Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

In previous work of the author and M. Culler, contractible simplicial complexes were constructed on which the group of outer automorphisms of a free group of finite rank acts with finite stabilizers and finite quotient. In this paper, it is shown that these complexes are Cohen-Macauley, a property they share with buildings. In particular, the link of a vertex in these complexes is homotopy equivalent to a wedge of spheres of codimension 1.

Asymptotic formulae are given for the distribution of Liusternik-Schnirelman eigenvalues of certain pairs of nonlinear functionals generalising the usual Weyl theory for linear pairs of elliptic operators. In particular an application is made to the von Kármán theory of buckled plates.

In this paper a characterization of the regular ω-semigroups whose congruence lattice is modular is given. The characterization obtained for such semigroups generalizes the one given by Munn for bisimple ω-semigroups and completes a result of Baird dealing with the modularity of the sublattice of the congruence lattice of a simple regular ω-semigroup consisting of congruences which are either idempotent separating or group congruences.

If is a category, an object of pro- is stable if it is isomorphic in pro- to an object of . A local condition on such a pro-object, called strong-movability, is defined, and it is shown in various contexts that this condition is equivalent to stability. Also considered, in the case is a suitable model category, is the stability problem in the homotopy category Ho(pro-), where pro- has the induced closed model category structure defined by Edwards and Hastings [6].

We study systems which in characteristic coordinates have the form

where A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).

The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.

Simple weak converge of the initial data does not imply weak convergence of the solutions.

A bilateral Laplace multiplier theory, based on Rooney's class , is developed for certain operators defined on the Fréchet spaces Dp,μ. The theory is applied to Riesz fractional integrals associated with the one-dimensional wave operator.

Suppose is holomorphic in Δ = {z:|z|<l} and (an)∈lp where 1≦p≦2. We prove that for k=1,2,…, and almost every θ. This result is sharp in the following sense: Let p∈[1,2] and ε(r) be a positive function defined on [0,1] such that limr→1-ε(r)=0. Then there exists a function holomorphic in Δ with (an)∈lp such that

for each k>1/p.

In this paper, for R a commutative ring, with identity, of characteristic p, we look at the group G(R) of formal power series with coefficients in R, of the form

and the group operation being substitution. The results obtained give the exponent of the quotient groups Gn(R) of this group, n∈ℕ.

We use the Mittag-Leffler partial fractions expansion of jv + 1(x)/Jv(x) to give simple proofs of some recent results due to S. H. Lehnigk concerning the number of positive roots of the equation ( −Br2 + A + q)Iq(r) + rIq,+ 1(r)=0, where A is real, B>0 and q>−1.