harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_i g_i\gamma_i =\sum_i\gamma_i \sum_{j:\,j\neq i}\frac{r_j\gamma_j}{D(Q_j, Q_i)^2}= \sum_j r_j \gamma_j \sum_{i:\,i\neq j}\frac{\gamma_i}{D(Q_i, Q_j)^2}&\le \sum_j r_j \gamma_j \sum_{i:\,i\neq j}\frac{r_i}{D(Q_i, Q_j)^2}\le A_0 \sum_j r_j \gamma_j r_j^{-1}=A_0\sum_j\gamma_j\,.
\eta^2 \cdot \tau^2{\cal{G}}\;\; :=\;\;{\cal{Y}}_{[ p_1 ,..., p_N ]}\circ \left({1 \over 4}\sum_{i,j,k,l =1}^{N}\left( \eta^{(ij)}\eta^{(kl)}+ 2 \eta^{(ik)}\eta^{(lj)}\right) \tau^{(ij)}\tau^{(kl)}\right){\cal{G}}
A_i=\sum_{m=0}^{N}(-y)^m e_{N+i-m}=\left[t^{N+i}\right](1-yt+\cdots +(-1)^Ny^Nt^N)(1+e_1t+\cdots+e_{N+1}t^{N+1}) &= (-1)^{N+i}\left[t^{N+i}\right]\frac{(1-(yt)^{N+1})}{1-yt}\cdot t^{N+1}P(1/t).
(T F)(t, x) = \left(\frac{\widetilde F}{|\tau+|\xi|^2|^{1/4}}\right)^\vee := \frac{1}{2\pi}\int d\xi\e^{i(\xi\cdot x+\tau t)}\frac{\widetilde{F}(\tau, \xi)}{|\tau+|\xi|^2|^{1/4}}
\Omega : t^k \longmapsto \, \left(\frac{z}{1-z}\right)(f(z,t)-1)\left(f(z,1)^{k}+ tf(z,1)^{k-1}+ \cdots + t^{k}\right) = \, \left(\frac{z}{1-z}\right)(f(z,t)-1)\left(\frac{f(z,1)^{k+1}-t^{k+1}}{f(z,1)-t}\right).
F_{\rm QAM}\left[ v(\theta) \right] = \frac{4}{\pi}\left( 1 - \frac{1}{\sqrt{M}}\right) \int_{0}^{\pi /2}v(\theta){\:{\rm d}\theta}- \frac{4}{\pi}\left( 1 - \frac{1}{\sqrt{M}}\right)^{2}\int_{0}^{\pi /4}v(\theta) \:{\rm d}\theta.
\sum_{\alpha = 1}^{4}n^{\alpha}\, n^{\alpha}\, = \, 1 , \quad \sum_{\alpha = 1}^{4}n^{\alpha}\, z^{\alpha}\, = \, 0 , \quad \sum_{\alpha = 1}^{4}n^{\alpha}\, z_{, i}^{\alpha}\, = 0 .
\begin{cases}k(x-1)(x-\lambda)(x-\alpha)^2-x(x-\beta)^2=k(x^2+ux+v)^2 k(x-1)(x-\lambda)(x-\alpha)^2-\lambda x(x-\beta)^2=k(x^2+u^{\prime}x+v^{\prime})^2\end{cases}
{\frac{\partial^{2}\psi}{\partial k_{y}^{2}}}\; + \;{\frac{\partial^{2}\psi}{\partial \kappa_{y}^{2}}}\; = \; 0 \qquad \qquad \left({\frac{\partial \;}{\partial p_{y}}}{\frac{\partial \psi}{\partial \overline{{p_{y}}}}}\; = \; 0 \right)
\sum_{l=0}^\infty(1+l)^{2s}\Big|\sum_{j=0}^\infty B_{l,j}f_j\Big|^2\le \sum_{l=0}^\infty (1+l)^{2s}\Big(\sum_{j=0}^\infty |B_{l,j}|\Big)^2 &\le C^2\sum_{l=0}^\infty(1+l)^{2s}R_0^{-2l-2}\le C(\alpha,s),
(Z_{1/\Omega}\circ H)g(t,\nu) \hskip-.75in = \Omega\,\sum_{q=0}^{L-1}c_{-q}\,\sum_{k}\sum_{m}\eta_H(t+k/\Omega+qT,\nu+m\Omega) \hskip.75in e^{-2\pi i(\nu+m\Omega)qT}\,e^{-2\pi i\nu k/\Omega}.
\mbox{epi}\varphi :=\{(x, \alpha)\in\mathbb{R}^n\times\mathbb{R}\,:\, \alpha\geq \varphi(x) \}, \mbox{hypo}\varphi :=\{(x, \alpha)\in\mathbb{R}^n\times\mathbb{R}\,:\, \alpha\leq \varphi(x) \}, \mbox{dom}\varphi := \{x\in \mathbb{R}^n \,:\, |\varphi(x)|<+\infty \}.
{\cal L}={\cal L}_{0}+{\cal L}_{I}= \left\{\frac{1}{2}\, ( \partial \Phi )^{2}- \frac{1}{2}\, m^{2}\Phi^{2}\right\}+ \left\{- \frac{\lambda}{4 !}\, \Phi^{4}+{\cal L}_{\mathrm{c t}}\right\},
\kappa_{n}=\frac{\lambda^{1/\nu}}{\nu^{n-1}}(1+\frac{(-1)^{n}(\nu^{2}-1)}{24\nu^{2}}\lambda^{-2/\nu}+\frac{(-2)^{n}(\nu^{2}-1)}{48\nu^{3}}\lambda^{-3/\nu}+O(\lambda^{-4/\nu})),
\delta_n(f)(x_1,x_2,\ldots,x_{n+1}) =x_1\cdot f(x_2,\ldots,x_{n+1}) +\sum_{i=1}^n (-1)^{i}f(x_1,\ldots,x_{i-1},x_ix_{i+1},x_{i+2},\ldots,x_{n+1}) + (-1)^{n+1}f(x_1,\ldots,x_{n}).
