harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{1}=\sum_{l}\partial_{\alpha,\beta}(w_{l}e_{p_{l}})w'_{l}F_{2}=\sum_{l}w_{l}\partial_{\alpha,\beta}(e_{p_{l}}w'_{l}) F_{3}=\sum_{l}D_{\alpha,\beta}(w_{l},p_{l},w'_{l}).
\mathcal F_{aabb}(v) =\mathcal F_{**00}(v)\cup\mathcal F_{00**}(v), \mathcal F_{abab}(v) =\mathcal F_{*0*0}(v)\cup\mathcal F_{0*0*}(v), \mathcal F_{abba}(v) =\mathcal F_{*00*}(v)\cup\mathcal F_{0**0}(v).
\rho_K(\xi) = \|\xi\|_K^{-1}= |D\cap \xi^\bot|= \frac 1{n-1}\int_{S^{n-1}\cap \xi^\bot}\|\theta\|_D^{-n+1}d\theta =\frac 1{n-1}R\left(\|\cdot\|_D^{-n+1}\right)(\xi),
\Psi_{2N+\nu}(a,b,c,d) =\sum_{i=0}^N \genfrac{[}{]}{0pt}{}{N}{i}_{Q}(-a;Q)_{i+\nu}(-abc;Q)_{i}\frac{(ac;Q)_{N+\nu}}{(ac;Q)_{i+\nu}}(ab)^{N-i}, \Phi_{2N+\nu}(a,b,c,d) =\frac{1}{(Q;Q)_{N}}\sum_{i=0}^N \genfrac{[}{]}{0pt}{}{N}{i}_{Q}\frac{(-a;Q)_{i+\nu}(-abc;Q)_{i}}{(ac;Q)_{i+\nu}}(ab)^{N-i},
l^{*},R^{*},t^{*}=argmin_{l>0,R\in SO(3),t\in\mathbb{R}^{3}}\sum_{i=1}^{N}min(\frac{1}{\sigma_{i}^{2}}\|s_{i}-lRm_{i}-t\|_{2}^{2},\overline{c}^{2})
i^{N}(h(x)-h(y))^{\alpha}K_{s,\phi,\theta}(x,y)=\frac{\phi(x)\theta(y)}{(2\pi)^d}\int_{\mathcal{O}}e^{ik\cdot (h(x)-h(y))}\nabla^{\alpha}_{k}K_s(k,\pi(x), \pi(y))dk.
E_\varepsilon[\varphi](x):=\frac{1}{\varepsilon^{2n}}\sum_{j=1}^l\mu_j\int_{\mathbb R^n}\Theta\biggl(\frac{y-x}{\varepsilon}\biggr)\int_{\mathbb R^n}\Theta\biggl(\frac{z-y}{j\varepsilon}\biggr)\varphi(z)\,dz\,dy, x\in\mathbb R^n.
\underline{\rm Hom}({\bf L}^{n},M(X))\otimes{\bf L}^n \rightarrow \underline{\rm Hom}({\bf L}^{n-1},M(X))\otimes{\bf L}^{n-1}\rightarrow \cdots \rightarrow \underline{\rm Hom}({\bf 1},M(X))=M(X).
{\frac{d_{o e}A_{e e}}{1 - d_{o o}A_{o o}}}= -{\frac{B_{e , k_{0}}D_{1}}{B_{o , k_{0}}D_{2}}}={\frac{B_{e , k_{0}}D_{2}}{B_{o , k_{0}}D_{1}}}, \Rightarrow D_{1}^{2}= - D_{2}^{2}
Ric(T,T) =-m\frac{\bar{u}_{\bar{t}\bar{t}}}{\bar{u}}-(n-1)\frac{\bar{g}_{\bar{t}\bar{t}}}{\bar{g}}\ge(n-1)l_i^2-m(\frac1N+\frac1{N^2})l_i^2 \ge\frac m2 l_i^2-m(\frac15+\frac1{25})l_i^2 >0,
\|x_0+h\|_1 = \|x_0+h_I\|_1+\|h_{I^c}\|_1\geq \|x_0\|_1-\|h_I\|_1+\|h_{I^c}\|_1 = \|x_0\|_1-2\|h_I\|_1+\|h\|_1 \geq \|x_0\|_1-2\sqrt{|I|}\|h_I\|_2+\|h\|_1 > \|x_0\|_1.
A_{4}= \int_{- L / 2}^{L / 2}\! \! \! \! d x \left[ \left. \mathrm{I m}\left( \phi \partial_{x}\bar{\phi}\right) \right|_{t = - T / 2}- \left. \mathrm{I m}\left( \phi \partial_{x}\bar{\phi}\right) \right|_{t = T / 2}\right] .
Hess V(\nabla V,\nabla V)(p) = 0, (\nabla Hess V)(\nabla V,\nabla V,\nabla V)(p) = Ric(\nabla V,\nabla V)(p) -\epsilon n, (\nabla^{2}Hess V)(\nabla V,\nabla V,\nabla V,\nabla V)(p) = 0.
\left( \begin{array}{c}{A_{k}^{\mu}\left( \theta \right)}{\tilde{A}_{k}^{\mu \dagger}\left( \theta \right)}\end{array}\right) ={\cal B}_{k}\left( \begin{array}{c}{A_{k}^{\mu}}{\tilde{A}_{k}^{\mu \dagger}}\end{array}\right) ,
\left\{x^{\alpha}, p_{\beta}\right\}_{D ( \Phi_{3 n}^{( 1 )})}= \delta_{\beta}^{\alpha}\, , \, \, \, \, \, \, \, \left\{\psi^{n}, \psi^{m}\right\}_{D ( \Phi_{3 n}^{( 1 )})}= \frac{i}{2}\eta^{n m}\, ,
(E_1 F_1 + E_2 F_2) \otimes 1 = -\frac{1}{4}(2 CC^{-}+ 2 C^{-}C + 3(h_1 + h_2) \otimes (E_1 F_1 + E_2 F_2) & + 2 E \otimes E_2 F_1 + (h_1 - h_2) \otimes (E_1 F_1 - E_2 F_2) + 2 F \otimes E_1 F_2).
