Dunkl operators and a family of realizations of osp(1|2)

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.Comment: 28 pages, some small changes, accepted in Trans. Amer. Math. So

Development of the interRAI Pressure Ulcer Risk Scale (PURS) for use in long-term care and home care settings

<p>Abstract</p> <p>Background</p> <p>In long-term care (LTC) homes in the province of Ontario, implementation of the Minimum Data Set (MDS) assessment and The Braden Scale for predicting pressure ulcer risk were occurring simultaneously. The purpose of this study was, using available data sources, to develop a bedside MDS-based scale to identify individuals under care at various levels of risk for developing pressure ulcers in order to facilitate targeting risk factors for prevention.</p> <p>Methods</p> <p>Data for developing the interRAI Pressure Ulcer Risk Scale (interRAI PURS) were available from 2 Ontario sources: three LTC homes with 257 residents assessed during the same time frame with the MDS and Braden Scale for Predicting Pressure Sore Risk, and eighty-nine Ontario LTC homes with 12,896 residents with baseline/reassessment MDS data (median time 91 days), between 2005-2007. All assessments were done by trained clinical staff, and baseline assessments were restricted to those with no recorded pressure ulcer. MDS baseline/reassessment samples used in further testing included 13,062 patients of Ontario Complex Continuing Care Hospitals (CCC) and 73,183 Ontario long-stay home care (HC) clients.</p> <p>Results</p> <p>A data-informed Braden Scale cross-walk scale using MDS items was devised from the 3-facility dataset, and tested in the larger longitudinal LTC homes data for its association with a future new pressure ulcer, giving a c-statistic of 0.676. Informed by this, LTC homes data along with evidence from the clinical literature was used to create an alternate-form 7-item additive scale, the interRAI PURS, with good distributional characteristics and c-statistic of 0.708. Testing of the scale in CCC and HC longitudinal data showed strong association with development of a new pressure ulcer.</p> <p>Conclusions</p> <p>interRAI PURS differentiates risk of developing pressure ulcers among facility-based residents and home care recipients. As an output from an MDS assessment, it eliminates duplicated effort required for separate pressure ulcer risk scoring. Moreover, it can be done manually at the bedside during critical early days in an admission when the full MDS has yet to be completed. It can be calculated with established MDS instruments as well as with the newer interRAI suite instruments designed to follow persons across various care settings (interRAI Long-Term Care Facilities, interRAI Home Care, interRAI Palliative Care).</p

A mixed methods pilot study with a cluster randomized control trial to evaluate the impact of a leadership intervention on guideline implementation in home care nursing

Abstract Background Foot ulcers are a significant problem for people with diabetes. Comprehensive assessments of risk factors associated with diabetic foot ulcer are recommended in clinical guidelines to decrease complications such as prolonged healing, gangrene and amputations, and to promote effective management. However, the translation of clinical guidelines into nursing practice remains fragmented and inconsistent, and a recent homecare chart audit showed less than half the recommended risk factors for diabetic foot ulcers were assessed, and peripheral neuropathy (the most significant predictor of complications) was not assessed at all. Strong leadership is consistently described as significant to successfully transfer guidelines into practice. Limited research exists however regarding which leadership behaviours facilitate and support implementation in nursing. The purpose of this pilot study is to evaluate the impact of a leadership intervention in community nursing on implementing recommendations from a clinical guideline on the nursing assessment and management of diabetic foot ulcers. Methods Two phase mixed methods design is proposed (ISRCTN 12345678). Phase I: Descriptive qualitative to understand barriers to implementing the guideline recommendations, and to inform the intervention. Phase II: Matched pair cluster randomized controlled trial (n = 4 centers) will evaluate differences in outcomes between two implementation strategies. Primary outcome: Nursing assessments of client risk factors, a composite score of 8 items based on Diabetes/Foot Ulcer guideline recommendations. Intervention: In addition to the organization's 'usual' implementation strategy, a 12 week leadership strategy will be offered to managerial and clinical leaders consisting of: a) printed materials, b) one day interactive workshop to develop a leadership action plan tailored to barriers to support implementation; c) three post-workshop teleconferences. Discussion This study will provide vital information on which leadership strategies are well received to facilitate and support guideline implementation. The anticipated outcomes will provide information to assist with effective management of foot ulcers for people with diabetes. By tracking clinical outcomes associated with guideline implementation, health care administrators will be better informed to influence organizational and policy decision-making to support evidence-based quality care. Findings will be useful to inform the design of future multi-centered trials on various clinical topics to enhance knowledge translation for positive outcomes. Trial Registration Current Control Trials ISRCTN0691089

On boundary value problems for some conformally invariant differential operators

We study boundary value problems for some differential operators on Euclidean space and the Heisenberg group which are invariant under the conformal group of a Euclidean subspace, respectively, Heisenberg subgroup. These operators are shown to be self-adjoint in certain Sobolev type spaces and the related boundary value problems are proven to have unique solutions in these spaces. We further find the corresponding Poisson transforms explicitly in terms of their integral kernels and show that they are isometric between Sobolev spaces and extend to bounded operators between certain L-p-spaces.The conformal invariance of the differential operators allows us to apply unitary representation theory of reductive Lie groups, in particular recently developed methods for restriction problems

Algebraic and analytic methods in representation theory

This book is a compilation of several works from well-recognized figures in the field of Representation Theory. The presentation of the topic is unique in offering several different points of view, which should makethe book very useful to students and experts alike.Presents several different points of view on key topics in representation theory, from internationally known experts in the fiel

Representations in L 2-spaces on infinite-dimensional symmetric cones

AbstractIn this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L2-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert–Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.

Invariant Differential Operators on H-Type Groups and Discrete Components in Restrictions of Complementary Series of Rank One Semisimple Groups

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups G to rank one subgroups G(1). For this we use the realizations of complementary series representations of G and G(1) on Sobolev-type spaces on the nilpotent radicals N and N-1 of the minimal parabolics in G and G(1), respectively. The groups N and N-1 are of H-type and we construct explicitly invariant differential operators between N and N-1. These operators induce the projections onto the discrete components. Our construction of the invariant differential operators is carried out uniformly in the framework of H-type groups and also works for those H-type groups which do not occur as a nilpotent radical of a parabolic subgroup in a semisimple group