harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta (f(G^{-1}(\bar b_i))^{2-n}-g) &=-\lambda f(G^{-1}(\bar b_i))^n(|\nabla\bar b_i|^2-g^{\frac{2n-2}{n-2}}) &+\lambda f(G^{-1}(\bar b_i))^ng\frac{\left(f(G^{-1}(\bar b_i))^{2-n}\right)^{\frac{n}{n-2}}-g^{\frac{n}{n-2}}}{f(G^{-1}(\bar b_i))^{2-n}-g}(f(G^{-1}(\bar b_i))^{2-n}-g).
{L}^{( 0 )}= \left[ m p_{i}( t ) - e A_{i}( t , \vec{q}) \right] \dot{q}^{i}( t ) - \theta \int d^{2}x \Pi_{i}( t , \vec{x}) \dot{A}^{i}( t , \vec{x}) - V^{( 0 )}
\delta^{\Psi\Psi}_{pp'qq'(j+1)}= \frac{\lambda_{p'q'j}^2}{\lambda_{pqj}^2}\, \delta^{\Phi\Phi}_{pp'qq'j}\qquad\Bar{\delta}^{\Phi\Phi}_{pp'qq'j}= \frac{\lambda_{p'q'j}^2}{\lambda_{pqj}^2}\, \Bar{\delta}^{\Psi\Psi}_{pp'qq'(j+1)}.
\sum_{n\geq 1,n\neq k}\frac{1}{j_{\nu,n}^4-j_{\nu,k}^4}=\frac{1}{2j_{\nu,k}^2}\sum_{n\geq 1,n\neq k}\frac{1}{j_{\nu,n}^2-j_{\nu,k}^2}-\frac{1}{2j_{\nu,k}^2}\sum_{n\geq 1,n\neq k}\frac{1}{j_{\nu,n}^2+j_{\nu,k}^2},
x_{2i,n-j}+ x_{2i,n-j+1}+ x_{2i+1,n-j}+ x_{2i+1,n-j+1}= S = x_{2i+1,n-j}+ x_{2i+1,n-j+1}+ x_{2i+2,n-j}+ x_{2i+2,n-j+1},
C'' + \biggl[ 3{\cal H}+ 2 \frac{{\cal H}'}{\cal H}- 2 \frac{\varphi''}{\varphi'}\biggr]C' + 3\biggl[3{\cal H}' - 2{\cal H}\frac{\varphi''}{\varphi'}\biggr] C - \partial_{\alpha}\partial^{\alpha}C =0.
\|u_i\|_2\leq\sqrt{\|u_i\|_0}\cdot \|u_i\|_\infty =\sqrt{k(t-1)-\ell}\cdot \|u_i\|_\infty \leq\sqrt{k(t-1)}\cdot \|u_i\|_\infty \leq\alpha\sqrt{k/(t-1)}.
\partial_{\mu}\Big ( \exp ( \frac{\phi}{f_{\phi}}) F^{\mu \nu}{}_{i}\Big ) + q \exp ( \frac{\phi}{f_{\phi}}) A_{\mu}{}^{j}f_{i j}{}^{k}F^{\mu \nu}{}_{k}= - q \overline{{{\psi}}}\gamma^{\nu}X_{i}\psi ,
\begin{aligned}((U_2^{-n}XU_1^n-X))\,U_1^k e, U_2^l\bar e)-((U_2^{k+l}XU_1^{-(k+l)}- U_2^n XU_1^{-n})\,U_1^k e, U_2^l\bar e)= =\sum_{m=1}^{k+l}(K U_1^{k-m-n}e, U_2^{l-m+1-n}\bar e). \end{aligned}
S_2^{*}=\frac{1}{(1-c)(cq)_N}\sum_{k=1}^N \frac{(cq)_k q^k}{(q)_k (1-q^k)}\sum_{j=0}^{N-k}\frac{(1/c)_j (cq)^j (cq)_{N-j-k}}{(q)_j(q)_{N-j-k}}= \frac{1}{(1-c)(cq)_N}\sum_{k=1}^N \frac{(cq)_k (cq)_{N-k}q^k}{(q)_k (1-q^k)(q)_{N-k}}\sum_{j=0}^{N-k}\frac{(1/c)_j q^j \left(q^{-(N-k)}\right)_j}{(q)_j \left(q^{-(N-k)}/c\right)_j},
u_k^\alpha=\prod_{\lambda=1}^r\left(\sum_{\mu=1}^r(T_{kj})^\lambda_\mu u_j^\mu+O(|u_j|^n)\right)^{\alpha_\lambda}=\sum_{|\beta|=|\alpha|}\tau_{kj, \beta}^\alpha\cdot u_j^\beta +O(|u_j|^{n+1})
X_{1}^{\prime}=\frac{k}{l^{3}}\frac{\rho}{\rho^{\prime}}(X_{1}+\epsilon\Sigma X_{1}\xi),Y_{1}^{\prime}=\frac{\rho}{\rho^{\prime}}\frac{Y_{1}}{l^{3}},Z_{1}^{\prime}=\frac{\rho}{\rho^{\prime}}\frac{Z_{1}}{l^{3}}
h^{ij}(\nabla_ih_{jl})u^l = h^{ij}\nabla_i(h_{lj})u^l = h^{ij}(\nabla_lh_{ij}-R_{iljk}N^k)u^l &= \frac{1}{2}\langle\nabla|h|^2,\nabla u\rangle - 2Hk_0\langle \mathbf{N},\nabla u \rangle - k_0h(\mathbf{N},\nabla u) = \frac{1}{2}\langle\nabla|h|^2,\nabla u\rangle,
g_{jk}: ={\alpha_{j-N, k-N}=}4\pi \eta_{-}^{j+k-2N-2}\frac{\varrho_{j}- \varrho_{k}}{(1 - \varrho_{j}- \varrho_{k}) \sqrt{1 - 2\varrho_{j}}\sqrt{1 - 2\varrho_{k}}}.
\|F\|_{\mathcal{G}^{*}}^{2}=\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}}e^{2\kappa(k,\xi)}|\widehat{F}(k,\xi)|^2\,\mathrm{d}\xi,\kappa(k,\xi)=\frac{(|k|+|\xi|)^{1/2}}{(\log^+(|k|+|\xi|))^{N_1}}.
< \chi^{0}( \nu_{1}^{\prime}, \nu_{2}^{\prime}, \nu_{3}^{\prime}, \nu_{4}^{\prime}) \mid \chi^{0}( \nu_{1}, \nu_{2}, \nu_{3}, \nu_{4}) > = \prod_{i = 1}^{4}\delta_{\nu_{i}\nu_{i}^{\prime}}.
\Big( \sum_{k=0}^\infty a_k \Big)^j = \sum_{k=0}^\infty \sum_{k_2=0}^{k}\dots \sum_{k_{j}=0}^{k_{j-1}}a_{k_j}\prod_{\ell = 1}^{j-1}a_{k_{\ell}-k_{\ell+1}}.
(A:B:C) = (\omega^k (\omega^2+\omega^3),\omega^{-k}(\omega+\omega^4),1),k=0,\ldots 4, (A:B:C) = (\omega^k (\omega+\omega^4),\omega^{-k}(\omega^2+\omega^3),1),k=0,\ldots 4.
|Y_t | = |\sup_n \inf_k Y_t^{n, k}| \leq \frac{1}{\gamma}\ln \mathbb{E}\Big[ \exp\Big(\gamma e^{\beta A_{t, T}}|\xi| + \gamma \int_t^T e^{\beta A_{t, s}}{\alpha_s dA_s}\Big)\Big| \mathcal{F}_t \Big].
u(t,x;u^*(g),g) =\lim_{s\to\infty}u(t,x;u(s,\cdot;u_0,g\cdot(-s)),g) &=\lim_{s\to\infty}u(t+s,x;u_0,g\cdot(-s)) &=\lim_{s\to\infty}u(t+s,x;u_0,(g\cdot t)\cdot(-t-s)) &=u^*(g\cdot t)(x)
\|\Lambda_{k,g}[\sigma,\rho]\| \lesssim |\lambda \sigma^{\big((n-3-\beta) + \beta m_0\big)+s_0 (m_1+\dots +m_{k})}\rho^{s_0}|^{-\frac 1 4}\sigma \Delta y^{\frac 1 4}\left (\frac{\Delta y}{\Delta x}\right)^{\frac 1 4}&\sim |\lambda \sigma^{(n-8 ) + (\beta-2) (m_0-1)+(s_0 -1)(m_1+\dots +m_{k})}\rho^{s_0-1}|^{-\frac 1 4}.
g_{n}(x)=\sum_{k=1}^{n}2^{k-1}(\begin{matrix}n-1\\n-k\end{matrix})(\begin{matrix}x\\k\end{matrix})=\sum_{k=0}^{n-1}2^{k}(\begin{matrix}n-1\\k\end{matrix})(\begin{matrix}x\\k+1\end{matrix})
\pi_I^{1 \pm}\times \pi_I^{2 \pm}, \quad\pi_I^{1 \pm}\times \pi_{III}^{2 \pm}, \quad\pi_{III}^{1 \pm}\times \pi_I^{2 \pm}, \quad\pi_{III}^{1 \pm}\times \pi_{III}^{2 \pm},
a_j^{m+1,0}=b_j^{m+1,0}, j=0,\dots,m+1, begin{align*}0.5em]a_j^{m+1,k}=b_j^{m+1,k}+c_j^{m+1,k}, \genfrac{}{}{0pt}{}{k=1,\dots,m,}{j=0,\dots,m+1-\left[\frac{k-1}{2}\right],}begin{align*}0.5em]a_j^{m+1,m+1}=c_j^{m+1,m+1}, j=0,\dots,\left[\frac{m+1}{2}\right],
\alpha_{j_1,\ell_1}=\sigma_{j_1}^{\ell_1}\int_{\mathbb{R}^{2j_1}}f\bigg(\sum_{s=1}^{j_1-1}t_se_s+i\sum_{s=1}^{j_1}t_s^{'}e_s+ t_{j_1}e_{j_1}+\zeta\bigg)H_{\ell_1}\Big(\frac{t_{j_1}}{\sigma_{j_1}}\Big) &\times d(\gamma_{\sigma_1}\otimes\gamma_{\sigma_1})(t_1,t_1^{'})\dots d(\gamma_{\sigma_{j_1}}\otimes\gamma_{\sigma_{j_1}})(t_{j_1},t_{j_1}^{'}).
+ \int\!\!d^D\!z_1 d^D\!z_2 d^D\!z_3 \left. \frac{\delta^2 W}{\delta j(x) \delta j(z_1)}\, \frac{\delta^2 W}{\delta j(x) \delta j(z_2)}\, \frac{\delta^2 W}{\delta j(x) \delta j(z_3)}\, \frac{\delta^3 \Gamma}{\delta \Phi(z_1) \delta \Phi(z_2) \delta \Phi(z_3)}\right] ,
2 \log \left( 1 + \lambda(1 - \lambda \rho) w_2 P' + \lambda \rho (1 - \lambda \rho) w_1 P' \right) \geq \log \left( 1 + \lambda (1 - \lambda \rho) w_2 P' \right) + \log \left( 1 + \lambda \rho (1 - \lambda \rho) w_1 P' \right).
J_x(u) :=\big((1-xH_u^2)^{-1}(1)\ |\ 1\big)=1+x\big((1-xH_u^2)^{-1}(u)\ |\u\big)\ , E_x(u) :=\big((1-xK_u^2)^{-1}(1)\ |\ 1\big)=J_x(u)-x\frac{\vert Z_x(u)\vert^2}{J_x(u)}\ , Z_x(u) :=\big(1\ |\ (1-xH_u^2)^{-1}(u)\big)\ .
r' (h_{i_{1}}\cdots h_{i_{t}}, h_{i}) = r' \big( h_{i_{1}}, (h_{i})_{(1)}\big) \cdots r' \big( h_{i_{t}}, (h_{i})_{(t)}\big) & = r'' \big( h_{i_{1}}, (h_{i})_{(1)}\big) \cdots r'' \big( h_{i_{t}}, (h_{i})_{(t)}\big) & = r'' (h_{i_{1}}\cdots h_{i_{t}}, h_{i})
I_2(n) \leq \frac 1{x^2}\left ( \int_{-Q(v)}^{Q(v)}\left \| \max_{2 \leq 2j \leq [n/q]}\left | \sum_{i=1}^j \tilde U_{2i}(t) \right |\right \|_2 dt \right )^2 \, .
\xi_n(x) = \frac{x^{\tfrac{n-1}{2}}(x^2 -1)}{2 + 2x^{\tfrac{-n+1}{2}}+ 2 x^{-n +1}+ x^{\tfrac{-3n+3}{2}}+ x^{-\tfrac{3n+1}{2}}}\ge \frac{x^{\tfrac{n-1}{2}}(x^2 -1)}{8},
\chi_{n,m}'(w, M, \cdot) := \chi_{n,m}(w, \cdot) \mathbb{1}_{X_1(M)}+ (\mathbb{E}_{n,m}[\chi_{n,m}(w', \cdot) | O(\chi_{n,m}'(w', M, \cdot)) = O(\chi_{n,m}'(w, M, \cdot))] + \mathcal{G}(0, \mathcal{K}_{n,m}^{w,M})(\cdot)) \mathbb{1}_{X_0(M)}
\left(\frac{|\langle (\mathrm{id}\otimes U_M)F_{\chi|\mathcal H \times \mathcal H}, g \times g \rangle|^2}{\langle g,g \rangle^4}\right)^{\sigma}= \frac{|\langle (\mathrm{id}\otimes U_M)F_{\chi|\mathcal H \times \mathcal H}^{\sigma}, g^{\sigma}\times g^{\sigma}\rangle|^2}{\langle g^{\sigma},g^{\sigma}\rangle^4},
[p\cdot\hat{H}, \widetilde{M}]_{\mu\nu}=\left[p\cdot\hat{H}, \,{(i/2)}\sum_{\rho\in\Delta_{+}}g_{|\rho|}\,{|\rho|^2}\,y (\rho\cdot q)\,\hat{s}_{\rho}\right]_{\mu\nu},\quad\mu,\nu\in{\cal R}
\int_{\Omega}(\epsilon^{-1}\nabla\times\widetilde{H_{\eta,\ell}})\cdot(\nabla\times\overline{\widetilde{H_{\eta,\ell}}})dx-k^{2}\int_{\Omega}\mu|\widetilde{H_{\eta,\ell}}|^{2}dx +\int_{\Omega}((\epsilon^{-1}-\epsilon_{0}^{-1})\nabla\times H_{\eta,\ell})\cdot(\nabla\times\overline{\widetilde{H_{\eta,\ell}}})dx=0,
\|BD^{-r}\Phi (\hat{x}-x)\|_2 = \|SR^T\Phi (\hat{x}-x)\|_2 \geq \sigma_L(BD^{-r}) \| R^T\Phi (\hat{x}-x)\|_2 & \geq \sqrt m\left(\frac{m}{L}\right)^{r/2-1/4}\sqrt L \|\frac{1}{\sqrt L}R^T\Phi(\hat{x}-x)\|_2.
[e_3,e_1]+\lambda J[e_3,e_1] + J[e_3,Je_1]-\lambda [e_3,Je_1] = -e_1-\lambda Je_1 + J[e_3,Xe_1+Ye_2]-\lambda [e_3,Xe_1+Ye_2] = -e_1-\lambda Xe_1-\lambda Ye_2 - X(Xe_1+Ye_2)-Y(Xe_1+Ye_2)-Y\frac{-1-X^2}{Y}e_1-Xe_2+ &+\lambda Xe_1+\lambda Ye_1+\lambda Y e_2
Pr \biggl( \lVert L^{-1}\sum_{l=1}^{L}\Pi\cdot\Pi_{X_l}\cdot X_l\cdot\Pi_{X_l}\cdot\Pi- \rho\rVert_1 \leq \epsilon +4 \sqrt{\epsilon}+24 \sqrt[4]{\epsilon}\biggr) \allowdisplaybreaks & \geq 1- 2D \exp \left( -\frac{\epsilon^3Ld}{2\ln 2D}\right)
(u_{t},\varphi) =(h,\varphi)+\int_{0}^{t}(Au_{s}, \varphi)ds+\int_{0}^{t}(F_{s}(u_{s}), \varphi)ds+\int_{0}^{t}(\varphi, \Sigma_{s}(u_{s})dW_{s}) &\quad+\int_{0}^{t}(\int_{Z}\Gamma_{s}(u_{s},z)\widetilde{N}(ds,dz), \varphi)
E_{1 \, non-disp.}=(\varepsilon - 1)^2 \Bigl(-\frac{23}{128}\frac{V}{\pi^2 \lambda^4}+ \frac{23}{384}\frac{S}{\pi^2 \lambda^3}+ \frac{23}{1536}\frac{1}{\pi a}\Bigr)\,\, + \, \mbox{const}
\dfrac{1+\psi(1-\lambda)-\psi(\lambda)}{2}f(t) =\dfrac{1}{\int_a^b \nu(t)\Delta t}\int_a^b K(s,t)f^{\Delta}(s)\Delta s + \dfrac{1}{\int_a^b \nu(t)\Delta t}\int_a^b \nu(s)f(\sigma(s))\Delta s & - \dfrac{\psi(\lambda)f(a)+\left(1-\psi(1-\lambda)\right)f(b)}{2},
\zeta_{\mathrm i n t}( s ) = - \zeta_{1 / 2 - | \alpha |}^{\mathrm ( i n t )}( s ) + \sum_{\nu = 1 / 2}^{\infty}\left[ \zeta_{\nu + \alpha}^{\mathrm ( i n t )}( s ) + \zeta_{\nu - \alpha}^{\mathrm ( i n t )}( s ) \right]
\mu^{( n = 2 )}( x_{\Delta}) = 2 \mu_{\mathrm{r e g}}^{( n = 1 )}+ \mu_{\mathrm{S}}( x_{\Delta}) + \mu_{\mathrm{b i n d}}^{2}= \mu^{2}+ \mu_{\mathrm{b i n d}}^{2}\ .
\lim_{r\to 0}r^{\delta_1}g'(r)=-\lim_{r\to 0}\frac{\tilde\alpha g^{1-m}(r)}{mr^{n-1-\delta_1}}\int_0^r\rho^{n-3+\frac{n-2-nm}{m}}g(\rho)d \rho =-\frac{\tilde\alpha\eta^{2-m}}{{n- 2-2m}}
T^{+ +}( x ) = \mathrm{t r}\left[ ( \partial_{-}\phi )^{2}+{\frac{1}{2}}\left( i \psi \partial_{-}\psi - i ( \partial_{-}\psi ) \psi \right) \right] = T_{B}^{+ +}( x ) + T_{F}^{+ +}( x ) .
\frac{P_m^{i,g}l^2}{2\tau}\sim D_{i,g},\frac{P_b^{i,g}}{2\tau}\sim \lambda_{i,g},\frac{P_d^{i,g}}{2\tau}\sim k_{i,g},P_b^{i,g}, P_d^{i,g}= O(\tau),\hbox{for}l\to0,\tau\to0.
\langle \mathcal{M}_{f, k_1}^N, \mathcal{M}_{g, k_2}^N \rangle(t)= &\int_0^t\mathcal{L}^N\left(\mu_{s,k_1}^N(f)\mu_{s,k_2}^N(g)\right)-\mu_{s,k_1}^N(f)\mathcal{L}^N\left(\mu_{s,k_2}^N(g)\right)-\mu_{s,k_2}^N(g)\mathcal{L}^N\left(\mu_{s,k_1}^N(f)\right) ds.
Z_{\lambda}= 1 + \frac{9}{2 \pi^{2}}\log ( \Lambda / M ) \; \lambda^{2}+ \left( \frac{8 1}{4 \pi^{4}}( \log ( \Lambda / M ) )^{2}- \frac{5 1}{4 \pi^{4}}\log ( \Lambda / M ) \right) \; \lambda^{3}.
\frac{1}{\prod\limits_{i<j}^n |x_i-x_j|^{\lambda 2^{j-i}}}\sum\limits_{i=1}^{n-1}\frac{\partial}{\partial x_i}\Bigl[ (x_i-x_n)\prod\limits_{i<j}^n |x_i-x_j|^{\lambda 2^{j-i}}\Bigr] = n-1 + \lambda\sum\limits_{i=1}^{n-1}\sum\limits_{k\neq i}^n 2^{|k-i|}\frac{x_i-x_n}{x_i-x_k}.
I_{L}(\hat\beta_0) = n^{-1}\sum_{t=1}^n \sigma_{t}^2(\hat\beta_0)\cdot \textrm{diag}\left( \sigma_{t-j_1}^{2-2\gamma}(\hat\beta_0), \ldots, \sigma_{t-j_L}^{2-2\gamma}(\hat\beta_0) \right).
g\to\begin{pmatrix}e^{-\lambda i/3}&e^{\lambda i/3}& 1\end{pmatrix}\begin{pmatrix}e^{\lambda i}-e^{-(\lambda+\mu)i}& e^{\mu i}\end{pmatrix}\begin{pmatrix}e^{2\lambda i/3}&e^{-2\lambda i/3}& 1\end{pmatrix}&= \begin{pmatrix}1 -e^{-\mu i}& e^{\mu i}\end{pmatrix}
\sum_{i\in I}(\operatorname{id}\otimes\varphi)\bigl(\Delta(c^*p_i)(1\otimes q_i^*d)\bigr)\longrightarrow \,(\operatorname{id}\otimes\varphi)\bigl(\Delta(c^*)[E(x\otimes1)](1\otimes d)\bigr) &=(\operatorname{id}\otimes\varphi)\bigl(\Delta(c^*)(1\otimes d)\bigr)\,x,
\Delta \coloneqq \begin{pmatrix}\Omega_0 (0 , \mathcal{I}_m) \Omega_1 (0 , \mathcal{I}_m) \bar{\Omega}(0 , \mathcal{I}_m) \Omega_2 (0 , \mathcal{I}_m) \Omega_0 (0 , \mathcal{I}_{\bar{m}}) \Omega_1 (0 , \mathcal{I}_{\bar{m}}) \bar{\Omega}(0 , \mathcal{I}_{\bar{m}}) \Omega_2 (0 , \mathcal{I}_{\bar{m}})\end{pmatrix}
\int_0^\infty e^{(c_2-\lambda)t}t^{-1/2}P_{s+\alpha t}f(x)dt= \int_0^\delta e^{(c_2-\lambda) t}t^{-1/2}P_{s+\alpha t}f(x)dt+ \int_\delta^\infty e^{(c_2-\lambda) t}t^{-1/2}P_{s+\alpha t}f(x)dt
g_{i\bar{j}}=\frac{1}{2\pi i}\oint_{|z|=1}\bigg\{\partial_i\Big( \log{h(z;\boldsymbol{\xi})}\partial_{\bar{j}}\log{\bar{h}(\bar{\xi};\bar{\boldsymbol{\xi}})}\Big)- \log{h(z;\boldsymbol{\xi})}\partial_i\partial_{\bar{j}}\log{\bar{h}(\bar{\xi};\bar{\boldsymbol{\xi}})}\bigg\}\frac{dz}{z}
{\cal C}= \left( \begin{matrix}{{1}}&{{0}}&{{- 1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{0}}&{{1}}\\{{0}}&{{- 1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{{\frac{1}{2}}}}&{{0}}&{{-{\frac{1}{2}}}}\\{{-{\frac{1}{2}}}}&{{0}}&{{-{\frac{1}{2}}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{- 1}}&{{0}}\\ \end{matrix}\right)
[x^{+},y^{+}]\otimes a^{+}=x^{+}\circ y^{+}\otimes\frac{[1^{-},a^{-}]_{A^{-}}}{2}+[x^{+},y^{+}]\otimes\frac{(1^{-}\circ a^{-})_{A^{+}}}{2}+(x^{+}\mid y^{+})\langle1^{-},a^{-}\rangle{}[x^{+},y^{-}]\otimes a^{-}=.x^{+}\diamond y^{-}\otimes\frac{[1^{-},a^{+}]_{A^{+}}}{2}+[x^{+},y^{-}]\otimes\frac{(1^{-}\circ a^{+})_{A^{-}}}{2}
\biggl < e^{S_{4}( \phi_{s}, \phi_{f})}\biggr >_{0 f}= \exp \biggl [ \biggl < S_{4}( \phi_{s}, \phi_{f}) \biggr >_{0 f}+ \frac{1}{2}\biggl ( \biggl < S_{4}^{2}( \phi_{s}, \phi_{f}) \biggr >_{0 f}- \biggl < S_{4}( \phi_{s}, \phi_{f}) \biggr >_{0 f}^{2}\biggr ) + \cdot \cdot \cdot \biggr ] .
|\varphi^{-1}(\widetilde{A})|^{\frac{1}{s}-\frac{1}{n}}\leq \|H_s \mid L_{\kappa}(\widetilde{A})\| |\widetilde{A}|^{\frac{1}{q}-\frac{1}{n}},\,\,{1}/{\kappa}={1}/{s}-{1}/{q},
X(n)_{s,t}=(1,\int_{s<s_{1}<t}dX(n)_{s_{1}},...,\int_{s<s_{1}<\cdot\cdot\cdot<s_{\lfloor p\rfloor}<t}dX(n)_{s_{1}}\otimes\cdot\cdot\cdot\otimes dX(n)_{s_{\lfloor p\rfloor}}).
s_{n_{k+1}+ n_{k}+1}= s_{n_{k+1}+ n_{k}+1}^{(n_{k}+1)}+ (-1)^{{\tfrac{(n_{k+1}- n_{k}-1)}{2}}}\left( \frac{t_{n_{k+1}}}{t_{n_{k}}}\right)^{\tfrac{1}{n_{k+1}- n_{k}}}\ ,
\sum_{k_{1}+\dots+k_{p}=n}g_{k_{1}}\dots g_{k_{p}}=\frac{\left(-1\right)^{n+p}}{n!}\sum_{k_{1}+\dots+k_{p}=n}\binom{n}{k_{1},\dots,k_{p}}\int_{0}^{1}\dots\int_{0}^{1}\prod_{i=1}^{p}\left(x+u_{i}-1\right)^{k_{i}}du_{1}\dots du_{p}=\frac{\left(-1\right)^{n+p}}{n!}\int_{0}^{1}\dots\int_{0}^{1}\left(px-p+u_{1}+\dots+u_{p}\right)^{n}du_{1}\dots du_{p}.
f_{\mathbf{x}}(x;\kappa) = G(\mathbf{x})\left[ 1+ \kappa^{i,j,k}h_{ijk}(\mathbf{x})/3! + \kappa^{i,j,k,l}h_{ijkl}(\mathbf{x})/4! + \kappa^{i,j,k,l,m}h_{ijklm}(\mathbf{x})/5! \right. & \left. + \left( \kappa^{i,j,k,l,m,n}+ 10\kappa^{i,j,k}\kappa^{l,m,n}\right) h_{ijklmn}(\mathbf{x})/6!\ldots \right]
(\Lambda_{N}(t)-\mid\gamma+t\mid^{2})(\Psi_{N,t},e^{i\left\langle\gamma+t,x\right\rangle})=\sum_{\gamma_{1}\in\Gamma(k+)}q_{\gamma_{1}}(\Psi_{N,t},e^{i\left\langle \gamma-\gamma_{1}+t,x\right\rangle}).
\sum_{i=1}^l\lambda_i\nabla f_i(x^0)+\sum_{j=1}^m\mu_j\nabla g_j(x^0)=0, &\mu=(\mu_1, \ldots, \mu_m)\geqq 0, \mu_jg_j(x^0)=0, &\lambda=(\lambda_1, \ldots, \lambda_l)\geqq 0, (\lambda, \mu)\neq 0;
|\epsilon(t)|=|N(t)-N(B(t),C(t))| \leq\int_{\mathbb{R}}|N(B(t)+y,C(t))-N(B(t),C(t))|p_{t,0}(y)dy &\leq\frac{1}{V_F}\int_{\mathbb{R}}|y|p_{t,0}(y)dy=\frac{1}{V_F}e^{-t}\int_{\mathbb{R}}|y|p_{}(y)dy.
f_{H_1,\lfloor H^{\beta}\rfloor}(\tau,\zeta) = \varphi_{\le \lfloor H^{\beta}\rfloor}(\tau-\omega(\zeta))f_{H_1}(\tau,\zeta), f_{H_1,L}(\tau,\zeta) = \varphi_L(\tau-\omega(\zeta)) f_{H_1}(\tau,\zeta),
\frac{1}{2}\inf_{a\in[0,1]}\left\{h_1^2(1-a)+\frac{a}{1-a}+\log(1-a)+\frac{(1-a)\partial_x\xi(a_0,b_0)}{\gamma}\right\}, \frac{1}{2}\inf_{b\in[0,1]}\left\{h_2^2(1-b)+\frac{b}{1-b}+\log(1-b)+\frac{(1-b)\partial_y\xi(a_0,b_0)}{1-\gamma}\right\}.
X^{\mu}( \sigma , \tau ) = X^{\mu}+ P^{\mu}\tau + 2 L^{\mu}\sigma + \frac{i}{2}\sum_{k \neq 0}\frac{1}{k}\left( \alpha_{k}^{\mu}\mathrm{e}^{- 2 i k \sigma_{+}}+ \tilde{\alpha}_{k}^{\mu}\mathrm{e}^{- 2 i k \sigma_{-}}\right)
\|v_{k}\|_{C_{T}L_{x}^{3}\cap L_{T}^{4}L_{x}^{6}}+\big\|\nabla|v_{k}|^{\frac{3}{2}}\big\|^{\frac{2}{3}}_{L_{T}^{2}L_{x}^{2}}&=g(T)\leq g_{-}(T)=\frac{1-\sqrt{1-4ab(T)^{2}C_{1}^{2}}}{2C_{1}b(T)}\leq\frac{4ab(T)^{2}C_{1}^{2}}{2C_{1}b(T)}&=2C_{1}\|v_{k}(\cdot,0)\|_{L^{3}}e^{C_{2}\int_{0}^{T}\|w\|_{L^{6}}^{4}dt}.
0\leq \lim_{k\to\infty}\dfrac{f_i(x^k)-f_i(x^0)}{f_1(x^0)- f_1(x^k)}\leq \lim_{k\to\infty}\dfrac{f_i(x^k)-f_i(x^0)}{-\frac12\delta t^2_k}&=-\frac{1}{\delta}[\langle\nabla f_i(x^0), v\rangle+\langle\xi^i, u\rangle]=0.
\int_{A_k}|v(x,t)|^r\, dxdt =r\int_0^\infty\lambda^{r-1}|\{(x,t)\in A_k: |v(x,t)|>\lambda\}|\, d\lambda &\lesssim 2^{-k}\int_0^\infty \frac{\lambda^{r-1}}{(1+\lambda)^p}\, d\lambda\simeq 2^{-k}.
0\neq 6F\left(\frac{2}{3},-\frac{8}{3},\frac{4}{3};1-\lambda\right)(\lambda^2-\lambda+1)-\frac{2}{3}F\left(\frac{2}{3},-\frac{5}{3},\frac{4}{3};1-\lambda\right)\left(\lambda+1 \right)(7\lambda^2-10\lambda+7)+ &2F\left(\frac{2}{3},-\frac{2}{3},\frac{4}{3};1-\lambda\right)\lambda(1-\lambda)^2,
L^{n,1}_\tau = n^{2\alpha+ 1}\int^{\tau}_0\gamma_s (\sigma'\sigma)^2(X_{\eta_n(s)}) (s-\eta_n(s))^\alpha (W_s-W_{\eta_n(s)}) \left( \int_0^s \psi_{n,1}(u,s) dW_u \right) ds.
\tilde \psi_{n0}= \chi_n\left(\varphi_{n,0}- \varphi_{n,1}+ \sum_k (\tilde Q_{n,0; k,0}\varphi_{k,0}- \tilde Q_{n,0; k,1}\varphi_{k,1}- \tilde Q_{n,1; k,0}\varphi_{k,0}+ \tilde Q_{n,1; k,1}\varphi_{k,1})\right), \tilde \psi_{n1}= \varphi_{n,1}+ \sum_k (\tilde Q_{n,1; k,0}\varphi_{k,0}- \tilde Q_{n,1; k,1}\varphi_{k,1}).
\Delta_E(v_i\cdot g) =(\lambda_i^{(0)}-2\mu)v_i\cdot g, \Delta_E(\nabla^2v_i) =(\lambda_i^{(0)}-2\mu)\nabla^2v_i, \Delta_E(\delta^*\alpha_i) =(\lambda_i^{(1)}-\mu)\delta^*\alpha_i,
f(x)-t_\alpha(x) = \left(f'(x_{j-1}) - \alpha h_j(\coth(\alpha h_j)+\tanh(\alpha x_j))f[x_{j-1},x_j] \right)(x-x_{j-1}) &\quad+O(\overline{h}^2) &= (f'(x_{j-1})-f[x_{j-1},x_j])(x-x_{j-1})+O(\overline{h}^2) &=O(\overline{h}^2).
\tilde{N}_p^t\tilde{S}_{\boldsymbol{t}}=\tilde{N}_p^t(\tilde{S}_{t_0}[t'_0]\tilde{S}_{t_1}[t'_1]\dots\tilde{S}_{t_{(p-3)/2}}[t_{(p-3)/2}']\tilde{S}_{t_{(p-1)/2}}[t_{(p-1)/2}']),
T r ( W^{\prime}[{\cal C}] ) = T r ( P \left\{e^{- \oint_{\cal C}A^{\prime}}\right\}) = T r ( e^{{{\frac{2 \pi}{l}}( r_{+}- r_{-})}T_{2}}) = 2 \cosh [{\frac{\pi ( r_{+}+ r_{-})}{l}}] .
\sum\limits_{n \leqslant x}{\sigma_{z_1,\beta}(n)\sigma_{z_2,\beta}(n)}=\frac{1}{{2\pi i}}\int_{c - iT_0}^{c + iT_0}{f(z_1,z_2,s;\beta )\frac{{{x^s}}}{s}ds}+E(z_1,z_2,\beta;x),
[X_1\wedge \dots \wedge X_p,Y_1\wedge\cdots \wedge Y_q] &= \sum_{i,j}(-1)^{i+j}[X_i,Y_j]\wedge X_1\wedge \cdots \wedge \hat{X}_i\wedge\cdots\wedge X_p\wedge Y_1\wedge \cdots\wedge \hat{Y}_j\wedge\cdots \wedge Y_q.
