harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left|\iint J_{3,1}dF_\tau(v)dF_\tau(w)\right| &= 2k \left|\int \sin^{2k-1}\frac{\xi^+\cdot v}{2}\cos \frac{\xi^+\cdot v}{2}dF_\tau(v) \int \Big( \sin \frac{\xi^-\cdot w}{2}-\frac{\xi^-\cdot w}{2}\Big) dF_\tau(w)\right|.
\dim \mathcal{W}_0' - \dim p_1(\mathcal{W}_0') = \dim p_1^{-1}(p_1(\mu)) \cap \mathcal{W}_0' &= \dim \pi_{| Y}^{-1}\left(g^{-1}\cdot (f' \circ p_1)(\mu)\right) &= \dim Y - \dim \pi(Y),
\frac{z_0^1(B^1;k)}{\frac{\sin\{kB^1\}}{k}}=1-\frac{\cot k B^1}{2k}Q(B^1)+\frac{[p(B^1)+p(0)-\frac{1}{2}Q^2(B^1)]}{4k^2}+O(\frac{1}{k^3}).
\{\nabla_{\alpha},\nabla_{\beta}\}=c_{\alpha \beta}^{\enspace \enspace \gamma}\nabla_{\gamma}\qquad\{\bar \nabla_{\dot \alpha},\bar \nabla_{\dot \beta}\}=c_{\dot \alpha \dot \beta}^{\enspace \enspace \dot\gamma}\bar \nabla_{\dot \gamma}
P_{0}T ( W^{( \mathbf{\beta}, \mathbf{\gamma})}( x_{1}) . . . W^{( \mathbf{\beta}, \mathbf{\gamma})}( x_{n}) ) P_{0}= P_{0}T ( W^{( \mathbf{0}, \mathbf{0})}( x_{1}) . . . W^{( \mathbf{0}, \mathbf{0})}( x_{n}) ) P_{0}+ \mathrm{d i v e r g e n c e s}
\sum_{n\ge0}\frac{(q^2;q^2)_nq^{n^2}}{(-q^2;q^2)_n(q;q^2)_{n+1}}= \frac{1}{2}\sum_{n\ge0}q^{n^2/2}(1+q^{2n+1})(1+q^{n+1/2})\sum_{|j|\le n}q^{j^2/2}
\begin{aligned}\int_{0}^T\left<\mathcal{K}_{\mathcal{I}_l}S\left(W_\epsilon(x_j,x_k)u_\epsilon\right),(1-P_R)v\right>dt=\int_{0}^T\left<\left(W_\epsilon(x_j,x_k)\mathcal{K}_{\mathcal{I}_l}u_\epsilon\right),S^*(1-P_R)v\right>dt.\end{aligned}
E(R)=\frac{1}{(1-\epsilon^{2})}(1-J_{0}^{2}(x)-J_{1}^{2}(x)+\epsilon^{2}[1-J_{0}^{2}(\epsilon x)-J_{1}^{2}(\epsilon x)]-4\epsilon\int_{0}^{x}\frac{J_{1}(t)J_{1}(\epsilon t)}{t}dt)
\gamma_{M+1}\left( \sum_{j=1}^M \gamma_j\|A_j\|^2\right)\leq \delta; \max_j\{\gamma_j\}\leq \frac{2(1-\sqrt{\delta})}{L}; \lambda \leq \frac{L\max_j\{\gamma_j\}}{4(1+\sqrt{\delta})}.
1-\overline{\Psi(w,x(\mu))}\Psi(z,x(\lambda))= 1-\cal{F}_{F(\mu)}(w)^*\cal{F}_{F(\lambda)}(z) = (1-\overline{w}z)\overline{\gamma(\mu,w)}\gamma(\lambda,z)+\eta(\mu,w)^*(I-F(\mu)^*F(\lambda))\eta(\lambda,z)
|a(n)| \leq \sqrt{\log k}\left(103\sum_{m=1}^{\ell-1}\frac{|a(m)|}{m^{\frac{k-1}{2}}}+ B(k) \sum_{m=1}^{\ell -1}|a(m)| e^{-7.288m}\right) d(n) n^{\frac{k-1}{2}},
\begin{array}{r}u(x,t)=1+\sum\limits_{k\in \mathbb{N}_+}\frac{(-1)^k\exp\left(2k(-at+2x)\right)}{(2k)!}\left( \cos2k\sqrt{-\Delta}t+\frac{a}{\sqrt{-\Delta}}\sin2k\sqrt{-\Delta}t\right) +\frac{(-1)^{k-1}\exp\left((2k-1)(-at+2x)\right)}{(2k-1)!(2k-1)\sqrt{-\Delta}}\sin(2k-1)\sqrt{-\Delta}t. \end{array}
d s^{2}= e^{2 A ( x )}\sum_{\mu , \nu = 0}^{q - 1}\eta_{\mu \nu}d y^{\mu}d y^{\nu}+ \sum_{i = q}^{D - s - 3}e^{2 F_{i}( x )}d y^{i}d y^{i}+ e^{2 B ( x )}\sum_{\gamma = D - s - 2}^{D - 1}d x^{\gamma}d x^{\gamma},
\Lambda = \frac{\max_{\boldsymbol{\alpha}}p\left(\textbf{y}_{\tau}|\boldsymbol{\alpha},\mathcal{H}_1\right) p\left(\boldsymbol{\alpha}|\mathcal{H}_1\right)}{p\left(\textbf{y}_{\tau}|\mathcal{H}_0\right)}\begin{matrix}\mathcal{H}_1 begin{align*}-1mm]> begin{align*}-2.5mm] < begin{align*}-1mm] \mathcal{H}_0 \end{matrix}\lambda_d,
\mathcal{D}_{k,T,n_{\mathrm{max}}}=\{\lbrace (m_1,l_1,c_1),\ldots,(m_n,l_n,c_n) \rbrace: \sum_{i=1}^n c_i \leq n_{\mathrm{max}},\, m_i + l_i/2 \leq T,\, i=1,\ldots,n\}
0 =\left(\partial^{N\otimes_BJ}\right)^2(\gamma') & = \partial^{N\otimes_BJ}(e\otimes \delta(b)) & = \partial^N(e)\otimes \delta(b)+(-1)^{|e|}e\otimes \partial^J\delta(b) & = (-1)^{|e|}e\otimes \partial^J\delta(b).
\int \frac{t}{\check\gamma_0\rho_0+\frac{(1-\rho_0)\check\gamma_0F(\check\gamma_0)}{1-\rho_0+\rho_0\check\gamma_0 F(\check{\gamma}_0)}t}\nu(dt) - \int \frac{t}{\check\gamma\rho+\frac{(1-\rho)\check\gamma F(\check\gamma)}{1-\rho+\rho \check\gamma F(\check{\gamma})}t}\nu(dt) = 0.
\limsup_{\alpha\downarrow 0}\limsup_{N\rightarrow\infty}p_{N,u,\alpha}\leq\mathcal{P}_u({x},{b},0)+\partial_\lambda\mathcal{P}_u({x},{b},0) \lambda+\frac{L}{2}\lambda^2 &=2\mathcal{P}({x},{b})-\frac{\delta f(u)^2}{L}\Bigl(1-\frac{\delta}{2}\Bigr) &<2\mathcal{P}({x},{b}).
I(z) = i(-1)^{\alpha}\int{d^{n}k\over(2\pi)^{n}}\,{e^{i(\bar{\theta}+z)\cdot k}\over(k^{2}+M^{2})^{\alpha}}= i(-1)^{\alpha}{\pi^{n\over 2}\over(2\pi)^{n}}\,{1\over\Gamma(\alpha) (M^{2})^{\alpha -{n\over2}}}\,2\left({|\bar{\theta}+ z|M\over 2}\right)^{\alpha -{n\over2}}\,K_{\alpha -{n\over 2}}(|\bar{\theta}+z|M)
\bar{\bf N}= - \sqrt{1 - \bar{U}^{2}}\frac{~ \partial}{\partial U}- \frac{\bar{U}}{\sqrt{1 - \bar{U}^{2}}}\left( T \frac{~ \partial}{\partial T}+ V \frac{~ \partial}{\partial V}+ X \frac{~ \partial}{\partial X}+ Y \frac{~ \partial}{\partial Y}+ Z \frac{~ \partial}{\partial Z}\right)
(-y)^{-\gamma\cdot \mathrm c_1(S)/2}\cdot \left\langle \gamma_1, \dots, \gamma_N \right\rangle^{\infty}_{g, \gamma}=(y^{1/2}-y^{-1/2{\tiny}})^{\gamma \cdot \mathrm c_1(S)}\cdot \left\langle \gamma_1, \dots, \gamma_N \right\rangle^{0^+}_{g, \gamma}.
F_{j}( \omega , \vec{k}) \; = \; \int \frac{d \omega^{\prime}}{2 \pi}\int \frac{d \vec{k}^{\prime}}{( 2 \pi )^{3}}\; \tilde{B}( \omega - \omega^{\prime}, \vec{k}- \vec{k}^{\prime}) \: C_{j - 1}( \omega^{\prime}, \vec{k}^{\prime}) \: F_{j - 1}( \omega^{\prime}, \vec{k}^{\prime})
\bigg|\sum_{i=1}^{k}k^{-m}\sum_{\sigma\in\{1,\ldots,k\}^{m}}f_{i}(\sigma)T_{\rho}f_{i}(\sigma)-\sum_{i=1}^{k}\int g_{i}T_{\rho}g_{i}d\gamma_{n}\bigg|\leq\epsilon.
\Omega(e_i) = f_i,\Omega(f_i)=e_i,\Omega(v^h)=v^{-h},\Omega(v)=v^{-1}, \Upsilon(e_i) = v_if_iK_i^{-1},\Upsilon(f_i)=v_i^{-1}K_ie_i,\Upsilon(v^h)=v^{-h},\Upsilon(v)=v^{-1}.
\left(\begin{array}{c}\varphi^{*}X_{1}\varphi^{*}X_{2}\end{array}\right) = A_{\varphi}\left( \begin{array}{c}X_{1}X_{2}\end{array}\right) + \left(\begin{array}{c}\sum_{j=1}^{n-2}g_{j}Y_{j}\sum_{j=1}^{n-2}h_{j}Y_{j}\end{array}\right)
\psi(g^{-1}(ab)^{q-1}aga) = \psi(g^{-1})\psi((abab)^{\frac{q-1}{2}})\psi(aga) &=(g^{-1}(ab)^{\frac{q-1}{2}}, g^{-1})((ba)^{\frac{q-1}{2}}, (ab)^{\frac{q-1}{2}})(g,(ba)^{\frac{q-1}{2}}g) &=(1,1).
\Delta =\frac{1}{p+1}\int y\psi_B \big{[}|\varepsilon+Q_b|^{p+1}-Q_b^{p+1}-(p+1)\varepsilon Q_b^p\big{]}'-p\int y\psi_B \varepsilon Q_b'Q_b^{p-1}&=-\frac{1}{p+1}\int (\psi_B-y\psi_B') \big{[}|\varepsilon+Q_b|^{p+1}-Q_b^{p+1}-(p+1)\varepsilon Q_b^p\big{]}&\quad-p\int \psi_B \varepsilon Q_b^{p-1}(yQ_b').
