harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z^{\mathbf{x},n}(\omega) = \lim_{s \to \infty}\mathbb{E}_{(e_s(\omega),x_s(\omega))}[ Z^{\mathbf{x},n}] = \mathbb{E}_{(e_0(\omega),x_0(\omega))}[ Z^{\mathbf{x},n}] = \mathbb{E}_{(e,x)}[Z^{\mathbf{x},n}].
\int_{0}^{\infty}\langle (\lambda-\Delta)^{\frac{1}{4}}u,\partial_t (\lambda-\Delta)^{\frac{1}{4}}\varphi \rangle dt = \int_{0}^{\infty}\langle (\lambda-\Delta)^{\frac{3}{4}}u,(\lambda-\Delta)^{\frac{3}{4}}\varphi \rangle dt & + \int_{0}^{\infty}\langle b \cdot \nabla u,(\lambda-\Delta)^{\frac{1}{2}}\varphi\rangle
w^r_k :=(x^{r}_1,\cdots, x^{r}_{k-1}, x^{r-1}_{k}, x^{r-1}_{k+1}, \cdots, x^{r-1}_K), \k=1,\cdots, K, \w^{r}_{K+1}:=x^r, w^{r}_{1}:=x^{r-1}.
\frac{B_{n}}{n!}=\sum_{\pi\in\mathcal{C}\left(n\right)}\frac{\left(-1\right)^{\vert \pi \vert}}{\left(\pi+1\right)!}=\sum_{\pi\in C\left(n\right)}\frac{\left(-1\right)^{\vert \pi \vert}}{\vert \pi \vert+1}\frac{1}{\pi!}.
\sum_{k=0}^{n}\binom{n}{k}\dbinom{m+k}{p}y^{n-k}xf_{m-p+k}^{\left( \alpha-1 \right)}\left( x\right) =\sum_{k=0}^{m}\binom{m}{k}\dbinom{n+k}{p}\left( -y\right)^{m-k}x f_{n-p+k}^{\left(\alpha-1 \right)}\left( x+y\right) .
\left\langle \begin{array}{l}{R_{1}}\\{J_{1}}\\ \end{array}\right|{\hat{\cal T}}_{J}^{R}\left| \begin{array}{l}{R_{2}}\\{J_{2}}\\ \end{array}\right\rangle = \left\langle \begin{array}{l l}{R_{1}}&{R}\\{J_{1}}&{J}\\ \end{array}\right. \left| \begin{array}{l}{R_{2}}\\{J_{2}}\\ \end{array}\right\rangle \langle R_{1}\|{\hat{\cal T}}^{R}\| R_{2}\rangle ,
(\begin{matrix}2&-1&0&0&-1&0\\ -1&3&-1&0&-1&0\\ 0&-1&2&-1&0&0\\ 0&0&-1&3&-1&-1\\ -1&-1&0&-1&3&0\\ 0&0&0&-1&0&1\\ \end{matrix})
{\fam2 O}= v^a\nabla_a+\sum\limits_{j=0}^\infty\gamma^{a_1}\ldots\gamma^{a_{2j}}w^j_{a_1\ldots a_{2j}}, w^j_{a_1\ldots a_{2j}}\in{\mathcal E}_{(a_1\ldots a_{2j})}
\frac{\frac{V_{+0,+0}}{V_{0+,0+}}+\frac{V_{+-,00}}{V_{-+,00}}}{\frac{V_{0+,0+}}{V_{+0,+0}}+\frac{V_{-+,00}}{V_{+-,00}}}=\frac{\left(V_{+-,-+}-V_{++,++}\right)\frac{V_{+0,+0}}{V_{0+,0+}}+V_{+-,+-}}{\left(V_{+-,-+}-V_{++,++}\right)\frac{V_{0+,0+}}{V_{+0,+0}}+V_{-+,-+}}\;.
{\cal A}_{N}^{0 , -}= < \prod_{i = 1}^{N - 1}{V}_{j_{i}, m_{i}}^{\omega_{i}}( z_{i}) \tilde{V}_{j_{N}, j_{N}}^{\omega_{N}( 0 ) , ( - )}( z_{N}) \prod_{n = 1}^{s_{+}}{\cal S}
-c(t,\phi_t(x)) u_0(x) \exp\{-\int_0^t c(r,\phi_r(x))dr \}= \frac{d}{dt}u(t,\phi_t(x)) = \partial_t u(t,\phi_t(x)) + \nabla u(t,\phi_t(x)) \cdot \dot{\phi}_t(x) = \partial_t u(t,\phi_t(x)) + \nabla u(t,\phi_t(x)) \cdot b(t,\phi_t(x)) + \nabla u(t,\phi_t(x)) \cdot \dot{X}_t .
\tilde{\Theta}_-(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi =\frac{1}{2}\tilde{\Theta}_-(D_a-UD_aU)\varphi &=-\frac{1}{2}(D_a-UD_aU)\tilde{\Theta}_-\varphi &=-(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\tilde{\Theta}_-\varphi=-(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\tilde{\Theta}_-\varphi.
\operatorname{Pr}\Bigg(\bigg|\bigg\langle u-v,\frac{1}{\sqrt{M}}\sum_{i=1}^M\epsilon_i\varphi_i\bigg\rangle\bigg|\geq\xi\Bigg)\leq e^{-\xi^2/2\sigma_\mathrm{max}^2\|u-v\|_2^2}\forall u,v\in\mathbb{R}^N.
M=\int d^{3}x_{1}e^{ip_{1}\cdot x_{1}}\langle\beta out|\overline{\Psi}_{in}(x_{1})\gamma^{0}u_{p_{1}}^{s_{1}}-\overline{\Psi}_{out}(x_{1})\gamma^{0}u_{p_{1}}^{s_{1}}|\alpha^{\prime}in\rangle.
