harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}I_{a}\leq k^- \int_0^t\bigg(\int_{\frac{|x|}{2}\le |y| \le 1}+\int_{|y|\le \frac{|x|}{2}}\bigg) \frac{w (x)\exp\Big(-\frac{|x-y|^2}{4s}\Big)}{\big(4\pi s \big)^{n/2}|y|^{2-\delta (m(l_u)-\nu)+\nu}}dyds =: k^- (I_a^-+I_a^+) \end{aligned}
I(m) :={}\sum \limits_{j = 1}^{\lfloor\frac{k + 1}{2}\rfloor}\big\{W_1(y_{2j - 1}) - W_1(y_1^*) \big\}+ \sum \limits_{j = 1}^{\lfloor\frac{k}{2}\rfloor}\big\{W_2(y_{2j}) - W_2(y_2^*) \big\}.
e_{f}e_{g}= \psi (f,\, g) e_{fg}, f, g \in \Gamma e_{f}v_{i}= \chi_{i}(f) v_{i}e_{f}, f \in \Gamma, \,\, i = 1, \dots, n v_{i}v_{j}- \chi_{j}(g_{i}) v_{j}v_{i}= \alpha_{ij}e_{g_{i}g_{j}}, 1 \leq i < j \leq n v_{i}^{2}= \xi_{i}e_{g_{i}^{2}}, i = 1, \dots, n
(1 - \mid{\tilde{Z_p}}\mid^2 ) = \frac{(1 -\mid a_p \mid^2)(1 - \mid Z_p\mid^2 )}{1 +\mid a_p\mid^2 \mid Z_p\mid^2 + 2\mid a_p\mid\mid Z_p \mid cos\chi_p}\quad \geq 0
\langle x , a \cdot (j_{E}\circ D ) (b) + (j_{E}\circ D ) (a) \cdot b \rangle = \langle x , j_{E}( a \cdot D (b) ) + j_{E}( D (a) \cdot b ) \rangle &= \langle x , a \cdot D (b) + D (a) \cdot b \rangle &= \langle x , D (a b) \rangle = \langle x , ( j_{E}\circ D ) (a b) \rangle \ ,
T_0:=\begin{bmatrix}1 1 u_1\bar u_2 u_1\bar u_3 \dots u_1\bar u_{n-1}1 1 1 u_2\bar u_3 \dots u_2\bar u_{n-2}\bar u_1u_2 1 1 1 \dots u_3\bar u_{n-3}\bar u_1u_3 \bar u_2u_3 1 1 \dots u_4\bar u_{n-4}\dots \bar u_1u_{n-1}\bar u_2u_{n-2}\bar u_3u_{n-3}\bar u_4u_{n-4}\dots 1\end{bmatrix}
\lim_{t\to\infty}-\tfrac1t\log\mathbb P(Z_t>0) = \begin{cases}1+\max\{0,-k(h'(1)-1)\}p=0, 1-p-\lambda(\mu-1) \nu\leq p, 1-\nu(1+\log\tfrac1\nu) p<\nu\leq1. \end{cases}
q=\frac{f_{0}(1+\frac{\beta_{0}y}{f_{0}}+\frac{\zeta}{f_{0}})}{H_{0}(1+\frac{\alpha_{0}y}{H_{0}}+\frac{\eta}{H_{0}})}=\frac{f_{0}}{H_{0}}(1+\frac{\beta_{0}y}{f_{0}}+\frac{\zeta}{f_{0}})(1-\frac{\alpha_{0}y}{H_{0}}-\frac{\eta}{H_{0}}+...),
0\longrightarrow H^i({{{{\mathfrak N}}}},V)\longrightarrow\bigoplus_{\lambda\in \nabla}H^i({{{{\mathfrak N}}}}[\lambda],V)\longrightarrow\ldots\longrightarrow \bigoplus_{E\subset \nabla\atop|E|=|\nabla|-i}H^i({{{{\mathfrak N}}}}[E],V)
V_{\mathrm{W_{B}}}( \theta ) = \frac{N T_{0}}{L \cos{\theta}}\Bigl ( 1 - \frac{L^{4}\sin^{4}{\theta}}{4 \sqrt{N^{2}- 1}}+ \frac{L^{8}\sin^{6}{\theta}\cos^{2}{\theta}}{3 2 ( N^{2}- 1 )}\Bigr ) ,
D_{\nu}D^{\nu}\psi +m^{2}\psi -\frac{i}{2}eF_{\mu \nu}S^{\mu}{}^{\nu}\psi -\frac{e}{2m}\left( \beta^{\mu}\beta^{\nu}\beta^{\alpha}+\beta^{\mu}\eta^{\nu \alpha}\right) D_{\nu}\left( F_{\alpha \mu}\psi \right)=0.
\hat{U}^{+}= \operatorname*{l i m}_{\tau \to - \infty}e^{\frac{i}{\hbar}\, \hat{H}_{0}\tau}e^{- \frac{i}{\hbar}\, \hat{H}\tau}= \int_{0}^{+ \infty}d k \, | k \rangle \, \langle \Psi_{k}^{-}| .
\Psi_1(0)+z\cdot\Psi_2(0)={}_2F_1\bigg[\begin{matrix}-a -a &1\end{matrix}\bigg|\,z\bigg]+z^{a+1}{}_2F_1\bigg[\begin{matrix}-a -a &1\end{matrix}\bigg|\,\frac1z\bigg] = (1+z)^{a+1}{}_2F_1\bigg[\begin{matrix}-\frac{a}2 -\frac{a}2+\frac12 &1\end{matrix}\bigg|\,\frac{4z}{(1+z)^2}\bigg]=(1+z)\Phi(0).
z(\cdot)= T_{\mathbb{A}+Q}(\cdot)x+\Phi_{\mathbb{A}+Q,J^{\mathbb{A},\mathbb{A}+Q}\mathbb{B}+\bar{Q}}(\cdot)u = T_{\mathbb{A}+Q}(\cdot)x+\Phi_{\mathbb{A}+Q,J^{\mathbb{A},\mathbb{A}+Q}\mathbb{B}}(\cdot)u+\Phi_{\mathbb{A}+Q,\bar{Q}}(\cdot)u\in D^p(Q^\mathbb{A}_\Lambda).
