Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2008
Volume
Vol. 28
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 28 (2008)
Projects
Pages
Articles
A note on a family of quadrature formulas and some applications
(2008) Bożek, Bogusław; Solak, Wiesław; Szydełko, Zbigniew
In this paper a construction of a one-parameter family of quadrature formulas is presented. This family contains the classical quadrature formulas: trapezoidal rule, mid-point rule and two-point Gauss rule. One can prove that for any continuous function there exists a parameter for which the value of quadrature formula is equal to the integral. Some applications of this family to the construction of cubature formulas, numerical solution of ordinary differential equations and integral equations are presented.
On a complete lattice of retracts of a free monoid generated by three elements
(2008) Foryś, Wit
We prove that the family of retracts of a free monoid generated by three elements, partially ordered with respect to the inclusion, is a complete lattice.
Application of Chebyshev and trigonometric polynomials to the approximation of a solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane
(2008) Karczmarek, Paweł
In this article Chebyshev and trigonometric polynomials are used to constructan approximate solution of a singular integral equation with a multiplicative Cauchy kernelin the half-plane.
A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices
(2008) Pchelintseva, Irina
We consider self-adjoint unbounded Jacobi matrices with diagonal $q_n = b_{n}n$ and off-diagonal entries $\lambda_n = n$, where bn is a $2$-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of $b_{1}b_{2} = 4$. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail.
On a multivalued second order differential problem with Hukuhara derivative
(2008) Piszczek, Magdalena
Let $K$ be a closed convex cone with the nonempty interior in a real Banach space and let $cc(K)$ denote the family of all nonempty convex compact subsets of $K$. Assume that continuous linear multifunctions $H,\Psi : K \to cc(K)$ are given. We consider the following problem $\begin{aligned}D^2\Phi(t,x) =& \Phi(t,H(x)), D\Phi(t,x)|_{t=0} =& \{0\}, \Phi(0,x) =& \Psi(x)\end{aligned}$ for $t \geq 0$ and $x \in K$, where $D\Phi(t,x)$ denotes the Hukuhara derivative of $\Phi(t,x)$ with respect to $t$.