harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{g}(Q_{\xi_X}(Y)+Q_{\xi_Y}(X)+\xi\cdot\xi_{\nabla_XY+\nabla_YX},Z){}={\bf B}((\nabla_X\xi)_Z-(\nabla_Z\xi)_X,\xi_Y)+{\bf B}((\nabla_Y\xi)_Z-(\nabla_Z\xi)_Y,\xi_X)-{\bf B}([\xi_X,\xi_Z],\xi_Y) \quad{}-{\bf B}([\xi_Y,\xi_Z],\xi_X)+{\bf B}((\nabla_Y\xi)_X+(\nabla_X\xi)_Y,\xi_Z){}=2\tilde{g}(\tilde{\nabla}_XY-\nabla_XY,Z).
\{\mathcal T_{+ +}^{\prime}( x ) , \mathcal T_{+ +}^{\prime}( x^{\prime}) \}= 4 \, ( \mathcal T_{+ +}^{\prime}( x ) + \mathcal T_{+ +}^{\prime}( x^{\prime}) ) \, \delta^{\prime}( x - x^{\prime}) - 1 6 \kappa \, n \, \delta^{\prime \prime \prime}( x - x^{\prime}) ,
\int{\cal D}[ \Phi_{s l o w}] e^{- ( S_{0}( s l o w ) + S_{i n t}( s l o w ) + S_{s l o w}^{\prime})}= \int{\cal D}[ \Phi^{\prime}] e^{- ( S_{0}[ \Phi^{\prime}] + S_{i n t}[ \Phi^{\prime}] )}.
\varphi^{+}( x , t ) = i \int d^{d - 1}\vec{x}^{\prime}( H t^{\prime})^{- d}G_{F}( x , t ; x^{\prime}, t^{\prime}) ( \overrightarrow{\partial_{t^{\prime}}}- \overleftarrow{\partial_{t^{\prime}}}) \varphi^{+}( x^{\prime}, t^{\prime})
D(\textbf{u},\bar{\textbf{u}}) = \left(\begin{array}{cccc}-Bk_{B}-Bk_{B}-Bk_{B}0 \frac{\epsilon \bar{E}(L-\bar{B})}{\bar{B}\bar{L}}\frac{-\epsilon (BL+\bar{B}E)}{\bar{B}\bar{L}}-\frac{\epsilon E}{\bar{L}}0 \frac{k_{N}D}{\bar{B}}0 -\frac{k_{N}B}{\bar{B}}0 0 0 0 \frac{\Gamma_{L}-M}{L(1-L)}\end{array}\right).
\mathcal{F}\left( Av\right) \left( \xi\right) =\frac{1}{n^{1/2}}\sum_{k\in\mathbb{Z}_{n}}Av\left( k\right) \exp\left( \frac{2\pi ik\xi}{n}\right) & =\frac{1}{n^{1/2}}\sum_{m\in\mathbb{Z}_{n}}v\left( m\right) \exp\left(\frac{2\pi i\left( m+1\right) \xi}{n}\right) & =\frac{1}{n^{1/2}}e^{\frac{2\pi i}{n}\xi}\mathcal{F}v\left( \xi\right) .
\bigcap_{i=1}^{d-m}H_i^{'}= \{(x_1,x_2,\ldots,x_d)\|\x_i=x_j\{\rm\if}\{i,j\}\subseteq B_k\{\rm for\some\}1\leq k\leq m, &{\rm and}\x_r=a_{r}\{\rm if}\r\in B_{m+1}\}.
S_{e e}= S_{e}\left( 1 \, + \, m_{e \mu}^{2}\, S_{\mu}S_{e}\, + \, m_{e \mu}^{4}\, S_{\mu}S_{e}S_{\mu}S_{e}\, + \, . . . \right) = S_{e}\left( 1 - m_{e \mu}^{2}\, S_{\mu}S_{e}\right)^{- 1}\, ,
L=lim_{b\rightarrow\infty}\sum_{k=0}^{\infty}\int_{0}^{b}x^{2}e^{-kx}=\sum_{k=0}^{\infty}lim_{b\rightarrow\infty}\int_{0}^{b}x^{2}e^{-kx}=\sum_{k=0}^{\infty}\frac{2}{k^{3}}=2\zeta(3)
B[\Phi,\Phi_1] :=\int_0^r \left[\phi'\phi'_1-\frac{z}{2}\phi'\phi_1+\frac{z}{2}e_\sigma\psi'\phi_1\right]+\int_0^r \left[\psi'\psi'_1-\frac{z}{2}\psi'\psi_1+\frac{z}{2}e_\sigma\phi'\psi_1\right]~
S_{\mathrm{h y p}}( \epsilon_{\kappa}) \stackrel{0 \leftarrow \tau / \sigma}{\longleftarrow}S_{\mathrm{c a t e n a r y}}( \epsilon_{\sigma}) \stackrel{\tau / \sigma \rightarrow \infty}{\longrightarrow}S_{\mathrm{3 / 2 - p a r a b}}( \epsilon_{\kappa}) ~ .
\int\limits_0^t G'(X_{s-})dX_s =\int\limits_0^t G'(X_{s-})\mu(X_{s-})ds+\int_0^t G'(X_{s-}) \sigma(X_{s-}) dW_s &\quad+\int\limits_0^t\int\limits_{\mathbb{R}}G'(X_{s-})\rho(X_{s-},y)\nu (dy,ds).
C_\Sigma^+ A_-\big(v_A v_B^{-1}- I\big)B_-^{-1}= C_\Sigma^- A_-\big(v_A v_B^{-1}- I\big)B_-^{-1}+ A_-\big(v_Av_B^{-1}- I\big) B_-^{-1}\hphantom{C_\Sigma^+ A_-\big(v_A v_B^{-1}- I\big)B_-^{-1}}{}= C_\Sigma^- A_-\big(v_A v_B^{-1}- I\big)B_-^{-1}+ A_+B_+^{-1}- A_-B_-^{-1}.
V_{i}\cdot V_{j}= \left( \begin{array}{c c c}{- 1}&{- 1 / 2}&{- 1 / 2}\\{- 1 / 2}&{- 1 / 3}&{- 1 / 3}\\{- 1 / 2}&{- 1 / 3}&{0}\\ \end{array}\right) .
dS_t^{t_0}= S_t^{t_0}\sqrt{V_t^{t_0}}dB_t^{t_0}, S_0^{t_0}= S_{t_0}, V_t^{t_0}= g_{t_0}(t) + \int_0^{t}K(t-s) \left( -\lambda V_s^{t_0}ds + \nu \sqrt{V_s^{t_0}}dW_s^{t_0}\right),
\sum_{k=N}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}= \frac{1}{2\pi i}\int_{c-i\infty \atop{{\rm Max}[1/2, N-1]< c={\rm Re}\, s< N}}^{c+i\infty}\frac{1}{(2\pi z)^{2s-1}}\; \frac{\Gamma(2s-1) \zeta(2s)}{\sin(s \pi)}\; ds & + \;\; \frac{1}{2\pi i}\int_{C}\frac{1}{(2\pi z)^{2s-1}}\; \frac{\Gamma(2s-1) \zeta(2s)}{\sin(s \pi)}\; ds \;\;.
\frac{1}{\prod\limits_{j=i+1}^k R(j)}\sum\limits_{x_{k-1}<x_k}\ldots\sum\limits_{x_{i+1}<x_i}f(x_i) = \frac{m(k,i)}{\prod\limits_{j=i+1}^k R(j)}\sum\limits_{x_{i}<x_{k}}f(x_i) =\frac{1}{R(k,i)}\sum\limits_{x_{i}<x_{k}}f(x_i).
\widehat{\psi_{p^{2k}}}(\mathbf{u})= \frac{1}{p^{2nk}}\sum_{\mathbf{c}\in (\mathbb{Z}/p^{2k}\mathbb{Z})^n}\psi_{p^{2k}}(f_\mathbf{c}) \exp\left( \frac{2\pi i c_1 u_1}{p^{2k}}\right).
X\bar z_1 = \alpha \bar w_2 X\bar z_2 = -\alpha \bar w_1 X\bar{w}_1 = \phi \bar{z}_2 X \bar{w}_2 =-\phi \bar{z}_1 T \bar{z}_1 = \psi \bar{w}_2 T\bar{z}_2 = -\psi \bar{w}_1 T\bar w_1 = \beta \bar z_2 T\bar w_2 = -\beta \bar z_1.
m= (\ell+1)(\ell'+1)-\#V=(\ell+1)(\ell'+1)-(\ell+1)\left[k^2\,+\,2k(k+r-1)\right] = (\ell+1)(\ell'+1-(k^2+2k(k+r-1)) = (\ell+1)(\ell-3k^2-2s_1k+s_2+1) = \ell^2 -\ell (3k^2+2s_1k-s_2-2) -3k^2-2s_1k+s_2+1
\left\{e^{\frac{1}{24}A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}-8+W_i)-e^{\frac{1}{24}A_1}A_1\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(10)}&=480\left\{e^{\frac{1}{24}A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(10)}.
a\ast_{\hbar}b(z)=\left( \tfrac{1}{4\pi\hbar}\right)^{2n}\int_{\mathbb{R}^{2n}}e^{\frac{i}{2\hbar}\sigma(z^{\prime},z^{\prime\prime})}a(z+\tfrac{1}{2}z^{\prime})b(z-\tfrac{1}{2}z^{\prime\prime})dz^{\prime}dz^{\prime\prime};
\int\limits_{\tau(\lambda_1)}^{\tau(\lambda_2)}f(r)\,dr\oint\limits_{\Gamma}\frac{d\mu}{\tau^{-1}(r)-\mu}=-\int\limits_{\lambda_1}^{\lambda_2}\,d\mu\oint\limits_{\tilde\Gamma}\frac{f(r)\,dr}{\tau^{-1}(r)-\mu}.
X^{\mu}= \bar{X}^{\mu}+ \tilde{X}^{\mu}, \qquad \bar{X}^{\mu}\equiv \left( \begin{array}{l l l l}{x_{1}^{\mu}}&{}&{}&{}\\{}&{x_{2}^{\mu}}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{x_{N}^{\mu}}\\ \end{array}\right) .
S_{\mathrm{p a r t}, 0}^{( \mathrm{N})}= \int d t \, \sum_{\alpha = 1}^{N}\left( \xi_{\alpha}^{\underline{{{a}}}}\left( E_{j , \alpha}^{\underline{{{a}}}}\dot{x}_{\alpha}^{j}+ E_{0 , \alpha}^{\underline{{{a}}}}\right) -{\frac{1}{2}}\xi_{\alpha}^{\underline{{{a}}}}\xi_{\alpha}^{\underline{{{a}}}}\right) \, .
h^{\mu}(z)=a^{\mu}(\theta,\bar{\theta})+\lambda(\theta,\bar{\theta})x^{\mu}+w^{\mu}_{~\nu}(\theta,\bar{\theta})x^{\nu}+2x{\cdot b}(\theta,\bar{\theta})x^{\mu}-x^{2}b^{\mu}(\theta,\bar{\theta})\,,
W_\beta = \left\{f \in R \left|\begin{array}{l}\dfrac{\partial f}{\partial z_1}(0, \rho) = \dfrac{\xi_1^2 \rho}{\xi_1 + \beta \rho}f(0, \rho) \dfrac{\partial f}{\partial z_2}(\rho, 0) = \dfrac{\xi_2^2 \rho}{\xi_2 + \beta \rho}f(\rho, 0)\end{array}\right.\right\}.
