harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_y = e_{\alpha}- \frac{\alpha^2}{L^3}+ \alpha \|\lambda_y\|^2 - \|\Phi^{\#}_y\|^2_{\#}+ O\Big(\frac{\alpha^2}{L^5}\Big) + O(\alpha^4 \log(\alpha^{-1}))+ O\big( \alpha^2 L e^{-L/2}\big).
\mathbf{y}_{\mathtt{U}}=\sum_{j=1}^{K_{\mathtt{U}}}\mathbf{h}_{\mathtt{U}_{j}}s_{\mathtt{U}_{j}}+\underbrace{\sum_{i=1}^{K_{\mathtt{D}}}\mathbf{H}_{\mathtt{SI}}\mathbf{w}_{\mathtt{D}_{i}}s_{\mathtt{D}_{i}}}_{\textrm{self-interference}}+\mathbf{n}_{\mathtt{U}}
0 \leqslant \int_{\mathbb{R}^{N}}[\mathrm{H}_{u}(u_k,v_k)u_k+ \mathrm{H}_{v}(u_k,v_k)v_k]\,dx &\leqslant \int_{\mathbb{R}^{N}}a|u_{k}|^{\theta(x)}+ b|v_{k}|^{\sigma(x)-1}|u_k|+a |u_{k}|^{\theta(x)-1}|v_{k}|+b |v_k|^{\sigma(x)}\,dx\longrightarrow 0
\frac{\partial^2v}{\partial\xi_{s+1}\partial\xi_s}=z_j(1-|w_j|^2)\left(\frac{\frac{\partial^2v^\prime}{\partial\xi_{s+1}\partial\xi_s}}{(1+\overline{w}_jv^\prime)^2}-\frac{2\overline{w}_j\frac{\partial v^\prime}{\partial\xi_{s+1}}\frac{\partial v^\prime}{\partial\xi_{s}}}{(1+\overline{w}_jv^\prime)^3}\right).
\|f_{s_1}-f_{s_2}\|_{K}^2=\|f_{s_1}\|_{K}^2+\|f_{s_2}\|_{K}^2-2\langle f_{s_1},f_{s_2}\rangle_{K}=\|f_{s_1}\|_{K}^2-\|f_{s_2}\|_{K}^2.
\int \prod_{j = 1}^{g}d \nu_{j}d \bar{\nu}_{j}d \hat{\nu}_{j}d \hat{\bar{\nu}}_{j}\, \, \mathrm{e}^{\int_{\Sigma_{g}}{\cal L}_{5}}= \left| \prod_{j = 1}^{g}\delta ( a_{j}) \, d a_{j}\right|^{2},
\begin{array}{ll}\displaystyle\sum_{i=1}^3\left(C_i(\lambda'_i\rho_0)-C_i(\lambda_i''\rho_0)\right)(\lambda_i'-\lambda_i'')>0 \displaystyle(\lambda'_1,\lambda'_2,\lambda'_3)\neq(\lambda''_1,\lambda''_2,\lambda''_3)\ \mbox{convex triples and for any}\rho_0>0.\end{array}
P_{\alpha}( \theta_{1}) \dot{P}_{\alpha}( \theta_{2}) = \nu P_{\alpha}( \theta_{1}) D_{\beta}( \theta_{2}) \Omega_{\beta \alpha}( \theta_{2}) - P_{\alpha}( \theta_{1}) V_{\beta}( \theta_{2}) \Omega_{\beta \alpha}( \theta_{2})
J(y)V_{\alpha,\overline{\alpha}}(z)=\frac{\alpha}{y-z}V_{\alpha,\overline{\alpha}}(z)+O(1),\overline{J}(y)V_{\alpha,\overline{\alpha}}(z)=\frac{\overline{\alpha}}{\overline{y}-\overline{z}}V_{\alpha,\overline{\alpha}}(z)+O(1)
E_{\exp}^{\left( 1\right)}=\left. -\frac \mu 2\frac d{dt}\left(\sum_{n=-\infty}^\infty e^{-t\left( \left( \frac{2n\pi}{L\mu}\right)^2+\left( \frac m\mu \right)^2\right)^{\frac 12}}\right)\right\rfloor_{t=0}
Ric(\partial_x,\partial_t)=\frac{1}{2}b_{xy}, \Ric(\partial_y,\partial_t)= \frac{1}{2}b_{yy}, Ric(\partial_z,\partial_t)=\frac{1}{2}b_{yz}, \ \Ric(\partial_t,\partial_t)= -b_{zx}+\frac{1}{2}bb_{yy}
\eta(\sigma)^*(y_{i_1,j_1}\cup\cdots\cup y_{i_k,j_k})=\prod_{\ell=1}^{k}\eta(\sigma)^{\ast}(y_{i_{\ell},j_{\ell}}) =\prod^k_{\ell=1}\left(\frac{c_{i_{\ell}}^{\sigma}}{c_{i_{\ell}}^{\{i_{\ell}\}}}x_{i_{\ell},j_{\ell}}\right) =\left(\prod^k_{\ell=1}\frac{c_{i_{\ell}}^{\sigma}}{c_{i_{\ell}}^{\{i_{\ell}\}}}\right)x_{\tau,\mathfrak{u}}.
L_{(\xi_0 n^{2/3},0)\to (\gamma^2 n,n)}\simeq \left(\sqrt{n}+\sqrt{\gamma^2n-\xi_0 n^{2/3}}\right)^2=(1+\gamma)^2 n-\tfrac{1+\gamma}{\gamma}\xi_0 n^{2/3}-\frac{\xi_0^2}{4\gamma^3}n^{1/3}+O(1).
\begin{aligned}\int u_k^s\Delta u_1^t\wedge\dots\wedge\Delta u_{k-1}^t\wedge\beta_n^{n-m}\wedge\Delta \omega=\int \Delta u_1^t\wedge\dots\wedge\Delta u_{k-1}^t\wedge\Delta u_k^s\wedge\beta_n^{n-m}\wedge\omega\geq 0,\end{aligned}
L_n^{(\alpha)}(x,y)=\frac{(q;q)_n}{(q^{\alpha+1};q)_n}y^n\mathcal{L}^{(\alpha)}_n (x/y), \,\,L_n^{(\alpha)}(x,1)=\frac{(q;q)_n}{(q^{\alpha+1};q)_n}\mathcal{L}^{(\alpha)}_n(x), \,\,L_n^{(\alpha)}(0,y)=y^n.
\dot{x}^{A}{}_{D^{\prime}}\acute{x}^{B C^{\prime}}\dot{x}_{A C^{\prime}}\acute{x}_{B}{}^{D^{\prime}}= - \dot{x}^{A}{}_{C^{\prime}}\dot{x}_{A D^{\prime}}\acute{x}^{B C^{\prime}}\acute{x}_{B}{}^{D^{\prime}}
R=\frac{r(\theta_1\,|\,\phi_0-\phi_1)\,r(\theta_3\,|\,\phi_3-\phi_0)}{r(\theta_1+\theta_3\,|\,\phi_3-\phi_1)}\frac{P(\theta_1\,|\,\phi_3-\phi_2)\,P(\theta_3\,|\,\phi_1-\phi_2)}{P(\theta_1+\theta_3\,|\,\phi_0,\phi_2)}\,,
\partial_\lambda \Big ( J_{\nu}(\sqrt{\lambda}\, r) \, J_{-\nu}(\sqrt{\lambda}\, r) \Big ) \ = \ -\frac{1}{2\sqrt{\lambda}}\, \Big(r\, J_{\nu+1}(\sqrt{\lambda}\, r)\, J_{-\nu}(\sqrt{\lambda}\, r') + r'\, J_{1-\nu}(\sqrt{\lambda}\, r')\, J_{\nu}(\sqrt{\lambda}\, r) \Big).
\begin{cases}v_{\tilde{t}}=Av+\tilde{f}_{\tilde{t}}(x,\tilde{t}), \ \tilde{Q}_1, \frac{\partial v}{\partial \nu_{A}}=0, \ \partial\Omega\times(0,2\delta_1), v(x,0)=\tilde{u}_{\tilde{t}}(x,0), \ \Omega.\end{cases}
\begin{array}{ll}T_{scs \hspace{.1 true cm}\theta}^{\theta}= -\frac{1}{2}\left\{e^{2(\psi - \gamma)}[\varphi'^2 + \sigma'^2] -\frac{e^{2\psi}}{\beta^2}\varphi^2P^2 + e^{-2\psi}\sigma^2 A^2 +\right. & -\left.\frac{e^{2(\psi-\gamma)}}{\beta^2}(\frac{P'}{q})^2 +e^{-2\gamma}(\frac{A'}{e})^2 + 2V(\varphi,\sigma, \Lambda) \right\}\end{array}
\begin{array}{rcll}{\rm tr}_d(ab) ={\rm tr}_d(ba){\rm tr}_d(1) = 1{\rm tr}_d(ag_n) = z\,{\rm tr}_d(a) ( ){\rm tr}_d(at_{n+1}^s) = x_s{\rm tr}_d(a) ( s = 1, \ldots , d-1) ,\end{array}
\sup_{0 \leq z < x}P_z^Z(Z_{\tau_{x^+}^Z}> x + y|\tau_{x^+}^Z < \tau_0^Z) \leq\begin{cases}C_3(1 + y^{2/3}x^{-1/3})e^{-C_4 y^{2/3}x^{-1/3}},~ \mbox{for}0 \leq y \leq x C_3(1 + y^{1/3})e^{-C_4 y^{1/3}},~~~~~~~~~~~~~~~~~ \mbox{for}y \geq x\end{cases}
\hbar^{k}W = \sum_{\beta \not = \gamma}\pi_{\beta}^{( k )}H^{( k )}\pi_{\gamma}^{( k )}= \sum_{\beta \not = \gamma}\pi_{\beta}^{( k )}( [ H^{( k )}, \pi_{\gamma}^{( k )}] \pi_{\gamma}^{( k )}.
\lim_{k\rightarrow\infty}\limsup_{n\rightarrow\infty}\sup_{x\in F_n}R_n(x,\phi^{n,k}(x))\leq \lim_{k\rightarrow\infty}\limsup_{n\rightarrow\infty}\left(2\varepsilon_k+\sup_{i=1,\dots,k}d_M(x_i^n,x_i)\right)=0.
\mathcal{K}f =\operatorname{Re}\bigl(\mathfrak{H}\bigr)f, \mathcal{K}^* f =-\operatorname{Re}\left( e^{i\theta}\mathfrak{H}( e^{-i\theta}f)\right)=-\operatorname{Re}\left( e^{i\theta}[\mathfrak{H}, e^{-i\theta}]f+\mathfrak{H}f\right).
H_{n+1}(x) =2xH_{n}(x)-2nH_{n-1}(x), K_{n+1}(x,p,N) =\frac{pN+n-2pn-x}{p(N-n)}K_{n}(x,p,N)-\frac{n(1-p)}{p(N-n)}K_{n-1}(x,p,N).
