harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(-1)^ks^k\partial^k(s^{\lambda_1-\lambda_2}\mu) =(-1)^ks^k\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}\frac{\Gamma(k-j+\lambda_2-\lambda_1)}{\Gamma(\lambda_2-\lambda_1)}s^{\lambda_1-\lambda_2+j-k}\partial^j\mu =s^{\lambda_1-\lambda_2}\sum_{j=0}^k\binom{k}{j}\frac{\Gamma(k-j+\lambda_2-\lambda_1)}{\Gamma(\lambda_2-\lambda_1)}(-1)^js^{j}\partial^j\mu\geq 0.
|a_{1, 1}^{(k+1)}|^2 \leq |a_{1, 1}^{(k)}|^4 + |\beta^{(k)}|^4 - \sum_{i=2}^{n+1}|a_{1, 1}^{(k)}|^2|a_{1, i}^{(k)}|^2(2-|u_1 - u_{i}|^2),
n^{\mu 2}\equiv \left( \frac{r \beta e^{- \psi}\cos \delta}{\sqrt{e^{2 \psi}\alpha^{2}- \beta^{2}r^{2}}}, 0 , - \frac{1}{r}\sin \delta , \frac{\alpha e^{\psi}\cos \delta}{r \sqrt{e^{2 \psi}\alpha^{2}- \beta^{2}r^{2}}}\right)
\widehat{C_{\lambda}^{a/q,1}}(\xi ) = \sum_{\ell \in \mathbb Z^{d}}G (a, \ell, q )\widetilde\psi_{q}(\xi - \ell /q) \widetilde\psi_{\lambda Q/N}(\xi - \ell /q) \widetilde{d \sigma_{\lambda}}(\xi - \ell /q).
\left( \frac{\mathrm{d e t^{\prime}}( \Delta_{v})}{\mathrm{d e t}( \Delta_{v}^{( 0 )})}\right)^{- \frac{1}{2}}\left( \frac{\mathrm{d e t}( \Delta_{A})}{\mathrm{d e t}( \Delta_{A}^{( 0 )})}\right)^{- 1}\left( \frac{\mathrm{d e t}( \Delta_{c})}{\mathrm{d e t}( \Delta_{c}^{( 0 )})}\right)^{+ 1}= \left( \frac{\mathrm{d e t^{\prime}}( \Delta_{-})}{\mathrm{d e t}( \Delta^{( 0 )})}\right)^{- 1}
V_S:=\underset{n\to\infty}\lim \left\{\sum_{h=0}^{(n-1)}\left(\frac{1}{n}\sum_{t=1}^{n}\textrm{Cov}(e_{t,\beta^\prime}^2, e_{t+h, \beta^\prime}^2)\right)+ \sum_{h=1}^{(n-1)}\left(\frac{1}{n}\sum_{t=h+1}^{n}\textrm{Cov}(e_{t,\beta^\prime}^2,e_{t-h, \beta^\prime}^2)\right)\right\}
S = \int dx^{+}dx^{-}\,{\rm tr}\Bigg[ \partial_{+}\phi\partial_{-}\phi+i\psi\partial_{+}\psi +\frac{g^2}{2}J^{+}\frac{1}{\partial_{-}^{2}}J^{+}-\frac{1}{2}ig^2 [\phi,\psi]\frac{1}{\partial_{-}}[\phi,\psi] \Bigg].
S_{\left\langle p,\phi \right\rangle}^{\left\langle q,\psi \right\rangle}=\frac{1}{|\Omega_{p}||\Omega_{q}|}\sum_{z\in \Omega}\sum_{x,y\in \Omega_{p}\cap \Omega_{zq}}\overline{\phi}(x,y)\overline{\psi}^{z}(y,x)\prod_{\xi \in \mathcal{O}(x,y)}\Lambda_{p_{\xi}}^{(zq)_{\xi}}\left( \frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
{\mathbf{Y}}_{{\rm{c}},m}={\mathbf{H}}_{m}(\sqrt{p_m}{\mathbf{p}}_{m}{\mathbf{s}}_{m}^{\mathsf{H}}+\sum\nolimits_{m'\neq m}\sqrt{p_{m'}}{\mathbf{p}}_{m'}{\mathbf{s}}_{m'}^{\mathsf{H}})+{\mathbf{N}}_{{\rm{c}},m},
({\bf x}\, t \vert{\bf x}^{\prime}t^{\prime}) = \frac{1}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}}\prod_{n = 1}^{N}\left[ \int_{- \infty}^{\infty}d x_{n}\right] \prod_{n = 1}^{N + 1}K_{0}^{\epsilon}( \Delta{\bf x}_{n}) ,
\log \|u_{1,m}\|_{v_1}=m+O(1), \log \|u_{1,m}\|_{v_2}=O(1), \log \|u_{1,m}\|_{v_i}=-\frac{m}{n-2}+O(1), i=3,\ldots, n, \log \|u_{2,m}\|_{v_1}=O(1), \log \|u_{2,m}\|_{v_2}=m+O(1), \log \|u_{2,m}\|_{v_i}=-\frac{m}{n-2}+O(1), i=3,\ldots, n.
\langle (n_{j_1})^{ja}_{b}|\sin(\phi^{ja}_{b}/L)|(n_{j_2})^{ja}_{b}\rangle = \frac{1}{2i}\left(\delta_{(n_{j_1})^{ja}_b, (n_{j_2})^{ja}_b + 1}-\delta_{(n_{j_1})^{ja}_b, (n_{j_2})^{ja}_b - 1}\right)
\underset{n\rightarrow +\infty}{\lim}(\Phi^{\prime}((u_{n},v_{n}))-\Phi^{\prime}(u_{0},v_{0}),(u_{n}-u_{0},v_{n}-v_{0})=\underset{n\rightarrow+\infty}{\lim}(f_{n},(u_{n}-u_{0},v_{n}-v_{0}))=0.
