harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{n}(c)=\sum_{t=c}^{n-1}[\prod_{s=c}^{t-1}(\frac{s-1}{s})](\frac{1}{t+1})+[\prod_{s=c}^{n-1}(\frac{s-1}{s})]\frac{1}{2}=\frac{2cn-c^{2}+c-n}{2cn}.
Q = \int_{0}^{t}\bigg\{\int_{0}^{1}\chi_{\{x+z\in[0,1]\}}\bigg|\int_{0}^{1}\chi_{\{|x-y|> |z|\}}\bigg(\frac{\partial G}{\partial y}(t-s,x+z,y)-\frac{\partial G}{\partial y}(t-s,x,y)\bigg) & \times v^{\delta+1}(s,y)\d y\bigg|^p\d x\bigg\}^{\frac{1}p}\d s.
\partial_\nu \hat u_k(y)=\delta_k^\frac{n}{2}\nabla u_k (\delta_k y+e_n)\cdot \nu(y)=\delta_k^\frac{n}{2}\nabla u_k (\delta_k y+e_n)\cdot \frac{\delta_k y+e_n}{|\delta_k y+e_n|}= \delta_k^2 \nabla u_k (x)\cdot \frac{x}{|x|}=0.
{\left< T_{\sigma^{-}\sigma^{-}}^{\left( i n \right)}\left( \sigma^{-}\right) \right>}_{i n}= 0 -{\frac{1}{2 4}}D_{\sigma^{-}}^{S}\left( y^{-}\right) ={\frac{\lambda^{2}}{4 8}}\left( 1 -{\frac{1}{{\left( 1 + \lambda \Delta e^{\lambda \sigma^{-}}\right)}^{2}}}\right) ,
\enspace U \in H^{2}_{0}(\Omega_{1D}), \enspace \enspace \mathcal{P}\in H^{1}_{0}({\omega}): \langle A(U \mathcal{P}), U \upsilon \rangle^{q}= \langle f, U \upsilon \rangle^{q}\enspace \forall \upsilon \in H^{1}_{0}({\omega}).
c_{X,n}^{(i,j)}(t) = \sum_{\pi \in T_n^{(i,j)}}\mathrm{sgn}(\pi) ~ t^{\frac{\delta(\pi)-1}{2}}\prod_{\mathclap{\substack{k\in [n], ~ k\neq i k+\pi(k) \;}}}o_{1,X,k+\pi(k)}(t) \prod_{\mathclap{\substack{r \in [n] r+\pi(r) \;}}}e_{1,X,r+\pi(r)}(t).
\sum_{m=0}^\infty \frac{\beta_m}{m!}z^m = \sum_{m=0}^\infty\sum_{k=0}^m \frac{1}{(k+1)!}\frac{(-1)^{m-k}}{(m-k)!}z^m = \frac{1}{z}(e^z-1)e^{-z}= \sum_{m=0}^\infty \frac{(-1)^m}{(m+1)!}z^m
A_1 = \left\{\left.\frac{n}{2d}\,\right| 3\leq n< d\right\}, A_2 = \left\{\left.\frac{2n+1}{4d+2}\,\right| d\leq n\leq 2d\right\}, A_3 = \left\{\left.\frac{n}{2d}\,\right| d+3\leq n\leq 2d\right\}.
\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\lim_{\mu_{f},\mu_{g}\to 0}\int_{0}^{\infty}d|{\bf p}|\:{\tilde f}_{{\mu}_{f}}(|{\bf p}|)\cdot\frac{\partial}{\partial |{\bf p}|}{\tilde g}_{{\mu}_{g}}(|{\bf p}|).
\sum_{p_{n+1}\le X}\frac{\log^{\alpha}d_n}{d_n}= \frac{X}{\log^2X}(1+o(1))\int_{\log 2}^{\log \left(\log X(\log\log X)^{-1}\right)}u^{\alpha}\,du &+\frac{X}{\log^2X}\left(o(1)+O\left((\log\log X)^{\alpha}\log\log\log X\right)\right).
\tilde{\phi}(s)=\frac{2\pi{}N}{\left(\alpha{}+1\right)}\frac{\Gamma\left(s\right)\Gamma\left(\frac{1-s}{1+\alpha{}}\right)}{\Gamma\left(\frac{\alpha{}+\theta{}-\alpha{}s-\theta{}s}{2\left(\alpha{}+1\right)}\right)\Gamma\left(\frac{2+\alpha{}-\theta{}+\alpha{}s+\theta{}s}{2\left(\alpha{}+1\right)}\right)}.
