We determine infinite products in the field of Laurent series with the property that the truncations of the product yield every second continued fraction convergent of the product. We mention some related examples and specialize to obtain numerical results.

The flow induced by an oscillating circular cylinder which may perform transverse, torsional and axial vibrations is considered. The steady streaming associated with purely transverse vibrations of the cylinder may be significantly modified by the presence of, and interaction with, torsional oscillations. Similarly the interaction between the transverse and axial vibrations introduces a modification to the axial flow, which results in a steady streaming motion in the axial direction.

Solutions of the differential equation

for large positive values of the parameter u, are considered for ζ in some domain Δ which includes the turning-point ζ = 0. The functions ψ(ζ) and ω(ζ) are holomorphic for ζ є Δ

One may perhaps doubt whether in the geometry of numbers any particular family of lattices deserves such an attention as, for example, BCH codes receive in coding theory. However, only recently a quite interesting family has emerged. The general case of these lattices considered by Rosenbloom and Tsfasman [5, Section 2] parallels Goppa's construction of codes from algebraic curves. Here we shall take a closer look at the case of genus zero where some special features of Goppa's early codes will show up again: There is a lattice Λ (L, g) in n-dimensional euclidean space associated with a subset L of the field, and a polynomial g satisfying g(є) ≠ for all λ є L. For g = zd previously known sphere packings are recovered and generalized. A nonconstructive argument shows that for n → ∞ and some irreducible polynomials g Minkowski's lower packing bound is met (this being not achieved in [5] where q is fixed, but the genus grows; cf. also [4]).

Using several transformation formulae from Ramanujan's second Notebook we achieve distribution results on random variables related to dynamic data structures (so-called “tries”). This continues research of Knuth, Flajolet and others via an approach that is completely new in this subject.

Fifty years ago Marcinkiewicz and Zygmund studied the circular structure of the limit points of the partial sums for (C, 1) summable Taylor series. More specifically, let

be a power series with complex coefficients, let

be the partial sums, and let

be the Cesàro averages. When the sequence σn(z) converges to a finite limit σ(Z), we say that the Taylor series is (C, 1) summable and σ(z) is the (C, 1) sum of the series. Concerning (C, 1) summable Taylor series Marcinkiewicz and Zygmund ([5], [6] Vol. II, p. 178) established the following theorem.

This article considers the effect of more than one quotient and improves a theorem of Tong which is a generalization of a theorem of Segre on asymmetric approximation.

The problem of finding rational points on varieties defined by two additive cubic equations has attracted some interest. Davenport and Lewis [12], Cook [8] and Vaughan [16] showed that the pair of equations

with integer coefficients a,, bt always has a nontrivial solution when s = 18, s = 17, and 5 = 16 respectively. Vaughan's result in s = 16 variables is best possible since there are examples of pairs of equations (1) with s = 15 which fail to vanish simultaneously in the 7-adic field. However if the existence of a 7-adic solution is assured then Baker and Briidern [2], building on work of Cook [9], showed that s = 16 could be replaced by s = 15, and recently Briidern [5] has obtained the result with s = 14.

In the part (16-3) of his extensive study on measurability in Banach spaces, Talagrand [12] considered the Banach space C(K) of continuous functions on a dyadic topological space K. He proved that C(K) is realcompact in its weak topology, if, and only if, the topological weight of K is not a twomeasurable cardinal (Theorem 16-3-1). Then he asked for an alternative to a rather complicated proof presented there (p. 214) and posed the problem whether C(K) is measure-compact whenever the weight of K is not a realmeasurable cardinal (Problem 16-3-2).

Let Q(x) = Q(x1,…, xn) є ęZ x1, …, xn] be a quadratic form. The primary purpose of this paper is to bound the smallest non-zero solution of the congruence Q(x) = 0 (mod q). The problem may be formulated as follows. We ask for the least bound Bn(q) such that, for any Ki > 0 satisfying

and any Q, the congruence has a non-zero solution satisfying

Soit Y un sous-ensemble algébrique de codimension 1 de R'. Une distribution globale (resp. partielle) de signes sur Rn est une application qui associe un signe a chaque composante connexe (resp. à certaines parmi les composantes connexes) de Rn – Y.

