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.

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.

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.

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.

The usual statements of the classical adjoint-functor theorems contain the hypothesis that the codomain category should admit arbitrary intersections of families of monomorphisms with a common codomain. The aim of this article is to formulate an adjoint-functor theorem which refers, in a similar manner, to arbitrary internal intersections of “families of monomorphisms” in the case where the categories under consideration are suitably defined relative to a fixed elementary base topos (in the usual sense of Lawvere and Tierney).

Let π be a convex lattice polygon with b boundary points and c (≥ 1) interior points. We show that for any given c, the number b satisfies b ≤ 2c + 7, and identify the polygons for which equality holds.

Sufficient conditions are obtained for all solutions of a class of second order nonlinear functional differential equations to be nonoscillatory.

The indecomposable representations in characteristic two of the groups PSL(2, q) where q is congruent to 3 or 5 modulo 8 are classified. For q = 3 or 5 the classification is obtained by explicit construction of modules, using the Green correspondence to prove completeness. For larger q, the classification is obtained using equivalences between appropriate categories of modules.

Initial value problems of the form x′ + A(t, x)x = f(t, x), x(0) = a, t ≥ 0, are considered in a real, separable, reflexive Banach space. Results concerning the existence of solutions on (0, ∞) are given by considering linear systems of the form x′ + A(t, u(t))x = f(t, u(t)). Here u(t) belongs to a suitable function space.

In this note we show that BVk, [a, b], the linear space of functions of bounded kth variation on [a, b], is a Banach space under the norm ‖ • ‖k, where

This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

We prove that most group algebras of free products have left zero divisors that are not right zero divisors.

To each associative (but not necessarily commutative) ring R we assign the complete distributive lattice R-tors of (hereditary) torsion theories over R-mod. We consider two ways of making this process functorial – once contravariantly and once covariantly – by selecting appropriate subcategories of the category of associative rings. Combined with a functor due to Rota, this gives us functors from these subcategories to the category of commutative rings.

This paper first assigns specific uniform convergence structures to the products and function spaces of pairs of uniform convergence spaces, and then establishes a bijection between corresponding sets of morphisms which yields cartesian closedness.

Let P*(α, β) denote the class of functions

analytic and univalent in |z| < 1 for which

where α є [0, 1), β є (0, 1].

Sharp results concerning coefficients, distortion theorem and radius of convexity for the class P*(α, β) are determined. A comparable theorem for the classes C*(α, β) and P*(α, β) is also obtained. Furthermore, it is shown that the class P*(α, ß) is closed under ‘arithmetic mean’ and ‘convex linear combinations’.

The fact that the group studied in the original paper, M.F. Newman, Bull. Austral. Math. Soc. 14 (1976), 293–297, is infinite follows immediately from a result in a 1940 paper of Coxeter. The computer aided methods give more detailed information about the group and some related groups.