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1.
A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π1(n) such that for every hypercube Q with dim(Q)?π1(n) there exists a family of pairwise vertex-disjoint paths Pi between Ai and Bi for i=1,2,…,n with if and only if {Ai,Bii=1,2,…,n} is a balanced set.  相似文献   

2.
3.
Let X1,X2,…,Xn be independent exponential random variables such that Xi has failure rate λ for i=1,…,p and Xj has failure rate λ* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by Di:n(p,q)=Xi:n-Xi-1:n the ith spacing of the order statistics , where X0:n≡0. It is shown that Di:n(p,q)?lrDi+1:n(p,q) for i=1,…,n-1, and that if λ?λ* then , and for i=1,…,n, where ?lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.  相似文献   

4.
Our starting point is the proof of the following property of a particular class of matrices. Let T={Ti,j} be a n×m non-negative matrix such that ∑jTi,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that Ti,lTj,l. Then, there exists a real vector k=(k1,k2,…,km)T,kikj,ij;0<ki?1, such that, if ij.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures {μ(·)(i)}iN, on the Borel σ-algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that
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5.
A quantum effect is a positive Hilbert space contraction operator. If {Ei}, 1?i?n, are n quantum effects (defined on some Hilbert space H), then their sequential product is the operator . It is proved that the quantum effects {Ei}, 1?i?n, are sequentially independent if and only if for every permutation r1r2rn of the set Sn={1,2,…,n}. The sequential independence of the effects Ei, 1?i?n, implies EnoEn-1ooEj+1oEjooE1=(EnoEn-1oEj+1)oEjooE1 for every 1?j?n. It is proved that if there exists an effect Ej, 1?j?n, such that Ej?(EnoEn-1oEj+1)oEjooE1, then the effects {Ei} are sequentially independent and satisfy .  相似文献   

6.
For any real number β>1, let ε(1,β)=(ε1(1),ε2(1),…,εn(1),…) be the infinite β-expansion of 1. Define . Let x∈[0,1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If is bounded, we obtain that for all x∈[0,1)?Q,
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7.
We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137-1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153-154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a1, … , an and b1, … , bn, respectively. Then for any scalars t and s
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8.
Let TRn×n be an irreducible stochastic matrix with stationary distribution vector π. Set A = I − T, and define the quantity , where Aj, j = 1, … , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ3 provides a sensitive measure of the conditioning of π under perturbation of T. Moreover, it is known that .In this paper, we investigate the class of irreducible stochastic matrices T of order n such that , for such matrices correspond to Markov chains with desirable conditioning properties. We identify some restrictions on the zero-nonzero patterns of such matrices, and construct several infinite classes of matrices for which κ3 is as small as possible.  相似文献   

9.
A collection A1A2, …, Ak of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer-Sethuraman on bounds for the dimension of the algebra generated by A1A2, …, Ak. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A1, …, Ak when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)  相似文献   

10.
Let A be an n×n complex matrix and c=(c1,c2,…,cn) a real n-tuple. The c-numerical range of A is defined as the set
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11.
A simple proof for a theorem of Luxemburg and Zaanen   总被引:1,自引:0,他引:1  
In this paper a simple proof for the following theorem, due to Luxemburg and Zaanen is given: an Archimedean vector lattice A is Dedekind σ-complete if and only if A has the principal projection property and A is uniformly complete. As an application, we give a new and short proof for the following version of Freudenthal's spectral theorem: let A be a uniformly complete vector lattice with the principal projection property and let 0<uA. For any element w in A such that 0?w?u there exists a sequence in A which satisfies , where each element sn is of the form , with real numbers α1,…,αk such that 0?αi?1 (i=1,…,k) and mutually disjoint components p1,…,pk of u.  相似文献   

12.
13.
The paper studies the eigenvalue distribution of some special matrices. Tong in Theorem 1.2 of [Wen-ting Tong, On the distribution of eigenvalues of some matrices, Acta Math. Sinica (China), 20 (4) (1977) 273-275] gives conditions for an n × n matrix A ∈ SDn ∪ IDn to have |JR+(A)| eigenvalues with positive real part, and |JR-(A)| eigenvalues with negative real part. A counter-example is given in this paper to show that the conditions of the theorem are not true. A corrected condition is then proposed under which the conclusion of the theorem holds. Then the corrected condition is applied to establish some results about the eigenvalue distribution of the Schur complements of H-matrices with complex diagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, … , n} are presented such that the Schur complement matrix A/α of the matrix A has eigenvalues with positive real part and eigenvalues with negative real part.  相似文献   

14.
Peter Borg 《Discrete Mathematics》2009,309(14):4750-4753
Families A1,…,Ak of sets are said to be cross-intersecting if for any AiAi and AjAj, ij. A nice result of Hilton that generalises the Erd?s-Ko-Rado (EKR) Theorem says that if rn/2 and A1,…,Ak are cross-intersecting sub-families of , then
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15.
Suppose that A=(ai,j) is an n×n real matrix with constant row sums μ. Then the Dobrushin-Deutsch-Zenger (DDZ) bound on the eigenvalues of A other than μ is given by . When A a transition matrix of a finite homogeneous Markov chain so that μ=1,Z(A) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, , of the iteration , to the stationary distribution vector of the chain.In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which |λ|=Z(A) for some stochastic matrix A.  相似文献   

16.
17.
Singular values, norms, and commutators   总被引:1,自引:0,他引:1  
Let and Xi, i=1,…,n, be bounded linear operators on a separable Hilbert space such that Xi is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a1?A?a2 and b1?B?b2 for some real numbers and b2, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b2-a1,a2-b1)(XX). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.  相似文献   

18.
Let An be the nth Weyl algebra and Pm be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {AnPm} is proved: an algebraAadmits a finite setδ1,…,δsof commuting locally nilpotent derivations with generic kernels andiffA?AnPmfor somenandmwith2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra AnPm (and for ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given, then (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given, then. Any automorphism is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for .One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [6]] problem 1) into a single question, (JD): is aK-algebra endomorphismσ:AnPmAnPman algebra automorphism providedσ(Pm)⊆Pmand? (Pm=K[x1,…,xm]). It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42(2) (2005) 435-452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].  相似文献   

19.
Let KE, KE be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product EE is KK-separable if it can be represented as finite sum , where xlK and for all l. Let S(n), H(n), Q(n) be the spaces of n×n real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further S+(n), H+(n), Q+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix AH(mn)=H(m)⊗H(n) is H+(m)⊗H+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m)⊗S(n), H(m)⊗S(n), and for m?2 in the space Q(m)⊗S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n)⊗S(2) is Q+(n)⊗S+(2)- separable if and only if it is positive semidefinite.  相似文献   

20.
Let K1,…,Kn be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,)×…×[0,)→[0,) of n variables, we define a non-negative matrix and consider the inequality
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