harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widetilde H_{n0}(t) \le \sqrt{\dfrac{2}{\pi}}\sum_{l=1}^\infty \left|{\langle g\left( \cdot,t \right),{\phi_{2ln}}\rangle}\right|\le \sqrt{\frac{2}{\pi}}\sum\limits_{l = 1}^\infty \frac{\widetilde E_2}{2^{\gamma}l^{\gamma}n^{\gamma}}\le \frac{C_2(\gamma, \widetilde E_2)}{n^{\gamma}}.
{{M_s^{-4}\pi^42}\over{3}}\int d\tau_2 \tau_2 d^{4}\eta \delta (1-\sum_i \eta_i)e^{{2\pi\tau_2 \over M_s^2}(s\eta_1\eta_3+t\eta_2\eta_4)}e^{-{2\pi\tau_2 \over M_s^2}\sum_i m_i^2\eta_i}\equiv{{\pi^2}\over{6}}B' \ .
\int_0^1tr \omega_R(\gamma(s)) =\int_0^1\partial \big(\frac{1}{4\pi}\int_0^{2\pi}\log (1-z_1(s)^2-z_2(s)^2-2z_1(s)z_2(s)\cos\theta)d\theta\big) &=\frac{1}{4\pi}\int_0^{2\pi}\big(\int_0^1 \frac{d}{ds}\log (1-z_1(s)^2-z_2(s)^2-2z_1(s)z_2(s)\cos\theta)ds \big)d\theta &=\frac{1}{4\pi}\int_0^{2\pi}\log (1-z_1^2-z_2^2-2z_1z_2\cos\theta)d\theta.
\frac{d^{n}f}{dx^{n}}(x)=\frac{\Delta_{h}^{n}[f](x)}{h^{n}}+o(h)=\frac{\nabla_{h}^{n}[f](x)}{h^{n}}+o(h)=\frac{\delta_{h}^{n}[f](x)}{h^{n}}+o(h^{2}).
(E-v(\theta))D_{E,11}(\theta) =e^{2\pi i \rho_{|s|}(E)}s(\theta) D_{E,11}(\theta+\alpha)+e^{-2\pi i \rho_{|s|}(E)}\tilde{s}(\theta-\alpha) D_{E,11}(\theta-\alpha), (E-v(\theta))D_{E,21}(\theta) =e^{-2\pi i \rho_{|s|}(E)}s(\theta)D_{E,21}(\theta+\alpha)+e^{2\pi i \rho_{|s|}(E)}\tilde{s}(\theta-\alpha) D_{E,21}(\theta-\alpha).
\begin{array}{ll}[M\cosh(\lambda \theta) + h\cosh(\lambda \theta +i\beta\phi_0)] [M\cosh(\lambda \theta) +{h}\cosh (\lambda \theta -i\beta\phi_0)]/(P_{\rm T}( \theta)P_{\rm T}(-\theta)) begin{align*}3mm] = \cos(\xi -i\lambda\theta)\cos(\xi +i\lambda\theta)[\cos^2(\xi) + \sinh^2(\lambda \theta) + k^2\sinh^2(\lambda \theta)\cosh^2(\lambda \theta)]^{-1}\end{array}
V_{l , k}^{m , n}( z ) = : \prod_{i = - l / 2}^{l / 2}V_{1 , 0}^{m_{i}, 0}\left( x^{2 i ( \xi + 1 )}z \right) \prod_{j = - k / 2}^{k / 2}V_{0 , 1}^{0 , n_{j}}\left( x^{2 j \xi}z \right) :
c^*(k):=\sup_{\xi\in Q_\ell^*}d^+_k(\xi)=\sup_{\xi\in Q_\ell^*}\inf_{\substack{V\textrm{subspace of}\Lambda^+ H^{1/2}_\xi \textrm{dim}V=k}}\sup_{u^+_\xi\in V\setminus\{0\}}\frac{\left(|D_\xi|u^+_\xi,u_\xi^+\right)}{\|u^+_\xi\|^2_{L_\xi^{2}}}.
\left\langle u_{1},v_{1}\right\rangle +\left\langle u_{2},v_{2}\right\rangle =\left\langle u,v\right\rangle \geq\cos\angle\left(v,V^{\bot}\right)\,\left\Vert u\right\Vert \left\Vert v\right\Vert \overset{\eqref{eq:definition-Psi}}{=}\left\Vert u\right\Vert \left\Vert v_{2}\right\Vert
G_{s}^{\overbrace{\kappa \cdots \kappa}^{m}\overbrace{w\cdots w}^{n}\mu_{1}\cdots \mu_{s-n-m}}=\kappa^{m-2}\left( -w\right)^{n}g_{s,(s-n-m)}^{\mu_{1}\cdots \mu_{s-n-m}},
4\Lambda^{4}\prod_{a=1}^{n_{c}-2}(x-\phi_{a})^{2}- 4\Lambda^{4}\prod_{i=1}^{2n_{c}-4}(x+m_{i}) = - 4\Lambda^{2}\alpha_{n_{c}-1}^{4}\prod_{a=1}^{n_{c}-3}(x-\alpha_{a})^{2}.
\nu(\{z(Q)\}) \geq \frac{1}{C'}\sum_{n=0}^\ell\mu_0([x |n])Z_{x |n}(r) &\geq \frac{1}{C'}G_{\mu_0}(x, r) \qquad\mbox{(by Lemma \ref{lem-gmu})}&=\frac{1}{C'}\int Z_{x\wedge y}(r)\; d\mu_0(y) &\geq \frac{1}{C'\cdot C_r(E)}\qquad\mbox{(by \eqref{e-inq'})}.
A_{\sc t}^{2}\simeq \frac{1}{1 5 0 \pi^{2}}\bigg ( \frac{1}{M_{P l}^{4}\sqrt{1 + 4 \alpha \Lambda_{b}/ 3 M^{5}}}\bigg ) V \left. \left[ 1 + \frac{V}{2 \Lambda_{2}^{+}}\right] \right|_{k = a H}\,
\langle \mathrm{v a c}; A ( t ) | \frac{d}{d{\alpha}_{\pm p}( t )}| \mathrm{v a c}; A ( t ) \rangle \equiv{\cal A}_{\mathrm{v a c}; \pm}( p , t ) = \pm ( 1 - \mathrm{N}^{2}) \frac{e_{+}^{2}\mathrm{L}^{2}}{8{\pi}^{2}}\frac{1}{p}{\alpha}_{\mp p},
Z_{\mathrm{g r}}[ \varphi , \mu , \bar{\mu}] = e^{K [ \varphi , \mu , \bar{\mu}]}\int{\cal D}T{\cal D}\overline{{{T}}}\, \, | \chi [ \mu , T ] |^{2}e^{-{\frac{1}{2}}\int d^{2}w \, \, e^{- \varphi}T \overline{{{T}}}},
\Omega_{+ q}^{- 2 i}= - 4 i \gamma_{q \dot{q}}^{i}\psi_{+ 2 \dot{q}}^{1 -}, \quad \Omega_{- q}^{+ 2 i}= 4 i \gamma_{q \dot{q}}^{i}\psi_{- 2 \dot{q}}^{2 +},
f^{-1}gf=f^{-1}\Big(x_1+P(x_2,x_3)+P'\big(x_2+Q(x_3), x_3+d\big),x_2+Q(x_3)+Q'(x_3+d),x_3+d+d'\Big) = \Big(x_1+P''(x_2,x_3)+P'\big(x_2+Q(x_3), x_3+d\big),x_2+Q''(x_3)+Q'(x_3+d), x_3+d'\Big)
= 2\left( \left\{\mathbf{A}^{-1}+ \mathbf{A}^{-1}\mathbf{B}\mathcal{H}^{-1}\mathbf{C}\mathbf{A}^{-1}\right\}x_{g,0}, x_{h,0}\right)_H-(S_0 g,h)_{\mathcal{H}},g,h\in\mathcal{H},\ \zeta\in \mathbb{D},
\|\psi_\ell-\psi\|^2=(\psi_\ell-\psi)^*(\psi_\ell-\psi_m) +(\psi_\ell-\psi)^*(\psi_m-\psi)\le\|\psi_\ell-\psi\|\,\|\psi_\ell-\psi_m\| +(\psi_\ell-\psi)^*(\psi_m-\psi).
\rho_{XX}(t_{1},t_{2})=\frac{K_{XX}(t_{1},t_{2})}{\sigma_{t_{1}}\sigma_{t_{2}}}=\frac{E[(X_{t_{1}}-\mu_{t_{1}})\overline{(X_{t_{2}}-\mu_{t_{2}})}]}{\sigma_{t_{1}}\sigma_{t_{2}}}.
\begin{aligned}v_1 =\sum_{i=1}^{minS(1)-2}\sqrt{f_i}e_i + \sqrt{a_1}e_{minS(1)-1}-\sqrt{\sigma_1-a_1}e_{minS(1)}, v_n =\sqrt{f_{minS(n-1)-1}-a_{n-1}}e_{minS(n-1)-1}+ \sqrt{f_{minS(n-1)}-\sigma_{n-1}+a_{n-1}}e_{minS(n-1)}+\sum_{i=minS(n-1)+1}^{minS(n)-2}\sqrt{f_i}e_i +\sqrt{a_n}e_{minS(n)-1}-\sqrt{\sigma_n-a_n}e_{minS(n)}, 1<n<N.\end{aligned}
\mathcal{\tilde L(\theta)}= w w_0 K_{12}-\frac{1}{2}w^2K_{22}&=\frac{P_{\infty}(\theta) w^2 \sigma_0^2w_0^2(2a+P_{\infty}(\theta)w^2)}{4a_0(a+P_{\infty}(\theta)w^2)(a+a_0+P_{\infty}(\theta)w^2)}&-\frac{P_{\infty}(\theta)^2 w^4}{4(a+P_{\infty}(\theta)w^2)}.
\int_{[0, \Lambda)}\lambda^{-2 r}d \mu(\lambda)\leq\int_{[0, \Lambda]}\lambda^{-2 r}d I(\lambda)=\left. \lambda^{-2 r}I(\lambda) \right|_{\lambda = 0}^{\lambda \searrow \Lambda}+2 r\int_{[0, \Lambda]}\lambda^{-2 r - 1}I(\lambda)d \lambda.
Q_{\lambda \kappa ; \sigma}+ Q_{\sigma \kappa ; \lambda}- \frac{2}{3}\left( g_{\sigma \lambda}Q^{\nu}{_{\kappa ; \nu}}+ g_{\kappa ( \lambda}Q_{\sigma )}{^{\mu}}{_{; \mu}}\right) ={\cal Q}_{\lambda \kappa \sigma}= O ( r^{- 1})
\begin{cases}d=1, -\frac12 \leq s-\ell \leq 1, \ 2 s \geq \ell + \frac12 \geq 0, d=2,3, \ell \leq s \leq \ell +1, \ \ell \geq 0, \ 2s-(\ell+1) \geq 0, d \geq 4, \ell \leq s \leq \ell +1, \ \ell > \frac{d}{2}- 2, \ 2 s -(\ell +1) > \frac{d}{2}-2.\end{cases}
{h_{a}^{( 1 )}}^{2}= \sigma^{( 1 )}={\frac{1}{t^{( 1 )}d_{1}+{\frac{\alpha^{2}}{4}}}},{h_{a}^{( 2 )}}^{2}= \sigma^{( 2 )}={\frac{1}{t^{( 2 )}d_{2}+{\frac{\beta^{2}}{4}}}}
\begin{aligned}\mathbb{E}\left[\left(B_{t_{i}}-B_{s}\right)\left(B_{t_{j}}-B_{s}\right)\right] \leq\mathbb{E}\left[\left(B_{t_{i}}-B_{s}\right)^{2}\right] =(t_{i}-s)^{2H}\leq(t-s)^{2H}.\end{aligned}
\rho_{r_n}\circ \ldots \circ \rho_{r_1}(\sigma(x)) = \rho_{r_n}\circ \ldots \circ \rho_{r_1}((x_{i+1})_{i=1}^{\infty}) &= \sigma^{q_1\cdot \ldots \cdot q_n}(\rho_{r_n}\circ \ldots \circ \rho_{r_1}(x_{i})_{i=1}^{\infty}).
(I-\frac{\Delta x_{k}y_{k}^{T}}{y_{k}^{T}\Delta x_{k}})H_{k}(I-\frac{y_{k}\Delta x_{k}^{T}}{y_{k}^{T}\Delta x_{k}})+\frac{\Delta x_{k}\Delta x_{k}^{T}}{y_{k}^{T}\Delta x_{k}}
I_{f_j,g_\ell}(\tau):= \ \int_{-\overline{\tau}}^{i\infty}\int_{w_1}^{i\infty}\frac{f_j(w_1)g_\ell(w_2)}{(-i(w_1+\tau))^{2-\kappa_1}(-i(w_2+\tau))^{2-\kappa_2}}dw_2dw_1.
B_{ji}(\sigma:0) = \tilde B_{ji}(\sigma:0) + B_{jj}(\sigma:k_L+1)B_{ji}(k_L)B_{ii}(k_L:0)\cr \geq \frac{1}{2}\eta_i (\eta_j - \varepsilon) (\delta-\varepsilon) + \eta_j \varepsilon \eta_i\cr =\frac{1}{2}\eta_i\varepsilon^2 + \frac{1}{2}\eta_i (\eta_j - \delta) \varepsilon + \frac{1}{2}\eta_i\eta_j\delta \geq \frac{1}{2}\eta_i\eta_j\delta,
G_{\alpha}({\bf x}) =-\Big[\sum_{j\not\in Q_{\alpha}}\tfrac{Z}{2}\Delta_{{\bf x}}(\sqrt{|x_j|^2+1}) -\sum_{j,k\not\in Q_{\alpha},j<k}\tfrac14\Delta_{{\bf x}}(\sqrt{|x_j-x_k|^2+1}) \Big] &=V({\bf x})-V_{\alpha}({\bf x}) - \Delta_{{\bf x}}F_{\alpha}({\bf x})\,.
\alpha(q,a) = \begin{cases}(q',b) (q,a,q',b) \in \Delta (q',a) (q,\pm 1, q') \in \Delta \end{cases}\beta_{+1}= \prod_{q' \mid \exists q, (q,+1,q') \in \Delta}\rho_{q'}\beta_{-1}= \prod_{q' \mid \exists q, (q,-1,q') \in \Delta}{\rho_{q'}}^{-1}
\begin{aligned}\mathbb{E}_n \bigg[ \sup_{0 \le t \le T}\bigg| \tilde{\mathcal{B}}^n_t (\varphi ) - \int_0^t \sum_{j \in \mathbb{Z}}\tau_j \mathcal{Q}^n_\rho (\ell; s ) \nabla^n \varphi^n_j (s) ds \bigg|^2 \bigg] \le C \bigg( \frac{\ell}{n^2}+ \frac{T}{\ell^{2}}\bigg) \int_0^T \sum_{j \in \mathbb{Z}}(\nabla^n \varphi^n_j ( t ) )^2 dt . \end{aligned}
\cot\sigma=\frac{\Vert\alpha^g_3(E_1,E_1,E_1)\Vert^2-\Vert\alpha^g_3(E_1,E_1,E_2)\Vert^2}{2\langle \alpha^g_3(E_1,E_1,E_1),\alpha^g_3(E_1,E_1,E_2) \rangle}.
\begin{cases}|\partial_\xi^\beta s_j^\ell(x,\xi)|\leq C_\beta \langle x \rangle^{\ell-1-\varepsilon}, |\partial_\xi^\beta p_j^\ell(y,\xi;t)|\leq C_\beta \langle |y|+t|v(\xi)| \rangle^{-\ell}.\end{cases}
\Psi ( \sigma_{1}, \sigma_{2}) = \sum_{s , t \in{\bf Z}}e^{i s \sigma_{1}/ ( n - m \theta )}e^{- i t \sigma_{2}/ n}\sum_{i = s b}^{n + s b}\sum_{j = m b t}^{n + m b t}c_{s , t , i , j}V^{i}U^{j}.
\mathrm{Aut}_G(\widetilde{\mathbb{X}}_b)(R)=\{\alpha\in \mathrm{Aut}_B(\widetilde{\mathbb{X}}_{b,R}), \beta\in \mathrm{Aut}(\widetilde{\mu}_{p^\infty,R})\mid \alpha\mathrm{\respects\the\polarization\up\to\}\beta\}\ .
\lim_{w\rightarrow z}\lim_{s\rightarrow 1}\bigg(\sum_{n\neq0}\frac{1}{|n|}W_{s}(n\sigma_{p}^{-1}z)\overline{V_{\overline{s}}(n\sigma_{p}^{-1}w)}-g_{\mathbb{H},s}(\sigma_{p}^{-1}z,\sigma_{p}^{-1}w)\bigg)= -\log\big(4\Im(\sigma_{p}^{-1}z)^{2}\big)+O_{z}(1).
L(s, S_f) := \sum_{n \geq 1}\frac{S_f(n)}{n^s}, L(s, S_f\times S_f) := \sum_{n \geq 1}\frac{S_f(n)\overline{S_f(n)}}{n^s}, L(s, S_f\times \overline{S_f}) := \sum_{n \geq 1}\frac{S_f(n) S_f(n)}{n^s},
( I_{3p}\otimes e_1 )^\top M_\Delta(M_+,M_-,M_0) ( I_{3p}\otimes e_1 ) =~ ( I_{3p}\otimes e_1 )^\top ( M_\Delta(m_+,m_-,m_0) \otimes I_p ) ( I_{3p}\otimes e_1 ) =~ ( I_{3p}\otimes e_1 )^\top ( M_\Delta(m_+,m_-,m_0) \otimes e_1 ) =~ M_\Delta(m_+,m_-,m_0).
\mathcal{L}_{\mathcal{I}^{({\rm CEU})}_{\rm u \rightarrow u}}(s|r_o)=\exp \left(- \pi \lambda \mathbb{E}_{P_{\rm{u}}}\left[ \sqrt{s P_{\rm{u}}}\arctan \left(\sqrt{\frac{s P_{\rm{u}}}{r_o^4}}\right) \right] \right),
\Delta \bar{u}(x) = \eta''(\rho_0 -\delta(x))|\nabla \delta|^2 - \eta'(\rho_0 -\delta(x))\Delta\delta(x) \le \eta''(\rho_0 -\delta(x)) + \frac{(N-1)}{\rho_0 -\delta(x)}\eta'(\rho_0 -\delta(x)) = - \frac{\mu}{\delta(x)^2}\eta + \eta^p = - \frac{\mu}{\delta(x)^2}\bar{u}+ \bar{u}^p \,.
\int_{0}^{2 \pi}d \theta \int_{0}^{\infty}d r \, r \Bigl ( \mathrm{e}^{-{\frac{\phi ({\bf r}_{i j})}{k T}}}- 1 \Bigr ) ={\frac{2}{N}}\Bigl \{b -{\frac{a}{( R T )^{2}}}\Bigr \}\ .
\Psi_g(w; \lambda) = (-1)^{r-p-q}\dfrac{(p w+a)_p(q w+b)_q((r-p)w-a)_{r-p}((r-q)w-b)_{r-q}}{(r w-1)_r (r w)_r}, begin{align*}2mm]\Psi_h(w; \lambda) = (-1)^{r-p-q}\dfrac{(p w+a)_p(q w+b)_q((r-p-q)w-a-b+1)_{r-p-q}}{(r w)_r},
\Delta Mf(u_n,t) =\frac{t-(x-u_n)}{t}M_Rf(x,t-(x-u_n))+2\frac{(x-u_n)}{t}M_Lf(x,x-u_n)-\frac{t+(x-u_n)}{t}M_Lf(x,t+(x-u_n)).
\sum_{j=1}^N \sum_{i=1}^{M-1}\left\{\sum_{k=1}^{H-1}(a_{ik}b_{kj}{-}a^0_{ik}b^0_{kj}){+}c_i \right\}^2 &=\sum_{i=1}^{M-1}x_i^2 + \sum_{j=2}^{N}\sum_{i=1}^{M-1}\left[ x_i +\sum_{k=1}^{H-1}\{(a_{ik}b_{kj}{-}a^0_{ik}b^0_{kj})- (a_{ik}b_{k1}- a^0_{ik}b^0_{k1})\}\right]^2.
\xi_{n,1}^{()}(1) = \frac{\xi\left(\frac{n-1}{2}\right)}{\xi\left(\frac{n+3}{2}\right)}\cdot\frac{\prod_{j=1}^{\frac{n-3}{2}}\xi(n-1-2j)}{\prod_{j=1}^{\frac{n-1}{2}}\xi(n+3-2j)}= \frac{\xi\left(\frac{n-1}{2}\right)}{\xi\left(\frac{n+3}{2}\right)}\cdot \frac{\xi (2)}{\xi (n+1)\,\xi(n-1)}.
b_{-1-l}(a_1-\lambda)+\sum_{j=0}^{l-1}\sum_{\alpha=0}^{l-j}\frac{1}{\alpha !}(\frac{\partial}{\partial\xi})^{\vec{\alpha}}b_{-1-j}(-i\frac{\partial}{\partial x})^{\vec{\alpha}}a_{1-l+\alpha+j}=0, \mbox{~~~~$l$=1,2,{\cal{D}}ots}
\frac{\mathrm{d}}{\mathrm{d}z}p + \frac{\mathrm{d}}{\mathrm{d}\bar{z}}\bar{p}= \frac{\mathrm{d}}{\mathrm{d}z}\frac{\partial{\cal L}}{\partial v}+ \frac{\mathrm{d}}{\mathrm{d}\bar{z}}\frac{\partial{\cal L}}{\partial \bar{v}}= \partial_{\varphi}{\cal L}.
\overline{{\Psi}}_{L}\partial \! \! \! / \Psi_{L}\to \overline{{\Psi}}_{L}e^{- i \beta ( \hat{C}_{+}^{\dagger}- \hat{C}_{-}^{\dagger})}e^{i \beta ( \hat{C}_{+}- \hat{C}_{-})}\partial \! \! \! / \Psi_{L},
\left\Vert \frac{\partial u}{\partial t}\right\Vert_{X_{\mathbf{p},q}}+\sum\limits_{\left\vert \alpha \right\vert =2l}\left\Vert D^{\alpha}u\right\Vert_{X_{\mathbf{p},q}}+\left\Vert Au\right\Vert_{X_{\mathbf{p},q}}\leq C\left\Vert f\right\Vert_{X_{\mathbf{p},q}}.
