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The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give…

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…

We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.

The characteristic polynomials of the adjacency matrix of line graphs of caterpillars and then the characteristic polynomials of their Laplacian or signless Laplacian matrices are characterized, using recursive formulas. Furthermore, the…

In this paper, we define some new notions of triangular Banach algebras and we investigate the derivations on these algebras.

Based on work presented in [4], we define $S^2$-Upper Triangular Matrices and $S^2$-Lower Triangular Matrices, two special types of $d\times d(2d-1)$ matrices generalizing Upper and Lower Triangular Matrices, respectively. Then, we show…

A triangulation of a polygon has an associated Stanley-Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals, and describe their separated models. More generally we do this for stacked simplicial…

The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…

We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the…

We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of first-order functional programs. We explain their semantics and prove that they form a…

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…

We improve upon the upper bounds for the cardinality of the value set of a multivariable polynomial map over a finite field using the polytope of the polynomial. This generalizes earlier bounds only dependent on the degree of a polynomial.

The paper gives a full description of the growth of the gradient of a polynomial in two complex variables at infinity near any fiber of the polynomial

In this paper we prove a conjecture about the dimension of linear systems of surfaces of degree d in P^3 through at most eight multiple points in general position.

We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…

We study the boundary of the nonnegative trigonometric polynomials from the algebraic point of view. In particularly, we show that it is a subset of an irreducible algebraic hypersurface and established its explicit form in terms of…

We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial…

This article gives a summary of the finite-dimesional irreducible representations of the $q$-Onsager algebra, which are treated in detail in our paper `The augmented tridiagonal algebra'.

In this paper, we consider the problem of representing any polynomial in terms of the ordered Bell and degenerate ordered Bell polynomials, and more generally of the higher-order ordered Bell and higher-order degenerate ordered Bell…