harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\b' = t \cdot \begin{pmatrix}-1 J_1 \end{pmatrix}+ \frac{1}{b_1^2 - d_1^2 J_1}\cdot \begin{pmatrix}a_1 b_1 + c_1 d_1 J_1 (a_1 d_1 + b_1 c_1) J_1 (a_1 d_1 + b_1 c_1) J_1 (a_1 b_1 + c_1 d_1 J_1) J_1 \end{pmatrix}.
\left[\hat X_{1,q},\; \hat X_{2,q}\right] = i\theta + (1-q^2)\left(\frac{i\hbar^2}{2(\lambda_1 +\lambda_2)^2}\right) \left(K_2 \tilde A_2^\dagger \tilde A_2 - K_1\tilde A_1^\dagger \tilde A_1\right);
^{(1)}\bar{u}^{n+1/2}_{j}=\theta^{(1)}\tilde{u}^{n+1}_{j}+(1-\theta)u^{n}_{j},^{(2)}\bar{u}^{n+1/2}_{j}=(1-\theta)^{(2)}\tilde{u}^{n+1}_{j}+\theta u^{n}_{j}.
(\Delta u+u_t+2)^2-4\mathrm e^F =\bigl((u_{yy}+u_{tt}+u_t+1)-(u_{xx}+1)\bigr)^2+u_{xy}^2+u_{xt}^2 &\ge\bigl((\Delta u +u_t+2)-2(u_{xx}+1)\bigr)^2,
\Upsilon =\sum_{N<x_1,x_2\le 2N}\biggl| \int_{\mathfrak m}\mathcal G(\alpha)^2e((x_1^k+x_2^k-n)\alpha)\,{\rm d}\alpha \biggr|^2 &=\int_{\mathfrak m}\int_{\mathfrak m}\mathcal G(\alpha)^2\mathcal G(-\beta)^2f_k(\alpha-\beta;N)^he(-n(\alpha-\beta))\,{\rm d}\alpha \,{\rm d}\beta .
Sq_{\mathbb{F}_4}(1) = 1, Sq_{\mathbb{F}_4}(\alpha \xi) = \alpha \xi + \alpha \xi^2, Sq_{\mathbb{F}_4}((\alpha + 1)\xi^2) = (\alpha + 1)\xi^2 + 2(\alpha + 1)\xi^3 + (\alpha + 1)\xi^4.
\bigsqcup_{\substack{h,\bar{h}, h\bar{h}=h_i h\in c(g, X), \bar{h}\in c(\bar{g}, X)}}(g_{t_1}\cdots g_{t_{i-1}})X_{g, h}\cap (g_{t_1}\cdots g_{t_{i-1}}g)X_{\bar{g},\bar{h}}\cap (g_{t_1}\cdots g_{t_{i-1}}g\bar{g})X_{t_{i+1}}&\quad{}=(g_{t_1}\cdots g_{t_{i-1}})X_{t_i}\cap (g_{t_1}\cdots g_{t_i})X_{t_{i+1}}.
0 =H(W_B|S_A^B,S_C^B) &=H(W_B|S_A^B)-I(W_B,S_C^B|S_A^B) &\geq H(W_B|S_A^B)-H(S_C^B) &\geq H(W_B|S_A^B) -(d-|A|)\beta_e &\geq H(W_B|W_A) -(d-|A|)\beta_e .
\sum_{n=1}^{\infty}e^{- \beta k E_n}= \frac{1}{\sqrt{\pi}}\int_0^{\infty}e^{- \frac{x^2}{4}- \frac{1}{x^2}{\beta}^2 k^2 m^2}\frac{1}{e^{\frac{2{\beta}^2 e B k^2}{x^2}}-1}\; .
\sum_{i=0}^jg''_i = \sum_{i=0}^{k-1-m}g''_i+\sum_{i=k-m}^jg''_i = \sum_{i=0}^{k-1-m}g_i+g_{-(i+1)}+\sum_{i=k-m}^jg_i+g_{-(i+1)}+g_{-(k-i)}=\sum_{-(j+1)}^jg_i+\sum_{i=k-j}^mg_{-i}\le (j+1)n
{\cal V}= - \frac{d^{2}}{d y^{2}}+ m^{2}\hspace{0 . 4 c m}, \hspace{0 . 4 c m}\bar{{\cal V}}= - \frac{d^{2}}{d x^{2}}+ 1 \hspace{0 . 4 c m}, \hspace{0 . 4 c m}{\cal V}= m^{2}\bar{{\cal V}}\hspace{0 . 4 c m}, \hspace{0 . 4 c m}\zeta_{\cal V}( s ) = \frac{1}{m^{2 s}}\zeta_{\bar{{\cal V}}}( s ) \, .
S_{Q2f}(M_{n,i}) = \sum_{i=0}^{L-1}\frac{1}{N}\mbox{Tr}\,(2M_{1,i}^2 + 2M_{2,i}^2 )+ \frac{2g}{N^2}\mbox{Tr}\, M_{1,i}^2 \mbox{Tr}\, M_{2,i}^2
-\frac{1}{4}(x-b)^{2}P_{n}^{(\alpha , \beta )}(x)=f_{n+1}\widehat{P}_{n+2}^{(\alpha , \beta )}(x)-2b \g_{n}\widehat{P}_{n+1}^{(\alpha , \beta )}(x)+h_{n}\widehat{P}_{n}^{(\alpha , \beta )}(x),
h_{SO(N)}(g)= \sum_{j=1}^M\sum_{i=1}^Q c_{ij}e^{ik_{ij}^1\phi_1}\cos^{m_{ij}^1}(\phi_2)\sin^{n_{ij}^1}(\phi_2)\cdots \cos^{m_{ij}^{N-1}}(\phi_{N-1})\sin^{n_{ij}^{N-1}}(\phi_{N-1}) &\cdot (h_{SO(N-1)})_{ij}(g_{SO(N-1)}),
M_P:=\left\{\begin{pmatrix}e^{i\theta}0 0 e^{-i\theta}\end{pmatrix}\colon\theta\in[0,2\pi]\right\};A_P:=\left\{\begin{pmatrix}t 0 0 t^{-1}\end{pmatrix}\colon t\in(0,\infty)\right\}.
\left[ \chi^{a b}, \chi^{c d}\right] \left( M_{n}^{\, \, \, \, \, m}\right) = \left( 1 - \Lambda \right)_{e f g h}^{a b c d}\, \left( \chi^{e f}\ast \chi^{g h}\right) \left( M_{n}^{\, \, \, \, \, m}\right) ={\mathbf{C}}_{e f}^{a b c d}\chi^{e f}\left( M_{n}^{\, \, \, \, \, m}\right) ,
0 = \tilde{\psi}_{\varepsilon}+ 2\pi a\varepsilon\frac{L_{a}^{-1}\tilde{u}_j}{\lambda_j - \lambda_{j,\varepsilon}}+ \frac{2\pi}{\varepsilon}L_{a}^{-1}\left( \mathcal{R}_\varepsilon + \mathcal{R}^{\lambda_j,\lambda_{j,\varepsilon}}\right) \tilde{\psi}_\varepsilon.
\sup_{|z|=R}\log|W(z)| < \frac12\left(R_{k-1}+\frac{C_1}{\sqrt k}\right)^2-\frac a8R_k\log^b R_{k}&= \frac{R_{k-1}^2}2+C_1\frac{R_{k-1}}{\sqrt k}+\frac{C_1^2}{2k}-\frac a8R_k\log^b R_{k}&\le \frac{R^2}2-\frac3p\log (R+1).
I_{\mu_1\cdots\mu_n}(x_1,\ldots,x_N) \equiv\int \frac{\mathrm{d}^4 k}{(2\pi)^4}\frac{k_{\mu_1}\cdots k_{\mu_n}}{\prod_{i=1}^{N}(k^2 + x_i )(q^2 + x_i )}.
\tilde{A}\!=\! \left[ \begin{matrix}A_c B_{c1}B C_2 A \!+\! BK \end{matrix}\right]\!\!, \tilde{B}\!=\! \frac{1}{2}\left[ \begin{matrix}B_{c2}\begin{array}{cc}E_1 0_{2, 1}\end{array}\end{matrix}\right] \left[ \begin{smallmatrix}1 0 0 1 0 1 1 0 \end{smallmatrix}\right]^{\top}\!\!.
(g\phi)(w) =\bigl((g\Phi)(w)\bigr)(1,1)=\bigl((g,1)\Phi(w)\bigr)(1,1)=\bigl((g,h_g)(1,h_g^{-1})\Phi(w)\bigr)(1,1) =\bigl((g,h_g)\Phi(h_g^{-1}w)\bigr)(1,1)=\Phi(h_g^{-1}w)(g,h_g)=(g,h_g)\bigl(\Phi(h_g^{-1}w)(1,1)\bigr) =(g,h_g)\phi(h_g^{-1}w).
\widetilde{D}^{s,p}(\Omega):=\Big\{u\in L^{p-1}_{\mathrm{\mathrm{loc}}}(\mathbb{R}^N)\cap L^{p^*}(\Omega) \,:\, \exists\, E\supset\Omega ,\, \mathrm{dist}(E^c, \Omega)>0 \mbox{and}\,[u]_{W^{s,p}(E)}<+\infty \Big\}.
+{\hbar}^{2}\frac{e_{+}^{2}}{4{\pi}^{2}}\sum_{\stackrel{q \in \cal Z}{q \neq ( 0 ; - p )}}{\cal F}_{+ -}( q ) \frac{1}{q ( p + q )}{\rho}_{\mathrm{t o t}}^{\mathrm{N}}( p + q ){\rho}_{\mathrm{t o t}}^{\mathrm{N}}( - q ) ,
G_ne_1 =\frac{D(f^n(p))}{D(p)}\Vert df^n_p v_0^s\Vert e_1=c_{1,n}(p)e_1, \; \; \mathrm{and}G_ne_2 =\frac{D(f^n(p))}{D(p)}\Vert df^n_p v_0^u\Vert e_2=c_{2,n}(p)e_2.
\begin{bmatrix}a_1^{(j-1)}a_2^{(j-1)}\ldots a_n^{(j-1)}\ \end{bmatrix}^{\mathcal{B}_{\rm cycl}}= \boldsymbol{V}^{-1}\begin{bmatrix}\frac{\tilde{S}_{j1}}{\tilde{S}_{11}}\frac{\tilde{S}_{j2}}{\tilde{S}_{12}}\ldots \frac{\tilde{S}_{jn}}{\tilde{S}_{1n}}\end{bmatrix}^{\mathcal{B}_{\rm int}}.
\frac{\theta_{k_0}'(t_1)}{\theta'(t_1)}=\frac{\theta_{k_0}'(t_1)}{\theta_{k_0-1}'(t_1)}\left( \frac{\theta'(t_1)}{\theta_{k_0-1}'(t_1)}\right)^{-1}> \frac{d}{\alpha}=M'\sqrt{d}.
x = [u^*_0,u^*_1,\dots,u^*_{N-2},u^*_{N-1}]^* \in \mathbb{R}^{Nr}U_l = [u_{2T-l},u_{2T+1-l},\dots,u_{N-l}] \in \mathcal{M}_{r \times \bar{N}}(\mathbb{R}).
