harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{E}_{\pi|_{(W_0 \cup W'_1 \cup W'_2)^c}}((T_1, \ldots, T_n)|_{(W_0 \cup W'_1 \cup W'_2)^c}) & = \mathcal{E}_{\pi|_{(W_0 \cup W'_1 \cup W'_2)^c \setminus V}}((T_1, \ldots, T_{\theta - 1}, T_\theta L_{\mathcal{E}_{\pi|_V}((T_1,\ldots, T_n)|_{V})}, T_{\theta + 1}, \ldots, T_n)|_{(W_0 \cup W'_1 \cup W'_2)^c\setminus V}).
w_{\rm spinor}= \frac{2(qE)^2}{(2 \pi)^3}\sum_{k = 1}^{\infty}\frac{1}{k^2}e^{- k a_f}+ \frac{2(qE)(qB)}{(2 \pi)^2}\sum_{k = 1}^{\infty}\frac{B_{2k}}{(2k)!}F_{\rm spinor}^{(2k-1)}(0),
\theta^t_\Gamma=\sum_{\alpha=1}^k\left((1-t)^{n+1}t^n dz_\alpha f_\alpha(z_1,\dots z_{n+k-2})+(1-t)^{n}t^{n+1}d\bar z_\alpha \bar f_\alpha(z_1,\dots z_{n+k-2})\right) dz_{k+1}d\bar z_{k+1}\cdots dz_{k+n-2}d\bar z_{k+n-2}+\cdots
\mathbf{E}_{h\cdot m}\left[\int_0^{t}f(X_s)dA^g_s\right] ={\bf E}_{h\cdot m}\left[\int_0^t f(X_s)g(X_s)ds\right] &=\int_0^t ds\,{\bf E}_{h\cdot m}[(fg)(X_s)].
M_{ijkl}^{(2)}={\bf{\lambda}}\delta_{ij}\delta_{kl}+2{\bf{\mu}}\delta_{i\hat{k}}\delta_{j\hat{l}}+v (\delta_{i\hat{k}}\epsilon_{j\hat{l}}+\delta_{j\hat{k}}\epsilon_{i\hat{l}}) +\delta_{ij}D_{kl}^{1}+\delta_{kl}D_{ij}^{2}+D_{ijkl},
\left[ - \left({\frac{\partial^{2}}{\partial \rho_{1}^{2}}}+{\frac{\partial^{2}}{\partial \rho_{2}^{2}}}+{\frac{\partial^{2}}{\partial \rho_{3}^{2}}}\right) +{\alpha^{2}}( \rho_{1}^{2}\rho_{2}^{2}+ \rho_{2}^{2}\rho_{3}^{2}+ \rho_{1}^{2}\rho_{3}^{2}) + \frac{{\hat{l}_{1}}^{2}}{\rho_{1}^{2}}+ \frac{{\hat{l}_{2}}^{2}}{\rho_{2}^{2}}+ \frac{{\hat{l}_{3}}^{2}}{\rho_{3}^{2}}\right] \Psi = 2 \, E \Psi
\begin{array}{c c}{{B M B^{- 1}= - M^{\dagger}\, , ~}}&{{~ B = \left( \begin{array}{c c c}{{0}}&{{1}}&{{0}}\\{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\ \end{array}\right) \, .}}\\ \end{array}
f(x) = (x - 2)^{-r}\sum\limits_{i=0}^{\lfloor -r/2 \rfloor}\delta_i \left(\frac{x}{x-2}\right)^{2i}= x^{-r}\sum\limits_{i=0}^{\lfloor -r/2 \rfloor}\delta_i \left(\frac{x-2}{x}\right)^{-r-2i}
\langle \rho(xb_ty)h,h'\rangle =\langle \rho(b_ty)h, \rho(x^*)h'\rangle =\langle \rho(b_ty)h, \sigma(x^*)h'+\sigma^\perp(x^*)h'\rangle =\langle\rho(b_ty)h, \sigma(x^*)h'\rangle =0,
\frac{2}{2n+1}\sum\limits_{i=0}^n{2n+1 \choose 2i+1}B_{2n-2i}\gamma_{2i+1}= \frac{2}{2n+1}\sum\limits_{i=0}^n{2n+1 \choose 2i+1}B_{2n-2i}\sum\limits_{j=0}^i{i\brace j}\gamma_{2j}.
\partial_\tau^\ell\big(P_t\ast g\big)=P_t\ast \partial_\tau^\ell g\quad\mbox{and}\partial_\tau^\ell\big(Q^{\rm I}_t\ast g\big)=Q^{\rm I}_t\ast \partial_\tau^\ell g\quad\mbox{in}\,\,{\mathbb{R}}^{n-1},
\begin{aligned}-\iint g^\epsilon \partial_y u_s h^\epsilon \leq L\Vert g^\epsilon_y\Vert_{L^2}\Vert y\partial_y u_s\Vert_{L^\infty}\Vert h^\epsilon_x\Vert_{L^2}\leq L \Vert yu_{sy}\Vert_{L^\infty}\Vert g^\epsilon_y\Vert_{L^2}^2,\end{aligned}
\Delta(x)=x\otimes 1+1\otimes x, &\Delta(y)=y\otimes 1+1\otimes y, &\Delta(z)=z\otimes 1+1\otimes z-2x\otimes y, &S(x)=-x, S(y)=-y, S(z)=-z-2xy, &\{x, y\}=y, \{y, z\}=y^2, \{x, z\}=z.
\frac{\ddot{a}}{a}( t ) = - \frac{4 \pi G_{N}}{3}\left[ 3 ( P_{\star}( t ) + P ( t ) e^{- 4 k b_{0}}+ \frac{\hat{P}_{B}}{k}( t ) ( 1 - e^{- 2 k b}) ) + \rho_{\star}( t ) + \rho ( t ) e^{- 4 k b}+ \frac{\hat{\rho}_{B}}{k}( t ) ( 1 - e^{- 2 k b_{0}}) \right] .
S = \int d^{3}x \left[ \frac 1 2 ( \nabla \chi )^{2}+ \frac 1 2 ( \nabla \psi )^{2}+ \frac{m_{H}^{2}}{2}\psi^{2}- 2 \zeta \mathrm{e}^{g_{m}\psi}\cos ( g_{m}\chi ) \right] \equiv \int d^{3}x{\cal L}[ \chi , \psi | g_{m}, \zeta ] .
\dfrac{n-1}{3}T(r) \leqslant (n+2)\sum_{u=1}^3\sum_{t=1}^{2n+2}N^{(1)}_{(f^u,H_t),>k_t}(r)\leqslant (n+2)\sum_{u=1}^3\sum_{t=1}^{2n+2}\dfrac{1}{k_t+1}N_{(f^u,H_t),>k_t}(r) &\leqslant (n+2)\sum_{t=1}^{2n+2}\dfrac{1}{k_t+1}T(r).
\mathcal{E}_v:= \int_{\Omega_\delta}|\nabla v|^2~dx, \mathcal{G}_i:= \int_{\Omega_\delta}|\nabla \psi_i|^2~dx=\int_{\partial \mathcal{B}_i}\frac{\partial \psi_i}{\partial n}~ds, i=1,2,
A = \sum_{k=1}^{\infty}2^{-k}E_k = \sum_{k=1}^{\infty}2^{-k}E_A\left(\bigcup_{j=1}^{2^{k-1}}\left(2^{-k}(2j-1),2^{-k}(2j)\right]\right) &= \sum_{k=1}^{\infty}2^{-k}\sum_{j=1}^{2^{k-1}}E_A\left(2^{-k}(2j-1),2^{-k}(2j)\right],
h^{(\lambda_1, \ldots, \lambda_d)}(x) = \prod_{i=1}^d \lambda_i^{i-d}e^{\sum_{i=1}^d \lambda_i x_i}\det\big((-1)^{d-j}\big(\frac{d}{dx}\big)^{d-j}e^{-\lambda_i x_j}\big)_{i, j = 1}^d.
\begin{aligned}D_{3}(\Pi^{3}_{M_1\cap M_2}v_\lambda, v_\lambda) = \frac{3 \lambda^{3}}{4}- \frac{\lambda^{2}\sqrt{\lambda^{2}+ 2 \lambda + 9}}{36}+ \frac{\lambda^{2}}{4}- \frac{\lambda \sqrt{\lambda^{2}+ 2 \lambda + 9}}{18}+ \frac{\lambda}{4}\qquad - \frac{\sqrt{\lambda^{2}+ 2 \lambda + 9}}{4}+ \frac{3}{4}\end{aligned}
S=\frac{1}{2}\int\left((e^{-\Phi}Q_Be^{\Phi})(e^{-\Phi}\eta_0e^{\Phi})-\int_0^1 dt(e^{-t\Phi}\partial_te^{t\Phi})\left\{(e^{-t\Phi}Q_Be^{t\Phi}),(e^{-t\Phi}\eta_0e^{t\Phi})\right\}\right) \ .
\sum_{i,j=1}^d \| \chi D_i D_j e^{-\mathcal{A}_D^0t}\|_{L_2(\mathcal{O})\rightarrow H^{l-1}(\mathbb{R}^d)}&\leqslant d C^{(6)}_{l-1}\sum_{s=0}^{l-1}\delta^{-(l-1-s)}\| e^{-\mathcal{A}_D^0t}\|_{L_2(\mathcal{O})\rightarrow H^{s+2}(\mathcal{O}'')}.
\omega_{4}^{3}= i \{\frac{1}{2}d \ln ( \frac{d \overline{{{g}}}_{2}}{d \overline{{{z}}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{{g}}}_{2}- \overline{{{g}}}_{2}d g_{2})}{( 1 + g_{2}\overline{{{g}}}_{2})}\},
E_m=\lim_{n \to +\infty}I(u_n) = \lim_{n \to +\infty}I(u_n(\cdot + y_n^1)) = \lim_{n \to +\infty}I(v_n^1(\cdot + y_n^1))+I(\omega_1) = \lim_{n \to +\infty}I(v_n^1) + I(\omega_1).
\left\langle \xi(x)\xi(x^{\prime})\right\rangle \equiv N(x,x^{\prime})=\left( \frac 18\right) \left[ \left\langle \left\{\Phi^2(x),\Phi^2(x^{\prime})\right\}\right\rangle_0-2<\Phi^2>_0^2\right]
S \ = \ \int d^4x \ \left( \ (\partial_\mu m )^2 + (\partial_\mu n)^2 \ + \ \frac{1}{e_m^2}(f^{abc}\partial_\mu m^b\partial_\nu m^c)^2 \ + \ \frac{1}{e_n^2}(f^{abc}\partial_\mu n^b\partial_\nu n^c)^2 \ \right)
H(k^{-1}\bar\rho) =\ln k + \frac{1}{k}\sum_{i=1}^kH(\rho_i)=2\ln k, E(\bar\rho) =\frac{d}{2}\ln\left[1-\frac{2}{k}c_\beta+\frac{1}{k^2}c_\beta^2\right]=d\ln\left[1-\frac{c_\beta}{k}\right].
\begin{aligned}I_{\nu+2}(z)-2I_{\nu+1}(z)+I_{\nu}(z)= \frac{z^{\nu}}{\sqrt{\pi}\,2^{\nu}\Gamma(\nu+1/2)}* \qquad\times \Big(\frac{2}{z}\int_{-1}^1e^{-zs}s(1-s^2)^{\nu-1/2}\,ds+ \int_{-1}^1e^{-zs}(1+s)^2(1-s^2)^{\nu-1/2}\,ds \Big).\end{aligned}
H_{(Z_+,D_{Z,+})}(y)=e^{\frac{t_+}{2\pi i}}\sum_{d\in\mathbb K_{+}}y^d\left(\frac{\prod_{i=1}^2 \Gamma(1+\frac{v_i}{z}+v_i\cdot d)}{\prod_{i\in M_0}\Gamma (1+\frac{\bar D_i}{z}+D_i\cdot d)}\right)\textbf{1}_{[d]}[\textbf{1}]_{(D_i\cdot d)_{i\in I_+},v_2\cdot d}.
\sum_{j=2}^{k}\frac1{|y-\xi_j|^{N-2}}\le \frac C{|y-\xi_1|^{N-2-\tau}}\sum_{j=2}^{k}\frac1{|\xi_j-\xi_1|^\tau}\le \frac{C k}{|y-\xi_1|^{N-2-\tau}}
{\cal L}^{( 0 )}= - \frac{1}{\rho^{2}}\left[ r \left( \partial_{\mu}\bar{w}\partial^{\mu}w + | m |^{2}| w |^{2}\right) + \frac{\theta}{2 \pi i}\varepsilon^{\mu \nu}\partial_{\mu}\bar{w}\partial_{\nu}w \right]
T_{j,j}(f) := (i\pi d_{j-1}+P_{j-1})T_{j,j-1}(r_{j-1})(f) = \prod_{i=1}^{j-1}(i\pi d_i +P_i) \widetilde{S}_{j,\mathbf{r}}(f) = \widehat{\widetilde{Q}_j(f)}.
p_{t-s}\bigl(c\,(t-s)^{1/2}-y\bigr) \leq p_{t-s}(1)=(t-s)^{-1/2}p_{1}\bigl((t-s)^{-1/2}\bigr) &=(2\pi)^{-1/2}(t-s)^{-1/2}e^{-1/2(t-s)}.
\begin{array}{rl}|\widetilde Z^* - \widetilde Z | \leq \left|\widetilde Z^* - \bar Z^* \right| + \left| \bar Z^* - \widetilde Z \right| & \leq{\displaystyle \max_{j\in S}}\left| \frac{1}{\sqrt{n}}\sum_{i=1}^ng_i\{\hat\psi_j(y_i,z_i) - \bar\psi_j(y_i,z_i)\}\right| + \left|\bar Z^* - \widetilde Z\right| \end{array}
a\sim\sum_{j\geq 0}a_{m'-\frac{j}{h+1}},a_{m'-\frac{j}{h+1}}(y,\l\eta)=\l^{m'-\frac{j}{h+1}}a_{m'-\frac{j}{h+1}}(y,\eta)\;\; \;\textrm{for any}\; \l>1,
tan\frac{\hat{\gamma}}{2}C=\frac{tan\frac{\hat{\beta}}{2}B+tan\frac{\hat{\alpha}}{2}A+tan\frac{\hat{\beta}}{2}tan\frac{\hat{\alpha}}{2}B\times A}{1-tan\frac{\hat{\beta}}{2}tan\frac{\hat{\alpha}}{2}B\cdot A}.
{\cal F}( x^{a}, \varphi , \tilde{p}_{a}, \tilde{p}_{\varphi}) = \varrho \int_{\varphi_{0}}^{\varphi}\frac{( \tilde{p}, \partial_{\varphi}n )}{( \tilde{p}, n )}d \varphi + \varphi \tilde{p}_{\varphi}+ x^{a}\tilde{p}_{a}\ ,
J_{{\rm F}}= \exp\{iS[g]+\frac{i}{4\pi}\int\!d^2 x\,[{\rm Tr}(B_{+}(\partial_{-}g)g^{-1}) +\frac{N}{2 \pi}\partial_-\theta\partial_+\theta + \frac{N}{\pi}C_+\partial_-\theta] \}\quad,
\Gamma_{[n]\setminus \{i\}}=\frac{1}{2\mu_k}\Big[\{\Gamma_{[n]\setminus \{i,j\}}, \Gamma_{\{j,k\}}\}-\Gamma_{[n]\setminus \{i,k\}}-2\mu_j \Gamma_{[n]\setminus\{i,j,k\}}\Big],
i \hbar \frac{\partial}{\partial t}\bigl < q | \phi ( t ) \bigr > = \Bigl [ - \hbar^{2}X ( t ) \frac{\partial^{2}}{\partial q^{2}}- i \hbar Y ( t ) \bigl ( q \frac{\partial}{\partial q}+ \frac{\partial}{\partial q}q \Bigr ) + Z ( t ) q^{2}\Bigr ] \bigl < q | \phi ( t ) \bigr > .
\begin{aligned}I =\sum\limits_{k=-\infty}^{\infty}D_{k}=\left(\sum\limits_{k=-\infty}^{\infty}D_{k}\right)\left(\sum\limits_{j=-\infty}^{\infty}D_{j}\right) &=\sum\limits_{|k-j| \leqslant N}D_{k}D_{j}+\sum\limits_{|k-j|>N}D_{k}D_{j}=T_{N}+R_{N},\end{aligned}
u(x)=O \left( \left(\frac{1}{|x|}\right)^{n-2}\right) v(x)= \begin{cases}O \left( \left(\frac{1}{|x|}\right)^{n-2}\right), \beta\leq n-2 o\left( \left(\frac{1}{|x|}\right)^{\frac{n-2}{n}(\beta+2)}\right), \beta>n-2. \end{cases}
K = \sum_\alpha\left(p^{\bar z}_\alpha \dot z_{\alpha}+p^{z}_\alpha \dot{{\bar z}}_{\alpha}+J_\alpha \cos \theta_\alpha \dot \phi_\alpha\right)+\int d^2 z{\kappa\over 2}\left(\dot A^a_z A^a_{\bar z}-\dot A^a_{\bar z}A^a_{z}\right)
D_{z}^{-}:H^{s}(\Sigma)\to H^{s-1}(\Sigma)\,,D^{-}_{z}:=\gamma_{1}^{-}P^{-}_{z}\,,s\ge -\frac12\,, D_{z}^{+}:H^{s}(\Sigma)\oplus H^{s}(\partial\Omega)\to H^{s-1}(\Sigma)\oplus H^{s-1}(\partial\Omega)\,,s\ge -\frac12\,, &D^{+}_{z}(\varphi,\psi):=\Big(\gamma_{1}^{+}P^{+}_{z}(\varphi,\psi),\gamma_{1}^{\partial}P^{+}_{z}(\varphi,\psi)\Big)\,,
J_{i}={\frac{1}{2}}\int d^{3}x \langle x | \mathrm{t r}\, \gamma_{5}\gamma_{i}{\frac{1}{\gamma \cdot D + M}}| x \rangle = -{\frac{1}{2}}\int d^{3}x \langle x | \mathrm{t r}\, \gamma_{5}\gamma_{i}( \gamma \cdot D ){\frac{1}{- ( \gamma \cdot D )^{2}+ M^{2}}}| x \rangle
\left( \begin{array}{c}{{\{M_{A q}^{f}\}^{Y^{1}q}}}\\{{\{M_{A q}^{f}\}^{Y^{2}q}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{0}}&{{i}}\\{{- i}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c}{{\{M_{A q}^{b}\}^{Y^{1}q}}}\\{{\{M_{A q}^{b}\}^{Y^{2}q}}}\\ \end{array}\right) .
\begin{cases}\langle\widehat{F}(\xi)\mu_1,\mu_1\rangle = \langle\widehat{F}(\xi)\mu_l,\mu_l\rangle, \mbox{for}l=2,3,\dots,n-1, \langle\widehat{F}(\xi)\mu_p,\mu_l\rangle = 0 \mbox{for}l,p=2,3,\dots,n-1, l\neq p, \sum_{j=1}^{n}\widehat{F}_{jk}\xi_j = 0, \mbox{for}k=1,2,\dots,n.\end{cases}
\hat{g}= \left({\frac{L}{u}}\right)^{2}\left( \begin{array}{l l l}{{}}&{{{\dot{u}}^{2}- u^{4}}}&{{\;{\dot{u}}{u^{\prime}}}}\\{{}}&{{{\dot{u}}{u^{\prime}}}}&{{{u^{\prime}}^{2}+ u^{4}}}\\ \end{array}\right) ,
2^{N-1}=2b_0+b_{2^{n-2}}+ b_{2^{n-1}}+\sum_{\substack{v_2(k)\neq 0,n,{n-1}b_k\in B_1}}b_k, 2^{N-1}=b_{3\cdot 2^{n-2}}+\sum_{\substack{v_2(k)\neq{n-2}b_k\in B_2}}b_k,
{\cal S}=\frac{1}{4\pi\alpha'}\int d\tau d\sigma\left\{\Phi^{1}_{(1)}\left( -\partial^2_\tau+\partial^2_\sigma+\frac{2}{\cos^2\tau}\right)\Phi^{1}_{(1)}+\Phi^{2}_{(1)}\left( -\partial^2_\tau+\partial^2_\sigma\right)\Phi^{2}_{(1)}\right\}
\sum_{k=-\infty}^{+\infty}\bigl\vert a(k)\widehat{f}(k)^2\bigr\vert\le\left\Vert a \right\Vert_r\bigl\Vert\widehat{f}^{\ 2}\bigr\Vert_s =\left\Vert a\right\Vert_r\bigl\Vert\widehat{f}\, \bigr\Vert_q^2\le\left\Vert a\right\Vert_r \left\Vert f \right\Vert_p^2.
s_4 s_3 s_5 s_2 s_3 s_1s_2 s_4 s_3 s_5 = \tau^{(1)}, s_3 s_4 s_2 s_1 s_3 s_2 = \sigma^{-1}\sigma^{\{1,2,3,5\}}s_2 s_1 s_3 s_5 s_2 s_3 = \tau^{(2)}, s_4s_3 = \sigma^{-1}\sigma^{\{1,3,5\}}s_3s_5= \tau^{(3)}, \s_1=\tau^{(4)}, \ \s_4=\tau^{(5)}.
\int_{P(\zeta,\delta)}\frac{\delta_{\Omega}^{\alpha-1}(z)}{\left|z-\zeta\right|}d\lambda(z)\lesssim\frac{\delta^{\alpha-1}}{\alpha}\tau_{n}(\zeta,\delta)\prod_{j=1}^{n-1}\tau_{j}^{2}(\zeta,\delta).
