harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lim_{\lambda \to \mathcal U}\|\varphi \circ \Phi - (\varphi \circ \Phi)(f_\lambda \, \cdot \, f_\lambda)\| \leq \lim_{\lambda \to \mathcal U}2|(\varphi \circ \Phi)((1 - f_\lambda)^2) |^{1/2}& \leq \lim_{\lambda \to \mathcal U}2|(\varphi \circ \Phi)(1 - f_\lambda) |^{1/2}= 0.
\|\delta(p_m)(v_mx_{n(m,k)}v_m^*+v_m^*x_{n(m,k)}v_m)p_mq_{m}\|_{\mathcal M}= \|q_{m}p_m(v_mx_{n(m,k)}v_m^*+v_m^*x_{n(m,k)}v_m)\delta(p_m)\|_{\mathcal M}\leq 5^{-1}2^{-m}(k+1)^{-1};
(T_{G,w}QT_{F,v}^*)(h) = \int_Xw(x)(QT_{F,v}^*h)(x)d\mu(x) & = \int_X\int_Xw(x)\langle T_{F,v}^*h,F_x(y)\rangle G_x(y)d\mu(y)d\mu(x) & = \int_X\int_Xw(x)v(x)\langle\pi_{F(x)}h,F_x(y)\rangle G_x(y)d\mu(y)d\mu(x) & = \int_X\int_Xw(x)v(x)\langle h,F_x(y)\rangle G_x(y)d\mu(y)d\mu(x) & = \int_X\int_X\langle h,v(x)F_x(y)\rangle w(x)G_x(y)d\mu(y)d\mu(x).
\overline{\mathbf{d}}=\mathbf{d}/\mathbf{d}\in[0,\infty]^{X\times X}x\overline{\mathbf{d}}z =\sup_{y\in Y}(x\mathbf{d}y-z\mathbf{d}y)_+. \underline{\mathbf{d}}=\mathbf{d}\backslash\mathbf{d}\in[0,\infty]^{Y\times Y}z\underline{\mathbf{d}}y =\sup_{x\in X}(x\mathbf{d}y-x\mathbf{d}z)_+.
\begin{array}{ll}(1-\frac{\eta}{\beta})\frac{u_x}{\tilde{u}_y}= \frac{1}{\beta}w_x +\frac{u}{\beta}\frac{\tilde{u}_{yyx}}{(\tilde{u}_y)^2}+ (1-\frac{2\eta}{\beta})u\frac{\tilde{u}_{yx}}{(\tilde{u}_y)^2}.\end{array}
V_{n}(R)=\int_{0}^{R}\int_{0}^{\pi}\cdot\cdot\cdot\int_{0}^{2\pi}r^{n-1}sin^{n-2}(\varphi_{1})\cdot\cdot\cdot sin(\varphi_{n-2})d\varphi_{n-1}\cdot\cdot\cdot d\varphi_{1}dr.
C_{\tilde{\mathcal H},\mathcal H}:= \begin{cases}\dfrac{\tilde{p}-p}p(|\Omega|+\|a\|_1)+\left(\dfrac{\tilde{p}-p}p\right)^{\frac{p-\tilde{p}}p}\quad \mbox{if}\dfrac{\tilde{q}}q\le\dfrac{\tilde{p}}p, \dfrac{\tilde{q}-q}q(|\Omega|+\|a\|_1)+\left(\dfrac{\tilde{q}-q}q\right)^{\frac{q-\tilde{q}}q}\quad \mbox{otherwise.}\end{cases}
(b\theta)(a_0\otimes\cdots\otimes a_n) =\theta(a_0a_1\otimes a_2\otimes\cdots\otimes a_n) &+\sum_{k=1}^{n-1}(-1)^k\theta(a_0\otimes a_1\otimes\cdots a_{k-1}\otimes a_{k}a_{k+1}\otimes a_{k+1}\otimes\cdots\otimes a_n) &+(-1)^n\theta(a_na_0\otimes a_1\otimes a_2\otimes\cdots\otimes a_{n-1}).
\frac{\partial x_{j}}{\partial\overline{x}_{k}}=\frac{\partial}{\partial\overline{x}_{k}}(\overline{x}_{i}(L^{-1})_{ij})=\frac{\partial\overline{x}_{i}}{\partial\overline{x}_{k}}(L^{-1})_{ij}=\delta_{ki}(L^{-1})_{ij}=(L^{-1})_{kj}
\frac{1}{\det(1_3-DA)}=\frac{1}{1+XZ+YZ+XY}=\sum_{k=0}^\infty(-1)^k(XY+YZ+ZX)^k &=\sum_{k=0}^\infty(-1)^k\sum_{a+b+c=k}\frac{k!}{a!b!c!}X^{a+c}Y^{a+b}Z^{b+c}.
e_{n}^{2}:= \left(\begin{array}{c}\frac{z^{n}}{\|z^{n}\|_{\lambda + 2m}}0\end{array}\right),\,\, n \geq 0\,\,\mbox{and}\,\,e_{-n}^{2}:= \left(\begin{array}{c}0 \frac{z^{n-1}}{\|z^{n-1}\|_{2-\lambda + 2k}}\end{array}\right),\,\,n \geq 1.
\varepsilon_k = \| \tilde{\mathbf{h}}_k \hat{\mathbf{Q}}^H \|^2 & = \| \tilde{\mathbf{h}}_k \hat{\mathbf{Q}}_{\boldsymbol{\pi}}^H\mathbf{G}_{K-1}^H \cdots \mathbf{G}_{1}^H\|^2 & \mathop = \limits^{(b)}\| \tilde{\mathbf{h}}_k \bar{\mathbf{Q}}_{\boldsymbol{\pi}}^H \|^2,
\delta_\Gamma(\vec x^{k+1},\vec x^k) =\delta_\Gamma(\vec z^{k+1},\vec z^k)=\sum_{i=1}^n\delta_\Gamma(\vec z^{k,i},\vec z^{k,i-1}) &=\sum_{i\in V(k)}\delta_\Gamma(\vec z^{k,i},\vec z^{k,i-1}),\;\;\;\Gamma\in\{F,\vec q,\vec p\}
G_{\alpha_{1}\ldots \alpha_{n}}( p_{1}, \ldots , p_{n}) \prod_{i = 1}^{n}J_{\alpha_{i}}^{i}( p_{i}) = \hat{G}_{\alpha_{1}\ldots \alpha_{n}}( p_{1}, \ldots , p_{n}) \prod_{i = 1}^{n}\hat{J}_{\alpha_{i}}^{i}( p_{i}) \ .
\sum_{1\leq i \leq n}\sum_{i+1<j\leq n}\mathbb{E}(X_iX_{i+1}X_jX_{j+1})= V_k(n)+O_{\alpha}\Big(\sum_{1\leq i \leq n}\sum_{i+1<j\leq n}\frac{\tau_3(i)\tau_3(i+1)\tau_3(j)\tau_3(j+1)}{\sqrt{i}}\Big) &+O_{\alpha}\Big(\sum_{1\leq i \leq n}\sum_{i+1<j\leq n}\frac{\tau_3(j)\tau_3(j+1)}{\sqrt{j-i-1}}\Big),
\begin{array}{l}\lambda_h^{m-n}(\{\alpha_n\}=\alpha)\!\!= \!\!\frac{2\lambda_n\lambda_m\beta_n^2f_uS}{\pi(\sum\limits_{i=1}^N\lambda_i\beta_i^2)^{1.5}}\!\int\limits_{0}^{+\infty}\!vf_v(v)dv\cdot\!\!\!\!\!\!\int\limits_0^{\min(1,\beta_n)}\!\![\sqrt{\frac{1\!-\!z^2}{\beta_n^2\!-\!z^2}}\!\!+\!\!\sqrt{\frac{\beta_n^2\!-\!z^2}{1\!-\!z^2}}]\!dz\end{array}
c_{g}^{\mathrm{e f f}}= \frac{6}{\pi^{2}}\sum_{a = 1}^{l}L \Bigl ( \frac{1}{1 + y_{a}}\Bigr ) \, , \qquad \mathrm{w h e r e}\quad \ln ( 1 + y_{a}) = \sum_{b = 1}^{l}( N_{a b}+ g \delta_{a b}) \ln ( 1 + y_{a}^{- 1}) \, .
\hat{\mathbf{u}}_n =\underset{\mathbf{u}\in \mathrm{\Delta_M}}{\operatorname{argmin\,}}I_3- 2I_2 &=\underset{\mathbf{u}\in \mathrm{\Delta_M}}{\operatorname{argmin\,}}\sum\limits_{i=1}^M\sum\limits_{j=1}^M c_{ij}u_i u_j - 2 \sum\limits_{m=1}^M c_m u_m,
z_1 z_2 = p_3(1- z_3) z_1 z_3 = -p_1 z_2 - (p_2 p_3)z_3 + 2 p_3 z_2 z_3 = -p_3 z_2 - (p_2 p_3)z_3 + 2 p_2 p_3 z_1^2 = p_1 p_3(1-z_3) z_2^2 = p_2 p_3 (1-z_3) z_3^2 = -6z_2 - 3p_2p_3 z_3 + 3p_3
P(\chi_{P}^{2}(\{p_{i}\})>T)=\sum_{\{k_{i}\}|\chi_{P}^{2}(\{k_{i}\},\{p_{i}\})>T}\frac{n!}{k_{1}!\cdot\cdot\cdot k_{m}!}\prod_{i=1}^{m}{p_{i}}^{k_{i}}
\tilde{\theta}=\theta_{0}+\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}Z_{i}[\hat{\gamma}_{\ell}(X_{i})-\gamma_{0}(X_{i})]+\frac{1}{n}\sum_{i=1}^{n}Z_{i}\gamma_{0}(X_{i})-\theta_{0}& =\theta_{0}+\frac{1}{n}\sum_{i=1}^{n}\zeta(W_{i})+o_{p}(n^{-1/2}).
{R_{ijkt}=W_{ijkt}+\frac{1}{n-2}(R_{ik}\delta_{jt}-R_{it}\delta_{jk}+R_{jt}\delta_{ik}-R_{jk}\delta_{it})-\frac{S}{(n-1)(n-2)}(\delta_{ik}\delta_{jt}-\delta_{it}\delta_{jk})}
\begin{aligned}a_1(a_2\rightharpoonup b) =a_1\circ (T(a_2)\rightharpoonup (a_3\rightharpoonup b)) &=a_1\circ ((T(a_2)\circ a_3)\rightharpoonup b)=a\circ (1\circ b)=a\circ b.\end{aligned}
\Lambda_1 =2\left| \sum_m\sum_{\substack{i<j i,j\in I_m\cup I_{m+1}}}(f(y_i)-f(y_j)) (x_{ij}h_{ji}-x_{ji}h_{ij})\right| \Lambda_2 =2\left|\sum_m\sum_{i\in I_m}\sum_{l\geq m+2}\sum_{j\in I_l}(f(y_i)-f(y_j)) (x_{ij}h_{ji}-x_{ji}h_{ij})\right| \Lambda_3 =2 \left| \sum_{m=1}^{k-1}\sum_{\substack{i<j i,j\in I_m}}(f(y_i)-f(y_j)) (x_{ij}h_{ji}-x_{ji}h_{ij})\right|.