\max_{1\leq k\leq2n}\left\vert \sum\nolimits_{j=1}^{k}a_{j}Q^{j}f(\xi_{0})\right\vert \leq E_{0}\max_{1\leq k\leq n}\left\vert \sum\nolimits_{j=1}^{k}a_{\j}E^{j}f(\xi_{0})\right\vert & +E_{0}\max_{1\leq k\leq n}\left\vert \sum\nolimits_{j=1}^{k}a_{2j+1}E^{j}f(\xi_{1})\right\vert .
[\delta_{\varepsilon_1},\delta_{\varepsilon_2}]X=\delta_{[\varepsilon_1,\varepsilon_2]}X\Longrightarrow[\check{\rho}(\varepsilon_1),\check{\rho}(\varepsilon_2)]_{T\Sigma}X=\check{\rho}([\varepsilon_1,\varepsilon_2]_{X^{**}Q})X.
P(\{k_{i},k_{j}\}|i\leftrightarrow j)\sim k_{i}^{-\alpha+1}k_{j}^{-\alpha+1}c(\frac{k_{i}^{-\alpha+2}}{k_{min}^{-\alpha+2}},\frac{k_{j}^{-\alpha+2}}{k_{min}^{-\alpha+2}}).
P_{error1\rightarrow any}\le M^{\rho}\sum_{y_{1}^{n}}(\sum_{x_{1}^{n}}Q(x_{1}^{n})[p(y_{1}^{n}|x_{1}^{n})]^{1-s\rho})(\sum_{x_{2}^{n}}Q(x_{2}^{n})[p(y_{1}^{n}|x_{2}^{n})]^{s})^{\rho}.
|\sum_{j=0}^{n-1}e^{2\pi ilj\alpha}|=|\sum_{j=0}^{n-1}(e^{2\pi il\alpha})^{j}|=|\frac{1-e^{2\pi iln\alpha}}{1-e^{2\pi il\alpha}}|\le\frac{2}{|1-e^{2\pi il\alpha}|},
V^-(z) V^+(w) =\exp\left(-[\sum_{m>0}\frac{1}{m}h_m z^{-2m}, \sum_{n>0}\frac{1}{n}h_{-n}w^{2n}]\right)\cdot V^+(w)V^-(z) &=\exp \left(\sum_{m>0}\frac{1}{m}\frac{w^{2m}}{z^{2m}}\right)\cdot V^+(w)V^-(z)=i_{z, w}\exp \left(-\ln \left(1-\frac{w^2}{z^2}\right)\right) \cdot V^+(w)V^-(z).
\mathfrak{a}[\mathbf{v}, \mathbf{v}] = \int_{\widetilde{\Omega}}\mathfrak{a}(\mathbf{k}) [\widetilde{\mathbf{v}}(\mathbf{k}, \cdot), \widetilde{\mathbf{v}}(\mathbf{k}, \cdot)] \, d \mathbf{k}.
2\Phi_{xx}= -\frac{1}{\Phi_{xq}}(\Phi_{xy}\Phi_{yq}+\Phi_{pq}\Phi_{xp}) -\frac{1}{\Phi_{xp}}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp}), 2\Phi_{qq}= -\frac{1}{\Phi_{xq}}(\Phi_{xy}\Phi_{yq}+\Phi_{pq}\Phi_{xp}) +\frac{1}{\Phi_{xp}}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp}).
r_{(P/A)^*}(X) =(r_{P/A})((E-A)-X)+||X||_{r_{P/A}}-(r_{P/A})(E-A) &=r_P(E-X)-r_P(A)+||X||_{r_{P/A}}-r_P(E)+r_P(A) &=r_P(E-X)-r_P(E)+\sum\limits_{a\in X}(r_{P/A})(\{a\}) &=r_P(E-X)-r_P(E)+\sum\limits_{a\in X}[r_P(A\cup \{a\})-r_P(A)].
\lim_{\varepsilon \rightarrow 0}\int_{0}^{T}\int_{\Omega}u_{\varepsilon}(x,t)v\left( x,t,\frac{x}{\varepsilon_{1}},\cdots ,\frac{x}{\varepsilon_{n}},\frac{t}{\varepsilon_{1}^{\prime}},\cdots ,\frac{t}{\varepsilon_{m}^{\prime}}\right) dxdt =\int_{0}^{T}\int_{\Omega}\int_{\mathcal{Y}_{n,m}}u_{0}(x,t,y^{n},s^{m})v(x,t,y^{n},s^{m})dy^{n}ds^{m}dxdt
\theta_{2,i}\delta(\varepsilon) + C_i(\varepsilon) = \theta_{2,i}\delta(\varepsilon)+ (1 - \theta_{2,i}) \frac{C_i(\varepsilon)}{1 - \theta_{2,1}}\le \theta_{2,i}\delta(\varepsilon) + (1 - \theta_{2,i}) \delta(\varepsilon) = \delta(\varepsilon).
1+W^{(2)}_{u_1 u_1 u_1}{u_2}_x=0\, , W^{(2)}_{u_1 u_1 u_1}{u_1}_x+W^{(2)}_{u_1 u_1 u_1 u_1}{u_2}_x=0\, , & u_1 +W^{(2)}_{u_1 u_1 u_1}{u_2}_t=0\, , 1 +W^{(2)}_{u_1 u_1 u_1}{u_1}_t+W^{(2)}_{u_1 u_1 u_1 u_1}{u_2}_t=0\, .
P(\mathcal S(h))_s = 2^{-|E|}\sum_{h' \in \mathcal S(h)}P(h')_s &= \frac{2^{-|E|}}{n}\sum_{h'\in \mathcal S(h)}\#\big\{k\le n :\ (\lambda_k(h')) = s \big \}&= \frac{2^{-|E|}}{n}\sum_{k=1}^{n}\# \big\{h' \in \mathcal S(h): \ (\lambda_k(h')) = s \big\}.
Y_t = Y_t^{b}\textbf{1}_{t<\tau}+ \xi^a_{\tau}\textbf{1}_{t\geq\tau}, Z_t = Z_t^{b}\textbf{1}_{t\leq\tau}, U_t = (\xi^a_t-Y_t^{b}) \textbf{1}_{t\leq\tau},
F_Y(\theta)=Y^{-\log(q+s)/\log{q}}\Bigl|\sum_{n<Y}\mathbf{1}_\mathcal{A}(n)e(n\theta)\Bigr|=\prod_{i=0}^{k-1}\frac{1}{q-s}\Bigl|\sum_{\substack{n_i<q n_i\notin\mathcal{B}}}e(n_i q^i\theta)\Bigr|.
S ={Tr^D_0 \over 2^D}\int \sqrt{g}\; \left( (1-q)\phi^{{2D\over D-2}}+{8 \over (D-2)^2}[(\partial\phi)^2 +{D-2\over 4(D-1)}\phi^2R] +{\cal O}(\phi^{{2(D-4)\over D-2}})\right).
\widehat{h}= \displaystyle \frac{\widehat{H}}{{\cal M}}= \displaystyle \frac{2 \pi}{l}\left( \widehat{L_{0}}+ \widehat{\overline{{{L}}}_{0}}- \displaystyle \frac{c}{1 2}\widehat{I}+ \kappa \displaystyle \frac{l^{2 - h}}{\left( 2 \pi \right)^{1 - h}}\widehat{B}\right)
det\vert A \vert = \prod\limits_{\scriptscriptstyle n=1}^{\infty}\vert \lambda_{\scriptscriptstyle n}\vert \hskip3pt, \hskip2cmdet\vert B \vert = \prod\limits_{\scriptscriptstyle n=1}^{\infty}\vert 1+\lambda_{\scriptscriptstyle n}\vert
\begin{aligned}0 = J'(u_*)(v) =\int_0^T \Big({}_0 D_t^{\alpha-1}({}_0^cD_t^{\alpha}u_*(t)) -{}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha}u_*(t)) &+ \int_0^t f(u_*(s))ds\Big) v'(t) dt, \end{aligned}
H(\textbf{q}_1,\textbf{p}_1,\cdots,\textbf{q}_N,\textbf{p}_N,t)=\sum_{j=1}^N \left( \frac{1}{2 m_j}\|\textbf{p}_j-e_j \textbf{A}_j(\textbf{q}_j,t)\|^2+e_j \phi_j(\textbf{q}_j,t) \right) + V(\textbf{q}_1,\cdots,\textbf{q}_N,t).
T_n(1,t) = [1a]_n(1,t) + \sum_{j=3}^\infty [j]_n(1,t) + \sum_{j=1}^\infty [j*]_n(1,t) &= t^2 (t+1)^{n-2}- t^n + t^n + t^{n-1}+ \sum_{j=2}^{n-1}\left( t^2 (t+1)^{n-j-1}- t^{n-j+1}+ t^{n-j}\right) &= t^2(t+1)^{n-2}+ t^{n-1}+ t^2\left(\frac{(t+1)^{n-2}-1}{(t+1)-1}\right) - t^{n-1}+ t &= t(t+1)^{n-1}.
- \ln Z_{1}= \int_{0}^{\infty}\frac{ds}{s}\left( \mbox{Tr}\, e^{sD^2}- \frac14 \mbox{Tr}\, e^{s (D^2{\bf 1}_{16}+i \Gamma .F/2)}- \frac12 \mbox{Str}\, e^{s \Delta_2}\right),
B_\mathbf{m}(s)=\int_{\mathbb{R}^3}\! \mathcal{D}_{\widetilde{P}_1'(\bar{t})s}^{(1)}f_{1,m_1}\left(x+\widetilde{P}_1(t),y\right)\mathcal{D}_{\widetilde{P}_2'(\bar{t})s}^{(2)}f_{2,m_2}\left(x,y+\widetilde{P}_2(t)\right) &\widetilde{\zeta}_\mathbf{m}(x,y,t,s)\,\mathrm{d}x\mathrm{d}y\mathrm{d}t.
L_0= (2n+1){|h|\over 2}+ |h|d_0^\dagger d_0 -{|h|\over 2}-|h|\sum_{k=1}^\infty\left( d^\dagger_k d_k- \tilde{d}^\dagger_k \tilde{d}_k+a^\dagger_k a_k -b^\dagger_k b_k \right) + L_0^{free}.
Q \cong\bigoplus_{\ell=1}^{n}P_2 J^{n-\ell}\oplus \bigoplus_{\ell'=1}^{n-1}P_{2+\ell'}J^{n-\ell'}\cong P_2 \oplus P_2 J \oplus \cdots \oplus P_2 J^{n-1}\oplus P_3 J^{n-1}\oplus P_4 J^{n-2}\oplus\cdots\oplus P_n J^2\oplus P_1 J.
