harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\eta_{( N = 3 )}^{\star}= 0 . 2 7 9 \pm 0 . 0 0 4 \, \, \, \, \, , \, \, \, \, \, \gamma = 1 . 5 9 \pm 0 . 1 0 \, \, \, \, \, , \, \, \, \, \,
\Phi = \tilde{\Phi},~~~\Psi ~= ~{\cal F}~ \tilde{\Psi},~(\tilde{\Psi}~= ~{\cal G}{\Psi}),~~~\Phi^i = e^i_\mu f^\mu_a \tilde{\Phi}^a,~(\tilde{\Phi}^a = f_\mu^a e^\mu_i{\Phi}^i).
E_{M}^{( 1 1 ) \, A}= \left( \begin{matrix}{{e^{-{\frac{\Phi}{3}}}e_{\breve{\mu}}^{( 1 0 ) \, \breve{\alpha}}}}&{{e^{{\frac{2}{3}}\Phi}B_{\breve{\mu}}}}\\{{0}}&{{e^{{\frac{2}{3}}\Phi}}}\\ \end{matrix}\right) ,
\beta^T (f_\theta)[A]\beta = \ \sum_{i,j=1}^{n}\beta_i\beta_j\int_{b/R}^\infty f(a_{ij}y^{-1}) \theta(y)\frac{dy}{y}= \ int_{b/R}^\infty \sum_{i,j=1}^{n}\beta_i \beta_j f(a_{ij}y^{-1})\theta(y)\frac{dy}{y}= \ \ \int_{b/R}^\infty \beta^T f[y^{-1}A] \beta \theta(y)\frac{dy}{y}.
\mu_1 = \mathcal{N}\left( 0, \left[\begin{array}{cc}2 -1 -1 4\end{array}\right] \right), \mu_2 = \mathcal{N}\left( 0, \left[\begin{array}{cc}4 -2 -2 4\end{array}\right] \right), \mu_3 = \mathcal{N}\left( 0, \left[\begin{array}{cc}2 -2 -2 4\end{array}\right] \right).
\mathbb{I}_{\mathrm{ooe}}^{(w)}\coloneqq\left\{(a,b,c)\in\mathbb{Z}_{>1}^{3}\mid(a,b,c)\equiv(1,1,0)\bmod{2},\,a+b+c=w+2\right\}, \mathbb{I}_{\mathrm{eee}}^{(w)}\coloneqq\left\{(a,b,c)\in\mathbb{Z}_{>1}^{3}\mid(a,b,c)\equiv(0,0,0)\bmod{2},\,a\leq b,\,a<c,\,a+b+c=w+2\right\},
\alpha\triangledown_{\mathcal{H}}\beta\overset{{\scriptstyle \mathrm{def}}}{\Leftrightarrow}\mathcal{H}_{\alpha}^{\beta}\in\mathfrak{G}(\mathbf{L})\mathcal{H}_{\beta}^{\alpha}\in\mathfrak{G}(\mathbf{L}),\triangledown := \bigcup_{\mathcal{H}\in\mathfrak{G}(\mathbf{L})}\triangledown_{\mathcal{H}}.
\int_\Omega|r|^2=\alpha^2\int_\Omega|x-x_0|^2\geq \frac{\pi}{2}\alpha^2s_0^4\rho_0^4 \geq\left(\frac{s_0^2}{4M_1}\right)^2\rho_0^2\int_\Omega|\nabla r|^2.
\max f^{-1}(1) =\max\{i\in [n]\mid \forall j\leq i, f(i)\leq f(j)\}&=\max\{i\in [n]\mid \forall j\leq i, g(i)\leq g(j)\}&=\max g^{-1}(1) &=\max(g^{-1}(\sigma(1)),
E \left [ P^{\ast}\left ( d_l^{-1}p^{-1/2}\left \lvert \sum_{j =1}^{pl}( H_{m}(X_j^{\ast})-E^{\ast}[H_{m}(X_j^{\ast})]) \right \rvert > \frac{\epsilon}{4}\frac{2^{K-1}}{\Lambda(+\infty)}\right ) \right ] \leq \C \frac{16}{\epsilon^2}\Lambda(+\infty)^2 2^{-2K+2}\leq \C l^{-2}p^{-1}d_l^2.
[T_{jk}(\lambda),T_{jk}(\mu)] = 0,j,k=1,2, A(\mu) B(\lambda) = f(\lambda,\mu) B(\lambda) A(\mu) + g(\mu,\lambda) B(\mu) A(\lambda), B(\mu) A(\lambda) = f(\lambda,\mu) A(\lambda) B(\mu) + g(\mu,\lambda) A(\mu) B(\lambda), D(\mu) B(\lambda) = f(\mu,\lambda) B(\lambda) D(\mu) + g(\lambda,\mu) B(\mu) D(\lambda), &
g(\sigma^1,\sigma^2):=\sum_{i=1}^n\sigma^1_i\Phi^2_i=\sum_{i=1}^n\Phi^1_i\sigma^2_i=\frac{1}{2\Delta x^2}\sum_{(i,j)\in E}g_{ij}(\rho)(\Phi_i^1-\Phi_j^1)(\Phi_i^2-\Phi_j^2),
f_{D 4}\sim \frac{1}{\sqrt{| x_{1}x_{2}x_{3}|}}e^{- \frac{| x_{1}x_{2}x_{3}|}{y z}}\quad \mathrm{o r}\quad \frac{1}{\sqrt{| x_{1}x_{2}x_{3}|}}e^{- \frac{| x_{1}x_{2}x_{3}|}{y \bar{z}}}
\lower 1pc\hbox to .8\linewidth{\vbox{\rightskip=.2\linewidth\noindent\textit{For $(i_1,\ldots,i_n),(j_1,\ldots,j_n)\in\ell K_i$, $\ell\geq1$, $\epsilon\geq0$, and $a\in I^{2n\ell+\epsilon}$, there is some $b\in R$ such that $ac_1^{i_1}\cdots c_n^{i_n}=bc_1^{j_1}\cdots c_n^{j_n}$.}}}
H=\frac{t}{2a}\sum_x\left( e^{iA}S^+(x+1)S^-(x)+e^{-iA}S^-(x+1)S^+(x)\right)+\frac{e_L^2a}{2}\sum_{x,y}S^3(x)V(x-y)S^3(y)+e_L^2E^2/2N
\sum_{i=0}^{j-1}\beta_i -j\beta_j = (\beta-\alpha)(\frac{j}{2j+\gamma}- \frac{j(j+1)}{2j+\gamma+2}+ \frac{j^2}{2j+\gamma}) = \frac{2j(j+1)(\beta-\alpha)}{(2j+\gamma)(2j+\gamma+2)}
i \frac{\partial F^{( \pm )}}{\partial t}= - \frac{1}{2}\frac{\partial^{2}F^{( \pm )}}{\partial q^{2}}+ \frac{q^{\prime}\pm q}{i t}\frac{\partial F^{( \pm )}}{\partial q^{\prime}}+ V ( q^{\prime}) F^{( \pm )},
D(P\|Q) \leq \frac{1}{2}\left( \frac{(1+\eta) \, \log(1+\eta)}{\eta}- (1-\eta) \log e \right)\, |P-Q| & \leq \frac{1}{2}\Bigl( (1+\eta) \log e - (1-\eta) \log e \Bigr) \, |P-Q| & = \bigl( \eta \log e \bigr) |P-Q|.
S_{0}= \sum_{l}{\frac{1}{2 \sigma_{l}^{2}}}\phi_{l}^{a}\phi_{- l}^{a}+ \sum_{l}{\frac{1}{2 \tau_{l}^{2}}}f_{l}^{a}f_{- l}^{a}+ \sum_{r}{\frac{1}{g_{r}}}\bar{\psi}_{r}^{a}\psi_{r}^{a}\, .
\mathbf{P}\Big( \{U^*_{n}(\theta_{n},c^{*}_{n})\neq\emptyset \}\cap \{\mathfrak{W}^*(c_{\pi^*})= \emptyset \}\Big)\le \eta/2, &\mathbf{P}\Big( \{U^*_{n}(\theta_{n},c^{*}_{n})= \emptyset \}\cap \{\mathfrak{W}^*(c_{\pi^*})\neq \emptyset \}\Big)\le \eta/2,
T_{m}(f_1,...,f_m)(x) =\int_{\mathbb R^{nm}}\frac{f_1(y_1)...f_m(y_m)}{(1+|y_1|^n+...+|y_m|^n)^m}dy_1...dy_m &=\int_{\mathbb R^n}...\int_{\mathbb R^n}\frac{f_1(|x|y_1)...f_m(y_m)}{(1+|y_1|^n+...+|y_m|^n)^m}dy_1...dy_m.
\|\pi(g)\|^{2}= \int\limits_{\Omega_{0}}|(g\circ \xi^{-1})(s)|^{2}d\nu(s) = \int\limits_{\Omega_{0}}|g(\xi^{-1}(s))|^{2}d\nu(s) &= \int\limits_{\Omega}|g(t)|^{2}d\mu(t) &= \|g\|^{2}.
\int_{\Omega}-\chi_t\eta-\chi_x q\,dtdx =\int_{\Omega}\chi Cdtdx+\mu\int_{\Omega}\chi_{xx}\eta dtdx &-\mu\int_{\Omega}\chi(\eta_{ll}(\partial_xl)^2+2\eta_{lm}\partial_x l\partial_x m+\eta_{mm}(\partial_x m)^2 )dtdx.
\frac{1}{\beta}log(\int_{\mathbb{R}}|g(y)|^{2\beta}dy)\le\frac{1}{2}log\frac{(2\alpha)^{1/\alpha}}{(2\beta)^{1/\beta}}+\frac{1}{\alpha}log(\int_{\mathbb{R}}|f(x)|^{2\alpha}dx).
\mathbb{E}\bigg[\big|\mathbf{h}_{n,k}\mathbf{w}_{n,k}\big|^{2}\bigg]=\mathbb{E}\bigg[\big|\sum\limits_{i=1}^{Z_{n}}(\mathbf{\Lambda}_{ii}+\sigma_{k}^{2})^{-1}|\boldsymbol{\delta}_{n,k,i}|^{2}\big|^{2}\bigg],
\sum_{|\alpha|=m, |\beta| = j,\beta \leq \alpha, |\gamma|={m-j-k},\gamma\leq \alpha-\beta}{\alpha \choose \beta~\gamma}a_\beta b_{\gamma}c_{\alpha -\beta-\gamma}& \qquad\leq{m\choose j~k}\left( \sum_{|\beta|=j}a_\beta \right) \left( \sum_{|\gamma|=k}b_\gamma \right) \left( \sum_{|\alpha|=m-j-k}c_\alpha \right).
\sum_{m=1}^M \sum_{j(\neq i_m)}^N \frac{z_{i_m}\, z_j}{z_{i_m j}\, z_{j i_m}}\, r_{i_m,j}\,= \frac{1}{2}\, \sum_{m=1}^M (z_{i_m}\, \partial_{z_{i_m}})^2- \frac{N}{2}\sum_{m=1}^M z_{i_m}\, \partial_{z_{i_m}}+ \frac{1}{12}\, M \, (N^2 - 1) \, ,
\langle P^{\prime}| J^{\mu}( 0 ) | P \rangle = \bar{u}( P^{\prime}) \, \Big [ \, F_{1}( q^{2}) \gamma^{\mu}+ F_{2}( q^{2}){\frac{i}{2 M}}\sigma^{\mu \alpha}q_{\alpha}\, \Big ] \, u ( P ) \ ,
C(1_{\lambda})=\bigg(-\frac{1}{8}+\sum_{i=1}^{k-1}\lambda(h_i)^2+2\sum_{i=1}^{k-1}\rho_{e,0}(h_i)\lambda(h_i)+4\sum\limits_{i=1}^{k-1}\delta(h_i)\lambda(h_i)\bigg)(1_{\lambda}).
{\{\pi_i,\pi_j\}}_{D(\Phi)}=gG_{ij}^a(x)I^a+ig\left(\nabla_k^{ab}G_{ij}^b(x)\right)I^a\xi_k({\cal P}_m\xi_m)\frac{(b\kappa+a)}{\tilde \beta(\omega+\tilde m)}=gG_{ij}^a(q)I^a_\varphi,
0=S_{ij}= 2\langle d\phi(e_i),d\phi(e_j)\rangle-g_{ij}|d\phi|^2+\frac{1}{2}\langle\psi,e_i\cdot\tilde\nabla_{e_j}\psi+e_j\cdot\tilde\nabla_{e_i}\psi\rangle -\frac{1}{6}g_{ij}\langle R^N(\psi,\psi)\psi,\psi\rangle.
\varphi_{\mathrm{rd}}(n+m;x_{0},\omega)=\varphi_{\mathrm{rd}}(n;\varphi_{\mathrm{rd}}(m;x_{0},\omega),\theta^{m}(\omega)),\quad\forall n,m\in\mathbb{N},\;\forall x_{0}\in\mathbb{R}^{d},\;\forall\omega\in\Omega_{0}.
\delta_{n}( f) (\lambda_0, \ldots, \lambda_{n}) = \f(\lambda_1, \ldots, \lambda_{n}) + (-1)^{n+1}f(\lambda_0, \ldots, \lambda_{n-1}) \cdot \lambda_{n}& + \sum_{i=1}^{n}(-1)^i f(\lambda_0, \ldots, \lambda_{i-1}\lambda_i, \ldots, \lambda_{n})
T_{{}^Q\nabla^\pm}(q_1,q_2)= -T_{\overline{\nabla}^\pm}(q_1,q_2), R_{{}^Q\nabla^\pm}(q_1,q_2)(q_3)= R_{\overline{\nabla}^\pm}(q_1,q_3)q_2-R_{\overline{\nabla}^\pm}(q_2,q_3)q_1 &+\overline{\nabla}^\pm_{q_3}T_{\nabla^\pm}(q_1,q_2)-T_{\overline{\nabla}^\pm}(\overline{\nabla}^\pm_{q_3}q_1,q_2)-T_{\overline{\nabla}^\pm}(q_1,\overline{\nabla}^\pm_{q_3}q_2).
C_{n}=(\begin{matrix}n\\ 3\end{matrix})+(\begin{matrix}n\\ 2\end{matrix})+(\begin{matrix}n\\ 1\end{matrix})+(\begin{matrix}n\\ 0\end{matrix})=\frac{1}{6}(n^{3}+5n+6)=\frac{1}{6}(n+1)(n(n-1)+6).
\frac{2}{\left( \alpha+m\right) \lambda}\mathcal{E}_{n+1}^{\left(\alpha+m\right)}\left( m;\lambda\right) =\frac{\left( 2\lambda^{-1}\right)^{m+1}}{\left( m+1\right) !}\binom{\alpha+m}{m+1}^{-1}\sum_{k=1}^{m+1}\left( -1\right)^{k}\genfrac{[}{]}{0pt}{}{m}{k-1}\mathcal{E}_{n+k}^{\left( \alpha\right)}\left( \lambda\right) .
y_- := \max_{k>0}-{{\textstyle{\frac{1}{k}}}}(1 + \lfloor y_1^{\textsc{d}}k \rfloor +(\lceil y_{1+}^{\textsc{g}}k \rceil - 1) ), y_+ := \min_{k>0}-{{\textstyle{\frac{1}{k}}}}(-1 + \lceil y_1^{\textsc{d}}k \rceil +(\lfloor y_{1-}^{\textsc{g}}k \rfloor + 1) ),
\varepsilon^{( 1 )}( x ) = \frac{g_{\varepsilon}\lambda^{2}}{4 \pi c \tau_{S}}\, \frac{\cos ( \sqrt{\omega_{S}^{2}/ c^{2}+ 1 / \lambda^{2}}\, | \vec{x}- \vec{x}_{S}| - \omega_{S}t )}{| \vec{x}- \vec{x}_{S}|}\, .
P_0=\int\!d\,^3x T^{00}_{e^+e^-}= \int\!d\,^3x[\overline{\Psi}(i \gamma_i\partial_i-e\gamma_i A_i +m)\Psi+e\overline{\Psi}\gamma_0 A_0\Psi] =\int\!d\,^3x \overline{\Psi}i\gamma_0 \partial_0\Psi\ ,
\langle G \vert J_{G B}^{i}\vert G \rangle = - \epsilon^{i l m}\int d^{3}x \langle G \vert x^{l}( i \partial_{( x )}^{m}[ \delta^{3}({\bf x - y}) ] ) \vert G \rangle = - \epsilon^{i l m}\int d^{3}x \left( i \partial_{( x )}^{m}[ \delta^{3}({\bf x - y}) ] \right) x^{l}
I(t_{0}^{\alpha}u(t_{0}\cdot),t_{0}^{\alpha}v(t_{0}\cdot)) <\mathop{\lim}\limits_{n\rightarrow\infty}\Big(\frac{1}{2}a(u_{n},v_{n})+\frac{1}{2}b(u_{n},v_{n})+\frac{1}{2}c(u_{n},v_{n})-\frac{1}{2p}F(u_{n},v_{n})\Big) &=c_{b},
\frac{1}{2}\, \sum_{m=1}^M (z_{i_m}\, \partial_{z_{i_m}})^2 + 2 \, \sum_{m < m'}^M \sum_{m\neq m'}^M \frac{z_{i_m}\, z_{i_{m'}}}{z_{i_m i_{m'}}\, z_{i_{m'}i_m}}- \frac{N}{2}\sum_{m=1}^M z_{i_m}\, \partial_{z_{i_m}}= E
\sum_{j_1+\cdots +j_m=n}\binom{n}{j_1,\ldots, j_m}B_{j_1}(x_1)\cdots B_{j_m}(x_m)=B_n(m;x) &=\sum_{k=0}^n \dfrac{s(m+k,m)}{k! \binom{m+k}{m}}\Delta^k I_n(x).
\begin{aligned}0 \geq F^{11}\left(\frac{\lambda_1^2}{40K}- 2\psi'^2\delta^{-1}K\right) +\left(\frac{1}{2}\psi'' + C_\epsilon\psi'\right) F^{kk}|u_k|^2 &- C_0 \mathcal{F}+ \epsilon C_0 \psi'\mathcal{F}-\psi' F^{kk}(\chi_{k\bar k}- g_{k\bar k}), \end{aligned}
\sum_{n_1 , ... , n_m = 0}^\infty \| V_{n_{m+1}}(A_{m+1};z_{m+1},z'_{m+1}) \cdots T_1(\zeta_1) V_{n_1}(A_1;z_1 , z_1') \Psi \| < \infty , & \sum_{n_1 , ... , n_m = 0}^\infty \| T_{m+1}(\zeta_{m+1}) V_{n_{m+1}}(A_{m+1};z_{m+1},z'_{m+1}) \cdots T_1(\zeta_1) V_{n_1}(A_1;z_1 , z_1') \Psi \| < \infty .
r(1,1)=(1,1), r(1,2)=(2,1), r(1,3)=(3,2), &r(2,1)=(1,2), r(2,2)=(2,2), r(2,3)=(3,1), &r(3,1)=(2,3), r(3,2)=(1,3), r(3,3)=(3,3),
\lim_{n\to\infty}\sup_{x\in\mathbb{R}}P\left(\left|\sup_{t\in[a_n,T-a_n]}|Z_n(t)|-x\right|\leq c_1v_n+8\varepsilon\right)=O(v_n\sqrt{\log n})+O\left(\frac{\log n}{(nh)^{1/6}}\right).
\frac{1}{h_n}\|Z_\nu^{h_n}(t)- X^*(t)-h_n Y_\nu(t))\|_{L^p}\leq (a_n^{(1)}+a_n^{(2)}+a_n^{(3)}) &+(C_x+C_m) \int_{s+h_n}^t\frac{1}{h_n}\|Z_\nu^{h_n}(\tau)-X^*(\tau)-h_nY_\nu(\tau))\|_{L^p}d\tau.
C_{\theta}( \tau ) = \langle{\cal A}( \tau ){\cal B}\rangle_{0}- \lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal A}}){\cal A}\rangle_{0}\lim_{\tau^{\prime \prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime \prime}}({\scriptstyle{\cal B}}){\cal B}\rangle_{0}\stackrel{\tau \rightarrow \infty}{\longrightarrow}
\mathbf{M}=\begin{pmatrix}a b c d\end{pmatrix}=\begin{pmatrix}\alpha\, p_k(r_1,\dots, r_k) \alpha\, p_{k-1}(r_1,\dots,r_{k-1}) \beta \, p_{k-1}(r_2,\dots, r_k) \beta\, p_{k-2}(r_2,\dots,r_{k-1})\end{pmatrix},
\nabla_w \cdot(\mu Q_h \textbf{v})={\cal Q}_h \nabla \cdot(\mu\textbf{v}), \textbf{v}\in H({\rm div}_\mu;\Omega), \nabla_w \times (Q_h \textbf{v})= \textbf{Q}_h (\nabla \times\textbf{v}), \textbf{v}\in H({\rm curl};\Omega).
\zeta_{\cal D}( s ) = \sum_{j = - \infty}^{\infty}\sum_{N = 0}^{\infty}{\frac{( N + 1 )^{2}}{[ ({\frac{2 \pi}{\beta}}j + i \mu )^{2}+{\frac{N ( N + 2 ) + 6 \xi}{a^{2}}}+ m^{2}]^{s}}}
u=a,v=u+\frac E{\sqrt{E-V}},p=\frac VE, D_{Y_{u,v,p}^2}=p u^2 +q v^2 - u^{2 p}v^{2q}, V_{Y_{u,v,p}^2}=pq(v-u)^2=V,E_{Y_{u,v,p}^2}=q(v-a)^2=q(v-u)^2=E,
\left[ a_{i}( K ) , a^{j}( L ) \right] = \delta_{i}^{j}\delta_{K L}\, , \qquad \left[ b_{i}( K ) , b^{j}( L ) \right] = \delta_{i}^{j}\delta_{K L}\, , \qquad \left[ c_{i}, c^{j}\right] = \delta_{i}^{j}\quad \mathrm{( i f ~ p r e s e n t )}\, ,
\vert \eta (B_n) - B_E( \eta) \vert = \left| \int_{E \cap R_n}\eta \circ f(t) d \mu (t) - \int_E \eta \circ f(t) d \mu (t) \right| \leq \int_{E\backslash R_n}| \eta \circ f(t)| \mu (t) \leq \| \eta \| \int_{\Omega \backslash R_n}\| f(t) \| d \mu (t) \leq \| \eta \| \int_{\Omega \backslash R_n}g(t) d \mu (t) . \stepcounter{equation}
\begin{array}{c c l}{( 0 0 - \frac{7}{2})}&{\rightarrow}&{( 0 - \frac{7}{2}) ( + \frac{7}{2}) \oplus ( 0 - \frac{5}{2}) ( + \frac{3}{2}) ,}\\{( 0 0 + \frac{3}{2})}&{\rightarrow}&{( 0 + \frac{5}{2}) ( - \frac{7}{2}) \oplus ( 0 + \frac{3}{2}) ( - \frac{3}{2}) .}\\ \end{array}
1-a = \dfrac{r(r-p-q)_p - (r-p)i+ q j}{r(r-p-q)_p}\ge \dfrac{r(r-p-q)_p - (r-p)(r-p-q)_p}{r(r-p-q)_p}= \frac{p}{r}> 0, 1+b = \dfrac{(r(r-p-q)_p)_q + (r-q)_p j -i}{(r(r-p-q)_p)_q}\ge \dfrac{(r(r-p-q)_p)_q -(r-p-q)_p}{(r(r-p-q)_p)_q}= \dfrac{r-q}{r}> 0.