\sum_{i=0}^{r}\overline \gamma_{a}(i) \; b_{a+i}^{r-i}= v^{-ar -{r+1 \choose 2}}\sum_{s=0}^p v^{2s(a+2s)}\frac{\prod_{k=1}^p (1-v^{2a+4k+2})}{\prod_{k=1}^s(1-v^{4k}) \prod_{k=1}^{p-s}(1-v^{4k})}\cdot \prod_{k=1}^{p-s}(1-v^{2a+4s+4k}).
\int_{B_{N}}\prod_{i=1}^{2}\|f_{i}\|_{L^2({B_{N^{{1}/{2}}}}(x))}^{\frac{s}{2}}\, dx \lesssim_{\epsilon,\nu}N^{\epsilon}N^{\frac{3s}{4}}\prod_{i=1}^{2}\|f_{i}\|_{L^2({w}_{B_{N}})}^{\frac{s}{4}}\prod_{i=1}^{2}\|f_{i}\|_{L^{s,N}({w}_{B_{N}})}^{\frac{s}{4}}.
A ~= ~{1\over 2}\;(\; X^{\dagger}_1 ~X_1 ~+~X^{\dagger}_2~ X_2\; ), ~~B ~= ~{1\over 3}~\sqrt{X^{\dagger}_3 ~X_3},~~x ~=~4~\sqrt{B}\;{\rm cos}\left(~{1\over 3}\;{\rm Arccos}~{~A~ \over B^{3/2}}~\right) \; .
n_\varepsilon(x,y) := B_0 \cdot \varrho\left(\sum_{i = 1}^d t_i(x_i) + \varrho \left(\frac{y}{B_0}\right) - d\right) - B_0 \cdot \varrho\left(\sum_{i=1}^d t_i (x_i) + \varrho\left(- \frac{y}{B_0}\right) - d\right).
\varphi^{-1}=(\sigma,(\varphi^{-1}_{\sigma^{-1}1},\varphi^{-1}_{\sigma^{-1}2}, \varphi^{-1}_{\sigma^{-1}3}))=(\sigma,(\varphi^{-1}_{\sigma^{-1}1},\varepsilon,\varphi^{-1}_{\sigma^{-1}3})) \in \prescript{\sigma}{}N_l^A.
\left( \sqrt{\tilde{x}^{2}+ \tilde{y}^{2}}- \widetilde{\rho}\frac{\operatorname{c o s h}\tau}{\operatorname{s i n h}\tau}\right)^{2}+ \tilde{z}^{2}= \frac{\widetilde{\rho}^{2}}{\operatorname{s i n h}^{2}\tau}.
\int_0^\infty g(r)\, r \, \mathrm{d}r \leq B_0 \int_0^\infty(1+\sqrt{B_0}\r)^{-2\beta}\, r\, \mathrm{d}r \leq B_0 \int_0^\infty (1+\sqrt{B_0}\r)^{1-2\beta}\, \mathrm{d}r &= \int_0^\infty (1+ s)^{1-2\beta}\, \mathrm{d}s = \frac{1}{2(\beta-1)}\,,
\left\{\alpha_{{\mathcal{R}},{\mathcal{T}}}^{\bar{\mathcal{R}}_i}\tilde{h}_{q,\mathcal{T}}^{\bar{\mathcal{R}}_i}m_{{\mathcal{R}},{\mathcal{T}},n}^{\bar{\mathcal{R}}_i}:\mathcal{R}\ni q,|\mathcal{R}|=r+1,|\mathcal{T}|=t,i\in[t],n\in[N^{\binom{N_R-1}{r+1}\binom{N_T}{t}}\right\}&\cup\{m:m\in\mathcal{M}_q[N+1]\}.
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^4}{n^5}}= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{-n}{30}+ \frac{n^3}{3}+ \frac{n^4}{2}+ \frac{n^5}{5}\right) \frac{z^n}{n^5}\right\}
ds_{\mathcal{M}^{}}^{2}=\frac{1}{\sigma^{2}}\left( \frac{1}{1-\rho^{2}}d\mu_{x}^{2}+\frac{1}{1-\rho^{2}}d\mu_{y}^{2}-\frac{2\rho}{1-\rho^{2}}d\mu_{x}d\mu_{y}+4d\sigma^{2}\right) .
b_{i,j}\left( z\right) =b_{i-1,j}\left( z\right) -b_{i-1,j}\varphi_{z_{i-1}}\left( z\right) , b_{i,j}=b_{i,j}\left( z_{i}\right) , b_{i,j}^{\ast}=b_{i,j}\left( z_{i+1}\right) ,
I (\xi,\eta) := \ \int (1+\tanh N) ( 1+ \tanh(\xi-N))(1+\tanh(\eta-N)) \, dN = \ \xi\left(1+\frac{1}{\tanh \xi}\right)\left( 1 + \frac{1}{\tanh(\eta-\xi)}\right) + \eta\left(1+\frac{1}{\tanh \eta}\right)\left( 1 + \frac{1}{\tanh(\xi-\eta)}\right),
w_{r}= 2b x_n^a (1-r^2)^{b-1}r w_{rr}= 2b x_n^a (1-r^2)^{b-2}[1-(2b-1) r^2] w_{x_n}=1 -a x_n^{a-1}(1-r^2)^b w_{x_n x_n}= a(1-a) x_n^{a-2}(1-r^2)^b w_{x_n r}=2ab x_n^{a-1}(1-r^2)^{b-1}r.