Let J = (s1, s2, … ) be a collection of relatively prime integers, and suppose that π(n) = |J∩{1,2,…, n}| is a regularly varying function with index a satisfying 0 < α < l. We investigate the “stationary random sieve” generated by J, proving that the number of integers less than k which escape the action of the sieve has a probability mass function with approximate order k-α/2 in the limit as k → ∞. This result may be used to deduce certain asymptotic properties of the set of integers which are divisible by no s є J, in that it gives new information about the usual deterministic (that is, non-random) sieve. This work extends previous results valid when si=pi2, the square of the ith prime.

All manifolds in this paper are assumed to be closed, oriented and smooth.

A contact structure on a (2n + l)-dimensional manifold M is a maximally non-integrable hyperplane distribution D in the tangent bundle TM, i.e., D is locally denned as the kernel of a 1-form α satisfying α ۸ (da)n ۸ 0. A global form satisfying this condition is called a contact form. In the situations we are dealing with, every contact structure will be given by a contact form (see [5]). A manifold admitting a contact structure is called a contact manifold.

Almost seventy years ago Jeffery [1] showed that a finite velocity can result at infinity when the biharmonic equation is solved for the titled problem. Here, we extend his calculations to show that finite vorticity is the more general conclusion, and then indicate a resolution of the apparent paradox.

On a convex surface S ⊂ Rd, two points x, y are conjugate if there are at least two shortest paths, called segments, from x to y. This paper is about the set of points conjugate to some fixed point xєS.

In this paper two expansions are obtained by contour integration methods for the velocity potential describing two-dimensional time-harmonic surface waves due to a free-surface wave source on water of infinite depth in the presence of surface tension. First the series expansion at r = 0 is found and then the asymptotic expansion as Kr®¥, where K is the wave number for progressive waves and r the radial distance from the source. The corresponding expansions for the more important submerged wave source in terms of the radial distance from the image source in the free surface may then easily be deduced. The latter are required in a number of surface wave problems, particularly those of a short-wave asymptotic nature, and are also relevant in obtaining expansions for finite constant depth.

A right S-system over a monoid S is a set A on which S acts unitarily on the right. That is, there is a function A such that (φ,1)φ and (a, st)φ = ((a, s) t)φ for all a є A and for all s, t є S. We shall refer to right S-systems simply as S-systems. It is clear what is meant by S-homomorphism, S-subsystem etc.; further details of the terms used in this Introduction are given in Section 2.

A. Bezdek and W. Kuperberg constructed a nonlattice packing of congruent ellipsoids in Euclidean 3-space E3 with density 0·7459 …, which exceeds the density σL2 = 0·74048… of the densest lattice packing of spheres and hence of ellipsoids in E3. G. Kuperberg improved this to 0·7533… We improve this slightly to 0·7549…. In our case the quotient of the largest and the smallest halfaxis of the ellipsoids is <42, so the ellipsoids are not too degenerate. If one combines G. Kuperberg's refinement and ours, one obtains a packing density of 0·7585…

Let X be a reflexive Banach space. This article presents a number of new characterizations of the topology of Mosco convergence TM for convex sets and functions in terms of natural geometric operators and functional. In addition, necessary and sufficient conditions are given for TM to agree with the weak topology generated by {d(x, C): x є X}, where each distance functional is viewed as a function of the set argument.

In typical linear programming problems, we are concerned with finding non-negative integers {x1,…, xn} that maximize a linear form c1x1 + … + cnxn, subject to a number of linear inequalities, for The maximum is necessarily attained at one of the vertices of the convex hull of integer points defined by the inequalities, so we have an interest in estimating the number M of these vertices. We give two results; one improving an upper bound result for M of Hayes and Larman concerning the Knapsack polytope, the other an example showing that, in 3-dimensions, it is possible to choose the coefficients aij to obtain a lower bound for M.