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The problem of selecting a small, yet high quality subset of patterns from a larger collection of itemsets has recently attracted lot of research. Here we discuss an approach to this problem using the notion of decomposable families of…

In this article, we study "questionable representations" of (partial or total) orders, introduced in our previous article "A class of orders with linear? time sorting algorithm". (Later, we consider arbitrary binary functional/relational…

This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each…

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this…

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from…

We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated…

In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally,…

A combinatorial characterization of measurable filters on a countable set is found. We apply it to the problem of measurability of the intersection of nonmeasurable filters.

Mathematica offers, by way of the package Combinatorics, many useful functions to work on graphs and ordered structures, but none of these functions was specific enough to meet the needs of our research group. Moreover, the existing…

We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological…

We consider fixed-point equations for probability distributions on isometry classes of measured metric spaces. The construction is required to be recursive and tree-like, but we allow loops for the geodesics between points in the support of…

Individual choices often depend on the order in which the decisions are made. In this paper, we expose a general theory of measurable systems (an example of which is an individual's preferences) allowing for incompatible (non-commuting)…

We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization…

We investigate nonlinear Dvoretzky's theorem for countably infinite metric spaces and analytic sets whose Hausdorff dimension are infinite.

We consider problems concerning the partial order structure of the set of spreading models of Banach spaces. We construct examples of spaces showing that the possible structure of these sets include certain classes of finite semi-lattices…

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of…

This paper demonstrates that every ultrametric space is homeomorphic to a clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore, it characterizes several topological properties of ultrametrizable spaces through the…

We consider finite point subsets (distributions) in compact metric spaces. Non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given in the case of general…

A problem posed by Erd\H{o}s in 1945 initiated the study of non-separable arrangements of convex bodies. A finite collection of convex bodies in Euclidean $d$-space is called a non-separable family (or NS-family) if every hyperplane…

While routinely used in other areas of dynamics, image sets are ill-defined objects in general non-invertible measurable dynamics. We propose a way of consistently working with image sets of null-preserving (and hence, in particular, of…