The cyclic tour property has previously been an equality for the expected time to complete a tour, compared with that for the reverse tour, for reversible Markov chains. We give a simple bijection to show that the equality can be extended to the distributions involved. The bijection is based on rotation of circular words.

A long time ago Ju. E. Vapne ([2], [3]) and, independently, the author ([4], [5]) classified those standard and complete wreath products that have faithful representations of finite degree over (commutative) fields. See [6] pages 37 40 & 150–154 for an account of this. Recently, in connection with finitary linear groups, I needed a more general wreath product. Somewhat to my surprise neither the classification nor the proof for these generalized wreath products was a straightforward translation from the standard case. The situation is intrinsically more complex and it seems worthwhile recording it separately.

We study the fraction of time that a Markov chain spends in a given subset of states. We give an exponential bound on the probability that it exceeds its expectation by a constant factor. Our bound depends on the mixing properties of the chain, and is asymptotically optimal for a certain class of Markov chains. It beats the best previously known results in this direction. We present an application to the leader election problem.

A blob of viscous Newtonian fluid is surrounded by inviscid fluid and sandwiched in the narrow gap between two plane parallel surfaces, so that initially its plan view occupies a simply connected domain. Recently Entov, Etingoff & Kleinbock (1993) produced some steady-state solutions for the blob placed in a quadrupole driven flow, and including the effects of surface tension. Here a numerical solution of the time-dependent problem using a Boundary Integral algorithm finds that for low values of the flow rate there exist two solutions. We find that one, which is close in shape to a circle, is stable, while the other, more deformed equilibrium, is unstable. The analysis also reveals that for certain flow strengths stable non-convex shapes also exist. If the flow strength is too large no stable equilibrium is possible.

In [CKR], Chan, Kim and Raghavan determine all universal positive ternary integral quadratic forms over real quadratic number fields. In this context, universal means that the form represents all totally positive elements of the ring of integers of the underlying field. This generalizes the usage of the term introduced by Dickson for the case of the ring of rational integers [D]. In the present paper, we will continue the investigation of quadratic forms with this property, considering positive quaternary forms over totally real number fields. The main goal of the paper is to prove that if E is a totally real number field of odd degree over the field of rational numbers, then there are at most finitely many inequivalent universal positive quaternary quadratic forms over the ring of integers of E. In fact, the stronger result will be proved that this finiteness holds for those forms which represent all totally positive multiples of any fixed totally positive integer. The necessity of the assumption of oddness of the degree of the extension for a general result of this type can be seen from the existence of universal ternary forms over certain real quadratic fields (for example, the sum of three squares over the field ℚ(√5), as first shown by Maass [M]).

The Pólya–Szegö dipole tensors are employed for the analysis of plane elasticity problems in non-homogeneous media. Classes of equivalence for defects (cavities and rigid inclusions) are specified for the Laplacian operator and elasticity equations: composite materials with defects of the same class have the same effective elastic moduli. Explicit asymptotic formulae for effective compliance matrices of dilute composites are presented.

Existence, uniqueness and regularity properties are established for monotone travelling wavesof a convolution double-obstacle problem

ut=J*u−u−f(u),

the solution u(x, t) being restricted to taking values in the interval [−1, 1]. When u=±1, the equation becomes an inequality. Here the kernel J of the convolution is nonnegative with unit integral and f satisfies f(−1)>0>f(1). This is an extension of the theory in Bates et al. (1997), which deals with this same equation, without the constraint, when f is bistable. Among many other things, it is found that the travelling wave profile u(x−ct) is always ±1 for sufficiently large positive or negative values of its argument, and a necessary and sufficient condition is given for it to be piecewise constant, jumping from −1 to 1 at a single point.

We investigate the distribution of the hitting time T defined by the first visit of the Brownian motion on the Sierpiński gasket at geodesic distance r from the origin. For this purpose we perform a precise analysis of the moment generating function of the random variable T. The key result is an explicit description of the analytic behaviour of the Laplace- Stieltjes transform of the distribution function of T. This yields a series expansion for the distribution function and the asymptotics for t →0.

For a fixed integer k ≥ 2 suppose we have functions Li (x) satisfying the following conditions.

Ramanujan's work on the asymptotic behaviour of the hypergeometric function has been recently refined to the zero-balanced Gaussian hypergeometric function F(a, b; a + b; x) as x→1.We extend these results for F(a, b; c; x) when a, b, c>0 and c<a + b.

The motion of an initially circular drop of viscous fluid surrounded by inviscid fluid in a Hele-Shaw cell withdrawn from an eccentric point sink is considered. Using a numerical algorithm based on a boundary integral equation, the solution for small, finite surface tension is observed. It is found that the zero-surface-tension formation of a cusp is avoided, and instead a narrow finger of inviscid fluid forms, which then rapidly propagates towards the sink. The scaling of the finger in the sink vicinity is determined.

For i = 1,…, n let ai be a homogeneous polynomial of degree ri(>0) in the graded polynomial ring R[x1, …, xm], or R[x] for short, where R is a commutative ring with unity and x1, …, xm are indeterminates of degree 1. Let of degree - 1 be a formal inverse of xj and let U denote the graded R[x]-module In [2, §2] we introduced a graded complex of r-modules.

The probability density for the solution yn of a stochastic difference equation is considered. Following Knessl et al. [1], it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a difference approximation to the Helmholtz equation in a random medium. A large deviation result is obtained for the probability density of the decay rate of a solution of this equation. Both the exponent and the pre-exponential factor are determined.

A random polytope, Kn, is the convex hull of n points chosen randomly, independently, and uniformly from a convex body It is shown here that, with high probability, Kn can be obtained by taking the convex hull of m = o(n) points chosen independently and uniformly from a small neighbourhood of the boundary of K.

We show that a measure on ℝd is linearly rectifiable if, and only if, the lower l-density is positive and finite and agrees with the lower average l-density almost everywhere.

Given a p-subgroup P of a finite group G we express the number of p-blocks of G with defect group P as the p-rank of a symmetric integer matrix indexed by the N(P)/P-conjugacy classes in PC(P)/P. We obtain a combinatorial criterion for P to be a defect group in G.

The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, its probability distribution decays geometrically, and the dynamics are characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented.