Related papers: A Graphical Approach to the Frobenius Number Probl…
In this work we will introduce the concept of ratio-covariety, as a nonempty family $\mathscr{R}$ of numerical semigroups verifying certain properties. This concept will allow us to: \begin{enumerate} \item Describe an algorithmic process…
We classify Frobenius forms, a special class of homogeneous polynomials in characteristic $p>0$, in up to five variables over an algebraically closed field. We also point out some of the similarities with quadratic forms.
We consider multicomponent local Poisson structures of the form $\mathcal P_3 + \mathcal P_1$, under the assumption that the third order term $\mathcal P_3$ is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two…
Recently, Chappelon and Alfons\'{\i}n defined the square Frobenius number of coprime numbers $m$ and $n$ to be the largest perfect square that cannot be expressed in the form $mx+ny$ for nonnegative integers $x$ and $y$. When $m$ and $n$…
We study the extended Frobenius problem for sequences of the form $\{l_a\}\cup\{l_a+l_n\}_{n\in\mathbb{N}}$ and $\{l_a+l_n\}_{n\in\mathbb{N}}$, where $\{l_n\}_{n\in\mathbb{N}}$ is the Lucas series and $l_a$ is a Lucas number. As a…
In this paper we study the Frobenius characters of the invariant subspaces of the tensor powers of a representation V. The main result is a formula for these characters for a polynomial functor of V involving the characters for V. The main…
We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a…
We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of…
The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of…
We compute the Frobenius number for sequences of triangular and tetrahedral numbers. In addition, we study some properties of the numerical semigroups associated to those sequences.
We study the problem of approximating a matrix $\mathbf{A}$ with a matrix that has a fixed sparsity pattern (e.g., diagonal, banded, etc.), when $\mathbf{A}$ is accessed only by matrix-vector products. We describe a simple randomized…
One important question in the theory of lattices is to detect a shortest vector: given a norm and a lattice, what is the smallest norm attained by a non-zero vector contained in the lattice? We focus on the infinity norm and work with…
We show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bound.
We consider the following list coloring with separation problem: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|= a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$…
Factor graph of an instance of a constraint satisfaction problem with n variables and m constraints is the bipartite graph between [m] and [n] describing which variable appears in which constraints. Thus, an instance of a CSP is completely…
We address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the…
For a real set $A$ consider the semigroup $S(A)$, additively generated by $A$; that is, the set of all real numbers representable as a (finite) sum of elements of $A$. If $A \subset (0,1)$ is open and non-empty, then $S(A)$ is easily seen…
For a subgroup $H$ of a finite group $G$, the Frobenius graph $\Gamma(G, H)$ records the constituents of the restrictions to $H$ of the irreducible characters of $G$. We investigate when this graph has diameter 3.
In 1990, Backelin showed that the number of numerical semigroups with Frobenius number $f$ approaches $C_i \cdot 2^{f/2}$ for constants $C_0$ and $C_1$ depending on the parity of $f$. In this paper, we generalize this result to semigroups…
In this paper, we address the accuracy of the results for the overdetermined full rank linear least squares problem. We recall theoretical results obtained in Arioli, Baboulin and Gratton, SIMAX 29(2):413--433, 2007, on conditioning of the…