harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2(N-K(N) )+O(\operatorname{nz}(N-K(N))) \le (\lambda_1^2 N+ 2 \lambda_1(\lambda_2\sqrt{2}-\lambda_1) N)(1+o(1)) = \lambda_1 (2\sqrt{2}\cdot \lambda_2 - \lambda_1)N(1+o(1)).
{1\over{2C_{\cal R}}}\sum_{\mu,\nu\in{\cal R}}\left(L^{\pm}L^{\mp}\right)_{\mu\nu}={1\over 2}p^{2}+{1\over2}\sum_{\rho\in\Delta_+}g_{|\rho|}(g_{|\rho|}-1) \,{|\rho|^{2}\over{(\rho\cdot q)^2}}+{\omega^2\over2}q^2 \mp \omega({r\over2}+\sum_{\rho\in\Delta_+}g_{|\rho|}),
{}^{M\!L\!\!}\textrm{F}_{p}(a,b;c;z)= \frac{2}{B(b,c-b)}\int_{0}^{\infty}\sinh^{2b-1}v \cosh^{2a-2c+1}v(\cosh^{2}v-z\sinh^{2}v)^{-a}&\times E_{\alpha, \beta}^{\gamma}\left(-p\cosh^{2}v\coth^{2}v\right)du.
\overline D \psi - D \psi = \sum_{i=1}^n e_i \cdot (\overline\nabla_{e_i}-\nabla_{e_i}) \psi &=- \frac12 \sum_{i=1}^n e_i \cdot e_0 \cdot W(e_i) \cdot \psi &= \phantom{-}\frac12 \sum_{i,j=1}^n g(W(e_i),e_j) e_i \cdot e_j \cdot e_0 \cdot \psi &=-\frac12 \sum_{i=1}^n g(W(e_i),e_i) e_0 \cdot \psi.
U_+:=\bigcap_{k=0}^\infty \alpha^k(U)=\prod_{\rho\geq 1}B^{E_\rho}_r(0)\;\;\mbox{and}\;\;U_-=\bigcap_{k=0}^\infty \alpha^{-k}(U)=\prod_{0<\rho\leq 1}B^{E_\rho}_r(0)
\sum_{i=1}^j\mu^{(p)}_i =\begin{cases}\sum_{i=1}^j\mu_i \sum_{i=1}^j\mu_i-1 \end{cases}\sum_{i=1}^j\lambda^{(q)}_i =\begin{cases}\sum_{i=1}^j\lambda_i \sum_{i=1}^j\lambda_i-1 \end{cases}
\rho^{[1..k-1]}=\sum_{\alpha}{(\lambda_{\alpha}^{[k-1]})}^{2}|\Phi_{\alpha}^{[1..k-1]}\rangle\langle\Phi_{\alpha}^{[1..k-1]}|=\sum_{\alpha}(\lambda_{\alpha}^{[k-1]})^{2}|\alpha\rangle\langle\alpha|.
(\rho^{k}\otimes (^{k}\rho))\Delta(\alpha) \Diamond \rho(z) =(\rho^{k}\otimes (^{k}\rho))(1 \otimes \alpha) \Diamond \rho(z) &=(\bar{e_{k}}\otimes p(\alpha)) \Diamond \rho(z) &=cyc(\rho(\alpha)\rho(z)) &=\rho(cyc(\alpha z)) &=\rho^{k}(\Delta(\alpha) \Diamond z)
\frac{y}{m}a_{2,1}+y^2a_{2,2}\left(1-\frac{1}{m}\right)=\frac{y}{m}a_{1,1}+y^2a_{1,1}\left(1-\frac{1}{m}\right) &\frac{y}{m}a_{2,1}+y^2a_{2,2}\left(1-\frac{1}{m}\right)=\frac{y}{m}+y^2\left(1-\frac{1}{m}\right) &
b_k^2 = e_1 e_2 e_k e_1 e_2 e_k = Z^3 e_1^2 e_2^2 e_k^2 =Z t_1 t_2 t_3 e_1 e_{2}e_k = Z t_1 t_2 t_3 \, b_k, b_k b_\ell = e_1 e_2 e_k e_1 e_2 e_\ell =Z e_1 e_2 e_\ell e_1 e_2 e_k = Z \, b_\ell b_k .
\Sigma_k \subseteq \left[\frac{q^{d}p^d \sigma_{p}}{[ (q+1)p]^d}\,, \frac{(q+1)^{d}p^d \sigma_{p}}{( qp)^d}\right] =\left[ \left(\frac{q}{q+1}\right)^{d}\sigma_{p}\,, \left(\frac{q+1}{q}\right)^{d}\sigma_{p}\right] ,
\begin{aligned}\mathcal{L}f(x) = \ \frac{1}{2}\sum_{j,k=1}^{d}a_{jk}(x) \frac{\partial^{2}}{\partial x_{j}\partial x_{k}}f(x) + \sum_{j=1}^{d}b_{j}(x) \frac{\partial}{\partial x_{j}}f(x) + \int_{U}[f(x+c(x, u)) - f(x) - c(x, u)\cdot Df(x)] M(\d u).\end{aligned}
\left\Vert x\frac{y_{0}^{u}}{x_{0}^{u}}\right\Vert = \left\Vert b_{i}x_{0}^{u-1}\frac{y_{0}^{u}}{x_{0}^{u}}\right\Vert= \left\Vert \frac{b_{i}y_{0}^{u}}{x_{0}}\right\Vert\geq x_{0}^{-1}.
v^+_0(x_1,0,t)=v^{(0)}_0(x_1,0,t), (x_1, t) \in (0,a) \times (0,T), \partial_{x_2}v^+_0(x_1,0,t) = h_0\, \partial_{x_2}v^{(0)}_0(x_1,0,t), (x_1, t) \in (0,a) \times (0,T).
\rho = - \frac{1}{2}\log \left[ \left( \frac{\pi T e^{2}}{\sqrt{2 p_{+}p_{-}}}\right)^{2}k_{( + )}k_{( - )}\frac{\partial_{+}b \partial_{-}\overline{{{b}}}}{\left( 1 + b \overline{{{b}}}\right)^{2}}\right]
per(\begin{matrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{matrix})=det^{2}(A_{11})per(\begin{matrix}A_{11}^{-1}&A_{11}^{-1}A_{12}\\A_{21}A_{11}^{-1}&A_{22}-A_{21}A_{11}^{-1}A_{12}\end{matrix})
\lambda(A_1\times S^n) ={\frac 1{\Omega_n}}\int_{A_1\times S^n}\nu(y)~d\sigma(x,y) &=\mu(A_1) \lambda(S^n\times A_2) ={\frac 1{\Omega_n}}\int_{S^n\times A_2}\nu(y)~d\sigma(x,y) &=\nu(A_2)
( - )^{\varepsilon_{A}}( S_{[ A}, S_{B C ]}) = S_{D [ C}f_{A B ]}^{D}+ \mathrm{\small{~ \frac{1}{3}~}}f_{A B C}^{D}S_{D}+ \mathrm{\small{~ \frac{1}{3}~}}( S , S_{A B C})
\underline \gamma_{y_+}= \frac12\left(\delta_{(x_+-y_+)^2}+ \frac{z_--y_+}{z_--y_-}\delta_{(x_--y_-)^2}+ \frac{y_+-y_-}{z_--y_-}\delta_{(z_--x_-)^2}\right) &\le_{cx}\frac12\left(\delta_{(x_--y_+)^2}+ \frac{z_--y_+}{z_--y_-}\delta_{(x_+-y_-)^2}+ \frac{y_+-y_-}{z_--y_-}\delta_{(z_--x_+)^2}\right)=\overline \gamma_{y_+}.
C_{2h}:=(t_{\alpha_1}t_{\alpha_2}\cdots t_{\alpha_{2h}})^{4h+2}t_d^{-1}, &C_{2h+1}:=(t_{\alpha_1}t_{\alpha_2}\cdots t_{\alpha_{2h+1}})^{2h+2}t_{d_2}^{-1}t_{d_1}^{-1}.
P_{|\Omega}^s(x,y)=-\frac{\partial}{\partial\nu_y}G^s_{|\Omega}(x,y)=\lim_{z\to y}\frac{G^s_{|\Omega}(x,z)}{\delta(z)}=-\frac1{\Gamma(s)}\int_0^\infty\frac{\partial}{\partial\nu_y}p_\Omega(t,x,y)\,t^{s-1}\,dt.
M^*(Z_p(\mu)) \leq C \begin{cases}\sqrt{p}1 \leq p \leq \sqrt{n}n^{-1/4}p \sqrt{n}\leq p \leq \sqrt{n}\log^2(1+n) \sqrt{p}\log(1+n) \sqrt{n}\log^2(1+n) \leq p \leq n / \log^2 (1+n) \frac{p}{\sqrt{n}}\log^2(1+n) n / \log^2 (1+n) \leq p \leq n \end{cases}~.
\tilde{P}= \sum_{n = 0}^{\infty}\sum_{\ldots}{\mathrm{c o n s t}}( n , \ldots ) C_{n , \ldots}\:{\cal{B}}\: C_{n - 1 , \ldots}\:{\cal{B}}\: \cdots \:{\cal{B}}\: C_{0 , \ldots},
\left< t r \frac{\delta}{\delta \gamma_{\mu}( s )}{\cal{F}}_{\mu}( \gamma ( 0 ) ) \right> = \frac{\partial^{2}W ( \gamma )}{\partial \gamma_{\mu}^{2}( 0 )}= - e^{2}\int \delta ( \gamma ( s ) - \gamma ( 0 ) ) W ( \gamma_{0 s}) W ( \gamma_{s 0}) \dot{\gamma_{\mu}}( 0 ) \dot{\gamma_{\mu}}( s ) d s
1 \otimes_{k+1}\xi_{k+1}^{i}\overset{(\ref{eq:xitosum}), (\ref{eq:Ytosum})}{=}(-1)^{i}\sum_{\ell=0}^{i}\sum_{p=0}^{i-\ell}(-1)^{p}\phi^*(Y_{i-\ell-p,k})\xi_{k+1}^p \otimes_{k+1}x_{\ell,k}.
R\cdot\left[R+C_{k+1}(P_{k}+Q) C_{k+1}^{H}\right]^{-1}=\big(C_{k+1}(P_{k}+Q)C_{k+1}^H R^{-1}\big)^{-1}\big(C_{k+1}P_{k+1}C_{k+1}^HR^{-1}\big)
Q = \begin{bmatrix}J_{c_1}(\lambda_1) \ddots J_{c_p}(\lambda_p) \check{I}_{d_1-1}\ddots \check{I}_{d_q-1}0_{(n-m-q)\times m}\end{bmatrix}, P = \begin{bmatrix}I_{c_1}\ddots I_{c_p}\hat{I}_{d_1-1}\ddots \hat{I}_{d_q-1}0_{(n-m-q)\times m}\end{bmatrix}.
\prod_{u=1}^{\beta_j}(q^{n-k}- q^{n_{i-1}+ p_{j-1}+u -1 - s_{j-1}}) = \prod_{u=1}^{\beta_j}(q^{n-k}- q^{(n_{i-1}-k) + k+ p_{j-1}+u -1 - s_{j-1}}) = \prod_{u=1}^{\beta_j}(q^{n-k}- q^{(n_{i-1}-k) + k - \beta_j - \cdots - \beta_{k_0+1}+ (u-1)}).
