Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

Reduced trabecular bone mineral density and cortical thickness accompanied by increased outer bone circumference in metacarpal bone of rheumatoid arthritis patients: a cross-sectional study

Introduction The objective of this study was to assess three-dimensional bone geometry and density at the epiphysis and shaft of the third meta-carpal bone of rheumatoid arthritis (RA) patients in comparison to healthy controls with the novel method of peripheral quantitative computed tomography (pQCT). Methods PQCT scans were performed in 50 female RA patients and 100 healthy female controls at the distal epiphyses and shafts of the third metacarpal bone, the radius and the tibia. Reproducibility was determined by coefficient of varia-tion. Bone densitometric and geometric parameters were compared between the two groups and correlated to disease characteristics. Results Reproducibility of different pQCT parameters was between 0.7% and 2.5%. RA patients had 12% to 19% lower trabecular bone mineral density (BMD) (P ≤ 0.001) at the distal epiphyses of radius, tibia and metacarpal bone. At the shafts of these bones RA patients had 7% to 16% thinner cortices (P ≤ 0.03). Total cross-sectional area (CSA) at the metacarpal bone shaft of pa-tients was larger (between 5% and 7%, P < 0.02), and relative cortical area was reduced by 13%. Erosiveness by Ratingen score correlated negatively with tra-becular and total BMD at the epiphyses and shaft cortical thickness of all measured bones (P < 0.04). Conclusions Reduced trabecular BMD and thinner cortices at peripheral bones, and a greater bone shaft diameter at the metacarpal bone suggest RA spe-cific bone alterations. The proposed pQCT protocol is reliable and allows measuring juxta-articular trabecular BMD and shaft geometry at the metacarpal bone.

Horizon shells and BMS-like soldering transformations

We revisit the theory of null shells in general relativity, with a particular emphasis on null shells placed at horizons of black holes. We study in detail the considerable freedom that is available in the case that one solders two metrics together across null hypersurfaces (such as Killing horizons) for which the induced metric is invariant under translations along the null generators. In this case the group of soldering transformations turns out to be infinite dimensional, and these solderings create non-trivial horizon shells containing both massless matter and impulsive gravitational wave components. We also rephrase this result in the language of Carrollian symmetry groups. To illustrate this phenomenon we discuss in detail the example of shells on the horizon of the Schwarzschild black hole (with equal interior and exterior mass), uncovering a rich classical structure at the horizon and deriving an explicit expression for the general horizon shell energy-momentum tensor. In the special case of BMS-like soldering supertranslations we find a conserved shell-energy that is strikingly similar to the standard expression for asymptotic BMS supertranslation charges, suggesting a direct relation between the physical properties of these horizon shells and the recently proposed BMS supertranslation hair of a black hole.

On the Closure of the Tame Automorphism Group of Affine Three-Space

We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of C3 is not closed, when the latter is seen as an infinite-dimensional algebraic group.

Local strain distribution in real three-dimensional alveolar geometries

Mechanical ventilation is not only a life saving treatment but can also cause negative side effects. One of the main complications is inflammation caused by overstretching of the alveolar tissue. Previously, studies investigated either global strains or looked into which states lead to inflammatory reactions in cell cultures. However, the connection between the global deformation, of a tissue strip or the whole organ, and the strains reaching the single cells lining the alveolar walls is unknown and respective studies are still missing. The main reason for this is most likely the complex, sponge-like alveolar geometry, whose three-dimensional details have been unknown until recently. Utilizing synchrotron-based X-ray tomographic microscopy, we were able to generate real and detailed three-dimensional alveolar geometries on which we have performed finite-element simulations. This allowed us to determine, for the first time, a three-dimensional strain state within the alveolar wall. Briefly, precision-cut lung slices, prepared from isolated rat lungs, were scanned and segmented to provide a three-dimensional geometry. This was then discretized using newly developed tetrahedral elements. The main conclusions of this study are that the local strain in the alveolar wall can reach a multiple of the value of the global strain, for our simulations up to four times as high and that thin structures obviously cause hotspots that are especially at risk of overstretching.