\tilde{D}_{T}^{[ r , 0 ]}( x_{M}^{2}) \rightarrow \frac{\chi_{r , 0}\Lambda^{\frac{1}{2}}}{\left( 2 \pi \sqrt{x_{M}^{2}}\right)^{\frac{3}{2}}}\, e^{- \gamma_{r , 0}\sqrt{x_{M}^{2}}}\cos \left( \omega_{r , 0}\sqrt{x_{M}^{2}}+ \varphi_{r , 0}\right) \, ,
\tilde{f}(\tilde{z}) = \sum_{b \in A}\tilde{f}(\tilde{z})\phi_b(x) & = \sum_{b \in A}\tilde{f}(\tilde{z})\phi_b(x) + \left( f(x) - \sum_{b\in A}\tilde{f}(\tilde{b})\phi_b(x) \right) & = \sum_{b \in A}(\tilde{f}(\tilde{z}) - \tilde{f}(\tilde{b}))\phi_b(x) + f(x).
\left[ - \frac{d^{2}}{d x^{2}}+ m^{2}\right] \xi_{\pm}( x ) = \omega^{2}\xi_{\pm}( x ) \quad ; \quad \xi_{\pm}^{\prime}( x ) |_{x = 0}= \mp \frac{m}{A}\xi_{\pm}( 0 ) \quad .
\begin{aligned}q \neq 1: \ \ \ \Delta(K)|J,m \rangle_q = q^{m}|J,m \rangle_q , \ \ \Delta(C_q) |J,m \rangle_q = [J]_q[J+1]_q |J,m \rangle_q , q= 1: \ \ \ \Delta(H)|J,m \rangle_{q=1}= m |J,m \rangle_{q=1}, \Delta(C_1) |J,m \rangle_{q=1}= J(J+1) |J,m \rangle_{q=1}.\end{aligned}
u(\mathsf{O}s) = \mathsf{O}(\overline{u}s), (\mathsf{O}s) u = \mathsf{O}(us u), u(\mathsf{O}s)u = \mathsf{O}(s u), u(s\mathsf{O}) = (us u)\mathsf{O}, (s\mathsf{O}) u = (s \overline{u})\mathsf{O}, u(s\mathsf{O})u = (us)\mathsf{O}.
\{\nabla_{\alpha}, \nabla_{\beta}\}= c_{\alpha \beta}^{\enspace \enspace \gamma}\nabla_{\gamma}\qquad \{\bar{\nabla}_{\dot{\alpha}}, \bar{\nabla}_{\dot{\beta}}\}= c_{\dot{\alpha}\dot{\beta}}^{\enspace \enspace \dot{\gamma}}\bar{\nabla}_{\dot{\gamma}}
DF_p(x,y,z) = x F_x(p) + y F_y(p) + z F_z(p), D^2F_p(x,y,z) = x^2 F_{xx}(p) + y^2 F_{yy}(p) + z^2 F_{zz}(p) + 2xy F_{xy}(p) + 2xz F_{xz}(p) + 2yz F_{yz}(p).
\int_{\mu_{1}}^{\infty}dx\, \left(x^{2}-\mu_{1}^{2}\right)^{\nu -1}e^{-\mu_{2}x}=\frac{1}{\sqrt{\pi}}\left(\frac{2\mu_{1}}{\mu_{2}}\right)^{\nu -1/2}\Gamma \left(\nu \right)K_{\nu -\frac{1}{2}}\left(\mu_{1}\mu_{2}\right),
\eta^{i}(X)=g(X,\xi_{i}),\begin{array}{cc}\begin{array}{c}\end{array}\end{array}g(X,\varphi Y)=-g(\varphi X,Y),\begin{array}{cc}\end{array}\varphi \xi_{i}=0,\begin{array}{cc}\end{array}\eta^{i}\circ \varphi =0.
Z_{n}^{j\ell}\bigg(\frac{R_{n+1}^{j\ell}}{\overline e_{n+1}^j.e_{n+1}^\ell-1}-\frac{R_{n}^{j\ell}}{\overline e_{n}^j.e_{n}^\ell-1}\bigg) = \frac{Z_{n}^{j\ell}}{\overline e_{n}^j.e_{n}^\ell-1}(R_{n+1}^{j\ell}-R_{n}^{j\ell}) + O^P({T}^{-2})
D_i^{\prime}=\bigoplus_{\substack{i+1 \leq j \leq n-1 1 \leq k \leq j-i}}D_j^{(k)}, D_i^{+}= \bigoplus_{\substack{i+2 \leq j \leq n-1 2 \leq k \leq j-i}}D_j^{(k)}, D_i^{-}= \bigoplus_{\substack{i+2 \leq j \leq n-1 1 \leq k \leq j-i-1}}D_j^{(k)}.
Y_\mathrm m = \left[\begin{array}{ccccc}y_\mathrm{m}(0) y_\mathrm{m}(1) \cdots y_\mathrm{m}(m-1) y_\mathrm{m}(1) y_\mathrm{m}(2) \cdots y_\mathrm{m}(m) \vdots \vdots \vdots y_\mathrm{m}(n-1) y_\mathrm{m}(n) \cdots y_\mathrm{m}(m+n-1) \end{array}\right],
u(\tau_2) - u(\tau_1) = v(\tau_2) / 2- v( \tau_1) / 4 \geq v(\tau_1) / 4\geq 2^{-2}\cdot 2^k \tau_1 \geq 2^{-2 + k}\cdot 2^{-3k - 3}= 2^{-2k -5}.
