harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(I-\frac{{\bf q}\cdot{\bf p}^{\rm H}}{{\bf p}^{\rm H}{\bf q}}\right)(A-\theta I)\left(I-\frac{{\bf q}\cdot{\bf p}^{\rm H}}{{\bf p}^{\rm H}{\bf q}}\right){\bf s}=-{\bf r}_q,{\bf s}\bot{\bf p},
\Delta^{( 2 )}{\cal V}= - \varepsilon^{2}\frac{3 m^{3}}{2{\sqrt{2}}\lambda}m^{2}t^{2}\int \frac{d z}{\cosh^{2}z}= - \varepsilon^{2}\frac{3 m^{5}}{\lambda{\sqrt 2}}t^{2}\equiv \frac{M V^{2}}{2}.
\Gamma_{[\ell+1]}K_{\ell}^{\pm}\Psi_{\mathbf{j}}^{s}(x_1,\ldots, x_{n}) =\Big[\{\Gamma_{[\ell+1]}, K_{\ell}^{\pm}\}-K_{\ell}^{\pm}\Gamma_{[\ell+1]}\Big]\Psi_{\mathbf{j}}^{s}(x_1,\ldots, x_{n}) &=\big[\pm 1-\lambda_{\ell+1}(\mathbf{j})\big] K_{\ell}^{\pm}\Psi_{\mathbf{j}}^{s}(x_1,\ldots, x_{n}).
\frac{1}{2}\frac{d}{dt}\Big(\|k\|^2+((H_1/J)\dot{D}h|\dot{D}h)\Big) + &+(\beta_1\dot{D}h|\dot{D}h)+(\beta_2\dot{D}h|h)+(\beta_3\dot{D}h|k)++(\beta_4h|k)+(\beta_5k|k)= &=((H_1/J)\dot{D}h|\dot{D}g_1)+(k|g_2),
[\Psi_G]_q\equiv q^2\Big([\Psi^{12},\Psi^1_2,\Psi^2_1,\Psi_{12}]_q- \sum_{i=1}^4 c_2(G\backslash i) - [\Psi^{13,24},\Psi^{14,23}]_q\Big) \mod q^3.
\{\lambda'_i(x)\}_{i\in S_n}=Eig^{\neq 0}(\bar{\Lambda}'(x))=Eig^{\neq 0}(V_1(x)\bar{\Lambda}'(x)V_1(x)^*) &=Eig^{\neq 0}(U_1(x)V_1(x)\bar{\Lambda}'(x)V_1(x)^*U_1(x)^*),~~\forall i\in S_n
W=W_\mu dx^\mu=\textsf{a}(1+\eta_{\mu\nu}x^\mu x^\nu)^{-\frac{3}{2}}[ (1+\eta_{\mu\nu}x^\mu x^\nu)d(x^0+x^1) &+(x^0+x^1)(x^0dx^0-x^1dx^1-x^2dx^2-x^3dx^3)].
\frac{P_{s_1}(t)}{P_{s_2}(t)}= \frac{Q_{s_2}(t)}{Q_{s_1}(t)}\le \bigg( \frac{1+s_2^2}{1+s_1^2}\bigg)^{1/2}\cdot \max \bigg(1, \frac{s_1^{1/2}}{s_2^{1/2}}\bigg)
M(p) = \frac{8}{\pi^2 N}\left[ \int_0^p dq\; \left( \frac{1}{p}+\frac{\xi\alpha}{2p^2}\right) \frac{q^2 M(q)}{q^2+M^2(q)}+\int_p^\alpha dq\; \left( q+\frac{\xi\alpha}{2}\right)\frac{M(q)}{q^2+ M^2(q)}\right].
(pr_1)_* (\tilde{h}_k\cdot pr_2^*(\underline\gamma)) = (pr_1)_*\big((pr_1^*(\eta) - pr_2^*(\eta)) \cdot pr_2^*(\underline\gamma)\big) =\eta\cdot (pr_1)_*pr_2^*(\underline\gamma) -(pr_1)_*pr_2^* (\eta\cdot \underline\gamma) &=\eta\cdot p_{i_k}^* (p_{i_k})_*(\underline\gamma) -p_{i_k}^* (p_{i_k})_* (\eta\cdot \underline\gamma)
\psi(f)^*\psi(h) =S^*\pi(f^*)\pi(h)S = S^*\pi(f^*\circ R_\gamma^{-1}\circ R_\gamma)\pi(h\circ R_\gamma^{-1}\circ R_\gamma)S = \pi(f^*\circ R_\gamma^{-1})S^*S\pi(h\circ R_\gamma^{-1}) = \pi((f^*h)\circ R_\gamma^{-1}).
\left[\begin{array}{ccc}\vrule begin{align*}-5pt]3\times3 \vrule \hspace{-4pt}X,Y\hspace{-2pt}begin{align*}-5pt] \vrule \hline \vrule begin{align*}-10pt]X,Y \vrule \hspace{-9pt}2\times2\hspace{-4pt}begin{align*}-10pt] \vrule \end{array}\right]
e^{w(\Lambda+\rho)-(\Lambda+\rho)}=\prod_{i=1}^rx_{\sigma(i)}^{-(r+d)k_{\sigma(i)}-\mu_i+i+\mu_{\sigma(i)}-\sigma(i)}q^{(\mu_i-i)k_{\sigma(i)}+\frac12(r+d)k_{\sigma(i)}^2}
\lambda^* :=\frac 1 2 \sup_{\varrho>0}\frac{\varrho}{\displaystyle\max_{{\footnotesize\begin{array}{c}x\in D |s|\leq \kappa\sqrt{\varrho}\end{array}}}\left|\int_0^{s}f(x,t)dt\right|}\in (0,+\infty]
\begin{array}{cc}{\rm v}_{\alpha\beta}=\textstyle{\frac{1}{2}}\epsilon_{\alpha\beta\gamma\delta}\tilde{{\rm v}}^{\gamma\delta}~~~~ ~~~~\tilde{{\rm v}}^{\alpha\beta}=\textstyle{\frac{1}{2}}\epsilon^{\alpha\beta\gamma\delta}{\rm v}_{\gamma\delta}\end{array}
I(f,g,h)(\xi) := e^{-i \xi^3}\iint_{\eta_1+\eta_2+\eta_3 = \xi}e^{i (\eta_1^3 + \eta_2^3 + \eta_3^3)}f(\eta_1) g(\eta_2) \bar h(-\eta_3) d\eta_1 d\eta_2.
\sum_{n=2^{k-1}}^{2^k-1}1_{[-B-t,B-t]}\left(\; \sum_{i=0}^{n-1}f(T^i x) \right)\ge\beta \sum_{n=0}^{2^k - 1}1_{[-B-t,B-t]}\left(\; \sum_{i=0}^{n-1}f(T^i x) \right).
\{\, x \in \P^1 \mid \lvert x - a_i \rvert \leq \alpha_{i,1}\, \}\cap \{\, x \in \P^1 \mid \alpha_{j,n_j}\leq \lvert x - a_j \rvert \leq \alpha_{j,n_j+1}\, \}= \{\, x \in \P^1 \mid \lvert x - a_i \rvert \leq \alpha_{i,1}\, \}.
\big\| \big(\mathcal{K}(y)\big)(t)-\big(\mathcal{K}(y)\big)(s)\big\|_V\leq\|U_Y(t,0)x-U_Y(s,0)x\|_V &+\Big\|\int_0^tU_Y(t,r)F\big(r,y(r)\big)\d r-\int_0^sU_Y(s,r)F\big(r,y(r)\big)\d r\Big\|_V=I_1+I_2.
\sum_{M\geq0}h(M)\,z^M =\sum_{M\geq0}z^M\sum_{0\leq 2j\leq M-1}(j+1)\binom{M-j-1}j &=\sum_{M,j\geq0}z^{M+2j+1}(j+1)\binom{M+j}j=\sum_{j\geq0}{z^{2j+1}(j+1)}{(1-z)^{-(j+1)}}&=\sum_{j\geq1}{z^{2j-1}j}{(1-z)^{-j}}=z^{-1}\frac{\frac{z^2}{1-z}}{\left(1-\frac{z^2}{1-z}\right)^2}=\frac{z-z^2}{(1-z-z^2)^2}.
\left( - \Delta + \{[ b^{2}- 2 a ( 2 m + 5 / 2 ) ] r^{2}+ 2 a b r^{4}+ a^{2}r^{6}\}- 2 ({\bf s \cdot l}) \cdot \{2 a r^{2}\}\right) \Psi ( x , y , z ) = E \Psi ( x , y , z )
\sum_{r\ge 1}y^r \sum_{b>0,\,a+b=r}\frac{1}{b}\binom{b}{a}w^a= \sum_{b>0}\frac{y^b}{b}\sum_a\binom{b}{a}(yw)^a = \sum_{b>0}\frac{y^b}{b}(1+yw)^b=\sum_{b>0}\frac{[y(1+yw)]^b}{b}= \log\frac{1}{1-y(1+yw)},
\varphi_q\left(z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right) \right) = 1.
\Big(\frac{d}{dt}\Big)^{s}\det(p_{i,j}(t))_{i,j=1,\ldots,k}=\sum_{\substack{l_1+\cdots+l_k=s l_1\geq 0,\ldots,l_k\geq 0}}\binom{s}{l_1,\ldots,l_k}\det \Big(p_{i,j}^{(l_{i})}(t)\Big)_{i,j=1,\ldots,k},
\Phi_{R}= \left( \begin{array}{c c c c c}{{- m}}&{{0}}&{{0}}&{{0}}&{{0 \nonumber}}\\{{0}}&{{- m}}&{{0}}&{{0}}&{{0 \nonumber}}\\{{0}}&{{0}}&{{- m}}&{{0}}&{{0 \nonumber}}\\{{.}}&{{.}}&{{.}}&{{.}}&{{. \nonumber}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{( k - 1 ) m}}\\ \end{array}\right) \, ,
\mathrm{i n d}^{D = 1 0}( \mathrm{E}_{6}) = 3 \quad \mathrm{i n d}^{D = 1 0}( \mathrm{E}_{7}) = 6 \quad \mathrm{i n d}^{D = 1 0}( \mathrm{E}_{8}) = 1 1
\begin{cases}i\partial_t u-\mathcal K_\lambda u=\epsilon |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, u(x,0)=u_0(x), (x,t)\in \mathbb{R}^n \times \mathbb{R},\end{cases}
E_{k,N}^{\chi}(Z,s; h) = j(g, I)^{k}E\Big(g,\frac{2s+k}{2n+1}-\frac 12, f^{(h)}\Big) = j(g, I)^{k}E\Big(gh^{-1},\frac{2s+k}{2n+1}-\frac 12, f\Big).
\mathcal{H}^n =\left(-1+\frac{\tau}{2}\eta^n-\frac{\tau^2}{2}\vartheta^n\right)\phi^{n-1}+\frac{\tau^2}{4}\Delta\phi^{n-1}\quad\;+2\phi^n+\tau^2\left(\Big(\vartheta^n+\chi^n-\sigma R^\prime(\|\phi^n\|^2-1)\Big)\phi^n-\frac{1}{2}\Delta\phi^n-G(\phi^n)\right)
^{\perp_{f}}A[>0] = \{Y \in\mathcal{R}(A) \mid \mathrm{Hom}_{\mathrm{D}(A)}(X, A[>0]) = 0\}&=\{Y \in \mathcal{R}(A) \mid \mathrm{H}^{i}(\mathrm{RHom}_{A}(X,A)) = 0~\textrm{for}~i>0\}.
y_n = s_n + \frac{1}{n+1}\sum_{k=1}^{n}\sum_{\ell=1}^k (-1)^\ell \binom{n+k}{k}\binom{k}{\ell}\frac{(k-\ell)!}{(n-q\ell)!}B_{n-q\ell,k-\ell}(\bar x) =s_n + \frac{1}{n+1}\sum_{\ell=1}^{\lfloor n/q\rfloor}\sum_{k=\ell}^n (-1)^\ell \binom{n+k}{k}\binom{k}{\ell}\frac{(k-\ell)!}{(n-q\ell)!}B_{n-q\ell,k-\ell}(\bar x).
(\mathcal{E}\phi)(z) =-\frac{1}{\pi}\int_{t-\pi}^{t+\pi}\log|e^{it}-e^{is}|(\psi(s)-\psi(t)-(s-t)\psi'(t))ds &=-\frac{1}{\pi}\int_{-\pi}^{\pi}\log(2|\sin(u/2)|)(\psi(u+t)-\psi(t)-u\psi'(t))du
\int_{\Omega}\rho\frac{\partial u}{\partial t}\cdot v+\int_{\Omega}\mu\nabla u\cdot\nabla v+\int_{\Omega}\rho(u\cdot\nabla)u\cdot v-\int_{\Omega}p\nabla\cdot v=\int_{\Omega}f\cdot v+\int_{\partial\Omega}(\mu\frac{\partial u}{\partial\hat{n}}-p\hat{n})\cdot v\forall v\in V.
T^{'}_t(u)=(Id_L+t[\mathfrak{X},-]) (T_t \Big(Id_V-tD(\mathfrak{X})+t^{2}D^{2}(\mathfrak{X})+\cdots+(-1)^{i}t^{i}D^{i}(\mathfrak{X})+\cdots)(u)\Big)) \quad\;\;\;\;\;=T(u)+t(T_1(u)-TD(\mathfrak{X})(u)+[\mathfrak{X},T(u)])+t^{2}T^{'}_{2}(u)+\cdots \quad\;\;\;\;\; =T(u)+t^{2}T^{'}_{2}(u)+\cdots(as\; T_1(u)=d_T(\mathfrak{X})(u)).
{\rm supp}\, \mathbb{B}[\nabla \varphi \cdot u_\lambda] \subset A\,,{\rm div}\, \mathbb{B}[\nabla \varphi \cdot u_\lambda] \, = \, \nabla \varphi \cdot u_\lambda\,, & \|\mathbb{B}[\nabla \varphi \cdot u_\lambda]\|_{W^{m+1,2}(\Omega)}\leq C \| \nabla \varphi \cdot u_\lambda \|_{W^{m,2}(\Omega)}\,,~~~~~m=0,1\,.
h(T_i,k_i)\sim e^{-\frac{2}{3}\pi k_i^2 T_i}\Bigl[\Theta_3(ik_iT_i,2i T_i)\Theta_3(ik_iT_i,6iT_i)+\Theta_2(ik_iT_i,2iT_i)\Theta_2(ik_iT_i,6iT_i)\Bigr]
\left( P_{b}^{i_{1}\ldots i_{p - 2 k - 1}}, b_{i_{1}\ldots i_{p - 2 k - 1}}\right) , \; \left( P_{b^{1}}^{i_{1}\ldots i_{p - 2 k - 1}}, b_{i_{1}\ldots i_{p - 2 k - 1}}^{1}\right) , \; k = 0 , \ldots , a ,
B = \Phi^{2}, \qquad B_{_{0 0}}= -{\frac{c^{4}}{c_{_\mathrm{P}}^{2}}}\, \Phi^{2}+{\cal O}\{\Theta^{4}\}\ , \qquad B_{_{1 1}}= \Phi^{2}+{\cal O}\{\Theta^{4}\}\ .
\sum_{i=0}^{L}\left\| D_{(m+1)\Delta}^{2L-i}F\right\|_{L^2(\Omega)}^{2}{\binom{L}{i}}^{4}\frac{i!}{\left( L!\right)^{2}}\Delta^{2L-i}\xrightarrow[L\rightarrow\infty]{}0,
{R^\lambda}_{\mu\nu\rho}=\partial_\rho{\Gamma_{\mu\nu}}^\lambda-\partial_\nu{\Gamma_{\mu\rho}}^\lambda+{\Gamma_{\mu\nu}}^\alpha\diamondsuit{\Gamma_{\rho\alpha}}^\lambda-{\Gamma_{\mu\rho}}^\alpha\diamondsuit{\Gamma_{\nu\alpha}}^\lambda.
x \ast A=dx x^{-1}+{\rm Ad}(x)A\frac{dz}{z}=\left((k+1)yz^{k+1}+ (A+[yz^{k+1},A])\mod z^{k+2}\mathfrak{g}(\mathcal{O})\right)\frac{dz}{z}&=(\sum_{i=0}^k b_i z_i) \frac{dz}{z}+ \left( (k+1)y+a_{k+1}+[y,b_0] \right) z^{k+1}\frac{dz}{z}\mod z^{k+2}\mathfrak{g}(\mathcal{O})\frac{dz}{z}.
\begin{array}{lll}x_i(1) = (\overline x_{r+\overline m}+\overline \alpha)^{\lambda_i}\overline x_i\mbox{for}1\le i\le r\mbox{and}x_{r+\overline m}(1) = (\overline x_{r+\overline m}+\overline \alpha)^{\lambda}-\overline\alpha^{\lambda},\,\,\alpha=\overline\alpha^{\lambda}\end{array}
\begin{cases}\alpha a+(\alpha-1)b+(3\alpha-2)c-\frac{p\alpha-1}{p}d=0; (1-2\alpha\mu)a+(1-2\mu(\alpha-1))b+3\big(1-\mu(6\alpha-4)\big)c-\big(1-\mu(2p\alpha-2)\big)d=0; \big(2\mu(\alpha-1)-1\big)b+2\big(\mu(6\alpha-4)-1\big)c-\frac{\mu(2p\alpha-2)-1}{p}d=0.\end{cases}
\mathcal{A}(x,t)= \left({\begin{array}{cc}\mathcal{A}_1(x,t) \mathcal{A}_2(x,t) \frac{t}{|t|}\frac{t^T}{|t|}\mathcal{A}_3(x,t) b(x,t)Id_{(n-d)\times (n-d)}\end{array}}\right),
\Lambda_{\nu}^{\mu}( x , y ) = \delta_{\nu}^{\mu}\delta^{3}( x - y ) - J^{\mu \alpha}\int d^{3}\! \rho d^{3}\! \sigma \delta_{\alpha ( x )}\varphi^{i}( \rho ) J_{i j}( \rho , \sigma ) \delta_{\nu ( y )}\varphi^{j}( \sigma )
A_1=2^{-1/2}\frac{b_1^{-1/8}}{\sqrt{n}}\biggl(\frac{\pi}{\epsilon}\biggr)^{1/4}\exp \biggl[-i(b_1)^{1/2}\biggl(\frac{n^2\epsilon}{4\pi}\biggr)\vert \eta_{\rm i}\vert^2-i\frac{\pi}{4}-i\frac{\phi_2}{2}\biggr], \quad A_2=0.
\int_K|F_{j}-(1-b(\Psi))fF^{1+\delta}|_{h}^2e^{v(\Psi)-\delta \max\{\psi+T,2\log|F|\}}c(-v(\Psi)) \le \bigg(\sup_{X_j}e^{-u(-v(\Psi))}\bigg)\int_{X_j}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|fF|^2_h.
\frac{\partial f_j}{\partial[\tilde{w}_i]_{\alpha,j}}=-\frac{1}{m|g|}\sum_{\substack{t=1 (t\neq j)}}^n[\tilde{v}_i]_{t,\alpha}\sum_{h=0}^{m-1}[\tilde{v}_{i+h+1}\tilde{w}_{i+h+1}]_{j,t}\frac{\lambda_t^{m-h-1}\lambda_j^h}{\lambda_t^m-\lambda_j^m}
\bar{\alpha}_i\cdot \bar{B}_2(\alpha_i)\cdot\bar{B}_2(\alpha_k) =\alpha_k^2\otimes \bigg[\binom{a_1}{2}+\binom{a_i}{2}\bigg]\gamma +\alpha_i^2\otimes \binom{a_1}{2}\gamma & +\binom{a_1}{2}\gamma\otimes\alpha_i^2+\bigg[\binom{a_1}{2}+\binom{a_i}{2}\bigg]\gamma\otimes \alpha_k^2
u(t)=\frac{^{\rho}I^{1-\gamma}u(0)}{\Gamma(\gamma)}\left(\frac{t^\rho}{\rho}\right)^{\gamma-1}-^{\rho}I^{\alpha}p(t)u(t)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{\alpha-1}s^{\rho-1}f(s,u(s),^\rho D^{\alpha,\beta}u(s))ds.
t_{n+2}=3t_{n+1}+3\sum_{j=0}^{n}(\begin{matrix}2n+2\\ 2j+2\end{matrix})t_{n-j}\sum_{k=0}^{j}(\begin{matrix}2j+1\\ 2k+1\end{matrix})t_{k}t_{j-k},t_{0}=1,t_{1}=3.
\log{n}\geq R(D_t) \geq h(z_t)+ \theta(D_t) &= h(F(z_{t-1},a_{t-1})+v_{t-1}) + \theta(D_t) &\geq h(F(z_{t-1},a_{t-1})+v_{t-1}|z_{t-1},a_{t-1}) + \theta(D_t) &= h(v_{t-1}) + \theta(D_t),
\left\vert \frac{\mathrm{d}w}{\mathrm{d}\theta}\right\vert = \frac{\left\vert \kappa \right\vert}{4}\left\vert \sin \left( \frac{\phi_{1}- \theta}{2}\right) \right\vert^{- \alpha_{1}}\left\vert \sin \left( \frac{\phi_{2}- \theta}{2}\right) \right\vert^{- \alpha_{2}}\ldots \left\vert \sin \left( \frac{\phi_{n}- \theta}{2}\right) \right\vert^{- \alpha_{n}}
\lambda^{a , \alpha \beta}\lambda_{a \delta}^{\gamma}- \lambda^{a , \gamma \alpha}\lambda_{a \delta}^{\beta}= \frac{2}{\sigma_{0}}\left( \Omega^{\alpha \beta}\delta_{\delta}^{\gamma}- 2 \Omega^{\beta \gamma}\delta_{\delta}^{\alpha}+ \Omega^{\gamma \alpha}\delta_{\delta}^{\beta}\right) ,
\alpha_{-1}(i,j,k,l) = \sum \alpha_0(2i-\delta_1+\delta_3, 2j-\delta_2+\delta_4, 2k,2l) &= \sum \alpha_0^{(3)}(2i-\delta_1+\delta_3, 2j-\delta_2+\delta_4, 2(k+l)) &= 4\alpha_0^{(3)}(i,j,k+l) = 4\alpha_0(i,j,k,l),
\mathbb{X}^{(\epsilon)}_{t_n,t_{n+2}}= \mathbb{X}^{(\epsilon)}_{t_n,t_{n+1}}+ \mathbb{X}^{(\epsilon)}_{t_{n+1},t_{n+2}}+ X^{(\epsilon)}_{t_n,t_{n+1}}\otimes X^{(\epsilon)}_{t_{n+1},t_{n+2}}.
\mu_{n}~\ddot{\theta}_{n}=\sum_{\ell =1,~\ell \neq n}^{N}\left[ \frac{\dot{\theta}_{n}~\left( \mu_{\ell}~\dot{\theta}_{\ell}+\eta_{\ell}\right)+\left( \mu_{n}~\dot{\theta}_{n}+\eta_{n}\right) ~\dot{\theta}_{\ell}~\cos \left( \theta_{n}-\theta_{\ell}\right)}{\sin \left( \theta_{n}-\theta_{\ell}\right)}\right] ~.
\begin{array}{r r}{{( 8 , 1 , 1 ; 1 , 1 , 8 ) ,}}&{{({\bar{8}}, 1 , 1 ; 1 , 8 , 1 )}}\\{{( 1 , 8 , 1 ;{\bar{8}}, 1 , 1 ) ,}}&{{( 1 , 1 , 8 ; 8 , 1 , 1 )}}\\ \end{array}
\begin{array}{l}{\delta \xi^{i}= \left({\frac{d z}{d t}}+ i z \right) C^{i j}\bar{\epsilon}_{j}+ i \overline{{W^{\prime}}}( \bar{z}) \varepsilon^{i j}\epsilon_{j}}\\{\delta z = i \epsilon_{i}\xi^{i}}\\ \end{array}
\frac{A_n^{(3)}(z)}{n^9 z}\frac{A_n^{(3)}(0)}{n^9 z}= \mathcal{O}\left(n^{-2}\right) \frac{A_n^{(3)}(0)}{n^9 z}= \mathcal{O}\left(n^{-\frac{1}{2}}\right).
m_{0n}(u \otimes w^{\otimes k}\otimes u \otimes w^{\otimes (n-2)-k}) = (-1)^ks_nu, 0 \leq k \leq n-2, m_{0n}(u \otimes w^{\otimes n-1}) = s_{n+1}w, m_{0n}(u \otimes w^{\otimes n-2}\otimes v ) = (-1)^{n-2}s_n v,
\gamma_k :=\left\{(x^{(j)})_{j=1,\dots,\ell}\in N;\,\begin{array}{ll}x^{(j)}=0, j \in J_{\prec k}\sqcup J_{\nparallel\, k}x^{(j)}\in \mathbb{R}^{n_j}, j \in J_{\succ k}\langle x^{(j)}, \xi^{(k)}\rangle > 0, j=k \end{array}\right\}.
\int \frac{d^{4}p^{\prime}}{\left( 2 \pi \right)^{4}}\left[{D}\left( p , p^{\prime}, P \right) +{K}\left( p , p^{\prime}, P \right) \right]{G}^{\left( 4 \right)}\left( p^{\prime}, p^{\prime \prime}, P \right) = \delta^{\left( 4 \right)}\left( p - p^{\prime \prime}\right) ,
g^{2}( t ) ={\frac{g^{2}}{\displaystyle 1 +{\frac{1 1 g^{2}t}{1 2 \pi^{2}}}}}, a_{1}( t ) = a_{1}, a_{3}( t ) = a_{3}-{\frac{6 2 t}{1 2 0 ( 4 \pi )^{2}}}.
Z^x_t := Y^x_0 + \sigma\, W_t+ \int_0^t L\left[x,\,P(x,C^b_s,C^l_s)- C^b_s r_s(x)+ \int_{\mathcal{X}}C^l_s r_s(x')\nu_s(x')\mu(\mathrm{d}x')\right] \,\mathrm{d}s,
A(2,\, (1,1)) ={2 \choose 1,1}\frac{1}{5\cdot 2}\int_{0}^{1}(1-s)s^2ds=\frac{2}{5!}A(2,\, (2,0)) ={2 \choose 2,0}\frac{1}{5}\int_{0}^{1}(1-s)^2sds= \frac{2}{5!}A(2,\, (0,2)) ={2 \choose 0,2}\frac{1}{5\cdot3}\int_{0}^{1}s^3ds=\frac{2}{5!}.
U_n(t) = \frac{1}{\sqrt{n}}\Big (\tfrac{nt-\lfloor nt \rfloor}{2}+\tfrac{n}{2}(1 - t)\log(1 - t) +\tfrac{n - \lfloor nt \rfloor}{2}(G_n - G_{n - \lfloor nt \rfloor + 1})\Big) + O\Big (\frac{\log(n)}{\sqrt{n}}\Big).
\pi^n_{k+1}\left( Z_{n}^{I; \vec{i}}\right) \cap \pi^n_{k+1}\left( Z_{n}^{J; \vec{j}}\right) = Z_{k+1}^{I; \vec{i}}\cap Z_{k+1}^{J; \vec{j}}&= Z_{k+1}^{I; i_1, \ldots, i_{k+1}}\cap Z_{k+1}^{J; j_1, \ldots , j_{k+1}}.