\gamma_{1}\left( \sum_{j=2}^M \gamma_j\|A_j\|^2 + \frac{1}{2N}\sum_{i=1}^N L_i\right)\leq \delta \lambda \leq \frac{(N+1)(1-\sqrt{\delta})M}{2 \left(1 + \sqrt{\delta}- \frac{1}{2\sqrt{\delta}N}\sum_{i=1}^N L_i\right)\left(\frac{M-1}{\frac{\gamma_1}{N}\sum_{i=1}^{N}L_i}+ 1\right)}.
\int_{-1/2}^{1/2}\log \left( 1 - \frac{e^{2 \pi i \frac{j + \theta_1/2}{m}}\left( e^{2 \pi i \phi/m}-1 \right)}{1- e^{2 \pi i \frac{j + \theta_1/2}{m}}+ \frac{x + iy}{m}}\right) \mathrm{d}\phi = O(1/j^2).
\widetilde{\eta}_{\sigma \tau}= \epsilon_{\sigma}^{\mu *}\left( k \right) \cdot \epsilon_{\tau , \mu}\left( k \right) \equiv \left( \begin{array}{c c c c}{- 1}&{0}&{0}&{0}\\{0}&{- 1}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{1}&{0}\\ \end{array}\right) .
\frac{\nu}{2}\int_\Omega|\mathbf{b}_{x}|^2dx+ \iint_{Q_t}|\mathbf{b}_t|^2 dxdt = \iint_{Q_t}\mathbf{w}_x\cdot\mathbf{b}_t dxdt - \iint_{Q_t}(u\mathbf{b})_{x}\cdot\mathbf{b}_t dxdt.
c_{\Omega_4}^{\Omega_4}= q^{-1/6}\varphi_1^{-4}(\chi_{3,4}^{1,2}\chi_{4,5}^{2,1}) = q^{-1/6}\varphi_1^{-4}(q^{1/24}V(q)^{-1})(q^{49/120}H_2V(q)^{-1}) = q^{17/60}\varphi_1^{-4}(\varphi_1^{-2}\varphi_2^2)H_2 = q^{17/60}\varphi_1^{-6}\varphi_2^2H_2.
LHS = ( - 1){}^t\varepsilon (t)\varepsilon (t + 1)\mathop \Pi \limits_{n = 0}^t \left({\widetilde{c}-{q^{'n}}{r^{{\textstyle{1 \over 2}}}}\omega +{q^{' - n}}{s^{{\textstyle{1 \over 2}}}}{\omega^{'}}}\right).
x_i= |\{(v_i,j)\in E_i \cap \bar{E}:j\in N\}|+ &1_{\{v_i\in S_{-c}\}}\cdot\sum_{u\in N_0}1_{\{(u,v_i) \in E_i\cap \tilde{E}\}}\cdot &1_{\{|\{(u,j)\in E_i :j\in N\}|>|N_{-c}| \}}.
\forall v,w, u \in V, \;\; \left(m\circ \left(\dot{+}\otimes \dot{+}\right)\right)(v\otimes u \otimes u\otimes w) = \delta_{v\dot{+}u, u\dot{+}w}(v\dot{+}u) = \dot{+}(u\otimes \delta_{v, w}v)
\sum_{n = 0}^{\infty}d_{p}(n,k,r)q^{n}= \frac{(q^{r};q^{p})_{\infty}}{(q;q)_{\infty}}\sum\limits_{n = 0}^{\infty}\frac{q^{k(pn + r)}}{1 - q^{pn + r}}\prod\limits_{j \neq n, j = 0}^{\infty}(1 + q^{pj + r}+ q^{2(pj + r)}+ \cdots + q^{(k - 1)(pj + r)})
C^{(+)}_r(m-l)=\frac{{m-l\choose r}(\tfrac{\beta}{2}-\lambda-m+l)_{m-l-r}(-\tfrac{\beta}{2}-m+l+1)_{m-l-r}}{{m\choose k}(\tfrac{\beta}{2}-\lambda-m)_{m-k}(-\tfrac{\beta}{2}-m+1)_{m-k}}C^{(+)}_k(m)
|A^{p+m}|^2 =A^{*m}|A^{p}|^2A^m\leq A^{*m}|A^{m+p}|^\frac{2p}{p+m}A^m &=f_{p+m,p}( \frac{p}{p+m})\leq f_{p+m,p}(1) &=|A|^\frac{2(p+m)}{p+2m},
\begin{cases}\partial_t \bar{\mathbb{L}}_1(A_t) + \bar{\mathcal{H}}(A_t,(\bar{\mathbb{L}}_1, \partial_x \bar{\mathbb{L}}_1, \partial_{xx}\bar{\mathbb{L}}_1)(A_t) ) \geq \frac{\delta}{t^2},~ t \in [0,T),~ A_t \in \Lambda \bar{\mathbb{L}}_1(A_T) = m(A_T) - \frac{\delta}{T},~ A_T \in \Lambda_T.\end{cases}
\sum_{k=0}^\infty\frac{(1-acq^{4k})(a;q)_k(q/a;q)_k (ac;q)_{2k}}{(1-ac)(cq^3;q^3)_k(a^2cq^2;q^3)_k(q;q)_{2k}}\,q^{k^2}=\frac{(acq^2;q^3)_\infty(acq^3;q^3)_\infty(aq;q^3)_\infty(q^2/a;q^3)_\infty}{(q;q^3)_\infty(q^2;q^3)_\infty(a^2cq^2;q^3)_\infty(cq^3;q^3)_\infty}.
Z_n(\ell)\to \frac{1}{\prod_{i=2}^{\infty}\zeta(i)}\prod_p Y(p,\ell) = \prod_p \frac{C_p}{[p,1]}\prod_p Y(p,\ell)= \prod_p \frac{C_p}{[p,1]}Y(p,\ell)= \prod_p Z(p,\ell)
\mathrm{(i)}\;L_{\alpha}^{ih}=\tfrac{\partial L^{i}}{\partial y_{h}^{\alpha}},\mathrm{(ii)}\;\tfrac{\partial L^{h}}{\partial y_{i}^{\alpha}}=\tfrac{\partial L^{i}}{\partial y_{h}^{\alpha}},1\leq \alpha \leq m,\;h,i=1,\dotsc ,n,
\pi(\partial_1 u_1)(t,0) = \sum_{100\le j\le M}\int_{\mathbb R^2}b(x) F(x) dx + Mt \int_{\mathbb R^2}b(x) (\nabla F)(x) \cdot (\nabla^{\perp}a)(x) dx + \sum_{100\le j\le M}\int_{\mathbb R^2}b(x) (\nabla F)(x)\cdot \int_0^t (t-\tau) 2^j (\nabla p)(\tau, 2^{-j}x) d\tau + \operatorname{error},
p_{\pm \i \pi}=|\alpha_{\pm \i \pi}\rangle\langle\beta_{\pm \i \pi}| |\alpha_{\pm \i \pi}\rangle = \binom{1}{-\displaystyle\frac{1}{\pm\i \pi-\Omega}|E\rangle}\langle\beta_{\pm \i \pi}| =\left( 1, \langle E|\frac{1}{\pm \i \pi-\Omega}\right)
\dfrac{(1+\delta k) \, (\tau+1)_{k}}{(\mu+1)_{k}}\sum_{\nu=k+1}^{\infty}a_{\nu}\leqq \sum_{\nu =k+1}^{\infty}\dfrac{(1+\delta\nu- \delta)) \, \Gamma(\nu+\tau)\Gamma{(\mu+1)}}{\Gamma(\nu+\mu)\Gamma(\tau+1)}\; a_{\nu},
\tilde{\theta}_\ell=w_{\ell j}r^{j+1/2}\eta_j +o(r^{j+1/2}), \ell =1, \ldots, j-1, \tilde{\theta}_j = r^{j+1/2}\eta_j +o(r^{j+1/2}), \tilde{\theta}_k = o(r^{j+1/2}), k=j+1, \ldots, J.
\Psi_{h,k}(z,w) =\frac{\Gamma(s_0+1)}{s_0 w^{s_0}}\sum_{j\in \mathbb{Z}_k}c(j)Li_{s_0+1}(e^{2\pi i \frac{hj}{k}}z), \omega_{h,k,n}(z) =e^{-2\pi i \frac{hn}{k}}\prod_{j\in \mathbb{Z}_k}(1-e^{2\pi i \frac{hj}{k}}z)^{-b(j)}, g_{h,k}(z,w) =\frac{1}{k}\sum_{r=1}^{k}z^r \frac{1}{2\pi i}\int_{\sigma_0-i\infty}^{\sigma_0+i\infty}J(s;z,r,h,k,w) ds.
\sigma_{ab}= r^2 \tilde \sigma_{ab}+ r^4\tilde \sigma^{(4)}_{ab}+ O(r^5) |H| = \frac{2}{r}+ h^{(1)}r + O(r^2) (\alpha_H )_a= (\alpha_H^{(2)})_a r^2 + O(r^3).
e^+_i(z):=\sum_{r\geq 0}e_{i,r}z^{-r},\e^-_i(z):=-\sum_{r<0}e_{i,r}z^{-r}, f^+_i(z):=\sum_{r>0}f_{i,r}z^{-r},\f^-_i(z):=-\sum_{r\leq 0}f_{i,r}z^{-r}.
E[X_n^p] \le \frac{1}{n^p}C^p E\bigg[\sum_{i,\ell \in [1,n]}|S_i-\tilde{S}_\ell |_+^{-2}\bigg]^p\le C^{2p}\Big[\sum_{k=1}^{p}k^p n^{-(p-k)/2}J_{k,n}\Big]^2, &J_{k,n}:=\frac{1}{n^k}\sum_{1 \le s_1 < \cdots < s_k \le n}\prod_{i=1}^k \Big(\frac{s_i}{n}-\frac{s_{i-1}}{n}\Big)^{-1/2}\,.
G=\begin{pmatrix}w^{40}w^{20}w^{22}w^3 w^{49}w^{55}w^{54}w^{65}w^{86}w^{27}w^{89}w^{64}w^{73}w^{23}w^{44}w^{79}w^{88}w^{103}w^{110}w^{97}w^{21}w^{51}w^{47}w^{70}\end{pmatrix}.
{\bf v}_{0}= \frac{\sqrt 2}l \left[ \begin{array}{c}{1}\\{a^{2}}\\ \end{array}\right] ,{\bf v}_{i}= \frac{{\sqrt 2}a}l \left[ \begin{array}{c}{1 / J_{2}( j_{i})}\\{- 1}\\ \end{array}\right] \ .
d_H(v,K^{(r)}_{s,t})\geq \delta(H,K^{(r)}_{s,t})\geq \delta_\ell=F_{r,s,t}\left(\left\lfloor\frac{r-1}{r}\ell\right\rfloor,\ell\right)\geq \Omega(\ell^{(r-1)s+t-1}).
X_t = \begin{pmatrix}(X^{(1)}_t)^T (X^{(2)}_t)^T ... (X^{(m)}_t)^T \end{pmatrix}^TX_k = \begin{pmatrix}(X^{(1)}_k)^T (X^{(2)}_k)^T ... (X^{(m)}_k)^T \end{pmatrix}^T
\epsilon_0\leq(\frac{\alpha_0\gamma_0}{80})^{\frac{1}{1-2\beta'}}\prod_{\mu=0}^{\infty}(2^{\mu}B_{\mu})^{-\frac{1}{3\kappa^{\mu+1}}},\quad\alpha_0\leq\min\{\frac{m_0}{10},\frac{M_{3,0}}{2}\}.