The well-known Banach Contraction Principle asserts that any self-map F of a complete metric space M with the property that, for some number k < 1, for all x, y,∈M, possesses a unique fixed point in M. some extensions and analogues have recently been given by Edelstein [1]. For the reader's convenlience we state here the result of Edelstein which we shall employ. It asserts that if F is a self-map of a metric space M having the property that for any two distinct points x and y of M, and if x0 is a point of M such that the sequence of iterates xn = Fn (x0) contains a subsequence which converges in M, then the limit of this subsequence is the unique fixed point of F.

This paper is concerned with two mani problems:

(a) the determination of the conjugacy classes in the finite-dimensional unitary, symplectic and orthogonal groups over division rings or fields;

(b) the determination of the equivalence classes of non-degenerate sesquilinear forms on finite-dimensional vector spaces.

One proof of Minkowski's fundamental theorem on lattice points in an n-dimensional parallelopiped depends upon a multiple Fourier expansion. Its terms involve multiple integrals which are easily evaluated and so are shown to have positive values. Then the expansion takes only positive values and the desired proof follows at once. It seems of interest to note a multiple integral which assumes only positive values.

The aim of this paper is to dervie two formulae for π(N) that need involve only a few of the smallest primes. Here m is a small integer, the b's are integers that will be found later, and Pij…k denotes the number of products figi… ≧ N, in which f, g,?…,h are unequal integers greater than 1 and prime to the first m primes. The suffixes run through all partitions of all integers.

Let Q denote the closed unit cube in Rn. The elementary area A(f) of a Lipschitz function f on Q is given by the formula.

Let G be a finite group. If N denotes a normal subgroup of G, a subgroup S of G is called a supplement of N if we have G = SN. For every normal subgroup of G there is always the trivial supplement S = G. The existence of a non-trivial supplement is important for the extension theory, i.e., for the description of G by means of N and the factor group G/N. Generally, a supplement S is the more useful the smaller the intersection S ∩ N. If we have even S ∩ N = 1, then S is called a complement for N in G. In this case G is a splitting extension of N by S.

The present note continues the discussion, begun in the first paper with the above title, of classes of spaces H which are extensions of L2 (0, ∞) and whose elements, which we call ‘sequence-functions’, exhibit some of the properties of distributions. The previous paper defined the spaces, and described how they can be used to extend the domain of definition of Waston transforms. Further related applications to transform theory are described in [2] and [3]. In this paper I pursue the analogy between these sequence- functions and other types of generalized functions further by discussing their local behaviour, the existence of ordinary and convolution-type products, and of derivatives and integrals.

In a recent paper on statistical fluid mechanics Professor J. Kampé de Fériet [1] employed several integrals of which the following is a typical example The function u(x, y, t), which it defines, formally satisfies the following three classical differential equations

A tetrad of Möbius tetrahedra consists of a set of 4 mutually inscribed and therefore circumscribed tetrahedra whose 16 vertices and 16 faces form a Kummer's 166 configuration (5; 11; 12; 21). As pointed out by the refere, fundamental to all work on the 166 figure are the 10 quadrics, called fundamental for the associated Kummer's quartic surface (13). To every quadric F correspond a matrix scheme of the 16 points or planes, arranged in 4 rows or columns, such that the 8 Rosenhain tetrahedra (7) formed of the rows and columns are all self-polar for F. The rows form one and the columns another tetrad of Mobius tetrahedra. Nine new schemes can be derived from one such scheme to make the total ten as explained by Baker (3, p. 133) leading to 80 Rosenhain tetrahedra in all. The 16 nodes (5; 8) or tropes of the Rummer's quartic are the 16 common elements of the 10 schemes such that the nodes and tropes are poles and polars for any one of the 10 quadrics. Each trope touches the quartic along a singular conic through the 6 points of the figure lying therein. The lines tangent to the surface at its each node N generate a quadric cone which is enveloped by the 6 tropes through N (12).

