We describe on the Heisenberg group Hn a family of spaces M(h, X) of functions which play a role analogous to the trigonometric polynomials in Tn or the functions of exponential type in Rn. In particular we prove that for the space M(h, X), Jackson's theorem holds in the classical form while Bernstein's inequality hold in a modified form. We end of the paper with a characterization of the functions of the Lipschitz space by the behavior of their best approximations by functions in the space M(h, X).

A non-empty subset I of a semigroup S is called an ideal if ab, ba ∈ I whenever a ∈I, b∈S. A subset R of S will be called a retract if there exsists a retraction of S onto R, that is a homomorphism of S onto R which leaves each element of R fixed. The purpose of this paper is to study semigroups in which every ideal is a retract. For convenience we shall call such semigroups retractable. Such semigroups seem to arise naturally; for example, it is easy to show that if the lattice of congruence relations on S is a complemented lattice then S is retractable.

Gleason [3] proved that in the category G of compact Hausdorff spaces and continuous maps, the projective objects are precisely the extremally disconnected spaces contained in the category. Strauss [7] generalised this and proved that in the category G of regular Hausdorif spaces and perfect maps the projective objects are again precisely the extremally disconnected ones. Observe that Gleason's category is a full subcategory of Strauss's category.

The theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

A class of totally disconnected groups consisting of partial direct products on an index set is examined. For such a group, the scale function is found, and for automorphisms arising from permutations of the index set, the tidy subgroups are characterised. When applied to the case where the index set is a finitely-generated free group and the permutation is translation by an element x of the group, the scale depends on the cyclically reduced form of x and the tidy subgroup on the element which conjugates x to its cyclically reduced form.

In 1962, O. Frink [2] showed that in a pseudo-complemented semilattice 〈P; ∧, *, 0〉, the closed elements form a Boolean algebra. We shall consider an extension of this result to arbitrary commutative semigroups with zero.

A semigroup is said to be congruence-free if it has only two congruences, the identity congruence and the universal congruence. It is almost immediate that a congruence-free semigroup of order greater than two must either be simple or 0-simple. In this paper we describe the semilattices of congruence-free inverse semi-groups with zero. Further, congruence-free inverse semigroups with zero are characterized in terms of partial isomorphisms of their semilattices. A general discussion of congruence-free inverse semigroups, with and without zero, is given by Munn (to appear).

The powerful starter-adder theorems for constructing Howell Designs are improved and consequently many types of Howell Designs that previously could only be constructed by multiplicative techniques are shown amenable to a modified starter-adder method. The existence question for Howell Designs of many new types H(s, 2n) is settled affirmatively. For prime powers pn, p ≧ 7, we reduce the entire existence question for designs of type H*(pn, 2r), pn + 1 ≦ 2r ≦ 2pn to the corresponding question for designs of type H*(p, 2m), p + 1 ≦ 2m≦2p. If these designs exist, s has no prime divisors ≤ 7 and t odd is “close” to 1, a design H * (s, s + t) is shown to exist.

A method is developed for obtaining compact, easily computed and statistically interpretable expressions for the generalized k-statistics associated with multiply-indexed arrays of random variables such as those which arise in variance component analysis. These expressions will be used in the next paper in this series to give formulae for variances and covariances of estimates of components of variance.

A universal power automorphism (Cooper [1]) of a group is an automorphism mapping every element x to a power xn for some fixed integer n. It is long known that a group admitting such an automorphism with n= −1, 2 or 3 must be Abelian. Miller [5] showed that for every other non-zero integral value of n there exist non-Abelian groups admitting a non-trivial universal power automorphism x→xn.

The Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

In this paper a general concept of regularity for rings is defined. It is shown that every regularity determines in a natural way a subradical and a radical for rings. A wide class of regularities is constructed: the class of polynomial regularities. All well-known regularities, such as the Perlis-Jacobson regularity, the von Neumann regularity and many others, belong to this class. Special attention is paid to regularities which are elementary in the sense that the so-called unic and nullic polynomial regularities can be thought of as intersections of the elementary ones.

It is shown that if G is a non-amenable group, then there are no non-zero translation invariant functionals on Lp(G) for 1 < p < ∞. Furthermore, if G contains a closed, non-abelian free subgroup, then there are no non-zero translation invariant functionals on C0(G). The latter is proved by showing that a certain non-invertible convolution operator on C0(G) is surjective.

A ℂ-linear map θ (not necessarily bounded) between two Hilbert C*-modules is said to be ‘orthogonality preserving’ if 〈θ(x),θ(y)〉=0 whenever 〈x,y〉=0. We prove that if θ is an orthogonality preserving map from a full Hilbert C0(Ω)-module E into another Hilbert C0(Ω) -module F that satisfies a weaker notion of C0 (Ω) -linearity (called ‘localness’), then θ is bounded and there exists ϕ∈Cb (Ω)+ such that 〈θ(x),θ(y)〉=ϕ⋅〈x,y〉 for all x,y∈E.

We refer the reader to Hanna Neumann [7] for notation and other undefined terms. Let and denote the varieties of groups defined by the laws (xy)n=xnyn, [x, y]n=1 and [x, yn]respectively, where n is an integer. (n)-groups were termed “n-abelian” by R. Baer [1] and have been a subject matter of investigation by various authors (see [3], [5], [6] and the references therein). Recently KaluŽnin [5] has shown that , thus clarifying the relationship between U(n) and the familiar varieties. From the elementary inequalities(n ≠ 0,1)it is easily deduced that(see for instance [5]). If G = Cm Wr C∞, then cleary but for any Thus and . It is also easy to see that in general (see for instance [6] § 5.1) and we are led to ask

The most elementary problem of the calculus of variations consists in finding a single-valued function y(x), defined over an interval [a, b] and taking given values at the end points, such that the integral is stationary relative to all small weak variations of the function y(x) consistent with the boundary conditions. Since y′ occurs in the integrand, it is clear that I is only defined when y(x) is differentiable and accordingly when y(x) is continuous. Usually y′(x) is also continuous. Occasionally, however, the boundary conditions can only be satisfied and a stationary value of I found, by permitting y′ (x) to be discontinuous at a finite number of points. The arc y = y(x) will then possess ‘corners’ and the well-known Weierstrass-Erdmann corner conditions [1]must be satisfied at all such points by any function y (x) for which I is stationary. Arcs y = y (x) for which y′(x) is continuous except at a finite number of points, are referred to as admissible arcs. In this paper, we shall extend the range of admissible arcs to include those for which y(x) is discontinuous at a finite number of points.

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G=PSL(2,q).

A generalised factorial function (z: k)! is defined as an infinite product similar to the Euler product for z!, but with the sequences of integers replaced by the roots of F(z) = sin πz+kπz. It is proved that, apart from poles in (z) < 0, (z: k)! is analytic in both variables, and that F(z) may be expressed in the form F(z) = πz/(z: k)!(—z: k)!

This paper is an outgrowth of my old battle with the open sentence problem for the theory of finite groups. The unsolvability of the word problem for groups (cf. [1] and [4]) entails the undecidability of the open sentence problem for the elementary theory of groups and thus strengthens the original undecidability result for this theory (cf. [7]). The fact that the elementary theory of finite groups is also undecidable (cf. [2] and [6]) therefore justifies my interest in the open sentence problem for that theory. This paper contains a construction of groups that might lead to a negative solution.

Kronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of bounded degree are explored.