Quantitative linear independence of an infinite product and its derivatives

We establish measures for the rational linear independence of 1 and the values of the product and its derivatives at finitely many rational points, q ≠ 0,±1 being a fixed integer. This is a quantitative improvement upon Bézivin’s very recent result in this journal. In contrast to his procedure, we use the method of Padé approximations of the second kind to get the above-mentioned improvement, some generalizations, and several irrationality measures.     

Note on the rings of integers of certain tame 2-galois extensions over a number field

For a number fieldF, we prove that any tame D4-extensionN/F has a normal integral basis (NIB) if and only if any tame (2, 2)-extensionK/F has a NIB.     

Some divisibilities amongst the terms of linear recurrences

Let (u_n)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesu _n is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)u _n , where τ(n) is the number of the positive integer divisors of n, provided thatu _n satisfies some mild technical conditions.     

Note on limits of finite graphs

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.     

Semigroup-valued solutions of the Gołąb-Schinzel type functional equation

Let (S, o) be a semigroup. We determine all solutions of the functional equation

On functions with the cauchy difference bounded by a functional. III

We are going to discuss special cases of a conditional functional inequality

The rate of convergence of GMRES on a tridiagonal Toeplitz linear system

The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛ X ⁻¹ and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over A’s spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both A’s spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This paper will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind.     

Partitions and functional Santaló inequalities

We give a direct proof of a functional Santaló inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santaló inequality. The argument combines a logarithmic form of the Prékopa-Leindler inequality and a partition theorem of Yao and Yao. Received: 21 August 2008     

On a conditional Gołab-Schinzel equation

Let We show that for every function satisfying the conditional equation

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

The saturation class for linear combinations of Bernstein operators

In this paper we study the saturation class for the linear combinations of Bernstein operators. The characterization of the saturation class involving the modulus of smoothness is proved under certain assumption. Received: 1 September 2007     

The symmetry axiom in Minkowski planes

We give a synthetic proof that in a symmetric Minkowski plane the rectangle axiom (G) holds. Dedicated to Professor Helmut Karzel     

Homogeneous symmetric means of two variables

Summary.  Let be given continuous functions on the interval I such that g ≠ 0, and is strictly monotonic (thus invertible) on I. Taking an increasing nonconstant function μ on [0, 1]

Semidirect products of lattices

The semidirect product of lattices is a lattice analogue of the semidirect product of groups. In this article we introduce this construction, show some basic facts and study a class of lattices closed under semidirect products. We also generalise this notion presenting the semidirect product of semilattices. Received February 22, 2005; accepted in final form August 29, 2006.