Let S be a bisimple semigroup and let Es denote its set of idempotents. We may partially order Es in the following manner: if e, f ∈ E s, e ≧ f if and only if ef = fe = e. We then say that Es is under or assumes its natural order. Let I0 denote the non-negative integers and let n denote a natural number. If Es, under its natural order, isomorphic to (I0)n under the reverse of the usual lexicographic order, we call S an ωn-bisimple semigroup. (See [9] for an explanation of notation.) We determined the structure of ωn-bisimple semigroups completely mod groups in [9]. The ωn-bisimple semigroups, the I-bisimple semigroups [8], and the ωnI-bisimple semigroups [9] are classes of simple semigroups except completely simple semigroups whose structure has been determined mod groups.

Let be a commutative Banach algebra over the complex field C, M an ideal of . Denote by M2 the set of all finite linear combinations of products of elements from M. M will be termed idempotent if M2 = M. The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.

Consider the following statement: For every positive integer n and every prime p there is a finite p-group of nilpotency class (precisely) c all of whose (n−1)-generator subgroups are nilpotent of class at most n.

Distributive pseudo-complemented lattices form an extensively studied class of distributive lattices. Examples are the lattice of all open sets of a topological space, the lattice of all ideals of a distributive lattice with zero and the lattice of all congruences of an arbitrary lattice. Lattice which are just pseudo-complemented have been studied in detail by J. Varlet [6], [7] where, however, the most interesting results require at least the assumption of modularity, sometimes distributivity.

M. H. Stone raised the problem ([1] Problem 70) of characterising the class of distributive pseudo-complemented lattices ℒ = 〈L; ∨, ∧, 0, 1〉 in which a* ∨ a** = 1 holds identically. Several solutions to this problem have now been offered — the first being by G. Grätzer and E. T. Schmidt [6], who gave this class of lattices the name Stone lattices. Later solutions were given by J. Varlet [11], O. Frink [4] and G. Grätzer [5]; see also G. Bruns [2].

To prove the statement given in the title take a set Σ1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈L; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L, a* is the pseudo-complement of a. Indeed, a ∧ a* 0; now, if a ∧ x = 0, then a* ∨ x* 0* = 1, and a** ∧ = 1* = 0; since a** is the complement of a*, the last identity implies x** ≦ a*, thus x ≦ x** ≦ a*, which was to be proved.

This paper continues an investigation of the complete integral closure of an integral domain which was begun in [2]. We recall that if D is an integral domain with quotient field K then an element x of K is said to be almost integral over D if there exists a nonzero element y of D such that yxn is an element of D for each positive integer n. The set D* of elements of K almost integral over D is called the complete integral closure of D and D is said to be completely integrally closed if D* = D.

In studying the compression of loops of fabric between flat parallel plates, the problem arises of the compression of bends. A ‘bend’ is illustrated in Figs. 1 and 2. Two bends are joined to form a loop. The material is postulated to yield in bending such that the bending moment where B is a bending rigidity, KI the current impressed curvature, and KR the ‘remanent’ curvature. This remanent curvature is the free, or natural curvature in the material caused by its past and present deformation. We study the case when the remanent curvature that is, the remanent curvature is proportional to the greatest previously impressed curvature. We call equations (1) and (2) the ‘Remanent Curvature Hypothesis’.

Some recent papers have revived interest in some questions concerning the motion of a simple pendulum which is oscillating with small angular amplitude under gravity, when the length of the pendulum changes with time in some prescribed manner.

A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a rational polygon?’ Many of the known results concerning this question are contained as special cases in theorem 1 below which was proved by one of us (cf. the references).

Let ω(x) be non-decreasing on the closed interval [a, b]. Outside the interval ω(x) is defined by ω(x) = ω(a) for x < a and ω(x) = ω(b) for x > b. Let S denote the set of points of continuity of ω(x) and D denote the set of points of discontinuity of ω(x). R. L. Jeffery [5] has defined the class U, of functions f(x) as follows:

Let (S, ≦) be a (non-void) partially ordered set with the property that for every (non-void) chain C (i.e., every totally ordered subset) of S, there exists in S the element sup C. Let SM be the set of all maximal elements s of S. ƒ:S/SM→S be a slowly increasing mapping in the sense that

Let us suppose that ƒ(x, y) is an indefinite binary quadratic form that does not represent zero. If P is the real point (x0, y0) then we may define where the infimum is taken over all integral x, y. The inhomogeneous minimum of the form ƒ is defined where the supremum taken over all real points P, need only extend over some complete set of points, incongruent mod 1.

A variety of groups is an equationally defined class of groups: equivalently, it is a class of groups closed under the operations of taking cartesian products, subgroups, and quotient groups. If and are varieties, then is the class of all groups G with a normal subgroup N in such that G/N is in ; is a variety, called the product of and . We denote by the variety generated by the unit group, and by the variety of all groups. We say that a variety is indecomposable if , and cannot be written as a product , with both and One of the basic results in the theory of varieties of groups is that the set of varieties, excluding , and with multiplication of varieties as above, is a free semi-group, freely generated by the indecomposable varieties. Thus one would like to be able to decide whether a given variety is indecomposable or not. In connection with this question, Hanna Neumann raises the following problem (as part of Problem 7 in her book [7]): Problem 1. Ifandprove that [] is indecomposable unless bothandhave a common non-trivial right hand factor.

Bieberbach's conjecture, proposed in 1916 and still unsolved, states that if ƒ(z) = z+a2z2+… is holomorphic and univalent in the disc ∣z∣ < 1 then ∣an∣ ≦ n for each n ≧ 2, with equality for some n only if ƒ(z) is the Koebe function of is obtained from this function by a rotation. Very recently Bombieri has succeeded in showing that if ƒ(z) is sufficiently close to the Koebe function, then with equality only if ƒ(z) = k(z). This had previously been proved by Garabedian, Ross and Schiffer [3] for even values of n.

Let E be a real infinite-dimensional Banach space. Let ℒ be the Banach algebra of all continuous linear mappings of E into itself with topology defined by the norm:

Let L denote a free Z-module of rank 2n and Φ an alternating bilinear mapping from L×L into the rational integers Z. Writing α · β for Φ(α, β), where α, β ∈ L, we have We shall assume that Φ is non-singular and unimodular (see Bourbaki [1]). L is now a (symplectic) lattice.

By the systematic use of Fourier transforms and suitable weight functions L. R. Volevich and B. P. Paneyakh brought many classes of spaces of distributions (including the Sobolev spaces) and their topological duals under the one unifying definition. The main purpose here is to demonstrate that the representation of multipliers between pairs of these spaces (that is, continuous linear operators from one space into another which commute with translations) may be related to the representation of multipliers between Lp and Lq.

Ramanujan was the first mathematician to discover some of the arithmetical properties of p(n), the number of unrestricted partitions of n. His congruence,

for example, is famous [2; 3]. Some progress has been made since then; it is known that the congruence,

has an infinitude of solutions for any arbitrary value of r [4]. This is a somewhat weak relation, and one would have liked to obtain, if possible, stronger results of the type,

for ‘almost all’ values of n, which in its turn is derivable from another stronger relation, viz.,

also established by Ramanujan [2], where r(n) is Ramanujan's function defined by

By a group A is meant throughout an additively written abelian group. A is said to have linear topology if there is a system of subgroups Ui (i∈I) of A such that, for a ∈ A, the cosets a+Ui(i∈I) form a fundamental system of neighborhoods of a. The group operations are continuous in any linear topology; the topologies are always assumed to be Hausdorff, that is, ∩iUi = 0. A linearly compact group is a group A with a linear topology such that if aj+Aj (j∈I) is a system of cosets modulo closed subgroups Aj with the finite intersection property (i.e. any finite number of aj+Aj have a non-void intersection), then the intersection ∩j(aj+Aj) of all of them is not empty.