harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sigma \big( D_0(b) \big) = \sigma_{\rm ess}\big( D_0(b) \big) = \big\lbrace \cup_{q=1}^{\infty}\big\lbrace \Lambda_q^- \big\rbrace \big\rbrace \bigcup \big\lbrace \cup_{q=0}^{\infty}\big\lbrace \Lambda_q^+ \big\rbrace \big\rbrace.
S_{\epsilon, \delta}(q) : = \{x \in A \ | \ \pi_x (B_\epsilon (q)) >0 \ \& \ \hbox{there exists $S\in{\mathcal A}_{\epsilon, \delta}(q)$ such that}\hbox{for each vertice $q_j$ of $S$,}\pi_x (B_\epsilon (q_j)) >0 \}.
H\left({\left. M \right|{{{\bf Y}}^{n-1}},{{{\bf S}}_0}={{{\bf{\tilde s}}}_0}}\right) \leq 1 + \Pr \left( \left.{M \ne \hat M}\right|{{{\bf S}}_0}={{{\bf{\tilde s}}}_0}\right)\cdot nR &\stackrel{(a)}{\leq}1 +{\epsilon_1}\cdot nR,
h(\vec{W}) = \phi^{-1}(\alpha^{(1)}(\vec{W}), \alpha^{(2)}(\vec{W}), \alpha^{(3)}(\vec{W}), \ldots) &\overset{(a)}{=}\phi^{-1}(X_1, X_2, X_3, \ldots) &\overset{(b)}{=}X
\begin{cases}x_j^{k+1}=\arg\min\{\mathcal{L}(x_1^{k+1},\dots,x_{j-1}^{k+1}, x_j,\tilde x_{j+1}^k,\dots, \tilde x_s^k, y^k)+\frac{\beta}{2\alpha_j}\|x_j-x_j^k\|_{2}^2: x_j\in\mathbb{R}^{n_j}\}, j\in\mathbb{N}_s, y^{k+1}=y^k+\beta(\sum_{i=1}^s{A_ix_i^{k+1}-b}), \tilde x_j^{k+1}=2x_j^{k+1}-x_j^k, j\in\mathbb{N}_s.\end{cases}
\langle \delta j^{\mu}( x ) \rangle \simeq \frac{1}{2}\int \frac{d^{\, 3}p}{( 2 \pi )^{3}}\left[ \left( 1 ,{\bf p}/ \omega_{+}\right)^{\mu}\frac{- i \Sigma_{o f f}^{( b a r e ) ( + )}}{| I m \Sigma_{R}^{( + )}|}- \left( 1 , -{\bf p}/ \omega_{-}\right)^{\mu}\frac{- i \Sigma_{o f f}^{( b a r e ) ( - )}}{| I m \Sigma_{R}^{( - )}|}\right] .
s_\gamma(x)-s_\delta(x) =\frac{\cosh(x_\gamma)\sinh(x_\delta)-\cosh(x_\delta)\sinh(x_\gamma)}{\sinh(x_\gamma)\sinh(x_\delta)}=\frac{\sinh(x_\delta-x_\gamma)}{\sinh(x_\gamma)\sinh(x_\delta)}&=\sinh[i\pi(\delta-\gamma)]r_\gamma(x)r_\delta(x)=-i\sin[\pi(\gamma-\delta)]r_\gamma(x)r_\delta(x).
\lambda\geq(p-1)C_7^{-1}\|(y_1,y_2)\|_{1,p}^{1-\frac{\kappa_1+\kappa_2+2}{p}}=(p-1)C_7^{-1}\|(y_1,y_2)\|_{1,p}^{\frac{p-(\kappa_1+\kappa_2+2)}{p}}.
\Phi_{m,n}(t):= \begin{cases}\displaystyle\sum_{j=1}^{m}e_j \otimes \Phi\left(\tfrac{t^{n}_{k}+t^{n}_{k+1}}{2}\right)e_j, t \in (t_k^n, t_{k+1}^n),\, k=0, \ldots, N_n-1, \displaystyle\sum_{j=1}^{m}e_j \otimes \Phi(t^{n}_k)e_j, t=t_k^n,\, k=0, \ldots, N_n, \end{cases}
S_{B H}= 2 \pi \left( \sqrt{N_{5}^{( 1 )}}+ \sqrt{\bar{N}_{5}^{( 1 )}}\right) \left( \sqrt{N_{5}^{( 2 )}}+ \sqrt{\bar{N}_{5}^{( 2 )}}\right) \left( \sqrt{N_{5}^{( 3 )}}+ \sqrt{\bar{N}_{5}^{( 3 )}}\right) \left( \sqrt{n}+ \sqrt{\bar{n}}\right)
{\tilde{\delta}}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] = \int ~{\cal D}\lambda_{a}~ e x p ~ i \int \lambda_{a}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] d t
\underset{p_k, q_k, \tau_k}{\max}\mathcal{U}_1=\sum\limits_{k=1}^K R^{ub}_{k}& L\sum\limits_{i=1}^l q_i \leq \sum\limits_{i=1}^l e^{r}_{i}, \forall l \in [1:K], & LT\sum\limits_{i=1}^l p_i \leq \sum\limits_{i=1}^l e^{t}_{i}, \forall l \in [1:K], & \tau_k\in[0,T),~p_k \geq 0,~q_k\geq0,~ ~\forall k \in [1:K].
