Related papers: Turning Borel sets into clopen sets effectively

We introduce a new technique for the study of the local extension property (LEP) for boolean algebras and we use it to show that the clopen algebra of every compact Hausdorff space $K$ of finite height has LEP. This implies, under…

Let $(G,\cdot)$ be a Polish group. We say that a set $X \subset G$ is Haar null if there exists a universally measurable set $U \supset X$ and a Borel probability measure $\mu$ such that for every $g, h \in G$ we have $\mu(gUh)=0$. We call…

Topological invariants, including the Chern numbers, can topologically classify parameterized Hamiltonians. We find that topological invariants can be properly defined and calculated even if the parameter space is discrete, which is done by…

It is proved that two useful and apparently different metrics on the set of Borel probabilities on countable products of Polish spaces of bounded diameters are equal. This paves the way for advances in their computation.

We consider stability theory for Polish spaces and more generally for definable structures (say, with elements of a set of reals). We clarify by proving some equivalent conditions for $\aleph_0$-stability. We succeed to prove existence of…

With every $\sigma$-ideal $I$ on a Polish space we associate the $\sigma$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing notions of Borel sets modulo the respective $\sigma$-ideals $I$ and $I^*$ and find connections…

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

We construct interpolation operators for functions taking values in a symmetric space -- a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized…

We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a…

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of…

We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.

Perturbative expansions in physical applications are generically divergent, and their physical content can be studied using Borel analysis. Given just a finite number of terms of such an expansion, this input data can be analyzed in…

We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an…

We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real…

$L^p$ to $L^p_{\beta}$ boundedness theorems are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional…

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball $\B \sub \C^n$ with its relative logarithmic capacity in $\C^n$ with respect to the same ball $\B$.…

The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of the weak order…

It is shown the construction of a module structure [2] with universe over a set of a particular kind of mathematical proofs, the base ring of this module will be built on a maximal consistent extension of a set of propositions, this…

We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call \emph{the Jelonek set}, is a subset of…

Given a compact metric space $X$ and a probability measure in the $\sigma-$algebra of Borel subsets of $X$, we will establish a dominated convergence theorem for ultralimits of sequences of integrable maps and apply it to deduce a…