Related papers: Borel and Continuous Systems of Measures

We construct a smooth nontrivial mixed partially hyperbolic system and explicitly identify its skeleton. This example shares characteristics with the classical examples. Moreover, the support of each physical measure contains three fixed…

This paper studies the measure-preserving homeomorphisms on compact groups and proposes new methods for constructing measure-preserving homeomorphisms on direct products of compact groups and non-commutative compact groups. On the direct…

Although extracted from several experiments using hadronic probes \cite{bor1}, narrow baryonic structures have been sometimes met with disbelief. New signatures are presented, which appear from already published data, obtained with hadronic…

This monograph introduces key concepts and problems in the new research area of Periodic Geometry and Topology for materials applications.Periodic structures such as solid crystalline materials or textiles were previously classified in…

Let $(X,d)$ be a compact metric space, and let an iterated function system (IFS) be given on $X$, i.e., a finite set of continuous maps $\sigma_{i}$: $ X\to X$, $i=0,1,..., N-1$. The maps $\sigma_{i}$ transform the measures $\mu $ on $X$…

We consider the problem of generating hypothesis from data based on ideas from logic. We introduce a notion of barcodes, which we call sequent barcodes, that mirrors the barcodes in persistent homology theory in topological data analysis.…

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…

We study Borel homomorphisms $\theta : G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$\theta_*(\mu )(\alpha )=\mu (\theta ^{-1}(\alpha ))\quad \text{for } \quad \alpha \subseteq H…

A framework to handle tree decompositions of the components of a Borel graph in a Borel fashion is introduced, along the lines of Tserunyan's Stallings Theorem for equivalence relations arXiv:1805.09506. This setting leads to a notion of…

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…

This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…

A survey of recent results concerning cardinal invariants of measure and category. Submitted as a chapter of the upcoming Handbook of Set Theory.

The richness of many complex systems stems from the interactions among their components. The higher-order nature of these interactions, involving many units at once, and their temporal dynamics constitute crucial properties that shape the…

We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of…

We study equivalent descriptions of the vague, weak, setwise and total-variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent…

We study stable conditional measures for a certain equilibrium measure for hyperbolic endomorphisms, on basic sets with overlaps; we show that these conditional measures are geometric probabilities and measures of maximal stable dimension.…

A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the…

In this work we explore the geometrical interpretation of gauge theories through the formalism of fiber bundles. Moreover, we conduct an investigation in the topology of fiber bundles, providing a proof of the Classification Theorem. In the…

We give a self-contained and enriched review about topology properties in the rapidly growing field of topological states of matter (TSM). This review is mainly focus on the beautiful interplay of topology mathematics and condensed matter…

We consider the differential entropy of probability measures absolutely continuous with respect to a given $\sigma$-finite reference measure on an arbitrary measurable space. We state the asymptotic equipartition property in this general…