It has been shown by Ratcliffe and White that the effective specific inductive capacity (ε) and the effective conductivity (σ) of soil placed, as dielectric, in a simple condenser vary in a marked manner with frequency. The range of frequency used was from 0·1 × 106 cycles per second to 10 × 106 cycles per second. It was found that at 6·1 × 106 cycles per second the specific inductive capacity was very large (about 40 e.s.u.) and that it decreased to a value of about 12 e.s.u. at 10 × 106 cycles per second. This latter value agrees with the value found in experiments on the propagation of short radio waves over the surface of the earth. The effective conductivity of the soil had a low value at 0·1 × 106 cycles per second but increased rapidly to a constant value for frequencies above 3 × 106 cycles per second.

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

Let denote a Hilbert space (real or complex), with inner product (|). In order to present our notation, we recall that if is a vector subspace of (and ‘vector sub-spaces’ will always be closed), with orthocomplement , a partial isometry V with initial domain is a linear operator in which preserves length, and so inner-product, in and is zero in is the final domain of V, and it is easy to verify that V*, the adjoint operator, is also a partial isometry, with initial domain and final domain .

H. Hilden(3, 4) and the author(9,10) proved independently the following result:

Theorem 1. Each closed, orientable 3-manifold is a 3-fold, dihedral covering of S3, branched over a knot.

The Young–Fibonacci graph [ ][ ] is an important example (along with the Young lattice) of differential posets studied by Fomin and Stanley. For every differential poset there is a distinguished central measure called the Plancherel measure. We study the Plancherel measure and the associated Markov chain, the Plancherel process, on the Young–Fibonacci graph.

We establish a law of large numbers which implies that the Plancherel measure cannot be represented as a nontrivial mixture of central measures, i.e. is ergodic. Our second result claims the convergence of the level distributions of the Plancherel measure to the GEM(½) probability law in the space of nonnegative series with unit sum, which is a particular example of distribution from the class of Residual Allocation Models.

In order to obtain the Plancherel process as an image of a sequence of independent uniformly distributed random variables, we establish a new version of the Robinson–Schensted type correspondence between permutations and pairs of paths in the Young–Fibonacci graph. This correspondence is used to demonstrate a recurrence property of the Plancherel process.

Surfaces of order n in space of n dimensions, for 3 ≤ n ≤ 9, were discussed by Del Pezzo, who showed that the prime sections of such a surface are represented on the plane by cubic curves through (9 − n) base points.

Whitham's exact averaged variational principle is found by a new method. Comparisons are drawn between this method and the method advanced by Whitham. Further results about the Euler equations from this variational principle are obtained.

A method of solution is given to the problem of internal wave motion in a non-homogeneous viscous fluid of variable depth. The approach is based on the inviscid theory of Keller and Mow(l) and on boundary-layer analysis. For internal progressive waves in uniform depths, it reduces essentially to the theory given by Dore(2). The present results are also applicable to surface waves when the fluid is homogeneous.

The head-on interaction, in the one-dimensional, unsteady isentropic flow of a perfect gas, of a simple compression wave and a simple expansion wave is studied by considering typical examples. The physical aspect of the problem is discussed in (1); in this note the possibility of shock formation is ignored, and the correspondence defined by the complete mathematical solution of the equations of isentropic flow between the x, t-plane and the plane of the characteristic variables is elucidated.

The solution is distinguished by the appearance of two limit lines and a second-order limit point where they meet. It is found that the image of the characteristic plane in the x, t-plane is four-sheeted; all sheets overlap each other, but each covers only part of the plane, and the only point common to all sheets is the second-order limit point, where both limit lines are cusped (§ 3·1).

The solution also contains an edge of regression, and a discussion of the properties of this type of singularity will be found in §§ 2 and 2·1.

1. Introduction. Suppose that K is a continuous function on the square Q = [ – 1, 1] x [– 1,1] satisfying , for – 1 ≤ s, t ≤ 1; then the Fredholm operator T on L2(-1,1)

is compact and symmetric. Suppose also that T is a positive operator, i.e.

then there is an eigenfunction expansion

where (λn) is a sequence of non-negative real numbers which decreases to 0 and (φn) is an orthonormal sequence in L2( – 1,1). In this paper we shall find asymptotic estimates for λn when K takes certain specific analytic forms. In all cases K will be real-valued on Q and analytic in a neighbourhood of Q in complex 2-space; for example

The success of the methods of [24] and [4] in investigating the structure of the weakly almost periodic compactification wℕ of the semigroup (ℕ, +) of positive integers has prompted us to see how successful they would be in another context. We consider wG, where G=[oplus ]i∈ωGi is the direct sum of a sequence of finite groups with its discrete topology. We discover a large class of weakly almost periodic functions on G, and we use them to prove the existence of a large number of long chains of idempotents in wG. However, the closure of any singly generated subsemigroup of wG contains only one idempotent. We also prove that the set of idempotents in wG is not closed. The minimal idempotent of wG can be written as the sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of βℕ, we find subsemigroups SF of wG which are ‘carried’ by closed subsets F of βω. wG contains the free abelian product of the semigroups SF corresponding to families of disjoint subsets F. Usually SF is a very large semigroup, but for some points p∈βω, Sp can be quite small.

We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces.

There are, in projective space Σ of three dimensions, two famous quartic surfaces: W, the Weddle surface with six nodes ((13), p. 69, footnote) and K, the Kummer surface with sixteen ((s), p. 246, (7) passim). They are in birational correspondence and have the same non-singular model: the octavic base surface F of the net of quadrics in [5], which contain a given line λ and for which a given simplex S is self-polar ((5); (6)). One naturally takes S, with vertices X0, X1, X2, X3, X4, X5 as simplex of reference for homogeneous coordinates x0, x1, x2, x3, x4, x5; F is invariant under the harmonic inversions hj in the vertices Xj and opposite bounding primes xj = 0 of S. These six hj, mutually commutative and having identity for their product, generate an elementary abelian group of order 32. This representation of throws into prominence what may, in this context, be called its positive subgroup , of order 16, consisting of identity and the 15 products hjhk = hkhj; these are harmonic inversions in the edges XjXk and opposite bounding solids xj = xk = 0 of S. The coset of consists of the six hj and their ten products in threes, complementary products being the same (h0h1h2 ≡ h3h4h5) because of the product of all six hj being identity. These ten products are harmonic inversions in the ten pairs of opposite plane faces of S.

1. Introduction. The problem of determining the distribution of stress in a medium with transverse isotropy (one axis of elastic symmetry) has recently been considered by Elliott (l, 2) and Shield (3). They have obtained solutions to certain special problems both with and without rotational symmetry. Elliott (2) has employed the method of dual integral equations to solve a number of axially symmetric crack and punch problems. (This crack problem is not to be confused with the torsional crack problem treated by Weinstein (4)). Shield (3) has reduced the general crack and punch problems for a medium with transverse isotropy to corresponding isotropic problems as formulated by Green (5), and has treated a few special examples. For the axially symmetric isotropic crack and punch problems the author (6) has shown that Green's formulation may be simplified, and has obtained in each case a simple usable solution of the general axially symmetric problem without the need for dual integral equations. As one might expect these simplifications may also be realized in the case of a medium with transverse isotropy. In this paper the author, guided by the work of Green (5) and Shield (3), presents a straightforward method for solving the general axially symmetric crack and punch problems for media with transverse isotropy. This method permits one to write down immediately in an appropriate system of curvilinear coordinates the solution to the general axially symmetric problem, whereas the method given by Shield requires an individual treatment of each problem. In fact it would appear to be extremely difficult if not impossible to obtain the complete solution to the general axially symmetric crack or punch problem by the methods suggested by Shield. Since the formulation given in this paper differs from that given by Shield, his equations for transforming the problem in transverse isotropy to an isotropic problem must be altered slightly. The solutions to the general axially symmetric problems for a medium with transverse isotropy then follow immediately from the solutions to the corresponding isotropic problems in (6). We shall therefore find it necessary to repeat some of the arguments and results of (6).

In this paper, we classify generic isotropic map-germs arising from the symplectic reduction process relative to a hypersurface (i.e. one-dimensional reduction), up to symplectic equivalence in the C category. These models are open Whitney umbrellas of arbitrary dimension and their suspensions. These singularities appear in the generalized Cauchy problem for HamiltonJacobi equations.

Let E be a Banach space, let (Ω, Σ, μ) a finite measure space, let 1 < p < ∞ and let Lp(μ;E) the Banach space of all E-valued p-Bochner μ-integrable functions with its usual norm. In this note it is shown that E contains a complemented subspace isomorphic to l1 if (and only if) Lp(μ; E) does. An extension of this result is also given.

1·1. Background. Throughout this note Q stands for a finitely generated multiplicative Abelian group of torsion-free rank n, R for a commutative ring with 1, and M for a finitely generated RQ-module. The geometric invariant ΣM of M was introduced in [BS1, BS2]. It can be viewed as a subset of the ℝ-vector space of all (additive) characters of Q, Q* = Hom(Q, ℝ) ≅ ℝn, as follows: for every character χ: Q → ℝ one considers the submonoid Qχ = {q ∈ Q [mid ] χ(q) [ges ] 0} of Q and puts

Note that 0 ∈ ΣM. It is often convenient to work with the complement ΣcM of ΣM in Q*.

The geometric invariant ΣM has been investigated for two reasons. Firstly, if R is a Dedekind domain, then ΣM turns out to be a polyhedral (i.e. a finite union of finite intersection of (open) vector half spaces) subset of Q*. This rather subtle fact was conjectured, for R a field, by Bergman [B] and established by Bieri and Groves in [BG2]; it opens the possibility for computations and imposes arithmetic restrictions on automorphisms of M. Secondly, for R = ℤ, ΣM contains interesting information on the (metabelian) groups G which are extensions of M by Q (i.e. G fits into a short exact sequence M [rarrtl ] G [Rarr ] Q). In [BS1] it is proved that G has a finite presentation if and only if ΣM ∪ −ΣM = Q*. A number of attempts have been undertaken to extend this result to a characterization of the higher dimensional finiteness property that G is of typeA group of G is of type FPm if the trivial G-module ℤ admits a free resolution F [Rarr ] ℤ with finitely generated m-skeleton. For metabelian groups G it is known, by [BS1], that FP2 is equivalent to finite presentability.FPm for m > 2, and they all revolve around the following:

FPm-Conjecture: G is of type FPmif and only if 0 ∈ Q* is not in the convex hull of m points of ΣcM.

The conjecture appeared in [BG1].

We classify smooth $n$-dimensional varieties $X_n\subset{\bb P}^{2n+1}$ with one apparent double point and of degree $d \leqslant 2n + 4$, showing that these are only the smooth irreducible divisors of type (2,1), (0,2) and (1,2) on the Segre manifold ${\bb P}^1\times{\bb P}^n \subset {\bb P}^{2n+1}$, a 3-fold of degree 8 and two Mukai manifolds, the first one of dimension 4 and degree 12, the second one of dimension 6 and degree 16. We also prove that a linearly normal variety $X_n \subset{\bb P}^{2n+1}$ of degree $d \leqslant 2n+1$ and with ${\rm Sec}(X_n) = {\bb P}^{2n+1}$ is regular and simply connected, that it has one apparent double point and hence it is a divisor of type (2,1), (0,2) or (1,2) on the Segre manifold ${\bb P}^1\times{\bb P}^n\subset{\bb P}^{2n+1}$. To this aim we study linear systems of quadrics on projective space whose base locus is a smooth irreducible variety and we look for conditions assuring that they are (completely) subhomaloidal; we also show some new properties of varieties $X_n\subset{\bb P}^{2n+1}$ defined by quadratic equations and we study projections of such varieties from (subspaces of) the tangent space.

The following theoretical investigation of the two-dimensional flow of an inviscid fluid past a keel and rudder, and of the consequent lateral force, follows experiments performed by Prof. T. B. Abell in the Department of Naval Architecture of the University of Liverpool, and we wish to acknowledge our indebtedness to him for the information given in many discussions.

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, then

A converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such that

for every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.