\omega^{2}_{\rho}(q) = q^{2}\frac{1 + \frac{2}{\pi}V_{(0)}^{00}+ \frac{1}{2 \pi^{2}}\{(U_{(0)}^{00})^{2}- 2 U_{(0)}^{00}U_{(1)}^{00}\}}{1 + \frac{2}{\pi}V_{(1)}^{00}- \frac{1}{2 \pi^{2}}(U_{(1)}^{00})^{2}},
\begin{array}{l}{{x^{i}x^{j}= q x^{j}x^{i}~ , ~ ~ \mathrm{f o r}~ i < j ~ ~ \mathrm{a n d}~ ~ i \not = - j ~ ,}}\\{{q x^{+ 2}x^{- 2}- q^{- 1}x^{- 2}x^{+ 2}= \frac{q^{1 / 2}- q^{- 1 / 2}}{q - 1 + q^{- 1}}\frac{1}{a^{2}}~ ,}}\\{{q x^{+ 1}x^{- 1}- q^{- 1}x^{- 1}x^{+ 1}= ( 1 - q^{2}) x^{+ 2}x^{- 2}+ q \frac{q^{1 / 2}- q^{- 1 / 2}}{q - 1 + q^{- 1}}\frac{1}{a^{2}}~ .}}\\ \end{array}
\frac{\partial}{\partial q_{t A}}= \frac{\partial r}{\partial q_{t A}}\frac{\partial}{\partial r}+ \frac{\partial e_{B}}{\partial q_{t A}}\frac{\partial}{\partial e_{B}}+ \frac{\partial E_{s}}{\partial q_{t A}}\frac{\partial}{\partial E_{s}}+ \frac{\partial y_{s B}}{\partial q_{t A}}\frac{\partial}{\partial y_{s B}}.
< F > = \frac{< p_{g . s .}| F | p_{g . s .}>}{< p_{g . s .}| p_{g . s .}>}= \frac{< g . s . | F | g . s . > + \epsilon^{2}< p_{g . s .}^{1}| F | p_{g . s .}^{1}>}{< g . s . | g . s . > + \epsilon^{2}< p_{g . s .}^{1}| p_{g . s .}^{1}>}
0 = \left.\frac{d}{dt}\right|_{t=0}\mathcal{G}\left((f,X)+t(u, Y)\right) &=\int_{\Omega}\rho_g (D\Phi^W_{(g,\pi)})^*(f, X) \cdot_g (D\Phi^W_{(g,\pi)})^*(u, Y) -\Pi_g (\psi, V)\cdot_g (u, Y) \, d\mu_g &=\int_{\Omega}\left( D\Phi^W_{(g,\pi)}\circ \rho_g (D\Phi^W_{(g,\pi)})^*(f, X) - \Pi_g(\psi, V) \right)\cdot_g (u, Y) \, d\mu_g\; ,
\sum_{i=1}^{s}\bigg(\frac{1}{\sigma_{i}}-\frac{1}{\widetilde{\sigma}_{i}}\bigg)^{2}+\sum_{i=s+1}^{r}\frac{1}{\sigma_{i}^{2}}\leq\|B^{\dagger}-A^{\dagger}\|_{F}^{2}\leq\sum_{i=1}^{s}\bigg(\frac{1}{\sigma_{i}}+\frac{1}{\widetilde{\sigma}_{i}}\bigg)^{2}+\sum_{i=s+1}^{r}\frac{1}{\sigma_{i}^{2}}.
2\tau\sum\limits_{i=0}^{m-1}h\zeta_k^iv_{i\bar t}^2(k) +\sum\limits_{i=0}^{m-1}hv_{ix}^2(k)-\sum\limits_{i=0}^{m-1}hv_{ix}^2(k-1)+ \tau^2\sum\limits_{i=0}^{m-1}hv_{ix\bar t}^2(k) = 2\tau\sum\limits_{i=0}^{m-1}hf_{ik}v_{i\bar t}(k)+2\tau p_kv_{m\bar t}(k)-2\tau g_k^nv_{0\bar t}(k) .
\pi_{\lambda^c}(z^{\gcd(n,b)}) = ((z^{\gcd(n,b)})^{q_b}, \lambda^c (z^{\gcd(n,b)})^{q_n}) = (z^b, \lambda^{kn + d}z^n) &= ((\lambda^{k\gcd(n,b)}z)^{q_b}, \lambda^d (\lambda^{k\gcd(n,b)}z)^{q_n})= \pi_{\lambda^d}((\lambda^k z)^{\gcd(n,b)}).
