harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2h_{ij}(q)\int_{\mathbb{R}_{+}^{n}}t\frac{\partial U}{\partial z_{i}}\frac{\partial v_{q}}{\partial z_{j}}dzdt-n\int_{\mathbb{R}^{n-1}}U^{\frac{2}{n-2}}v_{q}^{2}dz=-\int_{\mathbb{R}_{+}^{n}}|\nabla v_{q}|^{2}dzdt
\setlength{\abovedisplayskip}{3pt}\setlength{\belowdisplayskip}{3pt}\begin{cases}\partial_tu-\epsilon\Delta u-\kappa \Lambda\sigma=\mathcal{F}_1, \partial_t\sigma-\mu\Delta\sigma+\beta\sigma+\frac{\alpha}{2}\Lambda u=\mathcal{F}_2, \end{cases}
{d^{\dagger}}( \eta^n \cdot \tau^n{\cal{G}}) \;\; = \;\; \eta^n \cdot \tau^n \,{d^{\dagger}}{\cal{G}}+{\textsf{P}}_s (n) \; \eta^{n-1}\cdot \tau^{n-1}\,{d^{\dagger}}{\cal{G}}
\sqrt{\,\left\{\, X^\mu\ , X^\nu\,\right\}_{\mathrm{PB}}\left\{\, X_\mu\ , X_\nu\,\right\}_{\mathrm{PB}}\,}\longleftarrow \quad (?)\quad\longrightarrow\sqrt{\, \mathrm{Tr}\mathbf{F}_{\mu\nu}\, \mathbf{F}^{\mu\nu}\,}
\partial_{r}v_{2}(t,r) = \frac{c_{b}}{2}\int_{0}^{\infty}\sin(t \xi) \xi \left(J_{0}(r \xi)-J_{2}(r \xi)\right) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}d\xi
\lim_{\lambda\to\infty}\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\|_X=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{X}.
\P\left(\Omega_0 = \left\{\omega \in \Omega: \int_0^\eta |y_t|^2 dt \le \sqrt{\delta_1}\eta K^{2r}, \sup_{0 \le t \le \eta}|z_t-z_0| \le RK^{\frac{r-1}{3}}\right\}\right) \ge 1-\frac{\beta}{2}.
=\displaystyle\sum_{n=2}^{\infty}\Big|\sqrt{(1-\frac{h_{n}}{g_{n}}})\frac{A_{n}}{\sqrt{\lambda_{n}}}-\sqrt{(1+\frac{h_{n}}{g_{n-1}})}\frac{A_{n-1}}{\sqrt\lambda_{n}}\Big|^{2}\geq0.
_{\infty}D_{x}^{\alpha}\psi_{n}(x)\ =\frac{1}{\Gamma(2-\alpha)}\frac{d^{2}}{dx^{2}}\int_{x\geq a}^{\infty}(x^{\prime}-x)^{-\alpha+1}\psi_{n}(x^{\prime})dx^{\prime}& =0,x\geq a,
a_n(r,0)\equiv \prod_{i=0}^{m}a_{n_i}(r,s)+ 3\sum_{\nu=1}^m\left(\vphantom{\sum_{k_{\nu-1}=0}^2}\right.\underset{i\ne \nu-1,\nu}{\prod_{i=0}^m}a_{n_i}(r,s)\left.\vphantom{\sum_{k_{\nu-1}=0}^2}\right)f(n_{\nu-1},n_{\nu};r,0)(),
\displaystyle{\frac{\phi^{( 0 )}( a , b + 1 , c | u , v , w + 1 / 4 )}{\phi^{( 0 )}( a , b - 1 , c | u , v , w + 1 / 4 )}}= \displaystyle{\frac{W ( a , b + 1 | A_{0}) \, \overline{{{W}}}( b + 1 , c | B_{0})}{W ( a , b - 1 | A_{0}) \, \overline{{{W}}}( b - 1 , c | B_{0})}}.
D^{(n)}_q((d+1)q/2\pm u;t) D^{(n)}_{q,t}(d)= D^{(n)}_{q,t}(d+1) \prod_{1\le i\le n}\vartheta(z_i\pm u), D^{(n)}_{q,t}(d) D^{(n)}_q(-dq/2\pm u;t)= \prod_{1\le i\le n}\vartheta(z_i\pm u) D^{(n)}_{q,t}(d+1),
S_2\le C \sup_{t \in I}\Bigg[\sum_{\alpha=1}^\infty \lambda_\alpha^\beta \bigg(\int_0^t \phi_\alpha^2(t-s) ds\bigg) \Bigg]^\frac p2\le C \Bigg(\sum_{\alpha=1}^\infty \lambda_\alpha^{\beta-1}\Bigg)^\frac p2,
P_k \partial_j \Delta^{-1}\mathcal N(\phi_{k_1},\bar \phi_{k_2})= Q_{<k-Cm}P_k\partial_j \Delta^{-1}\mathcal N(Q_{<k-Cm}\phi_{k_1},Q_{<k-Cm}\bar \phi_{k_2})+ err(k,k_1,k_2)
\partial_{t}u(x,t)=a\left[ \mathcal{D}_{x}^{\alpha ,\theta}+\lambda t(I-\mathcal{O}_{1,x}^{\alpha ,\theta})\right] u(x,t)+\frac{\lambda}{\alpha}\left( I-\mathcal{O}_{1,x}^{\alpha ,\theta}\,\right) \,I_{1-\alpha}^{\alpha -1}\left[ xu(x,t)\right] ,
{\frac{d^{2}x^{i}}{d \tilde{\lambda}^{2}}}= -{\frac{F}{\sigma + \varrho}}\left[{\frac{1}{a_{0}^{2}}}\sum_{j}\left({\frac{d x^{j}}{d \tilde{\lambda}}}\right)^{2}+{\frac{2}{a_{0}^{4}}}\right]{\frac{d x^{i}}{d \tilde{\lambda}}}.
\begin{array}{l}H_{\rm VI}(q, p, t ; \tilde\alpha_{0}, \tilde\alpha_{1}, \tilde\alpha_{2}, \tilde\alpha_{3}, \tilde\alpha_{4}) =\dfrac{1}{t(t-1)}\{q(q-1)(q-t)p^2 begin{align*}4pt]-\{(\tilde\alpha_{0}-1)q(q-1)+\tilde\alpha_{3}q(q-t)+\tilde\alpha_{4}(q-1)(q-t)\}p+\tilde\alpha_{2}(\tilde\alpha_{1}+\tilde\alpha_{2})(q-t)\}.\end{array}
K_{m}-K_{m-1}=-\frac{(\alpha+\beta)(\gamma+\delta)}{1+\beta\gamma}\sum_{r=1}^{m-1}(-\delta)^{r-1}[r]!\,q^{-(m-2)r}e_{r}(1, q, \ldots , q^{m-2})\prod_{p=2}^{r}\frac{\alpha+\beta q^{p-1}}{1+\beta\gamma[p]}
\mathbf{S}_1(\gamma) =\mathbb{E}[\mathbf{S}_1(\gamma)|\mathbf{S}_1,S_2,S_2(\gamma)]+\mathbf{W}_1 &=\mathbf{A}_1\mathbf{S}_1+\mathbf{a}_2S_2+\mathbf{a}_3S_2(\gamma)+\mathbf{W}_1, S_2(\gamma) =\mathbb{E}[S_2(\gamma)|\mathbf{S}_1,S_2]+W_2 &=\mathbf{b}^T_1\mathbf{S}_1+b_2S_2+W_2,
r_{1}: = \sqrt{\vec{q}_{1}^{\; 2}}, \hspace{1 e m}r_{2}: = \sqrt{\vec{q}_{2}^{\; 2}}, \hspace{1 e m}x : = \cos < \hspace{- 0 . 7 5 e m}\mathrm{\raisebox{0 . 2 e x}{\scriptsize ~ )}}\; ( \vec{q}_{1}, \vec{q}_{2}) \hspace{1 c m}.
\Delta ( \tau_{i}^{\pm}) = \tau_{i}^{\pm}\otimes \tau_{i}^{\pm}, \ \Delta ( \tau_{2 1}) = \tau_{1}^{+}\otimes \tau_{2 1}+ \tau_{2 1}) \otimes \tau_{2}^{+}, \ \Delta ( \tau_{1 2}) = \tau_{2}^{-}\otimes \tau_{1 2}+ \tau_{1 2}) \otimes \tau_{1}^{+},
H^i \big( \operatorname{Sp}(A) , \mathcal{O}_{\operatorname{Sp}(A)}(<r) \big) = 0 \enspace \enspace i \neq 0 \enspace H^i \big( \operatorname{Sp}(A) , G \big) = 0 \enspace \enspace i \neq 0.
\mathcal{N}_{\lambda,\mu}^+:=\big\{(u,v)\in \mathcal{N}_{\lambda,\mu}: \varphi_{u,v}''(1) >0\big\}, & \mathcal{N}_{\lambda,\mu}^-:=\big\{(u,v)\in \mathcal{N}_{\lambda,\mu}: \varphi_{u,v}''(1) <0\big\}, &\mathcal{N}_{\lambda,\mu}^0:=\big\{(u,v)\in \mathcal{N}_{\lambda,\mu}: \varphi_{u,v}''(1) =0\big\}.
( \lambda - \mu ) ~ \left[ ~ T^{i j}( \lambda ) ~ , ~ T^{k l}( \mu ) ~ \right] ~ ~ = ~ ~ h{}~ \left( ~ T^{k j}( \mu ) ~ T^{i l}( \lambda ) ~ - ~ T^{k j}( \lambda ) ~ T^{i l}( \mu ) ~ \right) ,
\begin{pmatrix}\nabla\xi \nabla \phi \end{pmatrix}a \begin{pmatrix}\nabla\xi^T \nabla \phi^T \end{pmatrix}= \begin{pmatrix}\nabla\xi a\nabla\xi^T 0 0 \nabla\phi a\nabla\phi^T \end{pmatrix}= \begin{pmatrix}\mathcal{O}(1) 0 0 \mathcal{O}(\frac{1}{\delta}) \end{pmatrix}\,.
[ -{\frac{\delta_{\rho}}{k^{-}}}+ 2 \delta_{c}]{\frac{v ( X )}{X}}= [{\frac{\delta_{a}}{( k^{-})^{2}}}- 2{\frac{\delta_{\rho}}{k^{-}}}+ \delta_{c}( 3 -{\frac{k^{4}}{( k^{-})^{4}}}) ]{\frac{\partial v ( X )}{\partial X}},
\Pi_{0 0}^{\mathrm{t h e r m a l}}( k_{0}, \vec{k}) ={\frac{4 e^{2}}{( 2 \pi )^{4}i}}\int d^{3}p \int_{- i \infty + \epsilon}^{i \infty + \epsilon}d p_{0}\, n_{F}( p_{0}) \left[{\frac{m^{2}+ p^{2}+ p \cdot k - 2 p_{0}k_{0}- 2 p_{0}^{2}}{( p^{2}+ m^{2}) ( ( p + k )^{2}+ m^{2})}}+ ( k_{0}\leftrightarrow - k_{0}) \right]
{\rm \Lambda}_L= \underset{x\in[-1,\,1]}\max \sum_{j=1}^{N}w_j\Big| \sum_{\ell=0}^{L}\frac{1}{1+\lambda\mu_{\ell}^2}\tilde{\Phi}_{\ell}(x)\tilde{\Phi}_{\ell}(x_j)\Big| \leq \underset{x\in[-1,\,1]}\max \sum_{j=0}^{N}w_j \sum_{\ell=0}^{L}\frac{1}{1+\lambda\mu_{\ell}^2}\Big|\tilde{\Phi}_{\ell}(x)\tilde{\Phi}_{\ell}(x_j)\Big|.
\partial_s U_{\varphi_0}(s,t) =\int_{-\infty}^{\infty}\partial_s K(s-\xi,t)\varphi_0 (\xi)d\xi=\int_{-\infty}^{\infty}-\partial_\xi K(s-\xi,t)\varphi_0 (\xi)d\xi &=\int_{-\infty}^{\infty}K(s-\xi,t)\varphi_0^\prime (\xi)d\xi
\langle \, T_{--}( \sigma ) \, \rangle = \frac{1}{1 2 \pi}\, ( \partial_{-}p )^{1 / 2}\, \partial_{-}^{2}( \partial_{-}p )^{- 1 / 2}= - \, \frac{1}{2 4 \pi}\, \{p \, , \: \sigma^{-}\}_{S}\, .
\left\|\sum_{j=0}^{\infty}\mu\left(\prod_{k=1}^{j}B_k(\epsilon) \right)r_{j+1}(\epsilon) - \sum_{j=0}^{\infty}\mu\tilde{B}^j \tilde{r}- \epsilon \sum_{j=0}^{\infty}\sum_{i=1}^{j-1}i\mu\tilde{B}^i \tilde{B}^{(1)}\tilde{B}^{j-i}\tilde{r}- \epsilon\sum_{j=0}^{\infty}j\mu\tilde{B}^j\tilde{r}^{(1)}\right\| &\quad= O(\epsilon^2(\log(1/\epsilon))^w)
F(\mathbf{B}_B)= (\kappa-\nu)(\kappa-\nu-1)(\kappa-2a-(b>1)-2) + (\kappa-\nu)(a+(b>1))(\kappa-2a-(b>1)-1) + a(\kappa-\nu)(\kappa-2a-(b>1)-1)a(\kappa-\nu)(\kappa-2a-(b>1)).
(\mu^0\hat\oplus\mu^1)(c)=\begin{array}{c}\mu^0(\mathfrak p_1)\hat\oplus\mu^1(\mathfrak p_1)\leftrightarrow\cdots\leftrightarrow\mu^0(\mathfrak p_k)\hat\oplus\mu^1(\mathfrak p_k) \updownarrow \mu^0(\mathfrak q_1)\hat\oplus\mu^1(\mathfrak q_1)\leftrightarrow\cdots\leftrightarrow\mathfrak \mu^0(\mathfrak q_\ell)\hat\oplus\mu^1(\mathfrak q_\ell)\end{array}.
{\cal L}(1,\alpha-\frac{1}{2},g)= - i \epsilon_{\mu \nu \lambda}B^*_\mu ( \partial_\nu+ i a_\nu + i C_\nu +M ) B_\lambda + \frac{g}{4}(B^*_\mu B_\mu)^2- \frac{i}{8 \pi (\alpha - 1/2)}\epsilon_{\mu \nu \lambda}a_\mu \partial_\nu a_\lambda\;.
\left(\sum_{k\in\mathbb{Z}^d}\left|\langle f,\widetilde{\phi}_k\rangle\right|^p\right)^{\frac{1}{p}}+ \left(\sum_{i=1}^{2^{d}-1}\sum_{j\in\mathbb{N}_0}2^{j\left( s+d\left(\frac{1}{2}-\frac{1}{p}\right)\right) q}\left(\sum_{k\in\mathbb{Z}^d}\left|\langle f,\widetilde{\psi}_{i,j,k}\rangle\right|^p\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}< \infty,
\frac{\operatorname{p.v.}}{\pi}\int f(\beta,t)\frac{\xi_\beta(\beta,t)}{\xi(s,t)-\xi(\beta,t)}d\beta &= \frac{\operatorname{p.v.}}{L}\int_0^L f(\beta) \xi_\beta\cot \frac{\pi(\xi(s)-\xi(\beta))}{L}d\beta.
|\partial_{r}v_{5}(t,r)| \leq \begin{cases}\frac{C}{t^{7/2}\log^{3b-3+\frac{5N}{2}}(t)}, r \leq \frac{t}{2}\frac{C \log^{3}(t)}{\sqrt{r}t^{3/2}}, r > \frac{t}{2}\end{cases}
\int |\Phi(u_h)-\Phi(u)|\, \left|\frac{\delta_h u}{|h|^{\gamma+t}}\right|\,\eta^p\,dx \le L\, \int_{B_\frac{R+r}{2}}\left|\frac{\delta_h u}{|h|^\frac{\gamma+t}{2}}\right|^2\,dx\le C\,L^\frac{p}{p-2}\,R^{N}+C\,\int_{B_R}\left|\frac{\delta_h u}{|h|^\gamma}\right|^p\,dx,
\gamma_+(v^++tv^-) =I'(v^+)[v^+]+t^2\lambda\int_{\mathbb R^3}\phi_{v^-}(v^+)^2dx \leq I'(v^+)[v^+]+\lambda\int_{\mathbb R^3}\phi_{v^-}(v^+)^2dx=\gamma_+(v)<0.
y_{n_{r}}= p \left\{1 + C_{n_{r}}\, p^{- 2 / 3}+ \left( \lambda_{l}^{( \ast )}\right)^{1 / 2}\, \left[ - c + \ln \Theta_{l}\right] p^{- 1}+ O \left( p^{- 4 / 3}\right) \right\}\; .