A_{\overline{T}T\overline{T}T}(k_{1};k_{2};k_{3};k_{4};a_{i})\sim (2\pi )^{p+1}\delta (\sum k)\: \frac{1}{\alpha '}{\cal C}_{0}{\cal N}^{4}_{T}V_{\bot}\frac{s+t}{st}\left( 1+\alpha '(s+t)\right)
\int_{0}^{T}d t \frac{\delta}{{\delta s_{\mu \nu}( t )}}\exp ( - \frac{1}{4}\int_{0}^{T}{d t \dot{x}^{2}) = \frac{1}{2}}\int_{0}^{T}d t \omega_{\mu \nu}\left[ \dot{x}( t ) \right] \exp ( - \frac{1}{4}\int_{0}^{T}{d t \dot{x}^{2}}) ,
-\int_{0}^{T}\int_0^L (\dot u(t) \varphi_1(x) + u(t)\varphi'_1(x) + u(t)\varphi'''_1(x))y(t,x) dxdt +\int_{0}^{T}\int_0^L u(t)\varphi_1(x)(yy_x)(t,x) dxdt=0.
\clubsuit := \|(\dot{D}^{\ell_1}D^{j_1}u_1)\cdots(\dot{D}^{\ell_p}D^{j_p}u_p)\| &\lesssim\|u_1\|_{s_N+\ell_1+2j_1}\cdots\|u_{p-1}\|_{s_N+\ell_{p-1}+2j_{p-1}}\|u_p\|_{\ell_p+2j_p}.
{}_p F_q\left(\begin{matrix}a_{1},\cdots ,a_{p}c_{1},\cdots,c_{q}\end{matrix}\bigg| \xi\right)=\sum_{k=0}^{+\infty}\frac{(a_{1})_{k}\cdots (a_{p})_{k}}{(c_{1})_{k}\cdots (c_{q})_{k}}\frac{\xi^{k}}{k!},
N_0 = \frac{1}{(2 \pi)^3}\int \frac{d^3 k}{2 k^0}a_0^\dagger (\vec{k}) a_0 (\vec{k}) \, , \tilde{N}_0(t) = \frac{1}{(2 \pi)^3}\int \frac{d^3 k}{2 k^0}\tilde{a}_0^\dagger (\vec{k}) \tilde{a}_0 (\vec{k})
|\mathcal{Q}| \stackrel{(a)}{=}\frac{\mathcal{C''}_{1;2}}{C}&\stackrel{(b)}{=}\frac{\min\{\mathcal{C''}_{1;2,3},\mathcal{C''}_{2;1,3}\}}{C}&\stackrel{(c)}{=}\frac{\min\{\mathcal{C}_{1;2,3},\mathcal{C}_{2;1,3}\}}{C}-2|\mathcal{R}| &\stackrel{(d)}{\geq}{h}-2|\mathcal{R}|,
\begin{aligned}W_{j + i - 1}( W^-_{j + i -1}- W_{j + i -1}+ W^+_{j + i}- W_{j + i}) & = - ( W_{j + i -1}W_{j + i}- W_{j + i -1}W^-_{j + i - 1}) + ( W_{j + i -1}W^+_{j + i}- W_{j + i}^2 ) + ( W_{j + i}^2 - W_{j + i - 1}^2 ) .\end{aligned}
\omega(\Phi_t^1(y_1^0,y_2),y_2,t)=\frac{\rho_0(y_1^0,y_2)}{y_1^0}\int_0^{Q(t)}\frac{1}{1+(y_1^0)^{-1}\frac{y_2}{2\delta}Q^2(s)}d\underbrace{Q(s)}_{\int_0^s D(\tau)d\tau}= \frac{\rho_0(y_1^0,y_2)}{y_1^0}\sqrt{\frac{2\delta y_1^0}{y_2}}\arctan\left(\sqrt{\frac{y_2}{2\delta y_1^0}}\int_0^t D(s)ds\right).
\frac{t^3}{3}>\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}>\frac{\chi}{4(2\pi)^3}\sum_{n\in \mathbb{Z}-\{0\}}\frac{1}{|n|^3}=\frac{\chi\zeta(3)}{2(2\pi)^3}>0\;.
\mathcal{F}_1(v, x) = (x - \beta(v)) \displaystyle\prod_{w \in c(v)}\mathcal{F}_1(w, x) - \displaystyle\sum_{w \in c(v)}\left( \mathcal{F}_2(w, x) \displaystyle\prod_{t \in c(v), t \neq w}\mathcal{F}_1(t, x) \right) , \mathcal{F}_2(v, x) = \displaystyle\prod_{w \in c(v)}\mathcal{F}_1(w, x) ,
\|\delta_0^m(a)(D_0+i\lambda)^{-p-1}\|_1 \leq \|\delta_0(a)(D+i\lambda)^{-p-1}\|_1\left\|\left(\frac{D+i\lambda}{D_0+i\lambda}\right)^{p+1}\right\|_\infty = O(\lambda^{-1})
u_\varphi(t) = \left(\begin{array}{c c c}1 c_1 \, t^{p^{e_1}}c_2 \, t^{p^{e_2}}0 1 0 0 0 1\end{array}\right) (\, c_1, c_2 \in k , e_1, e_2 \geq 0 \,) ,
\begin{array}[c]{ll}dW_{x}(s,\bar{X}^{t,x;\bar{u}}(s)) &= \left\{W_{sx}(s,\bar{X}^{t,x;\bar{u}}(s))+W_{xx}(s,\bar{X}^{t,x;\bar{u}}(s))b(s)+\frac{1}{2}W_{xxx}(s,\bar{X}^{t,x;\bar{u}}(s))(\sigma(s))^{2}\right\}ds &\ \ +W_{xx}(s,\bar{X}^{t,x;\bar{u}}(s))\sigma(s)dB(s).\end{array}
S \simeq 2 \pi \sqrt{\frac{d}{6}\ \overline{N}}\left[ 1 - \frac{3 d}{4}\frac{1}{\overline{{\cal R}}}\right] \simeq 2 \pi \left( \frac{d \alpha^{\prime}}{6}\right)^{\frac{1}{2}}M \left[ 1 - \frac{3}{4}\frac{\ell_{s}^{2}}{R^{2}}\right] .
