harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\,\,\,f(\xi,\eta,\rho)=f_1(\xi,\rho)f_2(\eta,\rho)\,\,\,\,\,\,\|\mathcal{F}^{-1}f\|_{L^1(\mathbb{R}^3)}\lesssim \|\mathcal{F}^{-1}f_1\|_{L^1(\mathbb{R}^2)}\|\mathcal{F}^{-1}f_2\|_{L^1(\mathbb{R}^2)}.
\begin{array}{l l}{1 )}&{h = 3 / 8 , \quad a = 3 / 8 0 ,}\\{2 )}&{h = 7 / 8 , \quad a = 7 / 1 6 ,}\\{3 )}&{h = 3 / 8 + x , \quad a = 3 / 8 0 .}\\ \end{array}
\rho_{r}(S_{2}) = \dfrac{|S_{2}\cap B_r|}{|B_r|}= (2r+1)^{n(m-1)}\dfrac{|\pi_{2}(S_{2}\cap B_r)|}{(2r+1)^{n(k+m)}}= 1 - \rho_{r}(V_{k+1, n, k+1}(\mathbb{Z})).
\Delta A_{\mu}= ( \bar{\epsilon}_{2}\zeta^{( 9 )}- \bar{\epsilon}_{1}) \Gamma_{\mu}\lambda + ({\frac{1}{3}}\bar{\epsilon}_{1}- \bar{\epsilon}_{2}\zeta^{( 9 )}) \Gamma_{\rho}\lambda \bar{\lambda}\Gamma^{\rho}\partial_{\mu}\lambda + \xi^{\rho}\partial_{\rho}A_{\mu}+ \partial_{\mu}\xi^{\rho}A_{\rho},
\tag*{($\star$)}\renewcommand{\arraystretch}{1.2}\begin{array}{c}x_i\in A_1\cap B_2,\y_i\in A_2\cap B_1, (A_1-\{x_1,\dots,x_i\})\cup\{y_1,\dots,y_i\}\ (A_2-\{y_1,\dots,y_i\})\cup\{x_1,\dots,x_i\}\ \end{array}
\lim_{n\rightarrow \infty}\frac{a_ntQ'(a_nt)-a_nxQ'(a_nx)}{n(t^2-x^2)}&=B_{\alpha}^{-1}\lim_{n\rightarrow \infty}\frac{a_ntQ'(a_nt)-a_nxQ'(a_nx)}{a_nQ'(a_n)(t^2-x^2)}&=B_{\alpha}^{-1}\frac{t^{\alpha}-x^{\alpha}}{t^2-x^2}.
\sigma_\mathfrak{s}(\boldsymbol{J};n,p)=\sigma_\mathfrak{s}(\boldsymbol{J};q,p) \quad\ |n-q|\leq 1 \quad\sigma_\mathfrak{s}(\boldsymbol{J};n,p)=\sigma_\mathfrak{s}(\boldsymbol{J};n,m) \quad\ |p-m|\leq 1\,.
h^{0,0}-h^{0,1}+h^{0,2}=\frac{1}{12}\int_S c_1(S)^2+c_2(S), h^{1,0}-h^{1,1}+h^{1,2}=\frac{1}{6}\int_S c_1(S)^2-5c_2(S), h^{2,0}-h^{2,1}+h^{2,2}=\frac{1}{12}\int_S c_1(S)^2+c_2(S).
\sup_{y_1,\cdots,y_k\in [0,2\pi)}\phi_{k,n}(y_1,\cdots,y_k)\leq(n/4)^k(2\ln n)^{\frac{k}{2}}\prod_{j=1}^kD_n(F_n(x_j)/2) &=\prod_{j=1}^k(n(2\ln n)^{\frac{1}{2}}D_n(F_n(x_j)/2)/4)\to \prod_{j=1}^k\left(e^{c_0-x_j}/4\right), \,\,\, n\to+\infty,
Q_{i}=\{X_\ell\neq2\;\mbox{for}\;a_n\le\ell\le i-1,X_i=2\},\; a_n\le i\le b_n-1; Q'_{i}=\{X_\ell\neq2\;\mbox{for}\;a_n\le\ell\le b_n-1,X_\ell\neq2,3\;\mbox{for}\;b_n\le\ell\le i-1,X_i=2\},\;i\ge b_n; \mbox{and}\;\;Q''_{i}=\{X_\ell\neq2\;\mbox{for}\;a_n\le\ell\le b_n-1,X_\ell\neq2,3\;\mbox{for}\;b_n\le\ell\le i-1,X_i=3\},\;i\ge b_n.
\sigma(P_{1}) =\sigma(\psi^{-1}(z_{1}))=\psi^{-1}(\sigma(z_{1}))=\psi^{-1}(z_{1})=P_{1}\sigma(P_{2}) =\sigma(\psi^{-1}(z_{2}))=\psi^{-1}(\sigma(z_{2}))=\psi^{-1}(z_{1}\cdot{}z_{2})=P_{1}+P_{2}.
q(t^{\prime})^{-(1+r)}+2r\frac{t^{\prime\prime\prime}t^{\prime}-(t^{\prime\prime})^{2}}{(t^{\prime})^{2}}+\frac{(1+r)^{2}+4r^{2}}{4}\left(\frac{t^{\prime\prime}}{t^{\prime}}\right)^{2}-1=0\, .
\big\{(\epsilon, \emptyset, \emptyset), (1, \{\langle k-1,0\rangle \}, \emptyset), (12, \{\langle k-1,0,0 \rangle, \langle k-2,1,0 \rangle, \dotsc, \langle 0,k-1,0 \rangle \}, \{1\}), (21, \{\langle 0,1,0 \rangle, \langle k-2,0,0\rangle \}, \{1\}) \big\}
( I \otimes \breve{R}^{\Phi_{1}\Phi_{2}}( x ) ) ( \breve{R}^{\Phi_{1}\Phi_{3}}( x y ) \otimes I ) ( I \otimes \breve{R}^{\Phi_{2}\Phi_{3}}( y ) ) = ( \breve{R}^{\Phi_{2}\Phi_{3}}( y ) \otimes I ) ( I \otimes \breve{R}^{\Phi_{1}\Phi_{3}}( x y ) ) ( \breve{R}^{\Phi_{1}\Phi_{2}}( x ) \otimes I )
d\Pi_{\mu_1 \dots \mu_{p-1}\alpha_1}= a_1 \left((C \Gamma_{\nu\mu_1 \dots \mu_{p-1}})_{\beta \alpha_1}\Pi^\nu \Pi^\beta+{a_s \over a_0}(C\Gamma^\nu)_{\beta \alpha_1}\Pi_{\nu \mu_1\dots \mu_{p-1}}\Pi^\beta)\right)
\frac{\check{Z}^{\epsilon}_{0,n}}{Z_{0,n}^{\epsilon}}= P_{\epsilon,n}(\chi_1=n)= \mathcal{P}_{\epsilon}(\chi_1=n|\mathcal{A}_n)=\frac{\mathcal{P}_{\epsilon}(\chi_1=n)}{\mathcal{P}_{\epsilon}(\mathcal{A}_n)}
\begin{cases}\eta(0,t)=0,\,\,\eta(L,t)=0,\,\,\eta_{x}(0,t)=f(t), t \in (0,\infty), w(0,t)=0,\,\,w(L,t)=0,\,\,w_{x}(L,t)=g(t), t \in (0,\infty)\end{cases}
u(h_1,\dots,h_m) =u(\zeta(w_{1,2N}(x_1,\dots,x_m)),\dots,\zeta(w_{m,2N}(x_1,\dots,x_m))) &=\zeta(u(w_{1,2N}(x_1,\dots,x_m),\dots,w_{m,2N}(x_1,\dots,x_m))) &=\zeta(U(x_1,\dots,x_m)),
G_0(z,\omega_0) = \sum_{k=0}^3 \frac{\gamma_k \left( a \gamma_k + \omega_0 \right)^4}{z - i \left( a \gamma_k + \omega_0 \right)}\;\;\;\mbox{$\gamma_k = i e^{-3 i\pi/8}e^{ik \pi/4}$}.
\begin{aligned}\int_{0}^T \int_{\tilde \lambda_1}^{1}\Big( s \Theta a (v_x)^2 + s^3 \Theta^3 \frac{(x-x_0)^2}{a}v^2 \Big) e^{2s \varphi}\, dx dt \le C \int_{0}^T \int_{\omega}v^2 dxdt, \end{aligned}
\prod_{j=1}^d \langle v_{1,j}\rangle^{-1+\epsilon}\left| \nu( q + v_1) - E \pm i \eta \right|^{-1}\leq c_{{\rm (II)},d}(1 + \eta^{-1}) \prod_{j=1}^d \langle q_j \rangle^{-1 + \epsilon}.
u_\lambda v_\lambda\cdot x_{\alpha_4}= x_{\alpha_4}- \lambda z_2 + \lambda^2z_3, u_\lambda v_\lambda \cdot z_1 = z_1+\lambda z_2, u_\lambda v_\lambda \cdot z_2 = z_2, u_\lambda v_\lambda \cdot z_3 = z_3,
K(z,{z_0})= \left (1 - \frac{c^2}{2}\sum_{\alpha=1}^n w_\alpha(z) \overline{w_\alpha(z_0)}\right )^{\frac{-2}{c^2}}e^{f(T(z))+\overline{f(T({z_0}))}}, z, z_0\in U_p.
\left ( \! \! \begin{array}{c}\partial_t U_1 \partial_t U_2 \end{array}\! \! \right) =\left ( \! \! \begin{array}{c}(T-t)^{-1}U_2 - \frac{2}{p-1}(T-t)^{-1}U_1 (T-t) \Delta_r U_1 - (T-t)^{-1}\big (\frac{p+1}{p-1}U_2 - |U_1|^{p-1}U_1 \big ) \end{array}\! \! \right)
\ddot{\varrho}+\frac{4}{3t}\dot{\varrho}-\tilde{\kappa}t^{-2\gamma+\frac{2}{3}}\Delta\varrho-\frac{2}{3t^2}\varrho=(\gamma-1)\tilde{\kappa}t^{-2\gamma+\frac{2}{3}}\frac{D^i\varrho D_i\varrho}{1+\varrho}.
x^{3}p_{n,l+1}=p_{n+3,l}+I_{n,l}p_{n,l},\;\;\;I_{n,l}=\frac{h_{n,l+1}}{h_{n,l}}=\frac{Z_{n+1,l+1}Z_{n-2,l}}{Z_{n-2,l+1}Z_{n+1,l}}.
\sigma^{1^p}\star \sigma^{(n+1-m,0,\cdots, 0)}=\sigma^{1^p}\circ\sigma^{(n+1-m,0,\cdots, 0)},\mbox{for}1\leq p< m; \sigma^{1^m}\star \sigma^{(n+1-m,0,\cdots, 0)}=(\alpha_1+\cdots+\alpha_n)\sigma^{(n+1-m, 1, \cdots, 1)}+q.
\lambda_{\gamma}: \sum_{i=1}^n w^{\gamma}_i(\theta) \psi(x_i, \theta) = \sum_{i=1}^n \left(1 + \lambda_\gamma^T \psi(x_i, \theta) \right)^{-\frac{1}{\gamma + 1}}\psi(x_i, \theta) = 0 \,.
\begin{cases}\big\|\Theta[s]\big\|_\mathrm{HS}\le C N^{3\beta}, \mu[s] \le C, H[s](e,\overline{e}) - |\Theta[s](\overline{e},\overline{e})| \ge c \|e\|^2,e\in(\phi_s)_\perp, \end{cases}
b_j := \mathrm{Re}\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.
\hat P_n(x) =Q\begin{bmatrix}\frac{p_{n+1}(x)-\frac{p_{n+1}(x_1)}{p_n(x_1)}p_n(x)}{x-x_1}0 \dots 0 0 \frac{p_{n+1}(x)-\frac{p_{n+1}(x_2)}{p_n(x_2)}p_n(x)}{x-x_2}0 \vdots \ddots \ddots \vdots 0 0 \frac{p_{n+1}(x)-\frac{p_{n+1}(x_p)}{p_n(x_p)}p_n(x)}{x-x_p}\end{bmatrix}Q^\top,
R = \operatorname*{d e t}( B ) / \operatorname*{d e t}( C ) ={\frac{\operatorname*{d e t}( \left< W_{0}^{*}\psi_{\alpha}\; , \; \phi_{\beta}^{1}\right> )}{\operatorname*{d e t}( \left< \psi_{\alpha}\; , \; \phi_{\beta}\right> )}}.
0 =H^i(\hat{A},\phi_{L}^*(\mathcal{F})\otimes(\bigoplus^{h^0(\hat{A},L)}L^{p^e})\otimes Q) &=H^i(\hat{A},\phi_{L}^*(\mathcal{F})\otimes F^{e,*}\phi_{L}^*(\widehat{L}^{\vee})\otimes Q) &=H^i(\hat{A},\phi_{L}^*(\mathcal{F})\otimes\phi_{L}^*F^{e,*}(\widehat{L}^{\vee})\otimes Q) &=H^i(\hat{A},\phi_{L}^*(\mathcal{F}\otimes F^{e,*}(\widehat{L}^{\vee}))\otimes Q).
\phi\left( tx\right) \leq tx\left( \left\vert \ln t\right\vert +\left\vert\ln x\right\vert +1\right) \leq tx\left[ \left( 1/t^{1-p}-1\right) \left( \left\vert \ln x\right\vert+1\right) +\left\vert \ln x\right\vert +1\right] =t^{p}\phi\left( x\right)
\bigcup_{k\ge k_0}\bigcup_{m\in \varphi^{-1}(n)\cap J_k}\Big\{s\in [1,b_{m+1}-b_m);\; |v_{m}|\prod_{i=b_{m+1}-s}^{b_{m+1}-1}|w_{i}|\ge 2^{\delta^{(k)}-\tau^{(k)}}\Big\},
T_1 = [7 \to 6]^6 T_2 = [6 \to 5] \circ [5 \to 4] \circ [4 \to 3] \circ [7 \to 6] \circ [5 \to 4] \circ [4 \to 3] \circ [3 \to 2] \circ{}& [6 \to 5] \circ [4 \to 3] \circ [3 \to 2] \circ [2 \to 1] \circ [3 \to 4] \circ [4 \to 5] \circ [4 \to 3]
\delta\left\{\left(U_j, \sum_{\lambda, |\beta|=n}F_{j, \beta}^\lambda\cdot e_{j, \lambda}\*\otimes e_j^\beta\right)\right\}=\left\{\left(U_{jk}, \sum_{\lambda, |\alpha|=n}\left(h^\lambda_{1, jk, \alpha}-h^\lambda_{2, jk, \alpha}\right)\cdot e_{j, \lambda}^*\otimes e_j^\alpha\right)\right\}
\frac{d\mu_r}{d\mu}(n+m) =\left([n+m]_{|q|}!\right)^{-2}r^{-(n+m)}\le \left([n]_{|q|}!\,[m]_{|q|}!\right)^{-2}r^{-(n+m)}&=\frac{d\mu_r}{d\mu}(n)\frac{d\mu_r}{d\mu}(m).
F(t,q)\equiv-\partial_t\psi(t,q)-\nabla_qV(t,q)+\tilde F(t,q,\psi(t,q)), &\frac{1}{2}\left(\tilde\gamma^{-1}-(\tilde\gamma^T)^{-1}\right)^{ij}F_j= (\tilde\gamma^{-1})^{ki}H_{k\ell}(\tilde\gamma^{-1})^{\ell j}F_j, &\frac{1}{2}(S-S^\prime)^i=\beta^{-1}\partial_{q^j}\left((\tilde\gamma^{-1})^{ki}H_{k\ell}(\tilde\gamma^{-1})^{\ell j}\right).
T_{\eta}\xi^{\alpha}\! = \! \eta^{\alpha}( \xi ) , \, T_{\eta}^{- 1}\xi^{\alpha}\! = \! T_{\zeta}\xi^{\alpha}\! = \! \zeta^{\alpha}( \xi ) , \quad T_{\eta}f ( \xi ) = f ( \eta ( \xi ) ) , \, T_{\eta}^{- 1}f ( \xi ) = T_{\zeta}f ( \xi ) = f ( \zeta ( \xi ) ) ,
0 = \int_{N\mathcal{B}(1)}\nabla u \cdot \nabla \varphi + \hat \varepsilon^{-2}(|u|^2 - 1) u \cdot \varphi =\int_{\Omega}\Big(g^{ij}\partial_{y_i}\tilde u \partial_{y_j}\tilde \varphi + \hat \varepsilon^{-2}(|\tilde u|^2 - 1) \tilde u \tilde \varphi\Big) \sqrt{|\det(g)|}\,dy.
\tilde u_t = \frac{P_t}{1- P_t}\left[ \frac{1 - P_{t-2}}{P_{t-2}}\tilde u_{t-2}+ \frac{1}{P_{t-1}(1+p_{t-1})}u_{t-1}+ \frac{1}{P_t(1+p_t)}u_t\right] & = \ldots = \frac{P_{t}}{1-P_{t}}\sum_{i=1}^t \frac{1}{P_{i}(1 + p_i)}u_i,
\lambda \int_{M}| \mathrm{u}( z ) |^{r( z )}\,\, dv_{g}( z ) = \int_{M}| D \mathrm{u}( z ) |^{p( z )}\,\, dv_{g}( z ) + \int_{M}\mu ( z ) \, | D \mathrm{u}( z ) |^{q( z )}\,\,dv_{g}( z ) &- \int_{M}g( z ) \, | \mathrm{u}( z ) |^{1 - \gamma ( z )}\,\, dv_{g}( z ).
\begin{aligned}E_b =-\int_{t_b}^{t_m}F_-(\tau)d\tau \quad\,\,\left(\equiv -I_-(t_b,t_m)\right), 0 =\int_{t_m}^{t_f}e^{Bt}F_+(\tau)d\tau \quad\left(\equiv I_+(t_m,t_f)\right), E(t_b+1)-\frac{E(t_b+1)^2}{2\zeta}=\int_{t_f}^{t_b+1}F_-(\tau)d\tau \quad\,\,\left(\equiv I_-(t_f,t_b+1)\right).\end{aligned}
\mathrm{R}(\theta) \!\coloneqq\!\! \begin{bmatrix}\cos(\theta)\!\! -\sin(\theta) \sin(\theta)\!\! \cos(\theta)\end{bmatrix}\!,\;\; j \!\coloneqq\! \mathrm{R}(\pi/2),\;\; \mathrm{r}(\theta) \!\coloneqq\!\! \begin{bmatrix}\cos(\theta) \sin(\theta) \end{bmatrix}\!.
= \sum\limits_{n=0}^{N-1}\frac{\Lambda^+_n}{\sqrt{\pi}2^n}t^{-n/2}\int\limits_{\mu}^{+\infty}z^{-n}e^{-(z-\eta)^2}dz + \int\limits_{\mu}^{+\infty}R_N(z\sqrt{t}) e^{-(z-\eta)^2}dz,
\begin{array}{r c l c l}{{\dot{\varphi}^{0}}}&{{=}}&{{[ \varphi^{0}, H ]}}&{{=}}&{{\varphi_{1}^{0},}}\\{{\dot{\varphi}^{m}}}&{{=}}&{{[ \varphi^{m}, H ]}}&{{=}}&{{2 \varphi_{1}^{m},}}\\{{\dot{\varphi}_{1}^{0}}}&{{=}}&{{[ \varphi_{1}^{0}, H ]}}&{{=}}&{{- \partial_{m}\varphi_{1}^{m},}}\\{{\dot{\varphi}_{1}^{m}}}&{{=}}&{{[ \varphi_{1}^{m}, H ]}}&{{=}}&{{0 .}}\\ \end{array}
{\cal L}_1 = \psi^\dagger_\uparrow (i \partial_t - \epsilon (p))\psi_\uparrow + \psi^\dagger_\downarrow (i \partial_t - \epsilon (p))\psi_\downarrow + \sqrt{\lambda g}(\phi^*\psi_\downarrow \psi_\uparrow +\phi \psi_\uparrow^\dagger \psi^\dagger_\downarrow) +g \phi^* \phi\, \, \, \, .
{\mathbb E}{\left\{{{\bf{g}}_n^H{\sf diag}\left({{p_u}{\bf{G}}{{\bf{G}}^H}+{\bf{I}}}\right){{\bf{g}}_n}}\right\}}= M{\left({{\beta_n}{\rm{+}}{p_u}{\beta_n}\sum\limits_{i = 1}^N{{\beta_i}}+{p_u}\beta_n^2}\right)}.
\langle V_{N}^{( 0 )}| ={}_{1 \dots N}\langle 0 | \exp \left\{\frac{1}{2}\sum_{I , J = 1}^{N}\int_{- \infty}^{\infty}d \kappa a^{- ( I )}( \kappa ) \mu_{N}^{I J}( \kappa ) C a^{- ( J )}( \kappa ) \right\}.
\mathcal{F}(|x|^{-\alpha-2}x_{k}x_{l})=-d_{\alpha+2}(m-\alpha-2)(m-\alpha)\xi_{k}\xi_{l}|\xi|^{-(m-\alpha)-2}, \mathcal{F}(|x|^{-\alpha-2}x_{k}^2)=d_{\alpha+2}(m-\alpha-2)(|\xi|^{2}-(m-\alpha)\xi_{k}^{2})|\xi|^{-(m-\alpha)-2}.
\inf_{\theta\in N_1\cap \Theta_n^c}{\sf W}_p^{(\sigma)}(\hat \mu_n,\nu_{\theta}) >{\sf W}_p^{(\sigma)}(\hat \mu_n,\mu)+o_{\mathbb P}(n^{-1/2}) \geq\inf_{\theta\in N_1\cap \Theta_n}{\sf W}_p^{(\sigma)}(\hat \mu_n,\nu_\theta)+o_{\mathbb P}(n^{-1/2}).
