harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{( h^{\prime}| g_{0 0}| g - b^{2}) \left[ h^{\prime}g_{S}g ( C + E )^{2}- ( h^{\prime}| g_{0 0}| g - b^{2}) ( h^{\prime}g_{S}g^{3}e^{- 2 \phi}+ \ell^{2}) \right]}{4 h^{\prime}g_{r r}g_{S}g \left[ ( h^{\prime}| g_{0 0}| g - b^{2})^{2}g^{2}e^{- 2 \phi}+ b^{2}( C + E )^{2}\right]}}\left({\frac{g^{\prime}}{g}}\right)^{2}.
I \coloneqq (e_k(y_1,y_2,\ldots,y_n) \mid 1\le k\le n ), I' \coloneqq ((y_i-y_{i+1})e_k(y_1,\ldots,y_i) \mid 1\le i\le n-1,\ 1\le k\le \min\{i,n-i\}).
\bar{H}(\bar{x},\bar{p})=\lim_{t \rightarrow \infty}\sup_{\beta(\cdot)}\frac{1}{t}E\left[\int_0^{t}(|\sigma^T(\bar{x}, Z_s) \bar{p}|^2 -|\beta(s)|^2) ds\, |\, Z_0=z\right],
W \equiv \left| \begin{matrix}{{f_{1}}}&{{f_{2}}}&{{\cdots}}&{{f_{n}}}\\{{D f_{1}}}&{{D f_{2}}}&{{\cdots}}&{{D f_{n}}}\\{{\vdots}}&{{\vdots}}&{{}}&{{\vdots}}\\{{D^{n - 1}f_{1}}}&{{D^{n - 1}f_{2}}}&{{\cdots}}&{{D^{n - 1}f_{n}}}\\ \end{matrix}\right| \quad .
{\rm dist}((x,\lambda,v),\widetilde{\Phi}^{-1}(0,0,0)) ={\rm dist}((x,\lambda,v),\Phi^{-1}(0,0)) \le{\rm dist}((x,\lambda',v),\Phi^{-1}(0,0))+\|\lambda-\lambda'\| \le\kappa{\rm dist}((0,0),\Phi(x,\lambda',v))+\|\lambda-\lambda'\| \le\kappa\|(\xi',\eta)\|+\|\eta+\zeta\|\le\kappa\sqrt{4\gamma^2+3}\|(\xi,\eta,\zeta)\|.
\begin{array}{r c l}{{\frac{\partial^{n}}{\partial \lambda^{n}}\psi}}&{{=}}&{{( - 1 )^{n}\, n ! \, L ( \lambda )^{- n}\, \psi \, ,}}\\{{\frac{\partial^{n}}{\partial \lambda^{n}}\bar{\psi}}}&{{=}}&{{( - 1 )^{n}\, n ! \, L ( \lambda )^{- n}\, \bar{\psi}\, , \quad n \geq 1 \, ,}}\\ \end{array}
\|T_{\rho}f\|_q^q \le \sum_{S\subseteq [n]}\gamma^{q-2}\mathbb{E}_{x\sim \Omega^S}\|D_{S,x}T_{\frac{1}{\sqrt{2}}}f\|_2^2 = \gamma^{q-2}\sum_{S\subseteq [n]}\| L_S T_{1/\sqrt{2}}f\|_2^2 = \gamma^{q-2}\|f \|_2^2,
F_1=\{a^{(i)}=[(a_n^{(i)})_{n}]: 1\le i\le m\}, &F_2=\{b^{(i)}=[(b_n^{(i)})_{n}]: 1\le i\le m\}, &F_3=\{c^{(i)}=[(c_n^{(i)})_{n}]: 1\le i\le m\}.
d_{1}= 2 ( r_{m}- M + 2 H^{2}r_{m}^{3})^{- 1}, \; d_{2}= \frac{1}{6}( 1 + 6 H^{2}r_{m}^{2}) ( r_{m}- M + 2 H^{2}r_{m}^{3})^{- 1}.
X^{(p)}(\zeta)=\prod_{1\leq i<j\leq p}\left(\zeta_i\zeta_j\right)^{-1}\left(\zeta_i+\zeta_j\right)^{2}X(\zeta_1)\cdots X(\zeta_p)\Big|_{\zeta_1 =\zeta_2=\ldots=\zeta_p=\zeta}
\sum_{m=k}^{n}\binom{n}{m}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}\left( \frac{s}{s+n-m+j}\right)^{r}=\sum_{j=0}^{r-1}(-1)^{j}\frac{s^{j}}{j!}f^{(j)}_{n,k}(s)
d \, \langle \sigma_{1}, \sigma_{2}\rangle \; = \; \langle \, \nabla \, \sigma_{1}, \sigma_{2}\rangle \; - \; ( - 1 )^{\deg \, \sigma_{1}\, + \, \deg \, \sigma_{2}}\; \langle \, \sigma_{1}, \nabla \, \sigma_{2}\rangle .
\Gamma_{a a}^{( 2 )}( k ) = \coth{\frac{\beta k^{0}}{2}}\Bigl [ \Gamma_{a r}^{( 2 )}( k ) - \Gamma_{r a}^{( 2 )}( k ) \Bigr ] \, , \qquad \Gamma_{a r}^{( 2 )}( k ) = \Gamma_{r a}^{( 2 ) *}( k ) \, .
B = \left( \begin{matrix}{{2 \cos \alpha}}&{{\sqrt{2}\, \mathrm{e}^{- i \beta}}}&{{- 2 \sin \alpha}}\\{{- \sqrt{2}\, \mathrm{e}^{i \beta}}}&{{0}}&{{i \sqrt{2}\, \mathrm{e}^{i \beta}}}\\{{- 2 \sin \alpha}}&{{i \sqrt{2}\, \mathrm{e}^{- i \beta}}}&{{- 2 \cos \alpha}}\\ \end{matrix}\right) .
