In the first part of this paper we discuss the radiation from a single charged particle moving in an arbitrary central field of force and obeying Dirac's equation. We consider the electric quadripole and magnetic dipole radiation as well as the electric dipole. We derive the selection rules for the magnetic dipole radiation and collect together for reference the corresponding selection rules for the electric dipole and quadripole radiations. In the second part we discuss the relative intensities of the various types of radiation, treating in detail the cases where the selection rules for magnetic dipole and electric quadripole are simultaneously satisfied. Finally we show that these results have an important bearing on the theory of internal conversion of γ-rays. The internal conversion of soft γ-rays occurs with such high probability that the theory is unable to account for the experimental results unless it is assumed that the radiation is largely magnetic dipole in character. On the other hand, Fisk and Taylor (loc. cit.) were unable to account for the presence of magnetic dipole radiation in appreciable amounts. We show that this is due to the fact that, of the two possible transitions (a and e of § 2) in which both magnetic dipole and electric quadripole radiation can be emitted, Fisk and Taylor considered only the second. In the case of the second, corresponding to a transition between two distinct terms, we show that Fisk and Taylor were correct in predicting a negligible amount of magnetic dipole radiation, but in the case of the first, corresponding to a transition between two levels of one multiplet term, we find that there is indeed a high percentage of magnetic dipole radiation.

Combining Freiman's theorem with Balog–Szemerédi–Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ${\mathbb{R}^d$ and ${\mathbb{Z}^d$.

This paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.

The success of the methods of [24] and [4] in investigating the structure of the weakly almost periodic compactification wℕ of the semigroup (ℕ, +) of positive integers has prompted us to see how successful they would be in another context. We consider wG, where G=[oplus ]i∈ωGi is the direct sum of a sequence of finite groups with its discrete topology. We discover a large class of weakly almost periodic functions on G, and we use them to prove the existence of a large number of long chains of idempotents in wG. However, the closure of any singly generated subsemigroup of wG contains only one idempotent. We also prove that the set of idempotents in wG is not closed. The minimal idempotent of wG can be written as the sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of βℕ, we find subsemigroups SF of wG which are ‘carried’ by closed subsets F of βω. wG contains the free abelian product of the semigroups SF corresponding to families of disjoint subsets F. Usually SF is a very large semigroup, but for some points p∈βω, Sp can be quite small.

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemerédi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : G → [0, 1], there exists a subgroup H ⩽ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are small. Quantitatively, if one wishes to obtain that for 1 − ε fraction of the cosets, the nontrivial Fourier coefficients are bounded by ε, then Green shows that |G/H| is bounded by a tower of twos of height 1/ε3. He also gives an example showing that a tower of height Ω(log 1/ε) is necessary. Here, we give an improved example, showing that a tower of height Ω(1/ε) is necessary.

Let E be a Banach space, let (Ω, Σ, μ) a finite measure space, let 1 < p < ∞ and let Lp(μ;E) the Banach space of all E-valued p-Bochner μ-integrable functions with its usual norm. In this note it is shown that E contains a complemented subspace isomorphic to l1 if (and only if) Lp(μ; E) does. An extension of this result is also given.

The classification of affine cubic functions in the real case is a fairly easy corollary of that in the complex case (9). However as the results can be easily interpreted by diagrams, one can obtain a much richer understanding. For example, the question of which types of cubic curve occur as level curves of which types of function is now much less trivial. This will lead us first to re-examine the classification of cubic curves going back to Newton (4). Next the ‘dynamic’ approach of considering these curves as members of families leads to the diagrams associated with the umbilic catastrophes of Thorn (8). However the consideration of functions rather than of curves gives a 1-dimensional foliation of these diagrams which we describe next. We conclude by placing the results back in a protective setting.

In this paper we extend a result of Johnson and Lahr(3), which characterizes the multiplier algebra of L1(a, b) (the algebra of Lebesgue integrable functions on the interval of real numbers from a to b, under order convolution) to the L1 algebra of a general totally ordered semigroup. Similar work has been done in (l), but under more restrictive conditions.

The Langmuir-Frenkel theory of surface action has been shown to involve a simple relation between the partial pressure of the catalyst poison, and the reduction in reaction velocity which results. Very interesting observations have been made by Beebe showing that the heat of adsorption of carbon monoxide on a reduced copper catalyst falls considerably and reaches a limiting value as the surface approaches saturation. Pease and Stewart have shown that the presence of less than one percent, of the total carbon monoxide that a given copper catalyst can adsorb will reduce the reaction velocity of the combination of hydrogen and ethylene at the surface of the copper by 88 per cent.

