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.

In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ1 …, λr be any real numbers. Then there exists a real number α such that

for all integers u > 0 and for q = 1,…,r, where C = C(r) > 0. Thus all the numbers α+ λ1, …, α+ λr are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ1, …, λr and μ1, …, μr there exist α, β such that

for all integers u > 0 and for q=1,…, r, where again C = C(r) > 0. Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.

Let Q(x, y) = ax2 + bxy + cy2 be a positive definite quadratic form with discriminant d = b2 – 4ac. The Epstein zeta function associated with Q is given by

where Σ′ means the sum is over all pairs (x, y) of integers not both zero, and as usual, s = σ + it.

Let k be a knot of type K and with group πK. Let θ: nK → PSL(ℂ) = PSL (2, ℂ) be a parabolic representation (p-rep) as defined in [14]. We shall call the representation discrete when its image πKθ is a discrete subgroup of PSL(ℂ). It is known that PSL(ℂ) can be identified with the group of orientation preserving isometries of hyperbolic 3-space , and that each discrete subgroup of PSL(ℂ) acts discontinuously on . Hence each discrete p-rep θ has an associated orbit space . The present paper is a study of the general relations between the algebraic properties of a discrete image πKθ and the geometric properties of its orbit space.

The following well-known conjecture is generally attributed to Minkowski:

Let L1, …, Ln be n real homogeneous linear forms of determinant Δ ≠ 0 in the n variables x1, …, xn; and let (x1′, …, xn′) be any point. Then there exists a point (x1, xn) congruent to (x1′, …, xn′) (mod 1) at which

Let be k non-overlapping translates of the unit d-ball Bd in euclidean d-space Ed. Let Ck denote the convex hull of their centres and let Sk be a segment of length 2(k– 1). Furthermore, let Vd denote the d-volume. L. Fejes Tóth conjectured in [1], that, for d ≥ 5,

Let be the class of normalised polynomials of the form:

of degree n which are univalent in |z| < 1, and let be the class of “polynomials” of the form

of degree n which are univalent in 0 < |z| < 1. In this note we discuss the coefficients of polynomials in (k = 2, 3, 4) and (k = 2, 3).

For any fixed integer a and real variables x, y with y < x let Na(x, y) denote the number of primes p ≤ x for which p + a has at least one prime factor greater than y. As an elementary application of the following deep theorem of Bombieri on arithmetic progressions,

Theorem (Bombieri, [1]). For any constant A > 0, there exists a positive constant B such that ifwith l = log x, then for x > 1

where Φ(x; m, a) denotes the number of primes less than x which are congruent to a mod m;

we shall prove the following theorem:

Theorem 1. Let a be any fixed integer and let x > e. We then have

where the double sum is taken over primes p and q.

This paper treats the reducibility of the quasiperiodic linear differential equations

where A is a constant matrix with multiple eigenvalues, Q(t) is a quasiperiodic matrix with respect to time t, and ε is a small perturbation parameter. Under some non-resonant conditions, rapidly convergent methods prove that, for most sufficiently small ε, the differential equations are reducible to a constant coefficient differential equation by means of a quasiperiodic change of variables with the same frequencies as Q(t).

The purpose of this paper is to provide additional evidence to support our view that the modules of generalized fractions introduced in [8] are worth further investigation: we show that, for a module M over a (commutative, Noetherian) local ring A (with identity) having maximal ideal m and dimension n, the n-th local cohomology module may be viewed as a module of generalized fractions of M with respect to a certain triangular subset of An + 1, and we use this work to formulate Hochster's ‘Monomial Conjecture’ [2, Conjecture 1]; in terms of modules of generalized fractions and to make a quick deduction of one of Hochster's results which supports that conjecture.

The study of linear stability of a layer of stratified fluid in horizontal shearing motion leads, in the absence of diffusive effects, to a second order differential equation, often called the Taylor-Goldstein equation. This equation possesses a singularity at any critical point, i.e. at any point at which the flow speed, U, is equal to the wave speed, c. If c is complex, a similar singularity arises at any point at which the analytic extension of U into the complex plane is equal to c. Assuming the stratification is thermal in origin, the introduction of a small viscosity and heat conductivity removes this singularity, but leads to a governing equation of sixth order, four solutions being of rapidly varying WKBJ form. The circumstances in which the remaining two solutions can be uniformly represented in the limit of small viscosity and conductivity by the solutions of the Taylor-Goldstein equation are examined in this paper.

