harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle \nabla_j v_j , \nabla_j v_i \rangle_j (p)= \displaystyle \sum_{\mu}\frac{\partial_{\mu}v_j\bar{\partial}_{\mu}v_i}{\lambda_{\mu}}\leq \displaystyle \sqrt{\sum \frac{\vert\partial_{\mu}v_j \vert^2}{\lambda_{\mu}^2}}\sqrt{\sum \vert \partial_{\mu}v_i \vert^2}=\vert \nabla_j v_j\vert_i \vert \nabla_i v_i \vert_i.
- a_{- 1}\bar{Y}_{\lfloor s \rfloor}^{\frac{2 \theta}{\theta - 1}}+ a_{0}\bar{Y}_{\lfloor s \rfloor}^{\frac{2 \theta - 1}{\theta - 1}}+ \frac{1}{2}b^2 \theta \leq a_{0}+ \frac{1}{2}b^2 \theta =: K.
\widehat{S}_{( 0 )}p^{( 0 )}= p^{( 0 )}, \widehat{S}_{( 0 )}p^{( \widetilde{0})}= p^{( \widetilde{0})}, S_{( 1 )}^{2}p^{( 1 )}= p^{( 1 )}, S_{( 1 )}^{2}p^{( \widetilde{1})}= p^{( \widetilde{1})}.
\lim_{t \rightarrow \infty}e^{2\left[\gamma_{\beta}- \gamma_{\alpha}\right]t}\frac{\left|\cos \frac{1}{2}\omega_{\alpha}t\right|}{\left|\cos \frac{1}{2}\omega_{\beta}t\right|}= \frac{\left(\frac{1}{2}-\sigma\right)^2 + \omega^2}{\left(\frac{1}{2}+\sigma\right)^2 + \omega^2}.
h(\boldsymbol{\mu},D,\psi)=\int\left(\frac{1}{2M}\frac{|\boldsymbol{\mu}+ D\boldsymbol{A}|^2}{D}+ \frac{\hbar^2}{8M}\frac{(\nabla D)^2}{D}+ D\,\epsilon(\psi,\nabla\psi) \right)^3r\, .
f(\gamma) \triangleq \frac{\gamma \chi(\gamma) e^{- \gamma}}{e^{-\gamma}+\tau(1-e^{-\gamma})}&= \frac{\gamma \left( q_1 + \frac{q_2}{1!}\gamma + \frac{q_3}{2!}\gamma^2 + \cdots + \frac{q_M}{(M-1)!}\gamma^{M-1}\right) e^{- \gamma}}{e^{-\gamma}+\tau(1-e^{-\gamma})}
\left\Vert \Phi_N \right\Vert_{\mathcal{L}^1_{\beta,\mu}}=\sum_{s=1}^N \frac{1}{s!}\int_{\mathcal{D}_s}\left| \phi_N^{(s)}(Z_s) \right| e^{-\beta E_s(Z_s)}e^{-\mu s}dZ_s
Q_1 - \tilde{Q}_1 = \frac{3}{16 n \log^2n}\left(\begin{matrix}-1 i i 1\end{matrix}\right) + O\left(\frac{1}{n\log^3n}\right), Q_1^{(n)}- \tilde{Q}_1^{(n)}- \big(Q_1^{(n+1)}- \tilde{Q}_1^{(n+1)}\big) = \frac{3}{16 n^2 \log^2n}\left(\begin{matrix}-1 i i 1\end{matrix}\right) + O\left(\frac{1}{n^2\log^3n}\right).
\liminf_{\varepsilon\to 0^+}c^*_{\varepsilon}\geq 2\sqrt{a_{\inf}-\frac{\chi\mu a_{\sup}}{b_{\inf}-\chi\mu}}-\frac{\chi\mu\sqrt{N}a_{\sup}}{2\sqrt{\lambda}(b_{\inf}-\chi\mu)}:=c_{-}^*(a,b,\chi,\lambda,\mu)
\mathbb E\left[x^T(t) P^{-1}x(t)\right]= 2 \int_0^t \mathbb E\left[x^T(s) P^{-1}\left(A x(s)+ Bu(s) + \sum_{k=1}^m N_k x(s) u_k(s)\right)\right] ds &+\int_0^t \mathbb E\left[x^T(s)\sum_{i, j=1}^v H_i^T P^{-1}H_j k_{ij}x(s)\right] ds.
\ \u(m) = 1\; m_1 = M_1 \; \; 0 \leq m_j \leq M_j - 1 \; \; j \ne 1, & \; k = 2, \dots, g & \ \u(m) = 0\; m_k = M_k \; \; 0 \leq m_j \leq M_j - 1 \; \; \j \ne k.
\begin{aligned}\mathcal{S}_{\alpha\beta}(r,s)=\ \tau_{ji}^{(\alpha)}(r)e_{ij}^{(\beta)}(s)+\mu_{kji}^{(\alpha)}(r)\kappa_{kji}^{(\beta)}(s) - \sigma_{i}^{(\alpha)}(r)E_{i}^{(\beta)}(s)-Q_{ji}^{(\alpha)}(r)V_{ij}^{(\beta)}(s)\end{aligned}
\Phi^{\left( chiral)\right)}(x,y,\theta ,\bar{\theta})=\exp \left( \frac12\theta^{\alpha a}\sigma_{\alpha \dot{\beta}}^\mu \gamma_{a\dot{b}}^m\bar{\theta}^{\dot{\beta}\dot{b}}\partial_\mu \tilde{\partial}_m\right)F\left( x,y,\theta \right)
X((R_v, R_w),x,t) = \left(\begin{array}{c}(1-(v_0+V)^2 + \varphi_1) R_v -\frac{R_v^{3}}{3}-R_w -R_v^2 F_v-R_v F_v^2 -\frac{F_v^3}{3}+ \varphi_2 (R_v + F_v)^2 \varepsilon(R_v- \gamma R_w)\end{array}\right)
{\cal F}_{+ -}(p) \equiv \frac{d}{d{\alpha}_{-p}}{\cal A}_{+}-\frac{d}{d{\alpha}_{p}}{\cal A}_{-}= (1-{\rm N}^2) \frac{e_{+}^2{\rm L}^2}{4{\pi}^2{\hbar}^2}\frac{1}{p}.
