harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widetilde{\mu}(s) = \mathbb{I}_{\left[0,\frac{2\pi}{K-\sqrt{K}}\right]}(x) \widehat{\mu}(x) + \mathbb{I}_{\left(\frac{2\pi}{K-\sqrt{K}},\frac{2\pi}{\sqrt{K}}\right)}(x) + \mathbb{I}_{\left[\frac{2\pi}{\sqrt{K}},l_K\right]}(x) \left(1 - \widehat{\mu}\left(x - \frac{2\pi}{\sqrt{K}}\right)\right).
\frac{(q^{d+r-(d-2j-1)n};q^d)_k}{(q^{d-(d-2j)n};q^d)_k}=\frac{(q^{d-(d-2j)n+dk};q^d)_{(n+r)/d}}{(q^{d-(d-2j)n};q^d)_{(n+r)/d}},
t^{1-\alpha}\int_{0}^{t}(t-\tau )^{\alpha -1}||| E_{\alpha ,\alpha}((t-\tau )^{\alpha}A)||| \tau^{\alpha -1}d\tau \leq C_{1}t^{1-\alpha}\int_{0}^{t}(t-\tau )^{\alpha -1}\tau^{\alpha -1}d\tau & =C_{1}t^{1-\alpha}t^{2\alpha -1}B (\alpha ,\alpha ) & =C_{1}t^{\alpha}B (\alpha ,\alpha ) & \leq C_{1}T^{\alpha}B (\alpha ,\alpha ),
\widehat{G}(m,N,b,\varepsilon, s) \leq 2 \sum_{k=1}^{e^{(\log (m(1+\varepsilon)/2)^{1/b}}}k^{-ds}\sum_{j=0}^2 \widehat{G}((m-e^{(\log k)^b})(1+\epsilon)^j, N-1, b, \varepsilon, s).
{\cal W}^{( 1 )}= \frac{1}{2 \,{\cal Z}^{( 1 )}}\sum_{n_{1}\neq n_{2}}\frac{( n_{1}- n_{2}- 1 )}{( n_{1}- n_{2})}\exp \left[ - \frac{g^{2}A}{2}( n_{1}^{2}+ n_{2}^{2}) + \frac{g^{2}A}{2}( n_{1}+ n_{2}- \frac{1}{2}) - g^{2}A_{1}n_{1}\right] .
F''(s)+F'(s)\left[\frac{(2k_4+k_1)-sk_3(2k_4+2k_5+\frac{k_2}{k_3})}{s(1-k_3s)}\right] & -\left[\frac{2k_3k_4(k_4-1)+k_5k_3(2k_4+k_1)+k_4(k_1k_3+k_2)-B}{s(1-k_3s)}\right]F(s)=0,
S_{D}= - \int d^{4}\sigma \sqrt{- \mathrm{d e t}\Big ( G_{\mu \nu}+ \frac{1}{\sqrt{1 + C_{0}^{2}}}( \tilde{F}_{\mu \nu}+ C_{\mu \nu}+ C_{0}b_{\mu \nu}) \Big )}+ \int \Omega_{D},
\hat{H}_{I}^{( q )}( \hat{x}, \hat{p}) = \frac{f^{2}}{6}\bigl \{\hat{x}^{3}V^{\prime}( \hat{x}) \partial_{\hat{x}}^{2}+ \bigl [ \hat{x}^{3}V^{'^{\prime}}( \hat{x}) + 3 \hat{x}^{2}V^{'}( \hat{x}) \bigr ] \partial_{\hat{x}}+{\textstyle \frac{1}{3}}\hat{x}^{3}V^{'^{\prime \prime}}( \hat{x}) +{\textstyle \frac{3}{2}}\hat{x}^{2}V^{'^{\prime}}( \hat{x}) \bigr \}.
\textbf{Q}= \textbf{q}+h \sum_{i=1}^s b_i \textbf{k}_i (\textbf{q}) \textbf{P}= \left( \textbf{p}- h \sum_{i=1}^s b_i \nabla l_i(\textbf{q}) \right) \left(\textbf{I}+h\sum_{i=1}^{s}b_i \textbf{k}'_i(\textbf{q})\right)^{-1}
\tilde{I}_{1}^{\zeta}( t ) = I_{1}^{\zeta}( t ) - \left[ \left( a ( t ) \Lambda \vphantom{^2}\right)^{2}- M^{2}( t ) \ln \left( a ( t ) \Lambda \vphantom{^2}\right) \right] / ( 8 \pi^{2}) \, .
Q ( \xi ) = \frac{1}{1 6 \pi G}\int d \varphi \left\{\xi^{\perp}\left[ \frac{l}{r}+ \frac{r^{3}}{l^{3}}\left( g_{r r}- \frac{l^{2}}{r^{2}}\right) \right] + \frac{1}{l r}\left( \xi^{\perp}+ r \xi^{\perp},_{r}\right) \left[ g_{\varphi \varphi}- r^{2}\right] + 2 \xi^{\varphi}\pi_{\varphi}^{r}\right\}.