e \left( b \right) \equiv \sum_{a \, \in q \left( b \right) - \left\{b \right\}}\left[ m \left( a \right) - 1 \right] \; = \; \left[ \sum_{a \, \in q \left( b \right) - \left\{b \right\}}m \left( a \right) \right] - o \left( b \right) + 1
\mathrm{i i}) m_{\alpha}{\bf j^{\alpha}}\ne 0 , \qquad \qquad \frac{\partial}{\partial u}{\bf{\hat{C}}}= ( \log r ) \frac{1}{2}m_{\alpha}{\bf j^{\alpha}}( u , \phi ) \quad . \qquad \qquad \qquad \qquad
\epsilon_3(\beta, \lambda)= \frac{C}{2\lambda V}-\frac{1}{\beta V}\sum_{k=2}^{\infty}g(k)\biggl(\frac{\beta}{\lambda}\biggr)^k+\frac{\lambda}{4\beta^2 V}e^{-\frac{\lambda}{4\beta}}Ei\biggl(\frac{\lambda}{4\beta}\biggr)
\delta_{}\left(\tilde{\sigma}^{*}_{3}(t)I\right)=\int^{t_{2}}_{t_{1}}\left[-\dot{P}_{1}-Q^{1}\right]\delta Q^{1}dt+\left[-P_{1}+\dot{Q}^{1}\right]\delta P_{1}dt+\left[\Psi_{2}\delta\Xi^{2}+P_{1}\delta Q^{1}\right]^{t_{2}}_{t_{1}}
m_3 =-\frac{1}{3}\left[\mu+\frac{-1-i\sqrt{3}}{2}\left(\mu+\sqrt{3}\mu^{1/2}-\frac{3}{2}+O(\mu^{-1/2})\right)+\frac{-1+i\sqrt{3}}{2}\left(\mu-\sqrt{3}\mu^{1/2}-\frac{3}{2}+O(\mu^{-1/2})\right)\right] &=-\frac{1}{3}\left[\frac{3}{2}-3i\mu^{1/2}+O(\mu^{-1/2})\right] &=-\frac{1}{2}+i\mu^{1/2}+O(\mu^{-1/2}).
\binom{n}{k^*}=\binom{n}{cn}\sim \frac{(n/e)^n}{(cn/e)^{cn}((1-c)n/e)^{(1-c)n}}\frac{\sqrt{2\pi n}}{\sqrt{2 \pi n c}\sqrt{2 \pi n (1-c)}}= \frac{1}{c^{nc}(1-c)^{(1-c)n}}\frac{1}{\sqrt{2 \pi n c (1-c)}}.
S_{m+1}^{(k)}=\sum_{i=m-10}^{m+1}x_{i}^{k}=(\sum_{i=m-11}^{m}x_{i}^{k})-x_{m-11}^{k}+x_{m+1}^{k}=S_{m}^{(k)}-x_{m-11}^{k}+x_{m+1}^{k}.
\tilde{C}_{\mu}^{\alpha^{\prime}}M_{\alpha^{\prime}\beta^{\prime}}\tilde{C}^{\alpha^{\prime}\mu}= \sum_{( a )}\bar{\tilde{\phi}}_{V}^{( a )}\left[ - \tilde{g}_{3}\tilde{C}_{\mu}^{\alpha}\frac{\lambda_{\alpha}}{2}- \tilde{g}_{1}( \tilde{n}+ 1 / 3 ) \tilde{B}_{\mu}\right]^{2}\tilde{\phi}_{V}^{( a )},
{F_\lambda(0)}+2\sum_{m=1}^\infty{(H_1{F_\lambda})\bigr(m{t}^{1/\lambda}\bigr)}=f(0)+2\sum_{m=1}^\infty{(H_\lambda{f})\bigr(m^\lambda{t}\bigr)}=\frac{2}{{t}^{1/\lambda}}\sum_{m=1}^\infty\bigr(\widehat{H_1{F_\lambda}}\bigr)\left(\frac{m}{{t}^{1/\lambda}}\right)
F_1 = u_1 ; F_2 = u_2 \, , &F_{1,1}= u_{(2,0)}; \frac{1}{2}\left( F_{1,2}+ F_{2,1}\right) = u_{(1,1)}; F_{2,2}= u_{(0,2)}\, , &G_1^1 + G_2^2 = - q \, , &G_1^{(2,0)}+ \frac{1}{2}\, G_2^{(1,1)}= -p^1 ; \frac{1}{2}\, G_1^{(1,1)}+ G_2^{(0,2)}= -p^2 \, , &p^{(2,0)}- u_{(2,0)}= 0 ; p^{(1,1)}- 2u_{(1,1)}= 0 ; p^{(0,2)}- u_{(0,2)}= 0 \, .
\psi_{R}^{a}( 0 , x^{-}) = \frac{1}{\sqrt{2 L}}\sum_{n = \frac{1}{2}, \frac{3}{2}, \dots}^{\infty}\left( b_{n}^{a}e^{- i k_{n}^{+}x^{-}/ 2}+ d_{n}^{a \dag}e^{i k_{n}^{+}x^{-}/ 2}\right) ,
0=\sum_{d_1,d_2<D}\frac{1}{\sqrt{d_1d_2}}\sum_{Id_1, Jd_2<D}\alpha_{Id_1}\bar{\alpha}_{Jd_2}\sum_{L, M\geq 1}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}\times \sum_{f\in \mathcal{B}_{2k}(p)}\omega_f\lambda_f(Id_1)\lambda_f(Jd_2)\lambda_f(Ld_1)\lambda_f(Md_2).
\psi^2\left(\sum_{i=1}^n (-1)^{i-1}\left(\sum_{j=0}^{\lfloor\frac{k}{2}\rfloor}\frac{p_j(n)}{(k-2j)!}(n-2i)^{k-2j}\right)\delta(\bigwedge\nolimits^i\sigma_n)\right) = 2^{n-k}\sum_{i=1}^n (-1)^{i-1}\left(\sum_{j=0}^{\lfloor\frac{k}{2}\rfloor}\frac{p_j(n)}{(k-2j)!}(n-2i)^{k-2j}\right)\delta(\bigwedge\nolimits^i\sigma_n).