M_{4g}= q^{2g+1}\left(1-\frac{1}{q}\right) \sum_{\substack{f \in \mathcal{M}_{\leq 4g}f= \square}}\frac{d_4(f)}{|f|^{\frac{3}{2}}}\phi(f) \sum_{\substack{C \in \mathcal{M}_{\leq y}C | f^{\infty}}}\frac{1}{|C|^2},
T^*:=\mbox{sup}\left\{t>0, \varepsilon ||u_\psi||_{B^\frac 32}+||\partial_yu_\psi||_{B^\frac 12}+ \varepsilon ||(Q_{11}, \varepsilon Q_{12})_\psi||_{B^\frac 32}+||\partial_y (Q_{11}, \varepsilon Q_{12})_\psi||_{B^\frac 12}\le \delta e^{-\mathcal R t}\right\}
\begin{aligned}\frac{|\tilde{x}-\tilde{y}|^4}{\varepsilon^4}+\alpha(|\tilde{x}|^2+|\tilde{y}|^2) \leq L(|\tilde{x}-\tilde{y}|+1)+M(\tilde{t}+1) &- u(x_1, t_1)+v(x_1, t_1)+2\alpha|x_1|^2+{\lambda\over T-t_1}-{\lambda\over T-\tilde{t}}.\end{aligned}
m_{0}(x,y) = H_{0}(x,y) m_{1}(x,y) = e^{2 \pi i x}H_{1}(x,y) m_{2}(x,y) = e^{4 \pi i x}H_{2}(x,y) m_{3}(x,y) = e^{6 \pi i x}H_{3}(x,y)
b_0^{m+1,k}=2\,a_0^{m,k}, b_j^{m+1,k}=2\,(j+1)\,a_j^{m,k}+2\,(j-\lambda-m-1)\,a_{j-1}^{m,k},j=1,\dots,m-k/2, b_{m+1-k/2}^{m+1,k}=-(2\lambda+k)\,a_{m-k/2}^{m,k}, c_0^{m+1,k}=2\,(\lambda+m)\,a_0^{m,k-1}, c_j^{m+1,k}=2\,(\lambda+m)\,a_j^{m,k-1}-2\cdot2j\,a_j^{m,k-1}&=2\,(\lambda+m-2j)\,a_j^{m,k-1}j=1,\dots,m+1-k/2,
E=\langle\frac{\partial}{\partial u},\frac{\partial}{\partial u}\rangle=1F=\langle\frac{\partial}{\partial u},\frac{\partial}{\partial v}\rangle=\langle\frac{\partial}{\partial v},\frac{\partial}{\partial u}\rangle=0G=\langle\frac{\partial}{\partial v},\frac{\partial}{\partial v}\rangle=e^{u}
C_\lambda(x,y)=\frac{1}{2\alpha\sqrt{\lambda}}\begin{cases}\alpha e^{i\sqrt\lambda (x+y)}, x\geq 0,\,y\geq 0, -e^{\sqrt{\lambda}(ix+y)}, x\geq 0,\,y<0, e^{\sqrt{\lambda}(x+iy)}, x<0,\,y\geq 0, -\overline{\alpha}e^{\sqrt{\lambda}(x+y)}, x<0,\,y<0,\end{cases}
\widehat{F}=(I+F_{S_{A}}+F_{S_{A}}K^{\ast}K)^{-1}(I+F_{S_{A}}K^{\ast}K) &=I-(I+F_{S_{A}}+F_{S_{A}}K^{\ast}K)^{-1}F_{S_{A}}&=I-F_{S_{A}}(I+F_{S_{A}}+K^{\ast}KF_{S_{A}})^{-1}.
u(w) =\hbar^{-1}\log \left(\tfrac{1+w \,\hbar/2}{1-w\,\hbar/2}\right)=w+\frac{w^3 \hbar^2}{12}+\frac{w^5 \hbar^4}{80}+\dots, \gamma(w) =(1-\tfrac14w^2\hbar^2)^{1/2}.
Q_\lambda \le C \sum_{i=1}^{d-\ell-1}A_{i-1}^{-C}{A_i}^{1-\frac1p -\frac{{\beta_{d-\ell}(\alpha-\ell)}}{q}}Q_{\lambda}+ C A^{-C}_{d-\ell-1}\lambda^{-\frac{\alpha-\ell}q}.
\mathrm{~ H_{n m}^{\{{\bf ~ R}\}}( y_{1}, y_{2}) ~}= R_{1 2}^{m}\sum_{k = o}^{\mu_{m n}}\left( \frac{R_{1 1}}2 \right)^{\frac{n - k}2}\frac{( - 1 )^{k}n ! m ! y_{1}^{m - k}}{( n - k ) ! ( m - k ) ! k !}H_{n - k}\left( \frac{R_{1 1}y_{1}+ R_{1 2}y_{2}}{\sqrt{2 R_{1 1}}}\right) .
G(s, z) = \frac{\Gamma(s - \tfrac{1}{2}+ z)\Gamma(s - \tfrac{1}{2}- z)}{\Gamma(s)\Gamma(s+k-1)}\mathcal{Z}(s,w,z) = \frac{\zeta(s + w -\frac{1}{2}+ z)\zeta(s + w -\frac{1}{2}- z)}{\zeta^*(1+2z)}.
\lambda^{2}\left(\frac{\left\Vert x\right\Vert +\mu}{c_{2}^{2}}\gamma\left(\frac{\lambda}{c_{2}}\right)-\frac{\mu}{c_{1}^{2}}\gamma\left(\frac{\lambda}{c_{1}}\right)\right)=\left(\left\Vert x\right\Vert +\mu\right)\gamma\left(\frac{\lambda}{c_{2}}\right)-\mu\gamma\left(\frac{\lambda}{c_{1}}\right).
\frac{1}{2}\|w_k\|^2-\int_{-1}^{1}H(w_k(x, 0))dx=c+o(1)\; &\|w_k\|^2-\int_{-1}^{1}h_1(w_k(x, 0))w_{k_1}(x, 0)dx-\int_{-1}^{1}h_2(w_k(x, 0))w_{k_2}(x, 0)dx={o(\|w_k\|)}.
\hat{R}^*_\mu(;-a,-b;q,t,T;p) =\hat{R}^*_\mu(;a,b;q,t,T;p), \hat{R}^*_\mu(;p/a,p/b;1/q,1/t,1/T;p) =\hat{R}^*_\mu(;a,b;q,t,T;p).
\left[{\frac{\hbar^{2}}{2}}\left( \int_{0}^{1}d s \, \sqrt{x^{\prime \, 2}}\right)^{- 1}\int_{0}^{1}d s \, \sqrt{x^{\prime \, 2}}\,{\frac{\delta^{2}}{\delta \sigma^{\mu \nu}( s ) \delta \sigma_{\mu \nu}( s )}}- m^{4}\right] \Psi [ C ] = 0 \ , \qquad m^{2}\equiv 1 / 2 \pi \alpha^{\prime}\ .
