harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
W^{(N)}= W^{(N)}_0+\epsilon^{N+k}{u_1}_0{{y}}+\epsilon^{N+k-1}\left( -\frac{1}{2}({u_1}_0)^2+{u_2}_0\right){\tau}+\epsilon^{N+k+1}{W^{(N)}_k}^*{\upsilon}+o(\epsilon^{N+k+1})\, ,
0=|\begin{matrix}1&x_{1}&y_{1}&x_{1}^{2}+y_{1}^{2}\\ 1&x_{2}&y_{2}&x_{2}^{2}+y_{2}^{2}\\ 1&x_{3}&y_{3}&x_{3}^{2}+y_{3}^{2}\\ 1&x&y&x^{2}+y^{2}\end{matrix}|
\lVert Q(g_l, g_k-g_l,g_k,\dots,g_k) \rVert_X \le = \frac{1}{j!}\sum_{0<a+b+c \le j}N(a,b,c) \lVert T( ah^{-(j-1)}(g_l-g_k) + h(bg_l+cg_k)) \rVert_X & \le C(j)\left ( h^{-j(j-1)}\lVert g_l-g_k \rVert_{Y}^j + h^{j}M^j \right) <{\varepsilon}/{j},
y_{j+m}= y_{j+m}^{(m)}= y_{j+m}^{(1)}\prod_{k=1}^{m-1}\left(\frac{\prod_{(m-k+1) \rightarrow 1}(1+y_{j+k})}{\prod_{1 \rightarrow (m-k+1)}(1+y_{j+k}^{-1})}\right) &= y_{j+m}^{(1)}\frac{\prod_{(k+1) \rightarrow 1}(1+y_{j+m-k})}{\prod_{1 \rightarrow (k+1)}(1+y_{j+m-k}^{-1})}.
S_{\mathrm{a n o m a l o u s}}= -{\frac{1}{T_{H}}}\int_{\Sigma}e^{- \phi}{\cal L}_{E}\; d^{3}r = -{\frac{k}{\hbar}}\int_{\Omega}{\cal L}_{E}\; \sqrt{g_{E}}d^{4}x = - k \{\alpha \chi + \beta p \}.
\kappa_m\left(\sum_{i=1}^n c_i W B_{\gamma+3/2}(t_i) \right)= (m-1)! \left[\sum_{i_1,i_2=1}^n \frac{c_{i_1}c_{i_2}}{2}\left(|t_{i_1}|^{2\gamma+3}+|t_{i_2}|^{2\gamma+3}- |t_{i_1}-t_{i_2}|^{2\gamma+3}\right) \right]^{m/2}
{\cal L}_{0}^{\gamma full}= \chi_{x}\upsilon_{t}+\upsilon_{x}\chi_{t}- 2{\cal H}_0^E + \lambda \left( u - \frac{\chi_x^{\gamma-1}+ \upsilon_x^{\gamma-1}}{\gamma-1}\right) +\sigma \left( \rho - \chi_x \upsilon_x \right)
\begin{array}{ll}{[}J_i, J_j] = i{\epsilon_{ij}}^k J_k i,j,k\, =\, 1,2,3{[}J_i, S_j ] = i{\epsilon_{ij}}^k S_k{[}S_i, S_j] = i{\epsilon_{ij}}^k J_k\, (C_2 -2\{\vec{J}\,}^2)\ \{[}C_2, J_i ] = [C_2 , S_i] =0 \mbox{with}\{\vec{J}\,}^2 = J^2_1 + J^2_2 + J_3^2\end{array}
X(\alpha_1,\alpha_2,\alpha'_1,\alpha'_2)=\exp \left[ i\alpha_1 q^1{\partial\over\partial q^1}+ i\alpha_2 q^2{\partial\over\partial q^2}+i\alpha'_1 \overline{q}^1{\partial\over\partial \overline{q}^1}+i\alpha'_2 \overline{q}^2{\partial\over\partial \overline{q}^2}\right]
f_{n}^{\left[ j\right]}\left( \theta\right) =\sin^{2n-1}\left(\theta/2\right) g_{n}^{\left[ j\right]}\left(\theta\right) =\left( -4\right)^{n}\sin\left( \theta/2\right) \sum_{k=n}^{j+1/2}\frac{k}{n}\binom{k+n-1}{2n-1}\cos\left( k\theta\right) \text{\ \for}1\leq n\leq j+1/2\ .
\frac{C \log(t)}{t^{23/8}}\geq ||\partial_{r}N_{2}(f_{v_{5}})(t,r) + \frac{N_{2}(f_{v_{5}})(t,r)}{r}||_{L^{2}(r dr)}= ||\xi \widehat{N_{2}(f_{v_{5}})}(t,\xi)||_{L^{2}(\xi d\xi)}
\Lambda_{\rm{bc}}\left(a\right)=\sqrt{a}\left({\cal C}_{66}a+{\cal C}_{44}\right)\left[\left(e_{31}+e_{15}\right)\left(e_{15}a+e_{33}\right)+\left(\eta_{11}a+\eta_{33}\right)\left({\cal C}_{13}+{\cal C}_{44}\right)\right] ,
\begin{pmatrix}W_{t,1}W_{t,2}W_{t,3}\end{pmatrix}=(\Phi_t^*)^{-1}\begin{pmatrix}W_{0,1}W_{0,2}W_{0,3}\end{pmatrix}+\int_0^t(\Phi_t^*)^{-1}\Phi_s^*\begin{pmatrix}R^{\tilde{F}_s, \tilde{G}_s}_{s,1}R^{\tilde{F}_s, \tilde{G}_s, \tilde{F}_s}_{s,2}R^{\tilde{G}_s, \tilde{F}_s}_{s,3}\end{pmatrix}ds.