C = \frac{N-2B}{N}\cdot \frac{1-\frac{T}{N-2B}}{1-\left(\frac{T}{N-2B}\right)^M}=\frac{N-2B}{N}\cdot\left(1+\frac{T}{N-2B}+\frac{T^2}{(N-2B)^2}+\cdots+\frac{T^{M-1}}{(N-2B)^{M-1}}\right)^{-1}
\sum_{i\le k}c_id_i=\sum_{i\le k}c_i\Big(\sum_{j\le i}d_j-\sum_{j\le i-1}d_j\Big) =\sum_{i\le k-1}(c_i-c_{i+1})\sum_{j\le i}d_j \ + \c_k\sum_{i\le k}d_i.
P^{x_0(t)}\Big( R_t \leq \frac{\beta}{2}t + z(t) \Big) = P^{x_0(t)}\Big( R_t \leq \frac{\beta}{2}t + z(t) , \T_0 \leq \alpha t \Big) + P^{x_0(t)}\Big( R_t \leq \frac{\beta}{2}t + z(t) , \T_0 > \alpha t \Big)
\partial_{m}\partial_{n}\gamma^{i j}= - \partial_{m}\Gamma_{n j}^{i}- \partial_{m}\Gamma_{n i}^{j}+ \Gamma_{i m}^{h}\Gamma_{h n}^{j}+ \Gamma_{j m}^{h}\Gamma_{h n}^{i}+ \Gamma_{m h}^{i}\Gamma_{n h}^{j}+ \Gamma_{m h}^{j}\Gamma_{n h}^{i}
g_{\mathrm{m j}}\; \ddot{x}^{\mathrm{j}}+ \biggl ({\frac{\partial g_{\mathrm{m l}}}{\partial x^{\mathrm{k}}}}-{\frac{1}{2}}{\frac{\partial g_{\mathrm{k l}}}{\partial x^{\mathrm{m}}}}\biggr ) \dot{x}^{\mathrm{k}}\dot{x}^{\mathrm{l}}= -{\frac{\partial V}{\partial x^{\mathrm{m}}}}
\int_{M_{U}^{1 1}}\delta C^{\prime}\wedge K \wedge K = \frac{6}{\sqrt{2}}\frac{\bar{\kappa}^{2}}{\lambda^{2}}\int_{M_{U}^{1 1}}\epsilon ( x^{1 1}) \hat{I}_{4}\wedge d C^{\prime}\wedge \delta C^{\prime}= 0
T(B,A;\tau) \equiv \int_0^{2\pi}d\sigma W^{-1}B_{\mu \nu}W \frac{\partial x^{\mu}}{\partial \sigma}\frac{\partial x^{\nu}}{\partial \tau}=\frac12 \int_0^{2\pi}d\sigma W^{-1}B_{\mu \nu}W \frac{\partial (x^{\mu}, x^{\nu})}{\partial (\sigma, \tau)}
T^{AB}(\tau ,\vec \sigma )=-[{2\over{\sqrt{g}}}{{\delta S}\over{\delta g_{AB}}}](\tau ,\vec \sigma )=-\sum_{i=1}^N\delta^3(\vec\sigma -{\vec \eta}_i(\tau )){{m_i{\dot \eta}^A_i(\tau ){\dot \eta}^B_i(\tau )}\over{\sqrt{g_{\tau\tau}+2g_{\tau u}{\dot \eta}^u_i +g_{uv}{\dot \eta}^u_i{\dot \eta}^v_i}}}(\tau ,\vec \sigma ).
\rho :=\left[1 -\frac{1}{\kappa^2}\Big(\big(\sum_{i\in I\smallsetminus J}\frac{1+\alpha_i}{1-\alpha_i}\big) -(1-|J|)\min_{i\in I\smallsetminus J}\frac{1+\alpha_i}{1-\alpha_i}\Big)^{-1}\right]_+^\frac{1}{2(m-1+|J|)},
C(x,\rho )=\cos \rho (x\!-\!a)\!+\!\int\limits_{a}^{x}q_{1}(t)\sin \rho (x\!-\!t)dt\!+\!\int\limits_{a}^{x}q_{0}(t)\frac{\sin \rho (x\!-\!t)}{\rho}dt,
\vert z , \bar{z}\rangle = \exp \{z_{1}a_{1}^{\dagger}+ z_{2}a_{2}^{\dagger}+ z_{3}a_{3}^{\dagger}\cdots + \bar{z}_{1}b_{1}^{\dagger}+ \bar{z}_{2}b_{2}^{\dagger}+ \bar{z}_{3}b_{3}^{\dagger}+ \cdots \}\vert 0 \rangle ,
\delta_{\xi}( \widetilde g_{0}^{- 1}+ \widetilde V ) \ = \ ( \widetilde g_{0}^{- 1}+ \widetilde V ) \widetilde g_{0}( \delta_{\xi}\widetilde T ) \widetilde g_{0}( \widetilde g_{0}^{- 1}+ \widetilde V ) \ ,
\tilde{x}= \frac{\widetilde{\rho}\sinh \tau}{\cosh \tau - \cos \sigma}\cos \varphi , \tilde{y}= \frac{\widetilde{\rho}\sinh \tau}{\cosh \tau - \cos \sigma}\sin \varphi , \tilde{z}= \frac{\widetilde{\rho}\sinh \sigma}{\cosh \tau - \cos \sigma}
f|_{13}= \left[ 1 + \frac{\theta_{ij}- \delta_{ij}\theta}{2\theta^2}\left( C_i C_j - \delta_{ij}C^2 \right) + \frac{2}{5}\frac{q_k}{\rho \theta^2}C_k \left( \frac{C^2}{2 \theta}- \frac{5}{2}\right)\right] f_M,