\mathbb{E}e^{\lambda N_n}= \lambda \int_0^\infty e^{\lambda x}\mathbb{P}\left(N_n \geq x\right) \, \mathrm{d}x \leq e^{\beta_0 n}+ \lambda \int_{\beta_0 n/\lambda}^{\infty}e^{\lambda x}\mathbb{P}\left(N_n \geq x\right) \, \mathrm{d}x &\leq e^{\beta_0 n}+ n \int_{\beta_0}^\infty e^{-\Xi n}\mathrm{d}\Xi &\leq e^{\beta_1 n}
\begin{array}{rcl}\tilde{v}_{1}={V_{11}\over B_{1}}{cos^{4}\alpha \over cos2\alpha}-{V_{22}\over B_{2}}{sin^{4}\alpha \over cos 2\alpha}\tilde{v}_{2}= -{V_{11}\over B_{1}}{sin^{4}\alpha \over cos2\alpha}+{V_{22}\over B_{2}}{cos^{4}\alpha \over cos 2\alpha}\end{array}
C_{m_{d}}( \xi ) = \frac{1}{2 \pi}\int d^{2}p \hat{C}_{m_{d}}( p ) e^{i p \xi}= \frac{1}{( 2 \pi )^{2}}\int d^{2}p \frac{1}{p^{2}+ m_{d}^{2}}e^{i p \xi},
\mathrm{T a b l e I . C l a s s i f i c a t i o n o f}Q_{\alpha}^{a}\mathrm{a n d Z}_{\mu_{1}\cdots \mu_{p}}^{a b}\mathrm{u n d e r 1 1 D ( o r 1 2 D ) a n d}K \mathrm{.}
\tilde{V}( x ) : = \frac{2 \pi}{N ( \pi + g N )}\left( \mathrm{K}_{0}\left( \sqrt{\frac{e^{2}N}{\pi \! + \! g N}}| x | \right) + \ln \left( \frac{1}{2}\sqrt{\frac{e^{2}N}{\pi \! + \! g N}}\right) + \gamma \right) .
{\bf a}_t(\theta,f_k)\!=\! [ 1,\!e^{-\mathrm{j}v_t(\theta,f_k)}, \!\cdots\!,\!e^{-\mathrm{j}(N_t-1)v_t(\theta,f_k)}]^T,{\bf a}_r(\theta,f_k)\!=\! [ 1,\!e^{-\mathrm{j}v_r(\theta,f_k)}, \!\cdots\!,\!e^{-\mathrm{j}(N_r-1)v_r(\theta,f_k)}]^T,
f_{ij}^{(n)}(x_1,\ldots,x_{n-2},w,z,x;t) = \int_{\{0<t_1<\ldots<t_{i-1}<\theta<t_i<\ldots< t_{j-2}<r<t_{j-1}<\ldots<t_{n-2}<t\}}g_{n-j}(t_{j-1},x_{j-1},\ldots,t_{n-2},x_{n-2},r,z,t,x) & g_{j-i-1}(t_{i},x_{i},\ldots,t_{j-2},x_{j-2},\theta,w,r,z) f_{i-1}(t_1,x_1,\ldots,t_{i-1},x_{i-1},\theta,w) dt_1 \ldots dt_{n-2}drd\theta,
C\operatorname{-Gpd}_R(M) = \operatorname{Gpd}_{R\ltimes C}(M) C\operatorname{-Gid}_R(M) = \operatorname{Gid}_{R\ltimes C}(M) C\operatorname{-Gfd}_R(M) = \operatorname{Gfd}_{R\ltimes C}(M).
\tilde{\mathcal{S}}= \bigcup_{K \in \mathcal{K}}\bigcup_{J \in \tilde{\mathcal{J}}}\bigcup_{\mathbf{R}\in \mathcal{R}_{(K,J)}}{S}_{(K,J,R)}= \bigcup_{(K,J, \mathbf{R}) \in \tilde{\mathcal{M}}}{S}_{(K,J,R)}.
\sum_{|\alpha|\leq m}\sum_{j=1}^{+\infty}\mu_j||\partial^{\alpha}\psi_j||_{L^2}^2 = \sum_{j=1}^{+\infty}\mu_j ||\psi_j||_{H^m}^2 = \sum_{j=1}^{+\infty}\mu_j < + \infty
\bar{\nu}(x,t)\cdot\nabla\rho_\zeta^\varepsilon(x,t) = -k_d^{-1}\Bigl(\overline{V_\Gamma\zeta}\Bigr)(x,t)+d(x,t)\bar{\zeta}_2(x,t), (x,t)\in\overline{Q_{\varepsilon,T}}
\frac{d\mu}{d(\mu \circ \tau_\lambda^{-1})}|_{R_\lambda}= \frac{d( \mu \circ \tau_\lambda \circ \tau^n)}{d(\mu \circ \tau_\lambda^{-1}\circ \tau_\lambda \circ \tau^n)}|_{R_\lambda}= \left. \frac{d(\mu \circ \tau_\lambda)}{d\mu}\circ \tau^n\right|_{R_\lambda}= \Phi_\lambda \circ \tau^{n}|_{R_\lambda},
t_4:=\bar{X}_4, t_3=\bar{X}_3, t_2=\bar{X}_2-\frac{q^4}{(q^2+1)(q+q^{-1})}\bar{X}_3^2 \bar{X}_4^{-1}, t_1= \bar{X}_1 - \frac{q^2(q+q^{-1})}{q^2-1}\bar{X}_2 \bar{X}_3^{-1}.
|\nabla\psi|^2= \langle\psi,D^2\psi\rangle-\frac{R}{4}|\psi|^2+\frac{1}{2}\Delta|\psi|^2, \alpha({e_i})\langle\nabla_{e_i}\psi,\psi\rangle= |\alpha|^2|\psi|^2=\frac{|d|\psi|^2|^2}{4|\psi|^2}, \langle\beta(e_i)\cdot\nabla_{e_i}\psi,\psi\rangle= -\frac{|T|^2}{4|\psi|^2}.