\dfrac{\mathfrak{U^R}(x)}{\bigl(t-t_k+(\Lambda -1)\theta_k\bigr)^{\frac{1}{p-2}}}\Bigg\vert_{t=t_k+\theta_k}=\dfrac{\mathfrak{U^R}(x)}{(\Lambda CR^p)^\frac{1}{p-2}}h_k(x_k,t_k) &\leq \dfrac{\mathfrak{U^R}(x)}{(\Lambda CR^p)^\frac{1}{p-2}}\gamma h_k(x,t_k+\theta_k)\leq h_k(x,t_k+\theta_k)
\Psi_\kappa^{-1}x_1 =x_1, \Psi_\kappa^{-1}x_\ell = x_\ell - \sum_{j=1}^{\ell -1}\Big (\# W^u(x_\ell;P)\cap W^s(x_j;\tilde P) \ \ 2\Big ) \cdot \Phi_\kappa^{-1}x_j,\forall \ell\geq 2.
\begin{array}{l c l}{{\widehat \Sigma}_{m}^{( r , s )}}&{=}&{\oplus_{k = r}^{s}{\widehat \Sigma}_{m}^{( k , k )}( \theta )}\\{}&{=}&{\oplus_{k = r}^{s}\oplus_{l = 0}^{k}{\theta}^{l}{\Sigma}_{m}^{( k - l , k - l )}}\\ \end{array}
\mathcal L^{0}_{\mathcal P,t_{0}}:=\langle \{\textnormal{Re}(M_{0}^{n}): n\in\mathbb Z\}\rangle + \langle \{\textnormal{Im}(M_{0}^{n})i: n\in\mathbb Z\}\rangle \subset\mathcal L_{\mathcal P,t_{0}}
1\ge \liminf_{k\to\infty}\sum_{n={1}}^{\infty}f_{\lambda_{n_k}}^{(n)}(o,o)z_0^n\ge \sum_{n={1}}^{\infty}\liminf_{k\to\infty}f_{\lambda_{n_k}}^{(n)}(o,o)z_0^n=\sum_{n={1}}^{\infty}f_{\lambda_{0}}^{(n)}(o,o)z_0^n =U_{\lambda_0}(o, \, o \, | \, z_0).
\begin{matrix}{{A = \left( \begin{matrix}{{0}}&{{a}}&{{0}}&{{b}}\\{{c}}&{{0}}&{{d}}&{{0}}\\{{0}}&{{e}}&{{0}}&{{f}}\\{{g}}&{{0}}&{{h}}&{{0}}\\ \end{matrix}\right) ,}}&{{B = i \left( \begin{matrix}{{0}}&{{- f}}&{{0}}&{{e}}\\{{- h}}&{{0}}&{{- g}}&{{0}}\\{{0}}&{{- b}}&{{0}}&{{a}}\\{{d}}&{{0}}&{{c}}&{{0}}\\ \end{matrix}\right)}}\\ \end{matrix}.
\|x_{k_{j}}-x_{k_{n+1}}\| \geq |<z^{*}_{i},x_{k_{j}}-x_{k_{n+1}}>| &\geq 1-|<z^{*}_{i},x_{k_{n+1}}>|-|<z^{*}_{i},x_{k_{j}}-z_{i}>| &\geq 1-c-\|x_{k_{j}}-z_{i}\| &\geq 1-c-(1+c-\epsilon)=\epsilon-2c &\geq \epsilon-\delta
Z(s)=\left[\frac{x_0}{s-1}+\underset{k=0}{\overset{\infty}{\sum}}\beta_{2.k+1}.\frac{\Gamma(s+2.k+1)}{\Gamma(s)}-\sum_{n=1}^{n=N}\left(\frac{n}{x_0}\right)^{-s}\right].\exp\left(-s.\ln{x_0}\right)
G(z_1,z_2,\bar{z}_1 z_2 + \bar{z}_1^2 )=\overline{F_1}(\bar{z}_1)F_2(z_1,z_2,\bar{z}_1 z_2 + \bar{z}_1^2)+{\left(\overline{F_1}(\bar{z}_1) \right)}^2 .
b(0)=0; &b(x):=\P(X_{n+1}=x+1\| X_n=x)-\P(X_{n+1}=x-1\| X_n=x),\ \x>0; &b(x):=\P(X_{n+1}=x-1\| X_n=x)-\P(X_{n+1}=x+1\| X_n=x),\ \x<0;
(\begin{matrix}\rho_{x_{1}}\\ \rho_{x_{2}}\\ \rho_{x_{3}}\end{matrix})=(\begin{matrix}a_{1,x_{1}}&a_{1,x_{2}}&a_{1,x_{3}}\\a_{2,x_{1}}&a_{2,x_{2}}&a_{2,x_{3}}\\a_{d,x_{1}}&a_{d,x_{2}}&a_{d,x_{d}}\end{matrix})^{-1}(\begin{matrix}r_{x_{1}}\\r_{x_{2}}\\r_{x_{3}}\end{matrix})
F_{12}= f_1(x_1,x_2) y_1^2 + f_2(x_1,x_2) y_1 y_2 + f_3(x_1,x_2) y_2^2 F_{13}= g_1(x_1,x_2) z_1^2 + g_2(x_1,x_2) z_1 z_2 + g_3(x_1,x_2) z_2^2
K ={\o \mathrm{i}{2 \pi}}\partial{\bar{\partial}}\mathrm{l g}\left( \mathrm{i}\langle \Omega \vert{\bar{\Omega}}\rangle \right) \equiv{\o \mathrm{i}{2 \pi}}\partial{\bar{\partial}}\mathrm{l g}\left( \mathrm{i}\Omega^{T}\left( \begin{matrix}{{\bf 0}}&{{\bf 1}}\\{-{\bf 1}}&{{\bf 0}}\\ \end{matrix}\right){\bar{\Omega}}\right)
Z_{E H}={\frac{1}{4}}{\frac{1}{( \eta \bar{\eta})^{4}}}\int d p_{x}d p_{y}\, \, q^{p_{x}\bar{p}_{x}+ p_{y}\bar{p}_{y}}\sum_{i}\left|{\frac{\theta_{i}}{\eta}}\right|^{4}+{\frac{1}{4}}\sum_{i , j}\left|{\frac{\theta_{i}}{\theta_{j}}}\right|^{4}\, \, .
{\Delta}= \displaystyle \int_{x}\left[ \frac{{\delta}\bar{\Gamma}^{( 0 )}}{{\delta}A^{\mu a}}\frac{{\delta}}{{\delta}G_{\mu}^{a}}+ \frac{{\delta}}{{\delta}A^{\mu^{a}}}\frac{{\delta}\bar{\Gamma}^{( 0 )}}{{\delta}G_{\mu}^{a}}- \frac{{\delta}\bar{\Gamma}^{( 0 )}}{{\delta}c^{a}}\frac{{\delta}}{{\delta}L^{a}}- \frac{{\delta}}{{\delta}c^{a}}\frac{{\delta}\bar{\Gamma}^{( 0 )}}{{\delta}L^{a}}\right] \, ,
\pi^{\infty}_{a}\!\!= \!\!\sum_{j=1}^n{\!\left\{\delta^{v_j}\pi_{a}(\boldsymbol{\tau}^{D_j}; \hat{\boldsymbol{d}}_{j:n})+ \!\!\!\!\!\! \sum_{k=v_{j-1}+1}^{v_j-1}{\!\!\!\!\! \delta^k \pi_{a}(\hat{\boldsymbol{\tau}}_{j:n}; \hat{\boldsymbol{d}}_{j:n})}\right\}}
h_{\hat{\rho}_s}\left(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3 \right) = \frac{1}{2}\sum_{\pi \in \mathfrak{S}_3}\left( r^{(1)}_{\pi_1}- r^{(1)}_{\pi_2}\right) \left( r^{(2)}_{\pi_1}- r^{(2)}_{\pi_3}\right).
\sum_{r=1}^{m-1}\zeta(s,\frac{r}{m})cos\frac{2\pi rk}{m}=\frac{m\Gamma(1-s)}{(2\pi m)^{1-s}}sin\frac{\pi s}{2}\cdot\{\zeta(1-s,\frac{k}{m})+\zeta(1-s,1-\frac{k}{m})\}-\zeta(s)
2(s_1 - t_1) + \sum_{j = 2}^n (s_{1j}- t_{1j}) - \sum_{j = 2}^n (s_{2j}- t_{2j}) = 0, \;\;\vdots -\sum_{i = 1}^{n-2}(s_{i,n-1}- t_{i,n-1}) + \sum_{i = 1}^{n-1}(s_{in}- t_{in}) + 2(s_n - t_n) = 0.
\nabla_{j}u =h_{jl}\bar{g}(\lambda\partial_{r},e_{l}), \nabla_{i}\nabla_{j}u =\nabla_{i}h_{jl}\bar{g}(\lambda\partial_{r},e_{l})+\lambda'h_{ij}-h_{il}h_{jl}\bar{g}(\lambda\partial_{r},\nu) &=\nabla_{l}h_{ij}\bar{g}(\lambda\partial_{r},e_{l})+\bar{R}_{\nu jli}\bar{g}(\lambda\partial_{r},e_{l})+\lambda'h_{ij}-uh_{il}h_{jl}.
X(i,j)=[\begin{matrix}x(1,1)&x(1,2)&\cdot\cdot\cdot&x(1,j)\\x(2,1)&x(2,2)&\cdot\cdot\cdot&x(1,j)\\ \vdots&\vdots&&\vdots\\x(i,1)&x(i,2)&\cdot\cdot\cdot&x(i,j)\end{matrix}]
D_n(x-(I-R^{-1})^{-1}b) =\prod_{k=0}^{n}m_\tau(R^k(x-(I-R^{-1})^{-1}b)) &=\prod_{k=0}^{n}m_\tau(\mathcal{R}^kx-\mathcal{R}^k((I-R^{-1})^{-1}b)) &=\prod_{k=0}^{n}m_\tau(\mathcal{R}^kx-\mathcal{R}^k(\sum\limits_{i=0}^{\infty}R^{-i}b)) &=\prod_{k=0}^{n}m_\tau(\mathcal{R}^kx-(I-R^{-1})^{-1}b).
0 \xrightarrow{}R(-n) \xrightarrow{\phi_{n-2}}R(-n+2)^{b_{n-3}}\xrightarrow{\phi_{n-3}}R(-n+3)^{b_{n-4}}\xrightarrow{\phi_{n-4}}\ldots & \ldots\xrightarrow{\phi_{3}}R(-3)^{b_{2}}\xrightarrow{\phi_{2}}R(-2)^{b_{1}}\xrightarrow{\phi_{1}}R \xrightarrow{}0,
\begin{array}{c}\phi_\omega(s)=A_1{\cal J}_{l+1/2}(z_1)+ B_1{\cal Y}_{l+1/2}(z_1),{\rm \for \}s<a_1, \phi_\omega(s)= A_2{\cal J}_{l+1/2}(z_2) + B_2{\cal Y}_{l+1/2}(z_2),{\rm \for \}a_1<s<s_0, \end{array}
\frac{v_1(z)}{u_1(z)}= - \frac{f_1'(z)}{f_1(z)}= \widehat c_1 z^{\rho-1}+ \ldots , \frac{v_3(z)}{u_3(z)}= - \frac{f_k'(z)}{f_k(z)}= \widehat c_3 z^{\rho-1}+ \ldots ,
\begin{aligned}g(\nabla^{\widetilde{g}}_Z X,Y) =g(\nabla^g_Z X,Y)+d\sigma(Z)\underbrace{g(X,Y)}_{=0}+\underbrace{d\sigma(X)}_{=0}g(Z,Y)-g(^g\sigma,Y) &=\left(g(\nabla^g_Z X,\cdot) - d\sigma \right)(Y),\end{aligned}
N(J,t) = \left\{X^{\boldsymbol{b}}\; | \; \epsilon_j \leq b_j \leq N_j -1, \; 1 \leq j \leq m, \; \mathrm{and}\; \prod_{j=1}^m \left(b_j +1 - \epsilon_j \right) < t \right\},
\widetilde \nu ( x^{\mu}) = +{\frac{1}{2}}( \psi_{\downarrow}( x^{\mu}) + \psi_{\uparrow}^{c}( x^{\mu}) ) +{\frac{\gamma^{5}}{2}}( \psi_{\uparrow}( x^{\mu}) - \psi_{\downarrow}^{c}( x^{\mu}) ) .