A^{-1}= \begin{pmatrix}(\hat{U}_{1,1})^{-1}(\hat{L}_{1,1})^{-1}+ \hat{U}_{1,2}S_{2,2}^{-1}\hat{L}_{2,1}- \hat{U}_{1,2}S_{2,2}^{-1}-S_{2,2}^{-1}\hat{L}_{2,1}S_{2,2}^{-1}\end{pmatrix}.
f_{\bf k}({\bf x}, t ) ={\frac{n_{T}^{( m - 2 ) / 2}}{\left[ 2 \omega ( 2 \pi )^{m}\right]^{1 / 2}}}\left\{\Theta ( - x ) T_{l g}e^{- i \beta^{\prime}x_{1}}+ \Theta ( x ) \left[ e^{i n_{T}k_{1}x_{1}}+ R_{l g}e^{- i n_{T}k_{1}x_{1}}\right] \right\}e^{i n_{T}( k_{2}x_{2}+ \cdots + k_{m}x_{m})}e^{- i \omega t},
P_{N=ne}f(x_{N}) =P_{N=ne}P_{\left( x_{N}-ae\right)^{+}}^{d}e=P_{N=ne}\left( I-P_{\left( x_{N}-ae\right)^{+}}\right) e & =P_{N=ne}-P_{P_{N=ne}\left( x_{N}-ae\right)^{+}}e.
\textstyle \sum_v \big(3g_v-3+n_v-\sum_{l\in L_v}(k_l+b_l)-\sum_{e\in E_v}k_{(e,v)}\big)+\sum_{l\in L}k_l &\textstyle +\sum_{e=(v_1,v_2)\in E}(k_{(e,v_1)}+k_{(e,v_2)}+1)= 3g-3+n -\sum_{l\in L}b_l.
\lambda^2e^{\,u}\approx\begin{cases}\lambda^2\exp\left(U\left(\lambda\mu_\lambda t\right)+2\log\mu_\lambda-\log h_\gamma\right), |t|<\delta,\medskip \lambda^2\exp\left((\beta+2\log\beta) H_\gamma^\pm+\tilde H^\pm\right), \mathop{dist}(x, \gamma)>\delta.\end{cases}
\widetilde{\psi}_{1}^{(0^{2})}\big(x;\omega ,N,- \Vert \psi_{0}\Vert^{2}\big)=-E_{1}(\omega) \psi_{1}( x;\omega ,N ) -\frac{W(\psi_{0},\psi_{1}\,|\, x)}{\Psi_{0}^{(0)}\big(x;\omega ,N,- \Vert \psi_{0}\Vert^{2}\big)}.
E_k \triangleright (v_m \otimes f^m) = E_k \triangleright v_m \otimes K_k \triangleright f^m + v_m \otimes E_k \triangleright f^m & = q^{-\delta_{m,k}+ \delta_{m,k+1}}\delta_{m}^{k+1}q^{-1/2}v_{m-1}\otimes f^{m}- \delta_{m}^{k}q^{1/2}v_{m}\otimes f^{m+1}& = \delta^{k+1}_m q^{1/2}v_{m - 1}\otimes f^m - \delta^k_m q^{1/2}v_m \otimes f^{m + 1}.
||(\psi_1(x_{ij}))|| =\sup_{\eta\in (L^2(D))^n}\left[\sum_{ijk}\tau((\eta_jx_{ij})^*\eta_kx_{ik})\right]^{1/2}&=\sup_{\eta\in (L^2(D))^n}\left[\sum_{ijk}\tau(\eta_kE_D(x_{ik}x_{ij}^*)\eta_j^*)\right]^{1/2}&=||E_D(\sum_i x_{ik}x_{ij}^*)||^{1/2},
\left( \frac{f^{\prime}}{f^{2}}\right) \frac{g^{\prime \prime \prime}}{\left( g^{\prime \prime}\right)^{2}}-\frac{f^{\prime \prime \prime}}{\left( f^{\prime \prime}\right)^{2}}\left( \frac{g^{\prime \prime \prime}}{g^{\prime}g^{\prime \prime}}\right)^{\prime}=0.
\int_0^{\frac{\pi}{2}}x^n \log \left( \tan x \right) \d x = (-1)^{\lfloor \frac{n-1}{2}\rfloor}\frac{n!}{2^{n-1}}\left(1-\frac{1}{2^{n+2}}\right)\zeta(n+2) \delta(n) + \frac{n!}{2^{n}}\sum_{k=1}^{\lfloor{\frac{n}{2}}\rfloor}\frac{(-1)^{k-1}\pi^{n-2k+1}}{(n-2k+1)!}\left(1-\frac{1}{2^{2k+1}}\right) \zeta(2k+1),
\kappa_{d,p,N}^{(\infty)}= \frac{D^p\sqrt{A}r^p}{2^{p+1}p \log r}\log_+\Big(2(r^{-p}-r^{-2p})\sqrt{\frac N A}\Big) +\frac{D^p\sqrt{A}r^{2p}}{2^{p+1}(r^p-1)}= \frac{2^{p-1}}{p\log 2}\log_+\Big((2^{1-p}-2^{1-2p})\sqrt N \Big)+ \frac{2^{p-1}}{1-2^{-p}}.
