Riesz transform on manifolds and heat kernel regularity

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is Lp bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain Lp estimate in the same interval of p's.     

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

The FBI transform on compact manifolds

We present a geometric theory of the Fourier-Bros-Iagolnitzer transform on a compact manifold . The FBI transform is a generalization of the classical notion of the wave-packet transform. We discuss the mapping properties of the FBI transform and its relationship to the calculus of pseudodifferential operators on . We also describe the microlocal properties of its range in terms of the ``scattering calculus' of pseudodifferential operators on the noncompact manifold .

     

小波变换与矩心

利用多分辨分析的思想，研究了信号经波波变换后矩心的计算问题。     

Horospherical transform on Riemannian symmetric manifolds of noncompact type

We discuss I. M. Gelfand’s project of rebuilding the representation theory of semisimple Lie groups on the basis of integral geometry. The basic examples are related to harmonic analysis and the horospherical transform on symmetric manifolds. Specifically, we consider the inversion of this transform on Riemannian symmetric manifolds of noncompact type. In the known explicit inversion formulas, the nonlocal part essentially depends on the type of the root system. We suggest a universal modification of this operator.     

Wavelet transform associated with linear canonical Hankel transform

The main goal of this paper is to study about the continuous as well as discrete wavelet transform in terms of linear canonical Hankel transform (LCH‐transform) and discuss some of its basic properties. Parseval's relation and reconstruction formula of continuous linear canonical Hankel wavelet transform (CLCH‐wavelet transform) is obtained. Moreover, semidiscrete and discrete LCH‐wavelet transform are also discussed.     

Connections on manifolds and new characteristic classes

We give an extension of Maslov-Arnold classes to a certain class of symplectic manifolds. It is proved that any such generalized class of minimal surfaces is equal to zero for a large class of stable minimal surfaces. We describe some applications to pseudo-Riemannian geometry and to the investigation of completely integrable Hamiltonian systems.     

Comparative analysis of two asymptotic approaches based on integral manifolds

A comparative analysis of the two powerful asymptotic methods,ILDM and MIM (intrinsic low-dimensional manifolds; method ofinvariant manifold), is presented in the paper. The two methodsare based on the general theory of integral manifolds. The ILDMmethod is able to handle large systems of ODEs, whereas theMIM method treats systems with a limited number of unknown variables.The MIM method allows one to conduct analytical explorationof the original system and to obtain final expressions in compactform, whereas the ILDM method is a numerical approach that yieldsthe numerical form of the desired surface. The ILDM method workswell in a region where a rough splitting of the initial systemexists. Regions of the phase space where splitting does notexist are problematic for the ILDM method. In these regionsthe MIM method provides additional information regarding thedynamical behaviour of the system. A number of simple examplesare considered and analysed. It is shown that for the Semenovmodel (singularly perturbed system of ODEs) the ILDM methodgives a surface which appears close to the first order (withrespect to the corresponding small parameter) approximationof the stable (attracting) invariant manifolds. The complementaryproperties of the two asymptotic approaches suggests a feasiblecombination of the two methods, which is the subject of a futurework.     

TRANSPORTATION NETWORKS： OLD AND NEW

This paper shows a number of problems in pure and applied mathematics that are solved by constructing transportation networks Moreover, it also shows thatall the solutions are characterized by forbidden configuratlozz which are not minors.However, all the characterisations are much related to the graphic method which was found by Chinese for solving a kind of the transportation problem in the fifties.     

Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap

In this paper we study the Riesz transform on complete and connected Riemannian manifolds M with a certain spectral gap in the L² spectrum of the Laplacian. We show that on such manifolds the Riesz transform is Lp bounded for all p∈(1,∞). This generalizes a result by Mandouvalos and Marias and extends a result by Auscher, Coulhon, Duong, and Hofmann to the case where zero is an isolated point of the L² spectrum of the Laplacian.     

Cylindrical Kadomtsev-Petviashvili equation: Old and new results

We review results on the cylindrical Kadomtsev-Petviashvili (CKP) equation, also known as the Johnson equation. The presentation is based on our results. In particular, we show that the Lax pairs corresponding to the KP and the CKP equations are gauge equivalent. We also describe some important classes of solutions obtained using the Darboux transformation approach. We present plots of exact solutions of the CKP equation including finite-gap solutions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 304–320, August, 2007.     

Uncertainty principles for the continuous Hankel Wavelet transform

The aim of this paper is to prove Heisenberg-type uncertainty principles for the continuous Hankel wavelet transform. We also analyse the concentration of this transform on sets of finite measure. Benedicks-type uncertainty principle is given.     

A new class of quaternionic manifolds

Riemannian manifolds with structure group Sp(n)·Sp(1) are studied. Moreover for the Riemannian manifolds with structure group Sp(n) a new class is defined, and example of compact nilmanifolds in this class are constructed.     

A new construction of semi-free actions on Menger manifolds

A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .