harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(|\Xi - \xi|^\gamma + |\xi| + \Omega)^{-\alpha}= (|\Xi - \xi|^\gamma + |\xi| + \Omega)^{-\alpha+\epsilon(\Lambda)}(|\Xi - \xi|^\gamma + |\xi| + \Omega)^{-\epsilon(\Lambda)}&\leq (|\Xi - \xi|^\gamma + \Omega)^{-\alpha+\epsilon(\Lambda)}|\xi|^{-\epsilon(\Lambda)}.
\left|\int_1^{\frac{1}{\delta}}\left( \sum_{w\in\Lambda^*}e^{-\frac{\pi^2|w|^2}{t}}t^{\frac{s-d}{2}-1}-\pi^{-\frac{d}{2}}t^{\frac{s}{2}-1}\right)\mathrm{d}t\right|\leq \int_1^{\frac{1}{\delta}}C_1t^{\frac{s}{2}-1}e^{-l_0^2 t}\mathrm{d}t \leq \int_1^{\infty}C_1t^{\frac{s}{2}-1}e^{-l_0^2 t}\mathrm{d}t=:C_2(s).
\begin{bmatrix}\psi_0 \psi_1 \tilde s_1 \tilde s_2{\partial \psi_0}/{\partial x}{\partial \psi_1}/{\partial x}{\partial \tilde s_1}/{\partial x}{\partial \tilde s_2}/{\partial x}{\partial \psi_0}/{\partial Z_0}{\partial \psi_1}/{\partial Z_0}{\partial \tilde s_1}/{\partial Z_0}{\partial \tilde s_2}/{\partial Z_0}\end{bmatrix},
+ \sqrt{\frac{\kappa}{2}}M ( \psi_{+}^{+}\psi_{-}+ \psi_{-}^{+}\psi_{+}) + \frac{i}{4}\kappa f_{a b}( \overline{{\psi}}_{+}\gamma_{+}^{a b}\psi_{+}+ \overline{{\psi}}_{-}\gamma_{-}^{a b}\psi_{-}) \Big ) = 0 .
\Delta(f) = -\mathrm{div}(\nabla f) & =-\sum_j \sum_{i=1}^{k-1}\langle E_j,\nabla_{E_j}( E_i(f) E_i)\rangle & =-\sum_{i=1}^{k-1}E_i\circ E_i(f)+\sum_{i=1}^{k-1}E_i(f)\sum_{j=k}^{N}\langle\nabla_{E_j}E_j,E_i\rangle & =\Delta_T f+\langle \nabla f,\vec H\rangle,
\Pi_{1}^{\delta^{2}}= \delta^{2}\frac{\lambda T \eta^{2}}{8 \pi \Omega}\frac{1}{\sqrt{1 - r^{2}}}- \delta^{2}\frac{\lambda \eta^{2}}{1 6 \pi^{2}}\left[ - \frac{1}{\epsilon}+ \ln \left( \frac{4 \pi T^{2}}{M^{2}}\right) - \gamma \right] .
\frac{1}{{\nu^s (\nu + \mu )^t}}= \sum_{i = 0}^{s - 1}{\binom{t+i-1}i\frac{{( - 1)^i}}{{\nu^{s - i}\mu^{t + i}}}}+ \sum_{i = 0}^{t - 1}{\binom{s+i-1}i\frac{{( - 1)^s}}{{\mu^{s + i}(\nu + \mu )^{t - i}}}}\,,
\frac{1}{| f |^{2}}= \frac{1}{1 + | \beta |^{2}r^{2}+ | \lambda |^{2}r^{4}}- \frac{\lambda \gamma^{*}z^{2}+ \lambda^{*}\gamma z^{* 2}}{( 1 + | \beta |^{2}r^{2}+ | \lambda |^{2}r^{4})^{2}}+ O ( \vert \gamma \vert^{2}) .
R_{x}^{\alpha}f(x) =\mathcal{F}^{-1}\left\{-\left\vert \omega\right\vert^{\alpha}F(\omega)\right\}& =-\frac{1}{2\pi}\int_{-\infty}^{\infty}\left\vert \omega\right\vert^{\alpha}F(\omega)e^{i\omega x}d\omega.
T_{\mu \nu}( x ) = \sum_{a = 1}^{3}F_{\mu \nu}^{a}( x ) \widehat{\phi}^{a}( x ) - \frac{1}{\lambda}\sum_{a , b , c = 1}^{3}\epsilon^{a b c}\widehat{\phi}^{a}( x ) ( D_{\mu}\widehat{\phi}( x ) )^{b}( D_{\nu}\widehat{\phi}( x ) )^{c},
A^t(T-A)=A^tT-A^tA= \begin{bmatrix}0 n_1 -1 n_1-1 \ldots n_1-1 n_2 -1 0 n_2-1 \ldots n_2-1 \vdots \vdots \vdots \ddots \vdots n_d-1 n_d-1 n_d-1 \dots 0 \end{bmatrix}.
\delta_{13}=2\sqrt{2}r-4\sqrt{2}r^{3}+O\left( r^{5}\right) \frac{\delta_{12}+\delta_{23}}{1+\delta_{12}\delta_{23}}=2\sqrt{2}r-\frac{9}{2}\sqrt{2}r^{3}+O\left( r^{5}\right) .
\delta(t) =d(x,y)\le d(x,\Phi^b_a(x',s))+d(\Phi^b_a(x',s),\Phi^b_a(y',s))+d(\Phi^b_a(y',s),y) \le (1+\lambda)|t-s|/\Delta^b_a(M^b_a(\varphi))+\delta(s),
\Gamma_{\Lambda R_1}=\Gamma_{\Lambda R_1}^0+\hbar \sum_{n=0}\Lambda^{-n}\Gamma_{\Lambda R_1}^{\left( 1\right) n}+\hbar^2\sum_{n=-2}\Lambda^{-n}\Gamma_{\Lambda R_1}^{\left(2\right) n}+O\left( \hbar^3\right)
\frac{v_m(\lambda,r)}{W_m(\lambda)}\J_{|m+\alpha|}(\sqrt{\lambda}\, r') (1+i\cot(|m+\alpha|\, \pi)) = g_1(r,r')\, \delta_{m,k(\alpha)}+ \frac{\mathcal{O}(\lambda^{\frac{|m+\alpha|+1}{2})})}{1+|m|}\ , \lambda\to 0+
f(\mathbf{x}) = \sum_{k=0}^{\infty}(-1)^k\frac{\mathbf{c}(k,d)'}{k!}G^{(k)}(\mathbf{x}) = \sum_{k=0}^{\infty}\frac{\mathbf{c}(k,d)'}{k!}\mathbf{H}_k(\mathbf{x}; \mathbf{0},\mathbf{I}_d)
\delta_{\alpha,\beta}g (x) = \frac{d}{dx}\big( (1-x^2)^{1/2}g(x)\big)-\frac{\beta-\alpha-(\alpha+\beta+2)x}{2(1-x^2)}\big( (1-x^2)^{1/2}g(x)\big) \delta_{\alpha,\beta}^*g (x) =\frac{d}{dx}\big( (1-x^2)^{1/2}g(x)\big)+\frac{\beta-\alpha-(\alpha+\beta)x}{2(1-x^2)}\big( (1-x^2)^{1/2}g(x)\big).
E_{S}^{( 2 )}= E_{g . s .}^{( 2 )}+ 4 - \frac{\sum_{x = 1}^{N}< g . s . | \vec{S}_{x}\cdot \vec{S}_{x + 2}- \frac{1}{4}| g . s . >}{< g . s . | H_{J}| g . s . >}\quad .
W_{\pi}(g_{t,l,v})= q^{-\frac{t}{2}}\zeta_F(1)^{-2}\int_{\mathcal{O}^{\times}}\chi(y_1) \psi(y_1\varpi^{\frac{t}{2}})G(\varpi^{\frac{t}{2}}+y_1v\varpi^{-l},\chi)d^{\times}y_1.
\mathcal I(v_t)=\frac{t^{N-2}}2\int_{\mathbb R^N}|\nabla v_0|^2+\frac{t^N}2\int_{\mathbb R^N}|v_0|^2-\frac{t^{N+\alpha}}2\int_{\mathbb R^N}(I_\alpha*F(v_0))F(v_0),
S_{<d}(x_{s_1, \varepsilon_1}\dots x_{s_r, \varepsilon_r}) =S_{<d}\begin{pmatrix}\varepsilon_1 \dotsb \varepsilon_n s_1 \dotsb s_n \end{pmatrix}, \zeta_A (x_{s_1, \varepsilon_1}\dots x_{s_r, \varepsilon_r}) =\zeta_A\begin{pmatrix}\varepsilon_1 \dotsb \varepsilon_n s_1 \dotsb s_n \end{pmatrix}.
\frac{\partial}{\partial t}n(\vec{x},V_{b},t)+\frac{\partial}{\partial z}[n(\vec{x},V_{b},t)u_{b}(\vec{x})]+\frac{\partial}{\partial V_{b}}[n(\vec{x},V_{b},t)\frac{\partial}{\partial t}V_{b}(\vec{x},V_{b})]=S(\vec{x},V_{b},t)
\partial_M^n(f)(a_1 \otimes \cdots a_{n+1}) = a_1 f(a_2 \otimes \cdots \otimes a_{n+1}) + \sum_{i=1}^{n}(-1)^n f(a_1 \otimes \cdots \otimes a_{i-1}\otimes a_i a_{i+1}\otimes a_{i+1}\otimes \cdots \otimes a_{n+1}) &+ (-1)^{n+1}f(a_1 \otimes \cdots \otimes a_{n})a_{n+1}.
{\mathop{\sum\nolimits^*}\limits_{0 \leq d \leq c}}e\Big(\frac{d}{c}\Big) S(md, n_2; mcn_1^{-1})={\mathop{\sum\nolimits^*}\limits_{u\,(\, mcn_1^{-1})}}S(0, 1+un_1 ; c)e\Big(\frac{n_2 \bar{u}}{mcn_1^{-1}}\Big) .
S_{k^{\prime}k}^{Ac}=\frac{2\pi}{\hbar}Z_{DP}^{2}\frac{\hbar\omega_{q}}{2V\rho c^{2}}(N_{q}+\frac{1}{2}\pm\frac{1}{2})\delta_{k^{\prime},k\pm q}\delta[E(k^{\prime})-E(k)\pm\hbar\omega_{q}](16)
\Lambda_{p}(\Omega)\leq\frac{\left\Vert \nabla v\right\Vert_{p}^{p}}{\left\Vert v\right\Vert_{\infty}^{p}}=\frac{\Lambda_{p}(\Omega)v(x_{p})}{\left\Vert v\right\Vert_{\infty}^{p}}\leq\frac{\Lambda_{p}(\Omega)}{\left\Vert v\right\Vert_{\infty}^{p-1}}.
S=\frac{M_5^3}{2}\int_{{\cal M}_5}d^5 x \sqrt{-g}\left[ R-\frac{1}{2}\partial_\alpha \phi \,\partial^\alpha \phi -\frac{3}{2}e^{-2\phi}\alpha^2\right] -3\sum_{i=1}^2 M_5^3\alpha_i \int_{{\cal M}_4^{(i)}}d^4 \xi_{(i)}\sqrt{-g}\,e^{-\phi}\,.
