harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_{\alpha_{1},\alpha_{2}}\ldots_{\alpha_{m},\beta}(z)=\left[ \beta\int_{0}^{z}t^{\beta -1}\Pi_{i=1}^{m}\left( \frac{h_{i}(t)}{t}\right)^{\frac{1}{\alpha_{i}}}dt\right]^{\frac{1}{\beta}}~~~~~~~~~~~
\{\Psi_{1}( x_{n}, t ) , \overline{{{\Psi}}}_{1}( x_{m}, t ) \}= \frac{\delta_{m , n}}{\Delta x_{n}}- ( 1 - q^{- 2}) \overline{{{\Psi}}}_{2}( x_{m}, t ) \Psi_{2}( x_{n}, t )
z ( 1 - z ) \frac{d^{2}h}{d z^{2}}+ \left[ 2 \delta +{\frac{1}{2}}- ( 1 + 4 \delta ) z \right] \frac{d h}{d z}- \left[ 4 \delta^{2}- B_{l}^{2}\right] ~ h ( z ) = 0 .
\begin{array}{l}\vartheta(t;(\tau,\nu))= \displaystyle \int_{-\infty}^{+\infty}\widetilde{G}(t,s)f(s,y(s,\tau,\nu)+\vartheta(s;(\tau,\nu)),y(\gamma(s),\tau,\nu)+\vartheta(\gamma(s);(\tau,\nu)))\,ds\end{array}
[ X_{j}, [ X^{j}, X^{i}] ] +{\frac{i}{2}}\{b^{+ j}, b^{i}{}_{j}- b^{+ -}\delta^{i}{}_{j}\}= 0 , \qquad [ X_{i}, [ X^{i}, X^{-}] ] -{\frac{i}{4}}\{b^{i j}, b_{i j}\}= 0 .
\mathbf{t}(x_1)=(2\pi x_1)^{-\frac{1}{2}}[Kx_1^{i\epsilon}\mathbf{w}+\bar{K}x_1^{-i\epsilon}\bar{\mathbf{w}}+ K_3x_1^\kappa\mathbf{w}_3+K_4x_1^{-\kappa}\mathbf{w}_4],
\gamma_*\big( \gamma^*\O(x,y)\otimes \alpha^*\O_\rho(z) \big) = \gamma_* \O(x,y)((z-x)e) \O(x,y)\otimes \rho_* \O_\rho(z) - \O(x-3,y-1)\otimes \rho_* \O_\rho(z-1) \sum_{0 \leq h \leq z}\O(x-h,y)-\sum_{0 \leq k \leq -z -2}\O(1+x+k,y)-\sum_{0 \leq j \leq z-1}\O(x-3-j,y-1)+\sum_{0 \leq l \leq -z-1}\O(x+l-2,y-1)
= -{\frac{1}{\pi^{D / 2}\beta^{D - 1}}}\sum_{n = 0}^{\infty}\zeta_{R}\left( D - 2 n \right) \Gamma \left({\frac{D - 2 n}{2}}\right) \breve{c}_{n}\left({\frac{\beta}{2}}\right)^{2 n},
L = \sum_{\alpha}\left( p_{\alpha}^{\bar{z}}\dot{z}_{\alpha}+ p_{\alpha}^{z}\dot{{\bar{z}}}_{\alpha}+ i J_{\alpha}\frac{{\bar{\xi}}^{\alpha}\dot{\xi}^{\alpha}- \dot{\bar{\xi}^{\alpha}}\xi^{\alpha}}{1 + \mid \xi^{\alpha}\mid^{2}}\right)
U\cdot\left( \begin{array}{c}a_0 a_1 \vdots a_{c_X}\end{array}\right)=\left( \begin{array}{c}\frac{h^0(\mathcal{O}_X)}{\delta(0)}\frac{h^0(H)}{\delta(1)}\vdots \frac{h^0(c_X H)}{\delta(c_X)}\end{array}\right)\ ,
A^p\sigma_h B^p \ge A^{1/2}h(C^p)h(C)^{1-p}A^{1/2}& \geq \lambda_{\min}\left( \frac{h(C^p)}{h(C)^p}\right) A^{1/2}h(C)A^{1/2}& = \lambda_{\min}\left( \frac{h(C^p)}{h(C)^p}\right)(A\sigma_h B).
\sup_{u\in W^{1,n}_0(\Omega),\|\nabla u\|_{L^n(\Omega)}\leq 1}\int_\Omega e^{\alpha_n|u|^{\frac{n}{n-1}}}\, dx<C|\Omega|, \quad\alpha_n:=n\omega_{n-1}^{\frac{1}{n-1}}.
p(0)=0,p'(\rho)>0\ \forall\ \rho>0,\liminf_{\rho\rightarrow 0}\frac{p'(\rho)}{\rho}>0,p(\rho)\sim_{\rho\rightarrow \overline{\rho}-}|\overline{\rho}-\rho|^{-\beta},\mbox{for some}\beta > 5/2,
( \sigma_{\mu}{}^{\rho})_{\alpha \beta}[ \mathrm{A{\footnotesize \#}.{\footnotesize \#}}]_{\nu \rho}{}^{\alpha \beta}= b_{1}^{2}\sum_{k = 0}^{\infty}\frac{f_{\mathrm{A \# . \#}}( k )}{( k ! )^{2}}[ \mathrm{a{\footnotesize \# '}o r b{\footnotesize \# '}}]_{\mu \nu}( k )
J_{\alpha a}= \partial_{a}\phi_{\alpha}- 2 i ( \partial_{a}X^{\mu}) \Gamma_{\mu \alpha \beta}\theta^{\beta}- \frac{2}{3}( \partial_{a}\theta^{\beta}) \Gamma_{\beta \delta}^{\mu}\theta^{\delta}\Gamma_{\mu \alpha \epsilon}\theta^{\epsilon}\nonumber
=\biggr|\frac{\sum_{l=1}^\eta\sum_{m=l+1}^\eta c^{\bf a}_lc^{\bf a}_m\gamma^k_l\gamma^k_m(\gamma_l-\gamma_m)^2}{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+1}\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^k}\biggr|<\epsilon.
