harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta|\nabla L|^2= 2|\nabla\nabla L|^2+R|\nabla L|^2+2\langle\nabla Q,\nabla L\rangle-2(1+\frac{\alpha'}{2}R)\langle\nabla R,\nabla L\rangle = 2|\nabla\nabla L|^2-R|\nabla L|^2+2\langle\nabla Q,\nabla L\rangle
g_{\zeta_N\left( 1 \right)}\left( \tilde{\varphi}_i^{-1}\circ \tilde{\chi}_{\sigma_N}\right) = \delta\!\left( s_{N,0}\, e^{\pi i\left( -\frac{[k_N-1]_3}{3}- \frac{1}{6}+ \frac{1}{2}\right)}\right) \delta(\tilde{\alpha}) \tilde{t}_N \delta(\tilde{\alpha})^{-1}\ ,
\det R(z) =\exp\big(\frac{1}{4\pi}\int_0^{2\pi}\log |2z_1z_2|+\log|\cos\theta| d\theta\big) &=\sqrt{2|z_1z_2|}\exp \big(\frac{1}{4\pi}\int_0^{2\pi}\log |\cos\theta| d\theta\big) &=\sqrt{2|z_1z_2|}\exp \big(\frac{1}{\pi}\int_0^{\pi/2}\log \cos\theta d\theta\big) &=\sqrt{2|z_1z_2|}/ \sqrt{2}&=\sqrt{|z_1z_2|}.
Mf_{2}(y) \lesssim \Gamma^{-2\gamma}\frac{\int_{0}^{y}r^{\omega-1}e^{-r^{2}/4}dr}{\int_{0}^{y}r^{\omega+1}e^{-r^{2}/4}dr}\lesssim \Gamma^{-2\gamma}(1+\mathcal O(y^{-2}))=e^{-2\gamma\omega_{l}k\theta}(1+\mathcal O(y^{-2}))
\begin{array}{l c l}{\Delta X_{\pm ( i , j )}}&{=}&{X_{\pm ( i , j )}\otimes e^{- h H_{j ; n - 1}/ 2}+ e^{- h H_{i ; n - 1}/ 2}\otimes X_{\pm ( i , j )}+}\\{}&{}&{+ ( 1 - e^{\pm h}) \sum_{k = i + 1}^{j - 1}( X_{\pm ( i , k )}\otimes X_{\pm ( k , j )}) .}\\ \end{array}
e^{-it_n\Delta}({u}_n^J(t_n) - u_{n}(t_n)) = \sum_{j=1}^J e^{-it_n\Delta}v_n^j(t_n) - e^{ix\xi_n^j}\psi^j_{\{h_n^j \}}&= \sum_{j=1}^J e^{ix\xi_n^j}\bigl[(e^{-i(h_n^j)^2t_n\Delta}\Psi^j((h_n^j)^2t_n))_{\{h_n^j\}}-\psi^j_{\{h_n^j \}}\bigr],
t_{1}:=\sup \{t\in [t_{0},\frac{1}{s^{d-1}}] \mid \forall u\in [t_{0},t), K(h_{1})\subset \subset h_{1}^{-1}(K(g_{u}))\subset \subset g_{u}^{-1}(K(h_{1})) \subset \subset K(g_{u})\}.
1-m_N(g)=\frac{|gS^N\backslash S^N|}{|S^N|}\leq \frac{|S^{N+|g|}\backslash S^N|}{|S^N|}\leq \beta\frac{\sum_{i=N+1}^{N+|g|}i^{d-1}}{N^d}\lesssim \beta\frac{|g|}{N}, g\in \Gamma.
{\cal U}_{\cal A}= \frac{\sigma ( 1 ) \sigma ( \delta )}{p}\sum_{r , s = 0}^{p - 1}\mathrm{e}^{\displaystyle \frac{2 \pi \mathrm{i}}{p}[ b r^{2}+ ( d - a ) r s - c s^{2}] / 2 \delta}{\cal J}_{r , s}
A_2(r)^2 = \bigg( \sum_{k\ge 1}k^{3/2}\sqrt{c_k r^k}\cdot \sqrt{kc_kr^k}\bigg)^2 \le A_3(r)A_1(r) \quad A_1(r)^3 = \bigg(\sum_{k\ge 1}k (c_kr^k)^{1/3}\cdot (c_kr^k)^{2/3}\bigg)^3 \le A_3(r) A_0(r)^2.
{\bf P}_{\Omega_0}= Z_\nu \sum\limits_{k_1,k_2,k_3=1}^L{\cal W}_{\nu({\bf k})}{\bf P}_{\Omega_0}= Z_\nu \sum\limits_{k_1,k_2,k_3=1}^L \sum\limits_{q=1}^{R_{\cal N}}{\cal W}_{\nu({\bf k})}{\bf p}^{(1)}_{q}\otimes{\bf p}^{(2)}_{q}\otimes{\bf p}^{(3)}_{q}\in \mathbb{R}^{n\times n \times n},
\mathrm{\Phi}: \widehat{Q}\rightarrow Q=\mathrm{\Phi}( \widehat{Q}) \subset \mathbb{R}^{d+1}, \mathrm{\Phi}(\xi) = \sum_{{{i}\in\overline{\mathcal{I}}}}\widehat{R}_{{i},{p}}(\xi) \mathbf{P}_{{{i}}},
[-\frac{\hbar^{2}}{2m_{0}}\frac{d^{2}}{dr^{2}}+\frac{\hbar^{2}l(l+1)}{2m_{0}r^{2}}+\frac{1}{2}m_{0}\omega^{2}r^{2}-\hbar\omega(n+\frac{3}{2})]u(r)=0.
