Using lower bounds on distance spectrum components of a code on the Euclidean sphere, we improve the known asymptotical upper bounds on the cardinality of multiple packings of the sphere by balls of smaller radii. Let Rn be the n-dimensional Euclidean space, and Sn−1(r) ⊂ Rn be the (closed) Euclidean sphere of radius r with the center in the origin. Let further S̃n−1(r, ā) be the open ball of ra...

We give a characterization of Banach spaces X containing a subspace with a shrinking Markushevich basis {xγ , fγ}γ∈Γ. This gives a sufficient condition for X to have a renorming with Mazur’s intersection property. A biorthogonal system in a Banach space X is a subset {xγ , fγ}γ∈Γ ⊂ X×X such that fγ(xγ′) = δγγ′ for γ, γ ′ ∈ Γ. The biorthogonal system {xγ, fγ}γ∈Γ in X is called fundamental if X =...

This paper presents a novel projection-based adaptive algorithm for sparse signal and system identification. The sequentially observed data are used to generate an equivalent sequence of closed convex sets, namely hyperslabs. Each hyperslab is the geometric equivalent of a cost criterion, that quantifies “data mismatch”. Sparsity is imposed by the introduction of appropriately designed weighted...

Juggling patterns can be described by a closed walk in a (directed) state graph, where each vertex (or state) is a landing pattern for the balls and directed edges connect states that can occur consecutively. The number of such patterns of length n is well known, but a long-standing problem is to count the number of prime juggling patterns (those juggling patterns corresponding to cycles in the...

We prove a that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally, we give a short proof of topological entropy rigidity f...

In this paper, we provide in a general Hilbert space new characterizations of uniform prox-regularity involving outside but sufficiently close points considered sets. We show that the complement prox-regular set is nothing union closed balls with common radius. derive from given equivalent to semiconvexity property its distance function. Various estimates metric projection mapping are also esta...

Abstract The notion of the Urysohn d -width measures to what extent a metric space can be approximated by -dimensional simplicial complex. We investigate how local width bounds on Riemannian manifold affect its global width. bound 1-width in terms first homology and supremal unit balls. Answering question Larry Guth, we give examples n -manifolds considerable <m:math xmlns:m="http://www.w3.org/...