Related papers: Selection properties of the split interval and the…

A \emph{hull} of $A \subset [0,1]$ is a set $H$ containing $A$ such that $\lambda^*(H)=\lambda^*(A)$. We investigate all four versions of the following problem. Does there exist a monotone (wrt. inclusion) map that assigns a…

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and…

When split conformal prediction operates in batch mode with exchangeable data, we determine the exact distribution of the empirical coverage of prediction sets produced for a finite batch of future observables, as well as the exact…

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [PS05, PS06]. We show that under some natural assumptions on the inducing…

We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability…

Let $G$ and $H$ be locally compact groups with fixed two-side-invariant Haar measures. A polyhomomorphism $G\to H$ is a closed subgroup $R\subset G\times H$ with a fixed Haar measure, whose marginals on $G$ and $H$ are dominated by the Haar…

A result due to Cho, Miyaoka, Shepherd-Barron [CMSB] and Kebekus [Ke] provides a numerical characterization of projective spaces. More recently, Dedieu and H\"oring [DH] gave a characterization of smooth quadrics based on similar arguments.…

We prove that hereditarily Lindel\"of space which is $F_{\sigma\delta}$ in some compactification is absolutely $F_{\sigma\delta}$. In particular, this implies that any separable Banach space is absolutely $F_{\sigma\delta}$ when equipped…

We investigate $\mathcal F$-Borel topological spaces. We focus on finding out how a~complexity of a~space depends on where the~space is embedded. Of a~particular interest is the~problem of determining whether a~complexity of given space $X$…

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear…

We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is $D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed target…

Let $X$ be a Banach space with RNP, $(\vO,\vS,\mu)$ be a complete probability space and $\vG:\vO\to{cb(X)}$ (nonempty, closed convex and bounded subsets of $X$) be a multifunction. Assume that $\vX\subset\vS$ is a $\sigma$-algebra and the…

We show that for any countable group $ G $ equipped with a probability measure $ \mu $, there exists a randomized stopping time $ \tau $ such that $ (G, \mu _{\tau} )$ admits a strictly larger space of bounded harmonic functions than $…

Let $\mu$ be a Borel measure on a compactum $X$. The main objects in this paper are $\sigma$-ideals $I(dim)$, $J_0(\mu)$, $J_f(\mu)$ of Borel sets in $X$ that can be covered by countably many compacta which are finite-dimensional, or of…

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We…

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $\mu$ is an arbitrary $f$-invariant…

The Urysohn space is a complete separable metric space, universal among separable metric spaces for extending finite partial isometries into it. We present an alternative construction of the Urysohn space which enables us to show that…

It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular,…

We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2^\omega, then X has $\frak{c}$ types of countable dense subsets. We suggest a generalization of the \lambda-set for non-separable spaces. Let…

Let $X,Y$ be algebraic varieties defined over $\Bbb R$. Assume $Y$ is smooth and $X$ is Gorenstein. Suppose $\varphi:X\to Y$ is a flat $\Bbb R$-morphism such that all the fibers have rational singularities. We show that the pushforward of…