\lim_{\gamma_{-}\rightarrow 0^{+}}\inf h_{\gamma_{-}}\left( E_{1}\right)=\inf f_{(\Phi ,\mathfrak{a}_{\mathfrak{b}_{+}}-\mathfrak{a}_{\mathfrak{b}_{-}})}^{\flat}\left( E_{1}\right) =-\mathrm{P}_{(\Phi ,\mathfrak{a}_{\mathfrak{b}_{+}}-\mathfrak{a}_{\mathfrak{b}_{-}})}^{\flat}\ .
\ddot{z}^{+}- \dot{z}^{+}\frac{\ddot{z}^{-}}{\dot{z}^{-}}- 2\lambda \dot{z}^{+}(\dot{z}^{+}+\dot{z}^{-}) + 4\lambda \dot{z}^{+}\dot{z}^{-}= 0 \,,
\begin{array}{l}{A_{1}^{( n )}= A^{( 1 )}\otimes I \otimes \cdots \otimes I ,}\\{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots}\\{A_{n}^{( n )}= I \otimes I \otimes \cdots \otimes A^{( 1 )}.}\\ \end{array}
\prod_{i=1}^m\Big(\sum_{n}q_n\xi_m^{in}z^n\Big) =\prod_{i=1}^{m}Y_m(\xi_m^i z) =\exp\Big(\sum_{m\nmid n}\frac{(\xi_m^{n}+\xi_m^{2n}\dotsm+\xi_m^{mn})z^n}{n\epsilon_n}p_n\Big) =1,
\left\vert f(\zeta)-f(p_{m}/q_{m})\right\vert=\left\vert f(\zeta)-\frac{p_{m}^{\prime}}{q_{m}^{\prime}}\right\vert= \left\vert \frac{l_{1}}{l_{2}}-\frac{p_{m}^{\prime}}{q_{m}^{\prime}}\right\vert \geq \frac{1}{q_{m}^{\prime}l_{2}},
[G^{k_0 s - \ell_0 - 1}F^{\ell_0 + 1}]_{k_0}= \frac{1}{d(k_0 s - \ell_0)}Q(k_0, \ell_0) [G^{k_0 s - \ell_0}F^{\ell_0}]_{k_0}.
g_{1}(\omega V,\mathcal{T}_{U}\mathcal{B}X)+g_{1}(V,\phi U)\eta (X) =g_{2}((\nabla \pi_{\ast})(U,\mathcal{C}X),\pi_{\ast}\omega V)-g_{2}((\nabla \pi_{\ast})(U,X),\pi_{\ast}\omega\phi V)
(\Delta v)_{\sigma, \tau}=(v_{\sigma})[(c_{\sigma}) ({}^{\sigma}v_{\tau}) (c_{\sigma})^{-1}](v_{\sigma \tau})^{-1}&=(v_{\sigma}) (c_{\sigma})({}^{\sigma}v_{\tau}) [({}^{\sigma}\! c_{\tau})(c_{\sigma \tau})^{-1}](v_{\sigma \tau})^{-1}&=(b_{\sigma})({}^{\sigma}b_{\tau})(b_{\sigma \tau})^{-1}=(\Delta b)_{\sigma, \tau}
\begin{aligned}\mbox{Res}_{z_1}z_1^{k+\frac{j_2}{T}}z_0^{-1}\delta(\frac{z_1-z_2}{z_0})(\frac{z_1-z_2}{z_0})^{\frac{j_1}{T}}Y_{M^3}(u,z_1)I(w_1,z_2)w_2 = \mbox{Res}_{z_1}z_1^{k+\frac{j_2}{T}}z_1^{-1}\delta(\frac{z_2+z_0}{z_1})(\frac{z_2+z_0}{z_1})^{\frac{j_2}{T}}I(Y_{M^1}(u,z_0)w_1,z_2)w_2.\end{aligned}
\psi_{\alpha\beta w}(r,\theta)=e^{i\left(pr\cos\theta-\nu({\bar\theta}-\pi)\right)}-\sin\nu\pi e^{-i([\nu]+1)\theta}\int^{\infty}_{-\infty}{dt\over \pi}e^{ipr\cosh t}{e^{-\{\nu\}t}\over e^{-i\theta}-e^{-t}}\ ,
\lambda \left\{x \in \mathbb{I}: \left|\frac{k_n(x)}{n}- a\right| \geq \varepsilon\right\}=\ \lambda \left\{x \in \mathbb{I}: \frac{k_n(x)}{n}\geq a + \varepsilon\right\}+ \lambda \left\{x \in \mathbb{I}: \frac{k_n(x)}{n}\leq a - \varepsilon\right\},
\partial f_{n,m}=\frac{1}{nm P_{n,m}^{2}}(n\log(u\overline u)b+m\log(v\overline v)a), \partial\overline \partial f_{n,m}=\frac{2}{P_{n,m}^{3}}(b-a)\land (\overline a-\overline b), \partial f_{n,m}\land \partial\overline \partial f_{n,m}= \frac{2}{nmP_{n,m}^{4}}a\land b\land(\overline a-\overline b).
