harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}\mbox{\rm dim}\,\P= ~~\sqrt{{\rm Det}(1 - \hat\theta\theta)}, begin{align*}12pt]\omega = ~~\hat\theta ( 1 - \theta\hat\theta)^{-1}begin{align*}12pt]\hat{\omega}= ~~- \theta ( 1 - \hat\theta\theta)^{-1}begin{align*}12pt]\sigma = ~~( 1 - \hat\theta\theta)^{-1}begin{align*}12pt]\end{array}
\begin{array}{rcl}\beta(u,\epsilon,P) = \displaystyle\liminf_{R\to\infty}\frac{\log K_R(\epsilon, P)}{u(t)}; e(u,P) = \displaystyle\limsup_{\epsilon\to0}\beta(u,\epsilon,P); e((T_t),u) = \displaystyle \sup_{P}e(u,P).\end{array}
Z = \int d [ \psi , \bar{\psi ,}B_{\mu}] \exp i \int d^{3}x ( \bar{\psi}( i \partial \! \! \! / \, + m + B \! \! \! \! / \, ) \psi + \frac{1}{2 g^{2}}B^{\mu a}B_{\mu}^{a})
k_n = \Bigl(\frac{\alpha}{p(\xi+2\gamma)}W\Bigl(n^{\frac{p}{\alpha}}\frac{p(\xi+2\gamma)}{\alpha}\Bigr)\Bigr)^{1/p}= \Bigl(\frac{\log n}{\xi+2\gamma}+ O(\log\log n)\Bigr)^{1/p},
\sum_{n<x^2}\frac{\Lambda_{f\otimes f}(n)}{n^{1/2}}\log\left(\frac{x^2}{n}\right)=4(x-2+x^{-1})+O\left(\frac{x^{2a-1}(\log k)^2}{T}\right)+O(x^{1-\frac{3}{2}\eta}(\log k)^2).
\bigvee_{j = 1}^{i-1}(-u_i + z_j) \vee (-u_{i+1}+ 1 + z_i) \leq \bigvee_{j = i+1}^{n-2}(-u_j + z_j) \vee x_n \vee (-u_{n-1}) \; i = 1, \dots, n-2 \, .
(p_{m_{k_l+\beta_{l,i}+\kappa r_l+j}},p'_{m_{k_l+\beta_{l,i}+\kappa r_l+j}})= \tau_l^j (p_{m_{k_l+\beta_{l,i}+\kappa r_l}},p'_{m_{k_l+\beta_{l,i}+\kappa r_l}}).
\|p_k(t,g)\|_{{L^1(\mathbb{R})}}=\|e^{ik(\frac{2\pi}{V_F})g}(p_{t,k}*G_{t,k})(g)\|_{{L^1(\mathbb{R})}}\leq \|p_{t,k}(g)\|_{{L^1(\mathbb{R})}}\|G_{t,k}\|_{{L^1(\mathbb{R})}}&=\|p_{0,k}(g)\|_{{L^1(\mathbb{R})}}e^{-k^2(\frac{2\pi}{V_F})^2D(t)}.
deg_{L_0}\Big(\sum\limits_{i=0}^{l_n}\sum\limits_{j=0}^{h_m}c_{n,i}d_{m,j}Y^i_0L^j_0-\sum\limits_{i=0}^{l_n}\sum\limits_{j=0}^{h_m}c_{n,i}d_{m,j}(L_0+n)^jY^i_0\Big)=0.
\partial_{\eta_n}\xi_n|_{y_n=0}=\psi^{\prime\prime}_{x_n\eta_n}(x^\prime(y^\prime,0,\eta^\prime,\eta_n),0,\eta^\prime,\eta_n)\not=0\Rightarrow \psi^{\prime\prime}_{x_n\eta_n}(x^\prime,0,\eta^\prime,\eta_n)\not=0.
I_{2k}^{2}(\Pi;q) = \sum_{m=1}^k q^m I_{2m-2}^{2}(\Pi;q) I_{2k-2m}^{2}(\Pi;q) &=\sum_{m=1}^k q^m q^{m-1}C_{m-1}(\Pi;q) q^{k-m}C_{k-m}(\Pi;q)
\lim_{n}\frac{1}{\phi(n)}[S_{a_n,\phi(n)}(i,j;\omega)-S_{a_n,\phi(n)}(i;\omega)p(i,j)] = \lim_{n}\frac{1}{\phi(n)}\sum_{k=a_n+1}^{a_n+\phi(n)}\mathbf{1}_{\{i\}}(\xi_{k-1})[p_{k}(i,j)-p(i,j)]=0\ \a.e.
\bar{\Psi}^\downarrow = \psi :e^{\frac{i}{\sqrt{2}}(\varphi_c+\varphi_s)}: \,, \qquad \Delta^{\uparrow\uparrow}_s = \, :e^{i \sqrt{2}\varphi_s}: \,, \qquad \Delta_c = \, :e^{i \sqrt{2}\varphi_c}: \ ,
|P(A_{\Lambda})- \sum_{\boldsymbol{t}\in \Lambda}\Delta_{\boldsymbol{t}}|\le c_1(d,m) \sum_{\boldsymbol{s}\in \partial\Lambda}P(C_{\boldsymbol{t}}) &+c_2(d,m)\sum_{\genfrac{}{}{0pt}{1}{\boldsymbol{s},\boldsymbol{t}\in\Lambda}{\|\boldsymbol{s}-\boldsymbol{t}\|_{\infty}>m}}P(C_{\boldsymbol{s}}\cap C_{\boldsymbol{t}})
D_{GE}(p_{\hat{U}},\epsilon,u)=\underline{C}(p_{Y|X,S},p_S),u\in\mathcal{U}_+\cap\mathcal{G}_{\delta}, &D_{GE}(p_{\hat{U}},\epsilon,u)\leq\underline{C}(p_{Y|X,S},p_S),\quad\mbox{otherwise}
F_n^{(3)}=\displaystyle \frac{\lambda_1^n}{(\lambda_1 - \lambda_2)(\lambda_1 - \lambda_3)}+\displaystyle \frac{\lambda_2^n}{(\lambda_2 - \lambda_1)(\lambda_2 - \lambda_3)}+\displaystyle \frac{\lambda_3^n}{(\lambda_3 - \lambda_1)(\lambda_3 - \lambda_2)}
f(t) = \prod_{\substack{j=1}}^{\frac{n}{2}}(t - \lambda_{2j+1}) + (t - \lambda_{1}) \sum_{j=1}^{\frac{n}{2}}\frac{\sin^{2}\left[\frac{(2j + 1)\pi}{n + 3}\right]}{\sin^{2}\left(\frac{\pi}{n + 3}\right)}\prod_{\substack{\substack{m=1 m \neq j}}}^{\frac{n}{2}}(t - \lambda_{2m+1})
|H(x,\xi)| = (2\pi)^{-n}| V_{\phi_1}f (x,\xi)|\cdot |V_{\phi_2}g(x,\xi)| \leq C_{N,r}(1 + |\xi|^2)^{-(N-N_0)}e^{-(r_0-r) |x|^{1/s}}.
S_d \begin{pmatrix}\alpha r \end{pmatrix}S_d \begin{pmatrix}\beta s \end{pmatrix}= S_d \begin{pmatrix}\alpha\beta r + s \end{pmatrix}+ \sum \limits_{i + j = r + s}\Delta_{r,s}^{j}S_d \begin{pmatrix}\alpha\beta 1 i j \end{pmatrix},
U^{++}_{\underline{m}}\sigma^{\underline{m}}_{\underline{\mu}\underline{\nu}}= 2V_{\underline{\mu}q}^{~+}V^{~+}_{\underline{\nu}q}, \qquad U^{--}_{\underline{m}}\tilde{\sigma}^{\underline{m}\underline{\mu}\underline{\nu}}= 2V^{-\underline{\mu}}_{q}V_{q}^{-\underline{\nu}}, \qquad
I_{1}:= \int_{-x}^{-\delta}f(v+x)(1-v^{2})^{n}\dv, I_{2}:= \int_{-\delta}^{+\delta}f(v+x)(1-v^{2})^{n}\dv, I_{3}:= \int_{+\delta}^{1-x}f(v+x)(1-v^{2})^{n}\dv.
\Phi_{( r_{1}, \ldots , r_{n})}^{( n )}= \{\Phi_{r_{1}}^{( 1 )}, \ldots , \Phi_{r_{n}}^{( 1 )}\}= \Phi_{r_{1}}^{( 1 )}\otimes \cdots \otimes \Phi_{r_{n}}^{( 1 )},
f_{t}'(s) =\begin{cases}pt^{p-2}s |s|\leq t p|s|^{p-2}s \end{cases}, f_{t}''(s) =\begin{cases}pt^{p-2}|s|\leq t p(p-1)|s|^{p-2}\end{cases}, \frac{d}{dt}f_{t}'(s) =\begin{cases}p(p-2)t^{p-3}s |s|\leq t 0 \end{cases}.
\|x_{n+1}-x_n\|^2 = \ \|p_n-x_n+\lambda_n(Bx_n-Bp_n)+\alpha_{2,n}(x_n-x_{n-1})\|^2 \leq \ 2(1+\lambda_n\beta)^2\|x_n-p_n\|^2+2\alpha_{2,n}^2\|x_n-x_{n-1}\|^2,
( \frac{\partial{{L}_{D}}^{( k )}}{\partial q^{i ( k )}}- \Delta ( \frac{\partial{{L}_{D}}^{( k - 1 )}}{\partial \Delta q^{i ( k - 1 )}}) ) \Delta q^{i ( k )}+ \Delta{{H}_{D}}^{( k - 1 )}- \frac{\partial{H_{D}}^{( k )}}{\partial t_{k}}= 0 .
{\beta^*_{n}}= t_{n,1}^* + t_{n-1,1}^* + \cdots + t_{2,1}^* + t_{1,1}^* ,{\beta^*_{n-1}}= t_{n,2}^* + t_{n-1,2}^* + \cdots + t_{3,2}^* + t_{2,2}^*, \vdots{\beta^*_{n-k+1}}= t_{n,k}^* + t_{n-1,k}^* + \cdots + t_{k+1,k}^* + t_{k,k}^*, \vdots{\beta^*_{2}}= t_{n,n-1}^* + t_{n-1,n-1}^*,{\beta^*_{1}}= t_{n,n}^*\, ( = id)
\rho_{\mu_{1}\dots \mu_{p - 1}\alpha_{1}}^{( 1 )}= ( C \Gamma_{\nu \mu_{1}\dots \mu_{p - 1}})_{\beta \alpha_{1}}\Pi^{\nu}\Pi^{\beta}\ , \ \rho_{\mu_{1}\dots \mu_{p - 1}\alpha_{1}}^{( 2 )}= ( C \Gamma^{\nu})_{\beta \alpha_{1}}\Pi_{\nu \mu_{1}\dots \mu_{p - 1}}\Pi^{\beta}\quad .
