harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\ |I_{m,q}(\vec{c})|=\frac{mQ^2}{\sqrt{N}}\int h(\frac{q}{Q},y) \psi_1(y)^2 \psi_2(\frac{\tilde{\vec{u}}}{Q})e\big(\frac{\sum_{i=1}^{d-1}c_i \tilde{u}_i+\beta\left<\vec{c},\vec{v}\right>}{mq}\big) d\tilde{u}_1\dots d\tilde{u}_{d-1}dy.
\mathcal{U}\cdot W=\left(V_{\mathbb{Z}\beta}\otimes V_{\mathbb{Z}\beta}^{+}\right)\boxtimes_{V_{\mathbb{Z}\beta}\otimes V_{\mathbb{Z}\beta}^{+}}W\oplus\left(V_{\frac{1}{2}\beta+\mathbb{Z}\beta}\otimes V_{\frac{1}{2}\beta+\mathbb{Z}\beta}^{+}\right)\boxtimes_{V_{\mathbb{Z}\beta}\otimes V_{\mathbb{Z}\beta}^{+}}W.
\frac{\lambda}{p}\,\int_\Omega \frac{|u_n|^p}{d_\Omega^{s\,p}}\,dx=\frac{\lambda}{p}\,\int_\Omega \frac{|u|^p}{d_\Omega^{s\,p}}\,dx+\frac{\lambda}{p}\,\int_\Omega \frac{|u_n-u|^p}{d_\Omega^{s\,p}}\,dx+o(1),\mbox{as}n\to \infty,
\sum_{n=T}^{[\tau]}N_A\left(\frac{n+\xi}{T}\right) = \frac{T}{\phi(Q)}\sum_{n=T}^{[\tau]}\frac{e^{\frac{n+\xi}{T}}}{n+\xi}+ O\Bigg(\frac{T^2}{\phi(Q)}\sum_{n=T}^{[\tau]}\frac{e^{\frac{n+\xi}{T}}}{(n+\xi)^2}\Bigg).
{f^{(12,-12)}(\Omega,\bar\Omega) ={c\over \sqrt{\Omega_2}}\left({\partial \over \partial\alpha}\right)^{24}\sum_{m,n}\int{dt \over t}t^{23/2}\left.e^{-t|m+ n\Omega|^2 + \alpha (m+ n\bar \Omega)}\right|_{\alpha =0},}
{}_{L^2(H^{-1})}\langle \partial_su,\phi \chi^2\eta\rangle_{L^2(H^1_0)}+{}_{L^2(H^1_0)}\langle u\chi^2\eta,\partial_s\phi\rangle_{L^2(H^{-1})}+\int_a^b \int_\Omega \chi^2(x)u(s,x)\overline{\phi(s,x)}\eta'(s)\,{\rm d}x\,{\rm d}s = \int_a^b\partial_{s}\Bigl(\int_\Omega \bigl(u(s,x)\overline{\phi(s,x)}\chi^2(x)\eta(s)\bigr)\,{\rm d}x\Bigr)\,{\rm d}s = 0,
I =\left(\sum_{A\in\mathcal{P}}\left(1-\frac{\alpha'(A)}{\alpha(A)}\right)\alpha(A)\log\frac{\alpha(A)}{\beta(A)}\right)-H_{\alpha'\parallel\alpha}\left(\mathcal{P}\right) \le\sum_{A\in\mathcal{P}}\left(1-\frac{\alpha'(A)}{\alpha(A)}\right)\alpha(A)\log\frac{\alpha(A)}{\beta(A)}.
|R_j(x,y)-R_j(x,y')| \leqslant c |y_j-y_j'| \int_0^\infty |h_t(x,y)-h_t(x,y')|\frac{dt}{t\sqrt t}&\leqslant c|y_j-y_j'| \bigg(\int_0^{\|x-y\|^2}+\int_{\|x-y\|^2}^\infty\bigg)|h_t(x,y)-h_t(x,y')|\frac{dt}{t\sqrt t}&=: I\!I_1 +I\!I_2.
\sum_{n=M_-}^{M_+-1}(A_1+A_3)|C_n| \leq \frac{c_1}{(\gamma t)^{\frac d2+1}}\Big(\frac{(\gamma t)^{(\frac12+\varepsilon)d}}{M^d}\wedge 1\Big)\sum_{n=M_-}^{M_+}n^{d-1}\leq c_2\,(\gamma t)^{-1+\varepsilon d}\,,
Q^{i,k,j}_t=\sigma\left(\frac{1}{n}-\delta_{i,k}\right)\left(\frac{1}{n}-\delta_{i,j}\right)\phi_t, R^{i,k,j}_t= \left(\frac{1}{n}-\delta_{i,k}\right)\left(\frac{1}{n}-\delta_{i,j}\right)\phi_{t}\gamma^j_t ,
\ell(L_A)=r_y\sum_{j=1}^n \vec{y}_j=\frac{r_y+r_{y'}}{k}\leq \frac{|A^T|+|(A^c)^T|}{k}&\leq \frac{2(k/2)^n(n+1)^{(n+1)/2}}{k}&= (k/2)^{n-1}(n+1)^{(n+1)/2},
\tau_{\pi}= \gamma_3 \gamma_4 \gamma_1 \gamma_2 \gamma_4 \gamma_3 \gamma_2 \gamma_1 = (1,4)(1,5)(1,2)(1,3)(1,5)(1,4)(1,3)(1,2)= (1,5,3) \in S_5 \subseteq S_{\infty}.
