harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_{0,\tau(e_0)}\circ f_{e_0}= f_{e_0}= ad_{w_{e_0}}\circ f^{\prime}_{e_0}= ad_{\delta_{0}(e_0)}\circ f^{\prime}_{e_0}= ad_{\delta_{0}(e_0)}\circ f^{\prime}_{e_0}\circ H_{0,e_0 \,}.
\frac12 h(a) + \frac12 h(b) - h(r) = \frac12 h(sa_1) + \frac12 h(sb_1) - h(sr_1) &= s (\frac12 h(a_1) + \frac12 h(b_1) - h(r_1)) + (s\log s)( \frac12a_1 + \frac12 b_1 - r_1) &< 0
2{\widetilde a}_{1 , 2}={\frac{1}{g_{\mathrm{{\small Y M}}}^{2}}}+{\frac{\beta_{0}^{( 1 , 2 )}}{1 6 \pi^{2}}}~ \operatorname{l n}\left({\frac{\Lambda^{2}}{p^{2}}}\right) ~ ,
\int d^{2}\zeta_{SS}d^{2}\xi_{SS}\int d^4x_0\mbox{Tr}(\phi^3)=\frac{3}{2}\left(-\frac{g^2}{2^4\pi^2}\right)\left(\mbox{Tr}(A_1)-\frac{ig}{4\sqrt{2}\pi^2\rho^2}\sum_{k=3}^{N_c}(\xi_{Mk}\zeta_{Nk}+\xi_{Nk}\zeta_{Mk})\right).
E\max_{1\leq k\leq n}\left\vert \sum\nolimits_{j=1}^{k}a_{j}E^{j}X\right\vert^{p}\ll E\max_{1\leq k\leq n}|s_{k}E^{k}X|^{p}+E\left\vert \sum\nolimits_{j=1}^{n}a_{j}E^{j}X\right\vert^{p}.
\emptyset=\emptyset\cup\emptyset=(L_{1}\cap L_{2})\cup(R_{2}\cap R_{2})=(\bigcap_{i\in I}L_{i})\cup(\bigcap_{i\in I}R_{i})\ne\bigcap_{i\in I}(L_{i}\cup R_{i})=(L_{1}\cup R_{1})\cap(L_{2}\cup R_{2})=X\cap X=X.
(\delta_1 X_2)_{\rm extra}={\tau_1\tau_2\over 1 - \tau_1\tau_2}F_1 M^{-1}\eta_1^{\dagger}M X_2 +{t_1t_2\over t_1 t_2 - 1}X_2 M_0^{-1}\epsilon_1^{\dagger}M_0.
c_{4}^{( 1 )}= - \frac{5 4 0 0 2 2 9 3 8 2 6 3 7 1 9 2 7 2 1 0 4 6 5 6 5 9 2 8 0 0 4 8 5}{6 7 3 6 7 3 1 3 5 2 3 2 4 6 5 3 9 4 6 3 0 2 3 2 9 2 8 9 4 4 1 2 8}+ \frac{1 4 1 9 6 8 1 9}{1 0 4 8 5 7 6 0}\ln \frac{1 1}{1 6}\approx - 1 . 3 0 8 9 1 \, .
\prod_{u = 1}^{N}d v_{u}^{2}d v_{u}^{2 *}= d \xi_{1}^{2}d \xi_{1}^{2 *}\cdot 2^{N - 1}( \xi_{2}^{2})^{2 N - 3}d \xi_{2}^{2}d^{2 N - 3}\hat{\Omega}_{2}\ .
\mathbb{Q}''_{\mathbb{I}}:=\{Q''\in\mathbb{Q}'': Q''\subseteq\{\xi^{2}_{1}<\xi^{2}_{2}\}\}, \,\,\,\,\,\,\,\,\, \mathbb{Q}''_{\mathbb{II}}:=\{Q''\in\mathbb{Q}'': Q''\subseteq\{\xi^{2}_{1}>\xi^{2}_{2}\}\}.
\begin{array}{cc}\lambda^{a\alpha}=i\textstyle{\frac{1}{6}}\bar{D}^{a\beta}{\rm h}^{\alpha}{}_{\beta}\,,~~~~ ~~~~\bar{\lambda}_{a\alpha}=-i\textstyle{\frac{1}{6}}D_{a\beta}{\rm h}^{\beta}{}_{\alpha}\,,\end{array}
J (x^i \frac{\partial}{\partial x^j}-x^j\frac{\partial}{\partial x^i}) =-\frac{1}{8\pi}\lim_{r\to \infty}r^2 \int_{S^2}\epsilon^{ij}_{\,\,\,l}\tilde{x}^l\tilde{\epsilon}^{bc}\partial_b\pi_{rc}\, d\mu_{S^2}.
A_{j}^{(q)}(z) :=\sum_{n=0}^{\infty}\frac{A_{[n,j]}^{(q)}}{z^{n+1}}=\sum_{n\in\mathbb{Z}}\frac{A_{[n,j]}^{(q)}}{z^{n+1}}B_{j}^{(q)}(z) :=\sum_{n=0}^{\infty}\frac{B_{[n,j]}^{(q)}}{z^{n+1}}=\sum_{n\in\mathbb{Z}}\frac{B_{[n,j]}^{(q)}}{z^{n+1}}.
h_{\frac{2}{3}}(a_1,a_2,a_3,a_4) = \left( \prod_{1 \leq i < j \leq 4}\frac{1}{a_i^{2/3}+a_i^{1/3}a_j^{1/3}+a_j^{2/3}}\right) \cdot s_{(8,4,2,0)}(a_1^{1/3},a_2^{1/3},a_3^{1/3},a_4^{1/3}).
