harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \forall n,\; t_{j,n}=0 \right) \Longrightarrow \exists R_j>0,\;\forall n,\;\rho_{j,n}=\lambda_{j,n}R_j \lim_{n\to\infty}-\frac{t_{j,n}}{\lambda_{j,n}}\in \{\pm\infty\}\Longrightarrow \exists \rho_j,\;\rho_{j,n}=\left|\rho_j\lambda_{j,n}-t_{j,n}\right|\sim|t_{j,n}|n\to\infty.
L_{B}= -{\frac{1}{4}}H^{2}{}_{\mu \nu}( B ) +{\frac{m}{4 !}}\epsilon^{\mu \nu \rho \omega}B_{\mu}\partial_{[ \nu}A_{\rho \sigma ]}-{\frac{1}{2 \cdot 3 !}}F^{2}{}_{\mu \nu \rho}( A )
e_2(\{1,2,3\})+e_2(\{1,4,5\})+e_2(\{6,3,4\})+e_2(\{6,2,5\}) = e_2(\{1,3,4\})+e_2(\{1,2,5\})+e_2(\{6,2,3\})+e_2(\{6,4,5\}).
L_{\mu\nu}^{-1}(\tau,\tau^{\prime}) -\frac{{\cal F}_{\mu}^{\alpha}}{2}\int_{0}^1\epsilon(\tau-\tau^{\prime\prime}) L_{\alpha\nu}^{-1}(\tau^{\prime\prime},\tau^{\prime})d\tau^{\prime\prime}=\eta_{\mu\nu}\delta(\tau-\tau^{\prime})
ds^{2}=-{(r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})\over C^{2d}}dt^{2}+{r^{2d-2}C^{2}\over (r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})}dr^{2}+C^{2}d\Omega_{D-2}^{2}
_{r}\phi_s \left({{a_1, \ldots, a_{r}}\atop{b_1, \ldots,b_s}}; q, z \right) =\sum_{n=0}^\infty\frac{(a_1, \ldots, a_r; q)_n}{(q, b_1, \ldots, b_s; q)_n}\left((-1)^n q^{n(n-1)/2}\right)^{1+s-r}z^n.
|D^m U_k(x)|\le \sum_{j=0}^{N_k}|\gamma_j^{(k)}||D^m u_j(x)|\le C_m\sum_{j=0}^{N_k}|\gamma_j^{(k)}|R_0^{-j}(j+|m|)^{|m|}&\le C_m\Big(\sum_{j=0}^{N_k}|\gamma_j^{(k)}|^2\Big)^{1/2}\Big(\sum_{j=0}^{N_k}R_0^{-2j}(j+|m|)^{2|m|}\Big)^{1/2}\le C_m
\le \frac{2^{1/q}q}{q-1}\left\| \sum_{1 \le n \le m}\frac{4 w_n \mathbf{1}_{u_{n}}}{\mu u_{n}}\right\|_p= \frac{2^{1/q}4 q}{q-1}\left\| \sum_{1 \le n \le m}\frac{w_n \mathbf{1}_{u_{n}}}{\mu u_{n}}\right\|_p.
\sum_{k=0}^{(n-1)/2}(-1)^k q^{k^2}[4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}\equiv q^{(n-1)^2/4}[n] (-1)^{(n-1)/2}\pmod{[n]\Phi_n(q)^2}
\mathcal{G}^k_{\mathrm{Id}}(\vec{a};\vec{b})=\langle \vec{a},\vec{b}\rangle \int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\mathbf{e}\bigg(\frac{(i)^k\sqrt[k]{\sum \limits_{j=1}^{n}x^k_j}}{||\vec{a}||^{k+1}+||\vec{b}||^{k+1}}\bigg)dx_1dx_2\cdots dx_n.
T^{A B}( \tau , \vec{\sigma}) = - [{\frac{2}{\sqrt{g}}}{\frac{\delta S}{\delta g_{A B}}}] ( \tau , \vec{\sigma}) = - \sum_{i = 1}^{N}\delta^{3}( \vec{\sigma}-{\vec{\eta}}_{i}( \tau ) ){\frac{m_{i}{\dot{\eta}}_{i}^{A}( \tau ){\dot{\eta}}_{i}^{B}( \tau )}{\sqrt{g_{\tau \tau}+ 2 g_{\tau u}{\dot{\eta}}_{i}^{u}+ g_{u v}{\dot{\eta}}_{i}^{u}{\dot{\eta}}_{i}^{v}}}}( \tau , \vec{\sigma}) .
(\nabla_\xi \phi)(x) = \nabla_\xi(\phi x)-\phi(\nabla_\xi x)=\nabla_\xi(\xi\times x)-\xi\times \nabla_\xi x &=(\nabla_\xi\xi\times x)+(\xi\times \nabla_\xi x)-(\xi\times \nabla_\xi x)=\nabla_\xi\xi\times x,
\phi'(1) = \lim_{u \to 1}\frac{\phi'(u)}{u}= \lim_{x \to 0}\frac{\alpha(x)}{x \ln 2}&= \frac{1}{\ln 2}\cdot \lim_{x \to 0}\alpha'(x) = \frac{1}{\ln 2}\cdot \lim_{x \to 0}\frac{1}{1-x^2}= \frac{1}{\ln 2}
a_{\mathcal{F}_{k l,I}}(\xi^i_{\mathcal{F}_{k l,I}},v)=\lambda_{\mathcal{F}_{k l,I}}^i b_{\mathcal{F}_{k l}}(\xi^i_{\mathcal{F}_{k l,I}}, v)\forall v\in V_h^0(\mathcal{F}_{k l})
\psi_{m,\ell=0}(r)=N_mr^{-\frac{(D-1)}{2}}(\cosh{r})^{\frac{1}{2}}\frac{(1-\tanh{r})^{\frac{\lambda}{2}}(1+\tanh{r})^{\frac{\delta}{2}}}{[\frac{2Q}{(2P_1-2m-1)}\tanh{r}-(2P_1-2m-1)]}\hat{P}_{m+1}^{(\lambda-\frac{1}{2},\delta-\frac{1}{2})}(\tanh{r})
{H}_{\theta \theta}={\frac{{r}^{3}}{2}}({\frac{{H}_{rr}}{A}})'+{\frac{{r}^{2}{H}_{rr}}{A}}+{\frac{{r}^{3}B'{H}_{rr}}{4AB}}+{\frac{{r}^{3}B'{H}_{tt}}{4{B}^{2}}}.
\begin{array}{r c l c r c l}{\psi_{-}}&{\rightarrow}&{e^{- i \theta}g^{- 1}\psi_{-}}&{\quad \quad}&{\psi_{+}}&{\rightarrow}&{\psi_{+}\nonumber}\\{\psi_{-}^{\dagger}}&{\rightarrow}&{\psi_{-}^{\dagger}g e^{i \theta}}&{\quad \quad}&{\psi_{+}^{\dagger}}&{\rightarrow}&{\psi_{+}^{\dagger}}\\ \end{array}\quad ,
C \sum_{i \neq j}P_i \overline{Q}_{L,j}g (P_i \overline{Q}_{L,j})^{\dagger}+ \frac{1}{4}\mathcal Q_4^{}= C \widehat g(0) \frac{n_0 n_+^H}{\vert \Lambda \vert}+ \frac{1}{4}\mathcal Q_4^{}.
\frac1\tau\, \big\|(u_i - u_i^\star)_{i=0}^{k-1}\big\|_{L^p(L^q(\varOmega))}+\big\|( u_i - u_i^\star)_{i=0}^{k-1}\big\|_{L^p(W^{2,q}(\varOmega))}\le C\tau^k,
\begin{aligned}\lambda^{k+1}= \lambda^k-\gamma[ (1-\tau^{k})\beta^k(Ax^{k}-b)-(1-\tau^{k+1})\beta^{k+1}(Ax^{k+1}-b) ] &-\gamma\tau^k\beta^k(A\bar{x}^{k+1}-b),~\gamma\in(0,1].\end{aligned}
\begin{aligned}q^{-17}L_{1}(q^{32}) = \sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{(16n-1)^2 - 2(8j+3)^2}+ q^{(16n+1)^2 - 2(8j+3)^2}& + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{(16n+7)^2 - 2(8j+1)^2}+ q^{(16n+9)^2 - 2(8j+1)^2}.\end{aligned}
\ddot{x}^2\big[\acute{x}{}^2x\!\!\dot{}\,\acute{}\,{}^a -(\acute{x}x\!\!\dot{}\,\acute{}\,)\acute{x}{}^a\big] -\acute{x}^2(\ddot{x}x\!\!\dot{}\,\dot{}\,\acute{}\,)\dot{x}{}^a =0.
= 2g'\frac{Au_1}{w_1}\sum_{2\leq i\leq n}w_i\gamma^iu_{ii}-4u_1^2\sum_{2\leq i\leq n}w_iG_{ux_i}u_{ii}-\frac{4u_1}{w_1}\sum_{1\leq k\leq n}w_k\sum_{2\leq i\leq n}G_{x_kx_i}w_iu_{ii}&+\frac{c^{11}g'^2}{1-G_u}\frac{A^2u_1}{w_1}+AO(u_1^2).
G_{\varphi}(x) = \left(\begin{array}{cc}\frac{x^2 + 10}{x^2 + 1}\frac{6}{x^2 + 1}e^{i x \varphi}\frac{6}{x^2 + 1}e^{-i x \varphi}\frac{x^2 + 10}{x^2 + 1}\end{array}\right),\; x\in{\mathbb R}.
2\pi - 2 = (K_X + D) \cdot D = \sum_{i = 1}^r (K_X + D_i)\cdot D_i + 2\sum_{i < j}D_i \cdot D_j &= \sum_{i = 1}^r (2p_a(D_i) - 2) + 2\sum_{i < j}D_i \cdot D_j.
n>\frac{4\left( R+1\right) p}{\gamma\left( p-1\right)}A^{-p/\left(\gamma p-\gamma\right)}=\frac{\left( r+4\right) r^{4}p}{p-1}\left(\frac{r}{r-1}\right)^{r^{4}p/\left( p-1\right)}.
\int_0^x (x-t)^{\mu-1}E^\gamma_{\rho,\mu}(\lambda [x-t]^\rho) t^{\nu-1}E^\sigma_{\rho,\nu}(\lambda t^\rho)dt=x^{\mu+\nu-1}E^{\gamma+\sigma}_{\rho,\mu+\nu}(\lambda x^\rho).
f_{2n}(t)=d_nD_{n}e^{i\zeta_{n}t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t}+c_nC_{n}e^{i\omega_{n}t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}+E_ne^{-\eta t}\,.
T_{j,k-1}^{-1}T_{j+1,k}=T_{j+1,k}^\bullet(T_{j,k-1}^\bullet)^{-1}{\rm and}T_{j-1,k}T_{j,k-1}^{-1}=(T_{j,k-1}^\bullet)^{-1}T_{j-1,k}^\bullet
R_{(1,15)}(n) = 12\sigma(n) - 36\sigma(\frac{n}{3}) + 12\sigma(\frac{n}{15}) -36\sigma(\frac{n}{45}) + 144 \, W_{(1,15)}(n) + 1296\, W_{(1,15)}(\frac{n}{3}) - 432\, \biggl(\, W_{(1,5)}(\frac{n}{3}) + W_{(1,45)}(n)\, \biggr).
