let s be a locally compact foundation semigroup with identity and                          be its semigroup algebra. let x be a weak*-closed left translation invariant subspace of    in this paper, we prove that  x  is invariantly  complemented in   if and  only if  the left ideal  of    has a bounded approximate identity. we also prove that a foundation semigroup with identity s is left amenab...

Let S be a discrete semigroup, m (S) the space of all bounded real functions on S with the usualsupremum norm. Let Acm (S) be a uniformly closed left invariant subalgebra of m (S) with 1 c A. We say that A is extremely left amenable if there isamultiplicative left invariant meanon A. Let P = {h ?A: h =|g-1,g | forsome g ?A, s ?S}. It isshown that . A is extremely left amenable if and only ...

When a plane undergoes a deformation that can be represented by a planar linear vector eld, the projected vector eld on the image plane of an optical device is at most quadratic. This 2D motion eld has one singular point, with eigenvalues identical to those of the singular point describing the deformation. As a consequence, the nature of the singular point of the deformation is a projective inv...

(1) ut = u+ f (x; u;ru) ; x 2 ; t > 0; @u @ = 0 on @ ; on suitable Sobolev spaces has attracted much interest (see [A, H] and more recent references at the end of this note). Many e¤orts have been made to show the complexity of its dynamical behavior (see some survey papers and recent articles [DP, P1, P2, P3, Pr, PR, R] and the references therein). In particular, the following nice result was ...

We prove the following result: If G is a nite irreducible reeection group deened over a base eld k, then the invariant eld of G is purely transcendental over k, even if jGj is divisible by the characteristic of k. It is well known that in the above situation the invariant ring is in general not a polynomial ring. So the question whether at least the invariant eld is purely transcendental (Noeth...

We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the unimodular case, the Ricci soliton equation does not admit solutions in the set of left invariant vector fields. We prove that a left invariant soliton of gradie...

We use topological quantum eld theory to derive an invariant of a three-manifold with boundary. We then show how to use the structure of this invariant as an obstruction to embedding one three-manifold into another. 1. Introduction In the mid 1980's the Jones polynomial of a link was introduced 5], 6]. It was deened as the normalized trace of an element of a braid group, corresponding to the li...

In this paper we give some characterizations of topological extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative topological left invariant mean then T is topological left lumpy if and only if there is a multiplicative topological left invariant mean M on M(S)* ...

We find all left-invariant minimal unit vector fields and strongly normal unit vector fields on a Lie group which is isometric to the hyperbolic space.