Rotational integral geometry of tensor valuations

We derive a new rotational Crofton formula for Minkowski tensors. In special cases, this formula gives (1) the rotational average of Minkowski tensors defined on linear subspaces and (2) the functional defined on linear subspaces with rotational average equal to a Minkowski tensor. Earlier results obtained for intrinsic volumes appear now as special cases.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Characterization of granite matrix porosity and pore-space geometry by in situ and laboratory methods

Most available studies of interconnected matrix porosity of crystalline rocks are based on laboratory investigations; that is, work on samples that have undergone stress relaxation and were affected by drilling and sample preparation. The extrapolation of the results to in situ conditions is therefore associated with considerable uncertainty, and this was the motivation to conduct the ‘in situ Connected Porosity’ experiment at the Grimsel Test Site (Central Swiss Alps). An acrylic resin doped with fluorescent agents was used to impregnate the microporous granitic matrix in situ around an injection borehole, and samples were obtained by overcoring. The 3-D structure of the porespace, represented by microcracks, was studied by U-stage fluorescence microscopy. Petrophysical methods, including the determination of porosity, permeability and P -wave velocity, were also applied. Investigations were conducted both on samples that were impregnated in situ and on non-impregnated samples, so that natural features could be distinguished from artefacts. The investigated deformed granites display complex microcrack populations representing a polyphase deformation at varying conditions. The crack population is dominated by open cleavage cracks in mica and grain boundary cracks. The porosity of non-impregnated samples lies slightly above 1 per cent, which is 2–2.5 times higher than the in situ porosity obtained for impregnated samples. Measurements of seismic velocities (Vp ) on spherical rock samples as a function of confining pressure, spatial direction and water saturation for both non-impregnated and impregnated samples provide further constraints on the distinction between natural and induced crack types. The main conclusions are that (1) an interconnected network of microcracks exists in the whole granitic matrix, irrespective of the distance to ductile and brittle shear zones, and (2) conventional laboratory methods overestimate the matrix porosity. Calculations of contaminant transport through fractured media often rely on matrix diffusion as a retardation mechanism.

Mechanical constraints imposed by 3D cellular geometry and arrangement modulate growth patterns in the Arabidopsis embryo

Morphogenesis occurs in 3D space over time and is guided by coordinated gene expression programs. Here we use postembryonic development in Arabidopsis plants to investigate the genetic control of growth. We demonstrate that gene expression driving the production of the growth-stimulating hormone gibberellic acid and downstream growth factors is first induced within the radicle tip of the embryo. The center of cell expansion is, however, spatially displaced from the center of gene expression. Because the rapidly growing cells have very different geometry from that of those at the tip, we hypothesized that mechanical factors may contribute to this growth displacement. To this end we developed 3D finite-element method models of growing custom-designed digital embryos at cellular resolution. We used this framework to conceptualize how cell size, shape, and topology influence tissue growth and to explore the interplay of geometrical and genetic inputs into growth distribution. Our simulations showed that mechanical constraints are sufficient to explain the disconnect between the experimentally observed spatiotemporal patterns of gene expression and early postembryonic growth. The center of cell expansion is the position where genetic and mechanical facilitators of growth converge. We have thus uncovered a mechanism whereby 3D cellular geometry helps direct where genetically specified growth takes place.

How does the ascending aorta geometry change when it dissects?

OBJECTIVES The purpose of this study is to delineate changes in aortic geometry and diameter due to dissection. BACKGROUND Aortic diameter is the major criterion for elective ascending aortic replacement for dilated ascending aortas to prevent aortic dissection. However, recommendations are made on the basis of clinical experience and observation of diameters of previously dissected aortas. METHODS Six tertiary centers on 2 continents reviewed their acute aortic dissection type A databases, which contained 1,821 patients. Included were all non-Marfan patients with nonbicuspid aortic valves who had undergone computed tomography angiography <2 years before and within 12 h after aortic dissection onset. Aortic geometry before and after dissection onset were compared. RESULTS Altogether, 63 patients were included (27 spontaneous and 36 retrograde dissections, median age 68 [57; 77] years; 54% were men). In all but 1 patient, maximum ascending aortic diameter was <55 mm before aortic dissection onset. The largest increase in diameter and volume induced by the dissection were observed in the ascending aorta (40.1 [36.6; 45.3] mm vs. 52.9 [46.1; 58.6] mm, +12.8 mm; p < 0.001; 124.0 [90.8; 162.5] cm(3) vs. 171.0 [147.0; 197.0] cm(3), +47 cm(3); p < 0.001). Mean aortic arch diameter increased from 39.8 (30.5; 42.6) mm to 46.4 (42.0; 51.6) mm (+6.6 mm; p < 0.001) and descending thoracic aorta diameter from 31.2 (27.0; 33.3) mm to 34.9 (30.9; 39.5) mm (+3.7 mm; p < 0.001). Changes in thoracic aorta geometry were similar for spontaneous and retrograde etiology. CONCLUSIONS Geometry of the thoracic aorta is affected by aortic dissection, leading to an increase in diameter that is most pronounced in the ascending aorta. Both spontaneous and retrograde dissection result in similar aortic geometry changes.

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.

Geometry and kinematics of the Roisan-Cignana Shear Zone, and the orogenic evolution of the Dent Blanche Tectonic System (Western Alps)

The Dent Blanche Tectonic System (DBTS) is a composite thrust sheet derived from the previously thinned passive Adriatic continental margin. A kilometric high-strain zone, the Roisan-Cignana Shear Zone (RCSZ) defines the major tectonic boundary within the DBTS and separates it into two subunits, the Dent Blanche s.s. nappe to the northwest and the Mont Mary nappe to the southeast. Within this shear zone, tectonic slices of Mesozoic and pre-Alpine meta-sediments became amalgamated with continental basement rocks of the Adriatic margin. The occurrence of high pressure assemblages along the contact between these tectonic slices indicates that the amalgamation occurred prior to or during the subduction process, at an early stage of the Alpine orogenic cycle. Detailed mapping, petrographic and structural analysis show that the Roisan-Cignana Shear Zone results from several superimposed Alpine structural and metamorphic stages. Subduction of the continental fragments is recorded by blueschist-facies deformation, whereas the Alpine collision is reflected by a greenschist facies overprint associated with the development of large-scale open folds. The postnappe evolution comprises the development of low-angle brittle faults, followed by large-scale folding (Vanzone phase) and finally brittle extensional faults. The RCSZ shows that fragments of continental crust had been torn off the passive continental margin prior to continental collision, thus recording the entire history of the orogenic cycle. The role of preceding Permo-Triassic lithospheric thinning, Jurassic rifting, and ablative subduction processes in controlling the removal of crustal fragments from the reactivated passive continental margin is discussed. Results of this study constrain the temporal sequence of the tectono-metamorphic processes involved in the assembly of the DBTS, but they also show limits on the interpretation. In particular it remains difficult to judge to what extent precollisional rifting at the Adriatic continental margin preconditioned the efficiency of convergent processes, i.e. accretion, subduction, and orogenic exhumation.

SrGAP2-Dependent Integration of Membrane Geometry and Slit-Robo-Repulsive Cues Regulates Fibroblast Contact Inhibition of Locomotion

Migrating fibroblasts undergo contact inhibition of locomotion (CIL), a process that was discovered five decades ago and still is not fully understood at the molecular level. We identify the Slit2-Robo4-srGAP2 signaling network as a key regulator of CIL in fibroblasts. CIL involves highly dynamic contact protrusions with a specialized actin cytoskeleton that stochastically explore cell-cell overlaps between colliding fibroblasts. A membrane curvature-sensing F-BAR domain pre-localizes srGAP2 to protruding edges and terminates their extension phase in response to cell collision. A FRET-based biosensor reveals that Rac1 activity is focused in a band at the tip of contact protrusions, in contrast to the broad activation gradient in contact-free protrusions. SrGAP2 specifically controls the duration of Rac1 activity in contact protrusions, but not in contact-free protrusions. We propose that srGAP2 integrates cell edge curvature and Slit-Robo-mediated repulsive cues to fine-tune Rac1 activation dynamics in contact protrusions to spatiotemporally coordinate CIL.