harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(F_{B_R}g)(\hat{x}) =^{-i\frac{\pi}{4}}\int_{\mathtt{S}}\left\{\sqrt{\frac{k_p}{\omega}}u^{\infty}_{B_R}(\hat{x}; d, 1, 0)g_p(d) + \sqrt{\frac{k_s}{\omega}}u^{\infty}_{B_R}(\hat{x}; d, 0, 1)g_s(d)\right\}ds(d).
\sum_{m=m_1+1}^{+\infty}|S_n(f_m,t_l)|\leq\sum_{m=m_1+1}^{+\infty}\sum_{j=1}^{+\infty}\epsilon_{m,j}<\frac{\delta_1}3, &1\leq l\leq m_1,\,\,n\in\bigcup_{j=1}^{+\infty}\mathcal B_{m_1,j}.
\displaystyle\frac{2\nu^{2}(1-\nu^{2})}{(\mu-\lambda)^{2}}+2c\nu^{2}+\lambda\mu+\displaystyle\frac{c(b_{1}^{2}\lambda-b_{2}^{2}\mu)}{\mu-\lambda}-\displaystyle\frac{2(\lambda-\gamma)}{\mu-\gamma}\sum_{j=3}^{n}\langle\nabla_{X_{2}}X_{1},X_{j}\rangle^{2}=0.
\Gamma_1 =(\zeta^4+\zeta^3)x^2+(\zeta^5+\zeta^2)(\zeta^5+\zeta^2+1)xy+(\zeta^5+\zeta^2)y^2 +(-\zeta^5-\zeta^2+1)xz+(-\zeta^4-\zeta^3+1)yz+(\zeta^5+\zeta^2)(\zeta^4+\zeta^3-1)z^2,
A_1= r^2J(r)^{-2}\Delta^h+\beta \tfrac Jn r^2J(r)^{-1}r\partial_r , A_2= 2(1+\tfrac{J}{2n}r^2)J(r)^{-1}d, A_3= 2r^2(1+\tfrac{J}{2n}r^2)J(r)^{-3}\delta^h, A_4= r^2 J(r)^{-2}\Delta^h+(\beta+2)\tfrac Jn r^2 J(r)^{-2}(2+J(r)r\partial_r).
\tilde{r}_{a}(\Delta,h)=\lim_{\stackrel{N\to\infty}{h\;\mathrm{fixed}}}\frac{\tilde{m}_{a}}{\tilde{m}_{F}}=\lim_{\stackrel{N\to\infty}{h\;\mathrm{fixed}}}\frac{\tilde{G}_{a}}{\tilde{G}_{F}}\, ,\qquad a\in\{B1,B2,C\}
c_n = \frac{1}{2}\sqrt{\frac{k_n}{\kappa_j}}\left(1+\frac{\kappa_j}{k_n}\right) Y_{j,n}\qquad\mbox{and}d_n = \frac{1}{2}\sqrt{\frac{k_n}{\kappa_j}}\left(1-\frac{\kappa_j}{k_n}\right) Y_{j,n},
\begin{aligned}\int_X (\beta + \varepsilon \omega)^n - \varepsilon B \|\psi_\varepsilon\|_{L^\infty}^n C \leq \int_X (\beta + \varepsilon \omega + dd^c \psi_\varepsilon)^n &\leq\int_X (\beta+ \varepsilon \omega)^n + \varepsilon B\|\psi_\varepsilon\|_{L^\infty}^n C.\end{aligned}
A(\theta) =m_A{\rm e}^{\theta}- \ln 2 \cos \frac{\pi}{k+2}-\int \frac{d\theta'}{2\pi}\frac{1}{\cosh(\theta-\theta')}\ln(1+ B(\theta')^2), B(\theta) =m_B{\rm e}^{\theta}-\int \frac{d\theta'}{2\pi}\frac{1}{\cosh(\theta-\theta')}{\rm e}^{-A(\theta')}
0<\kappa:=K(0)L^{-\frac q2}(0)\le K(t)L^{-\frac q2}(t)\le \frac 1qL'(t)L^{-\frac q2}(t)=\frac{2}{(2-q)q}\Big(L^{\frac{2-q}{2}}(t)\Big)'.
\left|\frac{\sin (\alpha \omega)}{\tanh(d\, \omega)}\right|=\left|\frac{\sin (\alpha \omega)}{\omega}\right|\left|\frac{\omega}{\tanh(d\, \omega)}\right| & \leq \alpha\, \frac{\pi/(2 \alpha)}{\tanh(\pi d/(2 \alpha))}= \frac{\pi}{2 \tanh(\pi d/(2 \alpha))}.
| x^{\frac{1}{n}}- c^{\frac{1}{n}}| = \frac{| x^{\frac{1}{n}}- c^{\frac{1}{n}}| | x^{\frac{n - 1}{n}}+ x^{\frac{n - 2}{n}}c^{\frac{1}{n}}+ \cdots + x^{\frac{1}{n}}c^{\frac{n - 2}{n}}|}{| x^{\frac{n - 1}{n}}+ x^{\frac{n - 2}{n}}c^{\frac{1}{n}}+ \cdots + x^{\frac{1}{n}}c^{\frac{n - 2}{n}}+ c^{\frac{n - 1}{n}}|}
\sum_{R=0}^{N-1}T(R) \ge \sum_{R=1}^{N}\left( \frac{r(R)}{2}+1\right) + O\left( \operatorname{nz}(N)\right) = \frac{1}{2}\sum_{R=1}^N r(R) + N + O\left( \operatorname{nz}(N)\right).
\frac{{P}^{\xi_t, J_t}({\tau}^-_0>{\mathbf{e}_q})}{{P}^{x,i}({\tau}^-_0>{\mathbf{e}_q})}=\frac{{}^{q}U^-_{J_t}(\xi_t)}{{}^{q}U^-_{i}(x)}\leq \frac{{}^{q_0}U^-_{J_t}(\xi_t)}{U^-_{i}(x)}.
\int_{\Omega_{T}}-u^{\varepsilon}\left( x,t\right) v\left( x\right)\partial_{t}c\left( t\right) +a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon}\left( x,t\right) \right)\cdot \nabla v\left( x\right) c\left( t\right) dxdt =\int_{\Omega_{T}}f\left( x,t\right) v\left( x\right) c\left( t\right) dxdt
{{\bar{R}}_{\mu\nu\lambda}}^{\;\;\;\;\;\;\kappa}= \partial_\mu{\bar{\Gamma}}_{\nu\lambda}^{\;\;\;\;\kappa}- \partial_\nu{\bar{\Gamma}}_{\mu\lambda}^{\;\;\;\;\kappa}- [\bar{\Gamma}_\mu , \bar{\Gamma}_\nu{]_\lambda}^\kappa.
\widehat{(\Delta_{1})}=\frac{8}{p^{2}}\bigg[\frac{1}{n(n-1)}\{f^{(1)}(\widehat{\tau_{1}})\}^{2}\widehat{(\Sigma_{1}^{2})}+\frac{1}{m(m-1)}\{f^{(1)}(\widehat{\tau_{2}})\}^{2}\widehat{(\Sigma_{2}^{2})}+\frac{2}{nm}\{f^{(1)}(\widehat{\tau_{3}})\}^{2}\widehat{(\Sigma_{1}\Sigma_{2})}\bigg],
(x,u) = \lim_{k\to\infty}\Big( t_k x_k + \sum_{j=1}^{n_k}\big(\alpha^j_k e^j - \beta^j_k e^j\big), \frac{1}{2}t_k v_k + \sum_{j=1}^{n_k}\big(\alpha^j_k f^j_k - \beta^j_k g^j_k\big) \Big)
\begin{array}{r c l}{{f e}}&{{=}}&{{\langle e , S ( f ) \rangle \langle 1 ,{\bar{t}}^{- 1}\rangle 1 \cdot \bar{t}^{- 1}}}\\{{}}&{{}}&{{+ \langle t , S ( 1 ) \rangle \langle 1 , \bar{t}^{- 1}\rangle e f + \langle t , S ( 1 ) \rangle \langle e , f \rangle t \cdot 1}}\\{{}}&{{=}}&{{- \langle e , f \rangle{\bar{t}}^{- 1}+ e f + \langle e , f \rangle t ~ .}}\\ \end{array}
\begin{pmatrix}A B C D \end{pmatrix}= \begin{pmatrix}I_n BD^{-1}0 I_m \end{pmatrix}\begin{pmatrix}A - BD^{-1}C 0 0 D \end{pmatrix}\begin{pmatrix}I_n 0 D^{-1}C 0 \end{pmatrix}.
\begin{aligned}f(\rho) = \rho(\rho+1)\cdots(\rho+5) -p_{51}\rho(\rho+1)\cdots(\rho+4) +p_{42}\rho(\rho+1)\cdots(\rho+3) -p_{33}\rho(\rho+1)(\rho+2) +p_{22}\rho(\rho+1) -p_{11}\rho +p_0 \end{aligned}
|\int_{A_n}g\cdot f\;\mu-\int_{A_n}f\;\mu|\leqslant\int_{g^{-1}A_n\Delta A_n}|f|\;\mu =\int_{(g^{-1}A_n\Delta A_n)\cap A}|f|\;\mu &\leqslant\|f\|_{\infty}\;\mu((g^{-1}A_n\Delta A_n)\cap A).
\begin{cases}\nabla\times(\epsilon^{-1}\nabla\widetilde{H_{\eta,\ell}})-k^{2}\mu\widetilde{H_{\eta,\ell}}=-\nabla\times((\epsilon^{-1}(x)-\epsilon_{0}^{-1}(x))\nabla\times H_{\eta,\ell})\mbox{in}\Omega, \nu\times\widetilde{H_{\eta,\ell}}=0\mbox{on}\partial\Omega.\end{cases}
u \in\L_t\big(x_1\big(\textstyle\prod_{s=\ell}^{d-2}x_{n-st-1}\big)\big(\prod_{s=0}^{\ell-1}x_{n-st}\big),x_1\big(\prod_{s=0}^{d-2}x_{n-st}\big)\big), v =\big(\textstyle\prod_{s=\ell}^{d-1}x_{n-st-1}\big)\big(\prod_{s=0}^{\ell-1}x_{n-st}\big).