S=S_{K}-\int d^{4}x[g^{i\overline{j}}\partial_{i}W\partial_{\overline{j}}\overline{W}+\frac{1}{4}\psi^{i}\psi^{j}H_{ij}(W)+\frac{1}{4}\overline{\psi}^{\overline{i}}\overline{\psi}^{\overline{j}}H_{\overline{i}\overline{j}}(\overline{W})]
\tilde{T}_{m,n}=T_{2m,2n}&=\sum_{I\subset[N]}(-1)^{|I|}\tilde{\varphi}\left(2(m+n+\chi^I)\right) &=\sum_{I\subset[N]}(-1)^{|I|}\dot{\phi}\left(|m|+|n|+|I|\right) &=\sum_{k=0}^N \tbinom{N}{k}(-1)^k \dot{\phi}\left(|m|+|n|+k\right) &=\mathfrak{d}_1^N\dot{\phi}\left(|m|+|n|\right).
\begin{aligned}x_{1, l}= \max\{x \in \mathbb{Z}_{\ge 0}\mid \tilde{e}_{i_{1, l}}^x \tilde{e}_{i_{1, l-1}}^{x_{1, l-1}}\cdots \tilde{e}_{i_{1, 1}}^{x_{1, 1}}b \neq 0\}\end{aligned}
u_{t}=D_{1}\Delta u+f_{1}(u,v) \qquad\Omega\times(0,T) v_{t}=D_{2}\Delta v+f_{2}(u,v) \qquad\Omega\times(0,T) \partial_{n}u =0 = \partial_{n}v \qquad\partial\Omega\times(0,T)
\left\langle u_{t},\psi\right\rangle =\left\langle \Delta u,\psi\right\rangle +\left( (1-2u)\nabla u\nabla w+\mu\Delta w+g(u),\psi\right) \forall\psi \in H^{1}(\Omega)~\hbox{and a.a.}~t\in(0,T).
\bigg(\sup_{X_j}e^{-u(-v(\Psi))}\bigg)\int_{X_j}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|fF|^2_h \le \bigg(\sup_{X}e^{-u(-v(\Psi))}\bigg)\int_{X}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|fF|^2_h<+\infty.
x = i a - a + a - 6 \tilde{q}+ 2 r & \geq (i - 1) a + a - 6 k- 6 \left\lfloor \frac{j}{3}\right\rfloor & = (i - 1) a + a - \frac{a}{2}+ 2 - 6 \left\lfloor \frac{j}{3}\right\rfloor & = (3k+j-1) a + \frac{a}{2}+ 2 - 6 \left\lfloor \frac{j}{3}\right\rfloor,
\Big|(\mathcal{F}^{-1}m)(\mathbf x,\mathbf y)\Big| =c_k^{-1}\Big|\int_{B(0,1/t)}m(\xi)E(-i\xi,\mathbf{x})E(i\xi,\mathbf{y})\,dw(\xi)\Big| &\leq c_k^{-1}\|m\|_{L^{\infty}}\|E(i\cdot,\mathbf{x})\|_{L^2(B(0,1/t),dw)}\|E(i\cdot,\mathbf{y})\|_{L^2(B(0,1/t),dw)}&\leq C\|m\|_{L^{\infty}}w(B(\mathbf{x},t))^{-1/2}w(B(\mathbf{y},t))^{-1/2}.
\sum_{i \in I}v_{i}^{2}\Vert \pi_{uW_{i}}Cf\Vert^{2}\leq \Vert u\Vert^{2}\sum_{i \in I}v_{i}^{2}\Vert\pi_{W_{i}}u^{-1}Cf\Vert^{2}&=\Vert u\Vert^2\sum_{i\in I}v_i^2\Vert \pi_{W_i}C u^{-1}f\Vert^2 &\leq B \Vert u^{-1}\Vert^{2}\Vert u \Vert^{2}\Vert f\Vert^{2}.
\begin{array}{c c}{\left\{b ( \sigma , \tau ) , c ({\sigma}^{\prime}, \tau ) \right\}= 2 \pi \delta ( \sigma -{\sigma}^{\prime})}&{}\\{\left[ \beta ( \sigma , \tau ) , \gamma ({\sigma}^{\prime}, \tau ) \right] = 2 \pi \delta ( \sigma -{\sigma}^{\prime})}&{}\\ \end{array}
S [ \omega_{1}, \omega_{2}, F_{1}^{+ +}, F_{2}^{+ +}] ={\frac{1}{2 k^{2}}}\int d z^{( - 4 )}d u \lbrack ( F_{1}^{+ +})^{2}+ 2 F_{1}^{+ +}D^{+ +}\omega_{1}+ ( 1 \leftrightarrow 2 )
\widetilde{L}_{k,N}^{( 1^{2})}(z) = z^{N+1}( N+1-z)^{2}L_{k-1}^{N+1}(z) \hphantom{\widetilde{L}_{k,N}^{( 1^{2})}(z) =}{}+\left( kz^{N+2}- ( Nk+1 ) z^{N+1}+ ( k-1 )(N+1)!\sum_{j=0}^{N}\frac{z^{j}}{j!}\right) L_{k}^{N}(z) .
\sum_{s=1}^{k}\nu_s a_s(z)=0\quad\text{for some}\nu_s\in\mathbb{C}(q^{1/2})\setminus \left\{0\right\},\, a_s(z)\in\bigcup_{i=1}^{n}\mathcal{B}_{i,\psi}^{-}.
d s^{2}= \frac{k}{\cos ( p \ln g r ) \sin^{2}( p \ln g r )}( - d t^{2}+ d z^{2}) + \frac{3 k Z^{2}}{r^{2}}\cot^{2}( p \ln g r ) ( d r^{2}+ r^{2}d \Omega_{k}^{2}) \, ,
t\|Av''(t)\| + \|v''(t)\| + \|Av'(t)\| \leq C t^{\alpha-1}, \alpha=\min\limits_{1 \leq j \leq m}\{\alpha_j\}, t\|Au''(t)\| + \|u''(t)\| + \|Au'(t)\| \leq C e^{-\kappa t}t^{\alpha-1}, t \rightarrow 0^{+}.
\left(\mu_1^{\otimes(n_1 + m_1)}\otimes \ldots \otimes \mu_{k'}^{\otimes(n_{k'}+ m_{k'})}\right) = \left( \mu_1^{\otimes n_1}\otimes \ldots \otimes \mu_{k'}^{\otimes n_{k'}}\right) \otimes \left( \mu_1^{\otimes m_1}\otimes \ldots \otimes \mu_{k'}^{\otimes m_{k'}}\right),
\int_{\xi ( \pm \infty ) = 0}{\mathcal D}\xi \, \exp \{\frac{i}{\hbar}S [ \xi ] + i \int_{- \infty}^{+ \infty}d t \xi^{j}( t ) [ k^{j}\delta ( t - 1 ) + l^{j}\delta ( t ) ] \}= e^{- \frac{i \hbar}{2 b}\epsilon_{i j}k^{i}l^{j}}\; .
\langle (x_i,0), \tilde \Phi(-) (x_j,0)\rangle = \sum_{k=1}^{m_i}\sum_{l=1}^{m_j}\langle a_k^{(i)}\otimes b_k^{(i)}, (-)a_l^{(j)}\otimes b_l^{(j)}\rangle_E = \sum_{k=1}^{m_i}\sum_{l=1}^{m_j}b_k^{(i) \ast}\phi(a_k^{(i)\ast}(-) a_l^{(j)}) b_l^{(j)}.
[ E_0 \, \pi({F}_+) E_0, E_0 \, \pi({G}_+) E_0 ]= \lim_{T \rightarrow \infty}\! \mbox{$T^{-1}$}\! \int_0^T \! dt \, E_0 \, \pi ( [{F}_+, \alpha_t (G_+) ] ) E_0 = 0,
{\mathbf{\hat{h}}_{sd}^{(b)}}={\left(\textbf{R}_{d}^b\right)^{-1/2}\textbf{H}_{sd}^{}\textbf{w}_{0s}}, \quad\;{\mathbf{\hat{h}}_{sd}^{(m)}}={\left(\textbf{R}_{d}^m\right)^{-1/2}\textbf{H}_{sd}^{}\textbf{w}_{0s}}.
\frac{1}{\sqrt{- g}}{\cal L}= \frac{1}{2 \kappa}R - \frac{1}{2}g^{\mu \nu}\partial_{\mu}\phi \cdot \partial_{\nu}\phi - \frac{1}{2}\exp ( - 2 \frac{\phi}{f_{\phi}}) g^{\mu \nu}\partial_{\mu}a \cdot \partial_{\nu}a
{\cal{A}}_{n o n r e l a t}^{\, + + \, c l a s s}\, = \, i \frac{e^{2}}{m \theta}\frac{\vec{s}\wedge \vec{q}}{\vec{q}^{2}}\, + \, \frac{e^{4}}{4 \pi m \theta^{2}}\operatorname{l n}\Biggl ( \frac{- \Lambda^{2}}{\vec{p}^{2}}\Biggr )
\left . \frac{d}{d\gamma}\left(- \log \det(\Delta_{C_{\gamma}})\right) \right|_{\gamma = \alpha}= \left . \frac{d}{d\gamma}\left( - 2 \log \det(\Delta_{S_{\gamma}})\right) \right|_{\gamma = \frac \alpha 2}\frac 1 2 = \left . \frac{d}{d\gamma}\left( - \log \det(\Delta_{S_{\gamma}})\right) \right|_{\gamma = \frac \alpha 2}.
\int_{A+}^{(x-A)+}\overline{\xi}(x-u)\xi(du) \le c_1 \int_{A+}^{(x-A)+}g(x-u)\xi(du) \le c_1 g(x-A)\overline{\xi}(A+)-c_1\int_{A}^{x-A}g'(x-u)\overline{\xi}(u)du \le c_1^2g(x-A)g(A)+3c_1^2\int_{A}^{x-A}g(x-u)g(u)du.
C^{-1}=\frac{1}{3}\begin{pmatrix}2 1 1 2\end{pmatrix}C(z) =\begin{pmatrix}z+z^{-1}-1 -1 z+z^{-1}\end{pmatrix}\widetilde{C}(z) =\frac{1}{z^{2}+1+z^{-2}}\begin{pmatrix}z+z^{-1}1 1 z+z^{-1}\end{pmatrix}.
\O_{{C}_0}^{\oplus n_{j_0}}\xrightarrow{\varphi_{j_0}|_{{C}_0}}\mathcal{L}_{j_0}|_{{C}_0}\hookrightarrow \mathcal{L}_{j_0}|_{{C}_0}\otimes \O_{{C}_0}(\Sigma_{i=1}^l(\mathcal{L}_{j_0}|_{T_i})\cdot t_i).