\mathcal{L}_{m}( g ) = \frac{( - 1 )^{m}}{2^{m}}\delta_{\mu_{1}\nu_{1}\ldots \mu_{m}\nu_{m}}^{\rho_{1}\sigma_{1}\ldots \rho_{m}\sigma_{m}}{R^{\mu_{1}\nu_{1}}}_{\rho_{1}\sigma_{1}}\cdots{R^{\mu_{m}\nu_{m}}}_{\rho_{m}\sigma_{m}}\; ,
\lim_{m_{\pi}^2\rightarrow 0}m_{\pi}^2\int d \vec{x}\, \omega_{\pi}^{(1)}=-\lambda\int d \vec{x}\, (\varphi_{\pi}^3+\varphi_{\pi}\varphi_{\sigma}^2+2v\varphi_{\pi}\varphi_{\sigma})\ne 0.
(n-4) F + \langle J_{ij}, F\rangle \Omega_{ij}=\sum_{k\neq i,j}[J_{ki},\omega_{ki}] + [J_{kj},\omega_{kj}]+ \tfrac 12\langle J_{ij}, \omega_{ki}\rangle \Omega_{kj}- \tfrac 12 \langle J_{ij}, \omega_{kj}\rangle \Omega_{ki}.
0 = \phi(xy+yx-t^2) &= 2(a_1b_1 x^2 + a_2b_2 y^2) + (a_1b_2+a_2b_1)xy + (a_2b_1+a_1b_2)yx - c_3^2 t^2 &= 2(a_1b_1 x^2 + a_2b_2 y^2) + (a_1b_2 + a_2b_1-c_3^2)t^2.
\mathbf{T}(\tilde{Q}_{Y};J,{\cal Q}) =\min_{Q\in{\cal Q}:\, Q=\tilde{Q}_{Y}}\max_{\lambda\geq0}\left[I(Q)+\lambda J(Q)\right] =\max_{\lambda\geq0}\min_{Q\in{\cal Q}:\, Q=\tilde{Q}_{Y}}\left[I(Q)+\lambda J(Q)\right].
\frac{\lambda}{\alpha}\frac{d}{dt}\int_U u^{\alpha}dx = \int_U \nabla\cdot \left(K(|\nabla u|)\nabla u \right) u^{\alpha+\delta-1}dx &= - C_1\int_U K(|\nabla u|)|\nabla u|^2 u^{\alpha+\delta-2}dx +C_\varphi\Big( \int_U |u|^\alpha dx + \int_U |u|^{\alpha+\mu_0}dx\Big),
\mathcal{F}_{d,\theta}:= \biggl\{f \in \mathcal{F}_d: \mu_\alpha(f) \leq \nu, \|f\|_\infty \leq \gamma, \sup_{x:f(x) \geq \delta}M_{f,a,\beta}(x) \leq a(\delta) \ \forall \delta > 0\biggr\}.
\int_{\mathbb{R}^n}e^{i\psi(2^k\xi)}\varphi(\xi)d\xi =\int_{\mathbb{R}^n}e^{i2^k(x-y)\cdot\xi+i2^k(t-s)\sqrt{2^{-2k}+|\xi|^2}}\varphi(\xi) d\xi &\leq C (1+2^k|(x-y,t-s)|)^{-\frac{n}{2}}&\leq C2^{-\frac{n}{2}(k+j)}
\tfrac{X^{u+y\beta}_t-X^{u}_t}{y}=\int_{0}^{t}\Big(\tfrac{\gamma(s,X_{s{^-}}^{u+y\beta},u_s+y\beta_s)-\gamma(s,X_{s{^-}},u_s)}{y}\Big)dM_s, \ \t\in\left[ 0,T\right],
\frac{\sigma_{z}^{2}}{z^{2}}\approx\frac{(2xy)^{2}}{(x^{2}y)^{2}}\sigma_{x}^{2}+\frac{(x^{2})^{2}}{(x^{2}y)^{2}}\sigma_{y}^{2}+\frac{2(2xy)(x^{2})}{(x^{2}y)^{2}}\sigma_{x,y}
{\frac{1}{2}}\sum ( - 1 )^{| J | + | K | + | L |}{\binom{L}{M}}\int \Biggl [ D_{J + M}\theta_{\Omega}D_{K}\theta_{\Omega}E_{B}^{K}( g ) D_{L - M}\biggl ( E_{A}^{J}( f ) I_{A B}^{L}\biggr )
y^{2}= \prod_{k = 1}^{n_{c}}( x - \phi_{k})^{2}+ 4 \Lambda^{2 n_{c}- n_{f}}\prod_{j = 1}^{n_{f}}( x + m_{j}) , \quad S U ( n_{c}) , \, \, n_{f}\le 2 n_{c}- 2 ,
\Delta(X+Y) = \sum_{X =P+Q+R, Y=U+V+W}{L(R+W)\cdot (P+U+R+W,Q+V+R+W)}\hphantom{\Delta(X+Y)}{}\cong \sum_{X =P+Q+R}{L(R)\cdot (P+R,Q+R)\times \sum_{Y =U+V+W}{L(W)\cdot (U+W,V+W)}}\hphantom{\Delta(X+Y)}{}\cong \Delta X \times \Delta Y ,
\rho\frac{\partial\overline{u}_{i}}{\partial t}+\rho\overline{u}_{j}\frac{\partial\overline{u}_{i}}{\partial x_{j}}=\rho\overline{f}_{i}+\frac{\partial}{\partial x_{j}}[-\overline{p}\delta_{ij}+2\mu\overline{S}_{ij}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}]
\frac{h(\textbf{Y}_i^n|\Omega^n,W_{i+1},\ldots,W_K)}{r_i}- \frac{h(\textbf{Y}_{i+1}^n|\Omega^n,W_{i+1},\ldots,W_K)}{r_{i+1}}= \frac{r_{i+1}h(\textbf{Y}_i^n|\Omega^n,W_{i+1},\ldots,W_K)\!-\!r_ih(\textbf{Y}_{i+1}^n|\Omega^n,W_{i+1},\ldots,W_K)}{r_ir_{i+1}}
\chi_{n,0}(x_{1},...,x_{n}|\rho)\geq0, \underset{j}{\int_{-1}^{1}...\int_{-1}^{1}(}\prod_{s=1}^{n}\frac{1}{\pi\sqrt{1-x_{s}^{2}}})\chi_{n,0}(x_{1},...,x_{n}|\rho)dx_{1}...dx_{j}\allowbreak=\allowbreak\prod_{s=j+1}^{n}\frac{1}{\pi\sqrt{1-x_{s}^{2}}},
(v_{a}xv_{a}^{*})(v_{b}yv_{b}^{*}) =v_{a}xv_{s}v_{t}^{*}yv_{b}^{*}~~(\textrm{by Eq. \ref{isometries}}) =v_{a}v_{s}\alpha_{s}(x)\alpha_{t}(y)v_{t}^{*}v_{b}^{*}=v_{sa}\alpha_{s}(x)\alpha_{t}(y)v_{tb}^{*}=v_{c}\alpha_{s}(x)\alpha_{t}(y)v_{c}^{*}.
