harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\operatorname*{Op}\nolimits^{\mathrm{W}}(a) \psi(x)=\left( \tfrac{1}{2\pi\hbar}\right)^{n}\int_{\mathbb{R}^{2n}}a_{\sigma}(z_{0})\widehat{T}(z_{0})\psi(x)d^{2n}z_{0}
\boldsymbol{\lambda}(ab,c)=\mathbf{h}_{11}(a,bc)-(-1)^{|c|(|a|+|b|)}\mathbf{h}_{11}(ca,b)=\boldsymbol{\lambda}(a,bc)-(-1)^{|c|(|a|+|b|)}\boldsymbol{\lambda}(ca,b).
\sigma_1\leqslant a_2=\left(a_1^2-\left|\operatorname{det}\left(a_1^2-A^{H}A\right)\right|\left(\frac{n-1}{(n+1) a_1^2-\|A\|_{F}^{2}}\right)^{n-1}\right)^{1/2}\leqslant a_1
g(x) = x^{10}+ x^9 + x^7 + x^5 + x^4 + 1, x \, g(x) = x^{11}+ x^{10}+ x^8 + x^6 + x^5 + x, x^2 \, g(x) = x^{12}+ x^{11}+ x^9 + x^7 + x^6 + x^2, x^3 \, g(x) = x^{13}+ x^{12}+ x^{10}+ x^8 + x^7 + x^3, x^4 \, g(x) = x^{12}+ x^{10}+ x^7 + x^6 + x^5 + x^3 + x^2 + x + 1.
X_{T_i(t')^k}X_{T_i(t)^k}=q^{\frac{1}{2}\varLambda(d_i(t')^*,d_i(t)^*)}q^s \prod_{j\neq i}X_{T_j(t)^k}^{a_{ij}}+ q^{\frac{1}{2}(\varLambda(d_i(t')^*,d_i(t)^*)-1)}q^{s'}\prod_{j\neq i}X_{T_j(t)^k}^{b_{ij}}.
\int_0^1 u_0(1-t^2,\varphi)\frac{-1}{4(n+1)}\cdot t^{2n}\cdot 2tdt begin{align*}5pt] \, - \sum_{j=1}^{\infty}s_j F_j(1, \varphi)\cdot \frac{1}{2\pi}\int_0^{2\pi}\int_0^1 nt^{2n}\cdot \frac{2}{t}\int_t^1 u_0\Bigl(\frac{\rho^2-t^2}{\rho}, \theta\Bigr) \overline{Q_j(\rho,\theta)}\rho d\rho dt d\theta =0.
R_{}= R_{}+ R_{}+ R_{}&\;= I(A;\widetilde{X}) + I(U, V;\widetilde{X}|A)-I(U,V;Y|A)+6\epsilon &\; \overset{(a)}{=}I(A;\widetilde{X}) + I(V;\widetilde{X}|A)-I(V;Y|A)+6\epsilon &\;\overset{(b)}{=}I(A;\widetilde{X}) + I(V;\widetilde{X}|A,Y)+6\epsilon
\begin{aligned}\| t^{\gamma_1}\delta f_{n,1}\|_{L^{2r}(0,T;L^{p_3}_x)}+ \| t^{\gamma_2}\delta f_{n,1}\|_{L^{\infty}(0,T;L^{p_3}_x)}+ &+ \| t^{\beta}\nabla \delta f_{n,1}\|_{L^{2r}(0,T;L^{p_2}_x)}\leq\frac{1}{4}\big(\delta U_{n,\lambda}(T)+\delta U_{n-1,\lambda}(T)\big),\end{aligned}
D_{{\vec{c}}, p - 1}= D_{{\vec{c}}, 0}e^{- i 2 \pi \left( q \mid c \mid + N_{\Phi}c_{\gamma}\right) / \ell}e^{\frac{i \pi \tau}{\omega \kappa^{2}}\left( p - 1 \right)^{2}}e^{i \tau \beta N_{A}\left( p - 1 \right) /{\omega \kappa^{2}}}e^{- i \alpha N_{A}\left( p - 1 \right) /{\omega \kappa^{2}}}
{{f_{G_K}}\left( x \right) ={\rm E}_{T}\left\{{f_{\left. G_K \right|T}}\left( x|t \right)\right\}= \int\nolimits_0^\infty{{f_{\left. G_K \right|T}}\left({\left. x \right|t}\right){f_T}\left( t \right)dt}},
H_{i,j}= (n-1)\log(n-1), H^c_{i,j}= 2(n-1)\log(n-1), H^{av}= \left(\dfrac{(n-1)^2}{n}+ 1\right) \log (n-1), (t^{rel}+ 1) H(X) = \left(\dfrac{n-1}{n}+ 1\right) \log (n-1) \leq H^{av}\leq \left(\dfrac{(n-1)^2}{n}+ 1\right) \log n = ((n-1)t^{rel}+ 1) \log n.
i_1>\dots >i_r\ge 0, r\ge 0, a_1,\ldots ,a_r\ge 1, \tau\in P, k_1>\dots >k_t\ge 0, t\ge 0, c_1,\ldots ,c_t\ge 1, c_1+\dots +c_t-(a_1+\dots +a_r)=2(n-m),
a_{1}= k [ \tilde{H}]_{1}^{n - 2}( B_{0}^{2}A_{1}^{0}+ B_{1}^{1}A_{0}^{1}) + \frac{1}{\alpha}( - )^{n - 3}k{\overline{{A}}}_{1}^{*}B_{1}^{1}+ m_{1}A_{1}^{0},
L =\; m_2\left((u^2)^2+(v^2)^2\right)+m_3\left((u^3)^2+(v^3)^2\right)+v^2x^2+v^3x^3-u^2 y^2 - u^3 y^3 &-\left((x^1)^2+(y^1)^2\right)-\left((x^2)^2+(y^2)^2\right)-\left((x^3)^2+(y^3)^2\right).
