Numer czasopisma
Opuscula Mathematica
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ISSN: 1232-9274
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Data wydania
2006
Rocznik
Vol. 26
Numer
No. 1
Prawa dostępu
Dostęp: otwarty dostęp
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Prawa: CC BY 4.0 
Uznanie autorstwa 4.0 Międzynarodowe (CC BY 4.0)
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Opis
Rocznik czasopisma (rel.)
Rocznik czasopisma
Opuscula Mathematica
Vol. 26 (2006)
Artykuły numeru (rel.)
Artykuł
Dostęp ograniczony
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 γ2(G). We show that if G is a nontrivial connected cactus graph with k(G) even cycles (k(G) ≥ 0), then γ2(G) ≥ γt(G) - k(G), and if G is a graph of order n with at most one cycle, then γ2(G) ≥ (n + l - s)/2 improving Fink and Jacobson's lower bound for trees with l > s, where γt(G), l 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 ≥ 3, then γ2(T) ≤ β(T) + s - 1, where β(T) is the independence number of T.
Artykuł
Dostęp ograniczony
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 ∂tz(t, x) +Xn i=1 fi(t, x, z(t,x))∂xi z(t, x) = G(t, x, z(t,x)), z(t, x) = ϕ(t, x) ((t, x) ∈ [−h0, 0] × R n ) where z(t,x) : [−h0, 0] × [−h, h] → R is a function defined by z(t,x)(τ, ξ) = z(t + τ, x + ξ) for (τ, ξ) ∈ [−h0, 0]×[−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.
Artykuł
Dostęp ograniczony
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 χ=(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' ≥ k is called "the equitable chromatic threshold" of G and denoted by χ*=(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 [2] for Cartesian, weak and strong tensor products.
Artykuł
Dostęp ograniczony
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.
Artykuł
Dostęp ograniczony
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.
Artykuł
Dostęp ograniczony
A note on arbitrarily vertex decomposable graphs
(2006) Marczyk, Antoni
A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n₁, . . . , nk) of positive integers such that n₁ + . . . + nk = n there exists a partition (V₁, . . . , Vk) of the vertex set of G such that for each i ∈ {1, . . . , k}, $V_i$ induces a connected subgraph of G on ni vertices. In this paper we show that if G is a two-connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non- -adjacent vertices is at least n − 3, then G is arbitrarily vertex decomposable. We present another result for connected graphs satisfying a similar condition, where the bound n − 3 is replaced by n − 2.
Artykuł
Dostęp ograniczony
Bipartite embedding of (p, q)-trees
(2006) Orchel, Beata
A bipartite graph G = (L, R; E) where V (G) = L∪R, |L| = p, |R| = q is called a (p, q)-tree if |E(G)| = p + q − 1 and G has no cycles. A bipartite graph G = (L, R; E) is a subgraph of a bipartite graph H = (L ′ , R′ ; E ′ ) if L ⊆ L ′ , R ⊆ R ′ and E ⊆ E ′ . In this paper we present sufficient degree conditions for a bipartite graph to contain a (p, q)-tree.
Artykuł
Dostęp ograniczony
Proof of Milman's theorem on extension of M-basic sequence
(2006) Pličko, Anatolij Nikolaevič
We prove Milman's theorem on the extension, in a given direction, of M-basic sequence to M-basis in a separable Banach space.
Artykuł
Dostęp ograniczony
Some analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations. Part 1
(2006) Pytel-Kudela, Marzena; Prykarpatski, Anatolij
The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bilinear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.
Artykuł
Dostęp ograniczony
The Delsarte-Darboux type binary transformations, their differential-geometric and operator structure with applications. Part 1
(2006) Prikarpats'kìj, Ârema A.; Samojlenko, Anatolij Mihajlovič
The structure properties of multidimensional Delsarte-Darboux transmutation operators in parametric functional spaces are studied by means of differential-geometric and topological tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutative differential operator expressions related via a Delsarte-Darboux transformation and having a lot of applications in soliton theory.