harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{multlined}|(A\otimes B)(\vec{x})| \leq |A(\vec{x}\rvert_0)|\cdot |B(\vec{x}\rvert_1)| <C_1 \prod_{i\in a}\phi(\vec{x}(i)) \cdot C_2 \prod_{j\in b}\phi(\vec{x}(j)) = C_1C_2 \prod_{i\in a\sqcup b}\phi(\vec{x}(i))\;.\end{multlined}
i\hbar\partial_{t}(\begin{matrix}\psi_{1}\\ \psi_{2}\end{matrix})=c(\begin{matrix}\sigma\cdot\Pi\psi_{2}\\ \sigma\cdot\Pi\psi_{1}\end{matrix})+q\phi(\begin{matrix}\psi_{1}\\ \psi_{2}\end{matrix})+mc^{2}(\begin{matrix}\psi_{1}\\ -\psi_{2}\end{matrix}),
g^i\rightharpoonup c = b_{g^i}c b^{-1}_{g^i}, & g^i x \rightharpoonup c = b_{g^i}\bigl(b_x c - \alpha(c)b_x\bigr) b^{-1}_{g^{i+1}}\shortintertext{and}g^i x^2 \rightharpoonup c = b_{g^i}\bigl( b^2_x c + \xi b_x\alpha(c)b_x + \alpha^2(c)b^2_x\bigr) b^{-1}_{g^{i+2}},
|J_1-J^{(P)}_1| = \sum_{\frac{\eta_1 n}{2N}\leq k_1,k_2 \leq \frac{2\eta_2 n}{N}}|B_{l_1,l_2}- Poi(k_1;np_{l_1}) Poi(k_2;np_{l_2})| \Delta(k_1,q_{l_1}) \Delta(k_2,q_{l_2})
(p^{*}_{ll'}\Psi)(g_{2},g_{1}) = \Psi(g_{2}g^{-1}_{1}) = (\iota^{*}\Psi)(g_{1}g^{-1}_{2}) = (p^{*}_{l^{-1}l'}(\iota^{*}\Psi))(g_{2},g_{1}), begin{align*}0.25cm](p^{*}_{l^{-1}(l')^{-1}}\Psi)(g_{1},g_{2}) = \Psi(g_{1}g^{-1}_{2}) = (\iota^{*}\Psi)(g_{2}g^{-1}_{1}) = (p^{*}_{l(l')^{-1}}(\iota^{*}\Psi))(g_{1},g_{2}).
c_{\gamma}( r ) = \frac{6 r m}{\gamma \pi^{2}}\int_{0}^{\infty}\mathrm{d}\theta \, \cosh \theta \, \ln ( 1 + \gamma e^{- r m \cosh \theta}) = \frac{6 r m}{\pi^{2}}\sum_{k = 1}^{\infty}\frac{1}{k}( - \gamma )^{k - 1}K_{1}( k r m ) \, .
F(x,r)\simeq\frac{\pi K_{d}}{2}R^{4}\sum_{i,j=0}^{1}(\frac{(-1)^{i+j}}{(x+iL_{1}+jL_{2})^{2}}(1-\frac{3}{2}\frac{r^{2}}{(x+iL_{1}+jL_{2})^{2}}))
vol(pr(Y)) =\int_{(y_1,\dots,y_T)\in pr(Y)}1dy_1\dots dy_T &=|\det U_1|\int_{(x_1,\dots,x_T)\in X}1dx_1\dots dx_T &=|\det U_1|vol(X)=|\det U_1||\det C|^{-1}vol(Y).
{\cal L}_{\rho}= - \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}- \frac{1}{4}( G_{\mu \nu}^{a}+ f_{a b c}F_{\mu \nu}^{b}\rho^{c}) ( G^{a \mu \nu}+ f_{a b c}F^{b \mu \nu}\rho^{c}) + \frac{m}{2}\epsilon^{\alpha \beta \gamma}F_{\alpha \beta}^{a}\phi_{\gamma}^{a}
\sigma_\mu^A \sigma_\nu^B \sum_{j=1}^{N_v}\langle g_\mu^A \vert j \rangle \langle j \vert f_\nu^B \rangle \langle j \vert \upsilon \rangle = \tilde{\it \Phi}_{\mu\nu}\ , 1\leq \mu \leq M_A \ , \ \ 1 \leq \nu \leq M_B \ ,
g_{\mu \nu}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{\alpha^{2}\rho^{2}}}\\ \end{array}\right) \b_{\mu \nu}={\frac{1}{{l_{B}}^{2}}}\left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right)
{\psi}_m \propto\quad < \mbox{source}_{after}\vert<0\vert \frac{(a_i(\vec{k}))^{m+1}}{\sqrt{[m+1]_q!}}\vert \quad a^{\dagger}_i g_i \quad \vert\frac{(a^{\dagger}_i(\vec{k}))^m}{\sqrt{[m]_q!}}\vert 0>\vert \mbox{source}_{before}>
\frac{\partial}{\partial t^{\prime}}[ \omega_{a}^{Q H^{\prime}S}( t^{\prime}, t ) ]_{Q^{S}( t , z )}= z ( \bar{C}( \Psi , t^{\prime}, t )_{a}^{b}- \Psi ( 0 ) \delta_{a}^{b}) [ \omega_{b}^{Q H^{\prime}S}( t^{\prime}, t ) ]_{Q^{S}( t , z )}
2\cos\theta=\left\langle r', r \right\rangle =\left\langle a_1e_1+\cdots+a_ne_n, e_1+e_2+\cdots+e_{i-1}+2e_i+e_{i+1}+\cdots+e_n \right\rangle &=\frac{a_i(n+1)-\sum_{k=1}^na_k}{n+1}+\sum_{j=1}^n\frac{a_j(n+1)-\sum_{k=1}^na_k}{n+1}=a_i.
\begin{cases}\,\,\partial_t(|u|^{q-1}u)+(-\Delta)_p^su=0 \textrm{in}\Omega_\infty:=\Omega \times (0,\infty), \,\,u=0 \textrm{on}\!\!(\mathbb{R}^n \setminus \Omega)\times(0,\infty) , \,\,u(\cdot, 0)=u_0(\cdot ) \textrm{in}\Omega, \end{cases}
\frac{dE}{d\tau}=\frac{d}{d\tau}\int_{V}d^{3}\sigma\:[\frac{\vec{E}^{2}_{\perp}+\vec{B}^{2}}{2}](\tau,\vec{\sigma})=-\int_{S}d\Sigma\:\vec{n}\cdot(\vec{E}_{\perp}\wedge\vec{B})(\tau,\vec{\sigma}).
\frac{\theta_0(4\eta; 2\tau)}{\theta_0(-2\eta; 2\tau) \theta_0(-2\eta + \tau; 2\tau)}= \frac{\theta_0(4\eta; 2\tau)}{\theta_0(-2\eta; \tau)}= - e^{4\pi i \eta}\theta_0(2\eta + 1/2; \tau)
\sigma_{a^{\dagger}a}^{2}={\frac{1}{4}}\biggl ( \mu_{1}^{2}+ \mu_{4}^{2}+ ( m_{0}\omega_{0})^{2}\mu_{3}^{2}+{\frac{\mu_{2}^{2}}{( m_{0}\omega_{0})^{2}}}{\frac{m_{0}\omega_{0}}{\hbar}}\biggr ) ,
Q_{+}^{I}= \frac{\partial}{\partial \theta_{I}^{+}}- i \eta^{I J}\theta_{J}^{+}\frac{\partial}{\partial \sigma^{+}}, Q_{-}^{I^{\prime}}= \frac{\partial}{\partial \theta_{I^{\prime}}^{-}}- i \eta^{I^{\prime}J^{\prime}}\theta_{J^{\prime}}^{-}\frac{\partial}{\partial \sigma^{+}},
\frac{F^2}{w_+}(1 + r) - \frac{F^2}{w_+}(1 + \tilde{r}) + \frac{F^2}{w_-}(1 + r) - \frac{F^2}{w_-}(1 + \tilde{r}) = O\left(\frac{1}{n\log^3n}\right),
J^{(i)}=-\left(\setlength{\unitlength}{1mm}\begin{picture}(12,4)\thicklines\put(6,1){\circle{8}}\put(1.75,1){\circle*{1}}\put(10.25,1){\circle*{1}}\put(12.5,4){\makebox(0,0)[l]{$\scriptsize -\! 1$}}\end{picture}\right)\:\;\;\frac{\delta \Gamma^{(i)}}{\delta J^{(0)}}.
|x_{n+1}|^{\frac 1{3^{n+1}}}\le e^{-(n+1)}|x_0|+ \frac{\beta(e-1)}{e}[e^{-n}+e^{-n+1}+\dots+ e^{-1}+1] &=e^{-(n+1)}|x_0|+ \frac{\beta(e-1)}{e}\frac{1-e^{-n-1}}{1-e^{-1}}\le e^{-(n+1)}|x_0|+ \beta.
\left( m+1\right) \left\vert \Omega\right\vert =\frac{1}{2}\int_{\Sigma}\chi & \leq\frac{1}{2}\left( \int_{\Sigma}H_{b}\chi^{2}\right)^{1/2}\left(\int_{\Sigma}\frac{1}{H_{b}}\right)^{1/2}.
\Delta f^{-m}+ D_X f^{-m}+ 2P f^{-m}= -mf^{-m-1}\Delta f+ m(m+1)f^{-m-2}|\nabla f|^2 -mf^{-m-1}|\nabla f|^2+2P f^{-m}\leq-mf^{-m-1}(1-(m+1)|\nabla f|^2f^{-1}-\frac{1}{m+1}Rf).
\bar{\gamma}:= \{(x,y) \mid (1-x, 1-y) \in \gamma \cup (a,1] \times \{0 \}\cup \{0 \}\times (b, 1] \}, \bar{\gamma}_0 := \{(x,y) \mid (1-x, 1-y) \in \gamma_0 \cup (a_0, 1] \times \{0 \}\cup \{0 \}\times (b_0, 1] \}.
G_3(\varepsilon ;\zeta ):=\widetilde{g}^\varepsilon b(\mathbf{D})(B^0-\zeta \overline{Q_0})^{-1}+g^\varepsilon \bigl(b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon (B^0-\zeta \overline{Q_0})^{-1}.
\hat{P}_{k+1}(x)\mathcal{W}(x)=\hat{H}_{k+1}\begin{bmatrix}0 -A 0 0\end{bmatrix}^{\top}h_{k}^{-1}p_k(x)+\begin{bmatrix}(\omega_{k+1,k+2})_{11}(\omega_{k+1,k+2})_{12}(\omega_{k+1,k+2})_{21}(\omega_{k+1,k+2})_{22}\end{bmatrix}p_{k+1}(x)+\begin{bmatrix}0 A 0 0\end{bmatrix}p_{k+2}(x).
