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diff --git a/Master/texmf-dist/source/latex/elsarticle/contrib/mytools.tex b/Master/texmf-dist/source/latex/elsarticle/contrib/mytools.tex new file mode 100644 index 00000000000..f0c6f841739 --- /dev/null +++ b/Master/texmf-dist/source/latex/elsarticle/contrib/mytools.tex @@ -0,0 +1,83 @@ +\Clear{426.79134pt}{% +\let\la\lambda +\noindent\textbf{Remark 3.}\enspace +We remark that; when the ratio $h/\la$ tends to 0, the expression $\la +L(r,s)=-(s-r)/(4\left(\frac{h}{\la}\right)^2+(r-s)^2)$ tends to +$1/(r-s)$ which is a singular function. This means that the expression +$\la L(r,s)$ is not well behaved for the small values of +$h/\la$. Consequently, for the solution to converge, the integrals of +(10) and (11) must be evaluated with a large number of +nodes. In our numerical applications (cf. section 5), we +use 100 nodes to evaluate these integrals. With the smallest value of +$h/\la=0.02$, the convergence is good with $N=20$. +} +\Clear{426.79134pt}{% +\noindent\textbf{Theorem 2.}\enspace\itshape +For system (8), consensus can be achieved with +$\|T_{\omega z}(s)\|_{\infty}<\gamma$ if there exist a symmetric +positive definite + matrix $P\in \mathcal{R}^{(n-1)\times (n-1)}$ and a scalar $\mu>0$ satisfying +\setcounter{equation}{9} +\begin{eqnarray}\label{10} +\Gamma=\begin{bmatrix} +-\bar{L}^TP-P\bar{L}+U_1^TU_1+\mu \bar{E}&PU_1^TE_1&PU_1^T\\ +E_1^TU_1P&-\mu I&0\\ U_1P&0&-\gamma^2I\end{bmatrix}<0, +\end{eqnarray} +where $\bar{L}=U_1^TLU_1$ and $\bar{E}=U_1^TE_2^TE_2U_1.$ +} +\Clear{426.79134pt}{% +\noindent\textbf{Proof of Theorem 2.}\enspace Proof follows +straightforward from Lemma 3 and Theorem 1. However, it should be +emphasized that all possible $\bar{L}_{\sigma(t)}$ should share a +common Lyapunov function $V(\delta)=\delta^T(t)P\delta(t)$ (see +the proof of Lemma 3 in Appendix A). \hfill$\square$ +} +\Clear{426.79134pt}{% + \begin{enumerate}[1.] + \item The enumerate environment starts with an optional + argument `1.' so that the item counter will be suffixed + by a period. + \item If you provide parentheses to the number, the + output will have only one parentheses for all the item + counters. + \item You can use `(a)' for alphabetical counter and '(i)' for + roman counter. + \begin{enumerate}[a)] + \item Another level of list with alphabetical counter. + \item One more item before we start another. + \begin{enumerate}[(i)] + \item This item has roman numeral counter. + \item Another one before we close the third level. + \end{enumerate} + \item Third item in second level. + \end{enumerate} + \item All list items conclude with this step. +\end{enumerate} +} +\Clear{284.52756pt}{% +\lmrgn=4em + \begin{enumerate}[Step 1.] + \item This is the first step of the example list. +\item Obviously this is the second step. +\item The final step to wind up this example. + \end{enumerate} +} +\Clear{284.52756pt}{% +\centering + {\ttfamily\bs includegraphics[width=3in,angle=45]\lb + tiger.pdf\rb}\\ + \hspace*{-1cm} + \includegraphics[width=3in,angle=45,origin=c]{tiger.pdf}\\ + \raggedright + \textbf{Fig.~1.}~~More details on the usage of {\ttfamily\bs + includegraphics} can be found in the \textsf{grfguide.ps} of the + \LaTeX{} documentation. +} +\Clear{426.79134pt}{% +\raggedright +\begin{enumerate}[{[1]}] +\item Knuth, D.E., \emph{TeX: The Program}, Computers \& +Typesetting; B., 1995, Addisson-Wesley Publishing Co., Inc., New +York. +\end{enumerate} +} |