harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d^{p}H_{\lambda}( z )}{d z^{p}}= ( - 1 )^{p}p ! \frac{i}{2 \pi}\oint_{{\Gamma}_{1}}\oint_{{\Gamma}_{2}}d k_{1}\; d k_{2}\; \frac{k_{1}^{\Lambda}F ( k_{1}) k_{2}^{\Lambda}G ( k_{2})}{( z - k_{1}- k_{2})^{p}}
\Psi = \mathsf{G}_{[+\infty ,t_0]}(-\mathsf{V}+ \Xi)+ e^{-\tilde{\mathsf{D}}(t-t_0)}\Psi_0, \mathsf{G}=\begin{pmatrix}G 0 0 0 \bar{G}0 0 0 P\end{pmatrix}
L_{u}(u+1)= u \left (u+1 \right )^{q^2}+ (u+1)\left (u+1 \right )^q+ (u^2+u) (u+1) = u \left (u+1 \right )^{4}+ (u+1)\left (u+1 \right )^2+ (u^2+u) (u+1) = (u+1)^2 \left ( u^3+u+ u+1+u \right ) = 0.
\lbrack K ]_{q \bar{q}}= \left( \begin{array}{l l}{q \bar{q}+ q^{- 1}\bar{q}^{- 1}}&{- 1}\\{- ( 1 + \bar{q}^{2}+ \bar{q}^{- 2})}&{q \bar{q}^{3}+ q^{- 1}\bar{q}^{- 3}}\\ \end{array}\right)
\prod_{\ell=1}^n \mathbf{D}_\ell \mathbf{v}_{\min}\geq \rho(\mathbf{D}_{\min}) \prod_{\ell=1}^{n-1}\mathbf{D}_\ell \mathbf{v}_{\min}\geq \cdots \geq \rho(\mathbf{D}_{\min})^n\mathbf{v}_{\min},
w\left( f(A)X+X\bar{f}(A)\right) \leq \left({\frac{2}{d_{A}^{2}}}\int_{0}^{2\pi}d\mu (\alpha )\right) w(X-AXA^{\ast}) & =\left({\frac{2}{d_{A}^{2}}}f(0)\right) w(X-AXA^{\ast}) & ={\frac{2}{d_{A}^{2}}}w(X-AXA^{\ast}).
C(x) = IC[C(x)] = IC[\bigcup_{n\geq 0}C_n(x)] = IC[\bigcup_{n\geq 0}IC(\bigcup_{x' \sim_n x}\Gamma_{x'})] &= IC[\bigcup_{n\geq 0}\bigcup_{x' \sim_n x}\Gamma_{x'}] = IC[\bigcup_{x' \sim x}\Gamma_{x'}],
\Phi_{w_{m+1}}(z_{m+1}, x_{m+1}) = \Phi_{v^\iota_n}(z^\iota_n, x^\iota_n)+\sum_{j=n^2+k_n}^{m-1}\left(\frac{\sqrt{w_j}}{2}+o\left(\frac{1}{n^2}\right)\right) + \frac{\sqrt{w_m}}{2}+o\left(\frac{1}{n^2}\right) = \Phi_{v^\iota_n}(z^\iota_n, x^\iota_n)+\sum_{j=n^2+k_n}^{m}\left(\frac{\sqrt{w_j}}{2}+o\left(\frac{1}{n^2}\right)\right),
M_s(g(x))=\sum_{\ell=0}^s\sum_{m=1}^s mt_{\ell}t_mM_s(x^{\ell}, x^{m-1}), M_n(g(x))=\sum_{\ell=0}^s\sum_{m=1}^s mt_{\ell}t_mM_n(x^{\ell}, x^{m-1}),
f\left(x\right) =F_{\varphi}\left(x\right)=\int_{0}^{\frac{1}{2}}\varphi\left(y\right)F\left(x-y\right)dy =\int_{0}^{x}\varphi\left(y\right)\left(1-x+y\right)dy+\int_{x}^{\frac{1}{2}}\varphi\left(y\right)\left(1-y+x\right)dy.
\int_{{\cal F}\times S^{2}}\omega \ ( p , \zeta ) \langle p^{\prime}, \zeta^{\prime}| p , \zeta \rangle \langle p , \varsigma | p^{\prime \prime}, \zeta^{\prime \prime}\rangle = c \langle p^{\prime}, \zeta^{\prime}| p^{\prime \prime}, \zeta^{\prime \prime}\rangle
\begin{matrix}{{G_{\infty}}}&{{}}&{{\rightarrow}}&{{L_{\infty}}}\\{{}}&{{d}}&{{[ \cdot , \cdot ]}}&{{[ \cdot , \cdot , \cdot ]}}&{{\cdots}}\\{{\downarrow}}&{{\cdot}}\\{{\mathrm{c o m m .}~ A_{\infty}}}&{{\{\cdot , \cdot , \cdot \}}}\\{{}}&{{\vdots}}\\ \end{matrix}.
R_{N}(x^{N}(T))-R_{N}(x^{N}(0))+\frac{T(N-1)}{2}-\frac{2}{N}\sum_{i\neq j}A_{ij}^{N}(T) -\frac{1}{N}\sum_{i\neq j}\int_{0}^{T}b_{c(i)}(t,\,x_{i}^{N}(t))g'(x_{i}^{N}(t)-x_{j}^{N}(t))dt =\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{N}\left[\sum_{j:j\neq i}g'(x_{i}^{N}(t)-x_{j}^{N}(t))\right]d\beta_{i}(t)
\sum_{\substack{c \geq 1 \gcd(c,2) = 1}}\frac{1}{c^{2w}}\chi_k(c) H_h(c) = \prod_{\substack{p p \neq 2}}\Big( 1 + \frac{\chi_k(p) H_h(p)}{p^{2w}}+ \frac{\chi_k(p^2) H_h(p^2)}{p^{4w}}+ \cdots \Big).