F_{\mu}= \delta_{a}^{b^{\prime}}\delta_{a^{\prime}}^{b}\int \bigg \lbrack e^{i q \cdot ( z - y )}- e^{i q \cdot ( z - x )}\bigg \rbrack{\frac{( x_{\mu}- y_{\mu})}{q \cdot ( x - y )}}{\frac{d^{4}q}{( 2 \pi )^{4}}}+ F_{\mu}^{T},
\int_{0}^{T}{\bf c}^{\top}(t)\widehat{\bf w}(t)dt-\sum_{u=1}^{N}(t_{u}-t_{u-1})({\bf c}^{(u)})^{\top}{\bf w}^{(u)}\leq Np\bar{\epsilon}\parallel{\cal P}\parallel\frac{\tau}{\sigma}\cdot\exp\left (\frac{\eta T}{\sigma}\right ) & \leq \kappa Tp\bar{\epsilon}\cdot\frac{\tau}{\sigma}\cdot\exp\left (\frac{\eta T}{\sigma}\right )\mbox{(by (\ref{*clpeq74}))}
\begin{cases}nS^{m}-T^{m}=S^{m-k}U^{k}, nS^{m}+T^{m}=S^{m-k}V^{k},\end{cases}k\le m,\begin{cases}nS^{k}-S^{k-m}T^{m}=U^{k}, nS^{k}+S^{k-m}T^{m}=V^{k},\end{cases}k\ge m,
(\frac{u^{n+1}_{h}- u^{n}_{h}}{\Delta t},v_{h}) + b(u^{n}_{h},u^{n+1}_{h},v_{h}) + \nu (u^{n+1}_{h}, v_{h}) - (p^{n+1}_{h},\nabla \cdot v_{h}) = (f^{n+1},v_{h}) \; \; \forall v_{h}\in X_{h}, (\nabla \cdot u^{n+1}_{h}, q_{h}) \; \; \forall q_{h}\in Q_{h}.
\int \varphi^{k+t}(A)d\mu_n(A) = \int \sigma_1(A)\cdots \sigma_k(A)\sigma_{k+1}(A)^{t}d\mu_n(A) & =\int \varphi^{k+t_1}(A)^{\lambda}\varphi^{k+t_2}(A)^{1-\lambda}d\mu_n(A) &\leq \left(\int \varphi^{k+t_1}(A)d\mu_n(A)\right)^{\lambda}\left(\int \varphi^{k+t_2}(A)d\mu_n(A)\right)^{1-\lambda}
\begin{array}{c}\displaystyle{\cal A}_{S,x_0}(v(\xi))\cdot v(\xi)=-\sum_{j,k=1}\hat{\nu}(\sigma)\cdot\frac{\partial^2 s}{\partial\sigma_j\partial\sigma_k}(\sigma)\xi_j\xi_k \displaystyle=(A\xi,\xi)_{{\Bbb R}^2}\end{array}
\widetilde{F}^\delta\begin{pmatrix}z x \end{pmatrix}= \begin{pmatrix}z+ z^2 + a_3^\delta z^3 + a_4^\delta(x)z^4 +\dots b_0(x) + b_1^\delta(x)z+ b_2^\delta(x)z^2 +\dots \end{pmatrix},
\int_{X_{\epsilon}}\partial_{z}\phi ~ \partial_{\bar{z}}\phi ~ \frac{i d z \wedge d \bar{z}}{2}= \int ( \partial_{z}\phi_{M}\partial_{\bar{z}}\phi_{M}- 2 \alpha_{1}\phi_{M}\partial_{z}\partial_{\bar{z}}\phi_{B}) \frac{i d z \wedge d \bar{z}}{2}+
+ 1 3 4 6 4 4 4 3 1 6 0 a^{3}N^{4}- 2 6 9 2 8 8 8 6 3 2 a^{3}N^{5}- 2 0 8 0 8 6 2 2 8 0 0 a^{5}N + 1 1 8 5 2 1 4 0 5 0 0 a^{5}N^{2}
\mathbf{I}^{\dagger}( D_{\theta^{x}}) = - \mathbf{I}( D_{\theta^{x}}) \mathcal{R}_{x}\# , \qquad \mathbf{I}( D_{\theta^{x}}) = - \mathbf{I}^{\dagger}( D_{\theta^{x}}) \mathcal{R}_{x}^{- 1}\# .
\widehat{H}=\frac{2\pi}{L}\left( \widehat{L_{0}}+\widehat{\overline{L}_{0}}-\frac{c}{12}\widehat{I}+g\frac{L^{2-2\Delta_{V}}}{\left( 2\pi \right)^{1-2\Delta_{V}}}\widehat{B}\right) \: ,
\mu_{\boldsymbol\beta}\left( \mathbf{x}\right) = \left( \begin{array}{cccc}x^1_1 x^1_2 \ldots x^1_M x^2_1 x^2_2 \ldots x^2_M \vdots \vdots \ddots \vdots x^M_1 x^M_2 \ldots x^M_M \end{array}\right),
\|\widetilde{E}_{1,m-1}-\widetilde{E}_{1,m}\|_{C^{\ell}[0,1]}\leq |(C_m,S_m)-b_m(V_m^{(1)},V_m^{(2)})|(1+\log r_m)^\ell =b_m\left|\frac{1}{\sqrt{\sum_{j=1}^{r_{m+1}-r_m}c_{j,m}}}\sum_{j=1}^{r_{m+1}-r_m}H_{j,m}-(V_m^{(1)},V_m^{(2)})\right|(1+\log r_m)^\ell.
\tilde{A}_{\mu \nu}( k ) \sim \frac{\tilde{B}_{A}}{( k^{2})^{3 / 2 - \beta_{A}}}\left[ \eta_{\mu \nu}- ( 1 - \rho ) \frac{k_{\mu}k_{\nu}}{k^{2}}+ \vartheta \frac{\epsilon_{\mu \nu \sigma}k^{\sigma}}{( k^{2})^{1 / 2}}\right]
\left(\frac{1}{\sqrt{2\pi}}\right)^2 \int_{S_0}e^{-i[q^2 x^2 + q^3 x^3]}F_\alpha\left(\hat{H}(\rho), \hat{P}(\rho), x^2, x^3 \right)\dx^2dx^3 \otimes \rho(E^\alpha) = \hat{F}_\alpha\left(\hat{H}(\rho), \hat{P}(\rho), q^2, q^3 \right) \otimes \rho(E^\alpha).