\sum_{w\in W}\sum_{v\in\Delta_{\Lambda_{2}}}(-1)^{|w|}e^{i\left[w(\Lambda_{1})+v\right]\cdot x}=\sum_{w\in W}\sum_{\Lambda_{3}\in\Delta_{\Lambda_{1},\Lambda_{2}}}(-1)^{|w|}e^{iw(\Lambda_{3})\cdot x}.
L^{+}= \left( \begin{array}{l l}{{q^{S^{3}}}}&{{\Omega S^{-}}}\\{{0}}&{{q^{- S^{3}}}}\\ \end{array}\right) \ \ \ , \ \ \L^{-}= \left( \begin{array}{r r}{{q^{- S^{3}}}}&{{0}}\\{{- \Omega S^{+}}}&{{q^{S^{3}}}}\\ \end{array}\right)
{\frac{d}{\ell}}\approx{\frac{1}{- \nu_{2}\ell}}\operatorname{l n}\left({\frac{4 \ell^{- 1}\varphi_{( + )}\sqrt{1 +{\frac{M^{2}\ell^{2}}{4}}}}{\nu_{1}\varphi_{( - )}}}\right)
{P_{\ell,r}= \mathrm{S}_{\ell,r}\ltimes U_{\ell,r}}, &\mathrm{S}_{\ell,r}= \left\{\begin{pmatrix}A 0 0 B\end{pmatrix}|\A\in\mathrm{GL}_{\ell},\B\in\mathrm{GL}_{r}\right\}, &U_{\ell,r}= \left\{\begin{pmatrix}I_\ell X 0 I_r\end{pmatrix}|\X\in M_{\ell,r}\right\}.
E[\Delta Y_{\tau}/{\cal F}_{\tau^-}]= -\Delta A_{\tau}+ \Delta A'_{\tau}= -\Delta A_{\tau}{\bf 1}_{\{Y_{\tau^-}= \overline{\xi}_\tau\}\cap D}+ \Delta A'_{\tau}{\bf 1}_{\{Y_{\tau^-}=\underline{\zeta}_\tau\}\cap D'}{\rm a.s.}
\dfrac{d}{dt}\left(\dfrac{\partial\,\Theta}{\partial\dot{y}_k}\right)-\left(\dfrac{\partial\,\Theta}{\partial{y}_k}\right)+\displaystyle\sum_{\alpha=1}^{s_1}\left(\dfrac{\partial\,\Theta}{\partial{x}_\alpha}\right)\dfrac{\partial\,L_\alpha}{\partial\dot{y}_k}=0,
\frac{d^{2}P}{d z^{2}}+ \left[ - \frac{f^{\prime \prime}}{f^{\prime}}+ f^{\prime}\cot f \right] \frac{d P}{d z}+ f^{2}\left[ l ( l + 1 ) - \frac{m^{2}}{\sin^{2}f}\right] P = 0 .
\begin{aligned}\|\vartheta_1^\prime\|_\infty \leq C\|\vartheta_1^\prime\|_{1,2}\leq Cr\leq \sigma_0^{-3/4}\,, \|\vartheta_1\|_\infty \leq C \|\vartheta_1\|_{1,2}\leq Cr\leq \sigma_0^{-3/4}\,.\end{aligned}
{\cal K}^T M{\cal K}= \left( \begin{array}{cc}{\cal K}^{(1)}{\tilde{\cal K}}^{(1)}\end{array}\right) e^\varphi \left( \begin{array}{cc}e^{-2 \varphi}+ C^{(0)\, 2}C^{(0)}C^{(0)}1 \end{array}\right)\left( \begin{array}{c}{\cal K}^{(1)}{\tilde{\cal K}}^{(1)}\end{array}\right)\, ,
\widehat{\mathbf{H}}(\varphi)=\left\{h_{k+m}\right\}_{k,m=0}^{\infty},h_{p}=\frac{\left( -1\right)^{p+1}}{2\pi}\int_{-\infty}^{\infty}\frac{\varphi\left( \tau\right) z^{p+1}\left( \tau\right) d\tau}{\frac{1}{4}+\tau^{2}}.
P_{n}T^{\,j_0}\,e_{k}=\smash[b]{P_{n}T^{\,j_0-(b_{l+1}-k)}\ \Bigl(v_{l}\,\Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)\,e_{b_{\varphi (l)}}\Bigr),}\intertext{so that}\|P_{n}T^{\,j_0}\,e_{k}\| \smash[t]{\le|v_{l}|\,\Bigl(\prod_{s=k+1}^{b_{l+1}-1}|w_{s}|\,\Bigr)\,\sup_{j\ge 0}\ \|P_{n}T^{\,j}\,e_{b_{\varphi (l)}}\|.}
(E,h)>^*_m0 \iff (E^*,h^*)<_m0 \mathrm{and}i\Theta_{E,h}>^*_m0 \iff i\Theta_{E^*,h^*}=-i\Theta_{E,h}^\dag<_m0 \iff i\Theta_{E,h}^\dag>_m0.
