Extendible elements of the alternating groups

An element σ of A_n, the Alternating group of degree n, is extendible in S_n, the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Eigenvalue inequalities and equalities

We consider cases of equality in three basic inequalities for eigenvalues of Hermitian matrices: Cauchy's interlacing inequalities for principal submatrices, Weyl's inequalities for sums, and the residual theorem. Several applications generalize and sharpen known results for eigenvalues of irreducible tridiagonal Hermitian matrices.     

Global Support of a Scaling Vector

Multiwavelet decompositions are based on scaling vectors satisfying matrix refinement equations. The support and linear independence of scaling vectors play an essential role in the study of multiwavelets. In this paper we relate these properties with the coefficients in the matrix refinement equation satisfied by the scaling vector.     

Computing inertia sets using atoms

We consider the problem of computing inertia sets for graphs. By using tools for combining the inertia sets of smaller graphs we can reduce this problem to understanding the inertia sets for three-connected graphs that are not joins. We term such graphs atoms and give the inertia sets for all atoms on at most seven vertices. This can be used to compute the inertia sets for all graphs on at most seven vertices.     

Integral circulant graphs

In this note we characterize integral graphs among circulant graphs. It is conjectured that there are exactly 2^τ(n)-1 non-isomorphic integral circulant graphs on n vertices, where τ(n) is the number of divisors of n.     

Convergence domains for the campbell-baker-hausdorff formula

Three neighbourhoods of zero within which the Campbell-Baker-Hausdorff series for log(e^xe^y) converges are compared. Two are large enough to include the unit sphere under a norm compatible with Lie multiplication.     

Rank one perturbation and its application to the laplacian spectrum of a graph

Characterization of rank one perturbations of symmetric matrices which change only one eigenvalue are given. Then the result is applied to study how the Laplacian spectrum of a graph changes when adding an edge.     

Convexity of the upper complex plane part of the numerical range of a quaternionic matrix

The quaternionic numerical range of a matrix with quaternion entries has a convex intersection with the upper half complex plane. The quaternionic analog of the elliptical range theorem is proved.     

Possibilities for the number of invariant polynomials of the difference of two similarity classes

In this paper we describe all possible numbers of (nontrivial) invariant polynomials of the difference of two matrices with prescribed similarity classes over an algebraically closed field.