\varphi \left( x\right) =\mathrm{d}_{F}\left( x\left( t\right) \right) , \varphi \left( x,y\right) =\left( \sum_{i=0}^{N-1}\left\vert x\left(t_{i+1}\right) -x\left( t_{i}\right) \right\vert -g\left( y\right) \right)^{+},
\frac{1}{B}-1 >\frac{3\sqrt{3}\big( (d-1)(k-1)^2\sin^2 2v\sin^2 2w+(k-c)(k+c-2)\sin^2 2w\big)}{4(k-1)^2(\cos 2v-\cos 2w)}&>\frac{3\sqrt{3}(d-1)\sin^2 2v\sin^2 2w}{4(\cos 2v-\cos 2w)},
G_{\lambda\pm i\epsilon}(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ix\xi}}{\xi-(\lambda\pm i\epsilon)}~d\xi -\widetilde{G}_{\lambda\pm i\epsilon}(x) &= \pm i e^{\mp\epsilon x}e^{i\lambda x}\chi_{\real^{\pm}}(x) - \widetilde{G}_{\lambda\pm i\epsilon}(x)
\left(u^{~\underline{a}}_{\underline{m}}\right) =\left(u^{a}_{\underline{m}}, u^{~i}_{\underline{m}}\right) ~~ \in ~~ SO(1,D-1)~~~~\Leftrightarrow ~~~~u^{~\underline{a}}_{\underline{m}}\eta^{\underline{m}\underline{n}}u^{~\underline{b}}_{\underline{n}}= \eta^{\underline{a}\underline{b}}
\frac{\Gamma}{V_{3}}\simeq 2 \left( \frac{\Delta S_{E}}{2 \pi}\right)^{3 / 2}\exp \left[ \frac{4}{3}\pi R^{3}\Delta U_{e f f}- 4 \pi R^{2}\left({\cal S}_{0}+{\cal S}_{1}\right) \right] ,
\begin{array}{llll}\mbox{for}\phi=+1 ~:~ (i^k,1,\pm1,\pm1,\pm1,1) ,(k=0,...,3)~, \mbox{for}\phi=-1 ~:~ (i^k e^{i\pi/4},1,\pm1,\pm1,\pm1,1) ,(k=0,...,3)\; ,\end{array}
\|M_1(f_1, f_2, g)\|_r \lesssim \sum_{k \geq 0}2^{-k\left( \frac{1}{p}-\frac{1}{p'}\right)}\| \left( \sum_{|\omega|=2^{-k}}\vert BHT(P_{\omega_L}f_1, P_{\omega_L}f_2) \vert^{p}\right)^{1/p}\|_s.
\Pi \triangleq \begin{pmatrix}\pi^+(P_Y,0^+) - \pi^+(P_Y,0^-) \pi^-(P_Y,0^+) - \pi^-(P_Y,0^-) \pi^+(-P_Y,0^+) - \pi^+(-P_Y,0^-) \pi^-(-P_Y,0^+) - \pi^-(-P_Y,0^-)\end{pmatrix}.
\begin{pmatrix}1 0 0 -1 0 -1 0 1 0 4 5 8 0 0 1 -3 -5 -6 0 0 0 -3 -6 -7 0 0 0 -1 -1 -2 0 0 0 2 3 4\end{pmatrix}^{-1}\begin{pmatrix}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}\end{pmatrix}=\begin{pmatrix}1/2 0 1/2 0 0 x\end{pmatrix}\begin{pmatrix}1/2 0 1/2 0 1/2 x\end{pmatrix},
\overline{u_{i}u_{j}}=\overline{(\overline{u}_{i}+u_{i}^{\prime})(\overline{u_{j}}+u_{j}^{\prime})}=\overline{\overline{u_{i}}\overline{u}_{j}+\overline{u}_{i}u_{j}^{\prime}+u_{i}^{\prime}\overline{u}_{j}+u_{i}^{\prime}u_{j}^{\prime}}=\overline{u}_{i}\overline{u}_{j}+\overline{u_{i}^{\prime}u_{j}^{\prime}}
f(x)=\bigoplus_{i=0}^{n}(a_{i}\otimes x^{\otimes s_{i}})=a_{n}\otimes x^{\otimes s_{n}}\oplus a_{n-1}\otimes x^{\otimes s_{n-1}}\oplus\cdots\oplus a_{1}\otimes x^{\otimes s_{1}}\oplus a_{0}\otimes x^{\otimes s_{0}},
\int_{\mathbb{R}^3}(1-c_1^{-2})(y)\cdot \Delta^{-1}(c_1^{-2}f_1)(y)\cdot \Phi_0(x-y)\dy &+\frac{1}{8\pi}\int_{\mathbb{R}^3}(c_1^{-2}f_1)(y)\cdot |x-y|\dy = \int_{\mathbb{R}^3}(1-c_2^{-2})(y)\cdot \Delta^{-1}(c_2^{-2}f_2)(y)\cdot \Phi_0(x-y)\dy &+\frac{1}{8\pi}\int_{\mathbb{R}^3}(c_2^{-2}f_2)(y)\cdot |x-y|\dy,
\psi_{m,\ell=0}(r)=N_mr^{-\frac{(D-1)}{2}}\frac{e^{-\frac{1}{2}\alpha p_2r}}{(e^{-p_2r}+\alpha)}\exp{(-\frac{1}{2}e^{-p_2r})}\hat{L}_{m+1}^{\alpha}(e^{-p_2r})
u_\alpha(x',t)= \,\frac{4\mu}{3\mu+\lambda}\frac{1}{\omega_{n-1}}\int_{{\mathbb{R}}^{n-1}}\frac{t}{(|x'-y'|^2+t^2)^{\frac{n}{2}}}\,f_\alpha(y')\,dy' begin{align*}4pt] \,+\frac{\mu+\lambda}{3\mu+\lambda}\frac{2n}{\omega_{n-1}}\int_{{\mathbb{R}}^{n-1}}\frac{t\,(x'-y',t)_\alpha(x'-y',t)_\beta}{(|x'-y'|^2+t^2)^{\frac{n+2}{2}}}\,f_\beta(y')\,dy',
\phi_{b}^{( r )}( x ) = \rho_{b A}^{( r )}( \Sigma^{\prime}( x ) ) \zeta^{A}= \rho_{b c}^{( s )}( h ( x ) ) \rho_{c A}^{r}( \Sigma ( x ) ) \zeta^{A}= \rho_{b c}^{( s )}( h ( x ) ) \phi_{c}^{( r )}( x ) .
K = \left[{dT \over dz}\right]_{\bar{z}^+}- \left[{dT \over dz}\right]_{\bar{z}^-}= \left[{dT \over ds}\right]_{s(\bar{z})}\, \left[ \left({ds_2 \over dz}\right)_{\bar{z}^+}- \left({ds_1 \over dz}\right)_{\bar{z}^-}\right].
f(\mu)=\max_{\mathbf{x}\in \Delta}f(\mathbf{x})\;, \Delta=\left\{\mathbf{x}:\sum_{i=1}^{\binom{n}{2}}\mathbf{x}_i=1 \mathbf{x}_1,\ldots ,\mathbf{x}_{\binom{n}{2}}\geq 0\right\}\;.
\begin{aligned}m_1 =-\frac{1}{3}\left(a+C+\frac{D_0}{C}\right), m_2 =-\frac{1}{3}\left(a+\frac{(-1+i\sqrt{3})}{2}\,C+\frac{(-1-i\sqrt{3})}{2}\, \frac{D_0}{C}\right), m_3 =-\frac{1}{3}\left(a+\frac{(-1-i\sqrt{3})}{2}\,C+\frac{(-1+i\sqrt{3})}{2}\,\frac{D_0}{C}\right),\end{aligned}
x = (1 - \varepsilon/2) \Biggl(\frac{\log_2 T_k \log_4 T_k}{\sum_{T_{k-1}< p \leq T_k}\frac{1}{2 p^{1 + 2 \sigma_k}}\bigl( \frac{2 \log T_k}{\log p}\bigr)^2 \sin^2 \bigl( \frac{\log p}{2 \log T_k}\bigr) + \frac{1}{8p^{2 + 4 \sigma_k}}}\Biggr)^{1/2}\, ,
X'' + \frac{1}{r}X' = b^2\,s_+ X+ \frac{k^2s_+}{\sqrt{2}r^2}+ O(r^{-4}), Y'' + \frac{1}{r}Y' = \frac{1}{3}(4c^2\,s_+^2 - b^2\,s_+)\,Y + \frac{\sqrt{3}k^2\,s_+}{\sqrt{2}r^2}+ O(r^{-4}).
wp(ifE_{1}\rightarrow S_{1}[]...[]E_{n}\rightarrow S_{n}fi,R)=\begin{matrix}(E_{1}\vee...\vee E_{n})\\ \wedge(E_{1}\Rightarrow wp(S_{1},R))\\ ...\\ \wedge(E_{n}\Rightarrow wp(S_{n},R))\\ \end{matrix}
\delta_N(\mu_{k,q})=c_{n,k,q}\left(\frac{(k-2q)^2(2n-k+1)}{c_{n,k-1,q}(n-k+q+1)}N_{k-1,q}-\frac{q(2n-k+1)}{c_{n,k-1,q-1}}N_{k-1,q-1}\right),
f(s, t, w)(z) = L[A(0, 1)(z) A(s, t)(z)] w &= L \left[ \begin{bmatrix}0 -z z -1 \end{bmatrix}\begin{bmatrix}s z -s -t z t z -1 \end{bmatrix}\right] w =\dfrac{-t z^2 w + z}{(s t z^2 - t z)w -s z - t z^2 +1}.
=\rho\left[ \frac{\partial u_{2}}{\partial t}+u_{1}\frac{\partial u_{2}}{\partial x}+u_{2}\frac{\partial u_{2}}{\partial y}+\frac{K}{a(t)}\frac{y}{r}\frac{\partial}{\partial s}f(s)+\frac{y}{r}\frac{2\pi}{r}{\displaystyle\int\limits_{0}^{r}}\frac{e^{f(\frac{\eta}{a(t)})}}{a(t)^{2}}\eta d\eta\right]
0 = \frac{1}{\Gamma}\big[ G \Delta_0 H + 2 \delta^{\mu\nu}\{\partial_{\mu}G + 2 G (\partial_{\mu}\log \Gamma) \}\partial_{\nu}H + \{\Delta_0 G + 4\delta^{\mu\nu}(\partial_{\mu}G ) (\partial_{\nu}\log\Gamma) \}H \big]
\frac{\partial}{\partial t}|\nabla R|^2= (1+\frac{\alpha'}{2}R)\Delta|\nabla R|^2-2(1+\frac{\alpha'}{2}R)|\nabla^2R|^2+\alpha'\langle\nabla|\nabla R|^2,\nabla R\rangle +(4R-3r+\frac{5}{4}\alpha'R^2+\alpha'\Delta R)|\nabla R|^2.
\left( \frac{C^{2}\, e^{2 \, \alpha}\, n^{4}}{\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^{*}}\, n^{4}}{\left( \left( x_{0}+ y_{0}- i \, \epsilon \right)^{2}- \left( x_{s}+ y_{s}\right)^{2}\right)^{2}}\right.
\lim_{\rho_{0}\rightarrow \infty}D : = \lim_{\rho_{0}\rightarrow \infty}\left( \nu + | l | - \beta + 2 \beta{\frac{A_{1}}{B_{1}}}{\frac{\Phi ( A_{1}+ 1 , B_{1}+ 1 ; \beta )}{\Phi ( A_{1}, B_{1}; \beta )}}\right) = 0 .
\tilde{w}_s\Psi(\tilde{R}(u,v))\tilde{w}_s^{-1}\in \tilde{w}_s \left(\prod_{\omega\in C(r,r')}\tilde{U}_{[\omega]}\right)\tilde{w}_s^{-1}=\prod_{\eta\in w_s(C(r,r'))}\tilde{U}_{[\eta]}.
G_{r s}= \left( \begin{matrix}{{\frac{H_{0}}{H_{1}}}}&{0}\\{0}&{{\frac{H_{2}}{H_{3}}}}\\ \end{matrix}\right) ={\frac{H_{2}}{H_{3}}}\left( \begin{matrix}{( \mathrm{R e}U )^{2}}&{0}\\{0}&{1}\\ \end{matrix}\right) ,
\lambda_1 = (2,3,4,5,6,3,4,2), \lambda_2 = (3,6,8,10,12,6,8,4), \lambda_3 = (4,8,12,15,18,9,12,6), \lambda_4 = (5,10,15,20,24,12,16,8), \lambda_5 = (6,12,18,24,30,15,20,10), \lambda_6 = (3,6,9,12,15,8,10,5), \lambda_7 = (4,8,12,16,20,10,14,7), \lambda_8 = (2,4,6,8,10,5,7,4).
[ a ( q , \lambda ) , a^{\dagger}( q^{\prime}, \lambda^{\prime}) ] = \delta ( q^{+}- q^{\prime +}) \delta^{2}( q_{\perp}- q_{\perp}^{\prime}) \delta_{\lambda , \lambda^{\prime}}= \delta^{3}( q - q^{\prime}) \delta_{\lambda , \lambda^{\prime}}\; .
\begin{aligned}\int_X -u \Omega^{n}\leq \int_X -u \Omega_u \wedge \Omega^{n-1}+ & + B_\Omega \|u\|_\infty^2 C_n \int_X \left( \omega^2 \wedge \Omega^{n-2}+ \omega^3 \wedge \Omega^{n-3}\right)\end{aligned}
\log \Lambda_{n,m}&= \frac12 \left(h\left(\frac{n}{K}\right)-h\left(\frac{m}{K}\right)\right) +K \left(H\left(\frac{n}{K}\right)-H\left(\frac{m}{K}\right)\right) +\frac{1}{12K^2}\sum_{j=m}^{n-1}h''\left(\frac{\xi_j}{K}\right)
L_m R^*_k [\xi_{i,j}] = \frac{1}{\sqrt{N}}L_m \left(\left[ \sum^N_{y=1}r^*(h^k_{j,y}) \xi_{i,y}\right] \right) = \frac{1}{N}\left[\sum^N_{x,y=1}l(h^m_{i,x}) r^*(h^k_{j,y}) \xi_{x,y}\right],
\begin{array}{c c}{\left( \bar{C}_{0}\eta_{0}{\cal A}. . . \right) \left( \bar{C}_{1}\eta_{0}B . . . \right)}\\{\left( \bar{C}_{1}\eta_{1}{\cal A}. . . \right) \left( \bar{C}_{1}\eta_{0}{\cal B}. . . \right)}\\ \end{array}
\small\bar{\Delta}_n=\small\frac{1}{(2Re\lambda_n)^2}\left|\begin{array}{cccc}T \frac{1}{q_{1,n}}\ldots \frac{1}{q_{N-1,n}}\frac{1}{q_{1,n}}\frac{1}{2q_{1,n}}\ldots \frac{1}{q_{1,n}+q_{N-1,n}}\vdots \vdots \ddots \vdots \frac{1}{q_{N-1,n}}\frac{1}{q_{N-1,n}+q_{1,n}}\ldots \frac{1}{2q_{N-1,n}}\end{array}\right|+\Lambda_n(T),
\int_0^{\infty}\sum_{n\geq1}\chi(n)n\Gamma\left(-\frac12;2n^2t\right)e^{n^2t}t^{s-1}dt=L(2s-1,\chi)\int_0^{\infty}\Gamma\left(-\frac12;2t\right)e^tt^{s-1}dt.
1 - \rho_k = \frac{f(z_k) - f(x_k) - m_k(z_k)}{- m_k(z_k)}& \leq \frac{\alpha_0 \bar \varpi}{2\rho\lambda\delta}\left [ \frac{\|(\nabla^2 f(x_k) - H_k)[y_k]\|}{\|y_k\|}+ \frac{L_H}{3}\|y_k\| \right ] \frac{\|y_k\|^2}{g_k}\max \left \{\frac{1}{\chi}, \frac{A_4}{g_k}, \frac{A_5\sigma_k}{g_k}\right \},
\int \prod_{i > j}{\cal D}c^{i j}{\cal D}\bar{c}^{i j}{\cal D}A_{\mu}^{i j}e^{- S_{\mathrm{o f f - d i a g}}}= \exp \left\{{\frac{1}{8 \pi}}\int_{\cal M}d \mu R \sum_{i > j}\log | \lambda_{i}- \lambda_{j}| \right\}~ .
\sigma_i\displaystyle\frac{\rm d}{{\rm d}t}{\cal E}_i+\sum\limits_{j\in{\cal N}(i)}l_{(i,j)}\left<{\cal H}_{(i,j)},\boldsymbol{n}_{(i,j)}\right>=-\sigma_i{\cal K}_i|\boldsymbol{v}|^3_i,
|u^a_x|^2 \leqslant O(1) \sum_{k=1}^n |u^k_x|^2= O(1) \bigg( \sum\limits_{k=1 \atop k \neq p}^n |u^k_x|^2 + |u^p_x|^2 \bigg)\leqslant O(1) \bigg(\delta \sum\limits_{k=1 \atop k \neq p}^n |u^k_x| + |u^p_x|^2 \bigg).
\int_{0}^{t}dt^{\prime}\int d^{2}\xi^{\prime}K^{t-t^{\prime}}_{\xi\xi^{\prime}}e^{-\phi(\xi)}\left(-\frac{1}{2}(\xi^{\prime}-\xi)^{a}(\xi^{\prime}-\xi)^{b}\left(\partial_{a}\phi(\xi)\partial_{b}\phi(\xi)-\partial_{a}\partial_{b}\phi(\xi)\right)\right)\Delta^{\prime}K^{t^{\prime}}_{\xi^{\prime}\xi}
\Delta_{+}( x^{\prime \prime}- x^{\prime}) = \int_{- \infty}^{\infty}d N ( \tau^{\prime \prime}- \tau^{\prime}) \int{\cal D}p{\cal D}x \left( \prod_{\tau^{\prime}}^{\tau^{\prime \prime}}\Theta ( - p_{0}) \right) \exp \left( i \int_{\tau^{\prime}}^{\tau^{\prime \prime}}d \tau ( p \dot{x}- N{\cal H}_{0}) \right) .
\sqrt{\alpha}\left(\frac{1}{4}\gamma-\frac{1}{4}\log(4\beta)+\sum_{m,n=1}^{\infty}K_{0}(2mn\,\alpha)\right)=\sqrt{\beta}\left(\frac{1}{4}\gamma-\frac{1}{4}\log(4\alpha)+\sum_{m,n=1}^{\infty}K_{0}(2mn\,\beta)\right),
\int d^{8}z \, ( i a_{1}G \partial \bar{G}+ a_{2}K \bar{K}+{\mathrm{\small{~ \frac 1 2 ~}}}b_{1}G G \bar{K}+{\mathrm{\small{~ \frac 1 2 ~}}}\bar{b}_{1}\bar{G}\bar{G}K ) ,
f_j(\lambda_k) =\frac{1}{2}\Gamma\left(\frac{n_B-j+1}{2}\right)\,_1F_1\left(\frac{n_B-j+1}{2},\frac{1}{2};\left(\frac{\lambda_k}{a}-\frac{a}{2b}\right)^2\right) &+\left(\frac{\lambda_k}{a}-\frac{a}{2b}\right)\Gamma\left(\frac{n_B-j+2}{2}\right)\,_1F_1\left(\frac{n_B-j+2}{2},\frac{3}{2};\left(\frac{\lambda_k}{a}-\frac{a}{2b}\right)^2\right).
\mu_{\{b_{k}\},\{\tilde{D}_{k}\}}=\delta_{b_{1}^{-1}\frac{b_{1}}{b}D_{1}}* \delta_{b_{1}^{-1}b_{2}^{-1}\frac{b_{1}}{b}D_{2}}* \cdots * \delta_{b_{1}^{-1}b_{2}^{-1}\cdots b_{k}^{-1}\frac{b_{1}}{b}D_{k}}* \cdots &=\delta_{b^{-1}D_{1}}* \delta_{b^{-1}b_{2}^{-1}D_{2}}* \cdots * \delta_{b^{-1}b_{2}^{-1}\cdots b_{k}^{-1}D_{k}}* \cdots.
\|x\|_k=\left\|(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n\right\|_k=\left\|\sum\limits_{n=1}^{\infty}\left(c_{n}-C_{k}^{1}c_{n-1}+\dots +(-1)^{k}c_{n-k}\right) e_n\right\|,
\intertext{Use this asymptotic formula,}\sum_{d^r\le x}\mu(d)\frac{x^k}{d^{rk}}=x^k\left(\frac1{\zeta(rk)}+O\left((x^{1/r})^{-rk+1}\right)\right), &=\frac{x^k}{\zeta(rk)}+O(x^{1/r}).
e^{tA}\left(\begin{array}[c]{c}y z\end{array}\right) =\left(\begin{array}[c]{cc}\cos\sqrt{\Lambda}t \frac{1}{\sqrt{\Lambda}}\sin\sqrt{\Lambda}t -\sqrt{\Lambda}\sin\sqrt{\Lambda}t \cos\sqrt{\Lambda}t\end{array}\right) \left(\begin{array}[c]{c}y z\end{array}\right) ,t\in\mathbb{R},\;\; \left(\begin{array}[c]{c}y z\end{array}\right) \in H.
\left( \hat{V}^{( B = 0 )}\right)^{- 1}\partial_{\pm}\hat{V}^{( B = 0 )}={\frac{1 \mp t}{1 \pm t}}\left( \begin{array}{c c}{{- E^{- 1}\partial_{\pm}E}}&{{0}}\\{{0}}&{{E^{- 1}\partial_{\pm}E}}\\ \end{array}\right) ={\frac{1 \mp t}{1 \pm t}}\, P_{\pm}
\int_0^t <w_{p\tau}, w_p g^2>_{H^{-1}\times H^1}d\tau= \int_0^t\int_\mathbb{R}\big[ a^2 w_{pxx}- \lambda_p(u^a) w_{px}- w_{p\tau\tau}\big] w_p g^2 dxd\tau &+ \int_0^t\int_\mathbb{R}\Big(b_p - \sum_{j = 1}^n a_{pj}w_j\Big) w_p g^2 dxd\tau.
