image imagewidth (px) 4 2.95k | task_type stringclasses 17
values | answers stringlengths 1 2.94k |
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formulas_benchmark_complex_outline | 0= \int_{0}^{T}\int_{\mathbb{R}^{d}}\{-u_{0}(\hat{Y}^{x,n}_{t}( \omega))\rho^{{}^{\prime}}(t)\eta(x)\] \[+b_{n}(t,x+B^{H}_{t}(\omega))\cdot(\nabla u_{0})(\hat{Y}^{x,n}_{ t}(\omega))^{T}\frac{\partial}{\partial x}\hat{Y}^{x,n}_{t}(\omega)\rho(t)\eta(x)\} dxdt\text{, for a.a. }\omega.\] \[\Big{(}\left(t,x\right)\mapsto\h... | |
formulas_benchmark_complex_outline | \|[e^{it\partial_{x}^{2}}\tilde{w}_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{ t}}}\] \[\leq \|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}+ \sum_{j=1}^{M}\|[e^{it\partial_{x}^{2}}(-\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{ j}+e^{-it_{n}^{j}\partial_{x}^{2}}\phi^{j})](0)\|_{L^{q}_{\mathbb{R}_{t}}}\] \[\leq \|[e^... | |
formulas_benchmark_complex_outline | \frac{1}{2\tau}\left(\|\Phi^{k+1}-\Phi^{*}\|^{2}-\|\Phi^{k}-\Phi^{* }\|^{2}\right)\] \[\leq h\sum_{i=1}^{N-1}\Phi_{i}-h\sum_{i=0}^{N-1}(\Phi_{i})^{2}+ah \frac{\Phi_{m+1}(\Phi_{m+2}-\Phi_{m+1})}{\Phi_{m+2}-\Phi_{m+1}}+ah(1-2\Phi_{m+1})\] \[= h\sum_{i=1}^{N-1}\Phi_{i}+ah-ah\Phi_{m+1}-h\sum_{i=1}^{N-1}(\Phi _{i})^{2}\] \[... | |
formulas_benchmark_complex_outline | \mathbb{E}[H_{s}(\lambda_{t}-\mu)]-\mathbb{E}[H_{s}](\mathbb{E}[ \lambda_{t}]-\mu)\] \[=\mu\int_{0}^{s}\Phi(t-v)\left(1+\int_{v}^{s}\Psi(y-v)dy\right) \left(1+\int_{0}^{v}\Psi(w)dw\right)dv\] \[+\mu\int_{0}^{t}\int_{0}^{s\wedge u}\Phi(t-u)\Psi(u-v)\left(1+ \int_{0}^{v}\Psi(w)dw\right)\left(1+\int_{v}^{s}\Psi(y-v)dy\rig... | |
formulas_benchmark_complex_outline | z_{r}(\bm{Q}^{\text{RS}}) =\frac{1}{2}\sum_{\sigma^{0},\cdots,\sigma^{r}\in\{\pm 1\}}\exp \Big{[}\beta\lambda b\sum_{a=1}^{r}\sigma^{0}\sigma^{a}+\beta^{2}q\sum_{1\leq a <b\leq r}\sigma^{a}\sigma^{b}\Big{]}\] \[=\sum_{\sigma^{1}\cdots,\sigma^{r}\in\{\pm 1\}}\exp\Big{[}\beta \lambda b\sum_{a=1}^{r}\sigma^{a}+\frac{\beta... | |
formulas_benchmark_complex_outline | g\left(n_{\Sigma_{\tau}},n_{\Sigma_{\tau}}\right) =\begin{cases}-1-\dfrac{2Mr(r^{2}+a^{2})}{\Delta\rho^{2}}+\dfrac {\Delta}{\rho^{2}}\left(\dfrac{df}{dr}\right)^{2}\,,&r\geq 9M/4\\ -1-\dfrac{2Mr}{\rho^{2}}-\dfrac{4Mr}{\rho^{2}}\dfrac{df}{dr}+\dfrac{\Delta}{ \rho^{2}}\left(\dfrac{df}{dr}\right)^{2}\,,&r\leq 15M/8\end{ca... | |
formulas_benchmark_complex_outline | \left(\begin{array}{c}Z\\ \Phi\end{array}\right)_{t} =-\mathbf{A}_{kl}\left(\begin{array}{c}Z\\ \Phi\end{array}\right)+2\sum_{i,j=1}^{N_{kl}}\lambda^{kl}_{j}\left\langle \left(\begin{array}{c}\mathcal{L}_{kl}\mathbb{D}^{1}_{\gamma^{kl}_{i}}(\psi^{kl }_{i})\\ \mathbb{D}^{2}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\end{array}\ri... | |
formulas_benchmark_complex_outline | \frac{\mathrm{d}}{\mathrm{d}t}\|g\|_{\dot{H}^{\frac{1}{2}}}^{2} +C(M)\bigg{[}\log\left(4+||f_{1}||_{2,\frac{1}{3}}\right)^{-\frac{1 }{3}}+\log\left(4+||f_{2}||_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\bigg{]}\,\|g\|_ {\dot{H}^{1}}^{2}\] \[\lesssim_{M}\log\left(4+\|f_{1}\|_{2,\frac{1}{3}}\right)^{-\frac{ 1}{3}}\|f_{1}\|_{2... | |
formulas_benchmark_complex_outline | (A_{H,2})_{i,j} = \int_{\Omega}\zeta(u_{H}+\alpha\widetilde{u}_{h})^{2}\phi_{i,H} \phi_{j,H}d\Omega\] \[= \int_{\Omega}\zeta\big{(}(u_{H})^{2}+2\alpha u_{H}\widetilde{u}_ {h}+\alpha^{2}(\widetilde{u}_{h})^{2}\big{)}\phi_{i,H}\phi_{j,H}d\Omega\] \[= \int_{\Omega}\zeta(u_{H})^{2}\phi_{i,H}\phi_{j,H}d\Omega+2\alpha \int_{... | |
formulas_benchmark_complex_outline | \begin{split}&\hat{\mathbb{E}}[(\int_{0}^{T}n|(Y_{s}^{n})^{-}|^{ \alpha-1}(Y_{s}^{m})^{-}ds)^{\alpha^{\prime}}]\\ &\leq\hat{\mathbb{E}}[\sup_{s\in[0,T]}\{|(Y_{s}^{n})^{-}|^{( \alpha-2)\alpha^{\prime}}|(Y_{s}^{m})^{-}|^{\alpha^{\prime}}\}(\int_{0}^{T}n(Y _{s}^{n})^{-}ds)^{\alpha^{\prime}}]\\ &\leq(\hat{\mathbb{E}}[\sup_... | |
formulas_benchmark_complex_outline | \frac{\mathrm{d}}{\mathrm{d}s}\mathcal{Q}_{\bm{\eta}}^{2}\big{(} \dot{\gamma}(s)\big{)} =2\mathrm{tr}\big{\{}\ddot{\gamma}(s)^{\mathsf{T}}\dot{\gamma}(s) \big{\}}+\sum_{j=1}^{P}\eta_{j}\big{[}\mathrm{tr}\big{\{}\ddot{\gamma}(s)^{ \mathsf{T}}\bm{\Pi}_{j1}\dot{\gamma}(s)\bm{\Pi}_{j2}\big{\}}+\mathrm{tr}\big{\{} \dot{\gam... | |
formulas_benchmark_complex_outline | \mathrm{E}\,U_{1h}(x)^{m} =\mathrm{E}\,[(a+b)^{m}]\] \[=\mathrm{E}\,\Big{[}\sum_{k=0}^{m}\binom{m}{k}a^{m-k}b^{k}\Big{]}\] \[=\mathrm{E}\,\Big{[}a^{m}+b^{m}\Big{]}\qquad\text{because}\qquad a ^{m-k}b^{k}=0\;\forall\;k\in\{1,\ldots,m-1\}\] \[=\mathrm{E}\,\Big{[}t^{m}\,\bigg{(}\frac{1}{q(x)}\frac{1}{h^{2}} \,w^{\prime}\B... | |
formulas_benchmark_complex_outline | \int_{\mathbb{R}^{d}}\|\nabla V\|^{4}\,\mathrm{d}\rho,\quad\int_ {\mathbb{R}^{d}}(\Delta V)^{2}\,\mathrm{d}\rho,\quad\int_{\mathbb{R}^{d}}( \Delta\log\rho)^{2}\,\mathrm{d}\rho,\quad\int_{\mathbb{R}^{d}}\|\nabla\log \rho\|^{4}\,\mathrm{d}\rho,\] \[\int_{\mathbb{R}^{d}}\Big{\|}\nabla^{2}\log\rho(y)\nabla\frac{ \delta\mat... | |
formulas_benchmark_complex_matrix | \sum_{k=1}^{m}\left(\Sigma\left(\frac{\partial}{\partial Z(t)_{k} }\begin{pmatrix}0&\mathbf{GC}_{XY}\\ \mathbf{C}_{XY}{}^{\prime}\mathbf{G}&0\end{pmatrix}\right)Z(t)\right)_{k}\] \[= \frac{1}{T}\sum_{k=1}^{n}\mathbf{e}^{\prime}_{k}\mathcal{A}\left( z^{2}\mathbf{Ge}_{k}Y(t)^{\prime}\widetilde{\mathbf{G}}Y(t)+\mathbf{GC}... | |
formulas_benchmark_complex_matrix | \sum_{\begin{subarray}{c}p_{1},p_{2}\in[P_{1},P_{2}]\\ p_{1}\equiv p_{2}\equiv 1\ (\text{mod}\ D_{f})\end{subarray}}\frac{1}{p_{1}p_{2}} \sum_{\begin{subarray}{c}\mathbf{b}_{1}\in\mathbb{Z}^{4}\cap\mathcal{C}_{1}, \mathbf{b}_{2}\in\mathbb{Z}^{4}\cap\mathcal{C}_{2}\\ \mathbf{b}_{1}\equiv\mathbf{b}_{2}\equiv\mathbf{b}_{0... | |
formulas_benchmark_complex_matrix | \begin{array}{lcl}&\alpha^{n}\oint_{\infty}\frac{dz}{2i\pi}z^{-n-1}e^{-\sum _{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\tau_{u}( \boldsymbol{t}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G)\tau_{uq^{-n}t^{-1} }(\boldsymbol{t}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|G) \\ &+\oint... | |
formulas_benchmark_complex_outline | \mathcal{L}_{c}(p^{k+1},q^{k},\nu^{k+1})-\mathcal{L}_{c}(p^{k+1}, q^{k+1},\nu^{k+1})\] \[= G(q^{k})-G(q^{k+1})+\langle\nu^{k+1},Bq^{k+1}-Bq^{k}\rangle+\frac {c}{2}\|Ap^{k+1}-Bq^{k}\|^{2}-\frac{c}{2}\|Ap^{k+1}-Bq^{k+1}\|^{2}\] \[\geq \langle\nabla G(q^{k+1})-B^{T}\nu^{k+1},q^{k}-q^{k+1}\rangle- \frac{\omega_{G}}{2}\|Bq^... | |
formulas_benchmark_complex_outline | (\mathrm{D}) =\int_{B_{R}^{d}}\left|1-\frac{\pi_{\mathrm{loc}}(y)\overline{\pi }(x)}{\pi_{\mathrm{loc}}(x)\overline{\pi}(y)}\right|\frac{\overline{\pi}(y) \widetilde{Q}(y,x)}{\overline{\pi}(x)}\,\mathrm{d}y\] \[\leq(\exp(2\widetilde{\varepsilon})-1)\int_{B_{R}^{d}}\frac{ \overline{\pi}(y)\widetilde{Q}(y,x)}{\overline{\... | |
formulas_benchmark_complex_table | \begin{table}
\begin{tabular}{l|l l l} & **Performance** & **Offline time** & **Online time** \\ & **score** & (seconds) & (seconds) \\ \hline
**MPC** & 0.487 & - & 9.82 \(10^{-4}\) \\ \hline
**OLFC-10** & 0.506 & - & 1.14 \(10^{-2}\) \\ \hline
**OLFC-50** & 0.513 & - & 8.62 \(10^{-2}\) \\ \hline
**OLFC-100** & 0.510... | |
formulas_benchmark_complex_outline | \|N(\varphi)e^{i\mu\psi}\|_{\dot{H}^{\delta}}\] \[\lesssim\||D|^{\delta-1}((\nabla\varphi)\psi e^{i\mu\psi})\|_{L^{2} }+\||D|^{\delta-1}(\varphi(\nabla\psi)e^{i\mu\psi})\|_{L^{2}}+|\mu|\||D|^{ \delta-1}(\varphi\psi(\nabla\psi)e^{i\mu\psi})\|_{L^{2}}\] \[\lesssim\||D|^{\delta-1}\nabla\varphi\|_{L^{2}}\|\psi e^{i\mu \psi... | |
formulas_benchmark_complex_outline | \prod_{x\in X/S}F\left(1+\sum_{\underline{i}\in I}X_{\underline{ i}}\underline{t}^{\underline{i}}\right)_{x} = \prod_{x\in X/S}\left(1+\sum_{\underline{i}\in I}X_{\underline{ i},x}\prod_{k=1}^{n}s_{1}^{a_{k,1}i_{k}}\ldots s_{m}^{a_{k,m}i_{k}}\right)\] \[= \prod_{x\in X/S}\left(1+\sum_{\underline{i}\in I}X_{\underline{i... | |
formulas_benchmark_complex_table | \begin{table}
\begin{tabular}{|l||l|l|l|l|l|l|l||l|} \hline Objects & A & B & C & D & E & F & G & Results \\ \hline
1 & 2 & 2 & 2 & 3 & 2 & 4 & 3 & 1 \\ \hline
2 & 3 & 3 & 2 & 3 & 2 & 5 & 4 & 1 \\ \hline
3 & 3 & 3 & 1 & 4 & 2 & 4 & 4 & 1 \\ \hline
4 & 3 & 3 & 1 & 4 & 2 & 4 & 4 & 1 \\ \hline
5 & 2 & 2 & 1 & 3 & 2 & 5 & ... | |
formulas_benchmark_complex_outline | \Pi_{\bar{\mathcal{B}}}(\mathbf{t}_{N\backslash\{i\}}^{*},\mathbf{ s}_{N\backslash\{i\}}^{*}) =\mathbb{E}[V(D,\mathbf{s}_{N\backslash\{i\}}^{*}(D))-\sum_{j\in N ^{*}(\mathcal{C})}(\bar{P}_{j}(\mathbf{s}_{N\backslash\{i\}}^{*}(D))+\bar{R}_{ j}(\mathbf{t}_{N\backslash\{i\}}^{*}))\] \[\quad-\sum_{j\notin N^{*}(\mathcal{C}... | |
formulas_benchmark_complex_outline | M \lesssim\sum_{k=0}^{d-2}\frac{\rho^{k+2}}{\varepsilon}\int_{ \tilde{H}_{\epsilon_{1}}}...\int_{\tilde{H}_{\epsilon_{d-k}}}\ 1\wedge\left(\frac{\rho}{|\sum_{i=1}^{d-k}(x_{i}-y_{i})|} \right)^{d-k-\frac{3}{4}}d\nu_{k,\sum_{i=d-k+1}^{d}x_{i}}^{1}(y_{d-k},...,y_{ 1})\] \[+\sum_{k=0}^{d-2}\rho^{k+1}\mathbf{1}_{|\sum_{i=1}... | |
formulas_benchmark_complex_outline | \begin{split}\mathrm{(i)}&\sum\nolimits_{j}\int_{ \widetilde{\Omega}_{j}^{\eta,\gamma}}\big{|}\mathrm{sym}\big{(}(R_{j}^{\eta, \gamma})^{T}\nabla y-\mathrm{Id}\big{)}\big{|}^{2}\leq C_{0}\big{(}1+C_{\eta} \gamma^{-5d/q}\varepsilon\big{)}\int_{\Omega\setminus\overline{E}} \operatorname{dist}^{2}(\nabla y,SO(d))\,,\\ \ma... | |
formulas_benchmark_complex_outline | \Big{|}\mathcal{F}_{\alpha}(f)(m)\Big{|}= \Big{|}\frac{A_{\alpha}^{n}e_{\alpha}(m)}{(2\pi im_{j}\csc\alpha )^{s}}\int_{\mathbf{T}_{\alpha}^{n}}\partial_{j}^{s}(e_{\alpha}f)(x)e^{-2\pi im \cdot x\csc\alpha}dx\Big{|}\] \[\leq \Big{(}\frac{\sqrt{n}}{2\pi|\csc\alpha|m|}\Big{)}^{s}\frac{|A_{ \alpha}^{n}|}{2|\csc\alpha|^{n}}... | |
formulas_benchmark_complex_outline | a_{5}(15n+6,10,12,13)\equiv 0\pmod{3},\] \[a_{7}(21n+3,8,11,15,17,18)\equiv 0\pmod{3},\] \[a_{11}(33n+3,11,12,20,24,26,27,29,30,32)\equiv 0\pmod{3},\] \[a_{13}(39n+3,7,9,10,15,16,18,22,28,31,33,36)\equiv 0\pmod{ 3},\] \[a_{17}(51n+10,14,16,19,20,23,25,26,28,34,35,38,41,46,47,49)\equiv 0 \pmod{3},\] \[a_{19}(57n+7,14,16... | |
formulas_benchmark_complex_table | \begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \(i\) & \(d_{i}\) & \([r_{1}d_{i}]\) & \((r_{1},d_{i})\) & \((a_{r_{1}},u_{d_{i}})_{V}\) \\ \hline \(1\) & \((6,7,8)(9,10,11)\) & \(2^{2}\cdot 3^{2}\) & \(3\times 3\) & \(0\) \\ \hline \(2\) & \((4,5,6)(9,10,11)\) & \(2\cdot 3\cdot 4\) & \(3\times S_{4}\) & \(\frac{1}{3... | |
formulas_benchmark_complex_outline | \sum_{j_{1},j_{2},j_{3},j_{4}} \delta_{i_{1}j_{1}}\delta_{i_{2}j_{2}}\delta_{i_{3}j_{3}}\delta_ {i_{4}j_{4}}\mathbb{E}[Y_{j_{1}}Y_{j_{2}}Y_{j_{3}}Y_{j_{4}}]\] \[=\sum_{j_{1},j_{2},j_{3}}\delta_{i_{1}j_{1}}\delta_{i_{2}j_{2}} \delta_{i_{3}j_{3}}\left(\sum_{j_{4}\in\{j_{1},j_{2},j_{3}\}}\delta_{i_{4}j_{4 }}\mathbb{E}[Y_{... | |
formulas_benchmark_complex_matrix | \sum_{\begin{subarray}{c}q\leqslant Q\\ q(q,a_{1}a_{2})=1\\ q\equiv c_{0}\,(\text{mod}\ c)\end{subarray}}\sum_{\begin{subarray}{c}n\in \underline{\times}\\ n\equiv a_{1}\overline{a_{2}}\,(\text{mod}\ d)\\ n\equiv d_{0}\,(\text{mod}\ d)\end{subarray}}\Lambda(n)=\sum_{\begin{subarray}{ c}q\leqslant Q\\ (q,a_{1}a_{2})=1\\... | |
formulas_benchmark_complex_outline | ||\alpha_{\delta}^{1,0}-\alpha_{\delta^{*}}^{1,0}||\leq||\Delta_{ \delta}^{0,0}-\Delta_{\delta^{*}}^{0,0}||\] \[+\beta||\sum_{s^{\prime}}P(s^{\prime}|s,\delta^{*}(\zeta))\alpha_{ \delta^{*}}^{1,0}(\bar{\zeta}(s^{\prime},s_{2}^{L^{\prime}},s_{1}^{L^{\prime}}, \zeta,\delta^{*}(\zeta)))-\sum_{s^{\prime}}P(s^{\prime}|s,\de... | |
formulas_benchmark_complex_outline | \left|t_{P,H}(z)-\widetilde{t}_{P,H}(z)\right| =\left|\sum_{i=1}^{\infty}\alpha_{P,i}z^{M_{i}}-\sum_{i=1}^{ \infty}\sum_{H^{\prime}\in\mathcal{H}^{\prime}}\alpha_{P,H^{\prime},i}z^{M_{i }}\right|\] \[\leq\sum_{i=1}^{\infty}\left|\alpha_{P,i}-\sum_{H^{\prime}\in \mathcal{H}^{\prime}}\alpha_{P,H^{\prime},i}\right|\cdot\l... | |
formulas_benchmark_complex_outline | \beta_{0}^{l} =\frac{1}{4}(W_{i-2}-4W_{i-1}+3W_{i})^{2}+\frac{13}{12}(W_{i-2}-2 W_{i-1}+W_{i})^{2},\] \[\beta_{1}^{l} =\frac{1}{36}(W_{i-2}-6W_{i-1}+3W_{i}+2W_{i+1})^{2}+\frac{13}{12} (W_{i-1}-2W_{i}+W_{i+1})^{2}\] \[+\frac{1043}{960}(-W_{i-2}+3W_{i-1}-3W_{i}+W_{i+1})^{2}\] \[+\frac{1}{432}(W_{i-2}-6W_{i-1}+3W_{i}+2W_{... | |
formulas_benchmark_complex_outline | \alpha_{\xi(n)}\beta_{\xi(n)} = 1+\frac{bc}{a}\left(\frac{a}{b}\right)^{2\xi(n)}-\frac{bc^{3}}{a }\left(\frac{a}{b}\right)^{2\xi(n)}-\left(\frac{a}{b}\right)^{\xi(n)}bc\] \[+\left(b\left(\frac{a}{b}\right)^{\xi(n)}+\left(\frac{a}{b}\right) ^{2\xi(n)}\frac{bc}{a}\Delta\right)i\] \[+\left(\left(ab+2c\right)+\frac{bc}{a}\... | |
formulas_benchmark_complex_table | \begin{table}
\begin{tabular}{c c c c c c c} \hline n & r & CMILP+MTZ & CMILP+MTZ+2Clq & CMILP+DL & CMILP+SC & CMILP+2C \\ \hline
3 & 5 & 0.43 & 0.78 & 0.46 & 0.48 & 0.50 \\
3 & 10 & 0.71 & 0.78 & 0.72 & 0.88 & 1.10 \\
3 & 15 & 2.13 & 1.88 & 1.88 & 2.32 & 2.63 \\
3 & 20 & 25.96 & 27.39 & 22.46 & 4.82 & 5.89 \\
3 & 25 &... | |
formulas_benchmark_complex_outline | \Theta_{(0,0,0)} = \sum_{n=-\infty}^{\infty}(-1)^{n}Q_{\tau}^{\tfrac{3}{2}(n-\tfrac {1}{6})^{2}}Q_{1}^{n}Q_{2}^{-n+\tfrac{1}{3}}={\rm e}^{\tfrac{\pi{\rm i}}{6}}Q_ {1}^{\tfrac{1}{6}}Q_{2}^{\tfrac{1}{6}}\theta_{4}^{[-\tfrac{1}{6}]}(3\tau,t_{1} -t_{2})\,\] \[\Theta_{(-1,0,0)} = \sum_{n=-\infty}^{\infty}(-1)^{n}Q_{\tau}^{\... | |
formulas_benchmark_complex_outline | \begin{array}{ll}\Gamma_{m}^{+}(1_{1})\cap\Gamma_{m}^{+}(1_{0})=\emptyset,& \Gamma_{m}^{-}(1_{1})\cap\Gamma_{m}^{+}(1_{0})=\emptyset,\\ \Gamma_{m}^{+}(x_{m-1})\cap\Gamma_{m}^{+}(1_{0})=\{x_{0}\},&\Gamma_{m}^{-}(x_{ m-1})\cap\Gamma_{m}^{+}(1_{0})=\emptyset,\\ \Gamma_{m}^{+}(x_{0}^{-1})\cap\Gamma_{m}^{+}(1_{1})=\emptyset... | |
