harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\ddot{z}\left( t\right) +\gamma\left( t\right) \dot{z}\left( t\right)\leq\left\Vert \dot{x}\left( t\right) \right\Vert^{2}+\left\Vert x\left(t\right) -x^{\ast}\right\Vert \left\Vert g\left( t\right) \right\Vert=k(t).
a(u,\varphi):= \int_\Omega \varepsilon(x) \nabla u(x) \cdot \nabla \varphi(x) \, dx, b(u, \varphi):= \int_\Omega \Big( e^{u(x) - \hat{v}(x)}- e^{\hat{w}(x) - u(x)}\Big) \varphi(x) \, dx, f(\varphi):= \int_\Omega k_1(x) \varphi(x) \, dx.
S_{i j k l}( \vec{x}, \vec{y}) = \left( \begin{array}{c c}{0}&{- 1}\\{1}&{0}\\ \end{array}\right) \Bigl ( \eta_{i j , k l}\nabla^{2}+ \partial_{i j , k l}^{2}\Bigr ) \delta ( \vec{x}- \vec{y})
B_{n,p,q}(f;x)=\frac1{p^{\frac{n(n-1)}2}}\sum\limits_{k=0}^{n}\left[\begin{array}{c}n k\end{array}\right]_{p,q}p^{\frac{k(k-1)}{2}}x^{k}\prod \limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)~~f\left( \frac{[k]_{p,q}}{p^{k-n}[n]_{p,q}}\right) ,~~x\in \lbrack [0,1].
\langle \mathcal{N}_{S}\rangle_{\rho_t}= \sum_{j \in S}\langle a_j^* a_j \rangle_{\rho_t}= \sum_{j \in S}\langle c_j^* c_j \rangle_{\rho_t}=\sum_{j \in S}\langle \tau_t(c_j^* c_j) \rangle_{\rho}=\sum_{j \in S}\langle \tau_t(c_j^*) \tau_t(c_j) \rangle_{\rho}
\lambda_{\mathrm{min}}(\widehat{\mathbf{Y}}) = \lambda_{\mathrm{min}}(\mathbf{Y}) - \big(\lambda_{\mathrm{min}}(\mathbf{Y})-\lambda_{\mathrm{min}}(\widehat{\mathbf{Y}})\big) > 0.5\,\lambda_{\mathrm{min}}(\mathbf{Y}).
E_{x}|_{\pi_{1}^{-1}(o)}=\mathbb{P}_{2}^{2}\cup\mathbb{P}_{3}^{2},\;\;E_{y}|_{\pi_{1}^{-1}(o)}=\mathbb{P}_{3}^{2}\cup\mathbb{P}_{1}^{2},\;\;E_{z}|_{\pi_{1}^{-1}(o)}=\mathbb{P}_{1}^{2}\cup\mathbb{P}_{2}^{2}.
T_H(\lambda_q^{*^G})(\psi) =\int_G\psi_q(x)d\lambda_q^{*^G}(x) &=\int_G\psi_q(x^{-1})d\overline{\lambda_q}(x) &=\left(\int_G\overline{\psi_q(x^{-1})}d\lambda_q(x)\right)^{-}&=\left(\int_{G/H}T_H(\psi_q^{*^G})(xH)d\lambda(xH)\right)^{-}=\lambda^{\ast^{G/H}}(\psi).
C_n^{(q, q^{\theta})}\big(x_1/\big(q^{\theta}x_2\big)\big) = \frac{q^{\theta}x_2 - q^{2n}x_1}{q^{\theta}x_2 - x_1}\frac{1}{t^n}\prod_{i=1}^n{\left\{\frac{q^{\theta}- q^{i-1}}{1 - q^i}\frac{q^{\theta}x_2 - q^{i-1}x_1}{x_2 - q^ix_1}\right\}}.
\left\{\underline{g}\in\prod_{i=1}^{n}\mathcal{L}_{G}^{i}\middle|\overline{\left\langle \underline{g}\right\rangle}\neq G\right\}=\bigcup_{P\in\mathcal{P}}\left\{\underline{g}\in\prod_{i=1}^{n}\mathcal{L}_{G}^{i}\middle|\overline{\left\langle \underline{g}\right\rangle}=xPx^{-1}x\in G\right\}.
P_4^s = \gamma \sum_{t=1}^{\lfloor sT\rfloor-1}f(\frac{t}{T}) O( \eta+ d^{-1}T^{-1}+T^{-2}) = \frac{1}{T}\sum_{t=1}^{\lfloor sT\rfloor-1}f(\frac{t}{T}) O(d^{-1}\gamma+T^{-1}\gamma).
(I-N_j(\varphi)+P) f_l = (I-M_j(\varphi)+P) f_l &= \dfrac{1}{n}\sum_{i=1}^j (I-C_{\varphi_i})f_l+Pf_l &= \dfrac{1}{n}\sum_{i=1}^j (I+C_\varphi+...+C_{\varphi_{i-1}})(I-C_{\varphi})f_l+Pf_l &= \dfrac{1}{n}\sum_{i=1}^j (I+T_1+...+T_{i-1})(I-T_1)f_l+Pf_l\rightarrow 0,
\Pi(\mathcal{L}_{\Pi\eta_1}\eta_2)= \iota_{\mathcal{L}_{\Pi\eta_1}\eta_2}\Pi=\iota_{\mathcal{L}_{\Pi\eta_1}\eta_2}(X_1\wedge X_2) = \big(\iota_{\mathcal{L}_{\Pi\eta_1}\eta_2}X_1\big)\wedge X_2- X_1\wedge\big(\iota_{\mathcal{L}_{\Pi\eta_1}\eta_2}X_2\big)
P(\varepsilon z) =\sum_{k=0}^{m/2}\sum_{A\in \mathcal{A}_k}\sum_{l=k}^{m/2}\sum_{\gamma \in \Gamma_{A,l}}\sum_{\beta \in B_{A,l}}x_{2\beta+2\gamma+1_A}\varepsilon_A z^{2\beta+2\gamma+1_A}.
x = \frac{P_{\vec{n}+\vec{e}_k}(n^\gamma x)}{n^\gamma P_{\vec{n}}(n^\gamma x)}+ \frac{b_{\vec{n},k}}{n^\gamma}+ \sum_{j=1}^r \frac{a_{\vec{n},j}}{n^{2\gamma}}\frac{n^\gamma P_{\vec{n}-\vec{e}_j}(n^\gamma x)}{P_{\vec{n}}(n^\gamma x)}.
