Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when the domain bounded by a graph of a function defined on Rn is also homogeneou...

For a ring R, denote by SpecΓ(κ,R) the κ-spectrum of the Γ-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that SpecΓ(א1 , R) is full for a suitable von Neumann regular algebra R, but the techniques do not extend to cardinals κ > א1. By a direct construction, we prove that for any field F and any regular uncountable cardinal κ there is a...

When 2−1/Q<p≤2, we establish the Cloc0,1 and Cloc1,α-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in Heisenberg group Hn, where Q=2n+2 is homogeneous dimension Hn.

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...

We prove an analogue for homogeneous trees and certain affine buildings of a result Bourgain on pinned distances in sets positive density Euclidean spaces. Furthermore, we construct example non-homogeneous tree with Hausdorff dimension, subset thereof, which not all sufficiently large (even) are realised.

In the present paper we describe one family of closed Riemannian manifolds with positive sectional curvature. Now the list of known examples is not large (for instance, all known manifolds with dimension > 24 are diffeomorphic to compact rank one symmetric spaces) (we restrict ourselves only by pointing out simply connected manifolds) : 1) Berger described all normally homogeneous closed positi...