Related papers: Minimal relations and the Diophantine Frobenius Pr…
In the first part of this article we prove that one of the conditions required in the original definition of nearly Frobenius algebra, the coassociativity, is redundant. Also, we determine the Frobenius dimension of the product and tensor…
Let $a,b$ be positive integers. In this note, we study the numerical semigroup $H=\left<a,a+1,b\right>$ and and the associated numerical semigroup ring $R=k[[H]]$. Under the certain conditions, we provide explicit formulas for the Frobenius…
We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations.…
We introduce the notion of the \textit{principal element} of a Frobenius Lie algebra $\f$. The principal element corresponds to a choice of $F\in \f^*$ such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is…
In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to…
We characterize the permutations of $\mathbb{F}_q$ whose graph minimizes the number of collinear triples and describe the lexicographically-least one, affirming a conjecture of Cooper-Solymosi. This question is closely connected to…
In this paper we formulate and solve extremal problems in the d-dimensional Euclidean space and further in hypergraphs, originating from problems in stoichiometry and elementary linear algebra. The notion of affine simplex is the bridge…
We present a procedure to enumerate the whole set of numerical semigroups with a given Frobenius number F, S(F). The methodology is based on the construction of a partition of S(F) by a congruence relation. We identify exactly one…
We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential…
We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the…
In a previous paper we studied ``weakly primitive axial algebras'' with respect to more general fusion rules, for which at least one axis satisfies the fusion rules. In this continuation, a concise description is provided of the…
Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the…
A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible…
In this work we show that every minimal code in a semisimple group algebra $\mathbb{F}_qG$ is essential if $G$ is a simple group. Since the alternating group $A_n$ is simple if $n=3$ or $n\geq 5$, we present some examples of minimal codes…
Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, and let ${\mathcal S}$ denote the set of all nonnegative integer solutions of the equation $x_1a_1+\ldots +x_na_n=y_1b_1+\ldots +y_mb_m$. A solution…
Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical…
The main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in…
The classical Frobenius problem is to find the largest integer that cannot be written as a linear combination of a given set of positive, coprime integers using nonnegative integer coefficients. Prior work has generalized the classical…
We prove in this text a characterization of the possible entropy dimensions of minimal tridimensional subshifts of finite type with a computability condition, using Goldbach's theorem on Fermat numbers.
In this article we continue with the study started in [1] of nearly Frobenius structures in some representative families of finite dimensional algebras, as the radical square zero algebras, string algebras and the toupie algebras. We prove…