\begin{array}{c c c}{{( 0 ~ 0 ~ 0 ~ 0 ~ - \frac{7}{2}) ~ \rightarrow}}&{{}}&{{( 0 ~ 0 ~ 0 ~ - \frac{7}{2}) ( \frac{7}{2}) \oplus ( 0 ~ 0 ~ 0 ~ - \frac{5}{2}) ( - \frac{1}{2}) ,}}\\{{( 0 ~ 0 ~ 0 ~ 0 ~ - \frac{1}{2}) ~ \rightarrow}}&{{}}&{{( 0 ~ 0 ~ 0 ~ - \frac{1}{2}) ( \frac{1}{2}) \oplus ( 0 ~ 0 ~ 0 ~ + \frac{1}{2}) ( - \frac{7}{2}) .}}\\ \end{array}
E\left[T_1\right] = (\nu_1 + \lambda \nu_2) M \ln M \, + \, (\nu_1 + \lambda \nu_2) (\gamma + \, \ln \nu_1) M \, + \, \frac{\alpha_1}{2}\, + \, O\left(\frac{1}{M}\right),M \to \infty,
\lim_{x \rightarrow +\infty}2\beta e^{2\alpha x}\int_{x}^{+\infty}\! \! \! e^{-(\alpha-\mu )y}G(Q(y))\mathrm{d}y = \! \lim_{x \rightarrow +\infty}Q^{\prime}(x)e^{\lambda x}+\lambda Q(x)e^{\lambda x}=0
1 = \bar{u}_{+ i}\gamma_{+}u_{+ i}= \bar{u}_{- i}\gamma_{-}u_{- i}= 2 \bar{u}_{+ -}u_{--}= - 2 \bar{u}_{+ +}u_{- +}= - 2 \bar{u}_{- +}u_{+ +}= 2 \bar{u}_{--}u_{+ -}
|\varepsilon \boldsymbol{f}_\varepsilon ' |_{L^2(0,T;\boldsymbol{H})}= |\boldsymbol{f}-\boldsymbol{f}_\varepsilon |_{L^2(0,T;\boldsymbol{H})}\le M_{10}, \varepsilon^{1/2}|\boldsymbol{f}_\varepsilon|_{L^\infty (0,T;\boldsymbol{H})}\le M_{10},
\begin{cases}\frac{\partial}{\partial t}\Lambda(s',t)=\Sigma_{ij}a^\Lambda_{ij}(s',t)\frac{\partial^2\Lambda}{\partial s_i\partial s_j}(s',t)+b(s',t)+c^\Lambda(s',t)|\nabla_xv(X^0(s'),t)|^2 \Lambda(s',0)=0 .\end{cases}
i \lim_{r \rightarrow 0}\left( M r \right)^{\nu + 1}g_{n}( r ) \sin \left( \frac{\pi}{4}+ \frac{\Theta}{2}\right) = \lim_{r \rightarrow 0}\left( M r \right)^{- \nu}f_{n}( r ) \cos \left( \frac{\pi}{4}+ \frac{\Theta}{2}\right)
\|\Psi(b)\Psi(a)-\bar{h}\Psi(a)\| \overset{\eqref{eq: h3}}{=}\|\bar{h}\Psi(ba)-\bar{h}\Psi(a)\| \overset{\phantom{\eqref{eq: h3}}}{=}\|\bar{h}\Psi(ba-a)\| \overset{\eqref{eq: h2}}{=}\|ba-a\|
\Gamma_{\mathrm{e f f}}[ A , V ] = - \mathrm{l n}( \mathrm{d e t M}) = - \mathrm{T r}( \mathrm{l n}M ) = \int_{0}^{\infty}{\frac{\mathrm{d}T}{T}}\mathrm{T r}\; \mathrm{e}^{- T M}\; ,
\frac{1}{2}\vert \left({}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \left[\hat X_{1,q},\hat P_{1,q}\right] \vert J_1,\gamma_1; J_2,\gamma_2\rangle_q\right)\vert = \frac{\hbar}{4(\lambda_1 +\lambda_2)^2}\vert \lambda_2K_1\left(1-(1-q^2)J_1\right) + \lambda_1K_2\left(1 -(1-q^2)J_2\right)\vert;
P_{1}(\phi,\psi)(\xi) =\frac{1}{c}\int_{-\infty}^{\xi}e^{\frac{\beta(y-\xi)}{c}}F_{1}(\phi,\psi)(y) dy, \ \xi\in\mathbb R, begin{align*}0.2cm] P_{2}(\phi,\psi)(\xi) =\frac{1}{c}\int_{-\infty}^{\xi}e^{\frac{\beta(y-\xi)}{c}}F_{2}(\phi,\psi)(y) dy,\ \xi\in\mathbb R.
P = \begin{pmatrix}\hat{P}^{\{k\}}_0 \hat{P}^{\{k\}}_1 \cr\hat{P}^{\{k\}}_{-1}P^{\{k\}}_0 P^{\{k\}}_1 \cr P^{\{k\}}_{-1}P^{\{k\}}_0 P^{\{k\}}_1 \cr \ddots \ddots \ddots \end{pmatrix},
-\varphi_t -(-\Delta)^s\varphi - \frac{n-2s}{2}\alpha_{n, s}\frac{\dot{\mu}_j}{\mu_j}\frac{\mu^{-(n-2s)}_j}{\left(1+\left|\frac{x-\xi_j}{\mu_j}\right|^2\right)^{\frac{n-2s}{2}}}= 0 \mathbb{R}^n\times (t_0, +\infty).
S = \int d^{6}x \left[{\frac 1 6}F_{{\mu}{\nu}{\sigma}}( A ) F^{{\mu}{\nu}{\sigma}}( A ) + \frac{1}{2 ({\partial}_{\lambda}a ) ({\partial}^{\lambda}a )}{\partial}^{\mu}a{\cal F}_{{\mu}{\nu}{\sigma}}( A ){\cal F}^{{\nu}{\sigma}{\rho}}( A ){\partial}_{\rho}a \right] ,
\frac{\partial \overline{Z}_{i}}{\partial q_{j}}=\frac{\partial \overline{Z}_{i}}{\partial \overline{q}_{i}}\frac{\partial \overline{q}_{i}}{\partial q_{j}}=-\left[\frac{\left(1-\alpha\right)\overline{p}_{i}}{\left(\alpha \overline{p}_{i}+\left(1-\alpha\right)\overline{q}_{i}\right)^{2}}\right]\left[\frac{\delta_{ij}-\overline{q}_{i}}{\sum_{j}q_{j}}\right]
-\frac{1}{8\pi g^{2}}\int d^{2}x tr \sum_{\mu \neq 2,3,4,5}\left[ ((2M+1)/B+(2N+1)/B+U^{2}) \left(c_{\mu}c_{\mu}^{\dagger}+c_{\mu}^{\dagger}c_{\mu}\right) \right]_{\star}.
