harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\{\P_n \ | \ \deg P_{n}= n \ , \n \geq 0 \ \right\}= \left\{P_{n_{k}}\right\}_{0\leq k < m+1}= \left\{P_{n_{0}}\equiv 1 \, , \, P_{n_{1}}\, , \, \ldots \, , \, P_{n_{k}}\, , \, P_{n_{k+1}}\, , \, \ldots \ \right\}\ , begin{align*}0.2cm] \deg P_{n_{k}}= n_{k}\ , \ 0 \leq k < m+1\ .
\begin{array}{l c r}{\eta_{j}^{+}= u_{\alpha}^{+}\eta_{j}^{\alpha}; \qquad \overline{{\eta}}_{j}^{+}= u_{\alpha}^{+}\overline{{\eta}}_{j}^{\alpha}}{\eta_{j}^{\alpha}\overline{{\eta}}_{\alpha j}= 1 ; \qquad \eta_{j}^{\alpha}{\eta}_{\alpha j}= \overline{{\eta}}_{j}^{\alpha}\overline{{\eta}}_{\alpha j}= 0}\end{array}
\left( \begin{array}{c}{{\vec{x}}}\\{{\vec{p}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{{\mathbf a}}}&{{{\mathbf b}}}\\{{{\mathbf d}}}&{{{\mathbf c}}}\\ \end{array}\right) \left( \begin{array}{c}{{\vec{\alpha}}}\\{{\vec{\beta}}}\\ \end{array}\right)
\eta \left({{\mathbf{s}},\widehat{\mathbf{s}}}\right) = \frac{{{\mathbf{B}}\left({\mathbf{s}}\right){\mathbf{B}}{{\left({\widehat{\mathbf{s}}}\right)}^T}}}{{\left\|{{\mathbf{B}}\left({\mathbf{s}}\right)}\right\|\left\|{{\mathbf{B}}{{\left({\widehat{\mathbf{s}}}\right)}}}\right\|}},
E \hat{V}_{d-2}(Z\cap A)= e^{-\gamma EV_d(K)}\sum_{l=1}^{2^{n^d}-2}w_l^{(d-2)}\bigg(\gamma EQ(K,B_l,W_l) &+ \frac{\gamma^2}{2}\sum_{S\subseteq B_l}(-1)^{|S|}\bigg(E\int_{S^{d-1}}h(\check{S}\cup \check{W}_l,u) dS_{d-1}(K,du)\bigg)^2\bigg) + O(a).
d s^{2}= -{( \prod_{i = 1}^{4}H_{i})^{- 1 / 2}}f d t^{2}+ ( \prod_{i = 1}^{4}H_{i})^{1 / 2}\bigg ( f^{- 1}{d r^{2}}+ r^{2}d \Omega_{2 , k}\bigg ) \ ,
\tilde{S}_{Y}^{2}=\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2}=(\frac{1}{n}\sum_{i=1}^{n}Y_{i}^{2})-\overline{Y}^{2}=\frac{1}{n^{2}}\sum_{i,j:i<j}(Y_{i}-Y_{j})^{2}.
\sum_{m=0}^l c_o(m,l-m;s|-a,b,-q^{1/2}a,q^{1/2}b,q)={(b/a;q^{1/2})_l(s/a^2;q^{1/2})_l (s;q)_l\over (q^{1/2};q^{1/2})_l (q^{1/2}s/ab;q^{1/2})_l (s/a^2;q)_l}(q^{1/2}/b)^l.
Df_k(B)E_{ii}=\frac{1}{kf_k(B)^{k-1}}\sum_{\mbox{\tiny$\begin{array}{c}i\leq j_1<j_2<...<j_{k-1}\leq n, j_1,...,j_{k-1}\neq i \end{array}$}}\det{B_{(j_1,...,j_{k-1})}},
\widehat{S}(\boldsymbol{\theta}) = \begin{pmatrix}A\theta_1^2 + \frac{1}{4}C\theta_2^2 \left( B + \frac{1}{4}C \right)\theta_1 \theta_2 \left( B + \frac{1}{4}C \right)\theta_1 \theta_2 E\theta_2^2 + \frac{1}{4}C\theta_1^2 \end{pmatrix}.
\tfrac{1}{2}\|\phi_{n,2d}(y)- \phi_{n,2d}(z)\|^2_B = 1-\langle y,z\rangle^{2d}= (1-\langle y,z\rangle^2)(1+\langle y,z\rangle^2 + \cdots + \langle y,z\rangle^{2d-2}) \leq d(1-\langle y,z\rangle^2).
I_{2}= \frac{\pi^{3 / 2}}{1 2}\biggl ( \frac{\theta p}{2}\biggr )^{1 / 2}[ 1 5 \nu^{- 1 / 2}K_{1 / 2}( \nu \theta p ) - 2 0 \biggl ( \frac{\theta p}{2}\biggr )^{3 / 2}\nu^{1 / 2}K_{1 / 2}( \nu \theta p ) + 4 \biggl ( \frac{\theta p}{2}\biggr )^{5 / 2}\nu^{3 / 2}K_{3 / 2}( \nu \theta p ) ] ,
\sum_{\varrho_f}\phi(\varrho_f)=\frac{k}{12}\cdot \frac{3}{\pi}\iint_{\mathcal F}\phi(z) \frac{dx dy}{y^2}+\frac{1}{2\pi}\iint_{\mathcal F}\log (y^{k/2}|f(z)|) \Delta \phi(z) \frac{dx dy}{y^2}.
\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{\substack{j\in J\setminus J_{K(n)}}}\Phi(B_{t_{j}},t_{j}) (B_{t_{j+1}}-B_{t_{j}})= \int_0^T \Phi_{t}d B_{t}.
