harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int h\circ g(x^\infty,m_n)dQ(m_n) =\int h(\tilde G_{n,x^\infty}(\tilde \omega))d\tilde{\mathbf P}^*(\tilde\omega),~\forall h\in BL_1 \int h(\mathbb G_P(\omega))dP(\omega) =\int h(\tilde G_{P,x^\infty}(\tilde \omega))d\tilde{\mathbf P}^*(\tilde\omega),~\forall h\in BL_1,
\begin{array}{c}\left( \sum_ir_i^{(l)}\right)_L\times \left( \sum_ir_i^{(l)}\right)_R \times \left[\begin{array}{l}\{\left( 44+84\right)_B+128_F\}_L \times \{\left( 44+84\right)_B+128_F\}_R\end{array}\right]\end{array}
\tilde{\rho}(gh) =\tilde{\rho}(g)\tilde{\rho}(h)=\tau(s(g))\tau(s(h)) &=\tau(s(g)s(h))=\tau(z(g,h)s(gh)) &=\tau(z(g,h))\tau(s(gh))=\tau(z(g,h))\tilde{\rho}(gh)
{\bf Y}^{(a)}_m(\theta+\frac{\pi}{3}i ){\bf Y}^{(a)}_m(\theta+\frac{\pi}{3}i )=\frac{(1+{\bf Y}^{(a)}_{m-1}(\theta))(1+{\bf Y}^{(a)}_{m+1}(\theta))}{(1+({\bf Y}^{(a+1)}_{m}(\theta))^{-1})(1+({\bf Y}^{(a-1)}_{m}(\theta))^{-1})}
\bigg\|E\left[\sum_{i=1}^{n}\vartheta_i|\xi^i||\mathcal{F}_t\right]\bigg\|_{\infty}\leq\frac{1}{4}, 32\left(\sum_{i=1}^{n}\vartheta_i^2\right)\|\sum_{i=1}^n|\xi^i|^2\|_{\infty}<1.
F_{\Phi_{i}}(x) =1-\exp\left(-\frac{(\kappa^{2}_{R}m+N_{0}/p)x}{(\kappa^{2}_{T}(\omega+1)+\omega+1)}\right) &\times \sum^{n}_{k=1}\frac{A_{k}(x)\left(\frac{(\kappa^{2}_{R}m+N_{0}/p)x}{(\kappa^{2}_{T}(\omega+1)+\omega+1)}\right)^{k-1}}{(k-1)!},
\partial_{i}x^{j}=\tilde{J_{i}^{\alpha}}v_{\alpha}x^{j}= \partial_{i}(x^{j}) +\underbrace{\tilde{J_{i}^{\alpha}}O_{\alpha}{}^{\beta}(x^{j})J_{\beta}{}^{k}}_{L_{i}{}^{k}(x^{j})}\partial_{k}.
r_{s}(v) = \int_{\mathcal{S}}\frac{1}{\langle v, u \rangle^{-s}}\nu^*_{s}(du), r_{s}^*(v) = \int_{\mathcal{S}}\frac{1}{\langle v, u \rangle^{-s}}\nu_{s}(du), v \in \mathcal{S},
D[H(u,u)g_{ij}]:C^\infty(S^2(T^*TM))\rightarrow C^\infty(S^2(T^*TM)), D[H(u,u)g_{ij}](\frac{\partial g_{ij}}{\partial t})=D[H(u,u)](h_{ij})=\frac{\partial}{\partial t}\tilde{H}(u,u).
\varphi(x,t):=\frac{\theta(t-|x-y|)}{4\pi|x-y|}\xi(t-|x-y|)-\frac{m}{4\pi}\int_0^t\frac{\theta(s-|x-y|)J_1(m\sqrt{s^2-|x-y|^2})}{\sqrt{s^2-|x-y|^2}}\xi(t-s)ds.
\frac{\hat{A}_{n+1}(z)}{R_{n+1}(z)}- \frac{\hat{A}_{n}(z)}{R_{n}(z)}= \frac{1+c_1^2}{4d_1}\, (1- z^{-1})\left[\frac{Q_{n+1}(z)}{R_{n+1}(z)}- \frac{Q_{n}(z)}{R_{n}(z)}\right], n \geq 1,
A = \{j\in [n] \;:\; b_j=b_{n+j}=*\}, B = \{j\in [n] \;:\; b_j=0, \, b_{n+j}=*\}, C = \{j\in [n] \;:\; b_j=*, \, b_{n+j}=0\}, D = \{j\in [n] \;:\; b_j=b_{n+j}=0\}.
\begin{array}{r l}{{0}}&{{= \nabla_{\rho}h^{\rho z I_{3}\ldots I_{s}}=}}\\{{}}&{{= \partial_{z}h^{z z I_{3}\ldots I_{s}}+ \partial_{I}h^{z I I_{2}\ldots I_{s}}-{\frac{1}{z}}[ d + 2 s - 2 ] h^{z z I_{3}\ldots I_{s}}}}\\ \end{array}
(a_2 \cdot v_{(1,3)}) \cdot (a_1 \cdot v_{(2,3)}) = \frac{t^2}{2^5}(a_1 - a_{-1}+ a_2 - a_{-2}) + \frac{(2t - 1)t}{2^5}a_3 - \frac{(2t + 1)t}{2^5}a_{-3}+ \frac{t}{2^5}v_{(1,2)}+ \frac{t}{2^6}(v_{(1,3)}+ v_{(2,3)}) + \frac{t}{2^3}(a_1 \cdot v_{(1,2)}+ a_2 \cdot v_{(1,3)}- a_3 \cdot v_{(1,2)}) .
\frac{\left\| u(t) \right\|_{\frac{r(m_i+p_k-1)}{2}}^{\frac{m_i+p_k-1}{\vartheta}}}{C_S^{\ast\frac{2}{\vartheta}}\left\| u(t) \right\|^{\frac{(1-\vartheta)(m_i+p_k-1)}{\vartheta}}_{\frac{s(m_i+p_k-1)}{2}}}\le \int_{\Omega}\left| \nabla\!\left(u^{\frac{m_i+p_k-1}{2}}\right)\!(x,t)\right|^2 dx \forall t > 0 \, , \ \, i=1,2 \, .
\Gamma(1-\mu) \,\tau_0=1-\frac{I_-^\mu \big\{(1+x)^{-\mu}(\hat h_0(x)-1)\big\}\big|_{x=1}+\Gamma(1-\mu)}{I_-^\mu \big\{(1+x)^{1-\mu}\hat h_0(x)\big\}\big|_{x=1}}\,.
\begin{array}{c}-\beta \frac{\partial G}{\partial \mu}(\beta ,R,\mu )=Tr\{\frac \partial{\partial \mu}\ln \left( D(\beta ,R)-i\mu \gamma^0\right)_{bc}\}=Tr\{-i\gamma^0K_{bc}(t,x;t^{\prime},x^{\prime})\}, \end{array}
L_3^\prime(\lambda)=\begin{bmatrix}0 \lambda I_n -I_n 0 0 \lambda I_n -\lambda P_1+P_0 P_1 0 0 -I_n P_1 \lambda P_3+P_2 \lambda P_4 \lambda I_n 0 0 \lambda P_4 \lambda P_5-P_4 -I_n 0 0 \lambda I_n -I_n 0\end{bmatrix}
\P(L[(S(0) , \dots , S({\tau_D}))^-]=\omega^-) =\sum_{\omega^\prime \subset D : L[(\omega^\prime )^-]=\omega^-}\prod_{i=0}^{|\omega^\prime |-1}p(\omega^\prime_i, \omega^\prime_{i+1}) &=\sum_{\omega^\prime \subset D : L[\omega^\prime ]=\omega^-}\prod_{i=0}^{|\omega^\prime |-1}p^*(\omega^\prime_i, \omega^\prime_{i+1}),
\frac{\Gamma_2(z+1-k\,|\,\tau)}{\Gamma_2(z+1\,|\,\tau)}= \bigl(\frac{1}{2\pi\tau}\bigr)^{k/2}\tau^{\sum\limits_{j=0}^{k-1}(z-j)/\tau}\;\prod\limits_{j=0}^{k-1}\Gamma\bigl(\frac{z}{\tau}-\frac{j}{\tau}\bigr).
\min_{P \in \mathcal{P}_Q}D(P\|Q) begin{align*}0.1cm] \geq \varphi(\pi_Q) \; \min_{P \in \mathcal{P}_Q}|P-Q|^2 & = \varphi(\pi_Q) \, \left( \min_{a \in \mathcal{A}}\delta \, Q(a) \right)^2 begin{align*}0.1cm] = \varphi(\pi_Q) \, Q_{\min}^2 \, \delta^2 \triangleq E_{}& \geq \left(\frac{Q_{\min}^2 \; \log e}{2}\right) \, \delta^2
D_{i}=x_i\frac{\partial}{\partial x_i}+\beta\sum_{j\neq i,\sigma(i)=\sigma(j)}\frac{x_i}{x_i-x_j}\left( 1 - K_{ij}\right)+\beta\sum_{j,\sigma(i)\neq\sigma(j)}\frac{x_i}{x_i-x_j}\left( 1 - K_{ij}\right).
A=B + \begin{pmatrix}\mathbf{0}_{n-3}\mathbf{0}\mathbf{0}N_1 \end{pmatrix}+\dots +\begin{pmatrix}\mathbf{0}_{n-3}\mathbf{0}\mathbf{0}N_k \end{pmatrix}+ \left(T_u+\begin{pmatrix}\mathbf{0}_{n-3}\mathbf{0}\mathbf{0}K \end{pmatrix}\right)+\left(T_l+\begin{pmatrix}\mathbf{0}_{n-3}\mathbf{0}\mathbf{0}L \end{pmatrix}\right),
H_q(m)=\begin{cases}\prod_{j=1}^m\left(q^{-\frac j2}-q^{\frac j2}\right)^{m-j}, m=0,1,2,\dots, \prod_{j=1}^{m+\frac 12}\left(q^{\frac 14-\frac j2}-q^{\frac j2-\frac 14}\right)^{m+\frac 12-j}, m=-1/2,1/2,3/2,\dots.\end{cases}
\frac{d^2\phi}{d\theta^2}(\theta)=\mathsf{V}_\theta [(g(x,x')- \frac{d \phi}{d \theta}(\theta))+ \frac{d}{d\theta}\log \overline{P}^3_{\theta}(x)- \frac{d}{d\theta}\log \overline{P}^3_{\theta}(x')]>0,
\Delta_f R =R-2||^2, &\Delta_f \frac{||^2}{R}=\frac{||^2}{R}+2\frac{||^4}{R^2}+2\frac{|\nabla R -\nabla R|^2}{R^3}-4\frac{(,)}{R}. &\Delta_f R_{ij}=R_{ij}-2R_{ikjl}R_{kl}, &\nabla_k R_{jk}=R_{jk}f_k=\frac{1}{2}\nabla_j R,
+[\begin{matrix}0&0\\ 0&\delta z\\ \end{matrix}]=[\begin{matrix}\sigma_{xx}-\sigma_{hydrostatic}&\tau_{xz}\\ \tau_{zx}&\sigma_{zz}-\sigma_{hydrostatic}\\ \end{matrix}]+[\begin{matrix}\sigma_{hydrostatic}&0\\ 0&\sigma_{hydrostatic}\\ \end{matrix}]
\tilde{b}:= \tilde{\Psi}\circ (\tilde{h}- \tilde{\mathfrak{s}}) \circ \tilde{\psi}^{-1}: \tilde{U'}\rightarrow \tilde{U'}\lhd (\mathbb{R}^{k'}\times \mathbb{R}^n) \times \tilde{\mathbb{W}}
\varrho_0:= \left( \overline{\nu}\overline{\sigma}{\underline{\sigma}}^{-2}\underline{\eta}^{-2}\max \left\{8\overline \nu_1^2,16\overline \nu_2^2,32\overline \nu_3^2,64\overline \nu_4^2,128\overline \nu_5^2,128\overline{\nu}_0^2, \ 4\kappa_0^2\right\}\right)^{1/2}\, .
