harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{M}\; \mu \; ( 2 ( P^{j k}{}_{, i}D_{\mu j}^{l}+ P^{m k}{}_{, i j}A_{\mu m}P^{j l}) - D_{\mu i}^{m}P^{k l}{}_{, m}+ ( \epsilon^{\rho \nu}D_{\rho}X^{i}) \epsilon_{\mu \nu}P^{k l}{}_{, j i}) C_{l}C_{k}= 0 .
\lambda ( p^{\mu}) \equiv \left( \begin{array}{c}{\left( \zeta_{\lambda}\Theta_{[ j ]}\right) \phi_{_L}^{\ast}( p^{\mu})}\\{\phi_{_L}( p^{\mu})}\\ \end{array}\right) , \quad \rho ( p^{\mu}) \equiv \left( \begin{array}{c}{\phi_{_R}( p^{\mu})}\\{\left( \zeta_{\rho}\Theta_{[ j ]}\right)^{\ast}\phi_{_R}^{\ast}( p^{\mu})}\\ \end{array}\right) \quad ,
Pr(|X-\mu|\ge k\sigma)=Pr((X-\mu)^{2}\ge k^{2}\sigma^{2})\le\frac{\mathbb{E}[(X-\mu)^{2}]}{k^{2}\sigma^{2}}=\frac{\sigma^{2}}{k^{2}\sigma^{2}}=\frac{1}{k^{2}}.
\kappa(\lambda):= \Phi_\lambda(u)|_{E^+_\lambda\cap \partial B_\rho(0)}\geq\left(\frac{1}{8}-\frac{1}{2^{2p+1}}\right)\left(\frac{\beta_\lambda^p}{2C(N,\alpha,p)}\right)^{\frac{1}{p-1}}>0.
J_{D S}( x ) = \left( \begin{array}{c c c c c c}{{0}}&{{k}}&{{0}}&{{0}}&{{\cdots}}&{{0}}\\{{0}}&{{0}}&{{k}}&{{0}}&{{}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\ddots}}&{{\ddots}}&{{\vdots}}\\{{\vdots}}&{{\vdots}}&{{}}&{{\ddots}}&{{k}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{}}&{{0}}&{{k}}\\{{u_{N}}}&{{u_{N - 1}}}&{{u_{N - 2}}}&{{\cdots}}&{{u_{2}}}&{{0}}\\ \end{array}\right) .
e_1 e'_1 = q^0 ([e'_3 e_7] + q^1 [e_8 e_9 e_{10}]), e_2 e'_2 = q^0 ([e'_4 e_8] + q^2 (e'_3)^2), e_5 e'_5 = q^0 ([e'_3 e_{13}] + q e'_4), e_6 e'_6 = q^0 ([e'_4 e_{14}] + q^2 [e_{10}e_{12}e_{13}^2]).
\int_{\Omega}|u|^{\alpha-2p+2}|\nabla u|^{p}\Delta_{p,f}(|\nabla u|^{p})\,d\mu = -p(\alpha-2p+2)\int_{\Omega}|u|^{\alpha-2p+1}|\nabla u|^{3p-3}\langle\nabla u,\nabla|\nabla u|\rangle\,d\mu &-p^2\int_{\Omega}|u|^{\alpha-2p+2}|\nabla u|^{3p-4}|\nabla|\nabla u||^2\,d\mu
2 \frac{f_0^{1,0}(1,\lambda_0\pi)}{f_0^{0,1}(1,\lambda_0\pi)}\ = 2 \frac{M^\prime(a_0,1,z) \cdot \beta/\pi}{\frac{\partial M}{\partial a}(a_0,1,z) \cdot (-1/2\beta)}= c,
(i+1 \to X_{i,i+1}\to i) \stackrel{\mu_i}{\mapsto}\begin{cases}(1 \to X_{1,2}), (X_{1,2}\to 1^-), (X_{1,2},n^+) i=1, (i \to X_{i,i+1}\to i-1) i=2,\ldots,n-2, \end{cases}
\begin{aligned}\|\widetilde{\gamma}_{n}\|_{X^2_\infty}=\||D^0|\widetilde{\gamma}_{n}|D^0|\|_{\mathfrak{S}_{1,\infty}}=\||D^0|p_{n}\widetilde{\gamma}_{n}p_{n}|D^0|\|_{\mathfrak{S}_{1,\infty}}&\leq \|\widetilde{\gamma}_{n}\|_{Y}\||D^0|p_{n}\|_{\mathfrak{S}_{2,\infty}}^2\leq \|p_{n}\|_{X^2_\infty}\leq (q+M)(1-\kappa)^{-2}e^2.\end{aligned}
\tilde{\mathcal Z}_Q = \mbox{span}\Bigl \{(2-N)x_j Q + |x|^2 \partial_{x_j}Q - 2 x_j x \cdot \nabla Q, \partial_{x_j}Q, 1 \leq j\leq N, x_j \partial_{x_k}Q - x_k \partial_{x_j}Q, 1 \leq j < k \leq N, \frac{N-2}{2}Q + x \cdot \nabla Q \Bigr \}.
\prod_{j=1}^{\lambda_i}(\lambda'_j+\mu_i-i-j+1+x) = \prod_{r=i}^{\ell(\lambda)}\frac{1}{m_i-l_r+x}\frac{\Gamma(m_i-l_{r+1}+x)}{\Gamma(m_i-l_r+x)}&= \frac{\Gamma(m_i+\ell(\lambda)+1+x)}{\Gamma(m_i-l_i+x)}\prod_{r=i}^{\ell(\lambda)}\frac{1}{m_i-l_r+x}\ ,
\mathbb{P}_{0\to \mu}^{0,1}\left(\frac{g_1}{1-g_1}\leq t\right) =\mathbb{P}_{0\to \mu}^{0,1}\left(g_1\leq\frac{t}{1+t}\right) &=1-e^{-\frac{1}{2}\mu^2 t}.
