harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
( - i ) \tilde{S}( p ) ={\frac{p^{2}+ b^{2}- m^{2}+ 2 \left( b \cdotp + m \not \! b \right) \gamma_{5}- [ \not \! b , \not \!{\hat{p}}] \gamma_{5}}{\left( p^{2}+ b^{2}- m^{2}+ i \varepsilon \right)^{2}- 4 \left[ ( b \cdotp )^{2}- b^{2}\left( m^{2}- \hat{p}^{2}\right) \right]}}\left( \not \! p + m + \not \! b \gamma_{5}\right) \ ,
\int_{\xi (\pm \infty)=0}{\mathcal D}\xi \, \exp\{\frac{i}{\hbar}S[\xi] + i \int_{-\infty}^{+\infty}dt \xi^j(t) [k^j \delta(t-1)+l^j \delta(t)]\}= e^{-\frac{i \hbar}{2 b}\epsilon_{ij}k^i l^j}\;.
{\mathfrak S}_\geq^q[[a,b],\mathfrak B\cap[a,b];\ (s_j)_{j=0}^{m}] :=\left\{s(z)=\int_{[a,b]}\frac{d\sigma(t)}{t-z},\, \sigma\in{\mathcal M}_\geq^q[[a,b],\mathfrak B\cap[a,b]; (s_j)_{j=0}^{m}] \right\}.
[N^*(\{f=0\})]|_{T^\bullet U}=\int_{t\in\mathbb{R}^*}\delta^{n+1}_0\left(f(y),\frac{t}{\Lambda}d_yf-\eta\right)d_yf\wedge\bigwedge_{i=1}^nd\left(\frac{t}{\Lambda}\partial_{y^i}f-\eta_i\right)
\frac{M_{i}}{k^{2}- M_{i}^{2}+ i 0}\, \bar{D}^{2}\delta^{4}( \theta_{1}- \theta_{2}) \quad \mathrm{a n d}\quad \frac{M_{i}}{k^{2}- M_{i}^{2}+ i 0}\, D^{2}\delta^{4}( \theta_{1}- \theta_{2})
t\,\P(X_{1}>xb(t)) = t \frac 12 x^{{-2}}\frac{2}{t}\left(1+\frac1{\sqrt{2t}}\right)^{-2}\left(1+x^{-1}{\sqrt \frac 2t}\left(1+ \frac{1}{\sqrt{2t}}\right)^{-1}\right) = x^{-2}+ \sqrt{2}x^{-2}(x^{-1}-1) \frac 1{\sqrt t}+ o\left(\frac 1{\sqrt t}\right),
G_\omega(T_i)=L_{r \in T_i}^\omega\left(W_{d_r,T_i \backslash \{r\}}^{N(r,T_i \backslash \{r\})}\right), \mathbf{X}_\omega(S)=\sum_{T \subseteq S, |T|=t+1}{\mathbf{u}_{S}^{T}G_\omega(T)}.
\sum_{i=1}^{K}\frac{1}{{|h_{i}|^2}}\frac{(e^{\frac{r^*}{\alpha_i}}-1)\alpha_i}{(e^{r^*}-1)}\geq \sum_{i=1}^{K}\frac{1}{{|h_{i}|^2}}e^{(K-i)r^*}.
(1) \begin{cases}x_{t}\leq x_{1}, t=1, x_{t}=x_{t},t=2,\ldots,n, y_{t}=y_{t},t=1,\ldots,n-1, y_{t}\leq y_{n}-2,t=n. \end{cases}(2) \begin{cases}x_{t}\leq x_{2}, t=1, x_{t}=x_{t+1},t=2,\ldots,n-1, x_{t}=1,t=n, y_{t}=y_{t+1},t=1,\ldots,n-2, y_{t}=y_{n}-2,t=n-1, y_{t}\leq 1,t=n. \end{cases}
\sup_{f\in L^1(\Gamma_-,d\xi)\atop\|f\|_{L^1(\Gamma_-,d\xi)}=1}\Big|\int_{\Gamma_+}\phi(x,v)j_+K^{n+2}(I-K)^{-1}Jf(x,v)d\xi(x,v)\Big|\le C_{n+2}e^{(n+2)\|\tau\sigma_-\|_\infty}\|k\|_{\infty}^{n+2}\eta^{2n-2},
[\begin{matrix}\langle\hat{d}|\hat{a}\rangle&\langle\hat{e}|\hat{a}\rangle&\langle\hat{f}|\hat{a}\rangle\\ \langle\hat{d}|\hat{b}\rangle&\langle\hat{e}|\hat{b}\rangle&\langle\hat{f}|\hat{b}\rangle\\ \langle\hat{d}|\hat{c}\rangle&\langle\hat{e}|\hat{c}\rangle&\langle\hat{f}|\hat{c}\rangle\end{matrix}]
= \int_{0}^{\infty}d \omega \int d^{2}k \left[ n_{\omega}~ \phi_{\omega , k}^{*}( x ) \phi_{\omega , k}( x^{\prime}) + ( n_{\omega}+ 1 ) \phi_{\omega , k}( x ) \phi_{\omega , k}^{*}( x^{\prime}) \right] .
{}_2F_1(c,d,e; u) = \frac{\Gamma(e)}{\Gamma(c-d)\Gamma(d)}\int_0^1 (1-uz)^{-c}z^{d-1}(1-z)^{e-d-1}dz, e > d > 0, |u| < 1, c \in \mathbb{R},
\left[ \theta_{3 , 3 0}( q ) + \theta_{2 7 , 3 0}( q ) \right]{}~ \left[ \theta_{7 , 3 0}( q ) + \theta_{2 3 , 3 0}( q ) \right]{}~ \left[ \theta_{1 1 , 3 0}( q ) + \theta_{1 9 , 3 0}( q ) \right]{}~ \left[ \theta_{1 3 , 3 0}( q ) + \theta_{1 7 , 3 0}( q ) \right] ,
\sum_{j=1}^{M_v}\alpha_{Hj}^n(\Psi_j, \Psi_k)+\Delta t\sum_{j=1}^{M_v}\alpha_{Hj}^n a(\Psi_j, \Psi_k)=\sum_{j=1}^{M_v}\alpha_{Hj}^{n-1}(\Psi_j, \Psi_k)+\Delta t(f^n, \Psi_k).
\{\{. . . \{Q A_{0}( x ) \xi_{0}, \underbrace{S \}, S \}, . . . \}, S \}}_{n}= \left\{\begin{array}{l r}{{Q ( 2 i )^{n}\Phi_{n}( p , x , \vec{\xi}) \xi_{5}\xi_{0}}}&{{n - o d d}}\\{{}}&{{}}\\{{Q ( 2 i )^{n}\Phi_{n}( p , x , \vec{\xi}) \xi_{0}}}&{{n - e v e n}}\\ \end{array}\right. +{\cal O}( Q^{2}) ,
{\ddot{a}_0\over a_0}=-{{16\pi^2G^2_5\sigma}\over{9c^4\sqrt{I}}}\left(\varrho+3I{\wp\over c^2}\right)-{{32\pi^2G^2_5\sigma}\over{9c^6}}\left({\dot{\Phi}^2\over c^2}-V\right)+{{4\pi G_5}\over{3c^2}}\left(\Lambda+{{4\pi G_5}\over{3c^4}}\sigma^2\right)-{{{\cal C}c^2}\over a^4_0}.
\hat{A}= \hat{\Sigma}_X^{-{1\over 2}}\hat U_{[:, 1:r]},\hat A = [\hat{\alpha}^{(1)}, \hat{\alpha}^{(2)},\cdots, \hat{\alpha}^{(r)}], & \hat{B}= \hat{\Sigma}_Y^{-{1\over 2}}\hat V_{[:, 1:r]}, \hat B = [\hat{\beta}^{(1)}, \hat{\beta}^{(2)},\cdots, \hat{\beta}^{(r)}].
|b-x_{k_0}| \leq |b-x|+|x-x_{k_0}| \leq \frac{m(J)}{2}+ \frac{m(J_{k_0})}{2}< m(J_{k_0})+\frac{m(J_{k_0})}{2}= \frac{3}{2}m(J_{k_0}) < 2m(J_{k_0})
[A,B]_v = [vA,B]+[A,vB]-v[A,B] = \frac{1}{2}([A-\Gamma A,B]+[A,B-\Gamma B]-[A,B]+\Gamma [A,B]) = \frac{1}{2}([A,B]-[\Gamma A,B]-[A,\Gamma B]+\Gamma [A,B]) = \frac{1}{2}([A,B]-[A,B]_\Gamma ).
\int_{\underline C_k}(d\psi) a^t_k = \sum_\Gamma \Gamma \int_{\underline C_k}(d\psi)\! \left(\int_{\underline C_{n+k}/\underline C_k}\widetilde \theta^t_\Gamma \right)= \sum_\Gamma \Gamma \int_{\underline C_{n+k}}(d\psi) \widetilde \theta^t_\Gamma
\mu_p (\mathcal{T}_n(\ell)) =Z_n(p,\ell)\to \frac{C_p}{[p,1]}Y(p,\ell)=C_p \sum_{i=0}^{\ell}\frac{p^{-i^2}}{[p,i]^2}=: Z(p,\ell) \ \{\mathrm{as}}\ \n\to \infty\,,
A-2\sum_{j=1}^{j^*}{w}_{[j]}\leq 0< A-2\sum_{j=1}^{j^*-1}{w}_{[j]}, A-2\sum_{j=1}^{j_*+1}{w}_{[j]}<0\leq A-2\sum_{j=1}^{j_*}{w}_{[j]}.
D\frac{{\partial{C_{\rho\varphi}}(\rho,\varphi ,t|{\rho_{\rm tx}},\varphi_{\rm tx},{t_0})}}{{\partial \rho}}\mid_{\rho = \rho_c}= -k_{f}{C_{\rho\varphi}}(\rho_c,\varphi ,t|{\rho_{\rm tx}},\varphi_{\rm tx},{t_0}).
( \nabla_{a}E_{\alpha q}^{\underline{\alpha}}) E_{\underline{\alpha}, \beta r}={\frac{1}{4}}\hat{\Omega}_{a , b}^{i}m_{b}^{c}\gamma_{c \alpha \beta}( \gamma_{i})_{q r}+{\cal D}_{a}h_{\alpha \beta}C_{q r}.
\langle\sigma_{1}\rangle=\frac{trSA^{4}}{trA^{4}}=\frac{tr\alpha_{m}^{4}PSA^{4}P^{T}}{tr\alpha_{m}^{4}PA^{4}P^{T}}=\frac{trSA_{d}^{4}}{trA_{d}^{4}}.
z=-U_{i}(\sigma)/a 0<a\leq \lambda_{i}-c_{1}(L_{i})\cdot\mathfrak D, \langle a\rangle=\langle \lambda_{i}-c_{1}(L_{i})\cdot\mathfrak D\rangle=\langle \lambda_{i}\rangle=\hat{b}_{i},i\in \sigma,
\mathbb{E}_{wo}[\|x_k - x_* \|^2 ] \leq \|\mathcal{S}_d(\mathcal{A})\|\cdot \|x_0 - x_*\|^2 +\rho^2 \gamma^2 \Bigg(\underbrace{\sum_{l=1}^{k-1}\mathbb{E}_{wo}\Big [ \| A_k^*\cdots A_{l+1}^*A_{l+1}\cdots A_k \| \Big ]}_{(I_l)}+ 1\Bigg) \cdot \sup_{j \in [n]}\|a_j\|^2
H = \frac{1}{l_{4}}\frac{\sqrt{1 - 4 \omega_{4}M / l_{4}^{2}}\operatorname{s i n h}( 2 \tau / l_{4})}{1 - \sqrt{1 - 4 \omega_{4}M / l_{4}^{2}}\operatorname{c o s h}( 2 \tau / l_{4})},
\varepsilon^{a d}( P_{\pm})_{A c}^{B b}+ \varepsilon^{b d}( P_{\pm})_{A c}^{B a}- ( \sigma^{\alpha})^{a b}( \sigma_{\alpha})_{~ c}^{e}( P_{\mp})_{A e}^{B d}= - ( \sigma^{\alpha})^{a b}\bigr ( ( \sigma_{\alpha})_{~ c}^{d}\delta_{A}^{B}+ \delta_{c}^{d}( \sigma_{\alpha})_{~ A}^{B}\bigr )
(P_h\delta_\tau v^{m+1/2}, v^{m+1/2}) \le (S^m,v_i^{m+1/2}) = (S^m, v^{m+1/2}) & \le \|S^m\| \|v^{m+1/2}\| \le \sqrt{5}\|S^m\| \|v^{m+1/2}\|_P & \le \frac{\sqrt{5}}{2}\|S^m\| \left(\|v^{m+1}\|_P +\|v^{m}\|_P\right).
\Gamma_{+}^{2}= \frac{1}{( 2 \pi )^{2}}\int d t d^{2}{\bf x}\int d^{2}{\bf p}\left[ \frac{- e^{2}}{2}\pi \delta ( p_{1}, p_{2}) \epsilon_{i j}a_{j}\dot{a}_{i}\right] e^{i{\bf p}\cdot{\bf x}},
\prod_{1\leq i\leq l,~ 1\leq j \leq k'}L(s+\frac{1}{2}, \zeta_i\times \xi'_j)=\frac{\prod_{\substack{1\leq i\leq l 1\leq j \leq s}}L(s+\frac{1}{2}, \zeta_i\times \eta_j)}{\prod_{\substack{1\leq i \leq l 1\leq j\leq k}}L(s+\frac{1}{2},\zeta_i\times\xi_j)}
h_{\mathrm{Fal}}(A_n) = -\frac{L'(\chi_{-d},0)}{L(\chi_{-d},0)}- \frac{1}{2}\log d + \frac{1}{2}\log 2\pi &\quad\,+\frac{1}{2}\left(\sum_{p|n}\log p\left(r_p - \left(\frac{1-\left(\frac{p}{d}\right)}{p-\left(\frac{p}{d}\right)}\right)\left(\frac{1-p^{-r_p}}{1-p^{-1}}\right)\right)\right)
\left. \prod_{p \in Y_{j}}\, \frac{e \beta}{1- \frac{e \beta}{2}}\leq e^{- N_{p}(Y_{j}) \, \log (\frac{1}{2 e \beta})}\ \ \ \ \ \ \ \ \ \ \mbox{for}\ \ \ \beta < \frac{1}{e}\ , \right.
I_{\mu}^{\ \nu}\equiv \left[ \delta_\mu^\nu + \phi^{-1}\phi_{\mu\alpha}h^{\alpha\nu}\right] \;,\quad I^{-1}\_\mu^{\ \alpha}I_\alpha^{\ \nu}=\delta_\mu^\nu \;,\quad I\equiv detI_\mu^{\ \nu}\; .
\bar{Z}_{\gamma \; \; \; ( \tau )}^{\; ( \rho )}= \stackrel{( 0 )}{Z}_{\gamma \; \; \; ( \tau )}^{\; ( \rho )}+ g \stackrel{( 1 )}{Z}_{\gamma \; \; \; ( \tau )}^{\; ( \rho )}+ O \left( g^{2}\right) ,
\frac{\partial}{\partial x}\frac{\frac{\partial u}{\partial x}}{\sqrt{1+(\frac{\partial u}{\partial x})^{2}+(\frac{\partial u}{\partial y})^{2}}}+\frac{\partial}{\partial y}\frac{\frac{\partial u}{\partial y}}{\sqrt{1+(\frac{\partial u}{\partial x})^{2}+(\frac{\partial u}{\partial y})^{2}}}=0,
\int\limits_{A}^{t}\int\limits_{\mathbb{R}^{3}_{+}}\frac{\partial}{\partial y_{l}}G_{ij}^{2}(x,y,t-\tau)\Big[\frac{\partial}{\partial t}F^{(k)}_{jl}(y,\tau)+\delta_{jl}\frac{\partial}{\partial t}p^{1(k)}(y,\tau)\Big] dyd\tau =
\widetilde W_{\beta}^{\mathrm{c l}}[ \Phi , \Pi ] = \mathrm{e x p}\left[ -{\frac{1}{2}}\beta \int{\frac{d p}{2 \pi}}\; \Bigl ( \Pi^{*}( p ) \Pi ( p ) + E^{2}( p ) \Phi^{*}( p ) \Phi ( p ) \Bigr ) \right] \; .
k_j= \begin{cases}\sigma(j) (1\le j\le i+1) \min \{k_{2i+2-j}, \cdots , k_{i+1}\}(i+2\le j \le 2i) \min \{k_{j-2i},\cdots , k_{i+1}\}(2i+1\le j\le 3i). \end{cases}
( A N x^{2}\phi^{\prime})^{\prime}= 2 \gamma A e^{2 \gamma \phi}\left[ N ( K^{2}+ H^{2}) + \frac{1}{8 x^{2}}\left( \left( K^{2}+ H^{2}- 4 \right)^{2}+ 1 2 K^{2}H^{2}\right) \right] \ .
{f'}_{\lambda/\mu}=\sum\limits_{\lambda^-:\substack{\lambda\supseteq\lambda^-\supseteq \mu \ell(\lambda^-)<\ell(\lambda)}}{f'}_{\lambda^-/\mu}+2\sum\limits_{\lambda^-:\substack{\lambda\supseteq\lambda^-\supseteq \mu \ell(\lambda^-)=\ell(\lambda)}}{f'}_{\lambda^-/\mu}.
\langle\partial_{t}\rho_{n}(t), f \rangle_{*}= \langle - \partial_{x}(\rho_{n}(t) u_{n}(t)), f \rangle_{*}= (\rho_{n}(t) u_{n}(t), \partial_{x}f)_{L^{2}_{x}}begin{align*}1ex] \le \|\rho_{n}(t)\|_{L^{2}_{x}}\|u_{n}(t)\|_{L^{2}_{x}}\|f\|_{W^{1, \infty}_{0}}.
\begin{cases*}\partial_{\alpha}(a^{\alpha \beta}\partial_{\beta}\Psi^{(i)})=F(\Psi^{(i-1)},\partial \Psi^{(i-1)}) (\Psi^{(i)},\partial_t \Psi^{(i)})|_{\{t=0\}}=(\Psi_0, \Psi_1) \in \mathcal{H}. \end{cases*}
\biggl[\, G_{ijkl}(x) \frac{\delta}{\delta h_{kl}(x)}\frac{\delta}{\delta h_{ij}(x)}- \frac{1}{(16 \pi G)^2}\,\sqrt{h(x)}\Bigl(\!~ R(x) + 2 \Lambda \Bigr) \biggr]\, \Psi[h] = 0\ ,
Z_{m,n}(a):= Z_{m,n}^{\tilde{\omega}}=\sum_{x_{\cdot}\in \Pi_{m,n}}\prod_{i=1}^{m+n}\tilde{\omega}_{(x_{i-1},x_i)}=\sum_{x_{\cdot}\in \Pi_{m,n}}\prod_{i=1}^{t_1(x_{\cdot})\wedge r}\tilde{R}^1_{i,0}\prod_{i=t_1(x_{\cdot})\wedge r+1}^{m+n}\omega_{(x_{i-1},x_i)},
[ P_0,P_1]=x^{r+s-1}(sg'h-rh'g) &+x^{r+s-2}\left(\binom{s}{2}g''h-\binom{r}{2}h''g+(s-1)g'h_1-rh_1'g+sg_1'h-(r-1)h'g_1\right)+\dots
\bigg[ E \big[\boldsymbol{c}_{k+M}\big] \bigg]_i = \left( 1 - \mu \lambda_i\right)^M \bigg[ E \big[ \boldsymbol{c}_k \big] \bigg]_i &\;\;\;\; - \sum_{m=0}^{M-1}{\left( 1-\mu\lambda_i \right)^m\bigg[ \boldsymbol{w}^{\prime}_{k+M-m-1}\bigg]_i}
B_2 = \sum_{\substack{u \le n, v\le m (u,n), (v,m) \le T u/n \ne v/m}}\mu \Bigl( \Bigl(\frac{u - s/N}n, \frac{u+s/N}n \Bigr) \cap \Bigl(\frac{v - s/N}m, \frac{v+s/N}m \Bigr) \Bigr).
H_0 =1, H_{4k+1}= (-1)^{k}\frac{(2k)!^2}{2^{4k(2k-1)}}\prod_{j=1}^{2k-1}(2j+1)!^4, H_{4k+2}=0, H_{4k+3}= (-1)^{k+1}\frac{(2k+1)!^2}{2^{4k(2k+1)}}\prod_{j=1}^{2k}(2j+1)!^4, H_{4k+4}= (-1)^{k+1}\frac{(2k+1)!^2(4k+3)!^2}{2^{2(2k+1)^2}}\prod_{j=1}^{2k}(2j+1)!^4.
\begin{gathered}G(\rho)= f_3 (\rho) - \int_0^\infty u^{2\alpha}C_2(u)\dfrac{1}{1+w(u)}J_\nu (u\rho; q^2) \,d_q u + \int_0^\infty u^{2\alpha}C_1(u)\dfrac{w(u)}{1+w(u)}J_\nu (u\rho; q^2) \,d_q u. \end{gathered}
\eta_1(m):=-\frac{A}{m}\frac{E(m)}{K(m)},\eta_2(m):=\frac{A}{m}\Big(1-m-\frac{E(m)}{K(m)}\Big),\quad\eta_3(m):=\frac{A}{m}\Big(1-\frac{E(m)}{K(m)}\Big).
dl^2=\frac{1}{2f^2}\left|dE+\frac{2{\rm Im}\Phi}{{\rm Im}z}d\Phi-\left(\frac{{\rm Im}\Phi}{{\rm Im}z}\right)^2dz\right|^2-\frac{1}{f{\rm Im}z}\left|d\Phi-\frac{{\rm Im}\Phi}{{\rm Im}z}dz \right|^2+\frac{|dz|^2}{2({\rm Im}z)^2}.
M=\begin{pmatrix}1 0 0 0 0 0 0 0 0 \end{pmatrix}, S^1=\begin{pmatrix}0 0 0 0 s_{22}0 0 0 s_{33}\end{pmatrix}, S^2=\begin{pmatrix}0 0 0 0 s_{22}\frac{1}{2}|s_{22}| 0 \frac{1}{2}|s_{22}| 0 \end{pmatrix},s_{22}, s_{33}\neq0,
\int_0^t \phi_\alpha^2(t-s)ds=\begin{cases}\frac{1-e^{-2\lambda_\alpha t}}{2\lambda_\alpha},{\rm for}\ , \frac{\sqrt{\lambda_\alpha}t-\sin(2\sqrt{\lambda_\alpha}t)}{2\lambda_\alpha^{3/2}},{\rm for}\ .\end{cases}
(|D|^\alpha f)(x) = \frac{2^{\alpha}\Gamma \left(\frac{1}{2}(d+\ell-\sigma)\right) \Gamma\left(\frac{1}{2}(\ell+\sigma+\alpha )\right)}{\Gamma \left(\frac{\ell+\sigma}{2}\right) \Gamma \left(\frac{1}{2}(d+\ell-\sigma-\alpha )\right)}|x|^{-\alpha}\cdot |x|^{-\sigma}Y_{\ell,m}(\omega_x)
\sum_{F_1, F_2\colon e(v)\in E(F_1)\cap E(F_2)}q^{2i-(|E(F_1)\cap E(F_2)|-1)}=O\left(\sum_{J\colon e(v)\in E(J)}n^{2v(F)-2v(J)}q^{2i-(e(J)-1)}\right),
\psi^{}\left( k_{1},k_{2},t\right) =\left( N\right)^{-1/2}\left[ \psi_{1}\left( k_{1}\right) \psi_{2}\left( k_{2}\right)e^{-i\hbar\left( k_{1}^{2}+k_{2}^{2}\right) t/\left( 2m\right)}+\varepsilon\psi_{\mathrm{scat}}\left( k_{1},k_{2},t\right) \right] ,
G^{+}_{R}(x,x')=\langle 0_{R}\vert \varphi (x)\varphi (x')\vert 0_{R}\rangle= \frac{1}{\pi^2}\int \frac{d{\mathbf k}}{(2\pi )^{D-1}}\,e^{i{\mathbf k}({\mathbf x}-{\mathbf x'})}\int_{0}^{\infty}d\omega \sinh (\pi\omega ) e^{-i\omega (\tau -\tau ')}K_{i\omega}(\lambda \xi )K_{i\omega}(\lambda \xi ') .
\hat{c}_{4y}(i_1,i_2,i_3) = \ \hat{m}_{4y}(i_1,i_2,i_3) - \hat{m}_{2y}(i_1)\hat{m}_{2y}(i_2 - i_3) - \hat{m}_{2y}(i_2)\hat{m}_{2y}(i_3 - i_1) - \hat{m}_{2y}(i_3)\hat{m}_{2y}(i_1 - i_2)
\frac 1{2 \pi i}\int_{c - i \infty}^{c + i \infty}d s \frac{( - s )}{\mu}\left( \frac{t}{\mu}\right)^{- s - 1}\frac{2^{s - 1}}{\sqrt{\pi}}\Gamma \left( \frac{s + 1}2 \right) \left[ \mu^{- s}\Gamma \left( \frac s 2 \right) \zeta \left( \frac s 2 , \frac{D_{B}}{\mu^{2}}\right) \right] \, ,
M_{\partial D_j}=\frac{1}{2\pi}\int_{\partial D_j}|f_j|^2\bigg(\sum_{1\le k\le m_j}2\frac{\partial G_{D_j}(w_j,z_{j,k})}{\partial v_{w_j}}\bigg)^{-1}e^{-\varphi_j}|dw_j|.
(\chi + t'_j \chi + \tilde \chi + \sqrt{-1}\partial\bar\partial \varphi_{t'_j})^n = c (\chi + t'_j \chi + \tilde \chi + \sqrt{-1}\partial\bar\partial \varphi_{t'_j})^m \wedge \omega^{n - m}+ b_{t'_j}f_2 \omega^n .
\int_{\Omega}\sum_{i=1}^m \log c_{ih}^n (c^{n+1}_{ih}-c^{n}_{ih}) dx = - \Delta t \sum_{i=1}^m A_{c_{ih}^n}(p_{ih}^n- p_{ih}^n) -A_1(\psi_{h}^{n+1}-\psi_h^n, \psi_h^n).
\eta(\phi)_{uv}= \sum_{n=1}^p \frac{\nabla^n\eta (\phi_u)}{n!}(\phi_{uv})^{\otimes n}+ R^{\eta}_p(\phi_u,\phi_v) = \nabla\eta(\phi_u) R^{\phi}_{uv}+ \sum_{n=1}^p \nabla^n \eta(\phi_u) X_{uv}^{(n)}+ \Xi_{uv}
D_{\alpha}= \frac{\partial}{\partial \theta^{\alpha}}+ i \sigma_{\alpha{\dot{\alpha}}}^{\mu}{\overline{\theta}}^{\dot{\alpha}}\partial_{\mu},{\overline{D}}_{\dot{\alpha}}= - \frac{\partial}{\partial{\overline{\theta}}^{\dot{\alpha}}}- i \theta^{\alpha}\sigma_{\alpha{\dot{\alpha}}}^{\mu}\partial_{\mu}
c^{*}_e(A,q,\mu)\geq c^{*}_e(A,q,\overline{\mu})=_min_{e\cdot \xi>0}\frac{2}{T}\sqrt{\int_{0}^{T}\xi A \xi\int_{0}^{T}\mu}-\frac{1}{T}\int_{0}^{T}q\cdot \xi.
\mathcal{D}= \frac{A(p, 1)}{2 p^{\frac{1}{2}}}\sum_{m \geq 1}\frac{A(1, m)}{m}H_{m, p}- \frac{1}{2 p^{\frac{3}{2}}}\sum_{m \geq 1}\frac{A(1, m)}{m}H_{m p, p}.
f ( q , \mu ) = \sum_{\rho \neq \sigma \in \Phi_{+}, \, R_{\psi}= s_{\rho}s_{\sigma}}g_{| \rho |}\, g_{| \sigma |}\,{\frac{( \sigma^{\vee}\! \! \cdot \rho )}{| \sigma^{\vee}\! \! \cdot \rho |}}\, x_{| \rho |}( \rho \cdot q , ( \rho^{\vee}\! \! \cdot \mu ) \xi ) \, x_{| \sigma |}( \sigma \cdot q , ( s_{\rho}( \sigma )^{\vee}\! \! \cdot \mu ) \xi )
(\sigma_p^t)_{\perp}^2 = (\sigma_p^t)^2-(\hat{\gamma}_p^t)^*\tilde{E}_p^t=(\sigma_p^t)^2-(\tilde{E}_p^t)^*(\tilde{C}_p^t)^{-1}\tilde{E}_p^t, (\tau_p^t)_{\perp}^2 = (\tau_p^t)^2-(\hat{\alpha}_p^t)^*\breve{E}_p^t=(\sigma_p^t)^2-(\breve{E}_p^t)^*(\breve{C}_p^t)^{-1}\breve{E}_p^t.
\left[ \frac{\sqrt{3}}{2}( U_{0 \, \, 0}+ 1 ) + \frac{2 i}{\pi}\sum_{n = 0}^{\infty}\frac{( - )^{k}}{( 2 n + 1 )^{3 / 2}}U_{0 \, \, 2 n + 1}\right] \, .
\partial_x j(x;z)\partial_x \Phi_1(x,\bar{x};z,\bar{z}) = :\partial_x j(x;z)\partial_x \Phi_1(x,\bar{x};z,\bar{z}): +\frac{2}{k+2}\partial_x^2:(j(x;z)\Phi_1(x,\bar{x};z,\bar{z})):,
\mathbb{I}^\omega(k\ell)\mathbb{I}^\omega(k'\ell') = F_k(\mathbb{I}^\omega(\ell))F_{k'}(\mathbb{I}^\omega(\ell')) &= \left(\sum_{b=0}^kd_{k,b}\mathbb{I}^\omega(\ell)^b\right) \left(\sum_{b'=0}^{k'}d_{k',b'}\mathbb{I}^\omega(\ell')^{b'}\right) &= \sum_{b=0}^k\sum_{b'=0}^{k'}d_{k,b}d_{k',b'}\mathbb{I}^\omega(\ell)^{b}\mathbb{I}^\omega(\ell')^{b'}.
\Phi(x_{1}, \ldots, x_{n+1}) =\sum_{i=0}^{n}\dfrac{1}{\binom{n+1}{i}}\sum_{\mathrm{card}(I)=i}\left(\prod_{j\in I}x_{j}\right) \cdot f_{n+1-i}\left(\prod_{k\in \left\{1, \ldots, n+1\right\}\setminus I}x_{k}\right) \left(x_{1}, \ldots, x_{n+1}\in K\right).
\tau(q_1^{(n)}\oplus q^{(n)}_2\oplus\dots\oplus q^{(n)}_n) =\frac{(\eta_n)}{(\xi_n)}=\frac{\sum_{l=1}^n \kappa(k,l)}{\sum_{l=0}^n\sigma(l)}&=\frac{\sum_{l=1}^n \kappa(k,l)}{(n+1)!}.
[\kappa_1,\kappa_2]=\kappa_3, [\kappa_2,\kappa_3]=a_2\kappa_2^2+a_1\{\kappa_1,\kappa_2\}+c_1 \kappa_1+d \kappa_2 +e_1, [\kappa_3,\kappa_1]=a_1\kappa_1^2+a_2\{\kappa_1,\kappa_2\}+c_2 \kappa_2+d \kappa_1+e_2,
0=\langle J^{'}(\widehat{w}),\widehat{w}\rangle+\lambda_+\langle \varphi^{*}_{\widehat{w}^+},\widehat{w}\rangle+\lambda_-\langle \varphi^{*}_{\widehat{w}^-},\widehat{w}\rangle =\lambda_+\langle \varphi^{*}_{\widehat{w}^+},\widehat{w}\rangle+\lambda_-\langle \varphi^{*}_{\widehat{w}^-},\widehat{w}\rangle,
\widetilde \Phi_{M_n,n}( f)(x,t)= \dfrac{1}{n}\sum_{k=1}^n f(x_k,t)+\sum_{p=1}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}\left( \dfrac{\pi}{n}\sum_{k=1}^n f(x_k,t) \phi_p(x_k) \right) ds\Bigg] \phi_p(x).
A_{i}^{\mathrm{v p}}( x_{j}) = - \frac{\Phi}{2 \pi}\epsilon^{i j}\frac{x^{j}}{\rho^{2}}\Biggl [ \frac{\alpha}{4 \sqrt{\pi}}( m \rho )^{5}\, G_{1 3}^{3 0}\Biggl ( ( m \rho )^{2}\Bigg | \begin{array}{r r r}{{}}&{{0}}&{{}}\\{{- \frac{1}{2}}}&{{- \frac{1}{2}}}&{{- \frac{5}{2}}}\\ \end{array}\Biggr ) \Biggr ] .
S_{\Gamma}=-\frac{1}{\kappa}\int \sqrt{-g}e^{\alpha\phi/M_{p}}(\zeta +b_{g})g^{\mu\nu}R_{\mu\nu}(\Gamma) =-\frac{1}{\kappa}\int \sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}R_{\mu\nu}(\Gamma),
\lambda_1 L_{1}(s,z)+\lambda_2L_{1}(t,z) =\lambda_1 L_{1}(s,t)+(\lambda_1+\lambda_2)L_{1}(t,z) \lambda_1 L_{-1}(s,z)+\lambda_2L_{-1}(t,z) =\lambda_1 L_{-1}(s,t)+(\lambda_1+\lambda_2)L_{-1}(t,z) \mu_1 L_{0}(s,z)+\mu_2L_{0}(t,z) =\mu_1 L_{0}(s,t)+(\mu_1+\mu_2)L_{0}(t,z).
