harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(t^{a_1}\frac{\partial}{\partial t}t^{a_2}\right)^{\alpha}h_{\alpha}(x,t)= \frac{\partial^2}{\partial x^2}h_{\alpha}(x,t)+\frac{t^{-\alpha(1-a_1-a_2)-a_2}}{(1-a_1-a_2)^{-\alpha}}\frac{\delta(x)}{\Gamma(1-\alpha)}.
\frac{d}{dr}e^{C\psi(r)}\left(N(r) + C \psi(r)\right) = e^{C\psi(r)}\left(N'(r) + C \psi'(r) + C \psi'(r) N(r) + C^2 \psi(r) \psi'(r)\right) & \ge C^2 e^{C\psi(r)}\psi(r) \psi'(r) \ge 0,
\{x^{\mu}, g_{\mu \nu}( x ) , \psi_{\mu \nu . . .}( x ) \}\longrightarrow \{x^{\mu}, \Omega^{2}( x ) \tilde{g}_{\mu \nu}( x ) , \Omega^{- d}( x ) \tilde{\psi}_{\mu \nu . . .}( x ) \}.
K_{D}({\bf r}'',{\bf r}' ; t'',t') = \int_{{\bf r}(t') ={\bf r}'}^{{\bf r}(t'') ={\bf r}''}\;{\cal D}{\bf r}(t) \,\exp \left\{\frac{i}{\hbar}S \left[{\bf r}(t) \right] ({\bf r}'',{\bf r}' ; t'',t') \right\}\;
\theta ( \vec{x}) \mid U_{\pm}= \int_{N}\frac{- 2 i S_{0}}{1 + z_{\pm}( \vec{x}) \overline{{{z}}}_{\pm}( \vec{x})}\overline{{{z}}}_{\pm}( \vec{x}) \delta z_{\pm}( \vec{x}) d \mu ( \vec{x}) \, .
\lambda_1 = -1 -3^3-3^4-3^5-3^6-3^8+3^9+O(3^{10}), \lambda_2 = -3 -3^2 -3^3-3^4+3^6-3^8-3^9 + O(3^{10}), \lambda_3 = \mskip14mu 3^2 -3^3-3^5+3^6-3^8+3^9 + O(3^{10}).
Q_k(y) = \frac{1}{\Delta_{k+1}(a)}\begin{vmatrix}1 1 \cdots 1 a_1 a_2 \cdots a_{k+1}\vdots \vdots \ddots \vdots \varphi(y/a_1)/a_1 \varphi(y/a_2)/a_2 \cdots \varphi(y/a_{k+1})/a_{k+1}\end{vmatrix}
\int_{B_R}\frac{\phi(t)-\phi(0)}{|t|^{n+1+s}}\, d t =\frac12\int_{B_R}\frac{D^2\phi(0)[t^2]}{|t|^{n+1+s}}\, d t + O(1) &= \frac12 \frac{area(S^{n-1})}{n}\frac{R^{1-s}}{1-s}\Delta \phi(0) + O(1)
0 = \sum_{d=0}^m \left( -\frac{p}{1-p}\right)^d \langle h^{=d}, g \rangle \geq \mu_{p}(h) \mu_{p}(g) - \sum_{d=1}^m \left( \frac{p}{1-p}\right)^d | \langle h^{=d}, g \rangle | .
I_{\delta,\theta}\geq \frac{\rho}{2}\Big\{3(1-\theta)(1-\nu)\ln T_{tr}+(1-\theta)\nu\ln\det\Theta+3\theta \ln T_{\delta}+\delta\ln T_{\theta}-(3+\delta)\ln T_{\delta}\Big\}\cr = \frac{\rho}{2}\Big\{~(1-\theta)\big(3(1-\nu)\ln T_{tr}+\nu\ln\det\Theta-3 \ln T_{\delta}\big) +\delta\ln T_{\theta}-\delta\ln T_{\delta}\Big\}.
a(z,\psi) + b(z,\lambda_\psi) = (z, \widetilde \delta^{-1}(w - w_h)) \forall z \in \{z \in \prod_i H^1(\Omega_i)\,|\, z|_{\partial\Omega}= 0\}, b(\psi,q) = 0 \forall q \in \prod_{i}H^{1/2}_{00}(\gamma_i).
\begin{aligned}v_{2}(x) \leq M+C_1Mr_{k_0}^{s-n/p}\leq M+C_1Mr_{k_0}^{\beta}\leq \hat{C}Mr_{2k_0}^{\beta}\left(\frac{1}{\hat{C}\tau_1^{2k_0\beta}}+\frac{C_1}{\hat{C}\tau_1^{k_0\beta}}\right).\end{aligned}
cost=\frac{cott}{\pm\sqrt{1+cot^{2}t}}=\frac{-ma}{\pm\sqrt{m^{2}a^{2}+b^{2}}},sint=\frac{1}{\pm\sqrt{1+cot^{2}t}}=\frac{b}{\pm\sqrt{m^{2}a^{2}+b^{2}}}.
\sum_{i=1}^m\sum_{k\ge0}\frac{\lambda\theta_i^{-1}}{(k+\mu_i)^{1+a}}\le\sum_{i=1}^m\int_{-1}^\infty\frac{\lambda\theta_i^{-1}dt}{(t+\mu_i)^{1+a}}\le\sum_{i=1}^m\frac{\lambda}{\theta_i a(\mu_i-1)^a}=:\mathcal{A}_m.
\|y_v\|_1=\tau(y_v)=\frac{2^{v+1}}{A_j}\int_{2^v}^{2^{v+1}}\left\|\frac{\partial T_{s,j}(x_{v+j})}{\partial s}\right\|_2^2ds+2\sum_{k=0}^{A_j-1}\|T_{{\gamma_k}, j}(x_{v+j})\|_2^2
\begin{array}{lll}A_{ss}:= k_j + \lambda + \gamma_{k_j}, B_{ss}:= \mu - \nu, s\sim (j,m_j,k_j)\in I_1 A_{ss}:= \mu + \nu, B_{ss}:= -(k_j + \lambda - \gamma_{k_j}), s\sim (j,m_j,k_j)\in I_2, A_{st}=B_{st}=0, s \sim (j,m_j,k_j)\in I_1\cup I_2, \, 1\leq t\leq d, t\neq s. \end{array}
J^{i}( z e^{i 2 \pi n}) = e^{- 2 i \pi n \alpha_{0}^{-}\left( J_{0}^{0}+ J_{0}^{1}\right)}J^{i}( z ) \, e^{2 i \pi n \alpha_{0}^{-}\left( J_{0}^{0}+ J_{0}^{1}\right)}
\begin{array}{lcl}0 = s^{-1}\log_1(\pi_2) (v_1-v_2), - (\lambda_1+k^{-1}\mu_1 a_1) = v_1 , 0 = s^{-1}\log_1(\pi_2) (v_2-v_1), -(\lambda_2 +k^{-1}\mu_2 a_2) = v_2.\end{array}
\begin{aligned}\sup_{|\beta| \le 10}\sup_{\psi_1}\int_0^s \int_{\Sigma_t}|\partial^\beta \partial^\alpha (f g(d \phi,d \phi))| (|\partial_y \psi| + |\partial_z \psi| + |(\partial_t + \partial_x) \psi|) d x d t. \end{aligned}
H_{\mathrm{D i r a c}}= \int_{0}^{L}d x \left( \psi_{L}^{\dagger}( x ) ( i \nabla + e A ( x ) ) \psi_{L}( x ) - \psi_{R}^{\dagger}( x ) ( i \nabla + e A ( x ) ) \psi_{R}( x ) \right)
\int_s^t \int_s^u \Phi_{n-1}(t-r) f(r) dr du = \int_s^{t}\int_s^{v_n}\int_s^{v_{n-1}}\cdots \int_s^{v_2}\prod_{i=2}^{n}\Phi(v_{i}-v_{i-1}) f({v_1}) dv_1 \cdots dv_{n}.
X^{\mu}( \tau , \sigma ) = q^{\mu}+ a_{0}^{\mu}\tau + ( \frac{\pi}{2}- \sigma ) B_{\ \nu}^{\mu}a_{0}^{\nu}+ \sum_{n \neq 0}\frac{1}{n}e^{- i n \tau}( i a_{n}^{\mu}\cos n \sigma - B_{\ \nu}^{\mu}a_{n}^{\nu}\sin n \sigma ) .
|p_{0N}\rangle=\lim_{\tau \rightarrow -\infty}V_{T}(\sigma,\tau;p_{0}) |\mathbf{0}\rangle =e^{ip_{0\mu}\hat{x}_{0}^{\mu}}|\mathbf{0}\rangle~,\quad\langle -p_{0N}| =\langle \mathbf{0}| e^{ip_{0\mu}\hat{x}^{\mu}_{0}}~.
\int_{D_{\pm}}\nabla u_{1,\pm}\nabla v - \int_{\partial D_{\pm}}\frac{\partial u_{1,\pm}}{\partial \mathbf{n}}v = \lambda \int_{D_{\pm}}mu_{1,\pm}v + \mu_{1,\pm}\int_{D_{\pm}}u_{1,\pm}v \mbox{for all}\v \in \mathcal{C}^1(\overline{D_{\pm}}).
\left\vert \int_{\epsilon}^{\infty}r^{\alpha}e^{-\mu r}\mathrm dr \right\vert \leq \frac{2}{\Re\mu}e^{\frac{\epsilon}{2}\Re{\mu}}, & \int_{0}^{\epsilon}r^{\alpha}e^{-\mu r}\mathrm dr = \frac{\Gamma(\alpha+1)}{\mu^{\alpha+1}}+\mathcal O\left( \frac{2}{\Re{\mu}}e^{-\frac{\epsilon}{2}\Re{\mu}}\right),
\int_{\check{Y}_{\lambda,\epsilon}}{\int_{\check{Y}_{\lambda,\epsilon}}{\frac{|u_{\lambda,\epsilon}(x)-u_{\lambda,\epsilon}(y)|^2}{|x-y|^{n+2s}}\,dx}dy}=\sum_{\substack{i,j=1 i\neq j}}^k{\int_{\check{Y}_{\lambda,\epsilon}^i}\int_{\check{Y}_{\lambda,\epsilon}^j}\frac{|u_{\lambda,\epsilon}(x)-u_{\lambda,\epsilon}(y)|^2}{|x-y|^{n+2s}}\,dx\,dy}
A'(t) \geq \kappa_2\sum_{j=1}^{J_0}\theta_j\beta_j^2(t)+\frac{1}{C}\gamma^2(t), A'(t) \geq \frac{1}{\kappa_2}\sum_{j=1}^{J_0}\theta_j\left(\lambda_j'(t)\right)^2+\frac{1}{C}\gamma^2(t).
\tilde{J}_N^{\ast}(x) =\min_{\boldsymbol{u}}\sum_{k=0}^{N-1}\tilde{\ell}(x_k,u_k) + \tilde{F}(x_N) \mbox{s.\,t.}\ \{x}_{k+1}=A{x}_{k}+B{u}_{k}, \{x}_0=x\,,
\frac{\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (-s^{(1)}_{k})_{k=0}^{m-1}\right)}{b\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (1,0,\ldots,0) \right)}\stackrel{a\to0+}{\longrightarrow}\frac{\mathcal{D}\left((-s^{(1)}_{k})_{k=0}^{2m-3}, (-s^{(1)}_{k})_{k=m}^{2m-1}\right)}{b\mathcal{E}\left((-s^{(1)}_{k+1})_{k=0}^{2m-3}, (1,0,\ldots,0) \right)}>0.
\Big|\frac{1}{M}\sum_{k=0}^{R-1}e\big(\frac{m k}{M}\big) \Big| \leq \min\Big(1, \Big|\frac{e(m R/M) -1}{M(e(m /M)-1)}\Big|\Big) \leq \frac{|\sin(\pi m R/M)|}{\bar{m}}\leq \frac{1}{\bar{m}}.
I_{n , B}= \frac 1{( 4 \pi^{2})^{2}}\int_{0}^{\Lambda}\, | \varphi_{k}\dot{\varphi}_{k^{\prime}}- \dot{\varphi}_{k}\varphi_{k^{\prime}}|^{2}\frac{k^{2}k^{2}k^{2 n}}{k^{2}- k^{2}}\coth \frac{\beta W_{k}}2 \coth \frac{\beta W_{k^{\prime}}}2 \, d k d k^{\prime}\ .
T^{+ 2}= - 2 i e_{q}^{+}e_{q}^{+}- 2 i \pi_{q}^{2 +}\pi_{q}^{2 +}, \quad T^{- 2}= - 2 i \pi_{\dot{q}}^{1 -}\pi_{\dot{q}}^{1 -}- 2 i e_{\dot{q}}^{-}e_{\dot{q}}^{-}
\left(-\frac{\hbar^2}{2m}\left(\frac{d^2}{d\rho^2}+\frac{(D-1)}{\rho}\frac{d}{d\rho}-\frac{\cal{L}(\cal{L}+D-2)}{\rho^2}\right) +{\cal V}(\rho)-{\cal E}\right)\tilde R(\rho)=0,
\begin{array}{ll}\kappa_{+}=(f_{x}+ig_{x})+i(f_{y}+ig_{y}), \qquad \kappa_{-}=(f_{x}+ig_{x})-i(f_{y}+ig_{y}), \lambda_{+}=(f_{x}-ig_{x})+i(f_{y}-ig_{y}), \qquad \lambda_{-}=(f_{x}-ig_{x})-i(f_{y}-ig_{y}).\end{array}
\varphi^* \left(\begin{array}{c c}a b 0 d \end{array}\right) := \left(\begin{array}{c c c}a^{\ell_1}0 0 0 a^{\ell_1}a^{\ell_1}\, b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{- \ell_3}\end{array}\right) .
\begin{aligned}d_k\leq \frac{c^k d_0}{\phi(y_k)\cdot \cdots \cdot \phi(y_1)}+\frac{c^k\zeta Y_1(y_1)}{\phi(y_k)\cdot \cdots \cdot \phi(y_2)}+\frac{c^{k-1}\zeta Y_2(y_2)}{\phi(y_k)\cdot \cdots \cdot \phi(y_3)}+\cdots + \frac{c^2\zeta Y_{k-1}(y_{k-1})}{\phi(y_k)}+c\zeta Y_{k}(y_k). \end{aligned}
\left\langle \Phi_D^2\right\rangle_\phi =\left\langle \Phi_{ND}^2\right\rangle_\phi =\left\langle \Phi_D^2\Phi_{ND}^2\right\rangle_\phi =\left\langle \Phi_D^2\Phi_D^2\right\rangle_\phi \equiv 0 .
0 = \big\langle\omega,\mu_1^\mathrm{pp}\big\rangle_{Z_{1,\infty}}= \big\langle\omega,\mu_1^\mathrm{pp}\big\rangle_{Z_{1,R}}+ \big\langle\omega,\mu_1^\mathrm{pp}\big\rangle_{Y_{[R,\infty)}}.
\partial_t\tilde v - \Delta\tilde v + \tilde v\cdot\nabla_H\tilde v + v_3v_z + \bar v\cdot\nabla_H\tilde v = - \tilde v\cdot\nabla_H\bar v + \frac1h \int_{-h}^0 (\tilde v\cdot\nabla_H\tilde v + \mathrm{div}_H\,v\,\tilde v)\,dz + \frac1h v_z|_{\Gamma_b}\quad\;\Omega.
\int_{\mathbb{R}}F(x+\Delta)dF_{\lfloor n\tau\rfloor +1, \lfloor n\lambda\rfloor}(x)-\int_{\mathbb{R}}F(x+\Delta)dF(x) &=\int_{\mathbb{R}}F(x)dF(x+\Delta)-\int_{\mathbb{R}}F_{\lfloor n\tau\rfloor +1, \lfloor n\lambda\rfloor}(x)dF(x+\Delta).
\delta ~ ( b ) = ~ \frac{1}{4 \pi}~ \int_{0}^{\infty}\frac{d t}{t}e^{- b^{2}t / 2 \pi \alpha^{\prime}}~ [ \prod_{i = 1}^{l}\Theta_{3}( 0 , 8 i \pi^{2}\alpha^{\prime}t / L_{i}^{2}) ] ~ [ \prod_{i = p + 1}^{c}\Theta_{3}( 0 , i t L_{i}^{2}/ 2 \pi \alpha^{\prime}) ] \cdot ( B \times J ) ~ ~ .
2\big(N-\sqrt{N[1+(N-1)c]}\big) =2\Big(N-N\sqrt{\frac{1}{N}+\frac{N-1}{N}c}\Big) begin{align*}2 pt] >2\Big(N-N\sqrt{\frac{1}{N}+c}\Big) begin{align*}2 pt] >2(N-N\sqrt{2c}) begin{align*}2 pt] =2N(1-\sqrt{2c}),
\|y_k^1(t) - y_k^2(t)\| \leq \|y_k^{0,1}- y_k^{0,2}\| &+ \int_0^t \left(\|(K\star \mu^1(s))(y_k^1(s))- (K\star\mu^2(s))(y_k^2(s))\| +\|f_k(y^1(s))-f_k(y^2(s))\|\right)\,ds.
\left[ \frac{d^{2}}{d t^{2}}+ \vec{k}^{2}+ m_{R}^{2}( t ) + \frac{\lambda_{R}}{2}\left( \langle \Phi^{2}( t ) \rangle - \langle \Phi^{2}( 0 ) \rangle \right) \right]{\cal{U}}_{k}^{\pm}( t ) = 0
\tilde Q(x_1,x_2)f(k,l) =\sum_{(j,i)\in \mathcal M \times \mathcal M}\tilde q_{(k,l),(j,i)}(x_1,x_2)(f(j,i)-f(k,l)) &=\sum_{j\in \mathcal M}(q_{kj}(x_1)- q_{lj}(x_2))^+(f(j,l)-f(k,l)) & +\sum_{j\in \mathcal M}(q_{lj}(x_2)-q_{kj}(x_1))^+(f(k,j)-f(k,l)) & +\sum_{j\in \mathcal M}(q_{kj}(x_1)\wedge q_{lj}(x_2))(f(j,j)-f(k,l))
C(\beta)=2\,\int_{\mathbb{R}^N}\frac{\left||x|^{-\beta}-|y|^{-\beta}\right|^{p-2}\, \left(|x|^{-\beta}-|y|^{-\beta}\right)}{|x-y|^{N+s\,p}}\, dy,\mbox{for every}x\in\mathbb{S}^{N-1},
{\cal B}= \left( \begin{matrix}{H_{i j}}&{}&{0}\\{0}&{}&{- k^{- 2}}\\ \end{matrix}\right) \left( \begin{matrix}{\delta_{k j}}&{}&{H_{k l}^{- 1}U_{l}^{-}}\\{k^{2}U_{j}^{+}}&{}&{1}\\ \end{matrix}\right) ,
\left\vert\iint_{\mathbb{R}^{N+1}_+}\left\vert\nabla\omega_{\infty}\right\vert^{2}-\left\vert\nabla\omega^{R}\right\vert^{2}\right\vert =O(R^{N-1}e^{-2{m}(1-\varepsilon)R}), \left\vert\mathbb{D}(\omega_{\infty})-\mathbb{D}(\omega^{R})\right\vert =O(R^{N-1}e^{- \theta m (1-\varepsilon)R}).
\mathcal{C}_{i,j}= \mathbf{1}_{G_{j,l}\cap I_{j,l}}\int_{y_{j-1}}^{x_i}\Biggl| \sum_{\substack{n \leq x_i/t P(n) \leq y_{j-1}}}\frac{f(n)}{\sqrt{n}}\Biggr|^2 \sum_{\substack{t / (1 + 1/V) \leq m \leq t p|m \Rightarrow p \in (y_{j-1}, y_j]}}\frac{V \tau_q (m)}{m^2}dt \, .
\sum_{100 \le j\le M}\int_{\mathbb R^2}b(x) (\nabla F)(x) \cdot 2^j (\nabla p)(\tau, 2^{-j}x) dx = \int_{\mathbb R^2}b(x) (\nabla F)(x) \cdot \Delta^{-1}\nabla H dx = - \int_{\mathbb R^2}\Delta^{-1}\nabla \cdot (b(x) (\nabla F)(x)) H dx.
\begin{aligned}a_{m,n}(y, \varepsilon) := \max\{i \mid \max t_i^{m,n}\leq y+\varepsilon \}, b_{m,n}(y, \varepsilon) := \max\{i \mid \min t_i^{m,n}< y-\varepsilon \}c_{m,n}(y,\varepsilon) := \max\{b_{m,n}(y,\varepsilon) - a_{m,n}(y,\varepsilon), 0\}= \# \{i \mid \min t_i^{m,n}< y-\varepsilon \ \& \ \max t_i^{m,n}> y+\varepsilon \}. \end{aligned}
\begin{aligned}w = [ i_1,n ] \; t_{n+1}[ i_2,n ] \; t_{n+1}\dots [ i_p,n ] \; t_{n+1}( [ 0,n ] \; t_{n+1})^k \; w_r p>0 , w = ( [ 0,n ] \; t_{n+1})^k \; w_r p=0, \end{aligned}
\Lambda_{a}^{( \frac{1}{2})}\Lambda_{b}^{( \frac{1}{2})}= S_{a b}{\bf 1}+ \frac{1}{2}\, ( d_{a b c}+ i f_{a b c}) \, \Lambda_{c}^{( \frac{1}{2})}\quad \quad ( a , b , c = 1 , 2 , \dots 8 )
\nabla^{2}\Phi=\frac{1}{\sigma^{2}+\tau^{2}}[\frac{1}{\sigma}\frac{\partial}{\partial\sigma}(\sigma\frac{\partial\Phi}{\partial\sigma})+\frac{1}{\tau}\frac{\partial}{\partial\tau}(\tau\frac{\partial\Phi}{\partial\tau})]+\frac{1}{\sigma^{2}\tau^{2}}\frac{\partial^{2}\Phi}{\partial\varphi^{2}}
J_1:= -2E\left[\int_t^T\langle\partial_j(u-k_m)^+(s),a^{ij}\partial_i(u-k_m)^+(s)+\sigma^{jr}(s)v^{r,k_m}(s)\rangle \, ds|\mathcal{F}_t\right] \leq -\lambda E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]+\frac{1}{\varrho}E\left[\int_t^T\|v^{k_m}(s)\|^2\,ds|\mathcal{F}_t\right].
d^{2}f=(\frac{\partial^{2}f}{\partial x^{2}}(dx)^{2}+2\frac{\partial^{2}f}{\partial x\partial y}dxdy+\frac{\partial^{2}f}{\partial y^{2}}(dy)^{2})+\frac{\partial f}{\partial x}d^{2}x+\frac{\partial f}{\partial y}d^{2}y.
{\omega}_{\gamma_{i}}\sim{\frac{\rho^{3}R^{d_{i}}( \rho d^{4 - d_{i}}x_{0}+ x_{0}d \rho \, d^{3 - d_{i}}x_{0})}{( \rho^{2}+ x_{0}^{2})^{4}}}, \quad{\Omega}_{\gamma_{i}}\sim{\frac{R^{d_{i}}( 3 \rho^{2}+ x_{0}^{2}) x_{0}d^{3 - d_{i}}x_{0}}{( \rho^{2}+ x_{0}^{2})^{3}}}, \quad \mathrm{f o r}\, \, x_{0}\rightarrow \infty ,
F(k\mid x)\propto F(k)\sum_{d=1}^{\infty}M(d)\prod_{i=1}^{d}\bigg{\{}1+ \bigg{(}\frac{d}{i}\bigg{)}^{2k+1}\bigg{\}}^{-1/2}\exp\bigg{(}\sum_{i=1}^{d}\frac{x^{2}_{i}}{2\varepsilon^{2}}\frac{(d/i)^{2k+1}}{(d/i)^{2k+1}+1}\bigg{)}.
\begin{aligned}\left(uu_x,u_{xx}\right) \leq \|u\|_{L^2(I)}\|u_x\|_{L^\infty(I)}\|u_{xx}\|_{L^2(I)}&\leq \sqrt{2}\,\|u\|_{L^2(I)}\|u_x\|_{L^2(I)}^\frac{1}{2}\|u_{xx}\|^\frac{3}{2}_{L^2(I)}&\leq \epsilon^{-3}\,\|u\|^4_{L^2(I)}\|u_x\|^2_{L^2(I)}+ \frac{3\epsilon}{4}\|u_{xx}\|^2_{L^2(I)}. \end{aligned}
R_n(t;z) \prod_{k=0}^{n-1}\frac{1 + \mathcal{R}e(\rho_{k}(t)\alpha_{k-1})}{1 + \overline{\rho_{k}(t) \alpha_{k-1}}}= z\hat{S}_{n-1}(z) - \rho_{n-1}(t)\hat{S}_{n-1}^{\ast}(z), n \geq 1,
\dim\pi_1-\chi_{\pi_1}(h_1) =(q+1)(q^2+q+1)-(q+1)(1+\chi_{\pi_1}(-1)+\chi_{\pi_1}^{-1}(-1)) &=(q+1)(q^2+q+1)-(q+1)(1+2(-1)^j) &=(q+1)(q^2+q-2(-1)^j)
R_{k,j}^m=B_{\rm RF}\log_2 \left( 1 + \frac{p_{k,j}^m \left|G_{k,j}^m\right|^2}{{\sum\limits_{j'\ne j}\sum\limits_{k'\ne{{k}}}{p_{k',j'}^{{m}}{{{\left|G_{k',j}^m\right|^2}}}}}+ N_{\rm RF}B_{\rm RF}}\right), k\ne 0,
\frac{d}{dt}L=\frac{d}{dt}\sum_{i}^{N}L_{i}=\sum_{i}^{N}\frac{d}{dt}L_{i}=\sum_{i}^{N}\tau_{i}=\sum_{i}^{N}(\tau_{i}^{ext}+\tau_{i}^{int})=\tau^{ext}+\sum_{i}^{N}\tau_{i}^{int}
J(\nu;\nu_{\delta}) - J(\nu_{\delta};\nu_{\delta}) &= J(\nu;\nu_{\delta}) - J(\nu;\nu_{\ast}) + J(\nu_{\ast};\nu_{\ast}) - J(\nu_{\delta};\nu_{\delta}) + J(\nu;\nu_{\ast}) - J(\nu_{\ast};\nu_{\ast}) &\geq -\epsilon.
g_n(t,\omega):= \langle A(t, X_n(t,\omega)), X_n(t,\omega) - X(t,\omega) \rangle , F_n(t,\omega):= f(t)(2+\Vert X_n(t,\omega)\Vert_H^2) + C\Vert X(t,\omega)\Vert_V^{\alpha}& + C \Vert X_n(t,\omega)\Vert_H^{\beta(\alpha-1)}\Vert X(t,\omega)\Vert_V^{\alpha}.
\theta\left(\alpha\left(-n_{1}\right)\alpha\left(-n_{2}\right)\cdots\alpha\left(-n_{k}\right)\otimes e^{\beta+\lambda_{i}}\right)=\left(-1\right)^{k}\alpha\left(-n_{1}\right)\cdots\alpha\left(-n_{k}\right)\otimes e^{-\beta-\lambda_{i}}
-\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\cdot\frac{1}{e^{\lambda_{n}^{2}x}+1}=\frac{1}{4\left(1+\frac{1}{\pi p}\right)}+\frac{1}{2}\sqrt{\frac{\pi}{x}}\left(\sqrt{2}-1\right)\zeta\left(\frac{1}{2}\right)+\sqrt{\frac{\pi}{2x}}\,\sum_{n=1}^{\infty}\frac{G_{p}\left(\frac{\pi(2n-1)}{x}\right)}{\sqrt{2n-1}},
{\cal L}_{\mathrm{D u a l}}\approx \frac{\tau \kappa^{2}}{2}B_{m}B^{m}- \frac{( \tau \kappa^{2})^{2}}{2 m}\epsilon^{m n p}B_{m}\partial_{n}B_{p}+ \tau \kappa^{2}O ( \kappa^{2})
0 =\lim_{k\to\infty}\left\{||v_{n_k}||_{L^{2}(\mathbb{R}^{3})}+||b_{n_k}(\cdot+z_{k}e_z)-b||_{L^{2}(\mathbb{R}^{3})}\right\}&\geq\liminf_{k\to\infty}\left( ||v_{n_k}||_{L^{2}(\mathbb{R}^{3})}+\inf\left\{\ ||b_{n_k}-\tilde{b}||_{L^{2}(\mathbb{R}^{3})}\ \middle|\ \tilde{b}\in S_{h,\gamma}\right\}\right)\geq \frac{\varepsilon_0}{2}>0.
: e^{i 2 \sqrt{\pi}\; \varphi^{( b )}( x )}:_{Q^{W}}\; = \; : e^{i 2 \sqrt{\pi}\; \varphi^{( b )}( x )}:_{Q^{\mu}}\; \; \Big ( \mu \Big )^{\; \sum_{I = 2}^{N}( U_{I b})^{2}}\; .
\Sigma_1 : = p^{(w-1) h_p}\sum_{m=0}^{k_p-h_p-1}\sum_{n=0}^{\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m( \frac{1}{2}+s_1 -z)}p^{n(\frac{1}{2}+s_2+z)}}
{A}=\frac{1}{2(1-\eta)^2}\, \bigg[\frac{\Gamma(\frac{\eta}{2-2\eta})}{2\sqrt{\pi}\Gamma(\frac{1}{2-2\eta})}\bigg]^{\eta}\exp\bigg\{-\int_0^{\infty}\frac{dt}{t}\Big(\frac{\sinh(\eta t)}{\sinh(t)\cosh((1-\eta)t)}-{\eta}\, e^{-2 t}\, \Big) \bigg\}\, ,
\int_0^{2\pi}|\Phi(re^{i\theta})|^p \,d\theta &= \int_0^{2\pi}|w(re^{i\theta}) - T(f)(re^{i\theta})|^p \,d\theta &\leq \int_0^{2\pi}|w(re^{i\theta})|^p \,d\theta + \int_0^{2\pi}| T(f)(re^{i\theta})|^p \,d\theta &\leq ||w||^p_{H^p(D)}+ \int_0^{2\pi}| T(f)(re^{i\theta})|^p \,d\theta.
\begin{aligned}f(r,R;|b^j|)\coloneqq \frac{|b^j|^2}{8\pi}\frac{E}{1-\nu^2}\Big(2+\frac{r^2}{R^2}\Big(\frac{r^2}{R^2}-2\Big)-2\log R\Big) &+\frac{|b^j|^2}{32\pi}\frac{E}{(1-\nu)^2(1+\nu)}\frac{r^2}{R^2}\Big(\frac{R^2}{r^2}-1\Big)\Big(\frac{r^2}{R^2}\Big(\frac{R^2}{r^2}+1\Big)-2\Big)\,.\end{aligned}
\omega^{(g)}_{m,n}\coloneqq \Bigl(W^{(g)}_{m,n}+\delta_{(g,m+n),(0,1)}\Bigr) \prod_{i=1}^m \Bigl(-\frac{dx_i}{x_i}\Bigr)\prod_{i=m+1}^{m+n}\Bigl(-\frac{dy_i}{y_i}\Bigr).
\begin{array}{rl}R(3,G,2) = \displaystyle\sum_{\{a,b,c\}\textrm{is an A.P. in}G}p(x_a,x_b,x_c) &= \displaystyle\sum_{\{a,b,c\}\textrm{is an A.P. in}G}\frac{x_ax_b+x_ax_c+x_bx_c+1}{4}.\end{array}
(j^+_1)^{n_{1}}(j^+_2)^{n_{2}}\cdots (j^+_r)^{n_{r}}= 0, (j^-_1)^{n_{1}}(j^-_2)^{n_{2}}\cdots (j^-_r)^{n_{r}}= 0,{\rm for}~~ n_1+n_2+\cdots+ n_{r}= k+1.
\begin{array}{c}\varphi(t,y,x_t(\cdot)) - \varphi(\tau,z,w(\cdot)) = (t - \tau) \partial^{ci}_{\tau,w}\varphi(\tau,z,w(\cdot)) begin{align*}0.2cm]+ \langle y - z, \nabla_z \varphi(\tau,z,w(\cdot)) \rangle + o(|t - \tau| + \|y - z\|),\end{array}
|\psi(2^k{\bar \rho}(x,y))-\psi(2^k{\bar \rho}(x',y))| \le C2^k({\bar \rho}(x,x'))^\varepsilon [{\bar \rho}(x, y) +{\bar \rho}(x', y)]^{1-\varepsilon}&\le C2^k2^{-k(1-\varepsilon)}({\bar \rho}(x,x'))^\varepsilon &=C(2^k{\bar \rho}(x,x'))^\varepsilon\qquad\{\bar \rho}(x,x')\le (A_0)^32^{5-k}.
A_{x_\perp}^{21}=\int_{R^2}d^2z\,\psi_{-\frac{1}{2}}^\dagger(z+\omega)\frac{\partial}{\partial x_\perp}\psi_{\frac{1}{2}}(z-\omega)=\frac{\bar x_\perp}{2\pi |x_\perp|}e^{2ix\cdot\omega}(I+J).
\begin{array}{ll}\|\alpha\|_{\mathcal{C}_0^{k-1}}=\|\frac{u^s_{yy}}{u^s_y}\|_{\mathcal{C}_0^{k-1}}\leq C_{\ell}\|\frac{u^s_{yy}}{u^s_y}\|_{\mathcal{C}_{\ell}^{k-1}}= C_{\ell}\|\alpha\|_{\mathcal{C}_{\ell}^{k-1}}\leq C(T).\end{array}
\delta{\cal L}_{\mathrm{a n o m a l y}}\ = \ \frac{\theta_{i}}{3 2 \pi^{2}}\ \sum_{a}b_{a}^{\prime i}F^{a}\widetilde{F}^{a}\ - \ \frac{\theta_{i}}{7 6 8 \pi^{2}}\b_{g r a v}^{\prime i}R \widetilde{R}\ ,
\begin{array}{r c l}{{\hat{\cal K}}^{( 2 )}}{=}{d{\hat{\omega}}^{( 1 )}+ l_{p}^{- 2}\, \partial{\hat{X}}^{M}\partial{\hat{X}}^{N}( i_{\hat{k}}{\hat{C}}^{( 3 )})_{M N}\, ,}\end{array}
\sum_{n=0}^{\infty}p_{-6}\left(5^{j}n+\dfrac{3\times5^{j}+1}{4}\right)q^{n}\equiv b(j,1)\dfrac{E_{5}^{6}}{E_{1}^{12}}\equiv b(j,1)E_{5}^{4}\sum_{n=0}^{\infty}p_{-2}(n)q^{n}.
w^p(\theta_1,\cdots,\theta_{q};\xi_1,\cdots,\xi_{p-q})=w_1^q(\theta_1,\cdots,\theta_{q}) w_2^{p-q}(\xi_1,\cdots,\xi_{p-q}),\;\;\; \theta_i\in{\cal G}_1,\;\;\xi_i\in{\cal G}_2.
|{\rm Hess}\,u|_A^2 =\sum_{i,j}u_{ij}^2+2(p-2)\sum_{k=1}^{n}u_{kn}^2+(p-2)^{2}u_{nn}^2 &=(p-1)^2u_{nn}^2+2(p-1)\sum_{k=1}^{n-1}u_{kn}^2+\sum_{k,l=1}^{n-1}u_{kl}^2 &\geq(p-1)^2u_{nn}^2.
\sum_{n\ge 0}\frac{(a,b;q)_n}{(q,c;q)_n}z^n=\frac{(c/b,bz;q)_\infty}{(c,z;q)_\infty}\sum_{n\ge 0}\frac{(abz/c,b;q)_n}{(q,bz;q)_n}\left(\frac{c}{b}\right)^n.
\mathbb{E}\left\{\boldsymbol{\eta}_n^{(g)}\left(\boldsymbol{\eta}_{n'}^{(g)}\right)^H\right\}=\mathbf{R}^{(g)}_{\boldsymbol{\eta}}\delta_{nn'},\textrm{where}\mathbf{R}^{(g)}_{\boldsymbol{\eta}}\triangleq E_s \left(\sum_{g' \neq g}\gamma^{(g')}K_{g'}\sum_{l=0}^{L_{g'}-1}\rho_l^{(g')}\mathbf{R}_l^{(g')}\right)+N_0\mathbf{I}_N,
\sum_{{\bf{k}}}\Lambda_{{\bf{k}}}(-{\bf{q}})\frac{-i\langle T a^{n}_{{\bf{k}}}({\bf{q}})a^{\dagger}_{{\bf{k}}^{'}}({\bf{q}}^{'}) \rangle}{\langle T 1 \rangle}= G_{1}({\bf{q}},{\bf{k}}^{'},{\bf{q}}^{'}; n)
w^\uparrow_{\{1,2,3\}}(C_{(1, \{2,3\})})= 0, w^\uparrow_{\{1,2,3\}}(C_{(2, \{1,3\})})= 0, w^\uparrow_{\{1,2,3\}}(C_{(3, \{1,2\})})= 0 w^\uparrow_{\{1,2,3\}}(C_{(\{2,3\}, 1)})= 1, w^\uparrow_{\{1,2,3\}}(C_{(\{1,3\}, 2)})= 1, w^\uparrow_{\{1,2,3\}}(C_{(\{1,2\}, 3)})= 1
D_{22}= \frac{s_2 2^{-s_3}}{s_1+s_3-1}\int_{at_3}^\infty \frac{v-[v]-\frac{1}{2}}{v^{s_1+s_2+s_3}}dv &+ \frac{s_2 s_3}{s_1+s_3-1}\int_{at_3}^\infty \frac{v-[v]-\frac{1}{2}}{v^{s_2}}\int_v^\infty \frac{1}{u^{s_1}(u+v)^{s_3+1}}dudv &= D_{221}+D_{222},
{\cal H}_{B_{n}}^{t w i s t e d}={\frac{1}{2}}p^{2}+{\frac{g_{l}^{2}}{2}}\sum_{\alpha \in \Delta_{l}}\wp ( \alpha \cdot q ) +{g_{s}^{2}}\sum_{\lambda \in \Delta_{s}}\wp^{( 1 / 2 )}( \lambda \cdot q ) .
\max\{\, x_1-x_2-x_3, \, x_1+4, \, x_1+x_2+x_3 \,\}= -x_1+x_2+2 \max\{\, x_1+x_2+x_3, \, -x_1+x_2+2 \,\}= x_1-x_2-x_3 x_1+x_2+x_3 = x_1-x_2-x_3
T_{\lambda}:= \begin{tabular}{| c |}\hline $1 , \ldots , \lambda_1$ \hline$\lambda_1 + 1 , \ldots , \lambda_1 + \lambda_2 $ \hline$ \vdots$ \hline$\lambda_1 + \cdots + \lambda_{n-1}+1 , \ldots , d$ \hline\end{tabular}
\# \prod \mathrm{Cl}(O_K) = \prod h_K &=\prod \sqrt{D_K}\cdot \prod \frac{w_K}{R_K}\cdot \frac{1}{2^{r_1}(2\pi)^{r_2}}\cdot \mathrm{Res}_{s=1}\zeta_K(s) &\leq \left( \prod \sqrt{D_K}\left( \log D_K \right)^{d-1}\right) \cdot 500^g e^g.
