harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g''_2(B_n) = B_n^{-1}e^{-\left(\sqrt{B_n/2}-\sqrt{P/2}\right)^2}\cdot \left(-\frac{1}{4\pi}e^{-\left(\sqrt{B_n/2}-\sqrt{P/2}\right)^2}\right. & \left. + \frac{1}{4\sqrt{2\pi}}(1+{\rm{erf}}(\sqrt{B_n/2}-\sqrt{P/2}))\left( B_n^{1/2}- \sqrt{P}+ B_n^{-1/2}\right)\right).
\Phi(f_i) = \Phi(\Sigma - \zeta_1)\ldots(\Sigma - \zeta_{i-1})f_1 &= (\Sigma^q - \zeta_1)\ldots(\Sigma^q - \zeta_{i-1})\Phi(f_1) &= (\Sigma^q - \zeta_1)\ldots(\Sigma^q - \zeta_{i-1})m,
{\cal H}_{c}= \int d^d x\, \left\{{1\over 2}\sum_{i=1}^{N}\Bigl[ (\partial_\mu \Phi_i)^2 + r \Phi_i^2 \Bigr] +{1\over 4!}\Bigl[ u (\sum_i^N \Phi_i^2)^2 + v \sum_i^N \Phi_i^4 \Bigr]\right \}.
\int\limits_{\psi_0}^{\pi/2}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi &=\frac{(\cos\psi_0)^{n-3}}{|S^{n-3}|}\int\limits_{S^{n-2}}\int\limits_0^\infty \int\limits_{\mathbb{R}^{n-1}}f(\bar{x},z) \delta(\bar{x}\cdot (\cos\psi_0)\omega+z\sin\psi_0)d\bar{x}dz d\omega.
\tilde{\Gamma}_5(p,p')\to \gamma_5\left(1-\frac{\alpha}{2\pi}\right)-\gamma_5 \frac{\alpha}{2\pi}\, P^2\int\limits_{4m^2}^{\infty}\frac{dM^2}{M^2}\, \frac{a(M^2)}{M^2+P^2},
P_{d+1,x}=\{(t_0,\ldots,t_d)\in\mathbb R_{\geqslant 0}^{d+1}\,|\,t_0+\cdots+t_d\leqslant x\}, \Delta_{d,x}=\{(t_0,\ldots,t_d)\in\mathbb R_{\geqslant 0}^{d+1}\,|\, t_0+\cdots+t_d = x \}.
\sum_{c= \{W_1, \ldots, W_k\}\in F}(-1)^{|c|}\prod_{i=1}^k\prod_{a\in A(W_i)}x^{w(a)}= \sum_{c= \{W_1, \ldots, W_k\}\in U(v)}(-1)^{|c|}\prod_{i=1}^k\prod_{a\in A(W_i)}x^{w(a)}
\delta g_{\mu\nu}(x,\tau,\sigma,f) = (F \xi)_{\mu\nu}(x) + 2 [f^{\star}\delta \sigma \bar{g}_{\mu\nu}](x) + [f^{\star}\frac{\partial \bar{g}_{\mu\nu}}{\partial \tau_{i}}\delta \tau_{i}](x) \; .
\kappa^2 r^{2 - \frac{2}{\kappa}}\left[\varphi_{rr}+ \frac{1}{r}\varphi_{r}+ \frac{1}{r^2}\varphi_{\theta \theta}\right] = \psi_{ss}+ \frac{1}{s}\varphi_{s}+ \frac{m^2 \kappa^2}{s^2}\psi_{\sigma \sigma}.
\binom{r-\frac{1}{2}}{r-k}\binom{r-\frac{1}{2}}{k}=\frac{(2r-1)!!}{2^{r}(r-k)!}\frac{2^{k}}{(2k-1)!!}\frac{(2r-1)!!}{2^{r}k!}\frac{2^{r-k}}{(2r-2k-1)!!}&=\frac{1}{2^{r}(r-k)!k!}\frac{(2r-1)!!}{(2k-1)!!}\frac{(2r-1)!!}{(2(r-k)-1)!!}.
\mathcal{L}^su(x) =\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\mathcal{L}_\Omega}u(x)-u(x)\big)\,\frac{dt}{t^{1+s}}&=\operatorname{PV}\int_{\Omega}\big(u(x)-u(y)\big)K_s(x,y)\,dy+u(x)B_s(x),
t_{ij}=\frac{\det \left( r_{u_{i}u_{j}},r_{u_{1}},r_{u_{2}}\right)}{\sqrt{\det \left( g_{ij}\right)}},r_{u_{i}u_{j}}=\frac{\partial^{2}r}{\partial u_{i}\partial u_{j}},i,j\in \left\{1,2\right\}.
H_u(U_0( -\theta_0,x_*),V_0(-\theta_0,x_*))\, U_0'(-\theta_0,x_*) &+H_v(U_0(-\theta_0,x_*),V_0(-\theta_0,x_*))V'_0(-\theta_0,x_*) = \lim_{j\rightarrow\infty}\{H_u(w_1^j(0,0),w_2^j(0,0))\,n(p(x_j,t_j),t_j)\cdot\nabla w_1^j(0,0) &+H_v(w_1^j(0,0),w_2^j(0,0))\,n(p(x_j,t_j),t_j)\cdot\nabla w_2^j(0,0)\}=0.
w^{\prime \prime}+ ( \frac{\nu^{\prime}}{\nu}+ \frac{R^{\prime}}{R}- 2 \Phi^{\prime}) w^{\prime}- \frac{( w^{2}- 1 ) ( 4 \kappa + ( 4 R^{2}- 3 ) w + w^{3})}{2 \nu R^{4}}= 0 \, ,
\hat{K}=\left(\begin{array}{ccc}\frac{B}{4}\delta^{\mu\nu \alpha \beta}0 0 0 -\frac{B}{16}-\frac{B_j}{4}0 -\frac{B_i}{4}\frac{B_iB_j}{2B}-A \delta_{ij}\end{array}\right)\footnote{$\delta^{\mu\nu \alpha \beta}:= \frac{1}{2}\left( g^{\mu \alpha}g^{\nu \beta}+g^{\mu \beta}g^{\nu \alpha}\right)$}
a_k\leq Cb^{-\frac{s-s'}{Q}}\bigg(\int_{2B_0}\Big(\sum_{j\leq l-2}2^{s'jp}g_j^p\Big)\,d\mu\bigg)^{\frac{s-s'}{Q}}\sum_{n=k_0}^{k-1}2^{n\left(1-(s-s')\frac{p}{Q}\right)}+C'r_0^{s-s'}2^{k_0}.
\lim_{y \to 0}\left[ \phi_{n}^{1}( y ) \, \chi_{n}^{1}( y ) - \rho \, \phi_{n}^{2}( y ) \, \chi_{n}^{2}( y ) \right] = \phi_{n}^{1}( 0 ) \, \chi_{n}^{1}( 0 ) - \rho \, \phi_{n}^{2}( 0 ) \, \chi_{n}^{2}( 0 )
\lim_{m \to \infty}\frac{1}{m}\sum_{j = - m/2 + 1}^{m/2}\frac{e^{2 \pi i \frac{j + \theta_1/2}{m}\lfloor sm \rfloor}}{1 - \exp \left\{- 2 \pi i \frac{j + \theta_1/2}{m}\right\}+ \frac{z}{m}}=^{?}e^{\pi \theta_1 s}\sum_{j \in \mathbb{Z}}\frac{e^{2 \pi i j s}}{2 \pi i(j + \theta_1/2) + z},
I ( \phi ) \; = \; \int_{M^{\prime}\times M^{\prime}}d^{d}{\bf x}d^{d}{\bf y}\, \phi_{0}({\bf x}) \, G ({\bf x},{\bf y}) \, \phi_{0}({\bf y}) , \; G ({\bf x},{\bf y}) \; \propto \; \sum_{\gamma \in \Gamma}\frac{| \gamma^{\prime}({\bf y}) |^{\Delta}}{|{\bf x}- \gamma{\bf y}|^{2 \Delta}}.
\sum_{i=1}^n \mathbf{E}\Bigg[ \bigg| \frac{\phi_{i}}{\sqrt{\Sigma_{k=1}^n \phi_{k}^2}}\bigg|^{p}\Bigg] \leq \sum_{i=1}^n \mathbf{E}\Bigg[ \bigg| \frac{\phi_{i}}{\sqrt{\Sigma_{k=1}^n \phi_{k}^2}}\bigg|^{2}\Bigg] =1.
\begin{array}{c}(\xi_{i}-1) a^{2}_{i}=0 begin{align*}2mm]a_{i}a_{j}= e^{i(\lambda_{ji}-\lambda_{ij})}a_{j}a_{i}\qquad i \neq j begin{align*}2mm]a_{i}a^{+}_{j}=e^{i(\lambda_{ij}-\lambda_{ji})}a^{+}_{j}a_{i}, \qquad i \neq j begin{align*}2mm]a_{i}a^{+}_{i}=1+ \xi_{i}a^{+}_{i}a_{i}\end{array}
\begin{aligned}\dot{x}= -y + x^2 - y^2 + x \left(k_2 x^2 + 2 k_3 xy - k_2 y^2 \right), \dot{y}= x +2 xy +y \left(k_2 x^2 + 2 k_3 xy -k_2 y^2 \right),\end{aligned}
\Bigl\|\sum_{n=1}^N\varphi_n(T)x_n\Bigr\| =\|A^\ast\bigl(\oplus_{n=1}^Nx_n\bigr)\| \leq K C_{{\rm pol}, T}\mathop{}_{\zeta\in\mathbb T}\Bigr(\sum_{n=1}^N|\varphi_n(\zeta)|^2\Bigr)^{1/2}\Bigr(\sum_{n=1}^N\|x_n\|^2\Bigr)^{1/2}&\leq K C_{{\rm pol}, T}a^{1/2}\Bigr(\sum_{n=1}^N\|x_n\|^2\Bigr)^{1/2}.
\sum_{i=0}^{\tau_{1}-1}\sum_{l=0}^{t_{1}-1}\sum_{\tilde{\lambda}\in \tilde{\Lambda}_{ijl}}\left|\widehat{\nu}_{>1}\left(\frac{1}{q}(i+\tau_{1}j+\tau_{1}p_{1}l)+\tilde{\lambda}+x\right)\right|^{2}=1, (j=0,1, \cdots, p_{1}-1)
\int_a^b \psi(x) (~^{AB}I_b^\alpha\varphi(x)dx=\int_0^1 x (~^{AB}I_1^{1/2}(1-x)dx=\int_0^1x[\frac{1-x}{2}+\frac{2 (1-x)^{3/2}}{3\sqrt{\pi}}]dx=\frac{1}{12}+\frac{8}{105\sqrt{\pi}}.
\begin{aligned}\iota_{X_3}\omega= -da_j+\frac12 \sum_{s>j}a_j\frac{1}{G_{j+1}}\left(b_s da_s+a_s db_s \right) &+\frac12 \sum_{i<j}a_j\frac{G_j}{G_i G_{i+1}}\left(b_i da_i- a_i db_i \right)\,. \end{aligned}
{}^{M\!L\!\!}\textrm{F}_{p}( a,b;c;z)=\frac{1}{B(b,c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right)dt,
\int_0^1\left[ \int_0^x \pi^{(x)}(y) p_t(y,z)\,dy \right] f(z)\,dz = \int_{[0, 1]}\!\int_0^y\!\pi^{(y)}(z) f(z)\,dz\,P^*_x\,(X^*_t\in dy) &= \int_0^1 \left[ \int_{[z, 1]}\!\pi^{(y)}(z) P^*_x\,(X^*_t \in dy) \right] f(z)\,dz.