I\left( w_{i,j}\right) =\frac{1}{\Gamma(\alpha)}\int_{0}^{1}{\int_{0}^{x}w_{i,j}(x,y){dy}dx}& =\frac{1}{\Gamma(\alpha)}\sum_{\tau\subset\operatorname*{supp}(b_{i})}{\sum_{\sigma\subset\operatorname*{supp}(b_{j})}{\int_{\tau}{\int_{\sigma\cap\lbrack0,x]}w_{i,j}(x,y){dy}dx}}}
{\cal H}_{c}= \int d^{d}x \left\{{\frac{1}{2}}\sum_{i = 1}^{N}\Bigl [ ( \partial_{\mu}\Phi_{i})^{2}+ r \Phi_{i}^{2}\Bigr ] +{\frac{1}{4 !}}\Bigl [ u ( \sum_{i}^{N}\Phi_{i}^{2})^{2}+ v \sum_{i}^{N}\Phi_{i}^{4}\Bigr ] \right\}.
\displaystyle\int_{B(x_{0},\epsilon)}|a(x)|^{\frac{-qp(x)}{(p(x)-q)}}dx \leq\displaystyle \int_{B(0,\epsilon)}|x|^{\frac{-sqp^{+}}{(p^{-}-q)}}dx &= w_{n}\displaystyle\int_{0}^{\epsilon}r^{N-1}r^{\frac{-sqp^{+}}{(p^{-}-q)}}dr &= w_{n}\frac{\epsilon^{\alpha}}{\alpha},
w=p{\left(\frac{C_{\alpha{}}}{A\hslash{}\left(\alpha{}+1\right)}\right)}^{\frac{1}{(\alpha{}+1)}},~~y=\frac{1}{\hslash{}}\left(x-\frac{E}{A}\right){\left(\frac{C_{\alpha{}}}{A\hslash{}\left(\alpha{}+1\right)}\right)}^{\frac{-1}{(\alpha{}+1)}},
\displaystyle\sum_{i=1}^3 \left( [ \partial_1^2 v^{i}(\cdot, a_0) ]_{\alpha}+ [ \partial_2 v^{i}(\cdot, a_0) ]_{\alpha}+ \|v^i(\cdot, a_0) \| \right) \lesssim \sum_{i = 1}^3 \|f^i\|_{\alpha}\lesssim [ f ]_{\alpha -2},
\int_{t_0}^{t_0^*}dt \sqrt{\mu^2+\phi^2(t)}=\int_{t_0}^{t_0^*}dt{dS_0\over dt}\sqrt{\Omega+S_0^2}=\int_{i\sqrt{\Omega}}^{-i\sqrt{\Omega}}dS_0 \sqrt{\Omega+S_0^2}=-{i \Omega \pi \over 2}.
\widehat{\phi}(\mathfrak{C}((\omega^0,y^0),1_{H_1},1_{H_2}))(z_1,z_2) = \widehat{\Phi}(\mathfrak{C}((\omega^0,y^0),1_{H_1},1_{H_2})(z_1,z_2)) & = \widehat{\Phi}(y^0([\omega]((z_1,z_2),(1_{H_1},1_{H_2})))) & = \widehat{\Phi}(y(1_G,[\omega]((z_1,z_2),(1_{H_1},1_{H_2})))).
\frac{\beta_w}{\gamma_w}= -(c-1) \cdot \frac{\beta_v+\sqrt{-f_c(\beta_v,\gamma_v)}}{(c-1)\gamma_v+2\beta_v}\frac{\beta_w'}{\gamma_w'}= -(c-1) \cdot \frac{\beta_v-\sqrt{-f_c(\beta_v,\gamma_v)}}{(c-1)\gamma_v+2\beta_v}.
c(\gamma):=\int_0^{+\infty}\frac{\left(1+\tau^2+\frac{2}{\sqrt{N}}\tau\right)^{-\gamma/2}+\left(1+\tau^2-\frac{2}{\sqrt{N}}\tau\right)^{-\gamma/2}-2}{\tau^{1+2s}}\,d\tau\,.
A_2: =\int\limits_{\stackrel{1 \leq x\leq 1/\sqrt{\epsilon(r)}}{x+1/x\leq \frac{2\cos\theta}{\sqrt{1-r^4}}}}x^{2L-n-1}(x+\frac 1x)^j dx =\int_2^{\frac{2\cos\theta}{\sqrt{1-r^4}}}y^{2L-n-1+j}dy+\cdots
\begin{array}{r c l}{\displaystyle \left[ a_{q}, a_{q}^{\dagger}\right]}&{=}&{\left[ N + 1 + b \gamma \right]_{q}- \left[ N + b \gamma \right]_{q},}\\{\displaystyle \left[ N , a_{q}\right]}&{=}&{- a_{q}, \displaystyle \left[ N , a_{q}^{\dagger}\right] = a_{q}^{\dagger}.}\\ \end{array}
0 = \mathrm{p . v .}\int{d \beta}\left\{{{\textstyle{\frac{1}{2}}}\cot \left({{\frac{\alpha - \beta}{2}}}\right) - \sum_{n = 1}^{\infty}{\left({\zeta_{B}( x^{n}) - ( - 1 )^{n}\zeta_{F}( x^{n})}\right) \sin \left({n ( \alpha - \beta )}\right)}}\right\}\rho ( \beta )
{\cal W}\, ( \varphi_{1}, \varphi_{2}) = \frac{m^{2}}{4 \lambda}\, \varphi_{1}- \frac{\lambda}{3}\, \varphi_{1}^{3}+ \lambda \, \varphi_{1}\varphi_{2}^{2}+ \frac{\mu}{2}\, \varphi_{2}^{2}\, ,
\gamma_{x}= \left[ \begin{array}{c c}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right] \otimes 1 , \; \; \gamma_{y}= \left[ \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right] \otimes \left[ \begin{array}{c c}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right] , \; \; \gamma_{\bar{x}}= \gamma_{x}^{T}, \; \; \gamma_{\bar{y}}= \gamma_{y}^{T}
\mathrm{var}\big[\widehat{K}_{\mathrm{a-cpt-f}}^\mathbb{R}\big] =\upsilon^2\bigg(\!\Big(1\!-\!\frac{\pi}{4}\Big)\!\sum_{i\in\mathcal{K}}\!\Big(1\!+\!\frac{1}{N_1\bar{\gamma}_i}\Big)\!+\!\sum_{i\in\mathcal{K}}\!\frac{1}{2N_1\bar{\gamma}_i}\!+\!\frac{1}{2N_2\bar{\gamma}_c}\!\bigg),
\sum_{L = 0}^{M}\frac{( \rho_{1})_{L}( \rho_{2})_{L}( a q / \rho_{1}\rho_{2})^{L}}{( a q / \rho_{1})_{L}( a q / \rho_{2})_{L}}\frac{\alpha_{L}}{( q )_{M - L}( a q )_{M + L}}= \sum_{L = 0}^{M}\frac{( \rho_{1})_{L}( \rho_{2})_{L}( a q / \rho_{1}\rho_{2})^{L}( a q / \rho_{1}\rho_{2})_{M - L}}{( a q / \rho_{1})_{M}( a q / \rho_{2})_{M}( q )_{M - L}}\beta_{L},
A_{ij}=0, ~~~i,j\geq 4,~~ i\neq j, &A_{12}=\frac{C^1_{2,2}}{\mu}-\frac{E_3(B^1_{11,3})}{\mu}=\frac{C^1_{1,1}}{\mu}+\frac{E_3(B^1_{11,3})}{\mu}.
\frac{\sqrt{\lambda_1^A\lambda_N^A}}{\frac{1}{2}\left(\lambda_1^A+\lambda_N^A\right)}= \mu_1(A) > \frac{|p-2|}{p}\quad\Longleftrightarrow\left(\frac{1}{2}-\frac{1}{p}\right)^2\left(\lambda_1^A+\lambda_N^A\right)^2 < \lambda_1^A\lambda_N^A.
u^{+}=\int_{0}^{Y^{+}}\frac{2(1+K_{3}y^{+})}{1+[1+4(K_{1}y^{+})^{2}(1+K_{3}y^{+})(1-exp(-y^{+}\sqrt{1+K_{3}y^{+}}/K_{2}))^{2}]^{0.5}}dy^{+}
\begin{array}{ll}u~=~\cosh\mbox{\boldmath $\rho$}\cos\mbox{\boldmath $t$},~~~~~ v~=~\cosh\mbox{\boldmath $\rho$}\sin\mbox{\boldmath $t$}, w~=~\sinh\mbox{\boldmath $\rho$}\cos\mbox{\boldmath $\tilde{\theta}$},~~~~~ y~=~\sinh\mbox{\boldmath $\rho$}\sin\mbox{\boldmath $\tilde{\theta}$}.\end{array}
q =\pm R_{s;i_1,\cdots,i_k}q_{s,0}^{j_1+k-1}\cdots q_{s,s-1}^{j_s}S_s(m) &=\pm R_{s;i_1,\cdots,i_k}q_{s,0}^{j_1+k-1-n}\cdots q_{s,s-1}^{j_s}\frac{1}{L_s^{\frac{p-1}{2}\delta}}St_s(m).
E = \int d^2x \left({\kappa^2 - e^2 |\phi|^2 \over 2 e^2 |\phi|^2}B^2 +{1 \over 2}\left[ | \partial_i \phi |^2 +{\kappa^2 e^2 |\phi|^2 \over \kappa^2 - e^2|\phi|^2}\tilde{A}_i^2 \right] + V(\phi) \right) \, .
a_l^*-a_{l+i}^* =p^{-{l+i}}\left(p^ia_l-a_{l+i}-\frac{p^i-1}{p-1}\right) &\leq p^{-{l+i}}\left((p^i-1)a_l+2i-\frac{p^i-1}{p-1}\right) &\leq p^{-{l+i}}\left(2i-p\frac{p^i-1}{p-1}\right) &\leq 0.
\big(e\xrightarrow{\omega(\alpha\cdot{}\alpha',\beta'\cdot{}\beta)}u'\cdot{}u\cdot{}d\cdot{}y\cdot{}y'\big) =\big( e\xrightarrow{\omega(\alpha',\beta')}u'\cdot{}j\cdot{}y' \xrightarrow{u'\cdot{}\omega(\alpha,\beta)\cdot{}y'}u'\cdot{}u\cdot{}d\cdot{}y\cdot{}y'\big)
\mathbb{E}_{\rho,0}(e^{i\lambda \Theta_t}) = \frac{\sqrt{\pi}}{2}\sqrt{\frac{\rho^2}{2t}}e^{-\rho^2/(4t)}\left[{\it I}_{(|\lambda| -1)/2}\left(\frac{\rho^2}{4t}\right) +{\it I}_{(|\lambda| +1)/2}\left(\frac{\rho^2}{4t}\right)\right], \lambda \in \mathbb{R}.
