harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z_{n}^{I; \vec{i}}= \left\{z(i_1, \ldots, i_n)_j =0 \mid j \in [N] - \{i_1, \ldots, i_n \}\right\}\subseteq V^I_{\vec{i}}= \left( l^I(\vec{i}- i_n)_{i_n}\neq 0 \right).
\varphi(\Pi_g(X,Y))=\frac{1}{2}((\nabla_X\xi)_Y+(\nabla_Y\xi)_X)\in\mathfrak{m}(M), g(\pi_{N\ast}\Pi_g(X,Y),Z) =\frac{1}{2}({\bf B}(\xi_X,R_{\mathfrak{m}}(Y,Z))+{\bf B}(\xi_Y,R_{\mathfrak{m}}(X,Z))),
y_1=\frac{1}{\sqrt{\mu}}x_3, \, y_2=x_1+x_2, \, y_3=x_2; &[y_1,y_2]=\frac{1+w}{\sqrt{\mu}}y_2-\frac{2w}{\sqrt{\mu}}y_3, \,[y_2,y_3]=0, \,[y_3,y_1]=\frac{w-1}{\sqrt{\mu}}y_3.
\tilde{\textbf{y}}_{0}^{}= \mathbf{\tilde{h}}_{0}^{}\sqrt{P}U_{0}+\tilde{\textbf{z}}_{0},{\mathbf{\tilde{h}}_{0}^{}}={\left(\textbf{R}_{0}\right)^{-1/2}\textbf{H}_{0}^{}\textbf{w}_{0}}.
C_{1,k}(\cdot,\cdot,0)=0;\;\;\frac{\partial^{j}C_{1,q}}{\partial x^j_2}(\cdot,\cdot,0)=0, &\,\,\,j=1,...,k;\,\,q=1,...,k-j+1, &\,\,\,\mathrm{for\,\, the\,\, case}&\,\,\,\,\,\Delta_{p}\in D:=\{[...[[f,\underbrace{g],g],...,g}_{n \, \,}],n=1,...,L\}
\langle : \prod_{a = 1}^{r}O_{2}^{I_{a}}( x_{1}^{( a )}) : O_{2}^{I_{r + 1}}( x_{2}) : \prod_{b = 1}^{r}O_{2}^{J_{b}}( x_{3}^{( b )}) : O_{2}^{J_{r + 1}}( x_{4}) \rangle
C_0\left(\beta-i \alpha t\right) -C_0(\beta) + i\alpha t u=\int_0^1\left\{e^{(\beta-i\alpha t)x}-e^{\beta x}+i \alpha t x e^{\beta x}\right\}g(x) dx =\int_0^1 e^{\beta x}\left\{\frac{-\alpha^2 t^2 x^2}{2}+\frac{i^3 \alpha3 t^3 x^3}{3!}+\alpha^4 t^4 x^4\cdot O(1)\right\}g(x) dx
\begin{aligned}\mathcal{C}\left(C^{e}, C^{e^{\prime}}\right)(\tau) =\sum_{l=0}^{M-1}\mathcal{C}\left(\mathbf{c}_{l}^{e}, \mathbf{c}_{l}^{e^{\prime}}\right)(\tau) &= \begin{cases}MN, \tau=0, e=e^{\prime}, 0, 0 < |\tau| < Z, e=e', 0, |\tau| < Z, e\neq e', \end{cases}\end{aligned}
\partial_xU(t,x,\nu)=P_t,\quad\partial_{xx}U(t,x,\nu)=\partial_{x\nu}U(t,x,\nu)=0,\quad\partial_{\nu}U(t,x,\nu)=\partial_{\nu}\Phi(t,\nu),\quad\partial_{\nu\nu}U(t,x,\nu)=\partial_{\nu\nu}\Phi(t,\nu)
(m^*)^{it}\circ \mu_A = (m^*)^{it}\circ (\dot{+})^{it}\circ \bigotimes_{i=1}^l \mu_{a_{i}}= \left((\dot{+})^{it}\right)^{\otimes k}\circ w \circ \left((m^*)^{it}\right)^{\otimes l}\circ \bigotimes_{i=1}^l \mu_{a_{i}}
\left\{\begin{array}{ccc}l_1 l_2 l{}{}{L\over2}{L\over 2}{L\over 2}\end{array}\right\}\sim{(-1)^{l_1+l_2+l}\over\sqrt{L+1}}\left(\begin{array}{ccc}l_1 l_2 l{}{}0 0 0\end{array}\right) + O(L^{-3/2})
\{Q_{(\alpha)},Q_{(\beta)}\}= g[\mathcal{C}\Gamma_{s}(\hat{M}^{\hat{a}\hat{b}})]_{\alpha\beta}\hat{M}_{\hat{a}\hat{b}}= -i(\mathcal{C}\gamma^{a})_{\alpha\beta}P_{a}-{\textstyle\frac{g}{2}}(\mathcal{C}\gamma^{ab})_{\alpha\beta}M_{ab}\, ,
S_{5}= \int \! \! d^{3}x ~ \left( \alpha_{a b c d e}\varphi_{a}\varphi_{b}\varphi_{c}\varphi_{d}\varphi_{e}\right) , \quad S_{6}= \int \! \! d^{3}x ~ \left( \alpha_{a b c d e f}\varphi_{a}\varphi_{b}\varphi_{c}\varphi_{d}\varphi_{e}\varphi_{f}\right) ,
C(\beta)=\frac{2}{\mathcal{H}^{N-1}(\mathbb{S}^{N-1})}\,\int_{\mathbb{S}^{N-1}}\int_{\mathbb{R}^N}\frac{\left||x|^{-\beta}-|y|^{-\beta}\right|^{p-2}\, \left(|x|^{-\beta}-|y|^{-\beta}\right)}{|x-y|^{N+s\,p}}\, dy\, d\mathcal{H}^{N-1}(x).
