One-Point Suspensions and Wreath Products of Polytopes and Spheres

It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.Comment: 17 pages, 8 figure

Mechanics and force transmission in soft composites of rods in elastic gels

We report detailed theoretical investigations of the micro-mechanics and bulk elastic properties of composites consisting of randomly distributed stiff fibers embedded in an elastic matrix in two and three dimensions. Recent experiments published in Physical Review Letters [102, 188303 (2009)] have suggested that the inclusion of stiff microtubules in a softer, nearly incompressible biopolymer matrix can lead to emergent compressibility. This can be understood in terms of the enhancement of the compressibility of the composite relative to its shear compliance as a result of the addition of stiff rod-like inclusions. We show that the Poisson's ratio $\nu$ of such a composite evolves with increasing rod density towards a particular value, or {\em fixed point}, independent of the material properties of the matrix, so long as it has a finite initial compressibility. This fixed point is $\nu=1/4$ in three dimensions and $\nu=1/3$ in two dimensions. Our results suggest an important role for stiff filaments such as microtubules and stress fibers in cell mechanics. At the same time, our work has a wider elasticity context, with potential applications to composite elastic media with a wide separation of scales in stiffness of its constituents such as carbon nanotube-polymer composites, which have been shown to have highly tunable mechanics.Comment: 10 pages, 8 figure

Molecular Mechanism of the pH-Dependent Calcium Affinity in Langerin

The C-type lectin receptor langerin plays a vital role in the mammalian defense against invading pathogens. Its function hinges on the affinity to its co-factor Ca2+ which in turn is regulated by the pH. We studied the structural consequences of pro-tonating the allosteric pH-sensor histidine H294 by molecular dynamics simulations (total simulation time: about 120 μs) and Markov models. We discovered a mechanism in which the signal that the pH has dropped is transferred to the Ca2+-binding site without transferring the initial proton. Instead, protonation of H294 unlocks a conformation in which a protonated lysine side-chain forms a hydrogen bond with a Ca2+-coordinating aspartic acid. This destabilizes Ca2+ in the binding pocket, which we probed by steered molecular dynamics. After Ca2+-release, the proton is likely transferred to the aspartic acid and stabilized by a dyad with a nearby glutamic acid, triggering a conformational transition and thus preventing Ca2+-rebinding

Reconstructing a Simple Polytope from its Graph

Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling order of the facets of the dual polytope of P). Thereby, we derive polynomial certificates for both the vertex-facet incidences as well as for the abstract objective functions in terms of the graph of P. The paper is a variation on joint work with Michael Joswig and Friederike Koerner (2001).Comment: 14 page

WHO grade I meningiomas: classification-tree for prognostic factors of survival

World Health Organization (WHO) grade I meningiomas are intracranial extracerebral tumors, in which microsurgery as a stand-alone therapy provides high rates of disease control and low recurrence rates. Our aim was to identify prognostic factors of overall survival and time-to-retreat (OS; TTR) in a cohort of patients with surgically managed WHO grade I meningioma. Patients with WHO grade I meningiomas from a retrospectively (1990 to 2002) and prospectively managed (2003 to 2010) databank of Oslo University Hospital, Norway, were included. The mean follow-up was 9.2 ± 5.7 years, with a total of 11,414 patient-years. One thousand three hundred fifty-five patients were included. The mean age was 58 ± 13.2, mean Karnofsky Performance Status (KPS) 92.6 ± 26.1 and female-to-male ratio 2.5:1. The 1-year, 5-year, 10-year, 15-year, and 20-year probabilities were 0.98, 0.91, 0.87, 0.84, and 0.8 for TTR. Patient age (OR 0.92 [0.91, 0.94]), male sex (OR 0.59 [0.45, 0.76]), preoperative KPS ≥ 70 (OR 2.22 [1.59, 3.13]), skull base location (OR 0.77 [0.60, 1]), and the occurrence of a postoperative hematoma (OR 0.44 [0.26, 0.76]) were identified as independent prognostic factors of OS. Patient age (OR 1.02 [1.01, 1.03]) and skull base location (OR 0.30 [0.21, 0.45]) were independent predictors of decreased PFS. Using a recursive partitioning analysis, we suggest a classification tree for the prediction of 5-year PFS based on patient and tumor characteristics. The findings from this cohort of meningioma WHO I patients helps to identify patients at risk of recurrence and tailor the therapeutic management

Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure

On hydrogen bond correlations at high pressures

In situ high pressure neutron diffraction measured lengths of O H and H O pairs in hydrogen bonds in substances are shown to follow the correlation between them established from 0.1 MPa data on different chemical compounds. In particular, the conclusion by Nelmes et al that their high pressure data on ice VIII differ from it is not supported. For compounds in which the O H stretching frequencies red shift under pressure, it is shown that wherever structural data is available, they follow the stretching frequency versus H O (or O O) distance correlation. For compounds displaying blue shifts with pressure an analogy appears to exist with improper hydrogen bonds.Comment: 12 pages,4 figure

Totally Splittable Polytopes

A split of a polytope is a (necessarily regular) subdivision with exactly two maximal cells. A polytope is totally splittable if each triangulation (without additional vertices) is a common refinement of splits. This paper establishes a complete classification of the totally splittable polytopes.Comment: 15 pages, 7 figures; v2: major revision: corrections of some minor errors and some addition

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure