Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2010
Volume
Vol. 30
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 30 (2010)
Projects
Pages
Articles
A study of chaos for processes under small perturbations II - rigorous proof of chaos
(2010) Oprocha, Piotr; Wilczyński, Paweł
In the present paper we prove distributional chaos for the Poincaré map in the perturbed equation $\dot{z}=\left(1 + e^{i\kappa t} |z|^2\right)\bar{z}^2 - N e^{-i\frac{\pi}{3}}.$ Heteroclinic and homoclinic connections between two periodic solutions bifurcating from the stationary solution $0$ present in the system when $N = 0$ are also discussed.
Dominating sets and domination polynomials of certain graphs, II
(2010) Alikhani, Saeid; Peng, Yee Hock
The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x) = \sum _{i=\gamma(G)}^n d(G,i)x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$, and $\gamma (G)$ is the domination number of $G$. In this paper, we obtain some properties of the coefficients of $D(G,x)$. Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by $G^{\prime}(m)$, we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs $G^{\prime}(m)$, we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if $n \equiv 0,2(mod\, 3)$ and $D(G,x) = D(C_n, x)$, then $G = C_n$.
Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener
(2010) Azócar Bates, Luis Antonio; Guerrero, José Atilio; Matkowski, Janusz; Merentes Díaz, Nelson José
We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.
2-splittable and cordial graphs
(2010) Cichacz-Przeniosło, Sylwia
E. Miller and G. E. Stevens proved in [E. Miller, G. E. Stevens, <i>Some graphs for which even size is sufficient for splittability</i>, Congressus Numerantium 173 (2005), 137–147] the existence of certain families of $2$-splittable caterpillars. In this paper we characterize other families of $2$-splittable caterpillars. Moreover, we show that for some of them there exists a friendly labeling inducing two isomorphic subgraphs.
On some properties of the superposition operator on topological manifolds
(2010) Dronka, Janusz
In this paper the superposition operator in the space of vector-valued, bounded and continuous functions on a topological manifold is considered. The acting conditions and criteria of continuity and compactness are established. As an application, an existence result for the nonlinear Hammerstein integral equation is obtained.