\Psi_1 ={\kappa{p_u}\sum\limits_{i \ne n}^N{{{\left|{{\bf{a}}_n^H{{\bf{g}}_i}}\right|}^2}}\!+ \!{{\left\|{{{\bf{a}}_n}}\right\|}^2}\!+ \!\rho{p_u}{\bf{a}}_n^H{}\left({{\bf{G}}{{\bf{G}}^H}}\right){{\bf{a}}_n}}.
\begin{array}{lll}\displaystyle \int_{\mathbb{R}^2 \backslash B_{R_0}(0)}A(x)^r v^{2m}\leq C_5\left\|v\right\|^{2m}+ C_6\left\|v\right\|^{2r/\beta_0}\left\|v\right\|^{2(m\beta_0-r)/\beta_0}&= C_7\left\|v\right\|^{2m},\end{array}
W_{\mu}^{T}= W_{\mu}^{F}+ W_{\mu}^{G}= \int d^{4}\! x \, t r \big ( F_{\mu \nu}( x ){\frac{\delta}{\delta A_{\nu}( x )}}+ D_{\nu}A_{\mu}( x ){\frac{\delta}{\delta A_{\nu}( x )}}\big ) ,
w_{2p}'(t) \,\le\, -\frac{C}{2p}\min_{\beta = 2,4}\biggl\{\Bigl( \frac{w_{2p}(t)}{w_p(t)}\Bigr)^{\beta p}\biggr\}w_{2p}(t) + \alpha^2 \Bigl(1 + \frac{p}{2}M^2\Bigr)w_{2p}(t)~, t > 0~.
\int_{0}^{R}y(r)r^{-1}\,dr =\iint_{P(R)}w(x,t)\left(\int_{((|x|-1)^2+t)/R}^{\infty}\left[ \eta^*\left(s\right) \right]^{2p'}s^{-1}\,ds\right)\,dx\,dt &\leq (\log 2)\iint_{P(R)}w(x,t) \psi_R(x,t)\,dx\,dt.
c_0 = \frac{\beta}{2}\left(\frac{1}{12}- \zeta'(-1) \right) + \frac{\gamma}{6\beta}-\frac{\log2\pi}{4}- \frac{(\beta/2)}{2}\hphantom{c_0 =}{}+\left(\frac{17}{8}- \frac{25}{24}(\beta/2 + 2/\beta )\right)\log 2 +\int_{0}^{\infty}\frac{1}{e^{\beta t/2}- 1}\left(\frac{t}{e^t - 1}-1 +\frac{t}{2}-\frac{t^2}{12}\right)dt,
\frac{1}{4z(s)}=\frac{1}{2\sigma^2+\mathcal{O}(h^2+s\sigma^4)}&=\frac{1}{2\sigma^2}\cdot\frac{1}{1+\mathcal{O}(h^2\sigma^{-2}+s\sigma^2)}&=\frac{1}{2\sigma^2}+\mathcal{O}(h^2\sigma^{-4}+s)
\begin{array}{rcl}[v_1,z_{r_k}] = t_{r_{k,1}}, \left[v_2,z_{r_k}\right] = t_{r_{k,2}}, \mbox{and}\left[v_3,z_{r_k}\right] = -(t_{r_{k,1}}+t_{r_{k,2}}) \end{array}
\frac{\partial^2 \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_c \partial a_c}>0 ~{\rm and}~ \frac{\partial^2 \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_i \partial a_i}>0, ~ i=1,2, \ldots, K
\begin{aligned}\frac{d}{dt}\int(|\nabla^N n|^2+|\nabla^{N+1}n|^2)dx+\frac{N+2}{1+t}\int(|\nabla^{N}n|^2+|\nabla^{N+1}n|^2)dx &\lesssim \left(\frac{N+2}{1+t}\right)^2 \int(|\nabla^{N-1}n|^2+|\nabla^{N}n|^2)dx+(1+t)^{-(N+3)}&\lesssim (1+t)^{-(N+\frac{5}{2})}+(1+t)^{-(N+3)}&\lesssim (1+t)^{-(N+\frac{5}{2})}.\end{aligned}
a(n;i,j) =a(n-1;i,j),\qquad\mbox{if $1\leq i<j\leq n-2$}, a(n;n-1,i) =a(n-1;i),\qquad\mbox{if $1\leq i\leq n-2$}, a(n;i,i-1) =a(n-1,i-1),\qquad\mbox{$2\leq i\leq n$}, a(n;i,i+1) =a(i-1),\qquad\mbox{$1\leq i\leq n-1$}, a(n;i,j)+w_{i-3,j-1}=a(n;i+1,j),\qquad\mbox{$1\leq j<i-1<n-1$}, a(n;n-1,n) =a(n-2), a(n;n;i) =a(n-1,i),\qquad\mbox{if $1\leq i\leq n-1$}.
MT_{\sf d}^{KM(KN+KM)}+N(T_{\sf d}+1)^{KM(KN+KM)}+(T+1)^{KN(KN+KM)}&\leq (M+N)(T_{\sf d}+1)^{KM(KN+KM)}+(T+1)^{KN(KN+KM)}&=\frac{\lambda_2}{\lambda_1}\left(1+\frac{N}{M}\right)T^{KN(KN+KM)}+(T+1)^{KN(KN+KM)}&\leq \frac{1}{\lambda_1}(T+1)^{KN(KM+KN)},
\delta H ( x^{m}) = \frac{3 \alpha}{3 2 \pi^{4}a^{6}}\lbrace \ln ( \frac{R^{2}+ a^{2}}{R^{2}- a^{2}}) + 2 \tan^{- 1}( \frac{R^{2}}{a^{2}}) - \pi \rbrace .
I:=\int_{-\infty}^{\infty}dx^{\prime}\int_{-\infty}^{\infty}dy^{\prime}\frac{e^{ik_{0}|r-r^{\prime}|}}{|r-r^{\prime}|}=\int_{0}^{2\pi}d\varphi^{\prime}\int_{0}^{\infty}d\rho^{\prime}\frac{e^{ik_{0}R}}{R}=2\pi\int_{0}^{\infty}d\rho^{\prime}\frac{e^{ik_{0}R}}{R}.
3|l_{2}(\zeta_{1})|^{2}\leq\bar{l}_{2}(\zeta_{1})\|\zeta_{1}\|^{2}&3|\kappa_{2}(\zeta_{2})|^{2}\leq\bar{\kappa}_{2}(\zeta_{2})|\zeta_{2}|^{2}&3|\omega_{2}(\psi)|^{2}\leq\bar{\omega}_{2}(\psi)\|\psi\|^{2}.
K_{w}( t_{1}, t_{2}, \cdots t_{l}) = \prod_{j = 1}^{l}\Gamma ( - \beta_{j})^{- 1}\prod_{p = 1}^{N}\prod_{j = 1}^{n}\{P_{w}^{j}( x^{( p )}, z_{p}; t_{1}, t_{2}, \cdots , t_{l}) \}^{\mu_{j}^{( p )}}\prod_{j = 1}^{l}t_{j}^{- \beta_{j}- 1},
\rho_O(\mathcal D_O) =\begin{cases}\lbrace \iota(\gamma^\vee) \mid \gamma \in \Pi \setminus \Gamma^\perp \rbrace \ \sigma \notin 2\Pi; \lbrace \iota(\alpha^\vee)/2 \rbrace \cup \lbrace \iota(\gamma^\vee) \mid \gamma \in \Pi \setminus (\Gamma^\perp \cup \lbrace \alpha \rbrace) \rbrace \ \sigma = 2\alpha \in 2\Pi.\end{cases}
I_3 =C_{0}\int_{\left(\bar B_{r_2}(x_1)\setminus B_{r_1}(x_1)\right)\cap\Omega}\frac{v^p (y)}{|x_1-y|^2}\mathrm{d}y &\leq C_{0}\left(\frac{1}{M}\int_\Omega \ln\frac{1}{|x-y|}v^p(y)\mathrm{d}y \right)\frac{M}{r_1^2 \ln \frac{1}{r_2}}&\leq C_4 M^{1-\frac{2}{n}}N^{\frac{2p}{n}}q^{\frac{2}{n}-1}.
K_{\frac72}(x) = \frac{5}{x}K_{\frac52}(x)+K_{\frac32}(x) &= \frac{5}{x}\left( \frac{3}{x^2}+\frac{3}{x}+1 \right)K_{\frac{1}{2}}(x) +\left(\frac{1}{x}+1 \right)K_{\frac12}(x) = \left( \frac{15}{x^3}+\frac{15}{x^2}+\frac{6}{x}+1 \right) K_{\frac12}(x), x \neq 0.
B_{11}= P_5, B_{12}+B_{12}^T+A_{11}=P_4, B_{13}+B_{22}+B_{13}^T+A_{12}+A_{12}^T = P_3, B_{23}+B_{23}^T + A_{13}+A_{22}+A_{13}^T = P_2, B_{33}+A_{23}+A_{23}^T = P_1, \mbox{and}A_{33}= P_0,
(q_1,\dots,q_J) =\left(a\delta^Q_1,\dots,a\delta^Q_{J-1},a\left(1-\sum_{j=1}^{J-1}\delta^Q_j\right)\right), (e_1,\dots,e_{J-1}) =\left((L-a)\delta^{E}_1,\dots,(L-a)\delta^{E}_{J-1}\right),
\Xi_{1}( \kappa ) = \lim_{N \to \infty}\mu_{1 , N}^{1 1}( \kappa ) = \lim_{N \to \infty}\left[ \frac{\sinh \frac{\pi \kappa}{2}}{\sinh \frac{\pi N \kappa}{4}}e^{\frac{\pi N \kappa}{4}}- e^{\frac{\pi \kappa}{2}}\right] = - e^{- \frac{\pi | \kappa |}{2}}.
|C|V_q(n,d-1) \leq q^k \cdot (1+d-1) \binom{n}{\left\lfloor \frac{n}{2}\right\rfloor}q^{d-1}& = d \binom{n}{\left\lfloor \frac{n}{2}\right\rfloor}q^{k+d-1}& < q^{k+d}& \leq q^{n}.
