harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{C a s}( v ) = \frac{1}{2 \pi}\int_{0}^{\infty}\mathrm{l n}\Bigl ( 1 - e^{- 2 \pi x}\left( \frac{x - v}{x + v}\right)^{2}\Bigr ) d x - \frac{v}{\pi}\mathrm{l n}( v T ) - \frac{v}{\pi}\gamma .
4^{-(j+1)}\Bigg\vert\Bigg\vert \prod_{\ell=0}^j(S_{n_{k_{\ell}}}-I)S_{n_k}f\Bigg\vert\Bigg\vert\le 4 \cdot 4^{-(j+2)}\Bigg\vert\Bigg\vert \prod_{\ell=0}^j (S_{n_{k_\ell}}-I)(S_{n_k}-I)f \Bigg\vert\Bigg\vert &+4^{-(j+1)}\Bigg\vert\Bigg\vert \prod_{\ell=0}^j (S_{n_{k_\ell}}-I)f \Bigg\vert\Bigg\vert.
G_{\ell_i,\ell_j,k}=\binom{\ell_i+\ell_j-k}{k,\ell_i-k,\ell_j-k}\frac{\ell_i!\ell_j!}{(\ell_i+\ell_j-k)!}=\binom{\ell_i}{k}\binom{\ell_j}{k}k!.
R_{\ell,\ell'}= (R_{\ell,\ell'}^{k,k'})_{k,k'\leq \kappa},\,R^n = \bigl(R_{\ell, \ell'}\bigr)_{\ell,\ell'\leq n},\,C^\theta_{\ell,\ell'}=\prod_{j\leq m}\bigl( R_{\ell,\ell'}^{\circ p}\lambda^j,\lambda^j \bigr)^{n_j},
\Gamma_{W Z}[ \widetilde{\phi}_{1}\widetilde{\phi}_{2}] \ = \ \Gamma_{W Z}[ \widetilde{\phi}_{1}] + \Gamma_{W Z}[ \widetilde{\phi}_{2}] - \frac{i}{4 \pi}\int_{\Sigma}T r ( \phi_{1}^{- 1}d \phi_{1}\phi_{2}d \phi_{2}^{- 1}) \ .
\left( \phi_W \star \psi_W\right)\left( \hat{x}\right)\equiv\phi\left(\hat{x}\right) \psi\left( \hat{x}\right) = \frac{1}{\left( 2\pi\right)^n}\int d^n k d^nq\, e^{-i(k+q)\hat{x}}e^{ik\hat{x}}\tilde{\phi}\left( q+ k\right)\tilde{\psi}\left( -k\right) .
A_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,i\right)=\sum_{j=0}^{i}\left( -1\right)^{j}\binom{rp+\sigma +1}{j}\binom{a\left(i-j\right) +b}{r}^{p}\prod_{s=2}^{L}\binom{\alpha_{s}\left( i-j\right)+\beta_{s}}{r_{s}}^{p_{s}}.
V^{(0)}_{B}=\frac{T_{0}C_{2}Nf^{2}\beta^{2}}{\lambda^{2}}\left(\frac{M^{2}\lambda^{2}}{f^{2}}-\frac{4}{9}\right)\left(\frac{1}{2}+\frac{1}{1+3\sqrt{1-\frac{2M^{2}\lambda^{2}}{f^{2}}}}\right)
\sum^{T-1}_{t=0}E_t=\sum^{T-1}_{t=0}\bigg[ \frac{7M_xD_x+7M_yD_y}{2(t+3)}+\frac{Q_g+Q_r}{(t+3)^2}\bigg] \leq \ln(T+1)\left(3.5M_xD_x+3.5M_yD_y\right)+2Q_g+2Q_r.
\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{n(-1)^{n-1}q^{n(n+1)/2}}{(q)_n}+\sum_{n=1}^{\infty}\frac{q^{n(n+1)}}{(q)_n^{2}(1-q^n)}=\sum_{n=1}^{\infty}\frac{q^n}{(q)_n(1-q^n)}.
\widehat{\mu}^{\mathrm{asy}}(\overline M) = \frac{(\overline M^{d+1})_S}{(M^d)(\dim Y + 1)}\quad\widehat{\mu}^{\mathrm{asy}}(\pi^*(\overline M)) = \frac{(\pi^* (\overline M)^{d+1})_S}{(\pi^*(M)^d)(\dim X + 1)},
\begin{aligned}p_j(z) = (X_j(X(z))^{-1}) p(z) p_{jk}(z) = -(X_j (X(z))^{-1}X_k(X(z))^{-1})p(z) +(X_j(X(z))^{-1})(X_k(X(z))^{-1})p(z)\end{aligned}
F=(x_{1}^{m_{1,2}}) (x_{1}^{m_{1,3}}x_{2}^{m_{1,3}+m_{2,3}})\cdots(x_{1}^{m_{1,n}}\cdots x_{n-1}^{m_{1,n}}x_{2}^{m_{2,n}}\cdots x_{n-1}^{m_{2,n}}\cdots x_{n-1}^{m_{n-1,n}}).
(t_{1j}-t_{2j})_{N_2-l}^{N_2}= z_j^{N_2-l+1}(q(z_j)p_j(z_j)+r_j)+z_j^{N_2-l}(a_{1}-a_{2}) &=z_j^{N_2-l}(z_jq(z_j)p_j(z_j)+z_jr_j+(a_{1}-a_{2})).
\min_{x\in \mathbb{R}^2}\max_{y\in \mathbb{R}^2}\tfrac{1}{2}x^T\begin{pmatrix}L_x 0 0 \mu_x\end{pmatrix}x+x^T\begin{pmatrix}0 L_{xy}L_{xy}0 \end{pmatrix}y-\tfrac{1}{2}y^T\begin{pmatrix}L_y 0 0 \mu_y\end{pmatrix}y,
\sum_{n=-\infty}^\infty\;e^{-c^2n^2}=\theta_3(0,e^{-c^2})=\sqrt{\frac{\pi}{c^2}}\,\theta_3(0,e^{-\pi^2/c^2})=\sqrt{\frac{\pi}{c^2}}\sum_{n=-\infty}^\infty\,\exp[-\pi^2n^2/c^2].
