Suppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), where

and we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)

where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.

We prove that if q is an integer greater than one, r and s are any positive rationals, then

formula here

is irrational and is not a Liouville number.

The Mullineux map is an involutory bijection on the set of p-regular partitions of any given integer n, where a partition is called p-regular if no part of it is repeated p or more times. Many combinatorial properties of the Mullineux map make it reasonable to view this map as a p-analogue of the transposition map T on the set of all partitions.

Based on the work of Kleshchev [7], it has been shown [2, 5, 14] that the Mullineux map M has the following property.

If λ is a p-regular partition of n and Dλ is the p-modular representation of the symmetric group Sn labelled by λ (see [6]) then

formula here

where sgn is the sign representation of Sn. Thus, from a representation theoretic point of view as well, the Mullineux map is a p-analogue of the transposition map T. The Mullineux map plays a vital role not only in the representation theory of symmetric groups but also in other contexts. The definition of M as given in [9] is quite complicated and to study various questions involving this map it is desirable to have other descriptions. One alternative inductive description of M using the concept of good boxes in a p-regular partition was given by Kleshchev [7] and this was used by Walker to prove a result contained in Theorem 4·5 below; this work was motivated by the investigation of Schur modules. In this paper we study a third description of M based on the operator J on the set of p-regular partitions defined in [13].

In this paper we apply singularity theory methods to the classification of reversible-equivariant steady-state bifurcations depending on one real parameter. We assume that the group of symmetries and reversing symmetries is a compact Lie group Γ, and the equivalence is defined in order to preserve these symmetries and reversing symmetries in the normal forms and their unfoldings. When the representation of Γ is self-dual, we show that the classification can be reduced to the standard equivariant context. In this case, we establish a one-to-one association between the classification of bifurcations in the reversible-equivariant context and the classification of purely equivariant bifurcations related to them. As an application of the results, we obtain the classification of self-dual representations of Z2 ⊕ Z2 and D4 on the plane.

In (1) the authors introduced the idea of a syntopogenous pre-ordered space and, in (2), discussed the more general notion of a convex syntopogenous space. The former generalizes Nachbin's ‘uniform preordered space’ (4) and Singal and Lal's ‘proximity preordered space’ (7); the latter does the same for Nachbin's ‘convex topological space’ (4) and Redfield's ‘nearly uniform ordered space’ (5).

Let S be a (jointly continuous) topological semigroup, C(S) the set of all bounded complex-valued continuous functions on S and M (S) the set of all bounded complex-valued Radon measures on S. Let (S) (or (S)) be the set of all µ ∈ M (S) such that x → │µ│ (x-1C) (or x → │µ│(Cx-1), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S, and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x-1C and Cx-1 are as defined in Section 2.) When S is locally compact the set Ma(S) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S.

Complementary variational principles are developed for boundary-value problems with field equations φiv = f(φ) in (0, 1) subject to φ(0) = φ(1) = 0, φ′(0) = φ′(1) = 0. The theory is applied in the non-linear case f(φ) = – keφ and accurate variational solutions, which are simple polynomials in x, are obtained.

Some explicit formulae for the right coaction of A* on H*BO are given. As corollaries, closed formulae for χ(ξk) and other elements in A* are given.

We study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F. We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic ℤp-extension and certain correction terms. This extends our earlier work [16], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.

It is shown that a quartic equation may be found which expresses the grid characteristic of a valve accurately over a considerable range, and that for modern dull filament valves the grid current curve is exponential over only a very restricted length.

Expressions are given for calculating rectified current and the amplitude of harmonics produced by a simple grid rectifier when voltages of the following wave-forms are applied: E sin pt, (A + B sin nt), and (A + B cos mt + C cos nt)sin pt. Further, a formula is given from which we may calculate the rectified current produced by a cumulative grid rectifier when a voltage E sin pt is applied.

All these formulae are applicable if the grid characteristic, is parabolic, cubic, quartic or quintic in form.

It is also shown that when an acoustically modulated radio frequency voltage is applied to a cumulative grid rectifier, even if the valve has a parabolic grid characteristic, an infinite number of harmonics of the modulation frequency are produced. These harmonics are due to the. action of the grid leak and condenser. It appears, however, that in practice the amplitude of these harmonics will be very small compared with those produced, primarily, by the curvature of the grid characteristic.

A locally non-trivial holomorphic family of Riemann surfaces over a hyperbolic Riemann surface B determines a homomorphism (monodromy) of the fundamental group of B to the modular group. We give a necessary condition for a homomorphism to be a mondromy of a holomorphic family of Riemann surfaces. And we estimate the number of holomorphic families in terms of topological information for holomorphic families over Riemann surfaces of genus 0.

In the usual notation let

where (a)n = a(a + 1)(a + 2)…(a + n − 1), (a)0 = 1.

Dr L. J. Slater ((3), page 628) gave an expansion of the form

for all values of n and t, real or complex.

The theory of laminar boundary layers is employed to determine the rate of decay of small amplitude oscillations of a non-homogeneous viscous fluid which completely fills a stationary rectangular container. An application of Green's formula is invoked and an expression is obtained in terms of the (general) density and viscosity distributions.

Let A be a right Λ-complex and C be a left Λ-complex where Λ is a ring which is both left and right hereditary; a principal ideal domain is a special case of such a ring. This paper is concerned with the Künneth theorem expressing the homology H(A⊗C) of the product complex A⊗C (= A⊗ΛC) in terms of H(A) and H(C). Any notations we do not explain are those of (1).

It is well known that the Julia set J(f) of a rational map f: ℂ → ℂ is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: ℝn → ℝn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

Using arguments involving combinatorial enumeration and asymptotics we compute the probability that a point of a random tree is fixed. The method is also applied to homeomorphically irreducible trees to illustrate how it works for other species of trees. To the nearest per cent, the limiting probability of a fixed point in a randomtree is 70%, and for homeomorphically irreducible trees it is 20%.

In numerical solution of a Sturm-Liouville system, it is necessary to determine an eigenvalue by a method of successive approximation. A relation is derived between the estimated accuracy of an approximate eigenvalue and the accuracy at every point of its corresponding eigenfunction. A method is also described whereby the correction to a trial eigenvalue, required for convergence to its true value, can be automatically determined. This method has been successfully used in solving radial wave-function equations, both with and without ‘exchange’, arising from the Hartree-Slater-Fock analysis of Schrödinger's equation.

The deflexion at any point of a thin circular plate subjected to certain loadings along the circumference or over the area of an eccentric circle is obtained when the plate is supported at several points along the boundary, the supports being symmetrically situated with respect to the common diameter of the plate and circle.

Recent reports contain tables of a parameter K required in calculating the performance of an airscrew by a new method.

The method of calculating K (due to Goldstein) is unsuitable for large values of the pitch especially near the tip of the airscrew.

In the case of infinite pitch we fall back, for a two-bladed airscrew, on the problem of a rotating lamina in two dimensions.

The solution for a cross lamina (corresponding to a four-bladed propeller) is given below and the tables of K for four blades are completed.

A formula for the limit of K/Kp at the airscrew tip is given for a propeller with any number of blades, where Kp is an approximate value of K due to Prandtl.

K for any number of blades is given in the form of an infinite series. The case of three blades is discussed in detail.