\psi_{n \to f}( z ) ={\frac{1}{x}}\left[ C_{1}E_{1}( x_{+}^{2}- x_{-}^{2})^{( 1 - \sqrt{1 - \mu}) / 2}x^{\sqrt{1 - \mu}}+ C_{1}E_{2}( x_{+}^{2}- x_{-}^{2})^{( 1 + \sqrt{1 - \mu}) / 2}x^{- \sqrt{1 - \mu}}\right] ,
\aligned(T_i-q) (T_i+1)=0, T_iT_{i'}=T_{i'}T_i, T_iT_{i+1}T_i =T_{i+1}T_iT_{i+1}, T_ic_j =c_jT_i, T_ic_i=c_{i+1}T_i, T_ic_{i+1}=c_iT_i-(q-1)(c_i-c_{i+1}). \endaligned
\begin{array}{l l l l}{{\mathrm{f o r ~}\phi = + 1}}&{{~ : ~}}&{{( i^{k}, 1 , \pm 1 , \pm 1 , \pm 1 , 1 )}}&{{, ( k = 0 , . . . , 3 )}}\\{{\mathrm{f o r ~}\phi = - 1}}&{{~ : ~}}&{{( i^{k}e^{i \pi / 4}, 1 , \pm 1 , \pm 1 , \pm 1 , 1 )}}&{{, ( k = 0 , . . . , 3 ) \; ,}}\\ \end{array}
\left(\frac{R}{b_{i+1}}\right)^{\frac{3}{2}}\exp\left( \frac{2}{\sqrt{R}}\sqrt{b_{i+1}}\right) \left(\frac{b_{i}}{R}\right)^{\frac{3}{2}}\exp\left( \frac{-2}{\sqrt{R}}\sqrt{b_{i}}\right) = \left(\frac{b_i}{b_{i+1}}\right)^{3/2}\exp\left(\frac{2}{\sqrt{R}}(\sqrt{b_{i+1}}-\sqrt{b_{i}})\right)
\tilde{E}_i[\eta|t] = - \frac{2}{\bar{N}}\epsilon_{ij\rho\sigma}\dot{\eta}^j(t) \int \delta \xi ds \, \omega(\xi(s)) E^\rho[\xi|s] \omega^{-1}(\xi(s)) \dot{\xi}^\sigma(s) \dot{\xi}^{-2}(s) \delta(\xi(s)-\eta(t)),
|I\rangle_M:=\mu_I\exp\left(-{1\over2}a^\dagger Ca^\dagger\right)|0\rangle,\ \ \ \mu_I\mu_3^M=\det{}^{d\over2}\left(1-M_0\right)=\det{}^{d\over2}\left({(1-T)^2\over1-T+T^2}\right),
\sum_{\ell\in\mathbb{Z},\left\vert p\ell\right\vert\leq k\delta}\lbrace\frac{p}{n!}k^{n}\int_X(i\mathcal{R}^L_x-\frac{p\ell}{k}d\omega_0(x))^{n}\wedge\omega_0(x) \rbrace+o(k^{n}) &=\lbrace\frac{1}{n!}k^{n+1}\int_X\int_{[-\delta,\delta]}(i\mathcal{R}^L_x-sd\omega_0(x))^{n}\wedge\omega_0(x)ds\rbrace+o(k^{n+1}).
Q ( \zeta ) = \frac{1}{4}\left( \frac{1 - \mu_{1}^{2}}{\zeta^{2}}+ \frac{1 - \mu_{2}^{2}}{( \zeta - 1 )^{2}}+ \frac{1 - \mu_{1}^{2}- \mu_{2}^{2}+ \mu_{\infty}^{2}}{\zeta ( 1 - \zeta )}\right) .
\begin{aligned}J_0(B(r)\times B(2r)^c) \lesssim \int_{B(r)}|x|^{\alpha-d}dx\int_{B(2r)^c}\frac{|y|^{\alpha-d}}{|x-y|^{d+\alpha}}dy &\lesssim \int_{B(r)}|x|^{\alpha-d}dx\int_{B(2r)^c}|y|^{-2d}dy<\infty. \end{aligned}
\langle c^t(y_i\otimes y_j), x_h\otimes x_k\rangle = \langle y_i\otimes y_j, c(x_h\otimes x_k)\rangle = q_{hk}\langle y_i\otimes y_j, x_k\otimes x_h\rangle = q_{hk}\delta_{jk}\delta_{ih}= q_{ij}\langle y_j\otimes y_i, x_h\otimes x_k\rangle.
f(d) = \sum_{j=1}^d \frac{(n-j+1)\varepsilon + (\varepsilon+ 1+C) \cdot (j-1)}{n - j +1}= \sum_{j=1}^d (\varepsilon + (\varepsilon+1+C) \cdot \frac{j-1}{n-j+1}) \leq d\varepsilon + (\varepsilon +1+C) \sum_{j=1}^d \frac{j-1}{n-j+1}\leq d\varepsilon + (\varepsilon+1+C)\cdot \frac{d(d-1)}{n-d+1}
{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}\partial_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}\partial_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\rightarrow{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}{\cal D}_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}{\cal D}_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\, ,
g'_{N,q}(t)\exp_q(t)^2 =\dfrac{p-1}{\Gamma(p)}t^{p-2}\exp_q(t)-\dfrac{q-1}{\Gamma(q)}t^{q-2}\exp_p(t) &=\sum_{j=0}^\infty\left(\dfrac{p-1}{\Gamma(p)\Gamma(q+j)}-\dfrac{q-1}{\Gamma(q)\Gamma(p+j)}\right)t^{p+q-3}&=\sum_{j=0}^\infty\dfrac{(p-1)\prod_{i=0}^{j-1}(p+i)-(q-1)\prod_{i=0}^{j-1}(q+j)}{\Gamma(q+j)\Gamma(p+j)}t^{p+q-3}&>0,
\Delta E_{\mathrm{u p}}= \frac{1}{T^{( 0 )}}\frac{4 e^{S^{( 0 )}}\tan^{2}\frac{1}{2}( \phi_{\alpha}+ \phi_{\gamma}) + 4 e^{S^{( 0 )}}}{4 e^{2 S^{( 0 )}}\tan^{2}\frac{1}{2}( \phi_{\alpha}+ \phi_{\gamma}) - 1},
\varphi_{R\atop L}(\epsilon E_p, \epsilon \vec{p})=\Bigg[\sqrt{\frac{\gamma+1}{2}}\pm \vec{\sigma}.\epsilon \frac{\vec{p}}{p}\sqrt{\frac{\gamma-1}{2}}\Bigg]\varphi_{R\atop L}(\epsilon m, 0), \vspace{0.5in}
\int_{E}\tilde{\mathbf P}(\tilde V^I_n(\theta'_n,c)=\emptyset\cap\tilde{\mathfrak{W}}(c)\ne\emptyset)dP_n+\int_{E^c}\tilde{\mathbf P}(\tilde V^I_n(\theta'_n,c)=\emptyset\cap\tilde{\mathfrak{W}}(c)\ne\emptyset)dP_n\le \eta(1-\eta)+\eta\le 2\eta.
