Related papers: Modular binomials with an application to periodic …
For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the…
This paper addresses the decomposition of biochemical networks into functional modules that preserve their dynamic properties upon interconnection with other modules, which permits the inference of network behavior from the properties of…
Let $\underline{a}$ and $\underline{b}$ be primitive sequences over $\mathbb{Z}/(p^e)$ with odd prime $p$ and $e\ge 2$. For certain compressing maps, we consider the distribution properties of compressing sequences of $\underline{a}$ and…
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of…
Two nested classes of discrete-time linear time-invariant systems, which differ by the set of periodic signals that they leave invariant, are studied. The first class preserves the property of periodic monotonicity (period-wise…
We study Fourier transforms of regular holonomic D-modules. In particular we show that their solution complexes are monodromic. An application to direct images of some irregular holonomic D-modules will be given. Moreover we give a new…
We introduce a notion of strong periodicity of a module over a finite-dimensional algebra over a field. We prove that the existence of such modules over certain idempotent algebras is both a necessary and sufficient condition for the…
The field of automatic music composition has seen great progress in recent years, specifically with the invention of transformer-based architectures. When using any deep learning model which considers music as a sequence of events with…
In this article we further the study of non-commutative motives. We prove that bivariant cyclic cohomology (and its variants) becomes representable in the category of non-commutative motives. Furthermore, Connes' bilinear pairings…
The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…
Normal and composition series of modules enumerated by ordinal numbers are studied. The Jordan-Holder theorem for them is discussed.
In this paper we study the action of a generalization of the Binomial interpolated operator on the set of linear recurrent sequences. We find how the zeros of characteristic polynomials are changed and we prove that a subset of these…
Periodic patterns in dynamical behaviours of biological models described by simple form differential delay equations are studied. Mathematical models are given by a class of scalar delay differential equations with a multiplicative time…
Much is known about binomial coefficients where primes are concerned, but considerably less is known regarding prime powers and composites. This paper provides two conjectures in these directions, one about counting binomial coefficients…
We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply…
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is…
We introduce a sequence representation of a random variable $X$ supported on a compact interval $[a,b]$, which we call a primitive sequence. We construct this sequence by repeatedly antidifferentiating the associated cumulative distribution…
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
A novel representation of a quasi-periodic modified von Mangoldt function L(n) on prime numbers and its decomposition into Fourier series has been investigated. We focus on some particular quantities characterizing the modified von Mangoldt…