\sup_{t_s \in \Omega_s}||B \exp(t+t_a+t_s)v|| \geq \kappa_2 \sup_{|J|+d_1+...+d_{a_0}= d}||B\exp(t+t_a)\sum_{\zeta}\alpha_{J\sqcup \zeta}e_J\wedge e_\zeta||
\tau_{i}=\bigg(q^{W}_{i-1}(x_{i-\frac{1}{2}})-q_{i}(x_{i-\frac{1}{2}})\bigg)^2+\bigg(q^{W}_{i+1}(x_{i+\frac{1}{2}})-q_{i}(x_{i+\frac{1}{2}})\bigg)^2,
\int\limits_{\gamma(0,h)}\left( a_{k}(t)\Psi_{k,t}(x)+a_{-k}(t)\Psi_{-k,t}(x)\right) dt=\int\limits_{[-h,h]}\left( a_{k}(t)\Psi_{k,t}(x)+a_{-k}(t)\Psi_{-k,t}(x)\right) dt.
\begin{cases}\Delta u_1+\lambda e^{u_2}(1-e^{u_1})=\displaystyle 4\pi\sum_{i=1}^{N_1}\delta_{\epsilon p_{1i}}, \Delta u_2+\lambda e^{u_1}(1-e^{u_2})=\displaystyle 4\pi\sum_{i=1}^{N_2}\delta_{\epsilon p_{2i}},\end{cases}\mathbb R^2
\eta_{\vec{p}}\left( t \right) = \sqrt{\frac{\omega_{0}\left( \vec{p}\right)}{\Omega_{\vec{p}}\left( t \right)}}\operatorname{e x p}\left\{- i \, \int_{0}^{t}\, \Omega_{\vec{p}}\left( t^{\prime}\right) d t^{\prime}\right\}
[ j_{+}^{n}, j_{+ -}^{p q}] = j_{+ -}^{p - n , q}, \quad [ j_{-}^{n}, j_{+ -}^{p q}] = - j_{+ -}^{p , q - n}, \quad [ j_{+}^{n}, j_{- +}^{p q}] = - j_{- +}^{p , q + n}, \quad [ j_{-}^{n}, j_{- +}^{p q}] = j_{- +}^{p + n , q}
\mu \Bigl( (a_h(w^i) (t))^2 \Bigr) = \mu \Bigl( (a_h(w^i) (0))^2 \Bigr)+ 2 \int_0^t ds \mu \bigl(a_h(w^i) (s) \frac{1}{h}(w^i(s+h) - w^i(s))\bigr).
T^{i j}= \kappa \, \eta^{i k}\eta^{j l}h_{\mu \nu}(\xi)\, \partial_k \xi^\mu \partial_l \xi^\nu - \frac{\kappa}{2}\,\eta^{i j}\eta^{k l}h_{\mu \nu}(\xi) \, \partial_k \xi^\mu\partial_l \xi^\nu - \kappa \, \eta^{\mu \nu}V(\xi).
2 \pi \alpha^{\prime}S_{b}= \int d \tau \int_{0}^{2 \pi \alpha^{\prime}p^{+}}d \sigma \left({\frac{1}{2}}G_{M N}\partial_{a}X^{M}\partial^{a}X^{N}+ B_{M N}\partial_{\tau}X^{M}\partial_{\sigma}X^{N}\right) .
- \frac{\sqrt{3}\pi}{8}\frac{q^{2}\alpha_{0}^{2}}{m_{H}^{2}}( 1 - \frac{q^{2}\alpha_{0}^{2}}{m_{H}^{2}})^{1 / 2}\kappa r N_{1}( \sqrt{3}\kappa r ) \cos ( 2 \omega_{0}t + 2 \delta ) .
Q_{\vec \ell}(t,x) = Q \left (\left ( -\frac{t}{\sqrt{1 - |\vec \ell|^2}}+ \frac{1}{|\vec \ell|^2}\left ( \frac{1}{\sqrt{1 - |\vec \ell|^2}}- 1 \right )\vec \ell \cdot x\right ) \vec \ell + x\right )= Q_{\vec \ell}(0, x - t\vec \ell).
f (g) f (x_{i}) = a_{i}(f(g) - f(gg_{i})) + \sum_{j}b_{ji}f(g) x'_{j}& = a_{i}(f(g) - f(gg_{i})) + \sum_{j}b_{ji}\chi_{j}' \big( f(g) \big) x'_{j}f (g)
\sigma_{(a_{i}^{\dag}a_{k})}^{+}=<{\bf A}_{+}\mid\frac{1}{2}(a_{i}^{\dag}a_{k}+a_{k}a_{i}^{\dag})\mid{\bf A_{+}}>=\alpha_{i}^{*}\alpha_{k}tanh\mid{\bf A}\mid^{2}+\frac{1}{2}\delta_{ik},
\begin{aligned}\varepsilon^3\int_0^t{|(\Delta^h_k(v\partial_yv)_\Theta, \Delta^h_k\varepsilon^2w^2_\Theta)_{L^2}|dt'}\lesssim \varepsilon^3 d_k^22^{-k}||(\varepsilon^2 w^2)_\Theta||_{\tilde L^2_{t, \zeta'(t)}(B^1)}||u_\Theta||^\frac 12_{\tilde L^\infty_t(B^\frac 32)}||\partial_yu_\Theta||_{\tilde L^2_{t}(B^{\frac 32})}\end{aligned}
\begin{aligned}D_1(x,x,y) = x, D_n(x,y,y) = Q(x,y,y), Q(x,x,y) = y, \begin{aligned}D_i(x,y,x) = x 1\leq i\leq n, D_i(x,y,y) = D_{i+1}(x,x,y) 1\leq i<n. \end{aligned}\end{aligned}
A_k:=\|\tilde{u}\|_{L^2(R_k\times [0, 1])}, P_k:=\|y_1^{\beta}\partial_{y_1}\tilde{u}\|_{L^1(R_k\times [0, 1])}, Q_k:=\|y_1^{-\beta}\partial_{y_2}\tilde{u}\|_{L^1(R_k\times [0, 1])},
