Related papers: Rank Function Equations and their solution sets
The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let…
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…
We provide a determinantal formula for tau-functions of the KP hierarchy in terms of rectangular, constant matrices $A$, $B$ and $C$ satisfying a rank one condition. This result is shown to generalize and unify many previous results of…
We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…
In this article, I provide a solution to a rank computation problem related to the computation of the Hilbert-Kunz function for any disjoint-term trinomial hypersurface, over any field of characteristic 2. This rank computation problem was…
Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of…
We propose a representation of boolean bent functions by bent rectangles, that is, by special matrices with restrictions on rows and columns. Using this representation, we exhibit new classes of bent functions, give an algorithm to…
In this work, we study vector-valued functional equations with multiple recursive terms that arise naturally when we are dealing with vector-valued multiplicative Lindley-type recursions. We provide a detailed framework for the solution of…
Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical…
A class of parametric functions formed by alternating compositions of multivariate polynomials and rectification style monomial maps is studied (the layer-wise exponents are treated as fixed hyperparameters and are not optimized). For this…
This paper explores idempotent and nilpotent operators in bicomplex spaces, focusing on their properties and behavior. We define idempotent and nilpotent matrices in this framework and derive related results. Several theorems are presented…
A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum…
An explicit formula is given for a fundamental solution for a class of semielliptic operators. The fundamental solution is used to investigate properties of these operators as mappings between weighted function spaces. Necessary and…
Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan…
In this paper, we consider the rank-one tensor completion problem. We address the question of existence and uniqueness of the rank-one solution. In particular we show that the global uniqueness over the field of real numbers can be verified…
We define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild…
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 give an elementary characterization of rational functions among meromorphic functions in the complex plane.