harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{n=0}^{\infty}\left(\frac{1}{4}\right)^n \binom{2 n}{n}\frac{f_{n}}{4 n+1}\ \ \ \ \ \ \sum_{n=0}^{\infty}\left(\frac{1}{4}\right)^n \binom{2 n}{n}\frac{f_{n}}{4 n+3}.
\omega_{z,y}(s_K^*)=\omega_{z,y}((s(z)^*)^{\gamma}\,s_K^*\,s(y))=\sum_{|J_1|=\gamma m}\sum_{|J_2|=j}\overline{(z^{\otimes \gamma})_{J_1}}\,y_{KJ_1J_2}\, \omega_{z,y}(s_{J_2}).
\begin{array}{rcl}8 \partial \left[C^{(7)}+3\cdot 5\cdot 7 C^{(3)}B^{2}-3\cdot 7 C^{(5)}B \right] =\hspace{-4cm}& & ={}^{\star}G^{(2)}_{(0)}-28\left[{}^{\star}G^{(4)}_{(0)}B -30\partial C^{(3)}B^{2}\right]\, ,\end{array}
\gamma_T(t) =2\int_{x_0}^{\xi_T(t)}\Biggl[f'_T(u)\int_{0}^{u}\frac{g_T(v)}{f'_T(v)}\,dv-c_0 \Biggr]\,du, \eta^{(1)}_T(t) =\int_{0}^{t}\bigl[\varPhi'_T \bigl(\xi_T(s)\bigr)-2c_0 \bigr]\,dW_T(s), \gamma^{(3)}_T(t) = 2c_0\int_{0}^{t}\bigl[a_T \bigl(s,\xi_T(s) \bigr)-\hat{a}_T \bigl(\xi_T(s)\bigr) \bigr]\,ds,
\begin{aligned}z_k = 2\nu_k (x-\xi_k(t)), \xi_k(t) =2\mu_k t +\xi_{k,0}, \phi_k ={\mu_k \over \nu_k}z_k + \delta_k(t), \delta_k(t) =2(\mu_k^2+\nu_k^2) t +\delta_{k,0},\end{aligned}
\max_{S\in\mathcal{B}( \bigcap_{i=1}^m\mathcal{F}_i)}|S|\geqslant \max_{A\subseteq E}\biggl(2\cdot\sum_{i=1}^mr_i(E)-(m-1)|A|-\sum_{i=1}^m r_i(E-A \cap T_i^c)\biggr)
\frac{S}{A}( a_{t}) = \frac{\sigma}{2 \gamma \alpha_{\gamma}^{1 / 2}}B \left( \frac{\gamma - 1}{2 \gamma}, \frac{1}{2}\right) \left( \frac{\Lambda}{\alpha_{\gamma}}\right)^{\frac{2 - \gamma}{2 \gamma}}.
( \prod_{a}{H_{a}^{( 2 )}}^{- \Delta_{a i}^{( 2 )}}\prod_{b}{U_{b}^{( 2 )}}^{\Lambda_{a i}^{( 2 )}})^{\sigma^{( 2 )}}\sum_{m_{i}= 1}^{r_{i}}d z^{m_{i}}d z^{m_{i}}+ \sum_{\alpha}d x^{\alpha}d x^{\alpha}] ,
\left| \frac{\partial}{\partial m}(qf(mn+mh) - qf(mn)) \right| = \big| q [(n+h) f'(mn+mh)- n f'(mn) ] \big| = \frac{q}{m}|(t_1 f'(t_1) - t f'(t))| = \frac{q(t_1 -t)}{m}\left| \frac{d}{dt}(t f'(t))|_{t=t_0}\right| = qh |f'(t_0) + t_0 f''(t_0)|.
G^{B = 0}= \left( \begin{array}{c c}{{e^{\lambda^{( 0 )}+ \psi^{( 0 )}}}}&{{0}}\\{{0}}&{{e^{\lambda^{( 0 )}- \psi^{( 0 )}}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\xi_{1}}}&{{0}}\\{{0}}&{{\xi_{2}}}\\ \end{array}\right)
{\bf{P}}_{1}:= a^{3}\, \sum_{n}\frac{1}{1 + \eta \, \lambda_{n}^{(3)}}\int_{B}e_n^{(3)}(y)\,dy\otimes\int_{B}e_n^{(3)}(y)\,dy = a^{3}\, \sum_{n}\frac{1}{1 + \eta \, \lambda_{n}^{(3)}}\, \sum_{m}\int_{B}e_{n,m}^{(3)}(y)\,dy\otimes\int_{B}e_{n,m}^{(3)}(y)\,dy,
{\frac{\partial \qquad}{\partial Q_{I}( t )}}= \int_{0}^{\Delta t}d s \left[{\frac{\partial Q ( s , t )}{\partial Q_{I}( t )}}\right]{\frac{\delta \qquad}{\delta Q ( s , t )}}= \int_{0}^{\Delta t}d s{\frac{s^{I - 1}}{( I - 1 ) !}}{\frac{\delta \qquad}{\delta Q ( s , t )}},
{\delta_1^{(0)}L^{(3)}_1 = -i \,{\det\,e}\, \left(\bar \epsilon^* \gamma^\mu \psi_\mu^*\right)\, \lambda^{16}\,\left( 8 f^{(12,-12)}+ 6 \cdot 144\, D_{11}f^{(11,-11)}\right),}
\frac{-4b}{t^{2}\log^{b+1}(t)}\int_{t+\log^{(\alpha-1)b}(t)}^{t+\log^{(\alpha-1)b}(t)+\frac{1}{2}}\frac{dx}{(\log^{(\alpha-1)b}(t)+x-t)}| &= \frac{-4b}{t^{2}\log^{b+1}(t)}\left(\log(2\log^{(\alpha-1)b}(t)+\frac{1}{2}) - \log(2\log^{(\alpha-1)b}(t))\right) &=\frac{4b^{2}(\alpha-1) \log(\log(t))}{t^{2}\log^{b+1}(t)}+ E_{v_{3,ip},1b}
\Upsilon(p)\!:=\inf_{z\in\mathbb{X}}\sup_{u\in\mathcal{A}(F(\overline{x}),F'(\overline{x})w)}\Big\{\langle u,D^2F(\overline{x})(w,w)\!+\!p\rangle+\langle\nabla\! F(\overline{x})u\!-\!\overline{v},z\rangle\Big\}
\mathbf{P}(T_g>n,Z^*_{\nu_{m(n)}}>2B_{m(n)}) =\int_{2B_{m(n)}}^\infty\mathbf{P}(Z^*_{\nu_{m(n)}}\in dy)Q_{\nu_{m(n)},n}(y,0) &\leq\int_{2B_{m(n)}}^\infty\mathbf{P}(Z^*_{\nu_{m(n)}}\in dy)\frac{2y}{B_n}&=\frac{2}{B_n}\mathbf{E}[Z^*_{\nu_{m(n)}};Z^*_{\nu_{m(n)}}>B_{m(n)}].
\frac{t}{2 v_r}-\frac{t}{2 v_{r+1}}= \frac{tK_r}{2 v_rv_{r+1}}\ge \frac{t2^{\ell}}{2^{2\ell+3}u_0^2}\ge \frac{t}{2^{\ell+3}u_0^2}\ge \frac{\pi}{B2^{\ell+2}},
\varphi_{n}(\boldsymbol{\lambda}_{1},\boldsymbol{\lambda}_{2})\underset{L^{2}_{G_{\alpha}\otimes G_{\alpha}}(\mathbb{R}^{2d})}{=}\sum_{p=1}^{\infty}\gamma_{pn}\phi_{pn}(\boldsymbol{\lambda}_{1})\phi_{pn}(\boldsymbol{\lambda}_{2}),
\Delta P ( x , y ) \; = \; \left( - \frac{1}{8}\: h ( x ) \: - \: \frac{3}{8}\: h ( y ) \right) P ( x , y ) \: - \: i \frac{\partial}{\partial y^{k}}\Delta P [ \gamma^{j}h_{j}^{k}] ( x , y ) ,
\left( \begin{array}{c}{{( - 1 6 . 5 9 \mp 5 . 3 4 3 i ) \times 1 0^{- 4}}}\\{{( - 2 . 9 7 0 \pm 1 . 1 3 6 i ) \times 1 0^{- 3}}}\\{{- 0 . 3 4 8 5 \pm 0 . 3 3 9 2 i}}\\{{0 . 8 7 3 6}}\\{{0}}\\{{0}}\\{{0}}\\{{0}}\\{{\ldots}}\\ \end{array}\right) .
\phi([a,b]\cdot g')= \phi([a,b])+\phi(g')-\partial_2([a,b]\otimes g') = \phi(g')-\partial_2([a,b]\otimes g') + \partial_2\left( a\otimes a^{-1}+b\otimes b^{-1}-a\otimes b a^{-1}b^{-1}-b\otimes a^{-1}b^{-1}-a^{-1}\otimes b^{-1}+2 \cdot 1_G\otimes 1_G \right).
f(w; \check{\lambda}) = \check{C}\cdot \check{d}^w \cdot \dfrac{\prod_{i=0}^{r-p-q-1}\varGamma\left(w+\frac{i}{r-p-q}\right)}{\prod_{i=1}^{r-p-q}\varGamma\left(w+ \check{v}_i \right)},
\big( (m_{1}+2m_{2}+m_{3}) (\cot \langle \alpha_{2}, H\rangle )^{2}-(m_{1}+2n_{2}+m_{3})\big) ((\cot \langle \alpha_{2}, H\rangle )^{2}-1) = 4m_{3}(\cot \langle \alpha_{2}, H\rangle )^{2}.
S ( \bar{\gamma}_{n}) \cdot \bar{\gamma}_{n - 1}/ \gamma_{n}- \bar{\gamma}_{n}\cdot S ( \bar{\gamma}_{n - 1}/ \gamma_{n}) = - \bar{\gamma}_{n}\cdot \bar{\gamma}_{n - 1}/ \gamma_{n}+ \bar{\gamma}_{n}\cdot \bar{\gamma}_{n - 1}/ \gamma_{n}= 0 .
\sum_{0\leq i_1<...<i_k\leq n}(u_{i_1}+...+u_{i_k})\partial_{u_{i_1}}...\partial_{u_{i_k}}g+(n+2-k)\sum_{0\leq j_1<...<j_{k-1}\leq n}\partial_{u_{j_1}}...\partial_{u_{j_{k-1}}}g=0
\mathrm{d}\mu_{\, \mathrm{S U}( N )}= \frac{\pi^{2 N - 6}\, e^{- 8 \pi^{2}/ g^{2}}}{2^{8}\, ( N - 1 ) ! \, ( N - 2 ) !}\, ( M_{\mathrm{P V}})^{3 N}\left( \frac{8 \pi^{2}}{g^{2}}\right)^{N}\! \! \mathrm{d}\rho^{2}\, \mathrm{d}^{4}x_{0}\, \mathrm{d}^{2}\theta_{0}\, \mathrm{d}^{2}\bar{\beta}\, \prod_{i = 1}^{2 N - 4}\rho \, \mathrm{d}\bar{\xi}_{i}\, .
\textrm{GT}(\mathbf{k})=\textrm{GT}(k_1,k_2,\ldots,k_n)=\bigsqcup_{\mathbf{l}\in \underline{[k_1,k_2)}\times\cdots\times\underline{[k_{n-1},k_n)}}\textrm{GT}(\mathbf{l}).
