λ-Aluthge Iteration and Spectral Radius

Let (H) be an invertible operator on the complex Hilbert space H. For 0 < λ < 1, we extend Yamazaki’s formula of the spectral radius in terms of the λ-Aluthge transform where T = U|T| is the polar decomposition of T. Namely, we prove that where r(T) is the spectral radius of T and ||| · ||| is a unitarily invariant norm such that (B(H), ||| · |||) is a Banach algebra with ||| I ||| = 1. In memory of my brother-in-law, Johnny Kei-Sun Man, who passed away on January 16, 2008, at the age of fifty nine.     

Spectral decomposition of square-tiled surfaces

Absract  We consider N copies of a square S ₀ and define selfadjoint extensions of the Euclidean Laplacian acting on by choosing some boundary conditions that are parametrized by two unitary matrices H and V acting on Denoting by Sp(Δ _N ,H,V) the spectrum of such an operator we derive conditions on H and V so that the following spectral decomposition holds:

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice , spanning a Euclidean space V. Let m be a positive integer and be the arrangement of hyperplanes in V of the form for and . It is known that the number of bounded dominant regions of is equal to the number of facets of the positive part of the generalized cluster complex associated to the pair by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of and conjecture that the corresponding refinement of coincides with the $h$-vector of . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1. 2000 Mathematics Subject Classification Primary—20F55; Secondary—05E99, 20H15     

A note on morphic cohomology and algebraic cycles

We give examples of smooth projective complex varieties of dimension d ≥ 4 and primes ℓ such that the morphic cohomology group is infinite, and is not finitely generated as a rational vector space. In particular, for these examples the semi-topological K-group has infinite dimension.     

Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Let B _i be deterministic real symmetric m × m matrices, and ξ _i be independent random scalars with zero mean and “of order of one” (e.g., ). We are interested to know under what conditions “typical norm” of the random matrix is of order of 1. An evident necessary condition is , which, essentially, translates to ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S _N is : for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints , where F is quadratic in X = (X ₁,... ,X _k ). We show that when F is convex in every one of X _j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices . Research was partly supported by the Binational Science Foundation grant #2002038.     

Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions:

Semilinear subelliptic problems with critical growth on Carnot groups

We prove existence and multiplicity of solutions for the semilinear subelliptic problem with critical growth in Ω, u = 0 on ∂Ω, where is a sublaplacian on a Carnot group , 2* = 2Q/(Q − 2) is the critical Sobolev exponent for and Ω is a bounded domain of .     

Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).     

Threshold θ ≥ 2 contact processes on homogeneous trees

We study the threshold θ ≥ 2 contact process on a homogeneous tree of degree κ = b + 1, with infection parameter λ ≥ 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point and for it survives iff , where this critical density satisfies , . For large b, we show that the process on has a qualitatively similar behavior when λ is small, including the behavior at and close to the critical point . In contrast, for large λ the behavior of the process on is qualitatively distinct from that of the mean-field model in that the critical density has . We also show that , where 1 < Φ₂ < Φ₃ < ..., , and . The work of L.R.F. was partially supported by the Brazilian CNPq through grants 307978/2004-4 and 475833/2003-1, and by FAPESP through grant 04/07276-2. The work of R.H.S. was partially supported by the American N.S.F. through grant DMS-0300672.     

Weak convergence of diffusion processes on Wiener space

Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e _i } ⊂ H be an orthonormal basis of H. Suppose that μ _n = ρ _n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e _i , then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly.     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Convergence of the Iterated Aluthge Transform Sequence for Diagonalizable Matrices II: λ-Aluthge Transform

Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U|T|. Then the λ-Aluthge transform is defined by

Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

Suppose thatА is a nonnegative self-adjoint extension to { } of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {

Compactness results for divergence type nonlinear elliptic equations

Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F _α defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g _α and of coefficients a _α on M, we discuss the compactness problem of minimal energy type solutions of the equation

The Asymptotics of the Heat Semigroup for a General Bounded Domain with Mixed Boundary Conditions

Abstract Small-time asymptotics of the trace of the heat semigroup where {μ _ν } are the eigenvalues of the negative Laplacian in the (x ¹, x ²)-plane, is studied for a general bunded domain Ω with a smooth boundary ∂Ω, where a finite number of Dirichlet, Neumann and Robin boundary conditions, on the piecewise smooth parts Γ _i (i = 1, ..., n) of ∂Ω such that , are considered. Some geometrical properties associated with Ω are determined.     

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of and to characterize the elements of in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals of , j = 0,..., , such that is the linear span of sums of permutations with a common set of interior peaks and is the peak algebra. We extend the above results to , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation