When a stream is flowing along a wall against an adverse pressure gradient, the skin friction at the wall decreases to zero, at which point or at least very near it the flow seems to leave the surface altogether. Between the main stream and the wall there may be a slow back-flow. Upstream of separation the influence of the wall on the flow is satisfactorily explained by the boundary-layer theory, which among other things presupposes that this influence is confined to a narrow region near the wall. Such a theory is clearly inadequate downstream, however, because this layer rapidly increases in width and, further, the main stream is seriously affected, and consideration of the Navier-Stokes equations would seem to be necessary before a full theoretical explanation of separation could be given. Apart from a note by Dean (2) on the relations between the pressure and velocity at separation, there has, however, been no relevant attack on these equations.

In this paper upper bounds are established for the principal solution of the differential equation

and its derivative, for unrestricted values of the complex variable t and the complex parameter μ. The results may have little interest in their own right, but they are of great value in developing the asymptotic theory of linear second-order differential equations in a domain containing two turning points. Equation (1·1) is the simplest example of a differential equation of this type.

Introduction. The present paper is concerned with the conformal geometry of Hermitian spaces. In the first part we find a necessary and sufficient condition for a Hermitian space to be conformally Kähler, that is, conformal to some Kähler space. The condition is that a certain conformal tensor, , vanishes identically. Then, defining a Hermitian manifold as in Hodge (3), we consider such a manifold where the restriction is made that at every point the tensor is zero. This will be called a conformally Kähler manifold, and conditions under which it may be given a Kähler metric are obtained. It is found that any conformally Kähler manifold may be given a Kähler metric provided it is simply-connected or that its fundamental group is of finite order.

1. Notation. For an arbitrary set X, |X| will denote its cardinal number. Rn and Tn will denote the n-dimensional real vector space and the n-dimensional torus respectively with the usual topologies.

Atiyah(2) defined the geometrical dimension of an element to be less than k + 1 (g dim (x) ≤ k) if there is a k-dimensional bundle over X whose stable equivalence class is x. If ξ is an n-plane complex bundle over X, we say that it has r sections if there is an (n − r)-plane bundle η such that ξ is isomorphic to η ⊕ εr where εr is the trivial r-plane bundle over X. If X has dimension 2n or less, then ξ has r sections if and only if g dim (ξ − n) ≤ n − r.

For each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.

Let A be a commutative domain, finitely generated as an algebra over a field k of characteristic zero and write (A) for the ring of k -linear differential operators. Then A is an Ore domain with quotient division ring, say Q. Our main result is that A is a maximal order in Q if and only if (i) A = ∩{Ap: height (p) = 1} and (ii) A is geometrically unibranched. In this case A is also a Krull domain with no reflexive ideals. We also determine some conditions under which A is simple.

We give examples which (i) disprove a conjecture of Sheil-Small regarding the valence of harmonic mappings of the unit disc to bounded convex domains, and (ii) answer negatively a question of the author regarding the valence of harmonic mappings with polynomial analytic and co-analytic parts.

In classical general relativity, the timelike convergence condition is Ric(v, v) ≥ 0 for every timelike vector v. But recent astronomical observations show that the timelike convergence condition does not hold. In this paper, we obtain an explicit upper bound of the volume form and the distance to a focal point (such as Hubble distance) along timelike geodesic with integrals of the negative part of Ricci curvature, which entails singularity theorems.

This paper continues the study initiated in [10] of the Markov branching process (MBP) subject to an independent Poisson process of catastrophe times at which the population size is reduced by random decrements whose distribution is independent of the current population size. More generally, let (Xt:t ≥ 0) be the Feller process on the non-negative integers ℕ+ having the generator

The formation of 62Cu, 106Ag and 120Sb by deuteron bombardment of copper, silver and antimony respectively is reported. The energy-yield curves for these reactions were determined and it is concluded that in each case the process involved is taking place, in all probability, through the formation of a ‘compound nucleus’ of the Bohr type.

