harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{P}\left(f_{\mathcal{A}_h}(r_{n_1})=c_1,\ldots,f_{\mathcal{A}_h}(r_{n_k})=c_k\right)=\prod_{l=1}^k\mathbb{P}\left(f_{\mathcal{A}_h}(r_{n_l})=c_l\right).
( D\mathcal{G}(u)h)(r,\zeta)= \frac{\nu}{4\pi}\int_{-1}^1\int_0^{R_0}K(r,\zeta,r',\zeta')u_{\sharp}(r',\zeta')^{\nu-1}h(r',\zeta')r'^2dr'd\zeta' &-\frac{\nu}{2}\int_{-1}^1\int_0^{R_0}u_{\sharp}(r',\zeta')^{\nu-1}h(r',\zeta')r'dr'd\zeta',
Li_{s_{1},...,s_{d}}(\mu_{1},...,\mu_{d})=\sum_{k_{1}>\cdot\cdot\cdot>k_{d}>0}\frac{\mu_{1}^{k_{1}}\cdot\cdot\cdot\mu_{d}^{k_{d}}}{k_{1}^{s_{1}}\cdot\cdot\cdot k_{d}^{s_{d}}}
g(y_1, y_2) = \int_0^1 \int_0^1 x_1^{\nu}(1-x_1)^{m-n-\nu-1}x_2^{\nu}(1-x_2)^{m-n-\nu-1}f\left( \frac{y_1}{x_1}, \frac{y_2}{x_2}\right) \frac{dx_1}{x_1}\frac{dx_2}{x_2}.
\frac{\phi(x,s)}{s^r}= s^{p-r}\frac{\phi(x,s)}{s^{p}}\le s^{p-r}L_p\frac{\phi(x,t)}{t^{p}}= L_p\Big{(}\frac{s}{t}\Big{)}^{p-r}\frac{\phi(x,t)}{t^r}\le L_p\frac{\phi(x,t)}{t^r}.
vol(D_{1,a}\cap\mathfrak{D}(f,\sigma))=vol(D_{2,a}\cap\mathfrak{D}(f,\sigma)) =\begin{cases}2a a\le1/2, 1 1/2<a<1,\end{cases}vol(D_{3,a}\cap\mathfrak{D}(f,\sigma))=vol(D_{4,a}\cap\mathfrak{D}(f,\sigma)) =\begin{cases}0 a\le1/2, 2a-1 1/2<a<1.\end{cases}
\mathbb{E}[\|Z\|^{-2p}]\leq \eta_0+\int_{\eta_0}^\infty \mathbb{P}(\|Z\|^{-2p}>\eta)d\eta= \eta_0+ p\int_0^{\eta_0^{-1/p}}\epsilon^{-(p+1)}\mathbb{P}(\|Z\|^2<\epsilon)d\epsilon,
\varphi ( \omega , y ) = \frac{1}{1 6 \pi}\frac{1}{y}\int_{m}^{\infty}d t \Bigg ( \frac{1}{2 t - y}\frac{e^{\beta t}+ \omega}{e^{\beta t}- \omega}- \frac{1}{2 t + y}\frac{e^{\beta ( t + y )}+ \omega}{e^{\beta ( t + y )}- \omega}\Bigg ) + \mathrm{p . p . c .}
{\cal Z}_{_I}= \int D \overline{{\xi}}^{\prime}D \xi^{\prime}D \overline{{\chi}}D \chi \exp \left\{i \int d^{2}x \left[ - i \overline{{\chi}}\partial \! \! \! / \chi - \frac{1}{m}\overline{{\chi}}\partial \! \! \! / \partial \! \! \! / \chi + m \overline{{\xi}}^{\prime}\partial \! \! \! / \partial \! \! \! / \xi^{\prime}\right] \right\}.
\left \langle I + \begin{pmatrix}1 0 0 d \end{pmatrix}p, I + \begin{pmatrix}0 1 0 0 \end{pmatrix}p \right \rangle, \left \langle I + \begin{pmatrix}d 0 0 1 \end{pmatrix}I + \begin{pmatrix}0 1 0 0 \end{pmatrix}p \right \rangle.
-\widetilde{\Delta}\widetilde{J}+\frac{n-4}{2}\widetilde{J}^{2}=\frac{1}{2\left( n-1\right)}\left( -\widetilde{\Delta}\widetilde{R}+\frac{n-4}{4\left( n-1\right)}\widetilde{R}^{2}\right) .
0=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}r^{2}\sin{(\alpha_{j}-\alpha_{i})}}{(\sigma-\sigma(r^{2}\cos{(\alpha_{j}-\alpha_{i})}+\sigma-r^{2})^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\},
\mathrm{I}(\xi_{1},\xi_{n}|X,\eta_{p})=\int_{X}\mathrm{I}(\xi_{1}^{x},\xi_{n}^{x})d\eta_{p}(x)=\int_{Y}\int_{S_{y}}\mathrm{I}(\xi_{1}^{x},\xi_{n}^{x})d\eta_{p}^{y}(x)d\nu(y)
\varphi_{Z_t^H-Z_s^H}(u) ={\Bbb E}\bigl[ \exp \bigl( iu\bigl(Z_t^H-Z_s^H\bigr) \bigr)\bigr] = \exp \biggl( - \frac{u^2}{2}{\Bbb E}\bigl( Z_t^H- Z_s^H \bigr)^2 \biggr),
\left[C_{\gamma}(\mathcal{Z}_{m,n}^{\gamma}) \right](z) = \dfrac{(-1)^{n}(\gamma+1)_{m+n}(\gamma+m+2)_{n-1}(-m-1)_{n}}{(\gamma +1)_{n}(m-n+2)_{n-1}(m+1)}& \times z^{n-1}\overline{z}^{m}\left(1-|z|^2\right)^{\gamma+1}{_2F_1}\left( \begin{array}{c}-(n-1), - m -(\gamma+1)-m-(n-1) \end{array}\bigg |\frac{1}{|z|^2}\right).
\mathcal M \models |M|=|Y|=|X|=|M-X| = |M-Y| \mathcal M \models |M|=|Y|=|X|=|M-X| >|M-Y| \mathcal M \models |M|=|Y|=|X|>|M-X|\geq |M-Y| \mathcal M \models \omega\leq |X|=|Y| < |M| \mathcal M \models |X|=|Y|<\omega \leq |M| \mathcal M \models |X|=|Y|\leq|M|<\omega
I_{r}^{'}(x,t):=\int_{0}^{t}\int_{\partial\Omega_{r}^{'}}\big\langle\nabla_{x}\varepsilon_{n, a}(x-\xi^{'},b(t,\tau)),\nu(x_{0})\big\rangle\varphi(x_{0},\tau)a(\tau)dS_{\xi^{'}}d\tau,
\int_0^r E_1(x)E_1(r-x)\,dx = 2(\gamma+\ln r)e^{-r}+2\left(1-\gamma r-r\ln r\right)E_1(r)-r\left(\zeta(2)+(\gamma+\ln r)^2\right) &- 2 r \sum_{m=1}^\infty \frac{(-r)^m}{m!m^2},
{\mathcal A}_p ( (X_\lambda)_{\lambda \in \Lambda_{d,n}}) = \det_V \left(X_{(1,1)}f_V(p) \wedge \cdots \wedge X_{(j_*,1)}\cdots X_{(j_*,\kappa_{n}-1)}f_V(p) \right) \times \det_W\left(X_{(j_*+1,1)}\cdots X_{(j_*+1,\kappa_{n})}f_W(p) \wedge \cdots \wedge X_{(n,1)}\cdots X_{(n-1,\kappa_{n})}f_W(p)\right).
\Gamma_{k+1}(x^1,x^2,\lambda) = \log \int \exp \left(x^1 \sigma^1+ x^2 \sigma^2 +\lambda \sigma^1 \sigma^2 \right) d \nu^b (\sigma^1) d \nu^b(\sigma^2), \Gamma_{p}(x^1,x^2,\lambda) =\frac{1}{n_{p}}\log \mathbb{E}\exp n_{p}\Gamma_{p+1}(x^1+y_{p}^1,x^2+y_{p}^2,\lambda), \ 1 \leq p\leq k.
\idotsint \left( \prod_{i=1}^{n}f(x_{i}) \right) h(x_{1},\dotsc,x_{n}) \, dx_{1}\dotsm \, dx_{n}\geq \idotsint \left( \prod_{i=1}^{n}f^{*}(x_{i}) \right) h(x_{1},\dotsc,x_{n}) \, dx_{1}\dotsm \, dx_{n}.
S_{0}= - N \int d \tilde{t}\, S T r \left( - \frac{1}{2}\partial_{\tilde{t}}\tilde{\Phi}\partial_{\tilde{t}}\tilde{\Phi}+ \frac{1}{4}[ \tilde{\Phi}, \tilde{\Phi}]^{2}+{\frac{1}{2}}\tilde{\Theta}\dot{\tilde{\Theta}}-{\frac{1}{2}}\tilde{\Theta}\gamma^{i}[ \tilde{\Phi}_{i}, \tilde{\Theta}] \right)
\langle \delta j^{\mu}( X ) \rangle \simeq \frac{1}{2}\int \frac{d^{\, 3}p}{( 2 \pi )^{3}}\left[ \hat{\cal P}_{+}^{\mu}Z_{+}\frac{- i \Sigma_{o f f}^{( + )}}{| I m \Sigma_{R}^{( + )}|}- \hat{\cal P}_{-}^{\mu}Z_{-}\frac{- i \Sigma_{o f f}^{( - )}}{| I m \Sigma_{R}^{( - )}|}\right] ,
\psi_{N}^{-}= \frac{1}{[ 2 r ( r + r_{3}) ]^{\frac{1}{2}}}\left( \begin{array}{c}{- r_{1}+ i r_{2}}\\{r + r_{3}}\\ \end{array}\right) = \left( \begin{array}{c}{- e^{- i \phi}\sin \frac{\theta}{2}}\\{\cos \frac{\theta}{2}}\\ \end{array}\right)
E(t):=E(t,u(\cdot))=\frac 1 2 \Vert u_t\Vert_H^2+\frac 1 2\Vert A_1^{\frac 12}u\Vert_H^2- G(u)+\frac 12 \int_{t-\tau}^t\vert k\vert \Vert B^*u_t(s)\Vert_{W_2}^2 ds\,.
\partial^{\alpha}_t f(t) =\frac{1}{\Gamma(k-\alpha)}\int^t_0 (t-s)^{k-\alpha-1}f^{(n)}(s)ds &=\frac{1}{\Gamma(k-\alpha)}\frac{d}{dt}\int^t_0 (t-s)^{k-\alpha-1}\left[f^{(k-1)}(s)-f^{(k-1)}(0)\right] ds.
\int_{B^c}\dfrac{dy}{|x-y|^{n+sp}}\leq \int_{B_d(x)^c}\dfrac{dy}{|x-y|^{n+sp}}begin{align*}3mm] =C(n)\int_d^\infty \dfrac{d\rho}{\rho^{1+sp}}=C(n)\dfrac{d^{-sp}}{sp}.
- r^{2}\, d t^{2}+{\frac{d r^{2}}{r^{2}}}+ \cos^{2}\theta \, d \psi^{2}+ d \theta^{2}+ \sin^{2}\theta \, d \Omega_{3}^{2}+ r^{2}\, d y_{1}^{2}+{\frac{r^{2}}{1 + a^{4}r^{4}}}\, d y_{2}^{2}
(4)F_{EW}^{2}M_{\pi T}^{2}\approx\frac{g_{ETC}^{2}\langle\overline{T}T\overline{T}T\rangle_{ETC}}{M_{ETC}^{2}}\approx\frac{16\pi^{2}F_{EW}^{6}}{\Lambda_{ETC}^{2}}.
\begin{aligned}y_{t+L-1}= \sum_{k=0}^{L-2}a_ky_{t+k}+\sum_{k=0}^{L-1}b_ku_{t+k}, y_{t+L-1}= \sum_{k=0}^{L-2}\tilde{a}_ky_{t+k}+\sum_{k=0}^{L-1}\tilde{b}_ku_{t+k}.\end{aligned}
A ( x ) = \frac{1}{\left( 2 \pi \right)^{\frac{3}{2}}}\int \left( e^{- i p x}a ( p ) + e^{i p x}a^{*}( p ) \right) \frac{d^{3}p}{2 \omega}= A^{\left( - \right)}( x ) + A^{( + )}( x )
\min_{0{}\leq{}i{}\leq{}N}\big\{\|\nabla f(x_i)\|^2\big\}{}\leq{}\frac{2L\,\big[f(x_0)-f_*\big]}{1 + \sum\limits_{i=0}^{N-1}\left[2 h_i - \frac{h_i^2}{2}\max(1, h_i) \right]}
\dot{D}k =\frac{1}{J}\partial_t(\dot{D}h)+\frac{a_{01}}{J}\dot{D}\check{D}h+ &+\frac{1}{J}(\check{D}a_{01}+a_{00})\dot{D}h+\frac{\dot{D}a_{00}}{J}h-\frac{\dot{D}J}{J}h-\frac{\dot{D}J}{J}k-\frac{1}{J}\dot{D}g_1.
\mathcal{I}_{2,1,n}= \frac{\pi_1^n}{\mu}\left( \frac{N_0}{\Omega_{\mathrm{ST-SR}}P_{\mathrm{ST}}}+ \frac{N_0}{\Omega_{\mathrm{SR-SD}}P_{\mathrm{SR}}}\right) & \times \underbrace{\int_{0}^{\theta_{\mathrm{S}}}\frac{\exp({-Sz})}{\left( z + \pi_1 \right)^n (z + \tau)}}_{\mathcal{J}_{2,1,n}}\mathrm{d}z.
{f^{\phi}}_{0 \frac{1}{2}0}={f^{\phi}}_{1 \frac{1}{2}1}= 2^{( \pi - 2 \rho ) / 4 \pi}{f^{\phi}}_{\frac{1}{2}0 \frac{1}{2}}= 2^{( \pi - 2 \rho ) / 4 \pi}{f^{\phi}}_{\frac{1}{2}1 \frac{1}{2}}= \sqrt{K ( i \rho ) 2^{( \pi - \rho ) / 2 \pi}\cos \left( \frac{\rho - \pi}{4}\right)}
\begin{aligned}\int_M \left| \varphi_{i_j}- \varphi_{i_k}\right|^{q^*}\omega^n \leq \Vert \varphi_{i_j}- \varphi_{i_k}\Vert^{q^* - 1}_{L^\infty}\int_M \left| \varphi_{i_j}- \varphi_{i_k}\right| \omega^n &\leq C \int_M \left| \varphi_{i_j}- \varphi_{i_k}\right| \omega^n &\to 0 (j,k \to \infty) ,\end{aligned}
\left\{\mathrm{e}^{- \frac{A}{2}}\not \hspace{- 0 . 1 c m}{\partial}_{x}+ \Gamma^{r}\left( \partial_{r}+{\frac{D_{1}}{4}}A^{\prime}+{\frac{D_{2}}{4}}B^{\prime}\right) + \mathrm{e}^{-{\frac{B}{2}}}\not \hspace{- 0 . 1 c m}{\Delta}_{y}\right\}\Psi = 0 ,
\mathcal{B}\big( [\![0,T-1]\!]\times \mathcal{X}\times \mathcal{P}(\mathcal{X})^{T}\big) =\mathcal{B}( [\![0,T-1]\!]) \otimes \mathcal{B}( \mathcal{X})\otimes \mathcal{B}(\mathcal{P}(\mathcal{X})^{T}) &= 2^{[\![0,T-1]\!]}\otimes 2^{\mathcal X}\otimes \mathcal{B}( \mathcal{P(X)})^{T},
\bar R_{SD}-\bar R_{\rm conv}=\frac{1}{2}+ \frac{M}{2}\sum_{k=0}^{M-1}{M-1\choose k}\frac{(-1)^k\log_2(1+k)}{(1+k)}&-M \sum_{k=0}^{2M-1}{2M-1\choose k}\frac{(-1)^k\log_2(1+k)}{(1+k)}.
\pi_0^*(\pi(g_1)) = \Big(\tbinom{a+1}{2}+ a(\epsilon_b-1) \Big)\pi(S_1) +\big( a+ \epsilon_b -1 \big) \sum\limits_{i=2}^{b}(-1)^{i-1}\pi(S_i) & =\Big(\tbinom{a+1}{2}+ a(\epsilon_b-1) + \big( a+ \epsilon_b -1 \big)[\frac{b}{2}] \Big) \pi(S_1).
\mathrm{Ker\,}\widetilde D_{b,m}=\mathrm{Ker\,}\widetilde{\Box}^+_{b,m}\subset\Omega^{0,+}_m(X,E), \mathrm{Ker\,}\widetilde D^*_{b,m}=\mathrm{Ker\,}\widetilde{\Box}^-_{b,m}\subset\Omega^{0,-}_m(X,E).
\Big| \frac{1}{2 \varepsilon n^2}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \sum_{r=1}^{\varepsilon n}\int_{\Omega_1(x)}\overleftarrow{\eta}^{\ell}(x) [\eta(x+1)-\eta(x+1+r) ] [ f ( \eta_{1,x,r}) - f ( \eta_{2,x,r}) ] d\nu_{b}\Big|.
C = \begin{pmatrix}1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 \end{pmatrix}, D = \begin{pmatrix}0 0 0 0 d_1 d_2 0 0 0 0 0 0 0 0 d_3 d_4 \end{pmatrix}.
R_k =\frac 1 2 s_k=\frac 1 2 s=\frac{f^\#(c)}{6\varphi(f^\#(c))}=\frac{1}{6\varrho\varphi(1/\varrho)}=\frac{1}{6\varrho_k\varphi(1/\varrho_k)}.
\Lambda_{i}^{2}=\max_{\left\| t\right\| =1}\mbox{Var}(t_{1}\,\delta_{i,n}+t_{2}\,u_{i,n})\leq 2\,\mbox{Var}(\delta_{i,n})+2\,\mbox{Var}(u_{i,n}),i=1,\ldots ,s_{n},
\dot{r}= p_{r}\ , \ \dot{p}_{r}= \frac{1}{r^{3}}p_{\theta}^{2}- \omega^{2}r \ ; \ \dot{\theta}= \left( \frac{1}{r^{2}}+ g^{2}\right) p_{\theta}\ , d o t{p}_{\theta}= 0 \ ,
(2-q)c_0 m + (p-q)c_0^{p-1}a = mc_0 \left\{(2-q) - (p-q)c_0^{p-2}k \right\}= mc_0 \left\{(2-q) - (p-q) \frac{(2-q)\int_\Omega m}{(p-q)(-\int_\Omega a)}k \right\}= 0.
\delta S^{(2)}=\int^{t_{2}}_{t_{1}}\sum_{i=1}^{n}\left[\frac{\partial L^{(2)}}{\partial \dot{q}^{i}}-\frac{d}{dt}\left(\frac{\partial L^{(2)}}{\partial \ddot{q}^{i}}\right)\right]\delta \dot{q}^{i}dt+\left[\sum_{i=1}^{n}\left(\frac{\partial L^{(2)}}{\partial \dot{q}^{i}}\right)\delta\dot{q}^{i}\right]^{t_{2}}_{t_{1}}.
\begin{array}{lll}g([W, X_2], X_3)_o+g([W, X_3], X_2)_o+(a_3-a_2+b_3-b_2)g(W, [X_2, X_3])_o\ & \ \ +(b_2a_3-b_3a_2)g(W, [X_2, X_3])_o=0.\end{array}
S = \sum_{\mu \nu}\frac{\beta_{\mu \nu}}{2 N}\sum_{\{p_{\sigma}\mu \nu \}}R e T r ( U_{p_{\sigma}\mu \nu}+ U_{p_{\sigma}\mu \nu}^{+}) = \sum_{\mu \nu}\frac{\beta_{\mu \nu}}{2 N}\sum_{\{p_{\sigma}\mu \nu \}}T r ( U_{p_{\sigma}\mu \nu}+ U_{p_{\sigma}\mu \nu}^{+}) ,
\hat X_t = e^{tA}\,\hat x_0 + \int_0^t e^{(t-s)A}\,b(\hat X,\hat I_s)\,ds + \int_0^t e^{(t-s)A}\,\sigma(\hat X,\hat I_s)\,d\hat W_s &\ + \, \int_0^t\int_{U \setminus \{0\}}e^{(t-s)A}\,\gamma(\hat X, \hat I_{s-}, z)\, (\hat\pi(ds\,dz)-\lambda_\pi(dz)\,ds), 0\leq t\leq T.
\int_{\Omega}|\partial_{jl}(K_{i}\ast\rho)|^{p}=\int_{\Omega}|\partial_{jl}K_{i}\ast\rho|^{p}\leq\left\Vert \rho\right\Vert_{L^{p}(\Omega)}^{p}\left\Vert \partial_{jl}K_{i}\right\Vert_{L^{\frac{p}{p-1}}(\mathbb{R}^{d})}^{p-1}|\Omega| & =C\left\Vert \rho\right\Vert_{L^{p}(\Omega)}^{p}~\forall j\in\{1,...,d\}
\phi(-q^{3/2}x_3 x_1 x_2) =\sum \frac{q^{\frac{n_{1,4}^{2}}{2}+n_{1,4}}(x_3 x_1 x_2)^{n_{1,4}}}{(q)_{n_{1,4}}}&=\sum \frac{q^{-\frac{n_{1,4}^{2}}{2}}(x_2)^{n_{1,4}}(x_3)^{n_{1,4}}(x_1)^{n_{1,4}}}{(q)_{n_{1,4}}}.
S_1(a,\,m_1,\,d_1)+S_1(0,\,m_2,\,d_2) =S_1(b,\,n,\,d_1), S_2(a,\,m_1,\,d_1)+S_2(0,\,m_2,\,d_2) =S_2(b,\,n,\,d_1),
\Vert f\Vert_{U^d[N]}^{2^d}= \frac{1}{\vert R\vert}\sum\limits_{x,h_1,\dots,h_d}\prod\limits_{\boldsymbol{\omega}\in \{0,1\}^d}\mathcal{C}^{\vert \boldsymbol{\omega}\vert}f1_{[N]}(x+\mathbf{h}\cdot\boldsymbol{\omega}),
{{\bf Q}^{3 \nu}}_{Z}= \left[ - m_{Z}^{2}- \frac{g^{2}\sigma^{2}}{4 \cos \theta_{W}}-- \frac{g \sigma m_{Z}}{\cos \theta_{W}}+ \frac{i g}{2 \cos \theta_{W}}\left( \partial_{\mu}\sigma \right) I^{\mu}\right] Z^{\nu}+ \beta X_{0}( \partial_{\mu}K^{3 \mu}) A^{\nu}.
A ( x_{1}) B ( x_{2}) = \cdots + g_{A B C}\, \frac{1}{( x_{1 2}^{2})^{\frac{1}{2}( \eta_{A}+ \eta_{B}- \eta_{C})}}C^{\eta_{C}, \eta_{A}- \eta_{B}}( x_{1 2}, \partial_{2}) C ( x_{1}) + \cdots{}.
\lim_{\epsilon \downarrow 0}\frac{1}{2 \pi i}\int_{\mathbb{R}}\frac{\mathrm{e}^{i s(\xi-2x)}}{s- (z+i \epsilon)}d s=\lim_{\epsilon \downarrow0}\begin{cases}\mathrm{e}^{i(z+i \epsilon)(\xi-2x)}, \xi-2x>0 0, \xi-2x<0\end{cases}begin{align*}6pt] =\chi(\xi-2x)e^{iz(\xi-2x)},
\frac{1}{3 \cdot 4 ! \cdot 2 !}( - 1 8 ) ( \frac{2}{9}) \int d \tau_{1}\dots d \tau_{4}F ( \tau_{1}- \tau_{2}) F ( \tau_{3}- \tau_{4}) [ \Theta ( \tau_{1}\geq \tau_{3}\geq \tau_{2}\geq \tau_{4}) + \dots ] .
\Pr\Big ( \big ||A_e| - \mathbb{E}(|A_e|) \big | > a \mathbb{E}(|A_e|) \Big ) \leq 2e^{-a^2\mathbb{E}(|A_e|)/3}\leq 2e^{-(1+o(1))a^2t/3\ell}=o( 1/n^2).
c_1^{11}(A) = \frac{\int_Y a}{c-\int_Y a}c_1^{22}(A) = \frac{\int_Y a}{c}(Q_1+Q_2), c_2^{11}(A) = \frac{\int_Y a}{c-\int_Y a}c_2^{22}(A) = \frac{\int_Y a}{c}(Q_2-Q_1),
\begin{array}{l l}{{u ~ = ~ \cosh \mathrm{\boldmath ~ \r h o ~}\cos \mathrm{\boldmath ~ t ~}, ~ ~}}&{{v ~ = ~ \cosh \mathrm{\boldmath ~ \r h o ~}\sin \mathrm{\boldmath ~ t ~},}}\\{{w ~ = ~ \sinh \mathrm{\boldmath ~ \r h o ~}\cos \mathrm{\boldmath ~ \tilde{\t h e t a}~}, ~ ~}}&{{y ~ = ~ \sinh \mathrm{\boldmath ~ \r h o ~}\sin \mathrm{\boldmath ~ \tilde{\t h e t a}~}.}}\\ \end{array}
G^{(2)}_{\mathcal R}(x,y)= &\frac{ie^{ik_{+}\vert x \vert}}{4\pi}\left\{\int^{+\infty}_{-\infty}\left[g(s_b)\sqrt{s-s_b}+ \frac{g(s_b)-g(0)}{s_b}(s-s_b)^{\frac 32}\right ]e^{-\vert x \vert s^2}ds+ G^{(2)}_{\mathcal R,1}(x,y)\right\},
x = y + \sum_{k=2}^{N}X_{k}(v,\varepsilon) y^{k}, u = v + \varepsilon \sum_{k=1}^{N}U_{k}(v,\varepsilon) y^{k}, (y, v, \varepsilon) \in B_{\delta}^{2n}\times \mathcal{A}(N,\nu) \times (-\varepsilon_{0},\varepsilon_{0}),
\int_0^t u(s,x)ds = \sum_{k=1}^\infty \int_{\mathbb{R}^{d}}\int_{0}^{t}I_s^{\alpha-\beta}p(s,x-y)\int_0^{t-s}g^k(r,y)dw^k_r ds dy &=\sum_{k=1}^\infty \int_{\mathbb{R}^{d}}\int_{0}^{t}p(t-s,x-y)I_s^{\alpha-\beta}\int_0^s g^{k}(r,y)dw_{r}^{k}dy ds &= \int_{0}^{t}\int_{\mathbb{R}^{d}}p(t-s,x-y)w(s,y)dy ds.
\left \{\alpha \in X_n^k: \alpha(0)= 0, \P(\alpha) = d \right \}= \left \{\alpha \in X_n^k : \alpha \ \mbox{is $d$-additive}\right \}\setminus \bigcup_{p \in PD_d}\left \{\alpha \in X_n^k : \alpha \ \mbox{is $\frac{d}{p}$-additive}\right \}
(t\partial_z +z^{-1}(D^+_{\jmath_1}D^-_{\jmath_1}-2\jmath_0D^0_{\jmath_1})) -(\tilde{J}^+(z)D^-_{\jmath_1}+2\tilde{J}^0(z)D^0_{\jmath_1}+\tilde{J}^-(z)D^+_{\jmath_1})
I_{T O P}= \mathrm{T r}\int_{X}\left( \tau R^{+}\wedge R^{+}+ \overline{{\tau}}R^{-}\wedge R^{-}\right) = \mathrm{T r}\int_{X}\left( \tau R^{+}\wedge R^{+}+ \overline{{\tau}}\overline{{{R^{+}}}}\wedge \overline{{{R^{+}}}}\right) ,
\begin{bmatrix}y_1 \vdots y_N\end{bmatrix}=\begin{bmatrix}M_{1,1}\cdots M_{1,{2N}}\vdots \vdots \vdots M_{N,1}\cdots M_{N,{2N}}\end{bmatrix}\begin{bmatrix}x_1 \vdots x_{2N}\end{bmatrix}
R_\mathrm{bc}^{\rm J}= \log{|\mathbf{H}\mathbf{Q}_\mathrm{s}\mathbf{H}^\mathrm{H}+ \mathbf{D}+\mathbf{\Theta}_\mathrm{z}|}-\log{|\mathbf{D}+\mathbf{\Theta}_\mathrm{z}|}
\begin{aligned}C_{\leq m,\ell}=\sup_{\xi'\in Q_\ell^*}\int_{3Q_\ell^*}\frac{4\pi\,d\xi}{|\xi-\xi'|^2}=\frac{4\pi^2}{\ell}\int_{Q}\frac{d\xi}{|\xi|^2}.\end{aligned}
A^{-1}\omega_2 = (1,2,2,3,2,1) A^{-1}\omega_2 - \alpha_2 = (1,2,1,3,2,1) A^{-1}\omega_2 - \alpha_2 - \alpha_4 = (1,2,1,2,2,1) A^{-1}\omega_2 - \alpha_2 - \alpha_4 - \alpha_5 = (1,2,1,2,1,1) A^{-1}\omega_2 - \alpha_2 - \alpha_4 - \alpha_5 - \alpha_6 = (1,2,1,2,1,0).
