harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(A,B,C)\mapsto [A,B,C]_{\bullet_\hbar}\equiv \sum_{\sigma\in S_3}\epsilon(\sigma) \Delta_{\sigma_1}A\bullet_\hbar \Delta_{\sigma_2}B\bullet_\hbar \Delta_{\sigma_3}C\; ,\quad \forall A,B,C\in{\cal A}_\hbar.
\aligned \partial{\mathcal M}(L_1,L_2;a,c;2,E) &=\bigcup_{E_1 + E_2 = E}{\mathcal M}(L_1,L_2;a,c;1,E_1) \times{\mathcal M}(L_1,L_2;c,b;1,E_2),\endaligned
g(t) := 1 + \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)}t - \frac{c(r - 2)(6 - r)}{a(p - 2)(6 - p)}t^{\frac{r - p}{q - p}}h(t) := 1 + \frac{b(q - 2)}{a(p - 2)}t - \frac{c(r - 2)}{a(p - 2)}t^{\frac{r - p}{q - p}}.
\left. \begin{array}{l c l}{T_{z}^{z}}&{=}&{V ( \phi ) -{{{{{\phi^{\prime}}}^{2}}}{{e^{- 2 a}}}}}\\{T_{i}^{i}}&{=}&{V ( \phi ) +{{{{{\phi^{\prime}}}^{2}}}{{e^{- 2 a}}}},}\\ \end{array}\right.
D \psi_{n}(|x|) = \psi'_{n}(|x|) \frac{x}{|x|},\ \D^{2}\psi_{n}(|x|) =\psi''_{n}(|x|) \frac{xx^{T}}{|x|^{2}}+ \psi'_{n}(|x|) \biggl[\frac{I}{|x|}- \frac{xx^{T}}{|x|^{3}}\biggr].
\times \int_{0}^{\infty}\frac{d \tau}{\tau}e^{2 \pi \tau}e^{- 2 \pi \tau \alpha^{\prime}k^{2}}e^{- \frac{y^{2}\tau}{2 \pi \alpha^{\prime}}}\mathrm{e}^{2 \pi \tau}T r \left( \prod_{n = 1}^{\infty}e^{- 2 \pi \tau \alpha_{- n}\cdot \alpha_{n}}\right) =
M_{5,4}(\mathbf{Y}) = \begin{pmatrix}Y_{1}\imath Y_{2}\imath^{2}Y_{3}\imath^{3}Y_{4}0 \imath Y_{1}\imath Y_{2}\imath^{2}Y_{3}\imath^{3}Y_{4}0 2 \imath Y_{1}\imath Y_{2}\imath^{2}Y_{3}0 0 3 \imath Y_{1}\imath Y_{2}0 0 0 4 \imath Y_{1}\end{pmatrix}.
\begin{aligned}\frac{1}{\tau^k}[f(x)-f(x^{k+1})]-\frac{1}{\tau^{k-1}}[f(x)-f(x^{k})]+(x-\bar{x}^{k+1})^T[-A^T\lambda^k &+\beta^kA^T(A\hat{x}^{k}-b)+\beta^kD(x^{k+1}-\hat{x}^k) ]\geq0,~\forall x.\end{aligned}
( \exp t T_{3}) \circ [ g_{c}( z ) ]_{B}= [ g_{c}( z^{\prime}) ]_{B}= \left[ \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{z_{1}^{\prime}}}&{{1}}&{{0}}\\{{z_{2}^{\prime}}}&{{z_{3}^{\prime}}}&{{1}}\\ \end{array}\right) \right]_{B}.
K_n(x_1\vee\cdots\vee x_n)=\frac{1}{n!}\sum_{i=0}^{n-1}\sum_{\sigma\in S_n}\frac{\epsilon(\sigma)}{n-i}ipX_{\sigma(1)}\vee\cdots\vee ipX_{\sigma(i)}\vee X_{\sigma(i+1)}\vee X_{\sigma(n)}.
\widetilde{\psi}_{k}^{(0^{2})}(x;2,1,\lambda_{1})=( 1-z)^{5/4}(1+z)^{3/4}\frac{\widetilde{P}_{2,1,k}^{( 0^{2})}(z; \lambda_1)}{\lambda_{1}-(z+1)^{2}+\frac{2}{3}(z+1)^{3}-\frac{1}{8}(z+1)^{4}},
- \int_{0}^{\tau}\! \mathrm{d}t \left[ \sum \dot{\phi}_{j}^{2}+ H_{\mathrm{b o s}}( i \dot{\phi}( t ) , \phi ( t ) ) \right] = - \int \! \mathrm{d}^{4}x{\cal L}_{\mathrm{E u c l}}.
\times \left[ 1 + \sum_{N = 1}^{\infty}\frac{\zeta^{N}}{N !}\left( \prod_{a = 1}^{N}\int d^{3}z_{a}\sum_{\alpha_{a}= \pm 1 , \pm 2 , \pm 3}^{}\right) \exp \left( i g_{m}\int d^{3}x \vec{\chi}\vec{\rho}_{\mathrm{g a s}}\right) \right] .
\langle y_1, y_1 \rangle = \det(g)^{-1}\big( g_4^2 \langle x_1,x_1 \rangle -2 g_3 g_4 \langle x_1,x_2 \rangle + g_3^2 \langle x_2,x_2 \rangle\big) & = \det(g)^{-1}\big( g_4^2 a- g_3 g_4 b + g_3^2 c\big) &=a.
\lim_{y\to\infty}\frac{G^{-1}(y)}{(\log y)^{-1/\alpha}}=1, G^{-1}\in_\infty(0), \lim_{y\to\infty}\frac{\Gamma(y)}{\log^{-(\alpha+1)/\alpha}(y)/y}=\frac{1}{\alpha}, \Gamma=g\circ G^{-1}\in_\infty(-1).
\hat \mu_I = \frac1I\sum_{i=1}^I \delta_{X_i - \bar X_I}\mbox{with}\bar X_I=\frac 1I\sum_{i=1}^IX_i \quad\mbox{and}\hat \nu_J = \frac1J\sum_{j=1}^J \delta_{Y_j - \bar Y_J}\mbox{with}\bar Y_J=\frac 1J\sum_{j=1}^JY_j,
G_{cd}=\frac{1}{2}\epsilon_{abcd}F^{ab}=[\begin{matrix}0&B_{x}&B_{y}&B_{z}\\ -B_{x}&0&\frac{1}{c}E_{z}&-\frac{1}{c}E_{y}\\ -B_{y}&-\frac{1}{c}E_{z}&0&\frac{1}{c}E_{x}\\ -B_{z}&\frac{1}{c}E_{y}&-\frac{1}{c}E_{x}&0\end{matrix}]
\frac{\mathfrak{m}_{j,3}}{s_j^3}=\theta_j^0 \frac{E_{\theta_0}[\psi_j(X_1^{(j)})^3] -3E_{\theta_0}[\psi_j(X_1^{(j)})^2]E_j + 2E_j^3}{s_j^3}&=\theta_j^0\frac{\mu_{j,3}}{F_j s_j^3}+ O\big(\alpha/(nF_j)^{1/2}\big) &=\theta_j^0 \mu_{j,3}n^{3/2}\sqrt{F_j}\alpha^{-3}+ O\big(\alpha/(nF_j)^{1/2}\big)
\mathbf E[n^{-mp}D_k(n)_{\uparrow m}] = \frac{D_k(n_0)_{\uparrow m}}{n^{mp}}\prod_{\ell=n_0}^{n-1}\tfrac{\ell+mp}\ell \xrightarrow{n\to\infty}\frac{D_k(n_0)_{\uparrow m}\cdot\Gamma(n_0)}{\Gamma(n_0+mp)}= \mathbf E[D_k(\infty)^m].
\delta^{( 4 )}( \dot{-}k_{3}\dot{+}( \dot{-}p ) \dot{+}k_{3}\dot{+}p^{\prime}) \sim \delta ( p_{0}- p_{0}^{\prime}) \, \, \delta^{( 3 )}( e^{\lambda p_{0}}\vec{k_{3}}- \vec{p}+ \vec{k_{3}}+ e^{\lambda k_{3 , 0}}\vec{p^{\prime}}) ~ .
g=E^{-1}=\left[\begin{array}{ccc}\frac{1}{1-e^{-2(q_1-q_2)}}-\frac{e^{-(q_1-q_2)}}{1-e^{-2(q_1-q_2)}}0 -\frac{e^{-(q_1-q_2)}}{1-e^{-2(q_1-q_2)}}\frac{1-e^{-2(q_1-q_3)}}{(1-e^{-2(q_1-q_2)})(1-e^{-2(q_2-q_3)})}-\frac{e^{-(q_2-q_3)}}{1-e^{-2(q_2-q_3)}}0 -\frac{e^{-(q_2-q_3)}}{1-e^{-2(q_2-q_3)}}\frac{1}{1-e^{-2(q_2-q_3)}}\end{array}\right]\;,
K_{ijkl}=\frac{1}{2}\Big{(}(a_{ij}+a_{ji})\delta_{kl}-\frac{m}{2}(\delta_{ik}a_{lj}+\delta_{il}a_{kj})-\frac{m}{2}(\delta_{kj}a_{li}+\delta_{lj}a_{ki})\Big{)}&+\frac{-tr(a)\delta_{kl}\delta_{ij}}{m}+\frac{\delta_{ij}}{2}(a_{lk}+a_{kl}).
\mathcal{L}P = \lambda'(\sum_{i}F^{ii}-1)+u^{\frac{\alpha+1}{\alpha}}(F^{ij}h_{il}h_{jl}-F^{2})-u^{\frac{1}{\alpha}}F^{ij}\bar{R}_{\nu jli}\bar{g}(\lambda\partial_{r},e_{l}) &+\frac{1}{\alpha}\bar{g}(\lambda\partial_{r},\nabla \log u)-\frac{1}{\alpha}u^{\frac{\alpha+1}{\alpha}}F^{ij}\nabla_{i}\log u\nabla_{j}\log u.
P_N(x,t)\ge c\Big\{1+ aN^{1/4}(x-N)+(a^2N^{2/4}+aN^{-3/4})\frac{(x-N)^2}{2!}&+ (a^3N^{3/4}+a^2N^{-2/4}+a N^{-7/4})\frac{(x-N)^3}{3!}&+( a^4N^{4/4}+a^3N^{-1/4}+a^2N^{-6/4}+aN^{-11/4})\frac{(x-N)^4}{4!}\Big\}\,e^{aN^{5/4}}&\ge c\,e^{aN^{5/4}}>0,
\begin{array}{c c}{\mathrm{y}_{\pm}= \left( \begin{array}{c c}{0}&{- ( y_{\pm}{\cdot \sigma})^{t}}\\{y_{\mp}{\cdot \sigma}}&{0}\\ \end{array}\right)}&{y_{\pm}^{\mu}= y^{\mu}\pm i \bar{\phi}{}^{a}\tilde{\sigma}^{\mu}\phi_{a}= x_{\pm}^{\mu}}\\ \end{array}
(a\phi_{d,d_1}\,{\times}\,b\phi_{d,d_2})^*{\cal P}_{{\cal E}_d}\,{\cong}\, (1\,{\times}\, a\phi_{d_1,d}b\phi_{d,d_2})^*{\cal P}_{{\cal E}_d}{=}(1\,{\times}\, abc\phi_{d_1,d_2})^*{\cal P}_{{\cal E}_d}\,{\cong}\, \big((1\,{\times}\, \phi_{d_1,d_2})^*{\cal P}_{{\cal E}_d}\big)^{abc}
\binom{3^n}{\leq 3^n/d_1}\binom{m_1}{\leq m_1/d_2}2^{m_2+\frac{m_1}{d_2}+ \frac{3^n}{d_1}}\leq \left(e\cdot d_1\right)^{3^n/d_1}\left(e\cdot d_2\right)^{m_1/d_2}2^{m_2+\frac{m_1}{d_2}+ \frac{3^n}{d_1}}&\leq 2^{m_2 + O\left(\frac{3^n\log{n}}{d_1}+ \frac{m_1\log{n}}{d_2}\right)}
[ V_{\alpha}, V_{\beta}] = \epsilon_{\alpha \beta}^{\! \gamma}V_{\gamma}, [ V_{\alpha}, V_{m}^{a}] = V_{m}^{b}( \sigma_{\alpha})_{b}^{~ a}, \{V_{m}^{a}, V_{m}^{b}\}= - m^{2}( \sigma^{\alpha})^{a b}V_{\alpha},
\frac{1}{\frac{6q+6}{2q_0+1}}+\frac{1}{\frac{6q+6}{6q-2q_0+5}}=1,\;\frac{1}{\frac{6q+6}{2q_0+1}}+\frac{1}{6}+\frac{1}{2}+\frac{1}{\frac{6q+6}{2q-2q_0+1}}=1\frac{1}{\frac{6q+6}{q+2q_0+2}}+\frac{1}{2}+\frac{1}{\frac{6q+6}{2q-2q_0+1}}=1,
e^h\mathrm{div}\Big(e^{-h}(V\cdot\nabla)V \Big) =e^h\partial_j\Big(e^{-h}V^i\,\partial_i V^j \Big)=\partial_j V^i\,\partial_i V^j+V^i\,\partial_j\partial_i V^j-\partial_j h\,V^i\,\partial_i V^j =\mathrm{tr}((\nabla V)^2)+V^i\,\partial_j\partial_iV^j-\partial_j h\,V^i\,\partial_i V^j.
\hat{H}( \tau ) = \mathrm{b d i a g}\left( \hat{h}( \tau ) , \, - \sigma_{3}\hat{h}( - \tau ) \sigma_{3}\right) \, , \; \; \hat{h}( \tau ) = \left. \hat{h}( x^{0}) \right|_{x^{0}= \tau}, \; \; \hat{h}( x^{0}) = q A_{0}I + \hat{\omega}\; .
V_{( j , m )}^{( 1 )}( z , \bar{z}) ={\large [}- \frac{j}{\alpha_{+}}\partial \phi ( z ) \gamma^{( j - m )}- ( j - m ) \partial \gamma \gamma^{j - m - 1}( z ){\large ]}\times c . c . \times e^{2 j \phi ( z , \bar{z}) / \alpha_{+}}
F_p(p,p-1,b) =1+\dim(p-2,b)+\dim(p-1,b-1)+\dim(p+b-3,1)\cr =\dim(p-2,b)+\dim(p-1,b-1)+\dim(p+b-3,1)\cr \ \ \ \ +\dim(p+b-2)\cr =\dim(p-1,b)+\dim(p+b-2,1).
V=\frac{m}{2}(\begin{matrix}x_{a}&x_{b}\end{matrix})(\begin{matrix}\frac{g}{L_{a}}+\frac{k}{m}&-\frac{k}{m}\\ -\frac{k}{m}&\frac{g}{L_{b}}+\frac{k}{m}\end{matrix})(\begin{matrix}x_{a}\\x_{b}\end{matrix}).
\sigma_{2}^{2}\left( Z_{1}\right) =\frac{17-\sqrt{17^{2}-4\cdot3\cdot2\cdot3}}{2}=\frac{17-\sqrt{217}}{2}>\frac{17-15}{2}=1 \sigma_{2}^{2}\left( Z_{2}\right) =\frac{18-\sqrt{18^{2}-4\cdot2\cdot3\cdot3}}{2}=\frac{18-\sqrt{252}}{2}>\frac{18-16}{2}=1.
\frac{Q_{t-1}(u_{j}^{t}+2)Q_t (u_{j}^{t}-2)Q_{t+1}(u_{j}^{t})}{Q_{t-1}(u_{j}^{t})Q_t (u_{j}^{t}+2)Q_{t+1}(u_{j}^{t}-2)}=-1
{S^{(M)}}_{\lambda}^{-{\alpha_{\Re}}}:=\sum_{m=0}^M a_m\int_{W_a^+}(\mathrm{V}(w^+))^me^{-\mathrm{Q}(w^+)}(w^+)^{\alpha_{\Re}}\;\mathcal{D}_\lambda w^+\;.
\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)
\begin{aligned}\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}=\sum_{i=1}^{n}\left(\int_{\partial A}(1-x_{i}^{2})\langle X(x),N(x)\rangle\gamma_{n}(x)dx\right)^{2}&\qquad+2\sum_{i=1}^{n}\int_{\partial A}(-x_{i})X^{(i)}\langle X(x),N(x)\rangle d\gamma_{n}(x)(\int_{A}(1-y_{i}^{2})d\gamma_{n}(y)).\end{aligned}
\pi_\lambda(x_1,\cdots,x_n) :=\Delta(x_1,\cdots,x_{\lambda_1})\Delta(x_{\lambda_1+1},\cdots,x_{\lambda_1+\lambda_2})\cdots, \Delta(y_1,\cdots,y_m) :=\prod_{1\leq i<j\leq m}y_i-y_j.
V_{1,z}:= \{(x,h) \in S_z \ : \ \Delta^{k+2}_{(h,\dots,h)}f_z(x)\geq 0 \}V_{2,z}:= \{(x,h) \in S_z \ : \ \Delta^{k+2}_{(h,\dots,h)}f_z(x)\leq 0 \}.
\begin{cases}\Delta_g u_\alpha+m_0^2 u_\alpha=u_\alpha^{p-1}+\omega_\alpha^2(1-qv_\alpha)^2 u_\alpha, \Delta_g v_\alpha +(m_1^2+q^2u_\alpha^2)v_\alpha=qu_\alpha^2,\end{cases}
\left(\frac{1}{\beta}\left( \binom{n-1}{2}-1 \right), \frac1\nu ({n-2})\right), \left( \frac{1}{\beta}\left( \binom{n}{2}-1 \right) , 0 \right),\left( \frac{1}{\beta}\left( \binom{n-1}{2}-1 +\frac{n-1}{2}\right), 0 \right).
D^{0}( k ) D^{0}( k + p ) ={\frac{1}{\eta \cdot k}}{\frac{1}{\eta \cdot ( k + p )}}={\frac{1}{\eta \cdot p}}\biggl [{\frac{1}{\eta \cdot k}}-{\frac{1}{\eta \cdot ( k + p )}}\biggr ] .
I_{E}= 2 \pi \beta \, e^{( \Gamma - \Lambda ) / 2}\, [ \, ( \Gamma^{\prime}- \phi^{\prime}) \, r^{2}+ \frac{\alpha^{\prime}e^{\phi}}{g^{2}}\, ( 1 - e^{- \Lambda}) \, ( \Gamma^{\prime}+ \phi^{\prime}) - \frac{\alpha^{\prime}e^{\phi}}{g^{2}}\, r e^{- \Lambda}\Gamma^{\prime}\phi^{\prime}\, ] \, |_{r_{h}}^{\infty}
{\cal F}_{m_{1}m_{2}}^{\pm}( x , y , t ) = \frac{i}{{\hbar}^{2}}\frac{1}{({\varepsilon}_{m_{1}, \pm}\cdot \mathrm{s i g n}({\varepsilon}_{m_{1}, \pm}) +{\varepsilon}_{m_{2}, \pm}\cdot \mathrm{s i g n}({\varepsilon}_{m_{2}, \pm}) )^{2}}
A_1^{\dagger}= \frac{1}{\sqrt{6}}\left(a_1\, E_1 + a_2\, E_2 + a_4\, E_4 + a_5 \, E_5\right) = a_1 A_1 + a_2 A_2 + a_4 A_1 A_2 + a_5 A_2 A_1
\left( \tfrac{m}{n}\right)^{-2\alpha}\tfrac{n}{2\alpha+1}\left(\left( \tfrac{m}{n}\right)^{2\alpha+1}-1 \right) + \left( \left( \tfrac{m}{n}\right)^{-\alpha}- 1 \right)^2 \left( \tfrac{n}{2\alpha+1}\right) = \tfrac{n}{2\alpha+1}\left( 1 + \tfrac{m}{n}- 2 \left(\tfrac{m}{n}\right)^{-\alpha}\right).
{\mathfrak b}_\eta(u,v):=\int_{\mathbb T}\varepsilon_\eta^{-1}\bigl(\tfrac{\cdot}{\eta}\bigr)\,{\rm curl}\,u\cdot{\rm curl}\,v+\int_{\mathbb T}u\cdot v,\ \ \ \u,v\in [H^1_{\#}({\mathbb T})]^3,
\frac{d a}{d\tau}=\sum_{i=1}^m n_i q_i^2 \frac{a^2(\tau)}{b_i^2(\tau)}, & \frac{d b_i}{d\tau}= 2 p_i -q_i^2\frac{a(\tau)}{b_i(\tau)}, \,\,\, 1 \leq i \leq m.
0\rightarrow{\rm Im}\,\delta^b_{p+1}=\ker H(i_b)\stackrel{i}{\rightarrow}H^p(\mathcal{A}_b(M))\stackrel{H(i_b)}{\longrightarrow}H^p_b(M)\stackrel{H(L)}{\longrightarrow}H^{p+2}_b(M)\stackrel{\delta^b_{p+2}}{\longrightarrow}{\rm Im}\,\delta^b_{p+2}\rightarrow0.
A_j =\{\underline{y}=(y_1,\ldots,y_n,y_1',\ldots,y_n'):y_k,y_k'\in\{g_r,c_{r+1}x_{i_{r+1}}\}, 1\leq k\leq n, & \mathrm{Card}\{k: y_k=x_{i_{r+1}}\}+\mathrm{Card}\{k: y_k'=x_{i_{r+1}}\}=j \}.
\langle \delta\mathbf{N}, \mathbf{r}_j \rangle = \langle c^i \mathbf{r}_i, \mathbf{r}_j \rangle = c^i = -\langle \mathbf{N}, \delta\mathbf{r}_j \rangle = -\langle \mathbf{N}, u_i \mathbf{N}+ u\mathbf{N}_i \rangle = -u_i,
D_{- 1}\mapsto \left( \begin{array}{r r}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{array}\right) , \qquad D_{0}\mapsto \left( \begin{array}{r r}{{{\textstyle \frac 1 2}}}&{{0}}\\{{0}}&{{-{\textstyle \frac 1 2}}}\\ \end{array}\right) , \qquad D_{1}\mapsto \left( \begin{array}{r r}{{0}}&{{0}}\\{{- 1}}&{{0}}\\ \end{array}\right)
\int ( w(\tau_0) -Q_b ) \phi_{1, b, \beta}\frac{y^2}{2}\rho_\beta dy=\varepsilon_0(\tau_0)\int \phi_{0, b, \beta}\phi_{1, b, \beta}\frac{y^2}{2}\rho_\beta dy+\varepsilon_1(\tau_0)\int \phi_{1, b, \beta}^2 \frac{y^2}{2}\rho_\beta dy+o(b^{\frac{\alpha}{2}}).
