harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J(x)=\begin{cases}x ,\;x\geq x^*, \frac{1}{2}e^{cx^*}(x^*-\frac{1}{c})e^{-cx}+\frac{1}{2}e^{-cx^*}(x^*+\frac{1}{c})e^{cx},\;0\leq x<x^*, \frac{1}{2}e^{cx^*}(x^*-\frac{1}{c})e^{cx}+\frac{1}{2}e^{-cx^*}(x^*+\frac{1}{c})e^{-cx},\;-x^*< x<0, -x ,\;x\leq-x^*.\end{cases}
\frac{\partial}{\partial z^{\mu}}\, \Delta V_{j i}^{a \, \mu}( z , u , v ) = \frac{i g_{2}}{2}Z \tau_{j i}^{a}[ \delta^{4}( z - v ) - \delta^{4}( z - u ) ] R \, / \hspace{- 2 m m}\partial_{v}\, \delta^{4}( v - u ) \, .
\Gamma = - \frac{T^{2}}{8 \pi^{2}}\int d^{4}x M^{2}\left[ \frac{\pi^{2}}{3}+ \frac{1}{8}\frac{M^{2}}{T^{2}}\ln \frac{M^{2}}{T^{2}}+ \frac{\xi^{( 1 )}( - 2 ) M^{4}}{2 4 T^{4}}+ \cdots \right] ,
\begin{aligned}\sup_{t \in [0,T]}\left(\|u(t)\|^2_{H^1_a(0,1)}\right)+ \int_0^{T}\left(\left\|u_t\right\|^2_{L^2(0,1)}+\left\|(au_{x})_x\right\|^2_{L^2(0,1)}\right)dt &\le C\left(\|u_0\|^2_{H^1_a(0,1)}+\|h\|^2_{L^2(Q_T)}\right).\end{aligned}
\mathbf{Pr}\left( \bigg| \frac{d}{n\epsilon}\sum_{i=1}^{n}\big\langle J(u_0 + \epsilon v_{0,i})v_{0,i}v_0' \big\rangle - \big\langle\nabla \hat{J}(u_0),v_0' \big\rangle \bigg| \geq t \right) \leq 2\exp\left( - \frac{cnt^2}{K^2}\right),
\mathbf{E}[g(Y_k)|X_0,Y_1,\ldots,Y_{k-1}] =\mathbf{E}[f(X_k)|X_0,Y_1,\ldots,Y_{k-1}], \mathbf{E}[g(Y_k)|Y_1,\ldots,Y_{k-1}] =\mathbf{E}[f(X_k)|Y_1,\ldots,Y_{k-1}],
\nabla f(\mathbf{x}) + \mathbf{A}^T\boldsymbol{\lambda}^{\star}= \mathbf{0}\mathbf{B}^T\boldsymbol{\lambda}^{\star}= \mathbf{0}\mathbf{A}\mathbf{x}^{\star}+ \mathbf{B}\mathbf{z}^{\star}= \mathbf{0}.
z =\sqrt{\frac{a}{a+b}}\left(\frac{\sqrt{1-b}\, \cos(t)}{\sqrt{1+a}}+ \frac{i\sqrt{1+b}\, \sin(t)}{\sqrt{1-a}}\right), w = \pm\sqrt{\frac{b}{a+b}}\left(\frac{\sqrt{1-a}\, \sin(t)}{\sqrt{1+b}}- \frac{i\sqrt{1+a}\, \cos(t)}{\sqrt{1-b}}\right), t\in [0, 2\pi),
L \left( m^{2}( \beta ) , \beta \right) = \beta^{- D}\sum_{k = 0}^{\infty}g ( D , k ) \left( m ( \beta ) \beta \right)^{2 k + 2}\int_{m ( \beta ) \beta}^{\infty}d \tau \frac{\tau^{D - 5 - 2 k}}{e^{\tau}- 1}.
\mathcal{H}_t:= \mathcal{H}(t, X_t, u_t, \hat{p}_t, \hat{q}_t, \hat{w}_t), \hat{\mathcal{H}}_t :=\mathcal{H}(t, \hat{X}_t, \hat{u}_t, \hat{p}_t, \hat{q}_t, \hat{w}_t),
\rho(a\cdot X)(r) = \eta(a)\rho(X)(r) = a\cdot \rho(X)(r), \rho(X)(a\cdot r) = \rho(X)\big(\eta(a)r\big) = \rho(X)\big(\eta(a)\big)r + \eta(a)\rho(X)(r) = \eta\big(\omega(X)(a)\big)r+a\cdot \rho(X)(r) = \omega(X)(a)\cdot r + a\cdot \rho(X)(r)
(-1)^{u-i}{u\choose i}{\,}c_{r,u}c_{d,d}^u+\delta_{r,r_0}\delta_{u,\frac{qq_2}{d}}{\,}\chi_{\{\frac{q_1q_2}{d},\frac{q_2^2}{d}\}}(i){\,}c_{q,q_1}^{q_2+1}~=~ 0.
\Theta_\mathbf{v}(Z)=2^{4N}e\left(-2^gN(2^g-1)(2^g+1)\mathbf{v}_u^T\mathbf{v}_l\right)\frac{\displaystyle \prod_{\mathbf{a}\in S_-}\theta_{\mathbf{a}-\mathbf{v}}(Z)^{4N(2^g+1)}}{\displaystyle \prod_{\mathbf{b}\in S_+}\theta_\mathbf{b}(Z)^{4N(2^g-1)}}\quad(Z\in\mathbb{H}_g)
{\mathcal F}_x{\mathcal F}_y{\mathcal F}_z \,\,\tau_{ijk\ell}\,\,{\mathcal F}^{-1}_x{\mathcal F}^{-1}_y{\mathcal F}^{-1}_z= \begin{pmatrix}D_{ijk\ell}C_{ijk\ell}B_{ijk\ell}A_{ijk\ell}\end{pmatrix}.
R \cdot \int_{S^{1}}|\widehat{gd\sigma}(Rx)|^{2}\, d\mu x = R \cdot \int_{S^{1}}\left| \sum_{1 \leq 4^{p}\lesssim R^{2/3}}2^{-\beta p}2^{\beta p}\widehat{g_{p}d\sigma}(Rx) \right|^{2}\, d\mu x & \lesssim_{\beta}\sum_{1 \leq 4^{p}\lesssim R^{2/3}}2^{2\beta p}R \cdot \int_{S^{1}}|\widehat{g_{p}d\sigma}(Rx)|^{2}\, d\mu x.
\leq \sum_{k=1}^{\infty}\frac{1}{2^{2k}}|| z((\pi_n \otimes 1_{N_2})(u_k)) - ((\pi_m\otimes 1_{N_2})(u_k))z||^2_2 & < \frac{\delta\Big(J,\frac{\varepsilon}{\sqrt{2J}}\Big)}{2^{2J}}.
L_1 =\int_X\chi'(u_1-u_2)d(u_1-u_2)\wedge d^c(\psi_1-u_1)\wedge T_1 \leq C_1\varrho B (1-\tilde{\chi}(-1))Q\left(\varrho^{-1}\int_X\chi'(u_1-u_2)d(u_1-u_2)\wedge d^c(u_1-u_2)\wedge T_1\right),
|| v^0 - v^* ||_{\rho'}\le C\delta^{-4\tau}\epsilon, & | \lambda^0 - \lambda^* |\le C \delta^{-2\tau}\epsilon, & | \sigma^0 - \sigma^* |\le C \delta^{-2\tau}\epsilon, & || c^0 - c^* ||_{\rho'}\le C \delta^{-4\tau}\epsilon,
\epsilon^{-1}\Theta(c_1'+c_2+1,h) X^{c_1'}X^{c_2}\int_{-\infty}^{\infty}\min\Big( \frac{1}{|s_2|}, \frac{\epsilon^{-1}}{|s_2(s_2+1)|}\Big) dt_2 s_2=c_2+it_2 \ll \epsilon^{-2}\Theta(c_1'+c_2+1,h) X^{c_1'+c_2},
\delta \ge \frac{t}{6(t+2)}\sum_{i=1}^{t}\left(x_i-\frac{1}{t+2}\right)^2+\sum_{i=t+1}^{t+2}\left(y_i-\frac{1}{t+2}\right)^2 & + (\sigma_1-x_{t})\left(\frac{y_{12}-y_{34}}{2}\right)^2+x_{t}\left(\frac{y_{13}-y_{24}}{2}\right)^2.
\bar{\jmath}_{\alpha}= - \bar{X}_{\alpha}^{\mu}( \xi ) [ \pi_{\mu}+ \kappa \lambda_{\mu \rho}( \xi ) \partial_{x}\xi^{\rho}] - \kappa c_{\alpha \gamma}\bar{\theta}_{\rho}^{\gamma}( \xi ) \partial_{x}\xi^{\rho},
\| \partial_x^2 (\nabla V\cdot\nabla u_{\neq}^3)\|_{L^2}=\| \nabla V\cdot\nabla \partial_x^2u_{\neq}^3\|_{L^2}\leq \| \nabla V\|_{L^{\infty}}\|\nabla \partial_x^2u_{\neq}^3\|_{L^2}\leq C\|\nabla \partial_x^2u_{\neq}^3\|_{L^2},
[y_1, y_1]=\frac{\alpha_2\beta^2_1}{\alpha^2_3\gamma_6}y_5, [y_1, y_2]=y_4+\theta_1y_5, [y_2, y_1]=-y_4+\theta_2y_5, [y_2, y_2]=y_4, [y_3, y_3]=y_5.
\Vert g \Vert_{U^{d}(\mathbb{R},N)}^{2^{d}}\asymp \frac{1}{N^{d+1}}\int\limits_{(x,\mathbf{h}) \in\mathbb{R}^{d+1}}\prod\limits_{\boldsymbol{\omega}\in \{0,1\}^d}\mathcal{C}^{\vert \boldsymbol{\omega}\vert}g(x+\sum\limits_{i=1}^d h_i\omega_i) \, dx\,dh_1\cdots dh_d.
\ln\left({\det[-{\cal M}_{\rm FP}]\over \det[-{\cal M}^0_{\rm FP}]}\right)_{\rm reg}\approx-\lim_{s\to0}\left\{{d\over ds}\left({M^{2s}\over\Gamma(s)}\int_0^{t_0}dt t^{s-1}\left[\alpha_{{\cal M}_{\rm FP}}t^{-1/2}-\beta_{{\cal M}_{\rm FP}}t^{1/2}\right]\right)\right\}\;,
\mbox{vol}_{n-2}\Big( \mbox{relbd}\big( P\cap\xi^\perp \big) \Big)= \max_{t\in\mathbb{R}}\mbox{vol}_{n-2}\Big( \mbox{relbd}\big( P\cap\lbrace t\xi + \xi^\perp \rbrace \big) \Big) \ \ \forall \ \ \xi\in S^{n-1},
\{f\in E(G)|\e_1 \prec f\prec \bar{e}_1\}= \{e_2, e_3\}\sqcup\{f\in E(G)|\ \bar{e}_2\prec f\prec e_2\}\sqcup \{f\in E(G)|\ \bar{e}_3\prec f\prec e_3\}.
\nu_{1}=-\frac{13\gamma^{2}-7\gamma+12}{(3\gamma-1)(2\gamma+1)},\nu_{2}=\frac{5(\gamma-1)}{2\gamma+1},\nu_{3}=\frac{3}{2\gamma+1},\nu_{4}=-\frac{\nu_{1}}{2-\gamma},\nu_{5}=-\frac{2}{2-\gamma}.
S_{r+k}\,f(z) = S_{r+k}\,g(z) + \sum_{j=0}^{k-1}a_j \int_0^\infty \frac{x^j}{(x+z)^{r+k+1}}\, dx & = S_{r+k}\,g(z) + \sum_{j=0}^{k-1}\frac{j! \Gamma(r+k-j)}{\Gamma(r+k+1)z^{j+k+r}}\, \, a_j.
\int_{\mathbb{R}^n\backslash B_{\delta}(x)}\frac{e^{n\left(\tilde{v}(y)+p\left(\frac{y}{|y|^2}\right)+q(y)\right)}}{|x-y|^k|y|^{n\left(2+\beta+\frac{\Lambda}{\gamma_{n}}\right)}}dy\rightarrow 0|x|\rightarrow\infty.
\frac{\mathrm{d}^2}{\mathrm{d}t^2}(J_{\omega_{\psi}}- I )(\varphi_t)= \int ( \ddot{\varphi}- |\nabla \dot{\varphi}|_{\varphi_t}^2) (_{\varphi_t}\omega_{\psi}-n) \omega_{\varphi_t}^n + \int \dot{\varphi}_{,\alpha}\dot{\varphi}_{,\bar{\beta}}(\omega_{\psi})_{\bar{\alpha}\beta}\omega_{\varphi_t}^n >0.
| \phi_{s}\rangle \equiv \zeta^{i_{1}}\ldots \zeta^{i_{s}}\phi^{i_{1}\ldots i_{s}}\, , \qquad | \phi_{s - 1}\rangle \equiv \zeta^{i_{1}}\ldots \zeta^{i_{s - 1}}\phi^{i_{1}\ldots i_{s - 1}}\, .
\int_R^\infty \left( \frac{t}{V(t)}\right)^{1/(p-1)}dt \ge c\int_R^\infty \left( \frac{1}{t^{n-1}(\log t)^{\alpha(p-1)}}\right)^{1/(p-1)}dt = c\int^\infty \frac{1}{t^{(n-1)/(p-1)}(\log t)^{\alpha}}dt = \infty
\langle |b_1-\langle b_1\rangle_{J_k}|\rangle_{A, J_k}\langle |b_2-\langle b_2\rangle_{J_k}|\rangle_{B,J_k}=\langle |b_1|\rangle_{A, J_k}\langle |b_2|\rangle_{B,J_k}&\simeq k^{-\frac 12}\log(e+k)^{\frac{1+\alpha}2}k^{\frac 12}\log(e+k)^{-\frac 12}\log\log(e^e+k)^{-\frac 34}&\overset{k\rightarrow\infty}{\rightarrow}\infty.
\alpha_n = \frac{1-q^{2n+1}}{1-q}q^{(k+1)n^2+kn}\sum_{\nu=-n}^{n}(-1)^\nu q^{-\frac{1}{2}(2k+1)\nu^2 - \frac{1}{2}(2k - (2\ell-1))\nu}\intertext{and}\beta_n = H_n(k,\ell;0;q).
\prod_{j=x_1}^k |\cos(\pi \theta_j)|\leq e^{-(\ln 2-\varepsilon)|x_1-k|}c_{n,\ell-1}, \prod_{j=k}^{x_2}|\cos(\pi \theta_j)|\leq e^{-(\ln 2-\varepsilon)|x_2-k|}c_{n,\ell}.
0 \neq \mathrm{Hom}_E(\varphi_1|_E,\varphi_2|_E) =\mathrm{Hom}_F(\mathrm{Ind}_{E}^F(\varphi_1|_E),\varphi_2) &=\oplus_i \mathrm{Hom}_F(\rho_i \otimes \varphi_1,\varphi_2)^{\oplus \mathrm{deg}(\rho_i)}
\begin{cases}\Phi_{\epsilon_1}^c\preceq\Phi_{\epsilon_2}^c\ \{\rm if\}0< \epsilon_1\leq \epsilon_2\ll 1, \Phi_{\epsilon}^{c_1}\succeq \Phi_{\epsilon}^{c_2}\ \{\rm if\}0<c_1\leq c_2.\end{cases}
\frac{d}{d\tau}\,\left(\int_{M_4}\,n^2(x)\,dvol_{{\eta}_4}\right) =\int_{M_4}\,\frac{d}{d\tau}\,\left(n^2(x)\,dvol_{{\eta}_4}\right) & =\,\int_{M_4}\,\frac{d}{d\tau}\,\left(n^2(x)\right)\,dvol_{{\eta}_4}=0.
S=-\gamma \int d \tau \int d \sigma \sqrt{{\dot x\acute x}^{2}-{\dot x}^{2}{\acute x}^{2}}-\sum_{i=1}^2 m_i\int d \tau\sqrt{{\left(\frac{d x^{\mu}({\tau}_i,{\sigma}_i(\tau))}{d\tau}\right)}^{2}}
W=\int_{t_{1}}^{t_{2}}F\cdot vdt=m\int_{t_{1}}^{t_{2}}a\cdot vdt=\frac{m}{2}\int_{t_{1}}^{t_{2}}\frac{dv^{2}}{dt}dt=\frac{m}{2}\int_{v_{1}^{2}}^{v_{2}^{2}}dv^{2}=\frac{mv_{2}^{2}}{2}-\frac{mv_{1}^{2}}{2}=\Delta E_{k}
\dim C^r_k(\Delta) \geq\binom{k+3}{3}+\biggl[f_2^0\binom{k+2-r}{3}&-\sum_{i=1}^{f_1^0}\biggl[s_i\binom{k+2-r}{3}- b_i\binom{k+3- \Omega_i}{3}- a_i\binom{k+2-\Omega_i}{3}\biggr] &+f_0^0\binom{k+3}{3}-\sum_{i=1}^{f_0^0}\biggl(\sum_{j=0}^k F(\zeta_i,r+1,3)_j\biggl)\biggr]_+
J_{p}( s ) = \sqrt{\pi}e^{- 2 \pi i p \tilde{t}_{1}}\sum_{n_{2}}\int_{0}^{\infty}\frac{d t}{t}\frac{t^{s}}{\sqrt{t}}e^{- \frac{\tilde{n}_{2}^{2}}{| \tau |^{2}}t - \frac{\pi^{2}}{t}p^{2}}.
\widetilde{W}({\bf k}) = \frac{( 2 \pi )^{{D_{0}}/ 2}}{k^{{D_{0}}/ 2 - 1}}\, \int_{0}^{\infty}d r r^{{D_{0}}/ 2 - 2}J_{{D_{0}}/ 2 - 1}( k r ) \; .
p_{K''|K}(k'') =\frac{\binom{K}{k''}}{\binom{Q}{K}}\!\!\!\!\sum_{\substack{\forall \mathcal{I},\mathcal{J}\subset\mathcal{Q}: \mathcal{I}\cap\mathcal{J}=\emptyset, |\mathcal{I}|=k'',|\mathcal{J}|=K-k''}}\!\!\!\!\prod_{i\in\mathcal{I}}\xi_i\prod_{j\in\mathcal{J}}(1-\xi_j)
\underset{\tau\to+0}{\mbox{{\rm ess limsup}}}\,\left|\int_{{\bf R}^N}[G(y,t-\tau)-G(y,t)]u(y,\tau)\eta_n(y)\,dy\right| \le \sup_{y\in B(0,2nt^{\frac{1}{\theta}}),s\in(t/2,t)}\,|\partial_t G(y,s)|\,\underset{\tau\to+0}{\mbox{{\rm ess limsup}}}\,\biggr[\tau\int_{B(0,2nt^\frac{1}{\theta})}u(y,\tau)\,dy\biggr]=0.
\widetilde{K}_{\lambda,1}(z,w) =\sum_{n=0}^{\infty}\frac{(n+\lambda+3)(n+\lambda+2)}{2\lambda+2}\phi_{n}(z)\overline{\phi_{n}(w)}, \widetilde{K}_{\lambda,2}(z,w) =K_{\lambda,2}(z,w)(1-|z|^2)(1-|w|^2).
\iint G(p,q) d\mu(p) d\mu(q) \leq \iint \log \frac{1}{|z_0(p)-z_0(q)|}d\mu(p)d\mu(q) + C_1 = \iint \log \frac{1}{|z-w|}d\omega(z) d\omega(w) + C_1 \leq -\log \delta + \log 4 + C_1.
V ( \sigma ) = \frac{1}{2 \lambda_{0}}\sigma^{2}- i \mathrm{t r}\ \displaystyle \int_{0}^{\sigma}s d s \frac{1}{L}\sum_{n = - \infty}^{\infty}\int \frac{d^{D - 1}k}{( 2 \pi )^{D - 1}}\frac{1}{k^{2}- s^{2}}.
\mathbb{E}[x^{V'_b};C_2]=\frac{1}{3}\mathbb{E}[x^{V_b}|B_2\cup B_3]=\frac{1}{3}\Big{(}f\Big{(}\frac{x+1}{3}\Big{)}^2 +f\Big{(}\frac{x+2}{3}\Big{)}^3-f\Big{(}\frac{x+2}{3}\Big{)}f\Big{(}\frac{x+1}{3}\Big{)}^2\Big{)}
\begin{array}{llcll}V_3 \stackrel{C^{12}_3}{\longrightarrow}V_1\otimes V_2 \Pi_3 \downarrow \downarrow \Pi_1 \otimes \Pi_2 V_3 \stackrel{C^{12}_3}{\longrightarrow}V_1\otimes V_2 \end{array}
(a_{p+n-i,j}-a_{p+n-j,i})(a_{q+n-k,p}-a_{q+n-p,k})+{}(a_{p+n-j,k}-a_{p+n-k,j})(a_{q+n-i,p}-a_{q+n-p,i})+{}(a_{p+n-k,i}-a_{p+n-i,k})(a_{q+n-j,p}-a_{q+n-p,j})=0.
\max\big(-\mu_2-\frac 12,\, \frac 12-\mbox{Re}\, \lambda_2\big) < \beta < \min\big(\mu_2+\frac 12,\, \mbox{Re}\, \lambda_2+\frac 32\big), \ \ \beta\not= \pm \frac 12\, , \ \ \beta\not= \frac 52\, .
T(n_{i+1})+T(n_{i+2}) < \frac{1}{n_{i+1}}\cdot \frac{35}{9}< \frac{35}{9}\cdot \frac{16}{27}\cdot \frac{1}{n_i}&= \frac{560}{243}\cdot \frac{1}{n_i}, T(n_i)+ T(n_{i+1})+T(n_{i+2}) < \left(\frac{19}{9}+\frac{560}{243}\right) \cdot \frac{1}{n_i}=\frac{1073}{243}\cdot \frac{1}{n_i}&< 3 \cdot \frac{97}{54}\cdot \frac{1}{X_0}.
{\Delta_\Omega}({\textbf{p}}^i) = \frac{{\Omega({\textbf{p}}^i) - \mathop{\min}\limits_{j = 1, \cdots ,W}\Omega({\textbf{p}}^j)}}{{\mathop{\max}\limits_{j = 1, \cdots ,W}\Omega({\textbf{p}}^j) - \mathop{\min}\limits_{j = 1, \cdots ,W}\Omega({\textbf{p}}^j)}},
\partial_t f^{\varepsilon,a,n}+ \big\{\beta_\varepsilon(v) + [v\!-\!\beta_\varepsilon(v)] \eta_\varepsilon(x) \big\}\cdot\nabla_{\!x}f^{\varepsilon,a,n}- \mathbf{B}\cdot\nabla_{\!v}f^{\varepsilon,a,n}= Q^\varepsilon[f^{\varepsilon,a,n}]
= \{\chi^{\mu\nu}\}+\left(\begin{array}{cccc}((p^2 - q^2)a + 2pqb) (pa + qb) (pb -qa) ((p^2 - q^2)a + 2pqb) (pa + qb) 0 0 (pa + qb) (pb -qa) 0 0 (pb -qa) ((p^2 - q^2)a + 2pqb) ( pa + qb) ( pb -qa) ((p^2 - q^2)a + 2pqb)\end{array}\right).
- \left( 1 - \frac{w \kappa}{4}e^{2 \phi}\right) \partial^{2}\phi + 2 ( \partial \phi )^{2}+ \frac{\kappa}{4}e^{2 \phi}\partial^{2}Z + e^{2 \rho + 2 \phi}\left( \Lambda e^{- 2 \phi}- 1 \right) = 0 ;
\begin{bmatrix}-S_1' -S_2' 0 0 S_3' S_4' 0 0\end{bmatrix}= \begin{bmatrix}I - S_2G_3' -S_2G_4' S_3 + G_3' S_4 + G_4' G_1' -R_1S_2 + R_2 + G_2' S_3G_1' + S_4G_3' -R_3S_2 + R_4 + S_3G_2' + S_4G_4'\end{bmatrix},
\rho(d^X \circ g)(x) = \sum_{(x)}(d^X \circ g)(x_0)_{-1}S^{-1}x_{-1}\otimes (d^X \circ g)(x_0)_0 = \sum_{(x)}g(x_0)_{-1}S^{-1}x_{-1}\otimes d^X (g(x_0)_0) = \sum_{(x)}g_{-1}\otimes d^X (g_0(x)),
\frac{\left| \langle \eta(x_1),\mathbf{B}(x_1)-\mathbf{B}(x_2)\rangle\right|}{|x_1-x_2|}\lesssim \sum_{l=1,2}||\nabla \eta||_{L^\infty}\mathbf{I}_1( \mu)(x_l) +\mathbf{M}(|D^a\mathbf{B}|)(x_l),
d_{i,j+1}(d_{k,\ell}- c_{k,\ell}) = (d_{i,j}- c_{i,j}) d_{k,\ell + 1}d_{i,j+1}(q_\ell - p_\ell) = (d_{i,j}-c_{i,j})q_{\ell +1}q_{j+1}(q_\ell - p_\ell) = ( q_j - p_j)q_{\ell + 1}q_0 = 0,
\mathcal A_N(t)+\int_0^t\mathcal B_N(s)ds\leq(\alpha+1)^{N-1}\int_0^tm(s)\mathcal A_N(s)ds \leq (\alpha+1)^{N-1}\int_0^t\mathcal A_N(s)K(s)\log\mathcal A_N(s)ds \leq (\alpha+1)^{N-1}\int_0^tK(s)dsq_1(t)q_0(t)=Q(t),
\begin{array}{c}H=\frac{1}{2m}[\vec{p}-\vec{A}(\vec{x})]^{2}-\frac{1}{2m}\sigma_{3}B(\vec{x})\hspace{1.4cm}\vspace{0.2cm}*\hspace{1cm}=\frac{1}{2m}\left[\sigma_{1}\left(p_{1}-A_{1}(\vec{x})\right)+\sigma_{2}\left(p_{2}-A_{2}(\vec{x})\right)\!\frac{}{}\!\right]^{2},\end{array}
\int_{\mathbb{R}^{2N}}\frac{|G_{\alpha,M}(U(x))-G_{\alpha,M}(U(y))|^p}{|x-y|^{N+s\,p}}\,dx\,dy &\le \mathcal{S}_{p,s}\, \left[K_0^{\alpha-1}\,\int_{\mathbb{R}^N}U^{p^*}\,dx+\left(\int_{\{U\ge K_0\}}U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\,\left(\int_{\mathbb{R}^N}\left(U_{M}^{(\alpha-1)}\,U^{p}\right)^\frac{p^*}{p}\,dx\right)^\frac{p}{p^*}\right],
{{\boldsymbol{P}}_{k|k}}= \left({{\boldsymbol{I}}-{{\boldsymbol{K}}_k}{{\boldsymbol{H}}_k}}\right){{\boldsymbol{P}}_{k|k - 1}}{\left({{\boldsymbol{I}}-{{\boldsymbol{K}}_k}{{\boldsymbol{H}}_k}}\right)^{}}+{{\boldsymbol{K}}_k}{{\boldsymbol{R}}_k}{\boldsymbol{K}}_k^T.
\widetilde a_{ij}\partial_i\partial_j h=r^{-n}E_+(r)\left[\phi(0)+\int_0^r \rho^{n-1}E_-(\rho) B[v](\rho)\,d\rho\right]\,\psi_2 +B[v]\,\widetilde a_{ij}\theta_i\theta_j,
\ell_s = g^{-1/4}\ell_p^{(10)}= g^{-1/3}\ell_p \ \ \ ell_p^{(10)}= (\ell_p^{(10)})^{9/8}R_{11}^{-1/8}, \ \ \R_{11}= g \ell_s =g^{2/3}\ell_p^{(11)}.
P_{g}^{\frac{1}{2}}( k ) = \frac{2}{\sqrt{\pi}}2^{\nu - \frac{1}{2}}\frac{\Gamma ( \nu )}{\Gamma ( 3 / 2 )}( \nu - \frac{1}{2})^{\frac{1}{2}- \nu}\frac{H}{m_{p l}}|_{k = a H}
\tilde{H}=(2\pi\alpha^{\prime})d\tilde{\cal F}-(2\pi\alpha^{\prime}) G^{(6)}{\cal G}^{(1)}+3(2\pi\alpha^{\prime})^{2}{\cal H}^{(3)}d{\cal H}^{(3)}+{\textstyle\frac{m}{2}}{\cal G}^{(7)}\, .
