harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|u(x)-u(y)| = u(x)-u(y) \leq u_i(x)+\epsilon-u_i(y) &\leq d(x,y)^s \left(g_{i,k}(x)+g_{i,k}(y)\right) + \epsilon &\leq d(x,y)^s \left(g_k(x)+g_k(y)\right) + \epsilon
I(p^{2},m_{2}^{2})=(1-\frac{d}{2})N_{d}(-\frac{1}{\epsilon}+O(\epsilon))\sum_{k=0}^{\infty}C_{\frac{d}{2}-2}^{k}\frac{p^{2k}}{m_{2}^{2k}}\int^{1}_{0}dt(t(1-t))^{k}.
\sum_{r , s > 0}r s d \overline{{D}}_{r s}z^{r + s}\zeta^{- r - s - 2}+{\frac{d}{9}}{\frac{1}{\zeta^{2}}}= \sum_{r , s > 0}r s d{D_{r s}}^{\ast}z^{- r - s}\zeta^{r + s - 2}+{\frac{d}{9}}{\frac{1}{\zeta^{2}}}\, ,
y_{qm}=\sum_{k=1}^{qm}\frac{1}{k}\binom{qm+k}{k-1}\frac{k!}{(qm)!}B_{qm,k}(0,\dots,q!c_q,0,\dots) &=\frac{1}{m}\binom{(q+1)m}{m-1}\frac{m!}{(qm)!}\frac{(qm)!}{m!}c_q^m &= \frac{1}{qm+1}\binom{(q+1)m}{m}c_q^m.
K(\lambda, \tilde{f})=\inf \left\{\| g_D \|_{L^2(\mathbb R^n)}+ \lambda \| g_N \|_{\dot H^2(\mathbb R^n)}: \tilde{f}= g_D + g_N \in L^2(\mathbb R^n) + \dot H^2(\mathbb R^n) \right\}.
\lambda_{0}^{i}:= (x_1^2 + x_2^2 + x_{12}^2 - x_1x_2x_{12}- 2)x_i - x_1x_{1i}- x_2x_{2i}+ (x_1x_2-x_{12})x_{12i}, \lambda_{1}^{i}:= -x_1x_i - x_2x_{12i}+ x_{12}x_{2i}+ 2x_{1i}, \lambda_{2}^{i}:= -x_2x_i - x_1x_{12i}+ x_{12}x_{1i}+ 2x_{2i}, \lambda_{12}^{i}:= -x_1x_{2i}- x_2x_{1i}- x_ix_{12}+ x_1x_2x_i + 2x_{12i};
\begin{aligned}d\lambda_1\cdot r_1 =\frac{1}{2\sqrt{\alpha_+}}\Big[\underbrace{-\partial_v\alpha_+ \frac{v}{\alpha_+}(\alpha_+-\frac{\beta^2}{v})}_{I_1}+\underbrace{\partial_{B_2}\alpha_+ B_2}_{I_2}+\underbrace{\partial_{B_3}\alpha_+ B_3}_{I_3}\Big],\end{aligned}
S_{N}f(x)=(y_{0}+\frac{c}{2})+\frac{2c}{\pi}(sin(\omega(x-x_{0}))+\frac{1}{3}sin(3\omega(x-x_{0}))+\cdot\cdot\cdot+\frac{1}{N-1}sin((N-1)\omega(x-x_{0})))
\sum_{n=1}^{\infty}n^2\Vert f_n^d \Vert_{L^{1}(S_2)}\Vert I_{1n}\Vert_{L^{1}(S_2)}\leq C\sum_{n=1}^{\infty}n^4\Vert f_n^d \Vert_{L^{1}(S_2)}^2
\mathcal{U}_{o,k}= \big\{(\nu_k, \nu_{k+1}, \cdots, \nu_{K-1}) ~|~ \nu_j \in \mathcal{A}\,,~k\le j < K\big\}\,, \mathcal{U}_{f,k}= \big\{(\nu_k, \nu_{k+1}, \cdots, \nu_{K-1}) ~|~ \nu_j : \mathbb{X}_N \rightarrow \mathcal{A}\,,~k\le j < K\big\}\,.
\Pr\bigg(\frac{nN}{2}\leq S_N\leq \frac{nN}{2}+\frac{n-1}{2}\bigg)=\Pr\bigg(0 \leq \frac{S_N-nN/2}{\sigma \sqrt{N}}\leq \frac{n-1}{2\sigma\sqrt{N}}\bigg) &\leq \Pr\bigg(0 \leq \frac{S_N-nN/2}{\sigma \sqrt{N}}\leq \sqrt{\frac{3}{N}}\bigg)=\Phi\bigg(\sqrt{\frac{3}{N}}\bigg)-\frac{1}{2}+O\bigg(\frac{1}{\sqrt{N}}\bigg)=o(1).
{\cal L}_\eta|\Phi_{ph}\rangle=\Bigl(\eta\hat{\partial}+\frac{1}{2}\hat{\partial}^I\eta^JM^{IJ}+\frac{1}{2}\eta^IM^{zI}+\hat{\partial}^+\eta^I\alpha^I(\frac{\partial^J}{\partial^+}\bar{\alpha}^J-\frac{s+d-4}{\hat{\partial}^+}\bar{\alpha}^z)\Bigr)|\Phi_{ph}\rangle\,.
|N(a_1)|+\sum_{i=2}^q |N(a_i) \triangle N(a_{i-1})| + q \leq |N(a_1)| + \sum_{i=2}^q (|N(a_i)|-|N(a_{i-1})| + 2s-2) + q = |N(a_q)| + (2s-2)(q-1) + q &\leq n + (2s-2)n + n &= 2sn.
\frac{dR(g_\tau,T,\hat y,\hat\eta_\tau)}{d\tau}=(2 H_\tau(y,\hat\eta_\tau))^{-\frac{1}{2}}\Big( \big((g_1-g_0)\hat\eta_\tau,\hat\eta_\tau\big) +2\Big(g_\tau\hat\eta_\tau,\frac{\partial\hat\eta_\tau}{\partial \tau}\Big)\Big)
A_{\varphi}^{\theta, \alpha}C_{\alpha,\frac{\theta}{\alpha}}+\alpha A_{\varphi}^{\theta, \frac{\theta}{\alpha}}C_{\theta}=C_{\alpha,\frac{\theta}{\alpha}}A_{\bar{\varphi}}^{\alpha, \theta}P_\alpha+ C_{\theta}A_{\bar{\varphi}}^{\frac{\theta}{\alpha}, \theta}P_{\frac\theta\alpha}\bar{\alpha},
{\mathcal L}_{e f f}= \frac{h \rho_{0}q}{2}\epsilon_{k l}[ \dot{x}_{k}^{( 1 )}x_{l}^{( 1 )}- \dot{x}_{k}^{( 2 )}x_{l}^{( 2 )}] - \frac{h^{2}\rho_{0}q^{2}}{2 \pi m}\ln | \frac{x^{( 1 )}- x^{( 2 )}}{\xi}| ,
\Psi_E = \left(\begin{array}{c}\varphi_E\left(r,\theta\right) \chi_E\left(r,\theta\right)\end{array}\right) \qquad,\quad{\rm satisfy:}\quad\begin{array}{c}L \varphi_E = E \chi_E L^\dag \chi_E = E \varphi_E\end{array}
\mathrm{T r \, L n}\, [ m^{2}+ \nabla^{2}] = - \int d^{3}x \int_{1 / \Lambda^{2}}^{\infty}\frac{d s}{s}\, \mathrm{t r}\, \langle x | \, e^{- i s \, [ m^{2}+ \nabla^{2}]}| x \rangle .
\left\langle a \right| N \left( \psi_{L}^{\dagger}\psi_{L}\psi_{R}^{\dagger}\psi_{R}\right) \varepsilon^{L_{0}+ \overline{{{L}}}_{0}}\left| B , a \right\rangle = \frac{1}{4 \pi^{2}\left| z \right|^{2}}\left( \frac{\sinh^{2}\left( \frac{\alpha \log \left| z \right|}{2 \varepsilon}\right)}{\sinh^{2}\left( \frac{\log \left| z \right|}{\varepsilon}\right)}- \frac{\alpha^{2}}{4}\right) .
\mathfrak{I}_{\mathrm{p}}^{(\omega ,\eta \mathbf{A}_{l})}\left( t\right)=\eta^{2}l^{d}\int\nolimits_{t_{0}}^{t}\int\nolimits_{t_{0}}^{s_{1}}\mathbf{X}_{l}^{(\omega )}(s_{1},s_{2})\ \mathrm{d}s_{2}\mathrm{d}s_{1}+\mathcal{O}(\eta^{3}l^{d})\ ,
\vert \sum_{n\leq 3\sqrt{T/\pi}}\frac{1}{n^{{1}/{2}}}\int_{T}^{2T}e^{i F(n,t)}dt\vert = O\left( \sum_{n\leq{3\sqrt{T/\pi}}}\frac{1}{n^{1/2}}T^{1/2}\right) = O\left(T^{3/4}\right).
X^{l m 0}\partial_{i}\mathrm{D}^{l m}- \partial_{m}( X^{m l 0}\mathrm{D}_{l i}) - \partial_{l}( X^{m l 0}\mathrm{D}_{m i}) - q{}\partial_{i}\partial_{l}\partial_{m}X^{l m 0}= 0 .
\log\left(Z_{n}(\beta)\right) + \frac{1}{2}\log\left(1- 2\beta J\right)- (n-1)\beta^2 + \beta (J -J') - \beta C_{n,1}- &~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \sum_{k=2}^{m_{n}}\frac{2(2\beta)^{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- (2\beta)^{2k}}{4k}\stackrel{p}{\to}0.
D_{p}^{|k|}=\frac{\Gamma(\frac{n}{2}+\alpha)\Gamma^{1/p}(\frac{p}{q}(n+|k|+\alpha-1))\Gamma^{1/q}(\frac{q}{p}|k|)}{\Gamma(\frac{n+\alpha-1}{q}+|k|+\frac{|k|+\alpha+1}{2})\Gamma(\frac{n+\alpha-1}{q}+|k|+\frac{|k|+n+\alpha-1}{2})},
\begin{aligned}\int_0^x\frac{u_t}{r^2}dy= \left(\mu\rho(r^2u)_x-\frac{Ca}{2}\rho^\gamma\right)(x,t)-\left(\mu\rho(r^2u)_x-\frac{Ca}{2}\rho^\gamma\right)(0,t) = \left(\mu\rho(r^2u)_x-\frac{Ca}{2}\rho^\gamma\right)(x,t)+\left(\frac{Ca}{2}+\frac{2}{We}\right)R^{-3\gamma_0}-\frac{2}{We}R^{-1}-2\mu R^{-1}\frac{dR}{dt}.\end{aligned}
\frac{1}{\pi}=2\sqrt{2}\sum_{k=0}^\infty \frac{\left(\frac{1}{2}\right)_k\left(\frac{1}{4}\right)_k\left(\frac{3}{4}\right)_k}{(1)_k^3}\left(1103+26390k\right)\left(\frac{1}{99^2}\right)^{2k+1}
\operatorname*{l i m}_{p^{2}\rightarrow 0}\, \mathrm{I m}\, \Pi_{2}^{\mathrm{( n o n \mathrm{-}p l a n a r )}}= \operatorname*{l i m}_{p^{2}\rightarrow 0}\, \mathrm{I m}\, \Pi_{2}= 0 .
\mathrm{ph\ddot{o}l}_\delta(g,\overline{Q}) := \sup\left\{\frac{|g(x,t)-g(y,s)|}{\left(\|x-y \|+\sqrt{|t-s|}\right)^{\delta}}: \;(x,t), (y,s) \in \overline{Q}, \;(x,t) \not=(y,s)\right\}\in [0,\infty],
\langle\Phi^{n}\rangle=\frac{1}{Z}\beta^{-n}\partial_{q}^{n}Z(q)=\frac{1}{Z}\beta^{-n}\partial_{q}^{n}\sum_{i}e^{-\beta E_{i}(q)}=\frac{\sum_{i}\Phi_{i}^{n}e^{\beta E_{i}(0)+\beta q\Phi_{i}}}{\sum_{i}e^{\beta E_{i}(0)+\beta q\Phi_{i}}},
\big( \frac12 \|\cdot\|_K^2 \big)^* = \frac12\|\cdot\|_{K^\circ}^2,\;\;\; \int_{\mathbb{R}^n}e^{- \frac12 \| x \|_K^2}\, dx = \frac{(2\pi)^\frac{n}{2}}{|{\rm B}_2^n|}|K|,
-504\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(12)}=\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{L_R\otimes C}-\widetilde{L_R\otimes C}\otimes \widetilde{L_R\otimes C}+\widetilde{L_R\otimes C})\right\}^{(12)}.
\varrho_{P}(\theta,\tilde{\theta})\equiv \Big\|\big\{\big[Var_P\big(\sigma_{P,j}^{-1}(\theta)m_j(X,\theta)-\sigma_{P,j}^{-1}(\tilde{\theta})m_j(X,\tilde{\theta})\big)\big]^{1/2}\big\}_{j=1}^J\Big\|.
S = S_{c m}-{\frac{1}{2}}\, \int_{\Sigma_{p + 1}}d^{p + 1}\sigma \, \sqrt{\, \overline{{g}}\,}\, \left[ \, \overline{{{g}}}^{\, m n}\, \partial_{m}\, Y^{\mu}\, \partial_{n}\, Y_{\mu}- m_{p + 1}\, p \, \right] \ .
\begin{array}{l c r}{{q J_{1}J_{2}- q^{- 1}J_{2}J_{1}}}&{{=}}&{{( q^{2}- q^{- 2}) J_{3}}}\\{{q J_{2}J_{3}- q^{- 1}J_{3}J_{2}}}&{{=}}&{{( q^{2}- q^{- 2}) J_{1}}}\\{{q J_{3}J_{1}- q^{- 1}J_{1}J_{3}}}&{{=}}&{{( q^{2}- q^{- 2}) J_{2}}}\\ \end{array}
K_{S}^{( 1 )}( [ \omega_{1}] , [ \omega_{2}] ) = R e s \sum_{A = 1}^{d}\frac{1 / 2 ( P_{2}{\frac{\partial}{\partial X^{A}}}P_{1}- P_{1}{\frac{\partial}{\partial X^{A}}}P_{2}) d X^{1}\ldots d X^{d}}{\frac{\partial W}{\partial X^{A}}\prod_{B = 1}^{d}\frac{\partial W}{\partial X^{B}}}
{\cal L}(y,p_{1},p_{2}) = \left[\begin{array}{ll}y^{-1}c_{1}c_{2}Z_{1}-b_{1}b_{2}y Z_{1}^{-1}-y(b_{1}d_{2}Z_{1}^{-1}-c_{1}a_{2}Z_{1})X \vspace{-2mm}\omega_{1}y^{-1}X^{-1}(d_{1}b_{2}Z_{1}^{-1}-a_{1}c_{2}Z_{1}) y^{-1}d_{1}d_{2}Z_{1}^{-1}-a_{1}a_{2}\omega_{1}^{2}y Z_{1}\end{array}\right].
s_i(a_0 \otimes \cdots \otimes a_{i+1}) = a_0 \big( (\psi_{i+1}\bullet z)(a_1 \otimes \cdots \otimes a_{i})\big) a_{i+1}&= \sum_{k=1}^{i + 1}(-1)^{k-1}\varphi_{i+1}(a_0 \otimes \dots \otimes a_{k-1}\otimes z \otimes a_{k}\otimes \cdots \otimes a_{i+1})
\begin{array}{l r c l}{\pm e_{i}\pm e_{j}}&{( 2 4 )}&{\mathrm{w i t h}}&{i \neq j \mathrm{a n d}i , j = 1 , 2 , 3 , 4}\\{( 0 , 0 , 0 , 0 )_{a}}&{( 2 )}&{\mathrm{w i t h}}&{a = 1 , 2 .}\\ \end{array}
\mathcal{B}_\varepsilon =(\mathbf{D}-\mathbf{A}^\varepsilon (\mathbf{x}))^*g^\varepsilon (\mathbf{x})(\mathbf{D}-\mathbf{A}^\varepsilon (\mathbf{x}))+\varepsilon^{-1}v^\varepsilon (\mathbf{x})+\mathcal{V}^\varepsilon (\mathbf{x}).
C_{n-1}(y_{n-2})=\sum_{y_{n-1}\in\{0,1\}}\Big(\log\big(\frac{q_{n-1}(y_{n-1}|x_{n-1},y_{n-2})}{\nu^{\pi^*}_{n-1}(y_{n-1}|y_{n-2})}\big)+C_n(y_{n-1})\Big)q_{n-1}(y_{n-1}|x_{n-1},y_{n-2}),~\forall{x_{n-1}}.
\begin{tabular}[c]{l}$du\left( t\right) +\left( Au\right) \left( dt\right) \ni f\left(t,u\left( t\right) \right) dt+B\left( t,u\left( t\right) \right)dW_{s}\left( t\right) $\medskip $u\left( s\right) =\xi,\;\;t\in\left[ s,\infty\right) ,$\end{tabular}
V_{ij}^{\perp}= \nabla_{F_j}^{\perp}\nabla_{F_i}V = \nabla_{F_j}^{\perp}\nabla_{F_i}^{\perp}V - \nabla_{F_j}^{\perp}\left( A^V_{ik}F_k \right) = \nabla_{F_j}^{\perp}\nabla_{F_i}^{\perp}V - A^V_{ik}\, A_{jk}\, .
N^{+}( [ 1 , n ] ) = \sum_{k = 0}^{n}( - 1 )^{k}\sum_{\begin{array}{c}{{\scriptstyle j_{0}+ \dots + j_{k}= n + 1}}\\{{\scriptstyle j_{i}> 0}}\\ \end{array}}c_{j_{0}}\cdots c_{j_{k}}\, .
\mu(k\eta, q) \rightarrow \sqrt{\frac{\eta_c}{q(\eta+2\eta_c)}}( c_{-}e^{\frac{i}{2}q\frac{(\eta + 2\eta_c)^2}{\eta_c}}+c_{+}e^{-\frac{i}{2}q\frac{(\eta + 2\eta_c)^2}{\eta_c}})
|V_{Z_2}(t, x; \xi)|\leq \begin{cases}Ct^{(\gamma-1)\frac\alpha2+1}p(t, x-\xi)\,, \d=1\,; Ct^{\frac{\nu_0 \alpha}{2}+1-\alpha}|x-\xi|^{-d+\gamma-\nu_1+2-\nu_0}p(t, x-\xi)\,, \d\geq 2 \end{cases}
{\rm g.c.d.}(A^{[k_1]}_{i}, B^{[k_2]}_{j}) =\begin{cases}p_{i,j},\,\,\, \max (1, |i|)\cdot \max (1, |j|) \le \min (k_1,k_2)\cr1,\,\,\,\,\,\,\,\,\max (1, |i|)\cdot \max (1, |j|) > \min (k_1,k_2)\end{cases}
\mathcal{R}_5=\Bigl\{(u,v,w,t):\, \theta_2-\theta_1<t<w<v,\, \theta_2<u<1/2,\, u+2v<1-\theta_1, &u+v+2w<1,\, u+v+w+2t<1,\, \theta_2<u+v<1-\theta_2, &\{u+v,u+w,u+t,v+w,v+t,w+t\}\notin[\theta_1,\theta_2]\,\Bigr\}.
\tilde{Z}^{\Lambda , \Lambda_{0}}( \varrho ; J ) = \int d \mu_{\Lambda , \Lambda_{0}}( \phi ) e^{- \frac{1}{\hbar}\bigl ( \tilde{L}^{\Lambda_{0}, \Lambda_{0}}( \varrho ; \phi ) + \tilde{I}^{\Lambda_{0}, \Lambda_{0}}( \varrho ) \bigr ) + \frac{1}{\hbar}\langle \phi , J \rangle}.
\widetilde{\psi}_{k}^{( 1^{2})}\big(x;\omega ,1,- \Vert \psi_{1}\Vert^{2}\big)=-\frac{2\omega z^{3/4}\exp ( -z/2 )}{z^{3}-z^{2}+2z+2}\widetilde{L}_{k,1}^{( 1^{2})}(z) ,
\prod_{i\in R^+}(1- A_i) \prod_{i\in R^-}(1 - q\hbar^{-1}A_i ) - (\hbar a)^\alpha \prod_{i\in R^+}(1 - q\hbar^{-1}A_i ) \prod_{i\in R^-}(1 - A_i ) , R = R^+ \sqcup R^- : ,
a_{2}= f \sum_{i = 1}^{i_{3}}{\cal U}_{i}^{( 2 , 2 )}\; I_{i}^{( 2 )}+ f_{; N}\sum_{i = 1}^{i_{2}}{\cal U}_{i}^{( 2 , 3 / 2 )}\; I_{i}^{( 3 / 2 )}+ f_{; N N}\sum_{i = 1}^{i_{1}}{\cal U}_{i}^{( 2 , 1 )}\; I_{i}^{( 1 )}.
E_{\theta}(t) = \frac{1}{2}\int_{\Omega}\left(\frac{1}{\alpha_{0}}\,|\theta + \alpha_{0}\,\theta_{t}|^{2}+ \frac{(\alpha_{0}\,d - 1)}{\alpha_{0}}\,|\theta|^{2}+ \alpha_{0}\,k\,|\nabla \theta|^{2}\right)d\boldsymbol{x}.
0\geq(2-\frac{n-2}{n\gamma})\int_M |\nabla u^{\gamma}|^2+\gamma\frac{R}{n-1}\int_M u^{2\gamma}-\sqrt{\frac{n-2}{2(n-1)}}\gamma\int_M \left(|W|^2+\frac{2n}{n-2}u^2\right)^{\frac 12}u^{2\gamma}.
\eta(F_{R})-\eta(F_{L}) =\int_{F_{L}}^{F_{R}}\eta'(y)dy=-\int_{F_{L}}^{F_{R}}\eta''(y)(y-F_{L})dy+\left.(y-F_{L})\eta'\right|_{F_{L}}^{F_{R}}&=-\int_{F_{L}}^{F_{R}}\eta''(y)(y-F_{L}(t))dy+(F_{R}-F_{L})\eta'(F_{R}).
{\cal Z}_{\delta}( x , z ; p , \epsilon ) \equiv \langle \alpha | \delta \rangle \, \theta_{\delta}( 0 )^{- 1}\langle \psi ( x_{1}) \! \cdot \! \psi ( x_{2}) \prod_{i = 1}^{4}p_{i}\! \cdot \! \psi ( z_{i}) \, \epsilon_{i}\! \cdot \! \psi ( z_{i}) \rangle_{\delta}\, .
<\eta_i>_{\nu(\overrightarrow{\eta})}= \sum \eta_i \gamma(\eta_i) & =\sum \frac{\eta_i \theta^{\eta_i}\,\, \Gamma(m+\eta_i)}{Z_\theta\, \eta_i!\,\, \Gamma(m)}= \theta \frac{\partial}{\partial \theta}\log Z_\theta = \frac{m \theta}{1-\theta},
Y_m= \iint_{Q_m}(v-k_m)_+^p\,dv_0\,dt\ge(k_{m+1}-k_m)^p|E_m|=\frac{k^p}{2^{(m+1)p}}|E_m| \Rightarrow\quad|E_m|\le \frac{2^{(m+1)p}}{k^p}Y_m.
k^2 \int_{V}|\chi_{k}\overline{\partial}\xi|^2_{\omega}e^{-k\varphi}\omega_{n}\lesssim k^2 \int_{|y|< \frac{\log k}{\sqrt{k}}, (x, y) \in V}|y|^{2m}e^{-2k\pi|y|^2}dxdy = O(k^{2-m}).
{\cal O}_{1 / 4}^{[ p , 0 , q ]}= [ W^{1}( \theta_{2 , 3 , 4}, \bar{\theta}^{1}) ]^{p}\ [ \bar{W}_{4}( \theta_{4}, \bar{\theta}^{1 , 2 , 3}) ]^{q}, \qquad \ell = p + q
S_{\rho,n}(e)\approx s_{\rho,n}(e)-\frac{1}{2}\frac{\left(\frac{1}{n}\sum_{i=1}^n\rho^{(1)}\left(\frac{r_i(e)}{\sigma_n}\right)^2\right)\chi^2_{k-k(e)}}{\frac{1}{n}\sum_{i=1}^n\rho^{(2)}\left(\frac{r_i(e)}{\sigma_n}\right)}
=\sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}(\mathrm{Ind}^G_{P_b}\circ \otimes^k_{i=1}\mathrm{Mant}_{M_{b_i}, b'_i, \mu_{b i}}\circ \mathrm{Red}_b)\otimes [1][|\cdot|^{\langle \rho_G, \mu_b \rangle - \langle \rho_G, \mu \rangle}]
\det D\varphi_{2n}^\mathbb{R}= \prod \limits_{k = 1}^n \prod \limits_{j = 1}^{k - 1}\beta_j^2 = \prod \limits_{j = 1}^{n - 1}\prod \limits_{k = j + 1}^n \beta_j^2 = \prod \limits_{j = 1}^{n - 1}\beta_j^{2n - 2j}.
\bigl(1-q_K^{-1}\bigr)^2 \bigl(1-q_K^{-2}\bigr) \dotsb \bigl(1-q_K^{-(n-1)}\bigr) (P_{\mathcal R,\mathcal S,K}(\mathbf r, \mathbf s)-1) = Z_{\{1\}}\bigl(-\mathbf r, -\mathbf s,\!\sum\mathbf r +\sum\mathbf s - n -1\bigr),
{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}\partial_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}\partial_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\rightarrow{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}{\cal D}_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}{\cal D}_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}},
H_{4,j}\widetilde H_{1,j}^{-1}H_{3,j}= \big[ \widetilde H_{1,j}- a H_{1,j}\big] \widetilde H_{1,j}^{-1}\big[ - \widetilde H_{1,j}+ b H_{1,j}\big] = \big[ -\widetilde H_{1,j}+ b H_{1,j}\big] \widetilde H_{1,j}^{-1}H_{3,j}\big[ \widetilde H_{1,j}- a H_{1,j}\big] = H_{3,j}\widetilde H_{1,j}^{-1}H_{4,j}.
\widetilde{P}_{\pm}= \int_{0}^{x \pm}d y^{\pm}\widetilde{T}_{\pm \pm}( y^{\pm}) \qquad \qquad \widetilde{\Delta}_{\pm}= \int_{0}^{x \pm}d y^{\pm}y^{\pm}\widetilde{T}_{\pm \pm}( y^{\pm}) .
\|u_m\|_{{pq\over q-p},\Omega}\leq \|u_m\|_{{np\over n-p},\Omega}^\lambda \|u_m\|_{1,\Omega}^{1-\lambda},\lambda={np\over np+p-n}\left(1-{1\over p}+{1\over q}\right); \|u_m\|_{np/(n-p),\Omega}\leq S_{p,\ell-1}\left(\|\nabla u_m\|_{p,\Omega}+\|u_m\|_{\ell-1,\Gamma}\right),
\sup_{0<s<\frac{1}{8(1+|x|)^2}}K_{23}(t,x,y)|_{t=\log\left(\frac{1+s}{1-s}\right)}\lesssim \sup_{0<s<\frac{1}{8(1+|x|)^2}}\frac{|x|}{s^{(n-1)/2}}\exp\Big(-\frac{c}{s(1+|x|)^2}\Big)e^{|y|^2-|x|^2}&\lesssim|x|(1+|x|)^{n-1}e^{|y|^2-|x|^2}\lesssim(1+|x|)^n e^{|y|^2-|x|^2}, \, (x,y)\in N^c.
