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author | Norbert Preining <norbert@preining.info> | 2021-04-13 03:00:56 +0000 |
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committer | Norbert Preining <norbert@preining.info> | 2021-04-13 03:00:56 +0000 |
commit | 32940266507ccd0c4ba129c61e8932734370e283 (patch) | |
tree | b26a8b13953b035ea6e12d2d46ad0488c9636d06 /macros/latex/contrib/aomart/aomsample1.tex | |
parent | 72c29070e5c574da1a734cb90214754b2f3aa698 (diff) |
CTAN sync 202104130300
Diffstat (limited to 'macros/latex/contrib/aomart/aomsample1.tex')
-rw-r--r-- | macros/latex/contrib/aomart/aomsample1.tex | 18 |
1 files changed, 12 insertions, 6 deletions
diff --git a/macros/latex/contrib/aomart/aomsample1.tex b/macros/latex/contrib/aomart/aomsample1.tex index 29257baead..30a52e5691 100644 --- a/macros/latex/contrib/aomart/aomsample1.tex +++ b/macros/latex/contrib/aomart/aomsample1.tex @@ -421,7 +421,8 @@ D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix} \section{Application} \label{lincomp} -We consider here the applications of \fullref{Theorems}{th-info-ow-ow} \fullref{and}{th-weak-ske-owf} to a complete +We consider here the applications of \fullref{Theorems}{th-info-ow-ow} +\fullref{and}{th-weak-ske-owf} on \fullpageref[page]{th-weak-ske-owf} to a complete multipartite graph $K_{n_1\dots n_p}$. It can be shown that the number of spanning trees of $K_{n_1\dots n_p}$ may be written @@ -928,7 +929,8 @@ achieve the general result using \thmref{t:conl}. \begin{step} Assume that $n=1$. Since $S_u$ is at most countable, \eqref{sum-bij} -yields that $\abs{\wt{D}v}(S_u\backslash S_v)=0$, so that \eqref{e:st} and \eqref{e:barwq} imply that $Dv=\wt{D}v+Jv$ is +yields that $\abs{\wt{D}v}(S_u\backslash S_v)=0$, so that~\eqref{e:st} +and \eqref{e:barwq} imply that $Dv=\wt{D}v+Jv$ is the Radon-Nikod\'ym decomposition of $Dv$ in absolutely continuous and singular part with respect to $\abs{\wt{D} u}$. By \thmref{th-weak-ske-owf}, we have @@ -956,7 +958,8 @@ and \hat v(t)=f(\hat u(t))\qquad\forall t\in\mathbf{R}.\end{equation} Let $t\in\mathbf{R}$ be such that $\abs{\wt{D}u}(\interval{\left[t,s\right[})>0$ for every $s>t$ and -assume that the limits in \eqref{joe} exist. By \eqref{j:mark} and \eqref{far-d} we get +assume that the limits in \eqref{joe} exist. By \eqref{j:mark} +and~\eqref{far-d} we get \begin{equation*}\begin{split} \frac{\hat v(s)-\hat v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}&=\frac {f(\hat @@ -1017,7 +1020,8 @@ for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. We claim that }}(y+t\nu)=\frac{\wt{D}u_y} {\abs{\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R} \end{equation} -for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$. In fact, by \eqref{sum-ali} and \eqref{delta-l} we get +for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$. In fact, +by~\eqref{sum-ali} and \eqref{delta-l} we get \begin{multline*} \int_{\pi_\nu}\frac{\wt{D}u_y}{\abs{\wt{D}u_y}}\cdot\abs{\wt{D}u_y }\,d\mathcal{H}_{n-1}(y)=\int_{\pi_\nu}\wt{D}u_y\,d\mathcal{H}_{n-1}(y)\\ @@ -1041,7 +1045,7 @@ u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(y+t\nu))-f(\tilde u(y+t\nu))}{h} =\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(y+t\nu)\] -for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$, and using again \eqref{detK1}, \eqref{detK2} we get +for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$, and using again~\eqref{detK1}, \eqref{detK2} we get \[ \lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde @@ -1120,7 +1124,9 @@ A^{(\lambda)}_l =\sum_{I_l \subseteq\mathbf{n}}\per \mathbf{A} \end{thm} It is worth noting that $A_l ^{(\lambda)}$ of \eqref{A-l-lambda} is -similar to the coefficients $b_l $ of the characteristic polynomial of \eqref{bl-sum}. It is well known in graph theory that the coefficients +similar to the coefficients $b_l $ of the characteristic polynomial +of~\eqref{bl-sum}. It is well known in graph theory that the +coefficients $b_l $ can be expressed as a sum over certain subgraphs. It is interesting to see whether $A_l $, $\lambda=0$, structural properties of a graph. |