\frac{1}{2 \pi i}\int_{c - i T}^{c+i T}\frac{\Gamma(s)}{\zeta(k-2s)}x^{-s}{\rm d}s= \frac{1}{2 \pi i}\int_{\lambda - i T}^{\lambda+i T}\frac{\Gamma(s)}{\zeta(k-2s)}x^{-s}{\rm d}s - R_0 - \mathcal{R}(x),
\|\widehat{q_t}(\theta)-q_t(\theta)\|_\Theta\leq C\big( 1+ |X_t |+\|{f}_\theta^t\|_\Theta \big)\big(\|\widehat{M}_\theta^t-{M}_\theta^t\|_\Theta+\|\widehat{f}_\theta^t-{f}_\theta^t\|_\Theta\big).
n^{-3}\sum_{q\geq 0}d_{2q+1}^2 (2q+1)! \sum_{i,j,k,\ell = 1}^n \rho(i-k)^{2q+1}&= \sum_{q\geq 0}d_{2q+1}^2 (2q+1)! \sum_{\vert i \vert < n}\frac{n-\vert i\vert}{n}\rho(i)^{2q+1}\to \sum_{q\geq 0}d_{2q+1}^2 (2q+1)! \sum_{i\in\mathbb{Z}}\rho(i)^{2q+1},
f_i(q) = \frac{1}{i!n^i}\Big((q-1)^i G_{h_i,n}(q)- \sum_{m_1=0}^{i-1}\sum_{m_2=m_1}^{i}\binom{i}{m_2}f_{m_1}^{(i-m_2)}(q)R_{n,m_1,m_2}(q)(q-1)^{i-m_2}q^{i-m_2}\Big),
{\hat{\cal K}}^{(2)}=2\partial{\hat \omega}^{(1)}+l_p^{-2}\, D{\hat X}^{\hat \mu}D{\hat X}^{\hat \nu}(i_{\hat k}{\hat C})_{{\hat \mu}{\hat \nu}}-m l_p^2 \, \partial{\hat \omega}^{(0)}{\hat b}\, .
\frac{1}{\zeta(2)}\sum_{l<M^{1-\Upsilon}}\frac{V_1(l/q)}{l}\bigg( \delta_{n,l^2}- 2\pi \sum_{c\,\equiv\,0\,(\mod \hskip-3ptq)}\frac{S(l^2,n;c)}{c}J_{1}\left(\frac{4\pi l\sqrt{n}}{c}\right) \bigg) .
\chi_{r,s}^{(p,p+1)}(q) = \frac{q^{\Delta_{r,s}^{(p,p+1)}-c/24}}{(q)_{\infty}}\sum_{j=-\infty}^{\infty}\left\{q^{p(p+1)j^2+[(p+1)r-ps]j}-q^{p(p+1)j^2+[(p+1)r+ps]j+rs}\right\}
E\big[|I^{\prime}_2(t)|^2\big] = (1-\lambda)E\big[|I_2(t)|^2\big] \approx \frac{(1-\lambda)^2\alpha}{\lambda}\cdot \frac{M^2}{\tau}\cdot \beta_{1,k'}^2 + (1-\lambda) M \beta_{1,k'}\beta_{1,k}.
\begin{cases}(u_{h,t}(t), v_h)_H + a(u_h(t),v_h) + b(v_h, p_h(t)) = 0, \forall v_h \in V_h, b(u_h(t), q_h) = 0, \forall q_h \in Q_h, u_h(0) = P_{h,\sigma}u_0,\end{cases}
\mu\left|\int_{\Omega}\chi\partial_{xx}l \,dxdt\right| =\mu\left| \int_{\Omega}\chi_x\partial_xl \,dx dt\right| &\le \sqrt{\mu}\left(\int_{supp(\chi)}\mu (\partial_x l)^2\,dxdt \right)^{1/2}\left(\int_{\Omega}(\chi_x)^2 dxdt \right)^{1/2}&\to 0\qquad\,\mu\to0+.
\frac{1}{4\pi^2}\int_{^{(2)}_{\texttt{b}}}f(X_1,X_2) \frac{\sqrt{\rho_{\texttt{b}}(X_1,X_2)}}{O_{\texttt{b}}(X_1,X_2;q,q_0)}X_1X_2 &= \sum_{m\geq \lambda_1\geq \lambda_2\geq 0}f\bigl(\boldsymbol{X}^{(m,2)}_{\texttt{b};(\lambda_1,\lambda_2)}\bigr) \hat{\Delta}^{(m,2)}_{\texttt{b};(\lambda_1,\lambda_2)},
\begin{cases}|\mathcal{I}^-|\geq 2\ \Longrightarrow\ |\mathcal{I}^+|=1\ \ \bigcup\mathcal{I}^+=\widetilde{J}\setminus J=\max O_{\widetilde{J}}|\mathcal{I}^+|\geq 2\ \Longrightarrow\ |\mathcal{I}^-|=1\ \ \bigcup\mathcal{I}^-=\widetilde{J}\setminus J=\min O_{\widetilde{J}}. \end{cases}
(\Gamma_{|k|\geq K}X)(v)=\sum_{\mathbf{v}\in V}\sum_{|k|\geq K,i,\alpha,\beta}X^{(\mathbf{v})}_{k,i,\alpha,\beta}e^{\mathbf{i}k\cdot x}y^{i}z^{\alpha}\bar{z}^{\beta}\partial_{\mathbf{v}},\Gamma_{|k|<K}:=\mbox{Id}-\Gamma_{|k|\geq K}
\dfrac{d^{k}}{dx^{k}}\phi_{2\lambda}\left( x\right) =\dfrac{s\beta_{2}}{\delta}\cos \left( s\left( c_{1}-a\right) \right) \dfrac{d^{k}}{dx^{k}}\left( \sin s\left( x-c_{1}\right) \right) +O\left( e^{t\left( x-a\right)}\right) ,
Z(C_{m,n}) = Z(C(a_d,a_{d-1},\ldots,a_3,a_2,a_1,a_1,a_2,a_3,\ldots,a_{d-1},a_d)) = Z(C(a_d,a_{d-1},\ldots,a_3,a_2,a_1)\cdot Z(C(a_1,a_2,a_3,\ldots,a_{d-1},a_d) +Z(C(a_d,a_{d-1},\ldots,a_3,a_2))\cdot Z(C(a_2,a_3,\ldots,a_{d-1},a_d) = m^2+n^2.
