harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R(\psi,\epsilon c) =\lambda_{E/F}(\psi)\Delta_{E/F}(\epsilon c)=\lambda_{E/F}(\psi)\Delta_{E/F}(\epsilon)\Delta_{E/F}(c) =\Delta(\epsilon)\lambda_{E/F}(\psi)\Delta_{E/F}(c)=\Delta(\epsilon)\cdot R(\psi,c).
- 2 g(X,Y) \psi = X \cdot Y \cdot \psi + Y \cdot X \cdot \psi, \nabla_V^S(X \cdot \psi) = \nabla_V^E X \cdot \psi + X \cdot \nabla^S_V \psi, h(X \cdot \psi, \psi') = (-1)^{s+1}h(\psi, X \cdot \psi'),
n_k = 2{3k \choose k}+ 3k \leq 2.5 \frac{(3k)!}{k!(2k)!}<3\frac{\bigl(\frac{3k}{e}\bigl)^{3k}\sqrt{6k\pi}}{\bigl(\frac{k}{e}\bigl)^{k}\sqrt{2k\pi}\bigl(\frac{2k}{e}\bigl)^{2k}\sqrt{4k\pi}}=\frac{3^{3/2}}{2}\biggl(\frac{3^{3}}{2^2}\biggl)^k\frac{1}{\sqrt{k\pi}}.
\left[\begin{array}{cc}p_{n-1}p_{n}q_{n-1}q_{n}\end{array}\right]=\left[\begin{array}{cc}p_{n-2}p_{n-1}q_{n-2}q_{n-1}\end{array}\right]\left[\begin{array}{cc}0 1 1 a_{n}\end{array}\right],\left[\begin{array}{cc}p_{0}p_{1}q_{0}q_{1}\end{array}\right]=\left[\begin{array}{cc}0 1 1 a_1\end{array}\right]
\sum_{\ell = 0}^{\infty}{\frac{( - 1 )^{\ell}}{\ell + a}}\left( \begin{array}{c}{{- 1 / 2}}\\{{\ell}}\\ \end{array}\right) ={\frac{\Gamma ({\frac{1}{2}}) \Gamma ( a )}{\Gamma ( a +{\frac{1}{2}})}}.
Z^e_t(\phi):=(M^eL^e)^{-1}\sum_{z\in\Lambda^e_n}\sum_{w\sim z}{\bf 1}_{\{w,z\notin x_1\}}\int_0^t \xi_{s-}(z) \xi^c_{s-}(w) \phi_s(z) \, dQ^{z,w}_s.
\mod{\frac{I_1(w)}{G(w)}}=\mod{\frac{O(w^{-1})e^{-w^2+1+O(w^{-1})}+O(w^{-1})e^{O(w^{-1})}}{e^{-w^2+1}-1}}\leq\frac{C}{\mod{w}}\frac{\mod{e^{-w^2+1}}+1}{\mod{e^{-w^2+1}-1}},
\begin{array}{l}U_0(x,y) = \Theta^{-1/2}(x,y) U_{j+1}(x,y) = \Theta^{-1/2}(x,y) \int_0^1 s^j \Theta(x,x_s)^{1/2}\Delta_2 U_j(x, x_s) ds\end{array}
- 2\beta \sum\limits_{i \in B}{F^{ii}u_{ii}d_i^2}\geq - 2\beta \sum\limits_{i \in B, u_{ii}> 0}{F^{ii}u_{ii}d_i^2}\geq - \frac{2}{n}\sum\limits_{i \in B,u_{ii}> 0}{F^{ii}u_{ii}}\geq - \frac{2}{n}\sum\limits_{u_{ii}> 0}{F^{ii}u_{ii}}= - \frac{2}{n}[k-l- \sum\limits_{u_{ii}< 0}{F^{ii}u_{ii}}].
\begin{array}{l}{\Phi_{2}= ( \phi_{2}^{+})^{- 1}\{D_{2}\}_{p , q},}\\{\bar{\Phi}_{2}= \phi_{2}^{+},}\\{\Phi_{1}= ( \phi_{1}^{+})^{- 1}\{D_{1}\}_{p , q}p^{D_{2}},}\\{\bar{\Phi}_{1}= \phi_{1}^{+}q^{D_{2}}.}\\ \end{array}
n^{p}= \sum_{l = 0}^{p}( - 1 )^{p - l}\left( \begin{array}{c}{{p}}\\{{l}}\\ \end{array}\right) \left( n + \gamma m +{\frac{d - 1}{2}}\right)^{l}\left( \gamma m +{\frac{d - 1}{2}}\right)^{p - l}
\rho(h) = \frac{h^4 \big(2a_1^2(1/2-a_1)h^2+4a_1^2-6 a_1+1\big)^2}{8 \big(2-a_1 h^2\big) \big(2-(1/2-a_1) h^2\big) \big(1-a_1(1/2-a_1) h^2\big)}.
L_j{:=}\sum_{r=1}^{+\infty}(-1)^r \Bigg(\sum_{\substack{j_{l_1},\dots,j_{l_r}\in \mathbb{N}\setminus\{0\}j_{l_1}+\dots +j_{l_r}=j}}\bigg((\Lambda[0])^{(-1)}R_{j_1}\bigg)\cdot \dots \cdot \bigg((\Lambda[0])^{(-1)}R_{j_r}\bigg) \Bigg)(\Lambda[0])^{(-1)}\, .
y \tilde{p}_{n}= \frac{h_{n}}{h_{n - 1}}\tilde{p}_{n - 1}+ \sum_{j = 1}^{m_{1}}g_{3 j - 1}( n ) \frac{h_{n}}{h_{n - 1 + 3 j}}\tilde{p}_{n - 1 + 3 j}.
