The quantum theory of the heavy electron field is applied to calculate approximately the cross-sections for “photoelectric” absorption of heavy electrons by a bound nuclear particle and for emission of a heavy electron by a free nuclear particle on collision with a nucleus. The absorption cross-section per nuclear particle is of the order 10−26 cm.2 for heavy electrons with energies up to 108 e.V., but probably decreases rapidly at higher energies. Although the energy loss of a proton, of energy 108 e.V. or higher, due to heavy electron emission is found to be greater than that due to radiation, the cross-section is very small (⋍ 10−29 cm.2) and the phenomenon is unlikely to be observed in a cloud chamber. In the present state of theory it is impossible to decide whether radiative emission of heavy electrons by nuclear particles is capable of contributing appreciably to the heavy electrons observed at sea-level but, if so, the cross-section for the process must be very much greater at high energies than in the non-relativistic energy region which we have investigated.

A method of deriving power series expansions for the correlation functions of lattice systems is described. The concept of lattice constants for rooted graphs is introduced and it is shown how the correlation functions can be expanded in terms of the lattice constants for connected graphs only. The weight functions in the expansions depend on correlation functions for finite systems and may, therefore, be determined by computer methods. This work is an extension of a method previously developed for the free energy. Applications to spin systems, percolation problems and the lattice gas are considered and sum rules for the weight functions are given.

Various constructions are described for sets F = F(q, k, l) of residues modulo q such that F–F contains all residues while (k)F = F + F + … + F (k summands) omits l consecutive residues, in the cases k = 2, 3, 4 and 5. Let Q(k, l) denote the set of moduli q for which such a set F(q, k, l) exists. The following results are proved: q∈Q (2, 1) if and only if q ≥ 6; q∈Q(2,l) if q ≥ 4l + 2; 29, 31 and 33∈Q(3, 1) and if q∈Q(3, 1) if q ≥ 35; 24l + 21∈Q(3, l) if q ≥ 44l + 41. Upper bounds are also obtained for the least elements in Q(4, l) and Q(5, l).

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

Let ${\bb D}_p$ be a central simple ${\bb Q}_p$-division algebra of index 2, with maximal ${\bb Z}_p$-order $\Delta_p$. We give an explicit formula for the number of subalgebras of any given finite index in the ${\bb Z}_p$-Lie algebra $\mathcal{L}\colone \spl_1(\Delta_p)$. From this we obtain a closed formula for the zeta function $\zeta_\mathcal{L}(s) \colone \sum_{M \leq \mathcal{L}} |\mathcal{L}:M|^{-s}$. The results are extended to the $p$-power congruence subalgebras of $\mathcal{L}$, and as an application we obtain the zeta functions of the corresponding congruence subgroups of the uniform pro-$p$ group $\SL_1^2(\Delta_p)$.

Square integrable and local martingales on a family of σ-fields generated by a basic jump process are shown to have representations as stochastic integrals with respect to a family of martingales associated with the jump process by using the idea of an innovation projection and the associated Lévy system, which is a local characterization of the jumps.

The free product A* B of groups A and B can be described in two ways.

We can construct the set of reduced words in A and B. Define a binary operation on by concatenating two words and performing as many reductions as possible. Prove that is a group; the difficult step is the proof of associativity. Define A * B = .

1. Introduction. In a recent paper Clark (1) has dealt with the problem of the rotating disk, the material of which is such that the waves of dilatation in this particular material travel with the velocity of light. The material of the disk is supposed to be under an isotropic stress p when in a strained state, and the relation between stress p and the dilatation Δ is found to be connected by an expression

where a = density in the unstrained state, and Δ is given by

Ui (i = 1, 2, 3) are the components of the strain.

The complete set of matrix representations of a set of n quantities β1, β2,…,βn satisfying

is obtained for all values of n. It is found that if n = 2v or n = 2v + 1, there are v + 1 irreducible representations with the respective degrees 1, but if n = 2v + 1 there are in addition two conjugate representations of order , the symbol denoting the binomial coefficient. The explicit representations are given for n = 2, 3, 4, 5.

In two recent papers of Conner and Floyd ((2)) and ((3)), the additive structure of the SU-cobordism (or bordism) ring was completely determined. The object of this note is to point out that their results can also be used to determine the somewhat complicated multiplicative structure.

