harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\dot{y}_{2}=l(\dot{\theta}_{1}sin\theta_{1}+\frac{1}{2}\dot{\theta}_{2}sin\theta_{2})\rightarrow\dot{y}_{2}^{2}=l^{2}(\dot{\theta}_{1}^{2}sin^{2}\theta_{1}+\dot{\theta}_{1}\dot{\theta}_{2}sin\theta_{1}sin\theta_{2}+\frac{1}{4}\dot{\theta}_{2}^{2}sin^{2}\theta_{2})
\mathbf{y}_k = \overline{\mathbf{H}}\; \overline{\mathbf{x}}_k + \mathbf{n}_k \; k=1,2, \ldots ,N+J, \; \overline{\mathbf{H}}= [\mathbf{H}_{J}\: \mathbf{H}_{J-1}\ldots \mathbf{H}_0], \;\overline{\mathbf{x}}_k \!= \!\! \left[ \mathbf{x}_{k-J}^T \; \mathbf{x}_{k-J+1}^T \; \ldots \; \mathbf{x}_{k}^T \right]^T\!\!\!.
E^{(0)}=-\frac 12\int\limits_{M}^{\infty}\frac{d\kappa}{\pi}(\cos\pi s)(\kappa^{2}-M^{2})^{\frac 1 2-s}\frac{\partial}{\partial\kappa}\bigg{[}\ln\frac{\kappa-m/2}{\kappa+m/2}+\ln \frac{\kappa-m}{\kappa+m}\bigg{]}\,.
\boldsymbol{\varpi}_j =\exp\Big{\{}-M\log\det\big(1+M^{-1}V_j^*\hat{X}_j^{-1}V_j \Omega_jT_j^{-1}\hat{B}_j^{-1}T_j\Xi_j\big)\Big{\}}&=:\exp\Big{\{}-TrV_j^*\hat{X}_j^{-1}V_j \Omega_jT_j^{-1}\hat{B}_j^{-1}T_j\Xi_j\Big{\}}\; \exp\Big\{\sum_{\ell=2}^4\frac{(-1)^{\ell-1}}{\ell M^{\ell-1}}\Delta_{\ell,j}\Big\},
\mathcal{B}^{\boldsymbol{\alpha}}_{p,q}\equiv\begin{cases}\{f\in L_p([0,1]^d):\|f\|_{\mathcal{B}^{\boldsymbol{\alpha}}_{p,q}}<\infty\}, 1\leq p<\infty, \{f\in C_u([0,1]^d):\|f\|_{\mathcal{B}^{\boldsymbol{\alpha}}_{p,q}}<\infty\}, p=\infty\end{cases}
h\left(-\left(1+\frac{1}{\delta\gamma}\right)^{1/2}\right) = -\left(1+\frac{1}{\delta\gamma}\right)^{3/2}+ 3\left(1-\frac{1}{\gamma}\right)\left(1+\frac{1}{\delta\gamma}\right)^{1/2}+ 3\frac{\beta}{\gamma}, &= \left(1+\frac{1}{\delta\gamma}\right)^{1/2}\left( 2-\frac{3+1/\delta}{\gamma}\right) + 3\frac{\beta}{\gamma},
\tilde{Y}(z) = \mathcal{O}\begin{pmatrix}1 z^{-1/2}h_{\alpha + \frac{1}{2}}z^{1/2}h_{\alpha}(z) 1 z^{-1/2}h_{\alpha + \frac{1}{2}}z^{1/2}h_{\alpha}(z) 1 z^{-1/2}h_{\alpha + \frac{1}{2}}z^{1/2}h_{\alpha}(z) \end{pmatrix}.
\phi ( x^{0}\, , \, x^{1}) = \frac{1}{2 R}\sum_{n \geq 0}\left\{\left[ a_{n}( x^{0}) + i b_{n}( x^{0}) \right] e^{i \omega_{n + \frac{1}{2}}x^{1}}+ \left[ a_{n}( x^{0}) - i b_{n}( x^{0}) \right] e^{- i \omega_{n + \frac{1}{2}}x^{1}}\right\},
\begin{aligned}\sup\limits_{t \in [0,T_0]}\big(\|w_1(t,\cdot) - w_2(t,\cdot)\|_{C^{0}([0,1]}+\|z_1(t,\cdot) - z_2(t,\cdot)\|_{C^{0}([0,1])}\big) + \|h_1-h_2\|_{C^1([0,T_0])}+ \|g_1-g_2\|_{C^1([0,T_0])}&\leq (C_3+1)(\|h_1-h_2\|_{C^1([0,T_0])}+ \|g_1-g_2\|_{C^1([0,T_0])}) + C_3(\|z_{01}-z_{02}\|_{C^{2}([0,1])}+ \|w_{01}-w_{02}\|_{C^{2}([0,1])}) \end{aligned}
{\rm Kaz}_{em}^E(\sigma. t_{a_i\pi_\mu a_j^{-1}})={\rm Kaz}_{em}^E\big(t_{\sigma(a_i)\sigma(\pi_\mu)\sigma(a_j)^{-1}}\big)={\rm Kaz}_{em}^E\big(t_{c_i\pi_\mu c_j^{-1}}\big)=t_{c_i'\pi_\mu' c_j'^{-1}}.
\frac{\partial}{\partial \|\rho\|_2^2}E(\rho) =\frac{d}{2k^2}\frac{c_\beta^2}{1-2/kc_\beta+\|\rho\|_2^2/k^2c_\beta^2}=\frac{2k\ln k + O_k(\ln k)}{2k}O_k\left(\frac1k\right) \le\frac{\ln k}{k}\left(1+O_k\left(\frac{1}{k}\right)\right).
-\mathcal{L}_{\epsilon,i}^{u_i}\bigl(\cdot\bigr) \bigl( x \bigr) = \Bigl \langle \triangledown \bigl(\cdot\bigr),\, \bigl( A + \sum\nolimits_{j \neq i}B_j \gamma_j^{\ast}\bigr)x + B_i u_i\Bigr \rangle + \frac{\epsilon}{2}\operatorname{tr}\Bigl \{\sigma(x) \sigma^T(x) \triangledown^2 \bigl(\cdot\bigr) \Bigr\},
k=20(\frac{2}{\pi})^{3/2}\frac{(k_{B}T)^{5/2}k_{B}}{m_{e}^{1/2}e^{4}ln\Lambda}\approx\frac{T^{5/2}}{ln\Lambda}\times1.8\times10^{-10}Wm^{-1}K^{-1}
\begin{aligned}(1+\lambda)\|\partial_t U\|_{L_p (Q_r)}\le (1+\lambda) \big(\|\partial_t U - v \cdot D_x U\|_{L_p (Q_r)}+ r \|D_x U\|_{L_p (Q_r)}\big) \le N (1+\lambda) (\|U\|_{L_p (Q_{r_1})}+ \|D^{|\alpha| - 1}_v D^{1 +|\beta|}_x u\|_{L_p(Q_{r_1})}+ \|D_x U\|_{L_p (Q_{r})}) \le N \|u\|_{L_p (Q_R)}, \end{aligned}
(\omega(g)\phi,\phi)= ~\|\phi\|^2 (\bar{\xi}\xi')^{\nu_2+\frac{1}{2}}(\bar{\eta}\eta'\bar{\zeta}\zeta')^{\nu_1+\frac{1}{2}}(\bar{\gamma}\gamma')^{\beta-\frac{1}{2}}(\cosh t)^{-\nu_2-\beta-3}&\times ( \cos \theta \cos \theta' +\cosh^{-1}t \zeta\bar{\zeta}'\sin \theta \sin \theta')^r.
\begin{array}{@{}l@{}}{\displaystyle I_3^{(k)}= B(\mathbf{A}^{k-\frac{1}{2}};(I_{h}\overline{\Psi}^{k}-\overline{\Psi}^{k}),\overline{\theta}_{\Psi}^{k}) +B(\mathbf{A}^{k-\frac{1}{2}};(\overline{\Psi}^{k}-\Psi^{k-\frac{1}{2}}),\overline{\theta}_{\Psi}^{k}).}\end{array}
V^{1/r^8}_{\rm gravity}= -{105 R^2 r_i\over 4 r^9}\left(\hat{T}_{IJ}\tilde{T}^{IJ(i)}+\hat{T}^{IJ(i)}\tilde{T}_{IJ}-{1 \over 9}(\hat{T}^{I(i)}{}_I \tilde{T}^J{}_J +\hat{T}^I{}_I\tilde{T}^{J(i)}{}_J )\right)
\int_D\mathbb{I}_{\{-t_0-1<\Psi<-t_0\}}|f|^2e^{-\Psi-\phi}\leq e^{t_0+1}\int_{\{\Psi<-t_0\}}|f|^2e^{-\phi}=e\left(\frac{R}{r}\right)^{2N+2n}\int_{\{\Psi<-t_0\}}|f|^2e^{-\phi}.
\gamma ( z ) \sim 4 i \pi g_{s}^{B}\alpha^{\prime}\sum_{\epsilon = \pm 1}\left( \sum_{p = 1}^{N_{1}}\log \frac{( z + \epsilon z_{1 p})}{\Lambda}- \sum_{q = 1}^{N_{2}}\log \frac{( z + \epsilon z_{2 q})}{\Lambda}\right)
\sum_{a \in \mathcal{A}}\max\{0, x_a - x^{\mbox{\tiny \upshape (\itshape k\upshape)}}_a\}= \begin{cases}0, \mbox{if}\mathbf{x}= \mathbf{x}^{\mbox{\tiny \upshape (\itshape k\upshape)}}, h, \mbox{if}\mathbf{x}= \mathbf{x}^{\mbox{\tiny \upshape (\itshape j\upshape)}}, \; j \neq k\end{cases}
L_{ww}= 2 (w^1)^2w^1_{x_1}+ 2w^1w^2(w^1_{x_2}+ w^2_{x_1}) + 2(w^2)^2 w^2_{x_2}= 2(c_1+a_1x_1 + b_1x_2)^2a_1 + 2( c_1+a_1x_1 + b_1x_2)( c_2+a_2x_1 +b_2x_2)(b_1+a_2) &\:\:\:\: + 2 (c_2+a_2x_1 +b_2x_2)^2b_2,
\begin{array}{r l}{{\bar{\cal{H}}\, m_{( 2 , 1 ; 4 , 2 )}=}}&{{\varepsilon_{( 2 , 1 ; 4 , 2 )}\, m_{( 2 , 1 ; 4 , 2 )}+ 2 \beta \, m_{( 3 , 1 ; 3 , 2 )}- 4 \beta \, m_{( 3 , 2 ; 2 , 2 )}+ 4 \beta \, m_{( 2 , 1 ; 3 , 3 )}}}\\{{}}&{{+ 8 \beta \, m_{( 2 , 1 ; 3 , 2 , 1 )}+ 2 4 \beta \, m_{( 2 , 1 ; 2 , 2 , 2 )}+ 4 \beta \, m_{( 2 , 1 ; 4 , 1 , 1 )}\, .}}\\ \end{array}
|\dot{z}_j(t)|=|\bar{F}+\sum_{k\neq j}\frac{\lambda_k i}{2\pi}\frac{1}{\overline{z_j(t)-z_k(t)}}|\leq M_0+\frac{\sum_{j=1}^N|\lambda_j|}{2\pi}\tilde{d}_P(t)^{-1}=M_0+\frac{\tilde{\lambda}}{2\tilde{d}_P(t)}.
\left|f(x)-f(y)\right|\leq2\sum_{i=1}^{\lfloor\log_2(d_G(x,y)/d_0(G))\rfloor+1}p(d_0(G)2^{i+1})\psi^{-1}\left(\frac{\Gamma(f)}{v(d_0(G)2^{i-1})^2}\right)
\begin{aligned}\left|G_{\left[x_{1}, x_{2}\right]}\left(x_{1}, y\right)\right|=\left|\frac{\Delta_{y+1, x_{2}}(\theta)}{\Delta_{x_{1}, x_{2}}(\theta)}\right|, &\left|G_{\left[x_{1}, x_{2}\right]}\left(y, x_{2}\right)\right|=\left|\frac{\Delta_{x_{1}, y-1}(\theta)}{\Delta_{x_{1}, x_{2}}(\theta)}\right|.\end{aligned}
|(\nabla [h \cdot u]_{\ell,m})(x)|^2 = \left||x|^{-\sigma}u'(|x|)-\sigma|x|^{-\sigma-1}u(|x|)\right|^2|Y_{\ell,m}(\omega_x)|^2 + |x|^{-2\sigma-2}|u(|x|)|^2\, |x|^2|\nabla Y_{\ell,m}(\omega_x)|^2.
\|F\|^2_p \leq\int_{\Delta^n(0,2r_p)}e^{-2p(\psi_x^\prime+ \widetilde\psi_x)}\,\frac{\omega^n}{n!}&\leq (1+4C_1r^2_p)\exp\!\big(16C_2p\,r_p^3\big)\int_{\Delta^n(0,2r_p)}e^{-2p\psi^\prime_x}\,dm &\leq\left(\frac{\pi}{2}\right)^n\,\frac{(1+4C_1r_p^2)\exp\!\big(16C_2p\,r_p^3\big)}{p^n\lambda_1\ldots\lambda_n}\,\cdot
I_{T}^{\gamma}(\varphi(t)) = \frac{1}{2}\int_{0}^{T}\biggl[\frac{d\varphi(t)}{dt}- \bigl(A + \sum\nolimits_{i=1}^n B_i \gamma_i\bigr)\varphi(t) \biggr]^T \Bigl(\sigma(\varphi(t))\sigma^T(\varphi(t))\Bigr)^{-1}& \quad\quad \times \biggl[\frac{d\varphi(t)}{dt}- \bigl(A + \sum\nolimits_{i=1}^n B_i \gamma_i\bigr)\varphi(t) \biggr] dt,
K_{\Delta_k}(s;m) = \frac{-8 s^{-\frac{m}{2}}\left((m (s-1)-4 s) s^{m/2}+s (m (s-1)+4)\right) \Gamma \left(\frac{m}{2}+2\right)}{(m-2) m^2 (m+2) (s-1)^3},
J^{(N)}(N-1; 1, \ldots, 1) = \frac{1}{\Lambda^{(N)}}\;\left(\prod m_i^2 \right) \;\sum_{i=1}^N \frac{1}{m_i^2}\; F_i^{(N)}\;J_i^{(N)}(N-1; 1, \ldots, 1) .
{\pi^2 \over 8a}\int_{0}^{1}{dx{1 \over \sqrt{1 -x^2}\left(1 - \lambda^2 + \lambda^2 x^2\right)}\left[{(x^4 - 3\delta^2x^2) \over \lambda^2 (x^2+\delta^2)^3}+{\left(1 + \lambda^2 - \lambda^2 x^2\right)\over\left(1 - \lambda^2 + \lambda^2x^2\right)}{x^2 \over (x^2 +\delta^2)}\right]},
\chi_{i,S}(P) = \det \begin{pmatrix}t_i(P) u_i(P) v_i(P) w_i(P) t_i(S_1) u_i(S_1) v_i(S_1) w_i(S_1) t_i(S_2) u_i(S_2) v_i(S_2) w_i(S_2) t_i(S_3) u_i(S_3) v_i(S_3) w_i(S_3) \end{pmatrix}.
p_{1,k_1,k_2}(x,t)= \frac{(-1)^{k_2}\pi^{k_1+k_2}}{4^{n}(\pi\delta)^{n+k_1-1}\sqrt{2\pi\kappa}}e^{-\frac{1}{4}d(x,t)^2}\left[1+O\left(\frac{1}{\kappa}+\delta \right) \right].
\frac{\tau_2-\tau_1}{F^{-1}(\tau_2)-F^{-1}(\tau_1)}=\frac{\tau_2-\tau_1}{\frac{\mathrm{d}}{\mathrm{d}\tau}F^{-1}(\tau^{*})(\tau_2-\tau_1)}=f\bigl(z^*\bigr),
f_{n,r}(x+y)-f_{n,r}(x)-\frac{d}{dx}f_{n,r}(x)y1_{\{|y|\leq r\}}=f_n\left(\tfrac{x+y}{r}\right)-f_n\left(\tfrac{x}{r}\right)-f_n'\left(\tfrac{x}{r}\right)\tfrac{y}{r}1_{\{|\frac{y}{r}|\leq 1\}}&\leq \tfrac{1}{2}\|f_n''\|_\infty \left(\tfrac{y}{r}\right)^21_{\{|y|\leq r\}}+\|f_n\|_\infty1_{\{|y|>r\}}
\hat{\cal R}_{m n}={\cal D}_{[ m}\psi_{n ]}+ \frac{\kappa}{2}\gamma_{a b [ m}\psi_{n ]}v^{a b}+ 2 i \kappa \gamma_{[ m}\vec{\tau}\psi_{n ]}\vec{t}- \frac{\kappa}{\sqrt{3}}\gamma^{a}\psi_{[ n}\hat{F}_{m ] a}
\partial_y^{k+2}v =\sum_{i=0}^{\infty}2^{\frac{2i+1}{2}-k-2}(-1)^{k-i-1}(2k-2i+1)!!C_iy^{\frac{2i+1}{2}-k-2}&\quad\quad+ \sum_{j=k+2}^{\infty}2^jj(j-1)\dots (j-k-1)b_{2j}y^{j-k-2}.