formulas_benchmark_complex_outline | \hat{\mathcal{L}}_{l} \coloneqq\frac{1}{2}\|\hat{v}_{\mathbb{E}_{l}}\|_{2}^{2}+\frac{1}{2} \|\hat{v}_{\mathbb{T}_{l}}\|_{2}^{2}-\sigma_{\mathbb{T}_{l}}\mu_{\mathbb{T}_{l} }\sum\log(\hat{w}_{\mathbb{T}_{l}})-\sigma_{\mathcal{I}_{\mathcal{L},l-1}}\mu_{ \mathcal{I}_{\mathcal{L},l-1}}\sum\log(\hat{w}_{\mathcal{I}_{\mathcal... | |
formulas_benchmark_complex_outline | I_{q,t}^{2}\lesssim \sum_{k}\theta(2^{-q}k)^{2}\sum_{k_{1},k_{2},k_{3},k_{1}^{\prime}, k_{2}^{\prime},k_{3}^{\prime}\neq 0:k_{123}=k_{123}^{\prime}=k}\] \[\times(1_{k_{1}=k_{1}^{\prime},k_{2}=k_{2}^{\prime},k_{3}=k_{3}^{ \prime}}+1_{k_{1}=k_{1}^{\prime},k_{2}=k_{3}^{\prime},k_{3}=k_{2}^{\prime}}) \int_{[0,t]^{2}}e^{-|k... | |
formulas_benchmark_complex_outline | M_{2} =\int_{\Omega}[\boldsymbol{u}^{m}\cdot\nabla(\boldsymbol{v}^{n}- \boldsymbol{v}^{m})]\cdot(\boldsymbol{u}^{n}-\boldsymbol{u}^{m})\] \[=-\int_{\Omega}[\boldsymbol{u}^{m}\cdot\nabla(\boldsymbol{u}^{n} -\boldsymbol{u}^{m})]\cdot(\boldsymbol{v}^{n}-\boldsymbol{v}^{m})\] \[=-\int_{\Omega}[\boldsymbol{u}^{m}\cdot\nabla... | |
formulas_benchmark_complex_outline | Z(z_{\text{out}},x_{\text{in}})=\int\mathcal{D}z_{\text{fl}} \;\mathcal{D}x_{\text{fl}}\;e^{\frac{i}{\hbar}S^{f}\left(z_{\text{out}}^{-}+z _{\text{fl}},\widetilde{x_{\text{in}}}+x_{\text{fl}}\right)}\\ =\int\mathcal{D}z_{\text{fl}}\;\mathcal{D}x_{\text{fl}}\;e^{\frac {i}{\hbar}\left(i\int_{I}z_{\text{fl}}\;dx_{\text{fl... | |
formulas_benchmark_complex_outline | \bar{I}_{h}^{n}(t) =\bar{I}_{h}^{n}(0)+\bar{N}_{1}^{n}\left(\beta_{h}^{n}\int_{0}^{t} \bar{I}_{v}^{n}(u)\bar{S}_{h}^{n}(u)\mathrm{d}u\right)-\bar{N}_{2}^{n}\left( \gamma_{h}^{n}\int_{0}^{t}\bar{I}_{h}^{n}(u)\mathrm{d}u\right),\] \[\bar{S}_{v}^{n}(t) =\bar{S}_{v}^{n}(0)+\bar{N}_{3}^{n}\left(\gamma_{v}^{n}\int_{0}^{t }\b... | |
formulas_benchmark_complex_outline | -\int_{0}^{T^{\prime}}\int_{\Omega}\mathrm{n}_{\varepsilon}\cdot \partial_{t}\xi\left|\nabla\psi_{\varepsilon}\right|\mathrm{d}x\,\mathrm{d}t =\int_{0}^{T^{\prime}}\int_{\Omega}(\nabla\cdot\xi)(B\cdot\mathrm{ n}_{\varepsilon})\left|\nabla\psi_{\varepsilon}\right|\mathrm{d}x\,\mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\... | |
formulas_benchmark_complex_outline | 2\mathcal{D}_{\text{H}}^{2}\big{(}Q_{X,Y|\widetilde{Z}=m}, \widetilde{Q}_{X,Y|\widetilde{Z}=m}\big{)}\ =\ \int\int\big{(}\sqrt{q_{XY|\widetilde{Z}}}-\sqrt{q_{X \cdot|\widetilde{Z}}q_{\cdot Y|\widetilde{Z}}}\big{)}^{2}d\mu_{X}d\mu_{Y}\] \[= \int\int\frac{\big{(}q_{XY|\widetilde{Z}}-q_{X\cdot|\widetilde{Z}}q_{ Y|\widetil... | |
formulas_benchmark_complex_outline | \log T\int_{\theta_{0}}^{\theta_{-\eta}}1-\frac{\sigma(1+2\eta)} {2\eta}+\frac{\sigma+\eta}{2\eta}d\theta+\log\log T\int_{\theta_{0}}^{\theta_ {-\eta}}\frac{\sigma+\eta}{\eta}c_{2}d\theta\] \[+\int_{\theta_{0}}^{\theta_{-\eta}}-\frac{\sigma}{\eta}\log\frac{ 1+\eta}{c_{1}(2\pi)^{\eta}}+\log\frac{c_{1}}{\sqrt{2\pi}}d\the... | |
formulas_benchmark_complex_outline | w_{1}^{\left(2\right)}(z)=\frac{d}{dz}\ln\left(\frac{\tau_{0}^{ \left(2\right)}}{\tau_{1}^{\left(2\right)}}\right)=-\frac{1}{z},\quad\alpha_{2 }=1;\quad w_{2}^{\left(2\right)}(z)=\frac{d}{dz}\ln\left(\frac{\tau_{1}^{\left( 2\right)}}{\tau_{2}^{\left(2\right)}}\right)=\frac{4t_{1}-2z^{3}}{4t_{1}z+z^{ 4}},\quad\alpha_{2}... | |
formulas_benchmark_complex_outline | H(S|A)_{\{\Pi^{A}_{j^{\prime}},\Pi^{S}_{i^{\prime}}\}} = -\sum_{i^{\prime},j^{\prime}}P_{i^{\prime}j^{\prime}}log(\frac{P _{i^{\prime}j^{\prime}}}{P_{j^{\prime}}})\] \[= -\sum_{i^{\prime},j^{\prime}}(\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_ {i^{\prime}})P(k_{ij},k_{j^{\prime}}))log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ ij},n_... | |
formulas_benchmark_complex_outline | \begin{split}&\int_{\Omega}z_{\varepsilon}(\cdot,T)+c_{1}\int_{0}^{T }\int_{\Omega}z_{\varepsilon}|\nabla w_{\varepsilon}|^{2}\\ &\quad+\frac{4(p+1)}{p}\int_{0}^{T}\int_{\Omega}\left|\nabla z_{ \varepsilon}^{\frac{1}{2}}+\frac{2k+p(p+1)\frac{u_{\varepsilon}}{u_{ \varepsilon}+1}}{2\sqrt{p(p+1)}}z_{\varepsilon}^{\frac{1}... | |
formulas_benchmark_complex_outline | Ec_{2}^{*}s_{2}E =\frac{1}{4}E^{1/2}(D-E)E^{1/2}+\frac{1}{4}E^{1/2}(D-E)D^{-1}E^{3/2}\] \[=\frac{1}{\pi}\int_{0}^{\infty}\frac{E^{1/2}}{E^{2}+t^{2}}D^{1/2} KD^{1/2}\frac{t^{2}}{D^{2}+t^{2}}E^{1/2}\mathrm{d}t\] \[\quad+\frac{1}{\pi}\int_{0}^{\infty}\frac{E^{1/2}}{E^{2}+t^{2}}D^{ 1/2}KD^{1/2}D^{-1}\frac{t^{2}}{D^{2}+t^{2... | |
formulas_benchmark_complex_outline | s_{i}^{k+1}+e_{i}^{k+1}+x_{i}^{k+1}+r_{i}^{k+1}=s_{i}^{k}+e_{i}^ {k}+x_{i}^{k}+r_{i}^{k}+h\left(-\beta_{i}^{k}x_{i}^{k}s_{i}+\beta_{i}^{k}x_{i }^{k}s_{i}-\sigma_{i}^{k}e_{i}^{k}+\sigma_{i}^{k}e_{i}^{k}-\delta_{i}^{k}x_{i }^{k}+\delta_{i}^{k}x_{i}^{k}\right)\] \[+\frac{h}{N_{i}}\sum_{j\neq i}\left(F_{ij}^{k}(P(s_{i}|T_{... | |