\mbox{$\sum\limits_{h=1}^{\ell-1}|\lambda^{(h)}|+\sum\limits_{j=1}^i \lambda_j^{(\ell)}\le \sum\limits_{k=1}^{\ell-1}|\mu^{(h)}|+\sum\limits_{j=1}^i \mu_j^{(\ell)}$ for $ \ell\le k$ and all possible $i$,}
{}\left\langle X + \left( \dfrac{\sqrt{\left\langle Y_{\mathfrak{m}}, Y_{\mathfrak{m}}\right\rangle}- \left\langle X, Y_{\mathfrak{m}}\right\rangle}{\left\langle Y_{\mathfrak{m}}, Y_{\mathfrak{m}}\right\rangle}\right) Y_{\mathfrak{m}}, \left[Y, Z \right]_{\mathfrak{m}}\right\rangle =0.
\begin{array}{c}{{e_{1}e_{2}= e_{3}\ , \e_{2}e_{3}= e_{1}\ , \e_{3}e_{1}= e_{2}}}\\{{e_{3}e_{2}= - e_{1}\ , \e_{2}e_{1}= - e_{3}\ , \e_{1}e_{3}= - e_{2}\ .}}\\ \end{array}
\frac{\partial \phantom{iii}}{\partial x_i^{a_i}}\,X^{a_1 x_1\ldots a_i x_i \,\ldots\,a_nx_n}= \bigl( \,\delta(x_i-x_{i-1}) - \delta(x_i-x_{i+1}) \,\bigr) X^{a_1 x_1\ldots a_{i-1}x_{i-1}\,a_{i+1}x_{i+1}\ldots a_n x_n}
\kappa^{2}g^{2}\alpha^{\prime}\left( \frac{1}{2}( \frac{1 1}{3^{2}})^{2}- 2 6 \cdot \frac{1}{2}( \frac{5}{3^{2}\sqrt{2}})^{2}\right) \phi^{4}= - \kappa^{2}g^{2}\alpha^{\prime}\left( \frac{3 4}{2 7}\right) \phi^{4}.
\lambda\to{{\lambda\cos\delta-\sin\delta}\over{\lambda\sin\delta+\cos\delta}},\ \ \ \F_{\mu\nu}\to -(\lambda_1\sin\delta+\cos\delta)F_{\mu\nu}+\lambda_2\sin\delta\tilde{F}_{\mu\nu},\ \ \ \g_{\mu\nu}\to g_{\mu\nu}.
\|F\|_{L^{p(\cdot)}(\Omega)}\le \sum_{i=1}^N\|F_i\|_{L^{p(\cdot)}(E_i)}\le C\|f\|_{L^{p(\cdot)}(\Omega)}, &\|G\|_{L^{q(\cdot)}(\Omega)}\le \sum_{i=1}^N\|G_i\|_{L^{q(\cdot)}(E_i)}+\sum_{i=1}^N\|S_i-S\|_{L^{q(\cdot)}(E_i)}\le C\|g\|_{L^{q(\cdot)}(\Omega)},
\phi(\alpha) = \phi(\zeta^{-1}\zeta\alpha) & \leq (\phi(\zeta^{-1})^x + \phi(\zeta\alpha)^x)^{1/x}& = \phi(\zeta\alpha) & \leq (\phi(\zeta)^x + \phi(\alpha)^x)^{1/x}& = \phi(\alpha)
{\bf E}\Bigl[\Bigl(\int\limits_0^1 f(t)\,M_\mu(dt)\Bigr)^n\Bigr] = \lim\limits_{N\rightarrow \infty}\Biggl[n!\varepsilon^n \sum\limits_{j_1<\cdots< j_n}^N f(s_{j_1})\cdots f(s_{j_n}) \prod\limits_{p=2}^n \prod\limits_{k=1}^{p-1}|s_{j_p}-s_{j_k}|^{-\mu d(p-k)}\Biggr].
\delta^{\frac{n-2}{2}}| \nabla \eta_\delta(x',0)\cdot (\nu(x)+e_n))| \leq |\nabla \varphi_0(\frac{x'}{\delta},0)| \, |\nu(x)+e_n| & \leq \frac{C|(x',1-\sqrt{1-|x'|^2})|}{(1+|\frac{x'}{\delta}|)^{n-2}}=O\left(\frac{\delta^{n-2}|x'|}{(\delta+|x'|)^{n-2}}\right)
\sum_{j = 0}^k{( - 1)^j ( - 1)^{rj}\binom kj\left({\frac{{F_{n + r}}}{{F_n}}}\right)^j G_{rj}}= ( - 1)^k ( - 1)^{rk}\left({\frac{{F_r}}{{F_n}}}\right)^k G_{(n+r)k},n\ne 0\,,
\sigma_{p}^{\left( \nu \right)}\left( \ell \right) =\frac{(-1)^{\ell}}{2(p+\ell +1)\ell !}\left[ B_{2(p+\ell +1)}\left(\nu+\frac{1}{2}\right)-B_{2(p+\ell +1)}\left(\frac{1}{2}\right)\right].
\sum_{n=k+1}^{k+N}\!t_n\Big[C(T_{1n}^{r}T_{2n})-p\Big]\!-\!(I\!-\!P_{12}^{(p-r)})\otimes I\!=\! (\!I-\!P_{12}^{(p-r)})\otimes \Big[\!\!\sum_{n=k+1}^{k+N}\!t_n \Big(C(T_{1n})\!-\!p\Big)\!-\!I\Big]\!\to\!0.
P_{\lambda}(x,y)=\frac{\lambda^{\nu}}{2^{\nu-1}\pi^{n/2}\Gamma((1-\beta)/2)}\frac{y^{1-\beta}K_{\nu}\left(\lambda\sqrt{|x|^2+y^2}\right)}{\left(\sqrt{|x|^2+y^2}\right)^{\nu}},
a_{0}\int_{\Omega}h(x,|\nabla u_{n}|) \u_{n}\nabla u_{n}\ \nabla v dx\leq\int_{\Omega}\widetilde{\psi}_{q}(x)\varphi(x,u_{n})v u_{n}dx +\int_{\Omega}\psi_{p}(x)| u_{n}|^{p^{\ast}(x)}v dx-a_{0}\int_{\Omega}h(x,|\nabla u_{n}|) \ |\nabla u_{n}|^{2}\v dx+o_{n}(1).
\ \ \left(|z|^{p-2}z - |w|^{p-2}w\right)\cdot(z-w) \geq c(p) \begin{cases}|z-w|^2(|z|+|w|)^{p-2}\ \ \ \ 1<p<2 |z-w|^p \ \ \ \p\geq 2\end{cases}
\begin{array}{r l}{{\}}&{{\Phi ( E , x ) = \}}\\{{\}}&{{E^{2}- \left( \omega^{2}+ \frac{\lambda^{2}}{4}+ \lambda E \right) \left( 2 x + 2 u - 1 \right)^{2}+ \frac{\lambda^{2}}{4}\left( 2 x + 2 u - 1 \right)^{4}.}}\\ \end{array}
\|w^{m+1}\|_{2,\varphi}\le \sum_{\ell = 0}^{m}\|w^{\ell+1}-w^{\ell}\|_{2,\varphi}+ \|w^{0}\|_{2,\varphi}\le \|w^{1}-w^{0}\|_{2,\varphi}\sum_{\ell = 0}^{m}L^{\ell}+ \|w^{0}\|_{2,\varphi}&\le 6\|u^{n}\|_{2,\varphi}\frac{1}{1-L}+ \|u^{n}\|_{2,\varphi}= \frac{7-L}{1-L}\|u^{n}\|_{2,\varphi}= K \|u^{n}\|_{2,\varphi},
X_{\pm}(s)=\frac{\beta}{2}\sum_n\left[E_n^2+\frac{\lambda\phi^2}{3}+\mu^2\pm 2\mu E_n\pm\frac{\mu\lambda\phi^2}{3E_n}\pm\frac{\lambda^2\phi^4}{144\mu E_n}\mp\frac{\mu\lambda^2\phi^4}{36E_n^3}\right]^{-\frac{s}{2}}\;.
\frac{\det D(\varphi)}{\det D(0)}\;=\; \lim_{\Lambda \to \infty}\left[\frac{\langle + \mid \varphi + \rangle}{\mid \langle + \mid \varphi + \rangle\mid}\;\frac{\langle \varphi + \mid \varphi - \rangle}{\langle + \mid - \rangle}\;\frac{\langle \varphi - \mid - \rangle}{\mid \langle \varphi - \mid - \rangle\mid}\right]
V_{1}(t)=\int^{S(t)}_{S^{*}}\dfrac{F(x,I^{*})-F(S^{*},I^{*})}{F(x,I^{*})}dx+\int^{E(t)}_{E^{*}}\dfrac{x-E^{*}}{x}dx + \dfrac{m_{1}}{\sigma^{\alpha}}\left( \int^{I(t)}_{I^{*}}\dfrac{x-I^{*}}{x}dx \right).
Q_{I}[X+1][k(1-\alpha)]=\frac{(1-zx\alpha)^{k}(1-x^{2}\alpha)^{k^{2}+k}}{(1-zx)^{k+1}(1-x^{2})^{{k+1 \choose 2}}(1-x^{2}\alpha^{2})^{{k+1 \choose 2}}}.
\widetilde{\Lambda}_{j,1}= \frac{- \Gamma\left( -\beta_j \right)}{\Gamma\left( \beta_j + 1 \right)}\Lambda_{j}^{2}\mbox{and}\widetilde{\Lambda}_{j,2}= \frac{- \Gamma\left( \beta_j \right)}{\Gamma\left(1-\beta_j\right)}\Lambda_{j}^{-2}.
\mathrm{J}(f)_{x_{o,\lambda}}= \begin{pmatrix}- \frac{\left(p^2+q^2\right)^2}{\lambda p^2q^2}0 0 0 -{\frac{p^2+q^2}{\lambda q^2}}0 0 0 -{\frac{p^2+q^2}{\lambda p^2}}\end{pmatrix}.
\Psi(M_0) \geq\Psi(M_t)=\int_{\mathbb{S}^n}f(x)\log h(x,t)dx &\geq\log h_{\max}(t)\int_{\mathbb{S}^n}f(x)dx+\int_{\mathbb{S}^n}f(x)\log |x\cdot x^t_{\max}|dx &\geq C\log h_{\max}(t)-Cf_{\max}.
\int_D K_w(xH,A)\> d(\delta_{Hh_1H}*\delta_{Hh_2H})(w)= \int_H K_{Hh_1hh_2H}(xH,A)\> d\omega_H(h) &= \int_H (p_G(\delta_x*\omega_H*\delta_{h_1hh_2}*\omega_H))(A)\> d\omega_H(h) &=(\delta_x*\omega_H*\delta_{h_1}*\omega_H*\delta_{h_2}*\omega_H)(p_G^{-1}(A)),
p_1 =-\frac{1}{8\pi^2}\mathrm{tr}\,\Omega^2\in\Omega^4(M) p_2 =-\frac{1}{64\pi^4}\mathrm{tr}\,\Omega^4+\frac{1}{128\pi^4}(\mathrm{tr}\,\Omega^2)^2\in\Omega^8(M).