B_{e f f}^{( 1 )}( \vec{R}_{p_{1}}) = \frac{2 \pi \kappa \sigma_{1}}{q_{1}}\varepsilon_{i j}\partial_{i}v_{j}^{( 2 )}( \vec{R}_{p_{1}}) , \partial_{i}= \frac{\partial}{\partial R_{p_{1}}^{i}}.
0 =\big|\varphi_1(x_i) - \varphi_1(x_j)\big|^2 + 2\big\langle x_i - x_j, \xi_2 (x_i - x_j)\big\rangle + 2\big\langle x_i - x_j, \varphi_2(x_i) - \varphi_2(x_j)\big\rangle, &= \big|\varphi_1(x_i) - \varphi_1(x_j)\big|^2 + 2l_{ij}^2 \hat{b}_{ij}^T\xi_2 \hat{b}_{ij}+ 2 l_{ij}\big\langle \hat{b}_{ij}, \varphi_2(x_i) - \varphi_2(x_j)\big\rangle,
u(X) =\sum_{j=1}^{m}\int_{\partial D}\langle K_{j}(X, Q),n_{Q}\rangle \widetilde{f}_{j-1}(Q)d\sigma(Q), &=\sum_{j=1}^{m}M_{j}\widetilde{f}_{j-1}(X),\,\,\,\, X\in D,
(\delta+1)\int_{S_{\mathcal{I}_{t-2}},y_{t-1}\geq M}{y^2_{t-1}p(\mathcal{I}_{t-1}|z_t)d\mathcal{I}_{t-1}}\leq (\delta+1)\int_{S_{\mathcal{I}_{t-2}},0<y_{t-1}<\infty}{y^2_{t-1}p(\mathcal{I}_{t-1}|z_t)d\mathcal{I}_{t-1}}=(\delta+1)\frac{E_{t-1}[y^2_{t-1}|z_t]}{2}.
\left( \partial_{r}^{2}+{\frac{\partial_{r}}{r}}-{\frac{( j - 1 / 2 )^{2}}{r^{2}}}- \kappa^{2}- 2 e^{2}v^{2}f^{2}\right) h_{4}- i ( \kappa + 2 e N ) 2 e^{2}v^{2}f^{2}({\frac{h_{2}}{\sqrt{2}e v f}}) = 0 .
( - )^{{\bar{\alpha}}+{\bar{\beta}}+ \bar{\alpha}\bar{\beta}}\frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]}{{\bar{\eta}}}\rightarrow \frac{{\bar{\vartheta}[_{\bar{\beta}}^{\bar{\alpha}}]}^{1 3}}{{\bar{\eta}}^{1 3}}.
w_3=({\bf n}_2 \cdot{\bf n}_1),~~w_2=({\bf n}_3 \cdot{\bf n}_1),~~\bar{w}_3=({\bf n}_2 \cdot{\bf n}_4),~~\bar{w}_2=({\bf n}_3 \cdot{\bf n}_4),~~w_1=({\bf n}_2 \cdot{\bf n}_3),
v_p(f^2_{\alpha, \beta}(1)) \geq \min \{v_p(\alpha) + v_p(d-r) + d v_p(\alpha), v_p(\alpha) + (d-r) v_p(\alpha), v_p(\beta) \}= \min \{v_p(d-r) + (d+1) v_p(\alpha), (d-r+1) v_p(\alpha)\}.
a_{1/s(t)}m_t^{-1}\ell_ha_rma_{s(t)}m_t =a_{r/s(t)}m_t^{-1}\ell_{[a_{1/r}(h)]}a_{s(t)}mm_t &=a_rm_t^{-1}\ell_{[a_{1/(rs(t))}(h)]}mm_t
[\begin{matrix}1&\epsilon_{b}^{n}\\ 0&0\\ \vdots&\vdots\\ 0&0\\ \epsilon_{f}^{n}&1\end{matrix}][\begin{matrix}\alpha_{f}^{n}&\alpha_{b}^{n}\\ \beta_{f}^{n}&\beta_{b}^{n}\end{matrix}]=[\begin{matrix}1&0\\ 0&0\\ \vdots&\vdots\\ 0&0\\ 0&1\end{matrix}].
\begin{array}{l}\displaystyle\big[\sigma_j(a')-\sigma_j(a)\big]\big[\sigma_{k}(c)-\sigma_{k}(b)\big] -\big[\sigma_j(c)-\sigma_j(b)\big]\big[\sigma_{k}(a')-\sigma_{k}(a)\big]= \displaystyle=\Big\{\big[s_{j,k}(a')-s_{j,k}(a)\big]\wedge\big[s_{j,k}(c)-s_{j,k}(b)\big]\Big\}\cdot\mathbb{I}_2\ ,\end{array}
\frac{\partial}{\partial h}\lambda(y^u,y^h) =e^{\gamma}\log y\left(\int_{1}^{u/h}y^{ht-u}\omega(t)\,dt-y^{h-u}\right)-h^{-1}\lambda(y^{u-h},y^h) &=h^{-1}\lambda(y^u,y^h)-e^{\gamma}y^{h-u}\log y-h^{-1}\lambda(y^{u-h},y^h).
f_{\delta}=\prod_{j=1}^{m_1+m_2}(z_{n-j+1})\times\sum_{i=0}^{m_2}t^{i m_1}\prod_{j=1}^{i}\left(\frac{1-t^j}{1-qt^{m_1+j}}\right)e_i\Big(z_1,\dots,z_{n-m_1-m_2}\Big)e_{m_2-i}\Big(z_{n-m_2+1},\dots,z_n\Big)
e^{i S_{\Lambda - \delta \Lambda}( \phi_{+}, \phi_{-})}\equiv \int_{\mathrm{C T C}}\prod_{\Lambda_{0}> |{\bf q}| > \Lambda - \delta \Lambda}{\cal D}[ \phi_{+}({\bf q}, t ) ]{\cal D}[ \phi_{-}({\bf q}, t ) ] e^{i S_{\mathrm{c l}}[ \phi_{+}, \phi_{-}]}.