\textstyle \sum_{j=1}^\alpha c_j \sum_v(-1)^{w-v}{j+w-v-1\choose w-v}{r-\alpha+j\choose v}{s-\alpha+j-v \choose j-v}= \frac{(-1)^w(r-s)(r-\alpha)_w}{(s-\alpha)_{w+1}}+ \delta_{w = \alpha}\frac{(s-r)_{\alpha+1}}{(s-\alpha)_{\alpha+1}},
\sum_{j\in[b]}N_j = \Big|\big(\bigcup_{i\in [b]}X_i\big)\cap \big(\bigcup_{j\in [b]}Y_j\big)\Big| \ge \frac{b}{2}\cdot \frac{\log a}{3\log\log n}\ge \frac{b}{18}\cdot \frac{\log n}{\log\log n}= \frac{a}{18}
K = \left( \begin{matrix}{{0}}&{{0}}&{{0}}&{{1}}&{{1}}&{{2}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{0}}&{{1}}\\{{1}}&{{0}}&{{0}}&{{1}}&{{0}}&{{0}}\\ \end{matrix}\right)
P\ =\ \frac{n+1}{4\pi}\int_{S^{2}}\widetilde{W}_{P}(g\mathbf{n}_{0})K^{g}dS\ =\ \frac{n+1}{4\pi}\int_{S^{2}}\widetilde{W}_{P}(\mathbf{n})K(\mathbf{n})dS
Z^{\prime}=\begin{cases}(\underbrace{0,\dots ,0}_{J-1},\underbrace{a_{1},\dots ,a_{1}}_{s_{1}},\dots,\underbrace{a_{m},\dots ,(\pm )a_{m}}_{s_{m}}) 2J\geq s_{1}(\underbrace{0,\dots ,0}_{J},\underbrace{a_{1},\dots ,a_{1}}_{s_{1}-1},\dots,\underbrace{a_{m},\dots ,(\pm )a_{m}}_{s_{m}}) 2J<s_{1}\end{cases}.
R \Bigl(t,x, u_0+U, \frac{\partial u_0}{\partial x}+ \frac{\partial U}{\partial x}\Bigr) - R \Bigl(t,x, u_0, \frac{\partial u_0}{\partial x}\Bigr) =c_1(t,x)U + c_2(t,x)\frac{\partial U}{\partial x}+ \sum_{j+\alpha \geq 2}c_{j,\alpha}(t,x) U^j \Bigl(\frac{\partial U}{\partial x}\Bigr)^{\alpha}
= \mathcal{J}^{u,\aleph_{t^2_{|k^2|},\ldots,t^{n-1}_{|k^{n-1}|}},\mathcal{S}^n}_k(a(t^1,\ldots,t^{n-1},\cdot))(\mathcal{G}_{t^2_{|k^2|},\ldots,t^{n-1}_{|k^{n-1}|}}(\omega^{n})).
P\{Y=0 \vert \textbf{X}=\textbf{x}\}=\dfrac{f_{\textbf{X}\vert Y=0}(\textbf{x}\vert 0) P\{Y=0\}}{f_\textbf{X}(\textbf{x})}= \dfrac{\dfrac{(1-p)}{\mu{(\mathcal{C}_{0})}}}{\dfrac{p}{\mu{(\mathcal{C}_{1})}}+ \dfrac{1-p}{\mu{(\mathcal{C}_{0})}}}
= - \bigg [ \frac{( D - 2 ) \beta + 4 \gamma}{4 ( D - 3 )}\bigg ] E + \bigg ( \frac{D - 2}{D - 3}\bigg ) ( \beta / 4 + \gamma ) C^{2}+ \bigg [ \frac{4 ( D - 1 ) \alpha + D \beta + 4 \gamma}{4 ( D - 1 )}\bigg ] R^{2}\,
\langle M_{g^{*}}\phi, \psi \rangle_{\mathcal{P}}= \langle \phi , M_{g}\psi \rangle_{\mathcal{P}}\mathrm{or}\langle \phi g^{*}, \psi \rangle_{\mathcal{A}}= \langle \phi , \psi g \rangle_{\mathcal{A}}
{\cal H}_{3}={\mathrm e}^{-{\mathrm i}\frac{\pi}{2}{\cal S}_{2}}{\mathrm e}^{-{\mathrm i}\frac{\pi}{2}{\cal S}_{3}}{\cal H}_{1}{\mathrm e}^{{\mathrm i}\frac{\pi}{2}{\cal S}_{3}}{\mathrm e}^{{\mathrm i}\frac{\pi}{2}{\cal S}_{2}}={\cal S}_{3}^{2}-{\cal S}_{2}^{2}
H = \{(x_1, x_2, x_3)\in \mathbb R^3: x_1^2 + x_2^2 = 1, |x_3| = R\}K_0 = \{(x_1, x_2, x_3)\in \mathbb R^3: x_1^2 + x_2^2 = 1, |x_3| < R\}\,.
