Related papers: Elementary remarks about Pisano periods
The set of periodic distributions, with usual addition and convolution, forms a ring, which is isomorphic, via taking a Fourier series expansion, to the ring ${\mathcal{S}}'({\mathbb{Z}}^d)$ of sequences of at most polynomial growth with…
This survey article is the outgrowth of two talks given at the Journ\'ees X-UPS "P\'eriodes et transcendance" at \'Ecole polytechnique. Periods are complex numbers whose real and imaginary parts can be written as integrals of rational…
In this short note, we simply collect some known results about representing algebraic cycles by various kind of "nice" (e.g. smooth, local complete intersection, products of local complete intersection) algebraic cycles, up to rational…
We analyze and partially solve system of recurrences that can be derived from the properties of martingale orthogonal polynomials that characterize quadratic harnesses (QH). We also specify conditions for the existence of moments of one…
This work is devoted to the study of the existence and periodicity of solutions of initial differential problems, paying special attention to the explicit computation of the period. These problems are also connected with some particular…
The effects of an aperiodic order or a random disorder on phase transitions in statistical mechanics are discussed. A heuristic relevance criterion based on scaling arguments as well as specific results for Ising models with random disorder…
A somewhat pretentious presentation of number systems (N, Z, Q, R, C, Q_p, >...). The problem of a p-adic characterisation of good-reduction p-adic curves is posed.
We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the…
A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the…
A general description of the Vi\`ete coefficients of the gaussian period polynomials is given, in terms of certain symmetric representations of the subgroups and the corresponding quotient groups of the multiplicative group…
This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.
Elementary particles, i.e. the basic constituents of nature, are characterized by quantum recurrences in time. The flow of time of every physical system can be therefore decomposed in elementary cycles of time. This allows us to enforce the…
We investigate the dependence of Poincar\'e recurrence-times statistics on the choice of recurrence-set, by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct…
After the language of module and theirs morphisms, this short course presents matricial calculus and determinants in a commutative ring as appliction of ``remarquable identities'' in the ring of polynomials with integer coefficients with…
We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the…
Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to…
In this note, we consider a situation that is generally used as an intermediate technical step in proving the Artin-Rees lemma but otherwise is not much discussed in introductory accounts of commutative algebra. I hope to show in this note…
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
We introduce a method for describing Riordan matrices via recurrence relations along their diagonals. This provides a new structural description that complements the classical row-wise and column-wise constructions via the A-sequence. As an…