harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\prod_{1\le i<j\le n}\frac{1-t^{j-i+1}}{1-t^{j-i}}=\prod_{d=1}^{n-1}\prod_{i<j\atop j-i = d}\frac{1- t^{d+1}}{1-t^d}=\prod_{d=1}^{n-1}\Big(\frac{1- t^{d+1}}{1-t^d}\Big)^{n-d}&= \Big(\frac{1- t^n}{1-t^{n-1}}\Big)\Big(\frac{1- t^{n-1}}{1-t^{n-2}}\Big)^2\cdots \Big(\frac{1- t^2}{1-t}\Big)^{n-1}= \frac{(1-t^n)(1-t^{n-1})\cdots (1-t^2)}{(1-t)^{n-1}}= [n]!.
\psi_{P,\mathbf{a}}\left( \mathbf{B};\mathcal{G}\right) =\psi_{P,\mathbf{a}}\left( \mathbf{G},\mathbf{B};\mathcal{G}\right) +\sum_{k=0}^{p}r_{k}\left( \overline{\mathbf{A}}_{k},\overline{\mathbf{G}}_{k},\overline{\mathbf{B}}_{k};s_{\mathbf{a,}k}^{\ast},P\right)
\langle J^{a i}( x ) \rangle_{\tilde{A}, \phi}= - \frac{\delta S}{\delta \tilde{A}_{i}^{a}( r_{b}, x )}, \langle{\cal O}^{I}( x ) \rangle_{\tilde{A}, \phi}= - \frac{\delta S}{\delta \phi^{I}( r_{b}, x )}
\tilde{J}_{R}\left( \bar{z}\right) =-ikg^{-1}\left( \bar{\partial}g\right)=ikg_{R}^{-1}\left( \bar{\partial}g_{R}\right) =\tilde{J}^{a}\tau^{a}=\left(\begin{array}{cc}-\tilde{J}^{2}\tilde{J}^{0}+\tilde{J}^{1}-\tilde{J}^{0}+\tilde{J}^{1}\tilde{J}^{2}\end{array}\right)
\langle e^{t\Lambda}f^\mathrm{in},\varphi\rangle_{X,X'}\ = \ \langle f^\mathrm{in},P_t\varphi-c_n\rangle_{X,X'}& \leq \ \|f^\mathrm{in}\|_{L^1(m)}\|P_t\varphi-c_n\|_{X'}.
H_k(\tau)=f_k(\tau)+\sum_{\ell=1}^{k-1}\alpha_k(\ell)\cdot F(\tau)^\ell\theta(2\tau)^{4(k-\ell)+2}+\sum_{\ell=k}^{2k}\beta_k(\ell)\cdot F(\tau)^{2k-\ell}F(2\tau)^{\ell-k}\theta(2\tau)^2
\|\mathbf{b}(x)\| = \sum_{s=1}^{m-1}|w(x+\lambda_s h) - w(x) - R_m(x, \lambda_s h)| \leq (m-1) \left(2\|w\|_{L^{\infty}([0,1])}+ \frac{h^m}{m!}\|w^{(m)}\|_{L^{\infty}([0,1])}\right).
S =\frac{P_0 G_t g_0}{4 \pi R^{\alpha_d}}\frac{\sigma_c A_e g_0^{\prime}}{4 \pi R^{\alpha_d}}= \gamma_0 \gamma_1 g_0 g_0^{\prime}R^{-2 \alpha_d}= c_0 g_p R^{-2 \alpha_d}
{}_pF_{q}= \begin{pmatrix}a_1, a_2, \dots ,\, a_p ; z b_1, b_2, \dots ,\, b_q \end{pmatrix}=\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\dots (a_p)_n}{(b_1)_n(b_2)_n\dots (b_q)_n}\frac{z^n}{n!}.
\begin{cases}c_{s}=0 \vdots c_{D-k_1}=0 \end{cases}\begin{cases}\binom{\overline{s}+s}{0}c_{s}+\binom{\overline{s}+s}{1}c_{s-1}+\cdots+\binom{\overline{s}+s}{s}c_{0}=0 \vdots \binom{\overline{s}+D-k_2}{0}c_{D-k_2}+\binom{\overline{s}+D-k_2}{1}c_{D-k_2-1}+\cdots+\binom{\overline{s}+D-k_2}{D-k_2}c_{0}=0.\end{cases}
(a^\ast_x\psi)_n := \ \frac{1}{\sqrt{n}}\sum^n_{j=1}\delta(x-x_j) \psi_{n-1}(x_1, \ldots, \hat x_j, \ldots, x_n), (a_x\psi)_n := \ \sqrt{n+1}\psi_{n+1}(x, x_1, \ldots, x_n).
S(t) \le \frac{1}{2\pi}\int_{-\infty}^{\infty}\Big\{R^+(u) \!-\! \chi_{[-t,t]}(u) \Big\}\, \frac{\Gamma_{\mathbb{R}}'}{\Gamma_{\mathbb{R}}}(\tfrac{1}{2}\!+\!iu) \, \mathrm{d}u - \frac{1}{2\pi}\sum_{n=2}^\infty \frac{\Lambda(n)}{\sqrt{n}}\, \widehat{R}^+\!\left( \frac{\log n}{2\pi}\right) + R^+\!\left(\frac{1}{2i}\right) -1.
\mathrm{I m}\int_{r_{o}}^{r_{f}}P_{-}d r = - \pi \int_{0}^{- \omega}{\frac{d{\cal H}- \Phi d Q}{\kappa ( M +{\cal H}, Q - e )}}~ = ~ - \frac{1}{2}\int_{M , Q}^{M - \omega , Q - e}d S_{B H},
E\left(X^2\right) = E\left(Z_1^2\right) + iE(Z_1Z_2) + i E(Z_2Z_1) - E\left(Z_2^2\right) = 0, E\left(X \overline{X}\right) = E\left(Z_1^2\right) + iE(Z_1Z_2) - i E(Z_2Z_1) + E\left(Z_2^2\right)=c_{X,X}, E(XY) = E(Z_1Z_3) + i E(Z_2Z_3) + iE(Z_1Z_4) - E(Z_2Z_4) = 0, E\left(X\overline{Y}\right) = E(Z_1Z_3) + i E(Z_2Z_3) - iE(Z_1Z_4) + E(Z_2Z_4) = c_{X,Y},
[\begin{matrix}0&0\\ 0&0\end{matrix}],[\begin{matrix}1&0\\ 0&1\end{matrix}],[\begin{matrix}2&0\\ 0&2\end{matrix}],[\begin{matrix}0&1\\ 2&0\end{matrix}],[\begin{matrix}1&1\\ 2&1\end{matrix}],[\begin{matrix}2&1\\ 2&2\end{matrix}],[\begin{matrix}0&2\\ 1&0\end{matrix}],[\begin{matrix}1&2\\ 1&1\end{matrix}],[\begin{matrix}2&2\\ 2&1\end{matrix}]
\begin{cases}(-\Delta + q -k^2)u=0 u(x) = e^{i k \theta \cdot x}+ u_s(k,\theta,x) \lim_{|x| \to \infty}(\frac{\partial u_s}{\partial r}- iku_s)(x) = o(|x|^{-(n-1)/2}),\end{cases}
{\Phi}^{( I )}= - \frac{s \, e^{2}F ( 1 - F )}{2 \pi | m |}\left[ \frac{1}{6}( F - \frac{1}{2}) + \frac{1}{4 \pi}\int_{1}^{\infty}\frac{d v}{v{\sqrt{v - 1}}}\frac{C_{F}v^{F}- C_{F}^{- 1}v^{1 - F}}{C_{F}v^{F}+ 2 + C_{F}^{- 1}v^{1 - F}}\right]
\tilde{\rm v}^a =\tilde{\rm u}^a, \tilde{E}^{\rm V}_a(R^\mu) = 0, \;\; \tilde{\rm p}_a= \mu_a,{v}^i ={\dot x}^i, X_i^{\rm V}(R^\mu) =p_i,{\dot p}_i = X_i^{\rm C}(R^\mu) - \mu_a B^a_{ij}v^j.
\mathop{\lim}\limits_{M \to \infty}\mathop{\max}\limits_{1 \le k \le K}\left\{{\eta_{K,M}^{(3)}+ \eta_{K,M}^{(4)}}\right\}= \mathop{\max}\limits_{1 \le k \le K}\left\{{\mathop{\lim}\limits_{M \to \infty}\eta_{K,M}^{(3)}+ \mathop{\lim}\limits_{M \to \infty}\eta_{K,M}^{(4)}}\right\}
- C_{0}\sum_{i = 1}^{N}\frac{a_{i}^{2 N}}{2 \Delta_{i}^{2}}\left( 1 - \sum_{i \neq j}\frac{a_{j}}{a_{i}- a_{j}}\right) = \frac{C_{1}^{\prime}\pi^{2 N}}{2^{2 N + 1}}\sum_{i = 1}^{N}\sum_{i \neq j}\frac{( a_{i}+ a_{j})^{2 N}}{( a_{i}- a_{j})^{2}\gamma_{i}\gamma_{j}}+ \frac{1}{2}A_{1}^{( 1 ; 1 )}\sum_{i = 1}^{N}a_{i}^{2}.
\langle \Psi_y\,|\,H_y \,|\,\Psi_y\rangle = \langle \,H_y \rangle_{u_{\alpha}\otimes \Phi_y}+ \langle \,H_y \rangle_{R_y}+ 2 \mathrm{Re}\langle u_{\alpha}\otimes \Phi_y\,|\,R_y\rangle.
\mathrm{SIR}_0(P_i) = \frac{P_i H_0 R_i^{-\alpha}}{\sum_{j=1}^{N}{\sum_{X_{j_k}\in \Phi_j \setminus X_0}{P_j \tilde{H}_{j0}||X_{j_k}- Y_0 ||^{-\alpha}}}}&\stackrel{d}{=}\frac{P_i H_0 R_i^{-\alpha}}{\sum_{j=1}^{N}I_0(\eta_j^{\frac{\alpha}{2}}P_j)}.
D_{\underline{\xi}'}\underline{v}(\underline{\xi}) =\left(\sum_{k=1}^n\frac{\partial\xi_k}{\partial\underline{\xi}_j}D_{\xi_k}v(\xi)\right)_{1\leq j\leq n-1}=AD_{\xi'}v(\xi)+BD_{\xi_n}v(\xi) D_{\underline{\xi}_n}\underline{v}(\underline{\xi}) =CD_{\xi_n}v(\xi).
