harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
N_{(1,14)}(n) = ~ 8\sigma(n) - 32\sigma(\frac{n}{4}) + 8\sigma(\frac{n}{14}) -32\sigma(\frac{n}{56}) + 64\, W_{(1,14)}(n) + 1024\, W_{(1,14)}(\frac{n}{4}) - 256\, \biggl( W_{(2,7)}(\frac{n}{2}) + W_{(1,56)}(n) \biggr),
(t_i-q)(t_i+q^{-1})=0,t_it_j=t_jt_i,t_kt_{k+1}t_k=t_{k+1}t_kt_{k+1}, &c_i^2=0,c_ic_j=c_jc_i,c_{k+1}c_{k}c_{k+1}=-c_{k+1}, c_{k}c_{k+1}c_{k}=-c_{k}, &t_ic_j=c_jt_i,c_it_i=-q^{-1}c_i,t_ic_i=qc_i, &t_kc_{k+1}c_k=-t_{k+1}c_k+(q-q^{-1})c_{k+1}c_k, c_{k+1}c_{k}t_{k+1}=-c_{k+1}t_k+(q-q^{-1})c_{k+1}c_k,
J=\max_h\left\lbrace \left| \int_0^T\!\! \overline{b}_h^i(t)dt\right| +1\right\rbrace\mbox{and}I=\min_h\left\lbrace \int_0^T \!\sum_{i=1}^r\xi_h^i(t)dt+R-\int_0^T\!\! \hat{b}_h^i(t)dt\right\rbrace
d^n h^{\mathrm{an}}_{(D, g)}(\xi) = h^{\mathrm{an}}_{(D, g)}((f^{\mathrm{an}})^n(\xi))= h^{\mathrm{an}}_{(D, g)}((f^{\mathrm{an}})^m(\xi)) = d^m h^{\mathrm{an}}_{(D, g)}(\xi),
( e^{- 2 \phi})_{\mathrm{f i x}}={\frac{| q_{0}q_{1}+ p^{0}p^{1}|}{( q_{1})^{2}+ ( p^{0})^{2}}}\qquad ( a )_{\mathrm{f i x}}={\frac{p^{1}q_{1}- q_{0}p^{0}}{( q_{1})^{2}+ ( p^{0})^{2}}}\ .
\Delta_{\xi_d}=\frac{((\gamma(s)-\gamma(0))\cdot \nu(0)-(\gamma(0)-\gamma(s))\cdot\nu(s))}{|\gamma(s)-\gamma(0)|}&=\frac{(\gamma(s)-\gamma(0))\cdot(\nu(0)+\nu(s))}{|\gamma(s)-\gamma(0)|}.
|a \alpha \omega+b| =\dfrac{|(a(1+T+T^2)+bT^3)\omega+b(1+T)+a |}{|T^3 \omega +1+T|}& \geq 2^{-1}\left|(a(1+T+T^2)+bT^3)\omega+b(1+T)+a \right| &\geq 2^{-1}.
C =[\zeta^0]P(z_1-z_2)P(z_1-z_3)P(z_1-z_4)P(z_2-z_3)P(z_2-z_4)P(z_3-z_4) &=-384G_2^6 + 480G_4G_2^4 - 200G_4^2G_2^2 + \frac{250}{9}G_4^3
\frac{\prod_{j=1}^{A-1}(j+ \beta/\alpha)}{\prod_{k = A+1}^{2A-1}(k + 2\alpha/\beta)}= \frac{\prod_{i=1}^{A-1}(i+ \beta/\alpha)}{\prod_{i =1}^{A-1}(i + A + 2\alpha/\beta)}&\stackrel{(a)}\leq \frac{\prod_{i=1}^{A-1}(i+ u)}{\prod_{i =1}^{A-1}(i + A + 2u)}&= \frac{(A-1+ u)! (A+2u)!}{u!(2A-1+2u)!}.
q = \frac 1{3 2 \pi^{2}}\int \epsilon_{\alpha \beta \gamma \delta}\mathrm{t r}\hat{F}^{\alpha \beta}\hat{F}^{\gamma \delta}d^{4}q = \frac 1{3 2 \pi^{2}}\int \epsilon_{a b c d}\mathrm{t r}\hat{F}^{a b}\hat{F}^{c d}\sqrt{g}d^{4}q .
\mathcal{P}_1^{\mathrm{ULA}}\left({x}\right)=\exp \left\{- 2\pi{\lambda_e}\int_0^\infty \int_0^{2\pi}{\rm{\mathbf{1}}}\left({\max \{{r_e},d\}}< \big(\frac{{P_t}{G_e(\varphi_{t_{e,o}})}\beta}{x \sigma_e^2}\big)^{\frac{1}{{\alpha_{\mathrm{LoS}}}}}\right) \frac{{{f_{\Pr}}({r_e})}}{2\pi}{r_e}d{\varphi_{t_{e,o}}}d{r_e}\right\}
[ a_{i j}(k^+,{\bf k}^{\perp}), a_{k l}^{\dagger}({\tilde k}^+,{\tilde{\bf k}}^{\perp})] = \delta_{i k}\delta_{j l}\delta (k^+ -{\tilde k}^+) \delta ({\bf k}^{\perp}-{\tilde{\bf k}}^{\perp}).
f_3 =- \frac{i e^2}{2\pi^2}\ln\frac{\Lambda}{\mu}\,\sum\limits_i c_i\int \frac{d^4k}{(2\pi)^4}\,\frac{M_i^2}{(k^2-M_i^2)^2\Big((k+p)^2-M_i^2\Big)}\qquad\quad
\frac{1}{2}\sum_{x=0}^{x=w}A_x = \frac{1}{2}\left( \frac{A_0 + A_w}{2}+ \frac{A_1 + A_{w-1}}{2}+ \dots + \frac{A_w + A_0}{2}\right) = \frac{1}{2}\sum_{x=0}^{x=w}\frac{C_x C_{w-x}}{2^{2w+2}}.
\left| \Phi \right\rangle =\left\{\phi_0,\phi_1^{(\mu_1)}\left(p_1\right) ,\phi_2^{(\mu_1)(\mu_2)}\left( p_{1,}p_2\right) ,...,\phi_n^{(\mu_1)...(\mu_n)}\left( p_1,...,p_n\right) ,...\right\}\;\,,
\tilde Q_\lambda(x_1,\dots, x_n)= 2^l \sum_{\sigma \in S_n}\sigma \left(f_{\lambda_1}(x_{1})\dots f_{\lambda_n}(x_n)\prod_{i=1}^{n}\prod_{i<j}\frac{x_{i}+x_{j}}{x_{i}- x_{j}}\right),
_{a}\mathcal{D}_{x}^{\beta}f(x) &:=\frac{1}{\Gamma^{\alpha}_{F}(n-\beta)}(D_{F}^{\alpha})^n\int_{S_{F}^{\alpha}(a)}^{S_{F}^{\alpha}(x)}\frac{f(t)}{(S_{F}^{\alpha}(x)-S_{F}^{\alpha}(t))^{-n\alpha+\beta+\alpha}}d_{F}^{\alpha}t.
[E_i(z),E_j(w)]=0\,, [F_i(z),F_j(w)]=0\, (i\not\equiv j,j\pm1)\,, &d_{i,j}g_{i,j}(z,w)E_i(z)E_j(w)+g_{j,i}(w,z)E_j(w)E_i(z)=0, &d_{j,i}g_{j,i}(w,z)F_i(z)F_j(w)+g_{i,j}(z,w)F_j(w)F_i(z)=0.
\Vert \{\lambda_{j}\}\Vert_{l_{q}^{s,1}}=\begin{cases}\left( \sum_{j\in \mathbb{N}}2^{js}|a_{j}|^{q}\right)^{\frac{1}{q}}~ ~0<q<\infty , \sup_{j\in \mathbb{N}}\left( 2^{js}|a_{j}|\right) ~ ~q=\infty\end{cases}
\textstyle \gamma_{1}= \frac \gamma{\Delta - 1}= \frac{2^{- \Delta - 1}\pi^{- \frac d 2}}{\Delta - 1}\frac{\Gamma ( \Delta + 1 )}{\Gamma ( \Delta + 1 - \frac d 2 )}, \qquad \gamma_{2}= - \Delta \gamma_{1}- \gamma = - \gamma_{1}.
T''_{k_2, \tilde{k}_2, k_3, \tilde{k}_3}= \delta_{k_2 = \tilde{k}_2 = k_3 = \tilde{k}_3= \delta_2\delta_32^{\alpha+\beta}= 1}\frac{M}{\pi^2}\left\{1+ O\left(\frac{1}{\sqrt{M}}\right)\!\!\right\}+O_C(\sqrt{M}V^2(\log V)^2(\log M)^{-C}).
\lim_{N\in\mathcal{N}}\frac{E_{1}(\alpha_{N})-\frac{1}{\pi}N^{2}\log N}{N^{2}}=\lim_{N\in\widetilde{\mathcal{N}}}\frac{E_{1}(\alpha_{N})-\frac{1}{\pi}N^{2}\log N}{N^{2}}=\lim_{N\in\widetilde{\mathcal{N}}}\left(S_{N,M,1}+S_{N,M,2}+S_{N,M,3}+S_{N,M,4}\right)\leq L+\frac{1}{\pi}K(\theta_{1},\ldots,\theta_{M})+O(\epsilon)\leq L+\frac{\kappa}{\pi}+O(\epsilon).
\begin{aligned}Z_{s,s+k+1}\left[ Z_s,t+\tau; t_1+\tau,\dots,t_k+\tau,0;v_{s+1},\dots,v_{s+k},v_{s+k+1};\right. & \left.\omega_1,\dots,\omega_k,\omega_{k+1}; i_1,\dots,i_k,i_{k+1}\right] & \in \mathcal{K}_{s+k+1}\cap \mathcal{U}_{s+k+1}^\eta\end{aligned}
= ( \frac{3}{2}\times \frac{4}{3}\times \frac{5}{4}\times \frac{6}{5}\times \cdots \times \frac{2 0 1 4}{2 0 1 3}) \times ( \frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \cdots \times \frac{2 0 1 2}{2 0 1 3})
\mathrm{vol}(\mathcal B_1^+(m)) = \sum_{j\geq -m}\mathrm{vol}(\mathcal R_{2j+1}) = p^{-1}(1-p^{-1})^2\sum_{j\geq -m}p^{-2j}= p^{2m-1}(1-p^{-1})^2 \frac{1}{1-p^{-2}}= p^{2m-1}\frac{1-p^{-1}}{1+p^{-1}}.
\widehat \nu_h+\widehat\rho-(s_0s_1)^m(\widehat \nu_h+\widehat\rho)(x+d) =\widehat \nu_h+\widehat\rho-s_1(s_0s_1)^m(\widehat \nu_h+\widehat\rho)(x+d) &=(m (-km+2 r+1))=-km^2 +(2r+1)m,
U=\left(\begin{array}{cc}\textrm{sign}(\zeta_1) 0 0 \textrm{sign}(\zeta_2)\end{array}\right),\,\,\,\,\,v=\left(\begin{array}{cc}a b c d\end{array}\right):=\left(\begin{array}{cc}\eta_1 0 0 \eta_2\end{array}\right) B.
f_k(5^2+\underbrace{1^2+\dotsb+1^2}_{5}+ \underbrace{1^2+\dotsb+1^2}_{k-6}) = f_k(4^2 + a_1^2 + \dotsb + a_5^2 + \underbrace{1^2+\dotsb+1^2}_{k-6})
\partial_{t_1}\partial_{t_2}\delta\Phi(t_1,t_2,x) =\partial_{t_1}\partial_{t_2}\bigl(\Phi(t_1+t_2,x)-\Phi(t_1,\Phi(t_2,x))\bigr) &=\partial_{t_1}\bigl(\partial_t\Phi(t_1+t_2,x)-\partial_x\Phi(t_1,\Phi(t_2,x))\partial_t\Phi(t_2,x)\bigr) &=\partial_t^2\Phi(t_1+t_2,x)-\partial_t\partial_x\Phi(t_1,\Phi(t_2,x))\partial_t\Phi(t_2,x).
