The problem of statistical investigation is the description of a population, or Kollektiv, of which a sample has been observed. At best we can only state the probability that certain parameters of this population lie within assigned limits, i.e. specify their probability density. It has been shown, e.g. by von Mises(1), that this is only possible if we know the probability distribution of the parameter before the sample is taken. Bayes' theorem is based on the assumption that all values of the parameter in the neighbourhood of that observed are equally probable a priori. It is the purpose of this paper to examine what more reasonable assumption may be made, and how it will affect the estimate based on the observed sample.

It is trivial that the fundamental equations of electromagnetism are invariant under orthogonal transformations of space. This invariance can be brought into evidence by using the calculus adapted to the orthogonal group, viz. the vector-calculus. The equations can be written either in their integral form

or in their differential form

A train of long waves travelling along a canal of depth h2 encounters a vertical step at which the depth changes to h1. The wave-length is much greater than h1 or h2. The classical Long-Wave theory where vertical acceleration is neglected is clearly inapplicable, although a treatment has been given by Lamb. In the present paper the problem is treated rigorously by the linearized theory of surface waves. A singular Fredholm integral equation of the first kind is obtained for the horizontal velocity above the step and is converted into a regular equation of the second kind with a kernel which tends to zero as the wave-length tends to infinity. To achieve this we first take the formal limit λ = ∞ in the integral equation of the first kind. This corresponds to a fluid motion between rigid boundaries which can be solved explicitly by a conformal transformation. In this way we obtain the inverse operator corresponding to λ = ∞, which is then applied to the original equation where λ < ∞. An equation of the second kind results which has a kernel tending to zero as λ tends to infinity and which is soluble by iteration for large λ. Although the Long-Wave method of Lamb fails to describe the details of the motion correctly, his predicted reflexion coefficient appears as a first approximation.

Boundary effects in the classical bifurcation experiments of fluid mechanics have been the subject of much recent investigation, most notably for us the work of Benjamin(2) on bifurcating steady flows in the Taylor problem in a very short annulus. His paper broaches a fundamental conceptual issue: how does the number of cells in the primary flow, an integer, depend on the length L of the annulus, a real parameter? By the primary flow we mean the flow which develops at high Reynolds number R as the rotation speed is increased gradually from rest, L being held constant (say L = L*). The relevant part of Benjamin's data is presented in Fig. 0·1. For the very short annuli of his experiment the primary flow possessed either two of four cells only. (See Section 1 concerning the bias towards an even number of cells.) If L* > L2, L1 < L* < L2 or L* < L1 the primary flow has 4, 2 or 2 cells respectively; however when L1 < L* < L2 the evolution of the flow as R is increased is not smooth but suffers a jump as R passes the middle of the three intersections of the line {L = L*} with the cusped curve Γ in the figure. (It is now widely accepted that the development of cells in the flow occurs over a range of Reynolds numbers; this point is not the issue here.) The curve Γ divides the plane into two regions such that there are one or two stable, experimentally observable flows according as (R, L) lies in region I or region II, respectively. We may refer to the two stable states in region II as 2 cell or 4 cell, although their cellular structure is only partially developed at the moderate values of the Reynolds number occurring in the figure. One or the other of the two flows in region II loses stability across Γ, giving rise to the possibility of jumps when, as above, (R, L) crosses the boundary moving from region II to region I. In general if (R, L) is varied quasi-statically along a path which begins in region I, passes through region II, and re-enters region I, a jump will occur on re-entry if and only if the point H lies between the two points of Γ where the path crosses the boundary (between in the sense of distance along Γ).

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this paper is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt–Eccles operad. We also prove that differential graded algebras over the Barratt–Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt–Eccles operad which is equivalent to the classical little squares operad.

The aim of this paper is to prove:

Theorem. Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

. Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q1,q2,… of elements of Q, there exist i and j such that i < j and qi ≤ qi. A restricted transfinite sequence on Q is a function from a well-ordered set onto a finite subset of Q. If f, g are restricted transfinite sequences on Q with domains A, B respectively and there exists a one-to-one order-preserving mapping μ of A into B such that f(α) ≤ h(μ(α)) for every α ∈ A, we write f ≤ g. It is proved that this rule well-quasi-orders the set of restricted transfinite sequences on Q. The proof uses the following subsidiary theorem, which is a generalization of a classical theorem of Ramsey (4). Let P be the set of positive integers, and A(I) denote the set of ascending finite sequences of elements of a subset I of P. If s, t∈A(P), write s ≺ t if, for some m, the terms of s are the first m terms of t. Let T1,…,Tn be disjoint subsets of A(P) whose union T does not include two distinct sequences s, t such that s ≺ t. Then there exists an infinite subset I of P such that T ∩ A(I)is contained in a single Tj.

The two-dimensional configuration is considered of a fixed, semi-infinite, vertical barrier extending downwards from a fluid surface and having, at some depth, a gap of arbitrary width. A train of surface waves, incident on the barrier, is partly transmitted and partly reflected. The velocity potential of the resulting fluid motion is determined by a reduction procedure and also by an integral equation formulation. It is shown that the two methods lead to the same Riemann–Hilbert problem. Transmission and reflexion coefficients are calculated for several values of the ratio gap width/mean gap depth.

A family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

Some theoretical results on the effect of non-normality on the t test of significance, though it is stressed that they are incomplete and not perhaps of much quantitative value, nevertheless tend to agree with the result of sampling investigations in showing that for moderate departures from normality this test may still be used with confidence, particularly for testing differences in means of equal numbers of observations.