\sqrt{n}\delta I(U_j;Y_j|U_{j+1}) =\frac{\sqrt{n}\delta}{2}\log\left(1+\frac{\frac{\gamma}{n^{1/3}}}{\sigma_{n,j}^2+\frac{(j-1)\gamma}{n^{1/3}}}\right) &=\frac{n^{1/6}\delta\gamma/2}{\sigma_{n, j}^2+ \frac{(j-1)\gamma}{n^{1/3}}}- O(n^{-1/6})\,.
\begin{cases}\dot{X}_t(x,v) = \frac{V_t(x,v)}{\sqrt{1+V_t^2(x,v)}}\dot{V}_t(x,v) = - \nabla ( V * \widetilde{\rho}_t) ( X_t(x,v))\end{cases}\begin{cases}\dot{X}^k_t (x,v) = \frac{V^k_t(x,v)}{\sqrt{1+V^k_t(x,v)^2}}\dot{V}^k_t (x,v) = - \nabla ( V * \widetilde{\rho}^k_t) ( X^k_t(x,v))\end{cases},
K(u) =\displaystyle\int_{\Omega}\widetilde{b}(x)\left|u\right|^{q(x)}dx \geq \displaystyle \int_{\Omega_{u}}\widetilde{b}(x)\left|u\right|^{q(x)}dx &\geq \epsilon_{1}\left\|u\right\|^{q^{-}}m\left(\Omega_{u}\right)\geq\epsilon_{1}^{2}\left\|u\right\|^{q^{-}}.
P^{d}= \left( \begin{array}{ccc}\frac{1 + (2\alpha -1)^{d}}{2}\frac{1 - (2\alpha -1)^{d}}{2}\frac{1 - (2\alpha -1)^{d}}{2}\frac{1 + (2\alpha -1)^{d}}{2}\end{array}\right).
\frac{Eh^{3}}{12(1-\nu^{2})}\Delta^{2}w-h(\frac{\partial^{2}\varphi}{\partial x_{2}^{2}}\frac{\partial^{2}w}{\partial x_{1}^{2}}+\frac{\partial^{2}\varphi}{\partial x_{1}^{2}}\frac{\partial^{2}w}{\partial x_{2}^{2}}-2\frac{\partial^{2}\varphi}{\partial x_{1}\partial x_{2}}\frac{\partial^{2}w}{\partial x_{1}\partial x_{2}})=P
z^{\alpha -1}E_{\alpha, \alpha}(- \mu_{m,n}z^\alpha) = \frac{-1}{z \mu_{m,n}}(-\mu_{m,n}z^\alpha E_{\alpha, \alpha}(- \mu_{m,n}z^\alpha)) = \frac{-1}{\mu_{m,n}}z^{-1}E_{\alpha, 0}(-\mu_{m,n}z^{\alpha}).
\mathcal{D}_i \left[ \frac{1}{\bar{\rho}}\partial_r ( \bar{\rho}^\gamma d \tfrac{1}{r^2}\partial_r (r^2 H)) \right] = \mathcal{D}_i L_0 H =\mathcal{L}_i \mathcal{D}_i H + \sum_{j=0}^{i-1}q_{ij}\mathcal{D}_{i-j}H,
s_k(x) : = \sum_{j=1}^{k-1}\left( \sum_{m=0}^{k-j-1}d^m \sum_{\substack{\alpha < a(k) w(\alpha)=j+m}}xq^{\alpha}\left( (-x)^{m-1}{j+m-1 \brack m-1}_{q^N}+ (-x)^m{j+m \brack m}_{q^N}\right) \right) \times \prod_{h=1}^{j-1}\left(1-xq^{hN}\right) f_{a(1)}\left(xq^{jN}\right).
\alpha (x) (y D(z))-(xD(y))\alpha (z)-(-1)^{|x||y|}\alpha(y)(xD(z))+(-1)^{|x||y|}(y D(x)) \alpha (z) &+\alpha (x)D(yz)-(x y) D(\alpha (z))-(-1)^{|x||y|}\alpha(y)(D(x)z)+(-1)^{|x||y|}(y x) D(\alpha (z))=0
\Delta_k= (\lambda_{2k-1}+\lambda_{2k})^2-4[\lambda_{2k-1}\lambda_{2k}+(\lambda_{2k-1}-\lambda_{2k})c_k-2c_k^2+1] = (\lambda_{2k-1}-\lambda_{2k})^2+4 (-\lambda_{2k-1}+\lambda_{2k})c_k+8c_k^2-4.
