harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f(m,r)= \sum_{k=0}^m (-1)^k \binom{k-1}{r}\binom{m-k-1}{r}+\sum_{k=0}^m (-1)^k \binom{k-1}{r-1}\binom{m-k-1}{r}+ \sum_{k=0}^m (-1)^k \binom{k-1}{r}\binom{m-k-1}{r-1}+ \sum_{k=0}^m (-1)^k \binom{k-1}{r-1}\binom{m-k-1}{r-1}= -f(m-2,r)-f(m-2,r-1) +2\sum_{k=0}^m (-1)^k \binom{k-1}{r}\binom{m-k-1}{r-1},
\begin{array}{r l}\hat{g}_{ab}=u^{q-2}g_{ab}\hat{K}_{ab}=u^{-2}[\frac{1}{2N_{g,\omega}}(\mathcal{L}_gW)_{ab}+U_{ab}]+\frac{1}{n}u^{q-2}g_{ab}\left(\tau^*+\frac{1}{N_{g,\omega}}div(\tilde{V})\right).\end{array}
s(3\cdot 2^k,2^k+(-1)^kf)= \frac{3\cdot 2^k}{12(2^k+(-1)^kf)}+\frac{2^k+(-1)^kf}{36\cdot 2^k}-\frac{(-1)^k}{4}+\frac{1}{36\cdot 2^k\cdot (2^k+(-1)^kf)}&+(-1)^ks(1,3\cdot 2^k).
\mathbb{P}_{couple}\left(n^{-\frac{1}{\alpha}}\sum_{i=1}^{M_{1}\left(n\right)}Z_{i}>n^{-c'}\right) \leq M_{1}\left(n\right)\mathbb{P}_{couple}\left(Z>n^{\frac{1}{\alpha}-c'}\right) +\exp\left(1-\frac{A\left(n\right)}{n^{\left(\frac{1}{\alpha}-c'\right)}}-\log\left(\frac{n^{\left(\frac{1}{\alpha}-c'\right)}}{A\left(n\right)}\right)\right),
W_{\pi}(g_{t,l,v}) = \zeta_F(1)^{-1}q^{-\frac{a_1+t}{2}}q^{s(l_1-l_2)}\epsilon(\frac{1}{2},\chi_1^{-1})\sum_{\mu\in \mathfrak{X}_l}G(\varpi^{-l_2},\mu\chi_2)G(-vb_1\varpi^{-l}, \mu^{-1}).
\theta_k = \eta_k r^k, k =j+1, \ldots, J, \theta_j = \eta_j r^j + \frac{1}{1-w_{jj}}\sum_{i=j+1}^J w_{ji}\theta_i, \theta_\ell = w_{\ell j}\theta_j + \sum_{i=j+1}^{J}w_{\ell i}\theta_i, \ell=1, \ldots, j-1.
V(-k) = \Theta_2 V(k) \Theta_1^{-1}k \in E_1 \cup E_6, V(-k + e_1) = \tau_2(e_1) \Theta_2 V(k) \Theta_1^{-1}\tau_1(e_1)^{-1}k \in E_3 \cup E_4,
W_4(u,z_1)=z_1K_\mu(uz_1)\left(\sum_{s=0}^{N-1}\frac{A_s(z_1)}{u^{2s}}+g_3(u,z)\right) \hphantom{W_4(u,z_1)=}{}-\frac{z_1}{u}K_{\mu+1}(uz_1)\left(\sum_{s=0}^{N-1}\frac{B_s(z_1)}{u^{2s}}+z_1h_3(u,z)\right),
\int_M(\Delta_fw)^2 =-\int_{M}\langle\nabla(\Delta_fw),\nabla w\rangle+\int_{\partial M}\Delta_fw\, \langle\nabla w,\nu\rangle =\int_{M}w\, \Delta_f(\Delta_fw)-\int_{\partial M}w\, \langle\nabla(\Delta_fw),\nu\rangle+\int_{\partial M}\Delta_fw \,\langle\nabla w, \nu\rangle =q_1\int_{\partial M}\eta^2,
\begin{array}{l}d\bar{X}_t = -\frac{\alpha(t)}{m}\sum_{i=1}^m \nabla u_i(X^{(i)}(t)) dt+\sqrt{2\sigma\alpha(t)}dB_t = -\alpha(t) \nabla U(\bar{X}(t))dt-\alpha(t)\left(\frac{1}{m}\sum_{i=1}^m \nabla u_i(X^{(i)}(t))-\nabla U(\bar{X}(t))\right)dt+\sqrt{2\sigma\alpha(t)}d\bar{B}_t\end{array}
\frac{T_{13}(z)}{\lambda_{2}(z)h(\bar v,z)}\mathcal{B}_{a-1,b-1}(\bar u;\bar v) = C_1 +C_2 + C_3{}= \left(\frac{T_{13}^{(1)}(z)T_{11}^{(2)}(z)}{\lambda_2(z)h(\bar v,z)}+ \frac{T_{23}^{(1)}(z)T_{12}^{(2)}(z)}{\lambda_2(z)h(\bar v,z)}+ \frac{T_{33}^{(1)}(z)T_{13}^{(2)}(z)}{\lambda_2(z)h(\bar v,z)}\right) \mathcal{B}_{a-1,b-1}(\bar u;\bar v).
\left|\partial_x^{\alpha}\big(\mathfrak Z_{}(x,y)- \mathfrak Z_{}(x,0)\big)\right| =\left|\sum_{k=1}^3 \,\partial_x^{\alpha}\partial_{y_k}\,\mathfrak Z_{}(x,\theta y) \,y_k\right|\le \mathfrak C(S_1,S)\, (|x|s_\tau(x))^{-3/2-|\alpha|/2}
T_0^{(-2,1,0,0)}(x) =-\frac{T(x,\xi_1;\xi_0,\xi_0)}{4G(\xi_0,\xi_1)^2G(x,\xi_0)^2}, &=\frac{(2\zeta+1)^2(\zeta+2)}{\zeta^2 x^3}\left((\zeta^2+\zeta+1)x(2\zeta+1-x)+\zeta(2\zeta+1)^2\right).
A^{32}_n:=\sum_{k: n\leq |\widehat{a}_k|\leq 2n+1n\leq |\widehat{b}_k|\leq 2n+1}\frac{\left(\widehat{f}(\widehat{a}_k)\varphi_n(\widehat{a}_k)-\widehat{f}(\widehat{b}_k)\varphi_n(\widehat{a}_k)\right)^2}{|\widehat{b}_k-\widehat{a}_k|}\leq A^{31}_n\rightarrow 0,
\left( \frac{\partial x}{\partial\theta_{i}},\frac{\partial x}{\partial\theta_{j}}\right) =g_{i,j}\left( \theta_{1},\theta_{2}\right) \left\Vert x-x^{0}\right\Vert_{\mu}\leq1 \Vert x-x^{0}\Vert_{\mu+\alpha}\leq C\left\Vert g-g^{0}\right\Vert_{\mu+\delta-1}
K_{h_2}\circ K_{h_1}((x_1,x_2), A_1\times A_2)= \int_{X_2}K_{h_1}((x_1,y_2), A_1\times A_2)\> K_{h_2}^2(x_2,dy_2) &=\int_{X_2}\omega_{X_2}(A_2) K_{h_1}^1(x_1,A_1)\> K_{h_2}^2(x_2,dy_2) = K_{h_1}^1(x_1,A_1) \omega_{X_2}(A_2) &=K_{h_1}((x_1,x_2), A_1\times A_2)
\widehat{D}_{2\ell}(q)\vartheta_4^2(q)\vartheta_4^2(q^{2\ell})= \widehat{D}_{\ell}(q) \vartheta_4^2(q)\vartheta_4^2(q^{\ell})+\ell \widehat{D}_{2}(q^\ell) \vartheta_4^2(q^\ell)\vartheta_4^2(q^{2\ell}).
E^{(k)}_{a,b}(\tau) = a_0(E^{(k)}_{a,b}) + \sum_{\substack{m,n\geq 1 m \equiv a (N)}}\zeta_N^{bn}n^{k-1}q^{mn/N}+ (-1)^k \sum_{\substack{m,n\geq 1 m \equiv -a (N)}}\zeta_N^{-bn}n^{k-1}q^{mn/N}(q=e^{2\pi i\tau}).
\Gamma = \Gamma_{( o )}^{( 3 )}( r , \theta ) = \left\{\frac{1}{4}r^{2}\sin ( 2 \theta ) h_{1 2}+ \frac{1}{1 6}r^{4}\sin ( 4 \theta ) h_{1 4 c}+ \frac{1}{3 6}r^{6}\sin ( 6 \theta ) a_{1 1}\right\}\ + O ( r^{7}) \ .
\underset{l}{\lim\sup}|x^*_l(x_{l_1})| = \underset{l}{\lim\sup}|x^*_l(Qz_1)| \leqslant \underset{l}{\lim\sup}\|x^*_l-P^{\textsf{E}^*}_{I_l}x^*_l\| + \underset{l}{\lim\sup}|P^{\textsf{E}^*}_{I_l}x^*_l(z_1)| \leqslant \delta.
\Psi : M_Z \to \bigsqcup_{(\textbf{h}, \textbf{k})}\bigg( \prod_{i=1}^{24}M^{\textup{(N)}}_{x_i}(h_{x_i})\times \prod_{j=1}^{d-1}M^{\textup{(G)}}_{u_j}(h_{y_j}, k_{y_j}) \bigg),
\mathrm{S U G R A}_{d + 1}~ \mathrm{w i t h ~ I R ~ c u t o f f}~ z_{0}~ \Longleftrightarrow ~ \mathrm{Q F T}_{d}~ \mathrm{w i t h ~ U V ~ c u t o f f}~ \Lambda_{0}= z_{0}^{- 1}.
S = \frac{1}{2}\int d t \left[{\cal G}_{\alpha \beta}\left({\dot{z}}^{\alpha}{\dot{z}}^{\beta}+ i{\overline{\psi}}^{\alpha}\gamma^{0}D_{t}\psi^{\beta}\right) + \frac{1}{6}R_{\alpha \beta \gamma \delta}{\overline{\psi}}^{\alpha}\psi^{\gamma}{\overline{\psi}}^{\beta}\psi^{\delta}\right] - \frac{4 \pi v}{e^{2}}k ,
\sum_{i+j=3g-1}u_1^iu_2^j\langle\tau_i\tau_j\rangle_g=\sum_{i+k=g}\frac{(u_1^{3}+u_2^3)^{i}(u_1+u_2)^{k-1}(u_1u_2)^{k}}{24^{i}i!4^k(2k+1)!!}.
U=-G\int_{0}^{R}\frac{(4\pi r^{2}\rho)(\frac{4}{3}\pi r^{3}\rho)}{r}dr=-G\frac{16}{3}\pi^{2}\rho^{2}\int_{0}^{R}r^{4}dr=-G\frac{16}{15}\pi^{2}\rho^{2}R^{5}
\begin{gathered}g (a,t) = (\nabla u) (X(a,t), t) = \mathbb{L}_\nu (\nabla u_0) \circ X (a,t) + \mathbb{G}\left (\tau \circ X^{-1}\right ) \circ X (a,t) - \mathbb{U}\left (\nabla_x \left ( ( v \otimes v ) \circ X^{-1}\right ) \right ) \circ X (a,t).\end{gathered}
\Gamma^{(1)}= \int d^3 x \left( f^{(0)}u^4 + f^{(2)}u^3 \left[ \psi^2 \right] + f^{(4)}u^2 \left[ \psi^4 \right]+ f^{(6)}u \left[ \psi^6 \right]+ f^{(8)}\left[ \psi^8 \right] \right) ,
\renewcommand{\arraystretch}{1 . 3}\begin{array}{r r c}{T^{k}T^{k}\left| a \right> = t ( t + 1 ) \left| a \right> ,}&{T^{3}\left| a \right> = t^{z}\left| a \right> ,}&{F \left| a \right> = f \left| a \right> ,}\\{Q \left| a \right> = q \left| a \right> ,}&{}&{}\\ \end{array}
\eta^{2}(A\mathbf{z})=-\langle A\mathbf{z},\overline{A\mathbf{z}}\rangle=-\langle A\mathbf{z},A\mathbf{\overline{z}}\rangle=-\langle \mathbf{z},\mathbf{\overline{z}}\rangle=\eta^{2}(\mathbf{z}).
\begin{aligned}\phi(a_{1p^t_1}, \ldots, a_{\omega^t-1 p^t_{\omega^t-1}}, u_{\omega^t p^t_{\omega^t}}, a_{\omega^t+1 p^t_{\omega^t+1}}, \ldots, a_{dp^t_d}) & - \phi(a_{1p^t_1}, \ldots, a_{\omega^t-1 p^t_{\omega^t-1}}, u_{\omega^t p^t_{\omega^{t}}-1}, a_{\omega^t+1 p^t_{\omega^t+1}}, \ldots, a_{dp^t_d}),\end{aligned}
\partial_s \ell(t,\sigma) = L_0'(\sigma) - \frac{kt L_1'(\sigma)}{1+t L_1(\sigma)}\ , \sigma>\theta\ , \partial_s \Sigma(t,\sigma) = 1 + t L_0'(\sigma) + k\ \frac{L_1'(\sigma)}{L_1(\sigma)^2}\ \left[ \ln{(1+t L_1(\sigma))}- t L_1(\sigma) \right]\ , \sigma>\theta\ .
\widehat{R}_{k}( \mathrm{\boldmath ~ \l a m b d a ~ \u n b o l d m a t h}) = \left. \frac{1}{( 2 \pi )^{k}}\frac{\partial^{k}}{\prod_{p = 1}^{k}\partial J_{p}}Z_{k}( \mathrm{\boldmath ~ \l a m b d a ~ \u n b o l d m a t h}+{\bf J}) \right|_{{\bf J}= 0}\ ,
\nu\left(\overline{{\check{\sigma}}}\right):=\int_G \overline{{\check{\sigma}}}d \nu= \int_G \overline{\int_{\widehat{G}}\gamma(x) d\sigma(\gamma)}d\nu(x) = \int_{\widehat G}\widehat{\nu}\overline{d\sigma}=: \overline{\sigma}(\widehat{\nu}).
\psi = \left[ \begin{matrix}{0}\\{0}\\{q^{4}- i q^{3}}\\{q^{2}- i q^{1}}\\ \end{matrix}\right] = \left[ \begin{matrix}{0}\\{0}\\{M^{1}}\\{M^{2}}\\ \end{matrix}\right] , \quad \quad \gamma_{i}= \left( \begin{matrix}{0}&{i \sigma_{i}}\\{- i \sigma_{i}}&{0}\\ \end{matrix}\right) , \quad \gamma_{4}= \left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) ,
\left( \begin{matrix}{P^{I}}\\{i Q_{I}}\\ \end{matrix}\right) \rightarrow \Gamma \left( \begin{matrix}{P^{I}}\\{i Q_{I}}\\ \end{matrix}\right) = \left( \begin{matrix}{U}&{Z}\\{W}&{V}\\ \end{matrix}\right) \left( \begin{matrix}{P^{I}}\\{i Q_{I}}\\ \end{matrix}\right) ,
\begin{array}{lll}I(U_*) =I(u_0)+\displaystyle \frac{1}{2}\int_\Omega|DV|^2-\frac{k}{2^*}\int_S V^{2^*}+k\int_S V^{2^*-1}u_0 &\quad-\displaystyle \int_{\Omega}u_0^{2^*-1}V+k\int_{\Omega_1\setminus S}u_0^{2^*-1}V+ O(\frac{k}{\lambda^{\frac{N-2}{2}(1+\delta)}}).\end{array}
\mu_\omega(2B_\rho)\le \mu_\omega(B_\rho(\infty,3r))=C_{\beta,a}\, \mu(Z)\, r^{\tfrac{2\beta-a-1}{\beta-1}}= C_{\beta, a}\, \mu_\omega(B_\rho(\infty,r/2)) \le C_{\beta,a}\, \mu_\omega(B_\rho),
\begin{array}{rcl}A = \displaystyle <0|\varphi_1\varphi_2N\left(D_{34}\star\varphi_4\star\varphi_4+\varphi_4\star D_{34}\star\varphi_4+\varphi_4\star\varphi_4\star D_{34}\right)|0>.\end{array}
(t^{-1})' = t^{\mathrm{ij}}{\partial\pi^\alpha\over\partial p^{\mathrm{i}}}{\partial\pi^\beta\over\partial p^{\mathrm{j}}}\;{\partial\over\partial\pi^\alpha}\otimes{\partial\over\partial\pi^\beta}\;\in T_{\pi(\mathrm{p})}(\Sigma) \otimes T_{\pi(\mathrm{p})}(\Sigma)
P\left( \sum_{n\geq \tau_m^{\mathfrak{B}}}\left| \frac{1}{w(2\mathfrak{R}_n)}- \frac{1}{w(2\mathfrak{B}_n+1)}\right| \geq C_0 m \right) \leq C \sum_{i=m}^\infty \left( e^{-c(\log i)^2}+ e^{-c i^{\kappa'}}\right).
