harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d s_{E}^{2}= \breve{g}_{\mu \nu}d x^{\mu}d x^{\nu}= ( 1 + r^{2}\breve{\Omega}^{2}) d \tau^{2}- 2 \breve{\Omega}r^{2}d \tau d \tilde{\varphi}+ r^{2}d \tilde{\varphi}^{2}+ d r^{2}+ d z^{2}~ .
\hat{X}^{\Lambda}={\frac{1}{\sqrt 2}}\left( \begin{matrix}{{1 - T U}}\\{{- ( T + U )}}\\{{- ( 1 + T U )}}\\{{T - U}}\\ \end{matrix}\right) , \qquad \hat{F}_{\Lambda}= \left( \begin{matrix}{{S \hat{X}^{0}}}\\{{S \hat{X}^{1}}}\\{{- S \hat{X}^{2}}}\\{{- S \hat{X}^{3}}}\\ \end{matrix}\right) = S \eta_{\Lambda \Sigma}\hat{X}^{\Sigma}\ ,
1\leq\frac{r_{o}\left( t\right)}{r_{i}\left( t\right)}\leq\left(\sqrt{I\left( t\right)}+\sqrt{I\left( t\right) -1}\right)^{2}\leq\left( \sqrt{I\left( 0\right)}+\sqrt{I\left( 0\right) -1}\right)^{2}:=\sigma,\ \ \t\in\lbrack0,T)
\frac{1}{|A_N|}\sum\limits_{\substack{p \leq N p \equiv t \,\, \bmod q}}e\left( f(p) \right) = \frac{1}{|A_N|}\sum_{p \leq N}e \left( f(p) \right) \frac{1}{q}\sum_{j=1}^q e \left( \frac{(p-t)j}{q}\right) &= \frac{\pi(N)}{|A_N|}\frac{1}{q}\sum_{j=1}^q \frac{1}{\pi(N)}\sum_{p \leq N}e \left( f (p) + \frac{j}{q}(p-t) \right)
\xi_{1}= \frac{- 2 \beta_{1}\beta_{2}}{\beta_{1}^{2}+ \beta_{2}^{2}}, \qquad \xi_{2}= 0 \quad \mathrm{a n d \quad}\xi_{3}= \frac{\beta_{2}^{2}- \beta_{1}^{2}}{\beta_{1}^{2}+ \beta_{2}^{2}},
\begin{aligned}\, \|\partial_t^j D_x^l D_v^{m}u\|_{L_p (Q_{1/2})}& \le N (d, \delta, p, j, l, m) \big(\||D_v u|+\lambda^{1/2}|u|\|_{L_p (Q_1)}+ \sum_{k = 0}^{\infty}2^{-2k}\big(|(-\Delta_x)^{1/6}u|^p)^{1/p}_{Q_{1, 2^k}}\big).\end{aligned}
\sum_{i=t_0}^{t}n_i \leq 2n+ \sum_{i=t_0+2}^{t}n_i\leq 2n+(\lambda'+1)\binom{n}{2}- \sum_{i=1}^{t_0+1}n_i \leq 2n+(\lambda'+1)\binom{n}{2}-f(\lambda',n)= 2n+\binom{n}{2}+|I|<2\binom{n}{2}-2n,
[\begin{matrix}1&lnR_{1}&ln^{3}R_{1}\\ 1&lnR_{2}&ln^{3}R_{2}\\ 1&lnR_{3}&ln^{3}R_{3}\end{matrix}][\begin{matrix}A\\B\\C\end{matrix}]=[\begin{matrix}\frac{1}{T_{1}}\\ \frac{1}{T_{2}}\\ \frac{1}{T_{3}}\end{matrix}]
\int\limits_{0}^{\infty}e^{q_1}\, b_1 \, q_1 \, e^{b_1 x}\, e^{-q_1 \, e^{b_1 x}}\, q_2 \, e^{b_2 x}\, dx= e^{q_1}\, q_1 \, q_2 \, \int\limits_{0}^{\infty}b_1 \, e^{b_1 x}\, e^{-q_1 \, e^{b_1 x}}\, e^{b_2 x}\, dx = e^{q_1}\, q_1 \, q_2 \, \int\limits_{1}^{\infty}y^{\frac{b_2}{b_1}}\, e^{-q_1 y}\, dy
r = \|f\|_{L^\infty}^{-1/d}\log \left( 1 + \|f\|_{L^\infty}\right)^{-1/d}, R = \|f\|_{L^\infty}^{-1/d}\log \left( 1 + \|f\|_{L^\infty}\right)^{-(d+\gamma)/(\gamma d)}.
P_{N}(\alpha_{1},\alpha_{2};\beta_{1};q;z)=\sum_{m=0}^{N}\frac{(q^{-N};q)_{m}(\alpha_{1};q)_{m}(\alpha_{2};q)_{m}}{(q;q)_{m}(\beta_{1};q)_{m}}(-1)^{m}q^{m(1-m)/2}z^{m}.
\frac{\partial \Phi_{0 1}}{\partial y}- \varphi ( y ) \frac 1 i \frac{\partial \Phi_{0 1}}{\partial q}= \varphi ( y ) \bigl [ p - 2 i \bigl ( \delta ( q ) - \delta ( 1 - q ) \bigr ) \bigr ] \Phi_{0 1},
\sum_{r=1}^{m-1}\psi(\frac{r}{m})\cdot cos\frac{(2l+1)\pi r}{m}=-\frac{\pi}{m}\sum_{r=1}^{m-1}\frac{r\cdot sin\frac{2\pi r}{m}}{cos\frac{2\pi r}{m}-cos\frac{(2l+1)\pi}{m}},l\in\mathbb{Z}
|e_{k,N}| = \left|\frac{j_{0,k}}{j_{0,N+1}}- \frac{(k-\frac{1}{4})}{(N+\frac{3}{4})}\right| = \left|\frac{b_k(N+\frac{3}{4})-b_{N+1}(k-\frac{1}{4})}{((N+\frac{3}{4})\pi+b_{N+1})(N+\frac{3}{4})}\right|.
\rho ( x ) = \frac{1}{x^{2}}+ \sum_{\{q \}\neq 0}\{\frac{1}{( x - q )^{2}}- \frac{1}{q^{2}}- \frac{2 q \cdot x}{q^{4}}- \frac{1}{q^{6}}( 4 ( q \cdot x )^{2}- q^{2}x^{2}) \}
1 \ge \frac px \sum_{\substack{n \le x: \, p|n R_p(n)<\beta+\frac{t}{\sqrt{\log_2 x}}}}1 \ge \frac px \sum_{\substack{n \le x: \, p|n R_p(n)<\beta +\frac{\sqrt{\log_3 x}}{\sqrt{\log_2 x}}}}1 = \Phi\left(\sqrt{\frac{\log_3 x}{\beta(1-\beta)}}\right) + O\left(\frac1{(\log_2 x)^{1/3}}\right) = 1 + O\left(\frac1{(\log_2 x)^{1/3}}\right)
\frac{1}{A(q^2)}= 1 - \frac{e^2(\xi - \xi_0)}{16q}+ \left(\frac{e^2(\xi - \xi_0)}{8\pi q}\right)^2 - \frac{1}{3}\left(\frac{e^2(\xi - \xi_o)}{8\pi q}\right)^4 + \ldots
R_N(1,\xi_n)= \begin{pmatrix}\frac{(-1)^n}{2i\xi_n}\int_0^1\varphi_1r_{N+1}\,dt i(-1)^{n+1}\widehat{(\varphi_2^{(N)})}(n)\end{pmatrix}+ \frac{\ell^2_n}{n}
{\rm Id}= \left(\begin{matrix}1 0 0 1 \end{matrix}\right),g_1 = \left(\begin{matrix}i 0 0 -i \end{matrix}\right),g_2 = \left(\begin{matrix}0 i i 0 \end{matrix}\right),T = \left(\begin{matrix}0 -i i 0 \end{matrix}\right).
\|u\|_{[L^2,E]_{\frac{2}{3}}}={\sum_{i=1}^{d}\Big(\|\;|D_{q_i}|^{\frac{2}{3}}u\|_{L^2(\mathbb{R}^{2d})}^2+\|\;|\partial_{q_i}V(q_i)|^{\frac{2}{3}}u\|^2_{L^2(\mathbb{R}^{2d})}\Big)+\|u\|^2_{L^2(\mathbb{R}^{2d})}~.}
\begin{aligned}({\mathcal L}_{v^T}g)(X,Y) =g(\nabla_X v^T,Y)+g(\nabla_Y v^T,X) &=2g(X,Y)+2 g(A_{v^\perp}X,Y) &=2g(X,Y)+2\tilde g(h(X,Y),v^\perp)\end{aligned}
T_{m^2}^{[0]}(p) = \delta(p^2 - m^2)\: \Theta(-p^0) T_{m^2}^{[n]}(p) = \frac{(-1)^n}{(n-1)!}\:\frac{1}{4^n}\: \big( p^2 - m^2 \big)^{n-1}\: \Theta(p^2-m^2)\: \Theta(-p^0) \:.
\frac{2 \, J(r)^2 K(r)^2}{\mathcal{L}(r)^2}= \frac{\left[ M(r)^4 \mathcal{L}(r)^2 J'(r) \right]'}{M(r)^4 \mathcal{L}(r)^2}J(r) -\alpha J(r)^2 \left( J(r)^2-1\right) \, .
\dim_{\rm B}C =\limsup_{n\in\mathbb{N}}\frac{n\log 2}{-\log L_{n}}\geq \limsup_{k\in\mathbb{N}}\frac{\left(n_{k+1}-1\right)\log 2}{-\log L_{n_{k+1}-1}}= -\left(\frac{\log 2}{\log q}\right) \limsup_{k\in\mathbb{N}}\frac{\sum_{j=1}^{k+1}a_{2j-1}}{\sum_{i=1}^{2k+1}a_{i}}.
\Psi_{2 n}- \Phi_{2 n}= - T_{( 2 n )}( p ) ( \vec{p}\cdot \xi ) \vec{\xi}\cdot \left[ \frac{\partial \vec{A}( x )}{\partial x^{0}}+ \vec{\partial}A_{0}( x ) \right] = + T_{( 2 n )}( p ) ( \vec{p}\cdot \xi ) \vec{\xi}\cdot \vec{E}( x ) .
k_{1}^{\pm}(u)^{-1}X_{1}^{+}(v)k_{1}^{\pm}(u) =\frac{u_{\pm}-v+h}{u_{\pm}-v}X_{1}^{+}(v), k_{1}^{\pm}(u)X_{1}^{-}(v)k_{1}^{\pm}(u)^{-1}=\frac{u_{\mp}-v+h}{u_{\mp}-v}X_{1}^{-}(v).