formulas_benchmark_complex_outline | f_{deg}=\frac{2(1-36\beta)^{2}}{3}(52542675\beta^{3}+178185258 \beta^{2}-9896841\beta-47632)\beta^{4}\] \[-11(397050199920\beta^{5}-40790893923\beta^{4}+4055047758\beta^{3} -243771759\beta^{2}+6417616\beta\] \[-59392)\beta^{3}m_{1}+(5465578392450\beta^{6}+19309935720393\beta ^{5}-3995019640449\beta^{4}\] \[+32734048171... | |
formulas_benchmark_complex_table | \begin{table}
\begin{tabular}{|r|r r r r r r|} \hline \(r\)\(\sigma\) & \(0.1\) & \(0.3\) & \(1\) & \(3\) & \(10\) & \(30\) \\ \hline \(0.5\) & \(0.0863\) & \(0.0798\) & \(0.0502\) & \(0.0195\) & \(0.00582\) & \(0.00192\) \\ \(1\) & \(0.454\) & \(0.444\) & \(0.380\) & \(0.276\) & \(0.198\) & \(0.157\) \\ \(2.5\) & \(1.... | |
formulas_benchmark_complex_outline | \left(\mathchoice{{\vbox{\hbox{$-$}}\kern-7.499886pt}}{{ \vbox{\hbox{$-$}}\kern-6.374903pt}}{{\vbox{\hbox{$-$}} \kern-4.499931pt}}{{\vbox{\hbox{$-$}}\kern-3.749943pt}}\! \int_{2B^{\prime}}|\nabla w_{\varepsilon}|^{q}\right)^{1/q} \leq C\left(\mathchoice{{\vbox{\hbox{$-$}}\kern-7.499886pt}}{{ \vbox{\hbox{$-$}}\kern-6.37... | |
formulas_benchmark_complex_outline | \bigg{(}\frac{1}{P(t)}\bigg{)}^{(l)}=\bigg{(}\frac{1}{(1+\beta_{1 }t)(1+\beta_{2}t)\cdots(1+\beta_{n}t)}\bigg{)}^{(l)}\] \[=\sum_{\begin{subarray}{c}j_{1}+j_{2}+\cdots+j_{n}=l;\\ j_{1},j_{2},\ldots,j_{n}\geq 0\end{subarray}}\binom{l}{j_{1};j_{2};\ldots ;j_{n}}\bigg{(}\frac{1}{1+\beta_{1}t}\bigg{)}^{(j_{1})}\bigg{(}\fra... | |
formulas_benchmark_complex_outline | \mathcal{S}_{11}:= -2[D_{t}^{2}\zeta,\mathcal{H}_{\zeta}\frac{1}{\zeta_{\alpha}}+ \bar{\mathcal{H}}_{\zeta}\frac{1}{\bar{\zeta}_{\alpha}}]\partial_{\alpha}D_{t} \zeta+2[\tilde{D}_{t}^{2}\tilde{\zeta},\mathcal{H}_{\zeta}\frac{1}{\bar{\zeta }_{\alpha}}+\bar{\mathcal{H}}_{\tilde{\zeta}}\frac{1}{\bar{\zeta}_{\alpha}}] \par... | |
formulas_benchmark_complex_table | \begin{table}
\begin{tabular}{|l|l|l|} \hline degree & types & toric system \\ \hline \(9\) & \(\mathbb{P}^{2}\) & \(L,L,L\) \\ \hline \(8\) & \(\mathbb{F}_{0}\) & \(H_{1},H_{2},H_{1},H_{2}\) \\ \hline \(8\) & \(\mathbb{F}_{1}\) & \(L_{1},E_{1},L_{1},L\) \\ \hline \(8\) & \(\mathbb{F}_{2}\) & \(F,S-F,F,S-F\) (where \(F... | |
formulas_benchmark_complex_outline | E_{\mathcal{X}^{c}}\left(\sum_{j=1}^{M}(I-z_{j}A_{c})^{-1}B_{c}u _{j}\right)\] \[=\sum_{j=1}^{M}\sum_{k=1}^{M}\left\langle B_{c}^{*}(I-\overline{ z_{k}}A_{c}^{*})^{-1}(I-z_{j}A_{c})^{-1}B_{c}u_{j},u_{k}\right\rangle_{\mathcal{U}}\] \[=\sum_{j=1}^{M}\sum_{k=1}^{M}\left\langle B_{0}^{*}(I-\overline{ z_{k}}A_{0}^{*})^{-1}... | |
formulas_benchmark_complex_outline | \mathbb{E}\Big{[}\big{|}H\big{(}\widetilde{\pi}_{n}^{0},\ldots, \widetilde{\pi}_{n}^{(m-1)},\widehat{\pi}_{n}^{m}\big{)}-H\big{(}\widetilde{ \pi}_{n}^{0},\ldots,\widetilde{\pi}_{n}^{(m-1)},\overline{\pi}_{n}^{m}\big{)} \big{|}\Big{]}\] \[=\mathbb{E}\Big{[}\big{|}H\big{(}\widetilde{\pi}_{n}^{0},\ldots, \widetilde{\pi}_{... | |
formulas_benchmark_complex_outline | f(x_{t}) \leq f(x_{sm})-\frac{\eta}{2}\sum_{j=sm+1}^{t}\|\nabla f(x_{j-1}) \|^{2}-\big{(}\frac{1}{2\eta}-\frac{L}{2}\big{)}\sum_{j=sm+1}^{t}\|x_{j}-x_{j -1}\|^{2}\] \[\qquad+\frac{\eta}{2}\sum_{k=sm+1}^{t-1}\frac{C^{\prime 2}L^{2} \sum_{j=sm+1}^{k}\|x_{j}-x_{j-1}\|^{2}}{b}\] \[\leq f(x_{sm})-\frac{\eta}{2}\sum_{j=sm+1}... | |
formulas_benchmark_complex_matrix | \left\{\begin{array}{l}x_{-\frac{1}{2}}^{0}<\ x_{\frac{1}{2}}^{0}<x_{\frac{ 2}{2}}^{0}<\cdots<x_{J-\frac{1}{2}}^{0},\\ 0<\underline{c^{0}}=\min\limits_{j\in\overline{0,J-1}}c_{j}^{0}\leq \overline{c^{0}}=\max\limits_{j\in\overline{0,J-1}}c_{j}^{0}<\infty,\\ 0<\underline{\rho^{0}}=\min\limits_{j\in\overline{0,J-1}}\rho_... | |
formulas_benchmark_complex_outline | \begin{cases}\frac{\partial x_{1}}{\partial t}=D_{h}\Delta_{y}x_{1}+b_{1}-\mu x _{1}-\beta_{2}x_{1}x_{4}-\beta_{1}x_{1}x_{2}+\phi x_{2},&y\in\mathbb{R},t>0,\\ \frac{\partial x_{2}}{\partial t}=D_{h}\Delta_{y}x_{2}+\beta_{1}x_{1}x_{2}+ \beta_{2}x_{1}x_{4}-(\mu+\phi)x_{2},&y\in\mathbb{R},\ t>0,\\ \frac{\partial x_{3}}{\p... | |
formulas_benchmark_complex_outline | B_{1} =\{1_{2},1_{3}\ ;\ 2_{2},2_{3}\ ;\ 3_{3},3_{1}\}, B_{2} =\{1_{1},1_{3}\ ;\ 2_{1},2_{3}\ ;\ 3_{2},3_{1}\}, B_{3} =\{1_{1},1_{2}\ ;\ 2_{1},2_{2}\ ;\ 3_{2},3_{3}\},\] \[B_{4} =\{1_{1},1_{3}\ ;\ 2_{1},2_{2}\ ;\ 3_{3},3_{1}\}, B_{5} =\{1_{2},1_{3}\ ;\ 2_{1},2_{3}\ ;\ 3_{2},3_{3}\}, B_{6} =\{1_{1},1_{2}\ ;\ 2_{2},2_{3}... | |