[{{\omega}^{- 1}}] ={\frac{1}{a c - b{b^{\prime}}}}\left( \begin{matrix}{{0}}&{{- c}}&{{0}}&{{{b^{\prime}}}}\\{{c}}&{{0}}&{{- b}}&{{0}}\\{{0}}&{{b}}&{{0}}&{{- a}}\\{{-{b^{\prime}}}}&{{0}}&{{a}}&{{0}}\\ \end{matrix}\right)
a^\frac 1 2_3(A_\varepsilon)=-\frac{1}{i\pi}\log(\varepsilon)\t_{12}+\left(\tau^\frac 1 2\right)^{1,2}+\left(\tau^\frac 1 2\right)^{12,3}+\mathcal O(\varepsilon|\log\varepsilon|^\bullet),a^\frac 1 2_3(B_\varepsilon)=-\frac{1}{i\pi}\log(\varepsilon)\t_{23}+\left(\tau^\frac 1 2\right)^{2,3}+\left(\tau^\frac 1 2\right)^{1,23}+\mathcal O(\varepsilon|\log\varepsilon|^\bullet).
\mathbb P\begin{bmatrix}\begin{Vmatrix}1_{S^n}-{\frac 1{(2k)^l}}\displaystyle\sum_{s\in{\hat S}^l}H_t(x_0s, x)\end{Vmatrix}_{L^2}\leq{\frac{r^{\frac n2}\alpha_n}3}\end{bmatrix}\geq 1-\delta.
D_n(x|a_1,\dots,a_r) =x D_{n-1}(x-1|a_1,\dots,a_r) &+\frac{r}{n}\sum_{l=0}^n\binom{n}{l}c_l D_{n-l}(x-1|a_1,\dots,a_r) &-\frac{1}{n}\sum_{j=1}^r\sum_{l=0}^n \binom{n}{l}a_j c_l D_{n-l}(x+a_j-1|a_1,\dots,a_r,a_j)\,,
\delta(x, x') =\left\langle \hat{\Phi}_{v,w}(\Phi_{v,w}(e^{v,w}_x)),\,e^{v,w}_{x'}\right\rangle_{\mathcal{V}_{v,w}}= \left\langle \Phi_{v,w}(e^{v,w}_x),\, \hat{\Phi}^*_{v,w}(e^{v,w}_{x'})\right\rangle_{\mathcal{V}_{v,w}}
\left( \begin{array}{cc}\vec{h}_1 \vec{g}_1 \end{array}\right) = \Lambda_1\left( \left( \begin{array}{cc}u^0 v^0 \end{array}\right), \left( \begin{array}{cc}u^1 v^1 \end{array}\right) + \left( \begin{array}{cc}v \nu(T,u,v) \end{array}\right) \right),
S L ( 2 ,{\bf R})_{S}: \ \ \left( \begin{array}{l}{\vec{\hat{\alpha}}}{\vec{\hat{\beta}}}\end{array}\right) \to \left( \begin{array}{l l}{a}{c}{b}{d}\end{array}\right) \left( \begin{array}{l}{\vec{\hat{\alpha}}}{\vec{\hat{\beta}}}\end{array}\right) .
{||f||}^2\leq \displaystyle \sum_{m \in \mathfrak{D}}\displaystyle \sum_{p=0}^{q-1}|s_{p,m}| \displaystyle \sum_{v \in \mathbb{Z}_N^2}|\overline{w_{v}}||\langle{T_{Am}\varphi_p, F_{v}}\rangle|.
[p_1, \ldots, p_{n-1}] := \langle \rho_0, \ldots, \rho_{n-1}\mid \rho_i^2 = 1 \textrm{for $0 \leq i \leq n-1$}, & (\rho_{i-1}\rho_i)^{p_i}= 1 \textrm{for $1 \leq i \leq n-1$}, & (\rho_i \rho_j)^2 = 1 \textrm{for $i,j \in \{0,\ldots,n-1\}$ with $|i-j| \geq 2$}\rangle.
\frac{d}{ds}(\Phi_s^* \omega_s) = \Phi_s^* L_{\theta_s}\omega_s +\Phi_s^* \omega' = \Phi_s^* (d i_{\theta_s}+ i_{\theta_s}d)\omega_s+ \Phi_s^* \omega' = -\Phi_s^* d\alpha + \Phi_s^* \omega' = 0,
\Delta_g\pi_g x =\Delta_g \left(\sum_{h\in G}f_h(x) \tau_{gh}\right)=\sum_{h\in G}f_h(x) \Delta_g\tau_{gh}=\sum_{h\in G}f_h(x)\sum_{u\in G}f_{gu}(\tau_{gh}) \tau_u =\sum_{h\in G}f_h(x)\sum_{u\in G}f_{u}(\tau_{h}) \tau_u=\sum_{h\in G}f_h(x)\tau_h=x,\forall x \in \mathcal{X}
E_0(t)+E_1(t) \leq \delta_0+C (\sup_t E_0(t))E_1(t) &+CML^{-1}E_1(t)+CL \int_0^t \big{(}\int_{\mathbb{R}^3}(|\widehat{f}(\tau,\eta)|+|\widehat{g}(\tau,\eta)| )d\eta \big{)}E_0(\tau)d\tau.
