The scattering of high-frequency sound waves by two-dimensional curved boundaries has received much attention over the past few decades, with particular interest in the effects of tangential ray incidence. In the event that the radius of curvature is not small, an analysis near the point of tangency gives rise to the Fock–Leontovič equation for the local field amplitude which, in turn, matches the creeping field of Keller's geometrical theory of diffraction. If the radius of curvature is sufficiently small, however, then this analysis is not valid and it is necessary to solve the full Helmholtz equation in the presence of a parabolic boundary. Under these conditions, which are canonical for diffraction by a sufficiently slender body, results are presented for the case of a plane wave impinging upon an acoustically hard parabolic cylinder. This diffraction process engenders a creeping field at one tip of the slender body, which then propagates around the body to the other tip. Here its energy is partially reflected, partially transmitted and partially radiated out in a detached field. A full description of this is given, along with a discussion of the ‘blunt’ limit in which we show that not only do we get the traditional creeping field of Keller's geometrical theory of diffraction, but also an exponentially small backward-propagating creeping field not predicted by traditional ray methods.

Formal hyperasymptotic expansions of integrals are obtained from the Poincaré asymptotic expansion by re-expanding the remainder term. We show that for the integrals under consideration these hyperasymptotic expansions can be made accurate to any desired exponentially-small order.

This paper addresses short-time existence and uniqueness of a solution to the N-dimensional Hele–Shaw flow problem with surface tension as driving mechanism. Global existence in time and exponential decay of the solution near equilibrium are also proved. The results are obtained in Sobolev spaces Hs with sufficiently large s. The main tools are perturbations of a fixed reference domain, linearization with respect to these perturbations, a quasilinearization argument based on a geometric invariance property, and a priori energy estimates.

Analysing the motion of a driven, damped pendulum as a function of the amplitude of the driving force, we show, first, that for moderate values and larger of the amplitude, deviations from a simple motion with the period of the driving force are bounded by a constant times the inverse square root of the amplitude, for late times. For amplitudes above a larger threshold we are able to show that, for late times, the motion becomes a periodic motion with the period of the driving force. The manner in which this periodic motion is achieved with the passage of time is analysed.

An unforced oscillator which obeys the equation x¨+α (x2+x˙2−1)x˙+x=0 has a pure sine-wave limit cycle which is attained rapidly for a range of values of α. In this paper, free and forced oscillation of this equation are examined experimentally and compared with those for the Van der Pol oscillator. Asymptotic solutions for small α confirm the experimental results.

In a recent paper, the author showed that for certain symmetric bisuperlinear equations, cosine-like boundary behaviours will not yield symmetric solutions [1]. In this paper, we attack the adiabatic invariant problem by showing that, for these strongly nonlinear oscillators, the adiabatic invariant is intimately related to z′(0;∈) for a family of solutions.

Given a class of combinatorial structures [Cscr], we consider the quantity N(n, m), the number of multiset constructions [Pscr] (of [Cscr]) of size n having exactly m [Cscr]-components. Under general analytic conditions on the generating function of [Cscr], we derive precise asymptotic estimates for N(n, m), as n→∞ and m varies through all possible values (in general 1[les ]m[les ]n). In particular, we show that the number of [Cscr]-components in a random (assuming a uniform probability measure) [Pscr]-structure of size n obeys asymptotically a convolution law of the Poisson and the geometric distributions. Applications of the results include random mapping patterns, polynomials in finite fields, parameters in additive arithmetical semigroups, etc. This work develops the ‘additive’ counterpart of our previous work on the distribution of the number of prime factors of an integer [20].

We apply an idea of Székely to prove a general upper bound on the number of incidences between a set of m points and a set of n ‘well-behaved’ curves in the plane.

