harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi^{0}(z)=\sum_{m\in{\bf Z}}d^{0}_{m}z^{-m-\frac{1}{2}}~,\quad \widetilde{\psi}^{0}(\overline{z}) =\sum_{m\in{\bf Z}}d^{0}_{m}\overline{z}^{-m-\frac{1}{2}}~,
\begin{array}{l l l}{d = 1 0 :}&{\Gamma_{M}}&{M = 1 , \cdots , 1 0 ,}\\{d = 1 1 :}&{{\tilde{\Gamma}}_{\tilde{M}}}&{{\tilde{M}}= 1 , \cdots , 1 0 , 1 1 ,}\\{d = 1 2 :}&{{\hat{\Gamma}}_{\hat{M}}}&{{\hat{M}}= 1 , \cdots , 1 0 , 1 1 , 1 2 .}\\ \end{array}
p(t,x) \leq c_1\int_0^\infty r^{d-1}e^{-t\phi(r^2)}\,dr=c_1\int_0^\infty r^{d-1}\int_{t\phi(r^{2})}^\infty e^{-s}\,ds\,dr &=c_1\int_0^\infty e^{-s}\int_0^{\phi^{-1}(\frac{s}{t})^{1/2}}r^{d-1}\,dr\,ds= \tfrac{c_1}{d}\int_0^\infty e^{-s}\left[\phi^{-1}(\tfrac{s}{t})\right]^{\frac{d}{2}}\,ds
2 \sum_{i=1}^{N}\delta \alpha_{i}\lambda_{i}= \sum_{i=1}^{N}\delta\alpha_{i}\lambda_{i}+ \sum_{i=1}^{N}(\alpha_{i}- 1) ( \delta\lambda_{i}- \delta \lambda_{0}) =\delta \left[ \sum_{i=1}^{N}(\alpha_{i}-1)(\lambda_{i}-\lambda_{0})\right].
((\tau))=-\left(\frac{t^4}{3}+\frac{t\chi\zeta(3)}{(2\pi)^3}\right), (\tau_{ij})Z^i\overline{Z}^j=|Z^0|^2\left(-\frac{2t^3}{3}+\frac{\chi\zeta(3)}{(2\pi)^3}\right)
S_I[x,\vec{\alpha}] = \int^t_0{dt'}\left\{\frac{1}{2}\left( \vec{\alpha}\cdot\frac{d \vec{\alpha}^*}{dt'}- \vec{\alpha}^* \cdot\frac{d\vec{\alpha}}{dt'}\right)- \frac{i}{\hbar}(H_R - \dot{x}h_I) \right\}
\sum_{\substack{m\in \mathbb{Z}3m^2+m\leq 2n}}\left(m+\frac{1}{6}\right) c\left(f; n-\frac{3}{2}m^2-\frac{1}{2}m\right) =\frac{4}{3}\sigma(n) -\frac{16}{3}\sigma\left(\frac{n}{2}\right)-2 \sum_{\substack{a,b\in \mathbb{Z}ab=2n 6|3a+b-1}}d\left(N, \tilde{N}, \frac{1}{6}, \frac{1}{6}\right).
{\cal F}^{(0)}={1\over a^3}\int_{-\infty}^\infty{d\omega\over 2\pi}e^{-i\omega\tau}\sum_{l=1}^\infty{2l+1\over4\pi}\left\{x'[\tilde s_l'(x')\tilde e_l'(x')-\tilde e_l(x')\tilde s_l''(x')]-x[\tilde s_l'(x)\tilde e_l'(x)-\tilde e_l(x)\tilde s_l''(x)]\right\}.
\Gamma (p^{2},m^{2})=\frac{i\lambda^{2}\mu^{\epsilon}}{16\pi^{2}\epsilon}-\frac{i\lambda^{2}\mu^{\epsilon}}{32\pi^{2}}\left\{\gamma+\int_{0}^{1}dz\ln [\frac{p^{2}z(1-z)-m^{2}}{4\pi \mu^{2}}]\right\}.
{}_M \mathcal{X}_M^{\pm}= \{[\alpha_{(0,0)}], [\alpha_{(1,0)}^{\pm}], [\alpha_{(0,1)}^{\pm}], [\alpha_{(2,0)}^{\pm}], [\alpha_{(1,1)}^{\pm}], [\alpha_{(0,2)}^{\pm}], [\alpha_{(3,0)}^{(j)\pm}], [\alpha_{(2,1)}^{(j)\pm}], [\alpha_{(1,2)}^{(j)\pm}], \mbox{for}j=1,2 \},
\left\| \nabla^2 v \right\|_{L^2(A_m^+)}\lesssim \ \tau^2 \left\| |x|^{-2}v \right\|_{L^2(A_{m-1}^+\cup A_m^+ \cup A_{m+1}^+)}+ \left\| f \right\|_{L^2(A_{m-1}^+\cup A_m^+ \cup A_{m+1}^+)}.
ds^2 = \frac{\rho_1^2}{a(x_{1})}dx_1^2 +\frac{\rho_{2}^{2}}{a(x_{2})}dx_{2}^{2}+ \frac{(\rho_{1}\rho_{2})^{2}}{a(x)}dx^{2}+ d\rho_{1}^{2}+ d\rho_{2}^{2},
\left( \int_{\gamma_\rho^+}+\sum_{\sigma_k=iy_k,\lambda_{\nu ,k}}\int_{C_\rho (\sigma_k)}\right)\frac{\bar H^{(1)}_{\nu}(z)}{\bar J_{\nu}(z)}f(z)dz-\left( \int_{\gamma_\rho^-}+\sum_{\sigma_k=-iy_k,-\lambda_{\nu ,k}}\int_{C_\rho (\sigma_k)}\right)\frac{\bar H^{(2)}_{\nu}(z)}{\bar J_{\nu}(z)}f(z)dz,
\Lambda_1=\inf_{\eta\in H^1((0,1),r^{M-1}dr), \eta(1)=0\atop\eta\not\equiv0}\frac{\int_0^1 r^{M-1}|\eta'|^2\,dr-\int_0^1r^{M-1}a(r)\eta^2 \,dr}{\int_0^1 r^{M-3}\eta^2\, dr}<0.