\lambda_n (\sum\limits_{j=1}^{p}\vert \beta_{0,j}+ z_{j}/ \sqrt{n}\vert - \vert \beta_{0,j}\vert ) \rightarrow\lambda_{0}\sum\limits_{i=1}^{p}( z_{j}sgn(\beta_{0,j}) I(\beta_{0,j}\neq0) + \vert z_{j}\vert I(\beta_{0,j}= 0)),
{\left|A(e^{i\theta})g_0(\theta)+\sum\limits_{j=0}^{\infty}((\bold{B}^0)^{-1}\bold{R}^0\bold{a})_je^{ij\theta}\right|^2}=\gamma_1{(f_0(\theta)+g_0(\theta))^2}\left(f_0(\theta)\right)^{\beta - 1},
\int_{\substack{0 \le \vert y\vert <\infty \arg y = \pm2\pi/3}}\left\vert y\sup_{\left(y, \frac{(n+1)^2}{n^2}y\right)}\frac{{\rm d}}{{\rm d}x}\left( \begin{matrix}-\Psi_{12}\Psi_{22}\left(x\right) \Psi_{12}^2(x) -\Psi_{22}^2(x) \Psi_{12}\Psi_{22}(x)\end{matrix}\right)\right\vert \vert{\rm d}y\vert = O(1),
D_{i,i+j}(\xi_{1}^{i})=\P_{\bf U}\left(\tilde\xi^{(0)}_j\neq\tilde\xi^{(\ell(\xi_{1}^{i}))}_j\right)\leq \P_{\bf U}(\tau_{0,\ell(\xi_{1}^{i})}\ge j) \le \sup_{\ell=1,\ldots,i}\P_{\bf U}(\tau_{0,\ell}\ge j)
\gamma(\zeta) = \frac{(p^2;p^2)_\infty}{(p;p)_\infty^2}\frac{\Theta_{p^2}(q^2)\Theta_{p^2}(p\zeta^2)}{\Theta_{p^2}(q^2\zeta^2)}\frac{g(\zeta^2)}{\mbox{snh}(\lambda)},
c_{\hat X}[k] = \Bigg(\sqrt{\left(\frac{\alpha_r[k]-\alpha_e[k]}{2\alpha_r[k]\alpha_e[k]}\right)^2 + \frac{\alpha_r[k]-\alpha_e[k]}{\mu \cdot \alpha_r[k]\alpha_e[k]}}-\frac{\alpha_r[k]+\alpha_e[k]}{2\alpha_r[k]\alpha_e[k]}\Bigg)^+,
H(Y_1^L) = - \sum_{y_1^L \in \mathcal{Y}^L}P_{Y_1^L}(y_1^L) \log P_{Y_1^L}(y_1^L) &= - \sum_{Q \in \mathcal{Q}_L}\sum_{y_1^L \in \mathcal{T}_{Q}^{L}}P_{Y_1^L}(y_1^L) \log P_{Y_1^L}(y_1^L) &= \sum_{Q \in \mathcal{Q}_L}|\mathcal{T}_{Q}^{L}| P_{Y_1^L}(y_1^L) \log \frac{1}{P_{Y_1^L}(y_1^L)},
\begin{array}{ll}\delta F^{\nu}_{,z_j}=F^{\nu}=\sum_{i=1}^dv_iF^{\nu}_{,z_j}\ast G_{\nu,i}+\sum_{j=1}^{2d}W^S_j(F^{\nu},F^{\nu})\ast G_{\nu,z_j}.\end{array}
K[H_n,1] = \left( \frac{m \omega \sqrt{\xi \xi_0}}{i \hbar \sin\omega t}\right) \left( \frac{m \omega \sqrt{\eta \eta_0}}{i\hbar \sin \omega t}\right) J_n \left( \frac{m \omega \sqrt{\xi \xi_0}}{i \hbar \sin \omega t}\right) J_n \left( \frac{m \omega \sqrt{\eta \eta_0}}{i\hbar \sin \omega t}\right)
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 2^2\cdot 3(2^{n_{2,\nu_2}}-1).
\protect S = S_{G}+ \lambda S_{E}= \sum_{i , j , \mu}C_{i j}( X_{i}^{\mu}- X_{j}^{\mu})^{2}+ \lambda \sum_{\hat{k}, \hat{l}, \mu}C^{\hat{k}\hat{l}}( 1 - n_{\hat{k}}^{\mu}\cdot n_{\hat{l}}^{\mu}) .
\log{\sin{\frac{\pi}{2}(s+\kappa)}}:=\begin{cases}\displaystyle \frac{(1-s-\kappa)\pi}{2}i - \log{2}- \sum_{n=1}^\infty \frac{e^{\pi i(s+\kappa)n}}{n}, t>1; \displaystyle \frac{(s+\kappa-1)\pi}{2}i - \log{2}- \sum_{n=1}^\infty \frac{e^{-\pi i(s+\kappa)n}}{n}, t<-1.\end{cases}
g_1 = G_{0}- G_{\infty}= \frac{1}{3}+ q_N + 3q_N^2 + 4q_N^3 + 7q_N^4 + 6q_N^5+12q_N^6 + 8q_N^7+15q_N^8+\cdots, g_2 = G_{1}- G_{\infty}= (1-\zeta_N)q_N + 3(2 + \zeta_N)q_N^2 + 7(1-\zeta_N)q_N^4+6(2+\zeta_N)q_N^5 + \cdots, g_3 = G_{2}- G_{\infty}= (2+\zeta_N)q_N + 3(1 - \zeta_N)q_N^2 + 7(2+\zeta_N)q_N^4+6(1-\zeta_N)q_N^5 + \cdots,
\vec{p}^{\prime}(t)\cdot(\vec{p}(t)-\vec{f}_{0})=(\vec{f}_{1}-\vec{f}_{2}\frac{1}{t^{2}})\cdot(\vec{f}_{1}t+\vec{f}_{2}\frac{1}{t})=\vec{f}_{1}^{2}t-\vec{f}_{2}^{2}\frac{1}{t^{3}}=0
\dim_{\mathbb{C}}({\rm Sp}\Sigma_{\{i,j\}})=\begin{cases}\dim_{\mathbb{C}}(\Sigma_{\{p(i),p(j)\}}), p(i)+p(j)<2n+1 \dim_{\mathbb{C}}(\Sigma_{\{p(i),p(j)\}})-1, p(i)+p(j)>2n+1 \end{cases}
C_0 = \frac{1}{2}(J^+_0J^-_0 + J^-_0J^+_0 + 2 J^0_0 J^0_0) +\sum_{m=1}^{+\infty}J^+_{-m}J^-_m + J^-_{-m}J^+_m +2 J^0_{-m}J^0_m
\sum_{\ell = 2}^L N_\ell = \sum_{\ell=1}^L N_\ell - 1 = N(\Phi) - d - 1 > M(\Phi) = \sum_{\ell=2}^L (\|A_\ell\|_{\ell^0}+ \|b_\ell\|_{\ell^0}),
e_{B+a}x^h-x^he_{B+a}-\frac{\partial f^a}{\partial x^h}=0, a=1,\ldots,k e_\alpha x^h-x^he_\alpha-x^\mu\tau^{\mu h}(e_\alpha)=0, \alpha=1,\ldots,B t^\alpha_\ell e_\alpha=0, \ell=1,\ldots,B+k-n e_\alpha e_\beta-e_\beta e_\alpha-C^\gamma_{\alpha\beta}e_\gamma=0, e_\alpha\xi^i-\xi^ie_\alpha=0
\sum_{n\ge0}\frac{(q^2;q^2)_n(-1)^nq^{n^2}}{(-q)_{2n}}=\sum_{n\ge0}q^{2n^2}(1-q^{4n+2})\sum_{i\ge0}q^{i(i+1)/2}\sum_{2|j|\le i}(-1)^jq^{-j(j-1)+2nj},
a(w, z; \alpha) = \displaystyle \int_\Omega \mu \nabla w \cdot \nabla z \, d\Omega+ \int_\Omega \big({\bf b}\cdot \nabla w + \sigma w \big) z \, d\Omega, f(z; \alpha) = \displaystyle \int_\Omega fz \, d\Omega,
\int_{B_{N}}\prod_{i=1}^{2}(\sum_{\theta \in \mathcal{P}_{N^{-1}}}\|f_{i}*\Xi_{\theta}\|_{L^2(B_{cN^{{1}/{2}}}(x+c_{i}))}^2)^{\frac{p}{4}}\,dx \lesssim N^{-\frac{3p}{4}}\int_{B_{N}}\prod_{i=1}^{2}(\sum_{\theta \in \mathcal{P}_{N^{-1}}}|\tilde{f}_{i,\theta}|^2*\chi_{\theta_{0}^{*}}(x))^{\frac{p}{4}}\,dx.
\mathcal{C}=2\pi\lambda_{b}\int_{0}^{\infty}x_{11}\mathrm{e}^{-\left(\pi\lambda_{b}x_{11}^2+ \frac{x_{11}^{\alpha}\Theta N}{P}+ \pi\lambda_{b}x_{11}^2 \Theta^{\frac{2}{\alpha}}\int\displaylimits_{\Theta^{\frac{-2}{\alpha}}}^{\infty}\frac{1}{1+z^{\alpha/2}}\mathrm{d}z \right)}\mathrm{d}x_{11},
a_1=\frac{\mu t_1-1}{\left(L+\mu-\mu_p\right)t_1-2}, \ \ \ \ a_2=\frac{1-L t_1}{\left(L+\mu-\mu_p\right)t_1-2}a_3=-\frac{\left((Lt_1-2)(\mu t_1-2)-1\right)\mu_pt_1}{\left(L+\mu-\mu_p\right)t_1-2}\ \ \ \ a_4=-\frac{\mu_p t_1}{\left(L+\mu-\mu_p\right)t_1-2}.
\begin{cases}c_{p,q}^k=0, \forall q\in \{1,\cdots, k-1\}, p\in \{1,\cdots, q\}c_{p,q}^k=0, \forall q\in \{k+1,\cdots, n\}, p\in \{1,\cdots, k-1\}c_{p,q}^k=0, \forall q\in \{k+1,\cdots, n\}, p\in \{k+1,\cdots, q\},\end{cases}
\gamma_{\psi(z)}\circ \rho_L(g^{-1})(\theta) =\gamma_{\psi(z)}(T_g^*(\phi_{g^{-1}}\theta)) &=(T_g^*\phi_{g^{-1}}(z)T_g^*(\theta)(z)=\phi_{g}^{-1}(z)T_g^*(\theta)(z) &=\phi_{g}(z)^{-1}\theta(T_gz)=\phi_{g}(z)^{-1}\gamma_{\psi(T_gz)}(\theta),
B=\frac{l_0}{l_W}\sin \vartheta_WA^0+ \frac{l_1}{l_W}\cos \vartheta_W A^1,\;\;Z=-\frac{l_1}{l_W}\cos \vartheta_W A^0+\frac{l_0}{l_W}\sin \vartheta_W A^1,\;\;W^{+}=A^2,\;\;W^{-}=A^3.
\begin{array}{l}\lambda_0 = 2e^{J_1}\cosh J + 2 e^{-J_1}\cos \varepsilon \lambda_2 = 2e^{J_1}\cosh J - 2 e^{-J_1}\cos \varepsilon \lambda_1 = 2e^{J_1}\sinh J + 2 e^{-J_1}\sin \varepsilon \lambda_3 = 2e^{J_1}\sinh J - 2 e^{-J_1}\sin \varepsilon \;.\end{array}
deg_{LS}(I-(S+QN_{f}+KS) ,\Omega,0) & = deg_{B}\left(I-(S+QN_{f}+KS)\left|_{\overline{\Omega \cap \mathbb{R}^{2}}}\right. ,\Omega\cap \mathbb{R}^{2}, 0\right) &=deg_{B}(G,\Omega\cap \mathbb{R}^{2},0)\neq 0.
