A locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic investigations. The class ℒ ℐ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice ℒ(ℒℐ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CS of completely simple semigroups. An important consequence of our results is a determination of the join of CS with the e-variety of inverse semigroups – it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.

1. Let K1, K2, K3 be three conjugate* cubic algebraic number fields, of which we shall take K1 to be real and K2, K3 to be complex conjugate. Let ωi1, ωi2, ωi3 the conjugate basis for the integers of Ki. Write

so that ξi runs through the integers of Ki as the xj run independently through the rational integers, and the ξi (i = 1, 2, 3) are conjugate. The determinant of the ξi, regarded as linear forms in the xj, is ± √d, where d is the common discriminant of the fields.

In the course of their enumeration of all 4-dimensional space groups, Brown, Bülow, Neubüser, Wondratschek and Zassenhaus have introduced concepts appropriate to the study of n-dimensional crystallography for n ≥ 4 (see (2), (3)). One such concept is that of a crystal family. Families seem to be particularly useful as a framework within which to study higher dimensional crystallography, primarily because they determine a classification of all the standard crystallographic objects and overcome the traditional confusion over crystal systems (see (2); pp. 16–17, (9)). In this paper, techniques are developed for the determination of all rationally decomposable families in a given dimension from the indecomposable families of lower dimensions. These techniques place emphasis on three geometric invariants of families: the decomposition pattern; the canonical decomposition pattern; and the number of free parameters. This, it is felt, further reinforces their position as fundamental objects. The key result (Theorem 5.4) is:

a family uniquely determines, and is uniquely determined by, the constituent families in the canonical decomposition.

We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

By an H-space (X, μ) we will mean a topological space X having the homotopy type of a connected CW complex of finite type together with a basepoint preserving map μ: x × X → X with two sided homotopy unit. Let p be a prime and let /p be the integers reduced modp. Given an H-space (X, μ) then H*(X;/p) is a commutative associative Hopf algebra over the Steenrod algebra A*(p) and H*(X;/p) is iso- morphic, as an algebra, to a tensor product [⊗ , where each algebra At is generated by a single element ai (see Theorem 7.11 of (24)). The decomposition Ai is called a Borel decomposition and the elements {ai} are called the Borel generators of the decomposition. The decomposition ⊗ Ai and the resulting generators {at} are far from unique. Many choices are possible. Since A*(p) acts on H*(X;/p) an obvious restriction would be to choose the Borel decomposition to be compatible with this action. We would like the /p module generated by the Borel generators and their iterated pth powers to be invariant under the action of A * (p). More precisely we would like H*(X;/p) to be the enveloping algebra U(M) of an unstable Steenrod module M (see § 2). If H*(X;/p) admits such a choice then it is called a U(M) algebra. The fact that H*(X; /p) is a U(M) algebra has applications in homotopy theory. In particular there exist unstable Adams spectral sequences which can be used to calculate the homotopy groups of X (see (21) and (7)). However, the question of U(M) structures for modp cohomology seems of most interest simply as a classification device for finite H-spaces.

We study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU*(2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields non-vanishing (co)homology classes.

In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms.

A possible degree of heterogeneity in the structure of continua is investigated in this paper.

In many mechanical and other problems the following equation

is reached, where Jν(α) and Yν(α) are Bessel functions of the first and second kind of any real order ν and β is a positive parameter.

For example, equation (1) is reached in the case of determining the critical load Pcr, for a simply supported strut with a variable inertia moment by a power law, where the power m is any real number.

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

The classical Godement Theorem for manifolds, characterizing kernel pairs for submersions (cf. e.g. [15], LG IV §5), has been used by Pradines[12] as a crucial property for having a good theory of differentiable groupoids; in fact, he developed an axiomatic theory of categories in which Godement's and some other exactness properties hold, under the name of ‘Godement diptych’.

It is proved that if G is a compact connected Hausdorff group of uncountable weight, w(G), then G contains a homeomorphic copy of [0, 1]w(G). From this it is deduced that such a group, G, contains a homeomorphic copy of every compact Hausdorff group with weight w(G) or less. It is also deduced that every infinite compact Hausdorff group G contains a Cantor cube of weight w(G), and hence has [0, 1]w(G) as a quotient space.

The joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer p ≥ 1, there exists a pair of square matrices of dimension 2p(2p+1 − 1) for which every extremal sequence has subword complexity at least 2−p2np.

This paper discusses the Brauer–Manin obstruction on double covers of the projective plane branched along a plane section of a Kummer surface from both the practical and the theoretical points of view. Theoretical highlights include the determination of a complete set of generators for the Brauer group; on the practical side, we give several surfaces with a Brauer–Manin obstruction and verify that the Brauer–Manin obstruction is the only one for a collection of several thousand surfaces of this type.

In this paper we investigate certain generalized forms of third and fourth order non-homogeneous differential equations. Ezeilo (l) derived interesting results for the problem

.

We show similar results for:

,

and

.

We study homeomorphisms of finite exponentially integrable distortion of the unit ball Bn onto a domain Ω of finite volume. We show that under such a mapping the images of almost all radii (in terms of a gauge dimension) have finite discrete length. We also show that our dimension estimate is essentially sharp.

Since the foundational paper (10) by Northcott and Rees in 1954 there have been quite a few papers concerning reductions of ideals and the analytic spread of an ideal. One particular line of investigation concerning the analytic spread l(I) of an ideal I in a local ring (R, M) was begun in 1972 by Burch in (5), where it was shown that l(I) ≤ altitude R – min (grade R/In; n ≥ 1). This result was sharpened in 1980–81 by Brodmann in three papers, (2, 3, 4). Therein he showed that the sets {grade R/In; n ≥ 1} and {grade In−1/In; n ≥ 1} stabilize for all large n, and calling the stable values t and t*, respectively, it holds that t ≤ t* and l(I) ≤ altitude R – t* when I is not nilpotent. He then gave a case (involving R being quasi-unmixed) when equality holds. In 1981 in (20) Rees used two new approaches to Burch's inequality, and he proved two nice results which may both be stated as: l(I) ≤ altitude R – s(I) with equality holding when R is quasi-unmixed; here, s(I) = min {height P; P is a minimal prime divisor of (M, u) R[tI, u]}– 1 (in the first theorem), and s(I) is the length of a maximal asymptotic sequence over I (in the second theorem).

The purpose of this note is to present simple proofs of the basic results concerning semi-simple annihilator Banach algebras as well as new results (for terminology notation and the standard approach see [2]).

The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1’s in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1’s. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.