harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(-1)^{l+m-1}n^{l+2m}2^{l+2m}P_{l+m}(m) \frac{(l+2m)!}{(l+2m)!}\alpha_p(1)^{l+2m}= (-1)^{l+m-1}n^{l+2m}2^{l+2m}P_{l+m}(m),
\left( \begin{array}{c c}{0}&{A^{-}( \ell )}\\{A^{+}( \ell )}&{0}\\ \end{array}\right) \left( \begin{array}{c}{\chi_{E \ell}( r )}\\{\chi_{E \ell + 1}( r )}\\ \end{array}\right) = \sqrt{E + \frac{1}{2 ( \ell + 1 )^{2}}}\left( \begin{array}{c}{\chi_{E \ell}( r )}\\{\chi_{E \ell + 1}( r )}\\ \end{array}\right) ,
M_{a,b,c,d}=\begin{pmatrix}a^{2^{h}+1}a^{2^h}b ab^{2^h}b^{2^{h}+1}a^{2^h}c a^{2^h}d b^{2^h}c b^{2^h}d ac^{2^h}bc^{2^h}ad^{2^h}bd^{2^h}c^{2^{h}+1}c^{2^h}d cd^{2^h}d^{2^{h}+1}\end{pmatrix},
F_{n-1}F_{n+1}-F_{n}^{2}=det[\begin{matrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{matrix}]=det[\begin{matrix}1&1\\ 1&0\end{matrix}]^{n}=(det[\begin{matrix}1&1\\ 1&0\end{matrix}])^{n}=(-1)^{n}.
f^{( n_{0}+ 1 )}( \sigma^{2}) = - \sum_{k > 0}k C_{k}( f^{\prime}( \sigma^{2}) )^{- k}\frac{f^{\prime \prime}( \sigma^{2})}{f^{\prime}( \sigma^{2})}+ \sum_{l > 0}C_{l}l ( f^{\prime}( \sigma^{2}) )^{l}\frac{f^{\prime \prime}( \sigma^{2})}{f^{\prime}( \sigma^{2})}
I_{MN}= 1 + \sum_{d=1}^\infty (qz)^d \sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}s_\lambda(a_1,\dots,a_N)s_\lambda(b_1,\dots,b_N) \Omega_{\frac{1}{M}}^{-1}(\lambda) \Omega_{\frac{1}{N}}^{-1}(\lambda).
M(u(t),v(t)) := \|u(t)\|^2_{L^2}+ c\|v(t)\|^2_{L^2}= M(u_0,v_0), E(u(t),v(t)) := \frac{1}{2m}\|\nabla u(t)\|^2_{L^2}+ \frac{c}{4M}\|\nabla v(t)\|^2_{L^2}+ \emph{Re}(\lambda \langle v(t), u^2(t) \rangle ) = E(u_0,v_0),
f_{1}(x,m^{\hat v}_{t},\hat{v}_{1}(x,m^{\hat v}_{t}))+Du^{\hat v}_{1}(x,t).g_{1}(x,m^{\hat v}_{t},\hat{v}_{1}(x,m^{\hat v}_{t}))= H_{1}(x,m^{\hat v}_{t},Du^{\hat v}_{1}(x,t)) \, ,
\begin{aligned}\int_{\pi/2 - \delta}^{\pi/2 + \delta}|S_n(e^{it})|^2 w_\mathbb{T}(e^{it}) dt \leq \int_{\pi/2 - \delta}^{\pi/2 + \delta}|P_n(\cos t)|^2 w_\mathbb{T}(e^{it}) dt \leq \int_{-1}^{1}|P_n(x)|^2 w_{[-1,1]}(x) dx = \lambda_n(\mu_{[-1,1]},0).\end{aligned}
\begin{aligned}|f(a,x,N)|=\frac1{|a+xw_1+Nw_2|^k}\rightarrow 0, &\left|\frac{\partial f}{\partial y}(x,N)\right|=k\frac{|w_2|}{|a+xw_1+Nw_2|^{k+1}}\rightarrow 0 \quad N\to \infty,\end{aligned}
\sum_r c_{n,r}(\mu)d_r^{-1}= \sum_{t_1,\ldots,t_n}\mu(t_1)\cdots \mu(t_n) \Big(\sum_r m_{t_1,\ldots,t_n}^r\Big) \frac{1}{d_{t_1}\cdots d_{t_n}}&\leq \sum_{t_1,\ldots,t_n}\mu(t_1)\cdots \mu(t_n) \frac{\dim(U_{t_1})\cdots\dim(U_{t_n})}{d_{t_1}\cdots d_{t_n}}&= \Big(\sum_t \mu(t)\,\frac{\dim(U_t)}{d_t}\Big)^n.
{\bf \nabla}W = iW \int^\beta_0 dt^{\prime}[\nabla A_0 (t^{\prime},{\bf x}) -\partial_{t^{\prime}}{\bf A}(t^{\prime},{\bf x})] =-iW\int^\beta_0~{\bf E}(t^{\prime},{\bf x})~dt^{\prime}, \;\; E_i \equiv F_{0i}
{\Gamma^{1}}_{a}= \left\{\left( \begin{matrix}{0}&{\sigma_{1}}\\{- \sigma_{1}}&{0}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{0}&{- I}\\{- I}&{0}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{I}&{0}\\{0}&{- I}\\ \end{matrix}\right) \right\},
\begin{array}{l}\theta_1=\theta^u(v)-\theta^v(w)+\theta^u(w)-\theta^w(v) \theta_2=\theta^v(u)-\theta^w(v)+\theta^v(v)-\theta^u(u) \theta_3=\theta^v(w)-\theta^w(u)+\theta^v(u)-\theta^u(w) \theta_4=\theta^w(v)-\theta^u(w)+\theta^w(w)-\theta^v(v) \end{array}
\left|\binom{m}{k}a_{m}\right|_F = |\underbrace{a_{m}+ a_{m}+ \dotsb + a_{m}}_{\binom{m}{k}\;}|_F \le \max(\underbrace{|a_{m}|_F, \dotsc, |a_{m}|_F}_{\binom{m}{k}\;}) = |a_{m}|_F < c
\mathrm{A m p l i t u d e ~}= \frac{R_{- p}}{( \theta - i \theta_{0})^{p}}+ \frac{R_{- p + 1}}{( \theta - i \theta_{0})^{p - 1}}+ \cdots + R_{0}+ R_{1}( \theta - i \theta_{0}) + \cdots .
\Upsilon_\pi(x_1,\dots, x_n) = \frac{|\pi|}{4}\sum_{\begin{subarray}{c}S \subset \{1,\dots,|\pi|\}\end{subarray}}(|S|-1)! (|S^c|-1)!\ \bigg| \sum_{i \in \underset{j\in S}{\bigcup}\pi_j}x_i \bigg| \ .
\zeta^{( T )}\left( z , \beta , \sigma \right) = \sum_{n = 0}^{\infty}\sum_{l = 1}^{\infty}D_{n}^{T}( \Lambda_{n , l}^{T}( \sigma ) )^{- z}+ \sum_{n = 1}^{\infty}\bar{D}_{n}^{T}( \Lambda_{n , 0}^{T}( \sigma ) )^{- z},
\|\mathcal{H}_2\| \leq \bar{c}_1\Delta_{\max}\sum_{i \in [M]}\sum_{t=0}^{T-1}\exp\left(-\bar{c}_2 N_in_x \left(\frac{\alpha \rho^{(i)}\|\Sigma^{(i)}_t\|}{\rho^{(i)}\|\Sigma^{(i)}_t\| + \sqrt{n_x}}\right)^2\right) ,
K(p,N)\, \int_\Omega |\nabla u|^p\,dx = K(p,N)\, \mu\left(\bigcup_{i \in I}C_i\right) = K(N,p) \sum_{i \in I}\mu(C_i) &\leq (1+\varepsilon) \sum_{i \in I}\alpha(C_i) \leq (1+\varepsilon)\, \liminf_{k \to \infty}\sum_{i \in I}\alpha_k(C_i) &\leq (1+\varepsilon)\, \liminf_{k \to \infty}(1-s_k)\,[u_k]^p_{W^{{s_k},p}(\mathbb{R}^N)}.
G_{M N}^{( I )}d x^{M}d x^{N}={\frac{r^{2}}{\sqrt{Q_{e}Q_{p}}}}( - d t^{2}+ d x_{1}^{2}) +{\frac{\sqrt{Q_{e}Q_{p}}}{r^{2}}}d r^{2}+ \sqrt{\frac{Q_{e}}{Q_{p}}}\sum_{i = 2}^{5}d x_{i}^{2}+ \sqrt{Q_{e}Q_{p}}d \Omega_{3}^{2}.
d_Y(a,y)=\min\{\rho(a,y),\rho'(a,y)\}\leqslant\min\{\rho(a,b)+\rho(b,y),\rho'(a,b)+\rho'(b,y)\}&= d_B(a,b)+\min\{\rho(b,y),\rho'(b,y)\}.
\hat\phi '_1=\frac{\phi}{\sqrt{2{\rm Tr}\phi^2}},\ \ \hat\phi '_2=\frac{1}{\sqrt{2[-2({\rm Tr}\hat\phi '_1{}^3)^2+{\rm Tr}\hat\phi '_1{}^4]}}[-2({\rm Tr}\hat\phi '_1{}^3)\hat\phi '_1+\hat\phi '_1{}^2]
\arg \left(w-U \right) = \psi(w) + \arg(w) = \psi(z_+) + \frac{\pi}{4}-\frac h2 + O(hr) = \int_0^U \frac{dt}{(1-t)^2+1}+ \frac{\pi}{4}-\frac h2 + O(hr) = \frac{\pi}{4}-\frac h2+ \frac{U}{2}+ \frac{U^2}{4}+O(hr) + O(r^3).
E(z_1,z_2,\beta;x)\ll \sum_{\substack{x/2 < n < 2x n \ne x}}\sigma_{a_1,\beta}(n)\sigma_{a_2,\beta}(n)\min \bigg( 1, \frac{x}{T_0|x-n|}\bigg) + \frac{4^{c}+x^{c}}{T_0}\sum_{n=1}^{\infty}\frac{\sigma_{a_1,\beta}(n)\sigma_{a_2,\beta}(n)}{n^{c}}.
\begin{aligned}m,\d,\ \alpha_0,\ \alpha_1,\ \Vert g\Vert_{L_\infty},\ \Vert g^{-1}\Vert_{L_\infty};\ \ \textnormal{the constants}\ \mathfrak{C}_1, \mathfrak{C}_2 \ \textnormal{from}\ \eqref{Garding}; &\textnormal{the parameters of the lattice}\ \Gamma; \ \ \textnormal{the domain}\ \mathcal{O}.\end{aligned}
p\sigma + \frac{\beta}{2}|\mathbf{h}|^2= -\rho u^2\sigma + \varepsilon u_x\sigma+\alpha g'(1/\rho)h(|\psi|^2)\sigma - \left( \int_b^x \rho u \sigma d\xi \right)_t &+ \int_b^x\left[\left(\rho u^2 + p + \frac{\beta}{2}|\mathbf{h}|^2-\alpha g'(1/\rho)h(|\psi|^2)\right)\sigma_x - \varepsilon u_x\sigma_x \right]d\xi.
d(x,y;e^2)=d(t,y;e^2)d\left(\frac{x}{t},\frac{y}{t};e^2d(t,y;e^2)\right),\;s(x,y;e^2)=s(t,y;e^2) s\left(\frac{x}{t}, \frac{y}{t}; e^2d(t,y;e^2)\right)
\partial_t Q(t)^{-1}P^{-1}\begin{pmatrix}w \overline{w}\end{pmatrix}= Q(t)^{-1}P^{-1}\begin{pmatrix}S_2(t) + e^{3i\lambda_1 (\Phi_1-\theta_1)}R_2(t) \overline{S_2(t) + e^{3i\lambda_1 (\Phi_1-\theta_1)}R_2(t)}\end{pmatrix}.
L(\pi\otimes\pi',s) =\prod_p L_p(\pi\otimes\pi',s)=\prod_p\prod_{i=1}^{d}\prod_{j=1}^{e}\Bigl(\,1-\frac{\alpha_{\pi\otimes\pi',(i,j)}(p)}{p^s}\,\Bigr)^{-1}&=\sum_{n\geq 1}\frac{\lambda_{\pi\otimes\pi'}(n)}{n^s},\ \Re s>1
\Xi^{\lambda}(K_{a}\otimes F_{a}\otimes E_{a}K_{a}^{-1}\otimes K_{a})=\sum_{i,j,m,n,o,p}(2c_{j}^{i}-\delta_{j}^{i})\pi(K_{a})_{m}^{j}c_{n}^{m}\pi(F_{a}E_{a}K_{a}^{-1})_{o}^{n}c_{p}^{o}\pi(K_{a}K_{\lambda})_{i}^{p}.
\left(\alpha(q)\alpha(q^{23})\right)^{1/8}+\left((1-\alpha(q))(1-\alpha(q^{23}))\right)^{1/8}&+2^{2/3}\left(\alpha(q)\alpha(q^{23})(1-\alpha(q))(1-\alpha(q^{23}))\right)^{1/24}=1.
\Vert g - h\Vert_{\mathcal{C}(K_j)}\geq \Vert f - h \Vert_{\mathcal{C}(K_j)}- \Vert f - g \Vert_{\mathcal{C}(K_j)}& \geq 2 \, \alpha \, [M (K_j)]^n - \frac{\rho_j^{\, l}\}{(1 -\rho_j^2)^{1/2}}\geq \alpha \, [M(K_j)]^n \, ,
W^{\Omega^*_{z_1},\Omega_{z_2}}\left( \begin{array}{cc}c d b a \end{array}\right)=(-1)^{n-1+\mu +\kappa}W^{(n-1,1)}\left( \left. \begin{array}{cc}c d b a \end{array}\right| z_1 -z_2 \right),
B(v,e) =\begin{dcases}\frac{\prod_{u\in e}x_u}{\lambda^{(p)}(G[S]) x_v^p}, ~v\in e~~ v\in S,\, e\in E(G[S]), 0, ,\end{dcases}begin{align*}2mm]w(e) =\frac{r\prod_{u\in e}x_u}{\lambda^{(p)}(G[S])}.
\prod_{\gamma \in G}\left( \prod_{i=1}^n x_{\sigma(i)}^{a_{\gamma^{-1}(i)}}\right)^{t_{\gamma}}\leq \sum_{\gamma \in G}t_{\gamma}\left( \prod_{i=1}^n x_{\sigma(i)}^{a_{\gamma^{-1}(i)}}\right).
\int f(\vec{x}) L^{\prime(2n)}_{q^{-1/2}\sqrt{-1}}(\vec{x};t;p,q) \Delta^{(2n)}_S(\vec{x};t;p,q) = \int f\big({\pm}\sqrt{-\vec{z}}\big)\Delta^{(n)}_S\big(\vec{z};1,p,t,pt;t^2;p^2,q^2\big)
{\cal A}_{\alpha_{j k}}={\frac{n}{4 \kappa}}\, \oint_{\alpha_{j k}}{\frac{\displaystyle u^{n}\, d u}{\displaystyle \sqrt{1 - e^{- ( 1 + u^{n}) / \kappa}}}}= e^{i \pi ( j + k + 1 ) / n}\sin \bigl ( \pi ( j - k ) / n \bigr ) \,{\cal I}_{n}( \kappa )
\lim_{s \to s_0}\frac{\int_0^1 | \phi_{s_0}' |^2 P_s(t) \,dt}{\int_0^1 | \phi_{s_0}|^2 Q_s(t) \,dt}= \frac{\int_0^1 | \phi_{s_0}' |^2 P_{s_0}(t) \,dt}{\int_0^1 | \phi_{s_0}|^2 Q_{s_0}(t) \,dt}= \lambda_{1,2}(\sigma_{s_0})
\|f_k\|^{2}_{H^2}\leq 2 ( \|a_{k}^{\#}\|_{\infty}^{2}\|\widetilde{f}\|^{2}_{H^2}+ \widetilde{c}_{k}\|p\|^{2}_{H^2}) & \leq c_{k}(\|\widetilde{f}\|^{2}_{H^2}+ \|p\|_{H^2}^{2}) & = c_{k}\|f\|_{b}^{2},
\hat\rho = \mathbf{\Psi_r}\cdot \mathbf{\Psi_r}^*, \mathbf{\Psi_r}= \left[ \sqrt{p_1}|\psi_1 \rangle, \sqrt{p_2}|\psi_2 \rangle , \cdots, \sqrt{p_r}|\psi_r \rangle \right],
f_{j+i, m}F^{j+i}_{\bar{H}}(A^{j}(x))= f_{j+i, m}\circ \mathrm{pr}^{j+i}_{(m)}\circ F_{H}^{j+i}(A^{j}(x))=(\mathrm{pr}^{i}_{(m)}\circ F^{j+i}_{H})(A^{j}(x)),
\deg_{F_2' \sqcup \mathcal{A}_F' \sqcup E}(\iota^*_{E,{F}_2' \sqcup \mathcal{A}_{F'}\sqcup E}(x^{\mathcal{A}_{F'}})x^{F_1'}) = \deg_{F_2' \sqcup \mathcal{A}_F' \sqcup E}(\iota^*_{E,{F}_2' \sqcup \mathcal{A}_{F'}\sqcup E}(x^{{F}_1' \sqcup \mathcal{A}_{F'}})) = \deg_{F' \sqcup E}(\iota^*_{E, F' \sqcup E}(x^{\mathcal{A}_{F'}})).
A^{(1)}=\left( \begin{array}{ccc}A^{(1)ij,kl}A^{(1)ij,0}A^{(1)ij,b}A^{(1)0,kl}A^{(1)00}A^{(1)0b}A^{(1)a,kl}A^{(1)a0}A^{(1)ab}\end{array}\right)
A = \left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right) , \, R = \left( \begin{array}{c c c c}{{q}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{\lambda}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{q}}\\ \end{array}\right) .
F(s) =\sup_{r>0}\big((sr)^q-G(r)\big)^{1/q}=\sup_{\substack{r>0 (sr)^q\geq G(r)}}\left((sr)^q-2^q\left(\frac12 G(r)^{1/q}\right)^q\right)^{1/q}\leq 2\sup_{\substack{r>0 (2sr)^q\geq G(r)}}\left(sr-\frac12 G(r)^{1/q}\right) =\sup_{r>0}\left(sr-G\left(\frac r2\right)^{1/q}\right),
i D [ u ] = \left[ \begin{array}{c c}{{0}}&{{\sigma_{\mu}\partial_{\mu}- i g \sigma_{\mu}u_{\mu}( x )}}\\{{\left( \sigma_{\mu}\partial_{\mu}- i g \sigma_{\mu}u_{\mu}( x ) \right)^{\dagger}}}&{{0}}\\ \end{array}\right] \ ,
\begin{aligned}\|\eta_k-\eta_0\|_{C(B_R(0))}\Big(\rho_k\int_{B_R(0)}{\rm ln}(1+|x|^2)\big|\tilde u_{2,k}(x)+\tilde u_{1,k}(x)\big|{\rm d}x \quad+\int_{B_R(0)}\big|\tilde u_{2,k}(x) +\tilde u_{1,k}(x)\big|{\rm d}x \cdot\big(\|\tilde u_{2,k}\|_*^2+C\|\tilde u_{2,k}\|^2\big)\Big) < \frac{\epsilon}{2}\ \ \mbox{as}\ \k\to\infty.\end{aligned}
\int_D \psi_1(n_1; \lambda_1) \cdots \psi_{k-1}(n_{k-1}; \lambda_{k-1}) \, \psi_{k+1}(n_{k+1}; \lambda_{k+1}) \cdots \psi_g(n_g; \lambda_g)\, h(\lambda_2, \dots, \lambda_g) \, d\lambda_2 \cdots d\lambda_g & = 0
\mathbb{E}\Bigg[\sup_{n\in \mathbb{N}}\bigg( \int_{\mathcal{D}}U(x)\frac{\partial^{\alpha}\Phi_n(x)}{||\Phi_n||_{L^2}}dx\bigg)^2\Bigg] = \lim_{n\rightarrow \infty}\mathbb{E}[V_N] \leq C_{\alpha}< + \infty
G_r(s,x;\varphi)= \sum_{n=1}^\infty\frac{\lambda_f(n)(-\log n)^r}{n^s}\sum_{j=0}^l\varphi^{(j)}\left(\frac{n}{y}\right)\left(-\frac{n}{y}\right)^j\gamma_{j}^{(m-r)}\left(s,\frac{1}{|t|}\right)+ &+O(y^{1-\sigma}(\log y)^r(\log|t|)^{m-r}|t|^{-\frac{l}{2}}\|\varphi^{(l+1)}\|_1).
\langle \omega^*_{\overline{f}}, D_n \mathrm{exp}^* \left(\mathrm{loc}_p c^+_{\mathbb{Q}(\mu_n)}\right) \rangle_{\mathrm{dR}}= D_n \langle \omega^*_{\overline{f}}, \mathrm{exp}^* \left(\mathrm{loc}_p c^+_{\mathbb{Q}(\mu_n)}\right) \rangle_{\mathrm{dR}}
H^{2}_{0}(\Omega_{-}) :=\big\{u_{-}\in H^{2}(\Omega_{-}):(\gamma_0^{-}u_-,\gamma_1^{-}u_-)=(0,0) \big\}\,, H^{2}_{0}(\Omega_{+}) :=\big\{u_{+}\in H^{2}(\Omega_{+}):(\gamma_0^{+}u_+,\gamma_1^{+}u_+,\gamma_{0}^{\partial}u_{+},\gamma_{1}^{\partial}u_{+})=(0,0,0,0)\big\}\,.
\Pr(Y_{\infty}\not\in \tilde{H}') = \Pr(Y_{\infty}\not\in\tilde{H}',Y_{\infty}\in H) + \Pr(Y_{\infty}\not\in\tilde{H}',Y_{\infty}\not\in H) &\leq \Pr\left(\sup_{J \leq l <\infty, 0\leq k \leq 2^{l}-1}\left|\frac{N_{l,k}}{w_{l}}\right| \geq \tilde{c}_{4}\frac{J^{1/2}}{\overline{w}_{J}}\right)+ \Pr(Y_{\infty}\not\in H).
\Big|\int_{\mathbb{R}^3}\widehat{f_1}(\xi)\widehat{f_2}(\eta)\widehat{f_3}(\rho-\xi) \widehat{f_4}(-\rho-\eta)m(\xi,\eta,\rho)\,d\xi d\rho d\eta\Big| &\lesssim \|f_1\|_{L^{p_1}}\|f_2\|_{L^{p_2}}\|f_3\|_{L^{p_3}}\|f_4\|_{L^{p_4}}\|\mathcal{F}^{-1}m\|_{L^1}.
x^2 v_i = q^{2d-4i}x^2z^2v_i = q^{2d-4i}\left(1-q^2(q+q^{-1})\nu_y+q^4\nu_y^2\right)v_i, y^2 v_i = q^{2d-4i}y^2 z^2 v_i = q^{2d-4i}\left(1 - q^{-2}(q+q^{-1})\nu_x + q^{-4}\nu_x^2\right)v_i, \nu_zv_i = q^{2d-4i}\nu_z z^2 v_i = q^{2d-4i}\left(q^{-1}z^2 - q^{-1}+q^{-2}\nu_x + q^2 \nu_y - q \nu_x \nu_y\right)v_i.
T_{r}:=1+6\sum^{\infty}_{n=1}\frac{nq^{2n}}{1-q^{2n}}-30\sum^{\infty}_{n=1}\frac{nq^{10n}}{1-q^{10n}}=\frac{-1}{4}\left(P\left(q^2\right)-5P\left(q^{10}\right)\right),
\sum_{k\geq N}\langle X_n e_k,e_k\rangle= \sum_{k\geq N}\|X_n^{\frac 12}e_k\|^2\geq \sum_{k,l\geq N}\langle X_n^{\frac 12}e_k,e_l\rangle^2= \sum_{k,l\geq N}\langle X_n^{\frac 12}, e_k\otimes e_l\rangle_{\mathcal H}^2.
(a_0,a_1,a_2,a_3) =\left(m+\frac{a}{2},m+\frac{(b-1)}{2},m-\frac{a}{2},m-\frac{(b+1)}{2}\right), (b_0,b_1,b_2,b_3) =\left(m+\frac{(c-1)}{2},m+\frac{(d-1)}{2},m-\frac{(c+1)}{2},m-\frac{(d+1)}{2}\right),
\sigma_{\vec{\alpha}}( z ) = \vartheta \left[ \begin{array}{c}{{\alpha_{2}/ n + 1 / 2}}\\{{\alpha_{1}/ n + 1 / 2}}\\ \end{array}\right] ( z , \tau ) , \theta^{( j )}( z ) = \vartheta \left[ \begin{array}{c}{{1 / 2 - j / n}}\\{{1 / 2}}\\ \end{array}\right] ( z , n \tau ) ,
\mathcal{A}^- = \left\{\begin{aligned}(\tau,v_{s+k+1},\omega_{k+1}) \in[0,\infty) \times \mathbb{R}^d \times \mathbb{S}^{d-1}\textnormal{such that}& \omega_{k+1}\cdot \left(v_{s+k+1}- v_{i_{k+1}}^\prime (\tau;0) \right) \leq 0\end{aligned}\right\}
\Xi_2^2(t) \leq \frac{8 t^2}{m}\sum\limits_{n=1}^{\infty}\left(\frac{1}{n}\sum\limits_{j=1}^{n}\left|c_j-c_{j-1}\right| \right)^2 \leq \frac{32 t^2}{m}\sum\limits_{n=1}^{\infty}|c_{n}-c_{n-1}|^2 &\leq \frac{32 M t^2}{m}\left\| \sum\limits_{n=1}^{\infty}(c_{n}-c_{n-1}) e_n \right\|^2= \frac{32 M t^2}{m}\|x\|_1^2.
( b_{i})^{n}\psi_{p}= \sum_{{{1 \leq j_{k}\neq j_{l}\leq N \atop j_{k}, j_{l}\neq i}\atop ( 1 \leq k < l \leq n )}}\prod_{k = 1}^{n}\frac{1}{z_{i}- z_{j_{k}}}\psi_{p}\, ,
\lim_{s\to\infty}\left\|w(s)-m(\varphi)\psi_d\right\|_{C^2(\{R^{-1}\le|y|\le R\})}=0,\lim_{s\to\infty}\left\|(\partial_sw)(s)\right\|_{C^2(\{R^{-1}\le|y|\le R\})}=0.
H = \iint_S \eta_H(t,\nu)\, M_\nu T_t \dt\,d\nu = \iint_{S-(t_0,\nu_0)}\eta_H(t+t_0,\nu+\nu_0)\T_{t+t_0}M_{\nu+\nu_0}\dt\,d\nu = T_{t_0}M_{\nu_0}\widetilde H,
P_{\lambda}^{m}(z)=lim_{\mu\rightarrow m}P_{\lambda}^{\mu}(z)=\frac{(-\lambda)_{m}(\lambda+1)_{m}}{m!}[\frac{1-z}{1+z}]_{2}^{m/2}F_{1}(-\lambda,\lambda+1;1+m;\frac{1-z}{2}),
p(0,\,x;\,t,\,y) =q(0,\,x;\,t,\,y)\mathbb{E}^{Q}\exp\left\{-\int_{0}^{t}\hat{b}(s,\,Z_{s}^{x,y})dZ_{s}^{x,y}-\frac{1}{2}\int_{0}^{t}\hat{b}^{2}(s,\,Z_{s}^{x,y})ds\right\}.
n!{\|\tilde{h}_n(.,t,x)\|}_{\mathcal{H}^{\otimes n}}^2 \geq \frac{1}{n!}C_9 e^{-2\mu_1 t}\eta(t)^n t^{n\big(\frac{\alpha-\beta}{\alpha}\big)}C_{\alpha,\beta,\epsilon}^n\int_{[1/2, 1]^n}\prod_{i=1}^n (t_{i+1}-t_{i})^{-\beta/\alpha}d\textbf{t},
L = - \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}+ \frac{1}{2}\sum_{i = 1}^{2}\left( D_{\mu}A_{i}\right)^{a}\left( D^{\mu}A_{i}\right)^{a}- \frac{e^{2}}{2}f^{a b c}A_{4}^{b}A_{5}^{c}f^{a d e}A_{4}^{d}A_{5}^{e}+
\theta_{k,i}= 1 - \mu_{k,i}, k=2,3,\ldots,d, \; i=1,2,\ldots,r \theta_{1,i}= \theta_{2,i}^{-1}\cdots \theta_{d,i}^{-1}= 1 - \mu_{1,i}i=1,2,\ldots,r
\begin{array}{ll}B_{ij}(\partial)=C_{ij}(\partial)=(H_1)_{ij}(\partial)=\delta_{ij}\partial\,, i,j=1,\dots,\ell B_{\alpha i}(\partial)=(H_1)_{\alpha i}(\partial)=\alpha(x_i)e_{\alpha}\,, i=1,\dots,\ell,\alpha\in\Delta\,.\end{array}
S=S_{a_{f}}+ S_{\Phi}=\int L_{a_{f}}(\zeta_{f})\,d\zeta^{(1,1,1)}\wedge\cdots \wedge d\zeta^{(2,2,3)}+\int \sqrt{g}L_{\Phi}(\zeta)\,d\zeta^{(111)}\wedge\cdots \wedge d\zeta^{(223)},
g(a_1,a_2,a_3,\ldots,a_r)=a_1^{\sf w-deg}\cdot g^{\sf deh}\big(\frac{a_2}{a_1^{[a_2]}}, \frac{a_3}{a_1^{[a_3]}},\ldots,\frac{a_r}{a_1^{[a_r]}}\big)
\partial_2 g_{\beta, 2}'(x) = - \frac{X'_{\beta, 2}(x)}{f_{\beta,2}'(g_{\beta, 2}(x) )}+ X_{\beta, 2}\frac{f_{\beta, 2}''(g_{\beta, 2}(x))}{(f_{\beta, 2}' ( g_{\beta, 2}(x) ) )^3}.
H^F_D = \tilde{\lambda}^\mu(\tau){\cal H}_\mu(\tau) - \frac 12 \tilde{\lambda}^{\mu \nu}(\tau){\cal H}_{\mu \nu}(\tau) + \int{d^3\sigma \Big[ -A_\tau (\tau, \vec{\sigma}) \Gamma (\tau, \vec{\sigma}) + \mu_\tau (\tau,\vec{\sigma}) \pi^\tau (\tau, \vec{\sigma}) \Big]},
\eta(\eta(\tau_j, \tau_i),\eta(\tau_j,\tau_\ell)) =\eta((\tau_n^q\tau_1)^{n-i}, (\tau_n^q\tau_1)^{n-j}\tau_n^{q-1+D(n+\ell-j)}\tau_{n-j+\ell+1}) =(\tau_n^q\tau_1)^{i-j}\tau_n^{q-1+D(n+\ell-j)}\tau_{n+\ell-j+1}.
V_{\mathcal{D}_{d}}^{\mu[\nu_{N}]}(x) + \log w(x) = -K_{N}x \in [-\alpha_{N}, \alpha_{N}], & V_{\mathcal{D}_{d}}^{\mu[\nu_{N}]}(x) + \log w(x) \leq -K_{N}x \in \mathbf{R}\setminus [-\alpha_{N}, \alpha_{N}].
A_{4 , 1}= - \frac{1}{2}M_{1}V_{1}+ O ( M_{i}^{4}) , \qquad A_{4 , 0}= - \frac{1}{8}V_{\Phi}- \frac{1 3}{1 6}M_{1}V_{1}+ \frac{5}{6 P}e^{- \Phi_{\infty}}M_{2}V_{1}+ O ( M_{i}^{4}) , \, [ 2 m m ]
\sum_{p\le x}\frac 1p = \sum_{p\le x}\frac{\ln p}{p}\frac{1}{\ln p}= \sum_{n=2}^{x}\frac{f(n)-f(n-1)}{\ln n}&= \sum_{n=2}^x \frac{\ln n - \ln(n-1)}{\ln n}+ \sum_{n=2}^x \frac{r(n)-r(n-1)}{\log n}.
(E^{\Lambda_N}_{j+1}, U^{\Lambda_N}_{j+1}, K_{j+1}^{\Lambda_N}(\cdot ; 0)) = \Phi_{j+1}^{\Lambda_N}(E^{\Lambda_N}_j, U^{\Lambda_N}_j, K_j^{\Lambda_N}(\cdot ; 0)), 0\leq j \leq N-1 .
\widetilde{L}^{-1}=\left[ \begin{array}{cc}\widetilde{R}^{-T}0 \widetilde{R}^{-T}\widetilde{Q}G^{T}\end{array}\right], \widetilde{U}^{-1}=\left[ \begin{array}{cc}\widetilde{R}^{-1}-\widetilde{R}^{-1}0 G\widetilde{Q}^{-T}\end{array}\right]
\zeta^1 = \frac{1}{4}\left( \frac{y^1}{F}-(C_1+A_1x_1+B_1x_2)\right)(2F((A_5x_1+B_5x_2+C_5)y^1+(A_6x_1 &+B_6x_2+C_6)y^2)-(E(y^1)^2+Jy^1y^2+K(y^2)^2)-F^2\left(\sum_{l+k\leq 2}M_{lk}x_1^{l}x_2^k \right)) &-\frac{1}{4}F^2(A_3x_1+B_3x_2+C_3)-\frac{1}{2}FNy^2.
\frac{d}{ds}\big|_{s=0}\frac{1}{\Gamma (s)}\int_0^{+\infty}t^{s-1}\frac{1}{V_i}\int_{M_i}\left( \mathrm{tr}e^{-t \Delta^{(2)}}(\tilde x, \tilde x) - (\mathrm{tr}e^{-t \Delta_i}(x,x) -b_1 (M_i))\right) dx dt \to 0.
\begin{pmatrix}w(t) \overline{w(t)}\end{pmatrix}= P Q(t) P^{-1}\begin{pmatrix}w(1) \overline{w(1)}\end{pmatrix}& + P Q(t)\int^t_1 Q(\tau)^{-1}P^{-1}\begin{pmatrix}S_2 + e^{3i\lambda_1 (\Phi_1-\theta_1)}R_2 \overline{S_2 + e^{3i\lambda_1 (\Phi_1-\theta_1)}R_2}\end{pmatrix}(\tau)\d \tau
T_{M_2}l\int d^2\xi (2\pi\alpha^\prime)\epsilon^{ij}\partial_i \varrho\ \left({\tilde{\cal G}}^{(1)}_j-{\textstyle\frac{1}{2\pi\alpha^\prime}}{\tilde A}^{(1)}{}_{j}+{\textstyle\frac{m}{2}}b_{j}\right)\, .
\bigl ( \omega^{a b}\bigr ) = \left( \begin{array}{c c c c}{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\{{- 1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{0}}&{{0}}\\ \end{array}\right) \, \delta ( x - y ) \, .
\omega_{2}= \frac{1}{\sqrt{6}}\left( \frac{1}{8}\sin{\theta_{2}}\, d \theta_{2}\wedge d \phi_{2}- \frac{3}{1 6}\sin^{2}{\mu}\, \sigma_{1}\wedge \sigma_{2}+ \frac{3}{8}\sin{\mu}\cos \mu \, d \mu \wedge \sigma_{3}\right) .
S^\alpha{}_\beta=\begin{pmatrix}0 0 -a\sin(x^3+x^4) -a\sin(x^3+x^4) 0 0 \pm a\cos(x^3+x^4) \pm a\cos(x^3+x^4) -a\sin(x^3+x^4) \pm a\cos(x^3+x^4) a^2 a^2 a\sin(x^3+x^4) \mp a\cos(x^3+x^4) -a^2 -a^2\end{pmatrix}.
\Gamma_c=\left(\begin{array}{ccc}\displaystyle{\gamma^a_{cb}}\displaystyle{{\frac{1}{2}T_g^a{_c}}}\displaystyle{-\frac{{\dot g}_{ac}}{\sin^2\theta}}\displaystyle{\mbox{cot}\theta\partial_c\theta}\end{array}\right)_{a,b,c,=1\cdots M}
b_2 = a_1^2+4 a_2, b_4 = a_1 a_3+2 a_4, b_6 = a_3^2+4a_6, b_8= a_1^2 a_6-a_1a_3a_4+a_2 a_3^2+4a_2 a_6-a_4^2.
\left.\frac{\partial A}{\partial s}\right|_{s=0}\ =\ \frac{1}{2}+ \Bigl(\frac{\partial F_1}{\partial\theta}(\theta_0)\Bigr)^2\frac{r_{xx}(0,0)}{y_0}- \frac{1}{2}\frac{r_{xx}(0,0)r_y(x(0),y(0))}{r_{xx}(x(0),y(0))r_y(0,0)}.
\begin{array}{l}z^0=\alpha_0(Z)=\dfrac{1}{\Delta}(-2u_1p+u_0Dp), z_0=\beta_0(Z)=\dfrac{1}{u_0}Dz^0-\dfrac{2v_1}{\Delta}p+\dfrac{u_1}{u_0\Delta}Dp\end{array}
P^S = -(r^{1-n}S_r/\rho) Q^U_{2,l}= - S_r A_U f_{J_{2,l}}^{(l)}, P^\rho = - r^{1-n}D_r Q^U_{1,l}= -D_r( \rho A_U f_{J_{2,l}}^{(l)}) - \tfrac{n-1}{r}\rho A_U f_{J_{2,l}}^{(l)}, P^U = (r^{1-n}S_r/\rho) Q^S_{1,l}- D_r(r^{1-n}Q^\rho_{1,l}) = -( A_r +U_r A_U ) f_{J_{2,l}}^{(l)}
E_{1}\doteq \bigcap\limits_{x\in \mathbb{Z}^{d},A\in \mathcal{U}}\{\rho \in \mathcal{U}^{\ast}:\rho (\mathfrak{1})=1,\;\rho (|A|^{2})\geq0,\;\rho =\rho \circ \alpha_{x}\}\ ,
T=\frac{(\dot{x}(t),\dot{y}(t))}{\sqrt{\dot{x}(t)^{2}+\dot{y}(t)^{2}}}andN=\frac{(-\dot{y}(t),\dot{x}(t))}{\sqrt{\dot{x}(t)^{2}+\dot{y}(t)^{2}}}.
\int_{-\infty}^{\infty}\frac{d\epsilon}{2\pi}e^{\tau\epsilon/2\pi}\{\frac{1}{1+e^{\beta(\epsilon-\mu)}}-\theta(-\epsilon)\}=\frac{1}{\tau}\{\frac{(\frac{\tau T}{2})}{sin(\frac{\tau T}{2})}e^{\tau\mu/2\pi}-1\},0<\tau T/2\pi<1.
\Vert (S_{l}f)' \Vert_\infty \leq \sum_{k = 0}^{l - 1}k \,|b_k| \leq \bigg(\sum_{k = 0}^{l- 1}|b_k|^{2}w_{k}\bigg)^{1/2}\bigg(\sum_{k = 0}^{l - 1}k^2 w_{k}^{- 1}\bigg)^{1/2}& \leq \Vert f \Vert \, \bigg(\sum_{k = 0}^{l - 1}k^2 w_{k}^{- 1}\bigg)^{1/2}.
