_{TG},\mathcal{R}_{TG}\}$

-

$\textbf{W}_{8}\in\mathbb{R}^{d\times 2d}$

-

$\mathcal{Q}^{e}_{t}$

-

$\textbf{W}^{(l)}_{4}\in\mathbb{R}^{4d\times 4d}$

-

$\mathsf{LMS}$

-

$\textbf{W}_{7}\in\mathbb{R}^{1\times d}$

-

$\textbf{UE}_{t-k:t-1}=\mathrm{MEAN}(\sum_{i=t-k+1}^{t}\textbf{E}_{i})% -\textnormal{.}$

-

$\displaystyle+(1-\beta)\sum_{(s,o,t)\in\mathcal{Q}^{r}_{t}}y^{r}_{t}\log p(r|s% -,o,t,G_{t-1})\textnormal{,}$

-

$\textbf{E}_{t}=\mathrm{GRU}(\textbf{E}_{t-1},\textbf{G}_{t-1})\textnormal{.}$

-

$\textbf{W}^{(l)}_{5}$

-

$\mathbf{I}_{t}^{s,r}$

-

$p(o|s,r,t,G_{t-1})=\alpha p_{H}(o|s,r,t,G_{t-1})+(1-\alpha)p_{R}(o|s,r,t,G_{t-% -1})\textnormal{.}$

-

$\mathbf{W}^{(l)}_{2}$

-

$\textbf{o}^{(l+1)}=\sigma(\sum_{(s,r,o)\in UG}\alpha^{(l)}_{o,s}\textbf{W}^{(l% -)}_{5}\psi(\textbf{s}^{(l)}||\textbf{r})+\textbf{W}^{(l)}_{6}\textbf{o}^{(l)})% -\textnormal{,}$

-

$\textbf{GE}_{t}=\sigma(\textbf{W}_{7}\Theta_{e})\textbf{E}_{t}+(1-\sigma(% -\textbf{W}_{7}\Theta_{e}))\textbf{UE}_{t}\textnormal{,}$

-

$\mathcal{E}_{TG}$

-

$G=\{G_{1},G_{2},...,G_{\mathcal{T}}\}$

-

$p(o|s,r,t,G_{t-1})$

-

$1-A_{h}$

-

${\mathcal{X}_{1}}\in\mathbb{R}^{{1024}\times{{3}}}$

-

$P=\{{\mathcal{P}_{1}}\in\mathbb{R}^{{2}\times{{N}\times{C}}},{\mathcal{P}_{2}}% -\in\mathbb{R}^{{2}\times{{N_{1}}}\times{C_{1}}}$

-

${\mathcal{X}_{3}}\in\mathbb{R}^{{128}\times{{3}}}$

-

$\displaystyle+||(\Pi(\mathcal{J}(\mathcal{M}^{h}_{MANO}))-\Pi(\mathcal{J}(\hat% -{\mathcal{M}}^{h}_{MANO})))||_{2}.$

-

$X_{h}=\{X_{l}\in\mathbb{R}^{{N}\times{C}},X_{r}\in\mathbb{R}^{{N}\times{C}}\}$

-

$i -

$\mathcal{\hat{P}}=\mathcal{P}\odot\bm{\alpha}+\bm{\beta},(\bm{\alpha},\bm{% -\beta})=\psi(\mathcal{F}).$

-

$\mathcal{L}_{V}=\sum_{h\in\{L,R\}}||\mathcal{M}^{h}_{GCN}-\hat{\mathcal{M}}^{h% -}_{GCN}||_{1}+||\mathcal{M}^{h}_{MANO}-\hat{\mathcal{M}}^{h}_{MANO}||_{1}.$

-

${\mathcal{F}_{3}}\in\mathbb{R}^{{\frac{H}{4}}\times{\frac{W}{4}}\times{256}}\}$

-

${G_{out}}=\sum_{k=0}^{K-1}C_{k}(\hat{L})G_{in}W_{k}.$

-

$M=\{M_{l}\in\mathbb{R}^{{H}\times{{W}}},M_{r}\in\mathbb{R}^{{H}\times{{W}}}\}$

-

$\hat{P_{i}},NumPoints\_{i},BallRadius\_{i}$

-

$\mathcal{I}_{c}\in\mathbb{R}^{{H}\times{W}\times{3}}$

-

$num\_layers$

-

$\psi_{i}(\hat{F_{i}},P_{i})$

-

$\displaystyle\mathcal{L}_{rep}$

-

$cat(P_{i},PointNet(group(S_{i})))$

-

$BallRadius$

-

$(G\odot(\alpha+1)+\beta)$

-

$NumPoints$

-

$(P_{i}\odot(\alpha+1)+\beta)$

-

$\hat{P_{i}}$

-

$A_{h}\in[0,1]$

-

$F=\{{\mathcal{F}_{1}}\in\mathbb{R}^{{H}\times{{W}\times{3}}},{\mathcal{F}_{2}}% -\in\mathbb{R}^{{\frac{H}{2}}\times{\frac{W}{2}}\times{64}}$

-

$P_{ct}=\{P_{l}\in\mathbb{R}^{2},P_{r}\in\mathbb{R}^{2}\}$

-

$\psi_{i+1}(C,G)$

-

$\mathcal{G}\in\mathbb{R}^{{2}\times{1024}\times{1}}$

-

$\mathcal{I}_{d}\in\mathbb{R}^{{H}\times{W}\times{1}}$

-

$Fetch(F_{i}|u,v)$

-

$\mathcal{L}_{J}=\sum_{h\in\{L,R\}}||\mathcal{J}(\mathcal{M}^{h}_{MANO})-% -\mathcal{J}(\hat{\mathcal{M}}^{h}_{MANO})||_{1}.$

-

$K^{-1}X_{i}$

-

$\mathcal{L}_{m}=||M-\hat{M}||_{1},$

-

$[1,num\_layers]$

-

$W_{k}\in\mathbb{R}^{{C_{in}}\times{C_{out}}}$

-

$G_{in}\in\mathbb{R}^{{N}\times{C_{in}}}$

-

$PointNet(\hat{P_{i}})$

-

$\hat{L}\in\mathbb{R}^{{N}\times{N}}$

-

$\mathcal{L}_{root}=\sum_{h\in\{L,R\}}||Root^{h}-\hat{Root^{h}}||_{1}.$

-

$\mathcal{L}_{smooth}=\sum_{i=1}^{3}||e_{i}\cdot\hat{n}||_{1}+||e-\hat{e}||_{1},$

-

${\mathcal{P}_{3}}\in\mathbb{R}^{{2}\times{{N_{2}}}\times{C_{2}}}\}$

-

$\mathcal{G_{V}}\in\mathbb{R}^{{N}\times{C}},(N=63,126,252),(C=512,256,128)$

-

$\mathcal{L}_{c}=\sum_{h\in\{L,R\}}(1-A_{h})^{\gamma}\log(A_{h}),$

-

${\mathcal{X}_{2}}\in\mathbb{R}^{{512}\times{{3}}}$

-

$G_{out}\in\mathbb{R}^{{N}\times{C_{out}}}$

-

$\displaystyle=\sum_{h\in\{L,R\}}||(\Pi(\mathcal{M}^{h}_{MANO})-\Pi(\hat{% -\mathcal{M}}^{h}_{MANO}))||_{2}$

-

$F(t)=\frac{(1-t)(1-2t)-\sqrt{(1-t)(1-5t)}}{2t(2-t)}.$

- - - diff --git a/htmls/output_mathjax_example_10039.html b/htmls/output_mathjax_example_10039.html deleted file mode 100644 index 13fdb5f521ae4da2f070f558f6f5622ac806cfe8..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10039.html +++ /dev/null @@ -1,196 +0,0 @@ - - - - MathJax Example - - - - -

$\delta(R)$

-

$F(t)=F(t,1,1,1,1)=t^{5}C(t)^{4}$

-

$v_{n}\sim 4^{n+2}/\sqrt{\pi}n^{-3/2}$

-

$W(t)=\frac{1}{1-P(t)/t}$

-

$-(5t^{6}-16t^{5}+15t^{4}-28t^{3}+23t^{2}-8t+1)(t-1)^{2}F-(2t^{5}-5t^{4}+4t^{3}% --10t^{2}+6t-1)(t-1)^{2}=0$

-

$(2143,3\underline{41}2)$

-

$F(t,x_{1},x_{2},x_{3},x_{4})$

-

$F(t)=\frac{t(1-t)(1-7t+16t^{2}-11t^{3}+2t^{4})}{(1-4t+2t^{2})(1-3t+t^{2})^{2}}.$

-

$\mathrm{mod}\ 4$

-

$2\underline{14}3,3\underline{41}2$

-

$P_{1}=$

-

$(3,2,1,2)$

-

$\left.{D=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tA}{1-t}}% --1\right)A}\right\}$

-

$e_{m}:=[t^{m}]\prod_{i=1}^{4}(1+tx_{i})$

-

$F(t,x_{1},x_{2},x_{3},x_{4})=t^{5}\frac{1}{2\alpha}\left(\beta-\sqrt{\beta^{2}% --4\alpha e_{4}^{2}}\right)$

-

$(n+4)v_{n}-6(n+2)v_{n-1}+4(2n-1)v_{n-2}=0$

-

$\left.\left(\frac{1}{(1-t)^{2}}-1\right)D,\ \ {D=\frac{t^{2}}{1-t}+\left(\frac% -{1}{(1-t)^{2}}-1\right)A+\frac{S_{A}^{2}}{(1-t)^{2}}}\right\}$

-

$\alpha F^{2}-\beta F+e_{2}^{4}$

-

$\left.\frac{t^{2}}{1-t}+\frac{tS_{A}}{1-t}\right\}$

-

$\left.{S_{D}=\frac{t^{2}}{1-t}+\frac{tS_{A}}{1-t}}\right\}$

-

$V(t)=tC^{2}(t)(1+t^{2}C^{4}(t))$

-

$(1,2,3,1)$

-

${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A}$

-

$(45312,213)$

-

$F(t)=\frac{1-3t-t^{2}+2t^{3}-\sqrt{1-6t+7t^{2}+2t^{3}+t^{4}}}{2t^{2}(2-t)}.$

-

$\left\{S_{A}=\frac{t^{2}}{1-t}+\frac{tS_{D}}{1-t},\ \ S_{D}=\frac{(t+S_{A})^{2% -}}{1-(t+S_{A})}\right\}$

-

$O(\sqrt{n}\ln n)$

-

$F(t)=\frac{1-3t+t^{2}-\sqrt{1-6t+7t^{2}-2t^{3}+t^{4}}}{2t}.$

-

$\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D},\right.$

-

$F(t)=\frac{1-t-\sqrt{1-6t+t^{2}}}{2}$

-

$\alpha:=\prod_{i=1}^{4}(1-x_{i}+tx_{i}^{2})$

-

$P_{4}=\leavevmode\hbox to15.51pt{\vbox to15.51pt{\pgfpicture\makeatletter% -\raise-3.91434pt\hbox{\hskip 0.5pt\lower-0.5pt\hbox to 0.0pt{% -\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}% -\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}% -{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox -to% - 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} -{}{{}}{} -{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}% -\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{7.25558pt}\pgfsys@lineto{14.51118pt}{% -7.25558pt}\pgfsys@stroke\pgfsys@invoke{ } -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} -{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}% -\pgfsys@invoke{ }{}\pgfsys@moveto{10.15784pt}{7.25558pt}\pgfsys@lineto{10.1578% -4pt}{14.51118pt}\pgfsys@stroke\pgfsys@invoke{ } -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} -{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}% -\pgfsys@invoke{ }{}\pgfsys@moveto{4.35333pt}{0.0pt}\pgfsys@lineto{4.35333pt}{7% -.25558pt}\pgfsys@stroke\pgfsys@invoke{ } -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}% -\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}% -\lxSVG@closescope\endpgfpicture}}$

-

$P_{4}=$

-

$P_{3}=$

-

$2143,3\underline{41}2$

-

$\left\{{S_{A}=\frac{t^{2}}{1-t}+\left(\frac{1}{1-\frac{tS_{D}}{1-t}}\frac{1}{(% -1-t)^{2}}-1\right)S_{D}},\right.$

-

$t^{8}(t-2)^{2}F^{4}-t^{3}(t^{2}-3t+2)(t^{5}-7t^{4}+4t^{3}-6t^{2}+5t-1)F^{3}-t(% -t-1)(4t^{7}-22t^{6}+37t^{5}-42t^{4}+53t^{3}-35t^{2}+10t-1)F^{2}$

-

$\left\{{A=\frac{(t+D)^{2}}{1-(t+D)}},\ \ {D=\frac{(t+A)^{2}}{1-(t+A)}}\right\}$

-

$532642x_{1}=2x_{1}\times 11^{2}\times 31\times 71$

-

$(2143,3412)$

-

$\beta:=(2e_{4}t^{2}-4t(e_{4}-3e_{3}+2e_{2})+e_{4}-e_{3}+e_{2}-e_{1}+2)e_{4}$

-

$\left\{{S_{A}=\frac{t^{2}}{1-t}+\frac{S_{D}}{1-t}},\ \ S_{D}=\frac{t^{2}}{1-t}% -+\right.$

-

$tF^{3}+2tF^{2}+(2t-1)F+t=0.$

-

$\!,$

-

$\left\{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tD}{1-t}}% --1\right)D,\ \ D=\frac{(t+A)^{2}}{1-(t+A)}\right\}$

-

$C(t)=\frac{1-\sqrt{1-4t}}{2t}$

-

$2x_{1}x_{2}^{2}x_{3}x_{4}$

-

$2413,3142)$

-

$Z(t)=\frac{t(1-2t)}{1-4t+2t^{2}}$

-

$2\underline{14}3,45312$

-

$\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D},\ \ {D=\frac{% -(t+A)^{2}}{1-(t+A)}}\right\}$

-

$(2413,3142)$

-

$(2\underline{14}3,231)$

-

$P_{3}=\leavevmode\hbox to15.51pt{\vbox to15.51pt{\pgfpicture\makeatletter% -\raise-3.91434pt\hbox{\hskip 0.5pt\lower-0.5pt\hbox to 0.0pt{% -\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}% -\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}% -{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox -to% - 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} -{}{{}}{} -{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}% -\pgfsys@invoke{ }{}\pgfsys@moveto{7.25558pt}{0.0pt}\pgfsys@lineto{7.25558pt}{1% -4.51118pt}\pgfsys@stroke\pgfsys@invoke{ } -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} -{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}% -\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{4.35333pt}\pgfsys@lineto{7.25558pt}{4% -.35333pt}\pgfsys@stroke\pgfsys@invoke{ } -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} -{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}% -\pgfsys@invoke{ }{}\pgfsys@moveto{7.25558pt}{10.15784pt}\pgfsys@lineto{14.5111% -8pt}{10.15784pt}\pgfsys@stroke\pgfsys@invoke{ } -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope -\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}% -\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}% -\lxSVG@closescope\endpgfpicture}}$

-

$\delta(\mathcal{R})$

-

$(2143,45312)$

-

$d(f(t),g(t))=2^{-\operatorname{val}(f(t)-g(t))}$

-

$P(t)/t$

-

${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A+\frac{S_{D}^{2}}{(1-t% -)^{2}}}$

-

$F(t)=\frac{t(1-2t)}{1-4t+2t^{2}}.$

-

$(2143,231)$

-

$\!\}.$

-

$[1..b]$

-

$2143,3412$

-

$F(t)=t+A(t)+D(t)$

-

$21354,45312$

-

$2143$

-

$\operatorname{val}(f(t))=+\infty$

-

$V(t)=W(t)Z(t)=\frac{(1-2t)\big{(}\,1-4t+2t^{2}+(1-2t)\sqrt{1-4t}\,\big{)}}{2t^% -{3}}=tC^{2}(t)(1+t^{2}C^{4}(t)),$

-

$\left\{{S_{A}=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)S_{D}},\ \ S% -_{D}=\right.$

-

$A(t)=D(t)$

-

$\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D+\frac{S_{D}^{2% -}}{(1-t)^{2}}},\ \ {D=\frac{(t+A)^{2}}{1-(t+A)}}\right\}$

-

$\{\!$

-

$f(t)=\sum_{n\geq 0}f_{n}t^{n}$

-

$(2\underline{14}3,3\underline{41}2)$

-

$a_{i+1}>1$

-

$P_{2}=$

-

$(2\underline{14}3,45312)$

-

$P(t)=t^{5}C^{4}(t)\left(\frac{2}{1-\left(\frac{1}{(1-t)^{2}}-1\right)}-1\right).$

-

$2143,45312$

-

$t^{4}(t-2)^{2}F^{4}+t(t-2)(4t^{3}-7t^{2}+6t-1)F^{3}+(2t^{4}-t^{3}-2t^{2}+5t-1)% -F^{2}-(4t^{3}-7t^{2}+6t-1)F+t^{2}=0$

-

$2\underline{14}3$

-

$F(t)=\frac{1-t-\sqrt{1-6t+t^{2}}}{2}.$

-

$\left.\left(\frac{1}{(1-t)^{2}}-1\right)S_{A}+\left(\frac{1}{1-t}\frac{t^{2}}{% -1-2t}\right)^{2}\right\}$

-

$\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tD}{1-t}% -}-1\right)D},\ \ {D=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A+% -\frac{S_{A}^{2}}{(1-t)^{2}}}\right\}$

-

$1,2,\ldots,b$

-

$F(t,x_{1},11,31,71)$

-

$(21354,45312)$

-

${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tA}{1-t}}-1% -\right)A}$

-

$f_{n}\neq 0$

-

$\left\{A=\frac{t^{2}}{1-t}+\right.$

-

$(1,b,1,1)$

-

$A=\frac{(t+A)^{2}}{1-(t+A)}$

-

$F(t)=\frac{t(1-16t+11t^{2}-434t^{3}+1045t^{4}-1590t^{5}+1508t^{6}-846t^{7}+252% -t^{8}-30t^{9})}{(1-2t)^{4}(1-3t+t^{2})^{2}(1-4t+2t^{2})}.$

-

$t^{5}C^{4}(t)$

-

$\begin{array}[]{lcccl}\text{1.}\ (a,\ b,\ 1,\ 1)\ \longrightarrow\ (1,\ b+1,\ % -1,\ 1),&&&&\text{3.}\ (1,\ b,\ c,\ d)\ \longrightarrow\ (1,\ b,\ [1..c],\ d+1)% -,\\ -\text{2.}\ (1,\ b,\ c,\ 1)\ \longrightarrow\ (1,\ [1..b],\ c+1,\ 1),&&&&\text{% -4.}\ (a,\ b,\ c,\ d)\ \longrightarrow\ (a+1,\ b,\ c,\ [1..d]).\end{array}$

-

$\displaystyle\begin{split}&F(t,x_{1},x_{2},x_{3},x_{4})=t^{5}x_{1}x_{2}x_{3}x_% -{4}+tx_{1}x_{2}x_{3}x_{4}[x_{3}x_{4}]F(t,1,x_{2},x_{3},x_{4})\\ -&+tx_{1}x_{3}x_{4}\frac{[x_{1}x_{4}]F(t,x_{1},x_{2},x_{3},x_{4})-x_{2}[x_{1}x_% -{4}]F(t,x_{1},1,x_{3},x_{4})}{x_{2}-1}\\ -&+tx_{1}x_{4}\frac{[x_{1}]F(t,x_{1},x_{2},x_{3},x_{4})-x_{3}[x_{1}]F(t,x_{1},x% -_{2},1,x_{4})}{x_{3}-1}\\ -&+tx_{1}\frac{F(t,x_{1},x_{2},x_{3},x_{4})-x_{4}F(t,x_{1},x_{2},x_{3},1)}{x_{4% -}-1}.\end{split}$

-

$21354$

-

$(\mathbf{c},\sigma)=F_{\theta}^{c}(\mathbf{\gamma}(\mathbf{x}),\mathbf{\gamma}% -(\mathbf{d}))$

-

$\mathbf{f}_{z}\in\mathbb{R}^{4}$

- - - diff --git a/htmls/output_mathjax_example_1004.html b/htmls/output_mathjax_example_1004.html deleted file mode 100644 index f53863eb5f71380a6ea54cfe1cdaea0eb660253c..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1004.html +++ /dev/null @@ -1,121 +0,0 @@ - - - - MathJax Example - - - - -

$G\setminus\{a_{1},b_{1}\}$

-

$y\in V(F)$

-

$q_{1}=c$

-

$v\in V(G)\setminus V(H)$

-

$G[q_{j},\dots,,q_{t},b_{1},a_{1},a_{2}]$

-

$N_{G}(q_{t})\cap V(H)=B$

-

$G[S]=H[S]$

-

$\omega<2.3728596$

-

$x\in A$

-

$G_{i}-v_{i}$

-

$H\in\mathcal{C}$

-

$\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{1}}\}=\{2K_{1}\vee 2K_{1}\}=% -\{C_{4}\}$

-

$N_{G}(v)\cap V(H)\subseteq B$

-

$a_{2},a_{3},b_{1}$

-

$Q=q_{0},\dots,q_{t}$

-

$H\cong 2K_{1}$

-

$\mathcal{G}_{k}$

-

$V(G)\setminus V(H)$

-

$E(H_{1})\cup E(H_{2})\cup\{v_{1}v_{2}\mid v_{1}\in V(H_{1}),v_{2}\in V(H_{2})\}$

-

$G[S^{\prime}]\in\mathcal{C}$

-

$G_{A}:=G[A\cup C]$

-

$\bigcup_{v\in V(H)}X_{v}\subseteq V(2K_{1})$

-

$\mathcal{O}(n^{2}(n+m))$

-

$K_{2,3}$

-

$p_{1},\dots,p_{i},q_{j},\dots,q_{t}$

-

$K_{p,q}$

-

$\mathcal{F}_{\mathcal{C}}$

-

$S^{\prime}\subseteq S^{*}$

-

$G\in\mathcal{C}_{k}$

-

$\mathcal{O}(n^{4})$

-

$r_{1},\dots,r_{i},c$

-

$S=V(G)$

-

$\mathcal{O}(n^{2k+2})$

-

$S\subseteq A\cup C$

-

$S^{\prime}\subseteq S\cap(A\cup C)$

-

$S\subseteq V(G^{\prime})\setminus\{a,b,v^{xy}\}=V(G)\setminus\{a,b,x,y\}$

-

$A=\{a_{1},a_{2},a_{3}\}$

-

$G/e$

-

$p_{s}=c$

-

$S\subseteq V(G^{\prime})\setminus\{v^{xy}\}=V(G)\setminus\{x,y\}$

-

$a_{i},a_{j},a_{k}\in A$

-

$G[a_{1},a_{2},b_{1},b_{2},p_{0},\dots,p_{s}]$

-

$x,y\in V(G^{\prime})$

-

$S\subseteq V(G)\setminus\{a,b\}$

-

$\sum_{x\in X}w(x)$

-

$S_{v}=\{v,z^{v}\}$

-

$\mathcal{O}(n^{2+\epsilon})$

-

$p_{1},\ldots,p_{k}$

-

$r_{1}=x$

-

$K_{2,k+1}\notin\mathcal{G}_{k}$

-

$p_{1},\dots,p_{i-1},p_{j+1},\dots,p_{s}$

-

$P^{3}$

-

$p_{1},\dots,p_{i},q_{j+1},\dots,q_{t}$

-

$\chi(\overline{G})\leq k$

-

$r_{i},c\in C$

-

$G[S^{*}]\in\mathcal{C}$

-

$F\in{\mathcal{C}}$

-

$\overline{C_{6}}$

-

$|V(H)|-1$

-

$G\setminus S^{\prime}$

-

$S^{*}\cap V(H)\subseteq S$

-

$G^{\prime}[S]=G[S]$

-

$\overline{K_{2}\cup C_{2k+1}}\cong 2K_{1}\vee\overline{C_{2k+1}}$

-

$S\cap B=\emptyset$

-

$h_{1},\dots,h_{i}$

-

$y=b$

-

$a_{2},q_{j},b_{1}$

-

$K_{2,k}$

-

$a,b\in V(G)\setminus S$

-

$y\in B$

-

$N_{G}(q_{t})\cap V(H)=\{b_{1}\}$

-

$\mathcal{M}_{\mathcal{G}_{0}}=\{P_{3}\}$

-

$\mathcal{O}(n^{3+\epsilon})$

-

$K_{k-1}$

-

$\mathcal{O}(n^{2+\epsilon}).$

-

$p_{i}\in C$

-

$r_{k}=y$

-

$K_{2,k+1}$

-

$V(C_{1})$

-

$N_{G}[y]\setminus N_{G}[x]$

-

$p_{s}\in N_{B}$

-

${\mathcal{C}}$

-

$N_{G}[x]:=\{x\}\cup N_{G}(x)$

-

$N_{B}=\emptyset$

-

$\alpha(G^{*})=\alpha(G)+|E(G)|$

-

$G[v,a_{i},a_{j},a_{k}]$

-

$p_{1}\in A$

-

$G[x_{1},\dots,x_{t}]$

-

$G^{\prime}\in\mathcal{G}_{\mathcal{C}}$

-

$K_{\ell}\notin\mathcal{C}$

-

$V(H)\setminus U$

-

$G\in\mathcal{C}$

-

$uw\in E(H)$

-

$a_{2}\in N_{G}(p_{s})\cap V(H)\subseteq\{a_{2},b_{2}\}$

-

$G[A\cup B\cup\{q_{0},q_{1}\}]$

-

$\{a_{i},b_{i}\}$

-

$|N_{G}(v)\cap V(H)|\geq 2$

-

$a,b\in V(G^{\prime})\setminus S$

-

$X=\{x\}$

-

$E(H_{1})\cup E(H_{2})$

- - - diff --git a/htmls/output_mathjax_example_10040.html b/htmls/output_mathjax_example_10040.html deleted file mode 100644 index 613ae9dcb9aa07c706be204e23c61b1c7ca7d381..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10040.html +++ /dev/null @@ -1,143 +0,0 @@ - - - - MathJax Example - - - - -

$\tilde{z}^{i}_{t},{z}^{i}_{t}$

-

$\mathcal{L}_{rec}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-\hat% -{Z}^{i}(\mathbf{r})\right\|^{2}$

-

$\hat{Z^{i}}(\mathbf{r})$

-

$\mathcal{L}_{Mrec}$

-

$\mathcal{L}_{ref}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-% -\tilde{Z}^{i}(\mathbf{r})\right\|^{2}\hskip 2.84544pt,\textrm{where}\hskip 2.8% -4544pt\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i})$

-

$\hat{Z}^{i}(\mathbf{r})=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\gamma(t))% -\mathbf{f}_{z}(\mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544% -ptT(t)=\textrm{exp}\left(\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$

-

$t\mathbf{d}$

-

$\nabla\mathcal{L}_{MDDS}$

-

$\nabla_{\theta}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta}\mathcal{L}_{\mathrm{% -SDS}}(\mathbf{z},y_{src})-\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\hat{% -\mathbf{z}},y_{trg}),$

-

$\lambda_{im}$

-

$\mathcal{L}_{rtot}=\lambda_{rec}\mathcal{L}_{rec}+\lambda_{ref}\mathcal{L}_{ref}$

-

$\lambda_{ref}$

-

$y_{src}$

-

$\mathcal{L}_{MDDS}$

-

$\tilde{{Z}}^{i}$

-

${F_{\phi}(\cdot)}$

-

$\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta,\phi}\mathcal{L}% -_{\mathrm{SDS}}(\mathbf{z}^{i},y_{src})-\nabla_{\theta,\phi}\mathcal{L}_{% -\mathrm{SDS}}(\tilde{\mathbf{z}}^{i},y_{trg}).$

-

$(\mathbf{f}_{z},\sigma)=F_{\theta}(\mathbf{\gamma}(\mathbf{x}),\mathbf{\gamma}% -(\mathbf{d}))$

-

$\hat{C}(r)=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\mathbf{r}(t))\mathbf{c}(% -\mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544ptT(t)=\textrm{% -exp}\left(-\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$

-

$\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\mathbf{z},y_{trg},\epsilon,t)=% -\omega(t)(\epsilon_{\psi}\left(\mathbf{z}_{\mathbf{t}},y_{trg},t\right)-% -\epsilon)\frac{\partial\mathbf{z}_{\mathbf{t}}}{\partial\theta}$

-

$504\times 378$

-

$\lambda_{rec}$

-

$\mathcal{L}_{\mathrm{Mrec}}=\lambda_{im}\cdot\mathcal{M}\cdot\mathcal{L}_{% -\mathrm{rtot}}+\lambda_{om}\cdot(1-\mathcal{M})\cdot\mathcal{L}_{\mathrm{rtot}}.$

-

$I=\{I^{i}\}_{i=1}^{N}$

-

${z}:=\{{z}^{i}\}_{i=1}^{N}$

-

$\tilde{z}^{i}$

-

${z^{i}}=\mathcal{E}({{I^{i}}})\in\mathbb{R}^{64\times 64\times 4}$

-

$y_{trg}$

-

$\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{MDDS}}=\mathcal{M}\cdot(\nabla_{% -\theta,\phi}\mathcal{L}_{\mathrm{DDS}}),$

-

$[t_{near},t_{far}]$

-

$\mathcal{L}_{\mathrm{tot}}=\mathcal{L}_{\mathrm{MDDS}}+\mathcal{L}_{\mathrm{% -Mrec}}$

-

$\lambda_{om}$

-

$\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i}).$

-

$\mathcal{L}_{rtot}$

-

$\mathbf{r}(t)=\mathbf{o}$

-

$[T\cdot G,Q]$

-

$\displaystyle+R_{\text{upright}}+R_{\text{heading}}$

-

$205\pm 8$

-

$10P$

-

$\displaystyle+R_{\text{effort}}+R_{\text{act}}+R_{\text{dof}}$

-

$P=100,T=50,W=10$

-

$12+3\mathbb{A}+2\mathbb{F}$

-

$\mathbb{A},\mathbb{F}$

-

$\displaystyle+R_{\text{death}}\times\mathbf{1}_{\{\text{head\_height}\leq\text% -{termination\_height}\}}$

-

$\displaystyle\ R_{\text{progress}}+R_{\text{alive}}\times\mathbf{1}_{\{\text{% -head\_height}\geq\text{termination\_height}\}}$

-

$G=\lfloor(Q-P)/T\rfloor$

-

$(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\frac{1}{2}\Phi(p^{m-6i}))+1$

-

$v\in\mathbb{Z}_{p^{m}}$

-

$\tau^{\prime}\circ\tau_{i}(E)=\tau^{\prime}(E)=E^{\prime},\ \forall\ i\in\{1,2% -,3,...,t\}$

-

$j(E_{1})\neq 0,1728$

-

$\begin{split}\quad y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\end{split}$

-

$a^{\frac{p-1}{2}}\equiv 1\mod p)$

-

$N_{R}(\mathbb{Z}_{p})=\Phi(p^{2})$

-

$N_{R}(\mathcal{R})$

-

$\ \ \frac{\Phi(n^{4})}{|Aut(E)|}$

-

$32505856$

-

$E_{2}/\mathbb{Z}_{p^{m}}:y^{2}=x^{3}+\bar{a}x+\bar{b}$

-

$\displaystyle b_{6}$

-

$\Delta^{v}(p^{m})$

-

$2q-4+6+4=2q+6$

-

$E_{1}/\mathbb{F}_{q}:y^{2}=x^{3}+ax+b$

-

$|Aut(E)|=4$

-

$\{u^{\prime},\alpha u^{\prime},\alpha^{2}u^{\prime},-u^{\prime},-\alpha u^{% -\prime},-\alpha^{2}u^{\prime}\}$

-

$\{6,19\}$

-

$\begin{split}C_{R}(\mathbb{Z}_{n})=\sum_{\mathbb{E}_{k}}C^{(k)}_{R}(\mathbb{Z}% -_{n})\end{split}$

-

$34091302912$

-

$15400$

-

$(3^{-1})^{3}a^{3}\equiv-(2^{-1})^{2}b^{2}\ \mod p$

-

$\mathbb{Z}_{2^{m}}$

-

$k_{2}=4$

-

$6\nmid char(\mathbb{Z}_{n})$

-

$\displaystyle(E)=c_{4}^{3}/\Delta$

-

$\displaystyle b_{8}$

-

$3.5115653\times 10^{13}$

-

$\frac{q-1}{|Aut(E)|}$

-

$\{11,14\}$

-

$2p^{m}-4$

-

$28672$

-

$P(1,1)$

-

$N_{q}=2q+3+\left(\frac{-4}{q}\right)+2\left(\frac{-3}{q}\right)$

-

$N_{G}(\mathbb{Z}_{n})=\Phi(n^{5})$

-

$x=X/Z,y=Y/Z$

-

$Aut(E^{\prime})$

-

$y^{2}=x^{3}+3x$

-

$a=0\;\text{and}\;b\neq 0$

-

$|Aut(E)|=6$

-

$\tau^{\prime}\neq\tau_{i}$

-

$2p^{i-1}$

-

$|Aut(E_{1})|=6$

-

$q\equiv 1\mod k$

-

$E/\mathbb{F}_{q}$

-

$\{2,23\}$

-

$|Aut(\mathbb{E}_{k})|\;\text{over}\;\mathbb{Z}_{n}=\prod\limits_{i=1}^{l}|Aut(% -\mathbb{E}_{k_{i}})|\;\text{over}\;\mathbb{Z}_{p_{i}}$

-

$\mathbb{F}_{2^{m}},1\leq m\leq 10$

-

$F(X,Y,Z)=Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}-X^{3}-a_{2}X^{2}Z-a_{4}XZ^{2}-a_{6}Z^{3}=0$

-

$N^{\prime\prime}_{R}(\mathbb{Z}_{n})$

-

$y^{2}=x^{3}+4x+2$

-

$\mathbb{Z}_{p^{m}}^{*}$

-

$Aut(E)=\{\tau_{1},\tau_{2},...,\tau_{t}\}$

-

$\mathbb{Z}_{7^{m}}$

- - - diff --git a/htmls/output_mathjax_example_10041.html b/htmls/output_mathjax_example_10041.html deleted file mode 100644 index ab8c06c400b2ec8f8023160f2e3755a4131203cf..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10041.html +++ /dev/null @@ -1,139 +0,0 @@ - - - - MathJax Example - - - - -

$y^{2}=x^{3}+x$

-

$p^{2(i-1)}$

-

$2q-4$

-

$\tau:(x,y)\rightarrow(u^{2}x,u^{3}y),\;u\in\;\mathbb{Z}_{n}^{*}$

-

$2058$

-

$y^{2}=x^{3}-x+9$

-

$N_{R}(\mathbb{Z}_{n})$

-

$|Aut(E_{k})|=2,4$

-

$a=-3c^{2}$

-

$\begin{split}s_{1}^{3}+s_{2}^{2}&\equiv 0\mod p\\ -\implies s_{1}^{3}&\equiv-s_{2}^{2}\mod p\end{split}$

-

$\displaystyle=a_{1}^{2}a_{6}+4a_{2}a_{6}-a_{1}a_{3}a_{4}+a_{2}a_{3}^{2}-a_{4}^% -{2}$

-

$\displaystyle=a_{1}^{2}+4a_{2}$

-

$\displaystyle\implies k_{1}$

-

$\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),\;u\in\mathbb{K}^{*}$

-

$\Delta=-16(4a^{3}+27b^{2})$

-

$\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),u\in\;\mathbb{Z}_{n}^{*},r,s,% -t\in\mathbb{Z}_{n}$

-

$E(\mathbb{K})$

-

$\displaystyle=2a_{4}+a_{1}a_{3}$

-

$p^{m}=2j+1,\ i,j\in\mathbb{N}$

-

$\gcd(p,6)=1$

-

$\begin{split}E:Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}&=X^{3}+a_{2}X^{2}Z+a_{4}XZ^{2}\\ -&\ \ \ +a_{6}Z^{3}\end{split}$

-

$|A^{3}_{0}|=p^{\lfloor\frac{2m}{3}\rfloor}$

-

$3\mid\Phi(p^{m})$

-

$\begin{split}E:y^{2}+a_{1}xy+a_{3}y=&x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\\ -&\text{where}\ a_{i}\in\;\mathbb{Z}_{n}\end{split}$

-

$C_{G}(\mathbb{Z}_{n})$

-

$\left(.\right)$

-

$|A^{2}_{0}|=p^{\lfloor\frac{m}{2}\rfloor}$

-

$E_{2}/\mathbb{F}_{q}:y^{2}=x^{3}+\bar{a}x+\bar{b}$

-

$u^{4}\bar{a}=a$

-

$q^{5}-q^{4}$

-

$(\sum_{i=0}^{\left\lfloor\frac{m-1}{2}\right\rfloor}\frac{1}{2}\Phi(p^{m-2i}))% -+1.$

-

$Aut(E)$

-

$1\leq i\leq\left\lfloor\frac{m-1}{2}\right\rfloor+1$

-

$\gcd(p^{m},6)=1$

-

$\bar{a}=0$

-

$\displaystyle=c_{4}^{3}/\Delta$

-

$\frac{\sum\limits_{i=0}^{\left\lfloor\frac{m-1}{3}\right\rfloor}\Phi(p^{m-3i})% -}{3}+1.$

-

$N_{R}(\mathbb{Z}_{p^{m}})=\Phi(p^{2m})$

-

$\tau^{\prime}\circ\tau_{i}=\tau^{\prime}\circ\tau_{j}$

-

$A^{k}_{u}(p^{m})$

-

$|\#E(\mathbb{F}_{q})-q-1|\leq 2\sqrt{q}$

-

$\mathbb{Z}_{p}^{m}$

-

$y^{2}=x^{3}+1$

-

$16807$

-

$p^{m}\equiv 1\mod 4$

-

$\frac{\partial F}{\partial X},\frac{\partial F}{\partial Y},\frac{\partial F}{% -\partial Z}$

-

$n^{3}\Phi(n)=\Phi(n^{4})$

-

$2\leq l\leq 4$

-

$\begin{split}C^{(k)}_{R}(\mathbb{Z}_{n})&=\frac{\prod\limits_{i=1}^{l}N^{(k_{i% -})}_{R}(\mathbb{Z}_{p_{i}}).\prod\limits_{i=1}^{l}|Aut(\mathbb{E}_{k_{i}})|}{% -\prod\limits_{i=1}^{l}\Phi(p_{i})}\\ -&=\prod\limits_{i=1}^{l}\frac{N^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}}).|Aut(\mathbb% -{E}_{(k_{i})})|}{\Phi(p_{i})}\\ -&=\prod\limits_{i=1}^{l}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\end{split}$

-

$\displaystyle=-1728\frac{4a^{3}}{\Delta}$

-

$\frac{p-1}{6}$

-

$C^{(k)}_{R}(\mathbb{Z}_{n})$

-

$\mathbb{Z}_{p_{i}}(1\leq i\leq l)$

-

$\sum_{E_{k}}\frac{\Phi(n)}{|Aut(E_{k})|}=\Phi(n^{2})$

-

$y^{2}=x^{3}+3x+2$

-

$\Phi(n^{5})$

-

$\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),\ u\in\mathbb{Z}_{n}^{*}$

-

$u^{\prime}\in\mathbb{F}_{q}^{*}$

-

$A^{2}_{u}(25)$

-

$\begin{split}s_{1}^{3}+s_{2}^{2}&\equiv 0\mod(p^{m})\\ -\implies s_{1}^{3}&\equiv-s_{2}^{2}\mod(p^{m})\end{split}$

-

$p\equiv 3\mod 4$

-

$N_{G}(\mathcal{R})$

-

$|Aut(E_{1})|=4$

-

$n=p_{1}p_{2}\ldots p_{k}$

-

$\mathbb{F}_{(2g+1)^{n}}$

-

$y^{2}=x^{3}+x+2$

-

$\displaystyle=-16(4a^{3}+27b^{2})\in\mathbb{Z}_{n}^{*}$

-

$Aut(E^{\prime})=t$

-

$char(\mathbb{Z}_{n})\nmid 2,3$

-

$\bar{b}=0$

-

$\gcd(q,6)=1$

-

$\displaystyle(\frac{\Phi(p^{m})}{6})k_{1}$

-

$E(\mathbb{Z}_{n})\cong E(\mathbb{Z}_{p_{1}})\oplus E(\mathbb{Z}_{p_{2}})\oplus% -\ldots\oplus E(\mathbb{Z}_{p_{l}})$

-

$q\equiv 1\mod 12$

-

$p=3j+1$

-

$\displaystyle=b_{2}^{2}-24b_{4}$

-

$\Phi(n),n,n,n$

-

$p\equiv 2\mod 3$

-

$N_{R}(\mathbb{Z}_{p^{m}})$

-

$A^{k}_{0}(p^{m})=\{x\ |\ x^{k}\equiv 0\mod p^{m}\}$

-

$|Aut(E)|=2$

-

$q\equiv 1,5,7,11\mod 12$

-

$E_{1}/\mathbb{Z}_{p^{m}}:y^{2}=x^{3}+ax+b$

-

$p^{m}\equiv 1\mod 2$

-

$\begin{split}C_{R}(\mathbb{Z}_{n})&=\sum_{\mathbb{E}_{k}}C^{(k)}_{R}(\mathbb{Z% -}_{n})\\ -&=\sum_{\mathbb{E}_{k}}\prod\limits_{i=1}^{l}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}% -})\end{split}$

-

$(3^{-1}a,2^{-1}b)=(s_{1},s_{2})$

-

$r,s,t\in\mathbb{K}$

-

$\mathbb{Z}_{3125}$

-

$u^{6}\bar{b}=b$

-

$\tau^{\prime}(E)=E^{\prime}$

-

$j(E_{1})=0$

-

$4\mid\Phi(p^{m})$

-

$\mathbb{Z}_{p^{m}}$

-

$p^{m}=3j+1,\ i,j\in\mathbb{N}$

-

$C^{(k)}_{R}(\mathbb{Z}_{n})=\frac{N^{(k)}_{R}(\mathbb{Z}_{n}).|Aut(\mathbb{E}_% -{k})|}{\Phi(n)}$

-

$E:y^{2}=x^{3}+ax+b$

-

$\frac{q^{4}-q^{3}}{|Aut(E)|}$

-

$2\mid\Phi(p^{m})$

-

$\{3,22\}$

-

$\frac{p-1}{2}\mid(p-1)$

- - - diff --git a/htmls/output_mathjax_example_10042.html b/htmls/output_mathjax_example_10042.html deleted file mode 100644 index 8c550e36a90863bc8cd647774fa9d34a08962644..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10042.html +++ /dev/null @@ -1,128 +0,0 @@ - - - - MathJax Example - - - - -

$983040$

-

$\mod p$

-

$C_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}C_{R}(\mathbb{Z}_{p_{i}^{e_{i}}})$

-

$k_{1}+k_{2}+k_{3}=6+4+2p^{m}-4=2p^{m}+6$

-

$\forall e_{i}\geq 1$

-

$\#E(\mathbb{F}_{q})$

-

$p^{m}=4j+1,\ i,j\in\mathbb{N}$

-

$\mathbb{Z}_{5^{m}}$

-

$y^{2}=x^{3}+x+1$

-

$\displaystyle b_{4}$

-

$n=p_{1}p_{2}\ldots p_{l}$

-

$|A^{2}_{u}(25)|$

-

$\Delta\equiv 0\mod(p^{m})$

-

$P^{2}(\overline{\mathbb{K}})$

-

$d=\gcd(6,3j+1)=2$

-

$c=-3b/2a$

-

$A^{k}_{u}(p^{m})=\{x\ |\ x^{k}\equiv u\mod p^{m}\}$

-

$p=3j+2$

-

$p\equiv 1\ \mod 12$

-

$N_{G}(\mathbb{Z}_{n})$

-

$2\Phi(p^{m})$

-

$N^{\prime\prime}_{R}(\mathbb{Z}_{p^{m}})=(p^{2m}-\Delta^{0}(p^{m}))\geq N_{R}(% -\mathbb{Z}_{p^{m}})$

-

$n=p^{m}$

-

$1901$

-

$\{9,16\}$

-

$1056964608$

-

$\mathbb{Z}_{p_{i}}$

-

$3p^{2(i-1)}$

-

$|Aut(E_{k})|=6$

-

$\Phi(p^{m})$

-

$\{u^{\prime},\beta u^{\prime},\beta^{2}u^{\prime},\beta^{3}u^{\prime}\}$

-

$E:y^{2}=x^{3}+ax+b,\ char(\mathbb{K})\neq 2,3.$

-

$E(\mathbb{Z}_{{p_{i}}^{e_{i}}})$

-

$\Phi(p^{2m})-2\Phi(p^{m})=\Phi(p^{m})(p^{m}-2)$

-

$u\in{\mathbb{Z}_{p^{m}}^{*}}$

-

$m=3.20^{-1}$

-

$\Delta^{i}(n)$

-

$|Aut(E_{1})|$

-

$10^{e}$

-

$\begin{split}N_{R}(\mathbb{Z}_{n})&=\sum_{\mathbb{E}_{k}}N^{(k)}_{R}(\mathbb{Z% -}_{n})\\ -&=\sum_{\mathbb{E}_{k}}\prod_{i=1}^{l}N^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\end{split}$

-

$12500$

-

$\gcd(x^{2}+y^{2}+z^{2}+w^{2}+t^{2},n)=1$

-

$(\frac{p-1}{2}\times 1\times 2)+1=p$

-

$y^{2}=x^{3}+2x+1$

-

$p=2i+1$

-

$a\neq 0\ \text{and}\ b\neq 0\ (j(E)\neq 0,1728$

-

$\tau_{i}(E)=E$

-

$n=p_{1}^{e_{1}}p_{2}^{e_{2}}\ldots p_{l}^{e_{l}}$

-

$|Aut(E_{1})|=2$

-

$p^{m}\equiv 1\mod 3$

-

$x^{k}\equiv a\mod p$

-

$y^{2}=x^{3}+24x+1$

-

$char(\mathbb{F}_{q})\neq 2,3$

-

$E:y^{2}=x^{3}+ax+b,\ \ a,b\in\mathbb{Z}_{n}\ \text{and}\ \gcd(6,n)=1$

-

$6\nmid n$

-

$117012$

-

$\{12,13\}$

-

$\{1,24\}$

-

$1\leq i\leq\left\lfloor\frac{m-1}{3}\right\rfloor+1$

-

$\displaystyle=-16(4a^{3}+27b^{2})$

-

$|Aut(E)|$

-

$\frac{p-1}{6}\mid(p-1)$

-

$E(\mathbb{Z}_{n})$

-

$(\frac{p-1}{6}\times 3\times 2)+1=p$

-

$\frac{p-1}{2}$

-

$\displaystyle=\Phi(p^{m})$

-

$a^{\frac{p-1}{d}}\equiv 1\mod p$

-

$a\neq 0\ \text{and}\ b=0\ (j(E)=1728$

-

$d=\gcd(6,3j)=6$

-

$N^{(k)}_{R}(\mathbb{Z}_{n})$

-

$y(a-y)=x^{3}-x$

-

$\displaystyle=-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}$

-

$\mathbb{Z}_{25}$

-

$\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),u\in\;\mathbb{F}_{q}^{*}$

-

$1924$

-

$\Delta\in\mathbb{Z}_{n}^{*}$

-

$N_{R}(\mathbb{Z}_{n})=\Phi(n^{2})$

-

$\Delta\equiv 0\mod p$

-

$\mathbb{F}_{q}\ (q=p^{r})$

-

$\Delta=-16(27b^{2})$

-

$\bar{b}=u^{-6}b$

-

$N^{\prime}_{R}(\mathbb{Z}_{n})$

-

$p\equiv 1\mod 12$

-

$u\in\mathbb{F}_{q}^{*}$

-

$7^{1}$

-

$E_{1}/\mathbb{K}$

-

$(\sum_{i=0}^{\left\lfloor\frac{m-1}{3}\right\rfloor}\Phi(p^{m-3i}))+1$

-

$100842$

-

$\tau:(x,y)\rightarrow(u^{2}x,u^{3}y)$

-

$\mathbb{Z}_{3125},(5^{4}=3125)$

-

$\sum_{E_{k}}\frac{q-1}{|Aut(E_{k})|}=q^{2}-q$

-

$\sigma=\left\{\,\begin{array}[]{lll}2q+6&when&q\equiv 1\mod 12\\ -2q+2&when&q\equiv 5\mod 12\\ -2q+4&when&q\equiv 7\mod 12\\ -2q&when&q\equiv 11\mod 12\\ -\end{array}\right.\\$

-

$Q(21,4)$

-

$N^{(k)}_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}N^{(k_{i})}_{R}(\mathbb{Z}_{% -p_{i}})$

-

$u=-u^{\prime}$

-

$1.0952167\times 10^{12}$

-

$(x,y)\rightarrow(\frac{x-3b_{2}}{36},\frac{y}{216})$

-

$N^{{}^{\prime\prime}}_{R}(\mathbb{Z}_{p^{m}})$

-

$a\neq 0\;\text{and}\;b=0$

-

$y^{2}=x^{3}+4x$

- - - diff --git a/htmls/output_mathjax_example_10043.html b/htmls/output_mathjax_example_10043.html deleted file mode 100644 index 8e8865c7b3f82fe77ef06ebdf86be07ab2afc7df..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10043.html +++ /dev/null @@ -1,149 +0,0 @@ - - - - MathJax Example - - - - -

$p^{2m-1}$

-

$p\equiv 1\ \mod l$

-

$4a^{3}+27b^{2}=0$

-

$d=\gcd(k,p-1)$

-

$x^{m}\equiv 1\mod p$

-

$u\in\mathbb{Z}_{p^{m}}$

-

$C_{R}(\mathbb{Z}_{p_{i}})=\left\{\,\begin{array}[]{lll}2p+6&when&p\equiv 1\mod -1% -2\\ -2p+2&when&p\equiv 5\mod 12\\ -2p+4&when&p\equiv 7\mod 12\\ -2p&when&p\equiv 11\mod 12\\ -\end{array}\right.\\$

-

$N^{\prime\prime}_{R}(\mathbb{Z}_{p^{m}})$

-

$P(X,Y,Z)\in P^{2}(\overline{\mathbb{K}})$

-

$|Aut(E)|=t$

-

$(\tau^{\prime})^{-1}\circ\tau^{\prime}\circ\tau_{i}=(\tau^{\prime})^{-1}\circ% -\tau^{\prime}\circ\tau_{j}$

-

$\displaystyle=a_{3}^{2}+4a_{6}$

-

$y^{2}=x^{3}+2x$

-

$\mathbb{Z}_{3^{m}}$

-

$\begin{split}\Delta^{0}(p^{m})&=(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right% -\rfloor}\frac{1}{6}\Phi(p^{m-6i})\cdot 3p^{2i}\cdot 2p^{i})\\ -&+p^{\left\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{2m}{3}\right% -\rfloor}\\ -&=(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}p^{3i}\Phi(p^{m-6i}))+p^% -{\left\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{2m}{3}\right\rfloor}\\ -&=(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\Phi(p^{m-3i}))+p^{\left% -\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{2m}{3}\right\rfloor}\\ -\end{split}$

-

$p^{2}-p=\Phi(p^{2})$

-

$p\equiv 1\mod 3$

-

$C_{G}(\mathcal{R})$

-

$E/\mathbb{Z}_{n}$

-

$\{8,17\}$

-

$|Aut(E_{k})|=4$

-

$p\equiv 1,5,7,11\ \mod 12$

-

$r,s,t\in\mathbb{Z}_{n}$

-

$a^{\frac{p-1}{6}}\equiv 1\mod p$

-

$(x,y)\xrightarrow{}(x,y-\frac{a_{1}}{2}x-\frac{a_{3}}{2})$

-

$A^{2}(\overline{\mathbb{K}})=\overline{\mathbb{K}}\times\overline{\mathbb{K}}$

-

$\tau_{i}=\tau_{j}$

-

$\frac{p-1}{3}$

-

$j(E_{1})=j(E_{2})$

-

$m\mid(p-1)$

-

$\Delta\equiv i\mod n$

-

$b=2c^{3}$

-

$C_{R}(\mathbb{Z}_{n})$

-

$1.1248004\times 10^{15}$

-

$(\sum\limits_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\frac{1}{6}\Phi(p^{% -m-6i}))+1.$

-

$\tau:(x,y)\rightarrow(u^{2}x,u^{3}y),u\in\;\mathbb{Z}_{n}^{*}$

-

$u^{2}=\frac{\bar{a}b}{a\bar{b}}$

-

$p=4i+1$

-

$i\in\{1,2,3,...,t\}$

-

$a=0\ \text{and}\ b\neq 0\ (j(E)=0$

-

$E/\mathbb{K}$

-

$C_{R}(\mathbb{Z}_{p^{m}})=\left\{\,\begin{array}[]{lll}2p^{m}+6&when&p\equiv 1% -\mod 12\\ -2p^{m}+2&when&p\equiv 5\mod 12\\ -2p^{m}+4&when&p\equiv 7\mod 12\\ -2p^{m}&when&p\equiv 11\mod 12\\ -\end{array}\right.\\$

-

$\begin{split}C_{R}(\mathbb{Z}_{n})&=\prod\limits_{i=1}^{l}\sum_{\mathbb{E}_{k_% -{i}}}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\\ -&=\prod\limits_{i=1}^{l}C_{R}(\mathbb{Z}_{p_{i}}).\end{split}$

-

$\begin{split}N_{R}(\mathbb{Z}_{n})&=\prod_{i=1}^{l}\sum_{\mathbb{E}_{k_{i}}}N^% -{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\\ -N_{R}(\mathbb{Z}_{n})&=\prod_{i=1}^{l}N_{R}(\mathbb{Z}_{p_{i}})\end{split}$

-

$n=3^{m}$

-

$a\neq 0\;\text{and}\;b\neq 0$

-

$gcd(n,6)=1$

-

$n=p_{1}^{e_{1}}p_{2}^{e_{2}}\ldots p_{k}^{e_{k}}$

-

$\Phi(p^{m})=p^{m}-p^{m-1}$

-

$\{4,21\}$

-

$NQR$

-

$\frac{\Phi(n)}{|Aut(E)|}$

-

$\{7,18\}$

-

$N_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}N_{R}(\mathbb{Z}_{p_{i}})$

-

$j(E_{1})=1728$

-

$2310$

-

$C_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}C_{R}(\mathbb{Z}_{p_{i}})$

-

$C_{R}(\mathcal{R})$

-

$u,r,s$

-

$p\equiv 1\mod 2$

-

$N^{{}^{\prime}}_{R}(\mathbb{Z}_{p^{m}})$

-

$\bar{a}=u^{-4}a$

-

$p=3i+1$

-

$|Aut(E_{k})|=2$

-

$\Phi(p^{2m})$

-

$char(\mathbb{K})\neq 2,3$

-

$y^{2}=x^{3}+2$

-

$\displaystyle j(E)$

-

$y^{2}=x^{3}+4x+1$

-

$E_{2}/\mathbb{K}$

-

$\mathbb{Z}_{p^{m}},\ \gcd(p,6)=1$

-

$\{0,5,10,15,20\}$

-

$p\equiv 1\mod 4$

-

$j\in 2\mathbb{Z}$

-

$q^{2}-q-2(q-1)=(q-1)(q-2)$

-

$N^{\prime}_{R}(\mathbb{Z}_{p^{m}})$

-

$s(w,A,B)=\frac{1}{|A|}\sum_{a\in A}cos(\vec{w},\vec{a})-\frac{1}{|B|}\sum_{b% -\in B}cos(\vec{w},\vec{b})$

-

$T(V,W)=\frac{T^{V}_{obs}}{\sqrt{var(T^{V}_{random})}}-\frac{T^{W}_{obs}}{\sqrt% -{var(T^{W}_{random})}}$

-

$T^{V}_{obs}$

-

$cos(\vec{a},\vec{b})=\frac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\|\vec{b}\|}$

-

$\begin{split}T_{South}(1980-2009,1860-1889)=2.4774\\ -T_{Northeast}(1980-2009,1860-1889)=0.8116\end{split}$

-

$s(w,A,B)$

-

$\alpha\leq 0.05$

-

$s(X,Y,A,B)=\sum_{x\in X}s(x,A,B)-\sum_{y\in Y}s(y,A,B)$

-

$T^{W}_{obs}$

-

$s(w,\ A,\ B)$

-

$\text{Gini}(p)=1-\sum_{i=1}^{K}p_{i}^{2},$

-

$F^{s}_{in}$

-

$F^{b}_{in}$

-

$F_{out}^{s}$

-

$F_{i}^{s^{\prime}}=\mathrm{SAR}\left(F_{i}^{s},F_{i}^{b}\right).$

-

$F^{s^{\prime}}$

-

${A}={\left[\begin{array}[]{c}{a^{s}};{a^{b}}\end{array}\right]}={\sigma}(% -Aggregate([{F^{s}_{in}};{F^{b}_{in}}])),$

-

$Aggregate$

-

$F_{i+1}^{b}$

-

$a^{s}$

-

$L_{s_{out}}$

-

$F_{i}^{b}$

-

$P_{s}\in\mathbb{R}^{n\times h\times w}$

-

$A\in{[0,1]}^{2\times h\times w}$

- - - diff --git a/htmls/output_mathjax_example_10044.html b/htmls/output_mathjax_example_10044.html deleted file mode 100644 index 4ca4c6c837c521f5ab8d78b9655ed00d8b3c01d7..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10044.html +++ /dev/null @@ -1,150 +0,0 @@ - - - - MathJax Example - - - - -

$F_{in}^{s}$

-

$F^{u}=conv(F_{in}^{u}\bigodot a^{u})+F_{in}^{u},u\in\left\{s,b\right\},$

-

$L=L_{s_{out}}+\lambda_{s}L_{s_{i}}+\lambda_{b}L_{b_{i}},i\in\{1,2,3,4\},$

-

$L_{s_{i}}$

-

$F_{in}^{b}$

-

$P_{b}\in\mathbb{R}^{1\times h\times w}$

-

$F_{i}^{s^{\prime}}$

-

$Attention(q,k,v)=softmax(\frac{q*k^{t}}{\sqrt{d_{k}}})\cdot v,$

-

${F_{i}^{s}},{F_{i}^{b}}=\left\{\begin{matrix}\mathrm{SME}_{i}\left(F_{i+1},% -\left[F_{i+1};F_{1}\right]\right),&\mathrm{if}\;\;i=4,\\ -\mathrm{SME}_{i}\left(\left[{F_{i+1}^{s}};{F_{i+1}}\right],\left[{F_{i+1}^{b}}% -;{F_{1}}\right]\right),&\mathrm{else}.\end{matrix}\right.$

-

$L_{b_{i}}$

-

$F^{b}$

-

$\mathcal{V}_{rc}$

-

$\mathbf{f}_{u}$

-

$\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\mathrm{c}% -_{u}^{t}+\boldsymbol{B}_{a}\right)\right)-\mathbb{I}\left[u\notin\mathcal{A}^{% -t}\right]\infty,$

-

$\displaystyle=\operatorname{MLP}\left(\|_{k=1}^{K}\sigma\left(\sum_{j\in% -\mathcal{N}_{i}}\alpha_{ij}^{k}\boldsymbol{h}_{j}^{(t)}\mathbf{W}^{k}\right)\right)$

-

$\mathcal{A}^{t>0}=\{v|v\in\operatorname{ONE-HOP}(\hat{\mathcal{V}}_{lg}^{t}\}% -\cup\{\text{STOP}\}$

-

$\mathrm{EGAT}(\cdot)$

-

$\displaystyle s_{u}^{t}$

-

$\displaystyle\boldsymbol{h}_{i}^{(t+1)}$

-

$\hat{\mathcal{V}}_{lg}^{t}$

-

$\mathbb{I}[u\notin\mathcal{A}^{t}]\,(\mathbb{I}[\text{STOP}\notin\mathcal{A}^{% -t}])\mapsto\{0,1\}$

-

$\mathcal{V}_{hg}$

-

$\boldsymbol{h}_{i}^{(t)}\in R^{1\times d}$

-

$\mathbf{x}_{explored}$

-

$\boldsymbol{h}_{i}^{(0)}$

-

$\mathcal{A}^{t>0}=\{v|v\in\operatorname{ONE-HOP}(\hat{\mathcal{G}}_{rc}^{t})\}% -\cup\{\text{STOP}\}$

-

$\mathcal{G}_{hg}=(\mathcal{V}_{hg},\mathcal{E}_{hg})$

-

$\boldsymbol{W}_{b}$

-

$\boldsymbol{W}_{a}$

-

$\hat{\mathcal{V}}_{lg}^{T}$

-

$\mathbf{H}^{(t)}$

-

$\mathcal{E}_{rc}=\left\{(u,v)\mid u,v\in\mathcal{V}_{rc},(u,v)\in\mathcal{E}_{% -p}\right\}$

-

$\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\mathrm{c}% -_{\mathcal{G}}^{t}+\boldsymbol{B}_{a}\right)\right)-\mathbb{I}\left[\mathrm{% -STOP}\notin\mathcal{A}^{t}\right]\infty,$

-

$\hat{\mathcal{V}}_{rc}^{t=0}=\emptyset$

-

$A^{k}\in R^{3d\times d}$

-

$\mathbf{h}_{\mathcal{G}}^{t}$

-

$\mathbf{a}^{k}\in R^{1\times d}$

-

$\mathbf{f}_{ij}^{(t)}\in R^{1\times d}$

-

$\displaystyle e_{ij}$

-

$\mathbf{H}^{(t)}=[\boldsymbol{h}_{1}^{(t)};\boldsymbol{h}_{2}^{(t)};\cdots;% -\boldsymbol{h}_{n}^{(t)}]\in R^{n\times d}$

-

$\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\left(% -\mathbf{h}_{\mathcal{G}}^{t}+\mathbf{c}_{\mathcal{G}}\right)+\boldsymbol{B}_{a% -}\right)\right)-\mathbb{I}\left[\mathrm{STOP}\notin\mathcal{A}^{t}\right]\infty,$

-

$\left\{\mathbf{c}_{u}^{t}\right\}=\operatorname{EGAT}\left(\mathcal{G}_{p},% -\left\{\mathbf{x}_{u}+\mathbb{I}[u\in\hat{\mathcal{V}}_{rc}^{t}]\mathbf{x}_{% -explored}\right\},\left\{\mathbf{x}_{uv}\right\}_{v\in\mathcal{N}(u)}\right).$

-

$\mathcal{G}_{p}=(\mathcal{V}_{p},\mathcal{E}_{p})$

-

$\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\left(% -\mathbf{h}_{u}^{t}+\mathbf{c}_{\mathcal{G}}\right)+\boldsymbol{B}_{a}\right)% -\right)-\mathbb{I}\left[u\notin\mathcal{A}^{t}\right]\infty,$

-

$|\mathcal{E}_{hg}|$

-

$\hat{\mathcal{G}}_{rc}^{t}$

-

$\operatorname{ELU}(x)=\alpha(\exp(x)-1)$

-

$|\mathcal{V}_{p}|$

-

$\hat{\mathcal{V}}_{rc}^{t}$

-

$\mathcal{A}^{0}=\{v|v\in\mathcal{V}_{p}\}$

-

$s_{t}=\{\mathcal{G}_{p},\hat{\mathcal{V}}_{rc}^{t}\}$

-

$\displaystyle\boldsymbol{f}_{ij}^{(t+1)}$

-

$\left\{\mathbf{h}_{u}^{t}\mid u\in\mathcal{V}_{hg}\right\}$

-

$\mathcal{V}_{rc}\subseteq\mathcal{V}_{p}$

-

$\mathcal{A}^{0}=\{v|v\in\mathcal{V}_{hg}\}$

-

$\mathbf{h}_{\mathcal{G}}^{t}=\sum_{u\in\mathcal{V}_{hg}}\mathbf{h}_{u}^{t}$

-

$\boldsymbol{B}_{a}$

-

$\mathcal{G}_{rc}=(\mathcal{V}_{rc},\mathcal{E}_{rc})$

-

$\left\{\mathbf{c}_{u}^{t}\mid u\in\mathcal{V}_{p}\right\}$

-

$e_{ij}\in R$

-

$\mathbf{W}\in R^{d\times d}$

-

$\mathbf{D}_{v}\in R^{n\times n},\mathbf{D}_{e}\in R^{e\times e},\mathbf{W}\in R% -^{e\times e}$

-

$L^{CLIP}(\theta)=\hat{E}_{t}[\min(r_{t}(\theta)\hat{A}_{t},\operatorname{clip}% -\left(r_{t}(\theta),1-\epsilon,1+\epsilon\right)\hat{A}_{t})].$

-

$|\mathcal{V}_{hg}|$

-

$\mathbf{a}\in R^{1\times d}$

-

$\mathbf{f}_{explored}$

-

$s_{u}^{t}\in\mathbb{R}^{1},s_{s}^{t}\in\mathbb{R}^{1}$

-

$\mathcal{E}_{hg}$

-

$\boldsymbol{E}_{ -

$\displaystyle=\mathbf{a}\cdot{\boldsymbol{f}_{ij}^{(t+1)}}^{T},\alpha_{ij}=% -\frac{\exp\left(e_{ij}\right)}{\sum_{k\in\mathcal{N}_{i}}\exp\left(e_{ik}% -\right)},$

-

$\mathrm{HGNN}(\cdot)$

-

$\hat{\mathcal{V}}_{rc}^{T}$

-

$\hat{\mathcal{V}}_{lg}^{t=0}=\emptyset$

-

$\boldsymbol{E}=\left(e_{0},\cdots,e_{t},\cdots,e_{T}\right)$

-

$\mathcal{G}_{rc}$

-

$\mathbb{I}[u\in\hat{\mathcal{V}}_{rc}^{t}]\mapsto\{0,1\}$

-

$\mathbf{\Theta}^{l}\in R^{d^{l}\times d^{l+1}}$

-

$\displaystyle\mathbf{X}^{l+1}=\sigma\left(\mathbf{D}_{v}^{-\frac{1}{2}}\mathbf% -{HWD}_{e}^{-1}\mathbf{H}^{\top}\mathbf{D}_{v}^{-\frac{1}{2}}\mathbf{X}^{l}% -\boldsymbol{\Theta}^{l}\right)$

-

$\mathrm{SOFTMAX}$

-

$\mathbf{c}_{\mathcal{G}}^{t}=\sum_{u\in\mathcal{V}_{p}}\mathbf{c}_{u}^{t}$

-

$\mathbf{c}_{\mathcal{G}}$

-

$\displaystyle s_{s}^{t}$

-

$\mathbb{I}[u\in\hat{\mathcal{V}}_{lg}^{t}]\mapsto\{0,1\}$

-

$p(\boldsymbol{E}\mid\boldsymbol{p})=\prod_{t=1}^{T}p\left(e_{t}\mid\boldsymbol% -{p},\boldsymbol{E}_{ -

$\mathcal{V}_{lg}$

-

$\boldsymbol{f}_{ij}^{(0)}$

-

$\displaystyle=\text{ LeakyReLU }\left(\left[\boldsymbol{h}_{i}^{(t)}\mathbf{W}% -\left\|\boldsymbol{f}_{ij}^{(t)}\right\|\boldsymbol{h}_{j}^{(t)}\mathbf{W}% -\right]A\right),$

-

$A\in R^{3d\times d}$

-

$\mathrm{X}^{l}\in R^{n\times d^{l}}$

-

$s_{t}=\{\mathcal{G}_{p},\mathcal{G}_{hg},\hat{\mathcal{V}}_{lg}^{t}\}$

-

$\mathrm{H}\in R^{n\times e}$

-

$\mathbf{c}_{\mathcal{G}}^{t}$

-

$\mathbf{W}^{k}\in R^{d\times d}$

-

$\mathcal{G}_{hg}$

-

$\boldsymbol{E}=\left(e_{0}\right)$

-

$\alpha_{ij}\in R$

-

$\left\{\mathbf{h}_{u}^{t}\right\}=\operatorname{HGNN}\left(\mathcal{G}_{hg},% -\left\{\mathbf{f}_{u}+\mathbb{I}[u\in\hat{\mathcal{V}}_{lg}^{t}]\mathbf{f}_{% -explored}\right\}\right).$

-

$p(e_{0}\mid\boldsymbol{p})=1$

-

$\text{{TPR}}(f)=\frac{{\text{{TP}}(f)}}{{\text{{P}}}}$

-

$\epsilon=\frac{FP+FN}{TP+TN+FP+FN}\times 100\%$

- - - diff --git a/htmls/output_mathjax_example_10045.html b/htmls/output_mathjax_example_10045.html deleted file mode 100644 index d25ee03f1f0359d9af9c559762672279d860ecb1..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10045.html +++ /dev/null @@ -1,132 +0,0 @@ - - - - MathJax Example - - - - -

$prob(\text{positive-data})=\sigma(\sum_{j}y_{j}^{2}-\theta))$

-

$\text{ROC AUC score}=\int_{-\infty}^{\infty}\text{TPR}(f)\cdot\text{FPR}(f)% -\text{d}f$

-

$\text{{FPR}}(f)=\frac{{\text{{FP}}(f)}}{{\text{{N}}}}$

-

$insertBST(\sf T_{{\sf binNumber}},\sf gBinNumber[{\sf idNumber}(I_{t})],B-{\sf -cost% -}(I_{t}))$

-

${\sf idNumber}(I_{t^{\prime}})$

-

$[\ell_{q},r_{q}-\epsilon]$

-

$\sf IntervalGenerator$

-

$\sf gBinNumber[{\sf idNumber}(I_{t})]$

-

$I\in List_{i}$

-

$\sf VariableSizeDS$

-

$\alpha=\frac{B_{max}}{B_{min}}$

-

${\sf idNumber}=i(1\leq i\leq g)$

-

$L_{i}\leftarrow L_{i}+1$

-

${\sf binNumber}(I_{t^{\prime}})\leftarrow\sf idTobinNumber[{\sf idNumber}(I_{t% -^{\prime}})]$

-

$\sf insertHeap(\sf H^{min}_{F{\sf idNumber}},{\sf idNumber}(I_{t^{\prime\prime% -}}))$

-

${\sf cost}(q)=\min_{p\leq i -

$\sf H^{min}_{L}$

-

$\ell_{q}=T_{i^{\prime}}$

-

$l_{t^{\prime}}=t$

-

$\{(i,1),(i,2),\cdots,(i,L_{i})\}$

-

${\sf Color}$

-

$t=l_{t^{\prime}}$

-

$root.max<{\sf cost}(I_{t})$

-

$\sf deleteBST(\sf T_{U{\sf idNumber}},{\sf idNumber}(I_{t^{\prime\prime}}))$

-

$\sf OPT_{off}^{B}$

-

$\sf searchMap()$

-

$\sf T_{U{\sf idNumber}}$

-

${\sf binNumber}(I)\leftarrow L_{i}$

-

$\sf binToCapacity[{\sf idNumber}(I_{t^{\prime}})]\leftarrow\sf binToCapacity[{% -\sf idNumber}(I_{t^{\prime}})]-{\sf cost}(I_{t^{\prime}})$

-

${\sf binNumber}(I_{t^{\prime}})$

-

$({\sf idNumber}(I_{t^{\prime}}),{\sf binNumber}(I_{t^{\prime}}))$

-

$10-11$

-

$\sf extractMin()$

-

${\sf cost}(i,j)$

-

$p=(x_{p},y_{p})$

-

$r_{q}=T_{j^{\prime}}$

-

$\sf L_{i}$

-

$OPT\geq g$

-

$\sf insertHeap(\sf H^{min}_{L},(l_{t},t))$

-

$decreaseKey(T,node,c)$

-

$node.bin=\sf gBinNumber[{\sf idNumber}(I_{t})]$

-

$rem-{\sf cost}(I)$

-

$node.bin=1$

-

$node.left$

-

${\sf Color}[t]$

-

$\sf searchMap(\sf idTobinNumber,{\sf idNumber}(I_{t^{\prime}}))$

-

$\sf gBinNumber[{\sf idNumber}(I_{t})]\leftarrow\sf gBinNumber[{\sf idNumber}(I% -_{t})]+1$

-

$13-14$

-

${\sf Color}(I)=({\sf idNumber}(I),{\sf binNumber}(I))$

-

$I_{q}=[\ell_{q},r_{q}]$

-

$root.max -

${\sf Color}(I_{t})$

-

$r_{q}=\ell_{p}$

-

$\mathcal{S}=\{S_{1},S_{2},\cdots,S_{k}\}$

-

$\ell_{q}=T_{i}$

-

$root.left.max<{\sf cost}(I_{t})$

-

${\sf ALG}=\sum_{i=1}^{{\sf gIdNumber}}{L_{i}}\leq 2\cdot{\left(\sum_{i=1}^{{% -\sf gIdNumber}}{\frac{{\sf cost}(L_{i})}{B}}\right)}+{\sf gIdNumber}\\$

-

$root.bin$

-

${\sf binNumber}(I_{t})$

-

${\sf cost}(i,1)+2\cdot\sum_{j=2}^{L_{i}-1}{\sf cost}(i,j)+{\sf cost}(i,L_{i})>% -\sum_{j=1}^{L_{i}-1}B_{i}^{j}$

-

$B_{min}=\underset{1\leq i\leq g;1\leq j\leq L_{i}}{\min}B_{i}^{j}$

-

$r_{t^{\prime}}=t$

-

$S_{i}=(x_{i},y_{i})$

-

$root.left.max\geq{\sf cost}(I_{t})$

-

$1\leq j\leq L_{i}-1$

-

$\sf VariableSizeDS(\mathcal{I})$

-

$j\in M\quad and\quad i -

$l_{t^{\prime\prime}}\leftarrow\sf extractMin(\sf H^{min}_{L})$

-

$\sum_{i=1}^{g}\sum_{j=1}^{L_{i}-1}B_{i}^{j}<2\cdot\sum_{i=1}^{g}\sum_{j=1}^{L_% -{i}}{\sf cost}(i,j)$

-

$(r_{t},t)$

-

$5-6$

-

${\sf ALG}\leq 2\cdot{\left(\sum_{i=1}^{{\sf gIdNumber}}{\frac{{\sf cost}(L_{i}% -)}{B}}\right)}+{\sf gIdNumber}\leq 2\cdot\sf OPT_{off}^{B}+\sf OPT_{off}\leq 3% -\cdot\sf OPT_{off}^{B}\\$

-

$\{1,\cdots,{\sf gIdNumber}\}$

-

$I_{t}\neq I_{t^{\prime}}$

-

$root.rem\geq{\sf cost}(I_{t})$

-

$\sf insertBST(\sf T_{U{\sf idNumber}},{\sf idNumber}(I_{t^{\prime}}))$

-

$7-9$

-

$\sf binToCapacity$

-

$\sf insertMap(\mathcal{I},)$

-

${\sf Color}(I)$

-

$decreaseKey(\sf T_{{\sf binNumber}},root,{\sf cost}(I_{t}))$

-

$\sf T_{{\sf binNumber}}$

-

$rem\leftarrow b$

-

$\sum_{i=1}^{{\sf gIdNumber}}{W(i)}\leq 1.7\cdot\sf OPT_{off}^{B}$

-

$root.left=NULL$

-

$({\sf idNumber}(l_{t^{\prime\prime}}),$

-

$root.left\neq NULL$

-

$({\sf idNumber}(I_{t}),{\sf binNumber}(I_{t}))$

-

${\sf gIdNumber}\leq\sf OPT_{off}$

-

$List_{i}\leftarrow\emptyset$

-

$(I_{t^{\prime}})$

-

$\sf gBinNumber[{\sf idNumber}(I_{t})]=1$

-

$node.max$

-

$node.rem=B-{\sf cost}(I_{t})$

-

$\sf OPT_{off}$

-

$root.right$

-

$root.max\geq{\sf cost}(I_{t})$

-

$node.rem=c$

-

$(root,I_{t})$

-

${\sf ALG_{f}}$

- - - diff --git a/htmls/output_mathjax_example_10046.html b/htmls/output_mathjax_example_10046.html deleted file mode 100644 index 0280ea82aa0c7afd5b2825c66194d57f98eb7eac..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10046.html +++ /dev/null @@ -1,134 +0,0 @@ - - - - MathJax Example - - - - -

$\sf insertMap(\sf binToCapacity,<{\sf idNumber}(I_{t^{\prime}}),B-{\sf cost}(I% -_{t^{\prime}})>)$

-

$B_{max}=\underset{1\leq i\leq g;1\leq j\leq L_{i}}{\max}B_{i}^{j}$

-

$root.rem$

-

$I_{t}\neq NULL$

-

${\sf ALG_{f}}=\sum_{i=1}^{{\sf gIdNumber}}{L_{i}}\leq\sum_{i=1}^{{\sf gIdNumber% -}}{\left(W(i)+1\right)}=\sum_{i=1}^{{\sf gIdNumber}}{W(i)}+{\sf gIdNumber}\\$

-

$\sum_{i=1}^{g}L_{i}=\sum_{i=1}^{g}(L_{i}-1)+g\leq\sum_{i=1}^{g}\sum_{j=1}^{L_{% -i}-1}(\frac{B_{i}^{j}}{B_{min}})+g$

-

${\sf ALG}$

-

${\sf idNumber}=i$

-

$\sf insertHeap()$

-

$({\sf idNumber}(I_{t}),{\sf binNumber}(I_{t}))\neq({\sf idNumber}(I_{t^{\prime% -}}),$

-

$I_{t^{\prime\prime}}$

-

${\sf idNumber}(I_{t^{\prime}})\leftarrow\sf extractMin(\sf H^{min}_{F{\sf -idNumber% -}})$

-

$r_{t^{\prime\prime}}\leftarrow\sf extractMin(\sf H^{min}_{R})$

-

$r_{t^{\prime\prime}}=t$

-

$(l_{t},t)$

-

$L_{i}\leftarrow 0$

-

${\sf cost}(I)\leq rem$

-

$rem\leftarrow$

-

${\sf cost}(\cdot)$

-

$T_{1},\cdots,T_{k}$

-

$\sf idTobinNumber[{\sf idNumber}]$

-

$q=(x_{q},y_{q})$

-

$(S_{i},q,S_{j})$

-

${\sf binNumber}(l_{t^{\prime\prime}}))>)$

-

${\sf idNumber}(I_{t^{\prime}})\leftarrow{\sf gIdNumber}$

-

${\sf gIdNumber}$

-

$\sf DSusingBPC(I_{t}=[l_{t},r_{t}])$

-

$node.right$

-

$\sf insertBST()$

-

$l_{t}\geq t$

-

${\sf binNumber}$

-

$\sf insertHeap(\sf H^{min}_{R},(r_{t},t))$

-

$List_{i}$

-

$\frac{{\sf cost}(L_{i})}{B}\geq\frac{L_{i}-1}{2}$

-

$\sf deleteHeap()$

-

${\sf idNumber}$

-

${\sf cost}(L_{i})$

-

$r_{t^{\prime\prime}}==t$

-

$I_{t}=[l_{t},r_{t}]$

-

${\sf binNumber}=L_{i}$

-

$\sf binToCapacity[{\sf idNumber}]$

-

$L_{i}\leq W(i)+1$

-

$\sf binToCapacity[{\sf idNumber}(I_{t^{\prime}})]\leftarrow B-{\sf cost}(I_{t^% -{\prime}})$

-

$(i^{\prime},j^{\prime})=(i,j)$

-

${\sf ALG_{f}}\leq{\sf gIdNumber}+\sum_{i=1}^{{\sf gIdNumber}}{W(i)}\leq 1.7\sf -OPT% -_{off}^{B}+\sf OPT_{off}\leq 2.7\sf OPT_{off}^{B}\\$

-

${\sf binNumber}(I_{t^{\prime}}))$

-

$(root.left,I_{t})$

-

$\sf insertMap({\sf Color}, -

$\sf idTobinNumber[{\sf idNumber}(I_{t^{\prime}})]\leftarrow\sf idTobinNumber[{% -\sf idNumber}(I_{t^{\prime}})]+1$

-

$L_{i}\leq 2\cdot\frac{{\sf cost}(L_{i})}{B}+1$

-

$\sf insertMap()$

-

$insertBST(T,i,c)$

-

$OPT\geq\frac{\sum_{i=1}^{g}\sum_{j=1}^{L_{i}}{\sf cost}(i,j)}{B_{max}}$

-

$\sf H^{min}_{F{\sf idNumber}}$

-

$\sf insertMap(\sf idTobinNumber,<{\sf idNumber}(I_{t^{\prime}}),1>)$

-

$I_{t}\cap I_{t^{\prime}}\neq\emptyset$

-

$root.rem<{\sf cost}(I_{t})$

-

$\sf deleteHeap(\sf H^{min}_{L})$

-

$\{1,2,\cdots,g\}$

-

$\sf(I_{t^{\prime\prime}})$

-

$\sf gBinNumber$

-

$\sf W(i)$

-

$\sf OPT_{off}^{B}\geq\sum_{i=1}^{{\sf gIdNumber}}{\frac{{\sf cost}(L_{i})}{B}}$

-

$node.rem=node.max=B-{\sf cost}(I_{t})$

-

$t^{\prime\prime} -

${\sf cost}(q)>(S_{i},q,S_{j})$

-

$I_{t^{\prime}}=[l_{t^{\prime}},r_{t^{\prime}}]$

-

$[\ell_{p}+\epsilon,r_{p}]$

-

$node.rem$

-

$l_{t^{\prime\prime}}=t$

-

${\sf cost}(I_{t})$

-

$l_{t^{\prime\prime}}==t$

-

${\sf idNumber}(I_{t})$

-

$\sf binToCapacity[{\sf idNumber}(I_{t^{\prime}})]\geq{\sf cost}(I_{t^{\prime}})$

-

$node.bin$

-

$root.left$

-

${\sf gIdNumber}\leftarrow{\sf gIdNumber}+1$

-

$List_{{\sf idNumber}(I)}$

-

$\sf deleteHeap(\sf H^{min}_{R})$

-

$\sf deleteHeap(\sf H^{min}_{F{\sf idNumber}})$

-

${\sf cost}(q)=(S_{i},q,S_{j})$

-

$I_{p}=[\ell_{p},r_{p}]$

-

$\sf OPT_{off}\leq\sf OPT_{off}^{B}$

-

${\sf cost}(i,j)+{\sf cost}(i,j+1)>B_{i}^{j}$

-

$\sf DSusingBPC$

-

${\sf cost}(I_{q})={\sf cost}(q)$

-

$\sf H^{min}_{R}$

-

$r_{q}=T_{j}$

-

$\sf idTobinNumber$

-

$(root.right,I_{t})$

-

$\sf deleteBST()$

-

$O(k^{2}\cdot\mathcal{T})$

-

$\sum_{i=1}^{g}L_{i}\leq\frac{2\cdot\sum\limits_{i=1}\limits^{g}\sum\limits_{j=% -1}\limits^{L_{i}}{\sf cost}(i,j)}{B_{min}}+g\leq\frac{2\cdot\alpha\cdot\sum% -\limits_{i=1}\limits^{g}\sum\limits_{j=1}\limits^{L_{i}}{\sf cost}(i,j)}{B_{% -max}}+g\leq(2\alpha+1)OPT.\\$

-

${\bf q}_{orig}$

-

${\bf 2:}(3,8,7)$

-

${\bf 10:}(14,17,13)$

-

$\partial_{j}$

-

$\displaystyle C=\frac{1}{M}{\bf X}^{T}{\bf X}=\frac{1}{M}(V\Sigma U^{T})(U% -\Sigma V^{T})=\frac{1}{M}V\Sigma^{2}V^{T}.$

-

$\mu^{(h)}$

-

${\bf X}^{(h)}$

- - - diff --git a/htmls/output_mathjax_example_10047.html b/htmls/output_mathjax_example_10047.html deleted file mode 100644 index 36e2fe4c41771f1e018e90d162111263f1550fbc..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10047.html +++ /dev/null @@ -1,128 +0,0 @@ - - - - MathJax Example - - - - -

$N\cdot p_{h}$

-

$n_{\ell}\leq N$

-

${\cal\bf\cal R}=0.9$

-

${\cal O}(MN$

-

$EntrySize$

-

${\bf 4:}(12,9.2)$

-

$R^{(h)}$

-

$\displaystyle C=\frac{1}{M}{\bf X}^{T}{\bf X}=V\Lambda V^{T},$

-

${\cal\bf R}$

-

${\bf 11:}(7,14,12)$

-

$\lambda_{i}=\sigma^{2}_{i}/M\mbox{ and conversely }\sigma_{i}=\sqrt{M\lambda_{% -i}},1\leq i\leq N.$

-

$\displaystyle{\bf X}={\bf U}\Sigma V^{T},$

-

${F_{\ell}}\approx S/EntrySize(n_{\ell})$

-

$m_{1},\ldots,m_{H}$

-

$p=0.30$

-

${\cal O}(MN^{2})$

-

$H\cdot N$

-

$C({\bf q})=A({\bf q})\cap B({\bf q})$

-

$\partial_{j}^{th}$

-

$8\times(8.333/1000)\approx 0.067$

-

$D^{2}({\bf u},{\bf v})=||{\bf u}||^{2}+||{\bf v}||^{2}-2{\bf u}\times{\bf v}$

-

$\lambda_{i}^{(h)}m_{h},\forall{i},\forall{h}$

-

$\displaystyle=\frac{\sum_{h=1}^{H}m_{h}\sum_{j=n_{h}+1}^{N}\lambda_{j}^{(h)}}{% -\sum_{h=1}^{H}m_{h}\sum_{j=1}^{N}\lambda_{j}^{(h)}}.$

-

${\bf 3:}(9,10,8)$

-

$m_{h}\cdot p_{h}$

-

$\sigma_{i}\geq\sigma_{i+1},i=1,\ldots,N-1.$

-

${\bf Y}^{(}h)$

-

$({\bf X}_{i,j}-\bar{x}_{i})/s_{i},1\leq i\leq N\mbox{ where }s_{i}\mbox{is the% - std deviation },\forall{j}.$

-

${\bf 5:}(8,7,20)$

-

${\bf 9:}(11,11,15)$

-

$\kappa_{j}$

-

$\Lambda=\lambda_{1},\ldots,lambda_{N}$

-

${\bf 6:}(6.6.23)$

-

$A({\bf q})$

-

${\cal\bf P}$

-

${\bf 12:}(10,12,3)$

-

${\cal\bf R}=0.8$

-

${\bf Y}^{(h)},1\leq h\leq H$

-

${\bf 1:}(1,2,5)$

-

${\cal\bf R}\approx 1$

-

$\kappa_{j}^{th}$

-

${\bf X}^{(h)},\;h=1,\ldots,H$

-

${\bf 8:}(2,13.9)$

-

${\cal R}=0.8$

-

$\displaystyle NMSE=\frac{\sum_{i=1}^{M}\sum_{j=n+1}^{N}y^{2}_{i,j}}{\sum_{i=1}% -^{M}\sum_{j=1}^{N}y^{2}_{i,j}}=\frac{\sum_{j=n+1}^{N}{\lambda_{j}}}{\sum_{j=1}% -^{N}\lambda_{j}}$

-

${\cal O}(N^{3})$

-

$\displaystyle NMSE$

-

$D^{2}({\bf u},{\bf v})=\left[\sum_{i=1}^{n}|{u}_{i}-{v}_{i}|^{p}\right]^{1/p}.$

-

$\displaystyle{\bf Y}={\bf X}V.$

-

${\cal\bf P}=E[|C({\bf q})|/|B({\bf q})|].$

-

${\sum\nolimits_{k=1}^{n_{\ell}}{\lambda_{k}}}/{\sum\nolimits_{k=1}^{N}{\lambda% -_{k}}}\geq min(\ell\times p,1).$

-

${\cal\bf R}=E[|\frac{C({\bf q}|)}{|(B{\bf q})}|].$

-

$B({\bf q})$

-

$\max{\left\{\left[D({\bf q},\mu^{(i)}-R^{(i)}\right],0\right\}}.$

-

${\bf 7:}(0,3,27)$

-

$K^{*}(k)=k{\bf\cal R}/{\bf\cal P}$

-

$V=N\cdot H+\sum_{h=1}^{H}{\left(N\cdot p_{h}+m_{h}\cdot p_{h})\right)},$

-

$\displaystyle=\frac{\sum_{h=1}^{H}\sum_{i=1}^{m_{h}}\sum_{j=n_{h}+1}^{N}(y^{(h% -)}_{i,j})^{2}}{\sum_{h=1}^{H}\sum_{i=1}^{m_{h}}\sum_{j=1}^{N}(y^{(h)}_{i,j})^{% -2}}$

-

$1302_{1296}$

-

$450_{190}$

-

${}_{m}\textit{AoC}(2013)=9.54$

-

$7261_{4619}$

-

$475_{471}$

-

$2112_{885}$

-

$2974_{2357}$

-

$1530_{1522}$

-

$191_{190}$

-

$460_{452}$

-

$2355_{2344}$

-

$155_{153}$

-

$1306_{974}$

-

${}_{m}AoC(t)=\frac{1}{N}\sum_{j=1}^{N}\overline{\textit{AoC}}(x_{j})$

-

$13,775_{13,667}$

-

$t=2014,\ldots,2022$

-

$778_{770}$

-

$4979_{4724}$

-

$y_{t}{=}{{}_{m}\textit{AoC}(t)}$

-

$1209_{717}$

-

$1245_{1239}$

-

$4914_{3012}$

-

$396_{355}$

-

$8841_{4280}$

-

$3476_{3461}$

-

$2671_{2646}$

-

$0_{0}$

-

${}_{m}\textit{AoC}(2013)=8.56$

-

$1237_{1216}$

-

$\displaystyle\textit{$PoOC$}(x)=\frac{|\mathfrak{O}_{k}(x)|}{M}$

-

$x\mapsto 2\frac{x-\min{\mathbf{x}}}{\max{(\mathbf{x}})-\min{(\mathbf{x}})}-1$

-

$91_{91}$

-

$426_{417}$

-

$2598_{2589}$

-

$1193_{1167}$

-

$389_{388}$

-

$1441_{1436}$

-

$704_{694}$

-

$1383_{932}$

-

$2393_{2382}$

-

$1065_{1060}$

-

$999_{974}$

- - - diff --git a/htmls/output_mathjax_example_10048.html b/htmls/output_mathjax_example_10048.html deleted file mode 100644 index 03e53dc95c72cadc8e31f5e7dec8793c5fb1094a..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10048.html +++ /dev/null @@ -1,120 +0,0 @@ - - - - MathJax Example - - - - -

$\mathfrak{O}_{k}(x)=\{y\,|\,\textit{AoC}(x,y)\geq k\}$

-

$2013_{2007}$

-

$4623_{4259}$

-

$241_{240}$

-

$4516_{4287}$

-

$4751_{4432}$

-

$696_{405}$

-

$3202_{3194}$

-

$\approx 8k$

-

$219_{219}$

-

${}_{m}\textit{AoC}=6.15$

-

$2631_{2604}$

-

$978_{975}$

-

$4936_{4569}$

-

$1916_{1126}$

-

$994_{987}$

-

$23,965_{23,845}$

-

$1313_{1306}$

-

$386_{384}$

-

$2211_{4193}$

-

$2114_{1289}$

-

$2340_{1122}$

-

$47,364_{43,331}$

-

$179_{176}$

-

$30,867_{23,629}$

-

$662_{511}$

-

$137_{137}$

-

$2250_{2246}$

-

$85_{85}$

-

$\frac{y_{t}}{y_{t-1}}-1$

-

$\approx 4k$

-

$139,769_{110,024}$

-

$13,087_{6819}$

-

$1937_{1910}$

-

$4736_{2227}$

-

$1187_{1184}$

-

$1313_{1297}$

-

$2242_{2231}$

-

$\approx 3k$

-

$12,695_{9849}$

-

$601_{153}$

-

$2418_{2414}$

-

$1122_{537}$

-

${}_{m}\textit{AoC}(2022)=5.93$

-

$3882_{3798}$

-

$1238_{1231}$

-

$587_{355}$

-

$108,288_{102,532}$

-

$4574_{4445}$

-

$29964_{2951}$

-

$4571_{4447}$

-

$4797_{4708}$

-

$1004_{997}$

-

$14,625_{12476}$

-

$4598_{4454}$

-

$91_{90}$

-

$2130_{1273}$

-

$4006_{3990}$

-

$2160_{2134}$

-

$692_{176}$

-

$1859_{1166}$

-

${}_{m}\textit{AoC}=10.9$

-

$765_{759}$

-

$331_{331}$

-

$2339_{2314}$

-

$1053_{1050}$

-

$2843_{2806}$

-

$739_{418}$

-

$784_{775}$

-

$\textit{AoC}(x,y_{i})=\textit{YoP}(x)-\textit{YoP}(y_{i})$

-

$2541_{2511}$

-

$402_{373}$

-

$1922_{1425}$

-

$7133_{5640}$

-

$112_{112}$

-

$540_{529}$

-

$13,529_{8316}$

-

$2820_{2811}$

-

$1442_{862}$

-

$4642_{4193}$

-

${}_{m}\textit{AoC}(2022)=7.61$

-

$6578_{4876}$

-

$1318_{1295}$

-

$1337_{1323}$

-

$715_{704}$

-

$99_{99}$

-

$4170_{3198}$

-

$386_{382}$

-

$1662_{1652}$

-

${}_{m}\textit{AoC}(2022)=5.10$

-

${}_{m}\textit{AoC}(2022)=7.24$

-

$5582_{4109}$

-

$402_{399}$

-

$1523_{717}$

-

${}_{m}\textit{AoC}(2013)=17.61$

-

$3084_{1166}$

-

$2835_{2827}$

-

${}_{m}\textit{AoC}(2013)=10.91$

-

$1295_{1289}$

-

$13,754_{7949}$

- - - diff --git a/htmls/output_mathjax_example_10049.html b/htmls/output_mathjax_example_10049.html deleted file mode 100644 index 0adeb53993378f8a6ef62377ad5588087b9711e8..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10049.html +++ /dev/null @@ -1,129 +0,0 @@ - - - - MathJax Example - - - - -

$1055_{1051}$

-

$10,888_{10,786}$

-

$2789_{2589}$

-

$2279_{2243}$

-

$1963_{1950}$

-

$12,505_{12,435}$

-

$2407_{2403}$

-

$1319_{1314}$

-

$118_{117}$

-

$542_{534}$

-

$597_{529}$

-

$875_{554}$

-

$390_{386}$

-

$65,685_{48,391}$

-

$60,346_{59,822}$

-

$637_{629}$

-

$98_{98}$

-

$32,530_{32,229}$

-

$2467_{2467}$

-

$919_{916}$

-

$1356_{1351}$

-

$11,097_{5214}$

-

$2964_{2951}$

-

$11,000_{8546}$

-

$873_{867}$

-

$\displaystyle\textit{${}_{m}PoOC$}(t)=\frac{1}{N}\sum_{x=1}^{N}\textit{$PoOC$}% -(x)$

-

$10,752_{10,660}$

-

$8489_{6324}$

-

$838_{830}$

-

$1210_{721}$

-

$14,741_{14,531}$

-

$1962_{1955}$

-

$3952_{3815}$

-

$5174_{3506}$

-

$587_{474}$

-

$\displaystyle\overline{\textit{AoC}}(x)=\frac{1}{M}\sum_{i=1}^{M}\textit{AoC}(% -x,y_{i})$

-

$4_{4}$

-

$3670_{3667}$

-

$1096_{722}$

-

$766_{759}$

-

$57,935_{29,688}$

-

$663_{487}$

-

$m^{\rm rec}_{i}(t)\geq M$

-

$\Delta t=$

-

$2.3\times 10^{4}$

-

$R_{i}(t)\equiv m^{\rm rec}_{i}(t)/m^{\rm sent}_{i}(t)$

-

$\ell_{i}(t)$

-

$N_{model}$

-

$n^{\rm rec}_{i}(t)\geq T$

-

$Q_{i}(t)\equiv n^{\rm rec}_{i}(t)/n^{\rm sent}_{i}(t)$

-

$-97\%$

-

$\{M,\alpha,T,\beta,L,\tau\}$

-

$n_{i}^{\rm rec}(t)$

-

$R_{i}(t)\geq\alpha$

-

$N_{empirical}$

-

$m^{\rm sent}_{i}(t)$

-

$Q_{i}(t)\geq\beta$

-

$\ell_{i}(t)\geq L$

-

$m^{\rm rec}_{i}(t)$

-

$\phi_{i}(t)\leq\tau$

-

$n^{\rm sent}_{i}(t)$

-

$\phi_{i}(t)$

-

$2.3\times 10^{5}$

-

$\Delta t=30$

-

$u^{B}$

-

$u^{B}_{i}$

-

$\mathcal{L}=\mathcal{L}_{\text{nll}}+\mathcal{L}_{\text{complete}}+\mathcal{L}% -_{\text{consist}},$

-

$p^{g}_{j}$

-

$\mathcal{L}_{\text{complete}}=-\sum_{t}\log p(\mathcal{P}^{\text{new}}_{t}\mid% -\mathcal{P}^{\text{new}}_{ -

$\mathcal{P}^{g}=\{p^{g}_{1},p^{g}_{2},\cdots,p^{g}_{k}\}$

-

$S\in\mathbb{R}^{m\times k}$

-

$p_{B_{i}}$

-

$s_{i,j}=\text{sim}(E(p_{i}),E(p^{g}_{j})),$

-

$p^{g}_{I(i)}$

-

$\mathcal{L}_{\text{nll}}$

-

$\mathcal{P}^{\text{new}}$

-

$s_{i,I(i)}\geq\tau$

-

$\mathcal{P}^{g}_{A}$

-

$\mathcal{P}_{A}=\{p_{A_{1}},p_{A_{2}},\cdots,p_{A_{m}}\}$

-

$\mathcal{P}_{B}=\{p_{B_{1}},p_{B_{2}},\cdots,p_{B_{r}}\}$

-

$\mathcal{L}_{\text{gen}}=-\frac{1}{N}\sum^{N}_{t=1}\log p(u^{B}_{t}\mid u^{B}_% -{ -

$\mathcal{U}=\{u^{A}_{1},u^{B}_{1},u^{A}_{2},u^{B}_{2}\cdots,u^{A}_{n}\}$

-

$\mathcal{P}^{\text{con}}=\{p^{g}_{I(i)}\mid s_{i,I(i)}\geq\tau\}$

-

$\mathcal{L}_{\text{consist}}=-\sum_{p^{g}_{i}\in\mathcal{P}^{\text{con}}}\log{% -\frac{\exp(\text{sim}(h_{i},h_{\mathcal{P}}))}{\sum_{p^{g}_{j}\in\mathcal{P}^{% -g}}\exp(\text{sim}(h_{j},h_{\mathcal{P}}))}},$

-

$p_{A_{i}}$

-

$\mathcal{L}_{\text{complete}}$

-

$\mathcal{L}_{\text{consist}}$

-

$\mathcal{P}^{\text{con}}$

-

$\mathcal{L}_{\text{nll}}=-\sum_{t}\log p(\mathcal{P}_{t}\mid\mathcal{P}_{ -

$u^{B}_{n}$

-

$\mathcal{P}^{\text{new}}=\mathcal{P}^{\text{con}}\cup\mathcal{P}^{\text{miss}}.$

-

$\mathcal{P}^{\text{miss}}=\{p_{i}\mid\forall j\,s_{i,j}<\tau\}$

-

$\mathcal{P}^{\text{miss}}$

-

$h_{\mathcal{P}}$

-

$\mathcal{P}^{g}_{A}=\text{PESS}(\mathcal{U}_{A})=\{p^{g}_{A_{1}},p^{g}_{A_{2}}% -,\cdots,p^{g}_{A_{k}}\}$

-

$\mathcal{U}_{A}=\{u^{A}_{1},u^{A}_{2},\cdots,u^{A}_{n}\}$

-

$u^{A}_{n}$

-

$u^{A}_{i}$

-

$I(i)={\text{argmax}}_{j}s_{i,j}\,.$

-

$\mathcal{P}^{g}$

- - - diff --git a/htmls/output_mathjax_example_1005.html b/htmls/output_mathjax_example_1005.html deleted file mode 100644 index bb8fff824a7a702578f20e584f129c1bebb79753..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1005.html +++ /dev/null @@ -1,121 +0,0 @@ - - - - MathJax Example - - - - -

$i\in\{1,\dots,t-1\}$

-

$V(G)=V(G^{\prime})\setminus\{a,b\}$

-

$G\in\mathcal{G}_{\mathcal{C}}$

-

$\mathcal{M}_{\mathcal{G}_{\mathcal{C}}}=\{2K_{1}\vee H\mid H\in\mathcal{M}_{% -\mathcal{C}}\}$

-

$N_{G}(p_{0})\cap V(H)=\{a_{1},b_{1}\}$

-

$G_{A}\in\mathcal{G}_{\mathcal{C}}$

-

$\Big{(}\bigcup_{v\in V(H)}X_{v}\Big{)}\cap V(C)\neq\emptyset$

-

$(A,B,S)$

-

$V(H)\setminus\{a_{1},b_{1}\}$

-

$\mathcal{O}(|V(H)|+|E(H)|)$

-

$v\in C_{1}$

-

$\{2,\dots,k-1\}$

-

$G_{A}\setminus S$

-

$S\cap B$

-

$G\setminus S_{A}$

-

$\mathcal{M}_{\mathcal{G}_{2}}$

-

$p_{i},\dots,p_{s},a_{2},a_{3},b_{3},b_{1},p_{i}$

-

$v^{xy}=a$

-

$\{2,\dots,t-1\}$

-

$\{\overline{C_{2k+1}}\mid k\in\mathbb{N}\}$

-

$\mathcal{C}_{k}$

-

$C\neq C_{1}$

-

$(a^{\prime},b^{\prime})$

-

$F\in\mathcal{C}$

-

$q_{0}\neq q_{t}$

-

$\mathcal{O}(|V(H)|^{2})$

-

$C_{1},\ldots,C_{k}$

-

$p_{1},\dots,p_{s},q_{2},\dots,q_{t}$

-

$G\setminus S_{B}$

-

$v\in V(H)$

-

$X_{w}\cap V(C)$

-

$N_{G}(v)\cap V(H)\subseteq\{a_{i},b_{i}\}$

-

$v^{xy}\notin\{a,b\}$

-

$a^{\prime},b^{\prime}$

-

$p_{i-1},p_{j+1}$

-

$2\leq i\leq j\leq s-1$

-

$\{0,\dots,s\}$

-

$A,B\subseteq V(G)$

-

$G/xy$

-

$N_{G}(v)\cap V(H)=\{a_{i},b_{i}\}$

-

$uv\in E(H)$

-

$B=\{b_{1},\dots,b_{n}\}$

-

$p_{0}\in N_{A}$

-

$H[C]\in\mathcal{C}$

-

$S\setminus\{v^{xy}\}=S^{\prime}\setminus\{x,y\}$

-

$\{x,y\}\cap\{a,b\}\neq\emptyset$

-

$\mathcal{O}(|V(H)|)$

-

$a,b\in V(G)$

-

$w\in V(H)\setminus U$

-

$\mathcal{G}_{0}\subseteq\mathcal{G}_{1}\subseteq\mathcal{G}_{2}\subseteq\dots$

-

$B=\{b_{1},b_{2},b_{3}\}$

-

$\bigcup_{v\in V(H)}X_{v}\subseteq V(C)$

-

$G\in{\cal G}$

-

$r_{1},\dots,r_{k}$

-

$G[N_{G}(x)]$

-

$G\setminus(V(H)\cup V(P))$

-

$G[N_{G[v_{1},\ldots,v_{i}]}(v_{i})]$

-

$S\setminus\{a,b\}$

-

$X_{v}\subseteq V(C)$

-

$2K_{1}\vee H_{1}$

-

$C\neq C_{2}$

-

$A\setminus N_{G}(v)$

-

$H[B]$

-

$p_{1},p_{s}\in A\cup C$

-

$G[v,a_{i},a_{j},b_{i},b_{j}]$

-

$\mathcal{C}=\mathcal{C}_{2}$

-

$S\cup(V(G)\setminus V(H))$

-

$H[N_{H}(v)]$

-

$x,y\in V(G)$

-

$S\subseteq V(G^{\prime})\setminus\{v^{xy}\}$

-

$S^{\prime}\setminus\{y\}$

-

$N_{G}(v)\cap V(H)=A$

-

$S\subseteq V(G^{\prime})$

-

$H[S]=G[S]$

-

$i>j$

-

$\{x,y\}\cap\{a,b\}=\emptyset$

-

$\mathcal{O}(n(n+m))$

-

$\mathcal{M}_{\mathcal{C}_{0}}=\{K_{1}\}$

-

$X_{u}\cap V(C_{1})$

-

$x\in S^{\prime}$

-

$\{X_{v}\}_{v\in V(2K_{1}\vee H_{1})}$

-

$a_{2},q_{t},b_{1}$

-

$\mathcal{G}_{\mathcal{C}}=\mathcal{G}_{2}$

-

$X_{u}\subseteq V(C)$

-

$\bigcup_{v\in V(H_{1})}X_{v}\subseteq V(H_{2})$

-

$|N_{G}(v)\cap A|\geq 2$

-

$G[v_{1},\dots,v_{i}]$

-

$C_{1}\cup\{a\},C_{2}\cup\{b\},C_{3},\dots,C_{k}$

-

$p_{s}=y$

-

$F\setminus u$

-

$q_{j+1},\dots,q_{t}\in B$

-

$q_{j},\ldots,q_{t},b_{1},a_{1},q_{j}$

-

$(2K_{1}\vee H_{1})\setminus V(H)\cong H_{1}$

-

$u,v\in V(H)$

-

$S=S^{\prime}$

-

$K_{2,3}\in\mathcal{G}_{\mathcal{C}}$

-

$S_{B}:=S\setminus A$

-

$y\in A\cup C$

-

$D\neq C$

-

$F\in\mathcal{F}_{\mathcal{C}}$

- - - diff --git a/htmls/output_mathjax_example_10050.html b/htmls/output_mathjax_example_10050.html deleted file mode 100644 index 4b24688b9bc7b4b819d988c81184c09e606bc117..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10050.html +++ /dev/null @@ -1,138 +0,0 @@ - - - - MathJax Example - - - - -

$\mathbf{V_{M}}$

-

$\lambda_{us}$

-

$D_{S}(y|x)$

-

$\{<\mathbf{v_{i}},\mathbf{p_{i}}>|i=1,...,N,\mathbf{v}\in\mathbb{R}^{1\times d% -},\mathbf{p}\in\mathbb{R}^{L\times d}\}$

-

$err_{S}$

-

${D_{T}}$

-

$\displaystyle d_{\mathcal{H}}(S,T)$

-

$\mathcal{L}_{epa}(X,D)=\lambda_{1}\min D(\mathbf{X}_{i<|\mathbf{p}|\times M})+% -\lambda_{2}\max D(\mathbf{X}_{i\geq|\mathbf{p}|\times M})$

-

$\mathrm{L}1$

-

$\theta_{t}^{\prime}$

-

${D_{S}}$

-

$C\xrightarrow{}F$

-

$\mathcal{D}_{S}(x,y)$

-

$\displaystyle+\frac{1}{q-p}\sum_{j=p+1}^{q-p}D[\mathbb{U}(y_{j}=1)]])$

-

$\lambda_{epa}$

-

$\lambda_{dpa}$

-

$2e-06$

-

$\displaystyle=2(1-\varepsilon_{D_{S}}^{S}-\varepsilon_{D_{T}}^{T}) -

$\mathbf{V_{M}}=\mathrm{argmax}\sum_{i=1}^{M}\psi(\mathbf{V},\gamma(x))$

-

$d_{\mathcal{H}}$

-

$\theta_{t}^{\prime}=\alpha\theta_{t-1}^{\prime}+(1-\alpha)\theta_{t}$

-

$C\xrightarrow{}B$

-

$\mathbb{U}=\{x_{i},y=0\}_{i=1}^{p}\cup\{x_{j},y=1\}_{j=p+1}^{q}$

-

$\displaystyle=2(1-\varepsilon_{D}^{S}-\varepsilon_{D}^{T})$

-

$\mathcal{L}_{epa}$

-

$err_{T}$

-

$\displaystyle d_{{\mathcal{H}_{S},\mathcal{H}_{T}}}(S,T)$

-

$\mathcal{L}_{dpa}$

-

$\mathcal{L}=\lambda_{s}\mathcal{L}_{sup}+\lambda_{us}\mathcal{L}_{unsup}+% -\lambda_{epa}\mathcal{L}_{epa}+\lambda_{dpa}\mathcal{L}_{dpa}$

-

$\displaystyle=2(1-\min_{D\in\mathcal{H}}[\frac{1}{p}\sum_{i=1}^{p}D[\mathbb{U}% -(y_{i}=0)]$

-

$2e-04$

-

$S\xrightarrow{}C$

-

$\mathcal{D}_{T}(x)$

-

$\mathbf{L}\times C\times 2$

-

$\mathit{fineTuningCorpus}$

-

$\mathit{chaseCorpus}\cup\{\mathit{chasePromptResp}$

-

$D,\Sigma,G,\mathit{model},\mathit{nlpTask}$

-

$\mathsf{postprocess}(\mathit{chaseCorpus})$

-

$\mathit{chasePromptResp}$

-

$\textit{Position}(\textit{EGTech},0.3,37.2,1)$

-

$\mathsf{generate}(\mathsf{preprocess}(\mathit{verbPlan},\mathit{nlpTask}))$

-

$\mathsf{composeBack}(\mathit{step},\mathit{chase})$

-

$6.9\$$

-

$\mathit{Open}(\mathit{EGTech},0.3,1),\neg\mathit{MarketClose}(1)\to\mathit{% -Accepted}(\mathit{EGTech},0.3,1)$

-

$\displaystyle\textit{Accepted}(x,y,t_{1}),\textit{Price}(p_{1},t_{1}),k=y*p_{1}$

-

$\displaystyle\to\textit{Position}(x,y,k,t_{1})$

-

$\mathit{chase}$

-

$\mathit{verbChase}$

-

$\mathsf{verbalizeStep}(\mathit{step},\mathit{stepAggrContrib},G)$

-

$\mathsf{fineTune}(\mathit{model},\mathit{fineTuningCorpus})$

-

$\mathsf{map}(\mathit{tokenizedCorpus},\mathit{verbStep})$

-

$\mathit{tokenizedCorpus}$

-

$\mathit{chaseCorpus}\setminus\{\langle\mathit{prompt},\mathit{resp}\rangle$

-

$\displaystyle\textit{Close}(x,t_{2}),\textit{Price}(p_{2},t_{2}),\textit{% -Position}(x,y,k,t_{1}),$

-

$\mathit{ftModel}$

-

$\mathit{chaseCorpus}$

-

$\mathit{qualityScore}$

-

$\mathit{stepAggrContrib}$

-

$\mathsf{hasAggregate}(\mathit{step}.\mathsf{getRule}())$

-

$147\$$

-

$\mathit{verbChase}\cup\{\mathit{verbStep}$

-

$\textsc{Vadalog}.\mathsf{reason}(D,\Sigma)$

-

$\textit{Accepted}(\textit{EGTech},0.3,1)$

-

$\displaystyle t_{2}>t_{1},pl=y*p_{2}-k$

-

$\mathit{qualityScore}\leq\mathit{threshold}$

-

$y*p_{2}-k$

-

$\displaystyle\to\textit{Return(x,pl)}$

-

$\mathsf{verbalizePlan}(\Sigma.\mathsf{getLogicPlan}())$

-

$p_{1}*y$

-

$\displaystyle\textit{Open}(x,y,t_{1}),\neg\,\textit{MarketClosed}(t_{1})$

-

$\mathsf{checkQuality}(\langle\mathit{prompt},\mathit{resp}\rangle,\mathit{% -nlpTask},\mathit{verbChase})$

-

$\mathit{verbPlan}$

-

$\langle\mathit{prompt},\mathit{resp}\rangle$

-

$\displaystyle\to\textit{Accepted}(x,y,t_{1})$

-

$\mathit{verbStep}$

-

$\mathit{chaseCorpus}\cup\mathsf{paraphrase}(\langle\mathit{prompt},\mathit{% -resp}\rangle)$

-

$\Sigma(D)=D$

-

$\bar{\boldsymbol{l}}\in\mathbb{R}^{L\times 64}$

-

$\tilde{\boldsymbol{\alpha}}\in\mathbb{R}^{512}$

-

$\displaystyle\boldsymbol{l}=\text{MLPs}(\boldsymbol{\Phi}_{D}^{c}(\bar{% -\boldsymbol{l}}+\text{MLPs}(\hat{\boldsymbol{\alpha}}))).$

-

$(\boldsymbol{\theta}_{id}^{n},\boldsymbol{\theta}_{id})$

-

$\displaystyle\mathcal{L}_{con}=-\log\left[\frac{\exp\left(\mathcal{S}\left(% -\boldsymbol{\theta}_{id}^{p},\boldsymbol{\theta}_{id}\right)\right)}{\exp\left% -(\mathcal{S}\left(\boldsymbol{\theta}_{id}^{p},\boldsymbol{\theta}_{id}\right)% -\right)+\exp\left(\mathcal{S}\left(\boldsymbol{\theta}_{id}^{n},\boldsymbol{% -\theta}_{id}\right)\right)}\right].$

-

$\tilde{\boldsymbol{m}}=1-\boldsymbol{m}$

-

$\boldsymbol{m}_{i}=\boldsymbol{\mathcal{G}}_{MOD}(\hat{\boldsymbol{I}}_{i-1})$

-

$\hat{\boldsymbol{X}}$

-

$\boldsymbol{I}_{ad}^{i}$

-

$\boldsymbol{I}_{id}$

-

$\boldsymbol{I}_{id}^{i}=\hat{\boldsymbol{I}}_{1}$

-

$\hat{\boldsymbol{I}}_{i-1}$

-

$\mathcal{L}_{id}=\left\|\boldsymbol{\alpha}-\hat{\boldsymbol{\alpha}}\right\|_% -{2}$

-

$\boldsymbol{\mathcal{F}}_{MB}$

-

$\boldsymbol{\theta}_{id}\in\mathbb{R}^{512}$

-

$\displaystyle\hat{\boldsymbol{\beta}}=\text{MLPs}(\boldsymbol{\Phi}_{D}^{e}(% -\boldsymbol{\theta}_{e}+\bar{\boldsymbol{l}}+\text{MLPs}(\hat{\boldsymbol{% -\alpha}}))).$

-

$\boldsymbol{I}_{ad}$

-

$\boldsymbol{F}_{sc}=\boldsymbol{\mathcal{F}}_{SC}(\boldsymbol{\mathcal{E}}(% -\boldsymbol{I}_{rd})$

-

$(\boldsymbol{\theta}_{id}^{p},\boldsymbol{\theta}_{id})$

-

$\boldsymbol{\Phi}_{D}^{e}$

-

$\boldsymbol{\mathcal{F}}_{SC}(\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{rd}))$

-

$\bar{\boldsymbol{l}}$

-

$p\textless 0.8$

- - - diff --git a/htmls/output_mathjax_example_10051.html b/htmls/output_mathjax_example_10051.html deleted file mode 100644 index febb4019b31d36bbe89e6c37e46cf604c1dd9a49..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10051.html +++ /dev/null @@ -1,160 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\boldsymbol{\theta}_{e}=\text{Avg}(\boldsymbol{\Phi}_{E}^{e}(% -\text{MLPs}(\boldsymbol{A})+\text{PE})),$

-

$\hat{\boldsymbol{I}}_{1}=\boldsymbol{\mathcal{D}}(\text{CCF}(\boldsymbol{z}_{T% -},\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{rd}^{1}),\boldsymbol{Y})$

-

$\boldsymbol{\mathcal{G}}_{MOD}$

-

$\hat{\boldsymbol{I}}_{i}=\boldsymbol{\mathcal{D}}^{{}^{\prime}}(\text{CCF}(% -\boldsymbol{z}_{T},\boldsymbol{\mathcal{E}}([\boldsymbol{I}_{rd}^{i},% -\boldsymbol{I}_{id}^{i},\boldsymbol{I}_{ad}^{i}]),\boldsymbol{Y}),\boldsymbol{% -\mathcal{E}}(\boldsymbol{I}_{bg}^{i}),\boldsymbol{m}_{i})$

-

$\boldsymbol{\mathcal{D}}^{{}^{\prime}}$

-

$\boldsymbol{\alpha}\in\mathbb{R}^{80}$

-

$\mathcal{L}_{reg}=\left\|\boldsymbol{l}-\boldsymbol{\beta}\right\|_{2}$

-

$\boldsymbol{\theta}_{id}$

-

$\boldsymbol{F}^{k}_{ve}$

-

$\boldsymbol{I}_{rd}$

-

$\displaystyle\bar{\boldsymbol{l}}=\boldsymbol{\Phi}_{E}^{c}(\text{MLPs}(% -\boldsymbol{A})+\text{PE}),$

-

$i=2...H$

-

$\left\{\hat{\boldsymbol{Y}}_{1},\dots,\hat{\boldsymbol{Y}}_{N}\right\}$

-

$\boldsymbol{I}_{bg}^{i}=\boldsymbol{\mathcal{G}}_{IA}(\hat{\boldsymbol{I}}_{1})$

-

$\hat{\boldsymbol{V}}$

-

$\hat{\boldsymbol{\alpha}}\in\mathbb{R}^{80}$

-

$\boldsymbol{\mathcal{G}}_{IA}(\hat{\boldsymbol{I}}_{1})$

-

$\boldsymbol{\beta}\in\mathbb{R}^{L\times 64}$

-

$\boldsymbol{\Phi}_{D}^{c}$

-

$\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{ad}))$

-

$\hat{\boldsymbol{I}_{1}}$

-

$\bar{\boldsymbol{\alpha}}\in\mathbb{R}^{512}$

-

$\boldsymbol{I}_{id}^{i}$

-

$\bar{\boldsymbol{\theta}}_{id}\in\mathbb{R}^{L\times 512}$

-

$\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{rd}^{1})$

-

$\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{id})$

-

$\boldsymbol{\theta}_{e}\in\mathbb{R}^{512}$

-

$\boldsymbol{I}_{bg}^{i}$

-

$\boldsymbol{\Phi}_{E}^{id}$

-

$\displaystyle\mathbb{E}_{\boldsymbol{z}_{0},\boldsymbol{\varepsilon}\sim N(0,% -\boldsymbol{I}),t,\boldsymbol{F}_{sc}}\left\|\boldsymbol{\varepsilon}-% -\boldsymbol{\varepsilon}_{\theta}\left(\boldsymbol{z}_{t},t,\boldsymbol{Y},% -\boldsymbol{F}_{sc}\right)\right\|_{2}^{2},$

-

$\boldsymbol{I}_{bg}$

-

$\boldsymbol{F}^{k}_{vd}$

-

$\boldsymbol{\theta}_{e}$

-

$\boldsymbol{\mathcal{E}}$

-

$\boldsymbol{I}_{rd}^{i}$

-

$\displaystyle\hat{\boldsymbol{F}}_{vd}^{k}=\boldsymbol{F}_{vd}^{k}\otimes% -\boldsymbol{m}+\text{Conv}(\boldsymbol{F}_{ve}^{k})\otimes\tilde{\boldsymbol{m% -}},$

-

$\boldsymbol{\Phi}_{E}^{c}$

-

$\displaystyle\bar{\boldsymbol{\alpha}},\bar{\boldsymbol{\theta}}_{id}=% -\boldsymbol{\Phi}_{E}^{id}([\tilde{\boldsymbol{\alpha}},\text{MLPs}(% -\boldsymbol{A})]+\text{PE}),$

-

$\boldsymbol{\Phi}_{E}^{e}$

-

$\displaystyle\mathbb{E}_{\boldsymbol{z}_{0},\boldsymbol{\varepsilon}\sim N(0,% -\boldsymbol{I}),t,\boldsymbol{Y}}\left\|\boldsymbol{\varepsilon}-\boldsymbol{% -\varepsilon}_{\theta}\left(\boldsymbol{z}_{t},t,\boldsymbol{Y}\right)\right\|_% -{2}^{2}.$

-

$\boldsymbol{l}\in\mathbb{R}^{L\times 64}$

-

$\boldsymbol{I}_{ad}^{i}=\boldsymbol{\mathcal{M}}(\hat{\boldsymbol{I}}_{i-1})$

-

$\boldsymbol{\mathcal{G}}_{IA}$

-

$\boldsymbol{\mathcal{F}}_{SC}$

-

$\boldsymbol{\mathcal{F}}_{TI}$

-

$\displaystyle\hat{\boldsymbol{\alpha}}=\text{MLPs}(\bar{\boldsymbol{\alpha}}),% -\boldsymbol{\theta}_{id}=\text{Avg}(\bar{\boldsymbol{\theta}}_{id}),$

-

$\left\{\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{M}\right\}$

-

$\mathcal{L}_{lip}=-\boldsymbol{X}\text{log}P(\hat{\boldsymbol{X}}|\boldsymbol{% -V})$

-

$\boldsymbol{A}\in\mathbb{R}^{L\times 1280}$

-

$\left\{\boldsymbol{I}_{rd}^{i}\right\}_{i=1}^{H}$

-

$\hat{\boldsymbol{I}}_{1}$

-

$****$

-

$P(y|x,v)$

-

$\hat{y}_{test}$

-

$F_{S}(\boldsymbol{x}|\boldsymbol{P}_{j},\boldsymbol{B})$

-

$\begin{split}\mathcal{L}_{color}\left(\boldsymbol{x}\right)=\sum_{j}^{v}|\left% -(F_{S}\left(\boldsymbol{x}\mid\boldsymbol{P}_{j},\boldsymbol{S},\boldsymbol{C}% -_{\boldsymbol{T}}\right)-\boldsymbol{I}_{j}^{\textit{{r}}}\right)\odot\\ -F_{S}\left(\boldsymbol{x}\mid\boldsymbol{P}_{j},\boldsymbol{B}\right)|,\end{split}$

-

$\boldsymbol{x}=\left(\boldsymbol{E}+\lambda\boldsymbol{L}\right)^{-1}% -\boldsymbol{\mu},$

-

$\boldsymbol{C}_{\boldsymbol{T}}$

-

$\mathcal{L}\left(\boldsymbol{x}\mid\Theta,\boldsymbol{I}^{\textit{{r}}}\right)% -=\sum_{j}^{v}\left|F(\boldsymbol{x}\mid\Theta_{j})-\boldsymbol{I}_{j}^{\textit% -{{r}}}\right|.$

-

${\boldsymbol{T}}$

-

$\mathcal{L}_{depth}\left(\boldsymbol{x}\right)=\sum_{j}^{v}|\left(F_{D}\left(% -\boldsymbol{x}\mid\boldsymbol{P}_{j}\right)-F_{D}\left(\boldsymbol{\tilde{x}}% -\mid\boldsymbol{P}_{j}\right)\right)|,$

-

$\mathcal{M}=\left(\mathcal{X},\mathcal{E}\right)$

-

$\boldsymbol{L}\in\mathbb{R}^{n\times n}$

-

$\mathcal{L}_{normal}\left(\boldsymbol{x}\right)=\sum_{j}^{v}|\left(F_{N}\left(% -\boldsymbol{x}\mid\boldsymbol{P}_{j}\right)-F_{N}\left(\boldsymbol{\tilde{x}}% -\mid\boldsymbol{P}_{j}\right)\right)|,$

-

$\mathcal{L}=\mathcal{L}_{color}+\mathcal{L}_{depth}+\mathcal{L}_{normal}.$

-

$\boldsymbol{E}\in\mathbb{I}^{n\times n}$

-

$\boldsymbol{S}\in\mathcal{\mathbb{R}}^{9\times 3}$

-

$\boldsymbol{I}^{\textit{{r}}}\in\mathbb{R}^{w\times h\times c}$

-

$\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}}$

-

$\boldsymbol{P}_{j}\in\mathbb{PL}\left(3\right)$

-

$\boldsymbol{x}\leftarrow\boldsymbol{x}-\eta\left(\boldsymbol{E}+\lambda% -\boldsymbol{L}\right)^{-2}\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}}.$

-

$\boldsymbol{L}_{ij}=\begin{cases}-w_{ij},&\textrm{if}\quad(i,j)\in\mathcal{E}% -\\ -\sum_{(i,k)\in\mathcal{E}}w_{ik}&\mathrm{if}\quad i=j\\ -0&\mathrm{otherwise},\end{cases}$

-

$\boldsymbol{x}\in\mathbb{R}^{n\times 3}$

-

$\boldsymbol{p}(\textit{x},\textit{y})$

-

$\boldsymbol{I}(\boldsymbol{p}(x,y))=F(\boldsymbol{x};\Theta),$

-

$\boldsymbol{C}_{\boldsymbol{T}}\in\mathcal{\mathbb{R}}^{t\times t\times 3}$

-

$F_{S}:\mathbb{R}^{n\times 3}\rightarrow\mathbb{R}^{w\times h\times 3}$

-

$F_{D}:\mathbb{R}^{n\times 3}\rightarrow\mathbb{R}^{w\times h}$

-

$\boldsymbol{B}\in\left\{0,1\right\}^{t\times t\times 3}$

-

$F_{N}:\mathbb{R}^{n\times 3}\rightarrow\mathbb{R}^{w\times h\times 3}$

-

$\boldsymbol{\mu}\leftarrow\boldsymbol{\mu}-\eta\frac{\partial\boldsymbol{x}}{% -\partial\boldsymbol{\mu}}\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}},$

-

$\boldsymbol{\tilde{x}}\in\mathbb{R}^{m\times 3}$

-

$Loss=\sum_{i=1}^{N}BCE(v_{i},\hat{v}_{i})+\sum_{i=1}^{N}\sum_{j=1}^{N}\hat{v}_% -{i}\hat{v}_{j}sim(i,j)$

-

$sim(i,j)=h_{i}^{L}*h_{j}^{L}$

-

$\bigtriangledown_{\boldsymbol{z}}f(v)$

-

$a_{i}^{*}=\frac{\|\boldsymbol{a}^{*}\|}{\mu}\Big{(}\tau-\gamma\eta^{*}\psi_{i}% -(c_{i})-\log(1+e^{-y^{i}\cdot(\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{% -)}^{+},$

-

$\psi_{i}(c_{i})\leq p+\frac{1}{m^{k}}.$

-

$\displaystyle+\int_{y_{i}=z_{i}}\epsilon_{i}(y,z_{-i})dy_{i}\Big{)}f_{i}(z_{i}% -)dz_{i}f_{-i}(z_{-i})dz_{-i}.$

-

$t_{i}(0)\geq\int_{0}^{\infty}\epsilon_{i}(z)dz$

-

$c_{i}\cdot\epsilon_{i}(c)-t_{i}(c)$

-

$\Gamma(n,1)$

-

$\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(\boldsymbol{w}^{*})^{T}% -\boldsymbol{x}^{i}})$

-

$\boldsymbol{\epsilon}\in\mathbb{R}^{m}$

-

$p\gamma$

-

$\{\beta,\mu,\sigma,\gamma\}$

-

$t_{i}(D,\boldsymbol{c^{\prime}})=\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c^{% -\prime}}),\quad\Psi_{i}(c)=c+\frac{F_{i}(c)}{f_{i}(c)}$

-

$\displaystyle=\mathbb{E}_{S}\big{[}\sup_{\|w\|\leq\beta}(\mathbb{E}[% -\boldsymbol{w}]-\mathbb{\hat{E}}_{S}(\boldsymbol{w}))\big{]}$

-

$m\rightarrow\infty,$

-

$\lambda>\lambda_{conv}$

-

$\eta=\sum_{i=1}^{m}\epsilon_{i}=m\epsilon_{avg}$

- - - diff --git a/htmls/output_mathjax_example_10052.html b/htmls/output_mathjax_example_10052.html deleted file mode 100644 index 61831231869b8cdb60659e09a20fcc5f49d8f932..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10052.html +++ /dev/null @@ -1,210 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\qquad-\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}}% -)\big{)}\Big{]}\bigg{]}$

-

$V\subset\mathbb{R}^{n},$

-

$z_{i}=\log(\frac{1}{m})\ \forall i$

-

$\boldsymbol{w(D,\boldsymbol{c})}$

-

$\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})% -+2\boldsymbol{b}^{T}\boldsymbol{w}/\eta$

-

$\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+\sqrt{\sigma p\gamma+2||b||\sqrt{p% -\gamma}}<1.$

-

$\hat{\mathbb{L}}_{S}[\boldsymbol{w}]=\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot% -\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\boldsymbol{b^{\prime}}^{T}\boldsymbol{% -w},$

-

$\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\theta})$

-

$\displaystyle=\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(% -\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}\Big{(}\frac{\|% -\boldsymbol{a}^{*}\|}{\mu}\Big{)},$

-

$\psi_{i}(c_{i})\leq p+\frac{1}{m^{k}}$

-

$\epsilon_{i}(c)$

-

$\displaystyle\!\!-\!b(x_{1}^{i}y^{i})\begin{bmatrix}b(x_{1}^{i}y^{i})\!&\!\!-2% -\lambda_{0}^{i}\frac{x_{2}^{i}y^{i}}{x_{1}^{i}y^{i}}&\!\!-2\lambda_{0}^{i}% -\frac{x_{3}^{i}y^{i}}{x_{1}^{i}y^{i}}&\ldots&\!\!-2\lambda_{0}^{i}\frac{x_{n}^% -{i}y^{i}}{x_{1}^{i}y^{i}}\\ -b(x_{2}^{i}y^{i})&2\lambda_{0}^{i}&0&\ldots&0\\ -.&.&.&\ldots&.\\ -.&.&.&\ldots&.\\ -.&.&.&\ldots&.\\ -b(x_{n}^{i}y^{i})&0&0&\ldots&2\lambda_{0}^{i}\\ -\end{bmatrix}\!.$

-

$\boldsymbol{\epsilon},\boldsymbol{c^{\prime}},\boldsymbol{t}$

-

$\displaystyle\leq f(v^{t})+\langle\bigtriangledown f(v^{t}),v^{t+1}-v^{t}% -\rangle+\frac{K}{2}\|v^{t+1}-v^{t}\|^{2}$

-

$h(\boldsymbol{b^{\prime}}_{1})$

-

$\displaystyle=\mathbb{E}_{S}\big{[}\sup_{\|w\|\leq\beta}\mathbb{E}_{S^{\prime}% -}[\hat{\mathbb{E}}_{S^{\prime}}(\boldsymbol{w})-\mathbb{\hat{E}}_{S}(% -\boldsymbol{w})]\big{]}$

-

$\displaystyle\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T% -}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}$

-

$\displaystyle\frac{\mathbb{P}[\boldsymbol{w}(\boldsymbol{x}^{1},\ldots,% -\boldsymbol{x}^{m}=\boldsymbol{d},y^{1},\ldots,y^{m}=y)\in V]}{\mathbb{P}[% -\boldsymbol{w}(\boldsymbol{x}^{1},\ldots,\boldsymbol{x}^{m}=\boldsymbol{d^{% -\prime}},y^{1},\ldots,y^{m}=y^{\prime})\in V]}=\frac{h(\boldsymbol{b^{\prime}}% -_{1})}{h(\boldsymbol{b^{\prime}}_{2})}$

-

$z^{i}=(\boldsymbol{x}^{i},y^{i}),$

-

$\displaystyle\lim_{m\rightarrow\infty}\min_{\boldsymbol{w},\boldsymbol{% -\epsilon},\|\boldsymbol{w}\|\leq\beta}\mathbb{E}[\mathbb{I}_{\{sign(% -\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]+\gamma\sum_{i=1}\epsilon_{i}\Psi_{% -i}(c_{i})$

-

$\gamma K\leq 1$

-

$D=\{(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m},y^{m})\}$

-

$\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})% -\leq\hat{\mathbb{L}}(D,\boldsymbol{a},\boldsymbol{w},\eta)+\mu\|a\|+\frac{% -\sigma}{\eta}.$

-

$\sup_{\|\boldsymbol{w}\|\leq\beta}\Big{|}\mathbb{L}(D,\boldsymbol{c};% -\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})-\hat{\mathbb{L}}(D,% -\boldsymbol{a},\boldsymbol{w},\eta)\Big{|}\leq\mu\|a\|+\frac{\sigma}{\eta}.$

-

$0 -

$(\lambda-\lambda_{conv})$

-

$\{\boldsymbol{z}:\sum e^{z_{i}}=1,\eta e^{z_{i}}\leq\epsilon_{i}\}$

-

$\sum_{i=1}^{m}\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(% -\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}=\frac{\mu}{\|% -\boldsymbol{a}^{*}\|}.$

-

$\boldsymbol{\epsilon(D,\boldsymbol{c^{\prime}})}$

-

$\Psi_{i}(\boldsymbol{c_{i}})$

-

$f^{*}_{\eta}=\inf_{\boldsymbol{w},\boldsymbol{z}}f(\boldsymbol{w},\boldsymbol{% -z},\eta)$

-

$\boldsymbol{w}\leftarrow\boldsymbol{w}-\alpha\frac{d}{d\boldsymbol{w}}g(% -\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}}),$

-

$\displaystyle P(|\phi(S)-\mathbb{E}_{S}[\phi(S)]|>t)$

-

$\|\boldsymbol{w}\|\leq\beta,\eta>0,\boldsymbol{a}>0$

-

$\sum_{i}a_{i}=1,a\geq 0,\eta\geq 0,a_{i}\eta\leq\epsilon_{i}$

-

$\mathbb{E}[\mathbb{I}_{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y}]$

-

$\displaystyle a=e^{z_{i}}\log(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}\cdot y% -^{i}}),$

-

$e^{-\epsilon_{i}}\leq\frac{\mathbb{P}[\mathbb{A}(S)\in V]}{\mathbb{P}[\mathbb{% -A}(S^{\prime})\in V]}\leq e^{\epsilon_{i}}\quad\forall i\in\{1,2,\ldots,|S|\}.$

-

$v(t)=\sum_{i=0}^{n-1}\frac{(t/2)^{i}}{i!}e^{-\frac{t}{2}}$

-

$L_{1}>0$

-

$\eta=m\cdot\epsilon_{avg}$

-

$\epsilon_{i}(D,\boldsymbol{c})$

-

$\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})% -=\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]$

-

$\max(0,f(x)).$

-

$\boldsymbol{b^{\prime}}=\frac{2\boldsymbol{b}}{\eta}$

-

$\boldsymbol{b^{\prime}}_{1}$

-

$\mathrm{min}_{\boldsymbol{w}}\hat{\mathbb{L}}(D,\boldsymbol{w},\boldsymbol{a},\eta)$

-

$(\boldsymbol{a},\eta)\in\mathbb{F}$

-

$(f(x))^{+}$

-

$\min_{\boldsymbol{w},\boldsymbol{\epsilon}(\cdot)}\mathbb{E}_{\boldsymbol{c}}% -\bigg{[}\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\theta})% -+\gamma\cdot\sum_{i=1}^{m}\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c})\bigg{]}.$

-

$c_{i}\epsilon_{i}.$

-

$\displaystyle\qquad+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}+\mu\|% -\boldsymbol{a}\|+\sigma\frac{1}{\eta}\bigg{]}+\gamma\eta\sum a_{i}\Psi_{i}(c_{% -i})=0.$

-

$\displaystyle\qquad\leq\exp\Bigg{(}\frac{-2t^{2}}{\sum a_{i}^{2}\log^{2}(1+e^{% -\beta})+(\frac{2\beta v^{-1}(\delta^{\prime})}{\eta})^{2}}\Bigg{)}.$

-

$\displaystyle\qquad\qquad\bigg{(}\frac{1}{\log(1+e^{-\boldsymbol{w}^{T}% -\boldsymbol{x}^{i}y^{i}})}-\ln 2\cdot e^{\boldsymbol{w}^{T}\boldsymbol{x}^{i}y% -^{i}}\bigg{)}.$

-

$\displaystyle 2\lambda>\bigg{(}\frac{1}{\ln 2}\bigg{)}^{2}\cdot\max_{i}\|% -\boldsymbol{x}^{i}\|^{2}\bigg{(}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}% -y^{i}}}{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}\bigg{)}^{2}$

-

$\displaystyle+\mathbb{E}_{S^{\prime},\sigma_{i}}\bigg{[}\sup_{\|w\|\leq\beta}% -\Big{[}\sum_{i=1}^{m}-a_{i}\sigma_{i}\big{(}\log(1+e^{-y^{\prime i}\cdot% -\boldsymbol{w}^{T}\boldsymbol{x^{\prime}}^{i}})\Big{]}\bigg{]}$

-

$(\boldsymbol{w},\boldsymbol{z})$

-

$\epsilon_{avg}\in[0,L]$

-

$\displaystyle\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma\sum_{i=1% -}^{m}\epsilon_{i}\Psi_{i}(c_{i})\bigg{]},$

-

$\hat{\boldsymbol{w_{2}}}$

-

$t_{i}(D,\boldsymbol{c^{\prime}})=\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c^{% -\prime}}),$

-

$\|\boldsymbol{x}\|,\|\boldsymbol{w}\|$

-

$\eta=\frac{1}{m^{k^{\prime}}}$

-

$\displaystyle=f(v)-f(v^{*})+f(v^{*})-L$

-

$\mu(\delta,\beta)=\Big{(}\frac{3\ln\frac{1}{\delta}}{\sqrt{2}}\Big{)}\log(1+e^% -{\beta})+\frac{\beta}{\ln 2}$

-

$\mathbb{B}_{\boldsymbol{c}_{-i}}[t_{i}(D,c,\boldsymbol{c}_{-i})]$

-

$\sigma(\delta,\delta^{\prime},\beta)=\Big{(}\frac{6\ln\frac{1}{\delta}}{\sqrt{% -2}}+1\Big{)}\Big{(}2\beta v^{-1}(\delta^{\prime})\Big{)}$

-

$\displaystyle\!\!\begin{bmatrix}a&b(x_{1}^{i}y^{i})&0&\ldots&0\\ -b(x_{1}^{i}y^{i})&2\lambda_{0}^{i}+c(x_{1}^{i}y^{i})^{2}&-2\lambda_{0}^{i}% -\frac{x_{2}^{i}y^{i}}{x_{1}^{i}y^{i}}&\ldots&-2\lambda_{0}^{i}\frac{x_{n}^{i}y% -^{i}}{x_{1}^{i}y^{i}}\\ -b(x_{2}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{2}^{i}y^{i})&2\lambda_{0}^{i}&\ldots&0% -\\ -.&.&.&\ldots&.\\ -.&.&.&\ldots&.\\ -.&.&.&\ldots&.\\ -b(x_{n}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{n}^{i}y^{i})&0&\ldots&2\lambda_{0}^{i}% -\\ -\end{bmatrix}\!.$

-

$\displaystyle\qquad+\tau(1-\sum_{i=1}^{m}a_{i})-\sum_{i=1}^{m}\zeta_{i}a_{i}-% -\kappa\eta.$

-

$\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}\bigg{]}+% -\gamma\eta\sum a_{i}\Psi_{i}(c_{i})$

-

$\displaystyle\min_{\boldsymbol{w}}\Bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}% -\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\frac{\lambda}{2}\|\boldsymbol{w}% -\|^{2}+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}\Bigg{]},$

-

$\mathbb{F}=\{(\boldsymbol{a},\eta):\eta>0,\sum_{i}a_{i}=1,a_{i}>0,a_{i}\eta% -\leq\epsilon_{i}\ \forall i\}$

-

$h_{i}(c)$

-

$\displaystyle\leq\frac{\|\boldsymbol{w}\|}{\ln 2}\mathbb{E}_{\sigma}[\sum_{i}a% -_{i}\sigma_{i}\boldsymbol{x}^{i}]+\frac{\beta v^{-1}(\delta^{\prime})}{\eta}$

-

$\mathbb{I}_{\{\cdot\}}$

-

$\displaystyle\displaystyle\min_{\boldsymbol{a},\eta,\boldsymbol{w}}\bigg{[}% -\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})% -+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}$

-

$\eta=\sum_{i}\epsilon_{i}$

-

$\sigma/\eta$

-

$a_{i}=\epsilon_{i}/\eta$

-

$\{(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m-1},y^{m-1}),(% -\boldsymbol{d^{\prime}},y^{\prime})\}$

-

$\hat{\boldsymbol{w_{1}}}$

-

$(2\lambda_{0}^{i})^{(n-1)}(2\lambda_{0}^{i}+c\|\boldsymbol{x}\|^{2})$

-

$\|\boldsymbol{w}\|^{2}$

-

$\displaystyle\lim_{m\rightarrow\infty}\|a\|=0$

-

$\mu,\sigma,\eta\in\mathbb{R}_{+}$

-

$\{(\boldsymbol{x}^{i},y^{i}),c_{i}\},$

-

$\lim_{p\rightarrow 0}\lim_{m\rightarrow\infty}\min_{\begin{subarray}{c}% -\boldsymbol{w},\boldsymbol{\epsilon}\\ -\beta,\|\boldsymbol{w}\|\leq\beta\end{subarray}}\mathbb{E}[\mathbb{I}_{\{sign(% -\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]\!+\!\gamma\sum_{i=1}^{m}\epsilon_{% -i}\Psi_{i}(c_{i})\!\leq\!1.$

-

$\displaystyle\mathbb{E}\big{[}\mathbb{\hat{L}}_{S}[\boldsymbol{w}]\big{]}$

-

$a_{i}=\frac{1}{N}$

-

$\sum_{i=0}^{n-1}\frac{(\frac{\eta r}{2\beta})^{i}}{i!}e^{-\frac{\eta r}{2\beta% -}}=\delta^{\prime}.$

-

$\displaystyle\displaystyle\min_{\boldsymbol{a},\eta,\boldsymbol{w},\boldsymbol% -{\epsilon}}$

-

$(\boldsymbol{a^{*}},\boldsymbol{w^{*}},\eta^{*})$

-

$\big{|}\|\boldsymbol{b^{\prime}}_{1}\|-\|\boldsymbol{b^{\prime}}_{2}\|\big{|}<% -2a_{m}$

-

$\epsilon_{i}=a_{i}\eta.$

-

$\gamma\eta\sum_{i=1}^{m}e^{z_{i}}\psi_{i}(c_{i})$

-

$g(\boldsymbol{w},\boldsymbol{z},\eta)$

-

$\displaystyle\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c_{i},\boldsymbol{c}_{-i},z% -^{i};\epsilon_{i},t_{i})$

-

$\|\log(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}y})\|$

-

$\boldsymbol{b^{\prime}}_{2}$

-

$\displaystyle\!a\cdot\!\begin{bmatrix}2\lambda_{0}^{i}+c(x_{1}^{i}y^{i})^{2}&% -\!\!-2\lambda_{0}^{i}\frac{x_{2}^{i}y^{i}}{x_{1}^{i}y^{i}}&\!\!-2\lambda_{0}^{% -i}\frac{x_{3}^{i}y^{i}}{x_{1}^{i}y^{i}}&\ldots&\!\!-2\lambda_{0}^{i}\frac{x_{n% -}^{i}y^{i}}{x_{1}^{i}y^{i}}\\ -c(x_{1}^{i}y^{i})(x_{2}^{i}y^{i})&2\lambda_{0}^{i}&0&\ldots&0\\ -.&.&.&\ldots&.\\ -.&.&.&\ldots&.\\ -.&.&.&\ldots&.\\ -c(x_{1}^{i}y^{i})(x_{n}^{i}y^{i})&0&0&\ldots&2*\lambda_{0}^{i}\\ -\end{bmatrix}$

- - - diff --git a/htmls/output_mathjax_example_10053.html b/htmls/output_mathjax_example_10053.html deleted file mode 100644 index e1c31efb6b070e62658a83ee5685eb2736094402..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10053.html +++ /dev/null @@ -1,171 +0,0 @@ - - - - MathJax Example - - - - -

$2\boldsymbol{c}-e^{-4}$

-

$f(\boldsymbol{w}^{t},\boldsymbol{z}^{t},\eta)-f^{*}_{\eta}\leq\alpha^{t}(f(% -\boldsymbol{w}^{0},\boldsymbol{z}^{0},\eta)-f^{*}_{\eta}).$

-

$\displaystyle\leq\frac{\beta}{\ln 2}\sqrt{\sum_{i}a_{i}^{2}}+\frac{\beta v^{-1% -}(\delta^{\prime})}{\eta},$

-

$\displaystyle\leq\lim_{m\rightarrow\infty}\log(1+e^{-\delta m^{k^{\prime\prime% -}}})+\frac{2||b||m^{k^{\prime\prime}}}{m^{k^{\prime}}}+\sigma\frac{1}{m^{k^{% -\prime}}}+\gamma\frac{m^{k^{\prime}}}{m^{k}}$

-

$\beta=\frac{1}{\sqrt{p\gamma}}$

-

$\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma% -\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})\bigg{]}$

-

$\displaystyle\qquad\qquad\bigg{(}\frac{1}{\log(1+e^{-\boldsymbol{w}^{T}% -\boldsymbol{x}^{i}y^{i}})}-\ln 2\cdot e^{\boldsymbol{w}^{T}\boldsymbol{x}^{i}y% -^{i}}\bigg{)},$

-

$\displaystyle\mathbb{E}_{S}[\phi(S)]$

-

$\eta^{*}\rightarrow\infty$

-

$\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c^{\prime}_{i},\boldsymbol{c}_{-i},z^{i}% -;\epsilon_{i},t_{i})\leq 0\quad\forall i,c^{\prime}_{i},\boldsymbol{c}$

-

$\epsilon_{i}=a_{i}\eta$

-

$\sum_{i}t_{i}(D,\boldsymbol{c^{\prime}})$

-

$\displaystyle=e^{\eta(\|\boldsymbol{b^{\prime}}_{1}\|-\|\boldsymbol{b^{\prime}% -}_{2}\|)/2}\leq e^{a_{m}\eta}\leq e^{\epsilon_{m}},$

-

$\displaystyle\lim_{m\to\infty}m\cdot\mathbb{P}\big{(}\psi_{i}(c_{i})\leq p+1/m% -^{k}\big{)}\rightarrow\infty.$

-

$\|\boldsymbol{x}^{i}\|\leq 1\ \ \forall i$

-

$\alpha\in(0,1],\ g_{\min}=\infty$

-

$\displaystyle a_{i}^{*}$

-

$\quad\quad\quad\forall\boldsymbol{w}\ s.t.\ \|\boldsymbol{w}\|\leq\beta,\ (% -\boldsymbol{a},\eta)\in\mathbb{F}$

-

$t_{i}(0)\geq\int_{0}^{c_{i}}\epsilon_{i}(z)dz.$

-

$\displaystyle\lim_{m\rightarrow\infty}\min_{\begin{subarray}{c}\boldsymbol{a},% -\eta,\boldsymbol{w}\\ -\|w\|\leq\beta,\beta\end{subarray}}\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}% -\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}% -\boldsymbol{w}}{\eta}$

-

$(f(x))^{+}=\max(0,f(x))$

-

$\displaystyle\leq\lim_{m\rightarrow\infty}\min_{\beta,\eta}\bigg{[}\log(1+e^{-% -\delta\|x\|\cdot\|w\|})+\|\boldsymbol{b}\|\frac{2\beta}{\eta}$

-

$\epsilon_{i},$

-

$\mathbb{E}_{\boldsymbol{c}_{-i}}[\epsilon_{i}(D,c,\boldsymbol{c}_{-i})]$

-

$\|\boldsymbol{b}\|\sim\Gamma(n,1)$

-

$\displaystyle=\sum_{i}a_{i}\mathbb{E}[\log(1+e^{-y\cdot\boldsymbol{w}^{T}% -\boldsymbol{x}})]+\mathbb{E}[\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}]$

-

$(1-\delta)(1-\delta^{\prime})$

-

$\mathbb{U}[e^{-4},5e^{-4}]$

-

$\mu\|\boldsymbol{a}\|+\sigma/\eta$

-

$\displaystyle\Big{|}\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},% -\boldsymbol{\boldsymbol{w}})-\hat{\mathbb{L}}(D,\boldsymbol{a},\boldsymbol{w},% -\eta)\Big{|}$

-

$\frac{1}{1+e^{y^{\prime}\hat{\boldsymbol{w_{2}}}^{T}\boldsymbol{d^{\prime}}}}<1$

-

$\displaystyle\leq\frac{1}{\ln 2}E_{\sigma}[\sup_{\boldsymbol{w}}\sum_{i}a_{i}% -\sigma_{i}(-y^{i}\boldsymbol{w}^{T}\boldsymbol{x}^{i})]+\frac{\beta v^{-1}(% -\delta^{\prime})}{\eta}$

-

$\|\boldsymbol{b^{\prime}}_{1}-\boldsymbol{b^{\prime}}_{2}\|<2a_{m}$

-

$\displaystyle\leq\mu(\delta,\beta)\|a\|+\sigma(\delta,\delta^{\prime},\beta)% -\Big{(}\frac{1}{\eta}\Big{)}.$

-

$\epsilon_{\text{avg}}$

-

$a_{i}=e^{z_{i}}$

-

$\boldsymbol{z}_{\text{opt}}\leftarrow\boldsymbol{z},$

-

$z^{i}=(\boldsymbol{x}^{i},y^{i}).$

-

$\displaystyle c=e^{z_{i}}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}% -{(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}})^{2}}\frac{1}{\ln 2}.$

-

$\|\boldsymbol{x}^{i}\|\leq 1$

-

$(\boldsymbol{a},\boldsymbol{w})$

-

$\hat{\boldsymbol{w}}=\mathrm{argmin}_{\boldsymbol{w}}\hat{\mathbb{L}}(D,% -\boldsymbol{w},\boldsymbol{a},\eta)$

-

$\boldsymbol{z}\leftarrow\boldsymbol{z}-\alpha\frac{d}{d\boldsymbol{z}}g(% -\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}}),$

-

$\gamma\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})$

-

$R_{m}(\boldsymbol{w})$

-

$\sum_{i=1}^{m}\lambda_{0}^{i}=\lambda$

-

$\sum_{i}e^{z_{i}}=1,\eta\geq 0,\eta e^{z_{i}}\leq\epsilon_{i}$

-

$\lambda_{conv}$

-

$\boldsymbol{c}=[c_{1},c_{2},\ldots,c_{m}].$

-

$\big{|}\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}\big{|} -

$N\rightarrow m\mathbb{P}\big{(}\psi_{i}(c_{i})\leq\frac{1}{m^{k}}\big{)}$

-

$0 -

$\displaystyle\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma\eta\sum a% -_{i}\Psi_{i}(c_{i})$

-

$\eta r/\beta=t$

-

$\epsilon_{avg}$

-

$\mathbb{L}(D,\boldsymbol{c^{\prime}};\boldsymbol{\epsilon,\theta})$

-

$\boldsymbol{d^{\prime}}$

-

$\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+2\sqrt{\sigma p\gamma+2\|% -\boldsymbol{b}\|\sqrt{p\gamma}}<1.$

-

$\boldsymbol{b^{\prime}}_{1}-\frac{a_{m}\cdot\boldsymbol{d}y}{1+e^{y\hat{% -\boldsymbol{w_{1}}}^{T}\boldsymbol{d}}}=\boldsymbol{b^{\prime}}_{2}-\frac{a_{m% -}\cdot\boldsymbol{d^{\prime}}y^{\prime}}{1+e^{y^{\prime}\hat{\boldsymbol{w_{2}% -}}^{T}\boldsymbol{d^{\prime}}}}.$

-

$\displaystyle f(v)-L$

-

$\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma% -\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})\bigg{]},$

-

$\displaystyle=\mathbb{E}_{S,S^{\prime}}\bigg{[}\sup_{\|w\|\leq\beta}\Big{[}% -\sum_{i=1}^{m}a_{i}\sigma_{i}\big{(}\log(1+e^{-y^{\prime i}\cdot\boldsymbol{w}% -^{T}\boldsymbol{x^{\prime}}^{i}})$

-

$\|\boldsymbol{w}\|$

-

$a_{i}=1/m$

-

$\displaystyle\lim_{m\rightarrow\infty}\min_{\boldsymbol{a},\eta,\boldsymbol{w}% -,\|w\|\leq\beta,\beta}\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot% -\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}% -}{\eta}$

-

$\sum_{i=1}^{m}a_{i}=1$

-

$\displaystyle=f(v^{t})-\frac{\gamma}{2}(2-K\gamma)\|\bigtriangledown f(v^{t})% -\|^{2}$

-

$\psi_{i}(c_{i})$

-

$\displaystyle=\mathbb{E}_{c_{i}}[c_{i}\epsilon_{i}(c_{i})]+\mathbb{E}_{c_{i}}[% -\int_{c_{i}}\epsilon_{i}(z)dz]$

-

$R_{m}(\boldsymbol{w})=\mathbb{E}_{\sigma,S}[\sup_{\|w\|\leq\beta}\sum_{i=1}^{m% -}a_{i}\sigma_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})],$

-

$\displaystyle 2\lambda_{0}^{i}>\frac{b^{2}-ac}{a}\|\boldsymbol{x}^{i}\|^{2}$

-

$\epsilon_{\text{opt}}\leftarrow\epsilon_{\text{avg}},$

-

$\Psi_{i}(c_{i})=c_{i}+\frac{F_{i}(c_{i})}{f_{i}(c_{i})}\ \ \forall i\in\mathbb% -{N},c_{i}\in\mathbb{R}$

-

$\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c^{\prime}_{i},\boldsymbol{c}_{-i};% -\epsilon_{i},t_{i})=c_{i}\cdot\epsilon_{i}-t_{i}.$

-

$c_{i}\epsilon_{i}(c_{i})+\int_{c_{i}}\epsilon_{i}(z)dz$

-

$a_{i} -

$\|\boldsymbol{w}\|\leq\beta$

-

$\displaystyle=\Big{(}\frac{\sigma+2\boldsymbol{b}^{T}\boldsymbol{w}}{\gamma% -\sum_{i=1}^{m}\psi_{i}(c_{i})a_{i}^{*}}\Big{)}^{1/2}.$

-

$z^{i}=(\boldsymbol{x}^{i},y^{i})$

-

$\displaystyle\textrm{s.t.}\|\boldsymbol{w}\|\leq\beta,(\boldsymbol{a},\eta)\in% -\mathbb{F}.$

-

$\sqrt{\sigma p\gamma+2\|\boldsymbol{b}\|\sqrt{p\gamma}}/\gamma$

-

$\|\boldsymbol{x}^{i}\|\leq 1\ \forall i$

-

$\hat{\mathbb{L}}(D,\boldsymbol{a},\boldsymbol{w},\eta)$

-

$\exists\ k>0,$

-

$t_{i}(\boldsymbol{c})=\Psi_{i}(c_{i})\epsilon_{i}(\boldsymbol{c}),$

-

$\displaystyle\!\leq\!\hat{R}_{S}(\boldsymbol{w})\!+\!\sqrt{\frac{\ln\frac{1}{% -\delta}}{2}}\Bigg{(}\sqrt{\sum a_{i}^{2}\log^{2}(1+e^{\beta})}+\frac{2\beta v^% -{-1}(\delta^{\prime})}{\eta}\Bigg{)}.$

-

$\displaystyle\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})% -\neq y\}}]\leq\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\boldsymbol{w}^{T}\boldsymbol% -{x}^{i})})+\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}$

-

$\displaystyle=\mathbb{E}_{S,S^{\prime}}\big{[}\sup_{\|w\|\leq\beta}[\mathbb{% -\hat{E}}_{S^{\prime}}(\boldsymbol{w})-\mathbb{\hat{E}}_{S}(\boldsymbol{w})]% -\big{]}$

-

$\forall\ \delta>0$

-

$\|\bigtriangledown f(v^{t})\|^{2}$

-

$\displaystyle\mathbb{E}_{c_{i}}[t_{i}(c_{i})]$

-

$\epsilon_{i}(D,\boldsymbol{c^{\prime}})$

-

$\displaystyle\leq\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+\sqrt{\sigma p% -\gamma+2\|\boldsymbol{b}\|\sqrt{p\gamma}}.$

-

$\displaystyle g(\boldsymbol{w},\boldsymbol{z},\eta)=f(\boldsymbol{w},% -\boldsymbol{z},\eta)+\eta\gamma\sum_{i=1}^{m}e^{z_{i}}\Psi_{i}(c_{i}).$

-

$\epsilon_{i}(c_{i})\geq 0$

-

$\displaystyle\leq\delta+\frac{1}{2\mu}\|\bigtriangledown f(v)\|^{2}.$

-

$\mu||\boldsymbol{a}||$

-

$A+2\lambda_{0}^{i}I$

-

$\|w\|\leq\beta$

-

$\mathbb{N}(0,1)$

- - - diff --git a/htmls/output_mathjax_example_10054.html b/htmls/output_mathjax_example_10054.html deleted file mode 100644 index 01ed49261c38c32a79d788310f2ee40f3b7c9208..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10054.html +++ /dev/null @@ -1,167 +0,0 @@ - - - - MathJax Example - - - - -

$\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})$

-

$L\in\mathbb{R}_{+}.$

-

$\boldsymbol{\epsilon}=(\epsilon_{i})_{i=1}^{m}\in\mathbb{R}_{+}^{m}$

-

$\displaystyle\!\!\!f(\boldsymbol{w},\boldsymbol{z},\eta)$

-

$\mathbb{E}[\boldsymbol{w}]=0$

-

$c_{i}^{\prime}=c_{i}.$

-

$\displaystyle\sup_{\|\boldsymbol{w}\|\leq\beta}$

-

$\epsilon_{i}\leq k\epsilon_{avg}$

-

$\displaystyle\lim_{p\rightarrow 0}\lim_{m\rightarrow\infty}\min_{\boldsymbol{w% -},\boldsymbol{\epsilon}}\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}% -\boldsymbol{x})\neq y\}}]+\gamma\sum_{i=1}^{m}\epsilon_{i}\Psi_{i}(c_{i})$

-

$\displaystyle\qquad-e^{z^{2}}(\log(1+e^{-\boldsymbol{w^{2}}^{T}\boldsymbol{x}y% -})+\frac{e^{z^{2}}}{\|(e^{z^{2}})\|}\|^{2}.$

-

$a_{i}\eta=\epsilon_{i}\ \forall i$

-

$\displaystyle\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}% -\boldsymbol{x}^{i}})+\frac{\lambda}{2}\|\boldsymbol{w}\|^{2}+\frac{2% -\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}$

-

$\big{\{}(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m},y^{m})\big{\}}$

-

$(\boldsymbol{w}^{t},\boldsymbol{z}^{t})_{t\in\mathbb{N}}$

-

$\displaystyle\leq\delta+\langle\bigtriangledown f(v),v-v^{*}\rangle-\frac{\mu}% -{2}\|v^{*}-v\|^{2}$

-

$\displaystyle\|\bigtriangledown f(v^{1})-\bigtriangledown f(v^{2})\|^{2}$

-

$\mathbb{E}_{c_{i}}[t_{i}(c_{i})]$

-

$(\boldsymbol{a},\eta)$

-

$\boldsymbol{w}_{\text{opt}}\leftarrow\boldsymbol{w},$

-

$\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})$

-

$\displaystyle\min_{\boldsymbol{a},\eta,\boldsymbol{w}}$

-

$\|\boldsymbol{x}^{i}\|\leq 1,$

-

$c\|\boldsymbol{x}\|^{2}$

-

$\displaystyle\hat{R}_{S}(\boldsymbol{w})$

-

$\displaystyle b=e^{z_{i}}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}% -{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}\frac{1}{\ln 2},$

-

$m\rightarrow\infty.$

-

$\Psi_{i}(c_{i})=c_{i}+\frac{F_{i}(c_{i})}{f_{i}(c_{i})}$

-

$\mathbb{E}_{c_{i}}[t_{i}(c_{i})]=\mathbb{E}_{\boldsymbol{c}}[\Psi_{i}(c_{i})% -\epsilon_{i}(\boldsymbol{c})],$

-

$a_{i}=\frac{1}{m}$

-

$\displaystyle R_{m}(\boldsymbol{w})\leq\hat{R}_{S}(\boldsymbol{w})$

-

$\Psi_{i}(c_{i})$

-

$\frac{\lambda}{2}||\boldsymbol{w}||^{2}$

-

$\hat{R}_{S}(\boldsymbol{w})$

-

$f(v^{*})-L<\delta$

-

$\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})$

-

$\displaystyle\qquad+\|e^{z^{1}}(\log(1+e^{-\boldsymbol{w^{1}}^{T}\boldsymbol{x% -}y})+\frac{e^{z^{1}}}{\|(e^{z^{1}})\|}$

-

$\displaystyle\qquad+\Big{(}\frac{6\ln\frac{1}{\delta}}{\sqrt{2}}+1\Big{)}\Big{% -(}\frac{2\beta v^{-1}(\delta^{\prime})}{\eta}\Big{)}.$

-

$\displaystyle=\mathbb{E}[\log(1+e^{-y\cdot\boldsymbol{w}^{T}\boldsymbol{x}})]=% -\mathbb{L}[\boldsymbol{w}].$

-

$\|\boldsymbol{x}\|\leq 1$

-

$\bigtriangledown_{\boldsymbol{w}}f(v)$

-

$\displaystyle\leq\mathbb{E}_{S,\sigma_{i}}\bigg{[}\sup_{\|w\|\leq\beta}\Big{[}% -\sum_{i=1}^{m}a_{i}\sigma_{i}\big{(}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}% -\boldsymbol{x}^{i}})\Big{]}\bigg{]}$

-

$\mathbb{U}[p,q]$

-

$\displaystyle\phi(S)$

-

$\psi_{i}(c_{i})\in[p,q]$

-

$\displaystyle=\bigg{[}\sum_{i=1}^{m}e^{z_{i}}\log(1+e^{-y^{i}\cdot\boldsymbol{% -w}^{T}\boldsymbol{x}^{i}})+\frac{\lambda}{2}\|\boldsymbol{w}\|^{2}$

-

$\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]$

-

$\eta=\frac{\sqrt{(\sigma+2\beta||b||)}}{\sqrt{p\gamma}}$

-

$f(v^{t})-L\leq(1-\gamma\mu)^{t}(f(v^{0})-L)+\delta.$

-

$\sum_{i=1}^{m}\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(% -\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}=\frac{\mu}{\|% -\boldsymbol{a}^{*}\|},$

-

$\displaystyle=\int_{z_{-i}}\int_{z_{i}}\Big{(}z_{i}\epsilon_{i}(z_{i},z_{-i})$

-

$\displaystyle+\sqrt{\frac{\ln\frac{1}{\delta}\Big{(}\sum a_{i}^{2}\log^{2}(1+e% -^{\beta})+\big{(}\frac{2\beta v^{-1}(\delta^{\prime})}{\eta})^{2}\Big{)}}{2}}.$

-

$c=c_{i}$

-

$v^{*}\in\mathbb{R}^{m+n}$

-

$\displaystyle\qquad\qquad+\sqrt{\frac{\ln\frac{1}{\delta}\Big{(}\sum_{i}a_{i}^% -{2}\log^{2}(1+e^{\beta})+\big{(}\frac{2\beta v^{-1}(\delta^{\prime})}{\eta})^{% -2}\Big{)}}{2}}$

-

$\displaystyle\qquad\qquad+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}+\mu\|% -e^{\boldsymbol{z}}\|+\sigma\frac{1}{\eta}\bigg{]}.$

-

$\{\lambda,\mu,\sigma\}$

-

$h(\boldsymbol{b^{\prime}})\propto e^{-\frac{\eta}{2}\|\boldsymbol{b^{\prime}}\|}$

-

$\displaystyle\leq\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c^{\prime}_{i},% -\boldsymbol{c}_{-i},z^{i};\epsilon_{i},t_{i})\quad\forall i,c^{\prime}_{i},% -\boldsymbol{c}.$

-

$c_{i}\cdot\epsilon_{i}(c_{i})-t_{i}(c_{i})\leq c_{i}\cdot\epsilon_{i}(c^{% -\prime}_{i})-t_{i}(c^{\prime}_{i}).$

-

$\displaystyle\leq\min_{\beta}\log(1+e^{-\delta\beta})+\sqrt{\sigma+2\beta\|% -\boldsymbol{b}\|}\sqrt{p\gamma}$

-

$\big{|}\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}\big{|}$

-

$r=\frac{\beta v^{-1}(\delta^{\prime})}{\eta}.$

-

$y^{i}\in\{+1,-1\}$

-

$\{\lambda,\mu,\sigma,\gamma\}$

-

$t_{i}(c_{i})\geq c_{i}\epsilon_{i}(c_{i})+\int_{c_{i}}\epsilon_{i}(z)dz.$

-

$\eta^{*}=\Big{(}\frac{\sigma+2\boldsymbol{b}^{T}\boldsymbol{w}}{\gamma\sum_{i=% -1}^{m}\psi_{i}(c_{i})a_{i}^{*}}\Big{)}^{1/2}.$

-

$\displaystyle\qquad+\Bigg{[}\Big{(}\frac{3\ln\frac{1}{\delta}}{\sqrt{2}}\Big{)% -}\log(1+e^{\beta})+\frac{\beta}{\ln 2}\Bigg{]}\sqrt{\sum_{i}a_{i}^{2}}$

-

$\beta=m^{k^{\prime\prime}}$

-

$\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y}\}]\leq% -\mathbb{E}[\log(1+e^{-y\cdot\boldsymbol{w}^{T}\boldsymbol{x}})]=\mathbb{L}[% -\boldsymbol{w}].$

-

$\hat{\mathbb{L}}(D,\boldsymbol{w},\boldsymbol{a},\eta)$

-

$\boldsymbol{c^{\prime}}$

-

$\displaystyle\leq\lim_{m\rightarrow\infty}\min_{\beta,\eta}\bigg{[}\log(1+e^{-% -\delta\beta})+\|\boldsymbol{b}\|\frac{2\beta}{\eta}$

-

$\|\boldsymbol{b}\|$

-

$t_{i}(c_{i})=t_{i}(0)+c_{i}\epsilon_{i}(c_{i})-\int_{0}^{c_{i}}\epsilon_{i}(z)dz.$

-

$g_{\min}>g(\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}})$

-

$\displaystyle\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T% -}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}+\mu\|% -\boldsymbol{a}\|+\frac{\sigma}{\eta}\bigg{]}$

-

$\mathbb{E}_{\boldsymbol{c}}\bigg{[}\mathbb{L}(D,\boldsymbol{c^{\prime}};% -\boldsymbol{\epsilon,\theta})+\gamma\sum_{i}t_{i}(D,\boldsymbol{c^{\prime}})% -\bigg{]},$

-

$\displaystyle\begin{bmatrix}a&b(x_{1}^{i}y^{i})&\ldots&b(x_{n}^{i}y^{i})\\ -b(x_{1}^{i}y^{i})&2\lambda_{0}^{i}+c(x_{1}^{i}y^{i})^{2}&\ldots&c(x_{1}^{i}y^{% -i})(x_{n}^{i}y^{i})\\ -b(x_{2}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{2}^{i}y^{i})&\ldots&c(x_{2}^{i}y^{i})(x% -_{n}^{i}y^{i})\\ -.&.&\ldots&.\\ -.&.&\ldots&.\\ -.&.&\ldots&.\\ -b(x_{n}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{n}^{i}y^{i})&\ldots&c(x_{n}^{i}y^{i})^{% -2}+2\lambda_{0}^{i}\\ -\end{bmatrix},$

-

$\displaystyle\eta^{*}$

-

$N\rightarrow m\mathbb{P}\big{(}\psi_{i}(c_{i})\leq p+\frac{1}{m^{k}}\big{)}$

-

$\epsilon_{avg}=\sum_{i}\epsilon_{i}/m$

-

$\frac{1}{1+e^{y\hat{\boldsymbol{w_{1}}}^{T}\boldsymbol{d}}}<1$

-

$\mathbb{E}_{\boldsymbol{c}_{-i}}[\mathbb{L}(D,\boldsymbol{c};\boldsymbol{% -\epsilon},\boldsymbol{\theta})]$

-

$\hat{\mathbb{L}}(D,\boldsymbol{w},\boldsymbol{a},\eta)=\sum_{i=1}^{m}a_{i}\log% -(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\boldsymbol{b^{\prime}% -}^{T}\boldsymbol{w}$

-

$\sum_{i}a_{i}=1$

-

$(\boldsymbol{x}_{i},y_{i})$

-

$\boldsymbol{a^{*}},\boldsymbol{w^{*}},\eta^{*})$

-

$(2\lambda_{0}^{i})^{(n-1)}b^{2}\|\boldsymbol{x}\|^{2}$

-

$\displaystyle\leq f(v^{t})-\frac{\gamma}{2}\|\bigtriangledown f(v^{t})\|^{2}.$

-

$\displaystyle\leq\mathbb{E}_{S}[\phi(S)]$

-

$\|\boldsymbol{a}\|$

-

$t_{i}(D,\boldsymbol{c^{\prime}})$

-

$\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c^{\prime}})$

-

$h(\boldsymbol{b^{\prime}}_{2})$

-

$\boldsymbol{b^{\prime}}$

-

$\eta=m\min_{i}\epsilon_{i}$

-

$\phi(S)=\sup_{\boldsymbol{w}\in\mathbb{R}^{n}}(\mathbb{L}[\boldsymbol{w}]-% -\mathbb{\hat{L}}_{S}[\boldsymbol{w}])$

-

$\|(1/\|(e^{\boldsymbol{z}})\|)\|<\sqrt{m}$

-

$\epsilon_{i}.$

-

$\psi_{i}(c_{i})\leq\frac{1}{m^{k}}$

- - - diff --git a/htmls/output_mathjax_example_10055.html b/htmls/output_mathjax_example_10055.html deleted file mode 100644 index 42f7b4f821db33c23a7182f73fa5ab4ff665d6ed..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10055.html +++ /dev/null @@ -1,157 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\min_{\boldsymbol{z},\eta,\boldsymbol{w}}f(\boldsymbol{w},% -\boldsymbol{z},\eta),$

-

$a_{i}^{*}=\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(% -\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}\Big{(}\frac{\|% -\boldsymbol{a}^{*}\|}{\mu}\Big{)},$

-

$\|\boldsymbol{b^{\prime}}\|\sim\Gamma(n,\frac{2}{\eta})$

-

$F(c_{i})$

-

$\eta=m\epsilon_{avg}$

-

$\displaystyle=2R_{m}(\boldsymbol{w}),$

-

$\{(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m-1},y^{m-1}),(% -\boldsymbol{d},y)\}$

-

$\inf_{v}f(v)=L$

-

$\|\boldsymbol{b^{\prime}}\| -

$\boldsymbol{w^{*}}^{T}\boldsymbol{x^{i}}y^{i}\geq\delta\ \forall i\$

-

$\displaystyle\leq\lim_{p\rightarrow 0}\lim_{m\rightarrow\infty}\min_{% -\boldsymbol{a},\eta,\boldsymbol{w},\|w\|\leq\beta,\beta}\bigg{[}\sum_{i=1}^{m}% -a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})$

-

$\displaystyle=\|\bigtriangledown_{\boldsymbol{w}}f(v_{1})-\bigtriangledown_{% -\boldsymbol{w}}f(v_{2})\|^{2}$

-

$\displaystyle f(v^{t+1})$

-

$g_{\min}\leftarrow g(\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}})$

-

$\sum_{i=1}^{m}\bigg{[}e^{z_{i}}\log(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}% -\cdot y^{i}})+\frac{\lambda_{0}^{i}}{2}\cdot\|\boldsymbol{w}\|^{2}+\gamma\cdot -m% -\epsilon_{avg}e^{z_{i}}\Psi_{i}\bigg{]},$

-

$\theta(D,\boldsymbol{c^{\prime}})$

-

$\displaystyle 2\lambda_{0}^{i}>\|\boldsymbol{x}^{i}\|^{2}\bigg{(}\frac{1}{\ln 2% -}\bigg{)}^{2}\cdot e^{z_{i}}\bigg{(}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}% -^{i}y^{i}}}{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}\bigg{)}^{2}$

-

$\displaystyle\qquad\leq\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+2\sqrt{% -\sigma p\gamma+2\|\boldsymbol{b}\|\sqrt{p\gamma}},$

-

$\boldsymbol{w^{*}}$

-

$\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}\bigg{]}+% -\gamma\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})$

-

$\displaystyle=\mathbb{E}_{\sigma}[\sup_{\boldsymbol{w}\in R^{n}}\sum_{i}a_{i}% -\sigma_{i}\log(1+e^{-y^{i}\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\boldsymbol{b% -^{\prime}}^{T}\boldsymbol{w}]$

-

$\displaystyle\leq f(v^{t})-\gamma||\bigtriangledown f(v^{t})||^{2}+\frac{K% -\gamma^{2}}{2}\|\bigtriangledown f(v^{t})\|^{2}$

-

$\eta=\sum\epsilon_{i}$

-

$\Gamma(n,\frac{2}{\eta})$

-

$\displaystyle=\|\bigtriangledown_{\boldsymbol{w}}f(v^{1})-\bigtriangledown_{% -\boldsymbol{w}}f(v^{2})\|^{2}+||\bigtriangledown_{\boldsymbol{z}}f(v^{1})-% -\bigtriangledown_{\boldsymbol{z}}f(v^{2})\|^{2}$

-

$\displaystyle 2\lambda>\sum_{i}e^{z_{i}}\|\boldsymbol{x}^{i}\|^{2}\bigg{(}% -\frac{1}{\ln 2}\bigg{)}^{2}\cdot\bigg{(}\frac{e^{-\boldsymbol{w}^{T}% -\boldsymbol{x}^{i}y^{i}}}{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}% -\bigg{)}^{2}$

-

$\boldsymbol{z}\leftarrow\mbox{\text{Proj}}_{S}(\boldsymbol{z})$

-

$S=\{\boldsymbol{z}:\sum_{i=1}^{m}e^{z_{i}}=1\}$

-

$w_{i\rightarrow j,n}$

-

$<\delta_{avg}^{x}$

-

$S_{BA}$

-

$p(\mathbf{b}|\mathbf{I},\mathbf{x}_{q})$

-

$\mathbf{x_{s}}$

-

$\mathbf{X}=[\mathbf{a};\mathbf{b}]$

-

$p(\mathbf{a}|\mathbf{I},\mathbf{x}_{q})\!=\!\frac{\Gamma(\frac{1+S}{2})}{% -\Gamma(\frac{1}{2})\pi^{\frac{S}{2}}|\mathbf{\Sigma}_{a}|^{\frac{1}{2}}[1\!+\!% -(\mathbf{\mathbf{a}}\!-\!\bm{\mu}_{a})^{T}\mathbf{\Sigma}_{a}^{-1}(\mathbf{% -\mathbf{a}}\!-\!\bm{\mu}_{a})]^{\frac{1+S}{2}}}$

-

$\mathbf{\Phi}\leq Q(\gamma_{u})$

-

$\mathbf{V}=\mathcal{G}_{v}(\mathbf{F}^{K})$

-

$\displaystyle\text{LEAP}_{i\rightarrow j}(\mathbf{x}_{i,n})\|_{\rho},$

-

$\gamma_{v}=0.9,\gamma_{d}=0.9,\gamma_{u}=0.8,\gamma_{track}=3$

-

$\mathbf{F}=[\mathbf{f}_{1},\ldots,\mathbf{f}_{S}]$

-

$\mathbf{\Sigma}_{a}=K(\mathbf{F}_{a},\mathbf{F}_{a})+\sigma\mathbf{I}$

-

$\mathbf{Y_{s}}=\mathcal{F}(\mathbf{I}_{s})$

-

$(\mathbf{X},\mathbf{F})$

-

$S_{KF}=2$

-

$\mathbf{m}_{d}=\text{avgpool}(\mathcal{G}_{d}([\mathbf{X}^{K};\mathbf{X}^{K}_{% -A}],[\mathbf{F}^{K};\mathbf{F}^{K}_{A}])).$

-

$N_{a}=64$

-

$\mathbf{f}_{q}$

-

$\gamma_{track}$

-

$\mathbf{I}_{s}\in\mathbf{I}$

-

$\phi(\mathbf{x}_{s})=\mathbf{\Sigma}_{a}[s,s]+\mathbf{\Sigma}_{b}[s,s]$

-

$\displaystyle\mathcal{L}_{vis}=(1-\mathbf{V}^{*})\log(1-\mathbf{V})+\mathbf{V}% -^{*}\log\mathbf{V}.$

-

$\mathbf{V}^{*},\mathbf{m}_{d}^{*}$

-

$\mathbf{b}\in\mathbb{R}^{S}$

-

$\mathbf{I}=[\mathbf{I}_{1},...,\mathbf{I}_{S}],\mathbf{I}_{s}\in\mathbb{R}^{3% -\times H\times W}$

-

$[\bm{\mu}_{a};\bm{\mu}_{b}]=\mathbf{X}$

-

$\Delta\mathbf{D}^{(k)}$

-

$\mathbf{x}_{q}\in\mathbb{R}^{2}$

-

$\mathbf{x}_{s_{q}}=\mathbf{x}_{q}$

-

$\mathbf{I}_{s_{q}}$

-

$\mathbf{F}_{b}^{k}=\mathcal{G}_{b}(\mathbf{F}^{k})$

-

$\displaystyle\mathcal{L}_{dyn}=(1-\mathbf{m}_{d}^{*})\log(1-\mathbf{m}_{d})+% -\mathbf{m}_{d}^{*}\log\mathbf{m}_{d}.$

-

$\displaystyle\mathcal{L}_{main}=\sum_{k}^{K}\gamma^{K-k}\mathcal{L}_{NLL}(% -\mathbf{X}^{k},\mathbf{X^{*}},\mathbf{\Sigma}_{a}^{k},\mathbf{\Sigma}_{b}^{k}),$

-

$k=8,N_{a}=64$

-

$K_{BA}$

-

$S_{LP}=12$

-

$p(\mathbf{X}|\mathbf{I},\mathbf{x}_{q})=p(\mathbf{a}|\mathbf{I},\mathbf{x}_{q}% -)\cdot p(\mathbf{b}|\mathbf{I},\mathbf{x}_{q})$

-

$\mathbf{C}[\mathbf{X}^{k}]$

-

$\mathbf{V}=[v_{1},...,v_{S}]$

-

$(\mathbf{\Sigma}_{a},\mathbf{\Sigma}_{b})$

-

$\mathbf{X}_{i\rightarrow j}$

-

$\begin{split}(\Delta\mathbf{X},\Delta\mathbf{F})&=\text{Refiner}(\mathbf{F}^{k% -},\text{pos}(\mathbf{X}^{k}-\mathbf{x_{q}}),\mathbf{C}^{k}[\mathbf{X}^{k}]),\\ -\mathbf{X}^{k+1}&\leftarrow\mathbf{X}^{k}+\Delta\mathbf{X},\quad\mathbf{F}^{k+% -1}\leftarrow\mathbf{F}^{k}+\Delta\mathbf{F},\end{split}$

-

$\mathcal{A},|\mathcal{A}|=N_{a}$

-

$(\bm{\mu}_{a},\mathbf{\Sigma}_{a},\bm{\mu}_{b},\mathbf{\Sigma}_{b})$

-

$\mathbf{V},\mathbf{m}_{d}$

-

$\mathbf{X}=[\mathbf{x}_{1},...,\mathbf{x}_{S}],\mathbf{x}_{s}\in\mathbb{R}^{2}$

-

$\mathbf{Y}_{s}$

-

$S_{KF}$

-

$\mathbf{a}\in\mathbb{R}^{S}$

-

$\mathbf{m}_{d}$

-

$K(\mathbf{x},\mathbf{y})=\mathbf{x}^{T}\mathbf{y}$

-

$\text{LEAP}_{i\rightarrow j}$

-

$\Delta\xi^{(k)}\in\mathfrak{se}(3)~{}\textrm{(lie-algebra corresponding to}~{}% -\mathbf{T})$

-

$\mathbf{F}_{a}^{k}=\mathcal{G}_{a}(\mathbf{F}^{k})$

-

$\mathbf{\Sigma}_{b}=K(\mathbf{F}_{b},\mathbf{F}_{b})+\sigma\mathbf{I}$

-

$(\mathbf{X},\mathbf{V})=\text{TAP}(\mathbf{I},\mathbf{x}_{q},s_{q}).$

-

$\frac{N_{a}}{k^{2}}$

-

$(\mathbf{G}_{x},\mathbf{G}_{y})$

-

$\mathbf{I}_{t-S_{LP}:t-1}$

-

$w_{1}=1.0,w_{2}=0.5,w_{3}=0.5$

-

$\mathbf{I}_{t-S_{LP}+1:t}$

-

$S_{LP}=16$

-

$i,j\in[t-S_{LP}+1,t-1]$

-

$Q:[0,1]\rightarrow\mathbb{R}$

-

$\displaystyle\sum_{i}\sum_{j\in|i-j|\leq S_{BA}}\sum_{n}w_{i\rightarrow j,n}\|% -\mathcal{P}(\mathbf{T}_{i},\mathbf{T}_{j},\mathbf{K},d_{i,n})-$

-

$\displaystyle\mathcal{L}_{total}=w_{1}\mathcal{L}_{main}+w_{2}\mathcal{L}_{vis% -}+w_{3}\mathcal{L}_{dyn}.$

-

${\mathbf{x}_{q}}$

-

$v_{s}\in\{0,1\}$

-

$p(\mathbf{X}|\mathbf{I},\mathbf{x}_{q})$

-

$\mathbf{F}^{K}$

-

$\text{pos}(\cdot)$

- - - diff --git a/htmls/output_mathjax_example_10056.html b/htmls/output_mathjax_example_10056.html deleted file mode 100644 index 6a321ca209ca562e5f79b931a04fcc76eb8716ec..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10056.html +++ /dev/null @@ -1,128 +0,0 @@ - - - - MathJax Example - - - - -

$||{\mathrm{D}}||$

-

$({\cal H}_{Q^{\prime}},{\cal H}_{\mathcal{V}})$

-

$\langle a_{1},\ldots,a_{\rho}\rangle\in r^{\mathcal{A}}$

-

$(V,H)$

-

$\mathit{vars}(\mathit{atoms}(Q))$

-

$\mathit{color}(Q)$

-

$\{X,Y\}\subseteq(h\setminus\bar{W})$

-

${\cal FH}(Q_{0},\{A,B,C,D\})$

-

$\mathit{Fr}(A,\Lambda)=\mathit{Fr}(B,\Lambda)=\mathit{Fr}(C,\Lambda)=\emptyset$

-

${\cal H}\leq{\cal H}_{a}\leq{\cal H}^{\prime}$

-

${\cal H}_{\mathcal{V}_{0}}$

-

$h\subseteq V$

-

$r^{\mathcal{A}}\subseteq A^{\rho}$

-

$\mathit{vars}(A)$

-

$\mathrm{count}(Q,{\mathrm{D}})$

-

$\exists D,...,I\ \Phi\wedge\Phi_{f}$

-

$J\!T$

-

$\pi_{W_{1}}(S_{1}\bowtie S_{2})$

-

$\{D,G\}$

-

$\bar{X}=X_{1},...,X_{n}$

-

$\Phi=r_{1}({\bf u_{1}})\wedge...\wedge r_{m}({\bf u_{m}})$

-

$r_{1},...,r_{m}$

-

$\mathit{atoms}(Q)$

-

$X=X_{0},\ldots,X_{\ell}=Y$

-

$\mathsf{param}\textup{-}\textsc{\#Clique}[\mathbb{N}]$

-

${\cal H}_{Q_{0}}$

-

${\cal H}_{Q^{\prime}}$

-

$\mathit{form}(Q)$

-

$\mathit{Frontiers}(\mathbf{C})=\{{\cal FH}(Q^{\prime},\mathit{free}(Q))\mid Q% -\in\mathbf{C}$

-

$\mathit{Frontiers}(\mathbf{C})$

-

$\theta:\mathit{vars}(Q)\mapsto D$

-

$\theta^{\prime}(t)=t$

-

$\mathit{nodes}({\cal H})$

-

$\mathit{Fr}(Y,\bar{W},{\cal H})$

-

$\mathit{nodes}({\cal H})\setminus\bar{W}$

-

$\mathit{Fr}(A,\{D,E,G\},{\cal H}_{Q_{0}})=\{D,E\}$

-

$\Phi^{\prime}=mw(A,B,I)\wedge wt(B,D)\wedge wi(B,E)\wedge pt(C,D)\wedge st(D,F% -)\wedge rr(F,H)\wedge rr(D,H).$

-

$(I,\pi)\subseteq\Sigma^{*}\times\mathbb{N}$

-

$\bar{W}\subseteq\mathit{vars}(Q)$

-

${\cal FH}(Q_{0}^{\prime},\mathit{free}(Q_{0}^{\prime}))$

-

$st(D,G)\wedge rr(G,H)$

-

${\rm FPT}=\rm W[1]$

-

$Q\in\mathbf{C}$

-

$\{D,F\}$

-

$\mathsf{param}\textup{-}\textsc{Clique}[\mathbb{N}]$

-

${\cal FH}(Q_{0}^{\prime},\{A,B,C\})$

-

$\begin{array}[]{lll}\Phi_{f}&=&r_{A}(A)\wedge r_{B}(B)\wedge r_{C}(C).\\ -\end{array}$

-

$st(task,subtask)$

-

$A\cap\mathcal{U}$

-

$\deg_{{\mathrm{D}}}(F,v)$

-

$\exists\bar{X}\Phi$

-

$\mathit{bound}({\mathrm{D}},{\rm HD})$

-

$r(a_{1},...,a_{\rho})$

-

$\langle a_{1},...,a_{\rho}\rangle$

-

${\cal H}_{Q^{\prime}_{0}}$

-

${\cal H}\leq{\cal H}^{\prime}$

-

$\mathit{vars}(Q)$

-

${\cal H}_{\mathcal{V}}=(N,H)$

-

$h\in\mathit{edges}({\cal H})$

-

$\pi_{W_{2}}(\{\theta\})\subseteq S_{2}$

-

$rr(task,resource)$

-

$H=\{\mathit{Fr}(Y,\bar{W},{\cal H}_{Q^{\prime}})\mid Y\in\mathit{vars}(Q^{% -\prime})\}\cup\{e\in\mathit{edges}({\cal H}_{Q^{\prime}})\mid e\subseteq\bar{W}\}$

-

${\tt cores}(Q)$

-

$\rm\#P$

-

$\mathrm{\#CQ}[\mathbf{C}]$

-

$w_{q}\in\mathit{views}(Q)$

-

${\cal H}^{\prime}$

-

$\{G,H\}$

-

$r_{i}\in\tau_{Q}$

-

$\{\mathit{Fr}(Y,\Lambda,{\cal H}_{Q^{\prime}_{0}})\mid Y\in\{A,B,C,D,E,F,H,I\}% -\}\cup\{e\in\mathit{edges}({\cal H}_{Q^{\prime}_{0}})\mid e\subseteq\Lambda\}$

-

${\rm W[1]}$

-

$\mathit{edges}(C)$

-

${\rm HD}$

-

$\{X_{1},...,X_{n}\}$

-

$i\in\{0,...,\ell\mbox{-}1\}$

-

$W\subseteq\mathit{vars}(Q)$

-

$\langle A,B,C\rangle$

-

$\mathit{Fr}(Y,\bar{W},{\cal H})=\bar{W}\cap\mathit{nodes}(\mathit{edges}(C))$

-

${\cal H}_{Q_{0}^{\prime}}$

-

${\mathrm{D}}_{Q}$

-

$\mathit{vars}(Q^{\prime})\subseteq\mathit{vars}(Q)$

-

$\langle h(a_{1}),\ldots,h(a_{\rho})\rangle\in r^{\mathcal{B}}$

-

$\sigma_{\theta}(S)$

-

$(I,\pi)\in J$

-

$Y\in\bar{W}$

-

$\mathit{Fr}(H,\{D,E,G\},{\cal H}_{Q_{0}})=\{D,G\}$

-

$\textsc{Clique}[\mathbb{N}]$

-

$\mathit{free}(Q)=$

-

${\rm FPT}\subseteq{\rm W[1]}\subseteq{\rm W[2]}\subseteq\cdots$

-

$\begin{array}[]{ll}\Phi=&mw(A,B,I)\wedge wt(B,D)\wedge wi(B,E)\wedge pt(C,D)\ % -\wedge\\ -&st(D,F)\wedge st(D,G)\wedge rr(G,H)\wedge rr(F,H)\wedge rr(D,H).\\ -\end{array}$

-

${\cal H}_{1}\leq{\cal H}_{2}$

-

$D\subseteq\mathcal{U}$

-

$\pi_{\mathit{free}(Q)}(Q^{\mbox{\rm\tiny D}})$

-

$\mathit{nodes}({\cal H})\subseteq\mathit{vars}(Q)$

-

$\mathit{vars}(Q_{0}^{\prime})=\mathit{vars}(Q_{0})\setminus\{G\}$

-

$\mathit{free}(Q_{0}^{\prime})=\mathit{free}(Q_{0})=\{A,B,C\}$

-

$W_{1}\cup W_{2}$

-

$\rm FPT$

-

${\cal H}_{2}$

-

$r_{i}^{\mathcal{Q}}$

- - - diff --git a/htmls/output_mathjax_example_10057.html b/htmls/output_mathjax_example_10057.html deleted file mode 100644 index ce40c5a2b8659d40ce21692105ca3bc1242871f0..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10057.html +++ /dev/null @@ -1,125 +0,0 @@ - - - - MathJax Example - - - - -

$\mbox{\rm P}=\mbox{\rm NP}$

-

$\{h\in\mathit{edges}({\cal H})\mid h\cap C\neq\emptyset\}$

-

$\mathit{Fr}(Y,\bar{W},{\cal H})=\emptyset$

-

$w^{\mbox{\rm\tiny D}}$

-

$wt(worker\_id,task)$

-

$\tilde{O}(N^{\#subw})$

-

$Q_{0}^{\prime}=\Phi^{\prime}\wedge\Phi_{f}$

-

$({\cal H},{\cal H}^{\prime})$

-

$w^{\mbox{\rm\tiny D}}\supseteq\pi_{\mathit{vars}(w)}(Q^{\mbox{\rm\tiny D}})$

-

$\mathit{nodes}(\mathit{edges}(\{A,B,I\})=\{A,B,D,E,I\}$

-

$\mathit{color}(Q)\}$

-

$r_{v}=\ \pi_{\chi(v)}(\bowtie_{q\in\lambda(v)}q^{\mbox{\rm\tiny D}})$

-

${Q_{0}}$

-

$h(\langle a_{1},\ldots,a_{\rho}\rangle)$

-

${\cal FH}(Q_{0}^{\prime},\Lambda)$

-

${\cal H}_{\mathit{color}(Q_{0})}$

-

$\Lambda=\{A,B,C\}$

-

$(\mathit{vars}(Q^{\prime})\cup\bar{W},H)$

-

$r_{X}(X)$

-

$\deg_{{\mathrm{D}}}(B,wt)$

-

${\cal H}_{1}$

-

${\cal H}_{1}\leq{\cal H}_{a}\leq{\cal H}_{2}$

-

$S_{1}\ltimes S_{2}$

-

$\{B,C\}$

-

$\mathit{edges}(\{A,B,I\})$

-

$t\in\mathit{vars}(Q)$

-

${\cal FH}(Q^{\prime},\bar{W})$

-

$q\in\mathit{atoms}(Q)$

-

$({\cal H}_{1},{\cal H}_{2})$

-

$\mathit{free}(Q_{0})=\{A,B,C\}$

-

$\mathit{vars}(Q)\setminus$

-

$\deg_{{\mathrm{D}}}(C,pt)$

-

$\{D,F,G,H\}$

-

${\bf u_{1}},...,{\bf u_{m}}$

-

$S_{1}\bowtie S_{2}$

-

$\{D,G,H\}$

-

$\mathit{Fr}(A,\{D,E,G\},{\cal H}_{Q_{0}})=\{A,B,D,E,I\}\cap\{D,E,G\}=\{D,E\}$

-

$h:A\mapsto B$

-

$Q^{\mbox{\rm\tiny D}}$

-

$\bigcup_{h\in H}h$

-

$\{A,B,I\}$

-

$\pi_{F}(r_{v})$

-

${\mathit{bound}({\mathrm{D}},{\rm HD})}$

-

$Q_{0}^{\prime}$

-

$\mathit{edges}({\cal H})$

-

$\tau_{Q}$

-

$\mathit{nodes}(H)$

-

${\cal FH}(Q^{\prime},\mathit{free}(Q))\leq{\cal H}_{a}$

-

$r^{\mbox{\rm\tiny D}}$

-

$pt(project,task)$

-

$f(\pi)\times|I|^{c}$

-

$st(D,F)\wedge rr(F,H)$

-

$r_{i}({\bf u})\in\mathit{atoms}(Q)$

-

$mw(machine,worker\_id,machine\_hours)$

-

$X\in\mathit{free}(Q)$

-

$wi(worker\_id,worker\_info)$

-

$\pi_{W}(S)$

-

$\mathit{Fr}(Y,\bar{W})$

-

$\{D,E,G\}$

-

$\deg_{{\mathrm{D}}}(X,r)$

-

$\pi_{W_{1}}(\{\theta\})\subseteq S_{1}$

-

$\{F,H\}$

-

$\theta^{\prime}(t)=\theta(t)$

-

$\mathsf{param}\textup{-}\mathrm{\#CQ}[\mathbf{C}]$

-

$U_{Q}$

-

$w_{q}^{\mbox{\rm\tiny D}}\subseteq q^{\mbox{\rm\tiny D}}$

-

${\cal H}_{Q^{\prime}}\leq{\cal H}_{a}\leq{\cal H}_{\mathcal{V}}$

-

$J\subseteq\Sigma^{*}\times\mathbb{N}$

-

${\cal FH}(Q^{\prime},\mathit{free}(Q))$

-

$\langle h(a_{1}),\ldots,h(a_{\rho})\rangle$

-

$\mathit{color}(Q_{0})$

-

$Q_{0}=\exists D,...,I\ \Phi$

-

$\mathit{views}(Q)$

-

${\rm HD}=\langle T,\chi,\lambda\rangle$

-

$A\subseteq\mathcal{U}\cup\mathcal{X}$

-

$\{\theta^{\prime}\in S\mid\pi_{W}(\{\theta^{\prime}\})=\{\theta\})$

-

${\cal H}_{a}$

-

$\theta^{\prime}(r_{\alpha_{i}}({\bf u_{i}}))\in{\mathrm{D}}$

-

${\cal FH}(Q_{0},\{A,B,C\})$

-

$h(c)=c$

-

$\begin{array}[]{ll}\exists D,E,F,G,H,I&mw(A,B,I)\wedge wt(B,D)\wedge wi(B,E)% -\wedge pt(C,D)\ \wedge\\ -&st(D,F)\wedge st(D,G)\wedge rr(G,H)\wedge rr(F,H)\wedge rr(D,H).\\ -\end{array}$

-

$\mathrm{\#CQ}$

-

${\cal H}_{Q}$

-

$r^{\mathcal{A}}\subseteq r^{\mathcal{B}}$

-

$D\in(1,2)$

-

${(D_{i},D_{i+1})\subseteq(D^{\prime}_{j},D^{\prime}_{j+1})}$

-

$\displaystyle=\widetilde{V}(D)-V(D)$

-

$\widetilde{\Gamma}_{D}\cap\mathcal{F}_{1}=\emptyset$

-

${p_{2}=(e_{3},e_{7},e_{4})}$

-

${D=D_{i}}$

-

$\mathcal{R}^{\operatorname{act}}_{D}$

-

$\bar{\beta}=\beta^{\top}f^{\delta}$

-

${\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{vec}}(D_{i})+\delta C^% -{i}}$

-

${\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}=\mathcal{R}^{\operatorname{% -act}}_{D}\cap(\mathcal{S}^{\texttt{rem}})^{c}}$

-

$A_{\mathcal{Q}}$

-

$C_{p}(f^{T})=\delta\lambda^{M}T+\bar{\beta}$

-

$\delta\widetilde{\lambda}^{\widetilde{M}}>\delta\lambda^{M}$

-

${0 -

$\widehat{\Gamma}_{D}\cap\mathcal{F}_{1}\neq\emptyset$

-

$\displaystyle=C_{p}(f^{D})+\epsilon A_{p}f^{\delta}$

- - - diff --git a/htmls/output_mathjax_example_10058.html b/htmls/output_mathjax_example_10058.html deleted file mode 100644 index 8a565a859e9ebb7a8bba22975abfb87c830b5e49..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10058.html +++ /dev/null @@ -1,136 +0,0 @@ - - - - MathJax Example - - - - -

$\delta\lambda^{i}$

-

$C(f)=Af+\beta$

-

$\mathcal{Q},\mathcal{R},\widetilde{\mathcal{Q}},\widetilde{\mathcal{R}}% -\subseteq\mathcal{P}$