\int_0^tU(t,s)(\partial_tG)(y(s))ds = (\alpha-1)\int_0^tU(t,s)e^{(\alpha-1)W(s)}g(y(s))dW(s) & +(\alpha-1)^2\int_0^tU(t,s)\mu e^{(\alpha-1)W(s)}g(y(s))ds, |\nabla G(y)| \le |e^{(\alpha-1)W}|[(\alpha-1)|\nabla W||g(y)|+|\nabla g(y)|].
\Phi^\alpha(t)=\int_0^t \phi_\alpha^2(t-s)ds=\begin{cases}\frac{1-e^{-2\lambda_\alpha t}}{2\lambda_\alpha}, ; \frac{\sqrt{\lambda_\alpha}t-\sin(2\sqrt{\lambda_\alpha}t)}{2\lambda_\alpha^{3/2}}, .\end{cases}
\partial_{x}\theta ( x , t ) = \frac{m v}{\hbar}-{\frac{\lambda}{2}}\rho^{2}( x - v t ) \ \ \, L o n g r i g h t a r r o w \ \ \ \theta ( x , t ) = \frac{m v}{\hbar}x + \theta_{0}( t ) -{\frac{\lambda}{2}}\int_{- \infty}^{x - v t}~ d y ~ \rho^{2}( y ) ,
0<B_1:=\frac{1}{\gamma}-\frac{1}{2}\frac{1}{\gamma}\leq\frac{\log \Gamma_1(x)}{\log x}\leq \frac{1}{\gamma}+\frac{1}{2}\frac{1}{\gamma}=:B_2,\,x\in(0,g(\delta/2)], g(\delta/2)<1, g'(x)>0 x\in(0,\delta).
\Phi \left( p^{\mu}\right) =\Lambda \left( \dot p^{\mu}\to p^{\mu}\right)\Phi \left( \dot p^{\mu}\right) =e^{i(\vec{J}\cdot \vec{\vartheta}+\vec{K}\cdot \vec{\varphi})}~\Phi \left( \dot p^{\mu}\right)\, ,
\Omega_{n}^{\pm}(\lambda,b) :=\frac{1-b^{2}}{2b}\Lambda_{1}(\lambda,b)+\frac{1}{2}\Big(\Omega_{n}(\lambda)-\Omega_{n}(\lambda b)\Big) &\quad\pm\frac{1}{2b}\sqrt{\Big(b\big[\Omega_{n}(\lambda)+\Omega_{n}(\lambda b)\big]-(1+b^{2})\Lambda_{1}(\lambda,b)\Big)^{2}-4b^{2}\Lambda_{n}^{2}(\lambda,b)}
\mathbb{E}\left[e^{uZ_{t}}\right]=\exp\left\lbrace \int_{0}^{t}\int_{0}^{\infty}\left(e^{z\psi(s,u)}-1\right)\nu(\mathrm{d}z)\mathrm{d}s\right\rbrace ,\quad(t,u)\in\mathbb{R}_{\geqslant0}\times\mathcal{U}.
\mathcal{X}^{t+1}=\mathcal{X}^{t}-(\mathcal{A}_{(i,:,:)})^{T}*\left(\mathcal{A}_{(i,:,:)}*(\mathcal{A}_{(i,:,:)})^{T}\right)^{\dag}*\left(\mathcal{A}_{(i,:,:)}*\mathcal{X}^{t}-\mathcal{B}_{(i,:,:)}\right).
T_{\phantom{a}a}^{a}= -{\frac{1}{2 \alpha^{\prime}}}\beta_{\mu \nu}^{G}g^{a b}\partial_{a}X^{\mu}\partial_{b}X^{\nu}-{\frac{i}{2 \alpha^{\prime}}}\beta_{\mu \nu}^{B}\epsilon^{a b}\partial_{a}X^{\mu}\partial_{b}X^{\nu}-{\frac{1}{2}}\beta^{\Phi}R \ .
(n+\alpha+\beta+2)\phi_n^{(\alpha+1,\beta+1)}+(n+\alpha+1)\phi_{n-1}^{(\alpha+1,\beta+1)}=(2n+\alpha+\beta+2) \phi_{n}^{(\alpha+1,\beta)}, (n+\alpha+\beta+1)\phi_{n}^{(\alpha+1,\beta)}-(2n+\alpha+\beta+1) \phi_{n}^{(\alpha,\beta)}=(n+\beta)\phi_{n-1}^{(\alpha+1,\beta)}.
f^{(a,b)}\; \in \; L^{N-1}\Big( \mbox{I\hspace{-0.62mm}R}^{D ( n_a + n_b)}\; , \;\prod_{i=1}^{n_a}d^Dz_i^{(a)}\prod_{j=1}^{n_b}d^Dz_j^{(b)}\Big) \; \; ,
f_N' \left[ \phi (x) \right]={1 \over Z_0}\sum_{n=0}^{N}{\left( -1 \right)^n \over n!}e^{- S_0}\left({\lambda \over 4}\int{\rm d^d}x \phi^4 \right)^n{\rm Ch}(\Omega_N, \left\{\phi (x) \right\})
\begin{cases}\Delta V^t = -\Delta\dot h^t \mbox{in}\Omega^t V^t = 0 \mbox{on}\Gamma^t \lim_{x\to \infty}V^t(x) = \dot c^t, \lim_{x\to \infty}\frac{V^t(x)}{-\log |x|}= \dot c^t ( \n=2).\end{cases}
d_{0}=b_{1}-\frac{b_{0}}{c_{0}}c_{1}=\frac{c_{0}b_{1}-b_{0}c_{1}}{c_{0}};d_{1}=b_{2}-\frac{b_{0}}{c_{0}}c_{2}=\frac{c_{0}b_{2}-b_{0}c_{2}}{c_{0}};...(35)
Z = \int{\cal D}A_{\mu}e^{-{\frac{1}{4{\tilde{g}}^{2}}}\int d^{2}x [ \mathrm{T r}F^{\mu \nu}F_{\mu \nu}] + \mathrm{T r}\ln [ \gamma^{\mu}( \partial_{\mu}-{\bf A}_{\mu}) - m ]}.
