harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\zeta_s (t) = \frac{\gamma_{s,3}}{6 \gamma_{s,2}^{3/2}}+ \frac{\gamma_{s,4}\gamma_{s,2}- 3 \gamma_{s,3}^2}{24 \gamma_{s,2}^3}t + \frac{\gamma_{s,5}\gamma_{s,2}^2 - 10 \gamma_{s,4}\gamma_{s,3}\gamma_{s,2}+ 15 \gamma_{s,3}^3}{120 \gamma_{s,2}^{9/2}}t^2 + \ldots,
P|_{(0,0)}=\frac{1}{4}\begin{pmatrix}f_{1}f_{2}g_{1}g_{2}\end{pmatrix}\begin{pmatrix}f_{1}f_{2}g_{1}g_{2}\end{pmatrix}=\frac{1}{4}\begin{pmatrix}f_{1}^{2}+f_{2}g_{1}f_{1}f_{2}+f_{2}g_{2}f_{1}g_{1}+g_{1}g_{2}f_{2}g_{1}+g_{2}^{2}\end{pmatrix}.
{\cal H}_{I}( \phi_{r}, \Phi , \mu ) = -{\frac{1}{2}}\mu^{2}\phi_{r}^{2}-{\frac{m^{2}}{\beta^{2}}}\cos ( \beta ( \phi_{r}+ \Phi ) ) \exp \{{\frac{\beta^{2}}{2}}I_{( 1 )}({\cal M}^{2}) \}\; .
J_1 = \sum_{k=0}^\infty 2^{-k}\big(|P u - \overline{P u}|^2\big)^{1/2}_{Q_{2\nu r, (2^{k+1}/\delta^2) (2\nu r)}(z_0)}, J_2 = \sum_{k=0}^\infty 2^{-k}\big(|(a^{i j}- \overline{a}^{i j}) D_{v_i v_j}u|^2\big)^{1/2}_{Q_{2\nu r, (2^{k+1}/\delta^2) (2\nu r)}(z_0)}.
SM(f,g,h) = M(Sf,g,h) + M(f,Sg,h) + M(f,g,Sh) + \widetilde{M}(f,g,h) \widetilde{m}(\xi,\eta,\sigma) = -(\xi\partial_\xi +\eta \partial_\eta +\sigma\partial_\sigma) m(\xi,\eta,\sigma).
p(t_{1},t_{2},...,t_{N})=|\langle t_{1},t_{2},...,t_{N}|\psi_{out}\rangle|^{2}=\frac{|PermU_{S,T}|^{2}}{t_{1}!\cdot\cdot\cdot t_{N}!s_{1}!\cdot\cdot\cdot s_{N}!}.
\mu_m= \sum_{J_1,\ldots,J_{\ell}\in\mathcal I^m}\lambda_{J_1}^s\cdots \lambda_{J_{\ell}}^s (g_{J_{1}}\circ\cdots\circ g_{\ell})_* (\mu_m) = \sum_{I_i\in\mathcal S, \alpha_i\in A_{I_i}}\lambda_{I_1J_{\alpha_1}}^s\cdots \lambda_{I_{\ell}J_{\alpha_{\ell}}}^s (g_{J_{1}}\circ\cdots\circ g_{J_{\ell}})_{*}(\mu_m).
p \ =\ (4,\{9\}^{I},a_{m-I-1},\ldots,a_{I+1},\{9\}^{I+1})_{10}, r(p) \ =\ (\{9\}^{I+1},a_{I+1},\ldots,a_{m-I-1},\{9\}^{I},4)_{10}, p-2 \ =\ (4,\{9\}^{I},a_{m-I-1},\ldots,a_{I+1},\{9\}^{I},7)_{10},
[x_1,x_2,x_3]=-\epsilon(x_1,x_2)[x_2,x_1,x_3], \epsilon(x_3,x_1)[x_1,x_2,x_3]+ \epsilon(x_1,x_2)[x_2,x_3,x_1]+ \epsilon(x_2,x_3)[x_3,x_1,x_2]=0
\int_A \langle \nabla u(x,y), \nabla v(x,y)\rangle_{\rho,(x,y)}\mu_\omega(x,y)=\int_A\langle \nabla u(x,y), \nabla v(x,y)\rangle_{(x,y)}\, d(\mu\times\mathcal{L}^1)(x,y).
W\approx\prod_{i}\frac{(N_{i}+g_{i})^{N_{i}+g_{i}}}{N_{i}^{N_{i}}g_{i}^{g_{i}}}\approx\prod_{i}\frac{g_{i}^{N_{i}}(1+N_{i}/g_{i})^{g_{i}}}{N_{i}^{N_{i}}}
P_{\mu\lambda}Q^{\lambda\nu}+P_{\mu\lambda}^{\ast}\overset{\ast}{\left.Q^{\lambda\nu}\right.}=\frac{1}{4}\left( P_{\sigma\tau}Q^{\tau\sigma}+P_{\sigma\tau}^{\ast}\overset{\ast}{\left. Q^{\tau\sigma}\right.}\right)\delta_{\mu}^{\nu}& =\frac{1}{2}\left( \mathbf{E}\cdot\mathbf{D}-\mathbf{H}\cdot\mathbf{B}\right) \delta_{\mu}^{\nu}.
\begin{array}{lcr}\xi_a^{++}=i\zeta_a^+\overline{\zeta}^+_a=iu_{(\alpha}^+u_{\beta)}^+\zeta^{(\alpha}_a{\overline{\zeta}^{\beta)}_a}\zeta_a^{\pm}=u_{\alpha}^{\pm}\zeta^{\alpha}_a;\qquad\overline\zeta^{\pm}_a=u_{\alpha}^{\pm}\overline{\zeta}^{\alpha}_a,\end{array}
b_{1,j}(u,v)= \left|\{r\in u^{-1}([i-2])\,|\, r<j\}\right|-\left|\{r\in v^{-1}([i-2])\,|\, r<j\}\right|, begin{align*}5pt]b_{2,j}(u,v)= \left|\{r\in u^{-1}([i-1])\,|\, r<j\}\right|-\left|\{r\in v^{-1}([i-1])\,|\, r<j\}\right|, begin{align*}5pt]b_{3,j}(u,v)= \left|\{r\in u^{-1}([i])\,|\, r<j\}\right|-\left|\{r\in v^{-1}([i])\,|\, r<j\}\right|.