-

$D_{2}=2$

-

$\displaystyle\mymathbf{1}^{\top}f$

-

$[D^{\prime\prime}_{k-1},D_{i+1})$

-

$\displaystyle=\widetilde{\lambda}^{\operatorname{WE}}(\widetilde{D}_{% -\widetilde{M}})+(T-\widetilde{D}_{\widetilde{M}})\delta\widetilde{\lambda}^{% -\widetilde{M}}.$

-

$A_{i}\in{}^{1\times n}$

-

${D\in[D_{i},D_{i+1})}$

-

$f^{T_{i}}\in\mathcal{W}_{T_{i}}$

-

$[\frac{7}{2},\frac{35}{9}]$

-

$\lambda^{\operatorname{vec}}_{p}(D_{i})<\lambda^{\operatorname{vec}}_{r}(D_{i})$

-

$C_{p}(f):=\sum_{e_{k}\in p}C_{e_{k}}(f_{e_{k}}).$

-

$D^{\prime}\not=D$

-

$D^{+}\leq D$

-

$[D_{i},D_{i+1}]$

-

$\Gamma_{D}=\ker(A)\cap\mathcal{M}_{D}$

-

$\widecheck{(\cdot)}$

-

$\displaystyle\cdots,C_{n}(f)\big{)}^{\top},\text{ and }\,\mathcal{C}:=\{C_{e_{% -k}}\}_{e_{k}\in\mathcal{E}},$

-

$\widecheck{\mathcal{P}}:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}_{i,D}$

-

$f^{\delta}\in\ker(A)\cap\mathcal{M}_{D}$

-

$\widetilde{\Gamma}_{D}\subseteq\Gamma_{D}$

-

$f^{\top}Af+f^{\top}\beta\geq D(\delta\lambda^{M}D+\bar{\beta}D).$

-

$\displaystyle\int_{0}^{D}\widetilde{\lambda}^{\operatorname{WE}}(z)-\lambda^{% -\operatorname{WE}}(z)dz$

-

$\widetilde{f}^{\delta}\in\operatorname{SOL}(\widetilde{\mathcal{M}},A)$

-

$f^{\mu}\in\mathcal{W}_{T_{\mu}}$

-

$D_{M}\leq D -

$f^{*}_{p}=0$

-

$D^{-}\in(D_{i},D_{i+1})$

-

$f^{\delta}_{p}<0$

-

${\Gamma_{D}=\operatorname{SOL}(\mathcal{M}_{D},A)}$

-

$V(T)\leq\widetilde{V}(T)$

-

${[n]:=\{1,2,\cdots,n\}}$

-

$\mathcal{R}^{\operatorname{act}}_{D_{i+1}}$

-

$p^{\prime}\in\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{J}^{% -\operatorname{act}}_{i+1}$

-

$\displaystyle=(f^{\prime})^{\top}C(f),$

-

$(e_{i})_{j}=0$

-

$\displaystyle f^{\delta}\in\mathcal{F}_{1}.$

-

$f^{D}$

-

$\mu_{r}\in(0,1)$

-

$\mathcal{F}_{D}:=\Bigl{\{}f\in{\mathbb{R}}_{\geq 0}^{n}\;|\;\sum_{p\in\mathcal% -{P}}f_{p}=D\Bigr{\}}.$

-

$\widetilde{\mathcal{P}}\subseteq\mathcal{P}$

-

$\delta\widetilde{\lambda}^{j}<\delta\widetilde{\lambda}^{j+1}$

-

$\displaystyle\delta\lambda^{+}(D)$

-

$D\in[\widetilde{D}_{\widetilde{M}},\infty)$

-

$f^{r}_{r}>0$

-

$\displaystyle=\delta\lambda^{M}T+\bar{\beta}+(D-T)\delta\lambda^{M},$

-

$\widetilde{\mathcal{M}}=\mathcal{M}_{D}$

-

$\displaystyle\leq(f^{\delta_{1}}-f^{\delta_{2}})^{\top}A(f^{\delta_{1}}-f^{% -\delta_{2}})$

-

$\mathcal{M}_{D}\subseteq\mathcal{M}_{D_{i}}$

-

$\delta C^{i}=Af^{\delta}$

-

${f^{T}=f^{D}+(T-D)f^{\delta}}$

-

$f\in\mathcal{M}_{D}$

-

$j\notin\{i-1,i,i+1\}$

-

$\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$

-

$\lambda^{\prime\operatorname{WE}}(D_{i})=\lambda^{\operatorname{WE}}(D_{i})$

-

$f^{\mu}:=\operatorname{coco}_{\mu}(f^{D_{i}},f^{D_{i+1}})$

-

$\delta\widetilde{\lambda}^{i}$

-

${\mathcal{R}^{\operatorname{act}}_{D^{-}}=\mathcal{R}^{\operatorname{act}}_{D^% -{+}}}$

-

${f^{\delta}\in\ker(A)\cap\mathcal{M}_{D}}$

-

$f^{r}\in\mathcal{W}_{D}$

-

$\Gamma^{i}\subset\mathcal{H}_{1}$

-

$\widetilde{\mathcal{M}}^{-}\subseteq\mathcal{M}^{-})$

-

$\mathcal{S}^{\texttt{nec}}\subseteq\mathcal{P}$

-

$\delta C^{i}=\delta C^{j}$

-

$D^{\prime}_{j},D^{\prime}_{j+1}\in\mathcal{D}^{\prime}$

-

$D_{i+1}>0$

-

$\epsilon\in[0,\bar{\epsilon}]$

-

${\delta C^{M}_{p}=\min_{r\in\mathcal{P}}\delta C^{M}_{r}=\delta\lambda^{M}}$

-

$(f^{T_{i}}_{r}-f^{D}_{r})\geq-t$

-

$\Gamma_{D}=\Gamma^{M}$

-

$D_{j}>D_{j-1}$

-

$f^{D^{-}}_{p}>0$

-

${\operatorname{SOL}(\mathcal{M}_{D},A)\subseteq\Gamma_{D}}$

-

$f^{T}=f^{D_{i}}+(T-D_{i})f^{\delta}.$

-

$\widetilde{\mathcal{P}}=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}$

-

$\lambda^{\operatorname{vec}}_{r}(T)<\lambda^{\operatorname{vec}}_{p}(T)$

-

${\Gamma_{D}^{-}=\operatorname{SOL}(\mathcal{M}_{D}^{-},A)}.$

-

$\delta C_{p}^{M}=\delta\lambda^{M}$

-

$f^{D}_{\mathcal{R}^{\operatorname{use}}_{D}}>0$

-

$\widetilde{f}^{D}\in\mathcal{W}_{D}$

-

$\delta\widetilde{\lambda}^{\widetilde{M}}<\delta\lambda^{M}$

-

$\widetilde{\lambda}^{\operatorname{WE}}(D)=\frac{D}{2}+1.$

-

${f^{\delta}\in\Gamma_{D}}$

-

${i\in[M]_{0}}$

-

$\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{vec}}(D_{i})+(T-D_{i})% -\delta C^{i},$

-

${\lambda^{\operatorname{WE}}:{\mathbb{R}}_{\geq 0}\rightarrow{\mathbb{R}}_{% -\geq 0}}$

-

$\displaystyle\geq\lambda^{\operatorname{WE}}(D_{M}).$

-

$\mathcal{R}^{\operatorname{use}}_{D_{i+1}}\subseteq\mathcal{J}^{\operatorname{% -use}}_{i}\subseteq\mathcal{J}^{\operatorname{act}}_{i}$

-

$\mathcal{M}_{D}^{-}:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;f_{\mathcal{R}^{% -\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use}}_{D}}^{\delta}% -\geq 0,\enskip f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{\delta}=0\}.$

-

$\mathcal{S}^{\texttt{nec}}$

-

$\mathcal{M}^{-}_{D_{i}}$

-

$T>D_{M}$

-

$\widetilde{f}^{\delta}\in\mathcal{F}_{1}$

-

$\mathcal{M}=\mathcal{M}_{D}$

-

$(\mathcal{P}\setminus\mathcal{S}^{\prime},\mathcal{C})$

-

$\displaystyle\subseteq\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{J}% -^{\operatorname{act}}_{i}\subseteq\mathcal{R}^{\operatorname{act}}_{D_{i}},$

-

$(\mathcal{J}^{\operatorname{use}}_{i})^{c}\notin\mathcal{N}_{D}$

-

$\displaystyle=f^{D_{i}}+(T_{\mu}-D_{i})\frac{f^{D_{i+1}}-f^{D_{i}}}{{D_{i+1}}-% -D_{i}}$

-

$A_{p}f^{\delta}=\delta\lambda^{M}$

- - - diff --git a/htmls/output_mathjax_example_10059.html b/htmls/output_mathjax_example_10059.html deleted file mode 100644 index dd98ed2e980047dc35e1489cbf90088d4787617e..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10059.html +++ /dev/null @@ -1,146 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\leq(f^{\delta})^{\top}A\widetilde{f}^{\delta},$

-

$C_{p}(f^{T})\geq\delta\lambda^{M}T+\bar{\beta}$

-

$f^{D}\in\mathcal{W}_{D}$

-

$f^{\delta}_{p}>0$

-

${\delta C^{i+1}\not=\delta C^{i}}$

-

${C_{p}(f^{\prime})=\delta\lambda^{M}D+\bar{\beta}}$

-

$D_{i}=0$

-

$f^{T}=\operatorname{coco}_{\mu}(f^{D^{-}},f^{D^{+}})$

-

$\lambda^{\operatorname{WE}}(D)=\begin{cases}2D\quad&\text{if }0\leq D\leq 1,\\ -2\quad&\text{if }1\leq D\leq 2,\\ -\frac{D}{2}+1\quad&\text{if }2\leq D.\end{cases}$

-

$p\in\mathcal{J}^{\operatorname{act}}_{M}$

-

$\operatorname{SOL}(\mathcal{M}_{D},A)\subseteq\Gamma_{D}$

-

$W(D)\leq 0$

-

$f^{D}_{\mathcal{S}^{\texttt{nec}}}\neq 0\text{ for all }f^{D}\in\mathcal{W}_{D}.$

-

$p_{4}=(e_{3},e_{2})$

-

$\displaystyle\in\mathcal{H}_{1}\;|\;\exists f^{D}\in\mathcal{W}_{D},\hskip 2.0% -pt\bar{\epsilon}>0\text{ such that }f^{D}+\epsilon f^{\delta}\in\mathcal{W}_{D% -+\epsilon}\hskip 2.0pt\forall\epsilon\in[0,\bar{\epsilon}]\}.\hbox{$\bullet$}$

-

$\lambda^{\operatorname{vec}}(\cdot)$

-

$\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{D^{+}}$

-

$f^{D+\epsilon}_{p}>0$

-

$\displaystyle C_{e_{4}}(f_{e_{4}})=2f_{e_{4}},\hskip 8.0pt$

-

$D=D_{M}$

-

${(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta}$

-

$\mathcal{M}=\mathcal{M}^{-}_{D}$

-

$f_{e_{k}}\mapsto C_{e_{k}}(f_{e_{k}})$

-

$(D_{M},\infty)$

-

$\mathcal{I}_{1}\subseteq\mathcal{J}^{\operatorname{act}}_{M}$

-

${\widetilde{f}^{D}_{\mathcal{S}^{\texttt{rem}}}=0}$

-

$\lambda^{\operatorname{vec}}(D_{M})\geq\lambda^{\operatorname{WE}}(D_{M})% -\mymathbf{1}$

-

$r\in\mathcal{R}^{\operatorname{act}}_{D^{-}}$

-

$V(D)=\widetilde{V}(D)$

-

$\lambda^{\operatorname{WE}}(D)=\delta\lambda^{M}D+\bar{\beta}.$

-

$\mathcal{N}_{D}$

-

$\widetilde{\lambda}^{\operatorname{vec}}(D)=\lambda^{\operatorname{vec}}(D)$

-

$\widetilde{D}_{j},\widetilde{D}_{j+1}\in\widetilde{D}$

-

$C_{p}(f^{D})\leq C_{r}(f^{D})\quad\text{for all }r\in\mathcal{P}.$

-

$(D_{i-1},D_{i})$

-

$D^{-}>0$

-

${f^{\delta}\in\Gamma^{M}}$

-

$i\in[\widetilde{M}]_{0}$

-

$f^{0}=(-1,-1,1,1)^{\top}$

-

$\displaystyle C_{e_{1}}(f_{e_{1}})=2f_{e_{1}},$

-

$r\in\mathcal{Q}$

-

$\lambda^{\operatorname{vec}}_{p}(T)<\lambda^{\operatorname{vec}}_{r}(T)$

-

$\displaystyle=\mathcal{R}^{\operatorname{use}}_{D_{i}}\Rightarrow\delta\lambda% -^{i-1}>\delta\lambda^{i},\qquad\mathcal{J}^{\operatorname{use}}_{i}=\mathcal{R% -}^{\operatorname{use}}_{D_{i}}\Rightarrow\delta\lambda^{i-1}<\delta\lambda^{i},$

-

$f^{\delta-}$

-

$f^{\delta}$

-

$\displaystyle\lambda^{\operatorname{vec}}(D)=\begin{cases}\left(\begin{array}[% -]{cccc}1+D,&1+D,&2D,&2.1\end{array}\right)^{\top}&\text{for }D\in[0,1],\\ -\left(\begin{array}[]{cccc}2,&2,&2,&2.1\end{array}\right)^{\top}&\text{for }D% -\in[1,2],\\ -\left(\begin{array}[]{cccc}1+\frac{D}{2},&1+\frac{D}{2},&D,&2.1\end{array}% -\right)^{\top}&\text{for }D\in[2,2.2],\\ -\left(\begin{array}[]{cccc}2.1,&2.1,&2.2,&2.1\end{array}\right)^{\top}&\text{% -for }D\in[2.2,\infty).\end{cases}$

-

$f^{D}_{p}>0$

-

$\displaystyle=C\big{(}f^{D_{i}}+(T_{\mu}-D_{i})f^{\delta_{0}}\big{)}$

-

$D^{\prime\prime}_{k-1}>0$

-

$T\in(D,D_{i+1}]$

-

${C_{p}(f^{D+\epsilon})=\min_{r\in\mathcal{P}}C_{r}(f^{D+\epsilon})}$

-

$(D_{i+1},D_{i+2})$

-

$D\in(D_{i+1},D_{i+2})$

-

$D>D_{M}$

-

$\mathcal{J}^{\prime\operatorname{act}}_{j}=\mathcal{J}^{\prime\operatorname{% -use}}_{j}=\mathcal{P}^{\prime}$

-

${A_{p}f^{\delta}=\min_{r\in\mathcal{Q}}A_{r}f^{\delta}}$

-

$p\in\mathcal{R}^{\operatorname{use}}_{D_{i}}$

-

$\displaystyle f_{\mathcal{R}}$

-

$\mathcal{J}^{\prime\operatorname{use}}_{j}=\mathcal{P}^{\prime}$

-

$p\in\mathcal{J}^{\operatorname{use}}_{M}$

-

$\delta C^{1}=\delta C^{3}=\mymathbf{0}$

-

$(Af^{\delta_{0}})^{\top}(f-f^{\delta_{0}})\geq 0$

-

$0\leq D^{-} -

$\mathcal{P}=\mathcal{I}_{1}\cup\mathcal{I}_{2}\cup\mathcal{I}_{3}$

-

$C_{e_{6}}(f_{e_{6}}):=2.1.$

-

${f^{\delta}=(T-D)^{-1}(f^{T}-f^{D})}$

-

$D\in\real$

-

$(Af^{\delta})^{\top}(f-f^{\delta})=0$

-

$C(f)=A_{\mathcal{Q}}f+b_{\mathcal{Q}}$

-

$C_{p}(f^{T})\leq C_{r}(f^{T})$

-

$f^{\top}Af+f^{\top}\beta$

-

$A\in{\mathbb{R}}_{\geq 0}^{n\times n}$

-

$\mathcal{R},\mathcal{Q}\subseteq\mathcal{P}$

-

${f^{\delta}\in\operatorname{SOL}(\mathcal{F}_{1},A)}$

-

$f^{D^{\prime}}=\mu f^{D}+(1-\mu)f^{T}$

-

$\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{J}^{\operatorname{act}}_% -{i}$

-

$\mathcal{J}^{\operatorname{use}}_{M}=\mathcal{P}$

-

$D\leq{\widetilde{D}_{i+1}}$

-

$\mathcal{J}^{\operatorname{act}}_{i-1}=\mathcal{R}^{\operatorname{act}}_{D}% -\neq\mathcal{R}^{\operatorname{act}}_{D_{i}}$

-

$\displaystyle\forall r\in\mathcal{I}_{1},$

-

$\{p_{3}\}$

-

$\lambda^{\operatorname{WE}}(D)\leq\widetilde{\lambda}^{\operatorname{WE}}(D)$

-

$f_{\mathcal{R}^{\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use% -}}_{D}}^{D}=0$

-

$T_{i}>D$

-

$\displaystyle\mathcal{M}$

-

$\displaystyle C_{e_{3}}(f_{e_{3}})=f_{e_{3}}+1,$

-

$\mathcal{R}^{\operatorname{act}}_{D^{-}}\neq\mathcal{R}^{\operatorname{act}}_{% -D^{+}}$

-

$D^{\prime\prime}_{k}=0$

-

${f^{\delta_{1}},f^{\delta_{2}}\in\operatorname{SOL}(\mathcal{M},A)}$

-

$Af^{\delta_{0}}=\delta C$

-

$r\in\mathcal{P}$

-

$p\in\widetilde{\mathcal{P}}$

-

${p\in\mathcal{J}^{\operatorname{use}}_{M}\subseteq\mathcal{J}^{\operatorname{% -act}}_{M}}$

-

$\beta^{\top}f^{\delta}$

-

$T_{\mu}=\operatorname{coco}_{\mu}(D_{i},T)$

-

$(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta$

-

$\mathcal{R}^{\operatorname{use}}_{D}:=\{p\in\mathcal{P}\;|\;\exists f^{D}\in% -\mathcal{W}_{D}\text{ such that }f^{D}_{p}>0\}.$

-

$p^{\prime}\in\mathcal{J}^{\operatorname{act}}_{i}$

-

$f^{D}=\begin{cases}\left(\begin{array}[]{ccc}0,&0,&D\end{array}\right)^{\top}&% -\text{for }D\in[0,1],\\ -\left(\begin{array}[]{ccc}D-1,&D-1,&2-D\end{array}\right)^{\top}&\text{for }D% -\in[1,2],\\ -\left(\begin{array}[]{ccc}\frac{D}{2},&\frac{D}{2},&0\end{array}\right)^{\top}% -&\text{for }D\in[2,\infty).\end{cases}$

-

$T_{\mu}:=\operatorname{coco}_{\mu}(D,T)$

- - - diff --git a/htmls/output_mathjax_example_1006.html b/htmls/output_mathjax_example_1006.html deleted file mode 100644 index 053cff27cc7bfbf9592769f47aca29671d5cc7ee..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1006.html +++ /dev/null @@ -1,122 +0,0 @@ - - - - MathJax Example - - - - -

$\mathcal{M}_{\mathcal{C}_{1}}=\{2K_{1}\}$

-

$S^{*}\cap V(H)$

-

$x,y\notin S^{\prime}$

-

$p_{s}\in B$

-

$G^{\prime}[S]\in\mathcal{C}$

-

$p_{1},\dots,p_{s}$

-

$H\in\mathcal{M}_{\mathcal{C}}$

-

$N_{A}=\emptyset$

-

$V(H_{1})\cup V(H_{2})$

-

$\bigcup_{v\in V(2K_{1}\vee H_{1})\setminus V(H)}X_{v}\subseteq V(H_{2})$

-

$\bigcup_{v\in V(H_{1})}X_{v}\not\subseteq V(H_{2})$

-

$b_{1}b_{3}$

-

$B=V(G)\setminus A$

-

$j\in\{1,\dots,n\}$

-

$2K_{1}\vee H$

-

$q_{1},\dots,q_{t}$

-

$P^{1},P^{2},P^{3}$

-

$h_{0}h_{1},h_{0}h_{k-1},h_{i}h_{i+1},h_{i+1}h_{i+2}$

-

$A\cup C$

-

$S^{\prime}\subsetneqq S$

-

$(2K_{1}\vee H_{1})\setminus V(H)$

-

$K_{2,3}.$

-

$G,s$

-

$3K_{1}$

-

$h_{0}$

-

$\mathcal{G}_{\mathcal{C}}$

-

$\bigcup_{v\in V(H)}X_{v}=V(2K_{1})$

-

$H[N_{W}(v)]$

-

$\overline{H_{2}}$

-

$a_{2},b_{1},b_{3}$

-

$S\subseteq C$

-

$2K_{1}\vee H_{2}$

-

$e=xy$

-

$\mathcal{O}(n^{3+o(1)})$

-

$P=p_{0},\dots,p_{s}$

-

$N_{A}\cup N_{B}\neq\emptyset$

-

$H=h_{0},h_{1},\dots,h_{k-1},h_{0}$

-

$N_{G}(q_{0})\cap V(H)=A$

-

$G\setminus V(H)$

-

$\mathcal{M}_{\mathcal{G}_{1}}=\{C_{4}\}$

-

$\mathcal{O}(n^{\omega}\log n),$

-

$w:V\to\mathbb{Q}_{+}$

-

$\mathcal{M}_{\mathcal{C}_{2}}=\{\overline{C_{2k+1}}\mid k\in\mathbb{N}\}$

-

$N_{G}[y]\setminus N_{G}[x]=N_{\overline{G}}(x)\setminus N_{\overline{G}}(y)$

-

$G_{A}[S^{\prime}]=G[S^{\prime}]$

-

$w\in V(H)\setminus\{u\}$

-

$X_{w}\cap V(C)\neq\emptyset$

-

$X=\{x_{1},\dots,x_{t}\}$

-

$q_{1}=x$

-

$V(H)\subsetneqq V(G)$

-

$k\in\{1,\dots,n\}\setminus\{j\}$

-

$C_{1}\cup\{u\},C_{2},\ldots,C_{k}$

-

$H_{1}\vee H_{2}$

-

$v_{1},\dots,v_{n}$

-

$U=V(H)$

-

$\mathcal{O}(|V(G)|)$

-

$\mathcal{C}=\mathcal{C}_{k}$

-

$G^{\prime}\in\mathcal{G}_{k}$

-

$\overline{H_{1}}$

-

$r_{1},\dots,r_{i}$

-

$\{1,\dots,s\}$

-

$C_{1},\dots,C_{k}$

-

$N_{G}(p_{0})\cap V(H)=\{a_{1}\}$

-

$S^{*}\cap V(H)=S$

-

$G[S^{\prime}]$

-

$G[S]\in\mathcal{C}$

-

$C\setminus S$

-

$\mathcal{G}_{\mathcal{C}}=\mathcal{G}_{k}$

-

$.20{\scriptstyle\ \pm.01}$

-

$.29{\scriptstyle\ \pm.01}$

-

$\Big{(}\hat{p}_{i}-\underbrace{\frac{1}{|\mathcal{C}(i)|}\sum_{j\in\mathcal{C}% -(i)}\mathbf{1}(\hat{a}_{j}\text{ is correct})}_{\text{Cluster accuracy (target% -)}}\Big{)}^{2}.$

-

$.37{\scriptstyle\ \pm.01}$

-

$1.4\times 10^{-5}$

-

$\underline{\mathbf{.02}}{\scriptstyle\ \pm.00}$

-

$.26{\scriptstyle\ \pm.01}$

-

$.69{\scriptstyle\ \pm.02}$

-

$.79{\scriptstyle\ \pm.01}$

-

$.50{\scriptstyle\ \pm.01}$

-

$.39{\scriptstyle\ \pm.28}$

-

$.35{\scriptstyle\ \pm.01}$

-

$\underline{\mathbf{.73}}{\scriptstyle\ \pm.02}$

-

$.63{\scriptstyle\ \pm.02}$

-

$.25{\scriptstyle\ \pm.01}$

-

$\underline{\mathbf{.82}}{\scriptstyle\ \pm.02}$

-

$.49{\scriptstyle\ \pm.02}$

-

$5.12\times 10^{-5}$

-

$.54{\scriptstyle\ \pm.01}$

-

$.04{\scriptstyle\ \pm.01}$

-

$\underline{\mathbf{.18}}{\scriptstyle\ \pm.00}$

-

$.30{\scriptstyle\ \pm.01}$

-

$.00{\scriptstyle\ \pm.08}$

-

$.34{\scriptstyle\ \pm.01}$

-

$0.03327$

-

$.61{\scriptstyle\ \pm.02}$

-

$.20{\scriptstyle\ \pm.00}$

-

$.14{\scriptstyle\ \pm.00}$

-

$.69{\scriptstyle\ \pm.01}$

-

$.14{\scriptstyle\ \pm.01}$

-

$\underline{\mathbf{.82}}{\scriptstyle\ \pm.01}$

-

$.71{\scriptstyle\ \pm.03}$

- - - diff --git a/htmls/output_mathjax_example_10060.html b/htmls/output_mathjax_example_10060.html deleted file mode 100644 index 7e193d30d64a5a51d8566bb620a6a2ba83e73672..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10060.html +++ /dev/null @@ -1,129 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\delta\widetilde{\lambda}^{+}(D)$

-

$[\frac{1}{2},\frac{7}{2}]$

-

$f^{\prime D}\in\mathcal{W}_{D}$

-

$D\in[1,2)$

-

$v_{d}\in\mathcal{V}$

-

$\operatorname{SOL}(\mathcal{X},G)$

-

$\displaystyle=(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D_{M})$

-

$\displaystyle\widetilde{\mathcal{P}}$

-

$f^{\delta}\geq 0$

-

$\lambda^{\operatorname{WE}}(D)$

-

$\Gamma_{D}\subseteq\mathcal{M}_{D}$

-

$\delta C^{i}$

-

$\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{T}$

-

${f^{\delta-}\in\operatorname{SOL}(\mathcal{M}^{-},A)}$

-

${f^{\delta_{0}}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{M}_{D},A)}$

-

$Af^{\delta}=\delta C^{i}$

-

$\mathcal{R}^{\operatorname{use}}_{D_{i+1}}\neq\emptyset$

-

$\mathcal{R}^{\operatorname{act}}_{T}\neq\mathcal{R}^{\operatorname{act}}_{D^{-}}$

-

$p\notin\mathcal{R}^{\operatorname{use}}_{T}$

-

$f^{\mu}\geq 0$

-

$\displaystyle(f^{\delta})^{\top}\Big{(}A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}% -\big{)}+\beta\Big{)}$

-

$\widetilde{\Gamma}^{i}\neq\widetilde{\Gamma}^{i+1}$

-

$D\mapsto f^{D}$

-

$\displaystyle\geq\delta\lambda^{-}(D).$

-

$T=D_{i+1}$

-

$f^{D_{M}}\in\mathcal{W}_{D_{M}}$

-

${\delta C_{p}^{M}=\delta\lambda^{M}}$

-

$\mathcal{P}^{\prime\prime}\subset\mathcal{P}^{\prime}\subset\mathcal{P}$

-

$J(D)=0$

-

$C(f^{D})=C(\widehat{f}^{D})$

-

$\displaystyle\delta\lambda^{-}(D)$

-

$f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{T}=0$

-

$\mathcal{S}^{\prime\prime}\subset\mathcal{P}^{\prime}$

-

$D^{+}\in(D_{M},\infty)$

-

$\widetilde{\Gamma}_{D}\cap\mathcal{F}_{1}\neq\emptyset$

-

$f^{T}\geq 0$

-

$\displaystyle\quad\mymathbf{1}^{\top}f=D.$

-

$\displaystyle\widetilde{\lambda}^{\operatorname{vec}}(D)$

-

$D\in\{\frac{2}{3}\}\cup[2,\infty)$

-

$\widetilde{\mathcal{P}}=\mathcal{P}$

-

$f^{D_{M}}\in\mathcal{F}_{D_{M}}$

-

$b_{\mathcal{Q}}$

-

${\mathcal{D}:=\{D_{0},D_{1},\cdots,D_{M},D_{M+1}\}\subset{\mathbb{R}}_{\geq 0}% -\cup\{+\infty\}}$

-

$\lambda^{\operatorname{vec}}(D^{-})=\lambda^{\operatorname{vec}}(D_{i})+(D^{-}% --D_{i})\delta C^{i}$

-

$D^{\prime}_{j}=0$

-

$\Gamma_{D}\subseteq\operatorname{SOL}(\mathcal{M}_{D},A)$

-

$\widetilde{C}(f)=Af$

-

${\mathcal{J}^{\operatorname{act}}_{i-1}=\mathcal{R}^{\operatorname{act}}_{D_{i% -}}}$

-

${g:\real\rightarrow{}^{n}}$

-

$D -

$D^{\prime\prime}_{k}>0$

-

$f^{\delta}\in\operatorname{SOL}(\mathcal{F}_{1},A)$

-

$\widetilde{\mathcal{F}}_{D}\subseteq\mathcal{F}_{D}$

-

$\displaystyle C_{e_{7}}(f_{e_{7}})=f_{e_{7}}.$

-

$f^{D_{M}}\geq 0$

-

$f^{D+\epsilon}\in\mathcal{W}_{D+\epsilon}$

-

$f^{\delta,i}\in\Gamma_{D}$

-

$\mathcal{J}^{\operatorname{act}}_{M}$

-

$f^{\delta}\in\mathcal{M}_{D}$

-

$u_{\widetilde{\mathcal{P}},\widetilde{M}}$

-

$\vec{D}_{i}\in\vec{\mathcal{D}}$

-

$\lambda^{\operatorname{vec}}_{p_{4}}$

-

$T^{+}>T$

-

${\widetilde{\lambda}^{\operatorname{WE}}(T)\geq\lambda^{\operatorname{WE}}(T)}$

-

$D^{-} -

$\mathcal{P}^{\prime\prime}:=\mathcal{P}\setminus\mathcal{S}^{\prime\prime}$

-

$u_{\mathcal{P},i}$

-

${\lambda^{\operatorname{WE}}(T)=\lambda^{\operatorname{WE}}(D_{i})+(T-D_{i})% -\delta\lambda^{i}}$

-

$\displaystyle+(T-D_{M})\delta\lambda^{M}-(T-\widetilde{D}_{\widetilde{M}})% -\delta\widetilde{\lambda}^{\widetilde{M}}$

-

$D\in{\mathbb{R}}_{\geq 0}$

-

$(\frac{1}{2},\infty)$

-

${f^{\delta}\in\operatorname{SOL}(\mathcal{M},A)}$

-

$D\geq D^{\operatorname{BP}}$

-

${A_{pr}=\sum_{e_{k}\in(p\cap r)}\alpha_{e_{k}}}$

-

$f^{\delta}\in\Gamma^{i}\cap\Gamma^{i+1}$

-

$(Af^{\delta})^{\top}(f-f^{\delta})\leq 0$

-

$\displaystyle\widetilde{\mathcal{W}}_{D}$

-

$C_{p}(f^{D^{+}})\leq C_{r}(f^{D^{+}})$

-

${\lambda^{\operatorname{vec}}(D_{M})=Af^{D_{M}}+\beta}$

-

$(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D_{M})=\lambda^{\operatorname{% -WE}}(D_{M})$

-

$e_{k}\in p_{i}$

-

$\widetilde{f}_{p}>0$

-

$\lambda^{\operatorname{vec}}_{p}(D_{M})=\lambda^{\operatorname{WE}}(D_{M})% -\quad\text{ for all }p\in\mathcal{P}\text{ satisfying }f^{\delta}_{p}>0.$

-

$f^{\delta,i}\in\mathcal{H}_{1}$

-

$C_{e_{k}}(f^{D})=C_{e_{k}}(\widehat{f}^{D})$

-

$\tilde{\mathcal{M}}\subseteq\mathcal{M}$

-

$\min_{r\in\widetilde{\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}<\min_{r\in% -\mathcal{Q}}A_{r}f^{\delta}.$

-

$\Gamma_{D}=\operatorname{SOL}(\mathcal{M}_{D},A).$

-

$\vec{\mathcal{P}}=\mathcal{P}\setminus\{(e_{1},e_{4})\}$

-

$\alpha_{e_{k}},\beta_{e_{k}}\in{\mathbb{R}}_{\geq 0}$

-

$\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}=\widetilde{\mathcal{P}}$

-

$V(D)<\widetilde{V}(D)$

-

$\widetilde{\Gamma}_{D}$

-

$f^{T}\in\mathcal{F}_{T}$

-

$f^{\delta}\in\Gamma_{D}\cap\mathcal{F}_{1}$

-

$u_{\vec{\mathcal{P}},0}$

-

$T\in[D_{i},D_{i+1}]$

-

$D^{+}=0$

-

$\delta\widetilde{C}^{i}$

-

$\displaystyle\mathcal{M}^{-}$

- - - diff --git a/htmls/output_mathjax_example_10061.html b/htmls/output_mathjax_example_10061.html deleted file mode 100644 index 733ee6bf97f4d29283160d84cfd0a4a0414c5425..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10061.html +++ /dev/null @@ -1,139 +0,0 @@ - - - - MathJax Example - - - - -

$T\in(\widetilde{D}_{j},\widetilde{D}_{j+1})$

-

$\delta\lambda^{i}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D_{i}}}A_{r}f^{\delta}$

-

$\widecheck{\mathcal{P}}$

-

$\delta C^{i}_{p}=\delta C^{i}_{r}$

-

$\mymathbf{1}$

-

$\mathcal{Q}:=\mathcal{R}^{\operatorname{act}}_{D}$

-

$\displaystyle\widetilde{V}(D):=$

-

$\mathcal{Q}=\mathcal{R}^{\operatorname{act}}_{D}$

-

$\displaystyle C_{e_{4}}(f_{e_{4}})=f_{e_{4}},\hskip 8.0pt$

-

$f^{\delta_{0}}:=(D_{i+1}-D_{i})^{-1}(f^{D_{i+1}}-f^{D}).$

-

${f^{\delta}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)}$

-

$\displaystyle C(f):=\big{(}C_{1}(f),$

-

$\lambda^{\prime\operatorname{WE}}(\cdot)$

-

$0 -

$\displaystyle:=\{p\in\widetilde{\mathcal{P}}\;|\;\widetilde{\lambda}^{% -\operatorname{vec}}_{p}(D)\leq\widetilde{\lambda}^{\operatorname{vec}}_{r}(D)% -\text{ for all }r\in\widetilde{\mathcal{P}}\},$

-

$\mathcal{M}_{T}=\mathcal{M}$

-

$\mathcal{C}\subset\mathcal{K}$

-

$(\vec{\mathcal{P}},\mathcal{C})$

-

${T\in(D_{M},\infty)}$

-

$\widetilde{f}^{\delta}\in\widetilde{\Gamma}_{D}$

-

${\ker(A)=\{f\in\real\;|\;f_{1}=f_{2}=-f_{3}\}}$

-

$\mathcal{M}:=\{f^{\delta}\in\mathcal{H}_{T}\;|\;f^{\delta}_{\mathcal{Q}% -\setminus\mathcal{R}}\geq 0,\quad f^{\delta}_{\mathcal{Q}^{c}}=0\}.$

-

$f^{\delta}\in\Gamma^{M}$

-

$\displaystyle=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}% -\quad\hskip 10.0pt\text{for all }f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{% -D},A),$

-

$\displaystyle f_{r}=0\quad$

-

$\mathcal{R}_{i}:=\{r\in\mathcal{P}\;|\;f^{\delta,i}_{r}<0\}$

-

$\displaystyle:=\operatorname{SOL}(\widetilde{\mathcal{M}}_{D},A).$

-

$\widehat{f}^{D}\in\mathcal{F}_{D}$

-

$\vec{\mathcal{P}}\subseteq\mathcal{P}$

-

$\mathcal{P}^{\prime\prime}\subset\mathcal{P}^{\prime}$

-

$A\in{}^{n\times n}$

-

$f^{\delta,i}_{r}\leq nt(T_{i}-D)^{-1}+1$

-

$\mathcal{R}=\mathcal{R}^{\operatorname{use}}_{D}$

-

$\displaystyle C_{e_{2}}(f_{e_{2}})=f_{e_{2}}+1,$

-

$D^{+}\in(D^{-},D_{i+1}]$

-

$r^{\prime}\in\mathcal{J}^{\operatorname{act}}_{i+1}$

-

$\displaystyle\forall r\in\mathcal{I}_{2},$

-

$(D_{i},D_{i+1})$

-

$\mathcal{R}^{\operatorname{act}}_{T}\neq\mathcal{R}^{\operatorname{act}}_{D}$

-

$Af^{\delta}=\delta C^{M}$

-

$\frac{\partial^{+}}{\partial x}g(x):=\lim_{h\rightarrow 0^{+}}\frac{g(x+h)-g(h% -)}{h}$

-

$D\in(D^{\prime}_{j-1},D^{\prime}_{j})$

-

$(\cdot)^{\prime\prime\prime}$

-

${\frac{\partial}{\partial T}V(T)=\lambda^{\operatorname{WE}}(T)}$

-

$V(D)\leq\widetilde{V}(D)$

-

${f^{D}\in{\mathbb{R}}_{\geq 0}^{n}}$

-

$\widetilde{f}^{\delta}_{p}>0$

-

$\widetilde{f}\in\widetilde{\mathcal{F}}_{D}$

-

$\displaystyle=\{r\in\mathcal{P}\;|\;f^{\delta}_{r}=0,\hskip 2.0pt\delta C^{M}_% -{r}=\delta\lambda^{M}\}.$

-

$(f^{\delta})^{\top}Af^{D_{M}}=D_{M}\delta\lambda^{M}$

-

$f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{T},A)$

-

$\lambda^{\operatorname{vec}}_{p_{4}}(D)=\begin{cases}2&\text{for }D\in[0,2],\\ -\frac{1}{3}D+\frac{4}{3}&\text{for }D\in[2,\infty).\end{cases}$

-

$\displaystyle=\{r\in\mathcal{P}\;|\;f^{\delta}_{r}>0\},$

-

$T\in\real$

-

$\lambda^{\operatorname{vec}}_{r}(T)<\lambda^{\operatorname{WE}}(T)$

-

$\mathcal{J}^{\operatorname{act}}_{M}=\mathcal{R}^{\operatorname{act}}_{D^{+}}$

-

$\delta\lambda^{+}(D)$

-

$\widetilde{f}^{D}\in\widetilde{\mathcal{F}}_{D}$

-

$T\in[D_{i},D_{i}+\epsilon)$

-

$\mathcal{M}_{D}\neq\mathcal{M}_{D_{i}}$

-

$C_{p}(f^{*})=\delta\lambda^{M}D+\bar{\beta}\quad\text{for all }p\text{ such % -that }f^{*}_{p}\neq 0.$

-

$f^{\prime D}_{p}=0$

-

$D\in[D^{\prime}_{j},D^{\prime}_{j+1})$

-

$V(T)=\widetilde{V}(T)$

-

$\mathcal{R}^{\operatorname{use}}_{D^{-}}\neq\mathcal{R}^{\operatorname{use}}_{% -D^{+}}$

-

$f^{\delta}\in\operatorname{SOL}(\mathcal{M},A)$

-

${\mu\in[0,1]}$

-

$\displaystyle=\lambda^{\operatorname{WE}}(D_{M})-\widetilde{\lambda}^{% -\operatorname{WE}}(\widetilde{D}_{\widetilde{M}})$

-

${(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D)=\lambda^{\operatorname{WE}% -}(D)}$

-

$p_{3}=(e_{1},e_{5},e_{4})$

-

$Q_{k,k}=\alpha_{e_{k}}\geq 0$

-

$v^{\operatorname{in}}_{k},v^{\operatorname{out}}_{k}\in\mathcal{V}$

-

$\displaystyle:=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\mathcal{Q}% -\setminus\mathcal{R}}\geq 0,\quad f^{\delta}_{\mathcal{Q}^{c}}=0\},$

-

$\displaystyle\lambda^{\operatorname{vec}}(D^{+})=\lambda^{\operatorname{vec}}(% -D^{-})+(D^{+}-D^{-})Af^{\delta}.$

-

$\lambda^{\operatorname{vec}}(D)$

-

${f^{\delta}_{T}>0}$

-

$\mathcal{J}^{\prime\operatorname{use}}_{j-1}\subset\mathcal{P}^{\prime}$

-

$D\in(D_{i},D_{i+2})$

-

$\frac{\partial}{\partial T}\widetilde{V}(T)=\widetilde{\lambda}^{\operatorname% -{WE}}(T)$

-

$f^{T}_{p}\geq 0$

-

$D\in(2.2,\infty)$

-

$\enskip e_{6}$

-

$f^{\delta}\in\mathcal{M}$

-

$A_{r}f^{\delta} -

$f_{\mathcal{R}}$

-

$\Gamma_{D}\neq\Gamma_{D_{i}}$

-

$\widetilde{f}^{D}=\left(\begin{array}[]{cc}\frac{D}{2},&\frac{D}{2}\end{array}% -\right)^{\top},$

-

$\displaystyle u_{\mathcal{P}^{\prime\prime},k-1}$

-

$\displaystyle(\widetilde{f}^{\delta})^{\top}A\widetilde{f}^{\delta}=\min_{r\in% -\widetilde{\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}<\min_{r\in\mathcal{Q}}A_{r% -}f^{\delta}$

-

$e_{k}\in\mathcal{E}$

-

$f^{\delta}\in\mathcal{F}_{1}\cap\Gamma_{D}$

-

$[D^{-},D^{+}]$

-

$u_{\widetilde{\mathcal{P}},\widetilde{M}}(D)<\lambda^{\operatorname{WE}}(D)$

-

$\lambda^{\operatorname{WE}}(D_{M})=\min_{r\in\mathcal{P}}\lambda^{% -\operatorname{vec}}_{r}(D_{M})$

-

$\displaystyle C(f^{\prime})$

-

$\widetilde{\mathcal{W}}_{D}$

-

$\displaystyle\leq C_{r}(f^{T}),\quad\forall r\in\mathcal{P}.$

-

$\mathcal{R}\subseteq\mathcal{Q}$

-

$\{p_{1},p_{2}\}$

-

$\mathcal{J}^{\operatorname{act}}_{M}\cap\mathcal{I}_{2}=\emptyset$

- - - diff --git a/htmls/output_mathjax_example_10062.html b/htmls/output_mathjax_example_10062.html deleted file mode 100644 index bdceed8096d674e8ed404856102c84c4f96167c8..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10062.html +++ /dev/null @@ -1,137 +0,0 @@ - - - - MathJax Example - - - - -

$\delta C^{i}_{p^{\prime}}>\delta C^{i}_{r^{\prime}}$

-

${\delta\widetilde{\lambda}^{\widetilde{M}}=\delta\lambda^{M}}$

-

$\lambda^{\prime\operatorname{WE}}$

-

$\lambda^{\operatorname{WE}}(T)<\widetilde{\lambda}^{\operatorname{WE}}(T)\quad% -\text{for all }T\in(D^{-},D^{+}).$

-

$Af^{D_{M}}+\beta=\lambda^{\operatorname{vec}}(D_{M})$

-

$\widehat{\Gamma}^{\prime}_{D}\cap\mathcal{F}_{1}=\emptyset$

-

$f^{\delta}(D):=\frac{\partial^{+}}{\partial D}f^{D}=\begin{cases}\left(\begin{% -array}[]{ccc}0,&0,&1\end{array}\right)^{\top}&\text{for }D\in[0,1),\\ -\left(\begin{array}[]{ccc}1,&1,&-1\end{array}\right)^{\top}&\text{for }D\in[1,% -2),\\ -\left(\begin{array}[]{ccc}\frac{1}{2},&\frac{1}{2},&0\end{array}\right)^{\top}% -&\text{for }D\in[2,\infty).\end{cases}$

-

$\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{P}$

-

$f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{\delta}=0$

-

$p\in\mathcal{J}^{\operatorname{use}}_{i}$

-

$\beta:=(\beta_{p})_{p\in\mathcal{P}}$

-

${f^{D}+\epsilon f^{\delta}}$

-

$D^{-}\in(0,D^{+})$

-

$f^{T}\in\mathcal{W}_{T}$

-

$\displaystyle\widetilde{\Gamma}_{D}$

-

$D<\widetilde{D}_{\widetilde{M}}$

-

$\delta\widetilde{\lambda}^{-}(D^{+})\geq\delta\lambda^{-}(D^{+})$

-

$\{f^{\delta,i_{k}}\}_{k\in\mathbb{N}}$

-

$A_{p}\widetilde{f}^{\delta}=A_{r}\widetilde{f}^{\delta}$

-

$\widehat{\lambda}^{\prime\prime\operatorname{WE}}(D)<\widehat{\lambda}^{\prime% -\operatorname{WE}}(D)$

-

$p\in\mathcal{I}_{3}$

-

$\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{J}^{\operatorname{act}}_{i},% -\quad\mathcal{R}^{\operatorname{use}}_{D}=\mathcal{J}^{\operatorname{use}}_{i}.$

-

$\displaystyle=C(f^{T})+(D-T)Af^{\delta}.$

-

$\displaystyle C_{e_{1}}(f_{e_{1}})=f_{e_{1}},$

-

$T\in[D^{-},D^{+}]^{c}$

-

$\mathcal{J}^{\operatorname{act}}_{i-1}=\mathcal{R}^{\operatorname{act}}_{D_{i}}$

-

$\displaystyle=-\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}% -\quad\text{for all }f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D}^{-},A).$

-

$\mathcal{J}^{\prime\operatorname{use}}_{j-1}=\emptyset$

-

${0\leq D<\frac{2}{3}}$

-

$(\widecheck{\mathcal{P}},\mathcal{C})$

-

$\mathcal{D}:=(D_{0},D_{1},\cdots,D_{M},D_{M+1})$

-

$r\in\mathcal{R}^{\operatorname{use}}_{D}$

-

${\widetilde{\mathcal{P}}:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}=% -\mathcal{J}^{\operatorname{use}}_{i}}$

-

$\mathcal{J}^{\operatorname{use}}_{i}\neq\mathcal{J}^{\operatorname{use}}_{j}$

-

$p\in\mathcal{R}^{\operatorname{use}}_{T}=\mathcal{J}^{\operatorname{use}}_{M}$

-

$\Gamma_{D_{i}}\subseteq\Gamma^{i}$

-

$\displaystyle Af^{\delta}=\delta C^{M},$

-

$\displaystyle=z\big{(}\widetilde{V}(z)-V(z)\big{)}|_{0}^{D}-\int_{0}^{D}% -\widetilde{V}(z)-V(z)dz$

-

$\widehat{f}^{\delta}\in\Gamma_{D_{M}}$

-

$\mathcal{R}^{\operatorname{use}}_{T}\subseteq\mathcal{R}^{\operatorname{act}}_% -{D}$

-

${j\in[M+1]}$

-

$\displaystyle=\lambda^{\operatorname{vec}}(D^{-})+(D^{+}-D^{-})\delta C^{i},$

-

$\Gamma_{D}=\operatorname{SOL}(\mathcal{M}_{D},A)$

-

$\widehat{\lambda}^{\operatorname{WE}}(D)\leq u_{\mathcal{P},i}(D)$

-

$p\in\mathcal{R}^{\operatorname{act}}_{D_{i}}$

-

$u_{\widetilde{\mathcal{P}},\widetilde{M}}(D)<\lambda^{\operatorname{WE}}(D).$

-

$\widecheck{\lambda}^{\operatorname{WE}}_{i,D}(D)\leq u_{\widetilde{\mathcal{P}% -},i}(D).$

-

$\lambda^{\operatorname{WE}}(D)>\widetilde{\lambda}^{\operatorname{WE}}(D)$

-

$D\leq\widetilde{D}_{i+1}$

-

$A_{p}f^{\delta}=\min_{r\in\mathcal{P}}A_{r}f^{\delta}$

-

$\delta C^{M}_{p}>\delta\lambda^{M}$

-

$\delta\lambda^{i-1}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{i-1}}\delta C^% -{i-1}_{r}$

-

$\widetilde{(\cdot)}$

-

$\operatorname{SOL}(\mathcal{M},A)$

-

$\displaystyle(Af^{\delta_{2}})^{\top}(f^{\delta_{1}}-f^{\delta_{2}})$

-

$D\geq D_{M}$

-

$D_{1}=1$

-

$f:=f^{\delta}-\epsilon(e_{p}-e_{r})\in\mathcal{M}$

-

$f^{D+\epsilon}\geq 0$

-

$p\in\mathcal{R}^{\operatorname{use}}_{T^{+}}$

-

$\displaystyle\lambda^{\operatorname{vec}}(D^{+})$

-

$\displaystyle C_{e_{2}}(f_{e_{2}})=1,$

-

$\lambda^{\prime\operatorname{WE}}(D^{\prime}_{j})=u_{\mathcal{P},i}(D^{\prime}% -_{j})$

-

$\displaystyle C_{e_{5}}(f_{e_{5}})=0.$

-

$\mathcal{S}^{\prime}=(\mathcal{J}^{\operatorname{use}}_{i})^{c}$

-

$\enskip e_{4}$

-

$f^{D}=\sum_{r\in\mathcal{R}^{\operatorname{use}}_{D}}\mu_{r}f^{r}$

-

${D,T\in(D_{i},D_{i+1})}$

-

${\widetilde{\lambda}^{\operatorname{WE}}(D)<\lambda^{\operatorname{WE}}(D)}$

-

$\displaystyle f\geq 0.$

-

$\displaystyle\lambda^{\operatorname{WE}}(T)$

-

$D_{i},D_{i+1},D_{j},D_{j+1}\in\mathcal{D}$

-

$p\in\mathcal{R}$

-

${-t(T_{1}-D)^{-1}\leq f^{\delta,i}_{r}\leq-nt(T_{1}-D)^{-1}+1}$

-

$C_{p}(f^{D^{+}})=\delta\lambda^{M}D^{+}+\bar{\beta}.$

-

$\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{R}^{\operatorname{act}}_{T}$

-

$\Gamma_{D_{M}}\subseteq\Gamma^{M}$

-

$\Gamma^{M}$

-

$\displaystyle=\int_{0}^{D}V(z)-\widetilde{V}(z)dz.$

-

$f\in{}^{n}$

-

${T=D_{i+1}}$

-

$f^{D_{i}}$

-

${\widetilde{f}^{\delta-}\in\operatorname{SOL}(\widetilde{\mathcal{M}}^{-},A)}$

-

$\displaystyle\leq C_{r}(f^{D}),\quad\forall r\in\mathcal{P},$

-

$(\mathcal{P}^{\prime},\mathcal{C})$

-

$\delta\lambda^{-}(D)$

-

$\delta\lambda^{i}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{i}}\delta C^{i}_% -{r}$

-

$\widetilde{\mathcal{R}}\subseteq\widetilde{\mathcal{Q}}$

-

$T>D$

-

$[D_{M},\infty)$

-

$\displaystyle\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}$

-

$\widetilde{f}^{D}$

-

$B_{k,i}=1$

-

$f^{-},f^{+}\in{}^{p}$

-

$\displaystyle -

$u_{\mathcal{P},2}$

-

$f^{D}+(T-D)f^{\delta}$

-

$\widetilde{f}^{\delta}\in\operatorname{SOL}(\widetilde{\mathcal{M}}_{D},A)$

-

$\lambda^{\prime\prime\operatorname{WE}}$

-

$f^{D}+\epsilon f^{\delta,i}$

- - - diff --git a/htmls/output_mathjax_example_10063.html b/htmls/output_mathjax_example_10063.html deleted file mode 100644 index 1500a2e61d1c88a71924c6e974b27434c4b5afc4..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10063.html +++ /dev/null @@ -1,172 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle C_{e}(f)$

-

$p_{2}=(e_{3},e_{4})$

-

$\lim_{k\rightarrow\infty}T_{i_{k}}=\infty$

-

$\displaystyle=u_{\emptyset,i}(D_{i})+(D-D_{i})\delta\lambda^{i}.$

-

${\epsilon\in[0,\epsilon^{*})}$

-

$\displaystyle\quad f\in\mathcal{M}.$

-

${C_{p}(f^{D})=\min_{r\in\mathcal{P}}C_{r}(f^{D})}$

-

${\mathcal{H}_{D}:=\{f\in{}^{n}\;|\;\sum_{i\in[n]}f_{i}=D\}}$

-

$\mymathbf{0}_{n}$

-

$D\in[1,\frac{1}{2}]$

-

$\displaystyle\widetilde{\mathcal{M}}^{-}$

-

$\widetilde{f}^{\delta}\in\widetilde{\Gamma}_{D}\cap\mathcal{F}_{1}$

-

$f^{\delta,i}:=(T_{i}-D)^{-1}(f^{T_{i}}-f^{D}).$

-

${D\mapsto\lambda^{\operatorname{vec}}(D):=C(f^{D})}$

-

$\mathcal{H}_{-1}$

-

$(\widetilde{\mathcal{P}},\mathcal{C})$

-

$\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{D}$

-

$\displaystyle\widetilde{\mathcal{H}}_{D}$

-

$C(f)=Af+b=\left(\begin{array}[]{ccccc}1&0&1&0\\ -0&1&1&0\\ -1&1&3&0\\ -0&0&0&1\\ -\end{array}\right)f+\left(\begin{array}[]{c}1\\ -1\\ -0\\ -2\\ -\end{array}\right)$

-

$u_{\widetilde{\mathcal{P}},i}(D)$

-

$C(f)=Af+b=\left(\begin{array}[]{ccccc}1&0&1&0\\ -0&1&1&0\\ -1&1&2&0\\ -0&0&0&0\\ -\end{array}\right)f+\left(\begin{array}[]{c}1\\ -1\\ -0\\ -2.1\\ -\end{array}\right)$

-

$\displaystyle=f^{D_{i}}+(1-\mu)(f^{D_{i+1}}-f^{D_{i}})$

-

$f^{\prime}_{p}=0$

-

${p\in\mathcal{R}^{\operatorname{use}}_{D}}$

-

${C_{p}(f^{D^{+}})\geq\delta\lambda^{M}D^{+}+\bar{\beta}}$

-

$\widehat{\mathcal{S}}^{\prime}=(\widehat{\mathcal{R}}^{\operatorname{use}}_{D}% -)^{c}$

-

$\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{R}^{\operatorname{act}}_% -{D_{i}}$

-

$\mathcal{X}\subset{}^{n}$

-

$p\in PP^{\prime}$

-

$\displaystyle\frac{\partial}{\partial D}\big{(}\widetilde{V}(T)-V(T)\big{)}$

-

$\delta\widetilde{\lambda}^{-}(D^{+})>\delta\lambda^{-}(D^{+})$

-

$f^{T}_{p}=0$

-

$u_{\widetilde{\mathcal{P}},i}$

-

${\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}\subseteq\mathcal{R}^{% -\operatorname{use}}_{D}\cap(\mathcal{S}^{\texttt{rem}})^{c}}$

-

$p\in\mathcal{R}^{\operatorname{use}}_{D^{-}}$

-

$\displaystyle:=\operatorname{SOL}(\widetilde{\mathcal{F}}_{D},C),$

-

${\widetilde{\lambda}^{\operatorname{WE}}(D)=\widetilde{V}^{\prime}(D)}$

-

$f^{D}=\begin{cases}\left(\begin{array}[]{cccc}0,&0,&D,&0\end{array}\right)^{% -\top}&\text{for }D\in[0,1],\\ -\left(\begin{array}[]{cccc}D-1,&D-1,&2-D,&0\end{array}\right)^{\top}&\text{for% - }D\in[1,2],\\ -\left(\begin{array}[]{cccc}\frac{D}{2},&\frac{D}{2},&0,&0\end{array}\right)^{% -\top}&\text{for }D\in[2,2.2],\\ -\left(\begin{array}[]{cccc}1.1,&1.1,&0,&D-2.2\end{array}\right)^{\top}&\text{% -for }D\in[2.2,\infty).\end{cases}$

-

$\mathcal{R}^{\operatorname{use}}_{D_{i}}\subseteq\mathcal{R}^{\operatorname{% -use}}_{T}$

-

${C_{p}(f^{T})=\lambda^{\operatorname{WE}}(T)=\delta\lambda^{M}T+\bar{\beta}}$

-

$\widetilde{f}^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D},A)$

-

$\lambda^{\operatorname{vec}}_{p_{3}}$

-

$\delta\lambda^{i}\geq 0$

-

${(D-D_{M})(f^{\delta})^{\top}Af^{\delta}=(D-D_{M})\delta\lambda^{M}}$

-

$D\in[D_{i},D_{i+1})$

-

$\mathcal{S}^{\texttt{rem}}_{i,D}\subset\mathcal{P}$

-

$\delta\lambda^{M}=\min_{r\in\mathcal{P}}\delta C^{M}$

-

$\mathcal{S}^{\texttt{rem}}=\{p_{3}\}$

-

$\widetilde{\mathcal{M}}_{D}=\{\widetilde{f}^{\delta}\in\widetilde{\mathcal{H}}% -_{1}\;|\;\widetilde{f}_{\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}% -\setminus\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}}^{\delta}\geq 0,% -\enskip\widetilde{f}_{(\widetilde{\mathcal{R}}^{\operatorname{act}}_{D})^{c}}^% -{\delta}=0\}.$

-

$B\in{}^{q\times n}$

-

$\Gamma^{M}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$

-

${f^{\delta}\in\mathcal{H}_{1}}$

-

$\operatorname{coco}_{\mu}(f^{-},f^{+}):=\mu f^{-}+(1-\mu)f^{+}$

-

$\lim_{k\to\infty}f^{\delta,i_{k}}_{r}\geq 0$

-

$f^{D^{+}}:=f^{*}+(D^{+}-D)f^{\delta}\in\mathcal{W}_{D^{+}}.$

-

${\widetilde{V}(T)=V(T)}$

-

$\displaystyle\lambda^{\operatorname{vec}}(T_{\mu})$

-

$(\mathcal{P},\mathcal{C})$

-

$T\in(D^{+}-\epsilon,D^{+}]$

-

${C_{p}(f^{\prime})\geq\delta\lambda^{M}D+\bar{\beta}}$

-

$\widetilde{\lambda}^{\operatorname{WE}}(T)$

-

$\lambda^{\prime\prime\operatorname{WE}}(D)=u_{\mathcal{P}^{\prime},j-1}\quad% -\text{for all }D\in[D^{\prime\prime}_{k},D^{\prime\prime}_{k+1}),$

-

${\lambda^{\operatorname{WE}}(D)=\widetilde{\lambda}^{\operatorname{WE}}(D)}$

-

$C(f)=Af+b=\left(\begin{array}[]{ccccc}3&0&2&0\\ -0&4&2&1\\ -2&2&4&0\\ -0&1&0&2\\ -\end{array}\right)f+\left(\begin{array}[]{c}1\\ -1\\ -0\\ -6\\ -\end{array}\right).$

-

$D\in[0,2)$

-

$\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{J}^{\operatorname{act}}_% -{i+1}$

-

$D\in(\frac{2}{3},1)$

-

${T\in[0,D]}$

-

$f^{T}_{p}>0$

-

$Q\in{}^{q\times q}$

-

$f^{\delta}\in\mathcal{F}_{1}$

-

$\widetilde{f}^{\delta}\in\Gamma_{D}$

-

$\displaystyle\delta\widetilde{\lambda}^{-}(D)$

-

${f^{\mu}:=\operatorname{coco}_{\mu}(f^{D},f^{T})}$

-

$\displaystyle=(Af^{\delta_{1}})^{\top}(f^{\delta_{1}}-f^{\delta_{2}})-(Af^{% -\delta_{2}})^{\top}(f^{\delta_{1}}-f^{\delta_{2}})\leq 0,$

-

$f^{T}=f^{\mu}+(T-T_{\mu})f^{\delta}$

-

$[D,D_{i+1}]$

-

$\Gamma^{\prime}_{D}\cap\mathcal{F}_{1}=\emptyset$

-

$\displaystyle\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}$

-

$\begin{split}C_{1}(f)&=f_{1}+f_{3}+1,\\ -C_{2}(f)&=f_{2}+f_{3}+1,\\ -C_{3}(f)&=f_{1}+f_{2}+2f_{3},\end{split}$

-

$\displaystyle=Af^{T}+\beta+(D-T)Af^{\delta},$

-

$\displaystyle\Gamma_{D}:=\{f^{\delta}$

-

$\lambda^{\operatorname{vec}}_{p^{\prime}}(D)>\lambda^{\operatorname{vec}}_{r^{% -\prime}}(D)$

-

$\mathcal{R}^{\operatorname{use}}_{D_{i}}\subseteq\mathcal{J}^{\operatorname{% -use}}_{i}$

-

$\mathcal{S}^{\prime\prime}=(\mathcal{J}^{\prime\operatorname{use}}_{j-1})^{c}$

-

$V(2)=\widetilde{V}(2)$

-

$\widetilde{\lambda}^{\operatorname{WE}}(T)\geq\lambda^{\operatorname{WE}}(T)$

-

$\displaystyle\mathcal{R}^{\operatorname{use}}_{D_{i+1}}$

-

$D>6$

-

$\displaystyle\quad C_{r}(f)\geq\delta\lambda^{M}D+\bar{\beta}$

-

$f^{D^{\prime}}\in\mathcal{W}_{D^{\prime}}$

-

${f^{\delta}\in\mathcal{F}_{1}}$

-

$\mathcal{R}^{\operatorname{use}}_{D_{i+1}}\subseteq\mathcal{J}^{\operatorname{% -use}}_{i+1}\subseteq\mathcal{J}^{\operatorname{act}}_{i+1}$

-

$f^{\delta}\in\mathcal{H}_{1}$

-

$f^{D},f^{T}\geq 0$

-

$A\widetilde{f}^{\delta}=\delta C^{M}$

-

$\displaystyle:=\{\widetilde{f}^{\delta}\in\widetilde{\mathcal{H}}_{1}\;|\;% -\widetilde{f}_{\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}\setminus% -\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}}^{\delta}\geq 0,\enskip% -\widetilde{f}_{(\widetilde{\mathcal{R}}^{\operatorname{act}}_{D})^{c}}^{\delta% -}=0\},$

-

$\delta\widetilde{\lambda}^{-}(D^{+})=\delta\lambda^{-}(D^{+})$

-

$J(D)=\int_{0}^{D}\widetilde{\lambda}^{\operatorname{WE}}(z)-\lambda^{% -\operatorname{WE}}(z)dz$

-

$n=|\mathcal{P}|$

- - - diff --git a/htmls/output_mathjax_example_10064.html b/htmls/output_mathjax_example_10064.html deleted file mode 100644 index 7503d671cbff31ea44ca952f08bd709e655bdf33..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10064.html +++ /dev/null @@ -1,144 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\widetilde{\mathcal{M}}$

-

$\mathcal{M}^{-}_{D}$

-

$\displaystyle\min_{r\in\widetilde{\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}$

-

${T\in(T_{\mu},D_{i+1})}$

-

$f^{D_{M}}_{p}>0$

-

$\displaystyle=\widetilde{\lambda}^{\operatorname{WE}}(T)-\lambda^{% -\operatorname{WE}}(T)$

-

${p_{3}=(e_{1},e_{5},e_{4})}$

-

$T\notin\mathcal{D}$

-

$\lambda^{\operatorname{WE}}(T)>\widetilde{\lambda}^{\operatorname{WE}}(T)\quad% -\text{for all }T\in(D^{-},D^{+}).$

-

$\displaystyle:=\{\widetilde{f}\in{\mathbb{R}}_{\geq 0}^{n}\;|\;\sum_{p\in% -\widetilde{\mathcal{P}}}\widetilde{f}_{p}=D,\enskip\widetilde{f}_{\mathcal{S}^% -{\texttt{rem}}}=0\},$

-

$D,T\in(D_{i},D_{i+1})$

-

$\epsilon\in[0,T_{i}-D)$

-

$(f^{\delta})^{\top}Af^{D_{M}}=D_{M}\delta\lambda^{M}\quad\text{ for all }f^{% -\delta}\in\operatorname{SOL}(\mathcal{F}_{1},A).$

-

$D\in(\frac{1}{2},2)$

-

$A(f^{\delta_{1}}-f^{\delta_{2}})=0$

-

$J(D)\geq 0\text{ for all }D\in[0,\infty),$

-

$\displaystyle\text{(respectively},\quad\min_{r\in\widetilde{\mathcal{Q}}}A_{r}% -\widetilde{f}^{\delta-}$

-

$f^{*}\in f^{D_{M}}$

-

${0\leq D^{-}\leq D^{+}}$

-

$\displaystyle:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;f^{\delta}_{\mathcal{Q}% -\setminus\mathcal{R}}\geq 0,\quad f^{\delta}_{\mathcal{Q}^{c}}=0\},$

-

$p\in\mathcal{R}^{\operatorname{act}}_{D_{M}}$

-

$\lambda^{\operatorname{WE}}(1)=2$

-

$v^{\operatorname{in}}_{p_{1}}=v_{o}$

-

${\mathcal{R}^{\operatorname{use}}_{D}=\{p_{3}\}}$

-

${D\in(D_{i},D_{i+1})}$

-

$D^{+}>0$

-

$\displaystyle=D\delta\lambda^{M}+\beta^{\top}f^{\delta}.$

-

$\displaystyle\quad f_{r}\geq 0$

-

${\Gamma_{D}\cap\mathcal{F}_{1}=\emptyset}$

-

$f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D},A)$

-

$D\geq D_{M}=1$

-

$\mathcal{M}_{D}:=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f_{\mathcal{R}^{% -\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use}}_{D}}^{\delta}% -\geq 0,\enskip f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{\delta}=0\},$

