harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_0^{\frac{1}{2}}Q(x)\!\int_0^xK(x,t)\cos(2kt)dtdx\!+\!\int_{\frac{1}{2}}^bQ(x)\!\int_0^x\!K(x,t)\cos(2kt)dtdx &=\int_0^b\int_t^bK(x,t)Q(x)dx\cos(2kt)dt.
- 2P_1(1) P_3(1) \sideset{}{^+}\sum_{\chi(q)}\sum_n \frac{\chi(n)}{n^{\frac{1}{2}}}F(n) = -P_1(1) P_3(1)\sum_{w \mid q}\varphi(w) \mu(q/w)\sum_{\substack{n \equiv \pm 1 (w) (n,q)=1}}\frac{1}{n^{\frac{1}{2}}}F(n).
\|\Gamma(x)\|_\infty := \max_{1 \le i,j \le d}|\Gamma_{i,j}(x)| \le \max_{1 \le i \le d}\kappa ' (|x_{i-1}| + |x_{i+1}|)|x_i|+ \lambda' + \delta'|x_i| \le 2 \kappa' |x|_1^2 +\delta' |x|_1 +\lambda'.
0 = \int_0^T\left(i\frac{\partial u}{\partial t}(t) - H(t)u(t), v(t)\right)dt & = \int_0^T\left(u(t), i\frac{\partial v}{\partial t}(t) - H(t)v(t)\right)dt = \int_0^T \bigl(u(t), g(t)\bigr)dt,
\frac{1}{2 \pi}c^{( b )}: e^{\mp i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g N}}U_{1 b}\Phi^{( 1 )}( x )}:_{M^{( 1 )}}\; \prod_{I = 2}^{N}: e^{\mp i 2 \sqrt{\pi}U_{I b}\Phi^{( I )}( x )}:_{M^{( I )}}e^{\pm i \frac{\theta}{N}}\; ,
d^{S,T}_{[0,N)^2}(x, y) = \max_{\substack{0\leq m <N 0\leq n< N}}d\left(S^m T^n x, S^m T^n y\right), d^{S,T}_{[0,M)\times [0,N)}(x,y) = \max_{\substack{0\leq m <M 0\leq n <N}}d\left(S^m T^n x, S^m T^n y\right).
{{\cal A}}({\cal T}) \wedge{{\cal A}}({\cal N}) \; = \;{{\cal A}}({\widehat{\cal S}}) \; = \; 1 +{\frac{( 4 \pi^{2}\alpha^{\prime})^{2}}{1 9 2 \pi^{2}}}\; \mathrm{t r}\; ({\widehat{\cal R}}\wedge{\widehat{\cal R}}) + \cdots \ ,
\int_{C}{d^4}{\eta}D_{c}^{(4)}(\eta,{\eta}) = \int_{\partial C}d^3\eta K - 6a\int_{\partial C}d^3\eta KK' - \frac{2}{3}\frac{1}{a^2}\pi^{-\frac{1}{2}}\Gamma(\frac{5}{2})\int_{S_4}d^4\eta
\{\tilde \Theta_\alpha^\mu ,\tilde \Theta_\beta^\nu\}=\{\tilde \Theta_\alpha^\mu ,\tilde \Psi_l\}=\{\tilde \Theta_\alpha^\mu ,\tilde H\}= 0~~;~~\{\tilde \Psi_l ,\tilde \Psi_n\}\approx 0 ~;~\{\tilde \Psi_l ,\tilde H\}\approx 0.
\limsup_{N\rightarrow\infty}\left\Vert\left( f_N^{(s)}(0,Z_s)-f_0^{\otimes s}(Z_s)\right)\mathbf{1}_{Z_s \in \mathcal{K}_s \cap \mathcal{U}_s^{\eta(\varepsilon)}}\mathbf{1}_{E_s (Z_s) \leq R^2}\right\Vert_{L^\infty_{Z_s}}= 0
\int_{D}|\nabla u_\infty|^2\,dx\le \liminf_{n\to+\infty}\lambda_2(\Omega_n)\,,\int_{D}|\nabla v_\infty|^2\,dx\le \liminf_{n\to+\infty}\lambda_2(\Omega_n)\,, \int_{D}u_\infty^2\,dx=\int_{D}v_\infty^2\,dx=1\quad \int_{D}u_\infty v_\infty\,dx=0,
\begin{aligned}D(\lambda) =e^{2cz}\det\left[u_1(\lambda,z),u_2(\lambda,z),u_3(\lambda,z),u_4(\lambda,z)\right] &= -\det\left[\begin{array}{c c}\Omega(u_1(\lambda,z),u_3(\lambda,z)) \Omega(u_1(\lambda,z),u_4(\lambda,z)) \Omega(u_2(\lambda,z),u_3(\lambda,z)) \Omega(u_2(\lambda,z),u_4(\lambda,z))\end{array}\right]\end{aligned}.
Z_i = \frac{e^{\alpha}+1}{e^{\alpha}-1}W_i s(\theta_i - Y_i) \mbox{and}s(\theta_i - Y_i) \begin{cases}= \mathrm{sign}(\theta_i - Y_i), \theta_i \neq Y_i, \sim \mathrm{Unif}\{-1, 1\}, \theta_i = Y_i, \end{cases}
\begin{array}{c}{{\{\quad , \quad \}= i{\tilde{g}}^{A \bar{B}}\nabla_{A}\wedge{\bar{\nabla}}_{\bar{B}}+ g^{A \bar{B}}\frac{\partial}{\partial \theta^{A}}\wedge \frac{\partial}{\partial{\bar{\theta}}^{\bar{B}}}}}\\ \end{array}
\left.\left(\overline{\frac{\partial^{2}\left(g^{\star}_{ij}\left(Z,W\right) \right)}{\partial w_{aa}\partial w_{bb}}}+ \frac{\partial^{2}\left(g^{\star}_{ji}\left(Z,W\right) \right)}{\partial w_{aa}\partial w_{bb}}\right) \right\vert_{Z= O_{q\times N}\atop{W= O_{q\times q}}}=0,
\nu_i=\inf_{\scriptstyle\eta\in H^1_0(\Omega) ,\atop\scriptstyle\eta\perp span\left\{\psi_1,\psi_2,..,\psi_{i-1}\right\}}\frac{\int_\Omega|\nabla\eta|^2-\int_\Omega a(x)\eta^2}{\int_\Omega \frac{\eta^2}{|x|^2}}<0
\chi^+_1=-\int_r^\infty\frac{1}{s{U^+(s)}^2}\mathrm ds\int_0^s{\mathfrak h}_1^+(t){U^+(t)}^2\,\mathrm dt, begin{align*}1mm]\chi^-_1=-\int_r^\infty\frac{1}{s{U^-(s)}^2}\mathrm ds\int_0^s{\mathfrak h}_1^-(t){U^-(t)}^2\,\mathrm dt,
\pi(x, t) - c_{r,x_0}(t) = \frac{1}{3}|v(x, t)|^2 + \textnormal{p.v.}\int_{B_{3r}(x_0)}K(x - y) : (v \otimes v)(y, t) \,dy &+ \int_{\mathbb{R}^3 \setminus B_{3r}(x_0)}(K(x - y) - K(x_0 - y)) : (v \otimes v)(y, t) \,dy
R = - \frac{2}{C^{2}}\left( \frac{A^{\prime \prime}}{A}+ 3 \frac{B^{\prime \prime}}{B}+ 3 \frac{{B^{\prime}}^{2}}{B^{2}}+ 3 \frac{A^{\prime}B^{\prime}}{A B}- \frac{A^{\prime}C^{\prime}}{A C}- 3 \frac{B^{\prime}C^{\prime}}{B C}\right) = - h^{\prime \prime}( r ) - 6 \frac{h^{\prime}( r )}{r}- 6 \frac{h ( r )}{r^{2}}.
\frac{t W-\langle\nabla V_t,\nabla W\rangle}{W}(x)\leq\frac{n}{8t}-\frac{3|x|^2}{64t^2}+\frac{1}{16t^2}R\left(\frac{|x|^2}{2R}+\frac{R}{2}\right)=\frac{n}{8t}+\frac{R^2}{32t^2}-\frac{1}{64t^2}|x|^2,
\mathcal{J}(\mu\vert\nu) = \int \left|\nabla \log \rho\right|^2\,\rho\, d\nu=\int \frac{|\nabla \rho|^2}{\rho}\, d\nu,\mbox{if}\mu\ll \nu\, \mbox{and}\, \rho=\frac{d\mu}{d\nu},
\begin{array}{rcl}V_1^\kappa (p;p_o,v_o) = v_o - \kappa \, (p-p_o) \, F\left(\Pi_\kappa (p), \Pi_\kappa (p_o)\right) begin{align*}6pt] V_2^\kappa (p;p_o,v_o) = v_o + \kappa \, (p-p_o) \, F\left(\Pi_\kappa (p_o), \Pi_\kappa (p)\right) \end{array}
\begin{aligned}\dot{\xi}= \varepsilon \left[ F_{y}^{\prime}(y^{\ast}, 0) \xi + \Xi(\xi, \varphi, \varepsilon) \right], \dot{\varphi}= \bar{\omega}(y_{\ast}(\varepsilon), \varepsilon) + \varepsilon \Theta(\xi, \varphi, \varepsilon),\end{aligned}
G^{\Sigma^1}_{\Sigma^0, g}(x):= \begin{cases}\sup\{t\geq 0: \exp^g_x(s\nu^0(x))\in U^1 s\in [0, t] \}, \x\in U^1; \inf\{t\leq 0: \exp^g_x(s\nu^0(x))\in N\setminus U^1 s\in [t, 0]\}, \x\notin U^1. \end{cases}
\begin{aligned}\bar{h}_{ij}=h\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial x^{j}}\right)=h\left(\frac{\partial z^{k}}{\partial x^i}\frac{\partial}{\partial z^{k}},\frac{\partial z^{l}}{\partial x^j}\frac{\partial}{\partial z^{l}}\right)=\frac{\partial z^{k}}{\partial x^i}\frac{\partial z^{l}}{\partial x^j}h_{kl}.\end{aligned}
\exp \left[ L_{n}( x_{1}, x_{2}) \right] = \frac{2 r^{2}}{r_{+}^{2}}\left\{\cosh \left[ r_{+}( \Delta \phi + 2 \pi n ) \right] - \cosh ( r_{+}\Delta t ) \vphantom{A_{A}^{A}}\right\}+ O \left( 1 \right) \ \ ,
A_N^{-1}(w_N) = \prod_{i=0}^{N-1}A_1^{-1}(w_i) = \prod_{j=0}^{N-1-i}A_1^{-1}(w_j) \, \prod_{j=N-i}^{N-1}A_1^{-1}(w_j) = A_{N-1-i}^{-1}\, A^{-1}_i(w_{N-i})
H_j(t):\Omega^{0,+}(X,E) \rightarrow\Omega^{0,+}(X,E), u \rightarrow\int\chi_j(x)e^{-m\varphi_j(z)-im\theta}A_{B_j,+}(t,z,w)e^{m\varphi_j(w)+im\eta}\tau_j(w)\sigma_j(\eta)u(y)dv_X(y).
m = \langle \sigma_{0}\rangle = \frac{e^{h}g_{n}^{\gamma}( + ) - e^{- h}g_{n}^{\gamma}( - )}{e^{h}g_{n}^{\gamma}( + ) + e^{- h}g_{n}^{\gamma}( - )}= \frac{e^{h}x_{n}^{\gamma}- 1}{e^{h}x_{n}^{\gamma}+ 1}.
g(a\circ b) =g(\pi(\pi^{-1}(a)\pi^{-1}(b))=\eta f(\pi^{-1}(a)\pi^{-1}(b)) =\eta (f\pi^{-1}(a)f\pi^{-1}(b)) =\eta(\eta^{-1}g(a)\eta^{-1}g(b))=g(a)\circledast g(b).
\begin{array}{c}{\alpha ( a ( f ) ) = a ( U f ) + a^{*}( V f )}\\{U U^{*}+ V V^{*}= 1 = U^{*}U + V^{*}V , \quad V^{*}U + U^{*}V = 0 = U V^{*}+ V U^{*}}\\ \end{array}
f = \lim_{\kappa \to \infty}\mathop{\mathop{\sum_{Q^* \in \mathcal{D}}}_{\ell(Q^*) = 2^s}}_{Q^* \cap B(0, 2^N) \ne \emptyset}\mathop{\mathop{\sum_{Q \in \mathcal{D}}}_{Q \subset Q^*}}_{\ell(Q) > 2^{-\kappa}}\Delta_Q f.
\begin{pmatrix}\bar{g}_1(qx) \bar{g}_2(qx) \end{pmatrix}= \biggl( \overline{F}_{\infty}+\frac{\overline{F}_1}{1- \alpha_1 x}+\frac{\overline{F}_2}{1- \alpha_2 x}\biggr)\begin{pmatrix}\bar{g}_1(x) \bar{g}_2(x) \end{pmatrix}\! .
\varepsilon_{D}\left( L \right) = \frac{c ( D ) \pi^{( D - 1 ) / 2}}{2^{D - 1}L^{D - 1}}\Gamma \left( \frac{1 - D}{2}\right) \sum_{n = 0}^{\infty}( n + \frac 1{2})^{D - 1}.
a_{n+1}(x) =\Big(x-\omega(a)+\left(a-\beta \right)\left(a+p+\beta \right) - \alpha_{2}-\alpha_{1}\Big) a_n(x) -\Big( \alpha_{2}\alpha_{1}+\gamma\left(p+2\beta \right) \Big)a_{n-1}(x) -\gamma^2 a_{n-2}(x)\, ,\;
a_{n,2s+4,k}= \widetilde{A_n}(-k) = 2^{(6s+12)n}(-4k+2n)^{\delta}\left( \frac{(-k+1/4)_{n}(-k+3/4)_n}{k!^2(n-k)!^2}\right)^{s+2}&= (-4k+2n)^{\delta}\left( \frac{(4k)!(4n-4k)!}{(2k)!(2n-2k)!k!^2(n-k)!^2}\right)^{s+2}.
\{f ( z ) , g ( z ) \}_{\alpha}= \nabla_{i}f ( z ) ( \omega_{( \alpha ) i j}+ R_{i j k l}\theta^{k}\theta^{l})^{- 1}\nabla_{j}g ( z ) + \frac{\partial_{r}f ( z )}{\partial \theta^{i}}g^{i j}\frac{\partial_{l}g ( z )}{\partial \theta^{j}},
\int\limits_0^\infty{{x^{s - 1}}\sum\limits_{n = 0}^\infty{\phi \left({2n}\right)\frac{{{{\left({- 1}\right)}^{n}}}}{{\left({2n}\right)!}}{x^{2n}}dx = \phi \left({- s}\right)\Gamma \left( s \right)\cos \frac{{\pi s}}{2}}}.
\sum_{M\geq1}(g_1(M)-g_1(M-1))\,x^M =\frac{x^2-x^3}{(1-x-x^2)^2}=\sum_{M\geq1}(h_1(M)-h_1(M-1))\,x^M, \sum_{M\geq1}(g(M)-h(M))\,x^M =\frac{x^2-x^3}{(1-x-x^2)^2}.
\delta \varepsilon ( i | \omega | ) \delta ({\bf r}_{2}-{\bf r}) \omega^{2}D_{j k}( \varepsilon ,{\bf r},{\bf r}^{\prime}, i \omega ) + L_{j m}( \varepsilon ,{\bf r}, i \omega ) \delta D_{m k}({\bf r},{\bf r}^{\prime}, i \omega ) = 0 .
g(\gamma)=\begin{cases}\,\,1, \gamma \leq \eta^{\star}-\beta^{\star}\eta^{\star}, \,\,\frac{\gamma}{\eta^{\star}}+\beta^{\star}, \eta^{\star}-\beta^{\star}\eta^{\star}\leq \gamma \leq -\beta^{\star}\eta^{\star}, \,\,0, \gamma \geq -\beta^{\star}\eta^{\star}\end{cases}
\mbox{Aux}_{p}=\mbox{Aux}\left[p-2,\dots,0\right]\mbox{Aux}\left[p-3,1,\dots,0\right]\dots\mbox{Aux}\left[0,\dots,1,p-3\right] \mbox{Aux}\left[0,\dots,p-2\right],\quad\mbox{for all $p \geq 3$.}
L\varphi_{1}+ hB_{1}\varphi_{1}+ h^2A_{2}\varphi_{1}+ h^2\Phi\varphi_{1}+ h^2{\cal L}\varphi_{1}+ \eta h^2 B_2 (I + h A_1)^{-1}\varphi_{1}= L\varphi_{0}+ hLv_{0}+ hB_{1}\varphi_{0}+ h^2 B_2 (I + h A_1)^{-1}(\eta\varphi_{0}+\theta_{0}).
V_{n m}^{>}( B ) \propto \frac{- 1}{l^{3}}\int_{0}^{1}d \eta \int_{0}^{1}d u u^{2 ( n - 2 )}\left[ F ( u ; 0 ) \right]^{n - 2 m}[ \widetilde{G}( u ) ]^{m}.
A_k= \begin{pmatrix}x_0 x_1 \cdots x_k 0 0 \cdots 0 \cr0 x_0 x_1 \cdots x_k 0 \cdots 0 \cr \ddots \cr0 \cdots 0 x_0 x_1 \cdots \cdots x_k\end{pmatrix}.
\int_{\mathbb{T}^d\times[0,T)}\bar{f}\left(\frac{\partial \phi}{\partial t}+ u\cdot\nabla \phi\right) \; dxdt & = \lim_{i\to\infty}\int_{\mathbb{T}^d\times[0,T)}f^{\nu_i}\left(\frac{\partial \phi}{\partial t}+ u\cdot\nabla \phi + \nu_i \Delta \phi\right) \; dxdt & = - \int_{\mathbb{T}^d}f_0 \phi_0 \; dx,
[x_{1}, x_{2}] =\lambda_{1}x_{3}+\lambda_{2}x_{4},[x_{1}, x_{3}]=-\lambda_{1}x_{2}+\lambda_{4}x_{4}, [x_{2}, x_{3}] =\lambda_{1}x_{1}+\lambda_{3}x_{4},[x_{1}, x_{4}]=-\lambda_{2}x_{2}-\lambda_{4}x_{3}, [x_{2}, x_{4}] =\lambda_{2}x_{1}-\lambda_{3}x_{3},[x_{3}, x_{4}]=\lambda_{4}x_{1}+\lambda_{3}x_{2},
\begin{multlined}A_1\sim A_2 B_1\sim B_2 \Rightarrow A_2\otimes B_2=U(S_A\otimes A_1)\otimes U(S_B\otimes B_1) =U(U(S_A\otimes S_B)\otimes (A_1 \otimes B_1)) \Rightarrow A_1\otimes B_1\sim A_2\otimes B_2\;.\end{multlined}
\begin{cases}\bar{u}_{tt}-\sum_{j=1}^{n+1}\bar{u}_{x_jx_j}+\frac{\mu}{1+t}\bar{u}_t+\frac{\nu^2}{(1+t)^2}\bar{u}=\bar{f}(t,x,x_{n+1}), (x,x_{n+1})\in \mathbb{R}^{n+1}, \t>0, \bar{u}(0,x,x_{n+1})=\bar{u}_0(x,x_{n+1}), (x,x_{n+1})\in \mathbb{R}^{n+1}, \bar{u}_t(0,x,x_{n+1})=\bar{u}_1(x,x_{n+1}), (x,x_{n+1})\in \mathbb{R}^{n+1},\end{cases}
-\frac{A'(s_k)}{B(s_k)(s_k-s_l)^2}-\frac{A'(s_l)}{B(s_l)(s_k-s_l)^2}=\sum_n\frac{A(t_n)}{B'(t_n)(s_k-t_n)^2(s_l-t_n)^2}.
\lim_{u\to\infty}\frac{g^{-1}(u x^{-1/\beta})}{g^{-1}(u)}=x \lim_{u\to\infty}\frac{g(g^{-1}(u)x)-u}{q(u)}=\lim_{u\to\infty}\frac{g\left(g^{-1}(u)\frac{g^{-1}(u x^{-1/\beta})}{g^{-1}(u)}\right)}{u}-1=x^{-1/\beta}-1
\begin{aligned}U_{11}^t U_{11}+ U_{21}^t U_{21}= I U_{11}U_{11}^t + U_{21}U_{21}^t = I U_{11}^t U_{21}- U_{21}^t U_{11}= 0 U_{11}U_{21}^t - U_{21}U_{11}^t = 0 \end{aligned}
S_{a,a^\prime,n}(x) =\sum_{\substack{m\leq\sqrt{x}(m,an)=1}}\mu(m)T_{a,a^\prime m^{-2},n}(x/m^2) &=\frac{\phi(a)x}{an}\left(\sum_{\substack{m\leq\sqrt{x}(m,an)=1}}\frac{\mu(m)}{m^2}\right)+O(2^{\omega(a)}\sqrt{x})
\mathcal{A}^{\varepsilon}\varphi_{i}^{\varepsilon}=R_{i}\left(u^{\varepsilon}\right)-\sum_{m=0}^{M-2}\varepsilon^{m}\bar{R}_{i}\left(u_{m}\right)-\varepsilon^{M-1}\left(\mathcal{A}_{1}u_{i,M}+\mathcal{A}_{2}u_{i,M-1}\right)-\varepsilon^{M}\mathcal{A}_{2}u_{i,M}\;\;\Omega^{\varepsilon},
\big( \overline{l}_{A}^{c,d,\nu}\big)^{-1}( x ) = ( \nu^{-1}\otimes d\cdot x ) \big( \overline{r}^{c,d,\nu}\big)^{-1}( x ) =\big( c^{-1}\cdot x\otimes \nu^{-1}\big) , \qquad{}\ \x\in X.
\begin{aligned}\widetilde{B}_{n}(x) = \sum_{j=0}^{n-2}\big((n+j-1)x+n+j+1\big)x\,T_{n-1,j}(x) \ +\sum_{j=0}^{n-2}\big((2n-j-1)x+j+1\big)T_{n-1,n+j-1}.\end{aligned}
1 = \int \! D B_{+}^{1}\delta ( B_{+}^{1}) D B_{+}^{2}\delta ( B_{+}^{2}) \exp \{i \int \! d^{2}x [ \psi_{-}^{\dagger 1}B_{+}^{1}\psi_{-}^{1}+ \psi_{-}^{\dagger 2}B_{+}^{2}\psi_{-}^{2}] \}\quad ,
\begin{aligned}\int_{A}V(x,0)\mathrm{div}(\mathrm{div}(X(x))X(x))dx+\langle\nabla_{x}V(x,0),X(x)\rangle\mathrm{div}(X(x))dx &\qquad=\int_{\partial A}V(x,0)(\mathrm{div}(X(x)))\langle X(x),N(x)\rangle dx\end{aligned}
Gr_nMU^{C_2}_* = \Omega^{C_2}_{*-n\sigma}/ \Omega^{C_2}_{* - (n-1)\sigma}\cong \dfrac{\Omega^{C_2}_* \{u^n\}}{\begin{matrix}u^nd_{i,j+1}= 0 u^nq_{j+1}= 0 \end{matrix}}& \cong MU_*[d_{i,0}: i \geq 1] \{u^n\},
f(\mathbf{z}) ={\displaystyle \sum_{j=1}^{n}}4(x_jy_{n+1}-x_{n+1}y_{j})^{2}-\sum_{1\leq j<k \leq n}4(x_jy_{k}-y_{k}x_{j})^{2}& =4\langle\mathbf{x},\mathbf{y}\rangle^2-4\langle\mathbf{x},\mathbf{x}\rangle \langle\mathbf{y},\mathbf{y}\rangle,
\mathfrak{N}_{f, a}(L, L, H, H)-\mathfrak{N}_{f, a}(1, 1, 1, 1) \geq -\sum_{i=1}^t\theta(l_i)(q^{m/2}+n) -\sum_{i=1}^t\theta(l_i)(n+1)q^{m/2}-\sum_{i=1}^u\Theta(h_i)(2q^{m/2}+n) -\sum_{i=1}^u\Theta(h_i)(n+1)q^{m/2}-(2\epsilon_1+2\epsilon_2)\mathfrak{N}_{f, a}(1, 1, 1, 1).
S/I_\P \cong K \begin{bmatrix}r_1s_5t_2 r_2s_5t_2 r_3s_5t_7 r_4s_5t_6 r_5s_5t_1 r_6s_5t_1 r_7s_5t_1 r_1s_4t_2 r_2s_4t_2 r_3s_4t_7 r_4s_4t_6 r_5s_4t_1 r_6s_4t_1 r_7s_4t_1 r_1s_3t_8 r_2s_3t_8 r_5s_3t_5 r_6s_3t_5 r_7s_3t_5 r_1s_2t_1 r_2s_2t_1 r_3s_2t_3 r_4s_2t_4 r_5s_2t_2 r_6s_3t_2 r_7s_2t_2 r_1s_1t_1 r_2s_1t_1 r_3s_1t_3 r_4s_1t_4 r_5s_1t_2 r_6s_2t_2 r_7s_1t_2 \end{bmatrix}.
\left|{\int\limits_{t}^T E\left[f(s,y(s,x+W_{s-t}))\right]ds-\sum_{j=1}^Qw_{n,j}E\left[f(t_j, y(t_j,x+W_{t_j-t}))\right]}\right| =\dfrac{[Q!]^4(T-t)^{2Q+1}}{(2Q+1)[(2Q)!]^3}\left|F^{2Q}(\xi)\right| &\leq\dfrac{[Q!]^4(T-t)^{2Q+1}}{(2Q+1)[(2Q)!]^3}C_d=\epsilon_n(t)
\begin{cases}T_\pm : (\nu_1,\nu_2,\nu_3)\mapsto (\nu_1\pm 1,\nu_2\mp 1, \nu_3) \ , U_\pm : (\nu_1,\nu_2,\nu_3)\mapsto (\nu_1,\nu_2\pm 1, \nu_3 \mp1) \ , V_\pm : (\nu_1, \nu_2, \nu_3)\mapsto (\nu_1\pm 1, \nu_2, \nu_3\mp 1) \ ,\end{cases}
\|\sigma(f,v)-\sigma(f,w)\|^2 = \bigl\| (A(v))^{1/2}- (A(w))^{1/2}\bigr\|^2\le \|D(A^{1/2})\|_\infty^2 |v-w|^2 \leq C \bigl( 1 + C_{p}\| f\|_{L^p}\bigr)|v-w|^2 .
0<a_{j}^{*}<b_{j}^{*}<\infty, \, a_{j}^{*}\downarrow 0, \, (a_{j}^{*}, b_{j}^{*})\cap B(q)=\varnothing\quad\mbox{and}\quad\lim_{j\to\infty}\frac{b_{j}^{*}}{a_{j}^{*}}=\infty
[D_{\overline{\omega}}, \mathcal{R}_{\Phi}^{\varepsilon}] h=\varepsilon D_{\overline{\omega}}\Pi_S[(\beta_1)_x h]-\varepsilon \Pi_S[(\beta_1)_x D_{\overline{\omega}}h]=\varepsilon\Pi_S[(D_{\overline{\omega}}(\beta_1)_x) h]
\begin{array}{l}\Theta_{1}= pd^mx + p^i_\alpha d u^\alpha \wedge d^{m-1}x_i +p^{ij}_\alpha d u_i^\alpha \wedge d^{m-1}x_j \, , begin{align*}10pt]\Omega_{1}= -d p \wedge d^mx - d p^i_\alpha \wedge d u^\alpha \wedge d^{m-1}x_i -d p^{ij}_\alpha \wedge d u_j^\alpha \wedge d^{m-1}x_j \, .\end{array}
k_{t}^{\ast ,a}t^{-\alpha \left( \cdot \right)}f(x):=\sup_{y\in \mathbb{R}^{n}}\frac{t^{-\alpha \left( y\right)}\left\vert k_{t}\ast f(y)\right\vert}{\left( 1+t^{-1}\left\vert x-y\right\vert \right)^{a}},j\in \mathbb{N}_{0}.
S = \frac{1}{G}\int d x d t \: \sqrt{- \bar{g}}\: e^{- 2 \bar{\phi}}( \bar{R}+ 4 ( \bar{\nabla}\bar{\phi})^{2}+ 4 \lambda^{2}) - \frac{1}{2}\int d x d t \: \sqrt{- \bar{g}}\: \sum_{i = 1}^{N}( \bar{\nabla}f_{i})^{2},
( D_{l})_{\mu \nu}= \delta_{\mu , \nu}( D_{l})_{\mu}, \quad ( D_{l})_{\mu}= i \left[ g_{L_{1}}\wp ( 2 \mu \cdot q ) + g_{L_{2}}\wp ( 2 \mu \cdot q | \{\omega_{1}, 2 \omega_{3}\}) \right] .
L_{n}^{(\alpha^{\prime})}(x)=(\alpha^{\prime}-\alpha)(\begin{matrix}\alpha^{\prime}+n\\ \alpha^{\prime}-\alpha\end{matrix})\int_{0}^{x}\frac{t^{\alpha}(x-t)^{\alpha^{\prime}-\alpha-1}}{x^{\alpha^{\prime}}}L_{n}^{(\alpha)}(t)dt.
\mathfrak{g}_0 =\{x \in \mathfrak{g}\mid \operatorname{ad}(\eta) x = 0 \ \ \ \eta \in \mathfrak{t}\}, \mathfrak{g}_\alpha =\{x \in \mathfrak{g}\mid \operatorname{ad}(\eta)^2 x = - \langle \alpha, \eta \rangle^2 x \ \ \ \eta \in \mathfrak{t}\}.
\nu_\infty=\lim_{R\to\infty}\lim_{n\to\infty}\int_R^\infty\left(\exp_p(\mu|u_n|^{p'})-\dfrac{\mu^{p-1}}{\Gamma(p)}|u_n|^{p}\right)r^{\alpha_0}\mathrm dr+\lim_{R\to\infty}\lim_{n\to\infty}\dfrac{\mu^{p-1}}{\Gamma(p)}\|u_n\|^{p}_{L^{p}_{\alpha_0}(R,\infty)}.
Z_f(s, \chi, D)=q^{-ls}Z_{h_1}(s, \chi, D)=q^{-ls}\Big(c_1(\chi )+c_2(\chi )\frac{(1-q^{-1})q^{-s}}{1-q^{-1-s}}\Big):=\dfrac{H_{2, 1}(q^{-s}, \chi)}{1-q^{-1-s}}
t_1^2g\Big(\frac{t_2}{t_1^2}\Big)\frac{g(t_2^2)}{g(t_2)}+t_1t_2^2g\Big(\frac{t_1}{t_2^2}\Big)\frac{g\big(\frac{1}{t_1^2}\big)}{g\big(\frac{1}{t_1}\big)}+t_2g(t_1t_2)\frac{g\big(\frac{t_1^2}{t_2^2}\big)}{g\big(\frac{t_1}{t_2}\big)}=0
R(z)=\begin{cases}S(z)P^{(\infty)}(z)^{-1}, \mbox{for $z\in \mathbb C \setminus \left(\overline{U^{(0)}\cup U^{(b)}\cup \Sigma_S}\right)$,}S(z)P^{(0)}(z)^{-1}, \mbox{for $z\in U^{(0)}$,}S(z)P^{(b)}(z)^{-1}, \mbox{for $z\in U^{(b)}$.}\end{cases}
B_E((x_i),(r_i)) = \Big\{\sum_{i \in I}\alpha_i x_i : \lvert \alpha_i \rvert \le r_i \Big\}, B^-_E((x_i),(r_i)) = \Big\{\sum_{i \in I}\alpha_i x_i : \lvert \alpha_i \rvert < r_i \Big\}
(MU(t)f)(\tau, y) =\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-is\tau}(Ff)(s+t,y)ds &=\frac{e^{it\tau}}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-is\tau}(Ff)(s,y)ds &=e^{it\tau}(Mf)(\tau,y).
\langle \phi_{n,m}\left(z_{1}\right)\phi_{1,2}\left(z_{2}\right)\phi_{1,2}\left(z_{3}\right)\phi_{n,m}\left(z_{4}\right)\rangle=\left(\prod_{i<j}z_{ij}^{\gamma_{ij}}\right)\eta^{2\alpha_{n,m}\alpha_{1,2}-\gamma_{12}}\left(1-\eta \right)^{2\alpha_{1,2}^{2}-\gamma_{23}}Y\left(\eta \right)
< \overrightarrow{\mu}> = \frac{e}{m}\biggl \{1 + \frac{\alpha}{2 \pi}- \frac{m^{2}e^{2}| \widetilde{q}|^{2}}{4 8 \pi^{2}}\biggl [ \ln \frac{m^{2}| \widetilde{q}|^{2}}{4}- 1 + 2 \gamma_{E}\biggr ] \biggr \}\overrightarrow{s}+ \frac{i e^{2}( \widetilde{q \hspace{- 0 . 2 c m}/})}{4 \pi^{2}| \widetilde{q}|^{2}}\overrightarrow{\theta}
F_{s,\Lambda}(x):=\sum_{v\in\Lambda}\int_1^\infty e^{-|x+v|^2t}\frac{t^{\frac{s}{2}-1}}{\Gamma(\frac{s}{2})}\mathrm{d}t+\frac{1}{|\Lambda|}\sum_{w\in\Lambda^*\setminus\{0\}}e^{2\pi i w\cdot x}\int_0^1 \frac{\pi^{\frac{d}{2}}}{t^\frac{d}{2}}e^{-\frac{\pi^2|w|^2}{t}}\frac{t^{\frac{s}{2}-1}}{\Gamma(\frac{s}{2})}\mathrm{d}t,
h_{x}^{\left( r\right)}=\frac{x^{\overline{r}}}{x\Gamma\left( r\right)}\left( \psi\left( x+r\right) -\psi\left( r\right) \right) ,\text{}r,x,r+x\in\mathbb{C}\backslash\left\{0,-1,-2,...\right\}.
\widehat N^p_C(F(\bar x);\nabla F(\bar x)u) =N_C(F(\bar x);\nabla F(\bar x)u)=N_C(F(\bar x))\cap\{\nabla F(\bar x)u\}^\perp &=\left(\prod_{i=1}^\ell N_{\mathcal Q_{m_i}}(F_i(\bar x))\right)\cap\{\nabla F(\bar x)u\}^\perp\supset\prod_{i=1}^\ell N_{\mathcal Q_{m_i}}(F_i(\bar x))\cap\{\nabla F_i(\bar x)u\}^\perp.
n_{t,s,b,a}\cdot z_0 = \frac{1}{\frac{1}{2}(-is + |b|^2)+1}\left(\frac{t_1}{\sqrt{2k}},\ldots, \frac{t_k}{\sqrt{2k}}, -ib_1,\ldots,-ib_{n-1},\frac{1}{2}(is - |b|^2)\right).
\begin{aligned}\boldsymbol{\varphi}_{tt}+ 2\,\kappa\,\boldsymbol{\varphi}-\gamma\,\Delta\boldsymbol{\varphi}- (\alpha + \beta)\,\nabla\,{\rm{div}}\,\boldsymbol{\varphi}- \kappa\,(\nabla\times \boldsymbol{u}) &+ \int_{0}^{+\infty}g(s)\ \Delta\boldsymbol{\varphi}(\boldsymbol{x},\,t - s)\ds + \xi\,\boldsymbol{\varphi}_{t}=0, \end{aligned}
m \frac{d^{2}}{d t^{2}}R_{a}^{i}= k \, \sum_{j , b}M_{\left( i a \right) \left( j b \right)}R_{b}^{j}\; , \; \; M_{\left( i a \right) \left( j b \right)}= \varepsilon^{i j}\sum_{c = 1 , 2 , 3}\varepsilon^{a c b}N_{c}N_{b}
p\left( x,y\right) =p\left( r\cos\varphi,r\sin\varphi\right) ={\displaystyle\sum_{k=0}^{N}}{\displaystyle\sum_{\ell=1}^{a_{k}}}\widetilde{p}_{\left( k,\ell\right)}\left( r^{2}\right) r^{k}Y_{\left(k,\ell\right)}\left( \varphi\right) ;
\left| \langle \tau_t(\sigma_\xi^w) \sigma_\eta^{w'}\rangle_\alpha \right| = \left| \langle a^+_{1}(t) a^-_{1}(t) \cdots a^+_{\xi-1}(t) a^-_{\xi-1}(t) \a_\xi^{\#_w}(t) a_\eta^{\flat_{w'}}a^+_{\eta+1}a^-_{\eta +1}\dots a^+_{N}a^-_{N}\rangle_\alpha \right| \, .
\mathrm{P}_{m}^i = \sum^{m-1}_{l=0}\frac{(\lambda_c\left[\frac{\pi}{\bar{\tau}_i^2}- \pi\delta^2 \mathcal{R}_c^2\right])^l}{l!}e^{-\lambda_c\left[\frac{\pi}{\bar{\tau}_i^2}- \pi\delta^2 \mathcal{R}_c^2\right]},
\Bigl| S(\mathcal{A}{}_d,X^\delta)-\kappa_\mathcal{A}\frac{\#\mathcal{A}{}}{\#\mathcal{B}{}}S(\mathcal{B}{}_d,X^\delta)\Bigr|=\Bigl|S(\mathcal{A}{}_d,X^{\epsilon^4})-\kappa_\mathcal{A}\frac{\#\mathcal{A}{}}{\#\mathcal{B}{}}S(\mathcal{B}{}_d,X^{\epsilon^4})\Bigr|+O\Bigl(\frac{\#\mathcal{A}{}}{d\log{X}}\Bigr).
\mathcal{T}:\begin{pmatrix}x_1 , x_2 \end{pmatrix}\mapsto \begin{pmatrix}-x_1 , x_2 \end{pmatrix}, \mathcal{P}:\begin{pmatrix}x_1 , x_2 \end{pmatrix}\mapsto \begin{pmatrix}x_1 , - x_2 \end{pmatrix},
\chi^\pm_{\lambda\backslash\{0\}}(gx's')=(-1)^{\frac{q^2-1}{8}}\sum_{\mu\in M_q(\lambda_l)}(-1)^{L(c^{\lambda_l}_\mu)}\chi_{((\mu,\lambda\backslash\{l\})\backslash\{0\})}(x's').
\sum_{|d| \leq D}\Big(1 + \frac{d}{z}\Big)^{-k}= \sum_{|d| \leq D}\exp\Big(\frac{ikd}{y}- \frac{xdk}{y^2}- \frac{k d^2}{2y^2}\Big) + O\Big(\frac{kD^4}{|z|^3}\Big).
\begin{array}{ll}L(\psi) \leq \dfrac{2|\nabla g|}{|\nabla u|}+\sum\limits_{j=1}^l\dfrac{2nC_1}{|z-a_j||\nabla u|}+\sum \dfrac{|(\psi_0)_{\alpha}|^2}{\lambda_{\alpha}\psi_0^2}+\gamma'(u)(\dot{u}-n) +k'(t_0)-\gamma''(u)\sum\dfrac{|u_{\alpha}|^2}{\lambda_{\alpha}}-\eta\sum\dfrac{1}{\lambda_{\alpha}}. \end{array}
{\sf M}^{a b}( x , y ) = \left. \frac{\delta \chi^{a}[ A^{\Omega}] ( x )}{\delta \varphi^{b}( y )}\right|_{\varphi = 0}\equiv{\sf M}[ A ] \qquad ; \qquad \Omega = \exp ( - g \, \varphi^{a}T^{a})
c_{(2)}^{\ast}=\frac{(m-1)(m-3)\Gamma(1/2)}{(2\pi)^{1/2}m^{2}}\left[\int_{0}^{\infty}u^{1/2-1}\left(1+\frac{2t}{m}\right)^{-(m-3)/2}\mathrm{d}u\right]^{-1}=\frac{(m-1)(m-3)}{m^{5/2}B(\frac{1}{2},~\frac{m-4}{2})},~if~m>4,
\mathcal{J}_{\lambda_k}(u)= \dfrac{1}{2}\mathcal{B}_{\alpha}(u,u)-\dfrac{\lambda_k}{2}\int_{\Omega}|u(x)|^2 dx- \int_{\Omega}(F(x,u^-(x)+u^0(x))-F(x,u^0(x))) dx -\int_{\Omega}F(x,u^0(x)) dx
G(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}}=\frac{1}{(q;q^{5})_{\infty}(q^{4};q^{5})_{\infty}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdot\cdot\cdot
x - \frac{1}{V'(x)}- \sqrt{2W(x)}=- \frac{1}{V'(x)}+ \frac{x^2 - 2W(x)}{x + \sqrt{2W(x)}}& =- \frac{1}{V'(x)}- \frac{2 \log H(x)}{x + \sqrt{2W(x)}}.
\sum_{k=0}^\infty \frac{(\alpha-\beta)_k}k{k!}t^k\sum_{n=0}^\infty\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n=\sum_{n=0}^\infty\frac{(\alpha)_n}{n!}M_n(x;\alpha,c)t^n,
\ln k_{0}( 1 ) = 2 . 9 8 4 1 2 8 5 5 5 8 \ldots , \quad \ln k_{0}( 2 ) = 2 . 8 1 1 7 6 9 8 9 3 1 \ldots , \quad \ln k_{0}( 3 ) = \ldots \, .
d s^{2}= e^{- \phi / 2}g_{\mu \nu}d X^{\mu}d X^{\nu}+ e^{\phi}( d U^{1})^{2}+ 2 e^{\phi}\chi d U^{1}d U^{2}+ ( e^{- \phi}+ e^{\phi}\chi^{2}) ( d U^{2})^{2}\; .
\max_{\theta\in\Theta}L_n(X_0^n;\theta) =\max_{Q}\sum_{a,a'}n\hat{P}_{X_0X_1,n}(a,a')\log(Q(a'|a)) &=-n\min_{Q}\!\left\{\!D\left(\hat{P}_{X_0X_1,n}\Big\| Q\otimes\hat{P}_{X_0,n}\right)-\sum_{a,a'}\hat{P}_{X_0X_1,n}(a,a')\log\left(\frac{\hat{P}_{X_0X_1,n}(a,a')}{\hat{P}_{X_0,n}(a)}\right)\!\right\},
\theta^\star_{x,y}q_{ij}(x,y) =\frac{n(r-3)i}{k}q_{ij}(x,y)+ \frac{n(i+1)}{k}\frac{\binom{n}{i}}{\binom{k}{i}}K_{i+1}(x,k-y,r-1) H_j(y,n-i,k-i) &\quad+ \frac{n(r-2)}{k}\frac{\binom{n}{i}}{\binom{k}{i}}(k-y-i+1)K_{i-1}(x,k-y,r-1) H_j(y,n-i,k-i)\,.
\begin{aligned}G_k(x_0,\ldots,x_{n+1}) :=\left(x_0\frac{\partial}{\partial x_0}+\cdots+x_{n+1}\frac{\partial}{\partial x_{n+1}}\right)^{(k)}F(p) & =\sum_{i_0+\cdots+i_{n+1}=k}\frac{k!}{i_0!\cdots i_{n+1}!}x_0^{i_0}\cdots x_{n+1}^{i_{n+1}}\frac{\partial^kF}{\partial x_0^{i_0}\cdots \partial x_{n+1}^{i_{n+1}}}(p)\end{aligned}
h_n(x^i_n+x)= h_n(x^i_n+re^{i\theta}) = \frac14e^{-2i\theta}\Big(|\partial_r u_n|^2-r^{-2}|\partial_\theta u_n|^2-\frac{2i}{r}\langle\partial_r u_n,\partial_\theta u_n\rangle\Big)(x^i_n+re^{i\theta}),
\phi^s(n) =\sum_{k=n+1}^\infty (1+k)^{-s}\sum_{x\in X_n}\tau(x) |h_{k}(x)|^2 \phi^{p}_q(n) = \sum_{k=n+1}^\infty \mathrm{e}^{-q k^p}\sum_{x\in X_n}\tau(x) |h_k(x)|^2.
d_\mu =\sum_i -d_{\mu^i}=-(-1)^{l(\mu)-2}\frac{(l(\mu)-2)!}{m(\mu)!}\sum_i m_i\big(-d_i+\sum_k m_k d_k\big) =(-1)^{l(\mu)-1}\frac{(l(\mu)-2)!}{m(\mu)!}\big(-\sum_i m_id_i+l(\mu)\sum_k m_k d_k\big) =(-1)^{l(\mu)-1}\frac{(l(\mu)-1)!}{m(\mu)!}\sum_k m_k d_k.