{\bf D}_{\mu_2}(\sigma_2){\bf V}^{\mu_1}(\sigma_1)={\frac{\delta{\bf V}^{\mu_1}(\sigma_1)}{\delta z^{\mu_2}(\sigma_2)}}+\int d\sigma_3\,{\bf \Gamma}_{\mu_2\mu_3}^{\mu_1}(\sigma_1,\sigma_2,\sigma_3){\bf V}^{\mu_3}(\sigma_3) =(D_{\mu_2}V^{\mu_1})|_{z(\sigma_1)}\,\delta (\sigma_1-\sigma_2)
\mu(A\vartriangle As) \leq \mu(A\setminus As)+\mu(As\setminus A) =\Delta_G^{-1}(s^{-1})\mu(As^{-1}\setminus A)+\mu(As\setminus A) \leq 3\max_{s\in S}\mu(As\setminus A)\leq 3\mu(\partial A),
\phi_{2,p}(\kappa, \pi;v)= \int_{\mathcal{C}_{>}}\frac{dz}{2\pi i}e^{-\frac{1}{3}(z-\kappa)^3+ v(z-\kappa)}\frac{1}{\sqrt{2z}}\prod_{k = 1}^{p}\frac{z-\pi_{k}}{z+ \pi_{k}}.
a(s,s') =z(s)\big(1+\mathcal{O}(s\sigma^2)\big)+\widetilde z(s')\big(1+\mathcal{O}(s'h^2)\big) &=\frac{1}{2}\sigma^2+\frac{5}{2}h^2+\mathcal{O}(s\sigma^2+s'h^2) b(s,s') =\zeta(s)\big(1+\mathcal{O}(s\sigma^2)\big)+\widetilde\zeta(s')\big(1+\mathcal{O}(s\sigma^2)\big) &=-i(K-K_1+K_2-K_3+K_4-K_5)+\mathcal{O}(s\sigma^2), c(s,s') =\mathcal{O}(s).
\tilde{\psi}^{\dagger}_{\dot{\alpha}}\psi^{\dagger}_{\dot{\alpha}_{1}}\ldots \psi^{\dagger}_{\dot{\alpha}_{n}}\mid 0 \rangle = (-1)^{n}\psi^{\dagger}_{\dot{\alpha}_{1}}\ldots \psi^{\dagger}_{\dot{\alpha}_{n}}\psi^{\dagger}_{\dot{\alpha}}\mid 0 \rangle \vspace{-3pt}
\frac{-2\Delta_f u}{\int_Mu^2\ln u^2\,e^{-f}dv}-\frac{\int_M|\nabla u|^2\,e^{-f}dv}{(\int_Mu^2\ln u^2\,e^{-f}dv)^2}\left(2u\ln u^2+2u\right)+c_1u=0
\vartheta_{00}(x)^{-1/6}\vartheta_{01}(x)^{-2/3}[\frac{\vartheta_{00}(x)^{4}-\vartheta_{01}(x)^{4}}{16x}]^{-1/24}=\sum_{n=0}^{\infty}P(n)x^{n}=\prod_{k=1}^{\infty}\frac{1}{1-x^{k}}
\begin{array}{rcl}M_{0}= \left(\begin{array}{cccc}\rho_{*}0 0 \quad0 0 C^{-1}C^{-1}e \quad0 0 e^{*}C^{-1}\epsilon+e^{*}C^{-1}e \quad0 0 0 0 \quad\mu\end{array}\right), M_{1}= \left(\begin{array}{cccc}0 0 0 0 0 0 0 0 0 0 \sigma 0 0 0 0 0\end{array}\right).\end{array}
Q = \tilde{Y}= \left( \begin{array}{c c c}{0}&{0}&{0}\\{0}&{- 1}&{0}\\{0}&{0}&{- 1}\\ \end{array}\right) \: , \: \tilde{T}_{3}= \left( \begin{array}{c c c}{- 1}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{- 1}\\ \end{array}\right)
[\begin{matrix}M_{11}\\M_{12}\\M_{13}\end{matrix}]:=\int_{-b/2}^{b/2}\int_{-t/2}^{t/2}[\begin{matrix}x_{2}\sigma_{13}-x_{3}\sigma_{12}\\x_{3}\sigma_{11}\\ -x_{2}\sigma_{11}\end{matrix}]dx_{3}dx_{2}.
\|w\|_* \le \|a\|_* + \|w_1\|_* + \|w_2\|_* + \|w_3\|_* &\le\left(\frac{\|g\|_*- \|g_1\|_*}2+2\right) + 1 + \|w_2\|_* +\left(\frac{\|h\|_*- \|g_1^{w_2}\|_*}2+2\right) &\le\left(\frac{r- 1}2+2\right) + 1 + \|w_2\|_* +\left(\frac{r- 1}2+2\right) &= \|w_2\|_* + r+4.
\left( (5 h_{(p,r;q,s)}+1)\, h_{(p,r;q,s)}\; Q_{-2}- 12q_{(p,r;q,s)}\; L_{-1}^{2}+ 6 q_{(p,r;q,s)}\,(h_{(p,r;q,s)}+1)\; L_{-2}\right) \phi_{(p,r;q,s)}.
s_1' =-s_1~,~~ s_4' =s_4\ , s_2' =-s_2s_8s_9~,~~ s_5' =s_5s_{11}s_{12}\ , s_3' =-s_3s_8s_9s_{10}s_{11}~,~~ s_6' =s_6s_9s_{10}s_{11}s_{12}\ , s_7' =s_7~,~~ s' =s_8s_9s_{10}s_{11}s_{12}\ .
\frac{1-z^2}{\beta}\frac{d}{dz}\hat f_+ =\sum_{i=0}^{\infty}[2(1-4 a_{i+1}^2)z-2b_i](1-z^2)\frac{z^i p_i(x)}{\beta_i}&+\sum_{i=0}^{\infty}[(1-4 a_{i+1}^2)z^2-2b_i z]k_i(z).
\Pi_1(\varphi ,\upsilon ) = \sum_{k=2}^\infty S^{k-2}\varphi \cdot S_k \upsilon\, ,\Pi_2(\varphi ,\upsilon )=\sum_{k=0}^\infty (S_{k-1}\varphi + S_k \varphi+ S_{k+1}\varphi ) \cdot S_k \upsilon\, , &\mbox{and}\Pi_3(\varphi ,\upsilon )=\sum_{j=2}^\infty S_j \varphi \cdot S^{j-2}\upsilon=\Pi_1(\upsilon,\varphi) \, .
{\rm Re}\, \, I_n(\beta)=-\frac{M}{2{\pi^2}}\left( p^{2n}\beta + \frac{\beta^3}{3}p^{2n-2}+\ldots +\frac{\beta^{2n+1}}{2n+1}\right)+{p^{2n}}\frac{Mp}{4{\pi^2}}\log\left(\frac{\beta +p}{\beta-p}\right),
\Big(-A^{1,0}a_0\Big)_l = (-1)^{l-1}\Big( 2^{l+m}n^{l+m}P_l(m) &+ 2^{l+m-2}n^{l+m-2}P_{l-2}(m)(l+m-2)\alpha_p(3) &+ 2^{l+m-4}n^{l+m-4}P_{l-4}(m)(l+m-4)(l+m-5)\frac{\alpha^2_p(3)}{2!}&+ 2^{l+m-4}n^{l+m-4}P_{l-4}(m)(l+m-4)\alpha_p(5) + \cdots \Big).
L^{- 1}: = \overline{{L}}= \{\overline{{L}}_{M}^{A}\}= \left( \begin{array}{c c}{\overline{{L}}_{\alpha}^{a}}&{\overline{{L}}_{\alpha}^{i}}\\{0}&{\overline{{L}}_{\alpha}^{i}}\\ \end{array}\right) , \quad \overline{{L}}_{\mu}^{a}= 0 ,
\left|\left(\frac{zf_1'(z)}{f_1(z)}\right)^2-1\right| =\left|\left(1-\frac{\rho_1}{(1-\rho_1^2)}\left( \frac{u\rho_1^2+4\rho_1+u}{\rho_1^2+u\rho_1+1}+\frac{v\rho_1^2+4\rho_1+v}{\rho_1^2+v\rho_1+1}+\frac{q \rho_1^2+4\rho_1+q}{\rho_1^2+q \rho_1+1}\right)\right)^2-1\right|=1.
(\sigma^\alpha)^{ab}\bigr((\sigma_\alpha)^{b_0}_{~~\!c}\delta^d_{a_0}+\delta^{b_0}_c (\sigma_\alpha)^d_{~a_0}\bigr) = (\sigma^\alpha)^{ab}\bigr((\sigma_\alpha)^d_{~c}\delta^{b_0}_{a_0}+\delta^d_c (\sigma_\alpha)^{b_0}_{~~\!a_0}\bigr).
\xi_{2}( \mathrm{z}) = \vartheta [ \begin{array}{c}{\frac{1}{3}}\\{0}\\ \end{array}] ( \mathrm{z}, \tau ) \ \mathrm{o r \}\vartheta [ \begin{array}{c}{0}\\{\frac{1}{3}}\\ \end{array}] ( \mathrm{z}, \tau ) \ .
\langle a,b \rangle := b*a- a*b \langle a_1,\dots, a_m; b,c \rangle : = p(a_1,\dots,a_m;c;b) - p(a_1,\dots, a_m;b;c) \Phi(a_1,\dots,a_m; b_1,\dots, b_{n+1}) := &\hskip -2cm \frac{1}{m!(n+1)!}\sum_{\sigma \in S_{m}, \tau \in S_{n+1}}p(a_{\sigma(1)},\dots, a_{\sigma(m)}; b_{\tau(1)},\dots;b_{\tau(n+1)}).
\left(\begin{array}{c}{\rho}{\mbox{\scriptsize{U}}}{\mbox{\scriptsize{V}}}{\mbox{\scriptsize{S}}}\end{array}\right)=\left(\begin{array}{c}{\rho_i}{\mbox{\scriptsize{U}}_i}{\mbox{\scriptsize{V}}_i}{\mbox{\scriptsize{S}}_i}\end{array}\right)-{\epsilon}C(\beta)r^{-1/2}{\psi}^{1/2}\left(\begin{array}{c}{-\rho_0}{-c_0\kappa_0}{0}{0}\end{array}\right).
f^{S_0 \oplus S_1}_{(0, i)}= R_i - c_i + \beta \sum_{j \in S_1}p_{ij}f^{S_0 \oplus S_1}_{(1, j)}+ \beta \sum_{j \in N \setminus S_1}p_{ij}f^{S_0 \oplus S_1}(1, j) = f^{S_0 \oplus S_1}_{(1, i)}- c_i = f^{S_1}_i - c_i,
u^{\infty}_p(\hat{x}) = \sqrt{\frac{2}{\pi}}^{i\frac{\pi}{4}}\sum_{n \in \mathbb{Z}}i^{-n}A_n^{in\theta_x}, u^{\infty}_s(\hat{x}) = \sqrt{\frac{2}{\pi}}^{i\frac{\pi}{4}}\sum_{n \in \mathbb{Z}}i^{-n}B_n^{in\theta_x}.
\varrho = \varrho_{m}( 1 + \zeta e^{- 2 \psi}) + \varrho_{r}e^{- 2 \psi}+ \varrho_{\psi}, \wp = \textstyle{\frac{1}{3}}\varrho_{r}e^{- 2 \psi}c^{2}+ \varrho_{\psi}c^{2}.
\int_{0}^{1}(\eta''-\xi'')(\phi_{\nu}-\phi_{\mu})\,ds =\int_{0}^{1}-(\eta-\xi)'(\phi_{\nu}'-\phi_{\mu}')\,ds-\eta'(0)(\phi_{\nu}(0)-\phi_{\mu}(0)) =\int_{0}^{1}(\eta-\xi)(\phi_{\nu}''-\phi_{\mu}'')\,ds-\eta'(0)(\phi_{\nu}(0)-\phi_{\mu}(0))+\eta(0)(\phi_{\nu}'(0)-\phi_{\mu}'(0)).
f_{M}(x) = \max_{\alpha\in \Lambda_M^{min}}\{-\mu_\alpha\sin d(x,\partial\alpha)\}=\max_{\alpha\in \Pi'}\{-\mu_\alpha\sin d(x,\partial\alpha)\}=\max_{\alpha\in \Lambda_M}\{-\mu_\alpha\sin d(x,\partial\alpha)\}\geq f_{K'}(x).
C_{p+1/2}(d)=\sigma^{2}exp(-\frac{\sqrt{2p+1}d}{\rho})\frac{p!}{(2p)!}\sum_{i=0}^{p}\frac{(p+i)!}{i!(p-i)!}(\frac{2\sqrt{2p+1}d}{\rho})^{p-i},
d_X(v_1,v_2)=\min\bigg\{d_{L^2}(v_1,v_2)\, ;\, d_{L^2}(v_1,\mathcal{C}(z^-))+d_{L^2}(v_2,\mathcal{C}(z^-))\,;\,d_{L^2}(v_1,\mathcal{C}(z^+))+d_{L^2}(v_2,\mathcal{C}(z^+))\bigg\},
{\sqrt{2}\vartheta^2_2\widetilde\vartheta_3\over\vartheta_3}+\sqrt{\vartheta_3^2\widetilde\vartheta_3^2+\vartheta_2^2\widetilde\vartheta_4^2 -\vartheta_4^2\widetilde\vartheta_2^2}=0.
\| \mathcal{I}_{3,+}(t,x) \|_p^2 \leq k M_{p,t}^2 \|f\|_{L^q(\mathbb{R})}^4\Big(\frac{q}{q-1}\Big)^2 t^{2-2/q}+ 2t \|f\|_{L^2(\mathbb{R})}^4 \int_0^t U_s ds & = c^{(3)}_{p,t}+ c^{(4)}_{p,t}\int_0^t U_s ds,
ne_n =\sum\limits^d_{i=1}\sum\limits^\infty_{r=0}i \cdot e_{n-ip^r}a_i^{p^r}\big(\sum\limits_{j=1}^\ell (c_j\pi_j)^{p^r}\big) &\equiv\sum\limits^d_{i=1}\sum\limits^\infty_{r=0}i \cdot e_{n-ip^r}a_i^{p^r}\big(\sum\limits_{j=1}^\ell c_j\pi_j\big)^{p^r}\pmod{p}.
[K',E] \; = \; [K',F] \; = \; [K',K] \; = \; 0, \; KE \; =\; q^2 EK, KF \; = \; q^{-2}FK, q EF- q^{-1}FE \; = \; \frac{K^2-K'^{2}}{q-q^{-1}}.
C^1(E)|_U = C^1(E|_U), C^0(T^\vee(M) \otimes E)|_U = C^0((T^\vee(M) \otimes E)|_U) = C^0(T^\vee(M)|_U \otimes E|_U) & \cong C^0(T^\vee(M|_U) \otimes E|_U),
\|\hat{x}\|_{L_q(T_i(V),d\sigma)}=\|\hat{x}(T_i\cdot)\|_{L_q(V,d\sigma)}\lesssim_{p,q}\|U_{\theta_{T_i}}(\hat{x}(T_i\cdot))\|_{L_p(\mathcal{R}_{\theta_{T_i}}^2)}=\|\Psi_{T_i^{-1}}(U_{\theta_{T_i}}(\hat{x}(T_i\cdot)))\|_{L_p(\mathcal{R}_{\theta}^2)}=\|x\|_{L_p(\mathcal{R}_{\theta}^2)},
N_Q = \{r \in \Gamma_{M,N}\ | \ \forall y \in \mathbb Z_M, \ \exists f \in (k^{\mathbb Z_M})^{\times}\{\rm with}\ \rho_Q(y\cdot r \# 1)=f(h)1\}= \{r \in \Gamma_{M,N}\ | \ \forall y \in \mathbb Z_M, \ \exists \lambda \in k^* \{\rm with}\ \rho_Q(y\cdot r \# 1)=\lambda 1\}
F_{3,j}^\delta \left( \begin{array}{c}z x \end{array}\right) = \left( \begin{array}{c}z + \dots + a^\delta_j z^j + a^\delta_{j+1}(x) z^{j+1}+ \dots b^\delta_0(x) + b^\delta_1(x) z + b^\delta_2(x) z^2 + \dots \end{array}\right),
\begin{array}{ll}\| u_{+}\|_{H^{2}}\lesssim 1 \| v(t) \|_{H^{2}}\lesssim 1 \lim_{|t| \rightarrow \infty}\| u(t) \|^{2}_{H^{2}}- \| v(t) \|^{2}_{H^{2}}- \| e^{i t \triangle^{2}}u_{+}\|_{H^{2}}=0,\end{array}
2\left(\frac 1 p -\frac{1}{p^*_\alpha}\right)\frac{S_\alpha^{\frac{N-\alpha}{ps-\alpha}}}{\mu^{\frac{p}{p^*_\alpha-p}}}\leq \lim_{n}\left(\frac{\lambda}{p}-\frac{\lambda}{r}\right)\int_\Omega |v_n|^r\, dx +\left(\frac{1}{p}-\frac{1}{q_n}\right)\int_{\Omega}\frac{|v_n|^{q_n}}{|x|^\alpha}\, dx=\lim_{n}J_{\rho_n}(v_n).