R^*_{k,pt}\le 4R^{SC}_{k-l,pt}R^*_{l,pt}&=4\frac{R^*_{l,pt}}{R^*_{l}}\frac{R^*_{k+m}}{R^*_{k+m,pt}}\frac{R^{SC}_{k+m-l}R^*_l}{R^*_{k+m}}\frac{R^{SC}_{k-l,pt}}{\rho^{k-l}}\frac{\rho^{k+m-l}}{R^{SC}_{k+m-l}}\rho^{-m}R^*_{k+m,pt}
J=\int_{0}^{\pi}\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\theta}{|\rho \sin\theta+1|^{\tau}\rho^{q-1}}+\int_{\pi}^{2\pi}\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\theta}{|\rho \sin\theta+1|^{\tau}\rho^{q-1}}\doteq J_1+J_2.
\frac{F}{V}=\left(\frac{1}{4\pi^2\alpha'}\right)^5\int^{1/2}_{-1/2}d\tau_1\int_0^\infty\frac{d\tau_2}{\tau^6_2}\left|\frac{1}{\eta(\tau)}\right|^{24}2\sum_{\stackrel{n=1}{n\odd}}^{+\infty}e^{-\frac{\beta^2 n^2}{4\pi\alpha'\tau_2}}\left|\theta_2(0,\tau)\right|^8
c_{w}(z)= \begin{cases}\frac{\Gamma (2z ) \Gamma (\frac{1}{2}g_S+z)}{\Gamma (g_S+2z) \Gamma (\frac{1}{2}g_S+g_L+z)}\ \ \ , \frac{\Gamma (2z)}{\Gamma (\frac{1}{2}+g_S+z)}\ . \end{cases}
\sum_{i=1}^{\infty}\frac{16^{-i}\binom{2 i}{i}\left(\sum_{j=1}^i \frac{1}{1+2 j}-\frac{2}{9 (2i +1)}\right)}{(1+2 i)^3}=\frac{1}{216}\left(7 \pi^3 l_1+5 \pi \zeta_3-7 \pi^3+48\right)
\begin{array}{c}{{2 \, \Im m \, \displaystyle \int \displaystyle \frac{d x}{i}\varphi^{\prime}( x + i \epsilon ) \log ( 1 + f ( x + i \epsilon ) ) =}}\\{{= - 2 \, \Re e \, \displaystyle \int_{\Gamma}\displaystyle \frac{d u}{u}\log ( 1 + u ) - \displaystyle \frac{1}{2}\displaystyle \int d x \, G ( x ) \left( F^{2}( + \infty ) - F^{2}( - \infty ) \right)}}\\ \end{array}
S_\sigma([a_i, a_i + \ell_n(\Delta)]) = [a_i \varrho^n + a - a_i \varrho^n, a_i \varrho^n + a - a_i \varrho^n + \ell_n(\Delta)\varrho^n] = [a, a+\ell_n(\Delta) \varrho^n] = [a,b] = \Delta.
G_{lm,l^{\prime}m^{\prime}}\left( R\mathbf{\hat{e}}_{z}\right)=\delta_{m,m^{\prime}}\left( -1\right)^{m}\sum_{j=\left\vert l-l^{\prime}\right\vert}^{l+l^{\prime}}\left(\begin{array}[c]{ccc}l l^{\prime}j m -m^{\prime}0\end{array}\right) g_{l,l^{\prime}}^{j}\left( R\right) .
\bar{Z}_n^*= 2^{-1}\big(\bar{\tau}_n\bar{s}_n\big)^{-2}\Bigg[2\bar{\tau}_n \bar{s}_n^2\Big(n^{-1}\sum_{i=1}^{n}\psi'(\bar{\epsilon}_i)(G_i^*-\mu_{G^*})\Big) - \bar{\tau}_n^2 \Big(n^{-1}\sum_{i=1}^{n}\psi^2(\bar{\epsilon}_i)[(G_i^*-\mu_{G^*})^2-\sigma_{G^*}^2]\Big)\Bigg]
\hat{\Omega}\vert j_{1 2}, j_{1}, j_{2}> = \Omega \vert j_{1 2}, j_{1}, j_{2}> ={\frac{1}{2}}\left( j_{1 2}( j_{1 2}+ 1 ) - j_{1}( j_{1}+ 1 ) - j_{2}( j_{2}+ 1 ) \right) \vert j_{1 2}, j_{1}, j_{2}>
S_{2}= \frac{1}{2}\int d^{2}\xi \left[ \left( \partial_{\tau}\psi_{M}\right)^{2}+ \left( \partial_{\sigma}\psi_{M}\right)^{2}+ \frac{2}{\tau^{2}}( \psi_{M})^{2}+ \frac{c_{m}^{\prime}}{\tau a}\left( \psi_{m}\partial_{\sigma}\psi_{0}- \psi_{0}\partial_{\sigma}\psi_{m}\right) \right] \: .
