Using a thin-walled electroscope, absorption curves in paper and aluminium were obtained for the β-rays of actinium (B + C) and actinium C″. From these curves the range of the β-rays of actinium B was found to be 0·08 ± ·02 gm./cm.2 The energy required for this range is 300,000 ± 50,000 volts, which must be the end-point of the continuous spectrum.

On account of the overlapping of the β-ray spectra of actinium B and actinium C″ the range method can never yield an accurate value of the end-point of the former. Magnetic analysis appears to be the best means of finding this accurately, but the weakness and rapid decay of sources are formidable difficulties.

The purpose of this note is to extend Brumer's characterization of pro-p groups of cohomological dimension two ([1], corollary 5·3) to presentations more general than free ones. The result is then used to rid a proof in [7] of certain field theoretic ingredients. As a by-product we complete a result of Tsvetkov [6] about group extensions obtained by omitting relations.

This note deals with some relations between unmixed local rings and the notion of generalized analytic independence.

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequality

for almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, and f is a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint on f, the number of solutions of (1) is asymptotically equal to

as k tends to infinity.

In two recent papers in the Philosophical Magazine the statistical theory of collisions of electrons with atoms has been considered, and applied to explain the observed features of the capture and loss of electrons by α-particles. The basis of that discussion can however be improved. Firstly, interactions with the core of any atom entered by the α-particle were explicitly ignored. Such interactions—or rather interactions with the general intra-atomic field as distinct from individual electrons—might be expected to be (and are) important, and it has been found possible to include them in this paper. Secondly, the frequency laws for the processes concerned were based on the classical Thomson-Bohr theory of ionization by collision. This theory—it is well known—does not completely reproduce experimental facts at α-particle velocities, and a less restrictive form can be given to the frequency laws which appears to be in accord with all the known facts. The application to the capture and loss of electrons by α-particles proves to be unaffected.

In developing the theory of radiative transfer in expanding atmospheres (Chandrasekhar, 1945 a, b) the author has recently encountered certain novel types of boundary-value problems in hyperbolic equations which appear to merit consideration for their own sake. As related to the equation

the boundary-value problems which occur are of the following general type

i.e. along AB in Fig. 1. Along AD (x = 0 and 0 ≤ y ≤ l2) and BC (x = l1, 0 ≤ y ≤ l2) we are further given that

and

where φ(y) and ψ(y) are two known functions. The problem is to solve equation (1) in the rectangular strip ABCD satisfying the stated boundary conditions.

For a general family of weight functions, a relation between orthogonal polynomials on the unit sphere in ℝd+m and orthogonal polynomials on the unit ball in ℝd is discussed and used to show that the summability of the Fourier orthogonal series on the unit ball follows from the summability on the unit sphere. Convergence of the Cesàro summability is established for a number of weight functions on the sphere and on the ball.

This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over fields of characteristic p>2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized étale cohomology theory called étale BP2.

It is well known that charcoal possesses catalytic properties. It accelerates the decomposition of hydrogen peroxide, the removal of bromine from ββ-dibromo propionic acid and enables certain oxidations to take place in air at ordinary temperatures.

1.1. Introduction. In this paper, we continue with the theme of (1): the relationships holding between the Sπ (i.e. maximal π) subgroups of a locally finite group and the various local systems of that group. In (1), we were mainly concerned with ‘good’ Sπ subgroups – those which reduce into some local system (and are said to be good with respect to that system). Here, on the other hand, we are concerned with a very much more special sort of Sπ subgroup.

1·1. Es bezeichne für nichtnegative ganze n und k

das n-te Cesàro-Mittel der Ordnung k der Reihe μ0 + μ1 + … Insbesondere ist also die n-te Partialsumme dieser Reihe.

Introduction and statement of results. It is well known that if V is a vector space over a field k then its dual vector space Homk (V, k) has dimension not less than that of V, and that the dimensions are equal precisely when V is finite-dimensional. In this note we introduce a ‘dual module’ for a module over a principal ideal domain, and show that some though not all of the properties of the field case carry over.

This paper is devoted to the classical problem of surfaces of minimum area subtending a given closed contour.

Let andWe prove that there is an absolute constant c1 > 0 such thatfor every s ∈ (0, ∞) and n ≥ 9, where the supremum is taken for all f ∈ Gn withThis is what we call (an essentially sharp) Remez-type inequality for the class Gn. We also prove the right higher dimensional analog of the above result.

We construct two infinite series of Moufang loops of exponent 3 whose commutative centre (i. e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders 38 and 311 one of which can be defined as the Moufang triplication of the free Burnside group B(3, 3).

The family of Brownian bridge processes (ba)a∈R has a number of characterizations, the most fundamental being that ba: [0,1] → ℝ is Brownian motion conditioned to be at the point a at time 1. Equivalently, ba is a continuous process whose law Wa is that of Wiener measure conditioned on the set of paths with x1 = a. These ideas are not so easy to make precise, so that more down to earth and workable characterizations of the Brownian bridge such as the following are often used in practice (see [6] for example)

We show that the sequence 2n+n is a universal Hilbert sequence. That is, for each polynomial P(T, Y) irreducible in ℚ(T) [Y], the polynomial P(2n+n, Y) is irreducible in ℚ[Y] for all but finitely many n. This answers a question of M. Yasumoto. Other examples, like 2n+5n, are given. They are all obtained as special cases of a more general result which is proved from classical diophantine arguments.

Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$. Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant

\begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation}

$f\in\mathcal{H}(\overline{\rho})$

$\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$

$\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$

$\mathcal{H}(\overline{\rho})$

$\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$

For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

We construct characteristic classes for the change of rings spectral sequences. These have their values in appropriate Ext groups and provide descriptions of the first non-zero differentials in the spectral sequences. We use these classes to do some calculations in the cohomology spectral sequence of an extension of a finite cyclic group by a finite cyclic group.

In [1] B. J. Birch, solving in the affirmative a conjecture of Erdὅs, proved the following result:

Theorem 1. Let p and q be coprime integers greater than 1. Then every large natural number may be written as a sum of distinct terms of type paqb.

In fact Birch pointed out that, with similar arguments, one could obtain a stronger version where the exponent b of q can be bounded in terms of p and q. The proofs were entirely elementary.