harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{n}^{\left( \alpha+m\right)}\left( m\right) =\left( -1\right)^{n}E_{n}^{\left( \alpha+m\right)}\left( \alpha\right) =\frac{\left(2\right)^{m}}{m!\binom{\alpha+m-1}{m}}\sum_{k=0}^{m}\left( -1\right)^{k}\genfrac{[}{]}{0pt}{}{m}{k}E_{n+k}^{\left( \alpha\right)}.
K^{n,1}_\tau =n^{2\alpha+1}\int^{\tau}_{(\delta+\frac 1n) \wedge \tau}\gamma_s \left(\sigma'(X_s)\right)^2 \times \left[ \left( \int_{0}^{\eta_n(s)}\psi_{n,2}(u,s) \sigma(X_{\eta_n(u)})\, dW_u\right)^2 - \left( \int_{\eta_n(s)-\delta}^{\eta_n(s)}\psi_{n,2}(u,s) \sigma(X_{\eta_n(u)})\, dW_u\right)^2 \right] \, ds.
g_\infty = - \: \frac14 K^{-2}e^{- 2C_U}du^2 + \frac45 K^{- \: 2}e^{-\: 2 C_U}C_\infty u^{\frac32}d\widehat{\theta}^2 + e^{2C_U}d\widehat{x}^2 + e^{- \: 2 C_U}u \left( d\widehat{y}+ \frac{4}{K \sqrt{5}}C_\infty^{\frac12}u^{\frac14}d\widehat{\theta}\right)^2.
\hat{V}_{\mu \nu \rho}^{( 1 \, z ) \, A B}= 2 \cdot 2 \, \mathrm{T r}\, T^{A}T^{B}\, ( Y_{R}^{( d )}- Y_{L}^{( d )}) \, V_{\mu \nu \rho}^{( 1 )}= - 4 \, \mathrm{T r}\, T^{A}T^{B}\, V_{\mu \nu \rho}^{( 1 )}\, .
\mathfrak{B}_{k}^{\left[ j\right]}\left( \alpha\right) =\frac{\alpha^{k}}{\det\left( 1-2i\alpha~\boldsymbol{\hat{n}\cdot J}\right)}~\operatorname*{Trunc}_{2j-k}\left[ \det\left( 1-2i\alpha~\boldsymbol{\hat{n}\cdot J}\right) \right] \ ,
S_{-1}+ S_{0}\le \frac{4}{\lambda}\left[L^2 C^2 \frac{\left(1+\sqrt{\alpha}\right)^{2h}(2B_k)^h}{N^h}+ \frac{\left(1+\sqrt{\alpha}\right)^{2h}2^{h+2}(B_k)^{h}B_K L^2}{N^h}\frac{C^2 B_K}{\lambda^2}\right] = \frac{4L^2 C^2 \left(1+\sqrt{\alpha}\right)^{2h}(2B_k)^h}{\lambda N^h}\left[ 1 + \frac{4(B_K)^2}{\lambda^2}\right].
\psi(x,t)\,=\,\left( \begin{array}{c}\psi_{1}(x,t) \psi_{2}(x,t) \psi_{3}(x,t) \end{array}\right) \,=\,\left( \begin{array}{c}f_{1}(t) \phi_{1}(x) f_{2}(t) \phi_{2}(x) f_{3}(t) \phi_{3}(x) \end{array}\right)
\int_{\Omega}|(\mathbf{D}+ \mathbf{k}) \mathbf{u}|^2 d\mathbf{x}\ge \sum_{\mathbf{b}\in \widetilde{\Gamma}}| \mathbf{k}|^2 |\hat{\mathbf{u}}_{\mathbf{b}}|^2 = | \mathbf{k}|^2 \int_{\Omega}|\mathbf{u}|^2 d\mathbf{x}, \mathbf{u}\in \widetilde{H}^1(\Omega; \mathbb{C}^n), \; \mathbf{k}\in \widetilde{\Omega}.
\frac{\lambda_{i + 1}}{\lambda_i}= \frac{(i + 1) \cdot \Gamma(1 + \epsilon) \Gamma(i + 1 - \epsilon)}{\Gamma(1 - \epsilon) \Gamma(i + 2 + \epsilon)}\cdot \frac{\Gamma(1 - \epsilon) \Gamma(i + 1 + \epsilon)}{i \cdot \Gamma(1 + \epsilon) \Gamma(i - \epsilon)}&= \frac{i + 1}{i}\frac{i - \epsilon}{i + 1 + \epsilon}
H_t(x,y,p_x,p_y)=\,\sum_k \,\tilde{\beta}^\mu_{kx}\,\tilde{p}_{kx\mu}+\,\sum_k \,\tilde{\beta}^\mu_{ky}\,\tilde{p}_{ky\mu}=\,\,\sum_k \,\beta^\mu_{kx}\,p_{kx\mu}+\,\sum_k \,\beta^\mu_{ky}\,p_{ky\mu}.
k_{\gamma}^{\infty}=\frac{\sqrt{\frac{1}{2}\left(-e^{-\gamma_3}\dot{\gamma_1}+e^{\gamma_3}\dot{\gamma_2}\right)^2+\dot{\gamma}_3^2}}{|\omega(\dot{\gamma}(t))|},~~if ~~\omega(\dot{\gamma}(t))\neq 0,
C_{\mathrm{u}}(\mathbf{u}_{i},\mathbf{u}_{j})=\int C(\mathbf{x}_{i},\mathbf{x}_{j})\, p(\mathbf{x}_{i})\, p(\mathbf{x}_{j})\,\mathrm{d}\mathbf{x}_{i}\,\mathrm{d}\mathbf{x}_{j}
\frac{S(T_{2})}{S(T_{1})}=\frac{S_{b}(T_{2})(1-1/\rho_{i})}{S_{b}(T_{1})}\frac{\rho_{b}(T_{2})}{\rho_{b}(T_{1})}exp\{\frac{c}{\rho_{i}[S_{b}(T_{1})-S_{b}(T_{2})]}\}
\lim_id(x_i,\partial\Omega'_i))=\infty\;,\lim_{i,j}\|g-\phi_{ij}^*g\|_{C^{m_i},\Omega_i,g}=\lim_i\|g-\psi_i^*g\|_{C^{m_i},\Omega'_i,g}=0\;,
R_{i-k-l}^{(\alpha+2l,\beta+2k)}(x) = \sum_{h=k}^{n-l}a_{i-k-l,h-k}^{(n-k-l,\alpha+2l,\beta+2k)}B_{h-k}^{n-k-l}(x) (i=k+l,k+l+1,\ldots,n).
\begin{cases}L_h(\hat{u}_h^{l}(z))\leq f_{l}(z,u_h^{l}(z))\;\; z\in\Omega_h, L_h(\hat{u}_h^{l}(z))= f_{l}(z,u_h^{l}(z))\;\; z\in\{u_h^{l}(z)>0\},{u}_h^{l}(z)\geq 0\;\; z\in\Omega_h,{u}_h^{p}(z)\cdot{u}_h^{q}(z)=0,\;\;p\neq q,{u}_h^{l}(z)={\phi}_h^{l}(z)={\phi}_{l}(z),\;\; z\in\partial\Omega_h.\end{cases}
A= \begin{pmatrix}e^{-x_5}0 0 0 0 x_1 0 e^{x_5}0 0 0 x_2 -x_5e^{-x_5}0 e^{-x_5}0 0 x_3 0 -x_5e^{x_5}0 e^{x_5}0 x_4 0 0 0 0 1 x_5 0 0 0 0 0 1 \end{pmatrix}
E_5^{'}=\frac{2^{R_{52}^{'}+R_{51}^{'}}}{g_{51}}-\frac{1}{g_{52}}-2^{R_{52}^{'}}\left(\frac{1}{g_{51}}-\frac{1}{g_{52}}\right) &=\frac{2^{R_{61}^*}}{g_{61}}-\frac{1}{g_{62}}-2^{R_{62}^*}\left(\frac{1}{g_{61}}-\frac{1}{g_{62}}\right).
\sum_{n=0}^{\infty}\Delta_{2}(5n+4)q^{n}\equiv41\dfrac{E_{10}^{3}}{E_{1}E_{2}^{3}E_{5}}=41\dfrac{E_{2}^{2}}{E_{1}}\dfrac{E_{10}}{E_{2}^{5}}\dfrac{E_{10}^{2}}{E_{5}}\equiv \psi(q)\psi(q^{5})\pmod{5},
\epsilon(t,z,r,\theta, \phi)= e^{{\frac{i}{2}}\Gamma_4\theta}e^{{\frac{1}{2}}\Gamma_3\Gamma_4\phi}\Big((1+\Gamma_0\Gamma_1)r^{\frac{-1}{4}}-{\frac{(t-z)}{2(Z_m )_{{\rm cr}}^{\frac{3}{2}}}}r^{\frac{1}{4}}(\Gamma_0-\Gamma_1)\Gamma_2\Big)\kappa_0.
\lim_{\substack{L,N\rightarrow\infty N/L\rightarrow\rho}}\left\|\mu_{N,L}- \frac{1}{L}\sum_{x\in\Lambda}\left(\mu_{\varphi_c}^{\Lambda\setminus\{x_L\}}\ast\delta_{N-\sum_{\Lambda\setminus\{x_L\}}\eta}\right)\circ \sigma^{x,x_L}\right\|_{\rm{t.v.}}= 0,
|\varphi(c_1 (c_2 c_3))| \leq \varphi(c_1 c_1^{\ast})^{\frac{1}{2}}\varphi( c_2 c_3 c_3^{\ast}c_2^{\ast})^{\frac{1}{2}}= \varphi(c_1 c_1^{\ast})^{\frac{1}{2}}\varphi((c_2^{\ast}c_2) (c_3 c_3^{\ast}))^{\frac{1}{2}}&\leq \varphi(c_1 c_1^{\ast})^{\frac{1}{2}}\varphi((c_2 c_2^{\ast})^2)^{\frac{1}{4}}\varphi((c_3 c_3^{\ast})^2)^{\frac{1}{4}},
\phi_g^{\beta,\sigma}(x)\to \phi_g^{TF}(x)=\frac{1}{\sqrt{L}}, 0<x<L, &\mu_g(\beta,\sigma)\approx\frac{\beta}{L^{\sigma+1}}\to \infty,E_g(\beta,\sigma)\approx\frac{\beta}{(\sigma+1)L^{\sigma+1}}\to \infty.