\frac{d}{dt}\Big|_{t=s}\varphi_t^* \o_t = \frac{d}{dt}\Big|_{t=s}\varphi_t^* \o_s + \varphi_s^*\left( \frac{d}{dt}\Big|_{t=s}\o_t\right) = \varphi_s^*(\mathcal{L}_{X_s}\o_s) + \varphi_s^* (d \mu) = \varphi_s^* \left(d(\iota_{X_s}\o_s) + \iota_{X_s}d \o_s\right) + \varphi_s^* (d \mu) = -\varphi_s^*( d\mu) + \varphi_s^* (d \mu) =0,
\displaystyle\sum_{n=1}^{\infty}\Big(\frac{1}{\lambda_{n}}+\frac{1}{\lambda_{n+1}}\Big)\frac{\beta(1-\beta)}{2n^{2}}|A_{n}|^{2}< \displaystyle\sum_{n=1}^{\infty}w_{n}(\lambda,\beta)|A_{n}|^{2}\leq \displaystyle\sum_{n=1}^{\infty}\frac{|A_{n}-A_{n-1}|^{2}}{\lambda_{n}}.
\begin{array}{ccccccc}x_1 - 2x_2 + x_3 = 0\,, x_2 - 2x_3 + x_4 = 0\,, \vdots \vdots \vdots \vdots\,\, x_{k-2}- 2x_{k-1}+ x_k = 0\,.\end{array}
\phi \Big|_{k, m}A(\tau) = (c \tau + d)^{-k}e^{-2\pi i \frac{m c z^2}{c \tau + d}}\phi\left( \frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\right) = \sum_{n, r}c_A(n, r) q^n \zeta^r,
\left[\frac{\binom{K-2}{s-1}}{L_s^{(K)}}- \frac{\binom{K-2}{s-2}}{L_{s-1}^{(K)}}\right]\left[ \frac{D_s^{(K+1)}}{L_s^{(K+1)}}- \frac{D_s^{(K)}}{L_s^{(K)}}\right]= \left[\frac{\binom{K-2}{s-1}}{L_s^{(K)}}- \frac{\binom{K-1}{s-1}}{L_{s}^{(K+1)}}\right]\left[ \frac{D_{s-1}^{(K)}}{L_{s-1}^{(K)}}- \frac{D_s^{(K)}}{L_s^{(K)}}\right],
\sum_{1\le b_{1}<\cdots <b_{r+s}\le m}q^{\sum_{a=1}^{r+s}b_{a}}e_{s}(z_{b_{1}}, qz_{b_{2}}, \ldots , q^{r+s-1}z_{b_{r+s}}) =q^{s(s-1)/2}e_{r}(q^{s+1}, q^{s+2}, \ldots , q^{m})e_{s}(qz_{1}, q^{2}z_{2}, \ldots , q^{m}z_{m})
\left[ 1 + \frac{c}{n^{2\alpha}}\right]^d - 1\leq \exp\!\left( \frac{c\,d}{n^{2\alpha}}\right) - 1 \leq \frac{1}{1 - c\,d\, n^{-2\alpha}}- 1 = \frac{c\,d\, n^{-2\alpha}}{1 - c\,d\, n^{-2\alpha}}\leq \frac{c}{1-c_1}\, d \, n^{-2\alpha}
Q= 4\alpha H^{-\alpha-4}\nabla |A|^2\cdot\nabla H-2\alpha H^{-\alpha-3}|\nabla A|^2 -2\alpha(\alpha+1)H^{-\alpha-4}h_{ij}\nabla^iH\nabla^jH+2\alpha(\alpha-2)H^{-\alpha-5}|A|^2|\nabla H|^2.
P(a,b, X,Y) = \frac{1}{2}a(b^2 +4) |X|^2 \Im (X) + \frac{5}{2}a^2b \Im(X^2 \overline{Y}) - b^2 |X|^3 + 4a(1- b^2) |XY| \Im (X) + (4a^2 + 3b^2) |X^2 Y| - 10 ab |X| \Re (X \overline{Y}) + \frac{1}{2}b(4+a^2 + 5b^2) |X|^2 \Im (Y),
\mathcal{N}\bigg( \begin{bmatrix}\mathbf{m}_{j,}^k begin{align*}2mm] \mathbf{m}^k_{j,}\end{bmatrix}, \begin{bmatrix}Z^k_{j,}Z^k_{j,}begin{align*}2mm] Z^{k}_{j,}Z^k_{j,}\end{bmatrix}\bigg),
m_k := \frac{\lambda_+^{k+1}-\lambda_-^{k+1}}{\lambda_+-\lambda_-}, \lambda_+ := \frac{m_\xi+\sqrt{m_\xi^2+4m_\eta}}{2}, \lambda_- := \frac{m_\xi-\sqrt{m_\xi^2+4m_\eta}}{2}
I= \int^{\pi\over 2}_{-{\pi\over 2}}d\mu \, f^{-\tilde{\gamma}}\left[(K')^2 + (\tilde{\gamma}^2-9/4) K^2\right] >{-{9\over 4}}\int^{\pi\over 2}_{-{\pi\over 2}}d\mu \, f^{-\tilde{\gamma}}K^2\ ,
\left. \frac{d}{d \tau}\right|_{\tau = 0}x^{\alpha}( g_{A}( \tau ) g ) = \left. \frac{\partial x^{\alpha}( g_{A}( \tau ) g )}{\partial x^{M}( g_{A}( \tau ) )}\right|_{g_{A}= e}\cdot \left. \frac{d x^{M}( g_{A}( \tau ) )}{d \tau}\right|_{\tau = 0}= R_{M}^{\alpha}( g ) \delta_{A}^{M}= R_{A}^{\alpha}( g ) .
\Phi_h\bigg( \bigg( \sum_{i=0}^\ell r_i x_i \bigg)^h \bigg) = \frac1{h!}\sum_{\sigma\in\Sigma_h}\sum_{j_1=0}^\ell \cdots \sum_{j_h=0}^\ell r_{j_1}\cdots r_{j_h}z_{j_1j_2}\cdots z_{j_{h-1}j_h}.
\widetilde C = \begin{pmatrix}a 0 b 0 0 a b 0 0 0 c 0 0 0 0 c \end{pmatrix}C = \begin{pmatrix}1 0 1 0 0 1 1 0 0 0 1 0 0 0 0 1 \end{pmatrix}\overline C = \begin{pmatrix}1 1 0 1\end{pmatrix}.
\frac{m_{i}}{m_{j}}= \frac{\operatorname{s i n}\left[ ( \eta_{k}- \eta_{j}) \pi \vartheta_{h}+ ( \hat{\eta}_{k}- \hat{\eta}_{j}) \pi \vartheta_{H}\right]}{\operatorname{s i n}\left[ ( \eta_{i}- \eta_{k}) \pi \vartheta_{h}+ ( \hat{\eta}_{i}- \hat{\eta}_{k}) \pi \vartheta_{H}\right]}\; .
\int_{0}^{\infty}\int_{-\infty}^{\infty}u\eta_t\,dxdt+\int_{0}^{\infty}\int_{-\infty}^{\infty}u^m\eta_x\,dxdt=-\int_{0}^{\infty}\int_{-\infty}^{\infty}\varphi(u)\eta\,dxdt
\prod_{j=0}^{k-1}\left(1 + \frac{1}{x_j}\right) \leq \left(2^{p-1}\frac{(X_q(1) + 1)}{X_q(2^{p-1})}\right)^{k/p}= \left(2^{p-1}\frac{2q}{q^2}\right)^{k/p}= \frac{2^k}{q^{k/p}}.
\begin{array}{l l}{\hat{\delta}F^{0 + a b}= 4 \bar{\cal N}^{0 \Lambda}F_{\Lambda}^{+ a b},}{\hat{\delta}F_{i}^{+ a b}= 0 ,}{\hat{\delta}F^{0 - a b}= - 4{\cal N}^{0 \Lambda}F_{\Lambda}^{- a b},}{\hat{\delta}F^{i - a b}= 0 .}\end{array}
W_{A D S}= ( N_{c}+ 1 - N_{f}) \omega_{N_{c}+ 1 - N_{f}}( \frac{2^{N_{c}- 1}\Lambda_{N = 1}^{3 ( N_{c}+ 1 ) - N_{f}}}{\mathrm{P f}M})^{\frac{1}{N_{c}+ 1 - N_{f}}}
Z = \int d \mu{\cal D}_{e^{\phi}\hat{g}}X{\cal D}_{e^{\phi}\hat{g}}\phi{\cal D}_{e^{\phi}\hat{g}}b{\cal D}_{e^{\phi}\hat{g}}c \exp{\left\{- S_{P}[ X , \hat{g}] - S_{g h}[ b , c , \hat{g}] \right\}}
\begin{array}{rl}\displaystyle\frac{-\omega(1,q^b)^b}{(1-q^{-b})^2-bq^{-b}\omega(1,q^b)^b}> \displaystyle\frac{-(1-q^{-b})}{(1-q^{-b})^2-bq^{-b}(1-q^{-b})}&= \displaystyle\frac{-1}{1-q^{-b}- bq^{-b}}&= \displaystyle\frac{-1}{1-(1+b)q^{-b}}\end{array}
\sigma_{\b{a},n}(3)=1-\frac{3(\theta_{a_1}(3)+\theta_{a_2}(3)+\theta_{a_3}(3))}{\prod_{1\leq i \leq 3}{(2-\theta_{a_i}(3))}}+\prod_{1\leq i \leq 3}\Big(\frac{1+\theta_{a_i}(3)}{2-\theta_{a_i}(3)}\Big)
D^{j+1}= D\left(A^jP_j(K,\theta_K)\right) &=-jA^{j+1}\left(\frac{1+2K}{6(1-K)}\right)P_j(K,\theta_K) + A^{j}D_0\left(P_j(K,\theta_K)\right) &= A^{j+1}\left(\theta_K(P_j(K,\theta_K)) - j\left(\frac{1+2K}{6(1-K)}\right)P_j(K,\theta_K) \right),
G_2(Q) = \int \phi_m(x) p_0^m(x_1) \int \phi_m(x) \, p_0^m(x_2) Q(x_1,x_2) dx_2 dx_1 &\leq \bar{G}\int \phi_m(x) p_0^m(x_1) \int \Pi_{j=1}^{m}\phi_1( x^j_2) p_0^m(x_2) Q(x_1,x_2) dx_2 dx_1 \leq \bar{G}^2 \int \Pi_{j=1}^{m}\phi_1( x^j_2) p_0^m(x_2) \int \Pi_{j=1}^{m}\phi_1( x^j_1) p_0^m(x_1) Q(x_1,x_2) dx_1 dx_2.
E_1^{(12)}=\frac{3}{4t\left(D_{0}^{-2}r^{-1}(\eta)-1\right)}\tilde{T}_{\eta,1}^{(12)}-\frac{3}{4t\left(D_{0}^{-2}r^{-1}(-\eta)-1\right)}\tilde{T}_{-\eta,1}^{(12)}+\mathcal{O}(t^{-2}),
\| \lambda ( \phi ) \|_{K , m + n}= \sup_{\stackrel{| p | \leq m + n}{z \in K}}| D^{p}( \lambda ( \phi ) ) ( z ) | \, \, , \qquad D^{p}= \frac{\partial^{| q | + | r |}}{\partial x_{1}^{q_{1}}\cdots \partial x_{m}^{q_{m}}\partial \theta_{1}^{r_{1}}\cdots \partial \theta_{n}^{r_{n}}}\, \, .