\lVert F\rVert_{\mathbb{D}_{r}^{p}}=\left(\mathbb{E}\left[|F|^{p}\right]+\sum_{k=1}^{r}\mathbb{E}\left[\left|\lVert D^{k}F\rVert_{\mathcal{H}^{\otimes k}}\right|^{p}\right]\right)^{1/p}.
i{\tilde A}^{+}=\left [ \begin{array}{ll}<\psi^{(0)}_3|d\psi^{(0)}_3> <\psi^{(0)}_4|d\psi^{(0)}_3> <\psi^{(0)}_3|d\psi^{(0)}_4> <\psi^{(0)}_4|d\psi^{(0)}_4>\end{array}\right ]=i{\tilde A}_a dx^a .
(n+1) \begin{cases}x_{t}\leq 1, t=1, x_{t}=x_{1}-2,t=2, x_{t}=x_{t-1},t=3,\ldots,n, y_{t}=1,t=1, y_{t}=y_{t-1},t=2,\ldots,n-1, y_{t}\leq y_{t-1},t=n. \end{cases}(n+2) \begin{cases}x_{t}\leq x_{1}-2, t=1, x_{t}=x_{t},t=2,\ldots,n, y_{t}=y_{t-1},t=1,\ldots,n-1, y_{t}\leq y_{t},t=n. \end{cases}
0=\int_{M}d(\eta\wedge*\zeta)=\int_{M}(d\eta\wedge*\zeta-\eta\wedge*(-1)^{k+1}*^{-1}d*\zeta)=\langle\langle d\eta,\zeta\rangle\rangle-\langle\langle\eta,\delta\zeta\rangle\rangle,
M_{4}^{g}= \kappa^{2}( 4 s t u )^{- 1}t_{8}^{\mu_{1}\cdots \mu_{8}}t_{8}^{\nu_{1}\cdots \nu_{8}}R_{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}R_{\mu_{3}\mu_{4}\nu_{3}\nu_{4}}R_{\mu_{5}\mu_{6}\nu_{5}\nu_{6}}R_{\mu_{7}\mu_{8}\nu_{7}\nu_{8}}\equiv ( s t u )^{- 1}L_{4}^{g},
= 2 ( N_{c}^{2}- 1 ) V \int \! \frac{d^{3}k}{( 2 \pi )^{3}}\frac{| k |}{2}+ \begin{array}{c}{\mathrm{p e r t u r b a t i v e}}{\mathrm{c o r r e c t i o n s}}\end{array}
\|{h}-{h_0}\|_{L^1(D_{\frac{1}{4}})}\leq \|{h}-{h_0}\|_{L^1(D\left(z_0,\rho_0\right))}&=\|\widetilde{h}-\widehat{h}_0\|_{L^1(D_{1})}\leq C3^{m}C_{m-1}\|\tau(u)\|_{L^p(D_{1})}^{3^{1-m}}.
S ~ = ~ \oint ~ d \tau \int ~ d^{2}r_{\perp}~ \left({\frac{1}{2}}{\Omega}^{-}~{\vec{\nabla}}^{2}{\dot{\Omega}}^{+}-{\frac{1}{2}}{\Omega}^{+}~{\vec{\nabla}}^{2}{\dot{\Omega}}^{-}+{\dot{k}}^{+}{\Omega}^{-}-{\dot{k}}^{-}{\Omega}^{+}\right) .
S[q] = \lim_{T \rightarrow \infty}\left\{\int_{-T}^T dt \; \left[\frac12 m \dot{q}^2 - \frac14 m \omega^2_0 q^2\right] - \frac18 m \omega^2_0 \ell_0^2 T\sin\left[{2 \over T}\int_{-T}^T dt{q^2 \over \ell_0^2}\right] \right\}\; .
\iota_{{\sf j}[X,X']}\Omega_L={\sf d_v}\iota_{{\sf j}X}\iota_{{\sf j}X'}\Omega_L\pm\iota_{{\sf j}X}{\sf d_v}\iota_{{\sf j}X'}\Omega_L\pm\iota_{{\sf j}X'}{\sf d_v}\iota_{{\sf j}X}\Omega_L
R_{(a_1,a_2, \dots , a_k)^{\uparrow}}^{n}= \arrowvert A \arrowvert = \arrowvert C \arrowvert - \arrowvert B \arrowvert &=\binom{n+k-1}{k-1}- \sum_{m=1}^{k}(-1)^{m+1}\sum_{1\leq i_1 < i_2 < \dots <i_m \leq k}\binom{n-\sum_{l=1}^{m}a_{i_{l}}- m + k - 1}{k-1}\,.
\sum_{j=1}^{\infty}\gamma_j^{\frac{1}{2}}\, b^{w_j}= \sum_{j=1}^{\infty}j^{-\frac{a}{2}}\, b^{\lfloor c \log_b j \rfloor}\asymp\sum_{j=1}^{\infty}j^{-\frac{a}{2}}\, b^{c \log_b j}=\sum_{j=1}^{\infty}j^{c-\frac{a}{2}},
\overline{v}_\lambda(t) - \overline{v}_\mu(t) = \int_0^t S(t-s) \bigl( F_\lambda(s) \overline{v}_\lambda(s) - F_\mu(s) \overline{v}_\mu(s) \bigr)\,ds + \int_0^t S(t-s) \sigma'(u(s)) \bigl( \overline{v}_\lambda(s) - \overline{v}_\mu(s) \bigr) B\,dW(s),
{\rm dist}((x,\lambda,v),\Phi^{-1}(0,0)) ={\rm dist}((x,\lambda,v),\widetilde{\Phi}^{-1}(0,0,0)) \le\kappa{\rm dist}((0,0,0),\widetilde{\Phi}(x,\lambda,v))\le\|(\xi,\eta,-\eta)\| \le\sqrt{2}\kappa\|\Phi(x,\lambda,v)\|
2(P_pv,P_nv)_{L^2(\mathbb{T}\times(0,T))}^{2}=\int_{0}^{T}\int_{\mathbb{T}}s(\varphi_{tt}+2\frac{a}{b}\varphi_{xt}+\frac{a^2}{b^2}\varphi_{xx})v^2dxdt -\int_{\mathbb{T}}s(\varphi_t+\frac{a}{b}\varphi_x)v^2|_{0}^{T}dx--\int_{0}^{T}\frac{a}{b}s(\varphi_t+\frac{a}{b}\varphi_x)v^2|_{0}^{2\pi}dt.
d_{a,n+1}(t)-d_{a,n}(t) = \left(p_{a+1,n+1}(t)-p_{a,n+1}(t)\right)-\left(p_{a+1,n}(t)-p_{a,n}(t)\right) &=q_{a+1,n}(t)-q_{a,n}(t) &=(a+1)(1+t)^{n-a-1}(1+t^{a+1})-a(1+t)^{n-a}(1+t^a) &=(1+t)^{n-a-1}\left((1-at)+t^a(t-a)\right)\,.
\begin{array}{c}g(\vartheta |\vartheta_{k})=\displaystyle\sum^{N_{H}}_{k=1}\chi (\vartheta -h_{k})-\displaystyle\sum^{N_{S}}_{k=1}\left( \chi (\vartheta -\hat{y}_{k}+i\eta )+\chi (\vartheta -\hat{y}_{k}-i\eta )\right) + -\displaystyle\sum^{M_{C}}_{k=1}\chi (\vartheta -c_{k})-\displaystyle\sum^{M_{W}}_{k=1}\chi_{II}(\vartheta -w_{k})\end{array}
a_{34}^{4312}=a_{34}^{4321}=a_{34}^{4132}=a_{34}^{4123}=a_{34}^{4231}=a_{34}^{4213}= a_{34}^{1432}=a_{34}^{1423}=a_{34}^{1243}=a_{34}^{2431}=a_{34}^{2413}=a_{34}^{2143}
\underline x^t_{i_t}= (\tilde x^t + \tau_t \underline x^{t-1}_{i_t}) / (1 + \tau_t), \ \ \ \underline x^t_i = \underline x^{t-1}_i, \ \forall i \neq i_t; y_{i_t}^{t}= \nabla f_{i_t}(\underline x^t_{i_t}), \ \ \y_i^t = y_i^{t-1}, \ \forall i \neq i_t.
(-1)^{N+k/2}\frac{(2\pi)^k}{\Gamma(k)}N^{k-1}y_N^k \exp(-2 \pi N y_N)= \frac{(-1)^{N+\frac{k}{2}}}{N}\frac{(k/e)^k}{\Gamma(k)}(1 + \delta)^k \exp(-k\delta).
q''_1(\Delta_b,T) =\frac{(-1)^n}{(n!)^2}\cdot\frac{4(n+1)}{n+2}\prod_{l=0}^{n-1}(\Delta_b+(n+2-2l)iT) \intertext{and}q''_2(\Delta_b,T) =\frac{(-1)^n}{(n!)^2}\cdot\frac{4n}{n+2}\left(\prod_{l=0}^{n-2}(\Delta_b+(n+2-2l)iT)\right),
\widehat{w^{\varphi}}A_{j}^{\varphi}Y(x_{1})A_{l}^{\varphi}=\sum_{k=-\infty}^{\infty}w(k)P_{k}^{\varphi}A_{j}^{\varphi}Y(x_{1})A_{l}^{\varphi}&=\sum_{k=-\infty}^{\infty}w(k)A_{j}^{\varphi}Y(x_{1})A_{l}^{\varphi}P_{k+l-j}^{\varphi}
\frac{l_{k}^{z}\mu_{1}^{z}}{l_{1}^{z}M_{k}^{z}}\le \varphi_{m+n+k}\le \mathrm{\thinspace}\frac{l_{k}^{z}M_{1}^{z}}{l_{1}^{z}\mu_{k}^{z}}, k=1,\thinspace \mathellipsis ,\thinspace p,
(-1)^{t}\cdot \frac{(n+p)!}{t!}\cdot \binom{n+p+i}{n+p-t}~~~~~\mbox{and}~~~~~(-1)^{p}\cdot (-1)^{t-p}\cdot \frac{n!}{(t-p)!}\cdot \binom{n+i}{n+p-t}.
\int_{\gamma_{i_{n}}}G_{\mu_{1}}G_{\mu_{2}}. . . G_{\mu_{n}}| \Psi \rangle , \, \, \, \, \, \, \, \, \, i_{n}= 1 , . . . , d_{n}, \, \, \, \, \, \, \, n = 0 , . . . , d .
\langle{E_{k_1},E_{k_2}}\rangle = \displaystyle \sum_{m \in \mathfrak{D}^*}E_{k_1}(m) \overline{E_{k_2}(m)}= \frac{1}{|\mathfrak{D}^*|}\sum_{m \in \mathfrak{D}^*}e^{2 \pi i \langle{m, (Ak_1-Ak_2)}\rangle/N}.
t^N \langle 1 \cdots M|\mathcal{B}^\prime(z_1)\cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle/D^\prime = t^N \langle 1 \cdots M|\mathcal{B}^\prime(z_1)\cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle/D^\prime \Big|_{z_1 \longleftrightarrow z_1^{-1}},
S=\frac{1}{2}\int [(e^{-\Phi}G^+_0 e^{\Phi})(e^{-\Phi}\tilde G^+_0 e^{\Phi}) -\int_0^1 dt(e^{-t\Phi}\partial_t e^{t\Phi})\left\{e^{-t\Phi}G^+_0 e^{t\Phi},e^{-t\Phi}\tilde G^+_0 e^{t\Phi}\right\}].
