V. D. Milman proved in [14] that the product of two strictly singular operators on L p [0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying ...

Let X be a rearrangement-invariant space. An operator T : X → X is called narrow if for each measurable setA and each ε > 0 there exists x ∈ X with x = χA, ∫ xdμ = 0 and ‖Tx‖ < ε. In particular all compact operators are narrow. We prove that if X is a Lorentz function space Lw,p on [0,1] with p > 2, then there exists a constant kX > 1 so that for every narrow projection P on Lw,p ‖Id− P‖ ≥ kX ....

To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T [X]. Billera, Jia and Reiner recently introduced ...

Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε → ∆ X (ε) tending to zero with ε > 0 such that every operator T : L 2 → L 2 with T ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L 1 and on L ∞ must be of norm ≤ ∆ X (ε) on L 2 (X). We show that ∆ X (ε) ∈ O(ε α) for some α > 0 iff X is isomorphic to ...

We are concerned with the numerical study of a simple one-dimensional Schrödinger operator − 1 2 ∂xx + αq(x) with α ∈ R, q(x) = cos(x) + ε cos(kx), 0 ≤ ε ¿ 1 and k being irrational. This governs the quantum wave function of an independent electron within a crystalline lattice perturbed by some impurities whose dissemination induces long-range order only, which is rendered by means of the quasi-...

We present a new efficient localized algorithm to construct, for any given quasi-unit disk graph G = (V,E) and any ε > 0, a (1+ ε)spanner for G of maximum degree O(1) and total weight O(ω(MST )), where ω(MST ) denotes the weight of a minimum spanning tree for V . We further show that similar localized techniques can be used to construct, for a given unit disk graph G = (V,E), a planar Cdel(1+ε)...

Given A ⊆ ℝ , the Hattori space H(A) is topological ( τA ) where each a ∈ has -neighborhood base { − ε + : &gt; 0 } and b [ . Thus, may be viewed as hybrid of Euclidean topology lower-limit topology. We investigate properties spaces well other topologies on using various combinations discrete, left-ray, lower-limit, upper-limit, topologies. Since these generated by quasi-metric ℝ, we quasi-metr...

We consider several models of nonlinear wave equations subject to very strong damping and quasi-periodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under very general non-resonance conditions on the frequency, we show the ...

Using a recent result of Batson, Spielman and Srivastava, We obtain a tight estimate on the dimension of lp , p an even integer, needed to almost isometrically contain all k-dimensional subspaces of Lp. In a recent paper [BSS] Batson, Spielman and Srivastava introduced a new method for sparsification of graphs which already proved to have functional analytic applications. Here we bring one more...