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1.
Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p Wp(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ Ik) with ξ(p × k) unknown. Denote the columns of X by X(1) ,…, X(k) and set ψ(0)(S, X) = {(np + 2)!/(n + 2)!} | S |, ψ(i)(X, X) = min[ψ(i−1)(S, X), {(np + i + 2)!/(n + i + 2)!} | S + X(1) X(1) + + X(i) X(i) |] and Ψ(i)(S, X) = min[ψ(0)(S, X), {(np + i + 2)!/(n + i + 2)!}| S + X(1) X(1) + + X(i) X(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ(0)(S, X) is dominated by each of Ψ(i)(S, X), i = 1,…,k and for every i, ψ(i)(S, X) is better than ψ(i−1)(S, X). In particular, ψ(k)(S, X) = min[{(np + 2)!/(n + 2)!} | S |, {(np + 2)!/(n + 2)!} | S + X(1)X(1)|,…,| {(np + k + 2)!/(n + k + 2)!} | S + X(1)X(1) + + X(k)X(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.  相似文献   

2.
Let G be a finite abelian group of order n and Davenport constant D(G). Let S=0h(S)gGgvg(S)∈F(G) be a sequence with a maximal multiplicity h(S) attained by 0 and t=|S|?n+D(G)−1. Then 0∈k(S) for every 1?k?t+1−D(G). This is a refinement of the fundamental result of Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103].  相似文献   

3.
For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λkr(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (kr)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λkr(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ(A) is the intersection of all Λk(A) for k=1,2,…. If AμI is not compact for all μC, then the closure and the interior of Λ(A) coincide with those of the essential numerical range of A. The situation for the special case when AμI is compact for some μC is also studied.  相似文献   

4.
Let X be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space Λ(R,X) of all measurable functions x such that xhX for every a.e. converging series h=∑anrnX, where (rn) are the Rademacher functions. We study the situation when Λ(R,X) is a rearrangement invariant space different from L. Particular attention is given to the case when X is an interpolation space between the Lorentz space Λ(φ) and the Marcinkiewicz space M(φ). Consequences are derived regarding the behaviour of partial sums and tails of Rademacher series in function spaces.  相似文献   

5.
We are interested in the asymptotic integration of linear differential systems of the form x′=[Λ(t)+R(t)]x, where Λ is diagonal and RLp[t0,∞) for p∈[1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ωΛ. Our results include examples that are not covered by the Hartman-Wintner theorem.  相似文献   

6.
Let M be a II-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X(M,τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form IX(M,τ)=MX(M,τ) that are contained in Lp(M,τ) for some p>0. It is proved that an element T in IX(M,τ) is a finite sum of commutators of the form [A,B] with AIX(M,τ) and BM if and only if the function belongs to X, where νT is the Brown spectral measure of T and tλt(T) is the non-increasing rearrangement of the function λ→|λ| with respect to νT. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L1,∞(M,τ) is always zero.  相似文献   

7.
Let (ks) denote the set of all k-element-subsets of a finite set S. A k-simplical matroid on a subset E of (ks) is a binary matroid the circuit of which are simplicial complexes {X1,…Xm} ? E with boundary 0 (mod 2). The k-simplical matroid on (ks) is called the full simplicial matroid Gk(S). The polygon matroid on the edges of a finite graph is 2-simplicial. Polygon-matroids and their duals are regular. The dual of Gk(S) is Gn?k(S) if the cardinnlity of S is n. More details on simplicial matroids can be found in [3, Chapter 6] and also in [4, pp. 180–181].Welsh asked if every simplicial matroid is regular. We prove that this is not the case, for all full k-simplicial matroids Gk(S) with 3?k?n?3 are non-regular (n is the cardinality of S). This result has also been proved σy R. Cordovil and M. Las Vergnas recently. Their proof is different from our proof, which is somewhat shorter.  相似文献   

8.
Marcel Bökstedt 《Topology》2005,44(6):1181-1212
Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H*(ΛX;Z/p) and H*((ΛX)hT;Z/p). The E2-terms are certain non-Abelian-derived functors applied to H*(X;Z/p). When H*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.  相似文献   

9.
Let (X,μ) be a measurable topological space. Let S1,S2,… be a family of finite subsets of X. Suppose each xSi has a weight wixR+ assigned to it. We say {Si} is {wi}-distributed with respect to the measure μ if for any continuous function f on X, we have .Let S(N,k) be the space of modular cusp forms over Γ0(N) of weight k and let be a basis which consists of Hecke eigenforms. Let ar(h) be the rth Fourier coefficient of h. Let xph be the eigenvalue of h relative to the normalized Hecke operator Tp. Let ||·|| be the Petersson norm on S(N,k). In this paper we will show that for any even integer k?3, is -distributed with respect to a polynomial times the Sato-Tate measure when N→∞.  相似文献   

10.
We prove the inverse closedness of certain approximation algebras based on a quasi-Banach algebra X using two general theorems on the inverse closedness of subspaces of quasi-Banach algebras. In the first theorem commutative algebras are considered while the second theorem can be applied to arbitrary X and to subspaces of X which can be obtained by a general K-method of interpolation between X and an inversely closed subspace Y of X having certain properties. As application we present some inversely closed subalgebras of C(T) and C[−1,1]. In particular, we generalize Wiener's theorem, i.e., we show that for many subalgebras S of l1(Z), the property {ck(f)}∈S (ck(f) being the Fourier coefficients of f) implies the same property for 1/f if fC(T) vanishes nowhere on T.  相似文献   

11.
Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,BB(X) satisfy ABN(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:
(a)
There is a bijective bounded linear or conjugate-linear operator S:XX such that ? has the form A?S[f(A)A]S-1.
(b)
The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S−1.
If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra Mn of n × n complex matrices, then there exist a map f:MnC?{0}, a field automorphism ξ:CC, and an invertible S ∈ Mn such that ? has one of the following forms:
  相似文献   

12.
Let Mn be the semigroup of n×n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mn including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of AMn. Multiplicative maps ?:SMn satisfying rk(?(A))=rk(A) are characterized. From these results, one can deduce the structure of multiplicative preservers of Λk(A).  相似文献   

13.
Let AR be rings containing the rationals. In R let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a preorder of R (a proper subsemiring containing the squares) such that ST and I an A-submodule of R. Define ρ(I) (or ρS,T(I)) to be
ρ(I)={aR|sa2m+tI2m for some mN,sS and tT}.  相似文献   

14.
Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X1,…,Xn]. Let gk[X] and d?1. If for all 1?|α|?d the derivatives αg belong to ℘, then there exists ck such that g−c∈℘(d+1), the d+1th symbolic power of ℘. In particular, if ℘ is a complete intersection it follows that g−c∈℘d+1.  相似文献   

15.
Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I0(P) and K=Gp is the stabilizer of pP. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K0K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K0 is connected and the action of K0 on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H(X). In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.  相似文献   

16.
For an oriented graph D, let ID[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For SV(D), let ID[S] denote the union of all ID[u,v] for all u,vS. Let [S]D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with ID[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g(G)=min{g(D):DO(G)}, g+(G)=max{g(D):DO(G)}, h(G)=min{h(D):DO(G)}, and h+(G)=max{h(D):DO(G)}. By the above definitions, h(G)≤g(G) and h+(G)≤g+(G). In the paper, we prove that g(G)<h+(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g(G)−h(G)=a and g+(G)−h+(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].  相似文献   

17.
This paper extends to the Eisenstein integers a + b? (a, bτZ, ?2 + ? + 1 = 0) the problem of the existence of a bound on the size of a sequence of m consecutive kth powe residues of p, for all but a finite number of primes p and independent of p. The least such bound is denoted by ΛE(k, m). It is shown that ΛE(k, 2) is finite for k = 2, 3, 4 or 6n + 1. On the other hand, for every k, ΛE(2k, 3) = ΛE(3k, 4) = ΛE(k, 6) = ∞. Similar results are obtained for the related bound for m consecutives all in the same coset modulo the subgroup of kth power residues.  相似文献   

18.
A Hilbert space operator AB(H) is p-hyponormal, A∈(p-H), if |A|2p?|A|2p; an invertible operator AB(H) is log-hyponormal, A∈(?-H), if log(TT)?log(TT). Let dAB=δAB or ?AB, where δABB(B(H)) is the generalised derivation δAB(X)=AX-XB and ?ABB(B(H)) is the elementary operator ?AB(X)=AXB-X. It is proved that if A,B∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (dAB-λ)?1, and dAB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(dAB-λ)Y‖ for all X∈(dAB-λ)-1(0) and YB(H). Furthermore, isolated points of σ(dAB) are simple poles of the resolvent of dAB. A version of the elementary operator E(X)=A1XA2-B1XB2 and perturbations of dAB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for dAB.  相似文献   

19.
In this paper, we investigate the relation between the lower topology respectively the Lawson topology on a product of posets and their corresponding topological product. We show that (1) if S and T are nonsingleton posets, then Ω(S×T)=Ω(SΩ(T) iff both S and T are finitely generated upper sets; (2) if S and T are nontrivial posets with σ(S) or σ(T) being continuous, then Λ(S×T)=Λ(SΛ(T) iff S and T satisfy property K, where for a poset L, Ω(L) means the lower topological space, Λ(L) means the Lawson topological space, and L is said to satisfy property K if for any xL, there exist a Scott open U and a finite FL with xU⊆↑F.  相似文献   

20.
We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2 (1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2 (1) and the integrable highest weight module L(kΛ0) for A 1 (1) have the same parametrization of combinatorial bases and the same presentation P/I.  相似文献   

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