Related papers: A study concerning splitting and jointly continuou…
Suppose that $R$ is a local domain with fraction field $K$. If $R$ is Henselian then the $R$-adic topology over $K$ refines the \'etale open topology. If $R$ is regular then the \'etale open topology over $K$ refines the $R$-adic topology.…
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all…
We investigate the Baire classification of mappings $f:X\times Y\to Z$, where $X$ belongs to a wide class of spaces, which includes all metrizable spaces, $Y$ is a topological space, $Z$ is an equiconnected space, which are continuous in…
Given a finite graph G and a topological space Z, the graphical configuration space Conf(G, Z) is the space of functions V(G) -> Z so that adjacent vertices map to distinct points. We provide a homotopy decomposition of Conf(G, X x Y) in…
For a manifold $W$ and an $E_d$-algebra $A$, the factorisation homology $\int_W A$ can be seen as a generalisation of the classical configuration space of labelled particles in $W$. It carries an action by the diffeomorphism group…
We study algebraic varieties parametrized by topological spaces and enlarge the domains of Lawson homology and morphic cohomology to this category. We prove a Lawson suspension theorem and splitting theorem. A version of Friedlander-Lawson…
Several coupled maps models are sketched and reviewed in this short communication. First, a discrete logistic type model that was proposed for the symbiotic interaction of two species. Second, a model of many of these symbiotic species…
Clustering bifurcations are investigated by considering models of globally coupled map lattices. Typical classes of clustering bifurcations are revealed. The clustering bifurcation thresholds of the coupled system are closely related to the…
In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new…
This note presents an approach to studying the iterates of a mapping whose restriction to the complement of a finite set is continuous and open. The main examples to which the approach can be applied are piecewise monotone mappings defined…
Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and…
We study feebly compact topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$ is a semitopological semilattice and prove that for any shift-continuous $T_1$-topology $\tau$ on…
We are studying topological properties of the Julia set of the map $F(z, p)=((2z/(p+1)-1)^2, ((p-1)/(p+1))^2)$ of the complex projective plane $PC^2$ to itself. We show a relation of this rational function with an uncountable family of…
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…
Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…
Every open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed subsets of X so that their image by f is Y provided all fibers of f are infinite and C*-embedded in X. Applications are demonstrated for the…
It is shown that it is consistent with ZFC that every uncountable set can be continuously mapped onto a splitting family.
New tilings of certain subsets of $\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our…
In this paper we systematically describe relations between various structure sets which arise naturally for pairs of compact topological manifolds with boundary. Our consideration is based on a deep analogy between the case of a compact…
Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a…