\frac{\partial u}{\partial t}- \int_{0}^{t}\left[\frac{\beta_1(t-s)}{3}+ \frac{2\beta_2(t-s)}{3}\right] u_{xx}(x,s) ds = f(x,t), (x,t)\in \Omega\times (0,\infty), u(x,0) = u_0(x), x\in [0,1], u(0,t) = u(1,t) = 0, t \in(0,\infty).
\int_{G}^{}\int_{0}^{\infty}1_{L_1 (t_1)}(g') dt_1 \int_{0}^{\infty}1_{L_2 (t_2)}(g'^{-1}g) dt_2 dg' = \int_{0}^{\infty}\int_{0}^{\infty}\int_{G}^{}1_{L_1 (t_1)}(g') 1_{L_2 (t_2)}(g'^{-1}g) dg' dt_1 dt_2
U(t)\mathbf 1_{(u,v)}= \left(U_1(t)\otimes U_2(t)\right)\left(\mathbf 1_u\otimes \mathbf 1_x\right) &= U_1(t)\mathbf 1_u \otimes U_2(t)\mathbf 1_x &= \gamma_1\mathbf 1_v \otimes \gamma_2\mathbf 1_y &= \gamma \mathbf 1_{(v,y)}.
P_f(A,B+L_n) =(B+L_n)^{1/2}f\bigl((B+L_n)^{-1/2}A(B+L_n)^{-1/2}\bigr)(B+L_n)^{1/2}, P_{f_\delta}(A,B+L_n) =(B+L_n)^{1/2}f\bigl((B+L_n)^{-1/2}A(B+L_n)^{-1/2}+\delta I\bigr)(B+L_n)^{1/2}.
I(z)=I^{m,n}_{p,q}\Bigg[z\left|\begin{matrix}(a_i,A_i,\alpha_i)_{1,p}(b_j,B_j,\beta_j)_{1,q}\end{matrix}\right.\Bigg]=\frac{1}{2\pi i}\int_{C}\chi(s)z^{-s}\,ds,
E_\mathfrak{c}(z; s) = \delta_{\mathfrak{c}\infty}y^s + \pi^\frac12 \frac{\Gamma\left( s - \frac12 \right)}{\Gamma(s)}\varphi_{\mathfrak{c}, 0}(s) y^{1 - s}+ 2 y^\frac12 \frac{\pi^s}{\Gamma(s)}\sum_{n \neq 0}|n|^{s - \frac12}\varphi_{\mathfrak{c}, n}(s) K_{s - \frac12}(2 \pi |n| y) e(nx).
|\nabla_x y(s,t)-I| =|(\nabla_x q(s,t;y(s,t),\xi))^{-1}-I| &\leq C|\nabla_x q(s,t;y(s,t),\xi)-I| &\leq C\rho^{\varepsilon_0}\langle s\rangle^{-\varepsilon_1}.
\begin{array}{llll}\displaystyle a=\frac 12\left(\theta^2-\frac 13\right)\lambda, \displaystyle b=\frac 12\left(\theta^2-\frac 13\right)(1-\lambda) \displaystyle c=\frac 12\left(1-\theta^2\right)\mu, \displaystyle d=\frac 12\left(1-\theta^2\right)(1-\mu), \end{array}
\boldsymbol{K}_{\rm cil}=\left[\begin{array}{@{}c|c@{}}\boldsymbol{A}_{\rm cil}\boldsymbol{0}_3 \hline \boldsymbol{0}_3 \boldsymbol{A}_{\rm cil}\end{array}\right ] , \boldsymbol{A}_{\rm cil}=\left[\begin{matrix}1 1 1 1 1 1 1 1 1\end{matrix}\right ] .
\Biggl(\sum_{j=1}^\infty j^{1/2-1/\alpha}\delta_{k, j, \alpha}\Biggr)^\alpha \le\Xi_{\alpha, p}^{\alpha/\beta}\sum_{j=1}^\infty\lambda_j^\alpha\bigl(j^{1/2-1/\alpha}\delta_{k, j,\alpha}\bigr)^\alpha.
I(v)=\int_{B'}\Phi\left(x',v(x'),\frac{\left(-\nabla_{x'}v(x'),1\right)}{\sqrt{1+|\nabla_{x'}v(x')|^2}}\right)\sqrt{1+|\nabla_{x'}v(x')|^2}\dx' +\int_{B'}\int^R_{v(x')}\partial_t\Psi\left((x',x_N),t_i\right) dx_N dx'.