[y_1, y_2]=y_4+\theta_1y_5, [y_2, y_1]=\frac{\alpha_5}{\alpha_3}y_4+\theta_2y_5, [y_2, y_2]=\frac{\alpha_3\beta_2}{\beta_6\gamma_1}y_5, [y_1, y_3]=\frac{\beta_4}{\beta_6}y_5, [y_3, y_1]=y_5, [y_2, y_3]=y_4.
E\{(U^\top A U)^2\}=\sum_{i_1,j_1}\sum_{i_2,j_2}a_{i_1j_1}a_{i_2j_2}E(u_{i_1}u_{i_2}u_{j_1}u_{j_2}) &=\sum_i E(u_i^4)a_{ii}^2+2\sum_{i\neq j}a_{ij}^2+\sum_{i\neq j}a_{ii}a_{jj}&=\sum_i\delta_i a_{ii}^2+2(A^2)+\{(A)\}^2.
\lim_{t\to\infty}\left[\sup_{|x|\leq ct}|u(x,t;u_0)-\frac{a}{b}| + \sup_{|x|\leq ct}|\lambda_1 v_1(x,t;u_0)-\frac{a}{b}\mu_1| + \sup_{|x|\leq ct}|\lambda_2 v_2(x,t;u_0)-\frac{a}{b}\mu_2|\right]=0
F=b_1Y+\xi,\;\; X_2=\frac{B_{12,1}}{b_1-b_2}Y+Y_2,\;\;X_3=\frac{B_{13,1}}{b_1-b_3}Y+Y_3, &T=a_1Y+N-\frac{B_{12,1}}{b_1-b_2}Y_2-\frac{B_{13,1}}{b_1-b_3}Y_3-b_1\xi, &Q=2a_1+b_1^2+(\frac{B_{12,1}}{b_1-b_2})^2+(\frac{B_{13,1}}{b_1-b_3})^2.
dCov^{2}(X,Y)=\frac{1}{c_{p}c_{q}}\int_{\mathbb{R}^{p+q}}\frac{|\varphi_{X,Y}(s,t)-\varphi_{X}(s)\varphi_{Y}(t)|^{2}}{|s|_{p}^{1+p}|t|_{q}^{1+q}}dtds
\|x_n-x_m\|^4 =\|(x_n-x)-(x_m-x)\|^4 &\leq 2\|x_n-x\|^4+2\|x_m-x\|^4+12\|x_n-x\|^2\|x_m-x\|^2-\|(x_n-x)+(x_m-x)\|^4 &\leq 2\|x_n-x\|^4+2\|x_m-x\|^4+12\|x_n-x\|^2\|x_m-x\|^2-16d^4, ~\forall n , m \in \mathbb{N}.
I_{1}\ = \int \delta^{3}\! \left({\vec{r}_{1}- \vec{r}}\right) \delta \left( r_{1}- t \right) \frac{d^{3}r_{1}}{4 \pi r_{1}}= \frac{1}{4 \pi r}\delta \left( r - t \right) = \frac{1}{2 \pi}\delta \left( \tau^{2}\right)
\Phi^{*}_{2}\nu^{2}=\tfrac{1}{2}\sum_{p\neq q}( \psi_{p}(P_{1})\wedge\bar{\psi}_{q}(P_{1})\wedge\psi_{q}(P_{2})\wedge\bar{\psi}_{p}(P_{2}) - \psi_{p}(P_{1})\wedge\bar{\psi}_{p}(P_{1})\wedge\psi_{q}(P_{2})\wedge\bar{\psi}_{q}(P_{2})).
g_1''(\omega) = a''_{nm}(\omega) - (m^2 \omega^{m^2-1})^2 a_n''(\omega^{m^2}) - (m^2(m^2-1)\omega^{m^2-2})a_n'(\omega^{m^2}) &= \frac{m^4 b_n + m^2 c_n}{\omega^2}- \frac{m^4(b_n+c_n)}{\omega^2}- \frac{m^2(m^2-1)a_n'(1)}{\omega^2}=\frac{-m^2(m^2-1)(c_n+a_n'(1))}{\omega^2}
p(x,y)=[\begin{matrix}1&x&x^{2}&x^{3}\end{matrix}][\begin{matrix}a_{00}&a_{01}&a_{02}&a_{03}\\a_{10}&a_{11}&a_{12}&a_{13}\\a_{20}&a_{21}&a_{22}&a_{23}\\a_{30}&a_{31}&a_{32}&a_{33}\end{matrix}][\begin{matrix}1\\y\\y^{2}\\y^{3}\end{matrix}].
b_*=\left(\left(\begin{array}{cc}(1-z_1z_2)^{-1}z_2(1-z_1z_2)^{-1}&1\end{array}\right),\left(\begin{array}{cc}(1-z_1z_2)^{-1}z_1(1-z_1z_2)^{-1}&1\end{array}\right)\right), &h=\left(\left(\begin{array}{cc}1 -z_2 -z_1 1\end{array}\right),\left(\begin{array}{cc}1 -z_1 -z_2 1\end{array}\right)\right).