\mathcal{E}_{1}(\phi^{-1})\;=\; \int_{D^*}|D\phi^{-1}|\,d\mu &\;=\; \int_{D^*}|U_x|\,d\mu + \int_{D^*}|U_y|\,d\mu + \int_{D^*}|V_x|\,d\mu +\int_{D^*}|V_y|\,d\mu
=\frac{e^{4}}{(k-k^{\prime})^{4}}((\overline{v}_{k}\gamma^{\mu}v_{k^{\prime}})^{*}(\overline{u}_{p^{\prime}}\gamma_{\mu}u_{p})^{*})((\overline{v}_{k}\gamma^{\nu}v_{k^{\prime}})(\overline{u}_{p^{\prime}}\gamma_{\nu}u_{p}))
p^{2}\xi_{2 0}( p , q ) - p . q \xi_{1 1}( p , q ) ={\frac{1}{2}}\left\{- \left[{\frac{1}{2}}+ m^{2}\xi_{0 0}( p , q ) \right] +{\frac{q^{2}}{2}}\xi_{0 1}( p , q ) +{\frac{3 p^{2}}{2}}\xi_{1 0}( p , q ) \right\}
F(x)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(x-y)e^{-\frac{y^{2}}{4}}dy,
H=\left( \begin{array}{cc}\widetilde{\eta}_{\widetilde{\alpha}\widetilde{\beta}}\widehat{\eta}_{\widetilde{\alpha}\beta}\widehat{\eta}_{\alpha \widetilde{\beta}}\eta_{\alpha \beta}\end{array}\right) =\left( \begin{array}{cccccc}0 1 0 1 1 0 1 0 1 0 0 \beta 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 \beta 1 0 0 0\end{array}\right) .
\frac{1}{4}\triangle (w_1+\cdots+w_{n-1})\geq \frac{1}{\max_{i=1,\cdots,n-1}r_i}\cdot\frac{n(n^2-1)}{12}\big(\prod_{i=1}^{n-1}|\gamma_i|^{i(n-i)}\big)^{\frac{12}{n(n^2-1)}}\cdot e^{\frac{12}{n(n^2-1)}(w_1+\cdots+w_{n-1})}.
\delta u^{\natural}_{s \theta t}= A_{\theta t}^2 \delta u_{s \theta}+ A^1_{\theta t}( \delta u_{s \theta}- A_{s \theta}^1 u_s) & = (A_{\theta t}^2 + A_{\theta t}^1)\delta \mu_{s \theta}+ (A_{\theta t}^2 A_{s \theta}^1 + A_{\theta t}^2 A_{s \theta}^2 + A_{\theta t}^1 A_{s \theta}^2) u_s + (A_{\theta t}^1 + A_{\theta t}^2) u_{s \theta}^{\natural}
\sum_{r=1}^{\infty}\sum_{s=0}^{\theta-1}\left\{x^{\nu_r + \theta(r-1) + s}\prod_{\substack{i\geq 1 i \neq r}}{\frac{\big(q^{-\nu_r+\nu_i-s}t^{i-r+1}; q\big)_{\infty}}{\big(q^{-\nu_r + \nu_i - s}t^{i-r}; q\big)_{\infty}}}\times \prod_{\substack{\theta > j \geq 0 j \neq s}}{\frac{1}{(1 - q^{j-s})}}\right\}.
\sqrt{n!}=N^{\frac n2}(1-\xi N^{-2/3})^{\frac{N-\xi N^{1/3}}2}e^{-\frac N2-\frac{\xi N^{1/3}}2+o(1)}(2\pi N)^{\frac14}=(2\pi)^{\frac14}N^{\frac n2+\frac14}e^{-\frac N2+o(1)}.
\left(\begin{array}{c}t^{\prime}z^{\prime}\end{array}\right) =\left(\begin{array}{cc}0 -\frac{2\lambda}{\sqrt{\mu}}\frac{\sqrt{\mu}}{2\lambda}\frac{2\lambda}{\sqrt{\mu}}\end{array}\right) \left(\begin{array}{c}t z\end{array}\right) .
\begin{aligned}p_{r,s}(q):= \,\sum_{n\ge 1}q^n p_{n,r,s}=\frac{\prod_{i=1}^{r+s}(1-q^i)}{\prod_{j=1}^r(1-q^j)\prod_{k=1}^s(1-q^k)}= \,p(q)\cdot \prod_{j>r}(1-q^j)\cdot\prod_{k>s}(1-q^k)\cdot\prod_{i>r+s}(1-q^i)^{-1},\end{aligned}
\sum_{{B}_{=}}f(\mathrm{k})\le \binom{\ell}{2}\sum_{s_n<k_1,\dots,k_{\ell-1}\le n}f(k_1,k_1,k_2,\dots,k_\ell)\le \binom{\ell}{2}\sum_{s_n<k\le n}c_k^2 w_n^{2k}\cdot I_{1}^{\ell-2}.
y^{(m)}_{r_1,\ldots,r_k}=\Bigl(\prod_{i=1}^k\mu(r_i)g(r_i)\Bigr)\sum_{\substack{d_1,\ldots,d_k r_i\mid d_i\forall i d_m=1}}\biggl(\prod_{i=1}^k\frac{\mu(d_i)d_i}{\varphi(d_i)}\biggr)\sum_{\substack{a_1,\ldots,a_k d_i\mid a_i\forall i}}\frac{y_{a_1,\ldots,a_k}}{\prod_{i=1}^k\varphi(a_i)}.
M_{\texttt{b};\mu}(\boldsymbol{\xi}) := \frac{1}{N_{\texttt{b}; \mu}}\sum_{\substack{\sigma\in S_{n}\epsilon\in \{1,-1\}^n}}\exp (i \epsilon_1 \xi_{\sigma_1}\mu_1+\cdots +i \epsilon_n \xi_{\sigma_{n}}\mu_{n})
T_{R}(a_{ij})=a_{i+1,j},\ \T_{R}(b_{ij})=b_{i+1,j},\ \T_{R}(p_{ij})=p_{i+1,j+1},\ \T_{R}(q_{kl})=q_{kl}, &T_{C}(a_{ij})=a_{i,j+1},\ \T_{C}(b_{ij})=b_{i,j+1},\ \T_{C}(p_{ij})=p_{ij},\ \T_{C}(q_{kl})=q_{k+1,l+1}.
|e|_{1,\frac{1}{2}}\leq \frac{1}{\tilde \mu_1}\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}\frac{\mathcal{B}(e, (z,q))}{|(z,q)|_{1,\frac{1}{2}}}= \frac{1}{\tilde \mu_1}\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{|(z,q)|_{1,\frac{1}{2}}}.
( ( F^{s})^{I_{1}I_{2}I_{3}I_{4}})_{\mu \nu}= \sum_{\lambda = 0}^{2 I}\sigma_{\nu \lambda}( I_{1}, I_{2}, I_{3}, I_{4}) \sigma_{2 I - \lambda , \mu}^{- 1}( I_{3}, I_{2}, I_{1}, I_{4}) \ ,
\begin{aligned}G ={}\{v_1:[v_2, v_4, v_3, v_8], v_2:[v_1, v_4, v_3, v_6], v_3:[v_1, v_7, v_2, v_6], v_4:[v_1, v_2, v_8, v_5], v_5:[v_4, v_8, v_6, v_7], v_6:[v_2, v_3,v_5, v_7], v_7:[v_3, v_5, v_6, v_8], v_8:[v_1, v_7, v_4, v_5] \}\end{aligned}
\begin{array}{ll}g(x+h)=g(x)+\sum_{0<\alpha <l}\frac{(\partial^{\alpha}g)(x)}{\alpha !}+l\sum_{|\alpha|=m}\frac{h^{\alpha}}{\alpha !}\int_0^1(1-\theta)^{l-1}\left(\partial^{\alpha}g \right)(x+\theta h)d\theta ,\end{array}
\left|\mathbb{E}g_{\psi,x}(R^z)-\mathbb{E}g_{\psi,x}(R^{u})\right| & \leq C\Big( p \psi \Big)^{\frac{q}{q+1}}\Big( \sum_{k=1}^p \Theta_{M,k,q}^q \Big)^{\frac{1}{q+1}}+C \phi(M_x, M_y) \psi^2 p^4 +C\frac{(2M+1)^2}{\sqrt{n}}(\bar{m}^3_{x,3}+\bar{m}^3_{y,3})\psi^3 p^6 &+C\left(p \psi \varphi(M_x, M_y) \sqrt{\log (p/\gamma)}+\gamma \right).
\psi(x_1,x_2)=f\left(\frac{x_1}{\sqrt{\gamma}}+i\sqrt{\gamma}x_2\right)e^{-\frac{1}{\theta}\left(\left(\frac{x_1^2}{\gamma}+\gamma x_2^2\right)-z \left(\frac{x_1}{\sqrt{\gamma}}-i\sqrt{\gamma}x_2\right)\right)}
D_{1,r_1}(e^{it(\xi^3 + \xi \eta^2)}\,\widehat u_0)(\xi, \eta) = e^{it(\xi^3 + \xi \eta^2)}\, D_{1,r_1}\widehat u_0(\xi,\eta) + e^{it(\xi^3 + \xi \eta^2)}\Phi_{1, t,\alpha}(\widehat u_0)(\xi, \eta),
\left[ \begin{array}{c c c c}{{g_{2}+ w_{2}^{\ 2}h_{4}+ n_{2}^{\ 2}h_{5}}}&{{w_{2}w_{3}h_{4}+ n_{2}n_{3}h_{5}}}&{{w_{2}h_{4}}}&{{n_{2}h_{5}}}\\{{w_{2}w_{3}h_{4}+ n_{2}n_{3}h_{5}}}&{{g_{3}+ w_{3}^{\ 2}h_{4}+ n_{3}^{\ 2}h_{5}}}&{{w_{3}h_{4}}}&{{n_{3}h_{5}}}\\{{w_{2}h_{4}}}&{{w_{3}h_{4}}}&{{h_{4}}}&{{0}}\\{{n_{2}h_{5}}}&{{n_{3}h_{5}}}&{{0}}&{{h_{5}}}\\ \end{array}\right] ,
f(1)>f(y_{j_1}) \geq f(p), \j_{i_1}\in I_0=\{1,\dots, n_1\}&f(p)>f(y_{j_{2}})\geq 2f(p)-f(1), \j_{i_2}\in I_1=\{n_1+1,\dots, n_2\}&\dots &(k-1)f(p)-(k-2)f(1)>f(y_{i_k})>{f(0)\geq}kf(p)-(k-1)f(1), \j_{i_k}\in I_{k-1}=\{n_k+1,\dots, n_{k+1}=N\}.
=\delta_{\gamma,\gamma'}\delta_{n,n'}\left(\delta_{a,a'}V^{\gamma-1}\left(np^{-\gamma}+ap^{-\gamma}\right)-\dfrac{V_{\gamma-1}\left(np^{-\gamma}+ap^{-\gamma}\right)V_{\gamma-1}\left(np^{-\gamma}+a'p^{-\gamma}\right)}{V_{\gamma}\left(np^{-\gamma}\right)}\right).
\sigma_{(p,q,n)}(0) =\prod_{(a,b)}(-1)^\frac{N-1}{2}&=\prod_{(a,b)}\left((-1)^\frac{N-1}{2}\right)^{\frac{p(q-1)}{4}}&=(-1)^{\frac{(N-1)p(q-1)}{8}}.