\frac{(a;q)^n\prod\limits_{1\le i<j\le n}\big(a z_i^{\pm 1}z_j^{\pm 1};q\big)}{(ta;q)^n\prod\limits_{1\le i<j\le n}\big(t a z_i^{\pm 1}z_j^{\pm 1};q\big)}=\prod_{i<j}\frac{(a x_i x_j;q)}{(t a x_i x_j;q)}\biggr|_{\hat{x}=z_1,1/z_1,\dots,z_n,1/z_n},
U_{\mathrm{A P}}^{\mathrm{b f}}( \vec{x}; C ) = e^{- \mathrm{i}{\frac{\theta}{2}}\tau^{3}}U_{\mathrm{A P}}^{\mathrm{b f}}( D ( e^{\mathrm{i}{\frac{\theta}{2}}\tau^{3}}) \cdot \vec{x}; e^{\mathrm{i}{\frac{\theta}{2}}\tau^{3}}C e^{- \mathrm{i}{\frac{\theta}{2}}\tau^{3}}) e^{\mathrm{i}{\frac{\theta}{2}}\tau^{3}}
\begin{cases}\,(\phi(\tilde{u}'))'+f(\bar{u}+\tilde{u})\tilde{u}'+\lambda g(t,\bar{u}+\tilde{u})-\dfrac{\lambda}{T}\displaystyle \int_{0}^{T}g(\xi,\bar{u}+\tilde{u}(\xi)) \, d\xi =0, \,\tilde{u}\in\tilde{\mathfrak{D}}, \lambda\in\mathopen{[}0,1\mathclose{]}.\end{cases}
\pi_n \circ \Gamma^{2n}_n \circ P_n(t_\mu t^*_\nu) = \widetilde{\iota}_{n}(s_{\mu}) \Big(\sum_{p < n}\sum_{\alpha \in s(\mu) \Lambda^{p}}\Delta_{n,1}^{\mu, \nu}(p) S_{(\alpha, s(\alpha))}S_{(\alpha, s(\alpha))}^{*}\Big) \widetilde{\iota}_{n}(s_\nu^{*}).
\frac{\partial^2f}{\partial t_3\partial t_2}(0,t_1,0,t_1)=t_1^{-2}\left(1+\frac{t_1^2(z'(0))^{-1}z'''(0)}{6}-\frac{t_1^2((z'(0))^{-1}z''(0))^2}{4}+...\right)
I_{p}( 2 ) = \frac{1}{2 4}\sum_{n = 0}^{p - 1}\left( \begin{array}{c}{2 p + 1}\\{n}\\ \end{array}\right) ( 2 p + 1 - 2 n ) \{( 2 p + 1 - 2 n )^{2}- 1 \}.
D_2(L_2,S_2,U_2) &= (C_0^U+C_0^{\Delta_2})C_0^L + (1-C_0^S+C_0^{\Delta_2})\sigma_v^2/A^2 &= D_0(L,S,U)+C_0^LC_0^{\Delta_2}+C_0^{\Delta_2}\sigma_v^2/A^2.
h_{\hat{D}_l(k)}(\theta)=h_{{j}'_l(k\hat{r})}(\theta)+h_{\hat{y}_l(\hat{r};k)}(\theta)=(\hat{r}+\hat{B}(\hat{r}))|\sin\theta|,\,\theta\in[0,2\pi],\,\hat{r}_0\leq \hat{r}\leq\hat{r}_1,
\textstyle\prod\limits_{k=1}^{r}\left(\frac{\beta\,[\boldsymbol{\Lambda}]_{k,k}^2}{\beta+[\boldsymbol{\Lambda}]_{1,1}^2-[\boldsymbol{\Lambda}]_{k,k}^2}\right)=2^{{R}}\left(\frac{\sigma^2}{p_1^{(\rm id,a)}}\right)^{r}.
-\sin\psi\csc\eta\cdot\left(\frac{F[q_{0}](\frac{i}{\mu_{0}})w_{1}[q_{0}](\frac{i}{\mu_{0}})w_{3}[q_{0}](\frac{i}{\mu_{0}})}{w_{2}[q_{0}](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{2}\gamma^{0}\partial_{\phi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\sum\limits_{i=2}^n M_{Q_i}= \sum\limits_{i=2}^n \sum\limits_{Q\in \P[D]_1^*(Q_i)}(m_Q-m_{B^0})\cdot 1_Q+ (m_{B^0}-m_{Q_n}) \sum\limits_{i=2}^n \sum\limits_{Q\in \P[D]_1^*(Q_i)}1_Q+
\int_1^{\infty}\int_1^{x}\int_1^{x}\frac{dzdydx}{\left(x+y\ln x+z\ln^2 x\right)^{2\alpha_1}}+\int_1^{\infty}\int_1^{y}\int_1^{y}\frac{dzdxdy}{\left(x+y\ln y+z\ln^2 y\right)^{2\alpha_1}}&+\int_1^{\infty}\int_1^{z}\int_1^{z}\frac{dxdydz}{\left(x+y\ln z+z\ln^2 z\right)^{2\alpha_1}}< \infty.
\iota_1 :(b_1 \oplus b_2 ) \mapsto \begin{bmatrix}b_1 0 0 b_2 \end{bmatrix},E_1: \begin{bmatrix}b_1 b_2 b_3 b_4 \end{bmatrix}\mapsto \begin{bmatrix}b_1 0 0 b_4 \end{bmatrix}.
\frac{d r}{d \theta^{5}}= \tan \phi = \frac{\sin \theta^{5}- f \cos \theta^{5}}{\cos \theta^{5}+ f \sin \theta^{5}}, \quad \sin \phi = \frac{\sin \theta^{5}- f \cos \theta^{5}}{\sqrt{1 + f^{2}}}, \quad \cos \phi = \frac{\cos \theta^{5}+ f \sin \theta^{5}}{\sqrt{1 + f^{2}}}.
\begin{gathered}c_0(y,\eta,\lambda) = a_0(y)\sigma(y,\eta,\lambda) + a_1(y,\eta,\lambda)\sigma(y,\eta,\lambda), a_1(y,\eta,\lambda) = \sum_{1 \leq |\beta| \leq 2}a_{\beta}(y)\eta^{\beta}+ \lambda^2.\end{gathered}
\sum_{k,\ell}R^{pq}_{U, k \ell}(\sum_s Q_{ks}\otimes P_{si}) (\sum_{t}Q_{\ell t}\otimes P_{t j})= \sum_{k,\ell}R^{k \ell}_{V, ij}(\sum_s Q_{ps}\otimes P_{sk}) (\sum_t Q_{qt}\otimes P_{t\ell}) ,
{y^{[j]}}(n) -{y^{[j]}}({t_1}) = \sum\limits_{i = 1}^M{{{\bf{h}}^{[ji]}}(n){\bf{V}}_j^{[i]}(n){{\bf{s}}^{[ji]}}}- \sum\limits_{i = 1}^M{{{\bf{h}}^{[ji]}}({t_1}){{\bf{s}}^{[ji]}}}.
P(M_t<qt^{4\alpha(t)})= P(\max_{0\le u\le qt^{4\alpha(t)}}B(H(u))-B(H(qt^{4\alpha(t)}))>\max_{qt^{4\alpha(t)}\le u \le t}\tilde{B}(H(u))-\tilde{B}(H(qt^{4\alpha(t)}))) \le Q(t^{4\alpha(t)},t^{\frac{1}{2}-5\alpha(t)})+c_1\exp(-t^{c_2\alpha(t)}) = V( \frac{4\alpha(t)}{\frac{1}{2}-5\alpha(t)})+c_1\exp(-t^{c_2\alpha(t)}).
\begin{cases*}\partial^2_{t}\Lambda- \partial_x^2\Lambda=-2\sinh(2\Lambda)((\partial_x \phi)^2-(\partial_t\phi)^2), \partial_t^2 \phi-\partial_x^2 \phi=-\dfrac{\sinh(2\Lambda)}{\sinh^2(\Lambda)}(\partial_t \phi \partial_t \Lambda -\partial_x \phi \partial_x \Lambda). \end{cases*}
g = g_1g_0g_{-1}= \begin{pmatrix}1 0 0 \xi I 0 \frac{1}{2}\|\xi\|^2 \xi^T 1\end{pmatrix}\begin{pmatrix}\alpha 0 0 0 A 0 0 0 \alpha^{-1}\end{pmatrix}\begin{pmatrix}1 v^T \frac{1}{2}\|v\|^2 0 I v 0 0 1\end{pmatrix}
c_{2,5}(t)=4 a_7(t), c_{2,6}(t) = 10 a_8(t), c_{3,5}(t)= a_8(t), c_{2,7}(t)= 10 (34,78,118,148,168,178,181,178,168,148,118,78,34) \stackrel{?}{=}10 a_9(t).
\zeta_{T}( \nu | \beta ) ={\frac{\varrho^{- 2 \nu}}{\pi i}}\int_{\mu}^{\infty}d \omega \int_{C_{+}}d z \ln \left( 1 - e^{i \beta z}\right) \left[{\frac{2 \nu z \breve{\Phi}( \omega ; z )}{( z^{2}+ \omega^{2})^{\nu + 1}}}-{\frac{\partial_{z}\breve{\Phi}( \omega ; z )}{( z^{2}+ \omega^{2})^{\nu}}}\right] .
e_k^{s,N}= 2\bigg(t_1 \cos(\frac{2ks\pi}{N})+ t_2 \sin(\frac{2ks\pi}{N})\bigg)^2 \big|u_1^{s,N}(C_{0,0}) - u_1^{s,N}(B_{0,0})\big|^2.
\mathfrak{a}_{\Omega,4m,a,b,q}(f_{K,\Omega,j},f_{K,\Omega,j})) - \lambda \, \mathfrak{b}_{\Omega,2m,a,b,q}(f_{K,\Omega,j},f_{K,\Omega,j}) = (\lambda_{K,\Omega,j}- \lambda) \|f_{K,\Omega,j}\|_{L^2(\Omega)}^2 < 0,
L(A_\mu, \phi, \theta)={-d \over F(\phi)}G_{\mu \nu \rho \sigma}G^{\mu \nu \rho \sigma}-{\theta \over 4}\epsilon_{\mu \nu\rho \sigma}G^{\mu \nu \rho \sigma}-{1 \over 2}\partial^\mu\phi\partial_\mu\phi-{1 \over 2}m^2 \phi^2,
<\overline{\psi}_{L}^{({\cal N})}\ldots \overline{\psi}_{L}^{(1)}\psi_{R}^{(1)}\ldots \psi_{R}^{({\cal N})}>=(\frac{e^{\gamma}}{4\pi})^{\cal N}(\sqrt{\frac{\cal N}{\pi}}e_{c})^{\cal N}\quad .
\tau^0[G(U_N, U_N^*)]= \frac{1}{N}\sum_{\substack{\kappa :V \to [N] \kappa}}\mathbb{E}\left[ \prod_{(u,v)\in E_1}U_N(\kappa(v), \kappa(u)) \prod_{(r, s)\in E_2}\overline{U_N(\kappa(r), \kappa(s))}\right].
I_{3}= \frac{u_{a_{j}}( a_{i})}{(\frac{\lambda_{i}}{\lambda_{j}}+ \lambda_{i}\lambda_{j}\gamma_{n}G_{a_{j}}^{\frac{2}{2-n}}( a_{i}) )^{\frac{n}{2}}}\int_{\mathcal{A}}\frac{O(\frac{\lambda_{j}}{\lambda_{i}}\vert x \vert^{2})}{(1+r^{2})^{\frac{n+2}{2}}}
\pi_{bt}\overline{\gamma}\xi_{sa}(n) = c_{k}\displaystyle \sum_{v \in L(k)}v^{\ast}(ts) [bva]n &= c_{k}\left[b \displaystyle \sum_{v \in L(k)}v^{\ast}(ts)v a \right]n &=c_{k}[btsa]n &=c_{k}btsan &=c_{k}\pi_{bt}\beta \alpha \xi_{sa}(n)
N_{(1,9)}(n) = 8\,\sigma(n) - 32\,\sigma(\frac{n}{4}) + 8\,\sigma(\frac{n}{9}) - 32\,\sigma(\frac{n}{36}) + 64\,W_{(1,9)}(n) + 1024\,W_{(1,9)}(\frac{n}{4}) - 256\,\biggl( W_{(4,9)}(n) + W_{(1,36)}(n) \biggr).
\sum^{\infty}_{n,m=-\infty}q^{2n^2+2nm+2m^2}=\frac{3\eta(q)^2\eta(q^4)^2\eta(q^6)^{15}}{4\eta(q^2)^5\eta(q^3)^6\eta(q^{12})}+\frac{\eta(q^2)^{15}\eta(q^3)^2\eta(q^{12})^2}{4\eta(q)^6\eta(q^4)^6\eta(q^6)^5}
\begin{array}{ll}{\mathcal A}(c,d,\nu):=\{w \in W^{1,1}(Q_\nu;\mathbb R^d): w(y)= c \hbox{if}y\cdot \nu = \frac{1}{2}, w(y)= d \hbox{if}y \cdot \nu=-\frac{1}{2}, &w \hbox{is}1- \nu_1,\dots, \nu_{N-1}\, \}.\end{array}
\psi_{( 4 )}^{\alpha}= ( i \tau_{y}\psi )_{\alpha}= ( - i \psi^{T}\tau_{y})_{\alpha}, \quad{\bar{\psi}}_{( 4 )}^{\dot{\alpha}}= ( i \tau_{y}\psi^{*})_{\alpha}= ( - i \psi^{\dagger}\tau_{y})_{\alpha}.
\langle L^*_1(\hat R), y_0 \rangle = \langle \hat R, L_1(y_0) \rangle = \langle \hat R, V^TCV \rangle = V \hat R V^T \cdot C, \langle L^*_2( Z), y_0 \rangle = \langle Z, L_2(y_0) \rangle = \langle Z, U^TCV \rangle = U Z V^T \cdot C,
{\cal A}( g_{1}g_{2})^{\alpha \beta}\: = \:{\cal A}( g_{2})^{\alpha \beta}\: + \: g_{2}^{*}{\cal A}( g_{1})^{\alpha \beta}\: - \: \theta ( g_{1}, g_{2})^{\alpha}\: + \: \theta ( g_{1}, g_{2})^{\beta}\: + \: d \log \Omega_{\alpha \beta}^{g_{1}, g_{2}}
1 = \int \! D B_{+}^{1}\delta ( B_{+}^{1}) D B_{+}^{2}\delta ( B_{+}^{2}) \exp \{i \int \! d^{2}x \, [ \psi_{-}^{\dagger 1}B_{+}^{1}\psi_{-}^{1}+ \psi_{-}^{\dagger 2}B_{+}^{2}\psi_{-}^{2}] \}\quad ,
C_{\sigma^3 t}=\;C_{\sigma^2 t\sigma}-\frac{\bar{g}_t}{\bar{g}}C_{\sigma^3}=\;-\frac{1}{3}\Big(\beta\mu_{\sigma^2}+4\mu\beta_{\sigma^2}+3\beta\mu^2+5\beta_{\sigma}\mu_{\sigma}+3\alpha\mu_{\sigma}+\beta_{\sigma^4}\Big)C_{\sigma}\qquad\qquad-\mu\Big(\alpha+\beta_{\sigma}\Big)C_{\sigma^2}.
\tilde{F}^{( i )}( \frac{z_{1}}{z_{2}q^{2 n}}) = \mathrm{t r}_{V ( \Lambda_{i})}( q^{- 2 \rho}\tilde{\Phi}_{\Lambda_{i - 1}}^{\Lambda_{i}V^{*}}( z_{1}) \tilde{\Phi}_{\Lambda_{i}}^{\Lambda_{i - 1}V^{*}}( z_{2}q^{2 n}) )
W_{L}= \sum_{s , r = 1}^{N_{f}}( \lambda_{r , s}M +{m_{Q}}_{r , s}) V^{2 r - 1 , 2 s}+ ( \mathrm{P f}V )^{\frac{1}{N_{f}- 2}}( 2 - N_{f}){\Lambda_{L}}^{\frac{N_{f}- 6}{N_{f}- 2}}+ \sum_{k = 2}^{N_{c}}g_{r}{u^{c l}}_{r}, \nonumber
\mu_{a\rightarrow v}(x_{v})=\sum_{x_{a}^{\prime}:x_{v}^{\prime}=x_{v}}(f_{a}(x_{a}^{\prime})\prod_{v^{*}\in N(a)\backslash\{v\}}\mu_{v^{*}\rightarrow a}(x_{v^{*}}^{\prime}))
\delta E_{k}( \rho_{k}, n_{k}) = \frac{1}{2}\langle \psi_{0}^{( v k + 1 )}| - \frac{d^{2}}{d x^{2}}+ W_{v k}^{2}+ W_{v k}^{\prime}| \psi_{0}^{( v k + 1 )}\rangle .
