Related papers: Measure density for set decompositions and uniform…
In the first part we construct some Buck measurable sets. In the second part we apply the Niven theorem for Buck's measure density to certain sets.
The paper treats density measures as typical examples of finitely additive measures in $\mathbb{R}^n$. We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied…
The study of finite approximations of probability measures has a long history. In (Xu and Berger, 2017), the authors focus on constrained finite approximations and, in particular, uniform ones in dimension $d=1$. The present paper gives an…
We study finitely additive extensions of the asymptotic density to all the subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on $\mathbb{N}$ and…
We investigate some properties of density measures -- finitely additive measures on the set of natural numbers $\N$ extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the…
We apply random matrix theory to study the impact of measurement uncertainty on dynamic mode decomposition. Specifically, when the measurements follow a normal probability density function, we show how the moments of that density propagate…
Consider a positive Borel measure on a locally compact group. We define a notion of uniform density for such a measure, which is based on a group invariant introduced by Leptin in 1966. We then restrict to unimodular amenable groups and to…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
The notion of density of a finite set is discussed. We proof a general theorem of set theory which refines Bose-Einstein distribution.
We present a general approach to the study of the local distribution of measures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize, and simplify a number of recent results on local…
Porosity and dimension are two useful, but different, concepts that quantify the size of fractal sets and measures. An active area of research concerns understanding the relationship between these two concepts. In this article we will…
Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…
Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by…
We deal with finitely additive measures defined on all subsets of natural numbers which extend the asymptotic density (density measures). We consider a class of density measures which are constructed from free ultrafilters on natural…
Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using…
Measures play an important role in the characterisation of various function spaces. In this paper, the structure of density measures will be investigated. These are elements of the dual of the space of essentially bounded func- tions. The…
Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of…
We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first $k$…
Let $D$ be the ring of $S$-integers in a global field and $\hat{D}$ its profinite completion. We discuss the relation between density in $D$ and the Haar measure of $\hat{D}$: in particular, we ask when the density of a subset $X$ of $D$ is…
We introduce and investigate in this short report the new notion of uniform measure (distribution) on the arbitrary compact metric space. We consider also some possible applications of these measures in the theory of imbedding theorems and…