Let ρ,ρ0,ρ1 be positive, measurable functions on ℝN. For 1 ≤ t < ∞, consider the weighted Lebesgue and Sobolev spaces

It is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information about the number of ends of G in the sense of H. Abels can be provided by dH1(G, Bi(G)), where Bi(G) is the rational discrete standard bimodule of G, and dH•(G, _) denotes rational discrete cohomology as introduced in [6].

As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).

Equivariant shape theory is an improvement of equivariant homotopy theory which could be regarded as an equivariant version of Borsuk's shape theory. The main result in this paper is a description of the equivariant shape category ShG whose objects are equivariant spaces or G-spaces, i.e. topological spaces endowed with an action of a given topological group G, and whose morphisms are equivariant homotopy or G-homotopy classes of families of multi-valued functions which we call equivariant multi-nets or G-multi-nets. Previously equivariant shape theories have been described only under the assumptions that the group G is either finite or compact. We also study classes of G-spaces on which equivariant shape and equivariant homotopy coincide, look for conditions under which a G-map f: X → Y is an equivariant shape equivalence, and give some characterizations of G-spaces with trivial equivariant shape.

We illustrate a minor error in the biadjointness result for 2-categories of traced monoidal categories and tortile monoidal categories stated by Joyal, Street and Verity. We also show that the biadjointness holds after suitably changing the definition of 2-cells.

There are, in projective space Σ of three dimensions, two famous quartic surfaces: W, the Weddle surface with six nodes ((13), p. 69, footnote) and K, the Kummer surface with sixteen ((s), p. 246, (7) passim). They are in birational correspondence and have the same non-singular model: the octavic base surface F of the net of quadrics in [5], which contain a given line λ and for which a given simplex S is self-polar ((5); (6)). One naturally takes S, with vertices X0, X1, X2, X3, X4, X5 as simplex of reference for homogeneous coordinates x0, x1, x2, x3, x4, x5; F is invariant under the harmonic inversions hj in the vertices Xj and opposite bounding primes xj = 0 of S. These six hj, mutually commutative and having identity for their product, generate an elementary abelian group of order 32. This representation of throws into prominence what may, in this context, be called its positive subgroup , of order 16, consisting of identity and the 15 products hjhk = hkhj; these are harmonic inversions in the edges XjXk and opposite bounding solids xj = xk = 0 of S. The coset of consists of the six hj and their ten products in threes, complementary products being the same (h0h1h2 ≡ h3h4h5) because of the product of all six hj being identity. These ten products are harmonic inversions in the ten pairs of opposite plane faces of S.

We propose a way of associating to each finitely generated monoid or semigroup a formal language, called its loop problem. In the case of a group, the loop problem is essentially the same as the word problem in the sense of combinatorial group theory. Like the word problem for groups, the loop problem is regular if and only if the monoid is finite. We also study the case in which the loop problem is context-free, showing that a celebrated group-theoretic result of Muller and Schupp extends to describe completely simple semigroups with context-free loop problems. We consider right cancellative monoids, establishing connections between the loop problem and the structural theory of these semigroups by showing that the syntactic monoid of the loop problem is the inverse hull of the monoid.

Second order focusing can be obtained with an apparatus, which contains a circular magnetic field and differs little in its dimensions from the apparatus at present in use. The lay-out is shown to scale in Fig. 4: the essential data are collected at the end of Section V, and indications of the accuracy to be expected are given at the end of Section III.

A microcalorimeter is described accurate to 0·0005 cal. The elevation of temperature was measured by a series of iron-constantan thermo-couples, with one set of junctions making good thermal contact with the tube-where the heat was liberated and the others in a brass ring outside, kept in a thermostat. They were connected in series with a very sensitive moving coil galvanometer. The Tian multiple walled thermostat was used—three concentric copper cylinders insulated with kapok, the inner filled with water and the outer controlled by a mercury regulator. The constancy of temperature in the inner vessel was rather better than 1/500,000° C.

Continuous heat evolutions could also be measured by passing a current through another set of thermo-couples, when the Peltier cooling compensated for the heating and the temperature was kept down to its original value, thus avoiding cooling corrections.

Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

We study infinite families of quadratic and cubic twists of the elliptic curve E = X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complex L-series at s=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the same L-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

We discuss a confluence from Siegel–Whittaker functions to Whittaker functions on $Sp(2,{\mathbb{R})$ by using their explicit formulae. In our proof, we use expansion theorems of the good Whittaker functions by the secondary Whittaker functions.