Starters with adders in abelian groups of odd order have been used extensively in the construction and study of Room squares. It is possible to define these concepts in non-abelian groups, with similar applicability to Room squares. Examples are given of adders for the “patterned starter” in the non-abelian groups of order pq, p and q primes larger than 3, p ≡ q ≡ 3 (mod 4), q ≡ 1 (mod p).

We give a necessary and sufficient condition for a class of row-decreasing infinite matrices to be partially well-ordered with regard to the component-wise ordering. Then, using these matrices, we determine all the classes of abelian groups, closed under taking subgroups, direct limits, and isomorphic groups.

In this note 4-fold torsion theories (for categories of modules) are classified by means of orthogonal pairs of comaximal ideals. Among the applications are results of Kurata concerning lengths of n-fold torsion theories and an upper bound for the number of 4-fold torsion theories over a semiperfect ring.

From recent results of Lausch, it is easy to establish necessary and sufficient conditions for a Fitting class to be maximal in the class of all finite soluble groups. We use Lausch's methods to show that there are normal Fitting classes not contained in any Fitting class maximal in the class of all finite soluble groups. We also find conditions on Fitting classes and for to be maximal in .

Let p, q, u, and v be any four positive integers, and let further δ be a number in the interval 0 < δ ≤ 2. In this note an effective lower bound for q will be obtained which insures that In the special case when u = v = 1, it was shown by J. Popken, Math. Z. 29 (1929), 525–541, that Here c and C are two positive absolute constants which, however, were not determined explicity. A similarly non-effective result was given in my paper, J. reine angew. Math. 166 (1932), 118–150.

The method of this note depends again on the classical formulae by Hermite which I applied also op. cit.

The third order delay equation

y‴(t) + a(t)yτ(t) = 0

is studied for its nonoscillatory nature under the general condition in which a(t) has been allowed to oscillate. It is shown by way of a differential inequality that if g(t) is a thrice differentiable and eventually positive function then

g‴(t) + t2|a(t)|g(t) ≤ 0

is sufficient for this equation to have bounded nonoscillatory solutions.

Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp

(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and

(2) the (even) number of fixed points.

Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

The hyper-archimedean kernel Ar(G) of a lattice-ordered group (hence forth l–group) is the largest hyper-archimedean convex l–subgroup of the l–group G. One defines Arσ (G), for an ordinal σ as if a is a limit ordinal, and as the unique l–ideal with the property that Arσ(G)/Ar.σ–1(G) = Ar(G/Arσ–1(G)), otherwise. The resulting "Loewy"-like sequence of characteristic l–ideals, Ar(G) ⊆ Ar2(G) ⊆ … ⊆ Arσ (G) ⊆ …, is called the hyper-archimedean kernel sequence. The first result of this note says that each Arσ(G) ⊆ Ar(G)”.

Most of the paper concentrates on archimedean l–groups; in particular, the hyper-archimedean kernels are identified for: D(X), where X is a Stone space, a large class of free products of abelian l–groups, and certain l–subrings of a product of real groups.

It is shown that even for archimedean l–groups the hyper-archimedean kernel sequence may proceed past Ar(G).

Let V = mk + 1 be a prime power; we show for m even it is not possible to partition the Galois field GF(v) to give four (0, 1, −1) matrices X1, X2, X3, X4 satisfying:

(i) Xi * Xj = 0, i ≠ j, i, j = 1, 2, 3, 4;

(ii) is a (1, −1) matrix;

(iii) Thus this method of partitioning the Galois field GF(V), into four matrices satisfying the above conditions, cannot be used to find Baumert-Hall Hadamard arrays BH[4V] for v = 9, 11, 17, 23, 27, 29, ….

We introduce some methods of constructing stable graphs and characterize a few classes of stable graphs. We also give a counter example to disprove Holton's conjecture.

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form eO(m10gm) should hold, but I have not so far succeeded in proving such a result.

A star is a connected graph in which every vertex but one has valency 1. This paper concerns the question of when complete graphs can be decomposed into stars, all of the same order, which have pairwise disjoint edge-sets. It is shown that the complete graphs on rm and rm + 1 vertices, r > 1, can be decomposed into stars with m edges, if and only if r is even or m is odd.

Results are obtained which enable one to assess the extent to which the waiting times of queuing customers depend on the number of servers which provide them with service.

This paper presents an extension of Banach's contraction mapping principle to Hausdorff spaces, in fact to the larger class of topological spaces in which convergent sequences have unique limits. This is achieved by considering topologies on X generated by families of quasi-pseudo-metrics on X. An extension of the concept of Cauchy sequence to this non-metric setting is given.

Let X be a reflexive Banach space and be a family of weakly continuous operators which map X to X. Conditions are provided which guarantee the existence and the uniqueness to the Cauchy initial value problem

The object of this paper is to study common fixed points of mappings of a complete metric space into itself. The results obtained are generalizations of Ray Theorems.

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.

An ordinary second order differential equation is considered in which the coefficients are dependent on two parameters ω and F as well as the independent variable μ. The equation arises in the study of free oscillations of incompressible inviscid fluid in global shells. An asymptotic technique is presented which estimates the eigenvaiues (that is the values of ω for which the solution is bounded for all |μ| ≤ 1) as functions of F, as F → ∞. The agreement of the results with numerical computations is also discussed.

An algorithm is presented for computing the unique stationary distribution of an infinite regular stochastic matrix of a structural form subsuming both upper-Hessenberg and generalized renewal matrices of this kind. Convergence is elementwise, monotone from above, from information within finite truncations, of increasing order.

In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.