Related papers: On compacta not admitting a stable intersection in…

The Fermat--Steiner problem is to find all points of the metric space Y such that the sum of the distances from each of them to points from some fixed finite subset A = {A_1, ..., A_n} of the space Y is minimal. This problem is considered…

A family of permutations A \subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a family,…

We study the problem of when the continuous linear image of a fixed closed convex set $X \subset\mathbb{R}^n$ is closed. Specifically, we improve the main results in the papers \cite{Borwein2009, Borwein2010} by showing that for all, except…

Let $X$ be a compact metric space which is locally absolutely retract and let $\phi: C(X)\to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${\rm dim}Y\le 2.$ It is proved that there exists a sequence of $n$…

Let X and Y be CW-complexes, U be an abelian group, and f:[X,Y]->U be a map (a homotopy invariant). We say that f has order at most r if the characteristic function of the r'th Cartesian power of the graph of a continuous map a:X->Y…

If $G$ is a graph with vertex set $V$, let Conf$_n^{\text{sink}}(G,V)$ be the space of $n$-tuples of points on $G$, which are only allowed to overlap on elements of $V$. We think of Conf$_n^{\text{sink}}(G,V)$ as a configuration space of…

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines $\Pi^d\subset\mathbb R^m$ such that $|g^{-1}(\Pi^d)|\geq q$. Let also $\mathcal H(q,d,m,k)$ denote the maps $g\colon…

Let $X$ be a locally compact topological space, $(Y,d)$ be a boundedly compact metric space and $LB(X,Y)$ be the space of all locally bounded functions from $X$ to $Y$. We characterize compact sets in $LB(X,Y)$ equipped with the topology of…

A space X is kappa-resolvable (resp. almost kappa-resolvable) if it contains kappa dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X). Answering a problem raised by Juhasz, Soukup, and…

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a…

We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a…

Confirming a conjecture posed by Caro, it was shown by Chen and Yu that every graph $G$ with $n$ vertices and at most $2n-4$ edges has a stable cutset, which is a stable set of vertices whose removal disconnects the graph. Le and Pfender…

We say that a finite set S of points in R^d is in "strong general position" if for any collection {F_1,..., F_r} of r pairwise disjoint subsets of S (1 <= r <= |S|) we have: d-dim (the intersection of aff F_1,aff F_2,...,aff F_r) = min{d+1,…

One theorem of Nemhauser and Trotter ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in…

We consider smooth deformations of the $CR$ structure of a smooth $2$-pseudoconcave compact $CR$ submanifold $\textsf{M}$ of a reduced complex analytic variety $\textsf{X}$ outside the intersection $D\,{\cap}\,\textsf{M}$ with the support…

Let L be a compact convex set in R^n, and let 1 <= d <= n-1. The set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set…

Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of…

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n}…

The stability result, as stated, is incorrect. In particular, the 1-dimensional extended persistence diagrams of a finely-triangulated simplicial complex X equipped with a continuous real-valued function f, and its one-skeleton graph G…

The 2-sets convex feasibility problem aims at finding a point in the intersection of two closed convex sets $A$ and $B$ in a normed space $X$. More generally, we can consider the problem of finding (if possible) two points in $A$ and $B$,…