\tilde{k}_{\mu, \nu}^{\lambda}=\sum_{\sigma^{(1)}, \sigma^{(2)}, \sigma^{(3)}\in \mathcal{P}}\sum_{\rho^{(1)}, \rho^{(2)}, \rho^{(3)}\in \mathcal{P}}k_{\sigma^{(2)}, \sigma^{(3)}}^{\sigma^{(1)}}c_{\sigma^{(1)},\rho^{(2)}, \rho^{(3)}}^{\lambda}c_{\rho^{(1)}, \sigma^{(2)},\rho^{(3)}}^{\mu}c_{\rho^{(1)}, \rho^{(2)},\sigma^{(3)}}^{\nu}
R_1 = \sum\limits_{\alpha,\beta}\left(\sum\limits_{i,j}h_{ij\alpha}h_{ij\beta}\right)^2+\sum\limits_{i,j,\alpha,\beta}\left(\sum\limits_p h_{ip\alpha}h_{jp\beta}-h_{jp\alpha}h_{ip\beta}\right)^2 begin{align*}5pt] R_2 = \sum\limits_{i,j}\left(\sum\limits_{\alpha}H_{\alpha}h_{ij\alpha}\right)^2.
\textbf{I}= \sum_{k=1}^T \langle \lambda_k F(x_k), x_0 - x\rangle = \sum_{k=1}^T \langle s_{k-1}- s_k, x_0 - x\rangle = \langle s_0 - s_T, x_0 - x\rangle = \langle s_T, x - x_0\rangle.
\psi_{25}(m) = \sum_{p_1,p_2,p_3}\bigg\{\psi(m/(p_1p_2p_3), z) - \sum_{z \le p_4 < p_3}\psi(m/(p_1 \cdots p_4), z) \qquad\qquad\qquad\qquad+ \sum_{z \le p_5 < p_4 < p_3}\psi(m/(p_1 \cdots p_5), p_5) \bigg\}= \psi_{27}(m) - \psi_{28}(m) + \psi_{29}(m), .
\lambda_2^{(k)}= \gamma (2k-1), \phi_0^{(k)}(\sigma) = \frac{\gamma^{\frac 14}\, e^{-\gamma \sigma^2/2}\, H_{k-1}(\gamma^{\frac 12}\sigma)}{\pi^{\frac 14}\sqrt{2^{k-1}\, (k-1)!}}\, (k = 1,2,\ldots),
\sum_{a=0}^{b-1}{\rm id}_l(a)a = \sum_{a=0}^{b-1}(a \oplus l)a = \sum_{a=0}^{b-l-1}(a+l)a+\sum_{a=b-l}^{b-1}(a+l-b)a = \sum_{a=0}^{b-1}(a+l)a-b\sum_{a=b-l}^{b-1}a=\frac{b}{6}(1+2b^2+3l^2-3b(1+l)).
\| \nabla u(t,\cdot)\|_{L^2}= \||\xi| \widehat u(t,\cdot)\|_{L^2}\le \| U(t,\cdot)\|_{L^2}\lesssim \frac1{\lambda(t)}\big( \|u_0\|_{H^1}+ \|u_1\|_{L^2}\big) \| (1+t)^{-1}u(t,\cdot)\|_{L^2}\le \| U(t,\cdot)\|_{L^2}\lesssim \frac1{\lambda(t)}\big( \|u_0\|_{H^1}+ \|u_1\|_{L^2}\big)
G \left( z , \overline{{{z}}}; z^{\prime}, \overline{{{z^{\prime}}}}\right) = - \ln{\left| z - z^{\prime}\right|}= -{\frac{1}{2}}\ln{\left( z - z^{\prime}\right) \left( \overline{{z}}- \overline{{{z^{\prime}}}}\right)}\; .
\sum_{k=1}^{n-2}(n-2) p^{n-3}(p-1) (n-k-1) = (n-2)p^{n-3}(p-1) \left((n-2) + (n-3) + \ldots + 1\right) &= (n-2) p^{n-3}(p-1) \binom{n-1}{2}.
\sum_{\substack{\overline{\gamma}\in \overline{\Gamma}\ell(\gamma) \geq \ell_{0}(\widetilde{X})}}e^{- \ell(\gamma)\mathrm{Re}(s)}= \mathrm{Re}(s)\int_{\ell_{0}(\widetilde{X})}^{\infty}e^{-\mathrm{Re}(s)x}\Pi_{\Gamma}(x)dx \leq C \frac{\mathrm{Re}(s) e^{-(\mathrm{Re}(s)-\delta)\ell_{0}(\widetilde{X})}}{\mathrm{Re}(s)-\delta}.
\frac{n+\alpha+\beta}{n+\alpha}\phi_n^{(\alpha,\beta)}+A \phi_{n-1}^{(\alpha,\beta)}=\frac{(1+A)n+\alpha+\beta}{n+\alpha}\phi_n^{(\alpha,\beta-1)}+2\mu\frac{1-A}{n+\alpha}\left( \phi_n^{(\alpha,\beta-1)}\right)'.
Z_f(s, \chi, D) = q^{-ls}\int_D\chi(ac (\pi^{m-l}x^p+\alpha y^nz^l+\beta\pi^nz^{n+l}))|\pi^{m-l}x^p+\alpha y^nz^l+\beta\pi^nz^{n+l}|^s|dxdydz|.
\Phi_i= \Biggl\{\overrightarrow{\prod_{j \in I}}\Biggl\{\overrightarrow{\prod_{i \in I_{+}^{c}\atop i < j}}~u_{ij}^{-1}\Biggr\}\Biggr\}\Biggl\{\overrightarrow{\prod_{j \in I_{+}^{c}}}\Biggl\{\overrightarrow{\prod_{i \in I \atop i < j}}u_{ij}\Biggr\}\Biggr\}.
