Let Q denote the field of rational numbers, and let p be an odd prime number. Let K be a cyclic extension of Q of degree p, and let a be a generator of Gal (KQ). Let CK denote the p-class group of K (i.e., the Sylow p-subgroup of the ideal class group of K), and let for i = 1, 2, 3, . It is well known that is an elementary abelian p-group of rank tt1, where t is the number of ramified primes in KQ. So we focus our attention on . We let

A high Reynolds number theory is developed for a viscous fluid flowing through an elastic channel. Unlike the flow through rigid symmetric channels, the viscous flow through a symmetric elastic channel is found to admit free-interaction solutions, due solely to the interaction of the boundary layer with the elastic channel wall. The assumption of symmetry is found to be general providing that the streamwise extent of the channel collapse dilation is larger than O(K17) and the channel is allowed to deviate only slightly from a straight channel. These free-interactions are believed to be the viscous initiation of a sudden collapse or dilation of the channel, commonly observed in experiment. The collapse of the channel is found to occur over a wide range of possible streamwise length scales from O(l) to O(K). For a rigid channel which is coated with a thin elastic solid, the equations are found to reduce to the hypersonic strong interaction problem of triple-deck theory. The hypersonic triple-deck is known to admit both compressive and expansive free-interactions. The expansive free-interaction is found to correspond to a sudden collapse of the channel and an acceleration of the flow within the core of the channel. A cha nnel that is backed by a stagnant constant pressure fluid is also examined. For this problem, the pressure is proportional to the negativeof the fourth derivative of the channel wall displacement. This structure is also found to admit compressive andor expansive free-interactions, depending on whether the internal pressure within the channel is less than or greater than the constant pressure external to the channel. Terminal forms are developed for the expansive free-interaction and compared with numerical calculations.

Filters and אּ-complete filters can be used to produce set-theoretic extensions of direct sums and direct products. They can be applied to generalize theorems in module theory which involve these. For example, the theorem, stating that a ring is noetherian, if, and only if, direct sums of injectives are injective, can be generalized, provided we replace noetherian by Xa-noetherian and direct sums by אּ- complete filter sums with a suitable property.

The existence of inductive limits in the category of (topological) measure spaces is proved. Next, permanence properties of inductive limits are investigated. If (X, , ) is the inductive limit of the measure spaces (X, , ), we prove, for 1 p 221E;, that LP(X, , ) is embeddible into the projectilimit of Lp(X, ,) in the category Ban, for p <, respectively in the category C* in the case p = +. As an application, we exten existence theorems of strong liftings to inductive limits.

We begin by denning the notion of a tangential limit for a function f denned in the unit disc

Let be a positive continuous function on (0, 1) for which

Suppose B>0, -,and define

where The region . makes tangential contact with the boundary U of the unit disc at ei; when (r) = ( l - r2), for instance, (, , 1) is the disc with radiusand centre ei

This study extends earlier work on the characterization of the asymmetry of a section of a typical three-dimensional Brownian path using the moment of inertia tensor about the centre of mass. A new method for determining an upper bound on the ensemble average of the smallest eigenvalue is presented. This work has applications to polymer science, since single chain polymer molecules are often modelled as sections of Brownian paths.

The problem of illuminating the boundary of sets having constant width is considered and a bound for the number of directions needed is given. As a corollary, an estimate for Borsuk's partition problem is inferred. Also, the illumination number of sufficiently symmetric strictly convex bodies is determined.

The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.

In the application of electromagnetic methods to the non-destructive testing of electrically conducting materials for cracks or inclusions an electric current is applied to the specimen and the presence of a flaw is indicated by the perturbations it produces in the electromagnetic field. A number of different variants of the method can be used. The presence of a flaw may be observed by measuring either electric or magnetic field perturbations and the nature of the interrogating field will be sensitive to the choice of frequency chosen for the applied current. It is well known that when alternating current is applied to conductors the current tends to be confined to a surface layer whose depth, δ, is measured by the length 1/(ωσμ)½ where σ is the conductivity, μ is the magnetic permeability and ω the angular frequency. An important dimensionless parameter in the characterisation of the field perturbations is the ratio δ/l, where l is a length typical of the flaw dimensions. The electromagnetic field is described as a thin-skin or a thick-skin field according as this ratio is small or large respectively. In practical applications there is a need to model both thin and thick-skin fields. In the examination of surface fatigue cracks in large scale structures fabricated by welding together ferrous steel members surface fatigue cracks with depths of order 1–10 mm have been interrogated with currents at 5–6 KHz at which the skin depth is of order 0·1 mm (Dover, Collins and Michael [1]).

We shall deal throughout with regular matrix methods of summability defined by a stochastic matrix A = (ank). Thus

means

The concept of mixed invariant set is due to Bandt [1], Bedford [2], Dekking [3, 4], Marion [4] and Schulz [10]. An m-tuple B = (B1, …, Bm) of closed and bounded subsets Bi of a complete finitely compact (bounded and closed subsets are compact) metric space X is called a mixed invariant set with respect to contractions f1, …, fm and a transition matrix M = (mij), if, and only if,

for every i ∈ {1, …, m}. In the papers quoted an essential condition is that all mappings f1, …, fm be contractions. We will show that, under certain conditions, the construction of mixed invariant sets also works in cases where some of the mappings are isometries or even expanding mappings.

Let (R, ) be a commutative Noetherian local ring. We investigate conditions for a non-finitely generated R-module M to have a system of parameters. We prove that if

then any system of parameters for R/AnR (M) is a system of parameters for M. As an application we characterize by means of systems of parameters those balanced big Cohen–Macaulay R-modules M for which SuppR (M) = suppR (M).

Throughout the paper, let

be forms with rational integer coefficients of degrees d1, …, ds all at least 2 with n ≥ 4. Let p be a prime and Q a box in ℝn,

Let K be a convex compact body with nonempty interior in the d-dimensional Euclidean space Rd and let x1, …, xn be random points in K, independently and uniformly distributed. Define Kn = conv {x1, …, xn}. Our main concern in this paper will be the behaviour of the deviation of vol Kn from vol K as a function of n, more precisely, the expectation of the random variable vol (K\Kn). We denote this expectation by E (K, n).

We examine the asymptotic form of a fundamental set of solutions of the third-order equation

as x → ∞, where the leading coefficient q is nowhere zero in some interval [a, ∞). The equation is self-adjoint in the case when p, r and iq are real. However, our analysis is not confined to this case, and we generally take the coefficients to be complex-valued.

A study is made of the length L(h, k) of the Euclidean algorithm for determining the g.c.d. of two polynomials h, k in [X], a finite field. We obtain exact formulae for the number of pairs with a fixed length N which lie in a given range, as well as the average length and variance of the Euclidean algorithm for such pairs.

Let (C1, C2, …) be a sequence of convex bodies in n-dimensional euclidean space En, and let υ(Ci) denote the volume and d(Ci) the diameter of Ci. It is shown that the conditions

imply that the sets Ci can be rearranged by the application of rigid motions so that the resulting sets form a packing in En of density 1. A corresponding result for coverings of En is also proved.

One construction used to produce a random number table is to take a smooth function F(x) taking values between 0 and 1, to evaluate it at N points spaced 1/M apart, and to ignore the first t decimal digits. With T = 10t this corresponds to taking the fractional part of

where T>M>N. The grounds for assuming this sequence to be random are that it is so difficult to prove anything about it.