In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements of

corresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

1. Suppose that f is a homeomorphism of the Euclidean plane E2 onto itself. The set M ⊂ E2 is said to be invariant if f(M) = M and minimal if it is non-void, closed, invariant and irreducible with respect to these properties. In general, invariant and minimal sets in E2 can have a finite or infinite number of components.

Exact tests devised by Bartlett and Williams for the adequacy of a single discriminant function in the case of several groups are applicable to a wide range of problems in multivariate analysis. The appropriate likelihood criterion is factorized into direction and collinearity factors; and the exact distribution of these factors under the conditions of only one non-zero canonical correlation was first given by Bartlett in the general case of any numbers of dependent and independent variables, making use of geometrical arguments.

In this paper the factors are expressed directly in terms of the matrices occurring in the multivariate analysis of variance table; and the results are derived by an analytical method making use of matrix transformations and rectangular coordinates.

In this paper we shall show how a large number of higher-order cohomology operations (see (4)) may be calculated in suitable spaces by means of Chern characters: the method is an extension of that used by Adams in (1) to relate ‘mod p reductions’ of Chern characters and certain primary operations.

A. J. Miller has proposed a stochastic model for the study of highway traffic in one direction along a two-lane road in which the desired speed of a vehicle is sampled from a continuous probability distribution and queues of vehicles are Poisson distributed on the highway. Miller has derived the equilibrium expression for the relation between the spacial densities of all vehicles and the spacial densities of those which are freely travelling. This relationship takes the form of a non-linear integral equation which is left unsolved. The present model differs from the original only in the assumption of a discrete distribution of desired speeds. Although the derivations of the equilibrium equations are based upon different arguments from those of Miller, these relations transform to the corresponding integral equations in the limit of a continuous speed distribution. The main body of the paper is devoted to a detailed investigation of the case of two speeds and asymptotic results are obtained for three speeds.

0. Introduction. In a paper (1) with the same title, one of us attempted to classify the reducible k2-bundles on a quadric surface Q = P1 × P1 defined over an algebraically closed field k. This attempt suffered from two defects: first, the classification was given by a one-one correspondence rather than by an algebraic parameter variety, and, secondly, there is a mistake in the proof of Proposition 6 of (1), as a result of which Proposition 6, Proposition 7 and the exceptional clause of the Theorem in (1) are false. To correct these defects we use the definitions, notations and numbering of (1) without comment. Finally, we show how the classification of reducible k2-bundles on P1 × … × P1 (n factors) is determined by that of reducible k2-bundles on P1 × P1. The net effect of these corrections is to underline the moral of (1): while the classification of reducible k2-bundles (but with a distinguished factor of the product variety P1 × … × P1) is as nice as it could possibly be, the classification of reducible k2-bundles alone is as nasty as it could possibly be. Since every decomposable k2-bundle is just a sum of two line bundles, we consider only indecomposable reducible k2-bundles.

Throughout k denotes a field and k[X] = k[X1, …,Xn] the polynomial ring in n indeterminates over k.

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined by

where ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.

Explicit formulae are given for calculating the matrix elements of the irreducible representations of the three-dimensional pure rotation group by the direct method. In addition explicit formulae are derived to calculate the representations of the finite elements of any group when the eigenvalues of the matrix representing the corresponding infinitesimal elements are given.

1. Introduction. Let ℛn denote the set of all n × n matrices with real elements, and let denote the subset of ℛn consisting of all real, n × n, symmetric positive-definite matrices. We shall use the notation to denote that minor of the matrix A = (aij) ∈ ℛn which is the determinant of the matrix

The Schur Product (Schur (14)) of two matrices A, B ∈ ℛn is denned by

where A = (aij), B = (bij), C = (cij) and

Let ϕ be the mapping of ℛn into the real line defined by

for all A ∈ ℛn, where, as in the sequel, .

The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: A → A by

H is then fixed on the boundary of A. If now e: A → X is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on X − eA to a homeomorphism h:X → X. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.

1. Introduction. Over the past few years, the theory of single-server queues has grown substantially. Much of the effort expended has been applied to the study of transient multivariate queuing processes in continuous time, i.e. to the study of joint time-dependent distributions of two or more system variables given some known initial system configuration. A variety of methods have been employed in these studies to accommodate general interarrival and service time distributions, among them Erlang's method of fictitious phases, the method of supplementary variables, the extended imbedded chain method, and the semi-Markov method. In the first part of this paper an informal review of these transient studies and of the methods employed is given.

