harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
k_{0}(u,\omega):=\mathcal{J}\setminus\{u\}\cup\{\omega\},k_{0}(u,\mathfrak{g}_{c})=k_{0}(\mathfrak{g}_{c},u):=\mathcal{J}\setminus\mathfrak{g}_{r}\cup\{\alpha_{1},i_{3},u\}
R_n(\bar z)^* H_{1,n}^{-1}\left\{b u_{1,n}v_n^* - v_n u_{3,n}^* \right\}\Sigma_{3,n}= -R_n(\bar z)^* (I-bT_n^*) \Sigma_{3,n}= -(I-bT_n^*) R_n(\bar z)^* \Sigma_{3,n}.
\lim_{M\rightarrow\infty}\mathbb{E}\int_0^T \psi(t) \Big[ \Vert X (t\wedge\tau_u^M) \Vert_H^2 - \Vert x \Vert_H^2 \Big] dt = \lim_{M\rightarrow\infty}\liminf_{n\rightarrow\infty}\mathbb{E}\int_0^T \psi(t) \Big[ \Vert X_n (t\wedge\tau_u^M) \Vert_H^2 - \Vert P_n x \Vert_H^2 \Big] dt .
\left({\dot{a}_R\over a_R}\right)^2+2m_0\dot{R}{\dot{a}_R\over a_R}+m^2_0\dot{R}^2+{{kc^2\Omega^2_R}\over a^2_R}={{8\pi m_0G_5}\over{3(1-\Omega^2_R)}}\left[{{\tilde{\varrho}_1\Omega_R+\varrho_2\Omega^5_R}\over\sqrt{\Omega^2_R-\dot{R}^2/c^2}}+V_r/c^2\right].
d\Theta^2_{\underline\mu}=v^{+}_{\underline\mu q}\pi^{2-}_q+v^{-}_{\underline\mu\dot q}\pi^{2+}_{\dot q},=e^{-2}\psi^{2+}_{-2\dot q}v^-_{\underline\mu\dot q}+e^-_qv^{+}_{\underline\mu q},
- \frac{1}{2 \pi i ( u - v )}\left[ \left( e^{- i [ \theta_{p}( u ) - \theta_{p}( v ) + \mathrm{a r g}\, \Gamma ( 1 + i u ) - \mathrm{a r g}\, \Gamma ( 1 + i v ) ]}- 1 \right){\left( \frac{a}{2}\right)}^{i ( u - v )}\right.
[\lambda+1]_{qt}= \binom{\lambda+1}{\bar 1}_{\!\!\!q,t}= \prod_{s\in \bar 1}\dfrac{(1- q^{1+\lambda_{1+l'(s)}}q^{a'(s)}t^{n-1-l'(s)})}{(1- q^{1+a'(s)}t^{n-1-l'(s)})}
V= \sum_{d_1,\ldots,d_k \le x}h_{f,k}(d_1,\ldots,d_k) \Big(\Big(\frac{x}{d_1\cdots d_k}\Big)^{r+1}T_{k-1}\Big(\log \frac{x}{d_1\cdots d_k}\Big) + O\Big(\Big(\frac{x}{d_1\cdots d_k}\Big)^{r+\theta_{k}+ \varepsilon}\Big) \Big),
\int_{\mathbb{R}^2}|\nabla u_{(x, 0), r_i}|^2 \eta d\mathcal{H}^2 = - \int_{\mathbb{R}^2}u_{(x, 0), r_i}\nabla u_{(x, 0), r_i}\cdot \nabla \eta d\mathcal{H}^2 \rightarrow - \int_{\mathbb{R}^2}u_{\infty}\nabla u_{\infty}\cdot \nabla \eta d\mathcal{H}^2 = \int_{\mathbb{R}^2}|\nabla u_{\infty}|^2 \eta d\mathcal{H}^2.
P(\alpha,\beta+1) =\dfrac{4q}{K}P(\alpha,\beta)+P(\alpha,\beta-1), P(\alpha+2,0) =KP(\alpha+1,0)+q^2 P(\alpha,0), P(\alpha+2,-1) =\left(K-2q+\dfrac{4q^{2}}{K}\right)P(\alpha+1,0)-q^2 P(\alpha,1).
\begin{array}{rcrcl}e^{\pm iH_1}= \frac1{\sqrt{k}}(\psi^1 \mp i \psi^2)~, e^{\pm i(H_2 + \pi N_1)}={1\over 2\sqrt{k}}\chi^3 \mp{1 \over \sqrt{k}}\psi^3~, e^{\pm i(H_3 + \pi N_2 + \pi N_1)}= \frac1{2\sqrt{k}}(\chi^1 \mp i \chi^2)~,\end{array}
\mathrm{La}(\{B_{k,\ell}\}_{k\in \mathcal{K}}) = -\log_2\left(1+\frac{\sum_{k =1}^K(1- 2^{-\frac{B_{k,\ell}}{N_k-1}}) \alpha_{k,\ell}N_k}{{\hat{\sigma}_k}^2}\right) + \lambda(\sum_{k=1}^K B_{k,\ell}- B_{\ell})
B =\Big([f_-(-M_x)][g_+(H)]\Big)+\Big([g_-(-M_x)][f_+(H)]\Big) &=\Big(g(0)[f_-(-M_x)]+[f(0)g_+(H)]-f(0)g(0)\Big)+\Big(f(0)[g_-(-M_x)]+[g(0)f_+(H)]-f(0)g(0)\Big) &=C-2f(0)g(0).
\int K_{r}\left( x,y\right) ~dx\approx \int_{y_{1}-r}^{y_{1}}\left\{\int_{y_{2}-h_{x,y}}^{y_{2}+h_{x,y}}\frac{1}{h_{x,y}}dx_{2}\right\}dx_{1}\approx \int_{x_{1}}^{x_{1}+r}dy_{1}=r\ .
\tilde{g}=\sum_{c=0}^{R-1}\sum_{j\equiv c(\textrm{mod}R)}g_j \prod_{\substack{j'>j j'\equiv c(\textrm{mod}R)}}(1-G_{j'}) \textrm{with}G_j:=\sum_{\substack{t>0 t\equiv 0 (\textrm{mod}R)}}2^{-\alpha t}\omega_{j-t}.
2\xi\mathbf{F}_{x}+3\xi^{\prime}\mathbf{F}+\left(((A+\xi^{\prime}E)\mathbf{y}+\zeta)\cdot\nabla\right)\mathbf{F}-A\mathbf{F}=\xi^{\prime\prime\prime}\mathbf{y}+\zeta^{\prime\prime},
L [ \phi , \psi_{i}] = - \int_{0}^{\infty}\mathrm{d}r \left[ 4 \pi r^{2}\left\{\frac{1}{2}\left( \frac{\mathrm{d}\phi}{\mathrm{d}r}\right)^{2}+ U ( \phi ) \right\}+ \sum_{i}\sum_{K M}\psi_{i}^{\dagger}H_{f}\psi_{i}\right] ,
g(\tau + 1) = \begin{pmatrix}\zeta_{18}^{-1}0 0 0 \zeta_{18}^5 0 0 0 \zeta_{18}^{11}\end{pmatrix}g(\tau), g(-\frac{1}{\tau}) = \begin{pmatrix}\alpha_1 \alpha_2 \alpha_4 \alpha_2 -\alpha_4 -\alpha_1 \alpha_4 -\alpha_1 \alpha_2 \end{pmatrix}g(\frac{\tau}{3})
P =\left(\begin{array}{cccccc}1 \frac{q-1}{2}\frac{q-1}{2}1 \frac{-1+\sqrt{-q}}{2}\frac{-1-\sqrt{-q}}{2}1 \frac{-1-\sqrt{-q}}{2}\frac{-1+\sqrt{-q}}{2}\end{array}\right).
G_{4}( \Lambda ) = G_{4}( \mu ) \left( \frac{\Lambda}{\mu}\right)^{2}\frac{( 1 + b g^{2}( \mu ) \ln ( \Lambda / \mu ) )^{- c / b}}{1 + w G_{4}( \mu ) \int_{\mu}^{\Lambda}\frac{d \Lambda^{\prime}}{\Lambda^{\prime}}\left( \frac{\Lambda^{\prime}}{\mu}\right)^{2}( 1 + b g^{2}( \mu ) \ln ( \Lambda^{\prime}/ \mu ) )^{- c / b}},
x_{k}= -\frac{(1-r)}{8(1+r)^{2}}\left(-1 - 2 r + 2 k r - r^2 - \frac{(1-r)^{k}}{(1+r)^{k-2}}\right) y_{1}= -\frac{1}{4(1+r)^2}(3r+1), y_{k}= \frac{1}{4(1+r)}\left(-1 - 2 r + 2 k r - r^2 - \frac{(1-r)^{k}}{(1+r)^{k-2}}\right) k > 1, z_{k}= -\frac{1-r}{8(r+1)^2}\left(-1 - 2 r + 2 k r - r^2 - \frac{(1-r)^{k}}{(1+r)^{k-2}}\right),
\big\|\hat{u}\hat{v}\big\|_{L^{r', h}}\lesssim \big\|\hat{u}\big\|_{L^{q', \tilde{h}}}\big\|\hat{v}\big\|_{L^{\tilde{q}', \hat{h}}}= \big\|u\big\|_{\mathcal{L}^{q, \tilde{h}}}\big\|v\big\|_{\mathcal{L}^{\tilde{q}, \hat{h}}}.
- \left[ [(1+t^2)^r-1](1-t)^r +t(1-t)[(1-t)^r-1] \right] & = -[(1+t^2)^r(1-t)^r-(1-t)^r +t(1-t)[(1-t)^r-1] ] & = -(1+t^2)^r(1-t)^r +(1-t)^r -t(1-t)^{r+1}+ t(1-t) .
{v_{T}}\left( \lambda - \int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{1}{\sqrt{{\bf k}^{2}+ v_{T}^{2}}}\frac{\sinh ( \beta \sqrt{{\bf k}^{2}+ v_{T}^{2}})}{\cosh ( \beta \sqrt{{\bf k}^{2}+ v_{T}^{2})}+ 1}\right) = 0 \, \, .
H_{+}( x ) = \left( \begin{matrix}{h_{G}^{( 1 )}( x )}&{0}\\{0}&{h_{G}^{( 1 )}( x )}\\ \end{matrix}\right) \quad \mathrm{a n d}\quad H_{-}( x ) = \left( \begin{matrix}{h_{G + 1}^{( 1 )}( x )}&{0}\\{0}&{h_{G - 1}^{( 1 )}( x )}\\ \end{matrix}\right)
\mathbf{m}'(0) =\mbox{Cap}_{\mathcal{A}}(E_0)^{\frac{1}{n-p}-1}[\mbox{Cap}_{\mathcal{A}}(E_1)-\mbox{Cap}_{\mathcal{A}}( E_0)] &=\mathbf{m}(0)^{1-n+p}[\mathbf{m}(1)^{n-p}-\mathbf{m}(0)^{n-p}].
\Pr\left[|g|\ge n^{k-s}\right] =\Pr\left[g\ge \frac{1}{\sqrt{k2^{k^2}(h_*)^2}}\sqrt{n}\|g\|_2\right]\le \lambda^{{k\choose 2}}\exp\left(-\frac{{k\choose 2}}{2e}\lambda \left(\frac{t}{k2^{k^2}(h_*)^2}\right)^{\frac{2}{{k\choose 2}}}\right) &=O(e^{-\Omega(n^{\frac{2}{k^2}})})
\left. \mathcal{O}_{\widetilde{\mathcal{C}^{2}}}\left( \widetilde{C_{r}\times Z_{r}}\right) \right\vert_{\widehat{C}}=\mathcal{O}_{\widehat{C}}(\cdots+0\cdot\widehat{C_{k}}+0\cdot E+0\cdot\widehat{C_{l}}+\cdots).
