harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Theta = \frac{1}{\sqrt{2}}\begin{pmatrix}\theta_1 + i \theta_2 \theta_1 -i \theta_2 \end{pmatrix}\,,\quad\bar\Theta = \frac{1}{\sqrt{2}}\begin{pmatrix}\theta_1 - i \theta_2 \theta_1 + i \theta_2 \end{pmatrix}^T{\cal C}\, ,
\sum_{k=0}^n\frac{(a;q)_k (b;q)_k}{(q;q)_k}(-ab)^{n-k}q^{\frac{(n-k)(n+k-1)}{2}}=\sum_{k=0}^n\frac{(a;q)_{n+1}(-b)^k q^{k\choose 2}}{(q;q)_k (q;q)_{n-k}(1-aq^{n-k})}.
[E_1^{c_1}\cdots E_r^{c_r}, F_t] = E_1^{c_1}\cdots E_{t - 1}^{c_{t - 1}}[E_t, F_t] E_{t + 1}^{c_{t + 1}}\cdots E_r^{c_r}= E_1^{c_1}\cdots E_{t - 1}^{c_{t - 1}}\frac{K_t - K_t^{-1}}{q_t - q_t^{-1}}E_{t + 1}^{c_{t + 1}}\cdots E_r^{c_r}\in U_{q}(\mathfrak{l}).
S^{(1)}_{N}(x,y)=K_{N}(x,y)-\psi_{1}(x)\sum_{l=1}^{4}[AC(I-BAC)^{-1}]_{l1}\ \varepsilon \psi_{l}(y)-\psi_{2}(x)\sum_{k=1}^{4}[AC(I-BAC)^{-1}]_{k2}\ \varepsilon \psi_{k}(y),
\frac{t_{m-1}^*-t_{m}^*}{t_{m'-1}-t_{m'}}=\frac{\eta_*}{\eta_0}\cdot\frac{4(m')^2-1}{4m^2-1}\geq\frac{2}{3}\frac{\eta_*(m')^2}{\eta_0m^2}\geq\frac{1}{2}\frac{\eta_0}{\eta_*}\geq\frac{1}{3q}\geq\frac{1}{3n},
\mathcal{H}_N = \sum_{j=1}^N -\Delta_j + \sum_{1\leq i<j \leq N}v(x_i-x_j), &\Lambda = \Big[-\frac{\ell}{2}, \frac{\ell}{2}\Big]^d, \ell = \frac{K_{\ell}}{\sqrt{\rho \widehat{g}(0)}}.
x^*x =(\operatorname{id}\otimes\omega)\bigl((1\otimes c)W\bigr)^*\,(\operatorname{id}\otimes\omega)\bigl((1\otimes c)W\bigr) &\le\|\omega\|\,\bigl(\operatorname{id}\otimes|\omega|\bigr)\bigl(W^*(1\otimes c^*)(1\otimes c)W\bigr),
\partial_t c = \nabla \cdot(\nabla c+ c \nabla \psi) x\in \Omega,\; t>0, - \Delta \psi = c +\rho_0(x), x\in \Omega, t>0, c(0, x) =c^{\rm in}(x), x\in \Omega, \frac{\partial \psi}{\partial \textbf{n}}=\sigma, \frac{\partial c}{\partial \textbf{n}}+c \frac{\partial \psi}{\partial \textbf{n}}=0, x\in \partial\Omega,\; t>0.
\omega=a(r(t))\left(\theta^{2}\otimes e_{3}-\theta^{3}\otimes e_{2}+\theta^{4}\otimes e_{1}\right)+\left(b(r(t))-1\right)\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right)
dX_{t}^{i,N}= F_{A}\left( \frac{1}{N}\sum_{k=1}^{N}(K \ast V^{N})(X_{t}^{i,N}-X_{t}^{k,N})\right)\; dt + \sqrt{2}\; dW_{t}^{i}, t\leq T, 1\leq i \leq N,
\begin{array}{r c l}{{}}&{{}}&{{k_{i}X_{j}^{+}k_{i}^{- 1}= q_{i}^{1 / 2}X_{j}^{+}~ ,}}\\{{}}&{{}}&{{X_{i}^{+}X_{j}^{-}- q_{i}^{a_{i j}/ 2}X_{j}^{-}X_{i}^{+}= \delta_{i j}\displaystyle \frac{1 - k_{i}^{- 2}}{1 - q_{i}^{- 1}}~ .}}\\ \end{array}
g_{ii,\,00}-\frac{1}{g_{ii}}\,(g_{ii,\,0})^{2}- g_{ii,\,0}\big(\,\frac{1}{2}\,(\log|g_{00}|)_{,\,0}+\tau_{1}\sqrt{|g_{00}|}\,\big) +\frac2{n}\,\hat C\,|g_{00}|\,g_{ii}= 0
|\{(cl(i),cl(j)) \mid -j \in C_i, 1\le i,j\le l\}| &= \begin{cases}0 \mbox{if $1 \le l \le q^{(m+1)/2}-q$}, 2h \mbox{if $l=q^{(m+1)/2}-q+h$, $1 \le h \le q-2$}, 2(q-1) \mbox{if $q^{(m+1)/2}-1\le l\le q^{(m+1)/2}$}.\end{cases}
\sigma_1(S_g) =\frac{R_g}{2(n-1)}, \sigma_2(S_g) =\frac{1}{2}\left( \frac{n^2-n}{4(n-1)^2(n-2)^2}R^2_g - \frac{|\mathrm{Ric}_g|^2_{g}}{(n-2)^2}\right),
G^{(3)}= \frac{1}{(3u_3)^{1/3}}\exp\left(\frac{2{u_2}^3}{27{u_3}^2}- \frac{(u_1-{{u}}){u_2}^2}{3{u_3}}\right)\times\mathrm{Ai}\left( \frac{u_2}{(3u_3)^{4/3}}- \frac{u_1-{{u}}}{(3u_3)^{1/3}}\right)
\Psi \left( \mathbf{X}\right) =\mathbb{E}\left[ \Psi \left( \mathbf{X}^{\prime}\right) \right] +\left( \Psi \left( \mathbf{X}\right) -\mathbb{E}\left[ \Psi \left( \mathbf{X}^{\prime}\right) \right] \right) .
