Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2006
Volume
Vol. 26
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 26 (2006)
Projects
Pages
Articles
Bounds on the 2-domination number in cactus graphs
(2006) Chellali, Mustapha
A $2$-dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex not in $S$ is dominated at least twice. The minimum cardinality of a $2$-dominating set of $G$ is the $2$-domination number $\gamma_{2}(G)$. We show that if $G$ is a nontrivial connected cactus graph with $k(G)$ even cycles ($k(G)\geq 0$), then $\gamma_{2}(G)\geq\gamma_{t}(G)-k(G)$, and if $G$ is a graph of order n with at most one cycle, then $\gamma_{2}(G)\geqslant(n+\ell-s)/2$ improving Fink and Jacobson's lower bound for trees with $\ell>s$, where $\gamma_{t}(G)$, $\ell$ and $s$ are the total domination number, the number of leaves and support vertices of $G$, respectively. We also show that if $T$ is a tree of order $n\geqslant 3$, then $\gamma_{2}(T)\leqslant\beta(T)+s-1$, where $\beta(T)$ is the independence number of $T$.
Classical solutions of initial problems for quasilinear partial functional differential equations of the first order
(2006) Czernous, Wojciech
We consider the initial problem for a quasilinear partial functional differential equation of the first order $\partial_t z(t,x)+\sum_{i=1}^nf_i(t,x,z_{(t,x)})\partial_{x_i} z(t,x)=G(t,x,z_{(t,x)}),\\ z(t,x)=\varphi(t,x)\;\;((t,x)\in[-h_0,0]\times R^n)$ where $z_{(t,x)}\colon\,[-h_0,0]\times[-h,h]\to R$ is a function defined by $z_{(t,x)}(\tau,\xi)=z(t+\tau,x+\xi)$ for $(\tau,\xi)\in[-h_0,0]\times[-h,h]$. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.
Equitable coloring of graph products
(2006) Furmańczyk, Hanna
A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted by $\chi_{=}(G)$. It is interesting to note that if a graph $G$ is equitably $k$-colorable, it does not imply that it is equitably $(k+1)$-colorable. The smallest integer $k$ for which $G$ is equitably $k'$-colorable for all $k'\geq k$ is called the equitable chromatic threshold of $G$ and denoted by $\chi_{=}^{*}(G)$. In the paper we establish the equitable chromatic number and the equitable chromatic threshold for certain products of some highly-structured graphs. We extend the results from [Chen B.-L., Lih K.-W., Yan J.-H., Equitable coloring of graph products, manuscript, 1998] for Cartesian, weak and strong tensor products.
Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory
(2006) Grabowski, Piotr
The modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [12, 13] two groups of similarity criteria for a given hybrid closed-loop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions.
Rates of convergence for the maximum likelihood estimator in the convolution model
(2006) Majerski, Piotr
Rates of convergence for the maximum likelihood estimator in the convolution model, obtained recently by S. van de Geer, are reconsidered and corrected.