Variations on a theorem of Davenport concerning abundant numbers

Let \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/\sigma(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq x,~n/\sigma(n) \leq u} 1 exists for all u \in [0,1] and varies continuously with u. We study the behavior of the sums \sum_{n \leq x,~n/\sigma(n) \leq u} f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including \varphi(n), \tau(n), and \mu(n). They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport's result: For all u \in [0,1], the limit $$\tilde{D}(u):= \lim_{R\to\infty} \frac{1}{\pi R}\#\{(x,y) \in \Z^2: 0<x^2+y^2 \leq R \text{ and } \frac{x^2+y^2}{\sigma(x^2+y^2)} \leq u\}$$ exists, and \tilde{D}(u) is both continuous and strictly increasing on [0,1]

Policing Public Houses in Victorian England

This article examines the policing of that most important site for leisure and pleasure among the Victorian working-classes – the pub. It begins with an examination of how changes in policing arrangements from the late-eighteenth century into Victoria’s reign both reflected growing societal anxiety over the conduct of drinking places and led to increased action against them. It provides analyses of the overall incidence of prosecution of publicans in the period up to the important licensing legislation of 1869 and 1872. It examines that legislation and its effects and then turns its attention to the offences of permitting drunkenness and serving a drunken person as particularly indicative of the broader question of the conduct of public houses and of customers’ behaviour within them, setting out trends in their prosecution. It then analyses what underlay the trends revealed, taking in the key variables of the law, the practicalities of its enforcement by the police, the attitudes of the magistracy and the actual conduct of individual publicans and drinkers, within the context of economic, social and cultural changes. By the late-nineteenth and early twentieth centuries, it is argued, pubs were more orderly places, but the achievement of that end was the product of a much more complex set of variables than simply policing arrangements

Cabling in the Skyrme-Faddeev model

The Skyrme–Faddeev model is a three-dimensional nonlinear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimization scheme can be applied. We find the first known hopfions which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots

The dynamics of domain wall Skyrmions

It has recently been shown that Skyrmions with a fixed size can exist in theories without a Skyrme term, providing the Skyrmion is located on a domain wall. Here we numerically compute domain wall Skyrmions of this type, in a (2+1)-dimensional O(3) sigma model with a potential term. Moreover, we investigate Skyrmion dynamics, to study both Skyrmion stability and the scattering of multi-Skyrmions. We demonstrate that scattering events in which both Skyrmions remain on the same domain wall are effectively one-dimensional, and at low speeds are well-approximated by kink scattering in the integrable sine–Gordon model. However, more exotic fully two-dimensional scatterings are also presented, in which Skyrmions that are initially on different domain walls emerge on the same domain wall

Formulating Earthquake Resistant Design Criteria

The primary function of design criteria in general, and earthquake-resistant design criteria in particular, is to restate a complex problem that has unknowns and uncertainties into an unambiguous, simplified form having no uncertainties. The design criteria should provide clearly stated guidelines for the designers. For example, when actually designing a structure, an engineer needs to know the forces and deformations that the structure should be able to resist. Some of these forces, such as dead loads imposed by gravity, are well known, but others that result from transient actions of nature or man, such as earthquake, wind or live loads, are not known. This lack of knowledge must somehow be circumvented and a precise, unambiguous statement of the design conditions must be given to the design engineer. This is accomplished by means of the design criteria. The designer also needs to know the properties of the materials and structural elements that will be used, but as these are not precisely known, mainly because of imperfections in materials and workmanship, the design criteria must also take this into account. In the preparation of the design criteria, allowance must be made for the uncertainties, and it is necessary to be cognizant of all the unknowns for which allowances must be made

The Effect of Local Site Conditions on Recorded Strong Earthquake Motions

The subject of the effects o£ local soil conditions on strong ground motion is of particular interest now because of efforts underway to modify the building codes, new legislation concerning the aseismic design of hospitals, problems of earthquake-resistant design of nuclear power plants and a generally increasing sophistication of earthquake-resistant design procedures in structural engineering practice. As a result of these factors, many major projects now require geological and seismological studies, including some assessment of the expected earthquake motion at the site and an estimate of any probable effects of local site conditions on the expected motions

Comments on “Bedrock Intensity Attenuation and Site Factors from San Fernando Earthquake Records” by K. W. Campbell and C. Martin Duke

In the paper "Bedrock Intensity Attenuation and Site Factors from San Fernando Earthquake Records," by K. W. Campbell and C. Martin Duke, which appeared in the February, 1974 issue of the Bulletin, a correlation is made between a measure of intensity of recorded ground accelerations and four categories of soil, classified by increasing softness. The authors present different numerical site factors for each category and conclude that their results are sufficiently reliable to justify their use for purposes of zoning. The site factors and the conclusion are based mainly on the data presented in their Figure 5 and the lines of specified slope fitted by least squares to the data points. The set of lines in their Figure 6 present the authors' idealized relation between intensity, softness of ground, and distance from the source. The upper four lines in Figure 6 are taken from Figure 5

Memorial - Fritz Matthiesen (1926-1981)

R. B. "Fritz" Matthiesen died on 26 October 1981, at the age of 54, a victim of cancer, and his untimely death was a sad loss to his colleagues in earthquake engineering and seismology. We all miss his technical abilities, his sharp wit, and his irreverent ways of dealing with bureaucracy. Fritz had long been active in the affairs of the Seismological Society of America and was on the Board of Directors at the time of his death. His special technical interests were in the measurement and interpretation of strong ground motion and in the full-scale testing of structures such as buildings, dams, and nuclear reactors, and he was one of the world leaders in these fields. Until recently, he was Chief of the Seismic Engineering Branch of the U.S. Geological Survey, where he worked the past eight years of his career

Proceedings of the Fifth World Conference on Earthquake Engineering [Book Review]

The recently issued two-volume set of the Proceedings of the Fifth World Conference on Earthquake Engineering consists, essentially, of all the papers presented at the last world conference held in Rome in July of 1973. There are approximately 420 papers divided about equally between those 10 pages long and the shorter 4-page papers. The volumes include all the papers which were issued as preprints, a few submitted too late for preprinting, and some 30 discussions. Several of the discussions are categorized as free discussions, and are actually additional papers. Also included in the Proceedings is a list of the 850 participants from the 45 countries represented at the conference; an index of the papers, by session; an index of authors; and a brief section containing data about the International Association for Earthquake Engineering, the sponsoring organization. In the same section are the brief speeches given at the opening and closing ceremonies, administrative reports, and a few photographs. This general material comprises 168 pages, bringing the total for the volumes to nearly 3,200 pages. The massive volumes, 3½ in thick, are well-bound in maroon fabrikoid with gold lettering. The papers are reproduced photographically from originals supplied by the authors so there is considerable variation in typography. The quality of both the printing and the paper is very good, however, and the legibility of text, figures and photographs is at least equal to any of the Proceedings of previous world conferences