Lecture 1: Introduction to Clifford Algebras Pertti Lounesto 1.1 Introduction • 1.2 Clifford algebra of the Euclidean plane • 1.3 Quaternions • 1.4 Clifford algebra of the Euclidean space 3 • 1.5 The electron spin in a magnetic field • 1.6 From column spinors to spinor operators • 1.7 In 4D: Clifford algebra Cℓ 4 of 4 • 1.8 Clifford algebra of Minkowski spacetime • 1.9 The exterior algebra and contractions • 1.10 The Grassmann-Cayley algebra and shuffle products • 1.11 Alternative definitions of the Clifford algebra • 1.12 References Lecture 2: Mathematical Structure of Clifford Algebras Ian Porteous 2.1 Clifford algebras • 2.2 Conjugation • 2.3 References Lecture 3: Clifford Analysis John Ryan 3.1 Introduction • 3.2 Foundations of Clifford analysis • 3.3 Other types of Clifford holomorphic functions • 3.4 The equation D k ƒ = 0 • 3.5 Conformal groups and Clifford analysis • 3.6 Conformally flat spin manifolds • 3.7 Boundary behavior and Hardy spaces • 3.8 More on Clifford analysis on the sphere • 3.9 The Fourier transform and Clifford analysis • 3.10 Complex Clifford analysis • 3.11 References Lecture 4: Applications of Clifford Algebras in Physics William E. Baylis 4.

In this paper we study Clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure, or more generally, at least a pin structure. The examples treated here are manifolds that can be parametrized by U/Γ where U is a subdomain of either S n or R n and Γ is a Kleinian group acting discontinuously on U . The examples studied here include RP n and the Hopf manifolds S 1 × S n−1 . Also some hyperbolic manifolds will be treated. Special kinds of Clifford-analytic automorphic forms associated to the different choices of Γ are used to construct explicit Cauchy kernels, Cauchy integral formulas, Green's kernels and formulas together with Hardy spaces and Plemelj projection operators for L p spaces of hypersurfaces lying in these manifolds.

For α ∈ R, the class of α−order spherical harmonic functions in an open setis the spherical Laplace-Beltrami operator of order α and s is the radially independent part of the Laplace operator. We obtain a Green's integral formula for the functions in H α ( ) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented.

In this paper we study a generalization of the classical Rarita-Schwinger type operators and construct their fundamental solutions. We give some basic integral formulas related to these operators. We also establish that the projection operators appearing in the Rarita-Schwinger operators and the Rarita-Schwinger equations are conformally invariant. We further obtain the intertwining operators for other operators related to the Rarita-Schwinger operators under actions of the conformal group.

Retroperitoneal fibromatosis herpesvirus (RFHV), the macaque homolog of the human rhadinovirus, Kaposi's sarcoma-associated herpesvirus (KSHV), was first identified in retroperitoneal fibromatosis (RF) tumor lesions of macaques with simian AIDS. We cloned and sequenced the ORF73 latency-associated nuclear antigen (LANA) of RFHVMn from the pig-tailed macaque. RFHVMn LANA is structurally analogous to KSHV ORF73 LANA and contains an N-terminal serine-proline-rich region, a large internal glutamic acidic-rich repeat region and a conserved C-terminal domain. RFHVMn LANA reacts with monoclonal antibodies specific for a glutamic acid-proline dipeptide motif and a glutamic acid-glutamine-rich motif in the KSHV LANA repeat region. Immunohistochemical and immunofluorescence analysis revealed that RFHVMn LANA is a nuclear antigen which is highly expressed in RF spindloid tumor cells. These data suggest that RFHV LANA is an ortholog of KSHV LANA and will function similarly to maintain viral latency and play a role in tumorigenicity in macaques.

Two distinct lineages of rhadinoviruses related to Kaposi's sarcoma (KS)-associated herpesvirus (KSHV; Human herpesvirus 8), the causative agent of KS, have been identified. In macaques, the RV1 lineage is represented by retroperitoneal fibromatosis (RF) herpesvirus (RFHV), the homologue of KSHV, whilst the RV2 lineage is represented by rhesus rhadinovirus (RRV), a more distantly related virus. Real-time quantitative PCR was used to estimate the loads of RV1 and RV2 rhadinoviruses in simian acquired immunodeficiency syndrome-associated RF (SAIDS-RF), a neoplasm of macaques with similarities to AIDS-associated KS. Both RV1 and RV2 rhadinoviruses were detected in macaques with RF. The RV1 loads were 220-to 4300-fold higher in RF tumours than in spleen, showing a strong tumour association (mean loads of 1 800 000 vs 2900 copies per 10 6 cells in tumours and spleen, respectively). In contrast, RV2 loads in the RF tumours were 100-fold lower than RV1 loads and showed similar levels in tumours and spleen (mean loads of 16 000 vs 24 000 copies per 10 6 cells, respectively). Immunostaining with antibodies reactive against RFHV ORF73 latency-associated nuclear antigen (LANA) showed intense nuclear staining of the spindleoid RF tumour cells. Correlation of viral load and the number of LANA-positive cells indicated that RF tumour cells contained multiple copies of the RFHV genome per cell. This pattern of infectivity is similar to that seen in KS tumours latently infected with KSHV. Our study demonstrates similarities in the biology of KSHV and RFHV and supports a role for RFHV in the aetiology of SAIDS-RF.

Overall, findings of this study showed no evidence that the accuracy of expert radiologists is altered due to changing prevalence expectation rates. However, the time spent interpreting each image and the number of fixations increased at higher prevalence rates. Maintenance of diagnostic efficacy has been shown even when circumstances challenge normal observer behavior.

Complications may be encountered following surgical pancreatic biopsy, although the risk should be minimal with good surgical technique. Pancreatic biopsy may provide a useful contribution to case management but it is not clear whether a negative pancreatic biopsy should be used to rule out pancreatic disease. Dogs were more likely to have no significant pathology found on pancreatic biopsy than cats, where chronic pancreatitis was the most common finding.