harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\overline{F}_1 = \begin{pmatrix}B_1-\big(1-q^{\lambda}\big) B_2 0 0 \end{pmatrix}\!, \overline{F}_2 = \begin{pmatrix}0 0 B_1 B_2-\big(1-q^{\lambda}\big) \end{pmatrix}\!, \overline{F}_{\infty}= \begin{pmatrix}1 0 0 1 \end{pmatrix}\!.
\mathbf{e}(1,1) =(-1/\sqrt{2},-1\sqrt{2})_{1}=(-C_{1,0,1}^{1,1,1},C_{0,1,1}^{1,1,1})_{1}\mathbf{e}(1,0) =(1/\sqrt{2},0,-1/\sqrt{2})_{0}=(C_{1,-1,0}^{1,1,1},-C_{0,0,0}^{1,1,1},C_{-1,1,0}^{1,1,1})_{0}\mathbf{e}(0,0) =(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})_{0}=(C_{1,-1,0}^{1,1,0},-C_{0,0,0}^{1,1,0},C_{-1,1,0}^{1,1,0})_{0}
\begin{cases}\dot{u}=\log\det (u_{\alpha\bar{\beta}})-Au+g(z,t)\;\;\mbox{on}\;\; \Omega\times (0,T), u=\varphi\;\;\;\mbox{on}\;\; \partial\Omega\times [0,T), u(.,0)=u_0\;\;\;\mbox{on}\;\;\bar{\Omega}. \end{cases}
d Q_{n}( \zeta ) \left[ 1 - \sqrt{1 -{\frac{c_{1}^{( n )}}{d \zeta}}-{\frac{c_{2}^{( n )}}{d \zeta^{2}}}}\right] \; \equiv \; \sum_{i = 1}^{n}p_{i}\zeta^{i}+ p_{0}+ \sum_{j = 1}^{\infty}p_{- j}\zeta^{- j}\; ,
\begin{cases}(\bar{\mathbf{z}},t_1)\in\mathcal{E}, (\bar{\mathbf{z}}+\mathbf{p}(t_1)-\mathbf{p}(t_2),t_2)\in\mathcal{E}, (\bar{\mathbf{z}}+\mathbf{p}(t_1)-\mathbf{p}(t_2),t_3)\in\mathcal{E},\end{cases}
\sum_{(j_1, \ldots, j_n) \in i(D'_\pi)}\varphi_q\left(z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right) \right) = 0.
V(r)=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(r^{\prime})}{|r-r^{\prime}|}dr^{\prime}=-\frac{e}{4\pi\epsilon_{0}}\sum_{i\ne j}\int\frac{|\phi_{n_{j}}(r^{\prime})|^{2}}{|r-r^{\prime}|}dr^{\prime}
\big(F_1\Box_2F_2\big)^*(x^*,x)=\min_{y^*\in X^*}\left[F_1^*(x^*-y^*,x)+F_2^*(y^*,x)\right] &=\min_{y^*\in X^*}\left[F_A(x,x^*-y^*)+F_B(x,y^*)\right]=\big(F_A\Box_2F_B\big) (x,x^*).
P_k(x) ={}\frac{1}{\sqrt{2\pi}i}\int_{-i\infty}^{+i\infty}p_k(s) e^{\frac{1}{2}(x-s)^2}ds, Q_k(y) ={}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}q_k(t) e^{-\frac{1}{2}(y-t)^2}dt,
V_b=\tilde{\Phi}(\gamma_5,\gamma_3,\gamma_2,\gamma_1;V_a),Z_b=Z_a+\gamma_4, V_b=\tilde{\Phi}(\beta_5,\beta_3,\beta_2,\beta_1;V_m),Z_b=Z_m+\beta_4, V_m=\tilde{\Phi}(\alpha_5,\alpha_3,\alpha_2,\alpha_1;V_a),Z_m=Z_a+\alpha_4.
\begin{array}{rl}s_\beta s_\alpha \lambda(h_0) = \lambda(h_0) - 2(\lambda_\alpha+\lambda_\beta) + 2n_{\beta,\alpha}\lambda_\alpha & = \lambda_\alpha(\omega_\alpha(h_0) - 2(1-n_{\beta,\alpha})) + \lambda_\beta(\omega_\beta(h_0)-2) + \sum\limits_{\gamma\in\Pi\setminus\{\alpha,\beta\}}\lambda_\gamma\omega_\gamma(h_0) \;.\end{array}
D f ( x ) = \sum_{i = 0}^{k - 1}( D f )_{i}| i \rangle \, \, , \quad D f_{i}= d f_{i}+ \sum_{j = 0}^{k - 1}A_{i j}f_{j}\, \quad A_{i j}= \langle i | d | j \rangle \, \, .
t^{\star}(\{6\}^n)=\frac{(-1)^n\pi^{6n}}{2^{6n}}\sum_{|\boldsymbol{\ell}|=3n}\frac{E_{2\ell_1}E_{2\ell_2}E_{2\ell_3}}{(2\ell_1)!(2\ell_2)!(2\ell_3)!}\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)^{2\ell_2+4\ell_3}.
\left\langle f_v,\, \hat{\Phi}^*_{v,w}(e^{v,w}_x)\otimes \bigotimes_{w'\in\partial v\setminus \{w\}}\hat{\Phi}^*_{v,w'}(e^{v,w'}_0)\right\rangle_{\mathcal{V}_{\partial v}}=0 \left\langle \Phi_{v,w}(e^{v,w}_x)\otimes \bigotimes_{v'\in\partial w\setminus \{v\}}\Phi_{v,w}(e^{v',w}_0),\,g_w\right\rangle_{\mathcal{V}_{\partial w}}=0
\nabla_i\frac{A_{jk}}{\mathcal{H}^2}=\frac{1}{\mathcal{H}^2}\nabla_iA_{jk}-\frac{1}{\left(\mathcal{H}^2\right)^2}A_{jk}\nabla_i\mathcal{H}^2 &=\frac{1}{\left(\mathcal{H}^2\right)^2}(\mathcal{H}^2\nabla_iA_{jk}-A_{jk}\nabla_i\mathcal{H}^2),
\int_{\mathbb{T}}\left[\theta^{2}\Lambda \theta\right]dx=\int_{\mathbb{T}}\int_{\mathbb{T}}\frac{\left(\theta(x)-\theta(y)\right)^{2}}{\sin^{2}\left((x-y)/2\right)}\cdot \frac{\theta(x)+\theta(y)}{2}dxdy\ge 0.
A\alpha =\frac{b}{4a}(5c+8\alpha^2+9b^2c), Ab =\frac{a}{4\alpha}(5c-4\alpha^2-3ca^2), Va =Vb=0, V\alpha =-\frac{a}{2b}U\alpha, Ua = -\frac{a}{2\alpha}U\alpha, Ub =\frac{a^2}{2b\alpha}U\alpha.
a = d_2 d_3\xi_3 = \xi_1 d_{h,2}d_{h,3}b = d_0 d_3\xi_3 = \xi_1 d_{h,0}d_{h,3}c = d_0 d_1 \xi_3 = \xi_1 d_{h,0}d_{h,1}a + b = d_1 d_3\xi_3 = \xi_1 d_{h,1}d_{h,3}b + c = d_0 d_2\xi_3 = \xi_1 d_{h,0}d_{h,2}a + b +c = d_1 d_2\xi_3 = \xi_1 d_{h,1}d_{h,2}.
\alpha_{n,j}^{(N-1)}(f)=\frac{1}{\mathbb{P}_{n}^{(N-1)}(A_{j}^{(N)})}\alpha_{n}^{(N-1)}\left( (f-\mathbb{E}(f|A_{j}^{(N)}))1_{A_{j}^{(N)}}\right),
\mathfrak{P}:=\lbrace p:[0,1]\mapsto\mathbb{C}, \rbrace, \mathfrak{P}_0 :=\lbrace p\in\mathfrak{P}: p(0)=p(1)=0\rbrace, \mathfrak{P}_s :=\lbrace p\in\mathfrak{P}: p(t)=p(1-t)t\rbrace, \mathfrak{P}_{s,0}:=\mathfrak{P}_s\cap \mathfrak{P}_0.
\left[ 1 + \frac{g^{2}}{2 m^{2}}V^{( 2 )}( \phi ) \right] \frac{d p^{\mu}}{d \tau}= - \frac{g^{2}}{2 m}\left[ \partial^{\mu}V^{( 2 )}( \phi ) - \frac{p^{\mu}}{p^{2}}p \cdot \partial V^{( 2 )}( \phi ) \right] ,
\det Y =\det{X}\ (\det X^{r+1,\ldots,n}_{r+1,\ldots,n})^{r-1};Y=\big(y_{ij}\big)_{i,j=1}^{r},\ \y_{ij}=\det X^{i,r+1,\ldots,n}_{j,r+1,\ldots,n}.
\begin{aligned}\frac1{1+\eta}|T_{N}^{'}(z_0)| \leq |T_{N}^{'}(z)| \le (1+\eta) |T_{N}^{'}(z_0)| \frac{1}{1+\eta}|T_{N}^{'}(z_0)||z-z_0| \leq |T_N(z) - T_N(z_0)| \le (1+\eta) |T_{N}^{'}(z_0)||z-z_0|\end{aligned}
\mathcal I_1(n) = \int_{\mathcal A_1^+(n)}(cd,d)_p\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh + \int_{\mathcal A_1^-(n)}\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh.
\pi_{n-}^{\mu}(A) =\int_{\mathcal{X}}T(A|x)\pi_{n-1}^{\mu}(dx)=\int_{\mathcal{X}}\int_{A}\frac{dT(\cdot|x)}{d\bar{\mu}}(a)\bar{\mu}(da)\pi_{n-1}^{\mu}(dx) &=\int_{A}\left(\int_{\mathcal{X}}\frac{dT(\cdot|x)}{d\bar{\mu}}(a)\pi_{n-1}^{\mu}(dx)\right)\bar{\mu}(da)
lim_{x\rightarrow\infty}x^{n}\cdot e^{-x}=lim_{x\rightarrow\infty}\frac{x^{n}}{e^{x}}=lim_{x\rightarrow\infty}\frac{nx^{n-1}}{e^{x}}=n\cdot lim_{x\rightarrow\infty}\frac{x^{n-1}}{e^{x}}.
I_{\varphi}- I_{\alpha}= I - 4 \ln \ln{\frac{\Lambda_{0}}{\Lambda}}+ 4 \ln{\frac{\Lambda_{0}^{2}}{\Lambda^{2}}}- 4 \ln 2 +{\frac{4}{N - 1}}\sum_{i = 1}^{N - 1}\ln{\frac{\Lambda^{2}}{v_{i}}}\raise 2 p t \mathrm{,}
\tau_{H}=-m_{1}\cot \left( \frac{\vartheta}{2}\right) \alpha -m_{2}(\cot \vartheta ) 2\alpha &=-\left\{(m_{1}+m_{2})\cot \left( \frac{\vartheta}{2}\right) -m_{2}\cot \left( \frac{\vartheta}{2}\right) \right\}\alpha.
{A_{\theta}}^{a}= \frac{1 - H_{2}}{e}{v_{\varphi}}^{a}\ ,{A_{\varphi}}^{a}= - \frac{n}{e}\sin \theta \left( H_{3}{v_{r}}^{a}+ ( 1 - H_{4}){v_{\theta}}^{a}\right) \ .
w_T(z,\delta):=\sup_{0\leq t\leq T}\sup_{\substack{0\vee (t-\delta)\leq t_1\leq t_2\leq t_3\leq (t+\delta)\wedge T}}\inf_{0\leq \theta \leq 1}|z(t_2)-\left(\theta z(t_1)+(1-\theta) z(t_3)\right)|
\begin{aligned}\frac{|x_i^1u_{x_i}(P_1)-x_i^2u_{x_i}(P_2)|}{\rho^{\alpha}(P_1,P_2)}\leq C\eta^{1-\alpha}\|u\|_{C^{2+\alpha}_{WF}([0,T]\times\bar S_{n,m})}+ C\eta^{-m_0(1+\alpha)-\alpha}\|u\|_{C([0,T]\times\bar S_{n,m})}.\end{aligned}
-\frac{1+w}{u}=N(x,t_{m})\mathrm{exp}(w_{0}^{\infty+})\frac{\theta(\underline{z}(P_{\infty+},D_{\underline{\hat{\nu}}(x,t_{m})}))}{\theta(\underline{z}(P_{\infty+},D_{\underline{\hat{\mu}}(x,t_{m})}))},
V_{e}= \frac{e^{2 y_{0}}\left[ ( 1 6 n^{2}+ 1 6 n + 3 ) e^{4 n y}+ 3 2 n^{2}+ 8 0 n - 6 + ( 1 6 n^{2}+ 1 6 n + 3 ) e^{- 4 n y}\right]}{4 ( e^{2 n y}+ e^{- 2 n y})^{2 + \frac{1}{n}}}.
F_1 =\langle \nabla \cdot (a-1)\cdot w, |w|^{q-2}(-b\cdot w)\rangle + \langle (-\mu u - b\cdot w +f),|w|^{q-2}(-b\cdot w) \rangle & = \langle \nabla a \cdot w, |w|^{q-2}(-b\cdot w) \rangle & + c \langle \mathsf{f}\cdot (\mathsf{f}\cdot \nabla w), |w|^{q-2}(-b\cdot w) \rangle & + \langle (-\mu u - b\cdot w +f),|w|^{q-2}(-b\cdot w) \rangle.
\gamma^0S^+=\left(\begin{array}{cccc}2 c 0 \tilde l_1 c -c^* 0 -\tilde l_1c^* 2\tilde l_1 0 -l_1d 2 d l_1d^* -2l_1 -d^* 0\end{array}\right),\quad\gamma^0S^-=\left(\begin{array}{cccc}0 -c 2\tilde l_1 -\tilde l_1 c c^* 2 \tilde l_1c^* 0 -2l_1 l_1d 0 -d -l_1d^* 0 d^* 2\end{array}\right),
\int ( \nabla \theta \cdot D^{\gamma-1}\nabla^{\perp}D^s \theta) D^s \theta dx=- \frac 12 \int D^s \theta ( D^{\gamma-1}\nabla^{\perp}\cdot(D^s \theta \nabla \theta) &\quad- \nabla \theta \cdot D^{\gamma-1}\nabla^{\perp}D^s \theta) dx.
\Gamma_{11}^3 =\lambda_1 e^{-2\lambda_1 z}\Gamma_{22}^3 =\lambda_2 e^{-2\lambda_2 z}\Gamma_{13}^1 =\Gamma_{31}^1=-\lambda_1 \Gamma_{23}^2 =\Gamma_{32}^2=-\lambda_2 \Gamma_{ij}^k =0 \hbox{en todos los otros casos}
{\cal W}( x , x_{0}) = \exp \left\{- i \biggl [ \int_{x_{0}}^{x}d x_{j}A_{j}+ \frac{\theta}{2}\, \int_{x_{0}}^{x}d x_{j}\, \left[ m \, ({\bf v}\times \nabla A_{j}^{( 0 )})_{3}- ({\bf A^{( 0 )}}\times \nabla A_{j}^{( 0 )})_{3}\right] \biggr ] \right\}.
\frac{\mu^{2}}{2^{7}( \pi )^{\frac 5 2}( s - 1 )}\frac{\Gamma ( \frac 3 2 - s )}{\Gamma ( 2 - s )}\int d^{4}k \ \tilde{S}( k ) \tilde{S}( - k ) \left( \frac{k^{2}}{\mu^{2}}\right)^{- s + 1}.
\chi_{(\lambda_1,\lambda_2)}(s_1s_2)=\begin{cases}\chi^+_{\lambda_1}(s_1)\overline{\chi}_{\lambda_2}(s_2)+\chi^-_{\lambda_1}(s_1)\overline{\chi}_{\lambda_2}(s_2) \text{if}s_2\in\tilde{A}_{n_2}, 0 \end{cases}
\partial\cdot J=\partial^{\mu}\eta_{\mu\nu}J^{\nu}=\partial_{\nu}J^{\nu}=(\frac{\partial_{t}}{c},-\vec{\nabla})\cdot(\rho c,\vec{j})=\frac{\partial_{t}}{c}(\rho c)+\vec{\nabla}\cdot\vec{j}=\partial_{t}\rho+\vec{\nabla}\cdot\vec{j}=0
\mathcal{R}_X=\Bigl\{\mathbf{e}\in\mathbb{R}^\ell: e_i\in (a_i,a_i+\delta]1\le i\le\ell-1,\, \sum_{i=1}^\ell e_i\le 1,\,e_\ell\ge \max\Bigl(\frac{\eta}{4},1-\sum_{i=1}^{\ell-1}a_i-\ell\delta\Bigr)\Bigr\},
\int_{K_s}m_t \varphi^p m\, dz =\int_{K_s}\Big[\big(\varphi^p \frac{m^2}{2}\big)_t - \frac{p}{2}\varphi^{p-1}\varphi_t m^2\Big]\, dz=\frac12 \int_{B_\frac52}(\varphi^p m^2)(x,s)\, dx - \frac{p}{2}\int_{K_s}\varphi^{p-1}\varphi_t m^2\, dz,
A^{( 1 )}={\frac{\pi^{2}}{2}}\mathrm{t r}( T^{a_{1}}T^{a_{2}}T^{a_{3}}T^{a_{4}}){\frac{u ( \sin \pi{\frac{s}{2}}+ \sin \pi{\frac{t}{2}}+ \sin \pi{\frac{u}{2}})}{\Gamma ({\frac{s}{2}}+ 1 ) \Gamma ({\frac{t}{2}}+ 1 ) \Gamma ({\frac{u}{2}}+ 1 ) \sin ( \pi{\frac{s}{2}}) \sin ( \pi{\frac{t}{2}}) \sin ( \pi{\frac{u}{2}})}}
\sum_{m=1}^\infty\prod_{i=1}^m\mu([0,A(m)^{\rho i}]) \leq z_0-1+\sum_{m=z_0}^\infty\prod_{i=1}^m\mu([0,A(m)^{\rho i}]) &=z_0-1+\sum_{m=1}^\infty\prod_{i=1}^{m+z_0-1}\mu([0,A(m+z_0-1)^{\rho i}]) &=z_0-1+\sum_{m=1}^\infty\prod_{i=1}^{m+z_0-1}\mu([0,\tilde{A}(m)^{\rho i}]) &\leq z_0-1+\sum_{m=1}^\infty\prod_{i=1}^{m}\mu([0,\tilde{A}(m)^{\rho i}]).
C_\nu(t) = \begin{cases}C e^{-td^\nu}(t\ge 1) C t^{-\lambda_\nu}(0<t\le 1), \end{cases}\lambda_\nu \coloneqq \max\Big\{\frac{d}{\nu}\Big(\frac{1}{p_2}-\frac{1}{p_1}\Big),0 \Big\},
C = \left( \begin{array}{ccccc}0 0 \cdots 0 C^{1N_1}0 0 \cdots C^{2 (N_1-1)}C^{2N_1}\vdots \vdots{}\vdots \vdots 0 C^{(N_1-1) 2}\cdots C^{(N_1-1) (N_1-1)}C^{(N_1-1) N_1}C^{N_11}C^{N_12}\cdots C^{N_1 (N_1-1)}P^{N_1N_1}\end{array}\right)
3k_3(aK_{r+1}\cup C(b)) - (r-2)m &= 3\left(a\binom{r+1}{3}+ \binom{c}{3}+ \binom{d}{2}\right) - (r-2)\left(a\binom{r+1}{2}+ \binom{c}{2}+ d\right) &= a\binom{r+1}{2}+ (c-r)\binom{c}{2}+ d\left(\frac{3}{2}(d-1)-(r-2)\right) &\ge \binom{r+1}{2}+ (c-r)\binom{c}{2}+ d\left(\frac{3}{2}(d-1)-(r-2)\right)
\partial_t f = \frac{\tilde{H}\Delta |\tilde{A}|^2 - (2-\sigma) |\tilde{A}|^2 \Delta \tilde{H}}{\tilde{H}^{3-\sigma}}-\frac{\sigma}{2}\frac{\Delta \tilde{H}}{\tilde{H}^{1-\sigma}}- \frac{2}{\tilde{H}^{2-\sigma}}|\nabla \tilde{A}|^2 +\sigma |\tilde{A}|^2 f + O(\tilde{H}^{1+\sigma}).
H ( \omega , 0 ) = \frac{\phi_{1}f_{2}\partial_{r}( \phi_{2}^{( - )}/ A_{-}) - ( \phi_{2}^{( - )}/ A_{-}) f_{1}\partial_{r}\phi_{1}}{\phi_{1}f_{2}\partial_{r}( \phi_{2}^{( + )}/ A_{+}) - ( \phi_{2}^{( + )}/ A_{+}) f_{1}\partial_{r}\phi_{1}}\ .
a_{i}= \frac{1}{[i^2+(\alpha-2)i+(1-\alpha)]}\bigg\{(1-\delta_{i,2})\,\sum_{r=2}^{i-1}\,a_{r}\,a_{i+1-r}\,[r(1-\alpha)(i-r) &-r(r-1)]+\sum_{r=1}^{i-1}\sum_{s=1}^{i-r}\,a_{r}\,a_{s}\,a_{i+1-r-s}\,[r(r-\alpha)+s(\alpha+\beta-2) &\times(i+1-r-s)]\bigg\},
kC_k^\mu-t\Big(C_k^\mu\Big)' =kC_k^\mu-2\mu tC_{k-1}^{\mu+1}&=kC_k^\mu-\mu\Big(\frac{k}{k+\mu}C_k^{\mu+1}+\frac{k+2\mu}{k+\mu}C_{k-2}^{\mu+1}\Big) &=kC_k^\mu-k\frac{\mu}{k+\mu}\Big(C_k^{\mu+1}-C_{k-2}^{\mu+1}\Big)-2\mu C_{k-2}^{\mu+1},
d s^{2}= - ( 1 - \frac{2 m}{r^{d - 3}}- r^{2}) \d t^{2}+ \frac{1}{1 - \frac{2 m}{r^{d - 3}}- r^{2}}\d r^{2}+ r^{2}d \Omega_{d - 2}^{2},
B_{n,\mathbf{m}}^{(d)}(c\mathbf{z}\mid c{\boldsymbol{\omega}})= c^{|\mathbf{m}|-n}B_{n,\mathbf{m}}^{(d)}(\mathbf{z}\mid{\boldsymbol{\omega}}) \,\,\, (c\in \mathbb{C}^{*}).
\mathsf{d}_{\mathrm{com}}(\{X_{ni}\})=\sum_{n=1}^{d_1^2}\|X_{n1}\|_1+\|X_{n2}\|_1 =2\sum_{n=1}^{d_1^2}\|X_{n1}\|_1, \nu=\sqrt{\sum_{n=1}^{d_1^2}\|X_{n1}\|_1^2\|X_{n2}\|_1^2}=\sqrt{\sum_{n=1}^{d_1^2}\|X_{n1}\|_1^4}.
\begin{aligned}\sup_{t \in [0,T]}\left\{E^{(h)}(\eta^{(h)}(t)) + \frac{\rho}{2h}\int_{t - h}^t \|\partial_t \eta^{(h)}(s)\|_{L^2}^2\,ds \right\}\le C, \int_0^T R^{(h)}(\eta^{(h)}(s), \partial_t \eta^{(h)}(s))\,ds \le C.\end{aligned}
D_tV=\begin{bmatrix}v_t v_{tx}-\sum_{i=1}^N [v_x](x_i) \dot x_i \delta_{x_i}\end{bmatrix}=\begin{bmatrix}-\frac12 D_x(b)+\beta b -\frac12 D^2_x(b)-z\rho b \frac12 D_x(b) +\beta \end{bmatrix}V,
(Y_1)^l= \left(V\nabla_q \beta +\beta\tilde F\right)_i (\tilde\gamma^{-1})^{il}+(1+n/2)\beta^{-1}\partial_{q^j}\beta((\tilde\gamma+2\gamma I)^{-1})^j_{k}\delta^{kl}& +(n+2)\gamma\beta^{-1}\partial_{q^j}\beta ((\tilde\gamma+2\gamma I)^{-1})^j_{k}(\tilde\gamma^{-1})^{kl}
\int_{B_2}dy \int_{}dw\, \frac{g(|y|)}{|\bar X - Y^{N+1}(t)|^{2}}\chi\left(|w|\leq N+1\right)\leq &\frac94\int_{|\bar{X}-y|\geq 3R^{N+1}(t)}dy \int_{|w|\leq N+1}dw\, \frac{g(|y|)}{|\bar X -y|^{2}}\leq & CN^3\int_{|\bar{X}-y|\geq 3R^{N+1}(t)}dy\, \frac{g(|y|)}{|\bar X -y|^{2}}.
a_s :=\sup \{\Delta s\in \mathbb{R}: r\preceq s~\mbox{for all}~ r\in (s-\Delta s,s)\cap S(g) \}, b_s := \sup \{\Delta s\in \mathbb{R}: r\preceq s~\mbox{for all}~ r\in(s,s+\Delta s)\cap S(g) \},
[(\gamma, 1_G)]^*[(\delta, 1_G)] = [(\gamma, 1_G)]^*[(\gamma, 1_G)][(\gamma_1, 1_G)] = B_{|\gamma|}[(\gamma_1, 1_G)] & = [(\gamma_1, 1_G)]B_{|\gamma| + |\gamma_1|}= [(\gamma_1, 1_G)]B_{|\delta|}
\lim_{m \to \infty}I_m = \mathbf{1}_{\mathrm{Re}(z) > 0 , \delta = 0}- \mathbf{1}_{\mathrm{Re}(z) < 0 , \delta=1}+ \frac{1}{2}\frac{1 +(-1)^{\theta_1}e^{-z}}{1 - (-1)^{\theta_1}e^{-z}}- \frac{1}{2}+ \mathbf{1}_{\mathrm{Re}(z) < 0},
Z(t){}=\Sigma_1+\Sigma_2+\Sigma_3+\Sigma_4+\Sigma_5+\Sigma_6 &=\Bigl(q^{n+1}t-\tfrac{1}{t}-rq^{n-i}+rq^i &\qquad+\bigl(\tfrac{1}{t}-qt\bigr)\tfrac{(1-rq^it)(1+rq^{n-i}t)}{(1-rt)(1+rt)}\Bigr)E_i(-rt) E_{n-i}(rt)=0
\left\langle SO(n)\right\rangle_{SO(n+1)}=\left\langle SO(n)\right\rangle_{SO(n)\rtimes \mathbb{R}^{n}}=\left\langle SO(n)\right\rangle_{SO(n,1)}=\left\langle \varepsilon SO(n)\right\rangle_{G_{2}^{\circ}(n)}
\dim\big(3T^{(0)}(K_{n,n})^{ab}\big)= \sum_{k=0}^{n-1}(-1)^{k}(n+1-k)^{n-1}(n+1-k) !{n \brace n-k}\hphantom{\dim\big(3T^{(0)}(K_{n,n})^{ab}\big)}{}=\sum_{k=1}^{n+1}((k-1) !)^2{n+1 \brace k}^2 ,
a_{\alpha}({\bf x}) = - \left({\frac{\hbar \theta}{e^{2}\pi}}\right) \int d^{2}x^{\prime}{\frac{\epsilon_{\alpha \beta}\bigl ( x^{\beta}- x^{\prime \beta}\bigr )}{\vert{\bf x}-{\bf x}^{\prime}\vert^{2}}}\rho ({\bf x}^{\prime}) .
\sum_{j=0}^{M-1}(-1)^kc_j\left(\frac{q_j^2}{N}\right)^{s-m_0-\frac12}P_f(s-m_0;\delta+m_0,0)\Delta_f\!\left(s-m_0,\frac1{q_j},\cos^{(\delta+m_0+k)}\right) &-\eta\frac{(-2\pi N)^{m_0}}{m_0!}P_f(s-m_0;\delta+m_0,m_0)\Delta_{\bar{f}}\!\left(s,-\frac{q}{N},\cos^{(\delta)}\right)
\int_{\{-t'_3\le\Psi<-t'_4\}}|F_{t_0,\tilde{c}}|^2_h \ge \int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\}}|\tilde F|^2_h+\int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\backslash N\}}|F_{t_0,\tilde{c}}-\tilde F|^2_h &+\int_{\{z\in M: -\Psi(z)\in(t'_4,t'_3]\cap N\}}|F_{t_0,\tilde{c}}|^2_h.
|\partial^\alpha \Psi| \le C\biggl(\|\partial^\alpha f\|_{L^2_{p,\mathcal{I}}}+ \sum_{|{\alpha_1}|\le |\alpha|}\|\partial^{\alpha_1}f\|_{L^2_{p,\mathcal{I}}}\|\partial^{\alpha-\alpha_1}f\|_{L^2_{p,\mathcal{I}}}\biggl).