\begin{aligned}\left[f\right] =f^{+}(x^{*}) - f^{-}(x^{*}), \left[ f' \right] =f_x^{+}(x^{*}) - f_x^{-}(x^{*}), \left[ f'' \right] =f_{xx}^{+}(x^{*}) - f_{xx}^{-}(x^{*}), \left[ f''' \right] =f_{xxx}^{+}(x^{*}) - f_{xxx}^{-}(x^{*}),\cdots\end{aligned}
m_v((y,\omega),(z,\eta)) = \max\left\{\frac{v((y,\omega))}{v((z,\eta))},\frac{v((z,\eta))}{v((y,\omega))}\right\}, v((y,\omega)) := m_0((y,\omega),(x,\xi)) \cdot \max\{w(\Phi(\omega)),[w(\Phi(\omega))]^{-1}\},
H=-\sum_{\langle i,j\rangle}\hbar J_{ij}(\sigma_{x}^{i}\sigma_{x}^{j}+\sigma_{y}^{i}\sigma_{y}^{j}+\gamma\sigma_{z}^{i}\sigma_{z}^{j})-\sum_{i=1}^{N}\hbar B_{i}\sigma_{z}^{i}
8 \delta \tilde{C}= 0 \; , \delta ( \tilde{\Delta}\: \frac{1}{\sqrt{- g}}\: \tilde{C}) = 0 \; , \delta [ \tilde{\Delta}\frac{1}{\sqrt{- g}}\; \tilde{\Delta}\; . . . \; \frac{1}{\sqrt{- g}}\: \tilde{\Delta}C ) ] = 0 \; ,
{\mbox{\boldmath $L$}}(g_{\mu\nu},\varphi)=g^{1/2}\left\{\frac{m_{P}^{2}}{16\pi}R(g_{\mu\nu})-\frac{1}{2}\xi\varphi^{2}R(g_{\mu\nu})-\frac{1}{2}(\nabla\varphi)^{2}-\frac{1}{2}m^{2}\varphi^{2}-\frac{\lambda}{4}\varphi^{4}\right\},
\Lambda_\alpha (h) = \sum_{n\ge 0}\alpha_{n}\sum_{k=0}^n (a_1)_k (b_1)_{n-k}+ \sum_{n\ge 0}\alpha_{n}\sum_{k=0}^n (a_2)_k (b_2)_{n-k}= G_{\alpha}(a_1, b_1)+G_{\alpha}(a_2,b_2).
\begin{array}{c}K=h^{\mu \nu}\nabla_{\mu}u_{\nu}=-h^{\mu \nu}\Gamma_{\mu \nu}^{\lambda}u_{\lambda}=-\frac{1}{2\sqrt{f}}h^{\mu \nu}\left( \partial_{\nu}h_{\mu \tau}+\partial_{\mu}h_{\nu \tau}-\partial_{\tau}h_{\mu\nu}\right) =\frac{1}{2\sqrt{f}}h^{\mu \nu}\partial_{\tau}h_{\mu \nu}\end{array}
\begin{array}{l}\varphi_{1E}\left(r,\theta\right) =\sum_{n=-\infty}^\infty f_n\left(r\right) e^{i \left(n+\frac{1}{2}\right)\theta}\chi_{1E}\left(r,\theta\right) =\sum_{n=-\infty}^\infty g_n\left(r\right) e^{i\left(n+\frac{1}{2}\right)\theta}\end{array}
G_4(n,f) \ = \ 2 (-1)^{n+1}\binom{n}{f}\sum_{\ell = 0}^n (-1)^\ell \binom{f}{\ell}\sum_{c = 0}^\ell \binom{\ell}{c}= \ 2 (-1)^{n+1}\binom{n}{f}\sum_{\ell = 0}^n (-2)^\ell \binom{f}{\ell}.
\textbf{I}=2\int_{\mathbb{R}^n\times(\mathbb{R}^n\backslash B_\delta(0))}f_\lambda(x)f_\lambda(y)|x-y|^{\alpha-n}dxdy, \textbf{II}=\int_{(\mathbb{R}^n\backslash B_\delta(0))\times(\mathbb{R}^n\backslash B_\delta(0))}f_\lambda(x)f_\lambda(y)|x-y|^{\alpha-n}dxdy.
\varepsilon_5^N(\vec{f})= |\eta_0^N(\vec{f})|+K_6T\|\vec{f}\|_\infty^2+\frac{\gamma_NT}{N}K_7\|\vec{f}\|_\infty^3\exp\left(2\|\vec{f}\|_\infty\frac{\gamma_N}{N}\right) &+\frac{K_8T}{\gamma_N}\|\vec{f}\|_\infty+\max_{0\leq t\leq T}\mathcal{Z}_t^N(\vec{f}),
\frac{d^2}{dt^2}F_0(t)=\int_{\Omega}\left|u(x,t)\right|^p\phi_0(x)\,dx\geq \frac{|F_0(t)|^p}{\displaystyle \left( \pi C(t+R)^2\ln(t+R)\right)^{p-1}}=k[\ln(t+R)]^{-(p-1)}(t+R)^{-2(p-1)}|F_0(t)|^p
|\mathcal{T}_{\neq}| = \left| \iint A(\nabla_L^{\perp}\Delta_L^{-1}\ne{f}\cdot\nabla_Lf)Af\,dVdt \right| \lesssim \Vert \nabla_L^{\perp}\Delta_L^{-1}\ne{f}\Vert_{{L^2}H^{N}}\Vert \nabla_Lf\Vert_{L^{2}H^{N}}\Vert Af\Vert_{L^{\infty}L^{2}}.
u\oplus_{M}v=\frac{(1+\frac{2}{s^{2}}u\cdot v+\frac{1}{s^{2}}|v|^{2})u+(1-\frac{1}{s^{2}}|u|^{2})v}{1+\frac{2}{s^{2}}u\cdot v+\frac{1}{s^{4}}|u|^{2}|v|^{2}}
\begin{array}{ccl}J^\pm_n \rightarrow J^\pm_{n\pm w}J^3_n \rightarrow J^3_n -{k\over 2}w \delta_n \qquad \qquad w \in{\bf Z}L_n \rightarrow L_n - w J^3_n -{k\over 4}w^2 \delta_n.\end{array}
\varPsi(u)=\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left( \dfrac{|u(x)- u(y)|}{|x-y|^s}\right) \dfrac{dxdy}{|x-y|^N}+\int_{\Omega}\widehat{\varPhi}_{x}\left( |u(x)|\right) dx.