\theta_{m , k}\: ( \tau ) \; \theta_{m^{\prime}, k^{\prime}}\: ( \tau ) = \sum_{\ell = 1}^{k + k^{\prime}}\; \theta_{m k^{\prime}- m^{\prime}k + 2 \ell k k^{\prime}, k k^{\prime}( k + k^{\prime})}\: ( \tau ) \; \theta_{m + m^{\prime}+ 2 \ell k , k + k^{\prime}}\: ( \tau ) ,
\sum_{m_1=0}^{rp-1}\sum_{m_2=0}^{sp-1}\sum_{m_3=0}^{tp-1}\binom{m_1+m_2+m_3}{m_1,m_2,m_3}^2\equiv_{p^2}\sum_{m_1=0}^{r-1}\sum_{m_2=0}^{s-1}\sum_{m_3=0}^{t-1}\binom{m_1+m_2+m_3}{m_1,m_2,m_3}^2.
\big\| (-\Delta_v)^{\frac12 (1- \frac{n-1}2) + \varepsilon}X( |\widehat{gd\sigma}|^2) \big\|_{L^q_\omega L^2_v(\mathcal{M}_{1,n})}^q = \int_{\mathbb{S}^{n-1}}\Bigg( \int_{\langle \omega \rangle^\perp}|\eta|^{-(n-3) + 2\varepsilon}(gd\sigma) \ast (gd\sigma ) (\eta)^2\, d\lambda_\omega(\eta) \Bigg)^\frac q2\, d\sigma(\omega).
B_3=y_1^3\big(a_1^2 z_1+a_1 b_1 z_1+a_2 b_1^2 z_1 z_2+ a_1 b_1 z_1 z_3+b_1^2 b_2 z_1 z_2 z_3\big) \hphantom{B_3=}{}+y_1 y_2\big(2 a_1 \beta_1 z_1+b_1 \beta_1 z_1+b_1^2 \beta_2 z_1 z_2+b_1 \beta_1 z_1 z_3+a_1 \alpha_1 b_1 \alpha_1+\alpha_1 z_3\big) \hphantom{B_3=}{}+y_3\big(\beta_1 \alpha_1+ \beta_1^2 z_1+b_1^2 \alpha_2 z_1\big).
\frac{\partial}{\partial u_{j}^{( n )}}\, \exp \left({\sum_{j = 1}^{m + 1}\sum_{n = 1}^{\infty}E_{j j}^{( n )}u_{j}^{( n )}}\right) g = \left( \frac{\partial}{\partial u_{j}^{( n )}}g_{-}( u ) \right) g_{+}( u ) + g_{-}( u ) \frac{\partial}{\partial u_{j}^{( n )}}g_{+}( u )
I_{1}= 4r\int_{0}^{2\pi}\frac{\cos\left(\frac{x-y}{2}\right)\sin\left(\frac{x+y}{2}\right)(\cos(x)h(x)-\cos(y)h(y))}{((1+r^2)+(r^2-1)\cos(x+y))}dy & = 4r\int_{0}^{2\pi}\frac{\cos\left(\frac{2x-y}{2}\right)\sin\left(\frac{y}{2}\right)(\cos(x)h(x)-\cos(y-x)h(y-x))}{((1+r^2)+(r^2-1)\cos(y))}dy
\langle\otimes_{l = 1}^{m}\mathfrak{a}\rangle_{t}=\exp \biggl( -\frac{t}{2}\sum\limits_{j=1}^N \sum\limits_{i,p = 1}^m\otimes_{l = 1}^m(JK_{j})^{\delta_{il}+\delta_{pl}}\biggr) \langle\otimes_{l = 1}^{m}\mathfrak{a}\rangle_{0},
\sup_{a,b>0}\frac{\lambda-1 - \lambda(a+b) + (\lambda+2)ab}{a^2 b^2}= \sup_{x = (ab)^{-1/2}> 0}(\lambda-1)x^4 - 2\lambda x^3 + (\lambda+2) x^2 =: \sup_{x>0}\varphi(x).
K=\frac{1}{\sqrt{c_1}}\Big[T\max_{\overline{\Omega}}L(x,0)+2T\max_{\overline{\Omega}\times \mathcal{P}(\overline{\Omega})}|F|+2\max_{\overline{\Omega}\times \mathcal{P}(\overline{\Omega})}|G|+Tc_0\Big]^{\frac{1}{2}}
C_{n}=\frac{1}{n+1}\left( \begin{array}{c}3n n\end{array}\right) -\frac{1}{n+1}\sum_{j=0}^{n-1}\left( \begin{array}{c}n+j j\end{array}\right) \left( \begin{array}{c}2n-j-1 n-j\end{array}\right) .
\sum_{|\alpha|=m}\frac{(x)_\alpha}{\alpha!}= \sum_{k=0}^m\frac{(x_n)_k}{k!}\sum_{|\beta|=m-k}\frac{(x')_{\beta}}{\beta!}= \sum_{k=0}^m\frac{(x_n)_k(x_1+\cdots+x_{n-1})_{m-k}}{k!(m-k)!}= \frac{(x_1+\cdots+x_{n-1}+x_n)_m}{m!},
\phi_n^{(1)}(z)=\frac{(1-q^{n+\gamma})}{(1-q^{\gamma})}\frac{(q;q)_n}{(q^{\delta+1};q)_n}L_n^{(\delta)}(z;q) +\frac{q^{\gamma}(q^{n}-1)}{(1-q^{\gamma})}\ \frac{(q;q)_{n-1}}{(q^{\delta+1};q)_{n-1}}L_{n-1}^{(\delta)}(zq;q).
A_L = \begin{bmatrix}\tau_L 1 0 -\sigma_L 0 1 \delta_L 0 0\end{bmatrix}\;, A_R = \begin{bmatrix}\tau_R 1 0 -\sigma_R 0 1 \delta_R 0 0\end{bmatrix}\;, B = \begin{bmatrix}1 0 0\end{bmatrix}\;,
\left\{\begin{array}{lcl}y_{1}^{2}=(x-\mu_{4})(x-\mu_{6})(x-\mu_{7}) y_{2}^{2}=(x-1)(x-\mu_{5})(x-\mu_{6})(x-\mu_{7}) y_{3}^{2}=x(x-\mu_{4})(x-\mu_{5})(x-\mu_{6})\end{array}\right\}\subset{\mathbb C}^{4}.
\left( \begin{array}{c c}{f_{\alpha}^{I}}&{\bar{X}^{I}}\\ \end{array}\right) = e^{K / 2}\left( \begin{matrix}{- \frac{1}{z + \bar{z}}}&{1}\\{- \frac{i}{z + \bar{z}}}&{- i}\\ \end{matrix}\right) \ ,
\frac{a^{2}}{2}Q_{\nu+2}(a,b)=(\frac{a^{2}}{2}-\nu)Q_{\nu+1}(a,b)+(\frac{b^{2}}{2}+\nu)Q_{\nu}(a,b)-\frac{b^{2}}{2}Q_{\nu-1}(a,b).
\sum_{n=1}^{\infty}\Vert f_n^d(t)\Vert_{L^{1}(S_2)}\leq C \Big[\sum_{n=1}^{\infty}\dfrac{1}{n^2}\Big]^{1/2}\Big[\sum_{n=1}^{\infty}n^2\Vert f_n^d(t)\Vert_{L^{1}(S_2)}^2\Big]^{1/2}\leq C \sqrt{\xi (t)}
\Phi\Big(\psi((1-t)\, m_0+t\, m_1),(1-t)\, \phi_0+t\, \phi_1\Big) \le \Phi\Big((1-t)\, \psi(m_0)+t\, \psi(m_1),(1-t)\, \phi_0+t\, \phi_1\Big) &\le (1-t)\, \Phi(m_0,\phi_0)+t\, \Phi(m_1,\phi_1),
P\Big(z_2+z_2 b_{1k_2}P^{k_2}(z_2)(\alpha i-1)^{k_2}+z_2o(P^{k_2}(z_2))\Big)-P(z_2) &+\mathrm{Re}\Big(a_{0k_1}P^{k_1}(z_2)(\alpha i-1)^{k_1}+o(P^{k_1}(z_2))\Big)\equiv 0
\frac{1}{e^{2}( q^{2})}\cong \frac{1}{1 2 \pi^{2}}\, \ell n \frac{\Lambda^{2}}{q^{2}}\Rightarrow \frac{1}{1 2 \pi^{2}}\int_{q^{2}}^{\Lambda^{2}}R ( s ) \frac{d s}{s}
\left|\int g \circ (\sigma \times \sigma)\,dm - \int g\,dm\right| = \lim_{j \to \infty}\frac{1}{n_j}\left|\int g \circ (\sigma \times \sigma)\,dm_n - \int g\,dm_n\right| &=\lim_{j \to \infty}\left|\frac{1}{n_j}\left(g\left(\sigma^{n_j}x,\sigma^{n_j}z^{(n_j)}\right)-g\left(x,z^{(n_j)}\right)\right)\right| &\leq \lim_{j \to \infty}\frac{2|g|_\infty}{n_j}=0,
c_j = \sum_{n=1}^\infty \lambda_n \sum_{\rho = 1}^{L_n}\mu_\rho \sum_{\nu=1}^{d_n}v_{n\nu}^{(\rho)}c_{nj}^{(\nu)}, j \ge 0 , d_j = \sum_{n=1}^\infty \lambda_n \sum_{\rho = 1}^{L_n}\mu_\rho \sum_{\nu=1}^{d_n}w_{n\nu}^{(\rho)}c_{nj}^{(\nu)}, j \ge 0.
\begin{aligned}\int_{0}^{\infty}\frac{\cos\left(t_0\log x\right) + i \sin\left(t_0\log x\right)}{x^{1/2-\theta_0}\left(e^{x}+1\right)}dx=0 \int_{0}^{\infty}\frac{\cos\left(t_0\log x\right) + i \sin\left(t_0\log x\right)}{x^{1/2+\theta_0}\left(e^{x}+1\right)}dx=0 \end{aligned}
\gamma_s=\frac{\left( -1\right)^s}2\int_{\Sigma_s}\int_{\partial \Sigma_s={\bf c}}\frac{\left( \vec R-i\frac 12\vec \Gamma\right) \cdot d\vec S}{\left[\left(\vec R-i\frac 12\vec \Gamma \right)^2\right]^{\frac 32}},
\frac{\partial A}{\partial q_{j}}=0 &\frac{\partial X}{\partial q_{j}}=\beta(\beta-1)\left(\sum_{i}p_{i}q^{\beta-1}_{i}\right)^{\beta-1}p_{j}q^{\beta-2}_{j}&\frac{\partial Y}{\partial q_{j}}=\beta(1-\beta)\left(\sum_{i}q^{\beta}_{i}\right)^{-\beta}.\q^{\beta-1}_{j}
A_1=1+\frac{1}{2\pi}\int \frac{|D_tZ(\alpha,t)-D_tZ(\beta,t)|^2}{(\alpha-\beta)^2}d\beta-\sum_{j=1}^N \frac{\lambda_j}{\pi}Re\Big\{\frac{D_tZ-\dot{z}_j}{c_0^j(\alpha-w_0^j)^2}\Big\},
\frac{\partial u}{\partial\lambda}=2hc^{2}(\frac{hc}{kT\lambda^{7}}\frac{e^{hc/\lambda kT}}{(e^{hc/\lambda kT}-1)^{2}}-\frac{1}{\lambda^{6}}\frac{5}{e^{hc/\lambda kT}-1})=0,
GB2(y_{i};a_{i},b_{i},p_{i},q)=\frac{|a_{i}|y_{i}^{a_{i}p_{i}-1}}{b_{i}^{a_{i}p_{i}}B(p_{i},q)(1+(\frac{y_{i}}{b_{i}})^{a_{i}})^{p_{i}+q}}
\begin{aligned}v_t+(w+u_{\infty})\cdot \nabla v+\nabla p =(d+B_{\infty})\cdot \nabla b+\nu \Delta v, b_t+(w+u_{\infty})\cdot \nabla b+\nabla q =(d+B_{\infty})\cdot \nabla v+\mu \Delta b, \nabla \cdot v=\nabla \cdot b =0, w=\chi_{\varepsilon}*v,\d =\chi_{\varepsilon}*b.\end{aligned}
\mathbb{E}_{f_0,T}\max_k|Z_{jk}| =\mathbb{E}_T (\mathbb{E}_{f_0|T}(\max_k|Z_{jk}|) ) &\leq \mathbb{E}_T\big( \max_{k}|\mu_{n,m,k,T}| \big)+\sqrt{2(j+1)}\mathbb{E}_T\max_k\sigma_{n,m,k,T}&\leq c( \sqrt{(\log_2 n)\eta/n}+2^{j/2}\sqrt{j}e^{-c n^{\delta}})\leq C \sqrt{(\log_2 n)\eta/n},
H_{3,1}= U(N_{a,1}+N_{a,2}+N_{a,3}-N_{b,1})^2+\mu(N_{a,1}+N_{a,2}+N_{a,3}-N_{b,1}) &+t_{1,1}(a_{1}b_{1}^\dagger+a_{1}^\dagger b_{1})+t_{2,1}(a_{2}b_{1}^\dagger+a_{2}^\dagger b_{1})+t_{3,1}(a_{3}b_{1}^\dagger+a_{3}^\dagger b_{1})
e^{{\cal{G}}_{\rm c}\{J \}}= \frac{{\cal{Z}}\{J \}}{{\cal{Z}}_0}= \frac{\int{\cal{D}}\{\phi \}e^{- S \{\phi\}+ (J, \phi )}}{\int{\cal{D}}\{\phi \}e^{- S_0 \{\phi\}}}\; .
AC=[C^{-}]A=[\begin{matrix}c_{0}&-C_{3}&C_{2}&C_{1}\\C_{3}&c_{0}&-C_{1}&C_{2}\\ -C_{2}&C_{1}&c_{0}&C_{3}\\ -C_{1}&-C_{2}&-C_{3}&c_{0}\end{matrix}]\{\begin{matrix}A_{1}\\A_{2}\\A_{3}\\a_{0}\end{matrix}\}.
\beta_{\nu}^{( 1 )}= \varepsilon^{\mu , [ \mu \nu ]}+ \varepsilon^{[ \mu \nu ] , \mu}, \hspace{0 . 3 i n}\beta_{\nu}^{( \widetilde{1})}= \frac 1 2 e_{\mu \nu \rho \omega}\left( \varepsilon^{\widetilde{\mu}, [ \rho \omega ]}+ \varepsilon^{[ \rho \omega ] , \widetilde{\mu}}\right) ,
( \mathrm{M a s s})^{2}={\frac{1}{4 \lambda_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{L})^{2}+ N_{L}- 1 \right] ={\frac{1}{4 \lambda_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{R})^{2}+ N_{R}- \delta \right] ~ .
V_1^B=\left\langle \begin{pmatrix}1 0 \end{pmatrix}\right\rangle, V_2^B=\left\langle \begin{pmatrix}0 1 \end{pmatrix}\right\rangle, V_3^B=\left\langle \begin{pmatrix}1 1 \end{pmatrix}\right\rangle, V_4^B=\left\langle \begin{pmatrix}1 \beta \end{pmatrix}\right\rangle.
\times \;\prod_{i=1}^{m^\prime_b}\psi^{(b)}_2({y^\prime}_i^{(b)})\prod_{i=1}^{n_b}\overline{\psi}^{(b)}_1(x_i^{(b)}\!+ \hat{\tau})\prod_{i=1}^{m_b}\psi^{(b)}_2(y_i^{(b)}\!+ \hat{\tau})\prod_{i=1}^{n^\prime_b}\overline{\psi}^{(b)}_1({x^\prime}^{(b)}_i)\Big \rangle_{free}\; .
\sum\limits^n b_j^*\Gamma_N(a_j,a_k)b_k=\sum\limits_j^n b_j^*N(a_j^*)\sum\limits_k^na_k b_k-\sum\limits^n b_j^* N(a_j^*a_k)b_k+\sum\limits_j^nb_j^*a_j^*\sum\limits_k^n N(a_k)b_k \hphantom{\sum\limits^n b_j^*\Gamma_N(a_j,a_k)b_k}{}=- \sum\limits^n b_j^* N(a_j^*a_k)b_k.
{\mathrm{R e s}}_{z = 0}\frac{H_{\mu}^{( 1 )}( \lambda z )}{H_{\nu}^{( 1 )}( z )}F ( z ) ={\mathrm{R e s}}_{z = 0}\frac{J_{\nu}( z ) J_{\mu}( \lambda z ) + Y_{\nu}( z ) Y_{\mu}( \lambda z )}{J_{\nu}^{2}( z ) + Y_{\nu}^{2}( z )}F ( z )
\alpha_{1,1234}\circ i_{234_1}^{1234_1}=i_{234_1}^{1234_1}\circ\alpha_{1,234}, \alpha_{1,123}\circ i_{2_1}^{123_1}=i_{2_1}^{123_1}\circ\alpha_{1,2}, \alpha_{1,123}\circ i_{3_1}^{123_1}=i_{3_1}^{123_1}\circ\alpha_{1,3}
\{Q_{\pm}^{B}, Q_{\pm}^{C}\}= 2 \delta^{B C}P_{\mp}\hspace{2 c m}\{Q_{+}^{B}, Q_{-}^{C}\}= - ( - 1 )^{B}( \delta^{B C}2 T + \epsilon^{B C}2 \tilde{T})
\sum_{j=1}^J k_n^{(j)}v^{(j)}(n) \leq &(1-\epsilon) \left[ \frac{n}{2}\log \left( \sum_{j=1}^J k_n^{(j)}\right) - \sum_{j=1}^J \beta^{(j)}\ell_n H_2 \left( \alpha_n^{(j)}\right) \right]
D_{\beta,\delta}:=\sup_{x\ge C_{\beta,\delta}}\left(\frac1{x^{1/2}}+\frac{C_{\beta,\delta}(\mathrm{e}-1)(\log x)^{j^*_{\beta,\delta}\log 2}}{x^{1/2}}\,\left(1+\frac{\log\,x}{\log\,\omega^{-1}}\right)^{B+(\log 2)/\delta}\right)<\infty.
\log \left|F\left(a_{1}\right)\right|\leq\left(\lambda_1+\delta\right)\frac{T}{2}+\sum_{k=2}^{\infty}\log \left|\frac{a_{1}-a_{k}}{a_{1}-\bar{a}_{k}}\right|+\frac{\Im\left(a_{1}\right)}{\pi}\int_{-\infty}^{\infty}\frac{\log |F(s)|}{\left|s-a_{1}\right|^{2}}\mathrm{~d}s.
\kappa_n(w_{n}) =\begin{pmatrix}\sigma(2^{n-1}) \sigma(2^{n-1}+1) \end{pmatrix}\ \ \kappa_n(w_{n+1})=\kappa_n(w_{n+2})= \begin{pmatrix}\sigma(2^{n}) \sigma(1) \end{pmatrix},
I_{1}=\iint_{\Omega_{1}\times \Omega_{2}}|w(x_{1},y_{1})-w(x_{2},y_{2})|dx_{1}dy_{1}dx_{2}dy_{2}&\leq\iint_{\Omega_{1}\times \Omega_{2}}\left\vert\int_{y_{1}}^{y_{2}}w_{y}(x_{1},y)dy\right\vert dx_{1}dy_{1}dx_{2}dy_{2}
\left. \Im\int\bar{u}\psi' \partial_{r}u dx\right|_{t=0}^{t=T}\ge \int_{0}^{T}\int \left[ \frac{\epsilon r^{\epsilon}}{(1+r^{\epsilon})^{2}}\frac{|\nabla u|^{2}}{r}d + \frac{\epsilon r^{\epsilon}}{1+r^{\epsilon}}\frac{|u|^{2}}{r^{3}}\right] dxdt.
(\begin{matrix}\frac{E^{\prime}}{c}\\p_{x}^{\prime}\\p_{y}^{\prime}\\p_{z}^{\prime}\end{matrix})=(\begin{matrix}\gamma&-\gamma\beta&0&0\\ -\gamma\beta&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix})(\begin{matrix}\frac{E}{c}\\p_{x}\\p_{y}\\p_{z}\end{matrix})
h_{g}=\frac{0.026}{D_{*}^{0.2}}(\frac{\mu^{0.2}c_{p}}{Pr^{0.6}})(\frac{p_{c}}{c^{*}})^{0.8}(\frac{D_{*}}{r_{c}})^{0.1}(\frac{A_{t}}{A})^{0.9}\sigma
+ \frac{b_{0}^{( 3 )}}{L_{0}^{( 3 )}}Q^{( 3 )}| \Lambda_{a_{1}a_{2}}^{e}\rangle_{3}+ \frac{b_{0}^{( 3 )}}{L_{0}^{( 3 )}}[ \delta \varepsilon_{1}| \Omega_{1 a_{1}a_{2}}\rangle + \delta \varepsilon_{2}| \Omega_{2 a_{1}a_{2}}\rangle ]
( Z_{\bf m}^{N})_{\bf k l}\equiv \int d^{2}\sigma f_{\bf k}^{*}( \sigma ) Z_{\bf m}^{N}( \sigma ) f_{\bf l}( \sigma ) = - 2 i \sin \left( \frac{2 \pi}{N}{\bf m \times l}\right) \delta_{{\bf m}({\bf k - l})}.
\widehat{q}(t, x) =\widehat{q}(t,x; r, \alpha_0, \beta_0) &\equiv \lim_{\kappa \to \infty}q(t, x; r, \alpha_0, \beta_0, i \kappa)= \prod_{j=1}^5 \frac{\sin \zeta_j}{\sin \eta_j},
| \{(\xi_1, \eta_1) \ | \ (\tau_1, \xi_1, \eta_1) \in E(\tau, \xi,\eta) \}|= \int_{\theta_1}\int_{r_1}{\chi}_{E(\tau,\xi,\eta)}(|\xi_1|, \theta_1) r_1 d r_1 d \theta_1 \lesssim (N_1A)^{-1}\max (L_1, L_2).
~ \chi_{n}= \sum b_{\vec{n}, r , s , t}\lambda^{n_{1}}( \lambda y )^{n_{2}}( \lambda y_{c})^{n_{3}}( \lambda / \Delta_{-})^{n_{4}}e^{r \Delta_{+}( y - y_{c})}e^{s \Delta_{-}y}e^{t \Delta_{-}y_{c}},
t_{\mu\nu}=\frac{1}{8\pi{\cal G}}\left [ R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{\alpha}_{\;\;\;\alpha}-R^{(1)}_{\;\;\;\;\mu\nu}+\frac{1}{2}\eta_{\mu\nu}R^{(1)\,\alpha}_{\;\;\;\;\;\;\;\alpha}\right ] \,.
