harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
N_{1,\varphi}^{*+}(x) = \left(\begin{array}{cc}0 - \frac{i \left(e^{i \varphi x}- e^{- 2 \varphi}\right)}{x + 2 i}+ \frac{i \left(e^{i \varphi x}- e^{- 4 \varphi}\right)}{x - 4 i}\frac{i \left(e^{i \varphi x}- e^{- 4 \varphi}\right)}{x + 4 i}- \frac{i \left(e^{i \varphi x}- e^{- 2 \varphi}\right)}{x - 2 i}0\end{array}\right).
\hat{O}W_{n}= \frac{n + 1}{( n + 1 ) ! 0 !}\{\hat{O}^{0}x , \hat{O}^{n + 2}x \}+ \sum_{j = 1}^{\frac{n - 1}{2}}\frac{( n + 1 ) ( n - 2 j + 2 )}{( n + 2 - j ) ! j !}\{\hat{O}^{j}x , \hat{O}^{n + 2 - j}x \}+
~~~~~~~log\left(\theta_1\left(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\right)\right)=log(\epsilon_1(A))+\frac{1}{2}log(-i(c\tau+d))+\frac{\pi i c z^2}{c\tau+d}+log\left(\theta_1\left(z,\tau\right)\right).
a_{i}( x ) ={\frac{1}{\sqrt{2 \pi L_{1}}}}\sum_{n_{1}}\int d p_{2}{\frac{1}{p^{0}}}[ ({\frac{p_{i}}{\mu}}g ( \vec{p}) + i{\frac{\epsilon_{i j}p_{j}}{2}}h ( \vec{p}) ) e^{- i p \cdot x}+ c . c ]_{p^{0}= | \vec{p}|},
\Bigl(~_{s+1}\mathcal{F}_s({\bf a};{\bf b})(\alpha)^{q^{d-1}}\Bigr)^{(-1)}=\frac{\prod_{i=1}^{s+1}\Bigl((\theta-t)^{q^{d-2}}\mathbb{D}_{a_i-2}^{q^{d-a_i}}\Bigr)}{\prod_{j=1}^s\Bigl((\theta-t)^{q^{d-2}}\mathbb{D}_{b_j-2}^{q^{d-b_j}}\Bigr)}\alpha^{q^{d-2}}+~_{s+1}\mathcal{F}_s({\bf a};{\bf b})(\alpha)^{q^{d-1}}.
\begin{array}{c}a={i\sqrt{2}}\int\limits_0^{2\pi}\sqrt{u+\cos x^{10}}dx^{10}=i\left(4-{u-1\over 2}\log{u-1\over 2}+\ldots\right) a_D=\sqrt{2}\int\limits_{\cosh x^6<u}\sqrt{u-\cosh x^6}dx^6=2\pi\left({u-1\over 2}+\ldots\right)\end{array}
r^{-n}_{j}f_j (\xi) =-\Delta u(x_j +r_j \xi) \leq Cr^{-\alpha}_{j}\left( r^{2-n}_{j}\left[ \frac{1}{R^{n-2}}+(N_{2R}g_j )(\xi) \right] \right)^\lambda \leq Cr^{-\alpha-(n-2)\lambda}_{j}\left[ \frac{1}{R^{(n-2)\lambda}}+\left( (N_{2R}g_j)(\xi)\right)^\lambda \right] |\xi|<R,
M_{\mathrm{CsCl}}=\alpha (n)\left\lbrace \sum_{\mathbf{k}\in (2\mathbb{Z})^{n}\setminus \mathbf{\lbrace 0\rbrace}}\dfrac{1}{\Vert\mathbf{k}\Vert^{n-2}}-\sum_{\mathbf{k}\in (2\mathbb{Z}+1)^{n}}\dfrac{1}{\Vert\mathbf{k}\Vert^{n-2}}\right\rbrace ,
D_{t,q}\tilde w_{q+1,j,k}(x,t)=\chi'_{j}(t) a_{k,j}(x,t) b_k( \lambda_{q+1}\Phi_j(x,t) ) = \frac{\chi'_j(t)}{\chi_j(t)}\tilde w_{q+1,j,k}(x,t).
u_t =\Delta u -\chi\nabla v\cdot\nabla u +u(a(x,t)-u(b(x,t)-\chi\mu)-\chi\lambda v) &\leq \Delta u -\chi\nabla v\cdot\nabla u +u(a(x,t)-u(b(x,t)-\chi\mu)) &\leq \Delta u -\chi\nabla v\cdot\nabla u +u(a_{\sup}-(b_{\inf}-\chi\mu) u)
\frac{1}{\prod_{i\geq 1}(1-q^i)}=\sum_{\{s_i\}\in \mathcal{S}_m}q^{(m-1)\binom{s_1}{2}+ \sum s_i}\prod_{j\geq 1}\binom{s_j - s_{j+m}+ \chi (m-1 | j)}{s_{j+1}- s_{j+m}},
\begin{aligned}\|\eta(t) -\eta_a(t)\| = \|\eta(t)-\eta_a(t) \pm z(t)\| \le \|\eta(t)-z(t)\|+\|z(t)-\eta_a(t)\| \le \|\eta(t)-z(t)\| +\gamma \|\epsilon(\eta_a(t),t)\| \le \|\eta(t)-z(t)\| +\gamma \bar{k}_3(L_r,\delta).\end{aligned}
\tilde U^n({\boldsymbol \lambda}^{n};{\boldsymbol \beta}({\boldsymbol \lambda}))\;\triangleq\; \underbrace{U^n({\boldsymbol \lambda}^{n};{\boldsymbol \beta}({\boldsymbol \lambda}))}_{}\;-\;\underbrace{\sum\limits_{l = 1}^J{{p_{l}}\lambda_l^{n}(1 -{\beta_l}})}_{}\ .
\frac{1}{R(t)}=\lvert\partial_{r}u(0,t)\rvert=\frac{\lvert \partial_{y}f(0,s)\rvert}{\sqrt{T-t}}=\frac{U_{1}'(0)}{\varepsilon(s)\sqrt{T-t}}\approx \frac{\kappa}{(T-t)^{-\frac{1}{2}-\frac{\lambda_{l}}{\gamma}}}.
\mathcal{L}^{\rm (proj)}= \frac{1}{\sqrt{1-v^2}}\, \left( \begin{array}{c|c}1 \mathbf{v}^T \hline \mathbf{v}\sqrt{1-v^2}\,{\bf 1}\, + \, \frac{\mathbf{v}\mathbf{v}^T}{v^2}\, \left(1 -\sqrt{1-v^2}\,\right) \,\end{array}\right)~,
I_{q}(f) = \int_{\mathbb{Z}_p}f(x) d\mu_{q}(x) = \lim_{N \rightarrow \infty}\sum_{x=0}^{p^N-1}f(x) \mu_{q}(x+p^N \mathbb{Z}_p) &= \lim_{N \rightarrow \infty}\frac{1}{[p^N]_q}\sum_{x=0}^{p^N-1}f(x) q^x, (\,\,[13]).
\zeta^{(\delta,\ell,E,\tau)}_{n}(m,h)= \sum\limits_{k}\zeta^{(\delta,\ell,E,\tau)}_{n-1}(m,k)\Big(e^{-(h-1)\tau\varphi\big(D^{(\delta,\ell,E,\tau)}_{n-1,k},i_m\big)}-e^{-h\tau\varphi\big(D^{(\delta,\ell,E,\tau)}_{n-1,k},i_m\big)}\Big).
\begin{array}{llrr}\psi^{(0)}_k(z) =1, \psi^{(p)}_k(z) = -\rho \sum_{m=1}^N \frac{1}{2\pi i}\int_{L_m}\overline{n_m^2(t)}\;\overline{\psi^{(p-1)}_m(t)}E_1(t-z) \textrm{d}t+1, \; p=1,2,\ldots , \;z \in D_k(k=1,2,\ldots, N).\end{array}
\frac{\partial A_{i j}}{\partial q^{k}}- \frac{\partial A_{k j}}{\partial q^{i}}= - \frac{1}{2}\frac{\partial}{\partial \dot{q}^{j}}\left[ \frac{\partial}{\partial \dot{q}^{k}}( A_{i n}f^{n}) - \frac{\partial}{\partial \dot{q}^{i}}( A_{k n}f^{n}) \right] .
\tilde{\varphi}_{\omega_{0}, u_{0}}^{i n}( v , p ) = \int \! d \omega \e^{i \omega u_{0}}{\frac{e^{- ( \omega - \omega_{0})^{2}/ 2 \sigma^{2}}}{({2 \pi})^{1 / 4}\sigma^{1 / 2}}}e^{- i \omega v}{\tilde{\psi}}_{\omega}^{i n}( p )
\psi(k,v)=\psi(v,v)=\left(\frac{\sqrt{\lambda}}{1+\lambda}\right)^v\lambda^{-v/2}\frac{v!}{v!0!}=\frac{1}{(1+\lambda)^v}\leq c_4v^{-\frac{1}{2}}=c_4k^{-\frac{1}{2}}.
I_\Delta=(x_1x_2x_3, x_1x_2x_4, x_1x_3x_5, x_1x_4x_6, x_1x_5x_6, x_2x_3x_6, x_2x_4x_5, x_2x_5x_6, x_3x_4x_5, x_3x_4x_6).
\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)
| \i< \xi \otimes \eta, \theta_X^H (\sum_i a_i \otimes x_i ) \xi \otimes \eta' >| =| \i< \eta, \sum_i \i< \xi, \pi_X( a_i) \xi > x_i \eta' > |\leq \i< \eta, \Phi (\sum_i a_i \otimes x_i ) \eta'> | & \leq \|\Phi \| \| \theta_Y^H (\sum_i a_i \otimes x_i ) \| \| \eta \| \| \eta ' \| < \varepsilon \|\Phi \| \| \eta \| \| \eta ' \|.
\overrightarrow{{\bf X}^{(n)}}(x) = \overrightarrow{{\bf X}^{(0)}}(x) +{\bf Y}(x) \int_{x_0}^x{\bf Y}^{(-1)}(t)\overrightarrow{{\bf F}}(t,\overrightarrow{{\bf X}^{(n-1)}}(t))dt
\exp{\left\{- i ( \xi F_{1}+ \eta F_{2}) \right\}}= \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{\xi}}\\{{0}}&{{1}}&{{0}}&{{\eta}}\\{{\xi}}&{{\eta}}&{{1}}&{{( \xi^{2}+ \eta^{2}) / 2}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) .
s^P=\tilde{s}-\frac{3}{4}\sum_{i,j}\|R'(\tilde{e_i},\tilde{e_j})\|^2+\frac{1}{2}\sum_{i,a}\|Q_{\alpha_A}(\tilde{e_i})\|^2_{\tilde{g}}+\frac{1}{4}\sum_{A,B}\|[\alpha_A,\alpha_B]\|^2.
\Phi^* (S_{d,-r;\,1,0,l}) = S_{r,d;\,1,2,l+1}\,, \Phi^* (S_{d,-r;\,1,1,l}) = S_{r,d;\,2,1,l}\,, \Phi^* (S_{d,-r;\,2,1,l}) = -S_{r,d;\,1,1,l}\,, \Phi^* (S_{d,-r;\,1,2,l}) = -S_{r,d;\,1,0,l-1}\,.