H=\Delta\exp\left\{\frac{\alpha i}{4\hbar}\int d^4x\, G^a_{\mu\nu}{\cal M}^{-1}_{ab}\left( \frac{1}{2}\delta^b_c(\eta^{\mu\rho}\eta^{\nu\sigma}-\eta^{\mu\sigma}\eta^{\nu\rho})+\frac{\alpha}{2}\varepsilon^{\mu\nu\rho\sigma}{\cal C}^b_c\right)G^c_{\rho\sigma}\right\},
\Re\left(1-\left(\frac{\rho}{\rho-\tau}\right)^n\right) =\Re\left(1-\left(1+\frac{\tau}{\rho-\tau}\right)^n\right) & = -n\tau\Re\left(\frac{1}{\rho-\tau}\right)-\frac{n(n-1)\tau^2}{2}\Re\left(\frac{1}{(\rho-\tau)^2}\right) &\quad-\sum\limits_{j=3}^n\binom{n}{j}\Re\left(\left(\frac{\tau}{\rho-\tau}\right)^j\right) ,
K_G (\phi_s , \phi_t )_\phi = \frac{1}{2}\int_M |d a(s,t)|_{g_\phi}^2\frac{\omega_\phi^n}{n!}- \frac{1}{2}\int_M (d a(s,s) , d a(t,t))_{g_\phi}\frac{\omega_\phi^n}{n!}\, ,
sf(x,s)-2nF(x,s) =\frac{f(x,s)}{s^{2n-1}}s^{2n}-2nF(x,t)+2n\int_s^t f(x,\tau)d\tau &< \frac{f(x,t)}{t^{2n-1}}s^{2n}-2n F(x,t)+\frac{f(x,t)}{t^{2n-1}}(t^{2n}-s^{2n}) &\leq tf(x,t)-2nF(x,t),
\varpi_E^{-1}\tilde{\mathfrak{H}}^1 / \tilde{\mathfrak{J}}^0\cong \bigoplus_{j=0}^d\mathfrak{W}_{z,j},\quad\mathfrak{W}_{z,j}= \tilde{\mathfrak{P}}^{-1+(r_j/2)_+}_{\Lambda,E_{j+1}}/ (\tilde{\mathfrak{P}}^{-1+(r_j/2)_+}_{\Lambda,E_{j}}+ \tilde{\mathfrak{P}}^{r_j/2}_{\Lambda,E_{j+1}})
\int_{-\infty}^{\infty}f(x)dx=L\int_{0}^{\pi}\frac{f[Lcot(\theta)]}{sin^{2}(\theta)}d\theta\approx\frac{L\pi}{N}\sum_{n=1}^{N-1}\frac{f[Lcot(n\pi/N)]}{sin^{2}(n\pi/N)}.
\alpha_i=\displaystyle{\frac{\sum\limits_{j\in N_i, a_{ij}<0}a_{ij}}{\sum\limits_{\ell\in P_i, a_{i\ell}<0}a_{i\ell}}}\qquad\beta_i=\displaystyle{\frac{\sum\limits_{j\in N_i, a_{ij}>0}a_{ij}}{\sum\limits_{\ell\in P_i, a_{i\ell}>0}a_{i\ell}}}.
{\cal L}={\frac{k}{2}}\varepsilon^{\alpha \beta \gamma}A_{\alpha}\partial_{\beta}A_{\gamma}+ i \Psi^{\ast}( \partial_{t}+ i A_{0}) \Psi -{\frac{1}{2}}\left| ( \nabla - i{\bf A}) \Psi \right|^{2}+{\frac{g}{2}}\left| \Psi \right|^{4}\, .
\mathcal{W}={\mathop{\sum\nolimits'}\limits_{j}}k(t_j) \omega_j L\Big(\frac{1}{2}, f\times u_j\Big)+ \frac{1}{4\pi}\int_\mathbb{R}k(t) \omega(t)\Big| L\Big(\frac{1}{2}- it, f\Big) \Big|^2 ~dt.
\begin{array}{c c}{\displaystyle{\{D_{i \alpha}, D_{j \beta}\}= - 2 i \bar{\cal E}_{i j}\gamma_{\alpha \beta}^{A}\partial_{A}}}&{\displaystyle{\{\tilde{D}_{\tilde{\imath}}^{\alpha}, \tilde{D}_{\tilde{\jmath}}^{\beta}\}= - 2 i \bar{\tilde{\cal E}}_{\tilde{\imath}\tilde{\jmath}}\tilde{\gamma}^{A \alpha \beta}\partial_{A}}}\\ \end{array}
|\zeta_U(-2m-c_0+i\tau)| \leq D^{-\frac{2m+c_0}{2}}\sum_{k=0}^\infty \frac{|\Gamma(2m+c_0+1-i\tau)|}{k! |\Gamma(2m+c_0+1-k-i\tau)|}\frac{a^{2m+c_0}(|b|/a)^k}{\left|1- a^{2m+c_0-i\tau}(b/a)^k\right|}\ll \left(\frac{a}{\sqrt{D}}\right)^{2m+c_0}\sum_{k=0}^\infty \frac{(|\tau| |b|/a)^k}{k!}\ll \left(\frac{a}{\sqrt{D}}\right)^{2m+c_0}e^{|\tau|}.
