harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{(aq)_n (q)_n}+ \sum_{r=1}^n \frac{q^{r^2}(a^r q^r - a^{r-1}q^{-r})}{(aq)_{n+r}(q)_{n-r}}= -a^{-1}\frac{(1-a)}{(q)_n(a)_n}+ \frac{a^{-1}}{(aq)_{n}(q)_n}&= a^{-1}\left(\frac{-1}{(a)_n (aq)_{n-1}}+ \frac{1}{(aq)_n (q)_n}\right)= a^{-1}\frac{(-(1-aq^n)+1)}{(a)_n (aq)_{n}}= \frac{q^n}{(aq)_n (q)_n},
l_i(u^a) \nabla r_j(u^a) \big[\lambda_i(u^a)\mathbb{I}- f'(u^a)\big] u^a_x\leqslant O(1)\bigg(\delta \sum_{k=1 \atop k \neq p}^n |u^k_x|+|u^p_x| \chi_{\{i,j \neq p\}}+ e \bigg).
\int_{\mathcal{A}^l_m}\bigg|f_0^{'}\Big(U_{\mu^i_m,\xi^i_m}(x)\Big) \psi_{\mu^i_m,\xi^i_m}^h (x) u_m(x) \bigg|dx \leq C|\psi_{\mu^i_m,\xi^i_m}^h |_{\frac{2n}{n-2}}|u_m |_{\frac{2n}{n-2}}\Big(\int_{\mathcal{A}^l_m}U_{\mu^i_m,\xi^i_m}^{\frac{2n}{n-2}}dx\Big)^{\frac{2}{n}}= O\bigg(\Big(\frac{\epsilon}{|\ln\epsilon|^2}\Big)^{\frac{n}{n-2}}\bigg).
\Delta(\sigma_1(A)_{,ij}(p)x^ix^jr^2) = \Delta(\sigma_1(A)_{,ij}(p)x^ix^j)r^2+2\nabla_s(\sigma_1(A)_{,ij}(p)x^ix^j)\nabla_sr^2 +(\sigma_1(A)_{,ij}(p)x^ix^j)\Delta r^2 = 2\Delta\sigma_1(A)(p)r^2+8\sigma_1(A)_{,ij}(p)x^ix^j+2n\sigma_1(A)_{,ij}(p)x^ix^j+O(r^3) = -\frac{1}{6(n-1)}|W(p)|^2r^2+2(n+4)\sigma_1(A)_{,ij}(p)x^ix^j+O(r^3)
\frac{\partial^{2}\log S_{k}}{\partial \lambda_{i}\partial \lambda_{j}}h_{iip}h_{jjp}=\frac{k(k-1)\lambda_{i}^{k-2}}{S_{k}}h_{iip}^{2}-(\nabla_{p}\log S_{k})^{2}\geq \frac{k-1}{k}(\nabla_{p}\log S_{k})^{2}-(\nabla_{p}\log S_{k})^{2}=-\frac{1}{k}(\nabla_{p}\log S_{k})^{2}
\kappa_m \left( \sum_{i=1}^n c_i Y_\rho(t_i) \right) =\kappa_m \left( \sum_{i=1}^n c_i t_i(a_\rho X_1+b\rho X_2) \right)=\left(\sum_{i=1}^n c_i t_i\right)^m [\kappa_m(a_\rho X_1)+\kappa_m(b_\rho X_2)] &= \left(\sum_{i=1}^n c_i t_i\right)^m(a_\rho^m+b_\rho^m) \kappa(X_1)= 2^{m/2}(a_\rho^m+b_\rho^m)\left(\sum_{i=1}^n c_i t_i\right)^m \frac{(m-1)!}{2}.
B_{q_1, q_2}(p; x_1, x_2) - p(x_1, x_2) = \sum_{k = 0}^{q_1}\sum_{l = 0}^{q_2}\Bigg( p\left(\frac{k}{q_1},\frac{l}{q_2}\right) - c_{k,l}^{q_1, q_2}\Bigg) b_{k, l}^{q_1, q_2}(x_1, x_2).
F_K:=\frac{(\sin\Phi_1)^{n_1}\dots (\sin\Phi_K)^{n_K}(\sin\Phi_{K+1})^{n_{K+1}}}{\prod_{t=1}^T (l_t+\rho\sum_{j=1}^{K+1}b_j^t \sin\Phi_j)^{k_t}}e^{-i\rho\sum_q s_q\sin\Phi_q}.
\int_{0}^{\infty}t^{s - 1}e^{-{\frac{a}{t}}}e^{- \mu t}d t = 2 ({\frac{a}{\mu}})^{\frac{s}{2}}K_{s}( ( 4 a \mu )^{\frac{1}{2}}) = \Gamma ( s ) \mu^{- s}+ \Gamma ( - s ) ({a})^{s}+{\cal O}( \mu^{1 - s}, \mu )
-\Sigma=p!g^{p-1}\left(\frac{p-1}{p^2}\right)^{\frac{p-1}{2}}\frac{2^{\frac{p-1}{2}}}{\left(\frac{p-1}{2}\right)!}\prod_{i=1}^{\frac{p-1}{2}}\int\frac{d^Dk_i}{(2\pi)^D}p^{-\frac{1}{2}k_i^2}
\Phi_{\breve{\mathbf g}_p}(g) := \frac{\langle \tau_p(g)\breve{\mathbf g}_p, \breve{\mathbf g}_p\rangle}{||\breve{\mathbf g}_p||^2}, \Phi_{\mathbf h_p}(g) := \frac{\langle \tilde{\pi}_p(g)\mathbf h_p, \mathbf h_p\rangle}{||\mathbf h_p||^2}, \Phi_{\pmb{\phi}_p}(g) := \frac{\langle \omega_p(g)\pmb{\phi}_p, \pmb{\phi}_p\rangle}{||\pmb{\phi}_p||^2}
T_{2}\leq \mathrm{E}_{\theta_{0},\varepsilon^{2}}\bigg{[}1_{H(\theta_{0})}(X)\sum_{j=C}^{\infty}\sum_{k=1}^{N(j;c_{1})}\psi_{j,k}(X)\bigg{]}&\leq \sum_{j=C}^{\infty}\sum_{k=1}^{N(j;c_{1})}\exp\{-(j/8)(B/\varepsilon)^{2/(2\alpha_{0}+1)}\}&\leq \exp\{(2c_{1}+ 3 - C/8)(B/\varepsilon)^{2/(2\alpha_{0}+1)}\}.
\left(\begin{matrix}x=0 x=1 x=\infty \, \left( \frac12 \right) \begin{matrix}0 0 \theta^0 \end{matrix}\begin{matrix}0 0 \theta^1 \end{matrix}\overbrace{\begin{matrix}0 \sqrt{t_2}\theta^\infty_1/2 0 -\sqrt{t_2}\theta^\infty_1/2 -t_1 0 \theta^\infty_2 \end{matrix}}\end{matrix}\right),
{\cal P}_{x^{-}}e^{i \int_{- L}^{L}d y^{-}\big [ A_{-}( x^{+}, y^{-},{\vec{x}}_{\perp}) - i{\vec{\alpha}}_{\perp}\big ({\vec{\partial}}_{\perp}+ i{\vec{A}}_{\perp}( x^{+}, y^{-},{\vec{x}}_{\perp}) \big ) \big ]}= \pm 1 ,
\frac{dQ_n}{dP_n}(x_1x_2\dots)=\delta_{j,x_1}\frac{\lambda_{x_2}}{\lambda_{x_1}T_{x_1,x_2}^{(1)}}\prod_{k=1}^{n-2}\frac{T_{x_{k+1},x_{k+2}}^{(k)}}{T_{x_{k+1},x_{k+2}}^{(k+1)}}=:Z_n(x_1x_2\dots x_n)=:Z_n(x_1x_2\dots)
(\pi_{-}^{k})^{-1}(f_{\underline{i}}) = \frac{q^{- k (k - 1) / 2}}{[k]!}\mathbf{A}(f_{\underline{i}}), (\pi_{+}^{k})^{-1}(e_{\underline{j}}) = \frac{q^{- k (k - 1) / 2}}{[k]!}\mathbf{A}(e_{\underline{j}}).
X=\begin{pmatrix}\sqrt{a_1}0 0 -\sqrt{a_1}\end{pmatrix},Y=\sqrt{(1-a_1)}\begin{pmatrix}\frac{a}{2}\frac{1}{2}\sqrt{4-a^2}\frac{1}{2}\sqrt{4-a^2}-\frac{a}{2}\end{pmatrix},
u = x_0 \xleftarrow{\gamma_1^\lor}\cdots \xleftarrow{\gamma_{p-1}^\lor}x_{p-1}= \lfloor r_i x_{p}\rfloor\xleftarrow{z_{p+1}\gamma_{p+1}^\lor}\cdots \xleftarrow{z_r \gamma_r^\lor}\lfloor r_i x_r \rfloor =\lfloor r_i v \rfloor.
\begin{array}{l}R(\theta-\theta^{\prime})K^{+}_{1}(\theta^{\prime})^{t_{1}}M_{1}^{-1}R(-\theta -\theta^{\prime}-2\eta)K^{+}_{1}(\theta)^{t_{1}}M_{2}= K^{+}_{1}(\theta)^{t_{1}}M_{2}R(- \theta -\theta^{\prime}-2\eta)M_{1}^{-1}K_{1}^{+}(\theta^{\prime})^{t_{1}}R(- \theta-\theta^{\prime})\end{array}
E_x(x^n) = \sum_{k=1}^{p}{n \choose k}x^{n-k}(y^{p-1})^{(k-1)}&= \sum_{k=1}^{p-1}\frac{(x^n)^{(k)}}{k!}{{p-1}\choose{k-1}}(k-1)! y^{p-k}-{n \choose p}x^{n-p}= \sum_{k=1}^{p-1}\frac{(-1)^{k-1}}{k}(x^n)^{(k)}y^{p-k}-{n \choose p}x^{n-p}.
H= - \frac{1}{2}\sum_{j=1}^{L}\{( \sigma_{x}^{j}\sigma_{x}^{j+1}+ \sigma_{y}^{j}\sigma_{y}^{j+1})(1-U \tau_{z}^{j+1}) + ( \tau_{x}^{j}\tau_{x}^{j+1}+\tau_{y}^{j}\tau_{y}^{j+1}) (1- U \sigma_{z}^{j}) \}
{\rm div}(\widetilde{J}_{{\Bbb A}_{1}^{+}}^{*}\theta_{a,b}^{8})|_{U}=\sum_{\lambda\in L^{\lor}/\pm1,\,\lambda^{2}=-9/2,\,|\langle\lambda,d\rangle|=1,\,[\lambda]={\bf 1}_{L}}c_{a,b}(\lambda)\,H_{\lambda}.
\liminf_{n\to\infty}\frac{1}{a_c^{(n)}}\log P\left(\frac{n-A_n^*}{f_1(n)}\in O\right) \geq\liminf_{n\to\infty}\frac{1}{a_c^{(n)}}\log P\left(\frac{n-A_n^*}{f_1(n)}>\varepsilon\right) &=-J(x_0)=-\inf_{x\in O}I_1(x).
S = \frac{1}{2}\int d^{4}x \epsilon^{\mu \nu \alpha \beta}\epsilon_{a b c d}\bigg [ ({^+}\tau -{^-}\tau ) F_{\mu \nu}^{~ ~ a b}F_{\alpha \beta}^{~ ~ c d}+ ({^+}\tau +{^-}\tau ) F_{\mu \nu}^{~ ~ a b}\tilde{F}_{\alpha \beta}^{~ ~ c d}\bigg ] .
\boldsymbol{\Phi}_{\boldsymbol{\phi}}= \begin{bmatrix}\phi_0 \phi_{N_c-1}\cdots \phi_{2}\phi_{1}\phi_1 \phi_0 \cdots \phi_{3}\phi_{2}\vdots \ddots \vdots \phi_{N_c-2}\phi_{N_c-3}\cdots \phi_{0}\phi_{N_c-1}\phi_{N_c-1}\phi_{N_c-2}\cdots \phi_{1}\phi_{0}\end{bmatrix},
\gamma = 0 \cdot RLLLRR &= \varepsilon\left(\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\right) &= [0;2,1,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,2] &= \frac{19760}{51641}.
q =c\int_{\mathbb{R}^3}\int_0^\infty p \left(U^\nu p_\nu\right)F \left(1+\frac{\mathcal{I}}{mc^2}\right)\phi(\mathcal{I}) \,d\mathcal{I}\,\frac{dp}{p^0}\cr -\frac{1}{c}U \int_{\mathbb{R}^3}\int_0^\infty \left(U^\nu p_\nu\right)^2 F\left(1+\frac{\mathcal{I}}{mc^2}\right)\phi(\mathcal{I}) \,d\mathcal{I}\,\frac{dp}{p^0}.
\sup_{t \in [-T,T]}\| \big(e^{\frac{t}{k}A}Q\big)^k - e^{tQAQ}Q \| = \sup_{t \in [-T,T]}\|\big( e^{\frac{t}{k}\hat A}\hat Q\big)^k \hat x- e^{t\hat Q \hat A\hat Q}\hat Q \hat x\|,
\beta := \left(\alpha+ 1 + \frac{\mu(e_1)}{\mu(e_n)}\right) - 1- \mu(e_1) = \alpha + \frac{\mu(e_1)}{\mu(e_n)}- \mu(e_1) \gamma := \left(\alpha + \mu(e_1)-\frac{\mu(e_1)}{\mu(e_n)}\right) - 1 - \mu(e_1) = \mu(e_1)\left( 1- \sum_{i=1}^n \frac{1}{\mu(e_i)}\right).
\langle \Psi | \Psi^{\prime}\rangle = \int_{{\cal M}_{L}\times{\cal E}_{L}}{\cal D}^{e}e{\cal D}^{e}\widetilde{x}\left( \mathrm{V o l}_{e}{\cal D}_{L}\right)^{- 1}\overline{{\Psi}}\Psi^{\prime}.
\rho_{\iota_m}(z_1)\mu_2(z_2)\cdots\mu_n(z_n)\xi = W_{\iota_m}(1\otimes z_1)W_{\iota_m}^{-1}(\mu_2(z_2)\cdots\mu_n(z_n)\xi) &= W_{\iota_m}(1\otimes z_1)(\eta \otimes z_J) &= W_{\iota_m}(\eta \otimes z_1z_J),
A_n(s)= (S_n^{[s]})^TQ_n(s),s\in\{1,\dotsc,\gamma_r\}= \sum_{i=1}^{\ell_r^*}\left(\frac{1}{f_{g((s-1)\ell_r^*+i)}-\alpha_n}W_{\theta,i}^{[s]}\right)1_{\{i\in J_r^{[s]}\}}\quad+P_{\alpha_n}(\lfloor\frac{N}{2}\rfloor).
\varkappa(m,p_1,p_2):=\sum_{s=0}^{n}\sum_{\nu\in I_s}k^{\frac{sr}{6}(\frac{1}{p_1}-\frac{9}{4})}(\ln(k))^{sr}\prod_{\nu_j>k}[\nu_j^{\frac{r}{6}(\frac{1}{p_1}-\frac{9}{4})}\ln(\nu_j)^r ]\cdot |m(\nu)|^r<\infty,
D_{\mu \nu}^{R}( q ) = \delta_{\mu 0}\delta_{\nu 0}{\frac{1}{( \vec{q}^{2}+ M^{2})}}+ \delta_{\mu i}\delta_{\nu j}\left( \delta_{i j}-{\frac{q_{i}q_{j}}{( \vec{q}^{2}+ M^{2})}}\right){\frac{1}{q^{2}- M^{2}}},
\begin{aligned}|P(x,t)|=|v(x,t)|\leq Cx_n^{2+\alpha}, &|D P(x,t)|\leq \frac{C}{x_n}\left(\|v\|_{L^{\infty}(Q_{x_n}(p))}+x_n^2|\tilde{F}_{p}(0)|\right)\leq Cx_n^{1+\alpha}, &|D^2P(x,t)|\leq \frac{C}{x_n^2}\left(\|v\|_{L^{\infty}(Q_{x_n}(p))}+x_n^2|\tilde{F}_{p}(0)|\right)\leq Cx_n^{\alpha}. \end{aligned}
\sum\limits_{l\neq s}^{n}\sum\limits_{j\neq l}^n d(|j-l|) \frac{t_l-t_s}{t_l-t_j}= \sum\limits_{l\neq s}^{n}d(|s-l|) + \sum\limits_{l<j\neq s}^{n}\Bigl( d(j-l) \frac{t_l-t_s}{t_l-t_j}+ d(j-l) \frac{t_j-t_s}{t_j-t_l}\Bigr), = \sum\limits_{l<j}^{n}d(j-l).
|d_n| = \bigg| \frac{G(t_n)c_n}{A'(t_n) \mu_n^{1/2}}\bigg| = \bigg|\frac{a_n b_n t_n^2 \nu_n c_n}{\mu_n^{1/2}}\bigg| = |a_n t_n \nu_n^{1/2}c_n|.
