harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_3^{\varepsilon}:=\{|\Delta X_s|\leq C_{(\ref{eq:08_02_2})}(t-s)^{(1+\beta)^{-1}-\gamma}\;\;{\rm for\; all}\; s<t\}&\cap \{V\leq C_{(\ref{eq:08_02_3})}\}\cap \left\{\sup_{x_1,x_2\in (0,1), x_1\not=x_2}\frac{|X_t(x_1)-X_t(x_2)|}{|x_1-x_2|^{\eta_{\rm c}-\varepsilon}}\leq C_{(\ref{eq:08_02_20})}\right\}.
B_M = p(t) + O \left( \frac{t^{d-2}}{M_1 +1}+ \cdots + \frac{t^{d-2}}{M_d +1}\right) , \frac{1}{N^d}\sum_{M \in [0,N-1]^d}B_M = p(t) + O \left( t^{d-2}\frac{\log N}{N}\right) .
\int_0^t|v(s)|_{D^{3,q}}s\leq t^{\frac{1}{q_0}}\Big(\int_0^t |v(s)|^{p_0}_{D^{3,q}}s\Big)^{\frac{1}{p_0}}\leq Cc_1,\ &\int_0^t\|\nabla v(s)\|_{2}s\leq t^{\frac{1}{2}}\Big(\int_0^t |\nabla v(s)|^2_{2}s\Big)^{\frac{1}{2}}\leq C(c_1t+(c_1t)^{\frac{1}{2}})\leq Cc_1,
\eta^N_m:=\mathbb{P}_{N,m}\circ (\Phi_1^{N,m}, (\mu_t^{1,N,m})_{t=0}^T,(\xi_t^{1,N,m})_{t=1}^T,\widetilde \Phi_1^{N,m}, (X_t^{1,N,m})_{t=0}^T)^{-1}.
| V_{C K M}| = \left( \begin{array}{l l l}{{0 . 9 7 5}}&{{0 . 2 2 0}}&{{0 . 0 0 3}}\\{{0 . 2 2 0}}&{{0 . 9 7 4}}&{{0 . 0 4}}\\{{0 . 0 0 8}}&{{0 . 0 4}}&{{0 . 9 9 9}}\\ \end{array}\right) ,
\begin{array}{cc}\phi^E_\omega \mid_0 = \phi^L_\omega \mid_{2\pi}\hspace{10mm}\phi^E_\omega \mid_\theta =\phi^L_\omega \mid_\theta \partial_{\sigma}\phi^E_\omega \mid_0 = - \partial_{\sigma}\phi^L_\omega \mid_{2\pi}\hspace{10mm}\partial_{\sigma}\phi^E_\omega \mid_\theta = - \partial_{\sigma}\phi^L_\omega \mid_\theta\end{array}
h(x,t) =e^{-\pi i L a_{22}a_{21}t^2}& \hskip-.35in\sum_k \sum_{(q,m)\in\Gamma}\big[b_{(q,m),k}\,\widetilde{H}\widetilde g (\sqrt{L}(a_{22}t- (q-k)/L)) &e^{-2\pi i m(q-k)/L}\,e^{2\pi i (q-k)a_{21}t}& \hskip.35in \Phi_{(q,m)}(t , \, x -(q-k)a_{11})\big]
T \geq \int d^{2}x \left\{\pm \frac{1}{2}q a^{2}B \right\}= \pm \frac{1}{2}q a^{2}\Phi \, , \, \, \, \textrm{w h e r e}\, \, \, \Phi \equiv \int d^{2}x B = - \oint d l_{i}A_{i}
\left< W ( C ) \right> = \exp \left( \sum_{k = 1}^{\infty}\frac{( i g )^{k}}{k !}\int_{\Sigma}^{}d \sigma_{\mu_{1}\nu_{1}}( x_{1}) \cdots \int_{\Sigma}^{}d \sigma_{\mu_{k}\nu_{k}}( x_{k}) \left< \left< F_{\mu_{1}\nu_{1}}( x_{1}) \cdots F_{\mu_{k}\nu_{k}}( x_{k}) \right> \right> \right) .
\mathrm{Conf}_{2,0}^+\ni (z_1,z_2)\mapsto \phi^t(z_1,z_2)=\frac{1-t}{2\pi i}\log\!\left(\frac{z_1-z_2}{\overline z_1-z_2}\right)-\frac{t}{2\pi i}\log\!\left(\frac{\overline z_1-\overline z_2}{z_1-\overline z_2}\right)\in\mathbb C,
b^{2}(f_{1},f_{2})=\frac{|\langle F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{2})\rangle|^{2}}{\langle|F_{n}(f_{1})F_{n}(f_{2})|^{2}\rangle\langle|F_{n}^{*}(f_{1}+f_{2})|^{2}\rangle}
\Sigma = I + I_{e x t}= \int d t ( q_{i}( \partial_{t}a^{i}- h^{i}) + \bar{c}_{i}\partial_{t}c^{i}+ \bar{c}_{i}\partial_{j}h^{i}c^{j}+ J_{i}^{a}c^{i}+ i J_{\bar{c}}^{i}q_{i}) ,
(W_{ab}(t))=[\begin{matrix}0&2&1&1&6&0\\ 1&0&1&2&5&6\\ 0&7&0&0&0&0\\ 1&0&1&0&0&0\\ 1&0&7&5&0&0\\ 0&0&0&0&5&0\end{matrix}]
\begin{aligned}F_2(z_2,\bar{z}_1z_2+\bar{z}_1^2) = \bar{F}_1(\bar{z}_1)+\bar{F}_1(-\bar{z}_1-z_2), G(z_2,\bar{z}_1z_2+\bar{z}_1^2) = -\bar{F}_1(\bar{z}_1)\bar{F}_1(-\bar{z}_1-z_2).\end{aligned}
\lim_{R\rightarrow\infty}(R^{-4}\|e^{\tau\phi}x_{n+1}^{\frac{1-2s}{2}}|x|\tilde{u}\|_{L^{2}(A_{R,2R}^{+})}^{2}+R^{-2}\|e^{\tau\phi}x_{n+1}^{\frac{1-2s}{2}}|x|\nabla\tilde{u}\|_{L^{2}(A_{R,2R}^{+})}^{2})=0.
h(\mathbf{H}) = h(\mathbf{G}^{R_1})+h(\mathbf{G}^{R_2}) = \alpha_1+\alpha_2 \bar h(\mathbf{H}) \leq \bar h(\mathbf{G}^{R_1}) + \bar h(\mathbf{G}^{R_2})=\alpha_1+\alpha_2.
I = \langle T \mid q^{- J^{0}}\mid U \rangle = ( T^{a_{k}. . . a_{1}})^{\ast}q^{- J^{0}}U^{a_{1}. . . a_{k}}= \sum_{m = - j}^{+ j}( T^{j m})^{\ast}U^{j m}q^{- m}.
X^2 \ll \left( \sum_{2<p \leq X}(\log p)L(\tfrac{1}{2},f \otimes \chi_{8p}) \mathcal{M}(p)\right)^2 \leq \ \mathcal{N}\sum_{2<p \leq X}(\log p)\left| L(\tfrac{1}{2},f \otimes \chi_{8p}) \right|^2 |\mathcal{M}(p)|^2 \ll \mathcal{N}X.
\int_M \langle \nabla u, \nabla \varphi \rangle_{g}dv_g +m_0^2 \int_M u \varphi dv_g = \int_M u^{2^\star-1}\varphi dv_g +\omega^2 \int_M u(1-q \Phi(u))^2\varphi dv_g,
q_{r}\mapsto\sum_{A}\vec{q}_{r}^{A}\cdot(\vec{d}^{A}-\Delta\varphi(\vec{n}\times\vec{R}_{A}^{0}))=q_{r}-\Delta\varphi\vec{n}\cdot\sum_{A}\vec{R}_{A}^{0}\times\vec{q}_{r}^{A}=q_{r}.
+ \sqrt{\frac{\kappa}{2}}M ( \psi_{+}^{+}\psi_{-}+ \psi_{-}^{+}\psi_{+}) + \frac{i}{4}\kappa f_{a b}( \overline{{{\psi}}}_{+}\gamma_{+}^{a b}\psi_{+}+ \overline{{{\psi}}}_{-}\gamma_{-}^{a b}\psi_{-}) \Big ) = 0 .
\sum_{\{x,y\}\in G_{12}}(d_{G_{12}}(x) + d_{G_{12}}(y)) = \sum_{x \in V_1}d_{G_{12}}^2(x)+ \sum_{y \in V_2}d_{G_{12}}^2(y)\ge |\bar{H}_\pi| -\frac{n}{3}- \frac{t^2}{2}- 9 t.
\Omega_\delta:=\Big\{\lambda:{\Im}\sqrt[4]{\lambda}\geq - A - \delta{\rm log}(1 + |\lambda|^{1/4}), \,{\Re}\sqrt[4]{\lambda}\geq - A - \delta{\rm log}(1 + |\lambda|^{1/4}), |\lambda|^{1/4}\geq C_0\Big\},
\begin{cases}f(J_{\lambda_k}^fx_k,y)+\lambda_{k}\langle \overrightarrow{x_{k}J_{\lambda_k}^fx_k},\overrightarrow{J_{\lambda_k}^fx_ky}\rangle \geq 0 ,\ \ \ \ \ \forall y\in K, x_{k+1}=\alpha_k u\oplus(1-\alpha_k)J_{\lambda_k}^fx_k, \end{cases}
{}_{10}V_{9}\Big[a_0;a_1,a_2,a_3,a_4,a_5\Big] &=\dfrac{[\delta+a_0]_N[\delta+a_0-a_1-a_2]_N[\delta+a_0-a_1-a_3]_N[\delta+a_0-a_2-a_3]_N}{[\delta+a_0-a_1]_N[\delta+a_0-a_2]_N[\delta+a_0-a_3]_N[\delta+a_0-a_1-a_2-a_3]_N},
\begin{aligned}\|v\|_{L^\infty(D_T)}\leq C\big(1+\|v_0\|_{L^\infty}\big)\Big(1+\|g\|_{L^{2r_0}(D_T)}^{2(1+\frac{N}{2\kappa_1})}+\|f\|_{L^{r_0}(D_T)}^{1+\frac{N}{2\kappa_1}}\Big)\end{aligned}
\lim_{n\to \infty}A^{W,W}_\cdot (l_n) = \lim_{n\to \infty}\left ( \phi_n(\cdot, W) -\phi_n(0, W_0) - \int_0^\cdot \nabla \phi_n^* (r, W_r) \mathrm d W_r \right) & = 0.
\left( \mathbb E_{-\nu-1,1-\alpha\vphantom{\beta}}^{U}\mathbb E_{\nu,\beta}^{V}-\mathbb E_{\nu,1-\alpha\vphantom{\beta}}^{U}\mathbb E_{-\nu-1,\beta}^{V}\right)\log\left[ 1-\frac{1-\alpha}{\alpha}\frac{\beta}{1-\beta}(1-U)(1-V)\right]=0.
\begin{array}{lll}\gamma_0 = t_0; \gamma_{k}= \mu_{0,k}t_{0}^{(k)}+\mu_{1,k}t^{(k-1)}_1+\cdots +\mu_{k,k}t_k,\; 0\leq k<n-1;{\mathcal H}= \bigoplus\limits_{i=0}^{n-1}{\mathcal H}_i,\;\mbox{\rm where}\;{\mathcal H}_i=\bigvee\{t_i(w):w\in \Omega\},\; i\in\{0,1,\cdots,n-1\}.\end{array}
\langle z_\pm,\mu_{12},\epsilon_{12}|\bigtriangleup(X) = z\langle z_\mp,\mu_{12},\epsilon_{12}|, \langle z_\pm,\mu_{12},\epsilon_{12}|\bigtriangleup(R) = \pm \epsilon_{12}\langle z_\pm,\mu_{12},\epsilon_{12}| .
S^{(0)}_2={1\over2}x^i\Lambda_{ij}x^j+x^*_{ia}D^i_{1\alpha}c^{\alpha a}+y^*_{\mu a}D^\mu_{2\alpha}c^{\alpha a}+\bar{x}_iD^i_{3\alpha}B^\alpha+\bar{y}_\mu D^\mu_{4\alpha}B^\alpha-\varepsilon^{ab}c^*_{\alpha a|b}D^\alpha_{5\beta}B^\beta.
\Delta_{[w]}^{cyc}=2^4M^{cyc}_{(1,1,1,1)}+4\cdot 2^3M^{cyc}_{(2,1,1)}+3\cdot 2^2M^{cyc}_{(3,1)}+2\cdot 2^2M^{cyc}_{(2,2)}+2\cdot 2M^{cyc}_{(4)}.
\xi \left( x\right) =\left[ x_{n}+\left( x_{1}-1/2\right)^{2}/\omega^{2}+\displaystyle\sum \limits_{k=2}^{n-1}x_{k}^{2}+\frac{1}{4}\right]^{-\nu},\varphi_{\lambda}\left( x\right) =\exp \left( \lambda \xi \left(x\right) \right) ,
i \partial_{t}\Psi [ \xi , t ) = \int d^{2}x \biggl [ i \biggl ( A_{0}+{\frac{1}{\theta}}J_{i}\epsilon_{i j}{\frac{\partial_{j}}{\nabla^{2}}}\biggr ){\frac{\delta}{\delta \xi}}+ A_{0}\rho - \xi{\dot{\rho}}\biggr ] \Psi [ \xi , t ) ,
\int d^{1 0}\hat{x}\sqrt{| \hat{\jmath}|}\, \left[{\textstyle \frac{1}{2 \cdot 5 !}}\left( \hat{G}_{( 5 )}\right)^{2}+{\textstyle \frac{1}{2 \cdot 5 !}}\left( \hat{\overline{{{G}}}}_{( 5 )}\right)^{2}\right] \, .
F(\rho\xi_{k,q,n}) = e^{-\frac{1}{\rho a_{k,q}\xi_{k,q,n}}}\sum^{N-\tilde{M}}_{j=1}\frac{1}{j!\rho^j a_{k,q}^j\xi_{k,q,n}^j}= 1- e^{-\frac{1}{\rho a_{k,q}\xi_{k,q,n}}}\sum^{\infty}_{j=N-\tilde{M}+1}\frac{1}{j!\rho^j a_{k,q}^j\xi_{k,q,n}^j}\approx 1-\frac{1}{(N-\tilde{M}+1)!(\rho a_{k,q}\xi_{k,q,n})^{N-\tilde{M}+1}}.
\delta ( x - y ) \, \left( \frac{C^{2}\, n^{4}\, \Theta ( x_{0}- y_{0})}{\left( \left( x_{0}- y_{0}- i \, \epsilon \right)^{2}- \left( x_{s}- y_{s}\right)^{2}\right)^{2}}+ \frac{C^{2}\, e^{2 \, \alpha + 2 \, \alpha^{*}}\, n^{4}\, \Theta ( x_{0}- y_{0})}{\left( \left( y_{0}- x_{0}- i \, \epsilon \right)^{2}- \left( x_{s}- y_{s}\right)^{2}\right)^{2}}\right.
I(M_1; Y_{1, 0}^T|M_2) \leq \lim_{n \to \infty}\frac{1}{2}\sum_{i=1}^n \log (2 \pi e (\alpha P \delta_n^2 + N_1 \delta_n)) - \frac{1}{2}\sum_{i=1}^n \log (2 \pi e N_1 \delta_n) & = \lim_{n \to \infty}\frac{1}{2}\sum_{i=1}^n \log \left(1+ \frac{\alpha P \delta_n}{N_1}\right) &=\frac{\alpha P T}{2N_1}.
\begin{array}{r l}{\mathrm{g r a v i t y}:}&{( 3 , 3 ) + 2 ( 2 , 3 ) + ( 1 , 3 ) \nonumber}\\{\mathrm{t e n s o r}:}&{( 3 , 1 ) + 2 ( 2 , 1 ) + ( 1 , 1 ) \nonumber}\\{\mathrm{v e c t o r}:}&{( 2 , 2 ) + 2 ( 1 , 2 ) \nonumber}\\{\mathrm{h y p e r}:}&{2 ( 2 , 1 ) + 4 ( 1 , 1 ) .}\\ \end{array}
P_0 =\;\frac{81\Big(-4\sqrt{2}\xi({\xi}^2-81)-6{\xi}^2+2475\Big)}{4{\xi}^8}, P_1 =\;\frac{27\Big(-\sqrt{2}\xi(93-2{\xi}^2)+3{\xi}^2-603\Big)}{2{\xi}^6}, P_2 =\;\frac{9\Big(-12\sqrt{2}\xi(2{\xi}^2-43)+4{\xi}^4-36{\xi}^2+2781\Big)}{32{\xi}^4}, P_3 =\;\frac{9(-6\sqrt{2}\xi+2{\xi}^2-27)}{4{\xi}^2}.