The functions f defined by or by for c rational and less than + 1 map the set of rational numbers between 0 and 1 one-to-one onto itself; and they are the only fractional linear functions with this property. Miss Tekla Taylor recently raised the question * whether these are the only differentiable functions with the stated property. In the present note we show, by two different constructions, that the answer is negative; in each case much freedom remains, which could be used to make the functions in question have various additional properties.

Let L1 denote temporarily the usual Lebesgue space over the circle group (equivalently: the additive group of real numbers modulo 2π), and let H1 denote the Hardy space comprised of those f in L1 whose complex Fourier coefficients vanish for all negative frequencies, so tha D. J. Newman has settled a conjecture by showing [1] that there exists no continuous projuction of L1 onto H1, i.e. that there exists in L1 no topological complement to H1.

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).

Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

In [1] we considered various aspects of the quotient semigroup H. · H2 where H is an ℋ-class of a semigroup S. In particular, the action of the Schützenberger group of H upon SH was studied to obtain various results on the existence of subcontinua. Crucial in [1] was the notion of the (right handed) core of an ℋ-class which may be considered as a generalization of the notion of the core of a homogroup, [2].

Throughout this paper E, F and G denote separated locally convex spaces, F C G, the injection i: F → G being continuous (i.e. the topology on F is finer than that induced on it by the topology on G). E′, F′ and G′ denote the respective duals of E, F and G. i′ is the adjoint map of G′ into F', which is defined by restricting linear forms on G to F C G.

Certain integers have the property that they can be partitioned into distinct positive integers whose reciprocals sum to 1, e.g., and In this paper we prove that all integers exceeding 77 possess this property. This result can then be used to establish the more general theorem that for any positive rational numbers α and β there exists an integer r(α, β) such that any integer exceeding r(α, β) can be partitioned into distinct positive integers exceeding β whose reciprocals sum to α.

When a semi-infinite line load moves lengthways, at supersonic velocity, on the plane surface of an elastic solid, the resulting velocity field is conical. There are two characteristic cones, one associated with dilatation effects and the other with shear effects. The propagation process is more complicated than the well-known case of conical flow in supersonic aerodynamics not only because of the presence of two cones of discontinuity but also because the presence of a free surface implies interaction between shear and dilatation effects. It is the interaction process at the free surface which is examined in detail in this paper.

The results of this fundamental problem may be extended by the process of superposition to more general steadily moving loads. In particular by differentiating with respect to time, the potential of a steadily moving point load is obtained explicitly.

Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

First consider some familiar results, the inequality of the arithmetic and geometric mean is: Kantorovich's inequality (reference [1]) asserts that if 0 < A ≦f(x) ≦ B then: The Cauchy-Schwarz inequality is: This paper discusses a certain class of inequalities which includes the three above. Three theorems are proved which apply to any inequality of this class; then follow some examples. They are mainly to show how the general theory helps in the finding of inequalities, but the result of Example 1 seems worth reporting for its own sake.

Milne-Thomson has used the method of analytic continuation to solve boundary value problems of the annulus in plane elastostatics. However, his use of Cauchy integrals is incorrect, and it is shown in this note that the solution is obtained in terms of Laurent Series expansions. The solution is equivalent to that of Muskhelishvili, but is simpler to use in some applications.

A similar approach is used to solve the boundary value problem of the infinite strip, the solution being given in terms of functions of a complex variable expressed as Fourier integrals.

In recent years there has been extensive development in the theory and techniques of mathematical programming in finite spaces. It would be very useful in practice to extend this development to infinite spaces, in order to treat more realistically the problems that arise for example in economic situations involving infinitely divisible processes, and in particular problems involving time as a continuous variable. A more mathematical reason for seeking such generalisation is possibly that of obtaining a unification mathematical programming with other branches of mathematics concerned with extrema, such as the calculus of variations.