\hat \tau=\begin{cases}\displaystyle\min_{t>0}\{t \ \ | \ \ \|\mathbb{T}_t\psi\|\leqslant r\}\ \hbox{\rm if}\ \ \{t \ \ | \ \ \|\mathbb{T}_t\psi\|\leqslant r\}\neq \emptyset +\infty \ \hbox{\rm if}\ \ \{t \ \ | \ \ \|\mathbb{T}_t\psi\|\leqslant r\}= \emptyset.\end{cases}
\gamma_{n-2}=\frac{z^{2}\left( \frac{1}{2}-\frac{1}{2}n+\gamma_{n-1}\right)^{2}}{\gamma_{n-1}\left( \frac{1}{2}-n+\gamma_{n}+\gamma_{n-1}\right)}+n-\frac{3}{2}-\gamma_{n-1}.
B^{-1}\mathcal{L}_1 B=\rho\,\mathcal{L}_2, \mathcal{L}_2:=\Pi_S^{\perp}(\omega\cdot\partial_{\vartheta}+m_3 \partial_{yyy}+c_1 \partial_y+c_0)\Pi_S^{\perp}+\mathfrak{R}_2, &c_1:=\rho^{-1}(B^{-1}b_1), c_0:=\rho^{-1}(B^{-1}b_0), \mathfrak{R}_2:=\rho^{-1}B^{-1}\mathfrak{R}_1 B.
A:=\left(\begin{array}{ccccc}-\alpha_{k-1}/\alpha_k -\alpha_{k-2}/\alpha_k \ldots -\alpha_{1}/\alpha_k -\alpha_{0}/\alpha_k 1 0 \ldots 0 0 0 1 \ldots 0 0 0 0 \ddots \vdots \vdots 0 0 \ldots 1 0 \end{array}\right)
P(t)=\sum_{k>0}p_{k}(X)t^{k}=\sum_{k>0}\sum_{i=1}^{\infty}(X_{i}t)^{k}=\sum_{i=1}^{\infty}\frac{X_{i}t}{1-X_{i}t}=\frac{tE^{\prime}(-t)}{E(-t)}=\frac{tH^{\prime}(t)}{H(t)}
\mathcal{L}(\mathbf{p}_,\mu) =\sum_{k=1}^{K}\log\left(1+\frac{\alpha_k}{\hat{\lambda}^{2}_{_k}p_{_k}+ \beta_k}\right)+ &\mu(\sum_{k=1}^{K}p_{_k}-P_)+\sum_{k=1}^{K}\tau_k p_{_k},
L = \bar{\psi}i \gamma_{\mu}\partial_{\mu}\psi - j_{\mu}( A_{\mu}+ B_{\mu}+ \epsilon_{\mu \nu}\partial_{\nu}\theta ) + \frac{1}{\pi}A_{\mu}\epsilon_{\mu \nu}\partial_{\nu}\theta + \frac{1}{2 \pi}\partial_{\nu}\theta \partial_{\nu}\theta ,
\Delta ( h^{1}h^{2}) = \Delta ( h^{1}) \, \Delta ( h^{2}) : = \sum h_{1}^{1}h_{2}^{1}\otimes h_{1}^{2}h_{2}^{2}~ , \qquad \Delta ( h_{i}) = \sum h_{i}^{1}\otimes h_{i}^{2}~ ,
\mathfrak{a}_2^3 \mathfrak{a}_3 = (1 \otimes y_2^3- y_2 \otimes y_2^2+ y_2^2 \otimes y_2- y_2^3 \otimes 1) (1 \otimes y_3- y_3 \otimes 1) &= y_1y_2 \otimes y_2y_3+ y_2y_3 \otimes y_1y_2- y_1y_2y_3 \otimes y_2- y_2 \otimes y_1y_2y_3.
A = \frac{d^{l + m}}{d x^{l + m}}( 1 - x )^{l^{-}}( 1 + x )^{l^{+}}= \sum_{k = 0}^{l + m}{\binom{l + m}{k}}\frac{d^{k}}{d x^{k}}( 1 - x )^{l^{-}}\frac{d^{l + m - k}}{d x^{l + m - k}}( 1 + x )^{l^{+}}.
\mathcal{R}_{\xi,I}:=\left\{(x,t)| \xi<-\frac{C_{L}^2}{2}\right\}, \mathcal{R}_{\xi, II}:=\left\{(x,t)|-\frac{C^2_L}{2}<\xi<-\frac{C_{R}^2}{2}\right\}, \mathcal{R}_{\xi, III}:=\left\{(x,t)|\frac{-C_{R}^2}{2}<\xi<\frac{C_{R}^2}{2}\right\}, \mathcal{R}_{\xi,IV}:=\left\{(x,t)|\xi>\frac{C_{R}^2}{2}\right\}, \ \xi=\frac{x}{12t},
h(x) \le \Bigl(g_4(x)+\frac{2}{6^5}(2^5x-13)(102-3^5x) &\qquad+\frac{2}{6^6}(3^6 x - 304)(27-2^6x)\Bigr)\Bigr|_{x=\frac{88}{211}}+ \frac1{10\cdot 6^6}- \eta &=-\frac{56411054147338655852222279153714125703}{6316397576863411724254503567359990235792000}<0,
U_{\lambda}:=\left\{(x,y,t)\in \mathcal{Q}_{r_{1}}\,:\, |D^{\tau}d_{s}u(x,y,t)|\geq\lambda \right\}\subset\left(\bigcup\limits_{i\in\mathbb{N}}\mathcal{Q}_{5\rho_{{i}}}\left(z_{i}\right)\right)\bigcup \left(\bigcup\limits_{\mathcal{K}\times I\in\mathcal{A}}\mathcal{K}\times I\right)
\varepsilon_{k+1}= \frac{p-1}{p}\sigma^{-\frac{1}{p-1}}\frac{(A_{k+1}- A_k)^{\frac{p}{p-1}}}{\delta}\|\nabla f(x_{k+1})\|^{\frac{p}{p-1}}+ \frac{A_{k+1}}{\delta}(f(y_{k+1}) -f(x_{k+1})).
\| |D|^\frac12 \bar P[ w(1-Y) R_\alpha \bar R]\|_{L^2}\lesssim \ \| w(1-Y)\|_{L^2}\| |D|^\frac12 \bar P[R_\alpha \bar R]\|_{BMO}\lesssim_A \ \|w\|_{L^2}\| |D|^\frac12 R\|_{BMO}\|R_\alpha\|_{BMO}
\bar{\mathcal{A}}_{b,m}\doteq \{(\varphi_{k})_{k\in \mathbb{N}_{0}}:\varphi_{k}\in \bar{\mathcal{A}}_{+}\mbox{for each}k\in \mathbb{N}_{0}\mbox{such that for all}(\omega ,t,y)\in \Omega \times \lbrack0,T]\times \lbrack 0,1], & \frac{1}{m}\leq \varphi_{k}(\omega,t,y)\leq m\mbox{for}k\leq m\mbox{and}\varphi_{k}(\omega ,t,y)=1\mbox{for}k>m\}
F_j (u_j) - F(u) = \big( F_j (u_j) - F(u_j) \big)+ \big( F (u_j) - F_M(u_j) \big) & \hphantom{XXX}+ \big( F_M (u_j) - F_M(u) \big)+ \big( F_M (u) - F(u) \big).
\sum_{k=1}^\infty \frac{d_{i_k}}{q_{i_1}q_{i_2}\cdots q_{i_k}}= \frac{d_0}{q_0-1}+ \bigg(d_1-d_0\frac{q_1-1}{q_0-1}\bigg)\, \pi_{q_0,q_1}(i_1i_2\cdots).
\prod_{m=1}^{M}\left\{4 \sin^2\left[\frac{(2m-1)\pi}{2(2M+1)}\right]- 4 \sin^2\left[\frac{2n\pi}{2(2M+1)}\right] \right\}=(-1)^n\left[\cos\frac{\pi n}{2M+1}\right]^{-1}.
\int g(f-P_tf) d\mu = \int_0^t \int g \frac{\partial P_sf}{\partial s}d\mu ds = \int_0^t \int g \Delta P_sf d\mu ds = - \int_0^t \int \Gamma(P_sg,f) d\mu ds & \le \int_0^t \| \sqrt{\Gamma(P_sg)}\|_{\infty}\int\sqrt{\Gamma(f)}d\mu ds \le C_1 \sqrt{2 t}\int \sqrt{\Gamma(f)}d\mu.
d\left(f_{1}\circ g_{1},f_{2}\circ g_{2}\right)=\lim_{n\rightarrow\mathcal{U}}d\left(f_{1,n}\circ g_{1,n},f_{2,n}\circ g_{2,n}\right) \le \lim_{n\rightarrow\mathcal{U}}\left[d\left(f_{1,n},f_{2,n}\right)+d\left(g_{1,n},g_{2,n}\right)\right]=d\left(f_{1},f_{2}\right)+d\left(g_{1},g_{2}\right)
c_{i}=D_{i}\cdot d_{c,\sigma,j}=\int_{\mathbb{P}_{r_{1},r_{2}}}f^{*}D_{i}=\int_{\mathbb{P}_{r_{1},r_{2}}}^{\mathbb{T}}f^{*}U_{i}=\frac{U_{i}(\sigma)}{U_{j}(\sigma)/c}+\frac{U_{i}(\sigma^{\prime})}{-U_{j}(\sigma^{\prime})/c^{\prime}}=\frac{U_{i}(\sigma)}{U_{j}(\sigma)/c}+\frac{U_{i}(\sigma^{\prime})}{-U_{j}(\sigma)/c}
\frac{dc}{dt}=\mu hK_1\frac{b+c}{1+c}-\frac{\Gamma c}{K+c}+\lambda \theta=R_1(c,\theta,h;\mu,\lambda), \frac{d \theta}{dt}=-\theta+T(c)=R_2(c,\theta), \frac{dh}{dt}=\frac{1}{1+c^2}-h=R_3(c,h).
\mu_{\rho}= \frac{1}{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}\left[ \left( \frac{g^{2}}{2 \pi^{2}}p^{2}D ( p ) + 2 z \right)^{- 1}- \left( \frac{1}{\pi n}p^{2}+ 2 z \right)^{- 1}\right]
\frac{1}{N}\sum_{n=1}^N F_1(U^{an}(x, y, z)) F_2(U^{bn}(x, y, z))\exp \left[ -12\pi i (b-a)(bq_2y\alpha - 3bq_2) \alpha^2 n^2 \right] &= \frac{C_{y, z}}{N}\sum_{n=1}^N f_1(T^{an}x) f_2(T^{bn}x) \exp(2\pi i (12bq_2(b^2-a^2)) n^3 \alpha^2).
b_{\lambda}b_{(s)}=\psi_{w_{0}}y_1^{\lambda_1+n-1}y_2^{\lambda_2+n-2}\cdots y_n^{\lambda_n}\psi_{w_{0}}y_{\min}\psi_{w_{0}}y_1^sy_1^{n-1}y_2^{n-2}\cdots y_n\psi_{w_{0}}y_{\min}&=\psi_{w_{0}}y_1^{\lambda_1+n-1}y_2^{\lambda_2+n-2}\cdots y_n^{\lambda_n}\sum_{l_1+\cdots+l_n=s}y_1^{l_1}\cdots y_n^{l_n}\psi_{w_{0}}y_{\min}
\hat \Psi_{\alpha}(x) =\hat \Psi_{\alpha}(x)-\tau_{\alpha}(x)\hat \psi_{\alpha}(0)+\tau_{\alpha}(x)\hat \psi_{\alpha}(0) &=\tau_{\alpha}(x)\hat \psi_{\alpha}(0)+\sum_{k=0}^{\tau_{\alpha}(x)-1}\hat \psi_{\alpha}(f_{\alpha}^k(x))-\hat \psi_{\alpha}(0),
\hat{\mu}^p_{\Lambda,\epsilon}(d\phi, dt) := \frac{1}{Z^p_{\Lambda,\epsilon}(\alpha)}e^{-A^p(\phi, t)}\prod_{jk \in E(\mathbb{T}_N)}\Big(f_\alpha(e^{t_{jk}})e^{t_{jk}}dt_{jk}\Big)\prod_{j \in \Lambda}d\phi_j,
\nu_j(n, s) = -\frac{(s + \log_{1/p}(1 + (p/q)^s) + \psi(n) + 1)^2}{2}~ ~ ~~- \log_{1/p}n \log_{1/p}(1 + (p/q)^s) + \psi(n)^2/2 + o(\psi(n)^2).
N(X, 2r)^{1+\delta}< N(X-X, 2r) \le 4N(D, 4\times 2r) \le 4 C_\varepsilon \left(\frac{r\sqrt{2i-1}}{4\times 2r}\right)^{s+\varepsilon}=4C_\varepsilon \left( \frac{\sqrt{2i-1}}{8}\right)^{s+\varepsilon}
\xi_n(\tau_n) = -\frac{d}{d\tau_n}M_n(\tau_n) = -\frac{d}{d\tau_n}\ln\big(e^{-a_n\tau_n}+e^{-b_n\tau_n}\big) = \frac{a_ne^{-a_n\tau_n}+b_ne^{-b_n\tau_n}}{e^{-a_n\tau_n}+e^{-b_n\tau_n}},
\frac{1}{2}\int_{\varepsilon^{2}}^{\infty}\frac{d t}{t^{3}}\sum_{k = - \infty}^{\infty}\left[ e^{- k^{2}\pi t / R^{2}}- e^{- ( k + 1 / 2 )^{2}\pi t / R^{2}}\right] = \frac{1}{R^{4}}\frac{3 1}{1 6}\frac{3}{4 \pi^{2}}\zeta ( 5 )
V_b = \frac{e^2}{2\pi}\frac{1}{{q^+}^4}\int_0^{q^+}\frac{dk^+}{q^- - \frac{m^2}{2k^+}- \frac{m^2}{2(q^+-k^+)}}\stackrel{q^+\rightarrow 0}{\longrightarrow}- \frac{e^2}{\pi}\frac{1}{6m^2}\frac{1}{{q^+}^2}
\Sigma = m + \frac{4 g^{2}}{\beta}\sum_{n = - \infty}^{\infty}\int^{\Lambda}\frac{d^{d}{\bf q}}{( 2 \pi )^{d}}\frac{\Sigma}{{\bf q}^{2}+ ( \rho - \bar{\omega}_{n})^{2}+ \Sigma^{2}},
\prod_{k=1}^\infty(1-X^{2k})(1+X^{2k-1})^2 =\sum_{k=-\infty}^\infty X^{k^2}, \prod_{k=1}^\infty\frac{(1-X^k)^2}{1-X^{2k}}=\prod_{k=1}^\infty(1-X^{2k})(1-X^{2k-1})^2 =\sum_{k=-\infty}^\infty(-1)^k X^{k^2}, \prod_{k=1}^\infty(1-X^{2k})(1+X^{2k})^2 =\frac{1}{2}\sum_{k=-\infty}^\infty X^{k^2+k}=\sum_{k=0}^\infty X^{k^2+k},
\sum_{x=0}^n|M_{x,0}(m)|^2= \sum_{j_1,\ldots,j_4 =0}^n\Theta_m(\mu_{j_1},\mu_{j_2}) \Theta_m(\mu_{j_3},\mu_{j_4}) \prod_{k=1}^4\psi_{j_k}(0) \sum_{x=0}^n\prod_{k=1}^4\psi_{j_k}(x) .
\pi_1(X^{y,\Gamma}_n) \leqslant \pi_1(Z_{j_0-1})+\frac{\delta H}{5}< \pi_1(w_{j_0})+\frac{4\delta H}{5}+\frac{\delta H}{5}\mbox{using \eqref{e:localization_x_2}}=\pi_1(w_{j_0})+\delta H.
v_{m_{1}m_{2}\ldots m_{n}}^{( n )}= v_{m_{2}\ldots m_{n}m_{1}}^{( n )}, ( T r \; V )^{\dagger}= T r \; V , v_{m_{1}\ldots m_{n}}^{( n ) \; *}= v_{m_{n}\ldots m_{1}}^{( n )}
\mathrm{i}\begin{bmatrix}0 -1 1 0 \end{bmatrix}\begin{bmatrix}u' v' \end{bmatrix}+ \begin{bmatrix}0 0 0 \mathrm{i}\end{bmatrix}\begin{bmatrix}u v \end{bmatrix}= \mathrm{i}\begin{bmatrix}f 0\end{bmatrix}
\partial_t h = \ell s^{\ell - 1}H_0(y)^2 \ge\ell T_2^{\ell+\sigma-1}s^{-\sigma}H_0(y)^2, Cs^{-\sigma}H(\nabla h)^2 = 4C(1+s^\ell)^2 s^{-\sigma}H_0(y)^2 \le 4C(1+T_2^\ell)^2 s^{-\sigma}H_0(y)^2 \le \partial_t h,
f_{n}( y ) = N_{n}e^{2 \sigma}\left( J_{\nu}\left({\frac{m_{n}}{k}}e^{\sigma}\right) - \left({\frac{j_{\nu}({\frac{m_{n}}{k}})}{y_{\nu}({\frac{m_{n}}{k}})}}\right) Y_{\nu}\left({\frac{m_{n}}{k}}e^{\sigma}\right) \right)
\mathbb{P}(T_{m}>X_{(n)}) =\sum_{k=0}^{m-1}(-1)^{k}\frac{s^{k}}{k!}f^{(k)}(s) =\sum_{k=0}^{m-1}(-1)^{k}\frac{s^{k}}{k!}g^{(k)}(s)~~(\because f(s)=g(s))
\int_0^{2\pi}d k\,I[\omega(k)]=\int_0^{2\pi}d k\,\mathcal{P}\!\!\int_0^4 d\lambda\, \frac{\delta(\lambda)}{\lambda-\omega(k)}=\int_0^4 d\lambda\, \delta(\lambda)\,\mathcal{P}\!\!\int_0^{2\pi}\frac{d k}{\lambda-\omega(k)}=0.