S = -\frac{1}{2}\int\!d\sigma d\tau[\partial_\mu\chi_1 \partial^\mu\chi_1 + 2\chi_2\partial_\mu\chi_1 \partial^\mu\chi_3 + \frac{1}{1+{\chi_1}^2}\partial_\mu\chi_2 \partial^\mu\chi_2 + ({\chi_2}^2+\frac{1}{1+{\chi_1}^2})\partial_\mu\chi_3 \partial^\mu\chi_3 +\partial_\mu t \partial^\mu t].
m^{2}({\Delta u}^2+{\Delta v}^2)(x^2-2\langle q \rangle x)+ m^2{\Delta q}^2\bigg( v^2-2 \langle v \rangle v -\frac{\hbar^2}{m^2}\frac{\partial_{x}^2 \sqrt{\rho}}{\sqrt{\rho}}\bigg)+ \lambda_0 (t)- \partial_{t}\lambda_1-v\partial_{x}\lambda_1\,=\,0 \;\,.
\sup_{\mathcal{C},x}\left|\frac{df}{ds}\right| =\sup_{\mathcal{C},x}\lim_{p\to x,\;p \in \mathcal{C}}\left|\frac{f(x)-f(p)}{s(x)-s(p)}\right| =\sup_{\mathcal{C},x}\lim_{p\to x,\;p \in \mathcal{C}}\frac{|f(x)-f(p)|}{\mathrm{length}(\widehat{xp})}~.
\begin{array}{c c c}{{\delta \langle X \rangle}}&{{\equiv}}&{{\delta \langle A_{1}( x_{1}) , . . . , A_{N}( x_{N}) \rangle = \delta \int{\cal D}\varphi \, A_{1}. . . A_{N}\, e^{- S [ \phi ]}}}\\{{}}&{{=}}&{{\delta \int{\cal D}\varphi \, \left[ \sum_{k}A_{1}. . . \delta A_{k}. . . A_{N}\, e^{- S}- A_{1}. . . A_{N}\, e^{- S}\delta S \right]}}\\ \end{array}
\begin{aligned}\sum_{j=1}^{k-1}b_{ij}^{(m)}= \sum_{j=i}^{k-1}(-1)^{j-i}\binom{k-i}{j-i}= \sum_{j=0}^{k-i-1}(-1)^{j}\binom{k-i}{j}=0 - (-1)^{k-i}= -v_i,\end{aligned}
\sup\{|f_1(\lambda_0^n - s) - f_1(\lambda_0 - s)|\vee |f_2(\lambda_0^n - s) - f_2(\lambda_0 - s)| : s \in [0, \lambda_0 - \lambda_1]\}< \epsilon & \Gamma_n(f_1, f_2)(-s) = \Gamma(f_1, f_2)(-s)s\geq \lambda_0 - \lambda_1.
\begin{array}{l}{\{\Omega_{n}^{a}, \phi_{n^{\prime}}^{b}\}\approx 0 \a \ne b}\\{\{\Omega_{n}^{a}, \phi_{n^{\prime}}^{a}\}\approx 0 \n^{\prime}< n}\\{\{\Omega_{n}^{a}, \phi_{n}^{a}\}\approx ( - 1 )^{N_{a}- n}\eta^{a}( q , p )}\\ \end{array}
{\cal R}_{abc}^0=\partial_b\Omega_{ac}^0-\partial_c\Omega_{ab}^0+\Omega_{0b}^0\Omega_{ac}^0+\Omega_{db}^0\Omega_{ac}^d-\Omega_{0c}^0\Omega_{ab}^0-\Omega_{dc}^0\Omega_{ab}^d,
h^{\mu \nu}=\frac{e}{4f}\left( \epsilon^{\alpha \beta \sigma\nu}\partial_{\sigma}T_{(\alpha \beta )}{}^{\mu}+\epsilon^{\alpha \beta \sigma \mu}\partial_{\sigma}T_{(\alpha \beta)}{}^{\nu}\right) -g^{\mu \nu}\frac{ke}{2\left( f+4k\right)f}\epsilon^{\alpha \beta \rho \kappa}\partial_{\rho}T_{(\alpha \beta )\kappa}.
G_{MN}=\Phi_{\cal BD}^{-1}\left(\begin{array}{cc}g_{\mu\nu}\, 0 0 \,\Phi_{\cal BD}^{\frac{D-2}{D-4}}h_{mn}\end{array}\right)=\left(\begin{array}{cc}\tilde{g}_{\mu\nu}\, 0 0 \,\Phi_{\cal BD}^{\frac{2}{D-4}}h_{mn}\end{array}\right),
\widetilde{G}^{(d_1,d_2),n}_{p_1,p_2}(t)= \begin{cases}G^n_{p_1,p_2}(t), d_1>0, d_2 >0, G^{(1),n}_{p_1}(t), d_1>0,d_2=0, G^{(2),n}_{p_2}(t), d_1=0,d_2>0. \end{cases}
v\left(\frac{P(a_{\rho})}{Q(a_{\rho})}-\frac{P(a)}{Q(a)}\right)\ =\v\left(\frac{P(a_{\rho})Q(a)-P(a)Q(a_{\rho})}{Q(a_{\rho})Q(a)}\right) &=\v(P(a_{\rho})Q(a) - P(a)Q(a_{\rho})) - v(Q(a_{\rho})) - v(Q(a)).
q^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{2}}-p^{n_{2}^{\prime}-n_{s}^{\prime}}a_{n_{s}}N_{n_{2}}\frac{q^{n_{s}^{\prime}-n_{1}^{\prime}}}{N_{n_{s}}}=p^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{1}}\frac{N_{n_{2}}}{N_{n_{1}}}.