\nabla^G_{\xi}v^n(t,x) =\int_t^T\int_{H}e^{-(s-t){A}}GB^n(s,z+e^{(s-t){A}}x)\left\langle Q_{s-t}^{-1/2}e^{(s-t){A}}G\xi,Q_{s-t}^{-1/2}z\right\rangle \mu_{s-t}(dz) +\int_{t}^{T}\int_{H}e^{-(s-t){A}}\nabla^G v^n(s,z+e^{(s-t){A}}x)B^n(s,z+e^{(s-t){A}}x)\left\langle Q_{s-t}^{-1/2}e^{(s-t){A}}G\xi,Q_{s-t}^{-1/2}z\right\rangle \mu_{s-t}(dz)\,ds.
S_2 = \frac{1}{b^3}\sum_{\substack{k_1,k_2=1 k_1 \neq k_2}}^{n}b^{k_1-k_2}\left(\sum_{a_{k_1}=0}^{b-1}\sigma_{k_1}(a_{k_1})\right)\left(\sum_{a_{k_2}=0}^{b-1}a_{k_2}\right) =\frac{1}{b^3}\left(\frac{b(b-1)}{2}\right)^2\sum_{\substack{k_1,k_2=1 k_1 \neq k_2}}^{n}b^{k_1-k_2}.
\frac{e^{(1-a)x}}{e^x-z}- \frac{1}{1-z}= \lim_{N\to \infty}\sum_{n=N}^{-N}\biggl( \frac{z^{-a}e^{-2 \pi ina}}{x-2\pi in-\log z}+ \frac{z^{-a}e^{-2 \pi ina}}{2\pi in+\log z}\biggr) &=\sum_{n=\infty}^{-\infty}\frac{x z^{-a}e^{-2 \pi ina}}{(2\pi in+\log z)(x-2\pi in-\log z)}.
\mathcal{F}(\sqrt{\cdot}F_{4}(x,\cdot\lambda(x)))''(\omega \lambda(x)^{2}) &=\int_{0}^{\frac{2}{\sqrt{\omega}\lambda(x)}}\sqrt{r}F_{4}(x,r\lambda(x)) \partial_{2}^{2}\phi(r,\omega\lambda(x)^{2}) dr + \int_{\frac{2}{\sqrt{\omega}\lambda(x)}}^{\infty}\sqrt{r}F_{4}(x,r\lambda(x)) \partial_{2}^{2}\phi(r,\omega\lambda(x)^{2}) dr
\Big({\bf g}'_1(\overline{\kappa};(Q_1{\bf h}_1,Q_2{\bf h}_2)),{\bf g}'_2(\overline{\kappa};(Q_1{\bf h}_1,Q_2{\bf h}_2))\Big)=\Big(Q_1{\bf g}'_1(\overline{\kappa};({\bf h}_1,{\bf h}_2)),Q_2{\bf g}'_2(\overline{\kappa};({\bf h}_1,{\bf h}_2))\Big)\,,
\sum_{k=1}^{K}\left( (k + a)^2 d_{k+1}- (k -1)^2 d_k \right) \leq s\sum_{k=1}^{K}(k + a)^2\left( \theta_k - \theta_{k+1}\right) +s\sum_{k=1}^{K}(k + a)^2 \| g_k\| \| x_{k+1}-x_k \| .
\begin{pmatrix}b_j^+(t) b_j^-(t) \end{pmatrix}:= \begin{pmatrix}\tau_t(b_j^+) \tau_t(b_j^-) \end{pmatrix}= e^{2 t \Lambda_j}\begin{pmatrix}b_j^+ b_j^-\end{pmatrix}\, .
\hat{L}\left([f,g]\right)(x,y) =L\left([f,g]|_{x}^{y}\right)(x,y)=L\left([f|_x^y,g|_{x}^{y}]\right)(x,y) =\left([L(f|_x^y),g|_{x}^{y}]+[f|_x^y,L(g|_{x}^{y})]\right)(x,y) =\left([L(f|_x^y),g]+[f,L(g|_{x}^{y})]\right)(x,y).
C_\delta = 2 \hat h^2 - 2 \hat h + 1 = 2 \left(\hat h -\frac12 \right)^2 +\frac12 \le 2 \left( -\frac32 + \sqrt{1+\frac{2}\delta}\right)^2 + \frac12 = \frac4\delta + 7 - 6 \sqrt{1+\frac2\delta}.
e_{+}^{\prime\prime}(0)=e_{2,+}^{\prime\prime}(0)=\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T\left\{\frac12\int_{-t}^t\left<(\sigma_s-\langle\sigma\rangle_+)(\sigma-\langle\sigma\rangle_+)\right>_+\d s\right\}\d t.
\delta S ={\frac{1}{2}}\int_{{\cal M}^{2}}d^{2}\xi \sqrt{- g}\delta g^{m n}( \xi ) \left( \partial_{m}{\hat{X}}^{\underline{{{m}}}}\partial_{n}{\hat{X}}_{\underline{{{m}}}}-{\frac{1}{2}}g_{m n}g^{k l}\partial_{k}{\hat{X}}^{\underline{{{m}}}}\partial_{l}{\hat{X}}_{\underline{{{m}}}}\right) -
\prod_{k=1}^{\ell(s)}\left( 1 + c_1e^{c_1|s|-c_2k}\right) \leq \left( 1 + c_1e^{c_1|s|}\right)^{\ell(s)}& \leq \left( c_3 e^{c_1|s|}\right)^{\ell(s)}& \leq \exp\left( c_3 + c_4 |s| + c_5 |s|^2\right) & \leq \exp\left( c_6 + c_7 |s|^2 \right)
f_0(n) = q_0(n) \cdot f_1(n) + r_1(n) \cdot f_2(n), f_1(n) = q_1(n) \cdot f_2(n) + r_2(n) \cdot f_0(n+1), f_2(n) = q_2(n) \cdot f_0(n+1) + r_0(n+1) \cdot f_1(n+1),
\begin{array}{rcl}{\cal Q}_{e}^{I\ \prime}= (c\hat{a}_{0}+d){\cal Q}_{e}^{I}+c e^{-\hat{\phi}_{0}}{\cal Q}_{m}^{I}\, , &{\cal Q}_{m}^{I\ \prime}= -c e^{-\hat{\phi}_{0}}{\cal Q}_{e}^{I}+(c\hat{a}_{0}+d){\cal Q}_{m}^{I}\, . \end{array}
\frac{\pi_{m}(x+y)}{m!}=\sum_{r=0}^{m}\frac{L_{m-r}(x)\pi_{r}(y)}{(m-r)!r!}-\sum_{r=0}^{m-1}\frac{L_{m-r-1}(x)\pi_{r}(y)}{(m-r-1)!r!}.
S_{p h}+ S_{i n t}= \frac{1}{2}\int_{0}^{t}d s \sum_{k}\sum_{r}\sum_{l}( \dot{q}_{r l}^{2}({\bf k}) + k^{2}q_{r l}^{2}({\bf k}) + 2 q_{r l}({\bf k}) J_{r l}({\bf k})
G_{s;\mathcal B}= \left(\begin{smallmatrix}1 0 0 -1 -1 0 0 1 0 1 0 -1 0 0 1 0 0 0\end{smallmatrix}\right) G_{s';\mathcal B}=\left(\begin{smallmatrix}-1 0 0 1 1 0 0 1 0 0 -1 -1 0 0 1 0 0 0\end{smallmatrix}\right)
w^{(N)}=(M_i^{(N)}, Y_i^{(N)})_{i\in \mathbb{N}}\xrightarrow[\mathbb{P}-\textrm{a.s.}]{\mathcal{S}}w^{(\infty)}=(M_i^{(\infty)}, Y_i^{(\infty)})_{i\in\mathbb{N}}, \textrm{as}N\to \infty.
\left( \begin{array}{c c c c c}{0}&{\Delta_{1}}&{0}&{0}&{0}\\{\Delta_{1}}&{0}&{\Delta_{2}}&{0}&{0}\\{0}&{\Delta_{2}}&{0}&{\ddots}&{0}\\{0}&{0}&{\ddots}&{\ddots}&{\Delta_{k}}\\{0}&{0}&{0}&{\Delta_{k}}&{0}\\ \end{array}\right)
\int_{-1}^1 s^k\big(u_{1-}(s) + \hat{g}_2(s)\phi_+^{-n}\big){\rm d}s = 0 , \int_{-1}^1 s^kp_{n-1}(s){\rm d}s = -\int_{-1}^1 \big(C_{\Sigma}^- \hat{g}\phi_+^n\big)(s) +\hat{g}_2(s)\phi_+^{-n}{\rm d}s .
\left( \partial^{\gamma}\partial_{\alpha}T^{(\alpha \nu )\beta}-\partial^{\beta}\partial_{\alpha}T^{(\alpha \nu )\gamma}\right) =- \frac{D^{2}}{ae^{2}}\partial^2 T^{(\beta \gamma )\nu}+\frac{D}{2e^{2}}\partial^2 \epsilon^{\beta\gamma}{}_{\kappa\lambda}T^{(\kappa\lambda)\nu}.
{a}_{\bot}^{\ell}( x_{\bot}) = \frac{1}{L}\int d y^{-}d y_{\bot}d ( x_{\bot}- y_{\bot}) \nabla_{\bot}\left( \nabla_{\bot}\cdot A_{\bot}^{3}({y}_{\bot}, y_{-}) \right) \frac{\tau^{3}}{2}\, .
p_i =\sqrt{1-x_i^2},q_i=0 ,a_i=0, w_i(x_i) =\frac1{C(\alpha_i,\beta_i)}\,(1-x_i)^{\alpha_i}(1+x_i)^{\beta_i}dx_i, d\mu_i(x_i)=w_i(x_i)dx_i,X_i=(-1,1),
\rho_{m_{0}\rightarrow m_{1}+ m_{2}}^{2 \ell}\! = \frac{1}{2 \pi}\int d \mu \rho_{m_{0}\rightarrow m_{1 \mu}+ m_{2 \mu}}^{2 \ell - 1}\theta ( m_{0}- m_{1 \mu}- m_{2 \mu}) ,
\sum_{i=1}^{m}[a_i] \prod_{j \not= i}{{[x_i-x_j+a_j] \over [x_{i}-x_{j}]}}\prod_{1 \le k \le n}{{[x_i+y_k-b_k] \over [x_i+y_k]}}\qquad{}= \sum_{k=1}^{n}[b_k] \prod_{l \not= k}{{[y_k-y_l+b_l] \over [y_k-y_l]}}\prod_{1 \le i \le m}{{[y_k+x_i-a_i] \over [y_k+x_i]}}.
\ln_{q}m_{q}\left( n_{1},\cdots,n_{k};n,r_{1},\cdots,r_{k}\right) & =\ln_{q}\left[\begin{array}[c]{ccc}n n_{1}\cdots n_{k}\end{array}\right]_{q}+\frac{1}{2-q}\sum_{i=1}^{k}n_{i}^{2-q}\ln_{2-q}r_{i}+C_{q}
\sum_{\substack{\widehat{H}\subset H, |H| = s - 1 (H,S \smallsetminus \{i_k \},{R}) \textrm{positive}}}\det(A_{V_H, S \smallsetminus \{i_k \},{R}}) y^H (\prod_{i \notin S \smallsetminus \{i_k \}}y^{A_i}) .
