harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Q_{r,a}(X_1, \cdots,X_r) = \frac{p^a}{(p^a-1)(1-p^aX_r)}Q_{r-1,a-1}(p^{a+r-1}X_1,p^{a+r-2}X_2,\cdots,p^{a+1}X_{r-1}) &- \frac{1}{(p^a-1)(1-X_r)}Q_{r-1,a+1}(X_1,\cdots,X_{r-1}).
\tilde{\Gamma}_{t}^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) = \tilde{\Gamma}_{t , s = 0}^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) ,
z_1^2g_{1,0}\Big(\frac{z_2z_3}{z_1^2}\Big)f_1(z_2,z_3)+z_2^2g_{1,0}\Big(\frac{z_3z_1}{z_2^2}\Big)f_1(z_3,z_1)+z_3^2g_{1,0}\Big(\frac{z_1z_2}{z_3^2}\Big)f_1(z_1,z_2)=0
-\alpha_k := \sigma_d\left(\bar\sigma_f+\frac{S_k(\beta_{k-1}+S_{k-1}\sigma_f)}{\lambda_k^2}\right)-L \geq \sigma_d\bar\sigma_f + \beta\sigma_d + \frac{\sigma_d\sigma_f}{2^{p-1}(p+1)^2}k^2 > 0,~~~ k\geq 1,
(\alpha_{-1}- \beta_{-1}) \cdot \alpha_{-1}= -\frac{(2t - 1)t}{2^4}(a_1 - a_{-1}) -\frac{t - 1}{2^4}a_1 \cdot v_{(2,3)}+ \frac{3t - 2}{2^4}a_{-1}\cdot v_{(2,3)},
f(x;r,\sigma_{1}^{2},...\sigma_{M}^{2})=\prod_{m=1}^{M}\frac{1}{\sigma_{m}^{2r_{m}}}\sum_{k=1}^{M}\sum_{l=1}^{r_{k}}\frac{\Psi_{k,l,r}}{(r_{k}-l)!}(-x)^{r_{k}-l}e^{-\frac{x}{\sigma_{k}^{2}}},forx\ge0,
(\sigma_{\mathcal{L}}\sigma_{\mathcal{A}})_{\psi}\ge \frac{1}{2}|\langle\psi|[\mathcal{L},\mathcal{A}]|\psi\rangle| = \frac{1}{4}|\langle\psi|[a^{\dagger}b,b^{\dagger}a]-[b^{\dagger}a, a^{\dagger}b]|\psi\rangle| = \frac{\delta-\gamma}{4}|2\langle\psi|a^{\dagger}a|\psi\rangle - \gamma|.
\Phi \coloneqq \sum_{\substack{i, j=1 i < j}}^N (\lambda_i|\lambda_j) \log(z_i - z_j) - \sum_{i=1}^N \sum_{j=1}^m (\lambda_i|\alpha_{c(j)}) \log (z_i - w_j) &\qquad\qquad\qquad\qquad+ \sum_{\substack{i, j=1 i < j}}^m (\alpha_{c(i)}|\alpha_{c(j)}) \log (w_i - w_j).
\delta(\tilde{g}(A^{\omega}_{\nu}))\equiv\frac{\lim_{\alpha\rightarrow 0}e^{-\frac{1}{2\alpha}\int g(A^{\omega}_{\nu})dx}}{\lim_{\alpha\rightarrow 0}\int{\cal D}\Lambda e^{-\frac{1}{2\alpha}\int dx\Lambda^{2}}}
\phi_{L/K}^j(x) =\min\{h+(v_p(h+n)-j_0)v_L(p)+p^{j_0}x: 0\le j_0\le j,\;j_0\le v_p(h+n),\;a_h\not=0\}&=\min\left\{h+v_L\left(\binom{h+n}{p^{j_0}}\right)+p^{j_0}x:0\le j_0\le j,\,j_0\le v_p(h+n),\;a_h\not=0\right\}&=\min\left\{h+v_L\left(\binom{h+n}{p^{j_0}}\right)+p^{j_0}x:0\le j_0\le j,\;a_h\not=0\right\},
=|u|_{\overline{Q}}^{(0)}+\sum_{0<|\alpha|<m-n}|D^{\alpha}_{x}u|^{\gamma}_{\overline{Q}}+{\displaystyle\sum\limits_{j=0}^{j\leq n}}{\displaystyle\sum\limits_{|\alpha|=m-j}}|x_{N}^{n-j}D_{x}^{\alpha}u|_{\omega\gamma,\overline{Q}}^{(\gamma)}+|D_{t}u|_{\omega\gamma,\overline{Q}}^{(\gamma)}.
Z_{\Psi}= \int [ d A_{\mu}] [ d \bar{c}] [ d c ] [ d b ] [ d A_{\mu}^{\ast}] [ d \bar{c}^{\ast}] [ d c^{\ast}] \delta \Bigl [ \phi^{\ast}- \frac{\delta \Psi}{\delta \phi}\Bigr ] \exp \{i S \}.
\Phi_K^{+}=\Big\{e_3\,,e_4\,,e_1+e_2\,,e_1-e_2\,,e_3+e_4\,,e_3-e_4\,, \frac{1}{2}\,(e_1-e_2-e_3-e_4)\,, &\frac{1}{2}\,(e_1-e_2-e_3+e_4)\,,\frac{1}{2}\,(e_1-e_2+e_3-e_4)\,,\frac{1}{2}\,(e_1-e_2+e_3+e_4)\Big\}\,.
\begin{matrix}{{}}&{{M \ \mathrm{i s ~ t h e ~ c l o s u r e ~ o f ~ t h e ~ u n i o n ~ o f ~ a n ~ i n c r e a s i n g ~ s e q u e n c e ~ o f}}}\\{{}}&{{\mathrm{f i n i t e ~ d i m e n s i o n a l ~ a l g e b r a s .}\hfill}}\\ \end{matrix}
F(x) =1-2\ln \biggl(1+\frac1{2x}\biggr)^{2x}+8x^2\biggl[\frac1{2x}-\ln \biggl(1+\frac1{2x}\biggr)\biggr] &\to1-2\ln e+2\lim_{u\to0^+}\frac{u-\ln(1+u)}{u^2}&=0
q_{(w;y)}(\xi,\eta)=\frac{1}{4}\Bigg[\sum_{j=1}^n\xi_j^2+\sum_{i,j}a_{ij}\xi_i\xi_j+ 4\sum_{j,l}c_{jl}\xi_j\eta_l+ 4\sum_{l,m}d_{lm}\eta_l\eta_m\Bigg]
\left\|z\right\|_{C((0,t_{\max});X)}=\sup_{t\in(0,t_{\max})}\left\|z(t)\right\|_X &\le e^{\omega t_{\max}}\|z(0)\|_X+\left(\frac{e^{\omega t_{\max}}-1}{\omega}\right)\|r\|_{C((0,t_{\max});X)}.
E_{\mathbb{P}^d}g (t,x) = \sum_{\theta \in \mathcal{C}_{N_1^{-1}}}a_{N_1c_\theta}e^{i( x\cdot c_\theta + t|c_\theta|^2 )}&= \sum_{k \in \mathcal{S}_{N_1}}a_{k}e^{i( \frac{x}{N_1}\cdot k + \frac{t}{N_1^2}|k|^2 )}&= e^{i \frac{t}{N_1^2}\Delta}\phi(\frac{x}{N_1}).
\exp\left[L_n(x_1,x_2)\right]= \frac{2 r^2}{r_+^2}\left\{\cosh \left[r_+(\Delta \phi + 2\pi n ) \right] - \cosh(r_+\Delta t)\vphantom{A^A_A}\right\}+ O\left( 1 \right) \ \ ,
\sum_{j=1}^{12}\sum_{i=1}^{j}i=\sum_{j=1}^{12}\sum_{i=1}^{j}(\begin{matrix}i\\ 1\end{matrix})=\sum_{j=1}^{12}(\begin{matrix}1+j\\ 1+1\end{matrix})=(\begin{matrix}2+12\\ 2+1\end{matrix})=364.
er(g_1) = \|g_1-\hat g_1\|_{L^2[0,T]}\approx \Big( \frac{\Delta t}{2}\sum_{i=0}^N p_i \left| g_1(t_i)-\hat g_1(t_i) \right|^2 \Big)^\frac{1}{2}, er(|g_2|) = \big\| |g_2|- |\hat g_2| \big\|_{L^2[0,T]}\approx \Big( \frac{\Delta t}{2}\sum_{i=0}^N p_i \left( |g_2(t_i)|-|\hat g_2(t_i)| \right)^2 \Big)^\frac{1}{2},
R \equiv R_{\alpha}^{\alpha}= \frac{1}{2~A}\Biggl\{2 (\frac{\dot{A}}{A})^2+(\frac{\dot{B}}{B})^2 - (\frac{\dot{C}}{C})^2 - 2\frac{\ddot{A}}{A}- 4 \frac{\ddot{B}}{B}- 2(\frac{A'}{A})^2 -(\frac{B'}{B})^2 +(\frac{C'}{C})^2 + 2 \frac{A''}{A}+ 4 \frac{B''}{B}\Biggr\}.
\mathrel{\phantom{=}}\sum_{k=1}^{n}\sum_{\ell=1}^{d}\sum_{i=1}^{n}\sum_{j=1}^{d}a_{k\ell ij}^2(x,Y) \left(\sum_{k=1}^{n}\sum_{\ell=1}^{d}a_{k\ell k\ell}(x,Y)\right)^{-2}&\leq nd(nd-1)\lambda^2 - 2(nd-1) + 1
S^{n}(x)=\left[ \begin{array}{ccc}x^{2}x 1 1 0 0 0 1 0\end{array}\right]^{n}=\left[ \begin{array}{ccc}T_{n+1}(x) xT_{n}(x)+T_{n-1}(x) T_{n}(x) T_{n}(x) xT_{n-1}(x)+T_{n-2}(x) T_{n-1}(x) T_{n-1}(x) xT_{n-2}(x)+T_{n-3}(x) T_{n-2}(x)\end{array}\right],
E_1(x)< -\frac{459733}{20697600}\frac{1}{6 x^6}+\frac{13290749}{113836800}\frac{1}{7 (x - 1/2)^7}= -\frac{459733}{20697600}\frac{1}{6 (x+2)^6}-\frac{\Psi_4(11;x)(x-10)+470994290293217661904}{2390572800 x^6 (2 + x)^6 (-1 + 2 x)^7}< -\frac{459733}{124185600}\frac{1}{(x + 2)^6},
\mathsf{D}_{d}^{i-1}R_{T,n}^{(1)}(d)=(-1)^{i-1}(i-1)!\sum_{k=0}^{n-1}(k+d)^{-i}\mathsf{D}_{d}^{i-1}R^{(1)}(d)=-\psi^{(i-1)}(d).
\mathbb{E}\sum_{i_1,\cdots,i_{k}\}\prod_{j=1}^k(\tau_{i_j}-x_j)_+ = \mathbb{E}(n/4)^k(2\ln n)^{\frac{k}{2}}|\Sigma_k(F_n(x_1),\cdots,F_n(x_k))| = \int_{[0,2\pi)^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k.
\begin{cases}\mathcal{C}_{\rm s}=\Upsilon_{\rm PDE}\zeta\left(1+\varphi_{\rm AP}+\varphi_{\rm CT}\right)/E_{\rm ph}, begin{align*}2pt]\mathcal{C}_{\rm n}=(\vartheta_{\rm DCR}+\vartheta_{\rm B})\left(1+\varphi_{\rm AP}+\varphi_{\rm CT}\right),\end{cases}
\sum_{a=0}^t \pi_N(x)^a (\log \log x)^{t-a}L^{2r - a}L^{2r + x' - 2t}\sum_{(b_1,b_2,\dots,b_{t}) \atop{b_u \in \mathcal J_{\mathcal I(s_u)}\atop{a(\underline{b}) = a}}}L^{-a_3(\underline{b})}&\ll_r \sum_{a=0}^t \pi_N(x)^a (\log \log x)^{t-a}L^{2r - a}L^{2r + x' - 2t},
{\cal Q}( \sigma_{i}) = \{\Sigma_{i}, \, H \}= \left( \begin{array}{c c}{{2 M \sigma_{i}}}&{{V \sigma_{i}\pi^{*}\frac{\textstyle 1}{\textstyle \sqrt{V}}}}\\{{\sqrt{V}\pi \sigma_{i}}}&{{0 ~}}\\ \end{array}\right)
\begin{multlined}[t][12.5cm]\widetilde{\Pi}^{P}_{\nu_{G},6c+\nu_{1}+3\nu_{2}}\left((z_{0},z_{1},z_{2}),(z_{0},z_{1},z_{2})\right) =\frac{(3c+\nu_{1}+2\nu_{2}+2)!}{\pi^2(c+\nu_{2}+\nu_{1})!(c+\nu_{2})!c!}(|z_{0}|)^{2(c+\nu_{2}+\nu_{1})}(|z_{1}|)^{2(c+\nu_{2})}(|z_{2}|)^{2c},\end{multlined}
\sigma\left( A\left( \beta\right) \right) =\sigma_{0}\left( A\left(\beta\right) \right) \cup\sigma_{1}\left( A\left( \beta\right) \right),\sigma_{0}\left( A\left( \beta\right) \right) \cap\sigma_{1}\left( A\left( \beta\right) \right) =\emptyset.
f_0^{\lambda_n(x,y)}+f_n^{\lambda_0(x,y)}+\sum_{i+j=n,~i,j>0}f_i^{\lambda_j(x,y)}=\sum_{i+j=n,~i,j>0}[f_i^x,f_j^y]+[f_0^x,f_n^y]+[f_n^x,f_0^y].
~\epsilon\chi_{\mathbb{R}^+}(x)e^{i(\lambda+i\epsilon)x}*(m_1(\lambda+)-1) = ~\epsilon\int_0^{\infty}e^{i(\lambda+i\epsilon)y}(m_1(\lambda+)-1)(x-y)~dy = ~\int_0^{\infty}e^{i\lambda y/\epsilon}e^{-y}(m_1(\lambda+)-1)\left(x-\frac{y}{\epsilon}\right)~dy
\int x^{n-2}(q^2x^2;q^2)_\infty h_n(x;q) d_qx= \frac{x^n(x^2;q^2)_\infty}{[n-1]_q}\left(\frac{h_n(\frac{x}{q};q)}{x}-\frac{1}{q}h_{n-1}(\frac{x}{q};q) \right).
\lim_{t\rightarrow\infty}\frac{\sqrt{t-x}}{\sqrt{t}}= \lim_{t\rightarrow\infty}\sqrt{\frac{t-x}{t}}= \lim_{t\rightarrow\infty}\sqrt{1 - \frac{x}{t}}= \sqrt{\lim_{t\rightarrow\infty}\left(1 - \frac{x}{t}\right)}= 1.
B_{p}^{\left( \alpha ,\beta \right)}\left( x,y\right) =\int_{0}^{1}t^{x-1}(1-t)^{y-1}{}_{1}F_{1}(\alpha ;\beta ;\frac{-p}{t(1-t)})dt (Re(p) >0,Re(x)>0,Re(y)>0,Re(\alpha )>0,Re(\beta )>0).
\lambda_1-\lambda_2 \in L^{\varphi(p)(p^2-1)}\left({M^*}^{j_0}\lambda_0+\mathbb{Z}^2\right)=L^{\varphi(p)j_0}\left(L^{\varphi(p)(p^2-1-j_0)}({M^*}^{j_0}\lambda_0+\mathbb{Z}^2)\right).
\operatorname{rank}E(M_{k,\mathrm{in}}(N_{(k,-)})) + \operatorname{rank}{F((M_{k, \mathrm{out}}')^{-1}(N_{(-,k)}))}= \operatorname{rank}N_{(k,-)}= \sum_{j\in \Omega(k,-)}b_{kj}e_j.
C_{CSCC}^{L}(P) = \frac{1}{L}\left[ H(Y_1^L) - H(Y_1^L | X_1^L)\right] &= \frac{1}{L}\left[ \sum_{i=1}^{L}H(Y_i | Y_1^{i-1}) - \sum_{i=1}^{L}H(Y_i | X_i)\right] &\overset{(a)}{<}\frac{1}{L}\left[ \sum_{i=1}^{L}H(Y_i) - \sum_{i=1}^{L}H(Y_i | X_i)\right] &= C_{CCC}(P) ,
\|\mathbf{c}_X\|_2\leq \|\mathbf{c}_X\|_1=\|\mathbf{x}\|_1\leq \sqrt{NL}\|\mathbf{x}\|_2,\|\mathbf{c}_Y\|_2\leq \|\mathbf{c}_Y\|_1=\|\mathbf{y}\|_1\leq \sqrt{NL}\|\mathbf{y}\|_2,
\begin{pmatrix}\widehat{y}_{0}(qx) \widehat{y}_{1}(qx) \widehat{y}_{2}(qx) \end{pmatrix}= \begin{pmatrix}1 -B_1 -B_2 0 * * 0 * * \end{pmatrix}\begin{pmatrix}\widehat{y}_{0}(x) \widehat{y}_{1}(x) \widehat{y}_{2}(x) \end{pmatrix}\!,
< T_{\mu \nu}( x ) \, T_{\rho \sigma}( 0 ) \int T_{\mu}^{\mu}> \, \propto \tilde{c}( g ) \, X_{\mu \nu \rho \sigma}\left( \delta^{( 4 )}( x ) \right) + \mathrm{t r a c e \, \, t e r m s .}
\frac{\partial Z_{t}}{\partial\phi}= Y_{t-J_\phi}+\sum_{j\in J_\theta}\theta_{j}\frac{\partial e_{t-j}^I}{\partial\phi}, \frac{\partial Z_{t}}{\partial\theta}= e_{t-J_\theta}+\sum_{j\in J_\theta}\theta_{j}\frac{\partial e_{t-j}^I}{\partial\theta}.
{\bf E}\bigl[\beta^{-q}_{1,0}\bigl((\tau, 1+\tau(1+\lambda_1+\lambda_2)\bigr)\bigr] = \tau^{-\frac{q}{\tau}}\frac{\Gamma(-\frac{q}{\tau}+1+\lambda_1+\lambda_2+\frac{1}{\tau})}{\Gamma(1+\lambda_1+\lambda_2+\frac{1}{\tau})},
\begin{array}{rcl}(\check{R}^{V_{z_1}, V^{*}_{z_{2}}})^{i^* k}_{jl^*}= (\check{R}^{V_{z_2}, V_{z_1}})^{kl}_{ij}\displaystyle\prod_{p=2}^{n-1}\frac{h(-z_1 +z_2 +pw)}{h(w)}, (\check{R}^{V^{*}_{z_1}, V_{z_{2}}})_{i^* j}^{kl^*}= (\check{R}^{V_{z_2}, V_{z_1 +nw}})_{jl}^{ik}\displaystyle\prod_{p=1}^{n-2}\frac{h(-z_1 +z_2 -pw)}{h(w)}.\end{array}
-K(\theta) \geq E_x\left[\int_{0}^{\infty}C(X_t)e^{-rt}\,dt\right] = A'e^{\gamma x}E_0\left[\int_{0}^{\infty}e^{\gamma X_t-rt}\,dt\right] =A'e^{\gamma x}E^{\gamma}_0\left[\int_{0}^{\infty}e^{-(r-\psi(\gamma))t}\,dt\right] = Ae^{\gamma x}\int_{0}^{\infty}e^{-(r-\psi(\gamma))t}\,dt.
\left(\theta_{f(n)}-\theta_{f(n+1)}\right)\sum_{j=f(n)}^{f(n+1)}\lambda_j \le\left(\frac{q(d+1-p)}{1-p}n^{-\frac{dq}{1-p}-1}\right)\frac{d\cdot2^{d-1}n^{d-1}+(1-p)\cdot2^pn^{-\frac{dp}{1-p}}}{1-p}=\frac{2^{d-1}dq(d+1-p)}{(1-p)^2}n^{d-2-\frac{dq}{1-p}}+\frac{2^pq(d+1-p)}{1-p}n^{-\frac{d(p+q)}{1-p}-1}.
E_1 =\{(x_1,x_2,x_3)\in R: b(x_1,x_2,x_3)\geq m_{\tilde R}(b) \}, E_2 =\{(x_1,x_2,x_3)\in R: b(x_1,x_2,x_3)< m_{\tilde R}(b) \}.
{\cal L}= - \frac{1}{4 g^{2}}F_{\mu \nu}^{k}F_{\mu \nu}^{k}+ \frac{1}{g^{\prime}}F_{\mu \nu}^{k}\tilde{F}_{\mu \nu}^{k}+ \frac{2}{g^{\prime}}\left( D_{\mu}\cal{B}_{\mu}\right)^{k}\! \left( D_{\nu}\cal{A}_{\nu}\right)^{k}+{\cal L}_{F P},
\phi_{k+1}(x) - A_{k+1}^{-1}\tilde \varepsilon_{k+1}:= (1 - \tau_k) \Big(\phi_k(x) - A_k^{-1}\tilde \varepsilon_{k}\Big) + \tau_k f_k(x) = A_{k+1}^{-1}\Big(A_0(\phi_0(x) - \tilde \varepsilon_0) + \sum_{i=0}^k a_i f_i(x)\Big),
\begin{array}{lcl}0 = (\alpha_2-\alpha_1)\log_1(\pi_2)+\beta\log_1(u(K)), k\lambda_1+a_1\mu_1 = -s\alpha_1 , 0 = (\alpha_1-\alpha_2)\log_1(\pi_2)-\beta\log_1(u(K)), k\lambda_2+a_2\mu_2 = -s\alpha_2.\end{array}
\widetilde{S}= \left( \begin{array}{c c c}{2 S_{a , b}}&{S_{a , f}}&{S_{a , f}}\\{S_{f , b}}&{\frac{1}{2}}&{- \frac{1}{2}}\\{S_{f , b}}&{- \frac{1}{2}}&{\frac{1}{2}}\\ \end{array}\right) ,
\prod_{1\le i<j\le m}(x_{j}-x_{i})=\left| \begin{array}{ccccc}1 x_{1}x_{1}^{2}\dots x_{1}^{m-1}1 x_{2}x_{2}^{2}\dots x_{2}^{m-1}\vdots \vdots \vdots \ddots \vdots 1 x_{m}x_{m}^{2}\dots x_{m}^{m-1}\end{array}\right|
\phi_{(0)}^{\prime\prime}+3\alpha_{(0)}^{\prime}\phi_{(0)}^{\prime}+\beta_{(0)}^{\prime}\phi_{(0)}^{\prime}-\gamma_{(0)}^{\prime}\phi_{(0)}^{\prime}=-\sqrt{2}b\left(b^2-\frac{2}{3}\right)\sigma^2e^{2(\gamma_{(0)}-\sqrt{2}b\phi_{(0)})}.
-b(\varepsilon )\log \mathbb{\bar{E}}[e^{-F(Y^{\varepsilon})/b(\varepsilon)}]=\inf_{\varphi \in \bar{\mathcal{A}}_{+}}\mathbb{\bar{E}}\left[b(\varepsilon )\varepsilon^{-1}L_{T}(\varphi )+F\circ \mathcal{G}^{\varepsilon}(\varepsilon N^{\varepsilon^{-1}\varphi})\right] .
\Delta S= -\frac{1}{8}\left({1\over 4\pi}\right)^{3/2}\int d^3x{\left(\vec{\nabla}B\right)^2 \over B^{3/2}}\int_0^\infty{ds\over s^{1/2}}\,e^{-m^2s/B}\left({d\over ds}\right)^3\left[s\,coth(s)\right]
\widetilde{\Phi}_{1}^{\tau}:= \Phi_{1}^{\tau}- ik\tau\sum_{\ell = 1}^{3}a_{\ell}\Phi_{\ell}^{\tau}\widetilde{\Psi}_{j}^{\tau}:= \Psi_{j}^{\tau}- ik\tau \overline{a_{j}}\Psi_{1}^{\tau},
{\bf P}_{c}= \sum_{\nu=1}^{M_0}Z_\nu{\cal W}_{\nu}\widetilde{\bf P}_R =\sum_{\nu=1}^{M_0}Z_\nu \sum\limits_{q=1}^{R}{\cal W}_{\nu}^{(1)}{\bf p}^{(1)}_q \otimes{\cal W}_{\nu}^{(2)}{\bf p}^{(2)}_q \otimes{\cal W}_{\nu}^{(3)}{\bf p}^{(3)}_q\in \mathbb{R}^{n\times n \times n},
R(m,f_{\lambda})= \sum_{\Gamma_k \vec{h}\in H(m)}\frac{1}{|\Gamma_{k,\vec{h}}|}f_{\lambda}(\Gamma_k \vec{h})| f_{\lambda}|_{\infty}\sum_{\Gamma_k \vec{h}\in H(m)}\frac{1}{|\Gamma_{k,\vec{h}}|}\leq |f_{\lambda}|_{\infty}h(k,m),
\begin{array}{l l l l l l c c l}{{}}&{{p}}&{{s}}&{{t}}&{{u}}&{{v}}&{{x}}&{{y}}&{{w}}\\{{\lambda \}}&{{0}}&{{1}}&{{1}}&{{n}}&{{0}}&{{2 ( n + 2 )}}&{{3 ( n + 2 )}}&{{0}}\\{{\mu \}}&{{0}}&{{0}}&{{0}}&{{1}}&{{1}}&{{4}}&{{6}}&{{0}}\\{{\nu \}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{2}}&{{3}}&{{1}}\\{{\sigma \}}&{{1}}&{{1}}&{{0}}&{{q}}&{{0}}&{{2 ( q + 2 )}}&{{3 ( q + 2 )}}&{{0}}\\ \end{array}
\left\vert P\right\vert :=\sum_{k=0}^{d}\left\vert P_{k}\right\vert \text{where}\left\vert P_{k}\right\vert =\sum_{\mathbf{b}=\left( b_{1},\dots,b_{k}\right) \in\Lambda_{1}^{k}}\left\vert c_{k}\left( \mathbf{b}\right)\right\vert .
\begin{aligned}\frac{dA}{dt}=\frac{-A^3}{\lambda_1\lambda_2}\left(\frac{AD^2}{\lambda_1\lambda_4^2\lambda_5}+\frac{B}{\lambda_3D}\right) \frac{dB}{dt}=\frac{A^3D^2}{\lambda_1^2\lambda_2\lambda_4^2\lambda_5}\left(\frac{-B^4}{\lambda_1\lambda_2\lambda_3}+B\right) \frac{dD}{dt}=\frac{A^2B}{\lambda_1\lambda_2\lambda_3}.\end{aligned}
\eta^{}_{\pm}(z) \, = \, \eta^{}_{\pm}\bigl(\tfrac{z}{2}\bigr) \pm \tfrac{1}{2}\Bigl( \eta^{}_{\pm}\bigl(\tfrac{z-1}{2}\bigr) + \eta^{}_{\pm}\bigl(\tfrac{z+1}{2}\bigr) \Bigr),
F_{l m}^{0}= \frac{- i g}{\sqrt{L}}({\cal A}_{l}{\cal A}_{m}^{\dagger}-{\cal A}_{m}{\cal A}_{l}^{\dagger}) + \frac{i g}{\sqrt{L}}({\cal B}_{l}{\cal B}_{m}^{\dagger}-{\cal B}_{m}{\cal B}_{l}^{\dagger}) ;
d(x,y) =|I|^{H}d_{|I|}(x,y)\leq|I|^{H}\|\tilde{h}\|_{\bar{{\cal H}}([0,|I|])}=\lim_{m\rightarrow\infty}\left(\left(\sum_{k=1}^{m}|I_{k}|\right)^{H}\|\tilde{h}^{(m)}\|_{\bar{{\cal H}}([0,a_{m+1}])}\right),
\frac{2 \cdot 6 \cdots (2k-2) \cdot (2k+6)(2k+10) \cdots (4k+2)}{(k+2)(k+3) \cdots (2k+1)}= \frac{2^{k/2}(k-1)!!\, 2^{k/2}\frac{(2k+1)!!}{(k+1)!!}}{\frac{(2k+1)!}{(k+1)!}}= \frac{2^k (k-1)!!\,(k+1)!}{(k+1)!!}\cdot \frac{(2k+1)!!}{(2k+1)!}= 2^k k! \cdot \frac{1}{(2k)!!}= 1,
f\star g\ (\mathbf{n})\ =\ \iint_{S^{2}\times S^{2}}f(\mathbf{n}_{1})g(\mathbf{n}_{2})\mathbb{L}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n})d\mathbf{n}_{1}d\mathbf{n}_{2}\ .
\begin{aligned}({\rm Id}_q \otimes P^M)g(X^M)({\rm Id}_q \otimes P^M) =\sum_{i=1,j=1}^{m}B_{ij}\otimes P^M{X^M_i}{X^M_j}P^M =\sum_{i=1,j=1}^{m}B_{ij}\otimes M^s_{ij}\succeq 0. \end{aligned}
M\le\frac{1}{16D^4}\left[\sum_{j=1}^k\left([\ell(I_j)]^3R_j^2\right)^{1/5}\right]^{4}\left[\sum_{j=1}^k\left([\ell(I_j)]^3R_j^2\right)^{1/5}\right]=\frac{1}{16D^4}\left[\sum_{j=1}^k\left([\ell(I_j)]^3R_j^2\right)^{1/5}\right]^5.
\{f , g \}_{\kappa}= i \left( \frac{\partial^{R}f}{\partial \bar{z}^{A}}g_{\kappa}^{{\bar{A}}B}\frac{\partial^{L}g}{\partial z^{B}}- ( - 1 )^{( p ( A ) + \kappa ) ( p ( B ) + \kappa )}\frac{\partial^{R}f}{\partial z^{A}}g_{\kappa}^{{\bar{A}}B}\frac{\partial^{L}g}{\partial \bar{z}^{B}}\right) ,
\xi_k \circ \Delta_k^{r-1}\tilde{G}_{k,m,r}(z) = (4\pi)^{1-k}(1-k)^{r-1}\biggl(F_{2-k,-m,0}(z) + \sum_{\substack{A_{2-k}<n<0 (-1)^{\lambda_{2-k}}n \equiv 0,1(4)}}a_{2-k}(m,n)F_{2-k,n,0}(z)\biggr) &=(4\pi)^{1-k}(1-k)^{r-1}f_{2-k,m}(z).
\left(\sum_{r \geq 0}(-1)^r e_r(\mathbf{x}^{(\mathbf{1})}) \right)\left(\sum_{s \geq 0}(-1)^s e_s(\mathbf{y}^{(\mathbf{1})}) \right)=\left(\sum_{r \geq 0}(-1)^r e_r(\mathbf{x}^{(\mathbf{1})},\mathbf{y}^{(\mathbf{1})}) \right)
Z_{1}= \frac{| Q_{2}|}{r_{0}^{2}}\left( 2 \sinh ( \sqrt{c_{2}}( \tilde{c}- \tilde{c}_{2}) ) + e^{- \sqrt{c_{2}}( \tilde{c}- \tilde{c}_{2})}\frac{r_{0}^{2}}{r^{2}}\right) ,
\begin{array}{ll}U(Q_{(\lambda,t,u)}) \cong U(R_{\omega})(Q_{(\lambda,t,u)}) = \bigoplus_{(\tilde\lambda,\tilde{t},\tilde{u})}N_{(\lambda,t,u)}(R_{\omega})(Q_{(\lambda,t,u)}) \boxtimes W_{(\lambda,t,u)}, \end{array}
\{n \tilde{\tau}\,|\, n \in \mathrm{Gal}(E_{k+1}/E_k) \}= \{r_2 u n_v \widetilde{g_2}\gamma^{-1}\,|\, u \in R_{k+1,v}, \, n_v \in \mathrm{Gal}(E_{k+1,\dot{v}}/E_{k,\dot{v}}) \}, \mathrm{Gal}(E_{k+1}/E_k) = \{r_2 u n_v r_2^{-1}\,|\, u \in R_{k+1,v}, \, n_v \in \mathrm{Gal}(E_{k+1,\dot{v}}/E_{k,\dot{v}}) \}.
\sup_{|z| \leq 1}\log | F(z) | \leq \log{2}\frac{d-2}{N + 1}+ \sum_{n=1}^{N}\frac{1+q^{\frac{n}{2}}}{n}\leq \log{2}\frac{d-2}{N + 1}+ \log 2 N + 2 \frac{q^{\frac{N}{2}}}{N}
2 c_p (\alpha) ~ \|x_k(\beta) - x_k\|^{2}\leq D_{f_p}(x_k(\beta), x_k) + D_{f_p}(x_k, x_k(\beta)) = \langle J_p (x_k (\beta)) - J_{p}(x_k), x_k(\beta) - x_k \rangle \leq \beta ~ \|\nabla \Phi_{\alpha}(x_k) \| ~ \| x_k(\beta) - x_k \|,
\begin{aligned}\mathcal{A}_{s}^{A}: L^{2}(\mathbb{R}^{n}) \rightarrow L^{\frac{2n}{n-4s}}(\mathbb{R}^{n}) &\mathcal{A}_{s}^{A}: L^{\frac{2n}{n+4s}}(\mathbb{R}^{n}) \rightarrow L^{2}(\mathbb{R}^{n})\end{aligned}
u_t = D \Delta u+H(u) x\in \Omega, \quad 0<t<T v_t =\tilde D \Delta_{M}v+F(u,v) x\in M,\quad 0<t<T D\frac{\partial u}{\partial \eta}=G(u,v) x\in M, \quad 0<t<T u =u_0 x\in\Omega ,\quad t=0 v =v_0 x\in M , t=0
\gamma^{1,2,3}=(\begin{matrix}0&-i\sigma^{1,2,3}\\i\sigma^{1,2,3}&0\end{matrix}),\gamma^{4}=(\begin{matrix}I_{2}&0\\ 0&-I_{2}\end{matrix}),\gamma^{5}=(\begin{matrix}0&-I_{2}\\ -I_{2}&0\end{matrix})
| \alpha_\pm \rangle = \frac{1}{\sqrt{\langle h|(\pm z_0-\Omega)^{-2}|g \rangle}}\frac{1}{\pm z_0-\Omega}| g \rangle \langle\alpha_\pm' |= \frac{1}{\sqrt{\langle h|(\pm z_0-\Omega)^{-2}|g \rangle}}\langle h |\frac{1}{\pm z_0-\Omega}
L1_H \lesssim \| \nabla C \|_{L^2 H^N}\| \nabla g \|_{L^2 H^N}\lesssim \frac{C_1 \epsilon}{\nu^{3/2}}\frac{C_0 \epsilon}{\sqrt \nu}= \frac{C_0}{C_1}\left( \frac{C_1 \epsilon}{\nu}\right)^2 .