V = -\frac{1}{2}\phi^2 + g \kappa \phi^3 +\frac{1}{3 g^2 \kappa^2 \gamma^2}\left(1 -3g^2 \kappa^2 \beta \gamma \phi^2-(1-2 g^2 \kappa^2\beta \gamma \phi^2)^{3/2}\right)
{A}= \frac{1}{2 ( 1 - \eta )^{2}}\, \bigg [ \frac{\Gamma ( \frac{\eta}{2 - 2 \eta})}{2 \sqrt{\pi}\Gamma ( \frac{1}{2 - 2 \eta})}\bigg ]^{\eta}\exp \bigg \{- \int_{0}^{\infty}\frac{d t}{t}\Big ( \frac{\sinh ( \eta t )}{\sinh ( t ) \cosh ( ( 1 - \eta ) t )}-{\eta}\, e^{- 2 t}\, \Big ) \bigg \}\, ,
\max_{\mathbf{i}}\bigl(|\hat{\theta}_{\mathbf{i}}-\theta_{\mathbf{i}}|+\bigl|t^{T}(\hat{Z}_{\mathbf{i}}-Z_{\mathbf{i}})\bigr|\bigr)=\mathrm{O}_{p}\bigl(\epsilon_{\mathbf{n}}+\bigl(\tilde{\mathbf{n}}h_{\mathbf{n}}^{k}\bigr)^{-1/2}h_{\mathbf{n}}^{-1}\epsilon_{\mathbf{n}}\bigr)=\mathrm{O}_{p}(\tilde{\epsilon}_{\mathbf{n}}),
(\prod_{a}{H_{a}^{(2)}}^{-\Delta_{ai}^{(2)}}\prod_{b}{U_{b}^{(2)}}^{\Lambda_{ai}^{(2)}})^{\sigma^{(2)}}\sum_{m_{i}=1}^{r_{i}}dz^{m_{i}}dz^{m_{i}}+\sum_{\alpha}dx^{\alpha}dx^{\alpha}],
G(1,2)=T(1,2)A^{-1}=(\begin{matrix}1&0&0&0&0\\ 0&1&0&0&0\\ -\frac{1}{3}&0&\frac{4}{3}&0&0\\ 0&-1&0&2&0\\ \frac{7}{15}&0&-\frac{8}{3}&0&\frac{16}{5}\end{matrix}).
\psi\left(Lx\right) =b\left\Vert Lx\right\Vert_Y^{2}+\left\langle v,Lx\right\rangle_Y +d & =b\left\Vert x\right\Vert_{L}^{2}+\left\langle L^{*}v,x\right\rangle_X +d,
T_2^\varepsilon(q,n) = \varepsilon^n\sum_{1 \leq i \leq n}2^i \binom{n-1}{i-1}\binom{q}{i}= \sum_{1 \leq i \leq n}2^i \binom{n-1}{\varepsilon(n-i)}\binom{q}{\varepsilon i}
X(T) :=\Vert \sigma\Vert_{\widetilde{L}^{\infty}_{T}(\widetilde{B}^{\frac{N}{2}+1-\alpha,\frac{N}{2}})}+ \Vert u\Vert_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1})}+ \Vert \sigma\Vert_{L^{1}_{T}(\widetilde{B}^{\frac{N}{2}+1,\frac{N}{2}+2-\alpha})}+\Vert u\Vert_{L^{1}_{T}(\dot{B}^{\frac{N}{2}+1}_{2,1})}
\mathcal{Z}_{\Psi}(\mathbf{s}; \mathbf{m}) = \mathcal{Z}_{\Psi}(s_1,\ldots,s_r; \mathbf{m}) =\sum_{\substack{\mathbf{c}\in (\mathcal{O}_S / \mathcal{O}_{S}^{\times})^r \mathbf{c}= (c_1, \ldots, c_r)}}\dfrac{H^{(n)}(\mathbf{c}; \mathbf{m}) \Psi (\mathbf{c})}{|c_{1}|^{2 s_1}\ldots |c_{r}|^{2 s_r}}\, ,
F_T(x)=\frac{(1 - 2 x) \big( (1 - 5 x + 9 x^2 - 6 x^3)\sqrt{1 - 4 x}- (1 - 9 x + 29 x^2 - 38 x^3 + 18 x^4)\big)}{2 (1 - x)^2 x (1 - 7 x + 14 x^2 - 9 x^3)}\,.
\frac{Ma}{Re}=\frac{U_{\infty}/c_{s}}{\rho U_{\infty}L/\mu}=\frac{\mu}{\rho Lc_{s}}=\frac{\mu}{\rho L\sqrt{\frac{\gamma k_{B}T}{m}}}=\frac{\mu}{\rho L}\sqrt{\frac{m}{\gamma k_{B}T}}
T_{1}=\frac{a_{1}}{\sum_{h=1}^{H}a_{h}}T_{2}=\frac{a_{2}+a_{3}+a_{4}}{\sum_{h=1}^{H}a_{h}}T_{3}=\frac{\sum_{h=5}^{H}a_{h}}{\sum_{h=1}^{H}a_{h}}
\Gamma ( N / 2 ) [ \frac{g^{2}}{( 4 \pi )^{5 / 2}}]^{N}\prod \int d M_{i}^{2}\rho ( M_{i}^{2}) \int [ d x ]_{3 N}U^{- 5 / 2}( \phi / U + \sum x_{i}M_{i}^{2})^{- N / 2}.
I_{2}(t) = \int_{0}^{1}t^{-1+\beta/2}x^{-1/2 - 1/\beta}w(x^{-1/\beta}, \beta, 1)dx =\int_{0}^{1}t^{-1+\beta/2}x^{-1/2 - 1/\beta + 1 + 1/\beta}dx = 2t^{-1+\beta/2}/3.
\int_{B_1}-(\nabla{\hat\rho}\cdot h)\eta{\tilde d}^2\, d\|V_t\|=\int_{G_1(B_1)}S\cdot \left(\eta{\tilde d}^2\nabla^2{\hat\rho}+\nabla{\hat\rho}\otimes\nabla(\eta{\tilde d}^2)\right)\, dV_t(\cdot,S) \mbox{for a.e.}\; t.