D_0 = \nu_{\Sigma_0}^{(1)}(X) ( B_{21}D_{y_2}F - B_{22}D_{y_1}F ) - \nu_{\Sigma_0}^{(2)}(X) (B_{11}D_{y_2}F - B_{12}D_{y_1}F)+ \nu_{\Sigma_0}^{(3)}(X)
J(u)= 4n(n-1)c_{0}\alpha^{2}+\int L_{g_{0}}vv-\frac{32n(n-1)}{n-2}\int \varphi_{a, \lambda}^{\frac{4}{n-2}}v^{2}& +o_{\frac{1}{\lambda}}(1)+o(\Vert v \Vert^{2})
-\div(|\nabla u|^{p(x)-2}\nabla u) = V(x) |u|^{q(x)-2}u, -\div (|\nabla u|^{p(x)-2}\nabla u) + |u|^{p(x)-2}u = \lambda g(x) |u|^{\alpha(x)-2}u - h(x) |u|^{\beta(x)-2}u +K(x).
\frac{D_{m+1}(r,t,r-1-q)}{D_m(r,t,r-1-q)}=\frac{(q+m+1)(t-m)}{(m+1)(r-1+t-q-m)}=\frac{qt+mt+t-qm-m^2-m}{rm+r-2m-1+tm+t-qm-q-m^2}<1,
\hat{U}^{-1}(\theta_0)|A_{\bar{z}}(z,\bar{z})>=e^{-\frac{ik}{4\pi}\int_{T_2}dz\wedge d\bar{z}\left[\partial_{\bar{z}}\chi\partial_z\theta_0-\partial_{\bar{z}}\theta_0\overline{\partial_{\bar{z}}\chi}\,\right]}\|A_{\bar{z}}(z,\bar{z})+\partial_{\bar{z}}\theta_0(z,\bar{z})>\ \ \ ,
\begin{aligned}( \nabla f(x,s_1) - \nabla F(y))\cdot(x - y) = (\nabla f(x,s_1) - \nabla F(x) + \nabla F(x) - \nabla F(y))\cdot(x - y) \geq &(\nabla f(x,s_1) - \nabla F(x))\cdot(x - y) + m|x - y|^2 \geq \frac m2|x - y|^2 - \frac 1{2m}|\nabla f(x,s_1) - \nabla F(x)|^2. \end{aligned}
Re\left(\omega(s) e^{i\theta}\right) \overset{\eqref{concatenated_omega},\eqref{concatenated_s}}{=}-s Re\left( e^{i(\theta-\theta_2/2-\theta_1/2)}\right)= -s \cos(\theta-\theta_2/2-\theta_1/2) \overset{\eqref{concatenated_s},\eqref{concatenated_theta},\eqref{alpha}}{\leq}-s\cos\left(\alpha\right).
\int_{\alpha}^{\zeta}{\left({\tau^{2}-\alpha^{2}}\right)^{1/2}d\tau}=\int_{\sigma}^{x}{\left\{{f\left({\sigma,t}\right)}\right\}^{1/2}dt}=\int_{\sigma}^{x}{\left({\frac{t^{2}-\sigma^{2}}{1-t^{2}}}\right)^{1/2}dt}.
y_{tt}- (D_2 a(0,t,x) y_x + a(0,t,x) y_{xt})_x + D_1F(0,0) y_t + D_2F(0,0) y_{xt}= f_t 1_{\mathcal{O}}- \frac{1}{\mu_1}p^1_t 1_{\mathcal{O}_1}- \frac{1}{\mu_2}p^2_t 1_{\mathcal{O}_2}+ G_{t}.
f=x_1x_2+\sum_{i=0}^q\lambda_ix_2^ix_3^{w_1+w_2-iw_2}=\Bigl(x_1+\sum_{i=1}^q\lambda_ix_2^{i-1}x_3^{w_1+w_2-iw_2}\Bigr)x_2+\lambda_0x_3^{w_1+w_2}
x=\begin{pmatrix}0 1 0 0 c 1 0 0 0 -c 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1\end{pmatrix},\, c\neq 0,y=\begin{pmatrix}1 0 -1 0 b 0 0 -1 0 0 0 1 -1 0 0 0 0 0 0 -1 0 0 0 1 -1\end{pmatrix}.
\widetilde{h_{SO(n)}}(x_1,\ldots x_{\frac{(n-1)(n-2)}{2}},z_1,\ldots z_{n-1}):=\sum_{i,j}c_{ij}z_1^{k_{ij}^1}x_1^{m_{ij}^1}(1-x_1^2)^{\frac{n_{ij}^1}{2}}\cdots x_{n-1}^{m_{ij}^{n-1}}(1-x^2_{n-1})^{\frac{n_{ij}^{n-1}}{2}}&\cdot(\widetilde{h_{SO(n-1)}})_{ij}(x_n,\ldots,x_{\frac{(n-1)(n-2)}{2}},z_{n-2},\ldots,z_{n-1}).
\chi_{c}\: C \: i^{| I_{1}| + \cdots + | I_{k}|}\: ( - i )^{| K |}\; \gamma^{I}\; ( \partial^{I_{k}}V_{J_{k}, c_{k}}^{( k )}( x ) ) \cdots ( \partial^{I_{1}}V_{J_{1}, c_{1}}^{( 1 )}( x ) ) \: ( y - x )^{K}\; S^{\vee ( h )}( x , y ) ,
I^{0+}_\textup{CC}(\bar x) :=\{l\in\mathcal Q\,|\,\bar G_l(\bar x)=0\,\land\,\bar H_l(\bar x)>0\}, I^{+0}_\textup{CC}(\bar x) :=\{l\in\mathcal Q\,|\,\bar G_l(\bar x)>0\,\land\,\bar H_l(\bar x)=0\}, I^{00}_\textup{CC}(\bar x) :=\{l\in\mathcal Q\,|\,\bar G_l(\bar x)=0\,\land\,\bar H_l(\bar x)=0\}
\nabla_\xi p(t,s;x,\eta(t,s)) \partial_\xi^\beta\nabla_\xi \eta(t,s) &=-\sum_{0\lneq\beta^\prime \leq \beta}\binom{\beta}{\beta^\prime}\partial_\xi^{\beta^\prime}[\nabla_\xi p(t,s;x,\eta(t,s))] \partial_\xi^{\beta-\beta^\prime}\nabla_\xi \eta(t,s), \beta \neq 0.
\int\limits_0^{\pi}Cf (0,e_n,\psi)\sin\psi d\psi &=\frac{n-2}{|S^{n-3}|}\int\limits_{\pi/2}^\pi \int\limits_{S^{n-2}}Rf(((\sin \phi )\omega,\cos \phi),0) (-\cos\phi)(\sin\phi)^{n-2}d\omega d\phi.
f_{\ell}( k , r ) \equiv w_{\ell}( k r ) g_{\ell}( k , r ) \quad \mathrm{a n d}\quad \phi_{\ell}( k , r ) \equiv \frac{( - k )^{- \nu + \frac{1}{2}}}{2 \nu}\frac{h_{\ell}( k , r )}{w_{\ell}( k r )},
a \sharp b = ab + \frac{h}{2i}\{a, b\}+ \sum_{\substack{\alpha= (\alpha_1, \alpha_2) 2\le|\alpha|\le k}}\left(\frac{h}{2i}\right)^{|\alpha|}\frac{(-1)^{\alpha_1}}{\alpha!}\partial^{\alpha_1}_x\partial_{\xi}^{\alpha_2}a \, \partial_x^{\alpha_2}\partial_{\xi}^{\alpha_1}b + r_k \,,
S_{m}: = \left[ (-1)^{m+1}A_{n-m}\right] (\mathrm{rec}_{N,r}) &= d^{m-1}\sum_{\substack{\alpha < a(r) w(\alpha)=m-1}}q^{\alpha}+ d^{m}\sum_{\substack{\alpha < a(r) w(\alpha)=m}}q^{\alpha}+ \sum_{j=1}^r \sum_{k=0}^{\min(j-1,m-1)}c_{k,j}b_{m-k,j}q^{jN(n-m)}
y(x)=A \{\rm e}^{-\frac{x^2}{4}}\,{\rm M}\left(\frac{a}{2}+\frac{1}{4},\frac{1}{2}, \frac{x^2}{2}\right)+ B \x \{\rm e}^{-\frac{x^2}{4}}\,{\rm M}\left(\frac{a}{2}+\frac{3}{4},\frac{3}{2},\frac{x^2}{2}\right) \,,
M_0 := \iiint \! \Delta_{\delta_1}(\xi_1 + h) \Delta_{\delta_2}(\xi_2 + h) C_{\delta_1, \delta_2}\gamma_{\delta_1, \delta_2}(h) F(\xi_1, \xi_2, \eta) \, d\xi_1 d\xi_2 d\eta,
\frac{\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (as^{(1)}_{k-1})_{k=0}^{m-1}\right)}{ab\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (1,0,\ldots,0) \right)}\stackrel{a\to 0+}{\longrightarrow}\frac{\mathcal{D}\left((s^{(1)}_{k+1})_{k=0}^{2m-3}, (s^{(1)}_{k-1})_{k=0}^{m-1}\right)}{b\mathcal{D}\left((-s^{(1)}_{k+1})_{k=0}^{2m-3}, (1,0,\ldots,0) \right)}
xyz(1-tS(x,y,z))F(x,y,z,t) = xyz - t y(z^2+1) F(0,y,z) - t x(z^2+1) F(x,0,z) &- t(x^2y+y^2x+y+x)F(x,y,0) & + txF(x,0,0) + tyF(0,y,0),
D^{1}\omega_4(u) ={}2\omega_4 (u)D^1\log\frac{1}{u^2 (1-u)(9-u) ( 25-u)}{}+\frac{15\log u}{4u^2 (1-u)(9-u) ( 25-u)}\det\begin{pmatrix}D^0\nu^1_{2,2}(u) D^0\nu^1_{2,3}(u) D^0\nu^1_{2,4}(u) D^1\nu^1_{2,2}(u) D^1\nu^1_{2,3}(u) D^1\nu^1_{2,4}(u) D^2\nu^1_{2,2}(u) D^2\nu^1_{2,3}(u) D^2\nu^1_{2,4}(u) \end{pmatrix},
E \left( \beta \right) = E_{0}+ \widetilde{E}\left( \beta \right) = \frac{1}{2 r}\sum_{\ell = 1}^{\infty}\ell^{2}\left( \ell^{2}+ \mu_{e f f}^{2}\right)^{1 / 2}+ \frac{1}{r}\sum_{\ell = 1}^{\infty}\frac{\ell^{2}\left( \ell^{2}+ \mu_{e f f}^{2}\right)^{1 / 2}}{e^{\beta \left( \ell^{2}+ \mu_{e f f}^{2}\right)^{1 / 2}/ r}- 1},
\begin{aligned}\psi^{-}_{2}\left(\mathfrak{b}^{*}\otimes 1_{\mathbb{C}^{-}}\right)=\frac{\sqrt{2}}{2}\Big((j_{1}^{-1})^{*}\mathfrak{b}^{*}-(j_{2}^{-1})^{*}\mathfrak{b}^{*}\Big),\end{aligned}
\begin{aligned}U_2^pXU_1^{-p}-U_2^q XU_1^{-q}= \sum_{m=p}^{q-1}(U_2^{m}XU_1^{-m}-U_2^{m+1}XU_1^{-m-1}) = \sum_{m=p}^{q-1}U_2^{m}KU_1^{-m-1}.\end{aligned}
\|u\|_{L^p}=\Big(\int_{\mathbb{R}^{N}}|u(x)|^{p}\,dx\Big)^{\frac{1}{p}},u\in L^{p}(\mathbb{R}^{N}), \|u\|_{\infty}=\mbox{ess sup}_{x\in \mathbb{R}^{N}}|u(x)|, u\in L^{\infty}(\mathbb{R}^{N}).
