harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal E}= \frac{1}{- g^{tt}}\left\{- g^{t \phi}p_\phi + \left[ \left(g^{t \phi}p_\phi \right)^2 + ( - g^{tt}) \left( \mu^2 + g^{\phi \phi}p_\phi^2 + g^{rr}p_r^2 + g^{\theta \theta}p_\theta^2 \right) \right]^{1/2}\right\}- q A_t.
2P-I=\bigoplus_{v\in C_0}\frac{1}{2}J_2-I=\begin{pmatrix}\frac{1}{2}J_2-I begin{align*}3mm] \frac{1}{2}J_2-I begin{align*}3mm] \ddots begin{align*}3mm] \frac{1}{2}J_2-I\end{pmatrix},
(\mathrm{2SFPPA})\begin{cases}x_j^{k+1}= \mathrm{prox}_{\frac{\alpha_j}{\beta}f_j}(x_j^k-\alpha_j A_j^\top(\sum_{i=1}^{j-1}A_ix_i^{k+1}+A_jx_j^k + \sum_{i=j+1}^s A_i(2x_i^k-x_i^{k-1})-b)-\frac{\alpha_j}{\beta}A_j^\top y^k), ~~ j\in\mathbb{N}_s, y^{k+1}= y^k+\beta(\sum_{i=1}^s A_ix_i^{k+1}-b). \end{cases}
\mathbf{t}=\left(\mathbf{t}_{1}^{\prime},\mathbf{t}_{2}^{\prime}\right)^{\prime}=\left(t_{1}^{1},t_{1}^{2},\ldots,t_{1}^{n},t_{2}^{1},t_{2}^{2},\ldots,t_{2}^{m}\right)^{\prime}\in\mathbb{R}^{\left(n+m\right)\times1}
\begin{matrix}{{}}&{{M \ \mathrm{i s ~ c o m p l e m e n t e d ~ a s ~ a ~ s u b s p a c e ~ o f ~ t h e ~ n o r m e d ~ s p a c e ~ o f}}}\\{{}}&{{\mathrm{a l l ~ o p e r a t o r s ~ i n ~ a ~ H i l b e r t ~ s p a c e .}\hfill}}\\ \end{matrix}
S_1(z, q, N)=\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]_{q^2}\frac{(q;q^2)_{n-1}(q^2;q^2)_{n}(zq^2;q^2)_{N-n}z^nq^{2n-1}}{(zq;q^2)_{n}(zq^2;q^2)_{N}}.
{}d^{n-4}\,\hat{h}_\phi(\beta) =\lim_{\substack{h_X(P)\rightarrow\infty P\in I_{X,\phi}(K)}}d^{n-4}\; \frac{\hat{h}_{\phi_P}(\beta_P)}{h_X(P)}begin{align*}4pt] \leq\lim_{\substack{h_X(P)\rightarrow\infty P\in I_{X,\phi}(K)}}\frac{(\kappa_1+\kappa_3)\, h_X(P)+(\kappa_2+\kappa_4)}{h_X(P)}=\kappa_1+\kappa_3.
\begin{aligned}Vol(B(t)\setminus B(\frac{t}{2}+d),g^p) =\int_{B(t)\setminus B(\frac{t}{2}+d)}dV_{g^p}\leq e^{(n+1)d}\int_{B(t)\setminus B(\frac{t}{2}+d)}dV_{g^q}&=e^{(n+1)d}\cdot Vol(B(t)\setminus B(\frac{t}{2}+d),g^q) \leq e^{(n+1)d}\cdot Vol(D(t+d)\setminus D(\frac{t}{2}),g^q).\end{aligned}
Q_1(\lambda)=\begin{bmatrix}\lambda^2 P_5+\lambda P_4+P_3 -\lambda(I_n+P_8P_0) \lambda^2 P_8 \lambda^2 P_7+\lambda P_6 I_n P_0 -\lambda I_n \lambda^2 I_n 0 \lambda P_2+P_1 I_n 0 -\lambda^2 I_n \lambda^2 I_n 0 -I_n 0\end{bmatrix},
S=\int d^4x \sqrt{|g_4|}\left[ e^{-\Phi}\left( R+\left( \nabla \Phi \right)^2 -6 \left( \nabla \beta \right)^2 -\frac{1}{2}e^{2\Phi}\left( \nabla \sigma \right)^2 \right) -\frac{1}{2}Q^2 e^{-6\beta}\right] .
\Omega_4= G_m \sum_{z\in \mathbb{F}_p^*}\eta(z)\sum_{y\in \mathbb{F}_p^*}\zeta_p^{-y}\sum_{\delta \in \mathbb{F}_p^*}\zeta_p^{- \frac{A}{4z}\delta^2 -\big( \frac{By}{2z}+\rho \big)\delta}.
\bar{R}_{ikjl}(q)\int_{\mathbb{R}_{+}^{n}}z_{k}z_{l}\frac{\partial U}{\partial z_{i}}\frac{\partial U}{\partial z_{j}}dzdt =\bar{R}_{ikjl}(q)\int_{\mathbb{R}_{+}^{n}}\frac{z_{i}z_{j}z_{k}z_{l}dzdt}{\left[(1+t)^{2}+|z|^{2}\right]^{\frac{n}{2}}}=\frac{\alpha}{3}\left(R_{ikik}(q)+R_{ikki}(q)+R_{iijj}(q)\right)=0
|\mathcal{AC}+\mathcal{AC}||\mathcal{BC}+\mathcal{BC}| =|(\mathcal{AC}+\mathcal{AC})\times{(\mathcal{BC}+\mathcal{BC})}| &\geq{\sum_{(l_i,l_j)\in{E(T)}}|S_{ij}|}&=\sum_{(l_i,l_j)\in{E(T)}}|\mathcal{C}'|^2 &\gg{\sum_{(l_i,l_j)\in{E(T)}}|\mathcal{C}|^2},
\int \frac{d t}{t^{2}}\langle V ( z_{1}) . . V ( z_{l}) \oint \bf{1}\mathrm{\rangle_{\Sigma^{( 1 )}}\langle V ( z_{l + 1}) . . V ( z_{n}) \oint \bf{1}\mathrm{\rangle_{\Sigma^{( 2 )}}}}
\frac{1}{x_{n}}=\frac{2k_{t,d}+k_{t,c}}{{k_{p}}^{2}[M]^{2}}v_{p}+C_{M}+C_{S}\frac{[S]}{[M]}+C_{I}\frac{[I]}{[M]}+C_{P}\frac{[P]}{[M]}+C_{T}\frac{[T]}{[M]}
A_{\mu}d x^{\mu}= \frac{1}{2 e r}\left( t_{\varphi}^{n}( H_{1}d r + ( 1 - H_{2}) r d \theta - n ( t_{r}^{n}H_{3}+ t_{\theta}^{n}( 1 - H_{4}) ) r \sin \theta d \varphi \right)
\Pi_{2 , 2}^{0 , \beta}( m_{0}, i \Omega , \vec{k}= 0 ) = - \frac{\lambda^{2}I_{\beta}^{\epsilon}( m_{0})}{2 ( 4 \pi )^{2}}\left[ \frac{1}{\tilde{\epsilon}}- \ln \left( \frac{m_{0}^{2}}{\mu^{2}}\right) + 2 \right] + F ( \Omega^{2}) ,
Q_{3,\vec{v}_{s,t,f,0}^{(2)}}= \left(\left(\begin{array}{rrr}t^{2}+ s t 1 s t + 1 s 0 t 1 0\end{array}\right), \left(\begin{array}{rrr}t 1 0 s 0 1 1 0 0\end{array}\right), \left(\begin{array}{rrr}1 0 0 0 1 0 0 0 1\end{array}\right)\right).
\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_{conv}:=\left\{\psi:Y\to\mathbb{R}\ |\ \psi\left(y\right)= \left\langle v,y\right\rangle_Y + d ,v\in Y^{*}, d\in\mathbb{R}\right\}.
\epsilon ({A d S}) = \left[ \sqrt{f^{1 / 2}+ 1}- \sqrt{f^{1 / 2}- 1}\gamma_{\overline{{{r}}}}\right] e^{-{\frac{1}{2}}\gamma_{\overline{{{\theta r}}}}}e^{-{\frac{1}{2}}\gamma_{\overline{{{\phi \theta}}}}}e^{-{\frac{g t}{\sqrt{2}}}\gamma_{\overline{{{0}}}}}\epsilon_{0}.
K = \mathrm{T r}\left( \frac{1}{2}\bar{W}W + \tilde{q}^{+}\nabla^{--}q^{+}- q^{+}\nabla^{--}\tilde{q}^{+}\right) = \mathrm{T r}\left( \bar{\phi}\phi + \bar{q}^{i}q_{i}\right) + \theta \ \mathrm{t e r m s}\ .
\Pr \left \{X \in \bigcup_{i=1}^{j_0}B_{D}(y_{\pi (i)}) \right \}\overset{(a)}{\leq}\Pr \left \{X \in \bigcup_{i=1}^{j_0}B_{D}(y_i) \right \}& \overset{(b)}{=}\sum^{j_0}_{i=1}\Pr \{X \in A_D (y_i) \}& \overset{(c)}{=}\sum^{j_0}_{i=1}P_{\hat{Y}}(y_i) - \epsilon,
D\Theta_1(\varphi).v =\left.\frac{\partial}{\partial t}\right|_{t:=0}\Xi^{\omega}(\varphi+tv)=DF^{\omega}(\varphi,T\varphi).v= &=D_{.1}F^{\omega}(\varphi,T\varphi).v+D_{.2}F^{\omega}(\varphi,T\varphi).Tv\in\Lambda((TX^{0,1})^*),
\Gamma_{\mu \nu \lambda}^{k l n}( p , q , r ) = - ( 2 \pi )^{4}\left( \frac{2}{g^{\prime}}C_{k l}^{n}\right) \left[ \delta_{\mu \lambda}( r + q - p )_{\nu}+ \delta_{\nu \lambda}( q - r )_{\mu}+ \delta_{\mu \nu}( p - q - r )_{\lambda}\right] \, .
I(0,0,N-k)\,=\, 2\pi \int_0^{\infty}\frac{r}{(1+r^2)^{N-k+2}}\,dr\,=\, \frac{-\pi}{N-k+1}\ \bigg[\frac{1}{(1+r^2)^{N-k+1}}\bigg]_0^{\infty}=\, \frac{\pi}{N-k+1}
|\lambda \rangle \circledast |\mu \rangle =\sum_{\alpha \in \Pi_{n,k}}s_{\mu}(y_{\alpha}|t)~(\tilde{S}_{\lambda}Y_{\alpha})=\sum_{\alpha \in \Pi_{n,k}}(j_{\hat{w}^{\lambda}}s_{\mu}(y_{\alpha}|t))~Y_{\alpha}\;.
\Delta(t,x)\preceq \ (1-\epsilon) \big(\delta\big[\Phi'(x-\underline h(t))+\Phi'(-x-\underline h(t))\big]+2mL\epsilon_2\mathbf{1}\big) \preceq \ (1-\epsilon)\big(-\delta \epsilon_1 +2mL\epsilon_2 \big)\mathbf{1}= \mathbf{0}.
\mathbb{E}e^{-uM(T(t))}= \int_0^{\infty}\mathbb{E}e^{-uM(s)}\Pr \{T(t)\in ds \}&= \int_0^{\infty}e^{-sg(u)}\Pr \{T(t)\in ds \}&= e^{-\int_0^t f(g(u),\tau)d\tau}.
\int_{\mathbb{T}\times\mathbb{R}}(1+|y|)\cdot|\omega(0,x,y)|\,\mathrm{d}x\mathrm{d}y\leq\varepsilon,\quad\int_{\mathbb{T}\times\mathbb{R}}\omega(0,x,y)\,\mathrm{d}x\mathrm{d}y=0,
1 = S(\zeta) = \prod_{i=1}^h (\zeta^{p^{s_i}-1}+ \ldots + \zeta + 1) = \prod_{i=1}^h\frac{\zeta^{p^{s_i}}-1}{\zeta - 1}= \prod_{i=1}^h\frac{\zeta^{p^{t_i}}-1}{\zeta - 1}.
\langle v,v \rangle = \left\langle \sum^n_{j=1}\langle v, v_j \rangle v_j , v \right \rangle =\sum^n_{j=1}\langle v, v_j \rangle \langle v_j , v \rangle =\sum_{j=1}^{n}\|\langle v,v_j\rangle\|^2
C^{\lambda}_k(t) =\frac{1}{k}\left(2(k+\lambda-1)t\,C^{\lambda}_{k-1}(t)-(k+2\lambda-2)C^{\lambda}_{k-2}(t)\right),k\ge2, C^{\lambda}_0(t) =1,C^{\lambda}_1(t)=2\lambda t.
\psi (u) = \frac{1}{p}\iint_{M \times M}\frac{|u(x)-u(y)|^{p}}{(d_g(x,y))^{N+ps}}d\mu_g(x)d\mu_g(y) +\frac{1}{p}\int_{\Omega}\frac{|u(x)|^{p}}{(d_g(x,y))^{N+ps}}d\mu_g(x) - \int_{\Omega}F(x, u(x))d\mu_g(x) \geq \frac{1}{p}\|u\|^p_{W_0^{s,p}(M)}- \varepsilon C_{1}(p,q)\|u\|^p_{W_0^{s,p}(M)}- C C_{1}(p,q)\|u\|^q_{W_0^{s,p}(M)}.
I_{2 2}( y , a , b , d ) = \frac{1}{a}h ( d ) \sum_{n = 1}^{\infty}\int_{\alpha}^{\infty}d v ( v^{2}- m^{2}- ( \frac{n \pi}{a})^{2})^{\frac{d - 4}{2}}( \coth b v - 1 ) \cosh 2 v y .
A_{fN,\bar{d}_m}= \pm \chi(N) \cdot (\bar{d}_m,f) \cdot [k:k^{fN}] \cdot \prod_{q \mid N}\big( q- \chi(\sigma_q e_q) \big) &= \pm \chi(N) \cdot [k:k^{fN}] \cdot \prod_{q \mid (\bar{d}_m,fN)}(q- \chi(\sigma_qe_q)),
\mu_\mathbb{C}= \frac{4}{3}\biggl(\biggl[\frac{\sqrt{3}}{2}e_1+\frac{1}{2}e_2\biggr]^{\otimes 3}+ \biggl[-\frac{\sqrt{3}}{2}e_1+\frac{1}{2}e_2\biggr]^{\otimes 3}+ (-e_2)^{\otimes 3}\biggr)
\int_{\Omega_{2}}\partial_{1}\phi_{n}(x)\partial_{1}\phi_{m}(x)\,dx = &{4n_{1}m_{1}\over \pi^2}\int_{0}^\pi\int_{\ell_{2}}^{\ell_{2}+\delta_{2}}\cos(n_{1}x_{1})\cos(m_{1}x_{1})\, dx_{1}\sin(n_{2}x_{2})\sin(m_{2}x_{2})\, dx_{2}= 0.
I(f(X^n);Y_W^n)=I(f(X^n),X^n;Y_W^n)-I(X^n;Y_W^n|f(X^n)) &=I(X^n;Y_W^n)+I(f(X^n);Y_W^n|X^n)-I(X^n;Y_W^n|f(X^n)) &=I(X^n;Y_W^n) &\ \ \ -\sum_{m=1}^M \Pr(f(X^n)=m)I(X^n;Y_W^n|f(X^n)=m).
-{\cal A}_{\alpha}i \Omega ={\cal D}_{\alpha}{\cal V}\Omega{\cal V}^{T}= \left( \begin{matrix}{\langle{\cal D}_{\alpha}U_{\beta}, U_{\gamma}\rangle}&{\langle{\cal D}_{\alpha}U_{\beta}, \bar{U}^{\gamma}\rangle}\\{\langle{\cal D}_{\alpha}\bar{U}^{\beta}, U_{\gamma}\rangle}&{\langle{\cal D}_{\alpha}\bar{U}^{\beta}, \bar{U}^{\gamma}\rangle}\\ \end{matrix}\right)
\frac{\mathrm{t r}{\vphantom |}_{0}\bigl ({\cal K}^{+}( 0 ) \, \widehat{H}_{0 , N}\bigr )}{\mathrm{t r}\,{\cal K}^{+}( 0 )}= \frac{\mathrm{t r}{\vphantom |}_{0}\bigl ( q^{- 2 S_{0}^{3}}\, \widehat{H}_{0 , N}\bigr )}{\mathrm{t r}\, q^{- 2 S^{3}}}= - \mathrm{i}\frac{\sin \gamma}{2}S_{N}^{3}
Q =\left(\begin{array}{cccc}-1 1 0 0 0 -1 1 0 0 0 -1 1 1 0 0 -1 \end{array}\right)\mbox{and}D =\left(\begin{array}{cccc}\displaystyle0.01 0 \, \,0 \,\,\,\,0 0 0.01 \, \,0 \, \,\,\,0 0 0 \,\,1 \,\,\,\,0 0 0 \,\,0 \,\,\,\,1 \end{array}\right).
\tilde{D}_{k}^{[1]}=[\begin{matrix}I_{m}&0&V_{2k-1}^{(t)}\\ 0&I_{m}&V_{2k-1}^{(b)}&0\\ 0&W_{2k}^{(t)}&I_{m}&0\\ &W_{2k}^{(b)}&0&I_{m}\end{matrix}]
f = \sum_{i=1}^s \prod_{j=1}^d (\lambda_{i,j,0}x_0 + \lambda_{i,j,1}x_1 + \cdots + \lambda_{i,j,n}x_n) = \sum_{i=1}^s \ell_{i,1}\ell_{i,2}\cdots \ell_{i,d}.
b_n(x)= a_n(x)+\alpha_{2n+1}a_{n-1}(x)+\left(a+p+\beta_{2n+1}\right)R_n(x)+\gamma_{2n}R_{n-1}(x) &P_{n+1}(x)=R_{n+1}(x)+\alpha_{2n+2}R_n(x)+\left(a+p+\beta_{2n+2}\right)a_n(x)+\gamma_{2n+1}a_{n-1}(x)
\langle . . . \rangle = \frac{1}{Z ( t )}\int \frac{d^{4}k}{( 2 \pi )^{4}}\sum_{n = 0}^{\infty}e^{- \omega_{n}^{2}t}e^{- k^{2}t / 2}. . . e^{- k^{2}t / 2},
\sum_{\pi :\lambda_{\pi_{\infty}}\in t\Omega}\chi_{\pi_p}(\phi_\beta) =\Lambda_\Omega(t) \int_{\chi\in \mathcal{X}_p^{temp}/W}\chi(\phi_{\beta})\, d\nu_p(\chi)+O_\Omega \Big(\|\beta\|_{L^1}t^{d-\delta}\Big),
S\big(L_i(x)\big) = F_i\big(x, S(x)\big), = \alpha_i S(x) + \frac{A_i(1-\theta)^3 + B_i\theta (1- \theta)^2 + C_i \theta^2 (1-\theta) + D_i\theta^3}{1 + (r_i -3) \theta (1-\theta)}.
B^{0}=\{\tau(\beta)>t+h\}\cap\biggl\{\frac{1}{N}\sum_{j=1}^{N}\bigl(M_{t+h}^{j}\bigr)^{2}\leq h^{-1/8}\biggr\}&B^{2}=\biggl\{\frac{1}{N}\sum_{i=1}^{N}{\mathbf{1}}_{B^{i,2}}\leq h\biggr\}, B^{i,2}=\Bigl\{Ch\bigl(1+\sup_{s \in [0,t+h]}\vert Z_{s}^{i}\vert\bigr)+\sup_{s \in [t,t+h]}\vert W_{s}^{i}-W_{t}^{i}\vert\geq h^{1/16}\Bigr\}.
V_{eff}= -\frac{\left( \rho_1 + \lambda_1 \left(\frac{z_v}{z_h}\right)^{2-\nu}\right)^2}{2 (kz_v)^4 (\rho_2 + \lambda_2^{*(0)})}+ \frac{\alpha^2}{(k z_v)^4}P \left( (z_v/z_h)^{-\epsilon}, \left( \alpha \ln (z_v/z_h) \right) \right)
\left|\frac{\delta F}{\delta \mu}(\nu_t, \mu_t, y) - \frac{\sigma^2}{2}\log \left( \frac{\mu_t(y)}{\rho(y)}\right) + \frac{\sigma^2}{2}\operatorname{D_{KL}}(\mu_t|\rho)\right| &\leq 3C_{\mu}+ \frac{\sigma^2}{2}\left( \max\{|\log r_{1, \mu}|, \log R_{1, \mu}\}+ 2 \log R_{\mu}\right) =: C_{V, \mu}.
\left(\partial_{0}M_{0}+M_{1}+A\right)\left(\begin{array}{c}p \left(\begin{array}{c}v \eta_{1}\eta_{2}\end{array}\right)\end{array}\right)=\left(\begin{array}{c}f \left(\begin{array}{c}g h_{1}h_{2}\end{array}\right)\end{array}\right)
d(\sigma^2\mu_n)={}i_0\sigma\mu_n-i_1\sigma\mu_n+\sum_{p+q=n+1}\sigma^2\mu_p\{\mu_q\}&+\!\!\!\!\!\!\!\!\sum_{\substack{p,q\geq 0 u\geq p+q+1 \sum s_j+v+\sum t_j=n+p+q+1-u}}\!\!\!\!\!\!\!\!\mu_u\{i_0\sigma\mu_{s_1},\dots,i_0\sigma\mu_{s_p},\sigma^2\mu_v,i_1\sigma\mu_{t_1},\dots,i_1\sigma\mu_{t_q}\}.
\frac{1}{\rho(x)}=2q\big(\varphi_{1}'(x)\chi_{1}(x)-\varphi_{1}(x)\chi_{1}'(x)\big)+x\big(\varphi_{1}'(x)\big)^{2}-\varphi_{1}(x)\varphi_{1}'(x)-x\varphi_{1}(x)\varphi_{1}''(x).
Y_{t_{n}}^{t_{n},X^{n}}=\mathbb{\hat{E}}_{t_{n}}^{X^{n}}\left[ Y_{t_{n+1}}^{t_{n},X^{n}}+\int_{t_{n}}^{t_{n+1}}f_{t}^{t_{n},X^{n}}dt+\int_{t_{n}}^{t_{n+1}}g_{t}^{t_{n},X^{n}}d\langle B\rangle_{t}\right] .
\begin{array}{r c l}{{({\bf 1})_{a b c d}}}&{{=}}&{{S_{a c}S_{b d}}}\\{{( Q - 1 ) ( P_{0})_{a b c d}}}&{{=}}&{{S_{a b}S_{c d}}}\\{{\mathrm{a n d \; \;}( Q - 2 ) ( P_{1})_{a b c d}}}&{{=}}&{{Q \sum_{\alpha}S_{a \alpha}S_{b \alpha}S_{\alpha c}S_{\alpha d}- S_{a b}S_{c d}\; .}}\\ \end{array}
\sigma_n(D_1\cup D_2) \geq\mu_n\left(\overline{\bigcup_{j\in E}C_{p_j}\setminus C_{p_{j-1}}}\right) &=\mu_n\left(\overline{\bigcup_{j\in E_1}C_{p_j}\setminus C_{p_{j-1}}}\right)+\mu_n\left(\overline{\bigcup_{j\in E_2}C_{p_j}\setminus C_{p_{j-1}}}\right) &\geq\sigma_n(D_1)+\sigma_n(D_2)-\delta
0=v_{k+1}^{(0)}\cdots v_{k+1}^{(n_{k+1})}=u_{k+1}(u_{k+1}+y_1)\cdots (u_{k+1}+y_{n_{k+1}}) =u_{k+1}^{n_{k+1}+1}+u_{k+1}^{n_{k+1}}a_1+\cdots+u_{k+1}a_{n_{k+1}},
C_{1}(f)=\lim_{n\to\infty}(-1)^{n}\left(\frac{a}{q}\right)^{\! n/2}\! f_{n},\C_{2}(f)=\lim_{n\to\infty}\left((-1)^{n}f_{n}-C_{1}(f)\left(\frac{q}{a}\right)^{\! n/2}\right)\!(qa)^{-n/2}
\begin{cases}L(U^\pm, \Psi^\pm)U^\pm=0, ~\{y>0\}{\mathcal B}(U^+, U^-, \psi)=0, ~\{y=0\}U^\pm|_{x=0}=U_0^\pm(y,z), \psi|_{x=0}=\psi_0(z)\end{cases}
\begin{array}{lll}\overline x_i = (x_{r+\overline m}(1)+\alpha)^{\overline c\gamma_i}x_i(1)\mbox{for}1\le i\le r\mbox{and}\overline x_{r+\overline m}= (x_{r+\overline m}(1)+\alpha)^{\overline c}-\overline \alpha\end{array}
\sum_{j=0}^r (-1)^{\xi_{ij}}\binom{r}{j}_{qp}(qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij}= \sum_{b=0}^r (-1)^{a+b}\binom{r}{b}_{qp}(qp)^{\frac{(a+b)(a+b-1)}{2}+ar-ab}& = (-1)^a (qp)^{ar + \frac{a(a-1)}{2}}\sum_{b=0}^r (-1)^b \binom{r}{b}_{qp}(qp)^{\frac{b(b-1)}{2}},
\begin{array}{rcl}\tilde{f}_{1,i}= \tilde{f}_{L,i}\cos\theta_{\tilde{f}_i}+ \tilde{f}_{R,i}\sin\theta_{\tilde{f}_i}\tilde{f}_{2,i}= - \tilde{f}_{L,i}\sin\theta_{\tilde{f}_i}+ \tilde{f}_{R,i}\cos\theta_{\tilde{f}_i}\end{array}\quad .
I_{x, \delta}= \lim_{L \rightarrow \infty}I_{x,L, \delta}= \lim_{L \rightarrow \infty}\hat c_1 (\mathcal G) \int_{\delta L^2}^{L^2/\delta}\frac{1}{t}\phi_{\hat \sigma}\left( \frac{x L}{\sqrt t}, \frac{L}{\sqrt t}\right)dt
p\left({d_{p2}\leq \delta}\right) = p\left({d_{p2}^2 \leq \delta^2}\right) = M\iint\limits_\Omega \sin^{2\left({n - 4}\right)}\theta_1 \sin^{2\left({n - 4}\right)}\theta_2 \,\,\, \times \left({\sin^2 \theta_1 - \sin^2 \theta_2}\right)^2 d\sin^2 \theta_1 d\sin^2 \theta_2
\begin{cases}u^{\pm}\partial_{\tilde x}\Psi^{\pm}-v^{\pm}+w^{\pm}\partial_{\tilde z}\Psi^{\pm}=0,\tilde x>0 \Psi^\pm(0, \tilde y, \tilde z)=\pm \tilde y+\tilde{\Psi}^\pm_0(\tilde y,\tilde z),\end{cases}
(\nabla_i \nabla_i P) (Z) = \nabla_i [(\nabla_i P)(Z)] - \nabla P (\nabla_{e_i}e_i, Z) - (\nabla_i P)(\nabla_i Z) &= \nabla_i (\nabla_i (PZ)) - \nabla_i (P (\nabla_i Z))- \nabla P (\nabla_{e_i}e_i, Z) - (\nabla_i P)(\nabla_i Z) &= \nabla_i (\nabla_i (PZ)) &= \bar\nabla_{J_i}\bar\nabla_{J_i}X (0),
z =x_1x_{n-d+1}\cdots x_{n-\ell-2}\cdot x_{n-\ell}\cdots x_{n-1}x_n &=x_1\big(\textstyle\prod_{s=\ell+1}^{d-2}x_{n-s-1}\big)\big(\prod_{s=0}^{\ell}x_{n-s}\big).
\int^T_0 \varphi_n (t,\omega) dB(t) = \displaystyle \sum \limits_{i = 0}^{n-1}B(t_i)[B(t_{i+1}) - B(t_i)] & = \displaystyle \sum \limits_{i = 0}^{n-1}\frac{1}{2}[B(t_{i+1})^2 - B(t_{i})^2 - (B(t_{i+1}) - B(t_{i}))^2] & = \frac{1}{2}B(T)^2 - \frac{1}{2}\displaystyle \sum \limits_{i = 0}^{n-1}(B(t_{i+1}) - B(t_{i}))^2
\overline{{\zeta}}_{3}: = \frac{( d - 3 ) k^{4}}{6 ( g ( k^{2}) \pi^{d / 2})^{2}}\int \frac{d P_{1}d P_{2}d P_{3}~ \delta ( k - P_{1})}{P_{1}^{2}P_{2}^{2}P_{3}^{2}( P_{1}- P_{2})^{2}( P_{2}- P_{3})^{2}( P_{3}- P_{1})^{2}}
\frac{\partial}{\partial\mathbf{n}_{+}}u_{m}\left(z\right)=\frac{\partial}{\partial\mathbf{n}_{+}}g_{K_{m}^\pm}\left(z\right)-\frac{\partial}{\partial\mathbf{n}_{+}}g_{K}\left(z\right)=\pi\omega_{K_{m}^\pm}\left(z\right)-\pi\omega_{K}\left(z\right)
u_{d_{i}}(\{j:p(\sigma_{i}(g)(j))\ne \sigma_{i}(g)(p(j))\}) \leq 6\varepsilon+\sum_{r=1}^{L}|T_{r}\setminus g^{-1}T_{r}|\mu(\widetilde{V}_{r}) &\leq 6\varepsilon+\varepsilon\sum_{r=1}^{L}|T_{r}|\mu(\widetilde{V}_{r}) &\leq 6\varepsilon+\varepsilon\mu\left(\bigcup_{r=1}^{L}T_{r}\widetilde{V}_{r}\right) &\leq 7\varepsilon,
\mathbf{y}_r =\mathbf{H}_0\mathbf{w}_1{s}_1+\mathbf{n}_r, y_s^c =\mathbf{h}_{ps}^H\mathbf{w}_1{s}_1+n_s, y_{p,1}^c =\mathbf{h}_{pp}^H\mathbf{w}_1{s}_1+n_p,
[\Theta_L(\eta)*\iota_\epsilon](u) = (2\epsilon L)^{-d}\sum_{i\in\mathbb T_L^d}\mathbf 1_{[\frac{2i-1}{2L},\frac{2i+1}{2L})^d}(u) \sum_{j:|i-j|\le \lfloor\epsilon L\rfloor}\eta(j),
\widehat N^p_C(F(\bar x);\nabla F(\bar x)u) =\widehat N_{T_C(F(\bar x))}(\nabla F(\bar x)u) &=\widehat N_{\bigcup_{i\in J(F(\bar x))}T_{P_i}(F(\bar x))}(\nabla F(\bar x)u)=\bigcap\limits_{i\in J(\bar x;u)}N_{T_{P_i}(F(\bar x))}(\nabla F(\bar x)u)
\phi_{k,n}(y_1,\cdots,y_k) \leq(n/4)^k(2\ln n)^{\frac{k}{2}}\mathbb{P}(\xi^{(n)}(I_{n,k})=0) \leq(n/4)^k(2\ln n)^{\frac{k}{2}}\prod_{j=1}^k\mathbb{P}(\xi^{(n)}(I(y_j, F_n(x_j)))=0) =(n/4)^k(2\ln n)^{\frac{k}{2}}\prod_{j=1}^kD_n(F_n(x_j)/2).
S_{2}[ X , j ] = i g \int d^{4}y j_{\mu}( y ) \int_{0}^{T}d t \, \int d \theta \int \frac{d^{4}k}{( 2 \pi )^{4}}G^{\mu \nu}( k ) D X_{\nu}( t , \theta ) \exp \left\{\, - i k \cdot \left[ X ( t , \theta ) - y \right] \, \right\}.
Q^{( n )}= \sum_{\{r_{i}: \sum_{i = 1}^{N}r_{i}= n \}}\Bigg \{\prod_{i = 1}^{N}\left( \begin{matrix}{\frac{n}{N - 1}}\\{r_{i}}\\ \end{matrix}\right) u_{i}^{r_{i}}\Bigg \}.
v_i(x+y)^{s+r-2-2i}=\sum_{k=0}^i\sum_{j=0}^i\,\nu_k\binom{s+r-2-2i}{r-1-i-j+k}x^{r-1-j}y^{s-1-i+j}=\sum_{j=0}^i\left[\sum_{k=0}^i\,\nu_k\binom{s+r-2-2i}{s-1-i+j-k}\right]x^{r-1-j}y^{s-1-i+j}
X=ax+a\sum_{m\ge1}\binom{m+1}{1}(2b)^m\alpha_m(t), &T_1=\left(\alpha_1(t)+\sum_{m\ge2}\binom{m+1}{2}(2b)^{m-1}\alpha_m(t)\right)a^2, &T_2=\left(\alpha_2(t)+\sum_{m\ge3}\binom{m+1}{3}(2b)^{m-2}\alpha_m(t)\right)a^3, &\quad\ldots
k_1 P_{\nu_{23}, \nu_{12}}=-\frac{1}2\big\langle \nu_{23}, \nu| Q^{(12)}|\nu_{12}, \nu\big\rangle k_2 P_{\nu_{23}, \nu_{12}}=-\frac{1}2\big\langle \nu_{23}, \nu| Q^{(23)}|\nu_{12},\nu\big\rangle.
\sup_{(x,t) \in \Omega^{(j)}, \, \bar{t}-2^{\frac{j}{100}}\leq t \leq \bar{t}}|f_\alpha^{(j)}(x,t)| \leq \max \bigg \{\sup_{(x,t) \in \partial \Omega^{(j)}, \, \bar{t}-2^{\frac{j}{100}}\leq t \leq \bar{t}}|f_\alpha^{(j)}(x,t)|,\sup_{(x,t) \in \Omega^{(j)}, \, t=\bar{t}-2^{\frac{j}{100}}}|f_\alpha^{(j)}(x,t)| \bigg \}\leq C \, 2^{-\frac{j}{4}}
\prod_{r \in R_l/\{\pm 1\}}(Z_1 + z_2 + Z_3, r)= \prod_{r \in R_l/\{\pm 1\}}(Z_3, r) = \prod_{r \in R_l/\{\pm 1\}}(J'(Z_3), J'(r)) = \prod_{r \in R_l/\{\pm 1\}}(J'(Z_3), r).
\pi_I(\vartheta|X) = \bigotimes_{i=1}^n \mathrm{N}\Big(\frac{\tau_i^2(I) X_i+\sigma^2\mu_i(I)}{\tau_i^2(I)+\sigma^2}, \frac{\tau_i^2(I)\sigma^2}{\tau_i^2(I)+\sigma^2}\Big).
(c^{2}M_{3}^{2}- m_{2}^{2}G^{2})^{1/2}(c|M_{3}|)^{- 1}=( 1 + 4a^{2}(\omega c^{- 1})^{2}(1 - e^{2})^{- 1}(1 + (1 - (2a\omega c^{- 1})^{2})^{1/2})^{- 2})^{- 1/2}.
\int_0^t f\left (r, W_r, Y_r, \frac{\mathrm d[Y,W]_r}{\mathrm dr}\right) \mathrm dr = \int_0^t f\left (r, W_r, u(r, W_r),\nabla u (r, W_r) \right) \mathrm dr &= \int_0^t \tilde f(r, W_r) \mathrm dr,
\begin{array}{l}J^\prime_\pm\tilde{\psi}_{JM}({\bf x})=\sqrt{[J\mp M]_q[J\pm M+1]_q}\tilde{\psi}_{J,M\pm 1}({\bf x})~, begin{align*}2mm]J^\prime_3\tilde{\psi}_{JM}({\bf x})=M\tilde{\psi}_{JM}({\bf x})~.\end{array}
\begin{tabular}{| c | c | c | c | c |}\hline{n}&{1 , . . . , 1 5}&{1 6 , . . . , 2 1}&{2 2 , . . . , 2 7}&{2 8 , . . . , 3 3}\\ \hline{l}&{1}&{2}&{3}&{4}\\ \hline{}\\ \end{tabular}\ ,
\lim_{\lambda\rightarrow\infty}\frac{P_{\lambda p(p+1)+\rho+1,0}(z^{p+1})}{P_{\lambda p(p+1)+\rho,0}(z^{p+1})}=\widetilde{F}_{0}^{(\rho)}(z^{p+1}),z\in\mathbb{C}\setminus\Gamma_{0}.
\int_0^\infty \frac{1}{a}\exp\left(-\frac{x}{a}\right)\frac{1}{\sqrt{\pi b x}}\exp\left(- \frac{(u-x)^2}{b x}\right) dx &= \frac{1}{\sqrt{a (a + b)}}\exp\left( \frac{2}{b}\left[ u - |u| \sqrt{1+\frac{b}{a}}\right] \right).
{\mathcal W}^{1,2,p, \lambda}( (t_0, T) \times \mathbb{R}^d)) \ = \ \{\varphi \in L^p( t_0, T; W^{2, p, \lambda}(\mathbb{R}^d) | \frac{\partial \varphi}{\partial t}\in L^p( t_0, T; L^{p, \lambda}(\mathbb{R}^d) \}
\mathfrak{S}_{\delta}^{\Lambda*\mathbb{1}}(x, D)= \mathfrak{S}_{\delta, 1}^{\Lambda*\mathbb{1}}+ \mathfrak{S}_{\delta, 2}^{\Lambda*\mathbb{1}}+ \mathfrak{S}_{\delta, 3}^{\Lambda*\mathbb{1}}+ \mathfrak{S}_{\delta, 4}^{\Lambda*\mathbb{1}},
\begin{array}{r c l}{{\chi_{1}}}&{{=}}&{{A \Lambda_{1 , 2}+ B ( \partial \Lambda )_{1 , 2}\, ,}}\\{{\chi_{2}}}&{{=}}&{{\Lambda_{0 , 2}\, ,}}\\{{\chi_{3}}}&{{=}}&{{A^{\prime}\Lambda_{1 , 2}+ B^{\prime}( \partial \Lambda )_{1 , 2}\, ,}}\\{{\chi_{4}}}&{{=}}&{{\Lambda_{2 , 2}\, ,}}\\{{\chi_{5}}}&{{=}}&{{\frac{1}{2}\Lambda_{1 , 2}- \frac{1}{2}( \partial \Lambda )_{1 , 2}\, .}}\\ \end{array}
[D_t\zeta, \mathcal{H}]\frac{\partial_{\alpha}D_t\partial_{\alpha}^k\tilde{\sigma}}{\zeta_{\alpha}}=[D_t\zeta, \mathcal{H}]\frac{\partial_{\alpha}\partial_{\alpha}^kD_t\tilde{\sigma}}{\zeta_{\alpha}}+[D_t\zeta, \mathcal{H}]\frac{\partial_{\alpha}[D_t,\partial_{\alpha}^k]\tilde{\sigma}}{\zeta_{\alpha}}
G_{\perp\perp}=\frac{1}{2}\nu(\nu-1)\!\left(\frac{{a'}^2}{a^2}-\frac{\dot{a}^2}{n^2a^2}-\frac{k}{a^2}\right)+\nu\!\left(\frac{n'a'}{na}+\frac{\dot{n}\dot{a}}{n^3a}-\frac{\ddot{a}}{n^2a}\right).
