We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

In recent years many papers have dealt with the zero-dimensional multiple point sets of smooth immersions of closed manifolds in Euclidean spaces (see, for example, [2, 6, 7, 8, 9, 10, 13, 14, 15, 21]). In the present paper we deal with the case when the double points form a two-dimensional surface and consider the following question: Which surfaces can occur as double point surfaces of self-transverse immersions of closed n-manifolds in ℝ2n−2? We prove the following

1. The classical Kneser conjecture states that if M is a closed 3-manifold and π1(M) ≈ A * B, a free product, then there exist 3-manifolds MA, MB such that

and π1(MA) ≈ A, π1(MB) ≈ B. This has been proved by John Stallings(6).

Although the classification of affine cubic curves was undertaken by Newton(4), in one of the first major exercises ever in coordinate geometry (see Cayley(2) for a fuller account), a parallel study of cubic functions seems not to have been contemplated till recently. The essential difference is that a function f defines a pencil of curves – its level curves – which have to be considered simultaneously. The author's interest in the subject arose from problems in singularity theory (concerning canonical stratifications), and a later paper in the series will have applications of this kind. Here we study the simplest case as an introduction. Our techniques are entirely classical, but the results are hard to find elsewhere. In later papers we intend to study cubic functions on ℝ2, and on ℂ3.

Let X be a normed linear space and let W be a proper subspace of X. A projection is a surjective linear map P: X → W such that P is idempotent. It is immediately clear that P has norm at least unity. Thus the problem of calculating the number

has some interest. The number λ(W, X) is often called the relative projectiion constant of W in X. If the infimum is attained, any attaining projection is called a minimal projection. The problems of calculating λ(W, X) for a fixed X and W or finding a minimal projection turn out to be very dificult. For example, if X = C [0, 1] with the usual supremem norm and W is the subspace of polynominals of degree at most two then λ(W, X) remains unknown as does any example of a minimal projection. One of the few places where the problem shows much tractability is the case

1. The problem to be discussed in the present paper arises when the finite resolving time of a recording apparatus is taken into account. Events are divided into two classes, recorded and unrecorded. Any recorded event is followed by a dead interval of a certain length of time, during which any other event which occurs will be unrecorded. A typical example is an α-particle counter; a recorded particle causes the chamber to ionize, and no other particle can be recorded until the chamber has deionized. In the case when the dead interval is of constant length τ, if the observation time t lies between (n − 1)τ and nτ, the number of recorded events must be 0, 1, 2, …, n − 1 or n. We shall assume that the probability of an event occurring in the interval [t, t + dt] is λdt, λ being constant; this is the case of most practical interest. The probability distribution of recorded events for dead intervals of constant length has been determined by Ruark and Devol (2). The explicit expression of this distribution is fairly complicated, and it is therefore difficult to manipulate. The method employed in the present paper leads naturally to the Laplace transform of the distribution with respect to the time t, and this is relatively simple. The method can easily be generalized to deal with the case of dead intervals which are not all equal, but follow a probability distribution u(τ) dτ. Finally the Laplace transform is very convenient for determining the asymptotic behaviour of the distribution of recorded events for large times of observation.

We prove global and local weighted stability theorems for representations of abelian semigroups on Banach spaces under countable spectral conditions and certain growth assumptions on the weights. In the global case, we use a limit semigroup construction together with an extension theorem for weighted semigroup representations. In the local case, we adapt a definition of Albrecht to introduce a local spectrum of a representation which is no larger than the usual notion of local spectrum for representations of Z+ and R+, and we establish the corresponding local stability theorem. We give an application to the asymptotic theory of functions on abelian semigroups.

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex body.

A family ℳ of closed circular discs in the plane is called a saturated family or a saturated system of circles if (i) the infimum r of the radii of the discs in ℳ is positive and (ii) every closed disc of radius r in the plane intersects at least one disc in ℳ. For a saturated family ℳ, we denote by S the point-set union of the interiors of the members of ℳ and by S(l) the part of S inside the circular disc of radius l centred at the origin. We define the lower density ρℳ of the saturated family ℳ as

where V(S(l)) denotes the Lebesgue measure of the set S(l).

Lagrangian coordinates are used in conjunction with the self-similarity hypothesis in order to examine the problem of the vertical entry at constant speed of a wedge and also of a cone into an incompressible fluid initially at rest. Certain known properties of the free surface are recovered in a very direct and simple manner and new exact results concerning the inclination of the free surface and the angle of contact with the rigid surface are obtained.

Basis functions are derived and tabulated for the double-valued representations of the triclinic, monoclinic, orthorhombic, trigonal, tetragonal and hexagonal crystallographic point groups using the method of projection operators.

An interesting recent paper by Falk and Kaufmann[11] notes, with an element of surprise, that the percentile bootstrap applied to construct confidence intervals for quantiles produces two-sided intervals with coverage error of size n−½, where n denotes sample size. By way of contrast, the error would be O(n−1) for two-sided intervals in more classical problems, such as intervals for means or variances. In the present note we point out that the relatively poor performance in the case of quantiles is shared by a variety of related procedures. The coverage accuracy of two-sided bootstrap intervals may be improved to o(n−½) by smoothing the bootstrap. We show too that a normal approximation method, not involving the bootstrap but incorporating a density estimator as part of scale estimation, can have coverage error O(n−1+∈), for arbitrarily small ∈ > 0. Smoothed and unsmoothed versions of bootstrap percentile-t are also analysed.

The geometry of a desingularization Ym of an arbitrary subvariety of a generic hypersurface Xn in an ambient variety W (e.g. W = ℙn+1) has received much attention over the past decade or so. Clemens [5] has proved that for m = 1, n = 2, W = ℙ3 and X of degree d, Y has genus g [ges ] 1 + d(d − 5)/2; Xu [13, 14] improved this to g [ges ] d(d −3)/2 − 2 for d [ges ] 5 and showed that if equality holds for d [ges ] 6 then Y is planar; he also gave some lower bounds on the geometric genus pg(Y) in case m = n − 1. Voisin [10, 11] proved that, for X of degree d in ℙn+1, n [ges ] 3, m [les ] n − 2 then then pg(Y) > 0 if d [ges ] 2n + 1 − m and KY separates generic points if d [ges ] 2n + 2 − m (see also [1, 4]). For Xn a generic complete intersection of type (d1, …, dk) in any smooth polarized (n + k)-fold M, Ein [6, 7] proved that pg(Y) > 0 if d1 + … + dk [ges ] 2n + k − m + 1 and Y is of general type if d1 + … + dk [ges ] 2n + k − m + 2.

In [2] the first two authors applied the classical method of focal loci of families as in Segre [8] and Ciliberto–Sernesi [3] (really normal bundle considerations) to give a new proof of one of the main results of Xu in [13, 14] and to extend the lower bounds for the genus in the cases of general surfaces in a component of the Noether–Lefschetz locus in ℙ3 and of general projectively Cohen–Macaulay surfaces in ℙ4.

In this paper we introduce some notions of ‘filling’ subvarieties and use them to prove two new results which inter alia give another perspective on the above results, especially the genus bounds. The general philosophy is as usual that as Y moves with X, sections of KY (or some twist) can be produced through differentiation; but here this is implemented by exploiting and extending an elementary but perhaps surprising technique in the spirit of classical projective geometry which goes back to [2] and which gives a useful lower bound, depending on the dimension of the projective span of Y, on the number of independent sections of KY produced by the differentiation process.

As to our results, in Theorem 1 below we extend and refine in the above sense most of the above-quoted results (not including Voisin's), by giving a lower bound on the number of sections of a certain twist of KY involving KW (for a recent extension, including the proof of a conjecture of Clemens, see [15]). Our second result (Theorem 2), based on the notion of ‘r-filling’ subvariety, indicates an apparently new and unexpected direction as it deals with some higher-order tensors on Y, manifested in the form of an effective divisor on a Cartesian product Yr having certain vanishing order on a diagonal locus as well as on a ‘double point’ locus associated to the map Y → X. As one application, we conclude a lower bound on the number of quadrics (and higher-degree hypersurfaces) containing certain ‘adjoint-type’ projective images of Y (and even on the dimension of the kernel of certain ‘symmetric Gaussian’ maps) (Corollary 2·1 below). Our feeling, however, is that the latter is only the tip of an iceberg and we hope to explore further in this direction in the future.

The physical significance of the parameters which appear in the combinatory factors g(Ni) and g(Ni, Xij) in the statistical mechanics of mixtures is examined. Hitherto, closed ring molecules have been excluded from the jurisdiction of the formulae for these combinatory factors. The means by which this restriction on the validity of the formulae can be removed is indicated. It is concluded that the equations given by Guggenheim are formally applicable in all cases provided that appropriate values, which differ according as the molecules are simple and branched chain or closed ring molecules, are substituted for certain parameters which occur in the equations.

For any prime number p, let Γn, p denote the congruence subgroup of SLn(ℤ) of level p, i.e. the kernel of the surjective homomorphism fp: SLn(ℤ) → SLn(p) induced by the reduction mod p (Fp is the field with p elements). We define

using upper left inclusions Γn, p ↪ Γn+1, p. Recall that the groups Γn, p are homology stable with M-coefficients, for instance if M = ℚ, ℤ[1/p], or ℤ/q with q prime and q ╪ p: Hi(Γn, p; M) ≅ Hi(Γp; M) for n ≥ 2i + 5 from [7] (but the homology stability fails if M = ℤ or ℤ/p).

1. Introduction

In a recent paper [3], an extended Liouville–Green formula

formula here

was developed for solutions of the second-order differential equation

formula here

Here γM(x)∼Q−¼(x), QM(x)∼Q−½(x) and εM(x)→0 as x→∞, while M([ges ]2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M−1. The general form of (1·1) had been obtained previously by Cassell [5], [6], [7] and Eastham [10], [11, section 2·4]. In particular, the proof of (1·1) in [10] and [11] depended on the formulation of (1·2) as a first-order system and then on a process of M repeated diagonalization of the coefficient matrices in a sequence of related differential systems.

Classical solutions of the spherically symmetric Nordström–Vlasov system are shown to exist globally in time. The main motivation for investigating the mathematical properties of the Nordström–Vlasov system is its relation to the Einstein–Vlasov system. The former is not a physically correct model, but it is expected to capture some of the typical features of the latter, which constitutes a physically satisfactory, relativistic model but is mathematically much more complex. We show that classical solutions of the spherically symmetric Nordström–Vlasov system exist globally in time for compactly supported initial data under the additional condition that there is a lower bound on the modulus of the angular momentum of the initial particle system. We emphasize that this is not a smallness condition and that our result holds for arbitrary large initial data satisfying this hypothesis.

Let f:[0, 1]→R2 be a Jordan arc, and for t, u ∈ [0, 1] let d(t, u) = d(f(t), f(u)) denote the Euclidean length of the chord between f(t) and f(u), and l(t, u) = l(f(t), f(u)) the corresponding arc-length, when this is defined. We say that f has the increasing chord property if d(t2, t3) ≤ d(t1, t4) whenever 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ 1. In connexion with a problem in complex analysis, K. Binmore has asked (private communication, see (1)) whether there exists an absolute constant K such that

.

Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically bounded by the 4th root of the eigenvalue. It turns out that the number of nodal domains where the eigenfunction has an extremum of such order, remains bounded as the eigenvalue tends to infinity. We also observe that certain restrictions on the distribution of nodal extrema and a version of the Courant nodal domain theorem are valid for a rather wide class of functions on surfaces. These restrictions follow from a bound in the spirit of Kronrod and Yomdin on the average number of connected components of level sets.

1. The determination of potentials and wave functions defined in regions bounded by the natural coordinate surfaces of a cylindrical polar coordinate system leads to eigenvalue problems connected with Bessel's differential equation

We shall consider the annulus 0 < a ≤ r ≤ b and take first the boundary condition to be that of vanishing on both surfaces. It is then convenient to introduce the function

This function will automatically satisfy the condition of vanishing for r = a. It will also vanish on r = b if u and k are related by the condition that ψ(u, k, b) = 0. If we regard u as being fixed and real then there will be an infinite number of zeros kn(u)(n = 1,2,…).