5^N \leq 5^{\left( \log_\frac{4}{3}\frac{t_0}{D}\right) + 1}= 5 \times 5^{\log_\frac{4}{3}\frac{t_0}{D}}= 5 \times \left( \frac{t_0}{D}\right)^{\log_\frac{4}{3}5}&\leq 5 \left( \frac{\Sigma(T)}{C - 8 \delta - 4}\right)^{\log_\frac{4}{3}5}
z(P_n) = \sum_{l=0}^{r-1}\Big((a_l-a_{l+1})(p^l + \dots + p^l) + \dots a_r p^r\Big) + a_r p^r &= \sum_{l = 0}^{r-1}\Big( (l+1)(a_l - a_{l+1})p^l \Big) + (r+1) a_r p^r &= a_0 - a_1 + \left(\sum_{l = 1}^{r-1}a_l(l + 1)p^l\right) - \left( \sum_{l = 1}^{r-1}a_{l+1}(l + 1)p^l \right) + (r + 1)a_r p^r = a_0 + \sum_{l = 1}^{r}a_l\Big((l + 1)p^l - lp^{l-1}\Big),
\tilde{y}_{1}={{P}_{1}^{- 1}}y_{1}+{q P_{2}^{- 1}}\tilde{y}_{2}+ \varphi , \ \ \tilde{y}_{2}={{Q}_{2}^{- 1}}y_{2}, \ \ \tilde{t}= Q_{2}t , \ \p ={P}_{1}{P}_{2}, \ \p^{\prime}={Q}_{1}{Q}_{2}- q^{2}{P_{1}P_{2}^{- 1}}.
E_{\bf \alpha, z}F(Z(\S^2))=\frac{\mu^{\frac{\sigma}{\gamma}}}{\Gamma({\frac{\sigma}{\gamma}})}\int_0^{\infty}F(y)y^{{\frac{\sigma}{\gamma}-1}}e^{-\mu y}\,dy,\ \ \ \sigma:=\sum_i \alpha_i- 2Q
M_{2k}= \sqrt{\frac{(N-\beta-k)(\gamma+1+k)(N+\delta+1-k)(k+\beta+1)}{(N-\beta-2k)(2k-N+1+\beta)}}, M_{2k+1}= \sqrt{\frac{(\gamma+N-\beta-k)(k+1)(N-k)(k+\beta+\delta+2)}{(N-\beta-2k-2)(2k-N+1+\beta)}},
\|\zeta_0 (-\Delta_x)^{1/6}u_{\varepsilon}- (-\Delta_x)^{1/6}(u_{\varepsilon}\zeta_0)\|_{L_p (Q_{1, R})}& \le N (d) R^{-1}\sum_{k = 0}^{\infty}2^{-k - 3d k/p}\|u_{\varepsilon}\|_{L_{p}(Q_{1, 2^k R})}.
H(LT_B')_\sigma +H(M)_\sigma &\geq H(LT_B'M)_\sigma &= H(C_1\widehat{A}^n \widehat{L}\widehat{T}_B' )_\omega &= H(A_1 R^n E' )_\omega &\geq H(A_1)_\tau +H(R^nE')_\theta +f(\epsilon) &\geq H(A^n)_\theta-H(M)_\sigma +H(R^nE')_\theta +f(\epsilon).
\mathcal{S}_{\alpha\beta}\bigg\|(\Phi_{R}u_{k})^{\frac{\alpha(\cdot)}{q(\cdot)}}(\Phi_{R}v_{k})^{\frac{\beta(\cdot)}{q(\cdot)}}\bigg\|_{L^{q(\cdot)}(\mathbb{R}^{N})}\leqslant \|( \Phi_{R}u_k,\Phi_{R}v_k)\|.
\sum_{j=2}^k \binom{2k}{k-j}\left\{e^{-t(1+2y)}\left(1+\frac{1}{y}\right)\right\}^j\left[1-4y\cos\beta e^{i\beta}\right]^{j-2}= e^{-2t(1+2y)}\left(1+\frac{1}{y}\right)^2 \, \sum_{j=0}^{k-2}\binom{2k}{k-2-j}&\left\{e^{-t(1+2y)}\left(1+\frac{1}{y}\right)\left[1-4y\cos\beta e^{i\beta}\right]\right\}^{j}
{\rm ch}W_{-n}(x,q^2)=\sum_{k+s\le n}q^{k(k-1)}x^{-n+k+2s}\left(\begin{array}{c}n k,s,n-k-s \end{array}\right)_{q^2},{\rm ch}W^\sigma_{-n}(x,q)=\sum_{k+s\le n}q^{k^2}x^{-n+k+2s}\left(\begin{array}{c}n k,s,n-k-s \end{array}\right)_{q^2}.
\phi(t,y) = \phi(t,y)^* \psi(t,y) = \psi(t,y)^* , \phi(t,y)^2 = \phi(t,y) \psi(t,y)^2=\psi(t,y) , \phi(t,y)+ \psi(t,y) =_{U}, \sigma(t,y) = \sigma(t,y) \circ \phi(t,y ) \sigma(t,y) \circ \psi(t,y) = 0_{U}, \phi(t,y)\circ \psi(t,y) = 0_{U}= \psi(t,y)\circ \phi(t,y) .
ds^{2}=\frac{r^{2}}{L^{2}}\Big{[}-\Big{(}1-(\frac{r_{0}}{r})^{4}\Big{)}dt^{2}+(d\overrightarrow{x})^{2}\Big{]}+\frac{L^{2}}{r^{2}}\frac{dr^{2}}{\Big{(}1-(\frac{r_{0}}{r})^{4}\Big{)}}+L^{2}d\Omega_{5}^{2}
\sum_{j=0}^{N(x,h)-1}\big| \frac{D^2 f(f^j(\theta))}{Df(f^j(\theta))}\frac{\alpha(f^{N(x,h)}(\theta))}{Df^{N(x,h)-j}(f^j(\theta))}\big| \leq \sum_{j=0}^{N(x,h)-1}\frac{C}{\lambda^{N(x,h)-j}}\leq \frac{C\lambda}{\lambda-1}.