(I-\tilde{T})^2 = \begin{pmatrix}I I -I I-2T\end{pmatrix}\begin{pmatrix}I I -I I-2T\end{pmatrix}&=\begin{pmatrix}0 2(I-T) -2(I-T) -4T(I-T)\end{pmatrix}=2\begin{pmatrix}0 I -I -2T\end{pmatrix}\begin{pmatrix}I-T 0 0 I-T\end{pmatrix},
\overline{r}\triangleq \sum_{i=1}^L p_ir_i, a_i \triangleq \sum_{j=1}^i p_j -\sum_{j=i+1}^L p_j,~~ b_i \triangleq -\sum_{j=1}^i p_jr_j +\sum_{j=i+1}^L p_jr_j.
C_{+t}=\frac{1}{2}\begin{bmatrix}\widehat{X}_{+t}+A-JB 2\mathbf{x}_{+t}\left(\widehat{X}_{+t}-A+JB\right)J \mathbf{y}_{+t}^{\textsuperscript{T}}2c_{+t}\mathbf{y}_{+t}^{\textsuperscript{T}}J J\left(\widehat{X}_{+t}-A+JB\right) 2J\mathbf{x}_{+t}J\left(\widehat{X}_{+t}+A-JB\right)J\end{bmatrix}.
Q_{ab}:=2\Big(T_{ab}[H_1]-\frac{1}{2}V_1|H_1|^2g_{ab}\Big)+\Big(T_{ab}[H_2]-\frac{1}{2}V_2|H_2|^2g_{ab}\Big)-\left(\frac{2}{r^2}D H_1\cdot H_2 g_{ab}\right),
\left. \frac{\partial U_{4}}{\partial v}\right|_{v=v_{s}}= -\frac{1}{v_{s}^{3}}+v_{s}+\frac{3uv_{s}^{3}}{16\pi^{2}}[{\rm ln}\frac{v_{s}^{2}u}{M^{2}}-\frac{1}{2}]=0.
\Omega=\Omega_1\cup\Omega_2\cup\Omega_3, \Omega_1=\left\{y\in\Omega:\delta(y)>\frac32\delta(x)\right\}\Omega_2=\left\{y\in\Omega:\frac12\delta(x)\leq\delta(y)\leq\frac32\delta(x)\right\}\Omega_3=\left\{y\in\Omega:\delta(y)<\frac12\delta(x)\right\}.
V_{\beta}^{1 - l o o p}( \phi_{0}) = \int \frac{d^{\nu}k}{( 2 \pi )^{\nu}}{\frac{E}{2}}+{\frac{1}{\beta}}\int \frac{d^{\nu}k}{( 2 \pi )^{\nu}}\ln [ 1 - e^{- \beta E}] .
| B , \eta , V >_{o s c}^{f}= \exp{i \eta \sum_{n > 0}^{\infty}\{\frac{1}{2}( e^{- 2 v}\psi_{n}^{A \dagger}\tilde{\psi}_{n}^{A \dagger}+ e^{2 v}\psi_{n}^{B \dagger}\tilde{\psi}_{n}^{B \dagger}) - \psi_{n}^{T \dagger}\tilde{\psi}_{n}^{T \dagger}\}}| 0 > \; ,
H_p(T(2),T(2)) =H_p(\{0,1\},\{0,1\}) &=\max\left\{p(x,\{0,1\})x\in\{0,1\}\right\}&=\max\left\{p(0,0),p(1,1)\right\}&=(0,0) &\leq\frac{1}{4}(1,0) &=\lambda\left[p(T(2),2)+p(2,T(2))\right].
\mathrm{Res}_{x=0}\Big(\frac{(1- \alpha_i x)\prod_{p=1}^{d-1}\big(p +(d-p) \alpha_i x\big)\prod_{p=1}^{d-1}\big(p +(d-p) \alpha_k x\big)}{x^{4}(2- \alpha_i x- \alpha_k x)\prod_{q=0}^{n+1}(1- \alpha_qx)}\Big)
\tilde{\delta}_{\mu \nu}= \delta_{\mu \nu}- u_{\mu}u_{\nu}, \qquad \tilde{p}_{\mu}= p_{\mu}- ( u \cdot p ) u_{\mu}, \qquad \overline{{{u}}}_{\mu}= u_{\mu}- \frac{u \cdot p}{p^{2}}p_{\mu}
\begin{aligned}\ast F_{\mathsf{a}}= \mu_{1}\circ u \left(\frac{\partial \phi_{1}}{\partial x_{1}}- \frac{\partial \phi_{2}}{\partial x_{2}}\right) = \mu_{2}\circ u \left(\frac{\partial \phi_{1}}{\partial x_{2}}+ \frac{\partial \phi_{2}}{\partial x_{1}}\right) = \mu_{3}\circ u\end{aligned}
\left\langle \psi_{k}^{(j)}(X_j), \psi_{l}^{(j)}(X_j) \right\rangle \equiv \mathbb{E}\left[\psi_{k}^{(j)}(X_j) \, \psi_{l}^{(j)}(X_j) \right] \equiv \int_{\mathcal{X}_j}\psi_{k}^{(j)}(x_j) \, \psi_{l}^{(j)}(x_j)\, f_{X_j}(x_j) \, \mathrm{d}x = \gamma_k^2 \, \delta_{kl}\, ,
\delta'(\imath, r):= \delta(\imath, r)-\delta_{(\imath, r), (\imath_0, r_0+\frac{1}{2})}-\delta_{(\imath, r), (\imath_0, r_0-\frac{1}{2})}-\delta_{(\imath, r), (\imath_0-1, r_0)}+\delta_{(\imath, r), (\imath_0-1, r_0+\frac{1}{2})}+\delta_{(\imath, r), (\imath_0-1, r_0-\frac{1}{2})}+\delta_{(\imath, r), (\imath_0, r_0)}
a_{2k}=y_{k-1}y_{k},a_{2k+1}=y_{k-1}y_{k+1}, b_{2k}=x_{k-1},b_{2k+1}=y_{k}^{2}\frac{\theta^{2}x_{k-1}-\theta^{2}y_{k-1}}{x_{k-1}}\geq1.