\begin{aligned}\Delta R ={R_{{\rm{PL,upper}}}}-{R_{{\rm{QL,upper}}}}= \sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\frac{{{\sigma^2}+{\mathbb E}\left\{{{{\left|{\xi_k^\star}\right|}^2{p_k^\star}}}\right\}}}{{{\sigma^2}+{\mathbb E}\left\{{{{\left|{{\xi_k}}\right|}^2{p_k^\star}}}\right\}}}}.\end{aligned}
S =T \int d^{3}x \Phi [ -\frac{1}{2}\gamma^{ab}L^{\mu}_{a}L_{b \mu}- \frac{\varepsilon^{abc}}{\sqrt{-\gamma}}(L^{\mu}_{a}L^{\nu}_{b}L_{c \mu \nu}+ \frac{9}{10}L^{\mu}_{a}L^{\alpha}_{b}L_{c \mu \alpha}- \frac{1}{5}L^{\alpha}_{a}L^{\beta}_{b}L_{c \alpha \beta})]
\begin{aligned}r_n^2(t) =2(n+r_n(t))\alpha_n\alpha_{n-1}, r_n'(t) =2(n+r_n(t))(\alpha_{n-1}-\alpha_n), \frac{d}{dt}\ln D_n(t) =2tr_n(t)-2(n+r_n(t))(\alpha_n+\alpha_{n-1}), \frac{d^2}{dt^2}\ln D_n(t) =2r_n(t).\end{aligned}
g_{\infty}= \left(\begin{array}{cc}z\mathbf 1_2 0 0 z\mathbf 1_2\end{array}\right) \left(\begin{array}{cc}\mathbf 1_2 X 0 \mathbf 1_2\end{array}\right) \left(\begin{array}{cc}A 0 0{}^tA^{-1}\end{array}\right) \left(\begin{array}{cc}\alpha \beta -\beta \alpha\end{array}\right),
\sum_{k=1}^{m}k^{2H+1}|I_{k}|^{2(1-H)}\leq(C_{V,l}|I_{1}|)^{2(1-H)}\cdot\left(\sum_{k=1}^{m}k^{2H+1}(C_{V,l}|I_{1}|)^{2(1-H)(\alpha^{k-1}-1)}\right).
\mu = \lim_{k\to\infty}\frac{\|\theta_{k+1}- \theta^\star\|}{\|\theta_k - \theta^\star\|}&\leq \lim_{k\to\infty}\frac{c\cdot e^{-\gamma\times 0}\|\theta_k - \theta^\star\|}{\|\theta_k - \theta^\star\|}&= c,
A_{\mu}d x^{\mu}= \frac{1}{2 e}\left( \begin{array}{c c c c}{{3 \cos \theta d \phi}}&{{\omega_{1}\Theta}}&{{0}}&{{0}}\\{{\omega_{1}\bar{\Theta}}}&{{\cos \theta d \phi}}&{{\omega_{2}\Theta}}&{{0}}\\{{0}}&{{\omega_{2}\bar{\Theta}}}&{{- \cos \theta d \phi}}&{{\omega_{3}\Theta}}\\{{0}}&{{0}}&{{\omega_{3}\bar{\Theta}}}&{{- 3 \cos \theta d \phi \}}\\ \end{array}\right) \ ,
\mathcal{H}: \begin{cases}\dot{x}~~\in F(x) x\in \mathcal{F}_c:=\{x\in \mathcal{S}: (\hat{g},\hat{b}_a)\in \mathcal{F}_o\}x^+\in G(x) x\in \mathcal{J}_c :=\{x\in \mathcal{S}: (\hat{g},\hat{b}_a)\in \mathcal{J}_o\}\end{cases}
\frac{1}{a}\log (\int_0^t e^{a f(s)}ds ) \geq \frac{1}{a}\log(\int_{t_0}^{t_0+\delta}e^{a f(s)}ds)\geq \frac{1}{a}\log(\int_{t_0}^{t_0+\delta}e^{a (f(t_0)-\epsilon)}ds) = f(t_0) - \epsilon +\frac{1}{a}log(\delta).
\left( (q/p)^{(L+J)}u \right)^{1 - v -J\frac{\log(1/p)}{\log(p/q)}}\frac{(q/p)^{(R - v -J\frac{\log(1/p)}{\log(p/q)})(-J-L+1)}}{(-J-L+1)!}= \frac{((q/p)u)^{1 - v -J\frac{\log(1/p)}{\log(p/q)}}}{(-J-L+1)!}.
i(w^i,(w_1)_l) = \frac{l(C_1^l \cap w^i)\mathrm{sin}(\theta)}{\mathrm{height}(C_1^l)}i(w^j,(w_1)_l) = \frac{l(C_1^l \cap w^i)\mathrm{sin}(\theta_j')}{\mathrm{height}(C_1^l)}.
-y^2\left(\tfrac{\partial^2}{\partial x^2}+\tfrac{\partial^2}{\partial x^2}\right)\phi (\tau)=\tfrac{1-\nu_\infty^2}{4}\,\phi(\tau), &\phi(p\tau)+\sum_{j=0}^{p-1}\phi\left(\tfrac{\tau+j}{p}\right)=(p^{(\nu_p+1)/2}+p^{(-\nu_p+1)/2})\,\phi(\tau) .
K(x_{f},t_{f};x_{i},t_{i})=Qe^{\frac{iS_{c}}{\hbar}}\sqrt{\frac{\omega(t_{f}-t_{i})}{sin\omega(t_{f}-t_{i})}}=e^{\frac{iS_{c}}{\hbar}}\sqrt{\frac{m\omega}{2\pi i\hbar sin\omega(t_{f}-t_{i})}}.
\underline{{\underline{R}}}_{a 0}= - \frac 1{\ell^{2}}\underline{{\Theta}}_{a}\wedge \underline{{\Theta}}_{0}-{\frac{1}{r}}\left( d \underline{{\gamma}}_{a}+ \underline{{\omega}}_{a b}\wedge \underline{{\gamma}}^{b}\right) + O ( r^{- 3}) \qquad ,
P^{x_0(t)}\Big( R_t \leq \frac{\beta}{2}t + z(t) , \T_0 \leq \alpha t \Big) = P^{x_0(t)}\Big( \tilde{R}_{t - T_0}\leq \frac{\beta}{2}t + z(t) , \T_0 \leq \alpha t \Big)
\mathrm{d}\big(\mathrm{X}\mathrm{Y}\big)=\Big(\mathrm{X}\mathrm{d}\mathrm{Y}+ \big(\mathrm{d}\mathrm{X}\big)\mathrm{d}\mathrm{Y}+\big(\mathrm{d}\mathrm{X}\big) \mathrm{Y}\Big ) +\Big(\mathrm{X}\mathrm{d}\Lambda(\mathbf{L})\mathrm{d}\mathrm{Y}+ \big(\mathrm{d}\mathrm{X}\big)\mathrm{d}\Lambda(\mathbf{L})\mathrm{Y}\Big).
\tau^{j-1}\cdot n^{\l_j-2}\overset{\eqref{eq:taucheck}}{\geq}C_0^{\binom{\l_j}{2}-1}n^\frac{(\l_j-2)(k-\l_j)}{k+1}> n^\frac{k-3}{k+1}\overset{\eqref{n}}{\geq}k^{80k^4}R^{8k^2}\,,
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}\tilde{J}_{3}(t)g\right\|_{2}\le Ct^{1- \ell}(1+t)^{-n(\frac{1}{r_{1}}-\frac{1}{2})-(k-\tilde{k}_{3})}\| \nabla^{\tilde{k}_{3}}_{x}g \|_{r_{1}}& + C e^{-ct}t^{(1-\ell)-n(\frac{1}{r_{2}}-\frac{1}{2})-(k-\tilde{k}_{4})}\| \nabla^{\tilde{k}_{4}}_{x}g\|_{r_{2}},
\begin{aligned}0 \geq (1-\phi_u d)^2 F^{ii}U_{ii}^2 - \epsilon C_1 w^2 \sum_i F^{ii}- \frac{C}{\epsilon}w^{3/2}\sum_i F^{ii}& \geq \frac{w^2}{16nc_0^2}\sum_i F^{ii}- \epsilon C_1 w^2 \sum_i F^{ii}- \frac{C}{\epsilon}w^{3/2}\sum_i F^{ii}& \geq \frac{w^2}{32nc_0^2}\sum_i F^{ii}- C w^{3/2}\sum_i F^{ii},\end{aligned}
\sum_{\ell=4}^{n-1}\sum_{s=1}^{n-\ell}\left[\binom{n-s-1}{\ell-2}-(n-\ell-s+2)\right]=\sum_{\ell=4}^{n-1}\left[\binom{n-1}{\ell-1}-\binom{n-\ell+2}{2}\right] &=\sum_{\ell=3}^{n}\left[\binom{n-1}{\ell-1}-\binom{n-\ell+2}{2}\right]=2^{n-1}-n-\binom{n}{3}
\{x_{11}, x_{12}\}=\lambda x_{11}x_{12}, \{x_{11}, x_{21}\}=\lambda x_{11}x_{21}, \{x_{11}, x_{22}\}=2\lambda x_{12}x_{21}, \{x_{12}, x_{21}\}=0, \{x_{12}, x_{22}\}=\lambda x_{12}x_{22}, \{x_{21}, x_{22}\}=\lambda x_{21}x_{22}.
\rho_{1}= \Gamma_{1 1}\psi , \quad \rho_{3}={\frac{1}{6}}\psi^{3}+{\cal F}\Gamma_{1 1}\psi , \quad \rho_{5}={\frac{1}{1 2 0}}\Gamma_{1 1}\psi^{5}+{\frac{1}{6}}{\cal F}\psi^{3}+{\frac{1}{2}}{\cal F}^{2}\Gamma_{1 1}\psi , \quad \mathrm{e t c .},
\frac{T_{p + 2}}{\kappa}\int_{\mathcal{M}_{p + 3}}C_{p + 3}= \frac{T_{p + 2}}{\kappa}\int_{\mathcal{M}_{p + 3}}A_{p + 1}\wedge \omega_{2}= \frac{T_{p}}{2 \kappa}\int_{\mathcal{M}_{p + 1}}\frac{A_{p + 1}}{2 \pi^{2}\alpha^{\prime}}\, ,
\Pi_{\mu \nu}^{( I )}={\frac{e^{2}C^{( I )}}{2}}\, \int{\frac{d^{n}k}{( 2 \pi )^{n}}}\,{\frac{( 1 - \cos p \times k )}{k^{2}( k + p )^{2}}}\, N_{\mu \nu}^{( I )}; \; \; I = a , b , c
\sum_{\mu, \nu=1}^N\Big\| \|u_\mu\|_{W^{2,r}_x}\|u_\nu\|_{L_x^{r}}^{2p}\Big \|_{L^{q'}_{t>T}}\lesssim \sum^N_{\mu,\nu=1}\Big\| \|u_\mu\|_{W^{2,r}_x}\Big(\|u_\nu\|_{L_x^{r}}^{2p(1-\theta)}\|u_\nu\|_{L_x^{r}}^{2p\theta}\Big)\|_{L^{q'}_{t>T}}.
e^{-itH}D(\left( H+CI\right)^{\beta}) =D(\left( H+CI\right)^{\beta}) \left\Vert e^{-itH}\psi\right\Vert_{\left( H+CI\right)^{\beta}}=\left\Vert \psi\right\Vert_{\left( H+CI\right)^{\beta}}\forall~\psi\in D(\left( H+CI\right)^{\beta}).
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_I}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,s,k}R^d \eta^{-1}y\end{aligned}
\int_{\mathbb{R}_+}y(t,x,z)p(t,x,z)dx=\int_{\mathbb{R}_+}u(t,x,z)\Gamma(t,x)dx= &\int_{\mathbb{R}_+}\mathbb{E}_Q[y(T,x,z)U(x,z)\mathbb{E}_Q[\delta_Z(z)|\mathcal{F}^{G}_T]|\mathcal{F}^{G}_t]dx.
\int_a^b (T_\alpha^a f)(t) g(t) d_\alpha (t,a)=\int_a^b (T_\alpha^af)(t)~^{b}I_\alpha ~^{b}T_\alpha g(t) d_\alpha (t,a)+g(b) \int_a^b(T_\alpha^a f)(t)d_\alpha (t,a).
m \left(\left\{r>0 :\liminf\limits_{n \rightarrow \infty}(\inf\limits_{|z|=r}|z-z_n|^\frac{1}{n}) \leq (1-\epsilon)\right\}\right) \leq \lim\limits_{k\rightarrow \infty}\sum_{n=k}^{\infty}m\left(\left\{r:|z_n|-(1-\epsilon)^{n}\leq r \leq |z_n|+(1-\epsilon)^n\right\}\right) & \leq \lim\limits_{k \rightarrow \infty}\sum_{n=k}^{\infty}2(1-\epsilon)^n = 0
\sum_{0\leq j\leq n_{\texttt{c}}}| C_{\texttt{c}}(\boldsymbol{\xi}^{(1,n)}_{\texttt{c};\omega_{\texttt{c}; j}}) |^{-2}\Delta^{(1,n)}_{\texttt{c};\omega_{\texttt{c};j}}= \prod_{1\leq j\leq n}\frac{1-q}{1-q^j},
V_{p \Delta}= \int d^{4}a n_{+}^{- \Delta}( x + a , y ) e^{i p a}= y^{\Delta}e^{i p x}\int d^{4}a e^{i p a}\frac{1}{( a^{2}+ y^{2})^{\Delta}}= N ( \Delta ) p^{\Delta - 2}y^{2}K_{\Delta - 2}( p y ) e^{i p x}
c^{R'}(ij)=\begin{cases}L_r \textnormal{if}ij\in E(\tilde{W}_u) \textnormal{some}u\in [s] U_r \textnormal{if}ij\in E(\tilde{W_u},\tilde{W}_v) \textnormal{some}uv\in{[s]\choose 2}.\end{cases}
\sum_{l=0}^k s^l F_l \bigl( L(s;\alpha,\lambda_1), L'(s;\alpha,\lambda_1), \ldots, L^{(N-1)}(s;\alpha,\lambda_1) , ,\ldots, L(s;\alpha,\lambda_m), L'(s;\alpha,\lambda_m), \ldots, L^{(N-1)}(s;\alpha,\lambda_m) \bigr) \equiv 0.
\Delta(J_z^+)\subset J_z^+{}^\sharp U_z^0\otimes U^+_z+{}^\sharp U_z^{\geqq0}\otimes J_z^+, \Delta(J_z^-)\subset J_z^-\otimes{}^\sharp U^{\leqq0}_z+{}^\sharp U_z^{-}\otimes J_z^-{}^\sharp U_z^0, \varepsilon(J_z^\pm)= \{0\}, S(J_z^\pm)\subset J_z^\pm{}^\sharp U_z^0.
\begin{cases}\displaystyle a = \lim_{r \to 0}f_a(r) = \lim_{s\to\infty}s^{\frac{1}{\alpha}}w(s), \displaystyle L(a) = \lim_{r\to\infty}r^{\frac{2}{\alpha}}f_a(r) = \lim_{s\to 0}w(s) = w(0).\end{cases}
B = \frac{( n - 1 )^{2}}{4 \pi^{2}a^{4}}\frac{1}{4}\sum_{l = 1}^{\infty}\int_{0}^{x_{0}}x^{2}\frac{3 \nu^{9}- 4 2 x^{2}\nu^{7}+ 1 8 x^{4}\nu^{5}- 2 x^{6}\nu^{3}}{( x^{2}+ \nu^{2})^{6}},
\sum_{m=0}^{\infty}\sum_{\ell_1=0}^{\min\{n_1,m\}}\sum_{\ell_2=0}^{\min\{n_2,m\}}A_{m,\ell_1,\ell_2}=\sum_{\ell_1=0}^{n_1}\sum_{\ell_2=0}^{n_2}\sum_{m=\max\{\ell_1,\ell_2\}}^{\infty}A_{m,\ell_1,\ell_2}.
\langle T\phi, \psi\rangle = \langle K(x,y), \phi(y)\psi(x) \rangle = \langle K(x,y), \chi_0(y)\phi(y)\psi(x) \rangle = \langle K(x,y), \chi_0(y)[\phi(y)-\phi(x)]\psi(x) \rangle + \langle K(x,y), \chi_0(y)\phi(x)\psi(x) \rangle =: p+q,
\pi_s \big(\{x: \langle f, x \rangle \leq e^{- C_1 n}\}\big)= \int_{\mathbb S_+^{d-1}}(\mathbb{Q}_s^x)^{*n}\big( \langle f, X_n^x \rangle \leq e^{- C_1 n}\big) \pi_s(dx) \leq e^{- C_2 n},
\begin{array}{c c}{\gamma^{a}= \left( \begin{array}{c c}{0}&{\rho^{a}}\\{\bar{\rho}^{a}}&{0}\\ \end{array}\right) ,}&{\rho^{a}\bar{\rho}^{b}+ \rho^{b}\bar{\rho}^{a}= 2 \delta^{a b}.}\\ \end{array}
\mathbb{P}[S(\omega)<z \omega \le 0 \vert S(0)=x] =\mathbb{P}[\tilde{S}(\omega)<z \omega \ge 0 \vert \tilde{S}(0)=x] &=\mathbb{P}[\tilde{S}(\omega)<0 \omega \ge 0 \vert \tilde{S}(0)=x-z] &=\tilde{J}(0,x-z)
\psi(\mathbf{t}_{ij}(a))=t_{ij}(a), \psi(\mathbf{u}_{kl}(a))=u_{kl}(a), \psi(\mathbf{v}_{kl}(a))=v_{kl}(a), &\psi(\mathbf{w}_{kl}(a))=w_{kl}(a), \psi(\mathbf{f}_{i'k'}(a))=f_{i'k'}(a), \psi(\mathbf{g}_{k'i'}(a))=g_{k'i'}(a),
V_{j}^{- 1}V_{F}c^{F}\left( V_{0 F}^{- 1}V_{F}\Gamma{\cal P}^{- 1}\xi_{E}\right) V_{F}^{- 1}V_{j}= \left( \frac{d V_{0 j}^{- 1}V_{F}\Gamma{\cal P}^{- 1}\xi_{E}}{d V_{0 F}^{- 1}V_{F}\Gamma{\cal P}^{- 1}\xi_{E}}\right)^{- 1}c^{j}\left( V_{j}^{- 1}V_{F}\Gamma{\cal P}^{- 1}\xi_{E}\right) \ .
\epsilon=[\begin{matrix}\epsilon_{11}&\epsilon_{12}&\epsilon_{13}\\ \epsilon_{21}&\epsilon_{22}&\epsilon_{23}\\ \epsilon_{31}&\epsilon_{32}&\epsilon_{33}\end{matrix}];\sigma=[\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\end{matrix}]
0 \leq\int_{l_0}^L(1+l^2)(\cos(\alpha_1)\cos(\alpha_2))'+2l(\cos(\alpha_1)-\cos(\alpha_2))\,dl &=\int_{l_0}^L\frac{d}{dl}\Bigl((1+l^2)(\cos(\alpha_1)-\cos(\alpha_2))\Bigr)\,dl=\lim_{l\to l_0^+}(1+l^2)(\cos(\alpha_2(l))-\cos(\alpha_1(l))).
\leq \left\langle \prod_{b = 1}^{N}2 \cosh \left( - 2 \sum_{b = 1}^{N}\beta^{( b )}\int_{\Lambda}d^{2}x t ( x ) : \cos \Big [ 2 \sqrt{\pi}\varphi^{( b )}( x ) \Big ] :_{C^{1}}\right) \right\rangle_{C^{1}}.
H''_f(t) =-2\int_M (|\nabla v|^2+V_1v^2)\,dV_g +6t\int_M qv^2 \,dV_g & + 12t^2\int_M qvr_t\,dV_g+8t^3 \int_M qv\dot{r}_t\,dV_g+t^4\int_M qv\ddot{r}_t\,dV_g.
\mathbb{E}\sum_{k=1}^n(\tau_k-x)_+=\mathbb{E}\sum_{k=1}^n(m_k-F_n(x))_+(n/4)(2\ln n)^{\frac{1}{2}}=2\pi(n/4)(2\ln n)^{\frac{1}{2}}D_n(F_n(x)/2)\to(\pi/2)e^{c_0-x}.
\begin{array}{cccccccccc}\left\lceil \cfrac{n-2}{p}\right\rceil + 1 \leq v_{p, 2}(n) \leq \left\lceil \cfrac{n-2}{p}\right\rceil + 2 \textrm{for}n \not\equiv 2 \textrm{mod}p v_{p,2}(n) = \left\lceil \cfrac{n-2}{p}\right\rceil + 2 \textrm{for}n \equiv 2 \textrm{mod}p. \end{array}
\mathcal{L}_Z\theta_0=\frac{d}{dt}\big(\varphi^t_{(u,\eta)}\big)^*\theta_0|_{t=0}=\frac{d}{dt}\big({\rm id}^*+t\pi^*\circ(u,\eta)^*\big)\theta_0|_{t=0}=\frac{d}{dt}(\theta_0+t\pi^*\eta)|_{t=0}=\pi^*\eta,
\dot{V}\leq -\frac{\varepsilon}{2}\bar x^\top \bar x-2\sum_i(\Theta-l) ||d^\top P\hat x_{vi}||_1-\bar z^\top \bar z+2\sum\nolimits_{i=1}^N(\theta_i-\Theta)(\dot{\theta}_i-\Delta_i) &\leq -\frac{\varepsilon}{2}\bar x^\top \bar x-\bar z^\top \bar z\triangleq W(\bar x, \,\bar z)
\left(a;q\right)_{n}=\frac{\left(-aq^{-1/2}\right)^{n}}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}+inx\right)}{\left(q/a,-q^{1/2}/\left(ae^{ix}\right);q\right)_{\infty}}dx,
W_{(\alpha,\beta)}(n) = \sum_{\substack{{(l,k)\in\mathbb{N}_{0}^{2}}{\alpha\,l+\beta\,k=n}}}\sigma(l)\sigma(k) = \sum_{\substack{{(l,k)\in\mathbb{N}_{0}^{2}}{\delta\,\alpha_{1}\,l+\delta\,\beta_{1}\,k=n}}}\sigma(l)\sigma(k) = W_{(\alpha_{1},\beta_{1})}(\frac{n}{\delta}).
\rho\biggl(\prod_{\ell=1}^n \mathbf{D}_\ell\biggr) = \max_{\mathbf{x}>0}\min_i \frac{\displaystyle\biggl[\prod_{\ell=1}^n \mathbf{D}_\ell \mathbf{x}\biggr]_i}{[\mathbf{x}]_i}\geq \min_i \frac{\displaystyle\biggl[\prod_{\ell=1}^n \mathbf{D}_\ell \mathbf{v}_{\min}\biggr]_i}{[\mathbf{v}_{\min}]_i}\geq \rho_{\min}(\mathcal{D})^n\,.
\mathbb{E}(\|\hat{f}-f\|_2^2) \leq \sum_{j=1}^k\left\{2\varepsilon^2(B_0^2+\theta_j^2)+\frac{B^2}{n}\right\}+ \frac{r^2}{k^{2\beta}}\lesssim k\varepsilon^2B_0^2 +\varepsilon^2r^2 +\frac{B_0^2k^2}{n\alpha^2}+ \frac{r^2}{k^{2\beta}}\asymp k\varepsilon^2+\frac{k^2}{n\alpha^2}+ \frac{1}{k^{2\beta}},
T^{n,k}(x)=\Delta_{n,k}(a-x)+\delta_{n,k}^{1}+(1), T^{n,k}(x)=\Delta_{n,k}(a-x)+\delta_{n,k}^{1}+1-\Delta_{n,k}\int_{0}^{a-x}z \, f_{n,k}(z;0)dz+(1),
B_s^{(\bf p)}\,{\bf \overline{x}}=\prod_{k=1}^{s-2}\left( A - \gamma_1 \lambda_k I\right)\left( A - \gamma_1 \lambda_{s-1}I\right){\bf \overline{x}}=\prod_{k=1}^{s-2}\left( A - \gamma_1 \lambda_k I\right){\bf 0}={\bf 0}.
\mathcal{B - C}= \frac{(\xi L)^2}{2\pi^2}\sum_{j=1}^{\infty}\frac{A_j^2}{Z_j}\left[\psi^{(1)}\left(1+\frac{Z_j}{\pi}\right)+\frac{\pi^2}{2Z_j^2}- \frac{\pi}{Z_j}\right].
\varphi'_2(N) = \left(\frac{j (N-j+1)}{(2j+1) (N+2)}\right)^{1/2}\varphi'_+(N) - \left(\frac{(j+1) (N+j+2)}{(2j+1) (N+2)}\right)^{1/2}\varphi'_-(N).
\mathcal{L}[\beta] (k,2\pi i y) = \frac{1}{2}\textrm{P.V.}\int \frac{\widehat{\beta}^t(k,y-\tau)}{i \pi \tau}d\tau + \frac{1}{2}\widehat{\beta}^t(k,y) &=\frac{1}{2}\widehat{\beta}^t(k,y) - \frac{i}{2\pi}\int_{0}^{\infty}\frac{\widehat{\beta}^t(k,y-\tau)-\widehat{\beta}^t(k,y+\tau)}{\tau}d\tau.
g_\beta(x,t_1,t_2,\ldots,t_{d}) = \sum_{n_1=0}^\infty \cdots \sum_{n_d=0}^\infty \frac{t_1^{n_1}\cdots t_d^{n_d}}{n_1!\cdots n_d!}I_{\beta+ \sum_{i=1}^{d}(i+1)n_i}(2\sqrt{x}),
d(\tau,j)^{\delta}_{j}=\prod_{\tau\subseteq \sigma}(c(\sigma,j)^{\delta}_{j})^{(-1)^{|\sigma\setminus \tau|}}= \prod_{\tau\subseteq \sigma\subseteq\delta}(c(\sigma,j)^{\delta}_{j})^{(-1)^{|\sigma\setminus \tau|}}.
\int\limits_{0}^T\!\!\int\limits_{I}\big[d\tilde{u}\phi_{xx}+\kappa(\tilde{u})\phi_t\big]dxdt+\int\limits_{I}\kappa(\tilde{u}_0)\phi(0,x)dx+\int\limits_{0}^T\!\!\int\limits_{I}f(t,x,\tilde{u})\phi dxdt=0
{\rm ln}{\rm det '}\Delta = - \lim_{s \to 0}{{d}\over{ds}}\int_{\epsilon}^{\infty}{{dt t^{s-1}}\over{\Gamma(s)}}\int_{S}d^2 z{\sqrt{g}}[{\rm tr'}e^{-t(\Delta + m^2)}] - e^{-m^2 t},
g_\alpha(p) = (n-k-1)(n-2) p^{2n-k-5}&+ \left((n-k-1)(k-1) + 1 + (n-k-2)(n-k-3)\right) p^{n-3}(p-1) &+(n-k-2)p^{n-3}(p-1)^2 + (n-k-2)p^{n-2}(p-1) &+ (n-k-2)(n-k-3)p^{n-3}(p-1)(p-2).
{\cal S}_{C S}= \frac{1}{4 ( 2 \pi )^{3}}\int \mathrm{t r}_{q}\left( 2 \pi{\cal F}^{\prime}\wedge A^{( 2 )}+ \frac{1}{2}\, 2 \pi{\cal F}^{\prime}\wedge 2 \pi{\cal F}^{\prime}\wedge A^{( 0 )}\right) .
W^{(1)}-W^{(3)}= b_0y+(162b_0 + b_1)y^5+(11186b_0 + 116b_1 + b_2)y^9 &+(394498b_0 + 5081b_1 + 70b_2 + b_3)y^{13}&+(4628826b_0 + 65936b_1 + 1092b_2 + 24b_3 + b_4)y^{17}&+(-226397710b_0 - 2983519b_1 - 43758b_2- 781b_3 - 22b_4)y^{21}&+ \cdots.
q_c+q_{c+1}+\cdots+q_{d-1}=\sum_{\{i,c\}\in E(\mathsf{P}_{n+2}) \atop b\leq i<c}x_{ic}-\sum_{\{j,d\}\in E(\mathsf{P}_{n+2}) \atop c\leq j<d}x_{jd}
H^{2}( z ) = \left[ \frac{z^{4}/ \Lambda^{4}+ b^{4}}{z^{4}/ \Lambda^{4}- b^{4}}-{\frac{1}{\ln ( z / \Lambda )}}+{\frac{2 K}{\ln ( z / \Lambda )}}\left( \frac{z^{2}}{z^{4}- b^{4}\Lambda^{4}}\right) \right] .
\mathbb{E}[x^V;D^{(1)}_3]=\frac{1}{3}\sum_{k=0}^{\infty}\mathbb{P}(V'_b=k)\Big{(}\frac{1}{2}\Big{)}^k x^k=\frac{1}{3}\mathcal{H}[f]\Big{(}\frac{x}{2}\Big{)}
M_{bca}^{(s,t,r) \d}\equiv \hat{G}_{ab}\delta_c^d \delta_{r+s,0\hspace{.05in}mod\hspace{.04in}\lambda}+ \hat{G}_{ac}\delta_b^d \delta_{r+t,0\hspace{.05in}mod\hspace{.04in}\lambda}-f_{ab}^{\ \e}f_{ec}^{\ \d}.
L_{f}^m = q^{1/2}p_{\mu}\dot x^{\mu}+ \frac{i}{2}(\psi_{\mu}\;\dot \psi^{\mu}- \psi_{5}\dot \psi_{5})- \; \frac{e}{1+ q^2}(p^2 - m^2)+ i \chi (\psi_{\mu}p^\mu - \psi_{5}m ),
F(x,\sigma ) = \i\kappa [(\alpha_A\beta^A + x\alpha_A\acute{\alpha}^A )\bar{\alpha}^{B'}\bar{w}_{B'}(x,\sigma)- \alpha^A w_{A}(x,\sigma)(\bar{\alpha}_{B'}\bar{\beta}^{B'}+ x\bar{\alpha}_{B'}\acute{\bar{\alpha}}^{B'})].
\begin{cases}\,\,\eta_\epsilon\in H^1_0(\Omega)\,\,\, begin{align*}0.3cm] \int_{\Omega_\epsilon}\nabla \eta_\epsilon\nabla vdx-\int_{\Omega}\lambda_\epsilon \eta_\epsilon vdx=0\,\,\forall\,v\in H^1_0(\Omega) \end{cases}
{\rm HC}((A, M) / (R, P)) :={\rm HH}((A, M) / (R, P))_{hS^1},{\rm HC}^-((A, M) / (R, P)) :={\rm HH}((A, M) / (R, P))^{hS^1},{\rm HP}((A, M) / (R, P)) :={\rm HH}((A, M) / (R, P))^{tS^1},
{\widehat \gamma}^{1}= \left( \begin{array}{c c}{{\widehat E}_{2}}{0}{0}{-{\widehat E}_{2}}\end{array}\right) ,{\widehat \gamma}^{2}= \left( \begin{array}{c c}{0}{\sigma_{x}}{- \sigma_{x}}{0}\end{array}\right) ,
\frac{\partial}{\partial u_{j}^{( n )}}\Gamma = - ( \Theta \lambda^{n}E_{j j}{\Theta}^{- 1})_{-}\Gamma + \Theta ({\bf u}, \lambda ) \lambda^{n}E_{j j}D ({\bf u}, \lambda ) = ( \Theta \lambda^{n}E_{j j}{\Theta}^{- 1})_{+}\Gamma
\widetilde{b}_l(z) = \int_{\Sigma_z}(\mathcal{L}\xi_l)(x)\, d\mu_z(x) = \mathbf{E}_\mu\bigg[ (\mathcal{L}\xi_l)(x)\,\Big|\,\xi(x) = z\bigg] \,, 1 \le l \le m \,, \widetilde{\sigma}(z) = \bigg[\int_{\Sigma_z}\big(\nabla\xi a\nabla\xi^T\big)(x) \,d\mu_z(x)\bigg]^{\frac{1}{2}}\,,
\frac{\partial r_{0}}{\partial a_j}= \begin{cases}\displaystyle\frac{(r_{p_0}+a_0-a_{p_0-1})(a_{0}-r_{p_0})}{(r_0-a_{p_0-1})a_{0}}, begin{align*}5mm] \displaystyle\frac{a_j-a_{p_0-1}}{r_0-a_{p_0-1}}, \end{cases}
\frac{d s^{2}}{\sqrt{g_{s}}l_{s}}= \sqrt{\frac{u}{\sqrt{N}l_{s}}}\left[ - \left( 1 - \frac{u_{0}^{2}}{u^{2}}\right) d t^{2}+ \sum_{i = 1}^{5}( d y^{i})^{2}+ N l_{s}^{2}\left( \frac{d u^{2}}{u^{2}- u_{0}^{2}}+ d \Omega_{3}^{2}\right) \right] \ ,
\sum_{j=1}^{n+1}\| (\phi(x))^{4-t_j}\widetilde{U^j}\|_{C^{t_j, \alpha}(B_{\frac{1}{2}}(0))}&\le C\left( \sum_{j=1}^{n+1}\| (\phi(x))^{4+s_j}\widetilde{L_j U}\|_{C^{-s_j, \alpha}(B_1(0))}+ \sum_{j=1}^{n+1}\| (\phi(x))^{4-t_j}\widetilde{U^j}\|_{L^2(B_1(0))}\right),
e_{\mu}^{a}( x ) = - \frac{4 \pi G}{\lambda}p^{a}( \xi ) \epsilon_{\mu \nu}{\frac{\dot{X}^{\nu}( \xi )}{\dot{X}^{0}( \xi )}}\delta ( \sigma - X^{1}( \xi ) ) = - \frac{2 \pi G}{\lambda}p^{a}( \xi ) \partial_{\mu}\epsilon ( \sigma - X^{1}( \xi ) )
=[\sum_{i=1}^{N}2\frac{x_{i2}-x_{i1}}{2}\frac{x_{i2}-x_{i1}}{2}^{\prime}]^{-1}[\sum_{i=1}^{N}2\frac{x_{i2}-x_{i1}}{2}\frac{y_{i2}-y_{i1}}{2}]
\renewcommand{\arraystretch}{1}\sum_{\begin{array}{c}\scriptstyle\sigma_1,\ldots,\sigma_{L-1}=0,1 \scriptstyle\sigma_j\sigma_{j+1}=0\end{array}}q^{\;\; \displaystyle \sum_{k=1}^{L-1}k\sigma_k}\qquad\qquad \sigma_0=a, \; \sigma_L=0,
\left({\bf B}_{\mathcal{I}}^{t}\mathbf{\Sigma}_{\mathbf{\hat{z}}}^{-1}{\bf B}_{\mathcal{I}}\right)^{-1}-\frac{M{\bf I}_{JS}}{tr\left(\mathbf{\Sigma}_{\mathbf{\hat{z}}}^{-1}\right)}\xrightarrow[M\to+\infty]{a.s.}0.
