It is well-known (see e.g. [1, p. 5]) that a class ℳ of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:

(a)if ℳ, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈ℳ.

(b) If R∈ but R∉ℳ, then R has a non-zero ideal I∈, where ℳ = {K ∈ | every non-zero K/H∉ℳ}. In fact ℳ is the radical class whose semisimple class is ℳ. On the other hand, if ℘ is a radical class, then ℐ℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class ℳ is hereditary (that is, when R∈ℳ, then all its ideals are in ℳ), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2]). The condition (satisfied in all associative or alternative classes) is that ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].

Bellman [1], [2, p. 116] proved that, if all solutions of the equation

are in L2, ∞) and b(t) is bounded, then all solutions of

are also in L2(a, ∞). The purpose of this paper is to present conditions on the function f that guarantee that all solutions of

be in the class L2(a, ∞) whenever all solutions of the equation

have this property. It is assumed that r(t) >0, r and qare continuous on a half line (a, ∞) and f is continuous. Actually the continuity assumptions may be weakened to local integrability and L2 (a, ∞) may be replaced by Lp(a, ∞) for any p > 1.

Let A1 and A2 be sup-norm algebras, each containing the constant functions. Let P(A1, A2) denote the set of bounded linear operators from A1 to A2 which carry 1 into 1 and have norm 1. Several authors have considered the problem of describing the extreme points of P(A1, A2). In the case where A1 is the algebra of continuous complex functions on some compact Hausdorff space, and A2 is the algebra of complex scalars, Arens and Kelley proved that the extreme operators in P(A1, A2) are exactly the multiplicative ones (see [1]). It was shown by Phelps in [6] that if A1 is self-adjoint, then every extreme point of P(A1, A1) is multiplicative. In [4], Lindenstrauss, Phelps, and Ryff exhibited non-multiplicative extreme points of P(A, A) and P(H∞, H∞), where A and H∞ are, respectively, the disk algebra, and the algebra of bounded analytic functions on the open unit disk D. The extreme multiplicative operators in P(A, A) were described in [6]. Rochberg proved in [8] that, if T is a member of P(A, A) which carries the identity on D into an extreme point of the unit ball of A, then T is multiplicative and is an extreme point of P(A, A). Rochberg's paper [9] is a study of certain extremal subsets of P(A, A), namely, those of the form K(F, G) = {T∊P(A, A):TF=G}, where F and G are inner functions in A. We proved in [5] that, if F is non-constant, then K(F, G) contains an extreme point of P(A, A).

As in [4], a specification is a list (r, s, t1 h0, hmc (l), …, c(ho)) such that

(i) each of r, s, t1 h0, h∞ is a non-negative integer,

(ii) for each i, c(i) is a positive integer,

(iii) if h∞ = 0 then ho = ∞,

(iv) if h∞ = 1 and t1 + h0 is finite then t1 is even,

(v) r + s + t1 + h0 + h∞ = ∞.

G is a graph on n nodes with q edges, without loops or multiple edges. We write α = q/n and β for the maximum degree of any node of G. We write

and H for the number of Hamiltonian circuits (H.c.) in the complement of G, or, what is the same thing, the number of those H.c. in the complete graph Kn which have no edge in common with G. Our object here is to prove the following theorem.

It is well known that the category of finite groups has no non-trivial injective objects. In general, a group is said to be quasi-injective if for every subgroup H of G and homomorphism f:H → G there exists an endomorphism F:G → G such that F|H = G. In other words, a group is quasi-injective whenever each homomorphism from a subgroup into the group can be extended to the whole group.

Ito and Seidman in [5] define a BG space as a locally convex space in whichthere exists a bounded set with a dense span. In this note we extend the idea to a class of not necessarily locally convex linear topological spaces (l.t.s.). We note the link between the idea of a BG space and Weston’s characterization in [7] of separable Banach spaces. Finally we examine σ-BG spaces; here the bounded set in the definition of a BG space is replaced by the union of a sequence of bounded sets.

If Sis a regular semigroup then an inverse transversal of S is an inverse subsemigroup T with the property that |T ∩ V(x)| = 1 for every x ∈ S where V(x) denotes the set of inverses of x ∈ S. In a previous publication [1] we considered the similar concept of a subsemigroup T of S such that |T ∩ A(x)| = 1 for every x ∩ S where A(x) = {y∈ S;xyx = x} denotes the set of associates (or pre-inverses) of x ∈ S, and showed that such a subsemigroup T is necessarily a maximal subgroup Ha for some idempotent α ∈ S. Throughout what follows, we shall assume that S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y ∈ S). Under these assumptions, we obtained in [1] a structure theorem which generalises that given in [3] for uniquely unit orthodox semigroups. Adopting the notation of [1], we let T ∩ A(x) = {x*} and write the subgroup T as Hα = {x*;x ∈ S}, which we call an associate subgroup of S. For every x ∈ S we therefore have x*α = x* = αx* and x*x** = α = x**x*. As shown in [1, Theorems 4, 5] we also have (xy)* = y*x* for all x, y ∈ S, and e* = α for every idempotent e.

Let M be a differentiable manifold of dimension m. A tensor field f of type (1, 1) on M is called a polynomial structure on M if it satisfies the equation:

where a1, a2, …, an are real numbers and I denotes the identity tensor of type (1, 1).

We shall suppose that for any x ∈ M

is the minimal polynomial of the endomorphism fx: TxM → TxM.

We shall call the triple (M, f, g) a polynomial Lorentz structure if f is a polynomial structure on M, g is a symmetric and nondegenerate tensor field of type (0, 2) of signature

such that g (fX, fY) = g(X, Y) for any vector fields X, Y tangent to M. The tensor field g is a (generalized) Lorentz metric.

A one-relator product Gof groups A and Bis defined to be the quotient of their free product A * B by the normal closure, «W»A*B, of a single element W, which is assumed to be cyclically reduced and of length at least 2. For convenience, the group Gwill be denoted by (A * B)/W.

Let A and C be m × m matrices and let B and D be n × n matrices, all with elements in a field F. Let AT denote the transpose of A. A well-known theorem states that, if every m × m matrix X for which AX = XA also satisfies CX = XC, then C = φ(A) for some polynomial φ(λ). In this note we establish the following simple generalizations.

Theorem 1. Let A and B have the same minimal polynomial m(λ). If each m × n matrix X over F for which AX = XB also satisfies CX = XD, then C = φ(A) and D = φ(B) for a polynomial φ(λ) over F.

If X is any set and L ⊂ [ – ∞, ∞]x, the class ℬL of L-Baire functions is defined to be the smallest subclass of [ – ∞, ∞]x which contains L and is closed under the formation of monotone, pointwise, sequential limits, so that ℬL ∍ fn ↗ f or ℬL ∍ fn ↘ f ⇒ f ∊ ℬL.

In [6], Blyth and Varlet characterize those algebras having only principal congruences in some well known classes of algebras having distributive lattice reducts. In particular, they characterize those Stone algebras having only principal congruences. In this paper we characterize those quasi-modular p-algebras having only principal congruences and show on specializing that distributive p-algebras having only principal congruences can be described in exactly the same way as Blyth and Varlet described Stone algebras having the same property. The same problem is addressed for some distributive double p-algebras.

It is well-known [3; V.13.7] that each irreducible complex character of a finite group G is rational valued if and only if for each integer m coprime to the order of G and each g ∈ G, g is conjugate to gm. In particular, for each positive integer n, the symmetric group on n symbols, S(n), has all its irreducible characters rational valued. The situation for projective characters is quite different. In [5], Morris gives tables of the spin characters of S(n) for n ≤ 13 as well as general information about the values of these characters for any symmetric group. It can be seen from these results that in no case are all the spin characters of S(n) rational valued and, indeed, for n ≥ 6 these characters are not even all real valued. In section 2 of this note, we obtain a necessary and sufficient condition for each irreducible character of a group G associated with a 2-cocycle α to be rational valued. A corresponding result for real valued projective characters is discussed in section 3. Section 1 contains preliminary definitions and notation, including the definition of projective characters given in [2].

In [7], Z. Tang and H. Zakeri introduced the concept of co-Cohen-Macaulay Artinian module over a quasi-local commutative ring R (with identity): a non-zero Artinian R-module A is said to be a co-Cohen-Macaulay module if and only if codepth A = dim A, where codepth A is the length of a maximalA-cosequence and dimA is the Krull dimension of A as defined by R. N. Roberts in [2]. Tang and Zakeriobtained several properties of co-Cohen-Macaulay Artinian R-modules, including a characterization of such modules by means of the modules of generalized fractions introduced by Zakeri and the present second author in [6]; this characterization is explained as follows.

1. For any positive integral n and any positive real x1,…,xn we write

clearly. It is known [1,2] that

for n ≦ 6, and further [4, 5, 6] that (5) is false for even n ≧ 14 and for odd n ≧ 53. Mordell [2] conjectured that (5) is false for all n ≧ 7, but recently [3] stated that computations indicated that (5) is true for n = 7 and gave some calculations in support of (5) for n = 7.

Von Neumann's definition of the continuous sum of Hilbert spaces led Segal [3] to define the continuous sum of measures on a measurable space. In this note we employ Segal's definition to investigate the measure structures associated with a self-adjoint transformation of pure point spectrum and a self-adjoint transformation of pure continuous spectrum. While these transformations, as operators on separable Hilbert spaces, are the antithesis of each other we show that in their measure structure one is a particular case of the other.