Fourier, Gauss, Fraunhofer, Porod and the Shape from Moments Problem

We show how the Fourier transform of a shape in any number of dimensions can be simplified using Gauss's law and evaluated explicitly for polygons in two dimensions, polyhedra three dimensions, etc. We also show how this combination of Fourier and Gauss can be related to numerous classical problems in physics and mathematics. Examples include Fraunhofer diffraction patterns, Porods law, Hopfs Umlaufsatz, the isoperimetric inequality and Didos problem. We also use this approach to provide an alternative derivation of Davis's extension of the Motzkin-Schoenberg formula to polygons in the complex plane.Comment: 21 pages, no figure

Stability for Borell-Brascamp-Lieb inequalities

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs.Comment: to appear in GAFA Seminar Note

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.

Systematic Heterogeneity of Fractional Vesicle Pool Sizes and Release Rates of Hippocampal Synapses

AbstractHippocampal neurons in tissue culture develop functional synapses that exhibit considerable variation in synaptic vesicle content (20–350 vesicles). We examined absolute and fractional parameters of synaptic vesicle exocytosis of individual synapses. Their correlation to vesicle content was determined by activity-dependent discharge of FM-styryl dyes. At high frequency stimulation (30 Hz), synapses with large recycling pools released higher amounts of dye, but showed a lower fractional release compared to synapses that contained fewer vesicles. This effect gradually vanished at lower frequencies when stimulation was triggered at 20 Hz and 10 Hz, respectively. Live-cell antibody staining with anti-synaptotagmin-1-cypHer 5, and overexpression of synaptopHluorin as well as photoconversion of FM 1-43 followed by electron microscopy, consolidated the findings obtained with FM-styryl dyes. We found that the readily releasable pool grew with a power function with a coefficient of 2/3, possibly indicating a synaptic volume/surface dependency. This observation could be explained by assigning the rate-limiting factor for vesicle exocytosis at high frequency stimulation to the available active zone surface that is proportionally smaller in synapses with larger volumes

Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

DEAR project: Lunar dust surface interactions, risk and removal investigations

The DEAR project (Dusty Environment Application Research) investigates the interaction between lunar regolith and surfaces and components relevant for lunar exploration. Based on the TUBS regolith simulant which is representative in chemistry, size and shape properties to Moon soils to study the regolith transport, adhesion and strategies for cleaning. The regolith simulant will be applied to thermal, structural, optical sensor, sealing and other astronautic systems, providing input for requirements, justification and verification. The key applications are split in human space flight regolith investigations, wrinkled surface with random movement and hardware surfaces, flat material defined movement. The paper provides an overview of the DEAR project including a discussion of the first results, in particular vibration, shock and micro-vibration on regolith bearing surfaces. The investigation shall enable better understand the regolith layers interaction and the release mechanism, as well as potential cross contamination and cleaning strategies. The research is complemented by simulation of the regolith motion as parameter surface plasma interactions. The project is funded and supported by the European Space Agency (ESA). DEAR specifically addresses the development and testing of lunar dust removal strategies on optics, mechanisms and human space flight hardware (e.g., space suits). As the Moons regolith is known to be highly abrasive, electrically chargeable, and potentially chemically reactive, lunar dust might reduce the performance of hardware, such as cameras, thermal control surfaces and solar cells. The dust can cause malfunction on seals for on/off mechanisms or space suits. Of particular interest are risk assessment, avoidance, and cleaning techniques such as the use of electric fields to remove lunar dust from surfaces. Representative dust (e.g., regolith analogues of interesting landing sites) will be used in a dedicated test setup to evaluate risks and effects of lunar dust. We describe designs and methods developed by the DEAR consortium to deal with the regolith-related issues, in particular an electrode design to deflect regolith particles, cleaning of astronautical systems with CO2, design of a robotic arm for the testing within the DEAR chamber, regolith removal via shock, and regolith interaction with cleanroom textile

Orientations, lattice polytopes, and group arrangements II: Modular and integral flow Polynomials of graphs

We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law on the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers and answers a question by Beck and Zaslavsky.Regal Entertainment Group (Competitive Earmarked Research Grants 600703)Regal Entertainment Group (Competitive Earmarked Research Grants 600506)Regal Entertainment Group (Competitive Earmarked Research Grants 600608

Key Physiological Parameters Dictate Triggering of Activity-Dependent Bulk Endocytosis in Hippocampal Synapses

To maintain neurotransmission in central neurons, several mechanisms are employed to retrieve synaptically exocytosed membrane. The two major modes of synaptic vesicle (SV) retrieval are clathrin-mediated endocytosis and activity-dependent bulk endocytosis (ADBE). ADBE is the dominant SV retrieval mode during intense stimulation, however the precise physiological conditions that trigger this mode are not resolved. To determine these parameters we manipulated rat hippocampal neurons using a wide spectrum of stimuli by varying both the pattern and duration of stimulation. Using live-cell fluorescence imaging and electron microscopy approaches, we established that stimulation frequency, rather than the stimulation load, was critical in the triggering of ADBE. Thus two hundred action potentials, when delivered at high frequency, were sufficient to induce near maximal bulk formation. Furthermore we observed a strong correlation between SV pool size and ability to perform ADBE. We also identified that inhibitory nerve terminals were more likely to utilize ADBE and had a larger SV recycling pool. Thus ADBE in hippocampal synaptic terminals is tightly coupled to stimulation frequency and is more likely to occur in terminals with large SV pools. These results implicate ADBE as a key modulator of both hippocampal neurotransmission and plasticity