lim_{n\rightarrow\infty}\frac{2r^{2}}{n}\sum_{k=1}^{n}\sqrt{1-\frac{k^{2}}{n^{2}}}=2r^{2}lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\sqrt{1-\frac{k^{2}}{n^{2}}}
Q=[\begin{matrix}\sigma_{x}^{2}&\sigma_{xy}&\sigma_{xz}&\sigma_{xt}\\ \sigma_{xy}&\sigma_{y}^{2}&\sigma_{yz}&\sigma_{yt}\\ \sigma_{xz}&\sigma_{yz}&\sigma_{z}^{2}&\sigma_{zt}\\ \sigma_{xt}&\sigma_{yt}&\sigma_{zt}&\sigma_{t}^{2}\end{matrix}]
\left(\partial_{0,\nu}\begin{pmatrix}\frac{1}{2}0 0 \frac{1}{2}\end{pmatrix}+\begin{pmatrix}\frac{1}{2}0 0 \frac{1}{2}\end{pmatrix}+\begin{pmatrix}0 \partial_{1}\partial_{1,c}0\end{pmatrix}\right)\begin{pmatrix}u v\end{pmatrix}=\begin{pmatrix}f g\end{pmatrix}.
W_4(x,p) = \int_{-\infty}^\infty\! \frac{dy_0}{2\pi}\, e^{-ip_0y_0}\int \frac{d^3y}{(2 \pi)^3}\, e^{i\vec p \cdot \vec y}\left\langle \Psi \left(\vec x +{\textstyle{\vec y\over 2}},x_0 +{\textstyle{y_0\over2}}\right)\, \Psi^\dag \left(\vec x -{\textstyle{\vec y\over 2}},x_0 -{\textstyle{y_0\over2}}\right) \right\rangle \, .
\Psi ( x ) = \left( f ( x ) +{\cal F}_{-}^{\mu}( x ) \widehat{b}{}_{\mu}^{+}+ \epsilon_{\mu \nu \lambda}{\cal F}{}_{+}^{\mu}( x ) \widehat{b}{}^{+ \nu}\widehat{b}{}^{+ \lambda}+ \tilde{f}( x ) \epsilon_{\mu \nu \lambda}\widehat{b}{}^{+ \mu}\widehat{b}{}^{+ \nu}\widehat{b}{}^{+ \lambda}\right) | 0 \rangle ,
\tilde{K}(t) = \begin{pmatrix}t^2 + \cdots \gamma^{-1}\, K_{k+1}^y \, t^{k+1}+ \cdots\end{pmatrix}, & \tilde{K}(t) = \begin{pmatrix}t + \cdots \gamma^{-1}\, K_l^y \, t^l + \cdots\end{pmatrix},
\phi(\chi_{w_{n}}(u_{j}^{i}))=\begin{cases}\chi_{\vartheta_{n-1}}(v_{j}^{i}), \mbox{if}i \neq 1 \mbox{or}2n, \mbox{or}j \neq 1 \mbox{or}2n, \cr \delta_{ij}, \mbox{otherwise.}\cr\end{cases}
\Big ( \sum_{m \ge 1}\sup_{|a|_U =1}| \nabla_k \nabla_{a}^G P_t[\Phi_m](x)|^2 \Big)^{1/2}\le \frac{C}{t^{\frac{4 - 3\alpha}{2}}}|k|_K \, \| \Phi\|_{C^{\alpha}_b(H, J)}.
\begin{cases}x_{1}^{\left(2t+1\right)}=\;\delta_{1}x_{2}^{\left(2t\right)}y^{\left(2t\right)}\medskip x_{2}^{\left(2t+2\right)}=\;\gamma_{2}x_{1}^{\left(2t+1\right)}y^{\left(2t+1\right)}\medskip y^{\left(2t+1\right)}=\;\delta x_{2}^{\left(2t\right)}y^{\left(2t\right)}y^{\left(2t+2\right)}=\;\gamma x_{1}^{\left(2t+1\right)}y^{\left(2t+1\right)}.\end{cases}
\chi=\frac{\gamma}{E_{s}}\sqrt{[E(1-\frac{v\cdot k}{kc})]^{2}-2(1-\frac{v\cdot k}{kc})(\frac{E\cdot v}{kc})k\cdot E+(\frac{(E\cdot v)}{c}\frac{k}{k})^{2}-(\frac{E\cdot v}{c})^{2}}
\left( \ell_1,\dot \ell_1,h_1)(\ell_2,\dot \ell_2,h_2)=(\ell_1+h_1\ell_2h_1^{-1},\dot \ell_1+[\ell_1,h_1\ell_2h_1^{-1}]+h_1\dot \ell_2h_1^{-1},h_1h_2\right) \ .
\left\langle[\textup{ord}_p(q_E)^{-1}], [\exp_{\omega_E}^*(z^{\textup{BK}})] \right \rangle_{\textup{Nek},\rho}=E_p(1)^{-1}\textup{ord}_p(q_E)^{-1}\log_p(u_E)\exp_{\omega_E}^*(z^{\textup{BK}}) &=\langle \frak{C}_0, \textup{res}_p^{s}(\frak{z}_0^{\textup{BK}})\rangle_{\textup{Tate}}
\|U(x,t) - U(x,t_0)\|_{L^\infty(|x| \geq \frac{\epsilon_0}{4})}\leq \epsilon, \|\nabla U(x,t) - \nabla U(x,t_0)\|_{L^\infty(|x| \geq \frac{\epsilon_0}{4})}\leq \epsilon, &\|U(x,t_\epsilon) - U(x,t_0)\|_{L^\infty(|x| \geq \frac{\epsilon_0}{4})}= \epsilon \|\nabla U(x,t_\epsilon) - \nabla U(x,t_0)\|_{L^\infty(|x| \geq \frac{\epsilon_0}{4})}= \epsilon.
\binom{a+j}{j}\sum_{l=0}^{a}\binom{n-j+l}{l}\binom{n-j}{a-l}\bigl{(}\binom{n-(j+a-l)}{j+a-l}\sum_{k=a+j-l}^{n-(a+j-l)}\binom{n-2(a+j-l)}{k-(j+a-l)}G(n,k,a)\bigr{)}
\chi^{-1}|\chi_i|\leq \chi^{-1}(\sum_{i=1}^2\chi_i^2)^{\frac 12}\leq \lambda_2\frac{(1-\bar{g}(\partial_r,\nu)^2)^{1/2}}{\bar{g}(\partial_r,\nu)}=~ \lambda_2\sqrt{v^2-1}\leq ~ce^{-\lambda t}
\delta \tilde{\lambda}( t , s , \{\vec{k}\}) = [ i \delta ( 0 ) ] \frac{3}{2}a ( t )^{- 1}\tilde{\lambda}^{2}( t ) ( 2 \pi )^{- 3}\int d^{3}\vec{q}[ \vec{q}^{2}+ \tilde{m}^{2}( t ) - i \epsilon ]^{- 1}
\varphi_{1}(tw_{0}) =\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\Phi(tw_{0}(x)-tw_{0}(y))(tw_{0}(x)-tw_{0}(y))K(x,y)dxdy -\int_\Omega |tw_{0}|^{p^{\ast}_{s}}dx -\lambda \int_{\Omega}f(x,tw_{0})tw_{0}dx
f_p(s)=\sum_{b=0}^{\infty}\frac{a_{p^b}\sigma_{1,\textbf{1},\alpha}(p^{b+e_p})}{(p^{b})^{s + 1}}=\sum_{b=0}^{\infty}\frac{a_{p^b}\sigma_{1,\textbf{1},\alpha}(p^{b})}{(p^{b})^{s + 1}}+\sum_{b=0}^{\infty}\frac{a_{p^b}\alpha(p^b)p^b}{(p^{b})^{s+1}}(\alpha(p)p+\ldots+\alpha(p^{e_p})p^{e_p})
(X_{12}-X_{13})^i(X_{30}-X_{31})^i =(x_1-q^{d-2}I)(x_1-q^{d-4}I)\dots (x_1-q^{d-2i}I) &(x_3-q^{2i-2-d}I)\dots (x_3-q^{2-d}I)(x_3-q^{-d}I).
(-1)^i (-\Delta)^{N-i}\overline u (r) \leqslant (-1)^i (-\Delta)^{N-i}\overline u (0) + \sum_{l=1}^{i-1}\frac{(-1)^{i-l}(-\Delta)^{N-i+l}\overline u (0) r^{2l}}{\prod_{k=1}^{l}(2k) \prod_{k=1}^{l}[2N-1+2(k-1)]}
\pi(z) = \lim_{n \to \infty}\pi(h_{1,n}h_{2,n}h_{3,n}B^+/B^+) = \lim_{n \to \infty}h_{1,n}B^+ /B^+ \in U^-_{J,\ge 0}B^+/B^+ \bigcap U^+ \dot{u}_J \cdot B^+ / B^+.
\int_0^{\frac{\pi}{2}}\cos(4nx) \log\left( \tan x\right)\d x = \int_0^{\frac{\pi}{2}}\cos\left( 4n \left( \frac{\pi}{2}-x\right)\right) \log \left( \tan \left( \frac{\pi}{2}- x \right) \right) \d x = - \int_0^{\frac{\pi}{2}}\cos(4nx) \log \left( \tan x \right) \d x .
<\Omega_{2\hspace*{2mm}2\;\;....\;\;2\hspace*{6mm}2}^{\tilde{R}_1 S_1....\tilde{R}_{N/2}S_{N/2}}\mid m^2\alpha'\mid\Omega_{2\hspace*{2mm}2\;\;....\;\;2\hspace*{6mm}2}^{\tilde{R}_1 S_1....\tilde{R}_{N/2}S_{N/2}}>\approx \lambda N^2
\Xi^\lambda (K_a \otimes F_a\otimes 1 \otimes E_a) = \Xi^\lambda (K_a F_a \otimes 1 \otimes E_a K_a^{-1}\otimes K_a) = \sum_{i, j, k, \ell}c^i_j \pi(K_a F_a)^j_k c^k_\ell \pi(E_a K_\lambda)^\ell_i.
\int_\Omega\int_{\Omega_1\setminus\Omega}\sum_{{i,j}_{j \neq i}}\overline{u}_{i}(x)\phi_j(y)K(x,y)dydx=\int_\Omega\int_{\Omega_1\setminus\Omega}\sum_{{i,j}_{j \neq i}}\underline{u}_{i}(x)\phi_j(y)K(x,y)\,dy\,dx.
a\big(E_1+...+E_{\binom{b}{k}}\big)+E_{\binom{b}{k}+1}+...+E_{\binom{b+1}{k+1}}= \Big( \underbrace{a(a+1),...,a(a+1)}_{p}, \underbrace{2a,...,2a}_{q}, \underbrace{-2a,...,-2a}_{p}, \underbrace{-2,...,-2}_{q},..., \underbrace{(-1)^{b-1}2a,...,(-1)^{b-1}2a}_{p}, \underbrace{(-1)^{b-1}2,...,(-1)^{b-1}2}_{q}\Big)^t.
\begin{array}{cc}f_i(x_1^{(0)}+x_1^{(1)}t+\cdots+x_1^{(m)}t^m,\ldots,x_n^{(0)}+x_n^{(1)}t+\cdots+x_n^{(m)}t^m)= =\sum_{j=0}^mF_i^{(j)}(x_1^{(0)},\ldots,x_1^{(j)},\ldots,x_n^{(0)},\ldots,x_n^{(j)})\t^j\mbox{mod}t^{m+1}, \end{array}
[ \ell_{\mu}^{j}, q \cdot \! \nu ] = - i \left[ j \mu \cdot \nu \ell_{\mu}^{j - 1}+ \sum_{\rho \in \Delta_{+}}g_{| \rho |}( \rho \cdot \mu ) ( \rho^{\vee}\! \cdot \nu ) \frac{\ell_{\mu}^{j}- \ell_{s_{\rho}( \mu )}^{j}}{\ell_{\mu}- \ell_{s_{\rho}( \mu )}}\check{s}_{\rho}\right] ,
0<\Phi_{I,\lambda,P}(f)-\Phi_{I,\lambda,Q}(f) =f(t_{j})-f(t_{j-1})-\left[|f(t_{j})-f(y)|+|f(y)-f(x)|+|f(x)-f(t_{j-1})|\right]+2\lambda \le f(t_{j})-f(t_{j-1})-\left[f(t_{j})-f(y)+f(x)-f(y)+f(x)-f(t_{j-1})\right]+2\lambda =2[f(y)-f(x)+\lambda],
|M|_{s,\b+}=\sup_{a,b\in\L}(1+|w_a-w_b|)\Big(\frac{w(a,b)+|w^2_a-w^2_b|}{w(a,b)}\Big)^{\frac{s}{2}}(w_aw_b)^\beta\left\|M_{[a]}^{[b]}\right\| .
\frac{1}{\alpha}(\exp(\alpha(x(t)-t))-1)=\sum_{f=1}^{\infty}e^{ft}\sum_{\sigma_{f}\in P_{f}}(-1)^{l(\sigma_{f})}(\alpha+f)^{l(\sigma_{f})-1}S(\sigma_{f})\prod_{j=1}^{l(\sigma_{f})}\frac{b_{d_{j}}}{d_{j}}.
\langle{\cal W}( 1 ){\cal W}( 2 ) \rangle |_{\theta = 0}= \langle \phi ( 1 ) \phi ( 2 ) \rangle = \frac{z_{1 i}\delta^{i j}z_{2 j}}{x_{1 2}^{2}}\equiv \frac{( 1 2 )}{x_{1 2}^{2}}= \frac{( 2 1 )}{x_{2 1}^{2}}\; .
\psi_c^+(\overline{\omega_{\frac{n+3}{2}}}) = -\overline{\omega_{\frac{n+3}{2}}}\mbox{and}\phi_{\zeta}^+(\overline{\omega_{\frac{n+3}{2}}}) = \zeta^{n/2}\overline{\omega_{\frac{n+3}{2}}}= - \overline{\omega_{\frac{n+3}{2}}}.
\bar{z}_{DC,UPMF}=k_2 R_{ant}P M+k_4 R_{ant}^2 P^2 \frac{2N^2+1}{2N}M\left(M+1\right) &\stackrel{N,M\nearrow}{\approx}k_2 R_{ant}P M+k_4 R_{ant}^2 P^2 N M^2.
\|f_i\|_{L^p(M^n)}^p-\|f_i-f_*\|_{L^p(M^n)}^p-\|f_*\|_{L^p(M^n)}^p=o(1), &1-\|I_{\alpha}f_i-I_{\alpha}f_*\|_{L^q(M^n)}^q-\|I_{\alpha}f_*\|_{L^q(M^n)}=o(1).
\left({\frac{\Phi_{\alpha_{1}}\cdots \Phi_{\alpha_{N - 2}}{\cal B}_{\beta_{1}\dots \beta_{N - 4}}}{\Lambda^{3}}}\right)^{\frac{3}{2 N - 9}}= \left({\frac{\left[ \Phi_{\alpha_{1}}\cdots \Phi_{\alpha_{N - 2}}{\cal B}_{\beta_{1}\dots \beta_{N - 4}}\right]^{3}}{\Lambda^{9}}}\right)^{\frac{1}{2 N - 9}}~ .
\lim_{k\to\infty}\mathcal{F}_{1}\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_{k-1}\circ\mathcal{F}_k(A)=AP^{-1}\lim_{k\to\infty}\mathcal{F}_{1}\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_{k-1}\circ\mathcal{F}_k(P).
\begin{aligned}R(\tfrac{h}{2}\Omega) = \big( I - \tfrac{h}{2}\Omega \big)^{-1}\big( I + \tfrac{h}{2}\Omega \big) = I + 2 \kappa \big( \tfrac{h}{2}\Omega + \tfrac{h^2}{4}\Omega^2 \big).\end{aligned}
d(2x+y)+d(2x-y) &~=\frac{2d(x)d(y)}{\left(4d(y)^{\frac{1}{5}}-d(x)^{\frac{1}{5}}\right)^{10}}\Big[1024d(y)+11520d(x)^{\frac{1}{5}}d(y)^{\frac{4}{5}}+13440d(x)^{\frac{2}{5}}d(y)^{\frac{3}{5}}&\qquad\qquad\qquad\qquad+3360d(x)^{\frac{3}{5}}d(y)^{\frac{2}{5}}+180d(x)^{\frac{4}{5}}d(y)^{\frac{1}{5}}+d(x)\Big].
N = \frac{1}{2 \pi \alpha}\left[ \frac{1}{2}\ln \left( \frac{k^{2}( t ) r^{2 \alpha}}{\Lambda ( t )^{2 \alpha}}\right) - 1 + \frac{1}{r^{2 \alpha}/ \Lambda ( t )^{2 \alpha}- 1}\ln \left( \frac{k^{2}( t ) r^{2 \alpha}}{\Lambda ( t )^{2 \alpha}}\right) \right]
b(s',s_n,p^1,\dots, p^{n-1})= \Sigma_{k=1}^{n-1}p^ktrace(A^k)-trace(A^n) &-\frac{\Sigma_{k,l=1}^{n-1}p^kp^l(A^n\alpha^k)\cdot\alpha^l}{1+|\Sigma_{k=1}^{n-1}p^k\alpha^k|^2}+\frac{\Sigma_{k,l,m=1}^{n-1}p^kp^lp^m(A^m\alpha^k)\cdot\alpha^l}{1+|\Sigma_{k=1}^{n-1}p^k\alpha^k|^2}.
\begin{array}{l l}{{( T , X , Y , Z ) = L ( \sin t \cosh \frac{\varphi_{+}+ \varphi_{-}}{2},}}&{{\sin t \sinh \frac{\varphi_{+}+ \varphi_{-}}{2},}}\\{{}}&{{\cos t \sinh \frac{\varphi_{+}- \varphi_{-}}{2}, \cos t \cosh \frac{\varphi_{+}- \varphi_{-}}{2}) .}}\\ \end{array}
\frac{Z}{2}\; I = \int_X \left(\exp\big(-\Phi(u)\big) - \exp\big(- m^\Phi_N(u)\big)\right)^2 \mu_0(\mathrm{d}u) \leq 2 \left\|\Phi(u) - m^\Phi_N(u) \right\|_{L^2_{\mu_0}(X)}^2 .
\left\{-\frac{\partial^2}{\partial \xi^2}+\frac{\alpha}{\sin^2\xi}+ \Omega\left( \xi \right) + \Omega\left( \xi-\pi \right) \right\}f_l(\xi) = B_l^2 ~f_l(\xi),~{\rm where}~~ B_l=\frac{b_l}{3}~~.
\lambda_{\underbrace{00 \ldots 0}_{\frac{N}{2}}\underbrace{11 \ldots 1}_{\frac{N}{2}}}=\lambda_{\underbrace{11 \ldots 1}_{\frac{N}{2}}\underbrace{00 \ldots 0}_{\frac{N}{2}}}=\frac{1}{2}.
[Z_1^{\mu_1}\dots Z_n^{\mu_n}] = [Z_1^{2a_1}\dots Z_n^{2a_n}] &= \omega^{-\sum_{i<j}\varepsilon_{ij}(2a_i)(2a_j)}Z_1^{2a_1}\dots Z_n^{2a_n}&= \omega^{-4\sum_{i<j}\varepsilon_{ij}a_ia_j}Z_1^{2a_1}\dots Z_n^{2a_n}&= q^{-\sum_{i<j}\varepsilon_{ij}a_ia_j}X_1^{a_1}\dots X_n^{a_n}
\left.\begin{aligned}\int_{\Omega}\big( y_0^c \cdot z_0^c - \nu(\boldsymbol{x}) \nabla p_0^c \cdot \nabla z_0^c \big)\,d\boldsymbol{x}= \int_{\Omega}{y_d^c}_0 \cdot z_0^c\,d\boldsymbol{x}, \int_{\Omega}\big( \nu(\boldsymbol{x}) \nabla y_0^c \cdot \nabla q_0^c + \lambda^{-1}p_0^c \cdot q_0^c \big)\,d\boldsymbol{x}= 0,\end{aligned}\right \rbrace
l_{1}(x)=\frac{x-x_{0}}{x_{1}-x_{0}}\cdot\frac{x-x_{2}}{x_{1}-x_{2}}=\frac{x-2}{4-2}\cdot\frac{x-5}{4-5}=-\frac{1}{2}x^{2}+\frac{7}{2}x-5
\phi = \left( \begin{matrix}{{\phi^{+}}}\\{{\phi^{0}}}\\ \end{matrix}\right) = \frac{1}{\sqrt{2}}\left( \begin{matrix}{{\chi^{2}( x ) + i \chi^{1}( x )}}\\{{v + \psi ( x ) - i \chi^{3}( x )}}\\ \end{matrix}\right) ,
{\mathcal K}_{\psi}:=\overline{\operatorname{span}}^{\|\ \|}\bigl\{\Lambda_{\psi}((\operatorname{id}\otimes\varphi)[\Delta(r^*y)(1\otimes s)]):y\in{\mathfrak N}_{\psi},\,r,s\in{\mathfrak N}_{\varphi}\bigr\}.
\dot{\bar{x}}^*(t) = \bar{A}' \bar{x}^*(t) + \bar{B}' u(t) \dot{\bar{x}}_{n+1}(t) = -\varepsilon \bar{x}_{n+1}(t) + \displaystyle \sum_{j=1}^{n_{\rm in}}u_j(t) \bar{N}_j' \bar{x}^*(t) + \bar{H}'' ( \bar{x}^*(t) \otimes \bar{x}^*(t) )
\gamma_1(\omega_1,\omega_2)=\frac{\omega_2^2(\omega_1^2\omega_2^2+4\omega_1^2\omega_2+2\omega_1\omega_2+3\omega_1^2+3\omega_1+1)}{(\omega_1+1)(3\omega_1\omega_2^2+4\omega_1\omega_2+\omega_1+2\omega_2+1)}.
\hat{a}_{B D}( \zeta ) = \frac{e^{\pi \vert \zeta \vert / 2}}{( e^{\pi \vert \zeta \vert}- e^{- \pi \vert \zeta \vert})^{1 / 2}}\hat{a}_{H}( \zeta ) - \frac{e^{- \pi \vert \zeta \vert / 2}}{( e^{\pi \vert \zeta \vert}- e^{- \pi \vert \zeta \vert})^{1 / 2}}\hat{a}_{H}^{\dagger}( \zeta ) .
\langle x,y,z\rangle^{3}=\langle\frac{(3z^{2}-x^{2}-y^{2})x(x^{2}-3y^{2})}{x^{2}+y^{2}},\frac{(3z^{2}-x^{2}-y^{2})y(3x^{2}-y^{2})}{x^{2}+y^{2}},z(z^{2}-3x^{2}-3y^{2})\rangle.
\begin{array}[b]{lcl}\overline \kappa_{14}(c) = \max\{\psi(0;c,a^\prime_c,b^\prime_c,c_0),\psi(1;c,a^\prime_c,b^\prime_c,c_0)\}begin{align*}2mm] ={\varrho_0\overline{\nu}_0\sqrt{\overline{\nu}}}c^{-2}+\max\{{c}^{-1}, \left({\underline{\sigma}}\underline{\eta}/2+(c-c_0)\right)^{-1}\}\, .\end{array}
\sum_{n=0}^\infty p(13n+6)q^n \equiv \frac{(q^{13})_\infty}{(q)_\infty^2}+ q^5 \frac{(q^{13})_\infty^{11}}{(q)_\infty^{12}}+q^6\frac{(q^{13})_\infty^{13}}{(q)_\infty^{14}},
\begin{array}{ccccc}\mathcal{P}'(F)_\mathbf{C}\mathcal{P}(F)/(\mathcal{P}(F)\cap K) \longrightarrow \mathcal{P}'(F)_\mathbf{C}\mathcal{P}(F)/\mathcal{P}(F) \longrightarrow \mathcal{P}'(F) u\cdot p \mod \mathcal{P}(F)\cap K \longmapsto u\cdot p \mod \mathcal{P}(F)=u \longmapsto \Im{u}\end{array}
\int_{-\infty}^{+\infty}\frac{e^{-\theta^2+2m\theta}}{(yq^{1/2}e^{2ki \theta};q)_{\infty}(zq^{1/2}e^{-2ik\theta};q)_{\infty}}d \theta & =\sqrt{\pi}e^{m^2}\frac{(-yqe^{2mki};q)_{\infty}(-zqe^{-2mki};q)_{\infty}}{(yzq;q)_{\infty}}.
J_{\mathrm{Lie}}^{s}\overline{\eta}_{\mathrm{L}}\Lambda \overset{(\ref{Form:EpsEtaLbar})}{=}\overline{\eta}_{\mathrm{BrL}}^{s}J_{\mathrm{Lie}}^{s}\Lambda =\overline{\eta}_{\mathrm{BrL}}^{s}\Lambda_{\mathrm{Br}}J_{2}^{s}
+\frac{\sqrt{2}x^{2}}{\sqrt{1-x^{4}}}\{K(\frac{1}{2}\sqrt{2})-F[arccos(x);\frac{1}{2}\sqrt{2}]\}=\int_{0}^{1}\frac{2x^{3}(y^{2}+1)}{\sqrt{(1-x^{4})(1-x^{4}y^{4})}}dy
\partial_{r}^{2}v_{2}(t,r) = \frac{c_{b}}{4}\int_{0}^{\infty}\sin(t\xi) \xi^{2}\left(-3 J_{1}(r\xi) + J_{3}(r\xi)\right) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}d\xi
\begin{array}{cc}\left\{\,b(\sigma ,\tau ),c({\sigma}^{\prime},\tau )\right\}\,=2\pi\delta (\sigma -{\sigma}^{\prime}) \left[ \beta (\sigma ,\tau ),\gamma ({\sigma}^{\prime},\tau )\right] =2\pi\delta (\sigma -{\sigma}^{\prime}) \end{array}
\sum_{\alpha_{1}, \alpha_{2}, \alpha_{3}\in{\bf Z}}= \int d \alpha_{1}d \alpha_{2}d \alpha_{3}\sum_{n_{1}, n_{2}, n_{3}\in{\bf Z}}\prod_{i = 1}^{3}\delta ( \alpha_{i}- n_{i}) = \int d \alpha_{1}d \alpha_{2}d \alpha_{3}\sum_{m_{1}, m_{2}, m_{3}\in{\bf Z}}\exp \left( 2 \pi i \sum_{i = 1}^{3}\alpha_{i}m_{i}\right)
s =\sum_{i=0}^{r-1}\left\{\langle\frac{-2mp^i}{q-1}\rangle+\langle\frac{2mdp^i}{q-1}\rangle+\langle\frac{-2m(d-1)p^i}{q-1}\rangle\right\}&-\sum_{i=0}^{r-1}\left\{\langle\frac{-mp^i}{q-1}\rangle+\langle\frac{mdp^i}{q-1}\rangle+\langle\frac{-m(d-1)p^i}{q-1}\rangle\right\}.
z_1w_1+\cdots+z_n w_n = 1 S w_1\,dz_1+\cdots +w_n\,dz_n+z_1\,dw_1+\cdots+z_n\,dw_n = 0S w_1\,dz_1+\cdots +w_n\,dz_n = 0 H z_1\,dw_1+\cdots+z_n\,dw_n = 0 H.
Q_A V_k = U_{k+1}J_k, Q_A^T U_{k+1}= V_k J_k^T + \alpha_{k+1}v_{k+1}e_{k+1}^T, Q_B V_k = \widehat{U}_{k}\check{J}_k, Q_B^T \widehat{U}_{k}= V_k \check{J}_k^T + \check{\beta}_{k}v_{k+1}e_{k}^T,
\varphi(\gamma^{1/2}, q^{d_2})=\varphi(q^{d_2},\gamma^{1/2})=\varphi(q^{d_a},q^{d_b})=\varphi(\gamma^{1/2},\gamma^{1/2})=1\ \mathrm{for}\ 1\leq a,b\leq 2.
\frac{d}{dt}S_{WZ}(t) = i^{[\frac{p+1}{2}]}\, \frac{2 \, i}{p!}\int \phi^* \, \Pi^{a_1}(t) \wedge \cdots \wedge \Pi^{a_p}(t) \wedge \left( \bar{\nu}(t) \Gamma_{a_1}\cdots \Gamma_{a_p}\, t \, d\theta \right)
\frac{1}{4 \pi^{2}g^{2}N^{3}}\sum_{\sigma}\left( \partial A^{\mu}( \sigma ) \times \partial A^{\nu}( \sigma ) \right)^{2}\rightarrow \frac{1}{4 \pi^{2}g^{2}N}\int_{T^{2}}d^{2}\sigma \left( \partial A^{\mu}( \sigma ) \times \partial A^{\nu}( \sigma ) \right)^{2},
\Phi_{[a,b],\lambda}(f)=\sup_{k\ge0}(TV_{[a,b],k}(f)-\lambda k)=\sup_{k\ge0}\left(\max_{a=t_{0}<t_{1}<\cdots<t_{k+1}=b}\sum_{i=1}^{k+1}|f(t_{i})-f(t_{i-1})|-\lambda k\right).
\lambda_{\epsilon(1)}(T_1)E_{D_0}(\mu_2(T_2),\ldots, \mu_n(T_n)) = \left([L_{E(T_1)}(1-p_{\epsilon(k_{1})}) S_1 1_B] \otimes \cdots \otimes [(1-p_{\epsilon(k_{m})}) S_m1_B]\right) & \oplus \left([(1-p_{\epsilon(1)})T_1 1_B] \otimes [(1-p_{\epsilon(k_{1})}) S_1 1_B] \otimes \cdots \otimes [(1-p_{\epsilon(k_{m})}) S_m1_B]\right).
d'(x,y) \leq \bar{d}(x,y)= \sup\{d(\alpha(t,x),\alpha(t,y))\, :\, t \in G \}\leq \sup\{d(\alpha(t,x),\alpha(t,z)) +d(\alpha(t,y),\alpha(t,z))\,:\, t \in G \}\leq \sup\{d(\alpha(t,x),\alpha(t,z))\, :\, t \in G \}+\sup\{d(\alpha(t,y),\alpha(t,z))\, :\, t \in G \}= \bar{d}(x,z)+\bar{d}(y,z) = d'(x,z)+d'(y,z) \,.
K_{-1}:= \{|x - \frac{X}{t}| \geq \frac{1}{M\rho (0)^2 t}\}, &K_j := \{|x - \frac{X}{t}| \sim 2^{-j}\frac{1}{M\rho (0)^2 t}\}, 1\leq j \leq k , &K_{0}:= \{|x - \frac{X}{t}| \leq \frac{Y + M\rho (0)}{t}\},
\epsilon^{+}( z ) = \frac{z}{2 \pi}+ \frac{i}{\pi}\sum_{n > 0}\frac{e^{- i n ( z - i \varepsilon )}}{n}, ~ ~ \epsilon^{-}( z ) = \frac{z}{2 \pi}+ \frac{i}{\pi}\sum_{n < 0}\frac{e^{- i n ( z + i \varepsilon )}}{n}.
\{\begin{matrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{matrix}\}=\{\begin{matrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\end{matrix}\}=\{\begin{matrix}j_{1}&j_{3}&j_{2}\\j_{4}&j_{6}&j_{5}\end{matrix}\}=\{\begin{matrix}j_{3}&j_{2}&j_{1}\\j_{6}&j_{5}&j_{4}\end{matrix}\}=\cdot\cdot\cdot
\vartheta_4^4\!\left(e^{-\pi/\sqrt{N}}\right)=N\vartheta_2^4\!\left(e^{-\pi\sqrt{N}}\right)=N\vartheta_4^4\!\left(e^{-\pi\sqrt{N}}\right)\frac{\vartheta_2^4\!\left(e^{-\pi\sqrt{N}}\right)}{\vartheta_4^4\!\left(e^{-\pi\sqrt{N}}\right)}.
\begin{aligned}\tilde{g}_i =(x_i -i)\prod_{k=i+1}^{h(i)}(x_i-x_k -1)\cdot \prod_{\ell=i}^{h(i-1)}\left( \prod_{j=j_{\ell}}^{i-1}(x_j-x_{\ell}-1)\right).\end{aligned}
\sum_{0 \leq k < n}\frac{2a_k}{(k-n)}t^{(k-n)/2}|_{\Lambda^{-2}}^{\Lambda'^{-2}}= \sum_{0 \leq k < n}\frac{2a_k}{(k-n)}{\Lambda}^{n-k}( e^{-\tau(n-k)/2}- 1)
\begin{aligned}q^b(t,x,y) \geq \int_{B_{l-1}}\cdots\int_{B_{1}}q^b(t/l,x,z_1)\cdots q^b(t/l,z_{l-1}, y)\,dz_1\cdots dz_{l-1}&\geq (c_7t/l)^l\geq (c_8t/R)^{c_+R+1}\geq c_9(t/R)^{c_{10}R}&\geq c_9\left(\frac{t}{|x-y|}\right)^{c_{10}|x-y|}.\end{aligned}
\begin{aligned}|R_{nl}| \leq \|nl_h(u^{\Delta t}_{h},u^{\Delta t}_{h})\|_{H^{-1}}\|P_h(u^{\Delta t}_{h}\phi)-u^{\Delta t}_{h}\phi\|_{H^1}\leq c\,\sqrt{h}\|u^{\Delta t}_{h}\|_{2}\,\|u^{\Delta t}_{h}\|_{H^1}^{2}, \end{aligned}