S_{int}=8\Omega^2\sum_{m,p,q,r\in\mathbb{N}}i\phi_{pq}\phi_{qr}\phi_{mp}(a_r\delta_{m+1,r}-a_r\delta_{r+1,m})+4\Omega^2\sum_{m,n,k,r\in\mathbb{N}}\phi_{mn}\phi_{nk}\phi_{kr}\phi_{rm}.
\hat{\cal Y}_{P P}^{( P + 1 / 2 )}= \widehat{{\frac{d}{d \theta}}\mathrm{I m}\operatorname{l n}B}+ \widehat{{\frac{d}{d \theta}}\mathrm{I m}\operatorname{l n}f_{P}}+{\frac{1}{2}}\left( \hat{\xi}\right)^{2}
T^m_{+}(r,r') = \frac{4^{-|m+\alpha|}\, (r r')^{|m+\alpha|}}{\Gamma^2(|m+\alpha|+1)}\, , \qquad T^m_{-}(r,r') = \left( \frac{r r'}{4}\right)^{-|m+\alpha|}\, \Gamma^2(|m+\alpha|) \ \frac{\sin^2(|m+\alpha| \pi)}{\pi^2}T^m_{\pm}(r,r') = - (r/r')^{|m+\alpha|}\ \frac{\sin(|m+\alpha|\, \pi)}{\pi\, |m+\alpha|}\, , T^m_{\mp}(r,r') = T^m_{\pm}(r',r).
2^{k t}\|\tilde \Pi_1((S^{k-2}\varphi )^\Gamma, 1_{\Lambda,k}^\Gamma) \|_{B^s_{p,\infty}(\real^{d_s})}\le 2^{k t}\sum_{n=k-1}^{k+1}\|(\tilde S^{n-2}(S^{k-2}\varphi)^\Gamma) \cdot \tilde S_n(1_{\Lambda,k}^\Gamma)) \|_{B^s_{p,\infty}}\, .
H(v, s) = \exp\Big( \int_{\max\{v - R_{\tau}, R_{\tau}\}}^{v + R_{\tau}}\frac{2 y\lambda_J'(y)}{1 + \frac{y^{\alpha}}{s P_J}}dy \Big), &f(v) = 2 \pi \lambda_U^s(r, v) v \exp(-2\pi \int_{0}^{v}\lambda_U^s(r, y) y dy), &\lambda_J'(y) = \lambda_J\arccos(\frac{y^2 + v^2 - R_{\tau}^2}{2 y v}).
\widetilde{\varphi}_{\mu}( x ) = \frac 1{3 !}\epsilon_{\mu}^{\nu \rho \sigma}\varphi_{\nu \rho \sigma}( x ) , \widetilde{\varphi}( x ) = \frac 1{4 !}\epsilon^{\mu \nu \rho \sigma}\varphi_{\mu \nu \rho \sigma}( x )
\sqrt{\rho_g(x)}\left\langle \tilde{h}, \frac{1}{\sqrt{2}}\left[ \frac{v^2}{T_g}-1 \right] \mathcal{M}_{T_g}\right\rangle \leq \left(\int_{-\infty}^{\infty}\frac{\tilde{h}^2}{g^2}\rho_g(x) \mathcal{M}_{T_g}(v) \mathrm{d}v \right)^{1/2}.
|II| \leq C r \log(\log(t)) \sup_{x \geq t}|\lambda_{1}''(x)-\lambda_{2}''(x)|+ \frac{C r ||e_{1}-e_{2}||_{X}}{t^{2}\log^{b+1}(t) \sqrt{\log(\log(t))}}&\leq \frac{C r \log(\log(t)) ||e_{1}-e_{2}||_{X}}{t^{2}\log^{b+1}(t) \sqrt{\log(\log(t))}}+\frac{C r ||e_{1}-e_{2}||_{X}}{t^{2}\log^{b+1}(t) \sqrt{\log(\log(t))}}
\mathbb{E}\left[ \left( \int_{0}^{T}|g_{z}(t)\Delta(t)I_{E_{\epsilon}}(t)|dt\right)^{4}\right] \leq C\epsilon^{3}\int_{E_{\epsilon}}\mathbb{E}[|\Delta(t)|^{4}]dt=o\left( \epsilon^{2}\right) ,
U_{n, l}:= \left (T_{n-l}\times 2^{\frac{cn}{\ln \ln (n+3)}}\right)\times \left(j^l 2^{\frac{l^2}{4}}\right)\times \left(2^{(j-1)l}\right)\times{n \choose l}\times \left((n+1)^{2(j-1)}\right)
E\{y(n)\prod_{j=1}^{\overline{p}}x(n-\tau_{j})\}=E\{G_{\overline{p}}x(n)\prod_{j=1}^{\overline{p}}x(n-\tau_{j})\}=\overline{p}!A^{\overline{p}}k_{\overline{p}}(\tau_{1},...,\tau_{\overline{p}}).
l^\mathcal{L}(t,U,\Gamma) =\int_M\mathbf{i}_{\partial_t}\mathcal{L}\left(t,U(p),\operatorname{Hor}^\Gamma_p\right) &=\int_M\bar\ell\Big(t,[U(p)],\left[\operatorname{Hor}^\Gamma_p\right]\Big)\mu_M &=\int_M\bar\ell^\mathcal{A}\Big(t,\tilde{\mathcal{U}}(m),\left[p,\mathcal{A}_M\!\cdot\!\operatorname{Hor}^\Gamma_p\right]_G\Big)\mu_M,
V^{\mathrm{var}}(r_{k_0,l},w_l)\approx \sum_{i=1}^n r_{k_0,l}\int_{-1}^{1}P_l(r_{k_0,l}\xi) \phi(r_{k_0,l}\xi) \big[\cos\big(\frac{w_l\xi\theta_i}{\sigma_+}\big)-1\big] d\xi\ .
