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The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal…

We study semi-dynamical systems associated to delay differential equations. We give a simple criteria to obtain weak and strong persistence and provide sufficient conditions to guarantee uniform persistence. Moreover, we show the existence…

The affinely regular polygons in certain planar sets are characterized. It is also shown that the obtained results apply to cyclotomic model sets and, additionally, have consequences in the discrete tomography of these sets.

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which…

Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree $d$ whose Galois group is $S_d$. Let $(a_n)$ be a linearly recuresive sequence of integers which has $P$ as its characteristic polynomial. We prove, under…

A formal sequent system dealing with Menelaus' configurations is introduced in this paper. The axiomatic sequents of the system stem from 2-cycles of Delta-complexes. The Euclidean and projective interpretations of the sequents are defined…

We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.

We study relations between transitivity, mixing and periodic points on dendrites. We prove that when there is a point with dense orbit which is not an endpoint, then periodic points are dense and there is a terminal periodic decomposition…

We show that the set $\ell^{\infty}\setminus c_0$ is dense-lineable and dense-algebrable answering a question posed by Nestoridis and complementing a result by Garc\'ia-Pacheco, Mart\'in and Seoane-Sep\'ulveda.

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that…

We prove that characteristic equations of certain types of delay differential systems, under some mild conditions on their coefficients, can possess infinitely many complex roots.

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic…

In [8] the authors initiate the study of selective versions of the notion of $\theta$-separability in non-regular spaces. In this paper we continue this investigation by establishing connections between the familiar cardinal numbers arising…

Multivariate process satisfying affine stochastic recurrence equation with generic diagonal matrices is considered. We prove that the stationary solution is regularly varying. The results are applicable to diagonal autoregressive models.

We prove the existence of infinite dense free sets (in the usual topology) for set mappings on the reals, under reasonable assumptions.

We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets…

We give a number of results about families of Ulam sets. Generalizing behavior of Ulam sets U(1,n), we prove using an novel model theoretic approach that there is a rigidity phenomenon for Ulam sets U(a,b) as b increases. Based on this, we…

We prove that certain families of homogenous affine iterated function systems in $\mathbb{R}^d$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters.…

Let $\beta$ be a non-unit real algebraic integer greater than one and $\{a_{n}\}_{n \geq 0}$ be a sequence satisfying a linear recurrence relation $a_{n+3}=aa_{n+2}+ba_{n+1}+ca_{n}$. Under certain conditions, we prove that the number of…

A discrete set in the $p$-dimensional Euclidian space is {\it almost periodic}, if the measure with the unite masses at points of the set is almost periodic in the weak sense. We propose to construct positive almost periodic discrete sets…