\begin{cases}\u({\bf x},t)=u_0({\bf x}) \{\bf x}\in\partial\Omega, \t>0, \d({\bf x}, t)=d_0({\bf x}) \{\bf x}\in \partial B_1^2\times [0,1], \t>0, \frac{\partial d}{\partial z}({\bf x}, t)=0 \{\bf x}\in B_1^2\times \{0, 1\}, \t>0.\end{cases}
A_{p}\! = \! e^{\frac{\phi}{2}}[z_1 \, \gamma_{p}\otimes 1 \! \! 1 + z_{2}\Pi^{(1) j}\, h_{jp}\otimes \sigma_{3}+ z_3 \Pi^{(2) j}\, h_{jp}+ z_4 \Pi^{(3)}_{p}\otimes \sigma_1 + z_5{\tilde \gamma}^{j}h_{jp}\otimes \sigma_2]
\Gamma_{+}^{(1)}= \left(\begin{array}{ccc}1 0 0\end{array}\right), \Gamma_{0}^{(1)}= \left(\begin{array}{ccc}0 1 0\end{array}\right), \Gamma_{-}^{(1)}= \left(\begin{array}{ccc}0 0 1\end{array}\right).
|X_{\ell}(t) - x_{\ell}(t)| \leq H_{\ell}(t) + \sum_{j=0}^{\ell - 1}\frac{\mu^{\ell - j}}{(\ell - j - 1)! s^{\ell - j - 1}}\int_0^t H_j(u) e^{s \ell (t - u)}(1 - e^{-s(t - u)})^{\ell - j - 1}\: du.
{\cal L}U^p(x) + c_pU^p(x) = p\delta' \left(|x|_1 - \frac{d\lambda'}{\delta'}\right)^{2(p-2)}\left( \frac{2p-1}{V}\left(|x|_1 + \frac{d\lambda'}{\delta'}\right) - \frac{1}{2}\left(|x|_1 - \frac{d\lambda'}{\delta'}\right)^2\right)
t P_n^{( q)}(t) = P_{n+1}^{( q)}(t) +[n]_q(1 + q^{n-1})P_{n-1}^{( q)}(t), n= 2,3,4,\dots, & P_0^{(q)}(t)=1, P_1^{( q)}(t) =t, P_{2}^{( q)}(t)=t^2-1.
\{\left\Vert{h}\right\Vert}_{H_{s}^{r}}^{r}={\left\Vert{{\mathbb{E}}(fg)}\right\Vert}_{H_{s}^{r}}^{r}\leq C_{s}^{r}{\mathbb{E}}({\left\Vert{f}\right\Vert}_{H_{s}^{p}}^{r}{\left\Vert{g}\right\Vert}_{H_{s}^{q}}^{r})\leq C_{s}^{r}c_{q}^{r}{\left\Vert{\nu}\right\Vert}_{\ell^{q}(S)}^{r}{\mathbb{E}}({\left\Vert{f}\right\Vert}_{H_{s}^{p}}^{r}).
|| \sqrt{A_x}\, \partial_x w ||^2_{L^2_1}+ || \sqrt{A_z \over \varepsilon\left(z\right)}\, \partial_z w ||^2_{L^2_1}+\int_{\Omega_x}\frac{A_z\left(x,z_\iota\right)}{\varepsilon\left(z_\iota\right)}\partial_z\xi_2^\prime\left(x,z_\iota\right)w\left(x,z_\iota\right)\mathrm{d}x=0.
-{\frac{t_{1}}{t_{3}}}= ( E_{1}^{( B = 0 )})^{- 2}={\frac{1}{\xi_{1}}}, \qquad -{\, \frac{t_{2}}{t_{4}}}= ( E_{2}^{( B = 0 )})^{- 2}={\frac{1}{\xi_{2}}}
d_1 =\frac{\sigma_2+\sigma_1}{\sigma_2-\sigma_1}, d_2 =\sqrt{\frac{\frac{\sigma_2^2-\sigma_1^2}{\sigma_2^2}+2 C_s}{\frac{1}{\sigma_1^2}-\frac{1}{\sigma_2^2}}}\le \sqrt{\frac{\frac{\sigma_2^2-\sigma_1^2}{\sigma_2^2}+2 \bar{C}_s}{\frac{1}{\sigma_1^2}-\frac{1}{\sigma_2^2}}}.
\zeta_{(0,0),m}= 1 \otimes x_m, \zeta_{(1,0),\mu}= \gamma_1 \otimes \xi_{\mu}, \zeta_{(0,1),\mu}= \gamma_2 \otimes \xi_{\mu}, \zeta_{(1,1),m}= \gamma_1 \gamma_2 \otimes x_m,
\begin{array}{l l}{{{[}J_{i}, J_{j}] = i{\epsilon_{i j}}^{k}J_{k}}}&{{i , j , k \, = \, 1 , 2 , 3}}\\{{{[}J_{i}, S_{j}] = i{\epsilon_{i j}}^{k}S_{k}}}&{{}}\\{{{[}S_{i}, S_{j}] = i{\epsilon_{i j}}^{k}J_{k}\, ( C_{2}- 2 \{\vec{J}\,}^{2}) \ \}}&{{}}\\{{{[}C_{2}, J_{i}] = [ C_{2}, S_{i}] = 0}}&{{\mathrm{w i t h ~}\{\vec{J}\,}^{2}= J_{1}^{2}+ J_{2}^{2}+ J_{3}^{2}}}\\ \end{array}
\int_{S}\left| f(s) a(s) + \int_{-\infty}^{\infty}\left(\mathbb{I}_{[-1,1]}(w f(s))- f(s)\mathbb{I}_{[-1,1]}(w) \right)\nu(dw,s)\right| c(ds) < \infty, \int_S|f(s)|^2b(s) c(ds) < \infty, \int_{S}\int_0^{\infty}\min(1, |f(s)w|^2)\nu(dw,s) c(ds) < \infty.
\frac{\sum_{k=0}^n V_k - \frac{\mu_Q}{\mu_{\sigma}}n}{n^{1/2}}\geq \frac{\sum_{i = 1}^{i_n}Z_i}{n^{1/2}}+ \frac{\frac{\mu_Q}{\mu_{\sigma}}(\sigma_{i_n}- \sigma_0 - n)}{n^{1/2}}\equiv (I) + (II)
\frac{1}{h^2}\sum_{d=2m}c^{2m}p_m^2 H_{\rm con}^{1,1}\left(d;(m,m)\right) + \frac{1}{h}\sum_{d=2m-1}c^{2m-1}p_{2m-1}H_{\rm con}^{1,1}\left(d;(2m-1)\right)
\mathrm{\boldmath ~ \g a m m a ~}_{t}( s ) = \frac{d{\mathbf Z}_{t}}{d \ln \kappa}\,{\mathbf Z}_{t}^{- 1}= \left( \begin{array}{c c}{{0}}&{{- g_{B}^{2}\, K_{M m}( s )}}\\{{- g_{B}^{2}\, K_{M m}( s )}}&{{0}}\\ \end{array}\right) \: .
\begin{aligned}B_{n,n-i}= \sum_{j=i}^{n-1}\binom{3n-i-2}{j-i}(-1)^{j+n+1}B_{n,n-j}, i=-n,-n+1,\ldots,n-1, B_{n,n-i}= B_{n,n+i+1}, i=-n,-n+1,\ldots,n-1.\end{aligned}
CKN\left( N,\mu,\theta,s,p,q,r\right) =\left( \frac{N-p}{N-p-\mu}\right)^{\frac{1}{r}+\frac{p-1}{p}-\frac{1-a}{q}-\frac{p-1}{p}\left( 1-a\right)}GN\left( N,p,q,r,\mu,\theta,s\right) .
{\Bigl ( \partial^{2}- 2 i e F \partial -{\frac{B}{\bar{T}}}\Bigl )}^{- 1}={\Bigl ( \partial^{2}- 2 i e F \partial \Bigr )}^{- 1}+{\Bigl ( \partial^{2}- 2 i e F \partial \Bigr )}^{- 1}{\frac{B_{a b}}{\bar{T}}}{\Bigl ( \partial^{2}- 2 i e F \partial \Bigr )}^{- 1}+ \cdots ,
\sum_{t=0}^{k-1}\sum_{j=0}^{\ell-1}m_{t,j}(x)\zeta_{t}(z)\eta_j(y)p_j^{(t)}(x)=a(x,y,z)(x^s-\alpha) + b(x,y,z)(y^\ell-\beta)+c(x,y,z)(z^k-\gamma).
|\psi_l(\mathcal{R})f(x)| =\left|\int_{G}f(xz^{-1})k_{l}(z)dz \right| &=\left|\int_{G}(f(xz^{-1})-f(x))k_{l}(z)dz \right| &\leq \int_{G}|z|^s|k_{l}(z)|dz \Vert f \Vert_{\Lambda^s(G)}.
\mathbf{x}^{(i)}\circ \mathbf{s}^{(i)}=\left({\left(\mathbf{x}^{(i)}\right)}^T \mathbf{s}^{(i)}, \mathbf{x}_1^{(i)}\mathbf{s}_{2:n_i}^{(i)}+\mathbf{s}_1^{(i)}\mathbf{x}_{2:n_i}^{(i)}\right), \forall \mathbf{x}^{(i)},\mathbf{s}^{(i)}\in \mathcal{R}^{n_i}
a = c\, v_{1,\alpha^+}\; v_{-1,\alpha^-}\,\prod_{r=1}^R v_{\tau_r,\alpha^+_r}\; v_{\bar{\tau}_r,\alpha^-_r}\;, b = c\,d\, v_{1,\alpha^+}\; u_{1,\beta^+}\; v_{-1,\alpha^-}\; u_{-1,\beta^-}\;\prod_{r=1}^R v_{\tau_r,\alpha_r}u_{\tau_r,\beta_r}\; v_{\bar{\tau}_r,\alpha_r}u_{\bar{\tau}_r,\beta_r}\,.
T(n_i)+T(n_{i+1})+T(n_{i+2}) <\left( \frac{85}{27}+ \frac{64}{27}\right)\cdot \frac{1}{X_0}= \frac{149}{27}\cdot \frac{1}{X_0}&<8 \cdot \frac{3}{4}\cdot \frac{1}{X_0}= (k_i+k_{i+1}+k_{i+2}) \cdot \frac{3}{4}\cdot \frac{1}{X_0},
\Phi^{i_1\, i_2 \, \dots \, i_k}_{j_1 \, j_2 \, \dots \, j_k}\equiv \mbox{Tr}\left( u^{(i_1}\, v_{(j_1}\, u^{i_2}\, v_{j_2}\, \dots \, u^{i)_k}\, v_{j_k)}\right)
E_{j}^{( b ){\mathrm{t o t}}}= - \left( \frac{k_{D}z_{j}}{z_{2}- z_{1}}\right)^{D}\frac{2^{- D - 1}\pi^{- D / 2}}{\Gamma ( D / 2 + 1 )}{\mathrm{p . v .}}\int_{0}^{\infty}d t \, t^{D}\frac{d}{d t}\ln \left[ 1 - \frac{( B_{1}t + 1 ) ( B_{2}t + 1 )}{( B_{1}t - 1 ) ( B_{2}t - 1 )}e^{- 2 t}\right] .