\rho_{t_n}= \rho_{abs, t_n}+ \rho_{sal,t_n}= \rho_{abs,t_n}+ \sum_{k=1}^{M_{t_n}-1}\frac{s_1(t_n)}{Df^{k-1}_{t_n}(f_{t_n}(c))}H_{f^k_{t_n}(c)}
\hat{\cal P}{\cal H}=\hat{\cal H}{\cal H}=0,\qquad\hat{\cal K}{\cal H}=-{\cal P}, \hat{\cal P}{\cal P}=\hat{\cal H}{\cal P}=0,\qquad\hat{\cal K}{\cal P}=-\frac{1}{c^2}{\cal H}, \hat{\cal K}{\cal K}=0,\qquad\hat{\cal H}{\cal K}={\cal P},\qquad\hat{\cal P}{\cal K}=\frac{1}{c^2}{\cal H}.
\begin{cases}(e^{-1/2}+ o(1)) n!^2/n^{n-1}~d=2~~\Sigma G = 0, (1 + O_d(n^{2-d})) n!^d/n^{n-1}~d\geq 3~~(d+1)\Sigma G = 0, 0 .\end{cases}
[L_n+z,L_m+z']=(\{m\}-\{n\})L_{n+m}&[L_n+z,G_m+z']=(\{m+1\}-\{n\})G_{n+m}+c \frac{1}{q^{n-2}}\frac{1+q^{2}}{1+q^{n}}\frac{\{n+1\}\{n\}\{n-1\}}{\{12\}}\delta_{n+m,0},\ \forall n>0 &[G_n+z,G_m+z']=0.
\Pr[{\rm SIR}_{{\rm SVC},n,\ell}\geq \tau|d_0=x]= \mathbb E_{I_{n,\ell},\overline{I}_{n,\ell}}\left[\exp(-s(I_{n,\ell}+\overline{I}_{n,\ell}))\right]=\mathcal L_{I_{n,\ell}}(s) \mathcal L_{\overline{I}_{n,\ell}}(s),
(d_A\eta)(\xi_1,...,\xi_{n+1})= \sum_{i}(-1)^{i+1}\rho(\xi_i)(\eta(\xi_1,...,\hat{\xi_i},..,\xi_{n+1})) + \sum_{i<j}(-1)^{i+j}\eta([\xi_i,\xi_j],\xi_1,...,\hat{\xi_i},...,\hat{\xi_j},..,\xi_{n+1})
\big( H(b,V) - z \big) \begin{pmatrix}1 0 -(\mathcal{H}_{2}- z)^{-1}U (\mathcal{H}_{2}- z)^{-1}\end{pmatrix}& = \begin{pmatrix}\mathcal{H}_{1}- z - \overline{U}(\mathcal{H}_{2}- z)^{-1}U \overline{U}(\mathcal{H}_{2}- z)^{-1}0 1\end{pmatrix}.
c+o(1) = \left(\frac{1}{p}-\frac{1}{2\cdot p^\flat}\right) ([u_n]_{s,p}^p+a\|u_n\|_p^p) &\geq \left(\frac{1}{p}-\frac{1}{2\cdot p^\flat}\right) a\|u_n\|_p^p.
{\bf m}_{2}=\{m^{(-4k_{i}-2i+3)}_{0,\ldots, 0,\underset{i}{k_{i}},0, \ldots, 0}, m^{(-4k_{i}-2i+5)}_{0,\ldots, 0,\underset{i}{k_{i}},0, \ldots, 0}\mid i\in \{1,\ldots,n-1\}, \k_{i}\in \mathbb{Z}_{\geq0}\}.
\boldsymbol{L}_{JK}\left( \boldsymbol{P}_{M}u(x)\right) =\left(A_{JK}(\boldsymbol{P}_{M}u;K+p^{N}\mathbb{Z}_{p})\right)\Omega\left( p^{N}\left\vert x-J\right\vert_{p}\right) -A_{JK}\left( \boldsymbol{P}_{M}u\left( x\right) \right) \Omega\left(p^{N}\left\vert x-J\right\vert_{p}\right) .
g(u,y^{p},\theta )=\left( \begin{array}{ccc}-\frac{e}{2}u\, 1_{2}\frac{i}{\sqrt{2}}y^{p}\, \sigma_{p}\sqrt{e\,}\theta_{N}. \frac{e}{2}u\, 1_{2}. . \sqrt{e}\, \vartheta^{\dagger N}.\end{array}\right) =u{\cal C}+y^{p}{\cal N}_{p}+\theta^{\dagger N}Q_{N}+\overline{Q}^{N}\theta_{N}
W_{i_{1}j}^{( 1 )}( x , y ) = \left( \begin{array}{c c}{{\widetilde{W}_{i j}^{( 0 )}}}\\ \hline{{}}\\{{\begin{array}{c c c c c}{{0}}&{{\partial_{1}^{x}}}&{{- e}}&{{- ( a - 1 ) e}}\\ \end{array}}}\\ \end{array}\right) \delta ( x - y ) ,
-\frac{\partial L_{d}KLI(q\|p)}{\partial q_{j}}=-\sum_{j}p_{j}\sum_{i}\left[\frac{a}{a-b}\left( \frac{\overline{q}_{i}}{\overline{p}_{i}}\right)^{a-1}-\frac{b}{a-b}\left( \frac{\overline{q}_{i}}{\overline{p}_{i}}\right)^{b-1}\right]\frac{\partial \overline{q}_{i}}{\partial q_{j}}
\tau_1 . A(z) ={\tau_1}^2 \tau_2 \cdots \tau_{c-1}Q(z) ={\tau_1}^2 e(h_1,h_c,h_2,\ldots,h_{c-1},q,h_{c+1},\ldots,h_{m_k}) \tau_2 \cdots \tau_{c-1}Q(z) = \tau_2 \cdots \tau_{c-1}Q(z)
{\cal L}= \overline{{{D_{\mu}\phi^{i}}}}D^{\mu}\phi^{i}+ \sigma ( \phi^{i \dagger}\phi^{i}- N \chi ) + N ( m^{2}\chi +{\frac{\lambda}{2}}\chi^{2}+{\frac{\eta}{3}}\chi^{3}) + i N \epsilon_{\mu \rho \nu}A_{\mu}\partial_{\rho}A_{\nu}.
