harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
=\left\|\frac{f}{|x|^{\frac{\beta(1-\delta)-\gamma}{\delta}}}\right\|^{\delta}_{L^{\frac{\delta pQ}{\delta Q+p(\beta(1-\delta)-\gamma-a\delta)}}(\mathbb{G})}\left\|\frac{f}{|x|^{-\beta}}\right\|^{1-\delta}_{L^{q}(\mathbb{G})}.
{\bf s}\left({\bf x}\right) = \frac{m}{2g}\left[-\frac{1+ \cos\theta}{2g r \sin\theta}\hat{\varphi}\,\tau_{3}+\left((\hat{\varphi}+i m\hat{\theta})e^{\pm i\varphi}\tau_{+}+\mbox{h.c.}\right)\right] \quad , \quad m=\pm 1.
\begin{cases}\nabla \times{\bf H}=\dfrac{4\pi}{c}{\bf J}+\dfrac{1}{c}\dfrac{\partial{\bf D}}{\partial t}\nabla \cdot{\bf D}=4\pi \rho \nabla \times{\bf E}=-\dfrac{1}{c}\dfrac{\partial{\bf B}}{\partial t}\nabla \cdot{\bf B}=0,\end{cases}
\begin{cases}\phi_s''-\Delta \phi_s+ \sum_{r=1}^d\alpha_{rs}\phi_r=0 \hbox{in}(0, +\infty)\times \Omega, \phi_s=0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu\phi_s+\sum_{r=1}^d\beta_{rs}\phi_r=0 \hbox{on}(0, +\infty)\times \Gamma_1.\end{cases}
\frac{\mathbb{P}\left(\frac{\log Z_{n}-n\mu}{\sigma\sqrt{n}}\geq x\right)}{1-\Phi(x)}=1+o(1)\quad\frac{\mathbb{P}\left(\frac{\log Z_{n}-n\mu}{\sigma\sqrt{n}}\leq -x\right)}{\Phi(-x)}=1+o(1)
\begin{cases}\Delta w_{1n}+e^{w_{2n}}(1-r_n^2e^{w_{1n}})=\displaystyle 4\pi\sum_{i=1}^{N_1}\delta_{r_n\epsilon_n p_{1i}}, \Delta w_{2n}+e^{w_{1n}}(1-r_n^2e^{w_{2n}})=\displaystyle 4\pi\sum_{i=1}^{N_2}\delta_{r_n\epsilon_n p_{2i}},\end{cases}\mathbb R^2.
\overline{u}_{\tau,\varepsilon}(t):=u^{k}, t\in(t^{k-1},t^k],\,\underline{u}_{\tau,\varepsilon}(t):=u^{k-1}t\in[t^{k-1},t^k),\,\, k\in\mathcal{K}^0_\tau\,,
F^{ab}={1\over 2}({F^+}{}^{ab}+{F^-}{}^{ab})={1\over 2}({f_\beta^+}{}^\alpha{(\sigma^{ab})_\alpha}{}^\beta+{{f^-}^{\dot\beta}}_{\dot\alpha}{(\bar\sigma^{ab}){}^{\dot\alpha}}_{\dot\beta}),
I = ( \sigma^{2}i \partial_{\mu}\gamma^{\mu})_{\alpha \alpha_{j}}( i \partial_{\nu}\gamma^{\nu})_{\alpha \alpha_{i}}\Psi_{\alpha_{i}\alpha_{j}}^{( \pm )}= m^{2}\sigma_{\alpha_{i}\alpha_{j}}^{2}\Psi_{\alpha_{i}\alpha_{j}}^{( \pm )}
-L u_n + g_n \circ u_n = f \leq |f| = -L \tilde{u}_n + g_n\circ \tilde{u}_n \,\,\mbox{in}\,\, \Omega, u_n =0\,\,\,\,\,~~~~~~~~~~~~~~~~~~~~~~~~~~\, \tilde{u}_n= 0\,\,\mbox{on}\,\,\partial\Omega.
x_1^\prime = a_{11}x_1 + a_{12}x_2 \, , x_2^\prime = a_{21}x_1 + a_{22}x_2 \, ; y_1^\prime = b_{11}y_1 + b_{12}y_2 \, , y_2^\prime = b_{21}y_1 + b_{22}y_2 \, .
R_{a b}^{e f}( \vartheta - \vartheta^{\prime}) \left[ t_{e c}( \vartheta ) \right]_{\alpha \gamma}\left[ t_{f d}( \vartheta^{\prime}) \right]_{\gamma \beta}= \left[ t_{a e}( \vartheta^{\prime}) \right]_{\alpha \gamma}\left[ t_{b f}( \vartheta ) \right]_{\gamma \beta}R_{e f}^{c d}( \vartheta - \vartheta^{\prime})
w_{s,b,i}^{(3)}(1/2)=\lim_{\theta\rightarrow 1/2^-}u_{s,b,i}^{(3)}(\theta)/\|u_{s,b,i}^{(3)}(\theta)\|,w_{s,b,i}^{(4)}(1/2)=\lim_{\theta\rightarrow 1/2^+}u_{s,b,i}^{(4)}(\theta)/\|u_{s,b,i}^{(4)}(\theta)\|.
