Generalized intersection bodies

We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann–Petty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of generalized intersection-bodies. A second type was introduced by Koldobsky, who studied a different analytic generalization of this problem. Koldobsky also studied the connection between these two types of bodies, and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized Busemann–Petty problem. We show that these classes share many identical structural properties, proving the same results using integral geometry techniques for Zhang's class and Fourier transform techniques for Koldobsky's class. Using a functional analytic approach, we give several surprising equivalent formulations for the equivalence problem, which reveal a deep connection to several fundamental problems in the integral geometry of the Grassmann manifold.     

THE HOMOTHETIC MOTIONS IN THE LORENTZ 3-SPACE

In this article, the properties of the homothetic motions in three-dimensional Lorentz space are investigated. Also, some geometric results between velocity and acceleration vectors of a point in a spatial motion are obtained.     

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.     

Proving Theorems in Elementary Geometry with Clifford Algebraic Method

本文结合吴方法及平面几何的Ｃｌｉｆｏｒｄ代数表示，提出了几何定理机器证明的一种完备的方法．用这种方法证明定理时，三角化的过程及证明的过程通常较以前的方法更简短而且它们是可以几何解释的．     

Two‐dimensional anisotropic Cartesian mesh adaptation for the compressible Euler equations

Simulating transient compressible flows involving shock waves presents challenges to the CFD practitioner in terms of the mesh quality required to resolve discontinuities and prevent smearing. This paper discusses a novel two‐dimensional Cartesian anisotropic mesh adaptation technique implemented for transient compressible flow. This technique, originally developed for laminar incompressible flow, is efficient because it refines and coarsens cells using criteria that consider the solution in each of the cardinal directions separately. In this paper, the method will be applied to compressible flow. The procedure shows promise in its ability to deliver good quality solutions while achieving computational savings. Transient shock wave diffraction over a backward step and shock reflection over a forward step are considered as test cases because they demonstrate that the quality of the solution can be maintained as the mesh is refined and coarsened in time. The data structure is explained in relation to the computational mesh, and the object‐oriented design and implementation of the code is presented. Refinement and coarsening algorithms are outlined. Computational savings over uniform and isotropic mesh approaches are shown to be significant. Copyright © 2004 John Wiley & Sons, Ltd.     

When Do Some Geodesics of the Induced and the Metrical Connection Coincide?

In affine differential geometry, three natural connections can be considered, namely the induced connection, the Levi Civita connection of the affine metric and the conjugate connection. Quadrics are characterized by the conditions that all the pre-geodesies with respect to one of these connections are also pre-geodesies with respect to the other two connections. In the present paper we describe 3-dimensional affine hypersurfaces for which some of the geodesies or pre-geodesies for the different connections coincide.     

Extension of Fréchet valued real analytic functions from subvarieties of ℝd

It is shown that for an algebraic subvariety X of ℝ^d every Fréchet valued real analytic function on X can be extended to a real analytic function on ℝ^d if and only if X is of type (PL), i.e. all of its singularities are of a certain type. Necessity of this condition is shown for any real analytic variety. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inequivalent quantizations in multiply connected spaces

Summary The novel features which show up in the process of quantization of a dynamical system on a multiply (nonsimply)-connected configuration space are analysed in the present paper. After rediscussing the path integral approach to the problem, we show how one can give a fiber bundle classification of the inequivalent quantizations associated with nonsimply connected spaces. We discuss various examples, and the generalization to the case in which, due to internal symmetries, a system can admit of nonscalar quantizations. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

On Spin Structures and Dirac Operators on the Noncommutative Torus

We find and classify possible equivariant spin structures with Dirac operators on the noncommutative torus, proving that, similarly as in the classical case, the spectrum of the Dirac operator depends on the spin structure.