(\frac{\partial T}{\partial y})_{x}(\frac{\partial S}{\partial x})_{y}-(\frac{\partial P}{\partial y})_{x}(\frac{\partial V}{\partial x})_{y}=(\frac{\partial T}{\partial x})_{y}(\frac{\partial S}{\partial y})_{x}-(\frac{\partial P}{\partial x})_{y}(\frac{\partial V}{\partial y})_{x}
\left\vert W^{int}[\mathcal{T}_{n}]\right\vert \geq W(G_{1}(n,d_{1}^{\max},1,1))-W(G_{2}(n,d_{2}^{\min},x_{2}^{\max},-1))= & =\frac{n^{3}}{6}-\frac{\sqrt{n^{5}-n^{4}}}{\sqrt{2}}-3n^{2}+\frac{10}{3}\sqrt{2n^{3}-8n^{2}}+\frac{143n}{6}+ & +25\sqrt{2n-8}-25.
A := \begin{pmatrix}3 -1 1 1 -1 -1 2 0 0 2 1 0 2 1 1 1 0 1 3 0 -1 2 1 0 4\end{pmatrix},B := \begin{pmatrix}0 1 1 0 -1 1 1 0 0 1\end{pmatrix}, C := \begin{pmatrix}1 0 0 0 0 0 0 1 0 1\end{pmatrix}.
\int_{0}^{\infty}\frac{1}{\nu}(\frac{1}{2}e^{-\frac{|z|}{\nu}})(\frac{\alpha}{\Gamma(\frac{1}{\alpha})}\frac{1}{\nu}L_{\alpha}(\frac{1}{\nu}))d\nu=\frac{1}{2}\frac{\alpha}{\Gamma(\frac{1}{\alpha})}e^{-|z|^{\alpha}}
\sum_{k_3\in \mathbb{N}}\left(\frac{k_3\pi}{2h}\hat{u}_3(k)+\pi [\hat{u}_1(k)\frac{k_1}{\Pi_1}+\hat{u}_2(k)\frac{k_2}{\Pi_2}] \right)e^{i\frac{\pi k_3x_3}{h}}+ \sum_{k_3 \in \mathbb{N}}\left(\frac{k_3\pi}{2h}\hat{u}_3(k)-\pi [\hat{u}_1(k)\frac{k_1}{\Pi_1}+\hat{u}_2(k)\frac{k_2}{\Pi_2}] \right)e^{-i\frac{\pi k_3x_3}{h}}=0,
L_{\infty,k+1}\circ (i\circ p)^{\vee (k+1)}=\left(\sum_{\ell = 2}^{k+1}Q_{A,\ell}^1 \circ P^\ell_{k+1}- \sum_{\ell =1}^{k}P_\ell^1 \circ Q^\ell_{B,k+1}\right) \circ (i\circ p)^{\vee (k+1)}& =(Q_A)^1_{k+1}\circ p^{\vee (k+1)}- (Q_A)^1_{k+1}\circ p^{\vee(k+1)}=0.
R_{i t i t}= \frac{- a a_{t t}}{( 1 + \frac{K}{4}\vec{x}\vec{x})^{2}}, R_{i j i j}= \frac{a^{2}( K + a_{t}^{2})}{( 1 + \frac{K}{4}\vec{x}\vec{x})^{4}}; i \neq j
M(\xi)=-(\xi_1^2+\xi_2^2+\cdots+\xi_d^2)\begin{pmatrix}1 0 0 \epsilon\end{pmatrix}+i\xi_1 c\begin{pmatrix}1 0 0 1\end{pmatrix}+\begin{pmatrix}0 e^{-\kappa}0 -\kappa e^{-\kappa}\end{pmatrix},
\sum_{n=1}^{\infty}\Big((1)_{p,\lambda}+(2)_{p,\lambda}+\cdots+(n)_{p,\lambda}\Big)x^{n}=\sum_{n=1}^{\infty}x^{n}\sum_{k=1}^{n}(k)_{p,\lambda}&=\sum_{k=1}^{\infty}(k)_{p,\lambda}\sum_{n=k}^{\infty}x^{n}=\frac{1}{1-x}\sum_{k=1}^{\infty}(k)_{p,\lambda}x^{k}=\frac{1}{(1-x)^{2}}F_{p,\lambda}\Big(\frac{x}{1-x}\Big).
\frac{\partial f}{\partial t}=B\frac{\partial}{\partial v_{i}}(\int_{\mathbb{R}^{3}}dw\frac{(u^{2}\delta_{ij}-u_{i}u_{j})}{u^{3}}(\frac{\partial}{\partial v_{j}}-\frac{\partial}{\partial w_{j}})f(v)f(w))
\frac{1}{2\pi i}\int_{(a)}\breve{W}(s) N^s \sum_{U \subset A_s \atop |U| = j}\prod_{u\in U}X_d(1/2+u) \mathcal{B}^{(d)}(A_s - U + U^{-};\ell)\,ds \ll D^{-\epsilon}
{\cal F}= \det \left[ \sum_{r=-{\infty}}^{\infty}\sum_{p=-{\infty}}^{\infty}A_{r-N+j+p}A_{p}\left(t_{i}^{r+q}\pm t_{i}^{-(r+q)}-\frac{1}{2}\delta_{q0}\delta_{jN}(t_{i}^{r}+t_{i}^{-r})\right)\right].
\partial_{|x|}u =-\frac{\lambda''(s)}{2 \pi}\int_{0}^{t-s}\rho d\rho \int_{0}^{2 \pi}d\theta \frac{2(|x|+\rho \cos(\theta))}{(1+|x|^{2}+2|x|\rho \cos(\theta)+\rho^{2})((t-s)^{2}-\rho^{2})^{1/2}}
{}\int_X e_i^{2n}=c_X,{}\int_X e_i^{2n-2}e_j^2=\frac{c_X}{2n-1},{}\int_X e_i^{2n-4}e_j^4=\frac{3c_X}{(2n-1)(2n-3)},{}\int_X e_i^{2n-4}e_j^2e_k^2=\frac{c_X}{(2n-1)(2n-3)}.
0=\sum\limits_{j=1, j\neq i}^{n}\frac{m_{j}\sin{(\alpha_{i}-\alpha_{j})}}{(1-\cos{(\alpha_{i}-\alpha_{j})})^{\frac{3}2}(2-\sigma r^{2}(1-\cos{(\alpha_{i}-\alpha_{j})}))^{\frac{3}{2}}}
{\cal S}= \int d^5x\sqrt{-g}\left[{{\cal R}\over 16\pi G_N}-{1\over 2}\partial_\mu\Phi^a \partial^\mu\Phi^a-{\lambda\over 4}(\Phi^a\Phi^a -\eta^2)^2 \right]-\int_i d^4x\sqrt{-h}\sigma_i(\Phi^a)\,,
P_\alpha M_{\chi_D}(f)(z) = \Big (\frac{\alpha}{\pi}\Big )^n \int_{\mathbb{C}^n}e^{\alpha \langle z, w\rangle}\chi_D(w) e^{-\alpha |w|^2}f(w)dw &= \Big ( \frac{2}{p}\Big )^n \int_{\mathbb{C}^n}e^{\alpha \langle z,w\rangle - \frac{2\alpha - p\alpha}{2}|w|^2}\chi_D(w)f(w) d\mu_{p\alpha/2}(w).
(\bar{C}_\alpha,{\cal T}^\dagger_\beta)_\Omega=\frac{1}{2}[{\cal T}_\alpha,{\cal T}^\dagger_\beta],\quad({\cal T}_\beta,\bar{C}^\dagger_\alpha)_\Omega=\frac{1}{2}[{\cal T}_\beta,{\cal T}^\dagger_\alpha],
F_{\rm M2}= -\frac{2^{7/2}\pi^2}{3^4}N^{3/2}V_2 T^3 \left[ 1 + \frac{9}{8 \pi^2}\sum_{i=1}^4 \omega_i^2 - \frac{27}{128 \pi^4}\left( \sum_{i=1}^4 \omega_i^2 \right)^2 + \frac{81}{64 \pi^4}\sum_{i=1}^4 \omega_i^4 + \ldots \right]
\begin{matrix}\quad \vec \varphi(x) \quad \end{matrix}= \begin{pmatrix}c_1 \: \cos(r\,x) + c_2 \: \sin(r\,x) -c_1 \: \sin(r\,x) + c_2 \: \cos(r\,x) c_3 \: x \end{pmatrix}\;,
\int_{D 2}F = - k \int_{- \chi_{0}}^{\chi_{0}}d \tilde{\theta}- k \int_{- \chi_{0}+ \Delta \tilde{\theta}}^{\chi_{0}- \Delta \tilde{\theta}}d \tilde{\theta}= - 2 k \left( 2 \chi_{0}- \Delta \tilde{\theta}\right) \equiv - 2 k \chi_{0}^{\prime},
e^{-2\phi(x^+,x^-)}=\frac{N}{48}\frac{\displaystyle \left(C^{+}_{3}+ \frac{\Lambda}{8}C^{-}_{4}\right)x^{+}+\left(C^{-}_{3}+ \frac{\Lambda}{8}C^{+}_{4}\right)x^{-}}{\displaystyle 1+\frac{\Lambda}{8}x^{+}x^{-}}+a.
\int_Q (N_{Q, m}(x) )^{q + \epsilon}\, dx = \left(\int_{Q \backslash \left(\bigcup_{R \in D_1}R\right)}+ \sum_{R \in D_1}\int_{F_R}+ \sum_{R \in D_1}\int_{R \backslash F_R}\right) (N_{Q, m}(x) )^{q + \epsilon}\, dx \leq \tilde{C}|Q| + \sum_{R \in D_1}\int_{R}(N_{R, m}(x) )^{q + \epsilon}\, dx
g_{cr}\ge \frac{1}{1 -\frac{L\pi}{2\Lambda^2}+\frac{M^2}{3\Lambda^2}(\frac{\pi}{2L}- 1/\Lambda^2) -\frac{M}{\Lambda^2}\mbox{ln}2}\approx \frac{1}{1 -\frac{\pi|e||f_1|^{1/2}}{2\Lambda^2}-\frac{|e|}{\Lambda^2}(\frac{f_2^2}{|f_1|})^{1/2}\mbox{ln}2}.
{\frac{\delta{\cal E}( h )}{h^{2}}}= \alpha_{0}\beta_{1}\ln{\frac{h}{\Lambda}}+ \alpha_{0}{\frac{\beta_{2}}{\beta_{1}}}\ln \ln{\frac{h}{\Lambda}}+ \alpha_{1}+{\cal O}{\mathrm{\Large ~ (}}\ln \ln{\frac{h}{\Lambda}}{\mathrm{\Large ~ /}}\ln{\frac{h}{\Lambda}}{\mathrm{\Large ~ )}}\, .
T(a)(x) =\int_Q K(x,y)a(y)d\omega(y)=\int_Q [K(x,y)-K(x,x_Q)]a(y)d\omega(y) &\leqslant C\Big(\frac{\l(Q)}{\|x-x_Q\|}\Big)^\varepsilon \frac{1}{\omega(x, d(x,x_Q))}\|a\|_1.
\begin{cases}\phi_i'(x-\underline h(t))\leq -\epsilon_1 \mbox{for}x\in [\underline h(t)-K_0,\underline h(t)],\t\geq 0, \phi_i'(-x-\underline h(t))\leq -\epsilon_1 \mbox{for}x\in [-\underline h(t),-\underline h(t)+K_0],\t\geq 0.\end{cases}
P(n) \ge (n/2)! \cdot \frac{(n/2)!}{(n/2)^{n/2}}\cdot (n/2-2)^{n/2}& \ge \frac{1}{e^2}\left((n/2)!\right)^2(1-o(1)) & = \frac{\sqrt{2\pi}}{e^2}n^{3/2}\frac{(n-1)!}{2^{n+1}}(1-o(1)) \;.
(1-2\beta)^{p+1}\cdot\frac{\mathrm{d}}{\mathrm{d}\beta}\pounds_1\left(\frac{\beta}{2\beta-1}\right)=\frac{(1-2\beta)^p}{\beta}\cdot\pounds_0\left(\frac{\beta}{2\beta-1}\right)\equiv-\pounds_0(\beta)\pmod{p}.
T_{\pm 2 -}^{+ 2}= T_{\pm 2 +}^{+ 2}= T_{\pm 2 -}^{- 2}= T_{\pm 2 +}^{- 2}= 0 , T_{\pm \pm 2}^{+}= T_{+ - 2}^{+}= T_{\pm \pm 2}^{-}= T_{- + 2}^{-}= 0 ,
G(k,s,S_k)\sum_{\ell=0}^{k-2r-2s+2}V(k,s,r,\ell)+(S_r-1)H(k,s,r,S_k,S_r) \sum_{\ell=0}^{k-2r-2s+2}V(k,s,r,\ell)=0.
\frac{\partial \overline{T}_{i}}{\partial q_{j}}=\frac{\partial \overline{T}_{i}}{\partial \overline{q}_{i}}\frac{\partial \overline{q}_{i}}{\partial q_{j}}=\left(1-\alpha\right)\frac{\delta_{ij}-\overline{q}_{i}}{\sum_{j}q_{j}}
f(x)\equiv\cos(\kappa_j x) + \frac{\xi}{\kappa_j}\sin(\kappa_j |x|) =\frac{\xi L}{2}\left[ \frac{1}{Z_j^2}+ 2\sum_{n=1}^\infty \frac{\cos(k_n x)}{Z_j^2-(\pi n)^2}\right].
e^{H[s(t)]}\psi(z) e^{-H[s(t)]}= e^{-\sum_{q \geq 1}s_q(t) z^q}\psi(z) begin{align*}5pt]e^{H[s(t)]}\psi^\ast(z) e^{-H[s(t)]}= e^{\sum_{q \geq 1}s_q(t) z^q}\psi^\ast(z).
B^{[j]}= \left(\begin{array}{@{}c|c@{}}B_j 0 \hline 0 2^{k_j}u_1 \end{array}\right),B=\left(\begin{array}{@{}c|c@{}}B^{[j]}0 \hline 0 \begin{matrix}2^{k_j+1}u_2 0 0 \ddots \end{matrix}\end{array}\right).
g = \left( \begin{array}{c c}{{1}}&{{X}}\\{{}}&{{}}\\{{0}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c c}{{\exp ( \frac 1 2 \Phi )}}&{{0}}\\{{}}&{{}}\\{{0}}&{{\exp ( - \frac 1 2 \Phi )}}\\ \end{array}\right) \left( \begin{array}{c c}{{1}}&{{0}}\\{{}}&{{}}\\{{Y}}&{{1}}\\ \end{array}\right) \, ,
\rho(y) =\frac{e^{- \frac{|y|^2}{4}}}{(4 \pi)^{\frac{n}{2}}}= \prod_{j=1}^n \rho(y_j), \rho(\xi) = \frac{e^{- \frac{|\xi|^2}{4}}}{(4 \pi)^{\frac{1}{2}}},
\frac{\ell}{32\pi}+\frac{16\pi}{\ell}\tau\log\cos(\frac{1}{2}) = \ell \left( \frac{1}{32\pi}+16\pi\log\cos(\frac{1}{2})\frac{\tau}{\ell^{2}}\right) & = \ell \left( c_{1}-c_{2}\frac{\tau}{\ell^{2}}\right),
D_{m,n}(t) = \frac{1}{\sqrt{m+1}}\int_n^{n+1}\frac{v-n}{v^{3/2}}\left(1+i\log\frac{m+1}{v}\right)^{-t-1}dv E_{m,n}(t) = \int_n^{n+1}\frac{v-n}{v^{3/2}}\int_m^{m+1}u^{-1/2}\left(1+i\log\frac{u}{v}\right)^{-t-1}dudv,
\begin{aligned}\|\nabla^k \rho_t(t)\|_{H^{N-1-k}}+\|\nabla^k u_t(t)\|_{L^2}\le C(1+t)^{-\frac{5+2k}{4}}, &\|\nabla^l d_t(t)\|_{L^2}\le C(1+t)^{-\frac{7+2l}{4}}\end{aligned}
\{Q_{A}, Q_{B}\}= Z_{A B}= \left( C \Gamma^{\widetilde{\mu}_{1}\widetilde{\mu}_{2}}\right)_{A B}\, Z_{[ \widetilde{\mu}_{1}\widetilde{\mu}_{2}]}+ \left( C \Gamma^{\widetilde{\mu}_{1}\ldots \widetilde{\mu}_{6}}\right)_{A B}\, Z_{[ \widetilde{\mu}_{1}\ldots \widetilde{\mu}_{6}]}\, ,
\begin{aligned}\dot y_1 = f_1(y_1, y_2) = \frac{p_1}{1 - (y_2/p_2)}- p_3 y_1 + u, \dot y_2 = f_2(y_1, y_2) = \frac{- p_4}{1 + (y_1/p_5)}- p_{6}y_2, \end{aligned}
\mathcal{L}_{F}^{\alpha}\{~_{0}^{C}\mathcal{D}_{x}^{\beta}f(x) \}&=\mathcal{L}_{F}^{\alpha}\{_{0}\mathcal{I}_{x}^{m \alpha-\beta}(D_{x}^{\alpha})^{m}f(x) \}&=\frac{\mathcal{L}_{F}^{\alpha}[ (D_{x}^{\alpha})^{m}f(x)]}{s^{m\alpha-\beta}}
K(\Psi;M;\mu) = \prod_{j \in M}(1-L_j) \prod_{(i,j) \in \Delta^+\setminus\Psi}(1-R_{ij}) k_\mu = \prod_{j \in M \setminus y}(1-L_j)\prod_{(i,j) \in \Delta^+\setminus\Psi}(1-R_{ij}) (k_\mu - k_{\mu-\epsilon_{y}})\,,
|C|E_1\circ |C|E_i =\sum_{l=0}^4 K_1(\beta_l)K_i(\beta_l)A_l &=\sum_{l=0}^4 (n-i+1)K_{i-1}(\beta_l)A_l+\sum_{l=0}^4 (i+1)K_{i+1}(\beta_l)A_l &=(n-i+1)|C|E_{i-1}+(i+1)|C|E_{i+1}.