\mathbf{d}\widetilde\Omega =\widetilde{d\Omega}+ A^\alpha \widetilde{L_\alpha \Omega}- F^\alpha \widetilde{i_\alpha \Omega}=\widetilde{d\Omega}+\widetilde{A^\alpha L_\alpha \Omega}- \widetilde{F^\alpha i_\alpha \Omega}
\mathcal{G}_{n,m,\ell}={c_{\ell+n+2m}}(c_{\ell})^2-{c_{\ell+n+m}}{c_{\ell+m}}{c_{\ell}}-{c_{\ell+n}}{c_{\ell+2m}}{c_{\ell}}+{c_{\ell+n}}({c_{\ell+m}})^2, \mathcal{H}_{n,m,\ell}=c_{\ell+n+2m}(c_\ell)^2 -2c_{\ell+n+m}c_{\ell+m}c_\ell +c_{\ell+n}(c_{\ell+m})^2, \mathcal{I}_{n,m,\ell}=c_{\ell+n+2m}(c_\ell)^2-2c_{\ell+n}c_{\ell+2m}c_\ell +c_{\ell+n}(c_{\ell+m})^2.
C_{J}( \nu_{1}, \nu_{2}) = ( 2 J + \nu_{1}+ \nu_{2}+ 1 ) \frac{{\mit \Gamma}( J + 1 ){\mit \Gamma}( J + \nu_{1}+ \nu_{2}+ 1 )}{{\mit \Gamma}( J + \nu_{1}+ 1 ){\mit \Gamma}( J + \nu_{2}+ 1 )}.
w_{h,k+l}= 1, w_{h,k+l+1}= 1+\frac{(h-k)(2k+2l+\sigma+1)}{(k+l-n)(\beta+2k+1)}, begin{align*}1ex] w_{hi}= P_{h}(i)w_{h,i-1}+ S(i)w_{h,i-2}(i = k+l+2,k+l+3,\ldots,n),
\int_{\mathbb{R}^{3}}|u(x,t)|^{2}\phi(x,t) dx+2\int^{t}_{-T}\int_{\mathbb{R}^{3}}|\nabla u|^{2}\phi dxds &\leq \int^{t}_{-T}\int_{\mathbb{R}^{3}}|u|^{2}(\partial_{s}\phi+\Delta \phi)dxds+ \int^{t}_{-T}\int_{\mathbb{R}^{3}}u\cdot\nabla\phi (|u|^{2}+2\Pi)dxds,
d_K(x+y) = \inf_{z \in K}\| (x+y) - z \| = \inf_{z \in K}\| (x - z) + y \| \leq \inf_{z \in K}\big( \| x-z \| + \| y \| \big) = \inf_{z \in K}\| x-z \| + \| y \| = d_K(x) + \| y \|,
\int_0^{\frac{\pi}{2}}e^{2zx}\log\left( \tan x \right) \d x = \int_0^{\frac{\pi}{2}}e^{2z\left(\frac{\pi}{2}- x \right)}\log\left( \tan\left( \frac{\pi}{2}- x\right) \right) \d x = - e^{\pi z}\int_0^{\frac{\pi}{2}}e^{-2zx}\log\left( \tan x \right) \d x,
S=\frac{1}{2}\int_{-\pi}^{\pi}\int_{t_{1}}^{t_{2}}dxdt[(\partial_{t}\phi_{i}(x,t))^{2}-(\partial_{x}\phi_{i}(x,t))^{2}+F_{ij}\partial_{t}\phi_{i}(x,t)\partial_{x}\phi_{j}(x,t)],
\limsup_{n\to\infty}\|\gamma^{(d)}_{n}(A)\| =\max_{1\leq k\leq r}\|\Gamma'_k\|<\max_{1\leq k\leq r}\|\Gamma_k\| &\leq \max_{1\leq p\leq r}\|\gamma_{2p}^{(d)}(A)\|\leq \|A\|.
\begin{array}{l c r}{{J_{0}^{q}= \int{\frac{d z}{2 i \pi}}J^{q}( z )}}\\{{\bar{J}_{0}^{q}= \int{\frac{d{\bar{z}}}{2 i \pi}}{\bar{J}}^{q}( \bar{z}) .}}\\ \end{array}
g(X_{i,j},X_{i,k},X_{j,k})\quad(\{i,j,k\}\in\binom{\{0,1,2,3,4\}}{3}), & h(X_{i,j},X_{i,k},X_{i,l},X_{j,k},X_{j,l},X_{k,l})\quad(\{i,j,k,l\}\in\binom{\{0,1,2,3,4\}}{4}), & e_k(k\in\{1,2,3,4\}).
q_0(h)=\sum\limits_{r=0}^{\min(n_1,\Delta n)}\sum\limits_{s=0}^{n_1}\sum\limits_{t=0}^{\Delta n}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^r \mathbf{d'}\in [n_1+2,n_2]_<^t d_1\neq d'_1}}\det M_h,
V^{\tau}(R,l,E) = E - \frac{1}{2}\frac{(f_+)^{(3k+c_1-1)/2}}{(f_-)^{(3k+c_1+1)/2}}\left[ E^2 - \left(\frac{f_-}{f_+}\right)^{2k}- \frac{l^2}{r^2}\frac{(f_-)^{k-1/2}}{(f_+)^{k+1/2}}\right].
\partial_{t}T(y)(t,\omega) = -\int_{t}^{\infty}\cos((t-x)\sqrt{\omega})\left(F_{2}(y)(x,\omega)-\mathcal{F}(\sqrt{\cdot}F(x,\cdot \lambda(x)))(\omega \lambda(x)^{2})-\mathcal{F}(\sqrt{\cdot}F_{3}(x,\cdot\lambda(x)))(\omega \lambda(x)^{2})\right) dx
\Phi_{R,L}(x,t,k)\sim\Phi^{\pm\infty}(x,t,k), x\rightarrow \pm\infty, \Phi_{R,L}(x,k)e^{iX_{R,L}(k)x\sigma_3}=I+\mathcal{O}(k^{-1}), k\rightarrow\infty.
\eta =\lim\limits_{n\to\infty}\sum\limits_{i=1}^{m+1}\left[\Phi (x^{n}(t_i))+\int_{t_{i-1}}^{t_i}\mathcal{L}(t, x^{n}(t), u^n(t), v^n(t))dt\right] &\geq\sum\limits_{i=1}^{m+1}\left[\Phi (x^0(t_i))+\int_{t_{i-1}}^{t_i}\mathcal{L}(t, x^0(t), u^0(t), v^0(t))dt\right] &=\mathcal{J}(x^{0}, u^{0}, v^{0})\geq \eta,~ i=1,\ldots,m+1,
-\left(\frac{\partial^2\psi}{\partial\chi^2}+\cot \chi \frac{\partial\psi}{\partial\chi}+\frac{1}{^2\chi}\frac{\partial^2\psi}{\partial \xi^2}\right)+\frac{1}{1-2\lambda^2}\left[\frac{1}{^2\chi}\left(\frac{k_1^2}{^2\xi}+\frac{k_2^2}{^2\xi}\right)+\frac{k_3^2}{\cos^2\chi}\right]\psi=E\psi \,
\frac{1}{c_0^2}\left (\frac{\ddot a_0}{a_0}+\frac{\dot a_0}{a_0}\left(\frac{n-1}{2}\frac{\dot a_0}{a_0}-\frac{\dot c_0}{c_0}\right) \right) = \frac{1}{2{\cal R}^n}\frac{d}{d{\cal R}}\Big(H^2{\cal R}^{n+1}\Big),
\mathrm{M . S}: ~ \mathrm{\bf ~ T r}_{{\cal H}_{o}}\left[ \frac{\Omega}{2}\, ( - 1 )^{F_{s}}e^{- 2 \pi t \, \mathrm{H}_{o}}\right] - \mathrm{\bf ~ T r}_{{\cal H}_{o}}\left[ \frac{\Omega}{2}\, e^{- 2 \pi t \, \mathrm{H}_{o}}\right]
\left|\epsilon\int_0^{r/\epsilon}B(X_{\epsilon s}^\epsilon,y)(1_{\{Y_{s}=y\}}-\pi(y))ds\right|\leq\frac{\eta}{2}+2C\sum_{i=1}^{n^{\epsilon,r}+1}\epsilon\int_{0}^{\tau_i^\epsilon\wedge r/\epsilon}(1_{\{Y_{s}=y\}}-\pi(y))ds.
f^\lambda = \left(f_1 \delta_\sigma^\lambda g^{\mu \nu}+ f_2 g^{\mu \lambda}\delta^\nu_\sigma+ f_3 g^{\nu \lambda}\delta^\mu_\sigma \right)\bar{\Gamma}^\sigma_{~\mu \nu}+ G^{\lambda \rho \lambda ~ \alpha \beta}_{~~~~ \sigma}\nabla_\rho \nabla_\lambda \bar{\Gamma}^\sigma_{~\alpha \beta}
\rho_{\mathrm{mG}}(E)=\sum_{n=0}^N\sum_{k=1}^{n-1}f(n,k)\frac{1}{\sqrt{2\pi \sigma^2(n,k)}}\exp\left ( -\frac{(E-E(n,k))^2}{2\sigma^2(n,k)}\right )\ ,
\tau \stackrel{def}{=}g_1 g_2 \frac{\|R_{22}\|_2}{\|R_{22}\|_{1,2}}\frac{\left\|\widetilde{R}^{-T}\right\|_{1,2}^{-1}}{\sigma_{l+1}\left(A\right)}\mbox{and}\widehat{\tau}\stackrel{def}{=}g_1 g_2 \frac{\|R_{22}\|_2}{\|R_{22}\|_{1,2}}\frac{\left\|R_{11}^{-T}\right\|_{1,2}^{-1}}{\sigma_k\left(\Sigma_k\right)},
\frac{\partial^2 \zeta}{\partial X^2_1}=\left|\begin{array}{l}2 2 2 2\end{array}\right., \frac{\partial^2 \zeta}{\partial X^2_2}=\left|\begin{array}{l}2 2 2 2\end{array}\right., \frac{\partial^2 \zeta}{\partial X^2_3}=\left|\begin{array}{l}2 2 2 2\end{array}\right.
Q=(\begin{matrix}-\mu_{A}&\mu_{AG}&\mu_{AC}&\mu_{AT}\\ \mu_{GA}&-\mu_{G}&\mu_{GC}&\mu_{GT}\\ \mu_{CA}&\mu_{CG}&-\mu_{C}&\mu_{CT}\\ \mu_{TA}&\mu_{TG}&\mu_{TC}&-\mu_{T}\end{matrix})
(x^k-x^*)^T\left(\nabla^2f(\xi^k)+\sum\limits_{i=1}^n\lambda_i^*\nabla^2g_i(\xi^k)+\sum\limits_{i=1}^p\mu_i^*\nabla^2h_i(\xi^k)\right)(x^k-x^*) =\ (x^k-x^*)^T\nabla^2\ell(\xi^k)(x^k-x^*) \ =\ 2(\ell(x^k)-\ell(x^*)) \ \leq\ 0
\Delta_{l^{2}}\,u(t_{i})=\Vert u_{as}(\cdot ,t_{i})-u_{ns}(\cdot,t_{i})\Vert_{l^{2}},\;\;\delta_{l^{2}}\,u(t_{i})=\frac{\Delta_{l^{2}}\,u(t_{i})}{\Vert u_{as}(\cdot ,t_{i})\Vert_{l^{2}}},\;\;\Delta_{l^{\infty}}\,u=\Vert u_{as}-u_{ns}\Vert_{l^{\infty}},
E_0(x)= \mathrm{MC}_0(x)-\mathrm{MC}_0(x+1)-\frac{2x}{(x^2+r^2)^2}= \frac{-2 + 2 c_0}{x^3}- \frac{3 (c_0 + c_0 d_1)}{x^4}+\frac{4 c_0 + 6 c_0 d_1 + 4 c_0 d_1^2 - 4 c_0 d_2 + 4 r^2}{x^5}+O\left(\frac{1}{x^6}\right).
\sum_{i=0}^{N_1(u)}\mathbb{P}\Bigl( \sup_{(s,t)\in\overline{\triangle}_{i0}^T}X(s,t)>u \Bigr) = \frac{S}{S_1}u^{{2}/{\alpha}}\mathcal{H}_{Y_1}^{b}[S_1,T_1]\Psi(u) \bigl(1+o(1)\bigr)
x(t)=x_{0}+\frac{c^{2}}{g}[\frac{1}{\sqrt{1-\frac{v_{0}^{2}}{c^{2}}}}-\sqrt{\frac{1}{1-\frac{v_{0}^{2}}{c^{2}}}+\frac{g^{2}t^{2}}{c^{2}}-\frac{2v_{0}gt}{c^{2}\sqrt{1-\frac{v_{0}^{2}}{c^{2}}}}}]
\nabla_{i k}f^{( n )}( i ) = \nabla_{i k}f ( i ) \cdot \{f^{( n - 1 )}( k ) + f^{( n - 2 )}( k ) f ( i ) + \ldots + f ( k ) f^{( n - 2 )}( i ) + f^{( n - 1 )}( i ) \}
N_{m}=N_{m-1}+N_{m-3}&=(N_{m-3}+2N_{m-5}+N_{m-7})+(N_{m-5}+2N_{m-7}+N_{m-9}) &=(N_{m-3}+N_{m-5})+2(N_{m-5}+N_{m-7})+(N_{m-7}+N_{m-9}) &= N_{m-2}+2N_{m-4}+N_{m-6}.
\ln F\left( t,x\right) =\frac{2e^{-mt}}{\sqrt{1+\varepsilon^{2}}}i\arctan\left( \frac{\tan \left( \frac{x}{2}\right) +\varepsilon}{\sqrt{1-\varepsilon^{2}}}\right) +\frac{c}{m}e^{-mt}-\frac{1}{4m}e^{-2mt}
\|V_1^T\hat{V}\Lambda - \Lambda_1 V_1^T\hat{V}\|_{\mathrm{F}}= \|(\Lambda \otimes I_{p-d}- I_d \otimes \Lambda_1)\mathrm{vec}(V_1^T\hat{V})\| &\geq \min(\lambda_{r-1}- \lambda_r,\lambda_s - \lambda_{s+1})\|\mathrm{vec}(V_1^T\hat{V})\| &= \min(\lambda_{r-1}- \lambda_r,\lambda_s - \lambda_{s+1})\|\sin \Theta(\hat{V},V)\|_\mathrm{F},
( \Gamma_{( n )})_{A B}= - ( - 1 )^{\frac{1}{2}n ( n + 1 )}( \Gamma_{( n )})_{B A}\qquad \quad \Gamma_{( n )}{}^{T}= ( - 1 )^{\frac{1}{2}n ( n + 1 )}C \, \Gamma_{( n )}\, C^{- 1}\ .
\hat{\mathbf{h}}^{(i)}= \left(\check{\mathbf{F}}^{H}\mathbf{S}^{H}(\boldsymbol{\Phi}_{\boldsymbol{\phi}}^{(i-1)})^H\boldsymbol{\Phi}_{\boldsymbol{\phi}}^{(i-1)}\mathbf{S}\check{\mathbf{F}}\right)^{-1}\check{\mathbf{F}}^{H}\mathbf{S}^{H}(\boldsymbol{\Phi}_{\boldsymbol{\phi}}^{(i-1)})^{H}\mathbf{r}
\omega_{0}^{2}- k_{n}^{2}\left[{\frac{1}{g}}+{\frac{1}{2}}k_{n}^{D - 4}\int^{\Lambda / k_{n}}{\frac{d^{D}q}{( 2 \pi )^{D}}}\,{\frac{1}{q^{2}( q + 1 )^{2}}}\right] = 0 .
J(\psi)=\int_{\mathbb{S}^n}\left(\frac{1}{2}|(P_{g_0}^n)^{\frac{1}{2}}\psi|^{2}-\frac{(n-1)!\Lambda}{\Lambda_1}\psi\right)dg_0+ \frac{\Lambda}{n}\ln \left(\int_{\mathbb{S}^n}\tilde{K}e^{n(\psi-w_0\circ \pi)}dg_0\right).
V^{j}_L := \frac{n}{m}\epsilon_{s}^{2}\frac{\|\sqrt{\frac{n}{m}}\tilde{X}^{j}_L\|_2^2}{2(1 + \frac{n}{m}\epsilon_{s}^{2})}- 2^{L-1}\log(1 + \frac{n}{m}\epsilon_{s}^{2}) \leq \frac{\frac{n}{m}\epsilon_{s}^{2}}{2}\left( \|\sqrt{\frac{n}{m}}\tilde{X}^{j}_L\|_2^2 - 2^L \right)
\sum_{a=1}^n \{\langle\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi,\phi\rangle+\langle\phi,\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi\rangle\}\geq C |\kappa_B|^2.
