We prove that from a topological point of view most numbers fail to be normal in a spectacular way. For an integer $N \geq 2$ and $x \in [0, 1]$, let $x = \sum^\infty_{n=1} \varepsilon_{N,n}(x)/N^n$, where $\varepsilon_{N, n}(x) \in \{0, 1,\ldots, N - 1\}$ for all $n$, denote the unique non-terminating $N$-adic expansion of $x$. For a positive integer $n$ and a finite string ${\bf i} = i_1\cdots i_k$ with entries $i_j \in \{0, 1,\ldots, N - 1\}$, we write $$ \Pi_N (x,{\bf i};n) = \frac{|\{1 \le i \le n | \varepsilon_{N,i}(x) = i_1, \ldots, \varepsilon_{N,i+k-1} (x) = i_k\}|}{n} $$ for the frequency of the string ${\bf i}$ among the first $n$ digits in the $N$-adic expansion of $x$, and let $\Pi_N^k(x;n) \,{=}\, (\Pi_N(x,{\bf i};n))_{\bf i}$ denote the vector of frequencies $\Pi_N(x,{\bf i};n)$ of all strings ${\bf i} = i_1\cdots i_k$ of length $k$ with entries $i_j \in \{0, 1,\ldots, N-1\}$. We say that a number $x$ is extremely non-normal if each shift invariant probability vector in ${\mathbb R}^{N^k}$ is an accumulation point of the sequence $(\Pi_N^k(x;n))_n$simultaneously for all $k$ and all bases $N$, and we denote the set of extremely non-normal numbers by ${\mathbb E}$, i.e. \begin{eqnarray*} &&{\mathbb E} = \bigcap_N \bigcap_k \big\{x \in [0,1]|\, {\rm each}\, {\bf p} \in \Gamma^k_N\\ &&\quad\hbox{is an accumulation point of the sequence } (\Pi_N (x, {\bf i};n))_n\big\}, \end{eqnarray*} where $\Gamma^k_N$ denotes the simplex of shift invariant probability vectors in ${\mathbb R}^{N^k}$. Our main result says that ${\mathbb E}$ is a residual set, i.e. the complement $[0,1]\backslash {\mathbb E}$ is of the first category. Hence, from a topological point of view, a typical number in [0, 1] is as far away from being normal as possible. This result significantly strengthens results by Maxfield and Schmidt. We also determine the Hausdorff dimension and the packing dimension of ${\mathbb E}$.

Introduction. Let K be a connected simplicial complex, finite or infinite, its polyhedron ((2), page 45) being the space X. Then X is connected. Suppose further that X is simply connected. For any group G of simplicial transformations of X, H will denote the normal subgroup generated by elements which have a non-empty fixed-point set.

The main result of this paper establishes invariance under change of generators for automatic structures for monoids (a property that is well known to hold for automatic groups but fails for semigroups). This result is then applied to show that if a free product of two monoids is automatic, then so are both the free factors. Finally, the difference between automatic structures, in terms of monoid generating sets and semigroup generating sets, is discussed.

This is an exploration of the mechanical consequences of the symmetry inherent in a monatomic elastic crystal lattice. The investigation is based on a theorem by means of which the square of the right stretch tensor is uniquely decomposed. Generality is assured by the affine equivalence of all monatomic lattices. Response functions appropriate to perfect monatomic crystals are thereby constructed, and some simple deformation paths are considered.

The combined effects of uniform thermal and magnetic fields on the propagation of plane waves in a homogeneous, initially unstressed, electrically conducting elastic medium have been investigated.

When the magnetic field is parallel to the direction of wave propagation, the compression wave is purely thermo-elastic and the shear wave is purely magneto-elastic in nature. For a transverse magnetic field, the shear waves remain elastic whereas the compression wave assumes magneto-thermo-elastic character due to the coupling of all the three fields—mechanical, magnetic and thermal. In the general case, the waves polarized in the plane of the direction of wave propagation and the magnetic field are not only coupled but are also influenced by the thermal field, once again exhibiting the coupling of the three fields. The shear wave polarized transverse to the plane retains its magneto-elastic character.

Notation.

Hi = primary magnetic field components,

ht = induced magnetic field components,

To = initial thermal field,

θ = induced thermal field,

C = compression wave velocity.

S = shear wave velocity,

ui = displacement components,

cv = specific heat at constant volume,

k = thermal conductivity,

η = magnetic diffusivity,

μe = magnetic permeability,

λ, μ = Lamé's constants,

β = ratio of coefficient of volume expansion to isothermal compressibility.

Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.

We prove the existence and completeness of the wave operators for quantum-mechanical scattering by a potential which does not decrease to zero at infinity in two of the three space directions. We also obtain a new abstract result concerning the continuous dependence of the wave operators on the potential.

Two-dimensional elastic waves in a half-space 0 ≤ r < ∞, 0 ≤ θ ≤ π are examined under the assumption of dynamic similarity, so that the stresses depend only on r/t, θ. Analytic solutions are given for constant surface traction on θ = 0, 0 < r/t < V, where V is constant, the rest of the surface being unloaded, and for a concentrated load at r = 0.

Numerical results are quoted for the particular case V → ∞, corresponding to a load on half the bounding plane.

It has been shown by Chadwick and Lee(1), using a high-pressure ionization chamber, that if one neutrino is emitted by each disintegrating Ra E nucleus, then the neutrinos do not produce more than one pair of ions in 150 km. of air at N.T.P. Calculations based on the wave mechanics show that the ionization due to a neutrino having a magnetic moment of one Bohr magneton would be very easily detectable(2), whereas it has peen estimated that if the neutrino has no magnetic moment at all its encounters with nuclei will be as scarce as one in 1016 km. of water(3). I have investigated the matter again, using two Geiger-Müller counters, instead of an ionization chamber. The counters have the advantage of giving a discharge for a single pair of ions(4). The two counters of 15×5cm. were connected in parallel, and filled with air at 76mm. pressure. They were shielded on all sides by 45 mm. of lead. A source of 7 mg. of Ra (D, E, F) was placed at a distance of 11 cm. from the centres of the two counters, so that the total solid angle subtended by the counters at the source was 0·10, expressed as a fraction of the complete sphere. Assuming that both the disintegrating atoms of Ra D and Ra E give one neutrino, then the number of neutrinos crossing the counters per minute is 3·1 × 109. The average effective length of the path in the counter, reduced to N.T.P., is 0·3 cm., so in one min. the total range in air of all the neutrinos crossing the counters is 9·3 × 108 cm.

We have recently observed a curious phenomenon, undoubtedly well known, while mixing instant coffee in a mug. If the bottom of the mug was tapped repeatedly with the spoon as the powder was stirred into the water, the note emitted could be heard to rise in pitch by over an octave. This rise occurred in a matter of seconds, about the same time as it took the mug to heat up. Further experimentation showed that almost any powder could produce the effect in cold or hot water, but water alone would not. Vigorous stirring did not seem essential. A full mug showed as pronounced an effect as a half-filled one.

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power series

The problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.

In this article an estimate is obtained for the coefficients of the transformation polynomials for the elliptic modular function. This enables us to answer some questions raised by Mahler in [5] and [6].

Let ℒ be the class of discrete groups G which admit a finite complex X as classifying space. On ℒ we may define the Euler characteristic χ(G) = χ(X); that this is independent of X follows from the homological interpretation of characteristics.

Introduction and statement of results. We shall describe how, for successive integers N, the points {nξ}, with n = 0, 1, …,N – 1, are distributed in the closed unit interval U = [0, 1]; by showing how successive points {Nξ,} modify the partition of U produced by the previous points. The simple generalization to the k-dimensional sequence {nξ} = ({nξ(1)},{nξ(2)}, …,{nξ(k)}), in the unit hypercube Uk, is also made.

In a previous paper [6], the present authors established estimates for the continuous dependence of the solution on various data in the initial boundary value problem of linear elastodynamics on a bounded region of space. The main conclusion concerned continuous dependence on the body-force, but also it was shown how this result could be used to derive continuous dependence on the initial data, elasticities, boundary data and initial geometry. The method adopted was based upon logarithmic convexity arguments and hence led naturally to continuity in the sense of Hölder on compact sub-intervals of time. A special feature of the study entailed the lack of any sign-definiteness conditions on the elasticities which, of course, in the absence of any a priori constraint on the solution always gives rise to an ill-posed problem. (See, for instance, the comprehensive survey by Payne [10].)

We investigate the box dimensions of the horizon of a fractal surface defined by a function f ∈ C[0,1]2. In particular we show that a prevalent surface satisfies the ‘horizon property’, namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α, for α ∈ [2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

Let pc(d) be the critical probability for oriented percolation in ℤd and let μ(d) be the time constant for the first passage process based on the exponential distribution. In this paper we show that as d → ∞,dpc(d) and dμ,(d) → γ where γ is a constant in [e−1, 2−1] which we conjecture to be e−1. In the case of pc(d) we have made some progress toward obtaining an asymptotic expansion in powers of d−1. Our results show

The left hand side agrees, up to O(d−3), with a (nonrigorous) series expansion of Blease (1, 2):

We introduce a concept of unique factorization for elements in the context of Noetherian rings which are not necessarily commutative. We will call an element p of such a ring R prime if (i) pR = Rp, (ii) pR is a height-1 prime ideal of R, and (iii) R/pR is an integral domain. We define a Noetherian u.f.d. to be a Noetherian integral domain R such that every height-1 prime P of R is principal and R/P is a domain, or equivalently every non-zero element of R is of the form cq, where q is a product of prime elements of R and c has no prime factors. Examples include the Noetherian u.f.d.'s of commutative algebra and also the universal enveloping algebras of solvable Lie algebras. The latter class provides a rich supply of genuinely non-commutative examples.

Because of the symmetry of the problems encountered in the applications of matrices to physics, electrical engineering and numerical analysis, it frequently turns out that the matrix to be considered is a composite matrix, that is, a matrix whose elements are matrices. For example, the matrix may be where A1 and A2 are square matrices of order n which need not commute. It is easy to prove that the eigenvalues of this matrix of order 2n are the eigenvalues of the two matrices, A1+A2 and A1−A2 of order n.

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection

is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .