This paper is an investigation of the short-wave asymptotic motion due to a cylinder heaving at the surface separating two infinite liquids. The cylinder is assumed to have a smooth cross-section of an arbitrary shape that intersects the surface of separation at right angles. A non-rigorous argument is used to get the asymptotic expansion of the velocity potentials of the motion in the two liquids in the far field. The asymptotic evaluation of the coefficients describing wave-making, virtual-mass and damping are also obtained in terms of the limit potentials of the motions in the two liquids.

Let T be an operator on a complex Banach space such that sup {‖Tn‖: n ≥ 0} < ∞; such operators are said to be power-bounded. For such an operator one defines the peripheral spectrum to be the set

where σ(T) denotes the spectrum of T, and T is the unit circle in ℂ. Esterle proved in [3] that limnn→∞ ‖Tn−Tn+1‖ = 0 provided that ‖T‖ ≤ 1 and σ{T} = {1}. Later Katznelson and Tzafriri [6] proved the same result for power-bounded operators such that σΓ(T) = {1}, and in fact they proved a more general result. More precisely, let be an absolutely convergent Taylor series, i.e. , and assume that f is of spectral synthesis for σΓ(T) (in particular f|σΓ(T) ≡ 0). Then limn→∞ ‖f(T) Tn‖ = 0. Katznelson and Tzafriri prove an analogous result, in which T is replaced by a commuting k-tuple (T1, T2, …, Tk) of contractions. In this case they consider functions of the form f(T1, T2, …, Tk), where f is either an absolutely convergent Taylor series if k is odd, or an element of a smaller algebra if k is even. The result of Katznelson and Tzafriri was further extended in the case of a single contraction T by Esterle, Strouse and Zouakia[4], who consider functions f(T) where f is allowed to be in a more general function algebra. For instance, if T acts on a Hilbert space, they allow f to be an element of the disk algebra.

We characterize those countable rooted trees with non-trivial components whose full automorphism group has uncountable strong cofinality, and those whose full automorphism group contains an open subgroup with ample generics.

Asymptotic analysis and distribution theory are two very useful branches of Applied Mathematics. It is only recently, however, that the close ties between these theories have been studied in detail [5, 6, 11, 17].

Among the various approaches to the distributional theory of asymptotic expansions [6] gives an analysis based on the moment asymptotic expansion.

We construct certain algebraic integers $\alpha_n$ as special values of two variable theta functions in the ray class field of a certain quartic field modulo $2^n$, and study a property of prime ideals which appear in $\alpha_n$ in connection to the relationships between cyclotomic units and exponential functions and between elliptic units and elliptic theta functions.

We investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

Let (E, C) be a partially ordered vector space with positive cone C. The order-bound topology Pb(6) (order topology in the terminology of Schaefer(9)) on E is the finest locally convex topology for which every order-bounded subset of E is topologically bounded.

We shall understand the norm, h(A) of a regular matrix A = (amn) to be

The method has a norm, ‖ ‖, ‖ ‖ ≥ 1, given by ‖ ‖ = inf h(A) where the inf is taken over all the matrix methods equivalent to for bounded sequences (b-equivalent). These definitions are due essentially to Brudno(1) though his definition of

The theory of surfaces and of manifolds of higher dimension is greatly handicapped by the lack of exact information concerning the extension of the theorem of Riemann-Roch for curves. Allied to this is the difficulty of finding an expression for the arithmetic genus Pa of primals in terms of their defining elements. This paper seeks to examine certain particular applications of the theorem.

It is shown that the well-known connection between the second order linear differential equation h″ + B(z)h = 0, with a solution base {h1, h2}, and the Schwarzian derivativeof f = h1/h2, can be extended to the equation h(k) + B(z) h = 0 where k ≥ 2. This generalization depends upon an appropriate definition of the generalized Schwarzian derivative Sk(f) of a function f which is induced by k−1 ratios of linearly independent solutions of h(k) + B(z) h = 0. The class k(Ω) of meromorphic functions f such that Sk(f) is analytic in a given domain Ω is also completely described. It is shown that if Ω is the unit disc or the complex plane , then the order of growth of f ∈ k(Ω) is precisely determined by the growth of Sk(f), and vice versa. Also the oscillation of solutions of h(k) + B(z) h = 0, with the analytic coefficient B in or , in terms of the exponent of convergence of solutions is briefly discussed.

In Part I of this series it was shown that in a metal with overlapping energy bands, one of which is nearly full and the other nearly empty, there exists a critical temperature below which spontaneous magnetization will be present, Tc being given approximately by the equations

where kθ is the exchange energy term postulated by Stoner, b is the ratio of excess vacancies in the d-band to the maximum possible number of parallel spins, and ζd0 is the equivalent energy depth of the vacant levels in the d-band.