\sum_{v\in V^+}\langle \pi(y) p_v , p_v \rangle = \sum_{v\in V^+}\tau_V\circ E(y p_v p_v^*) = \sum_{v\in V^+}\tau_V\circ E(yp_v) & = \tau_V\circ E(y \sum_{v\in V^+}p_v) = \tau_V\circ E(y).
f_n(\tilde{\omega})=\frac{E[\prod_{i=0}^{\tau}F(S_i,\widehat{{S}}_i;S_{i+1},\widehat{S}_{i+1})]E[\prod_{i=\tau+1}^{n-1}F(S_i,\widehat{{S}}_i;S_{i+1},\widehat{S}_{i+1})]}{\{E\prod_{i=0}^{\tau}[\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda\rho(S_i)\rho(S_{i+1})}]\}^2\{E\prod_{i=\tau+1}^{n-1}[\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda\rho(S_i)\rho(S_{i+1})}]\}^2}.
\left\lvert \sum_{n=1}^{N}e^{2\pi ih f(n+c_0-1)/q}\right\rvert \leq \left(\frac{h}{q}\left\lvert f'(N+c_0-1)-f'(c_0)\right\rvert+2\right)\left(\frac{4}{\sqrt{h\rho_N/q}}+3\right)\,.
\frac{e^{-\tau t}- e^{-\kappa t}}{\kappa - \tau}\leq \begin{cases}\frac{e^{-\tau t}}{\kappa - \tau}, \kappa > \tau t e^{-\kappa t}, \kappa = \tau \frac{e^{-\kappa t}}{\tau - \kappa}, \kappa < \tau . \end{cases}
\pi_{\dot{q}}^{1 -}= e^{+ 2}\psi_{+ 2 \dot{q}}^{1 -}+ e^{- 2}\psi_{- 2 \dot{q}}^{1 -}+ e_{p}^{+}h_{+ p \dot{q}}^{1 -}+ e_{\dot{p}}^{-}\chi_{\dot{p}\dot{q}}^{1},
\|\nabla f(\hat{z}^{n+1})-\nabla f(\hat{z}^{n})\|_2 &= \|[\nabla F(P_I^T\hat{z}^{n+1})]_I - [\nabla F(P_I^T\hat{z}^{n})]_I\|_2 &\leq \|\nabla F(P_I^T\hat{z}^{n+1}) - \nabla F(P_I^T\hat{z}^{n})\|_2 &\leq L\|P_I^T\hat{z}^{n+1}- P_I^T\hat{z}^{n}\|_2= L \|\hat{z}^{n+1}- \hat{z}^{n}\|_2.
\varphi_{i}^{T}:\real\to[0,1]\quad\mbox{for}\lceil t_0\rceil+1 \le i\le \lfloor T\rfloor\intertext{and}\psi_{i}^{T}:\real\to[0,1]\quad\mbox{for}0\le i\le k(T):=\lceil \mu T/\log2\rceil
\begin{array}{rclcrcl}\psi^{\prime}_{-}= e^{i\theta (x)}\psi_{-}\quad \quad \psi^{\prime}_{+}= \psi_{+}\psi^{\prime \dag}_{-}= \psi^{\dag}_{-}e^{- i \theta (x)}\quad\quad \psi^{\prime \dag}_{+}= \psi^{\dag}_{+}\lambda_{-}= \lambda_{-}(\theta) \quad, \end{array}
(g_{31}y_1+g_{32}y_2+g_{33}y_3)\{g_{31}y_1 +g_{32}y_2+(g_{13}+g_{23}+g_{33})y_3)\}^{b-1}=g_{32}g_{23}^{b-1}y_2(y_1+y_2)^{b-1}+\alpha y_3(y_1+y_2+y_3)^{a-1},
+\sin\psi\csc\eta\cdot\left(\frac{F[p,q](i\mu_{0}+1)w_{1}[p,q](i\mu_{0}+1)w_{3}[p,q](i\mu_{0}+1)}{w_{2}[p,q](i\mu_{0}+1)}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\phi}u_{n}(\mu)\mid_{\mu=\mu_{0}-i}
\pi_{I}^{1 \pm}\times \pi_{I}^{2 \pm}, \quad \pi_{I}^{1 \pm}\times \pi_{I I I}^{2 \pm}, \quad \pi_{I I I}^{1 \pm}\times \pi_{I}^{2 \pm}, \quad \pi_{I I I}^{1 \pm}\times \pi_{I I I}^{2 \pm},
\phi(\lambda) =\int_0^M e^{-y}\mu(\lambda^{-1}y,\infty)\,dy+\int_M^\infty e^{-y}\mu(\lambda^{-1}y,\infty)\,dy &\leq 2e\int_0^M\phi(\lambda y^{-1})\,dy+\mu(M\lambda^{-1},\infty)\int_M^\infty e^{-y}\,dy &\leq 2e C_U\phi(\lambda)\int_0^M y^{-\delta}\,dy+2eH(M^{-1}\lambda)e^{-M}&\leq 2eC_U(1-\delta)^{-1}M^{1-\delta}\phi(\lambda)+2eM^{-2}H(\lambda),
X^{0}= t ( \tau , \sigma ) , \hspace{0 . 5 c m}X^{1}= r ( \tau , \sigma ) , \hspace{0 . 5 c m}X^{2}= \theta ( \tau , \sigma ) , \hspace{0 . 5 c m}X^{3}= \phi ( \tau , \sigma ) .
