harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\operatorname{Re}a_{K,\alpha}(f,f)\geq\left( \gamma\sum_{n=1}^{\infty}d_{n,n}c_{n,s}^{2}-C_{c}\sum_{\substack{n,m=1 n\neq m}}^{\infty}|d_{n,m}|C_{n,s}C_{m,s}\right) \rVert f\rVert_{H^{s}\left( \Omega\right)}^{2}\geq\tilde{\gamma}\rVert f\rVert_{H^{s}\left( \Omega\right)}^{2}.
\phi([p,qp,q][a,ba,b]) = \phi[r, sr,s] ~\ref{lem:SPidempotents}&= (Ps,rP) &= (Pb\cap Pq,pP\cap aP) &= (Pq,pP)(Pb, aP) &= \phi[p,qp,q]\phi[a,ba,a].
\sum_{d_1 \dots d_H \leq y}= \sum_{d_1 \leq X}\sum_{d_2 \leq X}\sum_{d_3 \dots d_H \leq y/(d_1 d_2)}+\sum_{d_1 \leq X}\sum_{d_3 \dots d_H \leq y/(d_1X)}\sum_{X \leq d_2 \leq y/(d_1 d_3 \dots d_H)}+\sum_{d_2 \dots d_H \leq y/X}\sum_{X \leq d_1 \leq y/(d_2 \dots d_H)}.
\mu(S_1 S_2)^{-1}\mu(S_1) \mu(S_2) \, \rho(x,\omega) \, \mu(S_2)^{-1}\mu(S_1)^{-1}\mu(S_1 S_2) = \rho((S_1 S_2)^{-1}S_1 S_2 (x,\omega)) = \rho(x,\omega).
\phi-\overline{\phi}\in C([0,T^*]; H^3), \ \phi_t \in C([0,T^*]; H^2), & u\in C([0,T^*]; H^{s'})\cap L^\infty([0,T^*]; H^3),\phi \nabla^4 u \in L^2([0,T^*] ; L^2), & u_t \in C([0,T^*]; H^1)\cap L^2([0,T^*] ; D^2)
\mathrm{H}_{\mathcal{H}_{\hbar,V}}^{s}:=\left\{u \in \mathrm{H}^{-\infty}_{\mathcal{H}_{\hbar,V}}: \left(I+\mathcal{H}_{\hbar,V}\right)^{s / 2}u \in \ell^{2}\left(\hbar \mathbb{Z}^{n}\right)\right\},
N^{I i}\equiv \frac{\gamma^{\tau \tau}\Pi_{\tau}^{i}+ ( \gamma^{\tau \sigma}\mp 1 / \sqrt{- \gamma}) \Pi_{\sigma}^{i}}{\gamma^{\tau \tau}\Pi_{\tau}^{+}+ ( \gamma^{\tau \sigma}\mp 1 / \sqrt{- \gamma}) \Pi_{\sigma}^{+}}~ ,
t\hat{\otimes}s=\frac{1}{(r+p)!}\sum_{\sigma\in S_{r+p}}sgn(\sigma)t^{i_{\sigma(1)}\cdot\cdot\cdot i_{\sigma(r)}}s^{i_{\sigma(r+1)}\cdot\cdot\cdot i_{\sigma(r+p)}}e_{i_{1}}\otimes e_{i_{2}}\otimes\cdot\cdot\cdot\otimes e_{i_{r+p}}.
\mathrm{vol}(\mathcal{I}\mathfrak{w}\mathcal{I})\sum_{l=0}^{\infty}(q^{2s_k+1})^lq^{-l}(\varpi^{-l},\varpi^{-l})_2^{k-1}\int_{\mathcal{O}^*}\gamma_{\psi'}^{-1}(\varpi^{-l}x)\psi^{-1}(\varpi^{-l}x)\, dx.
n ={\frac{\alpha}{\pi}}( x_{0}\Omega )^{2}\int \int d t d t^{\prime}\bigl [{\frac{( s i n \, t - s i n \, t^{\prime})^{2}}{( t - t^{\prime}+ i \epsilon )^{4}}}-{\frac{c o s \, t ~ c o s \, t^{\prime}}{( t - t^{\prime}+ i \epsilon )^{2}}}\bigr ]
f\left(x \right) = \sum_{i=1}^N \frac{w_i}{\sigma_i}k\left(\frac{x-\mu_i}{\sigma_i}\right), (\mu_i,\sigma_i)_{i=1}^N \Big|N \stackrel{}{\sim}G_0(\mu,\sigma) (w_1,\ldots,w_N) \Big| N \sim (a,\ldots,a), \;\; a>0 N \sim \pi(N) N =1,2,\ldots ,
-{\frac{1}{2}}\nabla^{2}F^{- 1}+ \partial^{i}\phi \partial_{i}F^{- 1}= 0 \ , \ -{\frac{1}{2}}\nabla^{2}K + \partial^{i}\phi \partial_{i}K + \partial_{u}\nabla_{i}{\cal A}^{i}= 0 \ ,
({\mathfrak A}^{1})'=\Big(\Delta_{s,\infty}^{1,r}=\lim_{n}\Delta_{s,n}^{1,r}:=T_{1s}^r\prod_{k=s+1}^\infty p^{-1}C(T_{1k})\mid 2\leq s,\,r\in{\mathbb F}_p\setminus\{0\}\Big)''.
\left\vert E, p_1^{\mbox{min}},\ldots, p_{N-1}^{\mbox{min}}\right>=\left\vert\begin{array}{c}E, \left[0\right] \left[ q \right] \end{array}\right>=\left\vert \begin{array}{c}E, 0, \ldots , 0 q_1,\ldots,q_k,\ldots,q_{N}\end{array}\right>.
\sum_{n \geq 0}\frac{(a)_n(b)_n}{(c)_n(q)_n}z^n = \frac{(c/b)_{\infty}(bz)_{\infty}}{(c)_{\infty}(z)_{\infty}}\sum_{n \geq 0}\frac{(\frac{abz}{c})_n(b)_n}{(bz)_n(q)_n}\left(\frac{c}{b}\right)^n.
P_\varepsilon -z=P_{\varepsilon ,\beta}-z+(V_0-V_{0,\beta})=\begin{cases}(P_{\varepsilon ,\beta}-z)(1+(P_{\varepsilon ,\beta}-z)^{-1}(V_0-V_{0,\beta})) \hbox{and also}(1+(V_0-V_{0,\beta}) (P_{\varepsilon ,\beta}-z)^{-1}) (P_{\varepsilon ,\beta}-z).\end{cases}
\langle{\cal O}[ \overline{{\psi}}, \psi , A , h ] \rangle = \frac{1}{Z}\sum_{n = 0}^{\infty}\frac{( - 1 )^{n}}{n !}\langle{\cal O}[ \overline{{\psi}}, \psi , A , h ] \Big ( S_{M}[ \overline{{\psi}}, \psi ] \Big )^{n}\rangle_{0},
\begin{aligned}\overline{f}(x) := \sup\left\{\limsup_{n \rightarrow \infty}R_n(\lambda)h_n(x_n) \, \middle| \, x_n \rightarrow x\right\}, \underline{f}(x) := \inf\left\{\liminf_{n \rightarrow \infty}R_n(\lambda)h_n(x_n) \, \middle| \, x_n \rightarrow x\right\},\end{aligned}
r_4 = r_3\frac{\big|\sqrt{2a^2-b}-\sqrt{2}|a|\big|^{\frac{|a|}{\sqrt{2(2a^2-b)}}+\frac{1}{2}}}{\big(\sqrt{2a^2-b}+\sqrt{2}|a|\big)^{\frac{|a|}{\sqrt{2(2a^2-b))}}-\frac{1}{2}}}
p^{G \times G}\circ \Psi^{G \times G}\circ \Omega(g) =p^{G \times G}\circ \Psi^{G \times G}(\tilde{g})=p^{G \times G}(\tilde{g}(0), \tilde{g}(1)) &=\tilde{g}(0) = (\tilde{g}_1(0), \tilde{g}_2(0)) = (g(0), g(1)) = \Psi^G(g).
W_{m,k,n}(z_M;z_K;z_N)=\sum_{\ell=1}^k\frac{1}{\ell!}\!\!\!\sum_{\substack{\sqcup_{i=1}^\ell M_i=M, \sqcup_{i=1}^\ell N_i=N, \sqcup_{i=1}^\ell K_i=K, K_i\ne\emptyset,~(|M_i|,|K_i|,|N_i|)\ne (0,1,0)}}\!\!\!\prod_{i=1}^\ell \tilde W_{|M_i|+|K_i|,|N_i|}(z_{M_i},z_{K_i};z_{N_i}),
9(n-1) - 3\sum_{j=1}^n g(e,Je_j)^2 + \sum_{\alpha=1}^3 \{(\sum_{i=1}^n g(e_i,J_{\alpha}Je_i))^2 + \sum_{j=1}^ng(\xi_{\alpha},Je_j)^2 \}&= 9(n-1).
\mathcal{E}Q_x(\mathcal{E}) =B(x)\bigl(Q_x(\mathcal{E})-Q_{x+1}(\mathcal{E})\bigr) +D(x)\bigl(Q_x(\mathcal{E})-Q_{x-1}(\mathcal{E})\bigr) \ \ (x=0,1,\ldots).
\left( \sum_{i=1}^n \sum_{h=0}^{h_0-1}\sum_{\substack{0 \le h_1 \le K+1-r 0 \le h_2 \le r-1 h_1 + h_2 = h}}A_{ij,h_1}y_{i, h_2+1}t^{h-h_0}\right) - \left(\sum_{h=0}^{h_0-1}\mathbf{b}_{j,h}t^{h-h_0}\right).
(b_1-a_1)x_1+\cdots+(b_p-a_p)x_p \geq \alpha + \beta, (b_1-a_1)x_1+\cdots+(b_p-a_p)x_p-x_{p+1}- (\alpha + \beta)x_{p+2}= 0, (b_1-a_1)x_1+\cdots+(b_p-a_p)x_p \geq 0.
\mathcal{P}_{n+1}(x) =\rho_n (x-c_n)\mathcal{P}_n(x)-d_n (x-a_n)(x-b_n)\mathcal{P}_{n-1}(x), n\geq 0, & \mathcal{P}_{-1}(x) = 0, \mathcal{P}_0(x) = 1,
(iv)h=\bar{T}-\bar{U}=\frac{1}{2}\dot{\rho}^{2}+\frac{\rho^{2}}{8}[\dot{\varphi}^{2}+(\sin^{2}\varphi)\dot{\theta}^{2}]+\frac{\omega^{2}}{2\rho^{2}}-\frac{U^{\ast}}{\rho}
F_p \left( \phi, \phi + \eta_1, \cdots, \phi + \eta_p \right) = \eta^{a_1}_1 \eta^{a_2}_2 \cdots \eta^{a_p}_p\, \, \partial^{(1)}_{a_1}\cdots \partial^{(p)}_{a_p}F_p (\phi, \phi, \cdots, \phi) + O \left(\eta^2_i \right)
\tilde{\psi}_{n}\left( \mathbf{p}\right) = \tilde{\psi}_{n}^{\left( 0 \right)}\left( \mathbf{p}\right) - T \sum_{m = - \infty}^{\infty}\tilde{S}_{F}^{n - m}\left( \mathbf{p}\right) \tilde{\eta}_{m}\left( \mathbf{p}\right) ,
{\cal S}= -{\frac{1}{2 \pi \beta^{\prime}}}\int d^{6}\sigma{\frac{1}{6 !}}\epsilon^{a b c d e f}\partial_{a}X^{M}\partial_{b}X^{N}\partial_{c}X^{P}\partial_{d}X^{Q}\partial_{e}X^{R}\partial_{f}X^{S}A_{M N P Q R S}.
