We consider the possibility of generalising the statement

to the case

Let W be a class of not necessarily associative rings which is universal in the sense that it is closed under homomorphic images and is hereditary to subrings. All rings considered will be assumed to belong to W. The notation I ⊲ R will mean I is an ideal of R. A relation σ on W will be called an H-relation if σ satisfies the properties:

(1) I σ R implies I is a subring of R.

(2) If I σ R and ø is a homomorphism of R, then IØ σ Rø.

(3) If I σ R and J is an ideal of R, then I ∩ J σ J.

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f∈C[0,1] mapping [0,1] into itself. If f∈C, fk will denote the kth iterate of f and we put Ck = {fk:f∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I, we let C(f) denote the set of points at which f is locally constant, i.e.

We let N denote the set of positive integers and NN denote the Baire space of sequences of positive integers.

We present a direct computation of the Rhodes expansions of the free objects in the varieties of bands, based in the manipulation of the invariants in introduced by Gerhard and Petrich [4] in the study of bands.

In this paper we consider the following semilinear elliptic equation

where n ≥ 3, and β ≥ 0, γ ≥ 0, q > p ≥ 1, μ and ν are real constants. We note that if γ = 0, β > 0 and ν ≥ 2, then the equation above is called the Matukuma-type equation. If β = 0, γ > 0 and ν > 2, then the complete classification of all possible positive solutions had been conducted by Cheng and Ni. If β > 0, γ > 0 and μ ≥ ν ≥ 2, then some results about the maximal solution and positive solution structures can be found in Chern. The purpose of this paper is to discuss and investigate the blow-up and positive entire solutions of the equation above for the μ ≥ 2 ≥ ν case.

It has already been shown that nine points can be found on a conic to form six triads all apolar to a given triad (ABC), the hessian lines of these six triads and of ABC being concurrent.

The construction of the corresponding system in the twisted cubic is easy. In the first place, the poles of the six triads P1Q2R3, etc., will lie on the plane of the triad ABC to which they are all apolar. Secondly, the hessian lines of these triads, which in the conic were concurrent, will now be generating lines of a quadric circumscribing the twisted cubic. Now the pole of a triad is in the osculating plane at any of the points of the triad, hence the osculating plane at Pl will contain the poles of the triads P1Q2R3 and P1Q2R3

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinates

where C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equation

in D can be approximated in the space by an entire Herglotz wave function

with kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B(X) (K(X)). Let N(A) and R(A) denote, respectively, the null space and the range space of an element A of B(X). Set R(A∞)=∩nR(An) and k(A)=dim N(A)/(N(A)∩R(A∞)). Let σg(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)=0} denote the generalized (regular) spectrum of A. In this paper we study the subset σgb(A) of σg(A) defined by σgb(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)<∞}. Among other things, we prove that if f is a function analytic in a neighborhood of σ(A), then σgb(f(A)) = f(σgb(A)).

Let ξ be the set of all finite groups that have efficient presentations. In this paper we give sufficient conditions for the standard wreath product of two ξ-groups to be a ξ-group.

In continuation of the results given in Vol. XI. of the Proceedings I note that the equations to AP, BQ, CR are respectively

and to AP′, BQ′, CR′ are

1. Consider the conic represented by the general equation

Differentiating three with respect to a we get

where (y) stands for .

§1. The object of this note is, in the first place, to show that Menelaus's Theorem, regarding the segments into which the sides of a triangle are divided by any transversal, is a particular form of the condition, in trilinear co-ordinates, for the collinearity of three points; and, in the second place, to point out an analogue of Menelaus's Theorem in space of three dimensions.