\begin{cases}u_t^{(n+1)}=\nabla u^{(n+1)}\cdot \nabla(-\triangle)^{-s}u_{\epsilon}^{(n)}-u_{\epsilon}^{(n)}(-\triangle)^{1-s}u^{(n+1)}; u^{(n+1)}(0)=\sigma_{\epsilon}*u_0,u^{(1)}=\sigma_{\epsilon}*u_0.\end{cases}
\Gamma(i+1) = \left\{(g_0,g_1,\ldots,g_{i+1}) \;\left|\; (g_0,\ldots,g_i)\in \Gamma(i), \; g_{i+1}^{-1}g_{i}\in \bigcap_{x\in P_1\cdots P_i}x P_{i+1}x^{-1}\right.\right\}.
\begin{array}{rcl}Q_6^{''}(x)\!\! \!\!\!=\!\!\! \!\! 60\pi^6 x^4+12(211\pi^6-453600\pi^2+4536000) x^2+6(529200\pi^3 begin{align*}1.0 ex]\!\! \!\!\! \!\!\! \!\!-5292000\pi) x-1620\pi^6-302400\pi^4+4536000\pi^2-13608000.\end{array}
c_{1b2}=(-81\nu^7+27\nu^6+1287\nu^5+13563\nu^4)-71211\nu^3+85929\nu^2-66699\nu+894 \ge -71211\nu^3 +85929\nu^2-66699\nu+89409> 80000 (-\nu^3+\nu^2-\nu+1)\ge 0.
\Longrightarrow xy(\theta)^2 +(x^2-(t^4+4t)xy)(\theta) -y^2=(t-\theta)^{n'}\theta^{m'}= (-1)^{n'}\left (\theta^2-\binom{n'}{1}t\theta +\binom{n'}{2}t^2 -\binom{n'}{3}t^3\theta^{-1}+\cdots \right ) \theta^{(n'+m'-2)}
0= \langle \tau_{H}, \alpha_{2}\rangle \{-(4m_{1}+2m_{2})(\cot \langle 2\alpha_{2}, H\rangle )^{2}& +(2m_{2}+4n_{3})(1-(\tan \langle 2\alpha_{2}, H\rangle )^{2})-4m_{1}\}(\alpha_{1}+2\alpha_{2}).
G_{\alpha}= \left( \begin{matrix}{q^{\Lambda \Sigma}{}_{\Pi \Delta}( \alpha )}&{p^{\Lambda \Sigma \Psi \Xi}( \alpha )}\\{p_{\Delta \Gamma \Pi \Delta}( \alpha )}&{q_{\Lambda \Sigma}{}^{\Psi \Xi}( \alpha )}\\ \end{matrix}\right)
\sum_{t<Y}F_{Y}\Bigl(\frac{t}{Y}\Bigr) \ll \frac{1}{(q-s)^k}\prod_{i=0}^{k-1}\Bigl|\sum_{t_i<q}\min\Bigl(q-s,\frac{q}{t_i}+\frac{q}{q-t_i}+s\Bigr)\Bigr| &= O\Bigl(\frac{q\log{q}+q s}{q-s}\Bigr)^k.
\sum_{\mathcal{A}}n_{\mathcal{A}}^{s_{r-1}}\le (c+o(1))\cdot n^{(a_1+\cdots+a_{r-1})-\frac{a_1\cdots a_{r-1}}{s_1 \cdots s_{r-2}}+s_{r-1}}, c=\frac{(s_r-1)^{\frac{a_1\cdots a_{r-1}}{s_1\cdots s_{r-2}}}}{\prod_{i=1}^{r-1}a_i!}.
{}_2F_1\biggl(\begin{matrix}r, \, 1-r 1 \end{matrix}\biggm|z\biggr)=\sum_{k=0}^\infty\frac{(r)_k(1-r)_k}{k!^2}z^k,r\in\big\{\tfrac13,\tfrac14,\tfrac16\big\}.
\begin{aligned}\bigcup_{\alpha\in H^1(\mathrm{Gal}(E/F),G(E))}\Pi_{(G_{\alpha}(F),\omega_{G_{\alpha}(F),E})}(G(E)) \xhookrightarrow{\iota}\Pi(G(E)) \xrightarrow{\mathrm{LLC}}\Phi(G(E))\xleftarrow{\mathrm{BC}}\Phi(G^{\mathrm{op}}(F)). \mathrm{Im}(\mathrm{LLC}\circ\iota) =\mathrm{Im}(\mathrm{BC}).\end{aligned}
\begin{array}{l l}{a^{\dagger}= \frac{x / m - i p_{x}}{\sqrt{2}},}&{a = \frac{x / m + i p_{x}}{\sqrt{2}},}\\{b^{\dagger}= \frac{y / n - i p_{y}}{\sqrt{2}},}&{b = \frac{y / n + i p_{y}}{\sqrt{2}}.}\\ \end{array}
e_{\hat{0}}^{\; \; \mu}e_{\alpha}^{\; \; \nu}\left. \frac{\partial \Gamma_{R}}{\partial g^{\mu \nu}}\right|_{b_{\mu \nu}}= -{\bar{e}}_{\alpha}^{\; \; i}{e}_{\hat{0}}^{\; \; 0}\frac{\partial \Gamma_{R}}{\partial v^{i}},
\phi_a=\begin{pmatrix}\sin f(r)~\sin(n\theta-\chi) \sin f(r)~\cos(n\theta-\chi) \cos f(r)\end{pmatrix}=\begin{pmatrix}sS sC c\end{pmatrix}=\begin{pmatrix}\phi_1 \phi_2 \phi_3 \end{pmatrix}
\frac{\mbox{d}}{\mbox{d}z}p+\frac{\mbox{d}}{\mbox{d}\bar{z}}\bar{p}=\frac{\mbox{d}}{\mbox{d}z}\frac{\partial{\cal L}}{\partial v}+\frac{\mbox{d}}{\mbox{d}\bar{z}}\frac{\partial{\cal L}}{\partial \bar{v}}=\partial_{\varphi}{\cal L}\,.