\big(T_{e}(c)\big)_{e'}= \begin{cases}c_{e'}- 2 \cdot q_{e,e'}e' = (u,w) \in A_{v}u \neq s c_{e'}- 2 \cdot q_{e,e'}+ c_{e}e' = (u,w) \in A_{v}u = s \end{cases},
\theta(n)=\begin{cases}t_i(j), n=k(i-1)+j,\;1\leq i\leq k,1\leq j\leq 2^d-1; \omega_b(s), n=k2^d+d(a-1)+s, \;1\leq s\leq d; 0, \mathrm{otherwise}.\end{cases}
D_t=(-1)^{t+3}\frac{1-u^{t+3}}{(1-u)(1+u)^{t+2}}, N_2^t=\frac{(-1)^{t+1}u^2(1-u^t)}{(1+u)^{t+3}(1-u)}, N_3^t= \frac{(-1)^{t+1}u(1-u^{t+2})}{(1-u)(1+u)^{t+3}}.
\alpha_1 = (-X_2w_{21}-X_3w_{31}) dx + (\partial_xw_{11}+X_3w_{21}-X_2w_{31}- \tilde \varphi) x^{-1}\theta^1 + (-X_3w_{11}+\partial_xw_{21}+xX_1w_{31}+xX_3\tilde \varphi) \theta^2 + (X_2w_{11}-xX_1w_{21}+\partial_xw_{31}-xX_2\tilde \varphi) \theta^3.
\Gamma_{+}C^{- 1}\tilde{\Gamma}_{+}= ( U_{3}+ \sum_{1}^{2}A^{( r )}U_{r}A^{( r ) T}) C^{- 1}( U_{3}^{- 1}+ \sum_{1}^{2}A^{( s )}U_{s}^{- 1}A^{( s ) T})
\begin{aligned}\frac{2}{t^2}\sum_{2\le|a|\le N}\int_0^ts(f,e^{isH}i[H^a,A^a]\tilde P^ae^{-isH}f)ds &\sim_{\epsilon_M}\frac{2}{t^2}\sum_{2\le|a|\le N}\int_0^ts(f,e^{isH}(\tilde P^a)^*i[H^a,A^a]\tilde P^ae^{-isH}f)ds.\end{aligned}
\frac{1}{n}\sum_{i=1}^K d_i^2 = n^{-\eta}\sum_{i=1}^K \left( \frac{C_F}{i}\right)^{2\alpha}L(n/i)^2+o(c_n^{-1})=c_n^{-1}\sum_{i=1}^K \left( \frac{C_F}{i}\right)^{2\alpha}+\varepsilon(c_n,K),
\Pi (x) = \pi (x) + \frac{1}{2}\pi (\sqrt[2]{x}) + \frac{1}{3}\pi (\sqrt[3]{x}) + \frac{1}{4}\pi (\sqrt[4]{x}) + \cdots x = 100 : \sqrt[2]{x}= 10, \sqrt[3]{x}= 4.6415 \, , \sqrt[4]{x}= 3.1622, \sqrt[5]{x}= 2.51188 \, , \sqrt[6]{x}= 2.15 \ldots, \sqrt[7]{x}= 1.930 \ldots < 2 \, .
R_{\lambda}=\mathbf{Q}[X_0,\dots,X_r,Y_0,\dots,Y_r,\mathbf{X}_1,\dots,\mathbf{X}_r]/\left(\tilde{P}_{j,k,\ell},\tilde{\phi}_i\right)_{1\leqslant j<k<\ell\leqslant n\atop 1 \leqslant i \leqslant r}
[u,e_3]+\lambda J[u,e_3] + J[u,Je_3]-\lambda [u,Je_3] = v+\lambda Jv + J[u,Ce_3]-\lambda [u,Ce_3] = v+\lambda Jv + CJv-\lambda C v = (1+\lambda Y+CY-\lambda C)v+(\lambda X+CX)u
\sigma_{(a,i)}^{(r+1)n_1}((b,j))=(b+(r+1)n_1\delta_{(a,i)},j+(r+1)n_1(r\delta_{(a,i)}+1))= &(b,j+rn_1+rn_1\delta_{(a,i)}+n_1)=(b+n_1(\delta_{(a,i)}+1),j+n_1(r(\delta_{(a,i)}+1)+1))= &\sigma_{\sigma_{(a,i)}((a,i))}^{n_1}((b,j)),
I_{Z}= (x_0x_n, x_1x_n, \ldots, x_{n-1}x_n, x_0x_1 \cdots x_{n-1}) = (x_0 \cdots x_{n-1},x_n) \cap (x_0, \ldots, x_{n-1}) = P_0 \cap P_1 \cap \cdots \cap P_{n-1}\cap P_n,
d s^{2}= \frac{d \xi^{0}d \xi^{0}}{1 + \xi^{0}\xi^{0}}- ( 1 + \xi^{0}\xi^{0}) \frac{d{\bf x}( I +{\bf x}^{\prime}{\bf x})^{- 1}d{\bf x}^{\prime}}{1 +{\bf x}{\bf x}^{\prime}},
\mathrm{s o l v}\bigl ({\frac{\mathrm{O}( 5 , Q )}{\mathrm{O}( 5 ) \times \mathrm{O}( Q )}}\bigr ) = \mathrm{s o l v}\bigl ({\frac{\mathrm{O}( 1 , Q )}{\mathrm{O}( Q )}}\bigr ) + \mathrm{s o l v}\bigl ({\frac{\mathrm{O}( 4 , Q )}{O ( 4 ) \times \mathrm{O}( Q )}}\bigr )
\Delta \phi_g = \cos^{-1}\left(\frac{(\mathbf{p}_g - \mathbf{p}_{d})\cdot \mathbf{\dot{p}}_{d}}{||\mathbf{p}_g - \mathbf{p}_{d}||~||\mathbf{\dot{p}}_{d}||}\right) \mathbf{z}_{\phi,g}= \mathrm{sign}\left(\left(\mathbf{\dot{p}}_{d}\times (\mathbf{p}_g - \mathbf{p}_{d})\right) \cdot \mathbf{z}_I\right)\mathbf{z}_I
(-\partial_x^2+B)B_0u_0 =-B_0\partial_xu-B_1u_0-B_2u-B_3\partial_xu+\partial_x^{m-1}\delta_L^{B_0'Av}+[B,B_0]u_0 &\in K^{(-\frac32-m)}(M,L)+K^{(\frac12-m-\epsilon)}(M,L)=K^{(-\frac32-m)}(M,L)\;.