\left| \sum\limits_{j=1}^{m}a_j (g_j, h_{l}^{(1)}h_{l}^{(2)})_{\Omega}\right| \leq C d(\mathcal{C}_1(f^{(1)}), \mathcal{C}_2(f^{(2)})) e^{C \epsilon^{-\mu}}\|h_l^{(1)}\|_{H^s_{\overline{\Omega}}}\|h_l^{(2)}\|_{H^s_{\overline{\Omega}}}& + \epsilon \|q_1 - q_2\|_{L^{\infty}(\Omega)}\|h_l^{(1)}\|_{H^s_{\overline{\Omega}}}\|h_l^{(2)}\|_{H^s_{\overline{\Omega}}}.
\begin{aligned}u_{\star, \lambda}(x, t)-u_{q, \lambda}(x, t) &\leq{1\over q}\max\left\{u(y_j, t), u(z_j, t)\right\}\max\{{\lambda}^{{1\over q}-1}(1-\lambda), (1-\lambda)^{{1\over q}-1}\lambda\}(1-r^q)+{1\over j},\end{aligned}
R_\mathrm{ma}^{\rm J}=\log\left|\mathbf{I}+\sum\limits_i\mathbf{H}_i\mathbf{Q}_i\mathbf{H}_i^\mathrm{H}(\mathbf{H}_\mathrm{z}\mathbf{Q}_\mathrm{z}\mathbf{H}_\mathrm{z}^\mathrm{H}+\sigma^2\mathbf{I})^{-1}\right|.
| D^\gamma u(x',s)| =| D^\gamma (u\eta_P)(x',s) | \leq C \sigma^{-n-j}\int_{B(x',\sigma)}|u(y',s)|dy'\,+C\int_{B(x',\sigma)}\frac{|D^k u(y',s)|}{|x'-y'|^{n-k+j}}dy' .
\partial_{\varphi}\left\{\partial_zS^{z}\partial_{\bar{z}}S^{\bar{z}}- \partial_{z}S^{\bar{z}}\partial_{\bar{z}}S^z+{\cal{H}}\right\}=\partial_z p+\partial_{\bar{z}}\bar{p}+\partial_{\varphi}{\cal H}=\partial_{\varphi}\left\{\partial_z S+\partial_{\bar{z}}\bar{S}+{\cal H}\right\}=0\,.
\sum_{\vec{z}+q\vec{y}\in P\mathcal{B}}e(\vec{\beta}\cdot\vec{f}(\vec{z}+q\vec{y}))=I(\mathcal{B},P^d\vec{\beta})\frac{P^n}{q^n}+O\left(\left(C|P^d\vec{\beta}|+1\right)\frac{P^{n-1}}{q^{n-1}}\right).
\ddot{a}=\left( \frac{\left( 2+r\right) r}{\left( 1+r\right)^{2}}\right)^{2}a^{\prime \prime}+\left( 2\frac{\left( 2+r\right)^{2}}{\left( 1+r\right)^{3}}-\frac{4+r}{\left( 1+r\right)^{2}}\right) a^{\prime}-\kappa^{2}r^{2}f^{2}a .
N_\alpha(\xi,t) = w^{(i)}_{\alpha}(0,t) + \xi_i \, \partial_{x_i}w^{(i)}_{\alpha -1}(0,t) + \mathcal{O}(\exp(-\beta_0\xi_i))\mbox{as}\ \ \xi_i\to+\infty, \ \ \xi \in \Xi^{(i)}, i\in \{1, 2, 3\}.
\eta(\eta(\tau_j, \tau_i), \eta(\tau_j, \tau_\ell)) =\eta((\tau_n^q\tau_1)^{n-i+1}, (\tau_n^q\tau_1)^{n-j}\tau_{n-j+\ell})=\eta(1, (\tau_n^q\tau_1)^{i-j-1}\tau_{n-j+\ell}) &=(\tau_n^q\tau_1)^{i-j-1}\tau_{n-j+\ell}.
a ( 1 , 2 , 3 ) = x_{1 2}^{2}g ( 1 , 2 , 3 ) + x_{1 2}^{2}x_{1 3}^{2}\; 2 \partial_{2}\cdot \partial_{3}f ( 1 , 2 , 1 , 3 ) + \mathrm{c y c l e}\ .
a\cdot y\cdot b=\phi_{\mathbf{X}'}^\mathcal{NT}(a)y\phi_{\mathbf{X}'}^\mathcal{NT}(b) \langle y,w\rangle_{\mathcal{NT}_{\mathbf{X}'}}=(\phi_{\mathbf{X}'}^\mathcal{NT})^{-1}(y^*w)
\alpha^{\prime}m_{\mathrm{v e c}}^{2}\equiv - \alpha^{\prime}\sum_{\sigma , \rho = 0}^{p^{\prime}}G^{\sigma \rho}k_{\sigma}k_{\rho}= 0 , \qquad \sum_{\sigma , \rho = 0}^{p^{\prime}}G^{\sigma \rho}k_{\sigma}\zeta_{\rho}( k ) = 0 .
\frac{\partial v''_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}=\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{12}\hat{Q}^{-1}+P_{N(\check{A}_{p})\cap S}\frac{\partial z_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]},
\frac{1}{2}\sum_{i=0}^{s}\min\left\{V^{(i+1)}_1+V^{(i+1)}_2-V^{(i)}_1-V^{(i)}_2,V^{(i+1)}_1-V^{(i+1)}_2-V^{(i)}_1+V^{(i)}_2\right\}.