(\Psi_{\gamma+t},e^{i\left\langle \gamma+\delta+t,x\right\rangle})=\sum_{\gamma_{1},\gamma_{2},...,\gamma_{m}}\left( \frac{q_{\gamma_{1}}q_{\gamma_{2}}...q_{\gamma_{m+1}}(\Psi_{\gamma+t},e^{i\left\langle\gamma+\delta-\gamma(m+1)+t,x\right\rangle})}{d(\gamma,\delta)d(\gamma,\delta-\gamma_{1})...d(\gamma,\delta-\gamma(m))}\right) .
\mathbf{\Sigma}^2(x_1)=\frac{(-x_1)^{-3/2}}{2\sqrt{2\pi}}\begin{pmatrix}-\beta(H_{24}^2-H_{22}H_{44})[(-x_1)^{i\epsilon}+ (-x_1)^{-i\epsilon}] i(H_{44}H_{12}-H_{14}H_{24})[(-x_1)^{i\epsilon}- (-x_1)^{-i\epsilon}] i(H_{14}H_{22}-H_{24}H_{12})[(-x_1)^{i\epsilon}- (-x_1)^{-i\epsilon}]\end{pmatrix},
d s_{6}^{2}={\frac{3{\tilde{R}}^{2}}{2^{2 / 3}}}K ( \tau ) [{\frac{1}{3 K^{3}( \tau )}}( d \tau^{2}+ ( g^{5})^{2}) + \cosh^{2}( \tau / 2 ) ( ( g^{3})^{2}+ ( g^{4})^{2}) + \sinh^{2}( \tau / 2 ) ( ( g^{1})^{2}+ ( g^{2})^{2}) ] .
\widehat{\mu}(t) = \exp \biggl[-\frac{a}{2}t^2 +\mathrm{i}\lambda t +\int_{\mathbb{R}}\bigl(\mathrm{e}^{\mathrm{i}tx}-1-\mathrm{i}tx 1_{D_b}(x) \bigr)\nu(\mathrm{d}x) \biggr], t\in{\mathbb{R}},
\mathrm{T r}\, \log ( 1 - \sum_{i = 0}^{N}A_{i}) ~ = ~ \mathrm{T r}\, \log ( 1 - \sum_{k = 1}^{N}\sum_{m = 0}^{k - 1}A_{k}\phi^{m}) + \mathrm{T r}\, \log ( 1 - \phi ) ~ .
D_{\mu\nu}^{(0)}(p) = \frac{1}{p^{2}+\kappa^{2}}\left[(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}) - \kappa \epsilon_{\mu\nu\lambda}\frac{p_{\lambda}}{p^{2}}\right] +\xi\,\frac{p_{\mu}p_{\nu}}{(p^{2})^{2}}
\begin{array}{rl}\Delta_{g_B}h =\Delta_{\overline{g}}h + m\frac{\overline{g}(grad_{\overline{g}}h,grad_{\overline{g}}f)}{f}= \varphi^2\varepsilon_{i_0}\left[h''-(n-2)\frac{\varphi'h'}{\varphi}\right] + m\varepsilon_{i_0}\varphi^2\frac{h'f'}{f},\end{array}
x^{n+1}+{\displaystyle\sum\limits_{k=1}^{\infty}}x\alpha_{n,k}\left( x\right) z^{2k}=x^{n+1}+{\displaystyle\sum\limits_{k=1}^{\infty}}\alpha_{n+1,k}\left( x\right) z^{2k}+x^{n-1}\gamma_{n}\left( z\right) +\gamma_{n}\left( z\right) \left({\displaystyle\sum\limits_{k=1}^{\infty}}\alpha_{n-1,k}\left( x\right) z^{2k}\right) ,
V_{e}\, = \, - 2 i \pi \, \hat{\theta}_{e}\, \int d p_{0}^{\prime}d p_{0}\delta ( p_{0}^{\prime}\! - \! \mu_{e}) \beta_{1}\beta_{2}K_{e}^{T}( p_{0}^{\prime}, p_{0}) \delta ( p_{0}\! - \! \mu_{e}) \, \hat{\theta}_{e},
\begin{aligned}F^{(n)}(t;\zeta,z)=\det\left(z\mathbb{I}_{\binom{p}{n}\times \binom{p}{n}}-\mathcal{B}^{(n)}(t;\zeta)\right)\ , G^{(n)}(t;\zeta,Q)=\det\left(Q\mathbb{I}_{\binom{p}{n}\times \binom{p}{n}}-\mathcal{Q}^{(n)}(t;\zeta)\right)\ , \end{aligned}
\varphi ( y ) = A^{\prime}\, e^{- i k y}F \left( - s , s + 1 ; 1 + k ; \frac{i \, e^{- i y}}{2 \, \sin y}\right) + B^{\prime}\, e^{i k y}F \left( - s , s + 1 ; 1 - k ; \frac{i \, e^{- i y}}{2 \, \sin y}\right) .