\lim_{\vec{p}\rightarrow 0}\frac{C_{3}^{*}\left( \left| \vec{p}\right| \right)}{4 p_{0}V}\sim \lim_{\vec{p}\rightarrow 0}\frac{C_{3}^{*}\left( \left| \vec{p}\right| \right) \left| \vec{p}\right|^{3}}{4 p_{0}}= 0 .
Q_{\lambda \kappa ;\sigma}+Q_{\sigma\kappa ;\lambda}-\frac{2}{3}\left( g_{\sigma \lambda}Q^{\nu}{_{\kappa ;\nu}}+g_{\kappa (\lambda}Q_{\sigma)}{^{\mu}}{_{;\mu}}\right) ={\cal Q}_{\lambda\kappa\sigma}=O( r^{-1})
\begin{aligned}\mathbb{E}[{H}_1(t)|\mathcal{F}_t^1]=\min\limits_{v_1\in U_1}\mathbb{E}[{H}_1^{v_1}(t)|\mathcal{F}_t^1], \mathbb{E}[{H}_2(t)|\mathcal{F}_t^2]=\min\limits_{v_2\in U_2}\mathbb{E}[{H}_2^{v_2}(t)|\mathcal{F}_t^2],\end{aligned}
\int_{S^2}W_0 \tilde X^i \tilde X^j dS^2 = \frac{8 \pi}{15}D_{ij}\int_{S^2}W_0^2 dS^2 =\frac{8 \pi}{15}\sum_{ij}D_{ij}^2 \int_{S^2}W_i \tilde X^j \tilde X^l dS^2 =\frac{4 \pi}{15}(\bar W_{0lij}+\bar W_{0jil}) \int_{S^2}W_0 W_i dS^2 = \frac{8 \pi}{15}\sum_{jl}D_{jl}\bar W_{0lij}.
\leq\frac{c}{k}\sum_{i,j=1}^{q}\left\Vert X_{i}X_{j}u\right\Vert_{L^{p}\left( B_{R}\right)}& +ck^{2+Q/p_{1}}\left\{\left\Vert Lu\right\Vert_{L^{p}\left(B_{R}\right)}+\left( a_{m+2,\gamma R}^{\sharp}\right)^{1/\beta p_{1}}\sum_{i,j=1}^{q}\left\Vert X_{i}X_{j}u\right\Vert_{L^{p}\left( B_{R}\right)}\right\}.
\nabla f_{ml}(x_{ip}) = \frac{z_{ip}\textrm{exp}(-\frac{\tilde{x}_{ip}^2}{2})}{\sqrt{2\pi}\sigma_v\phi (\tilde{x}_{ip})}-\frac{(1-z_{ip})\textrm{exp}(-\frac{\tilde{x}_{ip}^2}{2})}{\sqrt{2\pi}\sigma_v\phi (-\tilde{x}_{ip})},
0 =h'_{g_{u_{n}}}(1) &=-2\beta A(u_{n})+\left((1-\frac{3}{2}\beta)p+3\beta-2\right)C(u_{n})+(2-\beta)B(u_{n}) &\ \ \ +\frac{\beta}{4}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{-|x-y|}u_{n}^{2}(x)u_{n}^{2}(y)dxdy.
\delta_1(y)+\delta_1(y+h)=f(h,h)+1, \forall h\in H, \textrm{whenever}y, y+h\in B; & \delta_2(y)+\delta_2(h+\overline{y})=f(h,\overline{h})+f(\overline{e_0},\overline{h})+1, \forall h\in H, \textrm{whenever}y, h+\overline{y}\in B; & \delta_1(y)+\delta_1(\overline{y})+\delta_2(y)+\delta_2(\overline{y})=0, \textrm{whenever}y, \overline{y}\in B.
\widetilde{H}(\mathbf U,\mathbf V)_m=\frac{1}2\left\{(\delta_m^+U_{m})^2+(\delta_m^+V_{m})^2+(\delta_m^-U_{m})^2+(\delta_m^-V_{m})^2\right\}-\frac{1}2(U_{m}^2+V_{m}^2)^2.
\sum_{\mathbf{a}\in S_-}\mathbf{B}_2(\langle1/2-a_k+v_k\rangle)=\sum_{\mathbf{a}\in S_-}\mathbf{B}_2(\langle1/2-a_k+v_k'\rangle)\quad\textrm{for each}~k=1,\ldots,g.
\mathcal D=\bigg\{u\in L^2(X\times Y):\u^y\in \mathcal D_X\ \nu\textrm{-a.e.}y\in Y,\u_x\in \mathcal D_Y\ \mu\textrm{-a.e.}x\in X,\ \mathcal E(u,u)<\infty \bigg\},
\mathcal{E}_f(V)-\mathcal{E}_f(U_0) +\frac{3}{2}\,\|f\|_{W^{-1,2}(\Omega)}\,\|u-u_0\|_{W^{1,2}(\Omega)}&\ge\frac{1}{2}\left[\left(\int_\Omega|u_0|^{2p/(p+1)}\,dx\right)^{(p+1)/p}-\left(\int_\Omega|u|^{2p/(p+1)}\,dx\right)^{(p+1)/p}\right].
\frac{1}{J}\sum_{j=J+1}^{2J}\Big [ \frac{1}{n_{1}}\log \lVert M^{z}_{n_{1}}(x_{0}+j \omega)\rVert + \frac{1}{n_{1}}\log \lVert M^{z}_{n_{1}}(x_{0}+(-j-n_{1}) \omega)\rVert \Big ] = 2L_{n_{1}}(z) + O \Big ( \frac{1}{n_{1}}\Big).
(f_1\star (f_2\star f_3)-(-1)^{(m_1-1)(m_2-1)}f_2\star (f_1\star f_3)) = (-1)^{(m_1-1)(m_2+m_3)}f_1\star_0(f_2\star_0 f_3)+ f_1\star_1(f_2\star_1 f_3)- (-1)^{(m_1-1)(m_2+m_3)+(m_2-1)(m_1+m_3 )}f_2\star_0(f_1\star_0 f_3)-(-1)^{(m_1-1)(m_2-1)}f_2\star_1(f_1\star_1 f_3).
\begin{array}{lll}\|Y_{k}\tilde{P}Y^{-1}_{l}\| = \|S^{-1}_{k}X_{k}\tilde{P}X^{-1}_{l}S_{k}\| &\leq \|S^{-1}_{k}\|\cdot\|X_{k}\tilde{P}X^{-1}_{l}\|\cdot\|S_{l}\| &\leq K M^{2}\varepsilon^{|k|}\alpha^{k-l}\varepsilon^{l}\varepsilon^{|l|}&\leq K M_1^{2}(\varepsilon\alpha)^{k-l}\varepsilon^{l},\,\,k\geq l, \end{array}
L = P_{\mu}\Pi_{\tau}^{\mu}+ Z_{\alpha \beta}\Pi_{\tau}^{\alpha \beta}+ \bar{Z}_{\dot{\alpha}\dot{\beta}}\bar{\Pi}_{\tau}^{\dot{\alpha}\dot{\beta}}- \lambda_{v}h_{v}- \bar{\lambda}_{v}\bar{h}_{v}- \lambda_{u}h_{u}- \bar{\lambda}_{u}\bar{h}_{u}\, .
\begin{array}{rclrcl}F^{(1)m}= 2\partial A^{(1)m}\, , H = \partial B -{\textstyle\frac{1}{2}}A^{(1)m}F^{(2)}{}_{m}-{\textstyle\frac{1}{2}}A^{(2)}{}_{m}F^{(1)m}\, , & F^{(2)}{}_{m}= 2\partial A^{(2)}{}_{m}\, ,{\cal F}_{m}= F^{(2)}{}_{m}+ F^{(1)n}B_{nm}\, , \end{array}
\begin{aligned}\epsilon_{\mathrm{HM}}(t) =\epsilon^{-}_{j_{\pi}(T)(E),[g]}(t)\epsilon^{+}|_{j_{\pi}(T)(E),[g]}(t) &:=\det(\mathrm{Ad}(t)_{g G^{0}(E)_{[x]}g^{-1}\cap G(F)}) \prod_{i=0}^{d-1}\mathrm{sgn}_{k_{F}^{\times}}(\det \mathrm{Ad}(t)|_{\mathfrak{W}_i}).\end{aligned}
C = B - \sum_{l = 0}^{n - 1}\kappa^{( l )}D^{2 l + 1}( \frac{\gamma_{D}^{s t}}{\Delta}) - \sum_{l = 0}^{n}\kappa^{( l ) \prime}D^{2 l}( \frac{\gamma_{D}^{s t}}{\Delta}) ,
E_{\mathbb{Q}_{x,u}}(\frac{\phi_a(X_{t+r},t+r+u)}{\phi_0(X_{t+r},t+r+u)}\vert \mathcal{F}_r)=\frac{\phi_a(X_r,r+u)}{\phi_0(X_r,r+u)}e^{-\frac{a^2}{2}t}.
H_m (r) = \frac{d}{d\, r}\log \left( \sinh^2 r + \frac{2 \, m}{3 \, \sinh r}+ O (e^{- 2 \, r})\right) = 2 + 4\, e^{- 2 \, r}- 16 \, m \, e^{- 3 \, r}+ 4\, e^{- 4 \, r}+ O (e^{- 5 \, r}).
\sum_{n=0}^{\infty}\frac{q^{2n^2+2n}}{(q/z;q^2)_{n+1}(zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}b^2_\omega(m,n)z^mq^n =\sum_{n=0}^{\infty}\frac{z^{-n}q^n}{(zq;q^2)_{n+1}}
\ll \frac{1}{X^2}\sum_{n}\frac{\Lambda(n)}{\sqrt{n}}e^{c(t-\log n)}= \frac{e^{ct}}{X^2}\left[ -\frac{\zeta'}{\zeta}\left(\frac{1}{2}+c\right) \right] \ll \frac{t\cdot e^{t/2}}{X^2},
U(\prod_{i=1}^{g}\left(g_{a_i}^{~}g_{b_i}^{~}g_{a_i}^{-1}g_{b_i}^{-1}\right)) =e^{i\Theta_s^{\pm}}~,~V(\prod_{i=1}^{g}\left(g_{a_i}^{~}g_{b_i}^{~}g_{a_i}^{-1}g_{b_i}^{-1}\right)) =e^{i\Theta_j}~,
2 L_{E H}^{\prime}= |{D^{+ +}\omega}|^{2}-{m^{+ +}}^{2}\omega \bar{\omega}- V^{+ +}( \bar{\omega}D^{+ +}\omega - \omega D^{+ +}\bar{\omega}- \mu^{+ +}) +{V^{+ +}}^{2}\omega \bar{\omega}
S_{U V \psi}= N \mathrm{t r}_{N}\psi_{i}^{\dagger}\psi_{i}+ N \frac{1}{2}\mathrm{t r}_{N}( U^{2}+ \mathrm{e}^{2 \beta}V^{2}) - N g \mathrm{t r}_{N}( \sum_{i}\psi_{i}U \psi_{i}^{\dagger}+ \sum_{i , j}L_{i j}\psi_{j}V \psi_{i}^{\dagger}) .
\begin{aligned}L(\lambda) = \min\{b^T y - \lambda(B-c^T r)\mid A^T y + r \geq w, y\geq 0, r\geq 0\}=\min_{r\geq 0}\left(-\lambda B + c^T r + \max\{(w-r)^T x \mid Ax\leq b, x\geq 0\}\right) =\min_{r\geq 0}\left(-\lambda B + c^T r +\psi(r)\right),\end{aligned}
\mathcal{F}_{n+1}(z) = k_n(z-c_n)\mathcal{F}_n(z)-\lambda_n (z-a_n)\mathcal{F}_{n-1}(z), n\geq 0, & \mathcal{F}_{-1}(z) = 0, \mathcal{F}_0(z) = 1,
\begin{array}{l}\boldsymbol{d}(w)=\boldsymbol{d}(b)-\alpha^+-\alpha^-, \boldsymbol{d}(i^+)=\boldsymbol{d}(b)-\alpha^+, \boldsymbol{d}(i^-)=\boldsymbol{d}(b)-\alpha^-,\end{array}
(VI)' \hspace{7mm}\Psi_{\bf D}=\left(\begin{array}{c}a_1 +{\rm j}b_1 a_2 +{\rm j}b_2 a_3 +{\rm j}b_3 a_4 +{\rm j}b_4\end{array}\right) \hspace{5mm}\leftrightarrow \hspace{5mm}\Psi^{SR}_{\bf C}=\left(\begin{array}{c}-b_4+{\rm i}b_2 b_3-{\rm i}b_1 a_4-{\rm i}a_2 -a_3+{\rm i}a_1\end{array}\right)
\vartheta_{3}\left( \dfrac{\pi x_{l}}{2a}\right. \left| i\dfrac{\pi v}{a^{2}}\right)=\sum_{k_{l}=-\infty}^{+\infty}e^{-\pi^{2}k_{l}^{2}v/a^{2}}e^{i \pi k_{l}x_{l}/a}.
\mathcal{E}_{V}^{\theta}(\mu) = \frac{1-\lambda}{2}\int_{M}V \, d\mu + \int_{M \times M}g(x-y) d \mu_{x}d \mu_{y}+ \lambda \int_{M}V \, d\mu + \frac{1}{\theta}{\rm ent}[\mu] +\frac{1- \lambda}{2}\int_{M}V \, d\mu.
\limsup_{n\to\infty}\| u_n^1 - u_n^2\|^2 \leq \limsup_{n\to\infty}(1+C_0) \varepsilon \|u_n^1 - u_n^2\| &+ \limsup_{n\to\infty}D_\varepsilon |u_n^1 - u_n^2|_p + \limsup_{n\to\infty}C_1 |K|_\infty |u_n^1-u_n^2|_q^q = (1+C_0) \varepsilon \limsup_{n\to\infty}\|u_n^1 - u_n^2\|
\det{(e_z,e_{\bar{z}},w)}\! = \!\det\left|\left(\begin{array}{ccc}1 0 p 0 1 \bar{p}v \bar{v}{\cal H}\end{array}\right)\right|=\left({\cal H}-pv-\bar{p}\bar{v}\right)=-{\cal L}\neq 0\,.
\delta(H,\tau) = 2^{\binom{\ell}{2}-1}\sum_{j=2}^{\ell}2^{-\binom{j-1}{2}}\delta_j \leq 2^{\binom{\ell}{2}-1}t^t n^{-1/2}\sum_{j=2}^{\ell}2^{-\binom{j-1}{2}}\leq 2^{\binom{\ell}{2}}t^t n^{-1/2}.
J :=I + \left(\frac{F^2}{w_+}+ \frac{F^2}{w_-}- 2\right) \phi^{-2n}\left(\begin{matrix}\tilde{Q}_{12}\tilde{Q}_{22}-\tilde{Q}_{12}^2 \tilde{Q}_{22}^2 -\tilde{Q}_{12}\tilde{Q}_{22}.\end{matrix}\right)_-.
{\hat{\cal T}}_{J_{1}}^{R_{1}}{\hat{\cal T}}_{J_{2}}^{R_{2}}= \sum_{R , J}\left\langle \begin{array}{l}{{R}}\\{{J}}\\ \end{array}\right. \left| \begin{array}{l l}{{R_{1}}}&{{R_{2}}}\\{{J_{1}}}&{{J_{2}}}\\ \end{array}\right\rangle{\hat{\cal T}}_{J}^{R}.
\begin{array}{ccl}4\Delta_{b}W = (A_{11,\bar{1}\bar{1}}+A_{\bar{1}\bar{1},11})f-i(f_{1\bar{1}}-f_{\bar{1}1})W-i(f_{1}W_{\bar{1}}-f_{\bar{1}}W_{1}) = W_{0}f-i(if_{0})W-i(f_{1}W_{\bar{1}}-f_{\bar{1}}W_{1}) = -2W(W-\mu )+W_{0}f+\langle \nabla_{b}W,J(\nabla_{b}f)\rangle_{\theta}.\end{array}
k_1 =-\!a_5a_2\!-\!a_6a_3, k_2=-\!a_6a_3\!-\!a_4a_1, k_3 =-\!a_4a_1\!-\!a_5a_2, k_4=-\!a_5a_2\!-\!a_6a_3, k_5 =-\!a_6a_3\!-\!a_4a_1, k_6=-\!a_4a_1\!-\!a_5a_2.
\int \frac{d^{2}\vec{k}_{T}}{( 2 \pi )^{2}}e^{i \vec{k}_{T}\cdot \vec{b}_{T}}e^{- \frac{q^{2}}2 \tau}e^{- \frac{k^{2}}2 l^{\prime}}= \frac 1{2 \pi l}e^{\frac 1{2 l}( \vec{p}_{T}\tau - i \vec{b}_{T})^{2}}e^{- \frac{p^{( 1 ) 2}\tau + p^{( 2 ) 2}l^{\prime}}{2 \sinh^{2}( v_{1}- v_{2})}}\; .
u^{F,G}_{X,Y}(a(b_{i}\boxtimes c_{j})) =\sum_{l}c_{j}\boxtimes b_{l}\langle b_{l}\ |\ab_{i}\rangle &=c_{j}\boxtimes ab_{i}=c_{j}a\boxtimes b_{i}&=ac_{j}\boxtimes b_{i}=a(c_{j}\boxtimes b_{i}) &=au^{F,G}_{X,Y}(b_{i}\boxtimes c_{j}).
A=[\begin{matrix}3&1/7\\ 4&-1/7\end{matrix}][\begin{matrix}5&0\\ 0&-2\end{matrix}][\begin{matrix}3&1/7\\ 4&-1/7\end{matrix}]^{-1}=[\begin{matrix}3&1/7\\ 4&-1/7\end{matrix}][\begin{matrix}5&0\\ 0&-2\end{matrix}][\begin{matrix}1/7&1/7\\ 4&-3\end{matrix}].
M^{\left( \alpha\right)}\left( t;\mathbf{F}\right) :=\frac{1}{\overline{\mathbf{F}}\left( \mathbb{U}_{F^{\ast}}\left( t\right) \right)}\int_{\mathbb{U}_{F^{\ast}}\left( t\right)}^{\infty}\log^{\alpha}\left(x/\mathbb{U}_{F^{\ast}}\left( t\right) \right) d\mathbf{F}\left( x\right),
\tilde{g}= -{\cal F}( R ) \, d t^{2}+{\cal F}( R )^{- 1}d R^{2}+ R^{2}d \Omega^{2}, \qquad{\cal F}( R ) = 1 - \frac{2 M}{R}+ \frac{Q^{2}}{R^{2}}- H^{2}R^{2}
R^{n,3}_\tau =n^{2\alpha+1}\int^{\tau}_0\gamma_s (\sigma'\sigma)^2(X_s) \Xi^{n,1}_s ds &=n^{2\alpha+1}\sum_{i=0}^{M-1}\int_{\tau_i}^{\tau_{i+1}}\gamma_s (\sigma'\sigma)^2(X_s) \Xi^{n,1}_s ds &=: R^{n,M,1}_\tau + R^{n,M,2}_\tau,
X_{q}^{\pm \pm}= P_{2 1}^{\pm}( P_{1 2}A_{1}R_{2 1}A_{2}) P_{2 1}^{\pm}, X_{q}^{\pm \mp}= P_{2 1}^{\pm}( P_{1 2}A_{1}R_{2 1}A_{2}) P_{2 1}^{\mp}.
T_\alpha^{\Delta}\Big(-k(t)v(t)F^{p-\alpha+1}(t)\Big) \leq\beta k(t)v(t)r(t)f(t)\big(F^\sigma(t)\big)^{p-\alpha}+(p-\alpha+1)k(t)v(t)r(t)f(t)F^{p-\alpha}(t) &\leq(p+\beta+\alpha-1)k(t)v(t)r(t)f(t)F^{p-\alpha}(t).
\sqrt{\frac{\pi g_{\Lambda}}{2}}\sum_{l = 0}^{\infty}( 2 l + 1 ) I_{l + 1 / 2}( 1 / g_{\Lambda}) P_{l}({\vec{S}}{\vec{S}^{\prime}}) \equiv \sum_{l = 0}^{\infty}( 2 l + 1 ) F_{l}^{\Lambda}( 1 / g_{\Lambda}) P_{l}({\vec{S}}{\vec{S}^{\prime}}) \ \
p_{\mathcal{G},T}(x)=\underbrace{p_{G,r}(x)p_{H,T}(x)}_{}+\underbrace{p_{G,\emptyset}(x)p_{H,T\cup\{v\}}(x)}_{}-\underbrace{xp_{G,r}(x)p_{H,T\cup\{v\}}(x)}_{}.
k_{11}\int_{\Omega}{e^{u_0^{'}+v_1}}dx+k_{12}\int_{\Omega}{e^{u_0^{''}+v_2}}dx =|\Omega|-\pi N_1 k_{21}\int_{\Omega}{e^{u_0^{'}+v_1}}dx+k_{22}\int_{\Omega}{e^{u_0^{''}+v_2}}dx =|\Omega|-\pi N_2.
\frac{1}{(2\pi )^{n}}\int_{\mathbb{A}^{(n)}_{\texttt{b}}}f(\boldsymbol{\xi}) | C_{\texttt{b}}(\boldsymbol{\xi};q,q_0) |^{-2}\boldsymbol{\xi}=\sum_{\lambda\in\Lambda^{(m,n)}_{\texttt{b}}}f \bigl( \boldsymbol{\xi}^{(m,n)}_{\texttt{b};\lambda}\bigr) \hat{\Delta}^{(m,n)}_{\texttt{b};\lambda},
t^{\frac{\alpha}{2}}\big\||\xi|^{s+1-d}e^{-t\left| \xi \right|^2}1_{B^c_n}|\xi|^{d-1}\hat u_0(\xi)\big\|_{L^{\infty, r}_\xi}\leq \big\||\xi|^{s+1-d}e^{-\left|\xi \right|^2}\big\|_{L^{\infty}}\big\|1_{B^c_n}|\xi|^{d-1}\hat u_0(\xi)\big\|_{L^{\infty, r}}= C\big\|1_{B^c_n}|\xi|^{d-1}\hat u_0(\xi)\big\|_{L^{\infty, r}}< \frac{\epsilon}{2},
\mathbf{Var}_{i\sim \mathbf{u}}\left[f_{i}(\mathcal{X}^{t})\right] =\mathbf{E}_{i\sim \mathbf{u}}\left[f_{i}(\mathcal{X}^{t})^{2}\right]-\mathbf{E}_{i\sim \mathbf{u}}\left[f_{i}(\mathcal{X}^{t})\right]^{2}=\frac{1}{q}\sum_{i=1}^{q}\left(f_{i}(\mathcal{X}^{t})\right)^{2}-\frac{1}{q^{2}}\left(\sum_{i=1}^{q}f_{i}(\mathcal{X}^{t})\right)^{2}.