G_j = \begin{pmatrix}0 \gamma_{j,1}0 \overline{\gamma_{j,1}}0 \gamma_{j,2}0 \overline{\gamma_{j,2}}0 \ddots 0 & \ddots \ddots \gamma_{j,d_j-1}& 0 \overline{\gamma_{j,d_j-1}}0\end{pmatrix},
A(p_1, q_1, p_2, q_2, q_3, q_4) = \sum_{\mathbf i \in \mathcal J}\Psi_{\mathbf i}\cdot \varphi_2(\mathbf i_{1:4}) \varphi_2(\mathbf i_{\{1,2,5,6\}}) \varphi_2(\mathbf i_{\{1,2,7,8\}}),
wN_{-}\left(w^{-1}\right)\boldsymbol{B}_{+}\cap\boldsymbol{N}_{-}\boldsymbol{B}_{+}=wN_{-}\left(w^{-1}\right)\boldsymbol{B}_{+}\cap\boldsymbol{N}_{-}\boldsymbol{B}_{+}\cap G =wN_{-}\left(w^{-1}\right)\boldsymbol{B}_{+}\cap N_{-}\boldsymbol{B}_{+}.
\begin{aligned}\sum_{i=1}^{n}w_+(x_i,h_{+}) =\sum_{i=1}^{n}w_{-}(x_i,h_{-})=1, \sum_{i=1}^{n}x_{i}^{j}w_{-}(x_i,h_{-}) =\sum_{i=1}^{n}x_{i}^{j}w_{+}(x_i,h_{+})=0 j=1,\dotsc,p-1. \end{aligned}
\mathcal{B}= * B^{A}\left[ \begin{array}{c}{{v_{A}}}\\{{\overline{{{v}}}_{A}}}\\ \end{array}\right] - i B^{A}\left[ \begin{array}{c}{{v_{A}}}\\{{- \overline{{{v}}}_{A}}}\\ \end{array}\right] + b^{\alpha}\left[ \begin{array}{c}{{e_{\alpha}}}\\{{\overline{{{e}}}_{\alpha}}}\\ \end{array}\right] + i * b^{\alpha}\left[ \begin{array}{c}{{e_{\alpha}}}\\{{- \overline{{{e}}}_{\alpha}}}\\ \end{array}\right]
({\rm d}T_{n1},\dots,{\rm d}T_{nn}) \operatorname{diag}[\lambda_n-\lambda_1-1,\lambda_n-\lambda_2-1,\dots,-1]^{-1}P{}=\big(\tilde{\Omega}_{n1},\dots,\tilde{\Omega}_{nn}\big)P =-({\rm d}z_1,\dots,{\rm d}z_n) \operatorname{diag}[P_{n1},\dots,P_{nn}].
\int_{t}^{\infty}\frac{ds}{s^{\nu +1}}\,\exp \left( -\frac{x^2}{2s}\right) =\frac{1}{\nu t^{\nu}}\exp \left( -\frac{x^2}{2t}\right) +\frac{x^{2}}{2\nu}\int_{t}^{\infty}\frac{ds}{s^{\nu +2}}\, \exp \left( -\frac{x^2}{2s}\right) .
\left( \frac{\partial}{\partial ~t}\right) ~\psi_{N}\left( z,t\right)=-\psi_{N}\left( z,t\right) ~\sum_{m=1}^{N}\left[ \frac{\dot{z}_{m}\left(t\right)}{z-z_{m}\left( t\right)}\right] =-\sum_{m=1}^{N}\left\{\dot{z}_{m}\left( t\right) ~\prod\limits_{\ell =1,~\ell \neq m}^{N}\left[ z-z_{\ell}\left( t\right) \right] \right\}~,
S_{\phi_k}\circ\alpha_k(x) =\int_{0}^{1}\frac{d}{dt}\left(S_{\phi_k}\circ\alpha_k(\gamma_{x}(t))\right)dt\leq (A+1)\max_{t\in [0,1]}|d(S_{\phi_k}\circ\alpha_k)(\gamma_x(t))|_g &\leq (A+1)\max_{\Delta}|\alpha_k|_g.
\forall\psi_1\in\Omega^k(M),\quad\pi_{z_0}^{(k)}(\psi_1)=\sum_{a\in(f)}\sum_{j=1}^{m_a^{(k)}(z_0)}\left(\int_M\psi_1\wedge S_j(a,z_0)\right)U_j(a,z_0).
d\Phi=[\frac{9b_1b_2C_3^2}{(b_1-b_3)^2(b_2-b_3)^2}-R_{1212}-R_{1313}-R_{2323}]\omega_1\wedge\omega_2\wedge\omega_3 &+[\frac{9b_2b_3C_1^2}{(b_1-b_2)^2(b_1-b_3)^2}+\frac{9b_1b_3C_2^2}{(b_1-b_2)^2(b_2-b_3)^2}]\omega_1\wedge\omega_2\wedge\omega_3.
\begin{aligned}K A_{+}K^{-1}=q A_{+},K A_{-}K^{-1}=q^{-1}A_{-},KK^{-1}=1,P^2=1, [K,P]=0,[K^{-1},P]=0,\{A_{\pm}, P\}=0,\{A_{+},A_{-}\}=\frac{K-K^{-1}}{q^{1/2}-q^{-1/2}}. \end{aligned}
(\delta^nT)(a_1,..., a_{n+1}) :=a_1\cdot T(a_2,..., a_{n+1}) &+\sum_{k=1}^{n}(-1)^kT(a_1,..., a_ka_{k+1},..., a_{n+1}) &+(-1)^{n+1}T(a_1,..., a_{n})\cdot a_{n+1}, (T\in E_n, a_1,..., a_{n+1}\in \mathcal{A}).
S(z)= I_E + i \alpha({L^{\phantom{s}}\!\!\!}^{\scriptscriptstyle{-||}}-zI)^{-1}\alpha, z \in \rho({L^{\phantom{s}}\!\!\!}^{\scriptscriptstyle{-||}}),{L^{\phantom{s}}\!\!\!}^{\scriptscriptstyle{-||}}:=({L^{\phantom{s}}\!\!}^{\scriptscriptstyle{||}})^*.
\frac{d^{2n+1}}{dx^{2n+1}}\left[ (x^2+x)^{m}\right] = \sum\limits_{i=0}^m{m\choose i}(m+i)_{2n+1}x^{m-2n+i-1}= \sum\limits_{i=2n-m+1}^m{m \choose i}\frac{(m + i)!}{(m+i-2n-1)!}\:x^{m-2n+i-1}
\hat{R}^{*}_\lambda\big(z_1^{\pm 1},\dots,z_n^{\pm 1};a,b;q,t,t^n;p\big) \qquad{}= \begin{cases}\Delta^0_\lambda\big(t^{n-1}a/b|t^n;q,t;p\big)R^{*(n)}_\lambda(z_1,\dots,z_n;a,b;q,t;p), n\ge \ell(\lambda), 0, \ell(\lambda)>n.\end{cases}
R_{Tn}^{(2)}(d) =-\sum_{k=0}^{n-1}(k+d)^{-2}+(-\log T+\sum_{k=0}^{n-1}(k+d)^{-1})^{2}, R_{Tn}^{(3)}(d) =2\sum_{k=0}^{n-1}(k+d)^{-3}-3(-\log T+\sum_{k=0}^{n-1}(k+d)^{-1})\sum_{k=0}^{n-1}(k+d)^{-2}+(-\log T+\sum_{k=0}^{n-1}(k+d)^{-1})^{3}.
\begin{array}{l}\displaystyle x_j^2=\frac{\prod_{i=1}^D(\rho_i-e_j)}{\prod_{i\not=j}^D(e_i-e_j)}\enspace,\qquad(j=1,\dots,D) \displaystyle e_1<\rho_1<e_2<\dots<e_D<\rho_D\enspace.\end{array}
\leq \frac{1}{\sqrt{m}}\Biggl(\sqrt{M}\|x\|_k+ \sum\limits_{j=1}^{k}C_{k}^j\sqrt{\left(\mathcal{P}_j (|t|)\right)^2 \Omega_j}\Biggr) \leq \sqrt{\frac{M}{m}}\sum\limits_{j=0}^{k}2^j C_{k}^j \mathcal{P}_j (|t|) \|x\|_k,
E_x\left[ \int_0^{\infty}c_{\theta}(t,X_t,\theta_t)\int_{0}^{t}d(\eta_s-\theta_s)e^{-rt}\,dt\right] = E_x\left[\int_{0}^{\infty}\int_{s}^{\infty}c_{\theta}(t,X_t,\theta_t)e^{-rt}\,dt\,d(\eta_s-\theta_s)\right] &= E_x\left[\int_{0}^{\infty}p_{s}^{\theta}e^{-rs}\,d(\eta_s-\theta_s)\right]
S^2(A^{(4)}) = S^2(B^{(2)}) = S^2(B^{(1)})-\big{(}(b_{13}^{(1)})^2+(b_{24}^{(1)})^2\big{)}\leq \big{(}S^2(A)-a_{24}^2\big{)}- \frac{1}{2}(a_{14}^2+a_{23}^2) & \leq S^2(A) - \frac{1}{2}(a_{24}^2+a_{14}^2+a_{23}^2) = S^2(A)-\frac{1}{2}S^2(A) = \frac{1}{2}S^2(A).
T_{\alpha \beta}= \frac{2}{{\sqrt{- g}}}\frac{{\delta S}}{{\delta g^{\alpha \beta}}}= - \frac{2 K^{- 3}}{{g_{0}^{2}}}A_{\alpha \beta}( \widetilde \phi^{2}) - \frac{2}{g_{0}^{2}}C_{\alpha \beta}.
M_{1}= \left(\begin{array}{cccc}I 0 0 0 0 A -B 0 0 B A 0 0 0 0 I\end{array}\right) \: \: , \: \:I = \left(\begin{array}{cc}1 0 0 1\end{array}\right) \: \: ,A = \cos \frac{2 \pi}{n}I \:,\:B = \sin \frac{2 \pi}{n}I .
\div\, V (x) = \phi\big(\textstyle{\frac{|x|}{t}}\big)|\nabla u(x)|^2\,|x_{n+1}|^a+ \phi'\big(\textstyle{\frac{|x|}{t}}\big)u(x)\nabla u(x)\cdot\frac{x}{t\,|x|}\,|x_{n+1}|^a.
J(Z_u)\bullet_\hbar J(Z_v) = Z_u\bullet_\hbar^z Z_v+ \sum_{i}y^i ( Z_{\partial_i u}\bullet_\hbar^z Z_v + Z_u\bullet_\hbar^zZ_{\partial_i v})+\cdots\;,\quad\forall u,v\in N_1.
\lim_{n\rightarrow\infty}\left|\int_{\Omega}a(x)|u_n|^{p(x)-2}u_n(u_n-u_0)\,dx \right|\leq 2 \|a\|_{L^{\infty}(\Omega)}\lim_{n\rightarrow\infty}\left(\||u_n|^{p(x)-1}\|_{L^{p'(\cdot)}(\Omega)}\|u_n-u_0\|_{L^{p(\cdot)}(\Omega)}\right)=0.
{\cal L_{Q}}= -{\frac{1}{4}}\Bigl ( 1 +{\frac{a e^{2}}{\pi}}\Bigr ) F^{2}+{\frac{e^{2}}{2 \pi}}A^{2}+{\frac{1}{2}}\partial_{\mu}\Sigma \partial^{\mu}\Sigma -{\frac{e}{\sqrt \pi}}\partial_{\mu}\Sigma A^{\mu}-{\frac{e}{\sqrt \pi}}\widetilde{\partial}\cdot A \chi .
\int \varphi_{i}^{\frac{2n}{n-2}}= \int_{B_{c}(a_{i})}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}\gamma_{n}G^{\frac{2}{2-n}}_{a_{i}}})^{n}d\mu_{g_{a_{i}}}+O(\frac{1}{\lambda_{i}^{n}}).
{}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \tilde A_2^\dagger \vert J_1,\gamma_1; J_2,\gamma_2\rangle_q =\frac{\sqrt{J_2}}{E_q(J_1,J_2)}F_q(J_1,J_2,\gamma_2),
f_\mathrm{short}(\gamma) = \frac{1}{\bar{\gamma}_1}e^{-\gamma/\bar{\gamma}_1}+ \frac{1}{\bar{\gamma}_2}e^{-\gamma/\bar{\gamma}_2}- \left(\frac{1}{\bar{\gamma}_1}+\frac{1}{\bar{\gamma}_2}\right)e^{-(1/\bar{\gamma}_1 + 1/\bar{\gamma}_2)\gamma},
\hbox{N}_h(N,w_g; \xi)= c_h h^{-2}\ell^{-4}\sum_{k=1}^\infty k^{-4}S_{k}(0;\xi) I_{k,\ell}(0) &+O_{w,\ell,\theta,\epsilon}\big(\|g\|_E^\theta h^{3-(\theta-4)\delta}+\|g\|_E h^{3/2+3\delta+\epsilon}\big).
\langle \bar{O}_{m , - m}^{J}O_{0}^{J_{1}}O_{0}^{J_{2}}\rangle_{t o r u s}= Q_{1}^{'}+ Q_{2}^{'}+ Q_{3}^{'}= \frac{1}{6}( R_{1}^{'}+ R_{2}^{'}+ R_{3}^{'})
\dfrac{M_{2}(\mathcal{C}(G))}{|e(\mathcal{C}(G))|}- \dfrac{M_{1}(\mathcal{C}(G))}{|v(\mathcal{C}(G))|}=\dfrac{2q^{8}(q-5)+q^{5}(14q^{2}-13)+q^{3}(24q^{3}-q+4)+q(8q-7)+1}{q^{5}(q^{2}-2q-2)+q(3q^{3}-q-3)+(4q^{3}-1)}:= \dfrac{f(q)}{g(q)}.
\left\vert W^{int}[\mathcal{T}_{n}]\right\vert \geq(W(G_{3}(n,d_{3}^{\max},1,s))-W(G_{4}(n,d_{4}^{\min},x_{4}^{\max},s)))/2= & =\frac{n^{3}}{12}-\frac{\sqrt{n^{5}-3n^{4}}}{2\sqrt{2}}-n^{2}+\frac{5}{3}\sqrt{2n^{3}-6n^{2}}+\frac{83n}{12}+\frac{11\sqrt{n-3}}{6\sqrt{2}}-13.
{\bf A}\odot{\bf B}:=\sum\limits_{i=1}^{R_a}\sum\limits_{j=1}^{R_b}c_i c_j \left({\bf a}_i^{(1)}\odot{\bf b}_j^{(1)}\right) \otimes \ldots \otimes\left({\bf a}_i^{(d)}\odot{\bf b}_j^{(d)}\right).
\begin{aligned}a_N = (tA)^N \varphi_N(tA) u_0 + \sum\limits_{\ell=1}^N t^\ell (tA)^{N-\ell}\varphi_N(t A) g^{(\ell-1)}(0) \,\, + \sum\limits_{\ell=N+1}^\infty t^\ell \varphi_\ell (t A) W_N H_N^{\ell-1}e_1.\end{aligned}
(b-a)(\phi(x)-\phi(y)) =[(v_{n}-\widetilde{\Gamma})(x)-(v_{n}-\widetilde{\Gamma})(y)] [(v_{n}-\widetilde{\Gamma})^{+}(x)-(v_{n}-\widetilde{\Gamma})^{+}(y)] &\geq |(v_{n}-\widetilde{\Gamma})^{+}(x)-(v_{n}-\widetilde{\Gamma})^{+}(y)|^{2},
F_{n-1}(e_0) = \delta_{n-1}^{1/p}A_n, F_{n-1}(e_1) = \eta_{n-1}^{1/p}A_n, F_{n-1}(e_j) = a_j^{q/p}A_n \leq \alpha^{1/p}A_n j \in \Lambda_1, F_{n-1}(e_j) = t_j \alpha^{1/p}A_n j \in \Lambda_2^{(n-1)}, F_{n-1}(e_j) = 0 j \in \Gamma_{n-1}\setminus \{1\}.
S_{, i j}^{G}= -{\cal G}_{i j}^{- 1}+ \left\{\begin{array}{c}{k}{i \j}\end{array}\right\}{\cal E}_{k}+ \frac{\delta \chi^{\alpha}}{\delta \Phi^{i}}c_{\alpha \beta}\frac{\delta \chi^{\beta}}{\delta \Phi^{j}}.
\delta \lVert \alpha \rVert_{1}\leq \rho_{M}(T(\alpha)) =\sup_{\theta \in M}\lvert \theta(T(\alpha)) \rvert = \sup_{\theta \in M}\lvert (T^{*}(\theta))(\alpha) \rvert <\lvert \varphi(\alpha) \rvert \leq \lVert \varphi \rVert_{1}\lVert \alpha \rVert_{1}\leq\delta \lVert \alpha \rVert_{1},
[\begin{matrix}1-\frac{1}{2}\lambda^{2}&\lambda&A\lambda^{3}(\rho-i\eta)\\ -\lambda&1-\frac{1}{2}\lambda^{2}&A\lambda^{2}\\A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\end{matrix}]+O(\lambda^{4}).