{\widehat \Sigma}= \left( \begin{array}{c c}{{\Sigma_{R R},}}&{{\Sigma_{R A}}}\\{{\Sigma_{A R},}}&{{\Sigma_{A A}}}\\ \end{array}\right) \, , \qquad{\bar{\Sigma}}= \left( \begin{array}{c c}{{\Gamma_{a a}^{( 2 )},}}&{{\Gamma_{a r}^{( 2 )}}}\\{{\Gamma_{r a}^{( 2 )},}}&{{\Gamma_{r r}^{( 2 )}}}\\ \end{array}\right) \, .
s_{2}[ f ; m^{2}] = -{\frac{1}{1 6 \pi m^{2}}}\, \int \! d^{2}x \, \phi^{2}( x ) -{\frac{1}{9 6 \pi m^{4}}}\, \int \! d^{2}x \, \phi ( x ) \partial^{2}\phi ( x ) +{\cal O}( \partial^{4}) .
V_0^{s_{\ell_0}- \mu_1 j_{\ell_0}- \mu_2 k_{\ell_0}}V_1^{j_{\ell_0}}V_2^{k_{\ell_0}}=\sum \pi(\Phi'(Q^{t,\ell})) \hskip1ptV_0^{s_{\ell}- \mu_1 j_{\ell}- \mu_2 k_{\ell}}V_1^{j_{\ell}}V_2^{k_{\ell}}
R_{\alpha}^{j}\frac{\delta R_{\beta}^{i}}{\delta \phi^{j}}- R_{\beta}^{j}\frac{\delta R_{\alpha}^{i}}{\delta \phi^{j}}= C_{\alpha \beta}^{\gamma}R_{\gamma}^{i}+ M_{\alpha \beta}^{i j}\frac{\delta I}{\delta \phi^{i}}, M_{\alpha \beta}^{i j}= - M_{\alpha \beta}^{j i}
\Gamma(C) := \{\varphi(c_1,\ldots,c_m,d^1_{\varphi,\bar{c}},\ldots,d^n_{\varphi,\bar{c}}) : \varphi(\bar{x},\bar{y}) \in \Phi,~ |\bar{x}|=m,~ |\bar{y}|=n,~ \bar{c}\in C^m\}.
\varepsilon_{\pm}\left( \theta \right) =\mathrm{e}^{\pm \theta}+k_{0}-\int^{\infty}_{-\infty}\frac{d\theta '}{2\pi}\varphi \left( \theta -\theta '\right) \log \left( 1+\mathrm{e}^{-\varepsilon_{\pm}\left( \theta '\right)}\right)
\Big|\prod_{i=0}^l \operatorname{Sym}(D_n^{(i)})\Big| = \prod_{i=0}^l (d^{(i)}\cdot d^{n-1})! |\operatorname{Aut}_{U(F)}(B_n(v_0))| = \left( \prod_{i=0}^l d^{(i)}! \right) \cdot\left(\prod_{i=0}^{l}\frac{d^{(i)}!^{(d+1)}}{d^{(i)^{d^{(i)}}}}\right)^{\frac{d^{n-1}-1}{d-1}}.
T ( f(u) \cdot v ~+~ u \cdot f(v)) + f ( T(u) \cdot v + u \cdot T(v)) - T(u) f(v) - f(u) T(v) & + T \big( H ( T(u), f(v)) + H ( f(u), T(v)) \big) + f ( H (T(u), T(v)) ) = 0,~ u, v \in M.
E_{1}=m^{2}\left(\bar{p}+1+ \frac{2\tilde{r}-1+2\alpha}{2n}+ \frac{1-2\tilde{p}}{2m}\right)^{2}-\frac{1}{4}, E_{2}=m^{2}\left(\bar{p}+1 + \frac{1-2\tilde{r}+2\alpha}{2n}+ \frac{2\tilde{p}-1}{2m}\right)^{2}-\frac{1}{4},
i(G)=\sum_{r=0}^{q-1}(\begin{matrix}n\\r\end{matrix})+\sum_{(j_{s})_{s=1}^{q}}i(G[A(j_{1},...j_{q})])\le\sum_{r=0}^{q-1}(\begin{matrix}n\\r\end{matrix})+\sum_{(j_{s})}2^{|A(j_{1},...j_{q})|}
(\Delta)=\sum_{c/d \in \mathcal{C}_N}\frac{N}{d(d,N/d)}\textbf{a}_{\frac{c}{d}}, &(\Delta_N)=\sum_{c/d \in \mathcal{C}_N}\frac{d}{(d,N/d)}\textbf{a}_{\frac{c}{d}}.
\frac{1}{X}\sum_{2\nmid d}\nolimits^\flat L(\frac{1}{2},\phi\otimes \chi_{8d})\chi_{8d}(l)\Phi(\frac{d}{X}) = S(\chi_{8d}(l)B(d);\Phi) + \prod_{i=1}^3\frac{\Gamma(\frac{\frac{1}{2}+\gamma_i}{2})}{\Gamma(\frac{\frac{1}{2}-\gamma_i}{2})}\overline{S(\chi_{8d}(l)B(d);\Phi)},
\left(x^{\alpha+\beta}u_{x}v\right)_{x}= x^{\alpha+\beta}u_{x}v_{x}+x^{\beta}\left(\left(x^{\alpha}u_{x}\right)_x+\beta x^{\alpha-1}u_{x}+\frac{\mu}{x^{2-\alpha}}u\right)v -\frac{\mu}{x^{2-\alpha-\beta}}u v \in L^{1}(0,1).
I_{1}+I_{2}\lesssim e^{-2\lambda_{l}s}\begin{cases}e^{-4\gamma\omega_{l}s}(1+(Ke^{-\omega_{l}s})^{\omega-4\gamma}) \omega>4\gamma e^{-\omega\omega_{l}s}(1+K^{\omega-4\gamma}) \omega<4\gamma. \end{cases}
\Theta = \left \{\begin{pmatrix}\sigma_{1}^2 r \sigma_{1}\sigma_{2}r \sigma_{1}\sigma_{2}\sigma_{2}^2\end{pmatrix}, c_1 \leq \sigma_1^2, \sigma_2^2 \leq c_2, \ -d_1 \leq r \leq d_1 \right \},
J:L^2(E,\mathbb{C}^2) \to L^2(\mathbb{R}^2,\mathbb{C}^2),(J \varphi) (x_1,x_2):=\begin{cases}\varphi(x_1,x_2) \textup{if}x_2 \geq 0 (\sigma_1\varphi)(x_1,-x_2) \textup{if}x_2 < 0\end{cases}\; .
\bar{Y}^j( \alpha_j, \bar{X}^j) =\arg \min_{y^j}\left\{\sum_{i=1}^n(x_i^j-y_i^j)^2+\alpha_j\sum_{i=1}^{n-2}( y_{i+2}^{j}-2 y_{i+1}^{j}+ y_i^{j})^2\right\}.
\begin{array}{rl}r_1 u' - q^2 r_3 v + q^2 r_4 u - (q-q^3) r_5 u = u'r_1 - qvr_3 + ur_4, r_1 v' - q r_5 v + q^3 r_6 u = q^3 v' r_1 - q vr_5 + u r_6, q^3 r_2 u + r_3 v' - q r_5 u' = ur_2 + q^3 v'r_3 - qu'r_5, q^3 r_2 v + r_4 v' -qr_6 u' = v r_2 + q^2 v'r_4 - (q-q^3)v' r_5-q^2 u' r_6.\end{array}
\textrm{E}\left[ M^{(_m)}\right] \simeq \textrm{E}\left[ M' \right] = \frac{1-P_{M'}^{(1)}}{P_{M'}^{(2)}}&=\frac{1-e^{-\lambda T_{M}}\left[1-P_S\left(1-e^{-P_S\lambda T_S}\right)\right]}{e^{-\lambda T_{M}}\left[1-P_S\left(1-e^{-P_S\lambda T_S}\right)\left(1+P_S'-P_S\right) \right]}.
\big|a_nz+\alpha_{n-1}-(t_1-t_2)\alpha_n\big| \leq |a_n|t_1+|\alpha_{n-1}-(t_1-t_2)\alpha_n| &\leq t_1\alpha_n+t_1\beta_n+\alpha_{n-1}-(t_1-t_2)\alpha_n &=2t_1\beta_n+2(t_2\alpha_n+\alpha_{n-1}) &\qquad-(t_2\alpha_n+2t_1\beta_n+\alpha_{n-1}).
\sum_{t = t_{k,j}^- -1}^{t_{k,j}^* -1}\hat{d}_n(t) = \sum_{t = t_{k,j}^- -1}^{t_{k,j}^* -1}a_n^r(t) > Q_n^f(t_{k,j}^+ +1) - Q_n^f(t_{k,j}^-) - \sum_{t = t_{k,j}^-}^{t_{k,j}^* -1}a_n^{f}(t) \geq Q_n^f(t_{k,j}^+ +1) - Q_n^f(t_{k,j}^-) - \sum_{t = t_{k,j}^-}^{t_{k,j}^+}a_n^{f}(t) = \sum_{t = t_{k,j}^- -1}^{t_{k,j}^+}d_n(t),
\begin{array}{lcr}P(W_2)=(1+U_1+U_2+U_1^2U_2^3) + U_3( a_0U_1^2U_2^3+ a_1 U_1U_2+a_{n-1}U_1^{{(5-n)\over 2}}U_2^{{(6-n)\over 2}})+ U_3^2\sum\limits_{i=0}^{n-4}a_{i+1}U_1^{3-i}U_2^{4-i}. \end{array}
H_{k}^{M}=H_{p}^{M}+H_{q}^{M}=\frac{2E_{p}^{M}}{|p|}+\frac{2E_{q}^{M}}{|q|}>\frac{2(E_{p}^{M}+E_{q}^{M})}{|k|}=\frac{2E_{k}^{T}}{|k|}\ge\frac{2E_{k}^{M}}{|k|},
\left[ - \nabla_{\mathrm{\boldmath ~ \scriptstyle{~ \xi}~},{D}}^{2}- \lambda \,{\mathcal W}^{({D})}( \mathrm{\boldmath ~ \xi ~}) \right] \, \Phi ( \mathrm{\boldmath ~ \xi ~}) = \eta \, \Phi ( \mathrm{\boldmath ~ \xi ~}) \; .
|\partial_x F(\tilde{x}+iy(t),t)|\leq \frac{1}{2\pi}\Big(\int \frac{1}{|\tilde{x}+iy(t)-\zeta(\beta,t)|^4}d\beta\Big)^{1/2}\|\zeta_{\beta}\|_{L^{\infty}}\|\mathfrak{F}\|_{L^2}\leq C\epsilon \hat{d}_I(t)^{-3/2}, \forall~t\in [0,T_0],
\frac{i}{x_{1+2+3}}(-i\lambda_3) \left( -i \lambda_3 -ib_2 x_{1+2+3}\right) \left( \frac{i}{x_{1+2}}+ \frac{i}{x_{1+3}}+ \frac{i}{x_{1+4}}\right) +b_3 &= (-i \lambda_3) b'_2 (1+2+3)\left( \frac{i}{x_{1+2}}+ \frac{i}{x_{1+3}}+ \frac{i}{x_{1+4}}\right) +b_3 .