\left\vert \sum_{j=1}^{N+1}\psi_{N,j}^{\ast}(Q)\right\vert \leq \frac{C_{N}^{\ast}}{q},\left\vert \sum_{j=1}^{N+1}L_{N,j}^{\ast}(q)\right\vert \leq C_{N}^{\ast}, \qquad q>0,
\zeta ( \{2 p \}_{n}) = \frac{2 p \cdot ( 2 \pi )^{2 p n}}{( 2 p n + p ) !}\left( \frac{1}{2 \sin \frac{\pi}{2 p}}\right)^{2 p n + p}\bigg \{1 + \sum_{k = 2}^{N_{p}}R_{p , k}^{2 p n + p}\bigg \}\, ,
r^{2}( \lambda ) ={\frac{1}{( \widetilde{N}- \widetilde{M})^{2}}}\sum_{m , n = \widetilde{M}}^{\widetilde{N}}\left[{\frac{W_{\lambda m n}( N = 1 6 )}{W_{\lambda m n}( N = 2 0 )}}\, \, - \, \, 1 \right]^{2}\, ,
\begin{array}{cccc}\pi{^i}_{ab}+ \Theta{^i}_{ab}= 0 \hbox{(a)}\Theta{^i}_{ak}= 0 \hbox{(b)}\Theta{^i}_{jk}= 0 \hbox{(c)}\hbox{d}^\theta \pi_i = \frac{1}{2}\|\pi\|^2e^{(4)}\wedge \bar{\theta}^{(r-1)}_i \hbox{(d)}\end{array}
- \varphi^{\prime \prime}( z ) - \left( \frac{2 \alpha^{2}- m_{\mathrm{s c}}^{2}}{| 1 - \alpha^{2}z^{2}|}- \frac{\alpha^{2}}{| 1 + \alpha z |}\delta ( 1 + \alpha z ) - \frac{\alpha^{2}}{| 1 - \alpha z |}\delta ( 1 - \alpha z ) + M_{5}^{2}\right) \varphi ( z ) = 0 ~ ,
Q \bar{Q}Q = \left( \begin{array}{c c}{{- u_{1}d_{1}d_{2}+ u_{2}d_{1}^{2}}}&{{- u_{1}u_{2}d_{1}+ u_{1}^{2}d_{2}}}\\{{u_{2}d_{1}d_{2}- u_{1}d_{2}^{2}}}&{{u_{1}u_{2}d_{2}- u_{2}^{2}d_{1}}}\\ \end{array}\right)
\limsup_{n,m\to\infty}\mathbb{P}\left\lbrace \sup_{t\in[0,T]}\left\lvert X^{n,r}_t-X^{m,r}_t\right\rvert\ge \varepsilon\right\rbrace \le \lim_{R\to\infty}\limsup_{n,m\to\infty}\left[\mathbb{P}\left\lbrace T>\tau^{n,r}_R\right\rbrace + \mathbb{P}\left\lbrace T>\tau^{m,r}_R\right\rbrace + \frac{C(T,{\omega^\prime})I^{n,m}_{R,T}}{\varepsilon^2}\right]=0\, .
\lim_{\varepsilon\rightarrow 0}\varepsilon^{n/2}\int_{\mathbb{R}^n}T(Pw_{\varepsilon\alpha}) \overline{Q(x)}w_{\varepsilon \beta}(x)dx =(\pi/\beta)^{n/2}\int_{\mathbb{T}^n}e^{i2\pi( x,m)-i2\pi kx}a(x,m)dx.
{\left|A(e^{i\theta})f_0(\theta)-\sum\limits_{j=0}^{\infty}((\bold{B}^0)^{-1}\bold{R}^0\bold{a})_je^{ij\theta}\right|^2}={(f_0(\theta)+g_0(\theta))^2}\left(\varphi_1 (\theta)+\gamma_2 \right),
\begin{aligned}p=0 : w = (h(j,j+1) \; a_{n+1})^k \; w_r \ ( k>0), p>0 :w = h(i_1, r_1) \; a_{n+1}\dots h(i_p, r_p) \; a_{n+1}(h(j,j+1) \; a_{n+1})^k \; w_r , \end{aligned}
\underset{t\in[0,T]}{\sup}\int\limits_{-\infty}^{\infty}e^{a(t)\,x_{+}^{3/2}}\, |u(x,t)|^2\,dx \le c^{*}=c^{*}(a_0;\,\|u_0\|_2;\,\|e^{\frac12a_0x_{+}^{3/2}}u_0\|_2; T),
\int_0^t \int \Big(u \cdot w_t - (v\cdot \nabla) u \cdot w +\frac{A\gamma}{\gamma-1}\phi^{\frac{2\gamma-2}{\delta-1}}w \Big)xs &=-\int u_0 \cdot w(0,x)+\int_0^t \int \Big(\phi^2 Lu \cdot w-\nabla \phi^2 \cdot Q(v) \cdot w \Big)xs.
\sum_{l=1}^{2k-2}\biggl|\tilde{a}^{\infty}_{l}-\sum_{j=0}^{l}\frac{\alpha_{l-j}}{j!}\tau^i\biggr|+\biggl|\tilde{a}^{\infty}_{2k-1}-\sum_{j=1}^{2k-1}\frac{\alpha_{2k-1-j}}{j!}\tau^j\biggr| &+\sum_{j=1}^{2k-2}\biggl|\tilde{b}_{j}^{+,\infty}-\sum_{l=0}^{2k-2}\frac{\alpha_{2k-2-l}}{(l+j+1)!}\tau^{l+j+1}\biggr|=0.
y(0) = b_0E_{\alpha, 1}(\lambda\cdot 0^{\alpha})+b_1\cdot 0 \cdot E_{\alpha, 2}(\lambda x^{\alpha})+ \int_0^0 (0-t)^{\alpha-1}E_{\alpha, \alpha}[\lambda(0-t)^{\alpha}]f(t)dt = b_0.