\left.\frac{da}{dT}\right|_{T=T_c^+}=\left.\frac{da_1}{dT}\right|_{T=T_c^-}=-\frac{1}{4(1+c_1)^2}\,;\qquad\left.\frac{db}{dT}\right|_{T=T_c^+}=\left.\frac{db_2}{dT}\right|_{T=T_c^-}=\frac{1}{4(1-c_1)^2}\,,
q^{2}\xi_{0 2}- p . q \xi_{1 1}={\frac{1}{2}}\left\{- \left[{\frac{1}{2}}+ m^{2}\xi_{0 0}\right] +{\frac{p^{2}}{2}}\xi_{1 0}+{\frac{3 q^{2}}{2}}\xi_{0 1}\right\}
\frac{|-1+z_{j,n}|}{|z_{j,n}|}={\rm D}\left(\frac{|-1+z_{j,n}|}{|z_{j,n}|}\right)+{\rm tl}\left(\frac{|-1+z_{j,n}|}{|z_{j,n}|}\right)~~<~~ \frac{1 -\exp\bigl(-\frac{\pi}{a_{j,n}}\bigr)}{2 \exp\bigl(\frac{\pi}{a_{j,n}}\bigr) -1}
\mu'+\nu'+\frac{1}{4}-\frac{\mu'}{6}-\frac{5\nu'}{12}= \frac{7}{12}(\mu'+\nu')+\frac14+\frac{\mu'}{4}+o(1) \leq \frac{7}{12}(1+\delta)+\frac14+ \frac{1}{4}\Bigl(\frac12+2\delta+\kappa \Bigr)+o(1) =1-\frac{13}{12}(1/26-\delta)+\frac{\kappa}{4}+o(1).
T_{\mu \nu}= g_{\mu \nu}T - F_{\mu \rho}\mathrm{}^{a}F_{\nu}\mathrm{}^{\rho a}+ D_{\mu}\phi^{a}D_{\nu}\phi^{a}- \frac{1}{2}\ \phi^{2}\left( R_{\mu \nu}- \frac{1}{2}\g_{\mu \nu}R \right)
(K^{-1})^U_S = -\frac{z^{*3}(1-A|z|^2)P'(y)}{e^{\tilde{K}/2}P''(y)},\mbox{}(K^{-1})^U_T = \frac{(T+T^*)z^{*3}(1-A|z|^2)}{e^{\tilde{K}/2}(1-\frac{\bar{n}}{3}B(1-A|z|^2)\|\Pi\|)},
\frac{\sum_{i=1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}}=\frac{w_{1}x_{1}+w_{2}x_{2}+\cdot\cdot\cdot+w_{n}x_{n}}{w_{1}+w_{2}+\cdot\cdot\cdot+w_{n}}
c'_1 = b_0c_1+(a_1-d_1)a_2 c'_2 = b_0c_2+b_1c_1+(a_1-d_1)a_3-a_2d_2 c'_3 = b_0c_3+b_1c_2+b_2c_1+(a_1-d_1)a_4-a_2d_3-a_3d_2.
q^4_{\nu, \kappa}(\xi-\eta, \eta)= -q_2(\xi-\eta, \eta)\big( 1+c_{\nu}c_{\kappa}\frac{(\xi-\eta)\cdot \eta}{|\xi-\eta||\eta|}\big)= -q_2(\xi-\eta, \eta)(1-a_{\nu}a_{\kappa}\cos(\angle(\xi-\eta, \eta)).
\int_{\tau}^1 h(x)f^2(x)\mathbf{1}_{\{x\le a+\tau\}}\,\mathrm{d}x \le \frac{1}{1-\tau}\int^1_\tau h(x)f^2(x)\,\mathrm{d}x \int_\tau^1 \mathbf{1}_{\{x\le a+\tau\}}\,\mathrm{d}x &= \frac{a}{1-\tau}\int_\tau^1 h(x)f^2(x)\,\mathrm{d}x &\le 2a\int_0^1 h(x)f^2(x)\,\mathrm{d}x.
\delta W_{\mu}^{a}= - i \overline{{{\eta}}}\gamma_{\mu}\Lambda^{a}, \delta \Lambda^{a}= - i{}^{\star}{W^{\lambda}}^{a}\gamma_{\lambda}\eta , \delta W^{a}= \overline{{{\eta}}}\chi_{\vec{W}}^{a}, \delta A = \overline{{{\eta}}}\chi_{A}, \delta B = \overline{{{\eta}}}\chi_{B},
\|T_\mathcal{H}\|=\sup_{\|f\|_{A^{p(m-1)}}=1}\|T_\mathcal{H}(f)\|_{A^p}\mbox{and}\|F_\mathcal{H}\|=\sup_{\|f\|_{A^p}=1}\|F_\mathcal{H}(f)\|_{A^p}.
\sum_{n=0}^\infty T(p^n)t^n=\frac{P_{p}(t)}{Q_{p}(t)},\P_{p}(t)=1-p^2 R_p t^2, &Q_{p}(t)=1-T(p)t+\{T(p)^2-T(p^2)-p^2 R_p\}t^2-p^3 R_p T(p)t^3+p^{6}R_{p^2}t^4.
\L_{(a,b)}(x,y) = -\frac{\sqrt{a b}}{4 \pi}\log \left( \frac{16 \gamma^2}{a+b}\right) - \frac{\sqrt{a b}}{4 \pi}\log\left(a x^2 + b y^2 \right) - \frac{\sqrt{a b}}{24 \pi}\frac{a b \left( b x^4 - (a + b ) x^2 y^2 + a y^4 \right)}{(b\;x^2 + a\;y^2)^3}+ O\left(\frac{1}{(x^2 + y^2)^2}\right)
\mathcal{L}=\frac{4}{9}F_{(\alpha \beta \gamma )\nu}\,F^{(\alpha \beta\gamma )\nu}+\frac{2}{3}F_{(\alpha \beta \gamma )\nu}\,F^{(\alpha \beta\nu )\gamma}-F_{(\alpha \beta \mu )}{}^{\mu}\,F^{(\alpha \beta \nu)}{}_{\nu}\ ,
2 l_{1}^{e}+ l_{2}^{e}= l_{1}^{e}+ l_{0}^{e}+ l_{3}^{e}+1 \geq2l_{0}^{e}+ l_{2}^{e}+2, & 2 l_{2}^{e}+ l_{1}^{e}= l_{2}^{e}+ l_{0}^{e}+ l_{3}^{e}+1 \geq2l_{0}^{e}+ l_{1}^{e}+2, & l_{1}^{e}+ l_{2}^{e}+2 l_{3}^{e}\geq2 l_{0}^{e}+ l_{1}^{e}+ l_{2}^{e}+2 ,
\sum_{0\leq k\leq n}\vert c_{k,r}\vert \frac{L^n (2n)!}{n^n n!}e^{\frac{n}{L}(1+\log L)}\Big(\frac{(n-k)!k!}{n^n}L^n\Big)^{-1}\leq \max_{0\leq k\leq n}\vert c_{k,r}\vert e^{\frac{n}{L}(1+\log L)}\binom{2n}{n}2^n.
0 = \frac{\nabla_E w(x_0) - \nabla_E w(y_0)}{\bar{w}}- \frac{m}{\bar{w}}(\nabla_E\bar{w}), 0 \ge \frac{\nabla^2_{E,E}w(x_0) - \nabla^2_{E,E}w(y_0)}{\bar{w}}- \frac{m}{\bar{w}}\nabla^2_{E,E}\bar{w}.
f = \left( 1 - \frac{\rho_{0}}{\rho}\right)^{\frac{1 - k^{2}}{1 + k^{2}}}\; , \; \; r^{2}= \rho^{2}\left( 1 - \frac{\rho_{0}}{\rho}\right)^{\frac{2 k^{2}}{1 + k^{2}}}\; , \; \; e^{\phi - \phi_{0}}= \left( 1 - \frac{\rho_{0}}{\rho}\right)^{\frac{2 l}{1 + k^{2}}}
\sum_{\alpha_0 + \alpha_1 + \alpha_2 = 3}\frac{t_0^{\alpha_0}}{\alpha_0!}\cdots \frac{t_2^{\alpha_2}}{\alpha_2!}\int_{\mathbb{P}^2}H^{\alpha_1 + 2 \alpha_2}= \frac{1}{2}\left(t_0 t_1^2 + t_0^2 t_2 \right)
c_X(x,\xi) = (h_0(x,\xi) + 1)^{-\frac{1}{2}}(K_0(X,\xi) + 1)^{-1}\in S^{-3}, d_X(x,\xi) = (h_0(x,\xi) + 1)^{-\frac{1}{2}}(K_0(X,\xi) + 1)^{-2}\in S^{-5}.
h(\varsigma):=\left(\begin{matrix}h_{z\bar{z}}h_{z\bar{w}}h_{w\bar{z}}=\bar{h}_{z\bar{w}}h_{w\bar{w}}\end{matrix}\right) =\left(\begin{matrix}\frac{\mu}{P}\mu \frac{\eta}{P}\mu\frac{\bar{\eta}}{P}\frac{2k}{P^2}+\mu\frac{|\eta|^2}{P}\end{matrix}\right).
\begin{aligned}\partial_t u -\nabla\cdot (A\nabla u)= \F, (0,T] \times \Omega, u(0,\cdot) = \u_0, \, \Omega, u= \g_D^{}, (0,T] \times \Gamma_D^{}, \mathbf{n}\cdot A\nabla u= \g_N^{}, (0,T] \times \Gamma_N^{},\end{aligned}
de^{rt}\log f_t(X_t, Y_t) = \left( (\partial_t + L) \log f_t(X_t, Y_t) + r \log f_t(X_t, Y_t) \right) e^{rt}dt + dM_t = \left( (\partial_t \log \rho^y_t(x))(X_t, Y_t) + L^X \log f_t(X_t, Y_t) + L^Y \log f_t(X_t, Y_t) + r \log f_t(X_t, Y_t) \right) e^{rt}dt + dM_t
\mathbb{H}=\{v_{0}+v_{1}\mathbf{i}+v_{2}\mathbf{j}+v_{3}\mathbf{k}| \mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=\mathbf{ijk}=-1,v_{0}, v_{1}, v_{2}, v_{3}\in \mathbb{R}\}.
\zeta (\mu+\alpha_0) =-q \zeta (\mu), \zeta (\mu+\alpha_i) = -q^{(\alpha_i -\alpha_{-i}, \mu+\alpha_i)}\zeta (\mu), \zeta (\mu+\alpha_{-i}) = -q^{(\alpha_{-i}, \mu+\alpha_{-i}) - (\alpha_{i}, \mu)}\zeta (\mu).
\int_1^\infty \tau^{2s}\Big| \ln T(i\tau/2) +\sum_{j = 1}^{[s]}\tilde T_{2j}(i\tau/2) \Big| d\tau \lesssim \left(\frac1{(s+\frac12)^2}+ \frac{1}{[s]+1-s}\right) \| (u,v) \|_{\dot H^s}^2\|(u,v)\|_{l^2 DU^2}^{2[s]}.
\delta_n^k+d-\alpha(k+1-d)= \delta_n^k+d-\alpha\left (\left (\frac{k+1}{d}-1 \right )d \right)=\delta_n^k+d-\left (\frac{k+1}{d}-1 \right )\alpha\left (d \right) =\delta_n^k+d-\left (\frac{k+1}{d}-1 \right )d \frac{\delta_n^k}{k+1}= d+ d \frac{\delta_n^k}{k+1}=d+ d \frac{n-k+1}{k+1}= d \frac{n+2}{k+1}.
g(t) = \prod_{j=1}^{\frac{n-1}{2}}(t - \lambda_{2j}) + (t - \lambda_{n+1}) \sum_{j=1}^{\frac{n-1}{2}}\frac{\sin^{2}\left(\frac{2j\pi}{n + 3}\right)}{\sin^{2}\left[\frac{(n+1)\pi}{n + 3}\right]}\prod_{\substack{m=1 m \neq j}}^{\frac{n-1}{2}}(t - \lambda_{2m}).
\frac{1}{u_1}\frac{\partial}{\partial u_1},\,\,\,\,\,\frac{1}{u_1^{n+1}}\frac{\partial}{\partial \xi_1},\,\,\,\,\,\frac{1}{u_1^k}\frac{\partial}{\partial \xi_1}\in C^1(\mathcal{U},\Theta_S), \,\,\,\,\,k=1,...,n-1,
\tilde{L}_{5,h_j}(X_{k}^*-)=O(1)(dL^{c}_{h_j,\theta}(\Lambda_{c,k-1}^{*})+dQ_{h_j,\theta}(\Lambda_{k-1}^{*})+(e^{-lX_{k-1}^*}-e^{-lX_{k}^*})\|Z_{0}\|_{\infty}),
\sum_{\gamma\in\mathcal{U}_{[n,-1]}}w(\gamma) =\sum_{j=1}^{p}\sum_{k\in\mathbb{Z}}a_{0}^{(j)}A^{(1)}_{[k-1,j-1]}W_{[n-k-1,0]}& +\sum_{j=2}^{p}\sum_{s=1}^{j-1}\sum_{\ell\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}a_{-s}^{(j)}\, A_{[\ell-1,j-s-1]}^{(1)}B_{[k-\ell-1,s-1]}^{(1)}W_{[n-k-1,0]}.
G^{( \psi )}( z_{1}, z_{2}) = \left( \begin{array}{l l}{{G_{+ +}( z_{1}, z_{2})}}&{{G_{+ -}( z_{1}, z_{2})}}\\{{G_{- +}( z_{1}, z_{2})}}&{{G_{--}( z_{1}, z_{2})}}\\ \end{array}\right)
1+\sum_{\beta>0, n \in \mathbb{Z}}P_{n, \beta}q^n t^{\beta}&=\exp \left( \sum_{\beta>0, n>0}(-1)^{n-1}n N_{n, \beta}q^n t^{\beta}\right) \cdot \left( \sum_{\beta>0, n \in \mathbb{Z}}L_{n, \beta}q^n t^{\beta}\right).
\psi(2^{-1}(xuy^*+yux^*))=\psi(2^{-1}xuy^*)\psi(2^{-1}yux^*)=\psi(2^{-1}xuy^*)\psi(2^{-1}(x(yu)^*)^*)=\psi(xuy^*)=\chi(x,yu).
{\mathbf{y}}_k^{[j]}= \sum\nolimits_{l = 1}^J{{\mathbf{R}}_k^{[j]}{\mathbf{H}}_k^{[j]}{\mathbf{F}}_k^{[l]}{\mathbf{s}}_k^{[l]}}+{\mathbf{R}}_k^{[j]}{\mathbf{w}}_k^{[j]}
\mathbb{E}\int_0^T \psi(t) \Big[ \Vert X (t\wedge\tau_u^M) \Vert_H^2 - \Vert x \Vert_H^2 \Big] dt = \mathbb{E}\int_0^T \psi(t) \int_0^{t\wedge\tau_{u}^M}\Big[ 2\langle \mathcal{A}(s), X(s)\rangle + \Vert \mathcal{B}(s) \Vert_{L_2}^2 \Big] ds dt .
\frac{F_{1,1}(x,y)}{(x^2+y^2+z^2)^{3}}= \frac{2}{3}\pi^2 \frac{x^4+y^4}{(x^2+y^2+z^2)^{3}}, & \frac{\widetilde{F}_{1,1}(x,y)}{(x^2+y^2+z^2)^{3}}= \frac{2}{3}\pi^2 \frac{(x^2+y^2)^2}{(x^2+y^2+z^2)^{3}},
|\tilde{A}|^2_{\tilde{g}}+\widetilde{Ric}(\tilde{\nu},\tilde{\nu}) = e^{\frac{2f}n}\biggl(|A|_g^2+Ric_f(\nu,\nu)+\frac{1}n\Delta f -\frac{n-1}{n^2}|\nabla f|^2\biggr).
\Omega_{a_{1}a_{2}\ldots a_{2 m + 1}}\psi_{+}^{a_{1}}\psi_{+}^{a_{2}}\ldots \psi_{+}^{a_{2 m + 1}}\; , \qquad \Omega_{a_{1}\ldots a_{2 m}a_{2 m + 1}}\psi_{+}^{a_{1}}\ldots \psi_{+}^{a_{2 m}}j_{+}^{a_{2 m + 1}}\, ,
s_{\min}(Hg_0^*) = |s_{d_0}(Hg_0^*) - s_{d_0}(H\hat{g}^*)| \leq |s_1(H\Delta g^*)| \leq |H\Delta g^*|_{S_2}\leq \frac{5\sqrt{2}}{3\sigma_u^2}d_0^{1/2}\lambda T \leq \xi,
f(x)=2^{2^{-}}\pi^{-2^{-}}M^{3^{1}2^{-}}R^{-3^{1}2^{-}}T^{-3^{1}2^{-}}x^{2}e^{-2^{-}\pi^{-}M^{1}R^{-}T^{-}x^{2}}