harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_{D + d}= | | g_{M N}| | = \left( \begin{array}{c c c}{{g_{1}}}&{{g_{1}K}}&{{g_{1}{\cal A}}}\\{{K^{T}g_{1}}}&{{g_{2}+ K^{T}g_{1}K}}&{{g_{2}{\cal B}+ K^{T}g_{1}{\cal A}}}\\{{{\cal A}^{T}g_{1}}}&{{{\cal B}^{T}g_{2}+{\cal A}^{T}g_{1}K}}&{{e^{- \psi / 2}g_{D}+{\cal A}^{T}g_{1}{\cal A}+{\cal B}^{T}g_{2}{\cal B}}}\\ \end{array}\right) ,
\begin{multlined}(\mathcal M\rtimes\mathcal G)(a_1\dots a_n;a) :=\left\{(f,x)\;\middle|\;x\in\mathcal G(n),\f\in\mathcal M(a_{x^{-1}(1)}\dots a_{x^{-1}(n)};a)\right\}\ ;\end{multlined}
\hat{U}^{\dagger}( t_{2}, t_{1}) = \hat{U}^{- 1}( t_{2}, t_{1}) = \hat{U}( t_{1}, t_{2}) \ \ , \ \ \hat{U}( t_{3}, t_{2}) \, \hat{U}( t_{2}, t_{1}) = \hat{U}( t_{3}, t_{1}) \ ,
R^{e+}_h \circ \psi =\Phi^{\Gamma}_h \circ R^{e-}_h\circ\psi &=\Phi^{\Gamma}_h\circ \Psi \circ \tilde{R}^{e-}_h &=\Psi \circ\Phi^{\tilde{\Gamma}}_h\circ\tilde{R}^{e-}_h &=\Psi \circ \tilde{R}^{e+}_h
{\mathbf T}_1(E,\ell) = q^{\mp(\frac{1}{2}+\ell)}\frac{\mathbf{D}^{\mp}(E \Omega^{-2},\ell)}{\mathbf{D}^{\mp}(E,\ell)}+ q^{\pm(\frac{1}{2}+\ell)}\frac{\mathbf{D}^{\mp}(E \Omega^2,\ell)}{\mathbf{D}^{\mp}(E,\ell)}.
D_m=\frac{m}{t}\Bigg(\sum\limits_{i=0}^{q}\bigg(-\tau_i+\frac{3}{i+1}\tau_i-\frac{3t}{i+1}\frac{\partial\tau_i}{\partial t}\bigg)v^{i+1}-\frac{m}{2}t\sum\limits_{i=0}^{q}i\tau_iv^{i-1}&\quad\quad\quad\quad-\frac{m}{2}\bigg(\sum\limits_{i=0}^{q}\tau_iv^i\bigg)^2+3amtv-(am^2t+b)\sum\limits_{i=0}^{q}\tau_iv^i\Bigg)+\frac{\gamma_m}{t}.
\left({\frac{1}{2}}p^{2}+{\frac{1}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}( g_{| \rho |}- 1 ) | \rho |^{2}\, V ( \rho \cdot q ) + ({\frac{\omega^{2}}{2}}q^{2}) \right) \, e^{W}={\cal E}_{0}\, e^{W}.
\frac{\partial \tilde{N}_\pm (x; \pm{\bf p})}{\partial x_0}\pm \frac{{\bf p}}{\omega_\pm}\cdot \frac{\partial \tilde{N}_\pm}{\partial{\bf x}}\simeq \frac{- i \Sigma_{off}^{(bare)}(x; \omega_\pm,{\bf p})}{2 \omega_\pm}.
d{e_{i,t}}= d{y}_{i,t}-d{\overline{y}_{t}}= \sum_{j=1,j\neq i}^{N}\alpha_{ij}g(y_{i,t}-y_{j,t})\tilde{\gamma}dt-\nabla f(y_{i,t})\tilde{\gamma}dt+\frac{1}{N}\sum_{i=1}^N\nabla f({x}_{i,t})\tilde{\gamma}dt +\tau \tilde{\gamma}{dB_{i,t}}-\frac{\tau}{N}\tilde{\gamma}\sum_{i=1}^N{dB_{i,t}}.
\begin{aligned}|T'_1\cap T'_2| =|T'_1|+|T'_2|-|T'_1\cup T'_2| &\ge |T'_1|+|T'_2|-|A_{123}|-|A_{12}|+|T'_3| &> N/2-|A_{12}|-|U''_3|\ge N/4.\end{aligned}
\frac{{\partial \varphi}_{i}}{{\partial y}_{j}}=\frac{l_{i}^{x}}{l_{1}^{x}}\frac{\partial}{{\partial y}_{j}}\left( \frac{\sum\limits_{j=1}^n \sum\limits_{k=1}^p{e_{1jk}y_{j}z_{k}}}{\sum\limits_{j=1}^n \sum\limits_{k=1}^p{e_{ijk}y_{j}z_{k}}}\right).
W_{F_\lambda,l}:=F_\lambda^l\times\{0\}^{n-l}, 0\leq l\leq n W_{F_\lambda,l,t}:=\left\{\left(\begin{smallmatrix}z+\mu w t(z-\mu w) 0\end{smallmatrix}\right):z,w\in F_\lambda^{l}\right\}, 0\leq l\leq \lfloor\tfrac{n}{2}\rfloor
r=\frac{4-4{h^\prime}^2-2hh^{\prime\prime}}{h^2}E_1^\flat\wedge E_2^\flat-\left(\frac{5h^\prime h^{\prime\prime}+hh^{\prime\prime\prime}}{hh^\prime}+\frac{1}{8}{\textbf{H}^\prime}^2\right)E_3^\flat\wedge E_4^\flat+\frac{1}{8}{\textbf{H}^\prime}^2d\theta\wedge d\varphi.
\left.\mbox{Tr}K^{(0)}_{\delta}(s)\right|_{{\cal C}_\beta}={\beta \over 2\pi}\left.\mbox{Tr}K^{(0)}(s)\right|_{R^2}+{1 \over 8\pi is}\int_{0}^{\infty}rdr\int_{A}dz{\exp i{(\delta-\pi) \over \beta}z \over \sin{\pi \over \beta}z}\exp{r^2\sin^2{z \over 2}\over s}~~~,
\Lambda^{3 ( N_{c}+ 1 ) - ( N_{f}+ N_{c}^{\prime})}\tilde{\Lambda}^{3 ( \tilde{N}_{c}+ 1 ) - ( N_{f}+ N_{c}^{\prime})}= ( - 1 )^{N_{f}+ N_{c}^{\prime}- N_{c}- 1}\mu^{N_{f}+ N_{c}^{\prime}},
a_r^-(p,q,2N_1) = -\frac{1}{2}a_r^-(p,q,N_1) +\frac{1}{2}a_r^+(p,q^2,N_1) a_r^+(p,q,2N_1) = -\frac{1}{2}a_r^+(p,q,N_1) +\frac{1}{2}a_r^+(p,q^2,N_1)
K [ t , p , q , x ] = \frac{\pi^{2}}{2 t^{2}}\big ( \mathrm{e}^{- t ( p^{2}x - q^{2}x^{2})}- \mathrm{e}^{- t ( p^{2}x - q^{2}x^{2}) - p \circ p / t}\cos 2 x \theta ( p , q ) \big ) .
\{Q_{\beta}, Q_{\beta^{\prime}}\}= 2 i \delta_{\beta \beta^{\prime}}\overline{{{z}}}_{E}J_{E}\quad \quad \{Q_{\beta}, Q_{\beta^{\prime}}^{\dagger}\}= \delta_{\beta \beta^{\prime}}H \ + \ 2 \Gamma_{\beta \beta^{\prime}}^{j}x_{j E}J_{E}
d\langle \mathcal{M}^N_{f,k}, \Lambda^N_{F, G, H}\rangle_t=\Big(\mathcal{L}^N\left(\mu_{t,k}^N(f)V_{F,G,H}^N(t, \xi^N)\right)-V_{F,G,H}^N(t, \xi^N)\mathcal{L}^N\mu_{t,k}^N(f) &-\mu_{t,k}^N(f)\mathcal{L}^NV_{F,G,H}^N(t, \xi^N)\Big)dt.
\widetilde{\eta}=p^{2}\, \frac{\lambda}{\lambda_{_{\rm (gs)}}^{(\ast)}}\,\left\{1 +\left( \lambda_{_{\rm (gs)}}^{(\ast)}\right)^{1/2}\left[ \ln \pi + \gamma+ 2 \left( \lambda_{_{\rm (gs)}}^{(\ast)}\right)^{-1/2}- \ln \left| \frac{E}{\mu^{2}}\right|\right]\, p^{-1}+ O \left( p^{-2}\right)\right\}\; .
{\bf{E}}\Bigl[\Bigl(M_\mu^{(c=2)}(0,1)\Bigr)^n\Bigr] = n!\int\limits_{0<t_1<\cdots<t_n<1}\prod\limits_{k<p}^n |t_p-t_k|^{-\mu 2^{p-k-1}}\,dt,{\bf{E}}\Bigl[\Bigl(M_\mu^{(c=1/2)}(0,1)\Bigr)^n\Bigr] = n!\int\limits_{0<t_1<\cdots<t_n<1}\prod\limits_{k<p}^n |t_p-t_k|^{-\mu 2^{k-p-1}}\,dt.
I(\nu_{K})=\lim_{m\rightarrow\infty,\m\in{\mathcal{N}}}I\left(\nu_{K_m^{\pm}}\right)=\liminf_{m\rightarrow\infty,\m\in{\mathcal{N}}}I\left(\nu_{K_m^{\pm}}\right)\geq I\left(\nu\right)\geq I(\nu_{K}).
\sum_{\alpha=1}^\infty \frac{|\varphi_\alpha(y)-\varphi_\alpha(z)|^2}{\lambda_\alpha}\leq \sum_{\alpha=1}^\infty \frac{8\wedge 2\lambda_\alpha |y-z|^2}{\lambda_\alpha}\leq \int_0^\infty \frac{8}{\pi^2 u^2}\wedge 2|y-z|^2 du &\leq \int_0^{\frac{2}{\pi |y-z|}}2|y-z|^2du+ \int_{\frac{2}{\pi |y-z|}}^\infty \frac{8}{\pi^2 u^2}du=\frac{8}{\pi}|y-z|.
