arXiv subjects
Publications and source records attributed to Robert F. Tichy.
In this paper we consider an optimization problem for Cesàro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.
Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the study of decompositions of lacunary polynomials and the finiteness criterion for the equations of type f(x) = g(y).
Encouraged by the study of extremal limits for sums of the form $$\lim_{N\to\infty}\frac{1 }{N}\sum_{n=1}^N c(x_n,y_n)$$ with uniformly distributed sequences $\{x_n\},\,\{y_n\}$ the following extremal problem is of interest $$\max_γ\int_{[0,1]^2}c(x,y)γ(dx,dy),$$ for probability measures $γ$ on the unit square with uniform marginals, i.e., measures whose distribution function is a copula. The aim of this article is to relate this problem to combinatorial optimization and to the theory of optimal transport. Using different characterizations of maximizing $γ$'s one can give alternative proofs of some results from the field of uniform distribution theory and beyond that treat additional questions. Finally, some applications to mathematical finance are addressed.
In the present paper the authors construct normal numbers in base $q$ by concatenating $q$-adic expansions of prime powers $\lfloorαp^θ\rfloor$ with $α>0$ and $θ>1$.
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units. This has been investigated for several types of rings related to global fields, most importantly rings of algebraic integers. We also state some open problems and conjectures which we consider to be important in this field.