12,125 research outputs found

    Group Theoretical Foundations of Fractional Supersymmetry

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    Fractional supersymmetry denotes a generalisation of supersymmetry which may be constructed using a single real generalised Grassmann variable, θ=θˉ,θn=0\theta = \bar{\theta}, \, \theta^n = 0, for arbitrary integer n=2,3,...n = 2, 3, .... An explicit formula is given in the case of general nn 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 θ\theta.Comment: Plain Latex, 18 page

    Towards the disintermediation of creative music search: Analysing queries to determine important facets

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    Purpose: Creative professionals search for music to accompany moving images in films, advertising, television. Some larger music rights holders (record companies and music publishers) organise their catalogues to allow online searching. These digital libraries are organised by various subjective musical facets as well as by artist and title metadata. The purpose of this paper is to present an analysis of written queries relating to creative music search, contextualised and discussed within the findings of text analyses of a larger research project whose aim is to investigate meaning making in this search process. Method: A facet analysis of a collection of written music queries is discussed in relation to the organisation of the music in a selection of bespoke search engines. Results: Subjective facets, in particular Mood, are found to be highly important in query formation. Unusually, detailed Music Structural aspects are also key. Conclusions: These findings are discussed in relation to disintermediation of this process. It is suggested that there are barriers to this, both in terms of classification and also commercial / legal factors

    Optimally defined Racah-Casimir operators for su(n) and their eigenvalues for various classes of representations

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    This paper deals with the striking fact that there is an essentially canonical path from the ii-th Lie algebra cohomology cocycle, i=1,2,...li=1,2,... l, of a simple compact Lie algebra \g of rank ll to the definition of its primitive Casimir operators C(i)C^{(i)} of order mim_i. Thus one obtains a complete set of Racah-Casimir operators C(i)C^{(i)} for each \g and nothing else. The paper then goes on to develop a general formula for the eigenvalue c(i)c^{(i)} of each C(i)C^{(i)} valid for any representation of \g, and thereby to relate c(i)c^{(i)} to a suitably defined generalised Dynkin index. The form of the formula for c(i)c^{(i)} for su(n)su(n) is known sufficiently explicitly to make clear some interesting and important features. For the purposes of illustration, detailed results are displayed for some classes of representation of su(n)su(n), including all the fundamental ones and the adjoint representation.Comment: Latex, 16 page
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