Ordinary and strong density continuity of complex analytic functions

In the paper we prove that the complex analytic functions are (ordinarily) density continuous. This stays in contrast with the fact that even such a simple function asG:²²,G(x,y)=(x,y ³ ), is not density continuous [1]. We will also characterize those analytic functions which are strongly density continuous at the given pointa . From this we conclude that a complex analytic functionf is strongly density continuous if and only iff(z)=a+bz, wherea, b andb is either real or imaginary.     

On actually computable bijections between ℕ and Q +

We present three different ways of getting an actually computable enumeration of Q ⁺ in the sense of being able to know exactly which rational occupies a given position and vice versa. The first enumeration is based on the Pierce expansion model for representing real numbers. The other two are based on regular continued fractions.The first and third authors were supported by UPF Grant, Support d'Iniciació a la Recerca, # F3087613.     

Infinite analogues of block toeplitz matrices and related orthogonal functions

This paper concerns a class of infinite block matrices that are analogous to finite block Toeplitz matrices. Also studied are corresponding matrix-valued functions that are orthogonal for a matrixvalued inner product. An appendix presents basic results on orthogonalization in a Hilbert module.     

On functions behaving like additive functions

Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Majorization and some operator monotone functions

Let h(t) be a non-decreasing function on I and k(t) an increasing function on J. Then h is said to be majorized by k if k(A)≦k(B) implies h(A)≦h(B). f(t) is operator monotone, by definition, if f(t) is majorized by t. By making use of this majorization we will show that is operator monotone on [0,∞) for 0≦a,b<∞ and for 0≦r≦1; the special case of a=b=1 is the theorem due to Petz-Hasegawa.     

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ) such thatu ^–1 is operator monotone. We then look for another functionv such thatv(u ^–1) is operator monotone, namely,u(A)u(B) impliesv(A)v(B) for self-adjoint operatorsA andB.     

Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals     

The generalized cauchy equation for operator-valued functions

Aequationes mathematicae -     

Measurability implies continuity for solutions of functional equations - even with few variables

Summary. We prove that - under certain conditions - measurable solutions $f$ of the functional equation $f(x)=h(x,y,f(g_{1}(x,y)),\ldots,f(g_{n}(x,y))),\quad(x,y)\in D \subset \mathbb{R}^{s} \times \mathbb{R}^{l}$ are continuous, even if $1\le l\le s$. As a tool we introduce new classes of functions which - roughly speaking - interpolate between continuous and Lebesgue measurable functions. Connection between these classes are also investigated.     

Inequalities for sums of independent geometrical random variables

Summary We give a survey of known results regarding Schur-convexity of probability distribution functions. Then we prove that the functionF(p ₁,...,p_n;t)=P(X₁+...+X_n≤t) is Schur-concave with respect to (p ₁,...,p_n) for every realt, whereX _i are independent geometric random variables with parametersp _i. A generalization to negative binomial random variables is also presented.