\overline{C}(\alpha) := \sum_{K=1}^{N}\alpha_K\mu_K^{[2]}\left(\sum_{K=1}^{N}\alpha_K\kappa_K^{[1]}\right)^{\!-1}\mbox{and}\hat{C}(\alpha) := \frac{\hat{K}}{1-\sqrt{1-\varepsilon}}\sum_{K=1}^{N}\alpha_K\mu_K^{[1]}\left(\sum_{K=1}^{N}\alpha_K\kappa_K^{[1]}\right)^{\!-2},
\big( H(b,V) - z \big) \begin{pmatrix}I 0 -(H_+-z)^{-1}U (H_+-z)^{-1}\end{pmatrix}& =\begin{pmatrix}H_--z-\overline{U}(H_+-z)^{-1}U \overline{U}(H_+-z)^{-1}0 I \end{pmatrix},
\mathrm{P}^o_{1,k}= \sum^{k-1}_{p=0}{k-1 \choose p}(-1)^p \frac{K!}{(K-k)!(k-1)!}\times \int^{\epsilon_{1,k}^*}_{0}f_{\tilde{x}_k}(x)\left[F_{\tilde{x}_k}\left(x\right) \right]^{K-k+p}dx = \sum^{k-1}_{p=0}{k-1 \choose p}\frac{(-1)^p K!\left[F_{\tilde{x}_k}\left(\epsilon_{1,k}^*\right) \right]^{K-k+p+1}}{(K-k)!(k-1)!(K-k+p+1)}.
F_x(u) := \exp\{-uf(x)e^{\Psi(k)}\}\sum_{j=k}^\infty \frac{1}{j!}\bigl\{uf(x)e^{\Psi(k)}\bigr\}^j = e^{-\lambda_{x,u}}\sum_{j=k}^\infty \frac{\lambda_{x,u}^j}{j!},
\left[ \left({\frac{\dot{R}}{R}}\right)^{2}+ \xi_{+}^{2}R^{-{\frac{2}{3}}k^{2}}\right]^{\frac{1}{2}}- \left[ \left({\frac{\dot{R}}{R}}\right)^{2}+ \xi_{-}^{2}R^{-{\frac{2}{3}}k^{2}}\right]^{\frac{1}{2}}= -{\frac{\kappa_{5}^{2}}{3}}[ \lambda ( \phi ) + \rho ] .
a_n - \tilde{a}_n= \left(\frac{3}{16 n^2 \log^2n}\left(\begin{matrix}-1 i i 1\end{matrix}\right) + O\left(\frac{1}{n^2\log^3n}\right)\right)_{11}\hphantom{a_n - \tilde{a}_n}{}= - \frac{3}{16 n^2 \log^2 n}+ O\left(\frac{1}{n^2 \log^3 n}\right).
\delta_{(1)}^c{\cal L}_{APEGT}= \int d^4x{C_2 \sigma^2 g^2 \over 8\pi^2}*B_{\mu\nu}^i(x) I^{\mu\nu,\alpha\beta}\left[ f_0(\alpha) - f_1(\alpha){\partial_\lambda \partial^\lambda \over M_A^2}+ f_2(\alpha){(\partial_\lambda \partial^\lambda)^2 \over M_A^4}\right] *B_{\alpha\beta}^i(x) .
1 =\left\langle \nu^H,\frac{\xi^H}{|\xi^H|}\right\rangle^2+\left\langle \eta,\frac{\xi^H}{|\xi^H|}\right\rangle^2 &= \left(\frac{(2n-1)c}{2n+1}|\xi^H|^2\right)^2+\left(\frac{(2n-1)c}{2n+1}\left(t-t_1-\frac{2n+1}{(2n-1)c}\right)\right)^2
\left(\begin{array}{c}u_{-}(\alpha k+j+1, y,\lambda, E) u_{-}(\alpha k+j,y,\lambda, E)\end{array}\right)=T_{E,\omega}(\alpha k+j)\dots T_{E,\omega}(\alpha k)\left(\begin{array}{c}u_{-}(\alpha k,y,\lambda,E) u_{-}(\alpha k-1,y,\lambda,E)\end{array}\right),
=\int_{\frac{2\pi n}{\delta_1}}^{\frac{2\pi n}{\delta}}\left|\varphi_X\left(\frac{2\pi n}{t}\right)\right|^2\frac{(\log\frac{t}{2\pi})^\alpha}{t^\beta}dt+\int_{\frac{2\pi n}{\delta}}^T\frac{(\log\frac{t}{2\pi})^\alpha}{t^\beta}dt.
\phi = p_\tau + \frac{f(t_\tau)}{2m}(p_{1,\tau}^2 + p_{2,\tau}^2) + \frac{m\omega^2(t_\tau)f^{-1}(t_\tau)}{2}(x_{1,\tau}^2 + x_{2,\tau}^2)\; \sim 0.
