Those mixed boundary-value problems which can usefully be treated analytically often lead to the following mathematical problem. Two functions u(x), σ(x), defined over the interval ([0, ∞), take prescribed values over complementary portions of that interval; specifically, let

where p(x) is usually a simple function, for example a constant or a power of x. There exists a relation between u(x) and σ(x) which can be most simply expressed as a relation between their Hankel transforms. Using a circumflex to denote the Hankel transform, for example with

where Jv denotes as usual the Bessel function of the first kind of order v, we can state that relation between u and σ as follows:

where A(ξ) is a known function, determined at an earlier stage of the analysis. The problem is to derive u(x) for (xє [ 0, a), or σ(x) for x є (a, ∞).

1. Introduction and statement of results. Let f: Mn-1 → ℝn be an immersion into Euclidean space ℝn. Each unit vector v to ℝn determines a height function bv: ℝn → ℝ. The corresponding half-space Lv = b-1([0, ∞) has boundary Hv = (b−1). and L is a (globally) supporting half-space for M at m є M if (m) є H and f(M) ∩ Lv = f(M) ∩ Hv.

The following equation arose in a study of the evolution of a one-dimensional quantum system:

where α and κ are positive constants, and

It will be shown that this system provides an example of exactly exponential decay. This is of interest in view of recent papers discussing the validity of the exponential decay law (see, for example, Khalfin (1)and Martin (2)).

This paper investigates the ‘general position’ properties which ANR's may possess. The most important of these is the disjoint discs property of Cannon (5), which plays a vital role in recent striking characterizations of manifolds (5, 9, 12, 18, 19, 22). Also considered are the property Δ of Borsuk(2) (which ensures an abundance of dimension-preserving maps), and the vanishing of local homology groups up to a given dimension (cf. (9)). Our main results give relations between these properties, and clarify their behaviour under the stabilization operation of taking cartesian product with the real line. In the last section these results are applied to give partial solutions to questions about homogeneous ANR's.

The problem of completing the mathematical structures abstracted from the time ordering of the events in the space-times of general relativity is motivated and solved. The new construction is compared with that of Geroch, Kronheimer and Penrose (5).

Let K be a knot in the 3-sphere S3; let m be a meridian and 1 a longitude of K. In [1] we gave a detailed study of the ‘π-orbifold group’ O(K): π1(S3\K)/〈m2〉 of the knot, where 〈m2〉 is the subgroup normally generated by the square of the meridian. In particular, we found out to what extent the π-orbifold group classifies knots (and links) and determines the symmetry group of a knot. In the present paper, we consider the groups Gn(K): = π1(S3\K)/〈ln〉. Our main results are the following.

A considerable amount of experimental work has been done in recent years on the properties of the electrons which leave a metal surface when it is bombarded by an incident beam of electrons. Farnsworth, who first studied the energy distribution of the secondary beam systematically, showed that it is made up of two groups. The first has the same energy as the primary beam and may be regarded as consisting of reflected primary electrons; and the second has for the most part a very small energy, of the order of a few volts, and contains the true secondaries derived from the metal. The present writer in a preliminary note has described some measurements of the differential energy distribution by means of the magnetic spectrum method. Owing to an inadequate technique those experiments are considered to be valid only in the upper half of the energy range, where the form of the distribution has several features of particular interest when the mechanism of secondary emission and of electron reflection are considered. In this note, the author called attention to the peculiar depression in the curve immediately preceding the “reflected” peak. It was impossible then to insist on the reality of such an effect, owing to the probable presence of an instrumental cause (which was given there as the explanation). That it is real, however, has been demonstrated in a striking manner by Whiddington and Brown, who have succeeded in recording electrons as slow as 100 volts photographically on a specially sensitised film.

The object of this note is to study the properties of some functions self reciprocal in the Hankel transform. I denote a function f(x) as Rμ, if it is self reciprocal for Hankel transforms of order μ so that it is given by

where Jμ(x) is a Bessel function of order μ. If μ = ½, f(x) is denoted by Rs while f(x) is written as Rc when μ = − ½.