F(r\omega )=\frac{1}{\left( 1+r^{2}\right)^{(m+\nu )}}\sum_{\substack{0\leq p\leq m 0\leq q\leq m+2\nu}}r^{p+q}{}_{2}F_{1}\left( \begin{array}{c}p-m,q-m-2\nu n+p+q\end{array}\mid -r^{2}\right) \sum_{j=1}^{d(n,p,q)}a_{j}^{\nu ,p,q}h_{p,q}^{j}(\omega ,\bar{\omega}),
\frac{e^{(\alpha-\beta)\zeta_i(t)+\beta S(\zeta(t))}}{\sum_{k=1}^{n}e^{(\alpha-\beta)\zeta_i(t)+\beta S(\zeta(t))}}= \frac{e^{(\alpha-\beta)\zeta_i(t)}}{\sum_{k=1}^{n}e^{(\alpha-\beta)\zeta_k(t)}}
\begin{array}{l l}{{\partial X_{c l}( z ) = a \omega ( z ) ,}}&{{\partial \bar{X}_{c l}( z ) = \bar{a}\omega^{\prime}( z ) ,}}\\{{\bar{\partial}X_{c l}( \bar{z}) = b \bar{\omega}^{\prime}( \bar{z}) ,}}&{{\bar{\partial}\bar{X}_{c l}( \bar{z}) = \bar{b}\bar{\omega}( \bar{z}) ,}}\\ \end{array}
E_{a,b}^{(M)}E_{c,d}^{(N)}=\begin{cases}(-1)^{\overline{E}_{a,b}\overline{E}_{c,d}}q_bE_{c,d}E_{a,b}, M=N=1, b=d; (-1)^{\overline{E}_{a,b}\overline{E}_{c,d}}q_aE_{c,d}E_{a,b}, M=N=1,a=c; q_b^{MN}E_{c,d}^{(N)}E_{a,b}^{(M)}, .\end{cases}
\bar V_1^B=\left\langle \begin{pmatrix}0 -1 \end{pmatrix}\right\rangle, \qquad\bar V_2^B=\left\langle \begin{pmatrix}1 0 \end{pmatrix}\right\rangle, \qquad\bar V_3^B=\left\langle \begin{pmatrix}1 -1 \end{pmatrix}\right\rangle, \qquad\bar V_4^B=\left\langle \begin{pmatrix}\beta -1 \end{pmatrix}\right\rangle.
\mathbf{D}\left[:,:,0\right]=\left(\begin{array}{rr}\left(-x^{3}\right)^{\frac{1}{12}}0 \left(-x^{3}\right)^{\frac{1}{6}}0\end{array}\right),\:\mathbf{D}\left[:,:,1\right]=\left(\begin{array}{rr}0 \left(-x^{3}\right)^{\frac{1}{6}}0 1\end{array}\right).
\lim_{n\to\infty}\frac 1n E^Q_z N_n(1) = 1-\pi^Q_1(0) - (1-\lambda_C^{-1})\pi^Q_1(H_1)-\left ( \sum_{z^0=(0,b+1,b+2) \in H_2}\pi^Q (z^0) (1- \rho(b))\right).
\Gamma^{0}= \left( \begin{matrix}{{\gamma^{0}}}&{{0}}\\{{0}}&{{- \gamma^{0}}}\\ \end{matrix}\right) \ , \Gamma^{1}= \left( \begin{matrix}{{\gamma^{1}}}&{{0}}\\{{0}}&{{- \gamma^{1}}}\\ \end{matrix}\right) \ , \Gamma^{2}= \left( \begin{matrix}{{\gamma^{2}}}&{{0}}\\{{0}}&{{- \gamma^{2}}}\\ \end{matrix}\right) \ ,
\gamma_{\textrm{FDN}}^{\textrm{OFDMA}}=\frac{\left(1-\beta_{\textrm{F}}\right)P_\textrm{N}{|h_{\textrm{BF}}|^2}}{{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^2}\left( 1-\theta\right)}.
\Pi(B, F) =\Pi(B, \pi(F)\times \{0\}) =\Pi(\pi(B), \pi(F)):=\{\alpha\cdot \beta\in \mathbb F_q: \alpha\in \pi(B)\subset \mathbb F_q^{d-1}, ~\beta\in \pi(F) \subset \mathbb F_q^{d-1}\}
\dim\mu_{\lambda}=\lim_n\frac{H(\mu_{\lambda};\l^n)}{n\log\l^{-1}}=\lim_n\frac{H(\mu_{\lambda}^{(n)};\l^n)}{n\log\l^{-1}}\le\lim_n\frac{H(\mu_{\lambda}^{(n)})}{n\log\l^{-1}},
x^{*}=-\frac{1}{w}(\gamma_1 b_1+\gamma_2 b_2),\ \ (x^{*})^{T}x^{*}=\frac{1}{w^{2}}(\gamma_1 b_1+\gamma_2 b_2)^{T}(\gamma_1 b_1+\gamma_2 b_2)
\Delta_n^\mu-\Delta_n^\mathrm{proj}= \frac{1}{n}\sum_{i=1}^nE[X_{n,i}\varepsilon_{n,i}]E[\varepsilon_{n,i}X_{n,i}^{\prime}] &-\left(\frac{1}{n}\sum_{i=1}^n E[X_{n,i}\varepsilon_{n,i}]Z_{n,i}^{\prime}\right)\left(\frac{1}{n}\sum_{i=1}^n Z_{n,i}Z_{n,i}^{\prime}\right)^{-1}\left(\frac{1}{n}\sum_{i=1}^n Z_{n,i}E[\varepsilon_{n,i}X_{n,i}^{\prime}]\right).
\omega_{\pmb\eta}\geq \begin{cases}\dfrac{1-(1-\rho_n)\hat\omega}{\rho_n}\hskip 11mm\hat\omega\geq\dfrac{1-\sigma_m}{1-\rho_n}\dfrac{\sigma_m\vphantom{1^{\big|}}}{1-(1-\sigma_m)\hat\omega^{-1}}\hskip 6mm\hat\omega\leq\dfrac{1-\sigma_m}{1-\rho_n}\end{cases},
1 + S = \left( \begin{array}{c c}{{\frac{1}{2 V_{o o}}\sqrt{1 + V_{o o}}\left[ \sqrt{1 + V_{o o}}- \sqrt{1 - 3 V_{o o}}\right]}}&{{0}}\\{{0}}&{{\frac{1}{2 V_{e e}}\sqrt{1 + 3 V_{e e}}\left[ \sqrt{1 + 3 V_{e e}}- \sqrt{( 1 - V_{e e})}\right]}}\\ \end{array}\right) \, ,
2 \mathrm{Re}\langle u_{\alpha}\otimes \Omega\,|\, H_y\Phi^y_{\#}\rangle = - 8 \alpha \langle P u_{\alpha}\otimes A^+_y \Omega \,|\, h_{\alpha}- e_{\alpha}+ H_f^+\,|\,P u_{\alpha}\otimes A^+_y \Omega \rangle &= - 2 \|\Phi_{\#}^y\|^2_{\#}.
d(\overline{v}_\lambda - \overline{v}_\mu) + A(\overline{v}_\lambda - \overline{v}_\mu)\,dt = (F_\lambda \overline{v}_\lambda - F_\mu \overline{v}_\mu)\,dt + \sigma'(u) ( \overline{v}_\lambda - \overline{v}_\mu ) B\,dW
q^{\frac{3}{2}( \Lambda , \Lambda + 2 \rho )}= \sum_{\nu \in M ( \Lambda )}q^{\frac{1}{2}( \nu , \nu + 2 ( \Lambda + \rho ) )}\frac{D_{q}[ \Lambda + \nu ]}{D_{q}[ \Lambda ]}\sum_{s = 0}^{\infty}( m_{\nu - s \delta}^{+}- m_{\nu - s \delta}^{-}) \, q^{- s ( 3 g + 2 \kappa_{\Lambda})}\; .
( \frac{n \tau_{1}}{\epsilon}+ 1 )^{2}+ ( \frac{n \tau_{2}}{\epsilon}+ 1 )^{2}= 2 ( \frac{n \bar{\tau}}{\epsilon}+ 1 )^{2}+ \frac{n^{2}}{2 \epsilon^{2}}( \tau_{1}- \tau_{2})^{2},
u_{0 , l}( r ) = \left( \begin{array}{l}{\phi_{0 , l , 1}( r )}\\{0}\\ \end{array}\right) , \mathrm{}{}l \geq 1 ; u_{0}^{I I}( r ) = \left( \begin{array}{l}{\phi_{0 , 1}^{i r}( r )}\\{0}\\ \end{array}\right) ,{}l = 0 .
P_{\rm ang}:=2\partial^{(4)}_t\Delta_{\theta}-J_3\partial^{(3)}_t\Delta_{\theta}+J_2\partial^{(2)}_t\Delta_{\theta}-J_1\partial_t\Delta_{\theta}+J_0\Delta_{\theta}+3\partial^{(2)}_t\Delta^2_{\theta}-L_0\Delta^2_{\theta}+\Delta^3_{\theta}
\Phi(X_1,X_2) = \pi_p(X_1,X_2) + \frac{1}{p!}\sum_{\varepsilon\in\{1,2\}^{p+1}}X_{\varepsilon_1}\cdots X_{\varepsilon_{p+1}}\int_0^1 \partial_{\varepsilon_1}\cdots \partial_{\varepsilon_{p+1}}\Phi(tX_1,tX_2)\ (1-t)^p\dt &= \pi_p(X_1,X_2) + g_{0,0}(X_1,X_2).
\begin{array}{ll}(\nabla F_\ast) (X,Y) =-\frac{\lambda^2}{2}\left\{X (\frac{1}{\lambda^2})\tilde{Y}+ Y (\frac{1}{\lambda^2})\tilde{X}- g(X, Y) F_\ast (grad_\mathcal{H}\frac{1}{\lambda^2}))\right\},\end{array}
u(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\lambda x-\lambda^{2}t}Q(\lambda)d\lambda-\frac{1}{2\pi}\int_{\partial D^{+}}e^{i\lambda x-\lambda^{2}t}[Q(-\lambda)+2i\lambda\tilde{g_{0}}(\lambda^{2})]d\lambda,
S_N(x)+S_N(y) = \sum_{k=\gamma}^{N}a_k e^{\psi(k)}+ \sum_{k=\gamma}^{N}b_k e^{\psi(k)}&= \sum_{k=\gamma}^{N}(a_k+b_k) e^{\psi(k)}&= \sum_{k=\gamma}^{N}c_k e^{\psi(k)}+ r_{N}e^{\psi(N)}.
A_r^a = \frac{1}{L_{\perp}\sqrt{4 \pi}}\sum_{\lambda=\pm 1}\sum_{k=1}^\Lambda \frac{1}{\sqrt{k}}\left(a_{k,\lambda}^a \epsilon_r^{-\lambda}e^{-2 \pi i k x^-/L}+{a_{k,\lambda}^a}^{\dag}\epsilon_r^{\lambda}e^{2 \pi i k x^-/L}\right) \; .
\eta(\eta(\tau_j, \tau_i), \eta(\tau_j, \tau_\ell)) =\eta((\tau_n^q\tau_1)^{n-i}, (\tau_n^q\tau_1)^{n-j}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}) =(\tau_n^q\tau_1)^{i-j}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}.
V_{0}( m ) ={\frac{m^{2}}{2 g^{2}}}+{\frac{1}{8 \pi^{2}}}\left[{\frac{\Lambda^{4}}{2}}\Bigl ( 1 -{\frac{m^{2}}{\Lambda^{2}}}\Bigr ){\mathrm{e}}^{-{\frac{m^{2}}{\Lambda^{2}}}}+{\frac{m^{4}}{2}}E_{1}({m^{2}/ \Lambda^{2}}) \right] ,
\hat{h}(\omega_{i+\frac{1}{2},j}^{-}(y),\omega_{i+\frac{1}{2},j}^{+}(y),u_{i+\frac{1}{2},j}(y))=\frac{1}{2}[u_{i+\frac{1}{2},j}(y)(\omega_{i+\frac{1}{2},j}^{-}(y)+\omega_{i+\frac{1}{2},j}^{+}(y))-\alpha(\omega_{i+\frac{1}{2},j}^{+}(y)-\omega_{i+\frac{1}{2},j}^{-}(y))],
H(q_{1},\cdot\cdot\cdot,q_{n},p_{1},\cdot\cdot\cdot,p_{n})=\sum_{i=1}^{n}p_{i}\dot{q}_{i}-L(q_{1},\cdot\cdot\cdot,q_{n},\dot{q}_{1}\cdot\cdot\cdot,\dot{q}_{n})
U_{g}^{a \prime \prime}= V^{\prime}( y^{a}+ L^{a}) U_{g}^{\prime}V^{\prime \dagger}( y^{a}) = \left( \begin{array}{c c c c}{{\mathrm{e}^{- i \theta_{a , 1}}}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{}}&{{\mathrm{e}^{- i \theta_{a , N}}}}\\ \end{array}\right) \equiv U_{g}^{a : i n v}.
\mathrm{J_{0}\equiv \frac{1}{r + s}( a_{1}^{\dagger}a_{1}- a_{2}^{\dagger}a_{2}) \; , \; \mathrm{J_{+}\equiv ( a_{1}^{\dagger})^{s}a_{2}^{r}\; , \; \mathrm{J_{-}= a_{1}^{s}( a_{2}^{\dagger})^{r},}}}
\sqrt{r}K_{\nu}(2\nu r^{-1/(2\nu)}) = C\left(\frac{(2\nu)^{-2\nu}}{\Gamma(1-\nu)}r - \frac{1}{\Gamma(\nu+1)}+\mathcal{O}(r^{-1})\right) &= C\left(r - \frac{\Gamma(1-\nu)(2\nu)^{2\nu}}{\Gamma(\nu+1)}+ \mathcal{O}(r^{-1})\right).
\begin{array}{l}\displaystyle\displaystyle\tilde{\cal Q}^1_\alpha=(ip_{\alpha\beta}\tilde W^\beta-mW_\alpha)[1+{\rm q}^{cl}(bP_3-\sqrt{1-b^2}\,P_2-P_4)] begin{align*}1mm] \displaystyle\tilde{\cal Q}^2_\alpha=(ip_{\alpha\beta}\tilde V^\beta-mV_\alpha)[1+{\rm q}^{cl}(bP_3+\sqrt{1-b^2}\,P_2-P_4)]\,.\end{array}
R_1(\lambda;x,y) = \tfrac{i}{2\sqrt{\lambda}}\begin{cases}[e^{-ix\sqrt{\lambda}}-e^{ix\sqrt{\lambda}}]e^{iy\sqrt{\lambda}}y\geq x \geq 0, 0 y\geq 0\geq x, e^{-ix\sqrt{\lambda}}[e^{iy\sqrt{\lambda}}-e^{-iy\sqrt{\lambda}}] 0 \geq y\geq x,\end{cases}
\mathbb{P}\left(|L_n(z)|\leq e^{-n\epsilon}\right) \leq Q\left(\sum_{i=1}^{n}\frac{a_i}{z-z_i},e^{-n\epsilon}\right), \leq Q\left(\sum_{k=1}^{l_n}\frac{a_{i_k}}{z-z_{i_k}},e^{-n\epsilon}\right).
\left\| e^{ikx_1}\pi^k\right\|_{H^{-1}(\Omega)}\leq \left\| e^{ikx_1}\begin{bmatrix}\widetilde H_{1}^{k}\widetilde H_{2}^{k}\end{bmatrix}\right\|_{H^{-1}(\Omega)}+\left\| \nabla \left(e^{ikx_1}w^k\right) \right\|_{L^{2}(\Omega)}.
A_{j_1,\ldots,j_{2l}}= 2\sum_{r=1}^{j_1}C_{r-1}A_{j_1-r,j_2,\ldots,j_{2l}}+ \sum_{r=2}^{2l}(2j_r+1) C_{j_1+j_r}A_{j_2,\ldots,\widehat{j_r},\ldots,j_{2l}}.
u_{\beta\beta}(y) \le \epsilon \sup\limits_{B_{\frac{\delta R}{4}}(y)\cap \Omega}|D^2u| + C_\epsilon (1+ \sup\limits_{B_{\frac{\delta R}{4}}(y)\cap \partial\Omega}\sup\limits_{|\tau|=1,\tau\cdot \nu=0}|u_{\tau\tau}|) + \frac{C}{(\delta R/ 4)^2},
\#E(Q,Y_1)\le (4l-3)^q\prod_{i=1}^{4l-3}\#E(Q_i, Y_1) \le(4l-3)^q\prod_{i=1}^{4l-3}(128l^2)^{bq_i}\cdot (qb)^{4lq_i}\cdot\sigma^{12lq_i+8lb}&\le (4l)^q\cdot (128l^2)^{qb}\cdot (qb)^{4lq}\cdot\sigma^{12lq+32l^2b},
\det\mathbf M_{k-1}\det\mathbf M_k={}\frac{k [\Gamma (k/2)]^2(\det\mathbf N_{k-1})^2}{2 (2 k+1)}\prod_{j=1}^k \left[\frac{(2 j)^2}{(2 j)^2-1}\right]^{k-\frac{1}{2}}, \det\mathbf N_{k-1}\det\mathbf N_k={}\frac{2k+1}{k+1}\frac{(\det\mathbf M_{k})^2}{(k-1)!}\prod_{j=2}^{k+1}\left[\frac{(2 j-1)^2}{(2 j-1)^2-1}\right]^{k}.
G_{{\cal Q}^{4}}( x_{p}) = \frac{3^{3}\, \Gamma ( 1 1 )}{2^{7}\left( \pi^{3}\Gamma ( 4 ) \right)^{4}}\: g^{8}e^{- \frac{8 \pi^{2}}{g^{2}}+ i \theta}\, x_{1 2}^{4}x_{3 4}^{4}\prod_{p < q}\frac{\partial}{\partial x_{p q}^{2}}B ( x_{p q}) \, ,
\varrho (a,g(\upsilon)) = \lim_{i \to \infty}\varrho(a,g_i(\upsilon)) \leq \liminf_{i \to \infty}\mathbf{V}_0^\upsilon g_i = \upsilon, \varrho (g(\upsilon),x) = \lim_{i \to \infty}\varrho(g_i(\upsilon),g_i(b)) \leq \liminf_{i \to \infty}\mathbf{V}_\upsilon^b g_i = b-\upsilon,
p_{sr}= p_{rd}= \frac{1}{2}\left\{\frac{m^2\!-\!1}{n\!-\!1}\!-\!\frac{m^2}{n\!-\!1}\left(1\!-\!\frac{1}{m^2}\right)^n\!-\!\left(1\!-\!\frac{1}{m^2}\right)^{n\!-\!1}\right\}.
\prod_{c=1}^{\lambda_r}\frac{1-q^{c-1}t^{n-(r-1)}}{1-q^{\lambda_r-c}t^{\lambda_c'-r+1}}= \frac{(1-q^0t^{n-(r-1)})(1-q^1t^{n-(r-1)}) \cdots (1-q^{\lambda_r-1}t^{n-(r-1)})}{(1-q^{\lambda_r-1}t^{\lambda_1'-r+1})\cdots (1-q^1t^{\lambda'_{\ell-1}-r-1+1})(1-q^0 t^{\lambda_\ell'-r+1)}}
h_{1,k+2,2}(x) =\frac{1}{2}\left((1+\sqrt{x})^{2k+4}h_{1,k+1,2}(\sqrt{x})+(1-\sqrt{x})^{2k+4}h_{1,k+1,2}(-\sqrt{x})\right) &=\sum_{j=0}^k \frac{1}{2}a_{j,k}\left(\sqrt{x}^j(1+\sqrt{x})^{4k-2j+4}+(-\sqrt{x})^j(1-\sqrt{x})^{4k-2j+4}\right).
\ \ \ \ \frac{\ln(2)}2+\frac1{16\pi^{2}}{\displaystyle\int\limits_{-\pi}^{\pi}}{\displaystyle\int\limits_{-\pi}^{\pi}}\ln\left[ 7-3\cos(\theta)-3\cos(\varphi)-\cos(\theta)\cos(\varphi)\right]\,d\theta\,d\varphi \ =\frac G\pi+\frac12\ln(\sqrt{2}-1)+\frac1\pi\operatorname*{Ti}\nolimits_{2}(3+2\sqrt{2})=0.7866842753...
\ell_{i,j}^{2}=(s_i\hdots s_j)\ell_{i,j-1}(s_{j}\hdots s_{i})\ell_{i+1,j}&=(s_i\hdots s_j)\ell_{i,j}\ell_{i+1,j}&=(s_i\hdots s_j)(s_{j}\hdots s_{i})\ell_{i+1,j}^{2}.
P_N(\varphi) = \left( \prod_{j=1}^J P_{F_{n_j}}(\varphi, \varepsilon_j) \right) \left(\prod_{j=J+1}^m P_{F_{n_j}}(\varphi, \varepsilon_j)\right) \geq K_1^J \cdot 1^{m-J}\geq K_1^J.
\pi P_v f(u)\geq \int_{u-dv}^{u+dv}\frac{vf(t)dt}{(u-t)^2+v^2}\geq \frac{1}{1+d^2}\frac{1}{v}\int_{u-dv}^{u+dv}f(t)dt=\frac{2d}{1+d^2}Mf(u,dv),
[(\phi\wedge \psi)(x)]^{\beta[\frac{n}{x}]}=[\phi(x)]^{\beta[\frac{n}{x}]}\cap [\psi(x)]^{\beta[\frac{n}{x}]}&=i(n,N_\phi|A_\phi)\cap i(n,N_\psi|A_\psi) &=i(n,N_\phi|A_\phi \sqcap N_\psi|A_\psi)
w(r,t) = w(r_{0}(t),t)- \int_{r}^{r_{0}(t)}w_{r}(\rho,t)d\rho \leqslant K+\int_{r_{0}(t)}^{r}\vert w_{r}(\rho,t)\vert d\rho \leqslant K +r_{0}^{1-N}(t) \int_{r_{0}(t)}^{r}\rho^{N-1}\vert w_{r}(\rho,t)\vert d\rho \leqslant K +CKr_{0}^{1-N}(t).
P \left(d_\pi f\ast\chi_\pi \right) =\varphi\bigl((f\ast d_\pi\chi_\pi)^{*n},d_\pi\chi_\pi,\dotsc,d_\pi\chi_\pi\bigr) & =\varphi\bigl(f^{*n}\ast d_\pi\chi_\pi,d_\pi\chi_\pi,\dotsc,d_\pi\chi_\pi\bigr).
\chi_\Lambda(C_m^{\Lambda_0})=\sum_{(\lambda+\lambda_0',\kappa+\kappa_0,-s)\in D^+}\sum_{s=0}^\infty\,\alpha^m_{\lambda_0',s}(\Lambda)(I\otimes{\rm tr})\{(I\otimes\pi_{\Lambda_0}(q^{2h_\rho}))P[\lambda_0',s]\}
J_{\nu - 1}^{2}( i x ) - e^{2 \theta}J_{\nu}^{2}( i x ) \sim{\frac{( i x )^{2 ( \nu - 1 )}}{2 \pi}}\alpha_{\nu}e^{2 \alpha_{\nu}}e^{- 2 \nu \log ( \nu + \alpha_{\nu})}\Bigr [ \Sigma_{1}^{2}A_{\theta}( t ) + \Sigma_{2}^{2}+ 2 t \Sigma_{1}\Sigma_{2}\Bigr ] ,
F_{\theta}(x^{\star})-F_{\theta}(x_{\theta}^{\star})\leq F(x^{\star})-F_{\theta}(x_{\theta}^{\star})\leq F(x_{\theta}^{\star})-F_{\theta}(x_{\theta}^{\star})=g(x_{\theta}^{\star})-g_{\theta}(x_{\theta}^{\star})\leq \frac{L_0}{2}\theta.
\left(\sum_{j=i+2}^{d-3}(-1)^{j}v_jz^{3n^{i+2}+n^{i+1}+\cdots+n^2+n}\right)+(-1)^{d}(xz^{3n^{i+2}+n^{i+1}+\cdots+n^2+n}) & = -z^{3n^{i+2}+n^{i+1}+\cdots+n^2}f_3 + \left(\sum_{j=1}^{i+1}(-1)^{j+1}v_jz^{3n^{i+2}+n^{i+1}+\cdots+n^2+n}\right)+y^nz^{3n^{i+2}+n^{i+1}+\cdots+n^2}\in I_n.
{}\begin{aligned}(z_{n}+\frac{1}{n})^{\theta-\delta}\leq{(c\phi_1+n^{\frac{\delta+p-1}{p}})^{\frac{p(\theta-\delta)}{\delta+p-1}}}\leq (c\phi_{1})^{\frac{p(\theta-\delta)}{\delta+p-1}}\end{aligned}
\mathcal{R}_1^+= \sum_{m \geq 1}\sum_{n \geq 1}\frac{A(m,n)}{\sqrt{m^2 n}}g\Big(\frac{m^2 n}{N}\Big)\sum_{c \geq C_1/m}\frac{S(n,1;c)}{c}H_{m,n}^+ \Big( \frac{4\pi \sqrt{n}}{c}\Big),
G_Q^1 = \left\{\left(\begin{array}{cc}\left(-\frac{b(x,y)}{a(x,y)}\alpha+\beta\right) \left(-\frac{c(x,y)}{a(x,y)}\alpha\right) \alpha \beta \end{array}\right)\right\}, \ \G_Q^2 = \left\{\left(\begin{array}{cc}-\beta \left(\frac{c(x,y)}{a(x,y)}\alpha-\frac{b(x,y)}{a(x,y)}\beta\right) \alpha \beta \end{array}\right)\right\}.
{\zeta_{P\bar{\cal K}}}(s)=\zeta_{\bar{\cal V}}(s)+\frac{\Gamma(s+\frac{1}{2})}{\sqrt{\pi}\,\Gamma(s)}\left[\frac{2}{3^{s+\frac{1}{2}}}{}_2F_{1}[{\textstyle\frac{1}{2}},{s+\textstyle\frac{1}{2}},{\textstyle\frac{3}{2}},-{\textstyle\frac{1}{3}}]-\frac{1}{4^s}\frac{1}{s}\right]
\frac{ik}{2}\sum_{k_1 + k_2 +k_3 = k}^* e^{-it(\alpha k^3 - \alpha k_1^3 -\alpha k_2^3 - \alpha k_3^3)}\frac{(k_1 + k_2)n_{k_1}n_{k_2}n_{k_3}}{\alpha k^3 - (k_1 + k_2)^3 - \alpha k_3^3}+ ik \sum_{k_1 + k_2 + k_3 = k}^* e^{-it(\alpha k^3 - k_1^3 - k_2^3 -\alpha k_3^3)}\frac{(k_2+k_3)m_{k_1}m_{k_2}n_{k_3}}{\alpha k^3 - k_1^3 - \alpha(k_2+k_3)^3}.
I_{\pm}= 4 \pi \int_{0}^{\vert a \vert^{3 / 2}}\frac{1 + \vert \beta \vert^{2}\vert a \vert^{2}}{( 1 + \vert \beta \vert^{2}\vert a \vert^{2}+ 4 \vert \lambda \vert^{2}\vert a \vert^{2}\vert b \vert^{2})^{2}}d \vert b \vert^{2}.
D_{k+1}U^{k+1}_{i}= (1-\delta^k_{k-i})U^{k}_{i}+ \delta_{k-i}^k U^{k}_{i-1}D_{i}+ \sum\limits_{j=-1}^{k-i-1}U^k_{k-j-1}\Gamma_jU^{k-j-1}_{i}
q-|M_1|=\begin{cases}n - \sum_{i=1}^k a_i q - n + \sum_{i=1}^k a_i\ge 0 \sum_{i=1}^k a_i\le n; 0 q - n + \sum_{i=1}^k a_i\ge 0\sum_{i=1}^k a_i> n; q\le n - \sum_{i=1}^k a_i q - n + \sum_{i=1}^k a_i<0.\end{cases}
\tilde{R}= \frac{1 - 7 \operatorname{c o s}( 4 \rho ) + 8 \operatorname{s i n}( 2 \rho ) \operatorname{c o s}t}{4 \left( 1 + \operatorname{s i n}( 2 \rho ) \operatorname{c o s}t \right)^{2}}
\int_{0}^{x}f(t) h(t/q) \left( -\frac{q(1+q)+t}{q^2 (1-q)t}u(t/q) +\frac{1}{q(1-q)^2t}\right)y(t)\, d_qt &= \frac{f(x)}{q (1-q)}\sum_{n=0}^{\infty}(-q)^n \left(-1+q^2+q^nx\right) y(q^n x) = H(x).