H_{dN}= \left(\begin{smallmatrix}U_N^{(1)}A_N^{(1)}U_N^{(1)*}\ddots U_N^{(d)}A_N^{(d)}U_N^{(d)*}\end{smallmatrix}\right)W_{dN}\left(\begin{smallmatrix}V_N^{(1)}B_N^{(1)}V_N^{(1)*}\ddots V_N^{(d)}B_N^{(d)}V_N^{(d)*}\end{smallmatrix}\right)
\widehat{R}_{k}( \mathrm{\boldmath ~ \lambda ~ \unboldmath}) = \left. \frac{1}{( 2 \pi )^{k}}\frac{\partial^{k}}{\prod_{p = 1}^{k}\partial J_{p}}Z_{k}( \mathrm{\boldmath ~ \lambda ~ \unboldmath}+{\bf J}) \right|_{{\bf J}= 0}\ ,
\|D_{z,w}^2 F_R(t)\|_4 =\left\|\int_{B_R}D_{z,w}^2 u(t,x) dx \right\|_4 \leq \int_{B_R}\|D_{z,w}^2 u(t,x) \|_4 dx \leq C \int_{B_R}\widetilde{f}_2(w,z,x;t)dx= & C\int_{B_R}\left(\int_{0<\theta<r<t}G_{t-r}(x-z)G_{r-\theta}(z-w) dr d\theta + \int_{0<r<\theta<t}G_{t-\theta}(x-w)G_{\theta-r}(w-z) dr d\theta \right)dx.
f(\mu)f(x|\mu)=\pi(\mu)L(\mu)=\frac{1}{\sqrt{2\pi}\sigma_{m}}exp(-\frac{1}{2}(\frac{\mu-\mu_{0}}{\sigma_{m}})^{2})\prod_{j=1}^{n}\frac{1}{\sqrt{2\pi}\sigma_{v}}exp(-\frac{1}{2}(\frac{x_{j}-\mu}{\sigma_{v}})^{2}),
(\cdots,f_i-f_j,f_j-f_k,f_k-f_l,\cdots) &\qquad= (\cdots,(f_i-f_j)+(f_j-f_k),f_j-f_k,(f_k-f_l)+(f_j-f_k),\cdots) &\qquad= (\cdots,f_i-f_k,f_j-f_k,f_j-f_l,\cdots) &\qquad= -(\cdots,f_i-f_k,f_k-f_j,f_j-f_l,\cdots),
K_{n,-1}:=1,K_{n,p}:=1, K_{n,k}:=\left(\int_{a_{k}}^{b_{k}}P_{n,k}^{2}(\tau)\,\frac{|H_{n,k}(\tau)|\,d|\sigma_{n,k}|(\tau)}{|P_{n,k-1}(\tau)\,P_{n,k+1}(\tau)|}\right)^{-1/2},k=0,\ldots,p-1,
\mathrm{error}(h) := \max_{0 \le n \le N_h}\Big( \frac{1}{M}\sum_{m = 1}^M \big| X^{(m)}(h n) - X_h^{n,(m)}\big|^2 \Big)^{\frac{1}{2}}\approx \max_{0 \le n \le N_h}\big\| X(n h) - X_h^n \big\|,
t_\alpha{}^i:=\;\stackrel{{\rm B}}{\Sigma}_\alpha{}^{\!i}(\stackrel{{\rm B}}{X}) \qquad{\rm and}\qquad\stackrel{{\rm B}}{\tau}_{\alpha\beta}{}^i =0={\tau}_{\alpha\beta}{}^{\!i}- 2 \stackrel{{\rm B}}{X}_{[\alpha\beta]}{}^{\!i}
\frac{\pi}{n-1}\equiv \mathrm{Arg}(r^n) \mod 2\pi \equiv \mathrm{Arg}(t_0) + \mathrm{Arg}(r^{n-1}+1) \mod 2\pi \equiv \frac{\pi}{n-1}+ \mathrm{Arg}(r^{n-1}+1) \mod 2\pi.
\mu_{k,l}=\frac{\left| \langle a_l\cos\theta_l, a_{k}\cos\theta_{k}\rangle \right|}{\|a_l\cos\theta_l\|_{L^2}\|a_k\cos\theta_k\|_{L^2}},\delta_1= \sum_{k\neq l}\mu_{k,l}\frac{\|a_k\cos \theta_k\|_{L^2}}{\|a_l\cos \theta_l\|_{L^2}}.
2\omega_2-\omega_1-\omega_3-\omega_4 = 2(\epsilon_1+\epsilon_2) - \epsilon_1- \frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3-\epsilon_4) - \frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4) = \epsilon_2-\epsilon_3.
A_{i}^{\prime}= e^{- i g a_{0}x_{0}\tau_{3}}\Omega_{\mathrm{D}}A_{i}^{( 1 )}\Omega_{\mathrm{D}}^{\dagger}e^{i g a_{0}x_{0}\tau_{3}}+ e^{- i g a_{0}x_{0}\tau_{3}}s_{i}e^{i g a_{0}x_{0}\tau_{3}}+ \partial_{i}a_{0}x_{0}\tau_{3},
\partial_{\varphi}D(p||p_{\varphi,\lambda}) = - \kappa( -\sin(\varphi) \mathbb{E}_p[x_1] + \cos(\varphi)\mathbb{E}_p[x_2]), \partial_\lambda D(p||p_{\varphi,\lambda}) = -\frac{1}{\lambda}+ \mathbb{E}_p[x_3].
|z|^{k+\ell}\Delta_{k, \ell}(z) = |z|^k \Big(H_\ell(z) + O\Big(\frac{1 + \frac{y}{\sqrt{k}}}{y^2}\Big) \Big) + O((1 + k^{-1/2}y)(1 + \ell^{-1/2}y)),
\bigcap_{\delta>0}\overline{g^0_t(F_{t+\delta}\setminus F_t)}=\bigcap_{\delta>0}\overline{\iota_t \circ g_t(F_{t+\delta}\setminus F_t)}=\iota_t\left(\bigcap_{\delta>0}\overline{g_t(F_{t+\delta}\setminus F_t)}\right) &=\iota_t(\{\xi(t)\}) =\{U(t)\}.
P\phi\ |_{\gamma_{i}\left( c\right)}=\frac{\oint_{\gamma_{i}\left(c\right)}\frac{\phi\left( x,y\right)}{\left\vert \nabla\psi_{0}\right\vert}dl}{\oint_{\gamma_{i}\left( c\right)}\frac{1}{\left\vert\nabla\psi_{0}\right\vert}dl},
h_{j,j+1}= \sum_{n=1}^{N}F(n) X_j^{-n}X_{j+1}^n + \sum_{n,m=1}^{N}[ G_+(n,m) X_j^{-n}X_{j+1}^n Z_j^m + G_-(n,m) X_j^{-n}X_{j+1}^n Z_{j+1}^m ]
{}_{p}F_{q}\left(\ \begin{array}{lll}\alpha_{1},...,\alpha_{p}~;~ \beta_{1}, ..., \beta_{q}~;~\end{array}z\right)=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}... (\alpha_{p})_{n}}{(\beta_{1})_{n}... (\beta_{q})_{n}}\frac{z^{n}}{n!}~,\newline
I_{IR}(\rho)=\frac{1}{2\pi i}\int_{\tilde{C}}\frac{ds}{s}e^{\rho/s}F(s)=\frac{1}{2\pi i}\int_{\tilde{C}}\frac{ds}{s}e^{\rho/s}\tilde{F}(s^a)=\frac{k}{2\pi i}\int_{\tilde{C}'}\frac{ds}{s}e^{(\frac{\tilde{\rho}}{s})^k}\tilde{F}(s),
\log\Gamma_M(w|a) = \int\limits_0^\infty \frac{dt}{t^{M+1}}\Bigl(e^{-wt}\,f_M(t|a) - \sum\limits_{k=0}^{M-1}\frac{t^k}{k!}\,B_{M,k}(w|a)- \frac{t^M\,e^{-t}}{M!}\, B_{M, M}(w|a)\Bigr).
T(p)=\frac{p^{n}-ap^{n-1}+bp^{n-2}-cp^{n-3}+dp^{n-4}-ep^{n-5}+\cdot\cdot\cdot}{p^{n}+ap^{n-1}+bp^{n-2}+cp^{n-3}+dp^{n-4}+ep^{n-5}+\cdot\cdot\cdot}
\|C_{(m)}(C_{(m)}^2\star C_{(m)}^2)\|_{L^1}= \frac{1}{L^{d}}\sum_{k\in\Lambda^*}\frac{1}{m^2+\theta(k)}\Big(\frac{1}{L^{d}}\sum_{p\in\Lambda^*}\frac{1}{m^2+\theta(p)}\frac{1}{m^2+\theta(k-p)}\Big)^2.
\sum_{i=3,4}\big\| T_{k,j;n,l,r}^{\mu,m,i}(\mathfrak{m},E)(t,x, \zeta) + \hat{\zeta}\times T_{k,j;n,l,r}^{\mu,m, i}(\mathfrak{m},B)(t,x, \zeta) \big\|_{L^\infty_x}\lesssim \| \mathfrak{m}(\cdot, \zeta)\|_{\mathcal{S}^\infty}\big[ 2^{(1- 19.5\epsilon)M_{t^{\star}}}
e_{i_{1}i_{2}\cdot\cdot\cdot i_{p}}^{j_{1}j_{2}\cdot\cdot\cdot j_{q}}=e_{i_{1}}\otimes e_{i_{2}}\otimes\cdot\cdot\cdot e_{i_{p}}\otimes e^{j_{1}}\otimes e^{j_{2}}\otimes\cdot\cdot\cdot e^{j_{q}}
< \overline{{{T}}}( k_{1}) T ( k_{2}) \overline{{{A}}}_{i}( k_{3}) >_{t r u n c}= - i ( 2 \pi )^{p + 1}\delta \left( \sum k \right) \: i \overline{{{e}}}\left( \widehat{\overline{{{k}}}}_{1}- \hat{k}_{2}\right)_{i}
p_{n+2}(x) = \sum_{\substack{m=0 \textnormal{m even}}}^{n-1}\frac{i^{m+1}}{8\pi^2 m!}\int_{\Omega}(1-c^{-2})(y)L^{\frac{n-m+1}{2}}(f_2-f_1)(y)|x-y|^{m-1}dy \ \ \ \ -\frac{i^{n+2}}{8\pi^2(n+1)}\int_{\Omega}c^{-2}(y)(f_2-f_1)(y)|x-y|^n dy.
| (3,2) |^2 = \frac{9\, \omega^4 + 216\, \omega^2 + 3600}{\omega^6 + 9\, \omega^4 + 216\, \omega^2 + 3600}|G_{mn}^{\gamma}(\omega)|^2 = \frac{25 \left( \gamma - 1 \right)^4 \omega^4 + 75 \left( \gamma - 1 \right)^2 \omega^2 + 225}{\gamma^6 \omega^6 + 6\, \gamma^4 \omega^4 + 45\, \gamma^2 \omega^2 + 225}
( H_{1 2 3}^{\frac{1}{2}0 0})^{T}= \partial_{2}\partial_{3}H_{1 2 3}\partial_{2}\partial_{3}={\cal P}_{1 2 3}( x_{3 2}x_{2 1})^{- 1}H_{1 2 3}x_{3 2}x_{2 1}{\cal P}_{1 2 3}.
\Big \langle \frac{1}{4}\int d^{4}x d^{4}\theta \Big ( \phi^{*}e^{2 V}\phi + \tilde{\phi}^{*}e^{- 2 V}\tilde{\phi}\Big ) \Big \rangle = - \frac{1}{1 6 \pi^{2}}\mathrm{R e}\int d^{4}x d^{2}\theta W_{a}C^{a b}W_{b}.