P_{\vec{\sigma}, a, \vec{z}}(\eta,\theta) = \frac{e^{\sigma_1\eta}e^{\sigma_2\theta}\exp((1-a)(e^\eta + e^\theta))}{f_{\vec{\sigma},a,\vec{z}}(\vec{0}) (\exp(e^\theta)-z_2) (\exp(e^\eta + e^\theta)-z_1)}, \eta,\theta \in{\mathbb{R}}.
(\Delta_{\{1,2\}}(V), \Delta_{\{1,3\}}(V), \Delta_{\{1,4\}}(V)) = (2, 1, 4), & (\Delta_{\{1,2\}}(V), \Delta_{\{2,3\}}(V), \Delta_{\{2,4\}}(V)) = (2, 4, -6), & (\Delta_{\{1,3\}}(V), \Delta_{\{2,3\}}(V), \Delta_{\{3,4\}}(V)) = (1, 4, -11), & (\Delta_{\{1,4\}}(V), \Delta_{\{2,4\}}(V), \Delta_{\{3,4\}}(V)) = (4, -6, -11)
\begin{array}{c}{{a_{0}+ a_{1}^{\prime}- a_{1}^{\prime \prime}+ a_{1}^{\prime \prime \prime}= a_{1}+ a_{0}^{\prime}- a_{2}^{\prime \prime}+ a_{2}^{\prime \prime \prime}= 0 ,}}\\{{a_{2}- a_{2}^{\prime}+ a_{0}^{\prime \prime}+ a_{3}^{\prime \prime \prime}= a_{3}- a_{3}^{\prime}+ a_{3}^{\prime \prime}+ a_{0}^{\prime \prime \prime}= 0 ,}}\\ \end{array}
\left\langle \pi_\nu(a)v,\tilde{v}\right\rangle = \sum_{\Theta\subseteq \Delta,\underline{p'},\underline{p},\underline{\chi}}\chi_{A_\Theta^-\left(\varepsilon\right)}(a)\alpha_{\underline{p'},\underline{p},\underline{\chi}}\prod_{i=1}^{r_\Theta}\chi_i(a_i)\left|a_{i}\right|^{p'_{i}}\log_q^{p_{i}}\left|a_{i}\right|,
\left\{A,B\right\}= \left(\frac{\partial A}{\partial q^{i}}\frac{\partial B}{\partial p_{i}}+ \frac{\partial A}{\partial p_{i}}\frac{\partial B}{\partial q^{i}}\right)+ (-1)^{\sigma(A)}\left(\frac{\partial A}{\partial \theta^{i}}\frac{\partial B}{\partial \pi_{i}}+ \frac{\partial A}{\partial\pi^{i}}\frac{\partial B}{\partial\theta_{i}}\right)\ .
\Gamma_{ij}^{3}=\begin{cases}-\tfrac14{x_1 x_2}L, (i,j)=(1,1), \tfrac18 ({x_{1}^{2}-x_{2}^{2}}) L, (i,j)\in \{(1,2),(2,1)\}, -\tfrac14{x_1}L, (i,j)\in \{(1,3),(3,1)\}, \tfrac14{x_1 x_2}L, (i,j)=(2,2), -\tfrac14{x_2}L, (i,j) \in \{(2,3), (3,2)\}, 0, (i,j)=(3,3). \end{cases}
d^-Y (t) = f'(X(t))\alpha(t)dt+\int_{\mathbb{R}_0}\big(f(X(t^-) + \theta(t, z))-f(X(t^-))-f'(X(t^-))\theta(t, z)\big)\nu(dz)dt &+\int_{\mathbb{R}_0}\big(f(X(t^-) + \theta(t, z))-f(X(t^-))\big)\tilde{N}(d^-t, dz).
\beta_k(t) = \frac 1{2^{n+k-2}\pi^{\frac{n-1}2}\Gamma(\frac{n+2k-1}2)}t^{k-1}\left(-\frac d{dt}\right)^{n+k-1}\left(t^{-k+1}(1-t^2)_+^{\frac{n+2k-3}2}\right) dt.
1_{\phi^{-1}(\mathbb{R}_{> t})}^* = 1_{(\phi^*)^{-1}(\mathbb{R}_{> t})}= 1_{( - \mu ( \phi^{-1}( \mathbb{R}_{> t}) ) / 2 , \mu ( \phi^{-1}( \mathbb{R}_{> t}) ) / 2 )}
q^{-2}L_{5}(q^2) = 2\sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{3(2n)^2 - (2j)^2}+ 2\sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{3(2n+1)^2 - (2j+1)^2}.
\left\langle x u_1\,\bigg|\,\frac{(h_{1}-e_{1})}{(-i\alpha^2(h_{1}-e_{1})+w)}+ \frac{(h_{1}-e_{1})}{(i\alpha^2(h_{1}-e_{1})+w)}\,\bigg|\, x u_{1}\right\rangle = \left\langle x u_1\,\bigg|\,\frac{2 w (h_{1}-e_{1})}{(\alpha^4(h_{1}-e_{1})^2+w^2)}\,\bigg|\, x u_{1}\right\rangle
(\partial^{2(p+1)}_y \tilde{u}_0)(x, 0)=\sum^p_{q=2}\sum_{(\alpha, \beta)\in \Lambda_q}C_{\alpha,\beta}\prod\limits_{j=1}^q \partial_x^{\alpha_j}\partial_y^{\beta_j +1}\big( u^s_0 + \tilde{u}_0 \big)\big|_{y=0}\,,\, \,\,\forall x\in \mathbb{R},
\mathfrak{W}\cong \mathfrak{W}^{\otimes r}\otimes \mathbb{C}[x]^{\otimes (n-2r)}\gamma_Q = \gamma_{Q(\alpha_1)}\otimes\cdots \otimes\gamma_{Q(\alpha_r)}\otimes (\gamma_1)^{\otimes (p-2r)}\otimes (\gamma_0)^{\otimes (n-p)}.