S_{Q}^{( 1 )}= - \mu^{- \epsilon}\int d^{4}\theta d^{d}x \partial_{\alpha \dot{\alpha}}\sigma \partial^{\alpha \dot{\alpha}}\bar{\sigma}( \frac{1 8 \xi_{1}^{2}{( 4 \pi )}^{4}}{Q^{4}\epsilon}+ f i n ) \equiv S_{Q_{d i v}}^{( 1 )}+ S_{Q_{f i n}}^{( 1 )}
\begin{aligned}JA+A^TJ =\left(\begin{array}{c c}\lambda QS^{-1}-f'(\hat{u}) -cQ 0 -QS\end{array}\right)+\left(\begin{array}{c c}-\lambda S^{-1}Q+f'(\hat{u}) 0 cQ SQ\end{array}\right) & = -c\left(\begin{array}{c c}0 Q -Q 0\end{array}\right)=-cJ.\end{aligned}
\partial_t\phi= \dot{\Phi}\dot{F}^{ij}\phi_{ij}- \dot{\Phi}\dot{F}^{ij}\left(\phi_i\phi_j+2\phi_i(\chi^{-1}\chi_j-u_j)+2\chi^{-2}\chi_i\chi_j-2\chi^{-1}\chi_iu_j+(1-\coth u)u_iu_j\right) -2\dot{\Phi}\dot{F}^{ij}h_i^kh_{kj}+(1-\coth u)\dot{\Phi}\dot{F}^{ij}g_{ij}+\dot{\Phi}\dot{F}^{ij}h_{ij}v^{-1}+(1-\alpha){\Phi}{\chi}^{-1}\cosh u+vF^{-\alpha}.
\lambda_{\min}(\boldsymbol{\varphi}) := \min_{x \in \bar{G}}\min_{|\boldsymbol{\xi}| = 1}\boldsymbol{\xi}\cdot \big(\boldsymbol{\varphi}(\mathbf{x}) \boldsymbol{\xi}\big) \boldsymbol{\varphi}\in C^{0}\big(\bar{G}, \mathbb{R}^{3 \times 3}\big).
\prod_{i=1}^h \#Z({\bf s}^{\Lambda_{Q, j}}) = \prod_{i=1}^h \left\langle\prod_{\ell\in I_i}\tau^{(a_\ell,b_\ell)}_{d_\ell}\sigma_1^{k_1(i)}\sigma_2^{k_2(i)}\sigma_{12}\right\rangle^{\mathbf{s}^{\Gamma_{0,k_1(i),k_2(i),1,I_i}},o}
( \partial_{\sigma_{c}}^{2}- \partial_{\tau}^{2}) \delta x_{\parallel}^{( 2 )}= U_{\parallel \hspace *{. 5 m m}\parallel}^{( 2 )}\delta x_{\parallel}^{( 0 )}+ U_{\parallel \hspace *{. 5 m m}\parallel}^{( 1 )}\delta x_{\parallel}^{( 1 )}.
F_R(u;\sigma,\nu){=}\exp{\left(\!-\frac{\nu^2}{2\sigma^2}\!\right)}\sum_{k_1=0}^\infty\frac{\psi^{2k_1}}{4^{k_1}k_1!\beta^{k_1}}\,{-}\exp{\left(-\frac{u^2}{2\sigma^2}\right)}\sum_{k_1=0}^\infty\sum_{k=1}^{k_1+1}\frac{\psi^{2k_1}u^{2(k_1{+}1{-}k)}}{4^{k_1}k_1!(k_1{-}k{+}1)!\beta^{k-1}}.
\mathfrak{n}(A,m)=\begin{cases}p^k\cdot\frac{(p^{m-1}-1)(p^{m-2}-1)\cdots(p^{k+1}-1)}{(p^{m-k-1}-1)(p^{m-k-2}-1)\cdots(p-1)}m>k+1, p^k m=k+1, 0 m\leq k. \end{cases}
\max\{|S|:S\in \bigcap_{i=1}^{k+1}\mathcal{F}_i\}=\max\{|S|:S\in(\bigcap_{i=1}^{k}\mathcal{F}_i)\bigcap\mathcal{F}_{k+1}\}=\min\{\Bar{r}_k(X)+r_{k+1}(E\backslash X):X\subseteq E\}
\psi_{\mu m}=( \gamma_\kappa \partial^\kappa + M_{N}-i g_{\pi}\gamma^{5}\tau^n \pi_{n}-i g_{\rho}\gamma^{\kappa}\tau^n \rho_{\kappa n}-i g_{\eta}\gamma^{5}\eta-i g_{\omega}\gamma^{\kappa}\omega_{\kappa})^{-1}
\widetilde{Z}( R ( S ) , \langle \cdot \, , \cdot \rangle ) = \det ( \widetilde{T}^{2})^{- 1 / 4}\, \prod_{k = 1}^{N}\det ( \widetilde{T}_{k}^{*}\widetilde{T}_{k})^{\frac{1}{2}( - 1 )^{k - 1}}
0 \leq \sum\limits_{i = 1}^n{F^{ii}\widetilde{P}_{ii}}- \widetilde{P}_t \leq - \frac{2\beta}{{n}}F^{i_0 i_0}u_{i_0 i_0}+ (4\kappa_{max}+ \frac{2}{n}) \sum\limits_{u_{ii}> 0}{F^{ii}u_{ii}}+ [- (A + \frac{1}{2}M)c_6 + C_{23}](\sum\limits_{i = 1}^n{F^{ii}}+ 1).
G_2(k) = \sum_{h=1}^{\lfloor k/2\rfloor}\frac{1}{(1+k-h)^{2s}(1+h)^{2r}}\lesssim \frac{1}{k^{2r}}\sum_{h=1}^{\lfloor k/2\rfloor}\frac{1}{(1+k-h)^{2(s-r)}(1+h)^{2r}}\lesssim \frac{1}{k^{2r}}\sum_{h=1}^{\lfloor k/2\rfloor}\frac{1}{h^{2s}}
\Omega_{-10\ell}:=\{x\in\mathbb{R}^{2}:MM(\frac{\chi_{E_{1}}}{|E_{1}|})(x)>C2^{10\ell}\}\cup\{x\in\mathbb{R}^{2}:MM(\frac{\chi_{E_{2}}}{|E_{2}|})(x)>C2^{10\ell}\},
{\cal T}={\frac{f_{\pi}^{2}}{2}}{\cal R}_{0}\cdot{\cal R}_{0}-{\frac{1}{2 e^{2}}}\biggl [{\cal R}_{0}\cdot{\cal R}_{\mathrm{i}}{\cal R}_{0}\cdot{\cal R}_{\mathrm{i}}-{\cal R}_{0}\cdot{\cal R}_{0}{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{i}}\biggr ]
R^{(1)}_{m+1}(y) =(\tfrac{\beta}{2}-m+1)_{2m}\big[y-\tfrac{\beta}{2}(\tfrac{\beta}{2}+1)\big] \quad+\sum_{j=2}^{m+1}(\tfrac{\beta}{2}-m+j)_{2m+2-2j}\times \big[y-\tfrac{\beta}{2}(\tfrac{\beta}{2}+1) \big] R_{j-1}(y;m+1),
W_{1+s}^{x_0}(r,u):=\frac{1}{r^{n+1}}\int_{B_r(x_0)}|\nabla u_r|^2\,d\mu_a -\frac{1+s}{r^{n+1}}\int_{\partial B_r(x_0)}|u_r|^2\,|x_n|^a\,d\mathcal{H}^{n-1},
\eta^{\nu\alpha}\xi_{\sigma}\nabla_{\alpha}\xi^{\sigma}=- \eta^{\nu\alpha}\delta \xi_{\alpha}, \qquad \xi^{\sigma}\eta^{\nu\alpha}\eta_{\sigma}{^{\rho}}(\nabla_{\alpha}\xi_{\rho}+ \nabla_{\rho}\xi_{\alpha}) = - \xi_{\sigma}\delta\eta^{\nu\sigma},
f_{\Phi}\left(\textbf{Z}_{0}\right)=\frac{\exp\left(-\textrm{tr}\left(\textbf{Z}_{0}\textbf{P}_{0}^{-1}\textbf{Z}_{0}^{H}\right)\right)}{\pi^{N_t}\det\left(\textbf{P}_{0}\right)},
E_{x^{2}-y^{2},x^{2}-y^{2}}=\frac{3}{4}(l^{2}-m^{2})^{2}V_{dd\sigma}+[l^{2}+m^{2}-(l^{2}-m^{2})^{2}]V_{dd\pi}+[n^{2}+(l^{2}-m^{2})^{2}/4]V_{dd\delta}
\frac{d x_i}{dt}= u_i, \frac{du_i}{dt}= - \sum_{\substack{j=1 j\neq i}}^{N}\frac{1}{\sigma_N}\Phi' \left( \frac{|x_i(t) - x_j(t)|}{\sigma_N}\right) \frac{x_i(t) - x_j(t)}{|x_i(t) - x_j(t)|}, 1 \leq i \leq N.
{[}\, [ \, C_{km}\, , \, \sigma_{kl}\, ] \, , \, \sigma_{mn}\, ]~~ =~~- \, [ \, [ \, \sigma_{kn}\,,\, \sigma_{mn}\, ] \,,\, \sigma_{kl}\, ]~~ =~~[ \, [ \, C_{ln}\, , \, \sigma_{kl}\, ] \, , \, \sigma_{mn}\, ]~.~~~~~~
\begin{aligned}|m+j|^{-4}\le (|m|-|j|)^{-4}= |m|^{-4}(1-|j|/|m|)^{-4}&\le |m|^{-4}(1+{30}|j|/|m|) \le |m|^{-4}(1 +{30\cdot2^r}|m|^{r-1}).\end{aligned}
{\cal L}_{\rm{U(1)FP}}=i\overline{c}\raisebox{-0.5mm}{\hskip 0.6mm\rule{0.1mm}{3.1mm}\rule{3.1mm}{0.1mm}\rule{0.1mm}{3.2mm}\hskip-3.3mm\rule[3.1mm]{3.2mm}{0.1mm}\hskip 0.7mm}c-\kappa\widetilde{h}^{\mu\rho}i\partial_{\mu}\overline{c}\partial_{\rho}c+{\rm O}(\kappa^2).
\lim_{l\rightarrow\infty}\hat{\mathbb{E}}[\sum_{i=1}^{n}\xi_i{\varphi^{i,m}_{l}}\prod_{j=1}^{i-1}{\widetilde{\varphi}^{j,m}_l}] = \hat{\mathbb{E}}[\sum_{i=1}^{n}\xi_iI_{K^{i}_m}\prod_{j=1}^{i-1}I_{\widetilde{K}^{j}_m}] &\leq\hat{\mathbb{E}}[\sum_{i=1}^{n}\xi_iI_{K^{i}_m}] &\leq\hat{\mathbb{E}}[\sum_{i=1}^n\xi_iI_{A_i}].
{\varepsilon^*_{n-k+1,n}}\cdot \left(id - X_{\{k,\,k+1\}}\, t_{n,k+1}^*\right)=\left(id - X_{\{k,\,k+1,\,\dots ,\,n\}}\right) \cdot{\varepsilon^*_{n-k+1,n-1}}
1-\sum_{\nu =k+1}^{\infty}\dfrac{(1+\delta \nu- \delta ) \,\Gamma(\nu+\tau)\Gamma(\mu+1)}{\Gamma(\nu+\mu)\Gamma(\tau+1)}\; a_{\nu}\geqq \gamma (0 < \tau \leqq 1;\; 0 < \mu \leqq 1),
\begin{aligned}\displaystyle \sqrt{K}\left\| \sum_j (\Phi^j_I)^Tr^j\right\| > L\left\| x^*_U\right\| - \left\| x^*_U \right\| \left( \sum_j\delta^j_{K+1}(1+\epsilon_{3}L) \right) > L\left\| x^*_U\right\| \left( 1 - \delta^{max}_{K+1}(1+\epsilon_{3}L) \right).\end{aligned}
\mathcal{L}_{I_J}(s) \overset{(a)}{=}\mathbb{E}_{\Phi_J}\Big[\prod_{\mathbf{z}_i \in \Phi_J}\mathcal{L}_h(s P_J D_{\mathbf{z}_i, \mathbf{u}_0}^{-\alpha}) \Big] & = \exp\Big(-2 \pi \lambda_J \int_{0}^{\infty}\frac{y}{1 + \frac{y^{\alpha}}{s P_J}}dy\Big),
K_{S}: = \underset{n\to\infty}\lim \left\{\sum_{h=0}^{(n-1)}\left(\frac{1}{n}\sum_{t=1}^{n}E (e_{t,\beta'}e_{t+h,\beta'}^{2}) x_{nt}\right) + \sum_{h=1}^{(n-1)}\left(\frac{1}{n}\sum_{t=h+1}^{n}E (e_{t,\beta'}e_{t-h,\beta'}^{2}) x_{nt}\right)\right\}
{\widehat G}= \left( \begin{array}{c c}{G_{R R},}&{G_{R A}}\\{G_{A R},}&{G_{A A}}\\ \end{array}\right) = \left( \begin{array}{c c}{G^{\mathrm{r e t}},}&{0}\\{0 ,}&{G^{\mathrm{a d v}}}\\ \end{array}\right) ,
A_{k}( t , r , \theta , \varphi ) = \sum_{l = 1}^{\infty}\sum_{m = - l}^{l}\Bigr [ c_{l}( r ) \partial_{k}Y_{l m}( \theta , \varphi ) + T_{l}( r ) \varepsilon_{k p}\partial^{p}Y_{l m}( \theta , \varphi ) \Bigr ] e^{i \omega t}.
\frac{1}{l}\Biggl( \int^{t+l}_{t}\Bigl(\sup_{u\in K}\bigl\| f(s+\tau,u)-f(s,u)\bigr\|\Bigr)^{q}\, ds\Biggr)^{1/q}\leq \frac{c_{q}}{l}\Biggl[ \epsilon \Biggl(\int^{t+l}_{t}\bigl[ L_{f}^{q}(s+\tau)+L_{f}^{q}(s)\bigr] \, ds\Biggr)^{1/q}+\sum_{i=1}^{k}\Biggl(\int^{t+l}_{t}\bigl\| f(s+\tau,x_{i})-f(s,x_{i})\bigr\|^{q}\, ds \Biggr)^{1/q}\Biggr].
\|f^Z(t,\cdot)\|_{L^2}\le \Big(\sum_{i+j=0}^2\|\partial^i_x\partial^j_t u(t,\cdot)\|^2_{L^{\infty}}\Big) E^Z_1(t,u)\le \|u(t,\cdot)\|^2_{W^{t,3,\infty}}E^Z_1(t,u) \,,
\|p(t,v,g)-\frac{1}{V_F}p_0(t,g)\|_{L^1((0,V_F)\times\mathbb{R})}\leq \sum_{k\neq 0,k\in\mathbb{Z}}e^{-k^2(\frac{2\pi}{V_F})^2D(t)}= 2\sum_{k=1}^{+\infty}e^{-k(\frac{2\pi}{V_F})^2D(t)}&=2\frac{e^{-(\frac{2\pi}{V_F})^2D(t)}}{1-e^{-(\frac{2\pi}{V_F})^2D(t)}}.
\sum_{|k|\le N}\int_{\eta_{k}}^{\eta_{k+1}}\frac{v_1^{-p'}(x)}{\bigl[V_1^-(x)\bigr]^{p}}\biggl(\int_{a(x)}^{\eta_{k}}v_1^{-p'}(t)\bigl|G^{(\delta)}_{2,k}(t)\bigr|^{p'-1}\,dt\biggr)^p\,dx\lesssim\int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}\bigl|G_{2,k}^{(\delta)}\bigr|^{p'}=\bigl[\mathbf{G}^{(\delta)}_{2,N}(g)\bigr]^{p'-1},
\sum_n H(x_1+\omega_n,x_2+\omega_n) \simeq\left(\frac{z}{s+1}\frac{1}{x_1-x_2}+ \frac{1}{2}\right)\frac{\tanh(\mu-x_1)}{2\mu-x_1+x_2}+ X_{1,2}+ \hbox{p.p.c.}\,.
\displaystyle{S_{2}}(a,b)\, \equiv \,\sum_{k=0}^{\infty}{\frac{\Gamma \left( k+1-a \right) \Gamma \left( a+k \right)\left( \Psi \left( k+1-a \right)+\Psi(a+k)-2\Psi(k+1)\right)}{\left( a-b+k \right) \Gamma \left( k+1 \right)^{2}\mbox{}}}\,.
\lim_{n\to \infty}\mathbb{E}\int^T_0 \partial_y\log\gamma_{\mathbb{R}_+}(T-s,Y_{y_n}(s))ds = \mathbb{E}\int^T_0 \partial_y\log\gamma_{\mathbb{R}_+}(T-s,Y_{0}(s))ds<\infty.
\partial_\xi\Phi(\eta_1,\eta_2,\xi)=-\frac{1}{[1+(\xi-\eta_1-\eta_2)^2]^{3/2}}+\frac{1}{(1+\xi^2)^{3/2}}, \partial_{\eta_1}\Phi(\eta_1,\eta_2,\xi)=-\frac{1}{(1+\eta_1^2)^{3/2}}+\frac{1}{[1+(\xi-\eta_1-\eta_2)^2]^{3/2}}.