\tau_{t}\leq \tau_{t_{0}}\prod_{i={t_{0}}}^{t-1}\Big(1-\frac{c}{i}\Big)\leq \tau_{t_{0}}\prod_{i={t_{0}}}^{t-1}\Big(1-\frac{1}{i}\Big)\le \frac{(t_{0}-1)\tau_{t_{0}}}{t}\le \frac{2(t_{0}-1)\delta_{t_{0}}}{t}\enspace.
\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\int_{A(\eta,\delta_k)}\nabla \frak{h}^0_k \cdot \nabla\log(|x|) dx=\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}C^0_{\eta,k}\log(\frac{\eta^2}{\delta_k})<+\infty.
\begin{matrix}H_1 \subsetneq H_2 \subsetneq \cdots \subsetneq H_n = G begin{align*}.75mm]\mid \simeq \mid \simeq \mid \simeq begin{align*}.75mm]H_1' \subsetneq H_2' \subsetneq \cdots \subsetneq H_n' = G' \end{matrix}
Y_{j}(\mathbf{x})=\frac{|\mathcal{D}(T)|}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\exp\left(\mathrm{i}\left\langle \mathbf{x}, \boldsymbol{\lambda}\right\rangle\right)K\left( \boldsymbol{\lambda},\mathcal{D}(T)\right)f_{0}^{1/2}(\boldsymbol{\lambda})Z_{j}(d\boldsymbol{\lambda}),\mathbf{x}\in \mathcal{D}(T),j=1,\dots,r.
L^{-}= q \left( \begin{array}{c c c}{q^{- h_{2}}}&{0}&{0}\\{( q - q^{- 1}) \alpha^{-}}&{q^{- h_{1}- h_{2}}}&{0}\\{( q^{- 1}- q ) b^{-}}&{( q^{- 1}- q ) \beta^{-}}&{( - 1 )^{F}q^{- h_{1}- 2 h_{2}}}\\ \end{array}\right)
T_{4,5}= 8B\int e^{3i\eta\nu^2}\left(\eta-\frac{\nu}{2}\right)\frac{e^{i\beta(\eta+\nu)^3/8}}{(\eta+\nu)^3}\Bigg(e^{2ia\ln|(\eta+\nu)/2|}\chi'\left(\frac{\eta+\nu}{2}\right) &\qquad\qquad-e^{-2ia\ln|(\eta+\nu)/2|}\chi'\left(-\frac{\eta+\nu}{2}\right)\Bigg)S\left(\frac{\eta-\nu}{2}\right)d\nu.
\{x^i,x^j\}_\mathbb{R}=\frac{1}{4}{\rm Re}\,\{z^i,z^j\}\,,\,\{x^i,x^{n+j}\}_\mathbb{R}=\frac{1}{4}{\rm Im}\,\{z^i,z^j\}\,,\,\{x^{n+i},x^{n+j}\}_\mathbb{R}=-\frac{1}{4}{\rm Re}\,\{z^i,z^j\}
P_{e,1}\left( i \,| g \right) \leq \Pr\left( \sum_{t=1}^n{\bar{Z}_t}^2 - \sigma_Z^2 > \tau_n \right) \stackrel{(a)}{\leq}\frac{3\sigma_Z^4}{n\tau_n^2}\stackrel{(b)}{=}\frac{27\sigma_Z^4}{A^2 \gamma^4 n^{\kappa+b}}\leq e_1 \;,\,
\Vert e^{-\mathcal{A}_{\dag ,\varepsilon}t}-e^{-\mathcal{A}_{\dag}^0 t}-\varepsilon \mathcal{K}_\dag (t;\varepsilon )\Vert_{L_2(\mathcal{O})\rightarrow H^1(\mathcal{O}')}&\leqslant{C}(\delta) \varepsilon t^{-1}e^{-c_{\dag}t},t\geqslant \varepsilon^2,0< \varepsilon \leqslant \varepsilon_0.
r_1= 0, \ \r_2=i_2-i_1+2, r_{2s+1}= -i_1+2(i_2-i_3+\cdots -i_{2s-1}+i_{2s})-i_{2s+1},\ \s\ge 1, r_{2s+2}= -i_1+2(i_2-i_3+\cdots +i_{2s}-i_{2s+1})+i_{2s+2}+2,\ \s\ge 1,
|\mathcal{B}_{i,i-l}|=|\{(i-1,i-l,t,p)\mid(i-1,i-l,t,p)\in \mathcal{I}'_{k-1}\}|,\ \ \ \ \ &|\mathcal{B}_{i-l,i}|=|\{(i-l,i-1,t,p)\mid(i-l,i-1,t,p)\in \mathcal{I}'_{k-1}\}|.
\frac{D_{3,p}}{2\pi}=\int_{R\varepsilon^-_p\delta}^{2R\varepsilon^-_p\delta}\frac{\psi_p^2(R\varepsilon^-_p\delta)}{(R\varepsilon^-_p\delta)^2}\frac{(r-2R\varepsilon^-_p\delta)^2}r dr\leq\int_{R\varepsilon^-_p\delta}^{2R\varepsilon^-_p\delta}\frac{1}{R^{2+\gamma}}\frac1{(R\varepsilon^-_p\delta)^3}(r-2R\varepsilon^-_p\delta)^2 dr=\frac1{3R^{2+\gamma}}.
(a_1 d_1 - b_1 c_1) \cdot (a_1 b_1 + c_1 d_1 J_1) = a_1^2 b_1 d_1 + a_1 c_1 d_1^2 J_1 - a_1 b_1^2 c_1 - b_1 c_1^2 d_1 J_1 = a_1^2 b_1 d_1 - b_1 c_1^2 d_1 J_1 = \nu_1 b_1 d_1 \ne 0,
\frac{d^{\ell+1-m}}{dz^{\ell+1-m}}P_{\ell}(z) = \frac{1}{2^\ell}\sum_{p=0}^{r}\frac{(-1)^p(2\ell-2 p)!}{p! (\ell-p)!(\ell-2p)!}\frac{(m-2 p)}{\ell-2 p+1}(m+1-2 p)_{(\ell-m+1)}z^{m-2p-1}.
g_{m_{a}, m_{b}; \mu}( s ) = \sqrt{{\frac{\pm [ ( m_{a}+ m_{b})^{2}- \mu^{2}] ( m_{a}+ m_{b})^{2}}{[ k^{2}( s ) - ( m_{a}+ m_{b})^{2}] s}}}g_{m_{a}m_{b}; \mu}( 0 )
I_{0}( \frac{2 \sqrt{2}i m \omega}{\hbar \sin \omega \tau}( r_{a}r_{b}+ \vec{x}_{a}\vec{x}_{b})^{1 / 2}) = \int_{0}^{2 \pi}\frac{d \theta}{2 \pi}e x p \{\frac{2 \sqrt{2}i m \omega ( r_{a}r_{b}+ \vec{x}_{a}\vec{x}_{b})^{1 / 2}}{\hbar \sin \omega \tau}\cos \theta \}
\sum_{0<\gamma \le T}n(\gamma,k)^{2k}\le \frac{\log T}{2\pi}\sum_{\gamma\le T}\int_{J(\gamma)}n(t,k+1)^{2k}\,dt= \frac{\log T}{2\pi}\int_0^T n(t,k+1)^{2k}\Big(\sum_{\gamma\in I(t)}1\Big)\,dt\,.
P [ \psi ( x_{1}, x_{2}, x_{3}) ] = \gamma^{2}\tau^{1}\psi ( x_{1}, - x_{2}, x_{3}) , \qquad P [ \bar{\psi}( x_{1}, x_{2}, x_{3}) ] = - \bar{\psi}( x_{1}, - x_{2}, x_{3}) \gamma^{1}\tau^{1}] .
\lim_{t \to 0^+}t^n K \left( \frac{1}{\phi(t)^{n-1}}\right)= \lim_{s \to \infty}\frac1s K \left( \frac{1}{\phi\left(s^{-\frac1n}\right)^{n-1}}\right) = \lim_{s \to \infty}\frac{K \left( H^{-1}(s) \right)}{H \left( H^{-1}(s) \right)}= \infty,
F_{\alpha}(x,n,[u])=F_{\alpha}\left(x,n,u^{\beta}_{\bold{1}_i;\bold{0}},\ldots,u^{\beta}_{J_1;\bold{0}},u^{\beta}_{\bold{0},\bold{1}_i},u^{\beta}_{\bold{0};-\bold{1}_i},\ldots,u^{\beta}_{\bold{0};J_2}\right)
E_{2}( S ) = 1 - 2 4 \sum_{n}\frac{n e^{2 i \pi z}}{( 1 - e^{2 i \pi z})}= 1 - 2 4 \sum_{i = 1}^{\infty}\sigma_{1}( n ) q^{n}, \; E_{S}( S ) = \frac{d}{d S}\log ( \eta ( S ) ) ,
\begin{cases}\rho(x) w_{tt}- \operatorname{div}(K(x) \nabla w) + V_1(x,t)w =V_3(x,t)z ~\hbox{in}~ \Omega \times (0,T), \rho(x) z_{tt}- \operatorname{div}(K(x) \nabla z) + V_2(x,t)z=V_3(x,t)w ~\hbox{in}~ \Omega\times (0,T), w=z=0 ~\hbox{in}~ \omega\times(0,T),\end{cases}
{\mathbf a}_{\nu}^{\dagger}\: = \:{\cal U}_{\nu \alpha}\, \, \psi_{\alpha}\; \; \; \; \; \; \; , \; \; \; \; \; \; \;{\mathbf a}^{\rho}\: = \:{\frac{\partial}{\partial \psi_{\beta}}}\, \,{\cal U}_{\beta \rho}^{- 1}
\tau(\theta_{t_0}(a)a^\ast) = \langle U_{t_0}(a),a\rangle = \sum_{i,j=0}^\infty\langle U_{t_0}(p_i(a)),p_j(a)\rangle= \sum_{i=0}^\infty\cos(t_0)^i\|p_i(a)\|_2^2 &\geq \sum_{i=0}^\kappa\cos(t_0)^i\|p_i(a)\|_2^2\geq\frac{1}{2}2\delta= \delta\; .
w_0 =s_{l}\left(s_{l-1}s_ls_{l-1}\right)\left(s_{l-2}s_{l-1}s_{l}s_{l-1}s_{l-2}\right)\ldots \left(s_2\ldots s_{l-1}s_{l}s_{l-1}\ldots s_2\right)\left(s_1s_2\ldots s_{l-1}s_{l}s_{l-1}\ldots s_2 s_1\right),
\omega_{b \mu}^{a}( x ) \longrightarrow \omega_{b \mu}^{\prime a}( x ) = \Lambda_{c}^{a}( x ) \Lambda_{b}^{d}( x ) \omega_{d \mu}^{c}( x ) + \Lambda_{c}^{a}( x ) \partial_{\mu}\Lambda_{b}^{c}( x ) .
\frac1{x/p}\#\left\{n \le x: \, p|n, \frac{R_p(n)-\beta}{(\log_2 x)^{-1/2}}< t\right\}= \Phi\left(\frac{t}{\sqrt{\beta(1-\beta)}}\right) + O_{\epsilon}\left(\frac1{(\log_2 x)^{1/3}}\right),
[{\cal D}\mu] ={\cal D}A_\mu{\cal D}\phi{\cal D}\theta{\cal D}\rho{\cal D}\Pi^1{\cal D}\xi \prod^4_{\beta = 1}\delta\left(\Gamma_{\beta}[A_0 + \xi, A_1, \phi, \theta, \rho]\right) \det \mid \{\tilde{\Omega}_{\alpha}, \Gamma_{\beta}\}\mid.