Z\bar Z\log\sin^2\theta = Z\bar Zu_1^2 + Z\bar Zu_2^2 - 2Z\bar Z\log(u_1^2+u_2^2) = u_1^2 + \frac{2|Q|^2}{u_1^2u_2^2}+ u_2^2 - \frac{2}{u_1^2+u_2^2}[Z\bar Z\log (u_1^2+u_2^2)](u_1^2+u_2^2),
\eta\left(\frac{1-\alpha*q}{\alpha*q}+\eta^2\right)s^4+\left(3\eta^2\frac{1}{\alpha*q}-\eta^4-\eta\right)s^3 &+3\eta\frac{1-2(\alpha*q)}{\alpha*q}(\eta^2-1)s^2 &+\left(\frac{1-\alpha*q}{\alpha*q}\eta^4+1-3\frac{\eta^2}{\alpha*q}\right)s-\eta\left(1+\frac{1-\alpha*q}{\alpha*q}\eta^2\right)=0,
\alpha_1^\vee(\lambda_{\alpha_1})s_1(t)\cdot [p_{\gamma_2, \alpha_2}(z_2), \;p_{\gamma_3, \alpha_3}(z_3), \;\ldots, \; p_{\gamma_n, \alpha_n}(z_n)] & = [p_{\gamma_2, \alpha_2}^\prime(z_2), \;\ldots, \; p_{\gamma_n, \alpha_n}^\prime(z_n)] \in Z_{(s_1, \ldots, s_n)},
\Lambda (\beta) = -8(2\pi \alpha^\prime)^{-D/2}\int_{E}\frac{d^2\tau}{2\pi \tau_2^2}\; K_h (\bar{\tau}, \tau; D) \sum_{\ell \in Z;{\rm odd}}\exp \left[ - \frac{\beta^2}{4\pi \alpha^\prime \tau_2}\; \ell^2 \right] \quad,
\frac{1}{\mu}= \operatorname*{l i m}_{| \Pi | \rightarrow 0}\frac{1}{| \Pi |}\frac{\partial \Pi_{0}}{\partial | \Pi |}= \operatorname*{l i m}_{| \pi | \rightarrow 0}\frac{1}{| \pi |}\frac{\partial \pi_{0}}{\partial | \pi |}
{\gamma_{s,e}}= \mathop{\max}\limits_{{e_k}\in{\Phi_{s,e}}}\left\{{\frac{{{{\left|{{h_{{s_0},{e_k}}}}\right|}^2}{{\left|{{X_{{s_0},{e_k}}}}\right|}^{- \alpha}}}}{{\underbrace{{I_{s,e}}+{I_{ap,e}}}_{I{n_{s,e}}}+{{{\delta^2}}\mathord{\left/{\vphantom{{{\delta^2}}{{P_s}}}}\right. \kern-\nulldelimiterspace}{{P_s}}}}}}\right\},
dR^I_t= \sqrt 2 \sum_{i\in I}(X^{i,N}_t-\bar X^I_t)\cdot (dB^i_t-d\bar B^I_t) +2(|I|-1)dt &+\frac{\chi}N \sum_{i\in I}(X^{i,N}_t-\bar X^I_t)\cdot\Big[\sum_{j\ne i}K(X^{i,N}_t-X^{j,N}_t) - |I|^{-1}Z^I_t\Big]dt
\underset{t \to 0}{\lim}\left[P^{s,N}_{HP}(t)f\right](x)=\underset{t \to 0}{\lim}\left[\mathcal{S}^N(t)f\circ \mathsf{eval}_N\right](U^*xU)=\left[f\circ \mathsf{eval}_N\right](U^*xU)=f(x).
g^{- 1}= \left( \begin{array}{c c c}{{- 2 \lambda_{2},}}&{{- 6 \lambda_{3},}}&{{- 4 \lambda_{4}}}\\{{- 6 \lambda_{3},}}&{{4 \lambda_{3}( \lambda_{2}^{2}- 4 \lambda_{4}) ,}}&{{\lambda_{2}\lambda_{3}}}\\{{- 4 \lambda_{4},}}&{{+ \lambda_{2}\lambda_{3},}}&{{- 2 \lambda_{2}\lambda_{4}+ \frac 3 4 \lambda_{3}}}\\ \end{array}\right)
\gamma_{0}=-i\left( \begin{array}{cc}0 -1 1 0\end{array}\right) ,\qquad \gamma_{1}=-i\left( \begin{array}{cc}0 1 1 0\end{array}\right) , \qquad \gamma_{5}=\gamma_{0}\gamma_{1}=\left( \begin{array}{cc}1 0 0 -1\end{array}\right),
{\cal A}^*({\cal E})=\sum_{i}\sum_{j}\int\limits_{-\infty}^{\infty}d\tau\, e^{-i{\cal E}\tau}\, u_{i}[{\tilde x}(\tau)]\, v_{j}^*[{\tilde x}(\tau)]\; \langle 0\vert{\hat a}_{i}{\hat b}_{j}\vert\Psi\rangle.
\tilde{\zeta}( s ) = \frac{V_{q}}{( 4 \pi )^{q / 2}}\frac{\Gamma ( s - q / 2 )}{\Gamma ( s )}\Big \lbrace ( \mu_{c}^{2}- \mu^{2})^{q / 2 - s}+ \sum_{{\bf p}\ne{\bf 0}}( \sigma_{\bf p}^{d}+ m^{2}- \mu^{2})^{- s}\Big \rbrace \; ,
\int\limits_0^{\pi}Cf (0,e_n,\psi)\sin\psi d\psi &=\frac{n-2}{2|S^{n-3}|}\int\limits_0^\pi \int\limits_{S^{n-2}}Rf(((\sin \phi)\omega,\cos \phi),0) |\cos\phi|(\sin\phi)^{n-2}d\omega d\phi &=\frac{n-2}{2|S^{n-3}|}\int\limits_{S^{n-1}}Rf(\sigma,0) |\sigma\cdot e_n|d\sigma.
\omega_i + \Delta \omega \approx \tan(\mu_i/2) + \Delta \mu \left( \frac{\tan^2(\mu_i/2)}{2}+ \frac{1}{2}\right) &= \omega_i + \Delta \mu ~\frac{\omega_i^2+1}{2}\quad\textrm{and hence}\Delta \mu \approx \Delta \omega \frac{2}{\omega_i^2+1}.
\frac{1}{n_{2}}\sum_{t=1}^{n_{2}}E(X_t^2) = c \left[\frac{1}{n_{1}}\sum_{t=1}^{n_{1}}E(X_t^2) \right]+ (1-c)\left[\frac{1}{n_{2}-n_{1}}\sum_{t=n_{1}+1}^{n_{2}}E(X_t^2) \right] \leq P,
R(i) = B^{-}(i)-L^{+}(i)+n^{\gamma}[r(t_i)+g_r(t_i)]-n^\gamma g_r(0), R(i) = B^{+}(i)-L^{-}(i)+n^{\gamma}[r(t_i)-g_r(t_i)]+n^\gamma g_r(0).
a_n(G_{\lambda})=\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda^2}\mathcal{H}_{0}(g)(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})+\frac{1}{n^a\chi^{1/4}_{n,\alpha}(c)}O(\frac{1}{\lambda^b}).
\Gamma_{n,\omega,j}^{q,k}{:=}\sum_{z \in \mathbb{Z}^n \setminus \{0\}}\frac{1}{4 \pi^2 |Q| |q^{-1}z|^2}\Biggl[ -\delta_{j,k}+\frac{\omega}{\omega+1}\frac{(q^{-1}z)_j(q^{-1}z)_k}{|q^{-1}z|^2}\Biggr]E_{2 \pi iq^{-1}z}\forall (j,k) \in \{1,\dots,n\}^2\,,
\lim_{n \to +\infty}\int_{\mathbb{R}^N}\left| \left( -\Delta+m^2 \right)^\frac{1}{4}u_n \right|^2 + \int_{\mathbb{R}^N}V |u_n|^2 = \lambda_1 \int_{\mathbb{R}^N}|u_n|^2 =1.
\varphi(f)(t)=u(t)\begin{pmatrix}f(\lambda_1(t)) 0 \ldots 0 0 0 f(\lambda_2(t)) \ldots 0 0 \vdots \vdots \vdots \vdots \vdots 0 0 \ldots f(\lambda_a(t)) 0 0 0 \ldots 0 0_{m-na}\end{pmatrix}u(t)^{\ast}.
{j^{W}}^{( L )}( x ) = \left( \begin{array}{l}{{\theta ( x_{1}+ L ) \theta ( L - x_{1}) \; [ \delta ( x_{2}- L ) - \delta ( x_{2}+ L ) ]}}\\{{\theta ( x_{2}+ L ) \theta ( L - x_{2}) \; [ \delta ( x_{1}+ L ) - \delta ( x_{1}- L ) ]}}\\ \end{array}\right) \; .
\begin{aligned}\big| (t-s)^{\alpha-1}E_{\alpha,\alpha}\big[c\big(t- s\big)^{\alpha}\big] - \big(\tau(k-i)\big)^{\alpha-1}\big| \leq \big(\tau^{1/2}-\tau\big)^{\alpha-1}&\times E_{\alpha,\alpha}\big[c\big(\tau^{1/2}-\tau\big)^{\alpha}\big] - \big(\tau^{1/2}\big)^{\alpha-1}E_{\alpha,\alpha}\big[c\big(\tau^{1/2}\big)^{\alpha}\big] &\leq T_{\alpha,\tau}\end{aligned}
\vec{\Gamma}_{\mu}^{\prime}= \vec{e}_{3}( \vec{e}_{3}\vec{\Omega}_{\mu}) = ( \cos \theta - 1 ) \, \vec{e}_{3}[ \vec{e}_{3}\cdot \vec{e}_{\phi}\wedge \partial_{\mu}\vec{e}_{\phi}] = ( \cos \theta - 1 ) \partial_{\mu}\phi \; \vec{e}_{3}.
\int e^{i \vec{q}\cdot \vec{\varrho}}\frac{1}{\vec{q}^{2}( \vec{q}^{2}+ \mu^{2})}\frac{d^{2}\vec{q}}{( 2 \pi )^{2}}= - \frac{1}{2 \pi \mu^{2}}\left[ \ln ( \varrho ) + K_{0}( | \mu | \varrho ) \right] ,
||((\phi \otimes f_{n})*\psi)(X,s)-\phi(X)R_{a}(\psi)(X,s)|| &=\Big|\Big|\phi(X) \int_{t \in U_{n}}f_{n}(t)(R_{t}(\psi(X,s))dt-R_{a}(\psi)(X,s))dt \Big|\Big| & \leq ||\phi||_{\infty}~\epsilon \int f_{n}(t)dt \leq \epsilon ~||\phi||_{\infty}.
\Delta =-\partial^2_r+r^{-2}\left(-h^{-2}(r)\partial^2_\theta+\frac12\frac{2rh'(r)+r^2h''(r)}{h(r)}-\frac14\frac{(h(r)+rh'(r))^2}{h^2(r)}\right) &=:-\partial^2_r+r^{-2}A(r).
A_{\Lambda'}(\underline \omega_\Lambda)=A =\{\omega \in \{0,1\}^{\Lambda'}: \; \omega\eta_{\Lambda^c}\in \Omega_{\textrm{EP}}, \, \omega_{\Lambda}=\underline \omega_\Lambda\}, B_{\Lambda'}(\underline \omega_\Lambda)=B =\{\sigma \in \{-1,1\}^{\Lambda'_*}: \delta(\sigma_{\Lambda_{m+1,*}})=\underline \omega_{\Lambda}\}.
\lim_{\varepsilon \downarrow 0}P\left( \underset{t\in [0, \frac{\varepsilon}{\mu}|\log \varepsilon | ]}{\sup}\| u^\varepsilon (t,\cdot ) - u(t,\cdot) \|_{L^2(\mathbb{R})}\leq \varepsilon^{\kappa}\right) =1,
J^{(N)}\left(n; 1,\ldots, 1\right)= 2 \; \mbox{i}^{1-2N}\; \pi^{n/2}\;\Gamma\left(N \!-\!{\textstyle{n\over2}}\right) \;\left(\prod f_i \right) \;\int\limits_0^{\infty}\! \ldots \! \int\limits_0^{\infty}\frac{\prod \mbox{d}\alpha_i}{\left( \sum \alpha_i f_i \right)^{n-N}}\;\delta\left( \alpha^T \|C\| \alpha - 1 \right) .
\pi_i(\widetilde{\mathbf{x}}_j(\theta))=\pi_{i,j}\circ \pi_j(\widetilde{\mathbf{x}}_j(\theta))=\pi_{i,j}( \mathbf{x}_j(\theta))=\pi_{i,j}( x_{t_a(b)}^j)=x_{t_a(b)}^i.