h_{out}(n)=\frac{\mathbb{E}[k_{out}]}{n-1}\tilde{u}_{out}^{*n}(n-2),n>1,\tilde{u}_{out}=\frac{k_{out}+1}{\mathbb{E}[k_{out}]}\sum_{k_{in}\ge0}u(k_{in},k_{out}+1),
\rho_{\phi}(\beta_v)_{A+ha}= r_{\phi}(\beta_v)_{A+ha}+\rho (\beta_v)_{A+ha}= \underset{\neq0}{\underbrace{\varepsilon (v) \cdot c_{2}(E)}}+ \underset{0}{\underbrace{\rho(\beta_v)_{A}}},
(-q;q^2)_\infty \sum_{k=1}^\infty \frac{q^k}{1+q^k}= \left(\sum_{n=0}^{\infty}q_o(n) q^n\right)\left(\sum_{n=1}^{\infty}(p_{o^o}(n)-p_{e^e}(n)) q^n\right) = \sum_{n=1}^{\infty}\left(\sum_{m=0}^{n-1}q_o(m)p_{o^o}(n-m)-q_o(m)p_{e^e}(n-m)\right)q^n,
\begin{array}{lll}X_l := [u^*_{l-1},u^*_{l-2},\dots,u^*_{l-2T+1}]^* \in \mathbb{R}^{(2T-1)r}\bar{X}_l := [u^*_{l-2T},u^*_{l-2T-1},\dots,u^*_0,0,\dots,0]^* \in \mathbb{R}^{\bar{N}r}W_l := [w^*_{l-1},w^*_{l-2},\dots,w^*_0,0,\dots,0]^* \in \mathbb{R}^{Nd_0},\end{array}
X_{[bra]^{\ast}}(\rho(P)) =\displaystyle \lim_{n \to \infty}X_{[bra]^{\ast}}(\rho(P_{n})) &=\displaystyle \lim_{n \to \infty}\rho(Y_{[bra]}(P_{n})) &=\rho(\displaystyle \lim_{n \to \infty}Y_{[bra]}(P_{n})) &=\rho(w)
d_{\Omega}(p,q) = d_{\Omega}( \varphi_{\delta}(0), \varphi_{\delta}(\zeta_0/\delta)) \leq d_{\Delta}(0, \zeta_0/\delta) & \leq (1+\epsilon)d_{\Delta}(0,\zeta_0) = (1+\epsilon)d_{\Omega_n}(p,q).
I_{3}\lesssim c_0\nu \sum_{p> Q}\l_p^{1+s}\|\tilde w_p\|_2\sum_{Q<q\leq p+2}\lambda_q^{1+s}\|w_q\|_2(L\l_{q-p})^{1+s+\sigma}&\lesssim c_0\nu \sum_{p> Q}\l_p^{2+2s}\|w_p\|_2^2+c_0\nu \sum_{p>Q}\left(\sum_{Q<q\leq p+2}\lambda_q^{1+s}\|w_q\|_2(L\l_{q-p})^{1+s+\sigma}\right)^2 &\lesssim c_0\nu \sum_{p> Q}\l_p^{2+2s}\|w_p\|_2^2,
\begin{array}{r l}{{\Lambda \sqrt{\sigma_{r , 3}}=}}&{{\omega_{r , 0}- i \gamma_{r , 0}= \Lambda \sqrt{\sigma_{r , 1}^{*}}\, ,}}\\{{\varrho_{r , 2}i^{- \frac{3}{2}}( \sigma_{r , 3})^{\frac{1}{4}}=}}&{{\chi_{r , 0}\, e^{- i \varphi_{r , 0}}= \left[ \varrho_{r , 0}i^{\frac{3}{2}}( \sigma_{r , 1})^{\frac{1}{4}}\right]^{*}\, .}}\\ \end{array}
P_{\mu \nu}^{--}= U_{\mu}^{-}( T_{\mu}^{-}U_{\nu}^{-}) ( T_{\mu}^{-}T_{\nu}^{-}U_{\mu}^{+}) ( T_{\nu}^{-}U_{\nu}^{+}) , P_{\mu \nu}^{+ +}= P_{\mu \nu}^{--}( + \leftrightarrow - ) , \mu \neq \nu
\frac{1}{n!}\sum_{\pi\in S_{2n}}\prod_{j=1}^n |m_{\pi(2j-1)\, \pi(2j)}|= \sum_{\pi \in \mathcal{A}_{2n}}\prod_{j=1}^n |m_{\pi(2j-1)\, \pi(2j)}| \leq C^n \sum_{\pi\in \mathcal{A}_{2n}}e^{-\mu \sum_{j=1}^{n}|x_{\pi(2j-1)}-x_{\pi(2j)}|}= C^n\sum_{\substack{B\subset [2n] |B|=n}}\sum_{\pi\in \mathcal{A}_B}e^{-\mu \sum_{j=1}^{n}|x_{b_j}-x_{\pi(2j)}|}.
\int^{\infty}_{0}|h'(r)|^{2}r^{Q+1-2\beta}dr =\int^{\infty}_{0}|(Q-2\beta)s^{\frac{Q-2\beta-1}{Q-2\beta}}h'(s)|^{2}s^{\frac{Q-2\beta+1}{Q-2\beta}}\frac{s^{\frac{1}{Q-2\beta}-1}ds}{Q-2\beta}&=(Q-2\beta)\int^{\infty}_{0}s^{2}|h'(s)|^{2}ds.
C \| AB-A_0 B_0 \|^2 &\leqq \sum_{j=1}^N \sum_{i=1}^{M-1}\left\{\sum_{k=1}^{H_0-1}(a_{ik}b_{kj}-a^0_{ik}b^0_{kj}) + c_i \right\}^2 + \sum_{j=1}^N \sum_{i=1}^{M-1}\left( \sum_{k=H_0}^{H-1}a_{ik}b_{kj}\right)^2.
\prod_{i=1}^p(1+x_i)^t =\prod_{i=1}^p\sum_{n}\binom{t}{n}x_i^n &=\sum_{\lambda_1,\ldots,\lambda_p}\binom{t}{\lambda_1}\ldots\binom{t}{\lambda_p}x_1^{\lambda_1}\ldots x_p^{\lambda_p}\,.
T(f)(x) =\int_{\mathbb H^n}K(e,y)|y|_h^{-Q/p}dy|x|_h^{-Q/p}\|T(f)(x)\|_{L_{|x|_h}^pL_\theta^{\bar{p}_2}(\mathbb H^n)}=\omega_Q^{\frac{1}{\bar{p}_2}-\frac{1}{\bar{p}_1}}\int_{\mathbb H^n}K(e,y)|y|_h^{-Q/p}dy\|f_j\|_{L_{|x|_h}^pL_\theta^{\bar{p}_2}(\mathbb H^n)}
H_{\mathrm{CV}}(s,i,p,q,l) \coloneqq (s+i)lw+\beta(1-\theta l)^2si\left(q-p\right) +i\phi(i)\left(\frac{w}{r}+\chi\right)-(\gamma+\phi(i))iq &= \beta \theta^2 si (q-p) l^2 + \left[(s+i)w - 2\beta \theta si (q-p)\right] l &+ \beta si (q-p) + i\phi(i)\left(\frac{w}{r}+\chi\right)-(\gamma+\phi(i))iq\, ,
J_\nu(z) \sim \Big ({2 \over \pi z}\Big )^{1/2}\bigg ( \cos \omega \sum_{k=0}^\infty (-1)^k{a_{2k}(\nu) \over z^{2k}}- \sin \omega \sum_{k=0}^\infty (-1)^k{a_{2k+1}(\nu) \over z^{2k+1}}\bigg ),
\overline{\theta}\Gamma^N\partial_{\nu}\theta=(\theta_R+\theta_L)^T\Gamma^0\Gamma^N\partial_{\nu}(\theta_R+\theta_L)=\theta_R^T\Gamma^0\Gamma^N\partial_{\nu}\theta_R+\theta_L^T\Gamma^0\Gamma^N\partial_{\nu}\theta_L
TF_1(\zeta_1)\dots F_k(\zeta_k)\exp \left(\int_\Gamma d\zeta \,F(\zeta) \right)\Psi :=\sum_{n=0}^\infty \frac{1}{n!}\int_{\Gamma^n}d\zeta_{k+1}\dots d\zeta_{k+n}\,T\left(F_1(\zeta_1)\dots F_k(\zeta_k)F(\zeta_{k+1})\dots F(\zeta_{k+n})\right)\Psi.
\sum_{k=0}^n \binom{p+k}{p}\binom{q+n-k}{q}\binom{k}{c}\binom{n-k}{d}k^2 = \binom{n+p+q+1}{n-c-d}\binom{p+c}{c}\binom{q+d}{d}\cdot \frac{f_3(n,p,q,c,d)}{(p+q+c+d+2)(p+q+c+d+3)}
\mathcal L_{s\partial}(\psi) \!\!=\!\! \lambda_s \circ (\partial \otimes 1) \circ \psi + \rho_s \circ (1 \otimes \partial) \circ \psi - \lambda_s \circ \psi \circ \partial^\bullet - \psi \circ h \circ d - \underset{=0}{\underbrace{\psi \circ d}}\circ h\,.
k^{2}\varepsilon^{\mu \nu}- k^{\mu}k_{\alpha}\varepsilon^{\alpha \nu}- k^{\nu}k_{\alpha}\varepsilon^{\alpha \mu}+ k^{\mu}k^{\nu}\varepsilon + \eta^{\mu \nu}( - k^{2}\varepsilon + k_{\alpha}k_{\beta}\varepsilon^{\alpha \beta}) = 0 .
{\cal W}_{F}^{( 2 ; 3 g )}= 2 S_{1 1 2}^{F}+ 2 S_{1 2 3}^{F}+ 2 S_{1 2 4}^{F}+ 2 S_{2 3 3}^{F}+ 2 S_{1 4 4}^{F}+ 2 S_{3 4 4}^{F}.
\frac{\partial G}{\partial z_1}(z_1,z_2,\bar{z}_1 z_2 + \bar{z}_1^2 )=\overline{F_1}(-z_2-\bar{z}_1)\frac{\partial F_2}{\partial z_1}(z_1,z_2,\bar{z}_1 z_2 + \bar{z}_1^2) .
\alpha_j=\alpha_j'\dfrac{d(a_j,c_j)}{d(a_j,b_j)},\alpha_{m+j}=\alpha_j'\dfrac{d(c_j,d_j)}{d(a_j,b_j)},\alpha_{2m+j}=\alpha_j'\dfrac{d(d_j,b_j)}{d(a_j,b_j)}.
\dot{p}_{\mu}\circ \delta q^{\mu}=\frac{1}{2}\dot{q}^{\mu}(\partial_{\alpha}g_{\mu\nu}\delta q^{\alpha})\dot{q}^{\nu}-\delta q^{\mu}\partial_{\mu}U_{q}+\frac{1}{2}[\delta q^{\mu},\dot{g}_{\mu}]
\frac{\pi^{2}}{6}(\frac{2m}{2m+1})(\frac{2m-1}{2m+1})<\frac{1}{1^{2}}+\frac{1}{2^{2}}+\cdot\cdot\cdot+\frac{1}{m^{2}}<\frac{\pi^{2}}{6}(\frac{2m}{2m+1})(\frac{2m+2}{2m+1}).
\sum_{j=0}^{d-1}h_{S+j}(z) =\sum_{j=0}^{d-1}\prod_{i\in S}g_{i+j}(z)(g_{i+j+1}(z)-1) &=\sum_{j=0}^{d-1}z^{|S|}\omega^{\sum_{i\in S}(i+j+1)}&=z^{|S|}\omega^{\sum_{i\in S}(i+1)}\underbrace{\sum_{j=0}^{d-1}\omega^{j|S|}}_{1\leq|S|\leq \lfloor d/2\rfloor}&=0.
(\sigma_{\mu}{}^{\rho})_{\alpha\beta}\,[\mbox{A{\footnotesize\#}.{\footnotesize\#}}]_{\nu\rho}{}^{\alpha\beta}= b_1^2 \sum_{k=0}^\infty \frac{f_{\mathrm{A\#.\#}}(k)}{(k!)^2}\,[\mbox{a{\footnotesize\#$'$}or b{\footnotesize\#$'$}}]_{\mu\nu}(k)