\gamma^{\mu}\Psi_{, \mu}= - i \gamma^{\mu}\frac{e^{\gamma^{5}\alpha}}{\rho}\{I m ( \overline{{{\Psi}}}_{, \mu}\Psi ) - j_{\mu}- \gamma^{5}[ I m ( \overline{{{\Psi}}}_{, \mu}\gamma^{5}\Psi ) - g_{\mu}] \}\Psi
\int_0^{\pi}\, \mathrm{d}\theta \; \sin \theta \cos^2\theta \cos(2\rho y \cos\theta) = \frac{\sin(2 \rho y)}{\rho y}+ \frac{\cos(2 \rho y)}{\rho^2 y^2}- \frac{\sin(2\rho y)}{2 \rho^3 y^3}&= \left(\frac{-i}{2 \rho y}+ \frac{1}{2 \rho^2 y^2}+ \frac{i}{4 \rho^3 y^3}\right) e^{2 i \rho y}+ ,
\frac{2\sum_{n=0}^{k}\left( -1\right)^{n}\chi\left( n\right) \cosh\left(nt\right)}{\sinh(kt)}=\frac{2\sum_{n=0}^{k}\left( -1\right)^{n}\chi\left( n\right) \sinh\left( nt\right)}{\cosh(kt)+1}& =\sum\limits_{j=0}^{\infty}E_{2j+1,\overline{\chi}}\left( 0\right)\frac{t^{2j+1}}{(2j+1)!}.
\begin{array}{c}f(x)=\sum\limits_{i=1}^{N}\big(\|x\|^{2}+\|\varphi_{i}\|^{2}-2\langle x,\varphi_{i}\rangle\big)=2N-2\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}x_{j}\varphi_{ij}; begin{align*}2 pt]g(x)=\sum\limits_{i=1}^{M}x_{i}^{2}=1.\end{array}
\chi_H(\alpha \boxtimes \beta) = \chi_H(p_1^* \alpha \otimes p_2^*\beta) &=\chi_H(\alpha \otimes (p_1)_*p_2^* \beta) &=(\pi_1)_* \left(\alpha \otimes \pi_1^* ((\pi_2)_*\beta)\right) &= \chi_H(\alpha) \chi_H(\beta),
L_{\pi'}(Z) = \sum_{k\ge 0}\, (-1)^k s_{(1^k)}[s_{\pi'}](Z) = \sum_\nu \ell_{\pi'\nu'}\, s_{\nu'}(Z)\cr = \sum_{k\ge 0}\, (-1)^k (\,s_{(1^k)}[s_{\pi}](Z)\,)' = \sum_\nu \ell_{\pi\nu}\, s_{\nu'}(Z)
\mathcal{I}_{\mathrm{D}}(\delta_1, 0) = \sup_{\pi \in \Pi(\mu_1, \mu_2)}\inf_{\lambda_1 \in \mathbb{R}_{+}}\left[ \lambda_1 \delta_1 + \int g_{\lambda, 1}\, d \pi \right] = \sup_{\pi \in \Pi(\mu_1, \mu_2)}\inf_{\lambda \in \mathbb{R}^2_{+}}\left[ \langle \lambda , \left(\delta_1, 0 \right) \rangle + \int_{\mathcal{V}}g_{\lambda}\, d \pi\right].
\frac{Y(\mathcal{I}_{\alpha_{1}}^{i_{1}})_{\omega_{2}\omega_{1}}^{i_{2}m_{2}}}{Y(\mathcal{I})_{\omega_{2}\omega_{1}}^{i_{2}m_{2}}}=g_{\alpha_{1}\omega_{1}\omega_{2}}^{i_{1}i_{2}m_{2}}=\frac{Y(\mathcal{I}_{\omega_{2}}^{m_{2}})_{\omega_{1}\alpha_{1}}^{i_{1}i_{2}}}{Y(\mathcal{I})_{\omega_{1}\alpha_{1}}^{i_{1}i_{2}}}\in\mathbb{C}
\sum_{i=1}^qc_im_{P_i,S}(P) \leq (c_1-c_2)m_{P_1,S}(P)+c_2\sum_{i=1}^qm_{P_i,S}(P)+O(1) &\leq (c_1-c_2)h(P)+c_2\sum_{i=1}^qm_{P_i,S}(P)+O(1).
\mathcal{L}(\Lambda) =\mathcal{L}(\cup_{j=1}^{n}f_{j}(\Lambda)) &= \sum_{j=1}^{n}\mathcal{L}(f_{j}(\Lambda))-\sum_{1\leq i<j\leq n}^{n}\mathcal{L}(f_{i}(\Lambda)\cap f_{j}(\Lambda))+\sum_{1\leq i<j<h\leq n}\mathcal{L}(f_{i}(\Lambda)\cap f_{j}(\Lambda)\cap f_{h}(\Lambda)) - &\cdots +(-1)^{n-1}\mathcal{L}(\cap_{j=1}^{n}f_{j}(\Lambda)).
(\delta_\mathrm{mRBA})^2 (\chi, \Phi) =~ \delta_\mathrm{mRBA}\big( \delta_\mathrm{Hoch}(\chi), ~ - \widetilde{\delta}_\mathrm{Hoch}(\Phi) - \Psi^n (\chi) \big) =~ \big( (\delta_\mathrm{Hoch})^2 (\chi), ~ (\widetilde{\delta}_\mathrm{Hoch})^2 (\Phi) + \widetilde{\delta}_\mathrm{Hoch}\circ \Psi^n (\chi) - \Psi^{n+1}\circ \delta_\mathrm{Hoch}(\chi) \big) = 0.
\frac{C_{q,n}}{(k-2-q)!}= - 2 \sum_{m=2}^{q+1}a_{k+m-1-q}\biggl[(n-1) m_{(m-1,\dot{0}}+ \sum_{p=1}^{[(m-1)/2]}m_{(m-1-p,p,\dot{0})}\biggr] - \sum_{m=1}^{q}b_{k+m-1-q}m_{(m,\dot{0})}- \frac{n(n-1)}{k(k-1)}q(2k-q-1) a_{k-q}- \frac{n}{k-1}q b_{k-q-1}, l=0, 1, \ldots, k-3,
v^{(0)}=u^n, v^{(1)}=v^{(0)}+\Delta t L(v^{(0)}), v^{(2)}=\frac{3}{4}v^{(0)}+\frac{1}{4}(v^{(1)}+\Delta t L(v^{(1)})), v^{(3)}=\frac{1}{3}v^{(0)}+\frac{2}{3}(v^{(2)}+\Delta t L(v^{(2)})), u^{n+1}=v^{(3)}.
d s^{2}= -{\tilde{n}}^{2}({\tilde{x}}^{\mu},{\tilde{z}}) d{\tilde{\tau}}^{2}+{\tilde{a}}^{2}({\tilde{x}}^{\mu},{\tilde{z}}) d{\tilde{x}}_{i}^{2}+{\tilde{b}}^{2}({\tilde{x}}^{\mu},{\tilde{z}}) d{\tilde{z}}^{2}.
t^{m}\mathcal{MB}^D_{1/t}(f) - \mathcal{P}_{t}(M;\mathbb{Z}) = (1+t)\mathcal{Q}(t), \frac{1}{t^{m}}\mathcal{MB}^D_{t}(f) - \mathcal{P}_{1/t}(M;\mathbb{Z}) = \left(1+\frac{1}{t}\right)\mathcal{Q}\left(\frac{1}{t}\right), \mathcal{MB}^D_{t}(f) - t^{m}\mathcal{P}_{1/t}(M;\mathbb{Z}) = (1+t)t^{m-1}\mathcal{Q}\left(\frac{1}{t}\right).
\lim_{x \to y}|u(x)|= \lim_{x \to y}\frac{|\varphi(x) u(x) - \varphi(y)u(y)|}{|x - y|}\frac{|x-y|}{|\varphi(x)|}= |\nabla (\varphi u)(y)| |\nabla \varphi(y)|^{-1}.
\frac{d}{dt}|\big< \theta(t), f_{\lambda}(t) \big>|^{2}= 2 \big< \theta(t), f_{\lambda}(t) \big> \bigg( \big< \theta'(t), f_{\lambda}(t) \big> + \big< \theta(t), f_{\lambda}'(t) \big> \bigg).
H=(a,a,1/2,-x)={\pi^{{-1\over 2}}\over 2}\Gamma(a+{1\over 2})\Gamma(1-a)(1+x)^{-a}\left( P_{2a-1}[{x^{1\over 2}\over\sqrt{(1+x)}}]+P_{2a-1}[{-x^{1\over2}\over\sqrt{(1+x)}}] \right)
\langle E_{i}( \sigma_{-}) E_{j}( \sigma_{+}) \rangle ={\frac{4 \hbar}{\pi c^{3}}}\; \delta_{i j}\;{\frac{( a / c )^{4}}{\operatorname{s i n h}^{4}( a \tau / 2 c )}}. \hfill \qquad \qquad
\begin{array}{l l}{{A_{l}\, \rightarrow}}&{{A_{1}\, \, ,}}\\{{A_{\overline{{{l}}}}\, \rightarrow}}&{{\overline{{{A}}}_{1}\, \, ,}}\\{{A_{h}\, \rightarrow}}&{{A_{2}\, \, ,}}\\{{A_{\overline{{{h}}}}\, \rightarrow}}&{{\overline{{{A}}}_{2}\, \, ,}}\\{{A_{L}\, \rightarrow}}&{{B_{1}\, \, ,}}\\{{A_{H}\, \rightarrow}}&{{B_{2}\, \, \, ,}}\\ \end{array}
\hbox{$\frac{1}{2}$}( S_m, S_m )^a + V_m^a S_m =i \hbar \Delta^a S_m,\qquad\hbox{$\frac{1}{2}$}\{S_m, S_m \}_\alpha + V_\alpha S_m =i \hbar \Delta_\alpha S_m;
- 2 x^{+}= x^{+}(0) \left(\tau^{+}(1)\overline{\tau}^{-}\right) &= \left(x^{+}(0)\tau^{+}\right)(1)\overline{\tau}^{-}+\tau^{+}(1)\left(x^{+}(0) \overline{\tau}^{-}\right) &= 0+\tau^{+}(1) \overline{\tau}^{+}.
2 2 f ({\bf \eta}) * h ({\bf \eta}) = \exp [ i ( \hbar / 2 \epsilon^{2}) \epsilon^{i j}\partial_{i}^{a}\partial_{j}^{b}] f ({\bf \eta}_{a}) h ({\bf \eta}_{b}) |_{{\bf \eta}_{a}={\bf \eta}_{b}={\bf \eta}}.
\frac{E^{i}}{c}= - \frac{\partial}{\partial x^{i}}\frac{\Phi}{c}- \frac{1}{c}\frac{\partial}{\partial t}A^{i}= F^{i 0}\ \ , \ \B^{i}= \epsilon^{i j k}\left[ \frac{\partial}{\partial x^{j}}A^{k}- \frac{\partial}{\partial x^{k}}A^{j}\right] = - \epsilon^{i j k}F^{j k}\ \ , \ \i , j , k = 1 , 2 , 3 ,
\frac{\beta_{j^{\prime}; j}}{\beta_{j^{\prime}; j}^{( 1 0 )}}= b_{0}^{1 / 2}( \frac{2 ( s^{\prime}- s ) - | m | + | m^{\prime}| b_{0}}{( 2 s^{\prime}+ \frac{3}{2}) + ( | m^{\prime}| - | m | - 2 s - \frac{3}{2}) b_{0}}) ,
\theta'(x)=- \sin^2\theta(x)+(m-p)\frac{\sin 2\theta(x)}{2}\tanh x\frac{h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}&-\frac{m-1}{2}\cos^2\theta(x)\frac{(3-p)h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}\frac{\sin 2h(x)}{h(x)}.
r_{4}= r_{3}\frac{\big | \sqrt{2 a^{2}- b}- \sqrt{2}| a | \big |^{\frac{| a |}{\sqrt{2 ( 2 a^{2}- b )}}+ \frac{1}{2}}}{\big ( \sqrt{2 a^{2}- b}+ \sqrt{2}| a | \big )^{\frac{| a |}{\sqrt{2 ( 2 a^{2}- b ) )}}- \frac{1}{2}}}
\lim_{M\rightarrow\infty}a^{\dag}\left( t\right) \mathcal{P}_{M}\varphi =\lim_{M\rightarrow\infty}\left[ \delta\left( t\right) a\mathcal{P}_{M}\varphi+\overline{\gamma\left( t\right)}a^{\dag}\mathcal{P}_{M}\varphi\right] & =\overline{\delta\left( t\right)}\bar{a}\varphi+\overline{\gamma\left(t\right)}a^{\ast}\varphi.
Z=\sqrt{\pi}\int{\cal D}G{\cal D}\bar{A}{\cal D}\Omega{\cal D}\Lambda \,, exp \bigg( \int \{G^{\mu\nu}_a [-\alpha G_{\mu\nu}^a + F_{\mu\nu}^a (\bar{A})-2{\cal D}_\mu^{(\bar{A})}\Lambda_\nu^a]-{M^{-1}}^{\mu\nu}_{ab}\Omega_\mu^a\Omega_\nu^b\}dx \bigg) ,
\gamma\,=\,{8\Lambda\|\psi(\cdot,0)\|_{L^2}^2\,c_f^{3,2,2}}/(c_f^{1,2,0}c_f^{3,2,0}),D\,=\,{32\pi^2 D_r\,c_f^{3,2,2}}/{(c_f^{3,2,0})^2},
\frac{\Delta t}{\epsilon^2}\sum_{\ell=0}^{n-1}\Bigl(\frac{1}{(1+\frac{\Delta t}{\epsilon^2})^{n-\ell}}-e^{-\frac{(n-\ell)\Delta t}{\epsilon^2}}\Bigr)^2 \le C\frac{\Delta t}{\epsilon^2}\sum_{\ell=0}^{n-1}\frac{1}{(n-\ell)^2}\le C\frac{\Delta t}{\epsilon^2}.
\sup_{\boldsymbol{x}\in[0,1]^d}\Sigma_{\boldsymbol{J}_n}(\boldsymbol{x},\boldsymbol{x})\lesssim\|(\boldsymbol{\Psi}^T\boldsymbol{\Psi})^{-1}\|_{(2,2)}\sup_{\boldsymbol{x}\in[0,1]^d}\|\boldsymbol{\psi}_{\boldsymbol{J}_n}(\boldsymbol{x})\|^2\leq Q_2n^{-1}2^{\sum_{l=1}^dJ_{n,l}}
\phi (\theta )=E\left[\exp\left(\mathrm{i}\theta S_{\infty}^{\chi^{2}_{r}}\right)\right]=\exp \left( \int_{0}^{\infty}\left( \exp(\mathrm{i}u\theta )-1-\mathrm{i}u\theta \right) \mu_{\alpha/d}(du)\right),
\mathcal{D}_{\mathcal{S}}^2(\mathbb{X}, \mathbb{Y})^2 = \mathbb{E}_{X,X^\prime}[\langle S^2(\hat X^1), S^2(\hat X^{\prime,1})\rangle_{\mathcal{H}^2}] &+ \mathbb{E}_{Y,Y^\prime}[\langle S^2(\hat Y^1), S^2(\hat Y^{\prime,1}) \rangle_{\mathcal{H}^2}] & -2\mathbb{E}_{X,Y}[\langle S^2(\hat X^1), S^2(\hat Y^1)\rangle_{\mathcal{H}^2}].
\max_{P_M\in\mathcal{P}(\mathcal{M}_n)}I_Q(M;\mathbf{Z})\leq\max_{m\in\mathcal{M}}\mathsf{D}\Big(Q^{(\mathcal{B}_n)}_{\mathbf{Z}|M=m,I,\mathbf{U}}\Big|\Big|Q^n_{Z|U}\Big|Q^{(\mathcal{B}_n)}_{I,\mathbf{U}}\Big)\leq e^{-n\lambda},
Q^i :=Q_0\cap\{x\in \mathbb{R}^n\,:\, f_k(x)\leq f_k(x^0), k\in I\setminus\{i\}\}, \ \i \in I, Q :=Q_0\cap\{x\in \mathbb{R}^n\,:\, f_k(x)\leq f_k(x^0), k\in I\}.
S_N(x,t)\ge \; c\Big\{N^{1/4}(x-N) +(aN^{2/4}+N^{-3/4})\frac{(x-N)^2}{2!}&+( a^2N^{3/4}+aN^{-2/4}+a N^{-7/4})\frac{(x-N)^3}{3!}&+(a^3N^{4/4}+a^2N^{-1/4}+aN^{-6/4}+a N^{-11/4})\frac{(x-N)^4}{4!}\Big\}\ge 0.
\left\|\partial_{N}w^t_{21}- \frac 1 2 f \right\|_{C^{k-2,\alpha}(\Gamma^0)}\le C\epsilon \left\|w^t_{21}\right\|_{C^{k-1,\alpha}(\overline{\Omega^0})}+ \| \partial_{\nu^0, 0}w^t_{21}\|_{C^{k-2,\alpha}(\Gamma^0)}&\le C\epsilon \left\|f\right\|_{C^{k-2,\alpha}(\Gamma^0)}+ C\| f \|_{C^{k-3,\alpha}(\Gamma^0)}
T_{s_2}(\mathbf{T}_{(+,-,-,+)}+\mathbf{T}_{(-,+,+,-)}) = q(\mathbf{T}_{(+,-,-,+)}+\mathbf{T}_{(-,+,+,-)}), T_{s_2}(\mathbf{T}_{(-,+,-,+)}+\mathbf{T}_{(+,-,+,-)}) = (\mathbf{T}_{(-,+,-,+)}+\mathbf{T}_{(+,-,+,-)}+\mathbf{T}_{(+,-,-,+)}+\mathbf{T}_{(-,+,+,-)}),