\big(f_{2k}(x_{t+1}), f_{2k-1}(x_{t+1})\big) = (2, m-1), \big(f_{2k}(x_{t+m}), f_{2k-1}(x_{t+m})\big) = (2,0), \big(f_{2k}(x_{t+m+1}), f_{2k-1}(x_{t+m+1})\big) = (1,m-1).
\rho_{(1),\infty}^{(\rm cJ)}(x;\beta,p,q) \Big |_{\beta = 4 \atop q = 0}= \rho_{(1),\infty}^{(\rm cJ)}(2x;\beta,p,q) \Big |_{\beta = 2 \atop q = 0, p \mapsto 2p}- \pi p{J_{2p-1/2}(2x) \over x^{1/2}}\int_0^x s^{-1/2}J_{2p+1/2}(2s) \, ds,
[\begin{matrix}A&B\\C&D\end{matrix}]^{-1}=[\begin{matrix}A^{-1}+A^{-1}BS^{-1}CA^{-1}&-A^{-1}BS^{-1}\\ -S^{-1}CA^{-1}&S^{-1}\end{matrix}]=[\begin{matrix}I_{p}&-A^{-1}B\\ &I_{q}\end{matrix}][\begin{matrix}A^{-1}&\\ &S^{-1}\end{matrix}][\begin{matrix}I_{p}&\\ -CA^{-1}&I_{q}\end{matrix}]
\begin{aligned}A_n(z) =\left .\frac{P_n^2(y)w(y)}{h_n(y-z)}\right |_{y=a}^{y=b}+\frac{1}{h_n}\int_{a}^{b}\frac{v'(z)-v'(y)}{z-y}P_{n}^2(y)w(y)dy, B_n(z) =\left .\frac{P_n(y)P_{n-1}(y)w(y)}{h_{n-1}(y-z)}\right |_{y=a}^{y=b}+\frac{1}{h_{n-1}}\int_{a}^{b}\frac{v'(z)-v'(y)}{z-y}P_{n}(y)P_{n-1}(y)w(y)dy,\end{aligned}
\boldsymbol{\psi}= \left[\begin{matrix}1 \mathbf{x}_1 \end{matrix}\right]^T \otimes \mathbf{x}_0 = \left[\begin{matrix}\mathbf{x}_0 \mathbf{x}_0 x_{11}\cdots \mathbf{x}_0 x_{1K}\end{matrix}\right]^T\,,
h\log y\int_{2}^{u/h}y^{ht-u}t\rho(t)\,dt =uh^{-1}\rho(uh^{-1})-2\rho(2)y^{2h-u}-\int_{2}^{u/h}y^{ht-u}(t\rho(t))'\,dt &\le uh^{-1}\rho(uh^{-1})-2\rho(2)y^{2h-u}+\int_{2}^{u/h}y^{ht-u}\rho(t-1)\,dt.
\rho_{1}^{v_{2,n}}\psi_{1}^{v_{2,n}}(i)=\inf_{v_1\in V_1(i)}\biggl[\sum_{j\in S}\psi_{1}^{v_{2,n}}(j)\pi_{ij}(v_1,v_{2,n}(i))+c_1(i,v_1,v_{2,n}(i))\psi_{1}^{v_{2,n}}(i)\biggr],
|\psi_{\uparrow}|^2\rightarrow |\psi_{\uparrow,0}|^2=\dfrac{\bar{\rho}}{2}\qquad |\psi_{\downarrow}|^2\rightarrow |\psi_{\downarrow,0}|^2=\dfrac{\bar{\rho}}{2}\qquad|x|\rightarrow\infty.
{J}_{( \mu_{1}\dots \mu_{n}; \mu_{n + 1})}^{( n )}+ \frac{\partial{J}_{( \mu_{1}\dots \mu_{n}}^{( n )}}{\partial \psi^{\sigma}}\Gamma_{\mu_{n + 1}) \lambda}^{\sigma}\psi^{\lambda}= \frac{i}{2}\psi^{\rho}\psi^{\sigma}R_{\rho \sigma \nu ( \mu_{n + 1}}{{J}^{( n + 1 ) \nu}}_{\mu_{1}\dots \mu_{n})}
^{( 0 )}_{\psi , T , R}\langle D p , f , \eta_{1}| D p , f , \eta_{2}\rangle_{\psi , T , R}^{( 0 )}= \lim_{x \to 1}\;_{\psi , T , R}^{( 0 )}\langle D p , f , \eta_{1}| x^{2 ( F_{0}+ G_{0})}| D p , f , \eta_{2}\rangle_{\psi , T , R}^{( 0 )}
2 \mathrm{Re}\langle u_{\alpha}\otimes 2 \alpha^{3/2}\eta \tilde{\Phi}_*^1\,|\, \alpha V_y \Phi^y_{\#}\rangle \geq -C |\eta|^2 \alpha^4 \|\tilde{\Phi}_*^1\|^2 -C\alpha |\kappa|^2 \|V_y \Phi_{\#}^y\|^2 = O(\alpha^4).
F_{n}^{\mathcal{O}|\ldots \mu_{i}\mu_{j}\ldots}(\ldots \theta_{i},\theta_{j}\ldots )=F_{n}^{\mathcal{O}|\ldots \mu_{j}\mu_{i}\ldots}(\ldots\theta_{j},\theta_{i}\ldots )S_{\mu_{i}\mu_{j}}(\theta_{i,j})\,.
-\ \frac{1}{4\, s^2}\: \left( \partial^\mu s\, \partial_\mu s\, +\ \partial^\mu a\, \partial_\mu a \right)\ +\ \frac{\delta}{4\, s^2}\: \partial_\mu a\, A_X^\mu\ -\ \frac{\delta^2}{16\, s^2}\: A_X^\mu A_{X \mu}\ +\ \frac{\delta}{4\, s}\: D_X
0=\frac{1}{m}\frac{\delta S_{{\rm eff}}}{\delta z_{\rho}}= - \left[\frac{d^2 z^{\rho}}{d \tau^2}+ \Gamma^{\rho}_{\mu\sigma}\frac{d z^{\mu}}{d \tau}\frac{d z^{\sigma}}{d \tau}\right]+\frac{1}{m}\frac{\delta \Delta S_m}{\delta z_{\rho}},
I_2 = \sup_{(b,c,F) \in \Theta}\bigg\{\int_{|z|>1}|D_x\psi(\xi_z) z| \, F(dz) \bigg\}\leq \|D_{x}\psi\|_\infty \, \sup_{(b,c,F) \in \Theta}\bigg\{\int_{|z|> 1}|z| \, F(dz)\bigg\}
\sum_{\substack{\omega \, \mathrm{trivial}L(\omega, \chi^*) = 0}}\frac{1}{|\sigma+it-\omega|^2}\leq a(\chi) \sum_{k=0}^{\infty}\frac{1}{(\sigma+2k)^2 + t^2}+ b(\chi) \sum_{k=0}^{\infty}\frac{1}{(\sigma+2k+1)^2 +t^2}\leq n_K \sum_{k=0}^{\infty}\frac{1}{(\sigma+2k)^2}\leq \Big( \frac{1}{2\sigma}+ \frac{1}{\sigma^2}\Big) n_K
u_1(a' (ae_i)) = a' u_1 (a e_i) - u_2 (f_P(a' \otimes ae_i)) + f_Q(a' \otimes u_0(ae_i)) & = a' u_1 (a e_i) - u_2 (f(a' \otimes a)e_i) + f_Q(a' \otimes abe_j) & = a' u_1 (ae_i) - f(a' \otimes a)b + f(a' \otimes ab)e_j & = a' u_1(ae_i) - a' f(a \otimes b)e_j + f(a'a \otimes b)e_j,
(a_{-1}-a_1)^k\prod_{j=1}^k\chi_{\{\tau^{(n)}_{i_j}\geq a_{-1}\}}\leq\prod_{j=1}^k((\tau^{(n)}_{i_j}-a_{1})_+-(\tau^{(n)}_{i_j}-a_{-1})_+) =\sum_{\varepsilon_1,\cdots,\varepsilon_{k}\in\{\pm1\}}\prod_{j=1}^k\varepsilon_{j}(\tau^{(n)}_{i_j}-a_{\varepsilon_{j}})_+\leq(a_{-1}-a_1)^k\prod_{j=1}^k\chi_{\{\tau^{(n)}_{i_j}>a_{1}\}}.
\begin{aligned}q^{-9}L_4(q^{32}) = \sum_{\substack{n \geq 0 -n \leq j \leq n-1}}q^{(16n+3)^2 -2(8j+3)^2}+ \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{(16n+13)^2 -2(8j+3)^2}&+ \sum_{\substack{n \geq -1 -n-1 \leq j \leq n+1}}q^{(16n+21)^2 -2(8j+1)^2}+ \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{(16n+11)^2 -2(8j+1)^2}.\end{aligned}
\|e^{it\Phi(D)}f \|_{L_x^q(\mathbb R^n;L_t^r(\mathbb R))}\leq C_{m,\mu}\bigg(\sum_{k\in\mathbb{Z}}2^{2k(\frac{n}{2}-\frac{n}{q}-\frac{m}{r})}\| P_kf \|_2^2 \bigg)^{1/2}= C_{m,\mu}\| f\|_{\dot{H}^{\frac{n}{2}-\frac{n}{q}-\frac{m}{r}}(\mathbb R^n)}.
U_{1}( j + 1 , j ) = \sqrt{\frac{i}{\cal N}}\exp [ - \frac{i}{\hbar}Q_{j}Q_{j + 1}] , ~ ~ U_{2}( j + 1 , j ) = \sqrt{\frac{i}{\cal N}}\exp [ \frac{i}{\hbar}( Q_{j}^{2}- Q_{j}Q_{j + 1}+ Q_{j + 1}^{2}) ] .
\begin{aligned}\omega(x,y,z) = -2 x \dH^* \wedge dY^* +2y \dX^*\wedge dY^* - 2z \dX^* \wedge dH^* = -2 ( x \dH^* \wedge dY^* -y \dX^* \wedge dY^* + z\dX^* \wedge dH^* ). \end{aligned}
E_3 := K n^{\theta\frac{p}{2}}E \int_{t_0}^{u}\Big|\int^{s}_{\kappa(n,s)}\sigma_r^n(x^n_{\kappa(n,r)})dw_r\Big|^\frac{p}{2}ds & \leq K n^{\theta\frac{p}{2}}\int_{t_0}^{u}E \Big(\int^{s}_{\kappa(n,s)}|\sigma_r^n(x^n_{\kappa(n,r)})|^2dr\Big)^\frac{p}{4}ds
e^{-\frac{\Delta}{c_2}}\int_{\sigma_{\mathrm{w}}^2 + \sigma_{\mathrm{a}}^2}^{\infty}\left(\frac{1}{v}\right)^{\frac{n}{M}}e^{-\frac{z^{(0)}}{v}}e^{-\frac{v}{\zeta}}dv &\qquad\qquad=\int_{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}^{\infty}\left(\frac{1}{v}\right)^{\frac{n}{M}}e^{-\frac{(z^{(0)}+\Delta)}{v}}e^{-\frac{v}{\zeta}}dv.
{\bar{\theta}}( x_{k}, y_{k}; x ) \equiv \theta ( x_{k}, y_{k}; x ) - \frac{y_{k}- x_{k}}{L}= \frac{1}{L}\sum_{p \ne 0}\frac{1}{i p}\left( e^{i p y_{k}}- e^{i p x_{k}}\right) e^{- i p x}\, ,
M_{\beta}^{2}( \phi ) = m_{0}^{2}+ \frac{\lambda_{0}}{6}\left( \frac{\phi^{2}}{N}+ F_{\beta}( \phi ) \right) + \frac{\eta_{0}}{5 !}\left( \frac{\phi^{2}}{N}+ F_{\beta}( \phi ) \right)^{2},
G^{\epsilon}( z , \bar{z}) = \lim_{| z_{3}\bar{z_{3}}| \rightarrow \infty}| z_{3}\bar{z}_{3}|^{2 \Delta_{\alpha_{3}}}\langle \phi_{\alpha_{3}}\left( z_{3}, \bar{z}_{3}\right) \phi_{\alpha_{2}}\left( 1 \right) R_{\alpha}^{\epsilon}\left( z , \bar{z}\right) R_{\alpha_{1}}^{\epsilon}\left( 0 \right) \rangle
\begin{aligned}|N_{G}(v)\setminus V(S')| =d_{G}(v)-|N_{G}(v)\cap V(S')| \geq t-|Y_{S'}| \geq |Y|-|Y_{S}| \geq |N_{T}(\phi^{-1}(v))\cap (Y\backslash Y_{S})| \geq d_{T}(\phi^{-1}(v))-d_{S}(\phi^{-1}(v)). \end{aligned}
L_{k}u=\sum\limits_{i=0}^{m_{k}}\left[ \delta_{ki}u_{x_{k}}^{\left[ i\right]}\left( a_{k},x\left( k\right) \right) +\sum\limits_{j=0}^{N_{k}}\nu_{kij}u_{x_{k}}^{\left[ i\right]}\left( x_{kij},x\left( k\right) \right) \right] =0
\begin{array}{l}R(x,y,z,w) = \,[g(y,z)-2\,\eta(y)\eta(z)]g(x,w) - 2\,g(y,z)\eta(x)\eta(w) begin{align*}4pt]\phantom{R(x,y,z,w) = \,}-[g(x,z) -2\,\eta(x)\eta(z)]g(y,w) + 2\,g(x,z)\eta(y)\eta(w).\end{array}
F_\infty\mathcal O_{mod}(\tilde C)=F_\infty^{\mathbb C\Gamma_C}\mathcal O_{mod}(\tilde C)\subset F_\infty^{\mathbb C\mathbb Z}\mathcal O_{mod}(\tilde C)\subset(\varphi_{s,x}^*)^{-1}(F_\infty^{\mathbb C\mathbb Z}\mathcal O_{mod}(\tilde D^\times)) =(\varphi_{s,x}^*)^{-1}(\mathcal O_{mod}(D^\times)[e^*\mathrm{log}z])
C_{2}^{2}I_{5}+ 6 C_{2}^{2}E I_{3}+ 4 C^{2}C_{2}I I_{3}+ 1 2 C^{2}C_{2}E I^{2}+ 6 C^{3}C_{2}I^{2}I_{3}+ 1 6 C^{3}C_{2}E I^{3}+ 1 6 C_{2}^{2}E^{2}I + . . .
\begin{array}{lll}\partial^{\gamma,\mathbf{A}}_b p{_i}^{ab}= \partial_bp{_i}^{ab}- \mathbf{A}{^k}_bc^\ell_{ki}p{_\ell}^{ab}+ \gamma{^a}_{cb}p{_i}^{cb}+ \gamma{^b}_{cb}p{_i}^{ac}\partial^{\gamma,\mathbf{A}}_bp{_i}^{jb}= \partial_bp{_i}^{jb}- \mathbf{A}{^k}_bc^\ell_{ki}p{_\ell}^{jb}+ \mathbf{A}{^k}_bc^j_{k\ell}p{_i}^{\ell b}+ \gamma{^b}_{cb}p{_i}^{jc}\end{array}
\lim_{N\to\infty}\frac{2 \, \pi \, \left|\gamma_{s_{N}^{\alpha}}^{(N)}(K_{4_1})\right|}{N}=\lim_{N\to\infty}\frac{2 \, \pi \, \left|\gamma_{N-s_{N}^{\alpha}}^{(N)}(K_{4_1})\right|}{N}=\rm{Vol}\left(M_\alpha\right)
\frac 1 2 \int_{s_{0}}^{s_{1}}[ \frac 1 2 \left( \frac{d A_{1}}{d s}- \frac{d A_{2}}{d s}\right)^{*}\left( \frac{d A_{1}}{d s}- \frac{d A_{2}}{d s}\right) + \sum_{i = 1}^{2}\left( \frac{d Z}{d s}- i e A_{i}Z \right)^{*}\left( \frac{d Z}{d s}- i e A_{i}Z \right) ] d s
Res_{z=\frac{U_{j}(\sigma)}{c}}f_{(\sigma,b)}(z)dz=-q^{d_{c,\sigma,j}}Rec(c)^{(\sigma^{\prime},b^{\prime})}_{(\sigma,b)}f_{(\sigma^{\prime},b^{\prime})}(z)|_{z=\frac{U_{j}(\sigma)}{c}},
\phi(x,t) = \int_0^t\lim_{\lambda\to\infty}(\tfrac p\lambda \tau^{p-1}+ \min\{\phi_-'(x,\tau), p\lambda \tau^{p-1}\})\, d\tau &\le\liminf_{\lambda\to\infty}\int_0^t\tfrac p\lambda \tau^{p-1}+ \min\{\phi_-'(x,\tau), p\lambda \tau^{p-1}\}\, d\tau=\liminf_{\lambda\to\infty}\phi_\lambda(x,t).