\lim_{n\to \infty}\frac{1}{n}\sum^{n-1}_{j=0}\varphi \circ f_{\omega}^{(j)}(X_{s^{\prime \prime}}(\omega)) = \lim_{n\to \infty}\frac{1}{n}\sum^{n-1}_{j=0}\Phi_h \circ \Theta_{E_k}^j (\omega ,x),
J \cap K = ((A \setminus \{\bar{a}_n\}) \cup [\bar{b}_n \mapsto \bar{a}_n](G)) \cap (C_1 \cup [\bar{b}_n \mapsto \bar{a}_n](H)) &\subseteq ((A \setminus \{\bar{a}_n\}) \cup [\bar{b}_n \mapsto \bar{a}_n](G)) \cap ((C \setminus \{\bar{a}_n\})\cup [\bar{b}_n \mapsto \bar{a}_n](H))
\mathcal{Y}= \big\{v \in L^{2}_{\sigma}(\mathbb{R}^2) \big| v_F \in C^{\infty}_{\sigma, c}(\mathbb{R}^2) v_R\in\mathcal{R}v|_{\mathcal{F}}= v_F|_{\mathcal{F}}v|_{\mathcal{S}_0}= v_S|_{\mathcal{S}_0}\big\},
\partial_1u_3(y) = (\partial_1|u(y)|)\cos\theta_3(y) -|u(y)|\sin\theta_3(y)\,\partial_1\theta_3(y), \partial_3u_1(y) = (\partial_3|u(y)|)\sin\theta_1(y) +|u(y)|\cos\theta_1(y)\,\partial_3\theta_1(y)
{\bf y}_r= \sum\limits_{i=1}^3 \sum\limits_{j \neq i}{\bf H}_{i,r}{\bf V}_{i,j}^p{\bf s}_{i,j}^p+\sum\limits_{(i,j) \in{\cal S}}{\bf H}_{i,r}{\bf V}_{i,j}^c{\bf s}_{i,j}^c &+{\bf H}_{2,r}{\bf V}_{2,3}^r{\bf s}_{2,3}^r+{\bf H}_{3,r}{\bf V}_{3,1}^r{\bf s}_{3,1}^r+{\bf n}_r,
S_2 =\frac{2c(s)}{(2n)^{s+1}}\bigg( - \left(\frac{2^{s+1}-1}{n^2-1}+ \frac{1}{n+1}\right)\zeta(s+1) &~~~~~~ + \sum_{j=1}^{\infty}\frac{1}{j^{s+1}}\sum_{k=1}^{n/2}\frac{2}{4k^2 - 1}\sum_{q=1}^{\infty}\frac{(s+1)_{2q}}{(2q)!}\left( \frac{k}{jn}\right)^{2q}\bigg) + \mathcal{O}(n^{-s-2}).
\beta_{d,t}(x) \xleftarrow{\varepsilon \to 0}\frac{P(|Y_x^-(t)|\leq \varepsilon)}{C_d\varepsilon^d}&\leq \frac{1}{C_d}\prod_{i=1}^d \frac{P(|A_i(t)|\leq \varepsilon)}{\varepsilon}& \xrightarrow{\varepsilon \to 0}\frac{2^d}{C_d}\prod_{i=1}^d \beta_{1,t}(x_i)
\mathbb{E}_{x}\left[\int_{0}^{t}\int_{\lbrace z>1\rbrace}\left\vert V(X_{s-}+z)-V(X_{s-})\right\vert \nu(\mathrm{d}z)\mathrm{d}s\right] \quad\leqslant t\int_{\lbrace z>1\rbrace}\left(\log(2)+\log(z)\right)\nu(\mathrm{d}z) \quad=t\log(2)\nu(\lbrace z>1\rbrace)+t\int_{\lbrace z>1\rbrace}\log(z)\nu(\mathrm{d}z)<\infty,t\geqslant0,
\rho_i^{(0)}= \frac{b\,m}{d-b}, \rho_i^{(1)}= \frac{b\,d\,m\,(N+1)}{(d-b)^2 \,N+2\,(d-b) \,m-(d-b)^2}- \frac{2\,b\,d\,m \,i}{(d-b)^2 \,N+2\,(d-b) \,m-(d-b)^2},
b(\mathbf{D})= \begin{pmatrix}D_1 0 \frac{1}{2}D_2 \frac{1}{2}D_1 0 D_2 \end{pmatrix}, g(\mathbf{x})= \begin{pmatrix}1 0 0 0 g_2 (x_1) 0 0 0 g_3 (x_1) \end{pmatrix},
i\hbar\frac{\partial}{\partial t}\Delta\langle\hat{B}_{\omega}^{\dagger}\hat{B}_{\omega^{\prime}}\rangle=(\hbar\omega^{\prime}-\hbar\omega)\Delta\langle\hat{B}_{\omega}^{\dagger}\hat{B}_{\omega^{\prime}}\rangle+i\sum_{k}[F_{\omega^{\prime}}^{*}\Pi_{k,\omega}+F_{\omega}\Pi_{k,\omega^{\prime}}^{*}]
\Upsilon_{k'j'}\triangleq \begin{bmatrix}G_{k'j'}+G_{k'}'\sum_{i\in [j':k'-1]}H_{k'i}F_{ij'}G_{k'}'(H_{k'j'}F_{j'}'+H_{k'j'}') \sum_{i\in [j':k'-1]}H_{k'i}F_{ij'}H_{k'j'}F_{j'}'+H_{k'j'}' \end{bmatrix}.
