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1.
As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because is a non-associative algebra over ℝ. However since is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.  相似文献   

2.
For a fixed weight Δ(dx) onR 1 and a linear space ℋ ⊆L p(Δ) of entire functions that is closed under difference quotientsh(·)→(z−·) −1[h(z)−h(·)], theL p(Δ) closure of ℋ is studied and characterized in terms of the normsL(z), (z∈C 1 of the evaluation functionalsh→h(z),h∈ℋ. Partially supported by DA-ARO-31-124-71-6182 and NSF GP-43011.  相似文献   

3.
Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely ( k n−1 )k-faces, (3)H k (C)=0. We prove that for ,H k−1(C) is a finite group, and our main result is: . This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.  相似文献   

4.
In this paper we consider operators acting on a subspace ℳ of the space L 2 (ℝm; ℂm) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳs,k of specific Clifford basis functions of L 2(ℝm; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳs,k into a similar space ℳs′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.  相似文献   

5.
Summary A distribution on the unit sphere in q with a densityf(‖x v ) is considered where is ans(<q) dimensional subspace andx v is the part ofx in . For a large sample the estimation of , a test that and a test for rotational symmetry within is given. For several samples with possibly different subspaces but the samef, a test that is given. For all tests power functions for contiguous alternatives are given. For the special density proportional to expk‖x v 2, additional results are given. Research supported in part by a Contract with the Office of Naval Research N00014-81-K-0146 awarded to Princeton University, Princeton, New Jersey 08544.  相似文献   

6.
The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class.  相似文献   

7.
Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH n, ℂH n, ℝH 2 t+1, ℍP n, ℂP n and ℝP 2n+1 of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.  相似文献   

8.
Summary Piecewise polynomial and Fourier approximation of functions in the Sobolev spaces on unbounded domains Θ ⊂ Rn are applied to the study of the type of compact embeddings into appropriate Lebesgue and Orlicz spaces. It is shown that if Θ and s satisfy certain conditions, the embeddings , m/n+1/q−1/p>0 and , Φ being an Orlicz function subordinate to both φ(t)=|t|p exp |t|n/(n−m) and Φσ(t)=exp |t|σ−1, σ ⩾ 1, m/n>1/p, are of type ls. One result dealing with multiplications maps from into Lq(Θ) is also obtained. Entrata in Redazione il 14 ottobre 1976.  相似文献   

9.
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.  相似文献   

10.
This note proves that, forF = ℝ, ℂ or ℍ, the bordism classes of all non-bounding Grassmannian manifoldsG k(F n+k), withk <n and having real dimensiond, constitute a linearly independent set in the unoriented bordism group N d regarded as a ℤ2-vector space.  相似文献   

11.
We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S 1-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, ℂP 2, 2,S 2×S 2and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4,ℂP 2,S 2×S 2,ℂP 2#  2 or ℂP 2# ℂP 2. Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5,S 3×S 2, then on trivial S 3-bundle over S 2 or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.  相似文献   

12.
We provide conditions on the complex dilatation of a homeomorphismf of the upper half plane ℍ into ℂ, which guarantee thatf(ℍ) is a proper subset of ℂ and, in case wheref(ℍ) is a Jordan domain, thatf has a homeomorphic extension onto .  相似文献   

13.
Let Gn,k denote the oriented grassmann manifold of orientedk-planes in ℝn. It is shown that for any continuous mapf: Gn,k → Gn,k, dim Gn,k = dim Gm,l = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂGm,l → ℂGn,k,h: ℍGm,l → ℍGn,k, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained.  相似文献   

14.
The “Volterra relation” is the commutation relation [S,V]⊂V 2, where S is a not necessarily bounded operator, V is a bounded operator leaving D(S) invariant, and [⋅,⋅] is the Lie product. When S,V are so related, and in addition iS generates a bounded C 0-group of operators and V has some general property, it is known that S+α V (α∈ℂ) is similar to S if and only if α=0 (cf. Theorem 11.17 in Kantorovitz, Spectral Theory of Banach Space Operators, Springer, Berlin, 1983). In particular, SV is not similar to S. However, it is shown in this note that (without any restriction on V and on the group S(⋅) generated by iS), the perturbations (SV)+P are similar to S for all P in the similarity sub-orbit {S(a)VS(−a);a∈ℝ} of V. When S is bounded, the above perturbations are similar to S for all P in the wider similarity sub-orbit {e aS Ve aS ;a∈ℂ}.  相似文献   

15.
Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim r → ∞ e2r s(r) = 0, then (M, g) has to be isometric to ℍ n . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim r → ∞ r 2 s(r) = 0, then (M, g) is isometric to ℝ n , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if Ka on a geodesic ball B p (R) in M and K = a on ∂B p (R), then K = a on B p (R).  相似文献   

16.
Summary This paper is dealing with the oscillatory and asymptotic behavior of the bounded solutions of n-th order (n>1) differential equations with deviating arguments involving the so called r-derivatives D r (i)x (i=0, 1, ..., n) of the unknown function x defined by , where ri (i=1, 2, ..., n−1) are positive continuous functions on the interval [t 0 , ∞). The fundamental purpose is to find a necessary and sufficient condition in order to have at least one (bounded nonoscillatory) solution whose the limit at ∞ exists inR−{0}. Entrata in Redazione il 29 giugno 1977. This paper is a part of the author's Doctoral Thesis submitted to the School of Physics and Mathematics of the University of Ioannina.  相似文献   

17.
Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ n preserving the unit ball ℍ n , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ n and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ n which are tangent to n at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of n which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.  相似文献   

18.
We obtain optimal height estimates for surfaces in ℍ2 × ℝ and × ℝ with constant Gaussian curvature K(I) and positive extrinsic curvature, characterizing the extreme cases as the revolution ones. Moreover, we get a representation for surfaces with constant Gaussian curvature in such ambient spaces, paying special attention to the cases of K(I) = 1 in × ℝ and K(I) = −1 in ℍ2 × ℝ. The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant No. PAI-05-034. The authors are partially supported by MEC-FEDER, Grant No. MTM2007-65249.  相似文献   

19.
Sunto Si dànno delle condizioni necessarie affinchè due superficie algebriche F, G d’una V 3 algebrica non singolare possano presentare contatto d’ordine assegnato q −1 lungo una curva priva di punti multipli. Si approfondisce sopratuito il caso q=3, con applicazione al problema di determinare le superficie algebriche Gn di S 3 che presentano contatto del 3o ordine lungo una n con una superficie quartica F 4 .  相似文献   

20.
We study the growth of an entire functionƒ, whose Borel transformγƒ is holomorphic outside a bounded convex regionD ƒ with boundary curvature bounded away from 0 and ∞. The functionγƒ is assumed to belong to the Dirichlet space, i.e., it satisfies , wheredv(ξ) is the area element. It is shown that forγƒ to satisfy the above conditions, it is necessary and sufficient to have and the growth indicatrixh ƒ ofƒ satisfies the relation 0<mh″(ϕ)+h(ϕ)≤M<∞. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 58–65, July, 1996.  相似文献   

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