\frac{81 q^2 +220 q^4 + 86 q^6 - 4 q^8 + q^{10}}{\left(1 - q^2 \right)^5}< \left. \frac{81 q^2 +220 q^4 + 86 q^6 + q^{10}}{\left(1 - q^2 \right)^5}\right|_{q = e^{-\pi}}< 0.16.
|J_5|= \Big|-\varepsilon \! \int_0^t \!\int_\Omega\mathcal{C}(\mu_{\varepsilon}-\mu_A) \,\mathrm{d}x\,\mathrm{d}\varsigma \Big| \leq C\varepsilon^{\frac{7}{2}}\|(\mu_{\varepsilon}-\mu_A)\|_{L^2(\Omega \times (0,t))}\le C\varepsilon^7 +\eta \|(\mu_{\varepsilon}-\mu_A)\|_{L^2(\Omega\times(0,t))}^2.
\frac{\delta^{2}W}{\delta x_{\mu}(s)\delta x_{\mu}(s')}\mathop{=}_{s\to s'}\frac{1}{|s-s'|^{4}}(C_{1}+C_{2})(1-\dot{\phi}^{2})+\frac{1}{|s-s'|^{2}}\left( \frac{(C_{1}+C_{2})}{12}\dot{\phi}\stackrel{{\, ...}}{\phi}+\frac{C_{1}}{4}\ddot{\phi}^{2}\right) +O(\phi^{4})\; .
\chi(L_\xi(\lambda,\mu)) = \frac{\sum_{w\in\mathfrak{S}_{n+1}}(-1)^{l(w)}\xi^{\langle\lambda+2\rho,(w\bullet(-2\rho,2\rho))_2\rangle + \langle(w\bullet(-2\rho,2\rho))_1+2\rho,\mu\rangle}}{\xi^{\langle 2\rho,2\rho\rangle}\sum_{w\in\mathfrak{S}_{n+1}}(-1)^{l(w)}\xi^{\langle 2\rho, w\bullet 0\rangle}}.
Q^+_i = 2y_i\sqrt{|h_{ii}|^2p_i}-y^2_i\left(\sum\nolimits_{j\ne i}|h_{ij}|^2p_j+\sigma^2_i\right), Q^-_k = 2\tilde y_k\sqrt{\sum\nolimits_{j}|\tilde h_{kj}|^2p_j+\tilde\sigma^2_k}-\tilde y^2_k|\tilde h_{kk}|^2p_k.
v_it^m = m \partial_{\xi_{2\ell-i+1,m}}+(1-\delta_{i,\ell})\partial_{\xi_{2\ell-i,m}}, v_{\ell + i}t^m =-m\partial_{\xi_{\ell-i+1,m}}+(1-\delta_{i,\ell})\partial_{\xi_{\ell-i,m}},
\begin{array}{l l l l l l l}{{g_{r r}}}&{{=}}&{{\displaystyle \frac{l^{2}}{r^{2}}+ O ( r^{- 4}) \;}}&{{}}&{{g_{t t}}}&{{=}}&{{\displaystyle - \frac{r^{2}}{l^{2}}+ O ( 1 ) \;}}\\{{g_{t r}}}&{{=}}&{{O ( r^{- 3}) \;}}&{{}}&{{g_{\varphi \varphi}}}&{{=}}&{{r^{2}+ O ( 1 ) \;}}\\{{g_{\varphi r}}}&{{=}}&{{O ( r^{- 3}) \;}}&{{}}&{{g_{t \varphi}}}&{{=}}&{{O ( 1 ) \;}}\\ \end{array}
|\ell + m + u_{R}(\ell) - u_{R}(m)| \geq |\ell - m| -|u_{R}(\ell) - u_{R}(m)| \geq |\ell - m| - C_{1}\lambda |\ell - m|^{1/2}= |\ell - m|^{1/2}\,( |\ell - m|^{1/2}- C_{1}\lambda ) \geq \frak{m}|\ell - m|.
\lim_{n \to \infty}D(P^{\mu}|_{\mathcal{F}^{\mathcal{Y}}_{0,n}}\|P^{\nu}|_{\mathcal{F}^{\mathcal{Y}}_{0,n}}) =D(P^{\mu}|_{\mathcal{F}^{\mathcal{Y}}_{0,\infty}}\|P^{\nu}|_{\mathcal{F}^{\mathcal{Y}}_{0,\infty}})
C_J (\nu_1, \nu_2)=(2 J + \nu_1 + \nu_2 + 1)\frac{{\mit\Gamma}(J + 1){\mit\Gamma}(J + \nu_1 + \nu_2 + 1)}{{\mit\Gamma}(J + \nu_1 + 1){\mit\Gamma}(J + \nu_2 + 1)}\, .
P_{j}^{i}h_{i}^{j}( u , p ) = \frac{\sqrt{6}}{l^{2}}\left\{[ 1 6 - 6 \alpha ( \alpha - 2 ) ] A_{1}+ [ 4 + 3 \alpha ( \alpha - 2 ) ] B_{1}+ 2 4 \frac{L^{2}}{l^{2}}B_{2}\right\}+ \mathcal{O}( u ) ~ .
\partial_t \, c_A+\mathbf{\tilde{v}}_A \cdot \nabla c_A-\Delta c_A+\frac{1}{\varepsilon^2}f'(c_A)+\mathbf{w}|_\Gamma \, \nabla c_{A,0}=\mathcal{C}+\mathbf{w}|_\Gamma \, \mathbf{Q},
|\psi_R(x)-\phi_R(x)| =\frac{1}{r}\left| \int_0^{r}\phi_R(\tau) - \phi_R(r) d\tau\right| &\lesssim \frac{1}{r}\left|\int_0^r \int_0^1 \phi'_R(r+\theta(\tau-r)) (\tau-r) d\theta d\tau \right| &\lesssim \frac{|x|}{R}.
{\cal E}_{0}\left( \beta_{2 i}\right) = 4 ~ ,{\cal E}_{0}\left( \bar{\beta}_{2 i}\right) = 8 ~ ,{\cal E}_{0}\left( \beta_{4 i}\right) = 2 ~ ,{\cal E}_{0}\left( \bar{\beta}_{4 i}\right) = 1 0 ~ .
p^{(\alpha)}_\theta({\bf{x}}) = \Big({Z(\theta)^{\alpha}}\big/~{\|p_\theta\|^\alpha}\Big)~ \Big[h({\bf{x}}) + w(\theta)^\top f({\bf{x}})\Big]^{\frac{\alpha}{1 - \alpha}}=Z'(\theta)\Big[ h({\bf{x}}) + w(\theta)^\top f({\bf{x}})\Big]^{\frac{1}{\frac{1}{\alpha}-1}},
\Bigr [ p^{2}- m_{*}^{2}( X ) + \Pi_{\delta}( X ) + \mathrm{R e}\Pi^{+}( X , p ) \Bigl ] A ( X , p ) = 2 \, \mathrm{I m}\Pi^{+}( X , p ) \, \mathrm{R e}\Delta^{+}( X , p ) \; .
\widehat{\left(|u|^{p-1}u\right)}_k = 2\pi\sum_{k=k_1-k_2+\cdots +k_{p}}\widehat{u}_{k_1}\overline{\widehat{u}}_{k_2}\cdots\widehat{u}_{k_{p-2}}\overline{\widehat{u}}_{k_{p-1}}\widehat{u}_{k_p}= \frac{1}{L_t[u]}\widehat{\left(|v|^{p-1}v\right)}_k.
\int_{\mathbb R^n}x^\gamma g_k(x)\,dx=\sum_{\beta\in\mathbb{Z}_+^n,\,\beta\leq\gamma}\binom{\gamma}{\beta}\int_{Q_{0,k}}\left[\int_{\mathbb R^n}(x-y)^{\beta}\theta(x-y)\,dx\right]y^{\gamma-\beta}\eta(y)\,dy=0.
\begin{aligned}\frac{1}{2}\sum_{t=1}^\infty e^{\frac{-\lambda^2 - \mu_{\eta^,1}^2t^2 + 2\lambda\mu_{\eta^,1}t}{2\sigma_{\eta^,1}^2\xi t}}= \beta \frac{1}{2}\sum_{t=1}^\infty e^{\frac{-\upsilon^2 - \mu_{\eta^,0}^2t^2 + 2\upsilon\mu_{\eta^,0}t}{2\sigma_{\eta^,0}^2\xi t}}= \alpha.\end{aligned}
\gamma_{\alpha \beta}^{0}= \left( \begin{array}{c c}{1}&{0}\\{0}&{1}\\ \end{array}\right) , \quad \gamma_{\alpha \beta}^{1}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \quad \gamma_{\alpha \beta}^{2}= \left( \begin{array}{c c}{- 1}&{0}\\{0}&{1}\\ \end{array}\right) .
C'_{K,w}=n\sum_{v\in V(K)\cap N(w)}|z'_{wv}|+2\sum_{v\in N(w)}|z'_{wv}| &=n\sum_{v\in V(K)\cap N(w)\cap V}|z'_{wv}|+2\sum_{v\in N(w)\cap V}|z'_{wv}|\overset{\eqref{sunnight2}}{\leq}\frac{3nr}{|V|}C'_{K,j(w)},
{\frac{b c_{1 , 2}}{2 \pi^{2}}}\int_{k^{2}= p^{2}}{\frac{d^{2}k}{k^{2}}}-{\frac{N_{1 , 2}d^{2}}{\pi^{2}}}\int{\frac{d^{2}k}{k^{2}+ p^{2}}}= 0 .
W(D_{2r,r,1}) =2\sum_{i<j}\left(d(v_i,v_j)+d(v_j,v_i) \right) + 2\sum_{1 \le i,j \le r}d(v_i,w_j) &=2\sum_{1\le i<j\le r}\left((j-i)+1 \right) + 2r \sum_{1 \le j \le r}j &=2\sum_{2\le j\le r}\left(\binom{j+1}{2}-1\right) + r^2(r+1) &=2\binom{r+2}{3}-2r+ r^2(r+1)
P(\zeta_{k+1}= j \mid \zeta_k = i) = P\left( \mathfrak{B}^{+}_{\tau_{i+1}^{\mathfrak{R}^+}}= j \right), \forall i,j\geq 0. \shortintertext{and}P(\tilde\zeta_{k+1}= j \mid \tilde\zeta_k = i) = P\left( \mathfrak{B}^{-}_{\tau_{i}^{\mathfrak{R}^-}}= j \right), \forall i,j\geq 0.
\Phi_{L}\left( p^{\mu}\right) = \Lambda_{L}\left( \dot{p}^{\mu}\to p^{\mu}\right) \Phi_{L}\left( \dot{p}^{\mu}\right) = e^{i \vec{J}\cdot ( \vec{\vartheta}+ i \vec{\varphi})}~ \Phi_{L}\left( \dot{p}^{\mu}\right) .
\begin{cases}\partial_t \omega + \boldsymbol u\cdot \nabla \omega ={\mathrm{curl}\,}\boldsymbol f,{\mathrm{div}\,}\boldsymbol u=0,\boldsymbol u=\boldsymbol{\mathrm{K}}_\Omega\omega=\nabla^{\perp}\mathrm{G}_{\Omega}\omega, \omega(0)={\mathrm{curl}\,}\boldsymbol u_0,\end{cases}
e_n\left(x^2+\dfrac{f_n}{e_n}x-\dfrac{f_n}{e_n}\right) \mathcal{L}_{n+1}(x)=(p_nx^3+q_nx^2+r_nx+s_n)\mathcal{L}_n(x) +t_n(x^2+\omega^2)\left(x^2+\dfrac{u_n}{t_n}x-\dfrac{u_n}{t_n}\right)\mathcal{L}_{n-1}(x), n\geq 1.
a_1^2=a_2^{2^m}=a_i^4=1 \hskip5mm (3\leq i\leq n) a_{n-1}^2=b^2, & [a_1,b]=[a_i,a_j]=1 \;\;\;(1\leq i<j\leq n), & [a_n,b]=a_1, [a_{i-1},b]=a_i^2 \;\;\;(3\leq i\leq n).