Var(\hat{\pi}^a_{h^*, T_n}(x)) = \frac{2}{T_n^2}\int_0^{T_n}\int_0^{t}k(t,s) 1_{s < t}\, \big(1_{|t -s| \le h_1^* h_2^*}& + 1_{h_1^* h_2^* \le |t -s| \le (\prod_{j \ge 3}h_j^*)^{\frac{2}{d-2}}}+ 1_{(\prod_{j \ge 3}h_j^*)^{\frac{2}{d-2}}\le |t -s| \le D}+ 1_{D \le |t -s| \le T_n}\big) ds dt & = \sum_{j = 1}^4 \tilde{I}_j,
\overset{\longleftarrow}{\prod_{\mu \in \mathbb{Q}_{> 0}}}\mathbf{1}_{\mu}^{\theta}\star \mathbf{1}^{\sigma, \theta}= \mathbf{1}^{\sigma}= \overset{\longleftarrow}{\prod_{\mu \in \mathbb{Q}_{> 0}}}\mathbf{1}_{\mu}^{\theta^{\prime}}\star \mathbf{1}^{\sigma, \theta^{\prime}}.
\begin{array}{rcl}S_{g}={\displaystyle\int}d^{4}x\, e \left\{R(e,\omega) +6g^{2}+2 e^{-1}\epsilon^{\mu\nu\rho\sigma}\bar{\psi}_{\mu}\gamma_{5}\gamma_{\nu}\left(\hat{\cal D}_{\rho}+ig A_{\rho}\sigma^{2}\right)\psi_{\sigma}-{\cal F}^{2}\right. & & \left. +{\cal J}_{(m)}{}^{\mu\nu}({\cal J}_{(e) \mu\nu}+{\cal J}_{(m) \mu\nu}) \right\}\, ,\end{array}
\widetilde{\omega}^{- 2}= \omega^{- 2}\left[ 1 + \epsilon \eta \left( x^{i}, v \right) \right] \qquad \mathrm{a n d}\qquad \widetilde{B}= B \left( x^{i}, v \right) \left[ 1 + \epsilon \eta \left( x^{i}, v \right) \right]
\left|\frac{\partial\tilde{\varphi}}{\partial t}(x,t)\right| = \left|\frac{\partial\tilde{\varphi}}{\partial t}(x,t)-\frac{\partial\tilde{\varphi}}{\partial t}(y,t)\right| & = |u(x,t)-u(y,t)| & \leq \theta(t) & \leq Ce^{-\eta t},
\begin{aligned}V_{(1,1)}\otimes V_{(1,1)}= V_{(2,2)}\oplus V_{(3,0)}\oplus V_{(0,3)}\oplus V_{(0,0)}V_{(1,1)}\otimes V_{(1,0)}= V_{(2,1)}\oplus V_{(1,0)}\oplus V_{(0,2)}, V_{(1,1)}\otimes V_{(0,1)}= V_{(1,2)}\oplus V_{(0,1)}\oplus V_{(2,0)}. \end{aligned}
\Theta(Q_j) \leq \frac1{c_{db}}\,p(Q_j) = \frac1{c_{db}}\Biggl (\sum_{k=0}^{j-1}\frac{\ell(Q_j)}{\ell(Q_{j-k})}\,\Theta(Q_{j-k})+ \frac{\ell(Q_j)}{\ell(Q_0)}\,p(Q_0)\Biggr).
\Xi_{il}(\boldsymbol{\alpha}) \,\, := \,\,\,\, \alpha_{l}\, \varphi_{i}(q_{l}) \,, \hbox{for}i=1 \dots, d \hbox{and}l=1, \dots, K \, , \Theta_{ij}(\boldsymbol{\beta},\boldsymbol{\gamma}) \,:= \,\,\, \beta_{j}\, \Delta\varphi_{i}(p_{j}) \, + \,\gamma_{j}\, \varphi_{i}(p_{j}) \, , \,\hbox{for}i=1 \dots, d \hbox{and}j=1, \dots, N \, .
\alpha_t(\lambda) = \begin{pmatrix}1 0 0 1\end{pmatrix}, \beta_t(\lambda) = \begin{pmatrix}e^{-(2b+1)u_t+m\pi\sqrt{-1}}\frac{\sinh\bigl((2b+1)u_t\bigr)}{\sinh(u_t/2)}0 e^{(2b+1)u_t-m\pi\sqrt{-1}}\end{pmatrix}
W^{\mu \nu}= \frac{\partial^{2}L}{\partial \dot{x}_{\mu}\partial \dot{x}_{\nu}}= - \frac{m}{\sqrt{\dot{x}^{2}}}\left( g^{\mu \nu}- \frac{\dot{x}^{\mu}\dot{x}^{\nu}}{\dot{x}^{2}}\right) = - m \left( g^{\mu \nu}- \dot{x}^{\mu}\dot{x}^{\nu}\right) \; ,
0 = (D_1^6-5 D_1^3 D_3-5 D_3^2 +9 D_1 D_5 )(Z,Z)(\mathbf{0}) 0 = (D_1^8 + 7D_1^5D_3 -35D_1^2D_3^2 - 21D_1^3D_5 - 42D_3D_5 + 90D_1D_7)(Z,Z)(\mathbf{0})
\begin{aligned}\left| \int_\Gamma f\,d\gamma - \int_{\Gamma_h}f\circ\pi\,d\gamma_h \right| \le C\delta \|f\|_{L^1(\Gamma)}, \|f\|_{L^p(\Gamma(\delta))}\le C(\delta^{1/p}\|f\|_{L^p(\Gamma)}+ \delta \|\nabla f\|_{L^p(\Gamma(\delta))}), \|f - f\circ\pi\|_{L^p(\Gamma_h)}\le C\delta^{1-1/p}\|\nabla f\|_{L^p(\Gamma(\delta))},\end{aligned}
\frac{1}{T^{\rm on}_{\rm DR}(E)}=\frac{1}{C_{\rm DR}}+ \frac{2 \mu}{(4 \pi)^{w}}(-2 \mu E)^{w-1}\Gamma(1-w) + i \frac{2 \pi \mu}{\Gamma(w) (4 \pi)^{w}}(2 \mu E)^{w - 1}.