By a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance under the local moves. As an example of such invariants, we present the Dijkgraaf–Witten invariant of 3-manifolds.

Duskin [1, 2] showed that gerbes over a Grothendieck topos E can be interpreted by certain internal groupoids of E which he called bouquets. He proved that for any bouquet B of E the fibred category F ors(B) of B-torsors is a gerbe over E, and conversely that any gerbe G gives rise to a bouquet B satisfying G ≅ F ors(B). We want to give here a somewhat different approach to these results, based on a construction of gerbes given in [3].

We give an elementary construction of the compact real form of the Lie algebra 2. This construction exhibits the group 23·L3(2) as a group of automorphisms. We also show that there is a unique 14-dimensional real Lie algebra invariant under the action of this group.

The notion of cycle-free partial order (CFPO) was defined in [9] and the class of sufficiently transitive CFPOs was investigated, and in many cases, classified. In [10] a complete classification was given for the countable 3- or 4-CS-transitive CFPOs having a finite chain and embedding an infinite ‘alternating chain’ ALT. Here, the major case is that of the so-called ‘skeletal’ CFPOs, in which the CFPO itself forms a mere skeleton of the structure of the whole picture, which is more accurately provided by its Dedekind–MacNeille completion. It was asserted that a similar classification should also be possible in the infinite chain case (still with the countability restriction), and it is the object of the present paper to finish this task.

The overall plan of the work is similar in style to [10]. The CFPOs we study can all be construed as arising from a chain, suitable ‘adorned’ with instructions as to how to branch (and repeatedly). An obvious difference from the finite chain case is that, this time, many of the points in the chain will actually survive as elements of the structure; whereas in [10], since the chains of the structure had to have length 2, only the endpoints would. On the whole, the treatment is parallel, with just a small number of additional cases.

We recall the main ideas from [10], quote results that are needed, and concentrate on aspects of the structures which are new or specific to the infinite chain case. We conclude by remarking on some direct connections between the finite and infinite chain cases.

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [6] for basic definitions and elementary results.

In this paper, the author (1) compare subnormal closures of finite sets in a free group F; (2) obtains the limit for the series of subnormal closures of a single element in F; (3) proves that the exponential growth rate (exp.g.r.) $\lim_{n\to \infty}\sqrt[n]{g_H(n)}$, where gH(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (4) gives sharp estimates from below for the exp.g.r. of subnormal subgroups in free groups; and (5) finds the cogrowth of the subnormal closures of free generators.

In the early seventies, Steenrod posed the question: which polynomial algebras over the Steenrod algebra appear as the cohomology ring of a topological space? For odd primes, work of Adams and Wilkerson and Dwyer, Miller and Wilkerson showed that all such algebras are given as the mod-p reduction of the invariants of a pseudo reflection group acting on a polynomial algebra over the p-adic integers. We show that this necessary condition is also sufficient for finding a realization of such a polynomial algebra. We also show that, up to completion, every space with polynomial mod-p cohomology is equivalent to the product of the classifying space of a connected compact Lie group and some spaces determined by irreducible p-adic rational pseudo reflection groups.

It has been known for some time ((6), p. 270; (4), theorem 9·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite objects in is again a topos. For a non-Boolean topos , however, Kf need not be a topos, as can be seen when is the Sierpinski topos ((1), example 7·1); on the other hand, two other full subcategories of , coinciding with Kf when is Boolean, suggest themselves as candidates for a subtopos of finite objects. Of one of these, the category dKf of decidable K-finite objects in , the Main Theorem of (1) asserts that it is always a (Boolean) topos. The other is the category sKf of -subobjects of K-finite objects. The inclusions

dKf ⊆ Kf ⊆ sKf are clear.

are clear.

The following gives a brief summary of a paper, which it is hoped to publish in extenso later. It is a contribution to the theory of ferromagnetic crystals, and contains in particular a theoretical explanation of Webster's experimental results.