The holomorphic automorphism group of the complex disk

The group of all holomorphic automorphisms of the complex unit disk consists of Möbius transformations involving translation-like holomorphic automorphisms and rotations. The former are calledgyrotranslations. As opposed to translations of the complex Plane, which are associative-commutative operations forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations forming agyrogroup.     

Optimization over analytic functions whose founrier coefficients are constrained

This article gives necessary and sufficient conditions for local solutions to several very general constrained optimization problems over spaces of analytic functions.The results presented here have many applications, a particular instance of which is the sup-norm approximation of functions continuous on the unit circle in the complex plane by functions continuous on the circle and analytic on the open disk and whose Fourier coefficients satisfy prescribed linear relations.Also, the results in this article generalize Nevanlinna-Pick and Caratheodory-Fejer Interpolation results to allow values of arbitrary derivatives of functions to be assigned or merely bounded. Classically, NP and CF solve only problems with consecutive derivatives specified.In engineering, constraints on the Fourier coefficients of a frequency response function correspond to constraints on its time domain behavior. Indeed the central problems of control theory involve both time and frequency domain constraints. That is precisely what the results in this paper handle.Supported in part by the AFOSR and the NSF     

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     

Finely continuously differentiable functions

In Lávi?ka [A remark on fine differentiability, Adv. Appl. Clifford Algebras 17 (2007) 549–554], it is observed that finely continuously differentiable functions on finely open subsets of the plane are just functions which are finely locally extendable to usual continuously differentiable functions on the whole plane. In this note, it is proved that, under a mild additional assumption, this result remains true even in higher dimensions. Here the word “fine” refers to the fine topology of classical potential theory.     

Room designs and one-factorizations

The existence of a Room square of order 2n is known to be equivalent to the existence of two orthogonal one-factorizations of the complete graph on 2n vertices, where orthogonal means any two one-factors involved have at most one edge in common. DefineR(n) to be the maximal number of pairwise orthogonal one-factorizations of the complete graph onn vertices.The main results of this paper are bounds on the functionR. If there is a strong starter of order 2n–1 thenR(2n) 3. If 4n–1 is a prime power, it is shown thatR(4n) 2n–1. Also, the recursive construction for Room squares, to obtain, a Room design of sidev(u – w) +w from a Room design of sidev and a Room design of sideu with a subdesign of sidew, is generalized to sets ofk pairwise orthogonal factorizations. It is further shown thatR(2n) 2n–3.     

Hurwitz rational functions

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establish an analog of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants.     

On Darboux points of real functions

The nature and relationship of the sets of Darboux points and continuity points of real function are studied.     

Collections of paths and rays in the plane which fix its topology

A collection $\text{F}$ of proper maps into a locally compact Hausdorff space X fixes the topology of X if the only locally compact Hausdorff topology on X which makes each element of $\text{F}$ continuous and proper is the given topology. In I²=[-1, 1]×[-1, 1], neither the collection of analytic paths nor the collection of regular twice differentiable paths fixes the topology. However, in I², both the collection of C^∞ arcs and the collection of regular C¹ arcs fix the topology. In $\text{I}{}^2\text{=[?1,1]×[?1,1], the collection of polynomial rays together with any collection of paths does not fix the topology. However, in}\text{R}{}^2$, the collection of regular injective entire rays together with either the collection of C^∞ arcs or the collection of regular C¹ arcs fixes the topology.     

Riordan group involutions

We study involutions in the Riordan group, especially those with combinatorial meaning. We give a new determinantal criterion for a matrix to be a Riordan involution and examine several classes of examples. A complete characterization of involutions in the Appell subgroup is developed. In another direction we find several examples that generalize the RNA matrix but are of independent interest.     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

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.