B(x,y)=\int\limits_0^\infty\frac{a^{x-1}}{(a+1)^{x+y}}\,da=2\int\limits_0^\infty\frac{a^{x-y}}{(a+a^{-1})^{x+y}}\,d^\times a\qquad{\rm Re}(x),{\rm Re}(y)>0.
\mathrm{J_{0}\equiv \frac{1}{r + s}( a_{1}^{\dagger}a_{1}- a_{2}^{\dagger}a_{2}) , \mathrm{J_{+}\equiv ( a_{1}^{\dagger})^{s}a_{2}^{r}, \mathrm{J_{-}= a_{1}^{s}( a_{2}^{\dagger})^{r},}}}
\begin{array}[c]{r}{\displaystyle\int_{\Omega}}v(s)\left[ -\Delta u(s)+\gamma(s)u\right] \,ds={\displaystyle\int_{\Omega}}\left[ \triangledown u(s)\cdot\triangledown v(s)+\gamma(s)u(s)v(s)\right]\,ds\quad\medskip -{\displaystyle\int_{\partial\Omega}}v\left( s\right) \dfrac{\partial u(s)}{\partial n_{s}}\,ds.\end{array}
\iint_{X\times X}U(x,y)\frac{d\mu(x)d\mu(y)}{d^\gamma(x,y)}= \int_{d(x,x_0)<R}\int_{d(y,x_0)\geq 2R}U(x,y)\frac{d\mu(x)d\mu(y)}{d^\gamma(x,y)}& + \int_{d(x,x_0)<2R}\int_{d(y,x_0)<2R}U(x,y)\frac{d\mu(x)d\mu(y)}{d^\gamma(x,y)}& + \int_{d(x,x_0)\geq 2R}\int_{d(y,x_0)<R}U(x,y)\frac{d\mu(x)d\mu(y)}{d^\gamma(x,y)}&= I + II + III.
V^{(k)}\Big[a_0; a_1,\, \cdots, a_r\Big]=\dfrac{[a_0+2k\delta]}{[a_0]}\dfrac{[a_0]_k\ [a_1]_k\ \cdots\ [a_r]_k}{[\delta]_k[\delta+a_0-a_1]_k\cdots[\delta+a_0-a_r]_k}.
C_{l, k}= \frac{\langle \varphi(x,y), J_0(\gamma_l x) \sin(k\pi y) \rangle}{||J_0(\gamma_l x) \sin(k \pi y)||_2^2}= \frac{1}{||J_0(\gamma_l x) \sin(k \pi y)||_2^2}\int_0^1 \int_0^1 x\varphi(x,y) J_0(\gamma_l x)\sin(k \pi y) dx dy.
G_2(s) =(n+p)P_q^{n,p-q}+P_q^{n-1,p-q}-pP_{q-1}^{n,p-q}&=(n+p+q+1)P_q^{n-1,p-q}+(n+p)P_q^{n,p-q}-(n+p+q)P_q^{n-1,p-q}-pP_{q-1}^{n,p-q},
\delta_{ij}=\int_{0}^1 t^{i-1}p_j^{(m)}(t) d t=\sum_{k=1}^{m+1}a_{kj}^{(m)}\int_{0}^1 t^{i+k-2}d t=\sum_{k=1}^{m+1}\frac{1}{i+k-1}a_{kj}^{(m)},
\lim\limits_{\varepsilon\rightarrow 0}\int\limits_{\{s_1<\cdots< s_n\}}\bar{g}_\varepsilon(s) ds = \int\limits_{\{s_1<\cdots< s_n\}}\lim\limits_{\varepsilon\rightarrow 0}\bar{g}_\varepsilon(s) ds < \infty.
{\rm{\bf K}}\left(\frac{z_{i+\frac{1}{2}}^{p+1}-z_{i+\frac{1}{2}}^{p}}{\tau}\right)+{\rm{\bf L}}\left(\frac{z_{i+1}^{p+\frac{1}{2}}-z_{i}^{p+\frac{1}{2}}}{h}\right)=\nabla_{z}S_1(z_{i+\frac{1}{2}}^{p+\frac{1}{2}})+\nabla_{z}S_2(z_{i+\frac{1}{2}}^{p+\frac{1}{2}})\frac{\Delta W_{i+\frac{1}{2}}^{p+\frac{1}{2}}}{\tau},
\prod_{\mathit{v}\in M_\infty}|\rho_\mathit{v}(\alpha_i)\rho_{\mathit{v}}(qu)-p_i|_\mathit{v}=\prod_{j=1}^d\prod_{\mathit{v}\in S_j}|\rho_{\mathit{v}}(\alpha_i)\sigma_j(qu)-p_i|_v.
|\partial_{t}z(\bar{x},\sigma)|^{2}=\frac{1}{|I_{r}(y)|}\int_{I_{r}(y)}|\partial_{t}z(l,\sigma)|^{2}dl= \frac{1}{2r}\int_{I_{r}(y)\cap I}|\partial_{t}z(l,\sigma)|^{2}dl \leq \frac{1}{2r}\int_{I}|\partial_{t}z(l,\sigma)|^{2}dl
x_i = s q^{1-i}(1 \leq i \leq d), y_i = s q^{1-i}(1 \leq i \leq d), z_i = \begin{cases}q^d (q^i-1)(1- s^{-2}q^{i-2}) , - q^{i-1}(q^{d-i+1}-1)(1-s^{-2}q^{d+i-1}) \end{cases}(1 \leq i \leq d).
\bar\sigma(x)=TR_{\bar s(x)^{-1}}\!\cdot\! T_x\bar s\quad\quad\bar\sigma^\mathcal{A}(x)=\overline{\mathcal{A}}(x)+TR_{\bar s(x)^{-1}}\!\cdot\! T_x\bar s\in L(TU,\mathfrak{g}),
\left(\frac{1}{4}-\frac{\widetilde{C}\Delta t}{2}\right) \|w^{1}\|^{2}_{2,\varphi}\le \left(\frac{1}{2}+\frac{1}{4}+\frac{\widetilde{C}\Delta t}{2}\right)\|u^{n}\|_{2,\varphi}^{2}+ \frac{C_{2}}{2}\lambda^{2}\|u^{n}\|_{2,\varphi}^{4}.
L_{\phi}= - \frac{3}{4}c \int_{- R}^{R}e^{- 2 \sigma ( y )}\sqrt{- \gamma}\partial_{\mu}\phi \partial^{\mu}\phi \, d y = - \frac{3}{2}\frac{k R^{2}}{e^{2 k R}- 1}\eta^{\mu \nu}\partial_{\mu}\phi \partial^{\nu}\phi .
TG^{(2)}_\gamma \overset{\simeq}{\longrightarrow}\mathrm{pr}_1^*(TG^{r(\gamma)})=\mathrm{pr}_1^*(\ker dr)|_{G^{(2)}_\gamma},\ (\gamma_1,\gamma_2,X_1,X_2)\longmapsto (\gamma_1,\gamma_2,X_1).
M_{A_{0}Y_{1}}^{2}= 2 \left( \begin{array}{c c}{{- H}}&{{- 2 i v}}\\{{2 i v}}&{{H}}\\ \end{array}\right) M_{Y_{8}Y_{9}}^{2}= 2 \left( \begin{array}{c c}{{H}}&{{- 2 i c}}\\{{2 i c}}&{{H}}\\ \end{array}\right) .
Q_{nm}= \frac{n !}{(n+m)!}\sum_{k=0}^{m}\binom{m}{k}\frac{(n + k)!}{n!}\,(-s)^{m-k}P_{nm}=\frac{m !}{(n+m)!}\sum_{k=0}^{n}\binom{n}{k}\frac{(m + k)!}{m!}\, s^{n-k}.
\frac{(at_3)^{1-s_1}}{s_1+s_3-1}\sum_{n \leq at_3}\frac{1}{n^{s_2}(at_3+n)^{s_3}}\ll \begin{cases}t_3^{-\sigma_1-\sigma_3}(\sigma_2>1) t_3^{-\sigma_1-\sigma_3}\log t_3 (\sigma_2=1) t_3^{1-\sigma_1-\sigma_2-\sigma_3}(\sigma_2<1). \end{cases}
x^2 P_n(x^2) = \P_{n+1}(x^2)+(A_n+C_n-a^2)P_n(x^2)+A_{n-1}C_nP_{n-1}(x^2) \equiv \P_{n+1}(x^2)+b_nP_n(x^2)+u_nP_{n-1}(x^2).
n_0\mathbf{B}_2(\langle1/2+v_k\rangle)+n_{1/2}\mathbf{B}_2(\langle v_k\rangle)=n_0\mathbf{B}_2(\langle1/2+v_k'\rangle)+n_{1/2}\mathbf{B}_2(\langle v_k'\rangle)\quad\textrm{for each}~k=1,\ldots,2g.
df_{ik}\wedge\theta^k - f_{ij}\theta^j_k \wedge\theta^k = - df_t\wedge \theta^t_i + f_k\theta^k_t \wedge \theta^t_i - f_k\Theta^k_i= =-(f_{tk}\theta^k + f_k\theta^k_t) \wedge\theta^t_i +f_k\theta^k_t \wedge \theta^t_i - \frac{1}{2}f_kR^k_{ijt}\theta^j\wedge \theta^t,
\frac{1}{15}[\frac{l^{3}}{w^{2}}+\frac{w^{3}}{l^{2}}+d(3-\frac{l^{2}}{w^{2}}-\frac{w^{2}}{l^{2}})+\frac{5}{2}(\frac{w^{2}}{l}ln(\frac{l+d}{w})+\frac{l^{2}}{w}ln(\frac{w+d}{l}))]
\begin{gathered}\alpha(\varepsilon, \xi((t-\varepsilon)\wedge \tau_\delta))=\alpha(t,u)+\int^{(t-\varepsilon)\wedge\tau_\delta}_0 \nabla_u\alpha(t-s,\xi(s))dw(s) -1_{\tau_\delta<t-\varepsilon}\int^{t-\varepsilon}_{\tau_\delta}\frac{\partial \alpha}{\partial t}(t-s, \xi(\tau_\delta))ds.\end{gathered}
\underset{p_k, q_k, \tau_k}{\max}\sum\limits_{k=1}^K R_{k}& L\sum\limits_{i=1}^l q_i \leq \sum\limits_{i=1}^l e^{r}_{i},~ \forall l \in [1:K], & LT\sum\limits_{i=1}^l p_i \leq \sum\limits_{i=1}^l e^{t}_{i},~ \forall l \in [1:K], & \tau_k\in[0,T),~p_k\geq0,~~ q_k\geq0,\forall k \in [1:K].
\left|\left|\frac{1}{n}\sum_{i=1}^nf(\boldsymbol{X}_i)\right|-\left|\int_{[0,1]^d}f(\boldsymbol{x})d\boldsymbol{x}\right|\right|\leq C\frac{1}{n}\int_{[0,1]^d}\left|\frac{\partial^d}{\partial x_1,\cdots\partial x_d}f(\boldsymbol{x})\right|d\boldsymbol{x}.
e^{2 k}= e^{({\frac{\alpha^{2}+ 1}{2 \alpha^{2}}}) 2 k_{g}}e^{-{( \gamma q_{1}q_{2}-{\frac{\tau_{o}^{2}}{2 \alpha^{2}}})}2 k_{e}}e^{- ({\frac{\tau_{o}}{2 \alpha^{2}}}) 2 k_{s}}\equiv e^{2 K_{2}k_{g}}e^{2 K_{1}k_{e}}e^{2 K_{3}k_{s}},
\left\langle \psi_{a , b}^{\left[ A , B \right]}( x ) , \psi_{a^{\prime}, b^{\prime}}^{\left[ A , B \right] *}( y ) \right\rangle_{0}= P_{a , b ; \, a^{\prime}, b^{\prime}}( i \partial_{x}) i \Delta^{( + )}( x - y ) \,
\varkappa(m,p_1,p_2):=\sum_{s=0}^{n}\sum_{\nu\in I_s}k^{-\frac{sr}{6}(\frac{1}{p_2}+\frac{5}{4})}(\ln(k))^{sr}\prod_{\nu_j>k}[ \nu_j^{-\frac{r}{6}(\frac{1}{p_2}+\frac{5}{4})}(\ln(\nu_j))^{r}]|m(\nu)|^r<\infty,
\mathbb{D}^2_\lambda i.[(\nabla_\lambda i)^{-T}\nabla_\lambda g]=- k\,e^{\frac{k}{2}\log^2\frac{\lambda_1}{\lambda_2}}\left(\log \frac{\lambda_1}{\lambda_2}\right)\frac{1}{(\lambda_1-\lambda_2)}\left(\frac{1}{\lambda_1}+\frac{1}{\lambda_2}\right)\left(\begin{array}{cc}0 1 1 0 \end{array}\right)
\phi(\gamma) = \sum\limits_{i=0}^{\infty}\phi_i h_i(\gamma) \approx \sum\limits_{i=0}^{Q}\phi_i h_i(\gamma), \phi_i = \int\limits_{-\infty}^{+\infty}\phi(z) \frac{1}{i!}h_i(z) \exp(-z^2/2) dz,
\Xi_\Omega(E) =\frac{d{\Omega}(u)}{d u}\large|_{u=E}\star \,\frac{d\Phi_{\mathcal{E}}(t,E)}{dt}\large|_{t=d_{\mathbb{K}min}}-\,\frac{d{\Omega}(u)}{d u}\large|_{u=E}\star \,\frac{d\Phi^c_{\mathcal{E}}(t,E)}{dt}\large|_{t=d_{\mathbb{K}min}},
X_+(t)Y_+(t)= \sum \xi^+_i(t)\mathcal{E}^+_i(t)+\sum_{i:\gamma^-_i>\gamma}\xi^+_-(t)\mathcal{E}^+_-(t)+\sum_{i\in I_1}\xi_{i,1}(t)\mathcal{E}_{i,1}(t) +\xi_{i,1}(t)\mathcal{E}_{i,1}(t) &+\sum_{i\in I_2}\xi_{i,3}(t)\mathcal{E}_{i,3}(t) +\xi_{i,4}(t)\mathcal{E}_{i,4}(t).
\Re \Big( \Big( \frac{-\theta}{\sin^2 \theta}+ \frac{e^{i\theta}}{\sin \theta}\Big)^2 (1 - \frac{\sin\theta}{\theta}e^{-i\theta}) \Big) = \frac{1}{\sin^2\theta}\Big( \Big( \frac{\theta}{\sin\theta}- \cos\theta \Big)^2 + \sin^2\theta \Big) > 0,
\sum_{\lambda \atop l(\lambda) \leq r}\frac{1}{p^{n(\lambda)+|\lambda|}\prod_{i \geq 1}\prod_{j=1}^{\lfloor m_i(\lambda)/2 \rfloor}(1-1/p^{2j})}\cdot \frac{(1-1/p) \cdots (1-1/p^r)}{(1-1/p) \cdots (1-1/p^{r-l(\lambda)})}.
E[e^{(X_1+X_2+\cdots+X_k)t}] = \left( \frac{-t}{(1-t) \log(1-t)}\right)^k = \left( \frac{-t}{\log(1-t)}\right)^k (1-t)^{-k}&= \sum_{n=0}^\infty B_n^{(n-k+1)}(-k+1)(-1)^n \frac{t^n}{n!}.
\begin{array}{rrrr}((Q_0 \cdot Q_{A_{\psi_0}})(\iota(\pi)) | \mathcal{L}_{\psi_0}) = 0, begin{align*}2mm]\bigwedge^{2}(Q_{A_{\psi_0}}(\iota(\pi) | \mathcal{L}_{\psi_0})) = 0, begin{align*}2mm](Q_{A_{\psi}}(\iota(\pi)) | \mathcal{L}_{\psi}) = 0, \;\psi \neq \psi_0. begin{align*}3mm]\end{array}
\left(\frac{d}{2}\right)^{-\frac{p}{2}}\frac{\Gamma((d-k+1)/2)}{\Gamma((d-p-k+1)/2)}\left( \prod_{i=1}^{k-1}c_i'\beta_i\right)^{-1}\left( \frac{\det S_k}{\det S_{k-1}}\right)^{-\frac{p}{2}}\left(c_k'\beta_k\right)^{d-p-k-1}e^{\frac{\|x\|^2}{2}}.
{\rm Cut}_{(a_{1},\dots,a_{n};b_{1},\dots,b_{m})}(e_{s_{1}}\cdots e_{s_{t}})=\begin{cases}w' e_{s_{1}}\cdots e_{s_{t}}\e_{a_{1}}\cdots e_{a_{n}}w'e_{b_{1}}\cdots e_{b_{m}}{\rm otherwise}0.\end{cases}
I_2=\frac{\pi^{3/2}}{12}\biggl(\frac{\theta p}{2}\biggr)^{1/2}[15\nu^{-1/2}K_{1/2}(\nu\theta p)-20\biggl(\frac{\theta p}{2}\biggr)^{3/2}\nu^{1/2}K_{1/2}(\nu\theta p)+4\biggl(\frac{\theta p}{2}\biggr)^{5/2}\nu^{3/2}K_{3/2}(\nu\theta p)],
\psi_{w_{\mu_2}(\mu_1)}(w_{\mu_2}xw_{\mu_2}^{-1})=\tilde{w}_\theta\tilde{x_2}\tilde{w}_\theta^{-1},x_2=w_\theta^{-1}w_{\mu_2}xw_{\mu_2}^{-1}w_\theta \in \tilde{U}_{[w_\theta w_{\mu_2}(\mu_1)]}.
\left( \beta(\rho_1, \dots , \rho_n) - \gamma( \rho_1, \dots , \rho_n) \right) \cap x^{\rho_1}=\beta(\rho_1, \dots , \rho_n) \cap x^{\rho_1}- \gamma(\rho_1, \dots , \rho_n) \cap x^{\rho_1}& = \beta(\rho_2, \dots , \rho_n) - \gamma(\rho_2, \dots , \rho_n) = 0
f_{r a d}( x - y ) = \left( \begin{array}{c c c c}{0}&{{- 1}}&{0}&{\partial_{x}}\\{1}&{0}&{{- \partial_{x}}}&{0}\\{0}&{{- \partial_{x}}}&{0}&{0}\\{\partial_{x}}&{0}&{0}&{0}\\ \end{array}\right) \delta ( x - y )
\frac{\partial}{\partial t}\tilde{H}(u,u) =\frac{\partial}{\partial t}(\tilde{H}_{ij}\frac{y^i}{\tilde{F}}\frac{y^j}{\tilde{F}}), &=\tilde{F}^{-2}(\nabla_s\nabla_0h_0^s-\frac{1}{2}\nabla_s\nabla^sh_{00}-\nabla_0\nabla_0h^s_s+{})\tilde{g}_{ij}.
T_\Lambda\mathcal{T}^*\{g_i\}_{i\in \mathbb{Z}}=\sum_{i\in \mathbb{Z}}\Lambda_{i-1}^*g_i=\sum_{i\in \mathbb{Z}}(T^{i-1})^*\Lambda_0^*g_i =(T^{-1})^*\Big(\sum_{i\in \mathbb{Z}}(T^{i})^*\Lambda_0^*g_i\Big) &=(T^{-1})^*\Big(\sum_{i\in \mathbb{Z}}\Lambda_i^*g_i\Big) &=(T^{-1})^*T_\Lambda\{g_i\}_{i\in \mathbb{Z}}=0.
L2=-\frac{1}{N}\sum_{i=1}^{N}\log\left(1+h_i(\alpha_{b_i}c_i+d_{b_i})\tau/N_s\right)+\sum_{i=1}^{N}\lambda_i\left(\sum_{k=1}^{i}\left(\tilde{d}_{b_k}-\tilde{c}_{b_k}\right)\tau-B_0\right) & -\sum_{i=1}^{N}\omega_id_{b_i}+\sum_{i=1}^{N}\delta_i(d_{b_i}-D_p)-\sum_{i=1}^{N}\mu_i(\alpha_{b_i}-\alpha_{c_i})+\sum_{i=1}^{N}\nu_i(\alpha_{b_i}-1)
M_{ADM}^2 ={1\over 2}{\vec{\tilde\alpha}^T}\mu_+{\vec{\tilde\alpha}}+{{1\over 2}}e^{-4\Phi'_\infty}{\vec \beta}^T \mu_+{\vec \beta}+ e^{-2\Phi'_\infty}\left [({\vec \beta}^T\mu_+{\vec \beta})({\vec \alpha}^T\mu_+{\vec \alpha})-({\vec \beta}^T\mu_+{\vec \alpha})^2\right]^{1\over 2},
T^m(f,g)(x):= &\sum_{j>N}\int\left|f*\Phi_{aj+m}\left(x-\frac{t^a+\epsilon_P(t)}{2^{aj}}\right)g*\Phi_{bj+m}\left(x-\frac{t^b+\epsilon_Q(t)}{2^{bj}}\right)\rho(t)\right| \,dt.
\sum_{i=0}^ma_i\frac{d^i}{dx^i}\underline{U}(x,p,r)+\sum_{i=0}^nb_i \underline{U}(x,p,r)-\sum_{j=1}^i p^{j-1}b_i \underline{u}^{(i-j)}(x,0,r)+\underline{U}(x,p,r)+\underline{F}(x, t,r)=K(t)\underline{U}(x,p,r).
L_{m}( \alpha ) \equiv \left( \begin{array}{c c}{{e^{- i V_{m}^{-}}}}&{{e^{\alpha}e^{i V_{m}^{+}}}}\\{{e^{\alpha}e^{- i V_{m}^{+}}}}&{{e^{i V_{m}^{-}}}}\\ \end{array}\right) \, , \quad \bar{L}_{m}( \alpha ) \equiv \left( \begin{array}{c c}{{e^{- i V_{m}^{+}}}}&{{e^{- \alpha}e^{i V_{m}^{-}}}}\\{{e^{- \alpha}e^{- i V_{m}^{-}}}}&{{e^{i V_{m}^{+}}}}\\ \end{array}\right) \, ,
\left< F \right| V_{\mathrm{C D}}^{( r )}( m , \Lambda ) \left| I \right> = \left[ \langle F \vert \delta V^{( r )}\hspace{- . 0 7 c m}( m , \Lambda , \Lambda^{\prime}) \vert I \rangle - \sum_{s = 2}^{r - 1}B_{r - s , s}\left< F \vert V^{( r - s )}( m , \Lambda ) \vert I \right> \right]_{\Lambda \hspace{. 1 c m}\mathrm{t e r m s}}.
I_Q(V;Y,S|U)-I_Q(V;Z|U) =I_P(S;Y,S|T)-I_P(S;Z|T)=H_P(S|T,Z), I_Q(U,V;Y,S)-I_Q(U,V;S) =I_P(S,T;Y,S)-I_P(S,T;S)=I_P(T;Y|S),
\sum_{\imath=1}^{\infty}\frac{\varsigma(\imath)\Big(\sum_{\jmath=\imath}^{\infty}\varsigma(\jmath)\varrho(\jmath)\Big)^p}{\Big(\sum_{\jmath=\imath}^{\infty}\varsigma(\jmath)\Big)^\alpha}\leq \Big(\frac{p}{\alpha-1}\Big)^p\sum_{\imath=1}^{\infty}\varsigma(\imath)\varrho^p(\imath)\Big(\sum_{\jmath=\imath}^{\infty}\varsigma(\jmath)\Big)^{p-\alpha}.
\boldsymbol t_k(i)=\frac{\lambda^{-1}\boldsymbol Q_{\bar{\boldsymbol\omega}_k}(i-1)\bar{\boldsymbol\omega}_k(i-1)}{1+\lambda^{-1}\bar{\boldsymbol\omega}_k^H(i-1)\boldsymbol Q_{\bar{\boldsymbol\omega}_k}(i-1)\bar{\boldsymbol\omega}_k(i-1)}.
\left(\int_{\{|x|<R\}}\frac{dx}{|x|^{b}}\right)^{\frac{1}{q}}\left(\int_{\{2R<|x|\}}\left(\frac{|x|}{2}\right)^{(a-Q)p^{\prime}}dx\right)^{\frac{1}{p^{\prime}}}&\leqslant CR^{\frac{Q-b}{q}}R^{\frac{(a-Q)p'+Q}{p'}}&\leqslant C,
\pi_1\big(C' \setminus \mathcal B, q_0\big)= \big\langle \gamma_1, \ldots, \gamma_r, \alpha_1,\beta_1,\ldots, \alpha_{g'},\beta_{g'}~ \big\vert ~\gamma_1 \cdots \gamma_r \cdot \prod_{i=1}^{g'}[\alpha_i,\beta_i] \big\rangle.
\langle Q_{2n}(x),Q_{2m+1}(x)\rangle_{4,\omega}=u_n\delta_{n,m},\langle Q_{2n}(x),Q_{2m}(x)\rangle_{4,\omega}=\langle Q_{2n+1}(x),Q_{2m+1}(x)\rangle_{4,\omega}=0
\frac{1}{2}\frac{d}{dt}|\theta^{(j)}- \theta|_{L^2(\Omega)}^2 + \kappa |\nabla (\theta^{(j)}- \theta)|_{L^2(\Omega)}^2 = \tau \sum_{i=1}^N \int_{\Omega}(\nabla^{\delta_0}u_i^{(j)}\cdot \nabla \theta^{(j)}- \nabla^{\delta_0}u_i \cdot \nabla \theta) (\theta^{(j)}- \theta) dx \mbox{a.e on}[0,T] \mbox{for}j
B_{e f f}^{( 1 )}( \vec{R}_{p_{1}}) = \frac{2 \pi \kappa \sigma_{1}}{q_{1}}\varepsilon_{i j}\partial_{i}v_{j}^{( 2 )}( \vec{R}_{p_{1}}) \; \; , \; \; \partial_{i}= \frac{\partial}{\partial R_{p_{1}}^{i}}\; \; .
x=a_0+\sum_{\ell=1}^3a_\ell D_\ell+\sum_{a,b\in \{1,2,3\},a\ne b}(f_{a,b}D_aD_b+g_{a,b}D_aD_bD_{6-a-b}+h_{a,b}D_aD_bD_a)\ ,
W(k)\leq \left(\frac{1}{2}\right)^k\frac{2}{(\lambda\kappa T_1)^\frac{1}{p}}Y(0)^\frac{1}{p}+8\left(\frac{H(L+1)}{\lambda\kappa}\right)^\frac{1}{p}=:M(k),\k\in\{0\}\cup\mathbb{N}.
\mathbf T_{\mathbf{P}_w}= \sum_{j=1}^n \big(\mathbf T_{\partial_j(rw)/w}\big)^*\, \mathbf T_{\partial_j}= - \sum_{j=1}^n \big( \mathbf T_{r\partial_j}\big)^* \, \mathbf T_{\partial_j}= -\sum_{j=1}^n \mathbf T_{\partial_j}^* \, \mathbf T_{r}\, \mathbf T_{\partial_j}.
I \left( a \right) = \frac{\int_{{\cal F}_{a}}\left[{\cal D}\Psi{\cal D}\overline{{{\Psi}}}\right] e^{-{\cal A}_{M T M}}}{\int_{{\cal F}_{0}}\left[{\cal D}\Psi{\cal D}\overline{{{\Psi}}}\right] e^{-{\cal A}_{M T M}}},
\begin{array}{c}\delta_1=\left| \begin{array}{cc}f_0(c_g(t_1)) f_2(c_g(t_1)) f_0(c_g(t_2)) f_2(c_g(t_2))\end{array}\right|, \delta_2=\left| \begin{array}{cc}f_1(c_g(t_1)) f_0(c_g(t_1)) f_1(c_g(t_2)) f_0(c_g(t_2))\end{array}\right|.\end{array}
\mathbb{E}\left[1_{\mathcal{E}_\delta}\Vert\hat{\Pi}_{V_1}(f_2-\hat{\Pi}_{V_2}f_2)\Vert_n^2\right] \leq\frac{2}{(1-\delta)}\mathbb{E}\left[\sum_{j=1}^{d_1}\langle\phi_{j,\Pi_2},(1-\hat{\Pi}_{V_2})f_2\rangle_n^2\right] +\frac{2}{(1-\delta)}\mathbb{E}\left[\sum_{j=1}^{d_1}\langle\phi_j-\phi_{j,\Pi_2},(1-\hat{\Pi}_{V_2})f_2\rangle_n^2\right] ,
\begin{aligned}A=a X-a^{-1}Y+\frac{b c^{-1}\left(XY-YX\right)}{q^{1/2}+q^{-1/2}}, B=b Y-b^{-1}Z+\frac{c a^{-1}\left(YZ-ZY\right)}{q^{1/2}+q^{-1/2}}, C=c Z-c^{-1}X+\frac{a b^{-1}\left(ZX-XZ\right)}{q^{1/2}+q^{-1/2}}, \end{aligned}
\tau_k = \begin{cases}\inf \{t \leq T: \int_0^t g^2(s, W_0^s, Y_0^s) ds \geq k\}, \mbox{if}\int_0^T g^2(s, W_0^s, Y_0^s) ds \geq k T, \mbox{if}\int_0^T g^2(s, W_0^s, Y_0^s) ds < k.\end{cases}
\exp \left[ \frac{n^{\prime}}{( e{\cal P}^{\prime})}( e Q ) \sum_{q > 0}\frac{1}{q}\mathrm{e}^{- 2 i q \sigma_{+}^{\prime}}\right] = \left( 1 - \mathrm{e}^{- 2 i \sigma_{+}^{\prime}}\right)^{- \frac{n^{\prime}}{( e{\cal P}^{\prime})}( e Q )}
p_{N{\mathrm{(int)}}}^{(2)}(\xi_1,\xi_2)=\frac{A_d}{2\xi_2^2}\int_{0}^{\infty}dk k^{d-2}\int_{0}^{\infty}d\omega \frac{I'_{\omega}(k\xi_1)(1+\omega^2/k^2\xi_2^2)}{I'_{\omega}(k\xi_2)N'_{\omega}(k\xi_1,k\xi_2)}.
\nabla u\cdot\nabla v=\frac{\partial u}{\partial x}\cdot\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\cdot\frac{\partial v}{\partial y}=\frac{\partial u}{\partial x}\cdot\frac{\partial v}{\partial x}-\frac{\partial u}{\partial x}\cdot\frac{\partial v}{\partial x}=0
\Theta_q(a,b,n) =\frac{(b-q^n)(ab-1-a^2+aq^n)}{(a-b)(1-ab)}\frac{(bq)^{(n+1)/2}(q^{-2}/b;q^2)_{(n+1)/2}}{(bq^2;q^2)_{(n+1)/2}}begin{align*}5pt] +\frac{(1-aq^n)(a-q^n)}{(a-b)(1-ab)}\frac{(b;q^2)_2(q^{-1};q^2)_{(n+1)/2}^2}{(q^{-1};q^2)_2(q^2/a,aq^{2};q^2)_{(n+1)/2}}.
\frac{1}{\varepsilon_{k}^{n}}\int_{M}u_{k}^{2}\psi(u_{k})d\mu_{g}\le\sum_{i=1}^{m}\frac{1}{\varepsilon_{k}^{n}}\int_{B_{g}(\xi_{i},r)}u_{k}^{2}\psi(u_{k})d\mu_{g}+\sum_{i=m+1}^{l}\frac{1}{\varepsilon_{k}^{n}}\int_{I_{\xi_{i}}(r)}u_{k}^{2}\psi(u_{k})d\mu_{g}
\partial_tH^{1-\alpha}= \alpha H^{-\alpha-1}\Delta H^{1-\alpha}-\alpha(1-\alpha)H^{-2\alpha-2}|\nabla H|^2 -(1-\alpha)H^{-2\alpha}|\mathring{A}|^2-\frac{1-\alpha}2H^{-2\alpha}(H^2-4).
\frac{d}{dt}\frac{f(t)}{t^{1+M}}=\frac{1}{t^{1+M}}\left(f'(t)-(1+M)\frac{f(t)}{t}\right)\ \leq\ 0, \frac{d}{dt}\frac{f(t)}{t^{1+m}}=\frac{1}{t^{1+m}}\left(f'(t)-(1+m)\frac{f(t)}{t}\right)\ \geq\ 0:
\Big|\underbrace{c_{1}e^{i\xi_1 t}+\cdots +c_{i_*}e^{i\xi_{i*}t}}_{=:f_{AP}}\;+ \;\displaystyle t^{-d}\;\sum_{k,\xi \atop k<d}c_{k,\xi}t^k e^{i\xi t}\Big|\leq \frac{M}{|t|^d}.
\phi_{2}\ = \ \frac{1}{4}\bigg [ \frac{{\tilde{y}}^{( 4 )}}{8}+ \frac{1 1 + 6 p}{1 2}{\tilde{y}}^{( 3 )}\ + \ \frac{3 1 + 2 4 p + 4 p^{2}}{8}{\tilde{y}}^{( 2 )}\ +
D_\lambda(p_{X|Z}\|q_{X|Z}|r_Z) \triangleq\begin{cases}\frac{1}{\lambda-1}\log \sum\limits_{z\in\mathcal{Z}}r_Z(z) \sum\limits_{x\in\mathcal{X}}\frac{(p_{X|Z}(x|z))^\lambda}{(q_{X|Z}(x|z))^{\lambda-1}}D(p_{X|Z}\|q_{X|Z}|r_Z) \end{cases}
\mathcal{H}[\psi (x),\bar{\psi}(x)]=\psi^{\dag\alpha}_{a}(x)\ \!(\mathbf{a}.\mathbf{p})^{\phantom{\alpha}\beta}_{\alpha}\ \!\psi(x)^{\phantom{\beta}}_{\beta a}+ V(\psi^{\dag}(x)\gamma_{0}\psi(x))
\zeta(e_n) = -f_{n+1},\zeta(x_n)= -y_{n+\frac{1}{2}}, \zeta(h_{n}) -h_n +2\delta_{n,0}K, \zeta(K) = K, \zeta(y_n)= x_{n-\frac{1}{2}}, \zeta(f_n)= -e_{n-1}.