\Delta(\{a, b\})= \,\,\{\Delta(a), \Delta(b)\}= \,\,\{a, b\}\otimes 1+\{a, h\}\otimes b+ \{g, b\}\otimes a+gh\otimes \{a, b\}+\{u, \Delta(a)\}+\{a\otimes1+g\otimes a, v\}.
S = \int d^{d - 1}x \sqrt{| g |}\ \left[ R +{\textstyle \frac{1}{2}}\left( \partial \varphi \right)^{2}-{\textstyle \frac{1}{4}}e^{- a \varphi}F_{( 2 )}^{2}+{\textstyle \frac{1}{2}}\left({\cal D}\phi \right)^{2}-{\textstyle \frac{1}{2}}m^{2}e^{a \varphi}\right] \, ,
\tau_{\tilde{D}}^{\epsilon,\gamma}= \inf \Bigl\{t > 0 \, \bigl\vert \, \exp \Bigl\{\Bigl( A + \sum\nolimits_{i=1}^n B_i \gamma_i\Bigr)t\Bigr\}x_0 \notin \tilde{D}\cup \partial \tilde{D}\Bigr\}.
V_1(h_0,h_1,h_2,\dots)=(Ah_0,F_2^* D_P h_0 + F_1 h_1 , F_2^*h_1 + F_1 h_2 , F_2^*h_2 + F_1 h_3,\dots) & V_2(h_0,h_1,h_2,\dots)=(Bh_0 , F_1^* D_P h_0 + F_2 h_1 , F_1^*h_1 + F_2 h_2 , F_1^*h_2 + F_2 h_3,\dots) & V_3(h_0,h_1,h_2,\dots)=(Ph_0, D_P h_0,h_1,h_2,\dots).
P_{\lambda_{h,\beta}}=_{\mathfrak{S}_{d_{\beta}}}\left(w_{\beta,1}^{r_{\beta}(h,1)}\cdots w_{\beta,d_{\beta}}^{r_{\beta}(h,d_{\beta})}\prod_{1\leq i<j\leq d_{\beta}}\frac{w_{\beta,i}-v_{\beta}^{-2}w_{\beta,j}}{w_{\beta,i}-w_{\beta,j}}\right).
{\sf M}\left\{\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_1 i_2)q}\right)^{2n}\right\}\le C_{n,2}\left(\frac{(T-t)^2}{2}\left(\frac{1}{2}-\sum_{i=1}^q\frac{1}{4i^2-1}\right)\right)^n\ \to 0\ \ \ \hbox{if}\q\to\infty,
G(s) =(p-1)P_{q-1}^{n+1,p-q-1}+(n+p)P_q^{n+1,p-q-1}-\beta P_q^{n,p-q-1}-(p-1)P_q^{n+1,p-q-2}&-(n+p)sP_{q-1}^{p-q}-(n-\beta+1)P_q^{n,p-q-1}&=(p-1)(P_{q-1}^{n+1,p-q-1}-P_q^{n+1,p-q-2})+(n+p)P_q^{n+1,p-q-1}-(n+1)P_q^{n,p-q-1}-2s\big(P_q^{n,p-q-1}\big)',
\dim_{A}C_{a}=\inf :\Bigl\{\beta :\exists \k_{\beta},\n_{\beta}\ \ 2^{n}\leq \left( \frac{s_{k}}{s_{k+n}}\right)^{\beta}\forall k\geq k_{\beta},n\geq n_{\beta}\Bigr\}& =\limsup_{n}\left( \sup_{k}\frac{n\log 2}{\log (s_{k}/s_{k+n})}\right)
\mathcal{I}-d^{-}(p,H) = \mathcal{I}-d_{-}(p,H) = \mathcal{I}-\liminf x_n = \mathcal{I}-\liminf (1-y_n) = 1- \mathcal{I}-\limsup y_n = 1- \mathcal{I}-d^{-}(p,H^{c})
\left( \left( \frac{d}{du}\right)^{n}\frac{1}{(u+1) \sqrt{1-u^2}}\right) \, \Bigg|_{u=0}= \left(-\frac{1}{2}\right)^n (n+1)! \binom{n}{\left\lfloor \frac{n}{2}\right\rfloor}.
\mathrm{mod}_q(d\varphi)=-\lim_{\tau\to\infty}\sum_{[\{\alpha,\beta;\epsilon\}]\in\vec{\triangle}^{\eta^-_i}_{\xi^-_i}}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho_\tau)+\ell_{\beta}(\rho_\tau))}+(-1)^{\alpha\cdot\beta}\right)^{-1}.
R_k:CF(H^{(k)},J) \rightarrow CF(H^{(k)},J^{+\frac{1}{k}}) R_k\bigg(\sum_{z\in \widetilde{P}(H^{(k)})}a_z\cdot z\bigg)= \sum_{z\in \widetilde{P}(H^{(k)})}a_z\cdot R_k(z)
\frac{p}{p-q}=\begin{cases}\left[a_0+1\right]^- \left[a_0+2, [2]^{b_1-3}, a_1 + 2\right]^- \left[a_0+2, [2]^{b_1-3}, a_1+3, [2]^{b_2-3}, \dots, [2]^{b_k-3}, a_k + 2\right]^- .\end{cases}
\frac{\chi_{p}(E)}{4\pi}\int_{0}^{\infty}dE^{\prime}\nu_{p}(E^{\prime})\Sigma_{f}(r,E^{\prime},t)\phi(r,E^{\prime},t)+\sum_{i=1}^{N}\frac{\chi_{di}(E)}{4\pi}\lambda_{i}C_{i}(r,t)+
S_{\rm bulk}= \int d^6x \,\sqrt{-\det G}\left[\frac{1}{2\kappa_{orb}^2}{\cal R}(G) -\frac{1}{2}\partial_{\mu}\hat \varphi \partial^{\mu}\hat \varphi -\frac{1}{2}\partial_{\mu}\hat \eta_a \partial^{\mu}\hat \eta_a -\frac{1}{4}{\rm e}^{\kappa_{orb}\hat \varphi}\,\hat F^{\,2}\right]
N_{\exp_p({\hat v})}\pi^{-1}(C) = (Conv(T_{\exp_p({\hat v})}\pi^{-1}(C)))^\perp = (Conv((d\exp_p)_{\hat v}(T_{\hat v}V)))^\perp &= (d\exp_p)_{\hat v}(Conv(T_{\hat v}V))^\perp &= ((d\exp_p)_{\hat v}(W))^\perp.
J(f,g)^2 = \begin{cases}\frac{(2\pi)^{2-(c_1+c_2)}a^{c_1}b^{c_2}(1 - e^{-2t})^2}{\det \left( \begin{array}{cc}1+(1-e^{-2t}) c_1 (a-1) -e^{-t}begin{align*}1ex] -e^{-t}1+(1-e^{-2t}) c_2 (b-1) \end{array}\right)}+\infty \end{cases}
\Phi_{\mathrm{reg}}[\eta_n] =\frac{6\gamma}{\pi}\mspace{1.5mu}n^2\!\int_{\mathbb{R}^3}\mspace{-7.5mu}d\vec{p}\! \int_{\mathbb{R}^3}\mspace{-7.5mu}d\vec{q}\;\frac{\overline{\hat{f}(\vec{p})}\,\hat{f}(\vec{q})}{\lvert\vec{p}-\vec{q}\rvert^2}=n^2 \Phi_{\mathrm{reg}}[f].
_{q/2}\Bigl(\frac{ab}{1+a^2+b^2}\Bigr)\, =_{q/2}\Bigl(\frac{ab}{1+b}\Bigr)=_{q/2}\Bigl(a+\frac a{1+b}\Bigr)\cr =_{q/2}\Bigl(\frac{a^2}{(1+b)^2}\Bigr)=_{q/2}\Bigl(\frac b{1+b}\Bigr)=0.
\beta_j^{k,\lambda}(x) = \frac{\lambda^{-1 + \frac{2}{r_j}}}{\sqrt{k}}\sum_{\varepsilon_1, \varepsilon_2 = \pm 1}\varepsilon_1 \varepsilon_2 \rho(\lambda(x-x^\ast_\epsilon)) \sin{kx_1}
Q := \left(\begin{array}{cc}q_{1,1,1}q_{2,1,1}q_{1,2,1}q_{2,2,1}q_{1,1,2}q_{2,1,2}q_{1,2,2}q_{2,2,2}\end{array}\right), \tilde Q := \left(\begin{array}{cc}q_{2,2,1}q_{2,1,1}q_{1,2,1}q_{1,1,1}q_{2,2,2}q_{2,1,2}q_{1,2,2}q_{1,1,2}\end{array}\right).
\lim_{j \to \infty}\sum_{[\omega] \in \alpha^{n_0}}\inf_{t \geq t_j}\{-\mu_{tf}[\omega]\log(\mu_{tf}[\omega])\}= \sum_{[\omega] \in \alpha^{n_0}}\lim_{j \to \infty}\inf_{t \geq t_j}\{-\mu_{tf}[\omega]\log(\mu_{tf}[\omega])\}.
\mathcal E\bigl((\rho_t)_{t\in[0,T]}\bigr) = \frac 12\int_0^T\|v_t\|_{\chi(\rho_t)}^2\;\!\mathrm dt,\mathcal E^\star\bigl((\rho_t)_{t\in[0,T]}\bigr) = \frac 12\int_0^T\|w_t\|_{\chi(\rho_t)}^2\;\!\mathrm dt.
\frac{\varphi(z_n) - \varphi(\overline z_m)}{z_n-\overline z_m}= \frac{\varphi_n(z_n) - \varphi_n(\overline z_m)}{z_n-\overline z_m}+ \frac{\varphi_m(z_n) - \varphi_m(\overline z_m)}{z_n-\overline z_m}+ O(1)\quad\quad\eta\to 0+.
(f_{g_1}(g_i x),f_{g_2}(g_i x),\ldots,f_{g_n}(g_i x)) = F(g_ix)=\tau_2(g_i,F(x)) = (f_{g_i.g_1}( x),f_{g_i.g_2}(x),\ldots,f_{g_i.g_n}(x))
R \sqrt{\lambda}\left( \ln{\lambda}+ \sigma \right)^{\prime}\varepsilon_{u l}^{\pm}= i{\bf P}_{i}^{N K}\gamma^{i}\varepsilon_{l u}^{\mp}\pm{\frac{1}{2}}{\bf P}_{i j}^{R F}\gamma^{i j}\varepsilon_{l u}^{\pm}.
\bigg|\int_{[0,1]}u'h'\bigg| = \bigg|\int_{[0,1]}\big(u'h'+u h-u h\big)\bigg|=\big|\langle u,h\rangle_{H^1}-\langle u,h\rangle_{L^2}\big| \leq\big|\langle u,h\rangle_{H^1}\big|+\big|\langle u,h\rangle_{L^2}\big| \leq\|u\|_{H^1}\|h\|_{H^1}+\|u\|_{L^2}\|h\|_{L^2}\leq2\|u\|_{H^1}\|h\|_{H^1}\,,
H^{(n)}: Y^{2}= X^{3}+ A X^{2}+ \Bigl(B_{1}t^{n}+ B_{2}+ \frac{B_{3}}{t^{n}}\Bigr)X+\Bigl(C_{1}t^{n}+ \frac{C_{2}}{t^{n}}\Bigr)^{2}.
y_1=x_1, \, y_2=\frac{1}{\sqrt{\mu}}x_3, \, y_3=\frac{1}{\sqrt{\nu}}x_2; &[y_1,y_2]=-\frac{1}{\sqrt{\mu}}y_1, \,[y_2,y_3]=\frac{1}{\sqrt{\mu\nu}}y_1 + \frac{1}{\sqrt{\mu}}y_3, \,[y_3,y_1]=0.
\prod_{m=1}^{\infty}\frac{\left( 1 -t^m \right)^2}{\left(1 - qt^m \right)\left(1 - q^{-1}t^m \right)}= 1 + \left(q^{1/2}- q^{-1/2}\right) \sum_{d=1}^{\infty}\sum_{k=1}^{\infty}t^{d k}\left( q^{d-k/2}- q^{-d+k/2}\right).
V_{\eta,\kappa}= \{-k\!: k \in \mathbb{N}\cap[0,\kappa] \}\sqcup \{(i,j)\!:j\in\mathbb{N}_{1}, i\in \mathbb{N}\cap[1,\eta] \}, E_{\kappa}= \{(-k,-k+1)\!: k\in \mathbb{N}\cap[1,\kappa] \}\sqcup \{(0,(i,1))\!: i\in\mathbb{N}\cap [1,\eta]\}, E_{\eta,\kappa}= E_{\kappa}\sqcup \{((i,j),(i,j+1))\!: i\in\mathbb{N}\cap[1,\eta],j\in\mathbb{N}_{1}\},
(\partial^{2(p+1)}_y \tilde{u}_\epsilon)(t, x, 0)= \sum^p_{q=2}\sum^{q-1}_{l=0}\epsilon^l\sum_{(\alpha^l, \beta^l)\in \Lambda^l_q}C_{\alpha^l,\beta^l}&\qquad\times\, \prod\limits_{j=1}^q \partial_x^{\alpha^l_j}\partial_y^{\beta^l_j +1}\big( u^s(t, 0) + \tilde{u}_\epsilon(t, x, 0) \big),
\omega =\frac{2w_{0}}{\rho_{0}v_{0}}-\frac{1}{2}K_{0}\rho_{0}^{2}v_{0}h =\frac{1}{2}\rho_{1}^{2}+\frac{1}{8}\rho_{0}^{2}v_{0}^{2}+2\frac{w_{0}^{2}}{\rho_{0}^{4}v_{0}^{2}}-\frac{w_{0}K_{0}+u_{0}}{\rho_{0}}+\frac{1}{8}K_{0}^{2}\rho_{0}^{2}v_{0}^{2}
\mathbb{P}\left( \gamma_{\mathrm{u},ik}\geq \theta \right) \approx 1 -{2\beta \theta}= 1 - \frac{2 K \theta}{N}, \mathbb{P}\left( \gamma_{\mathrm{a},ik}\geq \theta \right) \approx 1 - \left({2 \beta \theta}\right)^2 = 1 - \frac{4 K^2 \theta^2}{N^2}.
\left\langle u,v\right\rangle_{\widehat{S}_p}=\frac{1}{p}\int\limits_S \sum_{j=0}^{p-1}e^{\frac{mj\pi i}{p}}u( \zeta )e^{\frac{-mj\pi i}{p}}\overline{v( \zeta )}\,d\sigma (\zeta)=\left\langle u,v\right\rangle_S.
\langle T_{--}( \sigma ) \rangle = - \frac{1}{1 2 \pi}\frac{( 1 + v )^{2}}{( 1 - v^{2})^{3 / 2}}\: \frac{d}{d t}\frac{v^{\prime}}{( 1 - v^{2})^{3 / 2}},
\bigl\| e^f \bigr\|= \bigl\| \bigl( e^f \bigr)' \bigr\|_{\infty}= \bigl\| e^f \cdot f' \bigr\|_{\infty}\leq \bigl\| e^f \bigr\|_{\infty}\| f'\|_{\infty}= \bigl\| e^f \bigr\|_{\infty}\|f\|,
h_k - h_{k+1}= \frac{1}{2}\|x_{k+1}-x_k \|^2 + \left\langle x_{k+1}-x^{*},y_k - x_{k+1}\right\rangle -\frac{k -1}{k + \alpha -1}\left\langle x_{k+1}-x^{*},x_k - x_{k-1}\right\rangle .
\lambda_{\underbrace{00 \ldots 0}_{\frac{N-1}{2}}\underbrace{11 \ldots 1}_{\frac{N-1}{2}}\underbrace{1}_{1}}=\lambda_{\underbrace{11 \ldots 1}_{\frac{N-1}{2}}\underbrace{00 \ldots 0}_{\frac{N-1}{2}}\underbrace{1}_{1}}=\frac{1}{2}.
|(f^n)'(z)| =\prod_{j=0}^{n-1}\left|f'(z_{j})\right|\leq C^n\prod_{j=0}^{n-1}|z_{j}|^{\rho-1}|z_{j}|= \frac{C^n|z_n|}{|z_0|}\prod_{j=0}^{n-1}|z_{j}|^{\rho}.
G_H(x,x')=\frac{\Delta^{1/2}(x,x')}{2(2\pi)^{n/2}}\frac1{\sigma^{n/2-1}(x,x')}\sum_{k=0}^{\frac{n-3}2}a_k(x,x')\sigma^k(x,x')\frac{\Gamma(n/2-k-1)}{2^k}\ ,
\lambda(t)\log_2 t +B_j\left(\sum_{p \in E_j \atop p \leq t}\frac{1}{p}- \rho_{E_j}(t;\tilde{g},T)\right) - \sum_{p \in E_j \atop p \leq t}\frac{|g(p)|}{p}&= \sum_{p \in E_j \atop p \leq t}(c_j-1)\sum_{p \in E_j \atop p \leq t}\frac{|g(p)|}{p}+ B_j\left(\sum_{p \in E_j \atop p \leq t}\frac{1}{p}- \rho_{E_j}(t;\tilde{g},T)\right)
\langle\alpha_1,\alpha_2,\alpha_3,\alpha_4 \rangle_{0,4,1}=\int_{[\overline{\mathcal{M}}_{0,4}(X,1)]}\mathrm{ev}_1^* \alpha_1\cup \mathrm{ev}_2^* \alpha_2\cup \mathrm{ev}_3^* \alpha_3\cup \mathrm{ev}_4^* \alpha_4.
c(\nu ,m,l)_{(j,l')(i,m')}:=a_{\nu ,i,j}\delta_{l'l}\delta_{m'm}=(A_{\nu}^\tau )_{j,i}(E_{lm}^{(k)})_{l',m'}=(A_{\nu}^\tau \otimes E_{lm}^{(k)})_{(j,l'),(i,m')}
\varphi^{(p,q)}\bigl(re^{i\theta}\bigr)=\frac{(-1)^{q+\max\{p,q\}}\max\{p,q\}}{q}(1-r^2)r^{|p-q|}P^{(1,|p-q|)}_{\min\{p,q\}-1}(2r^2-1)e^{i(q-p)\theta}.
P_n : = \{T_n (\xi \otimes e_n) \, : \, \xi \in \mathrm{dom}(D^n) \}= \{\xi \otimes e_n + \sum_{j=1}^n \frac{1}{j!}(iD)^j \xi \otimes e_{n-j}\, : \, \xi \in \mathrm{dom}(D^n) \,\},
Y_{n,m}:= \sum_{\substack{i \in [1,n], \ell \in [1,m]}}g_{n,\alpha}(i) G(S_i,\tilde{S}_\ell )\tilde{g}_{m,\alpha}(\ell) ,\underline{Y}_{n,m}= \overline{g}_{n,\alpha}\overline{g}_{m,\alpha}\sum_{\substack{i \in [1,n], \ell \in [1,m]}}G(S_i,\tilde{S}_\ell ) .
\bar{\lambda}_2 (n,D,K) -\bar{\lambda}_1 (n,D,K) = 3 \frac{\pi^2}{D^2}+ \frac{3(n-1)(n-3)}{32}\frac{D^2K^2}{\pi^2}+ \frac{(n-1)(n-3)}{480\pi^4}D^4K^3\left(15\pi^2-\frac{225}{2}\right) +O(K^4)
\begin{aligned}I =\sum_{x\in G}\Delta v_+(x,t)v_+(x,t)\,\eta^2(x)e^{\xi(x,t)}\mu(x) &\le \frac1{4}\sum_{x,y\in G}v_+^2(x,t)\left[1-e^{\xi(y,t)-\xi(x,t)}\right]^2\eta^2(x)e^{\xi(x,t)}\,\omega(x,y) &+ \sum_{x,y\in G}v_+^2(x,t)\left[\eta(y)-\eta(x)\right]^2e^{\xi(y,t)}\,\omega(x,y)\,.\end{aligned}
\| \nabla_{s}f(s)\|_{T_{s}S}^2 = \sum_{\rho,\mu}g_{\rho\mu}(s) \left(\sum_{k}g^{\rho k}(s)\partial_{k}f(s) \right)\left(\sum_{\ell}g^{\mu\ell}(s)\partial_{\ell}f(s) \right) =\sum_{\rho,\mu}g^{\rho\mu}(s) \partial_{\rho}f(s)\partial_{\mu}f(s) \mbox{with}\ (g^{\rho \mu})=g^{-1}.
b^{2}= \operatorname*{l i m}_{m \rightarrow 0}{\frac{1}{4 \pi}}( m +{\frac{6}{\rho}}\operatorname{l o g}m \rho -{\frac{6 \pi}{m \rho^{2}}}-{\frac{4 \pi}{m^{2}\rho^{3}}})
\frac{f ( - s \triangle ) - 1}{s \triangle}= \int_{0}^{1}\! \!{\mathrm{d}}\alpha_{1}\alpha_{1}( 1 - \alpha_{1}) \! \int_{0}^{1}\! \!{\mathrm{d}}\alpha_{2}{\mathrm e}^{\alpha_{1}( 1 - \alpha_{1}) \alpha_{2}s \triangle},
(g_{ij})_{_{\mu F_0}}=\frac{1}{2}\frac{\partial^2\big((\mu F_0)^2\big)}{\partial y^i y^j}=\frac{1}{2}\frac{\partial^2 (\mu^2 F_0^2)}{\partial y^i y^j}=\frac{1}{2}\mu^2 \frac{\partial^2 ( F_0^2)}{\partial y^i y^j}=\mu^2 (g_{ij})_{_{F_0}},
Na\sum_{k=1}^{k_-+ 1}\frac{(-1)^{k+1}(a^{-k}-{\lambda}^{k})}{k}N^{(\alpha-1)k}-1= Na\sum_{k=0}^{k_-}\frac{(-1)^{k}(a^{-(k+1)}-{\lambda}^{k+1})}{k+1}N^{(\alpha-1)(k+1)}-1\,.
e^{\pi i \theta_1 s}F(z + \pi i \theta_1 , s) = e^{\pi \theta_1 i s}\frac{e^{- (z + \pi i \theta_1 ) s}}{1 - e^{- z - \pi i \theta_1}}= \frac{e^{- z s}}{1 + (-1)^{\theta_1}e^{- z}}.