\geq \mathbb{E}(\liminf_{t \to \infty}\frac{1}{t}\sum_{j=0}^{t-1}\Pi_{C_\ell}(X_{C_1}(m),\ldots,X_{C_{\ell-1}}(m),\tilde{X}_{C_{\ell}}(m),X_{C_{\ell+1}}(m), \ldots,X_{C_{r}}(m))).
{f_{\left. G_K \right|T}}\left({\left. x \right|t}\right) = \left\{{{\mathcal M^{- 1}}\phi}\right\}\left( \left. x \right| t\right) = \frac{1}{{2\pi \rm i}}\int\nolimits_{c -{\rm i}\infty}^{c +{\rm i}\infty}{{x^{- s}}}\phi \left({\left. s \right|t}\right)ds,
\dfrac{d}{dr}\left(\int_{M\cap B_r}fS_1 dM\right) - \frac{m-1}{2}\sqrt{\kappa}(\cot\sqrt{\kappa}r)\int_{M\cap B_r}fS_1 dM & = (\sin \sqrt{\kappa}r)^{\frac{m-1}{2}}\dfrac{d}{dr}\left(\frac{1}{(\sin\sqrt{\kappa}r)^{\frac{m-1}{2}}}\int_{M\cap B_r}fS_1 dM\right),
u^{\Delta t}_{h}-v^{\Delta t}_{h}=\frac{1}{2}\,u_h^m+(1-\frac{1}{2})\,u_h^{m-1}-u_{h}^{m-1}-\frac{t-t_{m-1}}{\Delta t}(u_{h}^{m}-u_{h}^{m-1}) =\left(\frac{1}{2}-\frac{t-t_{m-1}}{\Delta t}\right)\left(u_h^{m}-u_{h}^{m-1}\right).
F(\Omega_{1},\Omega_{2},...,\Omega_{m})=\int_{-\infty}^{\infty}\cdot\cdot\cdot\int_{-\infty}^{\infty}f(t_{1},t_{2},...,t_{m})e^{-i\Omega_{1}t_{1}-i\Omega_{2}t_{2}\cdot\cdot\cdot-i\Omega_{m}t_{m}}dt_{1}\cdot\cdot\cdot dt_{m}
\begin{array}{rcl}\displaystyle{\sum_{u_1,\dots,u_m,n\geq 0}}\sharp \mathcal{D}(u_1,\dots,u_m,n) a_1^{u_1}\cdots a_m^{u_m}q^n = \displaystyle{\sum_{u_1,\dots,u_m,n\geq 0}}\sharp \mathcal{C}(u_1,\dots,u_m,n) a_1^{u_1}\cdots a_m^{u_m}q^n = (-a_1q;q)_{\infty}\cdots (-a_mq;q)_{\infty}\,\cdot\end{array}
\theta( (F(t_2) - F(t_1))\,t_1^{k_1}\cdots t_n^{k_n}) = -a_{(p,1,0, \dots, 0)}(t_1, \dots, t_n) =\ (a_{(p,0,\dots, 0)}\cdot e_1)(t_1, \dots, t_n) - a_{(q,0,\dots, 0)}(t_1, \dots, t_n) =\ q_{k_2}(t_1, \dots, t_n) \cdot e_1(t_1,\dots, t_n) - q_{k_2+1}(t_1,\dots, t_n)\ .
A(x,v) =x^2v^3+xvA(x,v)+xvB(x,v)+xvC(x,v), B(x,v) =x^2v^3+v^2b_2(x)+\frac{x}{1-v}(v^3B(x,1)-vB(x,v))+\frac{x}{1-v}(v^3A(x,1)-v^2A(x,v)), C(x,v) =x(B(x,v)-v^2b_2(x))+\frac{x}{1-v}(v^3C(x,1)-vC(x,v)),
\pi_i(x_i^k f) = \begin{cases}-\beta f (k=0), \left(\displaystyle\sum_{s=0}^{k-1}x_i^s x_{i+1}^{k-1-s}+ \beta \displaystyle\sum_{s=1}^{k-1}x_i^{s}x_{i+1}^{k-s}\right)f (k>0).\end{cases}
\begin{bmatrix}B_{1}^\top \otimes A_1 - D_{1}^\top \otimes C_1 & \ddots \ddots & B_{r-1}^\top \otimes A_{k-1}- D_{r-1}^\top \otimes C_{r-1}- (D_{r}^\top \otimes C_{r}) P_{n,n}B_{r}^\top \otimes A_{r}\end{bmatrix}{\cal X}={\cal E},
f\star g = \Omega_\psi\big(\Omega^{-1}_\psi (f)\, \Omega^{-1}_\psi (g)\big) f\, \tilde \star\, g = \Omega_{\tilde \psi}\big(\Omega^{-1}_{\tilde \psi}(f)\, \Omega^{-1}_{\tilde \psi}(g)\big)
N(\alpha) = w^4 + 2 w^3 x - w^2 x^2 - 2 w x^3 + x^4 + 2 w^2 y^2 + 2 w x y^2 + 3 x^2 y^2 + y^4 + 2 w^2 y z - 8 w x y z - 2 x^2 y z + 2 y^3 z + 3 w^2 z^2 - 2 w x z^2 + 2 x^2 z^2 - y^2 z^2 - 2 y z^3 + z^4.
K_{\lambda}\triangleright u_{b}^{a*}\triangleleft K_{\lambda^{\prime}}=(K_{\lambda}^{-1}\triangleright u_{b}^{a}\triangleleft K_{\lambda^{\prime}}^{-1})^{*}=q^{-(\lambda,\lambda_{b})}q^{-(\lambda^{\prime},\lambda_{a})}u_{b}^{a*}.