W_{\pi}(g_{t,l,v}) = \omega_{\pi}(-v^{-1})\chi_2(\varpi^{-t-2l}) \zeta_F(1)^{-1}q^{-\frac{t}{2}}\int_{1+\varpi^l\mathcal{O}}\omega_{\pi}(x)\psi(-xv^{-1}\varpi^{t+l})d^{\times}x.
\begin{cases}\zeta(s,\mathbb{A}^n/\mathbb{F}_1) = \frac{1}{s-n}Z(\mathbf{T}^{\times n},s) = \frac{s-n}{2 \pi}\zeta(s,\mathbb{P}^n/\mathbb{F}_1) = \prod_{k=0}^n \frac{1}{s-k}Z(\mathbb{P}^n_{\mathbb{F}_1},s) = \prod_{k=0}^n \frac{s-k}{2 \pi}\end{cases}
{\cal L}_{1 / m^{3}}^{( 2 + 1 ) s c a l}= - \frac{\alpha}{7 2 0 m^{3}}\left[ 6 F^{\nu \lambda}F_{\nu \lambda , \mu}^{\mu}+ 4 F^{\nu \lambda , \mu}F_{\nu \lambda , \mu}+ F_{\lambda}^{\nu \lambda ,}F_{\nu \mu ,}^{\mu}\right] ,
k^1_{n,i,j}= h\left(\left(cd(a-b)-\frac{b(e - a)(f - a)(g - a)}{(a - b - 1)}\right)\frac{(1+a-b)}{a(e-b)(f-b)(g-b)}-1\right), k^2_{n,i,j}= l\frac{b(e - a)(f - a)(g - a)(1+a-b)}{(a - b - 1)a(e-b)(f-b)(g-b)},
{}\sum_{i=1}^{p}\int_{0}^{T}w_{i}(t)\left [c_{i}(t)-\sum_{j=1}^{q}B_{ij}(t)z_{j}(t)+\sum_{j=1}^{q}\int_{0}^{t}K_{ij}(t,s)z_{j}(s)ds\right ]dt\geq 0.
\mathbb{P}\left(\bigcup_{A\in\mathcal{F}_1}\{f(A\widehat{+}A)=1\}\right) \leq \sum_{A\in\mathcal{F}_1}\mathbb{P}\left(f(A\widehat{+}A)=1\right)\leq \sum_{m\geq k-1}\sum_{A\in S_k^m\cap \mathcal{F}_1}\mathbb{P}\left(f(A\widehat{+}A)=1\right) &\leq \sum_{m\geq k-1}|S_k^m|2^{-(1-\delta)m}\leq \sum_{m\geq k-1}2^{-(\epsilon^3-\epsilon^4)m}=o(1).
\Xi_j^r := \frac{1}{e^{\pi i \tfrac{j+1}{r}}\hbar^{(j+1)/r}\Gamma(\tfrac{j+1}{r})(\zeta^{j+1}- 1)}\sum_{k = 0}^{r-2}(\zeta^{-k(j+1)}- \zeta^{j+1}) (\Psi_{k+1}- \Psi_k)
{\check \omega}^r(\theta^{\alpha},\dot \theta^{\alpha})=-{1\over 2}\epsilon^{ruv}[R^T\dot R]^{uv}(\theta^{\alpha},\dot \theta^{\alpha}) ={1\over 2}\epsilon^{ruv}[{\dot R}^TR]^{uv}(\theta^{\alpha},\dot \theta^{\alpha}).
d s_{E}^{2}= \breve{g}_{\mu \nu}d x^{\mu}d x^{\nu}= ( 1 + r^{2}\breve{\Omega}^{2}) d \tau^{2}- 2 \breve{\Omega}r^{2}d \tau d \tilde{\varphi}+ r^{2}d \tilde{\varphi}^{2}+ d r^{2}+ d z^{2}.
{\hat M}_{12}(p_1, q_1; p_1', q_1') = \frac 1{\pi^2}\chi_{[0,R]}(|q_1|)\frac{\widehat{|v|^{\frac 12}}\Bigl(p_1 + \frac{2}{\sqrt 3}q_1' +\frac{1}{\sqrt 3}q_1\Bigr) \widehat{|v|^{\frac 12}}\Bigl(\frac{1}{\sqrt 3}q_1'+ \frac{2}{\sqrt 3}q_1 - p_1'\Bigr)}{(2q_1' + q_1)^2 + 3q_1^2 + 3 z^2}\chi_{[0,R]}(|q_1'|) ,
\prod_{j=1}^N z_j^{N+1-j}(1+t^{-1}z_j^2)^{-1}\prod_{1 \le j<k \le N}(1+t^{-1}z_j z_k)^{-1}(t^{-1}+z_j z_k^{-1})^{-1}\times t^N\langle 1 \cdots M|\mathcal{B}^\prime(z_1) \cdots\mathcal{B}^\prime(z_N)| \overline{x_1}\cdots \overline{x_N}\rangle \Bigg|_{t=-1}=sp_{\overline{\lambda}}(\{z \}_N).
b = \left( \frac{N}{A}\right) \frac{\sinh ( N \tau ) + \sqrt{1 - \frac{A^{2}}{N^{2}}}\cosh ( N \tau )}{\cosh ( N \tau ) + \sqrt{1 - \frac{A^{2}}{N^{2}}}\sinh ( N \tau )}, \; \; e^{- \phi}= \cosh ( N \tau ) + \sqrt{1 - \frac{A^{2}}{N^{2}}}\sinh ( N \tau ) ,
\begin{array}[t]{c}{\displaystyle{\mathop{{\sum}'}}}{\scriptstyle 1\leq{n_1}<{n_2}< \cdots <{n_m}\leq D-1}\end{array}=\begin{array}[t]{c}{\displaystyle{\mathop{{\sum}}}}{\scriptstyle 1\leq{n_1}<{n_2}< \cdots <{n_m}\leq D-1}{\scriptstyle{l_j}{\not{|}}{n_i}}\end{array}\,\,\,.