{\mu}{}^{A}\lambda_{A}- \bar{\mu}{}^{\dot{A}}{\bar{\lambda}}_{\dot{A}}= 2 \l_{A}\l_{B}z^{A B}- 2 \bar{\l}_{\dot{A}}\bar{\l}_{\dot{B}}\bar{z}^{\dot{A}\dot{B}}+ 2 i \Theta^{A}\l_{A}\bar{\Theta}^{\dot{A}}\bar{\l}_{\dot{A}}
-\chi(-k)\int_{\{u_1 \le \phi -k\}}(\theta_{u_{1,k}}^n + \theta_{u_{2,k}}^n)= -\chi(-k)\int_{\{u_1 \le \phi -k\}}(\theta_{u_{1}}^n + \theta_{u_{2}}^n)\leq \int_{\{u_1 \le \phi -k\}}\mu.
\langle~ \delta_S{\bar Q}_{\dot \alpha}({\cal I}^+)~\rangle~=~\epsilon^{\beta}~\sum_k k_{\beta{\dot \alpha}}~ \sum_{n^B_k, n^F_k}~\rho_{n^B_k n^F_k,n^B_kn^F_k}~(-)^{n^F_k}~\left(~n^B_k~+~n^F_k~\right)~.
\mathsf{D}_{d}Z_{t}(d)=\sum_{n=0}^{t-1}\mathsf{D}_{d}(T^{1/2-d}\pi_{n}(d))\xi_{t-n}=T^{1/2-d}\sum_{n=0}^{t-1}(-\log T+\sum_{k=0}^{n-1}\frac{1}{k+d})\pi_{n}(d)\xi_{t-n}.
\mathcal{X}=L^{2}\left(\Omega,\mathbb{C}^{3}\right)\oplus\left(L^{2}\left(\Omega,\mathrm{sym}\left[\mathbb{C}^{3\times3}\right]\right)\oplus\tilde{X}\right)\oplus L^{2}\left(\Omega,\mathbb{C}^{3}\right)\oplus\left(L^{2}\left(\Omega,\left[\mathbb{C}^{3}\right]\right)\oplus X\right).
(\mathbb{I}_{2r+1}\otimes U^{-1})R^{(2r+1)}_{2,1}\left(t^{2r+1}/x\right)^{\tau_1}(\mathbb{I}_3\otimes U)R_{1,2}^{(2r+1)}(x)^{\tau_1}=\frac{(1-x) \left(t^{2r+1}-x\right)}{(t-x)\left(t^{2r}-x\right)}\mathbb{I}_{2r+1}\otimes\mathbb{I}_{2r+1},
\begin{cases}\tilde{\gamma}_i=\gamma_i+O(1)|\boldsymbol{\gamma}^{*}|Z_{a}h+O(1)|\gamma_4|h,i=1,2,3,5, &\tilde{\gamma}_4=(1-{\phi(T_b)h}/{u_b})\gamma_4+O(1)|\boldsymbol{\gamma}^{*}|Z_{a}h.\end{cases}
\underset{\{\mathbf{p}_i\}_{i=1}^N,\, \{\mathbf{q}_i\}_{i=1}^N}{\min}\sum_{i=1}^M C_i(\mathbf{p}_i,k) - \sum_{i=M+1}^N U_i(\mathbf{q}_i,k) & \underline{\mathbf{p}}_i^{[k]}\leq \mathbf{q}_i \leq \overline{\mathbf{p}}_i^{[k]},\, i = 1,\hdots,N & \sum_{i=1}^N \mathbf{q}_i = \mathbf{P}^{[k]}\mathbf{q}_i = \mathbf{p}_i,\, i=1,\hdots,N.
(ii) =\frac{1}{x_1}\log\frac{1+x_1}{1-x_1}+\log(x_1+1)(1-x_1)-2\frac{x_2}{x_1}\log\frac{2x_2+x_1}{2x_2-x_1}-\log(2x_2+x_1)(2x_2-x_1).
|\tilde{\Sigma}_s|^{\frac{4}{n-1}}\left(|H-(n-1)|^2+|\mathring{A}|^2\right) \leq Ce^{-\frac{4t}{n-1}}|\tilde{\Sigma}_s|^{\frac{6}{n-1}}|\nabla A|^2 \leq Ce^{-\frac{6t}{n-1}},
h_{\Delta}(P_{1},P_{2})=\mu(P_{1})+\mu(P_{2})-\tfrac{i}{2}\sum_{k=1}^{g}\left(\psi_{k}(P_{1})\wedge\bar{\psi}_{k}(P_{2})+\psi_{k}(P_{2})\wedge\bar{\psi}_{k}(P_{1})\right),
\int_{{\mathbb R}^d}\Phi^{(j)}(K \cap M^x, A\cap B^x)\,\lambda(dx)= \Phi^{(j)}(K,A) \lambda (M\cap B) & + \sum_{m=j+1}^{d-1}\Phi_{m,d+j-m}^{(j)}(K,M;A \times B) +\lambda (K\cap A)\Phi^{(j)}(M,B)
\left\lvert \ \int\limits_{|x-e|+|x|\leq 2\tau}\frac{q(x)}{|x||x-e|}dx\right\rvert =\left\lvert\frac{1}{2}\int\limits_{1}^{2\tau}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi}a(r)\sin\phi d\theta d\phi dr\right\rvert &\leq C\int\limits_{1}^{2\tau}|a(r)|dr.
\int\limits_{0}^{R}\left(\frac{\left(K(r)-\ln{(1+T')}\right)\sum\limits_{i\in \psi}r^\alpha d_i^{-\alpha}\left ( \underset{j \in \psi\setminus i}\prod{(1+T' r^\alpha d_j^{-\alpha}})\right)}{\left(\underset{j \in \psi}\prod{(1+T' r^\alpha d_j^{-\alpha})}\right)^2}\right)f_R(r)r=0,
\begin{aligned}\mathbb{G}_{L;0}(x,t)= \sum_{j=1}^{3}\int_{|\eta|<\delta}e^{i\eta x+\sigma(\eta)t}\vert e_j(\eta)\rangle\langle e_j(\eta)\vert d\eta = \sum_{j=1}^{3}\int_{|\eta|<\delta}e^{i\eta (x-a_j t)-A_j t|\eta|^2 + O(1) |\eta|^3 t}\left( \vert e_j(0) \rangle \langle e_j(0) \vert + \varepsilon_j(\eta)\right)d\eta,\end{aligned}
\frac{(4-7\delta_2)\varepsilon}{7}-\frac{7\delta_2}{2}< \frac{4-7\delta_2}{7}\cdot\frac{49\delta_2}{8-14\delta_2}-\frac{7\delta_2}{2}&= \frac{7\delta_2}{2}-\frac{7\delta_2}{2}= 0,
\begin{aligned}Q_{1}(f,g)- Q_{1}(h,h_{-}) +Q_{3}(f,g)- Q_{3}(h,h_{-}) &\qquad=Q_{1}(f-h,g-h_{-}) + Q_{1}(f,h_{-}) + Q_{1}(g,h) - 2Q_{1}(h,h_{-}) &\qquad+Q_{3}(f-h,g-h_{-}) + Q_{3}(f,h_{-}) + Q_{3}(g,h) - 2Q_{3}(h,h_{-}) &\qquad\stackrel{\eqref{qeq}}{=}Q_{1}(f-h,g-h_{-})+ Q_{3}(f-h,g-h_{-}).\end{aligned}
B(\varepsilon^+)= - 3(n-4) (1+ |y|^2 Q_b) e^{2\int_{\tau_0}^\tau \lambda_{M,b}d\tau'}\phi_{M,b}^2 f^2(y)- (n-4) |y|^2 e^{3\int_{\tau_0}^\tau \lambda_{M,b}d\tau'}\phi_{M,b}^3 f^3(y)
-\frac{\partial}{\partial t}(\nabla^{2}\psi)+\nabla\times(\nabla(\frac{(\nabla\times\psi)\cdot(\nabla\times\psi)}{2})+(\nabla\times(\nabla\times\psi))\times(\nabla\times\psi))=\nu(\nabla^{2}(\nabla(\nabla\cdot\psi))-\nabla^{4}\psi)
T_{F^t}(r) \geqslant N_{(F_1^{ti}-F_2^{ti})h_t}(r)\geqslant \sum_{\underset{v\ne i,t}{v=1}}^{2n+2}N^{(1)}_{(f,H_v),\leqslant k_v}(r)-N(r,S_i)-N(r,S_t).
\delta \left[ f \left[ A \right] \right]{\cal M}\left( x , y ; a , b \right) = - \delta \left[ f \left[ A \right] \right] \left( \partial_{3}^{y}\delta^{a b}\left( 1 - \epsilon^{a}+ \eta \epsilon^{a}\partial_{3}^{x}\right) - g f^{a b c}\epsilon^{c}A_{3}^{c}\left( y_{\perp}\right) \right) \delta^{4}\left( x - y \right) \ .
\omega_{n_{1}n_{2}}= \biggl ( \biggl ( \frac{n_{1}\pi}{L_{1}}\biggr )^{2}+ \biggl ( \frac{n_{2}\pi}{L_{2}}\biggr )^{2}\biggr )^{\frac{1}{2}}~ n_{1}, n_{2}= 1 , 2 , \dots ,
\left( \begin{array}{c c c c}{{0}}&{{0}}&{{\delta^{3}( x - y )}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\nabla^{- 2}}}\\{{- \delta^{3}( x - y )}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- \nabla^{- 2}}}&{{0}}&{{0}}\\ \end{array}\right) ,
A_C= \begin{bmatrix}a_1 a_n \dots a_{3}a_2 a_2 a_1 \dots a_{4}a_{3}\vdots \vdots \ddots \vdots \vdots a_{n-1}a_{n-2}\dots a_{1}a_{n}a_n a_{n-1}\dots a_{2}a_{1}\end{bmatrix}.
u(t)= T(t)u_{0}-\chi\underbrace{\int_{0}^{t}T(t-s)(\nabla u\nabla v)ds}_{I_{1}}-\chi\underbrace{\int_{0}^{t}T(t-s)(uv)ds}_{I_{2}}+ \underbrace{\int_{0}^{t}T(t-s)((a+1)u+(\chi-b)u^{2})}_{I_{3}}ds.
