Let a(n) denote the number of non-isomorphic Abelian groups of order n. It is well-known that

for a natural number k we define

and

It is shown that apart from

there are only finitely many arithmetic progressions with given differences of equal lengths ≥ 2 and with equal products and that they can be effectively determined.

We give a particularly short proof of the symmetrization of Cournot-Nash equilibrium distributions in large anonymous games with countable actions. Our proof relies on the Bollobás-Varopoulos extension of the marriage lemma.

Let d(n) denote the number of positive divisors of n, and let f(x) be a polynomial in x with integer coefficients, irreducible over ℤ. Erdös[3] showed that there exist constants λ1, λ2 (depending on f) such that

If λ > ω then there are 2λ graphs of cardinal λ none of them being a minor of another.

Let G and G′ be reductive algebraic groups defined over infinite fields k and k′ respectively. The purpose of this paper is to show that G and G′ have isomorphic root systems if their rational subgroups G(R) and G′(R′), where R and R′ are integral domains with R ⊇ k and R′ ⊇ k′, are isomorphic to each other, except in one particular case (see Theorem 3·4). This has been proved by R. Steinberg in [6, theorem 31] for simple Chevalley groups over perfect fields. In particular, when G and G′ are semisimple and adjoint, every isomorphism between G(R) and G′(R′) induces an isomorphism between their irreducible components (see Proposition 3·3). These results imply that, when G and G′ are semisimple k-groups and when both are either simply connected or adjoint, then they are isomorphic to each other as algebraic groups if and only if their rational subgroups over an integral domain that contains k are isomorphic to each other, except in one particular case (see Corollary 3·5).

Projecting a knot onto a plane – or, equivalently, looking at it through one eye – one sees a more or less complicated plane curve with a number of crossings (‘nodes’); viewing it from certain positions, some other more complicated singularities appear. If one spends a little time experimenting, looking at the knot from different points of view, then provided the knot is generic, one can convince oneself that there is only a rather short list of essentially distinct local pictures (singularities) – see Fig. 3 below. All singularities other than nodes are unstable: by moving one's eye slightly, one can make them break up into nodes. For each type X the following two numbers can easily be determined experimentally:

1. the codimension in ℝ3 of the set View(X) of centres of projection (viewpoints) for which a singularity of type X appears, and

2. the maximum number of nodes n into which the singularity X splits when the centre of projection is moved.

Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.

Simple singularities in positive characteristic

Simple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.

Let f: M → N be a smooth map of a closed n-manifold into a p-manifold (n ≥ p) having only Morin singularities [17]. We study the topology of such a map and obtain a modulo 2 congruence formula involving the Euler characteristics of M, N, the singular sets and the regular fibres of f. We also consider applications of this formula to the existence problem of maps having only fold singular points. Stable maps into 3-manifolds are also studied and we obtain a modulo 2 congruence formula involving the swallow tails and the number of triple points of the discriminant set.

It is shown that the family of representations {jt, t ∈ ℝ+} of the universal enveloping algebra U of the N-dimensional unitary group which is generated by the N-dimensional number process of quantum stochastic calculus can be expressed in the form

where ψ is a bijective linear map from U onto the space S of symmetric tensors over the Lie algebra, and It is the iterated (chaotic) integral on S. The chaotic product * is defined by the formula

and satisfies

This work generalizes and completes earlier results on the centre of U.

In recent work [5] which involved enumeration of singularity types of highly singular quintic curves, it was necessary to use rather detailed information on the geometry of quartic curves (for the case when the quintic consists of the quartic and a line). The present paper was written to supply this background. The cases of primary interest for this purpose are the rational quartics, and we concentrate on these.

For a compact Lie group G, E. Witten proposed topological invariants of a threemanifold (quantum G-invariants) in 1988 by using the Chern-Simons functional and the Feynman path integral [30]. See also [2]. N. Yu. Reshetikhin and V. G. Turaev gave a mathematical proof of existence of such invariants for G = SU(2) [28]. R. Kirby and P. Melvin found that the quantum SU(2)-invariant associated to q = exp(2π √ − 1/r) with r odd splits into the product of the quantum SO(3)-invariant and [15]. For other approaches to these invariants, see [3, 4, 5, 16, 22, 27].

In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

Let Q be a loop; then the left and right translations La(x) = ax and Ra(x) = xa are permutations of Q. The permutation group M(Q) = 〈La, Ra | a ε Q〉 is called the multiplication group of Q; it is well known that the structure of M(Q) reflects strongly the structure of Q (cf. [1] and [8], for example). It is thus an interesting question, which groups can be represented as multiplication groups of loops. In particular, it seems important to classify the finite simple groups that are multiplication groups of loops. In [3] it was proved that the alternating groups An are multiplication groups of loops, whenever n ≥ 6; in this paper we consider the finite classical groups and prove the following theorems

In this paper, we characterize the genus of an arbitrary torsion-free finitely generated nilpotent group of class two and of Hirsch length six by means of a finite number of arithmetical invariants. An algorithm which permits the enumeration of all possible genera that can occur under the conditions above is also given.

We study the second Chinburg invariant of a Galois extension of number fields. The Chinburg invariant lies in the class-group of the integral group-ring of the Galois group of the extension. A procedure is given whereby to evaluate the invariant in the case of the real cyclotomic case of regular prime power conductor and their subextensions of p-power degree. The invariant is shown to be zero in the latter cases, which yields new examples giving an affirmative answer to a question of Chinburg ([1], p. 358) which has come to be known as ‘Chinburg's Second Conjecture’ ([3], §4·2).

Let Sn denote the n-dimensional unit sphere. Recently, Bolton, Pedit and Woodward [1, 3] have begun to study a class of minimal surfaces in Sn, which are called superconformal. In this paper we will discuss the rigidity of superconformal minimal surfaces lying fully in S2m−1 among all superconformal minimal surfaces lying fully in odd-dimensional unit spheres.

In 1954, McLain [M] applied some well-known arguments of linear algebra on triangular matrices to establish the existence of characteristically simple, locally finite p-groups, now known as McLain groups [Rob, pp. 347–349]. His groups, having a trivial centre, illustrated sharply the difference between finite and locally finite p-groups. The construction of McLain groups depends on the dense linear ordering (ℚ,≤) and a field Fp of p elements. It was immediately clear that the parameters Fp, ℚ of the McLain group G(Fp, ℚ) could be replaced by other linearly ordered, dense sets S and by other fields F without doing much harm to the construction. If F has characteristic 0, then G(F, S) is still locally nilpotent but torsion-free. Wilson [W] investigated G(Fp, S) for other orderings and Roseblade[R] deeply studied the automorphism group Aut G(Fp, S) in his dissertation at Cambridge in 1963.

In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].