\mathfrak{M}(-l\,|\,\mu,\lambda_1,\lambda_2) =\frac{\Gamma\bigl(1+l/\tau\bigr)}{\Gamma^{-l}\bigl(1-1/\tau\bigr)}\,\prod\limits_{j=0}^{l-1}\frac{\Gamma\bigl(2+\lambda_1+\lambda_2+(l+j+2)/\tau\bigr)}{\Gamma\bigl(1+(j+1)/\tau\bigr)}\star \star\frac{1}{\Gamma\bigl(1+\lambda_1+(j+1)/\tau\bigr)\Gamma\bigl(1+\lambda_2+(j+1)/\tau\bigr)},
x\in\bigcap_{t=1}^d H_{nm}^{j_t}\iff n u_t - n \leq \pi_t(\iota^{-1}(x))\leq nu_t+n 1\leq t\leq d & \iff -n \leq \pi_t (\iota^{-1}(x-\iota(nu_1,\dots,nu_d))) \leq n 1\leq t\leq d & \iff x-\iota(nu_1,\dots,nu_d)\in H_n.
\begin{array}{lllll}f^2(x)\leq \lambda f(x)=|\lambda||f(x)|, begin{align*}2mm]f^4(x)\leq |\lambda||f^3(x)|<|\lambda|f^2(x), begin{align*}2mm]f^6(x)<|\lambda|f^4(x)<|\lambda|^2f^2(x), begin{align*}2mm]\dots\ \ \dots begin{align*}2mm]f^{2k}(x)\leq |\lambda|^{k-1}f^2(x)\leq |\lambda|^k|f(x)|.\end{array}
P^{(\lambda_{j})}(\lambda) = E_{\lambda_{j}}(\lambda) \Phi_{\mathrm{HG}}(r^{3/2}f_{\lambda_{j}}(\lambda);\beta_{j})(s_{j}s_{j+1})^{-\frac{\sigma_{3}}{4}}e^{-r^{3/2}g(\lambda)\sigma_{3}}.
(Comp1)\frac{A\Rightarrow_{amb}A^{\prime}}{A|B\Rightarrow_{amb}A^{\prime}|B};(Comp2)\frac{A\Rightarrow_{amb}A^{\prime}B\Rightarrow_{amb}B^{\prime}}{A|B\Rightarrow_{amb}A^{\prime}|B^{\prime}}
\begin{array}{ll}\partial_t^j u^s|_{y=0}=O(\frac{1}{\beta}), \partial_t^j\partial_y u^s|_{y=0}=O(1), 0\leq j\leq k, begin{align*}7pt]\big\|u|_{y=0}- u^s|_{y=0}\big\|_{A^k}=O(\frac{1}{\sqrt{\beta}}),\big\|\partial_y u|_{y=0}- \partial_y u^s|_{y=0}\big\|_{A^k}=O(\sqrt{\beta}),\end{array}
(a-c+\tilde{b}\eta)f(\chi_1,\ldots,\chi_n)^2 -\tilde{b}tf(\chi_1,\ldots,\chi_n)t^{-1}\eta f(\chi_1,\ldots,\chi_n) +f(\chi_1,\ldots,\chi_n)cf(\chi_1,\ldots,\chi_n) =0
g_R^2(\mu) ={g_R^2(\mu_0) \over 1+{22 \over 3}{C_2(G) \over (4\pi)^2}g_R^2(\mu_0) \ln{\mu \over \mu_0}}={1 \over{22 \over 3}{C_2(G) \over (4\pi)^2}\ln{\mu \over \Lambda_{\rm QCD}}}
\mathcal{F}\left\{R_{x}^{-\alpha}f(x)\right\}=\frac{(i\omega)^{-\alpha}+(-i\omega)^{-\alpha}}{2\cos\alpha\pi/2}F(\omega),\alpha>0,\alpha\neq1,3,..., & =\left\vert \omega\right\vert^{-\alpha}F(\omega),\omega.
\varphi(x) = \left| \frac{J^2}{s^2u}\right|^{\frac{1}{2}}\varphi_1'(x_1') \varphi_3'(x_3') \int_F \int_F \varphi_2'(x_2') \varphi_4'(x_4') \psi(x_2'y_2' + x_4' y_4') \, dx_2' \, dx_4' = \left| \frac{J^2}{s^2u}\right|^{\frac{1}{2}}\varphi_1'(x_1') \hat{\varphi}_2'(y_2') \varphi_3'(x_3') \hat{\varphi}_4'(y_4').
G_{y}^{y}= - \frac{p + 1}{p - 1}\Lambda_{p}e^{2 U}+ \frac{p + 1}{2}\Big [ 2 ( n - 1 ) U^{\prime}V^{\prime}+ p ( U^{\prime})^{2}\Big ] + \frac{( n - 1 ) ( n - 2 )}{2}\Big [ ( V^{\prime})^{2}- e^{2 V}\Big ]
A:=\left(\begin{array}{ccc}\frac{\partial v_1}{\partial x_1}\cdots \frac{\partial v_1}{\partial x_{n-1}}\vdots \ddots \vdots \frac{\partial v_{n-1}}{\partial x_1}\cdots \frac{\partial v_{n-1}}{\partial x_{n-1}}\end{array}\right)
\begin{array}{rcl}\psi_{2,1}\cong \rho_{1,1}= L_{-1}\psi_{1,1}\phi_{2,1}\cong \phi'_{1,1}= L_{-1}\phi_{1,1}\xi_{2,1}\cong \phi_{1,1}= \frac{1}{2}L_{1}\rho_{1,1}\,.\end{array}
J_{\lambda ,\alpha}\left( Q\right) =\frac{1}{2}\int\limits_{\Omega}\left[\Delta Q+F\left( \nabla Q\right) \right]^{2}\varphi_{\lambda ,\nu_{0}}^{2}dx+\frac{\alpha}{2}\left\Vert Q\right\Vert_{H^{3}\left( \Omega\right)}^{2},Q\in G\left( R,\Phi ,\Psi \right) .
h_{\nu}'(1) =\sqrt{\pi}2^{\nu}\Gamma\left(\nu+\frac{3}{2}\right)\left(\mathbf{H}_{\nu}(1)-\mathbf{H}_{\nu+1}(1)+\frac{1}{\sqrt{\pi}2^{\nu}\Gamma\left(\nu+\frac{3}{2}\right)}\right) &=1+\int_0^1(1-t^2)^{\nu-\frac{1}{2}}(2\nu+t^2)\sin(t)dt>1-\int_0^1(1-t^2)^{\nu+\frac{1}{2}}\sin(t)dt>0,
P(x) + P(-x) \leq |P(x)| + |P(-x)| = (x+1) \prod_{k=1}^{n-1}|x+\xi^k| + (-x+1) \prod_{k=1}^{n-1}|-x+\xi^k| \leq (x+1) \prod_{k=1}^{n-1}2 + (-x+1) \prod_{k=1}^{n-1}2 = 2^n.