\begin{aligned}\zeta_n(x)-\int_M \zeta_n\mathrm{d}\mu =\int_MG(y,x)f_n^*(y)\mathrm{d}\mu(y) &=\sum_{j=1}^m A_{n,j}G(x_{n,j}^{(1)},x)+\sum_{j=1}^m\int_{M_j}(G(y,x)-G(x_{n,j}^{(1)},x))f_n^*(y)\mathrm{d}\mu(y).\end{aligned}
{\cal H}_{3}= | \nabla \psi_{L}|^{2}+ | \nabla \psi_{R}|^{2}| + | \nabla \psi_{L}| | \nabla \psi_{R}| + m_{0}^{2}( | \psi_{L}|^{2}+ | \psi_{R}|^{2}) + g_{1}( | \psi_{L}|^{4}+ | \psi_{R}|^{4}) + g_{2}| \psi_{L}|^{2}| \psi_{R}|^{2}\, ,
\lVert \Delta^{p,+}_{a,b,\lambda,\mu}(h_p^- - h_q^-) \rVert +\lVert \Delta^{p,+}_{a,b,\lambda,\mu}(h_q^- - h_s^-) \rVert < \epsilon_q - \epsilon_p + \epsilon_s - \epsilon_q = \epsilon_s - \epsilon_p,
\lambda_{a|a|e|f}^{e|e|a|a}+ \lambda_{a|a|e|f}^{e|f|a|a}\lambda_{c|c|e|f}^{e|e|c|c}= \lambda_{c|c|e|f}^{e|e|c|c}+ \lambda_{c|c|e|f}^{e|f|c|c}\lambda_{a|a|e|f}^{e|e|a|a}.
{\cal L}= - \pi_{i}^{T}\dot{A}_{i}^{T}- \frac{1}{2}{\pi_{i}^{T}}^{2}- \frac{1}{4}F_{i j}^{2}( A^{T}) + \frac{1}{2}j_{0}\frac{1}{\partial^{2}}j_{0}+ j_{i}A_{i}^{T}+{\cal L}_{M}
\sum_{\Delta}\left( \frac{\sum_{\gamma=1}^{n-2}\det\Big( X_{i_1}(\xi), \ldots, \widetilde{X}_{i_\gamma}(\xi), \ldots, X_{i_{n-2}}(\xi), n_j,\xi \Big)}{(n-2)! \sqrt{1 - \langle n_j,\xi\rangle^2}}\right)= 0 ,
\rho ( x - y ) \; : = \; \lim_{\mu \rightarrow 0}\prod_{I = 1}^{N}\left\langle : e^{- i 2 \sqrt{\pi}\omega^{( I )}U_{I a}\Phi^{( I )}( x )}:_{M^{( I )}}: e^{+ i 2 \sqrt{\pi}\omega^{( I )}U_{I a}\Phi^{( I )}( y )}:_{M^{( I )}}\right\rangle_{K_{I I}^{\mu}}
\sum_{n=0}^{\infty}\frac{1}{(n+\frac{1}{2})(n-\frac{1}{2})}x^{n}=\frac{2(xtanh^{-1}(\sqrt{x})-tanh^{-1}(\sqrt{x})-\sqrt{x})}{\sqrt{x}}=-4+\frac{4x}{3}+\frac{4x^{2}}{15}+O(x^{3})
\lim_{x \rightarrow \infty}\prod_{i=1}^{\ell+L+1}\b{x}_i^{p_i}\times\mathcal{C}^{\lambda}_{\mu\nu}(x)={\rm Coeff}\left[\prod_{1 \leq i<j \leq \ell}(z_i-z_j)s_{\lambda/\mu}(z_1,\dots,z_\ell),\prod_{i=1}^{\ell}z_i^{\ell+\nu_i-i}\right]=c^{\lambda}_{\mu\nu},
|Ef(x)| \le \Big|\sum_{\tau \in \acute I}Ef_{\tau}(x) \Big| + \Big|\sum_{\tau \in \acute I^c}Ef_{\tau}(x) \Big| &\le \Big|\sum_{\tau \in \acute I}Ef_{\tau}(x) \Big| + 16\max_{L=L_{\parallel e_1}L_{\parallel e_2}}\bigg| \sum_{\tau:\bar\tau \subset L}Ef_{\tau}(x) \bigg| &\le |Ef_{\acute I}(x)| + 16 \alpha |Ef(x)|.