U_5(F)=F(1+4\cdot 5 t), &U_5(Ft^{-1})=F(-3-3\cdot 5^2 t-5^3 t^2), &U_5(Ft^{-2})=F(1+5^2t+0\cdot t^2 -5^5t^3), &U_5(Ft^{-3})=F(9\cdot 5+9\cdot 5^3 t+0\cdot t^2+0\cdot t^3-5^7t^4), &U_5(Ft^{-4})=F(-51\cdot 5-51\cdot 5^3t +0\cdot t^2+0\cdot t^3+0\cdot t^4-5^9 t^5).
d G = -{\frac{3 \sqrt 2}{2 \pi}}\left( \frac{\kappa_{1 1}}{4 \pi}\right)^{\frac{2}{3}}\sum_{i}\delta ( x_{5}- x_{5 , i}) ( t r F_{i}^{2}-{\frac{1}{2}}t r R^{2}) d x^{5}\ ,
{\cal G}= \frac{1}{2}M_{i j}d{\bf x}_{i}\cdot d{\bf x}_{j}+ \frac{g^{4}}{2 ( 4 \pi )^{2}}( M^{- 1})_{i j}( d \xi_{i}+{\bf W}_{i k}\cdot d{\bf x}_{k}) ( d \xi_{j}+{\bf W}_{j l}\cdot d{\bf x}_{l}) .
k_j = N_j\frac{1-\delta_{\{-j\}}}{1-\delta_{\{1,\dots,K\}}}+ \left(N_j\frac{1-\delta_{\{-j\}}}{1-\delta_{\{1,\dots,K\}}}\right)^{3/4}, \mbox{and}n_1 = \max_{j}\frac{k_j+k_j^{3/4}}{p_j}.
\delta_{i\star j}^{(2c)}a=\frac{\phi^{(2c)}_{i\star j}a-a^{p^{2c}}}{\pi}=\frac{\phi_i^{(c)}\phi^{(c)}_j a-a^{p^{2c}}}{\pi}= \frac{\phi_{ij}^{(2c)}a-a^{p^{2c}}}{\pi}=\delta_{ij}^{(2c)}a.
e^{-C\|h\|_\infty}\lesssim\mu_{N,\frac{\overrightarrow{\boldsymbol{N}}}{N},\beta}^{V;\boldsymbol{A}}(\exp(\sum_{i=1}^N ( h(\lambda_{i})-\int h(\eta)d\mu_{{\rm eq}, \frac{\overrightarrow{\boldsymbol{N}}}{N}}(\eta)))) \lesssim e^{C(\|h\|_{\mathcal{L}}^{2}+\|h\|_\infty)}\,.
-\frac{\partial}{\partial \nu}H^{\phi}_{\nu}(f) = \frac{1}{(1-\nu)^2}\int_{\mathbb{R}}z(x){\rm log}\left(\frac{z(x)}{\phi(x) f(x)}\right){\rm d}x= \frac{1}{(1-\nu)^2}\mathbb{D}_{KL}(z||\phi f).
f_{N,C}(x) = \frac14\sum_{n = 0}^N \prod_{i = 0}^n \left(\frac{1}{\log^{(i)}(x)}\right)^2 + C\prod_{i = 0}^{N + 1}\left(\frac{1}{\log^{(i)}(x)}\right)^2 &= \frac1{x^2}\left[\frac14 + \frac{1}{\log^2(x)}\left[\frac14 + \cdots + \left(\frac{1}{\log^{(N)}(x)}\right)^2\left[\frac14 + C\left(\frac{1}{\log^{(N + 1)}(x)}\right)^2\right]\cdots\right]\right]
| b_{+}b_{+}b_{+}\rangle = \int_{0}^{\infty}d k_{1}d k_{2}d k_{3}\hspace{1 m m}\delta ( \sum_{i = 1}^{3}k_{i}- P^{+}) f_{b_{+}b_{+}b_{+}}( k_{1}, k_{2}, k_{3}) \frac{1}{N^{3 / 2}}\mathrm{T r}[ b_{+}^{\dagger}( k_{1}) b_{+}^{\dagger}( k_{2}) b_{+}^{\dagger}( k_{3}) ] | 0 \rangle
S_{\infty}^{\chi^{2}_{r}}\underset{d}{=}\int_{0}^{\infty}\exp\left(-u\right)d\left(\sum_{k=1}^{\infty}\lambda_{k}(S_{\infty}^{\chi^{2}_{r}})A^{(k)}(u)\right)\underset{d}{=}\int_{0}^{\infty}\exp\left(-u\right)dZ(u),
\begin{aligned}[\rho_\pm]^2 =\sum_{g\in G}[g]\,, [\rho_\pm][\rho_\mp] =\sum_{g\in G}[g\tau]\,, [g][\rho_\pm] =[\rho_\pm][g] =[\rho_\pm]\,, [\tau][\rho_\pm] =[\rho_\pm][\tau]=[\rho_\mp]\,. \end{aligned}
\sum_{j=1}^{4s+1}\left[ \left(e^{ib_j\left(\frac{\alpha-\beta}{2}\right)}+e^{-ib_j\left(\frac{\alpha-\beta}{2}\right)}\right) \delta_0\, -\left(e^{ib_j\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{b_j}+e^{-ib_j\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{- b_j}\right) \right]^s \ast f_j.
pb_0b_{n-1}= (a_n-1)(a_1 -1) - (a_1^2-b_0^2p -2a_1+1) \sum_{k=0}^{n-1}a_k , b_0a_n = b_0+b_{n-1}(a_1-1)- (a_1^2-b_0^2p -2a_1+1)\sum_{k=0}^{n-2}b_k.
{\gamma_{t,ik}}= \frac{{Pd_{t,ik}^{- \beta}{{\left\|{{h_{t,ik}}}\right\|}^2}}}{{\sum\limits_{j \in{\mathcal{M}\mathord{\left/{\vphantom{\mathcal{M}{{\mathcal{M}_i}}}}\right. \kern-\nulldelimiterspace}{{\mathcal{M}_i}}}}{Pd_{t,ij}^{- \beta}{{\left\|{{h_{t,ij}}}\right\|}^2}+{\sigma^2}}}},
\sum_{j=1}^{j_0}(\Delta_{\tau,j}- \mu_s) < [ n^{\epsilon_1}+ 2n^{\frac{1}{2}(\epsilon_1 + \epsilon_2)}] - N_n^- \mu_s \leq 4n^{\frac{1}{2}(\epsilon_1 + \epsilon_2)}\mbox{and}&\sum_{j=1}^{j_0}(\Delta_{\tau,j}- \mu_s) > [ n^{\epsilon_1}- 2n^{\frac{1}{2}(\epsilon_1 + \epsilon_2)}] - N_n^+ \mu_s \geq - 4n^{\frac{1}{2}(\epsilon_1 + \epsilon_2)}
\dfrac{1}{2^{s}}\left(1-\dfrac{1}{|R_0|/m_0}\right)\prod_{i=1}^s\left(1-\dfrac{1}{|R_i|/m_i}\right)\left(\sqrt{|R_i|/m_i}+1\right)>1-\frac{1}{|R_0||R|}.
W_{j}:=\left\{\mathbf{x}=(x_{1},\cdots,x_{2p})\in\mathbb{R}^{2p}:|\overline{x}_{j}|\leq c|x_{j+1}|\right\},\ \j=1,\cdots,2p-1, &W_{2p}:=\left\{\mathbf{x}=(x_{1},\cdots,x_{2p})\in\mathbb{R}^{2p}:|\overline{x}_{2p}|\leq c|\overline{x}_{2p}-x_{1}|\right\}.
\mathcal{T}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})=(\mathbf{n}_{1}\!\!\cdot\!\mathbf{n}_{2})-(\mathbf{n}_{1}\!\!\cdot\!\mathbf{n}_{3})(\mathbf{n}_{2}\!\cdot\!\mathbf{n}_{3})-i[\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}] \ , \i=\sqrt{-1}\ ,
W^L \left(\pi(y_1^L) | \pi(x_1^L) \right) = \prod_{i=1}^{L}W \left( y_{\pi(i)}| x_{\pi(i)}\right) &= \prod_{i=1}^{L}W(y_{i}| x_{i}) = W^L \left( y_1^L | x_1^L \right)
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m+1)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 2^2\cdot\frac{(2^{n_{2,\nu_2}}-1)(2^{n_{2,\nu_2}-2}-1)}{2^{n+1}-1}.
{\cal G}_\omega(r,r',\theta)=\sum_{n=0}^\infty{2\nu \Gamma\left({D\over2}-1\right)\over8(\pi r r')^{D/2-1}\sin\pi\nu}C_n^{(D/2-1)}(\cos\theta)\left[J_\nu(kr_<)J_{-\nu}(kr_>)-\beta J_\nu(kr)J_\nu(kr')\right],
\begin{aligned}v (t,x) = \alpha \Theta (t,x) + w (t,x), \tilde{p}(t,x) = \alpha^2 \Pi (t,x) + q (t,x), \ell_v (t,x) = \ell_w (t,x) \omega_v (t,x) = \alpha g(t, 1) + \omega_w (t,x). \end{aligned}
\left|\widetilde{Ny_0}(e^{i\theta_0})\right| = \Big|\eta(e^{i\theta_0}) \Psi_2(y_0) +(1-\epsilon) (1- \eta(e^{i\theta_0})) \widetilde{Ty_0}(e^{i\theta_0})\Big| & = |\Psi_2(y_0)| & = 1.
\max_{P_i\ge 0,\forall i}~~ \mathbb{E}\left[\sum_{i=1}^K\ln\left(1+\frac{h_{i}P_i}{\sigma^2+\sum_{j=1,j\neq i}^K h_{j}P_j}\right)\right], \mbox{s.t.}~~~~ \eqref{Con-AIPC},~\eqref{Con-ATPC}.
\tau_0 = \frac 12\left(\begin{array}{cc}0 1 -1 0 \end{array}\right)\,, \quad\tau_1 = -\frac 12\left(\begin{array}{cc}0 1 1 0 \end{array}\right)\,, \quad\tau_2 = \frac 12\left(\begin{array}{cc}-1 0 0 1 \end{array}\right)\,
P_{n}\left( x \right) = \frac{1}{\sqrt{h_{n}h_{n + 1}}}\left( \begin{array}{l l l l}{{\gamma_{0}}}&{{\gamma_{1}}}&{{. . .}}&{{\gamma_{n}}}\\{{\gamma_{1}}}&{{\gamma_{2}}}&{{. . .}}&{{\gamma_{n + 1}}}\\{{. . .}}&{{. . .}}&{{. . .}}&{{. . .}}\\{{\gamma_{n - 1}}}&{{\gamma_{n}}}&{{. . .}}&{{\gamma_{2 n - 1}}}\\{{1}}&{{x}}&{{. . .}}&{{x^{n}}}\\ \end{array}\right) ,
{\frac{1}{i \beta}}\int_{A^{\prime}}{\frac{d z}{\sin{\frac{\pi}{\beta}}z}}{\frac{\cos \frac z 2}{\sin^{4}\frac z 2}}= -{\frac{1}{1 8 0}}( \gamma^{2}- 1 ) ( 7 \gamma^{2}+ 1 7 ) ,
\left( \begin{array}{c}{{\vec{F}^{( 1 )}}}\\{{- \vec{F}^{( 2 )}}}\\ \end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{c c}{{1}}&{{1}}\\{{- 1}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c}{{\vec{F^{R}}}}\\{{\vec{F}^{L}}}\\ \end{array}\right) \, .
\mbox{SIR}=\frac{M_\mathrm{t}M_\mathrm{r}R_0^{-\alpha_\mathrm{L}}}{\sum_{\ell:X_\ell\in\Phi\cap\mathcal{B}(0,R_\mathrm{B})}D_\ell R_\ell^{-\alpha_\mathrm{L}}}.
\frac{1}{p}\frac{d}{dt}\int_{\Omega}n_1^p = -\frac{1}{p}\int_{\Omega}u \cdot \nabla n_1^p + \int_{\Omega}n_1^{p-1}\Delta n_1 + \chi_1\frac{p-1}{p}\int_{\Omega}\nabla n_1^p\cdot\nabla c \quad\,+ \mu_1\int_{\Omega}n_1^p - \mu_1\int_{\Omega}n_1^{p+1}- a_1\mu_1\int_{\Omega}n_1^p n_2.
\begin{array}{r c l}{{\widetilde \phi_{\mathrm{f i x e d}}( u )}}&{{=}}&{{\widehat \phi_{\mathrm{f r e e}}( u / \sqrt{2}) \ ,}}\\{{\widetilde \phi_{\mathrm{f r e e}}( x )}}&{{=}}&{{\widehat \phi_{\mathrm{f i x e d}}( \sqrt{2}x ) \ .}}\\ \end{array}
\rho_s := \mathbf{P}[\{Y_t^v,\ldots,Y_t^{v+m}\}_{t\in[0,\delta]}\in\,\cdot\,|Y_{\le 0},\{Y_t^{<v}\}_{t\in[0,\delta]},X_{\le s}], \rho :=\mathbf{P}[\{Y_t^v,\ldots,Y_t^{v+m}\}_{t\in[0,\delta]}\in\,\cdot\,|Y_{\le 0},\{Y_t^{<v}\}_{t\in[0,\delta]}],
T_{k+1}= T_k^\prime - \lambda^2{\cal D}\left[ T_k^\prime \right] + T_{\rm exp}- P_A \left( P_A^* T_k^\prime P_B \right) P_B^* \ ,{\it \Lambda}_{k+1}={\it \Lambda}_k^\prime - \lambda^2{\cal D}\left[{\it \Lambda}_k^\prime \right] +{\it \Lambda}_{\rm exp}- Q_A \left( P_A^* T_k^\prime P_B \right) P_B^* \ .
Y_{m_1,m_2}(\alpha,\beta,\gamma) = \frac{1}{2}(-1)^{(\theta_1+m_1+1)(\theta_2+m_2+1)}\prod_{j \in \mathbb{T}_{m_1,m_2}}\left( \alpha + \beta \exp \left\{2 \pi i \frac{j_1 + \theta_1 /2}{m_1}\right\}+ \gamma \exp \left\{2 \pi i \frac{j_2 + \theta_2 /2}{m_2}\right\}\right).
x_{i}^{1-N}\prod\limits_{\ell\neq i,j}(x_{i}-x_{\ell}) =\prod\limits_{\ell\neq i,j}\left( 1-\frac{x_{\ell}}{x_{i}}\right)=\sum_{\ell=0}^{N-2}(-1)^{\ell}x_{i}^{-\ell}\sum_{J\subset E_{ij},\, \#J=\ell}x^{\varepsilon_{J}}.
H ( p ) = 4 \pi N \int \frac{d^{4}l}{{( 2 \pi )}^{4}}\left\{\frac{( l + p )^{2}- m^{2}}{{[ ( l + p )^{2}- m^{2}]}^{2}+ \varepsilon^{2}}+ ( p \to - p ) \right\}\delta ( l^{2}- m^{2}) \sin^{2}\theta ( l^{0}, \mu )
\begin{array}{l}\Theta\left[K_{\Omega}(\vec{p},\vec{R}) G_{\Omega}(\vec{p}',\vec{R}')\right]=1, \Theta\left[F^{+}_{\Omega}(\vec{p}',\vec{R}')K_{-\Omega}(-\vec{p},\vec{R})\right]=\Theta\left[\Delta^{*}(\vec{R})\widehat{\gamma}_{A}(\vec{p}',\vec{R}')\widetilde{G}_{\Omega}(\vec{p}',\vec{R}')\right]\end{array}
P\left( \theta+\alpha,\theta^{\ast}+\bar{\alpha}\right) =\left(\theta+\alpha\right) \left( \theta^{\ast}+\bar{\alpha}\right) \left(\theta+\alpha\right) +\left( \theta^{\ast}+\bar{\alpha}\right) \left(\theta+\alpha\right) \left( \theta^{\ast}+\bar{\alpha}\right) & =P_{0}+P_{1}+P_{2}+P_{\geq3}
II=-dN\cdot dP=\omega_{1}^{3}\odot\omega^{1}+\omega_{2}^{3}\odot\omega^{2}=(\begin{matrix}\omega^{1}&\omega^{2}\end{matrix})(\begin{matrix}ii_{11}&ii_{12}\\ii_{21}&ii_{22}\end{matrix})(\begin{matrix}\omega^{1}\\ \omega^{2}\end{matrix}).
(L_i)_{j,k}:=([L]_{i,i+1})_{j,k}=\big(p_j-c_iq_j^{-1}\big)\delta_{j,k}+|g|q_j^iq_k^{m-i-1}(q_j^m-q_k^m)^{-1}(1-\delta_{j,k}),
\langle Q_{2n}(x;q),Q_{2m+1}(x;q)\rangle_{4,\omega}=\frac{\tau_{2n+2}(q)}{\tau_{2n}(q)}:=u_n(q)\delta_{n,m}, & \langle Q_{2n}(x;q),Q_{2m}(x;q)\rangle_{4,\omega}=\langle Q_{2n+1}(x;q),Q_{2m+1}(x;q)\rangle_{4,\omega}=0.
