harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_B = L \int_{0}^{1}d\alpha \, \left[ 2x_B - \frac{1}{L}\sum_{j=-L/2 +1}^{L/2}\frac{2\cos (2\pi j/L)}{1+\alpha -2\alpha \cos (2\pi j/L) +\alpha gx_B}+\frac{gx_B^2}{2}-x_B \right]
(\beta V)(t,q_t)-(\beta V)(s,q_s) = \int_s^t \partial_r(\beta V)(r,q_r) dr+\int_s^t \nabla_q(\beta V)(r,q_r)\circ dq_r = \int_s^t \partial_r(\beta V)(r,q_r) dr+\int_s^t \nabla_q\beta(r,q_r) V(r,q_r)\circ dq_r+\int_s^t \beta(r,q_r)\nabla_qV(r,q_r)\circ dq_r,
\frac{1}{D(a, b)}\sum_{j=1}^{w}A(\nu-2j) = \sum_{j=1}^{w}\big(a^{\nu-2j+1}+a^{-\nu+2j-1}-b^{\nu-2j+1}-b^{-\nu+2j-1}\big) & = \frac{a^{\nu}-a^{-\nu}}{a-a^{-1}}-\frac{b^{\nu}-b^{-\nu}}{b-b^{-1}}- \frac{a^{\nu-2w}-a^{-\nu+2w}}{a-a^{-1}}+ \frac{b^{\nu-2w}-b^{-\nu+2w}}{b-b^{-1}}.
\mu_n[t+1]= \left[\mu_n[t]-\hat{\mu}_n[t]\left(\frac{T\log_2\left(1+\frac{c_n}{\frac{T}{1-T}}\right)}{d_n}-\mathcal{R}_{\mathrm{eq}}\right)\right]^+, &\,\forall n\in\mathcal{N},
\mathrm{\vert f_{n}\vert^{2}\left(x_{n}\right)=1=\underset{x\in\pi^{-1}\left(y_{n}\right)}{max}\,\left|f_{n}\right|^{2}=\underset{x\in\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|f_{n}\right|^{2}}.
f(x,y,t)=-e^{-t}x^2(1-x)^2y^2(1-y)^2 \ -\frac{2e^{-t}x^{2-\beta_1}y^2(1-y)^2}{\Gamma(3-\beta_1)}\Bigg(1 -\frac{6x}{3-\beta_1}+\frac{12x^2}{(3-\beta_1)(4-\beta_1)}\Bigg) \ -\frac{2e^{-t}x^2(1-x)^2 y^{2-\beta_2}}{\Gamma(3-\beta_2)}\Bigg(1 -\frac{6y}{3-\beta_2}+\frac{12y^2}{(3-\beta_2)(4-\beta_2)}\Bigg)
\Big|\frac{\hat\sigma^2_{n,j}(\theta)}{\sigma^2_{P,j}(\theta)}-1\Big|\le \Big|\frac{n^{-1}\sum_{i=1}^n m(X_i,\theta)^2-E[m_j(X_i,\theta)^2]}{\sigma^2_{P,j}(\theta)}\Big|+\Big|\frac{\bar m_{n,j}(\theta)^2-E[m_j(X_i,\theta)]^2}{\sigma^2_{P,j}(\theta)}\Big|.
k_{p}(\omega ) = \frac{-a_{2\sigma}\omega\cos(h\omega)+\left(\sigma a_\sigma+\omega^{2}\right)\sin(h\omega)}{b\omega e^{h\sigma}}\;, k_{i}(\omega ) = \frac{(\sigma^2+\omega^{2})\left(\omega\cos(h\omega)+a_\sigma\sin(h\omega)\right)}{b\omega e^{h\sigma}}\;.
\langle Ax,x|x|^{p-2}\rangle =-\langle \nabla x,\nabla (x|x|^{p-2})\rangle &=-(p-1)\langle \nabla x,\nabla x |x|^{p-2}\rangle &=-(p-1)\||^{\frac{p-2}{2}}\nabla x\|_{L^2}^2 &=-\frac{4(p-1)}{p^2}\|\nabla (|x|^{\frac{p}{2}})\|_{L^2}^2 &\le -\lambda_1\frac{4(p-1)}{p^2}\||x|^{\frac{p}{2}}\|_{L^2}^2 &=-\lambda_1\frac{4(p-1)}{p^2}\|x\|_{L^p}^p.
\frac{1}{|{\bf x}|}\equiv \sum_{n = - \infty}^{+ \infty}\frac{1}{\sqrt{\vec{x}^{2}+ ( \beta n )^{2}}}= 2 T \sum_{n = - \infty}^{+ \infty}K_{0}( 2 \pi T | \vec{x}{}| n ) \simeq - 2 T \ln ( \mu | \vec{x}{}| ) .
S_{e f f}= \Gamma^{\alpha \beta}\, h_{\alpha \beta}\left( 0 \right) + \frac 1 2 \int d^{4}k \ \Pi^{\mu \nu , \, \alpha \beta}\left( k \right) \h_{\mu \nu}\left( k \right) \h_{\alpha \beta}\left( - k \right) + \cdots \ ,
I_1(0) ={(23)^2\over x^4_{23}}\int du_5 \left[{(12)^2(35)\over (25)}\partial_2\cdot\partial_2 +{(13)^2(25)\over (35)}\partial_3\cdot\partial_3 + (12)(13)\; 2\partial_2\cdot\partial_3 \right]f(1,2,1,3) \ .
{\widetilde T}_{\mu \nu}= T_{\mu \nu}-{\widehat M}_{P}^{D - 4}\biggl [ - \partial^{\lambda}\partial_{\lambda}H_{\mu \nu}+ \partial_{\mu}\partial_{\nu}( \chi + g ) -{\frac{1}{2}}\eta_{\mu \nu}\partial^{\lambda}\partial_{\lambda}\chi \biggr ] .