Sturm's theorem was announced in 1829 and various proofs were given soon after (see [2], p. 363 for details). It is given in various textbooks on algebra, e.g. in [1], pp. 448–453, where its application to finding the number and location of the real zeros of a polynomial is explained. It has been applied, in conjunction with Givens's method, to finding the eigenvalues of a matrix (see, e.g. [3], p. 494), and has been used recently in a neat computational form by Takeuchi [4] for investigating algebraic fields.

Let $N$ be a prime number, and $X$ a curve that is intermediate to the cover $X_1(N)\rightarrow X_0(N)$. We study the automorphism group of $X$, and prove that in most cases it is generated by the Galois group of the cover $X\rightarrow X_0(N)$, and any lift of the Atkin–Lehner involution to $X$.

If a receiving aerial is placed in the field of a wave produced by a distant transmitter, currents are induced in the receiving aerial and these dissipate energy in the ohmic resistances of the aerial. This energy may be thought of as being absorbed directly from the incident wave, but this viewpoint gives no insight into the mechanism by which the absorption takes place. In this paper the following alternative way of regarding the phenomenon is considered. The currents induced in the aerial cause it to re-radiate a secondary wave. The electric and magnetic fields of this secondary wave combine with those of the unabsorbed incident wave to give resultant fields which have, in general, a direction different from those of the incident wave. These resultant fields give rise to energy flows which are different from those in the incident wave, both in magnitude and direction. In this way a flow of energy into the aerial is produced, by the superposition of the incident and the re-radiated fields. It is important to notice that if this viewpoint is adopted we must add the secondary re-radiated wave to the unabsorbed main wave, the addition, in itself, producing the requisite absorption.

The quantal quartic oscillator, characterized by two real and two complex conjugate transition points (simple zeros), is studied by means of the phase-integral method developed by Fröman and Fröman, and various quantization conditions are obtained. The main results, obtained in §3·3 and §4, are summarized below.

A correction to the generalized Bohr–Sommerfeld quantization condition, due to the complex conjugate transition points, is obtained from estimates of the. F-matrix for a path passing above the complex conjugate transition points. This quantization condition is closely related to the quantization condition for a double-oscillator. The correction is shown to be rigorously valid when the distance between the real transition points is large compared to the distance between the complex conjugate ones, which may lie at an unspecified distance from each other.

At sufficiently large distances from the cluster of four transition points the solutions of the Schrödinger equation can be characterized by the Stokes constants between anti-Stokes lines emerging from the cluster in six directions towards infinity. There are initially six non-trivial such Stokes constants in the problem under consideration, but, from the results in previous papers by N. Fröman and the present authors, these Stokes constants are linked by a number of algebraic relations, so that one or two Stokes constants suffice to describe the solutions far away from the cluster of transition points. The quantization condition is expressed in terms of these Stokes constants. Approximate values of the relevant Stokes constants are given in the three limiting cases when all four transition points coalesce (and the first-order phase-integral approximation is used), when the distance between the real transition points is sufficiently large (extreme double-oscillator), and in the Bohr-Sommerfeld limit.

We give examples of isospectral non-isometric surfaces of genus two and three with variable curvature, as well as hyperbolic orbifolds of genus two. We apply the first result to construct isospectral potentials on a Riemann surface of genus two.

In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.

The problem of classifying line fields or, equivalently, Lorentz metrics up to homotopy is studied. Complete solutions are obtained in many cases, e.g. for all closed smooth manifolds N, orientable or not, of dimension n ≡ 0(4) and, in particular, in the classical space-time dimension 4.

Our approach is based on the singularity method which allows us to classify the monomorphisms u from a given (abstract) line bundle α over N into the tangent bundle. The analysis of the transition to the image line field u(α) then centers around the notion of ‘antipodality’.

We express our classification results in terms of standard (co-)homology and characteristic classes. Moreover, we illustrate them for large families of concrete sample manifolds by explicit bijections or by calculating the number of line fields.

The topological theory of covering spaces may be used to prove results in group theory, for instance, the Kuros-Reidemeister-Schreier theorem (1). It seems likely that such methods can be applied to prove the Freiheitsatz (4) and the identity theorem (3), and also perhaps Lyndon's conjecture, that the normal closure in a free group F of a, single element r is freely generated by conjugates of r. However, although these problems may easily be stated in topological terms, no such proof is at present known. In this paper we prove a related result.