{\gamma \cdot D \Lambda ={e_{{\hat{L}}}}^{M}\gamma^{\hat{L}}\left( \partial_{M}+ \frac{1}{4}\omega_{M}^{\hat{M}\hat{N}}\gamma_{\hat{M}\hat{N}}\right) \Lambda = ( \rho \gamma^{\hat{5}}\partial_{5}+ \rho \gamma^{\hat{\mu}}\partial_{\mu}- 2 \gamma^{\hat{5}}) \Lambda ,}
\int_{0}^{t}d t^{\prime}\int d^{2}\xi^{\prime}K_{\xi \xi^{\prime}}^{t - t^{\prime}}e^{- \phi ( \xi )}\left( - \frac{1}{2}( \xi^{\prime}- \xi )^{a}( \xi^{\prime}- \xi )^{b}\left( \partial_{a}\phi ( \xi ) \partial_{b}\phi ( \xi ) - \partial_{a}\partial_{b}\phi ( \xi ) \right) \right) \Delta^{\prime}K_{\xi^{\prime}\xi}^{t^{\prime}}
l_i(u^a) \nabla r_j(u^a) \big[\lambda_i(u^p)\mathbb{I}- f'(u^p)\big] u^p_x= \lambda_i(u^p) l_i(u^a) \nabla r_j(u^a) u^p_x = \lambda_i(u^p) l_i(u^a)\Big\{\nabla r_j(u^p) + \big[\nabla r_j(u^a) - \nabla r_j(u^p)\big]\Big\}r_p(u^p)\rho_x \leqslant O(1) \big(|u^p_x| \chi_{\{i,j \neq p\}}+e\big),
||\phi_H||_U\leq||\phi_{H^{1,N}}||_U+\cdots+ ||\phi_{H^{n,N}}||_U & \leq C_U||H^{1,N}||+\cdots+ C_U||H^{n,N}||\leq N\cdot \frac{C_U}{N}||H||=C_U||H||.
\varphi^{d+1}g_e=b\varrho^{-1}t^{n_e+1-p^{d+1}}g_e \varphi^{d+1}g_f=b\varrho^{-1}t^{n_f+1-p^{d+1}}g_f \gamma(x)g_e-x^{s_e}g_e\in t\cdot k_{\mathcal E}^+\cdot g_e \gamma(x)g_f-x^{s_f}g_f\in t\cdot k_{\mathcal E}^+\cdot g_f.
U_n(z) = \left( \frac{2}{k A_k}\right)^{\frac{1}{k}\left(-n-\frac{\alpha}{2}\right) \sigma_3}Y_n\left( \left( \frac{2}{k A_k}\right)^{\frac{1}{k}}z \right) \left( \frac{2}{k A_k}\right)^{\frac{1}{k}\frac{\alpha}{2}\sigma_3},A_k = \prod_{j=1}^k \frac{2j-1}{2j},
v ~ = ~{\frac{r_{0}^{2}}{r^{2}}}; A ~ = ~{\frac{\omega^{2}}{4}}\left({\frac{r_{1}r_{5}r_{n}}{r_{0}}}\right)^{2}; ~ ~ B ~ = ~{\frac{\omega^{2}}{4}}\left({\frac{r_{1}r_{5}}{r_{0}}}\right)^{2}.
P_{e g}( \tau ) = \lambda^{2}| \left< e \right| m ( 0 ) \left| g \right> |^{2}\int_{0}^{\tau}\mathrm{d}\tau^{'}\int_{0}^{\tau}\mathrm{d}\tau^{'^{\prime}}e^{i E ( \tau^{'^{\prime}}- \tau^{'})}\left< \Phi_{i}\right| \varphi ( x ( \tau^{'}) ) \varphi ( x ( \tau^{'^{\prime}}) ) \left| \Phi_{i}\right> ,
{\cal Z}_{G U}={\displaystyle \int}{\cal D}( A^{\mu}, \pi_{\mu}, \lambda ) ~ e x p ~{i \displaystyle \int}d^{4}x ~ \left( \pi_{0}{\dot{A}_{0}}+ \pi_{i}{\dot{A}_{i}}-{\widetilde H_{G U}}- \lambda \chi \right) ,
(s_1,f_1) \circ (s_2,f_2) \cdot(x,y) = (s_1s_2,f_1f_2^{s_1}) \cdot (x,y) = (s_1s_2\cdot x,f_1(s_1s_2 \cdot x)f_2(s_2\cdot x) \cdot y),
f_{ij}\,^{k}=-(\chi_{i})_{j}\,^{k}=-(\mathcal{Y}^{k})_{ij}=-({(\mathcal{Y}^{k})^{t}})_{ji}=f'_{ji}\,^{k}=-(\chi'_{j})_{i}\,^{k}, \widetilde{f}^{ij}\,_{k}=-(\widetilde{\chi}^{i})^{j}\,_{k}=-(\widetilde{\mathcal{Y}_{k}})^{ij}=-(\widetilde{(\mathcal{Y}_{k})}^{t})^{ji}=\widetilde{f}'^{ji}\,_{k}=-(\widetilde{\chi'}^{j})^{i}\,_{k}.
b^2_{n,k-1}\leq a^2_{n,\lfloor \zeta q^{-n}\rfloor + k - \lfloor n^\delta q^{-n}\rfloor}\textrm{and}F_{n,k}\geq \tilde{F}_{n,k + \lfloor n^{\alpha}q^{-n}\rfloor}\textrm{for all}n\geq n_0.