\mathbf{C}_e(q,k) =\mathbb{E}[(\mathbf{\hat{h}}(q,k)-\mathbf{h}(q,k))(\mathbf{\hat{h}}(q,k)-\mathbf{h}(q,k))^H] \geq\frac{\partial \mathbf{h}(q,k)}{\partial \boldsymbol{\Theta}}^H\mathfrak{{F}}^{-1}(\boldsymbol{\Theta})\frac{\partial \mathbf{h}(q,k)}{\partial \boldsymbol{\Theta}}
\int\limits_{S^{n-1}}\int\limits_0^\pi Cf(u,\beta,\psi)\sin\psi d\psi d\beta =\frac{\pi}{|S^{n-1}|}\frac{2|S^{n-2}|}{n-1}\int\limits_{S^{n-1}}Rf(\omega,\omega\cdot u)d\omega =\frac{2\pi}{n-1}\frac{|S^{n-2}|}{|S^{n-1}|}\int\limits_{S^{n-1}}R^{\#}Rf(u) =\frac{\pi\Gamma(n)}{2^{n-1}\Gamma^2(\frac{n+1}{2})}R^{\#}Rf(u).
\int_\Omega(x\cdot\nabla u_1)Lu_2\, dx+\int_\Omega(x\cdot\nabla u_2)Lu_1\,dx=\frac{2s-n}{2}\int_\Omega u_1Lu_2\, dx+ &+\frac{2s-n}{2}\int_{\Omega}u_2Lu_1\, dx-\Gamma(1+s)^2\int_{\partial\Omega}\mathcal A(\nu)\frac{u_1}{d^{s}}\frac{u_2}{d^{s}}(x\cdot\nu)\, d\sigma.
\mathrm{Fitt}_{\Lambda_n / \omega^{\pm}_n}\left( \left( \mathrm{Sel}^{\pm}(\mathbb{Q}_n, E[p^\infty])[\omega^{\pm}_n] \right)^\vee \right) = \dfrac{\mathrm{Fitt}_{\Lambda_n}\left( \mathrm{Sel}^{\pm}(\mathbb{Q}_n, E[p^\infty])^\vee \right) + (\omega^{\pm}_n)}{(\omega^{\pm}_n)}
\sum_{d=0}^{\infty}q^d \chi\left(X^{[d]}, \wedge_{y}L^{[d]}\otimes (\wedge_{x}M^{[d]})^{\vee}\right) =(1-q)^{-\chi(\mathcal O_X)}(1+qy)^{\chi(L)}(1+qx)^{\chi(M^{\vee})}(1-qxy)^{-\chi(L\otimes M^{\vee})}.
\begin{array}{rcl}\psi(u) = sgn(g(t_1(u)))\frac{f(t_1(u))}{g(t_1(u))}+sgn(g(t_2(u)))\frac{f(t_2(u))}{g(t_2(u))}&= \frac{f(t_1(u))}{g(t_1(u))}-\frac{f(t_2(u))}{g(t_2(u))}&= \frac{1}{(1+nu)^{\frac{n-1}{n}}}[f((1+nu)^{\frac{1}{n}}) + f(-(1+nu)^{\frac{1}{n}})].\end{array}
{}_3F_2\bigg[\begin{matrix}\alpha \beta \gamma &2\beta \frac12(\alpha+\gamma+1)\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(\frac12)\Gamma(\frac12+\beta)\Gamma(\frac12+\frac12(\alpha+\gamma))\Gamma(\frac12+\beta-\frac12(\alpha+\gamma))}{\Gamma(\frac12+\frac12\alpha)\Gamma(\frac12+\frac12\gamma)\Gamma(\frac12+\beta-\frac12\alpha)\Gamma(\frac12+\beta-\frac12\gamma)}.
\begin{array}{c}{{\displaystyle{\frac{e^{\displaystyle{- \frac{i}{2}\, \pi \, \left( 1 + s \right)}}}{{\nu}^{1 + s}\, 8 \, \pi}}\, \left( \frac{8 \,{\nu}^{2}}{1 - s}+ \frac{4 \, \nu}{s}+ \frac{1 - 4 \,{\nu}^{2}}{1 + s}\right)}}\\ \end{array}
\lim_{s \rightarrow 1}\frac{d}{d~\bar{z_2}}\Xi_1(z_1,z_2,s)(z_2-\bar{z_2})^{2s-1}=-\sum_{ad-bc=1}\frac{1}{\mu_{\gamma}(z_1, \bar{z_2})^2}=-\omega_2(z_1, \bar{z_2}).