Favourable prognostic value of antibodies anti-HSP20 in patients with cystic echinococcosis: a differential immunoproteomic approach

Seeking biomarkers reflecting disease development in cystic echinococcosis (CE), we used a proteomic approach linked to immunological characterisation for the identification of respective antigens. Two-dimensional gel electrophoresis (2-DE) of sheep hydatid fluid, followed by immunoblot analysis (IB) with sera from patients with distinct phases of disease, enabled us to identify by mass spectrometry heat shock protein 20 (HSP20) as a potential marker of active CE. Using IB, antibodies specific to the 34 kDa band of HSP20 were detected in sera from 61/95 (64%) patients with CE, but not in sera from healthy subjects. IB revealed anti-HSP20 antibodies in a higher percentage of sera from patients with active disease than in sera from patients with inactive disease (81 vs. 24%; P = 10(-4)). These primary results were confirmed in a long-term follow-up study after pharmacological and surgical treatment. Herewith anti-HSP20 antibody levels significantly decreased over the course of treatment in sera from patients with cured disease, relative to sera from patients with progressive disease (P = 0.017). Thus, during CE, a comprehensive strategy of proteomic identification combined with immunological validation represents a promising approach for the identification of biomarkers useful for the prognostic assessment of treatment of CE patients.

Differential distribution of vasa vasorum in different vascular beds in humans

OBJECTIVE: Vasa vasorum (VV) have been implicated to play a role in the pathogenesis of atherosclerosis. This study was designed to describe the distribution of VV density in different vascular beds in humans and to investigate the association between VV density and the known distribution of atherosclerosis in human arteries. METHODS: Forty-two human arteries, harvested at autopsy or after explantation, were analyzed by three-dimensional microscopic-computed tomography (micro-CT). VV density, endothelial-surface-fraction (Sigma VV endothelial-surface-area/vessel-wall-volume) and vascular-area-fraction (Sigma VV area/vessel-wall-area) were calculated for coronary, renal and femoral arteries. Representatively five coronary, renal and femoral arteries were stained for endothelial cells (von Willebrand-Factor), macrophages (CD68), vascular endothelial growth factor (VEGF) and collagen (Sirius Red). RESULTS: Coronary arteries showed a higher VV density compared to renal and femoral arteries (2.12+/-0.26 n/mm(2) versus 0.61+/-0.06 n/mm(2) and 0.66+/-0.11 n/mm(2); P<0.05 for both) as well as a higher endothelial-surface-fraction and vascular-area-fraction. Histology showed a positive correlation between histologically derived VV density and CD68-positive cells/area (r=0.54, P<0.01), VEGF-immunoreactivity/area (r=0.55, P<0.01) and a negative correlation between VV density and collagen I content (r=0.66, P<0.05). CONCLUSION: This micro-CT study highlights a higher VV density in coronary than in peripheral arteries, supporting the relation between VV density and the susceptibility to atherosclerosis in different vascular beds in humans.

Two-dimensional supramolecular polymers: controlled formation and visualization on the surface

Self – assembly is a powerful tool for the construction of highly organized nanostructures. Therefore, the possibility to control and predict pathways of molecular ordering on the nanoscale level is a critical issue for the production of materials with tunable and adaptive macroscopic properties. 2D polymers are attractive objects for the field of material sciences due to their exceptional properties. [1] As shown before, amphiphilic oligopyrenotides (produced via automated solid-phase synthesis) form rod–like supramolecular polymers in water. [2] These assemblies form 1D objects. [3] By applying certain changes to the design of the oligopyrenotide units the dimensionality of the formed assemblies can be influenced. Herein, we demonstrate that Py3 (see Figure 1) forms defined supramolecular assemblies under thermodynamic conditions in water. To study Py3 self-assembly, we carried out whole set of spectroscopic (UV/vis, fluorescence, DLS) and microscopic experiments (AFM). The obtained results suggest that oligopyrenotides with the present type of geometry and linker length leads to formation of 2D supramolecular assemblies.

Infinite Transitivity on Affine Varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.