\begin{array}{l}{{f_{l k}^{( + )}= \exp \{\chi_{l k}^{R}- i \chi_{l k}^{J}\}, \quad f_{k l}^{( - )}={( f_{l k}^{( + )})}^{*},}}\\{{\chi_{l k}^{R}= \chi_{R}( \theta_{+ l}^{\prime}) - \chi_{R}( \theta_{+ k}) , \quad \chi_{l k}^{J}= \chi_{J}( \theta_{+ l}^{\prime}) - \chi_{J}( \theta_{+ k}) .}}\\ \end{array}
\Psi_{1,0}\big(C^{(+)}_1 A_2 C^{(+)-1}_1\big) = L^{(+)}_1 \widetilde{L}^{(+)}_1 L^{(+)}_2 L^{(-)-1}_2 \widetilde{L}^{(+)-1}_1 L^{(+)-1}_1 = L^{(+)}_1 L^{(+)}_2 L^{(-)-1}_2 L^{(+)-1}_1 &= R_{12}^{-1}L^{(+)}_2 L^{(+)}_1 R_{12}L^{(-)-1}_2 L^{(+)-1}_1 = R_{12}^{-1}L^{(+)}_2 L^{(-)-1}_2 R_{12}& = \Psi_{1,0}\big(R_{12}^{-1}A_2 R_{12}\big)
\small{H(b,0) = \begin{pmatrix}H_{1}(b) \otimes 1 + 1 \otimes \big( -\partial_3^2 \big) 0 0 H_{2}(b) \otimes 1 + 1 \otimes \big( -\partial_3^2 \big)\end{pmatrix}=: \begin{pmatrix}\mathcal{H}_{1}(b) 0 0 \mathcal{H}_{2}(b)\end{pmatrix}},
z_1^{\frac{j_2}{T}}(z_1-z_2)^{\frac{j_1}{T}}Y_{M^3}(u, z_1)I^{\circ}(w_1, z_2)w_2 - e^{\frac{j_1}{T}\pi i}z_1^{\frac{j_2}{T}}(z_2-z_1)^{\frac{j_1}{T}}I^{\circ}(w_1, z_2)Y_{M^2}(u, z_1)w_2 = \mbox{Res}_{z_0}z_0^{\frac{j_1}{T}}z_1^{-1}\delta(\frac{z_2+z_0}{z_1})(z_2+z_0)^{\frac{j_2}{T}}I^{\circ}(Y_{M^1}(u, z_0)w_1, z_2)w_2
B_{1}= \{2,5,6 \}, B_{5}= \{2,7,8 \}, B_{2}= \{3,5,7 \}, B_{6}= \{3,6,8 \}, B_{3}= \{4,5,8 \}, B_{7}= \{4,6,7 \}. B_{4}= \{2,3,4 \},
P\Big( \sum_{u=1}^p (Z_u-\underline{Z}_u) \ge \epsilon \bar h_4(n) \Big) \le \sum_{u=1}^p P(Z_u - \underline{Z}_u \ge \epsilon p^{-1}\bar h_4(n) ) \le C p^3 ( \epsilon \bar h_4(n) )^{-2}n^2 (\log n)^{-\alpha/2}\le C' \epsilon^{-2}(\log n)^{-c_1 \lambda}.
1+n_2(G)=2n_8(G) \Longleftrightarrow 2^{n+a+b}=2^{n+2a+2b-1}(2^b-1) &\Longleftrightarrow 1=2^{a+b-1}(2^b-1) &\Longleftrightarrow a=0 b=1.
\delta \phi (x,t) = \int \frac{d^3 k}{\sqrt{2 k}}\left( a_{\bf k}\delta \phi_{\bf k}(t) e^{i{\bf k}\cdot{\bf x}}+ a_{\bf k}^{\dagger}\delta \phi_{\bf k}^{*}(t) e^{-i{\bf k}\cdot{\bf x}}\right)
-{\partial^4 \ln[P_{\rm vac}(x^+)] \over \partial x^+ \partial x^- \partial^2x_{\bot}}={1 \over 4 \pi}\sum_{n =1}^{\infty}\left({e E(x^+) \over n \pi}\right)^2 \exp\left[-{n \pi m^2 \over \vert e E(x^+)\vert}\right] \; .
\frac{\log{\log{N}}}{N}\le\frac{3}{8}\frac{\log{N}}{N}<\frac{3}{8}\left(\frac{5\tau^2 A_F}{4}+1\right)\frac{\log{N}}{N}\le \frac{3}{8}\cdot\frac{1}{3}\log{R}=\frac{1}{8}\log{R}.
\int_{-\infty}^{\infty}\frac{e^{-2\pi i h x}}{\lvert z \rvert^{2w - k}z^k}dx = \frac{y^{-w}i^{-k}\pi^w \lvert h \rvert^{w-1}}{\Gamma(w + \frac{\lvert h \rvert}{h}\frac{k}{2})}W_{\frac{\lvert h \rvert}{h}\frac{k}{2}, \frac{1}{2}- w}(4\pi \lvert h \rvert y).
