harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum\limits_{i=0}^{m_{k}}\varepsilon^{\frac{i}{2}}\alpha_{i}u^{\left(i\right)}\left( 0,\varepsilon \right) =f_{1}\left( \varepsilon \right) ,\sum\limits_{i=0}^{m_{k}}\varepsilon^{\frac{i}{2}}\beta_{i}u^{\left( i\right)}\left( T,\varepsilon \right) =f_{2}\left(\varepsilon \right) ,
\begin{cases}\Delta u_1+ e^{u_2}(1-e^{u_1})=\displaystyle 4\pi\sum_{i=1}^{N_1}\delta_{\epsilon p_{1i}}, \Delta u_2+ e^{u_1}(1-e^{u_2})=\displaystyle 4\pi\sum_{i=1}^{N_2}\delta_{\epsilon p_{2i}},\end{cases}\mathbb R^2
\left\| R(\epsilon;t-s u_{(n)})\cdots R(\epsilon;t-su_{(1)}) -\left(\tilde{R}^n - \epsilon\sum_{i=0}^{n-1}s u_{(i+1)}\tilde{R}^{n-i-1}\tilde{R}^{(1)}\tilde{R}^{i}\right) \right\| & \le \epsilon^2d''s^3(1+\epsilon c's)^n \le \epsilon^2 d'' s^3(1+\epsilon c's)^{\frac{5}{4}\tilde{\lambda}s}
U=\begin{pmatrix}0 c_1^{21}c_1^{22}1 0 0 & 0 0 c_2^{21}c_2^{22}& c_1^{11}c_1^{12}0 0 & 0 0 c_3^{21}c_3^{22}& c_2^{11}c_2^{12}0 0 & \ddots \ddots \ddots \ddots\end{pmatrix},
\frac{d T}{d x}= - \frac{4 c \varepsilon^{2}H Z^{4}n}{\nu_{0}^{2}v^{2}}\int_{0}^{\infty}\xi d \xi \left\{1 - e^{- \alpha \xi \int_{\xi}^{\infty}\frac{d u}{u^{2}}B ( u )}\right\}\frac{\int_{\xi}^{\infty}\frac{d u}{u}B ( u )}{\alpha \xi \int_{\xi}^{\infty}\frac{d u}{u^{2}}B ( u )}.
S = \frac{1}{2}\int d^{4}x \int_{- 1 / \alpha}^{1 / \alpha}\left( - e^{- 2 \sigma ( z )}\delta^{I J}\partial_{I}\Phi \partial_{J}\Phi + \Phi \partial_{z}\left( e^{- 4 \sigma ( z )}\partial_{z}\Phi \right) + M_{5}^{2}e^{- 4 \sigma ( z )}\Phi^{2}\right)
G(\Omega_1,v|w|^2)=\left\{\left(\begin{array}{cc}\rho 0 0 \rho\end{array}\right),\,\,\left(\begin{array}{cc}0 \displaystyle\eta\frac{v_1}{v_2}\displaystyle\eta\frac{v_2}{v_1}0\end{array}\right)\,\,\hbox{with}\,\,\rho,\,\eta>0\right\},
(\Delta^e\otimes I)\Delta^em(a) =(\Delta^e\otimes I)(m\otimes m)\Delta(a)=\Delta^em(a_{(1)})\otimes m(a_{(2)}) &=(m\otimes m)\Delta(a_{(1)})\otimes m(a_{(2)})=m(a_{(11)})\otimes m(a_{(12)})\otimes m(a_{(2)}),
\int_{0}^{\infty}j_{k-\frac{1}{2}}(ixt)e^{-pt^{2}}t^{k+\frac{1}{2}}dt=\frac{\Gamma(k+\frac{1}{2})e^{\frac{x^{2}}{4p}}}{2p^{k+\frac{1}{2}}}=I_{k,p}(x).
M (\alpha,\beta,F, w) \, = \, ( |\alpha| +|\beta| ) \, \| \nabla w\|_{L^2 (\Omega)}+ |\beta |\, \| w\|_{L^\infty_{1}(\Omega)}& ~~~ + \| w\|_{L^\infty_{1}(\Omega)}\| \nabla w\|_{L^2 (\Omega)}|\log \| \nabla w \|_{L^2 (\Omega)}| \, + |\alpha| + \| F \|_{L^2 (\Omega)}\,.
F_3=\, 1+a_1^2+a_1^3+a_1^4+b+a_1 b+a_1^4 b+b^2+a_1 b^2+a_1^3 b^2+b^3+b^4+a_1^2b^4 &+b^5+a_1 b^5+b^6+a_1 b^6+b^7+(a_1^2 +a_1^4 +a_1^5 +a_1^2 b +a_1^4 b^2 +a_1^2 b^4+a_1^2b^5)k\cr +(a_1^4 +a_1^4 b +a_1^5 b +a_1^4 b^2 +a_1^5 b^2 +a_1^4 b^3)k^2+(a_1^6 +a_1^6 b)k^3,
\mathrm{d}X_i(t) = \mathfrak{b}_i(t,\mathfrak{X})\, \mathrm{d}t + \sum_{k}\mathfrak{s}_{ik}(t,\mathfrak{X}) \,\mathrm{d}W_k(t)\,, i=1,2,\dots,n\,,\mathfrak{X}(0) = x\,.
\mathbb{E}|R_{N,1}|^{\alpha'}\leq 2 \sum^N_{n=1}\mathbb{E}\Big|\sum^{\infty}_{j=N-n+1}K^{\infty}_{j,n}\Big|^{\alpha'}&\leq c_9\, \left(1+\sum^{N-1}_{n=1}\mathbb{E}\left| (N-n)^{1-\beta'}|\varepsilon_n|+(N-n)^{1-\alpha'\beta'}\right|^{\alpha'}\right) &\leq c_{10}\, N^{1+\alpha'(1-\beta')}
\sum \Gamma_{N}\doteq \int_{0}^{\infty}d u d t x t ( \frac{2 e}{u})^{1 / 2}e^{- u ( y + 1 )}[ e^{- t}- e^{- t_{0}( x , u )}] [ 1 - ( 2 e u )^{1 / 2}e^{- u}x t ]^{- 1}.
\mu_{np+r}(E(t)) = \sum_{|g_0| = r}\mu_r(g_0) \tau_{np}^{t(g_0)}(g_1: g_0g_1 \in E(t)) &= \sum_{|g_0| = r}\mu_r(g_0) \left(N(t,\sigma) +O\left( \frac{\log n}{\sqrt{n}}\right)\right) &= N(t,\sigma) +O\left( \frac{\log n}{\sqrt{n}}\right)
\begin{array}{cl}{\rm Res}_{z_{2}=z_{1}\tau^2}\cdots{\rm Res}_{z_{n}=z_{n-1}\tau^2}w_{\alpha_1 \cdots \alpha_n}(z_1,\cdots,z_{n}) = \delta_{\alpha_1 n}\delta_{\alpha_2 n-1}\cdots \delta_{\alpha_n 1}z_1^{n-1}(\tau^2 -1)^{n-1}\tau^{(n-1)^2}\left[ n-1 \right] ! ~~ v^{(n\cdots 21)},\end{array}
\frac{3}{2}\sum_{j=1}^3 \bigg\| \frac{L_jf}{|x|}\bigg\|_{2}^2 \leq \|\Delta_r f \|_{2}^2 + \bigg(1 - \frac{1}{\mathbf{C}_3}\bigg) \bigg\| \sum_{j=1}^3 L_j^2f \bigg\|_{2}^2 & - 2\bigg\langle \sum_{j=1}^3 L_j^2 \bigg(\partial_r f - \frac{f}{2|x|}\bigg), \partial_r f - \frac{f}{2|x|}\bigg\rangle.
H^d(X_\infty) \le \sum_{i=1}^{n}\left(\frac4{n^2\cdot 2^i}\right)^d + \sum_{i=n+1}^{\infty}\left(\frac4{i^2\cdot 2^i}\right)^d &\le \frac{4^d}{n^{2d}}\sum_{i=1}^{\infty}\frac1{2^{id}}=\frac{4^d}{n^{2d}(1-2^{-d})}
\lim_{\nu\rightarrow \infty}||G\circ T_\nu-G^{\nu+1}\circ T_\nu||_{p,\mu-1;\Omega_0\times \mathbb{T}^{n_2},\Pi_\gamma}=0{\rm for \,\, all}\,\, 0\leq p<l_1,\,0<\mu\leq \alpha
R(x_{1},x_{2})=y(T(x_{1}),x_{2})-y(T(x_{1}),x_{1})=x_{2}+F_{1}\frac{T^{2}(x_{2})}{2}+F_{1}T(x_{2})s+F_{2}\frac{s^{2}}{2}-x_{1}-F_{1}\frac{T^{2}(x_{1})}{2}=
\begin{array}{l l}{{A = A^{1}+ i A^{2},}}&{{A^{*}= A^{1}- i A^{2}}}\\{{\gamma = \gamma^{1}+ i \gamma^{2},}}&{{\gamma^{*}= \gamma^{1}- i \gamma^{2}}}\\{{\gamma_{\pm}= \gamma_{0}\pm \gamma_{3}}}&{{}}\\ \end{array}
\begin{pmatrix}g_2(z) g_3(z)\end{pmatrix}=\begin{pmatrix}\cos \theta -\sin \theta \sin \theta \cos \theta\end{pmatrix}\begin{pmatrix}f_2(z) f_3(z)\end{pmatrix}=\begin{pmatrix}f_2(z)\cos \theta-f_3(z)\sin \theta f_2(z)\sin \theta+f_3(z)\cos \theta\end{pmatrix}.
A_{3,1}^1 : = \sup_{t > 0}\bigg( \int_t^{\infty}[V_2^*]^{\frac{q}{p}}(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}V_*^{- 1}(t), A_{3,2}^1 : = \sup_{t > 0}W^{\frac{1}{q}}(t) \bigg( \int_t^{\infty}\bigg( \frac{V_2^*(\tau)}{V_*(\tau)}\bigg)^{\frac{1}{1-p}}v(\tau)\,d\tau \bigg)^{\frac{1 - p}{p}};
\begin{array}{r c l}{f e}&{=}&{\langle e , S ( f ) \rangle \langle 1 ,{\bar{t}}^{- 1}\rangle 1 \cdot \bar{t}^{- 1}}\\{}&{}&{+ \langle t , S ( 1 ) \rangle \langle 1 , \bar{t}^{- 1}\rangle e f + \langle t , S ( 1 ) \rangle \langle e , f \rangle t \cdot 1}\\{}&{=}&{- \langle e , f \rangle{\bar{t}}^{- 1}+ e f + \langle e , f \rangle t .}\\ \end{array}
\sum_{\substack{d_1+\dots+d_{n}=g-1 0\leq d_i \leq a_i+\delta_{i\ell}(a_{n+1}-1)}}\prod_{i=1}^{n}Q_{d_i+\delta_{i\ell}}(a_i+\delta_{i\ell}a_{n+1}) \prod_{i\not=\ell,n+1}\psi_i^{d_i}D_{i,\ell}\pi^*(\psi_\ell^{d_\ell})
\lambda_4 = \gamma_4 - \tfrac{5}{3}a_2^2, &\lambda_6 = \gamma_6 - \tfrac{4}{3}a_2 \gamma_4 - \tfrac{10}{27}a_2^3, &\lambda_8 = -2a_2 \gamma_6 - \tfrac{1}{3}a_2^2 \gamma_4 + \tfrac{20}{27}a_2^4, &\lambda_{10}= a_2^2 \gamma_6 + \tfrac{2}{3}a_2^3 \gamma_4 + \tfrac{8}{27}a_2^5. & 4\gamma_4^3+27 \gamma_6^2 \neq 0.