l^\mathcal{L}(t,U,\Gamma):=\int_M\mathbf{i}_{\partial_t}\ell\left(t,[U(p)],\left[\operatorname{Hor}^\Gamma_p\right]\right)=\int_M\mathbf{i}_{\partial_t}\ell^{\Gamma_0}\left(t,\tilde{\mathcal{U}}(m),\left[p,\Gamma_0\!\cdot\!\operatorname{Hor}^\Gamma_p\right]_G\right).
(-\Delta_p)^s \widetilde\Gamma (x)\leq \left(\frac{R}{R-1}\right)^{N+s\,p}\,\frac{2}{|x|^{N+s\,p}}\int_{B_1}\Gamma(y)^{p-1}\, dy=\frac{C}{|x|^{N+s\,p}},
F^{n-1}= -\delta u^{n-1}\cdot\nabla\varrho^{n-1}+\div(\delta\kappa^{n-1}\nabla\varrho^{n-1}), & H_e^{n-1}=\delta h^{n-1}-(\delta u^{n-1}-\delta\kappa^{n-1}\nabla\log\rho^n \, -\, \kappa^{n-2}\nabla \delta(\log\rho)^n )\cdot\nabla u^{n-1}- \delta\lambda^{n}\nabla\pi^{n-1},
f(w; k \lambda) = \check{C}\cdot d^{k w}\cdot \prod_{i=0}^{k-1}\dfrac{\varGamma\left(w+ \frac{u_1 + i}{k}\right) \cdots \varGamma\left(w+ \frac{u_m +i}{k}\right)}{\varGamma \left(w+ \frac{v_1 + i}{k}\right) \cdots \varGamma \left(w+ \frac{v_m +i}{k}\right)},
\mathcal{C}_{-1}\big(\mathcal{a}^{(x)}, \hat{\mathcal{n}}^{(x)}, j, J, J' \setminus \lbrace j \rbrace\big) = \mathcal{C}_{-1}\big(\alpha, \nu, p_{(x,j)}, \mathcal{Q}_{<x}\cup p^{(x)}_J, \mathcal{Q}_{>x}\cup p^{(x)}_{J' \setminus \lbrace j \rbrace}\big).
\left( \begin{array}{c}{\hat{{\bf b}}}\\{\hat{{\bf b}}^{\dagger}}\\ \end{array}\right) = \Omega \left( \begin{array}{c}{\hat{{\bf a}}}\\{\hat{{\bf a}}^{\dagger}}\\ \end{array}\right) + \left( \begin{array}{c}{{\bf d}}\\{{\bf d}^{*}}\\ \end{array}\right) , \qquad \Omega = \left( \begin{array}{c c}{\zeta}&{\eta}\\{\eta^{*}}&{\zeta^{*}}\\ \end{array}\right) ,
\varepsilon_{0}^{\beta}=\frac{|e|H}{(2\pi )^{2}}\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}dp \;\epsilon_{n}\left\{\frac{-2+{\rm e}^{\beta \epsilon_{n}}}{1-{\rm e}^{\beta \epsilon_{n}}}\right\}.
\mathbb{P}\bigg(\sum_{j=1}^L Z_j - \sum_{j=1}^L \mathbb{E}\big[Z_j\big] \geq \tau\bigg) \leq \exp\Big[-\frac{2\tau^2}{L}\Big] \mathbb{P}\bigg(\sum_{j=1}^L Z_j - \sum_{j=1}^L \mathbb{E}\big[Z_j\big] \leq -\tau\bigg) \leq \exp\Big[-\frac{2\tau^2}{L}\Big]\,,
\begin{matrix}a_{11}x_{1}+\cdot\cdot\cdot+a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+\cdot\cdot\cdot+a_{2n}x_{n}=b_{2}\\ \vdots\\a_{m1}x_{1}+\cdot\cdot\cdot+a_{mn}x_{n}=b_{m}\end{matrix}.
\left| \sum_{k = 0}^{k_1 - 1}\log \frac{|T'(x_k)|}{|T'(y_k)|}\right| = \left| \log \frac{(T^{k_1})' (x_0)}{(T^{k_1})' (y_0)}\right| \le \frac{C'}{|J|}| x_{k_1}- y_{k_1}|.
U_{\alpha}( t ) = \frac{1}{\sqrt{2}}\left( \begin{matrix}{{f_{\alpha}^{( - )}( t ) + f_{\alpha}^{( + )}( t )}}\\{{f_{\alpha}^{( - )}( t ) - f_{\alpha}^{( + )}( t )}}\\ \end{matrix}\right) , ~ ~ V_{\alpha}( t ) = \frac{1}{\sqrt{2}}\left( \begin{matrix}{{g_{\alpha}^{( - )}( t ) + g_{\alpha}^{( + )}( t )}}\\{{g_{\alpha}^{( - )}( t ) - g_{\alpha}^{( + )}( t )}}\\ \end{matrix}\right) ,
\frac{w_{p q}^{C P}( n )}{w_{p q}^{C P}( 0 )}= w_{p R ( q )}( n ) , \frac{\bar{w}_{p q}^{C P}( n )}{\bar{w}_{p q}^{C P}( 0 )}= \frac{1}{w_{p q}( - n )};
\frac{1}{\sqrt{N}}\| (A + V)^{-1}e_1 \| = \frac{\sqrt{1 + \| Q (g+b)\|^2}}{|g_0 + a - (g+b)^* Q (g+b)|}\leq \frac{1 + \| Q (g+b)\|}{|g_0 + a - (g+b)^* Q (g+b)|}
2 ( \dot{x}^{A A^{\prime}}\acute{x}^{B B^{\prime}}- \acute{x}^{A A^{\prime}}\dot{x}^{B B^{\prime}}) = v \bar{\pi}^{A}\bar{\pi}^{B}\epsilon^{A^{\prime}B^{\prime}}+ \bar{v}\pi^{A^{\prime}}\pi^{B^{\prime}}\epsilon^{A B}.
