We prove that any torsion-free, residually finite relatively free group of infinite rank is not א1-homogeneous. This generalizes R. Sklinos’ result that a free group of infinite rank is not א1-homogeneous, and, in particular, gives a new simple proof of that result.

Heisenberg groups are simply connected nilpotent Lie groups of class 2. A group is called almost homogeneous if its automorphism group acts with at most 3 orbits. Several open problems about the existence of embeddings between almost homogeneous Heisenberg groups have been posed in a previous paper by the second author. Most of these problems are solved. Mathematics Subject Classification 2000:...

We find a full strongly exceptional collection for the Cayley plane OP, the simplest rational homogeneous space of the exceptional group E6. This collection, closely related to the one given by the second author in [Man11], consists of 27 vector bundles which are homogeneous for the group E6, and is a Lefschetz collection with respect to the minimal equivariant embedding of OP.

In this article a review on the definition of the X- ray transform and some ofits applications in Nano crystallography is presented. We shall show that the X- raytransform is a special case of the Radon transform on homogeneous spaces when thetopological group E(n)- the Euclidean group - acts on ℝ2 transitively. First someproperties of the Radon transform are investigated then the relationship ...

An algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if A is a rational homogeneous algebra with dimA > 1 then A = 0.

A homogeneous family of subsets over a given set is one with a very “rich” automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of 2 א0 isomorphism types of homogeneous families over a countable set.