Related papers: Unit Reducible Fields and Perfect Unary Forms
We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an…
We devise a fairly general sufficient condition ensuring that the endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a Fra\"{\i}ss\'{e} limit) embeds all countable semigroups. This approach provides us not only…
We discuss the notions of indices of vector fields and 1-forms and their generalizations to singular varieties and varieties with actions of finite groups, as well as indices of collections of vector fields and 1-forms.
We analyze the unitarity of a non-relativistic non-commutative scalar field theory. We show that electric backgrounds spoil unitarity while magnetic ones do not. Furthermore, unlike its relativistic counterparts, unitarity can not be…
This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from…
This paper continues a study of field theories specified for the nonuniform lattice in the finite-dimensional hypercube with the use of the earlier described deformation parameters. The paper is devoted to spontaneous breakdown and…
A quadratic form over a non-archimedian local field of characteristic zero $F$ is called universal if it is integral and it represents all non-zero integers of $F$. Xu Fei and Zhang Yang determined all universal quadratic forms in the case…
We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of…
In our earlier article~\cite{CanSakran} we initiated a study of the complement-finite submonoids of the group of integer points of a unipotent linear algebraic group. In the present article, we continue to develop tools and techniques for…
We incorporate the concept of dimensional reduction at high energies within the perturbative formulation of quantum field theory. In this new framework, space and momentum integrations are modified by a weighting function incorporating an…
An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\mathbb{N}\to\{1,...,r\}$ of the natural numbers, there exist $x,y\in \mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More…
Chebotarev's theorem on roots of unity states that all minors of the Fourier matrix of prime size are non-vanishing. This result has been rediscovered several times and proved via different techniques. We follow the proof of Evans and…
Recent attempts at studying the Fermat equation over number fields have uncovered an unexpected and powerful connection with $S$-unit equations. In this expository paper we explain this connection and its implications for the asymptotic…
Let $ n \ge 2$ be an integer. We give necessary and sufficient conditions for an integral quadratic form over dyadic local fields to be $ n $-universal by using invariants from Beli's theory of bases of norm generators. Also, we provide a…
We study a dynamics of ultracold Fermi-gases near the unitary limit in the framework of Effective Field Theory. It is shown that, while one can obtain a reasonable description of the universal proportionality constant both in the narrow and…
The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite…
We investigate the possible structures of numbers (as physical quantities) over which accelerated observers can be modeled in special relativity. We present a general axiomatic theory of accelerated observers which has a model over every…
There exist numerous results in the literature proving that within certain families of totally real number fields, the minimal rank of a universal quadratic lattice over such a field can be arbitrarily large. Kala introduced a technique of…
We start an analysis of geometric properties of a structure relative to a reduct. In particular, we look at definability of groups and fields in this context. In the relatively one-based case, every definable group is isogenous to a…
We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here,…