harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{u=0}^{v}\varphi(u) =\sum_{u=0}^{v}\sum_{k=0}^{u+2}s_k\varphi (u+3-k) &-b_0c_0\varphi(2) \bigl({A}(v+1)-a_0\bigr) -b_0c_1\varphi(1) \bigl({A}(v+1)-a_0\bigr) &-c_0\varphi (1)\sum_{u=0}^{v}\sum_{k=0}^{u+2}a_k b_{u+2-k}.
f_j(u_{i+k})=\dfrac{(a+u_{i+k})_j}{(b+u_{i+k})_j}=\dfrac{(a+u_{i})_j}{(b+u_{i})_j}\dfrac{(a+u_{i+j})_k(b+u_i)_k}{(b+u_{i+j})_k(a+u_i)_k}.
I^{(0)}_j(\varphi) \lesssim 2^{-2js}\sum_{k=0}^{\gamma-1}2^{-2js(1-\alpha)k}\widetilde{I}_j^{(k)}(\varphi) + 2^{-2js(1-\alpha)\gamma}\widetilde{I}_j^{(\gamma)}(\varphi)\le 2^{-2js}\sum_{k=0}^{\gamma}\widetilde{I}_j^{(k)}(\varphi).
H_{\lambda =q^{-1}}(q = e^{i \pi /N})= -\sum^L_{j=1}\sum^{N-1}_{n=1}\left( \frac{1}{sin ( \frac{\pi n}{N})}\left[ \left( \Sigma^+_j\Sigma^-_{j+1}\right)^n + \left( \Sigma^-_j\Sigma^+_{j+1}\right)^n \right] +( 1- \frac{2n}{N}) e^{i \pi n/N}Z^n_j \right)
\omega_{-}^{\psi}= \left( \begin{array}{c c}{\Sigma_{-}^{( 1 )}- \chi}&{( \Sigma_{-}^{( 1 )}- \chi ) \chi - \chi \Sigma_{-}^{( 2 )}+ \partial_{-}\chi}\\{I_{n}}&{\Sigma_{-}^{( 2 )}+ \chi}\\ \end{array}\right) .
S_{f r e e}= \frac{1}{4 \pi \alpha^{\prime}}\int d^{2}z \partial^{\alpha}\xi_{\mu}\partial_{\alpha}\xi^{\mu}+ \frac{i}{4 \pi \alpha^{\prime}}\int_{z^{2}= 0}d z^{1}F_{\mu \nu}\xi^{\nu}\partial_{1}\xi^{\mu}.
\lim_{r\to 0}\frac{\int_0^r nH_\psi e^{-2\psi(\tau)}\xi^{n-1}(\tau)\,{\rm d}\tau}{e^{-3\psi(r)}\xi^{n-1}(r)}= \lim_{r\to 0}\frac{nH_\psi e^{\psi(r)}}{-3\psi'(r) + (n-1)\frac{\xi'(r)}{\xi(r)}}=0
C^{-2}_{y_k}|{\hat{C}}_{y_k}|^{2}=|{\bf a}^{H}_k{\bf s}|^{2}={\rm{Tr}}({\bf A}_k \bf S),C^{-2}_{w_k}|{\hat{C}}_{w_k}|^{2}=|{\bf b}^{H}_k{\bf s}|^{2}={\rm{Tr}}({\bf B}_k \bf S), C^{-2}_{z_j}|{\hat{C}}_{z_j}|^{2}=|{\bf a}^{'^{H}}_j{\bf s}|^{2}={\rm{Tr}}({\bf A}^{'}_j \bf S),C^{-2}_{q_j}|{\hat{C}}_{q_j}|^{2}=|{\bf b}^{'^{H}}_j{\bf s}|^{2}={\rm{Tr}}({\bf B}^{'}_j \bf S).
I_{N}= \frac{\eta_{-}}{\varphi(x)}\int_{\{z \in \mathcal{C}:\, \Re{z}> 1\}}\frac{dz}{2\pi i}\int_{\Sigma}\frac{dw}{2\pi i}e^{N(f(x;z) - f(x;w))}e^{\frac{\eta_{-}}{\varphi(x)}( v(z-\Re{z_{-}})-u(w-\Re{z_{-}}))}H(z, w)
\frac{q_1 + p_2q_2}{q_1+q_2}- \frac{p_2 q_2}{q_2+q_3}=\frac{q_1+q_2}{q_1+q_2}- \frac{q_2}{q_2+q_3}\leq \frac{q_1+q_3}{q_1+q_2+q_3}= \P( (G\cap H^c) \cup (H\cap G^c)\mid G\cup H).
T_\mu =\frac{(2\imath)^\frac{D-2}{2}}{(D-2)!}\epsilon_{\nu\mu\rho_2\ldots\rho_{D-1}}\pi^\mu\psi^{\rho_2}\ldots\psi^{\rho_{D-1}}+\frac{\alpha}{2^\frac{D-2}{2}}\pi_\nu=0\;.
f_1 = -B_{23}B_{32}C_{11}y^2 z + z^2( \,\, \cdots ) f_2 = -A_{33}B_{32}z^2 ( C_{11}x + C_{21}y + C_{31}z) f_3 = -A_{33}B_{23}z^2 ( C_{11}x + C_{12}y + C_{13}z)
R_{\ell+s}= \partial C_{\ell+s}\oplus \varphi( \partial R_{\ell}) \oplus R_\ell = \partial C_{\ell+s}\oplus \bigoplus_{0 \leqslant k \leqslant \ell, [k] = [\ell]}\partial C_{k}\oplus \varphi(R_{\ell+s}).
\sum^n_{\ell=0}t^{-\ell}\mathrm{STr\,}\alpha_\ell(x)dv_X(x) &=\frac{1}{2\pi}\left[\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)\wedge\mathrm{ch_b\,}(\nabla^{E},E)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0\right]_{2n+1}(x).
\vec{P}( \vec{\nu}, \{J \}) ={\cal I}( \vec{\nu}, \{J \}) -{\frac{1}{2}}\sum_{a = 1}^{n - 1}\sum_{j = 1}^{\ell}\sum_{\mu = 1}^{h_{j}^{( a )}}L G_{a 1}^{- 1}\delta_{j 1}.
\begin{cases}a \cdot \chi (b, c) - \chi (a \cdot b, c) + \chi (a, b \cdot c) - \chi (a, b) \cdot c = 0, R(a) \cdot \Phi (b) - S (a \cdot \Phi (b)) - \Phi \big( R(a) \cdot b + a \cdot R(b) \big) + \Phi (a) \cdot R (b) - S (\Phi (a) \cdot b) + \chi (R(a) , R(b)) - \kappa ~ \! \chi (a, b) - R \big( \chi(R(a), b) + \chi (a, R(b)) \big) = 0.\end{cases}
(N - 1)(|\lambda| - |\mu|) - (n(\lambda) - n(\mu)) \qquad{}=(N - 1)(|\lambda| - |\mu|) - ((N - 1)\lambda_N + (N - 2)(\lambda_{N-1}- \mu_{N-1}) + \dots + (\lambda_2 - \mu_2)){}\geq (N-1)(|\lambda| - |\mu|) - (N-1)(\lambda_N + \lambda_{N-1}- \mu_{N-1}+ \dots + \lambda_2 - \mu_2) \qquad{}= (N - 1)(\lambda_1 - \mu_1)\geq 0.
D(fg) = \sum_{j,k,l}\partial_j(f_k g_l) e_j e_k e_l = \sum_{j,k,l}(\partial_j f_k g_l + f_k \partial_j g_l) e_j e_k e_l &= (Df)g + \sum_{j,k,l}f_k \partial_j e_j e_k g_l e_l = (Df)g + \left(\sum_{j,k}f_k \partial_j e_j e_k \right)g,
\tilde{\tilde{F}}_{(2)}(x_1,\dots, x_n;t) =\frac{(1-t)^{n}}{\prod_{i=1}^{n-1}(1-t^i)}\sum_{\sigma \in S_n}\sigma\left((x_1^2+a)\prod_{i=1}^{n}\prod_{i < j}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right)= F_{(2)}+ a(1-t^n).
J_1[H_1]_a :=\Big(T_{ab}[H_1]-\frac{1}{2}V_1|H_1|^2g_{ab}\Big)T^b, J_1[H_2]_a :=\Big(T_{ab}[H_2]-\frac{1}{2}V_2|H_2|^2g_{ab}\Big)T^b.
=\sum_{\nu=0}^{n+|\textup{\textbf{m}}|-m_\alpha}b_{\nu,n}^{(\alpha)}\Phi_\nu(z)+\sum_{\nu=n+|\textup{\textbf{m}}|-m_\alpha+1}^{\infty}b_{\nu,n}^{(\alpha)}\Phi_\nu(z).
\sum_{i=1}^N \Lambda_X\big(\,\tau \omega_i(N)\big)= &\qquad\sum_{i=1}^N \Lambda_X\left(\tau \overline F\left(\frac{i-1}{N}\right)\right) +\sum_{i=1}^N \Lambda_X'\left(\tau \overline F\left(\frac{i-1}{N}\right)\right)\left[\frac{\tau}{2N}\overline F\,'\left(\frac{i-1}{N}\right)+O\left(\frac{1}{N^2}\right)\right]+O\left(\frac{1}{N^2}\right).
\begin{array}{ccc}W_1 = \left( \begin{array}{cc}1 0 0 1 0 0 \end{array}\right) W_2 = \left( \begin{array}{cc}1 0 0 0 0 1 \end{array}\right) W_3 = \left( \begin{array}{cc}0 0 1 0 0 1 \end{array}\right).\end{array}
\frac{g_u^{\ast 2}\ast f_r(x)}{g_u(x)}= e^{\Lambda_u}\frac{2}{\Lambda_u^2}\frac{f(x)}{g_u(x)}-\frac{2}{\Lambda_u^2}\frac{f_r(x)}{g_u(x)}-\frac{2}{\Lambda_u^2 g_u(x)}\sum_{n\neq 2}^\infty \frac{\Lambda_u^n}{n!}g_u^{\ast n}\ast f_r(x)
\left( ( D^{2})^{a b}+ g \epsilon^{a d c}A^{d}D^{c b}\right) f_{b}= j_{\mathrm{t o t}, 0}^{b}; ~ j_{\mathrm{t o t}, 0}^{b}= g \epsilon^{a b c}A_{i}^{a}E_{i}^{c}+ j_{0}^{b}~ ,
f(tx+\sum_{p \neq q}\phi^{(p)}\theta^{(p)}) + \sum_{i=1}^{N}\phi^{(q)}\theta^{(q)}_i \partial_i f(tx+\sum_{p \neq q}\phi^{(p)}\theta^{(p)}) = f(tx+\sum_{p=1}^{\mathcal{N}}\phi^{(p)}\theta^{(p)}).
\sinh \beta \sinh \gamma = \frac 1 2 \sinh^2(\alpha), \sinh \beta - \sinh \gamma = \frac{1}{\sqrt{2}}\sinh(2\alpha) \cos \theta, \tan(2\phi) = \frac{1}{\cosh \alpha \cos \theta}, \sin(2\omega_1) \cosh \beta = \frac{1}{\sqrt{2}}\sin \theta = - \sin(2\omega_2) \cosh\gamma.
\frac{\int_{0}^k |s|^{p-2}(s-k)_{-}\,ds}{\int_{k_\epsilon}^k |s|^{p-2}(s-k)_{-}\,ds}= 1 + \frac{\int_{0}^{k_\epsilon}|s|^{p-2}(s-k)_{-}\,ds}{\int_{k_\epsilon}^k |s|^{p-2}(s-k)_{-}\,ds}= 1 + I_{\epsilon}.
F_{\rm sing}(N, g_3 , g_4) \sim b\cdot \left[ \delta g_3 + \left( \frac{3^{1/4}}{4}- \frac{3^{7/4}}{2}\right) \delta g_4\right]^{5/2}+ b'\cdot [\delta g_4]^{-6}+ \cdots.
\Omega^{(r)}_{\lambda^+}f_{\lambda}(x_1,\dots,x_n)=\Omega^{(r-1)}_{(\lambda^*)^+}\sum_{\mu\in S_n\cdot (\lambda^{*})^{+}}T_{\lambda,\mu}(x_1,\dots,x_n) f_{\mu}(x_1,\dots,x_n)
C_{0}^{N}=(1+r)^{-N}\sum_{n=0}^{N}(\frac{q^{n}(1-q)^{N-n}}{\sum_{k=0}^{N}q^{k}(1-q)^{N-k}}){[S_{0}(1+b)^{n}(1+a)^{N-n}-K]}^{+}
K_{X,\mathrm{hyp}}^{\alpha,\delta}(t;z,w)= &K_{X,\mathrm{hyp}}(t;z,w)-\sum_{n:\,0\leq\lambda_{X,n}< \alpha}\varphi_{X,n}(z)\varphi_{X,n}(w)e^{-\lambda_{X,n}t}-\sum_{\gamma\in S_{\Gamma_{X}}(\delta;z,w)}K_{\mathbb{H}}(t;d_{\mathbb{H}}(z,\gamma w)).
\left< W_{R \times T}\right> = \frac{1}{N_{c}}\sum_{n}^{}\left< \Psi_{i j}( 0 ) \left| n \left> \right< n \right| \Psi_{j i}^{\dagger}( T ) \right> = \frac{1}{N_{c}}\sum_{n}^{}\left| \left< \Psi_{i j}( 0 ) | n \right> \right|^{2}\mathrm{e}^{- E_{n}T},
p_{H}-p_{0}=\frac{\Gamma}{V}(\frac{p_{H}\chi V_{0}}{2}-e_{0})or\frac{\rho_{0}C_{0}^{2}\chi}{(1-s\chi)^{2}}(1-\frac{\chi}{2}\frac{\Gamma}{V}V_{0})-p_{0}=-\frac{\Gamma}{V}e_{0}.
\frac{1}{2}[ Q ( \hat{K}_{L}( \hat{I}) , \hat{K}_{L}( \hat{I}) ] = - \frac{1}{2 \pi}\int_{0}^{\pi / 2}d \sigma c^{0}( \sigma ) \partial_{\sigma}F_{i k}^{I}( \sigma ) \eta_{I J}\partial_{\sigma}F_{k j}^{J}( \sigma )
A_n:=\begin{bmatrix}a_n -n^2\gamma^- \vphantom{\Big(}-n^2\gamma^+ b_n \vphantom{\Big(}\end{bmatrix}, B_n:=\begin{bmatrix}b_n^\star -n^2\gamma^- \vphantom{\Big(}-n^2\gamma^+ a_n^\star \vphantom{\Big(}\end{bmatrix}.
Z = \frac{{\displaystyle \int}{\cal D}{\bar \psi}{\cal D}\psi\,\,\,{\displaystyle e}^{iS ({\bar \psi}, \psi, A_-, A_+,A^5_-, A^5_+)}}{{\displaystyle \int}{\cal D}{\bar \psi}{\cal D}\psi\,\,\,{\displaystyle e}^{iS ({\bar \psi}, \psi)}},
:\chi(z)\chi(w): =\sum_{m, n\in \mathbb{Z}+1/2}:\chi_m \chi_n: z^{-m-1/2}w^{-n-1/2}= \sum_{m,n\in \mathbb{Z}}:\chi_{-m-\frac{1}{2}}\chi_{-n-\frac{1}{2}}:z^{m}w^{n},
\begin{aligned}\|(1+3x)^{\frac{2}{3}}(\tilde{u}_k)_x\|_{L^\infty_tL^2_x}\leq \|(1+3x)^{\frac{2}{3}}(u_k)_x\phi_k\|_{L^\infty_tL^2_x}+\|(1+3x)^{\frac{2}{3}}(\phi_k)_xu_k\|_{L^\infty_tL^2_x}\leq C(T)\|r_k^2(u_k)_x\|_{L^\infty_tL^2_x(Q_{k,T})}+\|(1+3x)^{\frac{2}{3}}(\phi_k)_xu_k\|_{L^\infty_tL^2_x}\leq C(T),\end{aligned}
\begin{aligned}x_{k}= \frac{\sin \frac{2k\pi}{n}\sin \left(\beta - (-1)^k\frac{\pi}{n}\right)}{\sin \frac{2\pi}{n}}, x_{k+\frac{n}{2}+1}= x_{k}+ (-1)^k \sin \frac{2k\pi}{n}, y_{k}= \frac{1}{2}+ \frac{\cos \frac{2k\pi}{n}\sin \left(\beta - (-1)^k\frac{\pi}{n}\right)}{\sin \frac{2\pi}{n}}, y_{k+\frac{n}{2}+1}= y_{k}+ (-1)^k \cos \frac{2k\pi}{n}.\end{aligned}
g_{\mu \nu}\, dx^{\mu}\, dx^{\nu}\simeq -dt^2 + \sum_{i=1}^d \, t^{2 p_i (x)}\, (\omega^i)^2 \, , \; \;\varphi \simeq p_{\varphi}\, (x) \, \ln \, t \, +\psi \, (x) \, ,
\int^{\infty}_0\!\!e^{i\varphi(x, y)t}\sum^\infty_{j=0}a_j(x, y)t^{1-j}dt =\lim_{\varepsilon\to0+}\int^{\infty}_0\!\!e^{-t\bigr(-i(\varphi(x, y)+i\varepsilon)\bigr)}\sum^\infty_{j=0}a_j(x, y)t^{1-j}dt.
\pi ( u ) = \frac{2 k - 1}{4 \pi}\int_{\tau = 0}^{\infty}d \tau \sinh \tau \int_{0}^{2 \pi}d \phi \left[ \cos \frac{\theta_{0}}{2}- i \sin \frac{\theta_{0}}{2}\hat{n}. \hat{r}\right]^{- 2 k}
\pi_n(a) \tau_n(\xi) = \pi^\infty(j_n(a)) u \tau^\infty(j_n(\xi)) = u \pi^\infty(j_{n+1}(a)) \tau^\infty(j_n(\xi)) &= u \tau^\infty(j_{n+1}(a) j_n(\xi)) = u \tau^\infty(j_n(a \xi)) = \tau_n(a\xi).
V_{long}= -\int \frac{ds}{s^{3/2}}\frac{e^{-b^2 s}}{8\sqrt{\pi}c_1 s c_2 s}(c_{1}^{2}-c_{2}^{2})^{2}s^4 (\frac{\sqrt{\pi}}{L_1 s^{1/2}})=-\frac{(c_{1}^{2}-c_{2}^{2})^{2}}{8 c_1 L_1 c_2 b^{2}}.
\begin{aligned}\frac{1}{\tau}\big\|(v_n-v_{n-1})_{n=k}^N\big\|_{L^p(X)}+ \big\|(v_n )_{n=k}^N\big\|_{L^p(D)}&\le C\Big(\big\|(f_n)_{n=k}^N\big\|_{L^p(X)}+\frac{1}{\tau}\big\|(v_i)_{i=0}^{k-1}\big\|_{L^p(X)}+ \big\|(v_i )_{i=0}^{k-1}\big\|_{L^p(D)}\Big) , \end{aligned}
{\frac{d_{o e}A_{e e}}{1 - d_{o o}A_{o o}}}= -{\frac{B_{e , k_{0}}D_{1}}{B_{o , k_{0}}D_{2}}}={\frac{B_{e , k_{0}}D_{2}}{B_{o , k_{0}}D_{1}}}, ~ \Rightarrow D_{1}^{2}= - D_{2}^{2}
\mathcal{A}_m(t)=\sum^N_{j=1}\chi_j(x)\Bigr(e^{-m\varphi_j(z)-im\theta}e^{-t\Box^+_{B_j,m}}(z,w)e^{m\varphi_j(w)+im\eta}\Bigr)\widetilde\chi_j(y), &\mathcal{P}_m(t)=\mathcal{A}_m(t)\circ Q^+_m.
f_1(x) = 1 + x^{4}+ x^{8}, f_2(x) = 4( x + x^{5}+ x^{6}), f_3(x) = 4( x^{2}+ x^{3}+ x^{7}), f_4(x) = 6( x^{2}+ x^{4}+ x^{6}),f_5(x) = 12( x^{3}+ x^{4}+ x^{5}).
\|\xi-\xi\chi_{A_n}\|_{L_2}\leq \|\xi\|_{L_2}\|\|\chi_{[0,\alpha)}-\chi_{A_n}\|_{L_2}=\|\xi\|_{L_2}m(A_n^c) &\leq \frac1n\|\xi\|_{L_2}\to 0n\to\infty.
\lbrack \frac{4}{5 ! ( D + 5 )}\epsilon_{\mu \nu \lambda}\eta_{\beta \gamma}^{T}\alpha_{- 1}^{\mu \nu \lambda \beta \gamma}+ \epsilon_{\mu \nu \lambda}( \alpha_{- 1}^{\mu}\alpha_{- 2}^{\nu \lambda}- \frac{4}{3}\alpha_{- 1}^{\mu \nu}\alpha_{- 3}^{\lambda}) ] \left\vert 0 , k \right\rangle ,
\mathrm{Var}(\hat{p}^{n,k}(x)) =\frac{\exp(2(x-a))}{n}\left((a-x)+\frac{\left[\left(a-x\right)^2+(a-x)\right](k-1)}{2n}+{\rm o}\left(\frac{1}{n}\right)\right), T^{n,k}(x) =n\left((a-x)\left[\frac{1}{k}-\frac{k-1}{2kn}\right]+\frac{3k-1}{2kn}+{\rm o}\left(\frac{1}{n}\right)\right),
\mathcal{R}_2\quad\equiv \begin{cases}f_1=y^1_{10}y^2_{01}-y^1_{01}y^2_{10}-1=0\,, f_2=y^1_{20}y^2_{01}+y^1_{10}y^2_{11}-y^1_{11}y^2_{10}-y^1_{01}y^2_{20}=0\,, f_3=y^1_{11}y^2_{01}+y^1_{10}y^2_{02}-y^1_{02}y^2_{10}-y^1_{01}y^2_{11}=0\,, f_4=y^1_{20}+ y^1_{02}=0\,, f_5=y^2_{20}+ y^2_{02}=0\,, \end{cases}
v_p \left(\lambda\right) =v_p \left((p^2-1)!\right)-2v_p\left(\left(\frac{p^2-1}{2}\right)!\right) =\frac{p^2-1-2(p-1)}{p-1}-2\cdot\left(\frac{\frac{p^2-1}{2}-2\left(\frac{p-1}{2}\right)}{p-1}\right)=p-1-(p-1)=0,
\bar{\mathbf{x}}_i={\mathbf{x}}_i+ \mathbf{Q}\mathbf{P}^{-1}_f \left( \mathbf{P}_1{\mathbf{d}}_{i-1}+\mathbf{P}_1\mathbf{A}^{-1}\mathbf{w}_{i-1}-\mathbf{P}_2\mathbf{d}_i \right).
