In (13) Wall classified up to diffeomorphism, PL-homeomorphism, topological homeomorphism, and homotopy equivalence all closed, oriented, 1-connected smooth (or PL) 6-manifolds with torsion-free homology and vanishing second Stiefel-Whitney class. This paper extends these classifications to all closed, oriented, 1-connected topological 6-manifolds with torsion-free homology. The method differs from that of (13) but uses two special cases of Wall's classification.

The effect of restricting the displacements of a quantum mechanical harmonic oscillator to semi-infinite and finite regions is studied. It is shown how the eigenvalue problem may be solved, both graphically, and from certain series expansions.

In this paper we consider reduced curves Γ of fixed degree d in the complex projective plane P2([Copf ]): we study the question of how singular such a curve can be.

The inequality which is proved here was established under different conditions in a paper entitled ‘Periodic points on an algebraic variety', which is to appear in the Annals of Mathematics. It is hoped that the new formulation will be more useful for applications. The present investigations use the rather sophisticated techniques of the ‘Theory of distributions’ which was developed by Weil (1). As the content of this theory may be unfamiliar to the reader, Weil's results are described at some length, though of course, his proofs are not given. The one result which we need and which is not given explicitly by Weil, is proved in detail.

Introduction. In (1–3,6) it is shown that homological algebra can be applied to stable homotopy-theory. In this application, we deal with A -modules, where A is the mod p Steenrod algebra. To obtain a concrete geometrical result by this method usually involves work of two distinct sorts. To illustrate this, we consider the spectral sequence of (1,2):

Here each group Extss, t which occurs in the E2 term can be effectively computed; the process is purely algebraic. However, no such effective method is given for computing the differentials dr in the spectral sequence, or for determining the group extensions by which is built up from the E∞ term; these are topological problems.

This note concerns a question of elementary calculus. Given a smooth composite function u = g o f [with values u(x) = g(f(x))], we write explicit formulae for its derivatives, of arbitrary order, in terms of derivatives of f and g. We consider (A) the general case,

in which E, F and G are Banach spaces, and U, V are open sets; (B) the finite-dimensional case E = ℝM and F = ℝN, where ℝM denotes real M-dimensional Euclidean space; and (C) the particular case of (B) (due to restricting g to part of an M-dimensional surface in ℝM + 1) in which N = M + 1 and u(x)= g(x, φ (x)), so that φ denotes a real-valued (scalar-valued) function of x = (x1, …, xM).

It has now been almost twenty years since Alperin introduced the idea of the complexity of a finitely generated kG-module, when G is a finite group and k is a field of characteristic p > 0. In proving one of the first major results in the area [1], Alperin and Evens demonstrated the connection of the study of complexity for modules to the group cohomology. That connection eventually led to the categorization of modules according to their associated varieties in the maximal ideal spectrum of the cohomology ring H*(G, k). In all of the work that has followed, two principles have proved to be extremely important. The first is that the associated variety of a module is directly related to the structure of the module through the rank variety which is defined by the matrix representation of the module. The second major result is the tensor product theorem which says that the variety associated to a tensor product M ⊗kN is the intersection of the varieties associated to the modules M and N. In this paper we generalize these results to infinitely generated kG-modules.

Alexander [1] showed that an oriented link K in S3 can always be represented as a closed braid. Later Markov [5] described (without full details) how any two such representations of K are related. In her book [3], Birman gives an extensive description, with a detailed combinatorial proof of both these results.

It is shown that the phenomena of double refraction—optical activity, Faraday rotation, Kerr effect, etc.—may be treated as scattering problems in quantum field theory. An expression for the scattering cross-section of an atom or molecule for photons may be easily obtained and from this expression using the idea of Stokes operators, equating their expectation values to the classical Stokes parameters, the usual quantum mechanical expressions for optical activity, etc. may be obtained. This approach does not require the ideas of refractive index or polarization of the medium, and emphasizes what are the actual observables in these experiments.

This paper gives an existence result for the discrete coagulation-fragmentation equations:

(If k = 1, the first and last sums are 0.)

