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-rw-r--r--macros/latex/contrib/aomart/aomsample1.tex18
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.