(1-x^m)A_{(m+1)p_n+r_n}=A_{(m+1)p_n-(m-r_n)}+x^{2m-2r_n}A_{(m+1)p_n-r_n}^{m \rightarrow m+1}(x)+x^{m-1}A_{(m+1)p_n+r_n-m-1}.
x_{+}(t).e_{2r}= \left(\sum_{j = 0}^{p-r-1}\binom{p-r-1}{j}t^{p-r-1-j}v_{1}^{p-2-j}v_{2}^{j}\right) \otimes f_{1}&= \sum_{m = r}^{p-1}\binom{p-r-1}{m-r}t^{m-r}e_{2m}.
R_{0}(\theta )=\prod_{l=1}^{\infty}\left[ \frac{\Gamma (4l\lambda +2i\lambda \theta /\pi )\Gamma (4(l-1)\lambda +1+2i\lambda \theta /\pi )}{\Gamma ((4l-3)\lambda +2i\lambda \theta /\pi )\Gamma ((4l-1)\lambda +1+2i\lambda \theta /\pi )}/(\theta \to -\theta )\right]
1=\frac{1}{Z}\sum_{\alpha,\alpha^{\prime}}e^{-\beta E_{\alpha}}(\langle\alpha|\psi_{k}|\alpha^{\prime}\rangle\langle\alpha^{\prime}|\psi_{k}^{\dagger}|\alpha\rangle-\zeta\langle\alpha|\psi_{k}^{\dagger}|\alpha^{\prime}\rangle\langle\alpha^{\prime}|\psi_{k}|\alpha\rangle).
\|\mathrm{F}_0u - u\|_{H^{k,0}(\mathcal{S}_b)}\leq \|\mathrm{F}_0u - u\|_{H^{k,0}(\mathcal{S})}= \Big{\|}\dfrac{\cosh{((z+1)\sqrt{\mu}|\mathrm{D}|)}}{\cosh{(\sqrt{\mu}|\mathrm{D}|)}}u - u\Big{\|}_{H^{k,0}(\mathcal{S})}\lesssim \mu | \nabla_Xu |_{H^{k+1}}.
E_Q =T_Q(E') \;\; (Q\in W_i^{*}), P_{B,Q}=T_Q(P_B') \;\; (B\in\Lambda, \;\; Q\in W_i^{*}), E =\bigcup_{Q\in W_i^{*}}E_Q, P_B =\bigcup_{Q\in W_i^{*}}P_{B,Q}\;\ (B\in\Lambda).
\Phi \left( p^{\mu}\right) = \Lambda \left( \dot{p}^{\mu}\to p^{\mu}\right) \Phi \left( \dot{p}^{\mu}\right) = e^{i ( \vec{J}\cdot \vec{\vartheta}+ \vec{K}\cdot \vec{\varphi})}\Phi \left( \dot{p}^{\mu}\right) ,
\uppercase\expandafter{\romannumeral1}\leq\sum\limits_{j=1}^{\infty}\vert \lambda_{j}\vert\vert T(a_{j})(x)\vert\chi_{Q_{j}^{*}}(x)+\sum\limits_{j=1}^{\infty}\vert\lambda_{j}\vert\vert T(a_{j})(x)\vert\chi_{(Q_{j}^{*})^{c}}(x)=:\uppercase\expandafter{\romannumeral1}_{1}+\uppercase\expandafter{\romannumeral1}_{2}.
\log N(Z_p(\mu),C_1 t \sqrt{p}B_2^n) \leq \begin{cases}C_2 \frac{n}{t^2}1 \leq t \leq \sqrt{n/p}C_3 \frac{\sqrt{n}\sqrt{p}}{t}\sqrt{n/p}\leq t \leq \sqrt{n/p}\log^2 (1+n) C_4 \frac{n \log^2(1+n)}{t^2}\sqrt{n/p}\log^2(1+n) \leq t \leq \sqrt{p}\end{cases}~.
[y_{(1\,p)},y_{(3\,l)}]_c =[y_{(1\,p)},[y_{(3\,p)},y_{(p+1\,l)}]_c]_c &=-\chi_{(p+1\,l)}(g_{(3\,p)})y_{(1\,l)}y_{(3\,p)}+\chi_{(3\,p)}(g_{(1\,p)})y_{(3\,p)}y_{(1\,l)}&=\chi_{(3\,p)}(g_{(1\,p)})(1-\chi_{(p+1\,l)}(g_{(3\,p)})\chi_{(3\,p)}(g_{(p+1\,l)}))y_{(3\,p)}y_{(1\,l)}
CF(r;x)-CF(r;x+1)=\sum_{k=0}^{\infty}\frac{2(x+k)}{\left((x+k)^2+r^2\right)^2}-\sum_{k=0}^{\infty}\frac{2(x+k+1)}{\left((x+k+1)^2+r^2\right)^2}=\frac{2x}{\left(x^2+r^2\right)^2}.
\mathrm{Hom}_{Sp_{4m}}(V_{\pi}, \mathrm{Ind}^{Sp_{4m}}_{Sp_{2n}\times Sp_{4m-2n}}(1)) = \ \mathrm{Hom}_{Sp_{2n}\times Sp_{4m-2n}}(V_{\pi}, 1) \neq \ 0
\|m^N_t - m^N_s\|_{BL}^\ast \leq \|m^N_{t}- m^N_{t_{k}}\|_{BL}^\ast + \|m^N_{t_{n+1}}- m^N_{s}\|_{BL}^\ast + \sum_{l=n+1}^{m}\|m^N_{t_{l+1}}- m^N_{t_{l}}\|_{BL}^\ast.
|\xi_\alpha|_X = |\xi_\alpha(c)| \leq |\xi_\alpha|_{X_n}\leq \frac{1}{|\alpha!|}\max_{\beta \leq \alpha}|y^{\alpha-\beta}|_{X_n}\cdot ||\varphi(X_n)|| \cdot |y^\beta|_{X_n}\leq \frac{||\varphi(X_n)||\cdot |\pi^{n|\alpha|}|}{|\alpha!|}