\partial_{+}\partial_{-}\Omega + \frac{\mu}{4}e^{2 \rho}\Omega^{1 - \lambda - \gamma / 4}+ \frac{1}{2}e^{- 2 \rho}\Omega^{( \gamma - 2 \epsilon ) / 4 + 1}F_{- +}^{2}= 0 ,
6-t_{za}=\frac{-2 p (M\mu+6) +M a+6 M+a+6}{M+1-2p}>\frac{-2 M\mu +M a+6 M+a-6}{M+1-2p}&>\frac{-2 M\mu +6 M-6}{M+1-2p}=\frac{2M(1- \mu)+4M-6}{M+1-2p}>\frac2{M+1-2p}>0.
\Delta_{h}\phi = \Delta_g\phi + \frac{1}{V}g(\nabla^{g}V,\nabla^{g}\phi) = (\partial_{s}^2\phi + \Delta_{s}\phi + H_{s}\partial_s \phi) + \frac{\partial_{s}V}{V}\partial_s\phi + \frac{1}{V}g^{ij}\partial_{i}V\partial_{j}\phi = \partial_{s}^2\phi + \frac{1}{s}\partial_s \phi + \Delta_{0}\phi + Q(\phi),
0 = \overline{\nabla}_a^\ast \mathrm{d}L (a) = \left( \operatorname{ad}_\xi + \frac{\mathrm{d}}{\mathrm{d}t}\right)^\ast \frac{\delta L}{\delta \xi}= \left( \operatorname{ad}_\xi^\ast - \frac{\mathrm{d}}{\mathrm{d}t}\right) \frac{\delta L}{\delta \xi},
b_i(\lambda^+,\lambda^-)=\begin{cases}a_0(\lambda)-a_i(\lambda) \textrm{if $0\leq i<i_0(\lambda)$}a_0(\lambda) \textrm{if $i=i_0(\lambda)$}a_0(\lambda)+a_i(\lambda) \textrm{if $i_0(\lambda)<i\leq h$}.\end{cases}
\omega_2 = \frac{1}{\sqrt{6}}\left( \frac{1}{8}\sin{\theta_2}\, d\theta_2 \wedge d\phi_2- \frac{3}{16}\sin^2{\mu}\, \sigma_1 \wedge \sigma_2+ \frac{3}{8}\sin{\mu}\cos \mu \, d\mu \wedge \sigma_3 \right).
( f_{\lambda}, f_{\lambda^{\prime}}) = \frac{2^{D - 1}\pi^{\frac{D}{2}}\Gamma ( \frac{N}{2})}{( r^{\prime})^{2}- r^{2}}\lim_{\sigma \rightarrow \infty}( 1 - \cosh^{2}\sigma ) W [ P_{- \frac{1}{2}- i r}^{\mu}( \cosh \sigma ) , P_{- \frac{1}{2}+ i r^{\prime}}^{\mu}( \cosh \sigma ) ] \: ,
L=\frac{\partial}{\partial\mu}[\frac{(1-\mu^{2})}{(\eta^{2}-\mu^{2})}\frac{\partial}{\partial\mu}]-\frac{1}{\eta^{2}-\mu^{2}}[-\frac{s}{\eta}\frac{(\eta^{2}+\mu^{2})}{(\eta^{2}-\mu^{2})}+\frac{s^{2}}{1-\mu^{2}}]
w_{s,b,i}^{(5)}(\theta)=\chi(\theta)w_{s,b,i}^{(1)}(\theta)+(1-\chi(\theta))w_{s,b,i}^{(3)}(\theta),\quad\quad\quad0\leq \theta\leq 1/2, &w_{s,b,i}^{(6)}(\theta)=\chi(1-\theta)w_{s,b,i}^{(2)}(\theta)+(1-\chi(1-\theta))w_{s,b,i}^{(4)}(\theta),1/2\leq \theta\leq 1.
P(\varepsilon_9^N>\epsilon) \leq \sum_{l=1}^{+\infty}\frac{T}{\delta}\exp\left(\frac{\gamma_N^2}{N}\left(K_{11}-\frac{10^l}{l}\frac{2T\epsilon}{e^{20T}\delta}\right)\right) &\leq \frac{2T}{\delta}\exp\left(\frac{\gamma_N^2}{N}\left(K_{11}-\frac{20T\epsilon}{\delta e^{20T}}\right)\right).
\left\Vert U_{\hbar}^{M}\left( t,s\right) \right\Vert_{\beta\rightarrow\beta}\leq e^{K\left( \beta,d\right) \sum_{k=1}^{d}\hbar^{k/2-1}\left(M+1\right)^{\left( k/2-1\right)_{+}}\int_{J_{s,t}}\left\vert P_{k}\left(\tau:\theta,\theta^{\ast}\right) \right\vert d\tau},
\mathcal{F}\psi_{q}^{\star}:= \mathcal{F}\psi_{q}- \sum_{l=0}^{q-1}\frac{[\mathcal{F}\psi_{q}, \mathcal{F}\psi_{l}^{\star}]}{[\mathcal{F}\psi_{l}^{\star}, \mathcal{F}\psi_{l}^{\star}]}\mathcal{F}\psi_{l}^{\star}.
\epsilon_{a_1, \dots , a_{h+k-1}, bc}(i_1, \dots , i_{h+k}) = \epsilon_{a_1, \dots , a_{h+k-1}, b}(i_1, \dots , i_{h+k}) = \epsilon_{a_1, \dots , a_{h+k-1}, c}(i_1, \dots , i_{h+k}) &= \epsilon_{a_1, \dots , a_{h+k-1}, 1}(i_1, \dots , i_{h+k})
v_3 =(r_2+r_1,r_3 +r_4,r_6-r_5), p_1 =v_3-\gamma(1,0,0)=(r_6-r_5-r_3-r_4,r_3 +r_4,r_6-r_5), p_2 =v_3-\gamma(0,1,0)=(r_2+r_1,r_6-r_5-r_1-r_2,r_6-r_5), p_3 =v_3+\gamma(0,0,1)=(r_2+r_1,r_3 +r_4,r_2+r_1+r_3 +r_4),
\left\{a \, \middle|\, 1 \leq a \leq l(\lambda),\j(\lambda)_a = i+1, \begin{array}{l}\bullet\, m(\lambda)_a < m(\lambda)_{a+1}; \bullet\, m(\lambda)_a = m(\lambda)_{a+1}j(\lambda)_{a+1}< i\end{array}\right\}
I\equiv\int_{0}^{r_{0}}\left(F(x_{1})-F(r_{0})\right)^{N}dx_{1}=\int_{0}^{r_{0}}\left(\int_{x_{1}}^{r_{0}}-F^{\prime}(s)ds\right)^{N}dx_{1}&=\int_{0}^{r_{0}}\left(\int_{0}^{r_{0}}\chi_{[x_{1},r_{0}]}(s)(-F^{\prime}(s))ds\right)^{N}dx_{1}
h_x(\bar L^{n,k+1}) = h_x(\bar L^{n}(t_k)) = h_x(\bar L^{n}(t_n-T + (t_k-t_n+T))) &= h_x(\hat M^n(t_k - (t_n-T))) \le h_x(\hat M(t_k - (t_n-T))) + \delta_1^{-1}\sup_{t \in [0,T]}\|\hat M^n(t)-\hat M(t)\|,
\phi_n^i(L^{\vee}) = Ber^{n-k}\tilde{\phi}_n^i(L^{\vee}) = Ber^{n-k}Ber^{-2(n-k)}\tilde{\phi}_n^i(L)^{\vee}= Ber^{-n+k}\otimes \tilde{\phi}_n^i(L)^{\vee}= (Ber^{n-k})^{\vee}\otimes \phi(L)^{\vee}= (Ber^{n-k}\otimes \tilde{\phi_n^i}(L))^{\vee}= \phi_n^i(L)^{\vee}.
ds^2 = -\frac{f^2}{l^2}\left( r_+ d \phi -r_-\frac{dt}{l}\right)^2 - \frac{f{'\;2}dr^2}{(\frac{f^2}{l^2}+1)}+ \left(\frac{f^2}{l^2}-1\right) \left( r_- d\phi - r_+\frac{dt}{l}\right)^2
\frac{1}{\sqrt{n}}\sum_{i=1}^n \big[ F(a_i) - F(b_i)\big] \xrightarrow[n\to+\infty]{\rm law}N(0, \beta) \frac{W^{(ab)}-\frac{n^2}{2}}{n^{3/2}}\xrightarrow[n\to+\infty]{\rm law}N(0, \beta) \, .
{\cal L}= \mathrm{T r}\left[ \frac{1}{2 r}( D_{t}X_{i})^{2}- \frac{T_{A}^{2}}{4 r}[ X_{i}, X_{j}]^{2}- \theta^{T}D_{t}\theta - \frac{T_{A}^{2}}{4}\theta^{T}\gamma_{i}[ \theta , X_{i}] . \right] ,
F^{\diamond}(W_{t_1}, \ldots, W_{t_K}) = \sum\limits_{k=0}^{\infty}a_{1,k}W_{t_1}^{\diamond k}\diamond \cdots \diamond \sum\limits_{k=0}^{\infty}a_{K,k}W_{t_K}^{\diamond k}, \ \ \max_{i \leq K}\sup\limits_{k}\sqrt[k]{k! |a_{i,k}|}< \infty.
m_{|u-c|}^\gamma(B(x,2^{-i})) \le C2^{-is}\big(m_{g}^{\gamma/C}(B(x,2^{-i+2}))+m_{g}^\gamma(B(x,2^{-i}))\big) &\le C2^{-is}m_{g}^{\gamma/C}(B(x,2^{-i+2})) &\le C2^{-is}\sum_{k=i-3}^i m_{g_k}^{\gamma/C}(B(x,2^{-i+2})),
\Psi_{k+1}(A,r) -\Psi_{k+1}(B,s)= (r-s)2^{k+1}y_\emptyset+ \Psi_k^{(j)}(F_k^{(j,+)}(A,r)) -\Psi_k^{(j)}(F_k^{(j,+)}(B,s)\big).
I_{\varepsilon}(u) =\left(\frac{1}{2}-\frac{1}{p}\right)\|u\|_{\varepsilon}^{2}+\omega\left(\frac{1}{4}-\frac{1}{p}\right)G_{\varepsilon}(u) =\left(\frac{1}{2}-\frac{1}{p}\right)|u^{+}|_{p,\varepsilon}^{p}-\frac{\omega}{4}G_{\varepsilon}(u) =\frac{1}{4}\|u\|_{\varepsilon}^{2}+\left(\frac{1}{4}-\frac{1}{p}\right)|u^{+}|_{p,\varepsilon}^{p}.
\begin{aligned}\|\varphi(T)\omega_n(T)\varphi_n(T) x_0\| \leq KC_{{\rm pol}, T}(1+\varepsilon_0)\frac{(2(1+c_1^2))^{1/2}}{1-c_1}\Bigl(\int_{\mathbb T}|\varphi|^2|\psi_{1n}|^2m\Bigr)^{1/2}\ \varphi\in H^\infty \n\in\mathbb N.\end{aligned}
P^\omega\left(A_t^c \cap S_{\alpha t}\right) = P^\omega\left(A_t^c \cap S_{\alpha t}\cap E_t^c\right) + P^\omega\left(A_t^c \cap S_{\alpha t}\cap E_t\right) &\leq P^\omega\left(E_t^c \mid S_{\alpha t}\right) + P^\omega\left(A_t^c \mid E_t\right) .
\langle \mathrm{s o l}| \frac{1}{2}\frac{( \delta m^{2})^{2}}{\lambda}\int_{- \infty}^{+ \infty}[ 1 - \cos \frac{\sqrt{\lambda}}{m}\phi_{\mathrm{s o l}}( x ) ] d x = \frac{\lambda}{8 m L^{2}}( \sum \frac{1}{\tilde{\omega}})^{2}
0 \geq \limsup_{n \to \infty}\left( \langle f_n, f_n \rangle - \sum_{i, \ell \colon \lambda_{i,\ell}^{(n)}= 0}\hat f_n(i,\ell)^2 - \left(\sqrt{\frac{k}{k'}\cdot{\sum_{i,\ell \colon 0< \lambda_{i,\ell}^{(n)}\leq k}\hat f_n(i,\ell)^2}}+ \sqrt{\sum_{i,\ell \colon \lambda_{i,\ell}^{(n)}> k}\hat f_n(i,\ell)^2}\right)^2 \right).
\left\|(|N|-\varepsilon)_+^nR_\delta^n\right\|\leq\sum_{k=1}^{m-1}\left(\frac{k}{m}\right)^n(1+\varepsilon)^n\left(\frac{m}{k}\right)^n=(m-1)(1+\varepsilon)^n\leq\frac{2}{\varepsilon}(1+\varepsilon)^n.
\sum_{\substack{x\leq n<2x n\equiv \nu_0\pmod{W^\prime}[d_i,e_i]\mid nM+h_i}}\chi_\mathbb{P}(nM+h_m) =\sum_{\substack{X+h_m\leq n<2X+h_m n\equiv b\pmod{qM}}}\chi_\mathbb{P}(n) &=\frac{\mathcal{P}_X}{\varphi(V)\prod_i\varphi([d_i,e_i])}+E(X,qM)+O(1),
D_{n,k}(1,y(1-y)) =\frac{k}{2^{i+1}}\,\Big[ \displaystyle\frac{(u+1)^{p^{l_1}+p^{l_2}+\cdots +p^{l_i}}(1-u)-(u+1)(1-u)^{p^{l_1}+p^{l_2}+\cdots +p^{l_i}}}{u}\Big]\cr +\frac{1}{2^i}\,\Big[(u+1)^{p^{l_1}+p^{l_2}+\cdots +p^{l_i}}+(1-u)^{p^{l_1}+p^{l_2}+\cdots +p^{l_i}}\Big]\cr
\begin{array}{l}W_{q}\simeq - \int dx \langle E[ \delta\left(\tilde{q}\left(1\right)\right)]\int_{0}^{\infty}\frac{d\tau}{\tau}\tau^{-\frac{2}{1+\gamma}}\Big( \exp(-\tau \alpha\phi_{c}^{2}(x) ) -1 +\tau\alpha\phi_{c}^{2}(x) \Big) \rangle \cr \simeq \int dx \langle v(x) \vert\phi_{c}\vert^{\frac{4}{1+\gamma}}\rangle\end{array}
{\frac{d \bar{w}}{d t}}\Pi^{T}{\frac{d w}{d t}}={\frac{1}{\kappa^{2}}}\sum_{r = 3}^{N}\left[{\frac{t_{1 r}\bar{t}_{1 r}}{b + r}}+{\frac{t_{2 r}\bar{t}_{2 r}}{b - r}}\right] .
\leq n!\cdot (nC\varepsilon^\alpha)^n+n!\sum_{i=1}^{n-1}\binom{n}{i}(nC\varepsilon^\alpha)^i(\|\mathbf Q\|_\infty+|\lambda|)^{n-i}\leq \tilde{C}\max_{1\leq k\leq n-1}( \|\mathbf Q\|_\infty+|\lambda|)^k\varepsilon^\alpha.
R^{12}\phi=i\int_{\widehat{k_0,\varsigma}}{h_{12}(\varsigma',\varsigma)}e^{{\frac{i}{2}}(\varsigma \overline{z}+\overline{\varsigma'}z)+\frac{t}{2}(\overline{\varsigma'}^2-\varsigma^2)}\phi(\varsigma ')d\overline{\zeta'}, \varsigma \in \partial D,~~\phi\in L^2(\partial D).
\Omega_2 =\sum\limits_{\delta \in \mathbb{F}_p^* \atop y=-a\delta}\zeta_p^{-y-\rho \delta}\sum_{x\in \mathbb{F}_r}\zeta_p^{\mathrm{Tr}\big((a\delta+y)x\big)}= r \sum_{\delta \in \mathbb{F}_p^*}\zeta_p^{(a-\rho )\delta}.
\tau_{jj}= E(t_{jj}) = E \left( \frac{p_{jj}\cdot 1 + \sum_{i \in S, i \ne j}p_{ji}(t_{ij}+ 1)}{1 - \sum_{i \not \in S}p_{ji}}\right) = \frac{p_{jj}+ \sum_{i \in S, i \ne j}p_{ji}(\tau_{ij}+1)}{1- \sum_{i \not \in S}p_{ji}}