\mbox [\hat{\Lambda}_{n},\hat{\Lambda}_{m}]= [\hat{\Lambda}_{n}^{\dagger},\hat{\Lambda}_{m}^{\dagger}]=0; \,\,\,[\hat{\Lambda}_{n}\,\,,\,\,\hat{\Lambda}_{m}^{\dagger}]= \frac{R}{\pi}\delta_{n,m}.
\frac{1}{\beta(\tau^k)^2}+\frac{\sigma''}{\tau^k}\geq\frac{1}{\beta(\tau^{k+1})^2}+\frac{\sigma''(1-\gamma)}{\tau^{k+1}},~\sigma''=\sigma'-\frac{\sigma'^2}{\sigma'+1},~\sigma'=\frac{\sigma_{\rm min}(A_mA_m^T)}{L},
\left[ e^{- t \bar{H}_{s}^{2}}\right]_{\mathrm{d i a g}}={\frac{1}{( 4 \pi t )^{3 / 2}}},{}\left[ e^{- t \bar{H}_{d}^{2}}\right]_{\mathrm{d i a g}}={\frac{r_{d}}{( 4 \pi t )^{3 / 2}}}\left[ 1 + \frac 1 2 \kappa^{2}t \right] .
\dim_{\mathrm{k}}(\mathcal{H}_{l}(A)a)=\dim_{\mathrm{k}}(\mathcal{H}_{l}(A))\quad \quad\dim_{\mathrm{k}}(a\mathcal{H}_{r}(A))=\dim_{\mathrm{k}}(\mathcal{H}_{r}(A)) .
X^{\alpha \beta}= - \partial_{\mu}\partial_{\mu}\delta^{\alpha \beta}+ r_{0}\delta^{\alpha \beta}- 2 g_{0}A_{\mu}^{\alpha \beta}\partial_{\mu}+ v_{0}T^{\alpha \beta}+ w_{0}S \delta^{\alpha \beta}+ g_{0}^{2}A_{\mu}^{\beta \gamma}A_{\mu}^{\alpha \gamma}.
\frac{\partial Y}{\partial x}=\left( \frac{A_3}{x}+A_2+A_1x+A_0x^2 \right)Y , \frac{\partial Y}{\partial t_1}=(A_0 x+B_{10})Y ,\frac{\partial Y}{\partial t_2}=\left( -A_0 x^2+B_{21}x+B_{20}\right)Y .
e^{- \lambda / 2}\sim f_{\mathrm{e f f}}( d ) = \left({\frac{1}{( D + 2 ) ( D + 4 )}}-{\frac{1}{( d + 2 ) ( d + 4 )}}+{\frac{2}{d}}\left({\frac{1}{d + 4}}-{\frac{1}{D + 4}}\right) \right) .
\partial_2(|\nabla u|^{-2}A_{11})-\partial_1(|\nabla u|^{-2}A_{12}) = |\nabla u|^{-2}(\partial_2 A_{11}-\partial_1 A_{12})+A_{11}\partial_2|\nabla u|^{-2}-A_{12}\partial_1 |\nabla u|^{-2}= |\nabla u|^{-2}(2A_{12}k_{13}-2A_{11}k_{23})+2A_{11}|\nabla u|^{-2}k_{23}-2A_{12}|\nabla u|^{-2}k_{13}= 0.
\sum_{i=1}^{n}w_{i}x_{i}\ge\prod_{i=1}^{n}x_{i}^{w_{i}}+\sum_{i=1}^{n}w_{i}(x_{i}^{\frac{1}{2}}-\sum_{i=1}^{n}w_{i}x_{i}^{\frac{1}{2}})^{2}
{\bf S}_{2,j}={\bf S}_2-\frac{1}{n_2}{\bf Y}_{\cdot j}{\bf Y}^*_{\cdot j}, ~~{\bf D}_j(z)=({\bf S}_2^{-\frac12}{\bf T}_p^{\frac12}) \left({\bf S}_{1}-\frac{1}{n_1}{\bf X}_{\cdot j}{\bf X}_{\cdot j}^*\right) ({\bf S}_2^{-\frac12}{\bf T}_p^{\frac12})^*-z\cdot{\bf I}_p~,
\frac{C_{q,n}}{(k-2-q)!}= - \sum_{t=0}^{q-1}[2(n-1) a_{k-t}+ b_{k-t-1}] m_{(q-t,\dot{0})}- \sum_{s=1}^{[q/2]}\sum_{t=0}^{q-2s}2a_{k-t}\, m_{(q-t-s,s,\dot{0})}- \frac{n(n-1)}{k(k-1)}q(2k-q-1) a_{k-q}- \frac{n}{k-1}q b_{k-q-1}, q=1, 2, \ldots, k-2,
\vec{\tilde{q}}= \left( \begin{array}{c}\tilde{q}^{(1)}\vdots \tilde{q}^{(N)}\end{array}\right)\, ,\hspace{1cm}\vec{\tilde{p}}= \left( \begin{array}{c}\tilde{p}^{(1)}\vdots \tilde{p}^{(N)}\end{array}\right)\, .