L = L_{0}( V ) + \frac{a F_{\pi}^{2}}{1 6}\left[ T r ( \log V - \log V^{\dagger}) \right]^{2}+ \frac{F_{\pi}}{2 \sqrt{2}}T r \left[ M ( \theta ) ( V + V^{\dagger}- \sqrt{2}F_{\pi}) \right] +
\frac{\partial f}{\partial\bar{z}}=-i|f|^2\left\{\mu_1|g|^2-\frac{\mu_2}{2}(1+\bar g^2)\right\}, \frac{\partial g}{\partial \bar{z}}=\frac{i}{2}\bar f\{\mu_1\bar g(1+g^2)-\mu_2g(1+\bar g^2)\}.
e^{-\frac{x}{2}}{}_{2}F_{2}(a,1+b;2a+1,b;x)={}_{0}F_{1}(;a+\frac{1}{2};\frac{x^{2}}{16})-\frac{x(1-\frac{2a}{b})}{2(2a+1)}{}_{0}F_{1}(;a+\frac{3}{2};\frac{x^{2}}{16}),
\sum_{n=0}^{\infty}q^{n^{2}}S_{n}\left(xq^{n};q\right)t^{n}=\sum_{k=0}^{\infty}\frac{q^{3k^{2}}\left(-xt\right)^{k}}{\left(q;q\right)_{k}}A_{q}\left(-tq^{3k}\right).
\hat{g}_{ij}(z) =\left( \gamma^{-1}_n\rho^{n-4}+ \alpha +O_4(\rho^{-1}) \right)^{\frac{4}{n-4}}g(\partial_{z^i},\partial_{z^j}), &= \left( 1 + \frac{\gamma_n\alpha}{\rho^{n-4}}+ O_4(\rho^{3-n}) \right)^{\frac{4}{n-4}}\left(\delta^i_k - 2\rho^{-2}z^iz^k \right)\left(\delta^j_l - 2\rho^{-2}z^jz^l \right)g_{kl}\left(\gamma_n^{\frac{2}{n-4}}\rho^{-2}z\right).
A\Delta_{{\bf T}_1}h +B\left[\begin{matrix}\Delta_{{\bf T}_2}T_{1,1}^*h 0 \vdots \Delta_{{\bf T}_2}T_{1,n_1}^*h 0\end{matrix}\right]=\left[\begin{matrix}\Delta_{{\bf T}_1'}T_{2,1}^*h \vdots \Delta_{{\bf T}_1'}T_{2,n_2}^*h\end{matrix}\right]
\mathrm{T r}e^{- \beta \, P \bar{\cal K}}= \frac{e^{- \beta}}{\sqrt{4 \pi}}\sum_{n = 0}^{\infty}a_{n}( \bar{\cal K}) \beta^{n - \frac{1}{2}}- 1 \hspace{0 . 4 c m}, \hspace{0 . 4 c m}a_{0}( \bar{\cal K}) = m L \hspace{0 . 4 c m}, \hspace{0 . 4 c m}a_{n}( \bar{\cal K}) = \frac{2^{n + 1}}{( 2 n - 1 ) ! !}\, .
\log N(\varepsilon, \,\mathcal{E}, \,\|\cdot\|_{\ell_2}) \geq m^{|I|}\log\Big(\frac{1}{\varepsilon}\Big) - \frac{1}{2}|I|\, m^{|I|}\log a - \frac{|I|\,m^{|I|+2}}{8a^2}& = m^{|I|}\log\Big(\frac{1}{\varepsilon\, a^{|I|/2}}\Big) - \frac{|I|\,m^{|I|+2}}{8a^2},
\begin{array}{l}{{\mathcal D}_{V}f = V^{a}\nabla_{a}f + \frac{n - 2}{2 n}( \nabla_{a}V^{a}) f}\\{\delta_{V}f = V^{a}\nabla_{a}f + \frac{n + 2}{2 n}( \nabla_{a}V^{a}) f .}\\ \end{array}
\begin{array}{l}\displaystyle I^{\phi_3}(f^{(n)}_{\alpha}) = \left({\rm log}\frac{(1-\alpha)(\alpha+\gamma)}{\alpha(2-\alpha-\gamma)}\right)^2 n^2 + C_1(\alpha,\gamma)n + C_2(\alpha,\gamma) + O \left(\frac{1}{n}\right)\end{array}
\| D \Delta^{-1}\nabla \times \partial_t \omega^C \|_{L_x^{\infty}(|x|<R_0/100)}\lesssim_{R_0}\; \| u\|_{\infty}(\|\omega^C \|_{1}+\| \frac{\omega^C}r \|_{1}) \lesssim_{R_0}\; \|u\|_{\infty}\| \frac{\omega^C(t=0)}r \|_{1}\lesssim_{R_0,u_{-1}}1.
