We obtain conditions for the existence and stability of de Sitter attractors in the phase space of homogeneous and isotropic cosmology in generalized theories of gravity (including non-linear and scalar-tensor theories). These conditions are valid for any form of the coupling functions of the theory. Stability with respect to inhomogeneous perturbations is analyzed using a covariant and gauge-i...

We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every n-dimensional homogeneous ANR is a topological n-manifold, whereas the Busemann Conjecture asserts that every n-dimensional G-space is a topological n-manifold. The key object in both cases are so-called generalized manifolds, i.e. ENR homology manifolds. We look at t...

A theory of gravitation is constructed in which all homogeneous and isotropic solutions are nonsingular, and in which all curvature invariants are bounded. All solutions for which curvature invariants approach their limiting values approach de Sitter space. The action for this theory is obtained by a higher derivative modification of Einstein's theory. We expect that our model can easily be gen...

A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands by the use of t-roots. We present new G-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics. 2000 Mat...

A theory of gravitation is constructed in which all homogeneous and isotropic solutions are nonsingular, and in which all curvature invariants are bounded. All solutions for which curvature invariants approach their limiting values approach de Sitter space. The action for this theory is obtained by a higher-derivative modification of Einstein s theory. We expect that our model can easily be gen...

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden’s identity, the expression fo...

Abstract. We study first some arrangements of hyperplanes in the n-dimensional projective space Pn(Fq). Then we compute, in particular, the second and the third highest numbers of rational points on hypersurfaces of degree d. As application of our results we obtain some weights of the Generalized Projective Reed-Muller codes PRM(q, d, n). And we also list all the homogeneous polynomials reachin...

in this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  we relate the cohomology of a normal subgroup $n$ of a topological group $g$ and the quotient $g/n$ to the cohomology of $g$. we get the inflation-restriction exact sequence. also, we obtain a seven-term exact cohomology sequence up to dimension 2. we give an interpretation of the first non-a...

In this paper, using the Steklov function, we introduce the generalized continuity modulus and denethe class of functions Wr;kp;' in the space Lp. For this class, we prove an analog of the estimates in [1]in the space Lp.

A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...