\det(\gamma^i e)_{i=0,1,\dotsc,n-1}= \det\bigl((-1+T)^i e\bigr)_{i=0,1,\dotsc,n-1}&= \det(T^i e)_{i=0,1,\dotsc,n-1}&= 2^{n(n-1)/2}\det(1-y)^{1-n}\det\bigl((1-y)^i e\bigr)_{i = n-1,n-2,\dotsc,0}&= 2^{n(n-1)/2}\det(1-y)^{1-n}\det(y^i e)_{i=0,1,\dotsc,n-1}.
|T(y) - \tau(y)| = \Big|T\Big(\sum_{p=1}^n \alpha_p u_p y u_p^*\Big) - \tau(y)\Big| & = \Big|T\Big(\sum_{p=1}^n \alpha_p u_p y u_p^* - \tau(y)I\Big)\Big| & \leq \Big|\Big|\sum_{p=1}^n \alpha_p u_p y u_p^* - \tau(y)I\Big|\Big| & < \frac{1}{3K}& < \frac{\varepsilon}{3}.
\left|\int_{\mathbb R}g'(x)f'(x)\overline{f(x)}\,\mathrm dx \right| \leq \|f\|_\infty \|f'\|_{L^2}\|g'\|_{L^2}\leq 4 \|q_-\|_{L^1}^{\frac{3}{2}}\|f\|_{L^2}^2 \sqrt{\frac{2}{\delta}}&\leq 16\cdot \sqrt{3}\|q_-\|_{L^1}^{2}\|f\|_{L^2}^2.
\begin{array}{l l l}{{\delta e^{a}= \frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{a}\psi}}&{{\hspace{1 c m}}}&{{\delta \omega^{a b}= - \frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{a b}\psi}}\\{{}}&{{}}&{{}}\\{{\delta \psi = D \epsilon}}&{{\hspace{1 c m}}}&{{\delta A^{a b c d e}= \frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{a b c d e}\psi .}}\\ \end{array}
\frac{1}{n}e^{n-1}_{i,g+1}+ \left(-1+\frac{1}{n}\right)e_{i,g}^{n-1}+\left(\frac{1}{n}-\frac{1}{n^2}-1\right)e_{i,g+1}^{n-2}+\left(-1+\frac{1}{n}\right)\sum_{\tau=2}^{n-2}e_{i,g+\tau}^{n-1-\tau}.
a^{(1)}= \pi_{X^+}(\sigma^{-1}b) = b_{-1}b_0b_1... a^{(2)}= \pi_{X^+}(\sigma^{-2}b) = b_{-2}b_{-1}b_0... \vdots a^{(n)}= \pi_{X^+}(\sigma^{-n}b) = b_{-n}b_{-n+1}b_{-n+2}... .
0=\left[ \Gamma_{\underline{01}}-E \Gamma_*\right](P-Q)\gamma_2\epsilon =- 2 \left[ \Gamma_{\underline{01}}-E \Gamma_*\right]\gamma_2 Q\epsilon=-2\gamma_2 Q\left[ \Gamma_{\underline{01}}+E \Gamma_*\right]\epsilon\,.
\varepsilon_s-(L\varepsilon)_y+b\Lambda\varepsilon= \bigg{(}\frac{\lambda_s}{\lambda}+b\bigg{)}(\Lambda Q_b+\Lambda\varepsilon)+\bigg{(}\frac{x_s}{\lambda}-1\bigg{)}(Q_b+\varepsilon)_y &+\Phi_b-b_sP_b-(R_b(\varepsilon))_y-(R_{NL}(\varepsilon))_y,
M_{a b , c}^{d}\equiv{\delta_{( a}}^{d}\hat{G}_{b ) c}+ \frac{1}{2}f_{e ( a}^{\d}f_{b ) c}^{\e}N_{a b , c}^{d e}\equiv \frac{i}{2}\delta_{( a}^{( d}f_{b ) c}^{\e )}.
\begin{array}{lll}{\displaystyle \sum_{k=1}^{M}{(a_{k}-a_{k-1})b_{k}}=a_{M}b_{M}-a_{0}b_{1}-\sum_{k=1}^{M-1}{a_{k}(b_{k+1}-b_{k})}.}begin{align*}2mm]{\displaystyle \sum_{k=1}^{M}{(a_{k}-a_{k-1})b_{k}}=a_{M}b_{M}-a_{0}b_{0}-\sum_{k=1}^{M}{a_{k-1}(b_{k}-b_{k-1})}.}begin{align*}2mm]\end{array}
\lim_{n \rightarrow +\infty}\left(\frac{\lambda_{2,n}}{\lambda_{1,n}}+ \frac{\lambda_{1,n}}{\lambda_{2,n}}+ \frac{|x_{1,n}- x_{2,n}|}{\lambda_{1,n}}+ \frac{|t_{1,n}- t_{2,n}|}{\lambda_{1,n}}\right) = + \infty.
Z = \int D \phi D \chi D \rho \mathrm{E x p}\left[ i \{\frac{1}{2}[ \partial_{\mu}\phi_{i}]^{2}- \frac{1}{2}\chi{\vec{\phi}\cdot \vec{\phi}}- N V [{\rho}] + \frac{N}{2}\rho \chi + \vec{j}\cdot \vec{\phi}+ N S \chi + N J \rho \}\right]
\theta ( g ) = \left[ \delta_{j}^{i}+ \frac{1}{2 !}c_{j k}^{i}g^{k}+ \frac{1}{3 !}c_{j k_{1}}^{h_{1}}c_{h_{1}k_{2}}^{i}g^{k_{1}}g^{k_{2}}+ \frac{1}{4 !}c_{j k_{1}}^{h_{1}}c_{h_{1}k_{2}}^{h_{2}}c_{h_{2}k_{3}}^{i}g^{k_{1}}g^{k_{2}}g^{k_{3}}+ \ldots \right] d g^{j}X_{i}\quad ,
\theta_{s,a^2}^2=\big(\bar{r}^2(s,a^2)-\bar{r}^2(s,a_s^2)\big)-\beta\left(\sum_{s'\in S}p_{s'}\bar{r}^2(s',a_{s'}^2) -\sum_{s'\in S}p^2(s'|s,a^2) \bar{r}^2(s',a_{s'}^2)\right),
\begin{aligned}{\mathbb F'}^h_r(B):= \mathbb F^{h}_r(a',b)+ \mathbb F^{h}_r(a,b')\subseteq \mathbb F^{h}_r(a,b)\subseteq H_r(Y) \mathbb F^h_r(B):= \mathbb F^{h}_r(a,b)/{\mathbb F'}^{h}_r(B).\end{aligned}
G_{\texttt{b}}(\xi_1,\xi_2)= \sum_{m\geq \mu_1> \mu_2\geq 0}e^{i\mu_1\xi_1+i\mu_2\xi_2}+\frac{1}{1+q}\sum_{m\geq \mu_1= \mu_2\geq 0}e^{i\mu_1\xi_1+i\mu_2\xi_2}.