\mathrm{\Pi_{\m u \n u}^{a f}( p ) = \frac 1{3 !}\int \! \! \int \frac{d^{4}q ~ d^{4}k}{( 2 \p i )^{8}}\, V_{\m u \l a m b d a \s i g m a \r h o}^{a b c d}\, G_{\r h o \a l p h a}^{d g}( q ) \, G_{\s i g m a \b e t a}^{c h}( p - k - q ) \, G_{\l a m b d a \x i}^{b e}( k ) \, V_{\a l p h a \b e t a \x i \n u}^{g h e f}~ ,}
\det(C_{>0,(n)}^+)=\sum_{\phi:\{1,2,\cdots,n\}\to \{1,2,\cdots,2n\}\atop \phi(1)<\phi(2)<\cdots<\phi(n)}\det((C_{(n)},I_n)(i,\phi(j)))_{1\le i,j\le n}&\cdot\det\left( \left(\begin{array}{c}I_n -C_{\le0,(n)}^+\end{array}\right)(\phi(i),j)\right)_{1\le i,j\le n}.
u_{k,0}^{[j],-}(y) := \frac{\partial^j}{\partial s^j}y^{1-\frac{k}{2}-s}\bigg|_{s=\frac{k}{2}}= (-1)^j(\mathrm{log}\y)^j y^{1-k}, u_{k,0}^{[j],+}(y) := \frac{\partial^j}{\partial s^j}y^{s-\frac{k}{2}}\bigg|_{s=\frac{k}{2}}= (\mathrm{log}\y)^j.
e_p^H(u,x) := e^{u p}\frac{1}{2 \pi i}\int_{|z|=1}e^{-(u/p) \Delta^X}(z x, x) z^{-p}\, \frac{dz}{z}&= e^{u p}\frac{1}{2 \pi i}\int_{|z|= e^{2 u}}e^{-(u/p) \Delta^X}(z x, x) z^{-p}\, \frac{dz}{z}&= e^{u p}\int_0^{2 \pi}e^{-(u/p)\Delta^X}(e^{i \theta + 2u}x, x) e^{-p ( i \theta + 2u)}d\theta.
{\mathcal S}_{t}\leq 2^p \int_X \frac 1{\mu(B(x,t))}\int_{B(x,t)}| u_{B(x,t)}- u(y)|^p \, d \mu(y) \, d \mu (x) & \leq c_P 2^p t^p \int_X \frac{\nu (B(x,t))}{\mu(B(x,t))}\, d \mu(x) = c_P (2t)^p \int_{X\times X}\frac{\chi_{\{d(x,y) \leq t\}}(x,y)}{\mu(B(x,t))}\, d \mu \otimes \nu (x,y).
\int_{B_{\frac{R+r}{2}}}\mathrm{Snail}\left(u_h;x;B_{R}\right)^{p}\,dx =C\,R^{s\,p}\,\int_{B_{\frac{R+r}{2}}-h}\int_{\mathbb{R}^N\setminus (B_{R}-h)}\, \frac{|u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy &\le C\,\left(\frac{R+r}{R-r}\right)^N\,\left(\frac{R+h_0}{R-r}\right)^{s\,p}\,\langle u\rangle^p_{\mathcal{X}^p_s(B_{\frac{R+r}{2}+h_0};B_{R+h_0})}.
\ln \left[F(\bar N)\right] \le 2 \cdot 3^{\bar N}\ln |x_0| + \frac{2}{3}\cdot 3^{\bar N}\sum_{j=0}^{\bar N}3^{-j}\ln h_{j}&= \frac 23 \cdot 3^{\bar N}\left[\ln |x_0|^3 + \sum_{j=0}^{\bar N}3^{-j}\ln h_{j}\right].
m \frac{d^2Y^\mu(\tau)}{d\tau^2}= - \frac{2 \tilde{g}}{\bar{N}}\int \delta \xi ds \epsilon^{\mu\nu\rho\sigma}\,\mbox{\rm Tr}\{\kappa[\xi|s] F_\nu[\xi|s] \}\frac{dY_\rho(\tau)}{d\tau}\dot{\xi}_\sigma(s) \dot{\xi}(s)^{-2}\delta(\xi(s)-Y(\tau)),
f \mapsto RF(f)(x):= \left( \sum_{k=1}^N \vert \int_{I_k}\hat{f}(\xi) e^{2 \pi i \xi x}d \xi \vert^2 \right)^{1/2}= \left( \sum_{k=1}^N \vert P_{I_k}f(x) \vert^2 \right)^{1/2},
B(f_{<j-2}g_j) -f_{<j-2}Bg_j = \; \int K_j(y) (f_{<j-2}(x-y) -f_{<j-2}(x) -\partial f_{<j-2}(x) \cdot (-y) ) g_j(x-y) dy & + \int K_j(y) \partial f_{<j-2}(x) \cdot (-y) g_j(x-y) dy.
q^{\mu}\Gamma_{\mu \nu}( q , q^{\prime}; p , p^{\prime}) = - e \mu^{\frac{\epsilon}{2}}\left\{\Gamma_{\nu}( q^{\prime}; p^{\prime}- q^{\prime}, p^{\prime}) - \Gamma_{\nu}( q^{\prime}; p , p + q^{\prime}) \right\}.