\zeta_N(s,\mathbb{X}_\alpha)= \sum_{n=0}^{N}\frac{\mathbb{X}_\alpha(n)}{(n+\alpha)^s}\quad\zeta_N(s,\mathbb{Y}_\alpha)= \sum_{n=0}^{N}\frac{\mathbb{Y}_\alpha(n)}{(n+\alpha)^s}
\sum_{| K | = 2}\int_{\Omega}E_{A}^{0}( f ) I_{A B}^{( K )}E_{B}^{K}( h ) = \sum_{| K | = 2}\int_{\Omega}D_{K}\biggl ( E_{A}^{0}( f ) I_{A B}^{( 0 )}E_{B}^{K}( h ) \biggr ) .
\varepsilon_{\psi}\sum_{l=h}^{\infty}(w-z)^{l-h}\langle\varphi,V(\psi,\zeta)\;V(S_{-l}\chi,z)\;\Omega\rangle -\sum_{m=-\infty}^{h-1}(w-\zeta)^{m-h}\langle\varphi, V(S_{-m}\psi,\zeta) V(\chi,z)\;\Omega\rangle;
q_{tot}\equiv\sum_{i=1}^{N}q_{i},P_{\alpha}\equiv\sum_{i=1}^{N}q_{i}r_{i\alpha},andQ_{\alpha\beta}\equiv\sum_{i=1}^{N}q_{i}(3r_{i\alpha}r_{i\beta}-\delta_{\alpha\beta}r_{i}^{2}),
{\cal H}^{( 0 )}= - \frac{2 \kappa}{\sqrt{- \gamma}}\frac{\delta I^{( 0 )}}{\delta \gamma_{\mu \nu}}\frac{\delta I^{( 0 )}}{\delta \gamma_{\lambda \rho}}\left( \gamma_{\mu \rho}\gamma_{\nu \lambda}- \frac{1}{d - 1}\gamma_{\mu \nu}\gamma_{\lambda \rho}\right) + \frac{\sqrt{- \gamma}}{\kappa}\Lambda = 0 ,
\left(\int_{(r_oB)^c}\left| K_j(g,\tilde g)\right|dg\right)\left|\int_{B}b\big(g'\big)a\big(g'\big)dg'\right|\leq C\left(\int_{(r_oB)^c}\left| K_j(g,\tilde g)\right|dg\right)\left|\int_{B}a\big(g'\big)dg'\right|=0,
\frac{1}{q}\partial_t \langle hw_\eta^q \rangle_{x,\eta}+ \frac{4(q-1)}{q^2}\langle h |\nabla w^{\frac{q}{2}}_\eta|^2\rangle_{x,\eta}& - \frac{2}{q}\langle b \cdot \nabla w_\eta^{\frac{q}{2}},h w_\eta^{\frac{q}{2}}\rangle_{x,\eta}- \sum_{i=1}^d \langle(\eta \cdot \nabla b_i)\partial_{\eta_i}w_\eta, hw_\eta^{q-1}\rangle_{x,\eta}=0,
\begin{cases}-\Delta \bar u_r=b(x)\bar u_r B_r(x_0)\cap \omega \bar u_r=g\circ u \partial (B_r(x_0)\cap \omega)\end{cases},\begin{cases}-\Delta \tilde u_r=b(x)\tilde u_r B_r(x_0)\cap \omega \tilde u_r=h\circ u \partial (B_r(x_0)\cap \omega)\end{cases}.
[\,x_{i-1}^-(0),\,y_{i-1\, i+1}^-(1)\,]_{s^{-1}}=0, 1<i<n, [\,x_{i}^-(0),\,y_{i\, i+1}^-(1)\,]_{(rs)^{-1}}=0,1\leq i \leq n-1, [\,x_2^-(0),\,y_{1\,4}^-(1)\,] =0, [\,x_{i}^-(0),\,y_{1\, i+2}^-(1)\,] =0, 1\leq i \leq n-2 [\,x_{i-1}^-(0),\,y_{1\, i+2}^-(1)\,] =0,2\leq i \leq n-2.
\frac{(S_{1}R_{2}(S_{3}+R_{3})+S_{1}S_{2}S_{3}+S_{2}S_{3}R_{1})(S_{4}+R_{4})}{(S_{1}R_{2}(S_{3}+R_{3})+S_{1}S_{2}S_{3})(S_{4}+R_{4})+S_{2}S_{3}S_{4}R_{1}}
-dN_{\varphi(s,t)}\varphi_s (s,t) = - \frac{d}{ds}N(\varphi(s,t))= - \frac{d}{ds}\left(- \sin \theta(s) E_2 + \cos \theta(s) E_3\right) = \cos \theta(s) \theta'(s) E_2 + \sin \theta(s) \theta'(s) E_3 = \theta'(s) \varphi_s (s,t). -dN_{\varphi(s,t)}\varphi_t (s,t) = - \frac{d}{dt}N(\varphi(s,t))=0.
q^{\alpha^{2}/2}A_{q}\left(q^{\alpha}z\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\left(zq^{1/2}e^{ix};q\right)_{\infty}\exp\left(\frac{x^{2}}{\log q^{2}}+i\alpha x\right)}{\sqrt{\log q^{-1}}}dx
z \in D \Longleftrightarrow I^{(\alpha)}(z) \coloneqq \sum\limits_{n \in \mathbb{Z}}\frac{\left|\left\langle \Gamma^{\infty}_{\alpha}(\cdot,z;a), \varphi_{D,\alpha}^{(n)}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda_{D,\alpha}^{(n)}\right|}< +\infty, \alpha=p, s.
\int\limits_T \| x^*(t)\| \| \tilde{y}_n -x(t)\| d\mu(t) =\int\limits_{A_n^c}\|\tilde{x}^*(t) \| \|\tilde{y}_n - \tilde{x}_n(t) \| d\mu(t) & + \int_{A_n}\| y^*(t) \|\big( \| \tilde{y}_n - x \| + \|y(t) - x \| \big) d\mu(t) &\leq \varepsilon^2/6 + \varepsilon^2/3 + \varepsilon^2/6 \leq \varepsilon.
\begin{cases}dX^i_t = \int_Ab(t,X^i_t,\widehat{\mu}^x_t,a)\Lambda^i_t(da)dt + \sigma(t,X^i_t,\widehat{\mu}^x_t)dW^i_t + \sigma_0(t,X^i_t,\widehat{\mu}^x_t)dB_t, \i=1,\ldots,n \widehat{\mu}^x_t = \frac{1}{n}\sum_{k=1}^n\delta_{X^k_t}.\end{cases}
{\rm det}\left[ \begin{array}{ll}1-p_+^2(k)-p_-^2(k) -2p_-(k)p_+(k) -2p_-(k)p_+(k) 1-p_+^2(k)-p_-^2(k) \end{array}\right] =\left[1-(p_+(k)+p_-(k))^2\right]\left[1-(p_+(k)-p_-(k))^2\right]\ge0.
\int_{0}^{T}\zeta (\nabla N(\partial_{t}f),\nabla u) \, dt = \left \langle \int_{0}^{T}\partial_{t}f \zeta \, dt, u \right \rangle = \left \langle -\int_{0}^{T}f \zeta' \, dt, u \right \rangle & = -\int_{0}^{T}\zeta' (\nabla N(f), \nabla u )\,dt.
\partial_x^2\mathbb{E}f\bigl(Y^n_t(x)\bigr) =\partial_x\mathbb{E}f' \bigl(Y^{n}_{t}(x)\bigr)+\frac{1}{2}\sqrt{t}\,\mathbb{E}\bigl(\xi\partial_x g_1\bigl(x, \xi\sqrt t, Y^{n-1}_{t}\bigr)\bigr) &=:P_2(t,x)+R_2(t,x).
u(x,t) = (p*u_0)(x,t) + \bigl(Qu\bigr)(x,t) = (p*u_0)(x,t) + \bigl(Q(p*u_0 + Qu)\bigr)(x,t) = (p*u_0)(x,t) + \bigl(Q(p*u_0)\bigr)(x,t) + Q^2\bigl((p*u_0) + Qu\bigr)(x,t) = \dots \textit{( by induction )}\dots = \sum\limits_{j=0}^n \bigl( Q^j(p*u_0) \bigr)(x,t) + \bigl( Q^{n+1}u \bigr)(x,t).
( H_{1 2 3})^{T}= \partial_{1}\partial_{2}\partial_{3}H_{1 2 3}( \partial_{1}\partial_{2}\partial_{3})^{- 1}= ( x_{1 2}x_{2 3}x_{3 1})^{- 1}H_{1 2 3}x_{1 2}x_{2 3}x_{3 1},
\sum_{j_{3}}(2j_{3}+1)\{\begin{matrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{matrix}\}\{\begin{matrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}^{\prime}\end{matrix}\}=\frac{\delta_{j_{6}j_{6}^{\prime}}}{2j_{6}+1}\{\begin{matrix}j_{1}&j_{5}&j_{6}\end{matrix}\}\{\begin{matrix}j_{4}&j_{2}&j_{6}\end{matrix}\}.
\frac{\delta F\left[f\right]}{\delta f(k)}=\lim_{\varepsilon\rightarrow 0}\frac{1}{\varepsilon}\left(F\left[f(p)+\varepsilon\delta^{(4)}_{D}(p\,\dot{+}(\dot{-}k))\right]-F\left[f(p)\right]\right) ~,
\begin{array}{l}{{A F ( z ) = q^{- j}F ( q z ) \, ,}}\\{{B F ( z ) = - \displaystyle{\frac{z}{q - q^{- 1}}}( q^{- 2 j}F ( q z ) - q^{2 j}F ( q^{- 1}z ) ) \, ,}}\\{{C F ( z ) = \displaystyle{\frac{1}{z ( q - q^{- 1})}}( F ( q z ) - F ( q^{- 1}z ) ) \, ,}}\\{{D F ( z ) = q^{j}F ( q^{- 1}z ) \, .}}\\ \end{array}
\omega(t^{tj})\prod_{i=0}^{r-1}\Gamma_p\left(\langle \frac{tp^ij}{q-1}\rangle\right)\prod_{h=1}^{t-1}\Gamma_p\left(\langle\frac{hp^i}{t}\rangle\right)=\prod_{i=0}^{r-1}\prod_{h=0}^{t-1}\Gamma_p\left(\langle\frac{p^ih}{t}+\frac{p^ij}{q-1}\rangle\right)