\MoveEqLeft\left|\int_{B(z,R)}G_1(x)\, dx\right|=\left|\int_{(z+Q_1)\setminus B(z,R)}G_1(x)\, dx\right|\leq \frac{1}{\pi}\left(1-\frac{4\pi\,R^3}{3}\right)^{1/2}\left(\sum_{k\in\mathbb{Z}^3\setminus\{0\}}\frac{1}{|k|^4}\right)^{1/2}.
\left( \begin{array}{c c}{2 A_{[{\tt i}}{}^{[{\tt I}}\delta_{{\tt j}]}{}^{{\tt J}]}}&{\Sigma_{\tt i j K L}}\\{\Sigma^{\tt k l I J}}&{2 A_{[{\tt k}}{}^{[{\tt K}}\delta_{{\tt l}]}{}^{{\tt L}]}}\\ \end{array}\right)
0\geq\left[(2-\frac{n-2}{n\gamma})\frac{n-2}{4(n-1)}\mu([g])-\sqrt{\frac{n-2}{2(n-1)}}\gamma\left(\int_M \left(|W|^2+\frac{2n}{n-2}u^2\right)^{\frac n4}\right)^{\frac 2n}\right]\left(\int_M u^{\frac{2n\gamma}{n-2}}\right)^{\frac{n-2}{n}}+\frac{4n\gamma+\frac{1}{\gamma}(n-2)^2-2n(n-2)}{4n(n-1)}R\int_M u^{2\gamma}.
\widehat{\Theta}^n_s = \sigma(X_s) \int_0^s \psi_{n,1}(u,s) dW_u + X^n_{\eta_n(s)}- X^n_s = \sigma(X_s) \int_0^s \psi_{n,1}(u,s) dW_u + \int^s_{\eta_n(s)}\left(s-\eta_n(u)\right)^{\alpha}\sigma(X_{\eta_n(u)})\, dW_u + \int^{\eta_n(s)}_0 \psi_{n,2}(u,s)\, \sigma(X_{\eta_n(u)})\, dW_u,
\sum_{n\geq 0}W_{k,a}(n)q^n= \frac{(-q^3;q^2)_\infty(q^{a+1},q^{2k+1-a},q^{2k+2};q^{2k+2})_\infty}{(q^2;q^2)_\infty}&+\frac{q(-q^3;q^2)_\infty(q^{a-1},q^{2k+3-a},q^{2k+2};q^{2k+2})_\infty}{(q^2;q^2)_\infty}.
(i s^*\rho_1 +j s^* \overline{\omega_3}) =|Z|^{-2}(\overline{h_1}\mathrm{d}h_1 + \overline{h_2}\mathrm{d}h_2)+R \frac{1}{\sqrt{2}a |Z|}(s^*\omega_1+j s^*\omega_2) =-h_2 h_1^{-1}\mathrm{d}h_1 + \mathrm{d}h_2,
\Bigl| \int u_{1,c_1}\overline{v}_{2,c_2}w_{3,\pm,c_3}dt dx \Bigr| & \lesssim C(N_1,N_2,N_3) \|u_{1,c_1}\|_{L_{t,x}^2}\|v_{2,c_2}\|_{L_{t,x}^2}\|w_{3,\pm,c_3}\|_{L_{t,x}^2},
\overline{U}_{2k+1,2a}(n)-\overline{U}_{2k+1,2a-2}(n) &=(xq)^{2a}U_{2k+1,2k-2a+1}(xq;q)+(xq)^{2a-2}U_{2k+1,2k-2a+3}(xq;q),
h_\mu\left(\alpha^W \right) = \inf_F \frac{1}{|F|}H_\mu\left(\alpha^{WF}\right) = \inf_F \frac{|WF|}{|F|}\left( \frac{1}{|WF|}H_\mu \left(\alpha^{WF}\right)\right) & \geq \inf_F \frac{|WF|}{|F|}h_\mu(\alpha) \geq r.
2^l(1-\kappa^2)^{j-l}\frac{C_l^{j-l+1/2}(\kappa)}{C_l^{j-l+1/2}(1)}= \frac{2^{j+1/2}(j-l)!}{\sqrt{\pi}\Gamma(j-l+1/2)}\int_0^{\arccos \kappa}[\cos(j+1/2)s](\cos s-\kappa)^{j-l}ds.
\langle x_v,h \rangle = \langle x_{v,0},x_{v,0}+x_{v,1}\rangle = p(0,0|v,v)+p(0,1|v,v) = p_A(0|v) = t, \|x_v\|^2 = \langle x_{v,0},x_{v,0}\rangle = \langle x_{v,0},h-x_{v,1}\rangle = \langle x_{v,0},h \rangle = t,
u(x,t)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu(k)}(r\rho)e^{it\rho^2}b^0_{k,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho,
\int_0^1\int_0^1\Big(1-\Gamma'(u+\sigma w)\cdot\Gamma'(u+\tau w)\Big)\,d\sigma d\tau=\frac 12\int_0^1\int_0^1|\Gamma'(u+\sigma w)-\Gamma'(u+\tau w)|^2\,d\sigma d\tau,
Minimize_{\mathbf{x}\in\mathbb{R}^{3n}}\ \ \sum\limits_{i=1}^n(a_J,a_J,b_{Ji})^T\mathbf{x}_i \sum\limits_{i=1}^n(a_k,a_k,b_{ki})^T\mathbf{x}_i+c_k\le0\ \ \forall\,k\in{1,...,m}\mathbf{x}_i\in\mathcal{L}_i
\omega =\omega^{r}e_{r}+\omega^{\theta}e_{\theta}+\omega^{z}e_{z}&=(-\partial_z u^{\theta})e_{r}+(\partial_{z}u^{r}-\partial_{r}u^{z})e_{\theta}+\Big(\partial_ru^{\theta}+\frac{u^{\theta}}{r}\Big)e_{z},
[M, N]_{\epsilon}=[E_{11}+E_{21},E_{12}+E_{22}]_{\epsilon}= [E_{11},E_{12}]_{\epsilon}+[E_{11},E_{22}]_{\epsilon}+[E_{21},E_{12}]_{\epsilon}+[E_{21},E_{22}]_{\epsilon}= [E_{21},E_{12}]_{\epsilon}=E_{21}.
\limsup \limits_{\Omega \ni z\to w}\Phi_{p}( z) \geq \limsup \limits_{\Omega \ni z\to w}\Phi_{p}( z) + \limsup \limits_{\Omega \ni z\to w}\log \frac{|K(z, p)|^2}{\mathcal C^2}= \limsup \limits_{\Omega \ni z\to w}\log \frac{K(z, z) K(p, p)}{\mathcal C^2}= \infty.
\sum_{k=0}^\infty\left[\binom{p}{k}\frac{(x-a)^{k-p}}{\Gamma(k-p+1)}\psi_1+\binom{q}{k}\frac{(x-a)^{k-q}}{\Gamma(k-q+1)}\psi_2\right]\frac{\partial^ku}{\partial x^k}=0.
{L}\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)=L_0\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=1}^M\lambda_j\,L_j\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=M+1}^N\lambda^0_j L_j\left(t,\textbf{x},\dot{\textbf{x}}\right),
\mathbb{G}_{M,\pi,m}(j_m\xi(x),j_m\eta(x))=\sum_{j=0}^{m}\mathbb{G}_{M,\pi}\left(\frac{1}{j!}D^j_{\nabla^M,\nabla^\pi}(\xi)(x),\frac{1}{j!}D^j_{\nabla^M,\nabla^\pi}(\eta)(x)\right).
\phi_\mu(e_1) = e_{\mu+1}-e_\mu, \phi_\mu(e_i) = e_{i-1}, 1< i \leq \mu, \phi_\mu(e_{\mu+1}) = e_{\mu+1}+ e_\mu, \phi_\mu(e_i) = e_i, \mu +1 < i \leq k,
\begin{cases}u_{tt}(x,t)-\Delta_x u(x,t)+bu_{t}(x,t)+mu(x,t)=f(u)(x,t), u(x,0)=u_{0}(x),\,\,\,\,x\in X, u_{t}(x,0)=u_{1}(x),\,\,\,\,x\in X, \end{cases}
\frac{1}{n_{j}}\sum_{m_{j}= 1}^{n_{j}}c o t h \frac{1}{2}K \bigl ( r \pm i ( t - c_{j}- 2 \pi k_{j}^{- 1}m_{j}) \bigr ) = c o t h \frac{1}{2}k_{j}\bigl ( r \pm i ( t - c_{j}) \bigr )
\begin{array}{rcl}\chi^{(p,kp+1)}_{1,k(r+1)+1}(q) = \displaystyle\sum_{n_1 \geq \cdots \geq n_{k}\geq 0}\frac{q^{n_1^2 +\cdots + n_{k-1}^2 + 2n_k^2 +r(n_1 + \cdots + n_{k-1}+2n_k )}}{(q)_{n_1 - n_2}\cdots (q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r}}\overline{F}^{(2n_k +r)}_{p-1;r}(q).\end{array}
\sum_{j=0}^n \frac{1}{(n-j)!}\frac{1}{j!}\mathrm{ad}(s)^j(b) s^{(n-j)}= \sum_{j=0}^n \frac{1}{(n-j)!}\frac{1}{j!}(L-R)^jR^{(n-j)}b = \frac{1}{n!}L^nb = \frac{1}{n!}s^nb.
\frac{1}{2}\int \frac{d k^{\prime}}{2 \pi}\frac{1}{\sqrt{k^{2}+ 4}}f ( k^{\prime}, x^{\prime}) f ( - k^{\prime}, x^{\prime}) = - \frac{\left( 1 - \coth^{2}2 ( x^{\prime}- x_{0}) \right)}{2 \pi}.
d\mu^X =\iota_X\omega = \omega(X)=\sum_{i=1}^3 dx_i\wedge dy_i(\sum_{j=1}^3a_j\dfrac{\partial}{\partial x_j}) =-\sum_{i=1}^3 dy_i\wedge dx_i (\sum_{j=1}^3a_j\dfrac{\partial}{\partial x_j}) =-\sum_{i=1}^3 a_idy_i.
\lim_{h\rightarrow 0^+}\frac{1}{2h}\int_{x-h}^{x+h}\vert f(t)-f(x)\vert dt= \lim_{h\rightarrow 0^+}\frac{1}{2h}\int_{x-h}^{x+h}dt-\frac{1}{2h}\int_{x-h}^{x+h}f(t)dt=0.
\begin{aligned}\sum_{m=-\infty}^{\infty}(q^{-\ell};q)_{m}w^{m}q^{\ell|m|/2}S_{\ell}\left(-q^{-|m|-\ell};q\right) =\frac{\left(1-q^{3\ell/2}/w\right)}{\left(q,w/q^{\ell/2};q\right)_{\ell}}\left(q^{1+\ell/2}/w;q\right)_{\infty}.\end{aligned}
\mathcal{B}(\tilde{f},\tilde{g},\tilde{h}) := \textup{p.v.}\int_{\mathbb{R}^3}\tilde{f}(x)\tilde{g}(y)\tilde{h}(z) \delta(ax+by+cz)\det\begin{pmatrix}1 1 1 a^{-1}b^{-1}c^{-1}x y z \end{pmatrix}^{-1}dxdydz
\tilde A_1\vert n_1,n_2\rangle_q = \sqrt{[n_1]_q}\,\vert n_1-1,n_2\rangle_q; \tilde A_2\vert n_1,n_2\rangle_q = \sqrt{[n_2]_q}\,\vert n_1,n_2-1\rangle_q \,.
