Lempert mappings and symplectic forms

We use Lempert's version of Riemann mapping to construct non-equivalent symplectic forms on an ellipsoid in .

     

Characterizations of Sobolev spaces in Euclidean spaces and Heisenberg groups

Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].     

Blow-up of regular submanifolds in Heisenberg groups and applications

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.     

Regular orthogonal basis on Heisenberg group and application to function spaces

In this paper, we construct some compactly supported orthogonal regular wavelet basis on Heisenberg group . Because of the regularity of wavelets, we could use such wavelets to characterize function spaces on , such as bounded mean oscillation space (BMO) space, Hardy space, Besov spaces and Besov–Morrey spaces. Copyright © 2014 John Wiley & Sons, Ltd.     

Fundamental groups of complements of plane curves and symplectic invariants

Introducing the notion of stabilized fundamental group for the complement of a branch curve in , we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these invariants and formulate conjectures supported by calculations on new examples.     

Basic differential forms for actions of Lie groups

A section of a Riemannian -manifold is a closed submanifold which meets each orbit orthogonally. It is shown that the algebra of -invariant differential forms on which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on which are invariant with respect to the generalized Weyl group of , under some condition.

     

Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups*

The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three-dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals–Gaveau–Greiner. Third, we prove that the uniform asymptotic behaviour at infinity (so the small-time asymptotic behaviour) of the heat kernel for Grushin operators, obtained by the first author, are still valid in two and three dimensions.     

Basic differential forms for actions of Lie groups. II

The assumption in the main result of Basic differential forms for actions of Lie groups (Proc. Amer. Math. Soc. 124 (1996), 1633-1642) is removed.

     

On harmonic and asymptotically harmonic homogeneous spaces

We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek–Ricci space. Independently, Y. Nikolayevsky has obtained the latter classification under the additional assumption of nonpositive sectional curvatures [N2]. Supported in part by DFG priority program “Global Differential Geometry” (SPP 1154). Received: September 2004; Revision: June 2005; Accepted: September 2005     

Witt kernels of quadratic forms for purely inseparable multiquadratic extensions in characteristic 2

The aim of this paper is to give a complete answer to the question of hyperbolicity of nonsingular quadratic forms over purely inseparable multiquadratic extensions in characteristic . This completes partial computations of Mammone and Moresi.

     

Extensions of the Heisenberg group by one‐parameter groups of dilations which are subgroups of the affine and the symplectic groups

Extensions of the multidimensional Heisenberg group by one‐parameter groups of matrix dilations are introduced and classified up to isomorphism. Each group is isomorphic to both, a subgroup of the symplectic group and a subgroup of the affine group, and its metaplectic representation splits into two irreducible subrepresentations, each of which is equivalent to a subrepresentation of the wavelet representation.     

A Newton basis for Kernel spaces

It is well known that representations of kernel-based approximants in terms of the standard basis of translated kernels are notoriously unstable. To come up with a more useful basis, we adopt the strategy known from Newton’s interpolation formula, using generalized divided differences and a recursively computable set of basis functions vanishing at increasingly many data points. The resulting basis turns out to be orthogonal in the Hilbert space in which the kernel is reproducing, and under certain assumptions it is complete and allows convergent expansions of functions into series of interpolants. Some numerical examples show that the Newton basis is much more stable than the standard basis of kernel translates.     

An integral recursive inequality and applications

An integral recursive inequality for two functions is obtained. It is used to describe the equality cases in the related inequalities. The applications involve some bi-Hermitian forms, integral transformations, and confluent hypergeometric functions.      

Random attractors for a stochastic hydrodynamical equation in Heisenberg paramagnet

This article studies the asymptotic behaviors of the solution for a stochastic hydrodynamical equation in Heisenberg paramagnet in a two-dimensional periodic domain. We obtain the existence of random attractors in˙ H 1 .     

On the conjugacy classes in the orthogonal and symplectic groups over algebraically closed fields

Let F$\mathbb{F}$ be an algebraically closed field. Let V$\mathbb{V}$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form B   over F$\mathbb{F}$. Suppose the characteristic of F$\mathbb{F}$ is sufficiently large  , i.e. either zero or greater than the dimension of V$\mathbb{V}$. Let I(V,B)$I(\mathbb{V},\mathbb{B})$ denote the group of isometries. Using the Jacobson–Morozov lemma we give a new and simple proof of the fact that two elements in I(V,B)$I(\mathbb{V},\mathbb{B})$ are conjugate if and only if they have the same elementary divisors.