By using the concept of tame embeddings of chains, a characterization is given of the subobjects of the lattice-ordered groups of order-automorphisms of the chains of rational and real numbers.

A group G is called semi-n-abelian, if for every g ∈ G there exists at least one a(g) ∈ G-which depends only on g-such that (gh)n = a-1(g)gnhna(g) for all h ∈ G; a group G is called n-abelian, if a(g) = e for all g ∈ G. According to Durbin the following holds for n-abelian groups: If G is n-abelian for at lesast 3 consecutive integers, then G in n-abelian for all integers and these groups are exactly the abelian groups. In this paper this problem is generalized to the semi-n-abelian case: If a finite group G is semi-n-abelian for at least 4 consecutive integers then G is semi-n-abelian for all integers and these groups are exactly the nilpotent groups, where the Sylow-2-subgroup is abelian, the Sylow-3-subgroup is any element of the Levi-variety ([[g, h], h] = e ∀ g, h ∈ G) and the Sylow-p-subgroup (p < 3) is of class <2. As a consequence we get a description of all finite (3-)groups, which are elements of the Levi-variety.

Barnes (1978, 1979) introduced the concept of a -extreme form, which is a Minkowski-reduced positive definite quadratic form having prescribed diagonal coefficients α1, α2, …, αn and providing a local minimum of the determinant of the form over all such forms. Here a class of forms which are -extreme for all α and all n is described.

Let G be LCA group with an algebraically ordered dual Ĝ. Suppose also that the semigroup P of positive elements in Ĝ is not dense in Ĝ. Subspaces (G) (1 < s < ∞) are defined analogous to the Hardy spaces on the circle group, and the question whether every multiplier from into can be extended to a multiplier from L3(G) into Lq(G) is investigated. If we suppose that s ≠ ∞, the it is shown that such an extension is possible if and only if (s, q) ∈ (1, ∞) × [1, ∞] ∪ {(1, ∞)}. (the negative result for (1, 1) was obtained in a previous paper.)

Given a measurable function k non-negative a.e. on the circle |z| = 1, when is the outer function Tk (see(1.3)) continuous on the disk |z| < 1 and further, Dirichlet-finite: We shall show, among other results, that the answer is in the positive if , with ess inf k > 0.

This paper generalizes the theory of imperfect bifurcation via singularity theory as developed by M. Golubitsky and D. Schaeffer to a Banach space setting. Like the parameter-free potential catastrophe theory, where similar generalizations have been discussed in the literature, Banach control spaces allow useful uniform control of function parameters through the universal unfolding. Among the results are tests for various germ properties and discussion of their reducibility under a Liapunov—Schmidt type splitting, as well as a generalization of the finite dimensional unfolding and germ classification theory.

A compact Hausdorff space is regular Wallman if it possesses a separating ring of regular closed sets, an s-ring. It was proved by P. C. Baayen and J. van Mill [General Topology and Appl. 9 (1978), 125–129] that if a locally compact Hausdorff space possesses an s-ring, then every Hausdorff compactification with zero-dimensional remainder is regular Wallman.

In this paper the reasoning leading to this result is modified to work in a more general setting. Iet αX be a Hausdorff compactification of a space X, and let be the family of those closed sets in αX whose boundaries are contained in X. A main result is the following: If contains an s-ring for some Hausdorff compactification γX, then every larger Hausdorff compactification αX for which is a base for the closed sets on αX — X, is regular Wallman. Various consequences concerning compactifications of a class of rim-compact spaces (called totally rim-compact spaces) are discussed.

A partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotone homomorphism θ of S onto a partially ordered group such that {} has a greatest member. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in which the imposed partial order extends the natural partial order on the idempotents. The main tool used is a local structure theorem which is introduced in Section 2. This local structure theorem applies to many other contexts as well.

Let m, n be infinite cardinals such that m < n, and let X be a set of cardinality m. Within the symmetric inverse semigroup on X the elements whose domain and range have complements of cardinality m form an inverse semigroup T. The closure Eω of the semilattice E of idempotents of T is a fundamental bismple inverse semigroup. Its maximum congruence is described. The quotient of Eο by this maximum congruence is a bisimple, congruence is a bisimple, congruence-free inverse semigroup.

In 1957 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a G when the marginal factor-group is of order Pn (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite p-group with respect to the variety of polynilpotent groups.

It is shown that non-trivial relations between certain values of the dilogarithm function can be obtained through the asymptotic comparison of coefficients of the expressions which appear in the Rogers-Ramanujan and Andrews-Gordon identities.

A short and easy proof that the minimum number of generators of the nth direct power of a non-trival finite group of order s having automorphism group of order a is more than logsn + logsa, n > 1. On the other hand, for non-abelian simple G and large n, d(Gn) is within 1 + e of logsn + logsa.

Frames have been defined as a certain type of generalization of Room square. Frames have proven useful in the construction of Room squares, in particular, skew Room squares.

We generalize the definition of frame and consider the construction of Room squares and skew Room squares using these more general frames.

We are able to construct skew Room squares of three previously unknown sides, namely 93, 159, and 237. This reduces the number of unknown sides to four: 69, 87, 95 and 123. Also, using this construction, we are able to give a short proof of the existence of all skew Room squares of (odd) sides exceeding 123.

Finally, this frame construction is useful for constructing Room squares with subsquares. We can also construct Room squares “missing” subsquares of sides 3 and 5. The “missing” subsquares of sides 3 and 5 do not exist, so these incomplete Room squares cannot be completed to Room squares.