harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\pi_1'' + \pi_1'\Bigl({6\over\lambda\phi^5}+{9\over\phi}\Bigr) + \pi_1 \Bigl({6\over\lambda\phi^6}+{15\over\phi^2}+{36\pi\over\lambda\phi^4}-{\lambda_1\over \pi \phi^6}\Bigl({6\pi\over\lambda}\Bigr)^{3/2}\Bigr) = 0\ .
\xi_{( 1 )}^{\mu}= [ \tau , \xi^{1}( \tau , \sigma ) , \xi^{2}( \tau , \sigma ) , \sigma ] , \xi_{( 2 )}^{\mu}= [ \tau , - \xi^{1}( \tau , \sigma ) , - \xi^{2}( \tau , \sigma ) , \sigma ] \; \; .
{\mathbb D}^{B}{}_{A}{\mathbb D}^{D}{}_{C}- \,{\mathbb D}^{D}{}_{C}{\mathbb D}^{B}{}_{A}=\tfrac{2}{3}\delta_{[A}{}^{(D}[{\mathbb D},{\mathbb D}]_{C]}{}^{B)}-\tfrac{2}{3}\delta_{(C}{}^{[B}[{\mathbb D},{\mathbb D}]_{A)}{}^{D]}.
\gamma_{o}(G) \geq \gamma_{o}(G_{0})+\gamma_{o}(T') & \geq \frac{n_{0}-l_{0}+s_{0}}{3}+\frac{n'-l'+s'+k}{3}& =\frac{n-l(G)-k+s(G)+1+k}{3}& =\frac{n-l(G)+s(G)+1}{3}
\hat{H}_{1}= \phi_{( \tilde{u}+ 2 , \bar{\tilde{u}})}^{7_{3}7_{3}}\phi_{( \tilde{u}, \overline{{{\tilde{u}+ 1}}})}^{7_{3}7_{3}}\phi_{( 1 , \bar{\tilde{u}})}^{3 7_{3}}\phi_{( \tilde{u}, \bar{2})}^{7_{3}3}M_{s}^{- 3}
I_{rr}= \frac{c_{b}}{4}\int_{0}^{\infty}\sin(t\xi) \xi^{2}\chi_{\leq 1}(r\xi) \left(-3 J_{1}(r\xi) + J_{3}(r\xi)\right) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}d\xi
{f_{\left. T \right|{\Lambda_{ex}},S}}(\left. t \right|{\lambda_{ex}},s) = \frac{{{{\left({{b_{gam}}}\right)}^{{m_{gam}}}}{t^{{m_{gam}}- 1}}{e^{-{b_{gam}}t}}}}{{\Gamma \left({{m_{gam}}}\right)}}u(t)
\operatorname{Ext}^1_{\mathcal{O}}(P^\circ,M^\circ) = H^1(\operatorname{Hom}(P^\circ,M^\circ)^{\mathcal{O}}) \rightarrow H^1(\operatorname{Hom}(P,M)^E) = \operatorname{Ext}^1_E(P,M)
\delta T_{0}^{0}=\delta \varepsilon =\varepsilon_{,X}\delta X+\varepsilon_{,\varphi}\delta \varphi =\frac{\varepsilon +p}{c_{S}^{2}}\left[ \left( \frac{\delta \varphi}{\dot{\varphi}}\right)^{.}-\Phi \right] -3H\left(\varepsilon +p\right) \frac{\delta \varphi}{\dot{\varphi}}
A :\Lambda_{N,\tilde{N}}\times\Lambda_{N,\tilde{N}}\longrightarrow\mathbb{R}, A\left( J_{1},J_{2}\right) =\left\langle J_{1}^{\prime}\left( b\right)-S_{\dot{\gamma}\left( b\right)}J_{1},J_{2}\left( b\right) \right\rangle.
\int f\{\rm d}\mu_\eta\otimes{\rm d}s_\eta^{\alpha_1}= \int (g-K)j\{\rm d}\mu_\eta\otimes{\rm d}s_\eta^{\alpha_1}= \int (g-K) \{\rm d}\mu_\eta\otimes{\rm d}s_\eta^{\sf H}= 0,
\tilde \ell_{k+d+1}- \tilde\ell_{k+d}= \tilde \ell_{k+d}\left( e^{\beta \varphi(m_{k+d+1}) - \alpha\varphi(m_{k+d})}-1 \right) &> \tilde \ell_{k+d}\left( \beta \varphi(m_{k+d+1}) - \alpha\varphi(m_{k+d}) \right) > \beta \tilde \ell_{k+d}> \frac{\alpha}{2}\tilde \ell_{k+d}.
\sup_{x\in \mathbb R}| \nabla u(t,x)| \leq \| u (t)\|_{C^{1,\alpha}}\leq \| u (t)\|_{H^{1+\delta}_p}\leq \sup_{t\in[0,T]}\| u (t)\|_{H^{1+\delta}_p}= \| u \|_{C([0,T];H^{1+\delta}_p)}=:C,
\beta^{-1}\lambda(X)|\xi|^2\leq \sum_{i,j=1}^{n+1}a_{ij}(X)\xi_i\xi_j\leq\beta\lambda(X)|\xi|^2\mbox{for all}(X,t)\in\mathbb R^{n+2},\ \xi\in\mathbb R^{n+1},
g = d \vec{x}_{a}\Phi_{a b}\cdot d \vec{x}_{b}+ \left( \frac{d \phi_{a}}{4 \pi}+ \vec{\Omega}_{a c}\cdot d \vec{x}_{c}\right) ( \Phi^{- 1})_{a b}\left( \frac{d \phi_{b}}{4 \pi}+ \vec{\Omega}_{b d}\cdot d \vec{x}_{d}\right) ,
\triangle(\tau',\tau)^{ab}=\theta^{ab}\frac{1}{2\pi}\sum_n ne^{in(\tau'-\tau)}\Rightarrow\triangle (\tau,\tau')^{-1}_{ab}=\theta_{ab}\frac{1}{2\pi}\sum_n\frac{1}{n}e^{in(\tau-\tau')}\ .
\int_{\Omega_{R,\delta}}\left( c_1 - u_1 \right) \omega \, dx = g\int_{B_R(0) \cap S}\eta \, dx' + \sigma \int_{\partial B_R(0) \cap S}\nu \cdot N \, ds + \sum_i \int_{\partial \tilde B^i_\delta}A \cdot N \, dS + \int_{\partial B_R(0) \cap \Omega}A \cdot N \, dS,
\kappa =\displaystyle\frac{2\Gamma (h/2)}{\pi \Gamma (1-h/2)}\left( \displaystyle\frac{\sqrt{\pi}}{2\Gamma \left( \displaystyle\frac{1}{2-h}\right) \Gamma \left( \displaystyle\frac{h}{4-2h}\right)}\right)^{2-h}\, .
\mathbb{V}^{\phi}_\theta(T) \geq \sum_{i,j=1}^{\nu}\left( g^{(i)}(\theta)-{\rm Q}_1^i + \tau{\rm Q}_2^i \right) \left( g^{(j)}(\theta)-{\rm Q}_1^j + \tau{\rm Q}_2^j \right){\rm J}^{\phi}_{ij}
m [ \: x^{\mu}, \dot{x}^{\nu}\: ] = - \mathrm{i}\hbar \{g^{\mu \nu}( x ) + h_{\: \rho}^{\mu \nu}( x ) \dot{x}^{\rho}+ k_{\: \rho \sigma}^{\mu \nu}( x ) \dot{x}^{\rho}\dot{x}^{\sigma}+ \cdots \}.
\psi_{y}(\xi_{1,0})=\prod_{j=0}^\infty\left(\prod_{i=0}^{c-1}(1-\frac{\theta^{q^i}}{\theta^{q^{c+\ell j}}}) \cdot \prod_{i=c}^{\ell-1}(1-\frac{\theta^{q^i}}{\theta^{q^{c+\ell+\ell j}}})\right)^{-1}.
\int_FM^*W (\left(\begin{smallmatrix}1 &1 &x 1 & 1\end{smallmatrix}\right))\psi(\frac12 x)\dx=\int_F L_W'(\left(\begin{smallmatrix}1 x &1 & 1 -x & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}1 -\frac12 &1 -\frac12 & 1 & 1\end{smallmatrix}\right))\psi(-x) \dx .
a_n = \sum_{k=\sum_{i=0}^r k_ip^i=0}^{n =\sum_{i=0}^r n_ip^i}\binom{\sum_{i=0}^r n_ip^i}{\sum_{i=0}^r k_ip^i}^2\binom{2k}{k}&\equiv \sum_{i=0}^r\sum_{k_i=0}^{p-1}\prod_{i=0}^r\binom{n_i}{k_i}^2\binom{2k}{k}\pmod{p}\hbox{(by Lucas formula)}
\Bigl(\frac{\partial f}{\partial z_{i}}\,d\bar z_{1}\wedge\cdots\wedge d\bar z_{n},\,\sum_{j=1}^{n}\frac{\partial g_{j}}{\partial z_{i}}\,d\bar z_{1}\wedge\cdots\wedge\widehat{d\bar z_{j}}\wedge\cdots\wedge d\bar z_{n}\Bigr).
[\sigma]_{(A_{p'})^{\frac{1}{p'}}(A^*_{\infty})^{\frac{1}{p}}}:= \sup\limits_{i\in \mathbb Z,Q\in \mathcal{F}^0_{i}}\Bigg(\mathop{\hbox{esssup}}\limits_{Q}(\mathbb E(\omega|\mathcal{F}_i)\mathbb E(\sigma|\mathcal{F}_i)^{p-1})\frac{\int_{Q}{^*M}_i(\sigma\chi_{Q})d\mu}{|Q|}\Bigg)^{\frac{1}{p}}.
P_{N}(X,Y)=\sum_{m=0}^{N-1}e^{-Y}\frac{Y^{m}}{m!}+\sum_{m=N}^{\infty}e^{-Y}\frac{Y^{m}}{m!}(1-\sum_{k=0}^{m-N}e^{-NX}\frac{(NX)^{k}}{k!}),
|G(V_n)-G(V)| \lesssim \|V_n-V\|_{L^2}^2+\|V_n-V\|_{L^2}\|\nabla (V_n-V)\|_{L^2}+\|V\|_{L^2}\|\nabla (V_n-V)\|_{L^2}+\|V_n-V\|_{L^2}\|\nabla V\|_{L^2}&\lesssim \|V_n-V\|_{\mathcal{H}^1}^2+\|V_n-V\|_{\mathcal{H}^1}\ \rightarrow\ 0
Z(s, f, \tilde{f},\varphi,\chi)=\int_{(G\times G)(F^+)\backslash (G\times G)({\bf A})}E(i_V(g,\tilde{g}), s, \varphi, \chi)f(g)\tilde{f}(\tilde{g})\chi^{-1}\circ\det(\tilde{g})dg d\tilde{g},
{\bf\hat{e}_{\rho}.\hat{e}'_{\mu}}={\bf\hat{e}_{\rho}.}L{\bf\hat{e}_{\mu}}=L^{\nu}_{\mu}{\bf\hat{e}_{\rho}.\hat{e}_{\nu}}=L^{\gamma}_{\mu}g_{\rho\gamma}\Rightarrow{L}^{\gamma}_{\mu}=g^{-1}_{\gamma\rho}{\bf\hat{e}_{\rho}.}L{\bf\hat{e}_{\mu}}.
