harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\times{{\left({\mathrm{D e t}_{D}}\hat{\Delta}\right)}^{- ( d - p - 1 ) / 2}}\exp \left( -{\frac{\mathrm{T}}{2}}\int{d^{2}}\xi \sqrt{\hat{g}}{\hat{g}^{a b}}{\partial_{a}}{Y_{c}^{i}}{\partial_{b}}{Y_{c i}}\right) .
= \Bigg |{\atop{\mathrm{d e t}\atop{\scriptstyle i , j = 1 , . . . , 2 n_{a}}}}\Bigg ( \frac{1}{w_{i}^{( a )}- z_{j}^{( a )}}\Bigg ) \Bigg | \times \Bigg |{\atop{\mathrm{d e t}\atop{\scriptstyle i , j = 1 , . . . , 2 n_{b}}}}\Bigg ( \frac{1}{w_{i}^{( b )}- z_{j}^{( b )}}\Bigg ) \Bigg |
{\rm e}^{-A_1(\theta)}=t^{(1)}_{1,2}(\theta) =3 \omega^3 w_E[\varphi(E\omega^{-1}) , \varphi(E\omega^{3})] \frac{d^2}{dE^2}\varphi(E \omega^{-1}),{\rm e}^{-A_2(\theta)}=t^{(2)}_{1,2}(\theta) =3 \omega^{-3}w_E[\varphi(E\omega^{-3}) , \varphi(E\omega)] \frac{d^2}{dE^2}\varphi(E \omega)
\int_{B_{1}}g_{\alpha}*\left( \phi\psi^{1+q}\partial_{s}(g_{1-\alpha,m}*\tilde{u}^{1-q}) \right)dx + (1-q)g_{\alpha}*\left[ \mathcal{E}(h_{m}*\tilde{u},\psi^{1+q}\tilde{u}^{-q}) \phi \right] & \leq q g_{\alpha}*\int_{B_{1}}\psi^{1+q}\tilde{u}^{1-q}g_{1-\alpha,m}\phi dx,
- \sum_{n}\, \log \left( \frac{v_{+}^{\dagger}( n ) v_{+}( n + \varphi )}{\mid v_{+}^{\dagger}( n ) v_{+}( n + \varphi ) \mid}\frac{v_{+}^{\dagger}( n + \varphi ) v_{-}( n + \varphi )}{v_{+}^{\dagger}( n ) v_{-}( n )}\frac{v_{-}^{\dagger}( n + \varphi ) v_{-}( n )}{\mid v_{-}^{\dagger}( n + \varphi ) v_{-}( n ) \mid}\right)
\Theta_{s}^0= [e_1(s\chi\eta_t\partial_z+s\chi\bar{\eta}_t\partial_{\bar{z}}) &+e_2(s\chi_{\bar{z}}\bar{\eta}\partial_z-s\chi_z\bar{\eta}\partial_{\bar{z}}) &+e_3(-s\chi_{\bar{z}}\eta\partial_z+s\chi_z\eta\partial_{\bar{z}})]
\begin{array}{l}{{< ( ( \phi_{\tau}, f_{1}) - E [ ( \phi_{\tau}, f_{1}) ] ) ( ( \phi_{\tau}, f_{2}) - E [ ( \phi_{\tau}, f_{2}) ] ) . . . .}}\\{{( ( \phi_{\tau}, f_{2 n}) - E [ ( \phi_{\tau}, f_{2 n}) ] ) > = \hbar^{2 n}\sum_{p a i r s}\prod_{( j , k )}\frac{1}{2 i}( f_{j},{\cal A}^{- 1}f_{k})}}\\ \end{array}
Z = \int d \left[ \psi , \bar{\psi}\right] \prod_{\mu = 0}^{2}d \left[ A_{\mu}\right] \prod_{c_{0}= 1}^{N - 1}d \left[ A_{3}^{c_{0}}\right] \Delta_{\mathrm{F P}}\left[ A \right] e^{i ( S \left[ A , \psi , \bar{\psi}\right] + S_{\mathrm{g f}}\left[ A \right] )}\ .
W_{i,j}=F_{i,j}+F_{i,j+1}-F_{i-1,j}-F_{i+1,j+1}&=\sharp \{1,2,\dots,j\mbox{'s in the}i^{\mathrm{th}}\mbox{row}\}-\sharp\{1,2,\dots,(j+1)\mbox{'s in the}(i+1)^{\mathrm{th}}\mbox{row}\}.
y^2 = \prod_{a=1}^{n_f/2+1}(x-\phi_a)^2 \prod_{k=1, k\neq (n_c-n_f/2+1)/2}^{n_c-n_f/2}\phi_k^2 - 4 \Lambda^{2n_c-n_f}\prod_{i=1}(x+m_i),
\mathcal{F}_1(M^\#)=\mathcal{F}_1(M/\mathfrak{I}^2(M))\cong\mathcal{F}_1(W_{2,2}^\#)\oplus\mathcal{F}_1(W_{2,2}^\#)\cong\mathcal{O}_{\mathbb{P}^1}(1)\oplus\mathcal{O}_{\mathbb{P}^1}(1).
\int_{S_r^{n-1}}|\widehat{f}|^2d\sigma=\int_{S_r^{n-1}}\widehat{f}\overline{\widehat{f}}d\sigma =\int_{\mathbb{R}^n}f(f\ast\widehat{d\sigma_r})dx &\leq\|f\|_{L^2(|V|^{-1})}\|f\ast\widehat{d\sigma_r}\|_{L^2(|V|)}&\leq rC(V)\|f\|_{L^2(|V|^{-1})}^2.
\bar{a}_{i}^{b}e_{b}^{\mu}D_{\mu L}| a \rangle = 0 \, , \quad ( a_{i}^{b}\bar{a}_{i}^{b}- h_{i}) | a \rangle = 0 \, , \quad \bar{a}_{i}^{b}\bar{a}_{i}^{b}| a \rangle = 0 \, , \quad \varepsilon^{i j}a_{i}^{b}\bar{a}_{j}^{b}| a \rangle = 0 \, ,
\begin{aligned}dq_t =p_t\,dt, dp_t =-\beta\, q_t\,dt-(A(Q_t)-A(\overline{Q}_t)\,dt-(B\ast\rho_t(Q_t)-B\ast\overline{\rho}_t(\overline{Q}_t))\,dt+\lambda\,z_t\,dt, dz_t =-\lambda\,p_t\,dt-\alpha\, z_t\,dt.\end{aligned}
\begin{cases}y_{k-11}+y_{k-10}+y_{k-9}'+y_{k-8}'=0; y_{k-10}+y_{k-9}'+y_{k-8}''=0; y_{k-11}+y_{k-10}+y_{k-9}''+y_{k-8}''=0; y_{k-10}+y_{k-9}''+y_{k-8}'=0.\end{cases}
(\widetilde{\chi}^{j})^{i}\,_{p}\widetilde{\chi}^{p}-\widetilde{\chi}^{j}\widetilde{\chi'}^{i}+\widetilde{\chi'}^{i}\widetilde{\chi}^{j}=0, (\widetilde{\chi}^{i})^{j}\,_{p}\widetilde{\chi}^{p}+(\widetilde{\chi}^{i})\widetilde{\chi}^{j}-\widetilde{\chi}^{j}\widetilde{\chi}^{i}=0.
h_{2}(\mathbf{z}_{1},\mathbf{z}_{2})h_{2}(\mathbf{z}_{2},\mathbf{z}_{3})h_{2}(\mathbf{z}_{3},\mathbf{z}_{1}) & =\frac{1}{4}(1+\mathbf{n}_{1}\cdot\mathbf{n}_{2}+\mathbf{n}_{2}\cdot\mathbf{n}_{3}+\mathbf{n}_{3}\cdot\mathbf{n}_{1}-i[\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}])
Q_y(x) := \frac{x(1 + y e^{-x})}{1 - e^{-x}}, \widehat{Q}_y(x) := \frac{x(1 + y e^{-x(1+y)})}{1 - e^{-x(1+y)}}= 1 + \frac{1-y}{2}x +\cdots
\phi = \phi_{e}+ \sum_{m=q_x}^{\infty}C_{m,0}\Delta x^{m}+ \sum_{m=q_t}^{\infty}C_{0,m}\Delta t^{m}+ \sum_{m=q_x+q_t}^{\infty}\sum_{n=q_t}^{m-q_x}C_{m-n,n}\Delta x^{m-n}\Delta t^{n}.
I_q(\Gamma, b) = -\frac12 \sum_{d\in X}(q^\frac12 - q^{-\frac12})q^{\mathrm{ind}_{\Gamma,b}(d)}+ \int_{S \setminus \tilde \Gamma}\frac{q^{\mathrm{ind}_{\tilde\Gamma,b}}-1}{q^{\frac12}- q^{-\frac12}}\, d\chi
\frac{\partial \overline{\mathcal{L}}(y, p_{_1}(y))}{\partial p_{_1}(y)}\!=\! \!-\!\log p_{_1}(y) \!-\!1 \!-\! \eta_0 \frac{p_{_0}(y)}{p_{_1}(y)}\!+\! \eta_1 \!+\! \eta_2 y^2.
tp_{\mu}(t)=\frac{1}{\pi}\frac{h^{\langle-1\rangle}(1/t)\sin f(h^{\langle-1\rangle}(1/t))}{|1-h^{\langle-1\rangle}(1/t)e^{if(h^{\langle-1\rangle}(1/t))}|^{2}},t\notin D_{\mu}.
p_{k}(t)=e^{-(\lambda+\mu)t}[\rho^{\frac{k-i}{2}}I_{k-i}(at)+\rho^{\frac{k-i-1}{2}}I_{k+i+1}(at)+(1-\rho)\rho^{k}\sum_{j=k+i+2}^{\infty}\rho^{-j/2}I_{j}(at)]
\begin{array}{lccl}\mathrm{MS}: \left(\frac{{R}[x]}{\langle F(x)\rangle}\,,\bullet\right ) \longrightarrow \left(\frac{{\mathcal R}[x]}{\langle f(x) \rangle}\,,\star\right ) begin{align*}1em] g(x) \mapsto \sum_{i=0}^{n-1}g({\alpha_i}+ y) x^i\end{array}
\partial_i\circ h_i(g_0\wedge\cdots\wedge g_i) =g_0\wedge\cdots\wedge g_i -\sum^i_{j=0}(-1)^j1\wedge g_0\wedge\cdots\wedge \hat{g_j}\wedge\cdots\wedge g_i &=g_0\wedge\cdots\wedge g_i - h_{i-1}\circ\partial_{i-1}(g_0\wedge\cdots\wedge g_i).
\Gamma_{2}= \frac{1}{C_{2}}= \frac{3}{a^{2}( 4 - a^{2})}, \ \Gamma_{4}= - \frac{C_{4}}{( C_{2})^{4}}= \frac{9 ( 1 - a^{2}) ( 5 - 2 a^{2})}{a^{4}( 4 - a^{2})^{4}}
\frac{\partial}{\partial A_{p}^{c}}{\cal A}_{n ( b )}^{m ( a )}= \frac{1}{2}\delta_{n}^{m}g^{2}f_{c b e}f_{d}^{a e}A_{p}^{d}+ \frac{1}{2}\delta_{n}^{m}g^{2}f_{e b d}f_{c}^{a d}A_{p}^{e}
\frac{e_{c}^{2}|\langle\varphi_{v}|e_{c}x|\varphi_{c}\rangle|^{2}D_{ph-e}[f_{e}^{o}(E_{e,v})-f_{e}^{o}(E_{e,c})]}{\epsilon_{o}^{2}m_{e,e}^{2}u_{ph}n_{\omega}\omega}
{Z}_{t n +i}= \begin{cases}\frac{f_i}{d_{\rm a,w}^{\alpha/2}}+ \frac{g_{t n + i}}{d_{\rm j,w}^{\alpha/2}}+{N}_{t n + i}^{\rm (w)}, \!\!\mbox{Alice transmits and}~t=0 \frac{g_{t n + i}}{d_{\rm j,w}^{\alpha/2}}+{N}_{t n + i}^{\rm (w)}, \!\!\mbox{else,}\end{cases}
(\alpha,1-\zeta{\pi'_n}^j)_{\rho,L,n}= (\dfrac{\zeta{\pi'_n}^j}{1-\zeta{\pi'_n}^j}\alpha, (\zeta{\pi'_n}^j)^{-1})_{\rho,L,n}= -j (\dfrac{\zeta{\pi'_n}^j}{1-\zeta{\pi'_n}^j}\alpha, \pi'_n)_{\rho,L,n}.
\Phi_n^{-1}={\mathbb I}_\bot + \check{\Psi}_n\,, \check{\Psi}_n := \begin{pmatrix}\check{\Psi}_{n, 1}\check{\Psi}_{n, 2}\overline{\check{\Psi}}_{n, 2}\overline{\check{\Psi}}_{n, 1}\end{pmatrix}.
|t+\tau|^{-\mu\sum\limits_{k=1}^n \sum\limits_{p=n+1}^{2n}d(p-k)}\leq\prod\limits_{\substack{k=1\cdots n p=n+1\cdots 2n}}^{2n}|t_p-t_k|^{-\mu\,d(p-k)}\leq |t-\tau|^{-\mu\sum\limits_{k=1}^n \sum\limits_{p=n+1}^{2n}d(p-k)}.
\Delta f = \frac{1}{\sqrt \rho}\left( \frac{1}{\epsilon}\nabla^{a}\frac{\partial^{L}}{\partial \theta^{a}}+ \frac{1}{\bar{\epsilon}}{\overline{{{\nabla^{a}}}}}\frac{\partial^{L}}{\partial{\bar{\theta}}^{a}}\right) ( \sqrt \rho f ) ,
S_0 = \left(\begin{array}{cccc}1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 \end{array}\right) \;\; B = \frac{1}{\sqrt{2}}\left(\begin{array}{rr}1 0 -1 0 0 1 0 -1 \ \end{array}\right).
\hat{U}(t_{1},t_{0})=1+\frac{1}{i\hbar}\int_{t_{0}}^{t_{1}}\hat{H}(t)dt+\frac{1}{(i\hbar)^{2}}\int_{t_{0}}^{t_{1}}dt^{\prime}\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\hat{H}(t^{\prime})\hat{H}(t^{\prime\prime})+\cdot\cdot\cdot.
{S_{L}}[ \varphi , \hat{g}] ={\frac{2 6 - d}{4 8 \pi}}\int{d^{2}}\xi \sqrt{\hat{g}}\left({\frac{1}{2}}{\hat{g}^{a b}}{\partial_{a}}\varphi{\partial_{b}}\varphi + \hat{R}\varphi \right) +{\frac{2 6 - d}{2 4 \pi}}\oint{d}\hat{s}{k_{\hat{g}}}\varphi .
\langle u, (H-\lambda )^{-1}w \rangle \geq \langle (T-\lambda)^{-1}u,( -ga^{\dagger}(v) (T-\lambda)^{-1})^{n_1}(-ga(v) (T-\lambda)^{-1})^{n_2}w \rangle &= \langle (T-\lambda)^{-1}( -ga(v) (T-\lambda)^{-1})^{n_1}u,(- ga(v)(T-\lambda)^{-1})^{n_2}w \rangle
\tau^-_{\Lambda_{t-\cdot}}:= \inf\{s\in[0,t+1]\!:(R_s-\Lambda_{t-s})(R_0-\Lambda_{t-})\leq0\}, &\tau_{\Lambda_{t+\cdot}}:= \inf\{s\geq 0\!:(R_s-\Lambda_{t+s})(R_0-\Lambda_t)\leq0\}, &\tau^-_{\Lambda_{t+\cdot}}:= \inf\{s\geq 0\!:(R_s-\Lambda_{t+s})(R_0-\Lambda_{t-})\leq0\},
\Omega^{E_k}_{n-1,w}\cong\bigoplus_{r+s=n-1}H_r(W;{\mathbb Z}_2)\otimes_{{\mathbb Z}_2}\Omega_s\cong\Omega^{E_k}_{n-1}/K^{E_k}_{n-1,w}\cong\Omega^{E_k}_{n-1,s}/K^{E_k}_{n-1,s,w}.
ds^2=\frac{X^{1/2}}{\rho}e^{2A(r)}dx_{//}^2 +\frac{X^{1/2}}{\rho}\left(dr^2+\frac{L^2}{\rho^2}\left[d\theta^2+\frac{\sin^2\theta}{X}d\phi^2+\frac{\rho^6\cos^2\theta}{X}d\Omega^2_3\right] \right),
\left| \frac{1}{(\varphi(x))^k}\mathrm{Pf}\Big[ K_{N}\Big(xN+\frac{u_i}{\varphi(x)}, xN+\frac{u_j}{\varphi(x)}\Big) \Big]_{i,j = 1}^{k}\right| \leq (2k)^{\frac{k}{2}}C^{k}\prod_{j=1}^{k}e^{-(a-v)u_{j}}.
I_1(v) \sim (1-\theta^2)w_1^2\int_0^1 \frac{xv^2}{2\pi(w_1^2 +(1-\theta^2)v^2x^2)^{3/2}}\,dx = w_1 \int_0^{v(1-\theta^2)^{1/2}/w_1}\frac{x}{2\pi(x^2+1)^{3/2}}\,dx \to w_1 \int_0^\infty\frac{x}{2\pi(x^2+1)^{3/2}}\,dx=w_1/(2\pi).
\langle w_0 \wedge w_1, v_1 \wedge v_0 \rangle = \langle w_{-1}\wedge w_0, v_0 \wedge v_{-1}\rangle = \frac{q^{-2}}{[2]_{q^2}}, \langle w_{-1}\wedge w_1, v_1 \wedge v_{-1}\rangle = \frac{1}{[2]_{q^2}},
\chi_m^{\ell,s}=\chi_{m+2k+4}^{\ell,s}=\chi_m^{\ell,s+4}=\chi_{m+k+2}^{k-\ell,s+2}\;\;{\rm and}\;\;\chi_m^{\ell,s}=0 \;\;{\rm unless}\;\; \ell+m+s\equiv 0 \;\;{\rm mod}\; 2~,
U(R)|j,k\rangle=|j,k\rangle-i\frac{\theta}{\hbar}\hat{n}\cdot\vec{J}|j,k\rangle+\sum_{k=2}^{\infty}\frac{(-i\frac{\theta}{\hbar}\hat{n}\cdot\vec{J})^{k}}{k!}|j,k\rangle=exp(-i\frac{\theta}{\hbar}\hat{n}\cdot\vec{J})|j,k\rangle
{\cal T}~ = ~ \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{- 1}}&{{- 1}}&{{- 2}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{0}}&{{- 1}}\\{{0}}&{{0}}&{{1}}&{{4}}&{{5}}&{{8}}\\ \end{matrix}\right)
y^d=\nu((16\pi^4t^3(t-1)^3-3t^3(2t-2)-18t^2(t-1)^2-6t(t-1)^3)cos(2\pi x))+ &\nu(3t^3(2t-2)+18t^2(t-1)^2+6t(t-1)^3)+t^3(1-t)^3(1-cos(2\pi x)).
\mathrm{T r}( e^{- \tau_{2}H_{\mathrm N S}^{0}}) = \frac{a_{0}( H_{\mathrm N S}^{0})}{\tau_{2}^{5}}+ \frac{a_{2}( H_{\mathrm N S}^{0})}{\tau_{2}^{4}}+ \cdots
\begin{array}[c]{rl}\tilde{Y}(t)= p(t)\tilde{X}(t)+\varphi(t), \tilde{Z}(t)= \left( 1-p(t)\gamma_{2}(t)\right)^{-1}\left[ \left(\alpha_{2}(t)p(t)+\beta_{2}(t)p(t)^{2}+q(t)\right) \tilde{X}(t)\right. & \left. +p(t)\beta_{2}(t)\varphi(t)+p(t)L_{2}(t)+\nu(t)\right] .\end{array}
\left\|u^{\Delta x}\right\|_{L^{2}(\mathbb{R})}\le \|u^{n-1/2}\|_{L^{2}(\mathbb{R})}+ \|u^{n+1/2}\|_{L^{2}(\mathbb{R})}\le C\left(\|u_{0}\|_{L^{2}(\mathbb{R})}\right),
\tau = \left( \begin{matrix}{{i T_{1 1}}}&{{i T_{1 2}}}\\{{i T_{2 1}}}&{{i T_{2 2}}}\\ \end{matrix}\right) , I = \left( \begin{matrix}{{- 1_{2}}}&{{0_{2}}}\\{{0_{2}}}&{{1_{2}}}\\ \end{matrix}\right) ~ ,
_{n^{\prime}}\langle a a | \hat{H}| a a \rangle_{n}= \Big [ \frac{I_{n}( \zeta )}{n}+ \frac{I_{K - n}( \zeta )}{K - n}\Big ] \, \Big ( \delta_{n}^{n^{\prime}}+ \delta_{K - n^{\prime}}^{n}\Big ) \, ,
0 \ = \ -\lambda\iota + \lambda a \ = \ (\pi^{*}K_X\ +\E)\ \cdot \ [\ell-(a\lambda)e] & = \ (\chi^*\pi'^{*}K_W \ +\t \widetilde{H}_b)\ \cdot \ [\ell-(a\lambda)e] & = \K_W\cdot (\pi'\circ \chi)_*[\ell-(a\lambda)e] \ + \t\lambda (b-a)<0.
\begin{aligned}\frac{d}{dt}(e^{itH}P_a^{{\varepsilon}_a}(t) e^{-itH}f) =e^{itH}\{i[H,P_a^{{\varepsilon}_a}(t)]+\partial_tP_a^{{\varepsilon}_a}(t) \}e^{-itH}f &=e^{itH}\frac{1}{t}Q_a^{{\varepsilon}_a}(t)^*Q_a^{{\varepsilon}_a}(t)e^{-itH}f+O(t^{-1-\delta})f\end{aligned}
D_t\partial_s\gamma=\sum_{i,j,k=1}^d\left(\partial^2_{ts}\,\gamma_k+\partial_s\gamma_i\,\partial_t\gamma_j\,\Gamma_{ij}^k\right)\partial_k D_s\partial_t\gamma=\sum_{i,j,k=1}^d\left(\partial^2_{st}\,\gamma_k+\partial_t\gamma_i\,\partial_s\gamma_j\,\Gamma_{ij}^k\right)\partial_k
w_i^g(s,\tau,\hat y')=v_i^g(s,\tau,\beta^{(i)}(s,\tau,\hat y')), \ \ \ \hat y'\in \Gamma_2, \tilde w_i^g(s,\tau,\hat y')=w_i^g(s,\tau,\hat y')e^{-b^{(i)}(s,\tau,\hat y')}.