{{\rm R}\hskip-0.3cm/}\hskip0.14cm_{IJ}-{1\over2}{{\rm R}\hskip-0.3cm/}\hskip0.14cm{\rm g}_{IJ}={\sl\Lambda}\!\{\rm g}_{IJ}-K\left({\rm B}_I^{\K}{\rm B}_{JK}-{1\over4}{\rm B}_{KL}{\rm B}^{KL}{\rm g}_{IJ}\right)
\mathbb{E}[\langle\phi(x),\phi(x^{\prime})\rangle]=\sum_{t,t^{\prime}\in T}(x_{t}x_{t^{\prime}}^{\prime})\cdot\mathbb{E}[\zeta(t)\zeta(t^{\prime})]\cdot\langle e_{h(t)},e_{h(t^{\prime})}\rangle
dV_t=V_{t-}u_t dt+ V^{1/2}_{t-}\left(\sum^{l}_{i=1}\frac{\psi^{(i)}_t}{V^{1/2}_{t-}}\Delta L^{(i)}_t+\sum^{l}_{i,j=1}\frac{c_{i,j}}{V^{1/2}_{t-}}\Delta L^{(i)}_t \Delta L^{(j)}_t\right),
Z_{S}=\Omega_{N_{p}}|H|^{-1/2}(\chi_{min}^{2})^{-\frac{N_{s}-N_{p}}{2}}\frac{\Gamma(\frac{N_{p}}{2})\Gamma(\frac{N_{s}-N_{p}}{2})}{\Gamma(\frac{N_{s}}{2})}
\nabla P_{E_{a_{S_j}}}[i] = \frac{1}{M\sqrt{2\pi}\sigma_{n}\sqrt{{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}}\sum^{M}_{j=1}exp\big(-\frac{c^2_j[i]}{2}\big)sgn(s_j)\Re[{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol h}_{SDA_j}[i]s_j],
L(2,\chi_{4D}) = \left(1 - \frac{1}{2^2}\right)L(2,\chi_{3}) = \frac{3}{4}\sum_{n=1}^\infty\frac{\chi_{3}(n)}{n^2}&\geq \frac{3}{4}\left(\sum_{n=1}^k\frac{\chi_{3}(n)}{n^2}- \sum_{n=k+1}^\infty\frac{1}{n^2}\right) &= \frac{3}{4}\left(-\sum_{n=1}^\infty\frac{1}{n^2}+ \sum_{n=1}^k\frac{\chi_3(n) + 1}{n^2}\right).
s_{k}^{0}=a_{1}+...+a_{2k-1},s_{k}^{o\ast}=\max_{1\leq j\leq k}|s_{j}^{o}| b_{k}^{0}=\max\left( k^{-1}\left( s_{4k}^{o\ast}\right)^{2},\left( s_{k}^{o}\right)^{2}-\left( s_{k-1}^{o}\right)^{2}\right)
P_{i,1}=(\dot{x}_i-\mu_i)\sigma_i\partial_y\widetilde{R}_i &P_{i,2}=-\bigg(\dot{\mu}_i+\sum^n_{\substack{j=1, j\not=i}}\frac{a_{ij}}{x^3_{ij}}+\sum^n_{\substack{k,j=1, j\not=i}}\frac{b_{ijk}\mu_k}{x^3_{ij}}\bigg)\frac{\sigma_i\Lambda\widetilde{R}_i}{1+\mu_i}.
F_{i j}^{a}= \partial_{j}A_{i}^{a}- \partial_{i}A_{j}^{a}- g{\epsilon}^{a b c}A_{i}^{b}A_{j}^{c}\, , \; \; \, F_{i 0}^{a}= \partial_{0}A_{i}^{a}+ \partial_{i}A_{0}^{a}+ g{\epsilon}^{a b c}A_{i}^{b}A_{0}^{c}\, ,
ad\left(\hat{G}_{\mu\nu}\right)\rightarrow 2\left(x_{\mu}\frac{\partial}{\partial x_{\nu}}-x_{\nu}\frac{\partial}{\partial x_{\mu}}-w_{\mu\lambda}\frac{\partial}{\partial w_{\lambda\nu}}+w_{\nu\lambda}\frac{\partial}{\partial w_{\lambda\mu}}\right).
\hat{U}(\theta_0)\,\hat{A}_i^{\pm\pm}(n_1,n_2)\,\hat{U}^{-1}(\theta_0)=\hat{A}_i^{\pm\pm}(n_1,n_2)\,+\,\Big[\frac{i}{\hbar}\int_{T_2}d^2\vec{x}\ \theta_0(\vec{x})\hat{\phi}(\vec{x})\ ,\ \hat{A}_i^{\pm\pm}(n_1,n_2)\Big]\ \ \ .
{\sf s}_{[i,j]}{\sf s}_{[i,j]}X ={\sf s}_{[i,j]}\{c:\exists a\in X, a\equiv_{ij}c\}&=\{b:(\exists a\in X\wedge c\in\eta(\Gamma)), a\equiv_{ij}c, c\equiv_{ij}b\}&=\{b:\exists a\in X, a=b\}&=X.
\limsup_{\varepsilon_1 \rightarrow 0^{+}}\limsup_{n \rightarrow \infty}\mathbb{E}_{\mu_n}\Big[ \int_0^T n^{\gamma-1}\sum_{x,y:|x-y| \geq \varepsilon n}[ \overleftarrow{\eta}_t^{\varepsilon_1 n}(x) \overrightarrow{\eta}_t^{\varepsilon_1 n}(x+1) - \eta_t^n(x) \eta_t^n(x+1) ] F(t, \tfrac{x}{n}, \tfrac{y}{n}) (c_{\gamma})^{-1}p (x-y) dt \Big] =0.