\int_{\Omega}{\widetilde{\widetilde{f}}}_{n}\cdot x_{n}d\mu =\int_{E^{c}_{2}}\widetilde{f}_{n}\cdot x_{n}d\mu &=\int_{\Omega}\widetilde{f}_{n}\cdot x_{n}d\mu-\int_{E_{2}}\widetilde{f}_{n}\cdot x_{n}d\mu &>\epsilon-\frac{\delta}{2}-\frac{\delta}{2^{2}}.
M \stackrel{d_{0}}{\longrightarrow}M \oplus M \stackrel{d_{1}}{\longrightarrow}M \oplus M \stackrel{d_{2}}{\longrightarrow}M \stackrel{d_{3}}{\longrightarrow}M \stackrel{d_{0}}{\longrightarrow}M \oplus M \stackrel{d_{1}}{\longrightarrow}\cdots
L_{e}= \int_{0}^{2 \pi}d \varphi \sqrt{g_{\varphi \varphi}}\mid_{x = x_{h}, \theta = \pi / 2}= 2 \pi x_{h}\sqrt{\frac{l ( x = x_{h}, \theta = \pi / 2 )}{f ( x = x_{h}, \theta = \pi / 2 )}}
W_{31}= O \biggl( \max \biggl\{\frac{k^{1/2+2\beta/d}}{n^{1+2\beta/d}}\, , \, \frac{\log n}{nk^{1/2}}\, , \, \frac{k^{-1/2 + \frac{2 \alpha}{\alpha+d}- \epsilon}}{n^{\frac{2 \alpha}{\alpha+d}- \epsilon}}\biggr\}\biggr).
\sum_{i+j=n}\big(m_{1,i}(x, (m_{1,j}(y, z))- m_{1,i}(m_{1,j}(x, y),z)+ m_{1,i}(m_{1,j}(x, z),y)\big)=0 &\sum_{i+j=n}\big(m_{2,i}(x, (m_{2,j}(y, z)) - m_{2,i}(m_{2,j}(x, y),z) + m_{2,i}(m_{2,j}(x, z),y)\big)=0.
(II) = \int P_{\alpha}^k(\partial_{\alpha}[\hat \Psi_{\alpha}h_{\alpha}]) \hat \Psi_{\alpha}\: dm + \int \partial_{\alpha}[P_{\alpha}^k](\hat \Psi_{\alpha}h_{\alpha})\hat \Psi_{\alpha}\: dm &=(IIa) + (IIb)
\left({1-z^{2}}\right) \frac{d^{2}y}{dz^{2}}-2z\frac{dy}{dz}+\left\{{\lambda_{n}^{m}\left({\gamma^{2}}\right) -\frac{m^{2}}{1-z^{2}}+\gamma^{2}\left({1-z^{2}}\right)}\right\}y=0.
D^c(x_1,t) \equiv -\frac{1}{(2\pi)^2}\int_{-\infty}^{\infty}dq_0 \int_{-\infty}^{\infty}dq_1 \frac{1}{q_0^2 - q_1^2 +i \epsilon}e^{-\frac{i}{\hbar}cq_0t}e^{\frac{i}{\hbar}q_1x_1}
\int_Y e^{-i\omega(r_k-r_j)}V_{\omega}\circ F^j\,\bar V_{\omega}\circ F^k\,d\mu_Y = \int_Y e^{-i\omega r_{k-j}\circ F^j}V_{\omega}\circ F^j\,\bar V_{\omega}\circ F^k\,d\mu_Y =\int_Y e^{-i\omega r_{k-j}}V_{\omega}\,\bar V_{\omega}\circ F^{k-j}\,d\mu_Y =\int_Y P_\omega^{k-j}V_{\omega}\,\bar V_{\omega}\,d\mu_Y.
P_{m}( x ) \equiv P_{I_{2}( m )}^{R_{m}}( x ) = \prod_{\mu \in{R_{m}}}( x - \mu \cdot \tilde{q}) = \prod_{j = 1}^{m}\left( x - \sin \Bigl ({\frac{2 j \pi}{m}}+{\frac{\varphi_{0}}{m}}\Bigr ) \right) ,
f_c(x) := \frac{1}{2mc}\left( \sqrt{(p+1)^2+m^2c^2}+ \sqrt{(p-1)^2+m^2c^2}\right)h(x), u_c(x) := f_c(x)\sin x, \lambda_c := c\left(\sqrt{1+m^2c^2}-mc\right), V_c(x) := \lambda_c - c\frac{\left(\sqrt{p^2+m^2c^2}-mc\right)u_c(x)}{u_c(x)}.
\left[\left(\bigotimes_{j=1}^n A_j\right) \cdot B \right]^{i_1, \ldots i_n}= \sum_{j_1=1}^{d}\cdots \sum_{j_n=1}^{d}\left(A_1\right)^{i_1}_{j_1}\left(A_2\right)^{i_2}_{j_2}\cdots \left(A_n\right)^{i_n}_{j_n}B^{j_1, \ldots, j_n}
\left|\sum_{|\alpha|>CL}a_{\alpha}\left(\frac{\Gamma(L+|\alpha|)}{\alpha!\Gamma(L)}\right)^{1/2}z^{\alpha}\right| \leq \sum_{m>CL}\left[\frac{\Gamma(L+m)}{\Gamma(L)m!}r^{2m}\right]^{1/2}\left(\sum_{|\alpha|=m}|a_{\alpha}|^2\right)^{1/2}.
\lim_{N \rightarrow \infty}\max_{\delta N_k \leq l \leq (1-\delta)N_k}\max_{\mu, \nu}[\mathbb{E}^V-\mathbb{E}^G]\theta[\frac{N}{\pi}\int_I \tilde{G}_{\mu \nu}(E+i\eta)q(\operatorname{Tr}f_E(Q_1))dE]=0.