\frac{1}{M}\int(x)dm=\frac{2}{\rho a^{2}}\int_{0}^{a}x(\rho x)dx=\frac{2}{\rho a^{2}}\int_{0}^{a}(\rho x^{2})dx=(\frac{2}{\rho a^{2}})(\frac{\rho a^{3}}{3})=\frac{2a}{3}
(2+n/2)G + xG_x - (n/2)(vG_v +\bar{v}G_{\bar v})+ GH_v- HG_v + \bar{G}H_{\bar v}- \bar{H}G_{\bar v}=0 &\i G + (2-n/2)H + xH_x -(n/2)(vH_v + \bar{v}H_{\bar v}) - HH_v - \bar H H_{\bar v}=kv^{1+2/n}\bar{v}^{2/n}
\sum_{n \le N}\pi(n)R_{X}^{n}x^{n}= \left( \log N+ B_{X}+ O \left( \dfrac{1}{N}\right) \right) x^{N}& - \log x \int_{1}^{N}x^t \left( \log t+ B_{X}+ O\left( \dfrac{1}{t}\right) \right) dt,
\| Z_{n+p}h-Z_nh \|^2 = \langle h, (Z_n-Z_{n+p})^2h\rangle &= \langle h, (Z_n-Z_{n+p})^{1/2}(Z_n-Z_{n+p}) (Z_n-Z_{n+p})^{1/2}h\rangle &\le \| Z_n-Z_{n+p}\|\langle h, (Z_n-Z_{n+p}) h\rangle.
E_1=\frac{1}{\sqrt{6}}\left(\begin{array}{ccc}-1 0 0 0 -1 0 0 0 2\end{array}\right)~;~~E_2=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}1 0 0 0 -1 0 0 0 0\end{array}\right)~;~~E_3=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}0 0 1 0 0 0 1 0 0\end{array}\right)
P_{\min}^{{\rm{S-N}}}\left({\overline S ,\overline \varepsilon ,\overline R}\right) = \bigcap{\left\{{P \in{{\mathbb{R}}^+}:\left({\overline S ,\overline R}\right) \in{\mathcal{R}}_{{\rm{SvB}}}^{{\rm{S-N}}}\left({W,P,K,\overline \varepsilon}\right)}\right\}}.
\int_{0}^{x}f^{\prime}(t)g^{\prime}(t)(1-t)^{-1}(1+t)^{\beta+1}dt=-f^{\prime}(0)g(0)+(1-x)^{-1}(1+x)^{\beta+1}f^{\prime}(x)g(x) & +\int_{0}^{x}m_{-2,\beta}[f](t)g(t)(1-t)^{-2}(1+t)^{\beta}dt-\int_{0}^{x}f(t)g(t)(1-t)^{-2}(1+t)^{\beta}dt.
M_\sigma \geq \Phi_\sigma (x_\delta ,y_\delta ,t_\delta ) &=\Phi (x_\delta ,y_\delta ,t_\delta )-\frac{\eta}{T-t_\delta}-\alpha (|x_\delta |^2+|y_\delta |^2)-\varepsilon |x_\delta -y_\delta | &\geq M-\delta - \frac{\eta}{T-t_\delta}-\alpha (|x_\delta |^2+|y_\delta |^2)-\varepsilon |x_\delta -y_\delta | &\geq \frac{M}{2}>0
\int_{\mathbb S^1}BD^m J_xD^mB\, dx= -\int_{\mathbb S^1}B_xD^m JD^mB\, dx-\int_{\mathbb S^1}BD^m JD^mB_x\, dx \int_{\mathbb S^1}JD^m B_xD^mB\, dx= -\frac12 \int_{\mathbb S^1}J_xD^m BD^mB\, dx
{Pr}_D(f,\sigma): =\lim_{X\to\infty}\displaystyle\frac{\#\{p\in{Spl}_X(f,\sigma) \mid (r_1/p,\dots,r_n/p)\in D\}}{\#{Spl}_X(f,\sigma)}&=\displaystyle\frac{{vol}({D}\cap{\mathfrak{D}}(f,\sigma))}{{vol}({\mathfrak{D}}(f,\sigma))}.
\begin{aligned}\mbox{for}~a<1,\quad\eta(U|U_l) \le a\eta(U|U_r) \quad\mbox{is equivalent to}&\eta(U) \le \frac{1}{1-a}(\eta(U_l)-a\eta(U_r) -\nabla \eta(U_l)\cdot U_l + a\nabla \eta(U_r)\cdot U_r +(\nabla \eta(U_l)-a\nabla \eta(U_r))\cdot U),\end{aligned}
\frac{\nabla_{g_C}}{dt}[(\nabla^k_{g_C}\Phi_\ast^t)(v_1,\ldots,v_{k+1})] = \nabla_{g_C}X|_{\Phi^t(p)}[(\nabla^k_{g_C}\Phi_\ast^t)(v_1,\ldots,v_{k+1})] + \Theta^{t}.
\begin{array}{l}\mathcal{V}_1=\displaystyle{\frac{-2+3S_1+9S_2}{3(-2+3S_1+9S_2)+27(S_1-S_2)S_2\cos(2\theta)}}, \mathcal{V}_2=\displaystyle{\frac{8 - 27 S_1^2 + 9 (8 - 3 S_2) S_2 + 6 S_1 (4 + 9 S_2) + 27 (S_1 - S_2)^2\cos(4 \theta)}{288\left(1+3S_1-9S_1\cos^2(\theta)-9S_2\sin^2(\theta)\right)^2}}.\end{array}
A_{N2}^R(1,\xi_n)= \begin{pmatrix}0 \frac{1}{2\xi_n}\int_0^1e^{i\xi_n(1-2t)}a_{N1}(t)dt\end{pmatrix}= \begin{pmatrix}0 \frac{\ell^2_n}{n}\end{pmatrix}\, .
s = \sqrt{\omega^2 - \lambda_1^2}e^{J_1 \theta_\omega}+ \sqrt{x_3^2 - \lambda_2^2}e^{-J_1 \theta_\lambda}J_2 + \sqrt{x_2^2 - \lambda_3^2}e^{J_1 \theta_x}j_2 + \sqrt{x_1^2 - t^2}e^{J_1 \theta_t}I ~,
=2 \int \left( \sum_{1 \le n \le N}\frac{w_n \mathbf{1}_{B^{cl}(x_{n}, t_n r_{n})}}{\mu B^{cl}(x_{n}, t_n r_{n})}\right) M h \d\mu\le2 \left\| \sum_{1 \le n \le N}\frac{w_n \mathbf{1}_{B^{cl}(x_{n}, t_n r_{n})}}{\mu B^{cl}(x_{n}, t_n r_{n})}\right\|_p\left\|M_{\mu}h \right\|_q.
\div \Big(\partial^2_{F}W(0,\mbox{Id}) (\nabla u -\mbox{Id})+ \partial_{\phi}\partial_F W(0,\mbox{Id})\phi\Big) = \div A, & \phi_t - \Delta \Big(\partial^2_{\phi}W(0,\mbox{Id})\phi+ \partial_{\phi}\partial_F W(0,\mbox{Id})(\nabla u - \mbox{Id})\Big) = \Delta B,
U_{E}=\frac{1}{2}\frac{1}{4\pi\epsilon_{0}}[\frac{Q_{1}Q_{2}}{r_{12}}+\frac{Q_{1}Q_{3}}{r_{13}}+\frac{Q_{2}Q_{1}}{r_{21}}+\frac{Q_{2}Q_{3}}{r_{23}}+\frac{Q_{3}Q_{1}}{r_{31}}+\frac{Q_{3}Q_{2}}{r_{32}}]
\frac{\langle \prod_{j=1}^n\sigma(z_j) \prod_{j=1}^k\sigma(w_j) \rangle_{D_a}}{\langle \prod_{j=1}^n\sigma(z_j) \rangle_{D_a}\langle \prod_{j=1}^k\sigma(w_j) \rangle_{D_a}}= \exp\Big[ 4 \lambda\tilde \mu_{D_a}\big(\gamma: \textstyle \sum_{j=1}^n N_{\gamma}(z_j) \textstyle \sum_{j=1}^k N_{\gamma}(w_j) \big) \Big]\ge 1,
L =\; m_2\left(r^2v_\theta^2+v_r^2\right)+m_3\left(\rho^2v_\theta^2+v_{\rho}^2\right) +r^2v_\theta+\rho^2v_\phi-r^2-\rho^2-\left((x^1)^2+(y^1)^2\right),
\frac{\kappa p+1}{(\kappa+1)^p}\|uu_h^{\kappa}\|_{1,p}^p \leq M_{23}\frac{\kappa p+1}{(\kappa+1)^p}\|uu_h^{\kappa}\|_{\tilde{q}_1}^p+M_{24}\|uu_h^{\kappa}\|_{\tilde{q}_2,\partial\Omega}^p +M_{25}\kappa.
\widetilde{\mathcal{N}_{0}}= Q(\psi, \psi)- Q(G_1,G_1)-Q(G_2,G_2), \quad\widetilde{\mathcal{N}}_{1}= Q_{1,2}(G_1,\psi), \widetilde{\mathcal{N}}_{2}= Q_{1,2}(G_2,\psi),
\lim_{n\to\infty}\,\frac{1}{\mu_n}\int_{\mu_n}^{2\mu_n}\int_{|x|<4t}\,\left(\partial_tu+\frac{x}{t}\cdot\nabla u+\left(\frac{d}{2}-1\right)\frac{u}{t}\right)^2\,dx\,dt=0,
\widehat{R}(e_{i}\otimes e_{j}- q e_{j}\otimes e_{i}) = e_{j}\otimes e_{i}+ (q - q^{-1}) e_{i}\otimes e_{j}- q e_{i}\otimes e_{j}& = -q^{-1}(e_{i}\otimes e_{j}- q e_{j}\otimes e_{i}).
2 1 \frac{d \bar{x}_{2}}{d t}+ \left( 2 \beta_{0}\bar{x}_{0}( t ) \right) \bar{x}_{2}= - \beta_{2}\bar{x}_{0}^{4}( t ) - 3 \beta_{1}\bar{x}_{0}^{2}( t ) \bar{x}_{1}( t ) - \beta_{0}\bar{x}_{1}^{2}( t ) , \bar{x}_{2}( 0 ) = 0 ,
\sum_{\lambda}G_\lambda(x_1,\ldots,x_n)\cdot g_\lambda(y_1,\ldots,y_n) = \sum_{\lambda}s_\lambda(x_1,\ldots,x_n)\cdot s_\lambda(y_1,\ldots,y_n) &=\prod_{i=1}^n\prod_{j=1}^n\frac{1}{1-x_iy_j},
1 =\max\left\{|\varphi_p(\alpha +A\beta)|: \alpha \in \mathfrak D_0, \beta \in \mathfrak D, 0\leq p \leq q-1 \right\}&= \max\{|\varphi_p(n)|: n \in{\mathbb{Z}}^2_N, 0\leq p \leq q-1 \},
\left\{\mathbb{J}_{\mathrm{th}}^{(\omega ,l)}\right\}_{k}:=\left\vert\Lambda_{l}\right\vert^{-1}\underset{x\in \Lambda_{l}}{\sum}\varrho^{(\beta ,\omega ,\vartheta ,\lambda )}\left( I_{\left( x+e_{k},x\right)}^{(\omega ,\vartheta )}\right) \ .