[\psi_i^\epsilon(z),\psi_j^{\epsilon'}(w)]=0 \,, \psi^\pm_{i,\mp b^\pm_i}\cdot (\psi^\pm_{i,\mp b^\pm_i})^{-1}= (\psi^\pm_{i,\mp b^\pm_i})^{-1}\cdot \psi^\pm_{i,\mp b^\pm_i}=1 \,,
Z_2(0)=\frac{1}{4}\,\Gamma\left(\frac{3}{2}\right)\Gamma\left(\frac{1}{2}\right) \frac{1}{8}\left(\lim_{s\to0}^{}\zeta(1+s)+\frac{1}{2}\right)=\frac{\pi}{64}\left(\left.\frac{1}{s}\,\right|_{s\to0}+\gamma\right)+\frac{\pi}{128}\,{.}
W_{N + K}( x_{1}, \cdots , x_{N}= 1 ) = \left. \frac{( - 1 )^{N + K}}{( N + K ) !}\frac{d^{N + K}}{d t^{N + K}}\log \left( \sum_{i = 0}^{N}( - 1 )^{i}x_{i}t^{i}\right) \right|_{t = 0},
G^t_a =xy^a+\sum_{s=1}^{a-1}(a-1-s)xy^{a-s}G^{bt}_s+q\sum_{s=1}^{a-1}xy^{a-s}G_s^{bt}+p\sum_{s=1}^{a-1}xy^{a-s}G_s^t &+pqxG_a^t+\sum_{s=2}^ax\frac{y^{a+1-s}-y^a}{1-y}G_s^t+\sum_{s\geq a+1}x\left(\frac{y-y^a}{1-y}+q\right)G_s^t.
h(z,w,\eta)=\Big(z_1(\frac{1-|\eta|^2}{1-w\bar \eta})^{\alpha_1},\dots,z_n(\frac{1-|\eta|^2}{1-w\bar \eta})^{\alpha_n}\Big), &l(z,w,\eta)=\Big(z_1e^{\gamma_1(w\bar{\eta}-|\eta|^2)},\dots,z_ne^{\gamma_n(w\bar{\eta}-|\eta|^2)}\Big).
\beta_{2}(m)=\lim_{v \to \infty}\dfrac{r_{2^v}^{\operatorname{Good}}(m)}{2^{3v}}+\lim_{v \to \infty}\dfrac{r_{2^v}^{\operatorname{Zero}}(m)}{2^{3v}}+\lim_{v \to \infty}\dfrac{r_{2^v}^{\operatorname{Bad-I}}(m)}{2^{3v}}.
{}^{M\!L\!}{\Phi}_{p}(b;c;z)=\frac{1}{B(b,c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}e^{zt}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right)dt,
\alpha \left( \beta+\gamma \right) +A\beta+A\gamma =\left( a+A\right)\left( \beta+\gamma \right) +A\beta+A\gamma & =a\left( \beta+\gamma \right) +A\left( \beta+\gamma \right)+A\beta+A\gamma & =a\beta+a\gamma+A\left( \beta+\gamma \right) +A\beta+A\gamma
I(X^n\rightarrow{Y}^n) = \sup_{{\cal S}_{0,n}\otimes{\cal R}_{0,n}}\sum^n_{t=0}\int_{{{\cal X}^{t}}\times{\cal Y}^{t}}\log\Bigg(\frac{d{r}_t(\cdot|x^{t-1},y^{t})}{dp_t(\cdot|x^{t-1},y^{t-1})}(x_t)\frac{ds_t(\cdot|y^{t-1},x^{t-1})}{d\nu_{t}^p(\cdot|y^{t-1})}(y_t)\Bigg){\bf P}^p(dx^t, dy^t)
\int_{\Gamma_t}c_t \, \mathrm{d}\mathcal{H}^d = \int_{\Gamma_t}(c+tw) \circ \theta_t^{-1}+ m_t \, \mathrm{d}\mathcal{H}^d &= \int_{\Gamma_t}(c+tw) \circ \theta_t^{-1}\, \mathrm{d}\mathcal{H}^d + \left( m - \int_{\Gamma_t}(c+tw) \circ \theta_t^{-1}\mathrm{d}\mathcal{H}^d \right) = m
\min_{\mathbf{x}\in X}\max_{\mathbf{c}\in \mathcal{C}}\ell(\mathbf{x}, \mathbf{c}) = \min_{\mathbf{x}, t}t \mbox{s.t.}\max_{\mathbf{c}\in \mathcal{C}}\Big( \ell(\mathbf{x}, \mathbf{c}) - t \Big) \leq 0 & \mathbf{x}\in X,
|\mathcal{DD}_{(s,s+d,s+2d)}|=\sum_{i=0}^{\lfloor(s-1)/2\rfloor}\binom{\lfloor (s+d-2)/2\rfloor}{\lfloor i/2 \rfloor, \lfloor (d+i)/2\rfloor, \lfloor(s-1)/2\rfloor -i}.
