Related papers: Dynamical systems defined by polynomials with alge…

This note is an introduction to the properties of stable polynomials in several variables with real or complex coefficients. These polynomials are defined in terms of where the polynomial is non-vanishing. We do not cover well-known topics…

Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in…

This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.

Let $K$ be a field and $\phi(z)\in K[z]$ be a polynomial. Define $\Phi(z) := \frac{1}{\phi(z)} \in K(z).$ For $n \in\mathbb{N}^* $, let the $n$-th iterate of $\Phi(z)$ be defined as $\Phi^{(n)}(z) = \underbrace{\Phi \circ \Phi \circ \cdots…

The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to a \textit{scalar product}, which we used to define \textit{orthogonals} in these…

Subject of research is complex networks and network systems. The network system is defined as a complex network in which flows are moved. Classification of flows in the network is carried out on the basis of ordering and continuity. It is…

We construct a ring of flows where we can decompose autonomous nonlinear dynamical systems into smaller parts, then solve each part and finally put everything together to obtain the exact solution of these systems.

For any complex polynomial P having all its zeros in the unit disk, we estimate the rate of change of the argument P (z) when the point z runs through the boundary of this disk.

For an associative algebra $A$ over a field of characteristic zero, let $P_n(A)$ and $P_n^z(A)$ denote the spaces of multilinear polynomials of degree $n$ modulo the polynomial identities and the central polynomials of $A$, respectively. We…

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions…

This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming,…

Let $f(X_1,\dots, X_n)$ be a nonzero multilinear noncommutative polynomial. If $A$ is a unital algebra with a surjective inner derivation, then every element in $A$ can be written as $f(a_1,\dots,a_n)$ for some $a_i\in A$.

Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We…

Let $\psi_1,...,\psi_k$ be periodic maps from $\Bbb Z$ to a field of characteristic p (where p is zero or a prime). Assume that positive integers $n_1,...,n_k$ not divisible by p are their periods respectively. We show that…

In this paper we investigate phase flows over $\mathbb{C}^n$ and $\mathbb{R}^n$ generated by vector fields $V=\sum P^{i}\partial_i$ where $P^{i}$ are finite degree polynomials. With the convenient diagrammatic technique we get expressions…

From a sequence $\left\{ a_{n}\right\} _{n=0}^{\infty}$ of real numbers satisfying a three-term recurrence, we form a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ whose coefficients are numbers in this sequence. We…

We obtain a family of polynomials defined by vanishing conditions and associated to tangles. We study more specifically the case where they are related to a O(n) loop model. We conjecture that their specializations at $z_i=1$ are {\it…

A polynomial transform is the multiplication of an input vector $x\in\C^n$ by a matrix $\PT_{b,\alpha}\in\C^{n\times n},$ whose $(k,\ell)$-th element is defined as $p_\ell(\alpha_k)$ for polynomials $p_\ell(x)\in\C[x]$ from a list…

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several…

Algebraic power series are formal power series which satisfy a univariate polynomial equation over the polynomial ring in n variables. This relation determines the series only up to conjugacy. Via the Artin-Mazur theorem and the implicit…