\partial_t\big(\begin{smallmatrix}\psi_1 \psi_2\end{smallmatrix}\big) =V\big(\begin{smallmatrix}\psi_1 \psi_2\end{smallmatrix}\big) V =\big(\begin{smallmatrix}u_x -4\lambda-2u 4\lambda^2-2u\lambda-2u^2+u_{xx}-u_x\end{smallmatrix}\big).
\mathbf{G}= \left[ \begin{array}{cc}\mathbf{0}_{N_t J \times N_t}\mathbf{I}_{N_t J}\mathbf{0}_{N_t \times N_t}\mathbf{0}_{N_t \times N_t J}\end{array}\right],\; \mathbf{F}= \left[ \begin{array}{cc}\mathbf{0}_{N_t J \times N_t}\mathbf{I}_{N_t}\end{array}\right]
\pi=\sum_{n=1}^{\infty}\frac{(-1)^{\epsilon(n)}}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{1}{10}+\frac{1}{11}+\frac{1}{12}-\frac{1}{13}+\cdot\cdot\cdot
\zeta_{1}=\lim_{N\to\infty}\,\left(\sum_{m\in\mathcal{M}(N)}\alpha(m)\sum_{k=1}^{\infty}\prod_{j=1}^{d(m)}\left(x_{k+j-1}x_{k+j}\right)^{m_{j}}\right)^{\!\!-1/(2N)}.
\binom{[\frac{k}{2}]+[\frac{n-k}{2}]}{[\frac{k}{2}]}_{t^4}=\frac{\prod_{i=1}^{[\frac{k}{2}]+[\frac{n-k}{2}]}(1-t^{4i})}{\prod_{i=1}^{[\frac{k}{2}]}(1-t^{4i})\prod_{i=1}^{[\frac{n-k}{2}]}(1-t^{4i})}
\! \! \! \! \! \! \! \! \! \! \! \!\mathcal{R}_{UL}(\alpha)= \int\limits_{0}^{\infty}\frac{B_{\rm{u}}+\alpha B}{\ln(2) (g+1)}\exp \left(- U(g) - \frac{\pi^2 \lambda}{2}\sqrt{\frac{g}{\rho}}\sqrt{P_{\rm{d}}|\mathcal{\tilde{C}}_{\rm{u}}(\alpha)|^2}-g \frac{{\beta}P_{\rm{d}}|\mathcal{\tilde{C}}_{\rm{u}}(\alpha)|^2}{\rho}\right)dg,
L^{pzm}_{Weyl}= \int d^3{x}\left[- \partial_i a_{-}\partial_{+}a_i + \frac 1 2 (\partial_{+}a_{-})^2 - \frac1 4 (\partial_{i}a_j - \partial_{j}a_i)^2+ J^\perp_{pzm}a_\perp + J^{-}_{pzm}a_{-}\right]\;,
\frac{1}{\mathrm{K}_p(1,x)}=\int_0^1 (1-(1-x^p)s)^{-\frac1p}\,ds &=\sum_{n=0}^\infty (-1)^n \binom{-\frac1p}{n}(1-x^p)^n \int_0^1 s^n\,ds &=\sum_{n=0}^\infty \frac{(\frac1p)_n}{(n+1)!}(1-x^p)^n &=F\left(1,\frac1p;2;1-x^p\right),
\begin{array}{@{}l@{}}{\displaystyle |\sum\limits_{k=1}^{m}\mathrm{Re}[J_2^{k,2}]|=|-V_0(\|\theta_{\psi}^{m}\|_{\mathcal{L}^2}^{2}-\|\theta_{\psi}^{0}\|_{\mathcal{L}^2}^{2})|\leq C\|\theta_{\psi}^{m}\|_{\mathcal{L}^2}^{2}+C h^{2r+2}.}\end{array}
L_{S S}= -{\frac{1}{8 \pi}}\int d^{2}\sigma \sqrt{- h}\bigg ( h^{a b}\partial_{a}X^{\mu}\partial_{b}X_{\mu}+ 2 i \bar{\psi}^{\mu}\gamma^{a}\partial_{a}\psi_{\mu}- i \bar{\chi}_{a}\gamma^{b}\gamma^{a}\psi^{\mu}\big ( \partial_{b}X_{\mu}-{\frac{i}{4}}\bar{\chi}_{b}\psi_{\mu}\big ) \bigg ) ,
\Psi ( \theta ) = \left( \begin{array}{l r}{\psi_{n , m - 1}( \theta )}\\{\psi_{n , m}( \theta )}\\{\psi_{n , m - 1}( \theta )}\\{\psi_{n , m}( \theta )}\\{\vdots}\\{\psi_{n , m - 1}( \theta )}\\ \end{array}\right)_{( 2 k + 1 ) \times 1}
\{x^i, x^j\}_{1}=0,\quad \{x^i, \theta^j\}_{1}= -\{\theta^{j}, x^{i}\}_{1}=\omega^{ij},\quad\{\theta^i, \theta^j\}_{1}=-\{\theta^j, \theta^i\}_{1}=\frac{\partial\omega^{ij}}{\partial x^k}\theta^k
\alpha_3^{\prime}=\left(\begin{array}{cccc}0 -i 0 0 i 0 0 0 0 0 0 i 0 0 ia 0\end{array}\right) ,\hspace{0.3in}\alpha_4^{\prime}=\left(\begin{array}{cccc}-i 0 0 0 0 -i 0 0 0 0 -i 0 0 0 0 ia\end{array}\right) ,
\begin{aligned}\chi_{10}(\tau) =\tilde{q}p q\prod\limits_{(r,s,t)> 0}(1-\tilde{q}^rp^s q^t)^{c(4rt-s^2)}&=\tilde{q}pq-2\tilde{q}q-16 \tilde{q}pq^2+\cdots \end{aligned}
\left(W_{n}^{M}\right)'(t) = \lim_{h \searrow 0}\left[ \frac{W_{n}^{M}(t+h) - W_{n}^{M}(t)}{h}\right] = \lim_{h \searrow 0}\left[ \frac{W_{n}(x_{t+h},t+h) - W_{n}(x_{t},t))}{h}\right] begin{align*}1ex] \ge \lim_{h \searrow 0}\left[ \frac{W_{n}(x_{t},t+h) - W_{n}(x_{t},t)}{h}\right] = \partial_{t}W_{n}(x_{t},t).