We consider arbitrary polynomial perturbations

formula here

of the harmonic oscillator. In (1), f and g are polynomials of x, y with coefficients depending analytically on the small parameter ε. Let us denote n = max (deg f, deg g), H = ½(x2 + y2). Using the energy level H = h as a parameter, we can express the first return mapping of (1) in terms of h and ε. For the corresponding displacement function d(h, ε) = [Pscr ](h, ε)−h we obtain the following representation as a power series in ε:

formula here

which is convergent for small ε. The Melnikov functions Mk(h) are defined for h[ges ]0. Each isolated zero h0∈ (0, ∞) of the first non-vanishing coefficient in (2) corresponds to a limit cycle of (1) emerging from the circle x2 + y2 = 2h0 when ε increases from zero. Our main result in this paper is the following.

A solution of Einstein's vacuum field equations, apparently new, is exhibited. The metric, which is homogeneous (that is, admits a three-parameter group of motions transitive on space-like hypersurfaces), belongs to Taub Type V. The canonical form of the Riemann tensor, which is of Petrov Type I, is determined.

This paper presents a uniqueness result for solutions to the general nonlinear coagulation-fragmentation equation

where

Equation (1·1) has many applications in the applied sciences (cf. [1, 3, 8, 13, 15]) and a brief physical interpretation can be found in Melzak [12] or the survey article by Drake[7]. c(x, t), for x ≥ 0, t ≥ 0, denotes the number of particles of size x at time t and the non-negative kernels K and F describe, respectively, the rates at which particles of size x coalesce with those of size y and particles of size (x + y) break-up into those of sizes x and y.

Let Γ be an undirected graph and G a subgroup of aut (Γ) acting transitively on the vertex set V(Γ) of Γ. Let x be an arbitrary vertex of Γ. We denote by T(x) the set of vertices adjacent to x and by G(x)Γ(x) the permutation group induced by the stabilizer G(x) of x in G on Γ(x); G(x)Γ(x) is called the subconstituent of G (with respect to Γ). Let G1(x) = {a ∈ G(x)|a ∈ G(y) for each y ∈ Γ(x)}. For each y ∈ Γ(x), let G(x, y) = G(x) ∩ G(y) and G1(x, y) = G1(x) ∩ G1(y). An s-path is an (s+ l)-tuple (x0, x1, …, xs) of vertices such that xi−1 ∈ Γ(xi) if 1 ≤ i ≤ s and xi−2 ≠ xi if 2 ≤ i ≤ s. Γ is called (G, s)-transitive if G acts transitively on the set of all s-paths but intransitively on the set of all (S+1)-paths in Γ.

Positive definite functions on metric spaces were considered by Schoenberg (26). We write σk for the unit hypersphere in (k + 1)-space; then σk is a metric space under geodesic distance. The functions which are positive definite (p.d.) on σk were characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.

A graph consists of a finite set of points and a set of lines joining some pairs of these points. At most one line is permitted to join any two points and no point is joined to itself by a line. A graph G′ is a subgraph of the graph G if the points and lines of G′ are also points and lines of G. The union of several graphs having the same set of points is the graph formed by joining two points in this set if they are joined in at least one of the original graphs. A graph is planar if it can be drawn in the plane (or equivalently, on a sphere) so that no lines intersect. The thickness of a graph G is defined as the smallest integer t such that G is the union of t planar subgraphs.

Let d be a Jordan derivation from a ring $\cal{A}$ into an $\cal{A}$-bimodule $\cal{M}$. Our main result shows that the restriction of d to the ideal of $\cal{A}$ generated by certain higher commutators of $\cal{A}$ is a derivation. This general statement is used for proving that under various additional conditions d must be a derivation on $\cal{A}$. Furthermore, several examples of proper Jordan derivations are given, $C^{\ast}$-algebras admitting proper additive jordan derivations are characterized, and the connections with the related problems on jordan homomorphisms and jordan $\cal{A}$-module homomorphisms are discussed.

It is shown that around each point of a C3 static vacuum space-time there is a neighbourhood in which the metric is, in a certain suitable coordinate system, analytic. If two of these neighbourhoods overlap, then the transformation betweea these suitable coordinate systems is analytic (in the overlapping domain).

A uniform asymptotic solution is obtained which describes the propagation near an isolated caustic of water waves on a slow irrotational current. Unlike earlier work there is no restriction to straight caustics or to steady currents.

We consider the Vlasov—Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular centre and have isotropic or anisotropic pressure, and solutions which have a Schwarzschild-singularity at the centre.