dq_t^\prime= \frac{1}{m}(p_t^\prime-\psi(t^*,q_t^\prime))dt, d(p_t^\prime)_i= \left(-\frac{1}{m}\gamma(t^*,q_t^\prime)((p_t^\prime)_i-\psi_i(t^*,q_t^\prime))-\partial_{q^i}V(t^*,q_t^\prime)\right. &\left.+\frac{1}{m}\partial_{q^i}\psi_k(t^*,q_t^\prime)\delta^{kj}((p_t^\prime)_j-\psi_j(t^*,q_t^\prime))\right)dt+\sigma(t^*,q_t^\prime)\delta_{i\rho}dW^\rho_t
s^{(1)}_m(y) =\sum_{k=0}^m D^{(1)}_k(m)R_k(y;0)-\sum_{k=0}^m D^{(1)}_k(m)(\tfrac{\beta}{2}-k+1)_{2k}=\sum_{k=0}^m D^{(1)}_k(m)R_k(y;0)-(\lambda)_m(\lambda-\beta)(\tfrac{\beta}{2})_m.
F_{\omega}(x)=e^{i\gamma_1x-\frac{\gamma_2}{2}x^2}\prod_{k=1}^{\infty}\frac{e^{-i\alpha_k^+x}}{1-i\alpha_k^+x}\prod_{k=1}^{\infty}\frac{e^{i\alpha_k^-x}}{1+i\alpha_k^-x}\ .
ch(Sym(\nu)) = exp\left(-2ln(1-z) -\sigma_1f +\frac{\sigma_{1,1}-\sigma_2}{2}(f+f^2) + \frac{\sigma_{2,1}}{6}(f+3f^2+2f^3))\right)= = \left(\frac{1}{1-z}\right)^2\left\{1-\sigma_1f +\frac{\sigma_{1,1}}{2}(f+2f^2) - \frac{\sigma_{2}}{2}f + \frac{\sigma_{2,1}}{6}(f+3f^2)+\frac{1}{12}\sigma_{2,2}f^2 \right\}.
a_{i}(t)=a_{i}(0)+\int_{0}^{t}\mathcal{F}_{i}(a)d\tau +\sum_{\left\vert \ell\right\vert =2,4,..}\frac{C_{\ell}}{\ell !}\int_{0}^{t}D^{\ell}\mathcal{F}_{i}(a)d\tau +\tilde{R}(t),
\varphi\circ D(ab)= \varphi\circ\psi(a) \varphi\circ D(b)+\varphi\circ\psi(b) \varphi\circ D(a) = \varphi(\psi(a)Db)+\varphi(\psi(b)Da) = \varphi(\psi(a)Db+\psi(b)Da).
D_{a}= \partial_{a}+{\frac{1}{4}}\partial_{a}x^{m}\big [ \ ( \omega_{\mu \nu m}-{\frac{1}{2}}H_{\mu \nu m}\rho_{3}) \Gamma^{\mu \nu}+ ({\frac{1}{3 !}}F_{\mu \nu \lambda}\Gamma^{\mu \nu \lambda}\rho_{1}+{\frac{1}{2 \cdot 5 !}}F_{\mu \nu \lambda \rho \kappa}\Gamma^{\mu \nu \lambda \rho \kappa}\rho_{0}) \Gamma_{m}\ \big ]
\begin{array}{l l}{{\bar{\bf s}X^{i *}= \psi^{i *}}}&{{{\bf s}X^{i *}= \xi^{i *}}}\\{{\bar{\bf s}\psi^{i}= - \frac{i}{2}\partial_{+}X^{i}}}&{{{\bf s}\psi^{i *}= - \frac{1}{2}F^{i *}}}\\{{\bar{\bf s}\xi^{*}= \frac{1}{2}F^{i *}}}&{{{\bf s}\xi^{i}= - \frac{i}{2}\partial_{-}X^{i}}}\\{{\bar{\bf s}F^{i}= i \partial_{+}\xi^{i}}}&{{{\bf s}F^{i}= - i \partial_{-}\psi^{i}\ ,}}\\ \end{array}
B= \underset{p,g}{\sum}\frac{[X_i(p),X_{ki}(g)]}{(z_i-p)(z_k-g\cdot p)}+\underset{p ,g}{\sum}\frac{-[X_i(p),X_{ki}(g)]}{(z_i-g^{-1}\cdot z_k)(z_k-g\cdot p)}+\underset{p,h}{\sum}\frac{[X_i(p),X_{ki}(g)]}{(z_i-g^{-1}z_k)(z_k-g\cdot \infty)}.
\|T_{\mu}^{n}x - T_{\mu}^{n}y\|^{2}= \langle T_{\mu}^{n}x - T_{\mu}^{n}y , x_{1}^{*}\rangle= \mu_{t}\langle T_{t}^{n}x - T_{t}^{n}y , x_{1}^{*}\rangle \leq \sup_{t}\|T_{t}^{n}x - T_{t}^{n}y\|\|T_{\mu}^{n}x - T_{\mu}^{n}y\| \leq \|x-y\|\|T_{\mu}^{n}x - T_{\mu}^{n}y\|,
J =\iint\limits_{[0,1]^{2}}\left|f_{1}^{2}(\alpha_{1})f_{2}^{2}(\alpha_{2})G^{4}(\alpha_{1}+\alpha_{2})\right|d\alpha_{1}d\alpha_{2}&=\sum\limits_{1\leq m_{1},\ldots m_{4}\leq L}\prod\limits_{i=1}^{2}t_{i}(2^{m_{1}}+2^{m_{2}}-2^{m_{3}}-2^{m_{4}}),
P(x;\mu_{N},\sigma_{N})=\frac{1}{x\sqrt{2\pi ln(1+\sigma_{N}^{2}/\mu_{N}^{2})}}exp(-\frac{[lnx-ln\frac{\mu_{N}}{\sqrt{1+\sigma_{N}^{2}/\mu_{N}^{2}}}]^{2}}{2ln(1+\sigma_{N}^{2}/\mu_{N}^{2})})
\footnotesize w_0(\alpha_1) = -s_{\alpha_1 + \alpha_2}(2\alpha_1 + \alpha_2) = -((4\alpha_1 + 2\alpha_2) - (3\alpha_1 + 2\alpha_2)) = -\alpha_1, w_0(\alpha_2) = s_{\alpha_1 + \alpha_2}(3\alpha_1 + 2\alpha_2) = 3(2\alpha_1 + \alpha_2) - 2(3\alpha_1 + 2\alpha_2) = -\alpha_2,
P_{\alpha}(\partial_{\alpha}[\hat \Psi_{\alpha}h_{\alpha}]) &=\sum_{n=1}^{\infty}(\partial_{\alpha}h_{\alpha}\circ F_{\alpha, n}^{-1}) (\hat \Psi_{\alpha}\circ F_{\alpha, n}^{-1}) G_{\alpha, n}+(h_{\alpha}\circ F_{\alpha, n}^{-1}) (\partial_{\alpha}\hat \Psi_{\alpha}\circ F_{\alpha, n}^{-1}) G_{\alpha, n}
v![w^v]\Bigl(\frac{w}{s(w)}\Bigr)^v\frac{w}{dw}\frac{ds(w)}{s(w)}(1+s(w)\,\hbar)^{\theta/\hbar}=v![s^v](1+s\,\hbar)^{\theta/\hbar}=v!\hbar^v\binom{\theta/\hbar}{v}
A_2 = \begin{pmatrix}1 0 \end{pmatrix}\begin{pmatrix}1 0 \end{pmatrix}, A_3 = \begin{pmatrix}1 1 \end{pmatrix}\begin{pmatrix}1 0 \end{pmatrix}, A_6 = \begin{pmatrix}0 1 \end{pmatrix}\begin{pmatrix}1 0 \end{pmatrix}, O = \begin{pmatrix}0 0 \end{pmatrix}\begin{pmatrix}1 0 \end{pmatrix},
\frac{|B_1|}{2}\leq W_{AC}(v_0;x,0+) & = \lim_{s\searrow 0+}\frac{|B_s(x)\cap \{v_0>0\}|}{s^n}& = \lim_{s\searrow 0+}\frac{|B'_s(x')\cap \{v_*>0\}|_{n-1}}{s^{n-1}}& = W_{AC}(v_*;x',0+).
x_{n,t}-x_{n+1,t}=\frac{t-\varkappa n}{\sqrt{n}}-\frac{t-\varkappa (n+1)}{\sqrt{n+1})}=\frac{\varkappa}{\sqrt{n+1}}+(t-\varkappa n)(\frac1{\sqrt{n}}-\frac1{\sqrt{n+1}}) =\frac{\varkappa}{\sqrt{n}}(1+ O(\frac1{\sqrt n})).
d(h(x),h(y))\left\|T(e)\right\| =\alpha\left(\delta_{(h(x),h(y))}\boxtimes T(e)\right) &=\alpha\left((h\boxtimes T)\left(\delta_{(x,y)}\boxtimes e\right)\right) &\leq\left\|h\boxtimes T\right\|\alpha\left(\delta_{(x,y)}\boxtimes e\right) &=\left\|h\boxtimes T\right\|d(x,y)\left\|e\right\|
[\hat{D}_{\alpha}, \hat{D}_{\beta}] f=-(\partial_{\alpha}A^{\mu}_{\beta}-\partial_{\beta}A^{\mu}_{\alpha})\partial_{\mu}f+(A^{\mu}_{\alpha}\partial_{\mu}A^{\nu}_{\beta}-A^{\mu}_{\beta}\partial_{\mu}A^{\nu}_{\alpha})\partial_{\nu}f\,.
\zeta_T(\nu|\beta)={\nu \varrho^{-2\nu}\over \pi i}\int_{\mu}^{\infty}d\omega \int_{C_+}dz{2z \over (z^2+\omega^2)^{\nu+1}}\ln\left(1-e^{i\beta z}\right)\breve{\Psi}(\omega;z) ~~~,
\limsup_{K\to\infty}\frac{\left|\sum_{k=1}^K\tilde{X}_{m,k}\right|}{\sqrt{2s_K^2\log(\log(s_K^2))}}\leq\sqrt{\frac{\Sigma_{f,M_n}+Cd^{1/2}\log(d)G^{-1/2}+1/m^2}{\Sigma_{f,M_n}-Cd^{1/2}\log(d)G^{-1/2}}}\quad\textnormal{a.s.}
\displaystyle \mathcal{D}_{t}=\displaystyle \sigma({\bf 1}_{\tau_{x}\leq u}, u\leq t) , \displaystyle\mathcal{F}_{t}^{Q}=\displaystyle \sigma(Q_{u}, u \leq t), \displaystyle\mathcal{G}_{t}= \displaystyle \mathcal{F}_{t}^{Q}\vee \mathcal{D}_{t},~~ \forall t \geq 0.