Tiffen and Lowe in two recent papers (4,5) have developed exact theories for the bending and stretching of generally loaded elastic plates using the moments of the fundamental equations of infinitesimal elasticity. The two theories depend on the choice of independent variables so that a pair of functions, denoted by and are independent. The choice is correct in (4) but not in (5). This paper pro-poses a modification to the forms of the functions and in (5) and then shows that the modified forms are independent. The subsequent alteration to (5), as will be seen, involves only the biharmonic function χ Apart from a minor algebraic adjustment the rest of the theory, containing the real and complex eigenfunctions and particular integral, is preserved. It should be noted that the modified theory like the original is obtained entirely from the moment equations.

Let G be a locally compact Abelian group; Γ the dual group of G; C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; CB(Γ) the continuous, bounded functions on Γ; M (G) the algebra of bounded Borel measures on G; L1(G) the algebra of absolutely continuous measures; and M(G)∩ the algebra of Fourier–Stieltjes transforms.

1. Introduction. The relationship between sufficient statistics and the exponential family was first investigated by Pitman (8) and Koopman (7). The treatments assumed a density function f(x; θ) which was at least differentiable with respect to both arguments.

In this paper we study the uniform boundedness of oscillatory singular integral operators with degenerate phase functions on the Hardy space H1. The H1 boundedness was previously known when the phase function is nondegenerate. Here we obtain a sufficient condition for H1 boundedness which allows the phase function vanishing to infinite order.

A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalise transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalise several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalisation, in the flag variety case, of a theorem of Knutson–Woo–Yong; we also generalise this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill–Noether varieties on elliptic curves, and a conjectural generalisation to higher genus curves.

A new analysis is made of the coefficients appearing in periodic Mathieu functions, and this is applied to establish the limiting properties of certain solutions of the equation

In his work on mercury, Benedicks (1) observed that the small temperature peak in the neighbourhood of the middle of the long tube containing the conductor was displaced in one direction or the other along the tube according to the direction of flow of the electric current by which the conductor was heated. Normally this effect increased with the cube of the current intensity, and was explained as due to Thomson heat. If the temperature was made as uniform as possible by means of controlled heaters at the ends of the tube, the effect became of opposite sign and increased linearly with the current. This was called the “electro-thermal effect” and was explained as due to an actual change of thermal equilibrium by the electric current, or the production of a temperature gradient by a flow of electricity.

In Part I of the work that follows solutions for the flow around a semi-infinite oscifiating plate are obtained, one by an operational method using parabolic coordinates and one in the form of a series solution. The latter is used to obtain an estimate of the skin friction distribution about the edge of the plate and hence the excess skin friction over the infinite oscilliating plate value. A valuable check on this result is provided by an equivalent consideration of the operational solution. Having demonstrated the usefulness of the series solution we proceed to extend its application to a wedge of arbitrary angle in Part II and eventually use the results to derive a general expression for the skin friction on an infinite cylinder of arbitrary cross-section oscifiating rapidly parallel to its length.

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formula

A linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:

(a) for each x in , the complex-conjugate function x* is in ;

(b) both and \ are invariant under the shift operators;

(c) every shift operator commutes with T.

(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)

1. Introduction. In a recent paper (2) Daykin has made the Conjecture. If t is real, then

where

He has proved (1, 2) that (i) Φ(t) is a convex function of t, (ii) Φ(t)≥ n if t ≥ 0 or 2, and (iii) Φ(t) takes values arbitrarily close to ½n if n is even, x2 = x4 = … = xn = 1, and xl = x3 = … = xn-1 and t have suitable small positive values. It is known (3) that (1) is true if t = 1.

An important problem in the topology of 3-manifolds is to classify manifolds obtained by Dehn surgeries on a knot in a closed 3-manifold, or equivalently, Dehn fillings of a 3-manifold M with boundary a torus.