\mathfrak{B}_{\mathcal{E}_3(q)}=\left \{\epsilon_I, \epsilon_I b_1, \epsilon_I b_2, \epsilon_I b_1 b_2 ,\epsilon_I b_2 b_1 , \epsilon_I b_1 b_2 b_1 \ \vert \I \in P(3) \right \}.
Sc\left(\left(f*{{\rm P}}^l\right)\left(0,0\right)\right)=\int_{\widehat{G^2}}{{{\Phi}_1({\alpha}^{-l}_i\left(\omega_i\right)){\Phi}_2({\alpha}^{-l}_j\left(\omega_j\right))\left|{\mathcal{F}}_r\left(f\right)\left(\omega_i,\omega_j\right)\right|}^2{d^2}_{{\mu}_{\widehat{G^2}}}\left(\omega_i,\omega_j\right)},
x(t) = \left\lbrace \begin{array}{rcl}\left( \frac{2}{3}\left(- t + \frac{3}{2}x_0^{\frac{2}{3}}\right) \right)^{\frac{3}{2}}\mbox{for}\left[0, \frac{3}{2}x_0^{\frac{2}{3}}\right] 0 \mbox{for}\left(\frac{3}{2}x_0^{\frac{2}{3}}, \infty \right) \end{array}\right.
\frac{d}{dt}\lambda_{j}^{2}\ln |\ln\lambda_{j}|=2\frac{d\lambda_{j}}{\lambda_{j}^{2}ds}\lambda_{j}\ln| \ln \lambda_{j}| (1+\delta(\alpha))=2\left(1+\delta(\alpha)\right)\frac{d\lambda_{j}}{\lambda_{j}ds_{j}}\ln |\ln \lambda_{j}|.
|\partial_{t}^{2}\left(\frac{16}{\lambda(t)^{2}}\int_{t}^{\infty}\lambda''(x) K_{1}(x-t,\lambda(t)) +\frac{16}{\lambda(t)^{2}}\int_{t}^{\infty}\lambda''(x) K(x-t,\lambda(t))\right)| &\leq \frac{C}{t^{3}\log^{1-2b}(t)}
\|D\Theta_1(\varphi).v\|_s \le c_s\left[\|D_{.1}F^{\omega}(\varphi,T\varphi)\|_s\|v\|_{s_0}+\|D_{.1}F^{\omega}(\varphi,T\varphi)\|_{s_0}\|v\|_s+\right. &\quad+\left.\|D_{.2}F^{\omega}(\varphi,T\varphi)\|_s\|Tv\|_{s_0}+\|D_{.2}F^{\omega}(\varphi,T\varphi)\|_{s_0}\|Tv\|_s\right]\le &\le K_s^1(\varphi)\|v\|_{s+1},
L \left( X_{\kappa}, \Theta_{\kappa}, A_{\kappa}\right) = L \left( X , \Theta , A \right) - \frac 1 2 L_{M N}\left[ \Omega_{g}^{M N}- \Omega^{M N}\left( \Theta \right) \right] - \frac 1 2 \left( A_{\kappa}^{i j}- A^{i j}\right) X_{i}\cdot X_{j}.
\sum\limits_{k=-\infty}^{n-1-\ell}K\exp\Big(-\sum\limits_{p=k+1}^{n}a_{p}\Big)\Delta_{k}+\sum\limits_{k=n+\ell}^{\infty}K\exp\Big(-\sum\limits_{p=n}^{k+1}a_{p}\Big)\Delta_{k}<\frac{\varepsilon}{2}\big(1-\frac{\theta}{3}\big),
\frac{\partial W_{M_n,n}}{\partial t}- \overline b(t) (-\Delta )^\beta W_{M_n,n}+a_0{\bf \overline R}_{n,\beta}W_{M_n,n}= \overline F_{Q_n}(W_{M_n,n}(x,t)) +\overline g_{M_n,n}(x,t).
U_{\geq 3}^5 :=[|\partial_x|,\partial_xT_{\partial_x\omega}](\widetilde{h}-h)+i|\partial_x|^{1/2}[T_{|\partial_x|^3\widetilde{h}}P_{\geq 1}\widetilde{h}-T_{|\partial_x|^3h}P_{\geq 1}h]+|\partial_x|M_2(\omega,h-\widetilde{h}),
C ( g , q ) = -{\frac{g \phi_{0}}{( g + q )}}-{\frac{g^{2}q}{2 ( g + q )}}\int_{0}^{\infty}d k C ( g , k ) [{\frac{1}{( k^{2}+ q^{2}+ k q )}}-{\frac{1}{2}}{\frac{1}{( k^{2}+ q^{2}- k q )}}] .
\ell(\tau) \leq \eta^{k}< \left( \frac{1}{2^k}\right)^{\alpha + \delta}\leq \left( \frac{2 \tau}{\tau_0}\right)^{\alpha + \delta}= \left( \frac{2^{\alpha + \delta}}{\tau_0^{\alpha + \delta}}\tau^\delta \right) \tau^\alpha.
\begin{array}{l}\bar{b}_{1}= \bar{b}_{3}, \qquad \bar{b}_{2}=\bar{b}_{4}, \qquad\bar{a}_{1}=\bar{a}_{3},\qquad \bar{a}_{2}=\bar{a}_{4}, \vspace{-2mm}d_{1}=d_{3}, \qquad \bar{d}_{2}=\bar{d}_{4}, \qquad c_{1}=c_{3},\qquad \bar{c}_{2}=\bar{c}_{4}.\end{array}
E[T_3]=I_{F}^2\frac{1}{2}\operatorname{tr}\bigl(V^{-2}\bigr) (A+D_i)+O\bigl(m^{-2}\bigr)=\frac{2(A+D_i)}{\operatorname{tr}(V^{-2})}+O\bigl(m^{-2}\bigr).