\begin{aligned}I_2 \leq c_{r_1,\beta}(1+|t|)(\| \widehat u_0\|_2 + \|\,(1 + \xi^2 + \eta^2)^{(r_1-\beta)}\widehat u_0\|_2) \leq c_{r_1,\beta}(1+|t|)(\| u_0\|_2+\| D_x^{2(r_1-\beta)}u_0\|_2 + \| D_y^{2(r_1-\beta)}u_0\|_2) & \leq c_{r_1,\beta}(1+|t|)(\|u_0\|_2 + \|\,D_x^{\beta+s}u_0\|_2 + \|\,D_y^{\beta+s}u_0\|_2). \end{aligned}
B := A^\ast\cup (A^\ast+c) C :=(A^\ast+7c-d)\cup (A^\ast+8c-d) D :=(A^\ast+10c-d)\cup(A^\ast+11c-d) E :=(A^\ast+17c-2d)\cup(A^\ast+18c-2d) F :=(A^\ast+30c-3d)\cup(A^\ast+31c-3d) G :=(A^\ast+37c-4d)\cup(A^\ast+38c-4d) H :=(A^\ast+40c-4d)\cup(A^\ast+41c-4d) I :=(A^\ast+47c-5d)\cup(A^\ast+48c-5d) J :=B\cup C\cup D\cup E\cup F\cup G\cup H\cup I
\delta^{a}{}_{b}\partial_{\nu}h_{\mu}= \textstyle{\frac{1}{2}}\left( \bar{D}^{a \alpha}( \bar{\lambda}_{b}\gamma_{\mu}\gamma_{\nu})_{\alpha}- ( \gamma_{\nu}\gamma_{\mu}D_{b \alpha}\lambda^{a})^{\alpha}\right) \, ,
\eta_{j,N}~ \dot = \sum_{\substack{k=2 k\,}}^{\min\{2j,4N-2j\}}\gamma^k \begin{pmatrix}j \frac{k}{2}\end{pmatrix}\begin{pmatrix}2N-j -1 \frac{k}{2}-1\end{pmatrix}&= \sum_{h=1}^{\min\{j,2N-j\}}\gamma^{2h}\begin{pmatrix}j h\end{pmatrix}\begin{pmatrix}2N-j -1 h -1\end{pmatrix}.
= ( \frac{3}{2}\times \frac{4}{3}\times \frac{5}{4}\times \cdots \times \frac{2 0 1 4}{2 0 1 3}) \times ( \frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \cdots \times \frac{2 0 1 2}{2 0 1 3})
g^{z^2}=g^{R(h)yR(c_{m-3})yR(h^{-1}c_{m-3})}&=(hh^{-1}gh)^{yR(c_{m-3})yR(h^{-1}c_{m-3})}&=\left(h(h^{-1}gh)^\tau\right)^{yR(h^{-1}c_{m-3})}&=h\left((h^{-1}gh)^\tau \right)^\tau c_{m-3}h^{-1}c_{m-3}&=h(h^{-1}gh)h^{-1}&=g
A=(p-1)\varphi(2)\Gamma_p\left(\frac{1}{4}\right)\Gamma_p\left(\frac{3}{4}\right)\left(1+\sum_{t\in\mathbb{F}_p^{\times}}\varphi(t(t-1)){_2G_2}\left[\begin{array}{cc}\frac{1}{4}, \frac{3}{4}0, 0\end{array}\mid\frac{t}{x}\right]\right).
I\lesssim P\left(\max_{|j-k|\geq\iota}\left|\frac{1}{\sqrt{n}}\sum^{n}_{i=1}u_{ij}u_{ik}\right|>\jmath'\right)+n^{-c/2}\lesssim\frac{\sqrt{n\log p}}{n^{3c/2-c_1}}+n^{-c/2}\lesssim n^{-c_1},
\int_{\mathfrak U}|\omega| ={\pi'_s}_!\int_{\mathfrak U'}|\omega|={\pi'_s}_!\Big({i'_s}^*\int_{\mathfrak X'}|\omega|\Big) &=(\mathfrak i_0^*\circ{\pi_s}_!)\int_{\mathfrak X'}|\omega| &=\mathfrak i_0^*\int_{\mathfrak X}|\omega|,
\int_{0}^{1}xJ_{\alpha}(xu_{\alpha,m})J_{\alpha}(xu_{\alpha,n})dx=\frac{\delta_{m,n}}{2}[J_{\alpha+1}(u_{\alpha,m})]^{2}=\frac{\delta_{m,n}}{2}[J_{\alpha}^{\prime}(u_{\alpha,m})]^{2}
\eta := \big( \| \nu - 1 \|_\infty + \|\bar{u}^h\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}\big)\exp\Big\{c_r \|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}^{4r}\Big\}\leq c_0.
m_{1,t}(x, (m_{1,t}(y, z)) = m_{1,t}(m_{1,t}(x, y),z)- m_{1,t}(m_{1,t}(x, z),y), &m_{2,t}(x, (m_{2,t}(y, z)) = m_{2,t}(m_{2,t}(x, y),z)- m_{2,t}(m_{2,t}(x, z),y).
e^{-\frac{1}{3}2^{n\left(R_1-I(U;W) -\epsilon^{(1)}_{\alpha,\delta_1}\right)}\frac{(c-1)^2}{c'}}\leq e^{-\frac{1}{3}2^{n\left(R_1-I(U;W) -\epsilon^{(1)}_{\alpha,\delta_1}\right)}\frac{(c-1)^2}{(c'+1)}}= e^{-\frac{1}{3}2^{n\frac{\delta_2-\delta_1}{2}}},
u^{4}+(\frac{-3B^{2}}{8A^{2}}+\frac{C}{A})u^{2}+(\frac{B^{3}}{8A^{3}}-\frac{BC}{2A^{2}}+\frac{D}{A})u+(\frac{-3B^{4}}{256A^{4}}+\frac{CB^{2}}{16A^{3}}-\frac{BD}{4A^{2}}+\frac{E}{A})=0.
t_0^{(1)}= 2 \left( \frac{\tau}{1+\rho e^{i \beta \pi}}\right)^{\frac{1}{2}}i -1 &\sim 2 \left( \frac{\tau}{\rho}\right)^{\frac{1}{2}}e^{i \frac{\pi}{2}(1- \beta)}-1, \rho \rightarrow \infty.
R^i_{jkl}= \frac{{\delta}\hat{\mathbb{G}}^i_{jl}}{{\delta}x^k}- \frac{{\delta}\hat{\mathbb{G}}^i_{jk}}{{\delta}x^l}+ \hat{\mathbb{G}}^i_{ks}\hat{\mathbb{G}}^s_{jl}- \hat{\mathbb{G}}^s_{jk}\hat{\mathbb{G}}^i_{ls}
\mathrm{SINR}_O = \displaystyle\frac{\mathrm{G}_{\mathrm{S,TX}}\,\mathrm{G}_{\mathrm{RX}}\displaystyle\sum_{i=1}^M \mathrm{P}_i \, h_i \, \ell^{\mathrm{(S)}}(d_{\mathrm{S},i})}{\mathrm{W}+ \mathrm{G}_{\mathrm{I,TX}}\,\mathrm{G}_{\mathrm{RX}}\, \mathrm{P}_\mathrm{I}\! \displaystyle\sum_{j \in \Phi}\, h_j \, \ell^{\mathrm{(I)}}(d_{\mathrm{I},i})},
({\rm Id}_4\otimes P)f(X^0)({\rm Id}_4\otimes P) =\begin{pmatrix}P (X_1^0 X^0_2 +X^0_2 X^0_1 -X^0_2 X^0_2) P 0 0 0\end{pmatrix}\oplus \begin{pmatrix}0 0 0 0\end{pmatrix}.
\| \mathcal{R}_\Omega [{\rm div}\, G'_\alpha (\vec{\beta},{\bf w})] \|_{L^{\infty}_{1}(B^{{\rm c}}_{4R_{0}})}& \le C \bigg ( |\alpha|^{-\frac{1}{2}}\| G'_\alpha (\vec{\beta},{\bf w}) \|_{L^2 (\Omega)}+ \big | \log |\alpha | \big | \| G'_\alpha (\vec{\beta},{\bf w}) \|_{L^\infty_{2}(\Omega)}\bigg ) \,.
C_{\mu \nu}^{- 1}= - \frac{1}{m^{2}}\left( \begin{array}{c c c c}{0}&{\partial_{1}^{x}}&{\partial_{2}^{x}}&{- 1}\\{\partial_{1}^{x}}&{0}&{- m}&{0}\\{\partial_{2}^{x}}&{m}&{0}&{0}\\{1}&{0}&{0}&{0}\\ \end{array}\right) \delta^{2}( x - y ) .
W \triangleq \begin{bmatrix}1 1 1 \ldots 1 1 \omega \omega^2 \ldots \omega^{K-1}\vdots \vdots \vdots \ddots \vdots 1 \omega^{K-1}\omega^{2(K-1)}\ldots \omega^{(K-1)(K-1)}\end{bmatrix},
\log F_{6}(t)=-\frac{1}{4}|t|^3+\frac{2\sqrt{2}}{3}|t|^{3/2}+\frac{1}{24}\log |t|+c_0+O\big(|t|^{-\frac{3}{2}}\big)\mbox{as \ \ $t\to-\infty$},
W_1W_1^\top+W_2W_2^\top =W_1^\top W_1+W_2^\top W_2=\frac{1}{2}(2k I_v+\lambda A+\mu(J_v-I_v-A)), W_1W_2^\top+W_2W_1^\top =W_1^\top W_2+W_2^\top W_1=\frac{1}{2}(\lambda A+\mu(J_v-I_v-A)).
\eta_{\underline{{{a b}}}}\, V_{m}^{\underline{{{b}}}}\, \eta^{i \ell_{1}}\, \epsilon^{m \ell_{2}\dots \ell_{d}}\, \epsilon_{\ell_{1}\dots \ell_{d}}\, + \, 2 \, ( d ! ) \, a_{1}\, \eta_{\underline{{{a b}}}}\, \Pi_{j}^{\underline{{{b}}}}\, h^{i j}= 0
Z_{one-loop}= \int{d^2 \tau \over (\hbox{Im}\tau)^{{D\over 2}+ 1}|\eta|^{24}}\sum_{\alpha , \beta}C\Big[ \begin{array}{c}\alpha \cr \beta\end{array}\Big]\, \prod_{f \in F'}\Theta^{1/2}\Big[\begin{array}{c}\alpha(f)\cr \beta(f)\end{array}\Big](\tau).
\Gamma_{WZ}[\bar g;\sigma] = b \lim_{d\rightarrow 4}{\int\,d^d\,\,x\,\sqrt{-g}\, F_d - \int\,d^d\,\,x\,\sqrt{-\bar g}\,\overline F_d\over d-4}+ b'\lim_{d\rightarrow 4}{\int\,d^d\,\,x\,\sqrt{- g}E_d - \int\,d^d\,\,x\,\sqrt{-\bar g}\,\overline E_d\over d-4}\,,
\partial_{\tau}(|A_1|^2+|A_2|^2) +c\cdot\nabla |A_1|^2 + c\cdot\nabla |A_2|^2 = 0, \partial_{\tau}(|A_1|^2+|A_3|^2) +c\cdot\nabla |A_1|^2 + c\cdot\nabla |A_3|^2 = 0.
|T_{t}u(x)|=\left\vert \int_{\mathbf{\mathbb{Q}}_{p}^{n}}Z_{t}(x-y)u(y)d^{n}y\right\vert \leq ||u||_{L^{\infty}}\int_{\mathbf{\mathbb{Q}}_{p}^{n}}Z_{t}(x-y)d^{n}y=||u||_{L^{\infty}}.