\left \|\sum\limits_{j=1}^{\infty}\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}+\left \|\sum\limits_{j=1}^{\infty}\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}<\infty,
- \int_0^T \, dt \int_\Omega \Bigl\langle v_h (x,t), \frac{\partial \phi}{\partial t}(x,t) \Bigr\rangle \, dx\, + \, \int_0^T \, dt \int_\Omega \langle D v_{\bar{h}}(x,t), D \phi (x,t) \rangle \, dx\; = \; 0 &\mathrm{for}\;^\forall \phi \, \in \, \dot{C}^\infty (\Omega\times(0,T)) \; \mathrm{with}\; \div \phi \, = \, 0,
\frac{R_{tt}}{Ae^{2\Omega}}+AR_{rr}+\frac{(d-2)R_{\theta_a \theta_a}}{r^2 \displaystyle{\prod_{b=1}^{a-1}\sin^2\theta_b}}=\frac{d-2}{r^{d-2}}\frac{d}{dr}\left[r^{d-3}(1-A)\right] \;.
-\frac{3c(-1)^i a b_j}{4}{}=\langle\bar{R}(U_1,U_2)U_i,\eta\rangle &{}=\langle\nabla_{U_1}S_\eta U_2,U_i\rangle-\langle\nabla_{U_1}U_2,S_\eta U_i\rangle-\langle\nabla_{U_2}S_\eta U_1,U_i\rangle+\langle\nabla_{U_2}U_1,S_\eta U_i\rangle &{}=-U_j\mu-\frac{3ca(b_j\lambda_i+2b_i\mu)}{2(\lambda_1-\lambda_2)}.
\sum_{j=0}^{n}(-1)^{j}\mathrm{dim\,}H_{b,m}^{j}(X)=\frac{1}{2\pi}\int_{X}\mathrm{Td_{b}\,}(T^{1,0}X)\wedge e^{-m\frac{d\omega_{0}}{2\pi}}\wedge\omega_{0}
\sum_{r}S_{r}(\omega, \psi, \chi, \varphi; z, \zeta)\lim_{\bar{z}\rightarrow 0}\;\bar{z}^{\Delta_{\bar{v}}+ \Delta_{\bar{r}}- \Delta_{\bar{x}}}\;\lim_{\bar{\zeta}\rightarrow 0}\;\bar{\zeta}^{\Delta_{\bar{w}}+\Delta_{\bar{u}}- \Delta_{\bar{r}}}\;\widehat{W}_{\bar{x},\bar{r}}^{2}(\bar{z},\bar{\zeta},0),
[ \Phi^{+ ( + ) [ + ]}( \theta^{+ +}, \theta^{( + + )}, \theta^{[ + ] \{\pm \}}) ]^{a_{3}}[ \Sigma^{+ ( + ) \{+ \}}( \theta^{+ +}, \theta^{( + + )}, \theta^{[ \pm ] \{+ \}}) ]^{a_{4}}
\log M_{{\rm sym}^1}^{-r}= 2r\log_2 r + 2(\gamma-2\log\zeta(2))r + O(r/\log r), \noalign{\vskip 1mm}\log M_{{\rm sym}^2}^{-r}= r\log_2 r + (\gamma-2\log\zeta(2))r + O(r/\log r), \noalign{\vskip 1mm}\log M_{{\rm sym}^m}^{r}= (m+1)r\log_2 r+(m+1)\gamma r + O(r/\log r)(m=1, 2),
q^2\xi_{02}(p,q) - p.q\xi_{11}(p,q) ={1\over2}\left\{-\left[{1\over 2}+ m^2\xi_{00}(p,q)\right] +{p^2\over 2}\xi_{10}(p,q) +{3q^2\over 2}\xi_{01}(p,q)\right\}
\int_0^1 |f-g| \, \mathrm{d}x \geq \frac{1}{8}\frac{3^{L^2}-(3^{L-1}-1)^L-2}{3^{L^2}}\geq \frac{1}{8}\frac{3^{L^2}-3^{(L-1)L}}{3^{L^2}}= \frac{1}{8}(1-3^{-L}) \geq \frac{1}{9}.
g^{(i)}(0) = 0, \: (g^{(i)})'(t) = r(\boldsymbol{\zeta}^{(i)}(g^{(i)}(t))){{1}}_{\{g^{(i)}(t) < \tau_{\boldsymbol{\zeta}^{(i)}}\}}+{{1}}_{\{g^{(i)}(t) \ge \tau_{\boldsymbol{\zeta}^{(i)}}\}},
\widehat{y}_1^{\,[K,L]}(x) = (1-q) x^{\lambda}\sum_{n=K}^{L}(q^n\xi )^{\mu' -\lambda}\frac{\big(x q^{-n}/ \xi, q^{1-n}/( \beta_1 \xi ), q^{1-n}/( \beta_2 \xi ) ; q\big)_{\infty}}{\big(x q^{-n-\lambda}/ \xi, q^{-n}/ (\alpha_1 \xi),q^{1-n}/ (\alpha_2 \xi) ; q\big)_{\infty}},
\widetilde B_0 = CD, \widetilde B_j = \begin{pmatrix}v_n^* T_n^{*(j-1)}0 0 u_{1,n}^* T_n^{*(j-1)}\end{pmatrix}\begin{pmatrix}I -I I -I \end{pmatrix}\begin{pmatrix}H_{1,n}^{-1}u_{1,n}0 0 H_{1,n}^{-1}v_{n}\end{pmatrix}CD, 1\le j \le n+1
\mathbb{E}(W | W_1 = 1) = \sum_{i=0}^n P(W = i | W_1 = 1) i &= \sum_{i=0}^n \frac{P(W=i)}{P( W_1 = 1)}P(W_1 = 1 | W = i) i &= \frac{1}{p}\sum_{i=0}^n P(W = i) i^2 &= n\frac{\mathbb{E}(W^2)}{\mathbb{E}(W)}\geq \frac{[\mathbb{E}(W)]^2}{\mathbb{E}(W)}= \mathbb{E}(W) = np.