-

$f^{D}\in\mathcal{F}_{0}$

-

$\frac{\partial}{\partial D}V(D)=\lambda^{\operatorname{WE}}(D).$

-

$\lambda^{\operatorname{WE}}(D)=\begin{cases}3D\quad&\text{for }D\in[0,\frac{1}% -{2}],\\ -\frac{1}{3}D+\frac{4}{3}\quad&\text{for }D\in[\frac{1}{2},\infty).\end{cases}$

-

$r\in(\mathcal{R}^{\operatorname{act}}_{D_{i}})^{c}$

-

$\widetilde{\lambda}^{\operatorname{WE}}(T)<\lambda^{\operatorname{WE}}(T)$

-

$p\in\mathcal{I}_{2}$

-

$\delta\lambda^{M}=\min_{r\in\mathcal{P}}\delta C^{M}_{p}$

-

$\mathcal{M}_{D}^{-}$

-

$p_{3}:=(e_{1},e_{5},e_{4})$

-

$\widetilde{\Gamma}_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$

-

$(f^{\delta})^{\top}Af^{D_{M}}$

-

$f^{D}+\epsilon f^{\delta}\in\mathcal{F}_{D+\epsilon}$

-

$\displaystyle\lambda^{\prime\prime\operatorname{WE}}(D)$

-

$\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{R}^{\operatorname{act}}_{D^{-}}=% -\mathcal{R}^{\operatorname{act}}_{D^{+}}$

-

$\displaystyle\min_{f\in\mathcal{F}_{D}}\sum_{e_{k}\in\mathcal{E}}\int_{0}^{f_{% -e_{k}}}C_{e_{k}}(z)dz,$

-

$\delta\widetilde{\lambda}^{\widetilde{M}}\geq\delta\lambda^{M}$

-

$\displaystyle=\epsilon(A_{p}f^{\delta}-A_{r}f^{\delta})<0.$

-

$f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{D}=0$

-

${T\in(D,D_{i+1}]}$

-

$D\in[D^{-},D^{+}]$

-

${T\in[D_{i-1},D_{i})}$

-

${C_{p}(f^{D^{-}})\leq C_{r}(f^{D^{-}})}$

-

$\mathcal{J}^{\prime\operatorname{use}}_{j-1}\neq\mathcal{J}^{\prime% -\operatorname{use}}_{j}$

-

$T\mapsto u_{\vec{\mathcal{P}},i}(T):=\vec{\lambda}^{\operatorname{WE}}(\vec{D}% -_{i})+(T-\vec{D}_{i})\delta\vec{\lambda}^{i},$

-

${f^{D}\in\mathcal{W}_{D}}$

-

$\mathcal{M}_{D}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{1},f^{\delta}_{% -2},f^{\delta}_{4}\geq 0\}$

-

$\displaystyle C_{p}(f)\leq C_{r}(f)\quad$

-

$\widetilde{\lambda}^{\operatorname{WE}}_{p}(D)=\widetilde{\lambda}^{% -\operatorname{WE}}_{r}(D)$

-

$\Gamma_{D}=\Gamma^{i}$

-

$\mymathbf{1}^{\top}f=D$

-

$D^{\prime}\in(D_{i},D_{i+1})$

-

$f\in\mathcal{F}_{D}$

-

$p\in\mathcal{R}^{\operatorname{act}}_{D_{i}}\cap\mathcal{R}^{\operatorname{act% -}}_{D_{i+1}}$

-

$D^{\operatorname{BP}}\geq 0$

-

$\mathcal{M}_{D_{i}}$

-

$\operatorname{coco}_{\mu}(f^{D},f^{T})\in\mathcal{W}_{\operatorname{coco}_{\mu% -}(D,T)}$

-

$(\mathcal{P}^{\prime\prime},\mathcal{C})$

-

$\displaystyle\mathcal{R}^{\operatorname{use}}_{D_{i}}$

-

$\mathcal{R}^{\operatorname{use}}_{D}$

-

$D\mapsto\mathcal{W}_{D}$

-

$C(f)=Af+\beta,$

-

$\operatorname{SOL}(\mathcal{M}_{D},A)$

-

$\delta\lambda^{i-1}<\delta\lambda^{i}$

-

$\left\lvert{\mathcal{P}}\right\rvert$

-

$\displaystyle=(f^{\delta})^{\top}\Big{(}A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}% -\big{)}+\beta\Big{)}$

-

$\operatorname{SOL}(\mathcal{M}_{D},A)=\operatorname{SOL}(\mathcal{F}_{1},A)$

-

$\operatorname{SOL}(\mathcal{F},A):=\operatorname{SOL}(\mathcal{F},C)$

-

$\displaystyle C_{p}(f^{\prime})$

-

$\displaystyle=C(f^{D_{i}})+(T_{\mu}-D_{i})Af^{\delta_{0}}$

-

$T\in(D^{-},D^{+})$

-

$\left\lvert{T-D}\right\rvert<\delta$

-

$\widetilde{\mathcal{M}}_{D}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{% -\widetilde{\mathcal{P}}^{c}}=0\},$

-

$\displaystyle C_{e_{5}}(f_{e_{5}})=0,$

-

$r\in(\mathcal{R}^{\operatorname{act}}_{T})^{c}$

-

$\lambda^{\prime\operatorname{WE}}(D) -

$\displaystyle=C(f^{\mu})$

-

${T_{\mu}=\operatorname{coco}_{\mu}(D_{i},{D_{i+1}})}$

-

$f^{D^{+}}_{p}=0$

-

$G(x^{*})^{\top}(x-x^{*})\geq 0,\enskip\forall x\in\mathcal{X}.$

-

$\widetilde{V}(T) -

$\displaystyle=\mathcal{R}^{\operatorname{act}}_{D_{i}}\Rightarrow\delta\lambda% -^{i-1}<\delta\lambda^{i},\qquad\mathcal{J}^{\operatorname{act}}_{i}=\mathcal{R% -}^{\operatorname{act}}_{D_{i}}\Rightarrow\delta\lambda^{i-1}>\delta\lambda^{i}.$

-

$p_{1}=(e_{1},e_{2})$

-

$\lambda^{\operatorname{vec}}(T)$

-

$u_{\mathcal{P},M}$

-

${A_{p}f^{\delta}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}}$

-

$\widetilde{\mathcal{P}}\subset\mathcal{P}$

-

$\lambda^{\operatorname{WE}}(D)=V^{\prime}(D)$

-

$\lambda^{\operatorname{WE}}(D)=\lambda^{\operatorname{WE}}(D_{M})+(D-D_{M})% -\delta\lambda^{M}$

- - - diff --git a/htmls/output_mathjax_example_10065.html b/htmls/output_mathjax_example_10065.html deleted file mode 100644 index 41d3bbe4408e8cac14542ec65af68c0ac86ef770..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10065.html +++ /dev/null @@ -1,145 +0,0 @@ - - - - MathJax Example - - - - -

$i\in[\widetilde{M}]$

-

$f_{e_{k}}:=\sum_{\{p\in\mathcal{P}\;|\;e_{k}\in p\}}f_{p}.$

-

$\displaystyle=(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D)$

-

${\mathcal{P}^{\prime}:=\mathcal{P}\setminus\mathcal{S}^{\prime}}$

-

$\lambda^{\prime\operatorname{WE}}(D)<\lambda^{\operatorname{WE}}(D)$

-

$D^{\operatorname{BP}}$

-

$f^{\delta}_{p}\neq 0$

-

$\alpha_{e_{k}}=0$

-

$\mathcal{P}^{\prime\prime}=\emptyset$

-

$f^{D}\in\operatorname{SOL}(\mathcal{F}_{D},C)$

-

$A_{p}f^{\delta}>A_{r}f^{\delta}$

-

$\displaystyle:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;f^{\delta}_{\widetilde{% -\mathcal{Q}}\setminus\widetilde{\mathcal{R}}}\geq 0,\quad f^{\delta}_{% -\widetilde{\mathcal{Q}}^{c}}=0\}.$

-

$[D^{+},D]$

-

$p,r\in\mathcal{J}^{\operatorname{act}}_{i}$

-

$f^{D^{+}}\in\mathcal{F}_{D^{+}}$

-

$\displaystyle\widetilde{\mathcal{F}}_{D}$

-

$f\in{\mathbb{R}}_{\geq 0}^{n}$

-

$\displaystyle C_{p}(f^{T})$

-

$\displaystyle=\epsilon(f^{\delta})^{\top}A(e_{p}-e_{r})$

-

$p_{2}:=(e_{3},e_{4})$

-

$C_{p}(f^{*})\geq\delta\lambda^{M}D+\bar{\beta}$

-

$f^{D^{-}}\in\mathcal{W}_{D^{-}}$

-

${f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D},A)}$

-

$f^{*}\in\mathcal{W}_{D_{M}}$

-

$\mathcal{J}^{\operatorname{act}}_{i}\cap\mathcal{J}^{\operatorname{act}}_{i+1}=\emptyset$

-

$\delta C^{M}=Af^{\delta}$

-

${\lambda^{\operatorname{vec}}(D_{M})\geq\lambda^{\operatorname{WE}}(D_{M})% -\mymathbf{1}}$

-

$D\in[D^{\prime}_{j-1},D^{\prime}_{j})$

-

${\widetilde{f}^{\delta}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)}$

-

$D\in(D_{i-1},D_{i})$

-

$f_{p_{4}}=0$

-

$f_{\mathcal{R}^{\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use% -}}_{D}}^{\delta}\geq 0$

-

$C_{p}(\widetilde{f}^{D})\leq C_{r}(\widetilde{f}^{D})\quad\text{for all }r\in% -\widetilde{\mathcal{P}}.$

-

$p\in\mathcal{I}_{1}\cup\mathcal{I}_{3}$

-

$\displaystyle\subseteq\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{J}% -^{\operatorname{act}}_{i}\subseteq\mathcal{R}^{\operatorname{act}}_{D_{i+1}}.$

-

$D\in(0,2)$

-

$f^{\delta}\in\Gamma_{T}$

-

$\beta_{p}=\sum_{e_{k}\in p}\beta_{e_{k}}$

-

$C_{p}(f)=\delta\lambda^{M}D+\bar{\beta}$

-

$\widetilde{\mathcal{F}}_{D}$

-

$\widetilde{\mathcal{P}}:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}$

-

$(\widetilde{f}^{\delta}-f^{\delta})^{\top}A(f^{\delta}-\widetilde{f}^{\delta})% -\leq 0.$

-

$\mathcal{D}=\{0,1,\infty\}$

-

$\widetilde{\lambda}^{\operatorname{WE}}(D)=u_{\widetilde{\mathcal{P}},% -\widetilde{M}}(D)$

-

${\mathcal{S}^{\texttt{rem}}:=(\mathcal{J}^{\operatorname{use}}_{i})^{c}}$

-

$C_{e_{k}}(f_{e_{k}}):=\alpha_{e_{k}}f_{e_{k}}+\beta_{e_{k}},$

-

$\widetilde{\lambda}^{\operatorname{WE}}(D)$

-

$(\mathcal{R}^{\operatorname{act}}_{D})^{c}=\mathcal{P}\setminus\mathcal{R}^{% -\operatorname{act}}_{D}$

-

$A_{p}f^{\delta}=\min_{r\in\mathcal{Q}}A_{r}f^{\delta}$

-

${[D^{\prime}_{j-1},D^{\prime}_{j})\subseteq[D^{\prime\prime}_{k},D^{\prime% -\prime}_{k+1})}$

-

${\widetilde{f}^{D}\in\mathcal{W}_{D}}$

-

$f^{D}_{e_{k}}=\widehat{f}^{D}_{e_{k}}$

-

$\displaystyle\text{ for all }p\in\mathcal{J}^{\operatorname{act}}_{M}\text{ % -and }r\in\mathcal{P},$

-

$f^{\delta_{0}}\in\mathcal{M}$

-

$\lambda^{\operatorname{vec}}_{p^{\prime}}(D)<\lambda^{\operatorname{vec}}_{r^{% -\prime}}(D)$

-

$(\widetilde{f}^{\delta})^{\top}\lambda^{\operatorname{vec}}(D_{M})=\lambda^{% -\operatorname{WE}}(D_{M}),$

-

$\widetilde{f}^{\delta}$

-

$\widetilde{f}^{\delta}\in\widetilde{\Gamma}_{T}$

-

${\widetilde{\mathcal{M}}_{D}\subseteq\mathcal{M}_{D}}$

-

$\{f^{\delta,i}\}_{i\in\mathbb{N}}\subset\Gamma_{D}$

-

$\displaystyle:=\{p\in\widetilde{\mathcal{P}}\;|\;\exists\widetilde{f}^{D}\in% -\widetilde{\mathcal{W}}_{D}\text{ such that }\widetilde{f}^{D}_{p}>0\},$

-

$\lambda^{\operatorname{vec}}_{p}(D_{M})=\lambda^{\operatorname{WE}}(D_{M})$

-

$T\in[0,D]$

-

$\displaystyle\mathcal{J}^{\operatorname{use}}_{i-1}$

-

$\operatorname{SOL}(\mathcal{M}_{D},A)=\ker(A)\cap\mathcal{M}_{D}$

-

$T\in(D,\infty)$

-

$\displaystyle\mathcal{W}_{D}:=\enskip\begin{cases}\left\{\left(\begin{array}[]% -{cccc}0&0&D&0\end{array}\right)^{\top}\right\}&\text{for }D\in[0,1],\\ -\{f\in\mathcal{F}_{D}\;|\;f_{1}+f_{3}=1,f_{2}+f_{3}=1\}&\text{for }D\geq 1.\\ -\end{cases}$

-

$\mathcal{M}_{D}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\mathcal{J}^{% -\operatorname{act}}_{i-1}\setminus\mathcal{J}^{\operatorname{use}}_{i-1}}\geq 0% -,\enskip f^{\delta}_{(\mathcal{J}^{\operatorname{act}}_{i-1})^{c}}=0\}.$

-

$\mathcal{P}^{\prime}=\mathcal{R}^{\prime\operatorname{use}}_{D}=\mathcal{R}^{% -\prime\operatorname{use}}_{T}$

-

$\Gamma_{D}\cap\mathcal{F}_{1}=\emptyset$

-

$\widetilde{f}_{\mathcal{S}^{\texttt{rem}}}=0$

-

$\displaystyle C_{e_{3}}(f_{e_{3}})=1,$

-

$\Gamma^{i+1}\cap\Gamma^{i}=\emptyset$

-

$e_{3}\enskip$

-

$\mathcal{R}^{\operatorname{use}}_{D}\subseteq\mathcal{R}^{\operatorname{act}}_% -{D}$

-

$v^{\operatorname{out}}_{pk}=v^{\operatorname{in}}_{pk+1}$

-

$f^{D}\in\mathcal{F}_{D}$

-

$\mathcal{W}_{D_{M}}$

-

$u_{\mathcal{P}^{\prime},j-1}$

-

${\mathcal{R}^{\operatorname{act}}_{D}=\{p_{1},p_{2},p_{3},p_{4}\}}$

-

$\delta C_{M}$

-

$\displaystyle V(D)=$

-

$f^{\delta}\in\mathcal{H}_{-1}$

-

$f^{D_{i}}\in\mathcal{W}_{D_{i}}$

-

$\displaystyle\quad\frac{1}{2}f^{\top}Af$

-

$(e_{p_{1}},\cdots,e_{p_{l}})$

-

$T\in(D_{M},\infty)$

-

$\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}\geq 0$

-

$k\in[l-1]$

-

$\widecheck{\mathcal{P}}\subseteq\widetilde{\mathcal{P}}$

-

$\widehat{\lambda}^{\operatorname{WE}}(D)<\widetilde{\lambda}^{\operatorname{WE% -}}(D)$

-

${(\widetilde{f}^{\delta})^{\top}A(f^{\delta}-\widetilde{f}^{\delta})\leq 0}$

-

$r^{\prime}\in(\mathcal{J}^{\operatorname{act}}_{i})^{c}\cap\mathcal{J}^{% -\operatorname{act}}_{i+1}$

-

$\displaystyle:=C(\widetilde{f}^{D})\text{ for any }\widetilde{f}^{D}\in% -\widetilde{\mathcal{W}}_{D},$

-

$\displaystyle:=\big{(}C_{p_{1}}(f),C_{p_{2}}(f),\cdots,C_{p_{m}}(f)\big{)}^{% -\top}.$

-

$\displaystyle C_{p}(f^{D+\epsilon})$

-

$f^{D^{+}}_{p}>0$

-

$(\widehat{\mathcal{P}}\setminus\widehat{\mathcal{S}}^{\prime},\mathcal{C})$

-

$T\in(D_{i},D_{i+1})$

-

$T\in[D_{i},D_{i+1})$

- - - diff --git a/htmls/output_mathjax_example_10066.html b/htmls/output_mathjax_example_10066.html deleted file mode 100644 index 4eb4f759e224758e80c6b5109afd443186a98826..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10066.html +++ /dev/null @@ -1,142 +0,0 @@ - - - - MathJax Example - - - - -

${A=B^{\top}QB}$

-

$\displaystyle\widetilde{\mathcal{M}}_{D}$

-

$(f^{\delta})^{\top}A(\widetilde{f}^{\delta}-f^{\delta})\geq 0,$

-

$T\geq D$

-

$\lambda^{\operatorname{WE}}(T)$

-

$Af^{\delta_{0}}=Af^{\delta}$

-

${f^{\prime}:=f^{T}+(D-T)f^{\delta}}.$

-

$\lambda^{\operatorname{WE}}(D):=\min_{p\in\mathcal{P}}C_{p}(f^{D})=C_{r}(f^{D}% -),\quad r\in\mathcal{R}^{\operatorname{act}}_{D}.$

-

$f^{*}+\epsilon f^{\delta}$

-

${D^{+}\in(D_{j},D_{j+1})}$

-

$(\mathcal{R}^{\operatorname{act}}_{D})^{c}\subseteq(\mathcal{R}^{\operatorname% -{use}}_{T})^{c}$

-

$\delta\lambda^{+}(T)=\delta\lambda^{-}(\bar{D})$

-

$Af^{0}=\mymathbf{0}$

-

$f^{D_{i+1}}\in\mathcal{W}_{D_{i+1}}$

-

$(e_{i})_{i}=1$

-

$\lambda^{\operatorname{WE}}(T)=\lambda^{\operatorname{WE}}(D_{i})+(T-D_{i})% -\delta\lambda^{i}.$

-

$\mathcal{Q}^{c}$

-

$\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta})\geq 0.$

-

$\mathcal{M}_{D}\subset\mathcal{M}_{D_{i}}$

-

$\displaystyle\geq\delta\lambda^{+}(D),$

-

$\delta\lambda^{+}(D)=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{% -\delta},\quad\delta\widetilde{\lambda}^{+}(D)=\min_{r\in\widetilde{\mathcal{R}% -}^{\operatorname{act}}_{D}}A_{r}\widetilde{f}^{\delta},$

-

$\displaystyle\widetilde{\lambda}^{\operatorname{WE}}(D)$

-

$\widetilde{D}_{i+1}$

-

${\delta\lambda^{M}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{M}}A_{r}f^{% -\delta}}$

-

$(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}$

-

$\displaystyle\geq\min_{r\in\mathcal{Q}}A_{r}f^{\delta},$

-

$\displaystyle\forall r\in\mathcal{I}_{3},$

-

$\widetilde{\lambda}^{\operatorname{WE}}(D)<\lambda^{\operatorname{WE}}(D)$

-

$T_{1}\leq T_{i}$

-

$D^{\prime\prime}_{k-1}\in\mathcal{D}^{\prime\prime}$

-

$\widetilde{V}(D)=V(D)$

-

${(f^{\delta})^{\top}Af^{\delta}=\min_{r\in\mathcal{Q}}A_{r}f^{\delta}}$

-

$\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta})+(Af^{\delta})^{\top}(f^{% -\delta}-f^{\delta_{0}})$

-

$u_{\vec{\mathcal{P}},i}$

-

$f^{\delta}:=\lim_{k\to\infty}f^{\delta,i_{k}}\geq 0.$

-

$p_{3}=(e_{1},e_{4})$

-

$f^{\delta}_{\mathcal{Q}^{c}}=0$

-

$p\in\mathcal{J}^{\operatorname{act}}_{i}$

-

$p\in\mathcal{R}^{\operatorname{use}}_{T}\subseteq\mathcal{R}^{\operatorname{% -act}}_{T}$

-

$f^{D_{M}}$

-

$\widetilde{\mathcal{R}}^{\operatorname{act}}_{T}=\widetilde{\mathcal{J}}^{% -\operatorname{act}}_{j}=\widetilde{\mathcal{P}}$

-

$\lambda^{\operatorname{WE}}(D)=\lambda^{\operatorname{vec}}_{p}(D)$

-

$\mathcal{M}_{D_{i}}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\mathcal{R}% -^{\operatorname{act}}_{D_{i}}\setminus\mathcal{R}^{\operatorname{use}}_{D_{i}}% -}\geq 0,\enskip f^{\delta}_{(\mathcal{R}^{\operatorname{act}}_{D_{i}})^{c}}=0\}.$

-

$\{\mathcal{J}^{\operatorname{act}}_{0},\mathcal{J}^{\operatorname{act}}_{1},% -\cdots,\mathcal{J}^{\operatorname{act}}_{M}\}$

-

$p\in\mathcal{I}_{1}$

-

$D\geq\widetilde{D}_{\widetilde{M}}$

-

$T\in(0,D)$

-

$A\widetilde{f}^{\delta}=\delta C^{i-1}$

-

$\frac{2}{3} -

${(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta=(f^{\prime})^{\top}C(% -f^{\prime})=D(\delta\lambda^{M}D+\bar{\beta})}$

-

${\lambda^{\prime\prime\operatorname{WE}}(D)<\lambda^{\prime\operatorname{WE}}(% -D)}$

-

$\mathcal{N}_{D}:=\{\mathcal{S}^{\texttt{nec}}\subseteq\mathcal{P}\;|\;f^{D}_{% -\mathcal{S}^{\texttt{nec}}}\neq 0\text{ for all }f^{D}\in\mathcal{W}_{D}\}.$

-

$A_{p}f^{\delta}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{M}}A_{r}f^{\delta}$

-

$\mathcal{J}^{\operatorname{use}}_{i}$

-

$\delta C^{i}=\delta C^{i+1}$

-

$\displaystyle=A\big{(}f^{T}+(D-T)f^{\delta}\big{)}+\beta,$

-

${u_{\mathcal{P},M}(D)<\lambda^{\operatorname{WE}}(D)}$

-

$\mathcal{J}^{\operatorname{act}}_{i}$

-

$(\mathcal{J}^{\operatorname{use}}_{i})^{c}$

-

$\{f^{T_{i}}\}_{i\in\mathbb{N}}$

-

$\mathcal{J}^{\operatorname{use}}_{M}$

-

$A_{p}f^{\delta}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{T}}A_{r}f^{\delta}$

-

$f^{D^{\prime}}_{p}>0$

-

$D\in(D_{i},D_{i+1})$

-

${\delta C^{i}=\delta C^{i+1}}$

-

$\displaystyle:=\{\widetilde{f}\in{}^{n}\;|\;\sum_{p\in\widetilde{\mathcal{P}}}% -\widetilde{f}_{p}=D,\enskip\widetilde{f}_{\mathcal{S}^{\texttt{rem}}}=0\}.$

-

${p,r\in\mathcal{J}^{\operatorname{act}}_{i}}$

-

$Af^{\delta}=\delta C$

-

$f^{D},\widehat{f}^{D}\in\mathcal{W}_{D}$

-

$D,T\in[D_{i},D_{i+1}]$

-

$\displaystyle\quad f^{\top}Af+f^{\top}\beta$

-

$p\in\mathcal{R}^{\operatorname{use}}_{D^{\prime}}$

-

$\delta\lambda^{\prime j-1}>\delta\lambda^{\prime j}$

-

$\delta C^{M}_{p}=\delta\lambda^{M}$

-

$\lambda^{\operatorname{WE}}$

-

$\delta\lambda^{i-1}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{i-1}}A_{r}f^{\delta}$

-

$\widetilde{\lambda}^{\operatorname{WE}}$

-

$\displaystyle:=(f_{p_{1}},f_{p_{2}},\cdots,f_{p_{m}})^{\top},$

-

$(f^{\delta})^{\top}\Big{(}A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}\big{)}+\beta% -\Big{)},$

-

$\displaystyle=\{r\in\mathcal{P}\;|\;f^{\delta}_{r}=0,\hskip 2.0pt\delta C^{M}_% -{r}>\delta\lambda^{M}\},$

-

$\displaystyle\quad C_{r}(f)=\delta\lambda^{M}D+\bar{\beta}$

-

$\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta_{0}})$

-

$\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}\in\mathcal{W}_{D+\epsilon}$

-

$\delta C^{M}_{p}>\min_{r\in\mathcal{P}}\delta C^{M}_{r}$

-

$\{\mathcal{J}^{\operatorname{use}}_{0},\mathcal{J}^{\operatorname{use}}_{1},% -\cdots,\mathcal{J}^{\operatorname{use}}_{M}\}$

-

$\widetilde{f}^{D}\in\mathcal{W}_{D}\cap\widetilde{\mathcal{F}}_{D}$

-

$\displaystyle\leq\min_{r\in\mathcal{P}}\bigl{(}C_{r}(f^{D})+\epsilon A_{r}f^{% -\delta}\bigr{)}$

-

${\operatorname{coco}_{\mu}(f^{-},f^{+})=\mu f^{-}+(1-\mu)f^{+}}$

-

$\widetilde{f}^{D}=(\frac{1}{2},\enskip\frac{1}{2})^{\top}$

-

$W(D)=\int_{0}^{D}z\big{(}\widetilde{\lambda}^{\operatorname{WE}}(z)-\lambda^{% -\operatorname{WE}}(z)\big{)}dz.$

-

$\mathcal{R}^{\operatorname{act}}_{D}:=\{p\in\mathcal{P}\;|\;C_{p}(f^{D})\leq C% -_{r}(f^{D}),\,\,\forall f^{D}\in\mathcal{W}_{D},\,\,\forall r\in\mathcal{P}\}.$

-

$Af^{\delta}=A\widetilde{f}^{\delta}$

-

$Af^{\delta}=\delta C^{i},$

-

$\delta C^{i-1}_{r}=Af^{\delta}$

-

${p_{4}:=(e_{6})}$

-

$C_{p}(f^{\mu})\leq C_{r}(f^{\mu})$

-

$\mathcal{S}^{\texttt{rem}}$

-

$D^{\prime}_{j-1}>0$

-

$\widetilde{\mathcal{M}}\subseteq\mathcal{M}$

-

${[n]_{0}:=\{0,1,\cdots,n\}}$

- - - diff --git a/htmls/output_mathjax_example_10067.html b/htmls/output_mathjax_example_10067.html deleted file mode 100644 index 9083de6bee46f3b03d1245a410b6fcfba79a1489..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10067.html +++ /dev/null @@ -1,150 +0,0 @@ - - - - MathJax Example - - - - -

$p\in\mathcal{R}^{\operatorname{use}}_{D}$

-

$B_{k,i}=0$

-

$\displaystyle(f^{\delta})^{\top}(Af^{D_{M}}+\beta)$

-

$\displaystyle(f^{*})^{\top}Af^{*}+(f^{*})^{\top}\beta=D(\delta\lambda^{M}D+% -\bar{\beta})$

-

$\Gamma^{-}_{D}$

-

$\displaystyle\mathcal{I}_{3}$

-

$\|f^{D}-f^{T}\|<\epsilon$

-

$v^{\operatorname{out}}_{p_{l}}=v_{d}$

-

$f^{\mu}\in\mathcal{F}_{T_{\mu}}$

-

$\vec{\lambda}^{\operatorname{WE}}$

-

$(\mathcal{J}^{\operatorname{use}}_{i})^{c}\notin\mathcal{N}_{D+\epsilon}$

-

$\delta C^{i}=\delta C^{i-1}$

-

$\delta C_{p}^{M}>\delta\lambda^{M}$

-

$\displaystyle:=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\widetilde{% -\mathcal{Q}}\setminus\widetilde{\mathcal{R}}}\geq 0,\quad f^{\delta}_{% -\widetilde{\mathcal{Q}}^{c}}=0\},$

-

$\displaystyle=\lambda^{\operatorname{WE}}(D_{M})+(T-D_{M})\delta\lambda^{M},$

-

$\displaystyle:=\big{(}C_{e_{1}}(f_{e_{1}}),\cdots,C_{e_{q}}(f_{e_{q}})\big{)}^% -{\top},$

-

$f^{\delta}\in\Gamma_{D_{M}}\cap\mathcal{F}_{1}$

-

$\Gamma_{D}\subseteq\Gamma^{M}$

-

$\widetilde{\mathcal{J}}^{\operatorname{act}}_{j}=\widetilde{\mathcal{R}}^{% -\operatorname{act}}_{\widetilde{D}_{j+1}}$

-

$\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{J}^{\operatorname{act}}_{i}$

-

$\displaystyle\lambda^{\operatorname{WE}}(D)$

-

$\mathcal{J}^{\operatorname{act}}_{i-1}\subset\mathcal{R}^{\operatorname{act}}_% -{D_{i}}$

-

$D^{\prime\prime}_{k-1} -

${\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{WE}}(D_{i})+(T_{\mu}-D% -_{i})Af^{\delta_{0}}}$

-

$\displaystyle=\int_{0}^{D}z\big{(}\widetilde{\lambda}^{\operatorname{WE}}(z)-% -\lambda^{\operatorname{WE}}(z)\big{)}dz$

-

${\mathcal{R}^{\operatorname{act}}_{D}=\{p_{1},p_{2},p_{4}\}}$

-

$\displaystyle=(f^{\delta})^{\top}Af^{\delta}$

-

$C_{e_{5}}(f_{e_{5}}):=f_{e_{5}},\quad C_{e_{6}}(f_{e_{6}}):=2+f_{e_{6}}.$

-

$(f^{*})^{\top}Af^{*}+(f^{*})^{\top}\beta=D(\delta\lambda^{M}D+\bar{\beta})$

-

$C(f)=Af+b=\left(\begin{array}[]{cccc}1&0&1&0\\ -0&1&1&0\\ -1&1&2&0\\ -0&0&0&0\\ -\end{array}\right)f+\left(\begin{array}[]{c}1\\ -1\\ -0\\ -2\\ -\end{array}\right).$

-

$r\notin\mathcal{R}^{\operatorname{act}}_{D^{+}}$

-

$\delta C^{M}$

-

$\mathcal{S}^{\texttt{rem}}_{T}\subset\mathcal{P}$

-

$\mathcal{R}^{\prime\operatorname{act}}_{D^{\prime}_{j}}=\mathcal{J}^{\prime% -\operatorname{act}}_{j}$

-

$D_{M}=1$

-

$\displaystyle\text{ for all }r\in(\mathcal{J}^{\operatorname{act}}_{M})^{c},$

-

$(\mathcal{P}\setminus\widehat{\mathcal{S}})$

-

$f^{D}=\begin{cases}\left(\begin{array}[]{cccc}0&0&D&0\end{array}\right)^{\top}% -&\text{for }D\in[0,\frac{1}{2}],\\ -\left(\begin{array}[]{cccc}\frac{8D-4}{12}&\frac{6D-3}{12}&\frac{7-2D}{12}&0% -\end{array}\right)^{\top}&\text{for }D\in[\frac{1}{2},\frac{7}{2}],\\ -\left(\begin{array}[]{cccc}\frac{4D}{7}&\frac{3D}{7}&0&0\end{array}\right)^{% -\top}&\text{for }D\in[\frac{7}{2},\frac{35}{9}],\\ -\left(\begin{array}[]{cccc}\frac{7D+15}{19}&\frac{3D+20}{19}&0&\frac{9D-35}{19% -}\end{array}\right)^{\top}&\text{for }D\in[\frac{35}{9},6],\\ -\left(\begin{array}[]{cccc}\frac{10D+27}{29}&\frac{4D+34}{29}&\frac{D-6}{29}&% -\frac{14D-55}{29}\end{array}\right)^{\top}&\text{for }D\in[6,\infty).\end{cases}$

-

${p_{1}:=(e_{1},e_{2})}$

-

$p\in\mathcal{R}^{\operatorname{act}}_{T}$

-

$G:{}^{n}\rightarrow{}^{n}$

-

$\displaystyle(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta$

-

$\alpha_{e_{k}}>0$

-

$f^{D^{+}}\in\mathcal{W}_{D^{+}}$

-

$(\mathcal{P},\mathcal{C},D)$

-

$f^{-},f^{+}\in{}^{n}$

-

$\displaystyle\mathcal{J}^{\operatorname{act}}_{i-1}$

-

$T\geq D^{\operatorname{BP}}$

-

$\widetilde{\lambda}^{\operatorname{vec}}$

-

$f^{\delta_{0}}\in\operatorname{SOL}(\mathcal{M},A)$

-

$D_{i}>0$

-

$\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta}+f^{\delta}-f^{\delta}_{0})$

-

$[D_{i},D_{i+1})$

-

$\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{vec}}(D)+(T-D)Af^{\delta}$

-

${Af^{\delta}=Af^{\delta_{0}}}$

-

$\mathcal{R}:=\mathcal{R}^{\operatorname{use}}_{D}$

-

$D^{-}\in[D_{i},D_{i+1})$

-

$f^{*}_{p}<0$

-

$f^{\delta}\in\Gamma_{T_{\mu}}=\Gamma^{i}$

-

$\displaystyle\quad f_{r}=0$

-

$C_{p}(f)\geq\delta\lambda^{M}D+\bar{\beta}$

-

${D\in{\mathbb{R}}_{\geq 0}}$

-

$\mathcal{R}^{\operatorname{use}}_{D_{i+1}}$

-

$\widehat{f}^{\delta}\in\mathcal{M}_{D_{M}}$

-

$\displaystyle=\min_{r\in\mathcal{P}}C_{r}(f^{D+\epsilon}).$

-

$\widetilde{f}\in{\mathbb{R}}_{\geq 0}^{\left\lvert{\widetilde{\mathcal{P}}}% -\right\rvert}$

-

$D^{\prime\prime}_{k},D^{\prime\prime}_{k+1}\in\mathcal{D}^{\prime\prime}$

-

$\widetilde{\lambda}(D)<\lambda^{\operatorname{WE}}(D)$

-

$\displaystyle:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}},$

-

$f^{\prime}_{p}\neq 0$

-

$\lambda^{\operatorname{vec}}_{p}(D)$

-

$\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}$

-

$\lambda^{\operatorname{vec}}_{p}(D)=\lambda^{\operatorname{vec}}_{r}(D)$

-

$V(0)=\widetilde{V}(0)=0$

-

$\delta C^{M}_{r}<\delta\lambda^{M}$

-

$\displaystyle\leq\min_{r\in\mathcal{Q}}A_{r}f^{\delta-}).$

-

$\displaystyle u_{\mathcal{P},i}(D)$

-

$p\in\mathcal{R}^{\operatorname{use}}_{D_{i+1}}$

-

$p,r\in\widetilde{\mathcal{P}}$

-

$n=\left\lvert{\mathcal{P}}\right\rvert$

-

$\widetilde{f}^{\delta}\in\widetilde{\mathcal{M}}\subseteq\mathcal{M}$

-

$(D^{-},D^{+})$

-

$D_{i},D_{i+1}\in\mathcal{D}$

-

$(e_{3},e_{2})$

-

$i\in[M]_{0}$

-

$\displaystyle -

$(2,\infty)$

-

$f^{D_{M}}+(D-D_{M})f^{\delta}\geq 0$

-

$f\in\ker(A)\cap\mathcal{M}_{D}$

-

$\displaystyle\beta^{\top}f^{\delta}$

-

$\displaystyle\min_{\widetilde{f}\in\widetilde{\mathcal{F}}_{D}}\sum_{e_{k}\in% -\mathcal{E}}\int_{0}^{\widetilde{f}_{e_{k}}}C_{e_{k}}(z)dz.$

-

$\mathcal{M}_{D_{i}}\subseteq\mathcal{M}_{D}$

-

$\displaystyle(Af^{\delta})^{\top}(f-f^{\delta})$

-

$\mathcal{J}^{\operatorname{act}}_{i}\cap\mathcal{J}^{\operatorname{act}}_{i+1}\neq\emptyset$

-

${r^{\prime}\in\mathcal{R}^{\operatorname{act}}_{D_{i+1}}}$

-

$\widetilde{f}^{D}\in\widetilde{\mathcal{W}}_{D}$

-

$C_{p}(f^{*})=\delta\lambda^{M}D+\bar{\beta}$

-

$T\in(D_{i},D_{i+1}]$

-

$f^{\delta}\in\Gamma_{D_{i}}$

-

$\displaystyle=\lambda^{\operatorname{vec}}(D_{i})+(D^{-}-D_{i})\delta C^{i}+(D% -^{+}-D^{-})\delta C^{i}$

- - - diff --git a/htmls/output_mathjax_example_10068.html b/htmls/output_mathjax_example_10068.html deleted file mode 100644 index 4ffbbd73768ccacf4ecb5aa3913b2c95ee8cb85b..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10068.html +++ /dev/null @@ -1,141 +0,0 @@ - - - - MathJax Example - - - - -

${D_{i},D_{i+1}\in\mathcal{D}}$

-

$1\leq i\leq j\leq l$

-

$\widetilde{D}_{\widetilde{M}}$

-

$r\in(\mathcal{J}^{\operatorname{act}}_{i})^{c}$

-

$T\in[D^{+},D]$

-

$\displaystyle=D(\delta\lambda^{M}D+\bar{\beta}).$

-

${\lambda^{\prime\operatorname{WE}}(D)=u_{\mathcal{P},i}(D)}$

-

$D^{\prime}=\mu D+(1-\mu)T$

-

$\lambda^{\operatorname{WE}}(D)=D\delta\lambda^{M}+\bar{\beta}.$

-

$\delta C^{i+1}=\delta C^{i}$

-

$\operatorname{VI}(\mathcal{X},G)$

-

$T\in{\mathbb{R}}_{\geq 0}$

-

$\delta\lambda^{i}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D_{i}}}\delta C^% -{i}$

-

$\mathcal{J}^{\prime\operatorname{use}}_{j-1}\subseteq\mathcal{P}^{\prime}$

-

${\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{D}}$

-

$r\in\mathcal{I}_{2}$

-

${f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D_{i}},A)}$

-

${\ker(A)\cap\mathcal{M}_{D}}$

-

$\delta C\in{}^{n}$

-

$\displaystyle=f^{D_{i}}+(T_{\mu}-D_{i})f^{\delta_{0}}.$

-

$(\widetilde{f}^{\delta})^{\top}A\widetilde{f}^{\delta}=\min_{r\in\widetilde{% -\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}$

-

$f^{D_{i+1}}$

-

$C_{p}(f^{D^{+}})\geq\delta\lambda^{M}D^{+}+\bar{\beta}.$

-

$\min_{r\in\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}}A_{r}\widetilde{f}^% -{\delta}\geq\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}.$

-

$\sum_{r\in\mathcal{R}_{i}}f^{\delta}_{r}\geq-nt(T_{i}-D)^{-1}.$

-

$f^{\delta}\in\mathcal{M}_{D_{M}}$

-

$\widetilde{\mathcal{P}}:=\mathcal{J}^{\operatorname{use}}_{i}$

-

$t:=\max_{r\in\mathcal{P}}f^{D}_{r}$

-

$\mymathbf{1}^{\top}f^{*}=D_{M}$

-

$\mathcal{V}=[N]$

-

$V(D):=\min_{f\in\mathcal{F}_{D}}\sum_{e_{k}\in\mathcal{E}}\int_{0}^{f_{e_{k}}}% -C_{e_{k}}(z)dz.$

-

$C(f+\epsilon f^{0})=C(f)$

-

$\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{P}$

-

$\lambda^{\operatorname{WE}}(D^{+})=\widetilde{\lambda}^{\operatorname{WE}}(D^{% -+})$

-

$(\widetilde{f}^{\delta})^{\top}A(f^{\delta}-\widetilde{f}^{\delta})>0.$

-

$v_{o}\in\mathcal{V}$

-

$\widecheck{\lambda}^{\operatorname{WE}}_{i,D}$

-

$\lambda^{\operatorname{WE}}(D_{i})=u_{\emptyset,i}(D_{i})$

-

$(D_{i+1}-\epsilon,D_{i+1}]$

-

$\displaystyle=(f^{\delta})^{\top}Af^{D_{M}}+(D-D_{M})(f^{\delta})^{\top}Af^{% -\delta}+\beta^{\top}f^{\delta}.$

-

$f^{D}=\begin{cases}\left(\begin{matrix}0,&0,&D,&0\end{matrix}\right)^{\top}&% -\text{for }D\in[0,\frac{1}{2}],\\ -\frac{1}{3}\left(\begin{matrix}2D-1,&2D-1,&2-D,&0\end{matrix}\right)^{\top}&% -\text{for }D\in[\frac{1}{2},2],\\ -\frac{1}{3}\left(\begin{matrix}D+1,&D+1,&0,&D-2\end{matrix}\right)^{\top}&% -\text{for }D\in[2,\infty).\end{cases}$

-

${p_{1}=(e_{1},e_{2})}$

-

${p_{4}=(e_{3},e_{6})}$

-

$\displaystyle=\widetilde{V}(z)|_{0}^{D}-V(z)|_{0}^{D}$

-

$f^{D^{+}}$

-

$D\in(\frac{2}{3},2)$

-

$p\notin\mathcal{R}^{\operatorname{act}}_{T}$

-

$\lambda^{\operatorname{vec}}$

-

$\displaystyle(Af^{\delta_{0}})^{\top}(f-f^{\delta_{0}})$

-

$\widehat{\mathcal{S}}^{\prime\prime}\subset\widehat{\mathcal{P}}^{\prime}$

-

${D\geq D_{M}=\max\big{(}\mathcal{D}\setminus\{\infty\}\big{)}}$

-

$\displaystyle=\lambda^{\operatorname{vec}}(D_{i})+(T_{\mu}-D_{i})Af^{\delta_{0% -}}.$

-

$C(f^{D^{+}})=C(f^{*})+(D^{+}-D)Af^{\delta}.$

-

$p\in\mathcal{R}^{\operatorname{act}}_{D}$

-

$q:=\left\lvert{\mathcal{E}}\right\rvert$

-

$\widetilde{\lambda}(D)=u_{\widetilde{\mathcal{P}},\widetilde{M}}(D)$

-

$D(\delta\lambda^{M}D+\bar{\beta})$

-

$\mathcal{S}^{\texttt{rem}}\subset\mathcal{P}$

-

$f^{D_{M}}+(D-D_{M})f^{\delta}\in\mathcal{W}_{D}$

-

${\frac{\partial^{-}}{\partial x}g(x):=\lim_{h\rightarrow 0^{-}}\frac{g(x+h)-g(% -h)}{h}}$

-

$f^{\delta}\in\Gamma_{D}$

-

$D\in[\widetilde{D}_{j},\widetilde{D}_{j+1})$

-

$f^{\delta,i}_{r}\geq-t(T_{i}-D)^{-1}$

-

$f^{*}_{p}>0$

-

$\delta\lambda^{+}(T):=\frac{\partial^{+}}{\partial D}\lambda^{\operatorname{WE% -}}(D)\Big{|}_{D=T},\quad\delta\lambda^{-}(T):=\frac{\partial^{-}}{\partial D}% -\lambda^{\operatorname{WE}}(D)\Big{|}_{D=T}.$

-

$\displaystyle\Gamma^{-}_{D}:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;\exists f^{D}% -\in\mathcal{W}_{D},\enskip\bar{\epsilon}>0\text{ such that }f^{D}+\epsilon f^{% -\delta}\in\mathcal{W}_{D-\epsilon}\enskip\forall\epsilon\in[0,\bar{\epsilon}]\}.$

-

$\displaystyle=\lambda^{\operatorname{WE}}(D_{i})+(D-D_{i})\delta\lambda^{i-1},$

-

$D=D_{i}$

-

$D_{M+1}=\infty$

-

$f^{\mu}=\operatorname{coco}_{\mu}(f^{D_{i}},f^{T})$

-

$\mathcal{J}^{\operatorname{act}}_{i-1}\subseteq\mathcal{R}^{\operatorname{act}% -}_{D_{i}}$

-

$\lambda^{\operatorname{vec}}_{r^{\prime}}(D_{i+1})\leq\lambda^{\operatorname{% -vec}}_{p^{\prime}}(D_{i+1})$

-

$p\notin\mathcal{J}^{\operatorname{act}}_{M}$

-

$f^{\mu}_{p}>0$

-

$D+\epsilon\in(D,D_{i+1})$

-

$\lim_{i\rightarrow\infty}T_{i}=\infty$

-

$\displaystyle=\delta\lambda^{M}D+\bar{\beta}.$

-

$D^{\prime\prime}_{k-1}=0$

-

$\delta C^{i}:=Af^{\delta_{0}}$

-

$\displaystyle:=\widetilde{\lambda}^{\operatorname{vec}}_{p}(D)\text{ for any }% -p\in\widetilde{\mathcal{R}}^{\operatorname{act}}_{D},$

-

$i\in[\vec{\mathcal{M}}]_{0}$

-

$\displaystyle W(D)$

-

$\mathcal{R}^{\operatorname{act}}_{T}=\mathcal{R}^{\operatorname{act}}_{D}$

-

$\sum_{r\in\mathcal{R}^{\operatorname{use}}_{D}}\mu_{r}=1$

-

$\mathcal{J}^{\operatorname{use}}_{i-1}=\mathcal{R}^{\operatorname{use}}_{D_{i}}$

-

$\mathcal{J}^{\operatorname{act}}_{i}\neq\mathcal{J}^{\operatorname{act}}_{i+1}$

-

$\{f^{\delta,i}\}_{i\in\mathbb{N}}$

-

$(e_{1},e_{4})$

-

$\delta\widetilde{\lambda}^{+}(D)\geq\delta\lambda^{+}(D).$

-

$f_{(\mathcal{R}^{\operatorname{use}}_{T})^{c}}^{T}=0$

-

$\displaystyle C_{p}(f^{D})$

-

$\operatorname{SOL}(\mathcal{F}_{1},A)$

-

$T=\operatorname{coco}_{\mu}(D^{-},D^{+})$

-

$f^{\mu}$

-

$T_{i} -

$f^{D+\epsilon}:=f^{D}+\epsilon f^{\delta}.$

-

${u_{\mathcal{P}^{\prime},j-1}(D) -

$f^{T_{i}}\geq 0$

-

$C(f)=Af$

-

$\lambda^{\prime\operatorname{WE}}(D)=\lambda^{\operatorname{WE}}(D),\quad\text% -{for all }D\in[D_{i},D_{i+1}].$

- - - diff --git a/htmls/output_mathjax_example_10069.html b/htmls/output_mathjax_example_10069.html deleted file mode 100644 index f8a125bfcbb51ea7cd6f815b498754655bb1dc5e..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10069.html +++ /dev/null @@ -1,138 +0,0 @@ - - - - MathJax Example - - - - -

$f^{\delta}_{p}=0$

-

$\widehat{\mathcal{P}}\subset\widetilde{\mathcal{P}}$

-

$A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}\big{)}+\beta=\lambda^{\operatorname{vec}% -}(D)$

-

${\mathcal{R}^{\operatorname{act}}_{D}=\{p_{1},p_{2},p_{3}\}}$

-

$(\widehat{\mathcal{P}}^{\prime}\setminus\widehat{\mathcal{S}}^{\prime\prime},% -\mathcal{C})$

-

${\mathcal{P}^{c}=\{p\in[n]\;|\;p\notin\mathcal{P}\}}$

-

$D^{\prime}_{j}>0$

-

$\lambda^{\operatorname{WE}}(D)>\widehat{\lambda}^{\operatorname{WE}}(D)$

-

$\mathcal{S}^{\prime}\cup\mathcal{S}^{\prime\prime}$

-

$\displaystyle\Gamma_{D}:=\begin{cases}\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{% -\delta}_{1},f^{\delta}_{2},f^{\delta}_{4}=0,\hskip 2.0ptf^{\delta}_{3}=1\},&% -\text{for }D\in[0,1),\\ -\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{1},f^{\delta}_{2},f^{\delta}_{4% -}\geq 0,\hskip 2.0ptf^{\delta}_{1}=f^{\delta}_{2}=-f^{\delta}_{3}\},&\text{for% - }D=1,\\ -\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{1}+f^{\delta}_{3}=0,\hskip 2.0% -ptf^{\delta}_{2}+f^{\delta}_{3}=0\},&\text{for }D\geq 1.\end{cases}$

-

$f^{\delta}\in\Gamma^{M}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$

-

$\displaystyle\widetilde{\lambda}^{\operatorname{WE}}(T)$

-

$C_{p}(f^{T})=\lambda^{\operatorname{WE}}(T)$

-

$\displaystyle(\mathcal{R}^{\operatorname{act}}_{D},\mathcal{R}^{\operatorname{% -use}}_{D})=\begin{cases}(\{p_{3}\},\emptyset)&\text{for }D=0,\\ -(\{p_{3}\},\{p_{3}\})&\text{for }D\in(0,1),\\ -(\{p_{1},p_{2},p_{3}\},\{p_{3}\})&\text{for }D=1,\\ -(\{p_{1},p_{2},p_{3}\},\{p_{1},p_{2},p_{3}\})&\text{for }D\in(1,2),\\ -(\{p_{1},p_{2},p_{3}\},\{p_{1},p_{2}\})&\text{for }D=2,\\ -(\{p_{1},p_{2}\},\{p_{1},p_{2}\})&\text{for }D\in(2,\infty).\\ -\end{cases}$

-

$\mathcal{R}^{\operatorname{use}}_{D^{-}}=\mathcal{R}^{\operatorname{use}}_{D^{% -+}}$

-

$\Gamma_{D_{i}}=\operatorname{SOL}(\mathcal{M}_{D_{i}},A)$

-

$\widetilde{\lambda}^{\operatorname{WE}}(1)=1.5$

-

${T>D}$

-

$\widetilde{f}^{D}_{\mathcal{S}^{\texttt{rem}}}=0$

-

$f^{\delta}\in\Gamma_{D^{-}}$

-

$\lambda^{\operatorname{WE}}(T)<\widetilde{\lambda}^{\operatorname{WE}}(T)$

-

$\Gamma_{D_{M}}=\operatorname{SOL}(\mathcal{M}_{D_{M}},A)$

-

$f^{T_{i}}$

-

$\lambda^{\operatorname{WE}}(T)=\widetilde{\lambda}^{\operatorname{WE}}(T)$

-

$f^{\delta}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$

-

$D^{\prime}_{j-1}=0$

-

$f^{D^{+}}=f^{D^{-}}+(D^{+}-D^{-})f^{\delta}.$

-

$\delta\lambda^{M}$

-

$\widetilde{\mathcal{M}}^{-}\subseteq\mathcal{M}^{-}$

-

$\displaystyle f^{\mu}$

-

$\displaystyle=\lambda^{\operatorname{WE}}(D)$

-

$\mathcal{W}_{D}$

-

$\delta C^{i}\neq\delta C^{i+1}$

-

$T\geq\max(D_{M},\widetilde{D}_{\widetilde{M}})$

-

$\Gamma_{D}\cap\mathcal{F}_{1}$

-

$\mathcal{M}=\mathcal{M}_{D_{i}}$

-

$\mymathbf{0}$

-

$\displaystyle C_{e_{6}}(f_{e_{6}})=f_{e_{6}}+5,$

-

$f^{D^{+}}\geq 0$

-

$\operatorname{coco}_{\mu}(f^{D^{-}},f^{D^{+}})\in\mathcal{W}_{T}$

-

${T_{\mu}\in(D_{i},D_{i+1})}$

-

$(v^{\operatorname{in}}_{k},v^{\operatorname{out}}_{k})$

-

$[\vec{D}_{i},\vec{D}_{i+1})$

-

$v^{\operatorname{out}}_{pj}\neq v^{\operatorname{in}}_{p_{i}}$

-

$\displaystyle C_{\mathcal{R}}(f)$

-

$\delta\lambda^{\prime j}=\delta\lambda^{i}$

-

$D\in(2,\infty)$

-

$C_{p}(f^{\prime})=\delta\lambda^{M}D+\bar{\beta}$

-

$f^{\prime}_{p}\geq 0$

-

$\mathcal{J}^{\operatorname{act}}_{i}\neq\mathcal{J}^{\operatorname{act}}_{j}$

-

$p\in\mathcal{R}^{\operatorname{use}}_{D_{M}}$

-

$\delta C_{p}^{M}<\delta\lambda^{M}$

-

$\mathcal{S}\notin\mathcal{N}_{D}$

-

$D^{-},D^{+}$

-

${\mathcal{R}=\{p_{1},p_{2},\cdots,p_{m}\}\subseteq\mathcal{P}}$

-

$\displaystyle(Af^{\delta_{1}})^{\top}(f^{\delta_{2}}-f^{\delta_{1}})$

-

$A\widehat{f}^{\delta}=\delta C^{M}$

-

$\mathcal{R}^{\operatorname{use}}_{D}=\mathcal{J}^{\operatorname{use}}_{i}$

-

$f^{\delta}\in\Gamma_{D_{M}}$

-

$\mathcal{S}^{\texttt{rem}}_{T}$

-

$\displaystyle=Af^{\prime}+\beta,$

-

$\lambda^{\operatorname{vec}}:{\mathbb{R}}_{\geq 0}\rightarrow{\mathbb{R}}_{% -\geq 0}^{n}$

-

${T\geq\max(D_{M},\widetilde{D}_{\widetilde{M}})}$

-

$\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}\in\widetilde{\mathcal{W}}_{D+\epsilon}$

-

$(b_{1},\dots b_{|V^{lca}|})\leftarrow\textsc{Sort}(V^{lca})$

-

$V^{olca}\leftarrow V^{olca}\cup\{\textsc{LCA}(v_{i_{j}},v_{i_{j+1}})\}$

-

$O\left(k\log k\right)$

-

$l=LCA(v,q)$

-

$d_{c}=\textsc{DistToClosest}(u)+w^{virt}(u,v)$

-

$x_{1},\dots,x_{i}$

-

$v\in V^{s}$

-

$t_{out}(v)\leftarrow timer$

-

$O_{opt}(I)=\operatorname{arg\,min}_{O\in\widetilde{O}(I)}cost(I,O)$

-

$O\left(k^{2}+k\log n\right)$

-

$\operatorname{dist}\left(1,\textsc{Parent}(v)\right)+1=\operatorname{dist}(1,v)$

-

$v_{i_{1}},\dots,v_{i_{k+1}}$

-

$V^{lca}\leftarrow V^{s}\cup V^{olca}$

-

$\textsc{Closest}(v)=v$

-

$v^{\prime}_{1},\dots,v^{\prime}_{k}$

-

$\textsc{Move}(\textsc{Closest}(r),\textsc{DistToClosest}(r))$

-

$\operatorname{dist}(l,v)\geq z$

-

$z\leq\operatorname{dist}(l,v)$

-

$|V^{lca}|\leq|V^{s}|+|V^{olca}|\leq|V^{s}|+|V^{s}|-1\leq 2|V^{s}|$

-

$\textsc{ComputeDistanceBase}(u)$

-

$u,v\in V^{s}$

-

$dist(x,u)+dist(x,v)=dist(u,v)$

-

$\textsc{UpdatePositions}(r,\textsc{DistToClosest}(r))$

-

$\operatorname{dist}(1,v)$

-

$\textsc{Move}(\textsc{Closest}(u),b)$

-

$t_{in}(v)\leftarrow timer$

-

$|V^{lca}|\leq 2|V^{s}|$

-

$z>\operatorname{dist}(l,v)$

-

$\textsc{UpdatePositions}(u,b)$

-

$O\left(k^{2}+k\cdot\log n\right)$

-

$V^{s}\subseteq V^{lca}$

-

$\textsc{Closest}(v)\leftarrow v_{c}$

-

$V^{lca}$

-

$\textsc{DistToClosest}(v)=0$

-

$v_{c}\leftarrow\textsc{Closest}(u)$

-

$\textsc{Children}(v)=\{u:\textsc{Parent}(u)=v\}$

- - - diff --git a/htmls/output_mathjax_example_1007.html b/htmls/output_mathjax_example_1007.html deleted file mode 100644 index ab62e81a043cbc7bab686244695c457a09dd15e3..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1007.html +++ /dev/null @@ -1,124 +0,0 @@ - - - - MathJax Example - - - - -

$.08{\scriptstyle\ \pm.01}$

-

$.61{\scriptstyle\ \pm.01}$

-

$\mathbf{\%}$

-

$.39{\scriptstyle\ \pm.02}$

-

$.03{\scriptstyle\ \pm.00}$

-

$.13{\scriptstyle\ \pm.01}$

-

$3.37\times 10^{-5}$

-

$\log\ \mathcal{U}[1\times 10^{-5},0.01]$

-

$.22{\scriptstyle\ \pm.01}$

-

$.04{\scriptstyle\ \pm.00}$

-

$.07{\scriptstyle\ \pm.00}$

-

$.60{\scriptstyle\ \pm.02}$

-

$\underline{\mathbf{.00}}{\scriptstyle\ \pm.00}$

-

$.48{\scriptstyle\ \pm.00}$

-

$\mathcal{C}_{m}$

-

$.49{\scriptstyle\ \pm.01}$

-

$.53{\scriptstyle\ \pm.01}$

-

$.62{\scriptstyle\ \pm.03}$

-

$9.58\times 10^{-5}$

-

$\mathbf{1}(\hat{y}_{i}=y_{i})$

-

$.27{\scriptstyle\ \pm.00}$

-

$0.005793$

-

$\mathbf{.12}{\scriptstyle\ \pm.01}$

-

$\mathbb{E}\Big{[}\big{|}\mathbb{P}\big{(}Y=\hat{y}\ |\ P=\hat{p}\big{)}-\hat{p% -}\big{|}\Big{]},$

-

$.16{\scriptstyle\ \pm.01}$

-

$.11{\scriptstyle\ \pm.00}$

-

$\mathbf{.18}{\scriptstyle\ \pm.01}$

-

$\tau=0.35$

-

$.05{\scriptstyle\ \pm.01}$

-

$\mathbf{.82}{\scriptstyle\ \pm.01}$

-

$.05{\scriptstyle\ \pm.00}$

-

$\hat{y}\in\mathcal{Y}$

-

$\mathbf{.01}{\scriptstyle\ \pm.00}$

-

$\mathcal{U}[1\times 10^{-4},0.05]$

-

$.02{\scriptstyle\ \pm.01}$

-

$0.01932$

-

$0,\ 0.33,\ 0.66$

-

$.70{\scriptstyle\ \pm.17}$

-

$8.84\times 10^{-5}$

-

$\hat{a}_{j}$

-

$.74{\scriptstyle\ \pm.01}$

-

$.81{\scriptstyle\ \pm.01}$

-

$.38{\scriptstyle\ \pm.03}$

-

$.11{\scriptstyle\ \pm.08}$

-

$.30{\scriptstyle\ \pm.00}$

-

$\underline{\mathbf{.03}}{\scriptstyle\ \pm.01}$

-

$.32{\scriptstyle\ \pm.00}$

-

$\hat{q}=\sigma(a\hat{p}+b)$

-

$.01{\scriptstyle\ \pm.00}$

-

$\hat{p}\in[0,1]$

-

$.22{\scriptstyle\ \pm.00}$

-

$0.008936$

-

$5.59\times 10^{-5}$

-

$.64{\scriptstyle\ \pm.02}$

-

$.32{\scriptstyle\ \pm.01}$

-

$.70{\scriptstyle\ \pm.01}$

-

$.55{\scriptstyle\ \pm.01}$

-

$\mathbf{.02}{\scriptstyle\ \pm.01}$

-

$\underline{\mathbf{.72}}{\scriptstyle\ \pm.02}$

-

$.15{\scriptstyle\ \pm.00}$

-

$0.73$

-

$.08{\scriptstyle\ \pm.00}$

-

$\mathbf{.03}{\scriptstyle\ \pm.01}$

-

$.06{\scriptstyle\ \pm.01}$

-

$.00{\scriptstyle\ \pm.12}$

-

$.26{\scriptstyle\ \pm.00}$

-

$.72{\scriptstyle\ \pm.02}$

-

$.07{\scriptstyle\ \pm.01}$

-

$\mathcal{B}_{m}$

-

$.48{\scriptstyle\ \pm.01}$

-

$.82{\scriptstyle\ \pm.01}$

-

$.19{\scriptstyle\ \pm.01}$

-

$.38{\scriptstyle\ \pm.00}$

-

$.09{\scriptstyle\ \pm.01}$

-

$0.01362$

-

$\sum_{m=1}^{M}\frac{|\mathcal{B}_{m}|}{N}\Big{|}\underbrace{\frac{1}{|\mathcal% -{B}_{m}|}\sum_{i\in\mathcal{B}_{m}}\mathbf{1}(\hat{y}_{i}=y_{i})}_{\text{Bin % -accuracy (target)}}-\underbrace{\frac{1}{|\mathcal{B}_{m}|}\sum_{i\in\mathcal{% -B}_{m}}\hat{p}_{i}}_{\text{Avg. bin confidence}}\Big{|},$

-

$.06{\scriptstyle\ \pm.00}$

-

$\mathbf{.02}{\scriptstyle\ \pm.00}$

-

$.25{\scriptstyle\ \pm.00}$

-

$.19{\scriptstyle\ \pm.00}$

-

$\mathbf{.07}{\scriptstyle\ \pm.01}$

-

$\mathbf{.03}{\scriptstyle\ \pm.00}$

-

$\underline{\mathbf{.06}}{\scriptstyle\ \pm.01}$

-

$\underline{\mathbf{.12}}{\scriptstyle\ \pm.01}$

-

$\mathcal{C}(i)$

-

$.39{\scriptstyle\ \pm.01}$

-

$.52{\scriptstyle\ \pm.02}$

-

$.47{\scriptstyle\ \pm.25}$

-

$\underline{\mathbf{.18}}{\scriptstyle\ \pm.01}$

-

$.65{\scriptstyle\ \pm.02}$

-

$1.62\times 10^{-5}$

-

$.45{\scriptstyle\ \pm.01}$

-

$.21{\scriptstyle\ \pm.00}$

-

$.02{\scriptstyle\ \pm.00}$

-

$.60{\scriptstyle\ \pm.14}$

-

$.75{\scriptstyle\ \pm.01}$

-

$0.03184$

-

$.24{\scriptstyle\ \pm.00}$

-

$.03{\scriptstyle\ \pm.01}$

-

$.72{\scriptstyle\ \pm.01}$

- - - diff --git a/htmls/output_mathjax_example_10070.html b/htmls/output_mathjax_example_10070.html deleted file mode 100644 index b977a1e9c05badee5d97b0da785d7925a49bb529..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10070.html +++ /dev/null @@ -1,122 +0,0 @@ - - - - MathJax Example - - - - -

$O(k\log k)+O(k)=O(k\log k)$

-

$q\in\{v^{\prime}_{1},\dots,v^{\prime}_{k}\}$

-

$x\in V^{lca}$

-

$i_{1},\dots,i_{k+1}$

-

$v\in V^{lca}$

-

$t_{in}(v)$

-

$V^{lca}=\{LCA(v,w):v,w\in V^{s}\}$

-

$|V^{s}|\leq|V^{lca}|$

-

$\sum_{i=1}^{k}\operatorname{dist}(v_{i},v^{\prime}_{i})$

-

$V^{s}\cup V^{olca}\subseteq V^{lca}$

-

$x\neq u,x\neq v$

-

$\textsc{Closest}(v)=\textsc{Closest}(u)$

-

$\textsc{DistToClosest}(v)=min\{\textsc{DistToClosest}(u)+w^{virt}(u,v):u\in% -\textsc{Children}(v)\}$

-

$i\in\{1,\dots,h-1\}$

-

$b\leftarrow\min\{\textsc{Closest}(r),\textsc{Closest}(u)\}$

-

$\textsc{ProcessingAQuery}(q,T,v_{1},\dots,v_{k})$

-

$V^{olca}=\{LCA(v_{i_{j}},v_{i_{j+1}}):1\leq j\leq k\}$

-

$d=dist(1,v)-z$

-

$V^{s}\cup V^{olca}$

-

$timer\leftarrow timer+1$

-

$Result\leftarrow\textsc{LA}(v,dist(1,v)-z)$

-

$M=|I|$

-

$v\in\{v_{1},\dots,v_{k}\}$

-

$V^{s}=\{v_{1},\dots,v_{k},q\}$

-

$O\left(k(\log n)^{2}\right)$

-

$\textsc{InitStack}({\cal S})$

-

$\textsc{Closest}(u)$

-

$\textsc{IsEmpty}({\cal S})=False$

-

$\textsc{LA}(v,d)$

-

$v\in T_{v}$

-

$\textsc{ConstructVirtualTree}(T,v_{1},\dots,v_{k},q)$

-

$\textsc{Move}(v,z)$

-

$v\in\{\textsc{Closest}(u):u\in\textsc{Children}(r)\}\backslash\{\textsc{% -Closest}(r)\}$