\begin{array}{@{}l@{}}{\displaystyle K_4^{(k)}=\left(f(\Psi^{k-1},\Psi^{k-1})-f(I_{h}\Psi^{k-1}, I_{h}\Psi^{k-1}),\;\mathbf{v}\right)}begin{align*}2mm]{\displaystyle +\left(f(I_{h}\Psi^{k-1}, I_{h}\Psi^{k-1})-f(\Psi^{k-1}_{h},\Psi^{k-1}_{h}),\;\mathbf{v}\right)\stackrel{\mathrm{def}}{=}K_4^{(k),1}+K_4^{(k),2}.}\end{array}
\cos(\beta({\frac{\delta}{\delta J_{r}}}+\Phi))\exp\{{\frac{1}{2}}Jf^{-1}J\}=\exp\{-{\frac{\beta^2}{2}}f^{-1}_{r r}\}\cos(\beta(\int d^2 r'f^{-1}_{r'r}J_{r'}+\Phi))\exp\{{\frac{1}{2}}Jf^{-1}J\}
\Phi_{Q,a}=:\left\{\phi:X\to\mathbb{R}\ |\ \phi\left(x\right)=a\left\Vert x\right\Vert_X^{2}+\left\langle u,x\right\rangle_X + c,u\in X^*,c\in\mathbb{R}\right\}, \Psi_{Q,b}=:\left\{\psi:Y\to\mathbb{R}\ |\ \psi\left(y\right)=b\left\Vert y\right\Vert_Y^{2}+\left\langle v,y\right\rangle_Y + d,v\in Y^*,d\in\mathbb{R}\right\},
G(\tau+t) =G(\tau)1_{\left\{Y(\tau)+\overline{\sigma_\tau Y}(t)<\bar{Y}(\tau)\right\}}+(\tau+\sigma_\tau G(t))1_{\left\{Y(\tau)+\overline{\sigma_\tau Y}(t)\ge\bar{Y}(\tau)\right\}}&=G(\tau)+ \Big(\sigma_{\tau}G(t)+(\tau-G(\tau)) \Big)1_{\left\{\overline{\sigma_\tau Y}(t)\ge\bar{Y}(\tau)-Y(\tau)\right\}}\,,
\frac{G(z)}{A(z)}\sum_n \frac{|c_n|^2}{z-t_n}= \bigg(\sum_n \frac{c_n\mu_n^{1/2}}{z-t_n}\bigg) \cdot\bigg(\sum_n \frac{G(t_n) \bar c_n}{A'(t_n)\mu_n^{1/2}(z-t_n)}\bigg).
g' =[(0,0),\ldots, (m-1,k),(m-1,k+1),(m,k+1),\ldots, (m,l),(m+1,l),\ldots, (c_i,c_{i+1}) ] g'' = [(0,0),\ldots, (m-1,k),(m,k),\ldots, (m,l-1) ,(m+1,l-1),(m+1,l),\ldots, (c_i,c_{i+1}) ]
\lim_{\alpha\to\infty}\left( \hat{v}^q_\alpha(0)+\left\vert\frac{\nabla \hat{v}_\alpha (0)}{\hat{v}_\alpha(0)}\right\vert^n+\left\vert\frac{\nabla^2\hat{v}_\alpha(0)}{\hat{v}_\alpha(0)}\right\vert^\frac{n}{2}\right)= 1.
\sum_{\substack{n \leq X}}g_{\mathcal{J}}(n) f(n)n^{-it}= \frac{X^{i(t_1-t)}}{1+ i(t_1-t)}\sum_{n \leq X}g_{\mathcal{J}}(n) f(n)n^{-it_1}& + O\left(X \frac{\log\log X}{\log X}\exp \left(\sum_{p \leq X}\frac{|1-f(p)p^{-it_1}g_{\mathcal{J}}(p)|}{p}\right)\right).
d s^{2}= \frac{\Delta - a^{2}\sin^{2}\theta}{\Sigma}\left( d t - \omega d \varphi \right)^{2}- \Sigma \left( \frac{d r^{2}}{\Delta}+ d \theta^{2}+ \frac{\Delta \sin^{2}\theta}{\Delta - a^{2}\sin^{2}\theta}d \varphi^{2}\right) ,
\begin{aligned}\chi_n(\tau) \sim \Sigma^2 \sum_{k=1}^{k(n)}\log^4 T_k + \Sigma^2\log^4 \bar T^n, \chi_n(\tau;p) = \sum_{k=1}^{k(n)}O(\log^{2p}T_k) + O\big(\log^{2p}\bar T^n\big), p>2.\end{aligned}
\Theta^{i j}= \left( \begin{array}{c c c c}{0}&{0}&{0}&{- B^{- 1}\sinh \gamma}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{- B^{- 1}\cosh \gamma}\\{B^{- 1}\sinh \gamma}&{0}&{B^{- 1}\cosh \gamma}&{0}\\ \end{array}\right) .
\begin{array}{rcl}\displaystyle{\frac{dw}{d\tau}}= \displaystyle{w''' + \tilde\beta z'' + 2\tilde\beta ww'+ \tilde\alpha w',}\displaystyle{\frac{dz}{d\tau}}= \displaystyle{w''w+K\tilde\beta w''-\frac12{w'}^2+2\tilde\beta wz'+\frac12\tilde\alpha w^2\,,}\end{array}
R + \frac{\gamma}{4}\left[ \frac{(D \Omega )^2}{\Omega^2}- 2\frac{D \cdot D \Omega}{\Omega}\right] + (1-\lambda)\mu\Omega^{-\lambda}+ \frac{1}{4}(1- \frac{\epsilon}{2}) \Omega^{- \epsilon /2}F^2 - \frac{\delta}{2}\Omega^{\delta -1}(Df)^2 = 0,
\sum_{n=1}^{N}\sum_{k=1}^{n}|E_{n,k}|\ll\sum_{n=1}^{N}(n^{11/8}+L_n^{1/2}n^{5/4})\ll N^{19/8}+\sum_{n=1}^{N}L_n^{1/2}n^{5/4}.
\mathcal{A}_{\mathbb{T}}^{\varepsilon}:=\nabla\cdot\left(-\mathbb{T}\left(\frac{x}{\varepsilon}\right)\nabla\right)=\frac{\partial}{\partial x_{i}}\left[-\tau_{ij}^{\alpha\beta}\left(\frac{x}{\varepsilon}\right)\frac{\partial}{\partial x_{j}}\right].
W[\Phi]={1\over{4\pi}}\int d^2x\:\Big{\{}\Big{(}\partial_+\Phi\partial_-\Phi + 2\,e\, A_+\partial_-\Phi - 2\,e\, A_-\partial_+\Phi\Big{)}+(a+b-2)\,e^2\,A_+\,A_-\Big{\}}
\|\psi\|_{L^3(\mathbb{R}^3\times\mathbb{R}^3)}\le C\|\psi\|_{H^1(\mathbb{R}^3\times\mathbb{R}^3)}, \|\psi\|_{L^2(\mathbb{R}^3\times\mathbb{R}^3)}\le C \|\psi\|_{H^1_\mathrm{trap}(\mathbb{R}^3\times\mathbb{R}^3)}.
J(t;\mathbf{v},\mathbf{w}): =\bigg(\rho\int_0^t(t-s)^{-\beta}e^{-\delta(t-s)}\nabla \mathbf{v}(s)ds,\nabla\mathbf{w}(t)\bigg),~~\forall \ \mathbf{v}(\cdot,t), \mathbf{w}(\cdot,t)\in\mathbf{X}
\{\hat{S}^{\mu \nu}, \hat{S}^{\lambda \rho}\}^{\prime}= \Delta^{\mu \lambda}\hat{S}^{\nu \rho}- \Delta^{\nu \lambda}\hat{S}^{\mu \rho}+ \Delta^{\nu \rho}\hat{S}^{\mu \lambda}- \Delta^{\mu \rho}\hat{S}^{\nu \lambda}
\left( \frac{d}{dt}-\Delta_{\omega_{\phi}}\right) \nabla_{\phi}U =\nabla_{\phi}\left( \frac{d}{dt}-\Delta_{\omega_{\phi}}\right) U+U \ast \nabla_{\phi}({\rm Rm}_{\phi}+\widetilde{\eta}+\nabla_{\phi}X)+{\rm Rm}_{\phi}\ast \nabla_{\phi}U,
{\bf Y}^{(a)}_2(\theta) =-1+ h y^{(a)}_{1,2}(\theta) + h^2 y^{(a)}_{2,2}(\theta)+ O(h^3),{\bf Y}^{(a)}_3(\theta) =-1+ h y^{(a)}_{1,3}(\theta) + h^2 y^{(a)}_{2,3}(\theta) + O(h^3) (a=1,2)
\begin{aligned}(i)~ \mbox{supp}~g(\omega) \subset [-M, M], K>\frac{M}{\varepsilon_0}\bigg(1+\frac{1}{\varepsilon_0}\bigg), & (ii)~ ||f_0||_{L^{\infty}}< \infty,{\mathcal M}(L^+_{\gamma_0}(0)) \geq{\mathcal M}_*(\varepsilon_0, \gamma_0).\end{aligned}
\bigg\{\sum_{j=1}^{\infty}\bigg(\int_{\mathbb{R}^n}|f_{j}(y)|dy\bigg)^{r}\bigg\}^{1/r}\leq C\int_{\mathbb{R}^n}\bigg\{\sum_{j=1}^{\infty}|f_{j}(y)|^{r}\bigg\}^{1/r}dy.
\left\lbrace \begin{array}{c}\phi(+a/2)=-\phi(-a/2) \partial_{x}\phi(+a/2)=-\partial_{x}\phi(-a/2) \partial_{x}^{2}\phi(+a/2)=-\partial_{x}^{2}\phi(-a/2) \partial_{x}^{3}\phi(+a/2)=-\partial_{x}^{3}\phi(-a/2)\end{array}\right. ,
\stackrel{\mathrm{H W}}{S^{\alpha \beta}}= \int \Psi^{\dagger}S_{\mathrm{o p}}^{\alpha \beta}\Psi d^{3}x , \quad S_{\mathrm{o p}}^{\alpha \beta}= \frac{1}{2}\, \sigma^{\alpha \beta}+ \frac{i}{2 m}\left( p^{\alpha}\gamma^{\beta}- p^{\beta}\gamma^{\alpha}\right) \, .
\sum_{i=0}^n\sum_{j=0}^i(2(s+1)(i-j)j + (n-i)i)\frac{x_{i-j}}{(i-j)!}\frac{x_j}{j!}\sum_{\ell=0}^n (s+\ell)!\frac{B_{n-i, \ell}(x_1, x_2, \ldots)}{(n-i)!}= 0
A =-\frac{\alpha\gamma+i\beta}{\beta^2+\gamma^2}C+\frac{\alpha\beta-i\gamma}{\beta^2+\gamma^2}D=-\kappa C+\sigma D, B =\frac{\alpha\beta-i\gamma}{\beta^2+\gamma^2}C+\frac{\alpha\gamma+i\beta}{\beta^2+\gamma^2}D=\sigma C+\kappa D,
\Phi_{0}= a_{0}^{- 1}\sum_{M \geq 0}\frac{x^{M}}{\Gamma^{2}( M + 1 )}\; \frac 1{2 \pi i}\int_{C}\frac{\Gamma ( 4 s + 1 ) \Gamma ( 2 M - s )}{\Gamma^{3}( s + 1 )}( e^{- \pi i}y )^{s}\, d s ,
{\cal L}_{1}= ( X + \bar{X}) R -{\frac{i}{2}}( X - \bar{X}) F - 2 \lambda_{-}\rho^{-}+ 2 \lambda_{+}\rho^{+}+ 2 \tilde{\lambda}^{-}{\tilde{\rho}}_{--}2 \tilde{\lambda}^{+}{\tilde{\rho}}_{+}+ 4 i ( M \bar{H}- \bar{M}H ) E^{z}E^{\bar{z}}.
\beta_{0}((I +(w))^h)=\sum_{\ell=1}^h \beta_{0}(I^{\ell})+1, \beta_{i}((I +(w))^h)=\sum_{\ell=1}^h \left[\beta_{i}(I^{\ell})+\beta_{i-1}(I^{\ell}), \right] i \in \mathbb{N}_{>0}.
\textbf{A}\otimes \textbf{P}^\alpha\textbf{C}+\textrm{diag}(\textbf{a})\textbf{K}_1\otimes \textbf{P}\textbf{C}+\textrm{diag}(\textbf{b})\textbf{K}_2\otimes \textbf{P}\textbf{C}=\textbf{F}
(\mathcal F_{1}\mathbf f)_{kl}(\varrho,s)=\frac{\varrho^\frac{n+1}2s^\frac{3-n}2}{(2\pi)^{\frac{n-1}2}}\mathbf H_\frac{n-3}2(\mathcal F_{1}\mathbf F)_{kl}(\varrho,s\varrho),\quad\mathbf F=\mathcal C\mathbf f.
U(r, \lambda)\downarrow^{D_r}_{D'_r}\cong \bigoplus_{\substack{\mu = \lambda-\square \mu = \lambda}}U'(r,\mu)\qquad U'(r+1, \lambda)\downarrow^{D'_{r+1}}_{D_r}\cong \bigoplus_{\substack{\mu = \lambda+\square \mu = \lambda}}U(r,\mu).
J_1 =-2\int_{\mathbb{R}^2}t\nabla\cdot H^{(\alpha,a)}\cdot \mu(t\partial_t\nabla\cdot H^{(\alpha,a)}-t\nabla\cdot f^2_{\alpha a})dx &=-2\int_{\mathbb{R}^2}t\nabla\cdot H^{(\alpha,a)}\cdot\mu(-r\partial_r\nabla\cdot H^{(\alpha,a)}+S\nabla\cdot H^{(\alpha,a)}-t\nabla\cdot f^2_{\alpha a})dx.
(I-Q^\dagger)F^{q,\dagger}_{vv}(v+\psi^\dagger(v,p),p)[v_1 + \psi^\dagger_v(v,p)v_1, v_2 + \psi^\dagger_v(v,p)v_2] \quad+ (I-Q^\dagger)F^{q,\dagger}_v(v + \psi^\dagger(v,p),p)\psi^\dagger_{vv}(v,p)[v_1,v_2] = 0.
\{b ( p , s ) , b^{\dagger}( p^{\prime}, s^{\prime}) \}= \delta ( p^{+}- p^{\prime + )}\delta^{2}( p_{\perp}- p_{\perp}^{\prime}) \delta_{s , s^{\prime}}= \delta^{3}( p - p^{\prime}) \delta_{s , s^{\prime}},
{\widetilde Q}^{( n )}= \frac{1}{2}\log ( u v ) \sum_{r + s = n}\left( \begin{matrix}{{n}}\\{{r}}\\ \end{matrix}\right) \left( \begin{matrix}{{n}}\\{{s}}\\ \end{matrix}\right) u^{r}v^{s}- \sum_{r + 2 = n}\left( \begin{matrix}{{n}}\\{{r}}\\ \end{matrix}\right) \left( \begin{matrix}{{n}}\\{{s}}\\ \end{matrix}\right) c ( r , s ) u^{r}v^{s}
\MoveEqLeft[8] (d_L \alpha)(x_0 , \dotsc, x_m) = \frac{1}{m+1}\sum_{k=0}^{m+1}(-1)^k \rho (x_k) \alpha (x_0, \dotsc, \widehat{x_k}, \dotsc , x_m ) begin{align*}1ex] + \frac{1}{m+1}\sum_{k < l}(-1)^{k+l+1}\alpha ([x_k , x_l], x_0 , \dotsc , \widehat{x_k}, \dotsc , \widehat{x_l}, \dotsc , x_m ).
\begin{array}{l}{\mu ( R ) = p + \frac{1}{2}a_{i j}\alpha^{i}\wedge \alpha^{j}+ q \alpha^{1}\wedge \alpha^{2}\wedge \alpha^{3}\wedge \alpha^{4},}\\{\mu ( \hat{R}) = q - \frac{1}{4}\epsilon^{i j k l}a_{i j}\beta_{k}\wedge \beta_{l}+ p \beta_{1}\wedge \beta_{2}\wedge \beta_{3}\wedge \beta_{4}}\\ \end{array}
\begin{array}{l}Cl(\eta_1)>0 \;\; \Longrightarrow Cl(\eta_1 \oplus \eta_2)=Cl(\eta_1) \otimes Cl(\eta_2) Cl(\eta_1)<0 \;\; \Longrightarrow Cl(\eta_1 \oplus \eta_2)=Cl(\eta_1) \otimes Cl(-\eta_2) \, . \end{array}
{\cal A}_{f i}^{( 1 )}= \int d{\bf x}\mathrm{e}^{- i{\bf p}^{^{\prime}}{\bf \cdot x}}\left[ \frac{m}{w_{{\bf p}}}H_{1}({\bf x},{\bf \nabla}) + V ({\bf x )}\right] \mathrm{e}^{i{\bf p \cdot x}}.
e^{- 1}{\cal L}= -{\frac{1}{2}}R -{\frac{1}{4}}G_{I J}F_{\mu \nu}{}^{I}F^{\mu \nu J}-{\frac{1}{2}}g_{i j}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{- 1}}{4 8}}\epsilon^{\mu \nu \rho \sigma \lambda}C_{I J K}F_{\mu \nu}^{I}F_{\rho \sigma}^{J}A_{\lambda}^{K}
\eta_{\Gamma}(t_{0},t;a,b):=\#\big\{\pi(t_{0}+t):\pi\in\Gamma , \varsigma_{\pi}\leq t_{0},\pi (t_{0})\in [a,b]\big\}, &\hat{\eta}_{\Gamma}(t_{0},t;a,b):=\#\big\{\pi(t_{0}+t):\pi\in\Gamma , \varsigma_{\pi}\leq t_{0},\pi (t_{0}+t)\in [a,b]\big\}.
q^{n}I_{q}\left(f_{n}\right)-I_{q}\left(f\right)=\frac{q-1}{\log q}\sum_{l=0}^{n-1}f^{\prime}\left(l\right)q^{l}+\left(q-1\right)\sum_{l=0}^{n-1}f\left(l\right)q^{l},
{}_1 F_1(a;b;z) \sim{e^z z^{a-b}\over \Gamma(a)}\sum_{s=0}^\infty{(1 - a)_s (b - a)_s \over s!}z^{-s}+{e^{\pi i a}z^{-a}\over \Gamma(b - a)}\sum_{s=0}^\infty{(a)_s (a-b+1)_s \over s!}(-z)^{-s}, z = i x
[x_{i, r+1}^{-}, x_{i+1, s}^{-}] - [x_{i, r}^{-}, x_{i+1, s+1}^{-}] = \varepsilon_1 x_{i, r}^{-}x_{i+1, s}^{-}+ \varepsilon_2 x_{i+1, s}^{-}x_{i, r}^{-},
L_{m,n,\beta}= \frac{1}{m\beta}\begin{pmatrix}c_1^2 c_1 d_1 c_1 d_1 c_2^2 + d_1^2 c_2 d_2 &\ddots \ddots \ddots & c_{n - 1}d_{n - 1}c_{n}^2 + d_{n - 1}^2 \end{pmatrix}.
\sum_{k=3}^{\infty}s_{b+1,k}- s_{b,3}= s_{b+1,3}- s_{b,3}+ \sum_{k=4}^{\infty}s_{b+1,k}> -\frac{5}{b^2}- \frac{y_b}{b^2 +1}+ \frac{2y_b}{b(b+1)}= -\frac{5}{b^2}+ \frac{b^2 - b +2}{b(b+1)(b^2+1)}y_b,
\langle \psi ( 0 , 0 ) \psi^{\dagger}( x , t ) \rangle_{T}\begin{array}{c}{{{}}}\\{{\longrightarrow}}\\{{\mathrm{\raisebox{2 m m}{~ x \to \infty \atop ~ t \to \infty ~}}}}\\ \end{array}\exp \left\{- \frac{2 \Delta^{+}\pi T}{v_{F}}| x - v_{F}t | - \frac{2 \Delta^{-}\pi T}{v_{F}}| x + v_{F}t | \right\}.
\gamma_{N,2}=\sum_{l=0}^{2}c_{l}\frac{2!\left( N-l\right) !}{N!\left( 2-l\right) !}=-\frac{1}{N^3}+\frac{6}{N^3}-\frac{6}{N^2 \left(N -1\right)}= \frac{5}{N^3}-\frac{6}{N^3 - N^2}.
\psi^2_{n,1}(s,\eta_n(t)) =[(\eta_n(t)- \eta_n(s))^\alpha- (\eta_n(t)-s)^\alpha]^2 \le (\eta_n(t)-s)^{2\alpha-2}(s- \eta_n(s))^2 \le( \delta/2)^{2\alpha-2}n^{-2},
\inf_{c}\|v-c\|^2_{L^2({\cal H}_{d})}=\inf_c\left({\cal H}_{d}(\partial\Gamma)\cdot c^2-2c\left(\int_{\partial\Gamma}vd{\cal H}_{d}\right)+\|v\|^2_{L^2({\cal H}_{d})}\right) =\|v\|^2_{L^2({\cal H}_{d})}-{\cal H}_{d}(\partial\Gamma)^{-1}\left(\int_{\partial\Gamma}vd{\cal H}_{d}\right)^2.
\begin{array}{c}{}[P_{\mu},K_{\nu}]=2i(M_{\mu\nu}+\eta_{\mu\nu}D)\,,{}{}[P_{\mu},S^{a}]=-\gamma_{\mu}Q^{a}\,,{}{}[K_{\mu},Q^{a}]=-\gamma_{\mu}S^{a}\,,{}{}\{Q^{a\alpha},\bar{S}_{b\beta}\}=-i\delta^{a}{}_{b}(2\delta^{\alpha}{}_{\beta}D+(\gamma^{[\mu}\gamma^{\nu]})^{\alpha}{}_{\beta}M_{\mu\nu})+2i\delta^{\alpha}{}_{\beta}A^{a}{}_{b}\,.\end{array}
J_S(z) = \begin{pmatrix}1 -(1 - \varphi_{n,t}(z)^{-1})z^n 0 1\end{pmatrix}, |z| = \rho_N J_S(z) = \begin{pmatrix}\varphi_{n,t}(z) 0 0 \varphi_{n,t}(z)^{-1}\end{pmatrix}, |z| = 1 J_S(z) = \begin{pmatrix}1 0 (1 - \varphi_{n,t}(z)^{-1})z^{-n}1\end{pmatrix}, |z| = \rho_N^{-1}
\mathcal{M}_{|\Delta_n|^2}(s) \approx \frac{\left({\alpha_n}/{b_{0,n}}\right)^{\alpha_n}}{2s\Gamma(\alpha_n)}& \times \int_0^{\infty}b^{\alpha_n-2}\exp\left(-\frac{\tilde{A}_n}{2b}-\frac{\alpha_n b}{b_{0,n}}\right)\mathrm{d}b.
\int_{0}^{\infty}e^{-\omega x}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|\eta x^{2})dx=\frac{1}{\sqrt{\pi}\omega}G_{p+2,q}^{m,n+2}(\begin{matrix}0,\frac{1}{2},a_{p}\\b_{q}\end{matrix}|\frac{4\eta}{\omega^{2}}),
|h(t)-g_0(t)|_{g_0(t)}+ \mathbf{r}|\nabla^{g_0(t)}h(t)|_{g_0(t)}= |s^{-1}Q^*\hat{g}(st) - s^{-1}Q^*\tilde{g}(st)|_{g_0(t)}+ \mathbf{r}|\nabla^{g_0(t)}h(t)|_{g_0(t)}\leq Q^*\left(|\hat{g}-\tilde{g}|_{\tilde{g}}+r_s|\nabla^{\tilde{g}}\hat{g}|_{\tilde{g}}\right)(st) < \delta,
Z_{n}^{( N_{f})}( M ) \ = \ \prod_{f = 1}^{N_{f}}\frac{( n + f - 1 ) !}{( N \Sigma^{2})^{n + f - 1}}\, \frac{C_{n}^{( N_{f})}( M )}{\Delta_{N_{f}}( - M^{2})}\: .
Q^{n,3}_\tau = n^{2\alpha+1}\int^{\tau}_0\gamma_s(s-\eta_n(s))^{2\alpha}\left( (\sigma'(X_s))^2- ( \sigma'(X_{\eta_n(s)}))^2 \right)\sigma^2(X_{\eta_n(s)}) \, & \times \left[\left(W_s-W_{\eta_n(s)}\right)^2 -(s-\eta_n(s))\right]\,ds
\nabla(\varpi_2)\otimes(-\varpi_2)=\begin{tabular}{|c|}\hline$\varepsilon$ \hline$\nabla^{P}(3\varpi_1-2\varpi_2)$ \hline$\nabla^{\alpha_1}(2\varpi_1-2\varpi_2)$ \hline$-\varpi_2$ \hline$\nabla^{\alpha_1}(3\varpi_1-3\varpi_2)$ \hline$-2\varpi_2$ \hline\end{tabular}.
\delta ( p + q ) = ( 2 \pi )^{4}\tilde{C}_{\alpha}^{\alpha_{0}}( q ) d e l t a_{\Phi_{c}( p )}\delta_{\Phi_{c}( q )}\Gamma_{0}^{\alpha , \alpha_{0}}( \Phi_{c}) \qquad \mathrm{f o r}\r = 0 \quad ,
H^{( 0 )}= \sum_{r = \pm 1}\sum_{\vec{p}, p^{0 2}= \vec{p}^{2}}| p^{0}| ( b_{\vec{p}, r}^{( 0 ) +}b_{\vec{p}, r}^{( 0 )}- d^{( 0 ) +}{}_{\vec{p}, r}d_{\vec{p}, r}^{( 0 )}) .
-\frac{f^{\prime\prime}(\xi)h^{3}N}{12}\le\frac{b-a}{N}[\frac{f(a)+f(b)}{2}+\sum_{k=1}^{N-1}f(a+k\frac{b-a}{N})]-\int_{a}^{b}f(x)dx\le\frac{f^{\prime\prime}(\xi)h^{3}N}{12}.
n a_{1}( \vec{q}^{\prime}, \vec{q}) + \sum_{\nu = 1}^{3}( q_{\nu}^{\prime}- q_{\nu}) \frac{\partial a_{1}( \vec{q}^{\prime}, \vec{q})}{\partial q_{\nu}^{\prime}}= a_{1}( \vec{q}^{\prime}, \vec{q}^{\prime}) = - V ( \vec{q}^{\prime}) ,
\{\bar q^{\dot\alpha}_i, \bar s^j_{\dot\beta}\}\Phi = \left[ -\delta^j_i (\sigma^{\mu\nu})^{\dot\alpha}{}_{\dot\beta}m_{\mu\nu}-2\delta^{\dot\alpha}_{\dot\beta}\left(2t^i_j +\delta^j_i(\ell+r) \right)\right] \Phi = 0\;.
(\iota_{t})_{*}(e_{1}\wedge \cdots \wedge e_{n})=(e^{1}(t) \wedge \cdots \wedge e^{n}(t))(e_{1}, \cdots, e_{n})) \cdot (\iota_{t})_{*}(e_{1}(t) \wedge \cdots \wedge e_{n}(t)),
q_{2,1}'(0)=\sum\limits_{r=1}^{\min(n_1,\Delta n)}\sum\limits_{s=1}^{n_1}\sum\limits_{t=0}^{\Delta n}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s c'_s=n_1}}\sum\limits_{\substack{\mathbf{d}\in [n_1+1,n_2]_<^r \mathbf{d'}\in [n_1+2,n_2]_<^t d_1=n_1+1}}\det M_0(\{r+s\}',\{2r+s+1\}'),
\int fd\mu_?-\epsilon \leq\int f_1d\mu_?=\lim_{X\rightarrow\infty}\int f_1d\mu_{?,X}&\leq\liminf_{X\rightarrow\infty}\int fd\mu_{?,X}&\leq\limsup_{X\rightarrow\infty}\int fd\mu_{?,X}&\leq\lim_{X\rightarrow\infty}\int f_2d\mu_{?,X}=\int f_2d\mu_? &\leq\int fd\mu_?+\epsilon.