formulas_benchmark_complex_matrix | \begin{pmatrix}g_{1}(\mathbf{x}_{1})&g_{1}(\mathbf{x}_{2})&\ldots&g_{1}(\mathbf{ x}_{m})\\ g_{2}(\mathbf{x}_{1})&g_{2}(\mathbf{x}_{2})&\ldots&g_{2}(\mathbf{x}_{m})\\ \vdots&\vdots&\ldots&\vdots\\ g_{\ell}(\mathbf{x}_{1})&g_{\ell}(\mathbf{x}_{2})&\ldots&g_{\ell}(\mathbf{x}_{m} )\end{pmatrix}\approx\begin{pmatrix}\eta_{1... | |
formulas_benchmark_complex_outline | \begin{split} 6\mathscr{F}_{\mathrm{IMS}}=&-(n_{\mathbf{2,1} }-4n_{\mathbf{2,4}}+8n_{\mathbf{3,1}}-8)\psi_{1}^{3}-8(n_{\mathbf{1,10}}+n_{ \mathbf{1,5}}-1)\phi_{1}^{3}-(8n_{\mathbf{1,10}}+n_{\mathbf{1,4}}-8)\phi_{2}^ {3}\\ &-3\phi_{1}^{2}\phi_{2}(4n_{\mathbf{1,10}}+n_{\mathbf{1,4}}-4n_{ \mathbf{1,5}}-4)+3(6n_{\mathbf{1,... | |
formulas_benchmark_complex_outline | \biggl{[}\frac{1-z_{1}/z_{3}}{1-z_{1}/(z_{3}t)}\cdot\frac{1-z_{2} /z_{3}}{1-z_{2}/(z_{3}t)}\biggr{]}_{3}\] \[= \biggl{[}(1+(t^{-1}-1)\sum_{k=1}^{\infty}t^{-(k-1)}z_{1}^{k}z_{3} ^{-k})\cdot(1+(t^{-1}-1)\sum_{l=1}^{\infty}t^{-(l-1)}z_{2}^{l}z_{3}^{-l}) \biggr{]}_{3}\] \[= [1]_{3}+(t^{-1}-1)\sum_{k=1}^{\infty}t^{-(k-1)}z_... | |
formulas_benchmark_complex_outline | \int_{B_{R}\setminus G}|\nabla\tilde{n}|^{2}-\int_{B_{R}\setminus G }|\nabla n|^{2}\] \[=\int_{B_{R}\setminus G}\left(2\nabla n\cdot\nabla(\tilde{n}-n)+| \nabla(\tilde{n}-n)|^{2}\right)\] \[=2\int_{\partial B_{R}}\partial_{r}n\cdot(\tilde{n}-n)+\int_{B_{R }\setminus G}\left(-2\Delta n\cdot(\tilde{n}-n)+|\nabla(\tilde{n... | |
formulas_benchmark_complex_outline | F_{3} \equiv \gamma_{14}e^{\frac{-(\eta_{1}-\eta_{4})+(\eta_{2}-\eta_{5})+( \eta_{3}-\eta_{6})}{2}}+\gamma_{25}e^{\frac{(\eta_{1}-\eta_{4})-(\eta_{2}-\eta _{5})+(\eta_{3}-\eta_{6})}{2}}+\gamma_{36}e^{\frac{(\eta_{1}-\eta_{4})+(\eta_{2 }-\eta_{5})-(\eta_{3}-\eta_{6})}{2}}+\] \[R_{14}e^{\frac{\eta_{1}-\eta_{4}-\theta_{14... | |
formulas_benchmark_complex_outline | \rho_{n,x}\left(X_{T}^{u}-xe^{r(T-n)}\right)=-\int_{n}^{T}m_{s}(u)\mathrm{d}s +\begin{cases}\varrho^{\tilde{L}_{1}}\left\|u_{s}^{T}\sigma_{s}ve^{r(T-s)} \right\|_{\mathcal{L}^{\alpha}((n,T]\times\mathbb{S}^{d},\mathrm{d}s\times \tilde{\sigma}(\mathrm{d}v))}&\text{if $\alpha<2\ \&\ 3a$}\\ \varrho^{\tilde{L}_{1}}\left\|u... | |
formulas_benchmark_complex_outline | (f\circ g)(x_{1},\ldots,x_{m+1})= (-1)^{(0)(m+1)}\frac{f(x_{1},\ldots,x_{m})}{x_{1}^{2}-x_{m}^{2}}\] \[\times\left(g(\bar{x}_{1},x_{m+1})+(-1)^{m+1}\frac{x_{m}}{x_{1}}g (\bar{x}_{1},x_{m+1})\right.\] \[\left.-g(x_{m},x_{m+1})+(-1)^{m+1}\frac{x_{m}}{x_{m+1}}g(x_{m},x_{ m+1})\right)\] \[+(-1)^{m+1}\frac{f(x_{2},\ldots,x_... | |
formulas_benchmark_complex_outline | \Delta_{0}(x,s) =u_{\gamma}^{0}\int_{x}^{x+i\delta}\phi(z)(\mathrm{Im}\,z)^{s-1/2} \,\mathrm{d}s(z)-|j(\gamma,x)|^{2s-1}\int_{\gamma x}^{\gamma x+i\eta(x)}\phi(z )(\mathrm{Im}\,z)^{s-1/2}\,\mathrm{d}s(z)\] \[=u_{\gamma}^{0}\int_{x}^{x+i\delta}\phi(z)(\mathrm{Im}\,z)^{s-1/2} \,\mathrm{d}s(z)-|j(\gamma,x)|^{2s-1}\int_{x}... | |
formulas_benchmark_complex_outline | \begin{array}{l}X_{13}\,+\,X_{24}\,-\,X_{14}\,=\,c_{13}\\ X_{24}\,+\,X_{3\bar{1}}\,-\,X_{\bar{1}1}\,=\,c_{24}\\ X_{2\bar{1}}\,+\,X_{14}\,-\,X_{24}\,-\,X_{1\bar{1}}\,=\,c_{13}\\ X_{2\bar{4}}\,+\,X_{1\bar{3}}\,-\,X_{2\bar{3}}\,=\,c_{2\bar{4}}\\ X_{2\bar{3}}\,+\,X_{3\bar{4}}\,-\,X_{2\bar{4}}\,-\,X_{3\bar{3}}\,=\,c_{2\bar{... | |
formulas_benchmark_complex_outline | \begin{split} Nat_{\text{pro-$\mathcal{O}$}}(E_{m}(-),E_{n}(-))& =[\{\operatorname{Free}_{\mathcal{O}}(S^{n}\otimes DE_{\alpha})\}, \{\operatorname{Free}_{\mathcal{O}}(S^{m}\otimes DE_{\beta})\}]_{\text{pro-$ \mathcal{O}$}}\\ &=\pi_{0}\lim_{\beta}\operatorname{colim}_{\alpha}\operatorname{ Map}_{\mathcal{O}}(\operatorn... | |
formulas_benchmark_complex_outline | -\frac{1}{4}\sum_{i=1}^{m}\left(\sum_{\alpha=1}^{n}\bar{\gamma} \left(\hat{A}^{\alpha}(e_{i})\cdot\gamma_{\alpha}\right)\right)^{2}|_{\Sigma^{ \pm}}\] \[= \frac{1}{4}\sum_{i=1}^{2}\sum_{\alpha,\beta=1}^{2}\bar{\gamma} \left(\hat{A}^{\alpha}(e_{i})\cdot\hat{A}^{\beta}(e_{i})\cdot\gamma_{\alpha} \cdot\gamma_{\beta}\right... | |
formulas_benchmark_complex_outline | \mathcal{G}f\left(x\right) \approx\sum_{i=1}^{d}\left(-\frac{\mu^{i}\left(x\right)}{2h}+ \frac{A^{i,i}\left(x\right)}{2h^{2}}-\sum_{j=1,j\neq i}^{d}\frac{\left|A^{i,j} \left(x\right)\right|}{2h^{2}}\right)f\left(x-he_{i}\right)\] \[\qquad+\sum_{i=1}^{d}\left(-\frac{A^{i,i}\left(x\right)}{h^{2}}+ \sum_{j=1,j\neq i}^{d}\... | |
formulas_benchmark_complex_outline | \left(\frac{1}{N}\sum_{i,j}s_{ij}^{6}(\hat{\alpha}^{0}_{N})\right) \sqrt{N}(\hat{\rho}^{0}_{N}-\rho_{0})\] \[=\frac{1}{\sqrt{N}}\sum_{i,j}s_{ij}^{3}(\hat{\alpha}^{0}_{N})\left( \hat{c}^{3}_{ij}-3s_{ij}(\hat{\alpha}^{0}_{N})\sigma_{ij}^{2}(\hat{\tau}^{0}_{N} )\hat{c}^{0}_{N}-s_{ij}^{3}(\hat{\alpha}^{0}_{N})\rho_{0}\righ... | |
formulas_benchmark_complex_outline | \frac{1}{n\tau}\log a_{n\tau}(f,K,D) \leq\frac{1}{\tau}\sum\limits_{j:\rho_{j}\geq 0}d_{j}\log( \left\lfloor e^{(\rho_{j}+\xi)\tau}\right\rfloor+1)+\frac{1}{n\tau}\sum_{i=0}^ {n\tau-1}f(\varphi(i,x^{0},u^{0}),u^{0}_{i})+\varepsilon\] \[\leq\frac{1}{\tau}\sum\limits_{j:\rho_{j}\geq 0}d_{j}\log(2e^{( \rho_{j}+\xi)\tau})+... | |
formulas_benchmark_complex_outline | \hat{\Psi}_{1}=\begin{bmatrix}\ell_{0}(x_{0})\ell_{0}(y_{0})&\ell_{0}(x_{0})\ell _{1}(y_{0})&\ldots&\ell_{0}(x_{0})\ell_{n}(y_{0})\\ \ell_{1}(x_{0})\ell_{0}(y_{0})&\ell_{1}(x_{0})\ell_{1}(y_{0})&\ldots&\ell_{1}(x _{0})\ell_{n}(y_{0})\\ \vdots&&\vdots&\\ \ell_{m}(x_{0})\ell_{0}(y_{0})&\ell_{m}(x_{0})\ell_{1}(y_{0})&\ldo... | |
formulas_benchmark_complex_outline | \int_{0}^{t}\|\eta^{\mathfrak{h},\mathfrak{l}}(s)\|^{6}_{L^{6}_{ x}}\,\mathrm{d}s+\int_{0}^{t}\|\eta^{\mathfrak{h},\mathfrak{g}}(s)\|^{6}_{L^{6}_{ x}}\,\mathrm{d}s+\int_{0}^{t}\|\eta^{\mathfrak{h},\mathfrak{t}}(s)\|^{6}_{L^{6}_{ x}}\,\mathrm{d}s,\] \[\int_{0}^{t}\|u(s)\|^{6}_{L^{6}_{x}}\,\mathrm{d}s+\int_{0}^{t}\| \eta... | |
formulas_benchmark_complex_outline | \begin{split}&\mathcal{J}_{N}(\pi_{1:N-1|0},p_{1:N-1|0})+\mathbb{E}_{ \pi_{1:N-1|0}}\left[\mathcal{D}_{\text{KL}}\left(\pi_{N|N-1}||p_{N|N-1}\right) \right]-\mathbb{E}_{\pi_{N-1:N-1}}\left[\bar{r}_{N}(\mathbf{X}_{N-1})\right]\\ &=\mathcal{J}_{N}(\pi_{1:N-1|0},p_{1:N-1|0})+\mathbb{E}_{\pi_{N-1: N-1}}\left[\mathcal{D}_{\... | |
formulas_benchmark_complex_outline | \lim_{s\to 0}\frac{\mathcal{F}[\bm{n}+s\delta\bm{n}]-\mathcal{F}[\bm{n}] }{s}\] \[=\int_{\Omega}k_{1}(\nabla\cdot\bm{n})(\nabla\cdot\delta\bm{n})+k_ {2}(\bm{n}\cdot\nabla\times\bm{n})(\delta\bm{n}\cdot\nabla\times\bm{n}+\bm{n} \cdot\nabla\times\delta\bm{n})\] \[\qquad+k_{3}(\bm{n}\times\nabla\times\bm{n})\cdot(\delta\b... | |
formulas_benchmark_complex_outline | x\rhd(y\rhd z) = x\rhd(y\rhd z)+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j}(\langle y,a_{j}\rangle)x\rhd z_{j}\] \[= x\rhd(y\rhd z)+\sum_{j=1}^{k}\langle y\rhd z,b_{j}\rangle f_{j}( \langle x,a_{j}\rangle)z_{j}+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j}(\langle y,a_{j}\rangle)z_{j},\] \[= x\rhd(y\rhd z)+\sum_{j=1}^{k}\l... | |
formulas_benchmark_complex_outline | \operatorname{Tr}(P_{g}L_{X}L_{Y}P_{g})=\sum_{i<j}\left<e_{i}e_{j} ^{T}-e_{j}e_{i}^{T},g^{-1}Xg(e_{i}e_{j}^{T}-e_{j}e_{i}^{T})Y\right>+\sum_{i<j} \left<e_{i}e_{j}^{T}+e_{j}e_{i}^{T},g^{-1}Xg(e_{i}e_{j}^{T}+e_{j}e_{i}^{T})Y\right>\] \[+\sum_{i}2\left<e_{i}e_{i}^{T},g^{-1}Xge_{i}e_{i}^{T}\right>\] \[=\sum_{i<j}2\bigl{(}\... | |
formulas_benchmark_complex_outline | \mathbf{C}^{-1}=\begin{pmatrix}\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2} }&0&\frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{2}}{ \tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}\\ 0&\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{\tau_{2}}{\tau^{2}-\tau_ {1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{1}}{\tau^{2}-... | |
formulas_benchmark_complex_outline | H^{1,\,0}_{\bar{\partial}}(X,\mathbb{C}) = \bigg{\langle}[\alpha]_{\bar{\partial}},\,[\beta]_{\bar{\partial}},\,[\gamma]_{\bar{\partial}}\bigg{\rangle},\ \ \ H^{0,\,1}_{\bar{\partial}}(X,\mathbb{C})=\bigg{\langle}[\overline{\alpha}]_{ \bar{\partial}},\,[\overline{\beta}]_{\bar{\partial}}\bigg{\rangle}=\pi^{\star }H^{0,... | |
formulas_benchmark_complex_outline | (\bm{I}-\bm{R})^{\top}(\bm{I}-\bm{R}) =2\bm{I}-\bm{R}-\bm{R}^{\top}=\begin{pmatrix}2\bm{I}-\bm{R}_{11}- \bm{R}_{11}^{\top}&-\bm{R}_{12}-\bm{R}_{21}^{\top}\\ -\bm{R}_{21}-\bm{R}_{12}^{\top}&2\bm{I}-\bm{R}_{22}-\bm{R}_{22}^{\top}\end{pmatrix}\] \[=\begin{pmatrix}(\bm{I}-\bm{R}_{11}^{\top})(\bm{I}-\bm{R}_{11})+ \bm{I}-\bm... | |
formulas_benchmark_complex_outline | \sup_{\gamma<xh_{0}}J_{1}(K,\gamma)=\begin{cases}-(K-xh_{0})^{2},&\text{ if }K>xh_{0};\\ +\infty,&\text{ if }K<xh_{0}\text{ and }P_{1,0}=h_{0}^{2};\\ \frac{P_{1,0}(K-xh_{0})^{2}}{h_{0}^{2}-P_{1,0}},&\text{ if }K<xh_{0}\text{ and }P_{1,0}<h_{0}^{2};\\ 0,&\text{ if }K=xh_{0},\end{cases}\] \[\sup_{\gamma\geqslant xh_{0}}J... | |
formulas_benchmark_complex_outline | -2y_{0}+x_{0}(x_{0}+s_{6}),\ -2y_{2}+x_{2}(s_{4}+x_{1}s_{7}),\] \[-y_{1}-y_{2}+(2x_{0}+s_{6})y_{0}+x_{0}s_{2},\ x_{3},...,\ x_{n}, \ x_{0}(x_{1}+s_{5}),\ x_{0}x_{2},\] \[(x_{1}+s_{5})^{2}+x_{2}^{2},\ 3x_{1}^{2}x_{2}-(x_{1}+s_{5})(s_{4}+x_ {1}s_{7})+x_{2}s_{3}+x_{2}^{2}s_{7},\] \[-\frac{1}{2}x_{0}(x_{0}+s_{6})(3x_{0}+s_... | |
formulas_benchmark_complex_outline | \int_{t+1}^{\infty}|\lambda^{\prime\prime}(s)||\partial_{2}K_{1}( s-t,\lambda(t))-\frac{\lambda(t)}{2(1+s-t)}|ds|\lambda^{\prime}(t)|\] \[\leq\frac{C}{t^{3}\log^{2b+2}(t)}\int_{t+1}^{\infty}|\partial_{2 }K_{1}(s-t,\lambda(t))-\frac{\lambda(t)}{2(1+s-t)}|ds\] \[\leq\frac{C}{t^{3}\log^{2b+2}(t)}\int_{1}^{\infty}dw\left(\... | |
