相关论文: The ultrafilter: A peerless tool
The ultrafilters on the partial order $([\omega]^{\omega},\subseteq^*)$ are the free ultrafilters on $\omega$, which constitute the space $\omega^*$, the Stone-Cech remainder of $\omega$. If $U$ is an upperset of this partial order (i.e., a…
This paper aims to provide an analysis of what it means when we say that a pair of theories, very generously construed, are equivalent in the sense that they are interdefinable. With regard to theories articulated in first order logic, we…
Several papers have been published recently, particularly in the field of topological insulators, that use a Fourier-transform technique in a way that can be misleading or otherwise inaccurate, and the processing is not always described in…
We prove that there exists a nonprincipal ultrafilter $\mathcal U$ on $\mathbb N$ such that for every countable (or separable) structure $B$ in a countable language the quotient map from the reduced product associated with the Fr\'echet…
It has been widely believed for half a century that there will never exist a nonlinear theory of generalized functions, in any mathematical context. The aim of this text is to show the converse is the case and invite the reader to…
The interrelations between various classes of convergence spaces defined by countability conditions are studied. Remarkably, they all find characterizations in the usual space of ultrafilters in terms of classical topological properties.…
This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular,…
Complex systems in nature and in society are often represented as networks, describing the rich set of interactions between objects of interest. Many deterministic and probabilistic clustering methods have been developed to analyze such…
We present the basic theory of central subsets of semigroups from the nonstandard perspective. A key feature of this perspective is the replacement of the algebra of ultrafilters with the algebra of elements of iterated hyperextensions, a…
Jacques Peyri\`ere investigated Riesz products associated with a given set of frequencies and the corresponding coefficients : mutual singularity or absolute continuity of the measures defined by two such products, Hausdorff dimensions of…
The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this…
We use the ultrafilter-convergence axiomatics for topological spaces to motivate in detail a gentle categorical introduction, first to Barr's Set-based relational T-algebras, and then to Burroni's T-preorders internal to a category C, here…
Following Baumgartner [J. Symb. Log. 60 (1995), no. 2], for an ideal $\mathcal{I}$ on $\omega$, we say that an ultrafilter $\mathcal{U}$ on $\omega$ is an $\mathcal{I}$-ultrafilter if for every function $f:\omega\to\omega$ there is $A\in…
We give several topological/combinatorial conditions that, for a filter on $\omega$, are equivalent to being a non-meager $\mathsf{P}$-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a…
We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin--Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF.…
We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of…
Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips…
In 2005, the paper "Fraiss\'e limits, Ramsey theory, and topological dynamics of automorphism groups" [KPT] by Kechris, Pestov and Todorcevic provided a powerful tool to compute an invariant of topological groups known as the universal…
These notes started to educate ourselves and to collect some background for our future work, with the hope that perhaps they will be useful to others also. Many if not all results are more or less elementary or available in the literature,…
We report a recent developement on the theory of upper conical densities. More precicely, we look at what can be said in this respect for other measures than just the Hausdorff measure. We illustrate the methods involved by proving a result…