\langle\overline{n_m,\ldots,n_1}|=\,(-1)^{\sum_{a}\bar{n}_a(a-1)}\langle\bar{n}_m,\ldots,\bar{n}_1|=\langle\bar{0}|(\hat{\eta}^m)^{n_m}\cdots(\hat{\eta}^1)^{n_1}
h h' = s + (2+\delta)A s^{1+\delta}+ (1+\delta)A^2s^{1+2\delta}h h'' = (1+\delta)\delta A s^\delta + (1+\delta)\delta A^2 s^{2\delta}(h')^2 = 1 + 2(1+\delta)As^\delta + (1+\delta)^2A^2 s^{2\delta},
L^2(M)_r \otimes_{h M'}(\overline{(.\#S^{(i)})_*[L^2(M,\eta_i)_{L^2(M)}+{}_{L^2(M)}L^2(M,\eta_i)]}^{M'-M'}\otimes_{h M'}L^2(M)_c &=(\overline{(.\#S)_*[(L^2(M,\eta_i)_c)+(L^2(M,\eta_i)_r)]}.
{\bf \Omega}_{{\bf x},{\bf x^{\prime}}}= m^{2}\delta_{{\bf x},{\bf x^{\prime}}}-{\frac{1}{\sigma^{2}}}\sum_{i = 1}^{3}\left( \delta_{{\bf x}+ \hat{\sigma}\left( i \right) ,{\bf x^{\prime}}}- 2 \delta_{{\bf x},{\bf x^{\prime}}}+ \delta_{{\bf x}- \hat{\sigma}\left( i \right) ,{\bf x^{\prime}}}\right) .
\psi_{0}(x) = \frac{\psi(x+0) + \psi(x-0)}{2}= x - \sum_{\rho}\frac{x^{\rho}}{\rho}- \frac{\zeta'}{\zeta}(0) - \frac{1}{2}\log\left( 1 - \frac{1}{x^{2}}\right).
\Big\Vert \sum_{n=M+1}^{N}\frac{a_{n}}{n^{L+\varepsilon}}n^{-s}\Big\Vert_{\mathfrak{X}(X)}\leq \frac{1}{N^{\frac{\varepsilon}{2}}}+ \frac{1}{M^{\frac{\varepsilon}{2}}}+ \vert L + \varepsilon \vert \sum_{n=M}^{N-1}\frac{1}{n^{1+\frac{\varepsilon}{2}}}\,
\Phi = \left( \begin{array}{c}{\Theta}{\chi}\end{array}\right) = \left( \begin{array}{c}{\Theta}{\frac{i E \sigma^{3}}{k_{z}- \lambda}\Theta}\end{array}\right) , \Theta = \left( \begin{array}{c}{\Theta_{1}}{\Theta_{2}}\end{array}\right)
\pi H_{v(s)}\varphi(s) =\int_{\vert s-t\vert\geq \eta s}\frac{(s-t)\varphi(t)dt}{(s-t)^2+v(s)^2}+\int_{\vert s-t\vert\leq \eta s}\frac{(s-t)\varphi(t)dt}{(s-t)^2+v(s)^2}&=I_1(s)+I_2(s).
i e_{0}^{2}\int \frac{d^{D}\! K}{( 4 \pi )^{D}k^{2}}\gamma_{\nu}\frac{1}{\not{p}+ \not{k}}\left[ f_{n}\gamma_{\mu}- g_{n}\frac{( \not{p}+ \not{k}) \gamma_{\mu}( \not{p}+ \not{k})}{( p + k )^{2}}\right] \left( \frac{\mu^{2}}{( p + k )^{2}}\right)^{n x}\frac{1}{\not{p}+ \not{k}}\gamma^{\nu}
\|(\lambda+A_{p'})^{-1}\|_{\mathcal{L}(L^{p'}(\Omega))}=\|((\lambda+A_p)^{-1})^\ast\|_{\mathcal{L}(L^{p'}(\Omega))}=\|(\lambda+A_p)^{-1}\|_{\mathcal{L}(L^p(\Omega))}\le\frac{C}{|\lambda|}.
\bigl\langle J_\mathcal{N}^{(1)}(k,\ell)\bigr\rangle:=\frac{1}{q}\sum_{m=1}^q \bigl\langle\Psi_{0,m}^{(N)},J_\mathcal{N}^{(1)}(k,\ell)\Psi_{0,m}^{(N)}\bigr\rangle.
\tilde{\delta}( k ) = 2 \operatorname{a r c t a n}\frac{\tilde{m}}{k}+ 2 \operatorname{a r c t a n}\frac{\tilde{m}}{2 k}\mathrm{~ a n d ~}\tilde{\delta}_{1}( k ) = \frac{3 \tilde{m}}{k}
\begin{aligned}W_{j}(t) := u_{j+1}(t)-u_{j}(t), Z_{j}(t) :=\nabla y_{j+1}(t)-\nabla y_{j}(t), Y_{j}(t) :=y_{j+1}(t)-y_{j}(t) , X_{j}(t) :=x_{j+1}(t)-x_{j}(t),\end{aligned}
\omega_E^\Omega = \bigcup\left\{E' \in \mathcal{E}_H|_{\Omega}\;:\; E \cap E' \neq \emptyset \right\}, \omega_E^{\Omega^c}= \bigcup\left\{E' \in \mathcal{E}_H|_{\Omega^c}\;:\; E \cap E' \neq \emptyset \right\}.
G_{\alpha\beta}^TH^{\beta}_i\overline{G_{\alpha\beta}}=H^{\alpha}_i =C^{\alpha}_{i}H^{\alpha}_{1}(\overline{C^{\alpha}_{i}}^{T})=C^{\alpha}_{i}G_{\alpha\beta}^TH^{\beta}_1\overline{G_{\alpha\beta}}(\overline{C^{\alpha}_{i}}^{T}).
\phi_1^{(2k+1)}(\pi) = \left( \hat{\phi}(t)\log\left( 2 (\sin(t/2))^2 \right) \right)^{(2k+1)}_{t = \pi}& = \sum_{j=0}^{2k+1}\binom{2k+1}{j}\hat{\phi}^{(j)}(\pi) \left(\log\left( 2 (\sin(t/2))^2 \right) \right)^{(2k+1-j)}_{t = \pi}& = 0,
c_{\lfloor \frac{r-1}{2}\rfloor, r-1}= \prod\limits_{w=0}^{r-k-1}(n-1+2w) \frac{(-1)^{k-1}\, (r-1)!}{2^{k-1}\, (k-1)! \, 1!}&= -\prod\limits_{w=0}^{r-k-1}(n-1+2w) \frac{(-1)^{k}\, r!}{2^{k}\, k!}\mbox{using}r=2k, &= - c_{k,r}=-c_{\lfloor \frac{r}{2}\rfloor, r}.
K^{\prime}+ \bar{K}^{\prime}= \frac{1}{2}\int_{0}^{T}d \tau \Biggl [ \frac{m_{1}^{2}}{\mu_{1}( \tau )}+ \mu_{1}( \tau ) \left( 1 + \dot{\bf z}^{2}( \tau ) \right) + \frac{m_{2}^{2}}{\mu_{2}( \tau )}+ \mu_{2}( \tau ) \left( 1 + \dot{\bar{\bf z}}^{2}( \tau ) \right) \Biggr ] ,
\begin{cases}x_{s}u^{j}_{s}+ y_{s}v^{j}_{s}+ z_{s}w^{j}_{s}= F^{j}, x_{s}u^{j}_{t}+ y_{s}v^{j}_{t}+ z_{s}w^{j}_{t}+ x_{t}u^{j}_{s}+ y_{t}v^{j}_{s}+ z_{t}w^{j}_{s}= G^{j}, x_{t}u^{1}_{t}+ y_{t}v^{1}_{t}+ z_{t}w^{1}_{t}= H^{j},\end{cases}
\Delta_nF(\Delta_nG, \Delta_{n-1}G\circ\pi_R, \dots, G\circ\pi_R) =\Delta_nF\circ K_nG &\simeq\pi_L\circ T^nF\circ T^nG\circ d_n^* \simeq \pi_L \circ T^n(FG)\circ d_n^*.
D_{i+\frac{1}{2}}:=\frac{1}{\partial_1(\Phi^a+\Psi_{i+\frac{1}{2}})}\partial_1\mathbb{L}({\mathbf U}^a+{\mathbf V}_{i+\frac{1}{2}},\Psi^a+\Psi_{i+\frac{1}{2}}),
\mathcal{D}_{z}= \left( \begin{array}{c c}{{\mathbf{a}+ \mathbf{b}\mathbf{x}}}&{{\mathbf{0}}}\\{{\mathbf{0}}}&{{\mathbf{a}^{\prime}+ \mathbf{b}^{\prime}\mathbf{x}^{\prime}}}\\ \end{array}\right) \equiv \left( \begin{array}{c c}{{D_{z}}}&{{\mathbf{0}}}\\{{\mathbf{0}}}&{{D_{z}^{\prime}}}\\ \end{array}\right) \, ,
V_q < \sum_{\substack{t t \le L}}\frac{1}{t!}- (1 - \beta) \sum_{\substack{t t \le L}}\frac{1}{t!}+ \gamma &= \sum_{t=0}^L \frac{(-1)^t}{t!}+ \beta \sum_{\substack{t t \le L}}\frac{1}{t!}+ \gamma &< e^{-1}+ \alpha + e \beta + \gamma &< e^{-1}+ \epsilon.
(1-t_1a_1)(1-\rho a_1(t_1)a_2(t_2))(1-t_2a_2) &= \frac{h_{a_1}(t_1)+h_{a_2}(t_2)-1}{h_{a_1}(t_1)h_{a_1}(t_2)}(1 - t(a_1+a_2)) &= \frac{h_{a_1+a_2}(t)}{h_{a_1}(t_1)h_{a_2}(t_2)}(1 - t(a_1+a_2)) &= \frac{t_1t_2}{tz}(1 - t(a_1+a_2).
\{x \}_{y}^{\pm}: = \prod_{k = 0}^{\infty}\left( 1 \pm q^{x + k y}\right) \, , \quad \left\{x_{1}; \ldots ; x_{n}\right\}_{y}^{\pm}: = \prod_{a = 1}^{n}\{x_{a}\}_{y}^{\pm}\, .
{\frac{d \tilde{n}^{( \lambda )}( \omega )}{d \omega}}={\frac{d n^{( \lambda )}( \omega )}{d \omega}}-{\frac{1}{4 \lambda}}\sum_{k}\left[ ( \phi_{\omega , k}^{( \lambda )}, \partial_{\lambda}H^{2}( \lambda ) \phi_{\omega , k}^{( \lambda )}) + ( \phi_{\omega , k}^{( \lambda )}, \partial_{\lambda}H^{2}( \lambda ) \phi_{\omega , k}^{( \lambda )})^{*}\right] ,
\bigg( \sum_{k=-\infty}^{M-1}\int_{x_k}^{x_{k+1}}\bigg( \int_{x_k}^t f^r v \bigg)^{\frac{q}{r}}u(t) dt \bigg)^{\frac{1}{q}}\leq \mathfrak{C''}\bigg( \sum_{k=-\infty}^{M-1}2^k \bigg( \int_{x_k}^{x_{k+1}}f \bigg)^p \bigg)^{\frac{1}{p}}
\mathcal{A}^{(-1)}:=\sum_{\alpha}\Big(\hat{X}_\alpha X_\alpha^{(0)}+X_\alpha^{(0)}\hat{X}_\alpha\Big), &\mathcal{A}^{(0)}:=\sum_{\alpha}\Big(\hat{X}_\alpha X_\alpha^{(1)}+X_\alpha^{(1)}\hat{X}_\alpha+\left(X_\alpha^{(0)}\right)^2+\frac{1}{2}R_{\gamma\beta\alpha}^{\beta}(q)x^\gamma \hat{X}_\alpha\Big).
\begin{dcases}\frac{d}{dt}\int_{\mathbb T}\zeta g_{t}~ d\theta =\int_{\mathbb T}\langle \nabla\zeta, \nabla\varphi \rangle g_t ~ d\theta, \forall \zeta \in C^\infty(\mathbb T), \partial_{t}\varphi+\frac{\mid\nabla\varphi\mid^{2}}{2}=0. .\end{dcases}
\prod_{\substack{(a,b)=1 a,b \geq 1}}\left( \frac{1}{1-y^a z^b}\right)^{\frac{1}{a^s b^t}}= \exp\left\{\left( \sum_{j=1}^{\infty}\frac{y^j}{j^s}\right) \left( \sum_{k=1}^{\infty}\frac{z^k}{k^t}\right) \right\}.
\frac{1}{x^n}= \sum_{k=1}^{\infty}(-1)^{n-k}e_{k-n}(a_1,\dots, a_{k-1})\frac{1}{(x|a)_{k}}&= \sum_{k,r=1}^{\infty}(-1)^{n-k}e_{k-n}(a_1,\dots, a_{k-1}) h_{r-k}(a_1,\dots, a_{k})\frac{1}{x^r}= \sum_{k,r=1}^{\infty}A_{-k,-n}A_{r,k}\frac{1}{x^r}.
q(z,s+h)-q(z,s) &\quad{}= \biggl(\int_{\mathbb R}{p}(t-s-h, x_1-y)\sigma(s+h,y)\,dy-\int_{\mathbb R}{p}(t-s, x_1-y)\sigma(s,y)\,dy \biggr) &\qquad{}- \biggl(\int_{\mathbb R}{p}(t-s-h, x_2-y)\sigma(s+h,y)\,dy-\int_{\mathbb R}{p}(t-s, x_2-y)\sigma(s,y)\,dy \biggr) &\quad{}:=D_1-D_2.
\|T\|^2 =\sup_{\substack{f\in \mathbb{E}^{p}_{q}\|f\|_{\mathbb{E}^{p}_{q}}\leq 1}}\left|\int_{-\infty}^\infty f(x) W(x)\mathrm{d}x-\sum_{x\in X_n}\omega(x) f(x) \right|^2 = \sum_{x,y\in X_n}\omega(x)\omega(y) \sum_{k=2n}^\infty e^{qk^p}\, h_{k}(x)h_{k}(y)\, .
\widetilde{B}^{a}= - \left[ \frac{k}{2 \pi}\frac{2 C_{\rho}}{d - 1}\left( u^{- 1}\partial_{-}u \right)^{a}+ \left( \Pi_{- \alpha}( \lambda^{a})^{\alpha \beta}\psi_{- \beta}\right) \right]
\mathrm{H}(X)= \ln\left\{\sqrt{2\pi}\sigma \mathrm{B}(\alpha,\beta)\right\}+\frac{1}{2\sigma^2}\left[\mathrm{E}(X^2)-2\mu \mathrm{E}(X) + \mu^2\right] &+\frac{1}{\mathrm{B}(\alpha,\beta)}\sum_{n=1}^{\infty}\frac{1}{n}\left[(\alpha-1)\mathrm{B}(\alpha,n+\beta)+(\beta-1)\mathrm{B}(n+\alpha,\beta)\right].
\int_0^\tau\int_\Omega p(\rho)b_\alpha(\rho)-I^\alpha_1=-\int_0^\tau\langle\nabla p(\rho), \mathcal{B}(b_\alpha (\rho)-\langle b_\alpha(\rho)\rangle\rangle =\sum_{j=2}^6 I^{\alpha}_{j}+J^{\alpha}(\tau),
\begin{gathered}3 \leq N_{\inf}:= \inf \{N^F : F \in \boldsymbol{F}\}\leq \sup \{N^F : F \in \boldsymbol{F}\}=:N_{\sup}< \infty, 0 < \ell_{\inf}:= \inf \{\ell^F : F \in \boldsymbol{F}\}.\end{gathered}
\begin{array}{rcl}A_{\mu}(x)\rightarrow A_{\mu}^{U_h}(x) = U_h^{\dagger}(x)A_{\mu}(x) U_h(x)+\frac{i}{g_{\circ}}U_h^{\dagger}(x)\partial_{\mu}U_h(x) & U_h(x)=\exp[ig_{\circ}h(x)] & A_{\mu}(x)-A_{\mu}^{U_h}(x)\simeq D_{\mu}(A)h(x)=\partial_\mu h(x)-i g_{\circ}[A_\mu(x),h(x)]\end{array}
\phantom{=}\;\;\L_{I_\rho}(s) =\exp\left(-\int_\rho^\infty (1-e^{-sr^{-1}}) \Lambda(\d r) \right) &=\exp\left(-\Big(s^\beta \int_0^{s \rho^{-1}}r^{-\beta}e^{r}\d r - \rho^\beta (1-e^{-s \rho^{-1}})\Big)\right),
\vert U \vert \leq \sum_{r=1}^{\infty}\sum_{\substack{k=1}}^{2^r}\vert I_k^r\vert\leq\sum_{r=1}^{\infty}\sum_{k=1}^{2^r}\frac{1}{2^{2r}}=\sum_{r=1}^{\infty}\frac{1}{2^{r}}=\frac{1}{2}.
L^{{\rm string}}_{{\rm fluc}}= \frac{\pi}{12 R_p}- \sum_{i=1}^2\frac{\omega v_i \bar\gamma_i}{\pi}\left[ \ln\left(\frac{Mm_i}{\sigma}\right) + 1 \right]+ \frac{1}{2}\omega + \omega f(v_1, v_2) \,.
K(p)>0\quad\forall p\in\overline{\Omega}\setminus\{p_1,\,\cdots\, , p_n\}\ \ \mathrm{and}\qquad \mathrm{order}_{p_j}(K) =2\,\mathrm{order}_{p_j}(e)=2\,\mathrm{order}_{p_j}(g)\ forall j\in\{1,\cdots ,n\}\, ,
\begin{aligned}\|\omega-\omega'\|_{\Omega}=\max_{\alpha=0,\cdots,N}\|f_\alpha-f'_\alpha\|_{\infty}\|\omega-\omega'\|_{\Omega, Y}=\max_{\alpha=0,\cdots,N}\sup_{y\in Y}|(f_\alpha-f'_\alpha)(y)|\quad(Y\subset X), \end{aligned}
\int_{n}^{n+1}B(h, R) ds = \frac{1}{2}\int_n^{n+1}\int_{\mathbb{R}^N}(L_+ h_1) R_1 dxds + \frac{1}{2}\int_n^{n+1}\int_{\mathbb{R}^N}(L_- h_2) R_2 dx ds & \lesssim \| \left\langle \nabla \right\rangle h \|_{S((n,+\infty),L^2)}\| \left\langle \nabla \right\rangle R \|_{S'((n,+\infty),L^2)}& \lesssim e^{-(c_0+ c_1)n},
x_{1}=\frac{|\begin{matrix}c_{1}&a_{2}\\c_{2}&a_{4}\end{matrix}|}{|\begin{matrix}a_{1}&a_{2}\\a_{3}&a_{4}\end{matrix}|}=\frac{c_{1}a_{4}-a_{2}c_{2}}{a_{1}a_{4}-a_{2}a_{3}}
f=\sum_{i\in\mathbb{Z}}\Lambda_i^*\Theta_i f=\sum_{i\in\mathbb{Z}}(T^*)^i\Lambda_0^*\Theta_0S^i f =T^*\Big(\sum_{i\in\mathbb{Z}}(T^*)^{i-1}\Lambda_0^*\Theta_0S^{i-1}\Big)Sf &=T^*\Big(\sum_{i\in\mathbb{Z}}\Lambda_i^*\Theta_i\Big)Sf=T^*Sf.
\begin{array}{c}{{( \xi_{i}- 1 ) a_{i}^{2}= 0}}\\{{a_{i}a_{j}= e^{i ( \lambda_{j i}- \lambda_{i j})}a_{j}a_{i}\qquad i \neq j}}\\{{a_{i}a_{j}^{+}= e^{i ( \lambda_{i j}- \lambda_{j i})}a_{j}^{+}a_{i}, \qquad i \neq j}}\\{{a_{i}a_{i}^{+}= 1 + \xi_{i}a_{i}^{+}a_{i}}}\\ \end{array}
F=\left(\begin{matrix}0 \lambda y 1 0 0 \lambda y 1 0 0\end{matrix}\right), G= \left(\begin{matrix}\tfrac{1}{3\lambda^2}-uu_x \tfrac{1}{\lambda}u_x-\lambda u^2y u_x^2 \tfrac{1}{\lambda}u -\tfrac{2}{3\lambda^2}-\tfrac{1}{\lambda}u_x-\lambda u^2y -u^2 \tfrac{1}{\lambda}u \tfrac{1}{3\lambda^2}+uu_x\end{matrix}\right).
f^{\prime\prime}(x)\approx\frac{\delta_{h}^{2}[f](x)}{h^{2}}=\frac{\frac{f(x+h)-f(x)}{h}-\frac{f(x)-f(x-h)}{h}}{h}=\frac{f(x+h)-2f(x)+f(x-h)}{h^{2}}.
\int_{0}^{\infty}P_{r}^{\bf V}( y ) P_{s}^{\bf V}( y ) \, y^{- 1}e^{- y}\, d y \, \propto \int_{0}^{\infty}L_{r - 1}^{( 1 )}( y ) L_{s - 1}^{( 1 )}( y ) y \, e^{- y}\, d y \, \propto \delta_{r \, s},
\Delta_{k}' = \displaystyle \sum_{sa_{1},b_{1}}\displaystyle \sum_{v \in L(k)}a_{1}^{\ast}v^{-1}(^{\ast}b_{1}) [b_{1}va_{1}] &= \displaystyle \sum_{sa_{1},b_{1}}\displaystyle \sum_{v,r \in L(k)}r^{\ast}(v^{-1}) a_{1}^{\ast}r (^{\ast}b_{1}) [b_{1}va_{1}]
\phi ( \tilde{\tau}) = \frac{1}{2 i}\left\{\frac{\pi \tilde{\tau}}{\omega_{2}}\; \frac{\theta_{1}^{\prime}}{\theta_{1}}( \frac{\pi a}{2 \omega_{2}}) + \log \mid \frac{\theta_{1}( \frac{\pi ( \tilde{\tau}- a )}{2 \omega_{2}})}{\theta_{1}( \frac{\pi ( \tilde{\tau}+ a )}{2 \omega_{2}})}\mid \right\},
\eta_{2}( z ) = \mathrm{}- \frac 1 2 \int_{0}^{\infty}\! \!{\mathrm{d}}t \left( \frac{\sin ( t )}{t^{2}}+ 3 \Big [{\frac{\cos ( t )}{t^{3}}}-{\frac{\sin ( t )}{t^{4}}}\Big ] \right) \left[ \frac{2 \pi}{z t}\, \frac{1}{{\mathrm t h}( 2 \pi t / z )}- \frac{1}{t^{2}}\right] .