Related papers: The Filter Dichotomy and medial limits

This is a translation into English of a paper written in French, published in Tatra Mountains Mathematical Publications, {L'ultrafiltre, un outil incomparable}, Tatra Mt. Math. Publ. {\bf 31} (2005), 131-176.It was also posted as…

In this text, the filtering unitary group method developed, among others, by S. Schochet is adapted to prove the existence and well-posedness of modulation equations describing the incompressible limit of the Euler-Maxwell Two-Fluid (EMTF)…

We prove that a function definable with parameters in an o-minimal structure is bounded away from infinity as its argument goes to infinity by a function definable without parameters, and that this new function can be chosen independently…

We study some limitations and possible occurrences of uniform ultrafilters on ordinals without the axiom of choice. We prove an Easton-like theorem about the possible spectrum of successors of regular cardinals which carry uniform…

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…

We consider the following conjecture (from Huang, et al): Let $\Delta^+$ denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic…

For every integer \(n\ge 3\), every \(1\le \ell\le n-2\), and every sufficiently large integer \(m\), we construct harmonic functions \(u_{m,\ell}\) on the unit ball \(B_1(0)\subset\mathbb{R}^n\) such that the frequency is bounded…

We prove a uniform version of the Dynamical Mordell-Lang Conjecture for \'etale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined…

We show that the continuum hypothesis implies there exists a Lindelof space X such that X x X is the union of two metrizable subspaces but X is not metrizable. This gives a consistent solution to a problem of Balogh, Gruenhage, and Tkachuk.…

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the…

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fr\'echet filter.…

We define a mixed topology on the fiber space $\cup_\mu \oplus^n L^n(\mu)$ over the space $\mathcal{M}(\Omega)$ of all finite non-negative measures $\mu$ on a separable metric space $\Omega$ provided with Borel $\sigma$-algebra. We define a…

For separable metrizable spaces $X,Y$ and a metrizable topological group $Z$ by $S(X\times Y,Z)$ we denote the space of all separately continuous functions $f:X\times Y\to Z$ endowed with the topology of layer-wise uniform convergence,…

In this note, as a particular case of a more general result, we obtain the following theorem: Let $\Omega\subseteq {\bf R}^n$ be a non-empty bounded open set and let $f:\overline {\Omega}\to {\bf R}^n$ be a continuous function which is…

We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points…

A simple \(P_\lambda\)-point on a regular cardinal \(\kappa\) is a uniform ultrafilter on \(\kappa\) with a mod-bounded decreasing generating sequence of length \(\lambda\). We prove that if there is a simple $P_\lambda$-point ultrafilter…

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an…

Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower $\ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for…

A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice)…

Limit and Pseudotopological spaces are two generalizations of topological spaces which are defined by indicating what filters converge under some axioms. In this article, we introduce covering spaces and set forth some necessary conditions…