\begin{array}{r| l c}C\vert\tilde{V}_\alpha(0)\vert \mu_\alpha^2\theta_\alpha^{-3}(x) n= 5 \vert\mathcal{L}_\xi R_\alpha(x) \vert\leq C\vert\tilde{V}_\alpha(0)\vert \mu_\alpha^2\theta_\alpha^{-3}(x)\ln\left(1+ \frac{\theta_\alpha(x)}{\mu_\alpha}\right) n=4, & C\vert\tilde{V}_\alpha(0)\vert \mu_\alpha^2\theta_\alpha^{-2}(x) n=3.\end{array}
R_{1 \bar{1}1 \bar{1}}=-32p^3K, &R_{1 \bar{2}1\bar{2}}=(p-1)f^{(1)}(0), &R_{1 \bar{1}2 \bar{2}}=-pf^{(1)}(0), &R_{2 \bar{2}2 \bar{2}}=\left(-3+\frac{1}{K}\right)\frac{f^{(1)}(0)^2}{16},
[M_{S'}, \mu_{S'}](I^G_{M_S}(\rho))=(\mathrm{Ind}^G_{P_{S'}}\circ [\mu_{S'}] )( \delta^{\frac{1}{2}}_{P_S}\otimes J^G_{P^{op}_{S'}}I^G_{M_S}(\rho))\otimes [1][|\cdot|^{\langle \rho_G, \mu_{S'}-\mu \rangle}]
\begin{aligned}(\tilde{\Delta}_E\tilde{h}_1,\tilde{h}_2)_{L^2(\tilde{g})}= -4(n-1)\lambda_i(\lambda_i-n)\int_0^{\infty}\varphi\psi\cosh \sinh^{n-2}dr, (\tilde{\Delta}_E\tilde{h}_2,\tilde{h}_3)_{L^2(\tilde{g})}= 4(n+1)\lambda_i\int_0^{\infty}\psi\chi \cosh \sinh^{n-2}dr.\end{aligned}
2h_{00}^{\frac{m+1}{2}}W_0^m\Phi^i_{\parallel l}= h_{00}^{m+1}A^i_{(1)\parallel l}- m h_{00}^{m+1}W_{0 \parallel l}A^i_{(1)}+ (h_{00})^{\frac{m+1}{2}}W_0^m A^i_{(2)\parallel l}+ & W_0^{2m}A^i_{(3)\parallel l}+ m W_0^{2m-1}W_{0 \parallel l}A^i_{(3)}.
u^n_t(y) =\, P^n_{t}u^n_0(y)+ \sum_e\frac{1}{L^e}\int_0^t u^n_s(x^e_1)\partial_s\phi_s(x^e_1)\,ds &+ \sum_e \Big(Y^e_t(\phi)+ Z^e_t(\phi)+E^{(1,e)}_t(\phi) + E^{(2,e)}_t(\phi)\Big) & + \sum_e \Big( T^e_t(\phi)+U^e_t(\phi)+ V^e_t(\phi)+ E^{(3,e)}_t(\phi)+ E^{(4,e)}_t(\phi)\Big)
F^{\prime}\left(g_{s,3,1}\right)= F\left(g_{s,3,1}\circ g_{e,1,3}\right)=F\left(g_{s,3,1}\circ g_{e,2,3}\circ g_{e,1,2}\right) = F\left(g_{e,3,1}\circ g_{s,2,3}\circ g_{e,1,2}\right)=F^{\prime}\left(g_{s,2,3}\right).
\{f,g\}_{\Lambda}= \omega^{ij}\left(\frac{\partial f}{\partial x^i}\frac{\partial_lg}{\partial \theta_j}-\frac{\partial_rf}{\partial \theta_i}\frac{\partial g}{\partial x^j}\right)+ \frac{\partial_r f}{\partial \theta^i}(\theta^k{\partial \omega^{ij}}/{\partial x^k})\frac{\partial_lg}{\partial \theta_j}
\frac{1}{2\pi}\int \frac{|D_tZ(\alpha,t)-D_tZ(\beta,t)|^2}{(\alpha-\beta)^2}d\beta= \sum_{1\leq j,k\leq N}\frac{\lambda_j\lambda_k}{(2\pi)^3}\frac{1}{(\alpha-z_j)\overline{(\alpha-z_k)}}\frac{2\pi i}{\overline{z_k}-z_j}= \sum_{1\leq j,k\leq N}\frac{\lambda_j\lambda_k}{(2\pi)^2}\frac{1}{(\alpha-z_j)\overline{(\alpha-z_k)}}\frac{i}{\overline{z_k}-z_j}.
\hat{p}(x_n|\boldsymbol{y},\hat{r}_n,\tau_n^r,\boldsymbol{\theta})= \frac{p(x_n|\boldsymbol{\theta})\mathcal{N}(x_n|\hat{r}_n,\tau_n^r)}{\int_xp(x_n|\boldsymbol{\theta})\mathcal{N}(x_n|\hat{r}_n,\tau_n^r)}
T M_{12}+ M_{12}T^* = \cos\theta M_{12}, T M_{1q}+ M_{1q}T^* = \frac12(1+\cos\theta) M_{1q}+ \frac12\sin\theta M_{2q},q>2, T M_{2q}+ M_{2q}T^* = \frac12(1+\cos\theta) M_{2q}- \frac12\sin\theta M_{1q},q>2,
D_1=\begin{pmatrix}m_1\end{pmatrix},D_2=\begin{pmatrix}1 m_1-1/2 m_1-1/2 m_2-m_1\end{pmatrix}, D_3=\begin{pmatrix}m_1 m_2-m_1/2 m_2-m_1/2 m_3-m_2\end{pmatrix}, D_4=\begin{pmatrix}1 m_1-1/2 m_2-m_1+1/2 m_1-1/2 m_2-m_1 m_3-3m_2/2+m_1/2 m_2-m_1+1/2 m_3-3m_2/2+m_1/2 m_4-2m_3+m_2\end{pmatrix},
G\left( h_{+}\right) \allowbreak = \dfrac{\sqrt{s^{2}+t^{2}}\sqrt{p_{1}}-\allowbreak \left\vert2wst-(t^{2}-s^{2})v\right\vert}{\sqrt{s^{2}+t^{2}}\sqrt{p_{1}}+\allowbreak\left\vert 2wst-(t^{2}-s^{2})v\right\vert}
\begin{aligned}u^\varepsilon (z_1) \leq \int_0^{t_1 + t_2}L \big( \xi^\varepsilon (s, z_1, \gamma ), \gamma (\xi^\varepsilon (s, x, \gamma )) \big) e^{-\lambda s}\, ds + u^\varepsilon (z_2) e^{-\lambda (t_1 + t_2)}\leq C(1 + \lambda ) \cfrac{4 \sqrt{h}}{c_0 \mu}+ u^\varepsilon (z_2),\end{aligned}
0\, =f(A(\alpha))-f(\alpha)=A(\beta_0+X_n^{q-1}u)-(\beta_0+X_n^{q-1}u)\cr =\bigl((X_n-X_1)^{q-1}-X_n^{q-1}\bigr)u=(X_1^{q-1}+X_1^{q-2}X_n+\cdots+X_1X_n^{q-2})u\ne 0,
\beta_{i j}^{( k )}\equiv \partial_{x}^{- 1}\left({\cal L}^{k}\, ( \Phi_{j}) \Psi_{i}\right) \; \; ; \; \; \beta_{i j}^{* \, ( k )}\equiv \partial_{x}^{- 1}\left( \Phi_{j}\,{\cal L}^{* \, k}( \Psi_{i}) \right)
\frac 1{2C}v_\epsilon''(t,\rho_t) = \int_{|\rho - \rho_t|\leq 1/2C}{\frac{v_\epsilon''(t,\rho_t)}2 d\rho}< \int_{-\infty}^{\infty}{v_\epsilon''(t,\rho) d\rho}= b_t - a_t = (1 + \alpha) (T-t)
\frac{d ( V_{i})}{d ( V_{3})}= \lim_{i \tau \nearrow 0}\frac{\sum_{j = 1}^{4}S_{j , i}\chi_{V_{j}}^{}( - \frac{1}{\tau})}{\sum_{k = 1}^{4}S_{k , 3}\chi_{k}^{}( - \frac{1}{\tau})}= \lim_{q \rightarrow 0}\frac{\sum_{j = 1}^{4}S_{j , i}\chi_{j}^{}( q )}{\sum_{k = 1}^{4}S_{k , 3}\chi_{k}^{}( q )}= \frac{S_{3 , i}}{S_{3 , 3}}.
W ( x ) = \exp \left( i \frac{x}{\Sigma}\right) \ \ \left( \begin{array}{c c c c c c}{{\mathrm{}^{\ddots}}}&{{e^{- 2 \pi i / N}}}&{{}}&{{}}&{{\phi}}\\{{}}&{{}}&{{1}}&{{e^{2 \pi i / N}}}&{{}}&{{}}\\{{}}&{{\phi}}&{{}}&{{}}&{{e^{4 \pi i / N}}}&{{\mathrm{}_{\ddots}}}\\ \end{array}\right)
u =u(x_m)=\frac{1}{4}\xi_{m-1}+\frac{3}{2}\xi_m+\frac{1}{4}\xi_{m-1}u'_m =u'(x_m)=\frac{-1}{h}\xi_{m-1}+\frac{1}{h}\xi_{m+1}u''_m =u''(x_m)=\frac{2}{h^2}\xi_{m-1}-\frac{4}{h^2}\xi_m+\frac{2}{h^2}\xi_{m+1}
b_4=\begin{pmatrix}s a 0 0 b 0 0 0 0 0 0 0\end{pmatrix}, b_5=\begin{pmatrix}0 -b 0 s a 0 0 0 0 0 0 0\end{pmatrix},b_6=\begin{pmatrix}0 0 0 0 0 0 s a 0 0 \pm b 0\end{pmatrix},b_7=\begin{pmatrix}0 0 0 0 0 0 0 \mp b 0 s a 0\end{pmatrix}
R_{21}=(d-ca^{-1}b)^{-1}([c,d]-ca^{-1}[c,b])\Delta^{-1}&=(ac^{-1}d-b)^{-1}(ac^{-1}[c,d]-[c,b])\Delta^{-1}&=((ac^{-1}d-b)^{-1}\Delta-c)\Delta^{-1},
\int_{\left[0,1\right]^d}\left(p(x) - \hat{p}\left(x\right) \right)^2dx = \int_{\left[0,1\right]^d}\hat{p}\left(x\right)^2 dx - 2\int_{\left[0,1\right]^d}p(y)\hat{p}(y) dy + \int_{\left[0,1\right]^d}p(z)^2 dz.
\overline{V_1(k,l)}=N(0,a,b,c)\ \ \ \ \{for}\ \ \a=0,\ \b=k+1-l,\ \c=k+1, & \overline{V_2(k,l)}=N(0,a,b,c)\ \ \ \ \{for}\ \ \a=b=l,\ \c=k+1, & \overline{V_3(k,l)}=N(-1,a,b,c)\ \ \{for}\ \ \a=b=l-1,\ \c=k+1, & \overline{V_4(k)}=N(0,a,b,c)\ \ \ \ \ \ \{for}\ \ \a=b=0,\ \ \c=k+1.
(h^{(j)})^{1-\alpha_j-\beta_j}\cdot\tau\left(\alpha_j,\beta_j,\xi^{(j)},(m^{(j)}+v_1^{(j)})\frac{y_j}{h^{(j)}}\right)=\tau\left(\alpha_j,\beta_j,\frac{\xi^{(j)}}{h^{(j)}},(m^{(j)}+v_1^{(j)})y_j\right).
\zeta_2 (s_1,s_2 \,; a) = a^{-s_1}\bigl( \zeta (s_2,a) - a^{-s_2}\bigr) + \frac{\zeta(s_1+s_2-1,a) - a^{1-s_1-s_2}}{s_2-1}+ \sum_{r=0}^l \frac{B_{r+1}(0)}{(r+1)!}(s_2)_r \bigl( \zeta(s_1+s_2+r,a) - a^{-s_1-s_2-r}\bigr)-\sum_{n=1}^\infty \frac{\Phi_l (s_2 \,|\, n,a)}{(n+a)^{s_1}}.
v_{\omega}\left( \sigma^{-}\left( y^{-}\right) \right) ={\frac{1}{\sqrt{2 \omega}}}\, e^{{\frac{i \omega}{\lambda}}\ln{\left( \lambda \Delta \left( e^{- \lambda y^{-}}- 1 \right) \right)}}\Theta \left( - y^{-}\right) \; ,