\frac{\textup{MMD}_{n,m}^{2}}{\surd{\textup{var}(\Delta_{1})}}=\frac{\Delta_1+\sum^{l-1}_{s=2}\{\Delta_s-E(\Delta_s)\}+\{\widetilde{\Delta}_l-E(\widetilde{\Delta}_l)\}+^2(P_X,P_Y)}{\surd{\textup{var}(\Delta_{1})}}.
\left| \bigcup_{i=1}^r L_i \right| = r(k+1) - \sum_{l=2}^r (-1)^l \binom{r}{l}(r-l)s = rs, \left| \bigcap_{j=1}^{r-1}L_{i_j}\right| = s \mbox{and}\bigcap_{j=1}^{r-1}L_{i_j}\cap \bigcap_{j=1}^{r-1}L_{i'_{j}}= \emptyset.
T_{\alpha\beta}= \partial_\alpha \Phi\partial_\beta \Phi -\frac{1}{2}g_{\alpha\beta}(g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi +V(\Phi))- \int_{\{g(p,p)=-\textbf{m}^2\}}\frac{\rho(x^\nu, p^\mu)p_\alpha p_\beta\sqrt{|g|}}{p^0}\;dp^1...dp^n.
D = \int_{- Q_{1}}^{Q_{1}}{\rho_{1}( k ) d k}, D_{m}= \int_{- B_{1}}^{B_{1}}{\sigma_{1}( \Lambda ) d \Lambda}, E_{1}/ N = D^{- 1}\int_{- Q_{1}}^{Q_{1}}{k^{2}\rho_{1}( k ) d k}.
C_2 = F \left( d, T, c_2,c_3, \|h\|_{\varrho(h)}, \frac{1}{\varrho(\gamma_{\infty}\circ h)},\frac{1}{\|\gamma_{\infty}\circ h\|_{\varrho(\gamma_{\infty}\circ h)}}, \frac{1}{\gamma_0^L}, \frac{1}{\gamma_{\infty}^L}, \frac{1}{c_1}\right);
P_{Q}^{n}\left(I_{1},\dots,I_{n-1},T\right)=\left\{v\in\mathbb{R}^{nd}:Q\left(v_{1}\right)-Q\left(v_{2}\right)\in I_{1},\dots,Q\left(v_{n-1}\right)-Q\left(v_{n}\right)\in I_{n-1},\left\Vert v\right\Vert \leq T\right\}.
T:\psi_{abc\cdot\cdot\cdot}(t,\vec{x})\mapsto\psi_{abc\cdot\cdot\cdot}^{\prime}(-t,\vec{x})={T_{a}}^{d}{T_{b}}^{e}{T_{c}}^{f}\cdot\cdot\cdot\psi_{def\cdot\cdot\cdot}(t,\vec{x})
S = \frac{1}{\gamma^{2}}\int_{- \infty}^{+ \infty}d \tau \int_{0}^{2 \pi}d \sigma \Bigl ( \frac{1}{2}\partial_{\alpha}\varphi \cdot \partial^{\alpha}\varphi - \mu^{2}\sum_{a = 1 , 2}\mathrm{e}^{\alpha^{a}\cdot \varphi}\Bigr ) ~ ,
0\leq f_{s,\mu,\lambda}\ast\rho_\nu =(||s||^2_{h_E\otimes h_L^\mu}-||s||^2_{h_E\otimes h_L^\lambda})\ast\rho_\nu =||s||^2_{(h_E\otimes h_L^\mu)\ast\rho_\nu}-||s||^2_{(h_E\otimes h_L^\lambda)\ast\rho_\nu}0\leq f_{s,\mu,+\infty}\ast\rho_\nu =||s||^2_{(h_E\otimes h_L^\mu)\ast\rho_\nu}-||s||^2_{(h_E\otimes h_L)\ast\rho_\nu}.
\begin{aligned}\varphi^{t}(p'') = h\circ\varphi_{0}^{t}\circ h^{-1}(p'') & = h\circ\varphi_{0}^{t}(0,y'',-1) & = h(0,\gamma''(t),-1) & = \big(0,\gamma''(t),\sqrt{r\circ\gamma''(t)}\big),\end{aligned}
\frac{dz}{d\lambda}=\mathcal{B}\frac{\omega(\zeta,\alpha)\omega(\zeta,\bar\alpha^{-1})}{\prod_{j=1}^{M}\omega(\zeta,\gamma_{1}^{(j)}) \omega(\zeta,\gamma_{2}^{(j)})}\prod_{j=0}^{M}{\prod_{k=1}^{n_j}\left[\omega(\zeta,a_{k}^{(j)})\right]^{\beta_k^{(j)}}},
{\cal U}_{\sigma}({\bf p}) ={\frac{m}{\sqrt{2}}}\left( \begin{matrix}{{D^{J}\left( \alpha ({\bf p}) \right) \xi_{\sigma}}}\\{{D^{J}\left( \alpha^{- 1 \, \dagger}({\bf p}) \right) \xi_{\sigma}}}\\ \end{matrix}\right) \quad ,
\hat{H}_{\hat{\mu}\hat{\nu}\hat{\rho}}= 3 \partial_{[ \hat{\mu}}\hat{B}_{\hat{\nu}\hat{\rho}]}-{\textstyle \frac{3}{2}}\hat{F}^{I}{}_{[ \hat{\mu}\hat{\nu}}\hat{A}^{I}{}_{\hat{\rho}]}\, .
\sum_{n=0}^{\infty}\frac{z^nq^{n^2+n}}{(zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}b^2_\nu(m,n)z^mq^n=\sum_{n=0}^{\infty}(-q;q^2)_n(zq)^n.
P_i=\int d^3x\: \left\{F_{0k}F_{ik}-{1\over 2}D_0\lambda\: D_i\phi-{1\over 2}D_0\phi\: D_i\lambda + iD_i\psi_j\: \chi_j +i D_i \eta\: \psi_0 -A_i\: G \right\},
|{\Psi_{\alpha}}_{2}( p ) \rangle_{t r e e}= \Big \{[ U_{2}^{( 2 )}( \alpha ) + U_{2}^{( 1 1 )}( \alpha ) ] | \Psi_{0}( p ) \rangle + U_{1}( \alpha ) | \Psi_{1}( p ) \rangle + U_{0}( \alpha ) | \Psi_{2}( p ) \rangle \Big \}_{t r e e}= | \Psi_{2}( p_{\alpha}) \rangle \; .
M = \mathrm{span}( u_1^1 + u_1^2 + u_1^3, v_1^1, v_1^2, v_1^3, v_2^1 + v_2^2 + v_2^3, w_1^1 + w_1^2 + w_1^3, 3w_1^3 + 2w_1^2 + w_1^3).
\langle 0 | q | 0 \rangle = \left( \begin{array}{c c c c c}{{q_{1}}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}&{{0}}&{{}}\\{{}}&{{}}&{{q_{\tilde{N}_{c}}}}&{{}}&{{}}\\ \end{array}\right) , \qquad \langle 0 |{\bar{q}}| 0 \rangle = \left( \begin{array}{c c c}{{\bar{q}_{1}}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{\bar{q}_{\tilde{N}_{c}}}}\\{{}}&{{}}&{{}}\\{{}}&{{0}}&{{}}\\ \end{array}\right) ,
M_{I}^{2}= e^{K}( K^{\varphi \bar{\varphi}})^{2}| \lambda A |^{2}= \frac{| \lambda A |^{2}e^{G / 3}}{9 ( s + \bar{s}- b G )}\left( 1 + \frac{b}{s + \bar{s}- b G}\right)^{- 2}.
\rho_X(x_1) = \dotsm = \rho_X(x_i) = y_1, \rho_X(x_{i+1}) = \dotsm = \rho_X(x_{2i}) = y_2, \hskip 4.2ex \vdots \rho_X(x_{(m-1)i+1}) = \dotsm = \rho_X(x_{mi}) = y_m, \intertext{and}\rho_X(x_{mi+1}) = \dotsm = \rho_X(x_n) = 0.
\begin{aligned}c\bigl(\iota_j\bigl(f\bigr)(1)\bigr) = \operatorname{Ad}(v_1^j) \Bigl( \operatorname{diag}\bigl[ \overbrace{c(f(0)), \dots, c(f(0))}^{a''}, &f\bigl(t_{n_1^0 + p'}^j(1)\bigr), f\bigl(t_{n_1^0 + 2p'}^j(1)\bigr), \dots, f\bigl(t_{k-n_1^1}^j(p)\bigr), &\qquad\qquad\qquad\qquad\quad\underbrace{c(f(1)), \dots, c(f(1))}_{b''}\bigr] \Bigr). \end{aligned}
\langle\langle B, \bar{V}\rangle\rangle_{k+1}=\langle b_q, v_{M+k+1}\rangle &= b^+_q p_{(M+k+1)}+b^-_q \bar{p}_{-(M+k+1)}+F_k(v_{(2M+1)},...,v_{M+k})= f_{k+1}\langle\langle B^*, \bar{V}\rangle\rangle_{k+1}=\langle \hat{b}_1, v_{M+k+1}\rangle &= \hat{b}^-_1 p_{(M+k+1)}+\bar{b}^+_1 \bar{p}_{-(M+k+1)}+G_k(v_{(2M+1)},...,v_{M+k})= \bar{f}_{-(k+1)}.
f_1 =\overrightarrow{e_1}\wedge \overrightarrow{e_2}, f_2 = \overrightarrow{e_1}\wedge \overrightarrow{e_3}, f_3 = \overrightarrow{e_1}\wedge \overrightarrow{e_4}, f_4 = \overrightarrow{e_2}\wedge \overrightarrow{e_3}, f_5 = \overrightarrow{e_2}\wedge \overrightarrow{e_4}, f_6 = \overrightarrow{e_3}\wedge \overrightarrow{e_4}.