\int^{\infty}_0 \|e^{-\xi t}T(t)x\|^2 dt = \frac{1}{2\pi}\int^{\infty}_{-\infty}\| R(\xi +i\eta,A)x\|^2 dt \int^{\infty}_0 \|t e^{-\xi t}T(t)x\|^2 dt = \frac{1}{2\pi}\int^{\infty}_{-\infty}\| R(\xi +i\eta,A)^2x\|^2 dt.
h_{22,1j}=h_{21,2j}=\mathring{h}_{21,2j},\forall~j=1,2, h_{11,1j}= \mathring{h}_{11,1j}+\frac 12H_{,1j}= \mathring{h}_{11,1j}+h_{22,1j}-\mathring{h}_{22,1j}= \mathring{h}_{11,1j}+\mathring{h}_{21,2j}-\mathring{h}_{22,1j},\forall~j=1,2.
\prod_{j = 1}^{g}\int_{\Sigma_{g}}\omega_{\bar{z}}^{j}G_{z}d^{2}z \cdot \delta \left( \int_{\Sigma_{g}}\omega_{\bar{z}}^{j}J_{z}d^{2}z \right) \sim \tilde{\Omega}_{g}\delta ({\cal V}_{g}) = c_{g}({\cal E}_{h o l}) .
\det_{1\leq j,k\leq m}\big((aq^{j-1},bq^{m-j};q)_{l_k}\big)=q^{\binom m3}b^{\binom m 2}\prod_{j=1}^m\frac{(a,b;q)_{l_j}(q^{j-m}a/b;q)_{j-1}}{(a,b;q)_{j-1}}\,\Delta(q^l).
\begin{aligned}\lim_k(g\psi_{n_k},gp\kappa_{n_k}) =\bigl(g(J_{\theta,1}^{-1}(p\gamma)-pJ_{\theta,1}^{-1}\gamma),gpJ_{\theta,1}^{-1}\gamma\bigr) & =(gJ_{\theta,1}^{-1}(p\gamma),gpJ_{\theta,1}^{-1}\gamma)-\|gpJ_{\theta,1}^{-1}\gamma\|^2. \end{aligned}
I=\frac{1}{2\kappa^{2}}\int d^{D}X\sqrt{-g}\left( R-\frac{1}{2}(\nabla\vec\phi)^2- \sum_{I=1}^{k}\frac{e^{-\vec\alpha^{(I)}\vec\phi}}{2(d_I+1)!}F^{(I)2}_{d_I+1}\right).
P\left(S_{n}>\alpha n\right) = P\big(\exp\left(tS_{n}\right)>\exp\left(\alpha tn\right)\big) & \le \frac{E\left[\exp\left(tS_{n}\right)\right]}{\exp\left(\alpha tn\right)}& = \exp\left(-\alpha tn\right)\left[q+pe^t\right]^{n}& = \left(e^{-\alpha t}\left[q+pe^t\right]\right)^{n}.
|Du(x)-Du(y)| \lesssim \sup_{y_0\in \Omega_{R/4}(x_0)}|Du(y_0)-\Theta_{y_0,\rho}|+\Phi(x,\rho)+\Phi(y,\rho) &\lesssim \sup_{y_0\in \Omega_{R/4}(x_0)}\Bigg(\sum_{j=0}^\infty \Phi(y_0, \kappa^j \rho)+\Phi(y_0, \rho)\Bigg) &\lesssim \sup_{y_0\in \Omega_{R/4}(x_0)}\sum_{j=0}^\infty \Phi(y_0, \kappa^j \rho)
\Pi_{\delta_\varepsilon(t),\xi}(u_0 - W_{\delta_\varepsilon(t),\xi}+\phi_{\delta_\varepsilon(t),\xi}-i^\ast (f_\varepsilon(u_0-W_{\delta_\varepsilon(t),\xi}+\phi_{\delta_\varepsilon(t),\xi})))=0,
K_{i\bar j}=-\frac12\mu_{ik}^r\mu_{\bar j\bar k}^{\bar r}+\frac14\mu_{\bar k\bar r}^{\bar i}\mu_{kr}^j+\frac12\mu_{ik}^k\mu_{\bar j\bar r}^{\bar r}- \frac{1}{2}\left(\mu_{rk}^k\mu_{\bar r \bar j}^{\bar i}+\mu_{ri}^j\mu_{\bar r\bar k}^{\bar k}\right).
\sum_{l^{\prime}}\bar{\Lambda}_{\gamma}( H^{- 1})_{l l^{\prime}}( 1 - e^{- 2 \beta_{l^{\prime}}}\Delta_{l^{\prime}}) = \bar{\Lambda}_{\gamma}( 1 - e^{- 2 \pi \gamma}\textit{\textbf{I}})^{- 1}\left( \begin{array}{c}{{- e^{- 2 \pi \gamma}\textit{\textbf{I}}- e^{- 2 \beta_{0}}\Delta_{0}}}\\{{1 + e^{- 2 \beta_{0}}\Delta_{0}}}\\ \end{array}\right) .
\begin{array}{l}2m\le\sum_i\deg_i(A)=\sum_\mu|\alpha_\mu|+\sum_\nu(2+|\beta_\nu|)=n+2v \le n+\sum_\nu|\beta_\nu|=2n-\sum_\mu|\alpha_\mu|\le 2n-u\le 2n.\end{array}
R_E =e^{\phi/2}\left(R+\frac{1}{2}(n-1)\Delta\phi-\frac{1}{16}(n-1)(n-2)|d\phi|^2\right) &=e^{\phi/2}\left(R+\frac{9}{2}\Delta\phi-\frac{9}{2}|d\phi|^2\right),
a_1 \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2 \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+a_3 \frac{d b_2}{b_2}\wedge \frac{d b_3}{b_3}= a_1'' \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+ \frac{d y_1}{y_1}\wedge \frac{d y_2}{y_2}+ \frac{d z_1}{z_1}\wedge \frac{d z_2}{z_2}
[ \tau ( \lambda ) Q ( \lambda ) \psi ] ({\vec{x}}) = \int d{\vec{y}}\prod_{i = N}^{1}w_{i}T r \left({\tilde{L}}( \lambda | x_{N}, \partial_{x_{N}}) \ldots{\tilde{L}}( \lambda | x_{1}, \partial_{x_{1}}) \right) \psi ({\vec{y}}) .
\frac{1}{| \Omega |}\sum_{z_{1}, \ldots , z_{N}\in \Omega}\sum_{x \in \bigcap_{i = 1}^{N}\Omega_{z_{i}p_{i}}}\sum_{\theta \in \Omega_{g}( x | z_{1}p_{1}, \ldots , z_{N}p_{N})}\prod_{i = 1}^{N}\frac{\overline{{{\phi_{i}}}}\left( c_{i}^{z_{i}}, x^{z_{i}}\right)}{| \Omega_{p_{i}}|}\prod_{\xi \in \mathcal{O}( x | \theta )}\mathcal{D}_{\xi}( z_{1}p_{1}, \ldots , z_{N}p_{N})
S = \int d^{4}x \sqrt{| g_{\mu \nu}|}\ \frac{e^{- \sigma}}{l_{s}^{2}}\left[{\frac{1}{2}}R +{\frac{1}{2}}( \nabla \sigma )^{2}+{\cal L}_{m}({\cal Y}, g_{\mu \nu}, \sigma ) \right] .
U^{( 1 )}= U^{( 0 )}\mid_{\Omega_{1}= 0}={\frac{1}{2}}\vec{\pi}^{2}-{\frac{1}{2}}c^{2}( \nabla \times \vec{u})^{2}+ \nu \vec{\pi}. \nabla^{2}\vec{u}- \vec{u}. \vec{J}+ \nu \vec{u}. \nabla n .
\|\Box_k^{\alpha_2}f\|_{M_2}= \|\Box_k^{\alpha_2}f\|_{L^{\infty}}\lesssim\sum_{l\in \Gamma_k^{\alpha_1,\alpha_2}}\|\Box_l^{\alpha_1}f\|_{L^{\infty}}\lesssim 2^{jn\alpha_1(1/p_1-1/p_2)}\sum_{l\in\Gamma_k^{\alpha_1,\alpha_2}}\|\Box_l^{\alpha_1}f\|_{L^{p_1}}\lesssim 2^{jn\alpha_1(1/p_1-1/p_2)}\|f\|_{M_1}.
I(u) =\frac{1}{bp^2}\left(a+b\int_0^T|{_0D_t^\alpha}u(t)|^pdt\right)^p-\int_0^TF(t,u(t))dt-\frac{a^p}{bp^2}&=\frac{1}{bp^2}(a+b\|u\|_{E^{\alpha,p}}^p)^p-\int_0^TF(t,u(t))dt-\frac{a^p}{bp^2}.
\mathcal{Z}\left(\hat{\mu}\right)\subset\bigcup_{i=1}^{r}\rho^{-i}\bigcup_{j=0}^{\infty}\frac{b_{j}}{p^{j}N_{j}},\ 1\leq i\leq r,\b_{j}\in\mathbb{Z}\setminus N_{j}\mathbb{Z}.
N_{it}=-\log\left(\max\left(\left\| \mathbf{r}_p(\mathbf{q}_w) \right\|,\left\| \mathbf{r}_d(\mathbf{q}_w) \right\|,\mu(\mathbf{q}_w)\right)/\epsilon_u\right)/\log\left(1-\delta/\sqrt{2k+1}\right).
T_0 =1, T_1 =1+x, T_2 =1+3x+x^2, T_3 =1+6x+5x^2+x^3, T_4 =1+10x+15x^2+7x^3+x^4, T_5 =1+15x+35x^2+28x^3+9x^4+x^5.
\frac{u_{( i + 1 , j )}- 2 u_{( i , j )}+ u_{( i - 1 , j )}}{{\Delta x_{1}}^{2}}+ \frac{u_{( i , j + 1 )}- 2 u_{( i , j )}+ u_{( i , j - 1 )}}{{\Delta x_{2}}^{2}}= f ( u_{( i , j )}) ,
\|\omega_n\|_X^2 =\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega_n(x,0)|^2\ln|\omega_n(x,0)|dx+\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_n(x,0)|^{2_\alpha^*}dx &\leq\lambda C(\|\omega_n\|_X^2+\|\omega_n\|_X^4)+C\|\omega_n\|_X^{2_\alpha^*}.
\mathcal{F}\left\{\mathcal{L}\left[p(\cdot,x)\right](s)\right\}(\xi) =\frac{s^{\alpha/2-1}}{2}\int_{-\infty}^{\infty}e^{-\textnormal{i}x\xi}\exp\{-s^{\alpha/2}|x|\}dx & = \frac{s^{\alpha/2-1}}{2}\cdot \frac{2s^{\alpha/2}}{s^{\alpha}+|\xi|^{2}}= \frac{s^{\alpha-1}}{s^{\alpha}+|\xi|^{2}}.
S_{j}^{i}:=\bigg\{x\in P_{\rho}:~\,\frac{\delta}{2}+\frac{(i-1)\delta}{2m_{j}}<\operatorname*{dist}\big(x,\partial P_{\rho}\big)<\frac{\delta}{2}+\frac{i\delta}{2m_{j}}\bigg\}~,
\|D_k(x,\cdot)\|_{\dot{\mathcal B}^{\alpha,q}_{p,\mathcal F}}=\bigg\{\sum_{j\in \Bbb Z}\Big(2^{j\alpha}\|D_j(D_k(x,\cdot))\|_{L^p_\mu}\Big)^q\bigg\}^{1/q}&\le C\frac 1{V_k(x)^{1-1/p}}\bigg\{\sum_{j\in \Bbb Z}2^{j\alpha q-|j-k|\varepsilon q}\bigg\}^{1/q}&\le C2^{k\alpha}\frac 1{V_k(x)^{1-1/p}},
\frac{1}{2\pi i}\int_C dsx^{-s}\frac{\Gamma(s)}{\Gamma(s+\frac{1}{2})}\;{}_2F_1(\frac{s}{2},\frac{s+1}{2};s+\frac{1}{2};z^2)=\frac{1}{\sqrt{\pi}}\frac{\Theta(1-x+\frac{x^2z^2}{4})}{\sqrt{1-x+\frac{x^2z^2}{4}}}.
g^{( + )}= \left( \begin{array}{c c c}{\frac{1}{2}\alpha^{( + ) 2}}&{}&{}\\{- \frac{i}{\sqrt{2}}\alpha^{+}}&{- \frac{1}{2}\alpha^{( + ) 2}}&{\tau_{m}}\\{- s^{( + )}\tau^{* n}}&{}&{}\\ \end{array}\right)
D(n,k,r,b,p)= \prod_{j=1}^{n+1}{((n-j+1)p +b+k+(j-1)(r-1) )! (n-j+1)! \over(k+(j-1)r )!((n-j+1)(p+1)+b )!}\hphantom{D(n,k,r,b,p)=}{}\times \prod_{1 \le i \le j \le n}((k-i+1)(p+1)+jr+(np+b)r).
f_1(v) = \frac{e^{-\frac{1}{2}(v - 2)^2}}{\sqrt{2 \pi}}+\frac{e^{-\frac{1}{2}(v + 2)^2}}{\sqrt{2 \pi}}, f_2(v) = \frac{e^{-\frac{1}{2}(v - 2)^2}}{\sqrt{2 \pi}}+\frac{e^{-\frac{1}{4}(v + 2)^2}}{\sqrt{4 \pi}}, f_3(v) = \frac{e^{-2(v - 2)^2}}{\sqrt{\pi/2}}+\frac{e^{-\frac{4}{3}v^2}}{\sqrt{3\pi/4}}+\frac{e^{-(v + 2)^2}}{\sqrt{\pi}}
[q^-_i , q^+_j] = (\mathbb{I}_2 - 2 N_{q_i})~\delta_{ij}, [N_{q_i}, q^+_j] = - \delta_{ij}q^+_j, [N_{q_i}, q^-_j] = + \delta_{ij}q^-_j [ q_i^- , q_j^- ] = [ q_i^+ , q_j^+ ] = 0.
\ddot{x}=-\dfrac{a\dot{x}}{\sqrt{\dot{x}^2+\dot{y}^2}}\dot{\tilde{\lambda}},\ddot{y}=-\dfrac{a\dot{y}}{\sqrt{\dot{x}^2+\dot{y}^2}}\dot{\tilde{\lambda}},\qquad\ddot{z}=-g+\dot{\tilde{\lambda}}.
\sum_{P \in \mathbb{P}_{\nu}(\mu,\lambda)}(-1)^{L(P)}W(P) = t^{12}(1-t)(1-t^2)+t^{12}(1-t)^2(1-t^2)+t^{12}(1-t)^2-t^{12}(1-t)^2 &=t^{12}(2-t)(1-t)(1-t^2).
\mathrm{J}=2\int \big(u_t-u_{txx}\big)u\Psi-2\int uu_{tx}\Psi' =2\int \big(3uu_xu_{xx}+u^2u_{xxx}-4u^2u_x\big)u\Psi +2\int (u^2u_{xx}+2uu_x^2+p_x*(3uu_xu_{xx}+2u_x^3+3u^2u_x))u\Psi' =4\int u^2u_x^2\Psi'+2\int u^4\Psi'+2\int \{p_x*(3uu_xu_{xx}+2u_x^3+3u^2u_x)\}u\Psi'.
\sigma_j(Ax) = \sigma_j\left( \lim_{k \to \infty}z_k \right) = \lim_{{k \to \infty}\atop{k \geqslant j}}\sigma_j(z_k - z_j)+\sigma_j(z_j) = \sigma_j(z_j) =\left( A_j \circ \sigma_j \right)(x),
\overline{\sigma}_1\ge\cdots\ge\overline{\sigma}_{k_0}\!>\overline{\sigma}_{k_0+1}=\!\cdots=\!\overline{\sigma}_{k}=\cdots=\overline{\sigma}_{k_1}:=\overline{\nu}>\overline{\sigma}_{k_1+1}\!\ge\cdots\ge\overline{\sigma}_m\!\ge 0.
\Delta^{\dagger}= a^{\dag}+ x b^{\dag}= \left( \begin{matrix}{I}&{B_{2}+ z_{2}}&{B_{1}+ z_{1}}\\{J^{\dagger}}&{- B_{1}^{\dagger}- \bar{z}_{1}}&{B_{2}^{\dagger}+ \bar{z}_{2}}\\ \end{matrix}\right) .
<u_1(t)>^{\widehat{}}(k)=\sqrt{\frac{2}{h}}\frac{h}{\Pi_1\pi k}\bigg((-1)^k-1\bigg) \int_{-\infty}^t e^{-\nu (\frac{\pi k}{h})^2(t-\tau)}p_1(\tau)d\tau,
\|\Psi_T(x)\|_{L_2(\mathcal{R}_{\theta_T}^2)}=|\det T|\|U_{\theta_T}(f(T\cdot))\|_{L_2(\mathcal{R}_{\theta_T}^2)}&=|\det T|\|f(T\cdot)\|_{L_2(\mathbb{R}^2)}=|\det T|^\frac{1}{2}\|x\|_{L_2(\mathcal{R}_{\theta}^2)}.
\phi_{n+1,\lambda}(x) =x\phi_{n,\lambda}(x)+\bigg(x\frac{d}{dx}-n\lambda\bigg)\phi_{n,\lambda}(x) &=x\sum_{k=0}^{n}\binom{n}{k}(1-\lambda)_{n-k,\lambda}\phi_{k,\lambda}(x).
{\rm Det'}^{-{1\over 2}}_P\Bigl[ -{\partial}^2_{\tau}+2ieF{\partial}_{\tau}\Bigr]={\lbrack 4\pi T\rbrack}^{-{D\over 2}}{\rm Det'}_P^{-{1\over 2}}\Bigl[{\bf I}-2ieF{{\partial}_{\tau}}^{-1}\Bigr]\quad
S \simeq \int d^{3}x \left[ \frac 1 2 ( \nabla \chi )^{2}- 2 \xi \cos ( g_{m}\chi ) \right] - 2 \xi^{2}\int d^{3}x d^{3}y \cos ( g_{m}\chi ({\bf x}) ){\cal K}({\bf x}-{\bf y}) \cos ( g_{m}\chi ({\bf y}) ) .
\begin{array}{r c l}{\Delta ( x - 1 \epsilon ( x ) )}&{=}&{x_{( 1 )}\otimes x_{( 2 )}- 1 \epsilon ( x ) \otimes 1}\\{}&{=}&{( x - 1 \epsilon ( x ) ) \otimes 1 + x_{( 1 )}\otimes ( x_{( 2 )}- 1 \epsilon ( x_{( 2 )}) )}\\ \end{array}
\limsup_{n\to\infty}\left\|M_n w^n \right\|_{L^\infty([-1, 1])}^{1/n}= \limsup_{n\to\infty}\left\|P_n^*w^n \right\|_{L^\infty([-1, 1])}^{1/n}\frac{1}{c_n^{1/n}\gamma_n^{1/n}a_n}= \frac{1}{2e^{1/\alpha}}\limsup_{n\to\infty}\left\|P_n^* w^n \right\|_{L^\infty([-1, 1])}^{1/n}=e^{-F_w}\limsup_{n\to\infty}\left\|P_n^* w^n \right\|_{L^\infty([-1, 1])}^{1/n}.
\tilde F_\lambda (x_1,\dots, x_n;t) =\frac{(1-t)^n}{\prod_{i=1}^{n-l}(1-t^i)}\sum_{\sigma \in S_n}\sigma\left(f_{\lambda_1}(x_{1})\dots f_{\lambda_n}(x_{n})\prod_{i=1}^{n}\prod_{i < j}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right),
\mathfrak{S}_{2,4,2}= -\sum_{(i,j,k,\ell)\in [n]_1^4}\sum_{\pi \in T_n^{(i,j)}\cap T_n^{(1,k)}\cap T_n^{(\ell,1)}}\mathrm{sgn}(\pi) \binom{k+1}{2}\mathcal{R}_{i+1}\mathcal{R}_{j+1}\mathcal{R}_{k-1}\mathcal{R}_{\ell+1}Q_{R}(\pi;1,i,\ell).
\hat{H}_{I}^{( q )}( \hat{x}, \hat{p}) = \frac{f^{2}}{6}\sum_{k = 1}^{\infty}\frac{V^{( k )}( 0 )}{k !}\hat{x}^{k}\biggl ( k \hat{x}^{2}\partial_{\hat{x}}^{2}+ k ( k + 2 ) \hat{x}\partial_{\hat{x}}+ \frac{1}{6}k ( k - 1 ) ( 2 k + 5 ) \biggr ) .
\Delta t < \frac{1}{\beta^n_1 +\beta_2^n}= \Big( \Big|\Big|J_{\mathcal A_{{t_{n}},~t_{n-1/2}}}\Big|\Big|_{L_{\infty}(\Omega_{t_n})}||\nabla \cdot \mathbf{w}_{h}||_{L_{\infty}(\Omega_{t_{n-1/2}})}&+ \Big|\Big|J_{\mathcal A_{{t_{n}},~t_{n+1/2}}}\Big|\Big|_{L_{\infty}(\Omega_{t_n})}||\nabla \cdot \mathbf{w}_{h}||_{L_{\infty}(\Omega_{t_{n+1/2}})}\Big)^{-1}
\sum_{k=N}^{\infty}|b_k| r^k \leq \sum_{n=N}^{\infty}\frac{|\tilde{b}_n|}{n}r^n = \int_{0}^{r}\frac{M_{t}^{N}( \tilde{G})}{t}dt & \leq \int_{0}^{r}\frac{M_{t}^{N}( G). M_{t}^{N}( \psi\circ \omega)}{t}dt & \leq \int_{0}^{r}\frac{M_{t}^{N}(\psi) t^{2N}}{t^2(1-t^2)}dt:= R^N(r).
\mathrm{b}(m \otimes a_1 \otimes \cdots \otimes a_n) := m a_1 \otimes \cdots \otimes a_n & + \sum_{i = 1}^{n - 1}(-1)^i m \otimes a_1 \otimes \cdots \otimes a_i a_{i + 1}\otimes \cdots \otimes a_n & + (-1)^n a_n m \otimes a_1 \otimes \cdots \otimes a_{n - 1}.
R_X(h) = \int_{\mathcal{X}}\left(\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{ih^{\top}u}|\widehat g(x,u)|^2du\right) p(dx) &=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{ih^{\top}u}\left(\int_{\mathcal{X}}|\widehat g(x,u)|^2 p(dx)\right) du=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{ih^{\top}u}\gamma(u) du.