S_{\Delta_n}((e_1-x)^2;x) =n\left(x-\frac{k-1}{n}\right)\left(\frac{k}{n}-x\right)\left[\left(\frac{k}{n}-x\right)-\left(\frac{k-1}{n}-x\right)\right] &=\left(x-\frac{k-1}{n}\right)\left(\frac{k}{n}-x\right),
\int_{\mathcal{A}_n^3}y^{1-2\alpha}|\nabla v_n|^2\,dxdy \le C\sigma_n\int_{\mathcal{A}_n^2}y^{1-2\alpha}|w_n|^{2}\,dxdy+C\int_{\mathcal{A}_n^2\times\{y=0\}}|w_n|^{2^*_\alpha}\,dx +C\int_{\mathcal{A}_n^2\times\{y=0\}}|w_n|^{2}\,dx.
\Phi ({\bf x}, \theta ) = \chi + \bar{\theta}\lambda + \frac 1 2 \bar{\theta}\theta F , ~ ~ D_{\alpha}= \frac{\partial}{\partial \bar{\theta}_{\alpha}}- ( \hat{\partial}\theta )_{\alpha}, ~ ~ \bar{D}_{\alpha}= \frac{\partial}{\partial \theta_{\alpha}}- ( \bar{\theta}\hat{\partial})_{\alpha}.
F_k(x,y) = \frac{1}{\pi}\int_{-\infty}^\infty F_k(t,0)\frac{y}{(x-t)^2 + y^2}\,dt f(x+i(y +y_k)) = \frac{1}{\pi}\langle F_k(\cdot, 0), P(x-\cdot,y)\rangle.
\chi ({\bf x}, t ) = \int \frac{d^{3}{\bf k}}{[ ( 2 \pi )^{3}2 \omega_{\bf k}]^{1 / 2}}[ c ({\bf k}) e^{i ({\bf k}_{n}\cdot x - \omega_{\bf k}t )}+ c^{\dagger}({\bf k}) e^{- i ({\bf k}_{n}\cdot x - \omega_{\bf k}) t )}]
\begin{aligned}\left\|R_t(\lambda) u\right\|_1^2\ \leq\ \left\|R_t(\lambda) u\right\|_0^2\,+\,\left\|B_t\,R_t(\lambda) u\right\|_0^2 &\leq\ \big(d+\lambda^2\big)^{-2}\,\|u\|_0^2\,+\,c_1^2\,\big(d+\lambda^2\big)^{-1}\,\|u\|_0^2\ \leq\ \big(d^{-1}+c_1^2\big)\,\big(d+\lambda^2\big)^{-1}\,\|u\|_0^2\,.\end{aligned}
P = P_{0}( l ) \left( \frac{2}{\lambda}\right)^{2 l + 3}\frac{( l + 1 ) !^{2}\Gamma^{4}( 1 + l / 4 )}{\pi}= P_{0}( l ) \left( \frac{2 \pi T}{\omega}\right)^{2 l + 3}\frac{( l + 1 ) !^{2}\Gamma^{4}( 1 + l / 4 )}{\pi}\, ,
\chi^{}_\mathrm{min}(k) = \log \sqrt{L}-\underset{n\rightarrow\infty}{\lim}\dfrac{1}{n}\log\left\Vert B(k)\cdot\ldots\cdot B(L^{n-1}k)\right\Vert \chi^{}_\mathrm{max}(k) = \log L- \chi^{}_\mathrm{min}(k)-m\left(Q-R\right),
(d^{[{\sf d}]}_1,\cdots,d_{N_1}^{[{\sf d}]},d^{[{\sf u}]}_1,\cdots,d_{N_2}^{[{\sf u}]})=\lim_{P\to\infty}\left(\frac{R^{[{\sf d}]}_1}{\frac{1}{2}\log P},\cdots,\frac{R^{[{\sf d}]}_{N_1}}{\frac{1}{2}\log P},\frac{R^{[{\sf u}]}_1}{\frac{1}{2}\log P},\cdots,\frac{R^{[{\sf u}]}_{N_2}}{\frac{1}{2}\log P}\right).
