This is the final paper of the series of three papers under the same title. The finite dimensional theory developed in the first of them 7 contains first of all:

(a) a calculus having among its consequences the calculi of convex subdifferentials and generalized gradients of Clarke (henceforth sometimes abbreviated C.g.g.) in the most general form which is partly due to the fact that in a finite dimensional space

for any convex function f and

for any SX (A means approximate, C means Clarke); and (b) a theorem stating that approximate subdifferentials are minimal (as sets) among all possible subdifferentials satisfying one or another set of conditions (usually very natural).

For a typical convex body in Ed a typical shadow boundary under parallel illumination has infinite (d - 2)-dimensional Hausdorff measurewhile having Hausdorff dimension d2.

By a well-known theorem of Szemerdi 8 any set of integers that has positive density contains arithmetic progressions of arbitrary length. One might expect that there are conditions of similar generality, under which an integer set contains arbitrarily long strings of consecutive integers, i.e., arithmetic progressions with 1 as common difference. Results of this kind would be of great importance because of potential applications to arithmetically interesting sets such as the set n: (n) = 1, where (n) is the Liouville function, or the sets

where P(n) denotes the greatest prime factor of n and 0<< 1. One naturally expects that such sets contain arbitrarily long strings of consecutive integers, but no results in this direction are known, and the problem seems to be a very difficult one, perhaps comparable in depth to the prime k-tuple conjecture.

The theorem of Aleksandrov-Fenchel-Jessen states that two convex bodies in n-dimensional Euclidean space En which, for some p l, , n - l 007D;, have equal area measures of order p (see Section 2 for a definition) differ only by a translation. Two independent proofs were given by Aleksandrov 1 and by Fenchel and Jessen 18 see also Busemann 5 (p. 70) and LeichtweiG 25 (p. 319), 26. If the boundaries of the two bodies are sufficiently smooth and of everywhere positive curvatures, then the assumption of the theorem is equivalent to saying that at points with parallel outer normals the p-th elementary symmetric functions of the principal radii of curvature of both boundary hypersurfaces are the same. For this case, Chern 6 gave a uniqueness proof by means of an integral formula.

The purpose of this paper is to take some first steps the investigation of the negative moments

where k>0 and12, and the related discrete moments

whereruns over the complex zeros of the zeta-function. We assume the Riemann hypothesis (RH) throughout; it then follows that Ik(, T) converges for every k > 0 when>but for no k =when =. We further note that Jk(T) is only defined for all T if all the zerosare simple and, in that case, Ik(, T) converges for all k<.

Ramsey's theorem implies that every function f:0, 1ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r). A typical result: if f is bounded from below and is not convex on any infiniteset then there exists an interval on which the graph of f can be covered by the graphs of countably many strictly concave functions.

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