(a_1\circ b)S(a_2)(a_3\circ c) =\pi(\pi^{-1}(a_1)\pi^{-1}(b))S(a_2)\pi(\pi^{-1}(a_3)\pi^{-1}(c)) &=a_1(\pi^{-1}(a_2)\rightharpoonup b)S(a_3)a_4(\pi^{-1}(a_5)\rightharpoonup c) &=a_1(\pi^{-1}(a_2)\rightharpoonup b)(\pi^{-1}(a_3)\rightharpoonup c)=a_1(\pi^{-1}(a_2)\rightharpoonup (bc)).
\sum_{n=-\infty}^{\infty}\omega^{2n}q^{n(n-1)}\left[\varphi_{k}\left(\omega q^{n}\right)\varphi_{\ell}\left(\omega q^{n}\right)+\omega^{2}q^{2n}\varphi_{k}\left(-\omega^{-1}q^{-n}\right)\varphi_{\ell}\left(-\omega^{-1}q^{-n}\right)\right] =2\|\varphi\left(\omega\right)\!\|^{2}\delta_{k,\ell},
\omega_{x}^{( 1 )}= - 9 6 \cdot 1 6 \pi^{2}{\frac{x^{2}m^{2}\, d m}{( x + 2 m )^{5}}}, \quad \quad \int_{0}^{\infty}\omega_{r}^{( 1 )}= - 1 6 \pi^{2}.
J_n \leq \sum_{k=1}^{n}\|f\|_\infty e^{k\|V\|_\infty}\|\mathcal{B}_n^V\textbf{1}\|_{\infty}~q^k \|u_0-u_0'\| +e^{n\|V\|_\infty}\|\nabla f\|_\infty q^n \|u_0-u_0'\|.
\psi = \left( \begin{array}{l}{{M_{1}\left( t \right) \, \, \, \, e^{i \, F_{1}\left( q , t \right)}}}\\{{M_{2}\left( t \right) \, \, \, \, e^{i \, F_{2}\left( q , t \right)}}}\\ \end{array}\right) . \,
\begin{cases}(\lambda\dot\lambda-\sigma) ( \Lambda_y{\sf Q}, \psi_2 )_{L_y^2(B_{4{\sf R}_{\sf in}})}+ \lambda^\frac{n-2}{2}( V (\eta^\frac{2}{1-q}\tilde v_1+\tilde w_1), \psi_2 )_{L_y^2(B_{4{\sf R}_{\sf in}})}= 0 t\in(0,T), \lambda(t)=0 t=T. \end{cases}
K_{(0,\hat{t})}[\varphi_\sigma](\hat{t},\hat{x}) =K_{(0,\rho)}[\varphi_\sigma](\hat{t},\hat{x})+K_{(\rho,\hat{t})}[\varphi_\sigma](\hat{t},\hat{x}) &\ge K_{(0,\rho)}[\varphi](\hat{t},\hat{x})+K_{(\rho,\hat{t})}[\varphi_\sigma](\hat{t},\hat{x})
\begin{array}{lllllllllll}0=\omega(C)-\omega(C) = |E_{g'=(0,0)}\cap E(C)|+|E(C)-E_{g'=(0,0)}|-\omega(C) &\le \frac{\omega(C)}{4}+\Omega+1+1+\Omega(7-1-2)-\omega(C) &=-2<0,\end{array}
\int \!{\cal D}\varphi \, \delta_{\eta}\phi{\frac{\delta S_{0}}{\delta \phi}}e^{- ( S_{0}+ S_{5})}= ( \delta_{\eta}^{( 0 )}\phi + \delta_{\eta}^{( 1 )}\phi ){\frac{\delta S_{0}[ \phi ]}{\delta \phi}}e^{- ( S_{0}[ \phi ] + S^{( 1 )}[ \phi ] )}
R(x,d,p_2)-R(x,d,p_1)\leq P(\beta<+\infty)=\sum_{l=1}^{+\infty}P(\beta=l) &=\sum_{l=1}^{+\infty}P(\beta>l-1,S_l(d,p_1)=\triangle,S_l(d,p_2)\neq\triangle) &=\sum_{l=1}^{\infty}P(S_{l-1}(d,p_1)\neq \triangle)(p_2-p_1) &=\sum_{l=1}^{\infty}p_1^{l-1}(p_2-p_1)=\frac{p_2-p_1}{1-p_1}.
\frac{M_{\beta}(s)}{M_{\beta}(s+1)}= \frac{\Gamma(1+a+\gamma s)\Gamma(1+b+\gamma s)\Gamma(1+\gamma(s-1))\Gamma(2+a+b+(s+1)\gamma)}{\Gamma(1+\gamma(s+1/2)+(a+b)/2)\Gamma(\gamma(s+1/2)+(a+b+3)/2)}& \times \frac{\pi}{2^{2(1+a+b)+(4s+1)\gamma}}\frac{1}{\Gamma(1+(a+b)/2+\gamma s)\Gamma((3+a+b)/2+\gamma s)}
s_\lambda(z_1,\dots,z_N)=\begin{vmatrix}e_{s}e_{s+1}e_{s+2}\cdots e_{s+d-1}e_{s+d}e_{s-1}e_{s}e_{s+1}\cdots e_{s+d-2}e_{s+d-1}\vdots \vdots \vdots \cdots \vdots \vdots e_{s-d+1}e_{s-d+2}e_{s-d+3}\cdots e_s e_{s+1}e_{s-d}e_{s-d+1}e_{s-d+2}\cdots e_{s-1}e_s \end{vmatrix}.