\tau_{e m}= \frac{1}{1 + b \chi + \chi^{2}}\left( \begin{array}{c c c}{{\chi^{2}}}&{{- 2}}&{{- \chi}}\\{{- \frac{1}{2}}}&{{( b + \chi )^{2}}}&{{- \frac{b + \chi}{2}}}\\{{2 \chi}}&{{4 ( b + \chi )}}&{{- 1 + b \chi + \chi^{2}}}\\ \end{array}\right)
\iota_\omega(\delta_{\mu k})=S_{\mu k}\otimes_\omega 1=S_{\mu k}S_k^*S_k\otimes_\omega 1=\sum_{l=1}^N A_{kl}S_{\mu k}S_lS_l^*\otimes_\omega 1 =\sum_{l=1}^N A_{kl}S_{\mu k}\otimes_\omega \omega(\chi_{C_l})=A_{kj}S_{\mu k}\otimes_\omega1.
I = - \nu{\frac{\Gamma ( - \nu )}{\Gamma ( \nu )}}\int{\frac{d^{d}k}{( 2 \pi )^{d}}}\tilde{A}_{i}({\bf k}) \left[ ( \delta_{i j}-{\frac{k_{i}k_{j}}{|{\bf k}|^{2}}}) +{\frac{\bar{c}}{c}}{\frac{k_{i}k_{j}}{|{\bf k}|^{2}}}\right] \left({\frac{|{\bf k}|}{2}}\right)^{2 \nu}\tilde{A}_{j}( -{\bf k}) .
\delta^{(2)}S_{\phi}=-\frac{3}{8\kappa_5^2}\int \sqrt{-g_0}\left(\frac{{\cal L}_{\phi}}{3}h_{ab}h^{ab}+ (\partial\delta\phi)^2 +\frac{1}{2}\frac{\partial^2 U}{\partial \phi^2}\delta\phi^2 + 2h^{ab}\partial_a\delta\phi\partial_b\phi\right)
d(\overline{x},\overline{y}) =d(gw^0_n, gw^l_n) \leq \sum\limits_{i=0}^{l-1}d(gw^i_n, gw^{i+1}_n) =\sum\limits_{i=0}^{l-1}t^i_n \to 0 \;\;{\rm as}\;\;n \to\infty,
v_{k}( \eta ) \, \simeq \, \frac{1}{\sqrt{2 k_{\mathrm{e f f}}( k , \eta )}}\operatorname{e x p}\biggl [ - i \int_{\eta_{\mathrm{i}}}^{\eta}k_{\mathrm{e f f}}\mathrm{d}\tau \biggr ] \, ,
{g(y;t_1,\cdots, t_N)=}\begin{cases}g(y-t_1), y<t_1^+, \min\{g(y-t_1),-g(y-t_2)\}, t_1^+=t_2^-<x<t_2^+, \min\{-g(y-t_2),g(y-t_3)\}, t_2^+=t_3^-<x<t_3^+, \cdots.\end{cases}
\lim_{n\rightarrow \infty}\mathbf{E}\left[ \prod_{l=1}^{p}\mathbf{S}_{l}^{\mathbf{Z}(m_{l},n)}\,\Big|\,\mathbf{Z}(n)\neq 0\right] =1-\left(1-\prod_{l=1}^{p}s_{l,i+1}\right)^{1/2^{i}}.
\left[ \left( f_{1}f_{5}\right)^{1 / 2}\Gamma^{0}\partial_{0}+ \Gamma^{1}\left( \partial_{r}+ \frac{3}{2 r}+ \frac{1}{8}d_{r}( \ln ( f_{1}f_{5}) ) \right) + \left( f_{1}f_{5}\right)^{1 / 2}\Gamma^{5}\partial_{x_{5}}+ \Gamma^{b}D_{b}\right] \chi + g ( r ) \chi = 0
K_{2}(f,\delta )=\inf_{C_B^2(\mathbb{R}^+)}\left\{\left( \parallel f-g\parallel_{C_B(\mathbb{R}^+)}+\delta \parallel g^{\prime \prime}\parallel_{C_B^2(\mathbb{R}^+)}\right):g\in \mathcal{W}^{2}\right\},
p_{Z}^{\dagger}(z)=\frac{1}{\sqrt{\pi}}\frac{p}{erf[q]}\frac{1}{r}\frac{1+\frac{p^{2}}{q^{2}}\frac{z}{r}}{(1+\frac{p^{2}}{q^{2}}[\frac{z}{r}]^{2})^{\frac{3}{2}}}e^{-\frac{p^{2}(\frac{z}{r}-1)^{2}}{1+\frac{p^{2}}{q^{2}}[\frac{z}{r}]^{2}}}
\frac{\partial}{\partial t}G(x,t) =-c_{1}\left(\lambda_1+\frac{\partial}{\partial x}\right)^{\alpha_1}G(x,t)-c_{2}\left(\lambda_2+\frac{\partial}{\partial x}\right)^{\alpha_2}G(x,t)+\lambda_1^{\alpha_1}{c_1}G(x,t)+\lambda_2^{\alpha_1}{c_2}G(x,t),
{\cal J}=\frac{1}{4 \pi}\int \left( \frac{(|R_z|^2 + |R_{{\bar z}}|^2)(1 + |z|^2)^2}{(1+|R|^2)^2}\right) \frac{2i~{\rm d}z{\rm d}{\bar z}}{(1 + |z|^2)^2}
\int_{|\nu|>20}e^{3i\eta\nu^2/4}\left(\eta-\frac{\nu}{2}\right)S'\left(\frac{\eta+\nu}{2}\right)S\left(\frac{\eta-\nu}{2}\right)d\nu = \sum_{i,j=1,2}T_{3,ij}T_{3,ij}:= \int_{|\nu|>20}e^{3i\eta\nu^2/4}\left(\eta-\frac{\nu}{2}\right)S_i'\left(\frac{\eta+\nu}{2}\right)S_j\left(\frac{\eta-\nu}{2}\right)d\nu.
