Diameters of sections and coverings of convex bodies

We study the diameters of sections of convex bodies in R^N determined by a random N×n matrix Γ, either as kernels of Γ^* or as images of Γ. Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in R^N has one well bounded k-codimensional section, then for any m>ck random sections of K of codimension m are also well bounded, where c?1 is an absolute constant. It is noteworthy that in the Gaussian case, when Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c=1.     

Efficient algorithms for a constrained k-tree core problem in a tree network

Let T=(V,E) be a free tree in which each vertex has a weight and each edge has a length. Let n=|V|. Given T and parameters k and l, a (k,l)-tree core is a subtree X of T with diameter l, having k leaves, which minimizes the sum of the weighted distances from all vertices in T to X. In this paper, two efficient algorithms are presented for finding a (k,l)-tree core of T. The first algorithm has O(n²) time complexity for the case that each edge has an arbitrary length. The second algorithm has O(lkn) time complexity for the case that the lengths of all edges are 1. The (k,l)-tree core problem has an application in distributed database systems.     

Unit Distances and Diameters in Euclidean Spaces

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large depending on d. As a corollary, we determine the exact maximum number of unit distances for all even d≥6 and the exact maximum number of diameters for all d≥4 and all n sufficiently large depending on d. This material is based upon work supported by the South African National Research Foundation.     

Etching of gold nanorods: The effects of diameter on analytical performances

Gold nanorods (GNRs) have been well employed for sensing/bio-sensing based on analytes modulated morphology or self-assembly states. Herein, we employed H₂O₂ based etching system (H₂O₂ molecules themselves and H₂O₂-Fe²⁺ Fenton reagents) as an example to study the diameters of GNRs on the analytical performances, for the colorimetric platforms using GNRs as reporters. We have found that the thinner GNRs possess a higher sensitivity; while the thicker ones bring more abound color presentation during the etching processes, which is especially for naked eye detection. In addition, a red shift of the plasmonic bands is observed for three kinds of thinner GNRs at the initial stage of the etching reaction, and the mechanism is also discussed.     

Etching of gold nanorods: The effects of diameter on analytical performances

The diameters of gold nanorods have profound effects on the analytical performances of the corresponding etching based sensing system.