On Lower Bounds for the Chromatic Number of Spheres

Mathematical Notes - Estimates of the chromatic numbers of spheres are studied. The optimality of the choice of the parameters of the linear-algebraic method used to obtain these estimates is...     

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation.     

Admissible decomposition for spectral multipliers on Gaussian $$L^p$$ L p

Mathematische Zeitschrift - This paper concerns harmonic analysis of the Ornstein–Uhlenbeck operator L on the Euclidean space. We examine the method of decomposing a spectral multiplier...     

Fundaments of Quaternionic Clifford Analysis I: Quaternionic Structure

Introducing a quaternionic structure on Euclidean space, the fundaments for quaternionic and symplectic Clifford analysis are studied in detail from the viewpoint of invariance for the symplectic group action.     

四元数Hurwitz环上的不可分解二次型

该文证明,变量不少于7个时存在不可分解的四元数整二次型     

Congruence Criteria for Finite Subsets of Quaternionic Elliptic and Quaternionic Hyperbolic Spaces

We investigate congruence classes of m-tuples of points in the quaternionic elliptic space ?P ⁿ . We establish a canonical bijection between the set of congruence classes of m-tuples of points in ?P ⁿ and the set of equivalence classes of positive semidefinite Hermitian m×m matrices of rank at most n+1 with the 1's on the diagonal. We show that with each m-tuple of points in ?P ⁿ is associated a tuple of points on the real unit sphere S ². Then we get that the congruence class of an m-tuple of points in ?P ⁿ is determined by the congruence classes of all its triangles and by the direct congruence class of the associated tuple on the sphere S ² provided that no pair of points of the m-tuple has distance π/2. Finally we carry out the same kind of investigation for the quaternionic hyperbolic space ?H ⁿ . Most of the results are completely analogous, although there are also some interesting differences.     

Functions Operating on Positive Semidefinite Quaternionic Matrices

 We study functions on the quaternionic unit ball which operate on positive semidefinite matrices in the sense that is positive semidefinite whenever is a positive semidefinite square matrix with entries . (Received 21 September 2000; in revised form 8 March 2001)     

Multidimensional Quaternionic Gabor Transforms

In this paper, we extend the Gabor transform to the quaternion valued functions on \({\mathbb{R}^{d}}\) in two different ways, where \({d\in \mathbb{N}}\) is arbitrary. We prove that the quaternionic Gabor transforms satisfy the properties including Parseval relation, inversion formula, linearity and uncertainity principle. We also present an extension of a quaternionic Gabor transform to Boehmians.     

Functions Operating on Positive Semidefinite Quaternionic Matrices

 We study functions on the quaternionic unit ball which operate on positive semidefinite matrices in the sense that is positive semidefinite whenever is a positive semidefinite square matrix with entries .     

$$L^p$$ L p -Multipliers Sensitive to the Group Structure on Nilpotent Lie Groups

Journal of Fourier Analysis and Applications - We propose new sufficient conditions for $$L^p$$ -multipliers on homogeneous nilpotent groups. The multipliers generalise the flag multipliers of...     

Quaternionic algebraic cycles and reality

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group . Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour's quaternionic -theory, and the other one classifies an equivariant cohomology theory which is a natural recipient of characteristic classes for quaternionic bundles over Real spaces .

     

Quaternionic pryms and Hodge classes

Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n−1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomology group. In general, it is not clear that these are cycle classes. In this paper we show that a particular 6-dimensional family of such 8-folds are Prym varieties and we use the method of Schoen to show that all Hodge classes on the general abelian variety in this family are algebraic. We also consider Hodge classes on certain 5-dimensional subfamilies and relate these to the Hodge conjecture for abelian 4-folds.     

Volume-Minimizing Foliations on Spheres

The volume of a k-dimensional foliation in a Riemannian manifold Mⁿ is defined as the mass of the image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing the construction by Gluck and Ziller (Comment. Math. Helv. 61 (1986), 177–192), ‘singular’ foliations by 3-spheres are constructed on round spheres S⁴ⁿ⁺³, as well as a singular foliation by 7-spheres on S¹⁵, which minimize volume within their respective relative homology classes. These singular examples, even though they are not homologous to the graph of a foliation, provide lower bounds for volumes of regular three-dimensional foliations of S⁴ⁿ⁺³ and regular seven-dimensional foliations of S¹⁵, since the double of these currents will be homologous to twice the graph of any smooth foliation by 3-manifolds.The second author was supported during this research by grants from the Universidade de Sāo Paulo, FAPESP Proc. 1999/02684-5, and Lehigh University, and thanks those institutions for enabling the collaboration involved in this work.Mathematics Subject Classifications (2000). 53C12, 53C38.     

Quadratic Interpolation on Spheres

Riemannian quadratics are C ¹ curves on Riemannian manifolds, obtained by performing the quadratic recursive deCastlejeau algorithm in a Riemannian setting. They are of interest for interpolation problems in Riemannian manifolds, such as trajectory-planning for rigid body motion. Some interpolation properties of Riemannian quadratics are analysed when the ambient manifold is a sphere or projective space, with the usual Riemannian metrics.     

关于齐次四元Monge-Ampère方程的一些注记北大核心CSCD

本文考虑了四元数空间Hn中齐次四元Monge-Ampère方程的狄利克雷问题解的正则性.首先,当区域是边界为C1,1的强拟凸域时,作者给出了解的Lipschitz估计.其次,考虑了四元MongeAmpère算子的收敛性.最后,讨论了齐次四元Monge-Ampère方程的粘性次解与F-次调和函数之间的关系.