\phi_{\gamma^{-1}\gamma_1,\gamma_2}=\sum_{h\in \Gamma}\beta^{\gamma}_{\gamma_1,h}\phi_{h,\gamma_2}=\sum_{h\in \Gamma}\phi_{\gamma_1,h}\beta^{\gamma}_{h,\gamma_2}=\phi_{\gamma_1,\gamma\gamma_2}\,,
\prod_{P_0,\dots,P_n}\hom_{\mathbb Z}\left(\bigotimes_{i=1}^n \mathbb Z\hom_R(P_i,P_{i-1}), \hom_R(P_n,P_0\otimes M)\right), \prod_{P_0,\dots,P_n}\hom_{R}\left(\bigotimes_{i=1}^n \mathbb Z\hom_R(P_i,P_{i-1})\otimes \hom_R(P_0,P_n), M\right),
\lambda_1-\omega_2-p = (\omega_1-\lambda_2)(\omega_1-\lambda_3)(\lambda_1-\omega_3)^{-1}= \left( \Bigl( \omega_1-\Re(\lambda_2) \Bigr)^2 +(\Im(\lambda_2))^2 \right) (\lambda_1-\omega_3)^{-1},
\frac{1}{8}\mathrm{Tr}\left[\left( M^{-1}\dot{M}\right)^2\right]=\frac{1}{4d}\left[\mathrm{Tr}\left(G^{-1}\dot{G}\right)\right]^2+\frac{1}{4}\left\{\mathrm{Tr}\left[\left(G^{-1}\dot{G}\right)_{t}^2\right]-\mathrm{Tr}\left[\left(G^{-1}\dot{B}\right)^2\right]\right\},
\lbrack \tilde{L}_{1}, [ \tilde{r}_{1 2}, \tilde{r}_{1 3}] + [ \tilde{r}_{1 2}, \tilde{r}_{2 3}] + [ \tilde{r}_{3 2}, \tilde{r}_{1 3}] + \{\tilde{L}_{2}, \tilde{r}_{1 3}\}- \{\tilde{L}_{3 ,}\tilde{r}_{1 2}\}] + \mathrm{c y c l . t e r m}= 0 .
(a;q)_n =\prod_{j=0}^{n-1}(1-aq^j), (a_1,\dots,a_m;q)_n =(a_1;q)_n\dotsm (a_m;q)_n,{}_{r+1}\phi_r\left(\begin{matrix}a_1,\dots,a_{r+1}b_1,\dots,b_r\end{matrix};q,x\right) =\sum_{k=0}^\infty\frac{(a_1,\dots,a_{r+1};q)_k}{(q,b_1,\dots,b_r;q)_k}\,x^k.
\Phi_\pi\left(Z^{\chi(1), \epsilon(1)}_{k_1}, \ldots, Z^{\chi(n), \epsilon(n)}_{k_n}\right) = \Phi_{\pi'}\left(Z^{\chi'(1), \epsilon'(1)}_{k'_1}, \ldots, Z^{\chi'(n), \epsilon'(n)}_{k'_n}\right).
Q_{p-1}\wedge E^b \wedge E^a ={1 \over (p+1)!}E^{a_0}\wedge E^{a_1}\wedge ... \wedge E^{a_p}\epsilon_{a_0 a_1 ... a_p}e^{-{p-3 \over 2}\phi}{\partial \over \partial F_{ab}}(\sqrt{- det (\eta + F)})
K^{\alpha,\beta,\gamma}_{m_1,m_2,\theta}(x,y) := (-1)^{\theta_1 \mathbf{1}\{x_1 = m_1-1, y_1 = 0\}+ \theta_2 \mathbf{1}\{x_2 = m_2 - 1, y_2 = 0\}}\left( \alpha \mathbf{1}_{y = x}+ \beta \mathbf{1}_{y = x + \mathbf{e}^1}+ \gamma \mathbf{1}_{y = x + \mathbf{e}^2}\right).
\Psi_{\alpha}^{-}=\int d\beta(\Psi_{\beta}^{+},\Psi_{\alpha}^{-})\Psi_{\beta}^{+}=\int d\beta|\Psi_{\beta}^{+}\rangle\langle\Psi_{\beta}^{+}|\Psi_{\alpha}^{-}\rangle=\sum_{n_{1}\sigma_{1}n_{2}\sigma_{2}\cdot\cdot\cdot}\int d^{3}p_{1}d^{3}p_{2}\cdot\cdot\cdot(\Psi_{\beta}^{+},\Psi_{\alpha}^{-})\Psi_{\beta}^{+},
U_{p,i}^\kappa(u,0) =Q_{p,i}^\kappa(u)U_{p,i}(u,0)=f^{-i}\left(\bigcup_{\kappa=0}^{\infty}Q_{p,0}^\kappa(u)\cup U_p^{(\infty)}(u)\right)=\{X_i>u\}R_{p,i}(u,0) =\{X_i>u, X_{i+1}\leq u,\ldots,X_{i+p}\leq u\}=Q_{p,i}^0(u)
\iint\limits_{\operatorname{supp}g^{s}_{\alpha}}\gamma(x,t)|\nabla g^{s}_{\alpha}(x,t)|\,dxdt \le C\sum\limits_{j=1}^{n(s,\alpha)}\mu_{n}(\check{Q}_{j}(s,\alpha))\frac{1}{\mu_{n+1}(Q_{j}(s,\alpha))}\iint\limits_{2Q_{j}(s,\alpha)}\gamma(x,t)\,dxdt.