\nabla{\hskip-0.3cm^{^{^{\psi}}}}_{X}\big(W\circ\psi\big)=\nabla^N{\hskip -0.4cm_{_{_{d\psi(X)}}}}\hskip -0.3cm W, \hskip 0.4cm \forall W\in\Gamma(F),~ \forall X\in\Gamma(TM).
\omega_{\exp(x)H}= \! \sum_{1\leq j_1,j_2 \leq r}\!\frac{1}{4}\frac{\partial^2 u}{\partial x_{j_1}\partial x_{j_2}}(x)i\gamma_{j_1}\wedge \bar{\gamma}_{j_2}+ \! \sum_{\alpha \in \Phi_{P^u}}\! \left< \alpha, \nabla u(x)/2-t_{\chi}\right>i\gamma_{\alpha}\wedge \bar{\gamma}_{\alpha}
\mathbb{E}_{f_0,T}\|f_0-\hat{f}\|_\infty 1_{\hat{j}> j^*}\leq \sum_{j=j^*+1}^{j_{\max}}\mathbb{E}_{f_0,T}^{1/2}\|f_0-\tilde{f}(j)\|_\infty^2\, \mathbb{P}_{f_0,T}^{1/2}(\hat{j}=j) & \lesssim \sum_{j=j^*+1}^{j_{\max}}2^{(3/2)j}\mathbb{P}_{f_0,T}^{1/2}(\hat{j}=j)\lesssim2^{j^*}e^{-c\tau^2 j^*}+2^{(3/2)j_{\max}}n^{-2}&=o( 2^{- j^*s}+1/\sqrt{n}),
F_{{x}_n}(x) \approx \gamma_n \sum^{n-1}_{j=0}{n-1 \choose j}(-1)^j \frac{\left(\frac{x^{N-M+1}}{(N-M+1)!}\right)^{K-k+j+1}}{K-n+j+1}\approx \frac{\gamma_n}{K-n+1}\left(\frac{x^{N-M+1}}{(N-M+1)!}\right)^{K-k+1},
p^{1/4r^2}z^{-1/r}(\log p)^{1/r}=\exp\bigg(\frac{\log p}{4r^2}-\frac{c(\log p\log\log p)^{1/2}}{r}+\frac{\log\log p}{r}\bigg) &=\exp\bigg(\!\!-c^2\log\log p+O\bigg(\frac{(\log\log p)^{3/2}}{(\log p)^{1/6}}\bigg)\bigg)\ll(\log p)^{-c^2}.
\,_pF_q^\alpha (a,b; X, Y) = \frac{(\epsilon)_m}{x^m \rho_m'}\frac{1}{2 \pi i}\int_\Gamma s^{\epsilon -1}\,_{p+1}F_{q+1}(a-m,1,b-m,\epsilon; xs) \prod_{j=1}^r \frac{1}{(s-y_j)^{1/\alpha}}{\rm d}s.
\begin{array}{cl}\underset{x \in \mathbb{R}^{m}, X \in \mathcal{S}^{m}}{\min}\langle Q,X \rangle & (x,X) \in \mathcal{F}& \langle Q_{i}, X \rangle = \langle c_{i}, x \rangle i = 1,\ldots,k.\end{array}
q_{\tilde{\Phi}_{\Psi,k}|X_{A,k},X_{k-1}}\left(\phi_y \big| |x_k|,x_{k-1}\right) &\equiv \int_{-\pi}^{\pi}p_{\Phi_{X,k}|X_{A,k},X_{k-1}}\left(\phi_x \big| |x_k|,x_{k-1}\right) q_{\tilde{\Phi}_{\Psi}|\Phi_{X}}(\phi_y | \phi_x)d\phi_x &= \frac{1}{2 \pi}.
\left( \begin{matrix}{\varpi_{D}^{( \bullet )}( z )}\\{\varpi^{( \bullet )}( z )}\\ \end{matrix}\right) = C_{\bullet}\left( \begin{matrix}{\varphi_{1}^{( \bullet )}( z )}\\{\varphi_{2}^{( \bullet )}( z )}\\ \end{matrix}\right) ,
\prod_{j=1}^{m-1}\prod_{\nu=b_{n(j)}+1}^{b_{n(j)+1}-1}\vert w_{\nu}\vert\leq A^{\sum\limits_{j=1}^{m-1}\Delta^{(k_j)}}\leq A^{\sum\limits_{i=0}^{k_m-1}\Delta^{(i)}}\quad\textrm{for every}m\ge 1.
H^{\prime\mspace{2mu}\textsc{cs}}+ \frac{\beta^2}{2}\, E^0 = \Phi_0^{-1}\Biggl( \frac{1}{2}\sum_{j=1}^N (z_j \, \partial_{z_j})^2 + \sum_{i<j}^N \frac{z_i \, z_j}{z_{ij}\,z_{ji}}\, \beta \, (\beta - s_{ij}) \Biggr) \, \Phi_0 \, .
