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.

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.

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.

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].

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.

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formula

is the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

Equivariant shape theory is an improvement of equivariant homotopy theory which could be regarded as an equivariant version of Borsuk's shape theory. The main result in this paper is a description of the equivariant shape category ShG whose objects are equivariant spaces or G-spaces, i.e. topological spaces endowed with an action of a given topological group G, and whose morphisms are equivariant homotopy or G-homotopy classes of families of multi-valued functions which we call equivariant multi-nets or G-multi-nets. Previously equivariant shape theories have been described only under the assumptions that the group G is either finite or compact. We also study classes of G-spaces on which equivariant shape and equivariant homotopy coincide, look for conditions under which a G-map f: X → Y is an equivariant shape equivalence, and give some characterizations of G-spaces with trivial equivariant shape.

In the study of polynomials acting on Banach spaces, the weak topology is not such a good tool as in the case of linear operators, due to the bad behaviour of the polynomials with respect to the weak convergence. For example,

is a continuous polynomial taking a weakly null sequence into a sequence having no weakly Cauchy subsequences. In this paper we show that the situation is not so bad for unconditional series. Recall that is a weakly unconditionally Cauchy series (in short a w.u.C. series) in a Banach space E if for every f ε E* we have that and is an unconditionally converging series (in short an u.c. series) if every subseries is norm convergent.

Let ρ,ρ0,ρ1 be positive, measurable functions on ℝN. For 1 ≤ t < ∞, consider the weighted Lebesgue and Sobolev spaces

For ξ ε (0, ½) let Eξ be the perfect symmetric set of constant ratio ξ and set

It was shown by the first author that if T is a contraction on the Hilbert space H with spectrum contained in Eξ, and if log ∥T−n∥ = O(nα) as n → ∞ for some α < b(ξ), then T is unitary. In the other direction, we show here that there exists a (non-unitary) contraction T on H such that SpT = Eξ, log ∥T-n∥ = O(nb(ξ)) as n → ∞, and lim supn → ∞ ∥T−n∥ = ∞.

Suppose that D is a plane domain and that f(z), g(z) are meromorphic in D and f(z) ╪ g(z) for all z in D. Then following Rubel and Yang [11], we say that f(z) avoids g(z) in D. A system of functions g1(z), …, gn(z) is said to be unavoidable if, whenever f is meromorphic in D, at least one of the equations f(z) = gv(z) has a root in D. Rubel and Yang [11] proved that if D is the open plane, then any two functions form an avoidable system, but three distinct polynomials a1, a2, a3 such that a1−a2 and a2−a3 are not both constant form an unavoidable system.

We show that the Liouville-Arnol'd theorem concerning knowledge of involutory first integrals for Hamiltonian systems is available for any system of second order ordinary differential equations. In establishing this result we also provide a new proof of the standard theorem in the setting of non-autonomous, regular Lagrangian mechanics on the evolution space ℝ × TM of a manifold M. Both the original theorem and its generalization rely on a certain bijection between symmetries of the system and its first integrals. We give two examples of the use of the theorem for systems on ℝ2 which are not Euler-Lagrange.

This paper shows that the localization theory for algebras of smooth functions and Fréchet modules provides a basis for a theory of differential spaces of finite type, as well as the usual localization theory for commutative rings and modules grounds the theory of schemes. Moreover, most of the elementary results and proofs in the theory of schemes remain valid for differential spaces.