-\frac{1}{\pi i}\int_{0}^{2\pi}f_{\beta}(\beta)\frac{\epsilon(\alpha)z(\beta)}{z(\alpha)-z(\beta)}d\beta+\frac{1}{\pi i}\int_{0}^{2\pi}f_{\beta}(\beta)\frac{\epsilon(\beta)z(\alpha)}{z(\alpha)-z(\beta)}d\beta=z[\epsilon,H]\left(\frac{f_{\alpha}}{z_{\alpha}}\right).
g = \begin{pmatrix}1 -v^T \frac{1}{2}\|v\|^2 0 I -v 0 0 1\end{pmatrix}\begin{pmatrix}\alpha^{-1}0 0 0 A^{-1}0 0 0 \alpha\end{pmatrix}\begin{pmatrix}1 0 0 -\xi I 0 \frac{1}{2}\|\xi\|^2 -\xi^T 1\end{pmatrix}= g_{-1}g_0 g_1
\sum_{n=0}^\infty b_n x^n = \sum_{j=0}^\infty \frac{(j!)^2 x^{2j}}{(1-x)^2 (1-2x)^2 \cdots (1-(j+1)x)^2}= \frac{1}{(1-x)^2}\sum_{j=0}^\infty f_j \left(2-\frac{1}{x}, 2-\frac{1}{x}\right),
\mathrm{l n d e t}\Delta = - \lim_{s \to 0}{\frac{d}{d s}}\sum_{n_{1}, n_{2}}[ ({\frac{4 \pi^{2}}{l^{2}}}) ( n_{1}^{2}+{\frac{( n_{2}+ \alpha )^{2}l^{2}}{4}}) ]^{- s}\quad .
T_{\ \sigma}^{\rho}= \Phi^{2}\left( \mu^{\rho}\mu_{\sigma}+{\cal K}s^{\rho}s_{\sigma}\right) + \left[{\cal L}+ c^{2}\Phi^{2}\left( s^{2}{\cal K}+ y^{2}{\cal A}\right) \right] g_{\ \sigma}^{\rho}\ .
(\begin{matrix}n_{1}+...+n_{p}\\m\end{matrix})=\sum_{k_{1}+\cdot\cdot\cdot+k_{p}=m}(\begin{matrix}n_{1}\\k_{1}\end{matrix})(\begin{matrix}n_{2}\\k_{2}\end{matrix})\cdot\cdot\cdot(\begin{matrix}n_{p}\\k_{p}\end{matrix}).
\begin{aligned}\|P_{\lambda_0}(\varphi_{\lambda_1,N_1}^\dagger\phi_{\lambda_2,N_2})\|_{L^2_tL^2_x}&\lesssim \lambda_0 \left(\frac{\min\{\lambda_0,\lambda_1,\lambda_2\}}{\max\{\lambda_0,\lambda_1,\lambda_2\}}\right)^\delta (\min\{N_1,N_2\})^{1-\eta}\|\varphi_{\lambda_1,N_1}\|_{U^2_{\theta_1}}\|\phi_{\lambda_2,N_2}\|_{U^2_{\theta_2}}.\end{aligned}
[D(m,s):D(m+1,n)]_q =q^{2(s-n)}[D(m,s-m-1):D(m+1,n-m-1)]_q &+q^{4s_1s_0}[D(m+\delta_{s_1,1},s-2s_0):D(m+1,n)]_q &+\delta_{n,m+1}\delta_{s_0,1}\delta_{s_1,2}q^{3m}+(1-\delta_{2s_0,m})\delta_{s_0,n}\delta_{s_1,1}q^{m+2s_0}.
\hat{G}_{( 5 )}^{-}={}^{\star}\tilde{\hat{G}}{}_{( 5 )}^{-}\, , \hspace{1 c m}\tilde{\hat{G}}{}_{( 5 )}^{-}= 5 \partial \tilde{\hat{C}}{}_{( 4 )}^{-}- 1 0 \hat{\cal H}\hat{C}_{( 2 )}^{-}\, ,
\int_\Omega |u_0^\epsilon|^2\le{\bf A},\E_\epsilon(d_0^\epsilon)=\int_\Omega \big(|\nabla d_0^\epsilon|^2+\frac{1}{2\epsilon^2}(1-|d_0^\epsilon|^2)^2\big)\le 2\pi m\log \frac{1}{\epsilon}+{\bf B}.
\begin{aligned}A'_{11}=2\zeta^4 + \zeta^2 qr -\frac{1}{4}q^2 r^2 +\frac{i}{2}\left(q_x r- q r_x \right) A'_{12}=2i\zeta^3q -\zeta q_x A'_{21}=2i\zeta^3r +\zeta r_x \ .\end{aligned}
\sum_{k=0}^\infty\epsilon_k= 2\sum_{k=0}^\infty\Big[(f+g)(x^k)-(f+g)(x^{k+1})\Big] \le2\Big[(f+g)(x^0)-\lim_{k\to\infty}(f+g)(x^{k+1})\Big] \leq 2\Big[(f+g)(x^0)-(f+g)(x_*)\Big]< +\infty.
h(\sigma^{N}, \alpha^{MN+1}\times I^{\infty}) = \lim_{l \to \infty}\frac{\log(N(\mathbf{H},(\vee_{i=0}^{l}\sigma^{-iN}(\mathcal{V})^{*}))}{l}& = \lim_{l \to \infty}\frac{\log(N(\star_{i=1}^{(M+l)N}H,\alpha^{(M+l)N+1}))}{l},
\mathcal{J}^{\rm rob}:= \mathcal{J}_{\rm CA}\cup \mathcal{J}_{\rm PCA}, \ \ \phi^{\rm rob}_0: = (Lfr_3 + 7 r^{\diamond})/n \ \ \mbox{and}\ \G^{\diamond}_4 : = G_1^{\diamond}- \bigcup \mathcal{J}^{\rm rob}.
A\cdot\delta(B) = A\cdot\lim\limits_{i\in \mathcal{I}}\delta(B\rho_i) = \lim\limits_{i\in \mathcal{I}}(AB\delta(\rho_i)+A\rho_i\delta(B)-AB\rho_i\xi) = \lim\limits_{i\in \mathcal{I}}AB\delta(\rho_i)+A\delta(B)-AB\xi.
|\alpha\langle V_y\rangle_{\Phi_{\#}^{y}}| \leq C\frac{\alpha}{y}\|\Phi^y_{\#}\|^2 + \langle \alpha V^>_y \rangle_{\Phi_{\#}^y}&\leq C \frac{\alpha^5}{L}\log(\alpha^{-1}) + \alpha \|\Phi_{\#}^y\| \|P \Phi_{\#}^y\| = O(\alpha^5 \log(\alpha^{-1})).
a_{w}(q,p)\star b_{w}(q,p)=a_{w}(q,p)\exp [\frac{i}{2}(\frac{\overleftarrow{\partial}}{\partial q}\frac{\overrightarrow{\partial}}{\partial p}-\frac{\overleftarrow{\partial}}{\partial p}\frac{\overrightarrow{\partial}}{\partial q})]b_{w}(q,p).
\mathrm{Spin}(2p)\cdot\mathrm{Spin}(2q+1)= \big\{\psi \in \mathrm{Spin}(2p+2q+1)\,;\, \psi=\varphi\,\phi\,, \qquad\varphi\in \mathrm{Spin}(2p)\,,\phi\in \mathrm{Spin}(2q+1)\big\}\,,
\mathcal{G}_{+ , l}( m e^{i \pi / 2}, r ) - \mathcal{G}_{- , l}( m e^{i \pi / 2}, r ) = \frac{i r}{\pi}\, e^{- i \pi | l |}( e^{2 i \delta_{l}^{+}}- e^{2 i \delta_{l}^{-}}) \, K_{W}^{2}( m r ) ,
M_{p l}^{2}={\frac{M_{X}^{3}}{k}}e^{- 2 k y_{1}- 2 c}\left({\frac{1}{2}}e^{- 2 k ( y_{2}- 2 y_{1})}- e^{- 2 k ( y_{2}- y_{1})}+ 1 -{\frac{1}{2}}e^{- 2 k ( \pi \rho - y_{2})}\right) . \,
F_{n}^{\mathcal{O}| \ldots \mu_{i}\mu_{j}\ldots}( \ldots \theta_{i}, \theta_{j}\ldots ) = F_{n}^{\mathcal{O}| \ldots \mu_{j}\mu_{i}\ldots}( \ldots \theta_{j}, \theta_{i}\ldots ) S_{\mu_{i}\mu_{j}}( \theta_{i , j}) \, .
H[n+1](e,\overline{e})-|\Theta[n+1](\overline{e},\overline{e})| \ge h[n+1](e,\overline{e})-\mu_{n+1}& - \frac{1}{N}|(\upsilon_N\Big(\frac{k_{n+1}}{\delta - k_{n+1}\circ \overline{k_{n+1}}}\Big) )(\overline{e},\overline{e})|,
(- \Delta_p)^s\, u(x):= 2\, \lim_{\varepsilon \searrow 0}\int_{\mathbb{R}^n \setminus B_\varepsilon(x)}\frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))}{|x - y|^{n+s\,p}}\, dy, x \in \mathbb{R}^n.
a^{(N)}_j(\lambda):= \frac{N! (-2)^{N-j}}{j!(2N-2j)!}\prod_{k=j}^{N-1}(2\lambda-4N+2k+n+1)a_N^{(N)}(\lambda), b^{(N)}_j(\lambda):= \frac{N! (-2)^{N-j}}{j!(2N-2j+1)!}\prod_{k=j}^{N-1}(2\lambda-4N+2k+n-1)b_N^{(N)}(\lambda),
<p_1'|{\cal T}^{(E)}|p_1> = \int\frac{d^3x}{(2\pi)^3}e^{i(p_1 - p_1') \cdot x}\,i\frac{d}{d\alpha}\, \exp \left[-\textstyle{\frac{i}{E}}\int_{\alpha E}^{\infty}d\tau V(x+2\textstyle{\frac{p}{E}}\tau,p)\right]\big|_{\alpha = 0}\, .
0=K_{\omega,\mathbf{c}}(\Psi)=\partial_{\lambda}S_{\omega,\mathbf{c}}(\lambda \Psi)\big|_{\lambda =1}=\sum_{j=1}^3\langle D_jS_{\omega,\mathbf{c}}(\Psi),\psi_j\rangle=\eta \sum_{j=1}^3\langle D_jK_{\omega,\mathbf{c}}(\Psi),\psi_j\rangle=\eta \partial_{\lambda}K_{\omega,\mathbf{c}}(\lambda \Psi)\big|_{\lambda =1}.
\mathrm{H}_{\mathcal{H}_{V}}^{s}:=\left\{f \in \mathcal{D}_{\mathcal{H}_{V}}^{\prime}\left(\mathbb{R}^{n}\right): \left(I+\mathcal{H}_{V}\right)^{s / 2}f \in L^{2}\left(\mathbb{R}^{n}\right)\right\},s\in\mathbb{R},
\gamma^{\mu\rho\nu}\nabla_{\rho}(\Psi^\mp_{\nu})^{A}-\gamma^{\mu\nu}\gamma^{5}\partial_{5}(\Psi^\pm_{\nu})^{A}-k\epsilon\tanh\left(k|y|-\frac{k\pi r_{c}}{2}\right)\gamma^{\mu\nu}\gamma^{\hat{5}}(\Psi^\pm_{\nu})^{A}-\frac{3}{2}k(\sigma_1)^{A}_{\;B}\gamma^{\mu\nu}(\Psi^\mp_{\nu})^{B}=0\ .
\langle\otimes_{l = 1}^{m}\mathfrak{c}\rangle_{t}=\exp \biggl( -\frac{t}{2}\sum\limits_{j=1}^N \sum\limits_{i,p = 1}^m\otimes_{l = 1}^m(EK_{j})^{\delta_{il}+\delta_{pl}}\biggr) \langle\otimes_{l = 1}^{m}\mathfrak{c}\rangle_{0},
Y_{1}Z_{2}Y_{3}=[\begin{matrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}\\c_{3}s_{2}&c_{2}&s_{2}s_{3}\\ -c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{matrix}]
\overset{1}{\mathbb{P}}:= \mathbb{L}^{2}\longrightarrow \mathbb{H}_{0}\left(\div=0 \right), \;\;\;\overset{2}{\mathbb{P}}:= \mathbb{L}^{2}\longrightarrow \mathbb{H}_{0}\left(Curl = 0 \right) \;\; \;\;\overset{3}{\mathbb{P}}:= \mathbb{L}^{2}\longrightarrow \nabla \mathcal{H}armonic.
\left\lbrace \begin{array}{cc}\partial_{\tau}A_1 +c_1\cdot\nabla A_1 = i \overline{A_2A_3}, \partial_{\tau}A_2 +c_2\cdot\nabla A_2 = i \overline{A_1A_3}, \partial_{\tau}A_3 +c_3\cdot\nabla A_3 = i \overline{A_1A_2}, \end{array}\right. \Omega,
{\prod_{j =1}^tx_j^{-\tau+\zeta_j}\prod_{i=t+1}^sh_{p^*_i}(i,\boldsymbol{x})}&= \tilde{K}\prod_{j=1}^{t}x_j^{-\tau+\zeta_j+|Q_j|}\prod_{i=t+1}^{s}x_{j_{p^*_i}}^{\tau-1-\zeta_i-d_{i,\bar{U}}-p^*_i,}
P_{0,s}^V(A_4) = P_{0,s}^V(A_4, i_{\max}> n) + P_{0,s}^V(A_4, i_{\max}\leq n) & \leq P_{0,s}^V(i_{\max}> n) + P_{0,s}^V(\exists 1 \leq i \leq n: F_i \mbox{occurs}) & \leq 2c_{10}n^{-\delta}+ n e^{-n^{\epsilon_1 \epsilon_2}}& \leq n^{-(\delta/2 + \delta \epsilon_2/4)}.
\hspace{. 7 4 5 i n}= - q ( \frac{z_{1}}{z_{2}})^{- \delta_{i 1}}r ( \frac{z_{1}}{z_{2}}) ( 1 \otimes q^{- 2 \overline{{\rho}}}) \overline{{R}}_{V^{*}V}( \frac{z_{1}}{z_{2}})
- (2\pi k_{f})\int_{-\infty}^{+\infty}\mbox{}\frac{dq_{1}}{2\pi}\mbox{}\frac{\Lambda_{k + q_{1}/2}(-q_{1})}{2\omega_{R}(q_{1})(\omega_{R}(q_{1}) + \omega_{k + q_{1}/2}(q_{1}))^{2}(\frac{m^{3}}{q_{1}^{4}})( cosh(\lambda(q_{1})) - 1 )}
\frac{\pi}{3T}\log B_{2T}^{(1)}(s, v) = \frac{\pi}{3T}\sum_{n=1}^{(s+v)T/2}\log \cos \left( \frac{\{2n+(3-2s)T-1\}\pi}{6T}\right) &\rightarrow \int_{(3-2s)\pi/6}^{(3-s+v)\pi/6}\log \cos u \, du\mbox{in $T \to \infty$}.
\rho_\leq (x) = \# \{(i,j):\ 1 \leq i \leq j \leq n\ \r_{i,j}=r_{i-1,j-1}\}, \rho_< (x) = \# \{(i,j):\ 1 \leq i < j \leq n\ \r_{i,j}=r_{i-1,j-1}\}.
P_{\pi}g=[\begin{matrix}e_{\pi(1)}\\e_{\pi(2)}\\ \vdots\\e_{\pi(n)}\end{matrix}][\begin{matrix}g_{1}\\g_{2}\\ \vdots\\g_{n}\end{matrix}]=[\begin{matrix}g_{\pi(1)}\\g_{\pi(2)}\\ \vdots\\g_{\pi(n)}\end{matrix}].
{\rm Det}^{\star}_n\begin{pmatrix}\lambda_1 0 \ldots \dots 0 0 \lambda_2 0 \ldots 0 \vdots \vdots \ddots 0 \vdots \vdots \vdots \vdots \ddots 0 0 \ldots 0 0 \lambda_n\end{pmatrix}=[\lambda_1\star \lambda_2\star \cdots \star \lambda_n]_{\overline{\mathbb{F}}^{\star}}.
[(\nabla)_{\beta}, \nabla_{\gamma_1}\nabla_{\gamma_2}\cdots \nabla_{\gamma_{m+1}}]W = [(\nabla)_{\beta}, \nabla_{\gamma_1}](\nabla)_{\gamma^{\prime}}W + \nabla_{\gamma_1}[(\nabla)_{\beta}, (\nabla)_{\gamma^{\prime}}]W
\mathbb{E}_{\mathbf{H}, \mathbb{W}}\left \{\| \tilde{\mathbf{h}}_k\hat{\mathbf{Q}}^H \|^2 \right\}= \mathbb{E}_{\mathbf{H},\mathbb{W}}\left( \left(K-1,n_T-1\right) \right) =\frac{K-1}{n_T-1},
[ \ell_{\mu}^{2}, q \cdot \! \nu ] = - i \left[ 2 ( \mu \cdot \nu ) \ell_{\mu}+ \sum_{\rho \in \Delta_{+}}g_{| \rho |}( \rho \cdot \mu ) ( \rho^{\vee}\! \cdot \nu ) \frac{\ell_{\mu}^{2}- \ell_{s_{\rho}( \mu )}^{2}}{\ell_{\mu}- \ell_{s_{\rho}( \mu )}}\check{s}_{\rho}\right] .