T\gamma\left(\frac{\partial}{\partial t}\right)=\frac{\partial}{\partial t}+\sum_{j=1}^n\frac{\partial \gamma^j}{\partial t}\frac{\partial}{\partial p_j},T\gamma\left(\frac{\partial}{\partial q^i}\right)=\frac{\partial}{\partial q^i}+\sum_{j=1}^n \frac{\partial \gamma^j}{\partial q^i}\frac{\partial}{\partial p_j},
\mathsf{D}_{d}^{m}W(r;d)=\int_{0}^{r}\mathsf{D}_{d}^{m}(\Gamma(d)^{-1}(r-s)^{d-1})\mathsf{d}W(s)=\Gamma(d)^{-1}\int_{0}^{r}R^{(m)}(d)(r-s)^{d-1}\mathsf{d}W(s).
\int_{s+\epsilon}^{\infty}e^{-\frac{1}{2}\int_{s}^{u}\phi'(\omega)^2 d\omega}\lim_{x\uparrow 0}\frac{\partial}{\partial x}\Upsilon(s,x,u)du=\int_{\epsilon}^{\infty}\frac{l^s(u)}{\sqrt{2\pi u^3}}du
I_1: =\int_{0}^{T}\int_{0}^{1}\chi_{\{|u_n|> M\}}|h_n(u_n)-h(u_n)|\d x\d t &\leq \int_{0}^{T}\big(|u_n|+|u_n|^{\delta+1}+|u_n|^{2\delta+1}\big)\d x\d t<+\infty,
\big|\partial_{\phi}\Psi[\lambda,\xi,\dot{\lambda},\dot{\xi},\phi][\bar{\phi}](x, t)\big|\lesssim \frac{1}{R^{a-2s}}\|\bar{\phi}(t)\|_{n-2s+\sigma, a}\left(\sum_{j=1}^k\frac{\mu_0^{\frac{n-2s}{2}+\sigma}(t)}{1+|y_j|^{a-2s}}\right).
\left[ \mathrm{T r}e^{- \bar{H}_{s}^{2}t}\right]_{\mathrm{d i v}}={\frac{\Gamma \left( 1 - \frac D 2 \right)}{( 4 \pi )^{( D - 1 ) / 2}}}{\frac{m^{D - 4}}{2 \kappa t^{3 / 2}}}\int_{\Sigma}\left[ \left( m^{2}- \left( \frac 1 6 - \xi \right) R \right) t -{\frac{P}{4 \kappa^{2}}}\right] ,
\varepsilon(L_{\pm 1}^{(n)}) =\delta_{n,0}, \varepsilon (L_{0}^{(n)}) =\delta_{n,0}, \Delta(L_{\pm 1}^{(n)}) =\sum_{i=0}^{n}L_{\pm 1}^{(n-i)}\otimes L_{\pm 1}^{(i)}, \Delta (L_{0}^{(n)}) =\sum_{i=0}^{n}L_0^{(n-i)}\otimes L_0^{(i)}
\bar{\eta}{}^{A}= \frac{i}{\sqrt{2}}\left[ \begin{array}{l}{{\cos \left( \frac{\tau}{2}- \frac{\pi}{4}\right) \mathrm{e}^{\hspace{2 m m}i \sigma / 2}}}\\{{\, \sin \left( \frac{\tau}{2}- \frac{\pi}{4}\right) \mathrm{e}^{- i \sigma / 2}}}\\ \end{array}\right] .
\frac{1}{n!\cdot m!}\cdot\sum_{i=0}^{m}\binom{m+n}{m-i}\cdot \frac{m!}{i!}\cdot \biggl(e^x D^{n+i}\left( e^{-x}x^n\right)\biggr)\cdot y^{i}=\sum_{i=0}^{m}\frac{1}{i!}\cdot \binom{m+n}{m-i}\cdot \biggl(\frac{1}{n!}e^x D^{n+i}\left( e^{-x}x^n\right)\biggr)\cdot y^{i}
{A'}^4_4 = -i\lambda_3 \left( -i\lambda_3 -i b_2 \left( x_1+x_2+x_3+x_4 \right) \right) \left( \frac{i}{x_{1+2}}+ \frac{i}{x_{1+3}}+ \frac{i}{x_{1+4}}\right) &+A_4.
\max \sum_{c,u,b,z}\pi_{cubz}X_{cubz}{\rm s.t.\}\sum_{u}X_{cubz}=1, \quad\forall \ (c,b,z)\in \mathcal{C}\times \mathcal{B}\times \mathcal{Z}, & \sum_{cb}X_{cubz}\leq 1,\forall \ (u,z) \in \mathcal{U}\times \mathcal{Z}, & X_{cubz}\in \{0,1\}, \forall \ (c,u,b,z) \in \mathcal{C}\times \mathcal{U}\times \mathcal{B}\times\mathcal{Z},
\begin{array}{ll}\mathcal{H}_{BV, f}= [i \cdot su(2) \oplus u(1)] \oplus i \cdot u(2) [1ex] \simeq \langle M_{1}, M_{2}, M_{3}, iE \rangle_{\mathbb{R}}\oplus \langle C_{1}, C_{2}, C_{3}, C_{4}\rangle_{\mathbb{R}}\ . \end{array}
S_{\tau}= \left( \begin{array}[c]{c}C_{\tau}D_{\tau}\end{array}\right) \;\;\; \; C_{\tau}= \int_{0}^{\tau}\frac{\sin \big ( \sqrt{\Lambda}(\tau -s)\big )}{\sqrt{\Lambda}}dW_s, \;\;\; D_{\tau}= \int_{0}^{\tau}{\cos \big ( \sqrt{\Lambda}(\tau -s)\big )}dW_s.