IS^{\mathrm{soft}}(\kappa, \pi;u,v) =IS^{(\mathrm{soft})}(\kappa;u,v) +\sum_{p=1}^{m}\Big( \phi_{2,p-1}(\kappa, \pi;u) \phi_{2,p}(\kappa, \pi;v)-\phi_{2,p}(\kappa, \pi;u) \phi_{2,p-1}(\kappa, \pi;v)\Big).
\begin{aligned}|\ll| = (m+s)\cdot(s-n) + (t_1-s+1)\cdot(t_1-z_n) + \left(\sum_{i=1}^{n-l-1}(t_{i+1}-t_i+1)\cdot(t_{i+1}-z_{n-i})\right) + (l-t_{n-l})\cdot(l-z_l) = (m+s)\cdot(s-n) + (t_1-s)\cdot(t_1-z_n) + \left(\sum_{i=0}^{n-l-1}(t_{i+1}-z_{n-i})\right) + \left(\sum_{i=1}^{n-l}(t_{i+1}-t_i)\cdot(t_{i+1}-z_{n-i})\right)\end{aligned}
z_2 \mu_j \equiv_{z_1 \mathcal{F}}(z_2 \alpha^j) \varphi = [x_1 \alpha^{j-i}, y_2 \alpha^i] \varphi = [x_1 a_{j-i}+ y_2 b_{j-i}+ \ldots, x_1 c_i + y_2 d_i + \ldots] \equiv_{z_1 \mathcal{F}}z_2 (a_{j-i}d_i - b_{j-i}c_i)
{Pr}_D(f,\sigma,\{k_j\}): =\lim_{X\to\infty}\displaystyle\frac{\#\{p\in{Spl}_X(f,\sigma,\{k_j\}) \mid (r_1/p,\dots,r_n/p)\in D\}}{\#{Spl}_X(f,\sigma,\{k_j\})}&=\displaystyle\frac{{vol}({D}\cap{\mathfrak{D}}(f,\sigma,\{k_j\}))}{{vol}({\mathfrak{D}}(f,\sigma,\{k_j\}))}.
B_{4}=(\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13})+(\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19})+(\frac{1}{101}+\frac{1}{103}+\frac{1}{107}+\frac{1}{109})+\cdot\cdot\cdot
H(J^{\{[[e_{\theta},\phi(v)],v]^{\natural}\}}_{0}v_{\nu,\ell_0},v_{\nu,\ell_0})=\sum_i([[e_{\theta},\phi(v)],v]|u_i)\nu(u_i)=\sum_i(e_{\theta}|[\phi(v),[v,u_i]])\nu(u_i).
C_l \approx \mathrm{E}\left[ \sum_{k_l=1}^{K_l}\log_2 \left(\xi_{l}g^2_{k_l k_l}\right) \right] = \chi_{l}+ \mathrm{E}\left\lbrace \log_2 \det \left[ \mathbf{B}_l \right] \right\rbrace,
F(x,y) - xF_1(y) =\sum_{b \geq 2}F_b(y) x^b = \frac{y}{1-y}\sum_{b \geq 2}x^bF_{b-1}(y) + \frac{y}{1-y}\sum_{b\geq 2}\sum_{\substack{0<i<b, i+j=b}}x^iF_i(y) \,x^jF_j(y) = \frac{y}{1-y}( x F(x,y) + \big(F(x,y)\big)^2 )
\int \frac{(q^2x^2;q^2)_\infty}{(q^{-(n+1)}x^2;q^2)_\infty}h_n(x;q) d_qx= & \frac{(x^2;q^2)_\infty}{[n+1]_q(q^{-(n+1)}x^2;q^2)_\infty}\left( x h_n(\frac{x}{q};q)- q^{n}(1-q^n) h_{n-1}(\frac{x}{q};q)\right),
\ \begin{aligned}y\left[ n \right] = \left({\sum\limits_{l = 1}^L{{\bf{h}}_l^H{{\bf{f}}_l}}}\right)s\left[{n -{n_{\max}}}\right]+ \sum\limits_{l = 1}^L{\sum\limits_{l' \ne l}^L{{\bf{h}}_l^H{{\bf{f}}_{l'}}s\left[{n -{n_{\max}}+{n_{l'}}-{n_l}}\right]}}+ z\left[ n \right]. \end{aligned}
\begin{array}{c}{{\rho_{i}= g_{i}\delta^{4}\left( x - z_{i}\right) , \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \rho_{i j}= g_{i j}\delta^{4}\left( x - z_{i j}\right) ,}}\\{{1 \leq j \leq l .}}\\ \end{array}
D_2 \leq \max_{\substack{(x,y) \in \{(r,s), (s,u) \}}}4n^2 \sum_{i=1}^{2n}|Z_{2n;i}(y) - Z_{2n;i}(x)|^3 \Big( 2n \sum_{j=1}^{2n}(Z_{2n;j}(y) - Z_{2n;j}(x))^2 \Big)^{1/2}\leq \max_{\substack{(x,y) \in \{(r,s), (s,u) \}}}\Big( n \sum_{i=1}^{2n}(Z_{2n;i}(y) - Z_{2n;i}(x))^2 \Big)^{3/2}\cdot \mathcal{O}_P(1),
\sum_{n,k = 1}^{N}[\sigma_n, \sigma_k] \int_s^t \beta_{s,u}^k\, d\beta_u^n \to V_{s,t}\sum_{n,k = 1}^{N}\partial_i[\sigma_n, \sigma_k] \int_s^t \beta_{s,u}^k\, d\beta_u^n \to Z_{s,t}^i
\beta_n = \beta_{n-1}\ \varepsilon_{n, 1}^{-a_{n,1}}\cdots \varepsilon_{n, n-1}^{-a_{n,n-1}}= \beta \ \prod_{j=1}^{n}\varepsilon_{n, j}^{-a_{1,j}}\prod_{j=1}^{n}\varepsilon_{n-1, j}^{-a_{2,j}}\cdots \prod_{j=1}^{n}\varepsilon_{1, j}^{-a_{n,j}}
{\cal R}= ( 2 S_{c l})^{3 k}\prod_{i = 1}^{3}\left( S_{i}\right)^{- k}\left( \frac{( S_{i}^{2}+ S_{c l}^{2})^{1 / 2}+ S_{c l}}{( S_{i}^{2}+ S_{c l}^{2})^{1 / 2}- S_{c l}}\right)^{- k / 2}
\beta_{a}(I_{u_\ell})=\beta_{a,b}(I_{u_\ell}) =\big|\big\{w\in\mathcal{L}_t^f(u):\min(u)=1\big\}\big|-\big|\big\{w\in\mathcal{L}_t^f(v)\setminus\{v\}:\max(w)=n\big\}\big| &=\big|\big\{u_\ell>u_{\ell+1}>\dots >u_{d-1}\}\big|-|\{v_{\ell+1}>\dots>v_d\}|=0,
2\left\langle R(t), \xi(t) \right\rangle =\Delta ||\xi(t)||^2 + 2 \int_0^1 \left( ||d_{e^{\textbf{i}s\xi(t)}A_0}\xi(t)||^2 + ||L_{e^{\textbf{i}s\xi(t)}u_0}\xi(t)||^2 \right)\, ds &\geq 2 ||\xi(t)|| \Delta ||\xi(t)||.
\sum_{n = 1}^{\infty}e^{- \beta k E_{n}}= \frac{1}{\sqrt{\pi}}\int_{0}^{\infty}d x e^{- \frac{x^{2}}{4}- \frac{\beta^{2}k^{2}m^{2}}{x^{2}}}\frac{1}{e^{\frac{2 \beta^{2}e B k^{2}}{x^{2}}}- 1}
\sum_{\substack{\mathbf{r}\in\mathcal{D}_k r_m=1}}\frac{(y_{\mathbf{r}}^{(m)})^2}{\phi_\omega(r)^2}\Bigl(\prod_{p|r}(\phi_L(p)-1)\Bigr)\sum_{A|r}\Bigl(\prod_{p|A}\frac{-1}{\phi_L(p)-1}\Bigr)\sum_{\substack{\mathbf{s}\in\mathcal{D}'_k A(\mathbf{r},\mathbf{s})=A}}1=\sum_{\substack{\mathbf{r}\in\mathcal{D}_k' r_m=1}}\frac{(y_{\mathbf{r}}^{(m)})^2}{\phi_\omega(r)}.
H = \frac{1}{L}\left[ \sqrt{\Bigl ( 1 + \frac{r^{2}}{L^{2}}\Bigr ) \Bigl ( P_{\phi}^{2}+ \frac{{\tilde{N}}^{2}r^{6}}{L^{6}}\Bigr )}- \frac{{\tilde{N}}r^{4}}{L^{4}}\right] ,
\left( I_{0{+}}^{\lambda ,\lambda^{\prime},\xi ,\xi^{\prime},\gamma}f\right) (x)=\frac{x^{-\lambda}}{\Gamma (\gamma )}\int_{0}^{x}(x-t)^{\gamma -1}t^{-\lambda^{\prime}}F_{3}\left( \lambda ,\lambda^{\prime},\xi,\xi^{\prime};\gamma ;1-\frac{t}{x},1-\frac{x}{t}\right) f(t)\,\mathrm{d}t
\big( (2m_{1}+m_{2}) (\cot \langle 2\alpha_{2}, H\rangle )^{2}-(m_{2}+2n_{3})\big) ((\cot \langle 2\alpha_{2}, H\rangle )^{2}-1) = -2m_{1}(\cot \langle 2\alpha_{2}, H\rangle )^{2}.
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m+1)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}\leq \frac{q_v^{n_{v,_nu_v}}-1}{q_v^{n+1}-1}.
< P O_{k}O_{l}> = \nonumber - \frac{1}{\pi}\mathrm{I m}\sum_{n , m}\frac{< n | O_{k}| m > < m | O_{l}| n >}{( E_{n}- t_{0}) ( E_{m}- t_{0})}\frac{( E_{n}- E_{m})^{2}}{p^{2}+ ( E_{m}- E_{n})^{2}}.
Z_{K_{+}}: = \frac{5 9^{3 / 2}}{( 2 \pi )^{2}}\frac{\zeta_{K_{+}}( 2 )}{2 \zeta ( 2 )}= 5 . 1 3 7 9 4 1 2 0 1 8 7 3 4 1 7 7 6 9 8 4 1 3 4 8 3 3 9 \ldots
\mathsf{C}_{j}\left( s\right)\mbox{\boldmath$ \psi$}_{j}=\left(\begin{array}[c]{c}\{\!\!\{u\}\!\!\}_{\operatorname*{D};j}\left( s\right) \{\!\!\{u\}\!\!\}_{\operatorname*{N};j}^{\operatorname*{ext}}\left( s\right)\end{array}\right) .
\begin{aligned}\left[\int_\Omega |\nabla u_p|^{(p-1)s}dx+\int_{\partial\Omega}\lambda|u_p|^{(p-1)s}d\mathcal H^{N-1}\right]^{\frac1s}&\le \left[\int_\Omega |\nabla u_p|^{p}dx+\int_{\partial\Omega}\lambda|u_p|^{p}d\mathcal H^{N-1}\right]^{\frac{p-1}p}\Lambda^{\frac1s-\frac{p-1}p}\le M(f,g,\lambda)\Lambda^{\frac1s}\end{aligned}
u_{t}+Uu_{x}+wU'+\epsilon(uu_{x}+wu_z)=-p_{x}\ \Omega_{\epsilon}, \delta^{2}(w_{t}+Uw_{x}+\epsilon(uw_{x}+ww_z))=-p_z \ \Omega_{\epsilon}, u_{x}+w_z=0 \ \Omega_{\epsilon}, w=\eta_{t}+(U+\epsilon u)\eta_{x}, \p=\eta \rho g \ z=1+\epsilon \eta, w=0 \ z=0.
