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1.
When and are given, we denote by the operator acting on the infinite dimensional separable Hilbert space of the form . In this paper, we first give some necessary and sufficient conditions for to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some , which extend the corresponding results in Cao et al. (2006) [4], Cao and Meng (2005) [5], Hwang and Lee (2001) [12] and Li and Du (2006) [15]. Then we present some counter-examples. 相似文献
2.
We consider one typical two-parameter family of quadratic systems of 2 × 2 conservation laws, and study the geometry of the
behaviour of the possible solutions of the Riemann problem near an umbilic point, following the geometric approach presented
by Isaacson, Marchesin, Palmeira, Plohr, in A global formalism for nonlinear waves in conservation laws, Commun. Math. Phys. (1992). The corresponding phase portraits for the rarefaction curves, shock curves and composite curves
are discussed.
Financial support from FCT and Calouste Gulbenkian Foundation. 相似文献
3.
Thawhat CHANGPHAS Klaus DENECKE 《数学学报(英文版)》2007,23(4):659-670
Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an M-hyperidentity for a subset M of the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type (2, 2), i.e., the order of the cyclic subsemigroup generated by that hypersubstitution of the monoid of all hypersubstitutions of type (2, 2). The main result is that the order is 1, 2, 3, 4 or infinite. 相似文献
4.
In this paper, we study the perturbation of spectra for 2 × 2 operator matrices such as M X = ( 0 B A X ) and M Z = ( Z B A C ) on the Hilbert space H ?? K and the sets $\bigcap\limits_{X \in \mathcal{B}(K,H)} {P_\sigma (M_X )} ,\bigcap\limits_{X \in \mathcal{B}(K,H)} {R_\sigma (M_X )} $ and $\bigcap\limits_{Z \in \mathcal{B}(H,K)} {\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {P_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {R_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {C_\sigma (M_Z )} $ , where R(C) is a closed subspace, are characterized 相似文献
5.
LetD be a division ring which possesses an involution a → α . Assume that
is a proper subfield ofD and is contained in the center ofD. It is pointed out that ifD is of characteristic not two, D is either a separable quadratic extension of F or a division ring of generalized quaternions
over F and that if D is of characteristic two,D is a separable quadratic extension ofF. Thus the trace map Tr:D → F, a → a + a is always surjective, which is formerly posed as an assumption in the fundamental theorem of n×n hermitian
matrices overD when n ≥ 3 and now can be deleted. WhenD is a field, the fundamental theorem of 2 × 2 hermitian matrices overD has already been proved. This paper proves the fundamental theorem of 2×2 hermitian matrices over any division ring of generalized
quaternions of characteristic not two
This research was completed during a visit to the Academy of Mathematics and System Sciences, Chinese Academy of Sciences. 相似文献
6.
Vesselin Drensky 《Algebras and Representation Theory》2003,6(2):193-214
We obtain defining relations of the algebra of invariants of the classical subgroups of GL
2(C) acting by simultaneous conjugation on m-tuples of 2×2 complex matrices. The sets of defining relations look uniformly for all m2 and are derived by translation of classical results on invariant theory of orthogonal groups in the language of 2×2 matrix invariants, combined with arguments of representation theory of the general linear group GL
m
(C) and ideas coming from the theory of algebras with polynomial identities. 相似文献
7.
We study movable singularities of the Malgrange isomonodromic deformation of a linear differential 2 × 2 system with two irregular singularities of Poincaré rank 1 and with an arbitrary number of Fuchsian singular points. 相似文献
8.
A. Ya. Belyankov 《Computational Mathematics and Mathematical Physics》2008,48(2):190-194
Algorithms for computing the commutator AB ? BA of 2 × 2 matrices A and B are proposed that involve five multiplications. 相似文献
9.
P. I. Katsylo 《Mathematical Notes》1998,63(4):464-470
Two results on the degrees of polynomial mappings 22 are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998. 相似文献
10.
Carla D. Martin 《Linear and Multilinear Algebra》2013,61(8):943-950
As computing power increases, many more problems in engineering and data analysis involve computation with tensors, or multi-way data arrays. Most applications involve computing a decomposition of a tensor into a linear combination of rank-1 tensors. Ideally, the decomposition involves a minimal number of terms, i.e. computation of the rank of the tensor. Tensor rank is not a straight-forward extension of matrix rank. A constructive proof based on an eigenvalue criterion is provided that shows when a 2?×?2?×?2 tensor over ? is rank-3 and when it is rank-2. The results are extended to show that n?×?n?×?2 tensors over ? have maximum possible rank n?+?k where k is the number of complex conjugate eigenvalue pairs of the matrices forming the two faces of the tensor cube. 相似文献
11.