S = \frac{k}{16\pi}\int dz d\bar{z}d\theta d\bar{\theta}\left[ -Dg^s\bar{D}g^{s\dagger}+ \int dt g^s\frac{\partial g^{s\dagger}}{\partial t}(Dg^s\bar{D}g^{s\dagger}+ \bar{D}g^sDg^{s\dagger}\right]
f(a \vee a') \to (f(a) \vee f(a')) = f(a \vee a') \to f(c) &= f((a \vee a') \to c) &= f((a \to c) \wedge (a' \to c)) &= (f(a) \to f(c)) \wedge (f(a') \to f(c)) &= (f(a) \vee f(a')) \to f(c) = 1,
\sum_{j,k}\sum_{j',k'}\Biggl(\sum^{+\infty}_{h=-\infty}\bigl |r^{jk,j'k'}_X(h)\bigr |\Biggr)^2<+\infty,\sum_{j,k}\sum^{+\infty}_{h=-\infty}\bigl |r^{jk}_X(h)\bigr |<+\infty
-\Delta(\frac{\phi}{\alpha})^{\alpha}=-\frac{1}{\alpha^{\alpha-1}}\left(\phi^{\alpha-1}\Delta\phi+\left( \alpha-1\right) \phi^{\alpha-2}\left\vert\nabla\phi\right\vert^{2}\right) & \leq-(\frac{\phi}{\alpha})^{\alpha-1}\Delta\phi=a\left( x\right)(\frac{\phi}{\alpha})^{\alpha q}\quad\Omega^{\prime}
u(t)= e^{-A(t-t_0)}u_0 -\chi\int_{t_0}^t e^{-A(t-s)}\nabla\cdot ( u(s) \nabla A^{-1}u(s))ds &+\int_{t_0}^t e^{-A(t-s)}u(s)\Big(1+ a_0(s,\cdot)-a_1(s,\cdot)u(s)-a_2(s,\cdot)\int_{\Omega}u(s)\Big)ds
\rho(x_1, x_2)\circ\rho(x_3, x_4) =\rho([x_1, x_2, x_3]_\mathfrak{g}, x_4)+\rho(x_3, [x_1, x_2, x_4]_\mathfrak{g})+\rho(x_3, x_4)\circ \rho(x_1, x_2), \rho(x_1, [x_2, x_3, x_4]_\mathfrak{g}) =\rho(x_3, x_4)\circ \rho(x_1, x_2)-\rho(x_2, x_4)\circ \rho(x_1, x_3)+\rho(x_2, x_3)\circ \rho(x_1, x_4).
\langle{\vec u}(\vec x).{\vec u}(0)\rangle =\sum_{i=j}^{3}iS_{ij}^{F}({\vec x}-{\vec{x^{\prime}}})=\int{{{\rm d}^{3}{\vec k}}\over{(2\pi)^{3}}}{{e^{i{\vec k}.({\vec x}-{\vec{x^{\prime}})}}\over{\omega_{k}}}}
\frac{k}{m}\leq \frac{\alpha_u + (m-\alpha_u)/2}{m}=\frac{1}{2}+ \frac{\alpha_u}{2m}\leq \frac{1}{2}+ \frac{m/s}{2m}= \frac{1}{2}+ \frac{1}{2s}= \frac{s+1}{2s}.
\sup_{s \in [0,T]}\| \Phi(s, \cdot )\|_{C^{\alpha}_b(H, K)}\le C_T \sup_{t \in [0,T]}\| B(t, \cdot )\|_{C^{\alpha}_b(H, U)}\;\; w(t,x)= \int_t^T R_{s-t}[ \Phi(s,\cdot)](x)\,ds,\;\;\; t \in [0,T],\; x \in H.
R_{\kappa \lambda \mu \nu}\, = \, R_{\kappa \mu}g_{\lambda \nu}- R_{\lambda \mu}g_{\kappa \nu}+ R_{\lambda \nu}g_{\kappa \mu}- R_{\kappa \nu}g_{\lambda \mu}- \frac{R}{2}\left( g_{\kappa \mu}g_{\lambda \nu}- g_{\kappa \nu}g_{\lambda \mu}\right) .
\delta m ~ = ~ ( \dots ) \int_{- \infty}^{\infty}d k ~ \sqrt{k^{2}+ 2 m^{2}}\left[ \int d x \frac{e^{i k x}}{\cosh^{2}\left( \frac{m x}{\sqrt{2}}\right)}\right] \left[ \int d y \frac{e^{- i k y}}{\cosh^{2}\left( \frac{m y}{\sqrt{2}}\right)}\right]
\Vert \hat h \Vert^{2}=\Vert \Pi_{W_{3}}\hat h\Vert^{2}+\Vert \Pi_{W_{3}^{\perp}}\hat h\Vert^{2},\;\Vert \Pi_{W_{3}^{\perp}}\hat h\Vert=o_{\varepsilon}(\Vert \hat h \Vert).
u(y,r) - \varphi(y,r + (t - s)) \leq u^{\theta}(y,r + (t - s)) + \frac{(t - s)^{2}}{2 \theta}- \varphi(y,r + (t - s)) &\leq u^{\theta}(x,t) - \varphi(x,t) + \frac{(t - s)^{2}}{2 \theta}&= u(x,s) - \varphi(x,t).
\phi_1 := \phi_{4A}(\tau, z) = 2 \zeta^{\pm 1}- 4 + (-4 \zeta^{\pm 2}+ 16 \zeta^{\pm 1}- 24)q + (2 \zeta^{\pm 3}- 24 \zeta^{\pm 2}+ 62 \zeta^{\pm 1}- 80) q^2 + O(q^3);
\mathbb{E}[(x_1^n-\mathbb{E}[x^n_1])^2] =\mathbb{E}\Big[\int_0^1\Big(\mathbb{E}[D_sx^n_1\vert \mathcal{F}_s]\Big)^2ds\Big] &\le\mathbb{E}\Big[\int_0^1\mathbb{E}[(D_sx^n_1)^2\vert \mathcal{F}_s]ds\Big]=\int_0^1\mathbb{E}[(D_sx^n_1)^2]ds.
\begin{cases}0<\underbar s_1^{n-1}\leq \underbar s_1^n\leq \bar s_1^n\leq \bar s^{n-1}_1\le \bar B_1\cr0<\underbar s_2^{n-1}\leq \underbar s_2^n\leq \bar s_2^n\leq \bar s_2^{n-1}\le B_2\end{cases}
\phi (U | \bar U;x,t) := (\nabla \bar A)^{-1}(A - \bar A ) - (U - \bar U)\;, G_1(U | \bar U;x,t) := \nabla G- \nabla \bar G - \nabla^2 \bar G \cdot (\nabla \bar A)^{-1}(A - \bar A )\;, G_2(U | \bar U;x,t) :=G_{x_\alpha}- \bar G_{x_\alpha}- \nabla \bar G_{x_\alpha}\cdot (\nabla \bar A)^{-1}(A - \bar A )\;,
x_1 \mapsto qx_1 x_2 \mapsto r^2(x_2 - 2 \sum\limits_{j=3}^{n}s_jx_j +t) x_j \mapsto r(x_j + s_j), \; j=3,\ldots,n x_{n+1}\mapsto qr^2(x_{n+1}+x_1\sum\limits_{j=3}^{n}s_j^2+ tx_1),
\begin{aligned}f_j(x_j)-f_j(\breve{x}_j^k)+(x_j-\breve{x}^k_j)^T(-A_j^T\lambda^k+\beta^kA_j^T(\sum_{i=1}^{j}A_i\breve{x}_i^k+\sum_{i=j+1}^{m}A_i\hat{x}_i^k-b))\geq 0.\end{aligned}
(D^n_1)_{n-i,n-j}=(-1)^{n-j}\sqrt{2}\sqrt{2(n-j)+1}\sum_{s=n-i}^{n-j}\sum_{k=0}^n\frac{(-n+j)_s(n-j+1)_s}{(1)_s(1)_s}\frac{d^i_{n,k}}{k+s+1}.
\int_{S^{n-1}}|x_1|^p dS_{B_2^n}(x)=(n+p)\int_{B_2^{n-1}}|z_1|^pdz=\frac{2\pi^\frac{n-1}{2}\Gamma\left(\frac{p+1}{2}\right)}{\Gamma\left(\frac{p+n}{2}\right)}.
\sum_{i=2n-1}^{2n}\sum_{j=1}^{2n-2}| D_iD_j u(p) - D_i D_j u(q) | + \sum_{i,j=1}^{2n-2}| D_iD_j u(p) - D_i D_j u(q) | \le C\left( d \sup_{B_\beta(0,1)}|u| + \int_0^d \frac{\omega(r)}{r}dr + d\int_d^1 \frac{\omega(r)}{r^2}dr \right),
\delta = 8 N_{-}^{2}\int_{- \infty}^{\infty}d x_{1}d x_{2}\frac{d k_{1}}{2 \pi}\frac{d k_{2}}{2 \pi}\int \frac{d s}{s}e^{- s r_{( x_{i}, k_{i})}^{2}}\frac{\sin^{4}( s v / 2 )}{s i n s v}
\frac{\delta S}{\delta \theta^{\alpha \beta}}= \frac{\beta}{2}\int d^{2}x{\mathrm{T r}}\left[ - \frac i 4 \{F_{\alpha \beta}, \Phi \}\star F_{0 1}- \frac i 4 \Phi \star \{F_{0 \alpha}, F_{\beta 1}\}+ \frac i 4 \Phi \star \{F_{0 \beta}, F_{\alpha 1}\}\right] \, .
-\hbar E_{N,N}(E_{1,1}-c) - \tfrac{\hbar}{2}h_0^2 = - \tfrac{\hbar}{2}\big(E_{N,N}^2 + (E_{1,1}-c)^2\big), -\hbar E_{i,i}E_{i+1,i+1}- \tfrac{\hbar}{2}h_i^2 = - \tfrac{\hbar}{2}\big(E_{i,i}^2 + E_{i+1,i+1}^2\big), i \neq 0.
\Vert A_{K,\alpha}u\Vert_{X_{\alpha/2}}^{2}\leqslant C\sum_{n=1}^{\infty}{|u_{n}|^{2}n^{-2\alpha}\mu_{n}^{\alpha}}=C\sum_{n=1}^{\infty}{|u_{n}|^{2}n^{-2\alpha}(\pi(n-1/2))^{\alpha}}& \leqslant C\pi^{2\alpha}\sum_{n=1}^{\infty}{|u_{n}|^{2}(\pi(n-1/2))^{-\alpha}}=C\pi^{2\alpha}\sum_{n=1}^{\infty}{|u_{n}|^{2}\mu_{n}^{-\alpha}}.
F(n, \ell+1) =\frac1{\binom{n}{\ell+1}}\sum_{0\le k<\ell+1}\binom{n}{k}=\frac{\binom{n}{\ell}}{\binom{n}{\ell+1}}+ \frac{\binom{n}{\ell}}{\binom{n}{\ell+1}}F(n, \ell) =\frac{\ell+1}{n-\ell}+ \frac{\ell+1}{n-\ell}F(n, \ell).
|\log|cr(\psi_\alpha(Q'))||= -\log\left(\frac{4}{((\sqrt 2+1)^\alpha-(\sqrt 2-1)^\alpha)^2}\right) = 2\log\left((\sqrt 2+1)^\alpha-(\sqrt 2-1)^\alpha\right)-\log 4 = \left((\sqrt 2+1)\log (\sqrt 2+1)-(\sqrt 2-1)\log (\sqrt 2-1)\right)(\alpha-1)+O(\alpha-1)^2 = \sqrt 2\left(\log (\sqrt 2+1)-\log (\sqrt 2-1)\right)(\alpha-1)+O(\alpha-1)^2~.
