Further results on nearly Kirkman triple systems with subsystems

AbstractIn this paper we further discuss the embedding problem for nearly Kirkman triple systems and get the result that: (1) For u≡v≡0(mod6), v⩾78, and u⩾3.5v, there exists an NKTS(u) containing a sub-NKTS(v). (2) For v=18,24,30,36,42,48,54,60,66 or 72, there exists an NKTS(u) containing a sub-NKTS(v) if and only if u≡0(mod6) and u⩾3v

Hypergraph matchings and designs

We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices

The life and work of Major Percy Alexander MacMahon

This thesis describes the life and work of the mathematician Major Percy Alexander MacMahon (1854 - 1929). His early life as a soldier in the Royal Artillery and events which led to him embarking on a career in mathematical research and teaching are dealt with in the first two chapters. Succeeding chapters explain the work in invariant theory and partition theory which brought him to the attention of the British mathematical community and eventually resulted in a Fellowship of the Royal Society, the presidency of the London Mathematical Society, and the award of three prestigious mathematical medals and four honorary doctorates. The development and importance of his recreational mathematical work is traced and discussed. MacMahon's career in the Civil Service as Deputy Warden of the Standards at the Board of Trade is also described. Throughout the thesis, his involvement with the British Association for the Advancement of Science and other scientific organisations is highlighted. The thesis also examines possible reasons why MacMahon's work, held in very high regard at the time, did not lead to the lasting fame accorded to some of his contemporaries. Details of his personal and social life are included to give a picture of MacMahon as a real person working hard to succeed in a difficult context

TeV astronomy of millisecond pulsars

This thesis is concerned with the detection of pulsed TeV γ-rays from millisecond pulsars. These stars appear to include some very efficient producers of high energy particles, but the mechanisms by which they produce TeV γ-rays are still a matter of debate. After an introductory section, there is a brief description of the principles used in the atmospheric Cerenkov technique. The design and operation of the University of Durham atmospheric Cerenkov telescopes are reviewed. The main analysis techniques used to search for periodic signals are then described. The effects on periodic signals of binary motion of a source are discussed. These are a particularly important consideration for observations of millisecond pulsars, where high timing accuracy is required. One of the problems of detecting TeV sources is the cosmic ray background. A means of rejecting background events in TeV γ -ray telescopes is considered in chapter 5. The technique is developed for the Durham Mark III telescope. Substantial rejection of the cosmic ray background is achieved, with minimal loss of source events. The evolutionary scenarios which lead to the formation of millisecond pulsars are outlined. Two models for 7-ray emission are discussed briefly and applied to six known millisecond pulsars. Empirical results on these and two other pulsars are also presented. In particular, a detection of PSR 1855+09 is reported, and an upper limit to the flux from PSR 1957+20 is derived. All the empirical fluxes are compatible with the emission models, but the 'polar gap' model may be favoured. The final chapter summarises the results obtained and suggests some directions for future work on the 7-ray emission from millisecond pulsars