=(0,1,1,1)^{\otimes|U|}[\begin{matrix}0&1\\ 1&0\end{matrix}]^{\otimes|E|}[\begin{matrix}0&1\\ 1&0\end{matrix}]^{\otimes|E|}([\begin{matrix}1\\ 0\end{matrix}]^{\otimes3}+[\begin{matrix}0\\ 1\end{matrix}]^{\otimes3})^{\otimes|V|}
\mathbb{E}_{H_f}[F^{(\tau)}\circ m] = \mathbb{E}_{\prod_{i=1}^n S_{r_i}}[F^{(\tau)}] = \prod_{i=1}^n \mathbb{E}_{S_{r_i}}[\prod_{\ell=1}^{\infty}(1+t_{d_i\ell})^{X_{\ell}}].
{\cal S}_{0}= \int d x^{0}d x^{3}\sum_{m = 0}^{\infty}\left( i \psi_{m}^{( L ) *}\partial_{+}\psi_{m}^{( L )}+ i \psi_{m}^{( R ) *}\partial_{-}\phi_{m}^{( R )}\right) + ( \mathrm{m a s s i v e m o d e s}) ,
\Lambda(z,\delta) = \sup \Big\{\oint_\gamma P_y(\gamma)dx-P_x(\gamma)dy\ :\ \gamma:[0,1] \to \mathbb{C}, \gamma(0)=\gamma(1)=z, |\gamma'(s)| \leq \delta\a.e., & \gamma'(s)=\alpha(s)+i\beta(s),\ \alpha,\beta \in FPWS[0,1] \Big\}.
\partial_{v_{b_{n}}}\partial_{\mu}\bigl[ \partial_{c}f(\mu,v_{1},...,v_{p}) \cdot e_{1}\otimes ... \otimes e_{i}\bigr](v_{p+1}) \cdot e_{i+1}\otimes e_{{n}}&=\partial_{\mu}\partial_{v_{b_{n}}}\bigl[ \partial_{c}f(\mu,v_{1},\cdots,v_{p}) , e_{1}\otimes ... \otimes e_{i}\bigr] (v_{p+1}) \cdot e_{{n}}\otimes e_{i+1},
m_d^{d-1}\left(R_{MN}-\frac{1}{2}g_{MN}R\right)+m_{d-1}^{d-2}g_M{}^\mu g_N{}^\nu\left(R^{(d-1)}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{(d-1)}\right)\delta(x^\perp)
\left. e^{- \tilde{E}_{\beta}\, ( a , q )}\leq e^{- E_{\beta}\, ( a , q )}\ \ \mathrm{w i t h}\ \ \E_{\beta}\, ( a , q ) = q a - \frac{n_{b}}{\beta}\, c \, e^{2 \pi | a |}\ . \right.
\int_{1}^{2}F\cdot ds=\int_{1}^{2}(F_{x}dx+F_{y}dy+F_{z}dz)=\int_{x_{1}}^{x_{2}}F_{x}dx+\int_{y_{1}}^{y_{2}}F_{y}dy+\int_{z_{1}}^{z_{2}}F_{z}dz,
D^{(1)}=4\sum_{1\leq a,b\leq r}\left(\Phi_1(a,b)-\Phi_3(a,b)\right)\left(\partial\Phi_1(a,b)-\partial\Phi_3(a,b)\right) &+2\sum_{1\leq a,b\leq r}\left(\Phi_2(a,b)-\Phi_2(b,a)\right)\left(\partial\Phi_2(a,b)-\partial\Phi_2(b,a)\right) &+2\sum_{1\leq a,b\leq r}\left(\Phi_4(a,b)-\Phi_4(b,a)\right)\left(\partial\Phi_4(a,b)-\partial\Phi_4(b,a)\right).
\hat{\tilde{p}}^\mu_j|x^\nu_a,y^\nu_a\rangle=\,\sum_a\sum_\rho \,\int_{TM}\,\tilde{\mu}_P(x',y')\,\tilde{\lambda}_l(x'^\nu_i,y'^\nu_i;x^\nu_a,y^\nu_a)\,|x'^\nu_i,y'^\nu_i\rangle .
(\theta, \boldsymbol u) \in \,\mathcal C([0, T], H^{s_1})\cap \mathcal C([0, T], H^{s_2}), (\theta_t, \boldsymbol u_t) \in L^2(0, T; H^{s_1-\beta})\cap L^2(0, T; H^{s_2-\alpha}).
w \equiv G^{-1}_{B_1 +B_2}(y) = R_{B_1 + B_2}+\frac{1}{y}=R_{B_1}+ R_{B_2}+\frac{1}{y}= G^{-1}_{B_1}(y) + G^{-1}_{B_2}(y)-\frac{1}{y}=2z-\frac{1}{y}
\begin{aligned}\frac{d}{dt}\int_0^\infty\left(1-\frac{r_1}{r_2}\right)^2dx\leq C(T)\int_0^\infty\left(1-\frac{r_1}{r_2}\right)^2dx+C(T)\int_0^\infty(u_1-u_2)^2dx.\end{aligned}
\lambda_\delta^{(2)}(x(t)) = \dfrac{\delta^2}{\left(x(t)^2+\delta^2\right)^{3/2}}= \dfrac{\delta^2}{\left(\xi_0^2t^2+\delta^2\right)^{3/2}}\left(1+\mathcal{O}\left(\dfrac{t^3}{t^2\xi_0^2+\delta^2}\right) \right) = \dfrac{\delta^2}{\left(\xi_0^2t^2+\delta^2\right)^{3/2}}+ \mathcal{O}(1).
\frac{\partial^{2}\bar{f}_{\epsilon}}{\partial \bar{t}^{2}}|_{P_{0}}= \frac{\partial^{2}\bar{f}_{\epsilon}}{\partial \bar{t}\partial \bar{x}}|_{P_{0}}= \frac{\partial^{2}\bar{f}_{\epsilon}}{\partial \bar{x}^{2}}|_{P_{0}}= 0
t_{v_1}^{\epsilon_1}t_{v_2}^{\epsilon_2}\cdots t_{v_n}^{\epsilon_n}[\mathcal{X}_1,\mathcal{Y}_1][\mathcal{X}_2,\mathcal{Y}_2] \cdots [\mathcal{X}_h,\mathcal{Y}_h] = \mathrm{id}.
I_{1 1}= \int d^{1 1}x \sqrt{- g}({\frac{R}{\kappa^{2}}}- 3 \hat{F}_{[ 4 ]}^{2}) +{2 8 8}\sigma \int \hat{F}_{[ 4 ]}\wedge \hat{F}_{[ 4 ]}\wedge \hat{A}_{[ 3 ]}
x \nu_y = q^2 \nu_y x, x^2\nu_z = q^{-2}\nu_z x, y \nu_z = q^2 \nu_z y, y^2\nu_x = q^{-2}\nu_x y, z \nu_x = q^2 \nu_x z, z^2\nu_y = q^{-2}\nu_y z.
V_{i}\cdot V_{j}= \left( \begin{array}{c c c}{{- 1}}&{{- 1 / 2}}&{{- 1 / 2}}\\{{- 1 / 2}}&{{- 1 / 3}}&{{- 1 / 3}}\\{{- 1 / 2}}&{{- 1 / 3}}&{{0}}\\ \end{array}\right) ~ .
\mathrm{P}\left( \bar{E}_{m,1}\right) = \mathrm{P}\left(\log\left(1+\frac{\alpha_m^2 x_m}{\beta_m^2x_m+\frac{1}{\rho}}\right)<R_{1,m}, \right. \log\left(1+\frac{\alpha_n^2 x_n}{\beta_n^2x_n+\frac{1}{\rho}}\right)>R_{1,n}, \left.\log\left(1+\beta_n^2 x_n\right)>R_{2,n}, \forall ~n\in \{1, \cdots, m-1\}\right).
\mathcal{L}_{I_B}(s) &= \prod_{t \in \mathcal{K}}\exp\Big( -2 \pi \lambda_t \frac{s P_t \Delta_{k, t}^{2 - \alpha}}{\alpha - 2}{}_2F_1(1, 1-\frac{2}{\alpha}; 2 - \frac{2}{\alpha}; -\frac{s P_t}{\Delta_{k, t}^{\alpha}})\Big),
\hat{K}\hat{K}^{-1}=1=\hat{K}^{-1}\hat{K},\hat{K}\hat{E}=q\hat{E}\hat{K},\hat{K}\hat{F}=q^{-1}\hat{F}\hat{K},\hat{E}\hat{F}-\hat{F}\hat{E}=\frac{\hat{K}^2-\hat{K}^{-2}}{q-q^{-1}}.
S_n := \int_{\mathbb{Z}_2}A_n^{(s)}\left(t+\frac{1}{4}\right) \mathrm{d}t ~\in \mathbb{Q}_2, T_n := \int_{\mathbb{Z}_2}B_n^{(s)}\left(t+\frac{1}{4}\right) \mathrm{d}t ~\in \mathbb{Q}_2.
\sigma =\frac{1+2\mu-\mu^2}{2},\lambda'=\sigma \log \varrho,H=\frac{h'}{\lambda '}+\frac{1}{\sigma}, \omega =2\left( 1+\sqrt{1+\frac{1}{4H^2}}\right),\theta=\sqrt{1+\frac{1}{4H^2}}+\frac{1}{2H}.
E(v;k,R) = \int_{B_{R}(0)}|\nabla v|^{2}+ \int_{B_{R}(0)}v^{10/3}- \int_{B_{R}(0)}\left( m_{k}* Y_{a_{k}}\right) v^{2}\bigg. & + \frac{1}{2}\int_{B_{R}(0)}\left( v^{2}\cdot \chi_{B_{R}(0)}* Y_{a_{k}}\right) v^{2}+ \int_{B_{R}(0)}\left( u_{k}^{2}\cdot \chi_{B_{R}(0)^{\rm c}}* Y_{a_{k}}\right) v^{2}.
W' = \left(\left(\left((q+j)_{j=0}^{r-1}\right)_{q=1}^{m-1}\right)_{p=1}^{r-1}, \left(\left((m+j)_{j=0}^{p-1},(j)_{j=p}^{r-1}\right)\right)_{p=1}^{r-1}, \left((q+j)_{j=0}^{r-1}\right)_{q=0}^{m-1},(m+j)_{j=0}^{r-1}\right).
\tilde Vf=-\dfrac{1}{x}\sum_{\ell =-\infty}^\infty x^{\ell}\sum_{r=0}^\infty\Bigl(\partial_\theta^r\theta^{-\ell}\bigm|_{\theta=1}\Bigr)[w^r y^{-\ell}](-y)^{r+1}e^{w/y}f(w,y^{-1}).
\omega_0(\Gamma^X_{\gamma_2}\Gamma^Z_{\widetilde\gamma}\Gamma^X_{\gamma_1})=-\omega_0(\Gamma^X_{\gamma_2}\Gamma^X_{\gamma_1}\Gamma^Z_{\widetilde\gamma})=-1\neq\omega_0(\Gamma^X_{\gamma_1}\Gamma^X_{\gamma_2}\Gamma^Z_{\widetilde\gamma}).
D_{- 1}\mapsto \left( \begin{array}{r r}{0}&{1}\\{0}&{0}\\ \end{array}\right) , \qquad D_{0}\mapsto \left( \begin{array}{r r}{{\textstyle \frac 1 2}}&{0}\\{0}&{-{\textstyle \frac 1 2}}\\ \end{array}\right) , \qquad D_{1}\mapsto \left( \begin{array}{r r}{0}&{0}\\{- 1}&{0}\\ \end{array}\right)
\mathbb{E}(K(x,t)^4) =\frac{\beta^4}{16N}\biggl[ \sum_{z,r}\Delta(z,r)^4 (\mu_4 - \mu_2^2) + \biggl(\sum_{z,r}\Delta(z,r)^2 \mu_2\biggr)^2\biggr] + o(N^{-1}) = \frac{\beta^4V}{16Nc}+ o(N^{-1}).
\begin{array}{ll}K_{N}(x,y) = \int \int \int \psi \left( \frac{\xi}{N}\right) \frac{\chi_{\frac{\mu_{4}}{4}}(z)}{\langle \eta \rangle^{100n}}e^{i \left( \xi \cdot x - \eta \cdot y - (\xi - \eta) \cdot z \right)}\, d \eta \, d \xi \, dz\end{array}
l^{2}\tilde{R}_{\, j}^{i}+ ( d - 2 ) h_{\, j}^{i}+ h_{\, k}^{k}\delta_{\, j}^{i}- y ( 2 \partial_{y}h_{\, j}^{i}+ h_{\, k}^{k}h_{\, j}^{i}) = 0
\gamma^{0}= \left( \begin{array}{r r}{0}&{- i}\\{i}&{0}\\ \end{array}\right) \gamma^{1}= \left( \begin{array}{r r}{0}&{i}\\{i}&{0}\\ \end{array}\right) \gamma^{2}= \left( \begin{array}{r r}{i}&{0}\\{0}&{- i}\\ \end{array}\right)
\|B^k\|_{L^{2}(0,T; H^{\frac52-\frac{\alpha}{2}})}\leq \C(T) \|B(0)\|_{H^{\frac{5}{2}-\alpha}}+C \|B^{k-1}\|_{L^{\frac{\alpha}{(s+\alpha-\frac52)}}(0,T; H^s)}\|B^{k}\|_{L^{\frac{\alpha}{(s+\alpha-\frac52)}}(0,T; H^s)}
\int fd(\mathfrak{T}^{n}\nu)=\sum_{\xi_0,\dots ,\xi_{n-1}}p_{\xi_{0}}p_{\xi_{1}}\ldots p_{\xi_{n-1}}\, \int f \, d T_{\xi_{0}*}T_{\xi_{1}*}\ldots_\ast T_{\xi_{n-1}*}\nu.
\left\{\left( \begin{array}{c}{u}\\{0}\\ \end{array}\right) , \left( \begin{array}{c}{v}\\{0}\\ \end{array}\right) \right\}= \left\{\left( \begin{array}{c}{0}\\{u}\\ \end{array}\right) , \left( \begin{array}{c}{0}\\{v}\\ \end{array}\right) \right\}= 0
e_k^{(0)}- f_k^{(0)}= \Pi^{(0)}f_k^{(0)}- f_k^{(0)}& = \frac{1}{2 \pi i}\int_\gamma (z - P^{(0)})^{-1}f_k^{(0)}dz - \frac{1}{2 \pi i}\int_\gamma z^{-1}f_k^{(0)}d z & = \frac{1}{2 \pi i}\int_\gamma (z - P^{(0)})^{-1}z^{-1}P^{(0)}f_k^{(0)}d z .
\begin{array}{r@{\;}c@{\;}l@{\qquad}r@{\;}c@{\;}l}Gz^\mu = \rho^\mu,\qquad\hfil G\bar z^{\bar\mu}= 0,\hfil G\bar\eta^{\bar\mu}= d\bar z^{\bar\mu}, G\chi_\mu = \kappa tg_{\mu\bar\mu}*d\bar z^{\bar\mu}-\Gamma^\lambda_{\mu\nu}\rho^\nu\chi_\lambda, G\rho^\mu = -\displaystyle \frac{1}{2}\Gamma^\mu_{\nu\lambda}\rho^\nu\rho^\lambda. \end{array}
H_{g o n i h e d r i c}^{3 d}= - \frac{1}{2}\sum_{\vec{r}, \vec{\alpha}, \vec{b e t a}}\sigma_{\vec{r}}\sigma_{\vec{r}+ \vec{\alpha}}\sigma_{\vec{r}+ \vec{\alpha}+ \vec{\beta}}\sigma_{\vec{r}+ \vec{\beta}}
\partial^T\Omega:=\cap_{i=1}^{n_0}\{x_i=0\}\cap\partial\Omega,\partial^{\pitchfork}\Omega:=\cap_{j=n_0+1}^n\{x_j=0\}\cap\partial\Omega,\underline{\Omega}:=\Omega\cup\partial^{\pitchfork}\Omega,
\left\langle\nabla F(X^*)^TX^*, I_n-X^TX^*\right\rangle = \left\langle (X^*)^T\nabla F(X^*), (X^*)^TX^*-X^TX^* \right\rangle &= \left\langle X^*\nabla F(X^*)^TX^*, X^*-X \right\rangle &= -\langle \nabla F(X^*), X-X^*\rangle.
\begin{aligned}\mathcal{RVW}:\mathcal{T}^r\times\mathcal{C}_k(P)\to\mathcal{C}_{k-1}'(P), \mathcal{RVW}(\tau^1,\tau^2,\tau^3,\theta_1,\theta_2,A,B,C) :=\begin{pmatrix}d_A^*B+d_AC+\tau^1(B\centerdot\theta_1+[B\centerdot B]\centerdot\theta_2) F_A^++\frac{1}{8}\tau^2[B\centerdot B]+\frac{1}{2}[B,C]+\tau^3B\end{pmatrix}, \end{aligned}
\hat{\mathrm{H}}_{\mathrm{F}}^{\mathrm{p h y s}}= \hat{\mathrm{H}}_{0 , +}+ \hat{\mathrm{H}}_{0 , -}- \frac{1}{2}( \xi_{+}+ \xi_{-}) - \frac{\pi}{\pi}{L}{\hbar}( 1 + \mathrm{N}^{2}) ( [ \frac{e_{+}b \mathrm{L}}{2{\pi}{\hbar}}] )^{2}
\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.
