Decomposition of the conjugacy representation of the symmetric groups

Consider the two natural representations of the symmetric groupS _n on the group algebra ℂ[S _n ]: the regular representation and the conjugacy representation (acting on the basis by conjugation). Letm(λ) be the multiplicity of the irreducible representationS ^λ in the conjugacy representation and letf ^λ be the multiplicity ofS ^λ in the regular representation. By the character estimates of [R1] and [Wa] we prove

Equi-Attraction and Continuous Dependence of Strong Attractors of Set-Valued Dynamical Systems on Parameters

Equi-attraction properties of the global attractors of a family of set-valued dynamical systems {G_λ, λ∈Λ} with respect to a parameter λ∈Λ, where Λ is a compact metric space, are investigated. In particular, under appropriate conditions, it is shown that the equi-attraction of the family is equivalent to the continuity of in λ with respect to the Hausdorff distance. An example involving differential inclusions is given. Mathematics Subject Classifications (2000) Primary 47H17, 58F03, 34C20. Desheng Li: Supported by NNSF of China (10251002) and NNSF of Gansu province (3ZS041-A25-006).     

Immersions of spheres and algebraically constructible functions

Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g _λ  =  g(· , λ) restricted to the sphere S ⁿ (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g _λ|S ⁿ (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.     

Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

The ‘window problem’ for series of complex exponentials

Under a suitable sparsity condition on the exponents Λ=(λ_k=τ_k+iσ_k), it is shown that the individual terms can be obtained from observation of the L² function through the ‘window’ t∈[0, δ]—with an l² estimate (uniform for such Λ) asymptotically as t, δ→0. Some applications are given to control theory for partial differential equations.     

Directional derivatives for the value-function in semi-infinite programming

For the problemP(λ): Maximizec ^T z subject toz∈Z(λ), whereZ(λ) is defined by an in general infinite set of linear inequalities, it is shown that the value-function has directional derivatives at every point such thatP( ) and its dual are both superconsistent. To compute these directional derivatives a min-max-formula, well-known in convex programming, is derived. In addition, it is shown that derivatives can be obtained more easily by a limit-process using only convergent selections of solutions ofP(λ _n ), λ _n → and their duals.     

Measured descent: a new embedding method for finite metrics

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to Bourgain (1985) and Rao (1999). We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion where α_X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an distortion embedding, where λ_X is the doubling constant of X. Since λ_X ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “low-dimensional,” improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)²). Received: April 2004 Accepted: August 2004 Revision: December 2004 J.R.L. Supported by NSF grant CCR-0121555 and an NSF Graduate Research Fellowship.     

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

A trace theorem and a linearization method for operator polynomials

Let p() = ⁿI - ^n-1 A₁ - A_n, where A_i are operators on a Banach space X with (1/i)^th power summable approximation numbers. Then the sum of the eigenvalues of the operator polynomial p() equals the trace of A₁. This is shown by a linearization method which is further used to generalize results on the completeness of the eigenvalues of p() under suitable conditions on the resolvent.     

The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.     

Absolute continuity of the distribution of some Markov geometric series

Let (∈n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift space ∑A defined by Fibonacci incident matrix A.We consider the measure μλ of the probability distribution of the random series ∑∞n=0 εnλn (0 ＜λ＜ 1).It is proved that μλ is singular if λ∈ (0,√5-1/2) and that μλ is absolutely continuous for almost all λ∈ (√5-1/2,0.739).     

Completeness of systems of entire functions in spaces of holomorphic functions

Letf be an entire function in . For a broad class of distribution densities of the set Λ, a scale of sufficient conditions for the completeness of the system of functions {f(λ×z):λ∈Λ},z∈E, where , in the spaceH(E) of holomorphic functions onE with respect to the topology of uniform convergence on compact subsets is given in terms of the mutual indicator of the functionf and the setE. These conditions are new already forn=1 even ifE is a disk. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 603–616, October, 1999.     

Systems with nonextendable convergence of quasipolynomials

The system , where Λ={λ _n } is the set of zeros (of multiplicitiesm _n ) of the Fourier transform

On Stabilized Point Spectra of Multivalued Systems

Let H be an infinite dimensional Hilbert space. Denote by Λ (E, F) the set of all for which the multivalued system 0 ∈ (F − λ E) (x) admits a nonzero solution x ∈ H. One says that Λ (E, F) is the point spectrum of the pair (E, F). It is well known that Λ (E, F) does not behave in a stable manner with respect to perturbations in the argument (E, F). The purpose of this note is to study the outer-semicontinuous hull (or graph-closure) of the mapping Λ.     

Convergence of mock Fourier series

For certain Cantor measures μ on ℝⁿ, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse ^2πiγ·x for λεΛ. a discrete subset of ℝⁿ called aspectrum for μ. For anyL ¹ functionf, we define coefficientsc _γ(f)=∝f(y)e ^−2πiγiy dμ(y) and form the Mock Fourier series ∑_λ∈Λ^cλ(f)e ^2πiλ·x . There is a natural sequence of finite subsets Λ_n increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by We prove, under mild technical assumptions on μ and Λ, thats _n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL ¹. Research supported in part by the National Science Foundation, Grant DMS-0140194.     

On the lowest eigenvalue of the Laplacian for the intersection of two domains

IfA andB are two bounded domains in _n and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B _x, ofB such that (AB _x)<(A)+(B). A similar inequality holds for .There are two corollaries of this theorem: (i) A lower bound for sup _x {volume (AB _x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW ^1,p ( _n ).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15     

On exact constant in a one-dimensional embedding theorem

In the case 1≤p<q≤∞, the question on the exact constant in the embedding of the space W _p ¹ (0,1) into the space L_q(0,1) is studied, i.e.,

Barrelledness of Generalized Sums of Normed Spaces

Let (E _i ) _iI be a family of normed spaces and a space of scalar generalized sequences. The -sum of the family (E _i ) _iI of spaces is

Statistical inference for a bivariate exponential distribution based on grouped data

Consider the bivarate exponential distribution due to Marshall and Olkin [2], whose survival function is F(x,y)=exp[-λ1x-λ2y-λ12 max(x,y)] (x≥0,y≥0)with unknown pexameters λ1＞0,λ2＞0 andλ12≥0. Based on grouped data, a new estimstor for ,λ1, ,λ2 and ,λ12 is derlved and its asymptotic perties are discussed.Bealdes, some test procedures of equalmarginals and independence are gven. A simulation result is given, too.     

A note on Sobolev orthogonality for Laguerre matrix polynomials

Let {Ln(A,λ)(x)}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0, ∞) by Ln(A,λ)(x)=n!/(-λ)n∑nk=0(-λ)κ/k!(n-1)! (A I)n[(A I)k]-1 xk,where A ∈ Cr×r. It is known that {Ln(A,λ)(x)}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re(z) ＞ - 1 for every z ∈σ(A).In this note we show that forA such that σ(A) does not contain negative integers, the Laguerre matrix polynomials Ln(A,λ) (x) are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases: the above matrix case and the known scalar case.