F_{in}=sin\gamma_{in}[\begin{matrix}0\\ 1+r_{10}^{TE}e^{i\Phi_{in}}\\ 0\end{matrix}]+cos\gamma_{in}[\begin{matrix}cos\theta_{1}^{in}(1-r_{10}^{TM}e^{i\Phi_{in}})\\ 0\\sin\theta_{1}^{in}(1+r_{10}^{TM}e^{i\Phi_{in}})\end{matrix}]
\mathbb{E}[ \sup_{0\leq u\leq t}|Y(t)|^q]\leq C\Big\{\| Y_0\|_{L^q(\Omega, \mathbb R^d)}^q+ \int_0^t \Big(\| F(s) \|_{L^q(\Omega,\mathbb R^d)}^q+\| G(s) \|_{L^q(\Omega, \mathbb R^{d\times m})}^q &+\| H(s) \|^q_{L^q(\Omega, L^q(\nu))}+\| H(s) \|^q_{L^q(\Omega, L^2(\nu))}\Big)ds\Big\}
\int_0^1 \mathrm{d}t \int_t^1 \mathrm{d}s \, F(t,s) = \int_0^1 \mathrm{d}s \int_0^s \mathrm{d}t \, F(t,s) \ , \int_0^1 \mathrm{d}t \int_0^t \mathrm{d}s \, F(t,s) = \int_0^1 \mathrm{d}s \int_s^1 \mathrm{d}t \, F(t,s)\ .
A[\begin{matrix}p_{1}&p_{2}&p_{3}&p_{4}\end{matrix}]=[\begin{matrix}p_{1}&p_{2}&p_{3}&p_{4}\end{matrix}][\begin{matrix}1&0&0&0\\ 0&2&0&0\\ 0&0&4&1\\ 0&0&0&4\end{matrix}]=[\begin{matrix}p_{1}&2p_{2}&4p_{3}&p_{3}+4p_{4}\end{matrix}].
b(a_0\otimes\cdots\otimes a_n) =a_0a_1\otimes a_2\otimes\cdots\otimes a_n &+\sum_{k=1}^{n-1}(-1)^ka_0\otimes a_1\otimes\cdots a_{k-1}\otimes a_{k}a_{k+1}\otimes a_{k+1}\otimes\cdots\otimes a_n &+(-1)^n a_na_0\otimes a_1\otimes a_2\otimes\cdots\otimes a_{n-1}.
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
W_r(r') = 1 + \sum_{w \in c_r(r')}W_r(w) = 1 + \sum_{w \in c_{r'}(r')- \{v\}}W_{r'}(w) = n+1 - W_{r'}(v) & \ge n+1 - \lfloor \frac{n+1}{2}\rfloor = \lceil \frac{n+1}{2}\rceil
\int_U(-\det D^2v)v^2\psi\,dx = - \frac32 \int_U (D|Dv|^2\cdot Dv) v \psi\,dx-\int_U |Dv|^4\psi \,dx &\quad-\int_U |Dv|^2 (Dv\cdot D\psi) v \,dx+\frac12\int_Uv^2\Big[ |Dv|^2\Delta\psi -D^2\psi Dv\cdot Dv\Big] \,dx.
\lim_{\substack{X\rightarrow P X\in \Gamma_{\gamma}(P)}}\left\langle\nabla\left(\int_{\partial D}\mathcal{K}_{m}(X, Q)f(Q)d\sigma(Q)\right),\,\, n_{P}\right\rangle =-\mathrm{K}^{*}_{m}f(P),
{\cal G}( x - x^{\prime}, 0 )_{\mu \nu , \gamma \delta}\, = \, \langle h_{\mu \nu}( x , 0 ) \, h_{\gamma \delta}( x^{\prime}, 0 ) \rangle = \int d^{N}m | \sigma_{m}( 0 ) |^{2}\langle \epsilon_{\mu \nu}^{( m )}( x ) \epsilon_{\gamma \delta}^{( m )}( x^{\prime}) \rangle \, .
\begin{array}{lll}d_v = 2kH+Nk+N(k+1)=2kH+(2k+1)N d_H = [\frac{2k(2k+1)}{2}2H-2H k(2k+1)]+ [N]+[2\frac{2k}{2}] & + [\frac{2k(2k+2)}{2}2N-Nk(2k+1)-N(k+1)(2k+1)] & = 2k. \end{array}
\mathbf{r}^{\prime \prime \prime}(s)=\left(\begin{array}{c}0,\cosh \left( \int \tau (s)\,ds\right) \kappa^{\prime}+\sinh \left( \int\tau (s)\,ds\right) \kappa (s)\tau (s), \kappa^{\prime}\sinh \left( \int \tau (s)\,ds\right) +\cosh \left( \int\tau (s)\,ds\right) \kappa (s)\tau (s)\end{array}\right).
\sum_{t\in Y^{n-1}}\mathbf{(1)}\ast t = \sum_{t\in Y^{n-1}}|\vee (|\ast t) + \sum_{\substack{t_1\in Y^{a},t_2\in Y^b a+b+1=n-1}}(\mathbf{(1)}\ast t_1)\vee t_2 =\sum_{t\in Y^{n-1}}|\vee t + \sum_{\substack{t_1\in Y^{a},t_2\in Y^b a+b+1=n-1}}(\mathbf{(1)}\ast t_1)\vee t_2
b_{jk}(z_k)=f_{jk}(z_k,0),\,\,\,\,\,\,\, \Lambda_{M_{0j}}^{\alpha,\beta}(z_j)=\Lambda_{M_j}^{\alpha,\beta}(z_j,0)=\Lambda_{N_j}^{\alpha,\beta}(z_j,0)
\Omega^{p}_\mathrm{abs}(X,F) =\big\{\sigma\in \Omega^{p}(X,F) \;:\; \left (i_{e_\mathfrak{n}}\sigma\right)|_{Y}=0 \big\}, & \Omega^{p}_\mathrm{rel}(X,F) =\big\{\sigma\in \Omega^{p}(X,F) \;:\; \left (e^\mathfrak{n}\wedge\sigma\right)|_{Y}=0 \big\}.
P_{}((G,G_1),t \varphi,X)=\lim_{n\to\infty}\frac{1}{n}\log \left(\frac{1}{m^n}\sum_{{\underline g \in G_n^*}}\inf_{\mathcal{U}}\sum_{U\in\mathcal{U}}e^{tS_{\underline g}\varphi(U)}\right),
q^{\sum_{1 \le i \le l_3}(\dim(W_{i-1}^\perp)+\dim(V_1))+l_3s}= q^{\sum_{1 \le i \le l_3}(\frac{kb}{2}-(l_2+2i-2)+\frac{kb}{4})+l_3s}= q^{l_3(\frac{3}{4}kb-l_2-l_3+1+s)}.
|J_k| \le \int_{-\infty}^{+\infty}\left |\frac{\Gamma(L-it-c)\Gamma(a-L+it+c)}{2\pi i\Gamma(b-L+it+c)}x^{c-L}\right |dt &=C(m,L,\delta) \int_{0}^{+\infty}\big((l_k + \frac{m}{2}+ L -\delta)^2 + t^2\big)^{\frac{2(L-\delta)+\frac{m}{2}}{2}}\exp(-\frac{\pi}{4}t)dtx^{c-L}&\le C(m,L,\delta)l_k^{2(L-\delta)+\frac{m}{2}}x^{c-L}.
-\psi(x) \left( \nabla^2 \psi \right)^{-1}\nabla \left(\log \det \nabla^2 \psi \right) = \left[ n+2 - \left \langle \left( \nabla^2 \psi \right)^{-1}\nabla \left(\log \det \nabla^2 \psi \right), \nabla \psi \right \rangle \right] x.
({}_a I_{x}^\rho u)(x)=\frac 1{\Gamma(\rho)}\int_{a}^x \frac{u(y)}{(x-y)^{1-\rho}}dy, ({}_x I_{b}^\rho u)(x)=\frac 1{\Gamma(\rho)}\int_{x}^b \frac{u(y)}{(y-x)^{1-\rho}}dy,
Z_f(s, \chi, A_7)= \dfrac{(1-5^{-1})^2}{1-5^{-1}t}\Big(-(5^{-1}+2\times5^{-2}-5^{-3}+5^{-4})t+1-5^{-1}-2\times5^{-2}\Big) + (5^{-2}-5^{-3})t^5Z_{g_7}(s, \chi),
Tr\circ\phi_f(x) =Tr(\sum_{g,h\in F}f(h^{-1}g)p_{go}xp_{ho})=\sum_{g\in F}Tr(f(g^{-1}g)p_{go}xp_{go}) &=\sum_{g\in F}Tr(p_{go}xp_{go})=Tr(x), x\in B.
V_{\mathrm{l o n g}}( r , u ) = -{\frac{u^{4}}{r^{7 - p}}}V_{p}2^{2 - 2 p}\pi^{( 5 - 3 p ) / 2}{\alpha^{\prime}}^{3 - p}\Gamma ({\frac{7 - p}{2}}) + O ( u^{6}) \quad .
\kappa^{(r)}\sum_{m=0}^{r-1}\int_{\mathbb{T}}\Delta_\epsilon^{(r)}(z,m;t_a,n_a)\frac{dz}{z}= \prod_{1 \leq a < b \leq 6}\Gamma^{(r)}(t_a t_b,n_a+n_b+\epsilon;p,q),
Z_{\mathrm{F}}[ A ] = \int \! D{\bar{\psi}}D \psi e^{- \int \! d^{2}x [ \psi_{1}^{\dagger}i \partial \psi_{1}+ \psi_{2}^{\dagger}i{\bar{\partial}}\psi_{2}- \psi_{1}^{\dagger}A \psi_{1}- \psi_{2}^{\dagger}{\bar{A}}\psi_{2}]},
\Sigma_{\sigma_2}^{1}\le 2 \int_{H_{\sigma_2}}^T \frac{x^{-\frac{1}{R\log t}}}{t}dN(\sigma_2,t)= 2\Big( \frac{x^{-\frac{1}{R\log T}}}{T}N(\sigma_2,T) - \int_{H_{\sigma_2}}^T N(\sigma_2,t){d}\Big( \frac{x^{-\frac{1}{R\log t}}}{t}\Big) \Big).
\frac{d\gamma}{dt}\frac{\partial\overline{V}}{\partial\overline{t}}+\frac{\sigma^{2}(t)}{2}\overline{S}^{2}\frac{\partial^{2}\overline{V}}{\partial\overline{S}^{2}}+(r(t)+\frac{d\alpha}{dt})\overline{S}\frac{\partial\overline{V}}{\partial\overline{S}}-(r(t)+\frac{d\beta}{dt})\overline{V}=0
I_1 \leq \frac{(k-2)!}{(4\pi m)^{k-1}}+ \frac{1}{(4\pi)^{k-1}}\sum_{n=\ell}^\infty \frac{|A_k^{(1)}(m,n)|^2}{n^{k-1}}\int_{2\pi \sqrt{3}n}^\infty u^{k-2}e^{-u}du.
\left[ A_{1}w_{1},w_{2}\right] =\left\langle L\left( I-P_{1}\right)JLw_{1},w_{2}\right\rangle =\left\langle LJLw_{1},w_{2}\right\rangle=-\left\langle Lw_{1},JLw_{2}\right\rangle & =-\left\langle Lw_{1},\left( I-P_{1}\right) JLw_{2}\right\rangle=-\left[ w_{1},A_{1}w_{2}\right] .