\mathcal{Q}(u_1,\dots, u_l) =\mathcal F(u_1,.., u_l; t)|_{t=-1}=\Gamma^+(u_1)|_{t=-1}\dots \Gamma^+(u_l)|_{t=-1}(1) &=\prod_{1\le i<j\le l}i_{u_i, u_j}\left( \frac{u_i-u_j}{u_i+u_j}\right)\prod_{i=1}^{l}Q(u).
B_{n}(y)=\sum\limits_{k=0}^{n}b_{nk}y_{k}=\sum\limits_{k=0}^{n}\frac{1}{r}\left( -\frac{s}{r}\right)^{n-k}\frac{f_{n+1}^{2}}{f_{k}f_{k+1}}a_{n}y_{k}=a_{n}x_{n}.
\Phi_{(P,Q)}{\bf v}= \begin{pmatrix}-\left(\gamma(\cdot,P,Q)v(\cdot)\right)_s-\mu(\cdot,P,Q)v Vf(Q)-\int_0^m F(s,P,Q)v(s)\,d s\end{pmatrix}, (P,Q)\in\mathbb{R}^2_+,
\tilde{I}_{\mu_{1}\ldots \mu_{n}}\sim \int_{\frac{- \pi}{a}}^{\frac{\pi}{a}}\frac{d^{4}l}{( 2 \pi )^{4}}\sum_{r = - \infty}^{+ \infty}\sum_{j = 0}^{n - 1}\frac{A_{j}^{\mu_{1}\ldots \mu_{n}}( l , Q , M_{r})}{P^{2}( l + Q_{j}) + M^{2}r^{2}}, \quad Q_{j}= p_{1}+ \ldots + p_{j}.
(\hat{\mathsf{R}}_1, \hat{\mathsf{R}}_2) = \left(\hat{\mathsf{R}}_{\mathrm{FO},1}- \frac{\hat{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{n}}, \hat{\mathsf{R}}_{\mathrm{FO},2}- \frac{\hat{\mathsf{R}}_{\mathrm{SO},2}}{\sqrt{n}}\right),
F'(s) =-\int_0^{+\infty}\frac{\ln|1 - \tau^2|\ln\tau}{\tau^{1 + s}}d\tau &=\frac2s\int_0^{+\infty}\frac{\tau^{1-s}\ln\tau}{1-\tau^2}d\tau-\frac1sF(s) &\leq-\frac2sI-\frac1sF(s),
\begin{array}{r c l}{{\Delta_{( p_{1}, p_{2})}}}&{{=}}&{{e^{- \phi_{0}}\left( P^{( 1 )}/ P_{0}\right)^{2}+ \left( P^{( 2 )}/ P_{0}+ \chi_{0}P^{( 1 )}/ P_{0}\right)^{2}e^{\phi_{0}}}}\\{{}}&{{=}}&{{\left( P^{( 1 )}/ P_{0}, P^{( 2 )}/ P_{0}\right){\cal M}_{0}\left( \begin{array}{c}{{P^{( 1 )}/ P_{0}}}\\{{P^{( 2 )}/ P_{0}}}\\ \end{array}\right) ,}}\\ \end{array}
\int_{\mathcal{L}_{p}\left(0, \,A \exp \left(B\, \omega(a)\right)\right)}G(x,y)\,|f(y)|\,dy = \int_{\mathcal{L}_{p}\left(0, \,(2A)^{-1}\exp(-B\omega(m_{0}))\right)}G(x,y)\,|f(y)|\,dy &\quad+ \int_{\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m_{0})),\,A \exp \left(B\, \omega(a)\right)\right)}G(x,y)\,|f(y)|\,dy \,.
\sup_{0<|h|<h_0}\left\|\frac{\delta_{h}(u_R\,\eta_{i_0-1})}{|h|^{\gamma_{i_0}}}\right\|^p_{L^p(\mathbb{R}^N)}\le \frac{Ci_0^{2p}}{s(1-\gamma_{i_0})^p}\,\left[\left(\frac{C_3}{s^2}\,\frac{i_0^{4\,(N+p)}}{(1-\gamma_{i_0})^p}\right)^{i_0}\,(1-s)\,\mathcal{A}_1+h_0^{-\gamma_{i_0}\,p}\,\left\|u_R\right\|^p_{L^p(B_1)}\right].
i \hbar{\frac{\partial \Phi}{\partial t}}( 1 , \cdots , N ) ={\frac{1}{2 m}}\sum_{p = 1}^{N}\{i \hbar \partial_{i}^{( p )}-{\frac{e}{c}}\sum_{r = 1 ( \ne p )}^{N}{\cal A}_{i}( p , r ) \}^{2}\Phi ( 1 , \cdots , N ) \, .
O_{\beta\mu_1\ldots\mu_n}=O^{\mathrm{tw2}}_{\beta\mu_1\ldots\mu_n}+O^{\mathrm{tw3}}_{\beta\mu_1\ldots\mu_n}+O^{\mathrm{tw4}}_{\beta\mu_1\ldots\mu_n}+\ldots \;,
\langle \hat\rho_v(\xi_\mu),\beta\rangle =\frac{1}{2}\sum_{\substack{\alpha\in R_0 \langle \alpha^\vee,\beta \rangle >0}}\bigl( v_\alpha(\langle \xi_\mu, \alpha\rangle)-v_\alpha (\langle s_{\beta^\vee}\xi_\mu, \alpha\rangle) \bigr) \langle{\hat\alpha},\beta \rangle.
\begin{aligned}|\nabla F(f)(t)|_p\leq c\mathcal{B}^2\eta_t\max\left\{1,\frac{4}{\theta_i^2},\i=1,2,...,N\right\}\int_0^t(t-s)^{-\frac{1}{p}}s^{-\gamma}|f(s)|_p|\nabla f(s)|_pds.\end{aligned}
\dot{X}_{0}= F_{0}(X_{0},e,\tilde{\chi},\mu) \dot{X}_{i}= F_{i}(X_{i},\check{\xi}_{i+1},\tilde{\chi},\mu),~i=1,\cdots,r-1 \dot{X}_{r}= F_{r}(X_{r},\tilde{\vartheta}_{r},\tilde{\chi},\mu)
\begin{array}[c]{c}\bigtriangledown^{T}H=\frac{1}{u+A}\bigtriangledown^{T}|\bigtriangledown^{T}u|^{2}-\frac{|\bigtriangledown^{T}u|^{2}}{(u+A)^{2}}\bigtriangledown^{T}u+\bigtriangledown^{T}tr_{\omega}\widehat{\omega}_{\infty}.\end{array}
X_{10}:=\frac{17}{161280}\left(E_{4,\mathbb{H}}E_{6,\mathbb{H}}-E_{10,\mathbb{H}}\right), &X_{12}:=\frac{21421}{203212800}\left(\frac{441}{691}E_{4,\mathbb{H}}^3+\frac{250}{691}E_{6,\mathbb{H}}^2-E_{12,\mathbb{H}}\right).
B(v,e) =\begin{dcases}\displaystyle\frac{\prod_{u\in e}x_u}{\lambda^{(p)}(G) x_v^p}, ~v\in e, 0, ,\end{dcases}begin{align*}2mm]w(e) =\frac{r\prod_{u\in e}x_u}{\lambda^{(p)}(G)}.
\partial_{\mu}\langle{\mathcal J}_{\ell}^{\mu}( x ) \rangle_{A}\; = \; 2 \, q \,{\mathcal A}( x ) ~ , ~ ~ \partial_{\mu}\langle{\mathcal J}_{r}^{\mu}( x ) \rangle_{A}\; = \; - 2 \, q \,{\mathcal A}( x ) ~ .
\widetilde D_{b,m}=\overline\partial_b+\overline{\partial}^*_b+A_m:\Omega^{0,\bullet}_m(X,E)\rightarrow\Omega^{0,\bullet}_m(X,E), &\widetilde D^{\pm}_{b,m}=\overline\partial_b+\overline{\partial}^*_b+A_m:\Omega^{0,\pm}_m(X,E)\rightarrow\Omega^{0,\mp}_m(X,E)
{\bf V}_{\mathrm{v e c}}( \xi^{1}, \theta ) \equiv V_{\mathrm{v e c}}^{( - 1 )}( \xi^{1}) + \theta V_{\mathrm{v e c}}^{( 0 )}( \xi^{1}) \equiv \frac{i}{2}\sum_{\mu = 0}^{p^{\prime}}: \zeta_{\mu}( k ){\bf \dot{X}}^{\mu}( \xi^{1}, \theta ) \, \exp \left( i \sqrt{\frac{\alpha^{\prime}}{2}}\sum_{\rho = 0}^{p^{\prime}}k_{\rho}{\bf X}^{\rho}( \xi^{1}, \theta ) \right) : ~ ,
\mathcal{S}=\mathcal{S}_{cs}+\int d^3x\;\left( b^a\partial A^a+\partial^\mu \overline{c}^a(D_\mu c)^a+\;\hat{A}_\mu^{*}(D^\mu c)^a-\frac12\;f_{abc}C^{*a}c^bc^c\right) \;,
g_{\mu \nu}=\frac{1}{2\pi i}\oint_{|z|=1}\partial_{\mu}\big(\log{h(z;\boldsymbol{\xi})}+\log{\bar{h}(\bar{z};\bar{\boldsymbol{\xi}})}\big)\partial_{\nu}\big(\log{h(z;\boldsymbol{\xi})}+\log{\bar{h}(\bar{z};\bar{\boldsymbol{\xi}}})\big)\frac{dz}{z}
\mathcal{d}_\mathrm{FR}(C_0,C_1)^2 := \inf \left\{\int_0^1 \int_\Omega\int_{\mathbb{S}^1_+}|r_t(x)|^2 \,\d C_t(x,\theta) \,\d t; \;{(C,r)\in \mathcal{A}[C_0,C_1]}\right\},
m{d\Omega_{BH}\over dm}\sim{mt_{eq}^{1/2}\over m_p^2t_f^{3/2}}\sim 10^{27}\left({\lambda \over e}\right)^{1/2}\left({M \over m_p}\right)^{5/2}\left({M \over em}\right)^{n-1 \over 2}\;,
\left( \frac{m^{2}}{\Lambda^{2}}\right)^{( d - 3 ) / 2}\Gamma{\left( \frac{3 - d}{2}\right)}\sim{\frac{2}{( 3 - d )}}+ \gamma -{\ln}\left( \frac{m^{2}}{\Lambda^{2}}\right) + O ( d - 3 ) \; ,
\partial_\varphi(\tau_ra)(rec(u)) = \mathrm{Res}_{\omega_L}(\tau_r a d\log g_{u,t_0}{(\omega_{LT})}) = \mathrm{Res}_{t_L}(\tau_r a d\log g_{u,t_0}{(\omega_{LT})}) = \mathrm{Res}_{t_L}( a t_L^{-r}d\log g_{u,t_0}{(\omega_{LT})}) = \mathrm{Res}_{t_L}( a t_L^{-r}\sum_{n\geq 1}n\psi_{CW}^n(u) t_L^{n-1}dt_L) = ar\psi_{CW}^r(u) \ .
\frac{2s_1-d}{2}|\nabla_{s_{1}}u|_{2}^{2}+\frac{2s_2-d}{2}|\nabla_{s_{2}}u|_{2}^{2}-\frac{d}{2}\int_{\mathbb{R}^{d}}V(x)|u|^{2}dx-\int_{\mathbb{R}^{d}}W(x)u^{2}dx-\frac{d\lambda}{2}|u|_{2}^{2}+d\int_{\mathbb{R}^{d}}G(u)dx=0.
\mu (I) = \frac{\lambda(I_{n(I)}(a))}{\lambda(J \cap F_k)}\mu (J) \leq \frac{e^{- (h - 4 \varepsilon) n (I)}}{e^{- (h + 4 \varepsilon) (1 + \iota) n (I)}e^{(h - 5 \varepsilon) n (I)}}\mu (J) = e^{- (h - 13 \varepsilon - \iota (h + 4 \varepsilon)) n (I)}\mu (J) \leq e^{- (h - 13 \varepsilon - \iota (h + 4 \varepsilon)) n (I)}\leq |I|^s,
\sum_{r , s > 0}r s d \overline{D}_{r s}z^{r + s}\zeta^{- r - s - 2}+{\frac{d}{9}}{\frac{1}{\zeta^{2}}}= \sum_{r , s > 0}r s d{D_{r s}}^{\ast}z^{- r - s}\zeta^{r + s - 2}+{\frac{d}{9}}{\frac{1}{\zeta^{2}}},
\tilde{\nabla}_i\Phi =2(\alpha-1)\frac{z^{i}}{\rho^2}+O_3(\rho^{-2}), \Delta_{\tilde{g}}\Phi =\frac{4(\alpha-1)}{\rho^2}+ O_2(\rho^{-3}), \tilde{\nabla}_i\Delta_{\tilde{g}}\Phi =-\frac{8(\alpha-1)}{\rho^3}\frac{z^i}{\rho}+ O_2(\rho^{-4}).
\lim_{L_1\to0}\tfrac{1}{L_1}\left(E(L_1,\ell_\nu,\ell_\mu)+E(L_1,\ell_\nu,\ell_{\mu'})\right) &=\lim_{L_1\to0}\tfrac{1}{L_1}(R(L_1,2\ell_\mu,\ell_\nu)+R(L_1,2\ell_{\mu'},\ell_\nu)-L_1) &=1-\frac{\sinh\frac{\ell_\nu}{2}(2\cosh\frac{\ell_\nu}{2}+\cosh\ell_\mu+\cosh\ell_{\mu'})}{(\cosh\frac{\ell_\nu}{2}+\cosh\ell_\mu)(\cosh\frac{\ell_\nu}{2}+\cosh\ell_{\mu'})}.
H = H_{e}+ H_{p}= \frac{\alpha_{f}\hbar c}{r_{0}}\int_{0}^{\infty}{d \rho}\left[ \frac{\sin^{4}\alpha}{2 \rho^{2}}+ ( \partial_{\rho}\cos \alpha )^{2}+ \rho^{2}\cos^{2 m}\alpha \right] .
d_I^2 f(x, y, z)= \circlearrowleft_{x,y,z}([f(x, y)\alpha(z)]+f([xy], \alpha(z))+g(x, y, z)), d_{II}^2 g(x, y, z, u)= \circlearrowleft_{x,y,z}(\{f(x, y)\alpha(z)\alpha(u)\}+g([xy], \alpha(z), \alpha(u))).
p_i = \frac{e_i^T A^{\lfloor c \log n \rfloor p}1}{\lambda^{\lfloor c \log n \rfloor p}}+ O\left( n^{-1/2}\right) \ \u_i = \frac{e_\ast^T A^{\lfloor c \log n \rfloor p}e_i}{\lambda^{\lfloor c \log n \rfloor p}}+ O\left(n^{-1/2}\right)
V_{0}[ \zeta ] = \sum_{n ={\frac{1}{2}}}^{\Lambda}\sum_{m ={\frac{1}{2}}}^{\Lambda}\Bigl ({\frac{u_{m}}{v_{n}}}-{\frac{v_{n}}{u_{m}}}\Bigr )^{2}w_{m + n}^{4}+ \sum_{l = 1}^{\Lambda + \frac{1}{2}}\sum_{m ={\frac{1}{2}}}^{\Lambda}\Bigl [ \Bigl ({\frac{w_{l}}{v_{m}}}-{\frac{v_{m}}{w_{l}}}\Bigr )^{2}v_{m + l}^{4}+ \Bigl ({\frac{w_{l}}{u_{m}}}-{\frac{u_{m}}{w_{l}}}\Bigr )^{2}u_{m + l}^{4}\Bigr ]
D_{0}= \frac{1}{2}\left( \begin{array}{c c}{{\sigma_{3}}}&{{0}}\\{{0}}&{{\sigma_{3}}}\\ \end{array}\right) \; , D_{+}= \frac{1}{2}\left( \begin{array}{c c}{{0}}&{{\sigma_{+}}}\\{{\sigma_{+}}}&{{0}}\\ \end{array}\right) \; , D_{-}= \frac{1}{2}\left( \begin{array}{c c}{{0}}&{{\sigma_{-}}}\\{{\sigma_{-}}}&{{0}}\\ \end{array}\right) ,
\langle\phi,\varphi\rangle_{\widetilde{H}\Gamma_n}=\frac{1}{|\widetilde{H}\Gamma_n|}\sum_{\tilde{x}\in \widetilde{H}\Gamma_n}\phi(\tilde{x})\varphi(\tilde{x}^{-1}) =\sum_{\rho\in \mathcal{OP}_n(\Gamma_{*})}\frac{1}{2^{l(\rho)}Z_{\rho}}\phi(D_{\rho}^{+})\overline{\varphi(D_{\rho}^{+})},
\Delta_{\beta}^{r}f^{+}= \sum_{\ell_{1},\ell_{2},\ell_{3}=0}^{m-1,n-1,k-1}\frac{\ell_{2}!}{(\ell_{2}-r)!}\frac{\ell_{3}!}{(\ell_{3}-r)!}\overline{\alpha}^{\ell_{1}}\overline{\beta}^{\ell_{2}-r}\beta^{\ell_{3}-r}h_{\ell_{1},\ell_{2},\ell_{3}}(\alpha)
\int_{-\infty}^\infty t_+^\lambda t\delta(t - f(x))\varphi(x)\,dt = \tilde\varphi(x,\lambda+1), &\int_{-\infty}^\infty t_+^\lambda\partial_t (\delta(t - f(x))\varphi(x))\,dt= -\int_{-\infty}^\infty \partial_t(t_+^\lambda)\delta(t - f(x))\varphi(x)\,dt= -\lambda\tilde\varphi(x,\lambda-1).
\begin{pmatrix}\theta y z\end{pmatrix}:=G_{\delta}\begin{pmatrix}\psi \eta w\end{pmatrix}:=\begin{pmatrix}\theta_0(\psi) y_{\delta}(\psi)+[\partial_{\psi}\theta_0(\psi)]^{-T}\eta+[(\partial_{\theta}\tilde{z}_0)(\theta_0(\psi))]^T\partial_x^{-1}w z_0(\psi)+w\end{pmatrix}
T^{(k)}_V(e)\sim \sqrt{2}\bigg(\begin{smallmatrix}-\frac{1}{2}~~k\end{smallmatrix}\bigg)\int_0^{\theta^\star}\frac{d\theta}{(e+\theta^4)^{k+\frac{1}{2}}}= \sqrt{2}\bigg(\begin{smallmatrix}-\frac{1}{2}~~k\end{smallmatrix}\bigg)\frac{1}{e^{k+\frac{1}{4}}}\int_0^{\theta^\star/e^{1/4}}\frac{d\theta}{(1+\theta^4)^{k+\frac{1}{2}}}
\frac{1}{4z(s)}=\frac{1}{2\sigma^2+6h^2+\mathcal{O}(s\sigma^4)}&=\frac{1}{2\sigma^2}\big(1-\frac{3h^2}{\sigma^2}+\mathcal{O}(s\sigma^2)\big) &=\frac{1}{2\sigma^2}-\frac{3h^2}{2\sigma^4}+\mathcal{O}(s).
\lim_{n\rightarrow \infty}\mathbf{E}\left[ \prod_{l=1}^{p}\mathbf{S}_{l}^{\mathbf{Z}(m_{l},n)}\,\Big|\,\mathbf{Z}(n)\neq 0\right] =\bar{X}_{R_{i}}\Big(\mathbf{y}_{1,p};\mathbf{S}_{1,p}^{(i)}\Big).
(\phi\rtimes G)\bigl(c\,i_G^\alpha(d)\bigr) =(\phi\rtimes G)(c)i_B^\beta(d) (\phi\rtimes G)\bigl(c\,i_A(a)\bigr) =(\phi\rtimes G)(c)i_B\circ\phi(a) (\phi\rtimes G)\bigl(c\,i_A\circ\mu(f)\bigr) =(\phi\rtimes G)(c)i_B\circ\nu(f).
\frac{dA}{dt}\left( t\right) =-\int_{0}^{2\pi}\frac{\Phi\left( k\left(\theta,t\right) \right)}{k\left( \theta,t\right)}d\theta,\ \ \ \ \ \frac{dL}{dt}\left( t\right) =-\int_{0}^{2\pi}\Phi\left( k\left( \theta,t\right) \right) d\theta.
\begin{aligned}\hat\lambda_j^h = \frac{80\sin^2(\frac{t_j}{2})}{h^2}\frac{2 - 18\eta + (1+24\eta) \cos(t_j) -6 \eta \cos(2t_j)}{33 + 26\cos(t_j) + \cos(2t_j)}, \hat{U}_{j}= c_j \big(\sin((k-1/2)t_j)\big)_{k \in\{1,\ldots, N\}}\end{aligned}
k_{\beta}(\mbox{\bf 1}, e^{\varphi^iT_i})= \frac{M}{(2\pi\beta)^{\frac{D}{2}}}\sum_{\nu}\prod_{\alpha>0}\frac{<\varphi+2\pi\nu,\alpha>/2}{\sin <\varphi+2\pi\nu,\alpha>/2}e^{-\frac{1}{2\beta}<\varphi+2\pi\nu,\,\varphi+2\pi\nu>}
v_\Sigma^{(n)}- v_\Sigma^{(n+1)}= \begin{cases}\left(\begin{matrix}0 0 \dfrac{F^2}{w}(z) \phi^{-2n}(z) \big(1 - \phi^{- 2}(z)\big) 0\end{matrix}\right) , 0 .\end{cases}
M'_{r}=\sum_{\pi \in \mathcal{DP}_2(2 r)}\int_{[0,1] \times [-\lambda,1]^{r}}\prod_{s=1}^{r}\chi_{[0,\lambda]}\left(x_0- \sum_{\ell=i}^{2s-1}\epsilon_\pi(i) x_{\pi(i)}\right)\chi_{[0,1]}\left( x_0- \sum_{i=1}^{2s}\epsilon_\pi(i) x_{\pi(i)}\right) \prod_{l=0}^{r}\mathrm{~d}x_l,
P(\lambda,\xi;k,A) =\det\left(M_\xi(k,A)-\lambda\left(\widetilde\Pi_\xi\Pi_\xi\right)(k,A)\right) &=c_3(\xi;k,A)\lambda^3+ic_2(\xi;k,A)\lambda^2+c_1(\xi;k,A)\lambda+ic_0(\xi;k,A)
\mathbb{E}\Big[ \varphi(X_n^x) \psi( \log|G_n x| - n( q_t + l )) \Big] & = \frac{r_t(x)}{\nu_t(r_t)}\frac{\exp \left( -n \Lambda^*(q_t + l) \right)}{\sigma_t \sqrt{2\pi n}}\left[ \nu_t(\varphi) \int_{\mathbb{R}}e^{-ty}\psi(y) dy + o(1) \right].
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_1-2e_3+\delta e_5,[e_3,e_{n+1}]=-e_3, [e_4,e_{n+1}]=e_2,[e_{j},e_{n+1}]=\left(j-4\right)e_{j}, \displaystyle[e_{n+1},e_1]=-e_1+2e_3-\delta e_5, [e_{n+1},e_i]=\left(4-i\right)e_i, (\delta=\pm1,3\leq i\leq n,5\leq j\leq n,n\geq5).\end{array}
f_{-1}=q^{-1}+ 5 + 11q - 54q^4 + 55q^5 + 44q^6 - 395q^9 + 340q^{10}+O(q^{11}), f_{-4}=q^{-4}+ 15 - 216q + 4959q^4 + 22040q^5 - 90984q^6 + 409944q^9 + 1388520q^{10}+O(q^{11}), f_{-5}=\frac{1}{2}q^{-5}+ 15 + 275q + 27550q^4 + 43893q^5 + 255300q^6 + 4173825q^9 + 4807100q^{10}+O(q^{11}).
\int_{0}^{1}\left( tb+\left( 1-t\right) a\right)^{p}dt=\int_{0}^{1}\left(ta+\left( 1-t\right) b\right)^{p}dt=\frac{b^{p+1}-a^{p+1}}{\left(b-a\right) \left( p+1\right)}=L_{p}^{p}\left( a,b\right)
GF\varepsilon_{\mathcal{A}}=G(\varepsilon_{\mathcal{B}}F\circ\Phi)=G\varepsilon_{\mathcal{B}}F\circ G\Phi=(\varepsilon_{C}G\circ\Psi)F\circ G\Phi=\varepsilon_{\mathcal{C}}GF\circ\Omega,
\left|\frac{\Gamma(s+w+\frac{k-1}{2})}{\Gamma(s+\frac{k-1}{2})}(e^{-i\frac{\pi}{2}{\rm sgn}(t)})^w\right| &\leq \begin{cases}c_5\dfrac{(1+|t+v|)^{\sigma+u-\frac{1}{2}+\frac{k-1}{2}}}{|t|^{\sigma-\frac{1}{2}+\frac{k-1}{2}}}, s\in D_1, \s+w\in D_1\setminus D_2, \ |t|\geq c_3, c_6|t|^u, s\in D_1, \ |w|\leq c_4|t|^{1/2}. \end{cases}
Q(t_0) =\frac{1}{q_{2}+q_{3}}\frac{\pi}{L_h}|X_{3}-X_{1}|^2 (t_0)+\frac{1}{q_{1}+q_{2}}\frac{\pi}{L_h}(X_2-X_{3})\cdot(X_2-X_{0})\ (t_0) &+\frac{1}{q_{3}+q_{4}}\frac{\pi}{L_h}(X_2-X_{1})\cdot(X_{2}-X_{4})\ (t_0).
{\overline{{Q}}}_{a}= \Big ( \frac{{\mathrm{\ae}}R^{- 1 / 2}P}{\sqrt{6}}+ i \frac{\sqrt{k}\sqrt{6 R}}{{\mathrm{\ae}}}\Big ){\overline{{\lambda}}_{a}}- 3 i{\mathrm{\ae}}( 6 R )^{- 3 / 2}({\overline{{\lambda}}}{\overline{{\lambda}}}) \lambda_{a},
{Q_{\pm}^{(j)}}^B =\int d x^1 \sum_{a,b}\left[\left[f^B\right]^{ab}(\partial_0 \phi^{(j)a}\mp\partial_1 \phi^{(j)a}) \psi_{\pm}^{(j)b}\pm\frac{\partial W^{(j)1}}{\partial \phi^{(j)a}}\psi_{\mp}^{(j)b}\right],
p^{\beta_{m+1}}_{1}(m+1,t) =-\lambda\left(I_t^{\beta_{m+1}}p^{\beta_{m+1}}_{0}(m+1,t)-I_t^{\beta_{m}}p^{\beta_{m}}_{0}(m,t)\right)=0, p^{\beta_{m+1}}_{2}(m+1,t) =-\lambda\left(I_t^{\beta_{m+1}}p^{\beta_{m+1}}_{1}(m+1,t)-I_t^{\beta_{m}}p^{\beta_{m}}_{1}(m,t)\right)=0.
\left( \mu_{3}^{A},\bar{\Phi}_{B}^{(2)}\right)_{1}=\delta_{B}^{A},\;\left( \bar{\Phi}_{A}^{(3)},\mu_{2}^{B}\right)_{1}=\delta_{A}^{B},\;\left( \nu_{1}^{A},\tilde{\Phi}_{B}\right)_{1}=\delta_{B}^{A}.
\displaystyle\lim_{\varepsilon \to 0+}\left\langle (sI+A_\varepsilon)^{-1}B\widehat u_0(s),\eta\right\rangle= \int_0^\infty{\rm e}^{-s t}\langle \widetilde w_0(t),\eta\rangle \,{\rm d}t\qquad(s>0,\ \eta\in X).
\frac{\pi}{2}=(\frac{2}{1}\cdot\frac{2}{3})\cdot(\frac{4}{3}\cdot\frac{4}{5})\cdot(\frac{6}{5}\cdot\frac{6}{7})\cdot(\frac{8}{7}\cdot\frac{8}{9})\cdot\cdot\cdot\cdot=\prod_{n=1}^{\infty}(\frac{4n^{2}}{4n^{2}-1}).
\int_{\Omega}\,|f(x)|^2\,dx=\sum_{i=1}^{N}\, \int_{\mathbb{R}^d}\,|\hat F_i(\lambda)|^2\,d\lambda\hat{f}(\xi)= \epsilon^{d/2}\,\sum_{i=1}^{N}\,\hat{F_i}(\epsilon \xi)\,e^{-2\pi i x_i\cdot\xi}.
I_1 = \mathbb{E}[\int_0^T\int_D\{H(t,x)-\widehat{H}(t,x)-\widehat{p}(t,x)(A_uY(t,x)-A_{\hat{u}}\hat{Y}(t,x)+\widetilde{a}(t,x)) - \widehat{q}(t,x)\widetilde{b}(t,x) -\int_{\mathbb{R}}\hat{r}(t,x,\zeta)\tilde{c}(t,x,\zeta)\nu(d\zeta)\}dx dt].
=\frac{im^2}{\pi^2}\Bigl( g^{\mu \nu}- \frac{k^\mu k^\nu}{k^2}\Bigr)\frac{1}{k^2 \sqrt{1-4m^2/k^2}}\log \frac{\sqrt{1-4m^2/k^2}+1}{\sqrt{1-4m^2/k^2}-1}\quad ,\quad k^2>4m^2,k^0>0.
\alpha_1 \beta_2 = 0, \alpha_1 \beta_3 = 0, \alpha_4 \beta_1 = 0, \alpha_4 \beta_2 = 0, - \alpha_2 \beta_1 = \alpha_2 \beta_2 = - \alpha_3 \beta_2 = - \alpha_3 \beta_1 = \alpha_4 \beta_3 , \alpha_1 \beta_1 = - \alpha_2 \beta_3 = \alpha_3 \beta_2 = - \alpha_3 \beta_3 .
d^i_{n,k}=(-1)^n\frac{\sqrt{2n-2i+1}}{(-n)_i}\frac{(-n)_k(n-i+1)_k}{(1)_k(1)_k}&\times\prod_{m=0}^{\frac{i-1}{2}}(n+k+1-2m)\prod_{m=0}^{\frac{i-3}{2}}(n-k-1-2m),\ 0\le k\le n,\i=0,\ \i
\begin{cases}|b_{x}(t,x,v)|+|b_{v}(t,x,v)|+|\sigma_{x}(t,x,v)|+|\sigma_{v}(t,x,v)|\leq L, |\ell_x(t,x,v)|+|\ell_v(t,x,v)|\leq L(1+|x|+|v|), |g_x(x)|+|f_x(t,x)|\leq L (1+|x|), |b(t,0,0)|+|\sigma(t,0,0)|+|\ell(t,0,0)|+|g(0)|+|f(t,0)|\leq L,\end{cases}
E(z, \b;x)\ll \sum_{\substack{x/2 < n < 2x n \ne x}}\sigma_{a,\beta}(n)\min \bigg( 1, \frac{x}{T|x-n|}\bigg) + \frac{4^{c}+x^{c}}{T}\sum_{n=1}^{\infty}\frac{\sigma_{a,\beta}(n)}{n^{c}}.
\pi_{ra}=A_a(p-1) r^{p-1}&\quad+ A_a\left[\frac{1}{2}p^2(3p+1) A^2+pA\tilde{\Delta}A+\frac{(p-1)}{2}|\tilde{\nabla}A|^2\right] r^{3p-3}+o_1(r^{3p-3}).
\widetilde z''(\omega_\beta(E_-))=\frac{F''_{\mu_\alpha}(\omega_\beta(E_-))}{F'_{\mu_\beta}(\omega_\alpha(E_-))}\big(1-F'_{\mu_\beta}(\omega_\alpha(E_-))\big)-\frac{F''_{\mu_\beta}(\omega_\alpha(E_-))}{F'_{\mu_\beta}(\omega_\alpha(E_-))}\big(F'_{\mu_\alpha}(\omega_\beta(E_-))-1\big)^2\,.
p_{1}((x_{1},x_{2}),(y_{1},y_{2}),(n_{1},n_{2}))=x_{1}^{2}-x_{2}^{2}-y_{1}^{2}+y_{2}^{2}-n_{1}^{2}+n_{2}^{2}; & p_{2}((x_{1},x_{2}),(y_{1},y_{2}),(n_{1},n_{2}))=x_{1}x_{2}-y_{1}y_{2}-n_{1}n_{2}
\begin{aligned}\alpha_1=\frac{1}{6}\left(-\nu_1^2+\nu_1\nu_2-\nu_2^2\right),\qquad\beta_1= -\frac{1}{54}\left(\nu_1-2 \nu_2\right)\left(2 \nu_1-\nu_2\right)\left(\nu_1+\nu_2\right).\end{aligned}
r= \begin{cases}\lfloor \alpha_{m,t}(1)\rfloor , \ \ \{\alpha_{m,t}(1)\}< 1/2; \lfloor \alpha_{m,t}(1)\rfloor \ \ \lfloor \alpha_{m,t}(1)\rfloor +1, \ \ \{\alpha_{m,t}(1)\}= 1/2; \lfloor \alpha_{m,t}(1)\rfloor +1, \ \ \{\alpha_{m,t}(1)\}> 1/2.\end{cases}
\Sigma_{\{6\}}= y^4 - y^3 + y^2, \Sigma_{\{i, 6\}}= (y-1)(y^2 - y + 1) i = 2, 4, &\Sigma_{\{3, 6\}}= (y-1)^3 + y^2, \Sigma_{\{i, 3, 6\}}= y^2 - 2y + 2 i = 2, 4, &\Sigma_{\{2, 4, 6\}}= y(y-1), \Sigma_{\{2, 3, 4, 6\}}= y - 1.
0 =\frac{d}{ds}[(\Delta-\lambda_s)\left(s-1\right)E^P_{s,\varphi}]= (\Delta-\lambda_s)\frac{d}{ds}[\left(s-1\right)E^P_{s,\varphi}]- r(r-1)(2s-1)(s-1)E^P_{s,\varphi}&= (\Delta-\lambda_s)\frac{d}{ds}[(s-1)E^P_{s,\varphi}]-r(r-1)(2s-1)[R+(s-1)E_1^*+O(s-1)].