\|\omega_n\|^2_X-\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega_n(x,0)|^2\ln|\omega_n(x,0)|dx-\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega_n(x,0)|^2dx & -(2_\alpha^*-1)\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_n(x,0)|^{2_\alpha^*}dx=o_n(1).
\overline{\mathfrak{f}}= \{\lambda \in E_r \mid \langle \lambda + \rho, \alpha^\vee\rangle = m_\alpha e \alpha \in R^0_+(\mathfrak{f}), & (m_\alpha-1)e \leq \langle \lambda + \rho, \alpha^\vee\rangle \leq m_\alpha e \alpha \in R^1_+(\mathfrak{f}) \}.
\frac{1}{|I_k|}\left|\int_{I_k}\exp(ib(\psi^s_{p,(-1,1)}(\tau;\varphi)+\alpha \tau)) d\tau\right|= \frac{1}{2}\left|\int^{1}_{-1}\exp(i\delta(k) b(\psi^s_{q'(k),(-1,1)}(\tau;\varphi)+\alpha' \tau)) d\tau\right|
\langle Q(f,f), M_{\Lambda}^2f \rangle =\langle Q(f,M_\Lambda f), M_{\Lambda}f \rangle+\langle M_\Lambda Q(f,f) - Q(f,M_\Lambda f), M_{\Lambda}f \rangle &\leq - \widetilde{C}_{f_0}\|M_{\Lambda}f\|_{H^{\nu}}^2+ C_{f_0}\|M_{\Lambda}f\|_{L^2}^2+ \langle M_{\Lambda}Q(f,f) - Q(f,M_{\Lambda}f), M_{\Lambda}f\rangle
I^{\lhd\lambda}X=I^{\lhd\lambda}I^{\le\lambda}X=\sum_{\mu\lhd\lambda}I^{\le\mu}I^{\le\lambda}X\subset\sum_{\mu\lhd\lambda}\sum_{\nu\le\lambda,\mu}I^{\le\nu}X\subset I^{<\lambda}X=0
\left||g_z+tg_{z,j}|^{q-2}\overline{(g_z+tg_{z,j})}g_{z,j}\right|=|g_z+tg_{z,j}|^{q-1}|g_{z,j}|\leq(|g_z|+|g_{z,j}|)^{q-1}|g_{z,j}|\in L^1(\Omega)
J^{\mathrm{2-loop}}(\phi_B)=\frac{\lambda_0\phi^3_B}{1+\frac{9\lambda_0}{16\pi^2}\ln \frac{\Lambda^2}{3\lambda_0\phi^2_B}}+\frac{\lambda_0^3\phi^3_B}{256\pi^4}\left[27\ln \frac{\Lambda^2}{3\lambda_0\phi^2_B}+ 54 + 432\Omega(1) \right]
{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z) =\frac{1}{B(b,c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}\exp(zt){}_{0}\Psi_{1}\left( \alpha,\beta;-\frac{p}{t(1-t)}\right)dt,{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z) =\frac{1}{B(b,c-b)}\int_{0}^{1}u^{c-b-1}(1-u)^{b-1}\exp(z(1-u)){}_{0}\Psi_{1}\left( \alpha,\beta;-\frac{p}{u(1-u)}\right) du,
X_{s,t}-X_{s,0}-X_{0,t}+X_{0,0}&=\int_0^t\int_0^sb(\xi,\zeta,X_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta+\int_0^t\int_0^sa(\xi,\zeta,X_{\xi,\zeta})\,\mathrm{d}W_{\xi,\zeta},\quad\forall\,(s,t)\in D.
k_\alpha(A^{m+n},K^{m+n})= \sup_{\substack{{\omega_0 \in \Sigma}\hat{p}\neq \hat{q}}}\int \frac{\delta(\hat{A}^{m+n}(\omega)\hat{p}, \hat{A}^{m+n}(\omega)\hat{q})^\alpha}{\delta(\hat{p},\hat{q})^\alpha}\;dK^{m+n}_{\omega_0}(\omega_1,\omega_2, \dots , \omega_{m+n}).
+\frac 58 F^{\lambda\nu}F^{\mu\rho}F_{\lambda\mu}F_{\nu\rho}+2F_{\mu\rho}~^{;\rho}F^{\mu\lambda}~_{;\lambda}-2(F_\rho~^\nu F^{\rho\mu})_{;\nu\mu}+(F_{\nu\rho}F^{\nu\rho})^{;\mu}~_{\mu}~~.