\min_{m}\{2\phi(m;\, n_0, d)\}&\geq \min_{m}\{\phi(m;\, n_0, d-1)+\phi(m;\, n_0, d+1)\}&\geq \min_{m}\{\phi(m;\, n_0, d-1)\}+\min_{m}\{\phi(m;\, n_0, d+1)\}.
U_{11}= - \frac{a}{\phi}|\nabla u|^2 + A_{11}+ \frac{1}{\nabla_1 u}\sum_{k \geq 2}\nabla_k u A^{1k}\leq - \frac{a}{\phi}|\nabla u|^2 + C (1 + |\nabla u| + |\nabla u|^{\gamma-2}).
\sum_{r, s\in \mathbb{Z}}b_{n,r,s}^{2}\le |\Gamma_n|\sum_{r, s\in \mathbb{Z}}\sum_{(j,k)\in \Gamma_n}a^2_{j+r, k+s}= |\Gamma_n|^2\sum_{r, s\in \mathbb{Z}}a_{r,s}^{2}.
\left. T_{e f f}^{( 2 )}\right.^{2}= \cosh ( 2 \rho ) \, \cosh^{2}\left( \frac{\chi}{\sqrt 3}+ \sigma \right) \pm \sqrt{\sinh^{2}( 2 \rho ) \, \cosh^{4}\left( \frac{\chi}{\sqrt 3}+ \sigma \right) + \sinh^{4}\left( \frac{\chi}{\sqrt 3}+ \sigma \right)}\quad .
- \frac{d - 1}{d - 2}\, ( \xi^{\prime})^{2}- 2 ( d - 1 ) \, \frac{\xi^{\prime}}{r}+ \frac{( d - 2 ) ( p + 1 )}{4}\left( Y^{\prime}\right)^{2}= F^{2}\, T ( x ) \, \delta^{( 5 - p )}( \vec{x}) ~ ~ .
\begin{array}{l l}{g_{_L}( \widehat{\mathbf{f}}_{\mu}, \widehat{\mathbf{f}}_{\nu}) = g_{\mu \nu}}\\{g_{_L}( \widehat{\mathbf{f}}_{\mu}, \mathbf{e}_{k}) = 0}\\{g_{_L}( \mathbf{e}_{k}, \mathbf{e}_{l}) = g_{0 k l},}\\ \end{array}
\varTheta_p(\eta, 1,\chi) \ll X^{-1}(X^{1/2}\log X)((X \log X)^{(1+\eta)/2}+ \eta X) & \qquad\qquad \max_{t \le 2(X\log X)^{1/2}}\left|S_p(\chi;t)\right| + 1.
\sum_{r,q}s_r s_q = \sum_{r,q \notin \{i,j\}}s_r s_q + s_i \sum_{r\notin \{i,j\}}s_r +s_j \sum_{r\notin \{i,j\}}s_r + s_i s_j < \underbrace{\sum_{r,q \notin \{i,j\}}s_r s_q + (s_i+1)\sum_{r\notin \{i,j\}}s_r +(s_j-1) \sum_{r\notin \{i,j\}}s_r + (s_i+1)(s_j -1)}_{=A}- 1.
\partial^{k}_{(j_1,...,j_k)}(P(Q_1,...Q_n))=\sum_{l=2}^k\sum_{n_1,...,n_l}\sum_{1\leq i_1<...<i_l=k}(\partial_{(n_1,...,n_l)}^{l}(P))(Q_1,...Q_n)\#(\partial_{(j_1,...,j_{i_1})}^{i_1}Q_{n_1},\partial_{(j_{i_1+1},...,j_{i_2})}^{i_2-i_1}Q_{n_2},...,\partial_{(j_{i_{l-1}+1},...,j_{k})}^{k-i_{l-1}}Q_{n_l}).
\Big\|\sum_{i}(x_i \cdot (x_i \mid x)_{C_0(E^0)}) - x\Big\|^2 = \sup_{v \in E^0}\sum_{s(e)= v}\Big|\sum_{i}(x_i \cdot (x_i \mid x)_{C_0(E^0)}) (e) - x(e)\Big|^2 \to 0.
\theta_L(z,w)=\frac{K_L(z,w)}{\sqrt{K_L(z,z)}\sqrt{K_L(w,w)}}=\frac{[1-|z|^2]^{L/2}[1-|w|^2]^{L/2}}{[1-\bar z w]^L}=\prod_{j=1}^n 1-\left|\frac{z_j-w_j}{1-z_j\bar w_j}\right|^2\ .
\vert\mathcal I_{1,2}\vert \le C_2\Vert\partial_1\dot{q}^+\Vert_{L^2(\Omega_t)}\left(\Vert\dot{H}^+_n\Vert_{L^2(\Omega_t)}+\Vert\partial_2\dot{H}^+_n\Vert_{L^2(\Omega_t)}\right) &\le C_2\left\{\int_0^t(I_{1,\ast}+I_{1,n})(s)ds\right\}\,.
\| \eta_T(t) e^{it\Delta}u_0 \|_{X^{0,b}}\lesssim T^{\frac12 - b}\| u_0\|_{L^2}\| \eta_T(t) e^{\pm itD}\phi \|_{Y^{s_1, b}}\lesssim T^{\frac12 - b}\Bigl(\| n_0\|_{H^{s_1}}+ \|n_1\|_{H^{s_1-1}}\Bigr).
\xi^{2}(t)=m\xi_{10}\left[ \cos \left( \frac{\sqrt{V^{\prime \prime}\left( x_{0}\right)}}{\sqrt{m}}t\right) -1\right] +\frac{\sqrt{m}\xi_{20}}{\sqrt{V^{\prime \prime}\left( x_{0}\right)}}\sin \left( \frac{\sqrt{V^{\prime \prime}\left( x_{0}\right)}}{\sqrt{m}}t\right) .
\left(\int_{2cQ}|\nabla Ef(x) |^s \,\d x \right)^\frac{p}{s}\ge \left( C (2c)^{n-s}\ell(Q)^{n-s}\right)^\frac{p}{s}= C' \ell(Q)^{n - p}\ell(Q)^{(\frac{p}{s}- 1)n}
(\pi^*a)(\pi^*b) = \tfrac1{k!}g(a, v^k) \tfrac1{l!}g(b, v^l) = g(a_1, v) \dotsm g(a_k, v) g(b_1, v) \dotsm g(b_l, v) = \tfrac1{(k + l)!}g(a \cdot b, v^{k + l}) = \pi^*(a \cdot b),
F_T(x)=\frac{2 - 11 x + 13 x^2 - 6 x^3 + (1 - x) x (1 - 6 x + 4 x^2) (1 - 4 x)^{-1/2}}{2 (1 - 6 x + 8 x^2 - 4 x^3)}\, .
P(Y_n=j, S_n \le t|Y_0=i_0,S_1=s_1,\ldots,S_{n-1}= s_{n-1},Y_{n-1}=i_{n-1}) =P(Z_{T_n}=j, U_{T_n-}\le t|Y_0=i_0,S_1=s_1,\ldots,S_{n-1}= s_{n-1},Y_{n-1}=i_{n-1}) =P(Z_{T_n}=j, U_{T_n-}\le t|U_{T_{n-1}}= 0,Z_{T_{n-1}}=i_{n-1}) =P(Y_n=j, S_n \le t|Y_{n-1}=i_{n-1}),
2^{\nu-\frac{1}{2}}z^{-\nu-\frac{1}{2}}\mathbf{\Phi}_{\nu}(\sqrt{2z})=\sum_{n\geq0}\frac{(-1)^n\left(\frac{z}{2}\right)^{2n}}{n!\Gamma(\nu+n+1)\Gamma(\nu+2n+2)}.
{}\frac{1}{|a_n|^{\frac{1}{2}+\epsilon^2}}\geq \frac{1}{|a_n|^{\frac{1+\epsilon}{2+\epsilon}}}=\frac{\;\;\;\,|a_n|^{\frac{1}{2+\epsilon}}}{|a_n|\;}\geq\frac{|b_n|}{|a_n|}=\rho(\phi^n(\alpha),\infty).
\sigma_{AB}(x,\xi) \sim u_{\xi}^{-1}(x)\int_{\Omega}K_{A}(x,y) \Big[\sum_{\alpha\geq 0}\frac{1}{\alpha!}q^{\alpha}(x,y)D^{(\alpha)}_{x}\sigma_{B}(x,\xi)\Big]u_{\xi}(z)dy &=\sum_{\alpha\geq 0}\frac{1}{\alpha!}\Big[ u_{\xi}^{-1}(x)\int_{\Omega}q^{\alpha}(x,y) K_{A}(x,y) u_{\xi}(y) dy \Big]D^{(\alpha)}_{x}\sigma_{B}(x,\xi)
\sum_{n=M+1}^{N}{\frac{a_{n}}{n^{L + \varepsilon+s}}}= \Big( \sum_{k=1}^{N}{\frac{a_{k}}{k^{s}}}\Big) \frac{1}{N^{L+\varepsilon}}- \Big( \sum_{k=1}^{M}{\frac{a_{k}}{k^{s}}}\Big) \frac{1}{M^{L+\varepsilon}}+ \sum_{n=M}^{N-1}\Big( \sum_{k=1}^{n}{\frac{a_{k}}{k^{s}}}\Big) \Big[\frac{1}{n^{L+\varepsilon}}- \frac{1}{(n+1)^{L+\varepsilon}}\Big] \,
A^h(\xi,\eta) = \frac{2i\eta J(\xi)\left(J(\zeta) - J(\xi) + J(\eta)\right)}{\Omega}, B^h(\xi,\eta) = -\frac{2i\zeta J(\xi)J(\eta)}{\Omega}, C^h(\xi,\eta) = -\frac{i\xi\eta\zeta\left(J(\zeta) - J(\xi) - J(\eta)\right)}{\Omega}.
\varphi\left( \frac{x}{x-1}\right) = C (-1)^m x^k (x - 1)^{q}(x - 2)^m \prod\limits_{i=1}^r (1 - \lambda_i)^{n_i}\left( x - \frac{\lambda_i}{\lambda_i - 1}\right)^{n_i}.
( x - x^{\prime})^{2}\, \int \! \mathrm{d}\rho^{2}\mathrm{d}^{4}x_{0}\, \frac{\rho^{8}}{\left[ ( x - x_{0})^{2}+ \rho^{2}\right]^{4}\left[ ( x^{\prime}- x_{0})^{2}+ \rho^{2}\right]^{4}}\, .
\langle\eta,\gamma_{X_{l}(t,x)}\eta\rangle_{\mathbb{R}^{N}}=\left\Vert D\left(\langle\eta,X_{l}(t,x)\rangle_{\mathbb{R}^{N}}\right)\right\Vert_{\bar{{\cal H}}}^{2}=\left\Vert D\left(\langle\eta,X_{l_{0}}(t,x)\rangle_{\mathbb{R}^{N}}\right)+D\left(\langle\eta,R_{t}\rangle_{\mathbb{R}^{N}}\right)\right\Vert_{\bar{{\cal H}}}^{2}.
\mathrm{Mant}_{G, b_1, \mu}( \mathrm{Red}_{b_1}(\pi))=\mathrm{Mant}_{G, b_1, \mu}(LJ(\delta^{\frac{1}{2}}_{P_{b_1}}\otimes J^G_{P^{op}_{b_1}}I^G_{M_{b_1}}(\rho_1 \boxtimes \rho_2))).
u_0 =\frac{(\alpha-1)(\beta-1)}{1-\alpha\beta}, u_{2j}=(-1)^j\cos(2j\theta)+O(R_0^{-2j}) \{|x|=R_0\},\,\,j\ge 1, u_{2j+1}=(-1)^j\sin(2j+1)\theta+O(R_0^{-2j-1})\{|x|=R_0\},\,\,\, j\ge 0.
{\sf M}\left\{\left(J^{*}[\psi^{(4)}]_{T,t}-\sum\limits_{j_1, j_2, j_3, j_4=0}^{p}C_{j_4 j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}\zeta_{j_4}^{(i_4)}\right)^2\right\}\le \frac{C}{p^{1-\varepsilon}}
\sum_{\substack{P^+(A)\le y \Omega(A) = J}}\frac 1A < z^{-J}\sum_{\substack{P^+(A)\leq y}}\frac{z^{\Omega(A)}}{A}= z^{-J}\left(1-\frac{z}{2}\right)^{-1}\prod_{3 \le p\leq y}\left(1-\frac{z}{p}\right)^{-1}\ll \frac{z^{-J}}{2-z}\exp\left(z\sum_{3 \le p \le y}\frac1p \right) \ll \frac{z^{-J}}{2-z}(\log y)^z.
\mbox{Med-bias}_{\theta_0}(\widehat{\theta}_n) = \left(\frac{1}{2}- \max\left\{\mathbb{P}\left(\sum_{i=1}^n \widehat{R}_{T,i}(\widehat{R}_{Y,i}- \theta_0\widehat{R}_{T,i}) \le 0\right),\,\mathbb{P}\left(\sum_{i=1}^n \widehat{R}_{T,i}(\widehat{R}_{Y,i}- \theta_0\widehat{R}_{T,i}) \ge 0\right)\right\}\right)_+.
(i)|\lambda_\mathbf{d}|\ll k^{-k}(\log{R})^k, &(ii)w_n\ll k^{-2k}(\log{x})^{2k}\prod_{i=1}^k\prod_{p|L_i(n),p\nmid B}4, &(iii)w_n\ll R^{2+o(1)}.
\|\Delta(|A(\eta_1+\eta_2)|) +\eta_1+\eta_2\|_X =\|\Delta(|A(\eta_2-\eta_1)|)-(\eta_2-\eta_1)+2\eta_2\|_X \geq2\|\eta_2\|_X-\|\Delta(|A(\eta_2-\eta_1)|)-(\eta_2-\eta_1)\|_X.
\langle P_gG_p,u \rangle =\kappa_g u(p) - \kappa_g \bar{u}+ \langle P_gU,u \rangle , &=\kappa_g u_p - \kappa_g\bar{u}- \langle Q_g,u \rangle + \langle \frac{\kappa_g}{\mathrm{vol}_g(M)},u \rangle =\kappa_g u_p - \langle Q_g,u \rangle.
\mathfrak{J}_2(\{\pmb{F}_k\}_{k=1}^K, \pmb{U}_{K+1}) = \sum\limits_{k=1}^{K}||\pmb{H}_{K+1,\ell}\pmb{F}_\ell - \pmb{U}_{K+1}\pmb{U}_{K+1}^H\pmb{H}_{K+1,\ell}\pmb{F}_\ell ||_\emph{F}^2
- \Bigl[ \left[ Q(\xi) , Q(\eta) \right] , b (k, \sigma) \Bigr] = \Bigl[ \left[ Q(\eta) , b (k, \sigma) \right] , Q(\xi) \Bigr] + \Bigl[ \left[ b (k, \sigma) , Q(\xi) \right] , Q(\eta) \Bigr]
\| \chi \left((S_\varepsilon -I) b({\mathbf D}) \widetilde{\mathbf u}_0\right) (\cdot,t) \|_{H^l({\mathbb R}^d)}\leqslant\| \chi \left(S_\varepsilon b({\mathbf D}) \widetilde{\mathbf u}_0\right) (\cdot,t) \|_{H^l({\mathbb R}^d)}&+\| \chi b({\mathbf D}) \widetilde{\mathbf u}_0 (\cdot,t) \|_{H^l({\mathbb R}^d)}.
\left\{\frac{\gamma}{4 N}\left( \sum_{1}^{N}- \frac{\partial^{2}}{\partial \alpha_{i}^{2}}- N ( N^{2}- 1 ) / 1 2 \right) + V ( \alpha ) \right\}\tilde{\psi}( \alpha ) ={\cal E}\tilde{\psi}( \alpha ) \; .
\psi_{\pi \varphi \ldots}( \stackrel{\rightharpoonup}{x_{1}}, \ldots , \stackrel{\rightharpoonup}{x_{N}}) \rightarrow ( - 1 )^{\# \pi - \mathrm{i n d i c e s}}\psi_{\pi \varphi \ldots}( - \stackrel{\rightharpoonup}{x_{1}}, \ldots , - \stackrel{\rightharpoonup}{x_{N}})
\frac{n}{2}\langle V_{TOT}\rangle_{\tau}=\langle\sum_{k=1}^{N}(\frac{1+\sqrt{1-\beta_{k}^{2}}}{2})T_{k}\rangle_{\tau}=\langle\sum_{k=1}^{N}(\frac{\gamma_{k}+1}{2\gamma_{k}})T_{k}\rangle_{\tau}.
\int_{\Omega}(\vec{b}\cdot \nabla u) \, v = \frac{1}{2}\left[\int_{\Omega}(\vec{b}\cdot \nabla u)v-\int_{\Omega}(\vec{b}\cdot \nabla v)u+\int_{\Omega}\nabla \cdot \vec{b}\, u \, v \right] \forall\u,v \in H^{1}_{0}(\Omega)
\langle \cdots \rangle_{\mid A}= \frac{1}{Z ( A )}\int \mathrm{~ \cal{D}~}_{g}\phi \mathrm{~ \cal{D}~}_{g}\sigma \, d \Lambda \, \delta \left( 1 - \frac{1}{A}\int d^{2}z \sqrt{g}\right) \, \delta \left( \Lambda A + 4 \pi \right) \, e^{- S_{g}}\left( \cdots \right) .
L_{\alpha}^{\prime ij}=X^{(1)}(L_{\alpha}^{ij})+\operatorname{div}(X^{\prime})L_{\alpha}^{ij}+A_{\alpha}^{ij}, L_{0}^{\prime}=X^{(1)}(L_{0})+\operatorname{div}(X^{\prime})L_{0}+T_{hk}^{\beta}L_{\beta}^{hk},
\begin{aligned}b_3-m_3 =\Upsilon'(0)\{M_x[g(a_1-1)-g(0)]-M_{\varphi, x}[g(a_1-1)-g(0)] \}begin{align*}1.5mm] +\Upsilon_{\geq 2}[M_x[g(a_1-1)-g(0)]]-M_{\varphi}[\Upsilon_{\geq 2}[M_x[g(a_1-1)-g(0)]]].\end{aligned}
\hat B_{x;\hat k,\hat l}+ \hat B_{x;\hat n,\hat k}+ \hat B_{x;\hat l,\hat n}+ \hat B_{x+\hat n;\hat l,\hat k}+ \hat B_{x+\hat l;\hat k,\hat n}+ \hat B_{x+\hat k;\hat n,\hat l}= 0 \ .
\sum_{\substack{k\geq1 \gcd(k,D)=1}}f_t(k)=\prod_{p\nmid D}\left(1+\sum_{\alpha\ge1}f_t(p^\alpha)\right)=\prod_{p\nmid D}\left(1-\sum_{\alpha\ge1}\frac{p^t-1}{p^{\alpha(t+1)-1}|\Gamma(p^\alpha)|(p-1)}\right).
\alpha_{0}>\underline{\alpha}\underline{q}<q<\overline{q},\underline{\alpha}=\underline{\alpha}\left( \beta_{0},2\right) ,~\underline{q}=\underline{q}\left( \alpha_{0},\beta_{0},2\right) ,~\overline{q}=\overline{q}\left( \alpha_{0},\beta_{0},2\right) .
\mathcal{F}_{n}\varphi:=\frac{1}{(2\pi)^{n/2}}\int\limits_{\mathbb{R}^n}\mathrm{e}^{-\mathrm{i}\langle\cdot,\mathbf{x}\rangle}\varphi(\mathbf{x}) \mathrm{d}\mathbf{x}.
\frac{(1+\lambda)^{n+1}}{\lambda}=\frac{\left(1+\lambda^{1-\frac{n+1}{j-1}}\lambda_{n,j}(\lambda)^{\frac{n+1}{j-1}}\right)^{n+1}}{\lambda^{1-\frac{n+1}{j-1}}\lambda_{n,j}(\lambda)^{\frac{n+1}{j-1}}}.
\sum_{n=1}^{\frac{m-1}{2}}[3n]\frac{(1-q)^2(q;q^2)_{n}(q^{m+2};q^2)_{n-1}^2 q^{m-{n+1\choose 2}-\frac{(2n+1)(m-1)}{2}}}{(q;q)_{n}^2 (q^2;q^2)_{n}}\equiv 0\pmod{\Phi_m(q)}.
\partial_s \zeta(t,s) = \frac{1}{\partial_s \Sigma(t,\zeta(t,s))}\;\; \partial_t \zeta(t,s) = - \frac{\partial_t \Sigma(t,\zeta(t,s))}{\partial_s \Sigma(t,\zeta(t,s))}\ , (t,s)\in (0,\infty)^2\ .
\gamma_{w_{0}}(G/P)^{-2}\mu(s, \tau, w_{0}) ={\gamma(s, \pi, r_0 \boxtimes \eta, \psi) \gamma(-s, \pi^\vee, r_0 \boxtimes \eta^{-1}, \bar{\psi})}= \gamma(s, (r_0 \circ \sigma) \boxtimes \eta, \psi) \gamma(-s, (r_0 \circ \sigma^\vee) \boxtimes \eta^{-1}, \bar{\psi}).
\mathbb{E}_0\left[\left( \bar{U}^{(12)}_h\right)^4\right] = \mathbb{E}_0\left[\left(\big(U^{(12)}_h\big)^2- \mu_h \right)^4\right] = \mathbb{E}_0\left[\left(U^{(12)}_h\right)^8 - 4 \mu_h \left(U^{(12)}_h\right)^6 + 6 \mu_h^2\left(U^{(12)}_h\right)^4 - 4 \mu_h^3 \left(U^{(12)}_h\right)^2 + \mu_h^4\right]
\ \tau \phi\big(\exp(H+\sum_{j=1}^mp_j\log A_j)\big) + (1-\tau)\phi\big(\exp(H+\sum_{j=1}^mp_j\log B_j)\big) \leq \ \sum_{j=1}^m\frac{p_j}{r}\Big(\tau\phi\big(\exp(L+r\log A_j-r\log C_j)\big) + (1-\tau)\phi\big(\exp(L+r\log B_j-r\log C_j)\big)\Big) \leq \ \sum_{j=1}^m\frac{p_j}{r}\phi\big(\exp(L)\big) = \ \phi\big(\exp(H+\sum_{j=1}^mp_j\log C_j)\big),
e^{\eta t}||e^{tA}y||_s =e^{\eta t}\left|\left|\sum_{i=1}^k\alpha_ie^{t\mu_i}h_{\mu_i}\right|\right|_s\le e^{\eta t}\left(\sum_{i=1}^ke^{t\Re(\mu_i)}||\alpha_ih_{\mu_i}||_s\right) & < e^{t(\eta+\delta_y)}\left(\sum_{i=1}^k||\alpha_i h_{\mu_i}||_s\right)
u(r,t)=Re\{\sum_{n=0}^{N}\frac{iP_{n}^{\prime}}{\rho n\omega}[1-\frac{J_{0}(\alpha n^{1/2}i^{3/2}\frac{r}{R})}{J_{0}(\alpha n^{1/2}i^{3/2})}]e^{in\omega t}\},
\frac{1}{128 \pi^4}\int_{C}\psi(\chi_{\rho_u}(\omega_1,\omega_2)) J_{x,y}(\omega_1,\omega_2)^2 \mathrm{d}\omega_1 \, \mathrm{d}\omega_2 = \frac{1}{16 \pi^4}\int_{\mathfrak{D}_{x,y}}\psi(u) |J_{x,y}(x,y)| \mathrm{d}x \, \mathrm{d}y,
\begin{array}{l l l}{{H \left| \Omega ( k , k + 1 ) \right\rangle}}&{{=}}&{{\left( Q + Q^{- 1}\right) \left| \Omega ( k , k + 1 ) \right\rangle + \left| \Omega ( k - 1 , k + 1 ) \right\rangle}}\\{{}}&{{}}&{{}}\\{{}}&{{+}}&{{\left| \Omega ( k , k + 2 ) \right\rangle}}\\ \end{array}
a_i(k_1)=\frac{k_2(\alpha_i-x)-(\gamma_i-Nx)}{k_2-k_1}=(\alpha_i-x)\frac{k_2-\frac{\gamma_i-Nx}{\alpha_i-x}}{k_2-k_1}=(\alpha_i-x)\frac{k_2-k}{k_2-k_1}.
| \psi > = \int_{- \infty}^{+ \infty}d A_{1}\, | A_{1}> < A_{1}| \psi > = \int_{- \infty}^{+ \infty}d A_{1}\, | A_{1}> \, \psi ( A_{1}) \ \ \ , \ \ \ \psi ( A_{1}) \equiv < A_{1}| \psi > \ \ \ ,
\frac{d}{d t}\biggl |_{t = 0}Z ( R ( S ) , \langle \cdot \, , \cdot \rangle^{A ( t )}) = \biggl ( \sum_{k = 0}^{N}\frac{1}{2}( - 1 )^{k}T r ( B_{k}) \biggr ) \, Z ( R ( S ) , \langle \cdot \, , \cdot \rangle ) \, .
N = (\theta_1|_F,\ldots,\theta_{d}|_F) \cdot M + (\theta_{j}|_{F}x^G: \enskip G \subset F \smallsetminus \mathcal{A}_F, \enskip \sigma(G \sqcup \mathcal{A}_F \sqcup E) = \{v_j\}^c).
\begin{array}{c}I=0,\,\alpha_i,\,\alpha_{i_1}\alpha_{i_2},\,\ldots,\,\alpha_{i_1}\!\cdots\alpha_{i_k},\,\ldots\,,\quad i_1<i_2<\cdots<i_k, e^0=1,\quad e^{\alpha_{i_1}\alpha_{i_2}\cdots\alpha_{i_k}}=\xi^{\alpha_{i_1}}\xi^{\alpha_{i_2}}\cdots\xi^{\alpha_{i_k}},\quad k=0,1, \ldots, n.\end{array}
\mathrm{s c h}V ( \Lambda ) \left( \mathrm{c h ~}S^{+}- \mathrm{c h}S^{-}\right) = \mathrm{s c h}\left( \wedge ( g_{- 1}/ r_{- 1}) \right) \sum_{c \in C}\mathrm{s g n}( c ) \mathrm{s c h}U ( c \bullet \Lambda ) .
f_{{\bf i},{\bf j}}:= \sum_{\substack{1 \leq p < q \leq m 1 \leq p' < q' \leq m}}\mathbb{E}[h^W_\tau ({\bf X}^{(pq)}_{i_1},\dots,{\bf X}^{(pq)}_{i_4}) h^W_\tau ({\bf X}^{(p'q')}_{j_1},\dots,{\bf X}^{(p'q')}_{j_4})] \geq 0,
f\left((z,h)+(z',h')\right) =f(z+z'-\beta(h,h'),h+h')= &=z+z'-\beta(h,h')+\alpha(h+h')= =z+z'+\alpha(h)+\alpha(h')= &=z+\alpha(h)+z'+\alpha(h')=f(z,h)+f(z',h').