(D_{\bar i}Q_i)({\overline Q}^j{\overline D}^{\bar j}) \epsilon_{jk}\epsilon_{{\bar j}{\bar k}}\left(C_{a_1 a_2}\Sigma^{i{\bar i}}_{a_1}\Sigma^{k{\bar k}}_{a_2}+ C^\prime_{a_1 a_2}\Sigma^{i{\bar k}}_{a_1}\Sigma^{k{\bar i}}_{a_2}\right)~,
\sum_{n_1,n_2 \geq 0}\frac{q^{2 n_1^2 + 6 n_1 n_2 + 6 n_2^2}}{(q;q)_{n_1}(q^3;q^3)_{n_2}}= (-q^2,-q^3,-q^4,-q^6; q^6)_\infty.
f(x) =\sum_{j=1}^N\sum_{k=0}^{m_j}a_{jk}x^{\alpha_j}\log^kx+x^pf_1(x) &=\sum_{j=1}^M\sum_{k=0}^{m'_j}b_{jk}x^{\beta_j}\log^kx+x^{-q}f_2(x),
I(G-N[v_0]) = x \cdot (2x+1)^{2k+5}&= x \cdot \bigg[\sum_{i=0}^{2k+5}\binom{2k+5}{i}(2x)^{i}\bigg] &= x \cdot [(2k)^{2k+5}+\dots] &= 2^{2k+5}x^{2k+6}+\dots
c_{\scriptscriptstyle G}^{\mathrm{e f f}}< c_{g}^{\mathrm{e f f}}\, , \qquad \mathrm{f o r}\qquad g_{a b}={\textstyle \frac{1}{G}}\delta_{a b}\quad \mathrm{a n d}\quad 0 < G < 1 \, .
U_{k_{1}, k_{1 z}; k_{2}, k_{2 z}; I_{z}, I_{z}^{\prime}}= \sum_{m , I_{z}}\langle k , m ; I , I_{z}| k_{2}, k_{2 z}\rangle D_{m , k_{1 z}}^{k_{1}}( \alpha , \beta , \gamma ) D_{I_{z}, I_{z}^{\prime}}^{I}( \alpha_{I}, \beta_{I}, \gamma_{I}) | I , I_{z}^{\prime}\rangle .
E = \{B(s) \leq b(s)+\Delta(s)\, \forall \, s\leq t\}A_1 =\{B(s) \ge b(s)-Kt^{1/6}\, \forall \, s\leq t\}\, , A_2 = \{\mathrm{Leb}(\{s\leq t:B(s)\leq b(s)\})\leq Ct^{1/3}\}\, .
\begin{array}{r c l}{{\psi \circ s_{h}: = \phi :}}&{{G / H}}&{{\longrightarrow{\bf C}^{n}}}\\{{}}&{{[ g ]}}&{{\longrightarrow \phi ( [ g ] ) = \psi ( s_{h}( [ g ] ) ) \equiv \psi ( g h )}}\\ \end{array}
\Omega_{\pm}= \frac 1{\sqrt{2}}\left( \begin{array}{c c}{{\exp \pm \frac{i \phi}2}}&{{- \exp \mp \frac{i \phi}2}}\\{{\exp \pm \frac{i \phi}2}}&{{\hphantom{-}\exp \mp \frac{i \phi}2}}\\ \end{array}\right) ; \qquad \Omega_{\pm}^{- 1}= \Omega_{\pm}^{\dagger};
- \int_{0}^{\sigma}d s \int_{- \infty}^{\infty}\frac{d \omega}{2 \pi}\frac{s}{( 4 \pi )^{\frac{D - 1}{2}}}( \omega^{2}+ s^{2})^{\frac{D - 3}{2}}\Gamma ( - \frac{D - 3}{2}) \mathrm{t r}\mathrm{\boldmath 1}.
\left( \begin{array}{c}{{\psi_{+}( z , t )}}\\{{\psi_{-}( z , t )}}\\ \end{array}\right) = \mathcal{F}\left( \begin{array}{c}{{\psi_{+}( z - \Delta t , t - \Delta t )}}\\{{\psi_{-}( z + \Delta t , t - \Delta t )}}\\ \end{array}\right) ,
Z = \int{\cal D}[ f ( \xi ) , g ( \xi ) , X ] \, \delta \left( g_{z z}^{f}\right) \delta \left( g_{\bar{z}\bar{z}}^{f}\right) \, \mathrm{d e t}\left( \frac{g_{z z}^{f}}{\delta f}\right) \mathrm{d e t}\left( \frac{g_{\bar{z}\bar{z}}^{f}}{\delta f}\right) \, e^{- S [ g^{f}, X ]}.
A(p, m p) = \sum_{d \mid (p, p m)}\mu(d) A\Big ( \frac{p}{d}, 1 \Big ) A \Big ( 1, \frac{p m}{d}\Big ) = A(p, 1) A(1, p m) - A(1, 1) A(1, m) = A(p, 1) A(1, p m) - A(1, m)
\|e^{2\pi imz}f_{1,m}(y)f_2\|_{L^2}^2 =\sum\limits_{k\in\mathbb{Z}}\|f_{1,m}f_{2,k}\|_{L^2}^2\leq\sum\limits_{k\in\mathbb{Z}}\|f_{1,m}\|_{L^2}^2\|f_{2,k}\|_{L^{\infty}}^2 \leq C\sum\limits_{k\in\mathbb{Z}}\|f_{1,m}\|_{L^2}^2\|f_{2,k}\|_{H^{1}}^2\leq C\|f_{1,m}\|_{L^2}^2\| f_2\|_{H^1}^2,
\int^{\infty}_{\lambda}\frac{\lambda}{\sigma^2}\sigma\bar{u}(\sigma)d\sigma=\frac{1}{1/\lambda}\int^{1/\lambda}_{0}\frac{1}{\gamma}\bar{u}(\frac{1}{\gamma})d\gamma=\frac{1}{\omega}\int^{\omega}_{0}\frac{1}{\gamma}\bar{u}(\frac{1}{\gamma})d\gamma.
S_{\omega_{\xi,\eta}}(\hat{T})(x)=\int_{K_2}\int_{K_1}k(g,h)\phi_{z(h)}(xh^{-1})\phi_{w(g,h)}(xh^{-1}g)T_{z(h),w(g,h)}\Delta(g)^{-1/2}\d\mu(g)d\mu(h).
\lim_{x\downarrow0}g''_0(x;a,b) =\lim_{x\downarrow0}g_1'(x;a,b)+4a^3f^{(4)}(A)\int_{-1}^1\int_0^s\,du\,s\,ds &\qquad+8a^5f^{(5)}(A)\int_{-1}^1\int_0^s\int_0^u\,dv\,u\,du\,s\,ds &=4af^{(3)}(A)+\frac{16a^3f^{(4)}(A)}{3}+\frac{16a^5f^{(5)}(A)}{15}.
\left( -{\frac{1}{r}}{\frac{\partial}{\partial r}}r{\frac{\partial}{\partial r}}+{\frac{\left( m - \delta{\frac{r^{2}}{R^{2}}}\right)^{2}}{r^{2}}}-{\frac{g}{2}}\delta{\frac{2}{R^{2}}}\right) \psi_{m}( r ) = \epsilon \psi_{m}( r ) .
\forall \pi \in S_n:\Q_{Y^n|X^n}W_{Y^n}[\pi] = W_{X^n}[\pi] Q_{Y^n|X^n}\iff \forall \pi \in S_n:\ \tilde{Q}_{X^nY^n}W_{X^nY^n}[\pi] = \tilde{Q}_{X^nY^n}\,.
\int_X|f|^2_{h_F^\chi\otimes h,\omega}dV_\omega =\int_X|f|^2_{h_F\otimes h,\omega}e^{-\chi\circ\psi}dV_\omega \mathrm{and}\int_X\langle B_{h_F^\chi,\omega}^{-1}f,f\rangle_{h_F^\chi\otimes h,\omega}dV_\omega \leq\int_X\langle B_{h_F,\omega}^{-1}f,f\rangle_{h_F\otimes h,\omega}e^{-\chi\circ\psi}dV_\omega
\Psi_{{\bf p}}^{(-)}({\bf x},t)={\rm e}^{+iw_{{\bf p}}t}\,\Psi_{{\bf p}}^{(-)}({\bf x})=\frac{1}{{2\pi}}\frac{w_{{\bf p}}+m}{\sqrt{4mw_{{\bf p}}}}\left( \begin{array}{c}\frac{m-w_{{\bf p}}}{m+w_{{\bf p}}}1\end{array}\right){\rm e}^{\,-i{\bf p}\cdot{\bf x}+iw_{{\bf p}}t}\;,
\left( \begin{array}{c c c c | c c}{{\times}}&{{\times}}&{{}}&{{}}&{{}}&{{}}\\{{\times}}&{{\times}}&{{}}&{{}}&{{}}&{{}}\\{{\times}}&{{\times}}&{{\times}}&{{\times}}&{{\times}}&{{\times}}\\{{\times}}&{{\times}}&{{\times}}&{{\times}}&{{\times}}&{{\times}}\\ \hline{{\times}}&{{\times}}&{{}}&{{}}&{{\times}}&{{}}\\{{\times}}&{{\times}}&{{}}&{{}}&{{\times}}&{{\times}}\\ \end{array}\right)
s_0 = T_\pi s_{n-1}T_\pi^{-1}= (y_1s_1\cdots s_{n-1})s_{n-1}(s_{n-1}\cdots s_1y^{-1}_1)=y_1 s_1\cdots s_{n-2}s_{n-1}s_{n-2}\cdots s_1 y^{-1}_1 = y_1y^{-1}_n s_1\cdots s_{n-2}s_{n-1}s_{n-2}\cdots s_1= y_1y^{-1}_n s_{n-1}\cdots s_2s_1s_2 \cdots s_{n-1}.
\begin{aligned}0={}{}b_j\bigl((-1)^jc^2(1-3a^2)^2+4c(1-3a^2)\lambda_i(\lambda_1-\lambda_2)+32(-1)^ik(\lambda_1-\lambda_2)^2\bigr) &{}+4b_i(\lambda_1-\lambda_2)\bigl(c(1-3a^2)-2(-1)^i(\lambda_1-\lambda_2)(\lambda_i-3\lambda_j)\bigr)\mu.\end{aligned}
\big(\prod_{k=1}^{s_1}(z-\alpha_{k}^{(1)})^{d_{k}^{(1)}},\cdots ,\prod_{k=1}^{s_1}(z-\alpha_{k}^{(m)})^{d_{k}^{(m)}}\big)\mapsto\big(\sum_{k=1}^{s_1}d_{k}^{(1)}\alpha_{k}^{(1)},\cdots ,\sum_{k=1}^{s_m}d_{k}^{(m)}\alpha_{k}^{(m)}\big),
\nabla_\alpha\eta^A-\frac{\sqrt{2}{\rm i}}{3}{\cal P}^{A}_{\;B}\gamma_\alpha\eta^B-\delta(y)\delta_\alpha^5(\omega_{0})^{A}_{\;B}\gamma_5\eta^B-\delta(y-\pi r_c)\delta_\alpha^5(\omega_\pi)^{A}_{\;B}\gamma_5\eta^B=0\ .
\tilde{\psi}^{(-)}_{n,m}(r) = r^{\ell/2}\exp(-\frac{1}{4}\omega r^2)\left[\frac{L_m^{\alpha+1}(-y)L_n^\alpha(y)-\partial_yL_n^{\alpha}(y)L_m^\alpha(-y)}{L_m^\alpha(-y)}\right]_{y=\frac{1}{2}\omega r^2}
\overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)p_{\alpha_i,\beta_j}) + \overline{\delta_k}(a_ip_{\alpha_i,b_j}Y_j) = \ \sigma_k(a_i)\sigma^{\alpha_i}(\delta_k(b_j))p_{\alpha_i, \beta_j}+ \sigma_k(a_i)p_{\alpha_i, \delta_k(b_j)}Y_j + \ \delta_k(a_i)\sigma^{\alpha_i}(b_j)p_{\alpha_i, \beta_j}+ \delta_k(a_i)p_{\alpha_i, b_j}Y_j.
\tilde{f}(s,y) =\alpha_{-1}(-\tilde{\theta}y^{2\tilde{\theta}+1})+\alpha_0y^{\tilde{\theta}+1}-\alpha_1\frac{y}{\tilde{\theta}}+\alpha_2 s^{2H-1}\frac{1}{\tilde{\theta}^{\rho}}y^{-\tilde{\theta}\rho+\tilde{\theta}+1}-\tilde{\sigma}Hs^{2H-1}y^{-1}(\tilde{\theta}+1),
{\rm tr}\left[ QP^{,\rho}\right] =0.\qquad{\rm tr}\left[ QP^{,\rho}P^{,\sigma}\right] =\frac{1}{k^{2}}P^{\rho \sigma}.\qquad{\rm tr}\left[QP^{,\rho \sigma}\right] =\frac{2}{k^{2}}P^{\rho \sigma}
\|c(\cdot,t)\|_{L^{\infty}(\Omega)}\leq \tilde C \left( 1 + \frac{2D_c \mu}{C^2}\right)^{2}\left( 1 + \frac{\sqrt{C^2+2\mu D_c}}{D_c}\right)^{2d}M(1) = \tilde{\tilde C}M(1).
\mathcal{R}_2\leq \frac{\sqrt{2}Cv^2_0L^{-2}T^2}{3}\frac{\widetilde{C}\ln(\lambda T+3)}{(\lambda T)^{3/2}}= 3^{-1}C_2{L}^{-2}_*{T}^{1/2}_*\ln({T}_*+3)\,,
\mathbb{E}\big[ \mathfrak{m}^{(p,p)}\big]= \mathbb{E}\big[(O_\prec(\hat{\Pi}^2)+O_\prec(\Psi \hat{\Upsilon}))\mathfrak{m}^{(p-1,p)}\big]+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{m}^{(p-2,p)}\big] &\qquad+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{m}^{(p-1,p-1)}\big]\,.
- \frac{\partial}{\partial t}K_{i^{\prime}j^{\prime}k l}( x^{\prime}, x ; t ) = - ({\nabla^{\prime}}\! \!_{p}{\nabla^{\prime}}^{p}+ \xi R ( x^{\prime}) ) K_{i^{\prime}j^{\prime}k l}( x^{\prime}, x ; t ) \ ,
\frac{d}{dt}\xi^{(k,\ell)}_{i,1}-2^{-1}m \xi^{(k,\ell)}_{i,1}+\beta_i\xi^{(k,\ell)}_{i,2}=\mathcal{E}^{(k,\ell)}_{i,1}, \frac{d}{dt}\xi^{(k,\ell)}_{i,2}-2^{-1}m \xi^{(k,\ell)}_{i,2}-\beta_i\xi^{(k,\ell)}_{i,1}=\mathcal{E}^{(k,\ell)}_{i,2},
I_{+}^{(m)}=\sum_{\mathbf{k}\in \mathcal{S}_m}\binom{m}{\mathbf{k}}\frac{w_1^{k_1}w_2^{k_2}w_3^{k_3}}{2^{k_1+k_2+k_3-1}}\mathcal{I}_{+}^{(m,\mathbf{k})},
\chi_1 (\zeta, a) := \sqrt{2}(1+i) e^{-\frac{i}{5}(\zeta^5 a^{-3}+ 5 \zeta^4 a^{-2}+10 \zeta^3 a^{-1}+ 10\zeta^2)}\mathcal{F}[e^{-\frac{i}{8}z^2 + i R(z, a)}\chi ] (\zeta).
\lim_{n\rightarrow+\infty}\frac 1n d(\xi_n^Y,\xi_{n\varphi^Y})=\lim_{n\rightarrow+\infty}\frac 1n \int_{\omega\in\Omega}d_\omega(\xi_n^Y,\xi_{n\varphi^Y})\,\nu(\mathrm{d}\omega).
\mathbb{P}\left( E \right) = \mathbb{P}\left( \bigcap_{k = 1}^n E_k \right) \ge \sum_{k = 1}^n \mathbb{P}(E_k) - (n - 1) & \ge \sum_{k = 1}^n \left( 1 - 2 (n - k + 1) \exp \left( - \frac{(\epsilon^2 - \epsilon^3) p}{4}\right) \right) - (n - 1) & \ge 1 - n (n + 1) \exp \left( - \frac{(\epsilon^2 - \epsilon^3) p}{4}\right),
(v_1'v_2-v_1v_2')^{-1}(r_1,z) = \left(\frac12r_1^{-1/2}I_{\nu}'(r_1z)+zr_1^{1/2}I'_{\nu}(r_1z)\right)r^{1/2}K_{\nu}(r_1z) &-r_1^{1/2}I_{\nu}(r_1z)\left(\frac12r_1^{-1/2}K_{\nu}'(r_1z)+zr_1^{1/2}K'_{\nu}(r_1z)\right) = r_1z\big(I_{\nu}'(r_1z)K_{\nu}(r_1z)-I_{\nu}(r_1z)K_{\nu}'(r_1z)\big)=1.
t_{s_h}^n = (t_xt_y)^nt_z^nt_a^{-n}t_b^{-n}t_c^{-n}&={}_{t_x}(t_y){}_{t_x^2}(t_y) \cdots{}_{t_x^n}(t_y) t_x^n t_z^n t_a^{-n}t_b^{-n}t_c^{-n}.