(M^{\ell,\ell-1}_e)^T(M^\ell_e)^{-1}M^{\ell,\ell-1}_e = R_e^TM^\ell_e(M^\ell_e)^{-1}M^{\ell}_eR_e = R^T_e M^\ell_e R_e = M^{\ell-1}_e.
\int_Xe^{\Phi(t)^*}e^{-t\psi}\omega_0^n \le\int_Xe^{-t\psi}\omega_0^n =c_0^{-n}\int_Xe^{-t\psi}(c_0\omega_0)^n \le c_0^{-n}C_{c_0\omega_0},
|G(j,k; z) - G_n(j, k; z)| = \left| \langle \delta_j, (H_{\infty, n}- z)^{-1}(H - H_{\infty, n}) (H-z)^{-1}\delta_k \rangle \right| \le \|H - H_{\infty, n}\| \|(H_{\infty, n}- z)^{-1}\delta_j\| \|(H - z)^{-1}\delta_k\|
D_r=(\gamma^\mu \otimes \mathbf{1})(\mathbf{f}_\mu- \frac{1}{4}\Gamma_{\mu\nu\rho}(x)\gamma^\nu\gamma^\rho) - \frac{1}{8}F_{\mu\nu k}(x) \omega \gamma^\mu \gamma^\nu \otimes \gamma^k+ \frac{1}{4}A_\mu^{\;m}(x) t_{mlk}\gamma^\mu \otimes \gamma^k \gamma^l + \frac{1}{4}t_{mlk}\omega \otimes \gamma^k \gamma^l \gamma^m \;,
\theta_{a b}= \left( \begin{array}{c c c c c}{{0}}&{{\theta_{1}}}&{{0}}&{{0}}&{{0}}\\{{- \theta_{1}}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\theta_{2}}}&{{0}}\\{{0}}&{{0}}&{{- \theta_{2}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{\theta_{i j}}}\\ \end{array}\right) , \i , j = 5 , \dots , 2 p \ .
{{\cal I}}_{\mathrm{1 - l o o p}}^{\mathrm{h e t}}= \frac{V^{( 8 )}T_{2}}{2^{8}\pi^{4}}\Delta_{F_{+}^{4 - l}F_{-}^{l}}t_{8}F_{+}^{4 - l}F_{-}^{l}\, ,
-\frac{\partial w^{\circledast}}{\partial r}=\Bigg(\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\varphi(x)}{|\nabla_{x}w|}\,d\mathcal{H}^{n-1}(x)\Bigg)^{-1}.
\tilde{a}^{\alpha}_{\tilde{n}}=\int\limits_{0}^{\pi}~\frac{d\sigma_{-}}{\pi}\exp\left[4i\tilde{n}\frac{e_{\mu}X^{\mu}_{R}(\sigma_{-})}{e_{\mu}P^{\mu}_{R}}\right] \xi^{\alpha}_{i}\partial_{-}X^{i}_{R}(\sigma_{-})
\big\|T_2(Z_n,Z_n) \big\|_1 =\frac14\big(\big\|T_1(S_n,S_n^*) \big\|_1+\big\|T_1(S_n^*,S_n) \big\|_1\big) \ge \frac14 \big\|T_1(S_n,S_n^*) \big\|_1\ge{\rm const}\ \log n.
H^{\frac52+\frac32}_{\mathrm{Gar},t_1}\left({t_1 \atop t_2};{q_1,p_1 \atop q_2,p_2}\right)= H_{\mathrm{II}}\left({0};t_1;q_1,p_1\right) -2p_2q_2q_1-q_2-\frac{t_2}{q_2}, t_2H^{\frac52+\frac32}_{\mathrm{Gar},t_2}\left({t_1 \atop t_2};{q_1,p_1 \atop q_2,p_2}\right)={p_2}^2{q_2}^2-p_1q_2+\frac{t_2}{q_2}\big(p_1-{q_1}^2-t_1\big).
\frac{f^{*}\left ( s+n\alpha \right )}{\Gamma \left ( s+n\alpha \right )}= \frac{f^{*}\left ( n\alpha +1 \right )}{\Gamma \left ( n\alpha +1 \right )}\frac{f^{*}\left ( s \right )}{\Gamma \left ( s \right )},\Re(s)>0,
\epsilon \simeq \frac{(2\pi)^2 g_s^2 F^2}{2 (M_s^2 A_T)^2}\left(\frac{M_P}{\Psi}\right)^2, \qquad\eta \simeq - \frac{2\pi g_s F}{(M_s^2 A_T)}\left(\frac{M_P}{\Psi}\right)^2.
H_r < \dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{2^r\n^r}+\dfrac{\widehat{C_2}(r)}{n^{r+\frac{1}{2}}}+\dfrac{r}{10^r\n^{r+\frac{1}{2}}}=\dfrac{C(r)}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{2^r\n^r}+ \underset{=C_2(r)}{\underbrace{\Bigl(\widehat{C_2}(r)+\dfrac{r}{10^r}\Bigr)}}\dfrac{1}{n^{r+\frac{1}{2}}}.
||\mathcal{F}(\sqrt{\cdot}L_{1}(u)(t,\cdot \lambda(t)))(\omega \lambda(t)^{2})||_{L^{2}(\rho(\omega \lambda(t)^{2})d\omega)}^{2}=\frac{1}{\lambda(t)^{2}}\int_{0}^{\infty}R (L_{1}(u)(t,R\lambda(t)))^{2}dR
\sum_{k=0}^n\binom{n}{k}e_kx^k=\sum_{n=0}^n\binom{n}{j}f_jx^j(1-x)^{n-j}=(1-x)^n\sum_{m=0}^n\binom{n}{j}f_j\left(\frac{x}{1-x}\right)^j.
U_1(j+1, j) = \sqrt{\frac{i}{\cal N}}\exp[-\frac{i}{\hbar}Q_{j}Q_{j+1}], ~~U_2(j+1, j) = \sqrt{\frac{i}{\cal N}}\exp[ \frac{i}{\hbar}(Q_{j}^{2}- Q_{j}Q_{j+1}+Q_{j+1}^{2})].
\begin{aligned}v_N =\sqrt{f_{minS(N-1)-1}-a_{N-1}}e_{minS(N-1)-1}+ \sqrt{f_{minS(N-1)}-\sigma_{N-1}+a_{N-1}}e_{minS(N-1)}&+ \sum_{i=minS(N-1)+1}^{\infty}\sqrt{f_i}e_i.\end{aligned}
\begin{array}{c}A^{ba}=(-)^{\epsilon_{a}+\epsilon_{b}+\epsilon_{a}\epsilon_{b}}A^{ab},\hspace{3mm}(A^{ab})^{*}=A^{ba}, (B^{a})^{*}=(-)^{\epsilon_{a}}B^{a},\hspace{3mm}C^{*}=C.\end{array}
\mu_{n}' \ = \ \sum_{\substack{2k_2 + 3k_3 + \cdots + nk_n = n k_{j}\geq 0}}\left(\frac{C_{2}}{2!}\right)^{k_{2}}\cdots \left(\frac{C_{n}}{n!}\right)^{k_{n}}\frac{n!}{k_{2}! \cdots k_{n}!}\,.
\mathbb{P}\left\{\left\vert \sum_{j=1}^ka_j(Z_j-1) \right\vert>r\right\}\leq 2\exp\left(-c\min\left\{\left(\frac{r}{\left\vert a \right\vert}\right)^2,\frac{r}{\left\vert a \right\vert_\infty}\right\}\right)
{\cal A}_{\mathrm Z M S}= \int{\mathrm d}^{2}x \left[ \frac{1}{2}\partial_{\nu}\Phi \partial^{\nu}\Phi + \frac{m_{0}^{2}}{\gamma}\Big ( \exp \left( 2 i \sqrt{\gamma}\Phi \right) + 2 \exp \left( - i \sqrt{\gamma}\Phi \right) \Big ) \right] .
\begin{array}{c}{{[ N_{i}, b_{j}] = - b_{i}\delta_{i j}, \quad \forall i , j \in S}}\\{{[ N_{i}, b_{j}^{+}] = b_{i}^{+}\delta_{i j}}}\\{{N_{i}= b_{i}^{+}b_{i}, \quad \forall i , j \in S .}}\\ \end{array}
q\cdot N_{d}^{\prime}=q^2+q\phi(b)+\frac{\phi(-1)}{q-1}\sum_{n=0}^{q-2}G_{-(d-1)n+\frac{q-1}{2}}G_{nd}G_{-n}G_{\frac{q-1}{2}}T^n\left(\frac{b}{a^d}\right).
\varrho_p(A_1,\ldots,A_N):=\lim_{n\to \infty}\left(\frac{1}{N^n}\sum_{i_1,\ldots,i_n=1}^N \left\|A_{i_1}\cdots A_{i_n}\right\|^p\right)^{\frac{1}{np}}=N^{-\frac{1}{p}}e^{\mathbf{M}((A_1,\ldots,A_n),p)/p}
R^{( 1 )}= R_{1 2}\mathrm{o r}R_{2 1}^{- 1}, \quad R^{( 2 )}= R_{2 1}, \quad R^{( 3 )}= R_{1 2}, \quad R^{( 4 )}= R_{1 2}^{- 1}\mathrm{o r}R_{2 1}.
\sum_{\epsilon_{l}\leq 0}u_{\epsilon_{l}l}\left( r \right) u_{\epsilon_{l}l}\left( r^{\prime}\right) + \int_{0 +}^{\infty}\frac{\mathrm{d}k}{2 \pi}u_{k l}\left( r \right) u_{k l}\left( r^{\prime}\right) = \delta \left( r - r^{\prime}\right) .
\mathrm{~ \cal ~ L ~}_{\mathrm{e f f}}= \frac{| e B |}{2 \pi}\sum_{n = 0}^{\infty}\int \frac{d k_{0}}{2 \pi i}\left[ \ln ( k_{0}- E_{n , +}+ \mu + \frac{b}{2}) + \ln ( k_{0}- E_{n , -}+ \mu - \frac{b}{2}) \right] .
\alpha(\omega)=\begin{cases}\alpha ,\omega\in[0,p_1) \beta ,\omega\in[p_1,1] \end{cases};\quad\varphi(\omega)=\begin{cases}\frac{\omega}{p_1},\omega\in[0,p_1) \frac{\omega-p_1}{p_2},\omega\in[p_1,1] \end{cases}.