F_{\ell,n}(p) =-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{p(x) \prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\}K_{n}(x) \mathrm{d}m(x) ,
S=2\pi\int_{-a}^{a}\sqrt{\frac{a^{2}b^{2}}{a^{2}}-\frac{b^{2}x^{2}}{a^{2}}}\sqrt{\frac{a^{2}-x^{2}}{a^{2}-x^{2}}+\frac{b^{2}x^{2}}{a^{2}-x^{2}}}dx
\int_0^{\infty}\exp\left(-sx-\frac{x+\tilde{A}_n^2}{2b}\right)I_0\left(\frac{\tilde{A}_n\sqrt{x}}{b}\right)\mathrm{d}x = & \frac{2b}{2sb+1}\exp\left[\frac{1}{2\tilde{A}_nb(2sb+1)}\right].
\Vert \partial_t S(t) \Delta g \Vert_{H_p^{\sigma-1}(\mathbb{R}^n)}= \Vert \partial_t \Lambda_1 (t) (I-\Delta) g \Vert_{H_p^{\sigma-1}(\mathbb{R}^n)}\leq C t^{-\frac n 2(1-\frac 2p)}\Vert (I-\Delta)g \Vert_{L^{p'}(\mathbb{R}^n)}\leq C t^{-\frac n 2(1-\frac 2p)}\Vert g \Vert_{H^2_{p'}(\mathbb{R}^n)},
\pi_{TM, X_V,Y_V,c}({\bf x},{\bf y})=\left( \begin{array}{ccc|ccc}0 0 0 0 0 0 0 0 0 0 0 x^1 0 0 0 0 -x^1 0 \hline 0 0 0 0 0 x^1 0 0 x^1 0 0 y^1+x^2 0 -x^1 0 -x^1 -y^1-x^2 0 \end{array}\right).
dS^{-1}_{jk}(t)= -S^{-1}_{jk}(t) dU_{jj}(t)- \sum_{1 \leq \ell \leq N: \ell \not=j}S^{-1}_{\ell k}(t)\frac{(S^{-1}(t) dM(t) S(t))_{j \ell}}{\Lambda_{\ell}(t)-\Lambda_j(t)},1 \leq j, k \leq N, \, t \geq 0.
\sup_{u \in C_{0}^{\infty}(\mathbb{B}^n), \int_{\mathbb{B}^n}|\nabla_{h}u|^n \,{\rm d}v_h \leq 1}\int_{\mathbb{B}^n}\left (e^{\alpha_n u^{\frac{n}{n-1}}}- \sum_{j=0}^{n-2}\frac{1}{j !}\alpha_n^j|u|^{\frac{jn}{n-1}}\right) \,{\rm d}v_{h}< \infty,
P(\Omega_{\{k,l,m\}})=\frac{2^{\frac{N(N-1)}{2}}\left( \frac{N}{N-1}\right)^{\frac{N(N-1)}{2}}\left( \frac{k}{k-1}\right)^{\frac{k(k-1)}{2}}\left( \frac{l}{l-1}\right)^{\frac{l(l-1)}{2}}}{\left( \frac{k+l}{k+l-1}\right)^{\frac{(k+l)(k+l-1)}{2}}}.
\left(\begin{array}{cc}\frac{\partial x_{1}}{\partial y}\frac{\partial x_{1}}{\partial z}\frac{\partial x_{2}}{\partial y}\frac{\partial x_{2}}{\partial z}\end{array}\right)\cdot\left(\begin{array}{cc}\frac{\partial y}{\partial x_{1}}\frac{\partial y}{\partial x_{2}}0 \mathrm{I}_{m-n}\end{array}\right)=\mathrm{I}_{m}.
\widehat{\phi}_{xy}^*(f)=f(x+h,x_2,\ldots,x_n)= \left(f+\frac{\partial{f}}{\partial{x}}\cdot h+\ldots +\frac{1}{i!}\frac{\partial^i{f}}{\partial{x^i}}\cdot h^i +\ldots\right)(x,x_2\ldots,x_n)
Z_{(k,l)}^0 = \left\{\left( [\xi_0^i : \xi_1^i] \right)_{i=1}^N \mid [\xi_0^k : \xi_1^k] =[\xi_0^l : \xi_1^l] = [0:1]\right\}Z_{(k,l)}^1 = \left\{\left( [\xi_0^i : \xi_1^i] \right)_{i=1}^N \mid [\xi_0^k : \xi_1^k] =[\xi_0^l : \xi_1^l] = [1:0]\right\}
|S_6|^2 \ll \frac{X^2 \log^3 u \, \cdot \log^5 X}{M}+ \frac{X^2 \log^3 u}{\log^5 X}&\ll X^{\frac{3}{2}}\log^{6}X + \frac{X^2}{\log^{9/2}X}\ll \frac{X^2}{\log^{9/2}X}.