U(\alpha)=\frac{\int_{\mathcal{A}}\,(r(\mathbf{x}_{*},\alpha)-r^{\mathrm{eff}}(\mathbf{x}_{*},\alpha))\,\mathrm{d}\mathbf{x}_{*}}{\int_{\mathcal{A}}\, r(\mathbf{x}_{*},\alpha)\,\mathrm{d}\mathbf{x}_{*}}\in[0,1).
\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}
K[\![t]\!]_0 =\left\{\sum_{i\in\mathbb{N}}a_it^i\in K[\![t]\!];a_i\in K,\ \sup_{i\in\mathbb{N}}{|a_i|}<\infty\right\}=\mathcal{O}_K[\![t]\!]\otimes_{\mathcal{O}_K}K, K\{t\}=\left\{\sum_{i\in\mathbb{N}}a_it^i\in K[\![t]\!];a_i\in K, \lim_{i\to\infty}{|a_i|\eta^i}=0\ (\eta\in (0,1))\right\}.
v^{S}(\lambda)=\sum\limits_{j\in\sigma}\lceil \lambda_{j}\rceil \rho_{j}+\sum\limits_{i\not\in\sigma,i\leq n}\lambda_{i}\rho_{i}+\sum\limits_{i=1}^{m}\lambda_{n+i}s_{i}\equiv \sum\limits_{i\not\in\sigma}\lambda_{i}b^{i}_\sigma \textbf{N}_{\sigma},
g((\nabla_XA)\phi \xi^\perp,Y)= \lambda||\xi^\perp||(\varepsilon+||\xi^\perp||) \{(u-\lambda)g(X,Y)-vg(\phi X,Y)\}g((\nabla_XA)\phi^2 \xi^\perp,Y)= \lambda||\xi^\perp||(\varepsilon+||\xi^\perp||) \{-pg(X,Y)+(q-\mu)g(\phi X,Y)\}.
\Gamma_{R}^{(n)}=\Gamma_{fin}^{1(n)}+\Gamma_{fin}^{3(n)}=(-i\lambda)^{2n+1}(i)^{3n}\prod_{i=1}^{n}\int \frac{d^{6}k_{i}}{(2\pi )^{6}}\Xi^{(n)}(k_{1},...,k_{n},p,p^{\prime},m^{2})\cdot
F_{\mathrm{s i n g}}( N , g_{3}, g_{4}) \sim b \cdot \left[ \delta g_{3}+ \left( \frac{3^{1 / 4}}{4}- \frac{3^{7 / 4}}{2}\right) \delta g_{4}\right]^{5 / 2}+ b^{\prime}\cdot [ \delta g_{4}]^{- 6}+ \cdots .
{\cal U}_{\Delta}^{\dagger}B_{i}^{3}\,{\cal U}_{\Delta}= \frac{1}{2}\epsilon_{i j k}\left\{\left( e^{- i g \Delta}F_{j k}e^{i g \Delta}\right)^{3}+ i g \,{\cal U}_{\Delta}^{\dagger}\left( [ A_{j}, A_{k}] \right)^{3}{\cal U}_{\Delta}\right\}\ .
\mathcal{L}_\mathrm{I,X}(s) = \exp\left(-\frac{2 \lambda_{\mathrm{I,X}}\, \pi^2 \, (\mathrm{G}_{\mathrm{I,TX}}\,\mathrm{G}_{\mathrm{RX}}\mathrm{P}_{\mathrm{I}})^{\frac{2}{\alpha_{\mathrm{X}}}}\, s^{\frac{2}{\alpha_{\mathrm{X}}}}}{\alpha_{\mathrm{X}}\sin\left(\frac{2 \pi}{\alpha_{\mathrm{X}}}\right)}\right).
\sum_{j=1}^{n}\binom{j+p}{p+1}j^{-1}= \sum_{j=1}^{n}j^{-1}\frac{1}{(p+1)!}\sum_{k=0}^{p+1}(-1)^{p+1-k}\begin{bmatrix}p+1 k \end{bmatrix}(j+p)^{k}& = \frac{1}{(p+1)!}\sum_{t=0}^{p+1}\sum_{k=t}^{p+1}(-1)^{p+1-k}p^{k-t}\binom{k}{t}\begin{bmatrix}p+1 k \end{bmatrix}\sum_{j=1}^{n}j^{t-1}.
\to \qquad \qquad \qquad \qquad \exp \left( - \frac{\mu^{2}s}{2 \pi}\int_{0}^{\pi}\mathrm{d}q \left( \frac{\left( \left| q \right| - \frac{q^{2}}{2 \pi}\right)}{\sin^{2}\left( \frac{q}{2}\right)}\right) \right)
d^{f(p)}\big(H_0(x)\cdot(\delta_tH_0(x))^{-1}\cdot\delta_t H_1(x),\exp_{f(p)}^{-1}\circ f(p)\big) &\geq d^{f(p)}\big(H_0(x),\exp_{f(p)}^{-1}\circ f(p)\big)-d^{f(p)}\big(H_0(x)\cdot(\delta_tH_0(x))^{-1}\cdot\delta_t H_1(x),H_0(x)\big) &=:I_1-I_2.
\sum_{B\in V}g(B)^{q}\mathcal{K}(B,B') =|B'|\sum_{n=0}^k \sum_{B\in V_n, \, 8B\cap 8B'\ne \emptyset}|B|^{-1+\alpha q}&\simeq |B'|\sum_{n=0}^k 2^{-nQ(-1+\alpha q)}\simeq |B'| 2^{-kQ(-1+\alpha q)}&\simeq 2^{-kQ\alpha q}\simeq |B'|^{\alpha q}\simeq g(B')^q
\begin{cases}\theta(z_1,[\mathfrak{X},z_2])D(\mathfrak{X})+\theta([\mathfrak{X},z_1],z_2)D(\mathfrak{X})+\theta([\mathfrak{X},z_1],[\mathfrak{X},z_2])=0, \theta([\mathfrak{X},z_1],[\mathfrak{X},z_2])D(\mathfrak{X})=0.\end{cases}
U_{i}^{(c)}=\int_{0}^{T}b_{c}(t,\,x_{i}^{N}(t))\left[dx_{i}^{N}(t)-d\tilde{A}_{i}^{N}(t)\right]-\frac{1}{2}\int_{0}^{T}b_{c}^{2}(t,\,x_{i}^{N}(t))dt V_{ij}^{(c_{1},\,c_{2})}=\int_{0}^{T}\log\left(1+\frac{\gamma_{c_{1},c_{2}}(t,\,x_{i}^{N}(t))}{\lambda N}\right)dJ_{ij}^{N}(t)-\int_{0}^{T}\gamma_{c_{1},c_{2}}(t,\,x_{i}^{N}(t))dA_{ij}^{N}(t)
a+c =(\beta+\lambda\eta)+(\beta+d^2\delta+e^2\delta b)=(d^2+d+e^2b)\eta+d^2\delta+e^2\delta b &=d^2\eta+d^2\delta+d\eta+e^2\eta b+e^2\delta b=(d\eta)^2+d\eta+(e\eta)^2b.
\varepsilon_{\mathrm{int}}^{\mathrm{elast}}= \tfrac{\sqrt{2}}{3}\Big((\operatorname{tr}\boldsymbol{\varepsilon}^{\mathrm{elast}})^{2}- \operatorname{tr}\big((\boldsymbol{\varepsilon}^{\mathrm{elast}})^{2}\big)\Big).
x_i' y_i' - x_i y_i = x_i' \left( \frac{2a_i}{b_i^2}x_i' -\frac{2a_i}{b_i}x_i \right) - x_i \left( \frac{2a_i}{b_i}x_i' - a_i x_i \right) = \frac{2 a_i}{b_i^2}(x_i')^2 - \frac{4a_i}{b_i}x_i x_i' + a_i x_i^2.
\Box^\perp r|_q =\sum_{i=1}^{2n-2}H(r)(E_i,E_i)|_q=\sum_{i=1}^{2n-2}I_\gamma(J_i,J_i)\leq \sum_{i=1}^{2n-2}I_\gamma(W_i,W_i) &=\frac{1}{(s_\lambda(r))^2}\int_0^r\Big((2n-2)(s'_\lambda(t))^2-\mathbf{Ric}^\perp(T)(s_\lambda(t))^2\Big)dt.
\sum\limits_{i= 1}^{r'}{\left|{{I_{{R_i}}}}\right|}=\sum\limits_{i = 1}^{r}{\left|{{I_{{R_i}}}}\right|}\le \sum\limits_{j = 1}^{b'}{\left|{{S_{{B_j}}}}\right|},\;\;\;\;\sum\limits_{j= 1}^{b'}{\left|{{I_{{B_j}}}}\right|}\le \sum\limits_{i = 1}^{r'}{\left|{{S_{{R_i}}}}\right|},
\frac{P_{\lambda}\big(q^{\mu_1}t^{N-1}, q^{\mu_2}t^{N-2}, \dots, q^{\mu_N}; q, t\big)}{P_{\lambda}\big(t^{N-1}, t^{N-2}, \dots, 1; q, t\big)}= \frac{P_{\mu}\big(q^{\lambda_1}t^{N-1}, q^{\lambda_2}t^{N-2}, \dots, q^{\lambda_N}; q, t\big)}{P_{\mu}\big(t^{N-1}, t^{N-2}, \dots, 1; q, t\big)}.
t_{j,j}=\sum_{k}z_{j,k}\left(\sum_{l}s_{k,l}z_{l,j}\right) &=\sum_{k,l}z_{j,k}s_{k,l}z_{l,j}&=\sum_{k}z_{j,k}s_{k,k}z_{k,j}+\sum_{k\neq l}z_{j,k}s_{k,l}z_{l,j}&=2^{-p}{\mathrm{Tr\,}}S +\sum_{k<l}\left( s_{k,l}z_{j,k}z_{l,j}+s_{l,k}z_{l,j}z_{k,j}\right) &=2^{-p}{\mathrm{Tr\,}}S.
\begin{aligned}\frac{\phi_2(h)}{h^2\sqrt{h}}=\frac{1}{2h^2\sqrt{h}}\int_{f(x)<h}g(x)dx =\frac{1}{2h^2}\int_{a\xi^2+g_3(\sqrt{h}\xi)<1}g(\sqrt{h}\xi)d\xi, \end{aligned}
\prod_{j=1}^r{\Gamma\Big({(t+1)n-r+j\over 2}\Big)\over\Gamma\Big({n-r+j\over 2}\Big)}\sim \prod_{j=1}^r e^{-{tn\over 2}}\Big({n\over 2}\Big)^{tn\over 2}(t+1)^{{(t+1)n\over 2}+{j-r-1\over 2}}&=e^{-{tnr\over 2}}\Big({n\over 2}\Big)^{rtn\over 2}(t+1)^{\big({(t+1)n\over 2}-{1\over 2}\big)r-{r(r-1)\over 4}}.
2\frac{\int_{b=\theta_{2}}^{b=\theta_{1}}e^{\frac{\rho rs\cos(b)}{1-\rho^{2}}}\,\d b}{2\pi\int_{0}^{2\pi}e^{\frac{\rho rs\cos a}{1-\rho^{2}}}\d a}\leq2\frac{\int_{b=0}^{b=\theta_{1}-\theta_{2}}e^{\frac{\rho rs\cos(b)}{1-\rho^{2}}}\,\d b}{2\pi\int_{0}^{2\pi}e^{\frac{\rho rs\cos a}{1-\rho^{2}}}\d a}<\frac{\theta_{1}-\theta_{2}}{2\pi^{2}}.