\varepsilon (0)=p_1(0)=p_2(0)=\frac{1}{8\pi^2a^4}\int_{0}^{\infty}{dzz^3\left[ \frac{K_0(z)}{I_0(z)}+\frac{K'_0(z)}{I'_0(z)}\right]}=-0.0168a^{-4}
\varphi_{\mathbf{L}}: \mathbb{C}[y_{\mathcal{I}}:\,\mathcal{I}\in\mathfrak{G}(\mathbf{L})]\longrightarrow\mathbb{C}[x_{\alpha}:\,\alpha\in[n]], \varphi_{\mathbf{L}}(y_{\mathcal{I}}) := \prod_{\alpha\in\mathcal{I}}x_{\alpha}.
\int_{\Omega}|u_k|^p\, dx = \int_{\Omega}\frac{|u_k|^p}{|x|^{\frac{p\alpha}{q}}}\, |x|^{\frac{p\alpha}{q}}\, dx \leq C\Big(\int_{\Omega}\frac{|u_k|^q}{|x|^\alpha}\, dx\Big)^{\frac{p}{q}}\leq C(1+[u_k]_{s,p}^{\frac{p}{q}}),
\left[P^{s,N}_{HP}(t)f\right](x)=\left[\mathcal{S}^N(t)f\circ \mathsf{eval}_N\right](H)=\left[\mathcal{S}^N(t)f\circ \mathsf{eval}_N\right](U^*xU)\ , \forall U\in \mathbb{U}(N),
S = - \gamma \int d \tau \int d \sigma \sqrt{{\dot{x}\acute{x}}^{2}-{\dot{x}}^{2}{\acute{x}}^{2}}- \sum_{i = 1}^{2}m_{i}\int d \tau \sqrt{{\left( \frac{d x^{\mu}({\tau}_{i},{\sigma}_{i}( \tau ) )}{d \tau}\right)}^{2}}
U=\int_{0}^{a}\frac{bD}{24}[12(\frac{d^{2}w_{x}}{dx^{2}})^{2}+b^{2}(\frac{d^{2}\theta_{x}}{dx^{2}})^{2}+24(1-\nu)(\frac{d\theta_{x}}{dx})^{2}]dx,
\mu_{\varphi}(\Psi)-\frac{3}{\pi}\mu(\Psi)= \sum_{k=1}^{\infty}\langle \Psi,u_{k}\rangle \langle u_k ,|\varphi|^2 \rangle+\frac{1}{4\pi}\int_{\mathbb{R}}\langle\Psi, E(\cdot,\tfrac{1}{2}+it)\rangle \langle E(\cdot,\tfrac{1}{2}+it) ,|\varphi|^2 \rangle dt.
\begin{aligned}J_3 \leq \Vert \langle y\rangle^{l+k}I^3_1\Vert_{L^2_y}\Vert \langle y\rangle^{l+k}D^\alpha u\Vert_{L^2_y}+\Vert \langle y\rangle^{l+k}I^3_2\Vert_{L^2_y}\Vert \langle y\rangle^{l+k}D^\alpha h\Vert_{L^2_y}.\end{aligned}
( H_{1}H_{2}. . . H_{n})^{4 u^{( \alpha )}/ r^{( \alpha )})}[ \sum_{i = 1}^{n}H_{i}^{- 2 / r^{( \alpha )}}d z_{i}^{m_{i}}d z_{i}^{m_{i}}+ d x^{\gamma}d x^{\gamma}] =
\varphi(u) =\sum_{k=0}^{u+2}s_{u+2-k}\ \varphi (k+1)-a_{u+1}b_0c_0\varphi(2)-a_{u+1}b_0c_1\varphi(1) &-c_0\varphi(1)\sum_{k=0}^{u+2}a_k b_{u+2-k}, u \in\mathbb{N}_0.
\left(p_\zeta^\lambda\right)^{(2)}(x) = -\frac{2(\lambda+1)\,e^{-\rho}\,(1-e^{-2\rho})}{\Sigma_n\,(1-2e^{-\rho}\cos\theta_1+e^{-2\rho})^{\lambda+2}}\,\cos\theta_1 &+\frac{4(\lambda+1)(\lambda+2)\,e^{-2\rho}\,(1-e^{-2\rho})}{\Sigma_n\,(1-2e^{-\rho}\cos\theta_1+e^{-2\rho})^{\lambda+3}}\,\sin^2\theta_1\cos^2\theta_2
\begin{aligned}u_h^n(q) = F_\tau^h(n;q)\big(\mathcal{I}_h v-D_h(q)\mathcal{I}_h^\partial b\big) + D_h(q)\mathcal{I}_h^\partial b + \tau \sum_{j=1}^n E_\tau^h(j;q) P_h f &= F_\tau^h(n;q)\big(\mathcal{I}_h v-D_h(q)\mathcal{I}_h^\partial b\big) + D_h(q)\mathcal{I}_h^\partial b + (I- F_\tau^h(n;q)) A_h(q)^{-1}P_h f,\end{aligned}
E^s(\lambda,0) =\{u(0)\in\mathbb{R}^{2n}:\,Ju'(t)+A(\lambda,t)u(t)=0,\, t\in\mathbb{R}; u(t)\rightarrow 0, t\rightarrow\infty\}, E^u(\lambda,0) =\{u(0)\in\mathbb{R}^{2n}:\,Ju'(t)+A(\lambda,t)u(t)=0,\, t\in\mathbb{R}; u(t)\rightarrow 0, t\rightarrow-\infty\},
s_{n,j}=\frac{a_{n,j}-\bar{a}_n}{\sqrt[]{\sum_{j=1}^n\left(a_{n,j}-\bar{a}_n\right)^2}}, S_n=\sum_{j=1}^n s_{n,j}T_{n,j}=\left[Z_{n,j}=s_{n,j}T_{n,j}\right]=\sum_{j=1}^{n}Z_{n,j}.