[\iota_t(\delta), f\partial_t]= \iota_t( \delta) (f)\partial_t , \,\,[\iota_t(\delta_1), \iota_t(\delta_2)]=\iota_t [\delta_1, \delta_2], \,\,\,\,[\iota_t(\delta), r\bar{C}]=\delta(r)\bar{C}\,\,.
\begin{array}{r c l}{\varphi_{1}^{+}\varphi_{2}^{+}\varphi_{4}^{-}}&{=}&{\varphi_{4}^{-}\varphi_{1}^{+}\varphi_{2}^{+}+ D_{2 4}\varphi_{1}^{+}+ D_{1 4}\varphi_{2}^{+},}\\ \end{array}
\vert u(t_{1},x_{1})-u(t_{2},x_{2})\vert \leq\underset{Q((t_{2},x_{2}),(1-\delta)^{K}R)}{Osc}u\leq\theta^{K-1}\underset{Q((t_{2},x_{2}),(1-\delta)R)}{Osc}u \leq\theta^{K-1}\underset{Q((t_{0},x_{0}),R)}{Osc}u=\theta^{-2}(\theta^{K+1})\underset{Q((t_{0},x_{0}),R)}{Osc}u.
\begin{array}{ccl}\mathbf u_k = [\begin{array}{ccc}u(0) \ldots u(n+2nu-1)\end{array}]^T \mathbf y_k = [\begin{array}{ccc}y(1) \ldots y(n+2nu)\end{array}]^T \mathbf r = [\begin{array}{ccc}r(1) \ldots r(n+2nu)\end{array}]^T\end{array}
\int_{\{\Psi<-t_1\}}|\tilde{F}_1-(1-b_{t_0,B}(\Psi))fF|^2_{\tilde{h}}e^{v_{t_0,B}(\Psi)}c(-v_{t_0,B}(\Psi)) \le \left(\int_{t_1}^{t_0+B}c(s)e^{-s}ds\right) \int_{M}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}{|fF|}^2_{\tilde{h}}.
\lim_{\epsilon \to 0}\limsup_{t\to \infty}\mathbb{P}\left(\exists u \in \mathcal{N}^2_t:X_u(t)\geq m_t-A, X_{u}(T(u))- \sqrt{2}T(u)\notin[-\frac{1}{\epsilon}\sqrt{t}, -\epsilon \sqrt{t}]\right)=0
\begin{gathered}\Big( \bigcap\limits_{j}V_j\big((a^1_{1}, \ldots, a^{k_1}_1), \ldots, (a^1_j,\ldots, a^{k_j}_j), \ldots, (a^1_r, \ldots, a^{k_r}_r)\big) \Big) \geq \Big(\varphi^{-1}(\underbrace{a,\ldots \ldots \ldots,a}_\mathrm{\sum\limits_{i,j}e^i_j -r})\Big) = \sum\limits_{i,j}e^i_j -r\end{gathered}.
\det(Y)^{k/2}|F(Z)| \ll_{n,\epsilon}\begin{cases}k^{\frac{3n(n+1)}{8}}\det(Y)^{\frac{n+1}{4}+\epsilon}k^{\frac{n(n+1)}{4}}\det(Y)^{\frac{3(n+1)}{8}+\epsilon}.\end{cases}
v''= 2 (|h'|^2 + \langle h'', h \rangle ) = 2 \Bigl(|h'|^2 + \gamma \langle h', h \rangle + \lambda |h|^2 + \langle e^{(\gamma-\delta)t}V(t) h, h\rangle\Bigr) &= 2|h'|^2 + \gamma v' + 2 \Bigl(\lambda v - \langle e^{(\gamma-\delta)t}V(t) h, h\rangle\Bigr) &\ge 2|h'|^2 + \gamma v' + 2 \Bigl(\lambda - e^{(\gamma-\delta)t}\|V\|_\infty \Bigr)v.
\underline{M}_{x}=L^{0}\left( \underline{N}_{x}L^{x}+ \sum_{y\leq^py<x}\underline{N}_{y}L^{y,x}+ \sum_{y\leq^1y<x}\sum_{i=1}^{\ell}\vartheta^i(\underline{N}_{y})L_i^{y,x}\right).
\min\limits_{x\in\mathcal{X}_+,s\in\mathcal{S}}\frac{p_{\hat{X}|S}(x|s)}{\sum_{s'\in\mathcal{S}}p_{\hat{X}|S}(x|s')}\geq\frac{e^{-1}}{e^{-1}+(|\mathcal{S}|-1)(1-e^{-1})}&=\frac{1}{(|\mathcal{S}|-1)e-|\mathcal{S}|+2}.
dZ_1(t) = \sigma_X(Z_1(t)) dB_X(t) + b(Z_1(t)) dt + d\Phi_1(t) dZ_2(t) = \sigma_Y(Z_2(t)) dB_Y(t) + \Big(\gamma(Z_2(t)) + \langle \rho(Z_2(t)), \nabla_{z_2}[\log \Lambda(Z_2(t), Z_1(t))] \rangle\Big) dt + d\Phi_2(t) + d\Phi(t),