Related papers: Some multiplicative functions over $\mathbb{F}_2$
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
We expand the theoretical background of the recently introduced superadditive and subadditive transformations of aggregation functions $A$. Necessary and sufficient conditions ensuring that a transformation of a proper aggregation function…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
In this article, we study algebraic decompositions and secondary constructions of almost perfect nonlinear (APN) functions. In many cases, we establish precise criteria which characterize when certain modifications of a given APN function…
The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result…
We obtain necessary and sufficient conditions on a function in order that it be the Laplace transform of an absolutely monotonic function. Several closely related results are also given.
We introduce the notion of Dunkl positive definite and strictly positive definite functions on $\mathbb{R}^{d}$. This done by the use of the properties of Dunkl translation. We establish the analogue of Bochner's theorem in Dunkl setting.…
In this paper, we give a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of…
This article is an extensive study of partitions with fixed number of odd and even-indexed odd parts. We use these partitions to generalize recent results of C. Savage and A. Sills. Moreover, we derive explicit formulas for generating…
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
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…
We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm…
We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.
Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure $\mathbf{A}$ is called…
In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…
We study existence and computability of finite bases for ideals of polynomials over infinitely many variables. In our setting, variables come from a countable logical structure A, and embeddings from A to A act on polynomials by renaming…
Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
The multiplicative structure of an odd perfect number $n$, if any, is $n=\pi^\alpha M^2$, where $\pi$ is prime, $\gcd(\pi,M)=1$ and $\pi\equiv \alpha\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that…