l A_{1}+ n B_{1}= l U_{l}^{n}S t r \left( \ldots \phi^{j_{l}}\ldots \underbrace{\partial_{a_{l}}\phi^{i_{2 k}^{\prime}}\phi^{i_{2 k - 1}^{\prime}}}\ldots \right) + n U_{l}^{n}S t r \left( \ldots \phi^{j_{l}}\ldots \underbrace{\phi^{i_{n}}\phi^{i_{2 k - 1}^{\prime}}}\ldots \partial_{a_{l}}\phi^{i_{2 k}^{\prime}}\ldots \right)
\alpha=\frac{(bd-b^2+bd\varepsilon+b^2\varepsilon^2)mN+2bm^2+(b^2-bd+bd\varepsilon-b^2\varepsilon^2)m}{((b-d)^2-b^2\varepsilon^2)N+(2d-2b)m+(b^2\varepsilon^2-(b-d)^2)}, \beta=\frac{-2bdm\varepsilon}{((b-d)^2-b^2\varepsilon^2)N+(2d-2b)m+(b^2\varepsilon^2-(b-d)^2)}.
\cdots{\frac{\buildrel Q_{n}}{\longrightarrow}}W_{- n + 2 P , n^{\prime}}{\frac{\buildrel Q_{P - n}}{\longrightarrow}}W_{n , n^{\prime}}{\frac{\buildrel Q_{n}}{\longrightarrow}}W_{- n , n^{\prime}}{\frac{\buildrel Q_{P - n}}{\longrightarrow}}W_{n - 2 P , n^{\prime}}{\frac{\buildrel Q_{n}}{\longrightarrow}}W_{- n - 2 P , n^{\prime}}{\frac{\buildrel Q_{P - n}}{\longrightarrow}}\cdots
\frac{\dot{a}^{2}+ k}{a^{2}}= \frac{8 \pi G}{3}\left[ \frac{1}{2}\frac{\dot{\phi}^{2}}{1 + 2 \eta f}+ \frac{\Lambda}{8 \pi G}\frac{1}{( 1 + 4 \eta f )}+{\rho_{b}( a )}\right]
P_p[f](x):=\left\langle f, P_p(\cdot,x) \right\rangle_{\widehat{S}_p}=\frac{1}{p}\sum_{j=0}^{p-1}\int_S f(e^{\frac{j\pi i}{p}}\zeta)\overline{P_p(e^{\frac{j\pi i}{p}}\zeta ,x )}\, d\sigma(\zeta)
S_2 =\frac{-1}{(r-t-1)!(t!)^2}\sum_{l=r-t-1-k}^{r-k-1}(t+l)!(r-l-1)!\binom{-r+l+k}{-r+t+1+l+k}&=-\sum_{l=r-t-1-k}^{r-k-1}(-1)^{r-t-1-l-k}\frac{(t+l)!(r-l-1)!(r-l-k-1-r+t+1+l+k)!}{(r-t-1)!(t!)^2(-r+t+1+l+k)!(r-l-k-1)!}&=-\sum_{m=0}^{t}(-1)^m\frac{(r-1-k+m)!(t+k-m)!}{(r-t-1)!t!m!(t-m)!}=:-\sum_{m=0}^{t}(-1)^m D_m(r,t,k).
T_{\mu\nu}^{\rm ren}=T_{\mu\nu}^{\rm reg}+ g_{\mu\nu}\left(\delta\Lambda+\frac{1}{2}\delta m^2 \phi^2(x)+\frac{\delta \lambda}{4!}\phi^4(x)\right) +A (g_{\mu\nu}\partial_\alpha\partial^\alpha-\partial_\mu\partial_\nu)\phi^2(x) \; .
I_3\le C\int^{1}_{0}\left( \int^{1}_{t}\dfrac{m\left( \tau \right)}{\tau^{2}}d\tau \right)^{2}dt\le4C\int^{1}_{0}\left( \dfrac{m\left( t\right)}{t}\right)^2dt=4C\mathcal W^\mu(0).
\sum_{n=1}^{\infty}\frac{(2n)^k(2x)^{2n}}{\binom{2n}{n}}= \frac{x}{(1-x^2)^{k+\frac{3}{2}}}\left(x\sqrt{1-x^2}p_k(x^2)+\arcsin(x) q_k(x^2)\right).
\hat{\delta}(g)(m \otimes \sigma):= d_N g(m \otimes \sigma) - (-1)^{|g|}g(d_M m \otimes \sigma) - (-1)^{|m| +|g|}g( m \otimes \partial(\sigma)) &+ \sum_{(\sigma)}(-1)^{|m| +|g|}g ( (m \cdot \tau(\sigma') )\otimes \sigma'') + \sum_{(\sigma)}(-1)^{|g|+|m| + |\sigma'|}g(m \otimes \sigma') \cdot \tau(\sigma''),
\{Q_{\alpha}, Q_{\beta}\}= ( C \Gamma^{\mu})_{\alpha \beta}P_{\mu}+ ( C \Gamma_{\mu}\Gamma_{1 1})_{\alpha \beta}{\hat{Z}}^{\mu}+ ( C \Gamma_{\mu \nu})_{\alpha \beta}{\hat{Z}}^{\mu \nu}+ ( C \Gamma_{1 1})_{\alpha \beta}{\hat{Z}}\quad
-2 (a_2 p+a_4 q+z)\cdot A(q) =-2(a_2 p+a_4 q+z)\cdot(A(q)-A(0)+A(0)) &\leq 2C_A(a_2 |p|+a_4|q|+|z|)|q|-2(a_2 p+a_4 q+z)\cdot A(0), &\leq C_A((2a_4+a_2+1)|q|^2+a_2|p|^2+|z|^2)+(\beta a_4 |q|^2+\lambda |p|^2+\alpha a_3 |z|^2) &\qquad+\left(\frac{a_4}{\beta}+\frac{a_2^2}{\lambda}+\frac{1}{\alpha a_3}\right)|A(0)|^2.