\begin{gathered}F_{B_1}= N_\lambda\otimes_S E_{\eta_1}, F_{B_\nu}=E_{\eta_\nu}, \,\,\, 2\leq\nu\leq m-1, F_{B_m}= E_{\eta_m}\otimes_S N_\rho, F_C = N_\lambda\otimes_S E_{A'}\otimes_S N_\rho. \end{gathered}
Q_{1,k}\leq C_{H}|I_k|\left(\sum_{j=1}^k|I_{k}|\right)^{1-2H}\|\dot{\bar{h}}_{k}\|_{\infty;[0,1]}^{2}\leq C_{H,l_0}\frac{|I_k|}{\left(\sum_{j=1}^m|I_{k}|\right)^{2H-1}},
\left[ x, p \right] = i, \quad\left[ a_0, a_0^\dagger \right] = 1, \quad\left[ a^I_n,{a^I_m}^\dagger \right] = \delta_{nm}, \quad\left[ a_n,{a_m}^\dagger \right] = \delta_{nm}, \quad\left[ \tilde{a}_n,{\tilde{a}_m}^\dagger \right] = \delta_{nm}
\lambda^{\left( p\right)}\left( G\right) \leq3!\left( \binom{n}{3}-k\binom{n/k}{3}\right)^{1-1/p}\left( \binom{n}{3}-k\binom{n/k}{3}\right)^{1/p}n^{-3/p}& =3!\left( \binom{n}{3}-k\binom{n/k}{3}\right) n^{-3/p}.
S_{BH}= \frac{2\pi A_9}{\kappa^2}= \frac{2\pi A_3}{\kappa^2_5}= \frac{8\sqrt{2}\pi^3m^{3/2}}{\kappa^2}S_{12}S_{34}S_{56}\times\cosh\alpha_1 \times \cosh \alpha_2 \times \cosh \alpha_3
p_{o}= 1-\!\sum_{\lambda_N=0}^\infty \sum_{\substack{\ell_{1},\ldots,\ell_{N-1}: \ell_{1}+\cdots+\ell_{N-1}=\lambda_N}}\!\!\prod_{j=1}^{N-1}\frac{(-\rho_j)^{\ell_j}}{\rho_N^{\ell_j}\ell_j!}I(\xi_N;\ell_1,\ldots,\ell_{N-1})
(f_d^{-1})^{(1)}(x) = \frac{-de^x}{(e^x-1)^2}, (f_d^{-1})^{(2)}(x) = \frac{de^x (e^x+1)}{(e^x-1)^3}, (f_d^{-1})^{(3)}(x) = \frac{-de^x (e^{2x}+4e^x+1)}{(e^x-1)^4}.
S = \int d^{4}x \sqrt{- \tilde{g}}\left[ \tilde{R}- \frac{1}{2}\left( \tilde{\nabla}\Phi \right)^{2}- \frac{1}{2}\left( \tilde{\nabla}y \right)^{2}- \Lambda_{\mathrm{M}}e^{2 \Phi}\cosh \left( \sqrt{3}y \right) \right] ,
\left[\sum_{i=1}^n \theta_0^i(\delta^i)^{t+\tau}\right]\left[\sum_{j=1}^n \theta_0^j(\delta^j)^{t+\Delta{t}+\tau+\Delta{\tau}}\right]-\left[\sum_{i=1}^n \theta_0^i(\delta^i)^{t+\tau+\Delta{\tau}}\right]\left[\sum_{j=1}^n \theta_0^j(\delta^j)^{t+\Delta{t}+\tau}\right].
\mathbb{E}[1-\mathcal{I}_M] \leq \sum_{j=1}^p \mathbb{P}(|\mathbf{Z}_j-\mathbf{Z}_j^M| \geq \Delta_M) \leq \frac{1}{\Delta_M^q}\sum_{j=1}^p \mathbb{E}|\mathbf{Z}_j-\mathbf{Z}_j^M|^q & \leq \frac{C}{\Delta_M^q}\sum_{j=1}^p \Theta^q_{M,j,q}.
\left\{\begin{array}{c c c}{{j_{1}}}&{{j_{2}}}&{{j_{1 2}}}\\{{j_{3}}}&{{j}}&{{j_{2 3}}}\\ \end{array}\right\}^{S}= \sum_{j_{1 3}}( - 1 )^{\Theta^{R}}\left\{\begin{array}{c c c}{{j_{1}}}&{{j_{3}}}&{{j_{1 3}}}\\{{j_{2}}}&{{j}}&{{j_{2 3}}}\\ \end{array}\right\}^{S}~ \left\{\begin{array}{c c c}{{j_{2}}}&{{j_{1}}}&{{j_{1 2}}}\\{{j_{3}}}&{{j}}&{{j_{1 3}}}\\ \end{array}\right\}^{S}\, .
{\rm tr}(\widetilde{Z}_{\alpha\alpha}) +{\textstyle\sum_{i=1}^{k-k_0}}\sigma_i\big([\widetilde{Z}_{\beta\beta}\ \ \widetilde{Z}_{\beta c}]\big) \le \tau \{\rm and}\{\rm tr}(\widetilde{Z}_{\alpha\alpha}) +\langle \overline{u}_{\beta},\sigma([\widetilde{Z}_{\beta\beta}\ \ \widetilde{Z}_{\beta c}])\rangle =\tau.
lim_{x\rightarrow A_{x}}y(x)=\frac{1}{2}(\frac{A_{y}+\sqrt{A_{x}^{2}+A_{y}^{2}}}{1-\frac{V_{t}}{V_{d}}}+\frac{A_{y}-\sqrt{A_{x}^{2}+A_{y}^{2}}}{1+\frac{V_{t}}{V_{d}}}).
\lim_{t \to \infty}e^{-\lambda t}S_j(t) = \frac{\nu q Y}{\lambda}\int_0^1 (1-py)^{-1}(1-y) y^{j-1}dy = \frac{\nu q Y}{\lambda}\sum_{k=0}^\infty \frac{p^k}{(j+k)(j+k+1)}, j\geq 1.
\sum_{p\in \mathcal{T}}h_{qp}(u)v_{{\mathcal{R}},{\mathcal{T}},p,n}^i(u)=0, \forall q\in \bar{\mathcal{R}}_i, \forall n\in[N^{\binom{N_R-1}{r+1}\binom{N_T}{t}}], \forall u\in[S]
\Big| \Big|G(\frac{i}{n}, \mathcal{F}_i)- G(\frac{i}{n}, \mathcal{F}_{i,j})-\sum_{k=1}^j \phi_{ik}(G(\frac{i-k}{n},\mathcal{F}_{i-k})-G(\frac{i-k}{n}, \mathcal{F}_{i-k,j-k})) \Big |\Big|_q \leq Cj^{-\tau}+ \sum_{k=1}^{j-1}\phi_{ik}(j-k)^{-\tau}.
\|X\|_{S_{2/3}}= \min_{U\in\mathbb{R}^{m\times d},V\in\mathbb{R}^{n\times d}:X=UV^{T}}\|U\|_{*}\|V\|_{F}= \min_{U\in\mathbb{R}^{m\times d},V\in\mathbb{R}^{n\times d}:X=UV^{T}}\!\left(\frac{2\|U\|_{*}+\|V\|^{2}_{F}}{3}\right)^{3/2}.
f[a{\bf 1}_{n \times n}+ u u^T]_{ij}= \ \sum_{m=0}^{k-1}c_m (a + u_iu_j)^m = \sum_{m=0}^{k-1}\sum_{l=0}^m c_m \binom{m}{l}a^{m-l}u_i^lu_j^l = \ \sum_{l=0}^{k-1}u_i^l u_j^l \sum_{m=l}^{k-1}c_m \binom{m}{l}a^{m-l}= \sum_{l=0}^{k-1}u_i^l u_j^l d_l,
\mathbb E\big[\sigma_j^{*\ell}(\{u\})-\sigma_{j-1}^{*\ell}(\{u\})|x_1,\cdots,x_{j-1}\big] &=N^{-d}\sum_{q=0}^{\ell-1}\binom{\ell}{q}(j-1)^q=N^{-d}(j^\ell-(j-1)^\ell).
\mathcal L_{i}(g) = M \int_{\mathbb R^d\times S^{d-1}}\phi(|v-v_{\ast}|)\partial^{p+1-i}b\, \mathcal M(v_{\ast}) \left[\frac{\partial^i g_{\ast}^{\prime}}{M_{\ast}^{\prime}}+\frac{\partial^i g^{\prime}}{M^{\prime}}-\frac{\partial^i g_{\ast}}{M_{\ast}}-\frac{\partial^i g}{M}\right] dv_{\ast}d\sigma\,,
\Psi(p,q,X) = \frac{\Phi(p,r_1)}{\Phi(q,r_1)}\cdot \frac{\Phi(p,r_2)}{\Phi(q,r_2)}\cdot ... \cdot \frac{\Phi(p,r_l)}{\Phi(q,r_l)}\times \Psi(p,q,X\backslash\{r_1,...,r_l\});
\|L-\zeta(X_1,X_2,...,X_m) \|^2=(L-\zeta(X_1,X_2,...,X_m))^T.P.(L-\zeta(X_1,X_2,...,X_m))= \zeta(X)^T.P.\zeta(X) -2L^T.P.\zeta(X)+L^T.P.L
\rho=\frac{-1}{p}\alpha_1\alpha_2 = \frac{1}{p}\widehat{\phi_1}\widehat{\psi_1}\widehat{\pi}\pi \phi_1^{(p)}\widehat{\phi_2^{(p)}}\psi_2 \phi_2 = \widehat{\phi_1}\widehat{\psi_1}\phi_1^{(p)}\widehat{\phi_2^{(p)}}\psi_2 \phi_2.
S_2(t,q) = -\frac{1}{q^2}\sum_{j=0}^{n-1}j \sum_{r \ge 1}(r-1) (-tq^j)^{r}&= -\frac{1}{q^2}\sum_{r \ge 1}(r-1)(-t)^r \sum_{j=0}^{n-1}j q^{jr}\implies \omega_n^{2i}[t^r]S_2(t,\omega_n^i ) = -(r-1)(-1)^r \sum_{j=0}^{n-1}j = -\binom{n}{2}(r-1)(-1)^r .
\mathbb{Q}\left(M\otimes\mathbb{I}_{|\mathcal{E}|^{n-1}}\right)\mathbb{Q}^{-1}=\mathbb{I}_{|\mathcal{E}|}\otimes\dots\otimes\mathbb{I}_{|\mathcal{E}|}\otimes M\otimes \mathbb{I}_{|\mathcal{E}|}\otimes\dots\otimes\mathbb{I}_{|\mathcal{E}|}
\|\hat{V}\hat{O}- V\|_{\mathrm{F}}^2 = \mathrm{tr}\bigl((\hat{V}\hat{O}- V)^T(\hat{V}\hat{O}- V)\bigr) = 2d - 2\mathrm{tr}(\hat{O}_2\hat{O}_1^T\hat{V}^TV) &= 2d - 2\sum_{j=1}^d \cos \theta_j \leq 2d - 2\sum_{j=1}^d \cos^2 \theta_j = 2\|\sin \Theta(\hat{V},V)\|_{\mathrm{F}}^2.
\mathbb{E}[N_c(d,n)] \le \frac{1+\frac{d}{n}2(d-2)+\frac{n-d}{n2^d}2(d-2)\sum_{j=1}^{d-1}\binom{d}{j}}{1-\frac{n-d}{n}2^{-(d-1)}}\log n &\le 2(d-1)\log n.
\gamma(\varphi(f\otimes e_{11})b\varphi(g\otimes e_{11}))^{\sigma}= \left( \psi(f\otimes e_{11})\gamma(b)\psi(g\otimes e_{11}) \right)^{\sigma}& =\psi(g\otimes e_{12})\gamma(b^\tau) \psi(f\otimes e_{21}) & =\gamma(\varphi(g\otimes e_{12})b^{\tau}\varphi(f \otimes e_{21})) & =\gamma((\varphi(f\otimes e_{11})b\varphi(g\otimes e_{11}))^{\sigma}).