{\cal T}={\frac{f_{\pi}^{2}}{2}}{\cal R}_{0}\cdot{\cal R}_{0}-{\frac{1}{2 e^{2}}}\biggl [{\cal R}_{0}\cdot{\cal R}_{\mathrm{i}}\;{\cal R}_{0}\cdot{\cal R}_{\mathrm{i}}-{\cal R}_{0}\cdot{\cal R}_{0}\;{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{i}}\biggr ]
\begin{aligned}\big(\int_{-\infty}^x dy\phi_3^{-2}(y)\int_y^a\phi_3(z)dz\big)\big(\int_x^a\phi_3^{-2}(t)dt\big)- &\big(\int_x^a dt\phi_3^{-2}(t)\int_t^a\phi_3(z)dz\big)\big(\int_{-\infty}^x\phi_3^{-2}(y)dy\big).\end{aligned}
\begin{aligned}\|P_\mu H_N(\varphi_{\lambda,N_1}^\dagger\gamma^0\psi_{\lambda,N_2})\|_{L^2_tL^2_x}&\lesssim \mu\left(\frac\mu\lambda\right)^{-\delta}(\|\varphi_{\lambda,N_1}\|_{V^2_{\theta_1}}\|\psi_{\lambda,N_2}\|_{V^2_{\theta_2}})^{1-\delta}(\lambda^{-1}\|\varphi_{\lambda,N_1}\|_{L^4_tL^4_x}\|\psi_{\lambda,N_2}\|_{L^4_tL^4_x})^\delta.\end{aligned}
= - i \frac{3 \hbar c}{2}\int d^{3}\vec{r}[ \epsilon_{0}( \varphi^{\dagger}\varphi + \chi^{\dagger}\chi ) + \varphi^{\dagger}( \vec{\sigma}\cdot \vec{\epsilon}) \chi + \chi^{\dagger}( \vec{\sigma}\cdot \vec{\epsilon}) \varphi ]{.}
|\varphi(x, y)-\varphi_{\lambda, v}(x, y)| =|(a(x),Py+y_0)-(\lambda^{-1}v, Py+y_0)| &=|(a(x)-\lambda^{-1}v, Py+y_0)|\leq\rho|a(x)-\lambda^{-1}v| &\leq \rho|a(x)-a(x_{\lambda, v})|+\rho|a(x_{\lambda,v})-\lambda^{-1}v| &\leq\rho\omega(\sqrt{k}2\delta_\lambda)+\rho\varepsilon_0/\lambda=2\rho\varepsilon_0/\lambda.
d\partial_x\Psi(s,X(s),\lambda) =\sqrt{\xi''(s)}\partial_{x}^2\Psi(s,X(s),\lambda)dW(s), d\partial_{x}^2\Psi(s,X(s),\lambda) =-\xi''(s)A(s)\bigl(\partial_{x}^2\Psi(s,X(s),\lambda)\bigr)^2ds+\sqrt{\xi''(s)}\partial_{x}^3\Psi(s,X(s),\lambda)dW(s).
S^{shogun}\le S(3;0,0)^8 S(3;2,2)^3 \le S(3;0,0)^8 S_U(3;2,2)^3 =: S_U^{shogun}\approx (6.6709 \times 10^{21})^8 \times (1.5976 \times 10^{11})^3 \approx 1.5993 \times 10^{208}.
F ( x , x^{\prime}) = F ( x ) - \frac{1}{2}F_{; \alpha}( x ) \sigma^{; \alpha}+ \frac{1}{2}F_{\alpha \beta}( x ) \sigma^{; \alpha}\sigma^{; \beta}+ \frac{1}{4}\{\frac{1}{6}F_{; \alpha \beta \gamma}( x ) - F_{\alpha \beta ; \gamma}( x ) \}\sigma^{; \alpha}\sigma^{; \beta}\sigma^{; \gamma}+ O ( \sigma^{2}) ~ ,
\widehat{w_d}(\xi,y_d)=i\int^{\infty}_0\frac{e^{-|\xi|y_d}- e^{-\omega_\lambda(\xi)y_d}}{\omega_\lambda(\xi)(\omega_\lambda(\xi)-|\xi|)}e^{-\omega_\lambda(\xi)z_d}\xi \cdot \widehat{f}'(\xi,z_d)dz_d
U^* \left(\!\!\begin{array}{cccc}\sigma_1 x_1 \sigma_2 x_2 \sigma_3 x_3 x_4 \end{array}\!\!\right) Q \left(\!\!\begin{array}{cccc}0 1 0 0 1 0 0 0 0 0 0 i 0 0 -i 0 \end{array}\!\!\right)Q \left(\!\!\begin{array}{c}\sigma_1 x_1 \sigma_2 x_2 \sigma_3 x_3 x_4 \end{array}\!\!\right) U = 0,
-G\left(A_t + \frac{1-\tilde{Q}_t(A_t, \dot{\varphi}_0)}{q_{\sigma,t}(1 + x)}\right) + \ln\left( \frac{x}{\mu}f_{\tilde{\sigma}_t(0)}(x) \right)+ \chi_t = \mu (\ln_2 t)^{\mu-1}h_{t,\dot{\varphi}_0}(x)
\mu_0(\varepsilon, B \setminus B_J)\&=\mu_0(\varepsilon, B \setminus B_J) + \mu_0(\varepsilon, B_J) \leq\nu_0(\varepsilon, B) \leq\nu_0(\varepsilon, B \setminus B_J) + \nu_0(\varepsilon,B_J) &=\nu_0(\varepsilon, B \setminus B_J)
|U|^{-\frac{1}{d}}\|w_U - w_E\|_\infty = |U|^{-\frac{1}{d}}\mu^{-1/2}\|u_U - u_E\|_\infty \leq C|U|^{\frac{d+2}{2d}-\frac{1}{d}}\|u_U - u_E\|_\infty = C|U|^{1/2}\|u_U - u_E\|_\infty
V(B_{p_{1},...,p_{n}}(L))=\frac{2^{n}\Gamma(\frac{1}{p_{1}}+1)\cdot\cdot\cdot\Gamma(\frac{1}{p_{n}}+1)}{\Gamma(\frac{1}{p_{1}}+\cdot\cdot\cdot+\frac{1}{p_{n}}+1)}L^{\frac{1}{p_{1}}+\cdot\cdot\cdot+\frac{1}{p_{n}}}.
{\Phi}^{( I )}= - \frac{s \, e^{2}F ( 1 - F )}{1 2{\pi}^{2}| m |}\int_{1}^{\infty}\frac{d v}{v{\sqrt{v - 1}}}\frac{( 1 + F ) ( 1 + A{v}^{F}) - ( 2 - F ) ( 1 + A^{- 1}v^{1 - F})}{A{v}^{F}+ 2 + A^{- 1}v^{1 - F}}.