\frak{m}^{(0,2)}=\bigg\{p_-(A): A=\begin{bmatrix}0 0 A_1 0 0 A_2{}^tA_1{}^tA_2 0\end{bmatrix}, A_i=\begin{bmatrix}x_i I_2 0 0 y_i\end{bmatrix}, x_i, y_i\in\mathbb{C}, i=1,2\bigg\}
\begin{array}{lll}\ \sum_{i=1}^{\infty}{\left(\sigma_i,\rho_j\right)}_{K.G.}a_{out}\left(\overline{\rho}_j\right)+{\left(\sigma_i,\overline{\rho}_j\right)}_{K.G.}a_{out}^{\dagger}\left(\rho_j\right) &= a_{out}\left(\overline{C\sigma}_i\right)-a_{out}^{\dagger}\left(\overline{D\sigma}_i\right)\;,\end{array}
\big|{y}_{2}(x)-\hat{y}_{2}(x)\big| \leq|{\epsilon}|\frac{(x-a)^{\gamma-1}}{\Gamma(\gamma)}+\frac{A}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}|y_{1}(t)-\hat{y}_{1}(t)|dt &\leq|{\epsilon}|(x-a)^{\gamma-1}\sum_{j=0}^{2}\frac{A^j(x-a)^{\alpha{j}}}{\Gamma(\alpha{j}+\gamma)}.
1=\int D\sigma d\vec{\pi}\exp\left[-i\int d^4 x{\mu^2\over 2}\left((\sigma +{g\over \mu^2}(\bar{q}q -{M_{0}\over G}))^2 + (\vec{\pi}+{g\over{\mu^2}}\bar{q}i \vec{\tau}\gamma_{5}q)^2\right)\right]\,,
\sum_{j=1}^{N}{v_j^k}= \sum_{j=1}^{N}(\hat v_j^{k-1}+x_j^{k}-x_j^{k-1}) = \sum_{j=1}^N \sum_{i=1}^N [W(k-1)]_{ji}v_i^{k-1}+\sum_{j=1}^{N}(x_j^{k}-x_j^{k-1}) = \sum_{i=1}^N v_i^{k-1}+\sum_{j=1}^{N}(x_j^{k}-x_j^{k-1}),
E_{n+1}^{{A_2^{(2)\dagger}}}(x,q^{-1},\infty)=\sum_{k_{22}+k_{21}+k_{11}=n}\left(c_2^{\dagger}(k_{22},k_{21},k_{11})+c_1^{\dagger}(k_{22},k_{21},k_{11})\right)x^{k_{11}-k_{22}+1}.
\frac{1}{k^2}(|E(G)| - |E(G - A)|) \, \leq\, \frac{1}{6}\left((\alpha + 1 - \sigma)^2+4(\alpha + 1 - \sigma)-2\right) + \frac{1}{2}(\alpha+1)\sigma\, =\, \frac{1}{6}(\alpha^2+\alpha \sigma+6 \alpha+\sigma^2-3 \sigma+3).
\big|[(v_i+w_j,A_i\cap B_j)_{i,j}]\big| \leq \big|[((v_i,A_i)_i]\big|+\big|[(w_j,B_j)_j]\big|, \big|\lambda[(v_i,A_i)_i]\big| =|\lambda|\,\big|[(v_i,A_i)_i]\big|, \big|g[(v_i,A_i)_i]\big| =|g|\,\big|[(v_i,A_i)_i]\big|,
{\mathcal C}_X(A,z) =\sum_{n=0}^\infty z^n=\frac 1{1-z},{\mathcal C}_X(L,z) = \sum_{n=1}^\infty \frac{z^n}n =-\ln(1-z),{\mathcal C}_X({\bf{N}}_s,z) = \sum_{n=1}^s \frac{z^n}n.
f(t,x_t) = f(0,x_0) + \int_0^t\left[\frac{\partial f}{\partial s}(s,x_s) + \frac{\partial f}{\partial x}(s,x_s)\vartheta_s + \frac{\partial f}{\partial x}(s,x_s)\varphi_s D^\phi_sx_s\right]\,\mathrm{d}{s}+ \int_0^t\frac{\partial f}{\partial x}(s,x_s)\varphi_s\delta B_s^{H}.
J_{G}(x)=[\begin{matrix}3&sin(x_{2}x_{3})x_{3}&sin(x_{2}x_{3})x_{2}\\ 8x_{1}&-1250x_{2}+2&0\\ -x_{2}exp(-x_{1}x_{2})&-x_{1}exp(-x_{1}x_{2})&20\\ \end{matrix}].
\frac{\langle n \rangle}{V}= \frac{1}{V}\lambda \frac{\partial}{\partial \lambda}\ln Z \left[ \lambda , T \right] = \lambda \langle \mathrm{T r}g^{\dagger}\mathrm{T r}g - 1 \rangle \rightarrow \lambda \left| \left< \mathrm{T r}g \right> \right|^{2}+ O ( N^{0}) ~ ~ ( \mathrm{a s}~ N \rightarrow \infty )
\tau \tau_{xxxx}\,\,-\,\,4\tau_{xxx}\tau_x\,\, +\,\, 3\tau_{xx}^2\,\,+\,\,4\tau_x\tau_t \,\,-\,\,4\tau\tau_{xt}\,\,+\,\,3\tau \tau_{yy}\,\, -\,\, 3\tau_y^2 \,\,=\,\, 0.
\eqref{eq:l-rec2}\leq c_0n^a\Big(f(n_0)+\sum_{k=n_0}^{n-1}\frac{g(k)}kc_1k^{-a}\Big) \leq c_2n^a\sum_{k=n_0}^{n-1}\frac{g(k)}{k^{a+1}}.
