Permutations of a semigroup that map to inverses

We investigate the question as to when the members of a finite regular semigroup may be permuted in such a way that each member is mapped to one of its inverses. In general this is not possible. However we reformulate the problem in terms of a related graph and, using an application of Hall’s Marriage Lemma, we show in particular that the finite full transformation semigroup does enjoy this property

Medical care in English prisons 1770-1850

A consensus exists amongst both contemporary and present day students of penology that the prison reform process of the late-eighteenth century achieved one of its main objectives: the removal of a high risk of death from disease whilst incarcerated as an additional penalty to that imposed by the courts. Most observers are also in agreement that much of this desirable outcome was achieved by improving the prison environment - principally in respect of better standards of hygiene and a reduction in overcrowding. No such consensus exists amongst modem commentators as to the role of prison medical officers in this amelioration; indeed there has been a tendency to impute a negative role to these men. They are accused of being deeply involved in a system designed to control and dominate the lower orders of society, conniving in resultant deleterious effects on the health of their patients. This thesis provides the first detailed analysis of the effects of the day-to-day work of prison medical officers in the period 1770 to 1850. I have amassed information from several sources including: (1) journals kept by prison medical officers, (2) parliamentary enquiries, and (3) reports of prison inspectors. Some of these sources have never been previously studied; others have never been subjected to scrutiny from an informed medical standpoint. I have concluded that the prime factor motivating prison medical staff was the provision of care; their participation in a control system, although unavoidable, did not have an adverse effect on the standard of care. This care was, so far as can be ascertained, of the same standard as that provided to their private patients and although some treatment methods may seem primitive or even cruel to the modem reader, these were standard at the time. The practice of medicine involves the exercise of power; these men exercised that power in a correct and professional manner, caring for their patients to the best of their ability and ensuring maintenance of as healthy a prison environment as was possible

Finite regular semigroups with permutations that map elements to inverses

We give an account on what is known on the subject of permutation matchings, which are bijections of a finite regular semigroup that map each element to one of its inverses. This includes partial solutions to some open questions, including a related novel combinatorial problem

The Biker-hiker problem

There are n travellers who have k bicycles and they wish to complete a journey in the shortest possible time. We investigate optimal solutions of this problem, showing they are characterized by a set of words in the Dyck language. Particular solutions with additional desirable properties are introduced and analysed

Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup

We continue the study of permutations of a fi nite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a cycle. Finally, as an application, we use involution matchings to prove that when the base set has at least four members, a fi nite full transformation semigroup has no cover by inverse subsemigroups that is closed under intersection

Micro Channel Cooler Performance Improvement by Insonation

The motivation for this work is the need to remove waste heat from laser diodes and high speed transistors in processes which are exponentially increasing past 1 kW/cm2 as anticipated by Moore\u27s Law. The hypothesis guiding the work is that ultrasonic insonation of micro coolers employed to dissipate these heat loads can improve heat removal. It is thought that the mechanism promoting the benefit is enhancement of the ability of the coolant to remove latent heat in two-phase operation by managing entrained bubble size near the cooler\u27s exit so as to forestall flow reduction or blockage caused by large bubbles, wedges and slugs accumulating there. Insonation experiments to prove the hypothesis have been done on several micro channel coolers in the range 4-80 kHz to quantify improvement in heat flux removal. In order to understand how insonation would produce benefit in heat removal, a research effort was undertaken to study the affect of 5-30 Pa acoustic fields on air bubbles rising in small aquariums. This involved developing a Faraday cage shielded acoustic probe, along with a force-beam calibration tool, for measuring field levels near a strongly electromagnetic-radiating ultrasonic source. Experiments were conducted on columns of pseudo monodisperse, sub-millimeter diameter air bubbles in water, and other fluids using bubble generators optimized for this purpose. A numerical analysis model based on energy balance of the acoustic work done on a bubble resulted in predicting mass transfer flux, and in quantifying bubble shrinkage and growth when irradiated on either side of its resonance. The model, and experiments show that bubble populations can be predictably altered by ultrasound. The research was concluded by identifying and quantifying micro channel cooler performance change when insonated in the range 4-80 kHz. It was discovered that 28 and 58 kHz radiation of exchangers having hydraulic diameters spanning 0.02 to 0.6 mm could produce heat flux removal improvements of 5 W/cm2 in devices normally removing less than 30 W/cm2, a factor of 17%. Peak thermal resistance improvement approaching 60 % has been observed