The mathematics of Arthur Cayley with particular reference to linear algebra

This thesis is principally concerned with Arthur Cayley's work on Invariant Theory, but also considers his contribution to matrix algebra and other algebraic systems, drawing on sources including unpublished letters between Cayley and his contemporary, J. J. Sylvester. The history of modern linear algebra and Cayley's part in its development has been extensively researched in the last decade by Thomas Hawkins. However, little has been written on Cayley's contribution to Invariant Theory, a subject to which he constantly reverted over a period of fifty years. In comparison, his work on Matrix Theory was a minor interest. The focal points in Cayley's passage through Invariant theory are investigated with reference being made, inter alia, to his correspondence with J. J. Sylvester which affords special insights into both the development of this Theory and the nature of their collaboration. Where appropriate, particulars of Sylvester's own work are given. Biographical details are included where these are believed to be unpublished or otherwise not generally available. A survey of Cayley's mathematical thought is offered in so far as it may be determined from his scattered remarks. Cayley pursued his algebraic researches on two distinct levels. First, he absorbed himself in calculation which led him to the combinatorial aspects of Invariant Theory and, secondly, he displayed a remarkable proclivity for systemisation, although this expressed itself in the classification of specific forms rather than in the development of an abstract theory as with the German algebraists. The basic text contains four chapters on Cayley's work in approximate chronological order followed by a final chapter on his general mathematical thinking. The Appendices include a statistical survey of his work, a bibliography of manuscripts, including, of course, his letters to Sylvester and a number of, little known photographs associated with Cayley and his times

Regional index of sustainable economic well-being development project: final report

This report presents results from a development project carried out by nef (the new economics foundation) on behalf of emda (the East Midlands Development Agency) and Natural England, to improve the methodologies used in the calculation of the R-ISEW (Regional Index of Sustainable Economic Well-Being)

Corporate social counterpositioning: How attributes of social issues influence competitive response

Research Summary Whilst existing research generally assumes corporate social responsibility (CSR) is seen as universally positive, firms increasingly adopt practices, and take stands, on highly polarizing social issues (e.g., gun-control, LGBTQ rights, abortion). To better understand this phenomenon, we develop a theory about when firms will emulate, ignore, or oppose each other's CSR efforts, based on attributes of the underlying social issue (its salience and polarization), the level of market competition, and the substantiveness of CSR. Our theory predicts several distinct equilibrium outcomes, including the potential for social counter-positioning, whereby rival firms take advantage of socio-political polarization to horizontally differentiate by taking opposing stances on a polarizing issue. Counterpositioning is more likely when salience is high, but agreement is low, when markets are competitive, and when CSR is largely symbolic. Managerial Summary Firms increasingly find themselves drawn, willingly or not, to taking stances on a controversial social issue (e.g., gun rights, abortion), though doing so risks alienating (some) stakeholders. In this paper, we develop a theory of why, when, and how firms should take a stance on a polarizing issue. We argue that firms profit from doing so when (1) the issue is salient, (2) markets are competitive, and (3) the actions are mostly symbolic. We also show that taking a stance on polarizing issues creates opportunities for the firms' competitors to counter their ideological positioning, strengthening weaker rivals in the process. Thus, in competitive markets, taking clear stances on polarizing, salient issues can segment the market, increasing the profits of all firms, and, potentially, intensifying polarization

Hip Fracture Types in Men and Women Change Differently with Age

BACKGROUND: Hip fractures are expensive and a frequent cause of morbidity and mortality in the elderly. In most studies hip fractures have been viewed as a unitary fracture but recently the two main types of fracture (intertrochanteric and subcapital) have been viewed as two fractures with a different etiology and requiring a different approach to prevention. The relative proportion of intertrochanteric fractures increases with age in women. In previous studies no particular pattern in men has been noted. In this study, we explored changes in the relative proportion of the two fracture types with age in the two genders. METHODS: Patients of 50 years and older, with a diagnosis of hip fracture, discharged from two local acute care hospitals over a 5 year period (n = 2150) were analyzed as a function of age and gender to explore the relative proportions of intertrochanteric and subcapital fractures, and the change in relative proportion in the two genders with age. RESULTS: Overall, for the genders combined, the proportion of intertrochanteric fractures increases with age (p = .007). In women this increase is significant (p \u3c .001), but in men the opposite pattern is observed, with the proportion of intertrochanteric fractures falling significantly with age (p = .025). CONCLUSIONS: The pattern of hip fractures is different in men and women with aging. It is likely that the pattern difference reflects differences in type and rate of bone loss in the genders, but it is conjectured that the changing rate and pattern of falling with increasing age may also be important. The two main hip fracture types should be considered distinct and different and be studied separately in studies of cause and prevention

Prevalence of secondary care multimorbidity in mid-life and its association with premature mortality in a large longitudinal cohort study

Objectives Multimorbidity is the coexistence of two or more health conditions in an individual. Multimorbidity in younger adults is increasingly recognised as an important challenge. We assessed the prevalence of secondary care multimorbidity in mid-life and its association with premature mortality over 15 years of follow-up, in the Aberdeen Children of the 1950s (ACONF) cohort. Method A prospective cohort study using linked electronic health and mortality records. Scottish ACONF participants were linked to their Scottish Morbidity Record hospital episode data and mortality records. Multimorbidity was defined as two or more conditions and was assessed using healthcare records in 2001 when the participants were aged between 45 and 51 years. The association between multimorbidity and mortality over 15 years of follow-up (to ages 60–66 years) was assessed using Cox proportional hazards regression. There was also adjustment for key covariates: age, gender, social class at birth, intelligence at age 7, secondary school type, educational attainment, alcohol, smoking, body mass index and adult social class. Results Of 9625 participants (51% males), 3% had multimorbidity. The death rate per 1000 person-years was 28.4 (95% CI 23.2 to 34.8) in those with multimorbidity and 5.7 (95% CI 5.3 to 6.1) in those without. In relation to the reference group of those with no multimorbidity, those with multimorbidity had a mortality HR of 4.5 (95% CI 3.4 to 6.0) over 15 years and this association remained when fully adjusted for the covariates (HR 2.5 (95% CI 1.5 to 4.0)). Conclusion Multimorbidity prevalence was 3% in mid-life when measured using secondary care administrative data. Multimorbidity in mid-life was associated with premature mortality

The relationship between weight loss and interleukin 6 in non-small-cell lung cancer.

Markers of the inflammatory response, interleukin 6, C-reactive protein, albumin and full blood count, were measured in non-small-cell lung cancer (NSCLC) patients (n = 21) with and without weight loss ( > 5%). There were significant increases in circulating C-reactive protein (P < 0.001), interleukin 6 (P < 0.01) and platelets (P < 0.01) in the weight-losing group. Moreover, there was a statistically significant correlation (r = 0.785, P < 0.001) between interleukin 6 and C-reactive protein concentrations. These results are consistent with interleukin 6 and the acute phase response promoting weight loss in NSCLC