\int_{t_1}^{t_2}\psi(t) dt = \int_{t_1}^{t_2}\Big( \frac{1}{\psi^{-1}_0 - \frac{2}{n}t}\Big) dt \displaystyle = - \frac{n}{2}\log ( \psi^{-1}_0 - \frac{2}{n}t) \Big|_{t_1}^{t_2}= \log \Bigg( \frac{\psi^{-1}_0 - \frac{2}{n}t_1}{\psi^{-1}_0 - \frac{2}{n}t_2}\Bigg)^{\frac{n}{2}}.
\nabla \sigma_\varepsilon(x,t) = (-1)^{i+1}\varepsilon\bar{g}(x,t)\{\bar{\nu}(x,t)-\varepsilon\bar{\tau}_\varepsilon^i(x,t)\}, x\in\Gamma_\varepsilon^i(t), \, t\in[0,T], \, i=0,1,
\begin{cases}\partial_{t}^2\alpha-\partial_{x}^2\alpha= 0 (\alpha, \partial_t \alpha)|_{t=0}= (1+\tilde \alpha_0, \alpha_1) \mbox{with}(\tilde \alpha_0, \alpha_1)\in C_c^{\infty}(\mathbb{R}) \times \mathcal{S}(\mathbb{R}). \end{cases}
\begin{cases}G_2 := |\partial_x|T_{|\partial_x|\phi}h-|\partial_x|(h|\partial_x|\phi )+\partial_xT_{\partial_x\phi}h-\partial_x(h\partial_x\phi), & H_2 := T_{|\partial_x|\omega}P_{\leq 0}|\partial_x|\omega+R(|\partial_x|\omega,|\partial_x|\omega)/2-R(\partial_x\omega,\partial_x\omega)/2,\end{cases}
\vert j(\tau_\sigma) \Delta(\tau_\sigma)\vert\leq \Big(1+240\Big\vert\sum_{n=1}^{+\infty}n^3\frac{q^n}{1-q^n}\Big\vert\Big)^3\leq \Big(1+240\sum_{n=1}^{+\infty}n^3\frac{e^{-\pi \sqrt{3}n}}{1-e^{-\pi\sqrt{3}n}}\Big)^3\leq 9.02.
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int |\nabla^{N+1}n|^2 dx+\!\int |\nabla^{N+2}n|^2 dx &=-\!\int \nabla^{N+1}(u \cdot \nabla n)\nabla^{N+1}n dx+\!\int \nabla^{N+1}(|\nabla n|^2(n+w_0))\nabla^{N+1}n \dx.\end{aligned}
\begin{array}{ll}\mathbb{P}(\mathcal{A}_1)=\mathbb{P}\left(\frac{Z_1}{Z_1+Z_2}>\Gamma_{RRT}^{\alpha}(k_0)\right)=\mathbb{P}\left(\frac{1-\Gamma_{RRT}^{\alpha}(k_0)}{\Gamma_{RRT}^{\alpha}(k_0)}\frac{Z_1}{n\sigma^2}>\frac{Z_2}{n\sigma^2}\right)\end{array}
0 = x_{t-1}^2 = R(x_t)R(x_t) = 2R\big(x_t x_{t-1}- x_t^2\big), R(x_{t-1}+x_tR(1)-x_t) = R(1)R(x_t) = R(1)x_{t-1}= (\lambda_{i_1}+t-2)x_{t-1}+ u,
S [ \phi ] \! \! = \! \! \int_{0}^{\beta}\! \! \! d \tau \int \! \! \! d^{3}x \! \left( \partial_{\nu}\phi_{a}\partial^{\nu}\phi_{a}\! \! + \! m_{o}^{2}\phi^{2}\! \! + \! \frac{\lambda_{o}}{4 N}\phi^{4}\! \! - \! \mu^{2}\phi^{2}\! \! + \! 2 i \mu ( \phi_{2}\partial_{\tau}\phi_{1}\! \! - \! \phi_{1}\partial_{\tau}\phi_{2}) \right)
\stackrel{{\rm HW}}{S^{\alpha\beta}}=\int \tau^{\alpha\beta0}_{\rm KG}\,d^3x\,=\frac{1}{4mi}\int \left[ \left(D^0\overline{\Psi}\right) \sigma^{\alpha\beta}\Psi - \overline{\Psi}\sigma^{\alpha\beta}\left( D^0 \Psi \right) \right] \,d^3x\,.
\begin{array}{ll}{\cal L}_K = \frac{1}{4}F_{\mu \nu}F^{\mu \nu}-(1-g^2 \zeta \eta^2) \tilde G_{\mu}\tilde G^{\mu}+2(m + g h \zeta \eta^2) A_{\mu}\tilde G^{\mu}+ &- 2h\zeta \eta^2 \Sigma \partial_{\mu}A^{\mu}+ h^2 \zeta \eta^2 A_{\mu}A^{\mu}+\zeta \eta^2 \partial_{\mu}\Sigma \partial^{\mu}\Sigma , \end{array}
L_{11}^{(\pm)}=q^{\pm \frac{1}{2}}q^{\pm J}=q^{\pm \left(\begin{array}{cc}1 0 0 0\end{array}\right)},\;\;L_{22}^{(\pm)}=q^{\pm \frac{1}{2}}q^{\mp J}=q^{\pm \left(\begin{array}{cc}0 0 0 1\end{array}\right)},\;\;
e_b \alpha_r = e_r \alpha_b,\e_b\beta_r = e_r\beta_b,\ \alpha_bf_r = \alpha_r f_b,\f_b f_r = f_r f_b,\ \beta_b g_r = \beta_r g_b,\ \beta_b h_r = \beta_r h_b, g_bg_r = g_r g_b,\g_b h_r = h_rg_b,\h_bg_r = g_rh_b, h_b h_r = h_r h_b.