Related papers: Some Notes on Finite Sets
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
The aim of this note is to give an easy example of a finitely presented group that cannot act without a fix point on a CAT(0) space of finite dimension. Such an example has been recently constructed by Arjantseva et al., using other…
In this paper we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them called here rated extremal principles. These developments are in the…
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…
We consider the problem of characterizing finite sets which tile the integers by translations. Coven and Meyerowitz (J. Algebra 1999) found necessary and sufficient conditions for a finite set A to tile the integers under the assumption…
Broadly speaking, a finiteness property of groups is any generalisation of the property of having finite order. A large part of infinite group theory is concerned with finiteness properties and the relationships between them. Profinite…
The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis. The goal of…
In constructive mathematics, several nonequivalent notions of finiteness exist. In this paper, we continue the study of Noetherian sets in the dependently typed setting of the Agda programming language. We want to say that a set is…
Mathematicians invented Mathematics to escape from words, but at last they depend on them just as much as everybody else. At the end, all basic definitions will be reliant on words, yet the mathematician believes that he's elevated from…
This paper illustrates the richness of the concept of regular sets of time bounds and demonstrates its application to problems of computational complexity. There is a universe of bounds whose regular subsets allow to represent several time…
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
According to Russell, strict uses of the definite article 'the' in a definite description 'the F' involve uniqueness; in case there is more than one F, 'the F' is used somewhat loosely, and an indefinite description 'an F' should be…
This document contains the notes of some of the lectures given at the first summer school on Finite Set Statistics held in Edinburgh from July 22, 2013 to July 26, 2013. The notes are mostly self contained and are accessible to everyone.
Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings,…
We introduce the notion of limiting theories, giving examples and providing a sufficient condition under which the first order theory of a structure is the limit of the first order theories of a collection of substructures. We also give a…
We discuss a formal system of mathematics. We use it to construct the natural numbers.
Sorin Popa initiated the study of Polish groups which are embeddable into the unitary group of a separable finite von Neumann algebra. Such groups are called of finite type. We give necessary and sufficient conditions for Polish groups to…
We give multiple descriptions of a topological universe of finitary sets, which can be seen as a natural limit completion of the hereditarily finite sets. This universe is characterized as a metric completion of the hereditarily finite…
This is an introduction to the theory of normal bases of finite fields. The first few chapters cover a wide range of topics on the theory of normal bases of finite fields. Most standard definitions and results, including proofs are given.…