L^{(1)}_{eff}=2\pi\kappa\frac{\sigma_{1}\sigma_{2}}{q_{1}q_{2}}\sum^{n_{1}}_{p_{1}=1}\sum^{n_{2}}_{p_{2}=1}\frac{d}{dt}Arg( \vec{R}^{(1)}_{p_{1}}- \vec{R}^{(2)}_{p_{2}}) \;\;.
\alpha_{\mu}=|E(uw)|^{2(n)}E(|w|^2) -(1+n)\mu^n|E(uw)|^{-2}\chi_{S_0}(E(|w|^2))^2E(|u|^2) &+n\mu^{n+1}|E(uw)|^{-2}\chi_{S_0}E(|w|^2).
\Gamma^{\sf t}_{ij}= a\left(\frac{da}{d{\sf t}}\right)\gamma_{ij}\;, ~~~\Gamma^{i}_{{\sf t}j}= \frac{1}{a}\left(\frac{da}{d{\sf t}}\right)\delta^i_j \;,~~~\Gamma^{i}_{jk}=\frac{\gamma^{il}}{2}(\partial_j\gamma_{lk}+\partial_k \gamma_{lj}-\partial_l\gamma_{jk}) \;.
\left( \sum_{i_{1},\dots ,i_{k}=1}^{\infty}\left\vert T\left(e_{i_{1}}^{n_{1}},\dots ,e_{i_{k}}^{n_{k}}\right) \right\vert^{\rho}\right)^{\frac{1}{\rho}}\leq M(k,m,p,\mathbb{K})\left\Vert T\right\Vert ,
R_{p n}^{m j}\widetilde{R}_{m q}^{i n}= \widetilde{R}_{p n}^{m j}R_{m q}^{i n}= \delta_{p}^{i}\, \delta_{q}^{j}\, , \ \ \, o v e r l i n e{R}{}_{p n}^{m j}\widetilde{\overline{{{R}}}}{}_{m q}^{i n}= \widetilde{\overline{{{R}}}}{}_{p n}^{m j}\overline{{{R}}}{}_{m q}^{i n}= \delta_{p}^{i}\, \delta_{q}^{j}\, .
\ln J = \alpha \left( \mathrm{T r}_{V}e^{t ( ( D_{\mu})^{2}\delta_{\mu \nu}- i F_{\rho \sigma}( M^{\rho \sigma})_{\mu \nu})}- \mathrm{T r}_{\lambda}e^{t{\not D}^{2}}- \mathrm{T r}_{\bar{\lambda}}e^{t{\not D}^{2}}\right) .
\begin{array}{r c l}{{Q}}&{{=}}&{{S_{+}C^{+}+ S_{-}C^{-}+ A_{1}C^{3}+ A_{2}\bar{C}_{-}C^{-}C^{3}+ A_{3}\bar{C}_{+}C^{+}C^{3}+ A_{4}\bar{C}_{3}C^{+}C^{-}}}\\{{}}&{{}}&{{+ A_{5}\bar{C}_{+}\bar{C}_{-}C^{+}C^{-}C^{3}~ ,}}\\ \end{array}
\begin{array}{l}\left\{e^{-1}n_\lambda \cdot \Gamma ~,~~e^{-1}n_y\cdot \Gamma~,~~e^{-1}n_A\cdot \Gamma ~,~~e^{-1}n_\psi \cdot \Gamma ~,~~e^{-1}n_\varphi\cdot \Gamma ~\right\}\,,\end{array}
\begin{aligned}\tau |S_\tau| |A| \le{I}(QA\times S_\tau , A^{-1}\times R) \ll |QA|^{5/8}|S_\tau|^{1/2}|A|^{27/16}+ |S_\tau|^{1/2}|A|^2 \cdot \sqrt{\max\{1, |QA|^2/p \}}& \ll|QA|^{5/8}|S_\tau|^{1/2}|A|^{27/16}\,.\end{aligned}
G=\beta(2)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=\frac{1}{1^{2}}-\frac{1}{3^{2}}+\frac{1}{5^{2}}-\frac{1}{7^{2}}+\frac{1}{9^{2}}-\cdot\cdot\cdot,
\Phi_{1}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{- \phi_{1}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- \phi_{2}}}\\{{\phi_{1}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\phi_{2}}}&{{0}}&{{0}}\\ \end{array}\right) , \qquad \Phi_{2}= \Phi_{1}( \phi_{1}\to - \phi_{3}, \phi_{2}\to - \phi_{4}) ,
(x_{1}+y_{1})^{n_{1}}\cdot\cdot\cdot(x_{d}+y_{d})^{n_{d}}=\sum_{k_{1}=0}^{n_{1}}\cdot\cdot\cdot\sum_{k_{d}=0}^{n_{d}}(\begin{matrix}n_{1}\\k_{1}\end{matrix})x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}...(\begin{matrix}n_{d}\\k_{d}\end{matrix})x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.
\begin{array}{l}{{\mathbf{F}= ( q \cos \theta + g \sin \theta ) [ ( \mathbf{E}\cos \theta + \mathbf{B}\sin \theta ) + \mathbf{v}\times ( - \mathbf{E}\sin \theta + \mathbf{B}\cos \theta ) ]}}\\{{= q \mathbf{E}+ g \mathbf{B}+ \mathbf{v}\times ( q \mathbf{B}- g \mathbf{E}) .}}\\ \end{array}
\widetilde{\varphi}_\mu (x)=\frac 1{3!}\epsilon_\mu^{\nu \rho \sigma}\varphi_{\nu \rho \sigma}(x),\hspace{0.5in}\widetilde{\varphi}(x)=\frac1{4!}\epsilon^{\mu \nu \rho \sigma}\varphi_{\mu \nu \rho \sigma}(x)
m_j^*= \sum \limits_{i = 0}^{\lfloor \frac{j - 1}{2}\rfloor}\binom{j - 1}{i}(z_1^*)^{i + 1}(z_2^*)^i (z_1^* + z_2^*)^{j - 1 - i}\frac{1}{i + 1}\binom{2i}{i}.
Z ( t ) ={\frac{1}{( 2 \pi \hbar )^{2}}}\left({\frac{2 \pi}{t}}\right)^{3 / 2}\sqrt{{\frac{2}{t v^{4}}}}\int_{0}^{\infty}d s e^{- s}s^{- 1 / 2}\left( 1 +{\frac{s}{2 z}}\right)^{- 1 / 2},
|\phi(0)|\leq |G_I(x_1, 0)|\cdot |\phi(x_1-1)|+|G_I(x_2, 0)|\cdot |\phi(x_2+1)| = \frac{|\tilde{P}_{x_2}(\theta_1)|}{|\tilde{P}_I(\theta_{x_1})|}\prod_{j=x_1}^0 |\cos(\pi(\theta_j))|\cdot |\phi(x_1-1)|+\frac{|\tilde{P}_{-x_1}(\theta_{x_1})|}{|\tilde{P}_I(\theta_{x_1})|}\prod_{j=0}^{x_2}|\cos(\pi(\theta_j))|\cdot |\phi(x_2+1)|.
6(-1)^{n+1}\sum_{\substack{a,b\in \mathbb{Z}ab=4n+3 12|3a-b-4}}d\left(N,\tilde{N}, \frac{1}{3}, \frac{1}{6}\right) = \sum_{\substack{a,b\in\mathbb{Z}^+ ab=4n+3 b\geq 3a+2 12|3a-b-4 \ \ 12|3a-b+4}}3a -\sum_{\substack{a,b\in\mathbb{Z}^+ ab=4n+3 1\leq b\leq 3a-1 12|3a-b-4 \ \ 12|3a-b+4}}b.
I^{(1)}_{\epsilon}(a^2)={{\pi}\over 2}{1\over{cos{{\pi}\over 2}\epsilon}}\left \{{1\over{a^{1+ \epsilon}}}+ 2 \sum_{n=1}^{\infty}{1\over{(2 \pi n)^{1+\epsilon}}}+ 2\sum_{n=1}^{\infty}{1\over{(2 \pi n)^{1+\epsilon}}}\; [{1\over{(1+{{a^2}\over{4 \pi^2 n^2}})^{{1+\epsilon}\over 2}}}-1 ] \right \}\; .
c_r = \zeta(r)\left(1 -2^{1-r}\right), d_r = -\left( \frac{\zeta\left(r-2\right)\left(1-2^{3-r}\right)}{2}+\rho_{C}\zeta(r-1)\left(1 - 2^{1-r}\right)\right) , d_r^{\prime}= -\left(\zeta\left(r-2\right)\left(1-2^{3-r}\right) +\frac{\rho_{R}}{2}\zeta(r-1)\left(1 - 2^{1-r}\right)\right) .
\cos\alpha_s^0 - \cos\alpha_t^0 \le \frac{a_2^2+a_1^2(1+r^2)-a_3^2}{2a_2a_1(1-r^2)}-\frac{a_2^2+a_1^2(1-r^2)-a_3^2}{2a_2a_1(1+r^2)}=\frac{r^2(2a_1^2+a_2^2-a_3^2)}{a_1a_2(1-r^2)(1+r^2)}=\frac{r^2}{1-r^4}\left(\frac{2a_1}{a_2}+\frac{a_2}{a_1}-\frac{a_3^2}{a_1a_2}\right) \le C(\delta)r^2.
\Psi^{[m]}(a_{1},...,a_{m};c;z_{1},...z_{m})=\sum\frac{(a_{1})_{k_{1}}\cdot\cdot\cdot(a_{m})_{k_{m}}z_{1}^{k_{1}}\cdot\cdot\cdot z_{m}^{k_{m}}}{(c)_{k}k_{1}!\cdot\cdot\cdot k_{m}!}.
2N\frac{(N-1)!(-4)^{N-j-1}}{j!(2N-2j-1)!}(j-2N+M+\frac 32)_{N-j}= = 2N\sum_{l=0}^M{M\choose l}\frac{(N-M-1)!(-4)^{N-M-j+l-1}}{(j-l)!(2N-2M-2j+2l-1)!}\times \times (j-l-2N+M+\frac 32)_{N-M-j+l-1},
\langle\xi^{*i}, u\rangle :=\max \left\{\langle \xi^i, u\rangle\,:\, \xi^i\in \partial_S^2f_i(x^0)(u)\right\},\ \i\in I, \langle\zeta^{*j}, u\rangle :=\max \left\{\langle \zeta^j, u\rangle\,:\,\zeta^j\in \partial_S^2g_j(x^0)(u) \right\},\ \j\in J.
\langle{\cal O}\rangle ={\frac{( a r_{+})^{\Delta}}{( a^{2}+ ( 1 + a^{2}) \sinh^{2}[ r_{+}( t + \phi ) / 2 ] )^{\Delta / 2}( a^{2}+ ( 1 + a^{2}) \sinh^{2}[ r_{+}( t - \phi ) / 2 ] )^{\Delta / 2}}}
\gamma_F(\frac{1}{2}\psi \circ q_3) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_2, \frac{1}{2}\psi) \cdot (-2 \nu t, J_2)_F = \gamma_F(J_2, \frac{1}{2}\psi) \cdot (2 \nu t, J_2)_F.
\begin{aligned}T(t) =e^{-t(\sqrt{A}+K_{V})}~, L^2 = L^2(\mathbb{R}^{2d})~, E = \lbrace u\in L^2(\mathbb{R}^{2d}), qu,\partial_qu\in L^2(\mathbb{R}^{2d})\rbrace\end{aligned}
\|W_{xx}(t)\|_1^2 + \|W_{xxt}( t)\|^2+ \int_0^t \big(\|W_{xxx}\|^2 + \|W_{xx\tau}\|^2\big) d\tau \leqslant O(1)\delta^{-1/4}\big( \|W(0)\|_3^2 + \|W_t(0)\|_2^2\big)+ O(1) \delta_0^{{1}/{4}}.
\Omega=2iJ\left[\frac{d{\bar\xi}\wedge d\xi}{1+\mid\xi\mid^2}-\frac{( \xi d{\bar\xi})\wedge({\bar\xi}d\xi)}{(1+\mid\xi\mid^2)^2}\right],\ \ \ \mid\xi\mid^2=\sum_p\mid\xi_p\mid^2.
( 1 - \gamma ) ( \partial_{y}+ 2 \ell^{- 1}) \, h |_{0_{+}}- \frac{2}{3}\, \gamma \, \ell \, \big ( \partial_{\lambda}\partial_{\rho}h^{\lambda \rho}- \partial_{\lambda}^{2}h \big ) |_{0_{+}}= \frac{1}{6 M_{*}^{3}}\, S_{\mu}^{\mu}\, .
W_j^v(n,s):=\bigg\{x\in V\setminus \bigcup_{k=0}^{j-1}W_k^v(n,s)\,\bigg|\, \exists k<j,\y\in W_k^v(n,s),\ 1\le i\le n(y): &s_{j-k}(y,i)=x\bigg\}
\begin{cases}\bigl|{\xi_2}+ \frac{1}{2}N + \frac{\sqrt{3}}{2}(\sqrt{2}-1) N\bigr|\leq 2^{12}{A'}^{-\frac{1}{2}}N, |\eta_2 - (\sqrt{2}+1)^{\frac{2}{3}}\xi_2 + (\sqrt{3}+1) (\sqrt{2}-1)^{\frac{1}{3}}N| \leq 2^{25}{A'}^{-1}N,\end{cases}
|\Theta_h(t,y^*)-\Theta_h(s,z^*)| = |y^*(h(t))-z^*(h(s))| &\leq |y^*(h(t))-y^*(h(s))|+|J_Y(h(s))(y^*)-J_Y(h(s))(z^*)| & \leq \|h(t)-h(s)\|+|J_Y(h(s))(y^*)-J_Y(h(s))(z^*)|,
( \Psi_{\pm})_{\mu}^{A}= \frac{1}{2}( \delta \pm \gamma_{5}\sigma_{3})_{B}^{A}\Psi_{\mu}^{B}\ , \quad ( \eta_{\pm})^{A}= \frac{1}{2}( \delta \pm \gamma_{5}\sigma_{3})_{B}^{A}\eta^{B}\ .
\{\circ , \circ \}_{D}\equiv \{\circ , \circ \}- \{\circ , \phi_{i}^{\alpha}\}\frac{1}{\{\phi_{i}^{\alpha}, \bar{\phi}_{\bar{j}\beta}\}}\{\bar{\phi}_{\bar{j}\beta}, \circ \}- \{\circ , \bar{\phi}_{\bar{j}\beta}\}\frac{1}{\{\bar{\phi}_{\bar{j}\beta}, \phi_{i}^{\alpha}\}}\{\phi_{i}^{\alpha}, \circ \}
E^j{\cal P}^{ip}_{jq}=E^p\delta^i_q\,,\ \ tilde{\varepsilon}_i{\cal P}^{ip}_{jq}=\tilde{\varepsilon}_q\,\delta^p_j\,,\ \ \ \varepsilon_p\,\tilde{E}^q\,{\cal P}^{ip}_{jq}=\tilde{E}^i\varepsilon_j\,,
\int_M s^H\frac{\omega_M^n}{n!}=s^H_S \frac{(p+c)^{n-1}- c^{n-1}}{p(n-1)}- \left[(P \varphi)'(x)P(x)^{n-2}\right]^1_0 + \int_0^1(P\varphi)'(x)P(x)^{n-2}dx &= s^H_S \frac{(p+c)^{n-1}- c^{n-1}}{p(n-1)}+ 2c^{n-1}+2(p+c)^{n-1}+ p\int_0^1 \varphi'(x)P(x)^{n-2}dx + \int_0^1 \varphi(x) P(x)^{n-1}dx
\vec f_{\phi, t}(g, e) = \left( \frac{mn}{d^2}\right)^t\sum_{o_2 = e_3, \dots, o_{2t}}\left(\prod_{s=1}^{t}M_{\iota(o_{2s}), \iota(o_{2s+1})}M_{\iota(o_{2s+2}), \iota(o_{2s+1})}\right) \phi_{\iota(o_{2t+2})},
\frac{\|x_{\delta}\|_2}{\|x\|_2}\leq \frac{\langle x_{\delta}, u(x)\rangle}{\langle x, u(x)\rangle}= \frac{\langle x-\Delta(x,\delta)u(x), u(x)\rangle}{\langle x, (u(x))\rangle}=1-\frac{\Delta(x,\delta)}{\langle x, u(x)\rangle}\leq 1-\Gamma_{\min}(\delta)\quad,
\langle \tilde{J}_\mu^{(\alpha)}(X),s_\mu \rangle = \sum_{k=0}^n (\ell \cdot \ell! h_k(B(c_1,\ldots,c_n)) + \ell! h_k(B(d_1,\ldots,d_n))){\alpha + k \choose n}& = \sum_{k=0}^n (\ell \cdot \ell! r_k(B(c_1,\ldots,c_n)) + \ell! r_k(B(d_1,\ldots,d_n))){\alpha \choose k}k!
\bar\nabla^{H}_{\frac{\partial}{\partial t}}D(\pi\circ\Phi)\;\big|_0 w= \pi^*\bar\nabla^{(\pi\circ\Phi)^{-1}TY}_{w}D(\pi\circ\Phi)\frac{\partial}{\partial t}\Big|_0 = \pi^*\bar\nabla^{\pi^{-1}TY}_{w}D\pi v .
\left[\begin{array}{c}p_n q_n\end{array}\right] =\left[\begin{array}{cc}0 1 1 a_1\end{array}\right]\left[\begin{array}{cc}0 1 1 a_2\end{array}\right]\cdots\left[\begin{array}{cc}0 1 1 a_{n-1}\end{array}\right]\left[\begin{array}{c}1 a_{n}\end{array}\right]
\mathbf{A}=\mbox{Prod}\left(\mbox{Prod}\left(\mathbf{U},\mathbf{D}_{0},\mbox{\ensuremath{\mathbf{D}}}_{0}^{\top}\right),\mbox{Prod}\left(\mathbf{V},\mathbf{D}_{1},\mbox{\ensuremath{\mathbf{D}}}_{1}^{\top}\right)^{\top^{2}},\mbox{Prod}\left(\mathbf{W},\mathbf{D}_{2},\mbox{\ensuremath{\mathbf{D}}}_{2}^{\top}\right)^{\top}\right)
\left[{dn_s(\omega|D) \over d\omega}\right]_{\tiny\mbox{div}}={\Gamma\left(1-\frac D2\right) \over (4\pi)^{D/2}}\,\,{m^{D-4}\over \kappa}\,\, \int_{\Sigma}\left[2\left(m^2-\left(\frac 16 -\xi\right)R\right)-{\omega^2\over \kappa^2}{\cal P}\right]~~~,
=[(\rho_n(x-c_n)-\alpha_{n+1})[ d_{n-1}(x^2+\omega^2)-\alpha_{n-1}\rho_{n-1}(x-c_{n-1})+\alpha_{n-1}d_n (x^2+\omega^2) ]]\mathcal{L}_{n}(x) &+d_{n-1}(x^2+\omega^2)(\alpha_n\rho_n(x-c_n)-\alpha_n \alpha_{n+1}-d_n (x^2+\omega^2)) \mathcal{L}_{n-1}(x).
R^{G}_{T_{w}}\theta_{w}(s)=\frac{1}{|C_G(s)^{F}|}\sum_{\{h\in G^{F}\mid hsh^{-1}\in T_{w}\}}Q^{C_G(s)}_{C_{h^{-1}T_{w}h}(s)}(1)\theta_{w}(hsh^{-1}).
R^{C}= F R^{A}F = \left( \begin{array}{r r r r}{{q}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{t}}&{{\Omega}}&{{0}}\\{{0}}&{{0}}&{{t^{- 1}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{q}}\\ \end{array}\right) \ .
g_{2k}(\frac{1}{2})=g_{2k+1}(\frac{1}{2})=\frac{1}{2}\frac{(2k-1)!!}{(2k)!!}=\frac{1}{2}\cdot\frac{1\cdot3\cdot\cdot\cdot(2k-1)}{2\cdot4\cdot\cdot\cdot(2k)}
v_{i,[\gamma]}^{j_{i}}=y_{i,[\kappa_{i}^{j_{i}}+\gamma]}^{j_{i},d}-\sum_{\alpha=\kappa_{i}^{j_{i}}}^{\kappa_{i}^{j_{i}}-1+\gamma}a_{i}^{j_{i},\alpha-\gamma}(v_{i,[\alpha-\kappa_{i}^{j_{i}}]}^{j_{i}}-y_{i,[\alpha]}^{j_{i},d})-\sum_{\beta=\gamma}^{\kappa_{i}^{j_{i}}-1}a_{i}^{j_{i},\beta-\gamma}(y_{i,[\beta]}^{j_{i}}-y_{i,[\beta]}^{j_{i},d})\,.
\Theta_{\Lambda}\left[ \begin{array}{c}{{a}}\\{{b}}\\ \end{array}\right] ( v^{I}+ \tau p^{I}+ q^{I}| \tau ) = e^{- i \pi \tau q^{2}- 2 \pi i \, p \cdot ( v + b ) + 2 \pi i a \cdot q}\Theta_{\Lambda}\left[ \begin{array}{c}{{a}}\\{{b}}\\ \end{array}\right] ( v^{I}| \tau ) .
\int_{\theta}^t f_1(s_1) \int_{\Delta_{\theta,s_1}^p}g_1(r_1) \dots g_p(r_p) dr_p \dots dr_1 ds_1 = \int_{\Delta_{\theta,t}^{p+1}}f_1(w_1) g_1(w_2) \dots g_p(w_{p+1}) dw_{p+1}\dots dw_1,
\Vert R_3^m(t)\Vert_{H^3(1+\rho_t)}^2 = \Vert R_3^m(t)\Vert_{H^1(1+\rho_t)}^2 + \Vert \partial_{xx}R_3^m(t)\Vert_{L^2(1+\rho_t)}^2 +\Vert \partial_{xxx}R_3^m(t)\Vert_{L^2(1+\rho_t)}^2
\int \delta ( x ( s ) - x ( s^{\prime}) ) \, f ( s^{\prime}) \, d s^{\prime}\, = \lim_{\Delta \to 4}( 4 - \Delta ) \int \frac{f ( s^{\prime}) \, d s^{\prime}}{| x ( s ) - x ( s^{\prime}) |^{\Delta}}= 0 \; ,
\tau = \int \frac{\sqrt{\dot{T}^{2}- T^{2}\dot{W}^{2}}}{T \ln^{2 / 3}T^{2}}d T , r_{c}y = \int \frac{\sqrt{\dot{T}^{2}- T^{2}\dot{W}^{2}}}{\ln^{2 / 3}T^{2}}d W ,
\Pr\left[\exists B_{ij},~\exists v\in B_{ij},~s.t.~d_B(v)> (1-\frac{\varepsilon}2)\left\lfloor{\frac{n}{k-1}}\right\rfloor\right] \leq \binom{k-1}2\cdot2\left\lfloor{\frac{n}{k-1}}\right\rfloor\cdot (1+o(1))e^{-\frac{\varepsilon^2}{48k}(1-2\varepsilon)n}=o(1).