\nabla_{\ \ \dot A}^B W_{ABC}=\frac{i}{2}\nabla_{\dot A}\nabla^B W_{ABC}+\frac{5}{2}i G_{\ \ \dot A}^B W_{ABC}= \nabla_{\dot A}\nabla^{\ \ \dot B}_{\underline{A}}G_{\underline{C}\dot B}+\frac{5}{2}i G_{\ \ \dot A}^B W_{ABC}
\log(t/z) = \log( \frac{(m t)^{1/3}}{\log^{1/3}m}) \geq \log( \frac{m^{1/3}}{\log^{1/3}m}) \approx \log m \log(m/z) = \log( \frac{m^{4/3}}{(\log m)^{1/3}t^{2/3}}) \geq \log( \frac{m^{4/3}}{(\log m)^{1/3}(m^{3/2})^{2/3}}) = \log( \frac{m^{1/3}}{\log^{1/3}m}) \approx \log m
\lim_{x_{n+1}\rightarrow0}\nabla'\overline{w}=0,\quad\lim_{x_{n+1}\rightarrow0}\Delta'\overline{w}=0\quad\quad\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2s}\partial_{n+1}\overline{w}=\eta\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2s}\partial_{n+1}\tilde{w}.
p^{r}_{j}(n)=\sum\limits_{w_{1}+\cdots+w_{k+1}= n}\frac{n!}{w_{1}!\times w_{2}!\cdots w_{m}!\cdots w_{k+1}!}(1)\times(1)\times\cdots\times(1)\times J^0_{w_{m}}\times J^0_{w_{m+1}}\times\cdots\times J^0_{w_{k+1}}
\sum_{j \in \mathbb{Z}}\tau_j \mathcal{Q}^n_\rho (\varepsilon n ; t ) \nabla^n \varphi^n_j (t ) = \frac{g^{\prime \prime}(0)}{2 n}\sum_{j \in \mathbb{Z}}\big( \tilde{\mathcal{X}}^n_t ({\textstyle \iota_{\varepsilon}( \frac{j- f_n t}{n}; \cdot )}) \big)^2 \nabla^n \varphi^n_j (t ) .
\mu_{i_1,\ldots,i_{a-a_1-a_2}}^{(1)}=\sum_{j_1,\ldots,j_{a_1}}E\bigg[\prod^{a-a_1-a_2}_{s=1}\{x_{i_s}-E(x_{i_s})\}\prod^{a_1}_{s=1}\{x_{j_s}-E(x_{j_s})\}^2\bigg],
\Delta (\underline{\theta})=\left\vert \begin{array}{cccc}s_{1}(\theta_{1}) s_{2}(\theta_{1}) \ldots s_{N}(\theta_{1}) s_{1}(\theta_{2}) s_{2}(\theta_{2}) \ldots s_{N}(\theta_{2}) \vdots \vdots \ddots \vdots s_{1}(\theta_{N}) s_{2}(\theta_{N}) \ldots s_{N}(\theta_{N})\end{array}\right\vert ~.
Im \, \Sigma_F (Q) \simeq \frac{g^2}{4 \pi}\, C_F \, T \, \gamma^0 \, \frac{g_\tau^{(i)}(q_0, q)}{|g_\tau^{(i)}(q_0, q)|}\ln \left( \frac{m_T}{\Gamma_q (Q)}\right) \, .
u_{0}( \theta^{\prime}) = -{\frac{c_{1}l^{d - 1}}{2^{d}}}( 2 k_{+}+ d ){\frac{1}{k_{+}}}\int_{S^{d}}( \sin^{2}{\frac{\gamma}{2}})^{k_{+}}u_{1}( \theta ) d \mu ( \theta ) ~ ~
\gamma_{g,9}\gamma_{S_1,9}\gamma_{g,9}^{-1}=\gamma_{S_2,9}~,~~~ \gamma_{g,9}\gamma_{S_2,9}\gamma_{g,9}^{-1}=\gamma_{S_1,9}~,~~~ \gamma_{g,9}\gamma_{S_3,9}\gamma_{g,9}^{-1}=-\gamma_{S_3,9}~.
\begin{bmatrix}\frac{2-q}{3}0 \frac{1}{3}\frac{q}{3}0 0 \frac{2}{3}0 0 \frac{1}{3}\frac{q}{3}\frac{1}{3}\frac{1-q}{3}0 \frac{1}{3}\frac{1}{3}\frac{q}{3}0 \frac{2-2q}{3}\frac{q}{3}0 0 \frac{q}{3}\frac{1}{3}\frac{2-q}{3}\end{bmatrix}
\mathcal Q_3^{(1)}+ (1- \varepsilon_K) \mathcal K^{\rm{diag}}_H = (1- \varepsilon_K) \sum_{k \in \mathcal{P}_H}\mathcal D_k c_k^\dagger c_k + \sum_{k \in \mathcal{P}_H}\mathcal T(k),
3[\frac{d^{2}}{dx^{2}}q(x)]^{2}-2[\frac{d}{dx}q(x)][\frac{d^{3}}{dx^{3}}q(x)]=\frac{\pi^{8}-4\pi^{4}(1+x^{2})^{2}K(x)^{4}}{16x^{4}(1-x^{2})^{4}K(x)^{8}}q(x)^{2}
\mathcal{L}_1\subset J^1_n(W)\,:\,\left\{\begin{array}{l}\omega_{rs}(x)+\sum_{1\le j\le n}(\bar\omega_{rj}(x)y^j_s-\bar\omega_{sj}(x)y^j_r) +\sum_{1\le i<j\le n}(y^i_ry^j_s-y^j_ry^i_s)\hat\omega_{ij}(x)=0 \end{array}\right\}_{1\le r<s\le n}.