formulas_benchmark_complex_matrix | \big{|}I_{k,\ell}^{1}\big{|}\lesssim\sum_{\begin{subarray}{c}|k^{ \prime}-k|\leq 4\\ |\ell^{\prime}-\ell|\leq 4\end{subarray}}\Big{(}2^{-k}\big{\|}S_{k^{\prime}-1}^{ \mathrm{h}}S_{\ell^{\prime}-1}^{\mathrm{v}}\nabla_{\mathrm{h}}v^{\mathrm{h}} \big{\|}_{L_{\frac{6p}{p+4}}^{\frac{6p}{p+4}}}\big{\|}\Delta_{k^{\prime}}^{ \... | |
formulas_benchmark_complex_matrix | \alpha^{P}_{s}\leq\min_{\begin{subarray}{c}\mathbf{u}_{\mathcal{A}} \in\mathbb{R}^{\mathcal{X}}_{+},\|\mathbf{u}_{\mathcal{A}}\|=1\\ \mathbf{v}_{\mathcal{S}}\in\mathbb{R}^{\mathcal{S}}_{+},\|\mathbf{v}_{s}\|=1 \end{subarray}}\mathbf{u}^{\top}_{\mathcal{A}}P_{0}(\cdot|s,\cdot)\mathbf{v}_{ \mathcal{S}} =\min_{\begin{suba... | |
formulas_benchmark_complex_matrix | \left(\begin{array}{ccccccccc}4\mathbb{A}_{1}&2\mathbb{A}_{3}&8\mathbb{A}_{1} &6\mathbb{A}_{1}&5\mathbb{A}_{1}&6\mathbb{A}_{1}&6\mathbb{A}_{1}\\ &2\mathbb{A}_{1}&6\mathbb{A}_{1}&(4\mathbb{A}_{1})_{I}&3\mathbb{A}_{1}&(4 \mathbb{A}_{1})_{I}&(4\mathbb{A}_{1})_{I}\\ &&4\mathbb{A}_{1}&2\mathbb{A}_{3}&5\mathbb{A}_{1}&6\mathb... | |
formulas_benchmark_complex_outline | I_{4,n}(t) =\frac{1}{2}\int_{0}^{t}\langle\varphi_{n}^{\prime\prime}(\rho_{1}(s )-\rho_{2}(s))\sum_{i=1}^{\infty}\left|\sigma_{i}(\cdot,g^{-1}(\rho_{1}(s)))- \sigma_{i}(\cdot,g^{-1}(\rho_{2}(s)))\right|^{2},\psi\rangle_{H}\,ds\] \[\leq\frac{c}{2}\int_{0}^{t}\langle\varphi_{n}^{\prime\prime}( \rho_{1}(s)-\rho_{2}(s))\le... | |
formulas_benchmark_complex_matrix | \mathcal{R}(\mathcal{X}^{(k+1)})=\mathcal{R}(\mathcal{X}^{(k)}+ \xi_{k}\mathcal{S}^{(k)})\] \[=(1-\xi_{k})\mathcal{U}^{(k)}\mathcal{D}\left(\mathcal{U}^{(k)} \right)^{T}+\xi_{k}\widetilde{\mathcal{W}}_{\ell}\widetilde{\mathcal{W}}_{ \ell}^{T}-\xi_{k}^{2}\Delta\widetilde{\mathcal{K}}^{(k+1)}\left(\Delta \widetilde{\math... | |
formulas_benchmark_complex_outline | E_{C_{\psi_{2}}}(g) =\frac{1}{2\pi}\int_{\alpha=0}^{2\pi}\int_{r=0}^{\infty}\exp\left[ \frac{r^{2}f^{2}(\alpha)}{C_{\psi_{2}}^{2}}-\frac{r^{2}}{2}\right]\,r\,d\alpha dr\] \[=\frac{1}{4\pi}\int_{\alpha=0}^{2\pi}\frac{2C_{\psi_{2}}^{2}}{2f^ {2}(\alpha)-C_{\psi_{2}}^{2}}\exp\left[\left(\frac{f^{2}(\alpha)}{C_{\psi_{2}} ^{... | |
formulas_benchmark_complex_outline | \|(h_{t*}-\phi_{t*}^{X})V(p)\| = \|Dh_{t}(h_{t}^{-1}(p))V(h_{t}^{-1}(p))-D\phi_{t}^{X}(\phi_{-t}^{ X}(p))V(\phi_{-t}^{X}(p))\|\] \[\leq \|Dh_{t}(h_{t}^{-1}(p))[V(h_{t}^{-1}(p))-V(\phi_{-t}^{X}(p))]\|\] \[+\|[Dh_{t}(h_{t}^{-1}(p))-Dh_{t}(\phi_{-t}^{X}(p))]V(\phi_{-t}^{ X}(p))\|\] \[+\|[Dh_{t}(\phi_{-t}^{X}(p))-D\phi_{t}... | |
formulas_benchmark_complex_outline | f_{3}(n+1) = 64f^{4}+384f^{3}g+192f^{3}h+32f^{3}t+960f^{2}g^{2}+312f^{2}h^{2}+ 12f^{2}t^{2}+1056f^{2}gh\] \[+192f^{2}gt+120f^{2}ht+1152fg^{3}+304fh^{3}+4ft^{3}+2064fg^{2}h+ 408fg^{2}t\] \[+1320fgh^{2}+204fh^{2}t+60fgt^{2}+48fht^{2}+552fght+552g^{4}+138h ^{4}+t^{4}\] \[+1416g^{3}h+304g^{3}t+708gh^{3}+144h^{3}t+12gt^{3}+... | |
formulas_benchmark_complex_outline | \begin{split} J_{n}&=\prod_{k=1}^{K\!-\!1}\!\!\left( \!1\!-\!\frac{p_{k}}{c}\!\!\int_{0}^{c}\!\!\!\frac{\xi s\,\mathrm{d}x}{s\!+\! \left(x\!+\!a_{k}\right)^{\!\eta}}\!\right)\!\!\approx\!\prod_{k=1}^{K\!-\!1} \!\!\left(\!1\!-\!\int_{0}^{c}\!\!\!\frac{\lambda\xi s\,\mathrm{d}x}{s\!+\! \left(x\!+\!a_{k}\right)^{\!\eta}}\... | |
formulas_benchmark_complex_outline | \begin{cases}(\rho^{(n+m+1)}-\rho^{(n+1)})_{t}+u^{(n+m)}v^{(n+m)}(\rho^{(n+m+1 )}-\rho^{(n+1)})_{x}=(u^{(n)}v^{(n)}-u^{(n+m)}v^{(n+m)})\rho^{(n+1)}_{x}+R^{ 1}_{n,m},\\ (m^{(n+m+1)}-m^{(n+1)})_{t}+u^{(n+m)}v^{(n+m)}(m^{(n+m+1)}-m^{(n+1)})_{x}=(u^{ (n)}v^{(n)}-u^{(n+m)}v^{(n+m)})m^{(n+1)}_{x}+R^{2}_{n,m},\\ (n^{(n+m+1)}-... | |
formulas_benchmark_complex_outline | \Lambda_{(1,-a)}.\Lambda_{(1,-a)}= -\sum_{\begin{subarray}{c}a^{\prime}\in\mathbb{Z}/p^{r}\mathbb{Z} \\ a^{\prime}\neq a\end{subarray}}\Lambda_{(1,-a^{\prime})}.\Lambda_{(1,-a)}-\sum_ {b\in\mathbb{Z}/p^{r-1}\mathbb{Z}}\Lambda_{(-pb,1)}.\Lambda_{(1,-a)}\] \[= -\deg\mathrm{S}(N)\sum_{\begin{subarray}{c}a^{\prime}\in \mat... | |
formulas_benchmark_complex_matrix | \frac{\operatorname{d}\!\pi}{\operatorname{d}\!x\!\operatorname{ d}\!y}(x,y) =\exp\left(\frac{f(x)+g(y)-\|x-y\|^{2}}{2\sigma^{2}}\right)\frac{ \operatorname{d}\!\alpha}{\operatorname{d}\!x}(x)\frac{\operatorname{d}\! \beta}{\operatorname{d}\!y}(y)\] \[=\omega\exp\left(\mathcal{Q}(\mathbf{A}^{-1}\mathbf{a}+\mathbf{u},\m... |
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