harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G(u,\tau, t) =\mathbb{E}u^{N(H(\tau +t))-N(H(\tau))}& =\mathbb{E}\bigl [\mathbb{E}\bigl(u^{N(H(\tau +t)-H(\tau))}| H(\tau), H(\tau +t) \bigr) \bigr] &=e^{-\int_{\tau}^{\tau +t}\lambda^{\alpha (s)}(1-u)^{\alpha (s)}ds}.
\tilde{\cal T}_{\mu}= \epsilon_{\mu \nu \lambda}p^{\nu}\gamma^{\lambda}\otimes \sigma_{1}= \frac{\sqrt{2}}{m}\epsilon_{\mu \nu \lambda}p^{\nu}\widehat{\Xi}{}^{\lambda}, \quad \tilde{\cal K}_{\mu}= \epsilon_{\mu \nu \lambda}p^{\nu}\gamma^{\lambda}\otimes \sigma_{2}= 2 i \tilde{\cal T}_{\mu}\widehat{N},
\widehat\psi_r(\beta_2) = \inf_{x\in[0,1]}J_r(x,\beta_2) \geq\, \inf_{x\in[0,1]}J_r(x,\beta_1)+\theta(x,\beta_1)(\beta_2-\beta_1) \geq\, \widehat\psi_r(\beta_1) + \inf_{x\in[0,1]}\theta(x,\beta_1)(\beta_2-\beta_1)>\widehat\psi_r(\beta_1).
\vec{p}(t_{0})=\vec{f}_{0}-\frac{\vec{f}_{1}\cdot\vec{f}_{2}}{2f_{2}^{2}}\vec{f}_{1}+\frac{(\vec{f}_{1}\cdot\vec{f}_{2})^{2}}{4(f_{2}^{2})^{2}}\vec{f}_{2}.
\psi_n(x)=\frac{\Psi_{\nu}(\lambda_nx)}{\|\Psi_{\nu}(\lambda_nx)\|}, &\Psi_{\nu}\Big(\frac{r^2}{4}\Big)=J_{\nu}(r)\Big(\frac{r}{2}\Big)^{-\nu},\nu=\frac{N}{2}-1, &\lambda_n=\Big(\frac{j_{\nu,n}}{2}\Big)^2,
\Gamma_{3-form}= \exp\left[ (|g-1|+\delta_{g,1})\left(2\pi\sigma r_s^2\sqrt{V(r_s)}\beta_H -\frac{q^2}{r_s r_+}(r_s-r_H)\beta_H+ \frac{s(r_s^3-r_H^3)\beta_H}{\l^2}\right) - \frac{1}{8\pi\sigma^2}\right]
\begin{dcases}\sum_{e\in E(G_i)}w_i(e)=1, \sum_{e\in E(G_i):\,v\in e}B_i(v,e)=1,\ \v\in V(G_i), w_i(e)^{p-r_i}\prod_{v\in e}B_i(v,e)=\alpha_i,\ \e\in E(G_i).\end{dcases}
\binom{kb+i}{ka}\binom{ka+i}{ka}^{-1}= \frac{(kb+i)(kb+i-1)\cdots (kb+i-ka+1)}{(ka+i)(ka+i-1)\cdots (i+1)}\geq \left ( \frac{kb+i}{ka+i}\right )^{ka}\geq \left ( \frac{b+a}{2a}\right )^{ka}
\prod_{\sigma\in\operatorname{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})}\left(\sigma\left(\sum_{i=0}^{2^m-1}\zeta^{l_i}\right)\right)\in2^{m|\operatorname{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})|}\mathcal{O}\cap\mathbb{Z}=2^{m|\operatorname{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})|}\mathbb{Z}.
- ({\frac{e}{c}}) \int d \vec{y}\psi^{( 2 ) +}( y ) ) K_{i}( y , x ) ( i \hbar \vec{\partial}_{y^{i}}- ({\frac{e}{c}}) A_{i}^{[ 1 ]}( y ) ) \psi^{( 2 )}( y ) \left. \right] \psi^{( 2 )}( x ) \, ,
\kappa^2 c^4 f^2\left(\frac{E}{\kappa c^2},\frac{\vec{P}^{\,2}}{\kappa^2 c^2}\right) - c^2 \vec{P}^{\,2}g^2\left(\frac{E}{\kappa c^2}, \frac{\vec{P}^{\,2}}{\kappa^2c^2}\right) \ = \inv. \ = \ \mu^2 c^4 .
\left\langle AB_1B_2\right\rangle-\left\langle AB_1\right\rangle\left\langle \tilde{B}_2\right\rangle-\left\langle AB_2\right\rangle\left\langle \tilde{B}_1\right\rangle+\left\langle A\right\rangle\left\langle \tilde{B}_1\right\rangle\left\langle \tilde{B}_2\right\rangle
U(r)=\Delta U=\int_{r_{0}}^{r}F_{ext}\cdot dr=\int_{r_{0}}^{r}\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}\cdot dr=-\frac{q_{1}q_{2}}{4\pi\epsilon_{0}}(\frac{1}{r_{0}}-\frac{1}{r})=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}}\frac{1}{r}
p_{B^{\prime}}^{}\left( x_{B}^{\prime}|\mu_{B}^{\prime},\sigma_{B}^{\prime}\right) =\frac{1}{\sqrt{2\pi\sigma_{B}^{\prime2}}}\exp\left( -\frac{\left( x_{B}^{\prime}-\mu_{B}^{\prime}\right)^{2}}{2\sigma_{B}^{\prime2}}\right) ,
\overline{\mathbb{K}}_{\psi_E}^{\pi_E}(w)\xi\{\chi,0\}=r_l\big(\epsilon(\widetilde{\chi}_0^{-1}\pi_E,\psi_E)\big)\xi\bigg\{\chi,\cfrac{-n(\widetilde{\chi}_0^{-1}\pi_E,\psi_E)}{e}\bigg\}
2^{-1}\int_0^T g_{\beta,T}(t) \|u(t)\|^2 \,dt+\int_0^T|u(t)|^2_1dt \leq \int_0^T \langle f(t),u(t)\rangle \,dt+\frac{1}{\Gamma(1-\beta)}\int_0^Tt^{-\beta}\langle u^0,u(t)\rangle\,dt,
\int_{|i-x|\leq1}\rho^N(x,t)\dx= \int_{|i-x|\leq1}dxdv f^N(x,v,t)= &\int_{|i-x|\leq1}dx\left[\int_{S_i}f^N(x,v,t)dv+\int_{S_{i}^c}f^N(x,v,t)dv\right].
