Scattered Subsets of Groups

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets.Розріджені підмножини групи визначено, як асимптотичні аналоги розраджених підпросторів топологічного простору. Доведено, що підмножина A групи G є розрідженою тоді i тільки тоді, коли A не містить кусково-зсунутих IP-підмножин. Показано, що для аменабельної групи G та розрідженого підпростору A групи G рівність μ(A)=0 виконується для кожної лівої інваріантної банахової міри μ на G. Встановлено, що кожну нескінченну групу можна розбити на ℵ0 розріджених підмножин

On the subset Combinatorics of G-spaces

Let $G$ be a group and let $X$ be a transitive $G$-space. We classify the subsets of $X$ with respect to a translation invariant ideal $\mathcal{J}$ in the Boolean algebra of all subsets of $X$, introduce and apply the relative combinatorical derivations of subsets of $X$. Using the standard action of $G$ on the Stone-$\check{C}$ech compactification $\beta X$ of the discrete space $X$, we characterize the points $p\in\beta X$ isolated in $Gp$ and describe a size of a subset of $X$ in terms of its ultracompanions in $\beta X$. We introduce and characterize scattered and sparse subsets of $X$ from different points of view

On better-quasi-ordering classes of partial orders

We provide a method of constructing better-quasi-orders by generalising a technique for constructing operator algebras that was developed by Pouzet. We then generalise the notion of $\sigma$-scattered to partial orders, and use our method to prove that the class of $\sigma$-scattered partial orders is better-quasi-ordered under embeddability. This generalises theorems of Laver, Corominas and Thomass\'{e} regarding $\sigma$-scattered linear orders and trees, countable forests and N-free partial orders respectively. In particular, a class of countable partial orders is better-quasi-ordered whenever the class of indecomposable subsets of its members satisfies a natural strengthening of better-quasi-order.Comment: v1: 45 pages, 8 figures; v2: 44 pages, 11 figures, minor corrections, fixed typos, new figures and some notational changes to improve clarity; v3: 45 pages, 12 figures, changed the way the paper is structured to improve clarity and provide examples earlier o

Topological radicals, V. From algebra to spectral theory

We introduce and study procedures and constructions in the theory of the joint spectral radius that are related to the spectral theory. In particular we devlop the theory of the scattered radical. Among applications we find some sufficient conditions of continuity of the spectrum and spectral radii of various types, and prove that in GCR C*-algebras the joint spectral radius is continuous on precompact subsets and coincides with the Berger-Wang radius

Extending the range of error estimates for radial approximation in Euclidean space and on spheres

We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of R^d and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error.Comment: 10 page

Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif

A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees

Scattered classes of graphs

For a class $\mathcal C$ of graphs $G$ equipped with functions $f_G$ defined on subsets of $E(G)$ or $V(G)$, we say that $\mathcal{C}$ is $k$-scattered with respect to $f_G$ if there exists a constant $\ell$ such that for every graph $G\in \mathcal C$, the domain of $f_G$ can be partitioned into subsets of size at most $k$ so that the union of every collection of the subsets has $f_G$ value at most $\ell$. We present structural characterizations of graph classes that are $k$-scattered with respect to several graph connectivity functions. In particular, our theorem for cut-rank functions provides a rough structural characterization of graphs having no $mK_{1,n}$ vertex-minor, which allows us to prove that such graphs have bounded linear rank-width.Comment: 42 pages, 5 figures. Adding a new section comparing these concepts with tree-depth, rank-depth, shrub-depth, modular-width, neighborhood diversity, et

Recent progress in subset combinatorics of groups

We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in $G$-spaces, new ideals in the Boolean algebra $\mathcal{P}_{G}$ of all subsets of a group $G$ and in the Stone-$\check{C}$ech compactification $\beta G$ of $G$ , the combinatorial derivation.Comment: Large, small, thin, thick, sparse and scattered subsets of groups; descriptive complexity; Boolean algebra of subsets of a group; Stone-$\check{C}$ech compactification; ultracompanion; Ramsey-product subset of a group; recurrence; combinatorial derivation. arXiv admin note: text overlap with arXiv:1704.0249

Ultrafilters on GG-spaces

For a discrete group $G$ and a discrete $G$-space $X$, we identify the Stone-\v{C}ech compactifications $\beta G$ and $\beta X$ with the sets of all ultrafilters on $G$ and $X$, and apply the natural action of $\beta G$ on $\beta X$ to characterize large, thick, thin, sparse and scattered subsets of $X$. We use $G$-invariant partitions and colorings to define $G$-selective and $G$-Ramsey ultrafilters on $X$. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on $\omega$, the $T$-points, and study interrelations between these ultrafilters and some classical ultrafilters on $\omega$

A generalization of Kantorovich operators for convex compact subsets

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussedComment: Research articl