\langle V_{g_1}f_1, V_{g_2}f_2 \rangle_{L^2 (\mathbb{R}^{2d})}= \langle f_1,f_2 \rangle_{L^2 (\mathbb{R}^d)}\overline{\langle g_1,g_2 \rangle}_{L^2 (\mathbb{R}^d)},
\begin{aligned}H^k =Q^k(N^k)^{-1}(P^k)^T=\frac{1}{\gamma}P^k(P^k)^T=\frac{1}{\gamma}\begin{pmatrix}\tau^k\beta^kJJ^T 0 0 \frac{1}{\tau^k\beta^k}I_l\end{pmatrix}, G^k =(Q^k)^T+Q^k-(M^k)^TH^kM^k\overset{(\ast)}{\succeq}(\frac{1}{\gamma^2}-\frac{1}{\gamma})(N^k)^TN^k,\end{aligned}
V_{\psi}=\gamma^i\gamma^j\frac{R_{ij}^{\widetilde{\nabla}}}{2}+\gamma^i(\overline{\nabla}_i\psi)+\psi^2-\frac{\gamma^i(\overline{\nabla}_i\psi)+(\overline{\nabla}_i\psi)\gamma^i}{2}+L^iL_i &=\gamma^i\gamma^j\frac{R_{ij}^{\widetilde{\nabla}}}{2}+\frac{1}{2}[\gamma^i,\overline{\nabla}_i\psi]+\psi^2+L^iL_i.
S_{k}(2^tq)=2^{k(t-1)}q^k+ 2^{t-1}\frac{S_{k}(2^{t-1}q)}{2^{t-2}}+2^{2t-1}\sum_{i=0}^{\frac{k-2}{2}}{\binom{k}{2i}}2^{(t-1)(k-2i-2)}q^{k-2i}S_{2i}(2^{t-1}q).
\frac{F(\tau_{\mathrm{min}})}{|\tau_{\mathrm{min}}|^{2+\frac{4s}{N}}}=\frac{N}{4s}\frac{\tilde{F}(\tau_{\mathrm{min}})}{|\tau_{\mathrm{min}}|^{2+\frac{4s}{N}}}>\frac{N}{4s}\frac{\tilde{F}(\tau_{\mathrm{max}})}{|\tau_{\mathrm{max}}|^{2+\frac{4s}{N}}}=\frac{F(\tau_{\mathrm{max}})}{|\tau_{\mathrm{max}}|^{2+\frac{4s}{N}}},
\mathbb{P}\left(N^{x}\geq R\right) =\mathbb{P}\left( e^{\frac{\beta^2\varepsilon^2}{4D^2}N^{x}}\geq e^{\frac{\beta^2\varepsilon^2}{4D^2}R}\right) \leq \mathbb{E}\left\{e^{\frac{\beta^2\varepsilon^2}{4D^2}N^{x}}\right\}e^{-\frac{\beta^2\varepsilon^2}{4D^2}R}\leq 2 e^{-\frac{\beta^2\varepsilon^2}{4D^2}R}.
\sum_{i = 1}^{N - 1}\epsilon_{i}\,{\frac{\mu_{i}^{N - 1 - j}\, d \mu_{i}}{\displaystyle 2 \sqrt{h_{N}- \mu_{i}}\, \prod_{j = 1}^{N - 1}( \mu_{i}- h_{j})}}= \delta_{j , 1}\, d x , \quad 1 \leq j \leq N - 1 , \quad \epsilon_{i}= \pm 1 .
\kappa_{\texttt{b};\pm}(q,q_0,q_1):= \frac{1}{2}\left( \frac{1-|q_0|}{1 + |q_0|}\right)^{\pm 1}+\frac{1}{2}\left(\frac{1-|q_1|}{1+ |q_1|}\right)^{\pm 1}&+ (n-1)\left( \frac{1-|q|}{1+|q|}\right)^{\pm 1}.
\|-\nabla_x \log \rho_{0|t}(x\,|\,y)\|^2 = \|-\nabla_x \log \rho_0(x)\|^2 &+ \frac{\alpha^2 e^{-2\alpha t}}{(1-e^{-2\alpha t})^2}\|e^{-\alpha t}x-y\|^2 &+ \frac{2\alpha e^{-\alpha t}}{1-e^{-2\alpha t}}\langle -\nabla_x \log \rho_0(x), \, e^{-\alpha t}x-y \rangle.
|F(z)-F_n(z)| = \left|\int_0^{\frac1n}s^{z-1}h(s)ds\right| \leq \int_0^{\frac1n}s^{\Re(z)-1}|h(s)|ds \leq \|h\|_{\infty}\int_0^{\frac1n}s^{\Re(z)-1}ds = \frac{\|h\|_{\infty}}{\Re(z)}n^{-\Re(z)}.
\Vert D_{m,n}\Vert_{p}^{(v,v)}:=\Vert D_{m,n}\Vert_{p}+\sup_{z_{1}\neq0,\,\,z_{2}\neq 0}\frac{\Vert D_{m,n}(x+z_{1},y+z_{2})-D_{m,n}(x,y)\Vert_{p}}{v(|z_{1}|)+v(|z_{2}|)}.
\frac{i}{2}\int_{X}F_{\mu \nu}^{D}( *_{\Omega}F )^{\mu \nu}= - \frac{3 i}{2}\int_{X}F_{\mu \nu}^{D +}F^{+ \mu \nu}+ \frac{i}{2}\int_{X}F_{\mu \nu}^{D -}F^{- \mu \nu}.