One of the classical models of applied probability is that which may be described as the covering of a line with a random collection of intervals of random length. It appears as such in ((9), pp. 23–25), but also as the problem of Type II counters with random dead time (1, 4, 10), as a model for a pedestrian crossing a busy road(5), and as the queue M/G/∞. If an excuse is required for returning to a problem which has been the object of so much research in the past, it is to be found in the fact, noted by Kendall (5), that the model gives rise to the general stable infinitely divisible p-function, and that the theory of the semigroup P of standard p-functions (6) requires deeper information than previous, practically motivated, studies have provided.

Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points.

Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers.

This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d ⩾ 1. They are (d + 2)-angulated categories, which belong to the subject of higher homological algebra.

We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.

Atomic wave functions calculated using the Hartree—Fock equation and a polarization potential are used to obtain mean square radii of atomic orbitals for the ions Si8+ and Ca+, and transition integrals for the Si IV and Ca II spectra. A table of radial wave functions for Ca+ is given. Our results are compared with those obtained from other wave functions, and in one case with an experimental value, and generally good agreement is obtained.

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

Given a compact, connected, oriented 3-manifold $M$ with boundary, and epimorphism $\chi$ from $H_1M$ to a free abelian group $\Pi$, two invariants $\beta$, $\tau \in \bb {Z}\Pi$ are defined. If $M$ embeds in another such 3-manifold $N$ such that $\chi_N$ factors through $\chi$, then the product $\beta\tau$ divides $\Delta_0(H_1\tilde {N})$.

A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein theoretic generalizations for $2n$-tangles provide invariants that persist in the corresponding invariants of links in which they embed. An example is given of a virtual 4-tangle for which Krebes's theorem does not hold.

Suppose that we have n objects, which may conveniently be represented by the integers 1, 2, 3, …, n. The total number of ways in which they can be ordered is n!. Let the order 1, 2, 3, …, n, be termed the normal order. Any other order can be classified according to the minimum number of interchanges (of one number with an adjacent one) required to restore the normal order.

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of such that G: = E/ is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E; G, λ) is a quotient of the C*-algebra of E obtained by completing the space of -equivariant functions on E. We show that C*(E; G, λ) is postliminal if and only if the orbit space of G is T0 and that C*(E; G, λ) is liminal if and only if the orbit space is T1. We also show that C*(E; G, λ) has bounded trace if and only if G is integrable and that C*(E; G, λ) is a Fell algebra if and only if G is Cartan.

Let be a second-countable, locally compact, Hausdorff groupoid with Haar system λ and continuously varying, abelian isotropy groups. Let be the isotropy groupoid and : = /. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(, λ) has bounded trace if and only if is integrable and that C*(, λ) is a Fell algebra if and only if is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).

In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

The use of artificially pulsed sources of particles in experiments in nuclear physics has already given rise to powerful methods of investigating short-period phenomena and of studying the effects of ‘slow’ neutrons of different energies. On the other hand, the use of such sources introduces complications in counting experiments which are not present when steady sources are employed and it is possible to assume that the time distribution of ionizing events in any counter is a strictly random one. Again, sources which are effectively steady in respect of one counting arrangement are not necessarily so in respect of all. Ordinary cyclotron sources (i.e. cyclotron sources which are not ‘artificially’ pulsed) belong to this class; they have justifiably been considered as steady in the majority of experiments hitherto performed, but the flux of primary particles reaching the target is definitely periodic in time (with the period of the R.F. Voltage applied to the dees), thus a cyclotron will appear as ‘naturally’ pulsed whenever it is used with a counting system for which the effective resolving time is sufficiently short (say < 10−7 sec.).

In the contour integral

f and g are regular functions of z in a neighbourhood of the contour C and of the complex parameters (α1, α2, …, αp) = α in a domain of . N is a positive parameter and asymptotic expansions are considered as N → + ∞. The method of steepest descents provides an asymptotic expansion for each fixed value of α, but this is non-uniform with respect to α near critical values α0 at which certain saddle points tend to coincidence. A more complicated expansion, involving Airy functions or generalizations thereof, is valid and uniform for α near α0. This expansion is known to be valid in a neighbourhood of α0 which does not depend on N, in contrast to certain other expansions having regions of validity which contract to a point or surface as N → ∞. It is thereby suggested that the actual region of validity has a definite geometrical shape determined by the functions f and g and the contour C, just as steepest descents expansions are valid in regions bounded by Stokes' lines. In this paper, procedures are derived for the determination of such regions, subject to assumptions concerning f, g and C. The validity of the expansion is established in regions so determined.