H = \Delta \exp \left\{\frac{\alpha i}{4 \hbar}\int d^{4}x \, G_{\mu \nu}^{a}{\cal M}_{a b}^{- 1}\left( \frac{1}{2}\delta_{c}^{b}( \eta^{\mu \rho}\eta^{\nu \sigma}- \eta^{\mu \sigma}\eta^{\nu \rho}) + \frac{\alpha}{2}\varepsilon^{\mu \nu \rho \sigma}{\cal C}_{c}^{b}\right) G_{\rho \sigma}^{c}\right\},
0 =\langle d_\nu G(\bar \nu),A(\mu)\bar\nu \rangle &=\int_{\mathbb S^{n-1}}\Big\{(G(\bar \nu)G(\nu)-\langle d_\nu G(\bar \nu),\nu\rangle\langle d_\nu G( \nu),\bar \nu\rangle\Big\}d\mu(\nu) &\ge \int_{\mathbb S^{n-1}}\Big\{G(\bar \nu)G(\nu)-\big|\langle d_\nu G(\bar \nu),\nu\rangle\big|\big|\langle d_\nu G( \nu),\bar \nu\rangle\big| \Big\}d\mu(\nu)
d{\tilde{\rho}}_{K}\left( X \right) = \sum_{k = 1}^{2 K}{\overbrace{{\tilde{\rho}}_{1 / 2}\left( 1 \right) \otimes \cdots \otimes{\tilde{\rho}}_{1 / 2}\left( 1 \right)}^{k - 1}\otimes d{\tilde{\rho}}_{1 / 2}\left( X \right) \otimes \overbrace{{\tilde{\rho}}_{1 / 2}\left( 1 \right) \otimes \cdots \otimes{\tilde{\rho}}_{1 / 2}\left( 1 \right)}^{2 K - k}}\ ,
E_{1}= \bigsqcup_{n\ge 1}\bigsqcup_{0\leq i < 2^{s-1}}E_1(n,i), E_{2}= \bigsqcup_{n\ge 2}\bigsqcup_{0\leq i < 2^{s}}E_2(n,i), E_{3}= \bigsqcup_{n\ge 2}\bigsqcup_{0\leq i < 2^{s}}E_3(n,i),
{\tilde g}= - \frac{1}{16}\frac{1 + 3 y}{\sqrt{y^3 (1 + y)^3}}- \frac{3}{16}C_1\frac{1 + 2y}{\sqrt{y^3(1+y)}}+ C_3 \left( \frac{1 + y}{y}\right)^{3/2}.
\varphi_{m}(x)=\frac{1}{\delta_{m}^{\frac{n-2}{2}}}\varphi\left(\frac{1}{\delta_{m}}\left(\psi_{q_{m}}^{\partial}\right)^{-1}(x)\right)\chi\left(\left(\psi_{q_{m}}^{\partial}\right)^{-1}(x)\right)x\in M.
\frac{\eta_{1,\delta}}{\eta_{0,\delta}}= \frac{\eta_{2,\delta}}{\eta_{1,\delta}}= \frac{\eta_{3,\delta}}{\eta_{2,\delta}}=\dots=\frac{\eta_{m-1,\delta}}{\eta_{m-2,\delta}}=\gamma^\delta \frac{\eta_{0,\delta}}{\eta_{m-1,\delta}}.
\begin{aligned}\partial_\tau g+ (v_1-v_2) \nabla_x g- \nabla f(v_1) \nabla_x E_2[g](x,v_2)- \nabla f(v_2) \nabla_x E_1[g](x,v_1) &= (\nabla_{v_1}- \nabla_{v_2})\left(f(v_1)f(v_2)\right)\nabla \phi (x). \end{aligned}
T_{rh}\approx \left( \frac{90}{\pi^2g_{*}}\right)^{\frac{1}{4}}min\left(\sqrt{H(\phi_e)},\sqrt{\Gamma}\right)\approx \left(\frac{30}{\pi^2g_{*}}\right)^{\frac{1}{4}}min\left(\Delta,\left( \frac{3}{8\pi^2}\right)^{\frac{1}{4}}p^{\frac{3}{2}}\kappa^{\frac{5}{2p}}\Delta^{3-\frac{5q}{2p}}\right)
\left(\frac{q^k-2(r\delta)^{r4^{r+1}+4^{r+1}}\binom{n+2 (r\delta)^{r4^{r+1}}}{k-1}q^{k-1}}{q^k}\right)^n=\left(1-\frac{2(r\delta)^{r4^{r+1}+4^{r+1}}\binom{n+2 (r\delta)^{r4^{r+1}}}{k-1}}{q}\right)^n \rightarrow (1-0)^n=1,
R^p_{\mathbf{s}_{\scriptscriptstyle{A}}}= E \left[ \log_2 \det\left( \mathbf{I}+ \hat{\mathbf{H}}^H_3 (\mathbf{N}+ \hat{\mathbf{H}}_4 \hat{\mathbf{H}}^H_4 )^{-1}\hat{\mathbf{H}}_3 \right)\right].
T_i(E_j) = \begin{dcases}-F_i L_i \i=j, \sum_{k=0}^{r}(-1)^kq_i^{-k}E_i^{(r-k)}E_j E_i^{(k)}, \end{dcases}T_i(F_j) = \begin{dcases}-K_i^{-1}E_i \i=j, \sum_{k=0}^{r}(-1)^kq_i^kF_i^{(k)}F_j F_i^{(r-k)}, \end{dcases}
G^{B = 0}= \left( \begin{array}{c c}{e^{\lambda^{( 0 )}+ \psi^{( 0 )}}}&{0}\\{0}&{e^{\lambda^{( 0 )}- \psi^{( 0 )}}}\\ \end{array}\right) = \left( \begin{array}{c c}{\xi_{1}}&{0}\\{0}&{\xi_{2}}\\ \end{array}\right)
\begin{aligned}\partial_{t}u(x,t) - \Delta u(x,t) = f(x,t) \, (x,t) \in Q := \Omega \times (0,T), u(x,t) = 0 \, \, (x,t) \in \Sigma := \partial \Omega \times (0,T), u(x,0) = u_{0}(x) \, \, \, x \in \overline{\Omega}, \end{aligned}
c(t)\leq A+(B+2)C_P \leq 128R^2+\left(\frac{18nR^2}{t}+\frac{6R^4}{t^2}-2+2\right)\left(8+\frac{36D}{e}\right)R^2\exp\left(17n+\frac{25R^2}{8t}\right) = 128R^2+12\cdot\frac{R^2}{2t}\left(3n+\frac{R^2}{t}\right)\left(8+\frac{36D}{e}\right)R^2\exp\left(17n+\frac{25R^2}{8t}\right).