In this paper we prove that the conjectures of Frisch and Parisi and Arneodo et al. (called the multifractal formalism for functions) may fail for some non-homogeneous selfsimilar functions on ℝ2. In these cases, we compute the correct spectrum of singularities and we show how the multifractal formalism must be modified.

In many technical problems on conduction of heat involving convection, radiation, or evaporation at the surface of a body, the flux of heat at the surface is known empirically as a function of the surface temperature with reasonable accuracy. The thermal properties of the body also vary with the temperature, but in many cases the nature of this variation is completely unknown, and in others it is slight over the range of temperature involved. Thus it seems worth while studying problems on conduction of heat in a medium with constant thermal properties and with heat transfer at its surface a given function of the surface temperature. Mathematically such problems occupy an interesting position between the classical linear theory and the general case in which both the differential equation and the boundary conditions are non-linear.

Our object is to describe all of the-primitive permutation groups of degree less than 1000 together with some of their significant properties. We think that such a list is of interest in illustrating in concrete form the kinds of primitive groups which arise, in suggesting conjectures about primitive groups, and in settling small exceptional cases which often occur in proofs of theorems about permutation groups. The range that we consider is large enough to allow examples of most of the types of primitive group to appear. Earlier lists (of varying completeness and accuracy) of primitive groups of degree d have been published by: C. Jordan (1872) [21] for d ≤ 17, by W. Burnside (1897) [5] for d ≤ 8, by Manning (1929) [34–38] for d ≤ 15, by C. C. Sims (1970) [45] for d ≤ 20, and by B. A. Pogorelev (1980) [42] for d ≤ 50. Unpublished lists have also been prepared by C. C. Sims for d ≤ 50 and by Mizutani[41] for d ≤ 48. Using the classification of finite simple groups which was completed in 1981 we have been able to extend the list much further. Our task has been greatly simplified by the detailed information about many finite simple groups which is available in the Atlas of Finite Groups which we will refer to as the Atlas [8].

Suppose that n runs through all integral values, that (øn) is a system of normal orthogonal functions for the interval (−∞, ∞), and that ψn is the Fourier transform of øn. Then, by Parseval's theorem for Fourier transforms,

and (ψn) is also a normal orthogonal system.

The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.

Summary of results. In this paper we shall study some properties of the algebra A(Rn) for n ≥ 1 relative to the unit sphere

A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new.

A theoretical treatment is given of the absorption of longitudinally polarized sound in dielectric crystals of high symmetry, due to interaction with the thermal phonous. The Q value for the absorption is calculated in terms of the deviation from Hooke's law, the result being applicable at low temperatures when the mean free path of the phonons is greater than the wavelength of the acoustic wave. The theory is applied to cubic crystals, and in particular to solid argon, where a minimum Q value of about 108–109 is calculated. It is suggested that measurement of the absorption (at present somewhat beyond experimental techniques) would supply information on the deviation from linear elasticity, as well as providing a satisfactory verification of the basic interaction theory. The effect of isotopes, and, in more complicated structures, of optic modes, is shown to be negligible.

Experiments are made on wave-lengths between 200 and 500 ms. for distances of transmission less than 200 kms. It is found that the variations of downcoming wave intensity are uncorrelated on two receivers separated by about one wave-length, and it is shown that this implies a considerable amount of lateral deviation of the waves. A special receiver is used to confirm the occurrence of lateral deviation, and an estimate of the angle of deviation is made. The possible causes of intensity variation are considered in the light of these experimental results, and it is suggested that a major cause of “fading” is the interference, at the ground, of waves “scattered” from a series of diffracting centres distributed over an area of radius at least 20 kms. in the present experiments. The possible results of such a mechanism are discussed.

Let B(t) be a two-dimensional Brownian motion. For 0 < α < 2π, set

and, for 0 ≥ β< 2π, let F(α,β) be F(α) rotated through an angle β about the origin.

For a compact Lie group G, E. Witten proposed topological invariants of a threemanifold (quantum G-invariants) in 1988 by using the Chern-Simons functional and the Feynman path integral [30]. See also [2]. N. Yu. Reshetikhin and V. G. Turaev gave a mathematical proof of existence of such invariants for G = SU(2) [28]. R. Kirby and P. Melvin found that the quantum SU(2)-invariant associated to q = exp(2π √ − 1/r) with r odd splits into the product of the quantum SO(3)-invariant and [15]. For other approaches to these invariants, see [3, 4, 5, 16, 22, 27].

In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

Consider $e_n = \inf \exp \int \log \parallel x - f(x) \parallel dP(x)$, where $p$ is a probability measure on $\real^d$ and the infimum is taken over all measurable maps $f{:}\ \real^d \rightarrow \real^d$ with $| f(\real^d)| \leq n$. We study solutions $f$ of this minimization problem. For absolutely continuous distributions and for self-similar distributions we derive the exact rates of convergence to zero of the $n$th quantization error $e_n$ as $n \rightarrow \infty$. We establish a relationship between the quantization dimension that rules the rates and the Hausdorff dimension of $P$.

A previous application of the theory, to the nonlinear boundary-value problem for steady flows of a viscous fluid in a bounded domain, is first retraced in order to verify a general theorem concerning the indices of multiple solutions. Then, in §5, the bearing of the theorem on questions of the bifurcation of steady flows is discussed, and the conclusion is drawn that a transcritical form of bifurcation is virtually universal in practice. In §6 it is proved that a flow represented by a solution with index i = – 1 is necessarily unstable, and hence it appears that lack of uniqueness generally implies the existence of an unstable steady flow.

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.

The concept of strong almost convergence was introduced in (2), where the matrices summing every strongly almost convergent sequence, leaving the limit invariant, were characterized.

A new set of stability criteria for linear systems is derived. This shows that about half of the Hurwitz criteria are redundant when certain of the coefficients of the characteristic equation are known to be positive. The theory is applied to obtain a very short derivation of the known aperiodicity criteria. The conditions for realizability of RC networks are shown to be closely related to the stability and aperiodicity criteria, and are stated as sets of criteria in terms of the polynomial coefficients. Two basic theorems are involved which give the necessary and sufficient conditions for the roots of two polynomial equations to be real and separated.

We exploit the correspondence between the three–dimensional Lorentzian Einstein–Weyl geometries of the hyper–CR type and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper–CR Einstein–Weyl structures by Kodaira deformations of the flat twistor space $T\mathbb{CP}^1$, and how to recover the pencil of Poisson structures in five dimensions illustrating the method by an example of the Veronese web on the Heisenberg group.

In this paper I shall define a deformation of hyperbolic structures on n-manifolds, called ‘bending’, by constructing a family of quasiconformal homeomorphisms of hyperbolic (n+ l)-space. The deformation can be applied to manifolds which contain totally geodesic submanifolds of codimension 1.

We show that if E is an analytic subset of ℝn then

for almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.