\begin{array}{lll}{\tt{Angle}}(x,y) = \mbox{Arccos}\left(\frac{x-x_P}{R_0}\right) \mbox{if}y \geq y_P \mbox{and}{\tt{Angle}}(x,y) = 2\pi - \mbox{Arccos}\left(\frac{x-x_P}{R_0}\right) \mbox{if}y<y_P.\end{array}
\varphi^\ast(\psi^\ast(wz)v\psi^\ast(z)^{-1}\psi^\ast(z)y)u\varphi^\ast(\psi^\ast(z)y)^{-1}&= (\psi\varphi^{vy})^\ast(wz)\varphi^\ast(vy)u\left((\psi\varphi^y)^\ast(z)\varphi^\ast(y)\right)^{-1}&= (\psi\varphi^y)^\ast(wz)\varphi^\ast(vy)u\varphi^\ast(y)^{-1}(\psi\varphi^y)^\ast(z)^{-1}\ ,
f_{Y_{[r_{1}:n]},\cdot\cdot\cdot,Y_{[r_{k}:n]}|X_{r_{1}:n}\cdot\cdot\cdot X_{r_{k}:n}}(y_{1},\cdot\cdot\cdot,y_{k}|x_{1},\cdot\cdot\cdot,x_{k})=\prod_{i=1}^{k}f_{Y|X}(y_{i}|x_{i})
j_{k}=\frac{1}{\varepsilon^{3}}\left(g_{ij}(\varepsilon_{k}z)\partial_{i}u_{k}(\exp_{q_{k}}(\varepsilon_{k}z))\partial_{j}u_{k}(\exp_{q_{k}}(\varepsilon_{k}z))|g_{q_{k}}(\varepsilon_{k}z)|^{\frac{1}{2}}\right)^{\frac{1}{2}}\mathbb{I}_{B_{g}(q_{k},r)}
A(\Phi)=\left(\begin{array}{c}-\partial_x^3 u+a_0\partial_x^5 u-a_1 \partial_x u-(-1)^k \displaystyle \int_0^\infty g(s)\partial_{x}^{2k}\eta^t(x,s)ds u-\partial_s\eta^t\end{array}\right)
[\phi (y) \, , \, \phi(y^\prime)]=-{\frac{i\pi \beta^2}{2}}s\left (\frac{y-y^\prime}{R}\right)\quad , \quad [\bar \phi (\bar y) \, , \, \bar \phi (\bar y^\prime )]={\frac{i\pi \beta^2}{2}}s \left (\frac{\bar y-\bar y^\prime}{R}\right) \quad ,
\left[ T_{\underline{{p}}}^{( N )}, T_{\underline{{q}}}^{( N )}\right] \ = \ \frac{N}{2 \pi M}\mathrm{s i n}\left( \frac{2 \pi M}{N}\left( p_{1}q_{2}- p_{2}q_{1}\right) \right) T_{\underline{{p}}+ \underline{{q}}( \mathrm{m o d}N )}^{( N )}\ .
\bar p=\underset{p \in \mathcal{C}(u)}{\mbox{argmax}}\ \frac{1}{r}\Vert d(p) \Vert^r -\Vert d(p) \Vert^{r-2}\Vert \mbox{proj}_{T_K(u)}\left( \frac{-p}{\Vert d(p) \Vert^{r-2}}\right) \Vert^2,
\frac{\partial^2\tilde\phi}{\partial \tilde t^2}-\frac{\partial^2\tilde\phi}{\partial \tilde x^2}+\tilde\eta_m\tilde\phi^2\frac{\partial\tilde\phi}{\partial \tilde t}=-\tilde\phi+\tilde\alpha\tilde\phi^2-\tilde\lambda\tilde\phi^3+\tilde\xi_m\tilde\phi.
R l(\alpha_\mu;\beta_\nu)\delta(\alpha^2-p^2) \delta(\beta^2-p^2)\delta\left( (\alpha\beta)-q\right) =l(\alpha_\mu; P_\rho^\sigma(\alpha)\beta_\sigma)\delta(\alpha^2-p^2) \delta(\beta^2-p^2)\delta\left( (\alpha\beta)-q\right)
\begin{cases}\,\,p\in H^1_0(\Omega)\,\,\,\, begin{align*}0.3cm] \int_{\Omega}\nabla v\nabla p dx-\int_{\Omega}k^2 v pdx=-2\int_{\Omega}(\nabla\eta-A)\nabla vdx-2\int_{\Omega}(\eta-\eta_0)vdx\;v\in H^1_0(\Omega). \end{cases}
e_0= \int_S\int_S \theta(z,t)\theta(\xi,t)\Gamma(z-\xi)d\xi dz=\sum_{n\in \mathbb{Z}}\sum_{m\in \mathbb{Z}}\int_{n<x<n+1}\int_{m<\xi_1<m+1}\theta(z,t)\theta(\xi,t)\Gamma(z-\xi)d\xi dz.
\int_0^b\|F(t,u,v)\|_{{\cal H}\times{\cal H}}dt = \int_0^b\sqrt{\|v\|^2+\|\gamma(t) v+\lambda(t)Bu\|^2}dt \leq \int_0^b\sqrt{(1+2\gamma^2(t))\|v\|^2+2\lambda^2(t)\|Bu\|^2}dt \leq \int_0^b\left((1+\sqrt{2}\gamma(t))\|v\|+\sqrt{2}\lambda(t)\|Bu\|\right)dt
R_{,u}= S_{\mathrm{IPLA}}(R_{\mathrm{IPLA},u}^*) = \sum_{i=1}^{N_{max}}\frac{R_{\mathrm{IPLA},u}^*}{i\cdot\pi}\int_{0}^{\infty}\bigg[ \mathcal{I}m \Big\{\Phi_{\mathrm{IPLA}}(t;i, R_{\mathrm{IPLA},u}^*) \quad\quad\quad- \Phi_{\mathrm{IPLA}}(t;i-1,R_{\mathrm{IPLA},u}^*) \Big\}\bigg] dt,
\varepsilon_1\cdot e^{\sigma^6k_0}\prod_{m=k_1+1}^{k_0}\bigg(\frac{\alpha k_0^2}{m^2}\bigg)\geq B_{k_1}\geq \varepsilon_1\cdot e^{-\sigma^6k_0}\prod_{m=k_1+1}^{k_0}\bigg(\frac{\alpha k_0^2}{m^2}\bigg),
\left[ \mathcal{V}^{(n)}+ \sum_{i=1}^n \mathcal{Q}^{(n)}_i \frac{\partial^{L-1}}{\partial x_i^{L-1}}\right] \bar{\mathcal{F}}_n (\bar{X}^{1,n}) = \Delta_{L-1}\bar{\mathcal{F}}_n ( \bar{X}^{1,n}) \; ,
\sum_{j=1}^k \sum_{d_i\in S_j}\frac{|d_i-d_{S_j}|}{d_id_{S_j}}\leq \sum_{j=1}^k n^2\|\tilde{D}^{-1}\|^2\|M\|^{3/4}\varepsilon^{1/4}\leq n^3\|\tilde{D}^{-1}\|^2\|M\|^{3/4}\varepsilon^{1/4}.
