On improved estimators of the generalized variance

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 W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + 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[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + 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.     

Higher rank numerical ranges and low rank perturbations of quantum channels

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)⊆Λk−r(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 (k−r)-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 Λk−r(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 A−μI is not compact for all μ∈C, 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 A−μI is compact for some μ∈C is also studied.     

A new result on Davenport constant

Let G be a finite abelian group of order n and Davenport constant D(G). Let S=⁰h(S)_∏g∈Ggvg(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].     

Rearrangement invariance of Rademacher multiplicator spaces

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 x⋅h∈X for every a.e. converging series h=∑anrn∈X, 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.     

A dynamical systems result on asymptotic integration of linear differential systems

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 R∈L^p[t₀,∞) for p∈[1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ω_Λ. Our results include examples that are not covered by the Hartman-Wintner theorem.     

Sums of commutators in non-commutative Banach function spaces

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 I_X(M,τ)=M∩X(M,τ) that are contained in L^p(M,τ) for some p>0. It is proved that an element T in I_X(M,τ) is a finite sum of commutators of the form [A,B] with A∈I_X(M,τ) and B∈M if and only if the function belongs to X, where ν_T is the Brown spectral measure of T and t→λ_t(T) is the non-increasing rearrangement of the function λ→|λ| with respect to ν_T. 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 L^1,∞(M,τ) is always zero.     

Non-regular simplicial matroids

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.     

A spectral sequence for string cohomology

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.     

Kuznietsov trace formula and weighted distribution of Hecke eigenvalues

Let (X,μ) be a measurable topological space. Let S₁,S₂,… be a family of finite subsets of X. Suppose each x∈S_i has a weight w_ix∈R⁺ assigned to it. We say {S_i} is {w_i}-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 a_r(h) be the rth Fourier coefficient of h. Let x_p^h be the eigenvalue of h relative to the normalized Hecke operator T′_p. 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→∞.     

Inverse closedness of approximation algebras

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 f∈C(T) vanishes nowhere on T.     

Maps preserving the nilpotency of products of operators

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,B∈B(X) satisfy AB∈N(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:X→X 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 M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Multiplicative maps preserving the higher rank numerical ranges and radii

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

Generalizations of a lemma of Freudenburg

Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X₁,…,X_n]. Let g∈k[X] and d?1. If for all 1?|α|?d the derivatives ∂^αg belong to ℘, then there exists c∈k 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.     

A semialgebraic closure for commutative algebra

Let A⊂R 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 S⊂T and I an A-submodule of R. Define ρ(I) (or ρ_S,T(I)) to be

ρ(I)={a∈R|sa^2m+t∈I^2m for some m∈N,s∈S and t∈T}.     

Equivariant cohomology of real flag manifolds

Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I₀(P) and K=Gp is the stabilizer of p∈P. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K₀⊂K 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.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(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 (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. 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 I_D[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):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(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].     

Notes on the products of the lower topology and Lawson topology on posets

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(S)×Ω(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(S)×Λ(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 x∈L, there exist a Scott open U and a finite F⊆L with x∈U⊆↑F.     

Bounds for consecutive kth power residues in the Eisenstein integers

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 Λ_E(k, m). It is shown that Λ_E(k, 2) is finite for k = 2, 3, 4 or 6n + 1. On the other hand, for every k, Λ_E(2k, 3) = Λ_E(3k, 4) = Λ_E(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.     

Normal covers of various products

Throughout this paper, we consider the following two problems: (A) When does a rectangular normal cover of a product X×Y (or an infinite product _∏λ∈ΛXλ) have a σ-locally finite rectangular cozero refinement? (B) What kind of a refinement makes a rectangular open cover of a product X×Y (or an infinite product _∏λ∈ΛXλ) be normal? We shall discuss these problems on various products listed below.