\begin{array}{r c l}{{\delta ( x_{i})}}&{{=}}&{{x_{i}\otimes 1 + q^{- h_{i}}\otimes x_{i}\ ,}}\\{{\delta ( y_{i})}}&{{=}}&{{y_{i}\otimes q^{h_{i}}+ 1 \otimes y_{i}\ ,}}\\{{\delta ( h_{i})}}&{{=}}&{{h_{i}\otimes 1 + 1 \otimes h_{i}\ ,}}\\{{\delta ( d )}}&{{=}}&{{d \otimes 1 + 1 \otimes d \ ,}}\\ \end{array}
Y_l^k(x)=A_l^k\prod_{\tau=1}^{n-1}C_{k_{\tau-1}-k_\tau}^{\frac{n-\tau}{2}+k_\tau}(\cos\vartheta_\tau)\sin^{k_\tau}\!\vartheta_\tau\cdot e^{\pm ik_{n-1}\varphi}
\lim_{\delta \to 0}\frac{I}{\delta}= - \frac{v_{\beta,2}'(g_{\beta,2}(x))}{( f_{\beta,2}'(g_{\beta,2}(x) ) )^2}= - \frac{X'_{\beta,2}(x)}{f_{\beta,2}'(g_{\beta,2}(x) )}.
\Vert P_\mathfrak{X}x\xi_\varphi\Vert_{L^2(M)}^2 \leq \big| (\pi(g)y\xi_\varphi |v_g y\xi_\varphi)_{L^2(M)}\big| + \frac{\varepsilon}{2}&\leq \sum_{k_1,k_2=1}^K \big| (\pi(g) w_{k_1}\xi_\varphi |v_g w_{k_2}\xi_\varphi)_{L^2(M)}\big| + \frac{\varepsilon}{2}
g_{n,R}(x_1,\ldots,x_n;t,s)= \int_{B_R}\big( f_n(x_1,\ldots,x_n,x;t) - f_n(x_1,\ldots,x_n,x;s) \big)dx & =\int_{B_R}\int_{T_n(t)}\prod_{j=1}^{n-1}G_{t_{j+1}-t_j}(x_{j+1}- x_j) \big( G_{t-t_n}(x-x_n) - G_{s-t_n}(x-x_n) \big) d\pmb{t}dx
\begin{array}{l}\alpha(\widehat{x}_i)\otimes 1 - 1\otimes\beta(\widehat{x}_i)= \widehat{x}_i\otimes 1 - 1\otimes e^{-\frac{\widehat{p}_0}{\kappa}}\widehat{x}_i, \alpha(\widehat{x}_i)\otimes 1 - 1\otimes\beta(\widehat{x}_i)= \widehat{x}_i\otimes 1 - 1\otimes\widehat{x}_0 + 1\otimes\frac{1}{\kappa}\widehat{p}_k\widehat{x}_k.\end{array}
c^k_{\mathrm{u2s},1}\left( N_{\mathrm{sc}}\right) = \mathsf{E}\left\lbrace B \sum^{S_{\mathrm{u2s}}\left( k \right)}_{s=1}\log_2\left[ \frac{\gamma_{\mathrm{u2s},s}\left( k \right)}{\gamma_{\mathrm{u2s},0}\left( k \right)}\right] \right\rbrace.
E^s_\lambda(\pm\infty) =\{x\in E: \exp(tJS_\lambda(\pm\infty))x\rightarrow 0\,\, t\rightarrow\infty\}=V^-(JS_\lambda(\pm\infty)) E^u_\lambda(\pm\infty) =\{x\in E: \exp(tJS_\lambda(\pm\infty))x\rightarrow 0\,\, t\rightarrow-\infty\}=V^+(JS_\lambda(\pm\infty)),
F_n(x) = G_{k,n}(x)+R_k(x) ~~~{\rm with}\,\, R_k(x) = O(\epsilon^k) ~~~{\rm and}G_{k,n}(x) = G(x)+\bigl\{\epsilon a_1(x) + \dots + \epsilon^{k-1}a_{k-1}(x) \bigr\}g(x),
g(t,s,\mathfrak{X}) := (\varphi_t-\mathcal{L}_{\alpha(s,\mathfrak{X})}\varphi)\big(t,\mathfrak{X}(s)\big)\ge \widehat{\mathcal{G}}\big(t,\mathfrak{X}(s)\big)\,,\forall\ (t,s) \in (0, \infty)\times [0, \infty)\,.
(v_{k}^{(s,t+1)}+V_{k+2}^{(s,t)}-v_{k+2}^{(s,t)}-V_{k+1}^{(s,t)})\tilde{\cal H}_{k+1}^{(s,t)}(z)+(v_{k}^{(s,t+1)}V_{k}^{(s,t)}-v_{k}^{(s,t)}V_{k+1}^{(s,t)})\tilde{\cal H}_{k-1}^{(s,t)}(z)=0.
a(z) k_{t\zeta_j, \ell}(z) = c_{\ell}\frac{z^{\ell}(z- \zeta_j)^{m_j}}{(1 - t\overline{\zeta_j}z)^{\ell + 1}}\prod_{k\neq j}(z-\zeta_k)^{m_k}& = c_{\ell}z^{\ell}(z - \zeta_j)^{m_j - (\ell + 1)}\left(\frac{z - \zeta_j}{1 - t\overline{\zeta_j}z}\right)^{\ell + 1}\prod_{k\neq j}(z-\zeta_k)^{m_k}.
\phi_1(\phi_2^\ell(0)) = \frac{1}{10}\left[1 - \left(\frac{3}{5}\right)^\ell\right] + \frac{1}{10}< \frac{3}{5}\left[\left(\frac{4}{5}\right)^{\ell}+ \frac{1}{2}\left(1 - \left(\frac{4}{5}\right)^\ell\right)\right] + \frac{1}{20}= \phi_2(\phi_1^\ell(1)),
B_{F}(\theta_{1}:\theta)+B_{F}(\theta_{2}:\theta)=B_{F}(\theta_{1}:\frac{\theta_{1}+\theta_{2}}{2})+B_{F}(\theta_{2}:\frac{\theta_{1}+\theta_{2}}{2})+2B_{F}(\frac{\theta_{1}+\theta_{2}}{2}:\theta)
\log \bigg( \frac{N \mu^{k^* + 1}e^{sk^* t^*}}{s^{k^* + 1}k^*!}\bigg) \leq -(k_N^+ - k_N) \log \bigg( \frac{s}{\mu}\bigg) + 2 k^* \log k_N - k^* \log k^* + k^* + o(k_N) &= - 2 k_N \log k_N + 2 k^* \log k_N - k^* \log k^* + k^* + o(k_N),
\Phi_{\widetilde{F}}( \widetilde{\boldsymbol{F}}_{n^2+k_n,n^2+\kappa_0}(z_0,x_0)) = \Phi_{\widetilde{F}}(z_0,x_0) +k_n-\kappa_0 + O\left(\frac{k_n^3}{n^2}\right) &= \Phi_{\widetilde{F}}(z_0,x_0) +k_n-\kappa_0 + o(1),
\omega^{G}( [ \Phi , \Phi^{*}] ) \approx \omega^{\prime}{}^{G}( [ \Phi , \Phi^{*}] ) \ \Leftrightarrow \, o m e g a^{\prime}{}^{G}- \omega^{G}={\cal S}\eta^{G - 1}( [ \Phi , \Phi^{*}] ) + \lambda \, \delta_{0}^{G}\ ,
a_{13}^{1342}=a_{13}^{1324}=a_{13}^{1432}=a_{13}^{1423}=a_{13}^{1234}=a_{13}^{1243}= a_{13}^{4132}=a_{13}^{4123}=a_{13}^{4213}=a_{13}^{2134}=a_{13}^{2143}=a_{13}^{2413}
{\cal L}^{\prime}\ = \{\cal L}\ - \ \left[ \frac{g^{2}}{3 2 \pi^{2}}\, ( T_{G}- T_{R}) \, \int d^{2}\Theta ~ 2{\cal E}~ \log P ~ W^{\alpha}W_{\alpha}\ + \ \mathrm{h . c .}\right] .
\Rightarrow\frac{\pi}{2}=\prod_{k=1}^{\infty}(\frac{2k}{2k-1}\cdot\frac{2k}{2k+1})=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\cdot\cdot\cdot
\begin{array}{r c l}{G ( x_{1}, x_{2})}&{=}&{\displaystyle - \frac{g^{2}}{2 !}\int d^{4}x_{3}\int d^{4}x_{4}A ,}\\{A}&{=}&{\displaystyle < 0 | T \left( \varphi_{1}\varphi_{2}( \chi \star \varphi \star \pi + \pi \star \varphi \star \chi )_{3}\right.}\\{}&{}&{\displaystyle \times \left. ( \chi \star \varphi \star \pi + \pi \star \varphi \star \chi )_{4}\right) | 0 > ,}\\ \end{array}
\mu_{k}^i\left(B\right) = \frac{1}{Z_{k}^i}\sum\limits_{\theta \in B}\prod\limits_{j=1}^{n}\mu_0^j\left(\theta\right)^{\left[A^{k}\right]_{ij}}\prod\limits_{t=1}^{k}\prod\limits_{j=1}^{n}\ell^j(s_{t}^j|\theta)^{\left[A^{k-t}\right]_{ij}}
\frac{1}{N}\sum_{\substack{n+n_{i_1},\dots,n+n_{i_p}\in M_x 1\leq n\leq N}}\mu(n)=\frac{1}{N}\sum_{1\leq n\leq N}\mu(n) (1-x(n+n_{i_1}))\cdots (1-x(n+n_{i_p})),
\Pi_{2}( k^{2}) = \frac{\beta_{2}a}{6 k^{4}}\left\{1 + \left( \frac{2 \beta_{3}}{\beta_{2}}\right){a}+ \left( \frac{3 \beta_{4}}{\beta_{2}}\right){a}^{2}+ O ({a}^{3}) \right\}
\partial_t \int_0^\infty g(s)\nabla u(t-s)ds =\frac{1}{\varepsilon}\partial_t \,\bigg(e^{-\frac{t}\varepsilon}\int_{-\infty}^t e^{\frac{s}\varepsilon}\nabla u(s)ds\bigg) &=\frac{1}{\varepsilon}\nabla u(t)-\frac{1}{\varepsilon}\int_0^\infty g(s)\nabla u(t-s)ds,
\bigg [ \prod_{I = 1}^{N}: e^{- i 2 \sqrt{\pi}\omega{( I )}U_{I a}\Phi^{( I )}( x )}:_{M^{( I )}}\; e^{+ i \frac{\theta}{N}}\; + \; \prod_{I = 1}^{N}: e^{+ i 2 \sqrt{\pi}\omega{( I )}U_{I a}\Phi^{( I )}( x )}:_{M^{( I )}}\; e^{- i \frac{\theta}{N}}\bigg ] \Bigg \rangle_{K^{\mu}}\; ,
|\tilde{\psi}(x) - \psi(x_0) - \nabla \psi(x_0) \cdot (x-x_0)|=|\tilde{\psi}(x) - \tilde{\psi}(x_0) - \nabla \tilde{\psi}\cdot (x-x_0)| \le \tilde{C}|x-x_0|^{2}
2 \langle b_n(x),x \rangle = 2 \langle b(\Phi_n(x)), \Phi_n(x) \rangle{\sqrt{1+ \big(1- \eta(\tfrac{x}{n}) \big) \, \tfrac{|x|^2}{n^2}}}\le 2 \langle b(\Phi_n(x)), \Phi_n(x) \rangle\Big (1 + \tfrac{|x|}{n}\Big),
M_{2}(\mathcal{NC}(G)) =\dfrac{19n(19n-1)^{3}}{2}+2\times \dfrac{(61n^{2}-19n)^{2}}{4}-3\times \dfrac{(61n^{2}-19n)}{2}(19n-1)^{2}~~~~~~~~~~~~~~+(19n-\frac{3}{2})(4n(4n-1)^{2}+15n(3n-1)^{2})-\dfrac{4n(4n-1)^{3}+15n(3n-1)^{3}}{2}&= \frac{1}{2}\times 74880n^{4}= 37440n^{4}.