\sum_{\delta\in\{0,1\}^{n}}u_{\delta}\prod_{i=1}^{n}x_{i}^{\delta_{i}}=\sum_{\delta\in\{0,1\}^{n}}v_{\delta}\,x_{n}^{\delta_{n}}\prod_{i=1}^{n-1}\left(x_{i}-x_{n}\right)^{\delta_{i}}
\sum_{d\leqslant \varepsilon_{1}^\frac{1}{2}\Xi(\alpha )^{-\frac{1}{2}}B^{1-\frac{1}{2r}}}O_{\varepsilon_i}\left(\frac{\varDelta(\alpha)\Xi(\alpha)^{-\frac{1}{2}}}{d}B^{1-\frac{1}{2r}}\log (B)\right) =O_{\varepsilon_i}\left(\varDelta(\alpha)\Xi(\alpha)^{-\frac{1}{2}}B^{1-\frac{1}{2r}}\log (B)\log (\Xi(\alpha)^{-\frac{1}{2}}B^{1-\frac{1}{2r}})\right).
F(g_{y}^{-1}g_{n}y)(g_{n}\cdot f) = D_{\pi}(g_{y}g_{y}^{-1}g_{n}y)(g_{n}\cdot f) = D_{\pi}^{(g_{n})}(y)(f) \leq \| f \|_{L^{\infty}(X,D_{\pi}(y))}- \delta.
\zeta_{EH}'(0;\alpha_1,\ldots,\alpha_d;\lambda_1,\ldots,\lambda_d) =\int_1^\infty\theta^\infty(t)\frac{dt}{t}+\int_0^1\Big(\theta^\infty(t)-\Big(\prod_{i=1}^{d}\alpha_i\Big)(4\pi t)^{-\frac{d}{2}}\Big)\frac{dt}{t}&-\frac{2}{d}\Big(\prod_{i=1}^{d}\alpha_i\Big)(4\pi)^{-\frac{d}{2}}.
[\underline{W_{k,r}^{(i)}}][\underline{W_{k,r+2r_i}^{(i)}}] = t^{\alpha/2}[\underline{W_{k+1,r}^{(i)}}] [\underline{W_{k-1,r+2r_i}^{(i)}}] + t^{\beta/2}[\underline{S_{k, r}^{(i)}}].
b_n-b_{n-1}=\lfloor\frac{\alpha n(n-1)}2\rfloor-\lfloor\frac{\alpha(n-1)(n-2)}2\rfloor &=\lfloor\frac{\alpha(n-1)(n-2)}2+\alpha(n-1)\rfloor-\lfloor\frac{\alpha(n-1)(n-2)}2\rfloor
\left. \left( -\psi_{+\mu}\delta \psi^\mu_+ +\psi_{-\mu}\delta \psi^\mu_-\right) \right|_{\sigma =0}= \left. \left( -\psi_{+\mu}\delta \psi^\mu_+ +\psi_{-\mu}\delta \psi^\mu_-\right) \right|_{\sigma =\pi}=0.
f(x) = H_{\delta}(\|Ax-b\|_2) + \frac{1}{2}\mu \|x\|^2 \nabla f(x) = \nabla H_{\delta}(\|Ax-b\|_2) + \mu x f \in \mathcal{F}_{\mu,\, \delta^{-1}\, \|A^T A\|_2 + \mu}
\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\varphi')^2(1+s^2)^{1/2}s^nds}{\int_{0}^{\infty}\varphi^2(1+s^2)^{1/2}s^{n-2}ds}\geq \inf_{\psi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\psi')^2s^nds}{\int_{0}^{\infty}\psi^2s^{n-2}ds}=\frac{(n-1)^2}{4},
\tilde{D}= \left( \begin{array}{c}{{D^{( 1 )}}}\\{{D^{( 2 )}}}\\ \end{array}\right) , ~ \tilde{E}= \left( \begin{array}{c}{{E^{( 1 )}}}\\{{E^{( 2 )}}}\\ \end{array}\right) , ~ \tilde{B_{\mu}}= \left( \begin{array}{c}{{B_{\mu}^{( 1 )}}}\\{{B_{\mu}^{( 2 )}}}\\ \end{array}\right) , ~ \tilde{\psi_{a}}= \left( \begin{array}{c}{{\psi_{a}^{( 1 )}}}\\{{\psi_{a}^{( 2 )}}}\\ \end{array}\right)
\int_1^\infty t^{\nu-\alpha-1/2}|df(t)|\lesssim \sum_{n=1}^\infty \Big( c_n^{\mu+\nu-\alpha-3/2}+(c_n+\varepsilon_n)^{\mu+\nu-\alpha-3/2}\Big) \lesssim \sum_{n=1}^\infty c_n^{\mu+\nu-\alpha-3/2}.
\limsup_{n\to\infty}\left\|M_n\right\|_{L^\infty([-1,1])}^{1/n}= \limsup_{n\to\infty}\left\|\frac{P_n^*}{c_n\gamma_na_n^n}\right\|_{L^\infty([-1,1])}^{1/n}= \limsup_{n\to\infty}\left\|P_n^*\right\|_{L^\infty([-1,1])}^{1/n}\frac{1}{|c_n|^{1/n}}\frac{1}{\gamma_n^{1/n}a_n}\le \frac{1}{2}
\hat{H}_{0}= \hbar g \omega \left[{\alpha_{i}^{a}}^{\dagger}\alpha_{i}^{a}+ \frac{1}{2}2 d \right] = \hbar g \omega \left[ \hat{N}+ d \right] \ \ , \ \ \hat{\phi}= i \hbar \epsilon^{a b}{\alpha_{i}^{a}}^{\dagger}\alpha_{i}^{b}\ \ .