\exp_{\hat p}^{\nabla}(v) = \cos(|v|) \hat{p}+ \sin(|v|) \frac{v}{|v|}, \exp_{\hat p}^{\nabla,-1}(p) = \cos^{-1}(\langle \hat p, p \rangle) \frac{p - \langle \hat p, p \rangle \hat p}{|p - \langle \hat p, p \rangle \hat p|}.
Y_{\lambda}(x,p)=\sum_{k=1}^{\infty}(k)_{p,\lambda}\sum_{n=k}^{\infty}\frac{x^{n}}{n!}=\sum_{k=1}^{\infty}(k)_{p,\lambda}\bigg(e^{x}-\sum_{l=0}^{k-1}\frac{x^{l}}{l!}\bigg).
\int_{\mathcal{S}(r)}e^{\frac{x}{w}q_r'Y q_r}({\rm d}q_r) = \frac{\Gamma(r/2)}{2 \pi i}\Bigl( \frac{w}{x}\Bigr)^{r/2 -1}\int_\Gamma e^{\frac{x}{w}s}\Delta_y(s){\rm d}s,
p(y_{k,b}|y_{[1:N],[1:b]}^{\mathrm{unf}},y_{[1:k-1],b+1}^{\mathrm{unf}},x_{[1:N],[1:b]}^{\mathrm{unf}},x_{[1:k-1],b+1}^{\mathrm{unf}})=p_{Y_k|Y_{[1:k-1]},X_{[1:N]}}(y_{k,b}|y_{[1:k-1],b},x_{[1:N],b})
\| \cdot \|_{L^{R^n}}= \big\| \big( \big\| \cdot \big\|_{L^{\tilde R^{n-1}}}^{r^{j_0}}\big)^{1/{r^{j_0}}}\big\|_{r^1}= \big \| \big\| \cdot \big\|_{L^{\tilde R^{n-1}}}^{r^{j_0}}\big\|_{L^{\frac{r^1}{r^{j_0}}}}^{\frac{1}{r^{j_0}}}.
S_{n}\left(-q^{-n+1/2};q\right)=\frac{q^{-\left(n^{2}-n\right)/4}}{\left(q^{1/2};q^{1/2}\right)_{n}},\S_{n}\left(-q^{-n-1/2};q\right)=\frac{q^{-\left(n^{2}+n\right)/4}}{\left(q^{1/2};q^{1/2}\right)_{n}}.
P\{\max_{0\le s\le t}\mathcal{T}(s)=0\ |\V(0) = -c_2,\N(t) = 2k\}\ =\ \binom{2k}{k}\frac{\alpha^k}{(1+\alpha)^{2k}}+ (1-\alpha)\sum_{j=0}^{k-1}\binom{2k}{j}\frac{\alpha^{j}}{(1+\alpha)^{2k}}
\delta v^i=dv^i+\Gamma^i_{j\alpha}v^jdz^\alpha,\ \Gamma^i_{j\alpha}={{\partial G_{j\bar k}}\over{\partial z^\alpha}}G^{\bar k i},\K_{i\bar j \alpha\bar\beta}=-G_{i\bar j \alpha\bar\beta}+G^{k\bar l}G_{i\bar l\alpha}G_{k\bar j\bar\beta}.
d|\nu_{n,k}|(\tau)=\frac{|h_{n,k}(\tau)|\,d|\sigma_{n,k}|(\tau)}{|P_{n,k-1}(\tau)\,P_{n,k+1}(\tau)|}=\varepsilon_{n,k}\,\frac{h_{n,k}(\tau)\,d\sigma_{n,k}(\tau)}{P_{n,k-1}(\tau)\,P_{n,k+1}(\tau)}.
G(R)=-R^{N-1}\int_{S^{N-1}}(Y\partial_r Y)(R,\theta)\,d\theta+\frac{N+2}{N-2}\int_R^{+\infty}\int_{S^{N-1}}|Q|^{\frac{4}{N-2}}Y^2\,d\theta\,r^{N-1}dr.
\gamma_{-}(w_{a})v=\sum_{i}\frac{\langle w_{i}^{(k-1)},\gamma_{-}(w_{a})v\rangle}{\langle w_{i}^{(k-1)},v_{i}^{(k-1)}\rangle}v_{i}^{(k-1)}=\sum_{i}\frac{\langle w_{i}^{(k-1)}\wedge w_{a},v\rangle}{\langle w_{i}^{(k-1)},v_{i}^{(k-1)}\rangle}v_{i}^{(k-1)}.