\dim(QP_k)_n = \sum_{\deg \omega = n}\dim QP_k(\omega) &\geqslant \dim QP_k(\omega_{(k,d)}) + \dim QP_k(\bar\omega_{(k,d)}) + \dim QP_k(\widetilde\omega_{(k,d)}) &> \dim QP_k(\omega_{(k,d)}) + \dim QP_k(\bar\omega_{(k,d)}) \geqslant c(k,d).
A_{( n )}^{* 0 k}\equiv \bar{C}_{( n )}^{k}, C_{( n )}^{* 0}\equiv \bar{C}_{( n )}, \eta_{( n )}^{* 0 k}\equiv \bar{\mu}_{( n )}^{k}, \sigma_{0}^{* ( n )}\equiv \bar{\mu}^{( n )}.
D^{2}={\vcenter{\mathrm{\hrule ~ h e i g h t . 4 p t \mathrm{\vrule ~ w i d t h . 4 p t h e i g h t 8 p t \kern 8 p t \vrule ~ w i d t h . 4 p t}\hrule ~ h e i g h t . 4 p t}}}- \partial^{\mu}a_{\mu}- 2 a^{\mu}\partial_{\mu}+ a^{\mu}a_{\mu}.
C_1\otimes_{eh}M_n(A)\otimes_{eh}C_2 \subset [C_1^*\otimes_h (T_n\widehat{\otimes}A^*)\otimes_h C_2^*]^* &\to (T_n\widehat{\otimes}[C_1^*\otimes_h A^*\otimes_h C_2^*])^* &=M_n (C_1^{**}\otimes_{eh}A^{**}\otimes_{eh}C_2^{**})\supset M_n (C_1\otimes_{eh}A\otimes_{eh}C_2)
R^{2s-N}\int_0^{\frac{r}{R^2}}\frac{t^{s-1}}{(t+1)^{\frac{N}{2}}}\dt=R^{-N}\int_0^{r}\frac{t^{s-1}}{(tR^{-2}+1)^{\frac{N}{2}}}\frac{R^{\varepsilon}}{R^{\varepsilon}}\dt =R^{\varepsilon-N}\int_0^{r}\frac{t^{s-1}}{(tR^{\delta-2}+R^\delta)^{\frac{N}{2}}}\dt.
\lim_{\phi \rightarrow \infty}\Phi_j(x,\bar x)=e^{\frac{2j}{\alpha_{+}}\phi}\delta^{(2)}(\gamma -x)+\frac{2j+1}{\pi}\left| \gamma-x\right|^{-4j-4}e^{\frac{-2j-2}{\alpha_{+}}\phi}+{\cal O}(e^{\frac1{\alpha_{+}}j\phi}),
\rho_{XX}(t_{1},t_{2})=\frac{K_{XX}(t_{1},t_{2})}{\sigma_{X}(t_{1})\sigma_{X}(t_{2})}=\frac{E[(X_{t_{1}}-\mu_{t_{1}})\overline{(X_{t_{2}}-\mu_{t_{2}})}]}{\sigma_{X}(t_{1})\sigma_{X}(t_{2})}
\nabla^0_2s =s \nabla^{k}_2s =\pi_2^*d_2\!\!\left(\frac{s}{\pi_2^*\sigma}\right)=_{\rm loc}d_2\nabla^{k-1}_2s-\nabla^{k-1}_2s\frac{d\pi_2^*\sigma}{\pi_2^*\sigma},\ \ \ \k\geqslant 0.
\int_{S_{y}}f_{n}(y)d\eta_{p}^{y}=\lim_{m\to\infty}\int_{S_{y}}f_{n}(y)d\eta_{p_{m}}^{y}\le\liminf_{m\to\infty}\int_{S_{y}}D\left(\alpha_{x,g}\parallel\alpha_{x,e}\right)d\eta_{p_{m}}^{y}.
T={\cal O}~W_1W_2W_3W_0W_1W_2\,,\qquad{\cal O}(\alpha_{0},\,\alpha_{1}, \,\alpha_{2},\,\alpha_{3},\,\alpha_{4}) = (\alpha_{3},\, \alpha_{4},\,\alpha_{0}, \,\alpha_{1},\,\alpha_{2})\,.
2 g_{\mu \nu}( X^{\alpha}) \partial_{a}\partial^{a}X^{\mu}+ 2 \partial_{a}X^{\mu}\partial^{a}X^{\rho}\partial_{\rho}g_{\mu \nu}( X^{\alpha}) + \partial_{a}X^{\mu}\partial^{a}X^{\eta}\partial_{\nu}g_{\mu \eta}( X^{\alpha}) = 0
e_{k}[g,h] = \frac1{k!}\cdot \left. \frac{\mathrm{d}^{k}}{\mathrm{d}\xi^{k}}( \hat g (\xi) \hat h^*(\xi) ) \right|_{\xi =0}= \frac1{k!}\cdot \sum_{j=0}^{k}\binom{k}j \hat g^{(j)}(0) \cdot \overline{\hat h^{(k-j)}(0)}.
\langle \chi,\chi\rangle=\frac{1}{|N_G(P)|}\left( (3^k(q-1))^2(1+q-1)+2\chi(T)\overline{\chi(T)}\frac{q(q-1)}{2}+\frac{q^2(q-1)}{3}\cdot(\varepsilon\cdot 3^k)^2(1+\omega\overline{\omega}+\overline{\omega}\omega)\right)=1.