\int_{0}^{\infty}{\frac{d s}{s}}\, G ( s ) ={\frac{1}{\pi}}\int_{0}^{\infty}d \lambda \,{\frac{\Im \, \widetilde E ( \lambda )}{\lambda^{2}}}= c_{1}={\frac{3}{1 6}}( 2 K^{2}+ 2 K + 1 ) ,
\|e_{m+1}\|^2 &\le \exp\Bigg(-(a-2\epsilon)(m+1)\tau +\int_{0}^{t_{m+1}}\|u(s)\|_{L_{\infty}}\|u_\tau^D(s)\|_{L_{\infty}}ds\Bigg)\|e_0\|^2 &\quad+\sum_{k=1}^{m+1}\sum_{i=0}^6\exp\Bigg(-(a-2\epsilon)(m+1-k)\tau +\int_{t_k}^{t_{m+1}}\|u(s)\|_{L_{\infty}}\|u_{\tau,m}^D(s)\|_{L_{\infty}}ds\Bigg)R_i^{k-1}.
d s^{2}= - H_{+}H_{-}^{2{\frac{d - 2}{\Delta}}- 1}d t^{2}+ H_{-}^{2{\frac{d - 2}{\Delta}}}\left( d y_{1}^{2}+ \dots + d y_{p}^{2}\right) \qquad \qquad \qquad \qquad
\sigma \epsilon^{p - 3}= \Phi \! \! \left( \! \! \begin{array}{c c}{{2}}&{{1}}\\{{1}}&{{3}}\\ \end{array}\! \! \right) = \Phi \! \! \left( \! \! \begin{array}{c c}{{p - 3}}&{{p - 3}}\\{{2}}&{{1}}\\ \end{array}\! \! \right) = \sigma^{p - 4}\bar{\sigma}^{p - 2}
\left\lVert{f}\right\rVert_{\mathrm{F}(X,Y)}:=\max\left(\underset{t\in\mathbb{R}}{\mathrm{sup}}\left\lVert{f(it)}\right\rVert_{X},\underset{t\in\mathbb{R}}{\mathrm{sup}}\left\lVert{f(1+it)}\right\rVert_{Y}\right)
m_{h}= \left( \begin{array}{c c c c}{0}&{0}&{\xi_{1}}&{\xi_{4}}\\{0}&{0}&{\bar{\xi}_{4}}&{\bar{\xi}_{1}}\\{\xi_{1}}&{\bar{\xi}_{4}}&{\bar{\Phi}_{1 2}}&{\phi_{2}}\\{\xi_{4}}&{\bar{\xi}_{1}}&{\phi_{2}}&{\Phi_{1 2}}\\ \end{array}\right) ,
\int_{\mathbb{X}\times \lbrack 0,T]}\ell (\varphi^{\varepsilon})d\nu_{T}\leq \kappa_{3}\int_{\mathbb{X}\times \lbrack 0,T]}(\varphi^{\varepsilon}-1)^{2}d\nu_{T}& =a^{2}(\varepsilon )\kappa_{3}\int_{\mathbb{X}\times \lbrack 0,T]}|\psi^{\varepsilon}|^{2}d\nu_{T}& \leq a^{2}(\varepsilon )M,
||f(z_0)-a||= || \int F(\gamma^{-1}z_{0},z_{0})\big(\alpha_{\gamma}\phi(s(\gamma))-a\big)d\lambda^{r(z_{0})}|| \leq \int_{\gamma \in U}F(\gamma^{-1}z_{0},z_{0})||\alpha_{\gamma}(\phi(s(\gamma))-a)||d\lambda^{(r(z_0))}\leq \epsilon \int F(\gamma^{-1}z_{0},z_{0})d\lambda^{r(z_0)}\leq \epsilon
R^{\lambda\nu}=\epsilon^{\lambda\nu\sigma\tau}F_{\sigma\tau}=\frac{1}{2}\left( \epsilon^{\lambda\nu\sigma\tau}-\epsilon^{\lambda\nu\tau\sigma}\right) F_{\sigma\tau}& =\frac{1}{4}\left( \epsilon^{\lambda\nu\sigma\tau}-\epsilon^{\lambda\nu\tau\sigma}+\epsilon^{\nu\lambda\tau\sigma}-\epsilon^{\nu\lambda\sigma\tau}\right) F_{\sigma\tau}
0 = \tilde{\pi}(\underline{f}) = \sum\limits_{i=0}^n (-1)^i y^{n-i}x^i f'_i = (-1)^{j_0}y^{n-j_0}x^{j_0}h(y) + \sum\limits_{i=j_0+1}^n (-1)^i y^{n-i}x^i f'_i = x^{j_0}\left((-1)^{j_0}y^{n-j_0}h(y) + \sum\limits_{i=j_0+1}^n (-1)^i y^{n-i}x^{i-j_0}f'_i \right).
\mathbf{\mathbf{e}}^{a}\mathbf{=}\theta^{a}-\tau^{a}\mathbf{m}-\overline{\tau}^{a}\overline{\mathbf{m}}+\zeta^{a}\mathbf{m}\overline{\mathbf{m}}; \mathbf{A}_{b}^{a}=\alpha_{b}^{a}-\beta_{b}^{a}\mathbf{m}-\overline{\beta}_{b}^{a}\overline{\mathbf{m}}+\gamma_{b}^{a}\mathbf{m}\overline{\mathbf{m}}.
J_{2,4}=\int_{3\pi -\theta_{0}}^{2\pi}\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\theta}{|\rho \sin\theta+1|^{\tau}\rho^{q-1}}= \int_{\pi}^{\theta_{0}}\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\varphi}{|\rho \sin\varphi+1|^{\tau}\rho^{q-1}}=J_{2,1}.
