Related papers: How many miles to $\beta\omega$? -- Approximating …

A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$.…

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism G^ --> D^ of the dual groups is a topological isomorphism. We introduce four conditions on D…

We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with…

We continue the research of an extension $\widetilde{\mid}$ of the divisibility relation to the Stone-\v Cech compactification $\beta N$. First we prove that ultrafilters we call prime actually possess the algebraic property of primality.…

We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function $Z$ of a…

Let $X$ be a ball Banach function space on ${\mathbb R}^n$. Let $\Omega$ be a Lipschitz function on the unit sphere of ${\mathbb R}^n$,which is homogeneous of degree zero and has mean value zero, and let $T_\Omega$ be the convolutional…

Fix a compact $M \subset \mathbb{R}^2$ and $r>0$. A minimizer of the maximal distance functional is a connected set $\Sigma$ of the minimal length, such that \[ max_{y \in M} dist(y,\Sigma) \leq r. \] The problem of finding maximal distance…

For a given class $\mathcal{F}$ of closed sets of a measured metric space $(E,d,\mu)$, we want to find the smallest element $B$ of the class $\mathcal{F}$ such that $\mu(B)\geq 1-\alpha$, for a given $0<\alpha<1$. This set $B$…

We consider shape optimization problems of the form $$\min\big\{J(\Omega)\ :\ \Omega\subset X,\ m(\Omega)\le c\big\},$$ where $X$ is a metric measure space and $J$ is a suitable shape functional. We adapt the notions of $\gamma$-convergence…

We study the complexity of computing (and approximating) VC Dimension and Littlestone's Dimension when we are given the concept class explicitly. We give a simple reduction from Maximum (Unbalanced) Biclique problem to approximating VC…

Let $\Omega\in\mathbb{R}^n$ be the region occupied by a body and let $\mathbf{x}_0$ be a flaw point in $\Omega$. Let $E(\cdot)$ be an energy functional (defined on some appropriate admissible set of deformations of $\Omega$). For $V>0$…

In the paper we are dealing with metric measure spaces of diameter at most one and of total measure one. Gromov introduced the sampling compactification of the set of these spaces. He asked whether the metric measure space invariants extend…

There are different definitions of homological dimension of metric compacta involving either \v{C}ech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to…

Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma}$ is a…

The Steiner Tree problem is a classical problem in combinatorial optimization: the goal is to connect a set $T$ of terminals in a graph $G$ by a tree of minimum size. Karpinski and Zelikovsky (1996) studied the $\delta$-dense version of…

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for…

A topological space $X$ is called submaximal if every dense subset of $X$ is open. In this paper, we show that if $\beta X$, the Stone-\v{C}ech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $\beta…

It was conjectured that if $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfies $\operatorname{rank} Df\leq m<n$ everywhere in $\mathbb{R}^n$, then $f$ can be uniformly approximated by $C^\infty$-mappings $g$ satisfying $\operatorname{rank}…

We study the approachability ideal I[\kappa^+] in the context of large cardinals properties of the regular cardinals below a singular \kappa. As a guiding example consider the approachability ideal I[\aleph_{\omega+1}] assuming that…

It is proved that a discrete group G is exact if and only if its left translation action on the Stone-Cech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete…