Semilinear elliptic partial differential equations of the type

1

will be considered throughout real Euclidean N-space, where m ≧ 2 is a positive integer, Δ denotes the N-dimensional Laplacian, and f is a real-valued continuous function in [0, ∞) × (0, ∞). Detailed hypotheses on the structure of f are listed in Section 3.

Our objective is to prove the existence of radially symmetric positive entire solutions u(x) of (1) which are asymptotic to positive constant multiples of |x|2m−2i as |x| → ∞ for every i = 1,…, m, N ≧ 2i + 1.

Consider the second order neutral differential equation

1

where the coefficients p and q and the deviating arguments τ and σ are real numbers. The characteristic equation of Eq. (1) is

2

The main result in this paper is the following necessary and sufficient condition for all solutions of Eq. (1) to oscillate.

THEOREM. The following statements are equivalent:

• (a) Every solution of Eq. (1) oscillates.

• (b) Equation (2) has no real roots.

Throughout the paper, G will denote an additively written, but not always abelian, group of finite order n; and X = (xij) will denote a square matrix of order n with entries from G and whose rows and columns are numbered 0, 1, …, n − 1. We call X a cartesian array (afforded by G) if

(1.1) The sequence {−xmi + xki, i = 0,…, n – 1} contains all elements of G whenever k ≠ m.

By a theorem of Jungnickel (see Theorem 2.2 in [5]), the transpose of a cartesian array is also a cartesian array. We call G a cartesian group if there is a cartesian array X afforded by G. In this case, we also call (G, X) a cartesian pair.

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.

If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]).

The mod torsion Hurewicz map

for compact Lie groups provides a useful and efficient means of studying G. In effect, it measures how far G fails to be a product of spheres. For the Hopf-Samelson theorem (see [17]) tells us that

In other words

Serre pointed out that there exists a map

inducing this Q isomorphism.

Let G be a domain in the complex plane and F a nonempty subset of G such that F is the closure in G of its interior F0. We will say f ∊ C1(F) if f is continuous on F and possesses continuous first partial derivatives in F which extend continuously to F as finite-valued functions. Let G* – F be connected and locally connected, f ∊ C1(F) be harmonic in F0, and E be a subset of ∂F ∩ ∂G (here G* denotes the one-point compactification of G and the boundaries ∂F, ∂G are taken in the extended plane). Suppose there is a sequence 〈hn〉 of functions harmonic in G such that

uniformly on F as n → ∞.

Let X and Y be 1-connected spaces having the homotopy type of cw-complexes.

Definition 0.1. A continuous map f:X → Y is Ω-split if Ωf:ΩX → ΩY admits a retraction up to homotopy.

In [6] we prove the following “mapping theorem”:

THEOREM 0.1. (a) If f is Ω-split, then cat(X) ≦ cat(Y);

(b) If π*(f) is split injective and ΩY has the homotopy type of a product of Eilenberg-MacLane spaces, then f is Ω-split.

Much of [2] was devoted to studying pairs of subrings A ⊂ B of a field with the property that A and B have the same prime ideals. In this paper, we continue that investigation, but we no longer assume that A and B are comparable. Interestingly, most of the results of [2] carry over to this more general context. Besides such extensions of [2], additional motivation for the more general context comes from the need to explicate some naturally occurring examples (see Examples 2.5, 3.6, and 4.3).

Section 2 begins by showing that we may reduce to the case in which R is a quasilocal domain with nonzero maximal ideal M and quotient field K. Proposition 2.3 establishes that the set C(R) of all subrings A of K with Spec(A) = Spec(R) forms a complete semilattice.

An H-relation, as introduced by Rossa and Tangeman [4], is a relation σ on the class of associative rings with their subrings satisfying the following conditions:

(1) IσR implies that I is a subring of R;

(2) if IσR and f is a homomorphism of R, then (If)σ(Rf);

(3) if IσR and J is an ideal of R, then (I∩J)σJ.

Puczylowski [3] imposes also the condition

(4) if J is an ideal of R, then JσR.

A further condition satisfied by many familiar H-relations is the following:

(5) if f is a homomorphism from a ring R onto a ring S and BσS, then there exists a subring A of R such that AσR and Af = B.

The isomorphism problem for modular group algebras of finite p-groups appears to be still far from a solution (see [7] for a survey of the existing results). It is therefore of interest to investigate the problem for special classes of groups.

The groups we consider here are the p-groups of maximal class, which were extensively studied by Blackburn [1]. In this paper we solve the modular isomorphism problem for such groups of order not larger than pp+1, having an abelian maximal subgroup, for odd primes p.

What we in fact do is to generalize methods used by Passman [5] to solve the isomorphism problem for groups of order p4. In Passman's paper the case of groups of maximal class is actually the most difficult one.

B. Sz.-Nagy and C. Foiaş gave the first examples of quasisimilar operators with different spectra [25]. Indeed, quasisimilar operators can even have different spectral radius [19] (see also [15]). Nevertheless, T. B. Hoover has shown in [19] that if T and S are quasisimilar, then σ(T)∩σ(S) ≠ ∅, where σ(R) denotes the spectrum of the (bounded linear) operator R. In [12], the author improved Hoover's result by showing that each component of σ(S) intersects σ(T), and viceversa. This, in turn, was further improved by L. A. Fialkow in [7]. (Actually, Fialkow's results were obtained independently of [12].)

L. A. Fialkow [7] and L. R. Williams [27] independently proved that σe(T)∩σe(S) ≠ ∅, where σe(R) denotes the essential spectrum of R. Several authors have raised the following natural question:

Is it also true that each component of σe(S) intersects σe(T), and viceversa? [7] [21] [26].

The upper light cone L in R3 is a homogeneous space of the 3-dimensional Lorentz group G. It may be identified with the space of horocycles in the upper hyperboloide H which is the symmetric space associated to G. There exists a duality between H and L (see [5] p. 144) and a general procedure leads to a generalized Radon transform:

and the dual Radon transform

These operations commute with the natural action of the group G.

In the first paper of this series [17], we set up some general machinery for studying ergodic actions of compact groups on von Neumann algebras, namely, those actions for which . In particular we obtained a characterisation of the full multiplicity ergodic actions:

THEOREM A. If α is an ergodic action of G on , then the following conditions are equivalent:

• (1) Each spectral subspace has multiplicity dim π for π in .

• (2) Each π in admits a unitary eigenmatrix in .

• (3) The W* crossed product is a (Type I) factor.

• (4) The C* crossed product of the C* algebra of norm continuity is isomorphic to the algebra of compact operators on a Hilbert space.