Related papers: On the Degree Growth in Some Polynomial Dynamical …
Reinforcement learning is a promising approach to learning robotics controllers. It has recently been shown that algorithms based on finite-difference estimates of the policy gradient are competitive with algorithms based on the policy…
We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves…
We study the convergence of a random iterative sequence of a family of operators on infinite dimensional Hilbert spaces, inspired by the Stochastic Gradient Descent (SGD) algorithm in the case of the noiseless regression, as studied in [1].…
Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
Positively graded algebras are fairly natural objects which are arduous to be studied. In this article we query quotients of non-standard graded polynomial rings with combinatorial and commutative algebra methods.
We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the M\"obius map $x \mapsto (ax + b)/(cx+d)$ over a finite field of $p$ elements at the moments of time that correspond to…
We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…
Iteration of the quadratic map produces sequences of polynomials whose degrees {\sl explode} as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree $4020$, while for the $52,377$ period-20…
This paper is motivated by the theory of sequential dynamical systems, developed as a basis for a mathematical theory of computer simulation. It contains a classification of finite dynamical systems on binary strings, which are obtained by…
The ability to generate synthetic data has a variety of use cases across different domains. In education research, there is a growing need to have access to synthetic data to test certain concepts and ideas. In recent years, several deep…
Many mathematicians have been studying various degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our main focus here is a new type of degenerate poly-Euler polynomials and numbers. This…
We study the probability distribution of the number of zeros of multivariable polynomials with bounded degree over a finite field. We find the probability generating function for each set of bounded degree polynomials. In particular, in the…
In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the…
This study covers an analytical approach to calculate positively invariant sets of dynamical systems. Using Lyapunov techniques and quantifier elimination methods, an automatic procedure for determining bounds in the state space as an…
The recently developed method (Paper 1) enabling one to investigate the evolution of dynamical systems with an accuracy not dependent on time is developed further. The classes of dynamical systems which can be studied by that method are…
Dynamical processes on complex networks such as information propagation, innovation diffusion, cascading failures or epidemic spreading are highly affected by their underlying topologies as characterized by, for instance, degree-degree…
The analysis in this paper helps to explain the formation of growing networks with degree distributions that follow extended exponential or power-law tails. We present a generic model in which edge dynamics are driven by a continuous…
This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial…
The degree sequence optimization problem is to find a subgraph of a given graph which maximizes the sum of given functions evaluated at the subgraph degrees. Here we study this problem by replacing degree sequences, via suitable nonlinear…