四元数矩阵的新特征值定位

Yin Caixia;Li Chaoqian(College of Mathematics and Statistics,Yunnan University,Kunming 650500,China)     

On the determinant of quaternionic polynomial matrices and its application to system stability

In this paper, we propose a definition of determinant for quaternionic polynomial matrices inspired by the well‐known Dieudonné determinant for the constant case. This notion allows to characterize the stability of linear dynamical systems with quaternionic coefficients, yielding results which generalize the ones obtained for the real and complex cases. Copyright © 2007 John Wiley & Sons, Ltd.     

Collars in Complex and Quaternionic Hyperbolic Manifolds

We use the complex and quaternionic hyperbolic versions of Jørgensen's inequality to construct embedded collars about short, simple, closed geodesics in complex and quaternionic hyperbolic manifolds. In general, the width of these collars depend both on the length of the geodesic and on the rotational part of the group element uniformising it. For complex hyperbolic space we are able to use a lemma of Zagier to give an estimate based only on the length. We show that these canonical collars are disjoint from each other and from canonical cusps. We also calculate the volumes of these collars.     

The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n‐tuples of commuting operators (called ?‐functional calculus) based on a new notion of spectrum, called ?‐spectrum, for the n‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ?‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.     

Totally real minimal 2-spheres in quaternionic projective space

Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(？) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(？)RPn(？) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (？) RPn(？) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.     

A Taylor–Wiles System for Quaternionic Hecke Algebras

Let &ell >3 be a prime. Fix a regular character of F&^{2 ×} of order &–1, and an integer M prime to &. Let fS ₂(₀(M&²)) be a newform which is supercuspidal of type at &. For an indefinite quaternion algebra over Q of discriminant dividing the level of f, there is a local quaternionic Hecke algebra T of type associated to f. The algebra T acts on a quaternionic cohomological module M. We construct a Taylor–Wiles system for M, and prove that T is the universal object for a deformation problem (of type at & and semi-stable outside) of the Galois representation ¯ _f over F¯_& associated to f; that T is complete intersection and that the module M is free of rank 2 over T. We deduce a relation between the quaternionic congruence ideal of type for f and the classical one.     

Dihedral and quaternionic homology and mapping spaces

We associate to the singular chains of the Moore loop space Y two dihedral chain complexes. The dihedral homology of these complexes is shown to be isomorphic to the O(2)-equivariant homology of the mapping spaces Y ^o(2) and . We also construct simple homotopy theoretic models of EO(2) x _O2 (Y)^G/BO(2) for G = O(2) or S¹, where denotes suspension. Quaternionic analogues of these results are also established.     

NEWTONIAN MECHANICS ON KAHLER MANIFOLD

ＮＥＷＴＯＮＩＡＮＭＥＣＨＡＮＩＣＳＯＮＫＡＨＬＥＲＭＡＮＩＦＯＬＤＺｈａｎｇＲｏｎｇｙｅ（张荣业）（Ｉｎｓｔｉｔｕｔｅｏｆ＇Ｍａｔｈｅｍａｔｉｃｓ．ＡｃａｄｅｍｉａＳｉｎｉｃａ．Ｂｅｉｊｉｎｇ１０００８０．Ｐ．Ｒ．Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｉｎ...     

LAGRANGIAN VECTOR FIELD ON KOHLER MANIFOLD

ＬＡＧＲＡＮＧＩＡＮＶＥＣＴＯＲＦＩＥＬＤＯＮＫ■ＨＬＥＲＭＡＮＩＦＯＬＤＺｈａｎｇＲｏｎｇｐｅ（张荣业）．（ＲｅｃｅｉｖｅｄＡｐｒｉｌ１６，１９９５；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＺｈｕＺｈａｏｘｕａｎ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ．ｗｅ...     

INVARIANT FORM AND INTEGRAL INVARIANTS ON KAHLER MANIFOLD

The important notions and results of the integral invariants of Poincare and Cartan-Poincare and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kahler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider and deeper results.