[m_{1,1}, m_{1,0}] + [m_{1,0}, m_{1,1}]+ [m_{1,0}, m_{1,0}] = 0, &[m_{2,1}, m_{2,0}] + [m_{2,0}, m_{2,1}] + [m_{2,0}, m_{2,0}] = 0, &[m_{1,1}, m_{2,0}] + [m_{1,0}, m_{2,1}] + [m_{1,0}, m_{2,0}] = 0.
g(z_{0}e^{i\epsilon})=\sqrt[3]{z_{0}e^{i\epsilon}}\frac{-1+i\sqrt{3}}{2}=\sqrt[3]{|z_{0}|e^{i(-\pi+\epsilon)}}\frac{-1+i\sqrt{3}}{2}=\sqrt[3]{|z_{0}|}e^{i(-\frac{\pi}{3}+\frac{\epsilon}{3})}e^{\frac{2\pi i}{3}}=\cdot\cdot\cdot
\frac{- 2}{X ( t ) r^{2}}\frac{\mathrm{d}M ( t )}{\mathrm{d}t}\left( \begin{matrix}{0}&{1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\ \end{matrix}\right)
{\cal K}^{T}M{\cal K}= \left( \begin{array}{c c}{{{\cal K}^{( 1 )}}}&{{{\tilde{\cal K}}^{( 1 )}}}\\ \end{array}\right) e^{\varphi}\left( \begin{array}{c c}{{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) \, ,
\limsup_a a^{-d-1}|\mathcal{F}(g_a)( a^{-1}\xi )|^2 =4 |\xi|^{-d+1}R^{d-1}|\mathcal{F}(f\circ \theta^H )(|\xi|)|^2 \liminf_a a^{-d-1}|\mathcal{F}(g_a)( a^{-1}\xi )|^2 = 0.
-\left\{\int_{0}^{T}\int_{\Omega}\left(H(u,k)\,\frac{\partial\phi}{\partial t}+ \displaystyle\sum_{j=1}^{d}Q_{i}(u,k)\,\frac{\partial\phi}{\partial x_{j}}\right)\right\}\,dx\,dt\leq \int_{\Omega}\,H(u_{0},k)\,\phi(x,0)\,dx + Md\,\int_{\Gamma}H(u,k)\,\phi\,dr.
W=n\sum_{\bf k}\left[\sum_{i}\ln\left(1-\frac{16\omega^2_i}{E_{\bf k}^2}\right) -\frac{1}{2}\sum_{\begin{array}{c}\scriptstyle s_1,\ldots,s_5=\pm 1 begin{align*}-2.5mm] \scriptstyle s_1\ldots s_5=1 \end{array}}\ln\left(1-\frac{2\sum\limits_{i}\omega_is_i}{E_{\bf k}}\right)\right].
R^+(x;t) =\frac{\sqrt{2x-1}}{2\sqrt{x}p(x)}\left(p^2(x)F_2(u(x,t))+F_1(u(x,t))\right), R^-(x;t) =\frac{-\sqrt{2x-1}}{2p(x)}(p^2(x)F_2(u(x,t))-F_1(u(x,t))).
\tau_N\left( Q_1(a_{i_1}^N) \dots Q_p(a_{i_p}^N)\right) = \tau_{N}\left( u_1^N u_2^N \dots u_{2p-1}^N u_{2p}^N \right) + \mathcal{O}\left(\frac{1}{N}\max\limits_{i\neq j}| y_i^N-y_j^N |^2 \right).
\begin{pmatrix}{P}^\prime{Q}^\prime{R}^\prime \end{pmatrix}= \begin{pmatrix}0 0 0 0 0 0 \lambda (h^{\lambda,m,0})^2 h^{\lambda,m,0}( \frac{m}{\lambda}+ \lambda p )h^{\lambda,m,0}\end{pmatrix}\begin{pmatrix}{P}{Q}{R}\end{pmatrix}.
Z_{N, \beta}^{V;\boldsymbol{B}}=\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}\prod_{i=1}^{N}d\lambda_{i}e^{-\frac{N\beta}{2}V(\lambda_{i})}\mathbf{1}_{\boldsymbol{B}}(\lambda_{i})\prod_{1\leq i<j\leq N}\left|\lambda_{i}-\lambda_{j}\right|^{\beta}.
\mathbf{PG}^1(\mathbb{Q}_p)\rightarrow\prod_{\ell\in\mathcal{S},\sigma\in\mathcal{T}(\ell)}\mathbf{PG}^1[\ell,\sigma]\backslash\mathbf{PG}^1(\mathbb{Q}_p)=\Gamma\backslash\prod_{\ell\in\mathcal{S},\sigma\in\mathcal{T}(\ell)}\mathbf{PG}^1(\mathbb{Q}_p)
\lambda_n (\sum\limits_{j=1}^{p}\vert \beta_{0,j}+ z_{j}/ \sqrt{n}\vert^q - \vert \beta_{0,j}\vert^q) \rightarrow \lambda_0 \sum\limits_{i=1}^{p}\vert z_j \vert^{q}I(\beta_{0,j}=0)
F(w) = \frac1{2\pi\mathrm{i}}\int_{\Gamma}F(\zeta) \Big( \frac1{\zeta-(\zeta_0+z)}- \frac1{\zeta-(\zeta_0-z)}\Big) \mathrm{d}\zeta = \int_\Gamma F(\zeta)K_z(\zeta,\zeta_0) \,\mathrm{d}\zeta.
\tilde{m}_{0, \mu}(\xi -\eta, \eta)= -\frac{c_{\mu}}{2 |\xi |^{2}}\Big(\frac{\xi \cdot (\xi - \eta)}{|\xi - \eta| |\eta|}- \frac{\xi \cdot \eta}{|\xi - \eta| |\eta|}\Big) \Big( (\xi - \eta) \times \eta \Big)^{2}- \frac{i}{2 |\xi | |\xi - \eta|}\Big( (\xi - \eta) \times \eta \Big)^{2},
K_M =\left[\frac{2q^2+3}{3}-\frac{r_-}{r_+}\right]\left\{\frac{1}{r_+}\left[\frac{2q^2+3}{3}+\frac{r_-}{r_+}\right]+\frac{2q^2+6}{2q^2+3}\frac{r_-^2}{r_+^3}\left[1-\frac{r_-}{r_+}\right]^{-1}\right\}^{-1}.
E_{i}^{\Lambda_{i}+ 1}(z) \Phi_{L.W.}^{(\lambda,\mu)}(w) = \frac{- [\Lambda_{i}+ \nu(\vec\alpha_{i}, \vec\beta) ]}{(z -w)}a_{ii}^{*^{\Lambda_{i}}}\Phi_{L.W.}^{(\lambda,\mu)}(w) + R.T..
\lim_{m\to \infty}\frac{1}{\sigma_1(m)}\left ( J_{N,1}(T_m . D_f) - e\left(\left(T_m . D_f\right)_{\; >1}\right) \right)=h_f\lim_{\epsilon''\to0}\int_{\mathcal{F}_{N}(\epsilon'')}J_{N,1}(z) d\mu(z).
\sum_{n=0}^{\infty}e_{q}^{-n}=1+\frac{1}{\left[ 2\right]_{q}}-\lim_{b\rightarrow\infty}\sum_{n=1}^{\infty}\left( \sum_{k=0}^{n}\left(\begin{array}[c]{c}n k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left( \left( 1+b\right)^{n-k}\left(-b\right)^{k}\right) \right) \frac{\mathit{\beta}_{n,q}}{\left[ n\right]_{q}!}
R^L(X_1,X_2)X_1=\frac{3}{4}LX_2+X_3,~~~ R^L(X_1,X_2)X_2=-\frac{3}{4}LX_1,~~~ R^L(X_1,X_2)X_3=-LX_1,~~~ &R^L(X_1,X_3)X_1=LX_2+\frac{3}{4}LX_3,~~~ R^L(X_1,X_3)X_2=-LX_1,~~~ R^L(X_1,X_3)X_3=(\frac{L^2}{4}-L)X_1,~~~ & R^L(X_2,X_3)X_1=0,~~~R^L(X_2,X_3)X_2=-\frac{L}{4}X_3,~~~ R^L(X_2,X_3)X_3=\frac{L^2}{4}X_2.
Z_{im}=\left[\begin{array}{cc}(Q_{ij}Q_{mj}-Q_{ij}Q_{mk}-Q_{ik}Q_{mj}+Q_{ik}Q_{mk}) 0 0 0\end{array}\right] &=\left[\begin{array}{cc}(u_{ij}u_{mj}-u_{ij}u_{mk}-u_{ik}u_{mj}+u_{ik}u_{mk}) 0 0 0\end{array}\right]
S(z)=1-\frac{z^{2}}{2!}-\frac{z^{4}}{4!}-3\frac{z^{6}}{6!}+17\frac{z^{8}}{8!}-9\frac{z^{10}}{10!}+111\frac{z^{12}}{12!}+\cdot\cdot\cdot,z\in\mathbb{C}
K_{0}\left( \frac{1}{f_{1}f_{1}^{\prime \prime}}\right) \left( \frac{1}{f_{2}^{\prime}}\right)^{2}=\frac{f_{2}f_{2}^{\prime \prime}}{\left(f_{2}^{\prime}\right)^{2}}-\frac{\left( f_{1}^{\prime}\right)^{2}}{f_{1}f_{1}^{\prime \prime}}.
\alpha ( y ){\frac{\partial^{2}L}{\partial y^{2}}}+ \alpha^{\prime}( y ){\frac{\partial L}{\partial y}}+ \beta ( y ) L = \alpha ( x ){\frac{\partial^{2}L}{\partial x^{2}}}+ \alpha^{\prime}( x ){\frac{\partial L}{\partial x}}+ \beta ( x ) L \, .
\textstyle \|v\|_{\dot Y}^{2}:= \sup\limits_{R>0}\frac{1}{R}\int_{\Omega_{\le R}}|v|^{2}dx, \|v\|_{Y}^{2}:= \sup\limits_{R>0}\frac{1}{\langle R\rangle}\int_{\Omega_{\le R}}|v|^{2}dx.
\nabla^{\ast}\nabla \omega =-\omega_{0i}^{\prime\prime}e^{0}\wedge e^{i}-\omega_{ij}^{\prime\prime}e^{i}\wedge e^{j}-(m-1)\coth(r)(\omega_{0i}^{\prime}e^{0}\wedge e^{i}+\omega_{ij}^{\prime}e^{i}\wedge e^{j}) &+\mathcal{D}\omega
I_{12}\leq\displaystyle\int_{\Omega_{a,-}}a(x)|\omega(x,0)|^2\ln\frac{|\omega(x,0)|}{\|\omega\|_X}dx &=\displaystyle\int_{\Omega_{a,-}}(-a(x))|\omega(x,0)|^2\ln\frac{\|\omega\|_X}{|\omega(x,0)|}dx &\leq C_{\gamma_0}\|\omega\|_X^{\gamma_0}\displaystyle\int_{\mathbb{R}^N}|a(x)||\omega(x,0)|^{2-\gamma_0}dx &\leq C\|\omega\|_X^2
G_n^r = \frac{1}{(\gamma + (r+1)^2 )^n}+\sum_{k=1}^{[(n-1)/2]}\ \sum_{m=k+1}^{n-k}{m-1 \choose k}\frac{1}{(\gamma + (r+1)^2 )^m}sum_{i_1 =1}^{\infty}\ldots \sum_{i_k =1}^{\infty}\delta_{i_1 + \ldots +i_k , n-m}G_{i_1}^{r+2}\ldots G_{i_k}^{r+2}
\times\!\bigg(\!\left(t_{C_{u-1}}\right)_{R_ur_uS_us_u}\!\left(t_B\right)_{V_uv_uW_uw_u}\delta_{S_uV_u}\delta_{W_uR_u}\!\left(\delta_{E_uS_u}\!\left(t_{\alpha_u}\right)_{s_uv_u}\!\delta_{w_ur_u}\!\!-\!\delta_{s_uv_u}\delta_{E_uW_u}\!\left(t_{\alpha_u}\right)_{w_ur_u}\right)\!\!\bigg)\!\times
\frac{4}{q^2\gamma^{2}n^2}\left(nq\left(1+\frac{\gamma^3}{64}\right) + n^2q^2\frac{\gamma^3}{16}\right) = \frac{4}{q\gamma^{2}n}\left(1+\frac{\gamma^3}{64}\right) + \frac{\gamma}{4}\leq \frac{\gamma}{2}
\prod_{j=1}^g\tau\left(\alpha_j,\beta_j,\xi_2^{(j)},\epsilon^{(j)}\xi_1^{(j)}y_j\right)=\prod_{j=1}^g|\epsilon^{(j)}|^{1-\alpha_j-\beta_j}\cdot\tau\left(\alpha_j,\beta_j,\epsilon^{(j)}\xi_2^{(j)},\xi_1^{(j)}y_j\right).