In this paper n always denotes an arbitrary but fixed positive integer. Let S be a subset of n-dimensional euclidean space En and (Si) = (S1, S2, …) a finite or infinite sequence of subsets of En. The sequence (Si) is called a covering of S if S ⊂ ⋃Si, and a packing in S if ⋃Si ⊂ S and int Si, ⋂ int Sj = Ø (for all i ≠ j). We say that (Si) permits an isometric covering of S or packing in S if there are rigid motions σi so that (σiSi) is a covering of S or a packing in S, respectively. If there are not only rigid motions but translations τi so that (τiSi) is a covering or packing, we express this by saying that (Si) permits a translative covering or packing. We consider sequences (Si) rather than sets {Si} not because the ordering is of any importance but because some of the sets Si may appear repeatedly.

In this paper we obtain matroid extensions of two important results in graph theory, namely the 4-colour theorem of Appel and Haken [1] and the 8-flow theorem of Jaeger [4]. As a corollary we prove that any bridgeless graph with no subgraph contractible to K3,3 has a nowhere zero 4-flow. These results depend heavily on a remarkable theory of splitters developed recently by Seymour [8], [9].

A recent paper [1] indicates that the beginnings of dynamic stall, near an aerofoil's leading edge, for instance, can be regarded as the finite-time nonlinear breakdown of a boundary layer subjected to an angle of attack above the critical value for the existence of a steady solution. The present theoretical study shows that the same non-linear breakdown can occur even in the below-critical regime. This happens particularly when reversed flow is present since short wavelength disturbances are then unstable and accumulate, for certain confined initial conditions, to force the finite-time collapse. A number of marginal cases with forward or reversed, subsonic or supersonic, oncoming motion are also noted and shed extra light on the instability and subsequent breakdown.

If an is a sequence of numbers between 0 and 1, then

has infinitely many integral solutions n, l either for almost all real x or for almost no real x[1,4]. Duffin and Schaeffer [2], improving on an earlier theorem of Khintchine [7], proved that for decreasing sequences an, (1) has infinitely many solutions a.e. if and only if Σan diverges. They also gave an example of a sequence an for which Σan diverges, but for which (1) has only finitely many solutions a.e. No general necessary and sufficient condition for (1) to have infinitely many solutions a.e. is known.

A random polytope is the convex hull of n random points in the interior of a convex body K. The expectation of the ith intrinsic volume of a random polytope as n → ∞ is investigated. It is proved that, for convex bodies of differentiability class Kk+1, precise asymptotic expansions for these expectations exist. The proof makes essential use of a refinement of Crofton's boundary theorem.

For a cell compex M decomposing the closed orientable 2-manifold Pg of genus g let pi (M) and νj(M) denote the number of i-gonal cells (faces, countries) and j-valent vertices of the graph (= 1 – skeleton) of M, respectively. It will be supposed that i, j ≥ 3. From Euler's formula follows

and

is even, where e is the number of edges of M. As seen, the relation above does not impose restrictions on the numbers p4 (M), ν4(M). In an attempt to characterize the vectors {pi(M)}, {νi(M)}which partially determine the combinatorial structure of the cell-decompositions, the first step could involve answering the question: Given sequences p = (p3, p5, …), ν = (ν3, ν5, …) of non-negative integers satisfying conditions

and

does there exist a cell-decomposition M of Pg, for which pi (M) = pi, νj(M) = νj for all i, j ≠ 4? (If so, the sequences p, ν are called realizable in the sequel. M itself is a realization of p, ν. For brevity's sake we will often use the word map instead of cell-decomposition in the sequel.)

For x, y ≥ 1, let Ψ(x, y) denote the number of positive integers less than or equal to x and free of prime factors greater than y. The behaviour of the function Ψ(x, y) has been the object of numerous articles (see e.g. Norton's memoir [5] and the bibliography there). It turns out that a good approximation to ψ(x, y)/x is given by ρ(log x/log y), where the function ρ(t) is defined for t ≥ 0 as the continuous solution of the equations

This functional equation for Dedekind's

has been established in many different ways; however, it seems that the following proof, a straightforward application of the theorem of residues, has not been observed before. Since

it suffices to prove that

C. M. Petty has conjectured the minimum value for a certain affine-invariant functional denned on the class of convex bodies. We give sharp bounds for this functional on a certain subclass of convex bodies, and we give a counterexample to an upper bound proposed by R. Schneider for the class of centrally symmetric convex bodies. We conjecture that the simplex provides the maximum on the class of all convex bodies, while the largest centrally symmetric subset of a simplex gives a sharp upper bound on the class of all centrally symmetric convex bodies.