G_{t, \bar{m}, \rho}:= \left\{x \in V_{k+1}: \mu_{\bar{m}}\left( \left(G_{t,\bar{m}}\right)_x \right) \geq \rho \right\}, G_{t,s, \bar{m}, \rho}:= \left\{x \in V_{k+1}: \mu_{\bar{m}}\left( \left(G_{t,s,\bar{m}}\right)_x \right) \geq \rho \right\}.
X_{1}=\begin{pmatrix}x_{1}0 \vdots 0\end{pmatrix}\in Hom(\mathbb{C}^{r},\mathbb{C}^{n-k})\implies\tilde{I}(X_{1})=X_{1}I^{\top}=\begin{pmatrix}x_{1}I^{\top}0 \vdots 0\end{pmatrix}\in R^{\prime},
f(Z)=\sum\limits_{\ell_{1}=0}^{m-1}\sum\limits_{\ell_{2}=0}^{n-1}\sum\limits_{\ell_{3}=0}^{k-1}{Z^{\ast}}^{\ell_{1}}\widetilde{Z}^{\ell_{2}}{Z^{\dagger}}^{\ell_{3}}H_{\ell_{1},\ell_{2},\ell_{3}}(Z)
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m+1)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 2^2\cdot 3\cdot\frac{(2^{(n_{2,\nu_2}+1)/2}+1)(2^{(n_{2,\nu_2}-1)/2}+1)}{2^{n+1}-1}.
G_\varphi (t) =\frac{e^{2Kt}-1}{K}\int_X\varphi |DT_t f|^2\,d\mu & = 2t \int_X \varphi |Df|^2 d\mu +2t^2 \left(K\int_X \varphi |Df|^2\,d\mu +2 \int_X \varphi Df\cdot DAf \,d\mu \right)+o(t^2).
\mathbb{E}\Big[H_{1v_1}(t)(v_1(t)-u_1(t))|\mathcal{F}_t^1\Big]=\Big(\frac{\partial}{\partial v_1}\mathbb{E}\big[H_1(t)|\mathcal{F}_t^1\big]\Big)(v_1(t)-u_1(t))\geq 0.
\| (1+|x|^2) \, |J|_2 \, |h|_2 \|_{L^1}\leq \| (1+|x|^2) \, |J|_2 \|_{L^2}\, \| |h|_2 \|_{L^2}\leq C \, \| J \|_{L^2}\, \| h \|_{W^{2,2}}\leq C \, \| U \|_{L^2}\, \left( \| U \|_{L^2}^2 + \| \phi_{U}\|_{L^2}\right) = C\,\| U\|_{L^2}^3+C\,\|U\|_{L^2}\,\|\phi_{U}\|_{L^2}\, .
\langle \lambda \vert \xi \rangle = \psi_{\xi}( \lambda ) = N \left( 1 - \vert \beta \vert \lambda^{2}\right)^{- 1 / 2}\left( 1 - \sqrt{\vert \beta \vert}\lambda \right)^{i \xi / 2}\left( 1 + \sqrt{\vert \beta \vert}\lambda \right)^{- i \xi / 2}
_{-\infty}D_{x}^{\alpha}f(x) =\frac{1}{\Gamma(n-\alpha)}\int_{-\infty}^{x}(x-x^{\prime})^{-\alpha-1+n}f^{(n)}(x^{\prime})dx^{\prime},\alpha>0, & =_{-\infty}\mathbf{I}_{x}^{n-\alpha}f^{(n)}(x)=_{-\infty}D_{x}^{\alpha-n}f^{(n)}(x).
H_{n}\left(\frac{B_{2k+1}\left(\frac{x+1}{2}\right)}{2k+1}\right) =\left(\frac{x^{3}-x}{24}\right)^{n+1}\quad\times\prod_{\ell=1}^{n}\left(\frac{(2\ell)^{2}(2\ell+1)^{2}(x^{2}-(2\ell+1)^{2})(x^{2}-(2\ell)^{2})}{16(4\ell-1)(4\ell+1)^{2}(4\ell+3)}\right)^{n+1-\ell}.
\sigma_{\sigma_{(a,i)}((a,i))}((b,j))=\sigma_{(a+\delta_{(a,i)},i+r\delta_{(a,i)}+1)}((b,j))= &(b-r(a+\delta_{(a,i)})+i+r\delta_{(a,i)}+1,j+r(i+r\delta_{(a,i)}+1)-r^2(a+\delta_{(a,i)})+1)= &(b-ra+i+1,j+ri+r-r^2a+1)=(b+\delta_{(a,i)}+1,j+r(\delta_{(a,i)}+1)+1).
\frac{1}{\gamma}( \partial_{i}\dot{A}_{i}^{a}+ \epsilon^{a b c}A_{i}^{b}\dot{A}_{i}^{c}) + \frac{k}{8 \pi}\epsilon^{i j}F_{i j}^{a}+ \frac{1}{2}\mathrm{T r}( \partial_{t}\phi [ T^{a}, \phi ] ) = 0 .
\|a-he_i-x_i\|_{l_2^d}^2-\|a-x_i\|_{l_2^d}^2 =\langle a-he_i-x_i,a-he_i-x_i\rangle-\langle a-x_i,a-x_i\rangle &=-2h\langle a,e_i\rangle+h^2\langle e_i,e_i\rangle+2h\langle e_i,x_i\rangle &=-2ha_i+h^2+2h x_{i,i}.
\left| \Phi \right\rangle = \left\{\phi_{0}, \phi_{1}^{( \mu_{1})}\left( p_{1}\right) , \phi_{2}^{( \mu_{1}) ( \mu_{2})}\left( p_{1 ,}p_{2}\right) , . . . , \phi_{n}^{( \mu_{1}) . . . ( \mu_{n})}\left( p_{1}, . . . , p_{n}\right) , . . . \right\}\; \, ,
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 , \mathrm{~ ~ l ~ = 1 , 2 ,{\cal{D}}o t s}
{C_{g}}^{i_{1}. . . i_{n}}={{C_{g_{1}}}^{i_{1}. . . i_{k}}}_{j}{C_{g_{2}}}^{j \, i_{k \! + \! 1}. . . i_{n}}= \sum_{j}{{C_{g \! - \! 1}}^{i_{1}. . . i_{n}j}}_{j}
\frac{dA}{dt}\left( t\right) =0,\ \ \ \frac{dL}{dt}\left( t\right)=-\int_{0}^{2\pi}k^{\alpha}\left( \theta,t\right) d\theta+\dfrac{2\pi}{{L}\left( t\right)}\int_{0}^{2\pi}k^{\alpha-1}\left( \theta,t\right)d\theta\leq0,\ \ \t\in\lbrack0,T).
T_{j}^{n+1}-T_{j}^{n}+\lambda(u^{n}T^{n})|_{j-\frac{1}{2}}^{j+\frac{1}{2}}+\lambda(\gamma-2)T_{j}^{n}u^{n}|_{j-\frac{1}{2}}^{j+\frac{1}{2}}=0
\int f\left(x-\tfrac{a_1}{1-a_2}y\right) k_{\lambda\frac{a_1a_3}{1-a_2}}(x) = \left(1+\lambda \tfrac{a_1a_3}{1-a_2}\Delta\right) \left.f\left(x-\tfrac{a_1}{1-a_2}y\right)\right|_{x=0}+ O(\lambda^2).
\left(\int f dx \right) \left( \int f_{(s)}^{\circ}dx \right) \leq \left( \int (1-s |x|^2)_+^\frac{1}{2s}dx \right)^2= \left(\frac{\pi}{s}\right)^n \frac{\left(\Gamma(1+\frac{1}{2s})\right)^2}{\left(\Gamma(1+\frac{n}{2}+\frac{1}{2s})\right)^2}.
{\cal L}_{\mathrm{S C S}}= \kappa \, \epsilon^{\mu \nu \rho}\, \mathrm{t r}\left( \tilde{A}_{\mu}\partial_{\nu}\tilde{A}_{\rho}- \frac 2 3 i \tilde{A}_{\mu}\tilde{A}_{\nu}\tilde{A}_{\rho}\right) .
x= \bigg ( u^{2/\gamma_0}\left(1-(\gamma_1 / \tilde{\gamma}_1)x^{1/\tilde{\gamma}_1}\right)\Big / \left(1- (\gamma_1 / \tilde{\gamma}_1)\right)\bigg )^{\tilde{\gamma}_1\gamma_0/(\tilde{\gamma}_1+\gamma_0)}.
e_{\Phi^{\mathfrak{b},\gamma}}\left( \rho \right) =\int_{\mathbb{S}\times \mathbb{D}_{\varepsilon}}e_{\mathcal{K}_{\gamma}\left( \Psi ,f\right)}\left( \rho \right) \mathfrak{b}\left( \mathrm{d}\left( \Psi ,f\right)\right) \ ,\rho \in E_{1}\ .
\gamma_1 =\{(w,u)\in C\mid u\in (-\infty,0), w\in \bold e(\dfrac{1}{2})\}, \gamma_2 =\{(w,u)\in C\mid u\in (\dfrac{1}{t},\infty), w\in \bold e(\dfrac{1}{6})\}, \gamma_3 =\{(w,u)\in C\mid u\in (0,1-x_1), w\in \bold e(\dfrac{1}{3})\}.
c_i(E^{\vee}) = (-1)^i c_i(E) \;\;{\rm and}\;\; c_i(E \otimes L) = \sum_{j=0}^i{r-j \choose i - j}c_j(E) c_1(L)^{i-j}, s_i(E^{\vee}) = (-1)^i s_i(E) \;\;{\rm and}\;\; s_i(E \otimes L) = \sum_{j=0}^i (-1)^{i-j}{r-1+i \choose r-1 + j}s_j(E) c_1(L)^{i-j}.
\gamma_F (x) := \{v \in V : \beta(y) - c(x,y) -\alpha (x) \geq v \cdot (y-x), \,\, \forall y \in Y_F \beta(y) - c(x,y) -\alpha (x) = v \cdot (y-x), \,\, \forall y \in F_x \}
\begin{array}{l}{\partial_{i}x^{j}= \delta_{i}^{j}+ q \hat{\cal R}_{i l}^{j k}x^{l}\partial_{k}~ ,}{\tilde{\partial}_{k}x^{v}= \delta_{k}^{v}+ q^{- 1}\left. \hat{\cal R}^{- 1}\right._{k j}^{v i}x^{j}\tilde{\partial}_{i}~ .}\end{array}
\theta(\tau_{1/2}z) =e^{-\pi/4}\big(\frac{z}{|z|}\big)^{1/2}\Big( \theta(z)+ \theta(z+1/2) -\theta(z) \Big)=e^{-\pi/4}\big(\frac{z}{|z|}\big)^{1/2}\theta(z+1/2).