K^*_0=\begin{bmatrix}q 1 -\alpha 0 -\overline\alpha 0 0 q\alpha 0 q\overline\alpha 0 \overline\alpha 0 \alpha J(2r,3r) 0 0 0 \overline\alpha J(r,r) 0 \alpha J(3r,3r) 0 \alpha 0 \overline\alpha J(2r,r) 0\end{bmatrix}
v^k(b_k,A_k)\triangleq \mathbb{E}_{h_k}\Big[\max_{p_k(\Gamma_k)\in{\cal D}_{A_k}(b_k)}\big\{\log(1 + p_k(\Gamma_k)h_k) + \gamma u^{k}(b_k,p_k(\Gamma_k),A_k)\big\}\Big]\ , k=1,2,\ldots,K\ ,
q_G(e)-\bar{q}_G(e) =q_G(e)+\bar{q}_G(e) &=\# \{f\in E^{\rm ori}(G)|\e\prec f\prec \bar{e}\}\{\rm mod\}2 &=\left(\sum_{f\in E(G)}\mu^2(f)\cdot \mu^2(e) \right)=(\xi^2_G\cdot \mu^2(e)),
L(u,v) = \frac{1}{\alpha}\left(\cos(\alpha u) \cos(\alpha v) w_1 + \cos(\alpha u) \sin(\alpha v) w_2 + \sin(\alpha u) w_3\right) e_3(u,v) = -\cos(\alpha u) \cos(\alpha v) w_1 - \cos(\alpha u) \sin(\alpha v) w_2 - \sin(\alpha u) w_3
- \frac{1}{2}\sum_{n_{1}}\log \frac{\cos [ \pi ( \varphi_{2}+ \tau_{1}( n_{1}+ \varphi_{1}) - i \tau_{2}\mid n_{1}+ \varphi_{1}\mid ) ]}{\cos [ \pi ( \tau_{1}n_{1}- i \tau_{2}\mid n_{1}\mid ) ]}.
\int_X\rho^{1-2\gamma}\langle\nabla\hat{u}_\alpha,\nabla\psi\rangle_g \,dv_g-\int_M|\hat{u}_\alpha|^{2^*-2}\hat{u}_\alpha\psi \,d\sigma_{\hat{h}}=o(1)\|\psi\|_{W^{1,2}(X,\rho^{1-2\gamma})}.
\Sigma_2 : = p^{(w-1) h_p}\sum_{m= \max\{0,k_p-h_p\}}^{\infty}\sum_{n=m+h_p-k_p+1}^{\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m( \frac{1}{2}+s_1 -z)}p^{n(\frac{1}{2}+s_2+z)}}.
G^{D}( z , w ) = \frac{1}{4 \pi}\ln \left| \frac{\theta_{1}( z - w | \tau )}{\theta_{1}( z - \overline{{{w}}}| \tau )}\right|^{2}- 2 \frac{( z - \overline{{{z}}}) ( w - \overline{{{w}}})}{I m \tau}
T^*=\frac{\eta \rho \sum_{n=1}^{N}g_n}{\eta \rho \sum_{n=1}^{N}g_n+\frac{\eta \rho \sum_{n=1}^{N}g_n-1}{W(\frac{\eta \rho \sum_{n=1}^{N}g_n-1}{e})}-1},
|-\lambda(t) \langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right)v_{5}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r=R\lambda(t)},\phi_{0}\rangle| \leq \frac{C}{t^{7/2}\log^{3b-3+\frac{5N}{2}}(t)}
I_{4k+1,k}\cap [4k a + 1, (4k+1)a] = [4ka + 1, 4 ka + 1 + 6k]_2, I_{4k+2,k}\cap [4k a + 1, (4k+1)a] = [4ka + 4, 4 k a + 4 + 6 k]_2.
\lvert f_2(gh)-f_2(g)-f_2(h)\rvert= \lvert f_2(gh)-f_1(gh)+f_1(gh) &-f_2(g)-f_1(g)+f_1(g) &-f_2(h)-f_1(h)+f_1(h)\rvert \leq \lvert f_1(gh)-f_1(g)-f_1(h)\rvert &+ \lvert f_2(gh)-f_1(gh)\rvert &+ \lvert f_2(g)-f_1(g)\rvert &+\lvert f_2(h)-f_1(h)\rvert \leq D(f_1)+3\lVert f_1-f_2 \rVert_\infty < \infty.
\varepsilon_{n}^2 :\,= \min_{\lambda\in\Lambda}\, \Big\{\min_{\varepsilon_\lambda>0}\Big\{\varepsilon^2_{\lambda}+ \Big(-\frac{1}{n}\log \Pi_\lambda(B_n(\theta^\ast,\,\varepsilon_\lambda))\Big)\Big\}+ \Big(-\frac{1}{n}\log \pi_\lambda\Big)\Big\}+ \frac{\log |\Lambda|}{n},
q^{\alpha^{2}/2}A_{q}(-q^{\alpha+n})=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{(-q^{1/2+n}e^{ix};q)_{\infty}\exp\left(\frac{x^{2}}{\log q^{2}}+i\alpha x\right)}{\sqrt{\log q^{-1}}}dx,
\mathcal{S}_{\alpha}:=\Bigg\{\bigcup_{i=1}^{d}S^{i}_{\beta}\colon d \leq S^{1}_{\beta}< S^{2}_{\beta}< \cdots < S^{d}_{\beta},\,\, \big\{S^{i}_{\beta}\big\}_{i=1}^d \subset \mathcal{S}_{\beta}d \in \mathbb{N}\Bigg\}\cup\big\{\emptyset\big\};
\left|(1-\chi(4r/r')-\chi(4r'/r) )(\mathcal{L}_V+1)^{-1}(\lambda z,\lambda z')\right|\lesssim\begin{cases}\lambda^{2-n}d(z,z')^{2-n}, d(z,z')\leq 1/\lambda; \lambda^{-N}d(z,z')^{-N}, d(z,z')\geq 1/\lambda.\end{cases}
y_{n}=\sum_{k=0}^{n}f^{(k)}(a)B_{n,k}(x_{1},...,x_{n-k+1})\Leftrightarrow x_{n}=\sum_{k=0}^{n}g^{(k)}(f(a))B_{n,k}(y_{1},...,y_{n-k+1}).