\psi(x)=(x+\delta)^2\ \ \ \ \ \ \ \ \ \ \ \ \ \x\in[\frac{\eta}{2},2\pi-\frac{\eta}{2}], \frac{d^k\psi}{dx^k}(0)=\frac{d^k\psi}{dx^k}(2\pi)\ \ \ \ \ \ \ \ \k=1,2,3,4,5,6,7, 2\delta\leq\frac{d\psi}{dx}(x)\leq2(2\pi+\delta)\ \ \x\in[0,2\pi].
I^{\pm}(z) =\sum_{i_1+j_1=q_1}(I_{D_{i_1}}^+(z)+I_{D_{i_1}}^-(z))=\sum_{i_1+j_1=q_1}\sum_{\sigma\in \mathfrak{S}_{q_1+k_1}}(I_{D_\sigma}^+(z)+I_{D_{\sigma}}^-(z)).
\mathrm{w i t h}\omega_{l , m , n}\equiv \sqrt{\Biggl ( \frac{l \pi}{2 L_{x}}\Biggr )^{2}+ \Biggl ( \frac{m \pi}{2 L_{y}}\Biggr )^{2}+ \Biggl ( \frac{n \pi}{2 L_{z}}\Biggr )^{2}},
\mathcal{C}_1 =\overline{\{O\big(C_2(\alpha)\big) \}_{\alpha \in\mathbb{C}}}= \{\mathbb{C}^{n+1}, B, C_1, C_2(\alpha), C_3 \}, \mathcal{C}_2 =\overline{O(D_{n+1})}= \{\mathbb{C}^{n+1}, B,C_1, D_3, \ldots, D_{n+1}\}.
E_{t-1}(H_{m,t}|z_t)= E_{t-1}(a_{0m}+a_{1m}y^2_{t-1}(1-w_{m,t-1})+a_{2m}y^2_{t-1}w_{m,t-1}+b_mH_{m,t-1}|z_t) = \underbrace{a_{0m}}_{I}+\underbrace{a_{1m}E_{t-1}[y^2_{t-1}|z_t]}_{II}+\underbrace{(a_{2m}-a_{1m})E_{t-1}[y^2_{t-1}w_{m,t-1}|z_t]}_{III}&\qquad\qquad\ \ \ \ \ \ \ \ \qquad\qquad+\underbrace{b_mE_{t-1}[H_{m,t-1}|z_t]}_{IV}.
\eta_0 = \{(z_1,z_2)\in \mathbb{C}^2 :1+ z_1^2 -z_2^2=0, Im(z_1 (1+\overline{z_2}))>0 \}\,\, & \eta_c =\{(z_1,z_2)\in \mathbb{C}^2 : |1+ z_1^2 -z_2^2|=c(1+|z_1|^2-|z_2|^2), Im(z_1 (1+\overline{z_2}))>0\}.
\left( \begin{array}{c}{{\xi_{1}^{2}( \theta )}}\\{{\xi_{2}^{2}( \theta )}}\\ \end{array}\right) ~ = ~ \sigma_{3}\left( \begin{array}{c}{{\xi_{1}^{1}( \theta )}}\\{{\xi_{2}^{1}( \theta )}}\\ \end{array}\right)
\|a\|_\sharp+\|x^\natural-a\|_\sharp \geq\|x^\natural\|_\sharp &\geq\|x^\star\|_\sharp &=\|x^\natural+(x^\star-x^\natural)\|_\sharp &=\|a+(x^\natural-a)+z_1+z_2\|_\sharp &\geq\|a+z_1\|_\sharp-\|(x^\natural-a)+z_2\|_\sharp &\geq\|a+z_1\|_\sharp-\|x^\natural-a\|_\sharp-\|z_2\|_\sharp &=\|a\|_\sharp+\|z_1\|_\sharp-\|x^\natural-a\|_\sharp-\|z_2\|_\sharp.
~~ I(Q_n;M), ~~ (p_1,p_2,\dots,p_{K-1}), ~~ -p_w \leq 0, \forall w\in 1:K-1, \begin{aligned}N\sum_{w=0}^{K-1}\binom{K-1}{w}(N-1)^w p_w -1 = 0. \end{aligned}
\Bigl\|A\Bigl( (\frac{1}{2}-\mu)(h_{S_0}+h^{(1)})-\frac{\mu}{2}u_i+\mu h\Bigr)\Bigr\|_2^2 = \Bigl\|A\Bigl( (\frac{1}{2}-\mu)(h_{S_0}+h^{(1)})-\frac{\mu}{2}u_i\Bigr) \Bigr\|_2^2 +2\Bigl\langle A\Bigl((\frac{1}{2}-\mu)(h_{S_0}+h^{(1)})-\frac{\mu}{2}u_i\Bigr), \mu Ah\Bigr\rangle +\mu^2\|Ah\|_2^2.
R_n <2R_1=:R, \tilde{M}_{n}\leq (\tilde{M}_{1}\frac{|\ln\epsilon_{1}|}{2\pi h_{1}})^{b_{3}\ln n}e^{b_{3}e^{4\sqrt{n}}}, \delta_{n}>\delta_{1}(\tilde M_1\frac{|\ln\epsilon_{1}|}{2\pi h_{1}})^{-b_{4}\ln n}e^{-b_{4}e^{4\sqrt{n}}}, c_{n}>(c_{1}\frac{2\pi h_{1}}{|\log\epsilon_{1}|})^{n^{b_{5}}}e^{-b_{6}e^{5\sqrt{n}}},