\lim\limits_{n\to+\infty}\frac{(2\ln n)^{\frac{1}{2}}}{n\sin(\alpha_n/2)}=\lim\limits_{n\to+\infty}\frac{(2\ln n)^{\frac{1}{2}}}{n\alpha_n/2}&=\lim\limits_{n\to+\infty}\frac{(2\ln n)^{\frac{1}{2}}}{n(1+z/\ln n)G_n(x)/4}=\lim\limits_{n\to+\infty}\frac{(32\ln n)^{\frac{1}{2}}}{nG_n(x)}=1,
X_{j,2}(t \wedge{\bar \rho}_j) = \int_{\xi_j}^{t \wedge{\bar \rho}_j}\mu X_{j-1}(u) e^{\int_u^{t \wedge{\bar \rho}_j}G_j(v) dv}\: du + e^{\int_{\xi_j}^{t \wedge{\bar \rho}_j}G_j(v) \: dv}Z_j'(t).
\gamma_{\mu}=\left(\begin{array}{cc}\sigma_{\mu}0 0 -\sigma_{\mu}\end{array}\right)\,,\quad\sigma_{\mu\nu}={1\over2i}[\gamma_{\mu},\gamma_{\nu}]=\epsilon_{\mu\nu\rho}\sigma_{\rho}\,;\quad\mu,\nu,\rho=1,2,3\,.
A(z) =F_{1}^{1}(z) -F_{0}^{2}(z) = \frac{1}{\sqrt{2\omega}}\exp (-z) \big( Q_{1}^{1}(z)-Q_{0}^{2}(z)\big) \hphantom{A(z)}{}= \frac{1}{\sqrt{2\omega}}\exp (-z) \left( \sum_{j=0}^{2}\frac{d^{j}}{dz^{j}}\big( z^{2}\big) -\sum_{j=0}^{3}\frac{d^{j}}{dz^{j}}\big( z ( 2-z )^{2}\big) \right) = \frac{z^{2}( 2-z )}{\sqrt{2\omega}}e^{-z}=B(z)
\sum_{\chi \in X_p}\chi(q_1)\overline{\chi}(q_2)\sum_{k=1}^{p-1}\vert S(k,\chi)\vert^2 = \sum_{\chi \in X_p}\chi(q_1)\overline{\chi}(q_2)\frac{p^2-1}{12}+ \frac{(p-1)p(2p-1)}{6}+ \frac{p^3}{2\pi^2}M_{q_1,q_2}(p) +O(p^2).
x^*_i> 0; \qquad\frac{x^*_{i}}{x^*_{i-1}}=O(i/d\Delta) \,\,\, ; 0\le \rho^*_{\tau}(i)\le 1\,\, ; \qquad\sum_{\tau\in\Gamma}\rho^*_{\tau}(i)\le 1.
({\bf a}_i\cdot{\bf a}_j)=({\bf b}_i\cdot{\bf b}_j)=({\bf c}_i\cdot{\bf c}_j)=\delta_{ij},\quad({\bf a}_i\cdot{\bf b}_j)=-\frac{1}{2},\quad({\bf a}_i\cdot{\bf c}_j)=\frac{1}{2},\quad
I ( s , t ) = \frac{3^{9}}{2^{1 2}}\int_{0}^{\infty}d T \; e^{T}\; \frac{\det ( 1 + S \tilde{X}( T ) )}{\det ( 1 - S \tilde{N}( T ) )^{1 3}}\exp \left( - \frac{1}{2}p_{i}Q_{i j}p_{j}\right)
E\left|Y(\omega)\right|^2={1 \over 2^{2(J-L)}}\prod_{j=L}^{J-1}\left|H(2^j\omega)\right|\cdot \left(\sum_{k=0}^{2^{J-L}-1}\prod_{j=L}^{J-1}\left|\widetilde{H}(2^j(\omega+k\pi/2^{J-L-1}))\right|\right).
p^i_\alpha(\omega_1) = p_\alpha^i([\omega_1]) = p_\alpha^i([\omega_2]) = p^i_\alpha(\omega_2) \, , p^I_\alpha(\omega_1) = p_\alpha^I([\omega_1]) = p_\alpha^I([\omega_2]) = p^I_\alpha(\omega_2) \, .
W_1(C)-(V_1(C)+V_2(C))\geq -1, (W_1(C)+W_2(C))-(V_1(C)+V_2(C)+V_3(C))\geq -1, &\vdots (\sum_{t=1}^{p-1}W_t(C))-\sum_{t=1}^{p}V_t(C))\geq -1.
M(q^2x,y,z;q) =\frac{xq}{yz}M(x,y,z;q)+z \cdot \frac{J_4^3j(q^2y^2z^2;q^4)}{j(q^2y^2;q^4)j(z^2;q^4)}&\ \ \ \ \ -\frac{xq}{y}\cdot \frac{J_4^3j(q^2x^2z^2;q^4)}{j(q^2x^2;q^4)j(z^2;q^4)}+\frac{q^2}{zy^2}\cdot \frac{J_4^3j(x^2y^2;q^4)}{j(q^2x^2;q^4)j(q^2y^2;q^4)}.