K_{d i v}( t ) = \frac{2 V \pi^{\frac{m - 1}{2}}}{( 2 \pi )^{m - 1}\Gamma \left( \frac{m - 1}{2}\right)}\int_{0}^{\infty}d k \, k^{m - 2}\, e^{- ( M^{2}- S^{2}+ k^{2}) t}\, .
\lim_{\lambda \to -0}\lim_{V \to \infty}\omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}}))= \lim_{\lambda \to +0}\lim_{V \to \infty}\omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}})) \ .
[G]_{i,j}= \sum_{n=1}^\infty \sum_{k=0}^\infty [Q_{0,0}^{(n-1)}]_{i,k}\, [ A_{-1}]_{k,j}= [N A_{-1}]_{i,j}, &[R]_{i,j}= \sum_{n=1}^\infty \sum_{k=0}^\infty [ A_1]_{i,k}[Q_{0,0}^{(n-1)}]_{k,j}= [ A_1 N]_{i,j}.
\textbf{w}= \left [ \sum_{n=1}^{N}G^{C}_{\sigma\,\sqrt{2}}(e_{n})\textbf{X}_{n}\textbf{X}_{n}^{H}\right ]^{-1}\left [ \sum_{n=1}^{N}G^{C}_{\sigma\,\sqrt{2}}(e_{n})\textit{d}_{n}^{*}\,\textbf{X}_{n}\right ]
\begin{array}{c}eC \subset \cap_{\alpha \in J}h^{-1}(\varphi_{\alpha}^{-1}d_{\alpha}\varphi_{\alpha}) h C =h^{-1}( \cap_{\alpha \in J}\varphi_{\alpha}^{-1}d_{\alpha}\varphi_{\alpha}) h C =h^{-1}c h C\,,\end{array}
\left\| \sum_{n=0}^{N}\frac{\gamma_n}{(n+\alpha)^s}-\sum_{n=0}^{N}\frac{\gamma_n}{(n+c)^s}\right\|\leq \rho \sqrt{K}\sum_{n=0}^{N}(n+c/2)^{-3/2}\leq \rho \sqrt{K}\zeta(3/2, c/2),
\begin{array}{lll}d=10: \Gamma_M M=1,\cdots ,10\, , d=11:{\tilde{\Gamma}}_{\tilde M}{\tilde M}= 1,\cdots ,10,11\, , d=12:{\hat{\Gamma}}_{\hat M}{\hat M}= 1,\cdots ,10,11,12\, .\end{array}
{\cal H}={1\over 2}\rho v_iv_i +R(\rho )~,~~\{v_i(x),\rho (y)\}={{\partial}\over{\partial x^i}}\delta (x-y)~,~~\{v_i(x),v_j(y)\}= -{{\partial_iv_j-\partial_jv_i}\over{\rho}}\delta (x-y).
\begin{pmatrix}3 1 0 0 0 1 7 -2 -3 2 0 -2 5 0 0 0 -3 0 1 0 0 2 0 0 -4\end{pmatrix}\mbox{and}\begin{pmatrix}0 2 2 2 1 2 -1 4 4 2 2 4 6 4 2 2 4 4 -5 2 1 2 2 2 3\end{pmatrix}.
\langle j_0(x)\rangle^{(1)}=i4\pi^2e^3g\int d^2y\int dp^0\,{\rm Tr}\,[\gamma_0\,\Sigma(p^0,{\bf x}-{\bf y})\,\gamma_0\,\Sigma(p^0,{\bf 0})\,\gamma_0\,\Sigma(p^0,{\bf y}-{\bf x})] \; .
\psi^2_{_{(\alpha)}}(x) = (\frac{\mu}{2\pi})^{^{1/2}}{\cal K}(\phi_1)\,{\cal K}(\phi_3)\, :e^{-i \sqrt{\pi}\{\int_{-\infty}^x dz^1 \pi_{_2}(z) + \gamma^5_{_{\alpha\alpha}}\phi_{_2}(x)\}}:,
{\mathcal B}_{\lambda, \mu}(\vec a;\sigma) :=\prod_{(i,j)\in \lambda}\frac{P(\vec a; i-j+\sigma)}{h_{\lambda}^2(i,j)(\lambda'_j+\mu_i-i-j+1+2\sigma)^2}\times &\times \prod_{(i,j)\in \mu}\frac{P(\vec a; i-j-\sigma)}{h_{\mu}^2(i,j)(\lambda_i+\mu'_j-i-j+1-2\sigma)^2}\ ,
\sum_{j = 1}^k{( - 1)^{j - 1}G_{2j - 1}}= - \frac{1}{5}G_0 + \frac{2}{5}G_1 + ( - 1)^k \frac{1}{5}G_{2k}- ( - 1)^k \frac{2}{5}G_{2k + 1}\,,
H^{-1}(t) \approx \frac{t^{\frac1p-1}}{\psi \left(t^{- \frac{1}{n}}\right)^{n-1}}= \frac{t^{\frac1p-1}}{\phi(1)^{n-1}\left(t^{- \frac{1}{n}}\right)^{n-1}}= \phi(1)^{1-n}t^{\frac{n-p}{np}}
\tau_1 = (1 \ldots e_1), \tau_2 = (e_1 \ldots (e_1+e_2-1)), \tau_3 = ((e_1+e_2-1) \ldots (e_1+e_2+e_3-2)), \vdots \tau_k =\Big(\big(\sum_{i=1}^{k-1}e_i-(k-2)\big) \ldots \big(\sum_{i=1}^{k}e_i-(k-1)\big)\Big), \vdots
\vert\vert (T^{n_p}-D^{n_p})x\vert\vert\le \vert\vert x \vert\vert \sum_{\ell=2}^{+\infty}\sum_{k=\max(1,\ell-n_p)}^{\ell-1}w_{\ell-1}\dots w_k\cdot\frac{\vert\lambda_\ell-\lambda_{j(\ell)}\vert}{\vert \lambda_k-\lambda_{j(k)}\vert}\cdot\big\vert s_{k,\ell}^{(n_p)}\big\vert.
g(T^*,T)(f(T^*,T)(B)) = \sum_{n=0}^{\infty}g_n T^{*n}\Big( \sum_{m=0}^{\infty}f_m T^{*m}B T^m \Big) T^n &= \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}g_n f_m T^{*n+m}B T^{n+m}& \stackrel{(\star)}{=}\sum_{k=0}^{\infty}\bigg( \sum_{n+m=k}f_m g_n \bigg) T^{*k}B T^k= h(T^*,T)(B),
1 < \frac{1 - \Phi(y_1)}{1 - \Phi(y)}< 1 + \frac{y^2 + 1}{y}e^{\frac{y^2}{2}}\frac{C_1}{\sigma \sqrt{n}}e^{-\frac{1}{2}(y - \frac{C_1}{\sigma \sqrt{n}})^2}= 1 + (y + \frac{1}{y}) \frac{C_1}{\sigma \sqrt{n}}e^{\frac{C_1 y}{\sigma \sqrt{n}}- \frac{C_1^2}{2 \sigma^2 n}}= 1 + O( \frac{y}{\sqrt{n}}).
\Delta_{\eta}(x)\stackrel{\rm def}{=}\sqrt{n}\left| f_{\eta}(x)-{\bf E\,}f_{n}(x)\right| =\frac{1}{\sqrt{n}\,h_{n}}\left| \,\sum_{i=1}^{\eta}K\left(\frac{x-X_{i}}{h_{n}}\right) -n\,{\bf E}\,K\left( \frac{x-X}{h_{n}}\right)\right| .
\sum_{j=1}^{n-1}\binom{n-1}{j}\cdot \left( -\frac{1}{2}\right)^j \cdot 1^{n-1-j}= \left( 1 - \frac{1}{2}\right)^{n-1}- 1 = \left(\frac{1}{2}\right)^{n-1}- 1 \enspace .
\frac{2d}{\pi^{2}+4d^{2}\omega^{2}}\frac{\omega}{\tanh(d\, \omega)}\leq \frac{2d}{\pi^{2}+ 0}\, \frac{\pi/(2d)}{\tanh(d\, \pi/(2 d))}=\frac{1}{\pi \tanh(\pi/2)}.
\partial_\nu \zeta_\delta(x)=O\left(\frac{\delta^\frac{4-n}{2}}{(1+\frac{|x\mp e_n|}{\delta})^{n-3}}\right) \partial B_1(0)\cap B_1(\pm e_n), \partial_\nu \zeta_\delta(x)=O(\delta^\frac{n-2}{2}) B_1(0)\cap B_1^c(\pm e_n).
v_{1}\otimes\cdot\cdot\cdot\otimes v_{i-1}\otimes(v_{i}\otimes v_{i+1})\otimes\cdot\cdot\cdot\otimes v_{n}\mapsto v_{1}\otimes\cdot\cdot\cdot\otimes v_{i-1}\otimes(\langle v_{i},v_{i+1}\rangle\sum_{k=1}^{d}(w_{k}\otimes w_{k}))\otimes\cdot\cdot\cdot\otimes v_{n}
var[Y_k] = \sum_{i=1}^k k\sum_{j=i}^k \frac{j-1}{(k-j+1)^2}& = k\sum_{j=1}^k \sum_{i=1}^j \frac{j-1}{(k-j+1)^2}& = k \sum_{j=1}^k \frac{j(j-1)}{(k-j+1)^2}.
X_{n}=\{\xi\in [0,\bar{x}]:(n(x)-n(\xi))(x-\xi)<0 \textnormal{for}x\neq \xi, x\in [0,\bar{x}] \}, X_{f}(y,v) =\{\xi\in [0,\bar{x}]:(f(x,y,v)-f(\xi,y,v))(x-\xi)<0 \textnormal{for}x\neq \xi, x\in [0,\bar{x}] \}, X =\cap_{y,v\in (0,\bar{y})\times (0,\bar{v})}X_{f}(y,v)\cap X_{n}.
( [ \hat{D}]_{q})_{i j}: = [ \ell_{i}]_{q}\delta_{i j}, \quad \quad ( [ I^{\omega}]_{q})_{i j}: = \left[ \sum_{n = 1}^{\ell_{i}}\hat{I}_{\omega^{n}( i ) j}\right]_{q}\; , \quad i , j \in \hat{\Delta}^{\omega}
\sum_{i=0}^ma_i\frac{d^i U(x,p)}{dx^i}+\sum_{i=0}^nb_i U(x,p)\ominus \sum_{j=1}^i p^{j-1}b_i u^{i-j}(x,0)+U(x,p)+L[f(x, t)]=L[k(t)]U(x,p).
\mathcal{G}_{\neq 0}(X) = \int_{\mathbb{R}}V(\xi)\, \mathcal{J}^+_1(N_0\xi,m,q)\,\, \overline{\mathcal{J}^+_1(N_0\xi ,m_1,q')}\, e\left(-{X\xi}\right)\, d\xi.
\mathcal E_1(X \xleftarrow f M \xrightarrow g Y) :=f_*g^*:\mathcal E_1(Y) \to \mathcal E_1(X), \mathcal E_2(X \xleftarrow f M \xrightarrow g Y) := f_*g^*:\mathcal E_2(Y) \to \mathcal E_2(X).
E\left\{\left( \mathbf{A}^{-1}\mathbf{w}_i \right)^{\rm H}\mathbf{A}^{-1}\mathbf{w}_i \right\}={\rm Tr}\left\{\hat{\mathbf{P}}\hat{\mathbf{P}}^{\rm H}+ \tilde{\mathbf{P}}\tilde{\mathbf{P}}^{\rm H}\right\}=2 \;{\rm Tr}\left\{\tilde{\mathbf{P}}\right\}=2 \;{\rm rank}\left\{\tilde{\mathbf{P}}\right\}=2(V+1).
\lim_{p \to \infty}W_f (\sqrt{p}\, \beta, p \Theta)= 1- \frac 4 \pi \sum_{k=0}^{+\infty}\frac{(-1)^k}{2k+1}\exp\left( -\frac{(2k+1)^2\pi^2 \Theta}{8 \beta^2}\gamma_f^2 \right),
\frac{f_1(x -y_1)}{\int_{\mathbb{R}^n}f_1 dx}= \frac{f_2( x -y_2)}{\int_{\mathbb{R}^n}f_2 dx}= \cdots \frac{f_k( x-y_k)}{\int_{\mathbb{R}^n}f_k dx}= e^{-\psi(x)},
\pi_{ij}= (\pi_{ij}\tilde{x}^i \tilde{x}^j) dr^2+2 r \pi_{ij}\frac{\partial \tilde{x}^i}{\partial u^a}\tilde{x}^j du^a dr+r^2\pi_{ij}\frac{\partial \tilde{x}^i}{\partial u^a}\frac{\partial \tilde{x}^j}{\partial u^b}du^a du^b & = \pi_{rr}dr^2 + 2 \pi_{ra}dr du^a + \pi_{ab}du^a du^b.
\Phi^{-1}|_{t=0}=\begin{pmatrix}\frac{1}{n-1}0 0 \dots 0 0 0 1 0 \dots 0 0 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots -\frac{21}{n-1}0 \dots \dots 0 1 \end{pmatrix}.
\Big|-\varepsilon\displaystyle\sum_{j=1}^{d}\displaystyle\lim_{\delta\to 0}\int_{\Omega_{T}}G_{\delta}^{'}(u^{\varepsilon})\frac{\partial\phi}{\partial x_{j}}B(u^{\varepsilon})\frac{\partial u^{\varepsilon}}{\partial x_{j}}\, dx dt.\Big|\to 0\,\,\mbox{as}\,\,\varepsilon\to 0.
(\Gamma_1,\ldots, \Gamma_n, \Gamma_{n + 1}) \equiv(\Delta_1,\ldots, \Delta_n, \Delta_{n + 1}) \Leftrightarrow(\Gamma_1 = \Delta_1 \& \ldots \& \Gamma_n = \Delta_n\& \& (\forall A \in Form^{Ag})(\Box A \in \Gamma_{n + 1}\Rightarrow A\in \Delta_{n + 1}).
\begin{array}{l}y_R^{[1]}= \sqrt{{P_1}}{h_1}{S_1}+ \sum_{k = 1}^{{L_R}}{\sqrt{{P_{I,R}}}{c_{R,k}}I_{R,k}^{[1]}}+ n_R^{[1]}, y_R^{[2]}= \sqrt{{P_2}}{h_2}{S_2}+ \sum_{k = 1}^{{L_R}}{\sqrt{{P_{I,R}}}{c_{R,k}}I_{R,k}^{[2]}}+ n_R^{[2]}, \end{array}
\begin{aligned}\|\nabla f(x_N)\|^2{}\leq{}\tfrac{2L\big[f(x_0)-f(x_N)\big]}{\min \big\{-1+(1-h)^{-2N}\,,\,{2 N h}\big\}}\|\nabla f(x_N)\|^2{}\leq{}\tfrac{2L\big[f(x_0)-f_*\big]}{\min \big\{(1-h)^{-2N}\,,\,{1 + 2 N h}\big\}}.\end{aligned}
\tau_{m}^{(0)}=\alpha_{2m-1}\alpha_{2m}=\frac{(2m-1)^{2}(2m)^{2}(x^{2}-(2m-1)^{2})(x^{2}-(2m)^{2})}{16(4m-3)(4m-1)^{2}(4m+1)}, \sigma_{m}^{(0)}=\alpha_{2m}+\alpha_{2m+1}=\frac{(8m^{2}+4m-1)r^{2}-(32m^{4}+32m^{3}+8m^{2}-1)}{4(4m+3)(4m+1)(4m-1)},
\mathcal{R}:=\bigoplus_{(d,\alpha,p)\in Q_{\tilde{\varphi}}}\mathrm{X}^p\otimes\mathcal{O}_\alpha\otimes\mathcal{M}^d\theta^d=\bigoplus_{(d,\alpha)\in S(X_\xi)}\mathcal{R}_{d,\alpha}\theta^d.
\sum_{i=1}^Na_i b_i = \sum_{\ell = 1}^L \langle \underbar{a}^{(i)}_{\ell}, -A^T_{\ell}\underbar{b}^{(o)}_{\ell}\rangle + \langle A_{\ell}\underbar{a}^{(i)}_{\ell}, \underbar{b}^{(o)}_{\ell}\rangle = 0.
\bar{\boldsymbol{\psi}}(t_{k+l}) \!=\! \bar{\boldsymbol{\psi}}(t_k) \!+\!\! \sum_{i=1}^{l}\!b_{S,k+i}\mathbf{f}_{\boldsymbol{\psi}}\left(\bar{\boldsymbol{\psi}}(t_{n+i\!-\!1})\right)\!+\! (\boldsymbol{\xi}_{k+l}\!-\! \boldsymbol{\xi}_{k})\!,\!
\begin{array}{c}(X, \mu ) := \coprod_{\alpha \in J}(X_{\alpha}, \mu_{\alpha}){}\left( \coprod_{\alpha \in J}X_{\alpha}, \coprod_{\alpha \in J}\kappa ( \mu_{\alpha}, k_{\alpha}) \right) := \coprod_{\alpha \in J}\left(X_{\alpha}, \kappa ( \mu_{\alpha}, k_{\alpha}) \right) \end{array}
\tilde{\Phi}_{\lambda m}^{\mu V^{( l )}}( z ) = g_{\lambda}^{\mu V^{( l )}}( z ) \int_{0}^{s_{1}\infty}d_{p}t_{1}J^{S}( t_{1}) \cdots \int_{0}^{s_{l - m}\infty}d_{p}t_{l - m}\, J^{S}( t_{l - m}) \phi_{m}^{( l )}( z ) ,
\rho_\sigma^{ab}= \frac{1}{\sqrt{\det\left[ 1+ W_{ab}^{cd,T}W_{ab}^{cd}\right]}}\equiv \frac{|J_{ab}^\sigma|}{\sqrt{\det\left[J_{ab}^{\sigma,T}J_{ab}^\sigma + J_{cd}^{\sigma,T}J_{cd}^\sigma\right]}}.
|F_{m+1}-F_m|_{S_{I,m+1}(\delta)}=|(\frac1{a_p})^{m+1}U_p^{nsp}(U_p^{sp})^m f|_{S_{I,m+1}(\delta)}&=|(\frac{U_p^{sp}}{a_p})^m F_1|_{S_{I,m+1}(\delta)}\leq p^{-m \mu}|F_1|_{S_{I,1}(\delta)}&\rightarrow 0 .
U(h_\lambda,h_{w^{-1}\lambda})^*U(h_\lambda,h_{w^{-1}\lambda}) =q^{-(\rho,\lambda-w^{-1}\lambda)}U(\bar h_\lambda\otimes h_\lambda,\bar h_{w^{-1}\lambda}\otimes h_{w^{-1}\lambda}) &=q^{-(\rho-w\rho,\lambda)}U(\bar h_\lambda\otimes h_\lambda,\bar h_{w^{-1}\lambda}\otimes h_{w^{-1}\lambda}).
\zeta(1-s,\frac{m}{n})=\frac{\Gamma(s)}{(2\pi n)^{s}}\sum_{k=1}^{n}[e^{\frac{\pi is}{2}}e^{-\frac{2\pi ikm}{n}}\zeta(s,\frac{k}{n})+e^{-\frac{\pi is}{2}}e^{\frac{2\pi ikm}{n}}\zeta(s,\frac{k}{n})]
\frac{\mathrm{d}}{\mathrm{d}t}\widetilde Y(t) = \frac{\mathrm{d}}{\mathrm{d}t}\bigg(\widetilde Y(t)^{\epsilon/(1+\epsilon)}\bigg)^{(1+\epsilon)/\epsilon}=\frac{1+\epsilon}{\epsilon}\widetilde Y(t)^{1/(1+\epsilon)}\frac{\mathrm{d}}{\mathrm{d}t}\big(\widetilde Y(t)^{\epsilon/(1+\epsilon)}\big),
E^s_\lambda(\pm\infty) =\{x\in E: \exp(tJS_\lambda(\pm\infty))x\rightarrow 0\,\,\,\, t\rightarrow\infty\}, E^u_\lambda(\pm\infty) =\{x\in E: \exp(tJS_\lambda(\pm\infty))x\rightarrow 0\,\,\,\, t\rightarrow-\infty\}.
c_{1}(E_{1}^{(0)}+H^{\prime}-E)\langle\Phi_{1}^{(0)}|\Phi_{1}^{(0)}\rangle+c_{2}(E_{2}^{(0)}+H^{\prime}-E)\langle\Phi_{1}^{(0)}|\Phi_{2}^{(0)}\rangle=0
\Big|\log[u^{p-1}]_h-\log u^{p-1}\Big|=\left|\log\left(\frac{[u^{p-1}]_h}{u^{p-1}}\right)\right| \leq \frac{\big|[u^{p-1}]_h-u^{p-1}\big|}{\min\{[u^{p-1}]_h,\,u^{p-1}\}}begin{align*}3mm] \leq \frac{1}{m^{2(p-1)}(1-e^{-\frac{t}{h}})}\big|[u^{p-1}]_h-u^{p-1}\big|
\sum_{j=1}^{p}\sum_{s=1}^{j-1}x^{-2j-2s}=\frac{x^{-4p+2}(1-x^{2p-2})(1-x^{2p})}{(1-x^2)(1-x^4)}&=\begin{cases}\displaystyle\sum_{j=1}^{p/2}\sum_{s=1}^{p-1}x^{-2p-4j+2s}(p: ), \displaystyle\sum_{j=1}^{p}\sum_{s=1}^{(p-1)/2}x^{-2p-2-2j+4s}(p: ).\end{cases}
\Bigl((\theta^{q^N}-t)^{c(N-n_1)q^{n_1+d-N}}\bigl(~_r\mathcal{F}_s(\alpha)\bigr)^{q^d}\Bigr)|_{t=\theta^{q^N}}=\sum_{i=1}^{l}\prod_{\substack{m=1 m\neq N}}^{n_i+d-1}(\theta^{q^m}-\theta^{q^N})^{c(m-n_i)q^{n_i+d-m}}\alpha^{q^{n_i+d}}
\ = \q^{2}\int_{q}d{\hat{z}}d{\hat{\bar{z}}}\ ({\bar{\partial}}_{q}{\hat{\xi}}_{k}) ({\hat{z}},{\hat{\bar{z}}}) \, ( \partial_{q}\Phi ) ({\hat{z}}q^{2 k + 1}) \, ( \partial_{q}\Phi ) ({\hat{z}}) \ .
\frac{\partial}{\partial r}=\frac{x}{\sqrt{x^{2}+y^{2}}}\frac{\partial}{\partial x}+\frac{y}{\sqrt{x^{2}+y^{2}}}\frac{\partial}{\partial y},and\frac{\partial}{\partial\theta}=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}.
P(k,k^{\prime})=\frac{2\pi}{\hbar}(D_{ac}\sum_{q}\sqrt{\frac{\hbar}{2MN\omega_{q}}}|q|\sqrt{n_{q}+\frac{1}{2}\mp\frac{1}{2}}I(k,k^{\prime})\delta_{k^{\prime},k\pm q})^{2}\delta[\epsilon(k^{\prime})-\epsilon(k)\mp\hbar\omega_{q}],
|\{w\in \O_n: w\}| =\begin{cases}C_m C_{m+1}n=2m, C_{m+1}^2 n=2m+1, \end{cases}|\{w\in \O_{2m}: w\}| =|\O_{2m,0}|=C_m C_{m+1}, |\O_n| =\begin{cases}(2m+1)C_m^2 n=2m, (2m+1)C_mC_{m+1}n=2m+1. \end{cases}
\Xi_n(P) = \left( p_1^*a_\lambda^*(\gamma_\lambda)\cdot p_2^*a_\lambda^*(\gamma_\lambda) \right) \cdot \Delta_*\left(a_\lambda^*(\gamma_\lambda)^{-1}\cdot i^*_{Y_\lambda}(P)\right) & = \Delta_*\left( a_\lambda^*(\gamma_\lambda)\cdot a_\lambda^*(P) \right)= \xi^{\lambda\lambda}_{(1,\gamma_\lambda P)},
F_{1}=\frac{1}{2}(m_{1}^{2}\phi_{1}+ \lambda_{1}\phi_{1}^{3}+ g \phi_{2}), \;\;\;\;\;\;F_{2}=\frac{1}{2}(m_{2}^{2}\phi_{2}+ \lambda_{2}\phi_{2}^{3}+ g \phi_{1})
u_{\varepsilon_i}\lfloor_{\partial \Omega}\to \tilde{u}\; \; \mathcal{H}^{n-1}\,a.e.\mbox{on}\partial \Omega, &\tilde{u}= \pm1 \; \; \mathcal{H}^{n-1}\, a.e.\mbox{on}\; \partial \Omega,
\prescript L{}{}\widetilde W{}^R \cap \left(\widetilde W_L x \widetilde W_R\right) = \left\{\prescript L{}{}x{}^R\right\}, \prescript{-L}{}{}\widetilde W{}^{-R}\cap \left(\widetilde W_L x \widetilde W_R\right) = \left\{\prescript{-L}{}{}x{}^{-R}\right\}.
\forall z \in C^2_e(S^{n-1}) \;\;\; \int_{S^{n-1}}z dV_K = 0 \;\; \Rightarrow \;\; \int_{S^{n-1}}(-L_K z) z dV_K \geq \frac{n-p}{n-1}\int_{S^{n-1}}z^2 dV_K .
\Delta_{\mu\nu}(p)={1\over Z(p^2)[p^2+m_{\rm ren}^2(p^2)]}\left(\delta_{\mu\nu}-{p_\mu p_\nu\over p^2}-m_{\rm ren}(p^2)\epsilon_{\mu\nu\rho}{p^\rho\over p^2}\right)+\xi{p_\mu p_\nu\over (p^2)^2}
F^0(N,g_j) =\sum^{\infty}_{n=0}a_{j_1 \cdots j_n}\delta g_{j_1}\cdots \delta g_{j_n}+ \sum_i \left( V_{ij}\delta g_j \right)^{2-\gamma_0^{(i)}}\sum^{\infty}_{n=0}b_{(i) j_1 \cdots j_n}\delta g_{j_1}\cdots \delta g_{j_n},
f(n,m) = \frac{1}{m}{2n \choose m-1}\sum_{k=0}^m{m \choose k}\frac{k}{n-m}{3(n-m) \choose n-m-k}&= \frac{1}{n-m}{2n \choose m-1}\sum_{k=1}^m{m-1 \choose k-1}{3(n-m) \choose n-m-1-(k-1)}&= \frac{1}{n-m}{2n \choose m-1}{3n-2m-1\choose n-m-1}
\begin{array}{lcl}X_n^{(II)}(\tau_0) ={\displaystyle i \sum_{m=-\infty}^{\infty}\frac{1- e^{im\sigma_0}}{m\sigma_0 - 2\pi n}X_m^{(I)}(\tau_0)}{\dot X}_n^{(II)}(\tau_0) ={\displaystyle i\sum_{m=-\infty}^{\infty}\frac{1- e^{im\sigma_0}}{m\sigma_0 - 2\pi n}{\dot X}_m^{(I)}(\tau_0)}\end{array}
U_{1}(x,y) = \begin{cases}\frac{f_{1}(x) - g_{1,[y,1]}(x)}{y - x}, x < y, -\infty, ,\end{cases}U_{2}(x,y) = \begin{cases}\frac{f_{2}(y) - g_{2,[0,x]}(y)}{y - x}, x < y, -\infty, ,\end{cases}