\mathrm{RW}(d) =\sup_{\lambda \ge 1}\left\{\inf_{\pi \in \Pi(\mu_{1,L+1}, \dotsc, \mu_{L, L+1})}\int_{\mathcal{V}}\min_{\ell \in [L]}\{y_\ell + \phi_{\lambda, \ell}(x_1, \dotsc, x_L) \}d \pi(s) - \langle \lambda, \delta \rangle \right\},
\mathcal{V}\left(\overline{u},\overline{v}\right)\leq d\overline{v}''-c\overline{v}'+s\overline{v}begin{align*}0.2cm] = -dhq(1)\lambda\left(2+\lambda z\right) e^{\lambda z}+chq(1)\left(1+\lambda z\right) e^{\lambda z}-shq(1)z e^{\lambda z}begin{align*}0.2cm] = -hq(1)e^{\lambda z}\left[ P(c,\lambda)z+(2d\lambda-c)\right]=0.
\nabla_{X\wedge X'}Y =\widetilde{\nabla}_{X\wedge X'}Y+F^{k,1}(X'',Y) &=X\wedge \widetilde{\nabla}_{X'}Y+(-1)^{k-1}X'\wedge \widetilde{\nabla}_XY+F^{k,1}(X\wedge X',Y) &=X\wedge \nabla_{X'}Y+(-1)^{k-1}X'\wedge \nabla_XY+F^{k,1}(X\wedge X',Y),
\smash{n=n_{j_{m}}+kd_{j_{m}}+k'\textrm{per}(x_{l}),\quad\textrm{where}\ 0\le k\le \dfrac{\alpha d_{j_{m}+1}}{d_{j_{m}}}-2\ \textrm{and}\ 0\le k'\le \dfrac{\alpha d_{j_{m}}}{\textrm{per}(x_{j})}}\cdot
\left(\Pi_{\lambda_1,\lambda_2}g\right)^{1-\lambda_1,1-\lambda_2}_{i-\frac{1}{2},j-\frac{1}{2}}= g^{1-\lambda_1,1-\lambda_2}_{i-\frac{1}{2},j-\frac{1}{2}}, \mbox{if}\left(i,j\right) \in (\Bbbk_2^-,\Bbbk_1^-).
P ( Z ) = U \left( \begin{array}{c c}{I}&{- Z^{\dag}}\\{Z}&{I^{\prime}}\\ \end{array}\right) \left( \begin{array}{c c}{I}&{}\\{}&{O^{\prime}}\\ \end{array}\right) \left( \begin{array}{c c}{I}&{- Z^{\dag}}\\{Z}&{I^{\prime}}\\ \end{array}\right)^{- 1}U^{\dag},
P_{n,r_0}^V(V_m \geq x|R_0 = r_0, \ldots, R_m = r_m) \geq P_{n,r_0}^V(V_m^- \geq x|R_0 = r_0, \ldots, R_m = r_m) & = P_{n,r_0}^V(V_m^- \geq x) \geq \prod_{j=1}^{m}P_{n,r_0}^V(V_j^- \geq V_{j-1}^-|V_{j-1}^- \geq \ldots \geq V_1^- \geq V_0^-) \geq q^m.
T(\sigma(a_1))\circ \sigma(a_2) =B_1(T(\sigma(a_1)))\sigma(a_3)S(B_2(T(\sigma(a_2)))) &=B_1(T(\sigma(a_1)))\sigma(a_2)S(B_2(T(\sigma(a_3)))) &=B_1(T(\sigma(a_1)))B_1(a_2)S(B_2(a_3))S(B_2(T(\sigma(a_4)))) &=\epsilon(a)1,
\hat{G}^{( \alpha \beta )}( x , y ) = - i \langle \hat{T}\left( \hat{\phi}^{( \alpha )}( x ) \; \displaystyle{\raisebox{0 . 9 e x}{\scriptsize{t}}}\mathrm{\hspace{- 0 . 1 e x}}\hat{\phi}^{( \beta )}( y ) \right) \rangle ,
V^{\alpha \dot{\alpha}}= \left( \begin{matrix}{V^{1 \dot{1}}}&{V^{1 \dot{2}}}\\{V^{2 \dot{1}}}&{V^{2 \dot{2}}}\\ \end{matrix}\right) ={\frac{1}{\sqrt 2}}\left( \begin{matrix}{V^{1}+ V^{4}}&{V^{2}- V^{3}}\\{V^{2}+ V^{3}}&{- V^{1}+ V^{4}}\\ \end{matrix}\right) \ ,
\beta_s : =\mathbb{I}_{\{\delta Y_s \neq 0 \}}\frac{f(s, Y_s, Z_s) - f(s, Y_s^\prime, Z_s)}{\delta Y_s}, \gamma_s : = \mathbb{I}_{\{\lambda_s\delta Z_s \neq \mathbf{0}\}}\frac{\big(f(s, Y_s^\prime, Z_s)-f(s, Y_s^\prime, Z_s^\prime)\big)\delta Z_s}{|\lambda_s \delta Z_s|^2},
\|t\nabla\cdot f_{\alpha a}^2\|^2_{L^2(r\leq \langle t \rangle/2)}\lesssim\sum\limits_{\tiny\begin{matrix}b+c=a \beta+\gamma=\alpha\end{matrix}}\big\|\langle t-r\rangle|\nabla^2 U^{(\beta,b)}| |\nabla U^{(\gamma,c)}|\big\|^2_{L^2(r\leq \langle t \rangle/2)}&\lesssim E_{|\alpha|+|a|+2}X_{[(|\alpha|+|a|)/2]+4}.
