Suppose that {Xn} is a countable non-homogeneous Markov chain and

If converges for any i, l, m, j with , then

whenever lim , whereas if converges, then

where and . The behaviour of transition probabilities between various groups of states is studied and criteria for recurrence and transience are given.

Let the stationary point process N(·) be the mixture of scaled versions of a stationary orderly point process N1(·) of unit intensity with mixing distribution G(·), so that

With

N(·) has finite or infinite intensity as is finite or infinite, and it is Khinchin orderly when the function γ(·) is slowly varying at infinity. Conditions for N(·) to be orderly involve both G(·) and the Palm distribution of N1(·).

Let K ⊂ ℝ3 be an open bounded strictly convex domain with smooth connected compact boundary ∂K. Set

We wish to study the filtered scattering amplitude, related to the transmission problem

Here w± are the limiting values of w on ∂Ω; from the Ω;± side, while ∂w±/∂n are the corresponding limiting values of the normal derivative ∂w?/∂n on ∂Ω;. The function α(x) ∈ C∞(K¯), called the index of refraction, has the property α(x) > 0 for x ∈ K¯.

Let Xi be iidrv's and Sn = X1 + X2 + … + Xn. Let (r)Sn be Sn minus the r terms of largest absolute value. If (Sn/Bn)− An converges to normality for some Bn, An, then so does ((r)Sn/Bn) − An. We show that the converse is true if Xi have a continuous symmetric distribution, and give some related results.

Roe (7) has recently given a striking characterization of the sine function as essentially the only function defined on the real line which has all its derivatives and successive ‘integrals’ bounded by a uniform bound, and this result has since been generalized by Burkill (5). In this note we relate this work to Banach algebra theory in a natural way by demonstrating the equivalence of Theorems 1 and 2 below, and also by giving a direct proof of Theorem 1. (Theorem 2 is Burkill's generalization of Roe's result stated in complex form.) We enunciate a Banach space version of the result (Theorem 3), and a further generalization of all these results is given in Theorem 4.

In this note we examine the influence of surface tension on surface waves incident against a fixed vertical plane wall. The motion is time harmonic and is determined by making the initial assumption that the free-surface slope at the wall is prescribed. From the unique solution obtained for the velocity potential, the parameter involved in this specification can be determined, for small laboratory-scale waves at least, using some longstanding experimental results on meniscus behaviour at a moving contact line. The effect of surface tension is to produce a motion wherein reflexion from the wall is not complete and there is a local disturbance, in contrast to the classical standing-wave motion in the absence of surface tension.

1. A series with partial sums {An} is said to be summable (e, c) (c > 0) if

exists, where it is to be understood that An+k = 0 when n+k < 0. The (e, c)-sum-mability method which is a regular method of summation was introduced by Hardy and Littlewood (3) (cf. also (5)) as an auxiliary method to prove the Tauberian theorem for Borel summability, viz, if Σan is summable (B) to A and an = 0(n−½), then it converges to A.

Let F be a distribution function with support confined to ( − π, π), W a Wiener process on (0, 1), and W0 a Brownian bridge. We examine the order of the approximation to W(F) and W0(F) by their Fourier series. The results we obtain are of theoretical interest since the Wiener process possesses the unusual property of satisfying a Lipschitz condition of order ∝ for each 0 < ∝ < ½, but not one of order ∝. = ½. The classical results on the rate of convergence of Fourier series are not sufficiently specific to provide an exact rate in the case of the Wiener process. The work also has practical significance because many statistical procedures can be closely approximated by functions of a Wiener process, and several procedures rely heavily on Fourier series methods. Thus, some of our results may be applied to obtain limit theorems for quantities such as trigonometric series estimates of density and distribution functions.

N. Martin. ‘Cobordism of homology manifolds’

A. W. Jessop(10) has pointed out that the proof of Lemma 4·5 is wrong as the N D condition enabling the inductive use of Theorem 4·3 will not hold in the case where r = 2m + 3.

In what follows Theorem 4·3 and Lemma 4·5 are restated with slightly different conditions but with essentially similar conclusions. References to diagrams are those in the original. Paper references are those of the original together with others listed at the end here.

Let M be a compact Riemannian manifold equipped with Laplace–Beltrami operator Δ. We use the Rademacher-Menchoff theorem and the asymptotics of the eigenvalues of Δ to show that if a function belongs to an L2 Sobolev space of positive index then its expansion, in terms of eigenfunctions of Δ, converges almost everywhere on M.

Every smooth, closed, orientable manifold of dimension 3 contains a fibred knot, i.e. an imbedded circle K such that M – K is the total space of a fibre bundle over S1 whose fibre F is standard near K. This means the boundary of the closure of F is K so that K is null-homologous in M. A natural problem suggested by Rolfsen (see (2), problem 3·13) is to determine which elements of π1M3 may be represented by fibred knots. We deal with this by proving the

Theorem. The collection of all elements of π1 M3 which can be represented by fibred knots is exactly the commutator subgroup [π1 M3, π1 M3].

N. Martin. ‘Transverse regularity for maps of homology manifolds’

An example given by Galewski and Stern (l) showed that Theorem A of the original paper was false as stated. The error is traceable to the codimension three condition which is not sufficient here.

E. M. Wright. ‘A quadratic recurrence of Faltung type.’

Middle of p. 197. The value of x( − ⅓) should be

x (− ⅓) = 0·351298985… = ei−1 × 0·95492965…

The error was mine and not Dr Ponting's.

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

The principal purpose of this note is to give an elementary proof of an inequality, due to Pisier (proposition 2·4 of (4)), for martingales taking values in a Banach space X. This proof yields an explicit estimate, sharp when X is a Hilbert space, for one of the constants involved. The notation follows (1, 4).

In finite elasticity a homogeneous strain is not always determined uniquely by the conjugate stress. The phenomenon is investigated for any constitutive work-function, and in both its global and local aspects. Uniqueness is lost in specific critical states. Basic properties of their aggregates in the stress and strain spaces are derived, and shown to be fundamental for the material response near individual critical states. The geometry of local branches is analyzed in detail. Further properties resulting from material isotropy are obtained. The general theory is illustrated by examples typifying the responses of metal crystals and rubberlike polymers.