\frac{(-4096)^{N-1}F(N-1,N-1)}{2N^2\binom{2N}{N}^2}=\frac{(-1)^{N+1}2^{4 N-5}\left(8 N^2-10 N+3\right) \binom{2 N-2}{N-1}^2 \binom{4 N-4}{2 N-2}}{N^2 \binom{2 N}{N}^2}&=\frac{(-1)^{N+1}2^{4 N-7}(4 N-3) \binom{4 N-4}{2 N-2}}{2 N-1}\in \mathbb{Z}
(x_{ij}, x_{jk})_{x_ix_jx_k}= x_{ij}x_k-x_ix_{jk}= (x_ix_j-(-1)^{|x_{i}||x_{j}|}x_jx_i)x_k-x_i(x_jx_k-(-1)^{|x_{j}||x_{k}|}x_kx_j) = -(-1)^{|x_{i}||x_{j}|}x_jx_ix_k+(-1)^{|x_{j}||x_{k}|}x_ix_kx_j \equiv -(-1)^{2|x_{i}||x_{j}|}x_ix_jx_k+(-1)^{2|x_{j}||x_{k}|}x_ix_jx_k\equiv 0~mod (J; x_ix_jx_k).
\varphi: \mathrm{Free}_2 \rightarrow SL(2,{\mathbb Z}), \ \ \ \ \ \varphi(a) = \left( \begin{array}{cc}1 -1 0 1 \end{array}\right), \ \ \ \ \ \varphi(b) = \left( \begin{array}{cc}1 0 1 1 \end{array}\right).
\beta (p,k) = \int_{-\infty}^{\infty}dx'\frac{q}{2}\frac{A_{p}(0,x')}{iq\pi\sqrt{\omega_p\omega_k}}\left\{\dot{\phi}_{1k}^{+ \ast}(t,x){\partial t\over\partial t'}+ \left(\partial_x\phi_{1k}^{+ \ast}(t,x)\right){\partial x\over\partial t'}-i\omega_p \phi_{1k}^{+ \ast}(t,x) \right\}_{t'=0}.
\begin{array}{r c l}{{{\cal L}_{\mathrm{g . f .}}}}&{{=}}&{{\displaystyle - \frac{1}{2 \xi_{1}}f_{1}f_{1}- \frac{1}{2 \xi_{2}}f_{2}f_{2},}}\\{{f_{1}}}&{{=}}&{{\partial^{\mu}G_{1 \mu}+ i g_{1}\xi_{1}( \phi^{\dagger}\phi_{0}- \phi_{0}^{\dagger}\phi ) ,}}\\{{f_{2}}}&{{=}}&{{\partial^{\mu}G_{2 \mu}- i g_{2}\xi_{2}( \phi^{\dagger}\phi_{0}- \phi_{0}^{\dagger}\phi ) .}}\\ \end{array}
E\sup_{t\in[0,T]}\|u(t)\|_m^2+E\int_{0}^T\left(\delta \|Du(t)\|_m^2+\sum_{k=1}^{d_1}\|L_ku(t)\|_m^2\right)\,dt &\leq C\left\{E\|\underline{u}_0\|_m^2 + E\int_{0}^T \left(\|f(s)\|_m^2+\|g(s)\|_m^2+\|h(s)\|_m^2\right)\,ds\right\},
1+\Big ( 1-x\Big )\sum_{n=0}^{\infty}q^{n+1}(q/x)_n(xq)_n = -\frac{1}{x}g_3(x^{-1};q)+\frac{\Theta(x^{-1};q)}{(q)_{\infty}}m(x^{-2},x;q) &= -\frac{1}{x}g_3(qx;q)-x\frac{\Theta(x;q)}{(q)_{\infty}}m(x^{2},x^{-1};q).
\kappa'\colon F_*((N \otimes \omega_R) / I(N \otimes \omega_R)) \to (N \otimes \omega_R) / I(N \otimes \omega_R) n \otimes m + I(N \otimes \omega_R) \mapsto \sum_{i=1}^t\phi_i(1)\gamma_N(n) \otimes \kappa_R(r_if^{p-1}m) + I(N \otimes \omega_R)
\displaystyle \int_{F^\times}\varphi(t) K^1_{f,R_a(t)f''}(hx,a(t)x) \psi(-t\lambda(h))d^\times t =\int_{H(F)}\int_{F^\times}K^A_{f,f''}(hx,h'hx)\varphi(t)\delta_H(a(t)) \psi\left(t\lambda(h')\right) d^\times t dh'
\mathcal{L}_{ii}= (i,i,1), (i,i,2), \ldots,(i,i,r), 1 \leq i \leq n, \mathcal{L}_{ij}= (i,j,1), (i,j,2), \ldots,(i,j,r),(j,i,1), (j,i,2), \ldots,(j,i,r), 1 \leq j<i \leq n;
V_{0}+ V_{v^{2}}\rightarrow - \frac{4 \alpha_{S}}{3 r}\left[ 1 - \frac{1}{2}\; \left( \bf v_{\mathrm{1}}\cdot v_{\mathrm{2}}+ v_{\mathrm{1}}\cdot \hat{r}\; v_{\mathrm{2}}\cdot \hat{r}\right) \right] .
\int_{R}^{2R}\big|J_{\nu(k)}( r\rho)\big|^2 \frac{\mathrm{d}r}{(1+\frac r M)^\beta}\lesssim\Big(\frac{M}{R}\Big)^\beta \int_{R}^{2R}\big|J_{\nu(k)}(r\rho)\big|^2 \mathrm{d}r\lesssim \Big(\frac{M}{R}\Big)^\beta.
f(x_1,x_2) = x_1^{-2/p}(\log x_1)^{-2/p}, g(y_1,y_2) = y_1^{-2/q}(\log y_1)^{-2/q}, h(z_1,z_2) = (-z_1)^{-2/r}\big(\log(-z_1)\big)^{-2/r}.