D^-\Bigl(\frac{g_1(\lambda)g_2(\lambda)}{H_\lambda}\Bigr) =g_1(\lambda)D^-\Bigl(\frac{g_2(\lambda)}{H_\lambda}\Bigr)+g_2(\lambda)D^-\Bigl(\frac{g_1(\lambda)}{H_\lambda}\Bigr) &-\sum_{\lambda^{-}}\frac 1{H_{\lambda^-}}\bigl(g_1(\lambda)-g_1(\lambda^-)\bigr)\, \bigl(g_2(\lambda)-g_2(\lambda^-)\bigr).
K_n(x,y): =K^{CUE(n)}(x,y)=\dfrac{1}{2\pi}\dfrac{\sin (n(x-y)/2)}{\sin ((x-y)/2)}&=\dfrac{1}{2\pi}\sum\limits_{k=0}^{n-1}e^{(k-(n-1)/2)i(x-y)}.
\sum_{t=1}^\infty \alpha_k^{t}= \infty,~~\sum_{t=1}^\infty (\alpha_k^{t})^2 < \infty,\forall k\in{\cal K}\sum_{t=1}^\infty \alpha^{t}= \infty,~~\sum_{t=1}^\infty (\alpha^{t})^2 < \infty.
\overline{H}(G_1 \overset{(\alpha, 1-\alpha)}{\sqcup}G_2) = \alpha \overline{H}(G_1) + (1-\alpha)\overline{H}(G_2) \Longleftrightarrow\: \overline{H}(G_1 \wedge G_2) = \overline{H}(G_1) + \overline{H}(G_2).
x^*_{l_2}(Qz_0) \geqslant \frac12x^*_{l_2}(x_{l_2}) - \frac12|x^*_{l_2}(x_{l_1})| - |x^*_{l_2}(v)| \geqslant \frac{1-2\delta}{2}-\delta-\delta=\frac12-3\delta.
S_{0}= \frac{1}{2}\int_{0}^{1}d \tau [ ~ \frac{\dot{x}^{2}}{e}+ e m^{2}+ \chi \psi_{a}\dot{x}^{a}/ e - \psi_{a}\dot{\psi}^{a}+ \psi_{S}\dot{\psi_{S}}- m \psi_{S}\chi ~ ] ,
\Phi_{1a}(2,2;\tau) =q^{1/3}\prod_{n=1}^{\infty}\frac{(1-q^{9n})}{(1-q^n)}&=q^{1/3}+q^{4/3}+2q^{7/3}+3q^{10/3}+5q^{13/3}+7q^{16/3}+\cdots,\intertext{and}\Phi_{1b}(2,2;\tau) =q\prod_{n=1}^\infty\frac{(1-q^{9n})(1-q^{9n-1})(1-q^{9n-8})}{(1-q^{n})(1-q^{9n-4})(1-q^{9n-5})}&=q+q^3+q^4+3q^5+3q^6+5q^7+6q^8+\cdots
\int \!\! \int d^dk_1~d^dk_2~\left( \frac{\partial}{\partial k_{1\mu}}+\frac{\partial}{\partial k_{2\mu}}\right)~\left[ k_{1\mu}~P_{k_1,m_1}^{\nu_1}P_{k_1-k_2,m_2}^{\nu_2}P_{k_2,m_3}^{\nu_3}\right] \equiv 0,
v_{1}^{\prime}= \frac{v_{2}\Delta ( v_{1})}{\omega v_{1}\Delta ( v_{2})}, ~ ~ \Delta ( v_{1}^{\prime}) = \frac{\Delta ( v_{2}/ ( \omega v_{1}) )}{\Delta ( v_{2})}, ~ ~ v_{2}^{\prime}= \frac{\Delta ( v_{1})}{\Delta ( v_{2})}, ~ ~ \Delta ( v_{2}^{\prime}) = \frac{\omega v_{1}\Delta ( v_{2}/ ( \omega v_{1}) )}{\Delta ( v_{2})};
\omega^M_{1010c}=-\frac{2}{3}e^{\phi/3}d\phi_c \omega^M_{10bc}=-\frac{1}{2}e^{4\phi/3}{G_2}_{bc}\omega^M_{a10c}=-\frac{1}{2}e^{4\phi/3}{G_2}_{ac}\omega^M_{abc}=e^{\phi/3}\omega^N_{abc}-\frac{1}{3}e^{\phi/3}(d\phi_b\eta_{ac}-d\phi_c\eta_{ab}).
\frac{d}{d\theta}G(\theta) = - \frac{1}{\alpha - 1}\big(1+\theta^{-(\alpha - 1)}\big)^{- \frac{1}{\alpha - 1}- 1}\frac{d}{d\theta}\theta^{-(\alpha - 1)}= \big(1+\theta^{-(\alpha - 1)}\big)^{- \frac{\alpha}{\alpha - 1}}\theta^{-\alpha}.