\begin{bmatrix}\dot{\Phi}^{(0)}_t \dot{\Psi}^{(0)}_t \end{bmatrix}=A \begin{bmatrix}\Phi^{(0)}_t \Psi^{(0)}_t \end{bmatrix}, \begin{bmatrix}\Phi^{(0)}_0 \Psi^{(0)}_0 \end{bmatrix}=\begin{bmatrix}0 1 \end{bmatrix},
\begin{array}{c}{{b^{-}~ = ~ l i m_{Q \to q}{\frac{Q^{\frac{\pm k N_{B}}{2}}{( B^{-})}^{k}}{( [ k ] ! )^{\frac{1}{2}}}}}}\\{{b^{+}~ = ~ l i m_{Q \to q}{{\frac{{( B^{+})}^{k}Q^{{\frac{\pm k N_{B}}{2}}}}{( [ k ] ! )^{\frac{1}{2}}}}}}}\\ \end{array}
{\bf a}= \left\| \begin{array}{c}{a^{1}}\\{a^{2}}\\{a^{3}}\\ \end{array}\right\| \quad{\bf b}^{T}= \left\| \begin{array}{c c c}{b_{1}}&{b_{2}}&{b_{3}}\\ \end{array}\right\| \quad r \in S O ( 3 )
S = \int d^{2}\sigma \sqrt{g}\left( \frac{1}{4 \pi \alpha^{\prime}}g^{\alpha \beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu \nu}( X ) + \frac{i}{2 \pi}\epsilon^{\alpha \beta}B_{\mu \nu}( X ) \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}- \frac{1}{4 \pi}\phi ( X ) R \right)
P^\eta(X_T^{\vec v}-X_1^{\vec v}\geq (v+\varepsilon)T-1) \leq P^\eta\Big(\sum_{m=1}^{T-1}\tilde Y_m\geq (v+\varepsilon)T-1\Big) +\sum_{m=1}^{T-1}\P^\eta(\mathcal E^c_{m-1}).
|g_k(x)| = \left | \sum_{n=1}^{[L-\delta]}\frac{a_n(1+a-b)_n}{n!}x^{-n}+ \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x) x^a \right | &\le C(m,\delta) l_k^{2\delta}x^{-\delta}.
\big (- \sum_{j=1,2}\sum_{l=1,2}\partial_{y_l}L (x,y)_{1j}\, F_{jl}\,, \, \, - \sum_{j=1,2}\sum_{l=1,2}\partial_{y_l}L (x,y)_{2j}\, F_{jl}\big )^\top \, = \, (F_{12}-F_{21})\frac{x^\bot}{4\pi |x|^2}
\frac{d}{dt}\|x_\nu^{(\frac{N}{2}+1)}(t)-x_\nu^{(\frac{N}{2}+2)}(t)\|^2 \leq-\alpha\nu\|x_\nu^{(\frac{N}{2}+1)}(t)-x_\nu^{(\frac{N}{2}+2)}(t)\|^2+\nu\|x_\nu^{(\frac{N}{2})}(t)-x_\nu^{(\frac{N}{2}+3)}(t)\|^2 &+\frac{1}{\nu}M_{T_1,T_2}^{\frac{N}{2}+1,\frac{N}{2}+2,5-\alpha}(\omega)
\left|\mathbb{E}\left(\widetilde{\alpha}_{t,\varepsilon}\right)^n\right|\leq\frac{1}{(2\pi)^n}\int_{[0,t]^{2n}}\int_{\mathbb{R}^n}\left|\mathbb{E}e^{\iota \sum_{i=1}^np_i(X^{H_1}_s-\widetilde{X}^{H_2}_r)}\right|dpdrds.
\mu_3 = -1 + \sqrt{1 + \frac{c_{1,\infty}}{c_{3,\infty}}- (1+\mu_1)^2\frac{c_{1,\infty}}{c_{3,\infty}}}\geq \sqrt{1 + \frac{c_{1,\infty}}{2c_{3,\infty}}}- 1.
\partial_t \rho_i = \nabla\cdot \left [ D_i(x) \left( \nabla \rho_i + z_i\rho_i \nabla \phi \right) \right ], i=1,\cdots, s, & -\nabla \cdot(\epsilon(x) \nabla \phi) = f(x)+\sum_{i=1}^{s}z_i\rho_i .
\begin{cases}-\Delta \psi_1(0) = \gamma_1(0) \psi_1(0) \mbox{in}\ \ \Omega, \frac{\partial \psi_1(0)}{\partial \mathbf{n}}= (q-1)b(x) w_0^{q-2}\psi_1(0) + \gamma_1(0) \psi_1(0) \mbox{on}\ \ \partial \Omega. \end{cases}
E^{x,i, \downarrow}\left[\int_0^\infty \mathbf{1}_{(\xi_t \geq z)}dt\right]=U^\downarrow((x,i),([z,0],E))=\sum_{j\in E}\int_{[z,0]}\frac{{U}^+_j(y)}{{U}^+_i(x)}U^\dagger((x,i), (dy,\{j\})),
Z [ A_{+}, A_{-}] = \int \! D \bar{\psi}D \psi e^{- \int \! d^{2}x [ \psi_{-}^{\dag}i \partial_{+}\psi_{-}+ \psi_{+}^{\dag}i \partial_{-}\psi_{+}- \psi_{-}^{\dag}A_{+}\psi_{-}- \psi_{+}^{\dag}A_{-}\psi_{+}]}
p \int \eta^2 u^{p-1}_k \partial_t u_k\;dx = - p\int (a\nabla u,\nabla (\eta^2 u_k^{p-1}))\;dx +p \int (u\nabla a,\nabla (\eta^2 u_k^{p-1}))\;dx = \widetilde{\textnormal{(I)}}+ \textnormal{(II)}.
B_2 \le 2 \int_0^{y/(m,n)}(m,n) \min \Bigl(\frac{2s}{Nn}, \frac s N \Bigl( \frac1n + \frac1m \Bigr) - \frac{(m,n)t}{mn}\Bigr) \d t & = 2 \int_0^y \min \Bigl(\frac{2s}{Nn}, \frac s N \Bigl( \frac1n + \frac1m \Bigr) - \frac{h}{mn}\Bigr) \d h.