\left[P_{ij}\right]=\begin{cases}B\left(n_{i},i-j,1-\epsilon\right) i\in[1,k],0<j\leq i \sum_{m=i}^{n_{i}}B\left(n_{i},m,1-\epsilon\right) i\in[1,k],j=0 1 i=0,j=0,\end{cases}
\mathcal{J}_{\lambda}(u) := \dfrac{1}{2}\int_{\Omega}|\nabla u|^2 \, dx + \dfrac{\alpha}{2}\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dxdy - \dfrac{\lambda}{2}\int_{\Omega}|u|^2 \, dx - \int_{\Omega}F(x,u)\, dx,
J(u_1^n ,u_2^n)=J(u_1^n(x', x_3-y_1^n), u_2^n(x', x_3-y_1^n)) &=J(u_1, 0)+ J(u_1^n(x', x_3-y_1^n)-u_1, u_2^n(x', x_3-y_1^n))+ o_n(1) &=J(u_1, 0)+ J(u_1^n(x', x_3-y_2^n)-u_1(x', x_3+y_1^n-y_2^n), u_2^n(x', x_3-y_2^n))+ o_n(1) &=J(u_1, 0)+ J(0, \tilde{u}_2) + J(u_1^n(x', x_3-y_2^n)-u_1(x', x_3+y_1^n-y_2^n), u_2^n(x', x_3-y_2^n)-\tilde{u}_2) + o_n(1).
2\pi\chi(\Sigma)=\int_{\Sigma}K_{g_{\phi_{\infty}}}d\mathrm{vol}_{g_{\phi_{\infty}}}+\int_{S^2}K_{g_{\vec{\chi}}}d\mathrm{vol}_{g_{\vec{\chi}}}+\int_{S^2}K_{g_{\vec{\Psi}}}d\mathrm{vol}_{g_{\vec{\Psi}}}.
\widehat A-\lambda \widehat B:= [U \ \, U_{\perp}]^*\,(A - \lambda B)\,[V \ \, V_{\perp}]= \left[\begin{array}{cc}U^*(A-\lambda B)V U^*(A-\lambda B)V_{\perp}begin{align*}1mm]U_{\perp}^*(A-\lambda B)V U_{\perp}^*(A-\lambda B)V_{\perp}\end{array}\right].
\begin{aligned}c_{0}= \frac{\Gamma(1-2\beta) \sin\pi\beta}{\pi}\cdot \frac{\tau_{-}(1)}{\tau_{+}(1)}c_{0}' = \frac{\Gamma(1+2\beta) \sin\pi\beta}{\pi}\cdot \frac{\tau_{+}(1)}{\tau_{-}(1)}\end{aligned}
{\cal H}=\frac{1}{2}\pi^2_\varphi+\frac{1}{2}\left(\partial_1\varphi\right)^2+\frac{1}{2}\mu^2\varphi^2+\frac{1}{2}\left(\frac{\phi}{\mu}\right)^2+\left(\frac{\phi}{\mu}\right)(\partial_1\varphi)\ ,
\lim_{n \to \infty}n^{-2 + \ell - \alpha}\dot{\Delta}_{P,\kappa_n}^{(\alpha, \beta; \,\ell)}(x^{(n)}) = 2^\alpha z^{\frac{-2 + \ell -\alpha}{2}}\widetilde{J}_{\alpha+1}^{(\ell)}( \kappa \sqrt{z}),
\frac{F}{V}= - \sum_{\stackrel{n = 1}{n ~ \mathrm{o d d}}}^{\infty}\sum_{k = 1}^{\infty}\sum_{s = 0}^{k - 1}\frac{1}{k^{2}}\left( \frac{1}{4 \pi^{2}\alpha^{\prime}\tau_{2}}\right)^{5}2^{9}\left| \theta_{4}\left( 0 , 2 \tau \right) \right|^{- 1 6}e^{- n^{2}\beta^{2}/ 4 \pi \alpha^{\prime}\tau_{2}}
\Lambda_{\{1\}}=A_1, \Lambda_{\{2\}}=A_{N_1},\Lambda_{\{3\}}=A_{N_2}, \Lambda_{\{4\}}=A_0, \Lambda_{\{1,2\}}=L_1\otimes 1, \Lambda_{\{2,3\}}=M_2, \Lambda_{\{3,4\}}=1\otimes K_1, \Lambda_{\{1,2,3\}}=L_2, \Lambda_{\{1,2,3,4\}}=A_2, \Lambda_{\{2,3,4\}}=K_2.
D^{\leq 0}:= \{E \in D(\mathcal A) \mid H^0(E) \in \mathcal T, H^i(E)=0 \}, D^{\geq 0}:= \{E \in D(\mathcal A) \mid H^{-1}(E) \in \mathcal F, H^i(E)=0 \}.
u^{\varepsilon,\delta}(t_\delta,x_\delta)=u^\varepsilon(t_\delta',x_\delta)-\delta^{-1}|t_\delta-t_\delta'|^2\quad &v_{\varepsilon,\delta}(s_\delta,y_\delta)=v_\varepsilon(s_\delta',y_\delta)+\delta^{-1}|s_\delta-s_\delta'|^2.
\sum_{\mathbf{i}\in\partial R_n^{u_n}\cap\mathbb{Z}^d}\bigl|\theta_n(\mathbf{i})\bigr|^2 \leq C(d){\lambda}_{n}^{d-1}u_n\biggl[\sum_{\mathbf{i}\in\mathbb{Z}^d}\bigl|\alpha(\mathbf{i})\bigr|\biggr]^2 = \mathrm{o}\bigl({\lambda}_{n}^{[3d -2\beta]}L({\lambda}_{n})^2\bigr).
\P(O_S = X_1) \P(O_S = X_2) = \prod_{v \in X_1 \cap X_2}p_v^2 \prod_{v \in X_1 \triangle X_2}p_v(1-p_v) \prod_{v \notin X_1 \cup X_2}(1-p_v)^2 &=\P(O_S = X_1 \cup X_2) \P(O_S = X_1 \cap X_2).
l_{\bar q}^\mu = \left(k_g + k_{\bar q}^\prime - k_{\bar q}\right)^\mu ={\vec k_{g_{\!\perp}}^{\,2}\over 2k_g^+}\ \eta^\mu \quad{\rm and}\quad l^\mu_q = \left(k_g + k_q - k_q^\prime\right)^\mu ={\vec k_{g_{\!\perp}}^{\,2}\over 2k_g^+}\ \eta^\mu
{\nabla'}_{X\wedge Y}Z =(-1)^k X\wedge{\nabla'}_Y Z+(-1)^{l(k-1)}Y\wedge{\nabla'}_X Z{\nabla'}_{X}(Y\wedge Z) =({\nabla'}_X Y)\wedge Z+(-1)^{l(k-1)}Y\wedge{\nabla'}_X Z
E_{13}=\int_K \nabla \Pi_1^{\nabla}g \cdot \nabla \overline{v_h}- \int_K \nabla g \cdot \nabla \overline{v_h}=\int_K \nabla(\Pi_1^{\nabla}g -g )\cdot \nabla \overline{v_h}begin{align*}1ex] =\int_K\nabla(\Pi_1^{\nabla}g -g )\cdot \nabla (\overline{v_h}-\tilde{v}_{\pi})\leq|\Pi_1^{\nabla}g -g |_{1,K}|v_h -\tilde{v}_{\pi}|_{1,K}begin{align*}1ex] \leq Ch_K|g |_{1,K}|v_h |_{2,K},
\Delta\Bigl( \frac{{x_1}^2 x^\alpha}{(\alpha_1 + 1)(\alpha_1 + 2)}\Bigr) = x^\alpha + \sum_{k=2}^n \frac{(\alpha_k - 1) \alpha_k}{(\alpha_1 + 1)(\alpha_1 + 2)}\frac{{x_1}^2 x^\alpha}{{x_k}^2}.
\gamma g+\beta u+\alpha Au =\gamma g+\beta(\frac{E}{G^2}g+u_{+}+u_{-}) &+ \alpha(\frac{2E}{G^2}g-\beta u_{+}+u_{-}) &=A^2u=\frac{4E}{G^2}g+\beta^2 u_{+}+u_{-}.
|\psi_1(x_1,\cdots,x_{i},x_{i+1})-\psi_2(x'_1,\cdots,x'_{i},x'_{i+1})| &\ \ \leq (L_\varphi(\sum_{j=1}^{i+1}|x_j-x'_j|+\hat{\mathbb{E}}[|B_{s_2}-B_{s_1}|]))\wedge (2C_\varphi) &\ \ \leq C_1(\sum_{j=1}^{i+1}|x_j-x'_j|\wedge 1+\sqrt{s_2-s_1}),
g^{- 1}= \left( \begin{array}{c c c}{- 2 \lambda_{2},}&{- 6 \lambda_{3},}&{- 4 \lambda_{4}}\\{- 6 \lambda_{3},}&{4 \lambda_{3}( \lambda_{2}^{2}- 4 \lambda_{4}) ,}&{\lambda_{2}\lambda_{3}}\\{- 4 \lambda_{4},}&{+ \lambda_{2}\lambda_{3},}&{- 2 \lambda_{2}\lambda_{4}+ \frac 3 4 \lambda_{3}}\\ \end{array}\right)
||\Omega||_{W^{\alpha_0,1}(B_2\backslash B_1)}:=\int_{B_2\backslash B_1}|\Omega|+\int_{B_2\backslash B_1}\int_{B_2\backslash B_1}\frac{|\Omega(x)-\Omega(y)|}{|x-y|^{d+\alpha_0}}dxdy\leq c_1,
\int_{a}^b d_{\alpha}r_k f(\{r_j\})\left(q_k g(\{r_j\})\right)=f_k(b)g_k(b)-f_k(a)g_k(a)-\int_{a}^b d_{\alpha}r_k g(r_1,\dots,\alpha r_k,\dots,r_N)\left(q_k f(\{r_j\})\right)
\begin{aligned}-\frac{1}{t^5}+ \frac{3}{t^8}-\frac{8}{t^{11}}< \theta < - \frac{1}{t^5}+\frac{3}{t^8}-\frac{7}{t^{11}}t + \frac{1}{t^5}-\frac{2}{t^8}< \theta' <t+\frac{1}{t^5}-\frac{1}{t^8}t^4 +3t - \frac{1}{t^8}< \theta'' < t^4+ 3t -\frac{1}{t^9}\end{aligned}
:\ \ \mathcal{D}=\{1\},Y=1; \ \mathcal{D}=\{2\},Y=1; \ \mathcal{D}=\{1,2\},Y=1; \ \mathcal{D}=\{1\},Y(\eta)=\eta, :\ \ \mathcal{D}=\{1\},Y=1; \ \mathcal{D}=\{2\},Y=1,
d s_{Q^{1 , 1 , 1}}^{2}= \frac{3}{8 \Lambda}( d \psi -{\cal A})^{2}+ \frac{3}{4 \Lambda}\sum_{i = 1}^{3}\left( d \theta_{i}^{2}+ \sin^{2}\theta_{i}d \phi_{i}^{2}\right) ,
\left( \begin{array}{l l}{a^{\prime \prime}}{b^{\prime \prime}}{c^{\prime \prime}}{d^{\prime \prime}}\end{array}\right) = \left( \begin{array}{l l}{a}{b}{c}{d}\end{array}\right) + \left( \begin{array}{l l}{a^{\prime}}{b^{\prime}}{c^{\prime}}{d^{\prime}}\end{array}\right)
\frac 1 3 ( \lambda^{0^{\prime}})^{2}\varepsilon^{i j}\sum_{\pm}\left( \begin{array}{l}{2 \bar{\varepsilon}_{\pm}\gamma^{m}\partial_{i}\theta_{\pm}\bar{\theta}_{\pm}\gamma_{m}\partial_{j}\theta_{\pm}}\\{- \bar{\varepsilon}_{\pm}\gamma^{m}\theta_{\pm}\partial_{i}\bar{\theta_{\pm}}\gamma_{m}\partial_{j}\theta_{\pm}}\\ \end{array}\right) ,
0= R_{S_0}(e_{\otimes}^{S_{\mathrm{int}}},\tilde s_1(x)S_{\mathrm{int}}+ \tilde s_0(x)\sum_{n\ge 2}S_n\lambda^n -\sum_{n\ge 2}\d j^{(n)}(x)\lambda^n)\ ,\quad x\in\mathcal{O}_1,
=t\cdot v-\stackrel[i=0]{n-1}{\sum}t^{i+1}l_{i}+\stackrel[i=2]{n-1}{\sum}t^{i}(\stackrel[j=i]{n-1}{\sum}T^{j-i+1}l_{j})-\stackrel[k=2]{n-1}{\sum}t^{k}(\stackrel[j=k]{n-1}{\sum}T^{j-k+1}l_{k})=
H(z, w) = \frac{1}{\sqrt{w^{2}-1}\sqrt{z^{2}-1}}\frac{1-zw}{w-z}\Big( \frac{z - \tau}{\tau z - 1}\frac{\tau w - 1}{w - \tau}\Big)^{n}\prod_{k=1}^{n}\frac{\sigma_{k}w - \tau}{\tau w - \sigma_{k}}\frac{\tau z - \sigma_{k}}{\sigma_{k}z - \tau}.
- \frac{7}{1 6}\kappa_{1 0}\left( \gamma_{P}\lambda{\bar{\psi}}_{M}\gamma^{P}\epsilon^{*}- \frac{1}{1 6 8 0}\gamma_{{P_{1}\ldots P_{5}}}\lambda{\bar{\psi}}_{M}\gamma^{P_{1}\ldots P_{5}}\epsilon^{*}\right)
\mathcal{P}_{n+1}(x)= \left(x-\dfrac{\theta}{\zeta+n}\right)\mathcal{P}_n(x)-\dfrac{n(2\zeta+n-1)}{4(\zeta+n)(\zeta+n-1)}(x^2+\omega^2)\mathcal{P}_{n-1}(x), n \geq 1,
\nabla_{\mu}\nabla^{\mu}\Phi = \frac{\kappa^{2}}{2}T_{3}\; e^{3 \Phi}V ( T ) \frac{3 + 2 e^{- 2 \Phi}\nabla_{\alpha}T \nabla^{\alpha}T}{\sqrt{1 + e^{- 2 \Phi}\nabla_{\alpha}T \nabla^{\alpha}T}},
\langle\rho^*(xy)\lambda,v\rangle= \langle\lambda,\rho(xy)^{-1}v\rangle = \langle\lambda,(\sigma(x,y)\rho(x)\rho(y))^{-1}v\rangle = \langle\lambda,\overline{\sigma(x,y)}\rho(y)^{-1}\rho(x)^{-1}v\rangle = \langle\sigma(x,y)\lambda, \rho(y)^{-1}\rho(x)^{-1}v\rangle = \langle\sigma(x,y)\rho^*(x)\rho^*(y)\lambda,v\rangle
\sum_m m^{-(1+\delta)}\left( \frac{m+b/a}{m+ (b-j)/a}\right)^2 \leq \sum_m m^{-(1+\delta)}\left(\frac{m+b/a}{m}\right)^2 = \sum_m m^{-(1+\delta)}\left(1+\frac{b}{am}\right)^2 & \lesssim \sum_m m^{-(1+\delta)}\left[ 1+ \frac{1}{m^2}\right] \coloneqq C_2 < \infty.
\varepsilon_{i j k}S_{j}v_{k}= \left[ \vec{S}, \vec{v}\right]_{i}= - \frac{S_{B}}{\omega}a_{i}, \quad \varepsilon_{i j k}S_{j}a_{k}= \left[ \vec{S}, \vec{a}\right]_{i}={S_{B}}{\omega}v_{i},
-(\frac{m}{e})^2 [\partial_{\mu}\hat{P}^{a}_{\nu}- \partial_{\nu}\hat{P}^{a}_{\mu}+ \frac{m}{e}\epsilon^{abc}\hat{P}^{b}_{\mu}\hat{P}^{c}_{\nu}] = m^2 \tilde{W}^{a}_{\mu\nu}.
S \approx \int d^{10}x \sqrt{|g|}\left[ e^{-\tilde{\Phi}}\left( R +|\nabla \tilde{\Phi}|^2 -| dB_2^{(2)}|^2 \right) -| dB_2^{(1)}|^2 -|dD_4|^2 \right] -\ldots
{\cal L}_{1 / m^{2}}^{( 3 + 1 ) s p i n}= \frac{\alpha}{7 2 0 \pi m^{2}}\left[ 1 8 F^{\nu \lambda}F_{\nu \lambda , \mu}^{\mu}+ 7 F^{\nu \lambda , \mu}F_{\nu \lambda , \mu}- 2 F_{\lambda}^{\nu \lambda ,}F_{\nu \mu ,}^{\mu}\right] ,
\otimes^L_{n}(F_1,\dots,F_n)X = \sum_{(U,V)\in \mathrm{mFlg}_nX}{L(U_1\backslash V_1) \otimes \dots \otimes L(U_{n-1}\backslash V_{n-1}}) \phantom{\otimes^L_{n}(F_1,\dots,F_n)X =}{}\otimes F_1 ( U_1\backslash V_0 ) \otimes \dots \otimes F_n ( U_n\backslash V_{n-1}) .
\lim_{R\rightarrow\infty}\lim_{i\rightarrow\infty}\Psi_{i}(x,R)=c_{n,\sigma p}\lim_{R\rightarrow\infty}\lim_{i\rightarrow\infty}\int_{B^{c}_{R}(0)}\frac{u^{p-1}_{i}(y)}{|y|^{n+\sigma p}}dy\geq0.
\frac{L}{2}- \sigma = \frac{R^{2}}{U_{0}}\int_{U / U_{0}}^{\infty}\frac{d y}{\sqrt{( y^{4}- 1 ) ( y^{4}- 1 + \epsilon )}}\approx \frac{R^{2}U_{0}^{2}}{3 U^{3}}\approx \frac{1}{3 b y^{3}},
S_{N}^{( 4 )}( x , y ) = K_{N}( x , y ) - \psi_{3}( x ) \sum_{l = 1}^{4}[ A_{0}C_{0 0}^{- 1}C_{0}]_{3 l}\ \varepsilon \psi_{l}( y ) - \psi_{4}( x ) \sum_{k = 1}^{4}[ A_{0}C_{0 0}^{- 1}C_{0}]_{4 k}\ \varepsilon \psi_{k}( y ) ,
D\left( \left\{t(j)\right\}_{j=1}^{N}\right) :=\sup_{s\in\left( 0,1\right)^{2},\;x\in\mathbb{R}^{2}}\left\vert\frac{1}{N}\sum_{j=1}^{N}\sum_{m\in\mathbb{Z}^{2}}\chi_{I\left( s,x\right)\cap\Omega}\left( t\left( j\right) +m\right) -\left\vert I\left(s,x\right) \cap\Omega\right\vert \right\vert \;.
h\varpi_p = \left(\begin{array}{cc}xp^{-1}y zp^{-1}t\end{array}\right) = \left(\begin{array}{cc}p^{-1}t^{-1}\det(h) y 0 t\end{array}\right) \left(\begin{array}{cc}1 0 t^{-1}p^{-1}z 1\end{array}\right),
\Gamma_{ij}:= (t_1t_3^{-1}-t_1^{-1}t_3)(c_{ij}^{0}+ c_{ij}^{1}) + (t_2t_3 - t_2^{-1}t_3^{-1})(c_{ij}^{2}+ c_{ij}^{3}) +(t_1^{-1}t_2^{-1}-t_1t_2)(c_{ij}^{4}+c_{ij}^5)
\cdot{_3 F}_{2}\left\{\begin{array}{l}{- n_{1}, - l + | m | , l + | m | + \delta_{1}+ \delta_{2}+ 1}\\{| m | + \delta_{1}+ 1 , - n + | m | + 1}\\ \end{array}\biggr | 1 \right\}E_{n_{1}n_{2}}^{l m}
\frac{\partial l}{\partial x_{i}^{(k)}}=\frac{\partial l}{\partial\hat{x}_{i}^{(k)}}\frac{1}{\sqrt{\sigma_{B}^{(k)^{2}}+\epsilon}}+\frac{\partial l}{\partial\sigma_{B}^{(k)^{2}}}\frac{2(x_{i}^{(k)}-\mu_{B}^{(k)})}{m}+\frac{\partial l}{\partial\mu_{B}^{(k)}}\frac{1}{m}
\sum_{\substack{p\leq n p\notin\mathcal{S}}}\dfrac{\log p}{p-1}\leq \dfrac{\log(2n^2+l)}{n-1}\left(2\log 4\dfrac{n}{\log n}+\sqrt{n}\right)+\dfrac{\log n}{\log(2n^2+l)}\sum_{i=1}^{r}\dfrac{\gamma_{l,p_i}(n)\log p_i}{n-1}+\dfrac{n\log 4}{n-1}.
\dot{\beta}(t, u^{1}, \dotsc, u^{m-2}) ={}\dot{\gamma}(t) + u^{1}\dot{X}_{1}(t) + \dotsb + u^{m-2}\dot{X}_{m-2}(t) &+ \dot{u}^{m-1}(t, u^{1}, \dotsc, u^{m-2}) X_{m-1}(t) &+ u^{m-1}(t, u^{1}, \dotsc, u^{m-2}) \dot{X}_{m-1}(t).
\nabla_{\phi m}\nabla_{\phi l}X^{\beta}=\nabla_{0m}\nabla_{0l}X^{\beta}-\nabla_{0p}X^{\beta}\cdot U_{ml}^p+\nabla_{0l}X^p \cdot U_{pm}^{\beta}+\nabla_{\phi m}X^k \cdot U_{lk}^{\beta}+X^k \nabla_{\phi m}U_{lk}^{\beta},
H_{G / H}= \frac{1}{2}\left( \pi_{\alpha}- \frac{i \hbar}{2}\Gamma_{\alpha}\right) g^{\alpha \beta}\left( \pi_{\beta}+ \frac{i \hbar}{2}\Gamma_{\beta}\right) = \frac{1}{2}\pi_{\alpha}g^{\alpha \beta}\pi_{\beta}+ V_{G / H}\, ,
\mathfrak{M}(l\,|\,\mu,\lambda_1,\lambda_2) =\frac{\Gamma\bigl(1-l/\tau\bigr)}{\Gamma^l\bigl(1-1/\tau\bigr)}\,\prod\limits_{j=0}^{l-1}\frac{\Gamma\bigl(1-j/\tau\bigr)\Gamma\bigl(1+\lambda_1-j/\tau\bigr)\Gamma\bigl(1+\lambda_2-j/\tau\bigr)}{\Gamma\bigl(2+\lambda_1+\lambda_2-(l+j-1)/\tau\bigr)}.
\psi_a(x'_1,x'_2) = \frac{16}{\pi^2}\sum_{k_1,k_2>0,\}\frac{1}{k_1}\,\frac{1}{k_2}\,\frac{1}{k^2_1+k^2_2+a^2}\, \sin k_1x'_1\,\sin k_2x'_2
\prod_{\alpha}{\frac{\sinh{\frac{1}{2}}( y_{i}- u_{\alpha}- i \gamma )}{\sinh{\frac{1}{2}}( y_{i}- u_{\alpha}+ i \gamma )}}= \prod_{j}{\frac{\sinh{\frac{1}{2}}( y_{i}- y_{j}- 2 i \gamma )}{\sinh{\frac{1}{2}}( y_{i}- y_{j}+ 2 i \gamma )}}{\frac{\sinh{\frac{1}{2}}( y_{i}- y_{j}+ i \gamma )}{\sinh{\frac{1}{2}}( y_{i}- y_{j}- i \gamma )}}
W^{j}_{6}=-\varepsilon\sum_{i=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{x_{j}}w^{1}-\partial_{x_{j}}w^{2})\left(\Theta_{p_{i}}(t,x,\mu^{2},Dw^{2})\partial^{2}_{x_{i}x_{j}}w^{1}-\Theta_{p_{i}}(t,x,\mu^{2},Dw^{2})\partial^{2}_{x_{i}x_{j}}w^{2}\right)\dx.
\frac{\partial\varphi}{\partial u}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial t}+\frac{\partial\varphi}{\partial x}\right),\ \frac{\partial\varphi}{\partial v}=\frac{1}{2}\left(-\frac{\partial\varphi}{\partial t}+\frac{\partial\varphi}{\partial x}\right).
(\ell+h^\vee)h- C_{U_1,U_2}=(\ell+h^\vee)(h_3-h_1-h_2)-\frac{1}{2}(C_{U_1\otimes U_2}-C_{U_1}\otimes 1_{U_2}-1_{U_1}\otimes C_{U_2}) = (\ell+h^\vee)h_3-\frac{1}{2}C_{U_1\otimes U_2}.
\left( \begin{array}{c}{\psi_{1}}{\psi_{2}}\end{array}\right) \longrightarrow \left( \begin{array}{c}{\widetilde \psi_{1}}{\widetilde \psi_{2}}\end{array}\right) = \left( \begin{array}{c}{a}{c}\end{array}\begin{array}{c c}{b}{d}\end{array}\right) \left( \begin{array}{c}{\psi_{1}}{\psi_{2}}\end{array}\right) ,
\mathbb{E}\{(\hat{\mu}- \mu)^2 \}= (\mathbb{E}\hat{\mu}- \mu)^2 + \mathrm{Var}\bigl( \mathbb{E}\bigl\{\hat{\mu}| J \bigr\}\bigr) + \mathbb{E}\bigl\{\mathrm{Var}( \hat{\mu}|J ) \bigr\}.
P_{1}=\frac{v_{\epsilon}(1)-v_{\epsilon}(0)}{2}\sum_{i\neq j}\int_{F_{ij}}U_{\epsilon,i}(x)\left(f_{ij}(x)+f_{ji}(x)\right)dS_{ij}(x) P_{2}=-\sum_{i=1}^{N}\int_{G_{N}}f(x)\nabla_{i}\left[U_{\epsilon,i}(x)V_{\epsilon,i}(x)\right]dx P_{3}=-\sum_{i=1}^{N}\int_{G_{N}}U_{\epsilon,i}(x)V_{\epsilon,i}(x)\nabla_{i}f(x)dx.
\begin{array}{r c l}{{K_{\nu}( t )}}&{{=}}&{{\displaystyle \frac{1}{2}\left( \frac{t}{2}\right)^{\nu}\int_{0}^{\infty}d \alpha ~ \alpha^{- 1 - \nu}\exp \left[ - \alpha - \frac{t^{2}}{4 \alpha}\right] , ~ | \arg t | < \frac{\pi}{2}, ~ \mathrm{R e ~}t^{2}> 0 .}}\\ \end{array}
V^{\nu}:=\{x\in\mathbb{R}^{n}:~|x\cdot\nu_{i}|<1/2i=1,\ldots,n-1\}~, Q^{\nu}:=\{x\in\mathbb{R}^{n}:~|x\cdot\nu_{i}|<1/2i=1,\ldots,n\}~,
dd^*(\kappa dr) = -\kappa'' dr + \left(\frac{m-1}{r^2}\kappa - \frac{m-1}{r}\kappa'\right)dr - \tilde{d}\kappa' - \frac{m-1}{r}\tilde{d}\kappa, d^*d(\kappa dr) = \tilde{d}\kappa' + \frac{1}{r^2}(\tilde{d}^*\tilde{d}\kappa)dr + \frac{m-3}{r}\tilde{d}\kappa.
\mathcal{E}_1(s)=2\,(u_x(0,s)\bar{u}_s(0,s))+\tfrac{\alpha}{\gamma}v_x(0,s)v_s(0,s), &\mathcal{E}_2(s)=\tfrac{\alpha}{2\gamma}\big[\,v_{xx}(0,s)+\tfrac12 v^2(0,s) -\gamma|u(0,s)|^2\,\big]^2.