\nu_{s_{m}\rightarrow\eta_{m}^{r}}(s_{m})\propto \nu_{h_{m}\rightarrow s_{m}}(s_{m})\times\nu_{h_{m+1}\rightarrow s_{m}}(s_{m})\times\nu_{\eta_{m}^{c}\rightarrow s_{m}}(s_{m}) = \pi_{s,m}^{r,out}\delta(s_{m}-1)+(1-\pi_{s,m}^{r,out})\delta(s_{m}),
\frac{\tilde{\phi}^{n+1}-2\phi^n+\phi^{n-1}}{\tau^2}+\eta^n\frac{\tilde{\phi}^{n+1}-\phi^{n-1}}{2\tau}&= \left(\frac12\Delta-\vartheta^n\right)\frac{\tilde{\phi}^{n+1}+\phi^{n-1}}{2}+\vartheta^n\phi^n-V\phi^n-\beta|\phi^n|^2\phi^n+\Omega L_z\phi^n+\lambda^n\phi^n, & \phi^{n+1}= \tilde{\phi}^{n+1}/\|\tilde{\phi}^{n+1}\|, n\geq1,
\tilde{\Delta}=\left({\rm tr}\;P\right)^{2}-4\det P=\frac{1}{16}\left[ \left({\rm tr}\;A\right)^{2}-2\det A\right]^{2}-\frac{1}{4}\left( \det A\right)^{2}=\frac{1}{16}\left({\rm tr}\;A\right)^{2}\left[ \left({\rm tr}\;A\right)^{2}-4\det A\right] .
V(k_2) \circ V(k_1) \, = \, i_{W(k_1, k_2)}\circ V(k_1 k_2),(k_1, k_2 \in K) begin{align*}1mm]W(k_1, k_2k_3) W(k_2, k_3) \, = \, W(k_1k_2, k_3) V(k_3)(W(k_1, k_2)),(k_1, k_2, k_3 \in K),
\|R\|=\|Q\|=|Q(w_L,w_R)|\leq \frac{\eta^3}{2^4}+|R(w_L,w_R)|\leq \frac{\eta^3}{2^4}+ 1 + \eta \bigl|\hat{T}(w_L,\hat{x}_R)\bigr| \bigl|\hat{T}(\hat{x}_L,w_R)\bigr|.
1 < \prod_{j=0}^{k-1}\left(1 + \frac{1}{x_j}\right) \leq \left(2^{5}\frac{5 \cdot 13 \cdot 33 \cdot 17 \cdot 9}{9 \cdot 25 \cdot 65 \cdot 33 \cdot 17}\right)^{k/5}= \left( \frac{2^5}{5^2}\right)^{k/5}= \frac{2^k}{5^{2k/5}},
\frac{d}{dt}V_{\hbar}\left( t\right) \psi =\Gamma_{\hbar}\left(t\right) V_{\hbar}\left( t\right) \psi \frac{d}{dt}V_{\hbar}^{\ast}\left( t\right) \psi =-V_{\hbar}^{\ast}\left( t\right) \Gamma_{\hbar}\left( t\right) \psi
\prod_{(i,j)\in \Delta_m}\frac{1-R_{ij}}{1+R_{ij}}= \sum_{s=1}^{m}(-1)^{s-1}\prod_{(i,j) \in \Delta_m \atop{i,j\in \{1,\dots,m\}\backslash \{s\}}}\frac{1-R_{ij}}{1+R_{ij}}.
\underset{t^{q_{i}-i}}{coef}\left[exp\left(\frac{-pt^{p}\theta}{e^{px}-t^{p}}\right)\cdot\frac{(1-t^{p}e^{-px})^{k+g_{Y}-1}}{(1-t^{p}e^{-2px})^{2g_{Y}-2}}\cdot \prod_{j=0}^{p-1}\left\{\frac{(1-\nu^{j}te^{-2x})^{m_{j,2}+m'_{p-j,-2}}}{(1-\nu^{j}te^{-x})^{m_{j,1}}}\right\}\right]\times
\lim_{\delta \downarrow 0}\frac{\tau_{\mu,\alpha}(h+\delta) - \tau_{\mu,\alpha}(h)}{\delta}= \lim_{\delta \downarrow 0}\int_{0}^{1}\frac{f(Y_{\alpha}^{h+\delta}(t)) - f(Y_{\alpha}^{h}(t))}{\delta}\mathrm{d}t & = \int_{0}^{1}\lim_{\delta \downarrow 0}\frac{f(Y_{\alpha}^{h+\delta}(t)) - f(Y_{\alpha}^{h}(t))}{\delta}\mathrm{d}t ,
\gamma^l = \frac{1}{\delta^1}\prod_{j=2}^l (1-\alpha^j)^{-1}= \frac{1}{\delta^1}\prod_{j=2}^l (\Lambda^{j}/\Lambda^{j-1})= \frac{1}{\delta^1}\frac{\Lambda^l}{\Lambda^1}= \Lambda^l .
|F(x)|=\Biggl|\sum\limits_{\theta\in\Theta}a_\theta\int\limits_0^x e^{i\theta t}dt\Biggr| \leq \sum\limits_{\theta\in\Theta}\Big|\frac{a_\theta}{i\theta}(e^{i\theta x}-1)\Big|\ll 1,\ \forall x\in\mathbb{R}.
\O_u \star \O_v = \O_{h_i}\star \mathcal{O}_{u'}\star \mathcal{O}_v = \sum_{w \in W^P, \beta_1 \in E(X)}N_{u',v}^{w, \beta}Q^{\beta_1}\O_{h_i}\star \mathcal{O}_w &= \sum_{w,\: w' \in W^P,\: \beta_1, \beta_2 \in E(X)}N_{u',v}^{w, \beta_1}Q^{\beta_1}N_{h_i,w}^{w', \beta_2}Q^{\beta_2}\mathcal{O}_{w'}
dHe^{\frac{H}{2\pi i}}\sum_{m\geq 1}(-1)^m\frac{(2\pi i)e^{\frac{2\pi im}{d}}}{e^{-H}-e^{\frac{2\pi im}{d}}}\frac{q^{-\frac m d}}{\Gamma(m)\Gamma(1+\frac{d^\prime}{d}m )\prod_{i=1}^N\Gamma(1-q_im)}.
\mathcal{F}(D(\lambda)f)(\xi) =\mathcal{F}(I_{\lambda+1}\circ M_{x_n}\circ I_{n-\lambda}f)(\xi) =-i c_{-2\lambda-2}c_{2\lambda-2n}r^{2\lambda+2-n}(\xi)\partial_n\big[r^{n-2\lambda}(\xi)\mathcal{F}(f)(\xi) \big].
F [ \Phi , X , \varphi , \chi ] = \int \! d x \; \left( \Phi^{j}J^{1 ~ j}[ \varphi ] - \frac i 2 ~ \overline{{X}}^{j}\gamma^{1}\chi^{j}\right) = \int d x \Bigl ( \Phi^{i}( \sqrt{1 -{\varphi}^{2}}\frac{\stackrel{\leftrightarrow}{\partial}}{\partial x}\varphi^{i}+ \varepsilon^{i j k}\varphi^{j}\frac \partial{\partial x}\varphi^{k}) - \frac i 2 \overline{{X}}^{j}\gamma^{1}\chi^{j}\Bigr ) ~ .
\left< T^{\mu}{}_{\mu}( x ) \right> ={\frac{a^{- N}}{2^{N - 1}\pi^{N / 2}\Gamma \left( \frac N 2 \right)}}\sum_{n = 0}^{\frac N 2 - 1}c_{2 n + 1}^{N}\left[ \frac{( - 1 )^{n + 1}}{2 ( n + 1 )}4^{- n - 1}- 2 H_{n}( 0 ) \right] .
\varphi_*(t)= C(t,0)x_0+S(t,0)y_0+\int_0^t S(t,s)f_*(s)ds+\int_0^t S(t,s)BB^{*}S^{*}(b,t)\Big[y_0-S(b,0)x_0 &-\int_0^b S(b,\eta)f_*(\eta)ds\Big](s)ds.
|D|^mx|D|^{-m}= \sum_{k=0}^m\binom{m}{k}\delta^{m-k}(x)|D|^{k-m}, |D|^{-m}x|D|^m = \sum_{k=0}^m(-1)^{m-k}\binom{m}{k}|D|^{k-m}\delta^{m-k}(x).
\eta_{\nu^{\pm}, \lambda}( r , \varphi ) = N_{\nu^{\pm}, \lambda}\left( \begin{matrix}{{\zeta^{\pm 1}J_{n - 1}( q r ) e^{i ( n - 1 ) \varphi}}}\\{{- i J_{n}( q r ) e^{i n \varphi}}}\\{{\pm \zeta^{\pm 1}J_{n - 1}( q r ) e^{i ( n - 1 ) \varphi}}}\\{{\pm i J_{n}( q r ) e^{i n \varphi}}}\\ \end{matrix}\right)
{\Delta_{3}}( p ) = - \frac{i}{2^{5}\pi^{2}}\left[ \frac{1}{\epsilon}+ + \left( \log \frac{4 \pi \mu^{2}}{- p^{2}}- 3 - 2 \gamma - 2 \log 2 \right) +{\cal O}( \epsilon ) \right] ,
I_B = \frac{1}{\widetilde C_B}(-1)^{\frac{S-\widetilde{S}}{2}}\:\sum_{m_j'=0}^{2k_j+3}\delta_{\frac{M-\widetilde{M}}{2}+\sum\frac{K'(m_j'+1)}{2k_j+4}}^{(K')}\prod_{j=1}^{r}\:n_{L_j \widetilde{L}_j}^{-m_j'-1}\:,
\left.\frac{d^2}{dv^2}\right|_{v=0}|\gamma_v|=\left.\frac{1}{2s_0}g(\gamma', \nabla_{V_i}V_i)\right.|^1_0+\frac{1}{2s_0}\int_0^1|(\nabla_{\gamma'}V_i)^{\bot}|^2-\langle R(\gamma', V_i)\gamma', V_i\rangle\ds.
\lim_{x \uparrow 0}\frac{\partial}{\partial x}m(s,x,s+\epsilon,y)=-\frac{2y}{\sqrt{2\pi \epsilon^3}}e^{-\frac{y^2}{2\epsilon}}\times \mathbb{E}\left[ \left (-\int_{s}^{s+\epsilon}\phi''(u)B(u)du \right ) \vert B(s)=0,B(s+\epsilon)=y\right]
J_1= -\sum_{\substack{i,j=1, i\not=j}}^n\Bigg[\partial_y\bigg(\frac{p\sigma_j (\tau_iQ)^{p-1}}{x_{ij}^2}-\frac{2p\kappa_0\sigma_j \partial_y[\tau_i(yQ^{p-1})]}{x_{ij}^3}\bigg)+\frac{a_{ij}\sigma_i\tau_i(\Lambda Q)}{x_{ij}^3}\Bigg].
2|\log\bar\alpha_{k}|\sum_{i=1}^k\sum_{j=i}^k\sup_{l=1,\ldots,k}\P_{\bf U}(\tau_{0,l}\ge j)|\log\bar\alpha_{j-1}| = 2|\log\bar\alpha_{k}|\sum_{j=1}^kj\sup_{l=1,\ldots,k}\P_{\bf U}(\tau_{0,l}\ge j)|\log\bar\alpha_{j-1}|.
\begin{aligned}AG\geq \frac{2}{n}\phi G^{2}-C_{4}\phi^{1/2}G^{3/2}-\frac{C_{1}^{2}}{C_{4}R^{2}}\phi^{1/2}G^{3/2}&-2KG\phi+(p-2)\lambda_{1}G\phi-p\lambda_{1}G\phi &+\lambda_{2}e^{b\omega}\phi(C_{4}\phi^{1/2}G^{1/2}-bG+b(p-1)\lambda_{1}\omega) &+\lambda_{1}\omega(-\frac{4}{n}\phi G(p-1)+C_{4}p\phi^{1/2}G^{1/2}).\end{aligned}
\left(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}\right)(u_{m}-u_{m-1}) begin{align*}2mm] \geq C(q) \big(|u_m| + |u_{m - 1}|\big)^{q - 1}|u_m - u_{m - 1}|^2.
