Related papers: Reduction of polynomial dynamical systems modulo p…
Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…
It is shown how the dimension of any arbitrary over-determined system of differential equations can be reduced, which makes the system suitable for numerical solution modeling. Specifically, over-determined equations of hydrodynamics are…
In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in…
A standard method for finding a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese remaindering, and then use Farey sequences for rational reconstruction.…
Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid…
We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
Discrete dynamical systems defined by the iteration of a polynomial map of the unit simplex to itself appear in the context of population genetic systems evolving under mutation, recombination and weak selection. Although exceptional…
One of the main open problems in the qualitative theory of real planar differential systems is the study of limit cycles. In this article, we present an algorithmic approach for detecting how many limit cycles can bifurcate from the…
We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
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 seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
We review how an algebraic formulation for the dynamics of a physical system allows to describe a reduction procedure for both classical and quantum evolutions.
We apply the Darboux integrability method to determine first integrals and Hamiltonian formulations of three dimensional polynomial systems; namely the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh-Rose model,…
We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the…
It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…
A set of algorithms is presented for efficient numerical calculation of the time evolution of classical dynamical systems. Starting with a first approximation for solving the differential equations that has a "reversible" character, we show…
We study the dynamics of a two-planet system, which evolves being in a $1/1$ mean motion resonance (co-orbital motion) with non-zero mutual inclination. In particular, we examine the existence of bifurcations of periodic orbits from the…