In 1913 Richardson published necessary and sufficient conditions for a system of three Sturm–Liouville equations, linked by three parameters, to possess eigenfunctions with arbitrarily many zeros. His work contains errors, but we give conditions of his type valid for k self-adjoint equations, with k parameters.

A geometric hypothesis is presented under which the cohomology of a group G given by generators and defining relators can be computed in terms of a group H defined by a subpresentation. In the presence of this hypothesis, which is framed in terms of spherical pictures, one has that H is naturally embedded in G, and that the finite subgroups of G are determined by those of H. Practical criteria for the hypothesis to hold are given. The theory is applied to give simple proofs of results of Collins-Perraud and of Kanevskiĭ. In addition, we consider in detail the situation where G is obtained from H by adjoining a single new generator x and a single defining relator of the form xaxbxεc, where a, b, c ∈ H and |ε| = 1.

Fibonacci algebras are groups equipped with an extra unary operation φ that satisfies a Fibonacci-type law. We described in an earlier paper the free objects in the resulting varieties, and here we do the same in the case when φ is assumed to be periodic. They turn out to be central extensions of Burnside groups with finite kernels whose orders can be expressed in terms of the resultants of certain polynomials.

If h is an outer function in H1 then it is shown that h = (q1 + q2)g where both q1 and q2 are inner functions with Im almost everywhere, and g is a strong outer function (equivalently, g/∥g∥1 is an exposed point of the unit ball of H1). If q1 + q2 is nonconstant then such an h is not strongly outer. Moreover a sum of two inner functions is studied.

For positive definite C1 kernels on a finite real interval the eigenvalues λn are known to be o(1/n2). In this paper this result is shown to be best possible in the best possible sense, namely that, given any decreasing sequence λn, which is o(1/n2), there exist positive definite C1 kernels whose eigenvalues are λn.

Maximal left ideals in matrix rings were studied by Stone [10]. Similar results are not necessarily valid in the general near-ring case and one of the objectives of this paper is to study these differences. Furthermore, although much is known about 2-primitivity in general matrix near-rings (Van der Walt [11]), quite the opposite is true for 0-primitivity and the other objective of this paper is to present some results on 0-primitivity in matrix near-rings in certain restricted cases.

We show that if G is a compact abelian group and U is a weakly continuous representation of G by means of isometries on a Banach space X, then holds for each measure µ in reg(M(G)), where π(µ) denotes the generalized convolution operator in B(X) defined by , σ the usual spectrum in B(X), sp(U) the Arveson spectrum of U, the Fourier-Stieltjes transform of µ and reg(M(G)) the largest closed regular subalgebra of the convolution measure algebra M(G) of G. reg(M(G)) contains all the absolutely continuous measures and discrete measures.

An asymptotic expansion is given for the series

as x→∞ in the sector |Argx|≦π/2–δ. Here δ, Re(a), and Re(s) are positive and r is a positive integer. In the case a = r = s = 1, this yields the nontrivial result

stated by Ramanujan in his notebooks [6].

Nous Présentons ici l'étude, dans l'optique exhibée dans [7], d'une catégorie d'algèbres à idempotents, commutatives, non associatives, définies par des relations. On y trouve une structure qui n'est en général ni pondérée ni de Jordan et où, cependant, ces conditions sont équivalentes.

Following a previous paper, we study here a class of commutative and non associative algebras with idempotents and defined by relations. We find a structure which is not, in general, baric or Jordan but where these notions are equivalent.

We prove the existence of a solution u(.,.;α,β) of the Darboux problem uxy∈F(x, y, u), u(x,0) = α(x), u(0, y) = β(y), which is continuous with respect to (α,β). We assume that F is Lipschitzean with respect to u but not necessarily convex valued.

We obtain a formula relating the Conway potential functions of links in S3 which are connected by a framed surgery operation. Using this formula we extend the theory of Conway potential functions to links in all oriented ℚ-homology 3-spheres.

The authors together with M. J. Karbe [Ill. J. Math. 33 (1989) 333–359] have considered Fitting classes of -groups and, under some rather strong restrictions, obtained an existence and conjugacy theorem for -injectors. Results of Menegazzo and Newell show that these restrictions are, in fact, necessary.

The Fitting class is normal if, for each is the unique -injector of G. is abelian normal if, for each. For finite soluble groups these two concepts coincide but the class of Černikov-by-nilpotent -groups is an example of a nonabelian normal Fitting class of -groups. In all known examples in which -injectors exist is closely associated with some normal Fitting class (the Černikov-by-nilpotent groups arise from studying the locally nilpotent injectors).

Here we investigate normal Fitting classes further, paying particular attention to the distinctions between abelian and nonabelian normal Fitting classes. Products and intersections with (abelian) normal Fitting classes lead to further examples of Fitting classes satisfying the conditions of the existence and conjugacy theorem.

We show that given a knot in a homology sphere there is a sequence of invariants with the property that if the nth invariant does not vanish, then this implies the existence of a family of irreducible representations of the fundamental group of the complement of the knot into SU(n).

We study Lie algebras with cone potential which play a prominent role in the Lie theory of semigroups. For these algebras, we obtain a uniqueness theorem for Levi complements and information on the fine structure of the root system.

Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.

J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.

The above result is then used to show that if R is a subring of rings Q1 and Q2 and the multiplicative subsemigroups of Q1 and Q2 are semigroups of left quotients of the multiplicative semigroup of R, then Ql and Q2 are isomorphic rings.

We answer a question raised in an earlier paper concerning the pure state space of a separable C*-algebra.

Let A be a finite-dimensional Bernstein algebra over a field K with characteristic not 2. Maximal subalgebras of A are studied, and they are determined if A is a genetic algebra. It is also proved that the intersection of all maximal subalgebras of A (the Frattini subalgebra of A) is always an ideal. Finally the structure of Bernstein algebras with Frattini subalgebra equal to zero is described.

We prove two inequalities which relate the Lp modulus of continuity of n-th order, ωn(f,·)p, of an Hp function f with the p-th mean values of the n-th derivative f(n). Using these inequalities we extend classical results of Hardy and Littlewood [5], Gwiliam [4], Zygmund [13] and Taibleson [12] as well as a recent result of Oswald [6].

Let be a class of finite groups. Then a c-group shall be a topological group which has a fundamental system of open neighbourhoods of the identity consisting of normal subgroups with -factor groups and trivial intersection. In this note we study groups which are existentially closed (e.c.) with respect to the class Lc of all direct limits of c-groups (where satisfies certain closure properties). We show that the so-called locally closed normal subgroups of an e.c. Lc-group are totally ordered via inclusion. Moreover it turns out that every ∀2-sentence, which is true for countable e.c. L-groups, also holds for e.c. Lc-groups. This allows it to transfer many known properties from e.c. L-groups to e.c. Lc-groups.

We describe for every natural n the class of rings R such that if R is an accessible (left accessible) subring of a ring then R is an n-accessible (n-left-accessible) subring of the ring. This is connected with the problem of the termination of Kurosh's construction of the lower (lower strong) radical. The result for n = 2 was obtained by Sands in a connection with some other questions.