Christoffel's problem, in its classical form, asks for the determination of necessary and sufficient conditions on a function φ, defined over the unit spherical surface Ω, in order that there exist a convex body K for which φ (u) is the sum of the principal radii of curvature at that boundary point of K where the outer unit normal is u. The figures Ω and K are in Euclidean n-dimensional space (n ≥ 3). It is assumed that φ is continuously differentiable and that K is of sufficient smoothness. A solution of Christoffe's problem was given in [6]. Yet that treatment is rather unsatisfactory in that the smoothness restrictions are set by the method rather than the problem, cf. [5; p. 60]. The present paper overcomes this defect. To do this it is first necessary to generalize the original problem so as to seek conditions on a measure M, defined over the Borel sets of Ω, in order that M be a first order area function for a convex body K. When K has sufficient smoothness, then φ is the Radon-Nikodym derivative of M with respect to surface area measure on Ω. It is this generalized Christoffel problem which is solved in what follows.

The temporal and spatial linear instability of Poiseuille flow through pipes of arbitrary cross-section is discussed for large Reynolds numbers (R). For a pipe whose aspect ratio is finite, neutral stability (lower branch) is found to be governed by disturbance modes of large axial wavelength (of order hR, where h is a characteristic cross-sectional dimension). By contrast, spatial instability for finite aspect ratios is governed by length scales between O(h) and O(hR). When the aspect ratio is increased to O(R1/7), however, these two characteristic length scales both become O(R1/7h) and a match with plane channel flow instability is achieved. Thus the general cross-section produces temporal and spatial instability if the aspect ratio is O(R1/7). Further, in the flow in a rectangular pipe neutral stability (lower branch) exists for some finite aspect ratios, while for the flow in any non-circular elliptical pipe spatial instability is possible. It is suggested that both temporal and spatial instability occur for a wide range of pipe cross-sections of finite aspect ratio. Part 2 (Smith 1979a), which studies the upper branch neutral stability, confirms the importance of the O(hR) scale modes in neutral stability for finite aspect ratios.

We determine all continuous translation invariant simple valuations on the space of convex bodies in d-dimensional Euclidean space.

Let N be a positive integer. We are concerned with the sum

Thus GN(N) is the ordinary Gauss sum. Previous methods of estimating such exponential sums have not brought to light the peculiar behaviour of GN(m) for m < N/2, namely that, for almost all values of m, GN(m) is in the vicinity of the point . A sharp estimate is given for max |GN(m)|, depending on the residue of N modulo 4. The results were suggested by graphs of GN(m) made for N near 1000. The analysis employs the Fresnel integrals and the Cornu spiral whose curvature is proportional to its arc length.

Let x, y, Q denote large real numbers, with x > y > Q. The object of this paper is to prove the following result

for arbitrary A > 0, with a larger value of Q than hitherto. The symbol denotes as usual the suppression of an absolute constant, π(N; q, a) denotes the number of prime numbers up to N which are congruent to a (mod q), and ϕ(q) denotes Euler's function.

Interactions between a finite number of bodies and the surrounding fluid, in a channel for instance, are investigated theoretically. In the planar model here the bodies or modelled grains are thin solid bodies free to move in a nearly parallel formation within a quasi-inviscid fluid. The investigation involves numerical and analytical studies and comparisons. The three main features that appear are a linear instability about a state of uniform motion, a clashing of the bodies (or of a body with a side wall) within a finite scaled time when nonlinear interaction takes effect, and a continuum-limit description of the body–fluid interaction holding for the case of many bodies.

We let K(a1a2, ak) denote the continuant formed from the positive integers a1 …, ak; that is,

Of course, K(a1, …, an) is the denominator of the continued fraction

We use the convention that K (empty set) = 1.

The object of this paper is to give a simple proof of Menger's famous theorem [1] for undirected and for directed graphs. Proofs of this theorem have been given by D. König [2], G. Nöbeling [3], G. Hajós [4], T. Gallai [5], P. Erdös [6] and 0. Ore [7]. The present proof is shorter, and formulated to apply to directed and undirected graphs equally. The term, graph is to be understood to mean either a finite undirected graph or a finite directed graph throughout. (A: B) denotes either an undirected edge between the two vertices A and B or a directed edge from A to B according to whether undirected or directed graphs are considered. G1 and G2 being two non-empty disjoint graphs, G1: G2-edge denotes an undirected edge between a vertex of G1 and a vertex of G2 or a directed edge from a vertex of G1 to a vertex of G2 as the case may be, and G1: G2-path denotes a path with one end-vertex in G1 and one in G2 having no intermediate vertex in G1 + G2, undirected or directed from the end in G1 to the end in G2, as the case may be. (A set of isolated vertices is a graph.)

Given a Banach space X and a norming subspace Z⊂X*, a geometrical method is introduced to characterize the existence of an equivalent σ(X, Z)-lsc LUR norm on X. A new simple proof of the Theorem of Troyanski: every rotund space with a Kadec norm is LUR renormable, and a generalization of the Moltó, Orihuela and Troyanski characterization of the LUR renormability, are provided without probability arguments. Among other applications, it is shown that a dual Banach space with a w*-Kadec norm admits a dual LUR norm.

We prove that an algebraic subvariety of a Shimura variety is weakly special if and only if analytic components of its preimage in the symmetric space are algebraic. We also prove an analogous result in the case of abelian varieties.

Let 0 < γ ≤ γ” denote the ordinates of consecutive nontrivial zeros of ζ(s) and set

The nonlinear interactions that evolve between a planar or nearly planar Tollmien-Schlichting (TS) wave and the associated longitudinal vortices are considered theoretically, for a boundary layer at high Reynolds numbers. The vortex flow is either induced by the TS nonlinear forcing or is input upstream, and similarly for the nonlinear wave development. Three major kinds of nonlinear spatial evolution, Types I-III, are found. Each can start from secondary instability and then becomes nonlinear, Type I proving to be relatively benign but able to act as a pre-cursor to the Types II, III which turn out to be very powerful nonlinear interactions. Type II involves faster streamwise dependence and leads to a finite-distance blow-up in the amplitudes, which then triggers the full nonlinear three-dimensional triple-deck response, thus entirely altering the mean-flow profile locally. In contrast, Type III involves slower streamwise dependence but a faster spanwise response, with a small TS amplitude thereby causing an enhanced vortex effect which, again, is substantial enough to entirely alter the mean-flow profile, on a more global scale. Concentrated spanwise formations in which there is localized focussing of streamwise vorticity and/or wave amplitude can appear, and certain of the nonlinear features also suggest by-pass processes for transition and significant change in the flow structure downstream. The powerful nonlinear 3D interactions II, III seem potentially very relevant to experimental and computational findings in fully fledged transition; in particular, it is suggested in an appendix that the Type-Ill interaction can terminate in a form of 3D boundary-layer separation which appears possibly connected with the formation of lambda vortices in practice.

1. A study of the recent papers of Roth and Bombieri on the large sieve has led us to the following simple result on the sum of the squares of the absolute values of a trigonometric polynomial at a finite set of points.

An analogy is drawn between the slow steady rotation of a solid of revolution about its axis in a viscous fluid and the motion, with constant velocity, of the solid along its axis in a perfect fluid. The rotation of a sphere in viscous fluid is then discussed using a method due in this case to Bickley [1] and here further developed.

Let be a positive definite quadratic form in n variables. Here x denotes a column vector with components x1, x2, …, xn, and we write A for the symmetric matrix {ajk} so that

In this note those quotient groups of the absolute class group of number fields are to be studied which can be described in terms of absolutely Abelian fields. This investigation will be based on a suitable generalization of the classical concepts of the principal genus, the genus group and the genus field. One possible description of the genus group in a cyclic field is that as the maximal quotient group of the absolute class group which is characterized by rational congruence conditions, i.e. in terms of rational residue characters. From this point of view, however, the restriction to cyclic—or Abelian—fields is quite artificial; the given description can thus be taken as the definition of the genus group in any finite number field. In general the genus field will then no longer be absolutely Abelian; it can now be described as the maximal non-ramified extension obtained by composing the given field with absolutely Abelian fields.

The two inter-related aspects of this laminar flow study are, first, the effects of indentations of length O(a) and height O(aK-⅓) on an otherwise fully developed pipeflow and, second, the manner in which such a pipeflow adjusts ahead of any nonsymmetric distortion to the downstream conditions. Here K is the typical Reynolds number, assumed large, and a is the pipewidth. The flow structure produced by the particular slowly varying indentation, or by a suitable distribution of injection, comprises an inviscid core, effectively undisplaced, and a viscous wall-layer, where the swirl velocity attains values much greater than in the core and where the nonlinear governing equations involve the unknown pressure force. Linearized solutions for finite-length, unbounded or point indentations, and for finite blowing sections (which model the influence of a tube-branching), demonstrate the upstream influence inherent in the nonlinear problem, for steady or unsteady disturbances. It is suggested that the upstream interaction caused there provides the means for the upstream response in the general case where the indentation, say, produces a finite constriction of the tubewidth.

Using the polynomial method of Dvir [On the size of Kakeya sets in finite fields. Preprint], we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties W over finite fields F. For instance, given an (n−1)-dimensional projective variety W⊂¶n(F), we establish the Kakeya maximal estimate for all functions f:Fn→R and d≥1, where for each w∈W, the supremum is over all irreducible algebraic curves in Fn of degree at most d that pass through w but do not lie in W, and with Cn,W,d depending only on n,d and the degree of W; the special case when W is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.

It is shown that, for any lattice polytope P⊂ℝd the set int (P)∩lℤd (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most 8d · (8l+7)22d+1. If, moreover, P is a simplex, then this bound can be improved to 8 · (8l+7 )2d+1. As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior.

In 1955 a programme of study of the first 10000 zeros of the Riemann Zeta-function

was completed. Use was made of the high-speed digital computer SWAC and a report of this programme has appeared recently [1]. More recently still, the programme has been extended to the first 25000 zeros. All these zeros have σ= ½ The purpose of this paper is to summarize the methods needed for this (and possibly future) work from the highspeed computer point of view.