12.
13.
Xiaoping Xu 《Frontiers of Mathematics in China》2011,6(4):759-774
Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module
that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all
the singular vectors of the polynomial representation of the simple Lie algebra of type F
4 over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly
described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic
polynomials with degree k ⩾ 2 is greater than or equal to 〚k/3〛 + 〚(k − 2)/3〛 + 2. 相似文献
14.
We consider operator matrices ${\rm{H = }}\left( {{_{B_{10} }^{A{_0}}} {_{A_{1} }^{B{_{01}}}} } \right)$ with self - adjoint entries Ai, i = 0,1, and bounded B10 = B10 acting in the orthogonal sum μ=μ0 ⊕ μ1 of Hilbert spaces μ0 and μ1. We are especially interested in the case where the spectrum of, say, A1 is partly or totally embedded into the continuous spectrum of A0 and the transfer function V1(z) = B10 where (χ - A0)?1 B01, admits analytic continuation (as an operator - valued function) through the cuts along branches of the continuous spectrum of the entry A0 into the unphysical sheet(s) of the spectral parameter plane. The values of χ in the unphysical sheets where M1?1(z) and consequently the resolvent (H - z)?1 have poles are usually called resonances. A main goal of the present work is to find non - selfadjoint operators whose spectra include the resonances as well as to study the completeness and basis properties of the resonance eigenvectors of M1(z) in μ1. To this end we first construct an operator - valued function V1(Y) on the space of operators in μ1 possessing the property: V1(Y)?1 = V1(z)?1 for any eigenvector ?1 of Y corresponding to an eigenvalue z and then study the equation H1 = A1 + V1(H1). We prove the solvability of this equation even in the case where the spectra of A0 and A1 overlap. Using the fact that the root vectors of the solutions H1 are at the same time such vectors for M1(z), we prove completeness and even basis properties for the root vectors (including those for the resonances). 相似文献
15.
Lu Qikeng 《数学学报(英文版)》1998,14(1):1-8
In the complexn-dimensional projective spaceCP
n
, let λ
p
(=4p(p+n)) be the eigen value of the Laplace-Beltrami operator andH
p
be the space of all eigen functions of eigen value λ
p
. The reproducing kernelh
p
(z, w) ofH
p
is constructed explicitly in this paper, and a system of complete orthogohal functions ofH
p
is constructed fromh
p
(z,w)(p=1,2, …).
Partially supported by NSF of China 相似文献
16.
N. Dunford and J.T. Schwartz (1963) striking Hilbert space theory about completeness of a system of root vectors (generalized eigenvectors) of an unbounded operator has been generalized by J. Burgoyne (1995) to the Banach spaces framework. We use the Burgoyne's theorem and prove n-fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. The theory will allow to consider, in application, boundary value problems for ODEs and elliptic PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions. 相似文献
17.
Nikolai Nikolski 《Journal of Approximation Theory》1999,98(2):61
This note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions f whose positive integer translates f(x−k), where k
+ are complete in the spaces Cl0(
), Lp(
), Wlp(
), 2<p<∞, l=0, 1, …, as well as in their weighted and/or vector-valued analogues. 相似文献
18.
We prove that universal cycles of 2-dimensional subspaces of vector spaces over any finite field F exist, i.e., if V is a finite-dimensional vector space over F, there is a cycle of vectors v1,v2,…,vn such that each 2-dimensional subspace of V occurs exactly once as the span of consecutive vectors. 相似文献
19.
Let (H, h) be a Riemannian manifold and assume f: H(0, ) is a smooth function. The Lorentzian warped product (a, b)
f
×H,–a<b, with metric ds
2=(–f
2 dt
2) h is called a standard static space-time. A study is made of geodesic completeness in standard static space-times. Sufficient conditions on the warping function f: H(0, ) are obtained for (a, b)
f
×H ×H to be timelike and null geodesically complete. In the timelike case, the sufficient condition is independent of the completeness of the Riemannian manifold (H, h). 相似文献
20.
Let H
3 be the root system associated with the icosahedron, and let M(H
3) be the linear dependence matroid corresponding to this root system. We prove , and interpret these automorphisms geometrically.
Dedicated to Thomas Brylawski. 相似文献