F_{B}( \beta ) = F_{B}( 2^{N}\beta ) + \sum_{k = 1}^{N}\int_{\cal S}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{- \frac{d - 2}{2}}\left[ \frac{\bar{\theta}_{4}^{4}}{\bar{\eta}^{1 2}}( z_{v}- z_{o}) - \frac{\bar{\theta}_{3}^{4}}{\bar{\eta}^{1 2}}( z_{v}+ z_{o}) \right] \theta_{2}\left( 0 \left| \frac{i ( 2^{k}\beta )^{2}}{2 \pi^{2}\tau_{2}}\right. \right)
F^{\#}_{T,R}(\nu_1, \nu_2) = \left( T^{\alpha_1^2/R^2}+ T^{\alpha_2^2/R^2}+ T^{\alpha_3^2/R^2}\right) e^{(\alpha_1^2 + \alpha_2^2 + \alpha_3^2)/T^2}\left(\prod \limits_{j=1}^3 \Gamma\left(\frac{2+R+3\nu_j}{4}\right) \Gamma\left(\frac{2+R-3\nu_j}{4}\right) \right).
\Gamma_1(N)\tilde{g}= \Gamma_1(N)\begin{pmatrix}0 -1 N 0\end{pmatrix}\begin{pmatrix}1 -\frac{n}{N}0 1\end{pmatrix}\begin{pmatrix}-A B C -D\end{pmatrix}\begin{pmatrix}N N'_S 0 0 N_S \end{pmatrix}^{-1}.
\psi(( \tilde{a}*h^*)*h) =\int_{s(V)}\psi_y\big(\big( \tilde{a}(U_y)(h^**h)(y)\big)\cdot\delta_{U_y}\big)\,d\mu(y) &=\int_{s(V)}|q(y)|^2\psi_y\big( \tilde{a}(U_y)\cdot\delta_{U_y}\big)\,d\mu(y).
\hat{B}( C , t ) | \vec{A}, \vec{E}; \phi_{1}, \phi_{2}{\rangle}_{0}\propto | \vec{A}+ L_{3}A^{A}( x_{1}, x_{2}) , \vec{E}; \phi_{1}+ L_{3}\phi_{1}^{A}( x_{1}, x_{2}) , \phi_{2}{\rangle}_{e_{3}}=
{\Psi}^{(w)}_{BL}\simeq N_w[\Psi^{(w)}_L+e^{-\pi w/\kappa}\Psi^{(-w)}_R],\qquad{\Psi}^{(w)}_{BR}\simeq N_w[\Psi^{(w)}_R+e^{-\pi w/\kappa}\Psi^{(-w)}_L],
a(x,y,z)= 3^a\cdot \frac{q-1}{s}\left( \sum\limits_{i=0}^{q-2}\alpha_i(x)\alpha_i(y)\alpha_i(z^{-1})\right)= 3^a\cdot \frac{q-1}{s}\left( \sum\limits_{i=0}^{q-2}\alpha_i(x)\alpha_i(yz^{-1})\right)
\Phi ={\frac{- 1}{2 i \pi}}G_{1}^{0}G_{2}^{0}G_{3}^{0}\beta_{1}\beta_{2}\beta_{3}\left[ V_{1 2}^{0}\psi_{1 2}+ V_{2 3}^{0}\psi_{2 3}+ V_{3 1}^{0}\psi_{3 1}\right]
\mathcal{M}_4(p;k,l,r) = \Bigg\{M = \begin{bmatrix}p^k 0 0 0 a_{21}p^l 0 0 a_{31}a_{32}p^r 0 0 0 0 1\end{bmatrix}\in \mathcal{M}_4(p) \Bigg\}.
G ( 1 , 2 ; 1 + 0 , 2 + 0 ) = 2 \alpha | \epsilon |^{- 4 / N^{2}}[ | z / \epsilon |^{4 ( N - 2 ) / N^{2}}- k^{2}| z / \epsilon |^{- 4 ( N + 2 ) / N^{2}}] + . . .
\|z\|_{L^{2\kappa}([\Omega\,\cap\, B_{H_0}(x,\rho)]\times(\frac{t}{4},t))}=\|z\|_{L^{2\kappa}(Q_0)}\le C[\rho^{-2}+t^{-1}]^{1/2\theta}\|z\|_{L^1(Q_\infty)}=C[\rho^{-2}+t^{-1}]^{1/2\theta}\|z\|_{L^1(B_{H_0}(x,2\rho)\times(\frac{t}{8},t))}
\overline{[A_{a,r}]}+ \sum^r_{i=1}v^{-\beta_{a}(i)}\begin{bmatrix}a + i i\end{bmatrix}\overline{[A_{a+i, r-i}]}= [A_{a,r}] + \sum^r_{i=1}v^{\beta_{a}(i)}\overline{\begin{bmatrix}a + i i\end{bmatrix}}[A_{a+i, r-i}]
\max_{\left\Vert \mathbf{a}\right\Vert = 1}n^{-1}\sum_{i=1}^{n}\left(\mathbf{a}^{\prime}\mathbf{R}_{n}\left(X_{i}\right)\right)^{2}\mbox{where}\mathbf{R}_{n}\left(\mathbf{t}\right)=n^{-1/2}\sum_{j=1}^{n}\left( \tau - \mathbb{I}\left[ U_{j}\left( \widehat{\beta}\right) <0\right] \right) Z_{j}\mathbb{I}\left(Z_{j}\leq\mathbf{t}\right)\, .
1-\theta_{n,p}(x)=\frac{(1-p)x^2}{2\log \frac{n}{2}}-\frac{(1-p)[3(1-p)x-4(1-2p)]x^{3}}{24(\log \frac{n}{2})^{2}}+o(\frac{1}{(\log \frac{n}{2})^{2}}).
\mathcal{B}_0\partial_t{\mathbf V}_{x_2}+\mathcal{B}_1\partial_1{\mathbf V}_{x_2}+\mathcal{B}_2\partial_2{\mathbf V}_{x_2}+(\partial_2\mathcal B_2+\mathcal{B}_3){\mathbf V}_{x_2}=\tilde{\mathcal F}_2\,,
\Delta_{L}{\tilde{A}}_{\mu}( x ) = - 8 \pi \int \delta \xi d s \Omega_{\xi}( s , 0 ){\cal D}_{\mu}( s ) \Lambda_{\nu}[ \xi | s ] \Omega_{\xi}^{- 1}( s , 0 ) \dot{\xi}^{\nu}( s ) \delta^{4}( x - \xi ( s ) ) ,
|F(\psi( s))|\lesssim y^{-2}= e^{-\lambda_l s}y^{2\lambda_{l}}(y^{-2\lambda_{l}-2}e^{\lambda_l s}) \le e^{-\lambda_l s}y^{2\lambda_{l}}(e^{-(1-2\sigma)\lambda_{l}s-2\sigma s}),
|u(x)| = \Big|u(0)+|x|\int_0^1u_\xi(tx)\,dt\Big| &\le |u(0)|+|x| \, |D u(0)|+|x|\int_0^1\frac{C}{\, 1-\alpha \,}\, |tx|^{1-\alpha}\,dt &\le |u(0)|+|x| \, |D u(0)|+\frac{C}{\, 1-\alpha \,}\, \frac{\, |x|^{2-\alpha}\,}{2-\alpha}.
\left\{\L_R \frac{\partial}{\partial \L_R}+ \beta \frac{\partial}{\partial g}-(n+\bar{n})\gamma_{\Phi}\right\}\Gamma^{\L}_{n,\bar{n}}\left[\Phi,\bar{\Phi}\right] = 2\gamma_{\Phi}{\cal T}^{\L}_{n,\bar{n}}\left[\Phi,\bar{\Phi}\right]
D^{2}( p + r ) = D^{2}( p ) + \frac{1}{2}r^{\alpha \beta}\left[ \frac{\partial}{\partial r^{\alpha \beta}}D^{2}( r ) \right]_{r = p}+ \frac{1}{2}\left[ \frac{1}{2}\frac{\partial^{2}}{\partial r_{\alpha \beta}\partial r^{\alpha \beta}}D^{2}( r ) \right]_{r = p}.
\Gamma_{\underline{{\alpha}}\underline{{\beta}}}^{a}= \left( \begin{array}{c c}{{- \gamma_{\alpha \beta}^{a}C_{q r}}}&{{0}}\\{{0}}&{{\tilde{\gamma}^{a \alpha \beta}C^{q r}}}\\ \end{array}\right) \, , \quad a = 0 , 1 , . . . , 5 \, ,
\begin{aligned}\frac{p(x)}{x^{k_1}}-\frac{p(x^{*})}{(x^{*})^{k_1}}=-\frac{1}{l_2}\frac{(1-\theta)}{\left(\frac{Kk_1}{k_1p(x^{*})-p'(x^{*})x^{*}}+\theta\right)}-\frac{l_1}{l_2}.\end{aligned}
\int_U W_1(x) |\nabla \bar P|^{2-a}dx\le \left( \int_U \mathcal{K}(x,t)|\nabla \bar P|^2dx \right)^{\frac{2-a}2}J_1^\frac a 2, J_1=\int_U \frac{W_1(x)^{\frac 2 a}}{\mathcal{K}(x,t)^{\frac{2-a}{a}}}dx.
\begin{array}{l l}{{{\cal L}_{\chi}}}&{{= ( 2 \pi R ){\cal L}_{5},_{\chi}= \tilde{g}^{M N}( \partial_{M}\chi )^{*}( \partial_{N}\chi ) - V}}\\{{}}&{{= e^{\sqrt{\frac 2 3}\kappa \varphi}g^{\mu \nu}( \partial_{\mu}\chi )^{*}( \partial_{\nu}\chi ) - V}}\\ \end{array}
\det \begin{pmatrix}b_{1,\varphi}(x,\xi',\xi_d=\rho^+_1,\tau) b_{2,\varphi}(x,\xi',\xi_d=\rho^+_1,\tau) begin{align*}4pt]b_{1,\varphi}(x,\xi',\xi_d=\rho^+_2,\tau) b_{2,\varphi}(x,\xi',\xi_d=\rho^+_2,\tau)\end{pmatrix}\neq 0.
f_1 = \mathcal{F}(\gamma(N_{\max}^{(5-2\alpha)+\varepsilon}t-n)\mathbf{1}_{[0,T]}(t) P_{N_1}u), \tilde{f}_2 = \mathcal{F}(\gamma(N_{\max}^{(5-2\alpha)+\varepsilon}t-n)\mathbf{1}_{[0,T]}(t) P_{N_2}\partial_x u), f_3 = \mathcal{F}(\gamma(N_{\max}^{(5-2\alpha)+\varepsilon}t-n)\mathbf{1}_{[0,T]}(t) P_N u),
\rho_{n}^2\rho_{n-1}^2(1-c_{n-1})^2-4\omega^2d_n\rho_{n}\rho_{n-1}= 0 \Longrightarrow \left(\dfrac{\zeta+n}{2\zeta+n}\right) \left(\dfrac{\zeta+n-1}{2\zeta+n-1}\right) \left(\dfrac{\zeta+n-\theta-1}{\zeta+n-1}\right)^2-\dfrac{n\omega^2}{2\zeta+n}=0.
f\left( u,v \right)=v\left({{\beta}_{y}}-{{\alpha}_{x}}\right)-u\left({{\beta}_{x}}+{{\alpha}_{y}}\right) g\left( u,v \right)=v\left({{\beta}_{x}}+{{\alpha}_{y}}\right)-u\left({{\alpha}_{x}}-{{\beta}_{y}}\right)
\Bigl(\mathbb E \Bigl\| \sum^N_{n=1}\varepsilon_n \,\sigma_n \phi_{n}(\sigma_1,\ldots,\sigma_{n-1})\Bigr\|^p\Bigr)^{\frac 1p}\geq \gamma_{p,X}^{\delta}\Bigl(\mathbb E \Bigl\| \sum^N_{n=1}\sigma_n \phi_{n}(\sigma_1,\ldots,\sigma_{n-1})\Bigr\|^p\Bigr)^{\frac 1p}.
K^{n,2}_\tau =n^{2\alpha+1}\int^{\tau}_{(\delta+\frac 1n) \wedge \tau}\gamma_s \left(\sigma'(X_s)\right)^2 \times \left[ \left( \int_{\eta_n(s)-\delta}^{\eta_n(s)}\psi_{n,2}(u,s) \sigma(X_{\eta_n(u)})\, dW_u\right)^2 - \left( \int_{\eta_n(s)-\delta}^{\eta_n(s)}\psi_{n,2}(u,s) \sigma(X_{\eta_n(s)-\delta})\, dW_u\right)^2 \right] \, ds.
\Vert [\widetilde{\Lambda}^\varepsilon ]\Pi_\varepsilon \Vert_{L_2\rightarrow L_2}\leqslant \vert \Omega \vert^{-1/2}(2r_0)^{-1}C_a n^{1/2}\alpha_0^{-1}\Vert g^{-1}\Vert_{L_\infty}=:\widetilde{M}_1,\ \varepsilon >0, &\Vert [\widetilde{\Lambda}^\varepsilon ]S_\varepsilon \Vert_{L_2\rightarrow L_2}\leqslant \widetilde{M}_1,\varepsilon >0.
P^{}_{n,p}(\lambda) = \frac{\prod_{i=n-2r+1}^n (1-1/p^i) \prod_{i=1}^{n/2-r}(1-1/p^{2i-1})}{p^{|\lambda|+4n(\lambda)}\prod_i \prod_{j=1}^{m_i(\lambda)}(1-1/p^{2j})}.
\sigma_{{a b}}^{{[ - 2 ]}}\equiv ( \sigma^{{0}}- \sigma^{{9}})_{{a b}}= \mathrm{{\it d i a g}}( 0 , 2 \delta_{{\dot{A}}{\dot{B}}}) = ( \tilde{\sigma}^{{0}}+ \tilde{\sigma}^{{9}})^{{a b}}= \tilde{\sigma}^{{[ + 2 ] a b}},
R_{2(\vartheta)}(\tau)=o\bigl(\tau^{\frac{d-1-\vartheta}{2m-n}+\vartheta}\bigr)\qquad\tau\to +\infty \intertext{and assuming that}R_{0(\vartheta)}(\tau)+ R'_{0(\vartheta)}(\tau)=O\bigl(\tau^{\frac{d-1-\vartheta}{2m-n}+\vartheta}\bigr)\qquad\tau\to +\infty
\frac{n^2G_n^2(x)}{32}=\frac{32\ln n}{32}+\frac{8x-5\ln(2\ln n)}{(2\ln n)^{\frac{1}{2}}}\frac{(32\ln n)^{\frac{1}{2}}}{32}+\frac{(8x-5\ln(2\ln n))^2}{32\cdot4\cdot(2\ln n)}&=\ln n+\frac{8x-5\ln(2\ln n)}{8}+o(1),
\mathcal I \leq p^{-1}\int_{B_{2r}}\partial_{0}(|u|^p)(r,x') r \zeta \,dx' & -{p}^{-1}\int_{B_{2r}}|u|^p(r,x')\zeta\,dx' +{p}^{-1}\int_{B_{2r}}|u|^p(0,x')\zeta\,dx' &+ C(\lambda_p,\Lambda,p,n) \|\mu'\|_{\mathcal{C}}\int_{B_{2r}}\left[\tilde{N}^{r}_{p,a}(u)\right]^p(u) \,dx' +\frac12\mathcal I &+ III_{32}+ IV.
\ln(\beta(t,q_t)/\beta(s,q_s))= \int_s^t \partial_r\ln(\beta(r,q_r))dr+\int_s^t \nabla_q\ln(\beta(r,q_r)) \circ dq_r = \int_s^t (\beta^{-1}\partial_r\beta)(r,q_r)dr+\int_s^t(\beta^{-1}\nabla_q\beta)(r,q_r) \circ dq_r = \int_s^t (\beta^{-1}\partial_r\beta)(r,q_r)dr-\int_s^t(\beta \nabla_q\beta^{-1})(r,q_r) \circ dq_r
v_i = \varepsilon \frac{(\theta_0-\theta_{1})(\theta_0-\theta_{2}) \cdots (\theta_0 - \theta_{d})}{\phi_{i+1}\phi_{i+2}\cdots \phi_{d}}(A^*-\theta^*_{i+1}I)(A^*-\theta^*_{i+2}I) \cdots (A^*-\theta^*_{d}I)u
\bigl({\mathfrak T}^{\tau,\mu}\; f(z)\bigr)^{\prime}= \frac{\tau}{\mu}- \sum_{\nu=k+1}^{\infty}\frac{\nu\Gamma(\nu+\tau)\Gamma(\mu)}{\Gamma(\nu+\mu)\Gamma(\tau)}\; a_{\nu}z^{\nu-1}(z\in \mathbb{U}),
Q^1=\left( \begin{array}{ccc}0 0 1 1 0 0 0 1 0\end{array}\right) ,Q^2=\left( \begin{array}{ccc}0 0 1 j 0 0 0 j^2 0\end{array}\right) ,Q^3=\left( \begin{array}{ccc}0 0 1 j^2 0 0 0 j 0\end{array}\right)