Let X be a non-locally convex F-space (complete metric linear space) whose dual X′ separates the points of X. Then it is known that X possesses a closed subspace N which fails to be weakly closed (see [3]), or, equivalently, such that the quotient space X/N does not have a point separating dual. However the question has also been raised by Duren, Romberg and Shields [2] of whether X possesses a proper closed weakly dense (PCWD) subspace N, or, equivalently a closed subspace N such that X/N has trivial dual. In [2], the space Hp (0<p<1) was shown to have a PCWD subspace; later in [9], Shapiro showed that ℓp (0<p<1) and certain spaces of analytic function have PCWD subspaces. Our first result in this note is that every separable non-locally convex F-space with separating dual has a PCWD subspace.

The structure theorem for completely 0-simple semigroups established by Rees [5] in 1940 has proved a very powerful tool in the investigation of such semigroups. In this paper the theorem is applied to an investigation of the subsemigroup of a completely 0-simple semigroup generated by its idempotents. Previous work on this problem has been carried out by Kim [4], but the present note offers a more direct approach.

Arrange any n integers around a circle. The following procedure can be used to obtain another circle of n integers. For each adjacent pair of the first integers, form the absolute value of their difference and place it between them; then remove the original numbers. This procedure can be repeated over and over. When n = 4 this always leads eventually to a circle of zeros. On the other hand when n = 3, unless the original numbers are equal, this never happens. We treat below the general case and related problems, using for convenience a slightly different formulation. Surprisingly there is enough structure to lead to some interesting mathematics.

Let s denote the space of all complex valued sequences and let E∞ be all eventually zero sequences. An FK space is a locally convex vector subspace of s which is also a Fréchet space (complete linear metric) with continuous coordinates. A BK space is a normed FK space. Some discussion of FK spaces is given in [11]. Well-known examples of BK spaces are the spaces m, c, c0 of bounded, convergent, null sequences respectively, all with and

The classical theory of Weierstrass points on a compact Riemann surface is well-known (see, for example, [3]). Ogawa [6] has defined generalized Weierstrass points. Let Y denote a compact complex manifold of (complex) dimension n. Let E denote a holomorphic vector bundle on Y of rank q. Let Jk(E) (k = 0, 1, …) denote the holomorphic vector bundle of k-jets of E [2, p. 112]. Put rk(E) = rank Jk(E) = q.(n + k)!/n!k!. Suppose that Γ(E), the vector space of global holomorphic sections of E, is of dimension γ(E)>0. Consider the trivial bundle Y × Γ(E) and the map

which at a point Q∈Y takes a section of E to its k-jet at Q. Put μ = min(γ(E),rk(E)).

In this paper we shall extend results obtained in [5] to the W*-algebra setting.

Let be a C*-algebra and let + denote the set of positive elements in . Given a fixed element A in , the Lyapunov transformation LA corresponding to A is the mapping of into itself which sends X to AX+XA*. We are interested in characterizing those Bin for which

In 1954 N. Kimura proved that each idempotent in a semigroup is contained in a unique maximal subgroup of the semigroup and that distinct maximal subgroups are disjoint [13] (or see [6, pp. 21–23]). This generalized earlier results of Schwarz [14] and Wallace [15]. These maximal subgroups are important in the study of semigroups. If G is a group, then the collection S(G) of nonempty complexes of G is a semigroup and it is natural to inquire what properties of G are inherited by the maximal subgroups of S(G). There seems to be very little literature devoted to this subject. In [5, Theorem 2], with certain hypotheses placed on an idempotent, it was shown that if G is a lattice-ordered group (“1-group”) then a maximal subgroup of S(G) containing an idempotent satisfying these conditions admits a natural lattice-order. The main result of this note (Theorem 1) is that if Gis a representable 1-group and E is a normal idempotent of S(G) and a dual ideal of the lattice G, then the maximal subgroup of S(G) containing E admits a representable lattice-order.

A conjecture of reputable vintage states that c(G)≤b(G) + l for a finite p-group G of class c(G) and breadth b(G). This result has been proved in a medley of special cases and in particular whenever b(G)≤3. We now prove it for b(G) = 4.

Let G be a finite group (with neutral element e) which operates trivially on the multiplicative group R* of a commutative ring R (with identity 1). Let H2(G, R*) denote the second cohomology group of G with respect to the trivial G-module R*. With every represented by the central factor system we associate the so called twisted group algebra (R, G, f) (see [3, V, 23.7] for the definition). (R, G, f) is determined by f up to R-algebra isomorphism. In this note we shall describe its representations in the case R is an algebraically closed field C of characteristic zero and G is an extra-special p-group P.

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.

In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].

Let G be a group. We define λ(G) to be the smallest integer n such that every element of the commutator subgroup G′ is a product of n commutators. Ito [4] has shown that λ (An)= 1 for all n. Thompson [7] has shown that λ (SLn(q))= 1 for all n and q. In fact, there is no known simple group G such that λ(G)>1. However, there do exist such perfect groups (cf. [7]).

The concept of acyclic coloring was introduced by Grünbaum [5] and is a generalization of point arboricity.

A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In [5], Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin [3] after the upper bound was improved three times in [7], [1] and [6]. In [2], we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the following

Theorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.

It is not known for any n>0 whether a(n)>H(n), the Heawood number [8].

If R is an integral domain with maximum condition for principal right ideals—right ACC1—every nonzero non-unit in Rhas irreducible factors, but is not necessarily a product of such factors. Using additional basic factors—called infinite primes in [1]—results about unique factorization in principal right ideal domains have been obtained in [1], [2], and [5].

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the form

where Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact that

is a modular form for a certain congruence subgroup of the modular group, and

with

where Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

In this paper we study several generalizations of the concept of unique factorization domain. An integral domain is called a π-domain if every principal ideal is a product of prime ideals. Theorem 1 shows that the class of π-domains forms a rather natural subclass of the class of Krull domains. In Section 3 we consider overrings of π-domains. In Section 4 generalized GCD-domains are introduced: these form an interesting class of domains containing all Prüfer domains and all π-domains.

to the paper

On a Dual Integral Equation with a Trigonometric Kernel

by D. C. Stocks

Professor W. E. Williams (Surrey University) has recently pointed out to me that my paper is incorrect.

The step from equation (5) to equation (6) implies that φ is independent of λ, which the solution (11) shows is not the case. In view of this, equation (12) is not a solution to the original equations unless

with φ defined by (11).