\mathcal{S}_{q}(z)=\frac{(q^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}}z^{1/2}J_{1/2}^{(3)}(z;q^{2}) \mbox{and}\mathcal{C}_{q}(z)=\frac{(q^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}}z^{1/2}J_{-1/2}^{(3)}(q^{-1/2}z;q^{2})
\begin{aligned}s ( \rho, q | \bar \rho , \bar q) = p (\rho | \bar \rho) \, , H ( \rho, q | \bar \rho , \bar q) = \rho \Big ( \frac{q}{\rho}- \frac{\bar q}{\bar \rho}\Big ) \otimes \Big ( \frac{q}{\rho}- \frac{\bar q}{\bar \rho}\Big ) \, , r ( \rho, q | \bar \rho , \bar q) = 0\, . \end{aligned}
C_1\Big( \int_0^T \sum_{l=0}^q || \mathcal{\mathbf{B}}_{n+l}^{\ast}U_{n+l}||_{G_{n+l}}^2 + \sum_{l=q+1}^p || \mathcal{\mathbf{B}}_{n+l}^{\ast}W_{n+l}||_{G_{n+l}}^2\,dt\Big)\le \displaystyle{\sum_{i=1}^n e_{1+i-n}(U_i)(0)}+ \sum_{j=1}^q e_1(U_{n+j})(0)+ \sum_{j=q+1}^p e_0(U_{n+j})(0)\,,
\Delta_{0}= (-1)^{\frac{(n_{0}+1)^{2}}{2}}s_{n_{0}}^{n_{0}+1}> 0 \ , \Delta_{k+1}= (-1)^{{\tfrac{(n_{k+1}- n_{k})^{2}}{2}}}\left(s_{n_{k+1}+ n_{k}+1}- s_{n_{k+1}+ n_{k}+1}^{(n_{k}+1)}\right)^{n_{k+1}- n_{k}}t_{n_{k}}^{2}> 0 \ , \ 0 \leq k < m-1 \ ,
L = i \overline{{{\psi}}}_{0}^{\prime}{\gamma}^{\mu}{\partial}_{\mu}{\psi}_{0}^{\prime}+ i \overline{{{\psi}}}_{0}^{\prime}e^{i{t}^{a}{\omega}_{a}}e^{- i{t}^{b}{\theta}_{b}}{\gamma}^{\mu}\partial_{\mu}\left( e^{i{t}^{c}{\theta}_{c}}e^{- i{t}^{l}{\omega}_{l}}\right){\psi}_{0}^{\prime}- m \overline{{{\psi}}}_{0}^{\prime}{\psi}_{0}^{\prime}.
\| g\|_{L^p_y}\le C \|g\|_{L^2}^{1-\delta}\|A_2 g\|_{L^2_y}^\delta, &\| g\|_{L^p_y}\le \frac{C}{|t|^{\delta}}\|g\|_{L^2}^{1-\delta}\|A_3(t) g\|_{L^2_y}^\delta,t\not =0,
4q_{p}(2)\equiv\sum_{k=\lfloor\frac{p}{10}\rfloor+1}^{\lfloor\frac{2p}{10}\rfloor}\frac{1}{k}+\sum_{k=\lfloor\frac{3p}{10}\rfloor+1}^{\lfloor\frac{4p}{10}\rfloor}\frac{1}{k}(modp).
W^0_{l+2}(p,p_1,\dots,p_{l+1})\ast W^0_{m+2}(p',p^{'}_{1}\dots,p^{'}_{m+1})= &\delta_{p,p'}\delta_{p_{l+1},p'_{1}}W^0_{k+2}(p,p_1,\dots,p_{k+1}),k=l+m.
\theta^0=2\varepsilon^{-1},\theta^\infty_1=-2\varepsilon^{-1},t_2=\varepsilon\tilde{t}_2,H_{t_2}=\varepsilon^{-1}\tilde{H}_2, q_2=\tilde{q}_2,p_2=\tilde{p}_2+\frac{1}{\varepsilon\tilde{q}_2},Y=\tilde{x}^{\varepsilon^{-1}}\tilde{Y}.
\left.\frac{\partial}{\partial t}\mathcal{F}_{1,\theta}\left[\Sigma_t\right]\right\vert_{t=0}=\int_\Sigma \left<P_1\nabla f,\nabla f\right>-2H_1\left(H_2+c\right)f^2\,d\mu_\Sigma+\int_{\partial\Sigma}\left\vert{P_1\nu}\right\vert\alpha_\theta f^2\,d\mu_{\partial\Sigma}.
(a\nabla (\eta u_k^{p/2}),\nabla (\eta u_k^{p/2})) = \frac{p^2}{4}u_k^{p-2}\eta^2 (a\nabla u_k,\nabla u_k)+p\eta u_k^{p-1}(a\nabla u_k,\nabla \eta)+u_k^{p}(a\nabla \eta,\nabla \eta),
P_{co}(l_k) = \mathbb{P}\left\{\frac{P_{S_k}|h_{{S_k},{D_k}}|^2/d_{S_k,D_k}^\alpha}{\sum_{J_j \in \Phi_J}P_{\bar{J}}|h_{J_j,D_k}|^2/d_{J_j,D_k}^\alpha}< \gamma_C \right\},
0\leq (\Delta f-f_t)_{nn}(x_0,t_0)-m\psi_{sss}(0,t_0)/2 \leq 2f_{nn}(x_0)^2-m\psi_{sss}(0,t_0)/2 = m^2\psi_s^2(0,t_0)/2-m\psi_{sss}(0,t_0)/2
\vert \{\lambda \}, \{\rho \}, \chi \rangle{}~ \equiv{}~ ( \alpha_{-{n_{1}}}^{0})^{\lambda_{1}}. . . ( \alpha_{-{n_{p}}}^{0})^{\lambda_{p}}( \epsilon_{- r_{1}}^{( \pm ) 0})^{\rho_{1}}. . . ( \epsilon_{- r_{s}}^{( \pm ) 0})^{\rho_{s}}\vert \chi ; q \rangle{}~ ,
L_{F}=\max\bigg\{\max_{1\leq j \leq n}\bigg(\bigg(1+\frac{M^{2}K^{2}b}{\delta^{j}}\bigg)\bigg(KK_{1}\gamma b+ KL_{\nu_{j}}\bigg)\bigg),\bigg( \bigg(1+\frac{M^{2}K^{2}b}{\delta^{0}}\bigg) &\qquad\qquad\qquad\bigg(K_{1}K\gamma b+KC_{\nu}\bigg)\bigg), \max_{1\leq j\leq n}L_{\nu_{j}}\bigg\}.