\int_\Omega (v(t)\vartheta(t) - v^{in}\vartheta(0))\ \mathrm{d}x + \int_0^t \int_\Omega \nabla v(s)\cdot \nabla\vartheta(s)\ \mathrm{d}x\mathrm{d}s + \int_0^t \int_\Omega (uv)(s)\vartheta(s)\ \mathrm{d}x\mathrm{d}s & = \int_0^t \int_\Omega v(s)\partial_t\vartheta(s)\ \mathrm{d}x\mathrm{d}s
\|\hat{\Pi}_{V_1}h_1-(\hat{\Pi}_V h_1)_1\|_n^2 =\|(\hat{\Pi}_V (h_1-\hat{\Pi}_{V_1}h_1))_1\|_n^2 &\leq \frac{(1+\delta)}{(1-\delta)}\frac{1}{(1-\rho_{0}^2)}\|h_1-\hat{\Pi}_{V_1}h_1\|_n^2.
\int_{C}{d^{n}}{\eta}= a^{n}\int_{0}^{2 \pi}d \theta_{1}\int_{0}^{\pi}d \theta_{2}s i n \theta_{2}\int_{0}^{\pi}d \theta_{3}s i n^{2}\theta_{3}. . . \int_{0}^{\theta_{n}^{0}}d \theta_{n}s i n^{n - 1}\theta_{n}
I_2 =\int_{\Omega_-^{\varepsilon}}-G_1^-\varphi_{x_3}+G_3^-\varphi_{x_1}\, dx=\int_{\partial \Omega^{\varepsilon}_-}\varphi (G_3^- n_1^{\varepsilon}-G_1^- n_3^{\varepsilon})\, d\sigma+\int_{\Omega^{\varepsilon}_{-}}\varphi((G_1^-)_{x_3}-(G_3^-)_{x_1})\, dx.
T_{nn}^{(2)}=-\frac{1}{2\pi^2}\sum_{m=-N}^{m=+N}\int d^3p\frac{\lambda_m^2}{{[{(\vec{p}- \vec{k})}^2 + \mu^2]}^2[p^2 - (K^2 -\frac{{(n-m)}^2}{R^2}) -i \epsilon]}.
T = 2 \pi \xi n + \int \! \mathrm{d}^{2}x \left\{\frac{1}{2 g^{2}}\left[ F_{1 2}+ g^{2}\left( | \varphi |^{2}- \xi \right) \right]^{2}+ \left| \left({\cal D}_{1}+ i{\cal D}_{2}\right) \varphi \right|^{2}\right\}.
-\lim_{y\to0^+}\frac{U(t,x,y)-U(t,x,0)}{y^{2s}}=\frac{|\Gamma(-s)|}{4^s\Gamma(s)}H^su(t,x)= -\frac1{2s}\lim_{y\to0^+}y^{1-2s}U_y(t,x,y),
\inf_{\omega \in I_1}| h_{\xi}(\omega)| = \left| h_{\xi}\left(\omega_{\xi}^{\ast}- \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}\right) \right| = \ldots = \frac{\frac{1-\alpha}{2}\xi^{\alpha}}{\xi - 1 + \frac{1}{2}\xi^{\alpha}},
\bar{D}(\tilde{r}_i) \leq \frac{N\left(1-\frac{1}{1+N+\cdots+N^i}\right)^2}{(N-1)+\frac{1}{1+N+\cdots+N^i}}=\frac{N(1+N+N^2+\cdots+N^i-1)^2}{(N-1)(1+N+\cdots+N^i)^2+(1+N+\cdots+N^i)}=\frac{N^2(1+N+\cdots+N^{i-1})^2}{N^{i}(1+N+\cdots+N^i)}.
= \int D S_{\mu \nu}\exp \left\{- \left[ ( \pi \eta )^{2}\int d^{4}x \int d^{4}y S_{\mu \nu}( x ) D_{m}^{( 4 )}( x - y ) S_{\mu \nu}( y ) + i \pi \int d^{4}x h_{\mu \nu}S_{\mu \nu}\right] \right\}.
sn[\frac{1}{3}K(\frac{1}{2});\frac{1}{2}]=tanh[\frac{1}{2}ln(\sqrt{2\sqrt{(\frac{4}{3})^{4/3}-(\frac{4}{3})^{2/3}+1}-(\frac{4}{3})^{2/3}+2}+\sqrt{(\frac{4}{3})^{2/3}+1})]
\sum_{\xi_{\Delta}\in \Omega_{\Delta}}\int_{\Omega}c_{\Delta}(\omega, \xi_{\Delta})f(\omega)g(\xi_{\Delta}\omega_{\Delta^c})\mu(d\omega) = \sum_{\xi_{\Delta}\in \Omega_{\Delta}}\int_{\Omega}\hat{c}_{\Delta}(\omega, \xi_{\Delta})f(\xi_{\Delta}\omega_{\Delta^c})g(\omega)\mu(d\omega),
\left( \int_{0}^{1}\left\vert \sum_{n=1}^{N}a_{n}r_{n}\left( t\right)\right\vert^{p}dt\right)^{\frac{1}{p}}\leq B_{p}A_{r}^{-1}\left(\int_{0}^{1}\left\vert \sum_{n=1}^{N}a_{n}r_{n}\left( t\right) \right\vert^{r}dt\right)^{\frac{1}{r}}
\Lambda^{6 ( N_{c}+ 1 ) - 2 ( N_{f}+ N_{c}^{\prime})}\Lambda^{\prime 6 ( N_{c}^{\prime}- 1 ) - 2 N_{c}}= \mu^{2 ( N_{f}+ N_{c}+ N_{c}^{\prime})}\tilde{\Lambda}^{\prime 4 ( N_{c}^{\prime}- 1 ) - 2 ( N_{f}+ \tilde{N}_{c})}.
