Segal-Bargmann-Fock modules of monogenic functions

In this paper we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extend it to Clifford algebra-valued functions. Then we apply the results to monogenic functions and prove that the Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform for monogenic functionals. This kernel is also the reproducing kernel for the monogenic Bargmann module.Comment: 11 page

Urinary myeloid IgA Fc alpha receptor (CD89) and transglutaminase-2 as new biomarkers for active IgA nephropathy and henoch-Schönlein purpura nephritis

Background: IgA nephropathy (IgAN) and Henoch-Schönlein purpura nephritis (HSPN) are glomerular diseases that share a common and central pathogenic mechanism. The formation of immune complexes containing IgA1, myeloid IgA Fc alpha receptor (FcαRI/CD89) and transglutaminase-2 (TG2) is observed in both conditions. Therefore, urinary CD89 and TG2 could be potential biomarkers to identify active IgAN/HSPN. Methods: In this multicenter study, 160 patients with IgAN or HSPN were enrolled. Urinary concentrations of CD89 and TG2, as well as some other biochemical parameters, were measured. Results: Urinary CD89 and TG2 were lower in patients with active IgAN/HSPN compared to IgAN/HSPN patients in complete remission (P < 0.001). The CD89xTG2 formula had a high ability to discriminate active from inactive IgAN/HSPN in both situations. : CD89xTG2/proteinuria ratio (AUC: 0.84, P < 0.001, sensitivity: 76%, specificity: 74%) and CD89xTG2/urinary creatinine ratio (AUC: 0.82, P < 0.001, sensitivity: 75%, specificity: 74%). Significant correlations between urinary CD89 and TG2 (r = 0.711, P < 0.001), proteinuria and urinary CD89 (r = -0.585, P < 0.001), and proteinuria and urinary TG2 (r = -0.620, P < 0.001) were observed. Conclusions: Determination of CD89 and TG2 in urine samples can be useful to identify patients with active IgAN/HSPN

Introductory clifford analysis

In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications

Hermite and Gegenbauer polynomials in superspace using Clifford analysis

The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an integration theory in superspace. Furthermore a lot of basic properties, such as orthogonality relations, differential equations and recursion formulae are proven. Finally, an interesting physical application of the super Clifford-Hermite polynomials is discussed, thus giving an interpretation to the super-dimension.Comment: 18 pages, accepted for publication in J. Phys.

Cardiovascular and pharmacological implications of haem-deficient NO-unresponsive soluble guanylate cyclase knock-in mice

Oxidative stress, a central mediator of cardiovascular disease, results in loss of the prosthetic haem group of soluble guanylate cyclase (sGC), preventing its activation by nitric oxide (NO). Here we introduce Apo-sGC mice expressing haem-free sGC. Apo-sGC mice are viable and develop hypertension. The haemodynamic effects of NO are abolished, but those of the sGC activator cinaciguat are enhanced in apo-sGC mice, suggesting that the effects of NO on smooth muscle relaxation, blood pressure regulation and inhibition of platelet aggregation require sGC activation by NO. Tumour necrosis factor (TNF)-induced hypotension and mortality are preserved in apo-sGC mice, indicating that pathways other than sGC signalling mediate the cardiovascular collapse in shock. Apo-sGC mice allow for differentiation between sGC-dependent and -independent NO effects and between haem-dependent and -independent sGC effects. Apo-sGC mice represent a unique experimental platform to study the in vivo consequences of sGC oxidation and the therapeutic potential of sGC activators

Spherical harmonics and integration in superspace

In this paper the classical theory of spherical harmonics in R^m is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.Comment: 22 pages, accepted for publication in J. Phys.

PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras

Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space related J-selfadjoint extensions for PTQM setups with ultra-localized potentials.Comment: 11 page