Volume 13, Issue 4

In order to detect formality of the DeRham algebra of a smooth manifold, we determine conditions that ensure that a general dδ-lemma holds; then, as an application of the previous constructions, we obtain a new characterization of Kähler structures.

Skew polynomial rings are used to construct finite semifields, following from a construction of Ore and Jacobson of associative division algebras. Johnson and Jha [10] constructed the so-called cyclic semifields, obtained using irreducible semilinear transformations. In this work we show that these two constructions in fact lead to isotopic semifields, show how the skew polynomial construction can be used to calculate the nuclei more easily, and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in [13] and implicitly by Dempwolff in [2].

We introduce the notion of CR quaternionic map and we prove that any such realanalytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold M of dimension 4k-1 of a quaternionic manifold N such that TM generates a quaternionic subbundle of TN| M of (real) rank 4k, there exists, locally, a quaternionic submanifold of N containing M as a hypersurface.

The aim of this paper is to give a characterization of a maximal curve given by the equation x n + y m = 1 over a finite field F q 2 .

Given a quasianalytic structure, we prove that the singular locus of a quasi-subanalytic set E is a closed quasi-subanalytic subset of E. We rely on some stabilisation effects linked to Gateaux differentiability and formally composite functions. An essential ingredient of the proof is a quasianalytic version of Glaeser’s composite function theorem, presented in our previous paper.

The Brunn-Minkowski theory in convex geometry relies heavily on the notion of mixed volumes. Despite its particular importance, even explicit representations for the mixed volumes of two convex bodies in Euclidean space ℝ d are available only in special cases. Here we investigate a new integral representation of such mixed volumes, in terms of flag measures of the involved convex sets. A brief introduction to (extended) flag measures of convex bodies is also provided.

We study the Tutte polynomial of two infinite families of finite graphs: the Sierpi´nski graphs, which are finite approximations of the well-known Sierpi´nski gasket, and the Schreier graphs of the Hanoi Towers group H (3) acting on the rooted ternary tree. For both of them, we recursively describe the Tutte polynomial and we compute several special evaluations of it, giving interesting results about the combinatorial structure of these graphs.

In this paper, we obtain inequalities between the Ricci curvature, the scalar curvature and the squared mean curvature of submanifolds in generalized Sasakian space forms such that each inequality is a lower bound for the Ricci curvature. Also, we obtain a sharp relationship between the scalar curvature, a Riemannian invariant and the squared mean curvature of anti-invariant submanifolds in generalized Sasakian space forms.

Let S k,m be the pseudosphere of signature ( k,m ). We show that the space ℒ 0 (S k,m ) of all oriented null geodesics in S k,m is a manifold, and we describe geometrically its canonical contact distribution in terms of the space of oriented geodesics of certain totally geodesic degenerate hypersurfaces in Sk;m. Further, we find a contactomorphism with some standard contact manifold, namely, the unit tangent bundle of some pseudo-Riemannian manifold. Also, we express the null billiard operator on ℒ 0 (S k,m ) associated with some simple regions in Sk;m in terms of the geodesic flows of spheres. For the pseudo-Riemannian product N of two complete Riemannian manifolds, we give geometrical conditions on the factors for ℒ 0 (N) to be manifolds and exhibit a contactomorphism with some standard contact manifold.

We characterize the combinatorial types of stacked d-polytopes that are inscribable. Equivalently, we identify the triangulations of a simplex by stellar subdivisions that can be realized as Delaunay triangulations

In the paper we prove some differential Harnack estimates for positive solutions of backward heat equations with potentials when the metric is evolving by an extended Ricci flow. We also give some applications of these estimates