\left(\kern-.1em\left(\kern-.1em\nabla\psi,\nabla\widetilde{\psi}\kern-.1em\right)\kern-.1em\right) =\left(\kern-.1em\left(\kern-.1emL_{\Omega}\left( I-\Pi_{h}^{E}\right) \Pi^{\operatorname*{comp},\ast}\mathbf{v}_{h},\nabla\widetilde{\psi}\kern-.1em\right)\kern-.1em\right) \qquad\forall\widetilde{\psi}\in H^{1}(\Omega).
(\sum_{k=1}^{P}u_{k})^{\mp d}\rightarrow\sum_{n_{1},...,n_{P}=0}^{\infty}\varphi_{n_{1},...,n_{P}}\prod_{k=1}^{P}u_{k}^{n_{k}}\frac{\langle\pm d+\sum_{j=1}^{P}n_{j}\rangle}{\Gamma(\pm d)}
P_h\delta_t v_i^{m+1/2}- \Delta_{2}v_i^{m+1/2}= S_i^m, 1\le i \le N-1 , 0\le m\le M-1, v_0^m = 0, v_M^m = 0, v_i^0 = v_0(x_i) , 0 \le i \le N.
\begin{array}{ll}l=(l_{12}+l_{13}+l_{23})/(2N), l'=(l_{14}+l_{15}+l_{45})/(2N) begin{align*}1.5mm]l''=(l_{24}+l_{26}+l_{46})/(2N), l'''=(l_{35}+l_{36}+l_{56})/(2N) \end{array}
\omega = \bar{z}_1 \xi+ \bar{z}_1^2+ r(z_1,\bar{z}_1,\xi,z_3,\ldots,z_n) , \omega = (-z_2-\bar{z}_1) \xi+ (-z_2-\bar{z}_1)^2+ r(z_1,(-z_2-\bar{z}_1),\xi,z_3,\ldots,z_n) .
0 \leq \sum_{i=1}^K (A_i X - \lambda X A_i )(A_i X - \lambda X A_i)^{\dagger}= \sum_{i=1}^K A_i X X^{\dagger}A_i^{\dagger}- \vert \lambda \vert^2 X X^{\dagger}- \vert \lambda \vert^2 XX^{\dagger}+ \vert \lambda \vert^2 X X^{\dagger}= 0.
Z ={\cal N}\int \prod_{x}\left\{\mathrm{d}v_{\mu}\mathrm{d}j_{5}^{\mu}\, \mathrm{d}\theta \right\}\, e^{\mathrm{i}\, \int \mathrm{d}^{2}x \,{\cal L}[ v_{\mu}, j_{5}^{\mu}, \theta ]},
\det_{1\leq j,k\leq s}\left[\sum_{m=0}^{r-1}m^{j+k-2}\binom{m+q}{m}\right]= \prod_{j=0}^{s-1}\frac{(j!(j+q)!)^2 (j+q+r)!}{q!(r-j-1)!(2j+q)!(2j+q+1)!}.
\prod_{i=1}^{N}\mathbb{E}^{-S_{ij}}\simeq1-\Omega^{-1/2}\sum_{i}S_{ij}\frac{\partial}{\partial\xi_{i}}+\frac{\Omega^{-1}}{2}\sum_{i}\sum_{k}S_{ij}S_{kj}\frac{\partial^{2}}{\partial\xi_{i}\partial\xi_{k}}+O(\Omega^{-3/2}).
\frac{2 ! - 1}{2 !}+ \frac{2}{3 !}+ \frac{3}{4 !}+ \cdots + \frac{n - 1}{n !}= 1 - \frac{1}{2 !}+ \frac{1}{2 !}- \frac{1}{3 !}+ \frac{1}{3 !}+ \cdots + \frac{1}{( n - 1 ) !}
\frac{\pi}{4}=(\prod_{p\equiv1(mod4)}\frac{p}{p-1})\cdot(\prod_{p\equiv3(mod4)}\frac{p}{p+1})=\frac{3}{4}\cdot\frac{5}{4}\cdot\frac{7}{8}\cdot\frac{11}{12}\cdot\frac{13}{12}\cdot\cdot\cdot,
\widehat{\mathcal{L}}u(t,x)\,:=\, \sup_{a\in \mathcal{A}(x)}\mathcal{L}_{a}u(t,x)\,, \mathcal{L}_{a}u(t,x)\,:=\,\sum_{i,j}x_i x_j a_{ij}\left(\frac{D^2_{ij}}{2}+ \frac{D_i}{||x||_1}\right) u(t,x)\,,
A_{\mu \nu}( x ) \sim \frac{B_{A}}{( x^{2})^{\beta_{A}}}\left[ \eta_{\mu \nu}+ \frac{2 \beta_{A}( 1 - \rho )}{( 2 - 2 \beta_{A}+ \rho )}\frac{x_{\mu}x_{\nu}}{x^{2}}+ i \vartheta h ( \beta_{A}) \frac{\epsilon_{\mu \nu \sigma}x^{\sigma}}{( x^{2})^{1 / 2}}\right]
r \int_{0}^{1}d\beta \int_{t}^{\infty}ds \int_{B_{s-t}(0) \cap(B_{\frac{s}{2}}(-\beta x))^{c}}\frac{dA(y)}{(s-t)}|\frac{v_{4,c}(s,|\beta x+y|)}{|\beta x+y|}|
(-\Delta f_\pm,g_\pm)_{L^2(\Omega_\pm)}-(f_\pm,-\Delta g_\pm)_{L^2(\Omega_\pm)}=\langle \gamma_D^\pm f_\pm,\gamma_N^\pm g_\pm\rangle - \langle \gamma_N^\pm f_\pm,\gamma_D^\pm g_\pm\rangle;
\iint \limits_{|(t,x,v)|>R}M_t^{(N)}(dx,dv)dt \leq \int_0^T\int_{\mathbb R^6}\dfrac{t^2+|x|^2+|v|^2}{R^2}M_t^{(N)}(dx,dv)dt \leq \dfrac{1}{R^2}\left(2B T+\dfrac{T^3}{3}\right).
