Wagner made the conjecture that given an infinite sequence G1, G2, … of finite graphs there are indices i < j such that Gi is a minor of Gj. (A graph is a minor of another if the first can be obtained by contraction from a subgraph of the second.) The importance of this conjecture is that it yields excluded minor theorems in graph theory, where by an excluded minor theorem we mean a result asserting that a graph possesses a specified property if and only if none of its minors belongs to a finite list of ‘forbidden minors’. A widely known example of an excluded minor theorem is Kuratowski's famous theorem on planar graphs; one of its formulations says that a graph is planar if and only if it has neither K5 nor K3, 3 as a minor. But several other excluded minor theorems have been discovered by now (see e.g. [7–9]).

The only finite-dimensional simple non-Lie Mal'cev complex algebra is given the structure of an H*-algebra and it is proved that this is the only topologically simple non-Lie Mal'cev H*-algebra.

A non-zero n-by-n matrix A is said to be completely positive if there exist non-negative vectors b1,…, bk, such that

The smallest such integer k is called the factorization index of (completely positive) A, and is denoted by ø(A). Completely positive matrices are important in the study of block designs [4], statistics and modelling of energy demand [3].

Throughout this paper we shall fix an algebraically closed field k. Consider the following:

Problem. Let (Y, L) be a normal polarized variety over k, i.e. a normal projective variety Y over k together with an ample line bundle L on Y. Then one may ask under which conditions the following statement holds:

(*) Every normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is L, is isomorphic to the projective cone over Y.

Smooth mappings defined on discriminant varieties of -versal unfoldings of isolated singularities arise in many interesting geometrical contexts, for example when classifying outlines of smooth surfaces in ℝ3 and their duals, or wave-front evolution [1, 2, 5]. In three previous papers we have classified various stable mappings on discriminants. When the isolated singularity is weighted homogeneous the discriminant is not a local smooth product, and this makes the classification of stable germs considerably easier than in general. Moreover, discriminants arising from weighted homogeneous singularities predominate in low dimensions, so such classifications are very useful for applications.

Several authors have considered relationships between individuals specified by the genes which they have in common, where two genes are regarded as the same if and only if they are identical by descent from some common ancestor. In particular Thompson [6] considered n pairs of genes, one pair from each of n individuals, and studied the number Nn of gene identity states, the number Mn, k of such states in which there are exactly k distinct genes and the number Dn of genetically distinct states (see Definitions 2 and 4 below). In this type of work the nature of the genes themselves is of no interest, but only which genes are identical to which, and precise definitions are best given in terms of orbits under group actions. Thompson did use some group theory in her work, but not the full machinery of Redfield-Pólya-de Bruijn enumeration.

Let H be a separable complex Hilbert space with orthonormal basis {ei: i ∈ ℕ}, let s be the unilateral shift defined by sei = ei+1 for each i and let K be the algebra of compact operators on H. The present paper classifies the involutory *-anti-automorphisms in the C*-algebra C*(sn, K) generated by K and a positive integral power sn of s. It is shown that, up to conjugacy by *-automorphisms, there are two such involutory *-antiautomorphisms when n is even and one when n is odd.

Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q + 1, q + 1)/S[U(q + 1, q) × U(1)], and SU(q + 1, q + 1)/SL(n, ℂ) ⋊ H(n) (p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

Let FH(X) denote the group of units of the classical cohomology ring H(X) = Πn≥0Hn(X; Z/2) of a CW-complex X. The total Stiefel–Whitney class can be viewed as a group homomorphism where is the reduced real K-theory of X. Both and FH( ) are representable functors, with representing spaces BO and FH, and thus w can be represented by a map w: BO → FH. By the Bott periodicity theorem, BO is an infinite loop space, and by a theorem of G. Segal[9] so is FH. However, it is well known that w is not an infinite loop map; this was first shown in [10]. The purpose of this paper is to prove the following:

Theorem 0·1. w: BO → FHis a loop map but not a double loop map.

The relationship between components and movability for compacta (i.e. compact metric spaces) was described by Borsuk in [5]. Borsuk proved that if each component of a compactum X is movable, then so is X. More recently Segal and Spiez[19], motivated by results of Alonso Morón[1], have constructed a (non-compact) metric space X of small inductive dimension zero and such that X is non-movable. The construction of Segal and Spiez was based on the famous space of P. Roy [16]. On the other hand, K. Borsuk gave in [5] an example of a movable compactum with non-movable components. The structure of such compacta was studied by Oledzki in [15], where he obtained an interesting result stating that if X is a movable compactum then the set of movable components of X is dense in the space of components of X. Oledzki's result was later strengthened by Nowak[14], who proved that if all movable components of a movable compactum X are of deformation dimension at most n, then so are the non-movable components and the compactum X itself.

1·1. The aim of the present paper is to prove the following

Theorem. Let Vi and Mn be closed smooth manifolds and i < n. Let f: Vi → Mn be a generic smooth map such that all of its singular points are of the type Σ;1, 0. (See [2].) Then there exists a non-zero integer N such that the bordism class N·[f]εΩi(Mn) contains an immersion.

A simple proof of Denjoy's theorem on topological conjugacy of circle maps with irrational rotation number to uniform rotation is presented.

We give a quite general geometric criterion for a function analytic in the unit disc to be a polynomial of a univalent function, and hence a criterion for multivalence. We believe that this is the essence why multivalent close-to-convex functions enjoy the latter decomposition property. As another application, we study, as suggested by T. Sheil-Small ‘9’, the geometry of classes of analytic functions which arise from his recent investigation of multivalent harmonic mappings.

Exact sequences which can be used for calculating the group (X) of homotopy classes of homotopy self-equivalences of a space X are known in the case of cellular decompositions (some references are given in § 1 of [8]), and in the case of Postnikov decompositions (for example 3·1 of [5]). Here we obtain exact sequences which can be used to calculate (X) using a simply-connected homology decomposition.

This paper is essentially concerned with the following problem. Let G be a topological group. AG-set is a set S equipped with a continuous action (where S is given the discrete topology). These G-sets form a category BG. It is well known that if G and H are discrete groups, BG is equivalent to BH iff G is isomorphic to H. However, this result cannot be extended much beyond the discrete case. The problem is: when is BG equivalent to BH, for topological groups G and H?

Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see [14]). The Archimedes integral introduced in [9] does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see [9]).

After the development of the modular theory one can construct non-commutative Lp-spaces associated with a von Neumann algebra M which is not necessarily semifinite (see [4], [12], etc.).

We give new proofs of transference theorems which allow the transfer of the bad behaviour of Fourier coefficients from one complete orthonormal system to the other. We also present some results on Carleman-type singularities.

In a 1928 paper [1], Hardy and Littlewood published the following inequality which is valid for all non-negative measurable functions ƒ and g on ( − ∞, ∞):

where 1 < p, q < ∞, 0 < λ < 1, l/p + 1/q + λ = 2 and Cλp, q depends only on λ, p, q; ‖ƒ‖p and ‖g‖q are the Lp and Lq norms of ƒ and g respectively. See also p. 288 of the monograph [2] by Hardy, Littlewood and Pólya for a discussion of this inequality and related matters.

In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.