\lim_{m \to \infty}\mathbb{P}[\exists \mathbf{x}\in \mathcal{F}^l ~\textrm{s.t.}~ \mathbf{D}\mathbf{x}= \mathbf{e}_i] \ge \lim_{m \to \infty}\dfrac{1}{1+\frac{1}{\mathbb{E}[Y_i]}}= 1.
\begin{cases}2e^{-1/|w|^2}\left(2\frac{e^{-1/|w|^2}}{|w|^6}+|z|^2\left(\frac{1}{|w|^6}-\frac{1}{|w|^4}\right)\right)=0 |z|^2+2e^{-1/|w|^2}=1.\end{cases}
\sum_{\pi\in B(r,r,t)/S_{2}}q^{|\pi|}=\prod_{i=1}^{r}[\frac{1-q^{t+2i-1}}{1-q^{2i-1}}\prod_{j=i+1}^{r}\frac{1-q^{2(i+j+t-1)}}{1-q^{2(i+j-1)}}]
\mathrm{(\widetilde{II})}~ \mathrm{MDTCov}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{Y})=\mathbf{\Sigma}^{\frac{1}{2}}\left[ \frac{\boldsymbol{\Omega}}{\overline{F}_{\mathbf{Z}}(\boldsymbol{\xi_{q}})}-\frac{\boldsymbol{\delta}\boldsymbol{\delta}^{T}}{\overline{F}_{\mathbf{Z}}^{2}(\boldsymbol{\xi_{q}})}\right] \mathbf{\Sigma}^{\frac{1}{2}},
[\bar{a}_n \mapsto \bar{b}_n](J) \cup [\bar{a}_n \mapsto \bar{b}_n](K) \cup [\bar{a}_n \mapsto \bar{b}_n](L) = [\bar{a}_n \mapsto \bar{b}_n](J \cup K \cup L) &=[\bar{a}_n \mapsto \bar{b}_n](\mathbb{N}) = \{\bar{b}_n\}
\dfrac{|\det D_1(z_0)|\cdot ||f(z_0)||}{\prod_{j=1}^{q}|(f,\tilde a_i)(z_0)|}=\dfrac{|\det D_i(z_0)|}{\prod_{\underset{j\ne i}{j=0}}^{q}|(f,\tilde a_j)(z_0)|}\cdot \biggl(\dfrac{||f(z_0)||}{|(f,\tilde a_i)(z_0)|}\biggl) &\le \Psi (z_0)\cdot \dfrac{|\det D_i(z_0)|}{\prod_{\underset{j\ne i}{j=1}}^{q}|(f,\tilde a_j)(z_0)|}.
J_2 \leq (C^{(1)}_2+24)T_*L^{-2}_*T^{-1/2}_*(1+T^{-3/2}_*)+3T_*L^{-2}_*T^{-1/2}_*(1+T^{-1/2}_*)+11T_*L^{-2}_*T^{-1/2}_*(1+T^{-3/4}_*) &\leq K_1 T_*L^{-2}_*T^{-1/2}_*(1+T^{-1/2}_*+T^{-3/4}_*+T^{-3/2}_*)\,.
\begin{array}{rl}{\rm E}\Big[\Big({\rm E}\Big[|\sum_{j\in B}\alpha_{ji}\epsilon_j c_j |^{\gamma_i}\mid c_j,\: j\in B \Big]^{\frac1{\gamma_i}}\Big)^{\gamma_i}\Big]^{\frac1{\gamma_i}}\leq 2\sqrt{e}{\rm E}\Big[\sum_{j\in B}\alpha_{ji}^2c_j^2 \Big]^{\frac12}, & & = 2\sqrt{e}\Big(\sum_{j\in B}\alpha_{ji}^2{\rm E}\big[c_j^2\big]\Big)^{\frac12}\end{array}
\sum_{z=0}^{L_{i_t}-1}{\overline{\alpha}_{i_t}^z}=\begin{cases}\frac{\overline{\alpha}_{i_t}^{L_{i_t}}-1}{\overline{\alpha}_{i_t}-1}, \overline{\alpha}_{i_t}\not=1, L_{i_t}\overline{\alpha}_{i_t}, \overline{\alpha}_{i_t}=1,\end{cases}
f_1|\gamma_2 =\frac{i}{256}\left(q_2 +3444q_2^3 + 313358 q_2^5 +O(q_2^7)\right),~f_1|\gamma_4 = -f_1, f_2|\gamma_2 =\frac{i}{256}\left(q_2 -2700 q_2^3-251890 q_2^5 +O(q_2^7)\right),~f_2|\gamma_4 = -f_2.
\begin{aligned}P_{n,k}(W_{\mathcal{S}(n,k;M)})=\sum_{l=1}^{k-1}\frac{\binom Ml(k)_l(k(n-1))_{M-l}}{(kn)_M}+ &k\sum_{j=1}^{n-1}\sum_{l=1}^k \frac{\binom Ml(k)_l(k(j-1))_{M-l}}{(kn)_M}\frac 1{k(n-j+1)-l}.\end{aligned}
L\left( - \frac{\ell^{s_1 + 2 s_2}}{1 + \ell^{s_1 + s_2}+ \ell^{2 s_1 + 2 s_2}}\right) \left[ R(\ell^{s_1}), L(\ell^{s_2}) \right] R\left( \frac{\ell^{2 s_1+s_2}}{1+\ell^{s_1+s_2}+\ell^{2 s_1+2 s_2}}\right) = D\left(\ell^{s_1+s_2}+\ell^{2(s_1+s_2)}\right),
I_{H}^{\pm}= \sum_{j = 1}^{N_{H}^{\pm}}I_{h_{j}}^{\pm}\, , \, I_{C}^{\pm}= \sum_{j = 1}^{M_{C}^{\pm}}I_{c_{j}}^{\pm}\, , \, I_{W}^{\pm}= \sum_{j = 1}^{M_{W}^{\pm}}I_{w_{j}}^{\pm}\, \mathrm{a n d}\, I_{S}^{\pm}= \sum_{j = 1}^{N_{S}^{\pm}}I_{y_{j}}^{\pm}
b_{4}= \frac{2}{3}\frac{\omega_{t}^{2}}{\omega_{y}^{2}}\frac{1}{\epsilon_{y}^{2}}+ \frac{3 \omega_{y}^{2}- \omega_{t}^{2}}{2 \omega_{y}^{2}+ 2 \omega_{t}^{2}}, \qquad c_{4}= - 2 \frac{\omega_{t}^{2}}{\omega_{y}^{2}}\frac{1}{\epsilon_{y}^{2}}.