H_{B}^{\mathrm{b a r e}}~ = ~ - \frac{1}{2}\sum_{k = 1}^{\infty}\Bigl ( \sigma_{k + 1}^{x}\sigma_{k}^{x}+ \sigma_{k + 1}^{y}\sigma_{k}^{y}+ \Delta \sigma_{k + 1}^{z}\sigma_{k}^{z}\Bigr ) + h \sigma_{1}^{z}~ .
\omega_\nu =e_{31}\otimes\big(\Upsilon_\nu e_{11}+\Upsilon_u (1-e_{11})\big) \;, \xi = e_{52}\otimes e_{11}\;, begin{align*}3pt]\omega_e =e_{42}\otimes\big(\Upsilon_e e_{11}+\Upsilon_d (1-e_{11})\big) \;, \eta =e_{56}\otimes 1 \;,
e_{j,J}^\bot = e_j\chi_J - \sum_{i=0}^J (e_j\chi_J,e_i)e_i = e_j\chi_J - (e_j\chi_J,e_j)e_j - \sum_{i\not=j}(e_j(\chi_J-1),e_i)e_i = e_j(\chi_J-1) + (e_j(1-\chi_J),e_j)e_j - \sum_{i\not=j}(e_j(\chi_J-1),e_i)e_i.
B_3 = \left\{\begin{array}{ll}E_{1,\ell_1,\ell_2,\ell_3,\ell_4}; E_{2,\ell_1,\ell_2,\ell_3,\ell_4}, E_{3,\ell_1,\ell_2,\ell_3,\ell_4}, \end{array}: \; 2 \leq \ell_1 \leq s, \; 0 \leq \ell_2, \ell_3, \ell_4 \leq s-2 \right\}.
u_{i + 1, j + 1}\oplus u_{i + 1, j - 1}= u_{i, j}\oplus u_{i + 2, j}&\leq (u_{i,j}\wedge u_{i + 1, j - 1}) \oplus (u_{i,j}\wedge u_{i + 1, j + 1}) \oplus (u_{i + 2,j}\wedge u_{i + 1, j - 1}) \oplus (u_{i + 2,j}\wedge u_{i + 1, j + 1}) &\leq u_{i + 1, j + 1}\oplus u_{i + 1, j - 1}.
\vert y(z)-x \vert = \vert x + z +f(z) -x \vert = \vert z +f(z) \vert \leq \vert z \vert + \vert f(z) \vert \leq \vert z \vert + w_x(\vert y(z)-x \vert) \cdot \vert y(z) -x \vert.
\lim_{n\rightarrow +\infty}(b_{n,k})^{-2}\sum_{x,y\in\mathbb Z^{d_0}}\left(\sum_{i,j= 1}^m \theta_{i,j}d_{i,n}(x)d_{j,n}(y)\right)^k=(K_k)^2\sum_{i,j=1}^m(\theta_{i,j})^k(t_i-t_{i-1})(t_j-t_{j-1})\a.s.,
\textbf{w}_{1,k'}= \frac{1}{\tau\lambda \sigma_{\hat{h}_{1,k}}^2}\textbf{h}_{1,k'}\Big(\sum_{k''\neq k'}^{K}\big(\textbf{p}_{1,k}\textbf{s}_{1,k''}^{H}\big)\textbf{h}_{1,k''}^H+\sum_{j\neq 1}^L\sum_{k''=1}^{K}\big(\textbf{p}_{1,k}\textbf{s}_{j,k''}^H \big)\textbf{h}_{j,k''}^H + \textbf{p}_{1,k}\textbf{N}_j^H+\textbf{h}_{1,k}\Big)\hat{\textbf{h}}~\!\!_{1,k}^H.
L^\infty_s (\Omega) \, = \, \big \{f\in L^\infty (\Omega)~|~ (1+|x|)^s f \in L^\infty (\Omega) \big \}\,, L^\infty_{s,0}(\Omega) \, = \, \big \{f\in L^\infty_s (\Omega) ~|~ \lim_{R\rightarrow \infty}{\rm ess.sup}_{|x|\geq R}|x|^s |f(x)| \, = \, 0\big \}\,.
\begin{array}{r c l}{\omega^{x}}&{=}&{2 f_{t t}d f_{t x}\wedge d f_{t t}+ \frac{3}{2}d f_{x}\wedge d f_{t x},}\\{\omega^{t}}&{=}&{2 f_{t x}d f_{t t}\wedge d f_{t x}- \frac{1}{2}d f_{x}\wedge d f_{x x}}\\ \end{array}
A_1=\begin{psmallmatrix}0 p_{2}0 0 0 p_{3}p_{1}0 0\end{psmallmatrix}\;\mbox{and}\;\;\;A_2=\begin{psmallmatrix}0 0 p_{3}p_{1}0 0 0 p_{2}0\end{psmallmatrix}\;.
L\Phi(x)= \sum_{i=1}^N c_i L \eta_i = \sum_{i=2}^{N-1}m c_i (\eta_{i-1}+\eta_{i+1}-2\eta_i)+ & c_1(bm+(b-d-m)\eta_1+m\eta_2)+c_N(bm+(b-d-m)\eta_N+m\eta_{N-1})