Z(Q_3(1,1,4)) = 10, Z(Q_3(1,2,3)) = 12, Z(\Theta_{3}(1,1,2,1)) =11, Z(\Theta_{3}(1,2,1,1)) =12, Z(\Theta_{3}(1,1,1,1,1)) =13, Z(Q_4(1,1,1,3)) =13, Z(K_{4}) =10, Z(K_{2,3}) =13, Z(Q_3(1,1,\dot 1)\vee Q_3(1,1,\dot 1)) = 12,
\langle q_{A}|\rho_{A}|q_{A}^{\prime}\rangle\propto exp(\frac{2(\omega_{+}-\omega_{-})^{2}q_{A}q_{A}^{\prime}-(8\omega_{+}\omega_{-}+(\omega_{+}-\omega_{-})^{2})(q_{A}^{2}+q_{A}^{\prime2})}{8(\omega_{+}+\omega_{-})})
S^{\prime}= \int d^2 \sigma \Big(- e^{-\phi}\sqrt{-{\rm det}\, (G_{\mu\nu}+{\cal F}_{\mu\nu})}+\frac{1}{2}\tilde{H}^{\mu\nu}(F_{\mu\nu}- 2 \partial_{\mu}A_{\nu})+ \frac{1}{2}e^{-\phi}\epsilon^{\mu\nu}C_{\mu\nu}-\frac{1}{2}C_0 \epsilon^{\mu\nu}F_{\mu\nu}\Big)
\tilde{\tau}_{i}^{2}\doteq\frac{1}{\sigma_{i}^{2}}\left(\frac{1}{m}\overset{n}{\underset{j=1}{\sum}}\sigma_{j}^{2}M^{\#}\left(\epsilon_{j}\right)\tilde{\tau}_{j}^{2}+\sigma^{2}\right),\; i\in[n].
(\rho_1,\rho_2)\left( \begin{array}{cc}\mathbb{E}[(\bar{S}_2(\gamma))^2] \mathbb{E}[\bar{S}_2(\gamma)X_a] \mathbb{E}[\bar{S}_2(\gamma)X_a] P_a+P_{d,1}+N_2 \end{array}\right) =(\mathbb{E}[S_2\bar{S}_2(\gamma)],\mathbb{E}[S_2X_a]),
\dot\Delta(\xi_n)= -\dot \theta_N(\xi_n)2 \sin \theta_N(\xi_n)+ \frac{2i(-1)^{n+1}}{(2i\xi_n)^{N+1}}\int_0^1\varphi_1(t)r_{N+1}(t)\,dt+\frac{\ell^2_n}{n^{N+1}}\,.
\begin{array}{l l}{K_{1}^{\theta}= - \sin \phi}&{K_{1}^{\phi}= - \cot \theta \cos \phi}\\{K_{2}^{\theta}= \cos \phi}&{K_{2}^{\phi}= - \cot \theta \sin \phi}\\{K_{3}^{\theta}= 0}&{K_{3}^{\phi}= 1 .}\\ \end{array}
\phi(y_{ij}) = y_{ij}, \phi(\xi_{ij}) = \sum_{k=1}^n y_{ik}\xi_{kj}(1\leq i, j \leq n), \psi(y_{ij}) = y_{ij}, \psi(\xi_{ij}) = \sum_{k=1}^n y_{ki}\xi_{kj}(1\leq i, j \leq n).
\frak{m}'_{\mathbb{C}}=\frak{m}'^{(1,1)}\oplus\frak{m}'^{(2,0)}\oplus \frak{m}'^{(0,2)}, &\frak{m}_{\mathbb{C}}=\frak{m}^{(1,1)}\oplus\frak{m}^{(2,0)}\oplus \frak{m}^{(0,2)},
[\begin{matrix}C_{11}&C_{12}\\C_{21}&C_{22}\end{matrix}]=[\begin{matrix}A_{11}B_{11}+A_{12}B_{21}&A_{11}B_{12}+A_{12}B_{22}\\A_{21}B_{11}+A_{22}B_{21}&A_{21}B_{12}+A_{22}B_{22}\end{matrix}].
\int_{B_{\mathcal{R}}}(|\tilde{u}_k(x)|^{t(x)}+ |\tilde{v}_k(x)|^{t(x)})\bigg(\int_{B_{\mathcal{R}}}\frac{| \phi (x)- \phi (y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy\bigg)\,dx \leqslant c_{1}(\mathcal{R})\int_{B_{\mathcal{R}}}(|\tilde{u}_k(x)|^{t(x)}+ |\tilde{v}_k(x)|^{t(x)})\,dx.
\overline{L}_{n}\phi_+(z)=\phi_+'(z)\left(\left[\left(z^{-2}\frac{\phi_+^{n+1}(z)}{\phi_+'(z)}\right)^*\right]_{++}-\frac12z\left((\frac{\phi_+(z)^{n+1}}{\phi_+'(z)})_1\right)^*\right).
\int{\frac{d^{d}q}{( 2 \pi )^{d}}}{\frac{1}{q^{2}+ m_{1}^{2}}}{\frac{1}{( q - k )^{2}+ m_{2}^{2}}}={\frac{1}{( 4 \pi )^{d}}}\big \{{\frac{2}{\epsilon}}- \gamma - b ( k^{2}, m_{1}^{2}, m_{2}^{2}) +{\cal O}( \epsilon ) \big \}
0={\rm tr.deg}_{\overline{k}(t)}\overline{k}(t)\bigl\{1,\_{s+1}\mathcal{F}_s({\bf a};{\bf b})(\alpha)^{q^{d}}\bigr\}={\rm tr.deg}_{\overline{k}}\overline{k}\bigl\{1,\_{s+1}F_s({\bf a};{\bf b})(\alpha)^{q^{d}}\bigr\}.
p_1 =\frac{1}{n}(a_1 +a_2 +a_3 + \cdots +a_n), p_2 =\frac{1}{\frac{n(n-1)}{2}}(a_1a_2 +a_1a_3 + \cdots + a_1a_n +a_2a_3 + \cdots + a_{n-1}a_n), \vdots p_n =a_1a_2 \cdots a_n.
\gamma(\|\mathbf{x}-\mathbf{y}\|)=\frac{\mbox{Cov}(\chi^{2}_{r}(\mathbf{x}),\chi^{2}_{r}(\mathbf{y}) )}{\mbox{Var}(\chi^{2}_{r}(\mathbf{0}))}=B^{2}(\|\mathbf{x}-\mathbf{y}\|),\beta =r/2.