\rho(g h g^{-1}) \eta_g(\rho) v = \rho(g h g^{-1}) \rho(g) v \rho(g^{-1}) = \rho(g h) v \rho(g^{-1}) = \rho(g) \rho(h) v \rho(g^{-1}) = \eta_g(\rho) \rho(h) v
\mathrm{Pr}\left(\{\mathfrak{W}(c)\neq \emptyset\}\cap \{\mathfrak{W}(c-\delta)=\emptyset\}\right)=\mathrm{Pr}\left(\{\mu'g\ge0,\forall\mu\in\mathcal M\}\cap \{\mu'(g-\delta\tau)<0,\exists\mu\in\mathcal M\}\right),
\exp\Big(n\sum_{k=1}^{\infty}\frac{p_{k}}{k}(1-(-y)^{k})x^{k}\Big)=\sum_{\lambda}\frac{p_{\lambda}}{z_{\lambda}}n^{l(\lambda)}x^{\left|\lambda\right|}\prod_{k=1}^{l(\lambda)}(1-(-y)^{\lambda_{k}}).
\chi(T_X)=2\chi(\mathcal{O}_X)+K_X^2-c_2(X), \chi(\mathcal{O}_X(\Delta))=\chi(\mathcal{O}_X)+1/2(\Delta^2-\Delta \cdot K_X), \chi(\omega_X^{-1}\otimes \mathcal{O}_X(-\Delta))= \chi(\omega_X^{2}\otimes \mathcal{O}_X(\Delta))= \chi(\mathcal{O}_X)+1/2\left((\Delta+2K_X)^2-(\Delta+2K_X)\cdot K_X\right)= \chi(\mathcal{O}_X)+1/2(2K_X^2+3K_X\cdot \Delta+\Delta^2).
{\mathcal{I}}= \frac{1}{2 x^{2}}\sum_{n = 1}^{\infty}l^{D}\int_{0}^{\infty}d z \, z^{D - 2}( 1 + z^{2}x^{2})^{1 / 2}e^{- 2 l [ \eta ( z ) - \eta ( z x ) ]}\left[ b^{( - )}( z x ) - b^{( + )}( z x ) \cos ( 2 l \phi ) \right] ,
\left({\frac{b^{\prime}}{b}}\right)^{2}+ ( k b^{2}- M ){\frac{b^{2}}{\phi^{2}}}={\frac{1}{6}}\left( A^{2}+{\frac{B^{2}}{\phi}}+{\frac{3 + 2 \omega}{2}}{\frac{\phi^{2}}{\phi^{2}}}\right) ,
\varphi_{\alpha}(t,\Phi(s)u) =\sup\limits_{\xi\geq t}e^{-\alpha(\xi-t)}\parallel U(\xi,t)(\Phi(s)u)(t)\parallel &=\sup\limits_{\xi\geq t}e^{-\alpha(\xi-t)}\parallel U(\xi,t)U(t,s)u(s)\parallel &=\sup\limits_{\xi\geq t}e^{-\alpha(\xi-t)}\parallel U(\xi,s)x\parallel.
| \Lambda , \Xi \rangle_{o r b}= c o n s t \sum_{\nu , \nu^{\prime}, \nu_{i}}( - 1 )^{\nu}( - 1 )^{\frac{s_{0}^{2}}{2}}| \lambda , \Xi + \nu \beta_{0}+ \nu^{\prime}\beta_{\gamma}+ \nu_{1}\beta^{( 1 )}+ \dots + \nu_{r}\beta^{( r )}\rangle .
\begin{array}{lrcl}\textrm{F-XII}u^{(iv}= - 4uu''' - 6u^2u'' - 4u^3u' - 12 uu^{\prime 2}- 10u'u'' \textrm{F-XVI}u^{(iv}= -5uu''' - 10u'uv'' - 15 uu^{\prime2}- 10u^2u'' - 10u^3u' - u^5 + A(x)(u''' + 4uu'' + 3u^{\prime 2}+ 6u^2u'+ u^4)+ B(x)(u'' + 3uu' + u^3(x)) + C(x)(u^2 + u')+ D(x)u + E(x) = 0. \,\end{array}
\begin{aligned}\frac{d}{dt}\int_{\Omega}\left ( \Psi(\varphi) + \frac{1}{2}|\nabla \varphi|^{2}+ \frac{1}{2}|\sigma|^{2}+ \chi \sigma (1-\varphi) \right ) \, dx & + \int_{\Omega}|\nabla \mu|^{2}+ |\nabla (\sigma - \chi \varphi)|^{2}\, dx \end{aligned}
\begin{array}{ll}[\pi_1\cdots\pi_i|\pi_{i+1}\cdots\pi_j|\pi_{j+1}\cdots\pi_k|\pi_{k+1}\cdots\pi_n], \chi\neq\rho,{[\pi_1\cdots\pi_i|\pi_{i+1}\cdots\pi_k|\pi_{k+1}\cdots\pi_n]}, \chi=\rho. \end{array}
{\cal L}_{\rm Y-M}^\prime ~=~ G^I_{\alpha\beta}G^I_{\gamma\delta}\epsilon^{\alpha\beta\mu\nu}\epsilon^{\gamma\delta\rho\tau}\nabla_\mu \phi^a \nabla_\nu \phi^b \nabla_\rho \phi^c \nabla_\tau \phi^d \eta_{ac}\eta_{bd}.
v_{d-3-k}^{N_k}\equiv \left(-v_{d-3-k}z^{t-n}+ \left(\left(\sum_{j = d-k-2}^{d-3}(-1)^{j+d+k}v_j \right)+(-1)^kx \right)^n-v_{d-k-3}z^n+v_{d-k-3}x^n \right)^{N_k}\equiv 0,
\mathcal{J}_{\lambda}:= (I + \lambda T)^{-1}, T_{\lambda}:= \frac{1}{\lambda}(I - \mathcal{J}_{\lambda}), x = \mathcal{J}_{\lambda}u, \; f = T_{\lambda}u & u = \mathcal{J}_{\lambda}u + \lambda T_{\lambda}u \forall \lambda > 0 T_{\lambda}u = f \in Tx = T \left ( \mathcal{J}_{\lambda}u \right ).
\prod_{j\le A}(\tau(j)n/j) \le\left(\frac1A\sum_{j\le A}\tau(j)\right)^A\frac{n^{A}}{A!}&\le(\log n)^{A}\exp\Big(A\log n-A\log A+A\Big) &\le\exp\Big(\frac32 A\log\log n+A\Big).
\int_{\mathcal{Y}^d}\widetilde{m}h_{y_j}h_{y_j}dy+\int_{\mathcal{Y}^d}\widetilde{m}\widetilde{w}_{y_iy_j}\widetilde{w}_{y_iy_j}dy=\int_{\mathcal{Y}^d}\widetilde{m}h_{y_j}V_{y_j}dy.
\begin{array}{r c l}{{( i_{\hat{k}}{\hat{\Omega}})_{\mu_{1}\dots \mu_{5}y}}}&{{=}}&{{- ( i_{k}\Omega^{( 6 )})_{\mu_{1}\dots \mu_{5}}}}\\{{}}&{{}}&{{}}\\{{( i_{\hat{k}}{\hat{\Sigma}})_{\mu_{1}\dots \mu_{5}y}}}&{{=}}&{{- ( i_{k}\Sigma^{( 6 )})_{\mu_{1}\dots \mu_{5}}\, .}}\\ \end{array}
\tilde{V}_{n , l}= \frac{Z_{n + 3 , l}Z_{n - 2 , l + 1}}{Z_{n , l + 1}Z_{n + 1 , l}}, 1 + \tilde{V}_{n , l}= \frac{Z_{n , l}Z_{n + 1 , l + 1}}{Z_{n + 1 , l}Z_{n , l + 1}}.
F(u,\mu)\sum_{s=0}^\infty \frac{a_s(z)}{u^{2s}}=\sum_{s=0}^\infty \frac{A_s(z)}{u^{2s}}, F(u,\mu)\sum_{s=0}^\infty \frac{b_s(z)}{u^{2s}}=\sum_{s=0}^\infty \frac{B_s(z)}{u^{2s}}.
{}\Delta_n \underset{\sim}{<}(\frac{\log T_n}{T_n})^{\frac{\bar{\beta}_3}{2 \bar{\beta}_3 + d - 2}(\frac{1}{\beta_1}+ \frac{1}{\beta_2})}\log T_n \mbox{for}\beta_2 < \beta_3, \Delta_n \underset{\sim}{<}(\frac{1}{T_n})^{\frac{\bar{\beta}_3}{2 \bar{\beta}_3 + d - 2}(\frac{1}{\beta_1}+ \frac{1}{\beta_2})}\mbox{for}\beta_2 = \beta_3.{}
\widetilde{g_n^{(t,x)}}(t_1,x_1,\ldots,t_n,x_n,s,y) & =\frac{1}{n+1}\Big[g_n^{(t,x)}(t_1,x_1,\ldots,t_n,x_n,s,y) & +\sum_{i=1}^{n}g_n^{(t,x)}(t_1,x_1,\ldots,t_{i-1},x_{i-1},s,y,t_{i+1},x_{i+1},\ldots,t_n,x_n,t_i,x_i)\Big]
\left(\beta_{F_{g, \nu}}(0)\Delta\right)(x) =L_1'(x, g(x), g'(x))h_{1, \Delta}(x) +L_2'(x, g(x), g'(x))h_{2, \Delta}(x) & + L_3'(x, g(x), g'(x))h_{2, \Delta}'(x) +L\left(x, g(x),g'(x)\right) \operatorname{tr}h_{3, \Delta}(x) & + L\left(x, g(x),g'(x)\right)\beta^\nu\left(h_{2, \Delta}(x), x \right).
T = \frac{i \gamma}{2}\sum_{j = 1}^{2}\int_{U_{j}}\left( 2 i ( \partial_{\bar{z}}\zeta_{z}- \partial_{z}\zeta_{\bar{z}}) + \partial_{t}( \partial_{z}h \partial_{\bar{z}}\chi + \partial_{\bar{z}}h \partial_{z}\chi ) - \dot{\chi}\Omega^{2}\right) d z \wedge d \bar{z}
\lim_{{\vec{k}}\rightarrow 0}k^{\mu}{\cal F}_{\mu}( k ) \equiv \lim_{{\vec{k}}\rightarrow 0}{\bar{\epsilon}}\int d^{4}x \sqrt{- g}k^{\mu}p_{k}( x ) < 0_{-}|{\cal T}{\cal S}_{\mu}( x ){\cal O}_{(}y ) | 0_{-}> ,
u^2(y_0) \le\frac{1}{|B_{c2^{-m-1}R_0}(y_0)|}\int_{B_{c2^{-m-1}R_0}(y_0)}u^2 &\le\frac{1}{|B_{c2^{-m-1}R_0}(y_0)|}\int_{E\cap (B_{S_m}(x_0)\backslash B_{S_{m+1}}(x_0))}u^2 &\le\frac{1}{|B_{c2^{-m-1}R_0}(y_0)|}M_mV_m &=\frac{1}{\omega_n}c^{-n}2^{n(m+1)}R_0^{-n}M_mV_m.
{\cal Z}_{\rm mon.}=\sum\limits_{N=1}^{+\infty}\sum\limits_{q_a=\pm 1}^{}\frac{\zeta^N}{N!}\prod\limits_{i=1}^{N}\int d^3z_i\exp\left[-\frac{\pi}{2e^2}\int d^3xd^3y \rho_{\rm gas}\left(\vec x{\,}\right)\frac{1}{\left|\vec x-\vec y{\,}\right|}\rho_{\rm gas}\left(\vec y{\,}\right)\right],
p(x_1, \ldots,x_{i-1}, \frac{x_i+x_j}{2}, x_{i+1}, \ldots, x_{j-1}, \frac{x_i+x_j}{2}, x_{j+1}, \ldots, x_m) & = p_1 + p_2\left(\frac{x_i+x_j}{2}+ \frac{x_i+x_j}{2}\right) + p_3\left(\frac{x_i+x_j}{2}\right)^2 & \ge p_1 + p_2(x_i+x_j) + p_3 x_ix_j= \lambda(p),
S_{a , b}( z ) = \frac{( - 1 )^{a + b - 1}}{( a - 1 ) ! \; b !}\! \int_{0}^{1}\mathrm{d}\xi \; \frac{\ln^{a - 1}\! \xi \ln^{b}( 1 \! - \! z \xi )}{\xi}\; .