\mathbb{E}\left[ \cos(\hat{\Phi}_{\Psi,k}-\Phi_{X,k}) \bigg| X_{A,k}=|x_k| \right] \geq 1 - \frac{1}{|x_k|^2 \Delta}\mathbb{E}\left[ \sin(\hat{\Phi}_{\Psi,k}-\Phi_{X,k}) \bigg| X_{A,k}=|x_k| \right] =0.
\vartheta_{3}(0\vert i)=\sqrt{\dfrac{2\textbf{K}}{\pi}}=2^{-1/2}\pi^{-3/4}\Gamma (1/4) \ \ \ , \ \ \ \vartheta_{2}(0\vert i)=\vartheta_{4}(0\vert i)=2^{-1/4}\vartheta_{3}(0\vert i)
x\begin{pmatrix}P_{0}(x) P_{1}(x) P_{2}(x) \vdots\end{pmatrix}=\begin{pmatrix}b_{0}1 a_{0}b_{1}1 a_{1}b_{2}1 \ddots \ddots \ddots \end{pmatrix}\begin{pmatrix}P_{0}(x) P_{1}(x) P_{2}(x) \vdots\end{pmatrix}
\mathrm{i}{\cal P}_{\Lambda}={\o{1}{2}}\left( k_{\Lambda}^{i}\partial_{i}G - k_{\Lambda}^{i^{\star}}\partial_{i^{\star}}G \right) = k_{\Lambda}^{i}\partial_{i}G = - k_{\Lambda}^{i^{\star}}\partial_{i^{\star}}G
S_{R_\epsilon}= \{z=(x,y) \in{\mathbb{R}_+^{N+1}}\mid{R_{\varepsilon}<\left\vert z \right\vert <R_{\varepsilon}+1}\}A_{R_\epsilon}= \{y \in{\mathbb{R}^{N}}\mid{R_{\varepsilon}<\left\vert y \right\vert <R_{\varepsilon}+1}\}.
Z^{(2)}_V(a_1,x_1;\ldots;a_L,x_L|b_1,y_1;\ldots;b_R,y_R;\tau_1,\tau_2,\epsilon) &= \sum_{u\in V}Z^{(1)}_V(Y[a_1,x_1]\ldots Y[a_L,x_L]u;\tau_1) Z^{(1)}_V(Y[b_R,y_R]\ldots Y[b_1,y_1]\overline{u};\tau_2).
\begin{array}{rcl}\hbox{Ein}(\mathbf{g}){_a}^b + \Lambda\delta{_a}^b = \frac{1}{2}(\mathbf{F}{^i}_{ac}\mathbf{F}{_i}^{bc}- \frac{1}{2}\|\mathbf{F}\|^2\delta{_a}^b) \partial^{\gamma,\mathbf{A}}_b\mathbf{F}{_i}^{ab}= 0 \end{array}
{\overline w}(r) \geq{\overline w}(2r) + \int_r^{2r}s^{1-N}\int_0^r t^{N-1}f(t) dt ds \geq{\overline w}(2r) + Cr^{2-N}\int_0^r t^{N-1}f(t) dt & \geq Cr^{2-N}\int_0^r t^{N-1}{\overline u}^{-p}(t) dt & \geq Cr^2{\overline u}^{-p}(r),
\Re{(\sum_{j=1}^{\Delta m/2}\tilde z_j)}= 0 \, , \quad \Im{(\sum_{j=1}^{\Delta m/2}\tilde z_j)}= \frac{1}{2}\Im{\left( \sum_{k=1}^{m^+}z_k - \sum_{l=m^+ + 1}^{m}z_l \right)}\, .
\gamma\int_\Omega \nabla u_n.\nabla T_k(u_n)T_k^{\gamma-1}(u_n) =\int_{\Omega}h_n\left(u_n+\frac{1}{n}\right)f_nT_k^\gamma(u_n) + \int_{\Omega}T_k^\gamma(u_n)\mu_n.
W(\gamma) =\frac{1}{2\pi i}\int_{L_x(\gamma)}\frac{1}{w}dw &=\frac{1}{2\pi i}\int_{0}^1\frac{1}{L_x(z(s))}dL_x(z(s)) &=\frac{1}{2\pi i}\int_{0}^1\frac{d}{ds}\log L_x(z(s))ds.
\mathcal{K}_+(x,k) = b_+^+(k) e^{ikx}+ b_+^-(k) e^{-ikx}, & \mbox{with}b^+_+(k) := \mathbf{1}_{+}(k)T(k) + \mathbf{1}_{-}(k), b_+^-(k) := \mathbf{1}_{-}(k)R_+(-k),
L_{1,1}(s) = \sum_{n=1}^{\infty}\frac{|\lambda_{\pi_1}(n)|^2|\lambda_{\pi_2}(n)|^2}{n^s}=L(s,(\pi_1\boxtimes\pi_2) \times(\widetilde\pi_1\boxtimes\widetilde\pi_2)) V_{1,1}(s)
\tau_{A}\frac{dx_{i}^{A}}{dt}=\sum_{j=1}^{N_{A-1}}\xi_{ij}^{(A,A-1)}g_{j}^{A-1}+\sum_{j=1}^{N_{A+1}}\xi_{ij}^{(A,A+1)}g_{j}^{A+1}-x_{i}^{A}
\begin{array}{l c r}{( | W_{1}|^{2}+ | Z_{2}|^{2}) - ( | Z_{1}|^{2}+ | W_{2}|^{2}) = R_{2}}\\{Z_{1}W_{1}+ Z_{2}W_{2}= \overline{{Z}}_{1}\overline{{W}}_{1}+ \overline{{Z}}_{2}\overline{{W}}_{2}= 0 ,}\\ \end{array}
b_{1, j+1}(us_j,vs_j)=b_{1, j+1}(u,v)-2\ \ \ \b_{1, k}(us_j,vs_j)=b_{1, k}(u,v),\ \ \k\neq j+1, begin{align*}5pt] b_{2, j+1}(us_j,vs_j)=b_{2, j+1}(u,v)-1\ \ \ \b_{2, k}(us_j,vs_j)=b_{2, k}(u,v),\ \ \k\neq j+1, begin{align*}5pt] b_{3, k}(us_j,vs_j)=b_{3, k}(u,v),\ \ .