\underline{{{\Omega}}}_{I J}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{\frac{1}{V}}}&{{- \underline{{{\Omega}}}_{C + +}}}\\{{0}}&{{- \frac{1}{V}}}&{{0}}&{{- \underline{{{\Omega}}}_{C --}}}\\{{- 1}}&{{\underline{{{\Omega}}}_{C + +}}}&{{\underline{{{\Omega}}}_{C --}}}&{{0}}\\ \end{array}\right) .
\|N v\|_{L^1(\partial B_{\rho})}= \|(N v)^{-}\|_{L^1(\partial B_{\rho})}+ \|(N v)^{+}\|_{L^1(\partial B_{\rho})}\leq 2 \|(N v)^{-}\|_{L^1(\partial B_{\rho})}+ \varepsilon \|\nabla u \|_{L^{1}(B_{\rho})}&\leq 2 \| N u \|_{L^1(\partial B_{\rho})}+ \varepsilon \|\nabla u \|_{L^{1}(B_{\rho})}.
\bigl( \mathcal{T}_{k_1}^{A,d}\times \mathcal{T}_{k}^{A,d}\bigr)\cap ( \mathfrak{D}_{j_1}^A \times \mathfrak{D}_{j}^A )\not= \emptyset, |\Phi(\xi_1, \eta_1, \xi, \eta)| \geq A^{-\frac{3}{2}}d^{-1}N_1^{3}\ \ \textnormal{for any}\ (\xi_1, \eta_1)\times (\xi,\eta) \in\mathcal{T}_{k_1}^{A,d}\times \mathcal{T}_{k}^{A,d}.
\chi_a \left(h(A_G)-h(A_{G'})\right) \chi_b = \frac{1}{2\pi i}\int_{\Gamma}\mathrm{d}z\, h(z) \chi_a R_z(A_{G}) \left( \chi_G A \chi_{G'\setminus G}\right) R_z(A_{G'})\chi_b,
\tilde{C}^{\alpha'}_\mu M_{\alpha' \beta'}\tilde{C}^{\alpha' \mu}= \sum_{(a)}\bar{\tilde{\phi}}^{(a)}_V \left[ -\tilde{g}_3 \tilde{C}^\alpha_\mu \frac{\lambda_\alpha}{2}- \tilde{g}_1 (\tilde{n}+1/3) \tilde{B}_\mu \right]^2 \tilde{\phi}^{(a)}_V,
\int\limits_{\psi_0}^{\pi-\psi_0}Cf(0, e_n, \psi)g(\psi_0,\psi)d\psi= \frac{(\cos\psi_0)^{n-3}}{|S^{n-3}|}\int\limits_{S^{n-2}}Rf((\cos\psi_0)\omega,\sin \psi_0),0)d\omega.
4|y_i-y_j|\geq |y_i+y_j|\cdot|y_i-y_j|=\left|y_i^2-y_j^2\right|=\left|(\gamma_n(u_i))^2-(\gamma_n(u_j))^2\right| = \left|(1+|u_j|/\ln n)^2-(1+|u_i|/\ln n)^2\right|S(I)^2 = \big||u_j|-|u_i|\big|/\ln n \cdot (2+|u_j|/\ln n+|u_i|/\ln n)S(I)^2 \geq \big||u_j|-|u_i|\big|/\ln n\cdot2S(I)^2\geq \varepsilon_1/\ln n\cdot2S(I)^2,
B_{0 i}( T ) \rightarrow \frac{- 2 v^{2}}{\pi \kappa}\epsilon_{0 i j}p^{j}T \frac{\partial}{\partial m_{\sigma}^{2}}\left[ \frac{m_{\sigma}^{2}\log ( M / m_{\sigma})}{m_{\sigma}^{2}- M^{2}}\right] \equiv \frac{1}{2}{\cal{\pi}}_{0 i}( p_{0}= 0 , T ) .
{\mathbf{A}}:=\frac{1}{\det(D\chi_{{\mathbf{x}}_{0}})}(D\chi_{{\mathbf{x}}_{0}})^{\intercal}(D\chi_{{\mathbf{x}}_{0}}),\qquad{\mathbf{B}}:=\left(\det(D\chi_{{\mathbf{x}}_{0}})\right) (D\chi_{{\mathbf{x}}_{0}})^{-1}(D\chi_{{\mathbf{x}}_{0}})^{-T};
\frac{s_k}{s_{k+1}}= \frac{k+1}{\AA (i,{i + k+1})}\frac{k-j+1}{k-j+2}\left[ \frac{\big(\AA (i,{i + k+1})\big)}{\big(\AA (i,{i + k})\big)}\right]^k \geq\frac{k+1}{2e(\ln k +2)}.
\lim_{r \to \infty}r^{-5}\int_{\Sigma_r}\tau div_{\sigma}\alpha_H d \Sigma_r = \int_{S^2}\left[\frac{4a^0}{3}W_0^2 +\frac{2}{3}a^iW_iW_0- (a^i \tilde{X}^i) |\beta|^2 \right]dS^2.
(I \!+\! N(\lambda \!+\! i \tau))^{-1}= \sum_{M = 0}^{M_0}(-1)^M \; N(\lambda)^M+ (-1)^{M_0 + 1}\; N(\lambda)^{M_0 + 1}(I\!+\! N(\lambda \!+\! i \tau))^{-1}.
\mathcal{A}^y_{h_L, \, F}(f_0) - \mathcal{A}^y_{h_L, \, F}(\tilde{f}_0) = \varphi_{h_L}(f_0, \, \tilde{f}_0) (f_0^x - \tilde{f}_0^x) + \psi_{h_L}(f_0, \, \tilde{f}_0) (f_0^y - \tilde{f}_0^y) & + \theta(f_0, \, \tilde{f}_0) (f_0 - \tilde{f}_0) \cdot h_L,
Y_n(z) = \begin{pmatrix}\frac{1}{\kappa_n}p_n(z) \frac{1}{\kappa_n}C(p_n w)(z) -2\pi i \kappa_{n-1}p_{n-1}(z) -2\pi i \kappa_{n-1}C(p_{n-1}w)(z)\end{pmatrix},
\{T_1^3,\, T_2^3,\, T_3^3,\,T_4^3, &T_1^2T_2,\,T_1^2T_3,\, T_1^2T_4,\, T_2^2T_3,\, T_2^2T_4,\, T_3^2T_4, &T_1T_2^2,\, T_1T_3^2,\, T_1T_4^2,\, T_2T_3^2,\, T_2T_4^2,\, T_3T_4^2\}
\left|\mathcal{I}\right| =\left|\{(\mathcal{P},C):C\in\mathcal{P}\mbox{is red}\}\right|+\left|\{(\mathcal{P},\{C,D\}):\{C,D\}\subseteq\mathcal{P}\}\right| & \leq\frac{16\cdot2^{2n}}{c^{3}n^{2+3\varepsilon}}\cdot p_{n}+\frac{8\cdot2^{3n}}{c^{4}n^{3+4\varepsilon}}.
|Df(e^{i\varphi})|^2 =(|f_z(e^{i\varphi})|+|f_{\bar z}(e^{i\varphi})|)^2 &=\lim_{z\to e^{i\varphi}}(|f_z(z)|+|f_{\bar z}(z)|)^2 &\le K \lim_{z\to e^{i\varphi}}(|f_z(z)|^2-|f_{\bar z}(z)|^2) =K(|f_z(e^{i\varphi})|^2-|f_{\bar z}(e^{i\varphi})|^2)=KJ_f(e^{i\varphi}).
\left<f,g\right>=\sum_{|k|=m}\int_{\mathbf{B}}(1-|x|^{2})^{2m}\frac{\partial^{m}f}{\partial x^{k}}(x)\overline{\frac{\partial^{m}g}{\partial x^{k}}(x)}d\tau(x),\enspace f,g\in B^{2}.
|G_r(r,z,\bar r,\bar z)| \, \le\, C\,\frac{|z-\bar z|}{r^{3/2}\,\bar r^{1/2}}\,\frac{r^{3/2}\,\bar r^{3/2}}{\bigl((r-\bar r)^2 + (z-\bar z)^2\bigr)^{3/2}}\, =\, C\,\frac{\bar r\,|z-\bar z|}{\bigl((r-\bar r)^2 + (z-\bar z)^2\bigr)^{3/2}}\,\le\, \frac{C}{\bigl((r-\bar r)^2 + (z-\bar z)^2\bigr)^{1/2}}~,
{\frac{T_{Q}}{2 . 7 3 \, \mathrm{K}}}\simeq \left({\frac{\kappa_{5}^{4}\lambda_{0}^{2}- 3 6 \xi^{2}}{2 \kappa_{5}^{4}| \lambda_{0}| \rho_{0}}}\right)^{\frac{3}{9 - k^{2}}}.
\|\varphi^{+}\|_{L^{m_{k}}(\partial B_1)}\leq(C m_0)^{\frac{1}{m_{0}}\sum_{j = k-l}^{k-1}\left(\frac{2}{2^{\star}}\right)^{j}}\left(\frac{2^{\star}}{2}\right)^{\frac{1}{m_{0}}\sum_{j = k-l}^{k-1}j\left(\frac{2}{2^{\star}}\right)^{j}}.
P\big(\cap_{n=1}^{2M}\big\{Z_n^{N,i}\in\{0,-1\}\big\}\big|\cup_{n=1}^M\{Z_{2n}^{N,i}=-1\}\big)\ge P\big(\cap_{n=1}^{2M}\big\{Z_n^{N,i}\in\{0,-1\}\big\}\big).
\frac{1+1/\tau'}{w_{i}+1}- \frac{w'_{s}-1/\tau'}{(w_i +1)w'_s}- \sum_{j=1}^{s-1}\frac{w'_{j}-1/\tau'}{w_i +1}\cdot \frac{\log q_{k'_j}}{w'_s \log q_{k'_{s}}},
\times \; \prod_{i = 1}^{m_{b}^{\prime}}\psi_{2}^{( b )}({y^{\prime}}_{i}^{( b )}) \prod_{i = 1}^{n_{b}}\overline{{{\psi}}}_{1}^{( b )}( x_{i}^{( b )}\! + \hat{\tau}) \prod_{i = 1}^{m_{b}}\psi_{2}^{( b )}( y_{i}^{( b )}\! + \hat{\tau}) \prod_{i = 1}^{n_{b}^{\prime}}\overline{{{\psi}}}_{1}^{( b )}({x^{\prime}}_{i}^{( b )}) \Big \rangle_{f r e e}\; .
\begin{cases}\|\partial_{x}^{\alpha}\Gamma_{\chi_{t},b,t,N,\omega}^{A}\|_{L^{2}(\Omega_{s}(\omega))}\leq c\tau^{|\alpha|-3/2}e^{-\tau(s-t)a_{A}}\|\partial_{x}^{\alpha}\Gamma_{\chi_{t},b,t,N,\omega}^{B}\|_{L^{2}(\Omega_{s}(\omega))}\leq c\tau^{|\alpha|-3/2}e^{-\tau(s-t)a_{B}}\end{cases}
\int_{b-iU}^{b+iU^{\prime}}\left( \sum_{n \leq x}f(n)^{-z}\right) \frac{y^z}{z}dz= \sum_{n \leq x}\int_{b-iU}^{b+iU^{\prime}}f(n)^{-z}\frac{y^z}{z}dz.
