Related papers: Results in descriptive set theory on some represen…
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…
We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions.…
We present a self-contained analysis of infinity from two mathematical perspectives: set theory and algebra. We begin with cardinal and ordinal numbers, examining deep questions such as the continuum hypothesis, along with foundational…
Using dynamics, Furstenberg defined the concept of a central subset of positive integers and proved several powerful combinatorial properties of central sets. Later using the algebraic structure of the Stone-\v{C}ech compactification,…
The main aim of this paper is to generalize the concept of vector space by the hyperstructure. We generalize some definitions such as hypersubspaces, linear combination, Hamel basis, linearly dependence and linearly independence. A few…
The main results of this paper are the construction, both rigourous and intuitive, of "the" intrinsic extension of the set of non negative integers N and the smallest over-field of R set which is continue (according to R.Dedekind). The aim…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
Across the natural and life sciences, images have become a primary measurement modality, yet the dominant analytic paradigm remains semantics-first. Structure is recovered by predicting or enforcing domain-specific labels. This paradigm…
In this paper we deal with unbounded composition operators defined in Orlicz spaces. Indeed, we provide some necessary and sufficient condition for densely definedness of composition operators on Orlicz spaces. Also, we will investigate the…
Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_\sigma$ set under it is…
The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…
The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for…
We present the effective version of the theorem about turning Borel sets in Polish spaces into clopen sets while preserving the Borel structure of the underlying space. We show that under some conditions the emerging parameters can be…
The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes…
This paper proves a representation theorem regarding sequences of random elements that take values in a Borel space and are measurable with respect to the sigma algebra generated by an arbitrary union of sigma algebras. This, together with…
We introduce a non real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field $R$. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of $R$. We show…
In this thesis, we develop various techniques for working with sets in machine learning. Each input or output is not an image or a sequence, but a set: an unordered collection of multiple objects, each object described by a feature vector.…
We develop a Hilbert space framework for a number of general multi-scale problems from dynamics. The aim is to identify a spectral theory for a class of systems based on iterations of a non-invertible endomorphism. We are motivated by the…
Expansions of the monadic second-order (MSO) theory of the structure $\langle \mathbb{N} ; < \rangle$ have been a fertile and active area of research ever since the publication of the seminal papers of B\"uchi and Elgot & Rabin on the…
In recent work by Khmaladze and Weil (2008) and by Einmahl and Khmaladze (2011), limit theorems were established for local empirical processes near the boundary of compact convex sets $K$ in $\R$. The limit processes were shown to live on…