\psi_{( m_{1}, n_{1})}\otimes \psi_{( m_{2}, n_{2})}= \sum_{k = | m_{1}- m_{2}| + 1}^{m_{1}+ m_{2}- 1}\; \sum_{l = | n_{1}- n_{2}| + 1}^{n_{1}+ n_{2}- 1}\; \left[ \psi_{( k , l )}\right] ,
\frac{\prod\limits_{k=1}^{n_f}{\mathcal{Z}}\left(J,m_f^{(k)}\right)}{\prod\limits_{l=1}^{n_b}{\mathcal{Z}}\left(J,m_b^{(l)}\right)}=\mbox{const}\;\int{\mathcal{D}}{\mathcal{F}}{\mathcal{D}}{\mathcal{B}}\;\exp\left\{-{\mathcal{A}}_{\mathcal{F}}(J)-{\mathcal{A}}_{\mathcal{B}}(J)\right\}
\left[ \Delta_{3}+ \partial_{z}^{2}- \frac{1 5 k^{2}}{4 \left( k \left| z \right| + 1 \right)^{2}}+ 3 k \delta \left( z \right) \right] \bar{V}= G \mu \, \delta^{( 3 )}\left( x \right) \delta \left( z \right)
u_{2C}(v) = \frac{1-\frac{1}{1+\frac{v^2}{\theta^2}}}{v^{2h}}= \frac{1-\frac{\theta^2}{\theta^2+v^2}}{v^{2h}}= \frac{\frac{v^2}{\theta^2+v^2}}{v^{2h}}= \frac{v^{2-2h}}{\theta^2+v^2\xrightarrow{v\rightarrow 0+}\theta^2},
V_G(\phi_0)={1\over V}\sum_{n=0}^{\infty}\frac{(-i)^n}{n!}\int_{-\infty}^0d t_n ... \int_{-\infty}^0 d t_n \langle 0 | T(H_I(0)H_I(t_1) ... H_I(t_n))|0{\rangle}_{conn}
\phi(x,E)+{\rm e}^{\frac{2\pi}{3}i}\phi({\rm e}^{\frac{2\pi}{3}i}x,{\rm e}^{\frac{2\pi}{3}i}E) +{\rm e}^{-\frac{2\pi}{3}i}\phi({\rm e}^{-\frac{2\pi}{3}i}x,{\rm e}^{-\frac{2\pi}{3}i}E) =0.
=\frac{T-t}{2}\left(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{j=1}^{\infty}\frac{1}{\sqrt{4j^2-1}}\left(\zeta_{j-1}^{(i_1)}\zeta_{j}^{(i_2)}-\zeta_j^{(i_1)}\zeta_{j-1}^{(i_2)}\right)\right).
\Phi(x;s|-a,b,-q^{1/2}a,q^{1/2}a|q)=\sum_{l,m\geq 0}{(a^2;q)_m(q^l s;q)_m\over (q;q)_m(q^{l+1}s/a^2;q)_m}(q x^2/a^2)^m &\times{(b/a;q)_l(s^2/a^4;q)_l (s;q)_l\over (q;q)_l (qs^2/a^3b;q)_l (s/a^2;q)_l}(qx/b)^l.
( \theta^{i j}) = \left( \begin{array}{c c c c}{\theta_{1}{\bf \epsilon}}&{0}&{\dots}&{0}\\{0}&{\theta_{2}{\bf \epsilon}}&{\dots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{\dots}&{\theta_{p}{\bf \epsilon}}\\ \end{array}\right)
\mathbf{p}_n =\sum_{k\in\mathcal{C}_n}\mathbf{P}_k - \mathbf{P}_n \mathbf{q}_n =\sum_{k\in\mathcal{C}_n}\mathbf{Q}_k - \mathbf{Q}_n \mathbf{v}_{\pi_n}- \mathbf{v}_n = \real\left\{\bar{\mathbf{Z}}_n(\mathbf{P}_n+j\mathbf{Q}_n)\right\}
\mathfrak{C}=\left\{x\in \mathcal{O}_{\mathbb{P}^1_{K}, \infty}\ | \x.\mathcal{O}_{\mathbb{P}^1_{K}, \infty}\subset \mathcal{O}_{C_K, P_{\infty}}\right\}=\mathcal{C}_{P_\infty}.
\sum \limits_{i=1}^\infty \theta_{B_R(0)}^\beta(r_i) \geq \frac 1{4^\beta}\sum \limits_{i=1}^\infty \frac 1{- \log(\frac{r_1}4) + \log (C^{i-1})}\geq \frac 1{4^\beta}\sum \limits_{i=1}^\infty \frac 1{- \log(\frac{r_1}4) + (i-1)\log (C)}= \infty.
\|T_kf_j - T_kf_{j^\prime}\|_{L^r(K)}= \|T_k(f_j - f_{j^\prime})\|_{L^r(K)}& \leq Cq^{-k}\|\Omega\|_{H^1(\mathfrak{D}^*)}\|f_j - f_{j^\prime}\|_{L^r(K)},
S^{(\mu_1)}_{\mu k}\overline{S_{\nu \mu_1}S^{(\mu_1)}_{i k}}(B^S)^{(\mu_1)}q((y^S)^{(\mu_1)})=(S_{\mu k}\overline{S_{i k}}B^S q(y^S))^{(\mu_1)}\overline{S}_{\nu \mu_1}.
