Identities for the gamma and hypergeometric functions: an overview from Euler to the present

A research report submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013Equations involving the gamma and hypergeometric functions are of great interest to mathematicians and scientists, and newly proven identities for these functions assist in finding solutions to differential and integral equations. In this work we trace a brief history of the development of the gamma and hypergeometric functions, illustrate the close relationship between them and present a range of their most useful properties and identities, from the earliest ones to those developed in more recent years. Our literature review will show that while continued research into hypergeometric identities has generated many new results, some of these can be shown to be variations of known identities. Hence, we will also discuss computer based methods that have been developed for creating and analysing such identities, in order to check for originality and for numerical validity

Applications of Special Functions in High Order Finite Element Methods

In this thesis, we optimize different parts of high order finite element methods by application of special functions and symbolic computation. In high order finite element methods, orthogonal polynomials like the Jacobi polynomials are deeply rooted. A broad classical theory of these polynomials is known. Moreover, with modern computer algebra software we can extend this knowledge even further. Here, we apply this knowledge and software for different special functions to derive new recursive relations of local matrix entries. This massively optimizes the assembly time of local high order finite element matrices. Furthermore, the introduced algorithm is in optimal complexity. Moreover, we derive new high order dual functions, which result in fast interpolation operators. Lastly, efficient recursive algorithms for hanging node constraint matrices provided by this new dual functions are given

Activation of the pro-resolving receptor Fpr2 attenuates inflammatory microglial activation

Poster number: P-T099 Theme: Neurodegenerative disorders & ageing Activation of the pro-resolving receptor Fpr2 reverses inflammatory microglial activation Authors: Edward S Wickstead - Life Science & Technology University of Westminster/Queen Mary University of London Inflammation is a major contributor to many neurodegenerative disease (Heneka et al. 2015). Microglia, as the resident immune cells of the brain and spinal cord, provide the first line of immunological defence, but can become deleterious when chronically activated, triggering extensive neuronal damage (Cunningham, 2013). Dampening or even reversing this activation may provide neuronal protection against chronic inflammatory damage. The aim of this study was to determine whether lipopolysaccharide (LPS)-induced inflammation could be abrogated through activation of the receptor Fpr2, known to play an important role in peripheral inflammatory resolution. Immortalised murine microglia (BV2 cell line) were stimulated with LPS (50ng/ml) for 1 hour prior to the treatment with one of two Fpr2 ligands, either Cpd43 or Quin-C1 (both 100nM), and production of nitric oxide (NO), tumour necrosis factor alpha (TNFα) and interleukin-10 (IL-10) were monitored after 24h and 48h. Treatment with either Fpr2 ligand significantly suppressed LPS-induced production of NO or TNFα after both 24h and 48h exposure, moreover Fpr2 ligand treatment significantly enhanced production of IL-10 48h post-LPS treatment. As we have previously shown Fpr2 to be coupled to a number of intracellular signaling pathways (Cooray et al. 2013), we investigated potential signaling responses. Western blot analysis revealed no activation of ERK1/2, but identified a rapid and potent activation of p38 MAP kinase in BV2 microglia following stimulation with Fpr2 ligands. Together, these data indicate the possibility of exploiting immunomodulatory strategies for the treatment of neurological diseases, and highlight in particular the important potential of resolution mechanisms as novel therapeutic targets in neuroinflammation. References Cooray SN et al. (2013). Proc Natl Acad Sci U S A 110: 18232-7. Cunningham C (2013). Glia 61: 71-90. Heneka MT et al. (2015). Lancet Neurol 14: 388-40