\mu(s\widetilde{\alpha},\sigma)=\gamma(G/P)^2q_F^{n(\sigma)+n(\sigma\otimes\omega)}\frac{(1-\omega(\varpi)q_F^{-2s})(1-\omega(\varpi)^{-1}q_F^{2s})}{(1-\omega(\varpi)q_F^{-1-2s})(1-\omega(\varpi)^{-1}q_F^{-1+2s})}
\Theta_h = p^1 \d u \wedge \d y - p^2 \d u \wedge \d x + \sqrt{1 - (p^1)^2 - (p^2)^2}\, \d x \wedge \d y \, , begin{align*}10pt] \Omega_h = -\d p^1 \wedge \d u \wedge \d y + \d p^2 \wedge \d u \wedge \d x &\qquad{}+ \displaystyle \frac{1}{\sqrt{1-(p^1)^2-(p^2)^2}}\left( p^1 \d p^1 \wedge \d x \wedge \d y + p^2 \d p^2 \wedge \d x \wedge \d y \right) \, .
\overline{V}R^k \overline{V}^{\,-1}=(\overline{V}R^l\overline{VR^l}^{\,-1}\overline{VR^l}R^l \overline{VR^{2l}}^{\,-1}\cdots \overline{VR^{(k-1)l}}^{-1}R^l \overline{V}^{\,-1})=(\overline{V}R^l\overline{V}^{\,-1})^m.
\begin{array}{rll}\sigma_2^* = \sup \lambda \textnormal{subject to}\displaystyle f(x) - \lambda= v_1^TQ_0v_1 + \sum_{i=1}^2 v_1^TQ_i^+v_1 (1+X_i)+ \sum_{i=1}^2 v_1^TQ_i^-v_1 (1-X_i), & Q_0,Q_1^+,Q_2^+,Q_1^-,Q_2^- \succeq 0.\end{array}
\partial_{L-u}f_u(x) = \{l_u\in L-u: \forall z\in X f_u(z)\geq f_u(x) + l_u(z) - l_u(x)\}= \{\begin{multlined}[t]l-u: \forall z\in X, l\in L, f(z) - u(z)\geq f(x) -u(x) + l(z) - u(z) -l(x) + u(x) \}\end{multlined}= \{l-u:l\in\partial_L f(x)\}
\triangle \left( \hat{Q}_{\pm}\right) = \hat{Q}_{\pm}\otimes I + q^{\pm T}\otimes \left( \hat{Q}_{\pm}- \hat{\epsilon}_{\pm}I \right) \in U_{q}( \widehat{s l_{2}}) \otimes \mathcal{B}_{q}^{\epsilon}( \widehat{s l_{2}}) \ .
{_0{\rm D}_x^{\alpha}}u_{\rm ex}= \int_0^x \frac{(x-z)^{-\alpha}}{\Gamma(1-\alpha)}u'_{\rm ex}(z) dz= \int_0^x \frac{(x-z)^{-\alpha}}{\Gamma(1-\alpha)}\left[\mu z^{\mu-1}(1-z)^\nu - \nu z^\mu(1-z)^{\nu-1}\right]dz.
|d_{\pm}| \leq \frac{4}{T_n}\left(\frac{|\delta_{\pm}|}{2}(T_n/2-R_n) + |c_{\pm}| \right) \leq \frac{4}{T_n}\left( \frac{T_n K_n R_n}{4}+ 3K_n g(R_n)\right) &= K_nR_n + \frac{12K_n g(R_n)}{T_n}& \leq \frac{13K_n g(R_n)}{T_n}.