QP=(\begin{matrix}1&2&3&4&5\\ 5&4&3&2&1\end{matrix})(\begin{matrix}1&2&3&4&5\\ 2&4&1&3&5\end{matrix})=(\begin{matrix}2&4&1&3&5\\ 4&2&5&3&1\end{matrix})(\begin{matrix}1&2&3&4&5\\ 2&4&1&3&5\end{matrix})=(\begin{matrix}1&2&3&4&5\\ 4&2&5&3&1\end{matrix}).
\frac{\partial^2 V}{\partial\lambda_i^2}= \frac{a}{\lambda_i^2}+ \beta \sum_{j \ne i}\frac{1}{(\lambda_i - \lambda_j)^2}, \frac{\partial^2 V}{\partial\lambda_i \partial \lambda_j}= - \beta \frac{1}{(\lambda_i - \lambda_j)^2}.
\left( \alpha(1-r^{2})-r_{1}^{2}\right)\left( \alpha(1-r^{2})-r_{2}^{2}\right)-\left( r_{12}(1-r^{2})+rr_{1}r_{2}\right)^{2}&= \frac{\alpha}{144}(\alpha^{2}-c_{m})(c_{m}^{2}+\alpha e_{m})z^{10}+O(m^{6}z^{9}+m^{15/2}z^{11}).
L_5^\prime(\lambda)=\begin{bmatrix}0 0 \lambda I_n -I_n 0 0 0 0 \lambda I_n -I_n \lambda I_n 0 -\lambda P_1+P_0 -\lambda P_2+P_1 P_2 -I_n \lambda I_n -\lambda P_2+P_1 -\lambda P_3+P_2 P_3 0 -I_n P_2 P_3 \lambda P_5+P_4\end{bmatrix}.
\psi_{3}(x,t)=c_{1}\psi_{1}(x,t)+c_{2}\psi_{2}(x,t)=\frac{1}{a}(\psi_{2}(x_{0},0)\cdot\psi_{1}(x,0)-\psi_{1}(x_{0},0)\cdot\psi_{2}(x,0))\cdot e^{-iE_{g}t/\hbar}.
\begin{array}{c l}{{}}&{{\displaystyle \prod_{\mu = 2}^{n - 2}( x_{\mu}- z_{1}\tau ) \Delta ( z_{1}\tau^{- 1}, x_{2}, \cdots , x_{n - 2}| z_{1}\tau^{- n}, z_{2}, \cdots , z_{n})}}\\{{=}}&{{\displaystyle \prod_{\mu = 2}^{n - 2}( x_{\mu}- z_{1}\tau^{- 1}) \Delta ( z_{1}\tau , x_{2}, \cdots , x_{n - 2}| z_{1}\tau^{n}, z_{2}, \cdots , z_{n}) .}}\\ \end{array}
\langle \hat u_j, u_k\rangle=\frac{\langle u_j, Eu_k\rangle}{\lambda_j-\lambda_k}+\frac{\langle \hat u_j-u_j, Eu_k\rangle}{\lambda_j-\lambda_k}+\frac{\lambda_j-\hat\lambda_j}{\lambda_j-\lambda_k}\langle \hat u_j, u_k\rangle,
F_{k,m,r}(z) = \frac{1}{r!}\frac{\partial^r}{\partial s^r}P_{k,m}(z,s)\bigg|_{s=1-k/2}&= \sum_{j=0}^r c_{m,j}^+ u_{k,m}^{[j],+}(y) e^{2\pi imx}+ \sum_{(-1)^{\lambda_k}n \equiv 0, 1(4)}\sum_{j=0}^r c_{n,j}^-u_{k,n}^{[j],-}(y)e^{2\pi inx},
\Lambda \frac{d}{d\Lambda}2\pi^2\int d^4q \frac{1}{q^2 (q+p)^2\Big(1+q^{2n}/\Lambda^{2n}\Big)^2}\sum\limits_i c_i \ln\frac{M_i}{q}= 4\pi^4 \ln \frac{\Lambda}{p}+ O(1)
\beta_{w',n-\ell}^*(P_{0,m-\ell})=\beta_{w',n-\ell}^*(P_{m-\ell,0})\leq \beta_{w'',n-m}^*(P_{(0,0)})\cdot \prod_{i=\ell+1}^{m}w_i =\prod_{i=\ell+1}^m w_i\;,
\lim_{n\to\infty}\frac{\log b_c^{(n)}}{\log b_c^{(n)'}}=\lim_{n\to\infty}\frac{\log n+(r-1)\log(n p_n)-n p_n}{\log n+(r-1)\log(n p_n)+n\log(1-p_n)}&=\lim_{n\to\infty}\frac{\log n/(n p_n)+(r-1)\log(n p_n)/(n p_n)-1}{\log n/(n p_n)+(r-1)\log(n p_n)/(n p_n)+\log(1-p_n)/p_n}=1
\vec{n}^{\prime}\vec{\Gamma}_{\mu}^{\prime}= \vec{n}^{\prime}\Omega \vec{\Gamma}_{\mu}+ \vec{n}^{\prime}\vec{\Omega}_{\mu}= \vec{n}\vec{\Gamma}_{\mu}+ \vec{n}^{\prime}\vec{\Omega}_{\mu}= \vec{n}^{\prime}\vec{\Omega}_{\mu}.
\begin{array}{lll}\mathfrak{Q}_1^{k-1}(x_{0}, \xi_1) = f_1(x_{[n_1]}^k, \Psi_1) + \mathcal{Q}_2^{k-1}(x_{[n_1]}^k)=F_1(x_{[n_1]}^k, \Psi_1)-\mathcal{Q}_2(x_{[n_1]}^k)+\mathcal{Q}_2^{k-1}(x_{[n_1]}^k).\end{array}
{{P}_{{\rm{DS}},k}}= \left\{\frac{{{\left(\sum\limits_{m = 1}^{{M_{\rm{R}}}}\left|[{{{\bf{\Xi}}_{{\rm{DS}},m}}}]_{k,k}\right|\right)^2}E}}{{{N_{\rm{R}}}{M_{\rm{R}}}{\sigma^2}}}<\gamma_{\rm th}\right\},
\mathcal{T}^\nu_{2,\mp}:= \mathcal{B}^{1,\nu}\circ \widetilde{\mathcal{F}}^\nu_{\mp}\circ \mathcal{R}^\nu & =\mathcal{B}^{1,\nu}\circ \widetilde{\mathcal{F}}^\nu_{\mp}\circ (\mathcal{B}^{1,\nu})^{-1}\circ Proj\circ (\mathcal{B}^{2,\nu})^{-1}& = \Gamma_{{\mp}i}\circ Proj\circ (\mathcal{B}^{2,\nu})^{-1}.
