Eigenvalues of matrix commutators ∗

We characterize the eigenvalues of [X,A]=XA?AX, where A is an n by n fixed matrix and X runs over the set of the matrices of the same size.     

Singular values of matrix exponentials ∗

The singular values of a matrix and those of its exponential are related via multiplicative majorization. Matrices giving some equalities in the majorization are characterized. As an application, a scalar inequality for the exponential function is generalized to a matrix-valued inequality and the case of equality is examined.     

The Moore–Penrose inverse of a companion matrix

Necessary and sufficient conditions are given for the Moore–Penrose inverse of a companion matrix over an arbitrary ring to exist.     

The applied geometry of a general Liénard polynomial system

In this work, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Liénard polynomial system with an arbitrary (but finite) number of singular points.     

The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

A note on the second largest eigenvalue of the laplacian matrix of a graph ∗

In this note, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second largest degree of the graph.     

Relative invariants of 3 × 3 matrix triples ∗

We present primary and secondary generators for the algebra of polynomial invariants of the direct product of two copies of the special linear group Sl ₃ acting naturally on triples of 3 × 3 matrices over a field of characteristic zero. We handle also the analogous problem for triples and quadruples of 2 × 2 matrices.     

∗-Polynomial Identities of a Nonsymmetric ∗-Minimal Algebra

Let F be an infinite field of characteristic ≠?2. We study the ?-polynomial identities of the ?-minimal algebra R?=?UT _?(F?⊕?F, F). We describe the generators of T _?(R) and a linear basis of the relatively free algebra of R. When char.F?=?0, these results allow us to provide a complete list of polynomials generating irreducible GL × GL-modules decomposing the proper part of the relatively free algebra of R. Finally, the ?-codimension sequence of R is explicitly computed.     

The minimal eigenvalue of the Lyapunov transform †

An (n m) hypergraph is a coupleH=(N E), where the vertex set N is {1,…n} and the edge set E is an m-element multiset of nonempty subsets of N. In this paper, we count nonisomorphic hypergraphs where isomorphism of hypergraphs is the natural extension of that of graphs. A main result is an explicit formula for the cycle index of the permutation representation of any permutation group P with object set N acting on the k-element subsets of N. By making a simple substitution in these cycle indices for P the symmetric group S_N and k=1,…,n, we obtain generating functions which enumerate various types of hypergraphs. Using the technique developed, we extend Snapper's results on characteristic polynomials of permutation representations and group characters from the case where the group has odd order to the general case.     

Hausdorff dimension of Julia set of a polynomial with a Siegel disk

Let f(z) = e2πiθz(1 z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of f(z) satisfies Dim(J(f))<2.     

The correlation numerical range of a matrix and Connes’ embedding problem

We define a new numerical range of an $n\times n$ complex matrix in terms of correlation matrices and develop some of its properties. We also define a related numerical range that arises from Connes’ famous embedding problem.     

The Mathematical Basis of Population Genetics∗

In this paper the topic known to biologists as population genetics is presented in terms of the Cartesian product. The gene, being a set of an ordered pair of elements (alleles), provides an example of a lattice in which the axes represent the composition of the gene in respective parents, and shows as co‐ordinates the possible genetic composition of individuals of the next generation with respect to that gene. When the axes are made to represent, respectively, the set of male and female parents of a population as a whole then it is the next generation as a whole that is represented by the co‐ordinates.

Population genetics is also shown to provide an example of the binomial expansion (p+q)ⁿ in which n is 2, while on those occasions when the gene can be composed of any two of a number of elements (alleles) it expands to (p+q + r+ ... + w)² with p,q,r.....w, each representing the frequencies of an allele of the gene in the population.

It is upon these mathematical models that the study of population genetics depends and the incidence of alleles in any population is established.

     

The inverse and determinant of a 2 × 2 uniformly distributed random matrix

Formulae are derived for the density of the determinant and the elements of the inverse of a 2 × 2 matrix, with entries which are independent random variables uniformlly distributed on [0,1]. Graphs of the densities are presented, and the relevance of the results to interval matrices is discussed.     

A note on maximizing the permanent of a positive definite hermitian matrix,given the eigenvalues ∗

As a step toward understanding the unsolved problem of determining how large the permanent of a positive semi-definite matrix can be, given the eigenvalues, we note that a necessary condition for A to be a permanent maximizing matrix is that A commute with its permanental adjoint.