-\frac{g_i^{(m+1)}(y_i)}{g_i(y_i)}=\frac{1}{g_i(y_i)}\sum_{\mu=0}^m\binom{m}{\mu}g_i^{(m-\mu)}(y_i)h_i^{(\mu)}(y_i) &=\sum_{\mu=0}^m\binom{m}{\mu}P_{m-\mu}\!\left(h_i(y_i),\ldots,h_i^{(m-\mu-1)}(y_i)\right)h_i^{(\mu)}(y_i)\qquad(1\leq i\leq s)
\phi_a=\left(\begin{array}{c}2(x_1x_3+x_2x_4) 2(x_2x_3-x_1x_4) x_1^2+x_2^2-x_3^2-x_4^2\end{array}\right),\quad |\phi|=r^2,\quad n=\phi/r^2\equiv n_{\rm H}.
c_\mathrm{nod}= \lim_{n \to \infty}\mathcal{A}(t_{n, +}u_n^+ + t_{n, -}u_n^-) &\ge \liminf_{n \to \infty}\mathcal{A}(t_{n, +}u_n^+) + \liminf_{n \to \infty}\mathcal{A}(t_{n, -}u_n^-) \ge 2 c_0,
f_{0}(k) = \frac{k}{\log k/2}+ \frac{k}{\log^{2}k/2}+ \frac{1.8 k}{\log^{3}k/2}- \frac{k}{\log k}- \frac{k}{\log^{2}k}- \frac{2.51 k}{\log^{3}k}- \log k, f_{n}(k) = \frac{k}{4(n+1)\log^{2}(nk +k)}- 1.118 \frac{nk +k}{(\log nk)^{3/4}}\exp( - \sqrt{\log(nk)/6.455}),
P(0,x_0;t,y)\le\frac{\pi^{(t)}(y)}{\pi^{(0)}(x_0)}\Big[\frac{2\pi^{(0)}(x_0)}{v(s)}+\frac{\pi^{(0)}(x_0)L_t}{\pi^{(t)}(y)^\alpha}\Big]=\frac{2\pi^{(t)}(y)}{v(s)}+\pi^{(t)}(y)^{1-\alpha}L_t.
\sqrt{- g}g^{\mu_{1}\nu_{1}}g^{\mu_{2}\nu_{2}}\cdots g^{\mu_{d - p}\nu_{d - p}}{\tilde{F}}_{\nu_{1}\nu_{2}\ldots \nu_{d - p}}= \frac{1}{p !}\epsilon^{\mu_{1}\mu_{2}\ldots \mu_{d - p}\nu_{1}\nu_{2}\ldots \nu_{p}}F_{\nu_{1}\nu_{2}\ldots \nu_{p}},
z_1\wedge e_{12}= 0 z_1\wedge e_{13}= -ye_{123}z_1\wedge e_{23}= xe_{123}z_2\wedge e_{12}= xe_{123}z_2\wedge e_{13}= 0 z_2\wedge e_{23}= 0 z_3\wedge e_{12}= ze_{123}z_3\wedge e_{13}= -xe_{123}z_3\wedge e_{23}= 0,
L_1 := t_{\alpha_1}^{-1}t_{\delta_1}^{-1}t_{\gamma_1}^{-1}t_{\beta_1}^{-1}t_{x_1}t_{y_1}t_{z_1}, L_1^\prime := t_{\beta_1}^{-1}t_{\gamma_1}^{-1}t_{\delta'_1}^{-1}t_{\alpha_1}^{-1}t_{x^\prime_1}t_{y^\prime_1}t_{z^\prime_1}.
\begin{array}{rlrl}1^* : a\mapsto \epsilon(a) \mu^*(f_1,f_2): a \mapsto f_1\otimes f_2\circ\Delta(a) \epsilon^*: f \mapsto f(1) \Delta^*(f): (a_1,a_2) \mapsto f(\mu(a_1,a_2)) S^*(f): a \mapsto f(S(a)) ~ ~\end{array}
\|f^{[j,m]}\|_{L^2(\mathbb{R}^{6})}^{2}(\sum_{\theta \in \mathcal{P}_{N^{-1}}}\|f^{[j,m]}*\Xi_{\theta}\|_{L^{\infty}(\mathbb{R}^{6})}^2)^{\frac{p-2}{2}}\lesssim (\sum_{\theta \in \mathcal{P}_{N^{-1}}}\|f*\Xi_{\theta}\|_{L^p(\mathbb{R}^{6})}^2)^{\frac{p}{2}}.
\lim_{J\to\infty}\limsup_{n\to\infty}\|w^J_{n}\|_{L^{\infty}_tL^{\frac{2k+2}{k}}_x\cap L^{\frac{k+2}{k}}_tL^{\frac{2k+4}{k}}_x(R^{2k+2}\times [0,\infty))}=0.
P^{\alpha+1}_{F,VB}\cap [p,q] = \emptyset \iff \exists [p',q']\supsetneq [p,q] F_{P^\alpha}[p',q'] P^{\alpha+1}_{F,VB_\ast}\cap [p,q] = \emptyset \iff \exists [p',q'] \supsetneq [p,q] F_{E,\ast}[p',q'], E= P^\alpha.
\Gamma = \frac{1}{2}\int d^{3}x d^{3}y \, A_{i}( x ) \left( K ( x - y ) + \frac{\lambda}{2}\, K^{2}( x - y ) \right) A_{i}( y ) + ( A_{i}\longleftrightarrow A_{i}+ \partial_{i}\varphi ) - \frac{\lambda}{4}\int d^{3}x \, F_{i j}F_{i j}.
