Related papers: Selmer Groups and Quadratic Reciprocity

We introduce and study the class of spherically ordered groups. The notions of spherically ordered groups and their spectra of spherical orderability are introduced. Values of these spectra are found for a series of natural groups.

We will show that an extension of quantum field theory, recently proposed to solve the hierarchy problem, can give an elegant explanation of quark/lepton family replications. This scenario prefers fermion mass models based on a family…

Using small cancellation for rotating families of groups, we construct new examples of aspherical polyhedra.

The article presents an algebra to represent two dimensional patterns using reciprocals of polynomials. Such a representation will be useful in neural network training and it provides a method of training patterns that is much more…

We study the Galois group $G_f$ of a random polynomial $f$ of height at most $H$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$, which arise in a wide variety of…

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…

We describe those group algebras over fields of characteristic different from 2 whose units symmetric with respect to the classical involution, satisfy some group identity.

We obtain Schur-Weyl dualities in which the algebras, acting on both sides, are semigroup algebras of various symmetric inverse semigroups and their deformations.

We investigate reciprocals of false theta functions, producing results such as congruences, simple asymptotic bounds, and combinatorial identities. Of particular interest is a connection between $1/\Psi(-q^2,q)$ and the truncated pentagonal…

We prove a convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool we introduce new semigroup of partial permutations. We…

For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the…

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to…

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…

Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show…

We survey some results relating noncommutative geometry to the class field theory of number fields. These results appear within the context of quantum statistical mechanics where some arithmetic properties of a given number field can be…

Two-dimensional linear spaces of symmetric matrices are classified by Segre symbols. After reviewing known facts from linear algebra and projective geometry, we address new questions motivated by algebraic statistics and optimization. We…

We reconsider some subtle points concerning the relativistic treatment of the gravitational fields generated by spherically symmetric structures.

We associate to a full flag $\mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a "symbol map" $\mu_{\mathcal{F}}:K(F_X) \to \Sigma^n K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(\cdot)$ is the…

We announce a very general statement involving the rational quartic residue symbol $(m/p)_4$ and, more generally, Legendre symbols of the type ${a+b\sqrt{m}/p$. We show how our main theorem can be used to produce many older results such as…

In this paper we study the relationships between the elementary abelian regular subgroups and the Sylow $2$-subgroups of their normalisers in the symmetric group $\mathrm{Sym}(\mathbb{F}_2^n)$, in view of the interest that they have…