\|u_i\|_{2\alpha}^2 \leq 2( \|u_i - \bar{u}_i\|_{2\alpha}^2 + \|\bar{u}_i\|_{2\alpha}^2) \leq c r_i^{2+m(\frac{1}{\alpha}-1)}\|\nabla u_i\|_2^2 + c r_i^{\frac{m}{\alpha}}\bar{u}_i^2
I_2 = \frac{C_u^2}{n^2}\left(nr - \frac{r^2}{2}- \frac{3r}{2}+1\right) \leq \frac{C_u^2}{n^2}\left(nr+1\right) =\frac{C_u^2r}{n^2}+ \frac{C_u^2}{n^2}\leq \frac{C_u^2 \epsilon}{n}+ \frac{C_u^2}{n^2}.
\begin{aligned}A= \mathbb{E}\int_0^T[l_1^{v_1}(t)-l_1(t)]dt, B= \gamma_1(y^{v_1}(0))-\gamma_1(y(0)), C= \mathbb{E}[\Phi_1(x^{v_1}(T))-\Phi_1(x(T))].\end{aligned}
\eta_{p}(s):=\sum_{k=1}^{\infty}\frac{(s,2\pi pk)_{k}}{k^{s}}=\left(2^{1-s}-1\right)\zeta(s)+\sum_{k=1}^{\infty}e^{2\pi pk}\sum_{\ell=1}^{k}\left(\begin{array}{c}k \ell\end{array}\right)\left(4\pi p\right)^{\ell}(-1)^{k-\ell}\,\intop_{k}^{\infty}t^{-s}e^{-2\pi pt}\,\frac{(t-k)^{\ell-1}}{(\ell-1)!}\,dt,
osc(\psi_1,B_\epsilon(x))= \underset{y\in B(x,\epsilon)\cap X}{\textrm{ess-sup}}\int_{B(y,r)}h(z)dz- \underset{\tilde y\in B(x,\epsilon) \cap X}{\textrm{ess-inf}}\int_{B(\tilde y,r)}h(z)dz.
\|\phi\|_{cb}=|c_+|+|c_-| + \begin{cases}\left(1-\frac{1}{d-1}\right)\left\|\left(1-\frac{1}{d-1}\tau\right)^{-1}H\right\|_{S_1}, 3\leq d <\infty, \|H\|_{S_1}, d=\infty,\end{cases}
\mathsf{A}=\mathsf{A}_1\,,\quad\quad\qquad\mathsf{B}=\mathsf{A}_{\mathsf{M}+1}\,,\quad\quad\qquad\mathsf{F}=\mathsf{M}:=\lfloor 2^{-10}\pmb{\eta}\,\mathsf{q}^{-1}\lambda^{-1}\rfloor\,,\quad\quad\qquad\phi = \frac{1}{2}\,.
-{2 l \alpha (l^2+r^2) \over (l^2+ \alpha^2)}d \beta^2 + \beta^2{2 (l^2 + \alpha^2) \over l \alpha (l^2+r^2)}\left[ \alpha d\chi +{(r^2 - \alpha^2) \over (1 + \alpha^2 l^{-2})}d\phi \right]^2.
\bigl\vert H_{X}(\eta) -H_{X_0^{(N)}}(\eta)\bigr\vert \leq2\sin\biggl(\frac{\pi}{2N}\biggr) \biggl(1+\frac{\sqrt{2}}{\sqrt{2}-1}\biggr) \operatorname{diam}(X) \Rightarrow \sup_\eta\bigl\vert\bigl(H_{X}(\eta)-H_{X_0^{(N)}}(\eta)\bigr)\bigr\vert =d_H\bigl(X,X_0^{(N)}\bigr) &\leq(6+2\sqrt{2})\sin\biggl(\frac{\pi}{2N}\biggr)\operatorname{diam}(X).
\mathcal{R}(g_{n_2}\circ g_{n_1}) = \mathcal{R}(g_{n_2}) \Tilde{\mathcal{R}}(g_{n_1}) + \mathcal{R}(g_{n_1}) - \Tilde{\mathcal{R}}(g_{n_1}) = (\mathcal{R}(g_{n_2})-1) \Tilde{\mathcal{R}}(g_{n_1}) + \mathcal{R}(g_{n_1}) \Tilde{\mathcal{R}}(g_{n_2}\circ g_{n_1}) = \Tilde{\mathcal{R}}(g_{n_2}) \Tilde{\mathcal{R}}(g_{n_1}).
K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)=K^{\pm}_{\lambda,\nu}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}-2K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}.
\sum_{i_{1}, . . . i_{l_{1}}; j_{1}, . . . , j_{l_{2}}= 1}^{J}\mathrm{t r}( ( Z \Omega )^{i_{1}}( Z^{\prime}\Omega ) ( Z \Omega )^{j_{1}- i_{1}}( W \Omega ) . . . . ( W \Omega )^{j_{l_{2}}}( Z \Omega )^{J - j_{l_{2}}})
\pi^+_x((D_1\#D_2)S)-\pi^+_x(D_1\#D_2) \pi_x^+(\psi(D_1)\# D_2)^{-1}=\pi^+\big((D_1\#D_2)(S(\psi(D_1)\# D_2)- 1)\big) \pi_x^+(\psi(D_1)\# D_2)^{-1}
v^{(1,2)}_0 = v^{(2,3)}_0 = v^{(2,4)}_0 \I_2\times(0, T), h_{1,2}\partial_{x_2}v^{(1,2)}_0 = h_{2,3}\partial_{x_2}v^{(2,3)}_0 + h_{2,4}\partial_{x_2}v^{(2,4)}_0 \I_2\times(0, T).