Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2009
Volume
Vol. 29
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 29 (2009)
Projects
Pages
Articles
On some quadrature rules with Gregory end corrections
(2009) Bożek, Bogusław; Solak, Wiesław; Szydełko, Zbigniew
How can one compute the sum of an infinite series $s := a_1 + a_2 + \ldots$? If the series converges fast, i.e., if the term $a_{n}$ tends to $0$ fast, then we can use the known bounds on this convergence to estimate the desired sum by a finite sum $a_1 + a_2 + \ldots + a_n$. However, the series often converges slowly. This is the case, e.g., for the series $a_n = n^{-t}$ that defines the Riemann zeta-function. In such cases, to compute $s$ with a reasonable accuracy, we need unrealistically large values $n$, and thus, a large amount of computation. Usually, the $n$-th term of the series can be obtained by applying a smooth function $f(x)$ to the value $n$: $a_n = f(n)$. In such situations, we can get more accurate estimates if instead of using the upper bounds on the remainder infinite sum $R = f(n + 1) + f(n + 2) + \ldots$, we approximate this remainder by the corresponding integral $I$ of $f(x)$ (from $x = n + 1$ to infinity), and find good bounds on the difference $I - R$. First, we derive sixth order quadrature formulas for functions whose 6th derivative is either always positive or always negative and then we use these quadrature formulas to get good bounds on $I - R$, and thus good approximations for the sum $s$ of the infinite series. Several examples (including the Riemann zeta-function) show the efficiency of this new method. This paper continues the results from [W. Solak, Z. Szydełko, <i>Quadrature rules with Gregory-Laplace end corrections</i>, Journal of Computational and Applied Mathematics 36 (1991), 251–253] and [W. Solak, <i>A remark on power series estimation via boundary corrections with parameter</i>, Opuscula Mathematica 19 (1999), 75–80].
On an evolution inclusion in non-separable Banach spaces
(2009) Cernea, Aurelian
We consider a Cauchy problem for a class of nonconvex evolution inclusions in non-separable Banach spaces under Filippov-type assumptions. We prove the existence of solutions.
A note on Radon-Nikodym derivatives and similarity for completely bounded maps
(2009) Gheondea, Aurelian; Kavruk, Ali Şamil
We point out a relation between the Arveson's Radon-Nikodym derivative and known similarity results for completely bounded maps. We also consider Jordan type decompositions coming out from Wittstock's Decomposition Theorem and illustrate, by an example, the nonuniqueness of these decompositions.
Continuous solutions of iterative equations of infinite order
(2009) Kapica, Rafał; Morawiec, Janusz
Given a probability space $(\Omega,\mathcal{A}, P)$ and a complete separable metric space $X$, we consider continuous and bounded solutions $\varphi: X \to \mathbb{R}$ of the equations $\varphi(x) = \int_{\Omega} \varphi(f(x,\omega))P(d\omega)$ and $\varphi(x) = 1-\int_{\Omega} \varphi(f(x,\omega))P(d\omega)$, assuming that the given function $f:X \times \Omega \to X$ is controlled by a random variable $L: \Omega \to (0,\infty)$ with $-\infty \lt \int_{\Omega} \log L(\omega)P(d\omega) \lt 0$. An application to a refinement type equation is also presented.
A note on the p-domination number of trees
(2009) Lu, You; Hou, Xinmin; Xu, Jun-Ming
Let $p$ be a positive integer and $G=(V(G),E(G))$ a graph. A $p$-dominating set of $G$ is a subset $S$ of $V(G)$ such that every vertex not in $S$ is dominated by at least $p$ vertices in $S$. The $p$-domination number $\gamma_p(G)$ is the minimum cardinality among the p-dominating sets of $G$. Let $T$ be a tree with order $n \geq 2$ and $p \geq 2$ a positive integer. A vertex of $V(T)$ is a $p$-leaf if it has degree at most $p - 1$, while a $p$-support vertex is a vertex of degree at least $p$ adjacent to a $p$-leaf. In this note, we show that $\gamma_p(T) \geq (n + |L_p(T)|-|S_p(T)|)/2$, where $L_{p}(T)$ and $S_{p}(T)$ are the sets of $p$-leaves and $p$-support vertices of $T$, respectively. Moreover, we characterize all trees attaining this lower bound.