-

$\textsc{ClosestComputing}(v)$

-

$\operatorname{dist}(1,1)\leftarrow 0$

-

$\textsc{Parent}(v)$

-

$d_{c}\leftarrow\textsc{DistToClosest}(u)+w^{virt}(u,v)$

-

$\textsc{Id}(v_{i})$

-

$d_{c}=NULL$

-

$\textsc{ClosestComputing}(r)$

-

$\textsc{Closest}(u)=\textsc{Closest}(v)$

-

$\textsc{Push}({\cal S},v)$

-

$A(I)=(y_{1},\dots,y_{M})$

-

$\textsc{LCA\_Preprocessing}()$

-

$z-\operatorname{dist}(l,v)$

-

$\operatorname{dist}(v,u)$

-

$LCA(v,v)=v$

-

$par=LCA(u,v)$

-

$\textsc{DistToClosest}(v)\leftarrow d_{c}$

-

$V^{olca}$

-

$T^{virt}$

-

$\textsc{ComputeDistance}()$

-

$cost(I,O)$

-

$\textsc{DistToClosest}(v)$

-

$dist(1,u)\leftarrow dist(1,v)+1$

-

$\textsc{ComputeDistance}(v)$

-

$O(\log^{3}n\log^{2}k)$

-

$O\in\widetilde{O}(I)$

-

$r\leftarrow\textsc{Root}(T^{virt})$

-

$\textsc{ComputeDistanceBase}(v)$

-

$d_{c}>\textsc{DistToClosest}(u)+w^{virt}(u,v)$

-

$(v_{i_{1}},\dots,v_{i_{k+1}})\leftarrow\textsc{Sort}(v_{1},\dots,v_{k},q)$

-

$\textsc{AddEdge}(T^{virt},\textsc{Pick}({\cal S}),v_{b_{j}})$

-

$par\in V^{olca}$

-

$\textsc{ComputeDistanceBase}(1)$

-

$v_{i_{j}}\in T_{v}$

-

$V^{s}=\{3,5,6,13,14\}$

-

$|V^{s}|,|V^{lca}|,|V^{olca}|=O(k)$

-

$\textsc{LCA}(u,v)$

-

$O=(y_{1},\dots,y_{M})$

-

$v_{c}\leftarrow NULL$

-

$par=LCA(v_{i_{j-1}},v_{i_{j}})$

-

$\textsc{DistToClosest}(r)$

-

$\textsc{UpdatePositions}(v,b)$

-

$d_{c}\leftarrow NULL$

-

$par\not\in V^{s}\cup V^{olca}$

-

$\textsc{Pick}({\cal S})$

-

$\textsc{LA}(v,dist(1,l)+z-\operatorname{dist}(l,v))$

-

$V^{olca}\subseteq V^{lca}$

-

$I\in{\cal I}$

-

$cost(I,A(I))\leq c\cdot cost(I,O_{Opt}(I))+\alpha$

-

$\textsc{Root}(T^{virt})$

-

$\textsc{DistToClosest}(v)\leftarrow 0$

-

$\textsc{LA}(v,dist(1,v)-z)$

-

$\textsc{Closest}(r)$

-

$u\in T_{u}$

-

$\textsc{ComputeDistanceBase}()$

-

$par\in V^{lca}$

-

$\textsc{Move}(u,z)$

-

$\textsc{IsAncestor}(\textsc{Pick}({\cal S}),v_{b_{j}})=False$

-

$t_{out}(v)$

-

$\cal{I}$

-

$t_{in}(u)\leq t_{in}(v)$

-

$\textsc{Closest}(v)\neq\textsc{Closest}(u)\ \textbf{then}$

-

$v_{i_{j-1}}\in T_{u}\cup T_{m}$

-

$\textsc{Closest}(v)\leftarrow v$

-

$\textsc{Pop}({\cal S})$

-

$w^{virt}(u,v)=dist(u,v)$

-

$|V^{olca}|\leq|V^{s}|-1$

-

$u\in\textsc{Children}(v)$

- - - diff --git a/htmls/output_mathjax_example_10071.html b/htmls/output_mathjax_example_10071.html deleted file mode 100644 index f50db6261a1206531e8b91a788b4db6f96798101..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10071.html +++ /dev/null @@ -1,144 +0,0 @@ - - - - MathJax Example - - - - -

$\textsc{Closest}(v)$

-

$\textsc{ClosestComputing}(u)$

-

$V^{lca}=V^{s}\cup V^{olca}$

-

$d=dist(1,l)+z-\operatorname{dist}(l,v)$

-

$\textsc{Push}({\cal S},v_{b_{j}})$

-

$t_{out}$

-

$LCA(u,v)\not\in V^{s}\cup V^{olca}$

-

$\textsc{Id}(v_{c})>\textsc{Id}(\textsc{Closest}(u))$

-

$b\leftarrow\min\{b,\textsc{DistToClosest}(v)\}$

-

$l=\textsc{LCA}(v,q)$

-

$\textsc{LA\_Preprocessing}()$

-

$V^{lca}\subseteq V^{s}\cup V^{olca}$

-

$t_{out}(v)\leq t_{out}(u)$

-

$|V^{s}|\leq|V^{lca}|\leq 2|V^{s}|$

-

$v_{1},\dots,v_{h}$

-

$\textsc{IsAncestor}(u,v)$

-

$Result\leftarrow\textsc{LA}(q,dist(1,l)+z-\operatorname{dist}(l,v))$

-

$\textsc{IsEmpty}({\cal S})$

-

$(v_{1},\dots,v_{h})$

-

$j\in\{2,\dots,|V^{lca}|\}$

-

$\textsc{Push}({\cal S},v_{b_{1}})$

-

$t_{in}(v_{i_{j}})\leq t_{in}(v_{i_{j+1}})$

-

$\textsc{AddEdge}(T^{virt},u,v)$

-

$\widetilde{O}(I)$

-

$V^{olca}\leftarrow\{\}$

-

$u\in\textsc{Children}(r)$

-

$dist(1,1)\leftarrow 0$

-

$I=(x_{1},\dots,x_{M})$

-

$\displaystyle p(\mathcal{A}_{i}|f$

-

$p_{k_{i}NN}$

-

$\mathcal{L}=\frac{\mathcal{L}_{1}}{N}+\frac{\mathcal{L}_{2}}{N}$

-

$k_{1},\cdot,k_{K}$

-

$\displaystyle\mathrm{softmax}(W_{2}^{T}[\mathrm{ReLU}(W_{1}^{T}\cdot f(\textbf% -{x},\textbf{y}_{1:i-1}))])$

-

$\mathcal{A}_{i}=1$

-

$z=f_{\mathcal{M}}(\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$\{(f(\textbf{x},\textbf{y}_{1:i-1}),y_{i})|i=1,\cdots,m\}$

-

$W_{2}\in\mathcal{R}^{d^{\prime}\times 2}$

-

$\textbf{y}=\{y_{1},\cdots,y_{m}\}$

-

$\displaystyle=\lambda p_{KNN}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{i-1})$

-

$q=f(\textbf{x},\hat{\textbf{y}}_{i-1})$

-

$\displaystyle+\sum_{j=1}^{K}p_{\text{Meta}}(k_{j})\cdot p_{k_{i}NN}(\hat{y_{i}% -}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$W_{1}\in\mathcal{R}^{d\times d^{\prime}}$

-

$\mathcal{L}_{2}=\sum_{i}[-\sum_{\hat{y}_{i}\in V}[y_{i}==\hat{y}_{i}]\log p(% -\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})]$

-

$p_{kNN}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$p_{KNN}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{i-1})=\mathrm{sofmtax}(\frac{% --d(q,k_{j})}{T}),j=1,\cdots,K$

-

$\displaystyle G\approx$

-

$p(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})=\left\{\begin{matrix}p_{MT}% -(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})&\mathcal{A}_{i}=0\\ -p_{\text{{combined}}}(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})&% -\mathcal{A}_{i}=1\end{matrix}\right.$

-

$p_{combined}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$\hat{\textbf{y}}_{1:i-1}$

-

$\mathcal{S}(z)==0$

-

$\displaystyle\hat{\textbf{y}}_{i-1})$

-

$g_{m}=-\log(-\log(u))$

-

$p_{MT}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$\textbf{x}=\{x_{1},\cdots,x_{n}\}$

-

$(\textbf{x},\hat{\textbf{y}}_{i-1})$

-

$p(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$\displaystyle\mathcal{L}_{1}$

-

$\displaystyle p_{\text{{combined}}}(\hat{y_{i}}|\textbf{x},$

-

$\displaystyle p_{\text{{combined}}}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{i% --1})$

-

$\displaystyle=p_{\text{Meta}}(f(\textbf{x},\textbf{y}_{1:i-1}))\cdot p_{MT}(% -\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$\mathcal{A}_{i}=\mathrm{argmax}_{\mathcal{A}}p(\mathcal{A}|f(\textbf{x},% -\textbf{y}_{1:i-1})),\mathcal{A}\in\{0,1\}$

-

$\displaystyle+(1-\lambda)p_{MT}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$

-

$f(\textbf{x},\textbf{y}_{1:i-1})$

-

$\hat{y}_{i}=\mathcal{M}(f(\textbf{x},\textbf{y}_{1:i-1}))$

-

$p_{MT}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})=\mathcal{M}(z)$

-

$\mathcal{A}_{i}\in\{0,1\}$

-

$\displaystyle(\textbf{x},\textbf{y}_{1:i-1}))$

-

$p_{\text{Meta}}$

-

$\displaystyle=\sum_{i}[-\frac{N}{B}[l_{i}=0]\log p(\mathcal{A}_{i}=0|f(\textbf% -{x},\textbf{y}_{1:i-1}))$

-

$\displaystyle\nabla_{W}\frac{\exp((\log p(\mathcal{A}|f(\textbf{x},\textbf{y}_% -{1:i-1});W))+g_{m}(\mathcal{A}^{\prime}))/\tau)}{\sum\limits_{\mathcal{A}^{% -\prime}\in\{0,1\}}\exp((\log p(\mathcal{A}^{\prime}|f(\textbf{x},\textbf{y}_{1% -:i-1});W))+g_{m}(\mathcal{A}^{\prime}))/\tau)}$

-

$\displaystyle-(1-\frac{N}{B})[l_{i}=1]\log p(\mathcal{A}_{i}=1|f(\textbf{x},% -\textbf{y}_{1:i-1}))]$

-

$next\_state_{\_}0(1,s)=3s+1$

-

$v\in V(\tau(\mathcal{A}))$

-

$\mathrm{f}_{\_}k(Int\ \tau(\Delta^{n}))$

-

$(v_{\_}p^{\prime},w_{\_}p^{\prime})\in G_{\_}p(\mathcal{A})$

-

$\mathrm{Ch}\ \mathcal{A}$

-

$next\_state$

-

$P(k,1)\in\Theta(f(k,r))$

-

$\textit{next\_state}_{\_}i:view,state,round\mapsto view$

-

$\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))\in\Theta% -\big{(}(2^{n-1}n)^{r}\big{)}$

-

$\sigma\in Skel^{i}\mathcal{A}$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))=\sum_{\_}{i=k% -}^{n}{\binom{n}{i}\sum_{\_}{j=1}^{k}{\binom{i}{j}\mathrm{f}_{\_}{k-j}(\mathrm{% -St}^{\circ}(\mathrm{Ch}\ \Delta^{i-j},v))}}$

-

$\mathrm{f}_{\_}k(K)$

-

$R(k,n)\in\Theta(f(k,n))\overset{\text{Lemma~{}\ref{lemma:ThetaBoundFkdelta}}}{% -\iff}\sum_{\_}{j=1}^{k}{\binom{n}{j}\frac{(k-j+1)^{i-k}(i-j)_{\_}{k-j}}{\ln(2)% -^{k-j-1}}}\\ -=\frac{(n)_{\_}k}{\ln(2)^{k-1}}\sum_{\_}{j=1}^{k}{\frac{(k-j+1)^{n-k}\ln(2)^{j% -}}{j!}}\in\Theta(f(k,n)\overset{\forall k\leq n}{\iff}\lim_{\_}{n\to\infty}{% -\sum_{\_}{j=1}^{k}{(\frac{k-j+1}{k+1})^{n-k}\frac{\ln(2)^{j}}{j!}}}=C>0\$

-

$|\operatorname{Im}\delta_{\_}p|\leq\Delta(G_{\_}p(\mathcal{A}))+1$

-

$\Xi_{\_}b$

-

$|\operatorname{Im}\textit{encode}|\leq\Delta(G)+1$

-

$\log_{\_}3(\lceil{\frac{1}{\epsilon}}\rceil)$

-

$(c,\sigma)$

-

$\mathrm{Ch}\ \Delta^{n}$

-

$\mathrm{f}_{\_}{k-i}(\Delta^{n-i})$

-

$(\mathcal{I},\mathcal{O},\Delta)$

-

$\Xi(\mathcal{I})$

-

$v^{\prime}\in V(\Delta^{i})$

-

$\textit{encode}:V(\mathcal{A})\rightarrow E$

-

$\mathrm{f}_{\_}k(\tau(Skel^{i}\mathcal{A}))=0$

-

$w_{\_}q$

-

$\mathrm{f}_{\_}k(K)+\mathrm{f}_{\_}k(\tau(Skel^{i-1}\mathcal{A}))=\mathrm{f}_{% -\_}k(\tau(Skel^{i}\mathcal{A}))$

-

$\frac{k-j+1}{k+1}<1$

-

$dim(\delta*v)=1$

- - - diff --git a/htmls/output_mathjax_example_10072.html b/htmls/output_mathjax_example_10072.html deleted file mode 100644 index af9484154719e634e37068fbd33e12bcefdd54cf..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10072.html +++ /dev/null @@ -1,157 +0,0 @@ - - - - MathJax Example - - - - -

$O(rn\log n)$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))\in\Theta(f(k,% -n))\overset{\text{IH}}{\iff}\sum_{\_}{i=k}^{n}{\binom{n}{i}\sum_{\_}{j=1}^{k}{% -\binom{i}{j}\frac{(k-j+1)^{i-k}(i-j)_{\_}{k-j}}{\ln(2)^{k-j-1}}}}\\ -\overset{\text{Lemma~{}\ref{lemma:A1}}}{=}\sum_{\_}{j=1}^{k}{\frac{\ln(2)^{j}}% -{\ln(2)^{k-1}j!}(k-j+2)^{n-k}(n)_{\_}k}=\frac{(n)_{\_}k}{\ln(2)^{k-1}}\sum_{\_% -}{j=1}^{k}{\frac{(k-j+2)^{n-k}\ln(2)^{j}}{j!}}\in\Theta(f(k,n))\\ -\overset{\forall k\leq n}{\iff}\lim_{\_}{n\to\infty}{\frac{\frac{(n)_{\_}k}{% -\ln(2)^{k-1}}\sum_{\_}{j=1}^{k}{\frac{(k-j+2)^{n-k}\ln(2)^{j}}{j!}}}{(n)_{\_}k% -(k+1)^{n-k}\ln(2)^{-k+1}}}=C>0\overset{\forall k\leq n}{\iff}\lim_{\_}{n\to% -\infty}{\sum_{\_}{j=1}^{k}{(\frac{k-j+2}{k+1})^{n-k}\frac{\ln(2)^{j}}{j!}}}=C>0$

-

$\tau(Skel^{i}\mathcal{A})$

-

$\mathrm{f}_{\_}k(\tau(Skel^{i-1}\mathcal{A}))=S(i-1)$

-

$f(k,n)$

-

$\delta_{\_}i(v)$

-

$\forall k\leq n$

-

$Int\ \tau(\sigma)\cap\tau(\sigma^{\prime})=\emptyset$

-

$\mathcal{A}\text{ is distinguishable under }\textit{encode}\iff\forall p\in\Pi% -,\textit{encode}\text{ is a proper vertex coloring of }G_{\_}p(\mathcal{A})$

-

$S(i)=\mathrm{f}_{\_}k(\tau(Skel^{i}\mathcal{A}))$

-

$(s,p_{\_}{\{0,1\}})$

-

$Int\ \mathrm{St}^{\circ}(\tau(\Delta^{i}),r)$

-

$|\operatorname{Im}\textit{encode}|\geq\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{A% -})}{\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))}$

-

$w_{\_}p\in\mathrm{Lk}(\mathcal{A},\mathrm{St}(\mathcal{A},v_{\_}p))\implies(v_% -{\_}p,w_{\_}p)\in G_{\_}p(\mathcal{A})$

-

$f(k,r):=\bigg{(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\bigg{)}^{r}$

-

$next\_state:\mathcal{S}\rightarrow\mathcal{S}$

-

$\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{% -Ch}^{r}\ \mathcal{A},v)))$

-

$\mathrm{Lk}(\mathcal{A},v_{\_}p)\raisebox{-2.15277pt}{$|$}_{q}$

-

$3s+i$

-

$\Xi^{r}_{\_}b$

-

$\mathcal{H}_{\_}{ij}=\{\{(p_{\_}0,v_{\_}i),(p_{\_}1,v_{\_}{ij})\},$

-

$view$

-

$\mathcal{S}\subset\mathbb{N}$

-

$dim(Skel^{i}\mathcal{A})\leq i$

-

$r\leq r\leq n$

-

$V(\mathcal{A})\cup V(\mathcal{B})$

-

$(c,\Delta^{n})$

-

$s_{\_}p,t_{\_}q$

-

$\Delta((p_{\_}1,1))={(p_{\_}1,1)}$

-

$\Xi_{\_}\epsilon^{r}$

-

$\sum_{\_}{i=k}^{n}{\binom{n}{i}\binom{i}{r}b^{i-\alpha}(i-r)_{\_}{k-r}}=\sum_{% -\_}{i=k}^{n}{\frac{n!}{(n-i)!r!(i-k)!}b^{i-\alpha}}=\frac{n!}{r!}\frac{(n-k)!}% -{(n-k)!}\sum_{\_}{i=0}^{n-k}{\frac{b^{i+k-\alpha}}{(n-k-i)!i!}}\\ -=\frac{(n)_{\_}k}{r!}b^{k-\alpha}\sum_{\_}{i=0}^{n-k}{\binom{n-k}{i}b^{i}}=% -\frac{b^{k-\alpha}}{r!}(b+1)^{n-k}(n)_{\_}k\squareforqed$

-

$\frac{T(k,n)}{f(k,n)}$

-

$T(k,n)$

-

$\mathrm{Ch}^{2}\ {\Delta^{2}}$

-

$\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathcal{A},v))$

-

$next\_state_{\_}i(v,s,k)$

-

$\Xi_{\_}b(\mathcal{I})\cong\mathrm{Ch}\ \mathcal{I}$

-

$(w_{\_}p,v_{\_}p)$

-

$\mathrm{St}^{\circ}(\tau(\Delta^{i})$

-

$\leq\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\tau^{r+1}(\mathcal{A}),v))$

-

$v_{\_}p\in V(\mathcal{A})\raisebox{-2.15277pt}{$|$}_{p}$

-

$\lim_{\_}{n\to\infty}{\sum_{\_}{j=1}^{k}{\big{(}\frac{k-j+2}{k+1}\big{)}^{n-k}% -\frac{\ln(2)^{j}}{j!}}}>\lim_{\_}{n\to\infty}(\frac{k+1}{k+1})^{n-k}\frac{\ln 2% -}{1!}=\ln 2>0\ \ \forall{k\leq n}$

-

$S(i-1)$

-

$P(k,r)\in\Theta(f(k,r))\overset{\text{HI}}{\iff}\sum_{\_}{i=k}^{n}{f(k,r-1)% -\sum_{\_}{j=1}^{k}{\binom{i}{j}T(k-j,i-k)}}\in\Theta(f(k,r))\\ -\overset{\text{Lemma~{}\ref{lemma:bound_intch}}}{=}f(k,r-1)\sum_{\_}{i=k}^{n}{% -\frac{(k+1)^{i-k}(i)_{\_}k}{\ln(2)^{k-1}}}=f(k,r-1)\frac{1}{\ln(2)^{k-1}}\sum_% -{\_}{i=k}^{n}{(k+1)^{i-k}(i)_{\_}k}\in\Theta(f(k,r))\\ -\iff f(k,r-1)\cdot f(k,1)\in\Theta(f(k,r))$

-

$n^{\prime}\in[0,n-1]$

-

$\mathrm{Ch}^{r}\ \mathcal{A}$

-

$\mathrm{St}^{\circ}(\tau(\mathcal{A}),v)$

-

$\displaystyle\overset{\text{Lemma~{}\ref{lemma:bound_intch}}}{\iff}\sum_{\_}{i% -=k}^{n}{P(i,0)\frac{(k+1)^{i-k}(i)_{\_}k}{\ln(2)^{k-1}}}\in\Theta(f(k,r))$

-

$V(\tau^{r+1}(\mathcal{A}))\setminus V(\tau^{r}(\mathcal{A}))$

-

$\mathcal{A}*\mathcal{B}$

-

$\mathrm{f}_{\_}n(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))\sim\frac{n!}{% -2\ln(2)^{n+1}}$

-

$\mathrm{f}_{\_}k(\tau(\mathcal{A}))=\sum_{\_}{i=k}^{n}{\mathrm{f}_{\_}i(% -\mathcal{A})\ \mathrm{f}_{\_}k(Int\ \tau(\Delta^{i}))}$

-

$view,state\mapsto decode_{\_}p(view,state)$

-

$\emptyset\in\mathcal{A}$

-

$v\in V(\mathcal{A})$

-

$\delta_{\_}1(0)=1$

-

$\raisebox{-2.15277pt}{$|$}_{p}$

-

$snapshot(M[k])$

-

$\textit{encode}:V(\mathcal{I})\rightarrow E$

-

$\delta*v$

-

$f(k,n):=\frac{(k+1)^{n-k}(n)_{\_}k}{ln(2)^{k+1}}$

-

$\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{% -Ch}^{r}\ \mathcal{A},v)))\in\Theta\bigg{(}\bigg{(}\frac{n!n^{n}}{\ln(2)^{n-1}}% -\bigg{)}^{r}\bigg{)}$

-

$\tau(\partial(\sigma))\subseteq\tau(Skel^{i-1}\mathcal{A})$

-

$V(\Delta)-1$

-

$\delta_{\_}i(s)$

-

$(s^{\prime},p_{\_}1)$

-

$\Xi_{\_}\epsilon^{0}(\mathcal{I})=\mathcal{I}$

-

$next\_state_{\_}1(\bot,s)=3s+1$

-

$s,t\in V(\mathcal{A})$

-

$\partial\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v)$

-

$\mathcal{A}\text{ is distinguishable under }\textit{encode}\implies|% -\operatorname{Im}\textit{encode}|\geq\max_{\_}{p\in\Pi}{\omega(G_{\_}p(% -\mathcal{A}))}$

-

$\Theta(\log_{\_}2((2^{n-1}n)^{r}))$

-

$v=input(i)$

-

$w_{\_}p,v_{\_}p\in\mathrm{Lk}(\mathcal{A},t_{\_}q)$

-

$v,w\in V(\mathcal{A})\raisebox{-2.15277pt}{$|$}_{p}:v\neq w$

-

$\mathrm{f}_{\_}k(Int\ \mathrm{St}^{\circ}(\tau(\Delta^{i}),v))$

-

$(i-1)(3s-1+2(s\mod 2))+i(3s+2(1-s\mod 2))$

-

$w_{\_}q\in\mathrm{Lk}(\mathcal{A},s_{\_}p)\arrowvert_{\_}q$

-

$\Xi\cong\mathrm{Ch}$

-

$\mathrm{f}_{\_}i(\mathcal{A})$

-

$\Xi_{\_}b(\mathcal{I})\ncong\mathrm{Ch}\ \mathcal{I}$

-

$G_{\_}\Pi(\mathcal{A})=\{G_{\_}p(\mathcal{A})\}_{\_}{p\in\Pi}$

-

$\Xi_{\_}b(\mathcal{I})\cong\Xi(\mathcal{I})\implies\Xi_{\_}b(\mathcal{I})\cong% -\mathrm{Ch}\ \mathcal{I}$

-

$h(k,n)^{r}$

-

$\mathrm{St}^{\circ}(\tau^{r}(\mathcal{A}),v)$

-

$\mathrm{f}_{\_}i(\mathcal{A})\cdot\mathrm{f}_{\_}k(Int\ \tau(\Delta^{i}))$

-

$(0,p_{\_}0)$

-

$(v,w)\in G_{\_}p(\mathcal{A})\iff\exists t\in V(\mathcal{A}):v,w\in V(\mathrm{% -Lk}(\mathcal{A},t))\raisebox{-2.15277pt}{$|$}_{p}$

-

$\alpha\cup\beta,\alpha\in\mathcal{A}$

-

$\lim_{\_}{n\to\infty}{\sum_{\_}{j=1}^{k}{\big{(}\frac{k-j+2}{k+1}\big{)}^{n-k}% -\frac{\ln(2)^{j}}{j!}}}<\lim_{\_}{n\to\infty}{\sum_{\_}{j=1}^{\infty}{\big{(}% -\frac{k-j+2}{k+1}\big{)}^{n-k}\frac{\ln(2)^{j}}{j!}}}<1\ \ \forall{k\leq n}$

-

$\lceil\log_{\_}3(\epsilon)\rceil$

-

$(n)_{\_}k$

-

$\mathrm{St}^{\circ}(\mathcal{A},v_{\_}p)$

-

$H=G_{\_}{p_{\_}0}(\mathcal{H})$

-

$\mathcal{I}=\{v_{\_}p,w_{\_}q,t_{\_}q,\{v_{\_}p,w_{\_}q\},\{v_{\_}p,t_{\_}q\},% -\{\}\}$

-

$\mathrm{f}_{\_}k(\tau(\sigma))=\mathrm{f}_{\_}k(\tau(\partial(\sigma)))+% -\mathrm{f}_{\_}k(\tau(Int\ \sigma))$

-

$\textit{encode}(s)=1$

-

$G_{\_}\Pi(\mathcal{A})$

-

$\log_{\_}2(|\operatorname{Im}\textit{encode}|)$

-

$\delta:\Xi(\mathcal{I})\rightarrow\mathcal{O}$

- - - diff --git a/htmls/output_mathjax_example_10073.html b/htmls/output_mathjax_example_10073.html deleted file mode 100644 index 3f9dc09f533812601b81f9ae6863ac8c73a60cbb..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10073.html +++ /dev/null @@ -1,168 +0,0 @@ - - - - MathJax Example - - - - -

$O(log(\epsilon))$

-

$v_{\_}p$

-

$|\operatorname{Im}\textit{encode}|$

-

$r\geq log_{\_}3(\epsilon)$

-

$\mathcal{H}=\bigcup_{\_}{(v_{\_}i,v_{\_}j)\in H}\mathcal{H}_{\_}{ij}$

-

$next\_state_{\_}i$

-

$p_{\_}0$

-

$(c^{\prime},\sigma^{\prime})$

-

$r\in V(\Delta^{i})$

-

$\mathcal{S}\subseteq\mathcal{A}$

-

$G_{\_}{p_{\_}0}(\mathcal{H})=H$

-

$\mathrm{St}^{\circ}(\mathcal{A},\mathcal{S})=\{\sigma\in\mathcal{A}:\mathcal{S% -}\subseteq\sigma\}$

-

$n,k,r\in\mathbb{N}$

-

$M[k]$

-

$t\in\mathrm{St}(\mathcal{A},v_{\_}p)$

-

$|\Pi|^{r}$

-

$\max_{\_}{p\in\Pi}{\Delta(G_{\_}p(\mathcal{A}))}=\max_{\_}{p\in\Pi}{\max_{\_}{% -v\in V(\mathcal{A})}{\mathrm{f}_{\_}0(\mathrm{Lk}(\mathcal{A},\mathrm{St}(% -\mathcal{A},v))\raisebox{-2.15277pt}{$|$}_{p})}}$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v)\in\Theta\bigg{% -(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\bigg{)}$

-

$\sum^{\infty}_{\_}{j=1}{\frac{\ln(2)^{j}}{j!}}=1$

-

$\delta_{\_}i:\mathcal{S}\rightarrow\{\frac{k}{\epsilon}:k\in[0,\epsilon]\},\ % -\delta_{\_}i(s)=\frac{2s+i}{\epsilon}$

-

$\Xi_{\_}\epsilon$

-

$\delta_{\_}1(\frac{3^{r}-1}{2})=\frac{3^{r}}{\epsilon}=1$

-

$\Xi_{\_}\epsilon(\mathcal{I})$

-

$(v_{\_}i,v_{\_}j)\in H$

-

$M[k,i]$

-

$\mathrm{St}(\mathcal{A},\mathcal{S})$

-

$\textit{encode}:\mathcal{S}\rightarrow\{1,2\},\ \textit{encode}(s)=2-(s\mod 2)$

-

$|\operatorname{Im}\textit{encode}|\leq\max_{\_}{p\in\Pi}{\Delta(G_{\_}p(% -\mathcal{A}))}+1$

-

$next\_state_{\_}0(\bot,s)=3s$

-

$\Xi_{\_}b^{r}(\mathcal{I})\cong\mathrm{Ch}^{r}\ \mathcal{I}\iff\forall r^{% -\prime}\in[0,r-1],\mathrm{Ch}^{r^{\prime}}\ \mathcal{I}\text{ is % -distinguishable under }\textit{encode}$

-

$K\cap\tau(Skel^{i-1}\mathcal{A})=\emptyset$

-

$|\operatorname{Im}\textit{encode}|\leq\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{A% -})}{\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(% -\mathrm{Ch}^{r}\ \mathcal{A},v)))}+1$

-

$\Delta((p_{\_}0,0))={(p_{\_}0,0)}$

-

$\{\mathrm{f}_{\_}n(\mathrm{St}^{\circ}(\mathrm{Ch}\ (\Delta^{n},v))\}_{\_}{n% -\geq 0}$

-

$\tau:\mathcal{A}\rightarrow B$

-

$\Delta:\mathcal{I}\rightarrow 2^{\mathcal{O}}$

-

$\sigma^{\prime}\in Skel^{i}\mathcal{A}:\sigma^{\prime}\neq\sigma$

-

$\mathrm{f}_{\_}k(\mathcal{A})=|\{\sigma\in\mathcal{A}:dim(\sigma)=k\}|$

-

$\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{% -Ch}^{r}\ \mathcal{A},v)))\in\Theta(f_{\_}n(k,r))\overset{\text{Lemma~{}\ref{% -lemma:f_linkstar}}}{\iff}\sum_{\_}{i=1}^{n}{\mathrm{f}_{\_}i(\mathrm{St}^{% -\circ}(\mathrm{Ch}^{r}\mathcal{A},v))}\in\Theta(f_{\_}n(k,r))\\ -\overset{\text{Theorem~{}\ref{theorem:finalBound}}}{\iff}\sum_{\_}{i=1}^{n}{% -\bigg{(}\frac{(i+1)^{n-i}(n)_{\_}i}{\ln(2)^{i-1}}\bigg{)}^{r}}\in\Theta(f(k,r)% -)\iff\lim_{\_}{n\to\infty}{\frac{\sum_{\_}{i=1}^{n}{\bigg{(}\frac{(i+1)^{n-i}(% -n)_{\_}i}{\ln(2)^{i+1}}\bigg{)}^{r}}}{\bigg{(}\frac{n!n^{n}}{\ln(2)^{n-1}}% -\bigg{)}^{r}}}\\ -=\lim_{\_}{n\to\infty}{\sum_{\_}{i=1}^{n}{\bigg{(}\frac{\ln(2)^{n-i}}{(n-i)!}% -\cdot\bigg{(}\frac{i+1}{n}\bigg{)}^{n}\cdot\frac{1}{(i+1)^{i}}\bigg{)}^{r}}}=C>0$

-

$\displaystyle P(k,1)\in\Theta(f(k,r))$

-

$\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{I})}{\mathrm{f}_{\_}1(\mathrm{St}^{% -\circ}(\mathrm{Ch}^{r}\ \mathcal{I},v))}\leq|\operatorname{Im}\textit{encode}|% -\leq\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{I})}{\mathrm{f}_{\_}0(\mathrm{Lk}(% -\mathrm{Ch}^{r}\ \mathcal{I},\mathrm{St}(\mathrm{Ch}^{r}\ \mathcal{I},v)))}+1$

-

$M[1],\ldots,M[r]$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))\in\Theta% -\bigg{(}\bigg{(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\bigg{)}^{r}\bigg{)}$

-

$\mathrm{f}_{\_}1(St^{\circ}(\mathrm{Ch}^{r}\ \Delta^{1},v))\in\Theta(1)$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))=\sum_{\_% -}{i=k}^{n}{\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathrm{Ch}^{r-1}\ \mathcal{A}% -,v))\sum_{\_}{j=1}^{k}{\binom{i}{j}\mathrm{f}_{\_}{k-j}(\mathrm{St}^{\circ}(% -\mathrm{Ch}\ \Delta^{i-j},v^{\prime}))}}$

-

$V(\mathcal{O})=\{(p_{\_}0,0),(p_{\_}1,\frac{1}{\epsilon}),(p_{\_}0,\frac{2}{% -\epsilon}),\dots,(p_{\_}0,\frac{\epsilon-1}{\epsilon}),(p_{\_}1,1)\}$

-

$\mathrm{f}(\mathcal{A})=(\mathrm{f}_{\_}{-1}(\mathcal{A}),\mathrm{f}_{\_}0(% -\mathcal{A}),\dots,\mathrm{f}_{\_}n(\mathcal{A}))$

-

$\mathcal{I}=\{0,1,\{0,1\}\}$

-

$\operatorname{Im}\textit{encode}_{\_}i$

-

${T(k^{\prime},n^{\prime}):k^{\prime}\in[0,k-1]\land n^{\prime}\in[0,n-1]}$

-

$T(k,n)\squareforqed$

-

$\Xi^{r}$

-

$\mathrm{f}_{\_}n(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))$

-

$v\in V(\Delta^{n})$

-

$\mathrm{Ch}\ {\Delta^{2}}$

-

$w_{\_}p$

-

$next\_state_{\_}p$

-

$\sum_{\_}{i=k}^{n}{\binom{n}{i}\binom{i}{r}b^{i-\alpha}(i-r)_{\_}{k-r}}=\frac{% -b^{k-\alpha}}{r!}(b+1)^{n-k}(n)_{\_}k$

-

$\Pi=\{p_{\_}1,\dots,p_{\_}{n+1}\}$

-

$G_{\_}p$

-

$\{(s,p_{\_}0),(s,p_{\_}1)\}$

-

$(s,p_{\_}0)$

-

$next\_state_{\_}i(v,s)$

-

$(3s,p_{\_}0)-(3s,p_{\_}1)-(3s+1,p_{\_}0)-(3s+1,p_{\_}1)$

-

$\mathrm{Ch}\ \mathcal{I}$

-

$\textit{encode}_{\_}i$

-

$\{G_{\_}p(\mathcal{A})\}_{\_}{p\in\Pi}$

-

$G_{\_}p(\mathcal{A})$

-

$\displaystyle\iff\frac{1}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{i-k}(i)% -_{\_}k}\in\Theta(f(k,r))$

-

$Int\ \mathcal{A}=\mathcal{A}\setminus\partial(\mathcal{A})$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ (\Delta^{n}),v))$

-

$\mathrm{f}_{\_}k(Int\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))=\sum_{\_% -}{i=1}^{k}{\binom{n}{i}\mathrm{f}_{\_}{k-i}(\mathrm{St}^{\circ}(\mathrm{Ch}\ % -\Delta^{n-i},v))}$

-

$\sum_{\_}{i=k}^{n}{(k+1)^{i-k}(i)_{\_}k}\in\Theta((k+1)^{n-k}(n)_{\_}k)$

-

$\Xi_{\_}b(\mathcal{I})$

-

$\mathrm{f}_{\_}k(Int\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))$

-

$f(k,r-1)\cdot f(k,1)=f(k,r)$

-

$s=next\_state_{\_}i(v,\bot,0)$

-

$p_{\_}1$

-

$\textit{encode}_{\_}i:state,round\mapsto state$

-

$R(k,n):=\sum_{\_}{j=1}^{k}{\binom{n}{j}T(k-j,n-j)}$

-

$\partial(\mathcal{A})=\{\sigma\in\mathcal{A}:\sigma\subset G\text{ for a % -unique facet }G\in\mathcal{A}\}\cup\{\emptyset\}$

-

$(\frac{3^{r}-1}{2},p_{\_}1)$

-

$\forall\tilde{v}\in V(\tau^{r}(\mathcal{A})),\mathrm{f}_{\_}k(\mathrm{St}^{% -\circ}(\tau^{r}(\mathcal{A}),\tilde{v}))\leq\mathrm{f}_{\_}k(\mathrm{St}^{% -\circ}(\tau^{r}(\mathcal{A}),v))$

-

$\frac{1}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{i-k}(i)_{\_}k}<\frac{1}{% -\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{n-k}(n)_{\_}k}\\ -=\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)}\implies P% -(k,1)\in O(f(k,1))\\ -\frac{1}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{i-k}(i)_{\_}k}>\frac{(k+% -1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}P(n,0)\implies P(k,1)\in\Omega(f(k,1))$

-

$Int\ \tau(\sigma^{i})$

-

$\mathrm{f}_{\_}k(\tau(\partial(\sigma)))$

-

$\Xi_{\_}b^{r}\cong\Xi^{r}$

-

$(p_{\_}0,v_{\_}i),(p_{\_}0,v_{\_}j)\in V(\mathrm{Lk}(\mathcal{H},(p_{\_}1,v_{% -\_}{ij})))\ \forall i,j$

-

$dim(\Delta)$

-

$|\operatorname{Im}{\textit{encode}}|$

-

$f_{\_}{-1}(\mathcal{A})=1$

-

$r-th$

-

$\sum_{\_}{j=1}^{k}{\binom{n}{j}\mathrm{f}_{\_}{k-j}(\mathrm{St}^{\circ}(% -\mathrm{Ch}\ \Delta^{n-j},v))}\in\Theta\bigg{(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln% -(2)^{k-1}}\bigg{)}$

-

$\{(s+1,p_{\_}0),(s,p_{\_}1)\}$

-

$k^{\prime}\in[0,k-1]$

-

$\mathrm{f}_{\_}{k^{\prime}}(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n^{\prime% -}},v))$

-

$v=\operatorname*{arg\,max}_{\_}{v\in V(\tau^{r}(\mathcal{A}))}{\mathrm{f}_{\_}% -k(\mathrm{St}^{\circ}(\tau^{r}(\mathcal{A}),v))}$

-

$v\in\Delta^{n}$

-

$S(i)=S(i-1)+\mathrm{f}_{\_}i(\mathcal{A})\cdot\mathrm{f}_{\_}k(Int\ \tau(% -\Delta^{i}))$

-

$\Delta(G_{\_}p(\mathcal{A}))+1$

- - - diff --git a/htmls/output_mathjax_example_10074.html b/htmls/output_mathjax_example_10074.html deleted file mode 100644 index 46e71f691870dc9f39c67220b1bf516c76364ffc..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10074.html +++ /dev/null @@ -1,137 +0,0 @@ - - - - MathJax Example - - - - -

$\Xi_{\_}b^{r}$

-

$q\in\Pi$

-

$\mathrm{Lk}(\mathcal{A},v_{\_}p)$

-

$v[(1-i)]=1$

-

$\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v)$

-

$\tau(\mathcal{A})$

-

$skel^{0}(\mathcal{A})=V(\mathcal{A})$

-

$\max_{\_}{p\in\Pi}{\omega(G_{\_}p(\mathcal{A}))}\geq\max_{\_}{p\in\Pi}{\max_{% -\_}{v\in V(\mathcal{A})}{\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathcal{A},v)% -\raisebox{-2.15277pt}{$|$}_{p})}}$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\tau(\mathcal{A}),v))=\sum_{\_}{i=k}^{n}{% -\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathcal{A},v))\ \mathrm{f}_{\_}k(Int\ % -\mathrm{St}^{\circ}(\tau(\Delta^{i}),v))}$

-

$\Theta(\log_{\_}2((\frac{n!n^{n}}{\ln(2)^{n-1}})^{r}))$

-

$\textit{encode}(v_{\_}p)=\textit{encode}(w_{\_}p)$

-

$\mathrm{Lk}(\mathcal{I},v_{\_}p)$

-

$Int\ \tau(\Delta^{i})$

-

$\Delta(\{(p_{\_}0,0),(p_{\_}1,1)\})=\mathcal{O}$

-

$\delta\circ\Xi(\mathcal{I})\subseteq\Delta(\mathcal{I})$

-

$|\delta_{\_}0(s)-\delta_{\_}1(s^{\prime})|=|\frac{2(s-s^{\prime})-1}{\epsilon}% -|\leq\frac{1}{\epsilon}$

-

$v_{\_}p\in V(\mathcal{A})$

-

$\delta_{\_}0(0)=0$

-

$-1\leq k\leq n$

-

$\mathrm{St}(\mathcal{A},v_{\_}p)$

-

$t_{\_}q\in V(\mathcal{A})$

-

$v[(1-i)]=\bot$

-

$\Xi_{\_}b(\mathcal{I})\cong\Xi(\mathcal{I})$

-

$G_{\_}p:\mathcal{A}\times\Pi\rightarrow(V(\mathcal{A}),V(\mathcal{A})\times V(% -\mathcal{A}))$

-

$t_{\_}q$

-

$next\_state_{\_}i(s,\bot)=3s+i$

-

$P(k,r):=\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))$

-

$decode$

-

$v_{\_}p,w_{\_}q\in V(\sigma)$

-

$\mathrm{St}^{\circ}(\mathcal{A},\mathcal{S})$

-

$|\operatorname{Im}\textit{encode}_{\_}i| -

$\delta\in\sigma:dim(\delta)=i-1$

-

$\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\tau^{r+1}(\mathcal{A}),v))$

-

$\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))$

-

$\forall\tilde{v}\in V(\tau^{r}(\mathcal{A})),\mathrm{f}_{\_}k(\mathrm{St}^{% -\circ}(\tau^{r+1}(\mathcal{A}),\tilde{v}))$

-

$\mathrm{f}_{\_}0$

-

$\Xi_{\_}b(\mathcal{I})\cong\mathrm{Ch}\ \mathcal{I}\iff\mathcal{I}\text{ is % -distinguishable under }\textit{encode}$

-

$p_{\_}i\in\Pi$

-

$M[r]$

-

$\Xi^{r}(\mathcal{I})\cong\Xi_{\_}b^{r}(\mathcal{I})$

-

$f_{\_}n(k,r):=\bigg{(}\bigg{(}\frac{n!n^{n}}{\ln(2)^{n-1}}\bigg{)}^{r}\bigg{)}$

-

$v[(1-i)]=2$

-

$K\cup\tau(Skel^{i-1}\mathcal{A})=\tau(Skel^{i}\mathcal{A})$

-

$P(k,0)$

-

$encode_{\_}i(s,k)$

-

$\mathrm{Lk}(\mathcal{A},\mathcal{S})$

-

$\beta\in\mathcal{B}$

-

$\textit{encode}(t_{\_}q)=\textit{encode}(w_{\_}q)$

-

$\frac{k-j+2}{k+1}\leq 1\ \forall k\leq n$

-

$dim(\delta*v)=k$

-

$\delta_{\_}p$

-

$\sigma\subseteq\sigma^{\prime}$

-

$\operatorname*{arg\,max}_{\_}{v\in V(\tau^{r}(\mathcal{A}))}{\mathrm{f}_{\_}k(% -\mathrm{St}^{\circ}(\tau^{r}(\mathcal{A}),v))}\in V(\tau(\mathcal{A})),\ \ \ % -\ \forall k\leq n,\forall r\geq 1$

-

$next\_state_{\_}1(1,s)=3s$

-

$(\ln 2,1)$

-

$K:=\tau(Skel^{i}\mathcal{A})-\tau(Skel^{i-1}\mathcal{A})$

-

$s-s^{\prime}\leq 1$

-

$T(0,n)\overset{\mathrm{def}}{=}1\in\Theta(1)=\Theta(f(0,n)),\forall n\in% -\mathbb{N}$

-

$skel^{l}(\mathcal{A})$

-

$s_{\_}p$

-

$\log_{\_}3(\epsilon)$

-

$\{(p_{\_}1,v_{\_}{ij}),(p_{\_}0,v_{\_}j)\},\{(p_{\_}0,v_{\_}i)\},\{(p_{\_}1,v_% -{\_}{ij})\},\{(p_{\_}0,v_{\_}j)\},\emptyset\}$

-

$T(k,n):=\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))$

-

${\max_{\_}{v\in V(\mathrm{Ch}^{r}\ \mathcal{A})}{\mathrm{f}_{\_}0(\mathrm{Lk}(% -\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{Ch}^{r}\ \mathcal{A},v)))% -\raisebox{-2.15277pt}{$|$}_{p}}}\in V(\mathrm{Ch}\ (\mathcal{A}))$

-

$decode_{\_}p(\textit{encode}_{\_}q(w_{\_}q),v_{\_}p)=w_{\_}q$

-

$Skel^{n}\mathcal{A}=\mathcal{A}$

-

$(v_{\_}p,w_{\_}p)\in G_{\_}p(\mathcal{A})$

-

$Int\ \mathrm{Ch}\ \Delta^{n}$

-

$\mathrm{St}(\mathrm{Ch}\ \mathcal{A},v_{\_}p)$

-

$\textit{encode}_{\_}q(w_{\_}q)=\textit{encode}_{\_}q(t_{\_}q)=e$

-

$\mathrm{f}_{\_}0(\mathrm{Ch}\ \mathcal{A},\mathrm{Lk}(\mathrm{St}(\mathrm{Ch}% -\ \mathcal{A},v_{\_}p))\raisebox{-2.15277pt}{$|$}_{p})=\sum_{\_}{i=1}^{n}{% -\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathcal{A},v_{\_}p)))}$

-

$f(k,n):=\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k+1}}$

-

$\mathrm{f}_{\_}0(Int\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))=0$

-

$g:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$

-

$c\rightarrow y$

-

$\{x^{(i)},y^{(i)},c^{(i)}\}_{i=1}^{n}$

-

$f:\mathbb{R}^{k}\rightarrow\mathcal{Y}$

-

$\{y^{(i)},c^{(i)}\}_{i=1}^{n}$

-

$\{x^{(i)},c^{(i)}\}_{i=1}^{n}$

-

$x\rightarrow c$

-

$\displaystyle t_{i}^{*}=(B-1)\left(t_{i}-t_{1}\right)/\left(t_{N}-t_{1}\right),$

-

$\hat{f_{j}^{i}}$

-

$\mathbf{V}_{0\xrightarrow{}4}$

-

$bins{=}13$

-

$\mathbf{V}_{0\xrightarrow{}5}$

-

$\mathbf{V}_{0\xrightarrow{}6}$

-

$\mathbf{V}_{0\xrightarrow{}8}$

-

$bins=21$

-

$\mathbf{V}_{0\xrightarrow{}7}$

-

$V_{0,i}$

-

$100{\times}\frac{i}{14}ms$

-

$dt{=}1$

-

${>}120dB$

-

$288{\times}384$

-

$\displaystyle k_{b}(a)=\max(0,1-|a|),$

-

$\mathrm{FWL}:=\frac{\sigma^{2}(I(E,\mathbf{V}))}{\sigma^{2}(I(E,0))}$

-

$640{\times}480$

-

$7.14ms$

-

$\displaystyle VG(x,y,t)=\sum_{i}p_{i}k_{b}\left(x-x_{i}\right)k_{b}\left(y-y_{% -i}\right)k_{b}\left(t-t_{i}^{*}\right),$

-

$60dB$

- - - diff --git a/htmls/output_mathjax_example_10075.html b/htmls/output_mathjax_example_10075.html deleted file mode 100644 index fbf4367b0f96adb7423873d2a969638274cb72cc..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10075.html +++ /dev/null @@ -1,132 +0,0 @@ - - - - MathJax Example - - - - -

$1{\times}1e^{-3}$

-

$f_{i}^{1}{\to}f_{4}^{N}$

-

$140Hz$

-

$dt{=}4$

-

$\mathbf{V}_{0\xrightarrow{}3}$

-

$B{\times}H{\times}W$

-

$20Hz$

-

$\mathbf{V}_{0,1}\to\mathbf{V}_{0,B-1}$

-

$I(E,\phi)=\left(\begin{array}[]{l}x_{i}^{\prime}\\ -y_{i}^{\prime}\end{array}\right)=\left(\begin{array}[]{l}x_{i}\\ -y_{i}\end{array}\right)+\left(t^{\prime}-t_{i}\right)\mathbf{V}(x_{i},y_{i}).$

-

$t_{b}-\tau -

$\mathbf{V}_{0,j}^{i}$

-

$\left\{\left(x_{i},y_{i},t_{i},p_{i}\right)\right\}_{i\in[1,N]}$

-

$E=(\left\{\left(x_{i},y_{i},t_{i},p_{i}\right)\right\}_{i\in[1,N]})$

-

$5{\times}1e^{-4}$

-

$4{:}1$

-

$\mathbf{V}_{0\xrightarrow{}13}$

-

$N{\times}B,H,W$

-

$\mathbf{V}_{0\xrightarrow{}14}$

-

$k_{b}(a)$

-

$\mathbf{V}_{0\xrightarrow{}12}$

-

$\mathbf{V}_{0\xrightarrow{}2}$

-

$50Hz$

-

$\mathbf{V}_{0\xrightarrow{}11}$

-

$bins{=}15$

-

$Loss=||\mathbf{F}_{gt}-\mathbf{F}_{pre}||_{1}.$

-

$bins{=}3$

-

$\Delta\mathbf{V}_{0,j}^{i}$

-

$\mathrm{RFWL}:=\sigma^{2}(\frac{I(E,\mathbf{V})}{\sum I(E,\mathbf{V})})/\sigma% -^{2}(\frac{I(E,0)}{\sum I(E,0)}).$

-

$93ms$

-

$(B-1)\times\tau$

-

$\displaystyle UVG=concat([Bin_{0},Bin_{1},...,Bin_{B-1}]).$

-

$\mathbf{V}_{0,B-1}$

-

$\mathbf{V}_{0,i}$

-

$10FPS$

-

$3.3ms$

-

$\mathbf{V}_{0,j}^{i-1}$

-

$346{\times}260$

-

$\mathbf{V}_{0\xrightarrow{}10}$

-

$bins{=}4$

-

$300Hz$

-

$\mathbf{V}_{0\xrightarrow{}1}$

-

$N,B,H,W$

-

$7.1ms$

-

$5ms$

-

$j^{i}$

-

$bins{=}9$

-

$Ht_{j}^{i-1}$

-

$\mathbf{V}_{0\xrightarrow{}9}$

-

$9.2ms$

-

$\displaystyle Bin_{b}(x,y)=\sum_{i}p_{i}k_{b}\left(x-x_{i}\right)k_{b}\left(y-% -y_{i}\right)k_{b}\left(\frac{t-t_{b}}{\tau}\right)$

-

$200Hz$

-

$Experts$

-

$Feedback$

-

$Probed\ system$

-

$Gating\ model$

-

$\bm{\lambda}^{\sf DMD}_{i,k}=G^{T}_{i}\bm{\lambda}_{k}=G^{T}_{i}A_{\lambda}% -\mathbf{y}_{k-1}\,.$

-

$h=1/N$

-

$\delta_{i}\approx 2h$

-

$A_{\lambda}(\bm{\mu}_{j})$

-

$\mathcal{F}_{i}(\bm{\lambda}_{k-1};\mathbf{u}_{1,k},\mathbf{u}_{2,k}):=G^{T}_{% -i}\bm{\lambda}_{k}=G^{T}_{i}{A}_{\lambda}\mathbf{y}_{k-1}\,,$

-

$\kappa_{i}>0$

-

$\bm{\Lambda}_{i,\beta_{i,k-1}}$

-

$\Delta_{128}t=1.67\times 10^{-3}$

-

$\mathbf{u}_{i,k+1}=\mathbf{u}_{i,k}+\Delta tM_{i}^{-1}\left(\mathbf{b}_{i,k}+(% --1)^{i}\bm{\lambda}_{i,k}\right)\,,\quad i=1,2\,.$

-

$q_{128}=3761$

-

$\bm{\mu}_{j}=\{\kappa_{1,j},\kappa_{2,j}\}$

-

$n_{l,\gamma}\times n_{{\delta_{1}},D}$

-

$u^{h}_{i,M}$

-

$\mathbf{u}_{i,k+1}=\mathbf{u}_{i,k}+\Delta tM_{i}^{-1}\left(\mathbf{f}_{i,k}-K% -_{i}\mathbf{u}_{i,k}+(-1)^{i}\bm{\lambda}_{i,k}\right)\,.$

-

$\dot{u}_{1}(\bm{x},t)=\dot{u}_{2}(\bm{x},t)\quad\mbox{on}\quad\gamma\times[0,T% -]\,.$

-

$n_{1,D}$

-

$A_{\lambda,u_{1}}$

-

$\bm{\mu}_{j}\in\mathcal{M}_{m}$

-

$2n^{2}_{1,\gamma}$

-

$\displaystyle(\dot{u}_{2},v_{2})_{0,\Omega_{2}}+(F_{2}(u_{2}),\nabla v_{2})_{0% -,\Omega_{2}}-\left<\lambda,v_{2}\right>_{\gamma}$

-

$\left_{\gamma}\,.$

-

$\bm{\lambda}_{i,k-1}$

-

$N_{\sf FS}=n_{l,\gamma}+n_{1,D}+n_{2,D}$

-

$\bm{\lambda}_{i,k}:=G^{T}_{i}\bm{\lambda}_{k}$

-

$\widetilde{y}_{i}=U^{T}_{k}y_{i}$

-

$\mathbf{y}_{k-1}=\left(\bm{\lambda}_{k-1},\mathbf{u}_{1,k}(\delta_{1}),\mathbf% -{u}_{2,k}(\delta_{2})\right)^{T}$

-

$\mathbf{U}_{i,\alpha_{i,k}}$

-

$u_{i,0}$

-

$u^{h}_{i,D}\in S^{h}_{i,D}$

-

$n_{\delta_{i},D}=O(n_{i,\gamma})$

-

$q=1866$

-

$\widetilde{A}_{k}:={U}_{k}^{T}A_{k}U_{k}\in\mathbb{R}^{k\times k}$

-

$E_{k}(Y)$

-

$\bm{\mu}=\{\kappa_{1},\kappa_{2}\}$

-

${H}^{1}(\Omega_{i})$

-

$u^{h}_{i}=u^{h}_{i,D}+g^{h}_{i}$

-

$B(\bm{\mu},R)$

-

${H}^{1}_{D}(\Omega_{i})$

-

$n_{1,D}=n_{2,D}$

-

$\{A_{\lambda}(\bm{\mu}_{j})\}_{j=1}^{m}$

-

$A=\mathbf{Y}^{\prime}\mathbf{Y}^{+}$

-

$\mathbf{u}_{1,k}$

-

$\displaystyle=(f_{i},v^{h}_{i})_{0,\Omega_{i}}+(-1)^{i}\left<\lambda,v^{h}_{i}% -\right>_{\gamma}\quad\forall v^{h}_{i}\in S^{h}_{i,D}\,.$

-

$G_{i}M_{i}^{-1}$

-

$u^{h}_{i,X}$

- - - diff --git a/htmls/output_mathjax_example_10076.html b/htmls/output_mathjax_example_10076.html deleted file mode 100644 index d38c15b4a4ba11a7beacfb9f68f95cc59751fcda..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10076.html +++ /dev/null @@ -1,139 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle=(f_{1},v_{1})_{0,\Omega_{1}}\qquad\forall v_{1}\in H_{D}^{1}(% -\Omega_{1})$

-

$\bm{\mu}\notin\mathcal{M}_{m}$

-

$O(n_{i,D}n_{l,\gamma})$

-

$H_{i,\gamma}=G_{i}{M}_{i,\gamma}^{-1}$

-

$\gamma^{h}_{i}$

-

$\mathbf{y}_{k+1}=A\mathbf{y}_{k}\,,$

-

$\mathbf{u}_{i,\gamma}\in\mathbb{R}^{n_{i,\gamma}}$

-

$\Omega^{h}_{1}$

-

$\Delta_{64}t=3.37\times 10^{-3}$

-

$\times 3.41$

-

$\displaystyle=\mathbf{f}_{2}$

-

$[\mathbf{Y}^{\prime}(\bm{\mu}_{i})]_{i=1}^{m}$

-

$\bm{\mu}_{i}\in\mathcal{M}_{m}$

-

$\lambda=F_{1}(u_{1})\cdot\bm{n}_{\gamma}=F_{2}(u_{2})\cdot\bm{n}_{\gamma}$

-

$\displaystyle(\dot{u}_{i},v_{i})_{0,\Omega_{i}}+(F_{i}(u_{i}),\nabla v_{i})_{0% -,\Omega_{i}}$

-

$\mathcal{M}=([1,2]\times[3,4])\times 10^{-3}$

-

$\mathcal{M}:=[\kappa_{1,\min},\kappa_{1,\max}]\times[\kappa_{2,\min},\kappa_{2% -,\max}]\subset\mathbb{R}^{2}\,.$

-

$\displaystyle=0\hskip 64.58313pt\forall\mu\in H^{-1/2}(\gamma)$

-

$\displaystyle M_{2}\dot{\mathbf{u}}_{2}+K_{2}\mathbf{u}_{2}-G_{2}^{T}\bm{\lambda}$

-

$\mathbf{u}_{i,m}$

-

$\displaystyle=(f_{i},v_{i})_{0,\Omega_{i}}+(-1)^{i}\left<\lambda,v_{i}\right>_% -{\gamma}\quad\forall v_{i}\in H_{D}^{1}(\Omega_{i})\,.$

-

$\mathbf{u}_{i,0}$

-

$\ell_{\bm{\mu}_{j}}$

-

$u_{i,0}(\bm{x})$

-

$n_{i,\Gamma}$

-

$\bm{v}=\left(0.5-y,x-0.5\right)$

-

$\left\{\begin{aligned} \dot{u}_{i}-\nabla\cdot F_{i}(u_{i})&=f_{i}\quad\mbox{% -in}\quad\Omega_{i}\times[0,T]\\[2.15277pt] -{u}_{i}&=g_{i}\quad\mbox{in}\quad\Gamma_{i}\times[0,T]\\[2.15277pt] -u_{i}(\bm{x},0)&=u_{i,0}(\bm{x})\quad\mbox{in}\quad\Omega_{i}\end{aligned}% -\right.\quad i=1,2,$

-

$q_{32}=918$

-

$\bm{x}_{i,r}\in\Gamma^{h}_{i}$

-

$\bm{\mu}=(1.5,2.5)\times 10^{-3}$

-

$\displaystyle E_{k}(Y):=\frac{\sum_{i=1}^{k}\sigma_{i}^{2}}{\sum_{i=1}^{n}% -\sigma_{i}^{2}}.$

-

$Y=U\Sigma V^{T}$

-

$\bm{\lambda}_{i,m}$

-

$\mathbf{u}_{i}=(\mathbf{u}_{i,\gamma},\mathbf{u}_{i,0},\mathbf{u}_{i,\Gamma})$

-

$S=\{(x,y)\in\Omega_{1}\,|\,0\leq x\leq 0.5\ \mbox{and}\ y=0.5\}$

-

$\displaystyle(\dot{u}_{1},v_{1})_{0,\Omega_{1}}+(F_{1}(u_{1}),\nabla v_{1})_{0% -,\Omega_{1}}+\left<\lambda,v_{1}\right>_{\gamma}$

-

$\mathcal{M}\subset\mathbb{R}^{M}$

-

$\mathcal{F}_{i}(\mathbf{u}_{1},\mathbf{u}_{2})=R_{j\mapsto i}K_{j,\gamma}% -\mathbf{u}_{j}$

-

$\bm{\mu}=(1.5,3.5)\times 10^{-3}$

-

${A}_{\lambda}(\bm{\mu}):=\sum_{\bm{\mu}_{j}\in\mathcal{M}_{m}(\bm{\mu},r)}\ell% -_{\bm{\mu}_{j}}(\bm{\mu}){A}_{\lambda}(\bm{\mu}_{j})$

-

$\lambda^{h}\in S^{h}_{l,\gamma}$

-

$\displaystyle=(f_{2},v_{2})_{0,\Omega_{2}}\qquad\forall v_{2}\in H_{D}^{1}(% -\Omega_{2})$

-

$\mathbf{Y}_{k}^{+}$

-

$(\cdot)_{\gamma}$

-

${V}_{k}$

-

$\langle\cdot,\cdot\rangle_{\gamma}$

-

$\mathbf{y}_{k-1}=\left(\bm{\lambda}_{k-1},\mathbf{u}_{1,k-1},\mathbf{u}_{2,k-1% -}\right)^{T}$

-

$F_{i}(u^{h}_{i})$

-

$\mathbf{b}_{i,k}=\mathbf{f}_{i,k}-K_{i}\mathbf{u}_{i,k}$

-

${M}_{i}=\mbox{diag}({M}_{i,\gamma},{M}_{0,\gamma})$

-

$B(\bm{\mu},r)\subset\mathcal{M}$

-

$\displaystyle=\mathbf{f}_{1}$

-

$F_{i}(u_{i})=\kappa_{i}\nabla u_{i}-\bm{v}u_{i},\quad i=1,2;$

-

$\Omega_{2}=[0.5,1]\times[0,1]$

-

$\mathbf{u}_{i,\Gamma}\in\mathbb{R}^{n_{i,\Gamma}}$

-

${M}_{i,\gamma}$

-

$q_{64}=1866$

-

$u^{h}_{M}$

-

$t_{k+1}=t_{k}+\Delta t$

-

$A_{k_{j}}(\bm{\mu}_{j})$

-

$\mathbf{u}_{i,k}$

-

$u^{h}_{i,L}$

-

$\bm{\lambda}^{\sf DMD}_{i,k}$

-

$\displaystyle\left_{\gamma}$

-

$(\cdot,\cdot)_{0,\Omega_{i}}$

-

$\mathbf{u}_{i,0}\in\mathbb{R}^{n_{i,0}}$

-

$\Omega^{h}_{i}$

-

$n_{1,\gamma}=n_{2,\gamma}$

-

$\Omega_{1}=[0,0.5]\times[0,1]$

-

$\mathcal{F}_{i,\bm{\mu}}(\bm{\lambda}_{k-1};\mathbf{u}_{1,k},\mathbf{u}_{2,k})% -:=G^{T}_{i}\bm{\lambda}_{k}=G^{T}_{i}{A}_{\lambda}(\bm{\mu})\mathbf{y}_{k-1}\,.$

-

$u_{i}(\cdot,t)\in H^{1}(\Omega_{i})$

-

$m=1,\ldots,4$

-

$\{A(\bm{\mu}_{j})\}_{j=1}^{m}$

-

$\ell_{i,r}(\bm{x}_{i,s})=\delta_{rs}$

-

$\bm{\mu}\in\mathcal{M}$

-

$\mathbf{y}_{k-1}:=\begin{bmatrix}\bm{\lambda}_{k-1}\\ -\mathbf{u}_{1,k}(\delta_{1})\\ -\mathbf{u}_{2,k}(\delta_{2})\end{bmatrix}\,,\quad k=1,2,\ldots\,.$

-

$A_{\lambda}(\bm{\mu})$

-

$2n_{1,\gamma}(2n_{1,\gamma}-1)\approx O(n^{2}_{1,\gamma})$

-

$\delta_{i}=5/2h$

-

$\text{dim}\,S_{i}^{h}=n_{i}$

-

$L^{2}(\Omega_{i})$

-

$\mathbf{y}_{i}:=\mathbf{y}(t_{i})$

-

$\mathbf{v}=\left(0.5-y,x-0.5\right)$

-

$\mathbf{u}_{i,k}(\delta_{i})=\left\{(\mathbf{u}_{i,k})_{j}\,|\,\exists\bm{x}_{% -j}\in\Omega^{h}_{i}\ \mbox{s.t.}\ d(\bm{x}_{j},\gamma)<\delta_{i}\right\}\in% -\mathbb{R}^{n_{\delta_{i},D}}$

-

$1-E_{k}(Y)\leq\epsilon\,,$

-

$\mathcal{M}_{m}:=\{\bm{\mu}_{j}\}_{j=1}^{m}$

-

$\Gamma^{h}_{i}$

-

$t_{1},t_{2},\ldots t_{n}$

-

$\mathbf{P}_{q}$

-

$S^{h}_{i,\Gamma}$

-

$R_{2\mapsto 1}$

-

$\epsilon=10^{-13}$

-

${M}_{0,\gamma}$

-

$\gamma^{h}_{1}$

-

$\mathbf{Y}(\bm{\mu}_{i})$

-

$\mathbf{Y}^{\prime}(\bm{\mu}_{i})$

-

$\mathcal{M}_{m}=\left\{(1,3),(1,4),(2,3),(2,4)\right\}\times 10^{-3}\,.$

-

$u^{h}_{i}\in S_{i}^{h}$

-

$n_{i,\gamma}$

-

$n_{i,D}\times n_{i,D}$

- - - diff --git a/htmls/output_mathjax_example_10077.html b/htmls/output_mathjax_example_10077.html deleted file mode 100644 index a40972ac4f05ee3b10f8f37cb73b1957103af475..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10077.html +++ /dev/null @@ -1,150 +0,0 @@ - - - - MathJax Example - - - - -

$S=G_{1}M_{1}^{-1}G_{1}^{T}+G_{2}M_{2}^{-1}G_{2}^{T}$

-

$\mathbf{c}_{k}=H_{1,\gamma}(\mathbf{b}_{1,k})_{\gamma}-H_{2,\gamma}(\mathbf{b}% -_{2,k})_{\gamma}\,,$

-

$\partial\Omega_{i}$

-

$N\in\{16,32,64,128\}$

-

$g^{h}_{i}\in S^{h}_{i,\Gamma}$

-

$\mathcal{F}_{i}(\mathbf{u}_{1},\mathbf{u}_{2}):=G^{T}_{i}\bm{\lambda}=G^{T}_{i% -}\left(G_{1}M_{1}^{-1}G_{1}^{T}+G_{2}M_{2}^{-1}G_{2}^{T}\right)^{-1}\left(G_{1% -}M_{1}^{-1}\mathbf{b}_{1}-G_{2}M_{2}^{-1}\mathbf{b}_{2}\right)\,.$