\beta\left(-e_k+\sum_{i=1}^m\max(0,b_{i,k})e_i,e_j\right)=-\lambda_{k,j}+\sum_{i=1}^m\max(0,b_{i,k})\lambda_{i,j}&\beta\left(-e_k+\sum_{i=1}^m\max(0,-b_{i,k})e_i,e_j\right)=-\lambda_{k,j}+\sum_{i=1}^m\max(0,-b_{i,k})\lambda_{i,j}
G_1(t)=m_{G_1}(t)-\frac{v_{G_1}(t)}{v_{G_2}(\rho_{G_1,G_2}(t))}m_{G_2}(\rho_{G_1,G_2}(t))+\frac{v_{G_1}(t)}{v_{G_2}(\rho_{G_1,G_2}(t))}G_2(\rho_{G_1,G_2}(t))
\widetilde{\omega}^\mathrm{zm}= \sum_{j=1}^m \widetilde{\omega}_{1,j}^\mathrm{zm}= \sum_{j=1}^m \Big( e^{-i\lambda_ju_1}\varphi_j +e^{i\lambda_ju_1}C_1(\lambda_j)\varphi_j \Big).
d(x_n,x_{n+k}) \leq s\{d(x_n,x_{n+1}) + d(x_{n+1},x_{n+2}) + \hdots + d(x_{n+v},x_{n+k})\}& < s \{d(x_0,x_1) + d(x_0,x_1) + \hdots + d(x_{n+v-1},x_{n+k-1})\}& \hdots & \hdots & < s \{v d(x_0,x_1) + d(x_0,x_{k-v})\}& < s \{v d(x_0,x_1) + M\}& = M_1, \mbox{say}.
\mathcal A_\rho \;(\delta \rho,\delta\b u):= \mathcal A_{p\mu\rho}(0,0,\delta\rho,\delta \b u)= \begin{pmatrix}\mathcal L_\rho \;(\delta \rho,\delta\b u) \mathcal B \;(\delta\b u)|_{\partial\Omega}\end{pmatrix}=\begin{pmatrix}\mathcal S 0\end{pmatrix}
G(n,k)=\frac{(-1)^{n+k}2 n^3 \binom{2 n}{n}^3 \binom{2 k+2 n}{k+n}\binom{2 n-2 k}{n-k}\binom{k+n}{n-k}}{\binom{2 k}{k}(2 k+2 n-1) 16^{-k+3 n-1}}.
\| \psi_{1}(t)-\psi_{2}(t)\|_{2}=\left\| \exp \left\{\frac{{\rm i}}{2}\int^{t}_{0}\big( \lvert \psi_{2}(0,\tau )\rvert^{2}-\lvert \psi_{1}(0,\tau )\rvert^{2}\big) \d\tau \right\}q_{1}(t)-q_{2}(t) \right\|_{2}begin{align*}3mm] \geq\inf_{\theta\in \mathbf{R}}\|\exp\{{\rm i}\theta \}q_{1}(t)-q_{2}(t)\|_{2}
\begin{cases}E\big(Cov_N(\delta_i, y_i) ; y_U\big) = \frac{1}{N}\sum\limits_{i\in U}p_i y_i - \big( \frac{1}{N}\sum\limits_{i\in U}p_i \big) \big( \frac{1}{N}\sum\limits_{i\in U}y_i \big) \rightarrow 0 E\big( E_N(\delta_i) \big) = \sum_{i\in U}p_i/N \rightarrow p > 0 \end{cases}
(\sum_{n=1}^{15}(-1)^{n}-1)(\frac{g^{2}}{32\pi^{2}})tr \epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}= -2 (\frac{g^{2}}{32\pi^{2}})tr \epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}
(4-\epsilon g_0 L^2)\int_0^T\|\partial_x u\|^2dt+5a_0\int_0^T\|\partial_x^2 u\|^2dt +2\int_0^T\int_0^La(x)xu^2 dxdt \leq L\|u_0\|^2+(a_1+g_0)\int_0^T\|u\|^2 dt+\left(1+\dfrac{1}{\epsilon}\right)\int_0^T\|\eta^t\|_{L_g}^2dt .
{\cal L}= \bar{\psi}_{i}( i{\not \! \partial}- e{\not \! \! A}- m ) \psi_{i}- \chi_{\mu}^{*}\bar{\psi}_{i}^{c}\gamma^{\mu}\tau \psi_{i}- \chi_{\mu}\bar{\psi}_{i}\gamma^{\mu}\tau^{T}\psi_{i}^{c}+ \frac{N}{g}\chi_{\mu}^{*}\chi^{\mu}.
\psi^{(t)}_-(y_{-k})(z) = -\pi\left( \sum_{m\geq 0}\frac{B_m}{m!}C^k_m(t) z^{m-k-1}\right) = -\sum_{m= 0}^k\frac{B_m}{m!}C^k_m(t) z^{m-k-1}= -C_0^k(t) z^{-k-1}
\left\langle J_{(ab,i)},P_{(c,j)}\right\rangle =\frac{1}{2}\left( \alpha_{0}K_{ij}{}^{0}+\alpha_{1}K_{ij}{}^{1}+\alpha_{2}K_{ij}{}^{2}+\alpha_{3}K_{ij}{}^{3}\right) \varepsilon_{abc},
\gamma^{I}_l = \sum_n A_n \gamma^{(n) I}_l=\gamma^{I}_l(A) = (\lambda +2l-1) \frac{\Gamma(\lambda +l-1)}{l! (n+1)! \, \Gamma(\lambda-2)}[1+\frac{1}{2}(-1)^l \frac{(l+1)!}{(\lambda-1)_{l-1}}]
\Big\{\sum_{i=1}^h u_i t_i :u_i=0,1,\dots,s_i-1; i=1,2,\dots,h\Big\}= \{0,1,2,\dots, t_{h+1}- 1\}, \Big\{\sum_{i=1}^{h+1}u_i t_i: u_i=0,1,\dots,s_i-1 i=1,2,\dots,h+1\Big\}= \{0,1,2,\dots, r! - 1\}.