D^n_{x_1}G = \left(\sum_{j=0}^n{n\choose j}\alpha_1^{n-j}D_{x_1}^j D_{x_m}^{n-j}F \right)\Biggr\rvert_{\substack(x_1,x_2,\ldots,\alpha_1 x_1+\cdots +\alpha_{m-1}x_{m-1})}.
\overset{2}{R}(X,AY,AZ,W) = \overset{g}{R}(X,AY,AZ,W) + \overset{2}{D}(X,AZ)g(Y,W) + \overset{2}{D}(AY,AZ)F(X,W), \overset{2}{R}(AX,Y,Z,AW) = \overset{g}{R}(AX,Y,Z,AW) - \overset{2}{D}(AX,Z)g(Y,W) + \overset{2}{D}(Y,Z)F(X,W),
\frac{\Delta tm_{n+1}^\epsilon}{\epsilon}=\Delta \beta_n+\epsilon(m_n^{\epsilon,\Delta t}-m_{n+1}^{\epsilon,\Delta t}) \zeta^\epsilon(t_{n+1})-\zeta^\epsilon(t_n) =\int_{t_n}^{t_{n+1}}\frac{m^\epsilon(t)}{\epsilon}dt=\beta(t_{n+1})-\beta(t_n)+\epsilon(m^\epsilon(t_n)-m^\epsilon(t_{n+1}))
~\mu[a,b)-(b-a)=\frac{1}{\pi}\sum_{n=1}^\infty\Im(\frac{\hat{\mu}(n)}{n}[e(nb)-e(na)])+\lim_{n\to \infty}\frac{1}{2n}\sum_{j=1}^n \hat{\mu}(j)[e(ja)-e(jb)].
\tilde{S}( x ) \; = \; \exp \left( - i \frac{x_{1}}{\sqrt{x_{1}^{2}+ \lambda^{2}}}\left[ \mathrm{a r c t a n}\left( \frac{x_{2}}{\sqrt{x_{1}^{2}+ \lambda^{2}}}\right) \; - \; \pi \Big ( \frac{1}{2}+ m \Big ) \right] \; \right) \; ,
\dot{\xi}_{\underline{c}}-{h}^{i}_{\ \underline{c}}\,{h}^{j}_{\ \underline{b}}\,T^{\underline{a}}_{ij}\, \xi_{\underline{a}}\, \xi^{\underline{b}}-{h}^{i}_{\ \underline{c}}\,T^{\underline{a}}_{i0}\, \xi_{{a}}=0,
\triangle_{p(x,t)}^Nu(x,t) &:=\bigg(\frac{p(x,t)-2}{|Du(x,t)|^2}\bigg)\sum_{i,j=1}^n\frac{\partial^2 u}{\partial x_i \partial x_j}(x,t)\frac{\partial u}{\partial x_i}(x,t)\frac{\partial u}{\partial x_j}(x,t)+\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}(x,t)
v = \left[{\frac{z_{0}^{2}+ z_{i}^{2}+ 1}{2 z_{0}}},{\frac{z_{0}^{2}+ z_{i}^{2}- 1}{2 z_{0}}},{\frac{z_{i}}{z_{0}}}\right] , , l = [ 1 , - 1 , \vec{0}] , ( v \cdot l ) ={\frac{z_{0}^{2}+ z_{i}^{2}}{z_{0}}}
-\frac{1}{\rho_{L}}\int_{P(R)}^{P(\infty)}dP=\int_{R}^{\infty}[\frac{1}{r^{2}}(2R(\frac{dR}{dt})^{2}+R^{2}\frac{d^{2}R}{dt^{2}})-\frac{2R^{4}}{r^{5}}(\frac{dR}{dt})^{2}]dr
y_3:=\frac{\sum_{j=1}^{n-1}w_{j,n-3}b_{j,n-3}}{\sum_{j=1}^{n-1}b_{j,n-3}^2}\,, z_3:=\frac{\sum_{j=1}^{n-1}w_{j,n-3}b_{j,n-2}}{\sum_{j=1}^{n-1}b_{j,n-2}^2}
\psi_{Q}= \left( \begin{array}{c}{\frac{\psi_{( 0 )}}{\sqrt{N_{f}}}I_{N_{f}}+ \psi_{( 1 )}}\\{\psi_{( 2 )}}\\ \end{array}\right) , \qquad \quad \psi_{\bar{Q}}= \left( \begin{array}{c c}{\frac{\psi_{( \bar{0})}}{\sqrt{N_{f}}}I_{N_{f}}+ \psi_{( \bar{1})}}&{\psi_{( \bar{2})}}\\ \end{array}\right)
S_2(...) = \frac{N^{3/4-i(t+\nu)}\,\eta_f(p_2)}{(p_1q)^{1/2}\,p_2^{1/4}}\sum_{\pm}\sum_{m \ll M_0}\frac{\lambda_f(m)}{m^{1/4}}e\left(-m \frac{\overline{(a+bq) p_2}}{q}\right) \mathcal{I}^{\pm}(m,u,q,p_1,p_2),
u(x_j) \ge \frac{A\psi(|x_j|)}{|x_j|^{n-2}}\left[\frac1{2n}\left(\frac{A\psi(|x_j|)}{|x_j|^{n-2}}\right)^\lambda\right]^{\frac1{1-\lambda}}&= \left(\frac{A\psi(|x_j|)}{2n}\right)^{\frac1{1-\lambda}}\frac1{|x_j|^{\frac{n-2}{1-\lambda}}}&\ge \left(\frac{A}{2n}\right)^{\frac1{1-\lambda}}\sqrt{\varphi(|x_j|)}\ |x_j|^{-\frac{n-2}{1-\lambda}}
d s_{\mathrm{h y p e r}}^{2}= d D^{2}+ D^{2}\sigma_{1}^{2}+ D^{2}\sigma_{2}^{2}+ D \tanh D \check{\sigma}_{1}^{2}+ D \coth D \check{\sigma}_{2}^{2}+ \check{\sigma}_{3}^{2}.
\left[ J^{\alpha \beta}( x ) , S_{+}^{\alpha^{\prime}\beta^{\prime}}( y ) \right] = \delta ( x - y ) \left( \delta^{\beta \alpha^{\prime}}S_{+}^{\alpha \beta^{\prime}}( x ) + \delta^{\beta \beta^{\prime}}S_{+}^{\alpha \alpha^{\prime}}( x ) \right) ,
\dfrac{d^{k}}{dx^{k}}P_{n}(x)={n \choose k}\dfrac{(n+1)_{k}}{2^{k}}\sum_{j=0}^{n-k}\dfrac{(-n+k)_{j}(n+1+k)_{j}}{(1+k)_{j}}\dfrac{z^{j}}{j!}.
\left |\left(z\frac{F'(z)}{F(z)}-1\right){\Big /}\left(z\frac{F'(z)}{F(z)}+1\right) \right| =\left |\left(z\frac{f'(z)}{f(z)}-1\right){\Big /}\left(z\frac{f'(z)}{f(z)}+1-2\beta\right) \right|.
s' : =s-\big(b_kp^k+(b_{k-1}-b_k)p^{k-1}+\cdots+(p-1-b_{k-m})p^{k-m-1}\big) &= \lfloor b_k \cdots b_{k-m}\underbracket[0.1ex]{(p-1) \cdots (p-1)}_{\ell-1}b_{k-m-\ell-1}\cdots b_0 \rfloor_p .
S_p = \frac{2 V_3N\sqrt{2m\lambda}l^4}{\pi \lambda^2 (2r_+^4 +3l^2r_+^2 +l^4)}\frac{l^2+2r_+^2}{l^2-2r_+^2}\frac{1}{\triangle_*}\left [\frac{m (l^2-2r_+^2)}{l^4}- \frac{u^2(u^2+l^2)-m}{u^2+l^2}-u^2\triangle_* \right],
\varphi^1\varphi^1_\omega-\varphi^1_{\omega\omega}-\frac{(\varphi^2)^2}{\omega}-\frac1{\omega^3}-\frac\omega4=0, \varphi^1\varphi^2_\omega-\varphi^2_{\omega\omega}+\frac{\varphi^1\varphi^2}{\omega}+2\frac{\varphi^2}{\omega^2}=0.
S = \frac{1}{2}\int d \tau t r \biggl \{\biggl ( D_{0}A^{I}\biggr )^{2}- \frac{1}{2}\biggl [ A^{I}, A^{J}\biggr ] ~ \biggl [ A^{J}, A^{I}\biggr ] + i{\bar{\lambda}}D_{0}\lambda + i{\bar{\lambda}}{\gamma}^{I}~ \biggl [ A^{I ,}\lambda \biggr ] \biggr \}
(D^s h)(x) = C(n,s)\, \mathrm{P.V.}\int_{\mathbb R^n}\frac{h(x+y)-h(x)}{|y|^{n+s}}\, dy & =C(n,s) \lim_{\varepsilon \searrow 0}\int_{|y|\ge \varepsilon}\frac{h(x+y)-h(x)}{|y|^{n+s}}\, dy,
\sum_{n\ge1}\frac{(-1)^nq^{n^2}}{(q;q^2)_n(1+q^{2n-1})}=\sum_{n\ge1}\frac{(-1)^nq^{1+3+\cdots+2n-1}}{(1-q)(1-q^3)\cdots(1-q^{2n-3})(1-q^{(2n-1)+(2n-1)})}.
A_{i_{1}\ldots i_{p}}^{( g )}( x_{0},{\bf x}) = x_{0}^{\frac{d}{2}}\int{\frac{d^{d}k}{( 2 \pi )^{d}}}e^{- i{\bf k}\cdot{\bf x}}a_{i_{1}\ldots i_{p}}({\bf k}) K_{\nu}( |{\bf k}| x_{0}) .
\begin{cases}\partial_t v(t,x)-\frac{1}{2}\partial_{xx}^2 v(t,x)=-\frac{1}{2}\bar A\left(v(t,x)\right)\left(\partial_x v(t,x),\partial_x v(t,x)\right)+\bar f\left(v(t,x),\partial_x v(t,x)\right), & v(0,x)=h(x),\ \ \ \t\in (0,T_1).\end{cases}
(-1)^k d_A ( *_\mu(P)) = *_\mu \bigg( \sum_{i=1}^{k}(-1)^i (_\mu X_i) (X_1 \wedge \cdots \wedge \hat{X_i}\wedge \cdots \wedge X_k) &+ \sum_{1 \leq i < j \leq k}(-1)^{i+j}[X_i, X_j]\wedge X_1 \wedge \cdots \wedge\hat{X_i}\wedge \cdots \wedge\hat{X_j}\wedge \cdots \wedge X_k \bigg),
P_n^{(a,b)}\bigl(\frac{z+z^{-1}}{2}\bigr)= \frac{c(z)z^n +c(z^{-1})z^{-n}}{((a-1)(b-1))^{n/2}}\quad\quad\>\> z\in\mathbb C\setminus\{0, \pm 1\}
\delta_i=\det(A_i) =\prod_{j=0}^{i-1}\frac{\binom{r+s-2i+j}{s-i}}{\binom{s-i+j}{s-i}}=\prod_{j=0}^{i-1}\frac{\binom{r+s-2i+j}{r-i+j}}{\binom{s-i+j}{j}}=\prod_{j=0}^{i-1}\frac{\binom{r+s-2i+j}{r-i}}{\binom{r-i+j}{j}}.
H = \int{d^{3}}x \left({- \frac{1}{2}\Pi_{i}^{a}\Pi_{a}^{i}+ \frac{1}{4}F_{i j}^{a}F^{a i j}+ c^{a}\left( x \right) \left({\partial^{i}\Pi_{i}^{a}+ g f_{a b c}A_{b}^{i}\Pi_{i}^{c}- J^{0}}\right)}\right) ,
(H_{1}-M)|\mathbf{s}\rangle=(1-q^{2})\sum_{\mathbf{s}' \in \mathcal{S}_{M}^{\mathrm{per}}\setminus\{\mathbf{s}\}}q(\mathbf{s}|\mathbf{s}')(|\mathbf{s}'\rangle-|\mathbf{s}\rangle),
b(g) = B_1 e^{\chi_2{\cal X}_2}e^{\chi_3{\cal X}_3}+ e^{-\chi_1{{\cal X}_1}^t}B_2 e^{\chi_3{\cal X}_3}+ e^{-\chi_1{{\cal X}_1}^t}e^{-\chi_2{{\cal X}_2}^t}B_3.
q_{\mu_{1}\boxtimes\mu_{2}}(x)=\frac{1}{\pi}\Im\frac{1}{1-\eta_{\mu_{1}}(re^{if(r)})}=\frac{1}{\pi}\frac{\Im\eta_{\mu_{1}}(re^{if(r)})}{|1-\eta_{\mu_{1}}(re^{if(r)})|^{2}},
\int_{0}^{\infty}d k Q_{\ell}({\frac{k^{2}+ q^{2})}{2 k q}}){\frac{q^{\ell + d_{1}}}{( c^{2}+ k^{2})^{d_{2}}}}B_{j}( k ) = q^{\ell + 1}\sum_{j = 0}^{3}a_{j}( q - p )^{j}.
(\begin{matrix}A_{x}\\A_{y}\\A_{z}\end{matrix})=(U^{*})^{-1}(\begin{matrix}A_{+}\\A_{-}\\A_{0}\end{matrix}),(U^{*})^{-1}=(\begin{matrix}-\frac{1}{\sqrt{2}}&+\frac{1}{\sqrt{2}}&0\\ -\frac{i}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{matrix}).
\frac{\partial V}{\partial \Lambda}= \frac{1}{\Lambda^{2}}\frac{\partial V}{\partial \psi_{\alpha a}^{\phantom{a}}}C^{\prime}( 0 ) \frac{\partial V}{\partial \bar{\psi}_{a}^{\alpha}}- \tau \Lambda^{D - 2}\frac{\partial^{2}V}{\partial \psi_{\alpha a}^{\phantom{a}}\partial \bar{\psi}_{a}^{\alpha}},
\tilde{\mathrm{E}}u (x',x_n) := \begin{cases}u(x',x_n)\, (x',x_n)\in\mathbb{R}^{n-1}\times(0,+\infty) u(x',-x_n)\, (x',x_n)\in\mathbb{R}^{n-1}\times(-\infty,0) \end{cases}
\partial_t\phi_N+\frac32\partial_x^5\phi_N+\frac54\, \frac{\big(\partial_x^3\phi_N\big)^2}{\partial_x\phi_N}= &\;\;=\Big\{a'\,x^{5/4}+k\, a^5 x^{5/4}+c_4\,a^4+c_3\,a^3x^{-5/4}+c_2\,a^2 x^{-10/4}+c_1\,a x^{-15/4}\Big\}\,\phi_N
e^{-\left|x\right|}=\int_{\mathbb{R}^{n}}2^{n}\pi^{\frac{n-1}{2}}\Gamma\left(\frac{n+1}{2}\right)\frac{e^{i\lambda x}}{\left(1+\left|\lambda\right|^{2}\right)^{\frac{n+1}{2}}}d\lambda_{1}\cdots d\lambda_{n}.
|M(z_{0})-1|^{2}= \left|\cosh e^{it/2}-1+\frac{k e^{i t/2}\sinh e^{it/2}}{2 ( \eta \cosh e^{it/2}+\gamma)}\right|^{2}&=\frac{A_{N}(\tau)}{A_{D}(\tau)}(-\pi/2 \leq \tau=t/2\leq \pi/2),
\eta = \eta(\xi,a) = \frac{\xi}{2\pi a}\left(\int_0^\infty \frac{y e^{-y}}{y e^{y}+ \frac{\xi a}{2}\sinh y}dy - \int_0^\infty \frac{y e^{-y}}{y e^{y}+ \frac{\xi a}{2}\cosh y}dy\right).
\ddot{x}= - 2 x y^{2}+{\frac{( p_{\alpha}^{2}+ p_{\beta}^{2}) x ( x^{4}+ 1 0 x^{2}y^{2}+ 5 y^{4}) - 2 p_{\alpha}p_{\beta}y ( y^{4}+ 1 0 x^{2}y^{2}+ 5 x^{4})}{\left( x^{2}- y^{2}\right)^{4}}}
\|g_i^{n_i}\|_* \ge \|h\|_*-\|a_i\|_*-\|b_i\|_*\ge \|h\|_* -A-B, \|g_i^{n_i}\|_* - \|g_i^{\alpha_i+\beta_i}\|_* \ge \|g_i^{n_i}\|_* - \|g_i^{\alpha_i}\|_* - \|g_i^{\beta_i}\|_* &\ge (\|h\|_* -A-B) - (A+1+r) - (B+1+r) & = \|h\|_* -2A-2B-2r-2 \ge (2V+1)r.