In the contour integral

the functions g( ) and f( ) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with the l parameters, denoted by α = (α1,α2,…,αl), and it may happen that two or more saddle-points coincide at a certain value of α. The asymptotic expansions given by the ordinary method of steepest descents are then non-uniform. The case of a single parameter and two nearly coincident saddle- points was studied earlier and leads to Airy functions. Here we are concerned with the case of l parameters and of m saddle-points which are nearly coincident when α is near 0. It is then known that f(z,α) can be transformed locally into a polynomial of degree m + 1, and accordingly we here restrict f(z,α) to be such a polynomial. The form of the resulting asymptotic expansion had already been conjectured (correctly as we now see), and involves certain special functions of m–1 variables. To establish the validity of the expansion we must show that the remainder after any number of terms is of smaller magnitude than the last term retained in the expansion. In other words, we must establish an inequality between integrals which is to hold for sufficiently small α and sufficiently large N, and which involves 1 parameters and m nearly coincident saddle-points.

Let 0 < p,pi ≤ ∞, 0 < q,qi < ∞ (i = 1, 2) such that

Let E be a quasi-Banach lattice which fails to contain c0 and whose α-convexity constant is equal to 1 for some 0 < α < ∞. Then for every f∈H(E(q)) there exist g∈Hp, 0(E(q0)), h∈Hp1(E(q1)) such that

Consequently, E is q-concave for some finite q if and only if E is uniformly H1-convexifiable in the sense of [24]. Analogous results are also obtained for symmetric spaces of measurable operators. Another result proved in the paper says that if E is a symmetric quasi-Banach function space on (0, ∞) having the analytic Radon–Nikodym property then LE(M, τ) also possesses this property for any semifinite von Neumann algebra (M, τ).

1. Introduction. Problems of unrestricted random walks on lattices have been considered by many authors and methods have been discovered for the exact enumeration of the number of walks between two points, for obtaining asymptotic estimates of the distribution functions and for evaluating special parameters such as the probability of ultimate retum to the starting point. By an unrestricted walk we mean one in which the probability of the next step at any stage is not influenced by the previous choice of steps. The corresponding problems for restricted (or correlated) random walks are more difficult and fewer results have been obtained. Restricted walks are, however, of considerable physical interest in connexion with the statistical behaviour of long chain molecules, the theory of cooperative phenomena in crystals and in other applications.

It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutations

generate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.

In a recent paper (1) Nagata proved that a (linear associative) algebra, not necessarily of finite dimension, over a field of characteristic 0 which satisfies the identical relation xn = 0 satisfies also the relation x1x2 … xN = 0, where N is an integer depending only on n. He remarked further that it is a corollary that the result remains true if the ground field is of prime characteristic p, provided that p is large enough compared with n; and he conjectured that the obviously necessary condition p > n is in fact sufficient. The object of this note is to prove Nagata's conjecture. To do this, we give a new proof of his theorem, and as a by-product we obtain a rather better bound for N than his, showing, namely, that we can take N = 2n − 1. The determination of the best possible value of N, or even of its order of magnitude, seems not to be easy; at any rate, the best I have been able to do in the opposite direction is to show that for large n we cannot take N as small as n2/e2, where e is the base of natural logarithms.

1. In a recent note I showed that Langmuir's adsorption isotherm

where θ is the fraction of the surface covered by adsorbed gas, p the gas pressure in equilibrium with it, and A (T) a specified function of the temperature, can be derived as a theorem in statistical mechanics without any appeal to the mechanism of deposition and re-evaporation. Necessary and sufficient assumptions for the truth of (1) are that the atoms (or molecules) of the gas are adsorbed as wholes on to definite points of attachment on the surface of the adsorber, that each point of attachment can accommodate one and only one adsorbed atom, and that the energies of the states of any adsorbed atom are independent of the presence or absence of other adsorbed atoms on neighbouring points of attachment. Under these assumptions the explicit form of (1) is

where m is the mass of the adsorbed atom or molecule, bg(T) the partition function for its internal states in the gas phase, and vs(T) the partition function for its set of adsorbed states. These sets of states are to be so specified that the energy zero is assigned tot the lowest state of each set in constructing bg(T) and vs(T), and then X is the energy required to transfer a molecule from the lowest adsorbed state tot the lowest gas state. Quite another adsorption isotherm was shown to hold when adsorption of a molecule takes place as atoms and requires two or more points of attachment.

A paranormed space X = (X, g) is a topological linear space in which the topology is given by paranorm g—a real subadditive function on X such that g(θ) = 0, g(x) = g(−x) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space X and continuity of multiplication means that λn → λ, xn → x(i.e. g(xn − x) → 0) imply λnxn → λx, for scalars λ and vectors x. We shall use the term semimetric function to describe a real subadditive function g on X such that g(θ) = 0, g(x) = g(−x). Two familiar paranormed sequence spaces, which have been extensively studied (3), are l(p) and m(p). For a given sequence p = (gk) of strictly positive numbers, l;(p) is the set of all complex sequences x = (xk) such that and m(p) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 < pk ≤ 1 so that natural paranorms would seem to be in m(p). In fact Simons showed that g1 was a paranorm for l(p), but that g2 did not satisfy the continuity of multiplication axiom.

Given a sequence of non-negative independent and identically distributed random variables, we determine conditions on the common distribution such that the sum of appropriately normalized and centred upper kn extreme values based on the first n random variables converges in distribution to a normal random variable, where kn → ∞ and kn/ n → 0 as n → ∞. The probabilistic problem is motivated by recent statistical work on the estimation of the exponent of a regularly varying distribution function. Our main tool is a new Brownian bridge approximation to the uniform empirical and quantile processes in weighted supremum norms.

The Lorenz attractor is a strange attractor which has been proposed as an explicit model for turbulence ((4), compare (5)). First studied by E. N. Lorenz as a truncation of the Navier-Stokes equations (2), it has since attracted the attention of mathematicians because of its particularly interesting dynamical properties.