0 \geq \limsup_{n \to \infty}\left( \langle f_n, f_n \rangle - \sum_{i, \ell \colon \lambda_{i,\ell}^{(n)}= 0}\hat f_n(i,\ell)^2 - \sum_{i,\ell \colon \lambda_{i,\ell}^{(n)}> k}\hat f_n(i,\ell)^2 \right) &= \limsup_{n \to \infty}\sum_{i, \ell \colon 0<\lambda_{i,\ell}^{(n)}\leq k'}\hat f_n(i,\ell)^2,
\psi_{x_n}^\pm=\frac{-\sum_{j=0}^{n-1}g^{nj}\psi_{x_j}^\pm+\sqrt{\big(\sum_{j=0}^{n-1}g^{nj}\psi_{x_j}^\pm\big)^2-g^{nn}\big(\sum_{j,k=0}^{n-1}g^{jk}\psi_{x_j}^\pm\psi_{x_k}^\pm\big)}}{g^{nn}(x)}
- r^{2 s - 1}{\frac{\Gamma ( 1 - s ) \sin ( \pi / \alpha )}{\Gamma ( s ) 8 \pi}}\int_{- \infty}^{\infty}{\frac{1}{( \cosh y / \alpha - \cos \pi / \alpha ) ( \cosh^{2}y / 2 )^{1 - s}}}d y
\begin{aligned}\mathbb E_{\nu_\rho}\Big[\big( \xi^{A}_{x+1}-\xi^{A}_{x}+D_2^+ (\xi^{B}_{x+1}-\xi^{B}_{x})\big)\big( \xi^{A}_{x+1}-\xi^{A}_{x}+D_2^- (\xi^{B}_{x+1}-\xi^{B}_{x})\big)\Big]=\frac{2}{9}(2-(D_2^++D_2^-)+2D_2^+D_2^-)=0\end{aligned}
\int_0^\infty \xi_\alpha(\lambda)d\lambda = \sum_{n=0}^\infty \alpha_n^2, \frac2\pi \int_0^\infty \left\{1-\exp\left(-\int_0^\infty \frac{\xi_\alpha(\lambda)}{\lambda+t^2}d\lambda\right)\right\}dt=\alpha_0.
F_{i_{1}, \dots , i_{n}}( \beta_{1}, \dots , \beta_{n}+ 2 \pi i ) = \exp [ 2 \pi i \omega ({\cal O}, \Psi ) ] F_{i_{n}, i_{1}, \dots , i_{n - 1}}( \beta_{n}, \beta_{1}, \dots , \beta_{n - 1}) .
\mathbf{A}(x) \leq C \sum_{j=0}^{\infty}{{|B_{j}|}\over{(2^{-j}\rho(x'))^{n}}}\bigg({1\over{|B'_{j}|}}\int_{B'_{j}}V(y')^{q}dy' \bigg)^{1\over q}\bigg({1\over{|B_{j}|}}\int_{B_{j}}f(y)^{q'}dy \bigg)^{1\over{{q'}}}&\leq C M_{q',}(f)(x) \sum_{j=0}^{\infty}(2^{-j}\rho(x'))^{2}\bigg({1\over{|B'_{j}|}}\int_{B'_{j}}V(y')dy'\bigg),
\begin{array}{ll}c_{11}c_{12}=qc_{12}c_{11}, c_{21}c_{22}=qc_{22}c_{21}, c_{11}c_{21}=qc_{21}c_{11}, c_{12}c_{22}=qc_{22}c_{12}, \left[c_{12},c_{21}\right]=0, \left[c_{11},c_{22}\right]=(q-q^{-1})c_{12}c_{21}, \multicolumn{2}{c}{c_{11}c_{22}-qc_{12}c_{21}=1.}\end{array}
p_{ij}= -G_{ij}(e_i^2+f_i^2) + G_{ij}(e_ie_j+f_if_j) -B_{ij}(e_if_j-e_jf_i) q_{ij}= \ \ \,B_{ij}(e_i^2+f_i^2) - B_{ij}(e_ie_j+f_if_j) - G_{ij}(e_if_j-e_jf_i).
\gamma\|h\|^2\leq \langle\big(\mathrm{Re}(KA-T)\big)h,h\rangle=\mathrm{Re}\langle(KA-T)h,h\rangle=\mathrm{Re}\langle \lambda Kh,h\rangle=\lambda \mathrm{Re}\langle Kh,h\rangle.
n\delta_q^{\textrm{Ric}}(p) \leq (n^2-n-2q)c+n^2H^2-\frac{1}{n-q}\sum_{r}(h_{11}^r+h_{22}^r+\cdots+h_{nn}^r)^2 &=(n^2-n-2q)c+\frac{n^2(n-q-1)}{n-q}H^2,
P^-=\frac{1}{2}\int dx^-d^2x^\perp : \left[ -\phi\partial^{\perp2}\phi +\sum_{i=1}^2\left( \bar\psi_{m}^{(i)}\frac{-\partial^{\perp2}+m^2}{i\partial^+}\gamma^+\psi_{m}^{(i)}+ 2g\phi \bar \psi_{m}^{(i)}\psi_{m}^{(i)}+g^2 \bar\psi_{m}^{(i)}\phi \frac{\gamma^+}{i\partial^+}\phi \psi_{m}^{(i)}\right)\right]:\;.
\frac{d}{dt}\mathbb{E}\left\vert u_{\varepsilon,\sigma}\left( t\right)-u_{\varepsilon,\tau}\left( t\right) \right\vert^{2}+\mathbb{E}\widetilde{G}_{\varepsilon}\left( u_{\varepsilon,\sigma}\left( t\right),u_{\varepsilon,\tau}\left( t\right) \right) =0,
\prod_{i=1}^N{\frac{\Gamma_q(\lambda_i + \theta(N-i)-z)}{\Gamma_q(\lambda_i + \theta(N-i+1)-z)}}\qquad{}= \frac{\Gamma_q(\lambda_1 + \theta(N-1) - z)}{\Gamma_q(\lambda_2 + \theta(N-1)-z)}\cdots\frac{\Gamma_q(\lambda_{N-1}+ \theta - z)}{\Gamma_q(\lambda_N + \theta-z)}\frac{\Gamma_q(\lambda_N - z)}{\Gamma_q(\lambda_1 + \theta N - z)},