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.     

Möbius Isoparametric Hypersurfaces in S n+1 with Two Distinct Principal Curvatures

A hypersurface x : M → S ⁿ⁺¹ without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ^?2∑ _i (e _i (H) + ∑ _j (h _ij ?Hδ _ij )e _j (log ρ))θ _i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S ^{n +1} without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ⁻²∑ _i (e _i (H) + ∑ _j (h _ij −Hδ _ij )e _j (log ρ))θ _i vanishes and its M?bius shape operator ? = ρ⁻¹(S−Hid) has constant eigenvalues. Here {e _i } is a local orthonormal basis for I = dx·dx with dual basis {θ _i }, II = ∑ _ij h _ij θ _i ⊗θ _i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S ^{n +1} is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S ^{n +1} with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S ^{n +1} can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002     

Off-line dynamic maintenance of the width of a planar point set

Agarwal, P.K. and M. Sharir, Off-line dynamic maintenance of the width of a planar point set, Computational Geometry: Theory and Applications 1 (1990) 65-78. In this paper we present an efficient algorithm for the off-line dynamic maintenance of the width of a planar point set in the following restricted case: We are given a real parameter W and a sequence Σ=(σ₁,...,σ_n) of n insert and delete operations on a set S of points in ², initially consisting of n points, and we want to determine whether there is an i such that the width of S the ith operation is less than or equal to W. Our algorithm runs in time O(nlog³n) and uses O(n) space.     

An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S^′, and set T, |T∩S| denotes the number of terms x of S with xT, and |S| denotes the length of S, and SS^′ denotes the subsequence of S obtained by deleting all terms in S^′. We first prove the following two additive number theory results.(1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A=A₁,…,A_n, such that

then there exists a subsequence S^′ of S, with length |S^′|≤max{|S|−n+1,2n}, and with an n-set partition, , such that . Furthermore, if ||A_i|−|A_j||≤1 for all i and j, or if |A_i|≥3 for all i, then .(2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a,bG such that . If |S|≥2m−1, then there exists an m-term zero-sum subsequence S^′ of S with or .Let be a connected, finite m-uniform hypergraph, and be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group ) of the vertices of the complete m-uniform hypergraph , there exists a subhypergraph isomorphic to such that every edge in is monochromatic (such that for every edge e in the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph of contains an edge with at least half of its vertices monovalent in , or if consists of two intersecting edges, then . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when is a single edge.     

The Weak Convergence for Functions of Negatively Associated Random Variables

Let {X_n, n1} be a sequence of stationary negatively associated random variables, S_j(l)=∑^l_i=1 X_j+i, S_n=∑ⁿ_i=1 X_i. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums are investigated. Applications to limiting distributions of estimators of Var S_n are also discussed.     

A constructive method for the solution of the stability problem

Summary In this paper it is shown that the problem of solving the Liapounov matrix equationSM +M ^T S = –I is greatly simplified when the given real matrixM is in upper Hessenberg form. The solution is obtained as a linear combinationS = p _i S _i ofn linearly independent symmetric matricesS _i , whereS _i M +M ^T S _i =2D _i and p _i D _i = –1/2I. Explicit formulae are given for the elements of theS _i , andD _i while determination of thep _i requires the solution of ann ×n linear system.     

Estimates for the closeness of successive convolutions of multidimensional symmetric distributions

Summary Let ξ₁, ξ₂,... be i.i.d random vectors in ℝ ^k with a common distribution ℒ(ξ_i),... = F, i = 1, 2,.... Let S _n = ξ₁+...+ξ _n . We investigate how small is the difference between ℒ(S _n ) and ℒ(S _{n+ m} ) in the case when ξ _i have symmetric distributions.     

Generalized Ham-Sandwich Cuts

Bárány, Hubard, and Jerónimo recently showed that for given well-separated convex bodies S ₁,…,S _d in R ^d and constants β _i ∈[0,1], there exists a unique hyperplane h with the property that Vol (h ⁺∩S _i )=β _i ⋅Vol (S _i ); h ⁺ is the closed positive transversal halfspace of h, and h is a “generalized ham-sandwich cut.” We give a discrete analogue for a set S of n points in R ^d which are partitioned into a family S=P ₁∪⋅⋅⋅∪P _d of well-separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n) ^d−3) algorithm which, given positive integers a _i ≤|P _i |, finds the unique hyperplane h incident with a point in each P _i and having |h ⁺∩P _i |=a _i . Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Bárány et al., and in addition, we strengthen the result via a partial converse.     

On the Degree of Caustics by Reflection

Given a point S ∈ ?²: = ?²(?) and an irreducible algebraic curve 𝒞 of ?² (with any type of singularities), we consider the lines ? _m obtained by reflection of the lines (S m) on 𝒞 (for m ∈ 𝒞). The caustic by reflection Σ _S (𝒞) is classically defined as the Zariski closure of the envelope of the reflected lines ? _m . We identify this caustic with the Zariski closure of Φ(𝒞), where Φ is some rational map. We use this approach to give general and explicit formulas for the degree (with multiplicity) of caustics by reflection. Our formulas are expressed in terms of intersection numbers of the initial curve 𝒞 (or of its branches). Our method is based on a fundamental lemma for rational map thanks to the notion of Φ-polar and on the computation of intersection numbers. In particular, we use precise estimates related to the intersection numbers of 𝒞 with its polar at any point and to the intersection numbers of 𝒞 with its Hessian curve. These computations are linked with generalized Plücker formulas for the class and for the number of inflection points of 𝒞.     

An asymptotic minimax risk bound for estimation of a linear functional relationship

We consider estimation of the parameter B in a multivariate linear functional relationship X_i=ξ_i+ξ_1i, Y_i=Bξ_i+ξ_2i, i=1,…,n, where the errors (ζ_1i^′, ζ_2i^′) are independent standard normal and (ξ_i, i ) is a sequence of unknown nonrandom vectors (incidental parameters). If there are no substantial a priori restrictions on the infinite sequence of incidental parameters then asymptotically the model is nonparametric but does not fit into common settings presupposing a parameter from a metric function space. A special result of the local asymptotic minimax type for the m.1.e. of B is proved. The accuracy of the normal approximation for the m.l.e. of order n^−1/2 is also established.     

A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

A Remark on Self-Normalization for Dependent Random Variables

Let (X _t , t ∈ Z) be a stationary process, and let S _n = ∑_{1⩽ i⩽n} X _i . In this paper, we consider the central limit theorem for the self-normalized sequence S _n /U _n , where U _n ² = ∑_1⩽j⩽N Y _j ² , Y _j = ∑_{(j−1)m<i⩽jm} X _i , n = mN. We show how such a self-normalization works for AR(1) and MA(q) processes.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 173–183, April–June, 2005.     

Schur functions and the invariant polynomials characterizing U(n) tensor operators

We give a direct formulation of the invariant polynomials _μG_q⁽ⁿ⁾(, Δ_i,;, x_{i,i + 1},) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions S_λ known as Schur functions. To this end, we show after the change of variables Δ_i = γ_i − δ_i and x_{i, i + 1} = δ_i − δ_{i + 1} that_μG_q⁽ⁿ⁾(,Δ_i;, x_{i, i + 1},) becomes an integral linear combination of products of Schur functions S_α(, γ_i,) · S_β(, δ_i,) in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}, respectively. That is, we give a direct proof that _μG_q⁽ⁿ⁾(,Δ_i,;, x_{i, i + 1},) is a bisymmetric polynomial with integer coefficients in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials _μ^mG_q⁽ⁿ⁾(γ₁,…, γ_n; δ₁,…, δ_m). These new symmetries enable us to give an explicit formula for both _μ^mG₁⁽ⁿ⁾(γ; δ) and ₁G₂⁽ⁿ⁾(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for _μ^mG_q⁽ⁿ⁾(γ; δ).     

Forbidden induced subgraph characterization of cograph contractions

Let S₁, S₂,…,S_t be pairwise disjoint non‐empty stable sets in a graph H. The graph H^* is obtained from H by: (i) replacing each S_i by a new vertex q_i; (ii) joining each q_i and q_j, 1 ≤ i # j ≤ t, and; (iii) joining q_i to all vertices in H – (S₁ ∪ S₂ ∪ ··· ∪ S_t) which were adjacent to some vertex of S_i. A cograph is a P₄‐free graph. A graph G is called a cograph contraction if there exist a cograph H and pairwise disjoint non‐empty stable sets in H for which G ? H^*. Solving a problem proposed by Le [ 2 ], we give a finite forbidden induced subgraph characterization of cograph contractions. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 217–226, 2004     

Strongly maximal antichains in posets

Given a collection S of sets, a set S∈S is said to be strongly maximal in S if |T?S|≤|S?T| for every T∈S. In Aharoni (1991) [3] it was shown that a poset with no infinite chain must contain a strongly maximal antichain. In this paper we show that for countable posets it suffices to demand that the poset does not contain a copy of posets of two types: a binary tree (going up or down) or a “pyramid”. The latter is a poset consisting of disjoint antichains A_i,i=1,2,…, such that |A_i|=i and x<y whenever x∈A_i,y∈A_j and j<i (a “downward” pyramid), or x<y whenever x∈A_i,y∈A_j and i<j (an “upward” pyramid).     

The short range expansion

Let V_i be short range potential and λ_i(ε) analytic functions. We show that the Hamiltonians H_ε = −Δ + ε⁻²∑_{i = l}ⁿλ_i(ε)V_i((· − x_i)/ε converge in the strong resolvent sense to the point interactions as ε → 0, and if V_i have compact support then the eigenvalues and resonances of H_ε, which remains bounded as ε → 0, are analytic in ε in a complex neighborhood of zero. We compute in closed form the eigenvalues and resonances of H_ε to the first order in ε.     

On the existence of a field of stress rates in a hardening elastic-plastic medium : PMM vol. 38, n 6, 1974, pp. 1114–1121

The boundary value problem for the stress rates and rates of change fields in the quasi-static motion of a volume V of an elastic-plastic medium [1] consists of finding the pairs σ_ij^., _ij^. related by the governing equations of an appropriate model; here the σ_ij^. should be statically admissible i.e. should satisfy the equations and boundary conditions σ_ij=−X/._i; /.σ_ijn_j|S_p=p_i and _ij should be kinematically admissible i.e. 2/._ij = v_ij + v_ji, where v_i|S_u = u_io Here S_p and S_u are nonintersecting parts of the boundary of the volume V, X_i, p_i, u_i/.o are specified functions. The question of the existence of a solution of this problem reduces to the question of the functional reaching the lower bound in a set of kinematically admissible /._ij^o and statically admissible σ_ij/.^/*. However, its lower bound may not be reached if in the minimization we limit ourselves only to smooth fields. It is proposed to augment the set of admissible fields σ_ij/.^/*,_ij/.^o by closing them in the norm L₂ (for v_i^o this corresponds to closure in the norm II¹). Some properties of the functional I (σ_ij^*,_ij/.) are considered in the augmented set of admissible fields. It is shown that the equivalence of the two problems is conserved, where I (σ_ij^*,_ij⁰ can be minimized in σ_ij^/*,_ij^o or in σ_ij^/*,_ij/.^o, The lower bound is reached in each of three cases, at a single point. From the fact that u_i^o belongs to the Sobolev space W₂⁽¹⁾, there results the absence of surfaces of velocity discontinuity. Variational principles have been used in plasticity theory to construct models [2] and to investigate the existence and properties of solutions [1, 3].     

The First Three Levels of an Order Preserving Hamiltonian Path in the Subset Lattice

Consider the lattice whose elements are the subsets of the set of positive integers not greater than n ordered by inclusion. The Hasse diagram of this lattice is isomorphic to the n-dimensional hypercube. It is trivial that this graph is Hamiltonian. Let be a Hamiltonian path. We say it is monotone, if for every i, either (a) all subsets of S _i appear among S ₁,...,S _i − 1, or (b) only one (say S) does not, furthermore S _i + 1 = S. Trotter conjectured that if n is sufficiently large, then there are no monotone Hamiltonian paths in the n-cube. He also made a stronger conjecture that states that there is no path with the monotone property that covers all the sets of size at most three. In this paper we disprove this strong conjecture by explicitly constructing a monotone path covering all the 3-sets.     

On the Common Substring Alignment Problem

The Common Substring Alignment Problem is defined as follows: Given a set of one or more strings S₁, S₂ … S_c and a target string T, Y is a common substring of all strings S_i, that is, S_i = B_iYF_i. The goal is to compute the similarity of all strings S_i with T, without computing the part of Y again and again. Using the classical dynamic programming tables, each appearance of Y in a source string would require the computation of all the values in a dynamic programming table of size O(nℓ) where ℓ is the size of Y. Here we describe an algorithm which is composed of an encoding stage and an alignment stage. During the first stage, a data structure is constructed which encodes the comparison of Y with T. Then, during the alignment stage, for each comparison of a source S_i with T, the pre-compiled data structure is used to speed up the part of Y. We show how to reduce the O(nℓ) alignment work, for each appearance of the common substring Y in a source string, to O(n)-at the cost of O(nℓ) encoding work, which is executed only once.     

Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

LetS be a bounded region inR ^N and let ℊ={S _i} _i ^=1/m be a partition ofS into a finite number of subsets having piecewiseC ² boundaries. We assume that whereC ² segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Letτ:S →S be piecewiseC ² on ℊ and expanding in the sense that there exists 0<σ< 1 such that for anyi=1, 2, ...,m, ‖Dτ _i ⁻¹ ‖<σ, whereDτ _i ⁻¹ is the derivative matrix ofτ _i ⁻¹ and ‖ ‖ is the euclidean matrix norm. The main result provides an upper bound onσ which guarantees the existence of an absolutely continuous invariant measure forτ. The research of the second author was supported by NSERC and FCAR grants.