R_1 = \begin{bmatrix}\frac{4b^2 - 3\varphi^2}{4b^4}-\frac{\varphi'}{b^2}-\frac{\varphi'}{b^2}-\frac{\varphi''}{\varphi}\end{bmatrix}, R_2 = R_3 = \begin{bmatrix}\frac{\varphi^2}{4b^4}\frac{\varphi'}{2b^2}\frac{\varphi'}{2b^2}0 \end{bmatrix},
[T^{p^v}\textbf{a}]_i \equiv\sum_{j=0}^{p^v}\binom{p^v}{j}\textbf{a}_{i+j}\equiv\sum_{j=0}^{p}\binom{p^v}{jp^{v-1}}\textbf{a}_{i+jp^{v-1}}\equiv \sum_{j=0}^{p}\binom{p}{j}\textbf{a}_{i+jp^{v-1}}\pmod{p^2},
- T_{2}({\bf{k}},{\bf{q}})c_{{\bf{k}}-{\bf{q}}/2}\frac{1}{\sqrt{n_{{\bf{k}}-{\bf{q}}/2}}}\frac{1}{\sqrt{n_{{\bf{k}}+{\bf{q}}/2}}}c^{\dagger}_{{\bf{k}}+{\bf{q}}/2}(\frac{n^{\beta}({\bf{k}}+{\bf{q}}/2)}{\langle N \rangle})^{\frac{1}{2}}(\frac{n^{\beta}({\bf{k}}-{\bf{q}}/2)}{\langle N \rangle})^{\frac{1}{2}}
\begin{aligned}\frac{1}{m^2}\left\|\sum_{i=1}^m g_i^k\right\|^2 = \frac{1}{m^2}\left\|\sum_{i=1}^m \left(g_i^k-\nabla f_i(\theta^{\ast})\right)\right\|^2 &\leq \frac{L^2}{m}\sum_{i=1}^m\left\| x_i^k- \theta^{\ast}\right\|^2=\frac{L^2}{m}\| x^k- x^{\ast}\|^2\end{aligned}
\Vert \varphi^a+\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}+\Vert\varphi^a_t +\tilde{\varphi}_{0\,,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}&\le C\left\{\Vert\varphi^{(0)}\Vert_{H^5(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^4(\mathbb T)}\right\}+\delta/C\,,
u_j^{n+1}=\sum\limits_{l=0}^{n}(w_{n-l}-w_{n-l+1})u_j^l + w_n u_j^0+\nu\alpha_0 (u_j^{n+1})_{xx}-\frac{\alpha_0}{3}\left( u_j^n(u_j^{n+1})_x+(u_j^nu_j^{n+1})_x\right).
|\varphi_i (z)-U_1(z)| \le C \begin{cases}m_i^{-2}, \mbox{if}5\le n\le 7, \max\{\Theta_i m_i^{-2}\log m_i,m_i^{-2}\}, \mbox{if}n=8, \max\{\Theta_i m_i^{-\frac{8}{n-4}},m_i^{-2}\}, \quad\mbox{if}n\ge 9,\end{cases}\forall~ |z|\le m_i^{\frac{p_i-1}{4}}.
\left( \begin{array}{c c}{\hat{h}+ \Sigma}&{\beta \Delta^{\dagger}\beta C}\\{- C \Delta}&{\hat{h}+ \Sigma}\\ \end{array}\right) \left( \begin{array}{c}{u}\\{v}\\ \end{array}\right) = E \left( \begin{array}{c}{u}\\{v}\\ \end{array}\right)
I_{\alpha}[\mu]\asymp 1+\sum_{k=1}^{\infty}\frac{|\hat{\mu}(k,0)|^2}{k^{\alpha}}+\sum_{l=1}^{\infty}\frac{|\hat{\mu}(0,l)|^2}{l^{\alpha}}+\frac{1}{2}\sum_{k \in \mathbb{Z}\setminus \{0\}}\sum_{l=1}^{\infty}\frac{|\hat{\mu}(k,l)|^2}{|k|^{\alpha}l^{\alpha}}.
H^{( a )}= 1 + \sum_{n}\frac{q_{n}^{( a )}}{| \vec{x}- \vec{x}_{n}|}\, , \hspace{1 c m}K^{( a )}= 1 + \sum_{n}\frac{p_{n}^{( a )}}{| \vec{x}- \vec{x}_{n}|}\, ,
(A'-A)^\gamma \Phi_k(x,x') =\sum_{|\mu|=k}\sum_{\tau+\sigma=\gamma}A^\tau\Phi_\mu(x)(A')^\sigma\Phi_\mu(x') &=\big(2(k+1)\big)^\frac{|\gamma|}2\sum_{|\mu|=k}\sum_{\tau+\sigma=\gamma}\Phi_{\mu+\tau}(x)\Phi_{\mu+\sigma}(x').
21782178=2\times 3^2\times 11^2\times 73\times 137 =198\times 110011 =2178\times(10^4+1) &217802178=2\times 3^2\times 11^3\times 9091 =198\times 1100011 =2178\times(10^5+1) &2178002178=2\times 3^2\times 11^2\times 101\times 9901 =198\times 11000011 =2178\times(10^6+1)