f_1 = e_1 f_2 = (\Sigma - \zeta_1)e_1 f_3 = (\Sigma - \zeta_1)(\Sigma - \zeta_2)e_1 \vdots f_n = (\Sigma - \zeta_1)(\Sigma - \zeta_2)\ldots(\Sigma - \zeta_{n-1})e_1.
W_{\Lambda}\left( \phi , \phi^{*}, \overline{{{\phi}}}, \pi \right) \equiv \overline{{{W}}}\left( \phi , \phi^{*}, \overline{{{\phi}}}, \pi , \varphi_{q}\left( \phi , \phi^{*}, \overline{{{\phi}}}, \pi \right) , \varphi^{*}= 0 , \overline{{{\varphi}}}= 0 , \pi_{\varphi}= 0 \right)
\begin{aligned}\lambda_1(t) = \gamma_1 \lambda_2(t) = \lambda_0 b_M (T - t) \lambda_3(t) = \lambda_0 \bigl( b_E (T - \tau_s) + \frac{1}{2}b_M k_M (\tau_s - t)^2 + ( b_E \mbox{}- a_M b_M k_M \frac{\gamma_1}{\lambda_0}+ b_M k_M (T - \tau_s)) (\tau_s - t) \bigr) \end{aligned}
\begin{cases}\lambda_{\max}(D, g) = \lambda_{\max}(D + (s), g - \log |s|), \lambda_{\max}^{\mathrm{asy}}(D, g) = \lambda_{\max}^{\mathrm{asy}}(D + (s), g - \log |s|), \hat{\mu}_{\mathrm{ess}}(D, g) = \hat{\mu}_{\mathrm{ess}}(D + (s), g - \log |s|).\end{cases}
\sqrt{- G}( R_{A B}- \frac{1}{2}G_{A B}R_{5}) = - k_{5}^{2}\sum_{\alpha}\sqrt{- g_{\alpha}}g_{m n}^{\alpha}\delta_{A}^{m}\delta_{B}^{n}{\cal L}_{\alpha}\delta ( y - y_{\alpha}) + 2 k_{5}^{2}\sqrt{- G}\Lambda G_{A B}.
\nabla : (\varprojlim_n L_n^\times) \otimes_\mathbb{Z}T^* \longrightarrow \mathbf{A}_L^{\psi=1}u \otimes at_0^* \longmapsto a \frac{\partial_\mathrm{inv}(g_{u,t_0})}{g_{u,t_0}}(\iota_{LT}(t_0)) \ .
J\approx\frac{\hat{z}\epsilon_{0}}{2\omega}\int(|E_{L}|^{2}-|E_{R}|^{2})d^{3}r+\frac{\hat{z}\epsilon_{0}}{2i\omega}\int\sum_{i=x,y,z}({E^{i}}^{*}\frac{\partial}{\partial\phi}E^{i})d^{3}r.
{\mathbf D}_{\lambda+N-1}\circ{\mathbf D}_{\lambda+N-2}\circ \dots \circ{\mathbf D}_\lambda \qquad{}= \pi^{n(N-1)}\left(\prod_{j=1}^{N-1}\Gamma(\lambda+j) \Gamma(n-\lambda-j)\right)^{-1}I_{-\lambda+n-N}\circ M^N\circ I_{\lambda}.
u_1(y+\rho\eta)=u_1(y)+A_1\rho+A_2\rho^{2}+o(\rho^{2}), &u_2 (z+\rho\eta)=u_2(z)+B_1\rho+B_2\rho^{2}+o(\rho^{2}), &u(x+\rho\eta)=u(x)+C_1\rho+C_2\rho^{2}+o(\rho^{2})
P(N\le y) =P(M\le y-1)+P(M=y,\; Z=0)+P(M=y,\; Z=1,\; Z_+=0) &+P(M=y+1,\; Z=0,\; Z_-=1) &=F_{y-1}+p_{y}(1-q)+p_{y}q(1-\pi_{y}^{(d)})+p_{y+1}(1-q)\rho_{y+1}^{(d)}&=F_{y}+ p_{y+1}(1-q)\rho_{y+1}^{(d)},
X_{n}\, =\, \sum_{0 \le \ell \le k \le n-1}{k \choose \ell}u^{\ell}( 1- a)^{\ell}a^{k-\ell}\varepsilon_{n-k, \ell}\, , \widetilde{X}_{n}\, =\, \sum_{0 \le \ell \le k \le n-1}{k \choose \ell}u^{\ell}( 1- a)^{\ell}a^{k-\ell}\varepsilon_{n-k, \ell+1}\, ,
\mathcal{T}^4\left\lbrace \begin{array}{l}\lbrack\widehat{p}_{\mu},\widehat{p}_{\nu}]=0, \Delta(\widehat{p}_{i})=\widehat{p}_{i}\otimes e^{-{\frac{\widehat{p}_{0}}{\kappa}}}+1\otimes\widehat{p}_{i}, \Delta(\widehat{p}_{0})=\widehat{p}_{0}\otimes1+1\otimes\widehat{p}_{0}, \end{array}\right.