\mathcal{R}_{}\! =\! \bigcup_{P_{U|X}}\!\Big\{\left(R_s,R_\ell,R_w\right)\!\colon\! &0 \leq R_s\leq I(U;Y), &R_\ell\geq I(U;X)-I(U;Y), &R_w\geq I(U;X)\Big\}
[A,B]_m = [\tilde A + \check A, \tilde B + \check B]_m = -J_6 ([J_6 \tilde{A},J_6 \tilde{B}]-[J_6\check{A},J_6\check{B}]+[J_6\tilde{A},J_6\check{B}]+[J_6\check{A},J_6\tilde{B}]) \,,
E_{k,3}= \sum_{\alpha_1+\alpha_2=2k-2}\int u^{(\alpha_1)}u^{(\alpha_2)}u dx+\sum_{\alpha_1+\alpha_2+2\alpha_3=k-2}(-1)^{k+1+\alpha_3}\int u^{(\alpha_1)}u^{(\alpha_2)}u^{(\alpha_3)}dx.
\begin{cases}\rho(x) w_{tt}- \operatorname{div}(K(x) \nabla w) =0 ~\hbox{in}~ \Omega \times (0,T), \rho(x) z_{tt}- \operatorname{div}(K(x) \nabla z) =0 ~\hbox{in}~ \Omega\times (0,T), w_t=z_t=0 ~\hbox{in}~ \omega\times(0,T).\end{cases}
\left( \begin{array}{c}{{t}}\\{{c}}\\{{u}}\\ \end{array}\right) ; \ \ \left( \begin{array}{c}{{b}}\\{{s}}\\{{d}}\\ \end{array}\right) ; \ \ \left( \begin{array}{c}{{\tau}}\\{{\mu}}\\{{e}}\\ \end{array}\right) ; \ \ \left( \begin{array}{c}{{\nu_{1}}}\\{{\nu_{2}}}\\{{\nu_{3}}}\\ \end{array}\right) .
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^\Phi(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^\Phi(\mathbb{R}^n)}.
Q_1(x_1, x_2) = M_L^{(1)}(x_1, x_2)\cdot P_1(x_1, x_2)\cdot M_R^{(1)}(x_1, x_2), &Q_2(x_1, x_2) = M_L^{(2)}(x_1, x_2)\cdot P_2(x_1, x_2)\cdot M_R^{(2)}(x_1, x_2).
\nu_0\left(\partial_{p^e}d_{j,0}x^j\right)=\nu_0\left(\binom{j}{p^e}d_{j,0}x^{j-p^e}\right) =\nu_0\left(\binom{j}{p^e}\right)+\nu(d_{j,0})+(j-p^e)\beta_0= = \nu(d_{j,0})+j\beta_0-p^e\beta_0=\nu_0(h)-p^e\beta_0.
Mf_{1}(y) =\Gamma^{\gamma}\frac{\int_{0}^{\Gamma e^{-\omega_{l}\tau}}r^{\omega-1}e^{-r^{2}/4}dr}{\int_{0}^{y}r^{\omega+1}e^{-r^{2}/4}dr}\lesssim \Gamma^{\gamma}\int_{0}^{\Gamma e^{-\omega_{l}\tau}}r^{\omega-1}e^{-r^{2}/4}dr
|u(t,x)|\leqslant \sum_{n=1}^{\infty}|u_n(t)||\psi_n(x)|\leqslant\frac{1}{2}\left(\sum_{n=1}^{\infty}\alpha_n^{2\beta}|u_n(t)|^2+\sum_{n=1}^{\infty}\alpha_n^{-2\beta}\psi_n^2(x)\right)
\mathbb E[U_{n,x,y}|\tilde{\mathcal F}]=a_n^{-\beta}\delta^{1-\beta}|\zeta_{n,x,y}|_-^\beta\left[(c_1-c_0)\frac{\beta}{\beta-1}+\epsilon_{a_n\delta|\zeta_{n,x,y}|^{-1}}\right].
(E_{l}^{\phi,\psi}|B_{d_1d}\cdot E_{k-l}^{\overline{\phi},\overline{\psi}}|B_{d_2d})|\gamma =E_{l}^{\phi,\psi}|B_{d_1d}\gamma\cdot E_{k-l}^{\overline{\phi},\overline{\psi}}|B_{d_2d}\gamma,
\prod_{k=1}^m \left(1 + q^{N(n-k)+a(r)}\right) = \prod_{k=0}^{m-1}\left(1 + q^{Nk+ N(n-m)+a(r)}\right) &= \left( 1+q^{N(n-m)+a(r)}\right) \prod_{k=1}^{m-1}\left(1 + q^{Nk+ N(n-m)+a(r)}\right) &= \left( 1+q^{N(n-m)+a(r)}\right) \sum_{k=0}^{m-1}q^{N \frac{k(k+1)}{2}+kN(n-m) +ka(r)}{m-1 \brack k}_{q^N},
\begin{aligned}D(t) \leq C_1p^2\int_{0}^{t}\int_{0}^{s}D(\tau)^{1-\frac{3}{p}}d\tau ds & +C_2(p^2+p)\int_{0}^{t}\int_{0}^{s}\int_{0}^{\tau}D(u)^{1-\frac{3}{p}}du d\tau ds & +C_3(p^2+p)\int_{0}^{t}\int_{0}^{s}\int_{0}^{\tau}\int_{0}^{u}D(w)^{1-\frac{3}{p}}dw du d\tau ds,\end{aligned}
\|D^* w' -u\| = \| D^* w' - (M_{n-k}-\tilde \lambda) \tilde v'' \| = \| D^* (w'-v') + D^*v' + (M_{n-k}-\lambda) v'' + (M_{n-k}-\lambda) (\tilde v''-v'') + (\lambda -\tilde \lambda) \tilde v''\| \le n^{\gamma'+\frac{1}{2}}\delta.