|-\lambda(t) \langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right)v_{4}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r=R\lambda(t)},\phi_{0}\rangle|\leq \frac{C}{\lambda(t)}\int_{0}^{\frac{t}{2 \lambda(t)}}\frac{R^{2}}{(1+R^{2})^{3}}|v_{4}(t,R\lambda(t))| dR &\leq \frac{C}{t^{2}\log^{3b+2N-1}(t)}
J_{p,f}(u)=\frac{1}{2}\int_\Omega|\nabla u|^2\,dx+\frac{1}{2}\left(\int_\Omega |u|^{2p/(p+1)}\,dx\right)^{(p+1)/p}-\langle f,u\rangle\qquad\forall u\in W^{1,2}_0(\Omega),
\big(\tbinom{n}{z}\cdot w \big)\ast\big( \tbinom{n'}{z'}\cdot w'\big) = \tbinom{n}{z}\cdot \big(w \ast \big(\tbinom{n'}{z'}\cdot w'\big)\big)+ \tbinom{n'}{z'}\cdot \big(\big( \tbinom{n}{z}\cdot w\big) \ast w'\big)+ \tbinom{n+n'}{z+z'}\cdot (w \ast w'),
S_{1}\stackrel{\rm d}{\equiv}\int{\rm d}^{4}x{\cal L}_{1}\left( \varphi_{A}, \partial_{\mu}\varphi_{A}, \partial_{\mu\nu}\varphi_{A};B^{a}_{\mu}, \partial_{\mu}B^{a}_{\nu}, \partial_{\mu\nu}B^{a}_{\rho};B^{a}_{\mu\nu}, \partial_{\mu}B^{a}_{\nu\rho},\partial_{\mu\nu}B^{a}_{\rho\sigma}\right).
A_{2,1}^1 : = \sup_{t > 0}\bigg( \int_0^t [V_3^*]^{q / p}(\tau) w(\tau)\,d\tau \bigg)^{1 / q}[V_1^*]^{- 1}(t), A_{2,2}^1 : = \sup_{t > 0}W_*^{\frac{1}{q}}(t) \bigg( \int_0^t \bigg( \frac{V_3^*(\tau)}{V_1^*(\tau)}\bigg)^{\frac{1}{1-p}}\{V_* \cdot [V_1^*]\}^{-2}(\tau)v(\tau)\,d\tau \bigg)^{\frac{1 - p}{p}};
h(q,a)h(q',a') =\bmatrix a 0 \sigma_q a^t\!a^{-1}\endbmatrix\,\bmatrix a' 0 \sigma_{q'}a' (^t\!{a'})^{-1}\endbmatrix =\bmatrix aa' 0 \sigma_q aa'+^t\!a^{-1}\sigma_{q'}a'^t\!a^{-1}(^t\!{a'})^{-1}\endbmatrix =\bmatrix aa' 0 (\sigma_q +^t\!a^{-1}\sigma_{q'}a^{-1})aa'^t\!(aa')^{-1}\endbmatrix
B_2:=\left\{\left.\alpha=\frac{\overbrace{(m+n_1)n_2+q+k}^{N_k}}{n_2\underbrace{(mn_1n_2+q)}_{D}}\right\vert0\leq k<n_2Dn_2\alpha,D\alpha\notin\mathbb{Z}\right\}.
\int_0^T \langle \dot{\tilde{p}}_k(t), \sigma_k(t) \rangle dt= \int_0^T \langle E \dot{\tilde{u}}_k(t), \sigma_k(t) \rangle dt-\int_0^T \langle \dot{\tilde{e}}_k(t), \sigma_k(t) \rangle dt.
n_{1}=1=\sum_{\alpha_{l}=1}^{\chi_{c}}(c_{\alpha_{l-1}\alpha_{l}})^{2}(\lambda_{\alpha_{l}}^{[l]})^{2}+\sum_{\alpha_{l}=\chi_{c}}^{\chi}(c_{\alpha_{l-1}\alpha_{l}})^{2}(\lambda_{\alpha_{l}}^{[l]})^{2}=S_{1}+S_{2}
|v_{l_1}^{-2}(m)-v_{l_2}^{-2}(m)| =|v_{l_1}^{-2}(m)v_{l_2}^{-2}(m)|\cdot|\mu_{n^{(l_1)}}^2-\mu_{n^{(l_2)}}^2| &\geq\frac{1}{2\pi^2}c_\star^2,
\Re \ell(\mathbf{b}_1)+\Re \ell(-\mathbf{b}_2) \geq \sum_{a=1,2}\sum_{j}\Big(\frac12(1+\mathfrak{s}_{jj}) \Re b_{j,a}^2+(-1)^{a+1}E\Im b_{j,a}- \log r_{j,a}\Big) &\geq c\sum_{a=1,2}\sum_{j}r_{j,a}^2-\sum_{a=1,2}\sum_{j}\log r_{j,a},
\mathbb{E}_{\Phi_J}\left\{\prod\limits_{J_j \in \Phi_J}f(z_{J_j})\right\}= \exp\left[-\lambda_J \int_{\mathbb{R}^2}1-f(z_{J_j}) dz_{J_j}\right] &= \exp\left[-2\pi\lambda_J \int_{0}^{\infty}(1-f(r))rdr\right],
\aligned d_0\circ \psi^{\epsilon}_{\alpha_2,\alpha_1}- \psi^{\epsilon}_{\alpha_2,\alpha_1}\circ d_0 &+\sum_{\alpha'_2}\frak m^{2,\epsilon}_{1;\alpha_2,\alpha'_2}\circ\psi^{\epsilon}_{\alpha'_2,\alpha_1}-\sum_{\alpha'_1}\psi^{\epsilon}_{\alpha_2,\alpha'_1}\circ\frak m^{1,\epsilon}_{1;\alpha'_1,\alpha_1}= 0.\endaligned
\frac{z D_{q}\mathcal{F}(z)}{\mathcal{F}(z)}= 1 +\frac{1-[2]_{q}}{3}a_{2}z + \left(\frac{[3]_{q}-1}{10}a_{3}+\frac{1-[2]_{q}}{9}a_{2}^{2}\right) z^2 + \dots
T(3,a,x)= G_{2}^3\,{}_3^0\left(x\left|\begin{matrix}0,0 -1,a-1,-1\end{matrix}\right.\right) = \frac{x^{a-1}{}_2\mathrm{F}_2\left(a,a; 1+a,1+a; -x\right)}{a^2}-\frac{\Gamma(a)\log(x)}{x}+\frac{\Gamma(a)\psi(a)}{x},
\left( \begin{array}{l l}{{0}}&{{x_{2}}}\\{{- x_{2}}}&{{0}}\\ \end{array}\right) \left( \begin{array}{l}{{\dot{x}_{1}}}\\{{\dot{x}_{2}}}\\ \end{array}\right) = \left( \begin{array}{l}{{\nu}}\\{{0}}\\ \end{array}\right) \; ,
B(t) =\sum_{\pm}\ \frac12 Op(Ad( O_{\pm})_{<0}) (t,x,D) e^{\pm it|D|}Op(Ad( O^{-1}_{\pm})_{<0})(D,0,y)( B_0 \pm i|D|^{-1}B_1) \ +Op(Ad( O_{\pm})_{<0})(t,x,D) \frac{1}{|D|}K^{\pm}Op(Ad( O^{-1}_{\pm})_{<0})(D,s,y) F
\psi_{\vec{z}}^{1^{k}}(\vec{x})=(1-q^{2})^{-k}\prod_{i=1}^{k}\frac{z_{i}^{M'-1}}{(1+z_{i})^{M}}\langle \prod_{1\le i \le k}C^{[M', M]}(z_{i}) \prod_{1\le i \le k}\beta_{x_{i}}^{*}\rangle_{[M', M]}\, (u_{1}\otimes \cdots \otimes u_{1}).
\Psi_{nlm}(r,\theta,\varphi;\delta_1,\delta_2) =\sum_{n_1=0}^{n-|m|-1}\widetilde W_{nlm}^{n_1}(\delta_1,\delta_2)\Psi_{n_1 n_2 m}(\mu,\nu,\varphi;\delta_1,\delta_2),
H = H_{+}+ H_{-}= \frac{\omega_{+}}{2}z_{i}^{2}+ \frac{\omega_{-}}{2}y_{i}^{2}= \frac{1}{2 m ( \omega_{+}+ \omega_{-})}\left( \frac{q_{i}^{2}}{m}+ k x_{i}^{2}\right)
W_f(g_1,g_2) =\mu_1(g_1\cdot f_1)\cdot\mu(g_2\cdot f_2)=\left(\int_F f_1(w\cdot m_x\cdot g_1) \Psi_1(-x)dx \right)\cdot \left(\int_F f_2(w\cdot m_x\cdot g_2) \Psi_2(-x)dx\right) &=W_{1,f_1}(g_1)\cdot W_{2,f_2}(g_2),
\sum_{j=1}^{\infty}d_{i,j}= r_i^{-p}\sum_{j\ge -\log_2 r_i}2^{(-j+1)p}\mu(B(y,2^{-j+1})) \mu(B(y,r_i))^{-1}&\le C_d r_i^{-p}\sum_{j\ge -\log_2 r_i}2^{(-j+1)p}&\le 2^p C_d,
[\sigma]_{ij}=\big(\nabla\Psi\otimes\nabla\Psi-\frac{1}{2}|\nabla\Psi|^{2}I\big)_{ij}=\partial_{x_{i}}\Psi\partial_{x_{j}}\Psi-\frac{1}{2}|\nabla\Psi|^{2}\delta_{ij}.
X\left(t,\xi\right) =e^{tA}\xi+ \int_0^te^{(t-s)A}b\left(X\left(s,\xi\right)\right) ds+\int_0^te^{(t-s)A}\sigma\left(X\left( s,\xi\right)\right) dW\left(s\right), \ \ \ \forall t \geq 0.
Y^{(a)}_m(v+i) Y^{(a)}_m(v-i) =\frac{(1+Y^{(a)}_{m-1}(v))(1+Y^{(a)}_{m+1}(v))}{(1+(Y^{(a+1)}_{m}(v))^{-1})(1+(Y^{(a-1)}_{m}(v))^{-1})}.
\begin{array}{l}T(x^{\prime \prime}) = L(l^{\prime})T(x)L^{+-1}(l^{\prime}) +T(x^{\prime}) + \phantom{T(x^{\prime \prime}) =}1/2i\left[\Xi(\xi^{\prime})U(u^{\prime})\Xi (\xi)L^{+-1}(l^{\prime}) - \right. \phantom{T(x^{\prime \prime}) = asdf}\left. L(l^{\prime})\Xi \left( \xi \right)U^{-1}(u^{\prime})\Xi^{+}(\xi^{\prime})\right]\end{array}
\mathbf{E}[\mathrm{Tr}[X^{2p}]] =\sum_{\mathbf{s}\in\mathcal{\tilde S}_{2p}}\sum_{\mathbf{u}:\mathbf{s}(\mathbf{u})=\mathbf{s}}\mathbf{E}[X_{u_1u_2}X_{u_2u_3}\cdots X_{u_{2p}u_1}] &\le\sum_{\mathbf{s}\in\mathcal{\tilde S}_{2p}}\sum_{\mathbf{v}\in[n]^{m(\mathbf{s})}}\prod_{e\in E(G(\mathbf{s}))}\mathbf{E}\big[|X_{v(e)}|^{k_e(\mathbf{s})}\big],
\begin{array}{ll}\int\limits_0^t \int\limits_{\mathbb{R}^3}- \tilde{v}\cdot\vec{\varphi}_2 - \frac{1}{k_1\tilde{\varrho}}\nabla\tilde{\xi}\cdot\vec{\varphi}_2\,\mathrm{d}x\mathrm{d}s = 0.\end{array}
\gamma(t) - \gamma(0) = \gamma'(0)t + \gamma''(0)\frac{t^{2}}{2!}+ \cdots + \gamma^{(d)}(0)\frac{t^{d}}{d!}+ \mathbf e(t) =: M^{\gamma,d}_{0}\gamma_{\circ}^d(t) + \mathbf e(t),
u_{n,m}=\hat{\chi}_{n,m}(u), &v_{(n,m)\to(n',m')}=\hat{\chi}_{n',m'}(L u_{n,m})=\hat{\chi}_{n',m'}\circ L\circ\hat{\chi}'_{n,m}(u_{n,m})\intertext{and}v_{n',m'}=\hat{\chi}_{n',m'}(Lu)=\sum_{(n,m)}v_{(n,m)\to(n',m')}
\limsup_{j\to+\infty}v_\varepsilon (p_j) = \limsup_{j\to +\infty}[w(p_j) -\psi(p_j)] & \leq w(p)-\psi(p)\leq \max(v_\varepsilon(p), f(p)) & \leq \max(v_\varepsilon(p), u(p) +\delta).
E_\pi(T_1, \ldots, T_n) = E\left( T'_{q_1}\cdots T'_{q_{j-1}}R_{b}L_{b'}\left(T_{\theta}X_1 \cdots X_p L_{E_{\pi|_{Y}}((T_1, \ldots, T_n)|_{Y})}\right) R_{b''}T'_{q_{j+1}}\cdots T'_{q_{k_{p+1}}}\right)
\theta_{w}:= \bigvee_{w' \in I_w}\varphi_{w'}, \varphi_w := \bigwedge_{p \in T_w}p \wedge \bigwedge_{q \in B_w}(q \to \theta_w) \wedge \bigwedge_{w' \in I_w}\left(\psi_{w'}\to \theta_w\right), \psi_w := \varphi_w \to \theta_w,
\hskip-5mm \mathcal{P}_{V, t}= \left\{P_{Y}: \begin{array}{ll}Y\in \mathbb{R}^{p_1\times p_2}, Y= X+Z, X , Z\overset{iid}{\sim}N(0, 1), X\in \mathcal{F}_{r, t}, VO, O\in \mathbb{O}_{r}\end{array}\right\}.
J_1(x,y,z) = x, J_{2n+1}(x,y,z) = z, \begin{aligned}J_{2i}(x,y,z) = P_i(x,y,z) 0\leq i<n, J_{2i+1}(x,y,z) = P_{i+1}(x,z,z) 0\leq i<n. \end{aligned}
D_T(0,x) = \frac{T}{2\pi}\sum_{n=-\infty}^\infty \int \frac{dk~e^{ikx}}{(2\pi nT)^2 + k^2 + \mu^2}= \frac{1}{4\pi}\int dk e^{ikx}\frac{1}{\sqrt{k^2+\mu^2}}\coth\frac{\sqrt{k^2+\mu^2}}{2T}
\hat{\tilde{\rm G}}_{\pm}^{\rm ex}(p) |{\rm phys};A;\rm ex \rangle\equiv (\hat{\rm G}_{\pm}^{\rm ex}(p) \pm \hbar \frac{e^2{\rm L}^2}{8{\pi}^2}\alpha_{\pm p}) |{\rm phys};A;\rm ex \rangle =0.
\bigg |\frac{1}{k}\int_{r}^{r+k}g(t)dt-g(r)\bigg | \leq \frac{1}{k}\cdot \frac{\epsilon}{2}\cdot m([r,r+k])+\frac{1}{k}\cdot 2M \cdot m([r,r+k]\setminus B_r) < \frac{1}{k}\cdot \frac{\epsilon}{2}\cdot k+\frac{1}{k}\cdot 2M \cdot \frac{\epsilon k}{4M}= \epsilon
\iint_{|s-s_0|+ |y-(\ell_k \mathbf{e}_1 + \beta \mathbf{e}_2)s|<B}|\partial_s v|^2 dyds & =\frac1{1-\beta^2}\iint_{|s-s_0|+ |y-(\ell_k \mathbf{e}_1 + \beta \mathbf{e}_2)s|<B}\left| \left( \tilde v_t - \beta \tilde v_{x_2}\right)\left(\frac{s-\beta y_2}{\sqrt{1-\beta^2}},y_1,\frac{y_2-\beta s}{\sqrt{1-\beta^2}},\overline{\overline y}\right)\right|^2 dyds.
Q_{n}^{(N,M)}(z) =\int_{\pi /2}^{x}d\xi \psi_{n}^{2}(\xi ;N,M)=-\frac{1}{2}\int_{-1}^{z}d\zeta (1-\zeta)^{N}(1+\zeta)^{M}\big( P_{n}^{(N,M)}( \zeta) \big)^{2}
\overline{P(u)p|_{\phi_1(r,s)}}, \ \overline{P(up|_{\phi_1(r,s)})}, \ \overline{P(u d( p|_{\phi_1(r,s)}))}<_n \overline{\phi_2(u,p|_{\phi_1(r,s)})}=\overline{\phi_2(u,v)}=w
\int_{t}^{t_2}{p(s) \, ds}\ge \int_{t}^{t_2}{\dfrac{\varphi'(s)}{\psi(s,\varphi(s))}ds}&= \int_{\varphi(t)}^{\varphi(t_2)}{\dfrac{dr}{\psi(t_2,r)}}& \mbox{}-\int_t^{t_2}\int_{\varphi(t)}^{\varphi(s)}\dfrac{\partial}{\partial s}\dfrac{1}{\psi(s,r)}\, dr \,ds.
(\kappa-1) \frac{1}{n^{k_1+k_2+1}}\sum_{i=1 \atop i^\prime=1}^{n}\left(\sum_{J \in \Pi_1(\pi)}g(i,i^\prime,J_1,J_2,p,n)+\sum_{J \in \Pi_2(\pi)}g(i,i^\prime,J_1,J_2,p,n)\right)+o(1),
g=\left(\begin{array}{ccc}1+{1\over 4}\eta\eta^\diamond -{1\over 2}\eta{1\over 2}\eta^\diamond -{1\over 2}(a\eta^\diamond-b^\diamond\eta) a(1-{1\over 8}\eta\eta^\diamond) -b^\diamond(1-{1\over 8}\eta\eta^\diamond) -{1\over 2}(b\eta^\diamond+a^\diamond\eta) b(1-{1\over 8}\eta\eta^\diamond) a^\diamond(1-{1\over 8}\eta\eta^\diamond)\end{array}\right)
A_n^\prime(\omega) := \frac{1}{n-\lfloor \sqrt{n}\rfloor}\sum_{k=\lfloor \sqrt{n}\rfloor+1}^n \left[2 \frac{c(\alpha) p_1^{-\frac{1}{\alpha}}+1}{n^{\frac{1}{\alpha}}}\right]^{\alpha(\varphi^{n-k}\omega) - \alpha}
- 2\pi \sum_{c\,\equiv\,0\,(\mod \hskip -3ptq)}\frac{S(1,n;c)}{c}J_{1}\left(\frac{4\pi\sqrt{n}}{c}\right) \ll \frac{\sqrt{n}}{q^{3/2-\varepsilon}}\sum_{r\geq1}r^{-3/2}\ll \frac{\sqrt{n}}{q^{3/2-\varepsilon}}.
c_{\gamma}(s_0^{-1}\gamma_i^{-1}s_0^{-1})\frac{u_\gamma(a_i)u_{s_0^{-1}\gamma s_0}(a_i)}{u_{\gamma}(a_0)u_{s_0^{-1}\gamma s_0}(a_0)}=c_{\gamma}(\gamma_i)c_{a_i,a_0}(s_0^{-1}\gamma s_0^{-1}\gamma_0^{-1})
\det\Big(M(\vec w,k)\Big) =\det\begin{pmatrix}c_{w_1}c_{w_1+p}\cdots c_{w_1+kp}c_{w_1-1}c_{w_1+p-1}\cdots c_{w_1+kp-1}c_{w_1-2}c_{w_1+p-2}\cdots c_{w_1+kp-2}\vdots \vdots \ddots \vdots c_{w_1-k}c_{w_1+p-k}\cdots c_{w_1+kp-k}\end{pmatrix}.
a(E_k^{(n)},T) = (-1)^{[(n+1)/2]}2^{n-[n/2]}\frac{k}{B_k}\prod_{i=1}^{[n/2]}\frac{2k-2i}{B_{2k-2i}}\prod_q F_q(T,q^{k-n-1}) &\times \begin{cases}\frac{B_{k-n/2,\chi_T}}{k-n/2}n 1 n \end{cases}
\;\|\tilde{w}_j\|_{L^\infty_T(D(A))\cap L^2_T(D(A^{1+s/2}))}\leq \;C(\nu)(\|f(\tilde{w}_{j-1}, u^{(j-1)})\|_{L^2_T D(A^{1-s/2})}+\|f(u^{(j-2)}, \tilde{w}_{j-1})\|_{L^2_T D(A^{1-s/2})}) \leq \;C_2M\|\tilde{w}_{j-1}\|_{L^\infty_T (D(A))\cap L^2_T (D(A^{1+s/2}))},
W_{\hbar}\left( t,s\right) \psi-W_{0}\left( t,s\right) \psi=i\int_{s}^{t}W_{\hbar}\left( t,\tau\right) \left[ H_{2}\left( \alpha\left(\tau\right) :\bar{a},a^{\ast}\right) -\overline{L_{\hbar}\left(\tau\right)}\right] W_{0}\left( \tau,s\right) \psi d\tau
T_{\delta}(T_{\varepsilon}(\varPhi))(x,y,z)= \varPhi\Bigl(\varepsilon\Bigl[\frac{x}{\varepsilon}\Bigr]_{Y}+\varepsilon\delta\Bigl[\frac{y}{\delta}\Bigr]_{Z}+ \varepsilon\delta z \Bigr) .
m_i + \sum_{s=j+1}^r \ell_{i+1,s}- \sum_{s=j}^r \ell_{is}=(\lambda_i^{r+1}-\lambda_{i+1}^{r+1}) + \sum_{s=j+1}^r (\lambda_{i+1}^{s+1}- \lambda_{i+1}^s) -\sum_{s=j}^r (\lambda_{i}^{s+1}- \lambda_{i}^s) = \lambda_i^{j}- \lambda_{i+1}^{j+1}.
\frac{n^2}{m^2}\sum_{l = 2}^m \mathbb{E}_0\left[(D^S_{nl})^2\middle| \mathcal{F}_{n, l -1}\right] , \frac{n^2}{m^2}\sum_{l = 2}^m \mathbb{E}_0\left[(D^T_{nl})^2\middle| \mathcal{F}_{n, l -1}\right] \underset{p}{\longrightarrow}k^4 (\zeta_1^h)^2 ,
{\bf \Omega}_{{\bf x},{\bf x'}}=m^2\delta_{{\bf x},{\bf x'}}-{1 \over \sigma^2}\sum^{3}_{i=1}\left(\delta_{{\bf x}+\hat{\sigma}\left(i\right),{\bf x'}}-2\delta_{{\bf x},{\bf x'}}+\delta_{{\bf x}-\hat{\sigma}\left(i\right),{\bf x'}}\right).
Q_0 :=\{(x,t)\in Q_T \colon \partial_t u\neq (\Delta u+f)_+\}, Q_1 :=\{(x,t)\in Q_T \colon \partial_t u= (\Delta u+f)_+>0\}, Q_2 :=\{(x,t)\in Q_T \colon \partial_t u= (\Delta u+f)_+=0\},
\partial_{2}\tau_{1}^{(n)}= (\frac{1-\dot{\lambda}}{2}) \tau_{4}^{(n+1)}, & \partial_{2}\tau_{2}^{(n)}= (1- \dot{\lambda}) \tau_{1}^{(n+1)}- ( 1+ \dot{\lambda}) \tau_{3}^{(n+1)}, \partial_{2}\tau_{3}^{(n)}= (\frac{1+ \dot{\lambda}}{2}) \tau_{4}^{(n+1)}.
Bf(w) =e^{-\pi i \xi \cdot \eta}e^{\frac{\pi}{2}|w|^2}V_{g_0}f(\xi, -\eta) & =e^{-\pi i \xi \cdot \eta}e^{\frac{\pi}{2}|w|^2}e^{-\pi i \xi \cdot (-\eta)}\langle f, \rho(\xi,-\eta,0) g_0 \rangle & = e^{\frac{\pi}{2}|w|^2}\langle f, \rho(\overline{w}, 0) g_0 \rangle.
H_{c}= \int(\frac{g^{2}}{64\pi^{2}}G^{a}_{ij}G^{a}_{ij}+\frac{8\pi^{2}}{g^{2}}(\pi^{a}_{i}+ \frac{1}{8\pi}\epsilon_{ij}A^{j,a})^{2}- A^{a}_{0}\wedge^{a})d^{2}x
\hat{h}_{\mathrm{r e d}}= \sum_{\vec{b}}\left\{- \frac{g^{2}L M}{4 \sqrt 2}\frac{1}{J}\frac{\delta}{\delta \varphi_{\vec{b}}}J \frac{\delta}{\delta \varphi_{\vec{b}}}+ \left( \frac{4 \sqrt 2}{g^{2}L M}\right) \sum_{\vec{\varepsilon}}( \varphi_{\vec{b}}- \varphi_{\vec{b}+ \vec{\varepsilon}})^{2}\right\}.
\oint_{\infty}d \Sigma^{r}\epsilon_{0}^{\ast}\frac{\tilde{\epsilon}}{r^{2}}= \oint_{\infty}d \Sigma^{r}\lbrace \epsilon_{0}^{\ast}\gamma^{r}\gamma^{m}\Gamma_{m}\epsilon_{0}+ \epsilon_{0}^{\ast}\frac{\gamma^{r}}{r^{3}}( \gamma^{\alpha}\partial_{\alpha}) \tilde{\epsilon}\rbrace
\partial_{\pm2}\alpha_{AB}=-\frac14\alpha_{AC}\Omega^{(i)(j)}_{\pm2}\gamma^{(i)(j)}_{CB},\ partial_{\pm2}\beta_{\dot A\dot B}=-\frac14\beta_{\dot A\dot C}\Omega^{(i)(j)}_{\pm2}\tilde \gamma^{(i)(j)}_{\dot C\dot B}.
5550 =2\cdot3\cdot5^2\cdot37, 55550 =2\cdot5^2\cdot11\cdot101, 555550 =2\cdot5^2\cdot41\cdot271, 5555550 =2\cdot3\cdot5^2\cdot7\cdot11\cdot13\cdot37, 55555550 =2\cdot5^2\cdot239\cdot4649, 555555550 =2\cdot5^2\cdot11\cdot73\cdot101\cdot137, 5555555550 =2\cdot3^2\cdot5^2\cdot37\cdot333667, \!\!\!\!55555555550 =2\cdot5^2\cdot11\cdot41\cdot271\cdot9091.
F(n,k) =(-1)^n(3n+2k+1)\frac{(\frac{1}{2})_n(\frac{1}{2}+k)_n^2(\frac{1}{2})_k}{(1)_n^2(1+2k)_n(1)_k}2^{3n}, begin{align*}5pt]G(n,k) =(-1)^n \frac{(\frac{1}{2})_n(\frac{1}{2}+k)_{n-1}^2(\frac{1}{2})_k}{(1)_{n-1}^2(1+2k)_{n-1}(1)_k}2^{3n-2}.
F(\lambda \phi(t,\frac x{\lambda})) = F(x) + (\nabla F)(x) \cdot \bigl( t \nabla^{\perp}a(x) +\lambda \int_0^t (t-\tau) (\nabla p)(\tau,\frac x{\lambda})d\tau & + O(t^3 A^3 M ) \bigr) & + O( \|D^2 F\|_{L_x^{\infty}{(|x|\sim 1)}}) \cdot O(t^2 +(t^2 M)^2 + t^6 A^6 M^2 ).
{Q_1}(M,P_i,\tau)= \left\{e^{\frac{1}{24}E_2(\tau)A_3}\widehat{L}(TM,\nabla^{TM}){\rm ch}\left[\bigotimes_{n=1}^{\infty}S_{q^n}(\widetilde{T_CM})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^m}(\widetilde{T_CM})\right]\right. &\left.\cdot\varphi(\tau)^{8}{\rm ch}(\mathcal{V}_i)\right\}^{(12)},
\mathbf{L}_r^S=\sqrt{\frac{{t+L-1 \choose t}}{(t+1){t+L \choose t+1}}}\mathbf{U}\left( \begin{array}{ccc}\mathbf{h}_r^H \mathbf{u}^{T_1}_S \dots 0 \vdots \ddots \vdots 0 \dots 0 0 \dots \mathbf{h}_r^H \mathbf{u}^{T_v}_S \end{array}\right),
( \Omega_{b})^{2}= \left( \begin{array}{c c c c c}{{r^{2}}}&{{- 2 i v}}&{{0}}&{{\cdots}}&{{0}}\\{{2 i v}}&{{r^{2}}}&{{0}}&{{\ddots}}&{{0}}\\{{0}}&{{0}}&{{r^{2}}}&{{\ddots}}&{{\vdots}}\\{{\vdots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{0}}\\{{0}}&{{0}}&{{\cdots}}&{{0}}&{{r^{2}}}\\ \end{array}\right)
\begin{aligned}\mathop{\max}\limits_{\{{\bf{\bar c}}_k\}_{k = 0}^{K - 1}}\ \frac{1}{K}\sum\limits_{k = 0}^{K - 1}{\log_2}\left({1 +{{\left|{\bf{\bar e}}_k^H{{\bf{\bar c}}_k}\right|}^2}}\right){\rm{s.t.}}\ \ \sum\limits_{k = 0}^{K - 1}{{{\left\|{{\bf{\bar c}}_k}\right\|}^2}}\le KP. \end{aligned}
\varrho_\mathcal H(\nabla w)=\int_\Omega(|\nabla w|^p+a(x)|\nabla w|^q)dx \le\liminf_{h\to\infty}\int_\Omega(|\nabla w|^{p_h}+a(x)|\nabla w|^{q_h})dx=\liminf_{h\to\infty}\varrho_{\mathcal H_h}(\nabla w).
w_R(t) = \prod_{|p_n| \geq R}\left(1- \frac{Rt}{p_n}\right)^2 & = \prod_{n \in S_1}\left(1- \frac{Rt}{p_n}\right)^2 \, \left(\prod_{n \in S_2 \cup S_3}\left(1- \frac{Rt}{p_n}\right)^2 \left(1- \frac{Rt}{p_{-n}}\right)^2 \right)
\int \xi^{n}\; d^{p}\xi \; d^{p}\xi^{*}\; e^{- \frac 1 2 [ \xi^{*}, \xi ]}\; ( \xi^{*})^{n}= ( n ! )^{2}{\binom{p}{n}}= \frac{n ! p !}{( p - n ) !}= \{n \}! .
\frac{2(1-z^{-1})}{1+z^{-1}}\widetilde{U}(z) + \sum\limits_{q=1}^{2}k^{1+\alpha_q}\widetilde{\mu}^{(q)}(z) \mathbf{B}_q\widetilde{U}(z) = k\widetilde{F}(z) - \frac{k}{1+z^{-1}}f^0 + \frac{2}{1+z^{-1}}U^0 + k \sum\limits_{q=1}^{2}\Theta^{(q)}(z) \mathbf{B}_qU^0,
\widehat{L}_{\mathrm{p a r t}}={\cal P}_{i}\dot{\eta}_{i}- \frac{1}{2}{\cal P}_{i}^{2}+ e ( \widehat{A}_{i}\dot{X}_{i}+ \widehat{A}_{0}+ \frac{e \theta}{2}\varepsilon_{i j}\widehat{A}_{i}\frac{d}{d t}\widehat{A}_{j}) - \frac{1}{2}\frac{d}{d t}( e \theta \varepsilon_{i j}{\cal P}_{i}\widehat{A}_{j}) \, .
\sigma_{3}= \left[ \begin{array}{c c}{- q}&{0}\\{0}&{q^{- 1}}\\ \end{array}\right] , \qquad \sigma_{-}= \left[ 2 \right]^{1 / 2}\left[ \begin{array}{c c}{0}&{0}\\{- q^{1 / 2}}&{0}\\ \end{array}\right] ,
\left( \begin{array}{rl}Q_{11}Q_{12}Q_{21}Q_{22}\end{array}\right) = \frac{1}{6}N \omega_1 R_1^2 \left( \begin{array}{rl}-\sin 2\omega_1 t \cos 2 \omega_1 t \cos 2\omega_1 t \sin 2 \omega_1 t \end{array}\right)
\log \left(\frac{F^2(1+r)}{F^2(1 +\tilde{r})}\right) = 2 \log \frac{F(1+r)}{F(1 + \tilde{r})}= \frac{a}{n\log\frac{2k}{\tilde{r}}}+ O\left(\frac{1}{n\log^3 n}\right),
(i)~~\inf_{0 \leq t < \infty}R(t) \geq R_0 > 0, \lim_{t \to \infty}(R(t),{\dot R}(t)) = (R_{\infty}, 0), \cr (ii)~|{\dot \phi}(t)| \leq K(1-R(t)), t \geq 0, \lim_{t \to \infty}|{\dot \phi}(t)| = 0.
(1+2\rho|\mathcal{N}_i|)\frac{\sum_{l=0}^{T-1}x_i^{k+l}}{T}-\rho|\mathcal{N}_i|\frac{\sum_{l=0}^{T-1}\mathcal{Q}_\delta(x_i^{k+l})}{T}-\rho\sum_{j\in\mathcal{N}_i}\frac{\sum_{l=0}^{T-1}\mathcal{Q}_\delta(x_j^{k+l})}{T}+\frac{\sum_{l=0}^{T-1}\alpha_i^{k+l}}{T}-r_i=0.
\begin{array}{r c l}{{L_{1}^{( m - 1 ) ( t - n )}\psi_{m , n}}}&{{=}}&{{\beta_{m , n}\; \xi_{m , n}\,}}\\{{L_{p}\psi_{m , n}}}&{{=}}&{{0 \, , \qquad \mathrm{f o r ~ p ~ \geq ~ 2 ~}\, .}}\\ \end{array}
\partial_t g_{ij}= g^{ab}\nabla_a \nabla_b g_{ij}& + \frac{1}{2}g^{ab}g^{pq}\big( \nabla_i g_{pa}\nabla_j g_{qb}+ 2 \nabla_a g_{jp}\nabla_q g_{ib}- 2\nabla_a g_{jp}\nabla_b g_{iq}\quad-2\nabla_j g_{pa}\nabla_b g_{iq}- 2 \nabla_i g_{pa}\nabla_b g_{jq}\big).
T_U^{r_{\varphi^{n}(X)}(\varphi^n(x))}\varphi^{n+1}(x) = \varphi (T^{r_{\varphi^{n}(X)}(\varphi^n(x))}\varphi^{n}(x) ) & = \varphi (T_{\varphi^{n}(X)}\varphi^{n}(x)) & = T_{\varphi^{n+1}(X)}\varphi^{n+1}(x),