[ a_{m}, \tilde{a}_{n}] = [ a_{m}^{\dagger}, \tilde{a}_{n}^{\dagger}] = [ \psi_{\dot{\alpha}}, \tilde{\psi}_{\dot{\beta}}]_{+}= [ \psi_{\dot{\alpha}}^{\dagger}, \tilde{\psi}_{\dot{\beta}}^{\dagger}]_{+}= 0
\zeta_{i,i+1}\left(\frac{z}{w}\right)=\frac{d^{-1}z-qw}{z-w}\,, \zeta_{i,i-1}\left(\frac{z}{w}\right)=\frac{z-qd^{-1}w}{z-w}\,, \zeta_{ii}\left(\frac{z}{w}\right)=\frac{z-q^{-2}w}{z-w}\,, \zeta_{ij}\left(\frac{z}{w}\right)=1 \mathrm{if}j\ne i,i\pm 1 \,.
\begin{array}{l l}{{( A ) :}}&{{n > 0 ,}}\\{{( B ) :}}&{{( i ) \k > 0 , \ \mathrm{o r}\ ( i i ) \k = 0 , \l > 0 ,}}\\{{( C ) :}}&{{( i ) \k > 0 , \ \mathrm{o r}\ ( i i ) \k = 0 , \l > 0 , \ \mathrm{o r}\ ( i i i ) \k = l = 0 , \b < 0 \, ,}}\\ \end{array}
F_{2}^{\mathrm{c l a s s}}\left( \phi_{0}, \phi_{1}, \phi_{2}\right) = \left[ a_{0}\left( \phi_{1}\right) - a_{0}\left( \phi_{0}\right) \right] \left[ a_{1}\left( \phi_{2}\right) - a_{1}\left( \phi_{1}\right) \right]
\lim_{|\mathcal{P}| \rightarrow 0}\sum_{[u,v] \in \mathcal{P}}\nabla\eta(\phi_{u}) \cdot \int_u^v b(r,\phi_r)dr = \lim_{|\mathcal{P}| \rightarrow 0}\int_0^t \sum_{[u,v] \in \mathcal{P}}\nabla\eta(\phi_{u})1_{[u,v]}(r) \cdot b(r,\phi_r) dr = \int_0^t \nabla\eta(\phi_r) \cdot b(r,\phi_r) dr
\left( \begin{array}{c c}{I}&{- Z^{\dag}}\\{Z}&{I^{\prime}}\\ \end{array}\right)^{- 1}= \left( \begin{array}{c c}{( I + Z^{\dag}Z )^{- 1}}&{0}\\{0}&{( I^{\prime}+ Z Z^{\dag})^{- 1}}\\ \end{array}\right) \left( \begin{array}{c c}{I}&{Z^{\dag}}\\{- Z}&{I^{\prime}}\\ \end{array}\right) ,
\begin{aligned}X\,f_{n}^{(e,\nu)}=e (-1)^{n}q^{-(n+\nu+1/2)}\Big(f_{n}^{(e,\nu)}-(1-q^{-1}) f_{n+1}^{(e,\nu)}\Big), Y\, f_{n}^{(e,\nu)}=e(-1)^{n}q^{n+\nu+1/2}\,f_{n}^{(e,\nu)}, Z\,f_{n}^{(e,\nu)}=e(-1)^{n}\Big(q^{-(n+\nu+1/2)}\,f_{n}^{(e,\nu)}+(q^{1/2}+q^{-1/2})\,\rho_n \,f_{n-1}^{(e,\nu)}\Big). \end{aligned}
\hat\varphi G_x-\hat\varphi_xu_xG_{u_x}+2\hat\varphi_xfG_f+\hat\varphi_{xx}u_xfG_g=\tilde\varphi^{\hat\varphi}_{\tilde x\tilde x}\tilde u_{\tilde x}F-\tilde\psi^{\hat\varphi}_{\tilde x\tilde x}F,
\mathcal{G}_{I}\left({{x}_{1}},{{x}_{2}}\right) ={{\mathcal{G}}^{\left(0\right)}}\left({{x}_{1}},{{x}_{2}}\right) +\int dz_{1}dz_{2}{{\mathcal{G}}^{\left( 0\right)}}\left({{x}_{1}},{z_{1}}\right) \Sigma^{\left( 1\right)}\left({z_{1}},{z_{2}}\right)_{2}{{\mathcal{G}}^{\left( 0\right)}}\left({z_{2}},x{_{2}}\right) \;,
A(x)=\sum_{k=0}^{\lfloor x\rfloor}a_k, B(x)=\sum_{k=0}^{\lfloor x\rfloor}b_k, &C(x)=\sum_{k=0}^{\lfloor x\rfloor}c_k, D(x)=\sum_{k=0}^{\lfloor x\rfloor}s_k, x\geqslant0.
q^5 (1+\beta)~\mathfrak{G}_{w}^{(\beta)}\big(1,q,q^2,q^3\big)= \operatorname{DET}\left |\begin{matrix}S_4(q;\beta) S_3(q;\beta) S_3(q; q \beta) S_2(q; q \beta)\end{matrix}\right | .
\mathcal{T}_{L}(t_n,x_i,y_j,\Delta t, \Delta x, \Delta y) = \frac{\Delta t}{2}\partial_{tt}f^{\epsilon}- a \frac{\Delta x}{2}\partial_{xx}f^{\epsilon}- b \frac{\Delta y}{2}\partial_{yy}q^{\epsilon}+ b \Delta t \partial_{yt}q^{\epsilon}+ \mathcal{O}(\Delta t^2) + \mathcal{O}(\Delta x^2) + \mathcal{O}(\Delta y^2).
S_{i_{1}i_{2}}^{k_{1}k_{2}}\left(\theta_{12}\right)S_{k_{1}k_{3}}^{j_{1}j_{3}}\left(\theta_{13}\right)S_{k_{2}i_{3}}^{j_{2}k_{3}}\left(\theta_{23}\right)=S_{i_{1}i_{3}}^{k_{1}k_{3}}\left(\theta_{13}\right)S_{k_{1}k_{2}}^{j_{1}j_{2}}\left(\theta_{12}\right)S_{i_{2}k_{3}}^{k_{2}j_{3}}\left(\theta_{23}\right)\,,