Related papers: A memo on bornologies and size functions

Let $G$ be a countable group. We study left-invariant metrics on $G$ that are not necessarily proper, introducing the notion of a \emph{bornological metric}: a metric $\rho$ such that for every $C>0$ there exists $S_C>0$ with the property…

Successive divisions of compact metric spaces appear in many different areas of mathematics such as the construction of self-similar sets, Markov partitions associated with hyperbolic dynamical systems, dyadic cubes associated with a…

The paper proposes a vector generalization of the basic concepts of the theory of complex variable: the concept of modulus and argument of complex number. The author introduces some generalizations of the notion of holomorphic functions and…

I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects (e.g., spacetimes) living in infinite-dimensional…

Uniform measures are defined as the functionals on the space of bounded uniformly continuous functions that are continuous on bounded uniformly equicontinuous sets. If every cardinal has measure zero then every countably additive measure is…

A method of constructing an entire function with given zeros and estimates of growth is suggested. It gives a possibility to describe zero sets of certain classes of entire functions of one and several variables in terms of growth of volume…

In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set X which takes values in a suitable non-Archimedean field, and satisfies the same formal properties of finite…

This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure…

In decision-making, maxitive functions are used for worst-case and best-case evaluations. Maxitivity gives rise to a rich structure that is well-studied in the context of the pointwise order. In this article, we investigate maxitivity with…

With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions…

We propose a novel method for the description of spatial patterns formed by a coverage of point sets representing galaxy samples. This method is based on a complete family of morphological measures known as Minkowski functionals, which…

Uniform measures are the functionals on the space of bounded uniformly continuous functions that are continuous on every bounded uniformly equicontinuous set. This paper describes the role of uniform measures in the study of convolution on…

In this short note it is shown that all invariant metrics and functions of bounded $\mathcal C^2$-smooth domain coincide on an open non-empty subset.

Some type-based approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls…

In the paper we are dealing with metric measure spaces of diameter at most one and of total measure one. Gromov introduced the sampling compactification of the set of these spaces. He asked whether the metric measure space invariants extend…

The concept of measurability of functions on a charge space is generalised for functions taking values in a uniform space. Several existing forms of measurability generalise naturally in this context, and new forms of measurability are…

Let $f: M \to M$ be a diffeomorphism defined on a compact boundaryless $d$-dimensional manifold $M$, $d\geq 2$. C. Morales has proposed the notion of measure expansiveness. In this note we show that diffeomorphisms in a residual subset far…

We give a new characterization of the space of functions of bounded variation in terms of a pointwise inequality connected to the maximal function of a measure. The characterization is new even in Euclidean spaces and it holds also in…

Topological measures and quasi-linear functionals generalize measures and linear functionals. We define and study deficient topological measures on locally compact spaces. A deficient topological measure on a locally compact space is a set…

Let $M$ be an open manifold of dimension at least $3$, which admits a complete metric of positive scalar curvature. For a function $v$ with bounded growth of derivative, whether $M$ admits a metric of positive scalar curvature with volume…