\int_{\Omega}\left(\nabla w_n \nabla v -\lambda_n m(x)w_n v -a(x)w_n^{p-1}v\right) -\lambda_n^{\frac{p-q}{p-2}}\int_{\partial \Omega}b(x)w_n^{q-1}v =0 \forall v \in H^1(\Omega)
\begin{aligned}\Delta \big(\mathbb{I}(a_0 ;a_1,\ldots,a_N;a_{N+1})\big) = \sum_{\substack{0\le k \le N i_0=0<i_1<\cdots<i_k<i_{k+1}=N+1}}\prod_{p=0}^k \mathbb{I}(a_{i_p};a_{i_p+1},\ldots,a_{i_{p+1}-1};a_{i_{p+1}}) &\otimes \mathbb{I}(a_0;a_{i_1},\ldots,a_{i_k};a_{N+1}),\end{aligned}
V ( y_{i}, \theta ) = - V_{p + 1}( i )^{p}\int_{0}^{\infty}\frac{d t}{t}( 8 \pi^{2}\alpha^{\prime}t )^{- \frac{p + 1}{2}}e^{- \frac{y^{2}}{2 \pi \alpha^{\prime}}t}1 6 t^{4}\simeq y^{p - 7}.
[n]\setminus\big(\underbrace{\bigcup_{i\in[k]}V(E_{P_i})}_{\supseteq I\cup J}\big)=(J\cup I)^c\setminus\big(\bigcup_{i\in[k]}V(E_{P_i})\big)=J^c\cap I^c\setminus\big(\bigcup_{i\in[k]}{V(E_{P_i})}\big)
\Omega_{i}\; = \; - \textstyle{\frac{1}{4 \sqrt{3}}}\gamma^{+}\left[ \lambda_{j}\gamma^{j}\gamma_{i}+ \lambda_{i}\right] \; , \; \Omega^{-}\; = \; \textstyle{\frac{1}{4 \sqrt{3}}}\lambda_{i}\gamma^{i}\left( \gamma^{+}\gamma^{-}+ 1 \right) \; .
(\gamma_{2})^{T}=\alpha(\begin{matrix}0.8834\\ 0.1166\end{matrix})\circ(\begin{matrix}0.3763\\ 0.6237\end{matrix})=\alpha(\begin{matrix}0.3324\\ 0.0728\end{matrix})=(\begin{matrix}0.8204\\ 0.1796\end{matrix})
\begin{pmatrix}A_1 a_1 b_1^* A_2 - \gamma_A b_2 b_1^* \end{pmatrix}\begin{pmatrix}x_1 x_2 \end{pmatrix}= \begin{pmatrix}y_1 y_2 \end{pmatrix}, (A_2 - \gamma_A b_2 b_1^*) x_2 = y_2, A_1 x_1 = y_1 - a_1 b_1^* x_2
\big(Q^{(n)}_{\gamma\alpha}\big)_{1\leq\alpha\leq M}=\, \tfrac{1}{2}\big(A_{nn}^{\top}\big)^{-1}\Big(\big(P_{\gamma\mu}\big)_{1\leq\mu\leq M}\Big) begin{align*}4pt] -\sum_{s=1}^{n-1}\big(A_{nn}^{\top}\big)^{-1}\,\Big(\big(a_{sn}^{\beta\mu}Q^{(s)}_{\gamma\beta}\big)_{1\leq\mu\leq M}\Big)\,\,\mbox{in}\,\,{\mathbb{R}}^{n-1}.
\int\limits_0^\infty{\sum\limits_{k = 0}^\infty{\phi \left({2k}\right){{\left({-{x^2}}\right)}^k}\left[{g\left({nx}\right) + g\left({- nx}\right)}\right]}}dx = \frac{\pi}{2}\sum\limits_{k = 0}^\infty{\frac{{\phi \left({- k - 1}\right)\psi \left( k \right){{\left({- n}\right)}^k}}}{{k!}}.}
\int_{0}^{\infty}{\frac{x^{\nu}J_{\nu}( x )}{e^{2 \pi x / b}- 1}F_{1}( x ) d x}= \frac{2}{\pi}\sum_{k = 0}^{\infty}{'}( b k )^{\nu}K_{\nu}( b k ) F_{1}( i b k ) - \frac{1}{\pi}\int_{0}^{\infty}{x^{\nu}K_{\nu}( x ) F_{1}( i x ) d x},
\begin{array}{l}{\bf X}^{a+1}_{a,a+2}{\bf U}^{'}_{a+1}+c_{2a}(u_{+}){\bf D}_{a+1}{\bf X}^{'a+1}_{a,a+2}=c_{2a}(u_{-}){\bf X}^{'a+1}_{a,a+2}{\bf U}_{a+1}\mbox{}+c_{2a}(u_{+})c_{2a}(u_{-}){\bf D}^{'}_{a+1}{\bf X}^{a+1}_{a,a+2}+c_{3a}(u^{'},u){\bf U}_{a-1}^{'}{\bf X}^{a+1}_{a,a+2}\end{array}
\ \sum_{k=4}^{n-4}\frac{(2n-8)!}{(2k-4)!(2n-2k-4)!}\frac{B_{2k}}{2k}\frac{B_{2n-2k}}{2n-2k}= -\frac{(n-6)(2n+1)(2n^2-11n+35)}{140(n-2)(n-3)(2n-5)(2n-7)}\frac{B_{2n}}{2n}-\frac{2n-11}{6}B_6 B_{2n-6}, n\ge 8.
\left( \begin{array}{c}3n n\end{array}\right) -\left( \begin{array}{c}2n n\end{array}\right) =\sum_{j=0}^{n-1}\left( \begin{array}{c}n+j j\end{array}\right) \left( \begin{array}{c}2n-j-1 n-j\end{array}\right) .
{\Phi}= \left( \begin{array}{c}l_{I}\cdot{\bf e}_{\mu}^{(R)}{\bar k}_{i}\cdot{\bf e}_{\mu}^{(R)}k^{j}\cdot{\bf e}_{\mu}^{(R)}\end{array}\right)={\frac{1}{2}}\left( \begin{array}{c}- A E^{*}DE^{*}E^{*}\end{array}\right)\; \in M(28,\;6,\;{\bf R})
\hat{U}( \theta_{0}) \, \hat{A}_{i}^{\pm \pm}( n_{1}, n_{2}) \, \hat{U}^{- 1}( \theta_{0}) = \hat{A}_{i}^{\pm \pm}( n_{1}, n_{2}) \, + \, \Big [ \frac{i}{\hbar}\int_{T_{2}}d^{2}\vec{x}\ \theta_{0}( \vec{x}) \hat{\phi}( \vec{x}) \ , \ \hat{A}_{i}^{\pm \pm}( n_{1}, n_{2}) \Big ] \ \ \ .
\begin{array}{ccc}\rho^* \circ F(W_1) \stackrel{\psi(W_1)}{\Longrightarrow}\rho^* \circ G(W_1) \makebox[0pt][r]{$\scriptstyle{\chi^F_{\rho}}$}\Downarrow \Downarrow \makebox[0pt][l]{$\scriptstyle{\chi^G_{\rho}}$}F(W_2) \circ \rho^* \stackrel{\psi(W_2)}{\Longrightarrow}G(W_2) \circ \rho^* \end{array}
\mathcal{L}_{os}=\frac{1}{2}\left( \partial_{\mu}\eta \partial^{\mu}\eta+2\mu^{2}\eta^{2}\right) -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{q^{2}v^{2}}{2}A_{\mu}^{\prime}A^{\prime \mu}~.
\partial_{\alpha}f^{j}=\left( -1\right)^{j}{\displaystyle\sum\limits_{k=0}^{j-1}}\frac{\left( -1\right)^{k}F_{\alpha}\left( f^{k},f^{k+1}\right)}{F_{y}\left( f^{k},f^{k+1}\right)}{\displaystyle\prod\limits_{i>k}^{j-1}}\frac{F_{x}\left( f^{i},f^{i+1}\right)}{F_{y}\left( f^{i},f^{i+1}\right)}.
\|\sigma\|_{(h\circ f)_*E}^2=(2\pi)^{-\dim Z}\int_{Z_b}|\sigma|^2 \frac{(\omega^Z)^{\dim Z}}{(\dim Z)!}, \|\sigma\|_{h_*f_*E}^2=(2\pi)^{-\dim Z}\int_{Y_b}\Big( \int_{X}|\sigma|^2 \frac{(\omega^X)^{\dim X}}{(\dim X)!}\Big) \frac{(\omega^Y)^{\dim Y}}{(\dim Y)!},
\delta_x^+ (f(\psi(x_j,t_n)) - f(\psi^n_j)) = \int_0^1 |\phi_{1,j}(\theta)|^2 \delta_x^+ \psi(x_j,t_n) -|\phi_{2,j}(\theta)|^2 \delta_x^+ \psi^n_j d\theta +\int_0^1 (\phi_{1,j}(\theta))^2 \delta_x^+ \overline{\psi(x_j,t_n)}- (\phi_{2,j}(\theta))^2 \delta_x^+ \overline{\psi^n_j}d\theta,
\mathcal I_p^{\sharp}(\mathbf h, \breve{\mathbf g}, \pmb{\phi}) = \frac{L(1,\pi_p,\mathrm{ad})L(1,\tau_p,\mathrm{ad})}{L(1/2,\pi_p\times \mathrm{ad}\tau_p)}\alpha_p^{\sharp}(\mathbf h, \breve{\mathbf g}, \pmb{\phi}).
0 \leq A := \frac{\varrho_1-\varrho_2}{(\varrho_1 - \varrho_2) + (p_1 - p_2)}\leq 1 0 \leq B := \frac{p_1-p_2}{(\varrho_1 - \varrho_2) + (p_1 - p_2)}\leq 1.
\delta_{ij}=\sum_{t=j}^{i}\frac{a_{i}}{a_{t}}\left(\prod_{k=i+1}^{N}\left(1-\frac{a_{k}}{a_{t}}\right)\right)\cdot\left(\overset{N}{\underset{\genfrac{}{}{0pt}{}{k=j}{k\neq t}}{\prod}}\frac{1}{1-\frac{a_{k}}{a_{t}}}\right),
\int_{0}^{\infty}d x \, x^{2}\log \left[ 1 - e^{- \left( x^{2}+ m_{e f f}^{2}\beta^{2}\right)^{1 / 2}}\right] = - \left( m_{e f f}\beta \right)^{2}\, \sum_{n = 1}^{\infty}\frac{1}{n^{2}}K_{2}\left( m_{e f f}\beta n \right) .
(\nabla_{X}f)Y =\nabla_{X}(fY)-f(\nabla_{X}Y), (\overline{D}_{X}h)Y =D_{X}(hY)-h(\nabla_{X}Y), (\overline{\nabla}_{X}t)V =\nabla_{X}(tV)-t(D_{X}V), (D_{X}s)V =D_{X}(sV)-s(D_{X}V).
\frac{h_0}{t+1}+ \frac{C}{2}= \frac{h_0}{\sqrt{t+1}}\left( \frac{1}{\sqrt{t+1}}+ \frac{C}{2h_0}\sqrt{t+1}\right) &\leq \frac{h_0}{\sqrt{t+1}}\left( \frac{1}{\sqrt{t+1}}+ \sqrt{\frac{C}{2h_0}}\right) &\leq \frac{h_0}{\sqrt{t+1}}\left( \frac{1}{\sqrt{t+1}}+ 1 \right) &\leq \frac{2h_0}{\sqrt{t+1}}.