\frac12\frac{d\|Y(t)\|^2}{dt}=\langle Y(t),f(Y(t)+Z(t)\rangle &\le \langle Y(t),f(Y(t)+Z(t))-f(Z(t))\rangle+\|Y(t)\|\|f(Z(t))\| &\le -\gamma\|Y(t)\|^2+\|Y(t)\|f(Z(t))\|.
H^{0 \frac{1}{2}}= \partial_{1}^{- 1}\log x_{1 2}\partial_{1}+ \log x_{1 2}+ \log \partial_{1}\partial_{2}- 2 \psi ( 1 ) = x_{1 2}\log \partial_{1}x_{1 2}^{- 1}+ \log \partial_{2}+ 2 \log x_{1 2}- 2 \psi ( 1 )
\rho_N(z)=q^{\frac{1}{N}-1}\frac{(q^2 z;q^{2N})_{\infty}(q^{2N-2}z;q^{2N})_{\infty}}{(z;q^{2N})_{\infty}(q^{2N}z;q^{2N})_{\infty}}\;.
\left[ - \omega^{2}\gamma^{t t}( r ) f ( r ) -{\frac{l ( l + n )}{r^{2}}}{\frac{f ( r )}{h ( r )}}+{\frac{\sqrt{- \gamma^{t t}f ( r )}}{U ( r )}}{\frac{d}{d r}}{\frac{U ( r )}{\sqrt{- \gamma^{t t}f ( r )}}}{\frac{d}{d r}}\right] \phi_{\omega}^{l}( r ) = 0 .
\left[\Re\frac{1+\eta_{\mu_{1}}(z)}{1-\eta_{\mu_{1}}(z)}\right]\int_{\mathbb{T}}\frac{|1-\eta_{\mu_{1}}(z)|^{2}}{|t-\eta_{\mu_{1}}(z)|^{2}}\,d\sigma(t)=\beta\Re\frac{1+\eta_{\nu_{1}}(z)}{1-\eta_{\nu_{1}}(z)}.
\frac{S_{(r, s)(r',s')}}{S_{(1, 1)(r', s')}}= (-1)^{(r+1)s'+(s+1)r'}\frac{\sin\left(\frac{\pi b}{a}rr' \right)\sin\left(\frac{\pi a}{b}ss' \right)}{\sin\left(\frac{\pi b}{a}r' \right)\sin\left(\frac{\pi a}{b}s' \right)}.
\begin{cases}s_{i-j}s_{n-j}\leq s_{i-j+1}s_{n-j-1}-s_{n-1}, \mbox{if}j \mbox{is even}s_{i-j}s_{n-j}+s_{n-1}\leq s_{i-j+1}s_{n-j-1}, \mbox{if}j \mbox{is odd}. \end{cases}
\int_{\mathbb{R}}\frac{1 + s^{2}}{s^{2}+ y^{2}}d\rho(s) = \frac{\Im{(F_{\mu}(iy)})}{y}- 1 &= \Im \left( \frac{i}{iy G_{\mu}(iy)}\right) - 1 &= \Im \left( \frac{i}{iy\left( \frac{r-1}{riy}+ \frac{G_{\nu}(iy)}{r}\right)}\right) - 1
\mathbb S^2 \big[\psi \,m_2(t) \big] = 4 m_1 m_2(t)\,\mathbb S\mathbb D (\psi) + 2 m_1^3\,\mathbb S \mathbb D (\psi) + (2m_1^2 + m_2(t)\,\mathbb S^2 (\psi) .
\delta A_{(3)}\;=\; d\Delta^{(2)}+{\textstyle\frac{1}{2}}\vec{\Sigma}_{(1)}^{\T}\eta\vec{F}_{(2)}-{\textstyle\frac{1}{2}}\vec{\Sigma}_{(0)}{}^{T}\eta \left[ \vec{F}_{(3)}-A_{(1)}\vec{F}_{(2)}\right] \; .
\Delta_1(t) = i \int_{S \wedge \mu}^{t \wedge \mu}\mathbf S(t-s) \left[ \mathbf N(\mathbf U(s)) - \mathbf N(\mathbf V(s)) \right] \, ds, \Delta_2(t) = i \int_S^t \mathbf S(t-s) \left[ \mathbf M(\mathbf U(s)) - \mathbf M(\mathbf V(s)) \right] \, dW(s).
p_0(\lambda) =2\arctan \left[\frac{\tanh(\lambda)}{\tan(\zeta/2)}\right],\theta(\lambda) = 2 \arctan \left[\frac{\tanh(\lambda)}{\tan(\zeta)}\right]\intertext{and}\epsilon_0(\lambda) = \frac{-2\sin^2 \zeta}{\cosh(2\lambda) -\cos \zeta}.
d_1S + \frac{1}{2}\{S,S\}= \frac{1}{2}\{S^0,S^0\}+ \Big(d_1S^0 - \{I^\nabla, S^0\}\Big) + \Big(-d_1I^\nabla + \frac{1}{2}\{I^\nabla,I^\nabla\}\Big) = S_R,
S=\frac{r_{h}^{2}}{90}{\tilde{\epsilon}}^{-2}+\left[\frac{\Delta_{h}''r_{h}^{2}}{180}+\frac{r_{h}^{2}\mu^{2}}{6}\right] \log{\tilde{\epsilon}}+{\mbox{terms finite as}}{\tilde{\epsilon}}\rightarrow 0.
\begin{aligned}\Big( (U_2^nXU_1^{-n}+U_2^{-n}XU_1^n)\,U_1^k e, U_2^l\bar e\Big)= =\Big((X+ U_2^{k+l}XU_1^{-(k+l)})\,U_1^k e, U_2^l\bar e\Big)+\sum_{m=1}^{k+l}(U_2^n K U_1^{-n}\, U_1^{k-m}e, U_2^{l-m+1}\bar e). \end{aligned}