{\Phi_R}= \left\{{\left.{\left({{{\hat x}_u},{{\hat y}_u}}\right)}\right|{{\hat x}_u}\ge k{{\hat y}_u}}\right\},{\Phi_L}= \left\{{\left.{\left({{{\hat x}_u},{{\hat y}_u}}\right)}\right|{{\hat x}_u}\le - k{{\hat y}_u}}\right\}
\sum_{m,n\geq 0}d^{m-\delta}z^{n-\delta}\langle\eta_m|U_3(\psi,w)|\eta_n\rangle=\sum_{m,n\geq 0}d^{m-\delta}z^{n-\delta}\langle P(\eta_m,\eta_n;w^\ast)|\psi\rangle\,.
\phi_\gamma=\phi_{\alpha}\phi_{\beta}-\eta_{\beta}\otimes\xi_{\alpha}=-\phi_{\beta}\phi_{\alpha}+\eta_{\alpha}\otimes\xi_{\beta},\quad \xi_{\gamma}=\phi_{\alpha}\xi_{\beta}=-\phi_{\beta}\xi_{\alpha}, \ \ \eta_{\gamma}=\eta_{\alpha}\circ\phi_{\beta}=-\eta_{\beta}\circ\phi_{\alpha},
\begin{aligned}\|u\|_{C^{k,2+\alpha}_{WF}([0,T]\times \bar S_{n,m})}\leq C\left(\|g\|_{C^{k,\alpha}_{WF}([0,T]\times\bar S_{n,m})}+ \|f\|_{C^{k,2+\alpha}_{WF}(\bar S_{n,m})}\right).\end{aligned}
\sum_{d\mid k/2}\sum_{e\mid d}\frac{1}{\varphi\left(d/e\right)}\sum_{\substack{a>B\sqrt{e/d}(a,d)=e}}\frac{\mu^2(a)}{\varphi(a^2)}\leq\sum_{a>B/\sqrt{k'}}\frac{\mu^2(a)}{\varphi(a^2)}\sum_{e\mid (a,k')}\sum_{\substack{(a,d)=e d\mid k'}}1 \leq 2^L\sum_{a>B/\sqrt{k'}}\frac{\mu^2(a)}{\varphi(a^2)},
\int_{\Sigma}\mathcal{K}^{\Sigma,L}\frac{1}{\sqrt{L}}d\sigma_{\Sigma,L}+\sum_{i=1}^n\int_{\gamma_i}k^{L,s}_{\gamma_i,\Sigma}\frac{1}{\sqrt{L}}d{s}_L=2\pi\frac{\chi(\Sigma)}{\sqrt{L}}.
\lim_{r\to\infty}r^{N+2s+k}D^k F_N(r) = \sum_{k \le 2j \le 2k}(-1)^j \,\alpha_{j,k}\,\lim_{r\to\infty}r^{N+2s+2j}F_{N+2j}(r) &= \sum_{k \le 2j \le 2k}(-1)^j \alpha_{j,k}\, \ell_{N+2j, \, 0}.
a_\Gamma = a_{\Gamma}^{(\mathrm{s})}\oplus a_{\Gamma}^{(\mathrm{as})}= a_{\Gamma}^{(1)}\oplus a_{\Gamma}^{(2)}\oplus a_{\Gamma}^{(\mathrm{as})}, begin{align*}4pt] A_\Gamma = A_{\Gamma}^{(\mathrm{s})}\oplus A_{\Gamma}^{(\mathrm{as})}= A_{\Gamma}^{(1)}\oplus A_{\Gamma}^{(2)}\oplus A_{\Gamma}^{(\mathrm{as})},
\{\hat U_{-b,-a}(x)\le y\}=\{\hat U_{-a-1}\circ\cdots\circ \hat U_{-b}(x)\le y\}=\{V_{a}\circ\cdots\circ V_{b-1}(x)\le y\}&=\{V_{a+1}\circ\cdots\circ V_{b-1}(x)\le U_a(y)\}=\ldots=\{x\le U_{b-1}\circ\cdots\circ U_a(y)\}=\{x\le U_{a,b}(y)\}.
\frac{\partial\vec{B}}{\partial t}+(\vec{v}\cdot\nabla)\vec{B}=\frac{D\vec{B}}{Dt}=(\vec{B}\cdot\nabla)\vec{v}-\vec{B}(\nabla\cdot\vec{v})+\frac{1}{\mu_{0}\sigma}\nabla^{2}\vec{B}
\left( F ._{\mathcal{P}}\xi \right) (k,[v],r,s) = \sum_{k_{1},k_{2}}F \left(k-k_{2}+k_{1}, [v+k_{1}\theta], r-k_{1}, s+k_{2}\right) &\hphantom{\sum_{k_{1},k_{2}}}\cdot \xi \left(k_{1}, [v+k_{1}\theta], k_{2}, [v+b (r+s)+(k_{2}-k)\theta]\right) .
(p_2)_\sharp \pi^z(E) =\pi^z(p^{-1}_2(E))=\mu(\Phi^{-1}(E))=\mu\left(\Phi^{-1}(E)\cap \left(\bigcup_{n\in\mathbb{N}}K_n\right)\right) =\mu\left(\bigcup_{n\in\mathbb{N}}\Phi^{-1}(E)\cap\left(K_n\setminus \cup_{1\leq j\leq n-1}K_j\right)\right) \leq \sum_{n\in\mathbb{N}}\mu(\Phi_n^{-1}(E))=0,
\mu(t+K) =\int_{K}e^{h(x-t)}dx=\int_{K}e^{h(x+t)}dx=\int_K \frac{1}{2}(e^{h(x-t)}+e^{h(x+t)})dx &\geq \int_K e^{(h(x-t)+h(x+t))/2}dx.