{\cal L}_{{B I}}^{{S U S Y}}= \frac{1}{4 e^{2}}\left[ \int \! d^{2}\theta W^{2}+ \int \! d^{2}\bar{\theta}\bar{W}^{2}\right] + \sum_{s , t = 0}^{\infty}a_{s t}^{( \beta )}\int \! d^{4}\theta W^{2}\bar{W}^{2}X^{s}Y^{t}
\{\Phi_x,\mathcal{K}\}=\{\Phi_x,\frac{1}{2}\int_0^1\!\!\Phi_y^2\mathrm{d}y\}=\int_0^1\!\!\Phi_y\{\Phi_x,\Phi_y\}\,\mathrm{d}y=0\,, \{\Pi_x,\mathcal{K}\}=\{\Pi_x,\frac{1}{2}\int_0^1\!\!\Phi_y^2\mathrm{d}y\}=\int_0^1\!\!\Phi_y\{\Pi_x,\Phi_y\}\,\mathrm{d}x=-\int_0^1\!\!\mathcal{E}_x(y)\Phi_y \mathrm{d}y\,,
\frak{a}(V,V^*) = \int_{\Omega}(A_{iJKj}u_{s,R}u^*_{i,J}+C_{iIJKLj}u_{j,LK}u^*_{i,IJ}) \ \mathrm{d}x &+\int_{\Omega}(M_{iJKL}\tau_{,L}u^*_{i,JK}+ M_{jLKI}u_{j,KL}\tau^*_{,I}+ K_{IJ}\tau_{,I}\tau^*_{,J})\ \mathrm{d}x,
(\xi-1-j)r_-+(\xi+j)r_+ <N_{Y_p}-(\xi+2+j)r_--(\xi-1-j)r_+,\; j=0,\dots,\xi-1, N_{Y_p}-(\xi+2+j)r_--(\xi-1-j)r_+ <(\xi-2-j)r_-+(\xi+1+j)r_+,\; j=0,\dots,\xi-2, N_{Y_p}-jr_--(2\xi-j)r_+ <(2\xi-j)r_-+jr_+,\; j=0,\dots,\xi-1, (2\xi-j)r_-+jr_+ < N_{Y_p}-(j+1)r_--(2\xi-j-1)r_+,\; j=0,\dots,\xi-1.
={\frac{\pi ( \Gamma ( \ell + 1 ) )^{2}}{2^{2 \ell + 2}( \Gamma ( \ell +{\frac{3}{2}}) )^{2}( \Gamma ( 2 \ell + 1 ) )^{2}}}\; k^{2 \ell - 1}r_{0}^{2 \ell + 1}e^{- 2 \pi k r_{0}}| \Gamma ( \ell + 1 + 2 i k r_{0}) |^{2}
\delta W = -{\frac{\alpha ( 1 + \alpha )^{2}}{1 5 3 6}}{\frac{\kappa^{4}}{\lambda^{6}}}\int_{M^{1 0}}d^{1 0}x \epsilon^{A_{1}A_{2}\cdots A_{1 0}}\mathrm{t r}\epsilon F_{A_{1}A_{2}}\mathrm{t r}F_{A_{3}A_{4}}F_{A_{5}A_{6}}\mathrm{t r}F_{A_{7}A_{8}}F_{A_{9}A_{1 0}}.
h^0(X,\mathcal O_X(D)) \leq h^0(X,\mathcal O_X(D))+h^2(X,\mathcal O_X(D)) &=p_a(X)+1+\frac{D^2-K_X\cdot D}2+ h^1(X,\mathcal O_X(D)) &\leq p_a(X)+1+\frac{D^2-K_X\cdot D}2+ \alpha h^0(X,\mathcal O_X(D)).
\frac{\alpha ( \sigma , \tau ) \buildrel{T > >{\frac{1}{H}}}}{= 2 H \, T ( \sigma , \tau ) + \log \left\{2 H^{2}\left[ ( A^{1}( \sigma )^{\prime})^{2}+ ( A^{2}( \sigma )^{\prime})^{2}\right] \right\}+ O ( e^{- 2 H T}) .}
\begin{aligned}(J^G_O \mathbf{n}_G, \mathbf{m}_G) =\sum_{i=1}^Nm_i\langle \mathbf{r}_i, \mathbf{n}_G\rangle_G \langle \mathbf{r}_i, \mathbf{m}_G\rangle_G &=\sum_{i=1}^Nm_i\langle \sqrt{G}\mathbf{r}_i, \sqrt{G}\mathbf{n}_G\rangle \langle \sqrt{G}\mathbf{r}_i, \sqrt{G}\mathbf{m}_G\rangle.\end{aligned}
\dot{z}= \begin{bmatrix}z_2-\hat{c}Nz_1+\hat{c}\sum_{j=1}^{N}x_{j_1}-\left( \tilde{\alpha}z_1^2+\tilde{\beta}z_2^2-\tilde{\gamma}\right)z_2 - \tilde{\omega}^2 z_1 - \hat{c}Nz_2 + \hat{c}\sum_{j=1}^{N}x_{j_2}\end{bmatrix}
\mu X_{\ell-1}(u) e^{-\int_{\tau_{\ell+1}}^u G_{\ell}(v) \: dv}\leq (1 + \delta) s e^{\int_{\tau_{\ell}}^u G_{\ell-1}(v) \: dv}e^{-\int_{\tau_{\ell+1}}^u G_{\ell}(v) \: dv}&= (1 + \delta) s e^{\int_{\tau_{\ell}}^{\tau_{\ell+1}}G_{\ell}(v) \: dv}e^{-s(u - \tau_{\ell})}&\leq \frac{2(1 + \delta) s^2}{\mu}\: e^{-s(u - \tau_{\ell})}.
u_{n+1}:= \begin{cases}\displaystyle{P_K \left[ u_n - \lambda_n \frac{\Theta_n (u_n)}{\| \Theta_n^{'}(u_n) \|^2}\Theta_n^{'}(u_n) \right]}\left(\Theta_n^{'}(u_n) \neq 0 \right), u_n \left(\Theta_n^{'}(u_n) = 0 \right),\end{cases}
\lim_{X \to \infty}\left(\frac{d^{n^2-1}}{ds^{n^2-1}}(\widetilde{\phi}(s)X^s)\big|_{s=1}\right)^{-1}\sum_{\pi}\mathrm{tr}(\pi)((\Phi^{S_1})^1b_{F'/F}(h_{S'_1})b_{E/F}(\Sigma_{\phi}^{S_0}(X)))
(I+ G_{0,0}V)(I-G_{0,0}vM_0^{-1}v^*) &=I-G_{0,0}vM_0^{-1}v^* +G_{0,0}V-G_{0,0}VG_{0,0}vM_0^{-1}v^* &=I-G_{0,0}vU(U+v^*G_{0,0}v)M_0^{-1}v^* +G_{0,0}V &=I,