s_3 s_4 s_5 s_6(s_{\alpha_7+\alpha_8}s_{\alpha_1 + 2\alpha_2 + 2\alpha_3 + 3\alpha_4 + 2\alpha_5 + \alpha_6 + \alpha_7})s_6 s_5 s_4 s_3 = s_{\alpha_3 + \alpha_4 + \alpha_5 + \alpha_6 + \alpha_7+\alpha_8}s_{\alpha_1 + 2\alpha_2 + 3\alpha_3 + 4\alpha_4 + 3\alpha_5 + 2\alpha_6 + \alpha_7}.
\mathbf{w}_{k,j}= \left(\!\!N_0 \mathbf{I}_{N_r(N\!+\!J)}+ \!\!\!\!\!\! \sum_{\substack{i=1, i \neq k}}^N \!\!\! v_{x_{i,j}}^{prio}\mathbf{h}_{i,j}\mathbf{h}_{i,j}^H + \mathbf{h}_{k,j}\mathbf{h}_{k,j}^H \!\!\right)^{-1}\!\!\!\!\! \mathbf{h}_{k,j}
H_1 = z+\frac{k}2(c_3\,x-b_3\, y)^2, H_2 =\frac{k}3\, \left( c_{3}x-b_{3}y \right)^{3}+\frac{c_1}{2}\,{x}^{2}+c_{2}\,xy+c_{3}\,xz-\frac{b_2}{2}\,{y}^{2}-b_{3}\,yz.
{\cal F}^{\rm B}_{\rm var}\left(\{R_{i}\},\{\Sigma_{i}\}\right)= \sum_{i}D_{\rm KL}(Q||P_0) +\sum_{\mu}D_{\rm KL}({\cal M}|| P_{\rm out}) &+ \frac 12 \sum_{\mu}\left( \log{2\pi V_\mu^*}+1+V_\mu^*\partial_{\omega}g_{\rm out}\right) ,
\|{\mathbb{T}}\|^\ell_{\dot{B}_{p,1}^{\gamma_3}}\le C\|{\mathbb{T}}\|^\ell_{\dot{B}_{2,1}^{\gamma_3+\frac n2-\frac np}}\le C\big(\|{\mathbb{T}}\|^\ell_{\dot{B}_{2,\infty}^{\sigma+1}}\big)^{\theta_{4}}\big(\|{\mathbb{T}}\|^\ell_{\dot{B}_{2,1}^{\frac n2}}\big)^{1-\theta_{4}},\theta_{4}=\frac{\frac np -\gamma_3}{\frac n2-1-\sigma}\in (0,1),
[M_{S'}, \mu_{S'}](I^G_{M_S}(\rho))=(\mathrm{Ind}^G_{P_{S'}}\circ [\mu_{S'}] )(\bigoplus_{w \in W_{\rho}}\delta^{\frac{1}{2}}_{P_{S'}}\otimes I^{M_{S'}}_{w(M_S)}(w(\rho)))\otimes [1][|\cdot|^{\langle \rho_G, \mu_{S'}-\mu \rangle}].
\big(I\odot\bar{I}\big)^{A}{}_{B}=\big(I^{A}{}_{R}\bar{I}^{R}{}_{B}+\bar{I}^{A}{}_{R}I^{R}{}_{B}\big)=[I,\bar{I}]^A{}_{B}+2\bar{I}^{A}{}_{R}I^{R}{}_{B},
- d_i\partial_r \big(v_i + \varepsilon^2 W_i\big)|_{r=1}- \varepsilon g_i({\bf R}^\varepsilon) = \varepsilon \big( g_i({\bf v}) - g_i({\bf R}^\varepsilon)\big), i\in \{1, 3, 4\}, - \partial_r v_i|_{r=1}= 0, i\in \{2, 5, 6\},
\Gamma_1' =\Gamma_1 + \{t,(t,c_{t,1}),(c_{t,1},c_{t,2}),\cdots,(c_{t,k_t-1},c_{t,k_t}) \}\Gamma_2' = \Gamma_2\setminus\{(t,c_{t,1}),(c_{t,1},c_{t,2}),\cdots,(c_{t,k_t 1},c_{t,k_t})\}
\begin{aligned}\frac{dB}{dt}=\frac{\lambda_1\lambda_2^2\lambda_3}{\lambda_4B^2}\left(\frac{1}{\omega}(B^2-k)\right)^{-1/2}B^2(B^2-k)^{1/2}\frac{dB}{dt}=\frac{\lambda_1\lambda_2^2\lambda_3\omega^{1/2}}{\lambda_4}.\end{aligned}
\Delta_{\gamma_4}\bigcap (\mathbb{N}^2\setminus \{0\})= \bigcup_{m=0}^{i_0-1}\Big\{\big(\frac{mj_0+n_m^{'}}{i_0}, m\big)+\big(aj_0+b, ai_0\big)|a, b\in \mathbb{Z}^{+}\Big\}= \bigcup_{m=0}^{i_0-1}\Big\{\big(\frac{mj_0+n_m^{'}}{i_0}+aj_0+b, m+ai_0\big)|a, b\in \mathbb{Z}^{+}\Big\}.
\begin{cases}dp(t,y) = - \pi(t,y) [b_0(t,y)p(t,y) + \lambda \gamma_0(t,y) r(t,y,1) ]dt+ r(t,y,1) d\tilde{N}(t); t\in[0, T] p(T,y) = U'(x(T,y))\mathbb{E}[\delta_Y(y)|\mathcal{F}_T]\end{cases}
\gamma^{0}=\left( \begin{array}{cc}\sigma^{3}0 0 -\sigma^{3}\end{array}\right) ,\;\gamma^{1}=\left( \begin{array}{cc}i\sigma^{2}0 0 -i\sigma^{2}\end{array}\right) ,\;\gamma^{2}=\left( \begin{array}{cc}-i\sigma^{1}0 0 i\sigma^{1}\end{array}\right) ,\;\gamma^{3}=\left( \begin{array}{cc}0 I -I 0\end{array}\right) \;.
S_{Liouv}[\varphi] ={1\over 8\pi}\int_D dx^2 \left[{1\over 2}(\partial_a\varphi)^2 +8\pi\mu b^2 e^\varphi \right] + \phi_\infty + 2\log{R}- \sum_i \left ( \eta_i \varphi_i + 2\eta_i^2 \log{\epsilon_i}\right).
\left( -\frac{d^{2}}{dr^{2}}+\omega^{2}r^{2}+\frac{\mu +A_{\ell,j,m,n_1,n_2}-\frac{1}{4}}{r^{2}}\right) F_{k,\ell,j,m,n_1,n_2}(r)=E_{k,\ell,j,m,n_1,n_2}\F_{k,\ell,j,m,n_1,n_2}(r) \ .
\mathcal{Z}(u_3,U_1,a_2,\gamma) =\exp\Big\{{-}\tfrac{1}{2}a_2\big(\gamma_4 +\tfrac{7}{3}a_2^2\big)u_3^2 - a_2^2 U_1 u_3 \Big\}\rho(u_3,U_1,a_2,\gamma)
m_{ij}= \left( \frac{1-R_{ij}}{1+R_{ij}-\beta T_i}\right)\cdot \left( (1-\beta T_i)^{k_s-i-\lambda_i}\cdot c_{\lambda_i}(i) \right) \left( (1-\beta T_j)^{k_s-j-\lambda_j}\cdot c_{\lambda_j}(j) \right)
\Phi_{n+1}= \frac{-1}{1 - \Phi_n^2}\left[ \frac{d \Phi_n}{dr}-\frac{n}{r}\Phi_n \right] = \Phi_{n-1}- \frac{2}{1 - \Phi_n^2}\frac{d \Phi_n}{dr},\;\;\;\;n \geq 1.
H=L_1^2+L_2^2+L_3^2+\big(x_1^2+x_2^2+x_3^2\big) \left(\frac{\mu_1(\mu_1-R_1)}{x_1^2}+\frac{\mu_2(\mu_2-R_2)}{x_2^2}+\frac{\mu_3(\mu_3-R_3)}{x_3^2}\right),
=\underset{t^{l}}{coef}\left[exp\left(\frac{-pt^{p}\theta}{e^{px}-t^{p}}\right)\cdot\frac{(1-t^{p}e^{-px})^{k+g_{Y}-1}}{(1-t^{p}e^{-2px})^{2g_{Y}-2}}\cdot \prod_{j=0}^{p-1}\left\{\frac{(1-\nu^{j}te^{-2x})^{m_{j,2}+m'_{p-j,-2}}}{(1-\nu^{j}te^{-x})^{m_{j,1}}}\right\}\right].
\|\tilde{f}_{ijk}^{\alpha a}\|_{L^2(r\leq \langle t\rangle/2)}\leq\sum_{\tiny\begin{matrix}\beta+\gamma=\alpha, b + c = a |\beta|+|b|,|\gamma|+|c|<|\alpha|+|a| \end{matrix}}\big\| |\nabla^2 U^{(\beta,b)}| |\nabla U^{(\gamma,c)}|\big\|_{L^2(r\leq \langle t\rangle/2)}.
\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{\beta}+n^{\beta}}\int_{B_{k,n}}\frac{dy}{(|y_1-k|^{\alpha_1}+|y_2-n|^{\alpha_2})(y_1^{\alpha_1}+y_2^{\alpha_2})},
B' =q^{n-i-3}\mu_{n-i-1}(q^{2})+2\mu_{n-i-2}(q^{2})\frac{q^{n-i-1}-1}{q^{2}-1}\;, C' =\mu_{n-i-2}(q^{2})\frac{q^{n-i-1}-1}{q^{2}-1}\;.
E(k)=E_{0}+\frac{\hbar^{2}}{2m_{x}^{*}}(k_{x}-k_{0,x})^{2}+\frac{\hbar^{2}}{2m_{y}^{*}}(k_{y}-k_{0,y})^{2}+\frac{\hbar^{2}}{2m_{z}^{*}}(k_{z}-k_{0,z})^{2}
{\bf{\hat{g}}}= \frac{1}{( N - 1 )^{2}}\Bigg [ ( 2 - N ) \sum_{i = 1}^{N}\Big ( \frac{d{\hat{u}}_{i}}{{\hat{u}}_{i}}\Big )^{2}+ \sum_{r \neq s}\frac{d{\hat{u}}_{r}}{{\hat{u}}_{r}}\frac{d{\hat{u}}_{s}}{{\hat{u}}_{s}}\Bigg ]
\{(2m_{2}+m_{1}+m_{3}) (\cot \langle 2\alpha_{1}, H\rangle )^{2}-n_{3}\}((\cot \langle 2\alpha_{1}, H\rangle )^{2}-1) = -(2m_{2}+m_{1})(\cot \langle 2\alpha_{1}, H\rangle )^{2}.
\mathcal{B}_{m_k}(t^{m}\hat{f}(t))(\tau) = \frac{\tau^{k}}{\Gamma(\frac{m}{k})}\int_{0}^{\tau^{k}}(\tau^{k}- s)^{\frac{m}{k}-1}\mathcal{B}_{m_k}(\hat{f}(t))(s^{1/k}) \frac{ds}{s}