J_{i+1}\leq C M_q(R) |A_{k_{i+1}, \rho_{i}}|^{\frac2n}\Big [ \frac{4^i}{(\sigma R)^2}J_i^{\frac{1}{q'}}+ \big( K^2 |A_{k_{i+1}, \rho_i}|\big)^{\frac{1}{q'}}\Big].
t_{\varepsilon}^{p-2}=\|u\|_{\varepsilon}^{2}+\frac{q\omega^{2}}{\varepsilon^{n}}\int_{M}\psi(t_{\varepsilon}u)u^{2}d\mu_{g}+\frac{q\omega^{2}}{2\varepsilon^{n}}t_{\varepsilon}\int_{M}\psi'(t_{\varepsilon}u)[u]u^{2}d\mu_{g}
\frac{1}{t^2}\sum_{p\le q}(2-\delta_{pq})\left(\frac{d(p+q)+2(p-q)-2pq}{4}-\frac{d(d-2)}{8}\right)\left(\frac{(m^{[p]})^2}{V_d~p!}\right)\left(\frac{(m^{[q]})^2}{V_d~q!}\right)
B=\begin{bmatrix}B_1 0 0 \ldots 0 B_2 0 \ldots 0 0 B_3 \ddots \vdots \ddots \ddots \ddots\end{bmatrix}, J_A=\begin{bmatrix}0 A_1 0 \ldots A_1^* 0 A_2 \ddots 0 0 A_2^* \ddots \vdots \ddots \ddots \ddots\end{bmatrix}.
S^{(1)}_{E_{i'j'}}= \Phi_1(p_{i'}\varphi(\varphi^{-1}(S^{(1)}_{E_{i'j'}})) p_{j'}) = \Phi_1(\varphi(p_i \varphi^{-1}(S^{(1)}_{E_{i'j'}})p_j)) = \Phi_1(\varphi(\sum^{\infty}_{n=m}b_nS^{(n)}_{E_{ij}})) & = \sum^{\infty}_{n=m}b_n\Phi_1(\varphi(S^{(n)}_{E_{ij}})) = 0
\sum^{N_0(n)}_{\ell=0}t^{-\ell}\Bigr(\mathrm{Tr\,}\alpha^+_\ell(x_0)-\mathrm{Tr\,}\alpha^-_\ell(x_0)\Bigr) &=\frac{1}{2\pi}\sum^N_{j=1}\chi_j(x_0)\Bigr(\mathrm{Tr\,}A_{B_j,+}(t,z_j,z_j)-\mathrm{Tr\,}A_{B_j,-}(t,z_j,z_j)\Bigr)+O(t),
\begin{array}{r c l}{{\psi^{\prime}( t , x )}}&{{=}}&{{e^{- i e \epsilon_{0}( t , x )}\psi ( t , x ) \ ,}}\\{{{A^{1}}^{\prime}( t , x )}}&{{=}}&{{A^{1}( t , x ) - \partial_{1}\epsilon_{0}( t , x ) \ ,}}\\{{\pi_{1}^{\prime}( t , x )}}&{{=}}&{{\pi_{1}( t , x ) \ , \}}\\{{{A^{0}}^{\prime}( t , x )}}&{{=}}&{{A^{0}( t , x ) + \partial_{0}\epsilon_{0}( t , x ) \ ,}}\\ \end{array}
\bar{\omega}_{1}\equiv \sum_{n = - \infty}^{\infty}\breve{\omega}_{1}( t , x^{1}, x^{2}, x^{3}, x^{4}, x^{5}- n ) \simeq -{\frac{2 \pi m_{0}a}{3 \bar{R}^{3}}}x^{2}\, , \qquad \bar{\omega}_{2}\simeq 0 \, ,
\sup_{K_j}\frac{1}{|x- \lambda - i\delta|}= \frac{1}{\sqrt{\frac{(M\rho (0) + Y )^2}{t^2}+ 2^{-2j}\frac{\mathcal{Y}^2}{M\rho(0)^4 t^2}}}\leq 2^{j+1}\frac{t}{\mathcal{Y}}M\rho (0)^2 \sim |K_j|^{-1}.
\begin{aligned}\theta_{i,m}^\star = \begin{cases}\angle\chi_{i,m}+ \pi, \angle\chi_{i,m}\in [0,\pi), \angle\chi_{i,m}- \pi, \angle\chi_{i,m}\in [\pi,2\pi), \end{cases}~~~~~~~~~~~\forall i \in \{\mathrm{t,r}\}, \forall m \in \mathcal{M}. \end{aligned}