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We propose several techniques to construct complete permutation polynomials of finite fields by virtue of complete permutations of subfields. In some special cases, any complete permutation polynomials over a finite field can be used to…
Unions of graph multiplier operators are an important class of linear operators for processing signals defined on graphs. We present a novel method to efficiently distribute the application of these operators. The proposed method features…
In this work, based on quantum operator Hermite polynomials and Weyl's mapping rule, we find a generation function of the two-variable Hermite polynomials. And then, noting that the Weyl ordering is invariant under the similar…
We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.
Motivated by the properties of the descent polynomials, which enumerate permutations of $S_n$ with a fixed descent set, we define descent polynomials for labeled rooted trees. We give recursive and explicit formulas for these polynomials…
We propose a new method for solution of the integrability problem for evolutionary differential-difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is…
Differential operators acting on functions defined on graphs by different studies do not form a consistent framework for the analysis of real or complex functions in the sense that they do not satisfy the Leibniz rule of any order. In this…
In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most…
We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are…
There is a relatively well-known description of the algebra of (higher order) left differential operators on commutative algebras. This note gives a construction of similar flavor for algebras of differential operators on not necessarily…
For a class of variational problems with linear differential operator, we obtain a convenient form of the deviation identity, i.e., the value of the distance between approximated solutions and the exact ones. We illustrate the result with…
For operators generated by a certain class of infinite band matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order finite difference equations. This…
We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…
A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.
We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices.
In this paper, we consider linear differential equations satisfied by the generating function for Hermite polynomials and derive some new identities involving those polynomials.
In this paper it is shown that it is possible to associate several polynomial ideals to a directed graph $D$ in order to find properties of it. In fact by using algebraic tools it is possible to give appropriate procedures for automatic…
We determine necessary and sufficient conditions on the ring of differential operators of a finite purely inseparable field extension of positive characteristic for determining whether the extension is modular.
This paper mainly studies problems about so called "permutation polynomials modulo $m$", polynomials with integer coefficients that can induce bijections over Z_m={0,...,m-1}. The necessary and sufficient conditions of permutation…
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.