Related papers: Some Notes on Finite Sets
Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…
We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the…
Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a…
In this paper it is stressed that there is no {\em physical} reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the…
We introduce the notion of almost finite dimensionality of algebras and study its connection with the classical finiteness conditions.
All possible products of all elements of an odd order finite group are considered. A set of all such products is called as a K-set. A hypothesis of K-set coincidence of any group of an odd order with its commutant is proposed and the…
This article, written in German language, gives a very elementary introduction to infinite sets. It is meant for interested non-mathematicians.
These are the notes of a series of lectures delivered by the author at the Graduate School of Mathematics of the University of Tokyo, during the month of October 2015. They were meant to be as self-contained as possible, taking into account…
The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.
The notion of cosilting module was recently introduced as a generalization of the concept of cotilting module. In this paper, it is introduced the notion of finitely cosilting module, i.e. a cosilting module with some finitness conditions,…
Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…
This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
This paper obtains a completeness result for inequational reasoning with applicative terms without variables in a setting where the intended semantic models are the full structures, the full type hierarchies over preorders for the base…
We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several equivalent statements} borrowed from algebra, real analysis, general topology, and non-standard analysis. We also…
The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for…
A finite W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. In this survey we review recent developments in the representation theory of W-algebras. We emphasize various interactions…
We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…
Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values…
These Course Notes provide an introduction to mathematical proofs for undergraduate students transitioning from computational calculus to abstract mathematics. Topics include propositional logic, proof techniques, mathematical induction,…