\begin{array}{l}{e_{a}\circ_{X}( e_{b}\circ_{X}( e_{c}\circ_{X}e_{d}) ) = \pm e_{7}}\\{\mathrm{i f e x a c t l y o n e o f}a , b , c , d = 0 \mathrm{o r}h = 7 ,}\\{e_{a}( e_{b}( e_{c}e_{d}) ) = \pm 1 \mathrm{o t h e r w i s e .}}\\ \end{array}
z = \frac{\Lambda^{3 N_{c}- N_{f}}S^{N_{f}- N_{c}}}{\mathrm{d e t}M - ( \tilde{B}^{A_{1}A_{2}\ldots A_{N_{f}- N_{c}}}M_{A_{1}}{}^{B_{1}}M_{A_{2}}{}^{B_{2}}\ldots M_{A_{N_{f}- N_{c}}}{}^{B_{N_{f}- N_{c}}}B_{B_{1}B_{2}\ldots B_{N_{f}- N_{c}}})}.
\int_0^k \int_0^1 \left[G(s,y)-1\right]^2\, dy ds = \int_0^k \int_0^1 G^2(s,y) \, dy ds -k &=\int_0^k \sum_{l\ge 1}\exp\left(-(2\pi l)^2 s \right) \, ds &\le C .
\begin{aligned}\begin{cases}Z(\sigma_k,k)=\inf_{u_k\in U_K}\mathbb{E}^\dagger\left[ Z(\Sigma^*(u_k,y_{k+1})\sigma_k, k+1) \right], Z(\sigma_K,K)= \int_{\mathbb{R}^n}\sigma_K(z)\left(\mu\Phi(z)\right)dz. \end{cases}\end{aligned}
\left\vert T(f)(x) \right\vert\leq\sum\limits_{j}\vert\lambda_{j}\vert\vert T(a_{j})(x)\vert+\sum\limits_{j}\vert\mu_{j}\vert\vert T(b_{j})(x)\vert=:\uppercase\expandafter{\romannumeral1}+\uppercase\expandafter{\romannumeral2}.
\frac{1}{2^{2l+1}}~t\left( 2n,2l\right) =\frac{1}{2}\sum_{m=1}^{2j+1}\left(V^{-1}\left[ j\right] \right)_{2l+1,m}\prod\limits_{k=0}^{n-1}\left(\left( j+1-m\right)^{2}-k^{2}\right) \ .
\begin{array}{l}\mbox{a)}\boldsymbol{u}_{\varepsilon ,\lambda}=0\mbox{on}\partial \Omega; \mbox{b)}f_{\varepsilon ,\lambda}(t,x,v)=f_{\varepsilon ,\lambda}(t,x,v^{\ast})\mbox{for}x\in \partial \Omega \mbox{with}v\cdot \nu(x)<0.\end{array}
\zeta_{r,d}^\bullet(\underbar{{\bf s}},A_r)=\left(\sum_{m_1=0}^\infty\cdots\sum_{m_d=0}^\infty\right)'\left(\sum_{m_{d+1}=1}^\infty\cdots\sum_{m_r=1}^\infty\right)\prod_{1\le i<j\le r+1}(m_i+\cdots+m_{j-1})^{-s(i,j)}.
\Phi_t^1(y_1^0,y_2)\in \bigg[\max\left\{\frac{1}{2}Y_2\delta,\delta\right\}D(t)^{-1}+\frac{y_2}{2\delta}H(t)D(t)^{-1},\frac{L\delta}{2}D(t)^{-1}+\frac{y_2}{2\delta}H(t)D(t)^{-1}\bigg]=[Lo,Up].
\varepsilon_2=\frac{(\tilde{\varepsilon}_2^2+2)G_1^2}{\sigma},~\varepsilon_3=2G_2^2\tilde{\varepsilon}_3,~\varepsilon_4=\frac{4\max\{1,G_2^2\tilde{\varepsilon}_4\}}{\min\{1,\frac{\sigma}{2\gamma_0}\}},~\varepsilon_{5}=\frac{n\tilde{\varepsilon}_2G_1G_2}{\sigma},~\varepsilon_{6}=\frac{\tilde{\varepsilon}_2G_2^2\gamma_{0}+\sigma}{\sigma}.
\frac{1}{2}\sum_{n \neq 0}\left\{\frac{\omega_{n}}{2}\chi_{n}^{2}+ \frac{1}{2 \omega_{n}}[ ( \partial_{\, z}+ F^{\prime}) \chi_{n}]^{2}- \frac{\omega_{n}}{2}\chi_{n}^{2}- \frac{1}{2 \omega_{n}}[ ( \partial_{\, z}+ F^{\prime}) \chi_{n}]^{2}\right\}\equiv 0 \, ,
V=\sum_{i=1}^{N}V_{1}(\vec{r}_{i})+\sum_{i,j=1}^{N}V_{2}(\vec{r}_{i},\vec{r}_{j})+\sum_{i,j,k=1}^{N}V_{3}(\vec{r}_{i},\vec{r}_{j},\vec{r}_{k})+\cdot\cdot\cdot
\int_{\mathbb{R}^{n-1}}[\phi]_\Gamma(\mathbf{c}\cdot\nabla)\eta~d\mathbf{x}=-\int_{\mathbb{R}^{n-1}}\eta(\mathbf{c}\cdot\nabla)[\phi]_\Gamma~d\mathbf{x}=-\int_{\mathbb{R}^{n-1}}\eta\mathbf{c}\cdot([\nabla_\mathbf{x}\phi]_\Gamma+[\phi_y]_\Gamma\nabla\eta)~d\mathbf{x}.
n = 2k+1 \implies \mathcal{T}(t) = v_0T_1+v_1\bigl(T_2-T_1\bigr) +\dots + v_0\bigl(T_{2k+1}-T_{2k}\bigr)+v_1\bigl(t-T_{2k+1}\bigr) &=(v_0-v_1)\bigl(T_1-T_2+T_3-\dots -T_{2k}+T_{2k+1}\bigr) +v_1t
\lim_{t\to+\infty}\left\|({P_*}(4bt,k)-\Pi(k)\cdot \chi_{E_s^c})\int_{2at}^{2bt}\frac{e^{iu^2/(4t)}}{\sqrt t}\widehat f_+(u/(2t))e^{-iku}du\right\|_{2,\sigma}= 0\,.
d s_{4}^{2}= -{\frac{1}{\sqrt{-{\frac{1}{6}}H^{0}C_{a b c}H^{a}H^{b}H^{c}}}}d t^{2}+ \sqrt{-{\frac{1}{6}}H^{0}C_{a b c}H^{a}H^{b}H^{c}}d \vec{x}^{2}.
\Delta(K; \alpha, \beta) = \Big[ \frac{(K^{\alpha + \beta}-1)^3}{\alpha + \beta}\Big]^2 - \frac{(K^{2\alpha}- 1)^3}{2\alpha}\cdot \frac{(K^{2\beta}- 1)^3}{2\beta}\leq \frac{K^{6\alpha+6\beta}}{(\alpha +\beta)^2}- \frac{K^{6\alpha}}{256|\alpha\beta|}.
\textnormal{Dom}(H)=\left\{\sum_{\nu\in\mathbb{N}_0^n}\langle f,\phi_\nu \rangle_{L^2}\phi_\nu: \sum_{\nu\in\mathbb{N}_0^n}|\lambda_\nu\langle f,\phi_\nu \rangle_{L^2}|^2 <\infty \right\}.
\vec{a}^{\vec{x}}a_j^{-k}= s_{1,2}^{x_1}s_{1,3}^{x_2}\cdots s_{1,j+1}^{x_j}\cdots s_{2,m+2}^{x_{m+1}}\cdots s_{1,m+2}^{x_{2m+1}}\cdot s_{1,j+1}^{-k}&=s_{1,2}^{y_1}s_{1,3}^{x_2}\cdots s_{1,j+1}^{x_j-k}\cdots s_{2,m+2}^{x_{m+1}}\cdots s_{1,m+2}^{x_{2m+1}+kx_{m+j}}.
\mathrm{\boldmath ~ \Psi ~ \unboldmath}= \left( \begin{array}{c}{\Psi_{+ 1}( x )}{\Psi_{- 1}( x )}\end{array}\right) \; , \; \; \Psi_{\zeta}( x ) = \left( \begin{array}{c}{\chi_{\zeta}( x )}{\varphi_{\zeta}( x )}\end{array}\right) \; ,
A_\mu=\mp{2i\over g}\,(H_1\,{v_\mu^1\over\rho_1}+g\,\kappa\,E_{13}^\pm\,R^1_\mu)\mp\,{2i\over g}\,(H_2\,{v_\mu^2\over\rho_2}+g\,\kappa\,E_{23}^\pm\,R^2_\mu).
F(0) = (k(\eta \sinh 2 + 2 \gamma \sinh 1) - 4(1-\cosh 1)\left(\gamma + \eta \cosh 1\right)^2)^2-32 \left(\gamma + \eta \cosh 1\right)^4\intertext{and}F\left(\frac{\pi}{2}\right) = (k(\eta \sin 2 + 2 \gamma \sin 1) - 4(\cos 1 - 1 )(\gamma +\eta \cos 1)^{2})^{2}- 32(\gamma +\eta \cos 1)^{4}.
{}D=\frac{2^{2^{n-1}}}{\# V},\;\;\;\; V:=\Big\{(e_1,e_2,\dots e_{2^{n-1}})\in\mathbb{F}_2^{2^{n-1}}:\; \prod_i\delta_i^{e_i}\in (K_{n-1}^*)^2\Big\}.
K\ni G(\mathcal{X}) \perp \mathcal{Y}\in K^{\circ}\Longleftrightarrow \mathcal{Y}\in\mathcal{N}_{K}(G(\mathcal{X}))\Longleftrightarrow G(\mathcal{X})-\Pi_{K}(G(\mathcal{X})+\mathcal{Y})=0,
\hskip -3mm (-q;q)_N^3\sum_{n=0}^{N}F(n,0) begin{align*}5pt] =(-q;q)_N^3\sum_{k=1}^{N}G\left(N+1,k\right) begin{align*}5pt] =(-1)^N\sum_{k=1}^{N}[N+1]{2N+2\brack N+1}{2N-2k+2\brack N}{2N-2k+2\brack N-k+1}\frac{(-q;q)_N^2 q^{N-2k+2}}{(1+q^{N+1})(-q;q)_{N-k+1}^2}.