S(z_1 ,z'_1 ;z_2 ,z_2 ')L(z_1 ,z_1 ')\otimes L(z_2 , z_2' )=L(z_2 , z_2' )\otimes L(z_1 ,z_1 ')S(z_1 ,z'_1 ;z_2 ,z_2 ').
I ={\frac{1}{1 6 \pi G}}\int d^{4}x \sqrt{- g}\biggl \{R -{\frac{1}{2}}\int_{0}^{1}d \xi_{1}\int_{\xi_{1}}^{1}d \xi_{2}\left[ \partial \sigma ( \xi_{1}) - \partial \sigma ( \xi_{2}) \right]^{2}- \int_{0}^{1}d \xi \; e^{- \sigma ( \xi )}F ( \xi )^{2}\biggr \},
{\cal G}_{k}( U , U^{\prime}) = - \frac{1}{N^{2}}\frac{1 + \gamma}{2 \gamma k^{2}}\frac{1}{U U^{\prime}}\left( \frac{U_{*}}{U_{>}}\right)^{- \gamma}\left[ ( \gamma - 1 ) \left( \frac{U_{*}}{U_{<}}\right)^{\gamma}+ ( \gamma + 1 ) \left( \frac{U_{*}}{U_{<}}\right)^{- \gamma}\right] ~ .
\int_{V}|\overline{\partial}\chi_{k}|^2_{\omega}e^{-k\varphi}\omega_{n}\lesssim \left(\frac{\sqrt{k}}{\log k}\right)^2 \int_{\frac{\log k}{2\sqrt{k}}< |y| < \frac{\log k}{\sqrt{k}}, (x, y) \in V}e^{-2k\pi|y|^2}dxdy \lesssim \left(\frac{\log k}{\sqrt{k}}\right)^{n-2}e^{-\pi (\log k)^2/2}= O(k^{-\infty})
\pi^{*}c^{h}_{{\rm top}}(T_{\sigma}) = c^{h}_{{\rm top}}\left (\bigoplus_{\alpha \in \Delta^{+}\setminus \Delta_{H}^{+}}L_{-\alpha}\right ) = \prod_{\alpha \in \Delta^{+}\setminus \Delta_{H}^{+}}c^{h}_{1}(L_{-\alpha}).
\rho_{p \rightarrow 1 + 2 + 3}= \frac{( 3 2 \pi )^{2 - 2 \ell}}{( \Gamma ( \ell - 1 / 2 ) )^{2}( p^{2})^{\ell - 1}}\int_{( m_{1}+ m_{2})^{2}}^{( \sqrt{p^{2}}- m_{3})^{2}}s^{1 - \ell}{\cal D}^{\ell - 3 / 2}\, d s ,
\sum_{i,j,k,l=0}^\infty \frac{(a_+a_-+b_+b_-)^i}{i!}\frac{(a_+b_-)^j(b_+a_-)^k}{j!k!}\frac{(i+j+k+l)!}{(i+j+k)!l!}Q^{2i+j+k+2l}.
\begin{bmatrix}0 -1 \alpha 0 \overline\alpha 0 q -q\alpha 0 -q\overline\alpha 0 -\overline\alpha q -\alpha J(T^{-2r},T^{-3r}) 0 0 0 -\overline\alpha J(T^{-r},T^{-r}) q -\alpha J(T^{-3r},T^{-3r}) 0 -\alpha 0 -\overline\alpha J(T^{-2r},T^{-r}) q\end{bmatrix}
\int \prod_{n}dX{n}^{I}dX{n}^{II}\frac{dk_{n}}{2\pi}\frac{dp_{n}}{2\pi}\frac{dq_{n}}{2\pi}\frac{dl_{n}}{2\pi}e^{-ik_{n}X_{n}^{I}-ip_{n}X_{n}^{I}-il_{n}X_{n}^{II}-iq_{n}X_{n}^{II}}e^{-\sum_{m>0}(\frac{k_{m}^{2}}{4 \omega_{m}}+ \frac{p_{m}^{2}}{4 \omega_{m}}+ \frac{l_{m}^{2}}{4 \omega_{m}}+ \frac{q_{m}^{2}}{4 \omega_{m}})}
i(u_1-{{u}}) \lambda -\lambda^2 u_2-i\lambda^3 u_3=\frac{2{u_2}^3}{27{u_3}^2}- \frac{(u_1-{{u}}){u_2}^2}{3{u_3}}+i\left( \frac{u_1-{{u}}}{(3u_3)^{1/3}}- \frac{u_2}{(3u_3)^{4/3}}\right)\lambda' -i \frac{\lambda'^3}{3}
\theta := \max_{\prec_\chi}\left(\left\{k \in W \, \mid \, k \prec_\chi \min_{\prec_\chi}(V)\right\}\right)\qquad\gamma := \min_{\prec_\chi}\left(\left\{k \in W \, \mid \, \max_{\prec_\chi}(V) \prec_\chi k\right\}\right),
P_{0}( l ) = \frac{4 \pi^{2}}{l !^{2}( l + 1 ) !^{2}}\left( \frac{\lambda \omega r_{0}}{4}\right)^{2 l + 2}= \frac{\pi^{2}\omega^{4}r_{1}^{2}r_{5}^{2}}{4}\frac{( \omega^{2}r_{1}r_{5})^{2 l}}{2^{4 l}l !^{2}( l + 1 ) !^{2}},
\begin{array}{c}{{\zeta_{\nu}( s ) = \displaystyle{- \frac{1}{\pi}}\, \Re \left\{x^{1 - s}\, e^{\displaystyle{- \imath \frac{\pi}{2}s}}\displaystyle{\int_{0}^{\infty}}( y - \imath )^{- s}\, \displaystyle{\frac{{\Delta_{\nu}^{T E}}^{\prime}\left( \imath x ( y - \imath ) \right)}{\Delta_{\nu}^{T E}\left( \imath x ( y - \imath ) \right)}}\d y \right\},}}\\ \end{array}
\begin{array}{lll}q_-(s) = \mathcal{K}_{s\tau}q_-(\tau) & +\displaystyle \int_{\tau}^{s}\mathcal{K}_{s s'}\left (P_-\left [\mathcal{N}(q)+\mathcal{D}_s(\nabla q)+\mathcal{R}_s (q)+b'(s')\mathcal{M}(q)\right ]\right )ds'.\end{array}
\hspace{1 5 m m}\times \prod_{i = 1}^{q}u_{i}^{a^{'}}\prod_{i = 1}^{q - 1}( u_{i}- 1 )^{b^{'}- 1}( u_{i}- z )^{c^{'}}\prod_{i = q - k + 1}^{q}( 1 - u_{i})^{b^{'}- 1}( z - u_{i})^{c^{'}}
\tau_{k + 1, \ell +1}^{(\alpha,\beta)}\tau_{k, \ell}^{(\alpha,\beta + 1)}=\tau_{k + 1, \ell + 1}^{(\alpha,\beta + 1)}\tau_{k, \ell}^{(\alpha,\beta)}-\tau_{k +1, \ell}^{(\alpha,\beta + 1)}\tau_{k, \ell + 1}^{(\alpha,\beta)},
\mathbf{P}\left((\eta^{N,Mar}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right)=\frac{\mathbf{E}\left(\nu(\eta^{Mar}_0)^{-1}\mathbf{1}_{\left\{(\eta^{Mar}_n)_{n=1}^{M}=(x_n)_{n=1}^{M},\:\eta^{Mar}_N=\eta^{Mar}_0,\:S_N>0\right\}}\right)}{\mathbf{E}\left(\nu(\eta^{Mar}_0)^{-1}\mathbf{1}_{\left\{\:\eta^{Mar}_N=\eta^{Mar}_0,\:S_N>0\right\}}\right)}.
Z_{k}\Big [ \begin{array}{c}{{\gamma}}\\{{\delta}}\\ \end{array}\Big ] ( \tau ) = \sum_{n = 0}^{k}e^{i \pi \delta n}\chi_{k}^{n}( \tau ) \left( \chi_{k}^{n + \gamma ( k - 2 n )}( \tau ) \right)^{*}
\det B_{n,m}(x): =\det \left[\binom{x+m+2i}{i-j+m}-\binom{x+m+2i}{i-j+m-1}\right]_{i,j=0}^{n-1}&=\prod_{i=1}^n\prod_{j=1}^m\frac{(x+i-j)(x+2i+j-2)}{(x+2i-j)(i+j-1)}.
\lim_{n\to \infty}\frac{1}{n}\sum^{n-1}_{j=0}\varphi \circ f_{\theta^{-j}\omega}^{(j)}(X_{s^{\prime \prime}}(\theta^{-j}\omega)) =\lim_{n\to \infty}\frac{1}{n}\sum^{n-1}_{j=0}\varphi \circ h(\omega) (E_k^j (x)) .
R = \frac{1}{2}\log\left(1+\gamma\right), R_b = \frac{1}{2}\log\left(1+3\gamma\right), R' = \frac{1}{2}\log\left(1+\frac{\gamma}{(1{-}\alpha)/2}\right), R_s = \frac{1}{2}\log\left(1+\frac{4\gamma+1{-}\alpha}{\alpha}\right).
0= \langle \tau_{H}, \alpha_{1}\rangle \{(m_{1}+2m_{2}+m_{3})(1-(\cot \langle \alpha_{1}, H\rangle )^{2}) &+(m_{1}+2n_{2}+m_{3})(1-(\tan \langle \alpha_{1}, H\rangle )^{2}) +4m_{3}\}(\alpha_{1}+\alpha_{2}).
e^{2 A}b \left( - \dot{b}\dot{\phi}- b \ddot{\phi}- 2 \dot{A}b \dot{\phi}\right) + 4 A^{\prime}\phi^{\prime}- \frac{b^{\prime}\phi^{\prime}}{b}+ \phi^{\prime \prime}= b^{2}\frac{\partial U}{\partial \phi}.
\mathcal{S}(\tau_1,\tau_2;h) = \prod_{p}\Big ( 1 - \frac{f_1(p) f_2(p)}{p^{\tau_2}}\Big ) \prod_{p^{\alpha}\mid \mid h}\sum_{j=0}^{\alpha}\Big( \frac{f_1(p^{j}) f_2(p^{j})}{p^{\tau_1 j}}- \frac{f_1(p^{j+1}) f_2(p^{j+1})}{p^{\tau_1 j + \tau_2}}\Big) \Big ( 1 - \frac{f_1(p) f_2(p)}{p^{\tau_2}}\Big )^{-1}.
\begin{array}{lll}\mu\circ (\eta\otimes(f\circ \eta)) = \mu\circ ( \eta\otimes id_M)\circ (id_I\otimes (f\circ\eta)) &= \lambda_M\circ (id_I\otimes (f\circ\eta)) &= (f\circ \eta)\circ \lambda_I. & \mbox{(by naturality of $\lambda$)}\end{array}
\frac{g^{k}_{\beta}}{K}\leq g\leq Kg^{k}_{\beta}\ (\textrm{quasi-isometric}),\g^{k}_{\beta}=\Sigma_{j=1}^{k}\frac{\beta_{j}^{2}}{|z_{j}|^{2-2\beta_{j}}}dz_{j}\otimes d\bar{z}_{j}+\Sigma_{j=k+1}^{n}dz_{j}\otimes d\bar{z}_{j}.
\|A+\delta K\| \geq \|\gamma_n^{(d)}(A+\delta K) \|=\| \gamma_n^{(d)}(A)+\delta R\| &\geq \|\gamma_n^{(d)}(A)\xi+\delta R\xi\|=(1+\delta)\|\gamma_n^{(d)}(A)\xi\| &=(1+\delta)\|\gamma_n^{(d)}(A)\|\geq (1+\delta)(1-\delta/2)\|A\| &>\|A\|.
C(t):=\sup\left\lbrace\bar{\omega}(t)\left(|P(t)||A(t-\tau,q,y_{\tau}(t),u_{\tau}(t))|+\frac{1}{2}|\dot{P}(t)|+|\bar{d}(t)||P(t)|\right):q\in Q(t-\tau)\right\rbrace.
n|H| = |G| = \, \sum_{\sigma = 1}^{s}|K g_\sigma H| & = m|H| \,+\, \sum_{\sigma = m+1}^{s}|H|\cdot (H : H \cap K^{g_\sigma}) &\geq\, m|H| \,+\, (s-m)p|H|.
\phi ( x^{\mu}) = \int_{( \infty )}\frac{d^{3}\vec{k}}{( 2 \pi )^{3}2 \omega ( \vec{k}\, )}\left[ a ( \vec{k}\, ) e^{- i k \cdot x}+ a^{*}( \vec{k}\, ) e^{i k \cdot x}\right] \ ,
1+9(\frac{1}{4})^{4}+17(\frac{1\times5}{4\times8})^{4}+25(\frac{1\times5\times9}{4\times8\times12})^{4}+\cdot\cdot\cdot=\frac{2\sqrt{2}}{\sqrt{\pi}\Gamma^{2}(\frac{3}{4})}.
V_{10}=-\varepsilon\bar{m}\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})\sum_{i=1}^{d}\left[\Theta_{p_{i}x_{i}}(t,x,\mu^{1},Dw^{1})-\Theta_{p_{i}x_{i}}(t,x,\mu^{2},Dw^{2})\right]\dx,
\sum_{i=0}^{n-3}\binom{n-2}{i}\frac{\partial^{i+1}t}{\partial u^{i+1}}\frac{\partial^{n-2-i}\tilde{B}}{\partial u^{n-2-i}}+\frac{\partial^{n-1}t}{\partial u^{n-1}}\tilde{B}.
H^s_h(\mathbb{R}) := \left\{v_h \in \mathcal{S}'(\mathbb{R})\, \Big| \, \sup_{h\in ]0,1]}\int_{\mathbb{R}}(1+|h\xi|^2)^s |\hat{v}_h(\xi)|^2 \, d\xi < + \infty \right\}\, .
\Lambda_{r,n}^{d}=\binom{n}{r}\frac{(2d+1)!!}{2^{d}(d+1)!}\sum_{j=0}^{d}(-1)^{d-j}\binom{d+\frac{1}{2}}{j}\binom{d+\frac{1}{2}}{d-j}\frac{\Gamma(r+j+\frac{3}{2})\Gamma(n+d-r-j+\frac{3}{2})}{\Gamma(n+d+3)},
Y_t = y + \int_0^t \tilde b ( Y_s) ds \int_0^t \int_{\{|z| \le r \}}g(Y_{s-}, z) \tilde N(ds, dz) + \int_0^t \int_{\{|z| > r \}}g(Y_{s-}, z) N(ds, dz), \;\; t \ge 0,
H_{t,\omega}(0)=\int_{B_{2C_1t^2}}-p_{t,\omega}(0,y)\log\left(p_{t,\omega}(0,y)\right)\;dy+\int_{\mathbb{R}^d\setminus B_{2C_1t^2}}-p_{t,\omega}(0,y)\log\left(p_{t,\omega}(0,y)\right)\;dy.
\phi^2(t, u -w_n) - 2(u -w_n) w_n'' \geq \sum_{j \in \mathcal{I}_n (t)}\left(\tilde{\phi}_j^2 (t, u-w_n)\right) - 2|u-w_n| |w_n''| & = \sum_{j \in \mathcal{I}_n (t)}\left(\psi_j^2 (t)|u-w_n|\right) - 2|u-w_n| |w_n''| & = |u-w_n| \left(\sum_{j \in \mathcal{I}_n (t)}(\psi_j^2 (t)) - 2|w_n''(t)|\right) & \geq | u -w_n|.
\frac{dy_i}{dt}= \frac{a_i x_i y_i}{1 + x_i}-d_i y_i+\rho_i(1-s)\left(\frac{a_ix_iy_i}{1+x_i}y_j\right)+ \rho_is( y_j-y_i) &\geq \frac{a_ix_iy_i}{1+x_i}-d_iy_i-\rho_isy_i \Rightarrow \frac{dy_i}{y_idt}\geq \frac{a_ix_i}{1+x_i}-d_i-\rho_is
\mathbf{r}_S^S = \mathbf{f}_S^S - \beta \mathbf{f}_S^S = (1-\beta) \mathbf{f}_S^S \mathbf{r}_{S^c}^S = \mathbf{R}_{S^c}+ \beta \mathbf{P}_{S^c N}\mathbf{f}^S - \mathbf{f}_{S^c}^S = \mathbf{R}_{S^c}+ \beta \mathbf{P}_{S^c N}\mathbf{f}^S.
S_{\lfloor nt \rfloor}+ \{nt\}X_{\lfloor nt \rfloor +1}= \sum_{k=1}^{\lfloor nt \rfloor}U^{*}Z_{k}+ \{nt\}U^{*}Z_{\lfloor nt \rfloor +1}= U^{*}\left( \sum_{j=-\infty}^{\lfloor nt \rfloor +1}a_{nj}(t) (U\varepsilon_{j})\right):=U^{*}\tilde{\zeta}_{n}(t)
E_1(T_{(2,8)})=(\Delta_1(T_{(2,8)}))\cdot(t_1-1,t_2-1)=(t_{1}^{3}+t_{1}^{2}t_{2}+t_{1}t_{2}^{2}+t_{2}^{3})\cdot(t_1-1,t_2-1).
W_{\bf a}[ R^{1}\times \Sigma_{g}^{2}] \simeq -{\frac{A ( \Sigma )}{8 \pi^{3 / 2}}}{\frac{\bf a}{\epsilon}}+{\frac{A ( \Sigma )}{3 2 \pi}}{\bf a}^{2}\ln ({\frac{|{\bf a}| \epsilon}{2}}) .
Q\xi(r) =\sum_{j\in\S,j\neq r}q_{rj}(\xi_j-\xi_r)=\sum_{j\notin F_i,j\in \S}q_{rj}(\xi_j-\xi_r) &=\sum_{k<i}\big(\sum_{j\in F_k}q_{rj}\big)(\xi_k^F-\xi_i^F)+\sum_{k>i}\big(\sum_{j\in F_k}q_{rj}\big)(\xi_k^F-\xi_i^F) &\leq \sum_{k<i}q_{ik}^F(\xi_k^F-\xi_i^F)+\sum_{k>i}q_{ik}^F(\xi_k^F-\xi_i^F)=\big(Q^F\xi^F\big)(i),
1-\frac{c^{2}}{2}+O(c^{4})=1-\frac{a^{2}}{2}-\frac{b^{2}}{2}+\frac{a^{2}b^{2}}{4}+O(a^{4})+O(b^{4})+cos(C)(ab+O(a^{3}b)+O(ab^{3})+O(a^{3}b^{3}))
2= q_1+q_2+q_3 > \frac{16}{27}\frac{1}{A+\sqrt{A^2+\frac{32}{27}(a_2+a_3)}}+\frac{1}{B+\sqrt{B^2+\frac{32}{27}(a_1+a_3)}}+ \frac{1}{C+\sqrt{C^2+\frac{32}{27}(a_1+a_2)}}\ge \frac{16}{3}\frac{1}{A+B+C+\sqrt{A^2+\frac{32}{27}(a_2+a_3)}+\sqrt{B^2+\frac{32}{27}(a_1+a_3)}+\sqrt{C^2+\frac{32}{27}(a_1+a_2)}}
g(1)-g(n) <g(1)-g(k)+g(l)-g(n)-(1-\alpha)\Big(3-\Big|\frac{1}{k}-\frac{1}{l}\Big|\Big) < 2+\frac{1}{k}+3-\frac{1}{n}+\frac{1}{l}-3+\Big|\frac{1}{k}-\frac{1}{l}\Big| +3\alpha = 2-\frac{1}{n}+\frac{1}{k}+\frac{1}{l}+\Big|\frac{1}{k}-\frac{1}{l}\Big|+\frac{2}{n}-\frac{2}{n+1}-3\alpha \le 2+\frac{1}{n}-3\alpha \le (1-\alpha)d(1,n).
\begin{gathered}w_{-1}\wedge w_{-1}=0,w_{0}\wedge w_{-1}=-q^{2}w_{-1}\wedge w_{0},w_{0}\wedge w_{0}=-q(q-q^{-1})w_{-1}\wedge w_{1}, w_{1}\wedge w_{-1}=-w_{-1}\wedge w_{1},w_{1}\wedge w_{0}=-q^{2}w_{0}\wedge w_{1},w_{1}\wedge w_{1}=0.\end{gathered}
H=2 \prod_{\alpha < \beta}^{M}(z_{\alpha}-z_{\beta})^{\frac{\kappa^2}{\lambda}}\prod_{i < j}^{N}(x_{i}-x_{j})^{\lambda}{\cal L}_{0}\prod_{i < j}^{N}(x_{i}-x_{j})^{-\lambda}\prod_{\alpha < \beta}^{M}(z_{\alpha}-z_{\beta})^{-\frac{\kappa^2}{\lambda}}=
Z [ j ] = e^{\frac{i}{\hbar}\int d x^{4}{\cal{L}}_{I}( \frac{\hbar}{i}\frac{\delta}{\delta j})}Z_{0}[ j ] \textrm{w i t h}Z_{0}[ j ] = \int_{\Lambda}{\cal{D}}\eta \; e^{\frac{i}{\hbar}\int d x^{4}\left( \frac{1}{2}[ ( \partial \eta )^{2}- m^{2}\eta^{2}] + j \eta \right)},
S(X; \pi, \hat{\phi}, W) := \sum_{(d,2)=1}\mu^2(d)\sum_{p}\frac{a_{\pi}(p)\log p}{\sqrt{p}}\left( \frac{8d}{p}\right) \hat{\phi}\left( \frac{\log p}{M\log X}\right) W \left( \frac{d}{X}\right).
|df_{n, p_j}(v_j)| \leq |\alpha_{q_j,p_j}| \cdot 2 \max\left\{\frac{1}{|\alpha_{q_j,n}|}, K_1 \cdot \frac{|\alpha_{n,p_j}|^{1-1/x}|\beta_{n,p_j}|}{|\alpha_{q_j,p_j}|}\right\}&= 2 \max\left\{|\alpha_{n,p_j}|, K_1 \cdot|\alpha_{n,p_j}|^{1-1/x}|\beta_{n,p_j}|\right\}.
(\pi_E (a) \otimes_B 1_A ) \xi = \sum_{i=1}^n a u_i \otimes u_i^* e^{-1/2}=\sum_{i=1}^n \sum_{j=1}^n u_j E(u_j^* a u_i ) \otimes u_i^* e^{-1/2}&= \sum_{j=1}^n u_j \otimes \sum_{i=1}^n E(u_j^* a u_i ) u_i^* e^{-1/2}=\sum_{j=1}^n u_j \otimes u_j^* a e^{-1/2}= \xi a,
A(x,y)=a_0+\sum_{k=1}^\infty \frac{b_k}{2}\big(\sqrt{2}\cos(2\pi x)\big)\big(\sqrt{2}\cos(2\pi y)\big)+\sum_{k=1}^\infty \frac{b_k}{2}\big(\sqrt{2}\sin(2\pi x)\big)\big(\sqrt{2}\sin(2\pi y)\big),
\biggl[\;\sum_{0\le i,j\le n-1}a_i^{q^j}X^{q^j(q^i-1)}\;\biggr]^{q-1}\equiv\;_{q^n/q}(a_0)^{q-1}+\Bigl[1-_{q^n/q}(a_0)^{q-1}\Bigr] X^{q^n-1}\cr \kern-3mm\pmod{X^{q^n}- X}.