\chi^{Fg}= \prod_{h \in F}(\chi_1^{-1})^{hg}\circ (\chi_2^{-1})^{hg}\circ \chi_1^{hg}\circ \chi_2^{hg}&= \prod_{h \in F}(\chi_1^{-1})^{hg}\circ \prod_{h \in F}(\chi_2^{-1})^{hg}\circ \prod_{h \in F}\chi_1^{hg}\circ \prod_{h \in F}\chi_2^{hg}&= [\chi_1^{Fg}, \chi_2^{Fg}]
C_N^{(\beta)}+\frac{1}{2}\sum_{p \in \Lambda_+^*}\left[ -F_p+\sqrt{F_p^2-G_p^2}\right] =\; 4\pi (N-1)\mathfrak{a}_N^{(\beta)}-\frac{1}{2}\sum_{p\in \Lambda_+^*}\left[ p^2+\widehat V(0)-\sqrt{|p|^4+2p^2 \widehat{V}(0)}-\frac{\widehat{V}^2(0)}{2p^2}\right] &+\mathcal{O}(N^{-\alpha})
a_j ={}(pq)^{j}(n-\alpha) - (\alpha(1+q) - (\sigma_1 + \sigma_2 q))\frac{(pq)^j - 1}{pq-1}={}(pq)^{j}\left(n-\alpha -\frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{pq-1}\right) + \frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{pq-1}{}\rightarrow a := -\frac{\alpha(1+q) - (\sigma_1 + \sigma_2 q)}{1-pq}< 0 \,\, j\rightarrow \infty.
-\int_{B_{1}}\partial_{s}(g_{1-\alpha,m}*[\phi\psi^{2}w^{2}])dx + (1-q)\frac{1}{2}\phi \cdot \leq \int_{0}^{s}\dot{g}_{1-\alpha,m}(s-\tau)(\phi(s)-\phi(\tau))\left( \int_{B_{1}}\psi^{2}\tilde{u}^{1-q}dx \right)(\tau)d\tau & + C(\delta,\Lambda)(1-q)(\rho-\rho')^{-2\beta}\phi(s)\int_{\rho B_{1}}w^{2}dx + R_{m}(s).
\int_{-\infty}^\infty \Big\langle \Lambda^TE(\cdot,1/2+ir) , \Lambda^TE(\cdot,1/2+ir) \Big\rangle\hat{g}(1/2 + ir) dr &=\int_{-\infty}^\infty \Big\langle \Lambda^TE(\cdot,1/2+ir) , \Lambda^TE(\cdot,1/2+ir) \Big\rangle|\hat{g}_0(1/2 + ir)|^2 dr
\Sigma^{a b}= \frac{1}{4}( \gamma^{a}\gamma^{b}- \gamma^{b}\gamma^{a}) = \frac{1}{4}\left( \begin{array}{c c}{{\sigma^{a}\bar{\sigma}^{b}- \sigma^{b}\bar{\sigma}^{a}}}&{{0}}\\{{0}}&{{\bar{\sigma}^{a}\sigma^{b}- \bar{\sigma}^{b}\sigma^{a}}}\\ \end{array}\right) .
\stackrel{( - 1 , 0 )}{\Phi}_{\alpha_{L - 2}}^{* ( 1 )}Z_{\; \; \alpha_{L - 1}}^{\alpha_{L - 2}}= \delta_{1}\left( \stackrel{( - 2 , 0 )}{\eta}_{\alpha_{L - 1}}^{* ( 1 1 )}- \stackrel{( - 2 , 0 )}{\gamma}_{\alpha_{L}}A_{\alpha_{L - 1}}^{\; \; \alpha_{L}}\right) .
W(z)-\frac\pi2 =\frac{|C|^2}2\left(\frac\pi{\pi+\delta}\|s'\|_2^2-\frac{\pi+\delta}\pi \|s\|_2^2\right)+\frac\delta2 &=\frac{|C|^2}2\left(\|s'\|_2^2-\|s\|_2^2\right)+\frac{\delta}{2}\left(-\frac{|C|^2}{\pi+\delta}\|s'\|_2^2-\frac{|C|^2}\pi \|s\|_2^2+1\right).
E_{\mu\nu}={\tilde E}_{\mu\nu}-\frac{1}{D-2}q_{\mu\nu}{}^{(D)}R_{\alpha\beta}n^\alpha n^\beta-\frac{1}{D-2}q_\mu^\alpha q_\nu^\beta{}^{(D)}R_{\alpha\beta}+\frac{1}{(D-1)(D-2)}q_{\mu\nu}{}^{(D)}R,
6 e^{2 \sigma ( \phi )}\left({\cal B}\sigma_{i}\sigma_{j}+ \frac{1}{2}\left({\cal B}_{i}\sigma_{j}+ \sigma_{i}{\cal B}_{j}\right) \right) = A ( \phi )_{i j}\; ,{\cal B}( \phi ) e^{2 \sigma ( \phi )}= B ( \phi ) \; , \frac{\lambda ( \phi )}{{\cal B}( \phi )}= \frac{\Lambda ( \phi )}{B ( \phi )}
Q_{E_8}=\begin{pmatrix}2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 -1 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 0 0 0 0 0 -1 0 0 2\end{pmatrix}.
\sigma_{ij}=-p_{\varepsilon\delta}\delta_{ij}+g \varepsilon\delta\dfrac{D_{ij}(u_{\varepsilon\delta})}{(D_{II}(u_{\varepsilon\delta}))^{\frac{1}{2}}}+2\mu\varepsilon^{2}\delta^{2}D_{ij}(u_{\varepsilon\delta}),
P_{0}( Z ) = \left( \begin{array}{c c}{I}&{- Z^{\dag}}\\{Z}&{I^{\prime}}\\ \end{array}\right) \left( \begin{array}{c c}{I}&{}\\{}&{O^{\prime}}\\ \end{array}\right) \left( \begin{array}{c c}{I}&{- Z^{\dag}}\\{Z}&{I^{\prime}}\\ \end{array}\right)^{- 1}.
\frac{(-1)^{n}}{2\pi i}\int_{|z|=\epsilon}\frac{(1+z)^{n}}{z^{n+1}}\frac{1}{2\pi i}\int_{|w|=\gamma}\frac{1}{w^{j+1}}\sum_{q=0}^{n}(\begin{matrix}n\\q\end{matrix})w^{q}(1+z)^{q}dwdz.
\lim_{z\rightarrow \infty}\frac{\overline{\mathbf{F}}\left( xz\right)}{\overline{\mathbf{F}}\left( z\right)}=x^{-1/\gamma_{1}}\lim_{z\rightarrow \infty}\frac{\overline{\mathbf{G}}\left( xz\right)}{\overline{\mathbf{G}}\left( z\right)}=x^{-1/\gamma_{2}}.
\int_{|i-x|\leq1}dx\int_{S_{i}^c}dvf^N(x,v,t)\leq\int_{|i-x|\leq Ca_i}\rho_0(x)\,dx\leq & C \sum_{\substack{k\in \mathbb{Z}^3 |k|\leq Ca_i}}\int_{| i+k-x|\leq 1}g(x)\,dx\leq C\sum_{\substack{k\in \mathbb{Z}^3 |k|\leq Ca_i}}\frac{1}{|i+k|^{2+\epsilon}}.
H_N(f)(x) = \sum_{\ell}\sum_{m = -\ell}^{\ell}\chi_{[N, 2N)}(\ell)\langle f(|x|\omega), Y_{\ell, m}(\omega)\rangle_{L^2_\omega(\mathbb S^2)}Y_{\ell ,m}\big(\frac{x}{|x|}\big), H_1(f)(x) = \sum_{\ell = 0, 1}\sum_{m=-\ell}^{\ell}\langle f(|x|\omega), Y_{\ell, m}(\omega)\rangle_{L^2_\omega(\mathbb S^2)}Y_{\ell, m}\big(\frac{x}{|x|}\big),
\left[ i \gamma_{\mu}\partial_{\mu}+{\frac{\mu_{4}^{2}c}{\mu_{3}\hbar}}{\frac{( 1 + \gamma^{5})}{2}}+{\frac{\mu_{3}c}{\hbar}}{\frac{( 1 - \gamma^{5})}{2}}\right] \tilde{\psi}= 0 \, .
G_0''(x)- \frac{11}{16 x^7}+ \frac{29}{8 x^8}=\frac{\Psi_1(15;x)}{96 x^8 (1 + x)^2 \Psi_2(12;x)}>0, &G_0''(x)- \frac{11}{16 x^7}+ \frac{29}{8 x^8}-\frac{9031}{800 x^9}=-\frac{\Psi_3(13;x)}{800 x^9 (1 + x)^2\Psi_2(12;x)}<0.
\big(\frac{v+nw-nt\tau/2}{w-t\tau/2}\big)>0, \big(\frac{v+nw-nt\tau/2}{w+t\tau/2}\big)>0, \big(\frac{v+nw-nt\tau/2}{-t}\big)>0, \big(\frac{-t\tau/2}{w-t\tau/2}\big)>0 , \big(\frac{-t\tau/2}{w+t\tau/2}\big)>0, \big(\frac{-t\tau/2}{-t}\big)=(\tau/2)>0 .
\mathbf{y}=\mathcal{T}_{\mathbf{A}^{(1)},\mathbf{A}^{(2)}}\left(\mathbf{x}\right)\Leftrightarrow\forall\;0\le k<n,y_{k}=\sqrt{\mbox{Prod}_{\mathbf{P}_{k}}\left(\mathbf{x}^{\top},\mathbf{x}\right)},
\limsup_{m\to\infty}\mathbb{P}_{n_{l_m}}\left( \left| \log(Y_{n_{l_m}}) - \sum_{k=1}^{K}\frac{2\mu_{k}W_{n_{l_m},k}-\mu_{k}^{2}}{2\sigma_{k}^{2}}\right|\ge \epsilon \right) \le \delta.
f_\alpha = \frac 12 |d((h_\alpha)_i+(h_\alpha)_j)| , \g_\alpha= \frac 12 |d((h_\alpha)_i-(h_\alpha)_j)|, f = \frac 12|d(h_i+h_j)|, g=\frac 12 |d(h_i-h_j)|.
\sum_{i = 1}^{N}\frac{\prod_{j = 1}^{N}\hat{q}_{j}^{( 0 )}}{\hat{q}_{i}^{( 0 )}}\cdot \hat{q}_{i}^{( 1 )}= \left( \sum_{i = 1}^{N}\frac{\hat{q}_{i}^{( 1 )}}{\hat{q}_{i}^{( 0 )}}\right) \det ( A - \lambda I ).
Q(x) =s q+Ce^{-\tilde{\lambda}x}+\underset{x\rightarrow+\infty}{o}\left(e^{-\tilde{\lambda}x}\right) =s -\tilde{\lambda}x+\log(C)+\log\left(1+\underset{x\rightarrow+\infty}{o}\left(e^{-\tilde{\lambda}x}\right)\right) =\log(s-q) Q^{-1}(s)=x =-\frac{\log(s-q)}{\tilde{\lambda}}+\frac{\log(C)}{\tilde{\lambda}}+\underset{x\rightarrow+\infty}{o}(1).
\eta(\sigma) = \begin{cases}Q_0 < \cdots < Q_m < Q_mR \mbox{if $i=m$}, Q_0 < \cdots < Q_i < Q_{i+2}< \cdots < Q_m \mbox{if $Q_iR = Q_{i+1}$, and}Q_0 < \cdots Q_i < Q_iR < Q_{i+1}< \cdots < Q_m \mbox{if $Q_iR < Q_{i+1}$}.\end{cases}
\tau: =\Phi_{k,n}( -\alpha-\gamma)+(k-n) \Phi_{k,n}(-\frac{\alpha+\gamma}{n-1})+(n-2)\frac{k-n}{k(n+1)}&=-(\alpha+\gamma)\frac{(k+1)(k-1)n}{(n+1)(n-1)k}+ \frac{(k-1)(k-n)}{k(n+1)}.
F_i(t,f_1(t),\dots,f_{n}(t))=P_i[ t,f_1(t),\dots,f_{n}(t), & j(iB(t)),j(iB(f_1(t))),\dots,j(iB(f_{n}(t))), & ij'(iB(t)), ij'(iB(f_1(t))),\dots,ij'(iB(f_{n}(t))), & j''(iB(t)),j''(iB(f_1(t))),\dots, j''(iB(f_{n}(t)))]=0.
f\left(x\right)=c_{0}h_{0}\left(x\right)+c_{1}h_{1}\left(x\right)+\sum_{n=1}^{\infty}\sum_{k\in S\left(n\right)}c_{n,k}h_{n,k}\left(x\right),\;\forall x\in\left[0,a\right];\mbox{where}
\Phi_{m}(z_{m})\circ\cdots\circ\Phi_{1}(z_{1}):M_{q}(\lambda_{1})\rightarrow M_{q}(\lambda_{m+1},k)\otimes ((V(0,\mu_{1})\otimes\cdots\otimes V(0,\mu_{m}))(z_{1},\ldots ,z_{m})
=b^n\sum_{k=0}^ma^kb^{m-k}+ \sum_{j=0}^{n}a^{m+j}b^{n-j}= b^n\sum_{k=0}^ma^kb^{m-k}+ a^m\sum_{j=0}^{n}a^{j}b^{n-j}=b^n(a+b)^m + a^m(a+b)^n.
\mathbb{E}J_n^2 = \frac{1}{n^2}\mathbb{E}(T_n - \hat{T}^{(n)}_n)^2 \leq \frac{1}{n^2}\left(\mathbb{E}T_n^2 + \mathbb{E}\left(\hat{T}^{(n)}_n\right)^2\right) \leq 2C
\{a_{1 , 1}^{x_{1 , 1}}, \ldots a_{1 , b_{1}}^{x_{1 , b_{1}}}; \ldots ; a_{i , 1}^{x_{i , 1}}, a_{i , 2}^{x_{i , 2}}\ldots , a_{i , b_{i}}^{x_{i , b_{i}}}, \ldots \}: = \prod_{i = 1}^{h - 1}\prod_{j = 1}^{b_{i}}\left( \frac{\sin \pi \tau ( a_{i , j}+ i )}{\sin \pi \tau i}\right)^{x_{i , j}}\, .
\int\limits^0_{A}\int\limits_{\mathbb{R}^{3}_{+}}U_{i(k)}(y,\tau)\partial_{j}\varphi(y,\tau) dyd\tau= -\int\limits^0_{A}\int\limits_{\mathbb{R}^{3}_{+}}\partial_{j}U_{i(k)}(y,\tau)\varphi(y,\tau) dyd\tau.
\prod \limits_{i=j}^{n-1}|c(\theta_i)p(x_i)| = \left( \prod \limits_{i=j}^{t_1 - 1}|c(\theta_i)p(x_i)| \right) \cdots\left( \prod \limits_{i=t_l}^{t_{l+1}- 1}|c(\theta_i)p(x_i)| \right) \cdots \left( \prod \limits_{i=t_r}^{n - 1}|c(\theta_i)p(x_i)| \right).
{\cal R}_{{\cal A}}{}^{{\cal B}}( z ; g ) = \left( \begin{array}{c l}{{R_{A}^{~ B}( z ; g )}}&{{i ( \Sigma^{j}( z ; g ) \tilde{\gamma}_{A}L ( z ; g ) )^{\beta}}}\\{{0}}&{{L_{\alpha}^{~ \beta}( z ; g ) U_{i}^{~ j}( z ; g )}}\\ \end{array}\right)
W^{(2)}(x;B', \psi_0, \psi_1) = \frac{1}{12}(B' x_0^{12}+ x_3^{12})+\frac{1}{3}x_4^3 +\frac{1}{2}x_5^2-\psi_0 \, x_0 x_3 x_4 x_5 - \frac{1}{6}\psi_1 \,x_0^6 x_3^6\ ,
V_{h}=[\begin{matrix}1/\sqrt{2}&-1/\sqrt{2}\\ 1/\sqrt{2}&1/\sqrt{2}\end{matrix}][\begin{matrix}1/\sqrt{2}&1/\sqrt{2}&0&0\\ 1/\sqrt{2}&-1/\sqrt{2}&0&0\end{matrix}]=[\begin{matrix}0&1&0&0\\ 1&0&0&0\end{matrix}]
\int_{0}^{\infty}| \partial_1^j \partial_2^l f^m_{t+T}| \, \textrm{d}T \lesssim (t^{\frac{1}{4}})^{-j - 2l}\int_{0}^{\infty}\| f^m_T \| \textrm{d}T \lesssim (t^{\frac{1}{4}})^{-j - 2l}\int_{0}^{1}(T^{\frac{1}{4}})^{\alpha -2}\, \textrm{d}T + \int_1^{\infty}e^{-T}\, \textrm{d}T < \infty,
A_n \leq 2 B \sum_{|\lambda | >n}|c_\lambda |^2 &\leq 2 B \sup_{|\lambda | >n}v(\lambda )^{-2}\, \sum_{|\lambda | >n}|c_\lambda |^2 v(\lambda )^2 & \leq 2 B \sup_{|\lambda | >n}v(\lambda )^{-2}\, \sum_{|\lambda | >n}|c_\lambda | v(\lambda ) = C \sup_{|\lambda | >n}v(\lambda)^{-2}\, .