\eta <\left\{\frac{1}{10k}, \frac{1}{(2k+2)(16k^2)}\cdot \left( 16k^2-1 - \frac{4(16k^3-1)}{5k}\right)\right\}\epsilon < \min\left\{\eta, \frac{\eta}{2}, \frac{1}{8k^3}, \frac{\eta}{32k^3+2}\right\}\alpha < \min \left\{\eta, \frac{\epsilon^2}{22k}\right\}.
\begin{cases}\partial_tV-\mu\Delta V-\nabla\cdot H=\nabla^\perp\cdot\nabla\cdot \Delta^{-1}\big(-\nabla^\perp V\otimes\nabla^\perp V+\nabla^\perp H\otimes\nabla^\perp H \big), begin{align*}-4mm] \partial_t H -\nabla V=\nabla^\perp H\nabla V, \end{cases}
F(\mathbf{v}) = \begin{bmatrix}D 0 \ell\sin\Psi -\cos\Psi 0 D -\ell\cos\Psi -\sin\Psi \frac{-\ell\sin\Psi}{R^2}-\kappa\ell D + \frac{\ell\cos\Psi}{R}\frac{\sin\Psi}{R}- \kappa U D^0_{-1}0 0 0 D^0_{1}0 0 0 0 0 D^0_{-1}0 0 0 D^0_{1}0 \end{bmatrix}\mathbf{v}.
C=(\begin{matrix}0&1&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&-1\\ 0&0&1&0&0&0&0&-4\\ 0&0&0&1&0&0&0&-4\\ 0&0&0&0&1&0&0&2\\ 0&0&0&0&0&1&0&4\\ 0&0&0&0&0&0&1&0\end{matrix}).
\mathbb P [(x_1, \dots, x_n) \in B ] \le\int_{[-M, M]^{n-1}}\int_{\frac{1}{a_n}( -\eta -d- \sum_{i=1}^{n-1}a_i x_i)}^{\frac{1}{a_n}( \eta -d- \sum_{i=1}^{n-1}a_i x_i)}\chi_{[-M, M]}dx_n \cdots d x_1\le (2M)^{n-1}\frac{2\eta}{|a_n|}.
K^{a,i,\tau}_{k}(T) \leq C_\tau e^{(\lambda_{1}a+5\tau(1+\lambda_1))r(0,k)}, C^{a,i,\tau}_{k}(T) \leq C_\tau e^{(\max\{\lambda_{i},\lambda_1 a\}+3\tau(1+\lambda_1))r(0,k)}.
G_{j}( x , y ) = y_{j}^{2}-{\frac{1}{4}}a_{j}x_{j}^{2}- x_{j}y_{j}y_{0}+{\frac{1}{4}}x_{j}^{2}x_{0}-{\frac{1}{4}}\sum_{k = 1 \atop k \neq j}^{2 m}{\frac{\left( x_{j}y_{k}- x_{k}y_{j}\right)^{2}}{a_{j}- a_{k}}}.
\textnormal{(I)}:= \frac{4(p-2)}{p}\int u^{p/2}(a\nabla (\eta u^{p/2}),\nabla \eta)\;dx+\frac{4}{p}\int u^{p}(a\nabla \eta,\nabla \eta)\;dx, \textnormal{(II)}:= \int u^p (\nabla a,\nabla (\eta^2))\;dx+(p-1)\int u\eta^2 u^p\;dx.
\mathcal{H}^{\rm LB}(D) =\Big\{\big[W^{\rm LB}(D, C')\big] ~\Big|~ \C' \in{\rm Col}_{X^2}^{\rm LB}(D) \Big\}&=\Big\{\mu_3^{\ast}\big(\big[W^{\rm SB}(D, C)\big]\big) ~\Big|~ \C \in{\rm Col}_{(B,X)}^{\rm SB}(D) \Big\}&=\mu_3^{\ast}\big(\mathcal{H}^{\rm SB}(D)\big).
\psi_{n}^{(1)}= \sum_{i \neq n}\frac{\langle \psi_{i}^{(0)}, H^{(1)}\psi_{n}^{(0)}\rangle}{\lambda_{n}^{(0)}-\lambda_{i}^{(0)}}\psi_{i}^{(0)}- \frac{1}{2}\langle \psi_{n}^{(0)}, G^{(1)}\psi_{n}^{(0)}\rangle \psi_{n}^{(0)}
\begin{array}{cc}\mathring{K}_4^p[H_1,H_2]\geq \frac{1}{C_1}(e^{p-1}[H_1]+e^{p-1}[H_2]) \textup{in}\ [R_0,\infty), \Big|\mathring{K}_4^p[H_1,H_2]\Big|\lesssim e[H_1]+e[H_2] \textup{in}\ [R_0-M,R_0], \mathring{K}_4^p[H_1,H_2]=0 \textup{in}\ [2M,R_0-M].\end{array}
A_{7,0}^{\#}= -494z^7 + 20748z^6 - 240084z^5 + 1200420z^4 - 3047220z^3 + 4113747z^2 - 2814669z + 767637 B_{7,0}^{\#}= -85z^7 + 7140z^6 - 115668z^5 + 722925z^4 - 2168775z^3 + 3346110z^2 - 2558790 z + 767637 K_{7,0}= -614125z^7 + 34819841z^6 - 471884616z^5 + 2626058475z^4 - 7227728820 z^3 + 10412142975 z^2 - 7521051447 z + 2148871842
\mathcal{E}(k\overline{D}_\psi+\overline{A})-\mathcal{E}(k\overline{D}'+\overline{A}) =\sum_{i=0}^n\int_X(k(\psi_D-\phi_{0,D}))dd^c(k\psi_D+\phi_A)^i dd^c(k\phi_{0,D}+\phi_A)^{n-i}&=k^{n+1}(\mathcal{E}(\overline{D}_\psi)-\mathcal{E}(\overline{D}'))+\int_X(\psi_D-\phi_{0,D})T,
\begin{aligned}R_n: H\tilde{C}_n (q) \longrightarrow H\tilde{C}_{n+1}(q) e_{\sigma_i}\longmapsto g_{\sigma_i}~~~ ~~~ ~~~ 0\leq i\leq n-1 e_{t_n}\longmapsto g_{\sigma_n}g_{t_{n+1}}g_{\sigma_n}^{-1}.\end{aligned}
L\left( A\right) =\int_{A}\gamma \left( s\right) c\left( \mathrm{d}s\right)+W\left( A\right) +\int_{A}\int_{\left\vert x\right\vert >1}xN\left(dxds\right) +\int_{A}\int_{\left\vert x\right\vert \leq 1}x\widetilde{N}\left( dxds\right) A\in \mathcal{B}_{b}\left( \mathbb{R}^{k}\right) ,
A_n := \frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}, begin{align*}1ex] C_n := \frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}.
\pi =\biggl(x_{m+v-1}\lambda, x_{m+v-2}\lambda, \ldots, x_{m+v-u}\lambda;\biggr. \biggl.\sigma_0, \sigma_1, \ldots, \sigma_{v-n-1}, \dfrac{q_{m+n}}{p_{m+n}}, \dfrac{q_{m+n-1}}{p_{m+n-1}}, \ldots, \dfrac{q_{m-n+1}}{p_{m-n+1}}, \sigma_{v+n}, \ldots, \sigma_u\biggr).
I_{eff}=\frac{1}{\pi}\int d^2z [ \frac{k}{2}\ \ \frac{\partial w\overline{\partial}w}{w^2}+ \frac{w^2}{w^2 - 4 \lambda}( \partial f \overline{\partial}f - c^2\partial y \overline{\partial}y)+ \frac{cw^2}{w^2 -4 \lambda}(\partial f \overline{\partial}y -\partial y \overline{\partial}f )]
t_k\left(1+ \int_\Omega q_0\psi_k^2\,dx-\lambda_1(q_0)\int_\Omega \psi_k^2\,dx\right) = &-\int_\Omega(q_0-q_k)\phi_1(q_k)\psi_k\,dx+(\lambda_1(q_0)-\lambda_1(q_k))\int_\Omega \phi_1(q_k)\psi_k\,dx,\,\, k=1,2,....
S = \left( \begin{array}{c c c c c c}{0}&{1}&{0}&{0}&{0}&{\rightarrow}\\{0}&{0}&{1}&{0}&{0}&{}\\{0}&{0}&{0}&{1}&{0}&{}\\{0}&{0}&{0}&{0}&{1}&{}\\{0}&{0}&{0}&{0}&{0}&{}\\{\downarrow}&{}&{}&{}&{}&{\ddots}\\ \end{array}\right)
E(v;k,R) = \int_{B_{R}(0)}|\nabla v|^{2}+ \int_{B_{R}(0)}v^{10/3}+ \int_{B_{R}(0)}\left( u_{k}^{2}\cdot \chi_{B_{R}(0)^{\rm c}}* Y_{a_{k}}\right) v^{2}& + \frac{1}{2}D_{a_{k}}\left( m_{k}- v^{2}\chi_{B_{R}(0)}, m_{k}- v^{2}\chi_{B_{R}(0)}\right) - \frac{1}{2}D_{a_{k}}\left( m_{k}, m_{k}\right).
\lim_{t\to\infty}p_{\ell,\delta}(t,y,x)=\frac{2\delta}{\ell}\frac{e^{-2\delta x/\ell}}{1-e^{-2\delta}},\lim_{t\to\infty}p_{\ell,0}(t,y,x)=\frac{1}{\ell},0\le x\le \ell.
S(\omega_{{\sf f}_1\cdots{\sf f}_m}\| \omega) = i \left. \frac{d}{dt}\right|_{t = 0}(\Omega_\omega,F_1 \cdots F_m \Delta^{it}F_m \cdots F_1 \Omega_\omega) = i \left. \frac{d}{dt}\right|_{t = 0}(\Omega_\omega,F_1 \cdots F_m F_m(t)\cdots F_1(t) \Omega_\omega) \,.
u_i^{\mathrm{a}}(\hat{\boldsymbol{\tau}}_i)= \sum_{k=0}^\infty{\delta^k}\pi_{a,i}( \hat{\boldsymbol{\tau}}_i;\boldsymbol{d}_i^*\!)= \frac{\pi_{a,i}( \hat{\boldsymbol{\tau}}_i;\boldsymbol{d}_i^*\!)}{1-\delta}\forall \; \hat{\boldsymbol{\tau}}_i \in \mathcal{A}_i^{\mathrm{s}}
V ( r ) ~ = ~ - ~{\frac{m_{1}~ m_{2}}{M_{*}^{3}}}~ \int{\frac{d^{3}~ \vec{p}}{( 2 \pi )^{3}}}~{\frac{\exp ( ~ i ~ \vec{p}\cdot \vec{r}~ )}{2 p ~ + ~ r_{c}~{p}^{2}}}~ ,
\sum_{\substack{1 \leq w \leq h 1 \leq z \leq s}}(-1)^{w + z}\sum_{\substack{y_1+...+y_w=h x_1+...+x_z=s}}\sum_{\substack{0 \leq p_w \leq ... \leq p_1 \leq i-hd 0 \leq v_z \leq ...\leq v_1 \leq r-i-sd}}\prod_{a=1}^w i_a \alpha^{i_a}_{r-(h-y_1-...-y_a)d-p_a}\prod_{b=1}^z x_b \alpha^{x_b}_{r-(s-x_1-...-x_b)d-v_b}.
\liminf_{k\to \infty}a_k + \liminf_{k\to \infty}b_k \stackrel{*}{\le}\liminf_{k\to \infty}(a_k+b_k) &\le \limsup_{k\to \infty}a_k + \liminf_{k\to \infty}b_k &\le \limsup_{k\to \infty}(a_k+b_k) &\stackrel{**}{\le}\limsup_{k\to \infty}a_k+\limsup_{k\to \infty}b_k.
\|P_{R_{j,k}}P_{N_1}e^{it\Delta}f_1 \cdot P_{N_2}e^{it\Delta}f_2\|_{L^2(M)}^2\lesssim N_2^{d-2r}\sum_{n\in\frac{2\pi}{\mathcal{T}}\mathbb{Z}}\left(\sum_{\substack{\lambda_1\in \Lambda^+\cap R_{j,k}, \lambda_2\in \Lambda^+, |\lambda_2|_\rho\sim N_2, |\lambda_1|^2_\rho+|\lambda_2|_\rho^2=n}}\|P_{\lambda_1}f\|_{L^2(M)}\|P_{\lambda_2}f_2\|_{L^2(M)}\right)^2,
S_3(X)=\sum_{1\le m_1,m_2,m_3\le X}\tau(m_1^2+m_2^2+m_3^2)=\sum_{\substack{1\le m_1,m_2,m_3\le X 1\le n\le 3X^2 m_1^2+m_2^2+m_3^2=n}}\tau(n).
|\Psi^*g_a - (a^2\sum_{j = 1}^3(dx_j)^2 + g_{0,0,0}+ |x|H_x + \frac{1}{a^2}\eta^2 )|_{g_0}&\leq C(\epsilon) \left(\frac{1}{\rho^2(m)}+ \frac{|x|}{\rho^3(m)}+ \frac{|x|^2}{\rho^4(m)}\right).
\frac{1}{\phi(2N+1)}\left(\frac{1}{|S_{m,N}|}\sum_{k\in S_{m,N}}|\xi^{m_0,N_0}_k|^p\right)^\frac1p \le \frac{1}{\phi(2N+1)}\left(\frac{1}{|S_{m_0,N}|}\sum_{k\in S_{m_0,N}}|\xi^{m_0,N_0}_k|^p\right)^\frac1p &= \frac{1}{\phi(2N+1)}\left(\frac{|S_{m_0,N}|}{|S_{m_0,N}|}\right)^\frac1p\le \frac{1}{C_1\phi(2N_0+1)}.
F(t,q) = \frac{1}{2}\left(\frac{1+t}{1-t}\right)^{q/2}\left(\vphantom{\frac{t+1}{t-1}}(1-t^2)^{-1/2}+(1-t^2)^{1/2}\right) & = \frac{1}{2}\left(\vphantom{\frac{t+1}{t-1}}(1+t)^{\frac{q-1}{2}}(1-t)^{-\frac{q+1}{2}}+ (1+t)^{\frac{q+1}{2}}(1-t)^{-\frac{q-1}{2}}\right).
\left(\frac{F^2}{w_+}(z) + \frac{F^2}{w_-}(z) - 2\right) \phi^{-2n}(z)= \frac{c}{\log^2 (\vert z -1\vert)}+ O\left(\frac{1}{\log^3(\vert z-1\vert)}\right), ,
\mathbf{T}_{K^\circ}=\mathrm{Res}_{K/\mathbb{Q}}(\mathbb{G}_m)\cap\mathbf{G}_{\mathbb{Q}},\mathbf{T}_{K^\circ}^1=\mathbf{T}_{K^\circ}\cap\mathbf{G}^1_{\mathbb{Q}}.
< \Delta v_{-}, t_{1}^{- 1}\otimes t_{2}^{- 1}> = \left( \begin{array}{c c c c}{0}&{0}&{0}&{0}\\{q^{2}}&{0}&{0}&{0}\\{q}&{0}&{0}&{0}\\{0}&{q^{3}}&{q^{2}}&{0}\\ \end{array}\right) .
\sup_{\lambda\in B^d_{n,\rho}}\| \mathbb G^b_{n,j+R_1}(\theta'_n)+\rho \hat D_{n,j+R_1}(\theta'_n)\lambda+\mathbb G^b_{n,j}(\theta'_n)+\rho \hat D_{n,j}(\theta'_n)\lambda \| =o_{P^*}(1),
\zeta(\{\{2\}^{m},1,\{2\}^{m},3\}^{n},\{2\}^{m})\stackrel{?}{=}\frac{1}{2n+1}\frac{\pi^{\mathrm{wt}}}{(\mathrm{wt}+1)!}\quad\left(=\frac{1}{2n+1}\zeta(\{2\}^{{\rm wt}/2})\right)
a_1(t) :=\frac{0.0196}{0.04962(J(t)+1.15)}\leq\frac{6\cdot 0.0196}{0.04962\log t}\leq \frac{0.526\log\log t}{\log t}, a_2(t) :=\frac{6(1.155\log\log t+\log(0.77)+0.685)}{\log t}\leq \frac{7.494\log\log t}{\log t}.
\rho_{u,\omega_1}(x) := \begin{pmatrix}e^{u/2}1 0 e^{-u/2}\end{pmatrix}, \rho_{u,\omega_1}(y) := \begin{pmatrix}e^{u/2}0 \omega_1+\omega_1^{-1}-2\cosh{u}e^{-u/2}\end{pmatrix}.
\log\Big|\dfrac{\partial}{\partial x}\phi(t,x)\Big| =\int_0^t\dfrac{\partial}{\partial x}b(s,\phi(s, x))\,ds &=\int_0^t \lim_{k\to\infty}v_k(s,\phi_k(s,x))\,ds &=\lim_{k\to \infty}\log\Big|\dfrac{\partial}{\partial x}\phi_k(t,x)\Big|.