Related papers: Aspects of Classical Descriptive Set Theory

The purpose of this paper is to give a survey on the notions of distance between subsets either of a metric space or of a measure space, including definitions, a classification, and a discussion of the best-known distance functions, which…

The article is devoted to a comparison of the \v{C}ech cohomology with the coefficients in a presheaf of Abelian groups and the topos cohomology of the sheaf generated by this presheaf for a poset with the Aleksandrov topology. The article…

The main purpose of this paper is to find the fixed point in such cases where existing literature remain silent. In this paper we introduce partial completeness, a new type of contraction and many other definitions. Using this approach the…

Physical systems are often neither completely closed nor completely open, but instead they are best described by dynamical systems with partial escape or absorption. In this paper we introduce classical measures that explain the main…

Lecture notes for a one-semester master-level course on analytical mechanics and classical field theory, covering: 0 Mathematical Introduction, 1 Lagrangian Mechanics, 2 Application: Motion in Central Fields, 3 Hamiltonian Mechanics, 4…

In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on…

Various spectral notions have been employed to grasp the structure of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we…

This paper examines a denumerable version of the nested-set theorem and derives from it a contradiction involving the formal consistency of the actual infinity assumed by the Axiom of Infinity.

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are…

This paper introduces an extension of descriptive intersection and provides a framework for descriptive unions of nonempty sets. Fibre bundles provide structures that characterize spatially near as well as descriptively near sets, their…

We establish a dynamical version of Kuratowski-Mycielski Theorem on the existence of "large" invariant dependent sets. We apply this result to the study of invariant chaotic sets in topological dynamical systems, simplify many known results…

We consider prediction theory for stationary stochastic processes in continuous time. We discuss prediction using the whole (infinite) past, and using only a finite section of the past. The solutions to both these classical problems have…

We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to…

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that in many cases,…

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem…

We present an approach to the study of stationary measures placing Tarski's foundational work in this area within a modern category theoretic context. Guiding this work is the notion that measurable spaces equipped with symmetries carry an…

We discuss the problems of incompleteness and inexpressibility. We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper…

Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better understand and efficiently solve those…

These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.

The concept of statistical convergence based on asymptotic density is introduced in this article through nets. Some possible extensions of classical results for statistical convergence of sequences are obtained in this article, with…