{\cal L}^{\mathrm{e f f}}( H ) = - \frac{1}{8 \pi^{2}}\int_{0}^{\infty}\frac{d s}{s^{3}}\left[ e H s \coth ( e H s ) - 1 - \frac 1 3 ( e H s )^{2}\right] \exp ( - m^{2}s ) .
\begin{cases}(1-\delta+\phi_*)^2 \phi''_*+\frac{z}{2}(1-\delta+\phi_*) \phi'_*-\frac{z}{2}\psi_* \psi'_* =(1-\delta+\phi_*)(|\psi_*'|^2-|\phi_*'|^2), (1-\delta+\phi_*)^2 \psi_*''+\frac{z}{2}(1-\delta+\phi_*) \psi_*'-\frac{z}{2}\psi_* \phi'_*=0, \phi_*(0)=0,\phi_*(\infty)=\delta, \psi_*(0)=0,\psi_*(\infty)=\sigma.\end{cases}
K_1 = k_1 = \frac 13 \left(2\varphi_1 - \varphi_2 - \varphi_3\right)~, ~~~~~ K_2 = \frac{\sqrt 3}{2}\left(k_1 + k_1\right) = - \frac{1}{2\sqrt 3}\left(2 \varphi_3 - \varphi_1 - \varphi_2 \right)~,
\begin{cases}z_{tt}- \displaystyle \left(x^\alpha z_{x}\right)_x=g, (t,x)\in Q, z(t,1)=0, (0,T), \begin{cases}z(t,0)=0, \alpha \in (0,1), & (x^\alpha z_{x})(t,0)=0, \alpha \in [1,2),\end{cases}t\in (0,T), z(0)=z^0 z_{t}(0)=z^1.\end{cases}
\Delta = r^6 s^2 - 6 r^4 s^2 t - 3 r^4 s + 2 r^3 s^2 t^2 + 2 r^3 s^2 t +3 r^3 s t + r^3 s + r^3 + 9 r^2 s^2 t^2 + 6 r^2 s t &- 6 r s^2 t^3 - 6 r s^2 t^2 - 9 r st^2 - 3 r s t - 3 r t + s^2 t^4 + 2 s^2 t^3 + s^2 t^2 + 2 s t^3 + 3 s t^2 + t^2 + t
\theta_{12}(t)=\theta\left(\begin{bmatrix}0 t 0 0 0 0 0 0 0 \end{bmatrix}\right), \theta_{13}(t)=\theta\left(\begin{bmatrix}0 0 t 0 0 0 0 0 0 \end{bmatrix}\right),\theta_{23}(t)=\theta\left(\begin{bmatrix}0 0 0 0 0 t 0 0 0 \end{bmatrix}\right)
\frac{1}{4\pi^2g^2N^3}\sum_\sigma \left(\partial A^\mu(\sigma)\times\partial A^\nu(\sigma)\right)^2 \rightarrow \frac{1}{4\pi^2g^2N}\int_{T^2}d^2\sigma \left(\partial A^\mu(\sigma)\times\partial A^\nu(\sigma)\right)^2,
\left(\begin{array}{c}v_k v_{ik}v_{ek}\tilde{s}_k\end{array}\right) :=\frac{1}{N(u_k,u_{ik},u_{ek},\lambda_k)}\left(\begin{array}{c}|\lambda_k| u_k\ |\lambda_k| u_{ik}|\lambda_k| u_{ek}s_k\end{array}\right).
\mathrm{s c h}V ( \Lambda ) \left( \mathrm{c h}S^{+}- \mathrm{c h}S^{-}\right) = \mathrm{s c h}\left( \wedge ( g_{- 1}/ r_{- 1}) \right) \sum_{c \in C}\mathrm{s g n}( c ) \mathrm{s c h}U ( c \bullet \Lambda ) .
b_{\pm}( \eta , z ) ~ = ~ \bar{b}~ \exp \left\{i k g_{\pm}( \eta ) ~ - ~ i k z ~ \mp ~ i \left[ \frac{p^{2}}{2}( h \eta ~ + ~ h_{0}) z^{2}) ~ + ~ O ( p^{4}) \right] ~ + ~ O \left( \frac 1 k \right) \right\}.
\begin{aligned}L(K,u^*) = \{tv\in \mathcal{U}:\G'(u^*)u^* + G'(u^*)v\in K,\ \forall\t > 0 \}& = \{tv\in \mathcal{U}:\G'(u^*)u^* + G'(u^*)v\in \mathcal{K}- G(u^*) + G'(u^*)u^*,\ \forall\t > 0\}& = \{tv\in \mathcal{U}:\G(u^*) + G'(u^*)v\in \mathcal{K},\ \forall\t > 0\}& = L(\mathcal{K}, u^*).\end{aligned}
c_1 = - \frac{\int_{z_1}^{z_2}h_0^n \varphi_2 dz}{\int_{z_1}^{z_2}h_0^n \varphi_1 dz}c_2,\;\;\;c_2 \neq 0,\;\;\;c_i = 0 \;\;\; (i \geq 3).
\frac{f^{2}}{6}E \int_{- \infty}^{\infty}d x \psi_{n}^{( 0 ) *}( x ) \bigl \{1 - 4 \mu x^{2}[ V ( x ) - E ] -{\textstyle \frac{2}{3}}x^{3}\mu V^{\prime}( x ) \bigr \}\psi_{n}^{( 0 )}( x ) = 0 .
B_{j,m}(f,g)(x):= 2^{-\frac{(b-a)j}{2}}\int \rho(t)f*\Phi\left(\frac{x}{2^{(b-a)j}}-2^m(t^a+\epsilon_P(t))\right) &g*\Phi(x-2^m(t^b+\epsilon_Q(t))) \, dt
g^{+}_v=v \psi, (\mbox{resp.}g^{-}_v=v \psi), r_v=v \psi_u, \psi_v\cdot \psi=\phi_v\cdot \phi, \psi_v\cdot g^{+}_u =\phi_v\cdot f_u (\mbox{resp.}\psi_v\cdot g^{-}_u=\phi_v\cdot f_u), \psi_v\cdot r_{u}=(\phi\cdot f_{uu})_v -\frac{v}2{(\phi\cdot \phi)_{uu}}+v (\psi_u\cdot \psi_u) ,
S\left ( \langle G_+ \rangle^{\perp}\right ) =\langle F_- \rangle^{\perp}S\left (\langle F_- \rangle^{\perp}\right ) =\langle G_+ \rangle^{\perp}S\left ( \langle G_- \rangle^{\perp}\right ) =\langle F_+ \rangle^{\perp}\theta\left ( \langle F_+ \rangle^{\perp}\right ) =\langle F_- \rangle^{\perp}
J_N(p_1, \ldots, p_m; k_1,\ldots, k_p)= & \max\left(0, \sum_{i=1}^{p_1}k_i, \sum_{i=1}^{p_1+p_2}k_i, \ldots, \sum_{i=1}^{p_1+\ldots+p_{m-1}}k_i\right)+\max\left(0, \sum_{i=1}^{p_1}(-k_i), \ldots, \sum_{i=1}^{p_1+\ldots+p_{m-1}}(-k_i)\right).
\begin{array}{ll}\nabla(\Phi)(X_0,\cdots,X_n) = \sum^n_{p=0}(-1)^p\nabla_{X_p}(\Phi(X_0,\stackrel{p\atop \vee}{\cdots\cdots}, X_n)) + \sum_{0\leq r<s\leq n}(-1)^{r+s}\ \Phi([X_r,X_s],X_0,\stackrel{r\atop \vee}{\cdots}\stackrel{s\atop \vee}{\cdots},X_n) \end{array}
\begin{aligned}u_1 = h_1(\overline{t}), u_2(s) = h_2(\overline{t}) + ah_1(st_1 +(1-s)t_0), u_3(s) = h_3(\overline{t}) + a h_2(st_1 + (1-s)t_0) + a^2 \epsilon(s),\end{aligned}
f_j(x) =\dfrac{[a\pm x]_j}{[b\pm x]_j}=\dfrac{[a+x]_j[a-x]_j}{[b+x]_j[b-x]_j}, g_j(x) =\dfrac{[c\pm x]_j}{[d\pm x]_j}=\dfrac{[c+x]_j[c-x]_j}{[d+x]_j[d-x]_j}\quad(j=0,1,2,\ldots)
U^*(0,t_0) \le U^*(y,s) - S(s,t_0) \phi_2(y)- a(s - t_0) - 2\delta |s - t_0| begin{align*}1.2mm] \le U^*(0,t_0) + S(s,t_0) \phi_1 (y) - S(s,t_0)\phi_2(y) - \delta |s- t_0| \le U^*(0,t_0) - \delta |s-t_0|,
\tau^*(2^2\cdot 3^2\cdot 5)=\tau(2^2\cdot 3^2\cdot 5)-2^{12}\tau(3^2\cdot 5) \equiv 0 \pmod{23}, & \tau^*(3^2\cdot 5)=\tau(3^2\cdot 5)\equiv 0 \pmod{23}, & \tau^*(2^2\cdot 5)=\tau(2^2\cdot 5)-2^{12}\tau(5)\equiv 0 \pmod{23}, & \tau^*(5)=\tau(5) \equiv 0 \pmod{23}.
\bar Y_t = \int_{t\wedge\tau}^{m\wedge\tau}f \bigl(s,Y^m_s,Z^m_s,V^m_s \bigr) ds - \int_{t\wedge\tau}^{m\wedge\tau}\bar Z_s dW_s begin{align*}-2pt] - \int_{t\wedge\tau}^{m\wedge\tau}\int_U \bar V_s(e) \widehat\pi(du,ds) - \bar M_{m\wedge\tau}+ \bar M_{t\wedge \tau}.
\mathbf{B}(x) \leq C \sum_{j=0}^{j_{0}+1}{{2^{-jk}}\over{(2^{j}\rho(x))^{n-2}}}\int_{B_{0}\cap C_{j}}V(y)f(y) dy \leq C \sum_{j=0}^{j_{0}+1}{{(2^{j}\rho(x))^{2}}\over{2^{jk}}}\Big({1\over{|B_{j}|}}\int_{B_{j}}V(y)^{q}dy\Big)^{1\over q}\Big({1\over{|B_{j}|}}\int_{B_{j}}f(y)^{q'}dy\Big)^{1\over q'},
\begin{aligned}\partial_k (\xi_k \wedge \cdots \wedge \xi_1 \otimes u) = \sum_{i=1}^{k}(-1)^{i+1}\xi_k \wedge \cdots \wedge \widehat{\xi}_i \wedge \cdots \xi_1 \otimes z_i u & + \sum_{1\leq i<j\leq k}(-1)^{i+j}\xi_k \wedge \cdots \wedge \widehat{\xi}_j \wedge \cdots \wedge \widehat{\xi}_i \wedge \cdots \wedge \xi_1 \wedge \widehat{[z_j,z_i]}\otimes u.\end{aligned}
\mathrm{G}_{d}= ( - 1 )^{n}\varepsilon_{\mu_{1}\nu_{1}\cdots \mu_{n}\nu_{n}}\varepsilon^{\alpha_{1}\beta_{1}\cdots \alpha_{n}\beta_{n}}\prod_{i = 1}^{n}R_{\alpha_{i}\beta_{i}}^{\mu_{i}\nu_{i}}~ ;
- \Re\Big\{\frac{\zeta_L'}{\zeta_L}(s) \Big\}= \frac{1}{2}\log d_L + \Re\Big\{\frac{1}{s-1}- \sum_{\rho}\frac{1}{s-\rho}+ \frac{1}{s}+ \frac{\gamma_L'}{\gamma_L}(s) \Big\},
\begin{array}{lcl}X^{(II)}(\sigma, \tau) ={\displaystyle X_0^{(II)}( \tau) +}&{\displaystyle\frac{\sigma_0}{2\pi}~ k~ \sum_{n \neq 0}^{}\left\{\frac{1}{|n|}~F_{\nu}\left( \frac{2\pi}{\sigma_0}|n| \tau ,\frac{2\pi}{\sigma_0}|n|\tau_0 \right) \phi_n(\sigma_0)~e^{i\frac{2\pi}{\sigma_0}n\sigma}\right\}}\end{array}
\tilde{\cal H}= \frac{1}{2}( \vec{\cal E}^{\, 2}+ \vec{\cal B}^{\, 2}) + \frac{1}{2}p^{\dagger}p + \frac{1}{2 ( e \chi )^{2}}( \vec{\nabla}\vec{\cal E})^{2}+ \frac{1}{2}( \vec{\nabla}\chi )^{2}+ \frac{1}{2}( e \chi )^{2}\vec{\cal A}^{\, 2}+ V ( \chi^{2}/ 2 ) \ .