\partial_{n}\big({( \lambda_0, \ldots, \lambda_{n-1}), a}\big) = \ \big({(\lambda_1, \ldots, \lambda_{n-1})}, a \big) + (-1)^n \big({(\lambda_0, \ldots, \lambda_{n-2})}, \lambda_{n-1}\cdot a\big) & + \sum_{i=1}^{n-1}(-1)^i \big({(\lambda_0,\ldots, \lambda_{i-1}\lambda_i, \ldots, \lambda_{n-1}), a}\big) \;,
n \geq |N_{G}(x)\cup \{x\}|+\sum_{y\in N_{G}(x)-X}|V(C_{y})| \geq (\delta (G)+1)+|N_G(x)-X|\delta(G) \geq (\delta(G)+1)+(\delta (G)-1)\delta (G) &=\delta (G)^{2}+1 \geq n+1,
\delta K = \delta \left( h^{a}{}_{b}\nabla_{a}n^{b}\right) = \delta h^{a}{}_{b}\nabla_{a}n^{b}+ h^{a}{}_{b}( \delta C )^{b}{}_{a c}n^{c}+ h^{a}{}_{b}\nabla_{a}\delta n^{b},
\left[\begin{array}{cccc}\eta^0 \eta^{1}\eta^{2}\eta^{3}\end{array}\right]=\left[\begin{array}{rrrr}1 0 0 0 0 1 0 0 0 0 1 0 c^3 0 0 1\end{array}\right]\left[\begin{array}{cccc}\eta^0 \eta^{1}\eta^{2}\theta^{3''}\end{array}\right],
P^{*}_{0}(\varepsilon)=\left(\begin{array}{l}P_{1 0}^{*}\ \ \ \ \ \ \ \ \varepsilon P_{2 0}^{*}\varepsilon \big(P_{2 0}^{*}\big)^{T}\ \ \varepsilon P_{3 0}^{*}\end{array}\right),\ \ \ \h_{0}(t,\varepsilon)=\left(\begin{array}{l}h_{1 0}(t) \varepsilon h_{2 0}(t)\end{array}\right).
\varGamma_{1}=\left\{\left(\omega_{1},\omega_{2}\right)\left|\omega_{1}T_{0}=\left(2k_{1}+1\right)\pi,\omega_{2}\left(T-T_{0}\right)=\left(2k_{2}+1\right)\pi,k_{1}\in\mathbb{Z},k_{2}\in\mathbb{Z}\right.\right\},
C_0:=\frac{p-q}{pq p^*_s}\frac{(p_s^\ast-q)^{\frac{p}{p-q}}}{(p_s^\ast-p)^{\frac{q}{p-q}}}|\Omega|^{\frac{p(p_s^\ast-q)}{p_s^\ast (p-q)}}S^{-\frac{q}{p-q}},
\begin{aligned}{\bar R_{{\rm{SM}},n}}\le{{{{\log}_2}\left({1 + \frac{{\mathbb E}\left\{|\beta_{n,{\rm R},l_n}|^2\right\}}{c_n}}\right)}}={{{{\log}_2}\left({1 + \frac{1}{c_n}}\right)}}={\bar R_{{\rm{SM}},n,{\rm upper}}},\end{aligned}
S_n' := \bigl(\delta_m + (m-1) k + mp){D_n^{(m)}}{}+ \sum_{i=1}^{m-1}\bigl(\delta_m \Delta_m +(m-1)k + (m-i) p\bigr)\frac{X_n^{(i)}}i{}+ \bigl(\delta_m+ (m-1) k \bigr) \, \frac{X_n^{(m)}}m + \sum_{i=2}^{m-1}\bigl(\delta_m+ (i-1) k \bigr) \, \frac{X_n^{(m+i-1)}}i.
\sum_{k\ge 1}\frac{k^{-(\beta-\gamma)}}{\gamma-\beta}= \int_1^\infty \frac{t^{-(\beta-\gamma)}}{\gamma-\beta}dt + \frac{1}{2(\gamma-\beta)}+ \int_1^\infty t^{-(\beta-\gamma+1)}P_1(t)dt.
\prod_{i=0}^{m-1}p_i \cdot \Bigg[\sum_{j=0}^{m-1}\sum_{k=0}^{m-1}\prod_{i_2=j+1}^{m-1}\rho_{i_2+k}+\sum_{j=0}^{m-1}\sum_{k=0}^{m-1}\prod_{i_2=j}^{m-1}\rho_{i_2+k}\Bigg].
|J_a = \epsilon, \; D_b = \lambda_b, \; D_c= \lambda_c, \; D_d = \lambda_d \rangle \equiv{1 \over 4}|a_{\epsilon}9 \rangle -{i \lambda_b \over 4 \sqrt{3}}|a_{\epsilon}b\rangle -{i \lambda_c \over 4 \sqrt{3}}|a_{\epsilon}c\rangle -{i \lambda_d \over 4 \sqrt{3}}|a_{\epsilon}d\rangle
I m \, \Sigma_{F}( Q ) \simeq \frac{g^{2}}{4 \pi}\, C_{F}\, T \, \gamma^{0}\, \frac{g_{\tau}^{( i )}( q_{0}, q )}{| g_{\tau}^{( i )}( q_{0}, q ) |}\ln \left( \frac{m_{T}}{\Gamma_{q}( Q )}\right) \, .
\begin{aligned}p_{abr}\approx p\{\beta^2 \rho_b \geq \alpha^2 \rho_a \geq \frac{1}{2}\beta^2\rho_b \cup \alpha^2\rho_a \geq \beta^2 \rho_b \geq \frac{1}{2}\alpha^2\rho_a \}p_{ar}\approx p\{\alpha^2 \rho_a \geq 2 \beta^2\rho_b \}p_{br}\approx p\{\beta^2 \rho_b \geq 2 \alpha^2\rho_a \}\end{aligned}