{\partial W_0\over \partial m^2}=-{AT\over2}\sum_{n=1}^{\infty}\int{dk_1dk_2dk_0\over (2\pi)^3}\int_0^{\infty}ds s^{\nu}e^{-is(-k_0^2+k_1^2+k_2^2 +{n^2\pi^2\over a^2}+m^2)}\,.
\MoveEqLeft m \wedge ( m \wedge H(m)) \cdot \delta E(m) = m \wedge ( m \wedge (-\delta E(m)) \cdot \delta E(m) + h m \wedge ( m \wedge e_1) \cdot \delta E(m) & = |m \wedge \delta E(m)|^2 - h (m \wedge e_1) \cdot (m \wedge \delta E(m)) & = (|\delta E(m)|^2 - |m \cdot \delta E(m)|^2) - h (m \wedge e_1) \cdot (m \wedge \delta E(m)).
(1-2^{1-s})\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}-2\sum_{n=1}^\infty \frac{1}{(2n)^s}&=(1+2^{-s}+3^{-s}+\dots) - 2(2^{-s}+4^{-s}+6^{-s}+\dots) &= 1-2^{-s}+3^{-s}-4^{-s}+\dots =,
L_1=\frac{1}{i}(x_2 \partial_{x_3}-x_3 \partial_{x_2}),L_2=\frac{1}{i}(x_3 \partial_{x_1}-x_1 \partial_{x_3}),L_3=\frac{1}{i}(x_1 \partial_{x_2}-x_2 \partial_{x_1}).
D^{\mu \nu \lambda \sigma}( x - y ) = 1 6 \pi G \left( \eta^{\mu \lambda}\eta^{\nu \sigma}+ \eta^{\mu \sigma}\eta^{\nu \lambda}-{\frac{2}{d - 2}}\eta^{\mu \nu}\eta^{\lambda \sigma}\right) \int{\frac{d^{d}k}{( 2 \pi )^{d}}}{\frac{e^{i k \cdot ( x - y )}}{- k^{2}}}\, .
\langle \overline{{{\phi}}}^{2}( x ) \rangle_{\beta}= \frac{1}{1 2 \beta^{2}}( 1 + \eta^{2}) - \eta^{2}\xi \frac{\sqrt{2}}{8 \pi^{2}r^{2}}S_{1}+ \eta^{2}\frac{1}{3 2 \pi^{2}r^{2}}S_{2},
{\rm OSL}_X(r)\, \subset\, H^0\left(X,\,{\rm Diff}^r_X({\mathcal L}^{\otimes (r-1)},\,{\mathcal L}^{-r-1})\right)\,=\, H^0(X,\,{\rm Diff}^r_X\left({\mathcal L}^{\otimes (r-1)},\, ({\mathcal L}^{\otimes (r+1)})^*)\right)
\int \!{\cal D}\varphi \delta_{\eta}\phi{\frac{\delta S_{0}}{\delta \phi}}e^{- ( S_{0}+ S_{5})}= ( \delta_{\eta}^{( 0 )}\phi + \delta_{\eta}^{( 1 )}\phi ){\frac{\delta S_{0}[ \phi ]}{\delta \phi}}e^{- ( S_{0}[ \phi ] + S^{( 1 )}[ \phi ] )}
F_{n}(t) = \frac{2}{l}\int_{0}^{t}f(\xi, t) \sin(\tfrac{\pi n}{l}x) \xi \mathrm{d}\xi + M_{n}(\mu_{1}, \mu_{2})(t), \Phi_{n}(t) = \frac{2}{l}\int_{0}^{l}\varphi(\xi, t) \sin(\tfrac{\pi n}{l}\xi) \mathrm{d}\xi + m_{n}(\mu_{1}, \mu_{2})(t)
\pi^{-1/2}c_k e^{-x^2/2}\int e^{-y^2}\,dy= \pi^{-1/2}c_k x^k e^{-x^2/2}\int e^{-y^2}\,dy= \pi^{-1/2}c_k x^k e^{-x^2/2}\pi^{1/2}= x^k c_k e^{-x^2/2}
A_3(z,\bar{z})=\frac{1}{\theta}\partial_{\bar{z}}\left(\frac{\chi_3}{\theta}\right)=-\frac{1}{2\cosh(l)}\,,\quad\bar{A}_3(z,\bar{z})=\frac{1}{\theta}\partial_{z}\left(\frac{\chi_3}{\theta}\right)=-\frac{1}{2\cosh(l)}\,.