\!\!\!\!\!\!\!\mathcal{O}_{\mathcal{UL}}(\alpha,\theta)=1- \exp \left( -{\rm U}(\theta)-\frac{\pi^2}{2}\lambda \sqrt{\frac{\theta P_{\rm d}}{\rho}}|\mathcal{\tilde{C}}_{\rm{u}}(\alpha)|-\theta \frac{{\beta}P_{\rm{d}}|\mathcal{\tilde{C}}_{\rm{u}}(\alpha)|^2}{\rho}\right),
-\Delta z_1=(m_+(x_0)-m_-(x_0))\delta_0-c_0\ \mbox{in $B_{r_0}$},z_1=b_0\ \mbox{on $\partial B_{r_0}$}, &-\Delta z_2=-s_-\leq 0\ \mbox{in $B_{r_0}$},z_2=0\ \mbox{on $\partial B_{r_0}$},
\sum_{i_1}\cdots\sum_{i_M}\sum_{j_1=1}^{s_1}\cdots\sum_{j_M=1}^{s_M}b_{j_1}^{(1)}\cdots b_{j_M}^{(M)}|z_{j_1\cdots j_M;i_1,\cdots,i_M}^{\rho+1}|^2 &=\sum_{i_1}\cdots\sum_{i_M}\sum_{j_1=1}^{s_1}\cdots\sum_{j_M=1}^{s_M}b_{j_1}^{(1)}\cdots b_{j_M}^{(M)}|z_{j_1\cdots j_M;i_1,\cdots,i_M}^{\rho}|^2,~a.s.,
L_{n}= c_{1}- \left(\tfrac{\pi n}{l}a_{1}\right)^{2}, D_{n}= \left(c_{2}- \left(\tfrac{\pi n}{l}a_{2}\right)^{2}\right) e^{-\left(c_{1}- \left(\tfrac{\pi n}{l}a_{1}\right)^{2}\right) \tau}.
\lim_{n \to \infty}\frac{1}{n}\log \lVert \delta_{A}^{W^{-}(n,\phi^{-1})}\rVert_{1}= -\alpha^{-}(\phi^{-1})\log \rho(\delta_{A}^{-1}) = \lvert \alpha^{-}(\phi^{-1}) \rvert \log \rho(\delta_{A}^{-1}).
\frac{\Delta(R)\Delta(R\cap (S\cap T)')\cap \Delta(R\cap S)\Delta(R)}{\Delta(R)\Delta(R\cap (S\cap T)')\Delta(R)+\Delta(R\cap S)\Delta(R\cap (S\cap T)')}=H_2\left(\frac{R}{R\cap (S\cap T)'}, \,\frac{\Delta(R)}{\Delta(R\cap S)}\right).
\nabla^{\mathrm{restr}}_{\partial z}(\tilde{v}_{2}) = \left( -\partial_{z}( \frac{\lambda +1}{2z}) +\frac{\eta}{z^{2}}(-\frac{\beta}{2}+\frac{\eta \gamma}{z}) +\frac{(\lambda +1)}{2\eta}( \frac{\dot{\eta}}{z}-\frac{2\eta}{z^{2}}) +\frac{(\lambda + 1)^{2}}{4z^{2}}\right) v_{1}& + \frac{1}{\eta}( \dot{\eta}- \frac{2\eta}{z}) \tilde{v}_{2}.
z(\lambda |\lambda_{j})=M\left( \sum^{N_{h}}_{h=1}e^{\pm \lambda_{h}}-\sum_{c=1}^{N_{c}}e^{\pm \lambda_{c}}+\Theta (1-p)\sum_{w}e^{\pm \lambda_{w}}\left( 1+e^{\mp i\pi (p+1)\mathrm{sign}(\Im m\lambda_{w})}\right) -2\sum^{N_{s}}_{s=1}e^{\pm \lambda_{s}}\right)
dX_{M,i}(t) =V_{M,i}(t)dt, 1\leq i\leq M, dV_{M,i}(t) =\Big(-u(X_{M,i}(t))+V_{M,i}(t)-\nabla U*c(X_{M,i}(t))+\frac{1}{M}\sum_{j\neq i}^{M}\nabla U(X_{M,i}(t)-X_{M,j}(t))\Big)dt + dW_t^i,
\int_{B}\exp \big( \alpha_n |u_\varepsilon|^{\frac n{n-1}+ |x|^\alpha}\big) dx &\geq |B| + |B| \exp \big( \sum_{i=1}^{n-1}\frac 1i \big) + O(R^{-\frac n{n-1}}) &+ \frac{c^{-\frac n{n-1}- a^\alpha}\alpha_n^{n-1}}{(n-1)!}\Big(\int_{B_a}\Big(-\frac n{\alpha_n}\log |x|\Big)^{n + (n-1) |x|^\alpha}dx + o(1)\Big).
E_{1}(\mbox{\mathversion{bold}$v$}(t))=E_{1}(\mbox{\mathversion{bold}$v$}_{0}) =E_{1}(\mbox{\mathversion{bold}$v$}^{R}+\mbox{\mathversion{bold}$\varphi$}_{0}) begin{align*}3mm] =E_{1}(\mbox{\mathversion{bold}$v$}^{R}) + E_{1}(\mbox{\mathversion{bold}$\varphi$}_{0})+R_{1}(\mbox{\mathversion{bold}$v$}^{R},\mbox{\mathversion{bold}$\varphi$}_{0}),
\frac{\partial l}{\partial\mu_{B}^{(k)}}=\sum_{i=1}^{m}\frac{\partial l}{\partial y_{i}^{(k)}}\frac{-\gamma^{(k)}}{\sqrt{\sigma_{B}^{(k)^{2}}+\epsilon}}+\frac{\partial l}{\partial\sigma_{B}^{(k)^{2}}}\frac{1}{m}\sum_{i=1}^{m}(-2)\cdot(x_{i}^{(k)}-\mu_{B}^{(k)})
\mathcal{L(}\mathbf{Z}\left( n+1\right) \;|\;\mathbf{Z}(n)=(z_{1},...,z_{p}),\Pi =(\mathbf{\Phi}_{0},\mathbf{\Phi}_{1},...))=\mathcal{L}\left( \sum_{i=1}^{p}\sum_{j=1}^{z_{i}}\mathbf{\xi}_{i}^{(j)}(n)|\mathbf{\Phi}_{n}\right) ,
\partial_r[w(\theta)]^\top\Big[\frac{\mathbb{E}_\theta [f(\textbf{X}) + c]}{h + c_0 + w(\theta)^\top \mathbb{E}_\theta [f(\textbf{X}) + c]}- \frac{\bar{f}+ c}{h + c_0 + w(\theta)^\top[\bar{f}+ c]}\Big] = 0,
\mathcal{C}_{1,\dots,r}e^{-w_{r}\mathcal{C}_{1,\dots,r}}= \sum_{n=0}^{\infty}\frac{\left(-w_{r}\right)^{n}}{n!}\cdot\frac{\left(-1\right)^{n+1}}{n+1}\left(\mathcal{C}_{1,\dots,r-1}+\mathcal{B}_{r}\right)^{n+1}= -\frac{1}{w_{r}}\left(e^{-w_{r}\left(\mathcal{C}_{1,\dots,r-1}+\mathcal{B}_{r}\right)}-1\right),
\left \langle \begin{array}{l}R_1 J_1 \end{array}\right |{\hat{\cal T}}_J^R \left | \begin{array}{l}R_2 J_2 \end{array}\right \rangle = \left \langle \begin{array}{ll}R_1 R J_1 J \end{array}\right. \left | \begin{array}{l}R_2 J_2 \end{array}\right \rangle \langle R_1 \|{\hat{\cal T}}^R \| R_2 \rangle,
\varepsilon = \{\varepsilon^{ij}\}= \left(\begin{array}{ccccc}0 0 0 0 0 0 0 \varepsilon^{12}\varepsilon^{13}0 0 -\varepsilon^{12}0 \varepsilon^{23}0 0 -\varepsilon^{13}-\varepsilon^{23}0 0 0 0 0 0 0\end{array}\right)
^{H}\mathbb{D}_{a+}^{\alpha ,\beta ;\psi}f\left( x\right) =I_{a+}^{\beta \left(n-\alpha \right) ;\psi}\left( \frac{1}{\psi^{\prime}\left( x\right)}\frac{d}{dx}\right)^{n}I_{a+}^{\left( 1-\beta \right) \left( n-\alpha\right) ;\psi}f\left( x\right)
p^{25}=\left[\begin{array}{ccccc}1 1 3 2 3 2 2 2 3 1 3 3 1 1 2 \end{array}\right],p^{26}=\left[\begin{array}{ccccc}1 2 3 1 2 2 1 2 3 3 3 3 1 2 1 \end{array}\right].
M_r=\begin{pmatrix}\frac{1}{\sqrt{2}}I_r 0 \frac{1}{\sqrt{2}}S_r 0 I_{n-2r}0 -\frac{1}{\sqrt{2}}S_r 0 \frac{1}{\sqrt{2}}I_r\end{pmatrix}M_rJ_rM_r^{-1}=\begin{pmatrix}-I_{r}0 0 I_{n-r}\end{pmatrix},
\int_{X}v(f)\omega^{n}&=C\left(\frac{\int_{X}\theta_{v}\Omega_{h}}{\int_{X}\Omega_{h}}-\frac{\int_{X}\theta_{v}\omega^{n}}{C}\right) &=C\left(\int_{X}\theta_{v}\left(\frac{\Omega_{h}}{\int_{X}\Omega_{h}}-\frac{\omega^{n}}{C}\right)\right).
\mathrm{Ker}\, d_3 =C\{1, y_1, v_0 z_1 \}\oplus D_0\{x_3^2, x_3^2 y_1, x_3z_1 \}, \mathrm{Im}\, d_3 =D_0 \{v_1 x_3^2, v_1 x_3^2 y_1,v_1 x_3 z_1 \}\intertext{and}\mathrm{Ker}\, d_3/\mathrm{Im}\, d_3 =C \{1, y_1, v_0z_1 \}\oplus D_1\{x_3^2, x_3^2 y_1, x_3z_1 \}.
Z_{2 \mathrm{i n s t}}\simeq - \frac{2 N}{\pi}[ \quad \frac{\pi}{2}- \operatorname{a r c s i n}( \frac{2 \pi^{2}m^{2}}{A}- 1 ) - \frac{2 \pi^{2}m^{2}}{A}\sqrt{\frac{A}{\pi^{2}m^{2}}- 1}\quad ] .