S^{\:n+\nu-1}=\left\{\left(x_{1},\ldots,x_{n},y_{1},\ldots,y_{\nu}\right)\in\mathbb{R}^{n+\nu}:x_{i}\geq0,y_{i}\geq0,\sum_{i=1}^{n}x_{i}+\sum_{i=1}^{\nu}y_{i}=1\right\}
\|du\|_p\approx\sum_{k=0}^n\|(du)^k\|_p=\sum_{\ell=0}^n\|d(u^\ell)\|_p\le \sum_{\ell=0}^n\|D_{{}^\Vert}(u^\ell)\|_p\approx\sum_{\ell=0}^n\bigl\|\sqrt{D_{{}^\Vert}^2}(u^\ell)\bigr\|_p = \sum_{\ell=0}^n\bigl\|(\sqrt{D_{{}^\Vert}^2}u)^\ell\bigr\|_p\approx \bigl\|\sqrt{D_{{}^\Vert}^2}u\bigr\|_p\approx \|D_{{}^\Vert}u\|_p.
t_{k}( u ) = Q \left( \textstyle u - \frac{( k + 1 ) \pi}{2 ( p + 1 )}\right) P \left( \textstyle u + \frac{( k + 1 ) \pi}{2 ( p + 1 )}\right) - Q \left( \textstyle u + \frac{( k + 1 ) \pi}{2 ( p + 1 )}\right) P \left( \textstyle u - \frac{( k + 1 ) \pi}{2 ( p + 1 )}\right)
F(w) = \frac{2^{\frac{a+2n}{2}}\pi^{n-\frac{1}{2}}\Gamma(n-\frac{1}{2})}{\Gamma(2n-1)}c_{n,a}\cdot F_1(w) = \frac{2^{\frac{a+2n}{2}}\pi^{n-\frac{1}{2}}\Gamma(n-\frac{1}{2})}{\Gamma(2n-1)}c_{n,a}\cdot{_2F_1}\left(\frac{a+2n}{4},\frac{a+2n}{4};n;1-w\right)
g^{-1}= \left( \begin{array}{ccc}- 2 \lambda_2, - 6 \lambda_3, - 4 \lambda_4 - 6 \lambda_3, 4 \lambda_3(\lambda^2_2 - 4 \lambda_4), \lambda_2 \lambda_3 - 4 \lambda_4, + \lambda_2 \lambda_3, -2 \lambda_2 \lambda_4 + \frac34 \lambda_3 \end{array}\right)
S_{c l}= \frac{1}{2 \pi}\int d^{2}x \; \sqrt{- g}[ e^{- 2 \phi}( R + 4 ( \nabla \phi )^{2}+ 4 \lambda^{2}) - \frac{1}{2}\sum_{i = 1}^{N}( \nabla f_{i})^{2}]
\mathbb{E}\left[ \frac{1}{t}\int_0^t \left( | \varphi_s(\cdot,x) |^2 - | a(\theta_s \cdot) |^2 \right) \, ds \right] & \leq \frac{1}{t}\, \mathbb{E}\left[ \int_0^{t_0}\left( | \varphi_s(\cdot,x) |^2 - | a(\theta_s \cdot) |^2 \right) \, ds \right]+ \frac{t-t_0}{t}\, \frac{\lambda_{top}}{2}
\begin{array}{lll}G_{\ell}(z,\hat{z},\phi) := \big(\hat{z}_{1},\ldots,\hat{z}_{\ell-1},z_{\ell},\hat{z}_{\ell+1},\ldots, \hat{z}_{M}, \quad\quad \phi_{1},\ldots, \phi_{\ell-1},b_{\ell},\phi_{\ell+1},\ldots,\phi_{M}\big).\end{array}
\sum_{j:j\neq i}\int_{F_{ij}}(f_{ij}+f_{ji})(x)dS_{ij}(x) \le\frac{1}{N}\sum_{u\neq v}\int_{F_{uv}}(f_{uv}+f_{vu})(x)dS_{uv}(x)+\sum_{k=1}^{N}\int_{G_{N}}\left|\nabla_{k}f(x)\right|dx.
Pu\neq O, \quad\left\langle u,v\right\rangle \geq\alpha\,\left\Vert u\right\Vert \left\Vert v\right\Vert \quad\mbox{and}\quad\left\langle Pu,\, Pv\right\rangle =\left(-1\right)\,\left\Vert Pu\right\Vert \left\Vert Pv\right\Vert \mbox{.}
TV_{\alpha^*}(df_{\infty}) \leq \liminf TV_{\alpha^*}(df_{t_{k_n}}) & = \liminf \int_M |\alpha^* + df_{t_{k_n}}| dV_g = \liminf \int_M |\beta_{t_{k_n}}| dV_g \leq \liminf \int_M \sqrt{t_{k_n}^{-2}+ |\beta_{t_{k_n}}|^2}dV_g & = \liminf E^{t_{k_n}}_{GMS}(\beta_{t_{k_n}})
\varepsilon^{i j k}{^{\pm}}A_{i}^{A B}\partial_{j}{^{\pm}}A_{k A B}= \varepsilon^{i j k}\big ({\frac{1}{2}}A_{i}^{A B}\partial_{j}A_{k A B}\mp{\frac{i}{2}}\varepsilon^{A B C D}A_{i A B}\partial_{j}A_{k C D}\big ) ,
\lambda_1 \int G_1 |u_{R,\delta}|^2 -\int G_1 |\nabla u_{R,\delta}|^2 +\frac12 \int \Delta G_1 \, |u_{R,\delta}|^2 = \Re\int \widetilde f_{R,\delta}\, G_1 \, \overline{u_{R,\delta}}\,.
Z_{N}^{i n f}( x_{1}, \cdots , x_{M}) = ( x_{1}+ \cdots + x_{M})^{N}= \sum_{{n_{i}}\atop{\Sigma n_{i}= N}}~{\frac{N !}{n_{1}! n_{2}! \cdots n_{M}!}}~ x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{M}^{n_{M}}\, \, \, \, .
\omega_{\varphi}^{2}\approx\frac{GM}{r_{outer}^{3}}=(\frac{r_{s}c^{2}}{2r_{outer}^{3}})=(\frac{r_{s}c^{2}}{2})(\frac{r_{s}^{3}}{8a^{6}})=\frac{c^{2}r_{s}^{4}}{16a^{6}}
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho |\theta_t|^2 x+\kappa\int_{\Omega}|\nabla \theta_t|^2x \leq C(|D(u)|_\infty+1)|\sqrt{\rho}\theta_t|^2_{2}+C(| u_t|^2_{D^1}+1).
\frac 1 s= \frac{(m+2)(n-1)}4 \left( \frac 1 2 -\frac 1 q\right) +\frac m{4 \mu}, \gamma=\frac{n+1}{2}\left( \frac 1 2 -\frac 1 q \right)-\frac m{2\mu(m+2)}.
ER_n=\pi\frac{n}{\log n}+O(\frac{n}{(\log n)^2}), &\mathrm{Var}R_n\sim \frac{cn^2}{(\log n)^4}, &\lim_{n \to \infty}\frac{R_n}{ER_n}=1{a.s.,}&\frac{(\log n)^2}{n}(R_n-ER_n)\to -4\pi^2(det\Theta)\gamma(l),
|\mathcal{F}_1| \ll q^{2g+1}\frac{1}{V_1^{2k}}\left( \frac{2k}{e}\right)^k \left( \frac{1}{2}\sum_{n=1}^g \frac{1}{n}+ \frac{1}{2}\sum_{n=1}^g \frac{\cos(2n \theta)}{n}\right)^k \ll q^{2g+1}\left( \frac{2k \mathcal{V}(u,g)}{e V_1^2}\right)^k .
\alpha_i^{v_l}= 2 ((\alpha_i^{v}/2) (\alpha^v_{i+N_{v}/2})), \ 1\leq i \leq N_v/2, &\alpha_i^{v_r}= \alpha^v_{i+N_v/2}+ (1-2\beta^{v_l}_i) \alpha^v_i, \ 1\leq i \leq N_v/2.
\gamma z\exp\left[\int_{\mathbb{T}}\frac{t+\eta_{\mu_{1}}(z)}{t-\eta_{\mu_{1}}(z)}\,d\sigma(t)\right]=\Psi(z)=\gamma z\exp\left[\beta\frac{1+\eta_{\nu_{1}}(z)}{1-\eta_{\nu_{1}}(z)}\right].
I_1 \leq \left|\int_X-\chi((1+\epsilon)\tilde{\varphi})dd^c(u_1-u_3)\wedge T\right|+ \left|\int_{X\setminus U}-\chi((1+\epsilon)\tilde{\varphi})dd^c(u_1-u_3)\wedge T\right| &:=I_3+I_4.
h_{ij}(x) = \tfrac{2}{n-2}c \delta_{ij}|x|^{2-n}+ O^{2,\alpha}(|x|^{\max\{-2q, 1-n \}}) v(x) = -c |x|^{2-n}+ O^{2,\alpha}(|x|^{\max\{-2q, 1-n \}})
\epsilon^{\alpha \beta \gamma \delta}\, e_{\alpha}^{a}\, e_{\beta}^{b}\, e_{\gamma}^{c}\, e_{\delta}^{d}= \det \left| \left| e_{\alpha}^{a}\right| \right| \, \epsilon^{a b c d}= ( \pm ) \sqrt{g}\, \epsilon^{a b c d}\quad \mathrm{a n d}\quad \sqrt{g}\, \epsilon^{a b c d}\, e_{a}^{\alpha}\, e_{b}^{\beta}\, e_{c}^{\gamma}\, e_{d}^{\delta}= ( \pm ) \epsilon^{\alpha \beta \gamma \delta}.
D^{(n)}_q(v,u/v;t) \bar{P}^{*(n)}_\lambda(;q,t,s){}= \sum_{\mu\subset\lambda}s^{-|\lambda/\mu|}C^0_{\lambda/\mu}\big(q^{1/2}t^{n-1}sv,q^{1/2}t^{n-1}su/v;q,t\big)C^0_{(1^n+\mu)/\lambda}\big(t^{n-1}u;q,t\big) \qquad{}\times d_{\lambda\mu}(q,t) \bar{P}^{*(n)}_\mu\big(;q,t,q^{1/2}s\big),
\frac{\hat\sigma_{n,j}(\theta_n)}{\hat\sigma^M_{n,j}(\theta_n)}-1=\frac{\hat\sigma_{n,j}(\theta_n)-\hat\sigma^M_{n,j}(\theta_n)}{\hat \sigma^M_{n,j}(\theta_n)}=\frac{(1-\hat \mu_{n,j}(\theta_n))O_{P_n}(\hat\sigma_{n,j}(\theta_n))}{(1+o_{P_n}(1))\hat \sigma_{n,j}(\theta_n)+(1-\hat \mu_{n,j}(\theta_n))O_{P_n}(\hat\sigma_{n,j}(\theta_n))}=o_{P_n}(1),
\frak{m}'^{(0,2)}=\bigg\{p_-(A): A=\begin{bmatrix}A_1 0 0 0 A_1 0 0 0 A_2\end{bmatrix}, A_i=\begin{bmatrix}0 Y_i^tY_i 0 \end{bmatrix}, Y_i\in M_{1\times 2}(\mathbb{C}), i=1,2\bigg\}
\mbox{\boldmath$\Psi$\unboldmath}=\left( \begin{array}{c}\Psi_{+1}({\bf x}) \Psi_{-1}({\bf x})\end{array}\right) \;,\;\;\Psi_{\zeta}({\bf x})=\left( \begin{array}{c}\chi_{\zeta}({\bf x}) \varphi_{\zeta}({\bf x})\end{array}\right) \;,
\phi ( x ) = - \int_{\sigma}{d \Sigma_{\nu}( x^{\prime \prime}) \left[{d ( \partial ) \Delta ( x - x^{\prime \prime}) \cdot \Gamma^{\nu}( \partial^{\prime \prime}, \mathord{\frac{\buildrel{\lower 3 p t \mathrm{\scriptscriptstyle \leftarrow}}}{- \partial^{\prime \prime}}}) \phi ( x^{\prime \prime})}\right]}.
(K|k)=\frac{(\begin{matrix}K\\k\end{matrix})(\begin{matrix}N-K\\n-k\end{matrix})}{(\begin{matrix}N\\n\end{matrix})}\frac{\frac{1}{N+1}}{\frac{1}{n+1}}=\frac{(\begin{matrix}K\\k\end{matrix})(\begin{matrix}N-K\\n-k\end{matrix})}{(\begin{matrix}N+1\\n+1\end{matrix})}
\begin{array}{l}\displaystyle C_1= \frac{1}{\alpha+\gamma}+\frac{1}{2-\alpha-\gamma}- \log \frac{\alpha(2-\alpha-\gamma)}{(1-\alpha)(\alpha+\gamma)}\left( \frac{\gamma}{\alpha(\alpha+\gamma)}- \frac{1-\gamma}{(1-\alpha)(2-\alpha-\gamma)}\right),\end{array}
K_{\omega,\mathbf{c}}(V_n-V) =-K_{\omega,\mathbf{c}}(V)+K_{\omega,\mathbf{c}}(V_n)-(K_{\omega,\mathbf{c}}(V_n)-K_{\omega,\mathbf{c}}(V_n-V)-K_{\omega,\mathbf{c}}(V)) &\rightarrow -K_{\omega,\mathbf{c}}(V)<0.
G = \langle x,y,\tilde{z},\tilde{n}\mid \tilde{n}^p= x^{p^2}=y^p=1, \tilde{z}^p=x^p, [x,\tilde{z}]=y, [x,y] = 1, & [y,\tilde{z}]=x^pn, [x,\tilde{n}]=[y,\tilde{n}]=[\tilde{z},\tilde{n}]=1\rangle.
\left(\mathrm{Log}\,\mathrm{Det}\,\mathrm{A}\right)^{-(\alpha+1)}_\lambda:=\frac{d}{d\alpha}\mathcal{M}_\lambda[\mathrm{Det}\,\mathrm{E}^{-\mathrm{A}};\alpha]\neq \widehat{\mathcal{M}}_\lambda[\mathrm{Det}\,\mathrm{E}^{-\mathrm{A}};\alpha]=:\left(\mathrm{Det}\,\mathrm{Log}\,\mathrm{A}\right)^{-(\alpha+1)}_\lambda
{\cal P}_{a}^{x}\, = \, \frac{1}{2}t r ( J^{x}a + J^{x}q t^{t}a ) \quad ; \quad{\cal P}_{b}^{x}\, = \, - \frac{1}{2}t r ( J^{x}q b^{t}) \quad ; \quad{\cal P}_{c}^{x}\, = \, - \frac{1}{2}t r ( J^{x}q c q^{t})
\delta_{n,m}=\sum_{\mu}\mathcal{C}_{n}\left( E_{\mu}\right) \mathcal{C}_{m}\left( E_{\mu}\right) +\int\limits_{\Omega_{c}\left( E\right)}\mathcal{C}_{n}\left( E\right) \mathcal{C}_{m}\left( E\right) dE
\delta \Psi_{\hat{a}}= \partial_{\hat{a}}\epsilon +{\frac{1}{8}}H^{- 1}( \partial_{\hat{a}}H ) \epsilon +{\frac{1}{8}}W H^{{\frac{m - n}{4}}}\gamma^{\underline{{{u x y}}}}\gamma_{\underline{{{\hat{a}}}}}( i \sigma_{2}) \epsilon = 0 \, .
\lambda_{2,K^4}= \frac{(n-1)^2(n-3)^2}{24^2}\left(\frac{8\pi^4-150\pi^2+315}{640}\right)+\frac{(n-1)(n-3)}{24}\left(\frac{17(4\pi^6-42\pi^4+210\pi^2-315)}{13440}\right).
\begin{aligned}\nabla \cdot (\mu A^{-2}\nabla \hat{u}) - \mu \mathfrak{c}^l \hat{u}= \mu f^l \textnormal{in}\mathcal{N}_\delta, & \frac{\partial \hat{u}}{\partial \nu}= 0\textnormal{on}\partial\mathcal{N}_\delta,\end{aligned}
n_w\cdot s(n_w^{-1}\cdot t\cdot x) = n_w\cdot s(n_w^{-1}tn_w\cdot x) \intertext{since $n_w\in H$,}= n_wn_w^{-1}tn_w\cdot s(x) \intertext{since $n\in N_K(T_s)$ and $s$ is $T_s$-invariant,}= \chi(n_w)t\cdot s(x) \intertext{where $\chi$ is the character of $H$ associated to $\mathcal{L}^{2k}$}= \chi(n_w) s(t\cdot x).
\sum_{n \leq x}f(n) = \sum_{n \leq x}\sum_{d \mid n}g(d) &= \sum_{d \leq x}g(d) \left \lfloor \frac{x}{d}\right\rfloor &= x \sum_{d \leq x}\frac{g(d)}{d}+ O \left( \sum_{d \leq x}g(d) \right) &= x \sum_{d=1}^\infty \frac{g(d)}{d}+ O \left(x \sum_{d>x}\frac{g(d)}{d}+ \sum_{d \leq x}g(d) \right).