S_T(b) = u\sqrt{T}\langle b_j-b_{0,j},a_\lambda \Phi_{\lambda,k}\rangle_{L^2}-u\sqrt{T}\langle b_j-b_{0,j},a_\lambda P_{V_J}[\Phi_{\lambda,k}/\mu_0] \rangle_{\mu_0}& = u\sqrt{T}a_\lambda \langle \mu_0(b_j- b_{0,j}), \Phi_{\lambda,k}/\mu_0 - P_{V_J}[\Phi_{\lambda,k}/\mu_0] \rangle_{L^2}.
\begin{aligned}\frac{1}{\sqrt{-c}}\bar{\nabla}_{aB+U+xZ}\bigl(bB+V+yZ\bigr)={}\Bigl(xy+\frac{1}{2}\langle U,V\rangle\Bigr)B-\frac{1}{2}\Bigl(bU+yJU+xJV\Bigr)\!\! &{}+\Bigl(-bx+\frac{1}{2}\langle JU,V\rangle\Bigr)Z,\end{aligned}
\mathcal{L}(p_{k,i},\lambda) = - \sum\limits_{k = 1}^K \sum\limits_{i=1}^{d}\omega_k \log_2 (1 + \frac{\pmb{\Lambda}_k^2(i,i)}{\sigma_k^2}p_{k,i}) + \lambda (\sum\limits_{k = 1}^K\sum\limits_{i = 1}^d p_{k,i}- P_{max})
I_{4}\equiv\det \left[\left( \begin{array}{c}\vec{\tilde{q}}\vec{\tilde{p}}\end{array}\right)\left( \begin{array}{cc}\vec{\tilde{q}}\ \ \ vec{\tilde{p}}\end{array}\right)\right]=\left(\vec{\tilde{q}}\cdot\vec{\tilde{q}}\right)^{2}\left(\vec{\tilde{p}}\cdot\vec{\tilde{p}}\right)^{2}-\left(\vec{\tilde{q}}\cdot\vec{\tilde{p}}\right)^{2}\, ,
\sigma_{t}^{2}=\omega+\alpha_{1}\epsilon_{t-1}^{2}+\cdot\cdot\cdot+\alpha_{q}\epsilon_{t-q}^{2}+\beta_{1}\sigma_{t-1}^{2}+\cdot\cdot\cdot+\beta_{p}\sigma_{t-p}^{2}=\omega+\sum_{i=1}^{q}\alpha_{i}\epsilon_{t-i}^{2}+\sum_{i=1}^{p}\beta_{i}\sigma_{t-i}^{2}
\widetilde{J}(p):=\sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p \left[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t\right. &\quad\left.-\exp(-r\tau_I)(k_1P(\tau_I)+k_0)-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\right].
\sqrt{\frac{1}{\left\vert B\right\vert}\int_{B}|u_{0}|^{2}}=e^{-b_{0}}\leq\tau \left\Vert \phi \right\Vert_{X}C\varepsilon\exp \left(-\frac{A_{N,\varepsilon}}{2}\left( \frac{\varphi (r)}{r}\right)^{\frac{1}{N-1-\varepsilon}}\right) =\eta_{N,\varepsilon}(r)\tau \left\Vert \phi \right\Vert_{X}\ ,
\int\limits_{\{|v_{j,n}|\geq |v_{j}|\}}\left(1+|\log|v_{j,n}|^2|\right)\left||v_{j,n}|^2-|v_{j}|^2\right|dx &\leq\left(\int\limits_{\{|v_{j,n}|\geq |v_{j}|\}}\left(1+|\log|v_{j,n}|^2|\right)^2(|v_{j,n}|+|v_{j}|)^2dx \right)^{\tfrac1{2}}\left(\int\limits_{\mathbb{R}_+}(|v_{j,n}|-|v_{j}|)^2dx\right)^{\tfrac1{2}}.
+|A_{kp}|\left\vert \frac{f^{(p)}(z+c_{k})}{f(z+c_{k})}\right\vert\left\vert \frac{f(z+c_{k})}{f(z+c_{l})}\right\vert+\sum_{j=1}^{m}|A_{lj}|\left\vert \frac{f^{(j)}(z+c_{l})}{f(z+c_{l})}\right\vert +\left\vert \frac{F(z)}{f(z+c_{l})}\right\vert .
\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}\left[\|\mathbf{B_m^*\Phi_m^*\Phi_m a_m^l}\|^2\right] \rightarrow inf, G_m(\lambda)=d_m(\lambda) (d_m(\lambda))^* -F_m(\lambda) \in D_G.
\frac{d}{dt}\varrho_t (x) = - m \varrho_t (x) + \int_{\mathbb{R}^d}a_t(x,y) \varrho_t (y) d y begin{align*}.2cm] a_t (x, y) := a^{+}(x-y) - \varrho_t(x) a^{-}(x-y).
\int_{\overline\Omega}\phi(x)\,(m^{\eta}(t_2,dx)-m^{\eta}(t_1,dx))=\int_{\Gamma}\Big[ \phi(e_{t_2}(\gamma))-\phi(e_{t_1}(\gamma))\Big] \,d\eta(\gamma) &=\int_{\Gamma}\Big[\phi(\gamma(t_2))-\phi(\gamma(t_1))\Big] \,d\eta(\gamma)\leq \int_{\Gamma}|\gamma(t_2)-\gamma(t_1)|\,d\eta(\gamma).