\frac{d^{n-3}}{dv^{n-3}}\left(F_\alpha\frac{d\alpha_+}{dv}\right) =\sum_{i=0}^{n-3}\binom{n-3}{i}\frac{d^iF_\alpha}{dv^i}\frac{d^{n-2-i}\alpha_+}{dv^{n-2-i}}=P_2, \frac{d^{n-3}}{dv^{n-3}}\left(F_\beta\frac{d\beta_+}{dv}\right) =\sum_{i=0}^{n-3}\binom{n-3}{i}\frac{d^iF_\beta}{dv^i}\frac{d^{n-2-i}\beta_+}{dv^{n-2-i}}=P_2.
\widetilde\Gamma^l_{ijk}\stackrel{(ijk)}{=}\partial_\lambda^l \widetilde\Gamma^\lambda_{\alpha\beta\gamma}\partial^\alpha_i \partial^\beta_j \partial^\gamma_k + 2\partial^l_\lambda \Gamma^\lambda_{(\alpha\beta)}\partial^\alpha_i \partial^\beta_{jk}+ \partial^l_\lambda \partial^\lambda_{ijk}
\Re(\lambda_p) =\Re(\alpha)+\Re(\beta\omega^p+\gamma\overline{\omega}^p)= &=\Re(\alpha)+\Re((\beta+\overline{\gamma})\omega^p)\leq &=\Re(\alpha)+|(\beta+\overline{\gamma})\omega^p|= &=\Re(\alpha)+|\beta+\overline{\gamma}|<0,
(f_1(x)+2f_2(x)+2uf_{14}(x))^* = f_1^*(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x) = f_1(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x) & = (f_1(x)+2f_2(x)+2uf_{14}(x))a(x)
\omega_1(s,t) = -s- t + (\beta +\gamma) \omega_2(s,t) = -\frac{1}{2}s-\frac{1}{2}t + \frac{1}{2}(\alpha -\beta) \omega_3(s,t) = \frac{1}{2}s + \frac{1}{2}t + \frac{1}{2}(\alpha -\beta) -\gamma \omega_4(s,t) = s \omega_5(s,t) = t.
(A_0 k)(\eta) = - m |\eta| k(\eta), begin{align*}.2cm](B k)(\eta) = - \int_{\mathbb{R}^d}E^{-}(y,\eta) k(\eta\cup y) dy + \int_{\mathbb{R}^d}\sum_{x\in \eta}a^{+}(x - y) k(\eta \setminus x \cup y) d y, begin{align*}.2cm](C k)(\eta) = - E^{-}(\eta) k(\eta) + \sum_{x\in \eta}E^{+}(x, \eta \setminus x) k(\eta \setminus x).
P(v_{i}^{k}=1|h)=\frac{exp(a_{i}^{k}+\Sigma_{j}W_{ij}^{k}h_{j})}{\Sigma_{k^{\prime}=1}^{K}exp(a_{i}^{k^{\prime}}+\Sigma_{j}W_{ij}^{k^{\prime}}h_{j})}
\left\| \prod_{0\le k \le m}P_{n-k}(\epsilon) - \left(\tilde{{P}}^{m+1}-\epsilon\sum_{j=1}^{m}j \tilde{P}^{m-j}\tilde{P}^{(1)}\tilde{P}^{j}\right) \right\| = O(\epsilon^2 m^3)
\bar{\mathsf{C}}_1 := p\mathsf{C}(g_1\mathsf{P}_1) + \bar{p}\mathsf{C}(h_1\mathsf{P}_1), \bar{\mathsf{V}}_1 := p\mathsf{V}(g_1\mathsf{P}_1) + \bar{p}\mathsf{V}(h_1\mathsf{P}_1).
{\cal F}_{{\rm F}=0}= (1-{\rm N}^2) \frac{e_{+}^2}{2{\pi}^2{\hbar}^2}\sum_{n>0}\frac{1}{n}\sin(\frac{2\pi}{\rm L}n(x-y)) = (1-{\rm N}^2) \frac{e_{+}^2}{2\pi{\hbar}^2}( \frac{1}{2}\epsilon(x-y)- \frac{1}{\rm L}(x-y) ).
\begin{array}{ll}\displaystyle\sum_{i=1}^{m}\overline{z_i}a_{ij}=\eta,\;\;\;j=1,2,...,n \displaystyle\sum_{i=1}^{m}\overline{z_i}=1 \lvert arg\overline{z_i}\rvert\leq\alpha,\;\;\;i=1,2,...,m \end{array}
slh[\frac{1}{2}aclh(x)]=\sqrt{\sqrt{x^{4}+1}+x^{2}-\sqrt{2}x\sqrt{\sqrt{x^{4}+1}+x^{2}}}=(\sqrt{x^{4}+1}-x^{2}+1)^{-1/2}(\sqrt{\sqrt{x^{4}+1}+1}-x)
g_{\delta_{1}\delta_{2}\delta_{3}}^{a_{1}a_{2}a_{3}}:=\frac{Y(\mathcal{I}_{\delta_{1}}^{a_{1}})_{\delta_{2}\delta_{3}}^{a_{3}a_{2}}+1}{Y(\mathcal{I})_{\delta_{2}\delta_{3}}^{a_{3}a_{2}}+1}.
\sup_{|a|\leq\delta}\int_{\Omega_\delta}|k(\varphi^\varepsilon (x+a,t) )-k(\varphi^\varepsilon (x,t))| \, dx \leq 2\sup_{|a|\leq\delta}\int_{\Omega_\delta}|\varphi^\varepsilon (x+a,t) -\varphi^\varepsilon (x,t)| \, dx \leq C \delta^{\frac{1}{2}}.
