Generalised Cesaro Convergence, Root Identities and the Riemann Hypothesis

We extend the notion of generalised Cesaro summation/convergence developed previously to the more natural setting of what we call "remainder" Cesaro summation/convergence and, after illustrating the utility of this approach in deriving certain classical results, use it to develop a notion of generalised root identities. These extend elementary root identities for polynomials both to more general functions and to a family of identities parametrised by a complex parameter \mu. In so doing they equate one expression (the derivative side) which is defined via Fourier theory, with another (the root side) which is defined via remainder Cesaro summation. For \mu a non-positive integer these identities are naturally adapted to investigating the asymptotic behaviour of the given function and the geometric distribution of its roots. For the Gamma function we show that it satisfies the generalised root identities and use them to constructively deduce Stirling's theorem. For the Riemann zeta function the implications of the generalised root identities for \mu=0,-1 and -2 are explored in detail; in the case of \mu=-2 a symmetry of the non-trivial roots is broken and allows us to conclude, after detailed computation, that the Riemann hypothesis must be false. In light of this, some final direct discussion is given of areas where the arguments used throughout the paper are deficient in rigour and require more detailed justification. The conclusion of section 1 gives guidance on the most direct route through the paper to the claim regarding the Riemann hypothesis

Forming contracts without offer and acceptance, Lord Denning and the harmonisation of English contract law

English contract law has traditionally used the requirement of a matching offer and acceptance as the means of identifying agreement. A more flexible approach was proposed by Lord Denning in several cases in the 1970s, but firmly rejected by the House of Lords in Gibson v Manchester City Council [1979] 1 WLR 294. More recently there have been some suggestions of a revival of Lord Denning’s approach, culminating in the decision of the Supreme Court in RTS Flexible Systems Ltd v Molkerei Alois Müller Gmbh [2010] UKSC 14, [2010] 1 WLR 753. This article analyses this development, discussing the implications for both practice and theory of a more flexible approach to formation of contracts. It also draws comparisons with the approach taken in other European jurisdictions and the United States. It concludes that provided the development is kept within identifiable limits, it provides a welcome harmonisation of English law with that of other jurisdictions

The Accounts of Society

Prize Lecture to the memory of Alfred Nobel, December 8, 1984National accounts;

Police powers and human rights in the context of terrorism

Purpose – The object of the paper is to analyse the justifications for the modification of police powers in response to terrorist threats, placing this issue in a European context. Design/methodology/approach – The paper consists of a critical examination of provisions relating to terrorism emanating from the European Union and the Council of Europe (European Convention on Human Rights (ECHR)), and the relevant English law on police powers of stop and search, arrest, and detention. Findings – Nothing in European law requires the amendments to police powers contained in English law; European law requires respect for human rights, even in dealing with terrorism; a shoot-to-kill policy is prohibited by the ECHR; and balance is an unsatisfactory method of resolving conflicts in this area. Research limitations/implications – The research was limited in its scope to certain areas of police powers, and to certain fundamental European documents. Future research should consider the issue in relation to wider areas. Originality/value – It challenges the idea of balance between liberty and security, proposing a test based on necessity instead

Research relative to the heavy isotope spectrometer telescope experiment

The Heavy Isotope Spectrometer Telescope (HIST) was launched during August 1978 on ISEE-3 (ICE). HIST was designed to measure the isotopic composition of solar, galactic, and interplanetary cosmic ray nuclei for the elements from H to Ni (1 less than or equal to Z less than or equal to 28) in the energy range from approximately 5 to approximately 200 MeV/nucleon. The results of these measurements have been used in studies of the composition of solar matter and galactic cosmic ray sources, the study of nucleosynthesis processes, studies of particle acceleration and propagation, and studies of the life-history of cosmic rays in the heliosphere and in the galaxy. On December 1, 1978, after 110 days in orbit, HIST suffered an electronic failure in its readout system. After that point, only one-half of the telemetry bits associated with the pulse heights measured by HIST were transmitted to Earth. As a result, the resolution of HIST was significantly degraded, and it served as an element rather than an isotope spectrometer. Fortunately, HIST was able to measure the isotopic composition of heavy nuclei in the 9/23/78 solar event (the largest solar energetic particle event since 1972) during the brief period that it operated at full resolution. This grant funded the analysis of data from the HIST instrument over the period from 12/1/85 to 11/30/92. In section 2 of this final report, we summarize the scientific accomplishments that have resulted from HIST measurements during this time period. A bibliography of tasks and papers that resulted is attached

Microbial diversity in the thermal springs within Hot Springs National Park

The thermal water systems of Hot Springs National Park (HSNP) in Hot Springs, Arkansas exist in relative isolation from other North American thermal systems. The HSNP waters could therefore serve as a unique center of thermophilic microbial biodiversity. However, these springs remain largely unexplored using culture-independent next generation sequencing techniques to classify species of thermophilic organisms. Additionally, HSNP has been the focus of anthropogenic development, capping and diverting the springs for use in recreational bathhouse facilities. Human modification of these springs may have impacted the structure of these bacterial communities compared to springs left in a relative natural state. The goal of this study was to compare the community structure in two capped springs and two uncapped springs in HSNP, as well as broadly survey the microbial diversity of the springs. We used Illumina 16S rRNA sequencing of water samples from each spring, the QIIME workflow for sequence analysis, and generated measures of genera and phyla richness, diversity, and evenness. In total, over 700 genera were detected and most individual samples had more than 100 genera. There were also several uncharacterized sequences that could not be placed in known taxa, indicating the sampled springs contain undescribed bacteria. There was great variation both between sites and within samples, so no significant differences were detected in community structure between sites. Our results suggest that these springs, regardless of their human modification, contain a considerable amount of biodiversity, some of it potentially unique to the study site

Eroding ribbon thermocouples: impulse response and transient heat flux analysis

We have investigated a particular type of fast-response surface thermocouple to determine if it is appropriate to use a one dimensional transient heat conduction model to derive the transient surface heat flux from the measurements of surface temperature. With these sensors, low thermal inertia thermocouple junctions are formed near the surface by abrasive wear. Using laser excitation, we obtained the impulse response of these commercially available devices. The response of particular sensors can vary if new junctions are created by abrasive wear. Furthermore, the response of these sensors was found to deviate substantially from the one dimensional model and varied from sensor to sensor. The impulse response was simulated with greater fidelity using a two dimensional finite element model, but three dimensional effects also appear to be significant. The impact of these variations on the derived heat flux is assessed for the case of measurements in an internal combustion engine. When the measured impulse response is used to derive the surface heat flux, the apparent reversal of heat flux during the expansion stroke does not occur

Light Play

An interactive playground experienc