V^I_n(\theta_n,c) \equiv \{\lambda\in B^d_{n,\rho}:p'\lambda=0\cap v^I_{n,j,\theta_n}(\lambda)\le c,j=1,\cdots,J\}, \mathfrak{W}(c) \equiv \big\{\lambda\in \mathfrak B^d_\rho: p^\prime \lambda = 0 \cap \mathfrak w_{j}(\lambda)\le c, \: \forall j=1,\dots,J\big\}.
Q(u) \, =\,{\varepsilon^2 \int_{\mathcal{D}}|\nabla u|^2 + \int_{\mathcal{D}}|u|^2\over (\, \int_{\mathcal{D}}|u|^{q})^{2\over q}},u\in H^1({\mathcal{D}})\setminus \{0\},
{\left[{\bf{A}}\right]_{11}}={\left[{{{\bf{F}}_2}}\right]_{11}},{\left[{\bf{A}}\right]_{kk}}={\left[{{{\bf{F}}_2}}\right]_{kk}}- \frac{{\left[{{\bf{G}}}\right]_{{{1k}}}}}{\omega}{\left[{{{\bf{F}}_2}}\right]_{k1}}, k\ge2,
E^0_{D,f,h}\left(e^{\Phi^h_{D}(\tilde{f})}\right)=\frac{E^0_{D,f,0}\left(e^{h\Phi^0_{D}(1_D)}e^{\Phi^0_{D}(\tilde{f})}\right)}{E^0_{D,f,0}\left(e^{h\Phi^0_{D}(1_D)}\right)}.
\Pi_{\alpha\beta}^j(h_j^1(z_j,t),...,h_{j}^n(z_j,t),t)=\sum_{r,s=1}^n g_{rs}^j(z_j,t)\frac{\partial h_{j}^{\alpha}}{\partial z_j^r}\frac{\partial h_{j}^{\beta}}{\partial z_j^s}.
\frac{g_{D_r,p}(z)}{g_{D_r,0}(z)}= \frac{\log\left|\frac{r^2-\overline{p}z}{r(z-p)}\right|}{\log\left|\frac{r}{z}\right|}\le \frac{\log \frac{R^2 + r_0r_1}{R_1(r_1-r_0)}}{\log \frac{R_1}{r_1}}
\begin{array}{l}{F = u_{0}+ v = \alpha + \bar{\beta},}\\{\quad q u a d \leftrightarrow \quad v = W = \alpha + \bar{\beta}- u_{0},}\\{\bar{F}= u + v_{0}= \bar{\alpha}+ \beta ,}\\{\quad \quad \leftrightarrow \quad u = \bar{W}= \bar{\alpha}+ \beta - v_{0}.}\\ \end{array}
\begin{aligned}\xi^{\prime}(x)+\eta^{\prime}(x) =\lambda\xi(x), x\in (0,1) , \eta^{\prime\prime}(x)+\eta^{\prime}(x)+\xi^{\prime}(x) =\lambda\eta(x), x\in (0,1), \xi(0) =\xi(1), \eta(0)=0,\ \ \eta(1) =0.\end{aligned}
(D_t^2-iA\partial_{\alpha})\theta\circ\kappa^{-1}=-2[D_t\zeta,\mathcal{H}\frac{1}{\zeta_{\alpha}}+\bar{\mathcal{H}}\frac{1}{\bar{\zeta}_{\alpha}}]\partial_{\alpha}D_t\zeta+\frac{1}{\pi i}\int_{-\infty}^{\infty}\Big(\frac{D_t\zeta(\alpha,t)-D_t\zeta(\beta,t)}{\zeta(\alpha,t)-\zeta(\beta,t)}\Big)^2(\zeta-\bar{\zeta})_{\beta}d\beta,
( \frac 1 2 [ \xi^{*}, \xi ] )^{n}= n ! \; \sum_{i_{1}< \cdots < i_{n}}\xi_{i_{1}}^{*}\cdots \xi_{i_{n}}^{*}\xi_{i_{1}}\cdots \xi_{i_{n}}, \; \cdots , ( \frac 1 2 [ \xi^{*}, \xi ] )^{p}= p ! \; \xi_{1}^{*}\cdots \xi_{p}^{*}\xi_{1}\cdots \xi_{p},
y\prime=\frac{y\cdot((y^{2}+(x-a)^{2})^{\frac{3}{2}}-(y^{2}+(x+a)^{2})^{\frac{3}{2}})}{(x+a)\cdot(y^{2}+(x-a)^{2})^{\frac{3}{2}}-(x-a)\cdot(y^{2}+(x+a)^{2})^{\frac{3}{2}}}
\mathbb{E}\left[e^{-Q_T(\phi,X_0,\mathbf{\varepsilon})}|X_0=z\right]= \frac{1}{\left(\sqrt{2\pi}\right)^T}\int_{\mathbb{R}^T}e^{-\mathbf{x^TA}_T\mathbf{x}-\mathbf{b^T}(\phi,z)\mathbf{x}-c(\phi,z)}\mathrm{d}\mathbf{x}.
\ln\det\mathcal{T}_{\nu,\theta}\geq\sum_{1\leq i\leq 3}\big\{(1-\theta)(1-\nu)\ln T_{tr}+(1-\theta)\nu \ln\Theta_i+\theta \ln T_{\delta}\big\}\cr = 3(1-\theta)(1-\nu)\ln T_{tr}+(1-\theta)\nu\ln \Theta_1\Theta_2\Theta_3+3\theta \ln T_{\delta}\cr = 3(1-\theta)(1-\nu)\ln T_{tr}+(1-\theta)\nu\ln\det\Theta+3\theta \ln T_{\delta}.