-

$\beta_{i,k-1}$

-

$\Omega^{h}_{2}$

-

$O(n_{1,\gamma}n_{1,D})$

-

$\mathcal{F}_{1}(\mathbf{u}_{1},\mathbf{u}_{2}):=K_{1,\gamma}\widehat{\mathbf{u% -}}_{1}=K_{1,\gamma}R_{2\mapsto 1}\mathbf{u}_{2}\quad\mbox{and}\quad\mathcal{F}% -_{2}(\mathbf{u}_{1},\mathbf{u}_{2}):=K_{2,\gamma}\widehat{\mathbf{u}}_{2}=K_{2% -,\gamma}R_{1\mapsto 2}\mathbf{u}_{1}\,,$

-

$\mathcal{M}_{m}(\bm{\mu},r)$

-

$K_{i,s}$

-

$\lambda(\cdot,t)\in H^{-1/2}(\gamma)$

-

$u^{h}_{i,D}$

-

$S^{h}_{i,D}$

-

$\mathbf{Y}^{\prime}:=[\mathbf{y}_{i}]_{i=1}^{s}$

-

$\displaystyle M_{1}\dot{\mathbf{u}}_{1}+K_{1}\mathbf{u}_{1}+G_{1}^{T}\bm{\lambda}$

-

$k=0,1,\ldots,N-1$

-

$u^{h}(\cdot,t)\in S^{h}_{i}$

-

$\mathcal{M}_{m}=\left\{(1,2),(1,3),(2,2),(2,3)\right\}\times 10^{-3}\,.$

-

$\mathcal{M}_{m}:=\{\bm{\mu}_{i}\}_{i=1}^{m}$

-

$\mathbf{Y}=U\Sigma V^{T}$

-

$S^{h}_{l,\gamma}$

-

$\mathcal{E}^{0}_{X}$

-

$\displaystyle=\mathbf{f}_{i}+(-1)^{i}\bm{\lambda}_{i}\,,\quad t\in(0,T]\quad% -\mbox{and}\quad\mathbf{u}_{i}(0)=\mathbf{u}_{i,0}\,,$

-

$n_{1,\gamma}\times n_{i,\gamma}$

-

$q_{16}=444$

-

$\left\{\begin{aligned} \dot{u}_{i}-\nabla\cdot F_{i}(u_{i})&=f_{i}&\text{in}\ % -&\Omega_{i}\times(0,T]\\[2.15277pt] -{u}_{i}&=g_{i}&\text{on}\ &\Gamma_{i}\times(0,T]\\[2.15277pt] -F_{i}(u_{i})\cdot\bm{n}_{i}&=(-1)^{i}\lambda&\text{on}\ &\gamma\times(0,T]\\[2% -.15277pt] -u_{i}(\bm{x},0)&=u_{i,0}(\bm{x})&\text{in}\ &\quad\Omega_{i}\end{aligned}% -\right.\,;\quad i=1,2.$

-

$\|\cdot\|_{1,\Omega_{i}}$

-

$\bm{\lambda}^{\sf DMD}_{i,k}=G^{T}_{i}\bm{\lambda}_{k}=G^{T}_{i}A_{\lambda}(% -\bm{\mu})\mathbf{y}_{k-1}\,.$

-

$\bm{\mu}_{j}=(\kappa_{1,j},\kappa_{2,j})$

-

$\kappa_{1}\neq\kappa_{2}$

-

$\mathbf{c}_{k}:=H_{1}\mathbf{b}_{1,k}-H_{2}\mathbf{b}_{2,k}$

-

$\mathbf{b}_{i}=\mathbf{f}_{i}-K_{i}\mathbf{u}_{i}$

-

$u^{h}_{i}$

-

$X\in\{C,L,D\}$

-

$\mathbf{Y}^{+}$

-

$u^{h}_{i,C}$

-

$\times 16.66$

-

$\mathcal{E}^{r}_{X}=\frac{1}{2}\sum_{i=1}^{2}\frac{\|u^{h}_{i,X}-u^{h}_{i,M}\|% -_{r,\Omega_{i}}}{\|u^{h}_{i,M}\|_{r,\Omega_{i}}}\,,r=0,1\,.$

-

$\{A_{k_{j}}(\bm{\mu}_{j})\,|\,\bm{\mu}_{j}\in\mathcal{M}_{m}(\bm{\mu},r)\}$

-

$O(n^{2}_{l,\gamma})$

-

$H^{-1/2}(\gamma)$

-

$K_{i,\gamma}$

-

$({u}^{h}_{i,0},v^{h}_{i})_{0,\Omega_{i}}=(u_{i,0},v^{h}_{i})_{0,\Omega_{i}}% -\quad\forall v^{h}_{i}\in S^{h}_{i}\,.$

-

$\mathbf{y}_{k}=\bm{\lambda}_{k}$

-

$\mathbf{y}_{k}=(\bm{\lambda}_{k},\mathbf{u}_{1,k+1}(\delta_{1}),\mathbf{u}_{1,% -k+1}(\delta_{2}))^{T}$

-

$R_{1\mapsto 2}$

-

$N_{\sf FS}=n_{l,\gamma}+n_{\delta_{1},D}+n_{\delta_{2},D}$

-

$g^{h}_{i}(\bm{x}_{i,r})=g_{i}(\bm{x}_{i,r})$

-

$[\mathbf{Y}(\bm{\mu}_{i})]_{i=1}^{m}$

-

$\Gamma_{i}:=\partial\Omega_{i}\backslash\gamma$

-

$\bm{\mu}=\{\kappa_{1},\kappa_{2}\}\in\mathcal{M}$

-

$\bm{\mu}\in\mathbb{R}^{m}$

-

$\mathbf{b}_{i,k}$

-

$\bm{\lambda}_{i,k}=\mathcal{F}_{i}(\bm{\Lambda}_{1,\beta_{1,k-1}},\bm{\Lambda}% -_{2,\beta_{2,k-1}};\mathbf{U}_{1,\alpha_{1,k}},\mathbf{U}_{2,\alpha_{2,k}})\,,% -\quad i=1,2\,.$

-

$\mathbf{Y}^{\prime}\approx{A}\mathbf{Y}.$

-

$u_{1}(\bm{x},t)=u_{2}(\bm{x},t)\quad\text{and}\quad F_{1}(u_{1})\cdot\bm{n}_{% -\gamma}=F_{2}(u_{2})\cdot\bm{n}_{\gamma}\quad\mbox{on}\quad\gamma\times[0,T]\,,$

-

$\bm{\lambda}_{i,k}$

-

$S^{h}_{i,\gamma}$

-

$S^{h}_{i}\subset H^{1}(\Omega_{i})$

-

$\mathcal{M}=([1,2]\times[2,3])\times 10^{-3}$

-

$\mathbf{y}_{k}\in\mathbb{R}^{N_{\sf FS}}$

-

$\bm{\mu}_{j}\in\mathcal{M}_{m}(\bm{\mu},r):=\mathcal{M}_{m}\cap B(\bm{\mu},r)\,.$

-

${U}_{k}$

-

$A_{\lambda}=\begin{bmatrix}A_{\lambda,\lambda}&A_{\lambda,u_{1}}&A_{\lambda,u_% -{2}}\end{bmatrix}\,.$

-

$n_{i,\gamma}\times n_{i,\gamma}$

-

$\displaystyle M_{i}\dot{\mathbf{u}}_{i}+K_{i}\mathbf{u}_{i}$

-

$\Omega\in\mathbb{R}^{\nu}$

-

$\Delta_{32}t=6.84\times 10^{-3}$

-

$\mathbf{u}_{i}(t)$

-

$S\bm{\lambda}=G_{1}M_{1}^{-1}\mathbf{b}_{1}-G_{2}M_{2}^{-1}\mathbf{b}_{2}\,,$

-

$\psi(x,y;x_{0},y_{0})=e^{-\frac{(x-x_{0})^{2}+(y-y_{0})^{2}}{2\sigma^{2}}}.$

-

$n_{1,\gamma}$

-

$N_{\sf FS}\times N_{\sf FS}$

-

$H_{i}:=G_{i}M^{-1}_{i}$

-

$O(n^{2}_{1,\gamma})$

-

$\bm{\lambda}_{1,k}(\bm{\mu}_{j})$

-

$\mathbf{u}_{2,k}$

-

$\mathbf{y}_{k-1}=(\bm{\lambda}_{k-1},\mathbf{u}_{1,k}(\delta_{1}),\mathbf{u}_{% -1,k}(\delta_{2}))^{T}$

-

$\mathbf{y}_{k+1}$

-

$\{u_{1}(\cdot,t),u_{2}(\cdot,t),\lambda(\cdot,t)\}\in H^{1}(\Omega_{1})\times H% -^{1}(\Omega_{2})\times H^{-1/2}(\gamma)$

-

$u_{i,0}\in H^{1}(\Omega_{i})$

-

$\text{dim}\,S_{i,\Gamma}^{h}=n_{i,\Gamma}$

-

$T=2\pi$

-

$u(x,y)=\left\{\begin{array}[]{ll}\displaystyle t(x+2y+3)&\mbox{if $(x,y)\in% -\bar{\Omega}_{1}$}\\[8.61108pt] -\displaystyle t\left(\frac{\kappa_{1}}{\kappa_{2}}x+2y+\frac{\kappa_{2}-\kappa% -_{1}}{2\kappa_{2}}+3\right)&\mbox{if $(x,y)\in{\Omega}_{2}$}\end{array}\right.\,,$

-

$\displaystyle(\dot{u}^{h}_{i},v^{h}_{i})_{0,\Omega_{i}}+(F_{i}(u^{h}_{i}),% -\nabla v^{h}_{i})_{0,\Omega_{i}}$

-

$\gamma^{h}_{2}$

-

$0<\delta_{i}<\mbox{diam}(\Omega_{i})$

-

$\mathbf{y}_{k-1}$

-

$\|\cdot\|_{0,\Omega_{i}}$

-

$\mathbf{y}_{k-1}:=\begin{bmatrix}\bm{\lambda}_{k-1}\\ -\mathbf{u}_{1,k}\\ -\mathbf{u}_{2,k}\end{bmatrix}\,,\quad k=1,2,\ldots\,.$

-

$\delta_{rs}$

-

$A\approx{A}_{k}:=\mathbf{Y}^{\prime}\mathbf{Y}_{k}^{+}=\mathbf{Y}^{\prime}{V}_% -{k}{\Sigma}_{k}^{+}{U}_{k}^{T}\,.$

-

$\mathbf{u}_{2,k}(\bm{\mu}_{j})$

-

$\mathbf{u}_{i,k+1}$

-

$f_{i}(\cdot,t)\in H^{-1}(\Omega_{i})$

-

${A}({\bm{\mu}})$

-

$S=LL^{T}$

-

$0=t_{0} - - - diff --git a/htmls/output_mathjax_example_10078.html b/htmls/output_mathjax_example_10078.html deleted file mode 100644 index 2925b61f8289495f79a045d5ed02608274f4fc00..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10078.html +++ /dev/null @@ -1,147 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle G_{1}\dot{\mathbf{u}}_{1}-G_{2}\dot{\mathbf{u}}_{2}$

-

$\bm{n}_{2}$

-

${\mathbf{y}}_{k-1}$

-

$\mathbf{u}_{i,D}=(\mathbf{u}_{i,\gamma},\mathbf{u}_{i,0},\mathbf{0})$

-

$\bm{n}_{\gamma}$

-

$\Delta_{16}t=1.42\times 10^{-2}$

-

$\nu=2,3$

-

$\kappa_{1}=1\times 10^{-3}$

-

$\lambda(\bm{x},t_{m})$

-

$\mathbf{Y}^{+}=V\Sigma^{+}U^{T}\approx{V}_{k}{\Sigma}_{k}^{+}{U}_{k}^{T}=:% -\mathbf{Y}_{k}^{+}\,,$

-

${A}_{\lambda}(\bm{\mu})$

-

$\widehat{\mathbf{u}}_{1}=R_{2\mapsto 1}\mathbf{u}_{2}\quad\mbox{and}\quad% -\widehat{\mathbf{u}}_{2}=R_{1\mapsto 2}\mathbf{u}_{1}\,,$

-

$\kappa_{2}=3\times 10^{-3}$

-

$\bm{x}_{i,r}$

-

$\text{dim}\,S^{h}_{i,\gamma}=n_{i,\gamma}$

-

$\mathbf{Y}:=[\mathbf{y}_{i}]_{i=0}^{s-1}$

-

$N_{\sf FS}(2N_{\sf FS}-1)$

-

$0\leq m\leq N$

-

$\mathbf{y}_{i+1}\approx A\mathbf{y}_{i}\,.$

-

$\mathbf{u}_{1,k}(\bm{\mu}_{j})$

-

$S_{i,D}^{h}$

-

$S^{h}_{i,D}\subset H^{1}_{D}(\Omega_{i})$

-

$\Delta t=3.37E-3$

-

$\Sigma_{k}^{+}$

-

$(N_{\sf FS}-n_{l,\gamma})(2N_{\sf FS}-1)$

-

$\kappa_{1}=\kappa_{2}=1\times 10^{-3}$

-

$A=\begin{bmatrix}A_{\lambda,\lambda}&A_{\lambda,u_{1}}&A_{\lambda,u_{2}}\\ -A_{u_{1},\lambda}&A_{u_{1},u_{1}}&A_{u_{1},u_{2}}\\ -A_{u_{2},\lambda}&A_{u_{2},u_{1}}&A_{u_{2},u_{2}}\\ -\end{bmatrix}$

-

$S_{i}^{h}$

-

$2n_{1,\gamma}(2n_{1,D}-1)$

-

$\mathbf{u}_{i,k}(\delta_{i})$

-

$q=2\pi/\Delta t$

-

$H^{1/2}(\gamma)$

-

$\mathcal{E}^{1}_{X}$

-

$n_{l,\gamma}$

-

$\bm{\lambda}_{k-1}$

-

$\mathbf{u}_{i}\in\mathbb{R}^{n_{i}}$

-

$u^{h}_{i}(\bm{x},t_{m})$

-

$\{{u}^{j}_{i,0}\}_{j=1}^{P}$

-

$g_{i}(\cdot,t)\in H^{1/2}_{00}(\Gamma_{i})$

-

$\mathbf{u}_{i,k+1}=\mathbf{u}_{i,k}+\Delta tM_{i}^{-1}\left(\mathbf{f}_{i,k}-K% -_{i}\mathbf{u}_{i,k}+(-1)^{i}\bm{\lambda}^{DMD}_{i,k}\right)\,.$

-

$\{\ell_{i,r}\}$

-

$\mathbf{y}_{k-1}(\bm{\mu}_{j})=\left(\bm{\lambda}_{1,k-1}(\bm{\mu}_{j}),% -\mathbf{u}_{1,k}(\delta_{1};\bm{\mu}_{j}),\mathbf{u}_{2,k}(\delta_{2};\bm{\mu}% -_{j})\right)^{T}$

-

$\mathbf{u}_{i,k}(\delta_{i})\subset\mathbf{u}_{i,k}$

-

$\text{dim}\,S_{i,D}^{h}=n_{i,D}:=n_{i}-n_{i,\Gamma}$

-

$(X_{*},\mathbf{y}_{*})$

-

$N_{*}\times 1$

-

$\phi_{i}(\mathbf{x})$

-

$\bm{\Phi}_{(\cdot)}$

-

$\bm{\mu}_{*}$

-

$\phi_{\mathbf{n}}(\mathbf{x})$

-

$\mathbf{y}_{*}=\{y_{*i}\in\mathbb{R}\,|\,i=1,\ldots,N_{*}\}$

-

$\nu\sim\mathcal{N}(0,\,\sigma_{n}^{2})$

-

$\displaystyle\phi_{\mathbf{n}}(\mathbf{x})$

-

$\displaystyle i=1,\ldots,n$

-

$\tilde{K}=(K+\sigma^{2}I)$

-

$\mathbf{f}_{*}|\mathbf{f},\mathbf{y},\bm{\theta}\sim\mathcal{N}(\bm{\mu}_{*},% -\Sigma_{*})$

-

$\displaystyle\approx m(X_{*})+W(\mathbf{y}-m(X))$

-

$p(\mathbf{f})=\mathcal{N}(\mathbf{0},\,k(X,X))$

-

$y=\sum_{i=1}^{p}\cos{x_{i}}+\nu$

-

$X=\{\mathbf{x}_{*i}\in\mathbb{R}^{p}\,|\,i=1,\ldots,N_{*}\}$

-

$y=f(\mathbf{x})+\varepsilon$

-

$\displaystyle\Sigma_{*}$

-

$n^{p}\times n^{p}$

-

$\displaystyle=\mathbf{\Phi}_{(X_{*})}\Lambda\mathbf{\Phi}_{(X)}^{T}\left(% -\Sigma_{N}^{-1}-\Sigma_{N}^{-1}\mathbf{\Phi}_{(X)}\bar{\Lambda}^{-1}\mathbf{% -\Phi}_{(X)}^{T}\Sigma_{N}^{-1}\right)$

-

$\Sigma_{n}^{-1}$

-

$\displaystyle\hskip 15.0pt\bm{\Phi}_{(X_{*})}\Lambda\bm{\Phi}^{T}_{(X)}\left(% -\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X)}+\sigma_{n}^{2}I\right)^{-1}\bm{\Phi}% -_{(X)}\Lambda\bm{\Phi}^{T}_{(X_{*})}$

-

$\Sigma_{*}$

-

$\displaystyle=\prod_{j=1}^{p}\phi_{n_{j}}(x_{j};\varepsilon_{j},\rho_{j})$

-

$\displaystyle=K_{**}-K_{*}(K+\sigma^{2}I)^{-1}K_{*}^{T}$

-

$9^{2}\times 9^{2}$

-

$k_{SE}^{ARD}(\mathbf{x},\mathbf{x^{\prime}})=\exp\left(-\varepsilon_{1}^{2}(x_% -{1}-x_{1}^{\prime})^{2}-\ldots-\varepsilon_{p}^{2}(x_{p}-x_{p}^{\prime})^{2}\right)$

-

$(X,\mathbf{y})$

-

$\displaystyle=\Lambda^{-1}+\mathbf{\Phi}_{(X)}^{T}\Sigma_{N}^{-1}\mathbf{\Phi}% -_{(X)}$

-

$\lambda_{n_{j}}(\varepsilon_{j},\rho_{j})$

-

$\Sigma_{n}=\sigma_{n}^{2}I$

-

$\displaystyle=\left(1+\left(\frac{2\varepsilon}{\rho}\right)^{2}\right)^{\frac% -{1}{4}},$

-

$k(\mathbf{x},\mathbf{x^{\prime}})\approx\sum_{i=1}^{n}\lambda_{i}\phi_{i}(% -\mathbf{x})\phi_{i}(\mathbf{x^{\prime}})$

-

$\displaystyle\lambda_{i}=\sqrt{\frac{\rho^{2}}{\rho^{2}+\delta^{2}+\varepsilon% -^{2}}}\left(\frac{\varepsilon^{2}}{\rho^{2}+\delta^{2}+\varepsilon^{2}}\right)% -^{i-1}$

-

$X=\{\mathbf{x}_{i}\in\mathbb{R}^{p}\,|\,i=1,\ldots,N\}$

-

$N_{*}\times N_{*}$

-

$\displaystyle=m(X_{*})+K_{*}(K+\sigma^{2}I)^{-1}(\mathbf{y}-m(X))$

-

$\varepsilon\sim\mathcal{N}(0,\,\sigma^{2})$

-

$K_{**}=k(X_{*},X_{*})$

-

$N_{*}\times N$

-

$K=k(X,X)$

-

$p(\mathbf{f})$

-

$\displaystyle\delta^{2}=\frac{\rho}{2}(\beta^{2}-1),$

-

$\displaystyle\approx\mathbf{\Phi}_{(X_{*})}\Lambda\mathbf{\Phi}_{(X_{*})}^{T}-% -W\mathbf{\Phi}_{(X)}\Lambda\mathbf{\Phi}_{(X_{*})}^{T}$

-

$N\times 9^{2}$

-

$m(X)=m(X_{*})=\mathbf{0}$

-

$\bar{\Lambda}=\left(\Lambda^{-1}+\bm{\Phi}^{T}_{(X)}\Sigma_{n}^{-1}\bm{\Phi}_{% -(X)}\right)$

-

$K_{*}=k(X_{*},X)$

-

$\displaystyle\approx m(X_{*})+\bm{\Phi}_{(X_{*})}\Lambda\bm{\Phi}^{T}_{(X)}% -\left(\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X)}+\Sigma_{n}\right)^{-1}(\mathbf% -{y}-m(X))$

-

$\Lambda=\begin{bmatrix}\lambda_{1}&&\\ -&\ddots&\\ -&&\lambda_{n}\end{bmatrix}$

-

$\mathbf{y}=\{y_{i}\in\mathbb{R}\,|\,i=1,\ldots,N\}$

-

$\geq 3.18$

-

$\bm{\Phi}_{(X)}=\begin{bmatrix}|&&|\\ -\phi_{i}(X)&\ldots&\phi_{n}(X)\\ -|&&|\end{bmatrix}$

-

$\approx 10.5$

-

$\displaystyle\gamma_{i}=\sqrt{\frac{\beta}{2^{i-1}\Gamma_{(i)}}},$

-

$\displaystyle\lambda_{\mathbf{n}}$

- - - diff --git a/htmls/output_mathjax_example_10079.html b/htmls/output_mathjax_example_10079.html deleted file mode 100644 index 3906a418e6b0b5611cc1232e827bacb8c6a883c1..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10079.html +++ /dev/null @@ -1,133 +0,0 @@ - - - - MathJax Example - - - - -

$\left(\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X)}+\Sigma_{n}\right)^{-1}=\Sigma_% -{n}^{-1}-\Sigma_{n}^{-1}\bm{\Phi}_{(X)}\left(\Lambda^{-1}+\bm{\Phi}^{T}_{(X)}% -\Sigma_{n}^{-1}\bm{\Phi}_{(X)}\right)^{-1}\bm{\Phi}^{T}_{(X)}\Sigma_{n}^{-1}$

-

$k(X,X^{\prime})\approx\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X^{\prime})}$

-

$\mathbf{f}=f(\mathbf{x})$

-

$\displaystyle\bm{\mu}_{*}$

-

$\displaystyle=\prod_{j=1}^{p}\lambda_{n_{j}}(\varepsilon_{j},\rho_{j})$

-

$k_{SE}^{ARD}{(\mathbf{x},\mathbf{x^{\prime}})}\approx\sum_{\mathbf{n}\in% -\mathbb{N}^{p}}\lambda_{\mathbf{n}}\phi_{\mathbf{n}}(\mathbf{x})\phi_{\mathbf{% -n}}(\mathbf{x^{\prime}})$

-

$\phi_{n_{j}}(x_{j};\varepsilon_{j},\rho_{j})$

-

$\displaystyle\bar{\Lambda}$

-

$\mathbf{x}=\begin{bmatrix}x_{1}&x_{2}&\ldots&x_{p}\end{bmatrix}\in\mathbb{R}^{p}$

-

$N\times n^{p}$

-

$\mathbf{\Phi}_{(X)}$

-

$\displaystyle\approx\bm{\Phi}_{(X_{*})}\Lambda\bm{\Phi}^{T}_{(X_{*})}-$

-

$k_{SE}(x,x^{\prime})=\exp\left(-\varepsilon^{2}(x-x^{\prime})^{2}\right)$

-

$k(\mathbf{x},\mathbf{x^{\prime}})=\sum_{i=1}^{\infty}\lambda_{i}\phi_{i}(% -\mathbf{x})\phi_{i}(\mathbf{x^{\prime}})$

-

$\lambda_{\mathbf{n}}$

-

$\displaystyle\phi_{i}(x)=\gamma_{i}\exp^{-\delta^{2}x^{2}}H_{i-1}(\rho\beta x)$

-

$\phi(a,b)=\frac{a^{T}b}{\left\lVert a\right\rVert\left\lVert b\right\rVert}$

-

$\mathcal{L}=-\sum_{i=1}^{T}\log\frac{e^{\phi(z_{i},z^{\prime}_{i})/\tau}}{\sum% -_{j\in D_{i}}e^{\phi(z_{i},z^{\prime}_{j})/\tau}}$

-

$WER=\frac{1}{|S^{\prime}|}\sum_{(x,z)\in S^{\prime}}\left\{\begin{array}[]{l l% -}0\text{ if }h(x)=z\\ -1\text{ otherwise }\end{array}\right.$

-

$a\in A,b\in B,x\neq a\in A$

-

$\tilde{X}=r(X,M)$

-

$z_{t}=h(x_{t})$

-

$D_{SSIMI}$

-

$ABX$

-

$d(a,x) -

$\mathcal{L}=\sum_{t\in M}\log p_{f}(z_{t}|\tilde{X},t)$

-

$(a,b,x)$

-

$PVQ$

-

$X=\left[x_{1},\cdots x_{T}\right]$

-

$t_{\operatorname*{DMN}}$

-

$|N_{v}\setminus U^{d}_{[v]_{\Pi_{d^{\prime}}}}|+|[v]_{\Pi_{d^{\prime}}}% -\setminus N_{v}|>d^{\prime}$

-

$t_{\text{$\operatorname*{DMN}$}}$

-

$6.7\cdot 10^{2}$

-

$\displaystyle=|N_{v}\setminus[v]_{\Pi}|+|[v]_{\Pi}\setminus N_{v}|$

-

$made\_join=True$

-

$\displaystyle\geq|N_{v}|-|N_{v}|/2=|N_{v}|/2>2d\enspace,$

-

$u^{\prime}\in C^{\prime}$

-

$\{u,v\}\in\binom{V}{2}$

-

$v\in N_{w}$

-

$\{u,v\}\in N_{w}$

-

$\varphi(\Pi)=d$

-

$\Pi=\{\{0,1\},\{2\},\{3\},\{4,5,6\}\}$

-

$U^{d}_{[v]_{\Pi_{d}}}$

-

$|N_{u}\triangle N_{u^{\prime}}|\leq 2d$

-

$U^{d}_{C}=\{v\in V\mid|N_{w}\triangle N_{u}|\leq 2d\;\forall w\in[v]_{\Pi_{d}}% -\;\forall u\in C\}$

-

$\displaystyle[v]_{\Pi_{d}}=U^{d}_{[v]_{\Pi_{d}}}=\{u\in V\mid|N_{u}\cap N_{v}|% ->|N_{v}|/2\}\enspace.$

-

$I_{\{u,v\}}$

-

$[v]_{\Pi_{d}}\subseteq[v]_{\Pi}\subseteq U^{d}_{[v]_{\Pi_{d}}}$

-

$1.996+\epsilon$

-

$C\in\Pi_{d}$

-

$\varphi(\Pi)\leq 2$

-

$6.4\cdot 10^{4}$

-

$4.1\cdot 10^{4}$

-

$1.7\cdot 10^{5}$

-

$U^{2}_{\{4\}}=\{0,4,5,6\}$

-

$|N_{w}\triangle N_{v}|$

-

$d=\max_{u\in C}|N_{u}\triangle C|$

-

$N_{v}:=\{v\}\cup\{w\in V\mid\{v,w\}\in E\}$

-

$|N_{u}\cap N_{v}|>2\varphi(\Pi)$

-

$C\subseteq U^{d}_{C}$

-

$x_{\{0,1\}}=x_{\{0,2\}}=\tfrac{1}{2}$

-

$E_{d}=\{\{u,v\}\in\binom{V}{2}\mid|N_{u}\cap N_{v}|>2d\}$

-

$x_{\{1,3\}}=x_{\{2,4\}}=\tfrac{3}{4}$

-

$1.2\cdot 10^{5}$

-

$x_{\{3,4\}}=x_{\{3,5\}}=x_{\{4,5\}}=\tfrac{1}{4}$

-

$U^{d}_{C}$

-

$\Pi_{2}=\{\{v\}\mid v\in V\}$

-

$t_{\operatorname*{DMN}}^{++}$

-

$\displaystyle=|(N_{u}\setminus N_{v})\cap C|+|(N_{u}\setminus N_{v})\setminus C|$

-

$\displaystyle|N_{4}\triangle[4]_{\Pi}|$

-

$\tfrac{1}{2}\sum_{w\in N_{u}\cap N_{v}}\min(\theta_{uw},\theta_{vw})$

-

$u\notin U^{d}_{[v]_{\Pi_{d}}}$

-

$\tfrac{7}{4}$

-

$|N_{u}\cap N_{v}|\leq 2\varphi(\Pi)$

-

$1.5\cdot 10^{6}$

-

$[u]_{\Pi}$

-

$E^{+}=E$

-

$U^{1}_{\{0\}}=\{0,1,2\}$

-

$-|N_{v}\triangle N_{w}|$

-

$|N_{v}|/4$

-

$[v]_{\Pi_{d}}$

-

$\mathcal{O}(n^{2}\log_{2}(\delta)+n\delta^{2})$

-

$|N_{w}|\leq\delta$

-

$\mathcal{O}(n^{2}\delta^{2})$

-

$N_{u}\cap N_{v}$

-

$I_{\{u,v\}}>2d$

-

$C=[u]_{\Pi}=[v]_{\Pi}$

-

$d=\varphi(\Pi)$

-

$x_{\{4,5\}}=0$

-

$\Pi_{1}=\{\{0\},\{1\},\{2\},\{3,4,5\}\}$

-

$\sum_{w\in N_{u}\cap N_{v}}\min(\theta_{uw},\theta_{vw})>2\varphi(\Pi)$

-

$|[v]_{\Pi_{d}}\setminus N_{v}|$

-

$\displaystyle=|N_{u}\triangle C|+|N_{v}\triangle C|$

-

$\displaystyle\min_{\Pi\in P_{V}}\underbrace{\max_{v\in V}\quad\left|[v]_{\Pi}% -\triangle N_{v}\right|}_{=:\ \varphi(\Pi)}$

-

$made\_join=False$

-

$x_{\{0,1\}}=x_{\{0,2\}}=x_{\{0,3\}}=x_{\{1,4\}}=x_{\{1,5\}}=x_{\{2,4\}}=x_{\{2% -,5\}}=x_{\{3,4\}}=x_{\{3,5\}}=\tfrac{1}{2}$

-

$\displaystyle|N_{u}\triangle N_{v}|$

-

$\mathcal{O}(n\delta^{2})$

-

$t_{\mathcal{A}^{*}}$

-

$C=\{u\in V\mid|N_{u}\cap N_{v}|>|N_{v}|/2\}$

- - - diff --git a/htmls/output_mathjax_example_1008.html b/htmls/output_mathjax_example_1008.html deleted file mode 100644 index ad0d5798a481f330151f421cff8851dca2b90780..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_1008.html +++ /dev/null @@ -1,134 +0,0 @@ - - - - MathJax Example - - - - -

$.08{\scriptstyle\ \pm.12}$

-

$0.03495$

-

$.80{\scriptstyle\ \pm.01}$

-

$\mathbf{.18}{\scriptstyle\ \pm.00}$

-

$.12{\scriptstyle\ \pm.00}$

-

$7.42\times 10^{-4}$

-

$2.96\times 10^{-5}$

-

$.15{\scriptstyle\ \pm.01}$

-

$\underline{\mathbf{.01}}{\scriptstyle\ \pm.00}$

-

$55\%$

-

$\underline{\mathbf{.83}}{\scriptstyle\ \pm.01}$

-

$.72{\scriptstyle\ \pm.03}$

-

$.52{\scriptstyle\ \pm.01}$

-

$.04{\scriptstyle\ \pm.02}$

-

$\mathbf{.81}{\scriptstyle\ \pm.01}$

-

$\mathbb{P}\big{(}Y=\hat{y}\ |\ P=\hat{p}\big{)}=\hat{p},$

-

$q\left(x_{t}\mid x_{t-1}\right)=N\left(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{% -t}I\right)$

-

$T_{o}\leq T$

-

$I^{k}_{s}$

-

$x_{o}$

-

$q\left(x_{1:T}\mid x_{0}\right)=\prod_{t=1}^{T}q\left(x_{t}\mid x_{t-1}\right)$

-

$v_{o}\in\mathbb{R}^{1\times 6}$

-

$T\rightarrow\infty$

-

$v_{c}=\Phi(I_{s})$

-

$x_{0},x_{1},\ldots,x_{T}$

-

$\sigma^{k}_{o}$

-

$t\sim\mathcal{U}(\{1,\ldots,T\})$

-

$k\in\{R,G,B\}$

-

$x_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}\left(x_{t}-\frac{1-\alpha_{t}}{\sqrt{1-% -\bar{\alpha}_{t}}}\epsilon_{\theta}(x_{t},I_{c},v_{c},t)\right)+\sigma_{t}z$

-

$L_{err}=\mathbb{E}_{x_{0},t,\epsilon\sim\mathcal{N}(0,I),I_{c},v_{c}}\left[% -\left\|\epsilon-\epsilon_{\theta}\left(x_{t},t,I_{c},v_{c}\right)\right\|\right]$

-

$2^{-16}$

-

$\forall t\sim\mathcal{U}(\{1,\ldots,T\})$

-

$x_{T_{o}}=I_{c}$

-

$v_{c}\in\mathbb{R}^{1\times 6}$

-

$\begin{split}I^{k}_{c}=\frac{I^{k}_{s}-\mu^{k}_{s}}{\sigma^{k}_{s}+\varepsilon% -}\sigma^{k}_{c}+\mu^{k}_{c},\end{split}$

-

$t=T_{o},T_{o-1},\dots,1$

-

${L}_{2}(\Theta)=\|v_{o}-v_{c}\|_{2},$

-

$p_{\theta}\left(x_{t-1}\mid x_{t}\right)=\mathcal{N}\left(x_{t-1};\mu_{\theta}% -\left(x_{t},t\right),{\sigma_{\theta}\left(x_{t},t\right)}^{2}I\right)$

-

$(I_{s},I_{o})\sim P$

-

$v_{c}$

-

$x_{t}=\sqrt{\bar{\alpha}_{t}}x_{0}+\epsilon\sqrt{1-\bar{\alpha}_{t}}\\$

-

$T_{o}$

-

$I^{k}_{c}$

-

$\sigma_{\theta}\left(x_{t},t\right)^{2}$

-

$\nabla_{\theta}\|\epsilon-\epsilon_{\theta}(x_{t},I_{c},v_{c},t)\|$

-

$x_{t}=\sqrt{\bar{\alpha}_{t}}I_{o}+\epsilon\sqrt{1-\bar{\alpha}_{t}}$

-

$\sigma^{k}_{c}$

-

$\mu^{k}_{o}$

-

$I_{c}=\text{ReNormalize}(v_{c},I_{s})$

-

$t\in\{T,\ldots,1\}$

-

$\mu_{\theta}\left(x_{t},t\right)$

-

$z\sim\mathcal{N}(0,I)$

-

$v_{o}$

-

$\mu^{k}_{c}$

-

$P=\{(I_{s}^{n},I_{o}^{n})\}_{n=1}^{N}$

-

$L_{err}=\mathbb{E}_{x_{o},t,\epsilon\sim\mathcal{N}(0,I)}\left[\left\|\epsilon% --\epsilon_{\theta}\left(x_{t},t\right)\right\|\right]$

-

$\mu^{k}_{s}$

-

$\sigma^{k}_{s}$

-

$I_{o}$

-

$q(x_{t-1}\mid x_{t})$

-

$x_{T}\sim\mathcal{N}(0,I)$

-

$s(A,B,X,Y)=\sum_{x\in X}s(x,A,B)-\sum_{y\in Y}s(y,A,B)$

-

$d=\frac{\mu(\{s(x,A,B)\}_{x\in X})-\mu(\{s(y,A,B)\}_{y\in Y})}{\sigma(\{s(t,X,% -Y)\}_{t\in A\cup B})}$

-

$s(w,A,B)\!=\!\frac{1}{|A|}\!\sum_{a\in A}\!\cos(w,a)\!-\!\frac{1}{|B|}\!\sum_{% -b\in B}\!\cos(w,b).$

-

$G=2$

-

$\mathbf{W}^{(x)}\in\mathbb{R}^{C^{\prime}\times C}$

-

$1\times{}1$

-

$\hat{\mathbf{q}}$

-

$\mathbf{g}_{:hw}=\textrm{softmax}\left(\mathbf{W}^{(g)}\cdot\textrm{ReLU}\left% -(\mathbf{W}^{(x)}\hat{\mathbf{x}}_{:hw}\odot\mathbf{W}^{(q)}\hat{\mathbf{q}}% -\right)\right),$

-

$C\cdot{}G+Q$

-

$\mathbf{g}\in\mathbb{R}^{G\times{}H\times{}W}$

-

$\hat{\mathbf{q}}\in\mathbb{R}^{Q}$

-

$\hat{\mathbf{x}}^{\prime}_{cghw}=\mathbf{g}_{ghw}\cdot\hat{\mathbf{x}}_{chw}% -\cdot\left(\mathbf{m}\downarrow_{H\times{}W}\right)_{hw},$

-

$\mathbf{W}^{(g)}$

-

$(\mathbf{q},\mathbf{m})$

-

$\hat{\mathbf{x}}$

-

$\hat{\mathbf{x}}^{\prime}$

-

$\mathbf{W}^{(g)}\in\mathbb{R}^{G\times C^{\prime}}$

-

$\hat{a}=\operatorname*{arg\,max}_{a\in\mathcal{A}}p_{\theta}(a\mid\mathbf{q},% -\mathbf{x},\mathbf{m}).$

-

$\hat{\mathbf{x}}^{\prime}=\textrm{att}(\hat{\mathbf{q}},\hat{\mathbf{x}},% -\mathbf{m})$

-

$\mathbf{W}^{(q)}\in\mathbb{R}^{C^{\prime}\times Q}$

-

$\mathbf{m}\downarrow_{H\times{}W}$

-

$\hat{\mathbf{x}}\in\mathbb{R}^{C\times{}H\times{}W}$

-

${:}hw$

-

$\mathbf{W}^{(x)}$

-

$Q_{g}=diag(q_{g})$

-

$q_{g}\in R^{K}$

-

$(M)_{i*}$

-

$p_{g}\in R^{D}$

-

$\in[0.2,0.4,0.6]$

-

$1\leq g\leq G$

-

$n_{c}=\lfloor a/((c-1)^{-\gamma}+b)\rceil$

-

$p\in[1,5,10,20,50]$

-

$\lfloor{.}\rceil$

-

$R_{D}$

-

$q_{g}$

-

$\begin{split}\Omega(W,P,Q)&=\lambda_{1}R_{W}(W,P,Q)+\lambda_{2}R_{D}(P,Q)+% -\lambda_{3}R_{E}(P,Q)\end{split}$

-

$\gamma\in\{2.0,1.0,0.6,0.20\}$

-

$l_{2,1}$

-

$p_{g}$

- - - diff --git a/htmls/output_mathjax_example_10080.html b/htmls/output_mathjax_example_10080.html deleted file mode 100644 index 92ba1ee733c35c86157f0eb1571ae48d5fe82a08..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10080.html +++ /dev/null @@ -1,131 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle\leq|C\setminus N_{v}|+|N_{u}\setminus C|+|C\setminus N_{u}|+|N_{% -v}\setminus C|$

-

$d=\operatorname*{CLB}(G)$

-

$[v]_{\Pi_{d}}\subseteq U^{d}_{[v]_{\Pi_{d}}}$

-

$\displaystyle=|N_{u}\cup N_{v}|-|N_{u}\cap N_{v}|$

-

$C=[v]_{\Pi}\cup[w]_{\Pi}$

-

$2\varphi(\Pi)$

-

$\displaystyle|N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}|=|\{u\in N_{v}\mid\{[u]_{% -\Pi_{d}},[v]_{\Pi_{d}}\}\notin E^{\prime}\}|\enspace,$

-

$|N_{u}\cap N_{v}|>|N_{v}|/2$

-

$t_{\operatorname*{CLB}}$

-

$\Pi\in P_{V}$

-

$x_{\{1,6\}}=x_{\{2,6\}}=x_{\{3,6\}}=\tfrac{3}{4}$

-

$|N_{v}|>4\varphi(\Pi)$

-

$U^{d}$

-

$U^{1}_{\{1\}}=\{0,1\}$

-

$V\setminus U^{d}_{C}$

-

$-|N_{w}\triangle N_{v}|$

-

$\varphi(\Pi)=0$

-

$I_{\{u,v\}}=0$

-

$U^{d}_{[v]_{\Pi_{d}}}\subseteq\{u\in V\mid|N_{u}\cap N_{v}|>|N_{v}|/2\}% -\subseteq[v]_{\Pi_{d}}$

-

$\mathcal{O}(n^{2}+n\delta^{2})$

-

$\mathcal{O}(n\delta)$

-

$\operatorname*{CLB}(G)$

-

$\operatorname*{DMN}$

-

$|N_{u}\cap N_{v}|\leq|N_{v}|/2$

-

$\varphi(\Pi)\geq|(N_{u}\cap N_{v})\setminus[v]_{\Pi}|$

-

$\displaystyle|N_{v}\triangle[v]_{\Pi}|$

-

$\displaystyle=|N_{u}\triangle[u]_{\Pi}|+|N_{v}\triangle[v]_{\Pi}|\leq 2\varphi% -(\Pi)\enspace.\qquad\square$

-

$|N_{v}|/2>2d$

-

$I_{\{u,v\}}=|N_{u}\cap N_{v}|$

-

$\tfrac{5}{4}$

-

$\mathcal{O}(\log_{2}(\delta))$

-

$\Pi:=\Pi\setminus\{[v]_{\Pi},[w]_{\Pi}\}\cup\{C\}$

-

$\varphi(\Pi)=1$

-

$4.1\cdot 10^{5}$

-

$\varphi(\Pi)\geq|N_{v}\triangle[v]_{\Pi}|>d^{\prime}$

-

$2.0\cdot 10^{5}$

-

$\operatorname*{CLB}$

-

$\Pi_{0}=\{V\}$

-

$\varphi(\Pi)\geq|N_{u}\triangle[u]_{\Pi}|\geq|N_{u}\setminus[u]_{\Pi}|\geq|(N_% -{u}\cap N_{v})\setminus[u]_{\Pi}|$

-

$\displaystyle=|N_{4}\setminus[4]_{\Pi}|+|[4]_{\Pi}\setminus N_{4}|$

-

$|N_{u}\triangle N_{v}|\leq 2\varphi(\Pi)$

-

$|N_{u}\triangle N_{v}|>2\varphi(\Pi)$

-

$d>|N_{v}\triangle[v]_{\Pi}|$

-

$0\leq\operatorname*{CLB}(G)\leq\delta$

-

$\Pi_{d}\in P_{V}$

-

$|N_{w}\cap N_{v}|$

-

$[v]_{\Pi}=\{u\in V\mid|N_{u}\cap N_{v}|>|N_{v}|/2\}$

-

$1.6\cdot 10^{3}$

-

$|N_{u}\cap N_{v}|\leq|(N_{u}\cap N_{w})\setminus[u]_{\Pi}|+|(N_{u}\cap N_{w})% -\setminus[v]_{\Pi}|\leq 2\varphi(\Pi)$

-

$[u]_{\Pi}\neq[v]_{\Pi}$

-

$\displaystyle\geq|N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}|+|[v]_{\Pi_{d}}% -\setminus N_{v}|\enspace.$

-

$\displaystyle\quad+|(N_{v}\setminus N_{u})\cap C|+|(N_{v}\setminus N_{u})% -\setminus C|$

-

$w\in\operatorname*{argmax}_{v\in V}|N_{v}\triangle[v]_{\Pi}|$

-

$\varphi(\Pi)=d^{\prime} -

$[v]_{\Pi}=[w]_{\Pi}$

-

$\min(\theta_{uw},\theta_{vw})$

-

$|N_{w}\cap N_{v}|-|N_{w}\triangle N_{v}|$

-

$U^{1}_{\{2\}}=\{0,2\}$

-

$|N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}|$

-

$(V,E^{+}\cup E^{-})$

-

$t_{\text{LP}}$

-

$U^{1}_{\{3,4,5\}}=\{3,4,5\}$

-

$made\_join$

-

$\left|[v]_{\Pi}\triangle N_{v}\right|$

-

$\varphi(\Pi)$

-

$C,C^{\prime}\in\Pi_{d}$

-

$f\in\{0,50,100,\dots,1000\}$

-

$\displaystyle\geq|N_{4}\setminus\{0,4,5,6\}|+|\{4\}\setminus N_{4}|=3$

-

$[u]_{\Pi}=[v]_{\Pi}$

-

$G_{d}=(V,E_{d})$

-

$\displaystyle\max_{v\in V}\;\;\Bigl{|}N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}% -\Bigr{|}+\Bigl{|}[v]_{\Pi_{d}}\setminus N_{v}\Bigr{|}\quad\leq\quad d\enspace,$

-

$|N_{v}|>4d$

-

$\displaystyle=|N_{u}\setminus N_{v}|+|N_{v}\setminus N_{u}|$

-

$7.06\times 10^{54})$

-

$\mu=1149$

-

$1\text{\times}{10}^{-25}$

-

$\sigma=334$

-

$\bm{\epsilon}\sim\mathcal{N}(\bm{0},\mathbf{Q})$

-

${\tilde{\mathcal{X}}}_{t}^{j}$

-

$\mathrm{DecSTER}$

-

$\mathcal{Z}^{j}_{t}$

-

$y_{i^{\prime}}\in\mathcal{Y}$

-

$\textbf{x}_{t,1}^{j},\dots,\textbf{x}_{t,\rho}^{j}$

-

$g(\mathbf{z}|\textbf{x},\mathbf{q})$

-

$\{\hat{\textbf{x}}_{1},\dots,\hat{\textbf{x}}_{\hat{n}_{\mathcal{G}}}\}$

-

$\textbf{x}_{t+1}=\mathbf{F}\textbf{x}_{t}+\bm{\epsilon}$

-

$\tilde{\rho}=100$

-

$(t,\mathbf{a}^{j}_{t},\mathcal{Z}^{j}_{t})$

-

$\hat{\mathcal{X}}_{t+1}^{j}$

-

$\displaystyle\psi_{\mathbf{z},\mathbf{q}}(\textbf{x})=g(\mathbf{z}|\textbf{x},% -\mathbf{q})p_{d}(\textbf{x}|\mathbf{q})$

-

${\hat{\mathcal{X}^{\prime}}}_{t+1}^{j}$

-

$\nu_{t}^{j}=\{(w_{1}^{j},\textbf{x}_{1}^{j}),\dots,(w_{\rho}^{j},\textbf{x}_{% -\rho}^{j})\}$

-

$p_{d}=0.9$

-

$\hat{n}=\sum_{i}w_{i},\forall\textbf{x}_{i}\in E$

-

$\bar{\nu}_{t+1}^{j}$

-

$\int_{E}\nu(x)dx$

-

$\{w_{1},\dots,w_{\rho}\}$

-

$n_{l}\times n_{w}=16\times 16$

-

$\bm{\omega}\sim\mathcal{N}(\bm{0},\sigma^{2}\mathbf{I})$

-

${w^{\prime}}_{i}$

- - - diff --git a/htmls/output_mathjax_example_10081.html b/htmls/output_mathjax_example_10081.html deleted file mode 100644 index 0f578c459a19ed4d3ed886314aa29ad97750a0aa..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10081.html +++ /dev/null @@ -1,142 +0,0 @@ - - - - MathJax Example - - - - -

$\mathbb{D}_{t}^{j}=\{(t^{\prime},\mathbf{a}^{\mathbf{q}_{j}^{\prime}}_{t^{% -\prime}},\mathcal{Z}^{j^{\prime}}_{t^{\prime}})\}_{t^{\prime} -

$v_{x},v_{y}\in[-v_{\text{max}},v_{\text{max}}]$

-

$\nu_{t+1}^{j}$

-

$\tilde{n}\sim\text{Poisson}(\hat{n}_{\mathcal{G}})$

-

$\mathbf{a}_{t}^{j}$

-

$\lambda\in\{0.005,0.04,1,5\}$

-

$\psi_{\mathbf{z},\mathbf{q}}(\textbf{x})$

-

$\text{OSPA}(\mathcal{X},\mathcal{Y})$

-

$\mathbf{q}=\begin{bmatrix}q_{x},q_{y}\end{bmatrix}^{\text{T}}$

-

$\tilde{\mathcal{X}}=\{\tilde{\textbf{x}}_{1},\dots,\tilde{\textbf{x}}_{\tilde{% -n}}\}$

-

${\hat{\mathcal{X}}}_{t}^{j}$

-

$|\mathcal{X}|=m\leq|\mathcal{Y}|=n$

-

$p_{s}(\textbf{x})$

-

$\hat{\mathcal{X}}_{t}^{j}$

-

$\mathcal{Z}=\{\mathbf{z}_{1},\dots,\mathbf{z}_{m}\}$

-

$\tilde{\mathcal{X}}_{t}^{j}$

-

$\textbf{x}=\begin{bmatrix}{l_{x},l_{y},v_{x},v_{y}}\end{bmatrix}^{\text{T}}$

-

$\mathbf{H}=\begin{bmatrix}1&0&0&0\\ -0&1&0&0\end{bmatrix}$

-

$t,\mathbf{a}^{\mathbf{q}_{j}}_{t},\mathcal{Z}^{j}_{t}$

-

${\nu}_{t}^{j}$

-

$\tilde{n}=\sum_{i=1}^{\tilde{\rho}}w_{i}$

-

$p_{d}(\textbf{x}|\mathbf{q})$

-

$\mathbb{D}_{t}^{j}$

-

$\mathbf{z}=h(\textbf{x})+\bm{\omega}$

-

$\mathcal{X}=\{\textbf{x}_{1},\dots,\textbf{x}_{|\mathcal{X}|}\}$

-

$\mathbf{z}=\mathbf{H}\textbf{x}+\bm{\omega}$

-

$\tilde{\mathcal{X}}_{R}$

-

$\tilde{n}\sim\text{Poisson}(\hat{n})$

-

$\mathcal{Z}_{t}^{j}$

-

$\bar{\mathcal{Z}}_{t}^{j}$

-

$\mathcal{Y}=\{\mathbf{y}_{1},\dots,\mathbf{y}_{|\mathcal{Y}|}\}$

-

$(l_{x},l_{y})\in[0,n_{w}]\times[0,n_{l}]$

-

$\{(w_{1},\textbf{x}_{1}),\dots,(w_{\rho},\textbf{x}_{\rho})\}$

-

$v_{\max}=0.1$

-

$\displaystyle\bar{\nu}_{t}(\textbf{x})=b(\textbf{x})+\int_{E}f(\textbf{x}|\bm{% -\xi})p_{s}(\bm{\xi})\nu_{t-1}(\bm{\xi})d\bm{\xi}$

-

$\displaystyle\sum_{i=1}^{\rho}\bar{w}_{i}+\frac{\alpha}{1-\alpha}\sum_{i=1}^{% -\rho}{w^{\prime}}_{i}-\frac{1}{1-\alpha}\sum_{i=1}^{\rho}{w^{\prime}}_{i}^{% -\alpha}\bar{w}_{i}^{1-\alpha}$

-

$\mathcal{Y}_{t}^{j}$

-

$\nu=\{(w_{1},\textbf{x}_{1}),\dots,(w_{\rho},\textbf{x}_{\rho})\}$

-

$f(\mathbf{x}|\bm{\xi})$

-

$\hat{n}_{\mathcal{G}}$

-

$\lambda_{\mathbf{q}}$

-

$\displaystyle\eta_{\mathbf{z}}(\nu)=\kappa(\mathbf{z}|\mathbf{q})+\int_{E}\psi% -_{\mathbf{z},\mathbf{q}}(\textbf{x})\nu(\textbf{x})d\textbf{x}$

-

$\mathbf{a}^{\mathbf{q}_{j}}_{t}$

-

$|\mathcal{A}|=340$

-

$\text{OSPA}(\mathcal{X},\mathcal{Y})=\big{(}\frac{1}{n}\min_{\pi\in\Pi_{n}}% -\sum_{i=1}^{m}d_{c}(x_{i},y_{\pi(i)})^{p}+c^{p}(n-m)\big{)}^{\frac{1}{p}}$

-

$\hat{n}=\sum_{i}w_{i}$

-

$E\subseteq\mathcal{G}$

-

$\displaystyle\mathbf{a}^{j}_{t}=\operatorname*{arg\;min}_{\mathbf{a}}\mathbb{E% -}_{\mathcal{Y}_{t}^{j}|\tilde{\mathcal{X}}_{t}^{j},\mathbf{a}}[\text{OSPA}(% -\tilde{\mathcal{X}}_{t}^{j},\mathcal{Y}_{t}^{j})]$

-

$\displaystyle\nu_{t}(\textbf{x})=(1-p_{d}(\textbf{x}|\mathbf{q}))\bar{\nu}_{t}% -(\textbf{x})+\sum_{\mathbf{z}\in\mathcal{Z}_{t}}\frac{\psi_{\mathbf{z},\mathbf% -{q}}(\textbf{x})\bar{\nu}_{t}(\textbf{x})}{\eta_{\mathbf{z}}(\bar{\nu}_{t})}$

-

$p_{s}=1$

-

$d_{c}(x,y)=min(c,||x-y||)$

-

$w_{t,1}^{j},\dots,w_{t,\rho}^{j}$

-

$\sum_{i=1}^{\rho}w_{i}$

-

$h(\textbf{x})=\begin{bmatrix}l_{x},l_{y}\end{bmatrix}^{\text{T}}$

-

$\nu_{t}^{j}=\{(w_{t,1}^{j},\textbf{x}_{t,1}^{j}),\dots,(w_{t,\rho}^{j},\textbf% -{x}_{t,\rho}^{j})\}$

-

${\nu^{\prime}}_{t+1}^{j}$

-

$\nu_{t}^{j}$

-

$\kappa(z)$

-

$\mathbf{a}_{\mathbf{q}}$

-

$\mathbf{F}=\begin{bmatrix}1&0&\Delta T&0\\ -0&1&0&\Delta T\\ -0&0&1&0\\ -0&0&0&1\end{bmatrix}$

-

$\{\textbf{x}_{i}\}_{i=1}^{\tilde{\rho}}$

-

$\tilde{\mathcal{X}}_{t}^{j}\sim\bar{\nu}_{t+1}^{j}$

-

$\{1,\dots,J\}$

-

$\bm{\omega}\sim\mathcal{N}(\mathbf{0},\sigma_{h}^{2}\mathbf{I})$

-

$\hat{n}=\int_{E}\nu(\textbf{x})d\textbf{x}$

-

$\hat{\mathcal{X}}\cup\tilde{\mathcal{X}}_{R}$

-

$p\in\{0.05,0.25,0.50,0.75,1\}$

-

$\Delta T=1$

-

$\mathbf{Q}=\begin{bmatrix}0.03&0&0.05&0\\ -0&0.03&0&0.05\\ -0.05&0&0.1&0\\ -0&0.05&0&0.1\end{bmatrix}$

-

$b(\textbf{x})$

-

$T_{b}\leftarrow$

-

$L(D)=\min_{h\in\mathcal{H}}(L(D\mid h)+L(h))$

-

$Z_{b}\in\mathbb{R}^{b\times max(|T|)\times d}$

-

$P\left(q_{3}\mid q_{3}\right)=P\left(q_{5}\mid q_{3}\right)=0<\frac{1}{5}$

-

$i,q\in enumerate(Q_{n})$

-

${ord}_{th}$

-

$clf$

-

$P\left(q_{3}\mid q_{4}\right)=\frac{3}{4}>\theta$

-

$\{a,X,b\}$

-

$X,Y,X,Y\cdots$

-

$q_{1}\ldots q_{x}q_{y}$

-

$Z_{n}\leftarrow$

-

$F=[Z||X]$

-

$q_{1},q_{2},\cdots,q_{m}$

-

$Z\leftarrow Z_{b}.\text{mean(axis=1,keepdim=False)}$

-

$\textit{result}\text{.append(}Z\text{)}$

-

$q_{1}q_{2}q_{3}$

-

$P\left(q_{3}\mid q_{3}\right)$

-

$\tau=1/|\operatorname{set}(S)|=1/5$

-

$P\left(q_{5}\mid q_{3}\right)$

-

$Z_{o}[\text{idx\_new}]\leftarrow Z_{n}$

-

$Z_{o}\leftarrow 0^{|Q|\times d}$

-

$q_{1},q_{2},\cdots,q_{n}$

-

$q_{2}q_{3}$

-

$P\left(q_{2}\mid q_{1}\right)=\frac{1}{2}<\theta$

-

$P\left(q_{4}\mid q_{3}\right)=\frac{2}{3}<\theta$

-

$P=\{p_{i}|i=1,2,\cdots,n\}$

-

$\begin{split}\mathbb{C}\left(S,M^{{ord}}\right)=2\times(\log{ord}+\log m+1)\\ -+m(({ord}+1)\log|\operatorname{set}(S)|+2\log|S|)\\ --\log\mathrm{P}\left(S\mid M^{{ord}}\right)\end{split}$

-

$x_{th}$

-

$Z_{o}\in\mathbb{R}^{b\times d}$

- - - diff --git a/htmls/output_mathjax_example_10082.html b/htmls/output_mathjax_example_10082.html deleted file mode 100644 index 2eb5a6ab8a73f221fdf15eef18b4a09f08b17133..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10082.html +++ /dev/null @@ -1,141 +0,0 @@ - - - - MathJax Example - - - - -

$P\left(q_{4}\mid q_{3}\right)=\frac{\operatorname{value}\left(n_{12}\right)}{% -\operatorname{value}\left(n_{3}\right)}=\frac{2}{3}>\frac{1}{5}$

-

$S_{T}\leftarrow len(T\;\text{for}\;T\;\text{in}T_{b})$

-

${max\_{ord}}=1$

-

$\mathrm{P}\left(S\mid M^{{ord}}\right)$

-

$\mathrm{P}\left(q_{x}\mid\mathrm{s}\right)=\frac{\operatorname{value}\left(n_{% -c}\right)}{\operatorname{value}\left(n_{p}\right)}$

-

$Z\leftarrow Z_{b}.\text{max(axis=1,keepdim=False)}$

-

$\operatorname{set}(S)$

-

$\begin{split}\mathrm{P}\left(S\mid M^{{ord}}\right)=P\left(q_{1}\right)\times P% -\left(q_{2}\mid q_{1}\right)\times P\left(q_{3}\mid q_{2}\right)\times P\left(% -q_{4}\mid q_{3}\right)\\ -\times P\left(q_{3}\mid q_{4}\right)\times\cdots\times P\left(q_{5}\mid q_{2}% -\right)\end{split}$

-

$Z_{o}\in\mathbb{R}^{|Q|\times d}$

-

$M^{ord}$

-

$\mathrm{P}\left(q_{y}\mid q_{1}\ldots q_{x}\right)\geq\theta$

-

$\begin{split}&P\left(X_{t}=x_{t}\mid X_{t-1}=x_{t-1},\ldots,X_{1}=x_{1}\right)% -\\ -&=P\left(X_{t}=x_{t}\mid X_{t-1}=x_{t-1},\ldots,X_{t-m}=x_{t-m}\right)\\ -\end{split}$

-

$\{c,Y\}$

-

$q\not\in D$

-

$n_{8}=2$

-

$query\_batch$

-

$max\_ord+1$

-

$Z_{o}$

-

$max(|T|)$

-

$S=q_{1}q_{2}q_{3}q_{4}q_{3}q_{4}q_{2}q_{3}q_{4}q_{3}q_{1}q_{3}\\ -q_{4}q_{3}q_{2}q_{5}$

-

$Z_{b}\leftarrow$

-

$q_{1}q_{2}...q_{1}q_{4}$

-

$P\left(q_{1}\mid q_{3}\right)$

-

$max(|T|)\times d$

-

$\textit{batch\_idx}\leftarrow\textit{batch\_idx}+b$

-

$A,B,C,\cdots$

-

$Z\in\mathbb{R}^{|T|\times d}$

-

$max\_ord$

-

$Z_{o}[i]\leftarrow D[q]$

-

$\textit{batch\_idx}<|Q|$

-

$max\_ord+2$

-

$\textit{batch\_end}=min(\textit{batch\_idx}+b,|Q|)$

-

$\mathrm{P}\left(q_{x}\mid\mathrm{s}\right)<\tau$

-

$P\left(q_{4}\mid q_{3}\right)$

-

$\textit{batch\_idx}\leftarrow$

-

$i,q\in\textsf{enumerate}(Q)$

-

$\tau=1/|\operatorname{set}(S)|$

-

$query\_batch=Q[\textit{batch\_idx}:\textit{batch\_end}]$

-

$clf(F)$

-

$P\left(q_{2}\mid q_{3}\right)=P\left(q_{1}\mid q_{3}\right)=\frac{% -\operatorname{value}\left(n_{10}\right)}{\text{ value }\left(n_{3}\right)}=% -\frac{1}{6}<\frac{1}{5}$

-

$\left(1-P\left(q_{4}\mid q_{3}\right)\right)/4=\frac{1}{12}$

-

$q_{4}q_{3}$

-

$D[q]\leftarrow Z_{n}[i]$

-

$\{Y,X\}$

-

$P\left(q_{2}\mid q_{3}\right)$

-

$\mathrm{P}\left(q_{x}\mid\mathrm{s}\right)\geq\tau$

-

$15-n$

-

$\displaystyle P_{\text{ana}}^{i}$

-

$\displaystyle=\bm{F_{agg}}^{i}\;(\bm{F_{ml}}^{i}(C_{i},E_{i}),\;\bm{F_{res}}^{% -i}(C_{i}))$

-

$P=CV^{2}$

-

$\text{MAPE}=1/n*\sum_{k=1}^{n}{|y_{k}-\hat{y_{k}}|/y_{k}}$

-

$1.1/0.8$

-

$\displaystyle=\bm{F_{agg}}^{i}\;(\textit{\#Hit, \#Miss, Energy per hit/miss})$

-

$\bm{F_{res}}^{i}\;(C_{i})$

-

$\hat{y_{k}}$

-

$P_{\text{ana}}^{i}$

-

$\displaystyle P_{\text{ana}}^{i}=\bm{F_{agg}}^{i}\;$

-

$\displaystyle=\textit{ICacheWay}*\textit{ICacheFetchBytes}$

-

$P_{\text{ml}}$

-

$\displaystyle=\bm{F_{ml}}^{i}(C_{i},E_{i})*\bm{F_{res}}^{i}\;(C_{i})$

-

$F_{res}=4,8,16$

-

$\bm{F_{res}^{i}}$

-

$P_{\text{PANDA}}^{i}=\bm{F_{ml}}^{i}(C_{i},E_{i})*\textit{ICacheWay}*\textit{ICacheFetchBytes}$

-

$/\bm{F_{res}}^{i}$

-

$P_{\text{ml}}=\bm{F_{ml}}\;(\{C,E\})$

-

$C=\{C_{i}\,|\,i\in[1,N]\}$

-

$F_{res}$

-

$40/28$

-

$f_{\text{ReserveStationNum}}$

-

$\bm{F_{res}}^{i}(C_{i})$

-

$\bm{F_{ml}}^{i}(C_{i},E_{i})$

-

$n\in[1,14]$

-

$\displaystyle\bm{F_{res}}^{i}(C_{i})$

-

$P_{\text{ml}}=\bm{F_{ml}}\;(\{C_{i}\,|\,i\in[1,N]\},\;\{E_{i}\,|\,i\in[1,N]\})$

-

$\displaystyle P_{\text{PANDA}}^{i}$

-

$P_{\text{ana}}=\sum_{i\in[1,N]}P_{\text{ana}}^{i}$

-

$E=\{E_{i}\,|\,i\in[1,N]\}$

-

$\bm{F_{res}}^{i}$

-

$40/28*(1.1/0.8)^{2}$

-

$\bm{F_{ml}^{i}}$

-

$\bm{F_{res}}^{i}\;(\textit{ICacheWay},\;\textit{ICacheFetchBytes})=\textit{% -Energy per hit/miss}$

-

$\bm{F_{agg}}^{i}$

-

$\bm{F_{ml}}^{i}$

-

$\displaystyle(E_{i},\;\bm{F_{res}}^{i}(C_{i}))$

-

$P^{i}_{\text{PANDA}}=\bm{F_{agg}}^{i}\;(\bm{F_{ml}}^{i}(C_{i},E_{i}),\;\bm{F_{% -res}}^{i}(C_{i}))$

-

$F_{res}=2$

-

$\displaystyle=\frac{\textit{\#Hit}*\textit{Energy per hit}+\textit{\#Miss}*% -\textit{Energy per miss}}{\textit{Total benchmark execution time}}$

-

$CV^{2}$

-

$\bm{F_{ml}}$

-

$j+\frac{D}{2}\ \mathrm{mod}\ D\in S_{J}$

-

$\displaystyle=e^{\frac{\imath 2\pi\sum_{i,k\in\left[d\right]}2^{i+k}a_{i}b_{i}% -}{D}}\left\lvert a\right\rangle_{h}=e^{\frac{\imath 2\pi ab}{D}}\left\lvert a% -\right\rangle_{h}$

-

$p=\Pr\left(ab\equiv c(\ \mathrm{mod}\ D)\right)$

-

$r_{4}=c_{1}-r_{2}r_{3}$

-

$2^{d-(d-d_{1})}$

-

$\forall x\in S_{X},\forall C\in S_{C},\Pr(X=x)=p_{x}=\frac{1}{\left\lvert S_{X% -}\right\lvert},\Pr(C=c)=p_{c}=\frac{1}{\left\lvert S_{C}\right\lvert}$