\sum_{m=1}^{\infty}N_{X\underset{x,y}{\star}Y}(m)u^m = \left(\sum_{m=1}^{\infty}N_X(m)u^m+\sum_{m=1}^{\infty}N_X(m)u^m\sum_{m=1}^{\infty}N_Y(m)u^m+\cdots\right) + \left(\sum_{m=1}^{\infty}N_Y(m)u^m+\sum_{m=1}^{\infty}N_Y(m)u^m\sum_{m=1}^{\infty}N_X(m)u^m+\cdots\right)
0= h_V\left( ev \circ T^2 \circ \left(\nabla ev^{\ast}\right)(v_1),v_2\right)=\left(T^2 \circ \left(\nabla ev^{\ast}\right)(v_1), ev^{\ast}(v_2)\right) = \left(\left(\nabla ev^{\ast}\right)(v_1), T^2 \circ ev^{\ast}(v_2)\right)=h_V\left(v_1, \left(\nabla ev\right) \circ T^2 \circ ev^{\ast}(v_2)\right),
x_1^{\mathsf{tr}}x_3=x_3^{\mathsf{tr}}x_1 \ , \ \ \x_2^{\mathsf{tr}}x_4=x_4^{\mathsf{tr}}x_2 \ , \ \ \x_1^{\mathsf{tr}}x_4-x_3^{\mathsf{tr}}x_2=I_g,
R^{\sf SCF}_{\sf sym}(\mathbf{H}) = \min_{\ell = 1, \dots, L}\log\left( \frac{{\sf SNR}}{{\sf SNR}L_{\ell, \ell}^2}\right) - \log\left( 2 \pi e G(\mathcal{L}') \right) - \log\left( \frac{\mu(\mathcal{L}, P_e)}{2 \pi e}\right)
K_{x}(t)=logM_{x}(t)=(\frac{\sigma^{2}t^{2}}{2}+\mu t)+log\{1-\Phi(-\frac{\mu}{\sigma}-\sigma t)+e^{\frac{\sigma^{2}t^{2}}{2}-\mu t}[1-\Phi(\frac{\mu}{\sigma}-\sigma t)]\}
\widetilde{\psi}_{k}^{( \nu_{1},\nu_{1})}(x) =( E_{\nu_{1}}-E_{k}) \psi_{k}(x) -\frac{\psi_{\nu_{1}}^{2}(x)}{\int_{x_{0}}^{x}dt\psi_{\nu_{1}}^{2}(t)-W_{0}}\psi_{k}^{( \nu_{1})}(x) .
\mathrm{l n}\mathrm{d e t^{\prime}}\Delta = - \lim_{s \to 0}{\frac{d}{d s}}\int_{\epsilon}^{\infty}{\frac{d t t^{s - 1}}{\Gamma ( s )}}\int_{S}d^{2}z{\sqrt{g}}[ \mathrm{t r^{\prime}}e^{- t ( \Delta + m^{2})}] - e^{- m^{2}t},
\|X\|_{S_{p}}=\|U^{*}\|_{S_{p_{1}}}\|V^{*}\|_{S_{p_{2}}}&\leq \|U_{1}\|_{S_{p_{1}}}\|V_{1}\|_{S_{p_{2}}}&\leq \|U\|_{S_{\widehat{p}_{1}}}\|U_{2}\|_{S_{q}}\|V_{1}\|_{S_{p_{2}}}&=\|U\|_{S_{\widehat{p}_{1}}}\|V\|_{S_{\widehat{p}_{2}}}
C=\begin{pmatrix}2c_{00}c_{01}\cdots c_{0(d-1)}c_{01}2c_{11}\cdots c_{1(d-1)}\vdots \vdots \ddots \vdots c_{0(d-1)}c_{1(d-1)}\cdots 2c_{(d-1)(d-1)}\end{pmatrix},
\widehat{u}_n^{(\ell)}= 2\mathbf{Im}\bigg\{\sum_{j=0}^{N-1}w_j^{(\ell)}F_\omega(\lambda_j^{(\ell)}) (1-\tau\lambda_j^{(\ell)})^{-[n-b_{\ell-1}^{(n)}-1]}y^{(\ell)}(\tau\lambda_j^{(\ell)})\bigg\},
\Gamma(N):=\bigg\{\bigg(\begin{matrix}a b c d\end{matrix}\bigg)\in\mathrm{SL}_{2}(\mathbb{Z})\,\bigg\vert\,a\equiv d\equiv 1\,\mathrm{mod}\,N,\,b\equiv c\equiv 0\,\mathrm{mod}\,N\bigg\}
\langle \Psi,\mathcal{H}\Psi\rangle = \frac{1}{\pi}\int_{\mathbb{C}}\langle \Phi(z),(\mathcal{K}(z) + \mathcal{Q}_3^{}+ \mathcal{Q}_4^{}+ \mathcal{R}_0)\Phi(z)\rangle_{}\mathrm{d}z.
{\cal A}= d y^{0}\wedge d y^{1}\wedge . . . \wedge d y^{q - 1}\wedge [ d z_{1}^{1}\wedge . . . \wedge d z_{1}^{r_{1}}h_{1}e^{C_{1}}+ . . . + d z_{n}^{1}\wedge . . . \wedge d z_{n}^{r_{n}}h_{n}e^{C_{n}}] ,
S_k(i) =\{(x_1,\ldots,x_n) \in L_{1k2}\mid x_1=\cdots=x_{i-1}=0, x_i=-1\}, T_k(i) =\{(x_1,\ldots,x_n) \in L_{1k2}\mid x_1=\cdots=x_{i-1}=0, x_i=1\}, U_k(i) =\{(x_1,\ldots,x_n) \in L_{1k2}\mid x_1=1, x_l=-1, x_j=1, \exists l\in \{2,\ldots, i\}, \exists j \in \{l+1,\ldots,n\}\}
|z(u_k)-z(u)| =\left|\int_{\Omega}\frac{|u_k(y)|^p-|u(y)|^p}{|x-y|^{n-2}}\,dy\right| &\leq C_{\Omega,p,q}\left(\int_{\Omega}||u_k|^p-|u|^p|^q\,dy\right)^{1/q}&\leq C_{\Omega,p,q}\left(\int_{\Omega}\xi^{(p-1)q}||u_k|-|u||^q\,dy\right)^{1/q},
R_k(x_1, \dots, x_k) = \left(\frac2\pi\right)^k \left(\prod_{p=1}^kx_p\right) \lim_{\{a_f\}\to\infty}\frac{\det[C_N(z_p,\zeta_q)]_{p,q=1,\dots,k+N_f}}{\det[C_N(a_f,im_g)]_{f,g=1,\dots,N_f}}
|S_{1}'| =\binom{n}{2}-\big| N_G{(S_1)}\cap E( K_n) \big| \leq \binom{n}{2}- \binom{s_1}{2}< \binom{n}{2}- \binom{\frac{5+\sqrt{{(2n-5)}^2 -48\alpha}}{2}}{2}\leq 32,
\begin{aligned}|w_i(x,z)| \leq \int_0^x|\partial_x w_i(s,z)|\mathrm{d}s \lesssim \int_0^x\left(\int_0^z |\partial_x w_i \partial_{xY}w_i|(s,\eta)\mathrm{d}\eta\right)^{1/2}\mathrm{d}s \lesssim \sqrt{x}\Vert \partial_x w_i\Vert_{L^2}^{\frac{1}{2}}\Vert \partial_{xY}w_i\Vert_{L^2}^{\frac{1}{2}}\lesssim \sqrt{L},\end{aligned}
\Upsilon(\alpha v+\beta v^b\bar{v}^a, \gamma v+\lambda v^b\bar{v}^a)= \nu v+\mu v^b\bar{v}^a , & \Psi(\alpha v+\beta v^b\bar{v}^a)=\nu v+\mu v^b\bar{v}^a + \epsilon v^{2b-1}\bar{v}^{2a}+ \kappa v^{a+b}\bar{v}^{a+b-1},
\langle -,-\rangle: \mathbf{N}\times \mathbf{M}\longrightarrow \mathbb{Z}, \mathbf{N}_\mathbb{R}:=\mathbf{N}\otimes_\mathbb{Z}\mathbb{R}, \mathbf{M}_\mathbb{R}:=\mathbf{M}\otimes_\mathbb{Z}\mathbb{R},
\delta x^{\alpha \alpha^{\prime}}= -{\frac{1}{2}}( \epsilon_{a}^{\alpha}\theta^{\alpha^{\prime}a}+ \epsilon^{\alpha^{\prime}a}\theta_{a}^{\alpha}) \ , \ \ \delta \theta_{a}^{\alpha}= \epsilon_{a}^{\alpha}\ , \ \ \delta \theta^{\alpha^{\prime}a}= \epsilon^{\alpha^{\prime}a}\ .
\int_{\widetilde{S}_\varepsilon(\xi)\cap B_R(0)}\frac{\lvert v(k)\lvert^2}{\omega(k)^2}d\lambda_\nu =\nu \lambda_\nu(\widetilde{S}_\varepsilon(\xi) \cap B_1(0))\int_{0}^\infty 1_{B_R(0)}(xe_1)\frac{\lvert v(ke_1)\lvert^2}{\omega(ke_1)^2}k^{\nu-1}d\lambda_1 &=\infty
B_\Omega(z,w) =B_\Omega(z,\sigma(\tau))+B_\Omega(\sigma(\tau),w) \geq B_\Omega(z,z_0)+B_\Omega(z_0,w) - 2B_\Omega(z_0,\sigma(\tau)) & \geq B_\Omega(z,z_0)+B_\Omega(z_0,w) - R
\big\{J_1^\xi,J_1^{\xi'}\big\}_1=0 ,\big\{J_2^\xi,J_2^{\xi'}\big\}_2=0 ,\big\{J_3^\xi,J_3^{\xi'}\big\}_3=-\big(\xi^a\xi'^{bc}+\xi'^{a}\xi^{bc}\big)h_{abc},
\eta_{H}( 0 ) = - 1 - 2 \, \mu ( \gamma ) , \ \ \ \zeta_{H}( 0 ) = - 1 , \ \ \, z e t a_{H}^{\prime}( 0 ) = 2 \ln \left| \frac{\sqrt{c}\, \ell [ \gamma ]}{2 \sin ( \sqrt{c}\, \ell [ \gamma ] )}\right| ,
m \simeq \frac{\alpha}{\pi^2}\int dq_3 dq_4 \int_0^\infty dr^2{\rm e}^{- r^2}\frac{1}{2 r^2 + L^2 (q_3^2 + q_4^2)}~\frac{m}{m^2 + (q_3^2 +q_4^2)}
T_{\phi \mu \nu}= B_{\mu \rho}B_{\nu}^{\rho}- \frac{1}{4}g_{\mu \nu}g^{\rho \sigma}g^{\lambda \tau}B_{\rho \lambda}B_{\sigma \tau}- 2 \frac{\partial V ( \phi )}{\partial g^{\mu \nu}}+ g_{\mu \nu}V ( \phi ) .
\frac{p_{r}^{2}}{g_{r r}}+ \frac{p_{\theta}^{2}}{g_{\theta \theta}}+ \frac{-{g^{\prime}}_{t t}}{- \cal D}\left( p_{\phi}+ \frac{g_{t \phi}+ \Omega_{0}g_{\phi \phi}}{{g^{\prime}}_{t t}}E \right)^{2}= \left( \frac{1}{-{g^{\prime}}_{t t}}E^{2}- \mu^{2}\right) ,
\psi([a,b]) =(b^{-1}a,1,a^{-1}b), \psi([a,b,a]) =((a^{-1}b)^2,b^{-1}a,b^{-1}a), \psi([a,b,a^2] =(a^{-1}b,a^{-1}b,(b^{-1}a)^2),
\int_XU(\bar \mu_t(x))dx= &\int_Y\frac{1}{\bar \nu(y)}\int_{T_0^{-1}(y)}U\Big(\frac{\bar \mu_0^y(x_0)\bar \nu(y)JT^y_t(\nabla\phi^y_t(x_0))}{\det(D^2\phi^y_t(x_0))}\Big)\frac{1}{JT^y_t(\nabla\phi^y_t(x_0))}\det(D^2\phi_t^y(x_0))d\mathcal H^{m-n}(x_0)dy
j=27\frac{(1+8\alpha)^3}{\alpha(1-\alpha)^3}\quad\mathrm{and}q=e^{2\pi i\tau}=\exp\Big(-\frac{2\pi}{\sqrt{3}}\frac{_2F_1(\frac{1}{3},\frac{2}{3},1,1-\alpha)}{_2F_1(\frac{1}{3},\frac{2}{3},1,\alpha)}\Big),
w_n^{\rm susy}\left(F_n\right)=\int \prod_{i=1}^nd^2x_i \prod_{i=1}^n d\xi_i\,\, w_n^{\rm susy}\left(x_1,\xi_1;\ldots;x_n,\xi_n\right) F_n\left(x_1,\xi_1;\ldots;x_n,\xi_n\right)\,\,.
f_i(x^*)= \int_{-\frac{1}{2a_{i,T}\lambda}}^{\frac{1}{2a_{i,T}\lambda}}\frac{1}{\sqrt{2 \pi}}\exp \left(\dfrac{-w^2}{2}\right) dw\le \int_{-\frac{1}{2a_{i,T}\lambda}}^{\frac{1}{2a_{i,T}\lambda}}dw = \frac{1}{|a_{i,T}|\lambda}\le \frac{1}{\min_{1 \le i \le m}|a_{i,T}|\lambda}.
\|\widetilde{V}_{2}^{\ast}V_{1}\Sigma_{1}\|_{F}^{2}=\|E\|_{F}^{2}-\|EB^{\dagger}B\|_{F}^{2}, \|\widetilde{\Sigma}_{1}\widetilde{U}_{1}^{\ast}U_{2}\|_{F}^{2}=\|E\|_{F}^{2}-\|AA^{\dagger}E\|_{F}^{2}.
\sum_{\substack{j,l \in \mathcal{J}j\not=l}}r(m_j)r(n_l)\Phi\left(|j-l|-1 \right) \le \sum_{\substack{j,l \in \mathcal{J}j\not=l}}r(m_j)^{2}\Phi\left(|j-l|-1 \right).
\lim_{M\rightarrow\infty}a\left( t\right) \mathcal{P}_{M}\varphi =\lim_{M\rightarrow\infty}\left[ \gamma\left( t\right) a\mathcal{P}_{M}\varphi+\delta\left( t\right) a^{\dag}\mathcal{P}_{M}\varphi\right] & =\gamma\left( t\right) \overline{a}\varphi+\delta\left( t\right)a^{\ast}\varphi
s_0^2=s_{2^{n-1}}^2=4,\s_{2^{n-2}}^2=s_{3\cdot2^{n-2}}^2=0 s_{2^{n-3}}^2=s_{3\cdot2^{n-3}}^2=s_{5\cdot2^{n-3}}^2=s_{7\cdot2^{n-3}}^2=2.
Y_{m}( \mu = 0 , y ) = \frac{\sqrt{m}\, \Gamma ( m y )}{m ! \Gamma ( m y + 1 - m )}e^{- m \tau_{0}}= \frac{\Gamma ( 1 + m y ) \Gamma ( 1 + m ( 1 - y ) )}{2 \, \Gamma ( 1 + m )}\, e^{- m \tau_{0}}B_{m}
z_1^n(\theta)= \frac{n^{1/\alpha}}{\delta x^{1/ \alpha}}\left( \frac{1}{n}(a_0-a) - \int_0^\frac{1}{n}(b_0 X_s - bx )ds + \delta_0 \int_0^\frac{1}{n}X_{s-}^{1/ \alpha_0}dL_s \right).
\left\|\Box_k^{\alpha_2}|~M_1\rightarrow M_2\right\| \gtrsim \frac{\|\Box_k^{\alpha_2}f_l^{\alpha_1}\|_{M_2}}{\|f_l^{\alpha_1}\|_{M_1}}\sim\frac{\|f_l^{\alpha_1}\|_{L^{p_2}}}{\|f_l^{\alpha_1}\|_{L^{p_1}}}\sim 2^{jn\alpha_1(1/p_1-1/p_2)}= 2^{jA_1(\mathbf{p},q,\alpha_1,\alpha_2)}
2^{\mu_*k_*t}\cdot 2^{‐m_t(s_*+1/q‐1/p_1)}= 2^{‐\alpha_*k_*t}\cdot 2^{‐m_t(1/q‐1/p_0)}\stackrel{\eqref{hat_theta_def}}{=}n^{‐q\hat \theta/2};
\begin{aligned}y\left[ i \right] = \sum\limits_{l = 1}^L{{\bf{h}}_l^H{\bf{\bar q}}\left[{i -{n_l}}\right]}+ z\left[ i \right] = \sum\limits_{l = 1}^L{\sum\limits_{{l'}= 1}^{L'}{{\bf{h}}_l^H{{\bf{{F}}}_{l'}}{\bf{d}}\left[{i -{n_l}-{\kappa_{l'}}}\right]}+ z\left[ i \right]}. \end{aligned}
(\rho^{+}\oplus\rho^{-})\otimes(\rho^{+}\oplus\rho^{-})\cong\bigoplus_{p=0}^{8}{\wedge}^{p}\rho_{v}\cong 2\left(1\oplus\rho_{v}\oplus{\wedge}^{2}\rho_{v}\oplus{\wedge}^{3}\rho_{v}\right)\oplus{\wedge}^{4}\rho_{v}.
\max_{\kappa,\lambda_{1},\lambda_{2}}\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)+\frac{\kappa_1}{x^2+\lambda_{1,1}x+\lambda_{1,2}+\frac{\kappa}{x^2+\lambda_1 x+\lambda_2}}}\right. &\quad\left.-U(x)\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{(x+1)^2+\lambda_{1,1}(x+1)+\lambda_{1,2}+\frac{\kappa}{(x+1)^2+\lambda_1 (x+1)+\lambda_2}}}-V(x)\right).
h\|B_{1}v_{n+1}\|_{H}^2 = h(B_{1}v_{n+1}, B_{1}v_{n+1})_{H}=-h(Lz_{n+1}, B_{1}v_{n+1})_{H}- h(A_{2}\varphi_{n+1}, B_{1}v_{n+1})_{H}- h(\Phi\varphi_{n+1}, B_{1}v_{n+1})_{H}\,\quad- h({\cal L}\varphi_{n+1}, B_{1}v_{n+1})_{H}+ h(B_{2}\theta_{n+1}, B_{1}v_{n+1})_{H}
(v_pv_q^*v_r)^* = v_r^*v_qv_p^* &= v_{r_1}v_q^*v_qv_p^* \eqref{eq:p*q1}&= v_{r_1}v_q^*v_qv_q^*v_{p_1}\eqref{eq:pq*1}&= v_{r_1}v_q^*v_{p_1}
g_{\mu \nu}\left( x \right) = \eta_{m n}\, e_{\mu}^{m}\left( x \right) e_{\nu}^{n}\left( x \right) \, \, , \quad g^{\mu \nu}\left( x \right) = \eta^{m n}\, e_{m}^{\mu}\left( x \right) e_{n}^{\nu}\left( x \right) \, \, ,
R^{\alpha - 2}\delta^{-\alpha - 2}= w_0^{\chi + 1}(\delta) \leq w_0^{\chi + 1}(r) \leq w_0^{\chi + 1}(0) = R^{\alpha - 2}\delta^{-\alpha - 2}\left(1 + \tfrac{\alpha(\alpha + 5)}{6}\right)^{\chi + 1},
\kappa \left( \left[ \begin{matrix}a_{0, 0}a_{0, 1}a_{1, 0}a_{1, 1}\end{matrix}\right] \right) = a_{0, 0}\otimes e_{0, 0}+ a_{0, 1}\otimes e_{0, 1}+ a_{1, 0}\otimes e_{1, 0}+ a_{1, 1}\otimes e_{1, 1}.
\nabla f = \epsilon_{N}\,g(\nabla f, N)N = \epsilon_{N}\frac{1}{| g_{00}|}\,\partial_{0}= \epsilon_{N}\frac{1}{\sqrt{| g_{00}|}}\,N, g(\nabla f, \nabla f) = \frac{g_{00}}{|g_{00}|^{2}}= \frac{\epsilon_{N}}{|g_{00}|}.
(p-q)P_{q+1}^{n-2,p-q}(x)+2s\Big(P_{q+1}^{n-2,p-q}(x)\Big)' =-(n+q)P_{q+1}^{n-1,p-q-1}(x)+(n+p+q+1)P_{q+1}^{n-1,p-q-1}(x) &=(p+1)P_{q+1}^{n-1,p-q-1}(x).
L:=\min\Big\{\frac{(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{-}+\chi_1\mu_1(\lambda_1-\lambda_2)_{-}}{\lambda_2(b+\chi_2\mu_2-\chi_1\mu_1-M)}, & \frac{(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{-}+\chi_2\mu_2(\lambda_1-\lambda_2)_{-}}{\lambda_1(b+\chi_2\mu_2-\chi_1\mu_1-M)}\Big\}.
X_{((T^\ast Q)_{\mathcal{O}_\mu},\omega_{\mathcal{O}_\mu},h_{\mathcal{O}_\mu},f_{\mathcal{O}_\mu},u_{\mathcal{O}_\mu})}=X_{h_{\mathcal{O}_\mu}}+\textnormal{vlift}(f_{\mathcal{O}_\mu})+\textnormal{vlift}(u_{\mathcal{O}_\mu}),
e(eS) =\underset{a\in S}{\cup}e(ea)=\underset{a\in S}{\cup}(ee)(ea)=\underset{a\in S}{\cup}(ae)(ee) =\underset{a\in S}{\cup}(ae)e=\underset{a\in S}{\cup}(ee)a=\underset{a\in S}{\cup}ea=eS.
\left|\mathrm{tr}\left( (D^{\frac{1}{2n}}g_r^*g_rD^{\frac{1}{2n}})^{n-1}D^{\frac{1}{2n}}g_r^*x_{i_{r+1}}D^{\frac{1}{2n}}\right)\right|\leq r^{2n-1}\cdot \dfrac{1}{2(r+1)r^{2n-1}}= \dfrac{1}{2(r+1)},
{\cal Z}_{2N}= K (2N)! \int_{\stackrel{\cdots}{R(-\infty,x_1,\ldots,x_{2N},\infty)}}\int \prod_{i=1}^{2N}e^{-\frac{\beta}{2}\sum_{i=1}^{2N}V(x_i)}\times[det (x_j^{i-1})]_{i,j=1,\ldots ,2N}
-\nabla_x \left(\boldsymbol{\nu}(y_1)NC\boldsymbol{\nu}^T(y_2)\right)- \sum_{r=1}^2 \partial_{y_r}\left( \Psi(x,y_r) \boldsymbol{\nu}(y_1)NC\boldsymbol{\nu}^T(y_2) \right)dy_r &=-\sum_{r=1}^2\omega(x,y_r)\boldsymbol{\nu}(x)NC\boldsymbol{\nu}^T(y_r) -\boldsymbol{\nu}(y_1)(\nabla_x N)C\boldsymbol{\nu}^T(y_2),
E_{y}=\frac{1}{j\omega\epsilon}\frac{dL}{dz}\frac{\partial T}{\partial y}^{TM}-L\frac{\partial T}{\partial x}^{TE}=\frac{-k_{z}}{\omega\epsilon}L\frac{\partial T}{\partial y}^{TM}-L\frac{\partial T}{\partial x}^{TE}(28)
\dfrac{\partial f_{1}(x,m^{\hat v}_{t},\hat{v}_{1}(x,m^{\hat v}_{t}))}{\partial v_{1}}+Du^{\hat v}_{1}(x,t).\dfrac{\partial g_{1}(x,m^{\hat v}_{t},\hat{v}_{1}(x,m^{\hat v}_{t}))}{\partial v_{1}}=0 \, .
E^{Q}\left(e^{\beta H_N(s^1)+\beta u L_N(s^1)}e^{\beta H_N(s^2)+\beta u L_N(s^2)}\right)=e^{\beta u L_N(s^1)}e^{\beta u L_N(s^2)}e^{\Phi(\beta)B_N(s^1, s^2)}e^{2\Lambda(\beta)N}.
U = \frac 1{2 e^{2}}\sum_{l = 1}^{s}D_{l}^{2}+ \sum_{i = 1}^{N}| F_{i}|^{2}+ 2 \sum_{k , l = 1}^{s}\sum_{i = 1}^{N}\bar{\sigma}_{k}\sigma_{l}Q_{i}^{( k )}Q_{i}^{( l )}| \phi_{i}|^{2},
i_{\eta \cup j}= (-1)^{p(\eta,j)-1}\left(\prod_{r=1}^{q-1}{D_{{\ell_{r}}-1}}\right)\left(-x_j\right)\left(\prod_{r=q+1}^{s}{D_{\ell_{r-1}}}\right)\left(\prod_{r=s+1}^{d+1}{D_{\ell_{r}-1}}\right).
T_{\lambda^2}:= 6\lambda^2 \Big[C\star \big(S^3-C^3\big)\star S + C\star \big({\bf 1}^\epsilon_0\|C^3\|_{L^1}-{\bf 1}^\epsilon_0\|C_\infty^3\|_{L^1}\big)\star S + C\star \psi \star S\Big],
\left( \sum^{m}_{k=0}\dfrac{\mu \left( B\left( 0,\rho_{k}\right) \right)}{\rho^{n-2}_{k}}\right)^{\frac{2}{n-2}}\le C\sum^{m}_{k=0}\left( \dfrac{\mu \left( B\left( 0,\rho_{k}\right) \right)}{\rho^{n-2}_{k}}\right)^{\frac{2}{n-2}},
G_1 = \left[\begin{array}{cc}I_{m-1}0\end{array}\right], F_1 = \left[\begin{array}{cc}0 I_{m-1}\end{array}\right], G_2 = \left[\begin{array}{cc}0 I_{n-1}\end{array}\right], F_2 = \left[\begin{array}{cc}I_{n-1}0\end{array}\right].
A_M = \sum_{m \in [-M_1 , M_1] \times \cdots \times [-M_d, M_d]}\frac{(-1)^d 2^d a_1 \cdots a_d}{(2 \pi i)^{d+1}}\int_{|z|=R}\frac{e^{-2 \pi i z t}}{z (z - m_1 a_1) \cdots (z- m_d a_d)}\, \mathrm{d}z .
({}_aD_g^{\alpha}f) (x) = \left(\frac{1}{g'(y)}\frac{d}{dy}\right) ({}_{a}I_g^{1-\alpha}f )(y) = \frac{\left(\displaystyle\frac{1}{g'(y)}\displaystyle\frac{d}{dy}\right)}{\Gamma (1-\alpha)}\int_a^x \frac{f(y) g'(y)}{(g(x)- g(y))^{\alpha}}dy.
\det\limits_{x,y \in \mathbb{T}_{m_1,m_2}}K_m^{A,\theta}(x,y) = \prod_{j \in \mathbb{T}_{m_1,m_2}}\left( \alpha + \beta \exp \left\{2 \pi i \frac{j_1 + \theta_1 /2}{m_1}\right\}+ \gamma \exp \left\{2 \pi i \frac{j_2 + \theta_2 /2}{m_2}\right\}\right).