S_{\mathrm{K K 1 1}}^{\mathrm{g r a v .}}= -{\frac{T_{\mathrm{K K 1 1}}}{2}}\int d^{7}\xi \sqrt{| \gamma |}\left[ k^{4 / 7}\gamma^{i j}D_{i}X^{\mu}D_{j}X^{\nu}g_{\mu \nu}- 5 \right] ,
\bar{q}(n,t)=\sum_{\Omega_{n}}\prod_{j=1}^{n}\frac{\left(\rho^{j}\binom{r+j-1}{j}\right)^{x_{j}}}{x_{j}!}\left(\frac{\lambda t (1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{z_{n}}e^{-\lambda t},
\mathbf{H}_k(\mathbf{x}; \mathbf{0},\mathbf{I}_d) = (2)^{\frac{k+d+1}{2}}(\pi)^{-d/2}\exp \left( \frac{1}{2}\mathbf{x}' \mathbf{x}\right) & \int_{\mathbb{R}^{d}}\mathbf{u}^{\otimes k}\exp \left( -\mathbf{u}'\mathbf{u}\right) \cos\left( \sqrt{2}\mathbf{x}'\mathbf{u}- \frac{k\pi}{2}\right) d \mathbf{u}
\left\Vert \mathbf{H}-\sum_{0\le t<\rho}\mbox{Prod}_{\boldsymbol{\Delta}^{(t)}}\left(\mathbf{X}^{(1)},\,\mathbf{X}^{(2)},\,\cdots,\mathbf{X}^{(m)}\right)\right\Vert =0.
\mathbb{E}\left[e^{\alpha M_{s,t}^{(n)}}\right]=\mathbb{E}\left[\exp\left(\alpha M_{s;t_{1},\cdots,t_{n}}\right)\right]\leq2\exp\left(\frac{\alpha^{2}}{2}\left[\gamma_{H}(t-s)^{2H}+(t-s)\right]\right).
\begin{aligned}\| v_{i}\|_{C^{(1+\alpha)/2,1+\alpha}}&\leq C\big(r^{-1-2\alpha}+r^{-2}+r^{-\frac{p}{2}(1+\alpha)}\big)\big(\|v_{0}\|_{C^{1+\alpha}}+\|f\|_{L^p(D_T)}+\|g \|_{C^{\alpha/2,\alpha}}\big).\end{aligned}
\frac{\partial}{\partial u}\left(\varphi^{-1}\frac{\partial\varphi}{\partial v}\right)+\frac{\partial}{\partial v}\left(\varphi^{-1}\frac{\partial\varphi}{\partial u}\right)-2U\left(\varphi^{-1}\frac{\partial\varphi}{\partial u},\varphi^{-1}\frac{\partial\varphi}{\partial v}\right)=0.
V_{F}= \frac{\tau_{p}( g_{s})}{4 N ( 2 \pi l_{s}^{2})^{2}}\left( | \Phi_{1}\Phi_{2}- \epsilon^{- 1}\Phi_{2}\Phi_{1}|^{2}+ | \Phi_{2}\Phi_{3}- \epsilon^{- 1}\Phi_{3}\Phi_{2}|^{2}+ | \Phi_{3}\Phi_{1}- \epsilon^{- 1}\Phi_{1}\Phi_{3}|^{2}\right) ,
\delta S_{\psi\psi\lambda\lambda}= \int d^{2}\sigma\left[\frac{1}{2}v^{\rho}\hat{D}^{(+)}{}_{\rho}F_{\mu\nu ab}-2 i F_{\mu\nu c b}\Sigma^{c}{}_{a}\right]\psi^{\mu}\psi^{\nu}\lambda^{a}\lambda^{b}\ .
(F_1F_2)(c_{r_1}^{(1)}\cdots c_{r_m}^{(m)}) = \left(\prod_{i \not\in I(F_1) \cup I(F_2)}c_{r_i}^{(i)}\right)\cdot F_1\left( \prod_{i\in I(F_1)}c_{r_i}^{(i)}\right)\cdot F_2\left( \prod_{i\in I(F_2)}c_{r_i}^{(i)}\right).
S_{2}( \phi , \phi ) = - \int d^{D}X \sqrt{- G}\left\{( D_{M}\phi )^{\dagger}D^{M}\phi + \frac{1}{2}\phi \frac{\partial^{2}U}{\partial \Phi^{2}}\phi + \frac{1}{2}\phi^{*}\frac{\partial^{2}U}{\partial{\Phi^{*}}^{2}}\phi^{*}+ \phi^{*}\frac{\partial^{2}U}{\partial \Phi \partial \Phi^{*}}\phi \right\}~ ,
T=\begin{pmatrix}t_{1,1}t_{1,2}t_{1,3}t_{2,1}t_{2,2}t_{2,3}t_{3,1}t_{3,2}t_{3,3}\end{pmatrix}\mapsto \Psi(T)=\begin{pmatrix}t_{1,1}\frac{1}{2}t_{1,2}\frac{1}{2}t_{2,1}\frac{1}{4}t_{2,2}+ \frac{3}{4}t_{3,3}\end{pmatrix}.
\sum_{K=1}^{\infty}\frac{(\log(\tilde{s}_K^2))^{10}}{(\tilde{s}_K^2)^2}\mathbb{E}\left[\tilde{X}_{m,K}^4\right]\leq c\sum_{K=1}^{\infty}\frac{(\log(K))^{10}}{K^2}<\infty.