\eta^h \;:=\; \mathbb{E}_0\left[ h\left({\bf R}^{(pq)}_{{\bf i}^1}\right)h\left({\bf R}^{(pq)}_{{\bf i}^2}\right)h\left({\bf R}^{(pq)}_{{\bf i}^3}\right)h\left({\bf R}^{(pq)}_{{\bf i}^4}\right) \right],
\mathcal{A}(a, b) = \sum_{i = 0}^{2n}(-1)^{n + i}\binom{2n}{i}^4 = (-1)^n\left[\binom{2n}{0}^4 - \binom{2n}{1}^4 + \binom{2n}{2}^4 - \binom{2n}{3}^4 + \cdots + \binom{2n}{2n}^4\right].
A^{( \ref{s d i r})}( p , \delta ) = 1 - \frac{g_{0}}{9}+ \frac{g_{0}}{3 p^{2}}\delta^{2}- \frac{g_{0}}{3 p^{3}}\delta^{3}\arctan \left( \frac{p}{\delta}\right) + \frac{g_{0}}{6}l n \left( \frac{p^{2}+ \delta^{2}}{\alpha^{2}+ \delta^{2}}\right) .
H_{z}=\frac{k_{o}^{2}\epsilon_{r}-k_{z}^{2}}{j\omega\mu}(Ce^{-jk_{x\epsilon}w}+De^{jk_{x\epsilon}w})e^{-jk_{xo}(x-w)}cos(\frac{m\pi}{a}y)e^{-jk_{z}z}(44)
{X^{i_{1}A_{1}i_{2}A_{2}\ldots i_{K}A_{K}}}_{p_{1}B_{1}p_{2}B_{2}\ldots p_{L}B_{L}}= a^{i_{1}A_{1}}a^{i_{2}A_{2}}\ldots a^{i_{K}A_{K}}a_{p_{1}B_{1}}a_{p_{2}B_{2}}\ldots a_{p_{L}B_{L}}
\chi(M)=(-1)^{r}\int_{M}\frac{\prod_{i}^{n}(1-e^{-x_{i}})}{\prod_{i}^{r}x_{i}}\prod_{i}^{n}\frac{x_{i}}{1-e^{-x_{i}}}(TM\otimes\mathbb{C})=(-1)^{r}\int_{M}(-1)^{r}\prod_{i}^{r}x_{i}(TM\otimes\mathbb{C})=\int_{M}e(TM)
W_{\pi}(g_{t,l,v}) = q^{\frac{n}{12}}\gamma_F(-2v,l)\chi^{-1}(4v^2) \psi(\frac{3}{4v}\varpi^{-\frac{n}{2}})_{\psi}(-16bv^3 \varpi^{\lceil \frac{l}{2}\rceil+2\{\frac{l}{2}\}-3\lfloor \frac{1}{2}\lceil\frac{l}{2}\rceil \rfloor};\Delta\varpi^{-\lfloor \frac{l}{2}\rfloor-\lfloor \frac{1}{2}\lceil\frac{l}{2}\rceil \rfloor}),
\sigma_w(\zeta^k) = \sum_{0 \leq i_1 < \cdots < i_w \leq n-1}\zeta^{k\left(q^{i_1}+ \cdots + q^{i_w}\right)}= \sum_{j \in \mathbb{Z}_{q^n-1}}\delta_w(j) \zeta^{kj}= \mathcal{F}_\zeta[\delta_w](k).
\begin{cases}h=\mathcal F^{-1}\varphi, \tilde h=\mathcal F^{-1}\chi, &u_q:=\Delta_qu=\mathcal F^{-1}(\varphi(\lambda_q^{-1}\xi)\mathcal Fu)=\lambda_q^n\int h(\lambda_qy)u(x-y)dy, \mbox{for}q\geq 0, & u_{-1}=\mathcal F^{-1}(\chi(\xi)\mathcal Fu)=\int \tilde h(y)u(x-y)dy.\end{cases}
|S_1||K_1|{\Pr}_r(S_1, K_1) = \underset{r \in [s, K_1]}{\underset{s \in S_1}{\sum}}|C_{K_1}(s)| \leq \underset{r \in [s, K_2]}{\underset{s\in S_2}{\sum}}|C_{K_2}(s)| = |S_2||K_2|{\Pr}_r(S_2,K_2).
J_{M_{g,k+1}}|_{\Omega_{\Lambda_{g,k+1}}^{0}\cup{\mathcal D}_{\Lambda_{g,k+1}}^{0}}=J_{M_{g,k}}\circ i|_{\Omega_{\Lambda_{g,k+1}}^{0}\cup{\mathcal D}_{\Lambda_{g,k+1}}^{0}}.
d s^{2}= \frac{1}{1 - \xi^{0}\xi^{0}}d \xi^{0}d \xi^{0}- ( 1 - \xi^{0}\xi^{0}) \frac{d{\bf x}( I -{\bf x}^{\prime}{\bf x})^{- 1}d{\bf x}^{\prime}}{1 -{\bf x}{\bf x}^{\prime}},
y_n = \sum_{k=1}^{n}\binom{n+k}{k-1}(k-1)!B_{n,k}(-a_1, -2!a_2, -3!a_3, \ldots) & = \sum_{k=1}^{n}\frac{(n+k)!}{(n+1)!}B_{n,k}(-a_1, -2!a_2, -3!a_3, \ldots) & = \frac{b_{n+1}}{n+1}
u_{k,n}= k! (-1)^k{-\frac{1}{2}\choose k}(4pq)^k\,[z^{n-k}]{1\over ({1-4pqz})^k\,(1-z)}= k (-1)^k{-\frac{1}{2}\choose k}(4pq)^k\,[z^{n-k}]{1\over (1-z)}\int_{0}^{\infty}x^{k-1}e^{-(1-4pqz)x}\,dx = (-1)^k{-\frac{1}{2}\choose k}(4pq)^k\,\int_{0}^{\infty}kx^{k-1}e^{-x}\left(\sum_{j=0}^{n-k}\frac{(4pqx)^j}{j!}\right)\,dx.
\sum_{i=0}^{4}(\begin{matrix}12\\i\end{matrix})x^{i}z^{12-i}+\sum_{i=5}^{9}(\begin{matrix}12\\i\end{matrix})12x^{i-5}z^{12-i}+\sum_{i=10}^{12}(\begin{matrix}12\\i\end{matrix})12^{2}x^{i-10}z^{12-i}=5
\div(\lambda^n \nabla\delta\pi^n) =\div H^{n-1}_e \,-\, \div(( u^{n-1}-\kappa^{n-1}\,\nabla\log\rho^n )\cdot\nabla\delta u^{n}), &=\div H^{n-1}_e \,-\,\div \Bigl( \delta u^{n}\cdot \nabla ( u^{n-1}-\kappa^{n-1}\,\nabla\log\rho^n )+\delta u^n \div(\kappa^{n-1}\nabla\log \rho^n) \Bigr).
\frac{b_1}2-\frac{\gamma_{11}^1}{4d_1}-\frac{\gamma_{22}^1}{2d_1}+\frac{\gamma_{22}^1}{4d_1}\frac{x_1^2}{x_2^2}=cz_1, \frac{b_2}2-\frac{\gamma_{22}^2}{4d_2}-\frac{\gamma_{22}^1}{2d_2}\frac{x_1}{x_2}=cz_2;
\partial_{s_{k}}\log \det (1-\mathcal{K}_{\vec{x},\vec{s}}) = \frac{e^{\pi i \alpha}}{2\pi i}\int_{-x_{k}}^{-x_{k-1}}[\Phi_{-}^{-1}(u;\vec{x},\vec{s})\partial_{u}\Phi_{-}(u;\vec{x},\vec{s})]_{21}du.
\begin{aligned}\frac{1}{N+1}\sum_{n=0}^N (U_{(\theta)1}^{-n}\varphi(U_{(\theta)c})U_{(\theta)1}^nx,x) = \frac{1}{N+1}\sum_{n=0}^N (W^{-n}\varphi(U_\nu)W^n\gamma,\gamma) &=\int_{\mathbb T}\varphi\frac{1}{N+1}\sum_{n=0}^N |W^n\gamma|^2\nu. \end{aligned}
4 \delta^{a b}( C_{1}C_{3}+{\frac{1}{4}}C_{1}C_{2}){\frac{g^{4}}{p^{2}}}\Lambda^{3 - \omega}\int{\frac{d^{\omega}k}{( 2 \pi )^{\omega}}}\int{\frac{d^{3}q}{( 2 \pi )^{3}}}{\frac{k \cdot p}{k^{2}q^{2}( k + p + q )^{2}}}.
\Lambda_{-}={\frac{1}{2}}\gamma^{0}\gamma^{-}={\frac{1}{4}}\gamma^{+}\gamma^{-}={\frac{1}{2}}\left( \begin{array}{c c}{{1}}&{{- \sigma_{3}}}\\{{- \sigma_{3}}}&{{1}}\\ \end{array}\right) \; .
\mathbf{1}^\top \tilde{\mathbf{P}}_0^*= \frac{\mathbf{1}^\top \breve{\mathbf{P}}_0^i}{1+3\nu^*}\mathbf{1}^\top \tilde{\mathbf{Q}}_0^*= \frac{\mathbf{1}^\top \breve{\mathbf{Q}}_0^i}{1+3\nu^*}.
\sum_{b \in (\mathbb{Z}/n\mathbb{Z})^\times}\chi(b) \cdot \left\langle \omega^*_{\overline{f}}, \mathrm{exp}^{*}\left( \mathrm{loc}_p \left( c_{\mathbb{Q}(\mu_n)}\right)^{\sigma_b}\right) \right\rangle_{\mathrm{dR}}= c \cdot d \cdot (c - \chi(c)) \cdot (d - \chi(d)) \cdot \frac{L^{(Np)}(f, \chi, 1)}{(-2 \pi i)\Omega^{\chi(-1)}_{f}}
\left( \partial_{r}^{2}+{\frac{\partial_{r}}{r}}-{\frac{( j - 1 / 2 )^{2}}{r^{2}}}- 2 e^{2}v^{2}f^{2}\right) ({\frac{h_{2}}{\sqrt{2}e v f}}) + i \kappa h_{4}= 0 ,
V_{( j , m )}^{\omega}( z , \bar{z}) ~ = ~ \gamma^{j - m}( z ) \bar{\gamma}^{j - m}( \bar{z}) e^{\frac 2{\alpha_{+}}j \phi ( z , \bar{z})}e^{i \sqrt{\frac{2}{k}}m X ( z , \bar{z})}e^{i \sqrt{\frac{2}{k}}( m +{\frac{k}{2}}\omega ) Y ( z , \bar{z})}
\begin{cases}{}^C D^\nu_{0^+}u \left( x,t \right) =H t^{2H-1}\frac{\partial^2}{\partial x^2}u \left( x,t \right),\nu \in \left( 0,1 \right], \: x \in \mathbb{R}, \: t >0, u \left( x,0 \right) = \delta \left( x \right),\end{cases}
\bigtriangledown{\bf \it J}_{\bar{n}}({p}_{1,\bar{n}},{p}_{2,\bar{n}}) =\frac{1}{\ln(2)}\Big({(1+{p}_{1,\bar{n}}{\vert{}{h}_{21,\bar{n}}\vert{}}^2+{p}_{2,\bar{n}}{\vert{}{h}_{22,\bar{n}}\vert{}}^2)}^{-1}&-{(1+{p}_{1,\bar{n}}{\vert{}{h}_{21,\bar{n}}\vert}^2)}^{-1}\Big){\vert{}{h}_{21,\bar{n}}\vert{}}^2.