\pi_{ra}=\frac{1}{\sqrt{1-|\bar{\nabla}f|^2}}\left[ f_{ra}-r^{-1}f_a+(f_{rr}+r^{-2}\tilde{\Delta}f+2r^{-1}f_r) f_r f_a\right. \left.+\frac{f_r f_a}{{1-|\bar{\nabla}f|^2}}(\bar{g}^{kp}\bar{g}^{lq}f_p f_q)(\bar{\nabla}_k\bar{\nabla}_l f)\right],
\lbrack w^{1-p^{\prime}}]_{A_{p^{\prime}}\left( B\right)}^{\frac{1}{p^{\prime}}}=|B|^{-1}\Vert w^{^{1-p^{\prime}}}\Vert_{L_{1}(B)}^{\frac{1}{p^{\prime}}}\Vert w^{\frac{1}{p}}\Vert_{L_{^{p}}(B)}.
\frac{d^2}{dt^2}\int \lambda |u|^2 dx=\frac{d}{dt}(\Im \int \partial_x \lambda\partial_x u \bar u dx) = \int \lambda'' |u'|^2 dx- \frac 14 \int \lambda^{iv}|u|^2 dx+q \lambda^{''}(0) |u(t,0)|^2 +\frac \alpha{\alpha+2}\int \lambda'' |u|^{\alpha+2}dx.
\begin{cases}c_1 g(|v|) \le |\rho(.,v)| \le c_2 g^{-1}(|v|)\,, |v| \le 1 \,, \mbox{a.e. on}\Omega \,, c_1 |v| \le |\rho(.,v)| \le c_2 |v| \,, |v| \ge 1 \,, \mbox{a.e. on}\Omega \,,\end{cases}
\Phi(u^{g_{1}}\lambda_{1}\cdots u^{g_{n}}\lambda_{n}u^{g_{n+1}}\lambda_{n+1}) = \Phi(\lambda_{1}\cdots\lambda_{n+1}u^{g_{1}}\cdots u^{g_{n}}\otimes u^{g_{n+1}}) & = \lambda_{1}\cdots\lambda_{n+1}u^{[g_{1}]}\cdots u^{[g_{n}]}\otimes u^{[g_{n+1}]}g_{1}\cdots g_{n+1}& = \lambda_{1}\cdots\lambda_{n+1}u^{[g_{1}]}\cdots u^{[g_{n}]}u^{[g_{n+1}]}g_{1}\cdots g_{n+1}
\AA^X = \frac{1}{2}\sum_{i = 1}^m \sum_{j = 1}^m a_{ij}(x) \partial_{x_i}\partial_{x_j}+ \sum_{i = 1}^m b_i(x) \partial_{x_i}\AA^Y = \frac{1}{2}\sum_{i = 1}^n \sum_{j = 1}^n \rho_{ij}(y) \partial_{y_i}\partial_{y_j}+ \sum_{i = 1}^n \gamma_i(y) \partial_{y_i},
\tilde{L}({\rm D}(w)) =\sum_{N\geq2}(-1)^{N}\zeta^{\mathfrak{m}}(N)\tilde{L}({\rm Cut}_{(\{0\}^{N-1};\emptyset)}(w))+\sum_{N\geq2}\zeta^{\mathfrak{m}}(N)\tilde{L}({\rm Cut}_{(\emptyset;\{1\}^{N-1})}(w)).
\left(r_{c}^2 - r^2 \right) \partial_{r}^2 R(r) + \left(\frac{2}{r}- \frac{4r}{r_{c}^2}\right) \partial_{r}R(r) +\left(\frac{\omega^2 r_{c}^2}{(r_{c}^2 - r^2)}- \frac{\nu(\nu+1)}{r^2}- m^2\right) R(r) =0,
b(z)=\sum_{j,l,k}\gamma(b)_{j,l}^{k}\frac{(1-|\zeta_{j,l}^{k}|^{2})^{M-\frac{1}{q}}\widehat{\eta}(\rho_{j})^{-\frac{1}{q}}}{(1-\overline{\zeta_{j,l}^{k}}z)^{M}},z\in\mathbb{D},
\psi^B_{i,i}= \partial_1+\cdots+\partial_i \ \ \ \ \ \ \ \ \ \ \ \ \ \ \{\rm for}\i=1,\ldots, n, &\psi^B_{i,j}= \psi^B_{i-1,j-1}+\alpha_{i,j}\psi^B_{i,j-1}\ \ \ \ \ \{\rm for}\ (i,j)\in \triangledown_n,
\sum_{j=2}^{k}\frac{P_j( \xi_1+\lambda^{-1}y)}{\lambda^{\frac{N-2}2}}\le C\sum_{j=2}^{k}\frac{1}{( 1+ |y - \lambda(\xi_j-\xi_1)|)^{N-2}}\le C\sum_{j=2}^{k}\frac{1}{| \lambda(\xi_j-\xi_1)|^{N-2}}\le \frac{C |\ln k|^{\sigma_N}k^{N-2}}{\lambda^{N-2}}\le\frac{C |\ln k|^{\sigma_N}}{\lambda^{\frac{N-2}2}},
S_n = \rho_{n,0}+ \sum_{2 \leqslant i \leqslant 2s+4 \atop i \equiv \delta \pmod{2}}\rho_{n,i}~ \zeta_2\left( i+s+1, \frac{1}{4}\right), T_n = \sigma_{n,0}+ \sum_{i=2}^{s+2}\sigma_{n,i}~ \zeta_2\left( i+s+1, \frac{1}{4}\right),
{{\bf{Y}}^{[j]}}({t_1}) ={{\bf{H}}^{[j1]}}({t_1}){{\bf{u}}^{[1]}}+{{\bf{H}}^{[j2]}}({t_1}){{\bf{u}}^{[2]}},{{\bf{Y}}^{[j]}}({t_2}) ={{\bf{H}}^{[j1]}}({t_2}){{\bf{v}}^{[1]}}+{{\bf{H}}^{[j2]}}({t_2}){{\bf{v}}^{[2]}},
\phi ={\small \left[ \begin{array}{*{3}c}-1 1 0 1 0 -1 0 -1 1\end{array}\begin{array}{*{3}c}(x_1-x_2)^{r+1}0 0 0 (x_1-x_3)^{r+1}0 0 0 (x_2-x_3)^{r+1}\end{array}\right]}
\sum_{j=0}^k |y_{(j)}| \sqrt{1+\tfrac{1}{2}c_g^2 \ln\tfrac{2m}{j}}\le \sqrt{\sum_{j=1}^k y_{(j)}^2}\sqrt{\sum_{j=1}^k (1+\tfrac{1}{2}c_g^2 \ln\tfrac{2m}{j})}=\sqrt{\sum_{j=1}^k y_{(j)}^2}\sqrt{k+\tfrac{1}{2}c_g^2 \ln\tfrac{(2m)^k}{k!}}.