\sum_{j=2}^t\lambda_{u+j}=\left(\frac{k^2-\theta^{t-3}}{k^2+k-k\theta^{t-3}}\right)\lambda_{u+1}+\left(\frac{k^2-k-1}{k^2+k-k\theta^{t-3}}\right)\left( \frac{1-\theta^{t-2}}{1-\theta}\right)\lambda_{u+1}&=\left( \frac{k^3-k-(k^3-2k^2+2)\theta^{t-3}}{k^2+k-k\theta^{t-3}}\right) \phi^u.
\frac{1}{\pi^2}\int_0^\infty dt\,\frac{\sinh(1+\nu)t}{\sinh t\sinh 2\nu t}\left(\cos\pi\nu\Bigl(\frac{t-\pi i}{\cosh\nu(t-\pi i)}+\frac{t+\pi i}{\cosh\nu(t+\pi i)}\Bigr)-\frac{2t}{\cosh\nu t}\right).
P^B_n(t) = t^n R_{n-1}(1/t) + \sum_{k=1}^{n}P_{k-1}^{A}(t) P^{B}_{n-k}(t) \, , t^n R_{n}(1/t) = P^B_n(t) + \sum_{k=1}^{n}t^{n-k}P_{k-1}^{A}(t)R_{n-k}(1/t) \, , P^A_n(t) = \sum_{k=1}^{n}t^k P^A_{k-1}(1/t) P^A_{n-k}(t) \, ,
{\cal M}_{T}\equiv{\frac{1}{T_{2}}}\left( \begin{matrix}{1}&{T_{1}}\\{T_{1}}&{| T |^{2}}\\ \end{matrix}\right) , \{\cal M}_{U}\equiv{\frac{1}{U_{2}}}\left( \begin{matrix}{1}&{U_{1}}\\{U_{1}}&{| U |^{2}}\\ \end{matrix}\right) .
(\sigma_{D}^{(0)}\cdot\mathrm{Sub}(D^2))(x_0,\xi) =-(\gamma^j\gamma^k\psi+\gamma^j\psi\gamma^k-2\gamma^jb_k)(x_0)\xi_j\xi_k, (\mathrm{Sub}(D^2)\cdot\sigma_{D}^{(0)})(x_0,\xi) =-(\gamma^k\psi\gamma^j+\psi\gamma^k\gamma^j-2b_k\gamma^j)(x_0)\xi_k\xi_j.
A =u^{\frac{e(p^3+p^2+p)}{p+1}}+u^{e(p^2+p)}, B=2u^{ep}-u^{\frac{e(p^3+p^2+p)}{p+1}}+u^{e(p^2+p)}, C =u^{\frac{e(p^3+p^2+2p+1)}{p+1}+r}+u^{e(p^2+p+1)+r}, D =u^{\frac{e(p^2+p+1)}{p+1}}-u^{e(p^2+1)}+u^{e(p+1)+r}+u^{e(p^2+p+1)+r}.
\widetilde{P}(x)=\left[\begin{array}{cc}R_{11}(x)R_{12}(x)-\mu_m R_{11}^2(x) R_{12}^2(x)-\mu_m R_{11}(x)R_{12}(x) -R_{11}^2(x) -R_{11}(x)R_{12}(x)\end{array}\right].
\prod_{v\in S_N}O_\gamma({\bf 1}_{K_v(N)K_{H_v}})\prod_{\substack{v \in S_\gamma v\notin S_N}}O_\gamma({\bf 1}_{K_v}) \ll N^{-\delta}D_{S_N}(\gamma)^{-C}\prod_{v\in S_N}O_\gamma({\bf 1}_{K_v}) \prod_{\substack{v \in S_\gamma v\notin S_N}}O_\gamma({\bf 1}_{K_v}) & = N^{-\delta}D_{S_N}(\gamma)^{-C}\prod_{v\in S_\gamma}O_\gamma({\bf 1}_{K_v}).
\pi(I|X)=\frac{\lambda_I \mathrm{P}_{X,I}}{\mathrm{P}_X}=\frac{\lambda_I \prod_{i=1}^n \phi\big(X_i,\mu_i(I), \sigma^2+\tau_i^2(I)\big)}{\sum_{J\in\mathcal{I}}\lambda_J\prod_{i=1}^n \phi\big(X_i,\mu_i(J), \sigma^2+\tau_i^2(J)\big)}.
S_1 =(-1)^{k-1+\lfloor\frac{k-2}{p}\rfloor}(k-2)!\left((k-1)\binom{q/p-1}{\lfloor (k-2)/p\rfloor}-q\binom{q/p-2}{\lfloor (k-2)/p\rfloor-1}\right) &=(-1)^{k-1+\lfloor\frac{k-2}{p}\rfloor}(k-2)!\left(\frac{q\{k-2\}_p -p(k-2)}{q-p}+1\right)\binom{q/p-1}{\lfloor (k-2)/p\rfloor}.
\eta 2_{2k}(n) = \frac{1}{(2k)!}\sum_{m\in\mathbb{Z}}g_k(m) N2(m,n), \mu 2_{2k}(n) =\frac{1}{(2k)!}\sum_{m\in\mathbb{Z}}g_k(m) M2(m,n), N2_{2k}(n) = \sum_{j=1}^{k}(2j)! S^*(k,j) \eta 2_{2j}(n), M2_{2k}(n) = \sum_{j=1}^{k}(2j)! S^*(k,j) \mu 2_{2j}(n).
\epsilon^{\delta \alpha \beta \gamma}D_{\delta}^{i}( w_{\alpha \ldots \alpha_{m_{1}}}w_{\beta \ldots \beta_{m_{2}}}w_{\gamma \ldots \gamma_{m_{3}}}) = 0 \ \rightarrow \ \ell = 6 +{\frac{1}{2}}( J_{1}+ 2 J_{2}+ 3 J_{3}) .