Two recent papers have discussed eigenvalue problems relating to second-order, self-adjoint differential equations from the point of view of the deferred approach to the limit in the finite-difference treatment of the problem. In both cases the problem is made definite by considering the differential equation

primes denoting differentiation with respect to x, with two-point boundary conditions

and given at the ends of the interval (0, 1). The usual finite-difference approach is to divide the range (0, 1) into N equal strips of length h = 1/N, giving a set of N + 1 pivotal values φn as the analogue of a solution of (1), φn denoting the pivotal value at x = nh. In terms of central differences we then have

and retaining only second differences yields a finite-difference approximation φn = Un to (1), where the pivotal U-values satisfy the equations

defined at all internal points, together with two equations holding at the end-points and approximately satisfying the end conditions (2). Here Λ is the corresponding approximation to the eigenvalue λ. A possible finite-difference treatment of the end conditions (2) would be to replace (1) at x = 0 by the central-difference formula

and use the corresponding result for the first derivative of φ, i.e.

whereq(x) = λρ(x) – σ(x). Eliminating the external value φ–1 between these two and making use of (1) and (2) we obtain the equation

where for convenience we write k0 = B0/A0. Similarly at x = 1 we obtain

withkN = B1/A1. If we neglect terms in h3 in these two they become what are usually taken to be the first approximation to the end conditions (2) to be used in conjunction with the set (4) (with the appropriate change φ = U, λ = Λ). This, however, results in a loss of accuracy at the end-points over the general level of accuracy of the set (4), which is O(h4), so there is some justification for retaining the terms in h3, e.g. if a difference correction method were being used they would subsequently be added as a correction term.

1. The following results are special cases of theorems of Irwin and Zeeman, respectively (see (4), (9), (10)).

(a) Let V be a (2n − m + 1)−connected piecewise-linear m−manifold (bounded or not), where m − n ≥ 3. Then any element of πn(V) can be represented by a piecewise-linear imbedding of Sn in V.

(b) Let M be a (2n − m + l)-connected closed piecewise-linear (m − 1)-manifold, where m − n ≥ 3. Then two piecewise linear imbeddings of Sn−1 in V are isotopic if and only if they are homotopic.

It has long been known (see Lévy (3), pp. 256, 260) that the sample paths of Brownian motion in the plane form an everywhere dense set of zero Lebesgue measure, with probability 1. In (7), a capacity argument was used to show that the Hausdorff measure with respect to tα is infinite for 0 < α < 2 with probability 1 so that the dimension of the path set is known to be 2. If one considers the initial part of the sample path Cω = C(1, ω) for 0 ≤ t ≤ 1, then it becomes interesting to ask if there is a measure function ψ(t) such that, with probability 1,

for suitable positive constants c1, c2. The corresponding problem for paths in k-space (k ≥ 3) has been solved. In this case, if φ1(t) = t2 log log t−1, Lévy (4) obtained the upper bound and Ciesielski and Taylor (1) obtained the lower bound. For the planar case, the path is recurrent, and the intricate fine structure makes the measure function φ1(t) inappropriate. In (2), Erdő and Taylor showed that the measure is finite with probability 1 with respect to φ2(t) = t2 log t−1, and at that time we thought that (1) might be true with ψ(t) = φ2(t). Recently Ray (5) has obtained the lower bound in (1)with

The purpose of the present note is to obtain the upper bound in (1) for the same measure function, thus showing that (2) defines the correct measure function for measuring planar Brownian motion.

In this paper we establish that a convex planar body C is centrally symmetric provided either one of the following conditions hold:

(1) Each line halving the circumference of the boundary γ of C is a diametral line.(A diametral line is a line intersecting C in a chord of maximal length in the family of parallel chords.)

(2) Each line halving the area of C is a diametral line.

All possible symmetries of the non-degenerate Harrison space-times are found. It is shown that if such a space-time admits an r-parameter group of motions, with r > 1, then it must admit a 2-parameter Abelian subgroup.

1. Mikusiński ((2), Problem 84) asks: can we express the surface of a sphere (in Euclidean 3-space) as the union of non-overlapping congruent great-circle arcs? We give a partial answer to this question.

Let f: X → X be a mapping of the metric space X into itself. We call f non-expansive if

An account is given of the application of Gaussian quadrature formulae to the numerical evaluation of certain definite integrals appearing in a recent paper by Craggs (2). Suitable changes of variable are used to effect a substantial reduction in the order of formula required. In comparing methods, use is made of an expression for the remainder as a contour integral, whose value is estimated with the aid of certain asymptotic formulae.