\phi-\overline{\phi}\in L^\infty([0,T_*];H^3), \phi_t \in L^\infty([0,T_*];H^2), & u\in L^\infty([0,T_*]; H^3), \phi \nabla^4 u\in L^2([0,T_*];L^2), & u_t \in L^\infty([0,T_*]; H^1)\cap L^2([0,T_*] ; D^2).
\begin{cases}\displaystyle \limsup_{t-s\to\infty}\int_\Omega u(t,x;s,u_0)\le M_0^*\cr\displaystyle \limsup_{t-s\to\infty}\int_\Omega u^{-p}(t,x;s,u_0)dx\le M_1^*\cr\displaystyle \limsup_{t-s\to\infty}\int_\Omega u^{q}(t,x;s,u_0)dx\le M_2^*.\end{cases}
{\cal L}_{Y . M .}^{( 5 )}= - \frac{1}{2 g_{5}^{2}}F_{M N}^{2}+ \frac{1}{g_{5}^{2}}\left( \left( D_{M}\Sigma \right)^{2}+ \overline{{\lambda}}i \Gamma^{M}D_{M}\lambda + \left( X^{a}\right)^{2}- \overline{{\lambda}}\left[ \Sigma , \lambda \right] \right)
\bf{A_1}=[L_1\psi_{j1}(x,t)\vert_{(x,t)=(x_i,t_i)}]_{i,j=1,2,\cdots},\,\,\bf{b_1}=[b_{11},b_{21},\cdots]^T, &\bf{C_1}=[F_1(x_1,t_1,y_1,p_1)+G_1(x_1,t_1), F_1(x_2,t_2,y_2,p_2)+G_1(x_2,t_2),\cdots]^T.
\mathbf{Q}_\alpha (\beta g) = \beta \, \mathbf{Q}_\alpha (g), \mathbf{Q}_\alpha (g) = \mathbf{Q}_\alpha (g \, \mathbf{Q}_\alpha (h)^{-1}h) = \mathbf{Q}_\alpha (g \, h\mathbf{Q}_\alpha (h^{-1})).
H_{F}\ = \ \left( \vec{q}_{1}+ i \, \vec{q}_{2}\right) \; \cdot \; \left( \vec{\lambda}\wedge \vec{\lambda}\right) - \left( \vec{q}_{1}- i \, \vec{q}_{2}\right) \cdot \left( \frac{\partial}{\partial \vec{\lambda}}\wedge \frac{\partial}{\partial \vec{\lambda}}\right) \ .
\begin{cases}\theta \ne x \ge 0, t\ge0, \left ( \mathcal{A}_{2}\cdot \mathcal{A}_{1}^{R_{k}}\cdot x \right )_{i}+t=\theta , if~x_{i}>\theta , \left ( \mathcal{A}_{2}\cdot \mathcal{A}_{1}^{R_{k}}\cdot x \right )_{j}+t\ge \theta , if~x_{j}=\theta, \end{cases}
\Delta_{\nu}= \partial_{\nu}\ln \psi , \Delta_{\nu \rho}= \partial_{\nu}\partial_{\rho}\ln \psi , \Delta_{\nu}^{2}= \partial_{\nu}^{2}\ln \psi , ( \Delta_{\nu})^{2}= ( \partial_{\nu}\ln \psi )^{2}, . . .
L=L\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)=L_0\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=1}^M\lambda_j\,L_j\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=M+1}^N \lambda^0_jL_j\left(t,\textbf{x}, \dot{\textbf{x}}\right)
{\cal T}^{(2n+1)}= P_{\ell-1}\big((\ell+2) P_{\ell-1}^2 - 2p (\ell+1) P_{\ell-2}\big)^n \hphantom{{\cal T}^{(2n+1)}=}{}+ \sum\limits_{j=0}^{n-1}c_j P_{\ell-1}^{2(n-j-1)}P_{\ell-2}^j P_{\ell-3}+ \sum\limits_{j=1}^n d_jP_{\ell-1}^{2n-2j+1}P_{\ell-2}^j,
J_{r}(Q_{j+1}, Q'_{j+1}\mid Q_{j}, Q'_{j}, \xi) = (\frac{i}{\cal N})^{1/2}\exp [ \frac{i}{\hbar}(2 r_{j}R_{j}+ 2 r_{j+1}R_{j+1}- r_{j}R_{j+1}- r_{j+1}R_{j}+ \xi r_{j+1})]
\frac{s!}{(n+s)!}\sum_k \binom{n}{k}\frac{(n+s)!}{(k+s)!}(1-Q((1-\gamma)\mu))^{n-k}(1-Q(\gamma\mu))^{k(k+s)}e^{k \theta}= \sum_k \binom{n}{k}\frac{s!}{(k+s)!}(1-Q((1-\gamma)\mu))^{n-k}(1-Q(\gamma\mu))^{k(k+s)}e^{k \theta}
(g-1)^3 \cdot (20 f_{2,5}-20 f_{3,4}+10 f_{4,3}) &=20 \sum_{k=0}^3 \binom{3}{k}f_{k-1,5-k}-20 \sum_{k=0}^3 \binom{3}{k}f_{k,4-k}+10 \sum_{k=0}^3 \binom{3}{k}f_{k+1,3-k}&=20(3 f_{1,3}+f_{2,2})-20 (3 f_{1,3}+3 f_{2,2}+f_{3,1})=10 (f_{1,3}+3 f_{2,2}+3 f_{3,1}) &=10 f_{1,3}-10 f_{2,2}+10 f_{3,1}&=(\det A_3(4,5)) x_3.
\sharp T_K(\varepsilon,\eta, d,B)=\frac{(\varepsilon-\eta)K^2}{2\alpha^2 d^2}B^{2-\frac{1}{r}}+O_{\varepsilon,\eta}\left(\frac{K^2 B^{2-\frac{1}{r}}}{N d^2}\right)+O\left(\varDelta(\alpha)N\log\left(KB\right)\right).
0 =L(e_{ii})e_{kl}-e_{kl}L(e_{ii})+e_{ii}L(e_{kl})-L(e_{kl})e_{ii}&=\left(C_{kk}^{ii}-C_{ll}^{ii}\right)e_{kl}+\left(C_{ik}^{ii}+C_{ik}^{kk}\right)e_{il}-\left(C_{li}^{ii}+C_{li}^{ll}\right)e_{ki}.
