Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping Δ: P G → P G defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: P X→ P X, A ↦ A d, where X is a topological space and A d is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ∇. For example, we show that if G is infinite and I is an ideal in...

Choquet and Deny considered on an abelian locally compact topological group the representation of a measure $P$ as the convolution product of itself and a finite measure $Q:P=P^{*}Q$.In this paper, we make an attempt to find, in the case of certain locally compact semigroups, those solutions $P$ of the above equation which are relatively invariant on the support of $Q$. A characterization of relatively invariant measures on certain locally compact semigroups is also presented. Our results on the above convolution...

Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{\Lambda }\left(G\right)$ or $L{¹}_{\Lambda }\left(G\right)$ and which quotients of the form $C\left(G\right)/C_{\Lambda }\left(G\right)$ or $L¹\left(G\right)/L{¹}_{\Lambda }\left(G\right)$ have the Daugavet property. We show that $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) if and only if $\Gamma \setminus {\Lambda }^{-1}$ is a semi-Riesz set. If $L{¹}_{\Lambda }\left(G\right)$ is a rich subspace of L¹(G), then $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C\left(G\right)/C_{\Lambda }\left(G\right)$ has the Daugavet property if Λ is a Rosenthal set, and that $L{¹}_{\Lambda }\left(G\right)$ is a poor subspace of L¹(G) if Λ is...

We survey results concerning the extent to which information about a convex body's projections or sections determine that body. We will see that, if the body is known to be centrally symmetric, then it is determined by the size of its projections. However, without the symmetry condition, knowledge of the average shape of projections or sections often determines the body. Rather surprisingly, the dimension of the projections or sections plays a key role and exceptional cases do occur but appear to...

Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow \widehat{D}$ given by $h\mapsto h|D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

For any topological group $G$ the dual object $\widehat{G}$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\widehat{G}$ is discrete. In an earlier paper we proved that $\widehat{G}$ is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when $G$ is an almost metrizable precompact group.

Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to ${ℝ}^k$, k>1. We consider a class of second order left-invariant differential operators on S of the form ${ℒ}_{\alpha }=L^a+{\Delta }_{\alpha }$, where $\alpha \in {ℝ}^k$, and for each $a\in {ℝ}^k,L^a$ is left-invariant second order differential operator on N and ${\Delta }_{\alpha }=\Delta -⟨\alpha ,\nabla ⟩$, where Δ is the usual Laplacian on ${ℝ}^k$. Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively) we obtain an...