\left\{\begin{array}{c}{( m - 1 ) \partial_{A}\partial_{B}x^{j}+ L^{i}{}_{j k}( x , y ) \partial_{A}x^{i}\partial_{B}x^{j}= 0}\\{( n - 1 ) \partial_{A}\partial_{B}y^{b}+ C^{a}{}_{b c}( x , y ) \partial_{A}y^{a}\partial_{B}y^{b}= 0}\\ \end{array}\right\}
\left(\sum_{n=0}^{\infty}\lambda_nP_n\right)^m=\sum_{n=0}^{\infty}\lambda_n^mP_n \quad \mbox{so that}\quad \frac{\partial V}{\partial\phi}\left(\sum_n\lambda_nP_n\right)=\sum_nP_n\frac{\partial V}{\partial\phi}(\lambda_n)=0.
0 = -\frac{d}{dz}\int_{-1}^1 \mu[p_1\psi_{1,1}(z,\mu)+p_2\psi_{2,1}(z,\mu)]d\mu & -\int_{-1}^1 [p_1\sigma_{a1}\psi_{1,0}(z,\mu)+p_2\sigma_{a2}\psi_{2,0}(z,\mu)]d\mu + (p_1q_1+p_2q_2).
(i)I\nabla_\lambda T-I!_\lambda T=\lambda(1-\lambda)(I-T)(T\nabla_\lambda I)^{-1}(I-T). &(ii)T^{\frac{1}{2}}\big(I\nabla_\lambda T^{-1}-I!_\lambda T^{-1}\big)T^{\frac{1}{2}}=\lambda(1-\lambda)(T-I)(I\nabla_\lambda T)^{-1}(T-I). &(iii)\quad(I!_\lambda T)^{-\frac{1}{2}}(I\nabla_\lambda T)(I!_\lambda T)^{-\frac{1}{2}}-I=\lambda(1-\lambda)(I-T)T^{-1}(I-T)
g_1 \in \mathcal{C}([0, \infty [ \, ) \mbox{or}g_1\in \mathcal{C}( \, ]m_{g1}, \infty [ \, ) , int_{m_{g1}}^{\overline{m}_a+1}g_1(m) dm = \infty , \ \m_{g1}\in [0, \overline{m}_a],
S_{-1,1,0}^{2n+1}=T_1H^{n+1}=T_{-1}S_1^{n+1}\subset{\mathbf{R}}_1^{n+2}\times{\mathbf{R}}_1^{n+2}, S_{-1}^{n+1}=H^{n+1}\colon \ ,S_1^{n+1}\colon \ ,
( \nu_{\alpha \beta}^{g_{2}}) \, g_{2}^{*}( \nu_{\alpha \beta}^{g_{1}}) \: = \: ( \nu_{\alpha \beta}^{g_{1}g_{2}}) \, h ( g_{1}, g_{2})_{\alpha}\, h ( g_{1}, g_{2})_{\beta}^{- 1}
\partial^2_\mu \Theta^i(\mu,\sigma) = 0, \partial_\mu \partial_\sigma \Theta^i(\mu,\sigma) = - 2 \sigma^{-1}\partial_\mu \Theta^i(\mu,\sigma), \partial^2_\sigma \Theta^i(\mu,\sigma) = - 3 \sigma^{-1}\partial_\sigma \Theta^i(\mu,\sigma).
{{y_{F}=}}\sqrt{P_F \beta_{F,k^*}}{\bf h}^H_{F,k^*}{\bf w}_{i^*}s_{F,k^*}+ \sqrt{P_M \beta_{M,k^*}}{\bf h}^H_{M,k^*}{\bf f}_{j^*}s_{M,k^*}+n_F
\sum_{j=1}^n [M_A (v)]_j = \sum_{j \in I_A}[M_A (v)]_j + \sum_{j \in J_A}[M_A (v)]_j &=\sum_{j \in I_A}v_j + \sum_{j \in J_A}\left( - \sum_{l \in a (j) \setminus \{j \}}v_l \right) &=\sum_{j \in I_A}v_j - \sum_{j \in J_A}\sum_{l \in a (j) \setminus \{j \}}v_l &\overset{(*)}{=}\sum_{j \in I_A}v_j - \sum_{l \in I_A}v_l &=0.
\zeta = \frac{a \tilde{\zeta}+ b}{- \overline{{b}}\tilde{\zeta}+ \overline{{a}}}, \qquad \eta = \frac{\tilde{\eta}}{( - \overline{{b}}\tilde{\zeta}+ \overline{{a}})^{2}}, \qquad | a |^{2}+ | b |^{2}= 1 ,
a_{k}(u,vs_j)= \left|\{r\in u^{-1}([i])\,|\, r<k\}\right|-\left|\{r\in (vs_j)^{-1}([i])\,|\, r<k\}\right| begin{align*}5pt]= \left|\{r\in u^{-1}([i-1])\,|\, r<k\}\right|-\left|\{r\in v^{-1}([i-1])\,|\, r<k\}\right| begin{align*}5pt]= a_{k}(u,v).
\left( \frac{r}{|s|}\right)^\alpha \exp(-(r+is)A) = \left[ \left( \frac{r}{|s|}\right)^\alpha ( 1 + rA )^{-\alpha}( 1 + |s| )^\alpha (1 + A)^\alpha \right] \times &\times \left[ ( 1 + |s| )^{-\alpha}(1 + A)^{-\alpha}e^{-isA}\right] \left[ ( 1 + rA)^\alpha \exp(-rA) \right].
(LF)(\gamma):= \sum_{x\in\gamma}\bigl[F(\gamma\setminus x) -F(\gamma)\bigr] &+ z \int_{{{\mathbb{R}}^d}}\bigl[F(\gamma\cup x) -F(\gamma)\bigr]\exp\bigl\{-E^\phi(x,\gamma)\bigr\}dx, \gamma\in\Gamma,
{}\displaystyle\sum_{n=1}^{\infty}\frac{|A_{n}-A_{n-1}|^{2}}{\lambda_{n}}=\displaystyle\sum_{n=2}^{\infty}\Big|\Big(\frac{n-1}{n}\Big)^{\frac{\beta}{2}}\frac{A_n}{\sqrt{\lambda_{n}}}-\Big(\frac{n}{n-1}\Big)^{\frac{\beta}{2}}\frac{A_{n-1}}{\sqrt{\lambda_{n}}}\Big|^{2}+\displaystyle\sum_{n=1}^{\infty}w_{n}(\lambda,\beta)|A_{n}|^{2}.
E\left[\left(S^k_{j,+}(t)-\bar{S}^k_{j,+,\Delta}(t)\right)^2\right] =\nu\Delta\sum_{\ell=0}^{\lfloor t/\Delta\rfloor}p^k_{j,+}(t-\ell\Delta)E[Z_0(\ell\Delta)] &+2\sum_{\ell_1<\ell_2}\left(I_1(\ell_1,\ell_2)-I_2(\ell_1,\ell_2)\right)+C_3\Delta e^{3\lambda t}&\leq C_1 \theta^kte^{\lambda t}+C_2\theta^kt^2e^{\lambda t}+C_3\Delta e^{3\lambda t}.
(-(b_1^2 - d_1^2 J_1) \nu_1 J_1, J_2)_F \cdot (u, J)_F = (\frac{\nu_1}{u}\cdot \nu_1 J_1, J_2)_F \cdot (u, J_2)_F = (\nu_1^2 J_1, J_2)_F = 1.
|A| \ge q^{K}- \binom{N}{1}q^{K - 1}+ \binom{N}{2}q^{K - 2}- \dots + (-1)^{K - g}\binom{N}{K - g}= q^{K}\left (1 - \dfrac{1}{q}\right )^{N}- \sum_{p = K - g + 1}^{N}(-1)^{p}\binom{N}{p}q^{K - p}.
\left.\frac{\partial}{\partial\zeta}R_{i,i+1}(\zeta)\right|_{\zeta=1}\propto h_{i,i+1}P_{i,i+1}+\mbox{const.}, \qquad \left.\frac{\partial}{\partial\zeta}K_i(\zeta;r)\right|_{\zeta=1}\propto f_i\sigma_i^z+\mbox{const.},
R_{a}(\varphi)=\frac{1}{\sqrt{K}}=\frac{1}{2\pi}\int_{0}^{2\pi}R_{c}(\alpha)d\alpha=\sqrt{MN}=\frac{a^{2}b}{(acos\varphi)^{2}+(bsin\varphi)^{2}}=\frac{a\sqrt{1-e^{2}}}{1-e^{2}sin^{2}\varphi}.
\psi ( \vec{p}\, ) \, = \,{\frac{1}{\sqrt{2}}}\left( \begin{array}{c c}{{\phi_{R}( \vec{p}\, ) \, + \, \phi_{L}( \vec{p}\, )}}\\{{\phi_{R}( \vec{p}\, ) \, - \, \phi_{L}( \vec{p}\, )}}\\ \end{array}\right) \quad ,
B_j \rho^j ={\rm max}_{j'\leq j}\{b_{j'}\rho^j\}={\rm max}\{{\rm max}_{j'<j_{\varepsilon,0}}\{b_{j'}\rho^{j'}\rho^{j-j'}\},{\rm max}_{j_{\varepsilon,0}\leq j' \leq j}\{b_{j'}\rho^{j'}\rho^{j-j'}\}\}< \varepsilon
\log \frac{A_{\sup}}{A_{\inf}}= \sum_i \left(- \log \left( \frac{1 + \rho_ie^{-\tau_i}}{1 + \rho_i}\right) \right) \le \sum_i \frac{\rho_i}{1 + \rho_i}\tau_i \le \frac{1}{3}\sum_i \tau_i = \frac{1}{3}\Sigma_*,
F_{m n}^{a}= - 4 \rho^{2}\frac{\overline{\eta}_{m n}^{a}}{( x^{2}+ \rho^{2})^{2}}+ 8 \rho^{2}\frac{( \overline{\eta}_{m k}^{a}x_{n}- \overline{\eta}_{n k}^{a}x_{m}) x_{k}}{x^{2}( x^{2}+ \rho^{2})^{2}}.
\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{|x|<r}\varepsilon_{1,k}|x|^{\beta_k - 2(N+2)}e^{a \hat{v}_k}= \frac{\beta_\infty (4+ a \beta_\infty - 2 a \beta)}{4(1-a(N+1))}
4(2\lambda-\beta+2)(\lambda-\beta)\omega^{(+)}_2 = 2(\lambda-\beta)R^{(1)}_1\omega^{(+)}_0 +2(\lambda-\beta+2)R^{(2)}_1\omega^{(+)}_0 \quad+2(2\lambda-\beta+2)(\lambda-\beta)\beta \tfrac{J}{2n}\omega^{(+)}_0
\langle \Psi_B(p,\boldsymbol{u})|\Psi_A(\boldsymbol{x})\rangle = i^{[A_X]}{\pi^N}{|p|^{-N-1}}[p]^{A_X}\prod_{k=1}^{N}a(\bar s-i\bar x_k) \prod_{j=1}^{N-1}q(x_k,u_j)\,.
\int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s+1}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda-1,j\in[k:\ell(n)], \int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda,j\in[\ell(n)+1:\eta(n)+\ell(n)-1], \int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda-1,j\in[\eta(n)+\ell(n):p-1],
1\leqslant i\leqslant s\nu_{i}=0\mu_{i}1\leqslant i,j\leqslant t\kappa_{i}=\pm\kappa_{j}\varepsilon_{i}=\varepsilon_{j}1\leqslant i\leqslant t\kappa_{i}=0\varepsilon_{i}=(-1)^{v}
\frac{i}{2}\mathrm{T r}\left( \frac{1}{- \partial^{2}- m^{2}}\hat{T}_{x}^{( 1 )}\right) = \frac{i}{2}\sigma ( x ) \int \frac{d^{d}k}{( 2 \pi )^{d}}\frac{1}{k^{2}- m^{2}}\, .
G_4(\Lambda) = G_4(\mu) \left(\frac{\Lambda}{\mu}\right)^2\frac{( 1+bg^2(\mu)\ln(\Lambda/\mu) )^{-c/b}}{1 + w G_4(\mu) \int_{\mu}^{\Lambda}\frac{d\Lambda'}{\Lambda'}\left(\frac{\Lambda'}{\mu}\right)^2( 1+bg^2(\mu)\ln(\Lambda'/\mu) )^{-c/b}},
I_{1 1}= \int d^{1 1}x \sqrt{- g}~ ({\frac{R}{\kappa^{2}}}- 3 \, \hat{F}_{[ 4 ]}^{2}) +{2 8 8}\sigma \int \hat{F}_{[ 4 ]}\wedge \hat{F}_{[ 4 ]}\wedge \hat{A}_{[ 3 ]}
\widetilde{S}_{0}^{(D_{0})}(k)= -\frac{1 + i \left(2 \omega_{_{\rm (gs)}}^{-1}/\pi - Q_{_{\rm (gs)}}\right)}{1 - i \left(2 \omega_{_{\rm (gs)}}^{-1}/\pi - Q_{_{\rm (gs)}}\right)}\; ,
\binom{n-k}{n-l}\frac{\prod_{j=1}^{n-k}{\omega_j}}{\bigl(\prod_{j=1}^{n-l}\omega_j\bigr)\bigl(\prod_{j=1}^{l-k}\omega_j)},\binom{l}{k}\frac{\prod_{j=1}^{l}{\omega_j}}{\bigl(\prod_{j=1}^k\omega_j\bigr)\bigl(\prod_{j=1}^{l-k}\omega_j)}
{\cal L}_{0}^{f u l l}={\cal L}_{1}^{m o d i f i e d}( \psi_{x}^{i}, \psi_{x x}^{i}. . . ) + \lambda_{i}[ u^{i}- M^{i}( \psi_{x}^{i}, \psi_{x x}^{i}. . . ) ]
\Pr\left[\sum_{k = 1}^m Q_{i, k}\leq Dm - \left(\sqrt{Dm}+ 1 \right) \log m \right] \leq \exp \left( -\frac{\left( \sqrt{Dm}+ 1 \right)^2 \log^2 m}{\frac{1}{2}\cdot Dm + \frac{1}{3}\cdot \left(\sqrt{Dm}+ 1\right) \log m}\right) \leq m^{-\log m}
\tilde{C}(w,z)=\left(q^{K(z)+2};q\right)_{\infty}\left(1+\sum_{j=1}^{\infty}\frac{(-1)^{j}q^{(j-1)j/2}}{\left(q^{K(z)+2};q\right){}_{j}}\frac{\left(q^{K(z)+1}w\right)^{j}}{\left(q^{K(z)+1}z;q\right){}_{j}}\right)\!.
\mp \frac{\pi i}{2}\zeta_{{D \! \! \! \! /}^{~ 2}+ m^{2}}( 0 ) + \frac{1}{2}\lim_{s \to 0}\frac{1}{\Gamma ( s )}\int_{0}^{1}~ d t ~ t^{s - 1}\sum_{n = 0}^{\infty}H_{n}t^{{( n - d )}/{2}}\left[ \frac{d K^{( 1 )}}{d s}\pm 2 m \sqrt{t}\frac{d K^{( 2 )}}{d s}\right] .
\tau_{\infty}\equiv \frac{\kappa_{+}\kappa_{-}(e^{\tau_{+}-\tau_{-}}\prod_{b=1}^{\mathsf{N}}\delta_{b}\gamma_{b}+e^{\tau_{-}-\tau_{+}}\prod_{b=1}^{\mathsf{N}}\alpha_{b}\beta_{b})}{\left( \zeta_{+}-1/\zeta_{+}\right) \left( \zeta_{-}-1/\zeta_{-}\right)}.
\sum_{j\ne i}\frac{2}{z_i-z_j}+ \frac{- 8b_2z_i^3 - 4(b_1-2b_2)z_i^2 + (4a+4b_1+3)z_i - 4a+2l+d-3}{2z_i(z_i-1)}= 0, i=1, 2, \ldots, n.
\sup_{1\leq k-t\leq s_0}r^*_{k,t}=\sup_{1\leq k-t\leq s_0}\sup_{\substack{0\le l\le L_k,0\le p\le L_t |n|\le N_k,|m|\le N_t}}|r_{x_k,x_t}(s_{k,l},\tau_{k,n},s_{t,p},\tau_{t,m})|\le\zeta<1.