{{\tilde{\psi}}}_{\alpha,s}\bigcirc_{w'=0}^s (\theta_{X_s}+\gamma_{r-s+w'})\bigcirc_{w=1}^{\ell}(\theta_{X_s}+\xi_w)(P)={\rm{Eval}}_{{{X_s\rightarrow \alpha}}}\bigcirc_{w=1}^{\ell}(\theta_{X_s}+\xi_w)(P)\enspace.
d s^{2}= \left({\frac{1 - | Z |^{2}}{1 - | Z_{\infty}|^{2}}}\right)^{- 1}d t^{2}- \left({\frac{1 - | Z |^{2}}{1 - | Z_{\infty}|^{2}}}\right) d \vec{x}^{2}\ .
(X_t - X_t')_+ = \int_0^t 1_{X_s > X_s'}d(X - X')_s + \frac{1}{2}L_t^0 = \int_0^t 1_{X_s > X_s'}(\sigma(X_s) - \sigma(X_s')) dB_s + \int_0^t1_{X_s > X_s'}(b(X_s) - b(X_s')) ds \phantom{=}+ \int_0^t 1_{X_s > X_s'}(d\Phi_s - d\Phi_s') - \int_0^t1_{X_s > X_s'}(d\Psi_s- d\Psi_s') + \frac{1}{2}L_t^0,
{\omega}^{2}={c}^{2}{k}^{2}+ 2 \,{\frac{{\alpha_{{0}}}^{2}{c_{{2}}}^{2}{c}^{2}{q}^{2}}{\gamma \, c_{{1}}}}+ 4 \,{\alpha_{{0}}}^{2}c_{{2}}{c}^{2}.
U_{\pi}^R = 2 \Re \sum_{\substack{\gamma \in \Gamma^S(\chi_0) \gamma\le T}}\frac{Z_\gamma}{\sqrt{\frac14+\gamma^2}}\quad V_{\pi}^R = 2 \Re \sum_{\substack{\gamma \in \Gamma^S(\chi_0) \gamma > T}}\frac{Z_\gamma}{\sqrt{\frac14+\gamma^2}}.
C^{( 1 )}+ 2 C^{( 2 )}r + 3 C^{( 3 )}r^{2}+ \dots = - \frac{d \psi}{d t}\bigl ( 1 + C^{( 1 )}r + C^{( 2 )}r^{2}+ \dots \bigr ) - i \bigl ( \frac{d C^{( 1 )}}{d t}r + \frac{d C^{( 2 )}}{d t}r^{2}+ \dots \bigr )
\tilde{\mathbf{E}}^{+}_{h}\ =\prod_{(\beta,s)\in\Delta^{+}\times\mathbb{Z}}\limits^{\rightarrow}\tilde{\mathbf{E}}_{\beta,s}^{+,(h(\beta,s))}, \tilde{\mathbf{E}}^{-}_{h}\ =\prod_{(\beta,s)\in\Delta^{+}\times\mathbb{Z}}\limits^{\rightarrow}\tilde{\mathbf{E}}_{\beta,s}^{-,(h(\beta,s))}.
{\cal L}_I = - \frac{2}{9}\Omega^{\rho,\mu\nu\alpha\beta}\Omega_{\mu,\rho\nu\alpha\beta}+ \frac{2}{9}\Omega^{\mu\nu\alpha}\Omega_{\mu\nu\alpha}- \frac{1}{9}\Omega^{\rho,\mu\nu\alpha\beta}E_{\mu\nu\alpha\beta,\rho}- \frac{4}{9}\Omega^{\mu\nu\alpha}E_{\mu\nu\alpha}
\int_{B_{\delta^{-1}}}|g_{I_1}\dots g_{I_k}|^2 \lesssim_{\kappa}(\delta^{k-1})^k \prod_{j=1}^k \int_{B_{\delta^{-1}}}|g_{I_j}|^2 = |B_{\delta^{-1}}|^{-(k-1)}\prod_{j=1}^k \int_{B_{\delta^{-1}}}\Big(\sum_{K_j \in P_{\delta}(I_j)}|g_{K_j}|^2\Big),
y_i=a_c s_c\left(\hat{\mathbf{h}}_{i,*}+\tilde{\mathbf{h}}_{i,*}\right)\mathbf{p}_c+\sum_{j=1}^{M}a_j s_j \left(\hat{\mathbf{h}}_{i,*}+\tilde{\mathbf{h}}_{i,*}\right)\mathbf{p}_j+ n_i.
\sum_{m=rJ+1}^{(r+1)J}(q(m) - q(rJ+1)) y(m) \leq \sum_{m=rJ+1}^{(r+1)J}(m-(rJ+1)) y^2_{max}= \frac{J(J-1)}{2}y^2_{max}\leq J(J-1)U.
E_{\mu \nu}={\tilde{E}}_{\mu \nu}- \frac{1}{D - 2}q_{\mu \nu}{}^{( D )}R_{\alpha \beta}n^{\alpha}n^{\beta}- \frac{1}{D - 2}q_{\mu}^{\alpha}q_{\nu}^{\beta}{}^{( D )}R_{\alpha \beta}+ \frac{1}{( D - 1 ) ( D - 2 )}q_{\mu \nu}{}^{( D )}R ,
q_1 \!\!\overset{\mathcal{R}^{j\!_1\cdots j\!_r}}{\sim}\!\!q_2 \Leftrightarrow \mbox{Col}_{q_1}\!(\!H\!L\!_{j_1}\!\!\cdots\!L\!_{j_r}\!)\!=\!\mbox{Col}_{q_2}\!(\!H\!L\!_{j_1}\!\!\cdots\!L\!_{j_r}\!).