\frac{1}{2}\frac{l^{''}}{l}+\frac{3}{2}\frac{l^{'}}{l x}-\frac{1}{4}\frac{(l^{'})^2}{l^2}=2\alpha^2 \left(\frac{l}{f}U(\Phi_1,\Phi_3,\Phi_4)+\frac{f}{lx^2}(K^{'})^2-\frac{K^2 \Phi_1^2}{x^2}\right)
\int_{\mathcal{H}}(\mathfrak{Im}\langle e_k, T^nx \rangle)^2\,(\mathfrak{Re}\langle e_\ell, x \rangle)^2\, dm(x)=\frac{1}{2}(\mathfrak{Im}\langle RT^{*n}e_k, e_\ell \rangle)^2+\sigma_k^2\sigma_\ell^2
\int \frac{e^{-a|y|}}{|y|^\sigma |x-y|}dy = \frac{4\pi}{|x|}\int_0^{|x|}e^{-ar}r^{2-\sigma}dr + 4\pi \int_{|x|}^\infty e^{-ar}r^{1-\sigma}dr &= 4\pi a^{2-\sigma}\Gamma(2-\sigma) + 4\pi \left( \frac{1}{|x|}\int_0^{|x|}e^{-ar}r^{2-\sigma}dr - \int_0^{|x|}e^{-ar}r^{1-\sigma}dr\right).
\begin{aligned}\left\|\varphi_{i,\delta}^j+\hat{f}_i^{j}\right\|_{C^{2,\alpha}\left(B_{R}(\hat{x},\hat{g}_{i,\delta}^*)\right)}\leq C_{5}(n,r,v,\rho,\alpha,Q)\left\|\varphi_{i,\delta}^j+\hat{f}_i^{j}\right\|_{C^{0}\left(B_{R}(\hat{x},\hat{g}_{i,\delta}^*)\right)}&\leq \Psi(i^{-1},\delta\,|\,n,r,v,\rho,\alpha,Q),\end{aligned}
A = \begin{pmatrix}I L_{21}I \vdots \vdots \ddots L_{N1}L_{N2}\cdots I \end{pmatrix}\begin{pmatrix}D_1 & D_2 & \ddots & D_N\end{pmatrix}\begin{pmatrix}I L_{21}^T \cdots L_{N1}^T & I \cdots L_{N2}^T & \ddots \vdots & I\end{pmatrix}.
\operatorname{ind}(A_{1},A_{2}):= \operatorname{dim}\operatorname{ker}(d_1)-\operatorname{dim}(\operatorname{ker}(d_2) \ominus d_{1}({\mathcal H}))+\operatorname{dim}({\mathcal H}\ominus d_{2}({\mathcal H\oplus \mathcal H})).
\mathcal{L}(\lambda)\begin{bmatrix}\Lambda_\epsilon(\lambda^\ell)\otimes I_n -(\widehat{\Lambda}_\eta(\lambda^\ell)^T\otimes I_m)(\lambda)M(\lambda)(\Lambda_\epsilon(\lambda^\ell)\otimes I_n)\end{bmatrix}=\begin{bmatrix}e_{\eta+1}0\end{bmatrix}\otimes P(\lambda),
\|f\|_{L^1(\Omega)}\le \alpha_N^{1/p'}\sum_{j=0}^J d_j^{N/p'}\|f\|_{L^p(\Omega\cap A_j)}, \sum_{j=0}^J d_j^{N/p'}\|f\|_{L^p(\Omega\cap A_j^{(1)})}\le (2^{N/p'}+ 1 + 2^{-N/p'}) \sum_{j=0}^J d_j^{N/p'}\|f\|_{L^p(\Omega\cap A_j)}.
F \left( \beta \right) = E_{0}+ \widetilde{F}\left( \beta \right) = - \frac{\pi^{2}a}{6 \beta^{2}}- \frac{1}{2 \beta}\log \left( \frac{\beta}{2 \pi a}\right) - \frac{1}{\beta}\sum_{p , n = 1}^{\infty}\frac{1}{n}e^{- 4 \pi^{2}n p a / \beta}.
S_{f}= - \frac{1}{2 g^{2}}T r \left( \sum_{i = 0}^{3}\overline{{{\psi_{i}}}}\Gamma^{\mu}[ A_{\mu}, \psi_{i}] + 2 \sum_{i = 1}^{3}\overline{{{( \psi_{i})^{c}}}}\bar{\Gamma}^{( i )}[ B_{i}^{\dagger}, \psi_{0}] + \sum_{i , j , k = 1}^{3}| \epsilon_{i j k}| \overline{{{( \psi_{i})^{c}}}}\Gamma^{( j )}[ B_{j}, \psi_{k}] + h . c . \right)
\begin{array}{rcl}h^{k}(\tilde{y}) = \frac{1}{2}\|d-(y_{1}+\cdots+y_{m+1})\|^{2}+\bigg[\underset{i=1}{\overset{m}{\sum}}\delta^{*}(y_{i},C_{i})\bigg]+\delta^{*}(y_{m+1},H_{m+1}^{k}) \bar{h}(\tilde{y}) = \frac{1}{2}\|d-(y_{1}+\cdots+y_{m+1})\|^{2}+\bigg[\underset{i=1}{\overset{m}{\sum}}\delta^{*}(y_{i},C_{i})\bigg]+\delta^{*}(y_{m+1},C).\end{array}
S^{n,M,2}_\tau = n^{\alpha+\frac 12}\sum_{i=0}^{M-1}\int_{\tau_i}^{\tau_{i+1}}(t-s)^\alpha [\sigma'(X_s)- \sigma'(X_{\tau_i})] \left( \int_0^s \psi_{n,1}(u,s) dW_u \right) ds