The problem of scattering of surface water waves obliquely incident on a fixed half immersed circular cylinder is solved approximately by reducing it to the solution of an integral equation and also by the method of multipoles. For different values of the angle of incidence and the wave number the reflection and transmission coefficients obtained by both methods are evaluated numerically and represented graphically to compare the results obtained by the respective methods.

A method of determining numerically to any degree of accuracy the eigen-values of Hamiltonians in the form of power series is presented. The case of a spherically symmetric potential function of the form V = ar2 + br4 + cr6 is treated in detail.

The problem of periodicity of solutions of the generalized Liénard equation

has attracted much attention. Many efforts have been made to give sufficient conditions to guarantee the existence and the uniqueness of periodic solutions (limit cycles) of (1·1). There are also some papers on the number of limit cycles of (1·1) (see, for example, [3, 5, 6, 13]). However, there are only a few results on non-existence of periodic solutions of (1·1).

1. Among the equations arising in the theory of natural selection is the finite difference equation

where k is a constant which may have any value between 1 and − ∞ inclusive, but is often small. Its solution is discussed in the accompanying paper(1). It is a particular case of the equation

where Φ(x) is a known one-valued function of x. When k is small this may obviously be solved approximately by treating it as a differential equation

whence

The exact solution for a trapezoidal potential barrier is given in terms of Airy functions and trigonometrical functions. Numerical values are found for the ratio of the amplitudes of the transmitted and reflected waves. These are compared with those given by the WKBJ approximation.

1. Mahler (1) has obtained a number of valuable existence theorems for homogeneous problems in the geometry of numbers. It appears that no analogous results for inhomogeneous problems have yet been published; and although none of them present great difficulty, it is convenient for applications to have them in print.

The determination of the number of zeros of a complex polynomial in a half-plane, in particular in the upper and lower, or right and left, half-planes, has been the subject of numerous papers, and a full discussion, with many references, is given in Marden (l) and Wall (2), where the basis for the determination is a continued-fraction expansion, or H.C.F. algorithm, in terms of which the number of zeros in one of the half-planes can be written down at once. In addition, determinantal formulae for the relevant elements of the algorithm can be obtained, and these lead to determinantal criteria for the number of zeros, including that of Hurwitz (3) for the right and left half-planes.

The order and asymptotic form of the error of a general class of numerical method for solving the initial value problem for systems of ordinary differential equations is considered. Previously only the convergence of the methods, which include Runge-Kutta and linear multistep methods, has been discussed.

An ordered group (o-group for short) is a group endowed with a linear (i.e. total) order such that for all x, y, z, xz ≤ yz and zx ≤ zy whenever x ≤ y. A group for which such an order exists is called an orderable group. A group G is said to be divisible if for each positive integer m and each g ε G, there is x ε G such that xm = g.

This paper describes a criterion which, if satisfied, guarantees uniqueness of solution for the problem involving the radiation or diffraction of small amplitude water waves by a totally submerged body. This criterion was originally stated by a Russian mathematician, V. G. Maz'ja, although his work does not appear to be widely known. For this reason a brief outline of Maz'ja's proof is given here, together with a discussion of the condition to be satisfied by the submerged body.

Maz'ja's criterion is not satisfied by all submerged bodies and so his result does not provide a general proof of uniqueness. However, the criterion is satisfied in some important cases and examples are presented here. The partial nature of this result is due entirely to the method of proof and not to any intrinsic difficulty with the underlying hydrodynamics.

Let (V,0) be the germ of an analytic variety in $\mathbb{C}^n$ and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce–Roberts number, μBR(V,f). Like the Milnor number of f, this number shows some properties of f and V. In this paper we investigate algebraic and geometric characterizations of the constancy of the Bruce–Roberts number for families of functions with isolated singularities on V. We also discuss the topological invariance of the Bruce–Roberts number for families of quasihomogeneous functions defined on quasihomogeneous varieties. As application of the results, we prove a relative version of the Zariski multiplicity conjecture for quasihomogeneous varieties.

The sheaves [Cscr ]*M and [Cscr ]d,*M are constructed for the microlocal analysis of singularities of ultradistributions and ultradifferentiable functions. The suppleness of these sheaves is proved.

This paper proves the existence of solutions for the discrete coagulation–fragmentation equation over all times, even when source and efflux terms are present. The hypotheses required cover most physical applications. Roughly speaking, the hypotheses ensure a finite flux of mass through the system. The techniques used, which extend those in I of this series, may apply to other infinite systems of differential equations.

In this paper we shall prove the following two results.

Theorem 1. Let ∊ > 0 and β a real number be given. Them, for almost all real a (in the sense of Lebesque measure), there are infinitely many pairs of square-free integers m, n such that

The irreversible heating of iron ammonium alum by an alternating field is shown to be proportional to the frequency of the field. The heating is therefore similar to that arising from hysteresis in ferromagnetics, but no remanence could be found by a ballistic method at the lowest temperature obtained, about 0·075° K. The rate of heating was much increased by the superposition of a steady magnetic field of a few hundred gauss.

For every Banach space X, denote by GL(X) the linear group of X, i.e. the group of all linear continuous invertible operators on X with the topology induced by the operator norm. One says that GL(X) is contractible to a point if there exists a continuous map F: GL(X) × [0, 1] → GL(X) such that F(A,0) = A and F(A, 1) = Id, for every A ∈ GL(X).

The advantages of using polynomial approximations for the purpose of constructing interpolable numerical tables on punched cards have been pointed out by Sadler (7). The object of this paper is to demonstrate the value of this method for ordinary published tables.

The Pompeiu problem has its origins in classical analysis in ℝn (see [2, 3, 4, 8] for a discussion and some history). In this context it may be stated as follows. Let D ⊂ ℝn be a bounded measurable set of positive Lebesgue measure and f a locally integrable function on ℝn. Then, if ∫σ(D)f = 0 for all rigid motions σ of ℝn, is f = 0?

Let ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.