Let E be an ordered Banach space with closed positive cone C. A base for C is a convex subset K of C with the property that every non-zero element of C has a unique representation of the form λk with λ > 0 and k ∈ K. Let S be the absolutely convex hull of K. If the Minkowski functional of S coincides with the given norm on E, then E is called a base norm space. Then K is a closed face of the unit ball of E, and S contains the open unit ball of E. Base norm spaces were first defined by Ellis [5, p. 731], although the special case of dual Banach spaces had been studied earlier by Edwards [4].

1. Introduction. Given a series we define , by the relations

The series Σan is said to be summable (C, k) to the sum s, if

as n→∞, and strongly summable (C, k), k>0, with index p, to the sum s, or summable [C; k, p] to the sum s, if

Birkhoff and Pierce [2] introduced the class f-rings—those lattice-ordered rings R which satisfy the additional condition that if a, b, and c are positive elements of R and if a ∧ b = 0, then ac ∧ b = 0 = ca ∧ b. They showed that f-rings may be characterized as lattice-ordered rings which are subdirect products of totally-ordered rings.

In [2], R. Loewy and H. Schneider studied positive linear operators on circular cones. They characterised the extremal positive operators on these cones and noticed that such operators preserve the set of extreme rays of the cone in this case. They then conjectured that this property of extremal positive operators is true in general.

This note is concerned with an inequality for even order positive definite hermitian matrices together with an application to vector spaces.

The abbreviations p.d. and p.s-d. are used for positive definite and positive semi-definite respectively. An asterisk denotes the conjugate transpose of a matrix.

We give a short and constructive proof of the general (multi-dimensional) Implicit Function Theorem (IFT), using infinitesimal (i.e. nonstandard) methods to implement our basic intuition about the result. Here is the statement of the IFT, quoted from [4];

Theorem. Let A ⊂ ℝn × ℝmbe an open set and let F:A → ℝ be a function of class Cp (p≥1). Suppose that (xO, yO) ε A with F(xO, yO) = 0 (xO ε ℝn, yO ε ℝm) and that the Jacobian determinantis not zero at (xO, yO). Then there is an open neighbourhood U of xO and a unique function f:U→ ℝmwith

F(x, f(x)) = 0

for all x ε U. Moreover, f is of class Cp.

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.

Throughout the following note R will denote an associative ring with unit element 1. We shall denote by R-mod [resp. mod-R] the category of all unitary left [resp. right] R-modules. Morphisms in these categories will be written as acting on the side opposite scalar multiplication. All other functions will be written as acting on the left. If is a category, we shall abuse notation and write “A∈ when we mean “A is an object of ”.

For a series Σan with partial sums An=a0 + a1 + … + an(n ≥ 0), supposed to be real in this note, we define, in a generally accepted notation ([2], pp. 7, 9, 94–98), the following transforms:

(C, α) sequence-transform,

(H, k)sequence-transform (k = 0, 1, 2, …):

(A: C, α)fnunction-transform,

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

For a given sequence {am} and p≠0, Schur (2) defined

In particular if p is a prime, a an integer and , then by Fermat's theorem

is integral. Schur proved that if p † a, then all the derivatives

are integral. Zorn (3) using p-adic methods proved Schur's results and also found the residue of Xm (mod pm), where and x = 1 (mod p). The writer (1) proved Zorn's congruences by elementary methods as well as certain additional results of a similar sort.

In a recent paper Cooke [1] obtained a solution of the integral equation

by using the identity

and the technique, first used by Copson, of interchanging the orders of integration and hence reducing the problem to that of the successive solution of two Abel integral equations. It is also shown in [1] that the above identity can also be used to solve the dual series equations

The kernel in equation (1) is a particular member of a general class of kernels which the author [6] has shown to be such that the resulting integral equation is directly soluble by using Copson's technique. The particular example of equation (1) is given in [6] and the identity of equation (2) was used by the author [7] to obtain the solution of equation (3).

In this paper, we show that a commutative Noetherian ring which satisfies the radical formula must be of dimension at most one. From this we give a characterization of commutative Noetherian rings that satisfy the radical formula.

There is a large body of literature on inverse semigroups. This literature contains a considerable amount of information concerning congruences on these semigroups, which is not surprising in view of the demonstrated fact that congruences on inverse semigroups play a decisive role in most of the existing structure theorems. In addition, for an inverse semigroup of known structure, finding its congruence lattice, or even certain properties of this lattice, often gives information about these semigroups not apparent in their structure theorems.

If f is a real function, periodic with period 1, we define

In the whole paper we write ∫ for , mE for the Lebesgue measure of E ∩ [0,1], where E ⊂ ℝ is any measurable set of period 1, and we also use XE for the characteristic function of the set E. Consistent with this, the meaning of ℒp is ℒp [0, 1]. For all real xwe have

if f is Riemann-integrable on [0, 1]. However,∫ f exists for all f ∈ ℒ1 and one would wish to extend the validity of (2). As easy examples show, (cf. [3], [7]), (2) does not hold for f ∈ ℒp in general if p < 2. Moreover, Rudin [4] showed that (2) may fail for all x even for the characteristic function of an open set, and so, to get a reasonable extension, it is natural to weaken (2) to

where S ⊂ ℕ is some “good” increasing subsequence of ℕ. Naturally, for different function classes ℱ ⊂ ℒ1 we get different meanings of being good. That is, we introduce the class of ℱ-good sequences as

J. Berman [2] initiated the study of a variety k of bounded distributive lattices endowed with a dual homomorphic operation paying particular attention to certain subvarieties km, n. Subsequently, A. Urquhart [8] named the algebras in k distributive Ockham algebras, and developed a duality theory, based on H. A. Priestley's order-topological duality for bounded distributive lattices [6], [7]. Amongst other things, Urquhart described the ordered spaces dual to the subdirectly irreducible algebras in Sif. This work was developed further still by M. S. Goldberg in his thesis and the paper [5]. Recently, T. S. Blyth and J. C. Varlet [3], in abstracting de Morgan and Stone algebras, studied a subvariety MS of the variety k1.1. The main result in [3]is that there are, up to isomorphism, nine subdirectly irreducible algebras in MS and their Hasse diagrams are exhibited. The methods employed in [3] are purely algebraic and can be generalized to show that, up to isomorphism, there are twenty subdirectly irreducible algebras in k1.1. In section 3 of this paper, we take a short cut to this result by utilizing the results of Urquhart and Goldberg. Our basic method is simple: the results of Goldberg [5] are applied to k1,1 to produce a certain eight-element algebra B1 in k1,1, whose lattice reduct is Boolean and whose subalgebras are, up to isomorphism, precisely the subdirectly irreducibles in k1.1. We then pick out of the list of twenty such algebras those belonging to the variety MS. In section 4, we sketch a purely algebraic proof along the lines followed by Blyth and Varlet in [3].