\rho_{n - 1}(s, X_{n - 2}(s, x)) - \rho_{n - 2}(s, X_{n - 2}(s, x)) &= \rho_{n - 1}(s, X_{n - 2}(s, x)) - \frac{\rho_0(x)}{1-\sigma_2s\rho_0(x)}= \rho_{n - 1}(s, X_{n - 2}(s, x)) - \rho_{n - 1}(s, X_{n - 1}(s, x)).
\lim_{h\to0}\frac{1}{h}p_{ii}(t,t+h,u,u+h) =\lim_{h\to0}\frac{1}{h}p_{ii}(t,t+h,u) &=\lim_{h\to0}\frac{1}{h}(p_{ii}(t,t+h,u)-1)+\frac{1}{h},
\Gamma_{\rm N, \{\nu_j \}}= \frac{\hbar}{2}\left( - \frac{\lambda}{4}\right)^{\rm N_1 + N_2}\left[\Phi \cdots\Phi \right]_{\star_{\rm N_1}}\left( - P \right) \widetilde{{\cal K}}_{{\rm N}- d/2}\left( P \circ P + 1/ \Lambda^2 \right) \left[ \Phi \cdots\Phi \right]_{\star_{\rm N_2}}(+ P ) ~ ,
g_5(C_1,C_2,q,\alpha_\eta) =g_0(C_1,C_2,q)+I_{\alpha_\eta,q}(X_1;X_2|U) &=C_1+C_2-(1-q) h_2(\alpha_\eta)-q+2h_2\left(\alpha_\eta\star\frac{q}{2}\right)-1.
J'_g(t,r,\xi) := \gamma_0 (\gamma_0 - 1) t \int_0^1 \big( \mathbf 1_{\gamma(\cdot) = \gamma_0}\kappa \cdot (\phi \eta_{ut})^{\gamma_0 - 1}g\big( ut,ru^{\frac{1}{\gamma_0 - 1}}, \cdot \big)^{\gamma_0 - 1}\big) (\xi_{(1-u)t}) du.
\int_{\lambda I_Q}\|R_j\partial_jg(\cdot,t)\|_{L^1(2\lambda Q_1)}dt \lesssim\int_{\lambda I_Q}\ell(Q)^{N/p}\|R_j\partial_jg(\cdot,t)\|_{L^q(2\lambda Q_1)}dt &\lesssim\int_{\lambda I_Q}\ell(Q)^{N/p}\|\partial_jg(\cdot,t)\|_{L^q(2\lambda Q_1)}dt\lesssim \ell(Q)^{N/p}\ell(Q)^{N/q}=\ell(Q)^N.
G_N(\omega)(x) - G_N(\omega)(y) =\,{DF_N(\omega)(x^*)}^{-1}[ DF_N(\omega)(x^*)(x-y) - (F_N(\omega)(x) - F_N(\omega)(y)) ] =\,{DF_N(\omega)(x^*)}^{-1}[DF_N(\omega)(x^*) - R_N(\omega)(x,y)] (x - y),
S_{l - e}= \int d^{2}x \sqrt{- G}e^{- 2 \Phi}\big \{R + 4 ( \nabla \Phi )^{2}+ \alpha^{2}-{\frac{1}{2}}e^{\epsilon \Phi}F^{2}-{\frac{1}{2}}( \nabla T )^{2}+ T^{2}\big \}.
\begin{aligned}\int_{B_{R}^{+}\backslash B_{r_{0}}^{+}}\eta^6|\nabla\omega|^{2}dxdy &=\frac{1}{2}\int_{B_{R}^{+}\backslash B_{r_{0}}^{+}}\omega^{2}(\Delta\eta^6+\boldsymbol{w}\cdot\nabla\eta^6)dxdy+\frac12\int_{B_{R}^{+}\backslash B_{r_{0}}^{+}}\nabla\eta^{6}\cdot(ay,b)\omega^{2}dxdy.\end{aligned}
\psi_{m}( r ) = r^{| m |}\,_{1}F_{1}\left({\frac{2 - g}{4}}+{\frac{| m | - m}{2}}-{\frac{\epsilon R^{2}}{2 \delta}}, 1 + | m | ,{\frac{\delta r^{2}}{R^{2}}}\right) e^{-{\frac{\delta r^{2}}{2 R^{2}}}}\, .
\Pi_{1 B}~ = ~ \frac{2 \pi^{2 \mu}}{( \mu - 1 )^{2}\Gamma^{2}( \mu )}\left[ 1 - \frac{\eta_{1}}{N}\left( \frac{2}{( \mu - 1 )}- \frac{3 \mu ( 2 \mu - 3 )}{2}\left( \Theta + \frac{1}{( \mu - 1 )^{2}}\right) \right) \right]
H^{k}_{\pi}(M)=\dfrac{ker \left(\delta_{\pi}: \mathcal{X}^k(M)\longrightarrow \mathcal{X}^{k+1}(M)\right)}{Im \left(\delta_{\pi}: \mathcal{X}^{k-1}(M)\longrightarrow \mathcal{X}^k(M)\right)}.
\kappa^{2}c^{4}f^{2}\left( \frac{E}{\kappa c^{2}}, \frac{\vec{P}^{\, 2}}{\kappa^{2}c^{2}}\right) - c^{2}\vec{P}^{\, 2}g^{2}\left( \frac{E}{\kappa c^{2}}, \frac{\vec{P}^{\, 2}}{\kappa^{2}c^{2}}\right) \ = \i n v . \ = \ \mu^{2}c^{4}.