\sigma_{\mu \nu}={\frac{1}{2}}( \sigma_{\mu}\bar{\sigma}_{\nu}- \sigma_{\nu}\bar{\sigma}_{\mu}) ~ , \hspace{0 . 8 c m}\bar{\sigma}_{\mu \nu}={\frac{1}{2}}( \bar{\sigma}_{\mu}\sigma_{\nu}- \bar{\sigma}_{\nu}\sigma_{\mu}) ~ ~ ;
M_{2^{k}}\left( t\right) =\left[ \bar{w}_{2^{k}}\left( t\right) ,\Lambda_{1}^{\left( 2^{k}\right)}\bar{w}_{2^{k}}\left( t\right) ,...,\Lambda_{2^{k}-1}^{\left( 2^{k}\right)}\bar{w}_{2^{k}}\left( t\right) \right]
\frac{|b_{14}^{(k)}+b_{23}^{(k)}|}{|b_{14}^{(k)}|+|b_{23}^{(k)}|}< \frac{2\sqrt{\epsilon}S(B)}{\sqrt{(b_{14}^{(k)})^2+(b_{23}^{(k)})^2}}= \frac{2\sqrt{\epsilon}S(B)}{S(B^{(k+1)})}< \frac{2\sqrt{\epsilon}S(B)}{(1-\epsilon )S(B)}= 2 \frac{\sqrt{\epsilon}}{1-\epsilon}<1,
\begin{gathered}\rho(x^2)=x^2 + y^2 + z^2 + w^2 = 0 5 \rho(x^4) + \rho(x^3y) + \rho(x^3z) -4\rho(x^2yw) -2 \rho(x^2zw) + 3\rho(x^2y^2) + 7\rho(xyzw) = 0 \end{gathered}
\vert R_n^+(z)\vert\le{2\vert\vert A_n'\vert\vert_1\over\vert (2\Lambda -1)(2\Lambda )^{n-1}\vert}e^{2\vert\vert g\vert\vert_1/\vert 2\Lambda -1\vert}, \vert R_n^-(z)\vert\le{2\vert\vert A_n'\vert\vert_1\over\vert (2\Lambda -1)[2(1-\Lambda )]^{n-1}\vert}e^{2\vert\vert g\vert\vert_1/\vert 2\Lambda -1\vert},
u\left(t^{(l)}_{m\,n}\right) =\delta_{m,n}\, q^{2(m-k)}u\left(t^{(l)}_{k\,k}\right),\quad v\left(t^{(l)}_{m\,n}\right) =\delta_{m,n}\, q^{2(k-m)}v\left(t^{(l)}_{k\,k}\right) .
\langle J^{'}(t_{\widehat{w}^+}\widehat{w}^{+}+s_{\widehat{w}^-}\widehat{w}^{-}),\widehat{w}^+\rangle=0\ \ \langle J^{'}(t_{\widehat{w}^+}\widehat{w}^{+}+s_{\widehat{w}^-}\widehat{w}^{-}),\widehat{w}^-\rangle=0.
\mathbb{P}_G\{|\xi^{v}_{n^{2\gamma}}|\geq \delta_0 n, |\hat{\xi}^{u}_{n^{2\gamma}}|\geq \delta_0 n\,|\,\xi^v_{n^{2\gamma}}\neq \emptyset, \hat{\xi}^u_{n^{2\gamma}}\neq \emptyset\}=(1-o(n^{-2}))^2=1-o(n^{-2}).
\sum_{a=1}^l (z_1^{s-k_a}g_{a,s}\Big(\frac{z_2z_3}{z_1^2}\Big)f_a(z_2,z_3)+z_2^{s-k_a}g_{a,s}\Big(\frac{z_3z_1}{z_2^2}\Big)f_a(z_3,z_1)+z_3^{s-k_a}g_{a,s}\Big(\frac{z_1z_2}{z_3^2}\Big)f_a(z_1,z_2))=0.
[\begin{matrix}A&B\\C&D\end{matrix}]^{-1}=[\begin{matrix}(A-BD^{-1}C)^{-1}&-(A-BD^{-1}C)^{-1}BD^{-1}\\ -D^{-1}C(A-BD^{-1}C)^{-1}&D^{-1}+D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1}\end{matrix}].
q(s,t,y(0,t,x,\xi),\xi) =q(s,t,q(t,0,x,\eta(t,0,x,\xi)),\xi) &=q(s,0,x,\eta(t,0,x,\xi)), p(s,t,y(0,t,x,\xi),\xi) =p(s,t,q(t,0,x,\eta(t,0,x,\xi)),\xi) &= p(s,0,x,\eta(t,0,x,\xi)).
0 >\cos{B\sqrt{H_2}}\geq \cos{D\sqrt{H_2}}&= \cos{B\sqrt{H_2}}\cdot \cos{C\sqrt{H_2}}+ \sin{B\sqrt{H_2}}\cdot \sin{C\sqrt{H_2}}\cdot \cos{\alpha}&\geq \cos{B\sqrt{H_2}}\cdot \cos{C\sqrt{H_2}}