Developed in 1999 by Akemann, Anderson, and Weaver, the spectral scale of an n × n matrix A, is a convex, compact subset of R that reveals important spectral information about A [4]. In this paper we present new information found in the spectral scale of a matrix. Given a matrix A = A1 + iA2 with A1 and A2 self-adjoint and A2 6= 0, we show that faces in the boundary of the spectral scale of A t...

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P . We construct a set of n points in general position in the plane with only ≈ 1.62n empty triangles, ≈ 1.94n empty quadrilaterals, ≈ 1.02n empty pentagons, and ≈ 0.2n empty hexagons.

Let D be a nonempty and convex subset of a Banach spaces E. The set D is called proximinal if for each x∈E, there exists an element y∈D such that ||x-y|| = d(x,D), where d(x,D) = inf{||x-z||: z∈D}. Let CB(D), CCB(D), K(D) and P(D) denote the families of nonempty closed bounded subsets, nonempty closed convex bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of D...

We give an alternative proof to the well known fact that each convex compact centrally symmetric subset of R2 containing the origin is a zonoid, i.e., the range of a two dimensional vector measure, and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a Chebyshev measure. We give a condition under which a two dimension...

Consider a finite group G and subgroups H,K of G. We say that H and K permute if HK = KH and call H a permutable subgroup if H permutes with every subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are permutable. We can define, for every finite group G, an arithmetic quantity that measures the probability that two subgroups (chosen uniformly at random with replacement) per...

This paper describes several speedups for computation in the order p + 1 subgroup of Fp2 and the order p 2 − p + 1 subgroup of Fp6 . These results are in a way complementary to LUC and XTR, where computations in these groups are sped up using trace maps. As a side result, we present an efficient method for XTR with p ≡ 3 mod 4.

In a previous paper, the second author established that, given finite fields F < E and certain subgroups C ≤ E×, there is a Galois connection between the intermediate field lattice {L | F ≤ L ≤ E} and C’s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product C⋊Gal(E/F ). However, the analysis when |F | i...

Recently K. Murota has introduced concepts of L-convex function and Mconvex function as generalizations of those of submodular function and base polyhedron, respectively, and has shown separation theorems for L-convex/concave functions and for M-convex/concave functions. The present note gives short alternative proofs of the separation theorems by relating them to the ordinary separation theore...