The purpose of this paper is to obtain quantitative results on invertible powers of (fractional) ideals in commutative separable algebras over dedekind domains. This is connected with the work of Dade, Taussky and Zassenhaus(2) on ideals in noetherian domains. We do not, however, make use of their paper, but rather draw on the general theorems on ideals in commutative separable algebras established by Fröhlich (3), in particular his qualitative result that some power of any given ideal is invertible. Our basic result (Theorem 1) concerns the invertibility of powers of a particular type of ideal, the componentwise dedekind ideals defined below. From this we deduce a general result (Theorem 2), which includes as a special case Theorem C of (2) for the case of separable field extensions; specifically, the (n – 1)th power of any ideal is invertible, where n is the dimension of the algebra. Although, as we show, it is possible to deduce Theorem 2 from (2), we have here an independent proof of one of the main results of (2) based entirely on the results in (3). As a further application of Theorem 1 we obtain a new result on ideals over the group ring of an abelian group over the ring of rational integers; the (t – 1)th power of such an ideal is invertible, where t is the maximum number of simple components of the rational group algebra of any Sylow subgroup. We also show that this is the best possible result when some Sylow subgroup whose rational group algebra has t components is cyclic.

In recent work on some topological problems (7), I was forced to adopt a complicated definition of ‘Hermitian form’ which differed from any in the literature. A recent paper by Tits(5) on quadratic forms over division rings contains a new and simple definition of these. A major objective of this paper is to formulate both these definitions in somewhat more general terms, and to show that they are equivalent.

The following problem arose in recent work of Stewart Robertson on finite subsets of Eucidean space admitting transitive groups of isometries (1): Determine the greatest number of involutions that can be possessed by a finite group of given order n.

Let G be a matrix. If there does not exist an infinite set of non-zero elements such that no two are on the same line (row or column), the matrix is said to be of finite term-rank. A set of lines such that each non-zero element belongs to some line of the set is said to be a covering of G.

In his fundamental paper (1) Chevalley showed how to associate with each complex simple Lie algebra L and each field K a group G = L(K) which is (in all but four exceptional cases) simple. If K is a finite field GF(q), G is a finite group of order

where l is the rank of L, m is the number of positive roots of L and d is a certain integer determined by L and K. The integers m1, m2,…,m1 are determined by L only and satisfy the condition

Let τ be the formally self-adjoint differential operator denned by

where the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely as

where ξk ∈K for each k. If this representation converges absolutely, i.e. if

for every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.

We have called a linear partial differential operator P almost hypoelliptic in some of the variables if the distribution u is strongly regular in those variables in every open subset where this is true of Pu. In an earlier paper (l) we have characterised almost hypoelliptic operators with constant coefficients. Following techniques of Hörmander (2) and Yoshikawa(5) we obtain in this paper some a priori inequalities for almost hypoelliptic operators and derive results about the solvability of the adjoint operator P*. Our notation for various standard distribution spaces and multi-indices follows Hörmander (2).

If one changes at random the signs of the Fourier coefficients of an L2 function the result is still the sequence of Fourier coefficients of an L2 function. L2 is the only LP space with this property; in fact if a sequence remains a Fourier transform for every insertion of ± signs then it is really the transform of an L2 function ((6), p. 215, Theorem 8·14). Our main theorem is a version of this principle, for sequences defined on suitable subsets of a group; it is phrased in terms of large sets of choices of ± signs rather than all sequences of ± signs.

Let 0 < λ1 < λ2 < … < λn → ∞ (n→∞). We write

Let ∑an be a given infinite series with the sequence {sn} for its nth partial sum. The (R, λ, 1) mean of the sequence {sn} is given by

Let be the given infinite series with {sn} as the sequence of partial sums and let be the binomial coefficient of zn in the power series expansion of the function (l-z)-σ-1 |z| < 1. Now let, for β > – 1,

converge for 0 ≤ x < 1. If fβ(x) → s as x → 1–, then we say that ∑an is summable (Aβ) to s. If, further, f(x) is a function of bounded variation in (0, 1), then ∑an is summable |Aβ| or absolutely summable (Aβ). We write this symbolically as {sn} ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, (Aα) ⊂ (Aβ). Note that for β = 0, (Aβ) is the same as Abel method (A). Borwein (2) also introduced the (C, α, β) method as follows: Let α + β ╪ −1, −2, … Then the (C, α, β) mean is defined by

Let f(t) be Lebesgue integrable in (-π, π) and be periodic with period 2π; let the Fourier series of f(t) at t = y be

We use some of the standard notation of analytic number theory; ψ(x) is defined in (1), page 12, and where μ(n) is the Möbius function.

In this paper a new generalization of the Meijer transform is given. It also generalizes the Mainra transform and the transform due to Varma. An inversion theorem is established and the result obtained has been illustrated by several examples.

The differential equations governing the propagation of waves of electric and magnetic types in a plane stratified isotropic plasma are suitably generalized, and we investigate the possibility of models for which the moduli of the reflexion coefficients are identical for the two modes. First, the models are examined without the necessity of finding general solutions, and, secondly, by using the circuit relations for the hypergeometric functions occurring in the explicit solutions.

The inequalities related to Mill's Ratio were conjectured true by Birnbaum (2) and were proved by Murty (4) for sufficiently large x > 0. In the present paper, these are proved for all finite x and have been used to calculate an upper limit for Mill's Ratio over the range x > – 1.

It is shown that given k samples of nj units from it is possible to construct simultaneous confidence intervals for two given linear functions of population means, (where cij are known constants), when population variances are not equal.

The method of steepest descents is widely used to find the asymptotic expansion of contour integrals of the form

where N is a large positive parameter, and f(z) and g(z) are analytic functions of z. (For the sake of simplicity it will here be supposed that f(z) and g(z) are regular except at ∞, but our arguments can be extended to include singularities at other points.) In the method of steepest descents the path of integration is deformed into a new equivalent path consisting of the paths of steepest descent through a and b, together with paths of steepest descent through certain saddle points, the relevant saddle points. (Paths of descent, etc., are defined in §1 below.) Watson's Lemma then shows that the complete asymptotic expansion depends on the local behaviour of f(z) and g(z) at the end-points and at the relevant saddle-points. On the other hand to determine which saddle-points are relevant is a global (non-local) problem and requires either the tracing of paths of steepest descent all the way from saddle-points and endpoints to ∞, or some other global process. It has often been noted that in applications of Watson's Lemma the paths of steepest descent can be replaced by a wider class of paths of descent. In this way we are led to global problems like the following: Given 2points z = α and z = β where Nf(a) > Nf(β), can apath of descent be drawn from α to β? If also Ff(α) = Ff(β), can a path of steepest descent be drawn from α to β?

In obtaining a solution of Laplace's equation in two dimensions by the method of conformal mapping, one first maps the points (x, y) of the Euclidean plane R2 into the algebra of complex numbers C by means of the real-linear function g: R2→C using the prescription g(x, y) = x + iy ≡ z. One then obtains solutions of Laplace's equation by allowing those mappings of C into itself that are expressed by analytic functions.

It is shown that a cosmological model of light propagation associated with a uniformly expanding universe provides a physical significance to the hyperbolic velocity space by which this model can be described. Some implications of this result are discussed.