K^W \; := \; \mbox{diag}\left(\frac{\pi}{\pi + gN}\frac{1}{-\triangle+\frac{e^2N}{\pi + gN}}\; , \;\frac{1}{-\triangle +1}\; , \; . \; . \; . \; . \; , \;\frac{1}{-\triangle +1}\right) \; .
\begin{array}{r l}{0}&{= \nabla_{\rho}h^{\rho z I_{3}\ldots I_{s}}=}\\{}&{= \partial_{z}h^{z z I_{3}\ldots I_{s}}+ \partial_{I}h^{z I I_{2}\ldots I_{s}}-{\frac{1}{z}}[ d + 2 s - 2 ] h^{z z I_{3}\ldots I_{s}}}\\ \end{array}
c_k:=e_k\big(\theta_1^{(n+m)},\ldots,\theta_{n}^{(n+m)}\big), 1 \le k \le n,{\overline{c}}_r:=e_r\big(\theta_{n+1}^{(n+m)},\ldots,\theta_{n+m}^{(n+m)}\big), 1 \le r \le m.
\left[X_j,Y_{j_1,\dots,j_k}^{(n-1,i)}\right], \tilde Y_{j_1,\dots,j_k}^{(n-1,i)}\in C_{n+1}\quad\mbox{as well as}\left[X_0,Y_{j_1,\dots,j_k}^{(n-1,i)}\right]+ \frac{1}{2}\sum_{j=1}^m\left[X_j, \left[X_j,Y_{j_1,\dots,j_k}^{(n-1,i)}\right]\right] \in C_{n+2}
\mathrm{P}(\mathcal{O}_{2}) = \sum^{N}_{l=1}\mathrm{P}\left(x_{n^*}<2R_2 ||\mathcal{S}_r|=l\right)\mathrm{P}\left( |\mathcal{S}_r|=l\right) = \sum^{N}_{l=1}\left(F(2R_2)\right)^l\mathrm{P}\left( |\mathcal{S}_r|=l\right).
V^b_{5(0)}\sim \frac{i}{2\sqrt{2}}\biggl[ (1-i)e^{iH_5}e^{iH_8}+(1+i)e^{iH_5}e^{-iH_8}+(1+i)e^{-iH_5}e^{iH_8}+(1-i)e^{-iH_5}e^{-iH_8}\biggr]
\left\vert f_{S}-f_{S_{j}}\right\vert =\left\vert \frac{1}{\mu(S_{j})}\int_{S_{j}}fd\mu-f_{S}\right\vert \leq\frac{1}{\mu(S_{j})}\int_{S_{j}}\left\vert f-f_{S}\right\vert d\mu & \leq\frac{c}{\mu(B(x_{j},3K_{n}r_{j}))}\int_{B(x_{j},3K_{n}r_{j})}\left\vert f-f_{S}\right\vert d\mu\leq c\lambda_{0}
\begin{array}{r c l}{{R_{\mu \nu}+ \frac{1}{2}\partial_{\mu}\varphi \partial_{\nu}\varphi + \frac{1}{1 2}{\cal V}_{0}e^{- \varphi}g_{\mu \nu}}}&{{=}}&{{0 \, ,}}\\{{}}&{{}}&{{}}\\{{\nabla^{2}\varphi + \frac{1}{2}{\cal V}_{0}e^{- \varphi}}}&{{=}}&{{0 \, .}}\\ \end{array}
{\delta^\dagger}^{(l)}_j(\boldsymbol{u}^{(k)})^{(k)}_i -{\delta^\dagger}^{(k)}_i(\boldsymbol{u}^{(l)})^{(k)}_j = 0 \quad\forall (i,j)\in B_e(k,l), \;\forall l\in{\mathcal{I}}_{\mathcal{F}}^{(k)},
\int_{\Omega}\delta ({\cal L}) \sqrt{g}\; d^{4}x = \int_{\Omega}J^{\mu \nu}{}_{\lambda \rho}\left\{- 2 \nabla_{\mu}\nabla^{\lambda}( \delta g )_{\nu}{}^{\rho}+ R_{\mu \nu}{}^{\sigma \lambda}( \delta g )_{\sigma}{}^{\rho}\right\}\sqrt{g}\; d^{4}x .
\left( \bar{a}_{\alpha}^{n^{\prime}}\bar{b}^{m^{\prime}\beta}- \frac{1}{C}\delta_{\alpha}^{\, \beta}\bar{a}^{n^{\prime}}\cdot \bar{b}^{m^{\prime}}\right) | 0 > , \quad \bar{a}_{\alpha_{1}}^{n_{1}^{\prime}}\cdots \bar{a}_{\alpha_{k}}^{n_{k}^{\prime}}| 0 > , \quad e t c .
-iq_{0}^{-1}\sin\psi\cot\eta\cdot\left(\frac{F[\frac{1}{q_{0}}](i\mu_{0})w_{1}[\frac{1}{q_{0}}](i\mu_{0})w_{3}[\frac{1}{q_{0}}](i\mu_{0})}{w_{2}[\frac{1}{q_{0}}](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}