In this paper, we announce results on the Dirichlet and periodic boundary-value problems for the equation -x″(t) = g(x(t)) – f(t) on [0, π]. We consider degenerate cases not covered by the author's earlier work.

In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.

We prove an approximation formula for the Riemann zeta function. We show that a classical theorem:

uniformly in the domain ½ ≤ σ < 1, is an immediate consequence of our approximation formula. Our method is real and free from complex analysis.

I show that a group of order-automorphisms of a linearly ordered set can be expressed as an unrestricted direct product in which each factor is either the infinite cyclic group or else a group of order-automorphisms of a densely ordered set. From this a couple of simple group embedding theorems can be derived. The technique used to obtain the main result of this paper was motivated by the Erdös-Hajnal inductive classification of scattered sets.

In a recent paper (Bull. Austral. Math. Soc. 13 (1975), 241–245), Tarafdar has considered nonexpansive self mappings on a subset X of a locally convex vector space E and proved an extension to E of a theorem of Göhde. The purpose of this paper is to show that the condition f: X → X, in Göhde-Tarafdar's Theorem in the above paper, may be weakened to f: X → E with f(∂X) ⊆ X. As a consequence, it is further shown that an extension to E of a well-known common fixed point theorem of Belluce and Kirk due to Tarafdar remains true on domains that are not necessarily bounded or quasi-complete.

Martin H. Schultz [Bull. Amer. Math. Soc. 76 (1970), 840–844] has investigated the spectral condition number of the Rayleigh-Ritz-Galerkin matrices that arise when normalized B-spline coordinate functions are used to approximate the solution of a class of linear, self-adjoint, elliptic boundary value problems in one dimension. This paper shows how results analogous to those of Schultz [op. cit.] can be established under weaker assumptions. We also extend the results to boundary value problems in higher dimensions.

All groups considered are finite. We enlarge the class of groups for which the conjecture below is known to hold, to include all nilpotent groups A such that every proper subgroup of A is Zp ˜ Zp free for all primes p.

This study is a continuation of an earlier paper on the regularity series of a convergence space. The notions of a R-Hausdorff series and the T3-modification of a convergence space are introduced, and their relationship with the regularity series is studied. The concept of a symmetric space is shown to be useful in studying T3-compactifications. Several examples are given; one being a Hausdorff convergence space with an arbitrarily large regularity series.

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)

have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

Let A = {ai} be a finite set of integers and let p and m denote the cardinalities of A + A = {ai+aj} and A - A {ai–aj}, respectively. In the paper relations are established between p and m; in particular, if max {ai–ai-1} = 2 those sets are characterized for which p = m holds.

Let H be a subgroup of a finite group G and let S be a set of generators of H. We prove that if G is soluble, then H is subnormal in G if and only if there exists an integer n such that for each g in G and a in S the commutator lies in H. This criterion for subnormality is also valid for soluble groups satisfying the maximal or the minimal condition on subgroups.

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. We say that groups G and H have isomorphic finite quotients (IFQ) if F(G) = F(H). In this note, we show that a finitely generated residually finite group G cannot have the same finite quotients as a proper homomorphic image (G is IFQ hopfian). We then obtain some results on groups with the same finite quotients as a relatively free group.

For a Banach space B let P denote the canonical projection of the third dual space of B onto the embedding of the first dual into the third. It is shown that if B = l1 then ‖I-P‖ = 2.

This fact shows to be mistaken a current belief in a statement which is equivalent to the statement that for all Banach spaces B the operator I – P is of norm one.

The author and Ira J. Papick have termed an integral domain R a going-down ring if R ⊂ T satisfies going-down for each domain T containing R. The present paper investigates conditions which, for an integral extension A ⊂ B of domains, imply that A (respectively B ) is going-down whenever B (respectively A ) is going-down. This explains the “descent” (respectively “ascent”) in the title. Two typical results (the first about descent, the second about ascent) are given next.

In an additive category A0, objects are said to be determined by their rings of endomorphisms if for each ring-isomorphism F of the rings of endomorphisms of two objects A, B in A0 there is an isomorphism f: A → B in A0 such that F(α) = fαf-1, for every endomorphism α of A. Considering.this problem in the context of closed categories (in Eilenberg and Kelly's sense), the author proves a general theorem which generalises results of Eidelheit (for real Banach spaces) and of Kasahara (for real locally convex spaces).

We call a topological space completely regular if points and closed sets can be separated by continuous real-valued functions, and we call a topological space Tychonoff or T3α if it is T1 and completely regular. It is well known that the Tychonoff spaces are precisely the gauge spaces, that is the topological spaces whose topologies are induced by separating families of pseudometrics. V.G. Boltjanskiĭ has given analogous characterizations for T0, T1, T2, regular, and regular-T1 spaces in terms of families of “metric-like” functions. The purpose of this note is to fill in a case missing in Boltjanskiĭ's paper, the completely regular spaces, and to relate Boltjanskiĭ's work to results on quasi-uniformization by William J. Pervin and results about semi-gauge spaces by J.V. Michalowicz.

The following theorem is proved.

Let X be a finite set of cardinality n ≥ 2, and let F be a family of subsets of X. Suppose that for F1, F2, F3 ∈ F we have |F1 ∩ F2 ∩ F3| ≥ 2. Then |F| ≤ 2n−2with equality holding if and only if for two different elements x, y of X, F = {F ⊆ X | x ∈ F, y ∈ F}.

A quotient algebra of a given algebra is said to be cogenerated by a quotient object if it is contained in the quotient object and is the biggest with this property. Triples T over a regular category are characterized which have the property that every quotient object of a T-algebra cogenerates some quotient algebra: these are precisely the right exact triples, preserving colimits of quotient chains. This improves a result of Michael Barr (J. Pure Appl. Algebra 4 (1974), 1–8) that every right exact, finitary triple has the investigated property. This result is related to categorical automata, since a triple has the above property iff triple machines admit a minimal realization of every behavior.

A continuous function mapping a compact interval of the real line into itself is called chaotic if the difference equation defined in terms of it behaves chaotically in the sense of Li and Yorke. The set of all such chaotic functions is shown to toe a dense subset of the space of continuous mappings of that interval into itself with the max norm. This result indicates the structural instability of nonchaotic difference equations with respect to chaotic behaviour.

Let G be a finite group. A nontrivial subgroup M of G is called a CC-subgroup if M contains the centralizer in G of each of its nonidentity elements. The purpose of this paper is to classify groups with a CC-subgroup of order divisible by 3. Simple groups satisfying that condition are completely determined.