\eta_k =\frac{\partial L}{\partial \dot z^k}(t,z_*,\dot z_*)+\int_t^{\omega}\frac{\partial L}{\partial z^k}(s,z_*,\dot z_*)ds &-\dot\gamma^j(t)\frac{\partial f_j}{\partial z^k}(t,z_*)-\int_t^\omega\dot\gamma^j(s)\frac{d}{ds}\Big(\frac{\partial f_j}{\partial z^k}(s,z_*)\Big)ds-\lambda_k\in Y.
z^{2}+A_{W}z^{2}=z^{2}+\mathfrak{p}_{-}z^{2}A_{W}1-A_{W}\mathfrak{p}_{+}z^{2}A_{W}1 & =z^{2}+\mathfrak{p}_{-}z^{2}\varphi_{W}-A_{W}\mathfrak{p}_{+}z^{2}\varphi_{W}& =\left( 1+\varphi_{W}\left( z\right) \right) \left(z^{2}-a_{2}-a_{1}m_{W}\left( z\right) +a_{1}^{2}\right)
E(u_{k};k,R) = \int_{B_{R}(0)}|\nabla u_{k}|^{2}+ \int_{B_{R}(0)}u_{k}^{10/3}- \int_{B_{R}(0)}\left( m_{k}* Y_{a_{k}}\right) u_{k}^{2}\bigg. & + \frac{1}{2}\int_{B_{R}(0)}\left( u_{k}^{2}\cdot \chi_{B_{R}(0)}* Y_{a_{k}}\right) u_{k}^{2}+ \int_{B_{R}(0)}\left( u_{k}^{2}\cdot \chi_{B_{R}(0)^{c}}* Y_{a_{k}}\right) u_{k}^{2}.
\begin{cases}\rho\geq0,\ \rho\in C([0,T]; W^{1,q}),\ \rho_t\in C([0,T]; L^{q}), \nabla\mathbf{u}\in L^{\infty}(0,T; H^1)\cap L^2(0,T;W^{1,q}), \sqrt{\rho}\mathbf{u}, \ \sqrt{\rho}\dot{\mathbf{u}}\in L^{\infty}(0,T; L^2), P\geq0,\P\in C([0,T];W^{1,q}),\P_t\in C([0,T];L^{q}),\end{cases}
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=\frac{\alpha\eta}{\tau}-\beta\rho\tau\leqslant -\frac{\beta(1+3\rho^2)}{2}\leqslant-\frac{\beta}{2},\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=\frac{\alpha\eta}{\tau}-\beta\rho\tau \leqslant -\frac{\beta(1+3\rho^2)}{2}\leqslant-\frac{\beta}{2},
\frac{\delta \omega(x,y)}{\lambda(x,y)}= \frac{2 \delta \omega(0,y)}{\lambda(0,y)}+ x \; 2\frac{\partial_x \delta \omega(0,y) \lambda(0,y) - \partial_x \lambda(0,y)}{\lambda(0,y)^2}+ \cdots \frac{\partial_y \delta \omega(x,y)}{\lambda(x,y)}= \frac{\partial_x \delta \omega(x,0)}{\lambda(x,0)}+ \cdots
\left[ \left( \mathbf{w},\mathbf{w}^{+}\right) \right]_{0,\Gamma}:=\gamma_{T}\mathbf{w-}\gamma_{T}^{+}\mathbf{w}^{+}\mathbf{,\quad}\left[\left( \mathbf{w},\mathbf{w}^{+}\right) \right]_{1,\Gamma}:=\gamma_{T}\operatorname{curl}\mathbf{w-}\gamma_{T}^{+}\operatorname{curl}\mathbf{w}^{+}.
I_{311}= -r\cos(kx)\sin(x)\int_{0}^{2\pi}\log\left(\sin^{2}\left(\frac{y}{2}\right)\right)\cos(x-y)dy = r\cos(kx)\sin(x)\cos(x)(2\pi) I_{312}= \cos(kx)\cos(x)\int_{0}^{2\pi}\log\left(\sin^{2}\left(\frac{y}{2}\right)\right)\sin(x-y)dy & = \cos(kx)\cos(x)\sin(x)(-2\pi)
S = \frac{1}{1 6 \pi}\int \mathrm{d}^{4}x \sqrt{- g}\left[ R - 2 \left( \nabla \phi \right)^{2}- \mathrm{e}^{- 2 \phi}F^{2}- \frac{1}{1 2}\mathrm{e}^{- 4 \phi}H^{2}\right] .
f(I_{B_2}) = s(B_2)(|c|+s(B_2)) f(I_{A_2}) = s(A_2)(|b_2|+s(A_2)) f(I_{B_1}) = s(B_1)(|c|+s(B_1)) f(I_{A_1}) = s(A_1)(|b_1|+s(A_1)).
B= R\psi^{\sigma_3}\hat{B}_0\psi^{-\sigma_3}R^{-1}+ R_tR^{-1}- \frac{x}{2}I +\frac{\kappa_t}{\kappa}I +\frac{\psi_t}{\psi}R\sigma_3R^{-1}\equiv -x^2J +x{B}_1 +{B}_0,
{}_2F_1\bigg[\begin{matrix}\alpha \alpha &1\end{matrix}\bigg|\,z\bigg]=(1-z)^{-\alpha}{}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12-\frac12\alpha &1\end{matrix}\bigg|\,-\frac{4z}{(1-z)^2}\bigg].
\mu_{i}=b_{i},\thinspace \thinspace K_{i}=\frac{b_{i}-1}{c_{ii}},\thinspace \thinspace \tilde{c}_{ij}=K_{i}c_{ij},\thinspace i\ne j,\thinspace \thinspace i, j=1,\thinspace \mathellipsis ,\thinspace d