W|_K^{-1}T W|_K P_K (S^ne_j) = P_K (S^{n+1}e_j) = (P_K S) (S^{n}e_j) &=(P_K S) (P_K S^{n}e_j) + (P_K S) (P_{K^\bot}S^{n}e_j) &\stackrel{(1)}{=}(P_K S) (P_K S^{n}e_j),
\int_{T_{i-1}}^{T_i}\frac{C_T}{t^{\frac{d}{p}}}\|f_t\|^2_{L^p(\mathbb{R}^d)}=0.1 1\leq i\leq m-1 , \int_{T_{m-1}}^{Tm}\frac{C_T}{t^{\frac{d}{p}}}\|f_t\|_{L^p(\mathbb{R}^d)}^2\leq 0.1.
\|\Phi_{j, \alpha}(f)\|_p = \left \|\,\lim_{\epsilon \to 0}\int_{|y| \geq \epsilon}\frac{e^{i\,t(\varphi(x+y\,\vec e_j)-\varphi(x))}-1}{|y|^{1+\alpha}}\,f(x+y\,\vec e_j)\, dy \, \right \|_p.
\frac{1}{8}\int (\psi(x)-\psi(y))^2 h(|x-y|)\,dx\,dy=\frac{1}{8}\int\frac{(\psi(x)-\psi(y))^2}{\sin^2((x-y)/2)}\,dx\,dy=\frac{1}{2}\int\frac{(\phi(z)-\phi(w))^2}{|z-w|^2}\alpha(dz)\alpha(dw)
L(y)=\exp\Big(\int_0^1\int_0^1b(\xi,\zeta,y_{\xi,\zeta}) \mathrm{d}y_{\xi,\zeta}-\frac{1}{2}\int_0^1\int_0^1|b(\xi,\zeta,y_{\xi,\zeta})|^2\mathrm{d}\xi \mathrm{d}\zeta\Big)
\kappa_I\epsilon^{ij}\nabla_j a^I_0({\bf r},t)+i\hbar\sum_p\frac{q_p^I}{2m_pc}\left[(D_i\Psi_p)^\dagger({\bf r},t)\Psi_p({\bf r},t) -\Psi_p^\dagger({\bf r},t)(D_i\Psi_p)({\bf r},t)\right]=0\,,
\Big\|u_{in}-\sum_{s=j+1}^{m}\phi_i(g_s)\,(w_{i}\circ g_s^{-1})(\,\cdot\,-g_s\xi_{rn})\Big\|^2=\Big\|u_{in}-\sum_{s=j}^{m}\phi_i(g_s)\,(w_{i}\circ g_s^{-1})(\,\cdot\,-g_s\xi_{rn})\Big\|^2+\|w_{i}\|^2+o_n(1),
P^{\vec{\mathcal{H}}}_{\Lambda,\eta,\beta}(\sigma)=\frac{1}{Z^{\vec{\mathcal{H}}}_{\Lambda,\eta,\beta}}\exp{\left[\beta\sum_{\{u,v\}}\sigma_u\sigma_v+\beta\sum_{\{u,v\}: u\in\Lambda, v\in\partial_{ex}\Lambda}\sigma_u\eta_v+\sum_{u\in\Lambda}\mathcal{H}_u\sigma_u\right]},
lA_1+nB_1=(U^n_l)\sum_{q=1}^{l}Str\left(\ldots \phi^{i_{2k-1}'}\ldots \underbrace{\partial_{a_q}\phi^{j_q}\phi^{i_{2k}'}}\ldots \right)+(U^n_l)\sum_{s=1}^{n}Str\left(\ldots \phi^{i_{2k-1}'}\ldots \underbrace{\phi^{i_s}\phi^{i_{2k}'}}\ldots \right)
\bar D_{3,a,5}= \frac{|X_1-X_2|}{|z|}\big| \big[ D_\xi Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,z+t_0(X_\tau))) &\qquad- D_\xi Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,t_0(X_\tau))) \big] D^2_{tt}\tilde h(X_\tau,z+t_0(X_\tau)) D_X t_0(X_\tau)|
\|\overline{u}\|_{L^\infty(\Omega_{R_1})}\le \left[\frac{C}{R_1-R_2}\left(1+\frac{\|f\|_q^\beta}{k^{\beta(m-1)}}\right)\right]^{1/(1-\chi)}\|\overline{u}\|_{L^{m\chi^{n_0}}(\Omega_{R_{n_0}})}.
c_j^{kl}(A) = \int_Y rAe_j\cdot \nabla v^{kl}&= \int_Y (r-1)(Ce_j\cdot \nabla v^{kl}) + \int_Y (ra-\bar{a})(Me_j\cdot \nabla v^{kl}) =m_{kl}\int_Y (M:D^2 w)(Ce_j\cdot \nabla w) - m_{kl}\int_Y (C:D^2 w)(Me_j\cdot \nabla w) &= 0,
\langle{\cal A}\rangle_{0}\lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal A B}}){\cal B}\rangle_{0}- \langle{\cal A}\rangle_{0}\lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal B}}){\cal B}\rangle_{0}= 0 .
\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{q^{Q+sp}(y^{-1}\circ x)}dxdy=\int_{\Omega}\omega(x)|u(x)|^{p-2}u(x)v(x)dx
\int_{0}^{\infty}\frac{x^{2n}}{coshax}dx=d_{2n}(\frac{\pi}{a})^{2n+1},d_{2n}=\frac{(-1)^{n}}{2}(\frac{1}{4^{n}}-\sum_{k=0}^{n-1}(-1)^{k}(\begin{matrix}2n\\ 2k\end{matrix})d_{2k}),d_{0}=\frac{1}{2}
L_{i_{1}}^{( 1 )}( x ) : = \int d^{2}y \left[ W_{i_{1}j}^{( 1 )}( x , y ) \ddot{\varphi}^{j}( y ) + \alpha_{i_{1}}^{( 1 )}( y ) \delta^{2}( x - y ) \right] = 0 , i_{1}= 1 , \cdots , 1 1 ,