\mathbb{E}[|\sum_{i=1}^N (X_i^2 - \mathbb{E}[X_i^2])a_{ii}|^p] = \mathbb{E}[|\sum_{i=1}^N (X_i^2 - \mathbb{E}[X_i^2])a_{ii}^+ - a_{ii}^-|^p] \le 2^{p-1}(\mathbb{E}[|\sum_{i=1}^N (X_i^2 - \mathbb{E}[X_i^2])a_{ii}^+|^p] + \mathbb{E}[|\sum_{i=1}^N (X_i^2 - \mathbb{E}[X_i^2])a_{ii}^-|^p]).
\begin{aligned}\max_{\mathcal{P}}Kr^*+\sum_{i=1}^K \Delta r_i \mathrm{s.t.}\sum_{i=1}^K P_i^e\leq P-P_N^*, \Delta r_i=\ln \Big(1+\frac{P_i^e}{n_i^e+ \displaystyle{\sum_{j<i}}P_j^e}\Big).\end{aligned}
(m_0 + 1) (\epsilon_0 + d -1) \leq\frac{(d-1)^2}{N-1}+ (d-1) + \frac{N -1}{4}& \leq \frac{(d-1)^2}{2}+ \frac{5 (d -1)}{4}= \frac{2d^2+d - 3}{4}.
d_{\mathbb{H}^3}(i_0(x),i_0(y)) \geq d_0(x,y_1)+d_0(y_m,y)+\sum_{j=1}^{m-1}d_0(y_j,y_{j+1})-m(2\log{(\sec{\frac{17\pi}{36}})}+1) & \geq \frac{9}{10}(d_0(x,y_1)+d_0(y_m,y)+\sum_{j=1}^{m-1}d_0(y_j,y_{j+1}))\geq \frac{9}{10}d_0(x,y).
\boldsymbol{\beta}_k = \frac{\left( \boldsymbol{\Sigma}_k + \sigma^2 \vec{I}\right)^{-1}\vec{g}_k}{\left\vert\left\vert{\left( \boldsymbol{\Sigma}_k + \sigma^2 \vec{I}\right)^{-1}\vec{g}_k}\right \vert\right \vert}.
{\cal S}={\frac{1}{2}}\mathrm{t r}\, [ \alpha \alpha^{t}] ={\frac{1}{2}}\mathrm{t r}[ a^{- 1}m b b^{t}m^{t}( a^{- 1})^{t}] ={\frac{1}{2}}m_{n}m_{k}\mathrm{t r}\, [ \alpha_{n}\alpha_{k}^{t}] = M^{t}G M ,
\nu^2(t) =\left(\sum_{x\in E}\sum_{y\in F: x\cdot y=t}1\right)^2 \leq |E| \sum_{x\in E}\left(\sum_{y\in F: x\cdot y=t}1\right)^2 &=|E| \sum_{x\cdot y=t=x\cdot y^\prime}E(x)F(y)F(y^\prime).
\hat{\lambda}_j^h h^2 = \frac{\hat{K}_{l,l}+ 2 \sum_{k=1}^p \hat{K}_{l,l+k}\cos(k \omega_j h)}{\hat{M}_{l,l}+ 2 \sum_{k=1}^p \hat{M}_{l,l+k}\cos(k \omega_j h)},
\sum_{i=0}^{n}(-1)^i\sum_{\mathrm{card}(I)=i}\left(\prod_{j\in I}x_{j}\right)\cdot A\left(\prod_{k\in \left\{1, \ldots, n+1\right\}\setminus I}x_{k}\right)=0 (x_{1}, \ldots, x_{n+1}\in R).
\big[\nabla u\big]^p_{W^{\tau,p}(B_r)}\le \frac{C\,(R/r)^N}{(\Gamma-1-\tau)\,\tau\,}\,\left(\frac{R}{h_0}\right)^{1+p}\left(\frac{h_0^{-\tau}}{(2-\Gamma)\,(\Gamma-1)}\right)^p\,\left[(1-s)\,\mathcal{M}_\gamma+h_0^{-\Gamma\,p}\,\|u\|^p_{L^p(B_{R+h_0})}\right].
K(\Lambda_1,\Lambda_2;t)) = \frac{1}{(2 \pi it)^{(m+n)/2}}\frac{1}{m!}\frac{1}{n!}det(e^{i(\lambda_{1,i}-\lambda_{2,j})^2 /2t}) det(e^{-i({\bar\lambda}_{1,\alpha}-{\bar\lambda}_{2, \beta})^2 /2t}),
\vert J_1,\gamma_1; J_2,\gamma_2\rangle = \frac{1}{\sqrt{\mathcal{N}(J_1,J_2)}}\sum_{n_1,n_2}^{\infty}\frac{J_1^{\frac{n_1}{2}}J_2^{\frac{n_2}{2}}e^{i(\gamma_1n_1 +\gamma_2 n_2)}}{\sqrt{n_1 ! n_2 !}}\vert n_1,n_2\rangle,
\{T_{\alpha}^{( 0 )}( a ) , T_{\beta}^{( 0 )}( a ) \}= T_{\gamma}^{( 0 )}( a ) U_{\alpha \beta}^{\gamma}( a ) , \quad \{H_{0}( a ) , T_{\alpha}^{( 0 )}( a ) \}= T_{\gamma}^{( 0 )}( a ) V_{\alpha}^{\gamma}( a )
\tilde u =\beta k\lambda_0 \ , \cr\tilde v =\alpha k\lambda_0 \ , \cr\tilde c =eg_\alpha(k\lambda_0,0)g_{\beta}(k\lambda_0,0)=\lambda_{-k}\lambda_k{\alpha k\lambda_0\over 1-e^{-\alpha k\lambda_0}}{\beta k\lambda_0\over 1-e^{-\beta k\lambda_0}}{c\over 12}(k^3-k) \ .