{\cal{Z}}_{r s}= P^{\mu}( C \Gamma_{\mu}^{( 1 1 )})_{r s}+ Z_{\bf{O}}^{\mu \nu}( C \Gamma_{\mu \nu}^{( 1 1 )})_{r s}= Z_{\bf{O}}^{[ \mu_{1}\ldots \mu_{5}]}( C \Gamma_{\mu_{1}\ldots \mu_{5}}^{( 1 1 )})_{r s},
\frac{d}{dt}\beta (\mathbf{r}(t))=\mathbf{\nabla}\beta (\mathbf{r}(t))\cdot \frac{d\mathbf{r}}{dt}=\mathbf{\nabla}\beta (\mathbf{r}(t))\cdot \mathbf{\vec{\omega}(r(}t\mathbf{))=}0
\frac{a^{-s-k}b^k}{1-a^{-s-k}b^{k}}=\frac{1-(s+\ell)\log a + O((s+\ell)^2)}{(s+\ell)\log a \left( 1-\tfrac{1}{2}(s+\ell)\log a + O(s+\ell)^2\right)},
\begin{aligned}\left\| \bar{\boldsymbol{\psi}}(t_{k+l}) - \tilde{\boldsymbol{\psi}}^k(t_{k+l})\right\|_2 \le L \sum_{i=1}^{l}b_{S,k+i}\left\| \bar{\boldsymbol{\psi}}(t_{k+i-1}) - \tilde{\boldsymbol{\psi}}^k(t_{k+i-1}) \right\|_2 + C,\end{aligned}
\begin{array}{ll}A := \int_{\{\inf_{1 \leq j \leq J_{0}}|x-x_{j}(t)| \geq \tilde{\mu}^{-1}_{2}\}\cap \{\inf_{1 \leq j^{'}\leq J_{1}}|x-z_{j^{'}}(t)| \geq \tilde{\mu}^{-1}_{1}\}}|v(t,x)|^{2}\, dx\end{array}
T_l(i,j) =T_{l}(i1,j1)+C_l(i1,j)+R_{l}(i,j1)+S_l(i,j), C_l(i,j) = C_l(i1,j) + S_l(i,j), R_l(i,j) = R_l(i,j1)+ S_l(i,j).
\varphi_{ij}\left( \theta +\frac{i\pi}{h}\right) +\varphi_{ij}\left(\theta -\frac{i\pi}{h}\right) =\sum\limits_{k=1}^{r}I_{ik}\varphi_{kj}\left( \theta \right) -2\pi I_{ij}\delta \left( \theta \right) \quad ,
\delta_0 \ge \eta \left[1+(\ell+\epsilon)M\right]^{-1}\inf_{t\in [0,T]}\big[1-(\ell+\epsilon)\omega(\cdot,\lambda_1)*p(t)\big] & \ge \eta\left[1+(\ell+\epsilon)M\right]^{-1}\big[1-(\ell+\epsilon)\sup_{t\in [0,T]}\omega(\cdot,\lambda_1)*p(t)\big] & = \eta \left[1+(\ell+\epsilon)M\right]^{-1}\big[1-(\ell+\epsilon)M\big]>0.
H\; =\; - \frac{2}{3\sqrt{3}}\; \sum_{j=1}^{M}\; \Gamma_{\, j}+\Gamma_{\,j}^{\, 2}+ \lambda\, ( \sigma_{\, j}\sigma_{\, j+1}^{\, 2}+ \sigma_{\, j}^{\,2}\sigma_{\, j+1})
\begin{array}{ll}(\nabla_XJ_\alpha )Y=\frac{1}{4n}\left[g(X,Y)p_\alpha^\top +\overline \theta_\alpha (Y)X+g(X,J_\alpha Y)J_\alpha (p_\alpha^\top )\right. \qquad\quad\left.+\overline \theta_\alpha (J_\alpha Y)J_\alpha X\right] , \alpha =2,3 ,\end{array}
\mu_k(\lambda, \tau + \delta) \; =\; \tilde{\mu}_k(\lambda, \tau + \delta) - (\tau + \delta) \; =\; \tilde{\mu}_k(\lambda, \tau) + \delta - (\tau + \delta)\; =\; \tilde{\mu}_k(\lambda, \tau) - \tau \; =\; \mu_k(\lambda, \tau)
\frac{n_0 \alpha'_{i + m(i)}}{i}= \frac{n_0 (\lfloor\gamma(i + m(i)) \rfloor+ 1)\alpha}{i(n + \lfloor\gamma(i + m(i)) \rfloor+ 1- i - m(i))}= \frac{n_0 \alpha}{n - m(i)}\le\alpha.
\left(\int_\Omega |u\varphi|^{2^*_s}\,dx\right)^{2/2^*_s}\leq \, C\left(\int_\Omega \varphi^2\,dx+\iint_Qu(x)u(y)\frac{(\varphi(x)-\varphi(y))^2}{|x-y|^{N+2s}}\,dxdy\right. &\left.+\int_\Omega u^2\varphi^2\,dx\right) \leq \, C\left(\iint_Qu(x)u(y)\frac{(\varphi(x)-\varphi(y))^2}{|x-y|^{N+2s}}\,dxdy+\int_\Omega u^2\varphi^2\,dx\right),
\frac{1}{N}\sum_{i=1}^N |\langle \Theta_i, \xi \rangle| \geq \frac{c_2}{\sqrt{n}}- \delta \textrm{and}\frac{1}{N}\sum_{i=1}^N \exp \left \{\left( \frac{\sqrt{n}|\langle \Theta_i, \xi \rangle| + 1}{2}\right)^{\alpha}\right \}\leq 3 C_3 + N^{1/4}\delta.
\eta_x \:=\:\begin{cases}1_{x\star_0\eta_{t_0(x)}\star_1\cdots\star_{n-1}\eta_{t_{n-1}(x)}}\tilde{\phi}_x \star_0 \eta_{t_0(x)}\star_1 \cdots \star_{n-1}\eta_{t_{n-1}(x)}\star_n \eta_{\tilde{x}}\end{cases}
f (y \, | \, \sigma_{1}, \sigma_{2}, \rho) = \tfrac{\big ( 2 \pi \sqrt{1 - \rho^{2}}\sigma_{1}\sigma_{2}\big )^{1-n}}{n}\exp \big ( \tfrac{-n}{2(1-\rho^{2})}\big [ \tfrac{s_{1}^{2}}{\sigma_{1}^{2}}- 2 \rho \tfrac{r s_{1}s_{2}}{\sigma_{1}\sigma_{2}}+ \tfrac{s_{2}^{2}}{\sigma_{2}}\big ] \big ),