\dot{\eta}'=\left\{\eta+\delta\eta,H\right\}^{(\eta,\Lambda)}+\left\{\eta,\frac{\partial \delta W}{\partial t}\right\}^{(\eta,\Lambda)}=\left\{\eta+\delta\eta,H+\frac{\partial \delta W}{\partial t}\right\}^{(\eta,\Lambda)}\;.
\varpi(x_{2i+1}) =x,\varpi(x_{2i+2})=x^{-1},i=0,...,(l-4)/2, \varpi(x_{l-1}) =x,\varpi(x_{l})=x \varpi(x_{l+1}) =x^{-2}y^{2},\varpi(d)=y^{-1}l
G_{l+\nu}\left(\left. r'', r' ; E \right| \lambda; a; \sigma\right)=G_{l+\nu}(r'', r' ; E| \lambda) -\frac{G_{l+\nu}(r'', a ; E| \lambda)\, G_{l+\nu}( a, r' ; E| \lambda)}{G_{l+\nu}(a,a;E|\lambda) - 1/\sigma}\; .
\Bigg \langle \! \exp \left( - S_{i n t}[ \Phi^{( I )}] - i e \Bigg [ \frac{1}{\sqrt{\pi \! + \! g N}}\Big ( A_{\mu}^{( 1 )}, \varepsilon_{\nu \mu}\partial_{\nu}\Phi^{( 1 )}\Big ) + \sum_{I = 2}^{N}\frac{1}{\sqrt{\pi}}\Big ( A_{\mu}^{( I )}, \varepsilon_{\nu \mu}\partial_{\nu}\Phi^{( I )}\Big ) \Bigg ] \right) \! \Bigg \rangle_{\! \{K^{( I )}\}}.
\int_0^rt^{a+n-1}[\log(2+t)]^b\,dt >\left[\log\left(2+\frac{r}{2}\right)\right]^b\int_{\frac{r}{2}}^rt^{a+n-1}\,dt &=\frac{1-2^{-(a+n)}}{a+n}r^{a+n}\left[\log\left(2+\frac{r}{2}\right)\right]^b &>\frac{1-2^{-(a+n)}}{2^b(a+n)}r^{a+n}[\log(2+r)]^b.
\sum_{p=n_{*}{\scriptscriptstyle (K)}+1}^\infty \frac{1}{\mu_{p}}=\frac{1}{K}\sum_{p=n_{*}{\scriptscriptstyle (K)}+1}^\infty \frac{1}{\frac{p}{K}\, \tilde\mu(\frac{p}{K})}\leq \int_{\frac{n_{*}{\scriptscriptstyle (K)}}{K}}^\infty \frac{\textup{d}x}{x\, \tilde \mu(x)}\leq I.
\mathcal{B}\mathcal{J}_{f,2n-1-f}= q^{\frac{1}{4}}\mathcal{J}_{2n-1-f,f}+\sum_{i=1}^{2n-1-f}\beta_{f,2n-1-f}^{2n-1-f-i,f+i}\mathcal{J}_{2n-1-f-i,f+i}
K_\xi^\mathfrak{E}(z) =\frac{E_+(z)E_+(\xi)^*-E_-(z)E_-(\xi)^*}{\rho_\xi(z)}& = F_+(z)\left[\frac{Q_+(z)Q_+(\xi)^*-Q_-(z)Q_-(\xi)^*}{\rho_\xi(z)}\right]F_+(\xi)^* & = F_+(z)\Gamma_\xi(z)F_+(\xi)^*,
\sum_{k=0}^{(p^r+1)/2}(-1)^k(4k-1)\frac{(-\frac{1}{2})_k^5}{k!^5}\equiv\frac{p^r(p^{2r}-p^{4r}-1)}{16(p^{2r}-1)}\sum_{k=0}^{(p^r-3)/2}\frac{(\frac{3}{2})_k^3}{k!(k+2)!^2}\pmod{p^{r+3}},
K_{\rm Airy}(x,y)=\frac{\operatorname{Ai}(x) \operatorname{Ai}'(y)-\operatorname{Ai}'(x) \operatorname{Ai}(y)}{x-y}= \int_{0}^\infty \operatorname{Ai}(x+a) \operatorname{Ai}(y+a) da.
K^{\alpha_\varepsilon}_{(0,x_\varepsilon -a)}[\varphi, p_\varepsilon](x_\varepsilon) =\int_0^{x_\varepsilon-a}\frac 1{z^{\alpha_\varepsilon}}\int_0^1\int_0^1 \varphi''(x_\varepsilon -st z)t\,ds dt.
