Roughly, the theorem says that each subgroup near enough to G* can be transformed into G* by an appropriate element of G. This result can be regarded as a generalization of the known fact that Lie groups cannot have arbitrarily small subgroups (other than the identity), although it was not from this point of view that our interest arose. To make our meaning clear, assume that G* is an invariant...

[Who doesn’t understand a glance, won’t understand a long explanation either.] – Arab proverb The problem of Similarity Searching consists in finding the elements from a set which are similar to a given query under some criterion. If the similarity is expressed by means of a metric, the problem is called Metric Space Searching. In this thesis we present new methodologies to solve this problem u...

A stationary space-time (M,g) is a 4-manifold M with a smooth Lorentzian metric g, of signature (−,+,+,+), which has a smooth 1-parameter group G ≈ R of isometries whose orbits are time-like curves in M . We assume throughout the paper that M is a chronological space-time, i.e. M admits no closed time-like curves, c.f. §1.1 for further discussion. Let S be the orbit space of the action G. Then ...

The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G′(V,E) as the representation of the metric database. A t-spanner is a subgraph G′(V,E) of a graph G(V,A), such that E ⊆ A and G′ approximates the s...

In this paper we prove that the length spectrum metric is topologically equivalent to the Teichmüller metric in Teichmüller space T (g,m, n) . This result solved a problem suggested by Sorvali [9] in 1972.

In the present work we study the dynamic behavior of the orbits of the natural action of the group G of isometries on a locally compact metric space X using suitable closed-open subsets of X . Precisely, we study the dynamic behavior of an orbit even in cases where G is not locally compact with respect to the compactopen topology. In case G is locally compact we decompose the space X into close...

This paper extends parts of the results from [14] for plane curves to the case of surfaces in Rn. Let M be a compact connected oriented manifold of dimension less than n without boundary. Then shape space is either the manifold of submanifolds of Rn of type M , or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M . We investigate the Sobolev Riemannian metrics on ...

The propagator W(t0,t1)(g,h) for the wave equation in a given space-time takes initial data (g(x),h(x)) on a Cauchy surface {(t,x) : t=t0} and evaluates the solution (u(t1,x),∂ tu(t1,x)) at other times t1. The Friedmann-Robertson-Walker space-times are defined for t0,t1>0, whereas for t0→0, there is a metric singularity. There is a spherical means representation for the general solution of the ...

We prove a related fixed point theorem for n mappings which arenot necessarily continuous in n fuzzy metric spaces using an implicit relationone of them is a sequentially compact fuzzy metric space which generalizeresults of Aliouche, et al. [2], Rao et al. [14] and [15].