\mathcal{G}(O_{n+1}) = \mathcal{G}(h_1) \oplus \dots \oplus \mathcal{G}(h_i) \oplus \dots \oplus \mathcal{G}(h_p) & = \mathcal{G}(h_1) \oplus \dots \oplus [\mathcal{G}(h_i-1) \oplus 1] \oplus \dots \oplus \mathcal{G}(h_p) \eqref{ob1}& = \mathcal{G}(O'_n) \oplus 1
-c^{2}d\tau^{2}=-\frac{1}{4}(3\sqrt{1-\frac{r_{g}^{2}}{R^{2}}}-\sqrt{1-\frac{r^{2}}{R^{2}}})^{2}c^{2}dt^{2}+(1-\frac{r^{2}}{R^{2}})^{-1}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d\varphi^{2}).
\left(M_{-}R\right)_{11}=e^{ic_+(x)\sigma_3}\left(M_{-}R\right)_{11}=r_2(z) e^{2 iz x}M_{+,21}, &\left(M_- R\right)_{21}=e^{ic_+(x)\sigma_3}\left(M_{-}R\right)_{21}=\bar{r}_1(z) e^{-2iz x}M_{-,11}.
\chi_{\Lambda}( C^{\Lambda_{0}}) = \sum_{( \lambda_{0}^{\prime}, \kappa_{0}, - s ) \in D^{+}}\, q^{2 ( \lambda_{0}^{\prime}, \lambda + \rho_{0})}\, \sum_{s = 0}^{\infty}n_{\lambda_{0}^{\prime}, s}q^{- 2 s ( \kappa + g )}
\sum_{i=r}^{k-1}\frac{i(i-1)\ldots (i-t) \Gamma(i-t)}{(i-r+1)\Gamma(i-t-\varepsilon)}\cdot \frac{\Gamma(r-1-t-\varepsilon)}{(r-1)(r-2)\ldots (r-1-t)\Gamma(r-1-t)}&\leq c_1 \sum_{i=r}^{k-1}\frac{i}{i-r+1}(i-1)(i-2)\ldots(i-t) (i-t)^{\varepsilon}&\leq r c_1 \sum_{i=r}^{k-1}i^{t+\varepsilon}&\leq r c_1 k^{1+t+\varepsilon}= r c_1 k^{1+\alpha}.
P(x;k,\lambda)\approx\Phi\{\frac{(\frac{x}{k+\lambda})^{1/3}-(1-\frac{2}{9f})}{\sqrt{\frac{2}{9f}}}\},wheref:=\frac{(k+\lambda)^{2}}{k+2\lambda}=k+\frac{\lambda^{2}}{k+2\lambda},
a_{1+p+q\,2t+3p-3q}=2c-(p+1)t-p(p+2), b_{1+p+q\,2t+3p-3q}=2c+(q+1)t-q(q+2), c_{2+p+q\,2t+3p-3q-3}=\frac{q+1}{p+q+2}, d_{2+p+q\,2t+3p-3q+3}=\frac{p+1}{p+q+2}.
g(x-k\gamma) = g(x) - \gamma\sum_{j=1}^k \frac{g(x-(j-1)\gamma) - g(x-j\gamma)}{\gamma}< g(x) - \gamma \sum_{j=1}^k(\Lambda+\varepsilon)^{-j}\frac{\Lambda - 1+ \delta}{\gamma}g(x) = g(x) \left( 1 - (\Lambda - 1 + \delta) \sum_{j=1}^k(\Lambda+\varepsilon)^{-j}\right) < 0,
\zeta(s)\sim\sum_{n=1}^{N}n^{-s}-\frac{N^{1-s}}{s-1}-\frac{N^{-s}}{2}+N^{-s}\sum_{m=1}^{\infty}\frac{B_{2m}s^{\overline{2m-1}}}{(2m)!N^{2m-1}}
( f )_{\mu}^{\sim}({\bf l}) = \int_{0}^{2 \pi R}d \theta_{1}\cdots \int_{0}^{2 \pi R}d \theta_{d}\exp \bigl [ i R^{- 1}\sum_{k = 1}^{d}l_{k}\theta_{k}\bigr ] ( f )_{\mu}( \mathrm{\boldmath{\theta}}) .
D_{\eta_0}^2 \setminus \bigcup_{\eta > 0}E_\eta = \{(x,x') \in D_{\eta_0}^2: \mbox{for all}\eta > 0 \mbox{there exist}\gamma_1,\gamma_2 \in G \mbox{such that}p(\gamma_1 x,\gamma_2 x') < \eta \}= \{(x,x') \in D_{\eta_0}^2: \mbox{there exist}\gamma_1,\gamma_2 \in G \mbox{such that}\gamma_1 x = \gamma_2 x' \}.
\frac{1}{\widetilde t_n-\widetilde s_n}\ln\frac{\|X(\widetilde t_n)\xi\|}{\|X(\widetilde s_n)\xi\|}= \frac{1}{t_{k_n}-s_{k_n}-n}\ln\frac{\|X(t_{k_n})\xi\|}{\|X(s_{k_n}+n)\xi\|}begin{align*}1ex] = \frac{1}{t_{k_n}-s_{k_n}-n}\left(\ln\frac{\|X(t_{k_n})\xi\|}{\|X(s_{k_n})\xi\|}+\ln\frac{\|X(s_{k_n})\xi\|}{\|X(s_{k_n}+n)\xi\|}\right).
p(\sigma_{i}(g)\sigma_{i}(h)(j))=p(\sigma_{i}(gh)(j))=p_{gh,r}(\sigma_{i}(gh)(j))=\sigma_{i}(gh)p_{e,r}(j) =\sigma_{i}(g)\sigma_{i}(h)p_{e,r}(j) &=\sigma_{i}(g)p_{h,r}(\sigma_{i}(h)(j)).