\int_{M\setminus\Sigma}|(\nabla\chi_n)\mathfrak{u}|^2\leq C^2n^2\Big(64\pi^2\frac{1}{n^2}\Big)\int_{N_{\frac{1}{n}}}|\nabla\mathfrak{u}|^2= 64C^2\pi^2\int_{N_{\frac{1}{n}}}|\nabla\mathfrak{u}|^2.
{\displaystyle \left\langle u,v\right\rangle_{\varepsilon}:=\frac{1}{\varepsilon^{n}}\int_{M}\varepsilon^{2}\nabla_{g}u\nabla_{g}v+(a-\omega^{2})uvd\mu_{g};\ \ \ \|u\|_{\varepsilon}=\left\langle u,u\right\rangle_{\varepsilon}^{1/2}.}
D = D(\xi_1,\xi_2) = (2\cdot\xi_1^{\,0}+3\cdot \xi_1^{\,1}- 4\cdot\xi_1^{\,3})\cdot\xi_2^{\,0}+(0\cdot\xi_1^{\,0}+3\cdot\xi_1^{\,1}-7\cdot\xi_1^{\,2})\cdot\xi_2^{\,1}+ (1-4\cdot\xi_1)\cdot\xi_2^{\,2}
\langle e_{a,1}(t,R \lambda(t)),\phi_{0}(R)\rangle_{L^{2}(R dR)}=-\frac{4}{\lambda(t)}\int_{t}^{\infty}\frac{\lambda''(s)}{1+s-t}ds + \frac{4 b}{\lambda(t)t^{2}\log^{b}(t)}+\frac{4 \lambda''(t) \log (\lambda(t))}{\lambda(t)}+f_{2}(\lambda(t),\lambda'(t),\lambda''(t))
\widetilde{S}_{\rho \sigma}^{( 3 )}={\lim_{\left| k_{2 \mu}\right| \rightarrow \infty}}\frac{k_{2}^{\lambda}\Gamma_{\rho \sigma \lambda}^{( 3 )}( p_{1}, p_{2}; q_{3})}{q_{3}^{2}- M^{2}+ i \varepsilon}
\int d\vec x\biggl(v^k\frac{\delta}{\delta A^{0k}}+\frac{1}{2}\,u^{kl}\frac{\delta}{\delta A^{kl}}+\frac{1}{2}\,\omega^{kl}\frac{\delta}{\delta\pi^{kl}}\biggr)\int d\vec y\,V^{(0)}=0
\psi_{j}(x)=\int \frac{dp_{j}^{1}}{\sqrt{4\pi}p_{j}^{0}}\left(a_{j}(p)u_{j}(p)e^{-ip_{j}\cdot x}+a_{_{\bar{\jmath}}}^{\dagger}(p)v_{j}(p)e^{ip_{j}\cdot x}\right) \,,\qquad
{\cal H}= -K\cdot \left({\cal K}\frac{\delta}{\delta K}\right)-K\cdot\left(\frac{\delta}{\delta K}\vee \frac{\delta}{\delta K}\right) - \frac{1}{N^2}K\cdot\left( K \cdot \left( \wedge \frac{\delta}{\delta K}\right)\right) - K\cdot T .
\int_{a}^{t}\int_{|y|\geq\eta}|K(t-s,y)|dyds = \int_{a}^{t}(t-s)^{-1-\frac{\alpha d}{2}}\int_{|y|\geq \eta}K(1,(t-s)^{-\alpha/2}y)dyds &=\int_{a}^{t}(t-s)^{-1}\int_{|y|\geq \eta(t-s)^{-\alpha/2}}K(1,y)dyds &\leq \left(\int_{a}^{t}\eta^{-2/\alpha}ds\right)\left(\int_{\mathbf{R}^{d}}|y|^{2/\alpha}K(1,y)dy\right) &\leq N(t-a)\eta^{-2/\alpha}.
\omega^{(\zeta,\theta)}(x)= \dfrac{2^{2\zeta-1}|\Gamma(e)|^2 e^{\theta \pi}(e^{-\cot^{-1}(\frac{x}{\omega})})^{2\theta}}{\Gamma(2\zeta-1)2\pi (\omega^2+x^2)^\zeta},
\{e_j,f_k\}=\sum_{i=0}^{m-1}\sum_{\alpha=1}^d\left(\frac{\partial e_j}{\partial[\tilde{w}_i]_{\alpha,j}}\frac{\partial f_k}{\partial[\tilde{v}_i]_{j,\alpha}}-\frac{\partial e_j}{\partial[\tilde{v}_i]_{j,\alpha}}\frac{\partial f_k}{\partial[\tilde{w}_i]_{\alpha,j}}\right).
P_{-}^{-1}(x) P_{+}(x) = \begin{pmatrix}1 0 0 0 0 i e^{-2\pi i \alpha}x_-^{-\beta}x_+^{\beta}0 -i e^{2\pi i \alpha}x_-^{-\beta}x_+^{\beta}0 \end{pmatrix}, x \in \Delta_2 \cap O_V.
\frac{\left( - 1 \right)^{k}}{\Gamma \left( \frac 1 2 \right) \Gamma \left( k \right)}\left( \frac t 2 \right)^{2 k - 1}a_{d + 2 k - 1}\left[ 2 \ln \left( \frac t 2 \right) - \left( \Psi \left( 1 \right) - \sum_{l = 1}^{k - 2}\frac 1{l - k + 1}\right) \right]