\begin{array}{l}e^{\pm,2}_{k}=\int_0^1L{''}_{(U^{a,\pm}+S_{\theta_k}V^{\pm}_k+\tau(1-S_{\theta_k})V^{\pm}_k;\Psi^{a,\pm}+S_{\theta_k}\Phi^{\pm}_k+\tau(1-S_{\theta_k})\Phi^{\pm}_k)}((\delta V^{\pm}_k,\delta \Phi^{\pm}_k),((1-S_{\theta_k})V^{\pm}_k,(1-S_{\theta_k})\Phi^{\pm}_k))d\tau\end{array}
n_{\ast}\left( \left( \frac{r}{K_{1}}\right)^{\frac{1}{2}},p_{0}U_{\perp}p_{0}\right) := n_{+}\left( \frac{r}{K_{1}}, \big( p_{0}U_{\perp}p_{0}\big)^{\ast}p_{0}U_{\perp}p_{0}\right) & = \phi(r) \big( 1 +o(1) \big), r \searrow 0,
e^{a}e^{b}\hat{\Omega}_{b , a}^{~ ~ i}= 0 \quad \rightarrow \quad \hat{\Omega}_{b , a}^{~ ~ i}= \hat{\Omega}_{a , b}^{~ i}={\nabla}_{b}E_{a}^{~ \underline{{b}}}u_{\underline{{b}}}^{~ i},
\mathbf{y}_{\mathrm{r}}\left [ l \right ] = \mathbf{H}_{\mathrm{r}}\mathbf{x}\left [ l \right ] + \mathbf{m}\left [ l \right ] = \alpha e^{j 2\pi \mathcal{F}_{D}lT}{\mathbf{b}}\left ( \theta \right ){\mathbf{a}}^{H}\left ( \theta \right )\mathbf{x}\left [ l \right ] + \mathbf{m}\left [ l \right ],
f(x)=\sum_{i=1}^{M}\frac{U_i}{x-\beta_{i,\xi}}-\sum_{i=1}^{N}\frac{P_i}{x+\gamma_{i,q}}-\sum_{i=1}^{M}\frac{H_i}{x-\beta_{i,q}}e^{\beta_{i,q}(b-y)}.
|\partial_{t}^{2}\left(\frac{-4 \lambda'(t)^{2}}{\lambda(t)^{2}}\int_{0}^{\infty}\chi_{\geq 1}(\frac{2R \lambda(t)}{\log^{N}(t)}) \frac{R^{2}\phi_{0}(R) dR}{(R^{2}+1)^{2}}\right)| \leq \frac{C}{t^{4}\log^{2+2b+2N}(t)}
R^{2,\operatorname{lin}}= \lbrack \beta(\rho_0+\nu+\nu_c)-\beta(\rho_0)\rbrack u_0 + [\beta(\rho_0+\nu+\nu_c)-\beta(\nu)]w, R^{2,\operatorname{corr}}=\beta(\rho_0+\nu+\nu_c)w_c, R^h = h_0 - h^\ast.
\widehat{I}_{j}(e^{\mu})=\frac{\alpha \, e^{\mu(v_{j}+v_{j+1})}+(1-q)e^{\mu(v_{j})}}{e^{\mu(v_{j})}-e^{\mu(v_{j+1})}}\left(e^{\mu}-e^{s_{j}\mu}\right) (\mu \in V^{*}).
|{\bf I}+{\bf W}_i{\bf R}| = |{\bf I}+{\bf W}_i{\bf U}_{\bot}{\bf U}_{\bot}^+{\bf R}{\bf U}_{\bot}{\bf U}_{\bot}^+| &= |{\bf I}+{\bf U}_{\bot}^+{\bf W}_i{\bf U}_{\bot}{\bf U}_{\bot}^+{\bf R}{\bf U}_{\bot}|
\lim_{t \to 0}u(t,x) = \lim_{t \to 0}t^{-\frac{1}{\alpha}}f_a \Bigl( \frac{|x|}{\sqrt t}\Bigr)= |x|^{-\frac{2}{\alpha}}\lim_{r\to\infty}r^{\frac{2}{\alpha}}f_a(r)= L(a)|x|^{-\frac{2}{\alpha}}.
\sum_{i=0}^{n}u(i) =\int_0^n u(x) dx+\frac{1}{2}\big(u(0)+u(n)\big)+\sum_{j=1}^{M}\frac{B_{2j}}{(2j)!}\big(u^{(2j-1)}(n)-u^{(2j-1)}(0)\big) &+\frac{1}{(2M+1)!}\int_0^n B_{2M+1}(x-[x]) u^{(2M+1)}(x)dx,
\begin{array}{cc}h_2(r)= 3C_0 \textup{in}\ [2M,R_1], h_2(r)= 1 \textup{in}\ [2R_1,\infty), |h_2'(r)|\leq \frac{3C_0}{\log 2}r^{-1}\textup{in}\ [R_1,2R_1].\end{array}
\lambda(T_1T_2)^*\rho(T_1T_2) =\lambda(T_2)^*\lambda(T_1)^*\rho(T_1)\rho(T_2) &=\lambda(T_2)^*\int_X c(T_1,x)d\mu(x)\rho(T_2) &=\int_X c(T_1,T_2(x))d\mu(x) \lambda(T_2)^*\rho(T_2) &=\int_Xc(T_1,T_2(x))c(T_2,x)d\mu(x),
\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}}=\sqrt{\frac{1}{N}(\sum_{i=1}^{N}x_{i}^{2})-\overline{x}^{2}}=\sqrt{(\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2})-(\frac{1}{N}\sum_{i=1}^{N}x_{i})^{2}},
\left( \beta_{\mu}^{( 0 )}\partial_{\mu}+ m_{0}\right) \Psi^{( 0 )}( x ) = 0 , \hspace{0 . 5 i n}\Psi^{( 0 )}( x ) = \left( \begin{array}{c}{{\psi_{0}( x )}}\\{{\psi_{\mu}( x )}}\\ \end{array}\right) .
w(r,t)=\sum_{n=1}^{\infty}C_{n}[J_{0}(\lambda_{n}r)-\frac{J_{0}(\lambda_{n}a)}{I_{0}(\lambda_{n}a)}I_{0}(\lambda_{n}r)][A_{n}e^{i\omega_{n}t}+B_{n}e^{-i\omega_{n}t}].
r(\mathcal{K}) = \limsup_{T \rightarrow \infty}\,\inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x_0}}\frac{1}{T}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t)) \, \Bigl \vert \, \varphi(t) \in D \cup \partial D,\,\, \forall t \in [0, T] \biggr\}.
e_{_{\mu}}^{a}= \left( \begin{array}{c c c}{1}&{0}&{0}\\{0}&{\rho \frac{1}{\beta}\cos \phi}&{\rho \frac{1}{\beta}\sin \phi}\\{0}&{- \rho \frac{\sin \alpha \theta}{\alpha}\sin \phi}&{\rho \frac{\sin \alpha \theta}{\alpha}\cos \phi}\\ \end{array}\right) .