V_{2r-2}\sum_{k=0}^{n-1}S_k^{(2r-1)}-V_{2r-2}n\sum_{j=0}^{n-1}(2j+1)^{2r-2}= \frac{V_{2r-2}}2\sum_{j=1}^{n-1}(2j+1)^{2r-2}\sum_{l=1}^j(2n-3l)\binom{n}l^2\binom{2l}l = \sum_{l=1}^{n-1}\frac{V_{2r-2}n^2(2n-3l)}{l^2}\binom{n-1}{l-1}^2\binom{2l-1}{l-1}\sum_{j=l}^{n-1}(2j+1)^{2r-2}.
\mathcal{K}_3(x) \leqslant\frac{A}{\sqrt{\Delta}}\sum\limits_{k>\frac{x-1}{c_5-1}}\frac{1}{\sqrt{k}}\mathbb{P}(\eta=\kappa+k) &\leqslant \frac{A}{\sqrt{\Delta}}\sqrt{\frac{c_5-1}{x-1}}\mathbb{P}\bigg(\eta>\kappa+\frac{x-1}{c_5-1}\bigg)
q_{3,3}'(0)=\sum\limits_{r=0}^{\min(n_1,\Delta n)}\sum\limits_{\substack{s=0 r+s\ge 1}}^{n_1}\sum\limits_{t=1}^{\Delta n}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^r \mathbf{d'}\in [n_1+1,n_2]_<^t d'_1=n_1+1}}\det (M_0|\mathbf{a}_2^\ast)_{(2r+s+1)}.
\lambda_k(\xi)\geq \inf_{\substack{V\,\textrm{subspace of}\, \Lambda_\xi^+ H_\xi^{1/2}\dim V=k}}\sup_{u_\xi\in V\setminus \{0\}}\frac{\left(D_{\gamma,\xi}\,u_\xi,u_\xi\right)}{\|u_\xi\|_{L^2_\xi}^2}.
\widehat{U}(\rho,\xi,y)= \frac{y^{2s}}{4^s\Gamma(s)}\int_0^\infty e^{-y^2/(4\tau)}e^{-\tau(i\rho+|\xi|^2)}\widehat{u}(\rho,\xi)\,\frac{d\tau}{\tau^{1+s}}.
|\partial^{\alpha_{1}}_{\bar{\xi}_{1}}\partial^{\alpha_{2}}_{\bar{\xi}_{2}}\widetilde{m}^{\varepsilon}(\bar{\xi})|\lesssim \prod_{i=1}^{2}\frac{1}{dist(\bar{\xi}_{i},\Gamma_{i})^{|\alpha_{i}|}}\cdot\langle\log_{2}dist(\bar{\xi}_{1},\Gamma_{1})\rangle^{-(1+\varepsilon)}
\left\langle T_{1 1}( \vec{x}) \right\rangle = - \frac{1}{3 2 \pi^{2}}F_{0}( \vec{x}) - \frac{1}{1 6 \pi^{2}}\left( D_{1}( \vec{x}) - D_{2}( \vec{x}) \right) + \frac{1}{6 4 \pi^{2}}\left( F_{1}( \vec{x}) - F_{2}( \vec{x}) \right)
\left(\frac{m+1}{m-1}\right) U(\kappa U^2)^{-1/(m+1)}\,{}_2F_1 \left[\frac{1}{m+1},\frac{m-1}{(m+1)n};1+\frac{m-1}{(m+1)n};\frac{\gamma}{\kappa}U^n\right] = \pm\xi + \xi_0.
f_{Cayley}(x_d) = \frac{1}{\prod_{i=1}^{d-1}(s_i-t_i)}\sum_{\sigma \in S_d}(-1)^\sigma \prod_{i=1}^d p_{\sigma_i}[x_d](t_1,\dots,t_{i-1},s_i,\dots,s_{d-1}),
J_{B \mu}'~=~F_{\mu \nu}\partial^{\nu}\frac{1}{\partial_{i}\partial^{i}}{\overline{\cal P}}+ m^{2}( A_{\mu}+ \partial_\mu \rho ) \frac{1}{\partial_{i}\partial^{i}}{\overline{\cal P}}+ B \partial_{\mu}\frac{1}{\partial_{i}\partial^{i}}{\overline{\cal P}}.
\cos^2 \phi_{Li, k}= \Vert U_i U_i^{\top}U_* e_k \Vert_F^2, \cos^2 \phi_{Ri, k}= \Vert e_k^{\top}V_*^{\top}V_i V_i^{\top}\Vert_F^2, &\sin^2 \phi_{Li, k}= \Vert (I-U_i U_i^{\top}) U_* e_k \Vert_F^2, \sin^2 \phi_{Ri, k}= \Vert e_k^{\top}V_*^{\top}(I-V_i V_i^{\top}) \Vert_F^2.
A=(\begin{matrix}-1&0&-1&1&1&3&0\\ 0&1&0&0&0&0&0\\ 2&1&2&-1&-1&-6&0\\ -2&0&-1&2&1&3&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ -1&-1&0&1&2&4&1\end{matrix})
r_1=-|A_0| h_{1212}^0, r_2=-2|A_0| h_{1213}^0, r_3=-2|A_0| h_{1223}^0, & r_4=-|A_0| h_{1313}^0, r_5=-2|A_0| h_{1323}^0, r_6=-|A_0| h_{2323}^0.