\Vert\widetilde x\Vert^2 = \Vert x_k \Vert_M^2, \Vert\widetilde r\Vert^2 = z_k^Tr_k = \Vert r_k \Vert_{M^{-1}}^2, \Vert\widehat b\Vert^2 = \Vert b \Vert_{M^{-1}}^2.
|M_n(t,k,\xi,\eta)|\leq (C\varepsilon_0)^n\int_{\zeta_1+\cdots+\zeta_n=\xi-\eta}\exp\bigg(-(C_0-3)\sum_{j=1}^n|\zeta_j|\bigg)\,\mathrm{d}\zeta_1\cdots\mathrm{d}\zeta_{n-1}\lesssim (C\varepsilon_0)^ne^{-(C_0-4)|\xi|},
K^{D,B(x_0,r)}(x_1,z) &=\int_{B(x_0,b_2 r)}G^{Y^D}_{B(x_0,r)}(x_1,y)J^{Y^D}(y,z)\, dy+\int_{A(x_0,b_2 r,r)}G^{Y^D}_{B(x_0,r)}(x_1,y)J^{Y^D}(y,z)\, dy &=I_1+I_2\, .
\log(k!) =(k+1)\log(k+1)-(k+1)-\frac{1}{2}\log\left(\frac{k}{2\pi}\right) + O(k^{-1}); 2\log(C_k) =k\log(2)+(k+1)\log(k+1)-(k+1)-\frac{1}{2}\log\left(\frac{k}{2\pi}\right) + O(k^{-1}).
\Pr_{x \sim \gamma_n}[p_s(x) \ge 0] - \sum_{s' \not =s}\Pr_{x \sim \gamma_n}[p_s(x) \ge 0 \wedge p_{s'}(x) \ge 0] \leq \Pr_{x \sim \gamma_n}[x \in \mathsf{Unique}(f,s)] \le \Pr_{x \sim \gamma_n}[p_s(x) \ge 0]
q^{\alpha^{2}/2}A_{q}\left(q^{\alpha}z\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\left(zq^{1/2}e^{ix};q\right)_{\infty}\exp\left(\frac{x^{2}}{\log q^{2}}+i\alpha x\right)}{\sqrt{\log q^{-1}}}dx,
\langle \hat{\rho}( x ) \hat{\rho}( y ) \rangle_{c}= -{\frac{1}{\pi^{2}N^{2}\beta}}{\frac{1}{( x - y )^{2}}}{\frac{a^{2}- x y}{[ ( a^{2}- x^{2}) ( a^{2}- y^{2}) ]^{1 / 2}}}.
q_{\rho}\left( x,t\right) =q\left( x,t\right) & +2\left[ \int\psi_{\rho}\left( x,t;\mathrm{i}s\right) \psi\left(x,t;\mathrm{i}s\right) \mathrm{d}\rho_{t}\left( s\right) \right]^{2}+4\int\psi_{\rho}\left( x,t;\mathrm{i}s\right) \psi^{\prime}\left(x,t;\mathrm{i}s\right) \mathrm{d}\rho_{t}\left( s\right) .
f(x) = \frac{1}{p(x)}\int_a^x \mathcal{L}^{\star}(g)(u) p(u)du = \frac{1}{p(x)}C_a^x(g, p) + \frac{1}{p(x)}\int_{a}^x g(u) \mathcal{L}(p)(u) du = \frac{1}{p(x)}C_a^x(g, p) =:F(g, p)(x)
{\int}d{\bf x}\, \left\{{\bar{\cal F}}^{b a}({\bf r},{\bf x}_{0}) \, + X_{i}\partial_{i}{\bar{\cal F}}^{b a}({\bf r},{\bf x}_{0}) + \frac{1}{2}\, X_{i}X_{j}\, \partial_{i}\partial_{j}{\bar{\cal F}}^{b a}({\bf r},{\bf x}_{0}) \; + \; \; \cdots \right\}{j}_{0}^{a}({\bf x})
\int_{0}^{\infty}\frac{x^{s}}{e^{x}-1}dx=\int_{0}^{\infty}x^{s}\frac{1}{e^{x}-1}dx=\int_{0}^{\infty}\frac{x^{s}}{e^{x}}\frac{1}{1-\frac{1}{e^{x}}}dx\wedge\frac{1}{1-r}=\sum_{n=0}^{\infty}r^{n}
\langle \bar{P}^{n}\cdots \rangle ={\frac{\partial^{n}}{\partial \bar{t}^{n}}}\langle \cdots \rangle = \left({\frac{\partial t_{s}}{\partial \bar{t}}}{\frac{\partial}{\partial t_{s}}}\right)^{n}= \left( -{\frac{1}{\langle P P \rangle_{0}}}{\frac{\partial}{\partial t_{s}}}\right)^{n}\ \langle \cdots \rangle \ ,
\lim\limits_{(\alpha_{1p}-\alpha_{jp})\downarrow 0}\frac{m_{j}\sin{(\alpha_{1p}-\alpha_{jp})}}{1-\cos{(\alpha_{1p}-\alpha_{jp})}}=\lim\limits_{(\alpha_{1p}-\alpha_{jp})\downarrow 0}\frac{m_{j}\cos{(\alpha_{1p}-\alpha_{jp})}}{\sin{(\alpha_{1p}-\alpha_{jp})}}=+\infty,
2k\zeta_{k-1}=2k\oint_\gamma\frac{u^{k-1}}{\sqrt{P(u)}}~du=\oint_\gamma\frac{u^kP'(u)}{P(u)^{3/2}}~du=\sum_{j=0}^nja_jI_{j+k-1},~~k=0,\ldots,n-1
\left| p_{0}^{1}( 4 , 2 ) , p_{1}^{1}( 4 , 2 ) , p_{2}^{1}( 4 , 2 ) , p_{3}^{1}( 4 , 2 ) , p_{0}^{2}( 4 , 2 ) , p_{1}^{2}( 4 , 2 ) \right\rangle \quad