k =\frac{\ell n (n-\ell-1)}{n (a^2+\ell)-(a-\ell)^2}, \lambda =\frac{n (n^2 (a^3+\ell^2)-2 (\ell+1) n (a^3+\ell^2)+(2 a \ell+a+\ell (\ell+2)) (a-\ell)^2)}{((a-\ell)^2-n (a^2+\ell))^2}, \mu =\frac{\ell n (a-\ell) (\ell-n+1) (a-\ell+n)}{((a-\ell)^2-n (a^2+\ell))^2}.
\delta^{a b}\, \delta_{x y}\, = \, \langle \, \frac{\delta S}{\delta \bar{c}_{x}^{a}}\, \bar{c}_{y}^{b}\, \rangle \, = \, \widetilde Z_{3}\, \langle \, i ( \partial D_{r}c )_{x}^{a}\, \bar{c}_{y}^{b}\, \rangle \, - \, \frac{Z_{\lambda}\lambda}{Z_{3}}\, \langle \, ( s_{r}B_{x}^{a}) \, \bar{c}_{y}^{b}\, \rangle \; .
\mathcal{L}_{kn}= 16\nu_0^3 e^{-\Delta_{kn}}(R_{kn}+R_{kn}^*)+\mathcal{O}(\epsilon^{3/2}), R_{kn}= e^{i(\widetilde{\delta}_n- \widetilde{\delta}_k)}\langle\vec{n}_k, \vec{n}_n\rangle, \widetilde{\delta}_k=\delta_k - 2\mu_0\xi_k, \; \Delta_{kn}=2s_{kn}\nu_0(\xi_k- \xi_n),
0\le \int_0^L (xw_{xx}+pw_x)^2dx = \int_0^L(x^2w^2_{xx}+ 2px w_xw_{xx}+ p^2 w_x^2) dx = \int_0^L x^2w_{xx}^2 dx + (p^2-p) \int_0^L w_x^2 dx + pLw_x^2(L).
I_S\left( k,T\right) =\frac 1{\left( 2\pi \right)^3}\int\frac{d^3q}{2Q}\frac{Q^S}{\exp \left( Q/T\right) -1}\left( \frac 1{k^2+2k\cdot q}+ \frac 1{k^2-2k\cdot q}\right) ,
( \nabla v_{\alpha q}^{\underline{\alpha}}) v_{\underline{\alpha}}^{~ \beta r}={\frac{1}{4}}\delta_{q}^{r}\hat{\Omega}^{a b}( \gamma_{a b})_{\alpha}^{~ \beta}+{\frac{1}{4}}\delta_{\alpha}^{~ \beta}\hat{\Omega}^{i j}( \gamma_{i j})_{q}^{~ r},
\mathcal{T}_{{\rm gph}\mathcal{M}_{x}}(v,\lambda) =\big\{(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}\ |\ \xi=\nabla g(x)\eta,\,\eta\in\mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda)\big\},
A_j = \mu_0^{n+4s}f(\mu_0^{-1}\mu, \xi, \mu_jy) + \frac{\mu_0^{2n-4s}}{1+|y_j|^{2s}}g(\mu_0^{-1}\mu, \xi,\mu_jy),y_j = \frac{x-\xi_j}{\mu_j},
\frac{\Phi(-y_2)}{\Phi(-y)}= \frac{1 - \Phi(y_2)}{1 - \Phi(y)}= 1 - \left( \int_{y}^{y + \frac{C_1}{\sigma \sqrt{n}}}e^{- \frac{t^2}{2}}dt \right) \left( \int_y^{\infty}e^{- \frac{t^2}{2}}dt \right)^{-1}.
[0 | M_{t-1}] \Lambda_t \mathbf{Q}_{t}^{-1}Q^*_t q^t_{\parallel}-\lambda_t m^{t-1}= \sum_{i=1}^{t-1}\lambda_{i}\gamma^t_{i}m^{i-1}-\lambda_t m^{t-1}&= - M_t \mathbf{M}_{t}^{-1}M_t^*\Big(\lambda_t m^{t-1}- \sum_{i=1}^{t-1}\lambda_{i}\gamma^t_{i}m^{i-1}\Big).
\widetilde{R}_j(t) := 2^{j(\alpha-1)}\Vert f_j(t) \Vert_{L^2}+ \Vert g_j(t)\Vert_{L^2}+ \Vert \widetilde{g}_j(t)\Vert_{L^2}+ \Vert \nabla \dot S_{j-1}u(t)\Vert_{L^\infty}X_j(t).
\mathcal{CM}_{e}= \left\{F: (F)=[1,+\infty),\overline F(e^t) t\in [0,+\infty)\right\}, \mathcal{P}_{2}= \Big\{F: (F)=[1,+\infty), \overline F(x) = Cx^{-\alpha}+ O(x^{-(1+\beta)\alpha}), & \alpha>0,\beta>0, C>0\Big\},
\det J = (\det D)\det{(A-BD^{-1}C)}&= e^{(n-1)t}\left(\cosh{t}- \frac{1}{2}|\tilde{s}^2|e^{t}- \sum_{i=1}^{n-1}(-s_ie^{t})e^{-t}(s_ie^{t})\right) &= e^{(n-1)t}(\cosh{t}+ \frac{1}{2}|\tilde{s}|^2e^{t}),