\left|T(f,g;x)\right| =\left|T(f-r+r,g-s+s;x)\right| \leq \left|T(f-r,g-s;x)\right|+\left|T(f-r,s;x)\right|+\left|T(r,g-s;x)\right|+\left|T(r,s;x)\right|.
\mathbf H_{n,m}\triangleq\begin{bmatrix}h_{n,m,1}h_{n,m,L}\cdots h_{n,m,2}h_{n,m,2}h_{n,m,1}\cdots h_{n,m,3}\vdots \vdots \ddots \vdots h_{n,m,L}h_{n,m,L-1}\cdots h_{n,m,1}\end{bmatrix}\in \mathbb{C}^{L\times L},
\mbox{If $M=0$ then}~{\bf P}_{Y_t|Y_{t-M}^{t-1},X_t}|_{M=0}={\bf P}_{Y_t|X_t},~\mbox{i.e., the channel is memoryless},~t=0,\ldots,n. &\mbox{If $N=0$ then}~{\gamma}_{t}(x_t,y^{t-1}_{t-N})|_{N=0}=\gamma_t(x_t),~t=0,\ldots,n.
G = \langle a, t \mid a^{2}= 1, [a, a^{t^i}]=1 \text{for $i \in \mathbb{Z}$}\rangle = \bigoplus\nolimits_{i \in \mathbb{Z}}\langle a_i \rangle \rtimes \langle t \rangle \cong C_2\wr C_\infty,
\chi^{I}= \sum_{a = 1}^{a = 2 s}\frac{q^{I}{}_{a}}{| \vec{x}- \vec{x}_{a}|}\ , \qquad \chi^{\alpha}={\frac{1}{\sqrt 2}}n^{\alpha}\ , \qquad ( n^{\alpha})^{2}= 1 \ ,
\tilde{D}_{3}^{2}= 0 , \ \tilde{D}_{3}^{\bar{2}}= - \frac{1}{3 \sqrt{2}}S^{+ + + -}, \ \tilde{D}_{3}^{\bar{1}}= \tilde{D}_{3}^{1}= \frac{1}{3 \sqrt{2}}B_{4}^{0}
I_{n_1,n_2,m_1,m_2}=2\int_{S^1}\int_{S^1}\int_0^{\pi/2}\sin\theta^{1+m_1+m_2}\cos\theta^{1+n_1+n_2}e^{(n_1-n_2)i\alpha}e^{(m_1-m_2)i\beta}d\theta d\alpha d\beta,
\frac{\partial^2\left(\log|f_X/f_Y|\right)}{\partial X\partial Y}= \frac{\partial}{\partial X}\left(\frac{1+\lambda}{-2\lambda X +(1+\lambda) Y}+ \frac{2}{-2Y +(1+\lambda) X}\right) &=\frac{2\lambda(1+\lambda)}{(-2\lambda X +(1+\lambda) Y)^2}- \frac{2(1+\lambda)}{(-2Y +(1+\lambda) X)^2}.
\omega=\frac{a-1}{2}, \kappa_n=\frac{n^2 \left(- (b1 - b2)^2+\left(a^2 - 2 (b1 + b2)\right) n^2 - n^4 \right)}{4 (2 n - 1) (2 n + 1)}, &\lambda_n=\frac{2 n^2 + 2 n + 1+2b_1+2b_2-a^2}{4}.
s_0(x_0,x_1,x_2) = (-x_0,x_0+x_1,x_0+x_2), s_1(x_0,x_1,x_2) = (x_0+x_1,-x_1,x_1+x_2), s_2(x_0,x_1,x_2) = (x_0+x_2,x_1+x_2,-x_2),
\mathfrak{M}= [\bar{\mathfrak{A}}_{2}, \mathfrak{B}_{2}] = [\bar{\mathfrak{A}}_{2}, \mathfrak{D}_{j_1}] = [\mathfrak{D}_{j_1}, \mathfrak{D}_{j_2}, \mathfrak{B}_{3}, \dots, \mathfrak{B}_{k}];
P^\chi_{out}(\tau_\chi) =\frac{2^{M_\chi+1}(K^R_h+1)^{M_\chi}(K^R_d+1)}{(2M_\chi+2)! \ (\Omega^R_h)^{M_\chi}\Omega^R_d}(\beta^R)^{-M_\chi/2}\cdot e^{-M_\chi K_s-K_d}(\tau_\chi)^{M_\chi+1}P_{BS}^{-(M_\chi+1)},