\theta = \frac{2\tau \sum_{d=1}^3 \Delta f_{i,2e_d}^{(m)}- \tau^2 \sum_{d=1}^3 (\Delta f_{i,e_d}^{(m)})^2 / (f_{i,0}^{(m)}+ \tau \Delta f_{i,0}^{(m)})}{3 (f_{i,0}^{(m)}+ \tau \Delta f_{i,0}^{(m)})}+ \theta_i^{(m)}.
\mathrm{(P1)}:\begin{cases}\underset{\{\textbf{u}_m\}_{m\in\mathcal{M}}}{\min}\ |\mathcal{M}| \ \underset{m\in \mathcal{M}}{\min}\|\mathbf w_k - \mathbf u_m\|\leq r, \ \forall k\in \mathcal{K},\end{cases}
\tilde G_k(n) = C_*(p) p^{j_0(j_0+1)/2}q^{j_0-1}n^{j_0}p^{j_0(k-j_0)}\frac{{\overline r_0}^{\overline r_1}}{\Gamma(\overline r_1+1)}\left( C(p, \overline r_0/\overline r_1, \langle \overline r \rangle) + o(1) \right).
Y_0 = 1 , Y_1 = x - 1 , Y_j = (x - c_{l(T)+1-j}- 1) Y_{j-1}- c_{l(T)+1-j}Y_{j-2}(\forall j \in \overline{2, l(T)-1}) , Y_{l(T)}= (x - c_1) Y_{l(T)-1}- c_1 Y_{l(T)-2},
M^{\beta\gamma}_{11}=\frac{T^{\beta\gamma}}{1-CI^{\beta\gamma\,-1}T^{\beta\gamma}+T^{\beta\gamma\,2}}=\frac{(CI^{\beta\gamma})^5-CI^{\beta\gamma}}{(1-(CI^{\beta\gamma})^{2})(1-3(CI^{\beta\gamma})^{2})}
X = \begin{bmatrix}u & u^{2}& 1\end{bmatrix}, Y = \begin{bmatrix}1 & u^2 & u \end{bmatrix}, A = \begin{bmatrix}-1 & -1 & 1\end{bmatrix}, B = \begin{bmatrix}1 & -1 & -1\end{bmatrix}.
\nabla^{*\perp}_{X}\widehat{W}= -\frac{1}{\overline{g}(L,L)}X(\overline{g}(L,L))\widehat{W}+ \frac{1}{\overline{g}(L,L)}\left[\{X(a) - a\tau(X)\}E \right. &\left. + \{X(b)+b\tau(X) + c\phi(X)\}N + \{X(c) + aD(X,E) + b\rho(X)\}W\right].
\bold{F}_{x,y}=\left[\begin{array}{c|c|c|c|c}\bold{0}\bold{A}_{x,y;1,1}\dots \bold{0}\bold{A}_{x,y;1,p_y}\hline\vdots \ddots \ddots \vdots \vdots \hline\bold{0}\bold{A}_{x,y;p_x,1}\dots \bold{0}\bold{A}_{x,y;p_x,p_y}\end{array}\right].
* \omega = \frac{\sqrt{g}}{r ! ( m - r ) !}\omega_{\mu_{1}\mu_{2}\dots \mu_{r}}\epsilon_{\nu_{r + 1}\dots \nu_{m}}^{\mu_{1}\mu_{2}\dots \mu_{r}}d x^{\nu_{r + 1}}\wedge \cdots \wedge d x^{\nu_{m}}.
E(\prod_{x,y}s_{x,y}^{N^{(1)}_{x,y}}\prod_x e^{-\sum_{x}\chi_{x}\hat{\mathcal{L}}_1^x})=E(e^{-\frac{1}{2}\sum_{x,y}C_{x,y}( s_{x,y}-1)\varphi_x \bar \varphi_y-\frac{1}{2}\sum \chi_{x}\phi_x \bar \phi_x}).
\lim\limits_{k\rightarrow\infty}\left( \sum_{i=1}^{N-1}\left\| A_i x_i^k - A_i x_i^{k+1}\right\|^2 + \left\| x_N^k - x_N^{k+1}\right\|^2 + \left\| \lambda^k - \lambda^{k+1}\right\|^2 \right)= 0.
\delta_0^2\|\varphi_0(T)x\|^2 \leq\|X\varphi_0(T)x\|^2=\int_\sigma|\varphi_0f|^2m =\int_{\sigma\setminus\tau}|\varphi_0f|^2m + \int_\tau|\varphi_0f|^2m &\leq\varepsilon^2\|X\|^2 + \int_\tau\gamma^2\delta_0^2\|\varphi_0(T)x\|^2|\psi_0|^2m \leq \varepsilon^2\|X\|^2 + \gamma^2\delta_0^2\|\varphi_0(T)x\|^2
\tilde{g}_{0 0}= \frac{g_{0 0}}{\left[ 1 + ( 1 + g_{0 0}) \sinh^{2}\alpha \right]^{2}}= - \frac{\left( 1 - \frac{2 \eta}{r}\right)^{\frac{m + \sigma}{\eta}}}{\left[ 1 + ( 1 - ( 1 - \frac{2 \eta}{r})^{\frac{m + \sigma}{\eta}}) \sinh^{2}\alpha \right]^{2}}