In his paper [1], Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is Kσ, the countable union of compact sets. He notes that the group of rationals under addition with the discrete topology is an example of a Polish group which is Kσ (since it is countable) but not compactly generated (since compact subsets are finite). Beros showe...

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space En = G/K where G is the semidirect product Rn · K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)–spherical functions by a certain affine algebraic variety, and of the positive definite on...

A generalization of the crust algorithm is presented that will reconstruct a smooth d-dimensional submanifold of R. When the point sample meets satisfy a minimal density requirement this reconstruction is homeomorphic to the original submanifold. In fact the reconstructed manifold is ambiently isotopic to the original via an isotopy that moves points a small distance. Also, bounds are given com...

In this paper, we define X-slant helix in Euclidean 3-space and we obtain helix, slant helix, clad and g-clad helix as special case of the X-slant helix. Then we study Darboux, tangential darboux developable surfaces and their singular points. Especially we show that the striction lines of these surfaces are singular locus of the surfaces.

Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed curve F in the complex plane, we introduce the notion of the co...