Related papers: Univariate amenable functions
We formulate some problems and conjectures about semigroups of rational functions under composition. The considered problems arise in different contexts, but most of them are united by a certain relationship to the concept of amenability.
The article discusses criteria for univalence of analytic functions in the unit disc. Various families of analytic functions depending on real parameters are considered. A unified method for creating new sets of conditions ensuring…
We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary…
In this article, we present univalence criteria for polyharmonic and polyanalytic functions. Our approach yields new a criterion for a polyharmonic functions to be fully $\alpha$--accessible. Several examples are presented to illustrate the…
We characterize left and right amenable semigroups of polynomials of one complex variable with respect to the composition operation. We also prove a number of results about amenable semigroups of arbitrary rational functions. In particular,…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
In this paper we study analogues of amenability for topological groups in the context of definable structures. We prove fixed point theorems for such groups. More importantly, we propose definitions for definable actions and continuous…
Replication of experimental results has been a challenge faced by many scientific disciplines, including the field of machine learning. Recent work on the theory of machine learning has formalized replicability as the demand that an…
We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show…
In the rapidly growing literature on explanation algorithms, it often remains unclear what precisely these algorithms are for and how they should be used. In this position paper, we argue for a novel and pragmatic perspective: Explainable…
Let $N$ be a positive integer. We say a non-constant rational function $U(x)\in{\mathbb C}(x)$ is $N$-\emph{unital} if all the zeros and poles of both $U(x)$ and $1-U(x)$ are either 0 or $N$-th roots of unity. These functions are called…
We derive normal approximation results for a class of stabilizing functionals of binomial or Poisson point process, that are not necessarily expressible as sums of certain score functions. Our approach is based on a flexible notion of the…
The stable matching problem sets the economic foundation of several practical applications ranging from school choice and medical residency to ridesharing and refugee placement. It is concerned with finding a matching between two disjoint…
Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical…
We establish common fixed point theorems for two pairs of weakly compatible self-mappings using an auxiliary function of two variables. Unlike classical results, our theorems do not assume continuity of the mappings and require completeness…
Functional data analysis is a fast evolving branch of statistics. Estimation procedures for the popular functional linear model either suffer from lack of robustness or are computationally burdensome. To address these shortcomings, a…
Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where…
Approximations of functions with finite data often do not respect certain "structural" properties of the functions. For example, if a given function is non-negative, a polynomial approximation of the function is not necessarily also…
A concept of "evolving categories" is suggested to build a simple, scalable, mathematically consistent framework for representing in uniform way both data and algorithms. A state machine for executing algorithms becomes clear, rich and…
Hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years. Recently, unexpected applications of hyperfiniteness were found in computer science in the context of testability…