B_n(v) =\sum_{i=1}^{n-2}b(n;i)v^{i-1}, B'_n(v)=\sum_{i=1}^{n-1}b'(n;i)v^{i-1}, A'_n(v) =\sum_{i=1}^na'(n;i)v^{i-1}, A''_n(v)=\sum_{i=1}^na''(n;i)v^{i-1}, A_n(v) =\sum_{i=1}^na(n;i)v^{i-1}.
~~~ \langle O(\alpha t) x, y \rangle \, O(\alpha t)^{\top}(O(\alpha t)x-y)\otimes(O(\alpha t)x-y) & \,=\, \frac{|x|^2}{2}x\otimes y + (\cos 2\alpha t)B_1(x,y) + (\sin 2\alpha t)B_2(x,y) + B_3 (x,y,\alpha t)\,,
B_0=-\langle D\rangle^{-1/2}[\langle D\rangle^{1/2},\widetilde{A}_1]\langle D\rangle^{-1/2}\cdot\nabla\langle D\rangle^{-1/2}+\widetilde{A}_1\langle D\rangle^{-1/2}\cdot\nabla\langle D\rangle^{-1/2}.
\widehat{\mu}(\lambda\omega)\underset{\lambda\rightarrow \infty}{\sim}\sum_{x\in \omega}\left(\frac{2\pi}{\lambda}\right)^{\frac{n}{2}}e^{i\frac{\pi}{4}(n_+-n_-)(x)}\frac{e^{-i\lambda\omega(x)}}{\sqrt{\vert\det \omega^{\prime\prime}(x)}\vert}\left[ 1+\sum_{j=1}^\infty b_j(x)\lambda^{-j}\right].
\langle \Theta ( x ) \ \Theta ( y ) \rangle ={\frac{2^{1 3}\cdot 3^{3}\cdot 5}{\pi^{3}}}\frac{\beta^{2}[ \lambda ( t ) ] f [ \lambda ( t ) ]}{| x - y |^{1 2}}, \mathrm{f o r}x \neq y ,
\sum_{p_1,\dots,p_k \leq z}g_1(p_1) \cdots g_k(p_k) 2^{\omega(p_1\cdots p_k)}\ll \sum_{p_1,\dots,p_k \leq z}\prod_{j=1}^k \log z = (\pi(z)\log z)^k \ll z^k = \sqrt x,
K_{X;X_0}(t,\rho)=K_{X;\mathfrak{P}_{\Gamma}}(t,\rho)-\sum_{P_j\in\mathfrak{P}_{\Gamma}}\log{t_{P_j}}\left(\sum_{k=0}^{n}\frac{(-1)^k\dim(\sigma_{\rho,k})e^{-t\lambda_{\rho,k}^2}}{\sqrt{4\pi t}}\right).
h = \frac{4 \mathcal H(F) / n}{C - \mathcal{G}(F) - \log \operatorname{rd}(K) + \frac{1}{hn}\sum_{\rho}\Phi(\rho) + \frac{2}{hn}\sum_{\mathfrak P}\sum_{m=1}^{\infty}\frac{\log N\mathfrak P}{N\mathfrak P^{m/2}}F(m \log N\mathfrak P)}
\{b_{r}^{\mu , j}, b_{s}^{\nu , j^{\prime}}\}= \eta^{\mu \nu}\delta^{j , j^{\prime}}\delta_{r , - s}; \{b_{r}^{\mu , k}, b_{s}^{\mu , k^{\prime}}\}= \eta^{\mu \nu}\delta^{k , k^{\prime}}\delta_{r , - s}: ~ ~ N S
\P_j^{k-1}:= \P_j^k - \sum_{|\nu|=k}\frac{\partial^{\nu}\P_j^k}{\nu!}x^{\nu}\tilde{\P}:= \P- \sum_{|\nu|=k}\frac{a^{\nu}}{\nu!}x^{\nu}{\cal D}'(\mathring{B}), j \rightarrow \infty.
\int\Big\{\frac{1}{2}\widetilde L_z\Gamma_1(\log\frac{p}{\pi}, \log\frac{p}{\pi})-\Gamma_1( \widetilde L_z\log\frac{p}{\pi}, \log\frac{p}{\pi})\Big\}p dx=\int \widetilde\Gamma_2^{z,\pi}(\log\frac{p}{\pi},\log\frac{p}{\pi}) p dx.
\left| \left( A_\theta(x)^{-1}\nabla^2 \varphi \left(\frac{x - x_j}h \right) \right) \right| &= \left| \left( \left( I_{d\times d}+ h^{\beta - 1}\sum\limits_{k=1}^M \theta_k \nabla^2 \varphi \left(\frac{x - x_k}h \right) \right)^{-1}\nabla^2 \varphi \left(\frac{x - x_j}h \right) \right) \right|.
k_{2}{\frac{\delta}{\delta k_{2}}}\bar{L}= \Lambda - \Lambda^{2}\left[{\frac{4 c_{1}}{D ( D - 1 )}}+{\frac{2 c_{2}}{D}}+ 2 c_{3}\right] - 1 2 c_{3}\Lambda [ ( \dot{H}+ 2 H^{2}+{\frac{k_{1}}{a^{2}}}) ] .