\bar{z}_\nu(t,\omega) =\bar{z}_\nu(T_1,\omega)+\frac{1}{N}\sum\limits_{j=1}^{N}\int_{T_1}^{t}e^{-O_s^{(j)}(\omega)}f^{(j)}(e^{O_s^{(j)}(\omega)}\bar{x}_\nu^{(j)}(s,\omega))ds &+\frac{1}{N}\sum\limits_{j=1}^{N}\int_{T_1}^{t}O_s^{(j)}(\omega)\bar{x}_\nu^{(j)}(s,\omega)ds
(f \bullet g) \bullet h = \sum^{m+n-1}_{j=1}\sum^{m}_{i=1}(-1)^{(j -1)(p -1)+(i -1)(n-1)}(f \bullet_i g) \bullet_j h f \bullet (g \bullet h) = \sum^{m}_{\xi=1}\sum^{n}_{\omega=1}(-1)^{(\xi-1)(n+p-2)+(\omega-1)(p-1)}f \bullet_{\xi}(g \bullet_{\omega}h)
\Phi(z)=\int_{\Omega_{2}}(Tf_{z})g_{z}d\mu_{2}=\sum_{j=1}^{m}\sum_{k=1}^{n}|a_{j}|^{\frac{u(z)}{u(\theta)}}|b_{k}|^{\frac{1-v(z)}{1-v(\theta)}}\gamma_{j,k}
\sum_{k=0}^\infty \binom{2s}{k}(-1)^{k}(k+w)!\frac{\Gamma (p+q-2 m)}{\Gamma (p+q-2m+k+w+1)}=\frac{\Gamma (w+1) \Gamma (p+q-2 m+2 s)}{\Gamma (p+q-2 m+2 s+w+1)}\,.
f=z^3\left(\begin{matrix}\alpha^{6}1 \alpha^{5}\alpha^{6}\end{matrix}\right) + z^2 \left(\begin{matrix}\alpha^{3}\alpha^{2}\alpha^{2}\alpha^{6}\end{matrix}\right) + z\left(\begin{matrix}\alpha^{4}\alpha^{4}1 0\end{matrix}\right) + \left(\begin{matrix}0 0 \alpha^{6}1\end{matrix}\right)
\begin{aligned}\|\nabla \delta_t r_h^1\|_0^2 =\|\nabla \delta_t(p(t_1)-\eta_p^1)\|_0^2\leq \|\nabla \delta_t\eta_p^1 \|_0^2+\|\nabla \delta_t p(t_1)\|_0^2 &\leq C (\Delta t)^2(\|\boldsymbol{u}\|_{W^{1,\infty}(t_0,T;W^{k_u+1,2})}^2+\|p\|_{W^{1,\infty}(t_0,T;W^{k_p+1,2})}^2).\end{aligned}
\Omega=\left\{\begin{array}[c]{c}\lambda_{1}Z_{1}^{\ast}+\lambda_{2}Z_{2}^{\ast}+\beta_{1}X_{1}^{\ast}+\beta_{2}X_{2}^{\ast}+\gamma Y^{\ast}\in\mathfrak{n}^{\ast}:\lambda_{1}\neq0 \left( \lambda_{1},\lambda_{2},\beta_{1},\beta_{2},\gamma\right) \in\mathbb{R}^{5}\end{array}\right\}
{\bf Model}\;{\bf1:}\;X_{j}=(Z_{j});\;{\bf Model}\;{\bf2:}\;X_{j}=(Z_{j},Z_{j-1}); &{\bf Model}\;{\bf3:}\;X_{j}=(Z_{j},Z_{j-1},Z_{j-2});\;{\bf Model}\;{\bf4:}\;X_{j}=(Z_{j},Z_{j-1},Z_{j-2},Z_{j-3}); &{\bf Model}\;{\bf5:}\;X_{j}=(Z_{j},Z_{j-1},Z_{j-2},Z_{j-3},Z_{j-4});\;\cdots.
\int_{Q}\textbf{curl}\, \boldsymbol{\tau}\cdot \boldsymbol{v}\, d\boldsymbol{x}\, dt = \int_{Q}\boldsymbol{\tau}\cdot \textbf{curl}\, \boldsymbol{v}\, d\boldsymbol{x}\, dt \forall \, \boldsymbol{v}\in \boldsymbol{H}^{\textbf{curl},0}_0(Q) \forall \,\boldsymbol{\tau}\in \boldsymbol{H}^{\textbf{curl},0}(Q),
\widetilde{H}_-^{\prime\mspace{2mu}\textsc{cs}}= \frac{1}{2}\sum_{j=1}^N (z_j \, \partial_{z_j})^2 + \frac{\beta}{2}\sum_{i<j}^N \frac{z_i + z_j}{z_i - z_j}\, \bigl(z_i \, \partial_{z_i}- z_j \, \partial_{z_j}\bigr) + \beta \sum_{i<j}^N \frac{z_i \, z_j}{z_{ij}\, z_{ji}}\, (1+P_{ij}) \, .
s_{i,l}^k = \tilde a_i^k+ l q_i^k,0\le l\le L^k_i, L^k_i = [1/ q_i^k],q_i^k = \theta_i^k \frac{\Delta(y_i^k)}{y_i^k},\theta_i^k = \left(m(y_i^k)\right)^{-4/\hat\alpha}, \tau_{i,n}^k = \tau(y_i^k) + n q_i^k,0\le |n| \le N^k_i,N^k_i = [\tau^*(y_i^k)/q_i^k].
\left| \frac{\partial ( a_{1}^{1}, a_{1}^{2}, a_{1}^{3}, a_{2}^{1}, a_{2}^{2}, a_{2}^{3})}{\partial ( r , \Theta , \Phi , \varphi^{1}, \varphi^{2}, \varphi^{3})}\right| = \frac{1}{8}\: r^{5}\sin ( 4 \Theta ) \; .