Solvability of systems of invariant differential equations on symmetric spaces G/K

We study the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type. We show how this can be used for questions of solvability of systems of invariant differential equations in analogy to Hörmander’s proof of the Ehrenpreis-Malgrange theorem. We get complete solvability for the hyperbolic plane H2 and partial results for finite products H2 × · · · × H2 and the hyperbolic 3-space H3.Mir studéieren Fourier Transformatioun fir Distributiounal Sektiounen vu Vektorbündelen u symmetresch Réim vun engem net-kompakten Typ. Mir bewéisen wéi et fir d’Léisbarkeet vu Systémer vun invarianten Differentialequatiounen an Analogie zu Hörmander’s Schätzungen, ugewand ka ginn. Mir kréien komplett Léisbarkeet fir hyperbolesch Pléng H2 a partial Résultater fir Produkter H2 ×· · ·×H2, wéi och fir hyperbolesch 3-Réim H3

How to solve invariant systems of differential equations on SL(2,R)?

In the Euclidean case, it is well-known, by Malgrange and Ehrenpreis, that linear differential operators with constant coefficients are solvable. However, what happens, if we genuinely extend this situation and consider systems of linear invariant differential operators, is still solvable? In the case of $\mathbb{R}^n$ (for some positive integer $n$), the question has been proved mainly by Hörmander. We will show that this remains still true for Riemannian symmetric spaces of non-compact type $X=G/K$, in particular for hyperbolic planes. More precisely, we will present a possible strategy to solve this problem by using the Fourier transform and its Paley-Wiener(-Schwartz) theorem for (distributional) sections of vector bundles over $\mathbb{H}^2=SL(2, \mathbb{R})/SO(2)$. This work was part of my doctoral dissertation supervised by Martin Olbrich

Diabetes mellitus and ischemic heart disease. the role of ion channels

Diabetes mellitus is one the strongest risk factors for cardiovascular disease and, in particular, for ischemic heart disease (IHD). The pathophysiology of myocardial ischemia in diabetic patients is complex and not fully understood: some diabetic patients have mainly coronary stenosis obstructing blood flow to the myocardium; others present with coronary microvascular disease with an absence of plaques in the epicardial vessels. Ion channels acting in the cross-talk between the myocardial energy state and coronary blood flow may play a role in the pathophysiology of IHD in diabetic patients. In particular, some genetic variants for ATP-dependent potassium channels seem to be involved in the determinism of IH

Ensuring sample quality for biomarker discovery studies - Use of ict tools to trace biosample life-cycle

The growing demand of personalized medicine marked the transition from an empirical medicine to a molecular one, aimed at predicting safer and more effective medical treatment for every patient, while minimizing adverse effects. This passage has emphasized the importance of biomarker discovery studies, and has led sample availability to assume a crucial role in biomedical research. Accordingly, a great interest in Biological Bank science has grown concomitantly. In biobanks, biological material and its accompanying data are collected, handled and stored in accordance with standard operating procedures (SOPs) and existing legislation. Sample quality is ensured by adherence to SOPs and sample whole life-cycle can be recorded by innovative tracking systems employing information technology (IT) tools for monitoring storage conditions and characterization of vast amount of data. All the above will ensure proper sample exchangeability among research facilities and will represent the starting point of all future personalized medicine-based clinical trials

A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on G/K

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type X = G/K. We prove a characterisation of their range. In fact, from Delorme’s Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener- Schwartz theorems for sections

Delorme’s intertwining conditions for sections of homogeneous vector bundles on two and three dimensional hyperbolic spaces

The description of the Paley-Wiener space for compactly supported smooth functions C^∞_c(G) on a semi-simple Lie group G, involves certain intertwining conditions, that are difficult to handle. In the present paper, we make them completely explicit for G = SL(2,R)^d (d ∈ N) and G = SL(2,C). Our results are based on a defining criterion for the Paley-Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley-Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces