harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}\Bigl[ (r + s)_{23}(x, \mu, \eta)~,~ (r + s)_{12}(x, \lambda, \eta)\Bigr] + \Bigl[ (r + s)_{23}(x, \mu, \eta)~,~ (r + s)_{13}(x, \lambda, \eta)\Bigr] begin{align*}2mm] + \Bigl[(r + s)_{13}(x,\lambda, \eta) ~,~ (r - s)_{12}(x, \lambda, \mu)\Bigr] + H^{(r+s)}_{1,23}(x, \lambda, \mu, \eta) - H^{(r+s)}_{2,13}(x, \mu, \lambda, \eta ) = 0 , \end{array}
\left( \frac{C^{2}e^{2 \alpha}n^{4}}{\left( \left( x_{0}+ y_{0}+ i \epsilon \right)^{2}- \left( x_{s}+ y_{s}\right)^{2}\right)^{2}}+ \frac{C^{2}e^{2 \alpha^{*}}n^{4}}{\left( \left( x_{0}+ y_{0}- i \epsilon \right)^{2}- \left( x_{s}+ y_{s}\right)^{2}\right)^{2}}\right.
\xi_{i_1,\ldots,i_n}(f_{i_n,\ldots,i_1,i_1,\ldots,i_n}) =\xi_{i_1,\ldots,i_n}(\psi^n_{i_n,\ldots,i_1,\ldots,i_n}(f_{i_n,\ldots,i_1,\ldots,i_n}))= &=\xi_{i_2,\ldots,i_n}(\delta^n_{i_n,\ldots,i_1,\ldots,i_n}(f_{i_n,\ldots,i_1,\ldots,i_n}))=\xi_{i_2,\ldots,i_n}(f_{i_n,\ldots,i_2,i_2,\ldots,i_n})= &=\epsilon(f).
\begin{aligned}v_t+(v+u_{\infty})\cdot \nabla v+\nabla p =(b+B_{\infty})\cdot \nabla b+\nu \Delta v, b_t+(v+u_{\infty})\cdot \nabla b =(b+B_{\infty})\cdot \nabla v+\mu \Delta b, \nabla \cdot v=\nabla \cdot b =0.\end{aligned}
\epsilon \mu(r_{n-1}) = \sum_{i=1}^n t_n^i \big[t_n^{-i}r_{n-1},x_n\big] & = t_n\big[\ell_{n-1},x_n] + \sum_{i=2}^{n-1}t_n^i \big[ [\ell(x_1,\ldots, x_{n-i}), r(x_{n-i+1},\ldots,x_{n-1})],x_n \big] + [r_{n-1},x_n] & = \big[\ell(x_2,\ldots,x_n),x_1] + \sum_{i=2}^{n-1}\big[ [\ell(x_{i+1},\ldots, x_n), r(x_1,\ldots,x_{i-1})],x_i \big] + [r_{n-1},x_n].
G =\langle \tilde{x},\tilde{y},\tilde{z},\tilde{n}\mid \tilde{n}^p= \tilde{x}^{p^2}=\tilde{y}^p=1, \tilde{z}^p=\tilde{x}^p, [\tilde{x},\tilde{z}]=\tilde{y}, [\tilde{x},\tilde{y}] = \tilde{n}^{-\alpha}, [\tilde{y},\tilde{z}]=\tilde{x}^p\tilde{n}^{\alpha+1}, [\tilde{x},\tilde{n}]=[\tilde{y},\tilde{n}]=[\tilde{z},\tilde{n}]=1\rangle.
\theta(E_{1,k_1},E_{2,k_2},E_{3,k_3})= \min_{m,n}2^{-|k_m-k_n|/3}\cdot \min_{\mu,\nu}\big(|E'_{\mu,k_\mu}|\,/\,|E'_{\nu,k_\nu}|\big)^{1/3}
\begin{aligned}\ll \int_{0}^{\cdot}\phi_1(s) W^1(s) + \int_{0}^{\cdot}\phi_2(s) W^2(s) \gg_t\, = \,\ll \int_{0}^{\cdot}\phi_1(s) W^1(s) \gg_t + \ll \int_{0}^{\cdot}\phi_2(s) W^2(s) \gg_t\end{aligned}
f\otimes f = \sum_{m,n=0}^\infty a_mt^m \otimes a_n t^n = \sum_{m,n=0}^\infty (a_m \otimes a_n) (t^m * t^n) =\sum_{n=0}^\infty (a_n \otimes a_n) t^n\shortintertext{and}\Delta f =\sum_{n=0}^\infty \Delta a_n t^n.
Q=\frac{1}{2\pi}\int_{0}^{2\pi}(t-\pi)\big(\mathrm{Fl}_{\Pi_{0,2}^{\sharp}\mathrm{d}J}^{t}\big)^{\ast}\mathrm{d}_{1,0}J\,\mathrm{d}t-\pi\langle\mathrm{d}_{1,0}J\rangle.
q_{{\bf b}}=\begin{cases}1{\bf b}=\emptyset -u{\bf b}=(1) -uv\cdot q_{{\bf b}'}{\bf b}=({\bf b}',1-z,1) 0{\bf b}=({\bf b}',1,1) (2u+v)\cdot q_{{\bf b}'}{\bf b}=({\bf b}',1-z).\end{cases}
\mathcal S^2:=-\varepsilon^2\partial_tv+\varepsilon^4\partial_x^2v+\varepsilon^2 \partial_y^2v -\varepsilon^2(U+\varepsilon u^{\varepsilon})\partial_xv^{\varepsilon}-\varepsilon^3 v^{\varepsilon}\partial_yv^{\varepsilon}
\pmb{W}_k ={\left(\sum\limits_{\ell = 1}^K{\pmb{H}_k{\pmb{F}_\ell}\pmb{F}_\ell^H\pmb{H}_k^H}+ \sigma_k^2{\pmb{I}_{{N_{{r_k}}}}}\right)^{- 1}}{\pmb{H}_k}{\pmb{F}_k}.
\sum_{m , m^{\prime}; m \pm m^{\prime}\mathrm{e v e n}}A_{M}^{B , m}\biggl \{\widetilde \Pi_{M N}^{+ B C}( m , \frac{m - m^{\prime}}{2}) + \widetilde \Pi_{M N}^{- B C}( m , \frac{m + m^{\prime}}{2}) \biggr \}A_{N}^{C , m^{\prime}}.
\frac{1+tH(x)}{1-t^{2}E(x)H(x)}=\frac{1}{1-t^{2}E(x)H(x)}+t\frac{H(x)}{1-t^{2}E(x)H(x)}=\sum_{\lambda}\frac{p_{\lambda}}{z_{\lambda}}2^{l(\lambda)}\frac{A_{l(\lambda)}(t^{2})}{(1-t^{2})^{l(\lambda)+1}}x^{\left|\lambda\right|}+t\sum_{\lambda}\frac{p_{\lambda}}{z_{\lambda}}\frac{B_{o(\lambda)}(t^{2})}{(1-t^{2})^{o(\lambda)+1}}x^{\left|\lambda\right|},
L_{3}( u ) \equiv \frac{d}{d u}\left( u \frac{d}{d u}\right) - \left( u \frac{d}{d u}+ \frac{5}{6}\right) \, \left( u \frac{d}{d u}+ \frac{1}{6}\right) = u \, \left( 1 - u \right) \frac{d^{2}}{d u^{2}}+ \left( 1 - 2 \, u \right) \, \frac{d}{d u}- \frac{5}{3 6}
\aligned16(\sum_{a=0}^8(q^*_a|_V)^2) =16G|_V(|x|^2+|y|^2+|z|^2)-\langle\nabla (G|_V),\nabla (G|_V)\rangle &-4c^2(\sum_{a=1}^4 p_a^*|_V\, z_a)^2,\endaligned
\begin{aligned}v_{j}(t) = i\int_{t}^{\infty}U(t-\tau)\left\{\tilde{{\mathcal N}}_{j}(v_{1}+u_{\mathrm{ap},1},v_{2}+u_{\mathrm{ap},2})-\tilde{{\mathcal N}}_{j}(u_{\mathrm{ap},1},u_{\mathrm{ap},2})\right\}(\tau)d\tau &+i\int_{t}^{\infty}U(t-\tau)\mathcal{E}_{j}(\tau)d\tau + \mathcal{R}_j.\end{aligned}
\frac{g \gamma_{d}^{\alpha}}{\sqrt{2}}\left( \begin{array}{c}{0}\\{i W_{\alpha}^{2}\psi_{2 L}+ Z_{\alpha}\psi_{1 L}/ \cos \theta_{w}}\\ \end{array}\right) + \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{\psi_{1 R}}\\{\psi_{1 R}}\\ \end{array}\right) .
- \frac{1}{2 \pi^{2}{\cal M}}\cdot \frac{\Lambda^{2}Y + \left[ q^{2}+ k^{2}( 1 - \gamma ) \right] / ( 2 - \gamma )}{\Lambda^{2}Y ( b \pm i q ) + ( 2 / \pi ) \Lambda ( 1 - Y ) ( b^{2}+ q^{2}) - \left( b^{3}\pm ( i q )^{3}\right) + O ( 1 / \Lambda )},
T'_{\tilde{k}_1,k_1, k_2, k_3}= \delta_{k_1=k_2=k_3=1}\frac{u(1,1,1 )M}{4\tilde{k}_1\pi^2}\prod_{p \mid \tilde{k}_1}\!\left(1+\frac{1}{p}\right)^{-1}\!\left\{1+ O\left(\tau(\tilde{k}_1)\sqrt{\frac{\tilde{k}_1}{M}}\right)\!\!\right\}\! &+O_C(\sqrt{M}V^2\log V(\log M)^{-C}).
E_{p o t}= \frac{e^{2}}{4 \pi}\left[ q^{2}\ln \frac{L^{2}}{4 \zeta^{2}}+ \left( \frac{1 - q}{2}\right)^{2}\ln \frac{L^{2}}{4 a_{1}^{2}}+ \left( \frac{1 + q}{2}\right)^{2}\ln \frac{L^{2}}{4 a_{2}^{2}}\right.
\nu_{m1}\delta_{m-1}^{n+1}+\nu_{m2}\phi_{m-1}^{n+1}+\nu_{m3}\delta_{m}^{n+1}+\nu_{m4}\phi_{m}^{n+1}+\nu_{m5}\delta_{m+1}^{n+1}+\nu_{m6}\phi_{m+1}^{n+1}=\nu_{m7}\delta_{m-1}^{n}+\nu_{m8}\delta_{m}^{n}+\nu_{m9}\delta_{m+1}^{n}
\begin{aligned}\Delta_N f\left(\frac{x}{N}\right) =N^2\left\{f\left(\frac{x+1}{N}\right)-2f\left(\frac{x}{N}\right)+f\left(\frac{x-1}{N}\right)\right\}\nabla_N f\left(\frac{x}{N}\right) =N\left\{f\left(\frac{x+1}{N}\right)-f\left(\frac{x}{N}\right)\right\}.\end{aligned}
\underline{v}=\overline{v}-rq(1)e^{\left(\lambda_1+\varepsilon\right)z}, begin{align*}0.2cm] \underline{v}'=\overline{v}'-rq(1) \left(\lambda_1+\varepsilon\right)e^{\left(\lambda_1+\varepsilon\right)z}, begin{align*}0.2cm] \underline{v}''=\overline{v}''-rq(1) \left(\lambda_1+\varepsilon\right)^2e^{\left(\lambda_1+\varepsilon\right)z}.
\int_{-\infty}^{\infty}f(x-\frac{u}{t})g(u)du=\int_{-\infty}^{-W}f(x-\frac{u}{t})g(u)du+\int_{-W}^{W}f(x-\frac{u}{t})g(u)du+\int_{W}^{\infty}f(x-\frac{u}{t})g(u)du
H = \frac{\sqrt{\alpha^\prime}}{2}g^{\mu \nu}(P_\mu - Q_{_{E}}{\cal A}_\mu - Q_{_{M}}{\tilde{\cal A}}_\mu)(P_\nu - Q_{_{E}}{\cal A}_\nu - Q_{_{M}}{\tilde{\cal A}}_\nu) + \frac{\sqrt{\alpha^\prime}}{2k^2}({Q_{_{E}}}^2 +{Q_{_{M}}}^2) ,
\dot{v}=\frac{X}{m}- g \sin (\theta) - q w, \dot{w}=\frac{Z}{m}+ g \cos (\theta) + q v, \dot{x}=w \sin (\theta) + v \cos (\theta), \dot{z}=-v \sin (\theta) + w \cos (\theta), \dot{\theta}=q, \dot{q}=\frac{M}{I_{yy}},
s\Delta n = -2 s_0 \mathbf{e_r}+ (r-1)\left[ 6 s_0 m_1 + 4s_0\mathbf{e_r}-2 s_1 \mathbf{e_r}\right] &+(r-1)^2 \left[ 12 s_0 m_2 + 6s_0 m_1 -6s_0\mathbf{e_r}+6 s_1 m_1 + 4 s_1 \mathbf{e_r}-2 s_2 \mathbf{e_r}\right] & + O((r-1)^3).
\begin{aligned}X_t^\sigma=X_0 + X_t^{\sigma,c}+ \int_0^{t \wedge \sigma}\chi(y)y \ \Big(\mu^{X^\sigma}(\cdot;ds,dy)-\nu^\sigma(\cdot;ds,dy)\Big) +\check{X}^\sigma(\chi)+B_t^\sigma(\chi).\end{aligned}
\eta_j = \left\lbrace \begin{array}{cc}2k + 1 , \;j=0 k , \;j=1 a_{x}2^{x-j+1}+ \cdots + a_{j-1}2^0 + \delta_j , 2\leq j\leq x+1 1 , \;x+2\leq j\leq k-1\end{array}\right.
\frac{\langle f_3,f_1\rangle}{||f_3||||f_1||}=-\frac{\overline{a}}{1+|a|^2}\sum_{j=0}^{\infty}\gamma_j\overline{\alpha_{j+1}}&=-\frac{\overline{a}}{1+|a|^2}\sum_{j=0}^{\infty}e^{i(\eta_1+j\eta_3)}\sqrt{(1-\rho)\rho^j}e^{-i(\eta_2+j\eta_3)}\sqrt{(1-\rho)\rho^j}&=-e^{i\theta_2}\frac{|a|}{1+|a|^2};
\begin{array}{r c r c l}{{e^{\pm i H_{1}}}}&{{=}}&{{\frac 1{\sqrt{k}}( \psi^{1}}}&{{\mp}}&{{i \psi^{2}) ~ ,}}\\{{e^{\pm i ( H_{2}+ \pi N_{1})}}}&{{=}}&{{{\frac{1}{2 \sqrt{k}}}\chi^{3}}}&{{\mp}}&{{{\frac{1}{\sqrt{k}}}\psi^{3}~ ,}}\\{{e^{\pm i ( H_{3}+ \pi N_{2}+ \pi N_{1})}}}&{{=}}&{{\frac 1{2 \sqrt{k}}( \chi^{1}}}&{{\mp}}&{{i \chi^{2}) ~ ,}}\\ \end{array}
R_n:=(L_n \tilde{\Pi}_n^{\perp}-\Pi_n^{\perp}L_n)\textbf{T}_n \Pi_n \mathcal{F}(U_n), Q'_n:=-\Pi_n (L_n \textbf{T}_n-\mathrm{I})\Pi_n \mathcal{F}(U_n).
0 \leq \frac{W_q(x) - W(x)}{W(x)}= W(x)^{-1}\sum_{j\geq 1}q^{j}W^{*(j+1)}(x)\leq x \sum_{j\geq 0}q^{j}W^{*(j+1)}(x) \leq x W_q(x)\xrightarrow{x \downarrow 0}0.
\widetilde{\tau}_{2}= 1 - 2 t_{2}Z_{2}+ \frac{3 ( t_{1}Z_{1})^{2}}{4}+ \frac{( t_{2}Z_{2})^{2}}{4}+ \frac{3}{1 6}\tan^{4}\left( \frac{\pi}{1 2}\right) ( t_{1}Z_{1}t_{2}Z_{2})^{2}
O_A^{(ba^k g)}= \, c_+^{(ba^k,g)}O_A^{(g)}+ \frac{c_+^{(ba^k,g)}- \left(c_+^{(ba^k,g)}\right)^{-1}}{\big(Q^2-Q^{-2}\big)^2}O_B^{(1)}O_B^{(1)}O_A^{(g)}, O_A^{(b^{-1}a^k g)}= \, c_-^{(b^{-1}a^k,g)}O_A^{(g)}+ \frac{c_-^{(b^{-1}a^k,g)}- \left(c_-^{(b^{-1}a^k,g)}\right)^{-1}}{\big(Q^2-Q^{-2}\big)^2}O_B^{(1)}O_B^{(1)}O_A^{(g)},
\{\textbf{S}_p(\textbf{y})\}_{11}= \mathrm{Var}(y_1 + y_2) = \mathrm{Var}(x_1 + x_2) = (\boldsymbol{\Sigma})_{11}+ 2 (\boldsymbol{\Sigma})_{12}+ (\boldsymbol{\Sigma})_{22}.
\frac{\partial}{\partial t}\begin{bmatrix}h k\end{bmatrix}+\begin{bmatrix}\mathfrak{a}_1 -J \mathcal{A}\mathfrak{a}_2\end{bmatrix}\begin{bmatrix}h k\end{bmatrix}=\begin{bmatrix}g_1 g_2\end{bmatrix}.
(\begin{matrix}\lambda&v^{i}\\w_{j}&a_{j}^{i}\end{matrix}),(v^{i})\in\mathbb{R}^{1\times n},(w_{j})\in\mathbb{R}^{n\times1},(a_{j}^{i})\in\mathbb{R}^{n\times n},\lambda=-\sum_{i}a_{i}^{i}
\mathrm{if}\; \Sigma\le P, \;\mathrm{then}\mathbb E\left( \prod_{i=1}^m f_i(X_i)\right) \le \prod_{i=1}^m\mathbb E \left( f_i(X_i)^{p_i}\right)^{\frac{1}{p_i}}, \mathrm{if}\; \Sigma\ge P, \;\mathrm{then}\mathbb E\left( \prod_{i=1}^m f_i(X_i)\right) \ge \prod_{i=1}^m\mathbb E \left( f_i(X_i)^{p_i}\right)^{\frac{1}{p_i}},
\frac{1}{4\pi L}\frac{\partial f_{i}}{\partial t}=\frac{\partial}{\partial v_{\alpha}}(-f_{i}\frac{\partial h_{i}}{\partial v_{\alpha}}+\frac{1}{2}\frac{\partial}{\partial v_{\beta}}(f_{i}\frac{\partial^{2}g_{i}}{\partial v_{\alpha}\partial v_{\beta}}))
f(X,Y) = \left( 1 - \frac{|\overline{X}\cap Y|}{n}\right)\left( 1 - \frac{|X \cap \overline{Y}|}{n}\right)\exp\!\left( - \frac{|X \cap Y|}{n}\right)= g(\overline{X}\cap Y) \,h(X \cap Y) \, ,
\pi_{n}\left(m_{1},\ldots,m_{k}\right)=\sum_{j=0}^{\infty}\binom{k+j}{j}\sum_{\underline{z}\in\left\{0,1\right\}^{k}}\pi_{n+1}(m_{1}+z_{1},\ldots,m_{k}+z_{k},\underbrace{1,\ldots,1}_{j}).
d_{n+1}= \frac{1-\sqrt{1-d_n^2}}{1+\sqrt{1-d_n^2}}, b_{n+1}= 2 b_n \, \frac{(1+d_{n+1})^3}{1-d_{n+1}}, a_{n+1}= a_n (1+d_{n+1})^2 + \frac{b_{n+1}d_{n+1}}{2(1+d_{n+1})}.
A = q_{h-2}q_{h+k-1}- q_{h-1}q_{h+k-2}, B = q_{h-1}r_{h+k-2}+ r_{h-1}q_{h+k-2}- r_{h-2}q_{h+k-1}- q_{h-2}r_{h+k-1}, C = r_{h-2}r_{h+k-1}- r_{h-1}r_{h+k-2}.
T_{f}^{\alpha}(g,h) =\langle gh,f\rangle_{A_{\alpha}^{2}}=\langle h,\overline{g}f\rangle_{L_{\alpha}^{2}}&=\langle P_{\alpha}(h),\overline{g}f\rangle_{L_{\alpha}^{2}}=\langle h,P_{\alpha}(\overline{g}f)\rangle_{A_{\alpha}^{2}}=\langle h,h_{f}^{\alpha}(g)\rangle_{A_{\alpha}^{2}},g,h\in H^{\infty}.
\begin{aligned}H(a,E,c) =\int_0^T\Big(\frac{u_x^2}{2}-F(u)\Big)~dx=\frac{\sqrt{2}}{2}\oint_\gamma\frac{E-V(u;a,c)-F(u)}{\sqrt{E-V(u;a,c)}}~du, M(a,E,c) =\int_0^Tu~dx=\frac{\sqrt{2}}{2}\oint_\gamma\frac{u~du}{\sqrt{E-V(u;a,c)}}, P(a,E,c) =\int_0^Tu^2~dx=\frac{\sqrt{2}}{2}\oint_\gamma\frac{u^2~du}{\sqrt{E-V(u;a,c)}},\end{aligned}
\left[\alpha_E,\left(\alpha_{\psi}^2\right)_E\right]_{\mathfrak{so}(\mathcal{T}(M))}\stackrel{!}{=}\left(\frac{1}{2}\alpha^2_{\alpha \cdot \psi, \psi}+ \frac{1}{2}\alpha^2_{\psi,\alpha \cdot \psi}\right)_E = \left(\alpha^2_{\alpha \cdot \psi, \psi}\right)_E.
\left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right) \quad \mathrm{~ a n d ~}\quad \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\ \end{array}\right) .
\frac{(x^3 + (1-x)^3)^{\beta_x}(y^3 + (1-y)^3)^{\beta_y}}{(x^2 + (1-x)^2)^{\beta_x}(y^2 + (1-y)^2)^{\beta_y}}\le \frac{(z^3 + (1-z)^3)^{\beta_x + \beta_y}}{(z^2 + (1-z)^2)^{\beta_x + \beta_y}}= \left( \frac{3s-1}{2s}\right)^{\beta_x + \beta_y}.
\frac{1}{2\pi}c^{(f)}: e^{\mp i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi +gN}}U_{1 f}\Phi^{(1)}(x)}:_{M^{(1)}}\; \prod_{I=2}^N: e^{\mp i 2 \sqrt{\pi}U_{I f}\Phi^{(I)}(x)}:_{M^{(I)}}e^{\pm i \frac{\theta}{N}}\; .
F{{w,k \!\!\!\!\!}\atop \bigg{|}}_{\left( \begin{matrix}a b c d \end{matrix}\right)}(\tau, z) = (c \tau + d)^{-w}\, e^{- \frac{\pi i kc (z|z)}{c \tau + d}}\F \left( \frac{a \tau + b}{c \tau + d}, \ \frac{z}{c \tau + d}\right).
\begin{aligned}a_{k, \delta}(x_a) := 2 \int_{0}^{1}y_\delta(x_a,t)\cos(k 2\pi t)\,dt b_{k,\delta}(x_a) := 2 \int_{0}^{1}y_\delta(x_a,t)\sin(k 2\pi t)\,dt\,.\end{aligned}
\widetilde{V}^{( 1^{2})}\big(x;\omega ,1,-\Vert \psi_{1}\Vert^{2}\big) =V ( x;\omega ,1 ) +6\omega\frac{z^{2}-4}{z^{3}-z^{2}+2z+2}-20\omega\frac{5z^{2}-4z-2}{\big( z^{3}-z^{2}+2z+2\big)^{2}}+ 2\omega,
\sum_{m\in \mathbb{Z}}\mu(T_m(\phi_1)) x^m = \left(\frac{1+x^{-1}}{2}\right) \left(\frac{x}{6}+ \frac{5}{6}\right) \left(\frac{27}{48}+\frac{21}{48}x^{-1}\right) &=\frac{35}{192}x^{-2}+ \frac{29}{64}x^{-1}+ \frac{61}{192}+ \frac{3}{64}x.
J_\gamma(t)=\frac 1{(2\pi \sqrt{-1})^n}\int\limits_{\Gamma_h}\frac 1{z^{\gamma+I}}\cdot\frac{dF}{F}=\frac 1{(2\pi \sqrt{-1})^n}\int\limits_{\Gamma_h}\frac{1}{z_1^{\gamma_1+1}\cdot\ldots\cdot z_n^{\gamma_n+1}}\cdot\frac{dF_1}{F_1}\wedge\ldots\wedge\frac{dF_n}{F_n},
g\cdot \zeta(\mu) = g\cdot \zeta(\mu_1+ \mu_2)= g\cdot \xi(\varphi(\mu_1+ \mu_2))= \xi(g\cdot \varphi(\mu_1+ \mu_2))= \xi(g\cdot (\varphi(\mu_1)+ \varphi(\mu_2))) & = \xi(g\cdot \varphi(\mu_1)+ g\cdot \varphi(\mu_2))= \xi(\varphi(\mu_1)+ \varphi(\mu_2))= \xi(\varphi(\mu_1+ \mu_2))= \zeta(\mu_1+ \mu_2)= \zeta(\mu),
\sup_{x \in\left(x_{0}, \infty\right)}\left\| \langle x\rangle \mathcal{P}^{+}\left(z^{-i}\bar{r}_1(z) e^{-2i z x}\right)\right\|_{L_z^2}\leq\left\|z^{-i}\bar r_1(z)\right\|_{H^1_z},i=0,1, &\sup_{x \in\left(x_0, \infty\right)}\left\|\langle x\rangle \mathcal{P}^-\left(r_2(z) e^{2iz x}\right)\right\|_{L_{z}^2}\leq\left\|r_2(z)\right\|_{H^1_z}.
H(q_{1},q_{2},...,p_{1},p_{2},...,t)=\sum_{i}\dot{q}_{i}p_{i}-L(q_{1},q_{2},...,\dot{q}_{1}(p_{1}),\dot{q}_{2}(p_{2}),...,t),p_{i}=\frac{\partial L}{\partial\dot{q}_{i}},
Z_{( a , m ) | ( a^{\prime}, m^{\prime})}( q ) = \sum_{( a^{\prime \prime}, m^{\prime \prime})}\hat{N}_{( a , m )^{*}( a^{\prime}, m^{\prime})}{}^{( a^{\prime \prime}, m^{\prime \prime})}\hat{\chi}_{( a^{\prime \prime}, m^{\prime \prime})}( q )
\mathcal{C}_u(\mathcal{S}) = \sum_{t \in \alpha_u(\mathcal{S})}\cfrac{N(\kappa(t))}{R(t)}+ \sum_{t \in \beta_u(\mathcal{S})}\cfrac{N(\kappa(t))}{R(t)}+ \sum_{t \in \gamma_u(\mathcal{S})}\cfrac{N}{R(t)},
R ( u_{1}, u_{2}, u_{3})_{i_{1}i_{2}i_{3}}^{j_{1}j_{2}j_{3}}\stackrel{\rho}{=}R ( ( \omega u_{2})^{- 1}, u_{1}, ( \omega u_{3})^{- 1})_{- j_{2}i_{1}- i_{3}}^{- i_{2}j_{1}- j_{3}}.
\frac{p_{r}^{2}}{g_{r r}}+ \frac{p_{\theta}^{2}}{g_{\theta \theta}}+ \frac{-{g^{\prime}}_{t t}}{- \cal D}\left( p_{\phi}+ \frac{g_{t \phi}+ \Omega_{0}g_{\phi \phi}}{{g^{\prime}}_{t t}}E \right)^{2}= \left( \frac{E^{2}}{-{g^{\prime}}_{t t}}- \mu^{2}\right) ,
\sqrt{- G}\, G^{M N}R_{M N}= \sqrt{- g}\, [ g^{\mu \nu}R_{\mu \nu}- \frac{D - 3}{4 ( D - 2 )}g^{\mu \nu}\partial_{\mu}\ln \Phi \cdot \partial_{\nu}\ln \Phi - \frac{\Phi}{4}f_{\mu \nu}f^{\mu \nu}] ,
a_{34}^{3412}=a_{34}^{3421}=a_{34}^{3142}=a_{34}^{3124}=a_{34}^{3241}=a_{34}^{3214}= a_{34}^{1342}=a_{34}^{1324}=a_{34}^{1234}=a_{34}^{2341}=a_{34}^{2314}=a_{34}^{2134}
\Pi_{S^{1}}( q^{2}; R ) = -{\frac{1}{8 \pi^{2}}}\int_{0}^{1}d x x \sqrt{1 - x^{2}}\ln \left[ 2 \sinh \left( \pi R \sqrt{q^{2}x^{2}/ 4 + m^{2}}\right) \right] .
\times \exp\left\{\frac{1}{(1-x)^3(1-y)^5}\sum_{n=1}^{\infty}(-nx^{n+2}y^{n+1}-nx^{n+2}y^{n+2}-ny^2x^{n+1}-y^2x^{n+1}) \frac{z^n}{n^4}\right\}
\sqrt{r}K_{0}(2\nu r^{-1/(2\nu)}) = C\left(\sqrt{r}\ln\left(2\nu r^{-1/(2\nu)}\right)+\mathcal{O}(r^{1/2})\right) &= C\left(\sqrt{r}\ln\left(((2\nu)^{-2\nu}r)^{-1/(2\nu)}\right)+\mathcal{O}(r^{1/2})\right).
(e_nx^2+f_nx+g_n)\mathcal{L}_{n+1}(x) =(p_nx^3+q_nx^2+r_nx+s_n)\mathcal{L}_n(x) +(t_nx^4+u_nx^3+v_nx^2+w_nx+z_n)\mathcal{L}_{n-1}(x), n\geq 1,
d\mathbf{P}_{\omega,T}(z_1,y_{1},\dots,y_{T})=dF\big(y_{1}-f_\omega(z_{1})\big)\prod_{i=2}^{T}dF\Big(y_{i}-f_\omega\big(\Phi_i(z_1, y_1,\dots,y_{i-1})\big)\Big).
\Phi_{\Xi}( \zeta , \zeta^{*}) = \left\langle \exp ( \zeta + \zeta^{*}) \mid \exp ( \varsigma + \tau^{*}) \right\rangle e^{- \left\langle \tau^{*}\mid \varsigma \right\rangle}= e^{\left\langle \zeta \mid \tau^{*}\right\rangle + \left\langle \zeta^{*}\mid \varsigma \right\rangle - \left\langle \tau^{*}\mid \varsigma \right\rangle}
\begin{cases}\partial_t\psi =-(-\Delta)^{s}\psi + V_{\mu, \xi}\psi+\tilde{\phi}(-(-\Delta)^{s})\eta_{R}+ \cdots, \quad\Omega\times (t_0, \infty), \psi = -u^*_{\mu, \xi}, \quad(\mathbb{R}^n\setminus\Omega)\times (t_0,\infty)\end{cases}
|Y_s^{n,t,x}(\omega)-Y_s^{t,x}(\omega)| \ = \ |u_n(s,X_s^{n,t,x}(\omega))-u(s,X_s^{t,x}(\omega))| &= \ |u_n(s,X_s^{n,t,x}(\omega))-u_n(s,X_s^{t,x}(\omega))| + |u_n(s,X_s^{t,x}(\omega))-u(s,X_s^{t,x}(\omega))|.
\widehat{\frac{d}{dt}\frac{\partial \Xi({x}^\mu_k,{y}^\mu_k)}{\partial x^\mu}}\,|x^\mu,y^\mu\rangle :=\,\frac{d}{d t}\,\frac{\partial \Xi({x}^\mu_k,{y}^\mu_k)}{\partial x^\mu}\,|x^\mu,y^\mu\rangle
\left( \beta x^{1-\alpha/\beta}D\right)^{n}\left[ x^{r/\beta}\left(1-x\right)^{s}\right] =x^{\left( r-n\alpha\right) /\beta}\left(1-x\right)^{s}w_{n}^{\left( -s\right)}\left( \frac{x}{1-x};\alpha,\beta,r\right)
\dfrac{\partial g_{ik}}{\partial \eta^i}\eta^j + \dfrac{\partial g_{ij}}{\partial \bar{\eta}^j}\bar{\eta}^j = 0; \dfrac{\partial g_{i\bar{k}}}{\partial \eta^j}\eta^j + \dfrac{\partial g_{i\bar{k}}}{\partial \bar{\eta}^j}\bar{\eta}^j = 0,
\varphi(z) = b + \frac az + \sum_{i=1}^{\varkappa}\left( \frac{\sigma_i/\lambda_i}{\lambda_i-z}+\frac{\sigma_i/\lambda_i}{z}\right) + \int_0^\infty\frac{d\sigma(t)}{t(t-z)}+ \frac 1z \int_0^\infty\frac{d\sigma(t)}{t(1+t^2)},
A_{k-1}^{(\ell+1)}=A_{k-1}^{(\ell)}(A_{k-1}^{(\ell)}+\widehat{B}_{k-1})^{-1}\widehat{A}_{k-1}, B_{k-1}^{(\ell+1)}=\widehat{B}_{k-1}(A_{k-1}^{(\ell)}+\widehat{B}_{k-1})^{-1}B_{k-1}^{(\ell)},
\theta_g\theta_\omega\{a_n\}_n =\theta_g \left(\sum_{n=1}^{\infty}a_n\omega_n\right)=\sum_{n=1}^{\infty}a_n\theta_g\omega_n &=\sum_{n=1}^{\infty}a_n\sum_{m=1}^{\infty}g_m(\omega_n)e_m= \sum_{n=1}^{\infty}a_ne_n &=\{a_n\}_n, \forall \{a_n\}_n \in \ell^p(\mathbb{N}).
{\mathcal I}_{411}= -\omega\Big[\big(\tilde{B}_{+,\omega}(t)\big)^{2}-\big(\tilde{B}_{-,\omega}(t)\big)^{2}\Big]- \frac{KR}{2}\Big[\big(\tilde{B}_{+,\omega}(t)\big)^{2}+\big(\tilde{B}_{-,\omega}(t)\big)^{2}\Big].
H_{min}(E\otimes (F_1[t]\times F_1[t]^{\vee})) =H_{min}(E\otimes F_1[t]\times E\otimes F_1[t]^{\vee}) &=\min H_{min}(E\otimes F_1[t], E\otimes F_1[t]^{\vee}) &\leq H_{min}(E\otimes F_1[t])=t H_{min}(E\otimes F_1)\leq tH_{min}(E)H_{min}(F_1)
\int_{- \infty}^{\infty}d q_{-}\frac{1}{q_{-}^{2}}\mathrm{e}^{i q_{-}x^{-}}= 2 \int_{0}^{\infty}d q_{-}\frac{\mathrm{c o s}q_{-}x^{-}}{q_{-}^{2}}= 2 \int_{0}^{\infty}d q_{-}\frac{1}{q_{-}^{2}}-{\pi}| x^{-}|
U_{d_0,d_1}(x, t_0) = \hat U^*(x)(1 - \chi_1(x,t_0)) & + \left\{s_0 + \ln \left[ \frac{A}{s_0^2}\left(d_0 + d_1\cdot x e^{\frac{s_0}{2}}\right)\chi\left(16xe^{\frac{s_0}{2}},s_0\right) + \psi_\alpha\left(xe^{\frac{s_0}{2}}, s_0\right)\right] \right\}\chi_1(x,t_0),
M(a,b,c) = \prod_{\substack{1 \le i \le a 1 \le j \le b 1 \le k \le c}}\frac{i+j+k-1}{i+j+k-2}= \prod_{i=1}^{c}\frac{(a+b+i-1) ! (i-1) !}{(a+i-1) ! (b+1-1) !}=\dim V_{(a^c)}^{{\mathfrak{gl}}_{b+c}}.
\begin{array}{c}Unknown (6,2,1) \end{array}\,\,\,d\hat{s}^{2}_{IIB}= H\left[H^{-1/2}\eta_{ij}dy^{i}dy^{j}-H^{1/2}dx^{2}\right]-H^{-1/2}dy^{2}-H^{-3/2}dz^{2}\, .
d\mu_t^\epsilon\,=\,\Bigl(\frac{\epsilon^{1-\kappa}}2\vert Dv_\epsilon\vert^2\,+\,\frac{F(v_\epsilon)}{2\epsilon^{1-\kappa}}\,-\,\frac{\epsilon^{1-\kappa}\dot\kappa}{K(\epsilon)}G(v_\epsilon)\Bigr)\,dx, & d\xi_t^\epsilon\,=\,\Bigl(\frac{F(v_\epsilon)}{2\epsilon^{1-\kappa}}\,-\,\frac{\epsilon^{1-\kappa}}2\vert Dv_\epsilon\vert^2\Bigr)\, dx,
\psi_{j',k'}* f(x_{I'},x_{J'},x_{S'}) &= \sum \limits_{j,k\in\mathbb Z}\sum_{R=I\times J\times S\in \mathcal R^N_{\frak z}(j,k)}|R| \psi_{j',k'}* \big(\widetilde \psi_{j,k}(\cdot_1, \cdot_2, \cdot_3,x_{I},x_{J},x_{S})\big) (x_{I'},x_{J'},x_{S'})\psi_{j,k}*f(x_I,x_J,x_S).
c_v(W_v,\tilde W_v, \Phi_v): = \left. \frac{\Psi_v (s, W_v, \tilde W_v, \Phi_v)}{L_v (s, \pi_v \times \tilde \pi_v)}\right|_{s=1}= \frac{\hat \Phi_v (0) \varTheta_v (W_v, \tilde W_v)}{L_v (1, \pi_v \times \tilde \pi_v)},
\sum_{n=0}^{\infty}(-1)^n q^n \left(c-1+q^2+q^n x \right)Ai_q(q^n x) &= \frac{qc+x}{q(1+q)}\,_1\phi_1 (0;-q^2;q,-x)-(1-q)Ai_q(\frac{x}{q}),
\mathbb{T'}_R =\begin{pmatrix}(db^{-1}a-c)^{-1}[d,c] (db^{-1}a-c)^{-1}\left( [a,d]+[c,b]+db^{-1}[b,a]\right) (d-ca^{-1}b)^{-1}[c,d] ( ca^{-1}b-d)^{-1}\left([a,d]+[c,b]+ca^{-1}[b,a]\right)\end{pmatrix}(\Delta')^{-1}.
\begin{aligned}\|\nabla u\|^2 = \int_{Q_T}|\nabla u|^2 \, d\boldsymbol{x}\,dt = T \, \|\nabla u_0^c\|_{\Omega}^2 + \frac{T}{2}\sum_{k=1}^\infty \|\nabla \boldsymbol{u}_k\|_{\Omega}^2 \geq \frac{1}{C_F^2}\left(T \, \|u_0^c\|_{\Omega}^2 + \frac{T}{2}\sum_{k=1}^\infty \|\boldsymbol{u}_k\|_{\Omega}^2 \right) = \frac{1}{C_F^2}\|u\|^2. \end{aligned}
0\leq c(D) = \inf \{|\langle Dx, x \rangle|: x \in \ell^2(\mathbb{Z}_{+}), \|x \| =1 \}\leq \inf\{|\langle De_i, e_i \rangle|: i \in \mathbb{Z}_{+}\}= \inf\{|\langle \lambda_ie_{i}, e_i \rangle|: i \in \mathbb{Z}_{+}\}= \inf_{i}{|\lambda_i|}= 0.
\begin{array}{l}{{X_{J_{i}^{\prime}}\rfloor \Omega_{q}= - d J_{i}^{\prime}~ ,}}\\{{[ X_{J_{i}^{\prime}}, X_{J_{j}^{\prime}}] = - X_{[ J_{i}^{\prime}, J_{j}^{\prime}]_{G P B}}~ ,}}\\{{\Omega_{q}( X_{J_{i}^{\prime}}, X_{J_{j}^{\prime}}) = [ J_{i}^{\prime}, J_{j}^{\prime}]_{G P B}~ .}}\\ \end{array}
r \, = \, \frac{e^{- 2 i k L}}{\Omega}\frac{( m + E + k ) B ( 2 \nu , - \mu - \nu - \lambda )}{( m + E - k ) B ( - 2 \nu , 1 - \mu + \nu + \lambda )}\times \left[ 1 - e^{4 i p L}\frac{B ( 2 \mu , - \mu - \nu + \lambda ) B ( 2 \mu , - \mu + \nu + \lambda )}{B ( - 2 \mu , \mu + \nu - \lambda ) B ( - 2 \mu , \mu - \nu - \lambda )}\right]
E_{(x,T)}[G^{\tau_2}\log G^{\tau_2}-G^{\tau_1}\log G^{\tau_1}]=E_{(x,T)}\int_{\tau_1}^{\tau_2}d (G^\tau\log G^\tau)=E_{(x,T)}\int_{\tau_1}^{\tau_2}\frac{1}{2G^\tau}\frac{d[G^\bullet]_\tau}{d\tau}d\tau.
\dot P_s(t) = \frac{st^2 (1-t)^{n-1}}{((1-t)^2+s^2t^2)^{3/4}(1+s^2)^{1/2}}- \frac{2s(1-t)^{n-1}((1-t)^2+s^2t^2)^{1/4}}{(1+s^2)^{3/2}}
|\bar{H}_\pi|= \sum_{x'\in V_i}\left(|V_j||V_k|-d_{j,k}(x')\right) \geq \sum_{x'\in S_x}\left(|V_j||V_k|-d_{j,k}(x')\right) \geq |S_x| \cdot \frac{1}{2}cn^2\geq \frac{1}{2}cn^{\frac{5}{2}},
{\cal I}( M^{2}) = - \mathrm{T r}\gamma_{5}{\frac{M^{2}}{- ( \gamma \cdot D )^{2}+ M^{2}}}= - \int d^{3}x \langle x | \mathrm{t r}\gamma_{5}{\frac{M}{\gamma \cdot D + M}}| x \rangle .
\chi_{+}(t) = c_{+}- \delta_{+}((x_n + R_n) + T) + \int_{-T}^t q_{+}(s)\, ds & = c_{+}+ \int_{x_n + R_n}^t q_{+}(s) \, ds & = - \int_{x_n + R_n}^{T}q_{+}(s)\, ds + \int_{x_n + R_n}^t q_{+}(s)\, ds & = - \int_t^{T}q_{+}(s)\, ds,
C_{1}=\frac{(\frac{t^{\prime}}{t})^{2}}{(\frac{1}{4}\cdot\frac{l}{t}-\frac{t^{\prime}}{t}+\frac{3}{4})\cdot(\frac{5}{8}\cdot\frac{l}{t}-\frac{t^{\prime}}{t}+\frac{3}{8})}
\psi_{j}( x ) = \int \frac{d p_{j}^{1}}{\sqrt{4 \pi}p_{j}^{0}}\left( a_{j}( p ) u_{j}( p ) e^{- i p_{j}\cdot x}+ a_{_{\bar{\jmath}}}^{\dagger}( p ) v_{j}( p ) e^{i p_{j}\cdot x}\right) \, , \qquad
\beta\to \int \left(\frac{\beta}{2\pi}\right)^{n/2}e^{-\beta\|z\|^2/2}z_{i_1}...z_{i_k}dz = \beta^{-k/2}\int \left(\frac{1}{2\pi}\right)^{n/2}e^{-\|u\|^2/2}u_{i_1}...u_{i_k}du
\mathbb{A}(R_0, h_2(d))=R_0^2 \mbox{Arccos}\left(\frac{R_0^2 + R_1^2 -d^2}{2 R_0 R_1}\right)- \frac{R_0^2 + R_1^2 - d^2}{2 R_1}\sqrt{R_0^2 -\left(\frac{R_0^2 + R_1^2 - d^2}{2 R_1}\right)^2}.
\int_{\partial \Omega}\sigma (\psi_h^{n+1}-\psi_h^n)ds +\int_{\Omega}( \rho_h^{n+1}\psi_h^{n+1}-\rho_h^n \psi_h^n) dx=A_{1}(\psi_{h}^{n+1}-\psi_h^n, \psi_h^{n+1}+\psi_h^n).
R[S_\Gamma(x,n)\leftrightarrow S_\Gamma(x,r)] \ge\frac{\alpha n^2}{10\deg(\Gamma)\beta_\Gamma(n)}=\frac{n^2}{10\deg(\Gamma)\beta_\Gamma(n)}\left(\frac{r}{n}-1\right) \ge\frac{n^2}{10\deg(\Gamma)\beta_\Gamma(n)}\log\frac{r}{n},