(\int_{0}^{x_{1}}\frac{dt}{t})^{n_{1}-1}\int_{0}^{x_{1}}\frac{dt}{1-t}(\int_{0}^{x_{2}}\frac{dt}{t})^{n_{2}-1}\int_{0}^{x_{2}}\frac{dt}{1-t}\cdot\cdot\cdot(\int_{0}^{x_{k}}\frac{dt}{t})^{n_{k}-1}\int_{0}^{x_{k}}\frac{dt}{1-t}
\Bigg{|}\int\Lambda \mathcal{Q}_p\varepsilon(1-\zeta_B)\Bigg{|}=\Bigg{|}\int\Lambda \mathcal{Q}_p \varepsilon\bigg{(}1-\zeta_B+\frac{y}{B}\bigg{)})\Bigg{|}&\lesssim e^{-\frac{\kappa B}{20}}\|\varepsilon\|_{L^{\infty}}\leq b_c^{\frac{7}{2}}.
\begin{aligned}n_{2r,7}(0,1)= \frac{7-7^{2r+1}}{24},\;\;\;n_{2r,5}(4,-3)=\frac{17\cdot 7^{2r}-17}{24}, n_{2r,7}(8,-7)= \frac{41\cdot 7^{2r}-41}{24}. \end{aligned}
\tilde u = \cos\left(\frac f2t\right)u_0(p)+\sin\left(\frac f2t\right)v_0(p)+ \frac f2y, \tilde v =-\sin\left(\frac f2t\right)u_0(p)+\cos\left(\frac f2t\right)v_0(p)- \frac f2x, \tilde\omega = 0,\tilde\phi =\phi(p) - \frac{f^2}{8}(x^2+y^2),\tilde T=T(p),
\left(\forall 1\leq i < N+1\right) \beta_{i1}:= \frac{N(1-\sqrt{\delta})}{2(N+1)\gamma_1^2 L_i}; \left(\forall i < N+1\right),\left(\forall j > 1\right) \beta_{ij}\equiv 0; \left(\forall j\right) \qquad\beta_{(N+1)j}:= \frac{1-\sqrt{\delta}}{\gamma_j}.
\int_{\mathbb{R}}x^k d\mu^{(\ell)}(x) (P_n^{(\ell)}(x))^\top=\int_{\mathbb{R}}x^{k+\ell}d\mu(x)(P_n^{(\ell)}(x))^\top=\langle x^\ell P_n^{(\ell)}, x^k\rangle=0,k=0,\cdots,n-1,
\|\tilde{h}\|_{\bar{{\cal H}}([0,T_{2}])}=\|K^{-1}\tilde{h}\|_{L^{2}([0,T_{2}])}=\left(\frac{T_{1}}{T_{2}}\right)^{H}\|K^{-1}h\|_{L^{2}([0,T_{1}])}=\left(\frac{T_{1}}{T_{2}}\right)^{H}\|h\|_{\bar{{\cal H}}([0,T_{1}])},
A=\int \frac{d^3p}{(2\pi )^3}\ \overline{\Psi}(p)S^{-1}(p)\Psi (p)\ +\int \frac{d^3p\d^3q}{(2\pi )^3}\ \overline{\Psi}(p+\frac{\ 1}{\ 2}\q)\Gamma(p,q)\Psi (p-\frac{\ 1}{\ 2}\q),
\zeta_{F_{n,\delta},\textup{top}}(s) = \prod_{i=0}^{n-1}\frac{s-2i}{s-2(n+i+\delta) + 1}, \zeta_{G_n,\textup{top}}(s) = \prod_{i=0}^{n-1}\frac{s-i}{s- n-i}, \zeta_{H_n,\textup{top}}(s) = \prod_{i=0}^{n-1}\frac{s-i}{s- \frac{n+i+1}{2}}.
\nu = \frac{1}{\mu^{2}}\left( 1 - \frac{K}{2^{5 / 2}\, x^{3 / 4}}e^{- 2 x + 2 \Phi_{\infty}}( 1 + . . . . ) \right) \, , \quad x \equiv \mu ( r + r_{\infty}) \, .
M'(t) = \lim_{h\to 0+}\frac{f(x(t+h),t+h) - f(x(t), t+h)+f(x(t), t+h)- f(x(t),t)}{h}&\geq \lim_{h\to 0+}\frac{f(x(t), t+h)- f(x(t),t)}{h}= \partial_t f(x(t),t).
\begin{array}{lcl}\phi_a(b) = \int a^{-n/2}\bar\psi\left(\frac{x-b}{a}\right)\phi(x)d^nx, \phi(x) = C_\psi^{-1}\int a^{-n/2}\psi\left(\frac{x-b}{a}\right)\phi_a(b)\frac{dadb}{a^{n+1}},\end{array}
\Gamma_{\Lambda}( \bar{\psi}_{\phantom{j}}^{c}, \psi_{\phantom{j}}^{c}) \ + \, b a r{\psi}_{\phantom{c}a}^{c \alpha}\ \Delta_{I R \alpha}^{- 1 \phantom{\alpha}\beta}\ \psi_{\phantom{c}\beta a}^{c}\ = \ - W ( \bar{\zeta}, \zeta ) \ + \, b a r{\psi}_{\phantom{c}a}^{c \alpha}\ \zeta_{\alpha a}^{\phantom{\rho}}\ + \, b a r{\zeta}_{a}^{\alpha}\ \psi_{\phantom{c}\alpha a}^{c}
\begin{aligned}\left\| u(t_2) - \overline{u}_0 \right\|_{p}^{p}+ p \, (p-1) \, \int_{t_1}^{t_2}\int_{\Omega}\left| u(x,t)-\overline{u}_0 \right|^{p-2}\phi^\prime(u(x,t)) \left| \nabla u(x,t)\right|^2 dx dt= \left\| u(t_1) - \overline{u}_0 \right\|_{p}^{p}\end{aligned}
f(r|H=h,T=t)=\frac{(\begin{matrix}N\\h\end{matrix})r^{h}(1-r)^{t}}{\int_{0}^{1}(\begin{matrix}N\\h\end{matrix})p^{h}(1-p)^{t}dp}=\frac{r^{h}(1-r)^{t}}{\int_{0}^{1}p^{h}(1-p)^{t}dp}.
{\mathbb E}\left[{{\mathrm{SINR}}_i^{{}}}\right] = \frac{{1 -{\mathbb E}\left[{{\bf{u}}_i{\mathbf{\tilde H}}_i{\mathbf{\tilde H}}_i^H{\bf{u}}_i}\right]}}{{K\gamma}}= \frac{{1 - \left({K - 1}\right){\mathbb E}\left[{x^2}\right]}}{{K\gamma}}= \frac{{N - K + 1}}{{NK\gamma}}.
v_{n}( ( - 1 )^{n}a_{n}^{+}+ ( - 1 )^{n - 1}n ( n - 1 ) a_{n - 2}^{+}+ 2 n b_{n - 1}^{+}+ 2 n c_{n - 1}+ 2 ( - 1 )^{n}d_{n}) = 0 \,
\begin{array}{r c l}{{E}}&{{=}}&{{E_{0}+ E_{q ( J ) - \mathrm{v i b}}( 0 )}}\\{{}}&{{=}}&{{E_{0}+ \displaystyle h c \nu_{\mathrm{v i b}}\frac{1}{2 \sinh ( \gamma ( J ) / 2 )}\sinh \left( \gamma \left( J \right) \left({\frac{1}{2}}+ c ( J ) \right) \right)}}\\{{}}&{{}}&{{+ \displaystyle \frac{h^{2}}{8 \pi^{2}I}J ( J + 1 ) ~ ,}}\\ \end{array}
f_L(v)=C_L1_{r_1\leq v_1\leq r_2}e^{-\frac{|v_2|^2}{2}}e^{-\frac{|v_3|^2}{2}},f_R(v)=C_R1_{-r_2\leq v_1\leq -r_1}e^{-\frac{|v_2|^2}{2}}e^{-\frac{|v_3|^2}{2}},
A_{p a r e n t}^{\mathrm{I V}}= - \frac{T_{p}}{2}\int d^{p + 1}{\xi}\left[ \sqrt{- g}\left( -{\Psi}^{( p + 3 ) / 2}g_{i j}h^{i j}+ ( p + 3 ){\Psi}^{( p + 1 ) / 2}\right) +{\Lambda}\left({\Psi}-{\Phi}\right) \right] .
\mathcal{O}' = \big{(}\min\{\mathsf{q}(i_0),\mathsf{q}(j_0)\},\max\{\mathsf{q}(i_0),\mathsf{q}(j_0)\}\big{)},\ldots & \ldots,\big{(}\min\{\mathsf{q}(i_{N-1}),\mathsf{q}(j_{N-1})\},\max\{\mathsf{q}(i_{N-1}),\mathsf{q}(j_{N-1})\}\big{)}.
T_+(m, t_0, f,g, u_0) = \sup \{\tau \geq t_0 \colon \exists \, G_m \eqref{EquationNonlinearIBVP}[t_0, \tau]\}, T_{-}(m, t_0, f,g, u_0) = \inf \{\tau \leq t_0 \colon \exists \, G_m \eqref{EquationNonlinearIBVP}[\tau, t_0]\}.
(\det^{\mathrm{pl}})^* \, \Xi (S_{j,k,l}) = \frac{1}{l!}\biggl( \sum_{\substack{(j',k') \in J (j',k') \neq (1,0)}}\epsilon_{j',k'}\boxtimes \Xi (S_{j',k',1}) \biggr)^l \setminus \epsilon_{j,k}\, .
\frac{D_{d}}{c}\frac{(\vec{\theta}-\vec{\beta})^{2}}{2}+\frac{D_{ds}}{c}\frac{[(\vec{\theta}-\vec{\beta})\frac{D_{d}}{D_{ds}}]^{2}}{2}=\frac{D_{d}D_{s}}{D_{ds}c}\frac{(\vec{\theta}-\vec{\beta})^{2}}{2}.
\int d^{2}\vec{\vartheta}\, F ( \vec{\vartheta}) \, e^{i \varphi}( \partial_{\vartheta}+ i \cot \vartheta \, \partial_{\varphi}) \, G ( \vec{\vartheta}) = - \int d^{2}\vec{\vartheta}\left[ e^{i \varphi}( \partial_{\vartheta}+ i \cot \vartheta \, \partial_{\varphi}) F ( \vec{\vartheta}) \right] G ( \vec{\vartheta}) .
(R \otimes 1_L)(\alpha \otimes \varphi_L) \Delta(l) =(R \otimes 1_L)(\alpha \otimes \varphi_L)(l_1 \otimes l_2) &=(R \otimes 1_L)(\alpha(l_1) \otimes \varphi_L(l_2)) &=R(\alpha(l_1)) \otimes \varphi_L(l_2) &\overset{\eqref{eq3.3}}{=}\alpha(R(l_1)) \otimes \varphi_L(l_2) &=(\alpha \otimes \varphi_L)(R \otimes 1_L)\Delta(l).
\sum_{\substack{m\in \mathbb{Z}3m^2+2m\leq n}}\left(m+\frac{1}{6}\right) c\left(\omega; n-3m^2-m\right) =(-1)^{n+1}\sum_{\substack{a,b\in \mathbb{Z}ab=4n+3 12|3a-b-4}}d\left(N,\tilde{N}, \frac{1}{3}, \frac{1}{6}\right).
T_{\varepsilon,\eta,d,k,B}=\frac{(\varepsilon-\eta) kK^2}{\alpha^2 d^2}\frac{B^{2-\frac{1}{r}}}{N^2}+\frac{\eta K^2}{\alpha^2 d^2}\frac{B^{2-\frac{1}{r}}}{N^2}+O\left(\frac{k^\sigma K^\sigma B^\sigma}{d^\sigma N^\sigma}\right).
\|\widehat{u}f \|_{L^2 (\mathbb{R}^2)}\leq \left( 2 \| f \|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{u}f \|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{v}f\|_{L^2 (\mathbb{R}^2)}^2\right)^{\frac{1}{2}}= \| f \|_{\alpha}
\hat{R}F(v_{1}\otimes v_{0}) =[2]^{1/2}(\hat{R}(v_{0}\otimes v_{0})+\hat{R}(v_{1}\otimes v_{-1})) =[2]^{1/2}(v_{0}\otimes v_{0}+q^{-2}(q^{2}-q^{-2})v_{1}\otimes v_{-1}+\hat{R}(v_{1}\otimes v_{-1})).
Q_2(M,P_i,\tau)=\left\{e^{\frac{1}{24}E_2(\tau)A_3}\left(\prod_{j=1}^{6}\frac{x_j\theta'(0,\tau)\theta_2(x_j,\tau)}{\theta(x_j,\tau)\theta_2(0,\tau)}\right)\right. &\left.\frac{1}{2}\left(\prod_{l=1}^8\theta_1(y_l^i,\tau)+\prod_{l=1}^8\theta_2(y_l^i,\tau)+\prod_{l=1}^8\theta_3(y_l^i,\tau)\right)\right\}^{(12)}.
\begin{cases}\square_{s,y}\tilde{v}= (|y|^2 - s^2)_+^\frac{2}{d-2}|\tilde{v}|^{\frac{4}{d-2}}\tilde{v}, \tilde{v}|_{s = 0}= v_0, \partial_s \tilde{v}|_{s=0}= v_1,\end{cases}
K^{(2r+1)}_0 (x)=\sum_{i=-r}^{r}E^{(ii)}+\frac{q-x^2}{h_0(a,c,x)}\Big(\sum_{0 < i \leq r}t_0 E^{(-i,-i)}+E^{(r+1-i,r+1-i)}{}-\sum_{0 < i \leq r}q^{-i}E^{(-i,r+1-i)}+q^{r+1-i}t_0E^{(r+1-i,-i)}\Big).
C_{u x u x}={\frac{- 3{{\left( \cos ({\frac{u}{2 a}}) + \sin ({\frac{u}{2 a}}) \right)}^{2}}}{{a^{2}}{{\left( \cos ({\frac{u}{2 a}}) - \sin ({\frac{u}{2 a}}) \right)}^{4}}}}
\liminf_{k\to\infty}H_{\delta}(z^{\ell(k)-i})\ge\lim_{k\to\infty}H_{\delta}(z^{\ell(k)}) \ \{\rm and}\ \ \lim_{k\rightarrow\infty}\|z^{\ell(k)-i}-z^{\ell(k)-i-1}\|=0.
\sup_{s\in [t_{0},t]}u(s\wedge\tau_{k})|Y_{s}^{\tau_{k}}|_{V_{s}^{\tau_{k}}}^{p}\leq u(t_{0})|Y_{t_{0}}|_{V_{t_{0}}}^{p}+ \int_{t_{0}}^{t\wedge\tau_{k}}|Y_{s}|_{V_{s}}^{p-2}\big(\dot{u}(s)|Y_{s}|_{V_{s}}^{2}+ u(s)p\hat{Z}_{s}\big)^{+}\,ds &+ \sup_{s\in [t_{0},t]}I_{s}^{\tau_{k}}\quad
y = \sum_{m=1}^\infty \lambda^{p^{-1}+\cdots+p^{-m}}t^{-p^{-1}-p^{-1-m}}, x = \sum_{m,n=1}^\infty \lambda^{p^{-1-n}+\cdots+p^{-m-n}}t^{-p^{-1-n}-p^{-1-m-n}},
\int_{0}^{r/2}d\rho \frac{(\rho^{2}+r^{2}+1)}{((\rho^{2}-r^{2}+1)^{2}+4r^{2})^{3/2}}\frac{1}{\sqrt{1+\rho^{2}}}\leq \frac{C}{r^{4}}\sinh^{-1}\left(\frac{r}{2}\right)
A_{1,s-2}= \sum_{r=1}^{n}\frac{S_{0rr}-S_{00r}}{r^2}- \sum_{r=2}^{n}\sum_{l=1}^{r-1}\frac{S_{0lr}+S_{0rl}}{r-l}\left(\frac{1}{r}-\frac{1}{l}\right),
\Omega_kU_k =\left(P_{\nu_k^{-1}}^T\widetilde{\Omega}_kP_{\nu_k^{-1}}\right)P_{\nu_k^{-1}}^T\widetilde{U}_k=P_{\nu_k^{-1}}^T\widetilde{\Omega}_k\widetilde{U}_k &=P_{\nu_k^{-1}}^T\left(-\lambda_k \widetilde{\partial G}(U_k)\right)=-\lambda_kP_{\nu_k^{-1}}^TP_{\nu_k^{-1}}\cdot \partial G(U_k) &=-\lambda_k\partial G(U_k),
\psi (t)=e^{\varepsilon^{-2}t\mathcal{A}}\psi (0)+\frac{1}{\varepsilon}\int_{0}^{t}e^{\varepsilon^{-2}(t-\tau )\mathcal{A}_{s}}\mathcal{F}^{s}(a+\varepsilon \psi )d\tau +\mathcal{Z}^{s}(t),
\begin{aligned}L_{12}(q) = -2 + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 2n - 4j^2 - 3j}+ q^{6n^2 + 10n + 4 - 4j^2 - 3j}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{6n^2 + 16n + 11 - 4j^2 - j}+ q^{6n^2 + 8n + 3 - 4j^2 - j}.\end{aligned}
\Delta M = \frac{1}{2}\sum_{i}\omega_{i}+ \frac{\left< V \right>}{4 \pi}+ \frac{1}{2}\int_{- \infty}^{\infty}\frac{d k}{2 \pi}\omega ( k ) \frac{d}{d k}\delta ( k ) - \frac{\left< V \right>}{4}\int_{- \infty}^{\infty}\frac{d k}{2 \pi}\frac{1}{\sqrt{k^{2}+ m^{2}}}\; .
d s_{T N}^{2}( m_{1}, x_{1}) = ( 1 + \frac{4 m_{1}}{r_{1}})^{- 1}( d \psi_{1}+ \cos \theta_{1})^{2}+ ( 1 + \frac{4 m_{1}}{r_{1}}) ( d r_{1}^{2}+ r_{1}^{2}d \Omega_{2}^{2}) ,
Q_1(\lambda,x) =\frac{1}{4\lambda(\lambda-1)}\left [4ab+i-i\left (\lambda-\frac{1}{2}\right )\frac{H'}{H}\right], Q_2(\lambda,x) =\frac{1}{2\lambda(\lambda-1)}\left [F_1(s)+F_2(s)-\left (\lambda-\frac{1}{2}\right )F_2(s)\frac{H'}{H}\right ],
\omega^{- 1}( \eta ( t ) ) \tilde{W}[ \eta | t ] \omega ( \eta ( t ) ) = - 8 \pi \dot{\eta}^{\mu}( t ) \int \delta \xi d s \epsilon_{\mu \nu \rho \sigma}W^{\rho \sigma}[ \xi | s ] \dot{\xi}^{\nu}( s ) \delta ( \xi ( s ) - \eta ( t ) ) ,
\sum_{j=1}^k \sum_{d_i\in S_j}\frac{|d_i-d_{S_j}|}{d_id_{S_j}}\|\Re(P_M(d_i))-\Re(P_{M_{\varepsilon}}(e_i))\| \leq \sum_{j=1}^k \sum_{d_i\in S_j}\frac{|d_i-d_{S_j}|}{d_id_{S_j}}.
(\varphi(u'))' = N_f (u), \u'(T)=bu'(0), \u'(T)=bu'(0) &\Leftrightarrow (D_{\varphi}D)(u)=N_{f}(u),\u\in{\rm dom}(D) &\Leftrightarrow u = (D_{\varphi}D)^{-1}N_{f}(u), \u\in C^{1}.
\mu\iint_{Q_t}|\mathbf{w}_{xx}|^2\frac{\omega_\delta^n}{\rho}dxds =\iint_{Q_t}\mathbf{w}_t\cdot \mathbf{w}_{xx}\omega_\delta^n dxdt+\iint_{Q_t}u\mathbf{w}_x\cdot \mathbf{w}_{xx}\omega_\delta^n dxds \quad-\iint_{Q_t}\mathbf{b}_x\cdot \mathbf{w}_{xx}\frac{\omega_\delta^n}{\rho}dxds.
S_{T}[ \overline{{{\psi}}}, \psi ] = \frac{1}{2}\; g \; \int d^{2}x \; \sum_{b = 1}^{N}{j^{( b )}}_{\mu}^{T}( x ) \sum_{c = 1}^{N}{j^{( c )}}_{\mu}^{T}( x ) \; ,
\hat{T}( z ) \hat{T}( w ) = \frac{\hat{c}/ 2}{( z - w )^{4}}+ \frac{2 \hat{T}( w )}{( z - w )^{2}}+ \frac{\partial_{w}\hat{T}( w )}{( z - w )}+ O ( ( z - w )^{0}) \hspace{. 1 i n}.
\det\left(\frac{y_j^{i-1}}{(i-1)!}\right)_{1\le i,j\le n_2}=\frac{(-1)^{\frac{n_2(n_2-1)}2}}{(2\pi i)^{n_2}}\int_{\gamma_{\tau_2}^{n_2}}d^{n_2}\omega\prod\limits_{j=1}^{n_2}e^{y_j\omega_j-\frac 12\mu_1\omega_j^2}\det\left(\frac 1{\omega_j^{n_2+1-i}}\right)_{1\le i,j\le n_2},
dX^{\varepsilon}(u,t) =(LX^{\varepsilon}(u,t)+\beta (X^{\varepsilon}(u,t)))dt+\varepsilon \int_{\mathbb{X}}G(X^{\varepsilon}(u,t-),u,y)N^{\varepsilon^{-1}}(ds,dy) X^{\varepsilon}(u,0) =x(u),u\in O\subset \mathbb{R}^{d}.
\frac{F(x_{N})-F^*}{a_N^2}+\frac{\tilde \mu}{2}\|x^*-v_N\|^2\leq \frac{\tilde\mu}{2}\|x^*-v_0\|^2+\rho M^2\sum_{i=1}^N\frac{1}{a_i}+\frac{NrM^2}{2}+\sum^N_{i=1}\frac{\delta_i}{a_i}&-\frac{\tilde\mu-\mu}{2}\sum_{i=1}^N\frac{\|x_i-y_i\|^2}{a_i^2}+2\sum_{i=1}^N\frac{\varepsilon_i}{a_i^2}+\sqrt{2\tilde\mu}\sum_{i=1}^N \|x^*-v_{i}\|\cdot\sqrt{\frac{\delta_i}{a_i}}.
S_{d}= \int d^{d}x [ \sqrt{{\rm det}(\delta_{\mu\nu}+2\pi\alpha' \tilde{F}_{\mu\nu})}+ \frac{1}{2}\mid D_{\mu}\varphi\mid^{2}+ \lambda (\mid\varphi\mid^{2})^{2}+ \frac{1}{2}i\Lambda^{\mu\nu}(\tilde{F}_{\mu\nu}-2\partial_{\mu}A_{\nu}) ],
\left|\left\langle \left(\frac{1}{2k}\sum_{j=1}^k{\left(\omega_j\pi(x_j) + \omega_j^{-1}\pi(x_j)^*\right)}\right)^{2r+1}v,w\right\rangle\right| \leq \epsilon + MO\left(\frac{r}{k}\right)^r.
\frac{1}{2m}\int_{R_0}^r\rho\left[1-\left(\frac{R_0}{\rho}\right)^{2m}\right] d\rho \leq\int_{R_0}^r \rho\left(1-\frac{R_0}{\rho}\right) d\rho = \frac{(r-R_0)^2}{2}, \forall\; r \geq R_0.
S_{massive}=S+ \int dx^{+}dx^{-}\,{\rm tr}\left[-\frac{1}{2}m_{b}^{2}\phi^{2}+\frac{i}{2}m_{f}^{2}\psi\frac{1}{\partial_{-}}\psi+m_{f}g\psi \frac{1}{\partial_{-}}[\phi , \psi]\right].
\frac{6 0 0 0 \times ( 1 0 0 \times 0 . 8 5 + 8 0 \times 1 + 4 0 \times 1 . 2 5 + 4 0 \times 1 . 5 + 3 0 \times 1 . 7 5 + 1 0 \times 2 )}{3 0 0}= 6 9 5 0 (
\frac{1}{2}\Vert \nabla{\theta}_{\phi}^{m}\Vert_{\mathbf{L}^{2}}^{2}\leq Ch^{2r}+ C\tau\sum_{k=0}^{m}\Vert \nabla{\theta}_{\phi}^{k}\Vert_{\mathbf{L}^{2}}^{2}+ \sum_{k=1}^{m}J_2^{k}.
\begin{aligned}\frac{2}{m}\frac{\lambda_{n,1}^{(1)}e^{-\lambda_{n,1}^{(1)}}}{h^2(x_{n,1}^{(1)}) e^{G_1^*(x_{n,1}^{(1)})}}\ell(\mathbf{q})+O(e^{-\lambda_{n,1}^{(1)}})=\frac{2}{m}\frac{\lambda_{n,1}^{(2)}e^{-\lambda_{n,1}^{(2)}}}{h^2(x_{n,1}^{(2)}) e^{G_1^*(x_{n,1}^{(2)})}}\ell(\mathbf{q})+O(e^{-\lambda_{n,1}^{(2)}}), \end{aligned}
Z_{\scriptscriptstyle MCS}= \frac{1}{N_{\scriptscriptstyle MCS}}\int\limits_{\Omega^{\scriptscriptstyle 1}(M)}\mathcal{D}A \hskip 4pt e^{-i \bigl\langle A, \hskip 4pt \left(\ast d_{\scriptscriptstyle 1}+(\ast d_{\scriptscriptstyle 1})^{\scriptscriptstyle 2}\right)A \bigr\rangle}
G_{M N}^{( 1 1 )}= \left( \begin{matrix}{e^{-{\frac{\hat{\Phi}}{3}}}G_{\hat{\mu}\hat{\nu}}^{( 1 0 )}}&{e^{{\frac{2}{3}}\hat{\Phi}}B_{M}}\\{e^{{\frac{2}{3}}\hat{\Phi}}B_{N}}&{e^{{\frac{2}{3}}\hat{\Phi}}}\\ \end{matrix}\right) ,
||x^*||_1\,\varphi(t^*,x^*) - \mathbb{E}\left[L(\rho)X(\rho)\varphi \big(t^*-\rho,\mathfrak{X}(\rho)\big)\right] = \mathbb{E}\left[\int_0^{\rho}L(s)X(s)\, g\big(t^*-s,s,\mathfrak{X}\big)\, \mathrm{d}s \right] <0\,.
\begin{cases}\displaystyle{\rm d}v^n(t)+\left[P^{(n)}A v^n(t)+P^{(n)}B_n(v^n(t),v^n(t)) \right]{\rm d}t =G_n(v^n(t))\,{\rm d}W(t), t \in [0,T] \nabla \cdot v^n(t)=0, t \in [0,T] \displaystyle v^n(0)=v^n_0 \end{cases}
\nabla_{z_{\alpha}}^{0}\left( \begin{array}{c}{{\langle v |}}\\{{\hat{V}}}\\ \end{array}\right) = \frac{1}{\theta_{\alpha}}\left( \begin{array}{c}{{\langle v | \widehat{\bar{z}}_{\alpha}}}\\{{- \lbrack \widehat{\bar{z}}_{\alpha}, \hat{V}\rbrack}}\\ \end{array}\right) \, .
1 \geq \lim_{n \rightarrow \infty}\dfrac{\ell(N/I_nN)}{\ell(R/I_n)}\geq \lim_{n \rightarrow \infty}\dfrac{t}{\ell(R/I_n) \leq 3({t+(3n^{d-2}+n^{d-3}+\cdots+n^2)})+(8(2n)^{d-2})^d}=1/3
\psi_{-}(\vec{x}) = \frac{1}{\sqrt{2}}\frac{1}{i\partial_{-}-e V_{-}}\ast \xi (\vec{x}) - \frac{e}{\sqrt{2}}\frac{1}{\Delta_{\perp}}\ast (J_{pzm}^i (x_\perp) - q_{i})\alpha^i \frac{1}{i\partial_{-}- e V_{-}}\ast\psi_{+}(\vec{x}) \;,
\lim_{N\to \infty}\mu^{v_N}_{B_N+\theta A_N}= \left(\lim_{N\to \infty}\mu^{v_N}_{\theta A_N}\right)\rhd\left(\lim_{N\to \infty}\mu^{v_N}_{B_N}\right)=\delta_\theta \rhd \mu_{sc}.
