harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{i}' =\begin{cases}V_{i}\cap N(u_{t+1}) \cap N(u_{t+2}) i = 1, V_{i}\cap N(v) \cap N(u_{t+2}) i = t+1, V_{i}\cap N(v) \cap N(u_{t+1}) \cap N(u_{t+2}) i\in [2,t]\cup \{t+4\}.\end{cases}
D = \begin{bmatrix}-\frac{\sqrt{n}}{2}I_n \left( \frac{-\sqrt{n}}{2}+1\right) I_n \left( -\frac{\sqrt{n}}{2}+1\right) I_n -\frac{\sqrt{n}}{2}I_n\end{bmatrix}= \begin{bmatrix}-\frac{\sqrt{n}}{2}\left( -\frac{\sqrt{n}}{2}+1\right) \left( -\frac{\sqrt{n}}{2}+1\right) -\frac{\sqrt{n}}{2}\end{bmatrix}\otimes I_n,
2\alpha n > e(\widetilde{H}) \ge e(\widetilde{H}-X)+ \left( \sum_{x \in X}d_{\widetilde{H}}(x) \right) -2 & \ge e(H''-X)-1 +|X|(3 \gamma n/ 5 -2)-2 & \ge e(H'')-\sum_{x \in X}d_{H''}(x)+|X|(3 \gamma n/ 5 -2)-3 & =e(H'')-\sum_{x \in X}(d_{H''}(x)-3 \gamma n/ 5 +2)-3
\vartheta^{(n)}_\sigma (w|\zeta )=\Theta_{x^2}\left( -\sigma \prod_{a\in A}w_a^{-1}\prod_{j=1}^{2n}\zeta_j \right) \prod_{j=1}^{2n}z_j^{-n}\prod_{a,b\in A\atop a<b}w_a^{-1}\Theta_{x^2}(w_a /w_b ).
|n\tilde{\nabla}(\varphi v_i dr\otimes dr)|^2_{\tilde{g}}=n^2(\varphi')^2v_i^2+n^2\varphi^2f^{-2}|\nabla v_i|^2+2n^3\varphi^2f^{-2}(f')^2v_i^2, \langle n\tilde{\nabla}(\varphi v_i dr\otimes dr),\tilde{\nabla}(\varphi f^2v_i g)\rangle_{\tilde{g}}=-2n^2 f'\cdot f^{-1}v_i^2\varphi^2.
\mathfrak{M}_{\ell}(X; \gamma, \psi): = \{\eta \in \mathfrak{M}(X), \psi(\gamma(x,y))\in L^{1}(|\eta|\times |\eta|) , \gamma(x,y)^{\ell}\in L^{1}(|\eta|\times |\eta|) & \eta(X)=0, \int_{X}\int_{X}K_{-\gamma}(x,y)^{j}d\eta(x)d\eta(y)=0, 1\leq j \leq \ell -1\}
\sum_{i=2}^{h-1}i\leqslant n-1 \;\;\Longleftrightarrow \;\;\frac{h(h-1)}{2}-1\leqslant n-1\;\; \Longleftrightarrow \;\;h\leqslant\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor.
{\lambda_{\ell}^{2}\frac 1 2 A_{77,\ell}=\lambda_{\ell}^{2}\left[ \frac{32}{\ell}+ \frac{3 \cdot 2^6}{\pi^2}\frac{1}{\ell^2}\log \ell - \frac{2^8}{\pi^2}\frac{1}{\ell^2}\log \ell+O(\ell^{-2}) \right] = 32 \ell^3 - \frac{2^6}{\pi^2}\ell^2 \log \ell +O(\ell^{2}).}
\hat{H}_{ph}=\frac{1}{2}\sum_{n,\alpha}[w_{1}(\hat{u}_{n+1,\alpha}-\hat{u}_{n,\alpha})^{2}+w_{2}(\hat{u}_{n,\alpha+1}-\hat{u}_{n,\alpha})^{2}+\frac{\hat{p}_{n,\alpha}^{2}}{M_{n,\alpha}}]
f(x) = 3x^5-7x^4+4x^3 = x^3 ( x-1)(3x-4 ) f'(x) =15x^4-28x^3 + 12 x^2 = x^2 \left(3x-2 \right) \left(5x-6 \right) f''(x) = 60x^3 - 84 x^2 + 24 x = 12x(x-1)(5x-2) f^{(3)}(x) = 180 x^2 - 168 x + 24 = 12 \left( 15x^2 - 14 x +2 \right) f^{(4)}(x) = 360 x - 168 = 24(15x-7) f^{(5)}(x) = 360.
\frac d{dt}\sum\limits_{|\alpha|=0}^{3}(\|\sqrt{B}\partial_\tau^\alpha u\|_{L^2}^2 +\|\sqrt{A}\partial_\tau^\alpha\tilde{h}\|_{L^2}^2)+C_1\|\partial_y(u, \tilde{h})\|_{H^{3, 0}}^2 &\le CD(t)^{\frac14}E(t)^{\frac54}+CD(t)^{\frac 12}E(t)+CD(t)E(t)^{\frac12}.
J_{1}^{k}| \Psi^{a b}\rangle = J_{1}^{k}J_{- 1}^{( a}| \phi^{b )}\rangle = - f_{d}^{k ( a}f_{e}^{b ) d}| \phi^{e}\rangle ~ + ~{\frac{k}{2}}g^{k ( a}| \phi^{b )}\rangle .
\|v_1(0, \cdot)\|_{L^2((0,1))}\leq N \big( \|v_1\|_{L^2((0,1) \times (0,1))}+ \|\nabla v_1\|_{L^2((0,1) \times (0,1))}\big) & \leq N \|\nabla v_1\|_{L^2((0,1) \times (0,1))}
[A, F_{0,-1}]_{q}\equiv (-d)^{n-3}[[\cdots [F_{n-1,1},F_{n-2,0}]_{q},\cdots,F_{2,0}]_{q},F_{0,-1}]_q &=(-d)^{n-3}[\cdots [[F_{n-1,1},F_{0,-1}]_q,F_{n-2,0}]_{q},\cdots,F_{2,0}]_{q}&=(-d)^{n-2}[\cdots [[F_{0,0},F_{n-1,0}]_q,F_{n-2,0}]_{q},\cdots,F_{2,0}]_{q}\,.
2T(r)\geqslant \sum_{i=j,l}(2\sum_{u=1}^3N^{(n)}_{(f^u,H_i),\leqslant k_i}(r)-3nN^{(1)}_{(f,H_i),\leqslant k_i}(r))+\sum_{i\ne j,l}(4N^{(1)}_{(f,H_i),\leqslant k_i}(r)-N(r,\nu_i)).
( i \frac{\partial}{\partial t}- \omega_{{\bf{k}}}({\bf{q}}) ) \frac{- i \langle T a_{{\bf{k}}}^{t}({\bf{q}}) a_{{\bf{k}}^{'}}^{\dagger}({\bf{q}}^{'}) \rangle}{\langle T 1 \rangle}= \delta_{{\bf{k}},{\bf{k}}^{'}}\delta_{{\bf{q}},{\bf{q}}^{'}}\delta ( t )
d s^{2}~ = ~ \alpha^{\prime}\left( \frac{u^{2}h ( u )}{R^{2}}~ e^{\gamma A}d x_{0}^{2}+ \frac{u^{2}}{R^{2}}~ e^{\gamma C}d x_{i}^{2}+ \frac{R^{2}}{u^{2}h ( u )}~ e^{\gamma B}d u^{2}+ R^{2}~ e^{\gamma D}d \Omega_{5}^{2}\right) ~ ,
\sigma_3=\begin{pmatrix}1 0 0 -1\end{pmatrix}, \ \ \ \P=\begin{pmatrix}0 u -\bar{u}0\end{pmatrix}, &\eta=\sqrt{\alpha}\left(k-\frac{\beta}{2k}\right),H= \alpha kU_x+\frac{i\alpha\beta^2}{2}\sigma_3\left(\frac{1}{k}P-P^2\right).
f_a = 3x+3y-3x^{2}-3y^{2}+z^{2}+x^{3}+y^{3}+w^{4}g_a = 4x+4y-6x^{2}-6y^{2}+4x^{3}+4y^{3}-x^{4}-y^{4}-z^{4}-w^{4},
\partial_t\dot{\Delta}_j u-\mathrm{div}\big((1+\dot{S}_m a)\dot{\Delta}_j\nabla u\big)= \dot{\Delta}_j\mathbb{P}(f+\dot{E}_m-\dot{S}_m a\nabla\Pi) &+\dot{\Delta}_j\mathbb{Q}\big(-\nabla \dot{S}_m a\cdot\nabla u-\dot{S}_m a\Delta u\big)+\mathrm{div}([\dot{\Delta}_j,\dot{S}_m a]\nabla u),
\begin{aligned}\sum_{\begin{subarray}{c}0{\le}m_1<_{c_1}\cdots<_{c_{p-1}}m_p<\infty m_i\in\mathbb{Z}\end{subarray}}\frac{(\alpha)_{m_1}}{{m_1}!}\frac{{m_p}!}{(\alpha)_{m_p}}\left\{\prod_{i=1}^{p}\frac{1}{(m_i+\alpha)^{a_i}(m_i+\beta)^{b_i}}\right\},\end{aligned}
\begin{bmatrix}PA+A^{T}P PB-A^{T}C^{T}B^{T}P-CA -(CB+B^{T}C^{T})\end{bmatrix}=\begin{bmatrix}-L^{T}L -L^{T}W -W^{T}L -W^{T}W\end{bmatrix}\leq 0.
\begin{array}{lll}\psi = \exp{\phi}\varphi ={\displaystyle\frac{d\phi}{dz}}\end{array}\hskip3truecm\begin{array}{rrc}\phi = \phi(z,\overline{z}) \overline{\varphi}={\displaystyle\frac{d{\phi}}{d\overline{z}}}\end{array}
B(h) =[0,2\pi)\cap Q_{h}=[h,\pi-h]\cup\lbrack\pi+h,2\pi-h], h <t_{1}<t_{2}<\cdot\cdot\cdot<t_{p}<\pi-h<\pi+h<t_{p+1}<t_{p+2}<\cdot\cdot\cdot<t_{s}<2\pi-h.
\|w(x) - \partial_x \rho\| = \|g_1(v (x) - \partial_x \rho) + g_2(\mu(x) - \partial_x \rho)\| \leq \varepsilon + \|\mu(x) - \partial_x \rho\| & \leq \varepsilon + \|\mu(x) - \mu(y)\| + \|\mu(y) - \partial_y \rho\| + \| \partial_y \rho - \partial_x \rho\| & \leq \varepsilon+ \varepsilon + c \varepsilon + \varepsilon = (3 + c) \varepsilon.
\Re \nabla \cdot Q + \Im\partial_{t}\{\bar{u}\psi' \partial_{r}u\}= \textstyle 2\psi''|\partial_{r}u|^{2}+2\frac{\psi'}{|x|}|\nabla^{T}u|^{2}-\frac12 \Delta^{2}\psi|u|^{2}-\psi'\partial_{r}c |u|^{2}.
\begin{cases}\Delta (xf)=\frac{k^2(\lambda\lambda'(\rho))'(\rho)}{\sigma^2}\,(xf), \frac{k f_\theta\lambda'(\rho)x}{\sigma^2\lambda}=0, x=\Delta \rho-\frac{k^2\lambda\lambda'(\rho)}{\sigma^2},\end{cases}
\langle \pi_t, \psi\rangle = \langle \pi_0 , e^{t\mathcal{A}}\psi\rangle - \int_0^t \! \mathrm{d}s \, \left\langle G\left(\frac{{\boldsymbol{\tilde{\varphi}}}^{\pi_s}}{{\boldsymbol{{\varphi}}}^{\pi_s}}\right) \pi_s, \partial_v \left( e^{(t-s) \mathcal{A}}\psi \right) \right\rangle \,.
\begin{aligned}\alpha_{\tau\sigma}(hM)=\frac{\tau-\sigma}{2}\varphi_{1}((\tau-\sigma) hM),\ \ \beta_{\tau}( hM )=(1-\tau)\varphi_{1}((1-\tau)hM),\ \ \gamma_{\tau}( hM )=\varphi_{0}((1-\tau)hM).\end{aligned}
X_i(t) \leq \kappa(t) X_i(t^*) e^{\int_{t^*}^t G_i(v) \: dv}= \kappa(t) \cdot \frac{X_i(t^*) e^{-s(k^* - i)(t - t^*)}}{X_{k^*}(t^*)}\cdot X_{k^*}(t^*) e^{\int_{t^*}^t G_{k^*}(v) \: dv}.
\sup_{\xi \in Q}\left\langle a + \frac{\xi}{L}\right\rangle^{-\alpha}= \left( \inf_{\xi \in Q}\left\langle a + \frac{\xi}{L}\right\rangle \right)^{-\alpha}\leq \left( c_d \sup_{\xi \in Q}\left\langle a + \frac{\xi}{L}\right\rangle \right)^{-\alpha}= c_d^{-\alpha}\inf_{\xi \in Q}\left\langle a + \frac{\xi}{L}\right\rangle^{-\alpha}.
\begin{array}{lll}\mbox{Bregman primal Condat--V\~{u}algorithm}\tilde x=2x^{(k+1)}-x^{(k)}\tilde z=z^{(k)}\mbox{Bregman dual Condat--V\~{u}algorithm}\tilde x=x^{(k)}\tilde z=2z^{(k+1)}-z^{(k)}.\end{array}
\zeta (s,a) = \frac{(-\pi i) (2\pi)^{s-1}}{\Gamma(s) \sin \pi s}\Biggl( e^{\pi is/2}\sum_{n=1}^\infty \frac{e^{2\pi ina}}{n^{1-s}}-e^{-\pi is/2}\sum_{n=1}^\infty \frac{e^{-2\pi ina}}{n^{1-s}}\Biggr),
p_{\zeta}'(x) = (1 - \varphi_{\zeta}(x))p'(x) + \varphi_{\zeta}(x) q'(x) + \varphi_{\zeta}'(x)(q(x) - p(x)), p_{\zeta}''(x) = (1 - \varphi_{\zeta}(x))p''(x) + \varphi_{\zeta}(x) q''(x) + 2\varphi_{\zeta}'(x)(q'(x) - p'(x)) &\qquad+ \varphi_{\zeta}''(x)(q(x) - p(x)).
\hat{{\mathbf U}}^{\pm}\in W^{3,\infty}(\Omega_T),\quad\hat{\varphi}\in W^{4,\infty}(\Gamma_T), ||\hat{{\mathbf U}}^{\pm}||_{W^{3,\infty}(\Omega_T)}+||\hat{\varphi}||_{W^{4,\infty}(\Gamma_T)}\leq K,
\varepsilon \leq \|\varphi_{km_{j_0}}-\rho\|_\infty =\|\varphi_{kj}\circ \varphi_{k(m_{j_0}-j)}-\rho\circ \varphi_{k(m_{j_0}-j)}\|_\infty &=\sup_{z\in \mathbb{B}_n}|(\varphi_{kj}-\rho)( \varphi_{k(m_{j_0}-j)}(z))| & \leq \|\varphi_{kj}-\rho\|_\infty.
\mathcal{B}_{M^{(2)},M^{(1)}}(P^{(1)},P^{(2)})=\!\!\sum_{i,j=1}^{\ell,k}\!\! B_{ij}\phi_{\ell-i}(y)\phi_{k-j}(x)\!:=\frac{M^{(2)}(y)P^{(2)}(x)-M^{(1)}(y)P^{(1)}(x)}{x-y}.
\begin{aligned}(X,i,f,j,g) \mapsto (Y,\varepsilon j, g) (X,i,f,j,gh) \xrightarrow{\smash h}(X,i,f,j,g) \mapsto (Y,\varepsilon j, gh) \xrightarrow{\smash h}(Y,\varepsilon j, g)\rlap{,}\end{aligned}
\mathcal{P}_{2 j +2}:=\left\lbrace \prod_{k=1}^{j+1}\partial_r V_k : V_k \in \{D_r, \tfrac{1}{r}\}\right\rbrace , \mathcal{P}_{2 j +1}:= \left\lbrace V_{j+1}\prod_{k=1}^{j}\partial_r V_k : V_k \in \{D_r,\tfrac{1}{r}\}\right\rbrace ,
\gamma_1 = 2 \alpha_0\alpha_1 -2 \beta_1 =-2q^2+4q^3+4q^3s-4q^3s^2-4q^4+2q^4s^2 + \left(-4q^3s+4q^3s^2+2q^4-2q^4s^2\right) e^{t\left(1-q\right)^2}\overset{( a)}{\le}4q^3 +2 e^{t(1-q)^2}q^4 \overset{( b)}{\le}6q^3 \, ,
(T)=T_{0\overline{0}}T_{1\overline{1}}-(T_{0\overline{1}})^2=-\frac{t^4}{3}-\frac{t\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\left(\frac{3(m\tau_1+n)^2}{|m\tau+n|^5}-\frac{1}{|m\tau+n|^3}\right)\,.
\sum_{i=1}^{\mu}w_{i}\frac{x_{i:\lambda}-m_{k}}{\sigma_{k}}(\frac{x_{i:\lambda}-m_{k}}{\sigma_{k}})^{T}=argmax_{C}\sum_{i=1}^{\mu}w_{i}logp_{N}(\frac{x_{i:\lambda}-m_{k}}{\sigma_{k}}|C)
\gamma^{\textbf{ext}}_{1}\mathrm{U}_{\mathrm{e}}= -\mathbb{H}_{\alpha_\mathrm{m}}\Big[\gamma^{\textbf{ext}}_{0}\mathrm{U}_{\mathrm{e}}\Big](\mathrm{x},t) + \Big(\frac{1}{2}I_{d}- \mathcal{K}_{\alpha_\mathrm{m}}^{*}\Big)\Big[\gamma^{\textbf{ext}}_{1}\mathrm{U}_{\mathrm{e}}\Big](\mathrm{x},t).
\int_0^\infty |v(x)-v(y)|^p\frac{x^2+y^2}{(x^2-y^2)^2}\, dy &=\int_I |u(r_0)-u(y)|^p\frac{x^2+y^2}{(x^2-y^2)^2}\, dy &\leq \int_I |u(x)-u(y)|^p\frac{x^2+y^2}{(x^2-y^2)^2}\, dy.
\vert a_{\nu}\vert^{2}= 4 \frac{\langle Q_{0}\rangle + 4}{\langle Q_{1}\rangle}\left[ \frac{H^{2}e^{\pi \nu}\Gamma ( - \sigma ) \Gamma ( - \sigma^{*})}{2^{5}\pi^{4}m^{2}}\right] = 4 \frac{\langle Q_{0}\rangle + 4}{\langle Q_{1}\rangle}\left[ \frac{H^{2}\nu^{2}+ H^{2}/ 4}{2^{4}\pi^{3}m^{2}}\right] \, .
\left( \frac{\dot{a}_{0}}{a_{0}}\right)^{2}+ \frac{K}{a_{0}^{2}}= \frac{\rho_{0 M}+ \rho_{\frac 1 2 M}e^{2 k b}- \rho_{0 M}\bar{\rho}_{\frac 1 2 M}e^{k b}\sinh ( k b )}{3 ( M^{3}/ k ) \left[ e^{k b}- 1 - \bar{\rho}_{\frac 1 2 M}e^{k b}( \cosh ( k b ) - 1 ) \right]}.
L_{ij}=\left(\begin{array}{cccc}0 k_{1}-k_{3}-k_{1}+k_{2}k_{3}-k_{2}-k_{1}+k_{3}0 k_{1}-k_{3}k_{1}-k_{2}-k_{1}0 k_{2}-k_{3}+k_{2}k_{3}-k_{2}0\end{array}\right) .
R(\lambda) \otimes R(\mu) = \bigoplus_{|\nu| = |\lambda| + |\mu|, k(\nu) \leq n}(c_{\lambda \mu}^{\nu})^2 R(\nu) \oplus \bigoplus_{|\nu| < |\lambda| + |\mu|, k(\nu) \leq n}d_{\lambda \mu}^{\nu}R(\nu)
\mathrm{Pf}\left[M_{k,l}\right]_{k,l=1}^{2N}\doteq \frac{1}{2^{N}N!}\sum_{\pi \in \mathcal{S}_{2N}}\left(-1\right)^{\pi}\prod\limits_{j=1}^{N}M_{\pi \left( 2j-1\right) ,\pi\left(2j\right)}
\mathbf{e}(1,1)=\left(\begin{array}[c]{cc}0 -1 0 0\end{array}\right) \ \mathbf{e}(1,0)=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}1 0 0 -1\end{array}\right) \ \mathbf{e}(1,-1)=\left(\begin{array}[c]{cc}0 0 1 0\end{array}\right)
\int\limits_{0}^{\infty}f_1(x \mid a_1, d_1, p_1) \, \left( \frac{x}{a_2}\right)^{p_2}\, dx = \frac{\Gamma \left(\frac{d_1+p_2}{p_1}\right)}{\Gamma \left(\frac{d_1}{p_1}\right)}\, \left( \frac{a_1}{a_2}\right)^{p_2}.
\lim_{a\to\infty}\frac{\mathbb{P}_y(\tau^{(-a,a)^{{\rm c}}}<\tau^{\{0\}})}{\mathbb{P}_x(\tau^{(-a,a)^{{\rm c}}}<\tau^{\{0\}})}= \lim_{a\to\infty}\frac{\mathbf{P}_{0,{\rm sign}(y)}(T^+_{\log (a/|y|)}<\infty)}{\mathbf{P}_{0,{\rm sign}(x)}(T^+_{\log (a/|x|)}<\infty)}= \frac{v_{{\rm sign}(y)}(\theta)}{v_{{\rm sign}(x)}(\theta)}\left|\frac{y}{x}\right|^\theta, x,y\in\mathbb{R}.
\dfrac{\Big( \hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'}) \Big) \Big( \hbar'(p - i -1) - (a_s - a_{s'}) \Big)}{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}& \times \dfrac{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}{\Big( \hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'}) \Big) \Big( \hbar'(p - i -1) - (a_s - a_{s'}) \Big)}& = 1.
a_1 =\frac{1}{2}-\frac{3}{\widehat{h}^2}\pm \frac{\sqrt{9-\widehat{h}^2}}{\widehat{h}^2},b_1 = \frac{3}{\widehat{h}^2}\pm \frac{\sqrt{9-\widehat{h}^2}}{\widehat{h}^2},0< \widehat h\leq 3.
\frac{(\frac{c}{d})_{\infty}(dq)_{\infty}}{(q)_{\infty}(cq)_{\infty}}\sum_{k=1}^{\infty}\frac{d^kq^{k(k+1)}}{(dq)_k(q)_k(1-q^k)}\sum_{j=0}^{\infty}\frac{(dq)_j(cq^k/d)^j}{(dq^{k+1})_j(q)_j}=\frac{(dq)_{\infty}}{(q)_{\infty}}\sum_{k=1}^{\infty}\frac{(c/d)_k(dq)^k}{(q)_k(1-q^k)}\sum_{j=0}^{\infty}\frac{(dq)_jc^jq^{(j+k)^2}}{(dq)_{j+k}(cq)_{j+k}(q)_j}.
V_{4}( \varphi_{c}) = \frac{1}{2 v_{s}^{2}}+ \frac{1}{2}v_{s}^{2}+ \frac{3 u^{2}v_{s}^{4}}{6 4 \pi^{2}}[ \mathrm{l n}\frac{v_{s}^{2}u}{M^{2}}- 1 ] ,
\Bigg \lfloor \frac{\Big \lceil \frac{\lceil d \rceil}{2}\Big \rceil}{2}\Bigg \rfloor = \Bigg \lfloor \frac{\lceil d \rceil}{4}\Bigg \rfloor = \Bigg \lfloor \frac{\lceil d \rceil + 1}{4}\Bigg \rfloor,
\psi_\eta^{\ell}(u)=N(\rho,\alpha)\frac{(1-|\eta|^2)^{\ell+1}}{\sqrt{\Gamma(\ell+1)}}u^{\frac{\ell}{2}}e^{-\frac{\varpi}{2}u}\frac{e^{\frac{u\eta}{\eta-1}}}{(1-\eta)^{1+\ell}}e^{i\theta_{n,\ell}(t)}.
\frac{d}{ds_1}\mathrm{sn}(x,y) = \frac{d}{ds}\left(\frac{[\gamma_{\partial B}(s),y]}{||y||_a}\right) = \frac{[b(x),y]}{||y||_a||b(x)||}= \mathrm{sn}(b(x),y) \ \mathrm{and}\frac{d}{ds_2}\mathrm{cm}(x,y) = \frac{d}{ds}\left([\gamma_{\partial B}(s),b(x)] \right) = \frac{[b(y),b(x)]}{||b(y)||}= \mathrm{cm}(x,b(y)).
\mathcal{E}^\mathrm{H}[u]:=\; \int_{\mathbb{R}^d}|\nabla u(x)|^2dx+ \int_{\mathbb{R}^d}V_\mathrm{DW}(x)|u(x)|^2dx &+\frac{\lambda}{2}\iint |u(x)|^2 w(x-y)|u(y)|^2dxdy,
\sum_{j=1}^{N^m-1}\beta''_j(t)\int^{b}_{a}\phi_i(x)\phi_j(x)\,d\mu=-\sum_{j=1}^{N^m-1}\beta_j(t)\int^{b}_{a}\phi'_i(x)\phi'_j(x)\, dx,1\le i\le N^m-1.
f_{X_l}(x_l)=\frac{\sigma_l\delta_l^{\sigma_l^{-1}}}{\chi_l(\sigma_l^{-1})}I^{m_l,n_l}_{p_l,q_l}\Bigg[\delta_l x_l^{\sigma_l}\left|\begin{matrix}(a_{li},A_{li},\alpha_{li})_{1,p_l}(b_{lj},B_{lj},\beta_{lj})_{1,q_l}\end{matrix}\right.\Bigg],\ \x_l>0,
F_{F}( \beta ) = \int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{- \frac{d - 2}{2}}\frac{\bar{\theta}_{2}^{4}}{\bar{\eta}^{1 2}}( z_{s}+ z_{c}) \theta_{4}^{'}\left( 0 \left| \frac{i \beta^{2}}{2 \pi^{2}\tau_{2}}\right. \right)
a_{k-i,0}^{(1)}= (-1)^i i! \frac{\Gamma\bigl(\frac{n}{2}+j\bigr)}{\Gamma\bigl(\frac{n-2k}{2}+i+j\bigr)}, a_{0,k-j}^{(1)}= 0 , a_{0,k-j}^{(2)}= (-1)^i i! \frac{\Gamma\bigl(\frac{n}{2}+i\bigr)\Gamma\bigl(k-i\bigr)}{\Gamma\bigl(\frac{n-2k}{2}+i+j\bigr)\Gamma\bigl(k-j\bigr)}, a_{k-i,0}^{(2)}= 0 .
\begin{aligned}A^{0}_{1}(u,v)\partial_{t}u+A^{i}_{11}(u,v)\partial_{i}u+A^{i}_{12}(u,v)\partial_{i}v =g_{1}(U,D_{x}v), A^{0}_{2}(u,v)\partial_{t}v+A^{i}_{21}(u,v)\partial_{i}u+A^{i}_{22}(u,v)\partial_{i}v-B^{ij}(u,v)\partial_{i}\partial_{j}v =g_{2}(U,D_{x}U),\end{aligned}
\begin{aligned}W_Y(z_--\varepsilon \eta) =W_Y(z_-)+C\varepsilon^{1-\nu/2}T_{2-\nu}\left(\sqrt{\eta}\right)-\varepsilon\frac{2\eta}{4-q}+\mathcal{O}(\varepsilon^{1+\nu/2})\ ,\end{aligned}
{\Phi}= \frac{1}{2}\left( \begin{array}{c}{{- A_{I j}E_{\nu}^{j}}}\\{{D_{i j}E_{\nu}^{j}}}\\{{E_{\nu}^{i}}}\\ \end{array}\right) ={\frac{1}{2}}\left( \begin{array}{c c c}{{{\cal E}_{I}^{M}}}&{{0}}&{{0}}\\{{0}}&{{T_{i}^{\mu}}}&{{0}}\\{{0}}&{{0}}&{{T_{i}^{\mu}}}\\ \end{array}\right){\hat{\Phi}}
\tilde{\alpha}_i[w]:= h_{i2}\alpha_i[w]+p_{i12}w(\tau_i):= \int_0^1 w(s)\,d \tilde{A}_i(s),\,\,i=1,2, \bar{\alpha}_i[w]:= h_{i1}\alpha_i[w]+p_{i11}w(\tau_i):= \int_0^1 w(s)\,d \bar{A}_i(s),\,\,\,i=1,2,
\delta_{\Lambda}c_{k}= ( \frac{\sum_{\tilde{i}= 2}^{\infty}\tilde{i}( \sum_{l^{\prime}= k}^{\tilde{i}+ k - 2}( \tilde{i}+ k - l^{\prime}) h_{\tilde{i}+ k - l^{\prime}}c_{l^{\prime}- k + 1})}{\sum_{\tilde{i}}\tilde{i}h_{\tilde{i}}}c_{k}+ \dot{c_{k}}c ) \Lambda
(0\leq)\ \C_{p,R}= \int_{\widetilde{\Omega}_p\cap\{|x|> R\}}e^U \widetilde{\varphi}_{1,p}^2\leq \sup_{|x|>R}e^{U(x)}\int_{\widetilde{\Omega}_p}\widetilde{\varphi}_{1,p}^2=\sup_{|x|>R}e^{U(x)}=\sup_{|x|>R}\left( \frac{1}{1+\frac{|x|^2}{8}}\right)^2\leq \frac{64}{R^2}.
\nabla_i^\perp(\nabla^\perp\cdot H ) =\nabla_l^\perp H_2 \nabla_l \nabla_i^\perp H_1-\nabla_l^\perp H_1 \nabla_l \nabla_i^\perp H_2 &=\nabla_l^\perp H_2 \nabla_l \nabla_i^\perp H_1+\nabla_l H_1 \nabla_l^\perp \nabla_i^\perp H_2 &=\nabla_i^\perp(\nabla_l^\perp H_2 \nabla_l H_1),
v^2 \frac{\partial}{\partial\tau}Q_{-\overline{\tau}}= \frac{i}{2}Q(-\overline{\tau},1), v^2 \frac{\partial}{\partial\tau}Q_{\tau}= \frac{i}{2}Q(\overline{\tau},1), v^2 \frac{\partial}{\partial\tau}\frac{Q(\tau,1)}{v^2}= i Q_{\tau}.
U^{\rm gh}_c = V_c^{\rm gh}\times \prod_{\scriptstyle i=1\atop{\scriptstyle i\neq b,c\atop\scriptstyle i\neq E}}^{N_1}\sum_{r=-1}^1E_{-1r}(V_i)b_r^i\times \sum_{\scriptstyle j=1\atop\scriptstyle j\neq F}^{N_2}\sum_{r=-1}^1E_{-1r}(V_j)b_r^j \times \prod_{\scriptstyle i=1\atop{\scriptstyle i\neq d,h\atop \scriptstyle i\neq F}}^{N_2}\sum_{r=-1}^1E_{-1r}(V_i)b_r^i\ .
L_{+}= \epsilon^{\mu \nu \rho \sigma}\bigg [ \frac{1}{4 \tau^{+}}G_{\mu \nu}^{+ ~ ~ a b}G_{\rho \sigma a b}^{+}- i G_{\mu \nu}^{+ ~ ~ a b}R_{\rho \sigma a b}^{+}( \omega^{+}) \bigg ] ,
\sum_{n=0}^\infty\sum_{|m|\leq n/2}(-1)^mq^{n^2-2m^2}(1+q^{2n+1}) = qg_{1,3,1}(q^6,q^6;q^4) + g_{1,3,1}(q^2,q^2;q^4) +q^4g_{1,3,1}(q^{10},q^{10};q^4)+q^9g_{1,3,1}(q^{14},q^{14};q^4),
\begin{array}{c}{8 t_{2}< O_{2}O_{k_{1}}. . . O_{k_{p - 1}}P >_{g}}\\{+ 4 t_{0}< O_{k_{1}}. . . O_{k_{p - 1}}P P >_{g}+ ( 4 ( p - 1 ) + 4 \Sigma ) < O_{k_{1}}. . . O_{k_{p - 1}}P >_{g}= 0 ,}\\ \end{array}
J_{\mathcal{G}_{f,g},g}(x) =\inf \biggl\{I_{\mathcal{G}_{f,g},g}\biggl(\frac{z}{1-x},z\biggr):z>0 \biggr\}&=\inf \biggl\{\frac{z}{1-x}I_f \biggl(\frac{\frac{z}{1-x}-z}{\frac{z}{1-x}}\biggr)+I_g(z):z>0 \biggr\}&=\inf \biggl\{\frac{z}{1-x}I_f(x)+I_g(z):z>0\biggr\}&=-\sup \biggl\{-z\frac{I_f(x)}{1-x}-I_g(z):z>0 \biggr\};
T_{\mu\nu}^{eff}=\frac{1}{2}\left(1+\frac{b_{k}}{b_{g}}\right)\left(\phi_{,\mu}\phi_{,\nu}-K\tilde{g}_{\mu\nu}\right)-\frac{\Delta^{2}Ke^{2\alpha\phi/M_{p}}}{2b_{g}M^{4}}\left(\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}K\tilde{g}_{\mu\nu}\right)+\tilde{g}_{\mu\nu}\frac{sM^{4}}{4b_{g}}e^{-2\alpha\phi/M_{p}}
Q(\hat{u}):= (1+|\nabla\hat{u}|^2)\left(g^{ij}-\dfrac{\nabla^i\hat{u}\nabla^j\hat{u}}{|\nabla \hat{u}|^2+1}\right)\nabla_{ij}\hat{u}&-s\sqrt{1+|\nabla\hat{u}|^2}^3\left(g^{ij}-\dfrac{\nabla^i\hat{u}\nabla^j\hat{u}}{|\nabla \hat{u}|^2+1}\right)K_{ij}&+\frac{s}{\varepsilon}(1+|\nabla \hat{u}|^2)-\kappa\hat{u}\sqrt{1+|\nabla \hat{u}|^2}^3,
2 N^{2}{\frac{g^{2}( M )}{M}}\int_{0}^{M}{\frac{d^{3}k}{( 2 \pi )^{3}}}{\frac{1}{k^{2}+ \sigma^{2}}}={\frac{N^{2}g^{2}( M )}{\pi^{2}}}\left( 1 -{\frac{\sigma}{M}}\arctan{\frac{M}{\sigma}}\right) = N
\begin{aligned}g_{_{X,f}}(d_1) =\sup_a Ef(a+d_1X) = \sup_a Ef\left(\frac{d_1a}{d_2}+d_1X\right) &= \sup_a Ef\left(\frac{d_1}{d_2}(a+d_2X)\right) \geq \sup_a Ef(a+d_2X) = g_{_{X,f}}(d_2). \end{aligned}
\begin{array}{r c l}{{{\cal R}^{( 3 )}}}&{{=}}&{{3 \partial a^{( 2 ) \prime}+{\textstyle \frac{1}{2 \pi \alpha^{\prime}}}C^{( 3 )}- 3{\textstyle \frac{m}{2}}( 2 \pi \alpha^{\prime}) b \partial b - 3{\textstyle \frac{m}{2}}( 2 \pi \alpha^{\prime}) B b}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{- \left( C^{( 1 )}-{\textstyle \frac{m}{2}b}\right) \left({\cal F}+{\cal L}^{( 2 )}\right) \, ,}}\\ \end{array}
D^\alpha (g^{jk}-g^{jk0})=o\bigl(|x|^{-|\alpha |}\bigr),\tag*{$\textup{(\ref*{23-2-39})}_1$}&D^\alpha (V_j-V_j^0)=o\bigl(|x|^{m_1+1-|\alpha |}\bigr),\tag*{$\textup{(\ref*{23-2-39})}_2$}&D^\alpha (V-V^0)=o\bigl(|x|^{2m-|\sigma |}\bigr) \forall \alpha : |\alpha |\le 1.\tag*{$\textup{(\ref*{23-2-39})}_3$}
\frac{1}{4\epsilon^{2}}\mathrm{trace}(\mathrm{OP}_{\hbar}^{T}(f(q))\mathrm{OP}_{\hbar}^{T}((2\pi\hbar)^{d}\nu))=\frac{1}{4\epsilon^{2}}\underset{\mathbb{R}^{2}\times\mathbb{R}^{2}}{\int}|q|^{2}\nu(dqdp)+\frac{\hbar}{4\epsilon^{2}}.
\mathcal{S}^* = \arg \min_{\mathcal{S}\in \mathbf{S}}\mathbf{T}(\mathcal{S}) = \arg \min_{\mathcal{S}\in \mathbf{S}}\sum_{t=1}^{\mathcal{C}(\mathcal{S})}\sum_{u \in \mathcal{U}}T_u(t).
d(z) = \sum_{\alpha \in S_+}:X^\alpha \varphi^\alpha: -\frac{1}{2}\sum_{\alpha, \beta, \gamma \in S_+}{f^{\alpha \beta}}_\gamma :\varphi_\gamma \varphi^\alpha \varphi^\beta: + \sum_{\alpha \in S_+}(f | q^\alpha) \varphi^\alpha + \sum_{\alpha \in S_{\frac{1}{2}}}:\varphi^\alpha \Phi_\alpha:.
A= \frac{(n-1)(k-1)^2}{nk-k-2n+3-\widehat{\lambda}_{n}(\underline{\xi}_{n})},B= \frac{(n-1)\lambda_{n}(\underline{\xi}_{n})+\widehat{\lambda}_{n}(\underline{\xi}_{n})+n-2}{1-\widehat{\lambda}_{n}(\underline{\xi}_{n})}.
\begin{array}{c c l}{{L_{a - b}- \! L_{a + b}= \displaystyle 2 \; \sqrt{k + 2}\; \exp \left( \! - \frac{i \pi ( a^{2}+ b^{2})}{4 ( k + 2 )}\right) \sin \left( \frac{\pi a b}{2 ( k + 2 )}\right) E_{k + a + b}\ .}}\\ \end{array}
\begin{array}{c}\frac{\partial P}{\partial x}(x)f(x) - \frac{1}{2}\frac{\partial P}{\partial x}(x)g(x)g^T(x)\frac{\partial^T P}{\partial x}(x) begin{align*}2mm] + \frac{1}{2}h^T(x)h(x) =0 \end{array}
clog\frac{\Omega_{xy}+\sqrt{\Omega_{xy}^{2}-\Omega_{xx}\Omega_{yy}}}{\Omega_{xy}-\sqrt{\Omega_{xy}^{2}-\Omega_{xx}\Omega_{yy}}}=2ic\cdot arccos\frac{\Omega_{xy}}{\sqrt{\Omega_{xx}\cdot\Omega_{yy}}}
\sigma^{0}\left( m_{2}^{2}\right) = \frac{1 2 \pi \tilde{\Gamma}_{e , 2}\tilde{\Gamma}_{f , 2}}{m_{2}^{2}\Gamma_{2}^{2}}\left( 1 - \frac{\Gamma_{2}^{2}}{m_{2}^{2}}\right) + \sigma_{\mathrm{B}}^{0}\left( m_{2}^{2}\right) ,
J^{0}= - \frac{1}{2}i \left[ \psi^{*}( \partial_{\tau}\psi - i \varphi \psi ) - ( \partial_{\tau}\psi^{*}+ i \varphi \psi^{*}) \psi \right] \ , \ \vec{J}= \frac{1}{2}i \left[ \psi^{*}( \vec{\partial}\psi + i \vec{a}\psi ) - ( \vec{\partial}\psi^{*}- i \vec{a}\psi^{*}) \psi \right] ,
\qquad \qquad \qquad \left. -{\frac{1}{6}}\sum_{M \parallel i}\partial_{a}\operatorname{l n}\tilde{E}_{M}+{\frac{1}{3}}\sum_{M \perp i}\partial_{a}\operatorname{l n}\tilde{E}_{M}\right] \Gamma_{\hat{a}}\eta = 0
m_{ap(\alpha)}(\{b,a-b+1\}) =m_{ap(\beta)}(\{b,a-b+1\})+1 m_{ap(\beta)}(\{b-1,a-b+1\}) =m_{ap(\alpha)}(\{b-1,a-b+1\})+1 m_{ap(\alpha)}(\{x,y\}) =m_{ap(\beta)}(\{x,y\})\hbox{otherwise.}
\tau_\pi(r_1)j_{r_1}=\tau_\pi(s_1)j_{s_1}=\tau_\pi(r_2)j_{r_2}=\tau_\pi(s_2)j_{s_2}\tau_\pi(r_1)j_{r_1}=\tau_\pi(s_1)j_{s_1}=-\tau_\pi(r_2)j_{r_2}=-\tau_\pi(s_2)j_{s_2}.
\frac{d \sigma_{c}}{d E_{+}}\approx \frac{4 \pi^{2}Z^{2}e^{4}}{\omega^{2}a_{2}a_{3}}\sum_{m , g_{y}}\frac{2 g_{y}^{2}+ R^{- 2}}{g_{m \parallel}^{2}( g_{y}^{2}+ R^{- 2})^{3 / 2}}\left[ \frac{\omega^{2}}{2 E_{+}E_{-}}- 1 + 2 \frac{\delta}{g_{m \parallel}}\left( 1 - \frac{\delta}{g_{m \parallel}}\right) \right] J_{m}^{2}( g_{y}u_{0}) ,
\tilde{A}_{1}^{( 1 )}( x ) = - \frac{1}{e}\frac{x_{2}}{x_{1}^{2}+ x_{2}^{2}+ \lambda^{2}}\left[ 1 + \frac{x_{1}^{2}}{( x_{1}^{2}+ \lambda^{2})^{\frac{3}{2}}}\right]
T^t_0 h_{3 k,3 k + 1}(q,J) T_0 = \frac{J}{4}\xi^2[\sigma'^x_k \sigma'^x_{k+1}+ \sigma'^y_k \sigma'^y_{k+1}+ \frac{q + q^{-1}}{2}\sigma'^z_k \sigma'^z_{k+1}
\left(Z_i(u),\frac{1}{T_i(v)}\right)_{u\geq 0,v\geq0,i\in V}=\left( \theta_iZ^*_i(u),\frac{1}{T_i^{\infty}}+\frac{1}{\theta_i^2T_i^*(v)}\right)_{u\geq 0,v\geq 0,i\in V}.
\frac{J ( \nu , z )}{\phi ( \nu / 2 )}= M e_{\nu}( z , h^{2}) , ( - 1 )^{n}\frac{\phi ( n + \nu / 2 )}{\phi ( \nu / 2 )}= \frac{c_{2 n}^{\nu}( h^{2})}{c_{0}^{\nu}( h^{2})}
c_L(\theta) =\mu+\mathbf{r}_L(\theta)'\mathbf{R}_L^{-1}(\mathbf \Upsilon-\mu \mathbf 1) \varsigma^2s^2_L(\theta) = \varsigma^2\biggl(1-\mathbf r_L(\theta)'\mathbf R_L^{-1}\mathbf r_L(\theta)+\frac{(1-\mathbf 1'\mathbf R_L^{-1}\mathbf r_L(\theta))^2}{\mathbf 1'\mathbf R_L^{-1}\mathbf 1}\biggr).
\quad\sum_{k=0}^{\infty}(20k^2+8k+1)\frac{\binom{2k}{k}^5}{(-2^{12})^k}=\frac{8}{\pi^2}, begin{align*}1mm] \sum_{k=0}^{\infty}(820k^2+180k+13)\frac{\binom{2k}{k}^5}{(-2^{20})^k}=\frac{128}{\pi^2}.
Q_2=\mbox{\bf Q}^{-1}Q_1\mbox{\bf Q}+\mbox{\bf Q}^{-1}\Delta \mbox{\bf Q}+2\mbox{\bf Q}^{-1}A_1\partial_z\mbox{\bf Q}+2\mbox{\bf Q}^{-1}B_1\partial_{\bar z}\mbox{\bf Q}\quad\mbox{in}\,\,\Omega .
M_n^{(a,b)}=M_n^{(2a-a, b)}=\sum_{k=0}^n\binom{n}{k}M_k^{(-a,b)}(2a)^{n-k}=(-1)^n\sum_{k=0}^n\binom{n}{k}M_k^{(a,b)}(-2a)^{n-k}.
\frac{\partial^{2}V_{f e r m .}^{( 1 )}}{\partial \mu^{2}}= \frac{e H}{\pi^{2}}\beta^{2}\frac{\partial}{\beta^{2}}\sum_{l = 1}^{\infty}( - 1 )^{l + 1}\int_{0}^{\infty}\frac{d s}{s}e x p ( - m^{2}s - \beta^{2}l^{2}/ 4 s ) c o t h ( e H s ) .