\beta_1^*:= \sup \{\beta > 0:{\mathcal D}^{(\beta)}_K \cap C(K) \hbox{is dense in}C(K)\}, \beta_2^*:= \sup \{\beta>0: \dim{\mathcal D}^{(\beta)}_K = \infty\}\beta_3^*:= \sup \{\beta > 0:{\mathcal D}^{(\beta)}_K \hbox{contains nonconstant functions}\},
\hbox{$X$-symmetrizer}p^X_0 = \sum_{w\in S_n}\xi_w\ , \hbox{$X$-antisymmetrizer}e^X_0 = \sum_{w\in S_n}(-1)^{\ell(w)-\ell(w_0)}\xi_w, \hbox{$Y$-symmetrizer}p_0^Y = \sum_{w\in S_n}\eta_w\ , \hbox{$Y$-antisymmetrizer}e_0^Y = \sum_{w\in S_n}(-1)^{\ell(w)-\ell(w_0)}\eta_w.
\sqrt{\lambda_n}= n - L + \frac{\sigma}{\pi n}+ \ell_2 \left( \frac{1}{n}\right), \sqrt{\lambda_n}- \sqrt{\vphantom{\lambda_n}\mu_n}= (n - L)^{-2 r - 1}\left( \nu + \ell_2 \left( \frac{1}{n}\right) \right)
(t+r+1)\cdots (t+n)\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_i+r-1}{r}+\dim_K(I_{\alpha})_t \right]=(t+r+1)\cdots (t+n) \binom{t+r}{r}&=(r+1)\cdots (n-1)n \binom{t+n}{n}=(r+1)\cdots (n-1)n\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_i+n-1}{n}+ \dim_K I_t \right].
\left[\begin{array}{c}\rho(\alpha z) u_{02}\end{array}\right]=R\left[\begin{array}{c}\alpha z u_{01}\end{array}\right]\mu(\alpha z)^{-1}=R\alpha\left[\begin{array}{c}z u_{01}\end{array}\right]\lambda(\alpha,z)^{-1}\mu(\alpha z)^{-1}=
\sum_{n\in D}w(n)|y_n| \ =\ \sum_{n\in D\cap (A\cup B)^c}w(n)|x_n| + t\sum_{n\in D\cap B}w(n) &\ \leqslant\ \sum_{n\in D\cap (A\cup B)^c}w(n)|x_n| + t\sum_{n\in E}1 &\ =\ \sum_{n\in G}w(n) |z_n|\ \leqslant\ \|z\|_\omega.
\|u\|_{Y_{b,s}^{\beta,\gamma}}=\left(\sum_{k\in\mathbb{Z}}\left(\int_{\mathbb{R}}\langle k\rangle^{s}\langle \tau-p^{\beta,\gamma}(k)\rangle^{b}|\widehat{u}(\tau,k)|d\tau\right)^2\right)^{\frac{1}{2}},
\int Md{\sigma}{\Delta_1}^{-1}{\Delta_2}^{-1}= 2{\pi}i D^{-1}({\hat q});\quad D({\hat q}) = 4{\hat \omega}({\hat \omega}^2 - M^2/4);\quad{\hat \omega}^2 ={m_q}^2 +{\hat q}^2
h(0,a,b,\theta ) =g_0(0,a, b)h(0,a,b, \theta ) +\mathbb{E}\left(e^{-q [\tilde{T}_0^- + \tau_a^- \circ \Theta_{\tilde{T}_0^-}] - \theta [a- X(\tilde{T}_0^- + \tau_a^- \circ \Theta_{\tilde{T}_0^-})]}; \tilde{T}_0^-<(l_{T_{E_B}}+\mathbf{e}_{r}) \wedge\tau_b^+\right).
Y^{(n)}(t_{n, i+1})=Y^{(n)}(t_{n, i})+ \int_{t_{n, i}}^{t_{n, i+1}}g(t_{n, i}, W_0^{(n), t_{n, i}}, Y_0^{(n), t_{n, i}}) ds + B(t_{n, i+1})-B(t_{n, i}),
\int_{\Sigma}(\dfrac{n}{2}+H^2)u^2e^{\frac{|x|^2}{4}}d\sigma \leq \int_{\Sigma}|\nabla u|^2e^{\frac{|x|^2}{4}}d\sigma=\lambda_1\int_{\Sigma}u^2e^{\frac{|x|^2}{4}}d\sigma.
I\circ \tilde{f}= f, (\Gamma\otimes_A I) \circ (\Gamma\otimes_A \tilde{f}) = \Gamma\otimes_A f, (\Gamma\otimes_A f) \circ \Psi_M = \Psi\circ f, \mbox{\ \and}\Psi\circ I = (\Gamma\otimes_A I) \circ \Psi_{L_i}.
Z(\mathcal{F}_{\varphi_v}, f_{1,v}, f_{2,v}) = \int_{G_{1,v}}\langle \omega_{\psi}(g_{1,v}) \varphi_{1,v}, \varphi_{2,v}\rangle \langle \pi_{B,v}(g_{1,v}) f_{1,v}, f_{2,v}\rangle \, d g_{1,v}.
\sigma_{\mathcal{L}}\sigma_{\mathcal{A}}= \sqrt{\langle\mathcal{L}^2\rangle - \langle\mathcal{L}\rangle^2}\sqrt{\langle\mathcal{A}^2\rangle - \langle\mathcal{A}\rangle^2}= \frac{1}{4}(\delta-\gamma)|\gamma|.
\dot{u}^i :=\frac{d u^i}{d t}=\,\frac{\partial H_t(u,p)}{\partial p_i}=\,\beta^i(u), \dot{p}_i:=\frac{d p_i}{d t}=\,-\frac{\partial H_t(u,p)}{\partial u^i}=\,-\sum^{8N}_{k=1}\,\frac{\partial \beta^k(u)}{\partial u^i}p_k,i,k=1,...,8N,