Related papers: Analytic varieties as limit periodic sets

We prove that a one-parameter family of rationally connected varieties (over an algebraically closed field of characteristic 0) always has a section.

In the present paper, we study the limit sets of the almost periodic functions $f(x)$. It is interesting that the values $r=\inf|f(x)|$ and $R=\sup|f(x)|$ may be expressed in the exact form. We show that the ring $r\leq |z|\leq R$ is the…

Necessary and sufficient conditions for the existence of limits of the form {equation*} \lim_{(x,y)\rightarrow (a,b)}\frac{f(x,y)}{g(x,y)} {equation*} are given, under the hipothesis that $f$ and $g$ are real analytic functions near the…

We present exact calculations of reliability polynomials $R(G,p)$ for lattice strips $G$ of fixed widths $L_y \le 4$ and arbitrarily great length $L_x$ with various boundary conditions. We introduce the notion of a reliability per vertex,…

With every real polynomial $f$, we associate a family $\{f_{\epsilon r}\}_{\epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $\epsilon>0,r\in N$, and such that $\Vert f-f_{\epsilon r}\Vert_1\to 0$ as…

We study bifurcations in finite-parameter families of vector fields on $S^2$. Recent papers by Yu. Ilyashenko, N. Goncharuk, Yu. Kudryashov, I. Schurov, and N. Solodovnikov provide examples of (locally generic) structurally unstable…

There exists a family $\{B_{\alpha}\}_{\alpha<\omega_1}$ of sets of countable ordinals such that o $\max B_{\alpha}=\alpha$, o if $\alpha\in B_{\beta}$ then $B_{\alpha}\subseteq B_{\beta}$, o if $\lambda\leq \alpha$ and $\lambda$ is a limit…

Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertex graph $G$ with $q$ colors, and considering the limiting function $W(\{G\},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, a fundamental question in graph theory is the…

In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y) = 0$ with variables $X$ and $Y$ over a finite field $\mathbb{F}_q$ of odd characteristic, we define a digraph by choosing the elements in…

We prove that every heteroclinic saddle loop (a two-saddle cycle) occurring in an analytic finite-parameter family of plane analytic vector fields, may generate no more than a finite number of limit cycles within the family.

For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non existence of periodic orbits not contained in this given…

For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

In 2020, Marques and Moreira proved that every subset of $\overline{\mathbb{Q}} \cap B(0,1)$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many transcendental analytic functions with…

Consider an analytical function $f:V\subset\mathbb R^2\rightarrow\mathbb R$ having $0$ as its regular value, a switching manifold $\Sigma=f^{-1}(0)$ and a piecewise analytical vector field $X=(X^+,X^-)$, i.e. $X^\pm$ are analytical vector…

We consider a system of the form x'=P_n(x,y)+xR_m(x,y), y'=Q_n(x,y)+yR_m(x,y), where P_n(x,y), Q_n(x,y) and R_m(x,y) are homogeneous polynomials of degrees n, n and m, respectively, with n<=m. We prove that this system has at most one limit…

Answering a question asked by Hsia and Tucker in their paper on the finiteness of greatest common divisors of iterates of polynomials, we prove that if $f, g \in \mathbb{C}(X)$ are compositionally independent rational functions and $c \in…

In this paper we consider the limit cycles of the planar system $$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and $\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$ respectively. We prove that…

We conclude an investigation of Abrishami, Esperet, Giocanti, Hamman, Knappe and M\"oller studying the existence of periodic colourings of locally finite graphs. A colouring of a graph $\Gamma$ is periodic if the resulting coloured graph…

The authors' previous results on the arity gap of functions of several variables are refined by considering polynomial functions over arbitrary fields. We explicitly describe the polynomial functions with arity gap at least 3, as well as…