I =x^{1-\beta}\Gamma(1-\beta)\lambda+f(\beta)+x^{-\beta}\int_{-\beta-i\infty}^{-\beta+i\infty}|\Gamma(s)f(s+\beta)|\mathrm ds &=x^{1-\beta}\Gamma(1-\beta)\lambda+f(\beta)+O_\varepsilon\left\{x^{-\beta}(q_1q_2)^{1+\varepsilon}\int_0^\infty t^{2+\varepsilon}|\Gamma(-\beta+it)|\mathrm dt\right\}
\langle \Omega_{- \sqrt{( 2 p + 1 ) / 2}}^{*}( N ) , \prod_{i = 1}^{N}V_{( 0 , 0 | 1 , 1 )}( z_{i}, \bar{z}_{i}) \Omega_{0}\rangle = \prod_{i < j}( z_{i}- z_{j})^{2 p + 1}\exp \left( - \frac{1}{2}\sum_{i}| z_{i}|^{2}\right) \, ,
t_{j}= \frac{1}{2 \cos^{2}( \frac{\pi j}{2 H})}\prod_{i = 0}^{j - 1}\frac{\tan ( \frac{\pi ( j - 1 - 2 i + w_{+})}{2 H}) \tan ( \frac{\pi ( j - 1 - 2 i + w_{-})}{2 H})}{\tan^{2}( \frac{\pi ( i + 1 )}{2 H})}.
D = \frac{1}{1-\sum_{i=1}^m \alpha_i}\begin{pmatrix}1+\frac{1-\sum_{i=1}^m \alpha_i}{\alpha_1}1 \cdots 1 1 1+\frac{1-\sum_{i=1}^m \alpha_i}{\alpha_2}\cdots 1 \vdots \vdots \ddots \vdots 1 1 \cdots 1+\frac{1-\sum_{i=1}^m \alpha_i}{\alpha_m}\end{pmatrix}.
\theta_i(b)="\mbox{No}\varphi_{\alpha_j}\;(j<i) \mbox{encodes}\varphi_{\alpha_i}(\bar{x},b)"\mbox{and}\varphi'_{\alpha_i}(\bar{x},y)=\varphi_{\alpha_i}(\bar{x},y)\wedge\theta_i(y).
H_1H_2a(x_1,x_2) = \int_{R}\frac{a(y_1,y_2)}{(y_1-x_1)(y_2-x_2)}dy_1dy_2 &=\int_{R}\bigg( \frac{1}{(y_1-x_1)(y_2-x_2)}- \frac{1}{(x_I-x_1)(x_J-x_2)}\bigg) a(y_1,y_2) dy_1dy_2.
\int\limits_{0}^{R}\left(\frac{\left(K(T,r)-\ln{(1+S_{th})}\right)\sum\limits_{i\in \psi}r^\alpha d_i^{-\alpha}\left ( \underset{j \in \psi\setminus i}\prod{(1+S_{th}r^\alpha d_i^{-\alpha}})\right)}{\left(\underset{j \in \psi}\prod{(1+S_{th}r^\alpha d_j^{-\alpha})}\right)^2}\right)f_R(r)r=0
0 =\nabla_{X_{k+1}}(\Omega_{k-1}(X_0,\ldots,X_k)\varphi)=\nabla_{X_{k+1}}\Omega_{k-1}(X_0,\ldots,X_k)\varphi-\Omega_{k-1}(X_0,\ldots,X_k)\circ\omega(X_{k+1})\varphi =\Omega_k(X_0,\ldots,X_{k+1})\varphi,\quad\forall X_i \in\Gamma(TL),
_1 \langle \varepsilon_{1}| \,_{2}\langle \varepsilon_{2}| \,_{3}\langle \varepsilon_{3}| \exp [ \sum_{r = 1}^{3}N_{0 0}^{r r}( \varepsilon_{r}^{2}- 2 \sqrt{2}\varepsilon_{r}) + \sum_{r , s = 1}^{3}\sum_{n = 1}^{\infty}N_{0 n}^{r s}\varepsilon_{r}\phi_{n}^{( s )}] ,
M_h=\left(\begin{matrix}\alpha^{4}z^{2}+ \alpha^{4}z \alpha^{2}z^{3}+ z^{2}+ \alpha z \alpha^{3}z^{2}+ \alpha z + \alpha^{6}\alpha z^{3}+ \alpha z^{2}+ \alpha^{3}z z \alpha^{5}z^{2}+ \alpha^{2}z 0 1\end{matrix}\right) M_s=\left(\begin{matrix}0 \alpha \alpha^{4}z + \alpha^{2}0 0 0 0 1\end{matrix}\right).
(e^{-2 \eta})_{\rm fix}=\left | \frac{p_2 \, p_4}{q_1 q_3}\right | \ ,\qquad (e^{-2\sigma})_{\rm fix}=\left | \frac{p_2 \, q_3}{q_1 p_4}\right| \ , \qquad (e^{-2\rho})_{\rm fix}=\left | \frac{p_4 \, q_3}{q_1 p_2}\right| \ .
\eta\rfloor\omega={1\over p!}\eta_i\omega_{j_1\ldots j_p}g^{ij_1}dx^{j_2}\cdots dx^{j_p}-{1\over p!}\eta_i\omega_{j_1\ldots j_p}g^{ij_2}dx^{j_1}dx^{j_3}\cdots dx^{j_p}+ \cdots.
A_{Q(K)}(x_0 \otimes \eta_0) = -A_K(\eta_0)-2 = \tau(K) -2 A_{Q(K)}(x_2 \otimes \xi_0) = -A_K(\xi_0)-2 = -\tau(K) -2 A_{Q(K)}(y_2 \otimes \xi_0) = -A_K(\xi_0)-1 = -\tau(K) -1.
J_+ (x) :=\int_0^1 \rho_{g^T}(x+y) \int_0^{\infty}|v| \frac{e^{-(1-y)/\kappa |v|}}{\kappa (1-e^{-1/\kappa |v|})}\left( \alpha\mathcal{M}_{T(x+y)}(v) + (1-\alpha) \mathcal{M}_{\tau(x+y)}(v) \right) \mathrm{d}v.
{{J_*}\over{A_*}}={\displaystyle{\int_0^l \exp\left\{-\int_0^s\displaystyle{\frac{\mu(r,J_*,A_*) +\gamma_s(r,J_*,A_*)}{\gamma(r,J_*,A_*)}}\,dr\right\}\,ds}\over \displaystyle{\int_l^m \exp\left\{-\int_0^s\displaystyle{\frac{\mu(r,J_*,A_*)+\gamma_s(r,J_*,A_*)}{\gamma(r,J_*,A_*)}}\,dr\right\}\,ds}}, R(J_*,A_*)=1,
(\sum_{j=1}^sa_j(f_j-1))(h\otimes 1) =\sum_{j=1}^sa_j(f_j(h\otimes 1)-1(h\otimes 1)) &=\sum_{j=1}^sa_j(h\otimes (f_j)1-h\otimes 1) &=\sum_{j=1}^sa_j(h\otimes 1-h\otimes 1) &=0,
\frac{1}{n}\sum_{t=1}^T\sum_{j=1}^n\|[g_{t}(x_{j,t})]_+\|^2\le\varepsilon_3+\sum_{t=1}^T\sum_{i=1}^n2(\|[g_{i,t}(x_{i,t})]_+\|^2+G_2^2\tilde{\varepsilon}_4\|\epsilon^z_{i,t}\|^2).
\mathrm{p g h}\left( V_{\alpha ( \lambda )}\right) = \mathrm{p g h}\left( A_{\alpha}^{\; \; ( \lambda )}\right) = 0 , \; \mathrm{p g h}\left( \eta_{( \lambda )}\right) = \mathrm{p g h}\left( C^{( \lambda )}\right) = 1 ,
\text E({\bf v}_k) = \sum_{j\neq k}\text E (x_j){\bf e}_j,{\text C}({\bf v}_k) = \sum_{j\neq k}{\text C}(x_j){\bf e}_j{\bf e}_j^T + N_0({\bf H}^H{\bf H})^{-1},{\underline{\text C}}({\bf v}_k) = \sum_{j\neq k}{\underline{\text C}}(x_j){\bf e}_j{\bf e}_j^T,
\delta (t) = \frac{\delta (0)}{1 - \frac{\delta (0)}{E^{(0)}_{d}}\int_{0}^{t}\left( \partial^{0}\tilde{\phi}^{(1)}_{d \, (\rightarrow)}\, \partial^{\xi}\tilde{\phi}^{(1)}_{d \, (\rightarrow)}\right) \mid_{\xi = 1 + \delta}dt'}.
\|I_4\|_{L^1(\mathcal G)}=\int_{(r_oB)^{c}}\left|\int_{B}K_j(g,\tilde g)(b(g')-b_B)a(g')dg'\right|dg &=\left(\int_{(r_oB)^c}\big| K_j(g,\tilde g)\big|dg\right)\left|\int_{B}b(g')a(g')dg'\right|.
d C^{( n )}= F^{( n + 1 )}= ( - 1 )^{\frac{n ( n + 1 )}{2}}\hat{\star}F^{( 9 - n )}= ( - 1 )^{\frac{n ( n + 1 )}{2}}\hat{\star}d C^{( 8 - n )}= ( - 1 )^{\frac{n ( n - 1 )}{2}}d \hat{\star}C^{( 1 0 - n )},
\|D z_N\|=\|(D_j x_j)_{j\le N}\|_{l_{q}(Y)}=\left ( \sum_{j\le N}\|D_j x_j\|_Y^q\right )^{1/q}\ge \left ( \sum_{j\le N}(1-\epsilon)\|D_j\|^q\right )^{1/q}
\| s^{n'}{s'}^{n}\|_{(h \otimes h)^{nn'}, \hat{\kappa}(\xi)}\leqslant\left(\| s \|_{h^{n}, \hat{\kappa}(\xi)}\right)^{n'}\left(\| s' \|_{h^{n'}, \hat{\kappa}(\xi)}\right)^{n}& \leqslant e^{nn'\epsilon}(|s|_{h^{n}}(\xi))^{n'}(|s'|_{h^{n'}}(\xi))^n & = e^{nn'\epsilon}|s^{n'}{s'}^n|_{(h\otimes h')^{nn'}}(\xi).
\varphi_{A_{1}. . . A_{n}a_{1}. . . a_{m}}( y ) = \int_{{\cal F}\times S^{2}}\omega ( p , \zeta ) \ \widetilde{\varphi}( p , \zeta ) \Phi ( p , \zeta | y )_{A_{1}. . . A_{n}a_{1}. . . a_{m}}\ \in{\cal H}_{E_{o}, s}
(y-y^{q^4})x^{q^3}+(x-x^{q^4})x^{2q^3}=(y-y^{q^2})x^{q}+(x-x^{q^2})x^{2q}+(y-y^{q^2})^{q^2}x^{q^3}+(x-x^{q^2})^{q^2}x^{2q^3}.
\mu_1 =\frac{N\lambda^d}{2\bar{\gamma}d(d+1)\theta^d}\sum_{n=0}^{N-1}\binom{N-1}{n}\left(-1 \right )^n \left( \frac{\lambda}{\theta}+\frac{n+1}{\bar{\gamma}}\right )^{-d-2}&\times_2F_1\left(1,d+2;3;\frac{\left(n+1 \right )/\bar{\gamma}}{\lambda/\theta+\left(n+1 \right )/\bar{\gamma}}\right ).
\lambda_{1,d}^{-1/2}= \sqrt{\frac{d(d-1)}{2}}t + O(t^3) \lambda_{2,d}^{-1/2}= \sqrt{2d}+ O(t) \lambda_{3,d}^{-1/2}= \sqrt{d(d-1)}\, t \lambda_{4,d}^{-1/2}= \sqrt{{\frac{d({d}^{2}-3\,d+2)}{12}}}\, t^2 + O(t^3).