\Bigr [ ( Q_{m}{\varepsilon}_{l n}Q_{l}^{*}Q_{n}^{*}) , ( Q_{m}{\varepsilon}_{l n}Q_{l}^{*}Q_{n}^{*}Q_{a}Q_{a}^{*}) , ( Q_{a}Q_{a}^{*}Q_{b}Q_{b}^{*}Q_{m}^{*}) \Bigl ]
\begin{cases}\bigl|{\xi_2}+ \frac{1}{2}N - \frac{\sqrt{3}}{2}(\sqrt{2}-1) N\bigr|\leq 2^{12}{A'}^{-\frac{1}{2}}N, |\eta_2 - (\sqrt{2}+1)^{\frac{2}{3}}\xi_2 + (\sqrt{3}-1) (\sqrt{2}-1)^{\frac{1}{3}}N| \leq 2^{25}{A'}^{-1}N,\end{cases}
\zeta_{e = - 2 p k , n}= - \sum_{j = 0}^{\infty}\zeta_{e = - 2 p k , n - j}, \mathrm{~ a n d ~}\zeta_{e = 2 p k , n}= \sum_{j = 0}^{\infty}\zeta_{e = 2 p k , n + 1 + j}
\frac{\mathrm{d}P_{\boldsymbol{\theta}}}{\mathrm{d}\lambda}\left(\boldsymbol{x}\right) =\frac{\exp\left(\boldsymbol{\theta}\cdot\boldsymbol{x}\right)}{Z\left(\boldsymbol{\theta}\right)}=\exp\left(\boldsymbol{\theta}\cdot\boldsymbol{x}-A\left(\boldsymbol{\theta}\right)\right).
\{\; \stackrel{1}{U^{\prime}}(x,\lambda)\;,\; \stackrel{2}{U^{\prime}}(y,\mu)\;\}= -\frac{1}{4}\frac{\sigma^{a}\otimes \sigma^{b}}{(1 -\lambda^{2})(1-\mu^{2})}\{\; \lambda J^{a}_{0}(x) + J^{a}_{1}(x)\; , \; \mu J^{b}_{0}(y)+J^{b}_{1}(y)\;\}=
G^{(n)}_\sigma (\zeta_1 , \cdots , \zeta_{2n})=(\sqrt{-1})^{-n}\sum_{k_0 , k_1 , \cdots k_{2n}}t^{k_0}_{k_1}(u_1 -u_0 ) \otimes \cdots \otimes t^{k_{2n-1}}_{k_{2n}}(u_{2n}-u_0 ) F^{(n)}_\sigma (\zeta_1 , \cdots , \zeta_{2n})^{k_0 k_1 \cdots k_{2n}}.
m =K(a_1,\ldots,a_d)=a_1K(a_2,\ldots,a_d)+K(a_3,\ldots,a_d) =a_1n+K(a_3,\ldots,a_d), m =K(b_1,\ldots,b_e)=b_1K(b_2,\ldots,b_e)+K(b_3,\ldots,b_e) =b_1n+K(b_3,\ldots,b_e).
\tilde{L_{1}}( t ) ( \alpha^{*}, \alpha ) = \frac{g^{2}}{2}\left( \frac{1}{3 !}V_{a b c}W^{a b c}+ \alpha_{a}^{*}( V_{c d}^{a}W_{b}^{c d}+ V_{b c d}W^{c d a}) \alpha^{b}\right)
\sum_{m=1}^\infty\frac{1}{m^s}\sum_{n = \lceil f(m) \rceil}^{\lfloor g(m) \rfloor}\frac{1}{n^s}\geq\sum_{m=1}^\infty\frac{1}{m^s}\left(-1+\frac{1}{-s+1}\left(g(m)^{-s+1}- f(m)^{-s+1}\right)\right) &=-\zeta(s)+\frac{1}{-s+1}\sum_{m=1}^\infty\frac{1}{m^s}\left(g(m)^{-s+1}- f(m)^{-s+1}\right).
\frac{{\partial [\frac{{\sigma_k (A )}}{{\sigma_l (A)}}]}}{{\partial a_{11}}}\geqslant \frac{n}{k}\frac{{k - l}}{{n - l}}\frac{1}{{n - k + 1}}\sum\limits_{i = 1}^n{\frac{{\partial [\frac{{\sigma_k (A )}}{{\sigma_l (A)}}]}}{{\partial a_{ii}}}}, \forall 0 \leq l < k,
(\mathsf{M}_{m}^{n})^{*}=\mathsf{M}_{n}^{m},\quad\mathsf{M}_{m}^{n}\mathsf{M}_{o}^{p}=\delta_{o}^{n}\mathsf{M}_{m}^{p},\quad\mathrm{Tr}(\pi(K_{2\rho}^{-1})\mathsf{M}_{m}^{n})=\delta_{m}^{n}q^{-(2\rho,\lambda_{m})}.
e m b^{*}\left( W Z \right) = \frac 1{\left( p + 1 \right) !}d x^{\mu_{1}}\cdots d x^{\mu_{p + 1}}\cdot \Pi_{, \mu_{1}}^{A_{1}}\cdots \Pi_{, \mu_{p + 1}}^{A_{p + 1}}\cdot \left( W Z \right)_{A_{p + 1}\cdots A_{1}}\left( \phi \right) \quad ;
\Gamma^{\ast}\left( s,\chi\right) =e^{-s\ell\left( 1,\chi\right)}{\displaystyle\prod\limits_{n=1}^{\infty}}\left[ \left( 1+s/n\right)^{-1}e^{s/n}\right]^{\left( -1\right)^{n}\chi\left( n\right)},
d(\sigma\mu_{n}) =i_0\mu_n-i_1\mu_n-\sum_{p+q=n+1}\sigma\mu_p\{i_1\mu_q\}+\sum_{\substack{1\leq v\leq u t_1+\cdots+t_v=n+v-u}}i_0\mu_u\{\sigma\mu_{t_1},\dots,\sigma\mu_{t_v}\}.
\Bigl \{b^{r}( k , \sigma ) , b^{s \dagger}( p , \sigma^{\prime}) \Bigr \}= \Bigl \{d^{r}( k , \sigma ) , d^{s \dagger}( p , \sigma^{\prime}) \Bigr \}= 1 6 \pi^{3}k^{+}\delta_{r s}\delta_{\sigma \sigma^{\prime}}\delta^{3}( k - p ) .
G_n^{(k)}= G_0^{(k)}\displaystyle \sum_{i=1}^{k}A(i;k)\alpha_i^{n-2}+ \displaystyle \sum_{m=0}^{k-3}\left( G_{m+1}^{(k)}\displaystyle \sum_{j=0}^{m+1}\displaystyle \sum_{i=1}^{k}A(i;k)\alpha_i^{n-2-j}\right) + G_{k-1}^{(k)}\displaystyle \sum_{i=1}^{k}A(i;k)\alpha_i^{n-1}
\tilde{V}( x ) : = \frac{2 \pi}{N ( \pi + g N )}\; \left( \mathrm{K}_{0}\left( \sqrt{\frac{e^{2}N}{\pi \! + \! g N}}| x | \right) + \ln \left( \frac{1}{2}\sqrt{\frac{e^{2}N}{\pi \! + \! g N}}\right) + \gamma \right) \; .
T_{\omega}((x_0,x_1)) =(T_{\omega}(x_0),T_{\omega}(\delta(x_0))+\xi T_{\omega}(x_1)), T_s((x_0,x_1)) =(T_s(x_0),T_s(x_1)), T_{t}((x_0,x_1)) =(T_{t}(x_0),T_{t}(x_1))\quad\mbox{for}t\in\overline{T}.
v'(\delta)(R-\delta)^{N-1}-v'(0)R^{N-1}+\beta\int_0^\delta (R-s)^{N-1}\{2\frac{v'}{s}-\frac{N-1}{R}\frac{v}{s}\}\:ds & =\int_0^\delta(R-s)^{N-1}\{v^p s^{\beta(p-1)}+ \beta \frac{N-1}{R(R-s)}v\}\:ds.
f_{\scriptscriptstyle{L, L}}(a)=\frac{1}{2 \pi \, a^L}\int_0^a dr\,r^L\int_0^{2 \pi}d \theta\,e^{i (1 -L)\theta}\,e^{2 \phi(r,\theta)}\sum_{m=0}^{\infty}b_{\scriptscriptstyle{m, L}}\,r^m\,e^{i m \theta}\,.
\varprojlim_{n\in\Delta}\mathcal F(Y_n) = \varprojlim_{n\in\Delta}\varprojlim_{k\in\Delta}\mathcal F(\tilde Y_{n,k}) = \varprojlim_{k\in\Delta}\left(\varprojlim_{n\in\Delta}\mathcal F(\tilde Y_{n,k})\right).
\left. \frac{d}{dt}\frac{F(t)}{|t|^{2+ \frac{4s}{N}}}\right|_{t=\tau}=\frac{f(\tau)\tau-\left(2+\frac{4s}{N}\right)F(\tau)}{|\tau|^{3+\frac{4s}{N}}\operatorname{sgn}(\tau)}=0.
\tilde{\zeta}( s ) = \frac{V_{q}}{( 4 \pi )^{q / 2}}\frac{\Gamma ( s - q / 2 )}{\Gamma ( s )}\Big \lbrace ( \mu_{c}^{2}- \mu^{2})^{q / 2 - s}+ \sum_{{\bf p}\ne{\bf 0}}( \sigma_{\bf p}^{d}+ m^{2}- \mu^{2})^{- s}\Big \rbrace ,
{\widetilde{E}_{m}^{[\alpha,\beta],\rho,\varepsilon}=\left\{x\in\mathbb{R}^d : \ \underline\dim_{{\rm loc}}(m,x)\in[\alpha,\beta] \and \ \ \forall r\leq \rho, \m(B(x,r))\leq r^{\underline{\dim}_{\mathrm{loc}}(m,x)-\varepsilon}\right\}}
a_{j}=\dfrac{\cosh(\alpha x_{j-1})}{\cosh(\alpha x_j)}E_{j-1}=\dfrac{h_j}{6}+O\left((\alpha \overline{h})^2\right); \alpha\to 0, c_{j}=\dfrac{\cosh(\alpha x_{j+1})}{\cosh(\alpha x_j)}E_{j}=\dfrac{h_{j+1}}{6}+O\left((\alpha \overline{h})^2\right); \alpha\to 0, d_{j}= \dfrac{y_{j+1}-y_j}{h_{j+1}}- \dfrac{y_{j}-y_{j-1}}{h_j}.
\begin{aligned}\sum_{k\in I}\frac{F^{kk}- F^{11}}{\lambda_1 - \lambda_k}|\nabla_1g_{k\bar 1}|^2 \geq \sum_{k\in I}\frac{(1-\delta )F^{kk}}{\lambda_1 - \lambda_k}|\nabla_1g_{k\bar 1}|^2 &\geq \frac{1-2\delta}{\lambda_1}\sum_{k\in I}F^{kk}|\nabla_1g_{k\bar 1}|^2.\end{aligned}
\|e^{-iT_1(-\Delta)^\sigma}u(T_1)-e^{-iT_2(-\Delta)^\sigma}u(T_2)\|_{H^{s_c}}&=\Big\|\int_{T_1}^{T_2}e^{-is(-\Delta)^\sigma}(|u|^{p-1}u)(s)ds\Big\|_{H^{s_c}}&\leq \|u(t)\|_{L_{t\in[T_1,T_2)}^{p-1}L_x^\infty}^{p-1}\|u(t)\|_{L_{t\in[T_1,T_2)}^\infty H_x^{s_c}}\to 0
E_{Cas}(v) = \frac{1}{2\pi}\int_{0}^{\infty}{\rm ln}\Bigl( 1 - e^{-2\pi x}\left( \frac{x-v}{x+v}\right)^{2}\Bigr) dx - \frac{v}{\pi}{\rm ln}(vT) - \frac{v}{\pi}\gamma \, .
\!\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v}) =\Big\{(d,w)\in\mathbb{X}\times\mathbb{X}\ |\w\in \nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d+\nabla\!g(\overline{x}) D\mathcal{N}_K(g(\overline{x})|\overline{\lambda})(g'(\overline{x})d)\Big\}.
f_1(q,z_2^s) =-\frac{q\xi(q)}{\xi'(q)}-q^2\log\frac{q\xi'(q)-\xi(q)}{\xi(q)}+q^2-\frac{2\xi(q)q}{\xi'(q)}+\frac{q[q\xi'(q)-\xi(q)]}{\xi'(q)}&=\frac{-4q\xi(q)+2q^2\xi'(q)}{\xi'(q)}-q^2\log\Big(1+\frac{q\xi'(q)-2\xi(q)}{\xi(q)}\Big) &< \frac{-4q\xi(q)+2q^2\xi'(q)}{\xi'(q)}-q^2\frac{\frac{2[q\xi'(q)-2\xi(q)]}{\xi(q)}}{2+\frac{q\xi'(q)-2\xi(q)}{\xi(q)}}=0.
P_{m}\left( x,A_{c}\right) =\chi_{1}\left( -c\right) k^{m-1}\sum\limits_{v=0}^{k-1}\left( -1\right)^{v}\chi_{1}\left( v\right)P_{m}\left( \dfrac{v+x}{k}\right) =\chi_{1}\left( -c\right) P_{m}^{\ast}\left( x,\overline{\chi_{1}}\right)
S=\frac{1}{4\pi}\int d^2x \left(\frac{1}{2}\partial_\mu \phi(x)\partial^\mu \phi(x)-4\pi \frac{\tilde{\lambda}}{2}e^{-i\beta \phi(x)/\sqrt{4\pi}}+\sqrt{2}i \alpha_0 \phi (x)R(x)\right)
\frac{1}{(1 - x_1) (1 - x_2) \cdots (1 - x_d) - x_1 x_2 \cdots x_d}= \sum_{\boldsymbol{n}\in \mathbb{Z}_{\geq 0}^d}Y_d (\boldsymbol{n}) \boldsymbol{x}^{\boldsymbol{n}},
\overline{\rho_{0,4}}= \underset{j=1}{\stackrel{d}{\textstyle{\prod}}}(w_{01}w_jw_{02}^{-1}w_j^{-1})^{c_{0j}}{[w_j w_{02}w_j^{-1}, w_{01}]^{c_{00j}}}& \cdot \underset{0<a<b}{\textstyle{\prod}}(w_{02}^{-1}w_a^{-1}w_{01}w_a)^{2c_{0a}c_{0b}}{[(w_aw_b^{-1})^2,w_{01}]^{c_{0ab}}}
\zeta_{i}= - w_{i}^{[ b e r n ]}+ ( \partial_{i}\ln | \varpi^{- 1}\Omega_{\lbrack 0 ]}| ) \left( \ln | \Omega_{\lbrack 1 ]}| \right)^{\ast}+ \left( \Omega_{\lbrack 1 ]}^{\ast}\right)^{- 1}\partial_{i}\Omega_{\lbrack 1 ]},
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^{r,\tau}(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^{r,\tau}(\mathbb{R}^n)}.
\hat{R}_{\mu \nu \alpha \beta}= R_{\mu \nu \alpha \beta}= \partial_{\mu}\Omega_{\nu , \alpha \beta}- \partial_{\nu}\Omega_{\mu , \alpha \beta}+ \Omega_{\mu , \alpha}^{\ \ \ \gamma}\Omega_{\nu , \gamma \beta}- \Omega_{\nu , \alpha}^{\ \ \ \gamma}\Omega_{\mu , \gamma \beta}\hspace{1 . 5 c m}
\phi_1=\frac{W+W^{*}}{1+W W^{*}}\,\,\,\,\,\,\,\,\,\phi_2=i \frac{W-W^{*}}{1+W W^{*}}\,\,\,\,\,\,\,\,\,\phi_3=\frac{1-W W^{*}}{1+W W^{*}}\,\,\,\,\,\,\,\,\,
\int_{\Omega}\{[L(\alpha^{A},{\alpha^{A}}_{,\nu},x^{\mu})-L(\varphi^{A},{\varphi^{A}}_{,\nu},x^{\mu})]+\frac{\partial}{\partial x^{\sigma}}[L(\varphi^{A},{\varphi^{A}}_{,\nu},x^{\mu})\delta x^{\sigma}]\}d^{4}x=0.
\langle \phi, \mathcal{P}_{\omega}^{(\mathrm{c})}\psi \rangle = \lim_{J \rightarrow \infty}\lim_{N \rightarrow \infty}\frac{1}{2N+1}\langle \phi, \mathcal{E}_{\omega}^{-N}(I - P_{J}) \mathcal{E}_{\omega}^N \psi \rangle,
R_T^{(r)}(E,R_0)=2\pi\sum_{m=1}^{\infty}\int_{0}^{\infty}dq \int_{-\infty}^{\infty}dk\,N_2^2\,J_m^2(qR_0)\, \delta\left(E-\frac{m}{R_0}\sinh\Omega R_0 +\omega\cosh\Omega R_0\right).
\alpha =\omega \, \displaystyle\frac{p+1}{p}+\chi_{\infty}\left( \left\lfloor \displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}+\displaystyle\frac{\omega}{\pi}\right\rfloor -\left\lfloor \displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}-\displaystyle\frac{\omega}{\pi}\right\rfloor \right)
\begin{aligned}\frac{dx_i}{dt}=r_i x_i\left(1 - \frac{x_i}{K_i}\right) - \frac{a_i x_i y_i}{1 + x_i}\frac{dy_i}{dt}= \frac{a_i x_i y_i}{1 + x_i}-d_i y_i+\rho_i(1-s)\left(\frac{a_ix_iy_i}{1+x_i}y_j\right)+ \rho_is( y_j-y_i) \frac{dy_j}{dt}= -d_j y_j-\rho_j(1-s)\left(\frac{a_ix_iy_i}{1+x_i}y_j\right)- \rho_js( y_j-y_i)\end{aligned}
a_l^k(g_\rho^d) =(-1)^k\sqrt{\frac{2l+1}{4\pi}}\cdot\mathcal O\!\left((\rho l)^de^{-\rho l}\,\sum_{j=0}^d\binom{d}{j}(-1)^j\delta_{k,2j-d}\right) &=(-1)^{k+\frac{d+k}{2}}\binom{d}{\frac{d+k}{2}}\sqrt{\frac{2l+1}{4\pi}}\,\mathcal O\left(\gamma_d(\rho l)\right),\qquad\rho\to0,
\int_Md\mu =2\pi\int_{-a}^aA(t)dt &=\frac{\omega(F)^2}{8}\left[4\frac{\omega(c_+)+\omega(c_-)}{\omega(F)}+\sum_j\frac{1}{r_js_j}\left(\frac{\omega(r_jE_j')-\omega(-s_jE_j)}{\omega(F)}\right)^2+c_1(Y)[\Sigma_a]-c_1(Y)[\Sigma_{-a}]\right],
d\overline{\Pi}_{\psi}(e^+)f([\epsilon, x]) =\frac{d}{dt}\Big\{\Pi_{\psi}([e^{te^+}, 1]) f([\epsilon, x])\Big\}\Big|_{t=0}=\frac{d}{dt}\Big\{e^{\pi i \epsilon x^2 t}f([\epsilon, x]) \Big\}\Big|_{t=0}&=\pi i\epsilon x^2 f([\epsilon, x]).
\gamma_{1,i}^{(k+1)}= \prod_{j=2}^d \bigl( \gamma_{j,i}^{(k)}- v_{j,i}^{(k)}\bigr)^{-1}= \prod_{j=2}^d \bigl(1 - \epsilon_{j,i}^{(k-1)}- v_{j,i}^{(k)}\bigr)^{-1}= \prod_{j=2}^d \bigl(1 - \epsilon_{j,i}^{(k)}\bigr)^{-1},
(-\Delta)^s v(x) +c(x)v(x) = -(-\Delta)^s u(x) - c(x) u(x) +c(x) \tau d^{-n-2}\bigg (1- \frac \theta 2 \bigg )^{-n-2}\zeta (x) &\geqslant -x_1 -C \tau d^{-n-2}\|c^-\|_{L^\infty(B_\rho^+)}\bigg (1- \frac \theta 2 \bigg )^{-n-2}\zeta (x).
F(u)-F(w)=(\beta+1)(u^{2}-w^{2})-(u^{3}-w^{3})+P\left( \boldsymbol{D}\right)(u^{m}-w^{m}) =(\beta+1)(u-w)\left( u+w\right) -(u-w)(u^{2}+uw+w^{2})+P\left(\boldsymbol{D}\right) ((u-w)q(u,w)),
\mu_2( Z(\eta) ): = \begin{cases}\prod_{0\leq n \leq N}( 1/2 + \alpha_{2n}^\eta), \;\; r(\eta) \in \{u, w\}\prod_{0\leq n\leq N-1}( 1/2 + \alpha_{2n+1}^\eta), \;\; r(\eta) = v \end{cases}
\sum_{i=1}^{[n/q]+1}{\mathbb E}(\|U_i\|_H^2)={\mathbb E}(\|U_{[n/q]+1}\|_H^2) +\sum_{i=1}^{[n/q]}{\mathbb E}(\|U_i\|_H^2)\leq 2nM\sum_{k=0}^{q-1}\theta (k)\, .
\hskip6mm\tilde{P}(0, x, z')-\tilde{P}(\sigma, x, z') &=\tilde{P}(0, x, z')-\tilde{P}(0,\sigma_2, x, z')+\tilde{P}(0, \sigma_2, x, z') -\tilde{P}(\sigma, x, z') &=\sigma_2 \int_0^1 \partial_{\sigma_2}\tilde{P}(0,t\sigma_2, x, z')\,dt+\sigma_1 \int_0^1 \partial_{\sigma_1}\tilde{P}(t\sigma_1,\sigma_2, x, z')\, dt.
\int_{\mathbb{R}^{N}}\Big[\Phi(\vert\nabla v_{n}\vert)-\Phi(\vert \nabla t_{n}v_{n}\vert)\Big]dx = o_{n}(1), \int_{\mathbb{R}^{N}}\Big[\Phi(\vert v_{n}\vert)-\Phi(\vert t_{n}v_{n}\vert)\Big]dx = o_{n}(1)
{\xi_k}={{\mathbb P}_{FA}}+{{\mathbb P}_{MD}}= \Pr \left({{\kappa^2}+ D_{jw}^{-{\alpha_{jw}}}{P_{jk}}h_{jw}^2 >{\varepsilon_k}}\right) &+ \Pr \left({D_{aw}^{-{\alpha_{aw}}}{P_{ak}}h_{aw}^2 +{\kappa^2}+ D_{jw}^{-{\alpha_{jw}}}{P_{jk}}h_{jw}^2 <{\varepsilon_k}}\right),
E\big[2^{\sigma_1(d)}1_{\{\tau_1(d)<+\infty\}}\big]= \sum_{l=0}^{+\infty}2^lP(2\leq \theta(d))<+\infty)^l\frac{1}{d}\leq \frac{1}{d}\sum_{l=0}^{+\infty}(\frac{2C}{d^2})^l=\frac{d}{d^2-2C}
\frac{d\mu_{r,r-1}}{dx}(x)=\frac{2^{r+1}\sin(\varphi)\left(3\sin(\varphi)-\rho_r(\varphi)\sin(2\varphi)\right)}{\pi \sin(r+1)\varphi ~\rho_r(\varphi)^{r-1}\left(4-4\rho_r(\varphi)\cos(\varphi)+\rho_r(\varphi)^2\right)},
\frac{1}{4\Delta t}(\mathbb{H}[u_h^{n}]-\mathbb{H}[u_h^{n-1}])+\|\nabla u_h^{n-\theta}\|^2+\gamma\|\sigma_h^{n-\theta}\|^2\leq -(f^{n-\theta}(u_h),u_h^{n-\theta})+(g^{n-\theta},u_h^{n-\theta}) \leq \frac12\|f^{n-\theta}(u_h)\|^2+\|u_h^{n-\theta}\|^2+\frac12\|g^{n-\theta}\|^2.
\begin{array}{rcl}y_1(0) = 0 \, , \lim\limits_{r\to\infty}y_1(r) = 1 \, , \end{array}\quad \begin{array}{rcl}y_2(0) = 1 \, , \lim\limits_{r\to\infty}y_2(r) = 0 \, , \end{array}\quad \begin{array}{rcl}y_3(0) = 1 \, , y_7(0) = 0 \, , \end{array}\quad \begin{array}{rcl}y_4(0) = 0 \, , y_8(0) = 1 \, . \end{array}\quad
\left(1-q^{nN}\right) A'_n = \sum_{m=1}^{r}\sum_{j=0}^{r-1}\left(d^{m-1}\sum_{\substack{\alpha < a(r) w(\alpha)=j+m-1}}q^{\alpha}+ d^{m}\sum_{\substack{\alpha < a(r) w(\alpha)=j+m}}q^{\alpha}\right){j+m-1 \brack m-1}_{q^N}q^{jN(n-m)}& \times (-1)^{m+1}\prod_{k=1}^m \left(1 + q^{N(n-k)+a(r)}\right) A'_{n-m}.
(u_{n}^{\prime},\psi)+(\mu_{\varepsilon}(u_{n})\nabla v_{n},\nabla\psi) =(g_{n}^{1},\psi)\forall\psi\in V_{n}v_{n}=P_{n}(K\ast(1-2u_{n})+f_{\varepsilon}^{\prime}(u_{n})) g_{n}^{1}=P_{n}\left( g^{1}(u_{n})\right) u_{n}(0) =u_{0n}
\int_{\widehat{G}/\mathcal{L}^\perp}p_\psi(\xi) d\xi = \int_{\widehat{G}/\mathcal{L}^\perp}\!\Big(\sum_{m\in\mathcal{L}^\perp}|\widehat{\psi}(\xi+m)|^2\Big) d\xi & = \int_{\widehat{G}}|\widehat{\psi}(\zeta)|^2 d\zeta = \|\widehat{\psi}\|_{\textup{L}^2(\widehat{G})}^2 = \|\psi\|_{\textup{L}^2(G)}^2 < +\infty.
\#E(Q_i,Y_1) \le(qb\sigma)^{iq_i}\prod_{x\in Q_i}2^b\cdot\sigma^{2i}\cdot (2i+1)^b\cdot \sigma^{(2i+1)b_{i,x}}(2i+3)^{q_ib_{i,x}}&\le(qb\sigma)^{2lq_i}\prod_{x\in Q_i}2^b\cdot\sigma^{4l+4lb_{i,x}}\cdot(4l)^{b+q_ib_{i,x}}&\le(qb\sigma)^{2lq_i}\cdot2^{bq_i}\cdot\sigma^{4lq_i+4lb}\cdot(4l)^{bq_i+bq_i}&=(32l^2)^{bq_i}\cdot (qb)^{2lq_i}\cdot\sigma^{6lq_i+4lb}.
p^* (L_{a,j})=p^* (X_{a,j}L_{a,j+1}) =\left(\lambda_{b-j}\frac{A_{a,j+1}A_{a-1,j-1}}{A_{a,j}A_{a-1,j}}\right)\left(\frac{A_{a-1,j}}{A_{a,j+1}}\prod_{k=i-a}^{b-j-1}\lambda_k\right) & =\frac{A_{a-1,j-1}}{A_{a,j}}\prod_{k=i-a}^{b-j}\lambda_k.
\int\limits_{\mathbb{R}^{N}}\left\vert v\right\vert^{r}\frac{dx}{\left\vert x\right\vert^{s}}=d^{1+\frac{p-1}{p}r}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert D_{N,d,p}v\right\vert^{r}}{\left\vert x\right\vert^{N+sd-Nd}}dx
\begin{array}{ccc}(\rho_1 \rho_2 \rho_3)^* \Longrightarrow \rho_3^* \circ ( \rho_1 \rho_2)^* \Downarrow \Downarrow (\rho_2 \rho_3)^* \circ \rho_1^* \Longrightarrow \rho_3^* \circ \rho_2^* \circ \rho_1^*\end{array}
e_q(x) =\frac 1{(x;q)_\infty}=\sum_{k=0}^\infty\frac{x^k}{(q;q)_k}, E_q(x) =(-x;q)_\infty=\sum_{k=0}^\infty\frac{q^{k(k-1)/2}x^k}{(q;q)_k}.
\mathbb{A}^{\emph{r}}:= \begin{cases}\{x\in\mathbb{R}^n \mid x_1>x_2>\cdots >x_n>0\}\ \ \emph{r}=\emph{bc}, \{x\in\mathbb{R}^n \mid x_n>0\}\ \ \emph{r}=\emph{t}, \{x\in\mathbb{R}^n \mid x_1>x_2>\cdots >x_n \}\ \ \emph{r}=\emph{cs},\end{cases}
\left[ e^{i \gamma \varphi}\right]_{n}( \theta_{1 ,}\theta_{2 ,}\dots , \theta_{n}) = \left[ e^{i \gamma \varphi}\right]_{1}( \theta_{1}) \, \left[ e^{i \gamma \varphi}\right]_{n - 1}( \theta_{2 ,}\dots , \theta_{n}) + O ( e^{- \theta_{1}}) \, .
\mathbb{P}[\omega_M >t, M \in dz \vert S(s)=x] =\mathbb{P}[\max\limits_{s \le u \le t}S(u) < z, \max\limits_{u \ge t}S(u) \in dz \vert S(s)=x] &=\int_{-\infty}^{z}f(s,x-z,t,y-z)\mathbb{P}[\max_{u \ge t}S(u) \in dz \vert S(t)=y]dy
E[I_L] = E \bigg[ \displaystyle \sum \limits_{i = 0}^{n-1}B(t^{*}_i)[B(t_{i + 1}) - B(t_{i})] \bigg] & = \displaystyle \sum \limits_{i = 0}^{n-1}E \Big[ B(t_i)[B(t_{i + 1}) - B(t_{i})] \Big] & = \displaystyle \sum \limits_{i = 0}^{n-1}E[B(t_i)] E[B(t_{i + 1}) - B(t_{i})] & = 0,
I_{\mathcal{W}^p}(S^2(r))(u,u) = \delta^2 \int_{S^2(r)}H^p\,dS \geq \int_{S^2(r)}\frac{2p^2-3p+4}{2r^2}\,u^2 \, dS \geq C(p,r)\int_{S^2(r)}u^2\, dS,
\begin{array}{ll}\underset{\sigma^2 \rightarrow 0}{\lim}\mathbb{P}(\mathcal{A}_2) =\underset{\sigma^2 \rightarrow 0}{\lim}\mathbb{P}(\{RR(k)>\Gamma_{RRT}^{\alpha}(k),\ \forall k\}) \leq \underset{\sigma^2 \rightarrow 0}{\lim}\mathbb{P}(\{RR(k_0)>\Gamma_{RRT}^{\alpha}({k_0})\})=0. \end{array}
(x_i^n - x_i^*) - \mu ([\nabla F(x^n)]_i - [\nabla F(x^*)]_i) &= (x_{i}^{n+1}- x_{i}^{*}) + sign(x_{i}^{*}) \lambda \mu( \phi'(|x_{i}^{n+1}|)- \phi'(|x_{i}^{*}|)).
{\frac{\delta ( \Phi_{+}^{( 1 )}- \Phi_{-}^{( 1 )})}{\delta \theta ( x )}}={\frac{\Lambda}{| \Lambda |}}{\frac{1}{4 8 \pi}}( \partial^{2}\delta_{\mu \nu}- \partial_{\mu}\partial_{\nu}) h_{\nu \mu}( x )
\begin{array}{ll}\beta_{1}(r,\theta) ={(-2 z_{st}\cos \theta - z_{tt}\sin \theta)}, \gamma_{1}(r,\theta) ={- z_{tt}\cos \theta}, \beta_{2}(r,\theta) ={z_{ss}\cos \theta}, \gamma_{2}(r,\theta) ={- z_{tt}\sin \theta}, \beta_{3}(r \theta)={z_{ss}\sin \theta}, \gamma_{3}(r,\theta) ={z_{ss}\cos \theta + 2 z_{st}\sin \theta}.\end{array}
L_{1}=[\begin{matrix}x_{1}\\y_{1}\end{matrix}]+t[\begin{matrix}x_{2}-x_{1}\\y_{2}-y_{1}\end{matrix}],L_{2}=[\begin{matrix}x_{3}\\y_{3}\end{matrix}]+u[\begin{matrix}x_{4}-x_{3}\\y_{4}-y_{3}\end{matrix}]
f_k(r) = c_1 e^{\frac{r^2}{4}}r^{2l_k}y(-\frac{r^2}{4}) &= c_1 e^{\frac{r^2}{4}}r^{2l_k}(-\frac{r^2}{4})^{1-(2l_k + \frac{m}{2})}(a_0 + a_1(-\frac{r^2}{4}) + \cdots) &= (-4)^{2l_k + \frac{m}{2}- 1}c_1 e^{\frac{r^2}{4}}r^{2 - (2l_k + m)}(a_0 + a_1(-\frac{r^2}{4}) + \cdots),
{\cal L}_{bosonic}=-\frac{1}{4 l^2}\partial_{\mu}l \partial^{\mu}l+\frac{1}{4 l^2}h^{\mu}h_{\mu}-G_{i{\bar j}}\partial_{\mu}z^i\partial^{\mu}{\bar z}^{\bar j}-\frac{i}{2}(G_{ij}\partial z^j-G_{i{\bar j}}\partial_{\mu}z^j -G_{i{\bar j}}\partial_{\mu}{\bar z}^{\bar j})h^{\mu}.
\left. + \left( \alpha ( 2 - \alpha ) ( \frac 4 3 + \frac 2 3 \alpha + \alpha^{2}) + 2 \alpha ( 2 - \alpha ) ( k_{1}^{2}+ k_{2}^{2}) - ( k_{1}^{2}- k_{2}^{2})^{2}\right) \left( \frac{g^{2}}{4 \pi}\right)^{3}\zeta_{3}+ O ( g^{8}) \right) ,
- \sum_{i=1}^m \int_{\Omega}q_i \nabla_x \psi \cdot \nabla_x c_i dx =\int_{\Omega}(\Delta \psi) \sum_{i=1}^m (q_ic_i)dx - \int_{\partial \Omega}\sigma \sum_{i=1}^m (q_ic_i)ds =-\int_{\Omega}|\Delta \psi|^2dx.
\widehat{L}( \lambda ) = \frac{1}{\sin \gamma}\left( \begin{array}{c c}{\sinh [ \gamma ( \lambda + \mathrm{i}S^{3}) ]}&{\mathrm{i}S^{-}\sin \gamma}\\{\mathrm{i}S^{+}\sin \gamma}&{\sinh [ \gamma ( \lambda - \mathrm{i}S^{3}) ]}\\ \end{array}\right) .
\frac{1}{N}\sum_k^{(i)}\frac{\partial \|\mathbf{g}_i\|^{-1}}{\partial g_{ik}}\mathbf{e}_k^* X_i G\mathbf{e}_i=-\frac{1}{2N}\frac{1}{\|\mathbf{g}_i\|^3}\sum_{k}^{(i)}\bar{g}_{ik}\mathbf{e}_k^* X_i\mathbf{e}_i=-\frac{1}{2N}\frac{1}{\|\mathbf{g}_i\|^2}\mathring{\mathbf{h}}_i^* X_iG\mathbf{e}_i=O_\prec(\frac{1}{N}),
\begin{aligned}\Delta\left(\frac{x_i-x_j+m}{\ell}\right) =\int_\mathbb{R}\chi\left(\frac{x_i+\ell x_0-x_j+m}{\ell}\right)\chi(x_0)\,dx_0 =\frac{1}{\ell}\int_\mathbb{R}\chi\left(\frac{x_i-x_0'+m}{\ell}\right)\chi\left(\frac{x_j-x_0'}{\ell}\right)\,dx_0'\end{aligned}
\Omega_i^{(2; 4-2\varepsilon)}=\int_0^{\tau_{0i}}\frac{\mbox{d}\theta}{\cos^{2-2\varepsilon}\theta}= 2 \tan\tau_{0i}\;\;_2F_1\left( \left.\begin{array}{c}1/2, \; \varepsilon 3/2 \end{array}\right| -\tan^2\tau_{0i}\right) ,
u\geq e^{-C_1s_0^{-\theta/2}(s-s_0)}\hat{u}-Cs_0^{-\theta/4}\geq e^{-C_1s_0^{-\theta/2}\log A}\hat{u}-Cs_0^{-\theta/4}&\geq \left(1-2C_1s_0^{-\theta/2}\log A\right)\hat{u}-Cs_0^{-\theta/4}.
A_{\lambda}(\boldsymbol{n}; k) = \prod_{j = 1}^{\ell}\binom{n_{s (j) + 1}+ \ldots +_{}n_{s (j) + \lambda_j}- (\lambda_j - 1) k}{n_{s (j) + 1}- k, \ldots, n_{s (j) + \lambda_j}- k, k}.
\Phi=\left(\begin{matrix}{}\Phi_{vv}\Phi_{v\b{v}}\Phi_{v\b{v}}\Phi_{\b{v}\b{v}}\end{matrix}\right)=\left(\begin{matrix}{}tq \frac{\phi_0}{2}+\frac{t^2}{2}\left(\frac{|q|^2}{\phi^2_0}+\alpha\right)\phi_0 \frac{\phi_0}{2}+\frac{t^2}{2}\left(\frac{|q|^2}{\phi^2_0}+\alpha\right)\phi_0 t\o{q}\end{matrix}\right)+O(t^4),