J_{t}|_{Q_t}(w) =\pi_{t}\circ J\circ \pi_{t}^{-1}(w) &=\pi_{t}\circ J(w+\zeta_{t}(w)V) &=J(w+\zeta_{t}(w)V)-\zeta_{t}(J(w+\zeta_{t}(w)V)) &=J(w)-\zeta_{t}(w)U-\zeta_{t}(J(w))V,
\sum_{\ell(\lambda)\leq K\atop \ell(\mu)\leq K}{\mathcal B}_{\lambda, \mu}([\ ];\sigma)e^{u(|\lambda|+|\mu|)}=\frac{\Gamma(2\sigma)^{2K}\Gamma(1-2\sigma)^{2K}}{t^{K(K-1)}}\times &\times\frac{1}{2\pi i}\oint_{|q|=1}\left[\det\left(\psi^{(i+j-2)}(u)\right)_{i,j=1}^{2K}\right]\frac{dq}{q}\ ,
\delta^{( 1 )}( A_{j})_{\overline{{{m_{j}^{\prime}- 1}}},{\bf m}_{\perp}, n}= i g{\frac{1}{2}}[ \delta \omega_{m_{j},{\bf m}_{\perp}, n}+ \delta \omega_{m_{j}- 1 ,{\bf m}_{\perp}, n}, ( A_{j})_{\overline{{{m_{j}^{\prime}- 1}}},{\bf m}_{\perp}, n}] ,
\Omega=[\begin{matrix}\Omega_{XX}&\Omega_{XY}&\Omega_{XZ}\\ \Omega_{YX}&\Omega_{YY}&\Omega_{YZ}\\ \Omega_{ZX}&\Omega_{ZY}&\Omega_{ZZ}\end{matrix}]=[\begin{matrix}P_{11}&P_{12}\\P_{21}&P_{22}\\ \end{matrix}]
\mathsf{D}_{d}Z_{\left\lfloor Tr\right\rfloor}(d) \Rightarrow-\psi(d)W(r;d)+A(r;d)=\Gamma(d)^{-1}\int_{0}^{r}(-\psi(d)+\log(r-s))(r-s)^{d-1}\mathsf{d}W(s) & =\int_{0}^{r}\mathsf{D}_{d}(\Gamma(d)^{-1}(r-s)^{d-1})\mathsf{d}W(s)=\mathsf{D}_{d}W(r;d).
\begin{aligned}\mu_{1,1}^T(u)= \|u\|_{L_T^\infty H^s}+\|D_x^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}+\|D_y^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}&+\|\partial_x u\|_{L_T^2 L_{xy}^\infty}+\|u\|_{L_x^{2}L_{yT}^{\infty}}.\end{aligned}
\iota_{r,e}\circ i_p^r(S)(xy\otimes e) = (i_p^r(S)(xy))\otimes e = (Sx)y \otimes e = ((Sx)\otimes g)(\beta_{g^{-1}}(y)\otimes g_0)\ (g_0:=(g|_{g^{-1}\cdot (p^{-1}r)})^{-1}) = \left(\iota_{p,g}(S)(x\otimes g)\right)(\beta_{g^{-1}}(y)\otimes g_0) = i_{(p,g)}^{(r,e)}\circ \iota_{p,g}(S)(xy\otimes e).
\mathbb{L}'_e(\hat{{\mathbf U}},\hat{\Psi})\dot{{\mathbf U}}:=\left[\begin{array}{c}\mathbb{L}'_e(\hat{{\mathbf U}}^+,\hat{\Psi}^+)\dot{{\mathbf U}}^+ \mathbb{L}'_e(\hat{{\mathbf U}}^-,\hat{\Psi}^-)\dot{{\mathbf U}}^-\end{array}\right]\,.
\frac{1}{n}\sum_{i=1}^nq_{i,t+1}^\top g_{i,t}(x_{i,t})+\frac{1}{n}\sum_{i=1}^n(l_{i,t}(x_{i,t})-l_{i,t}(y))\le \frac{1}{n}\sum_{i=1}^nq_{i,t+1}^\top g_{i,t}(y)+\frac{\tilde{\Delta}_t}{n}+\frac{1}{n}\sum_{i=1}^n\Delta_{i,t}(y),
\textbf{p}(t) \frac{\partial \bar{\textbf{g}}}{\partial \textbf{x}}(t,t_0,\textbf{x})+ \frac{\partial \bar{c}}{\partial \textbf{x}}(t,t_0,\textbf{x})=\textbf{p}(t_0) \frac{\partial \bar{\textbf{g}}}{\partial \textbf{x}}(t_0,t_0,\textbf{x})+ \frac{\partial \bar{c}}{\partial \textbf{x}}(t_0,t_0,\textbf{x}).
\mathcal{I}_{\mathrm{D}}^\star (\lambda) = \sup_{\gamma \in \mathcal{P}_{\mathrm{D}}}I_{\mathrm{D}, \lambda}[\gamma] = \sup_{\pi \in \Gamma(\Pi, \varphi_\lambda )}\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda \, d\pi = \sup_{\pi \in \Pi(\mu_1, \ldots, \mu_L)}\int_{\mathcal{V}}\, g_\lambda d\pi.
Q^{(l_{0}^{e},l_{1}^{e},l_{2}^{e},l_{3}^{e})}(E)=Q^{(l_{1}^{e},l_{0}^{e},l_{3}^{e},l_{2}^{e})}(E)=Q^{(l_{2}^{e},l_{3}^{e},l_{0}^{e},l_{1}^{e})}(E)=Q^{(l_{3}^{e},l_{2}^{e},l_{1}^{e},l_{0}^{e})}(E) & =Q^{(l_{0},l_{1},l_{2},l_{3})}(E)=Q^{(l_{1},l_{0},l_{3},l_{2})}(E)=Q^{(l_{2},l_{3},l_{0},l_{1})}(E)=Q^{(l_{3},l_{2},l_{1},l_{0})}(E).
\dim A\big(\mathcal{W}(p)^{D_m}\big)=\dim A_{0}\big(\mathcal{W}(p)^{D_m}\big) + \dim A_1\big(\mathcal{W}(p)^{D_m}\big) \phantom{\dim A\big(\mathcal{W}(p)^{D_m}\big)}\le m^2 p + 5 p -1 + 3p=\big(m^2 + 8\big) p -1.