P^1_1=-\frac{1}{2}(A_1A_5x_1^2+A_1B_5x_1x_2+B_1A_5x_1x_2+B_1B_5x_2^2+(A_5C_1 & +A_1C_5)x_1 +(C_1B_5+B_1C_5)x_2+C_1C_5)-\frac{1}{4}\left(\sum_{l+k\leq 2}M_{lk}x_1^{l}x_2^k \right), &L_1^1=\frac{1}{2}(A_5x_1+B_5x_2+C_5)+\frac{1}{4}(A_1Ex_1+B_1Ex_2+C_1E),
\Phi ({\bf r}_{1}^{( 1 )}, \ldots ,{\bf r}_{n_{M}}^{( M )}, t ) = e^{- \frac{i}{\hbar c}\Lambda}\bar{\Phi}({\bf r}_{1}^{( 1 )}, \ldots ,{\bf r}_{n_{M}}^{( M )}, t ) \, ,
p(N^+[S]) \ge p(N^+[v])+ p\big(N^+[S']\cap N^o(v)\big) &= p(v) + p(N^+(v)) + p(N_{G[N^o(v)]}^+[S']) & \geq p(v) + \big(p(N^+(v))+p(N^-(v)\big)/2 + p(N^o(v))/2 = p(V)/2.
- \partial_{0}\left( \begin{array}{c}{{-{\frac{\kappa}{2 e}}{\frac{\delta A_{+}}{\phi^{*}}}}}\\{{\delta \phi}}\\ \end{array}\right) = D_{( \infty )}\left( \begin{array}{c}{{-{\frac{\kappa}{2 e}}{\frac{\delta A_{+}}{\phi^{*}}}}}\\{{\delta \phi}}\\ \end{array}\right) .
P_n^{(\ell)}(x)=\left|\begin{array}{cccc}m_{\ell}m_{\ell+1}\cdots \mathbb{I}_p \vdots \vdots \vdots m_{\ell+n-1}m_{\ell+n}\cdots x^{n-1}\mathbb{I}_p m_{\ell+n}m_{\ell+n+1}\cdots \boxed{x^n\mathbb{I}_p}\end{array}\right|,m_{\ell+k}=\int_{\mathbb{R}}x^kd\mu^{(\ell)}(x),d\mu^{(\ell)}(x)=x^\ell d\mu(x).
u(t,x)= \ \mathbf E\left[\phi_0\left(X^{x,\alpha}(\tau_0(t))\right)\mathbf 1_{\{\tau_0(t)<\tau_\Omega(x)\}}\right] &\ +\mathbf E \left[\int_0^{\tau_0(t)\wedge \tau_\Omega(x)}f \left(-X^{t,\beta}(s),X^{x,\alpha}(s)\right)ds\right],
2^{jn\alpha_2/2}2^{jn(\alpha_1-\alpha_2)(1/q-1/2)}=2^{j\widetilde{A_2}(\mathbf{p},q,\alpha_1,\alpha_2)}=2^{j\widetilde{A_3}(\mathbf{p},q,\alpha_1,\alpha_2)}
\|\nabla^lE_\Omega u-(\nabla^lu)_{Q_0^*}\|_{\infty,Q_0}\leq\left\|\nabla^{l}\sum_{\emptyset\neq Q_0\cap Q\in\mathcal{W}_3}\varphi_Q(\pi_{Q^*}-\pi_{Q_0^*})\right\|_{\infty,Q_0}&+\|\nabla^l\pi_{Q_0^*}u-(\nabla^lu)_{Q_0^*}\|_{\infty,Q_0}=\mathbf{I}+\mathbf{II}.
\|\varphi\|_{*}\leq\Bigg(o(1)+\|h\|_{**}+\frac{\sum_{j=1}^{k}\frac{1}{(1+\lambda|y-\xi_{j}|)^{\frac{N-2}2+\tau+\theta}}}{\sum_{j=1}^{k}\frac{1}{(1+\lambda|y-\xi_{j}|)^{\frac{N-2}2+\tau}}}\Bigg).
\chi_{n}([g, \epsilon] [k(t),1]) =\chi_{n}([gk(t), \epsilon]) &=\chi_{n}([p_g, \epsilon][k_gk(t), 1]) &=\chi_{n}([p_g, \epsilon])\chi_{n}([k_g, 1])\chi_{n}([k(t), 1]) &=\chi_{n}([g, \epsilon]) \chi_{n}([k(t), 1]).
\frac{\int_{0}^{1}\frac{f(e^{2\pi ix})}{1-ze^{-2\pi ix}}\,d\mu(x)}{\int_{0}^{1}\frac{1}{1-ze^{-2\pi ix}}\,d\mu(x)}=\left(\sum_{n=0}^{\infty}\langle f,e_n\rangle z^n\right)\left(\sum_{m=0}^{\infty}\alpha_mz^m\right) &=\sum_{n=0}^{\infty}\left(\sum_{i=0}^{n}\langle f,\overline{\alpha_{n-i}}e_i\rangle\right)z^n &=\sum_{n=0}^{\infty}\langle f,g_n\rangle z^n.
\gamma^{\mu}{\stackrel{\leftrightarrow}{\nabla}}_{\mu}= e_{m}^{\mu}[ \gamma^{m}( \stackrel{\leftrightarrow}{\partial}_{\mu}- i g A_{\mu}^{a}\tau_{a}) +{\frac{i}{2}}\Gamma_{n \mu p}\{\gamma^{m}; \Sigma^{n p}\}] ,
\lim_{l\to\infty}\nu_{\mathcal P_l}\left(\bigg\{\frac{1}{k_l(n_l+M_l)}\big(S^{k_l(n_l+M_l)}h -\mathbb{E}_{\mathcal P_{l}}(S^{k_l(n_l+M_l)}h)\big)\leqslant k_l^{-1/2+\eta}\bigg\}\right) &=\lim_{l\to\infty}\mathcal N(k_l^{1/2+\eta}(n_l+M_l)/s_l)=1
Q^{a}= - i \frac{2 J}{1 + \epsilon \vert \xi \vert^{2}}\left( T_{0 0}^{a}+ T_{0 i}^{a}\xi_{i}+ \epsilon T_{i 0}^{a}\bar{\xi}_{i}+ \epsilon T_{i j}^{a}\bar{\xi}_{i}\xi_{j}\right) .
g_m ={\sf P}\big({\sf br}_{\S(\varphi,T)}(v)=m \,\big|\,\S(\varphi,T)\not=\phi, \,{\sf br}_{\S(\varphi,T)}(v)\not= 1 \big) &=(1-p_t)^{-m}p_t^{m-1}{\sum\limits_{k = m}^\infty \binom{k}{m}p^k q_k \over 1-(1-p_t)^{-1}\sum\limits_{k = 2}^\infty k p^k q_k}~={p_t^{m-1}\over m!}Q^{(m)}(1-p_t) \,(1-Q'(1-p_t))^{-1}.
\theta \circ \bar{H}_{*}(t_{E}) = \theta( \delta(\bar{E}) t_{E}\delta(E)^{-1}) = \delta(\bar{E}) t_{e}\gamma_{E}^{-1}\delta(E)^{-1}& = \delta(\bar{e}) t_{e}\delta(e)^{-1}H_{*}(\gamma_E)^{-1}& = H_{*}(t_{e}\gamma_E^{-1}) & = H_{*}\circ \theta(t_{E})
M^{2}\left( \begin{matrix}{{\phi^{\prime}}}\\{{{\overline{{{\phi}}}}^{* \prime}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{m^{2}}}&{{m^{2}x}}\\{{m^{2}x}}&{{m^{2}}}\\ \end{matrix}\right) \left( \begin{matrix}{{\phi^{\prime}}}\\{{{\overline{{{\phi}}}}^{* \prime}}}\\ \end{matrix}\right) \ .
F(\{x_j\};\{s_j\};r) =\sum_{l_1,\ldots,l_r=1}^\infty \prod_{j=1}^{r}\left(1-e^{-\sum_{\nu=j}^r x_\nu}\right)^{l_j-1}\frac{1}{\prod_{j=1}^{r}\Gamma(s_j)}& \times \int_0^\infty\cdots\int_0^\infty \prod_{j=1}^{r}\left\{t_j^{s_j-1}\exp\left(-(\sum_{\nu=1}^{j}l_\nu)t_j\right)\right\}\prod_{j=1}^{r}dt_j.
\begin{array}{rl}\displaystyle E(L)= -\displaystyle\frac{\pi}{6L}\left( c-12(\Delta_{+}+\Delta_{-})\right) +\ldots \: , P(L)= \displaystyle\frac{2\pi}{L}\left( \Delta_{+}-\Delta_{-}\right) +\ldots \: .\end{array}
\begin{aligned}|G_n'(M(t)+xf(t,s))-G_n'(M(t))| = |G_n''(z_0)|\cdot|xf(t,s)| \le \tilde C (1+|z_0|^{q})\cdot|xf(t,s)| \le \tilde C(1+ |M(t)|^{q}+ |M(t)+ x f(t,s)|^{q})\cdot|xf(t,s)|.\end{aligned}
|\nabla U|=\left|\nabla \frac{|Rc|^2}{R}\right|=\left| \frac{R|Rc| \nabla |Rc|-|Rc|^2 \nabla R}{R^2}\right| \leq C \frac{|Rc|^2|\nabla Rm|}{R^2}=CU \frac{|\nabla Rm|}{R}\leq CLU,
d_p(f)e\langle x_1\wedge\ldots\wedge x_p\rangle =\sum_{k=1}^{k=p}(-1)^{k+1}e(x_k-f(x_k))\langle x_1\wedge\ldots \wedge\Hat{x_k}\wedge\ldots \wedge x_p\rangle &+\sum_{1\le k<l \le p}(-1)^{k+l}e\langle[x_k,x_l]\wedge x_1\wedge\ldots\wedge \Hat{x_k}\wedge\ldots\wedge \Hat{x_l}\wedge\ldots\wedge x_p\rangle,
\|\phi(t)-\overline{\phi}\|^2_3 \leq \Big(\|\phi_0-\overline{\phi}\|^2_{3}+C\int_0^t \big(\|\nabla \psi\|^2_2\|v\|^2_3+|\phi\nabla^4 v|^2_2\big)s\Big) \exp\Big(C\int_0^t \big(\| v(s)\|_{3}+1\big)s\Big),
p\left( x_{2j-1}x_{2j}|\mu_{2j-1}\sigma_{2j-1}\right)\overset{}{=}\frac{1}{2\pi\sigma_{2j-1}^{2}}\exp\left[-\frac{\left( x_{2j-1}-\mu_{2j-1}\right)^{2}+\left[ x_{2j}-\mu_{2j}\left(\mu_{2j-1}\sigma_{2j-1}\right) \right]^{2}}{2\sigma_{2j-1}^{2}}\right]
<{\cal O}_1\cdots{\cal O}_k \,{\rm exp}\left\{\sum_{k=0}^4\sum_{p=4-k}^{4n}\sum_{\beta_k=1}^{b^k({\cal M})}\sum_{\alpha_{p}=1}^{b^p({\cal N})}s^{p,k}_{\alpha_p,\beta_k}{\cal D}^{\alpha_p,\beta_k}_{p,k}\right\}>.
\|r_h^+ u - r_h u\|_{0,2}\le C h^{\sigma'}\, \Big[ h\|r_h^+ u - u\|_{1,\eta}+ h\|r_h u - u\|_{1,\eta}+ \|r_h^+ u - u\|_{0,\eta}+ \|r_h u - u\|_{0,\eta}&+ \left(h^\mu \|r_h u - u\|_{\mu,\eta}\right)^{1/2}\left(\|r_h^+ u - u\|_{0,\eta}+ \|r_h u - u\|_{0,\eta}\right)^{1/2}&+ h^\mu \|r_h u - u\|_{\mu,\eta}\Big],