\int_{D_{j_{0}}}\overset{3}{\mathbb{P}}\left(E^{T}_{j_{0}}\right)(x) \, dx - \eta \, \sum_{j=1 \atop j \neq j_{0}}^{\aleph}\mathcal{C}_{j_0}\cdot \Upsilon_{k}(z_{j_{0}},z_{j}) \cdot \int_{D_{j}}\overset{3}{\mathbb{P}}\left(E^{T}_j\right)(y)\, dy = \mathcal{C}_{j_0}\cdot E^{Inc}_{j_{0}}(z_{j_{0}}) + \mathcal{O}\left(a^{\min(6;8-h-3t)}\right).
0 = \frac{1}{\Gamma}\big [ G \Delta_{0}H + 2 \delta^{\mu \nu}\{\partial_{\mu}G + 2 G ( \partial_{\mu}\log \Gamma ) \}\partial_{\nu}H + \{\Delta_{0}G + 4 \delta^{\mu \nu}( \partial_{\mu}G ) ( \partial_{\nu}\log \Gamma ) \}H \big ]
\left( \frac{\delta A}{\delta y ( s )}\right)^{2}+ \left( \frac{\delta A}{\delta \vec{x}( s )}\right)^{2}= \frac{1}{y^{4}( s )}\left\{\left( \frac{d y}{d s}\right)^{2}+ \left( \frac{d \vec{x}}{d s}\right)^{2}\right\}\; .
\tilde{Hess}( \varphi)(X,V)= \tilde{g}(\tilde{\nabla}_X tan(grad_{\tilde{g}}\varphi),V)+\tilde{g}(\tilde{\nabla}_X nor(grad_{\tilde{g}}\varphi),V) = hdh(V)g(X,tan(grad_{\tilde{g}}\varphi))=0.
\Phi_{S_0}(gx)=\Phi_{\eta_2}(gx)=g_2\Phi_{\eta}(g^{-1}_2gx)=g_2\Phi_{\eta}(g^{-1}_2gg_1 g^{-1}_1x)=g_2g^{-1}_2gg_1\Phi_{\eta}( g^{-1}_1x),
\begin{aligned}\frac{d}{dt}\int_{-\infty}^t a(t-s)u(s) \,{\rm d}s = A u(t) + f(u(t)) + g\, \left[r(u(t),\gamma(t)) + \dot W(t)\right], t \in [0,T] u(t) = u_0(t), t \le 0. \end{aligned}
I(X^n\rightarrow{Y}^n)=\int_{{\cal X}_{0,n}\times{\cal Y}_{0,n}}\log \Big( \frac{d{\overrightarrow Q}_{0,n}(\cdot|x^n)}{d\nu_{0,n}(\cdot)}(y^n)\Big)({\overleftarrow P}_{0,n}\otimes{\overrightarrow Q}_{0,n})(dx^n,dy^n)\equiv{\mathbb{I}}_{X^n\rightarrow{Y^n}}({\overleftarrow P}_{0,n},{\overrightarrow Q}_{0,n})
E ( a_{0}, a_{1}, \beta ) = \frac{2 \pi B}{\beta^{2}}( 2 a_{+}- 1 ) - \frac{1}{2}B + 1 \equiv 2 a_{+}\left( 1 - \frac{B}{2}\right) = ( a_{0}+ a_{1}) \left( 1 - \frac{B}{2}\right) ,
| 1 \rangle , J_{+}( \vec{k}) | 1 \rangle , J_{+}( \vec{k}) J_{+}( \vec{k}^{\prime}) | 1 \rangle \mathrm{f o r}\vec{k}\not = \vec{k}^{\prime}, ( J_{+}( \vec{k}) )^{2}| 1 \rangle , \cdots
\mathrm{T r}~ e^{- ( L_{i}+ m_{i}^{2}g_{\tau \tau}) t}\simeq{\frac{n_{i}}{( 4 \pi t )^{{\frac{D - 1}{2}}}}}\int_{\widetilde{\cal M}^{\prime}}d \tilde{V}^{\prime}e^{- m_{i}^{2}g_{\tau \tau}t},
\widetilde{\mathcal{Z}}_\beta = W^\alpha_\beta\left(\mathcal{Z}_\alpha - \rho^h_\alpha\dfrac{\partial M^\gamma_\varepsilon}{\partial z^h}W^\tau_\gamma u^\varepsilon \mathcal{V}_\tau\right), \widetilde{\mathcal{V}}_\beta = W^\alpha_\beta\mathcal{V}_\alpha,
\Sigma_\lambda^- (t) :=\Big\{x \in B_{\sigma\rho}(x_0): w(x,t) >\lambda \Big\}\textrm{for}t\in(t_0-(\sigma \rho)^p,\rho^p)\shortintertext{and}\Sigma_\lambda^+ (t) :=\Big\{x \in B_{\sigma\rho}(x_0): w(x,t) <-\lambda \Big\}\textrm{for}t\in(t_0,t_0+(\sigma \rho)^p),
\, : \! \varphi ( x ) \! : \quad \leftrightarrow p ( \underline{{{\theta}}}, \underline{{{l}}}) = \frac{\pi \nu}{\beta}\left( \frac{2}{F ( i \pi ) \sin \pi \nu}\right)^{\frac{n}{2}}\sum_{i = 1}^{n}( - 1 )^{l_{i}}