Group Theoretical Foundations of Fractional Supersymmetry

Fractional supersymmetry denotes a generalisation of supersymmetry which may be constructed using a single real generalised Grassmann variable, $\theta = \bar{\theta}, \, \theta^n = 0$, for arbitrary integer $n = 2, 3, ...$. An explicit formula is given in the case of general $n$ for the transformations that leave the theory invariant, and it is shown that these transformations possess interesting group properties. It is shown also that the two generalised derivatives that enter the theory have a geometric interpretation as generators of left and right transformations of the fractional supersymmetry group. Careful attention is paid to some technically important issues, including differentiation, that arise as a result of the peculiar nature of quantities such as $\theta$.Comment: Plain Latex, 18 page

Dismantling the signposts to public health? NHS data under the Health and Social Care Act 2012

The Health and Social Care Act 2012 will replace the administrative structure of the NHS in England, currently based on the resident populations of defined geographical areas, with one that relates instead to the shifting populations of individuals registered with specific general practices at given points in time.1 This will radically change the longstanding basis for collecting data routinely about the health needs of local populations, making it difficult to monitor the effect of new legislation on the health of the population locally or nationally.2 3 We discuss some of the implications of the act for existing routine data systems and the production of routine statistics that underpin essential NHS functions, including monitoring healthcare provision and ensuring equity of access, allocation of resources, and measurement of outcomes

An experimental comparison of a genetic algorithm and a hill-climber for term selection

Purpose – The term selection problem for selecting query terms in information filtering and routing has been investigated using hill-climbers of various kinds, largely through the Okapi experiments in the TREC series of conferences. Although these are simple deterministic approaches which examine the effect of changing the weight of one term at a time, they have been shown to improve the retrieval effectiveness of filtering queries in these TREC experiments. Hill-climbers are, however, likely to get trapped in local optima, and the use of more sophisticated local search techniques for this problem that attempt to break out of these optima are worth investigating. To this end, we apply a genetic algorithm (GA) to the same problem. Design/Methodology/Approach – We use a standard TREC test collection from the TREC-8 filtering track, recording mean average precision and recall measures to allow comparison between the hillclimber and GA algorithms. We also vary elements of the GA, such as probability of a word being included, probability of mutation and population size in order to measure the effect of these variables. Different strategies such as Elitist and Non-Elitist methods are used, as well as Roulette Wheel and Rank selection GA algorithms. Findings – The results of tests suggest that both techniques are, on average, better than the baseline, but the implemented GA does not match the overall performance of a hill-climber. The Rank selection algorithm does better on average than the Roulette Wheel algorithm. There is no evidence in this study that varying word inclusion probability, mutation probability or Elitist method make much difference to the overall results. Small population sizes do not appear to be as effective as larger population sizes. Research limitations/implications – The evidence provided here would suggest that being stuck in a local optima for the term selection optimization problem does not appear to be detrimental to the overall success of the hill-climber. The evidence from term rank order would appear to provide extra useful evidence which hill-climbers can use efficiently and effectively to narrow the search space. Originality/Value – The paper represents the first attempt to compare hill-climbers with GAs on a problem of this type

Dynamical Symmetries in q-deformed Quantum Mechanics

The dynamical algebra of the q-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a well-defined algebra SU$_q$(1,1) with consistent conjugation properties and comultiplication. We obtain non lowest weight representations of this algebra.Comment: 19 pages, latex, no figure

Compilation of relations for the antisymmetric tensors defined by the Lie algebra cocycles of su(n)su(n)

This paper attempts to provide a comprehensive compilation of results, many new here, involving the invariant totally antisymmetric tensors (Omega tensors) which define the Lie algebra cohomology cocycles of $su(n)$, and that play an essential role in the optimal definition of Racah-Casimir operators of $su(n)$. Since the Omega tensors occur naturally within the algebra of totally antisymmetrised products of $\lambda$-matrices of $su(n)$, relations within this algebra are studied in detail, and then employed to provide a powerful means of deriving important Omega tensor/cocycle identities. The results include formulas for the squares of all the Omega tensors of $su(n)$. Various key derivations are given to illustrate the methods employed.Comment: Latex file (run thrice). Misprints corrected, Refs. updated. Published in IJMPA 16, 1377-1405 (2001