T_{fg}\equiv \frac{1}{2}(Q_{0})_{f\beta_{0}}(Q_{1})_{\alpha_{1}\beta_{1}}(Q_{2})_{\alpha_{2}\beta_{2}}\cdots (Q_{r})_{\alpha_{r}g}\epsilon^{\beta_{0}\alpha_{1}}\epsilon^{\beta_{1}\alpha_{2}}\cdots \epsilon^{\beta_{r-1}\alpha_{r}}.
\tau=\underset{E_0\subset \Omega, \mathbb P(E_0)=0}{sup}(\underset{\Omega-E_0}{inf}\sigma)\leq \underset{\Omega_{\delta}:open, 0<\mathbb{P}(\Omega_{\delta})\leq \delta\leq \frac{1}{3}}{sup}(\underset{\Omega-\Omega_{\delta}}{inf}\sigma).
\int_{\varphi(\omega)}^{\varphi(\beta)}{}f\circ \theta^H(r) \cos(2\pi |\xi|{b^{-1}}(R+ar) + \nu_d)dr ={}\cos(2\pi |\xi|b^{-1}R + \nu_d) \int_{\varphi(\omega)}^{\varphi(\beta)}f\circ \theta^H(r) \cos(2\pi |\xi|b^{-1}ar)dr &- \sin(2\pi |\xi|b^{-1}R + \nu_d) \int_{\varphi(\omega)}^{\varphi(\beta)}f\circ \theta^H(r) \sin(2\pi |\xi|b^{-1}ar)dr.
\delta_{t}U^{1}_{ij}-\mu\Delta_{h}U^{1}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}U^{1}_{ij}=b^{1}_{ij}+\frac{g(U^1_{ij})+g(U^{0}_{ij})}{2},(x_i,y_j)\in\Omega_{h},
\Phi_1^X(s) :=\Phi\big(a^1,b(s),c_1,d(s);(X(s),s),Du_m(X(s),s),D^2u_m(X(s),s)\big), \Phi_1^x(s) :=\Phi\big(a^1,b(s),c_1,d(s);(x,s),Du_m(x,s),D^2u_m(x,s)\big).
\|g\cdot\eta_n-\eta_n\|_2 \leqslant \|1_{gA_n}-1_{A_n}\|_2+\|1_{gB}-1_B\|_2 &=\sqrt{2|gA_n\setminus A_n|}+\sqrt{2|gB\setminus B|}&\leqslant\sqrt{2|(gA_n\cap B)\setminus A_n)|+2|gB\setminus B|}+\sqrt{2|gB\setminus B|}.
J=\frac{\partial(x_{s+1}^{+},x_{s+1}^{-})}{\partial(x_{s}^{+},x_{s}^{-})}=\frac{\partial x_{s+1}^{+}}{\partial x_{s}^{+}}\frac{\partial x_{s+1}^{-}}{\partial x_{s}^{-}}=(\frac{x_{s+1}^{+}}{x_{s}^{+}})^{2}
\|A\|_\mathrm{F}= \Biggl\|\begin{pmatrix}U^T U_1^T\end{pmatrix}A \begin{pmatrix}W W_1\end{pmatrix}\Biggr\|_\mathrm{F}\geq \Biggl\|\begin{pmatrix}U^T U_1^T\end{pmatrix}AW\Biggr\|_\mathrm{F}\geq \|U^TAW\|_\mathrm{F}.
\frac{t_1t_2}{tz(w+1)}\cdot \frac{1}{H_{12}}= \frac{1}{H_1H_2}\left( 1 - \frac{w+1}{w}\right) - \frac{t_1t_2}{z^2w(w+1)}+ \frac{1}{H_1}\cdot \frac{t_2}{zw}+ \frac{1}{H_2}\cdot \frac{t_1}{zw}\,.
F^{(pr)}(t) =b_p+\left(f(-t)- \sum_{k=1}^{p}b_k\frac{t^{kr}}{(kr)!}\right)^{(pr)}(t) &=b_p+\frac{1}{2\pi i}\int_{1-\infty\,i}^{1+\infty\,i}e^{-tz}(-z)^{pr}\left(\frac{\varphi(z)}{z}- \sum_{k=1}^{p}\frac{(-1)^{kr}b_k}{z^{kr+1}}\right)\,dz,
H^2= \frac{\rho_m^2}{36}+\frac{U_B\rho_m}{12}-\frac{1}{16a^4}\int d\tau\frac{da^4}{d\tau}(\dot\phi^2-2U)-\frac{1}{12a^4}\int d\tau a^4\rho_m \frac{dU_B}{d\tau},
\mathcal{E}_{m}^{2}(u,v,w):=\sum_{|\alpha|=0}^{m}\langle A_{1}^{0}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u\rangle+\sum_{|\alpha|=0}^{m}\langle A_{2}^{0}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u\rangle+\sum_{|\alpha|=0}^{m}\langle A_{3}^{0}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u\rangle.
\left\| \left( p(t)*u_0\right)(-x) - \left( p(t) *u_0\right)(-z) \right\|_{L^k(u_0)}\le \mathbb{E}\left( \left\| u_0(x+X(t)) - u_0(z+X(t))\right\|_{L^k(u_0)}\right) &\le C^{1/k}\|x-z\|^\eta,
\begin{aligned}hv_{-}= hyv_{+}=[h,y]v_{+}+yhv_{+}=-2\ell^s\cdot yv_{+}+y(\mu_{+}v_{+}) +O(n) =(\mu_{+}-2\ell^s)v_{-}+O(n) = (-\ell^s+O(n-\alpha-2s-4v))v_{-}+ O(n) = -\ell^s v_{-}+O(n-\alpha-2s-4v),\end{aligned}