\vartheta_{1}(\beta,q)=2q^{1/4}(q^{2};q^{2})_{\infty}\sin(\beta)\,\left|\left(e^{2i\beta}q^{2};q^{2}\right)_{\infty}\right|^{2}=\sum_{k=0}^{\infty}(-1)^{k}q^{(2k+1)^{2}/4}\sin(\beta(2k+1)).
\hat H_{\partial M}=tr_{\partial M}(\hat\nabla\mathbf{\hat n}) &=tr_{\partial M}[\nabla d\Phi_f(\mathbf{\hat n})] &=tr_{\partial M}[\nabla(b\mathbf{N}+a\mathbf{n})] &=btr_{\partial M}(\nabla \mathbf{N})+atr_{\partial M}(\nabla\mathbf{n}) &=btr_{\partial M}K+aH_{\partial M}.
\phi^0(E) := \frac{\pi}{1+\frac{1}{M}}\log\left(\frac{1}{M+1}P_f^2 + H_f - E \right) \phi^I(E) := \int \! dp \, dq \: a_p^* \: \frac{1}{\frac{1}{M}(P_f + p + q)^2 + H_f + p^2 + q^2 - E}\: a_q.
\frac{A_z\left(x,z_\iota\right)}{\varepsilon\left(z_\iota\right)}\partial_z\xi^\prime_2\left(x,z_\iota\right)=-\int_{\Omega_z^2}\partial_x\left(A_x \partial_xw\left(x,z_\iota\right)\right) \mathrm{d}z -\int_{\Omega_z^2}\partial_x\left(A_x \partial_x\xi_2^\prime\right)\mathrm{d}z.
S_{a , b}( z ) = \frac{( - 1 )^{a + b - 1}}{( a - 1 ) ! \; b !}\! \int_{0}^{1}\mathrm{d}\xi \; \frac{\operatorname{l n}^{a - 1}\! \xi \operatorname{l n}^{b}( 1 \! - \! z \xi )}{\xi}\; .
C_0(E,d, L, \eta,f) := \| f \|_{*,\infty}\int\limits_{\| q \|_\infty \leq \sqrt{2 E + 1}}|q^2 - E - i 2^{-1/2}\eta |^{-1}dq + (E +1 )^{-1}\| f \|_{*,1}.
\frac{\max_{1\leq j\leq n}E^{0}(X_{j}^{2})}{n}\leq\varepsilon^{2}+\frac{{\sum\nolimits_{j=1}^{n}}E^{0}(X_{j}^{2}{I(|X_{j}|>\varepsilon}\sqrt{n}))}{n}& \leq\varepsilon^{2}+\frac{{\sum\nolimits_{j=1}^{n}E^{0}(X_{j}^{2}I(|X_{j}|>\varepsilon}\sqrt{M}{)}}{n}.
\left. \begin{array}{l c}{\mathrm{t a c h y o n}}&{| \Omega \rangle_{N S}\cong c e^{- \phi},}\\{\mathrm{m a s s l e s s N S}}&{\psi_{- 1 / 2}^{\mu}| \Omega \rangle_{N S}\cong \psi^{\mu}c e^{- \phi},}\\{\mathrm{m a s s l e s s R}}&{| \vec{s}\rangle_{R}\cong c e^{- \phi / 2}\Theta_{\vec{s}},}\\ \end{array}\right.
L^* := \mathbb{E}_{\P^\ast}\left[U^+\left(1+| X | \right)\right] =\frac{1}{2d}\sum_{i=1}^d 2\left(\left(1- \alpha \right)\mathbb{E}_{\P}\left[U^+\left(1+| X | \right)\right] + \alpha U^+(2)\right) & = \left(1- \alpha \right)\mathbb{E}_{\P}\left[U^+\left(1+| X | \right)\right] + \alpha U^+(2)<\infty.
\overline{w}_1 \Gamma^{(1)}S_F^{(1)}\Gamma^{(2)}\ldots \Gamma^{(n)}w_2 = (-1) \cdot \overline{w^c_2}\, \Gamma^{\prime\,(n)}\ldots S_F^{\prime\,(1)}\Gamma^{\prime\, (1)}w^c_1 ,
U \cup V = D((X - a_1) \dots (X - a_k)) \rlap{,}U = D((X - a_1) \dots (X - a_k) (X - b_1) \dots (X - b_l)) \rlap{,}V = D((X - a_1) \dots (X - a_k) (X - c_1) \dots (X - c_m)) \rlap{,}U \cap V = D((X - a_1) \dots (X - a_k) (X - b_1) \dots (X - b_l) (X - c_1) \dots (X - c_m)) \rlap{,}
E[T^p|\mathbf{a^{(0)}}=\mathbf{Vd}] =\sum_{m=1}^\infty \frac{1}{N}\sum_{k=2}^N s_k\lambda_k^{m-1}m^p &\sim\sum_{k=2}^N \frac{1}{N}s_k \frac{p!}{(1-\lambda_k)^{p+1}}&=p![N(N-1)]^{p+1}\sum_{k=2}^N \frac{s_k}{[k(k-1)]^{p+1}}.
Y_1(T_0+t)-Y_1(T_0) = \sqrt{snr}\int_{T_0}^{T_0+t}X_1(s) ds + \sqrt{inr}\int_{T_0}^{T_0+t}X_2(s) ds + B_1(T_0+t)-B_1(T_0) = \sqrt{snr}\int_0^t X_2(s) ds - \sqrt{inr}\int_0^t X_1(s) ds + B_1(T_0+t)-B_1(T_0).