\big [{\frac{d^{2}}{d x^{2}}}+ 1 6 x^{N - 2}( x + B )^{2}+ 8 \lambda_{n + 1}+ \sum_{i = 1}^{n}{\frac{\lambda_{i}}{x + A_{i}}}\big ] \Psi_{j}( x ; \lambda_{1}\ldots \lambda_{n + 1}) = 0
\gamma^{\mu \rho \nu}\nabla_{\rho}\Psi_{\nu}^{A}- \gamma^{\mu \nu}\gamma^{5}\partial_{5}\Psi_{\nu}^{A}+ k \epsilon \gamma^{\mu \nu}\gamma^{\hat{5}}\Psi_{\nu}^{A}- \sqrt{2}\left( g_{R}R \epsilon ( \sigma_{3})_{B}^{A}+ g_{S}S ( \sigma_{1})_{B}^{A}\right) \gamma^{\mu \nu}\Psi_{\nu}^{B}= 0 \ ,
y^{p^2}-y =y_1^{p^2}+\mu^{p^2}y_2^{p^2}-y_1-\mu y_2=(y_1^{p^2}-y_1)+\mu(y_2^{p^2}-y_2) &=[(y_1^p-y_1)^p+(y_1^p-y_1)]+\mu[(y_2^p-y_2)^p+(y_2^p-y_2)] &=T^p+T+\mu T^{2p}+\mu T^2=T^p(1+\mu T^p)+(T+\mu T^2)=\gamma,
\log Z [{v},{v}^{\dagger}] = \int d^{d}{x}d^{d}{y}{v}^{\dagger}({\mathbf{x}}) \left( K \gamma^{0}\frac{\gamma \cdot ({\mathbf{x}}-{\mathbf{y}})}{|{\mathbf{x}}-{\mathbf{y}}|^{d + 2 m + 1}}\right){v}({\mathbf{y}}) ,
\chi_L (\xi) = \begin{cases}1, |\xi| \leq \frac{\rho}{2}, 0, |\xi| \geq \rho, \end{cases}\chi_H (\xi) = \begin{cases}1, |\xi| \geq 2, 0, |\xi| \leq 4, \end{cases}\chi_M (\xi) = 1- \chi_L (\xi) - \chi_H (\xi).
\begin{array}{lll}\displaystyle{\lim_{k \to+\infty}\lim_{n\to +\infty}\int_Q v_{n,k}(y)\varphi (y)dy=\lim_{k \to +\infty}\varepsilon_k^{\frac{1}{p}}\int_Q v(x_0+ \varepsilon_k y)\varphi(y)dy=0,}\end{array}
V^{(r)}\hspace{-.07cm}(m,\Lambda)-V^{(r)}\hspace{-.12cm}\left(m,\Lambda^{\prime}\right)={\delta V^{(r)}\hspace{-.07cm}(m,\Lambda,\Lambda^{\prime})-\sum_{s=2}^{r-1}B_{r-s,s}V^{(r-s)}\hspace{-.07cm}(m,\Lambda)},
\widehat{\mathcal{P}_If}(\xi)+ \widehat{\mathcal{P}_Jf}(\xi)=[b_I^2(\xi)+b_J^2(\xi)(\xi)]\hat{f}(\xi) &+ b_J(\xi)b_J(2\alpha'-\xi)\hat{f}(2\alpha'-\xi)-b_I(\xi)b_I(2\alpha'-\xi)\hat{f}(2\alpha'-\xi).
\sum_{\mu \in{\cal R}}\widetilde{M}_{\mu \nu}= \sum_{\nu \in{\cal R}}\widetilde{M}_{\mu \nu}= - i{\cal D}\equiv{\frac{i}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}| \rho |^{2}\, y ( \rho \cdot q ) = -{\frac{i}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}| \rho |^{2}\, V ( \rho \cdot q ) .
2 T_{0}^{0}= w \Phi^{2}+ \frac{{F}}{r^{2}}e^{- 2 \Phi}+ \frac{4 \beta}{r^{2}}\left\{2 w ( w - 1 ) ( \Phi^{\prime \prime}- 2 \Phi^{2}) + \Phi^{\prime}w^{\prime}( 3 w - 1 ) \right\}e^{- 2 \Phi}\; .
\sigma^{ab}n_{ab}= \frac{1}{r}+ \frac{r}{2}[(\rho + (div_{\sigma}\eta)^{(0)}] + \frac{r^2}{3}[ D \rho+ (div_{\sigma}\eta)^{(1)}]+ \frac{r^3}{4}\Big [ \frac{1}{2}(D^2 \rho -\frac{2}{3}|\beta|^2) &+ (div_{\sigma}\eta)^{(2)}-\frac{1}{9}|\beta|^2+ \frac{1}{15}|\alpha|^2 \Big ] + O(r^4).