W_{I}= \lambda_{1}\epsilon_{a b c}{\mathrm{T r}}( P_{a}P_{b}Q_{c}) + \lambda_{2}{\mathrm{T r}}( Q_{1}R_{2}\Phi_{3}+ Q_{2}R_{3}\Phi_{1}+ Q_{3}R_{1}\Phi_{2}) + \lambda_{3}{\mathrm{T r}}( R_{1}R_{2}R_{3}) + \cdots ~ .
W = - 4 ( \frac{\pi}{2})^{\frac{p}{2}}\, ( 3 - l ) ! \, \prod_{a = 1}^{p - 1}\left( \frac{n^{\frac{1}{l}}}{L_{a}}\right) \omega_{l}^{3}\frac 1{r^{8 - p}}+ \cdot \cdot \cdot ,
\nabla_{\!x}=\left(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\ldots,\frac{\partial}{\partial x_n},0\right)\,,\nabla_{\!y}=\left(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\ldots,\frac{\partial}{\partial x_n},\frac{\partial}{\partial x_{n+1}}\right)\,.
\overline{N^{1}}(^{\ast}b)(u_{\sigma(b)}(n)) =(-\pi_{1}^{1}p_{1}\xi_{be_{k}}(u_{\sigma(b)}(n)),-\gamma_{1}\xi_{be_{k}}(u_{\sigma(b)}(n)),0) &=(-\pi_{1}^{1}p_{1}u_{out}(\xi_{be_{k}}(n)),-\gamma_{1}u_{out}(\xi_{be_{k}}(n)),0) &=(-\pi_{1}^{1}u_{1}p\xi_{be_{k}}(n)-\pi_{1}^{1}\rho_{2}\gamma\xi_{be_{k}}(n),-u_{2}\gamma\xi_{be_{k}}(n),0)
P^\omega\left(|\widehat{Z}_s|< e^{-\sqrt{\beta/2}\,\widehat{R}(s)}p_s e^{\beta s}\:\big|\: C_t \right) \leq P_{y_0}\left(\big| Z_s^{\widehat{B}_s}\big|< e^{-\sqrt{\beta/2}\,\widehat{R}(s)}p_s e^{\beta s}\right) &=\exp\left[-\sqrt{\beta/2}\,\widehat{R}(s)(1+o(1))\right].
C'/\langle \sigma\rangle \colon a_1x^2+a_2y^4+a_3xy^2+a_4x+a_5y^2+a_6=0, C'/\langle \tau\rangle \colon a_1x^4+a_2y^2+a_3x^2y+a_4x^2+a_5y+a_6=0, C'/\langle \sigma\circ\tau\rangle \colon a_1x^4+a_2+a_3x^2+a_4x^2y+a_5y+a_6y^2=0.
2\, [Z_j] = [Z_{j-1}]\cdot \left( c_{\lambda_j}( D_{j-1}^\perp/D_{j-1}- F_{q_j}/D_{j-1}- D_j/D_{j-1})\right. \left. - e_{\lambda_j}( D_j/D_{j-1}, F_{q_j}/D_{j-1}) \right) = [Z_{j-1}] \cdot \left( c_{\lambda_j}( D_{j-1}^\perp/D_{j-1}- F_{q_j}/D_{j-1}- D_j/D_{j-1}) \right. \left. + (-1)^{j}e_{\lambda_j}(D_j,F_{q_j})\right) ,
\left.\frac{d{P}^{x,i,\downarrow}}{d{P}^{x,i}}\right|_{\mathcal{F}_t}=\frac{U^+_{J_t}(\xi_t)}{U^+_i(x)}\mathbf{1}_{(t<\tau^+_0)}, t\geq 0, i\in E, x<0,
S^{- 1}\left( p \right) = Z_{2}\left( p \hspace{- 0 . 2 c m}/ - m_{0}\right) - i Z_{1}e^{2}\int^{\Lambda}\frac{d^{4}k}{\left( 2 \pi \right)^{4}}\gamma^{\mu}S \left( k \right) \gamma^{\nu}D_{\mu \nu}\left( k - p \right)
\vec{a}= (a_1,\ldots,a_D) = R(c_1 - \lambda_1 h_{11}- c_\infty(1-\rho_\infty),\ldots,c_D - \lambda_D h_{D1}- c_\infty(1-\rho_\infty))^T = R[C - \Lambda \bar{H}_1]\vec{e}- c_\infty(1-\rho_\infty)R\vec{e}= R[C - \Lambda \bar{H}_1]\vec{e}- \vec{r}
V(x_{1},x_{2})=-4\pi \ \Psi \overset{x_{1},x_{2}}{\ast}\rho =\log \left|\dfrac{\vartheta_{3}\left( \left.\frac{\pi z}{2a}\right| i\right)}{\vartheta_{1}\left( \left.\frac{\pi z}{2a}\right| i\right)}\right|^{2}
q=-{{a_0\ddot{a}_0}\over \dot{a}^2_0}={\dot{I}\over{2HI}}+{1\over 2}(I^{-3/2}\Omega_{\varrho}+I^{-1}\Omega_{\Phi})+{{16\pi^2G^2_5\sigma}\over{3c^6H^2}}(I^{-1/2}\wp+I^{-1}\wp_{\Phi}+2I^{-1/2}\wp_{\Lambda_4})+I^{-1}\Omega_{\cal C},