\Delta_{\kappa}( \vec{x}, x_{0}) \sim{\frac{- i \kappa \cosh \left({\frac{\pi \kappa x^{0}}{2}}\right) \Gamma \left( 1 + i \kappa x^{0}\right)}{\left( 2 \pi \right)^{2}\sqrt{1 +{\frac{m^{2}}{\kappa^{2}}}}| \vec{x}| \left( \kappa | \vec{x}| \right)^{1 + i \kappa x^{0}}}}
I_{p_{X_{\mathcal{I}}}\tilde q_{\tilde Y_{T^c}|X_\mathcal{I}}}(X_T; \tilde Y_{T^c}|X_{T^c}) \stackrel{}{=}I_{p_{X_{\mathcal{I}}}q_{Y_{T^c}|X_\mathcal{I}}}(X_T; Y_{T^c}, Y_d|X_{T^c}) \stackrel{}{=}I_{p_{X_{\mathcal{I}}}q_{Y_{T^c}|X_\mathcal{I}}}(X_T; Y_{T^c}|X_{T^c}).
\mathcal{Z}_{m,n}^{\gamma}(z,\overline{z}) =\frac{(-1)^{m}(\gamma+m+1)_{n}m!}{2\pi i}\left(1-\mid{z}\mid^{2}\right)^{-\gamma}\oint_{\mid t\mid=1}\dfrac{t^{n}}{(t-z)^{m+1}}\left(\sum_{j=0}^{+\infty}\dfrac{(-\gamma-m)_{j}}{j!}{t^{j}\overline{z}^{j}}\right) dt,
\mathbf{\Delta}_{\nu}(x)=\frac{\left(\frac{x}{2}\right)^{\nu-1}}{\sqrt{\pi}\Gamma\left(\nu+\frac{1}{2}\right)}\left[\mathbf{H}_{\nu}(x)-\frac{x}{2\nu+1}\mathbf{H}_{\nu-1}(x)\right]-\frac{1}{x}\mathbf{H}_{\nu}^2(x)\left[\frac{x\mathbf{H}_{\nu}'(x)}{\mathbf{H}_{\nu}(x)}\right]'.
( e^{- 2 \eta})_{\mathrm{f i x}}= \left| \frac{p_{2}\, p_{4}}{q_{1}q_{3}}\right| \ , \qquad ( e^{- 2 \sigma})_{\mathrm{f i x}}= \left| \frac{p_{2}\, q_{3}}{q_{1}p_{4}}\right| \ , \qquad ( e^{- 2 \rho})_{\mathrm{f i x}}= \left| \frac{p_{4}\, q_{3}}{q_{1}p_{2}}\right| \ .
\sum_{n=0}^\infty e(X^{[n]})(t \tilde t)^{-n}q^n=\frac{1}{((t \tilde t)^{-1}q)_\infty(t \tilde t^{-1}q)_\infty(q)_\infty^{20}(t^{-1}\tilde t q)_\infty(t\tilde t q)_\infty}
d_l =(\epsilon'_0-\epsilon_0)d_0+\dots+ (\epsilon'_{l-1}-\epsilon_{l-1})d_{l-1}\leq d_0+\dots+d_{l-1}\leq \sum_{i=0}^{l-1}\left(\frac1r\right)^{l-i}d_l =\frac{1-\left(\frac1r\right)^{l}}{r-1}d_l <d_l,
S_{0}^{( l )}[ Q ] ={\frac{l}{2 g^{2}\beta}}\sum_{{\vec{n}}}\sum_{{\hat{e}}}\varphi_{{\vec{n}}}[ \varphi_{{\vec{n}}+ 2{\hat{e}}}- 2 \varphi_{{\vec{n}}+{\hat{e}}}+ \varphi_{\vec{n}}] \; .
\mathrm{d}X_i(t) = X_i(t)\sum_k \sigma_{ik}(t,\mathfrak{X}) \big(\vartheta_k(t,\mathfrak{X})\, \mathrm{d}t + \mathrm{d}W_k(t) \big)\,,\ \ \ \i = 1,\dots,n\,,\ \ \ \ \mathfrak{X}(0)=x\,.
\exp\left({\sum_{j=1}^{m+1}\sum_{n=1}^{\infty}E^{(n)}_{jj}u^{(n)}_j}\right)=\sum_{j=1}^{m+1}E_{jj}e^{\sum_{n=1}^{\infty}\lambda^{n}u^{(n)}_j}= D ({\bf u}, \lambda)
(a_0 \star q)(m_1m_2) = \,\,\star &=q(m_1m_2a_0)a^{-1}_0 \,\, (\textsc{L}3) &=q(m_1q(m_2a_0))a^{-1}_0 &=q(m_1q(m_2a_0)a^{-1}_0a_0)a^{-1}_0 \,\, \star &= (a_0 \star q)(m_1q(m_2a_0)a^{-1}_0) \,\,\star &=(a_0 \star q)(m_1 (a_0 \star q)(m_2)).
(1+T)\int_0^Te_1(U_{n+j})\,dt \ge Te_1(U_{n+j})(0) - \displaystyle{D_{n,j}(T)\sum_{l=0}^{j-1}\int_0^T \| \mathcal{\mathbf{B^{\ast}_{n+l}}}(U_{n+l}) \|_{G_{n+l}}^2\,dt\,.}
( \phi, \phi^a, \phi^{ab}, \phi^{abc}, \ldots ) = ( \phi_1 \phi_2, \phi_1 \phi_2^a + \phi_1^a \phi_2, \phi_1 \phi_2^{ab}+ \phi_1^a \phi_2^b + \phi_1^{ab}\phi_2, \phi_1 \phi_2^{abc}+ \ldots, \ldots )
M^{(c)}_{\gamma_1}(s)=\frac{\xi^2A r \mu_r}{\Gamma(\alpha)B^r}\sum_{k=1}^{\beta}\frac{b_k}{\Gamma(k)}{\rm \mathcal{H}}_{4,3}^{1,4}\Biggl[\!\frac{\mu_r}{B^r}s\Bigg\vert \{(\delta_2,\Delta_2)\atop (\lambda_2,\Lambda_2)}\Biggr].