A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

Multi-Resolution Analysis of Multiplicity d: Applications to Dyadic Interpolation

This paper studies the Multi-Resolution Analyses of multiplicity d (d *), that is, the families (V_n)_n of closed subspaces in ²( ) such that V_n V_{n + 1}, V_{n + 1} = DV_n, where Dƒ(x) = ƒ(2x), and such that there exists a Riesz basis for V₀ of the form {φ_i(· − k), i = 1, . . . , d,k }, with φ₁, . . . , φ_d V₀. Using the Fourier transform, we prove that (λ) = ^t[ ₁(λ), . . . , _d(λ)] = H(λ/2) (λ/2), where H is in the set _d of continuous 1-periodic functions taking values in (d, ). If d = 1, the definition corresponds to the standard Multi-Resolution Analyses, and one can characterize the regular 1-periodic complex-valued functions H (called, then, scaling filters) which yield a Multi-Resolution Analysis. In this paper, we generalize this study to d ≥ 2 by giving conditions on H _d so that there exists = ^t[ ₁, . . . , _d] in ²( , ^d) solution of (λ) = H(λ/2) (λ/2), and so that the integer translates of φ₁, . . . , φ_d form a Riesz family. Then, the latter span the space V₀ of a Multi-Resolution Analysis of multiplicity d. We show that the conditions on H focus on the zeros of det H(·) and on simple spectral hypotheses for the operator P_H defined on _d by P_HF(λ) = H(λ/2)F(λ/2)H(λ/2)* + H(λ/2 + 1/2)F(λ/2 + 1/2)H(λ/2 + 1/2)*. Finally, we explore connections with the order r dyadic interpolation schemes, where r *.     

Second-Order Properties of a Two-Stage Fixed-Size Confidence Region for the Mean Vector of a Multivariate Normal Distribution

We consider the classical fixed-size confidence region estimation problem for the mean vectorμin theN_p(μ, Σ) population where Σ is unknown but positive definite. We writeλ₁for the largest characteristic root of Σ and assume thatλ₁is simple. Moreover, we suppose that, in many practical applications, we will often have available a numberλ*(>0) and that we can assumeλ₁>λ*. Given this addi- tional, and yet very minimal, knowledge regardingλ₁, the two-stage procedure of Chatterjee (Calcutta Statist. Assoc. Bull.8(1959a), 121–148;9(1959b), 20–28;11(1962), 144–159) is revised appropriately. The highlight in this paper involves the verification ofsecond-order propertiesassociated with such revised two-stage estimation techniques, along with the maintenance of the nominal confidence coefficient.     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

Middle diamond

Under certain cardinal arithmetic assumptions, we prove that for every large enough regular λ cardinal, for many regular κ < λ, many stationary subsets of λ concentrating on cofinality κ has the “middle diamond”. In particular, we have the middle diamond on {δ < λ: cf(δ) = κ}. This is a strong negation of uniformization.I would like to thank Alice Leonhardt for the beautiful typing. This research was partially supported by the Israel Science Foundation. Publication 775.     

Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δv_n⁻¹) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andv_n → ∞ slower thann^1/2. Let_n denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifn_n²(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT_(δjδ)^1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)^1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofT_μ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.     

The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations

Given two self-adjoint operators A and V = V₊  − V₋ , we study the motion of the eigenvalues of the operator A(t) = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantities N₊ (V; λ, α), N₋ (V; λ, α), and N₀(V; λ, α) defined as the number of eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α > 0. We study asymptotic characteristics of these quantities as α → ∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys.193 (1998), 233–243] are extended and given new applications to differential operators.     

Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability

The basic result of the paper states: Let F₁, …, F_n, F₁^′,…, F_n^′ have proportional hazard functions with λ₁ ,…, λ_n , λ₁^′ ,…, λ_n^′ as the constants of proportionality. Let X₍₁₎ ≤ … ≤ X_(n) (X₍₁₎^′ ≤ … ≤ X_(n)^′) be the order statistics in a sample of size n from the heterogeneous populations {F₁ ,…, F_n}({F₁^′ ,…, F_n^′}). Then (λ₁ ,…, λ_n) majorizes (λ₁^′ ,…, λ_n^′) implies that (X₍₁₎ ,…, X_(n)) is stochastically larger than (X₍₁₎^′ ,…, X_(n)^′). Earlier results stochastically comparing individual order statistics are shown to be special cases. Applications of the main result are made in the study of the robustness of standard estimates of the failure rate of the exponential distribution, when observations actually come from a set of heterogeneous exponential distributions. Further applications are made to the comparisons of linear combinations of Weibull random variables and of binomial random variables.     

Lambda-determinants and domino-tilings

Consider the 2n-by-2n matrix with m_i,j=1 for i,j satisfying |2i−2n−1|+|2j−2n−1|2n and m_i,j=0 for all other i,j, consisting of a central diamond of 1's surrounded by 0's. When n4, the λ-determinant of the matrix M (as introduced by Robbins and Rumsey [Adv. Math. 62 (1986) 169–184]) is not well defined. However, if we replace the 0's by t's, we get a matrix whose λ-determinant is well defined and is a polynomial in λ and t. The limit of this polynomial as t→0 is a polynomial in λ whose value at λ=1 is the number of domino-tilings of a 2n-by-2n square.     

Recursive constructions for optimal (n,4,2)‐OOCs

In [ 3 ], a general recursive construction for optical orthogonal codes is presented, that guarantees to approach the optimum asymptotically if the original families are asymptotically optimal. A challenging problem on OOCs is to obtain optimal OOCs, in particular with λ > 1. Recently we developed an algorithmic scheme based on the maximal clique problem (MCP) to search for optimal (n, 4, 2)‐OOCs for orders up to n = 44. In this paper, we concentrate on recursive constructions for optimal (n, 4, 2)‐OOCs. While “most” of the codewords can be constructed by general recursive techniques, there remains a gap in general between this and the optimal OOC. In some cases, this gap can be closed, giving recursive constructions for optimal (n, 4, 2)‐OOCs. This is predicated on reducing a series of recursive constructions for optimal (n, 4, 2)‐OOCs to a single, finite maximal clique problem. By solving these finite MCP problems, we can extend the general recursive construction for OOCs in [ 3 ] to obtain new recursive constructions that give an optimal (n · 2^x, 4, 2)‐OOC with x ≥ 3, if there exists a CSQS(n). © 2004 Wiley Periodicals, Inc.     

Bounds and Constructions on (v, 4, 3, 2) Optical Orthogonal Codes

In this paper, we are concerned about optimal (v, 4, 3, 2)‐OOCs. A tight upper bound on the exact number of codewords of optimal (v, 4, 3, 2)‐OOCs and some direct and recursive constructions of optimal (v, 4, 3, 2)‐OOCs are given. As a result, the exact number of codewords of an optimal (v, 4, 3, 2)‐OOC is determined for some infinite series.     

2-dimensional optical orthogonal codes from singer groups

We present several new families of (Λ×T,w,λ) (2D) wavelength/time optical orthogonal codes (2D-OOCs) with λ=1,2. All families presented are either optimal with respect to the Johnson bound (J-optimal) or are asymptotically optimal. The codes presented have more flexible dimensions and weight than the J-optimal families appearing in the literature. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q). This finite geometries framework gives structure to the codes providing insight. We establish that all 2D-OOCs constructed are in fact maximal (in that no new codeword may be added to the original whereby code cardinality is increased).     

Matrices dont deux lignes quelconque coïncident dans un nombre donne de positions communes

Le nombre maximal de lignes de matrices seront désignées par:

1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;

2. (b) S⁰(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y₁, y₂, y₃, ou y₁ = y₂ ≠ y₃;

3. (c) T⁰(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.

La fonction T⁰(k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que T⁰(k, λ) max(λ + 2, (k − λ)² + k − λ + 1). La fonction S⁰(k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant T⁰(k, λ), S⁰(k, λ), dans la remarque 9 on généralise le problème concernant R(k, λ). La fonction R(k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que R(k, λ) S⁰(k, λ) T⁰(k, λ) d'où découle en particulier: R(k, λ) λ + 2 pour λ k + 1 − (k + 2)^1/2; R(k, λ) = 0(k²) pour k − λ = 0(k); R(k, λ) (k − 1)² − (k + 2) pour k 1191.     

Uniform Estimates for a Class of Evolution Equations

L^∞ estimates are derived for the oscillatory integral ∫⁺₀^∞e^{−i(xλ + (1/m) tλm)}a(λ) dλ, where 2 ≤ m and (x, t) × ₊. The amplitude a(λ) can be oscillatory, e.g., a(λ) = e^{it (λ)} with (λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)^k with 0 ≤ k ≤ (m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations.     

A new characterization of supercompactness and applications

We give a new characterization of λ-supercompact cardinal κ in terms of (κ,λ)-Solovay pairs. We give some applications of (κ,λ)-Solovay pairs.     

Strong asymptotics in Lagrange interpolation with equidistant nodes

In this paper we prove three conjectures of Revers on Lagrange interpolation for f_λ(t)=|t|^λ,λ>0, at equidistant nodes. In particular, we describe the rate of divergence of the Lagrange interpolants L_N( f_λ,t) for 0<|t|<1, and discuss their convergence at t=0. We also establish an asymptotic relation for max_|t|1| |t|^λ−L_N( f_λ,t)|. The proofs are based on strong asymptotics for |t|^λ−L_N( f_λ,t), 0|t|<1.     

A Generalization of Hankel Operators

We introduce a class of operators, called λ-Hankel operators, as those that satisfy the operator equation S*X−XS=λX, where S is the unilateral forward shift and λ is a complex number. We investigate some of the properties of λ-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of λ-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to λ-Hankel operators for certain λ. We also study some other operator equations involving S.     

A bijective proof of the hook-length formula

A well-known theorem of Frame, Robinson, and, Thrall states that if λ is a partition of n, then the number of Standard Young Tableaux of shape λ is n! divided by the product of the hook-lengths. We give a new combinatorial proof of this formula by exhibiting a bijection between the set of unsorted Young Tableaux of shape λ, and the set of pairs (T, S), where T is a Standard Young Tableau of shape λ and S is a “Pointer” Tableau of shape λ.     

A new class of symmetric polynomials defined in terms of tableaux

A new class of symmetric polynomials in n variables z = (z₁,…, z_n), denoted t_λ(z), and labelled by partitions λ = [λ₁ … λ_n] is defined in terms of standard tableaux (equivalently, in terms of Gel'fand-Weyl patterns of the general linear group GL(n,C)). The t_λ(z) are shown to be a -basis of the ring of all symmetric polynomials in n variables. In contrast to the usual basis sets such as the Schur functions e_λ(z), which are homogeneous polynomials in the z_i, the t_λ(z) are inhomogeneous. This property is reflected in the fact that the t_λ(z) are a natural basis for the expansion of certain (inhomogeneous) symmetric polynomials constructed from rising factorials. This and several other properties of the t_λ(z) are proved. Two generalizations of the t_λ(z) are also given. The first generalizes the t_λ(z) to a 1-parameter family of symmetric polynomials, T_λ(α; z), where α is an arbitrary parameter. The T_λ(α; z) are shown to possess properties similar to those of the t_λ(z). The second generalizes the t_λ(z) to a class of skew-tableau symmetric polynomials, t_λ/μ(z), for which only a few preliminary results are given.     

Curves of positive solutions of boundary value problems on time-scales

Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem

(2)

(4)

u(a)=u(b)=0,