-

$\displaystyle\left\lvert a\right\rangle_{h}\left\lvert 0\right\rangle_{t}% -\overset{\mathcal{BMUL}(b)_{(h,t)}}{\longrightarrow}\left\lvert a\right\rangle% -_{h}\left\lvert ab\right\rangle_{t}\overset{\mathcal{BMUL}(-b^{-1})_{(t,h)}}{\longrightarrow}$

-

$-b\equiv D-b\ (\ \mathrm{mod}\ D)$

-

$S_{\mathbf{c}}=\left[D\right]^{3}\times\left[D\right]^{o}$

-

$h,t_{1},t_{2},g$

-

$ja_{2}$

- - - diff --git a/htmls/output_mathjax_example_10083.html b/htmls/output_mathjax_example_10083.html deleted file mode 100644 index 395cc525f6e9006bd1defa3626df724090e1e014..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10083.html +++ /dev/null @@ -1,179 +0,0 @@ - - - - MathJax Example - - - - -

$\rho_{s_{i}}=\left\lvert\psi_{s_{i}}\right\rangle\left\langle\psi_{s_{i}}\right\lvert$

-

$\displaystyle=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}+2^{d-l}\left(\sum_{i=0}^{l-1}% -a_{i+d-l}^{(l)}2^{l}+wa_{d-1-l}-k2^{l}\right)$

-

$\mathcal{CP}(i):\left\lvert a\right\rangle\left\lvert b\right\rangle% -\rightarrow e^{\frac{\imath 2\pi 2^{i}}{D}ab}\left\lvert a\right\rangle\left% -\lvert b\right\rangle$

-

$d_{1}+d_{2}\geq d$

-

$\displaystyle\rho=\sum_{s_{i}\in\left[D\right]^{o}}\frac{2}{D}\rho_{s_{i}}$

-

$H(X_{B}|\hat{M})=H(X_{B}|M)$

-

$s_{i}=2a+1$

-

$\mathcal{BROT}$

-

$tr_{P}(\cdot)$

-

$\displaystyle=\frac{4v+4\sum^{n}_{i=1}x_{i}y_{i}\ \mathrm{mod}\ D}{4}=\frac{4(% -v+\sum^{n}_{i=1}x_{i}y_{i})-a2^{d}}{4}$

-

$\displaystyle\overset{\mathcal{SUM}(r_{4})_{g}}{\longrightarrow}\frac{1}{\sqrt% -{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right\rangle_{h}% -\left\lvert j+c_{1}\right\rangle_{t_{1}}$

-

$\frac{d^{2}+d-2}{2\cdot 2^{d}}=\frac{5}{8}$

-

$\displaystyle=\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left% -\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp% -_{i}+c_{2}\right\rangle_{t_{2}}$

-

$\left\lvert r_{3}\right\rangle_{e_{1}}\left\lvert r_{4}\right\rangle_{e_{2}}$

-

$p_{s_{i}}=\frac{2}{D}$

-

$\displaystyle=H(X_{B})-H(X_{B}|M)-H(u).$

-

$\displaystyle=v+\sum^{n}_{i=1}x_{i}y_{i}\ \mathrm{mod}\ D=u.$

-

$\displaystyle\overset{\mathcal{SUM}(-c_{1})_{t_{1}}\mathcal{SUM}(-c_{2})_{t_{2% -}}}{\longrightarrow}$

-

$\left[2^{d-d_{3}+d_{1}}\right]$

-

$H(u|X_{B})=0$

-

$\rho\left\lvert\phi_{j+}\right\rangle=0\left\lvert\phi_{j+}\right\rangle,\rho% -\left\lvert\phi_{j-}\right\rangle=\frac{2}{\left\lvert S_{J}\right\lvert}\left% -\lvert\phi_{j+}\right\rangle.$

-

$\displaystyle\equiv a+\sum_{i=1}^{d-1}2^{i}wa_{i-1}(\ \mathrm{mod}\ 2^{d}).$

-

$4v_{i}\equiv\hat{M_{i}}-4x_{i}y_{i}(\ \mathrm{mod}\ D)$

-

$\rho=\frac{1}{\left\lvert S_{J}\right\lvert}\sum_{j\in S_{J}}\left(\left\lvert -j% -\right\rangle-\left\lvert j+\frac{D}{2}\right\rangle\right)\left\langle j% -\right\lvert.$

-

$\mu(m_{X})=\Omega(1/poly(m_{X}))$

-

$j\in S_{J}$

-

$\displaystyle I_{B}=H(M:X_{B}|u)=H(X_{B}:M,u)-H(X_{B}:u).$

-

$D|(a-b)$

-

$\hat{M}_{i}\equiv M_{i}-2x_{i}\equiv 4v_{i}+4x_{i}y_{i}(\ \mathrm{mod}\ D)$

-

$\displaystyle\overset{\mathcal{BSUM}_{(t_{1},g)}\mathcal{BSUM}_{(t_{2},g)}}{\longrightarrow}$

-

$\mathcal{ROT}(b)$

-

$w_{1}^{-1}\in\left[2^{d-d_{3}}\right]^{o}$

-

$h(x,c),f(j,x,c)$

-

$\displaystyle\left\lvert a\right\rangle_{h}\left\lvert 0\right\rangle_{t}% -\overset{\mathcal{SUM}(2^{0}b)^{h_{0}}_{t}}{\longrightarrow}\left\lvert a% -\right\rangle_{h}\left\lvert 2^{0}a_{0}b\right\rangle_{t}\overset{\mathcal{SUM% -}(2^{1}b)^{h_{1}}_{t}}{\longrightarrow}$

-

$\mathcal{U}_{h}$

-

$\displaystyle Output=\frac{\sum^{n}_{i=1}\left(M_{i}-2x_{i}\right)\ \mathrm{% -mod}\ D}{4}$

-

$\left\lvert S_{J}\right\lvert=0$

-

$\mathcal{MUL}(b)_{h}:\left\lvert a\right\rangle_{h}\rightarrow\left\lvert ab% -\right\rangle_{h}.$

-

$k_{1},k_{2},k_{3}\in\left[D\right]^{o}$

-

$\forall j\in S_{J}$

-

$\left\lvert\psi\right\rangle$

-

$\displaystyle=\frac{\sum^{n}_{i=1}\left(s_{i}+p_{i}q_{i}-2x_{i}\right)\ % -\mathrm{mod}\ D}{4}$

-

$r_{2}=c_{1}k_{1}+c_{2}k_{2}+c_{4}k_{3}$

-

$\mathcal{BSUM}_{(h,t)}=\mathcal{QFT}^{\dagger}_{t}\mathcal{BROT}_{(h,t)}% -\mathcal{QFT}_{t}$

-

$\mathcal{SUM}(b)$

-

$\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i}}=\frac{D}{2}$

-

$\left\lvert\psi\right\rangle_{h}$

-

$\displaystyle=\frac{1}{D\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]% -}\alpha_{bx}\left\lvert b\right\rangle\left\langle b\right\lvert_{Q}.$

-

$\displaystyle=\begin{pmatrix}\sum_{i=1}^{n}{a_{1i}b_{i1}}&\sum_{i=1}^{n}{a_{1i% -}b_{i2}}&\cdots&\sum_{i=1}^{n}{a_{1i}b_{in}}\\ -\sum_{i=1}^{n}{a_{2i}b_{i1}}&\sum_{i=1}^{n}{a_{2i}b_{i2}}&\cdots&\vdots\\ -\vdots&\vdots&\ddots&\vdots\\ -\sum_{i=1}^{n}{a_{ki}b_{i1}}&\sum_{i=1}^{n}{a_{ki}b_{i2}}&\cdots&\sum_{i=1}^{n% -}{a_{ki}b_{in}}\end{pmatrix}+\mathbf{V}$

-

$\left\lvert S_{J}\right\lvert=\frac{2^{d}}{2^{d-d_{l}}}=2^{d_{l}}$

-

$d_{l} -

$\displaystyle=\frac{2^{d-d_{l}-1}}{2^{d-1}}=\frac{1}{2^{d_{l}}};$

-

$u=\mathbf{x}\cdot\mathbf{y}+v\ \mathrm{mod}\ N=\sum_{i=1}^{n}{x_{i}y_{i}}+v\ % -\mathrm{mod}\ N$

-

$\left(k_{1}-r_{3}^{-1}\right)b_{1}+k_{2}b_{2}+k_{3}b_{3}+r_{4}r_{3}^{-1}=0.$

-

$\displaystyle=-{N}^{n+1}{N}^{n-1}\frac{1}{N^{n-1}}\frac{1}{N^{n+1}}\log_{2}{% -\frac{1}{N^{n}}}=nm.$

-

$\left[N\right]^{o}$

-

$\displaystyle\overset{\mathcal{MUL}(p_{i}^{-1})_{t_{2}}}{\longrightarrow}\frac% -{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right% -\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert j\right\rangle_{t_{2}}$

-

$\displaystyle=\log_{2}{N}-\left\lvert Im(g_{B})\right\lvert\frac{DN\log_{2}% -\left(DN\right)-ND\log_{2}D}{DN\frac{D^{4}}{2}}$

-

$\left\lvert\psi_{s_{i}}\right\rangle=\frac{1}{\sqrt{\left\lvert S_{J}\right% -\lvert}}\sum_{j\in S_{J}}\omega^{js_{i}}\left\lvert j\right\rangle.$

-

$\mathbf{y}=(y_{1},y_{2},\cdots,y_{n})$

-

$\mathcal{T}:\left\lvert a\right\rangle\left\lvert b\right\rangle\left\lvert c% -\right\rangle\rightarrow\left\lvert a\right\rangle\left\lvert b\right\rangle% -\left\lvert c\oplus a\cdot b\right\rangle$

-

$\left[D\right]^{o}$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{js_{i}}\left% -\lvert j\right\rangle_{t_{1}}\frac{1}{\sqrt{D}}\sum_{j^{\prime}\in\left[D% -\right]}\omega^{j^{\prime}q_{i}}$

-

$\displaystyle\rho_{s_{i}}=\left\lvert\psi_{s_{i}}\right\rangle\left\langle\psi% -_{s_{i}}\right\lvert=\frac{1}{\left\lvert S_{J}\right\lvert}\sum_{j^{\prime},j% -\in S_{J}}\omega^{(j^{\prime}-j)s_{i}}\left\lvert j^{\prime}\right\rangle\left% -\langle j\right\lvert$

-

$D=2^{d}$

-

$\left\langle f(j^{\prime},x,c)\lvert f(j,x,c)\right\rangle=\delta_{j^{\prime}j}$

-

$d_{r}=d$

-

$I_{B}=(n+1)m-nm-m=0$

-

$\displaystyle\overset{\mathcal{ROT}(s_{i})_{t_{1}}\mathcal{ROT}(q_{i})_{t_{2}}% -}{\longrightarrow}\frac{1}{\sqrt{D}}\omega^{(j+c_{1})s_{i}}\omega^{(jp_{i}+c_{% -2})q_{i}}$

-

$\mathbf{V}=\left(v_{ij}\right)_{k\times n},v_{ij}\in\left[N\right]$

-

$X,C$

-

$v\in\left[N\right]$

-

$\displaystyle\ \left\lvert S_{X}\right\lvert\frac{1}{\left\lvert S_{X}\right% -\lvert}\left(\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\frac{1}{D% -\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}% -\alpha_{bx}\right)$

-

$p_{\left\lvert S_{J}\right\lvert}=\left\{\begin{matrix}\frac{1}{\left\lvert S_% -{J}\right\lvert},&\ \mathrm{if}\ \left\lvert S_{J}\right\lvert -

$I_{B}=O\left(\frac{d^{2}}{2^{d}}\right)$

-

$\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert j+c_{1}% -\right\rangle_{g}$

-

$\mathcal{CP}(i+k)_{(h_{i},t_{i})}$

-

$a_{d-l-1}=1$

-

$2^{d-(d_{3}-d_{1})}$

-

$\displaystyle\left\lvert g(j^{\prime},x,c)\right\rangle\left\langle g(j,x,c)% -\right\lvert_{Q}$

-

$\left\lvert Im(g_{B})\right\lvert=D^{3}\cdot\frac{D}{2}=\frac{D^{4}}{2}$

-

$p(X_{B})=\frac{1}{N^{n+1}}$

-

$\displaystyle=\frac{\sum_{b\in\left[L\right]}\alpha_{bx}}{D\left\lvert S_{C}% -\right\lvert}\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\frac{1}{D% -\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}% -\alpha_{bx}$

-

$t_{1},t_{2},g,e$

-

$2^{d_{1}+d_{2}}w_{1}w_{2}=2^{d_{3}}\left(w_{3}+k2^{d-d_{3}}\right)$

-

$\displaystyle=H(M,u)-H(M,u|X_{B})-H(u)$

-

$S\left(\rho_{s_{i}}\right)=0$

-

$\mathcal{MUL},\mathcal{BSUM}$

-

$\displaystyle=m-(d+m)+d=0.$

-

$\displaystyle\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i}}=% -\sum_{a\in\left[\frac{D}{2}\right]}e^{\frac{\imath 2\pi}{D}\cdot(j^{\prime}-j)% -(2a+1)}$

-

$\displaystyle+\mathbf{V}=\mathbf{A}\cdot\mathbf{B}+\mathbf{V}.$

-

$\displaystyle\overset{\mathcal{QFT}_{h}}{\longrightarrow}\frac{1}{\sqrt{D}}% -\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{h}\left\lvert 0\right% -\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}\left\lvert 0\right\rangle_{g}$

-

$d_{3}=d$

-

$X_{B}=(\mathbf{y},v)$

-

$\displaystyle=\Pr\left(jk_{1}r_{3}+j^{\prime}k_{2}r_{3}+r_{4}=j\right)$

-

$a,b,c\in\left[D\right]$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{js_{i}}\left% -\lvert j\right\rangle_{t_{1}}\left\lvert(jk_{1}r_{3}+r_{4})\oplus j\right% -\rangle_{g}.$

-

$2^{d_{1}+d_{2}}w_{1}w_{2}=(k+1)2^{d}\equiv c(\ \mathrm{mod}\ D)$

-

$w=\frac{b-1}{2}$

-

$u=\sum_{i=1}^{n}{x_{i}y_{i}}+v\ \mathrm{mod}\ N$

- - - diff --git a/htmls/output_mathjax_example_10084.html b/htmls/output_mathjax_example_10084.html deleted file mode 100644 index 9771a3a9b347bc3eb87c389a0493cb5c4d4df01a..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10084.html +++ /dev/null @@ -1,178 +0,0 @@ - - - - MathJax Example - - - - -

$\displaystyle a^{(l+1)}=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}$

-

$\mathcal{H}:\left\lvert a\right\rangle\rightarrow\frac{\left\lvert 0\right% -\rangle+(-1)^{a}\left\lvert 1\right\rangle}{\sqrt{2}}$

-

$1-k_{1}k_{3}=2^{d_{l}}w_{l}$

-

$\displaystyle\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{h}\left% -\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2% -}}\left\lvert jr_{1}+r_{2}\right\rangle_{g}$

-

$O(2\epsilon^{-2}+n^{2})$

-

$\displaystyle=\omega^{j^{\prime}-j}\sum_{a\in\left[\frac{D}{2}\right]}e^{\frac% -{\imath 2\pi}{D}\cdot 2(j^{\prime}-j)a}=\omega^{j^{\prime}-j}\frac{1-e^{\imath -2% -\pi(j^{\prime}-j)}}{1-e^{\frac{\imath 2\pi}{D}\cdot 2(j^{\prime}-j)}}$

-

$h=\left(h_{d-1},h_{d-2},\cdots,h_{0}\right)$

-

$\alpha_{bx_{i}}=D$

-

$\rho^{Q}_{xc}=\frac{1}{D}\sum_{j\in\left[D\right]}\left\lvert g(j,x,c)\right% -\rangle\left\langle g(j,x,c)\right\lvert_{Q}.$

-

$\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert jr_{1}+r% -_{2}\right\rangle_{g}$

-

$jk_{1}r_{3}+r_{4}=j$

-

$g^{-1}(b)$

-

$S\left(\rho\right)=-\frac{\left\lvert S_{J}\right\lvert}{2}\cdot\frac{2}{\left% -\lvert S_{J}\right\lvert}\log_{2}\frac{2}{\left\lvert S_{J}\right\lvert}=\log_% -{2}\left\lvert S_{J}\right\lvert-1.$

-

$j-j^{\prime}\equiv\mathrm{odd}(\ \mathrm{mod}\ D)$

-

$c_{1},c_{2},c_{4}\in\left[D\right]$

-

$\displaystyle=\Pr\left(2^{d-d_{1}}\lvert b\right)=\frac{2^{d-(d-d_{1})}}{2^{d}% -}=\frac{1}{2^{d-d_{1}}}.$

-

$h(x,c)$

-

$\displaystyle S\left(\rho^{B}\right)=-\sum_{b\in\left[L\right]}\frac{\beta_{b}% -}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert}\log_{2}\frac{% -\beta_{b}}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert}$

-

$\displaystyle p_{2}=\Pr\left(ab\equiv c(\ \mathrm{mod}\ D)\lvert d>d_{3}\geq d% -_{1}\right)=p_{2a}\cdot p_{2b}$

-

$\left\lvert\mathbf{x}\right\lvert$

-

$\displaystyle-\frac{\sum_{b\in\left[L\right]}\beta_{b}\log_{2}{\beta_{b}}-% -\left\lvert S_{x_{i}}\right\lvert\sum_{b\in\left[L\right]}\alpha_{b{x_{i}}}% -\log_{2}{\alpha_{b{x_{i}}}}}{D\left\lvert S_{x_{i}}\right\lvert\left\lvert S_{% -\mathbf{c}}\right\lvert}$

-

$\mathcal{SWAP}_{(h,t)}=\mathcal{XOR}_{(h,t)}\mathcal{XOR}_{(t,h)}\mathcal{XOR}% -_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right\rangle_{t}% -\rightarrow\left\lvert b\right\rangle_{h}\left\lvert a\right\rangle_{t}$

-

$\beta_{b}=DN$

-

$b=b_{1}\parallel b_{2}\parallel b_{3}\parallel r_{3}\parallel r_{4}$

-

$\mathcal{MUL}(b)$

-

$H(X)=m_{X}$

-

$\displaystyle=\frac{2^{d_{1}}}{2^{d-d_{3}+d_{1}-1}}=\frac{1}{2^{d-d_{3}-1}}.$

-

$w_{1}w_{2}=w_{3}+k2^{d-d_{3}}$

-

$\displaystyle\overset{\mathcal{XOR}_{(h,t_{1})}\mathcal{XOR}_{(h,t_{2})}% -\mathcal{XOR}_{(h,g)}}{\longrightarrow}$

-

$w_{3}=1$

-

$\displaystyle\ \ \ \ +2^{d-l}\left(\sum_{i=0}^{l-1}a_{i+d-l}^{(l)}2^{l}+wa_{d-% -1-l}\ \mathrm{mod}\ 2^{l}\right),$

-

$\left\lvert 0\right\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}$

-

$\frac{1}{2^{d-d_{l}}}$

-

$\hat{M}=(\hat{M}_{1},\cdots,\hat{M}_{n})$

-

$\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{1}}% -\left\lvert ja_{1}\right\rangle_{t_{2}}\left\lvert ja_{2}\right\rangle_{g}$

-

$\displaystyle Output=(Output_{ij})_{k\times n}.$

-

$\mathcal{P}(i):\left\lvert a\right\rangle\rightarrow e^{\frac{\imath 2\pi 2^{i% -}}{D}a}\left\lvert a\right\rangle$

-

$Output_{ij}$

-

$\mathcal{SUM}^{h_{i}}$

-

$2^{d_{3}-d_{1}+1}\nmid b$

-

$\displaystyle\cdots=\sum_{i=0}^{d-l-1}a_{i}^{(0)}2^{i}=\sum_{i=0}^{d-l-1}a_{i}% -2^{i}.$

-

$\left\lvert 0\right\rangle_{t}$

-

$\mathcal{U}_{\varepsilon}:\left\lvert j\right\rangle_{t}\left\lvert 0\right% -\rangle_{e}\rightarrow\sqrt{\eta_{j}}\left\lvert j\right\rangle_{t}\left\lvert% -\varepsilon(j)\right\rangle_{e}+\sqrt{1-\eta_{j}}\left\lvert V(j)\right\rangle% -_{(t,e)},$

-

$\displaystyle(1-k_{1}r_{3})\left(j+\frac{D}{2}\right)\equiv(1-k_{1}r_{3})j% -\equiv r_{4}(\ \mathrm{mod}\ D).$

-

$Output=\frac{\sum^{n}_{i=1}\left(M_{i}-2x_{i}\right)\ \mathrm{mod}\ D}{4}$

-

$2^{d_{3}-d_{1}}\lvert b$

-

$\displaystyle\left\lvert 0\right\rangle_{h}\left\lvert 0\right\rangle_{t_{1}}% -\left\lvert 0\right\rangle_{t_{2}}\left\lvert 0\right\rangle_{g}$

-

$H(Z_{B}:X_{A})=0$

-

$\omega^{j(a_{1}b_{1}+a_{2}b_{2})}$

-

$c_{3}=c_{4}=0$

-

$\mathcal{ROT}(b_{2})$

-

$\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{P}\left% -\lvert ja_{1}\right\rangle_{Q_{1}}\left\lvert ja_{2}\right\rangle_{Q_{2}}$

-

$\left\langle\mathbf{x}\lvert\mathbf{y}\right\rangle$

-

$i\in\left[d\right]$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left% -\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp% -_{i}+c_{2}\right\rangle_{t_{2}}$

-

$r_{4}=2^{d_{r}}w_{r}$

-

$\displaystyle=\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\frac{1}{D% -\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}% -\alpha_{bx},$

-

$\displaystyle\ \ \ \frac{\sum_{b\in\left[L\right]}\beta_{b}\log_{2}{\beta_{b}}% --\left\lvert S_{X}\right\lvert\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}{% -\alpha_{bx}}}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert}.$

-

$\displaystyle=\frac{1}{D}\frac{(d-1)(d+2)}{2}=\frac{d^{2}+d-2}{2\cdot 2^{d}}=O% -\left(\frac{d^{2}}{2^{d}}\right).$

-

$\left\lfloor a\right\rfloor,\left\lceil a\right\rceil$

-

$H(M,u|X_{B})=H(u|M,X_{B})+H(M|X_{B})=H(M,X_{B})-H(X_{B})$

-

$1-\frac{\left\lvert S_{J}\right\lvert}{D}$

-

$x_{i},y_{i}\in\left[N\right]$

-

$O(n\epsilon^{-2})$

-

$j\in\left[D\right]$

-

$d_{1}+d_{2}\neq d_{3}$

-

$p_{\left\lvert S_{J}\right\lvert}=\frac{2}{2^{d-1}}$

-

$r_{3}=r_{1}^{-1}$

-

$\displaystyle\sum_{i=0}^{d-1}a_{i}^{(l+1)}2^{i}=a^{(l)}+wa_{d-1-l}2^{d-l}\ % -\mathrm{mod}\ D$

-

$\displaystyle\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}% -\right\rangle_{t_{2}}\left\lvert jc_{3}+c_{4}\right\rangle_{g}.$

-

$\displaystyle Output=v+\sum^{n}_{i=1}x_{i}y_{i}-a2^{m}$

-

$\displaystyle\left\lvert j^{\prime}\right\rangle_{t_{2}}\left\lvert(jk_{1}r_{3% -}+j^{\prime}k_{2}r_{3}+r_{4})\oplus j\right\rangle_{g}.$

-

$4v\in\left[D\right]$

-

$\displaystyle=\Pr\left(d_{1}+d_{2}\geq d_{3}\right)=\Pr\left(d_{2}\geq d-d_{1}\right)$

-

$H(Z:X)$

-

$2^{d_{3}-d_{1}}$

-

$1-k_{1}r_{3}=0$

-

$\displaystyle=2^{d-d_{3}}(2^{d_{1}}-l)-2,$

-

$\frac{2}{\left\lvert S_{J}\right\lvert}$

-

$\omega^{ja_{1}b_{1}}$

-

$p(\hat{M}|X_{B})=\frac{1}{N^{n-1}}$

-

$ab\equiv c(\ \mathrm{mod}\ D)$

-

$\displaystyle\frac{\sum_{b\in\left[L\right]}\beta_{b}\log_{2}{\beta_{b}}-\left% -\lvert S_{X}\right\lvert\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}{\alpha_{% -bx}}}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert},$

-

$\mathcal{SUM}(b\ \mathrm{mod}\ 2^{2})$

-

$0=D=2^{d}\times 1$

-

$0\leq i\leq d-l-1$

-

$2^{d_{1}+d_{2}}w_{1}w_{2}=(k+1)2^{d}$

-

$\displaystyle\mathcal{QFT}_{h}:\left\lvert a\right\rangle_{h}\rightarrow\frac{% -1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{aj}\left\lvert j\right\rangle_{h},$

-

$\displaystyle=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}+\sum_{i=d-l}^{d-1}a_{i}^{(l)}% -2^{i}+wa_{d-1-l}2^{d-l}-k2^{d}$

-

$X_{A}=x_{i}$

-

$\displaystyle=\frac{2}{D\left\lvert S_{J}\right\lvert}\sum_{j\in S_{J}}\sum_{j% -^{\prime}\in S_{J}}\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i% -}}\left\lvert j^{\prime}\right\rangle\left\langle j\right\lvert.$

-

$\mathcal{MUL}(r)$

-

$w_{2}=w_{1}^{-1}w_{3}+k2^{d-d_{3}}\in\left[2^{d-d_{3}+d_{1}}\right]^{o},$

-

$O\left(n2^{4m}\right)$

-

$\displaystyle I_{B}=H(Z_{A}:s_{i})\leq\sum_{\left\lvert S_{J}\right\lvert}% -\frac{\left\lvert S_{J}\right\lvert}{D}p_{\left\lvert S_{J}\right\lvert}\log_{% -2}\left\lvert S_{J}\right\lvert-1.$

-

$S_{x_{i}}=\left[N\right]$

-

$\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert jr_{1}+r% -_{2}\right\rangle_{g},$

-

$g:\left[D\right]\times S_{X}\times S_{C}\to\left[L\right]$

-

$\left[\begin{array}[]{c|c}\begin{matrix}1&0&0&0\\ -0&1&0&0\\ -0&0&j&1\\ -0&0&k_{3}&0\\ -k_{1}-r_{3}^{-1}&k_{2}&0&k_{3}\end{matrix}&\begin{matrix}b_{1}-j\\ -b_{2}-jp_{i}\\ -b_{3}\\ -r_{3}^{-1}-k_{1}-p_{i}k_{2}\\ --r_{4}r_{3}^{-1}\end{matrix}\end{array}\right].$

-

$\displaystyle\ \ \frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jh(x,c)}% -\left\lvert f(j,x,c)\right\rangle_{P}\left\lvert g(j,x,c)\right\rangle_{Q},$

- - - diff --git a/htmls/output_mathjax_example_10085.html b/htmls/output_mathjax_example_10085.html deleted file mode 100644 index 638d4cd76172d69722645510ff1cc6258d56142f..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10085.html +++ /dev/null @@ -1,169 +0,0 @@ - - - - MathJax Example - - - - -

$O\left(d^{2}\right)$

-

$M_{i}=s_{i}+p_{i}q_{i}$

-

$2^{d_{1}+d_{2}}w_{1}w_{2}=2^{d_{3}}w_{3}+k2^{d}$

-

$\displaystyle=\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{h}\left\lvert jk_{1}+c_{1}k_{1}\right\rangle_{t_{1}}\left\lvert jp_{i% -}k_{2}+c_{2}k_{2}\right\rangle_{t_{2}}$

-

$\left\lvert j\right\rangle_{g}\to\left\lvert 0\right\rangle_{g}$

-

$\left\lvert\mathbf{y}\right\lvert=\Theta\left(2^{m}\right)$

-

$I_{B}<\frac{d^{2}+d-2}{2\cdot 2^{d}}=O\left(\frac{d^{2}}{2^{d}}\right)$

-

$\displaystyle=\frac{1}{D}\sum_{j\in\left[D\right]}\left\lvert g(j,x,c)\right% -\rangle\left\langle g(j,x,c)\right\lvert_{Q}.$

-

$\mathcal{QFT}$

-

$v_{i},c_{1},c_{2},c_{3},c_{4},k_{1},k_{2},k_{3}$

-

$\left[D\right]$

-

$\displaystyle p_{1}=\Pr\left(ab\equiv c(\ \mathrm{mod}\ D)\lvert d=d_{3}\right)$

-

$1-\frac{1}{D}$

-

$d>d_{3}\geq d_{1}$

-

$b\in Im(g_{B})$

-

$\mathcal{BMUL}(b)_{(h,t)}$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{h}\left\lvert jk_{1}+c_{1}k_{1}\right\rangle_{t_{1}}\left\lvert jp_{i% -}k_{2}+c_{2}k_{2}\right\rangle_{t_{2}}$

-

$4\sum_{i=1}^{n}{v_{i}}\equiv 4v(\ \mathrm{mod}\ D)$

-

$\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert 0\right% -\rangle_{g},$

-

$\displaystyle\overset{\mathcal{MUL}(k_{1}^{-1})_{t_{1}}\mathcal{MUL}(k_{2}^{-1% -})_{t_{2}}}{\longrightarrow}\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left% -\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}$

-

$\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{1}},% -\left\lvert 0\right\rangle_{t_{2}},\left\lvert 0\right\rangle_{g}$

-

$\displaystyle\mathbf{x}_{i}=\left(a_{i1},a_{i2},\cdots,a_{in}\right),$

-

$q_{i}=2y_{i}+1$

-

$\left\{b\lvert g(a)=b,a\in A\right\}$

-

$\displaystyle\left\lvert jr_{1}+r_{2}\right\rangle_{g}$

-

$\displaystyle\left\lvert 0\right\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{% -2}},$

-

$\displaystyle ja_{2}\to jk_{1}+jk_{2}a_{1}+jk_{3}a_{2}=j(k_{1}+k_{2}a_{1}+k_{3% -}a_{2}),$

-

$X\in S_{X}$

-

$\displaystyle=\frac{4\sum^{n}_{i=1}v_{i}+4\sum^{n}_{i=1}x_{i}y_{i}\ \mathrm{% -mod}\ D}{4}$

-

$p=\frac{1}{2^{d-d_{1}}}$

-

$\displaystyle Output_{ij}=\mathbf{x}_{i}\cdot\mathbf{y}_{j}+v_{ij}.$

-

$\displaystyle\left[\begin{array}[]{c|c}\begin{matrix}1&0&0&0\\ -0&1&0&0\\ -0&0&1&0\\ -0&0&0&1\\ -0&0&0&0\end{matrix}&\begin{matrix}b_{1}-j\\ -b_{2}-jp_{i}\\ -k_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\\ -b_{3}-jk_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\\ -\left(k_{1}-r_{3}^{-1}\right)b_{1}+k_{2}b_{2}+k_{3}b_{3}+r_{4}r_{3}^{-1}\end{% -matrix}\end{array}\right].$

-

$1-k_{1}r_{3}$

-

$\displaystyle\frac{1}{D}\sum_{j,j^{\prime}\in\left[D\right]}\left\lvert j% -\right\rangle_{t_{1}}\left\lvert j^{\prime}\right\rangle_{t_{2}}\left\lvert 0% -\right\rangle_{g}\rightarrow$

-

$\mathbf{B}=\left(b_{ij}\right)_{k\times n}$

-

$(h_{d-1},h_{d-2},\cdots,h_{d-l})$

-

$\ \mathrm{mod}\ N$

-

$\left\lvert\sum_{i\in\left[2\right]}2^{i}a_{i+d-2}+b\ \mathrm{mod}\ 2^{2}\right\rangle$

-

$s_{i}=4v_{i}-2y_{i}-1\ \mathrm{mod}\ D$

-

$\forall b_{1},b_{2},b_{3}\in\left[D\right],r_{3}\in\left[D\right]^{o}$

-

$\displaystyle\Pr\left((jk_{1}r_{3}+j^{\prime}k_{2}r_{3}+r_{4})\oplus j=0\right)$

-

$I_{B}=0$

-

$\frac{2}{D}$

-

$\displaystyle\mathbf{A}=\left(a_{ij}\right)_{k\times n}=\begin{pmatrix}a_{11}&% -a_{12}&\cdots&a_{1n}\\ -a_{21}&a_{22}&\cdots&\vdots\\ -\vdots&\vdots&\ddots&\vdots\\ -a_{k1}&a_{k2}&\cdots&a_{kn}\end{pmatrix}$

-

$a_{i}^{(l+1)}=a_{i}^{(l)}$

-

$M_{i}\equiv p_{i}q_{i}+s_{i}\equiv 4x_{i}y_{i}+4v_{i}(\ \mathrm{mod}\ D)$

-

$H(X_{B}:u)=H(u)$

-

$O(kn\cdot nm^{2})=O(kn^{2}m^{2})$

-

$\displaystyle=\frac{2^{d-(d_{3}-d_{1})}2^{-1}}{2^{d}}=\frac{1}{2^{d_{3}-d_{1}+% -1}},$

-

$\omega^{c_{1}s_{i}+c_{2}q_{i}}$

-

$\left\lvert 0\right\rangle_{g}$

-

$\displaystyle=\frac{\sum^{n}_{i=1}\left(4v_{i}+4x_{i}y_{i}\right)\ \mathrm{mod% -}\ D}{4}$

-

$\displaystyle\frac{1}{D}\sum_{j,j^{\prime}\in\left[D\right]}\omega^{js_{i}+j^{% -\prime}q_{i}}\left\lvert j\right\rangle_{t_{1}}\left\lvert j^{\prime}\right% -\rangle_{t_{2}}\left\lvert jk_{1}+j^{\prime}k_{2}\right\rangle_{g}\rightarrow$

-

$2^{d-d_{1}}$

-

$O\left(4^{m}n\log^{2}n\right)$

-

$\mathcal{P}(i+k)_{h_{i}}$

-

$v=u-\mathbf{x}\cdot\mathbf{y}$

-

$\alpha_{bx}=\left\lvert g_{x}^{-1}(b)\right\lvert=\left\lvert\left\{(j,c)% -\lvert g(j,x,c)=b\right\}\right\lvert$

-

$H(Z:X)=0$

-

$\frac{\left\lvert S_{J}\right\lvert}{2}$

-

$a^{(0)}=a$

-

$4(v+\sum^{n}_{i=1}x_{i}y_{i})-a2^{d}\in\left[D\right]$

-

$\left\lvert a_{d-1}\right\rangle\left\lvert a_{d-2}\right\rangle$

-

$\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{1}},% -\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{2}}$

-

$\displaystyle\overset{\mathcal{XOR}_{(t_{1},g)}}{\longrightarrow}\frac{1}{% -\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right\rangle_{h% -}\left\lvert j+c_{1}\right\rangle_{t_{1}}$

-

$O(kn\cdot nm)=O(kmn^{2})$

-

$\mathcal{CNOT}:\left\lvert a\right\rangle\left\lvert b\right\rangle\rightarrow% -\left\lvert a\right\rangle\left\lvert b\oplus a\right\rangle$

-

$\displaystyle\rho^{Q}_{x}=\sum_{c\in S_{C}}\frac{1}{\left\lvert S_{C}\right% -\lvert}\rho^{Q}_{xc}$

-

$j^{\prime}-j\equiv\frac{D}{2}(\ \mathrm{mod}\ D)$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left% -\lvert j\right\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert jp_{i}% -\right\rangle_{t_{2}}$

-

$\mathcal{SUM}(b)_{h}:\left\lvert a\right\rangle_{h}\rightarrow\left\lvert a+b% -\right\rangle_{h}.$

-

$a^{(l+1)}=a^{(l)}+wa_{d-1-l}2^{d-l}\ \mathrm{mod}\ D$

-

$e^{\frac{\imath 2\pi}{D}}=\cos(\frac{2\pi}{D})+\imath\sin(\frac{2\pi}{D})$

-

$\left\lvert j\right\rangle_{t}$

-

$H(u)=m$

-

$H(X_{B})=(n+1)m$

-

$c_{3}\in\left[D\right]^{o}$

-

$p(X_{B}|\hat{M})=\frac{1}{N^{n}}$

-

$l2^{d-d_{3}}+1\leq w_{1}^{-1}w_{3}\leq(l+1)2^{d-d_{3}}-1,$

-

$1\leq i\leq k,1\leq j\leq n$

-

$\ \mathrm{mod}\ 2^{l}$

-

$\displaystyle Output=\begin{pmatrix}\mathbf{x}_{1}\cdot\mathbf{y}_{1}+v_{11}&% -\cdots&\mathbf{x}_{1}\cdot\mathbf{y}_{n}+v_{1n}\\ -\mathbf{x}_{2}\cdot\mathbf{y}_{1}+v_{21}&\cdots&\vdots\\ -\vdots&\ddots&\vdots\\ -\mathbf{x}_{k}\cdot\mathbf{y}_{1}+v_{k1}&\cdots&\mathbf{x}_{k}\cdot\mathbf{y}_% -{n}+v_{kn}\end{pmatrix}$

-

$1-k_{2}r_{3}\equiv 0(\ \mathrm{mod}\ D)$

-

$\displaystyle H(Z_{B}:X_{A})\leq\log_{2}{\left\lvert S_{x_{i}}\right\lvert}$

-

$Q=(t_{1},t_{2},g,e_{1},e_{2})$

-

$a^{(d)}=a^{(d-1)}+wa_{0}2^{0}\ \mathrm{mod}\ D=ab$

-

$g_{B}(j,x_{i},\mathbf{c})=j+c_{1}\parallel jp_{i}+c_{2}\parallel jc_{3}+c_{4}% -\parallel r_{1}^{-1}\parallel c_{1}-r_{2}r_{3},$

-

$\displaystyle=-\sum_{X_{B}\in\left[N\right]^{n+1}}\sum_{v_{1},v_{2},\cdots,v_{% -n-1}\in\left[N\right]}p(\hat{M}|X_{B})p(X_{B})\log_{2}{p(X_{B}|\hat{M})}$

-

$b\in\left[D\right]^{o}$

-

$\mathcal{ROT}(b_{1})$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left% -\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}$

-

$2^{d_{1}}$

-

$\displaystyle I_{B}=H(M:X_{B}|u)=H(X_{B}:M,u)-H(X_{B}:u)$

-

$\displaystyle H\left(Z_{A}:s_{i}|\rm{Alice\ passes\ the\ test}\right)$

-

$\omega^{bc}$

-

$O\left(2^{4m}\right)$

-

$\mathcal{MUL}$

-

$\displaystyle\leq 2^{d-d_{3}+d_{1}}-1-w_{1}^{-1}w_{3}\leq 2^{d-d_{3}+d_{1}}-1-% -l2^{d-d_{3}}-1$

-

$w_{2}\equiv w_{1}^{-1}w_{3}(\ \mathrm{mod}\ 2^{d-d_{3}})$

-

$\mathcal{ROT}_{h}=\prod_{i,k\in\left[d\right]}\mathcal{P}(i+k)^{b_{i}}_{h_{i}}$

- - - diff --git a/htmls/output_mathjax_example_10086.html b/htmls/output_mathjax_example_10086.html deleted file mode 100644 index 3a382270d524f437cc74810a77329848b3ded5ba..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10086.html +++ /dev/null @@ -1,187 +0,0 @@ - - - - MathJax Example - - - - -

$v_{n}=\frac{4v-4\sum_{i=1}^{n-1}{v_{i}}\ \mathrm{mod}\ D}{4}$

-

$d_{3} -

$\displaystyle p_{2b}=\Pr\left(w_{2}=w_{1}^{-1}w_{3}+k2^{d-d_{3}}\lvert d_{2}=d% -_{3}-d_{1}\right)$

-

$\displaystyle S\left(\rho^{Q}_{x}\right)=-\sum_{b\in\left[L\right]}\frac{% -\alpha_{bx}}{D\left\lvert S_{C}\right\lvert}\log_{2}\frac{\alpha_{bx}}{D\left% -\lvert S_{C}\right\lvert}$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert j\right\rangle_{t_{2}% -}\left\lvert j\right\rangle_{g}$

-

$t=\left(t_{d-1},t_{d-2},\cdots,t_{0}\right)$

-

$\left\lvert A\right\lvert$

-

$\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert j+r_{2}r% -_{3}\right\rangle_{g}$

-

$2^{d_{3}-d_{1}+1}$

-

$d=1+2=3$

-

$H(Z:X|F)$

-

$\sum_{b\in\left[L\right]}\alpha_{bx}=\left\lvert\left[D\right]\times S_{C}% -\right\lvert=D\left\lvert S_{C}\right\lvert$

-

$I_{A}=0$

-

$\displaystyle=\frac{1}{D}\sum_{j^{\prime},j\in\left[D\right]}\omega^{(j^{% -\prime}-j)h(x,c)}\left\lvert f(j^{\prime},x,c)\right\rangle\left\langle f(j,x,% -c)\right\lvert_{A}$

-

$I=H(Z:X)$

-

$p_{\left\lvert S_{J}\right\lvert}$

-

$\epsilon=\Theta(2^{-2m})$

-

$\displaystyle=\frac{4(v+\sum^{n}_{i=1}x_{i}y_{i}-a2^{m})}{4}=v+\sum^{n}_{i=1}x% -_{i}y_{i}-a2^{m},$

-

$\mathcal{BROT}_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right% -\rangle_{t}\rightarrow\omega^{ab}\left\lvert a\right\rangle_{h}\left\lvert b% -\right\rangle_{t}$

-

$l=0,1,\cdots,d-1$

-

$j=2^{d_{j}}w_{j}$

-

$f:\left[D\right]\times S_{X}\times S_{C}\to\left[2^{l_{1}}\right]$

-

$\mu(m_{X}):\mathbb{N}\to[0,1]$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert jp_{i}\right\rangle_{% -t_{2}}\left\lvert jc_{3}\right\rangle_{g}$

-

$\displaystyle=\sum_{d_{l}=1}^{d-1}\frac{2^{d_{l}}}{D}\cdot\frac{1}{2^{d_{l}}}% -\left(\log_{2}2^{d_{l}}-1\right)+\frac{2^{d}}{D}\cdot\frac{2}{2^{d}}\left(\log% -_{2}2^{d}-1\right)$

-

$\mathcal{SUM}(b\ \mathrm{mod}\ 2^{l})$

-

$\mathcal{XOR}$

-

$\alpha_{b{x_{i}}}=\beta_{b}=0$

-

$w_{1}^{-1}w_{3}$

-

$\displaystyle=\frac{1}{D}\sum_{d_{l}=1}^{d-1}(d_{l}-1)+\frac{2}{D}(d-1)=\frac{% -1}{D}\frac{(d-1)(d-2)+4(d-1)}{2}$

-

$\forall y_{i},v\in\left[N\right]$

-

$Output_{ij}=\mathbf{x}_{i}\cdot\mathbf{y}_{j}+v_{ij}$

-

$\mathcal{BSUM}$

-

$\displaystyle\overset{\mathcal{MUL}(k_{1})_{t_{1}}\mathcal{MUL}(k_{2})_{t_{2}}% -\mathcal{MUL}(k_{3})_{g}}{\longrightarrow}$

-

$\mathcal{SUM}(b)=\mathcal{QFT}^{\dagger}\mathcal{ROT}(b)\mathcal{QFT}$

-

$\displaystyle\left\lvert g(j^{\prime},x,c)\right\rangle\left\langle g(j,x,c)% -\right\lvert_{Q}.$

-

$\displaystyle=\sum_{i=0}^{d-1}a_{i}^{(l)}2^{i}+wa_{d-1-l}2^{d-l}\ \mathrm{mod}\ D$

-

$a_{ij},b_{ij}\in\left[N\right]$

-

$D|(1-k_{1}r_{3})\frac{D}{2}$

-

$\displaystyle=\frac{1}{D\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]% -}\sum_{g(j,x,c)=b}\left\lvert b\right\rangle\left\langle b\right\lvert_{Q}$

-

$k_{1}r_{3}$

-

$\displaystyle\left\lvert jc_{3}k_{3}+c_{4}k_{3}\right\rangle_{g}$

-

$\displaystyle\mathcal{QFT}^{\dagger}_{h}:\left\lvert a\right\rangle_{h}% -\rightarrow\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{-aj}\left\lvert j% -\right\rangle_{h}.$

-

$\displaystyle\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i}}=% -\sum_{s_{i}\in\left[D\right]^{o}}e^{\imath\pi s_{i}}=\sum_{s_{i}\in\left[D% -\right]^{o}}(-1)=-\frac{D}{2};$

-

$w_{1}\ \mathrm{mod}\ 2^{d-d_{3}}$

-

$\displaystyle I_{B}\leq\left(1-\frac{1}{2^{d-1}}\right)O\left(\frac{d^{2}}{2^{% -d}}\right)+\frac{1}{2^{d-1}}d\leq O\left(\frac{d^{2}}{2^{d}}\right).$

-

$\displaystyle\rho^{(P,Q)}_{xc}=\left\lvert\psi(x,c)\right\rangle\left\langle% -\psi(x,c)\right\lvert_{(P,Q)}$

-

$\displaystyle=H(M)-H(M,X_{B})+H(X_{B})-H(u)$

-

$d_{l}=d$

-

$b^{-1}\ \mathrm{mod}\ D$

-

$-l\leq k\leq 2^{d_{1}}-l-1$

-

$a^{(l)}=\sum_{i=0}^{d-1}a_{i}^{(l)}2^{i}$

-

$\displaystyle=\frac{1}{\left\lvert S_{X}\right\lvert}\sum_{x\in S_{X}}\frac{1}% -{\left\lvert S_{C}\right\lvert}\sum_{c\in S_{C}}\frac{1}{D}\sum_{j\in\left[D% -\right]}\left\lvert g(j,x,c)\right\rangle\left\langle g(j,x,c)\right\lvert_{Q}$

-

$\displaystyle\left\lvert j\left(k_{1}+p_{i}k_{2}+c_{3}k_{3}\right)+\left(c_{1}% -k_{1}+c_{2}k_{2}+c_{4}k_{3}\right)\right\rangle_{g}$

-

$p_{c}=\frac{1}{\left\lvert S_{C}\right\lvert}$

-

$d_{3}\geq d_{1}$

-

$\displaystyle p_{2a}=\Pr\left(d_{2}=d_{3}-d_{1}\right)$

-

$\displaystyle\cdots\mathcal{SUM}(w\ \mathrm{mod}\ 2^{2})^{h_{d-3}}\mathcal{SUM% -}(w\ \mathrm{mod}\ 2^{1})^{h_{d-2}},$

-

$\displaystyle\rho^{Q}=\sum_{x\in S_{X}}\frac{1}{\left\lvert S_{X}\right\lvert}% -\rho^{Q}_{x}$

-

$d_{2}=d_{3}-d_{1}$

-

$M=(M_{1},M_{2},\cdots,M_{n})$

-

$b=\sum_{k\in\left[d\right]}b_{k}2^{k}$

-

$\displaystyle\prod_{i,k\in\left[d\right]}\mathcal{P}(i+k)^{b_{i}}_{h_{i}}\left% -\lvert a\right\rangle_{h}=\prod_{i,k\in\left[d\right]}e^{\frac{\imath 2\pi 2^{% -i+k}a_{i}b_{i}}{D}}\left\lvert a\right\rangle_{h}$

-

$\displaystyle:X)\leq\log_{2}{\left\lvert S_{X}\right\lvert}-$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}% -\right\rangle_{t_{2}}\left\lvert jc_{3}+c_{4}\right\rangle_{g}$

-

$m,N=2^{m}$

-

$\mathcal{SUM}(c)$

-

$\displaystyle=\begin{pmatrix}\mathbf{x}_{1}\cdot\mathbf{y}_{1}&\mathbf{x}_{1}% -\cdot\mathbf{y}_{2}&\cdots&\mathbf{x}_{1}\cdot\mathbf{y}_{n}\\ -\mathbf{x}_{2}\cdot\mathbf{y}_{1}&\mathbf{x}_{2}\cdot\mathbf{y}_{2}&\cdots&% -\vdots\\ -\vdots&\vdots&\ddots&\vdots\\ -\mathbf{x}_{k}\cdot\mathbf{y}_{1}&\mathbf{x}_{k}\cdot\mathbf{y}_{2}&\cdots&% -\mathbf{x}_{k}\cdot\mathbf{y}_{n}\end{pmatrix}+\mathbf{V}$

-

$\left\lvert ja_{1}+c\right\rangle_{Q_{1}}$

-

$\displaystyle H(X_{B}|\hat{M})=-\sum_{X_{B}\in\left[N\right]^{n+1}}\sum_{\hat{% -M}\in 4\left[N\right]^{n}}p(X_{B},\hat{M})\log_{2}{p(X_{B}|\hat{M})}$

-

$H(M,u)=H(M)$

-

$C\in S_{C}$

-

$\displaystyle\cdots\overset{\mathcal{SUM}(2^{d-1}b)^{h_{d-1}}_{t}}{% -\longrightarrow}=\left\lvert a\right\rangle_{h}\left\lvert ab\right\rangle_{t},$

-

$\mathcal{QFT}^{\dagger}$

-

$a\ \mathrm{mod}\ D$

-

$\displaystyle=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ -a_{21}&a_{22}&\cdots&\vdots\\ -\vdots&\vdots&\ddots&\vdots\\ -a_{k1}&a_{k2}&\cdots&a_{kn}\end{pmatrix}\cdot\begin{pmatrix}b_{11}&b_{12}&% -\cdots&b_{1n}\\ -b_{21}&b_{22}&\cdots&\vdots\\ -\vdots&\vdots&\ddots&\vdots\\ -b_{k1}&b_{k2}&\cdots&b_{kn}\end{pmatrix}$

-

$I_{A}=H(Z_{B}:X_{A})=0$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}\left\lvert 0\right\rangle_{g% -}\rightarrow\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}\left\lvert jk_{1}\right% -\rangle_{g},$

-

$\left\{a\lvert g(a)=b,a\in A\right\}$

-

$I_{B}=H(Z_{A}:s_{i})=O\left(\frac{d^{2}}{2^{d}}\right)$

-

$\mathcal{ROT}(b)_{h}:\left\lvert a\right\rangle_{h}\rightarrow\omega^{ab}\left% -\lvert a\right\rangle_{h}.$

-

$\mathcal{ROT}_{h}$

-

$j(1-k_{2}r_{3})\equiv r_{4}(\ \mathrm{mod}\ D)$

-

$r_{1}=k_{1}+p_{i}k_{2}+c_{3}k_{3}$

-

$\left\{\begin{matrix}c_{1}=b_{1}-j\\ -c_{2}=b_{2}-jp_{i}\\ -jc_{3}+c_{4}=b_{3}\\ -k_{3}c_{3}=r_{3}^{-1}-k_{1}-p_{i}k_{2}\\ -\left(k_{1}-r_{3}^{-1}\right)c_{1}+k_{2}c_{2}+k_{3}c_{4}=-r_{4}r_{3}^{-1}\end{% -matrix}\right..$

-

$\displaystyle\left\lvert 0\right\rangle_{h}\left\lvert ab\right\rangle_{t}% -\overset{\mathcal{SWAP}_{(h,t)}}{\longrightarrow}\left\lvert ab\right\rangle_{% -h}\left\lvert 0\right\rangle_{t},$

-

$\mathbf{y}\in\left[N\right]^{n}$

-

$\displaystyle p_{\left\lvert S_{J}\right\lvert}=\Pr\left(2^{d_{l}}\lvert(1-k_{% -1}r_{3}),\ 2^{d_{l}+1}\nmid(1-k_{1}r_{3})\right)$

-

$a^{(l+1)}=\sum_{i=0}^{d-1}a_{i}^{(l+1)}2^{i}$

-

$a=2^{d_{1}}w_{1},b=2^{d_{2}}w_{2},c=2^{d_{3}}w_{3}$

-

$d_{r} -

$S_{J}=\{j|j(1-k_{1}r_{3})=r_{4},j\in\left[D\right]\}$

-

$X_{B}=(y_{i},v_{i})$

-

$\displaystyle\overset{\mathcal{MUL}(r_{3})_{g}}{\longrightarrow}\frac{1}{\sqrt% -{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right\rangle_{h}% -\left\lvert j+c_{1}\right\rangle_{t_{1}}$

-

$\displaystyle=\log_{2}{\left\lvert S_{X}\right\lvert}-$

-

$\left\lvert\phi_{j\pm}\right\rangle=\left\lvert j\right\rangle\pm\left\lvert j% -+\frac{D}{2}\right\rangle$

-

$\displaystyle=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}$

-

$p=p_{1}=p_{2}=\frac{1}{2^{d-d_{1}}}$

-

$\displaystyle\sum_{i=0}^{d-l-1}a_{i}^{(l+1)}2^{i}=\sum_{i=0}^{d-l-1}a_{i}^{(l)% -}2^{i}=$

-

$a_{1},a_{2}\in\left[D\right]^{o}$

- - - diff --git a/htmls/output_mathjax_example_10087.html b/htmls/output_mathjax_example_10087.html deleted file mode 100644 index bb3f78d35c3213322c75f7f0bc29781de0824ae8..0000000000000000000000000000000000000000 --- a/htmls/output_mathjax_example_10087.html +++ /dev/null @@ -1,175 +0,0 @@ - - - - MathJax Example - - - - -

$\left\{0,1,\cdots,N-1\right\}$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{js_{i}}\left% -\lvert j\right\rangle_{t_{1}}\left\lvert jk_{1}\right\rangle_{g}\rightarrow$

-

$\displaystyle ab\equiv a(2w+1)\equiv a+2w\sum_{i=0}^{d-1}2^{i}a_{i}\equiv a+% -\sum_{i=1}^{d}2^{i}wa_{i-1}(\ \mathrm{mod}\ 2^{d})$

-

$bra{j(k_{1}+k_{2}a_{1}+k_{3}a_{2})}_{g}\to\left\lvert j\right\rangle_{g}$

-

$\forall\hat{M_{i}}\in 4\left[N\right],y_{i}\in\left[N\right]$

-

$\mathbf{y},v$

-

$d=O(m)$

-

$\displaystyle\leq S\left(\rho\right)-\sum_{s_{i}\in\left[D\right]^{o}}p_{s_{i}% -}S\left(\rho_{s_{i}}\right)=\log_{2}\left\lvert S_{J}\right\lvert-1$

-

$d,D=2^{d}$

-

$\left\{\begin{matrix}j+c_{1}=b_{1}\\ -jp_{i}+c_{2}=b_{2}\\ -jc_{3}+c_{4}=b_{3}\\ -\left(k_{1}+p_{i}k_{2}+c_{3}k_{3}\right)^{-1}=r_{3}\\ -c_{1}-\left(k_{1}c_{1}+k_{2}c_{2}+k_{3}c_{4}\right)r_{3}=r_{4}\end{matrix}% -\right.,$

-

$\mathbf{U}=\mathbf{A}\cdot\mathbf{B}+\mathbf{V}$

-

$\left\{\begin{matrix}c_{1}=b_{1}-j\\ -c_{2}=b_{2}-jp_{i}\\ -c_{3}=k_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\\ -c_{4}=b_{3}-jk_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\end{matrix}% -\right..$

-

$\mathcal{XOR}_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right\rangle% -_{t}\rightarrow\left\lvert a\right\rangle_{h}\left\lvert b\oplus a\right% -\rangle_{t}.$

-

$\mathcal{P}(i+k)^{b_{i}}_{h_{i}}$

-

$\displaystyle=\log_{2}\left(D\left\lvert S_{X}\right\lvert\left\lvert S_{C}% -\right\lvert\right)-\frac{1}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}% -\right\lvert}\sum_{b\in\left[L\right]}\beta_{b}\log_{2}\alpha_{b}-$

-

$Im(g)$

-

$b\in\left[D\right]$

-

$H(A,B),H(A:B),$

-

$\displaystyle=\Pr\left(j^{\prime}=(k_{2}r_{3})^{-1}(j(1-k_{1}r_{3})-r_{4})% -\right)=\frac{1}{D}.$

-

$I_{A},I_{B}$

-

$\mathcal{BSUM}_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right% -\rangle_{t}\rightarrow\left\lvert a\right\rangle_{h}\left\lvert b+a\right% -\rangle_{t}.$

-

$L=2^{l_{2}}$

-

$\displaystyle\rho^{Q}_{xc}=tr_{P}\left(\rho^{(P,Q)}_{xc}\right)$

-

$j,ja_{1}$

-

$I_{A}=H(Z_{B}:X_{A})$

-

$v_{1},v_{2},\cdots,v_{n-1}\in\left[N\right]$

-

$r_{4}=-r_{3}\left[\left(k_{1}-r_{3}^{-1}\right)b_{1}+k_{2}b_{2}+k_{3}b_{3}% -\right],$

-

$1\leq w_{1}^{-1}w_{3}+k2^{d-d_{3}}\leq 2^{d-d_{3}+d_{1}}-1$

-

$d_{1}+d_{2}=d_{3}$

-

$p_{i}=2x_{i}+1$

-

$v+\sum^{n}_{i=1}x_{i}y_{i}-a2^{m}\in\left[\frac{D}{4}\right]=\left[N\right]$

-

$Z_{A},Z_{B}$

-

$d=m+2$

-

$4v_{n}\equiv 4v-4\sum_{i=1}^{n-1}v_{i}(\ \mathrm{mod}\ D)$

-

$w_{1}w_{2}\equiv w_{3}(\ \mathrm{mod}\ 2^{d-d_{3}})$

-

$a\equiv b(\ \mathrm{mod}\ D)$

-

$r=(k_{1}+k_{2}a_{1}+k_{3}a_{2})^{-1}$

-

$\mathcal{U}_{\varepsilon}$

-

$t_{1},t_{2},g$

-

$k_{1}+p_{i}k_{2}=r_{3}^{-1}$

-

$\omega^{ja_{2}b_{2}}$

-

$j(1-k_{1}r_{3})=r_{4}$

-

$poly(m_{X})$

-

$\displaystyle\mathbf{y}_{j}=\left(b_{1j},b_{2j},\cdots,b_{nj}\right).$

-

$\displaystyle=\frac{1}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right% -\lvert}\sum_{b\in\left[L\right]}\beta_{b}\left\lvert b\right\rangle\left% -\langle b\right\lvert_{Q}.$

-

$\displaystyle=\log_{2}\left(D\left\lvert S_{X}\right\lvert\left\lvert S_{C}% -\right\lvert\right)-\frac{1}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}% -\right\lvert}\sum_{b\in\left[L\right]}\beta_{b}\log_{2}\alpha_{b},$

-

$d_{1},d_{2},d_{3}\in\left[d+1\right],w_{i}\in\left[2^{d-d_{i}}\right]^{o}$

-

$\forall(j,x_{i})\in\left[D\right]\times\left[N\right]$

-

$d_{r}\geq d_{l}$

-

$\displaystyle=\frac{1}{D}\sum_{j^{\prime},j\in\left[D\right]}\omega^{(j^{% -\prime}-j)h(x,c)}\delta_{j^{\prime}j}\left\lvert g(j^{\prime},x,c)\right% -\rangle\left\langle g(j,x,c)\right\lvert_{Q}$

-

$f(j,x,c),g(j,x,c)$

-

$t_{1},g$

-

$1-k_{2}r_{3}\not\equiv 0(\ \mathrm{mod}\ D)$

-

$d_{1}+d_{2}\geq d_{3}$

-

$\mathcal{U}_{\varepsilon}:\left\lvert j\right\rangle_{t}\left\lvert 0\right% -\rangle_{e}\rightarrow\left\lvert j\right\rangle_{t}\left\lvert\varepsilon(j)% -\right\rangle_{e}$

-

$\rho=\sum_{y_{i}\in\left[N\right]}\frac{1}{N}\rho_{y}$

-

$\displaystyle H(Z$

-

$\frac{\left\lvert S_{J}\right\lvert}{D}$

-

$4v\equiv 4\sum_{i=1}^{n}v_{i}(\ \mathrm{mod}\ D)$

-

$\displaystyle=\frac{1}{\left\lvert S_{C}\right\lvert}\sum_{c\in S_{C}}\frac{1}% -{D}\sum_{j\in\left[D\right]}\left\lvert g(j,x,c)\right\rangle\left\langle g(j,% -x,c)\right\lvert_{Q}$

-

$\displaystyle-(l+1)2^{d-d_{3}}+1\leq-w_{1}^{-1}w_{3}\leq k2^{d-d_{3}}$

-

$\displaystyle I_{B}\leq\sum_{\left\lvert S_{J}\right\lvert}\frac{\left\lvert S% -_{J}\right\lvert}{D}p_{\left\lvert S_{J}\right\lvert}\log_{2}\left\lvert S_{J}% -\right\lvert-1$

-

$\displaystyle\overset{\mathcal{XOR}_{(h,t_{1})}\mathcal{XOR}_{(h,t_{2})}}{% -\longrightarrow}\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}% -\left\lvert j\right\rangle_{h}$

-

$\displaystyle\overset{\mathcal{SUM}(c_{1})_{t_{1}}\mathcal{SUM}(c_{2})_{t_{2}}% -\mathcal{SUM}(c_{4})_{g}}{\longrightarrow}\frac{1}{\sqrt{D}}\sum_{j\in\left[D% -\right]}\left\lvert j\right\rangle_{h}$

-

$(t_{1},t_{2},g)$

-

$O\left(nm^{2}\right)$

-

$w_{3}+k2^{d-d_{3}}$

-

$\displaystyle\mathcal{MUL}(b)=\mathcal{SUM}(w\ \mathrm{mod}\ 2^{d-1})^{h_{0}}% -\mathcal{SUM}(w\ \mathrm{mod}\ 2^{d-2})^{h_{1}}$

-

$\left\lvert a_{d-1}\right\rangle\cdots\left\lvert a_{d-l}\right\rangle\ % -\mathrm{mod}\ 2^{l}$

-

$\forall j^{\prime},j\in\left[D\right],\left\langle f(j^{\prime},x,c)\lvert f(j% -,x,c)\right\rangle=\delta_{j^{\prime}j}$

-

$\sum_{b\in\left[L\right]}\beta_{b}=\left\lvert\left[D\right]\times S_{X}\times -S% -_{C}\right\lvert=D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert$

-

$\mathcal{ROT}$

-

$\beta_{b}=\left\lvert g^{-1}(b)\right\lvert=\left\lvert\left\{(j,x,c)\lvert g(% -j,x,c)=b\right\}\right\lvert$

-

$\frac{1}{2^{d-1}}$

-

$I_{A}=H(Z_{A}:X_{B})=0$

-

$\displaystyle=\sum_{b\in\left[L\right]}\frac{\alpha_{bx}}{D\left\lvert S_{C}% -\right\lvert}\left(\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\log_{2% -}\alpha_{bx}\right)$

-

$\forall u\in\left[N\right]$

-

$q_{i},s_{i}$

-

$\displaystyle\left\lvert\psi(x,c)\right\rangle_{(P,Q)}=$

-

$\displaystyle\frac{1}{D}\sum_{j,j^{\prime}\in\left[D\right]}\left\lvert jk% -\right\rangle_{t_{1}}\left\lvert j^{\prime}\right\rangle_{t_{2}}\left\lvert jk% -_{1}+j^{\prime}k_{2}\right\rangle_{g}.$

-

$a=\sum_{i\in\left[d\right]}a_{i}2^{i}$

-

$p_{i}=k_{2}^{-1}\left(r_{3}^{-1}-k_{1}\right)$

-

$O\left(\sqrt{2^{m}}\right)$

-

$p_{x}=\frac{1}{\left\lvert S_{X}\right\lvert}$

-

$\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right% -\rangle_{h}\overset{\mathcal{QFT}^{\dagger}_{h}}{\longrightarrow}\left\lvert M% -_{i}\right\rangle_{h}\ \mathrm{mod}\ D,$

-

$\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right% -\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}% -\right\rangle_{t_{2}}\left\lvert jr_{1}+r_{2}\right\rangle_{g}$

-

$\mathbf{c}=(c_{1},c_{2},c_{3},c_{4})$

-

$\displaystyle H(Z:X)\leq S\left(\rho^{B}\right)-\sum_{x\in S_{X}}p_{x}S\left(% -\rho^{B}_{x}\right)$

-

$\displaystyle\ \ \ \ +2^{d-l}\left(\sum_{i=0}^{l-1}a_{i+d-l}^{(l)}2^{l}+wa_{d-% -1-l}^{(l)}\ \mathrm{mod}\ 2^{l}\right),$

-

$\displaystyle\left\lvert jr_{1}+r_{2}\right\rangle_{g},$

-

$\displaystyle\overset{\mathcal{MUL}(p_{i})_{t_{2}}\mathcal{MUL}(c_{3})_{g}}{\longrightarrow}$

-

$\mathcal{CNOT}$

-

$\displaystyle=\frac{1}{D}\sum_{j^{\prime},j\in\left[D\right]}\omega^{(j^{% -\prime}-j)h(x,c)}tr\left(\left\lvert f(j^{\prime},x,c)\right\rangle\left% -\langle f(j,x,c)\right\lvert_{P}\right)$

-

$\displaystyle=\mathcal{ROT}_{h}\left\lvert a\right\rangle_{h}.$

-

$u=\mathbf{x}\cdot\mathbf{y}+v\ \mathrm{mod}\ N$

-

$\displaystyle=\frac{1}{2^{d_{3}-d_{1}+1}}\cdot\frac{1}{2^{d-d_{3}-1}}=\frac{1}% -{2^{d-d_{1}}}=p_{1}.$

-

$p(u)=\frac{1}{N}$

-

$|\psi(\theta_{h},n)\rangle$

-

$\displaystyle\langle\omega(\theta_{h},5)|Z_{P(size)}|\omega(\theta_{h},5)\rangle,$

-

$\displaystyle W^{(127)}_{10}$

- - -