A collection H of integers is called an affine d-cube if there exist d+1 positive integers x0,x1,…, xd so that

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We address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1,2,…,n} not containing an affine d-cube. In 1892 Hilbert published the first Ramsey-type result for affine d-cubes by showing that, for any positive integers r and d, there exists a least number n=h(d,r) so that, for any r-colouring of {1,2,…,n}, there is a monochromatic affine d-cube. Improvements for upper and lower bounds on h(d,r) are given for d>2.

Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.

The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.

Consider first-passage percolation on the square lattice. Welsh, who together with Hammersley introduced the subject in 1963, has formulated a problem about mean first-passage times, which, although seemingly simple, has not been proved in any non-trivial case. In this paper we give a general proof of Welsh's problem.

It is known that any k-uniform family with covering number t has at most ktt-covers. In this paper, we deal with intersecting families and give better upper bounds for the number of t-covers. Let pt(k) be the maximum number of t-covers in any k-uniform intersecting families with covering number t. We prove that, for a fixed t,

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In the cases of t=4 and 5, we also prove that the coefficient of kt−1 in pt(k) is exactly (t2).

Let T be a semicomplete digraph on n vertices. Let ak(T) denote the minimum number of arcs whose addition to T results in a k-connected semicomplete digraph and let rk(T) denote the minimum number of arcs whose reversal in T results in a k-connected semicomplete digraph. We prove that if n[ges ]3k−1, then ak(T)=rk(T). We also show that this bound on n is best possible.

We study the number of comparisons in Hoare's Find algorithm. Using trivariate generating functions, we get an explicit expression for the variance of the number of comparisons, if we search for the jth element in a random permutation of n elements. The variance is also asymptotically evaluated under the assumption that j is proportional to n. Similar results for the number of passes (recursive calls) are given, too.

Let S be a generating subset of a cyclic group G such that 0=∉S and [mid ]S[mid ][ges ]5. We show that the number of sums of the subsets of S is at least min([mid ]G[mid ], 2[mid ]S[mid ]). Our bound is best possible. We obtain similar results for abelian groups and mention the generalization to nonabelian groups.

It is shown that every partially ordered set with n elements admits an endomorphism with an image of a size at least n1/7 but smaller than n. We also prove that there exists a partially ordered set with n elements such that each of its non-trivial endomorphisms has an image of size O((n log n)1/3).

Consider the complete n-graph with independent exponential (mean n) edge-weights. Let M(c, n) be the maximal size of subtree for which the average edge-weight is at most c. It is shown that M(c, n) makes the transition from o(n) to Ω(n) around some critical value c(0), which can be specified in terms of a fixed point of a mapping on probability distributions.

We show that if S1 is a strongly complete sum-free set of positive integers, and if S0 is a finite sum-free set, then, with positive probability, a random sum-free set U contains S0 and is contained in S0∪S1. As a corollary we show that, with positive probability, 2 is the only even element of a random sum-free set.

The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is satisfied. We obtain an explicit formula that gives the dependence of the perturbation growth rate upon the equilibrium considered and the initial data for the perturbation. Secondly, we obtain the upper and lower bounds for growing solutions of the linearized problem, and we identify the initial data corresponding to the most unstable mode (i.e. to the perturbation with maximal growth rate). All results are obtained in quite a general formulation: a nematic is inside a three-dimensional domain of an arbitrary shape and strong anchoring on the boundary is supposed; the standard equations of nematodynamics are employed as the governing equations.

An initial value problem for the functional differential equation

y′(t)=Ay(t)+By(qt)+Cy′(qt)+f(t), t ≥ t0 > 0

where A, B, C are complex matrices, q∈(0, 1), and f is a vector of continuous functions, is considered in this paper. Its solution is represented in terms of the fundamental solution via the variation-of-constants formula. For some special cases, the fundamental solutions are formulated as piecewise Dirichlet series. The variation-of-constants formula is used to analysis the asymptotic behaviour of the solutions of some scalar equations, including one with variable coefficients related to coherent states of the q-oscillator algebra in quantum mechanics.