| \Omega_{\theta}> = \sum_{\ell = - \infty}^{+ \infty}e^{i \ell \theta}e^{- \sum_{n = 1}^{\infty}A_{+ , n}^{\dagger}A_{- , n}^{\dagger}}| n_{+}= \ell , n_{-}= \ell ; 0 >_{\Phi}\ ,
\begin{aligned}\nu(\xi) =\frac{(n-1)\cos^2(\xi)}{n}\Big\{(1-\xi^2)[(\tan(\xi)-\sqrt{n-1})^2+\sec^2(\xi)]+ -n\xi(\tan(\xi)-\sqrt{n-1})+(n-2)(1-\xi^2)\tan(\xi)(\tan(\xi)-\sqrt{n-1})\Big\}+ -\frac{n-2}{n}\mu(\xi), \end{aligned}
A(z) = Q_{1}^{(1,1)}(z) -Q_{0}^{( 2,2)}(z) =\frac{1}{2}\int_{-1}^{z}d\zeta \big( \big( 1-\zeta^{2}\big)^{2}-4\zeta^{2}\big( 1-\zeta^{2}\big) \big) \hphantom{A(z)}{}=\frac{1}{2}\int_{-1}^{z}d\zeta \big( 1-6\zeta^{2}+5\zeta^{4}\big) =\frac{1}{2}z\big( 1-z^{2}\big)^{2}
\max_{B^{\Sigma}_r(p)\times [-\eta, \eta]}u_k\leq C\max_{\partial B^{\Sigma}_r(p)}u_k, \max_{\partial B^{\Sigma}_r(p)}u_k>0, \max_{B^{\Sigma}_r(p)\times [-\eta, \eta]}u_k\leq 0,\quad\quad \max_{\partial B^{\Sigma}_r(p)}u_k\leq 0,
\hat{y}_{j}=\frac{l_{j}^{y}}{l_{1}^{y}}\frac{\sum\limits_{i=1}^m \sum\limits_{k=1}^p{e_{i1k}x_{i}z_{k}}}{\sum\limits_{i=1}^m \sum\limits_{k=1}^p{e_{ijk}x_{i}z_{k}}},
\Sigma^{+}=W^{+}=\{\alpha_{1}, \alpha_{2}, \alpha_{1}+\alpha_{2}, 2\alpha_{1}+\alpha_{2}, 3\alpha_{1}+\alpha_{2}, 3\alpha_{1}+2\alpha_{2}\}, &\langle \alpha_{1}, \alpha_{1}\rangle=1, \ \langle \alpha_{1}, \alpha_{2}\rangle =-\frac{3}{2}, \ \langle \alpha_{2}, \alpha_{2}\rangle=3, tilde{\alpha}=3\alpha_{1}+2\alpha_{2},
a_{j_1,\dots, j_k}^{(k)}=\frac{\big\langle f_k,:\mathfrak{Re}\langle \mathfrak{e}_{j_1},\cdot\rangle\dots\mathfrak{Re}\langle \mathfrak{e}_{j_k},\cdot\rangle:\big\rangle_{L^2(m)}}{\mathrm{var}_m\big[:\mathfrak{Re}\langle \mathfrak{e}_{j_1},\cdot\rangle\dots \mathfrak{Re}\langle \mathfrak{e}_{j_k},\cdot\rangle:\big]}
d_1(\varphi(x_1,y),\varphi(x_2,y)) = f(\varphi(x_1,y))=|f(\varphi(x_1,y))-f(\varphi(x_2,y))| & = |f\otimes 1(\varphi(x_1,y),\tau(y))-f\otimes 1(\varphi(x_2,y),\tau(y))| & = |(U(f\otimes 1))(x_1,y)-(U(f\otimes 1))(x_2,y)| & \le L(U(f\otimes 1))d_2(x_1,x_2).
\mathcal{T}_\star^{p,r}:= \underbrace{\Omega_\star\otimes_{\mathcal{X}_\star}\ldots\otimes_{\mathcal{X}_\star}\Omega_\star}_{p}\otimes_{\mathcal{X}_\star}\underbrace{\Xi_\star\otimes_{\mathcal{X}_\star}\ldots\otimes_{\mathcal{X}_\star}\Xi_\star}_{r}
\mathcal{A}= \mathcal{F}\oplus \mathcal{C}\mathcal{F}\cdot \mathcal{F}\subseteq \mathcal{F}, \mathcal{F}\cdot \mathcal{C}\subseteq \mathcal{C}, \mathcal{C}\cdot \mathcal{F}\subseteq \mathcal{C}, \mathcal{C}\cdot \mathcal{C}\subseteq \mathcal{F}
0= d (L_p(\kappa,s(\kappa))) ={\log_p(u)}^{-1}\left.\frac{\textup{d}}{\textup{d}s}L_p(\kappa,s)\right\vert_{s=s(\kappa)}d({s(\kappa)}) + \left.d(L_p(\kappa,s))\right\vert_{s=s(\kappa)}.
S_{n}=2M_{n-1}^{(3,2)}=2M_{n-1}^{(3+\sqrt{2}-\sqrt{2},2)}=\sum_{k=0}^{n-1}(-1)^k2^{\frac k2+1}(3+\sqrt{2})^{n-1-k}\binom{n-1}{k}M_{k}.
\begin{array}{rrl}v_{RLT_1}:= \underset{x \in \mathbb{R}^{m}, X \in \mathcal{S}^{m}}{\min}\langle Q,X \rangle begin{align*}1ex] (x,X) \in \mathcal{F}begin{align*}1ex] e - x \geq 0 begin{align*}1ex] J - xe^{\mathrm T}- ex^{\mathrm T}+ X \geq 0 begin{align*}1ex] xe^{\mathrm T}- X \geq 0,\end{array}
\sup_{0\leq t\leq \varepsilon}\| \varphi(t)\|_{\infty}= \sup_{0\leq t\leq \varepsilon}\|I_{1}(t) + I_{2}(t) + I_{3}(t)\|_{\infty}begin{align*}.3cm] \leq \|\varphi_o\|_{\infty}+ \varepsilon\,\sup_{0 \leq s \leq T}\|f_{+}(s)\|_{\infty}+ \|\nabla v\|_{\infty}\,\sqrt{\varepsilon}\, \sup_{0 \leq s \leq T}\|\varphi(s)\|_{\infty}\,.