\bar{\alpha}_t\alpha_t=q_tg_t^{-1}\bar{q}_tg_t^{-1}=g_t^{\frac{1}{2}}g_t^{-\frac{1}{2}}q_tg_t^{-1}\bar{q}_tg_t^{-\frac{1}{2}}g_t^{-\frac{1}{2}}
{\cal L}_{eff}= \frac{1}{4\pi}{\rm Im}\left[ \int{d^{4}\theta\frac{\partial{\cal F}(A)}{\partial A}\bar{A}}+\int{d^{2}\theta \frac{1}{2}\frac{\partial^{2}{\cal F}(A)}{\partial A^{2}}W^{\alpha}W_{\alpha}}\right] ,
\begin{array}{l l}{\delta_{( \xi )}^{S}M^{\mu \nu}={\cal L}_{\xi}( \sqrt{g}g^{\mu \nu}) ,}{}{\delta_{( \xi )}^{S}L^{\mu \nu}={\cal L}_{\xi}\lbrack M^{\mu \nu}- s ( \sqrt{g}g^{\mu \nu}) \rbrack .}\end{array}
-\int_0^1 S^{\wedge, (1)}_{[\nu/\alpha]}(0,y) \:d\alpha = -2 \int_0^1 \Theta\Big(\xi^2- \frac{\nu}{\alpha}\Big) \:H^{(1)}_{[\nu/\alpha]}(0,y) \:d\alpha &= -\frac{1}{8 \pi}\int_0^1 \Theta\Big(\xi^2- \frac{\nu}{\alpha}\Big) \:d\alpha= -\frac{1}{8 \pi}\Big( 1 - \frac{\nu}{\xi^2}\Big)
\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (-s^{(1)}_{k})_{k=0}^{m-1}\right) \stackrel{a\to0+}{\longrightarrow}\mathcal{D}\left((-s^{(1)}_{k+1})_{k=0}^{2m-3}, (-s^{(1)}_{k})_{k=0}^{m-1}\right) = (-1)^{m}\mathcal{D}\left((-s^{(1)}_{k})_{k=0}^{2m-3}, (-s^{(1)}_{k})_{k=m}^{2m-1}\right)
\Psi^{[B;M,m,\epsilon]}_{a,b;\epsilon^\prime}(\tau,z_1,z_2,t)=q^{\frac{2m+1}{2M}ab}e^{\frac{2\pi i(2m+1)}{2M}(bz_1+az_2)}\Phi^{[B;m]}(M\tau,z_1+a\tau+\epsilon,z_2+b\tau+\epsilon,\frac{t}{M}),
\hat{p}\left(s_{m}^{r}|\boldsymbol{y}\right)=\frac{\pi_{s^{r},m}^{in}\pi_{s^{r},m}^{out}}{\pi_{s^{r},m}^{in}\pi_{s^{r},m}^{out}+(1-\pi_{s^{r},m}^{in})(1-\pi_{s^{r},m}^{out})},\forall m,
\Phi(r)\equiv\frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\cdot F(r^{\prime})}{|r-r^{\prime}|}dV^{\prime}-\frac{1}{4\pi}\oint_{S}\hat{n}^{\prime}\cdot\frac{F(r^{\prime})}{|r-r^{\prime}|}dS^{\prime}
\mathcal G_{(\varphi,g)}=\bigcup_{p\in M}\left\{(X_1, \ldots, X_r, Y_1, \ldots ,Y_s) \in \mathcal F_p (M) \colon \begin{array}{c}X_1, \ldots , X_r \in (D_{\phi})_p, Y_1, \ldots , Y_s \in (D_{\bar\phi})_p, g(X_i, X_j)=\delta_{ij}, i, j=1, \ldots, r, g(Y_i, Y_j)=\delta_{ij}, i, j=1, \ldots, s, g(X_i, Y_j)=0, i,=1, \ldots r, j=1,\ldots, s \end{array}\right\},
|T_n|^r =\sum_{i=1}^n Y_i\int_0^1 \left (\psi'_r(T_{i-1}+t Y_i)-\psi'_r(T_{i-1})\right ) \, dt+ \sum_{i=1}^n Y_i \left ( \sum_{j=1}^{i-1}\psi'_r(T_{j})-\psi'_r(T_{j-1}) \right) &=\sum_{i=1}^n Y_i\int_0^1 \left (\psi'_r(T_{i-1}+t Y_i)-\psi'_r(T_{i-1})\right ) \, dt+ \sum_{i=1}^{n-1}\left ( \psi'_r(T_{i})-\psi'_r(T_{i-1}) \right) (T_n-T_i).
w(t,x) = w_0(x) + \int_0^t \partial_t w(s,x) \mathrm{d}s\begin{cases}\leq w_0(x) + \left( -g\big(c_0(x)\big) \frac{1}{w_0(x)}+ \varepsilon \right) \cdot t, \geq w_0(x) + \left( -g\big(c_0(x)\big) \frac{1}{w_0(x)}- \varepsilon \right) \cdot t\end{cases}
\sum_{\l \leq L}\vert x_{\l}\vert^2 \ll \sum_{\l_1\leq L\atop (\l_1,r) =1}\vert \mu_f (\l_1)\vert^2 \sum_{\l_2 \leq L/\l_1\atop (\l_2, r) =1}\vert \mu_f (\l_2)\vert^2 \ll \sum_{\l_1\leq L\atop (\l_1,r) =1}\sum_{\l_2 \leq L/\l_1\atop (\l_2, r) =1}\vert \lambda_f (\l_2)\vert^2 \ll \sum_{\l_1\leq L\atop (\l_1,r) =1}(L/\l_1) \ll L,
-{\rm dim\tot}_u\phi(p_L,j_!{\cal F}) =({\rm Char}_{\cal L}(\bar j_!{\cal F}),[dp_L])_{T^*X,u}, -{\rm dim\tot}_{u'}\phi(p_{L'},j'_!{\cal F}') =({\rm Char}_{{\cal L}'}(\bar j'_!{\cal F}'),[dp_{L'}])_{T^*X',u'}.
{\cal L}={m\over2}\,\epsilon^{\mu\nu\lambda}A_{\mu}^{a}(\partial_{\nu}A_{\lambda}^{a}+{g\over 3}f^{abc}A_{\nu}^{b}A_{\lambda}^{c}) -F^{a}(\partial_{\mu}A^{\mu a}) +\partial^{\mu}\overline{c}^{a}(D_{\mu}c^{a})
h^0(Y_c, \mathcal O_{Y_c}(mK_{Y_c})) =\chi (Y_c, \mathcal O_{Y_c}(mK_{Y_c})) & =\deg f_c\cdot \chi (X_c, \mathcal O_{X_c}(mK_{X_c})) & =\deg f_c \cdot h^0(X_c, \mathcal O_{X_c}(mK_{X_c}))
{\prod_j}^{(1)}=\prod_{j=1}^{N_1},~~{\prod_{i,j}}^{(1)}=\prod_{i,j=1}^{N_1},~~{\prod_{i<j}}^{(1)}=\prod_{1\le i<j\le N_1}, &{\prod_j}^{(2)}=\prod_{j=N_1+1}^{N},~~{\prod_{i,j}}^{(2)}=\prod_{i,j=N_1+1}^{N},~~{\prod_{i<j}}^{(2)}=\prod_{N_1+1\le i<j\le N},~~{\prod_{i,j}}^{(3)}=\prod_{i=1}^{N_1}\prod_{j=N_1+1}^N.
f ( s , t ) ~ ~ \sim ~ ~{\frac{1}{t}}{\frac{\Gamma \left( 1 - i ( G s + \alpha{\frac{k_{\perp}^{2}}{2 s}}) \right)}{\Gamma \left( i ( G s + \alpha{\frac{k_{\perp}^{2}}{2 s}}) \right)}}~ ~ .
A = \big\{(x,y) \mapsto (\lambda_1 x + \lambda_2 y + \lambda_3,\ \mu_1 x + \mu_2 y + \mu_3) : \lambda_1\mu_2 \neq \lambda_2\mu_1, \lambda_i,\mu_i \in k\big\}, B = \big\{(x,y) \mapsto (\lambda x + \mu,\ \eta y + f(x)) : \lambda, \eta \in k^{\times}, \mu \in k, f(x) \in k[x]\big\};
\omega_{a}^{b}\big{(}[X_{\beta}, X_{\gamma}\big{]}\big{)}= g\big{(}X_b, \nabla_{[X_{\beta}, X_{\gamma}]}X_a \big{)}g \big{(}\nabla_{X_{\beta}}X_b, \nabla_{X_{\gamma}}X_a \big{)}=g \big{(}X_{\rho},\nabla_{X_{\beta}}X_b\big{)}g\big{(}X_{\rho},\nabla_{X_{\gamma}}X_a \big{)}= -\omega^{b}_{\rho}(X_{\beta}) \omega_{a}^{\rho}(X_{\gamma}),
g = \displaystyle \prod_{j=1}^{n}g_j^{(1)}= \displaystyle \prod_{j=1}^{n}\Biggl({{a_0^{(j)}+e^{-{k_j}(r+it)}}\over{\bar a_0^{(j)}+ e^{{k_j}(r+it)}}}\Biggr)
a(u,JIu-Jz_{\mathrm{nc}})+a_{\mathrm{pw}}(z_{\mathrm{nc}}, z_{\mathrm{nc}}-Iu) = b(\lambda u, JIu-Jz_{\mathrm{nc}}+ (1+\delta)(z_{\mathrm{nc}}-Iu)) =\lambda b( u, (J-1)(Iu-z_{\mathrm{nc}}))+\lambda b(\delta u, z_{\mathrm{nc}}-Iu).
(u_1,u_2,u_3) =(|x_1|^2-|x_2|^2, 2 \mathrm{Re}(x_1 \bar{x_2}), 2 \mathrm{Im}(x_1 \bar{x_2})) &=(r_1^2-r_2^2, 2 r_1r_2 \cos(\theta_1 - \theta_2), 2r_1r_2 \sin(\theta_1-\theta_2)).
\frac{1}{n}\mathrm{Tr}_C\left[\hat{H}_C\,\Phi(\hat{\rho}_A)\right] = \frac{\eta}{n}\mathrm{Tr}_A\left[\hat{H}_A\,\hat{\rho}_A\right] + |1-\eta|\,E_0 + \frac{\eta+|1-\eta|-1}{2}&\le \eta\,E + |1-\eta|\,E_0 + \frac{\eta+|1-\eta|-1}{2}\;.
f(x,y) \,=\, \prod_{i=1}^d (t_i x - s_i y)^{\lambda_i}\hbox{and}g(u,v) \, = \sum_{i=1 \atop \lambda_i \neq 1}^d (s_i u + t_i v)^{n-\lambda_i+2}\cdot g_i(u,v),
\begin{array}{rl}u_{10}= u_0 + \frac{p^+}{\sqrt{2}}v_{10}= \frac{1}{u_0+\frac{p^+}{\sqrt{2}}}\left\{1-\frac{[1- (u_0+\frac{p^{+}}{\sqrt{2}})(v_0+p^{-}\sigma_{per}^{+})]\times [1-(u_0+\frac{p^{+}}{\sqrt{2}})(v_0+p^{-}\sigma_{per}^{-})]}{[1-(u_0+\frac{p^{+}}{\sqrt{2}})(v_0-\frac{p^{-}}{\sqrt{2}})]}\right\}\end{array}
E(n) \leq\exp\left (-\frac{42n+43}{64}+\frac{33(6n+4)}{128\pi\left(1-\frac{3\pi^2}{2048288}\right)}\right ) begin{align*}5pt] =\exp\left (\left(-\frac{21}{32}+\frac{99}{64\pi\left(1-\frac{3\pi^2}{2048288}\right)}\right)n-\frac{43}{64}+\frac{33}{32\pi\left(1-\frac{3\pi^2}{2048288}\right)}\right ) begin{align*}5pt] <\exp\left (-0.163n-0.343\right ).
E_{\xi}(\Sigma_2)={}E_{\xi}(\Sigma_1), \int_{\Omega}\vert \beta_{a}\vert^2_{1,}+\frac{2 M}{25 r^4}\vert Z_{ab}\vert^2_{2}d\mu_{\Omega}\leq{}\frac{72}{5}E_{\xi}(\Sigma_1),
\bar a_{i,j}\big(\sqrt{\mu}\partial_{v_j}g\big) = a_{i,j}* \Big( \partial_{v_j}(\sqrt{\mu}g) -g \partial_{v_j}\sqrt{\mu}\Big)=\big(\partial_{v_j}a_{i,j}\big)* \big(\sqrt{\mu}g\big)+\frac12 a_{i,j}* \big( v_j \sqrt{\mu}g\big),
D_{\varphi(t)}\ddot{\varphi}(t) - B_{\varphi(t)}(\dot{\varphi}(t), \dot{\varphi}(t)) - \langle \partial \dot{\varphi}(t), \bar{\partial}D_{\varphi}(\dot{\varphi}(t)) \rangle_{\varphi(t)}= 0.
\textstyle\mu_2^*=\left({P_T+\sum\limits_{k=1}^{r_s}\frac{\sigma^2}{[\boldsymbol{\Lambda}]_{k,k}^2}}\right)\left({r_s\sum\limits_{k=1}^{r_s}\frac{1}{\left(\nu_2^*-[\boldsymbol{\Lambda}]_{k,k}^2\right)\ln2}}\right)^{-1},
\,_{c}\mathbb{A}^{-1}_L={(\det\mathbb{A})}^{-1}\left(\begin{matrix}d -b -c a\end{matrix}\right)\;\;\;\;\,_{c}\mathbb{A}^{-1}_R=\left(\begin{matrix}d -b -c a\end{matrix}\right){(\det\mathbb{A})}^{-1},
\begin{aligned}C_{\bullet}(W^{u}_{Y}, F^{*})=\bigoplus_{y\in B_{\partial}}F^{*}_{y}\otimes o^{u}_{y}C_{i}(W_{Y}^{u}, F^{*})=\bigoplus_{y\in B_{\partial}^{i}}F^{*}_{y}\otimes o^{u}_{y},\end{aligned}
e^n_s \bar b^n_s=(x_s-x_{\kappa(n,s)}^n) (b_s(x_s)-b_s(x_{\kappa(n,s)}^n))+ (x_s-x_{\kappa(n,s)}^n) (b_s(x_{\kappa(n,s)}^n)-b_s^n(x_{\kappa(n,s)}^n)) &+ (x_{\kappa(n,s)}^n-x_s^n)(b_s(x_s)-b_s(x_{\kappa(n,s)}^n)) & +(x_{\kappa(n,s)}^n-x_s^n)(b_s(x_{\kappa(n,s)}^n)-b_s^n(x_{\kappa(n,s)}^n))
\int \frac{d^{4}q}{( 2 \pi )^{4}}\: \hat{V}( q ) \: e^{- i \: \frac{q}{2}\: ( x + y )}\: \int_{- \infty}^{\infty}d \tau \; \epsilon ( q^{2}) \: \delta^{( \alpha )}( z ) \: H_{z}^{( \beta )}( x , y ) \:{\mathcal{P}}( \tau ) \; e^{- i q \tau ( y - x )}
I \left( a \right) = \begin{cases}\infty a \in \left( -\infty,0 \right); \ell \left( a_{-}; a \right) a \in \left[ 0,a_{-}\right]; 0 a \in \left[ a_{-}, a_{+}\right]; \ell \left( a_{+}; a \right) a \in \left[ a_{+},\infty \right) \end{cases}
\varepsilon (r) = 1 - \frac{\sin(\pi r)}{\pi r}> 0\quad \nu(r) = - \left (\frac{6}{\pi^3 r^3}- \frac{3}{\pi \,r}\right ) \sin( \pi r) + \frac{6\cos(\pi r)}{\pi^2 r^2}> 0.
|F|_{L^{\infty}[0,t]}:=\sup_{\{0\leq t'\leq t\}\times \mathbb{R}^{3}\cap V_{T}^{c}}\left(\tau_{+}^{5/2}\tau_{-}^{-1/2}|\alpha|+{\tau_{+}^{7/4}\tau_{-}^{1/4}|\rho|}+\tau_{+}\tau_{-}|\underline{\alpha}|\right),
d Y^{N, \kappa}_t = \big[ A Y^{N, \kappa}_t + F_{\kappa}\big( P_N( Y^{N, \kappa}_t ) \big) \big] \, dt + B_{\kappa}\big( P_N( Y^{N, \kappa}_t ) \big) \, dW_t , t \in [0,T] , Y^{N,\kappa}_0 = \xi .
\int \frac{x^2 (x^2;q^2)_\infty}{(r_n x^2;q^2)_\infty}p_n(x;-1;q) d_qx= & \frac{q^{n+2}(\frac{x^2}{q^2};q^2)_\infty}{[n]_q[n+1]_q(\frac{r_nx^2}{q^2};q^2)_\infty}\left( \frac{q^{n+1}(1-\frac{x^2}{q^2})}{[n+1]_q(1-q^n)}\,_3\phi_2 ( q^{-n}, q^{n+1},x;q;-q;q,q)-x p_n(\frac{x}{q};-1;q)\right).
\left( L_{\varepsilon}+\lambda \right) u=\ -\varepsilon u^{\left( 2\right)}\left( x,\varepsilon \right) +Au\left( x,\varepsilon \right) +Bu^{\left(1\right)}\left( x,\varepsilon \right) +\lambda u\left( x,\varepsilon\right) =f\left( x\right) ,x\in \left( 0,T\right) ,
\frac{d \rho}{d t}< \frac{\rho^{7 / 2}}{d_{0}^{1 / 2}e}\frac{( 1 - \frac{\rho_{0}^{7}}{\rho^{7}})}{( 1 + \frac{\rho_{0}^{7}}{\rho^{7}})^{2 / 7}}\, .