\begin{aligned}\int_{\Omega}dN_{\phi}(e_{\phi}^{i})\wedge\ast[\delta N_{\beta}(\omega),dN_{\phi}(e_{\phi}^{j})]_{1}=(-1)^{n-1}\int_{\partial\Omega}\langle dN_{\phi}(e_{\phi}^{i}),\delta N_{\beta}(\omega)\rangle_{\Lambda^{1}}\wedge e_{\phi}^{j}, \end{aligned}
2^{(k-1)(2k-1)}\lim_{u\to0^+}u^{k(2k-1)/2}\varOmega_{2k-1}(u)=(-1)^{\frac{(k-1)(k-2)}{2}}\frac{k[\Gamma(k/2)]^{2}}{2(2k+1)}(\det\mathbf N_{k-1})^2
u(z) =1-\alpha ze^{\lambda_1z}-\beta e^{\lambda_2z}+h.o.t., begin{align*}0.2cm] w(z) =0-\alpha \left[\lambda_1z+f(1)/(2\lambda_{1}-c)\right] e^{\lambda_1z}-\beta \lambda_2 e^{\lambda_2z}+h.o.t., begin{align*}0.2cm] v(z) =0-\alpha e^{\lambda_1z}-\beta\psi\left(\lambda_{2}\right) e^{\lambda_2z}+h.o.t..
\Psi_{\{[\{1,2\},3]\}}\Psi_{\{[\{1\},4],[\{2\},1]\}}= \Psi_{\{[\{1,2\}, 3], [\{3\}, 4], [\{4\}, 1]\}}+ \Psi_{\{[\{1,3\}, 3], [\{2\}, 4], [\{4\}, 1]\}}&+ \Psi_{\{[\{1,4\}, 3], [\{2\}, 4], [\{3\}, 1]\}}+ \Psi_{\{[\{2,3\}, 3], [\{1\}, 4], [\{4\}, 1]\}}& + \Psi_{\{[\{2,4\}, 3], [\{1\}, 4], [\{3\}, 1]\}}+ \Psi_{\{[\{3,4\}, 3], [\{1\}, 4], [\{2\}, 1]\}}
Q = \left( 1 + \frac{N + 1}{\ell}\right)^{N}\left( 1 + \frac{N}{\ell + 1}\right)^{- \frac{1}{2}}\left[ \left( 1 + \frac{N}{\ell + 1}\right)^{\ell + 1}e^{- N}\right] e^{J ( N + \ell + 1 ) - J ( \ell + 1 )},
\lim\limits_{a\to 0}E \hat{V}_{d-2}(Z\cap A) = e^{-\gamma V_d(K_0)}\bigg(\gamma \lim\limits_{a\to 0}E \hat{V}_{d-2}(K_0) &+ \frac{\gamma^2}{2}\sum_{l=1}^{2^{n^d}-2}w_l^{(d-2)}\sum_{S\subseteq B_l}(-1)^{|S|}\bigg(\int_{S^{d-1}}h(\check{S}\cup \check{W}_l,u) dS_{d-1}(K_0,du)\bigg)^2\bigg)
\alpha^{\prime \prime}+ ( n - 1 ) A \alpha^{\prime}- \frac 1 2 ( n - 1 ) C ( A + C ) \alpha - \left[ \frac{( n - 1 )^{2}}{4 R^{2}}+ m^{2}\right] \alpha = m \frac{\delta ( x - x^{\prime})}{\sqrt{g ( x )}},
\dim (z) = \dim \langle \nu_{z, \sigma_1^P}, \dots, \nu_{z, \sigma_k^P}, (0,\dots,0,1) \rangle &= \dim \langle (\nu_{z, \sigma_1}, 0), \dots, (\nu_{z, \sigma_k},0), (0,\dots,0,1) \rangle = \dim(z_s) +1.
\begin{aligned}R^{xy,v}_{x^4y}= [\varphi_{x^4y}] - [\gamma_{x^4y,x^4v}] - [\gamma_{x^4y,x^3yv}] R^{xy,w}_{x^4y}= [\varphi_{x^4y}] - [\gamma_{x^4y,x^4w}] - [\gamma_{x^4y,x^3yw}] R^{xy,z}_{x^4y}= [\varphi_{x^4y}] - [\gamma_{x^4y,x^4z}] - [\gamma_{x^4y,x^3yz}] \end{aligned}
{J}_1(\mathbf{k},\tau):= f \cos (\tau \mathcal{A}(\mathbf{k})^{1/2}) f^{-1}- f_0\cos (\tau \mathcal{A}^0(\mathbf{k})^{1/2})f_0^{-1}, &{J}_2(\mathbf{k},\tau):= f \mathcal{A}(\mathbf{k})^{-1/2}\sin (\tau \mathcal{A}(\mathbf{k})^{1/2}) f^* - f_0 \mathcal{A}^0(\mathbf{k})^{-1/2}\sin (\tau \mathcal{A}^0(\mathbf{k})^{1/2})f_0.
\mathcal{A}^{\rm wt}(\chi (\gamma)^{k-\kappa})\stackrel{\kappa:=k+\frac{\log(1+w)}{\log(1+p)}}{=}\mathcal{A}^{\rm wt}(\chi (\gamma)^{-\frac{\log(1+w)}{\log(1+p)}}) = \chi (\gamma)^{k-\kappa}=1+\log\chi(\gamma)\cdot (k-\kappa) + O((\kappa-k)^2)
\int_{S(I)}|\nabla\hat{f}(z)|^{2}(1-|z|^{2})^{p-2+2\beta}dm(z) \lesssim \int_{J}\int_{J}\frac{|f(e^{it})-f(e^{is})|^{2}}{|e^{it}-e^{is}|^{4-p-2\beta}}dtds+|I|^{2\beta+p}\left(\int_{|t|\geq2|J|/3}\frac{|f(e^{i(t+s_{0})})-f_{J}|}{t^{2}}dt\right)^{2},
A_{\lambda,r_1}(r_2,\tilde{r}_2,\theta_1,\theta_2,\theta) = a_\lambda(G(r_1,r_2,\theta_2-\theta))\overline{a_\lambda}(G(r_1,\tilde{r}_2,\theta_1-\theta))\eta(\theta_2-\theta)\eta(\theta_1-\theta).
((z,h)+(z',h'))+(z'',h'')=(z+z'-\beta(h,h'),h+h')+(z'',h'')= &=(z+z'+z''-(\beta(h,h')+\beta(h+h',h'')),h+h'+h'')= &=(z+z'+z''-(\beta(h',h'')+\beta(h,h'+h'')),h+h'+h'')= &=(z,h)+(z'+z''-\beta(h',h''),h'+h'')=(z,h)+((z',h')+(z'',h'')),
\liminf_{N\rightarrow+\infty}\frac{N}{\gamma_N^2}\log P(\frac{N}{\gamma_N}(\tau_c^N-\tau_c)>x) \geq -\left(\frac{d}{dt}\sum_{k=1}^3\mu_{t,k}(f_k)\Bigg|_{t=\tau_c}\right)^2x^2J_{contra, \tau_c}(1) &=-J_{hit}(x).
p\left({d_{pF}^{}\leq{\delta^{}}}\right) = p\left({d_{pF}^2 \leq{\delta^2}}\right) & = \int{\left({n - 1}\right){{\left({d_{pF}^2}\right)}^{n - 2}}d\left({d_{pF}^2}\right)}& ={\left( \delta \right)^{2n - 2}}.
{\mathbb R}_+^K \supset R_s \doteq \left( \bigcap_{n=1}^{|s|}\bigcap_{k_1, k_2 \in s(n)}\left\{t_{k_1}= t_{k_2}\right\}\right) \cap \left\{t_{s(1,1)}< t_{s(2,1)}< \cdots < t_{s(|s|,1)}\right\}.
\left| \frac{e^{i t \ln n}- e^{i t \ln (n-1)}}{t}- i\ln n + i \ln (n-1) \right|^2=\frac{\left(\cos\left(t \ln n\right)-\cos\left(t \ln (n-1)\right) \right)^2}{t^2}&+\left(\frac{\sin\left(t \ln n\right)-\sin\left(t \ln (n-1)\right)}{t}-\ln n + \ln(n-1)\right)^2= \widetilde{\theta}_n(t)+\widetilde{\widetilde{\theta}}_n(t).
\breve{M}(x,t,k)=\begin{pmatrix}1 0 0 \frac{k-i\kappa}{k}\end{pmatrix}+\frac{\breve{M}^r_{1}(x,t)}{k}+\frac{i\kappa}{k-i\kappa}P(x,t)+O(k^{-2}),{\rm as}\k\rightarrow\infty,
\xi_{2}=\frac{1}{\sqrt{F^2+K^2}}\left(y^i\frac{\partial}{\partial y^i}+p_i\frac{\partial}{\partial p_i}\right)\,,\,\xi_1=\phi(\xi_2)=\frac{1}{\sqrt{F^2+K^2}}\left(p^i\frac{\partial}{\partial y^i}-y_i\frac{\partial}{\partial p_i}\right),
\hat{k}_d(\Delta) = \prod_{q,i}x_{q,i}^{e(q,i)}\prod_{q=1}^{d+1}\prod_{\rho\in V'_1\times\cdots\times\widehat{V'_q}\times\cdots\times V'_{d+1}}D_{q,n_q}&= \prod_{q=1}^{d+1}\left[ \left(\prod_{i=1}^{n_i}x_{q,i}\right)^{e(q,i)}(x_{q,1}+\cdots+x_{q,n_q})^{\prod_{r\neq q}(n_{r}-1)}\right]
\frac{\mathcal{P}_X}{\varphi(V)}\sum_{u_1,\ldots,u_k}\Bigl(\prod_{i=1}^k\frac{\mu(u_i)^2}{g(u_i)}\Bigr)\sideset{}{^*}\sum_{s_{1,2},\ldots,s_{k-1,k}}\Bigl(\prod_{\substack{1\leq i,j\leq k i\neq j}}\frac{\mu(s_{i,j})}{g(s_{i,j})^2}\Bigr)y^{(m)}_{a_1,\ldots,a_k}y^{(m)}_{b_1,\ldots,b_k},
\int \prod_{j = 1}^{g}d \nu_{j}d \bar{\nu}_{j}d \hat{\nu}_{j}d \hat{\bar{\nu}}_{j}\mathrm{e}^{\int_{\Sigma_{g}}{\cal L}_{5}}= \left| \prod_{j = 1}^{g}\delta ( a_{j}) d a_{j}\right|^{2},
\begin{array}{l}{{\frac{M}{A_{8 - p}}= \frac{8 - p}{2}r_{H}^{7 - p}+ \frac{7 - p}{2}\alpha ,}}\\{{\left( \frac{Q_{0}}{( 7 - p ) A_{8 - p}}\right)^{2}= \alpha \left( \alpha + r_{H}^{7 - p}\right) ,}}\\ \end{array}
\int_{\mathcal{T}^d}\frac{1}{(1-r\langle x,u \rangle)^{|\boldsymbol{\lambda}|}}\prod_{i=1}^d u_i^{\lambda_i-1}\, du \qquad = \sum_{n_1=0}^\infty \cdots \sum_{n_d=0}^\infty \frac{(|\boldsymbol{\lambda}|)_{|\mathbf{n}|}}{n_1!\cdots n_d!}r^{|\mathbf{n}|}x_1^{n_1}\cdots x_d^{n_d}\int_{\mathcal{T}^d}\prod_{i=1}^d u_i^{\lambda_i+n_i-1}\, du .
\frac{dx}{dt}= 0.05x^2yz+0.05x^2y-0.05x^2z-0.05x^2+0.05xyz+0.05xy-0.05xz-0.05x+0.125y^3z-0.125y^3 &+0.125y^2z-0.125y^2+0.2yz^5+0.2yz^4-0.2z^5-0.2z^4; \frac{dy}{dt}= 0.125y^2z-0.125y^2+0.125yz-0.125y+0.2z^5+0.2z^4 \frac{dz}{dt}=-0.1z^2-0.1z
G =[[0,8n^2-8n+3],[1/2,-16n^4]] =[[1,8n^2-8n+7],[-1/2,-16n^4]]=[[8/9,8n^2-8n+19],[1/2,-16n^4]] =[[209/225,8n^2-8n+39],[-1/2,-16n^4]] =[[10016/11025,8n^2-8n+67],[1/2,-16n^4]]\;,
f_4^{(1)}:=u_{1,2}u_{2,3}u_{3,4}=u_{2,3}u_{3,4}u_{1,4}+u_{3,4}u_{1,4}u_{1,2}+u_{1,4}u_{1,2}u_{2,3}, f_4^{(2)}:=u_{2,3}u_{3,4}u_{1,2}u_{2,3}=u_{1,2}u_{3,4}u_{2,3}u_{1,4}+u_{1,2}u_{2,3}u_{1,4}u_{1,2}+u_{2,3}u_{1,4}u_{1,2}u_{3,4}\hphantom{f_4^{(2)}:=}{}+u_{3,4}u_{2,3}u_{1,4}u_{3,4}-u_{1,4}u_{1,2}u_{3,4}u_{1,4}.
\bar{a}_{11}^i = - ({a^i_{12}}+{a^i_{13}}+{a^i_{21}}) , \bar{a}_{12}^i = a^i_{23}-a^i_{13}, \bar{a}_{21}^i ={a^i_{21}}-{a^i_{31}}, \bar{a}_{22}^i = - ({a^i_{23}}+{a^i_{31}}+{a^i_{32}}) .
V ( z ) ={\frac{6 \bar{K}^{2}}{( 9 a^{2}- 4 )^{2}}}{\frac{1 0 - 9 a^{2}}{( 1 + \bar{K}| z | )^{2}}}-{\frac{1 2 \bar{K}}{4 - 9 a^{2}}}\delta ( z ) .
H = - \frac{e}{2}\pi^{2}+ \imath \pi_{\mu}\psi^{\mu}\chi - \left( \frac{1}{8 !}\epsilon_{\nu \mu \rho_{2}\ldots \rho_{9}}\pi^{\mu}\psi^{\rho_{2}}\psi^{\rho_{9}}+ \frac{\alpha}{2^{8}}\pi_{\nu}\right) b^{\nu}.
V_{a b c_{n}d_{n}}^{n + 1}( p , T , M ( T ) ) = \frac{1}{2}\rho^{n + 1}\Bigl ( \frac{\Sigma_{r e n .}^{1}( p , M ( T ) ) + \Sigma_{T}^{1}( p , M ( T ) , T )}{2}\Bigr )^{n}v_{a b c_{n}d_{n}}^{( n + 1 )}.
\pi_{\omega_j}(m') =m'+\omega_j(m')=m'+M^{-1}(\omega(M(m')+v_j)) &=M^{-1}(\underset{m}{\underbrace{M(m' ) +v_j}}+\underset{\omega(m)}{\underbrace{\omega(M(m')+v_j)}})-M^{-1}(v_j)=M^{-1}(\underset{\pi_\omega(m)}{\underbrace{m+\omega(m)}})- M^{-1}(v_j) &=M^{-1}(M(n) + v_j)-M^{-1}(v_j)=M^{-1}(M(n))=n.
\widetilde{T}(z_{0}): \mathcal{S}(\mathbb{R}^{2n}) \to \mathcal{S}(\mathbb{R}^{2n}), &\widetilde{T}(z_{0})\Psi(z) = e^{-\frac{i}{\hbar}\sigma(z,z_{0})}\Psi(z-\tfrac{1}{2}z_{0}).
\sum\limits_{\left\{i,j\right\}\in E\left( G^{\prime}\right)}x_{i}x_{j}=\sum\limits_{\left\{i,j\right\}\in E\left( G\right)}x_{i}x_{j}-x_{k+s-1}x_{k+s}-x_{k+s+2}x_{k+s+3}+x_{k+s}x_{k+s+2}+x_{k+s-1}x_{k+s+3}& =\sum\limits_{\left\{i,j\right\}\in E\left( G\right)}x_{i}x_{j}+\left( x_{k+s-1}-x_{k+s}\right)^{2}.