{m^2(S) \over m^2(V)}={1 \over 6 (4\pi)^2}\left[36e^2+{\lambda^2 \xi^2 \over e^2}\left(15+{3 \over 4\omega^2}-{9 \xi \over \omega^2}+{27 \xi^2 \over \omega^2}\right)\right] .
m_{b}=\frac{5r_{b}^{8}+(\ell^{2}-28N^{2})r_{b}^{6}+5N^{2}(14N^{2}-\ell^{2})r_{b}^{4}+5N^{4}(3\ell^{2}-28N^{2})r_{b}^{2}+5N^{6}(\ell^{2}-7N^{2})}{10\ell^{2}r_{b}}
G(x,y) \leq \exp(- \langle\alpha, y-x\rangle) G_\alpha(x,y) \leq~ \exp(- \langle\alpha, y-x\rangle) \P_x(X_\alpha(t) = y \; \; t \geq 0) G_S(0,0) \leq~ \exp(- \langle\alpha, y-x\rangle) G_S(0,0),
c_j r = \sum_{i=j}^n r \left( X^{i-j}{i \choose j}b_i \right) \stackrel{[X \in N_m(D)]}{=}\sum_{i=j}^n \left( r X^{i-j}\right){i \choose j}b_i &=\sum_{i=j}^n \left( \sum_{k=0}^{i-j}X^{i-j-k}(-1)^k{i-j \choose k}\delta^k(r) \right){i \choose j}b_i \stackrel{[X \in N_l(D)]}{=}
\prod_{k \in F}f^{-1}\circ (\chi^{-1})^k \circ f \circ \chi^k = \prod_{k \in F}((\chi^{-1})^k)^f \circ \chi^k &= \prod_{k \in F}((\chi^{-1})^k)^f \circ \prod_{k \in F}\chi^k &= [f, \chi^F],
\mathcal{D}_z = \left( \begin{array}{@{\,}cc@{\,}}\mathbf{a}+\mathbf{b}\mathbf{x}\mathbf{0}\mathbf{0}\mathbf{a}'+\mathbf{b}'\mathbf{x}'\end{array}\right)\equiv \left( \begin{array}{@{\,}cc@{\,}}D_z \mathbf{0}\mathbf{0}D_z^{\prime}\end{array}\right)\,,
\varepsilon(\widetilde{X},\pi^\ast \mathcal F)=\varepsilon(X,\mathcal F)\cdot \varepsilon(Y,i^\ast\mathcal F)^{r-1}\cdot q^{-\frac{r(r-1)}{2}\chi(Y_{\bar k},i^\ast\mathcal F)}\quad{\rm in}~\Lambda^\times.
{dn(\omega) \over d\omega}\simeq{2 \omega^{D-2}\over (4\pi)^{(D-1)/2}}\sum_{n=0}^\infty\left[{c_n\over \Gamma\left({D-1 \over 2}-n\right)}\omega^{-2n}+{d_n \over \Gamma\left({D-2 \over 2}-n\right)}\omega^{-(2n+1)}\right]~~~,
\frac{c^{-\nu}_{2}}{c^{-\nu}_{0}}=\frac{1}{\displaystyle h^{-2}[s^2-(-\nu+2)^2]-\frac{1}{\displaystyle h^{-2}[s^2-(-\nu+4)^2]-\frac{1}{\displaystyle h^{-2}[s^2-(-\nu+6)^2]\cdot\cdot\cdot}}}
\inf_{\substack{\eta \in H^{1,1}_{0,per}(Q_T),\boldsymbol{\tau}\in H(\emph{div},Q_T) v \in L^2(Q_T), \alpha, \beta > 0}}\mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v) = \mathcal{J}(y(u),u).
{}_pF_q\left(\left.\begin{matrix}a_1,\dots,a_p b_1,\dots,b_q\end{matrix}\right|x\right):=\sum_{k\ge0}\frac{(a_1)_k\dotsm(a_p)_k}{(b_1)_k\dotsm(b_q)_k}\frac{x^k}{k!},
y_{r, \mathrm{D}}= \sqrt{E_\mathrm{R}}\, \alpha_{r, \mathrm{D}}\, x_r + \sqrt{E_\mathrm{I}}\sum\limits_{l = 1}^L{\beta_{l, \mathrm{D}}}\, d_{l, \mathrm{D}}+ w_\mathrm{D},
Hom^{G}(M_{\mathbb C}(n+1),Poly_{\mathbb{C}}(S^{2})_{\leq n})\simeq{\displaystyle\prod\limits_{l=0}^{n}}Hom^{G}(M_{\mathbb C}(\varphi_{l}),Poly(\varphi_{l}))\simeq\mathbb{C}^{n+1}
\mathrm{d}Y_\alpha(t) = \frac{b_\alpha}{2}\big[S_\alpha Y_N - (1-S_\alpha)Y_\alpha\big] \mathrm{d}t + \sqrt{\kappa_\alpha Y_\alpha Y_N}\mathrm{d}W_\alpha(t), \alpha=1,\dots,N-1,
\begin{aligned}f(x) \,\to\,\min g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 j\in\mathcal P \tilde G_l(x)\tilde H_l(x) \,=\,0 l\in\mathcal Q \end{aligned}
\MoveEqLeft \left|\iint_{Q_\ell\times Q_\ell}G_\ell(x-y)\left(\rho_{\gamma_{\xi'}}(y)\phi_\xi^*(x)\psi_\xi(x)-\phi_\xi^*(x)\gamma_{\xi'}(x,y)\psi_\xi(y)\right)\,dxdy\right| \leq C_G\sum_{n\geq 1}|\lambda_n(\xi')|\|\phi_\xi\|_{L^2_\xi}\||D_\xi|\psi_\xi\|_{L^2_\xi}=C_G\|\gamma_{\xi'}\|_{\mathfrak{S}_1(\xi')}\|\phi_\xi\|_{L^2_\xi}\||D_\xi|\psi_\xi\|_{L^2_\xi},
\left |{\mathrm{Cov}}({\bf 1}_{X_0\leq t},{\bf 1}_{X_k\leq t})\right | \leq \|{\mathbb E}({\bf 1}_{X_k\leq t}|{\mathcal F}_0)-F(t)\|_1 \leq \alpha_{1, \bf X}(k) \, , \left |{\mathrm{Cov}}({\bf 1}_{X_0\leq t},{\bf 1}_{X_k\leq t})\right | \leq{\mathrm{Var}}({\bf 1}_{X_0\leq t})=B(t) \, .
u^s_{A,q}(x,y)= \sum_{\substack{(\ell_1,m_1)\in \Gamma (\ell_2,m_2)\in \Gamma}}\gamma_{\ell_1m_1 \ell_2m_2}h^{(1)}_{\ell_1}(k\vert x\vert)Y^{m_1}_{\ell_1}\left(\widehat{x}\right) h^{(1)}_{\ell_2}(k\vert y\vert)Y^{m_2}_{\ell_2}\left(\widehat{y}\right).
\left\langle \left(\delta_{ij}-\sum_{k=1}^n y_{ki}y_{kj}\right)\mu, \varphi \right\rangle = \left\langle \mu, \left(\delta_{ij}-\sum_{k=1}^n y_{ki}y_{kj}\right)\varphi\right\rangle &=\int_{SO_n}\left(\delta_{ij}-\sum_{k=1}^n y_{ki}y_{kj}\right)\varphi(y) \mu(dy)=0.
\delta \Gamma_{e f f}= t r \int d^{d}x \, ( D_{\mu}\epsilon^{a}) \frac{\delta \Gamma_{e f f}}{\delta A_{\mu}^{a}}= - t r \int d^{d}x \, \epsilon^{a}D_{\mu}\left( \frac{\delta \Gamma_{e f f}}{\delta A_{\mu}^{a}}\right) \; .
( \lambda - 3 ) v ( \phi^{2}) + \phi^{2}v^{\prime}( \phi^{2}) = \frac{( N - 1 ) v^{\prime}( \phi^{2})}{\left( 2 + 2 V^{\prime}( \phi^{2}) \right)^{3 / 2}}+ \frac{v^{\prime}( \phi^{2}) + 2 \phi^{2}v^{\prime \prime}( \phi^{2})}{\left( 2 + 2 V^{\prime}( \phi^{2}) + 4 \phi^{2}V^{\prime \prime}( \phi^{2}) \right)^{3 / 2}}
\sum_{|W|,|Z|}\binom{n}{|W|}\binom{n}{|Z|}\cdot 2 \exp\Big(-\frac{p|W||Z|}{12}\Big) \le \sum_{|W|,|Z|}2\cdot \big(\tfrac{en}{|Z|}\big)^{2|Z|}\exp\big(-4|Z|\log\tfrac{en}{|Z|}\big) \le 2n^2\exp\big(-n^{-1/2}\big)\,,
\textstyle \sum_{j = 1}^\mu F_{\omega_j}\left({\omega_j \choose u-1}- \delta_{u=\alpha+1}((-1)^\alpha\delta_{j=1}+\delta_{j=\mu}\delta_{L \equiv_{p-1}\alpha})\right) \textstyle + \sum_{l=\alpha-m}^\alpha \sum_{r - m > i(p-1)+\alpha > m}C_l{r-\alpha+l \choose i(p-1) +l}{i \choose u-1}= \delta_{u=\alpha+1},
X\leq Y \mbox{iff}\exists \; e\in \mathcal{A}\; e : X\leadsto Y & X \wedge Y = \{pab : a\in X \; \wedge \; b\in Y\}& X \vee Y = \{pka \ : a\in X\}\cup \{p\breve{k}b : b\in Y\}& \bot = \emptyset & \top = \mathcal{A}& X\Rightarrow Y = \{e\in \mathcal{A}: e:X\leadsto Y\}
h_{\mu \nu}( t ,{\bf x}) = - 8 \pi{\cal G}\frac{1}{( 2 \pi r )^{( D - 2 ) / 2}}\partial_{t}^{( \frac{D - 4}{2})}\left[ \int d^{D - 1}{\bf x^{\prime}}S_{\mu \nu}( t - |{\bf x - x^{\prime}}| ,{\bf x^{\prime}}) \right] \, ,
\varphi( x_1,\cdot) \in L^1(S_2,d|\mu_{x_1}|) \eta x_1 \in S_1, x_1 \to \int_{S_2}\varphi(x_1,x_2) \,d\mu_{x_1}(x_2) \ \in L^1(S_1, d\eta), \int_{S_1 \times S_2}\varphi( x_1,x_2) \,d\mu(x_1,x_2) = \int_{S_1}\left( \int_{S_2}\varphi(x_1,x_2) \,d\mu_{x_1}(x_2) \right) \,d\eta(x_1).
V_{\mathrm{d i p o l e}}= -{\frac{1 5}{3 2}}\pi^{3}i \bar{\Theta}_{1}\Gamma^{0 i j}\Theta_{1}\bar{\Theta}_{2}\Gamma^{0 i k}\Theta_{2}\partial_{j}\partial_{k}\left\{{\frac{1}{r^{7}}}\right\}\; ,
s(x,y)= \left(\left(\sqrt{x\left(1-\frac{x}{4}\right)},\sqrt{y\left(1-\frac{x}{2}-\frac{y}{4}\right)},1-\frac{x+y}{2}\right),\right. \phantom{coucou}\left.\left(-\sqrt{x\left(1-\frac{x}{4}\right)},\sqrt{y\left(1-\frac{x}{2}-\frac{y}{4}\right)},1-\frac{x+y}{2}\right)\right).
\left[\begin{array}{c}\mathbf{x}^{T}V^{T}\end{array}\right] A_{k}^{-1}\left[\begin{array}{cc}\mathbf{x}V\end{array}\right] =\left[\begin{array}{cc}\varepsilon_{k}^{-1}\mathbf{b}_{k}^{T}0 L_{k}^{-1}\end{array}\right]\mbox{\ \with$\mathbf{b}_k^T=-\frac{\mathbf{c}^TL_k^{-1}}{\varepsilon_{k}}$},
2 9 I_{A}[ \sigma ; g_{\mu \nu}] = \int d^{D}x [ \textstyle{\frac{1}{2}}\: \sigma \Delta_{D}\sigma + \sigma \bar{{\cal E}}_{D}] , \delta I [ \sigma + \phi ; \: 2 \phi g_{\mu \nu}] / \delta \phi = \bar{\cal E}_{D}.
{\overline{{\phi}}}( p , z ) = - \int{\frac{d^{2}k}{( 2 \pi )^{2}}}e^{- i k \cdot z}\left[ b L^{2}{\frac{{\cal O}_{4}( p )}{k^{2}+ p^{2}}}+ ( 2 \pi )^{4}c L^{- 2}{\frac{\delta^{( 4 )}( p )}{k^{2}}}\right] ~ .
Y_{t}^{i}=\zeta^{i}+\int_{t}^{T}g^{i}\left( Y_{s}^{i},Z_{s}^{i},\mathbb{P}_{\left( Y_{s}^{i},Z_{s}^{i}\right)}\right) \mathrm{d}s-\int_{t}^{T}Z_{s}^{i}W_{s}\mathrm{d}s,i=1,2,
\mathcal{L}:= \bigoplus_{i\in\mathbb{N}}\mathbb{Z}_{n_i}= \mathbb{Z}_{n_1}\oplus \mathbb{Z}_{n_2}\oplus \mathbb{Z}_{n_3}\oplus \cdots & \,= \big\{(a_j)_{j=1}^{\infty}\in{\textstyle\prod_{j=1}^{\infty}\mathbb{Z}_{n_j}}: a_j \big\},
w = \begin{pmatrix}u_{00}+ u_{01}(1-v_{11}u_{11})^{-1}v_{11}u_{10}u_{01}(1-v_{11}u_{11})^{-1}v_{12}v_{21}(1-u_{11}v_{11})^{-1}u_{10}v_{22}+ v_{21}(1-u_{11}v_{11})^{-1}u_{11}v_{12}\end{pmatrix}.
(e_{i}f_{j}-f_{j}e_{i})T(R) = \sum_{r=1}^{j}\sum_{s=1}^{i}\Phi(R,-\delta^{jr})f_{jr}(R)e_{is}(R+\delta^{jr})T(R-\delta^{jr}+\delta^{is}) - \sum_{r=1}^{j}\sum_{s=1}^{i}\Phi(R,\delta^{is})e_{is}(R)f_{jr}(R+\delta^{is})T(R-\delta^{jr}+\delta^{is}).
\langle \mathrm{` `}\partial_{\mu}j^{\mu}\mathrm{'^{\prime}}\rangle = -{\frac{e}{2 h^{4}}}[ ( t_{2})_{2 1}( t_{2})_{1 1}+ ( t_{3})_{2 1}( t_{3})_{1 1}- ( t_{2})_{2 0}( t_{2})_{1 0}- ( t_{3})_{2 0}( t_{3})_{1 0}] .
F_{\varphi}(x,y) =\chi\bigl(b(\rho(y))d(x,y)\bigr)\varphi(y),{\mathcal{R}}(\varphi)(x) =\int_{M}F_{\varphi}(x,y) dm(y),\{\mathcal{P}}(\varphi) =\frac{{\mathcal{R}}(\varphi)}{{\mathcal{R}}(1)},
J_{n,i,22}^{22}(\theta) = - \partial_\alpha f_\alpha (z^n_i(\theta)) - \frac{\ln \left( n / X_{\frac{i-1}{n}}\right)^2}{\alpha^4}\left(z^n_i(\theta) k'_\alpha (z^n_i(\theta)) \right) + \frac{\ln \left( n / X_{\frac{i-1}{n}}\right)}{\alpha^2}\left( -\frac{2}{\alpha}k_\alpha (z^n_i(\theta)) + 2z^n_i(\theta) f'_\alpha (z^n_i(\theta)) \right).
\Delta_{H}(x;y,z) =\frac{1}{\left\Vert k_{x}\right\Vert}\sup\left\{\operatorname{Re}j(x):j\in Hj(y)=j(z)=0,\left\Vert j\right\Vert_{H}\leq1\right\}. \Delta_{G}(x;y,z) =\sup\left\{\operatorname{Re}m(x):m\in A\text{,}m(y)=m(z)=0,\left\Vert m\right\Vert_{A}\leq1\right\}
| \alpha_x \rangle = \frac{1}{\sqrt{C_1^2 + \pi^2C_2^2}}(C_1(x)| \delta_x \rangle + h(x)A(x)) \langle\alpha_x' |= \frac{1}{\sqrt{C_1^2 + \pi^2C_2^2}}(C_1(x) \langle\delta_x | + g(x)A'(x))
f_{1/4}(\tau) :=t_4(\tau)=\frac{\eta(\tau)^8}{\eta(4\tau)^8}, f_{1/2}(\tau) :=\frac{1}{t_4(\tau)+16}=\frac{\eta(\tau)^8\eta(4\tau)^{16}}{\eta(2\tau)^{24}}, f_{1/1}(\tau) :=\frac{1}{t_4(\tau)}=\frac{\eta(4\tau)^8}{\eta(\tau)^8}