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).

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 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).

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.

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.

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 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.

Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.

The pseudocharacter of the space YX of continuous functions from X to Y is studied. For certain topologies on YX sufficient conditions, which are shown in some cases also to be necessary, for YX to have a specified pseudocharacter are given. An Ascoli theorem is a theorem which characterizes the compact subsets of YX. Several Ascoli theorems are obtained, including ones which utilize results on pseudocharacter.

It has recently been shown that discontinuous functional calculi exist for certain commutative Banach algebras. Such an algebra thus possesses two distinct calculi so that there exist analytic functions whose action on the algebra is not uniquely determined. In this note a method is given for constructing commutative Banach algebras which admit two inequivalent complete norm topologies and the result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined.

A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn(G) is the minimal number of paths necessary to form a path decomposition of G.

We show that pn(G) ≥ max{0, od(v)-id(v)} the sum taken over all vertices v of G, with equality holding if G is acyclic. If G is a subgraph of a tournament on n vertices we show that pn(G) ≤ with equality holding if G is transitive.

We conjecture that pn(G) ≤ for any digraph G on n vertices if n is sufficiently large, perhaps for all n ≥ 4.

It is proved that the subvarieties of the variety are CREAM in the sense of Higman when m, n are coprime and n is an odd integer not divisible by q4 for any prime q.

The object of the present paper is to study the delay differential equation of arbitrary order namely y(n)(t) + a(t)yτ(t) = f(t), n≥2 (an integer) and prove a nonoscillation theorem under the general situation in which a(t) and f(t) are allowed to oscillate arbitrarily often on some positive half real line. This is accomplished by way of two differential inequalities of nth order.

The two generator restricted Burnside group of exponent five is shown to have order 534 and class 12 by two independent methods. A consistent commutator power presentation for the group is given.