Reverse and dual Loomis-Whitney-type inequalities

Various results are proved giving lower bounds for the $m$th intrinsic volume $V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in terms of the $m$th intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when $m=1$ and $m=n-1$. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume $V_1(K)$, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke and McMullen made in 1983

Intersections of Dilatates of Convex Bodies

We initiate a systematic investigation into the nature of the function ∝K(L,ρ) that gives the volume of the intersection of one convex body K in Rn and a dilatate ρL of another convex body L in Rn, as well as the function ηK(L, ρ) that gives the (n - 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρ L) of ρL. The focus is on the concavity properties of αK (L, ρ). Of particular interest is the case when K and L are symmetric with respect to the origin. In this situation, there is an interesting change in the concavity properties of αK (L, ρ) between dimension 2 and dimensions 3 or higher. When L is the unit ball, an important special case with connections to E. Lutwak\u27s dual Brunn-Minkowski theory, we prove that this change occurs between dimension 2 and dimensions 4 or higher, and conjecture that it occurs between dimension 3 and dimension 4. We also establish an isoperimetric inequality with equality condition for subsets of equatorial zones in the sphere S2, and apply this and the Brunn-Minkowski inequality in the sphere to obtain results related to this conjecture, as well as to the properties of a new type of symmetral of a convex body, which we call the equatorial symmetral

Conceptual modeling for genomics: Building an integrated repository of open data

Many repositories of open data for genomics, collected by world-wide consortia, are important enablers of biological research; moreover, all experimental datasets leading to publications in genomics must be deposited to public repositories and made available to the research community. These datasets are typically used by biologists for validating or enriching their experiments; their content is documented by metadata. However, emphasis on data sharing is not matched by accuracy in data documentation; metadata are not standardized across the sources and often unstructured and incomplete. In this paper, we propose a conceptual model of genomic metadata, whose purpose is to query the underlying data sources for locating relevant experimental datasets. First, we analyze the most typical metadata attributes of genomic sources and define their semantic properties. Then, we use a top-down method for building a global-as-view integrated schema, by abstracting the most important conceptual properties of genomic sources. Finally, we describe the validation of the conceptual model by mapping it to three well-known data sources: TCGA, ENCODE, and Gene Expression Omnibus

Oversampling errors in multimodal medical imaging are due to the Gibbs effect

To analyse multimodal 3-dimensional medical images, interpolation is required for resampling which - unavoidably - introduces an interpolation error. In this work we consider three segmented 3-dimensional images resampled with three different neuroimaging software tools for comparing undersampling and oversampling strategies and to identify where the oversampling error lies. The results indicate that undersampling to the lowest image size is advantageous in terms of mean value per segment errors and that the oversampling error is larger where the gradient is steeper, showing a Gibbs effect

Biological Strategies to Enhance Healing of the Avascular Area of the Meniscus

Meniscal injuries in the vascularized peripheral part of the meniscus have a better healing potential than tears in the central avascular zone because meniscal healing principally depends on its vascular supply. Several biological strategies have been proposed to enhance healing of the avascular area of the meniscus: abrasion therapy, fibrin clot, organ culture, cell therapy, and applications of growth factors. However, data are too heterogeneous to achieve definitive conclusions on the use of these techniques for routine management of meniscal lesions. Although most preclinical and clinical studies are very promising, they are still at an experimental stage. More prospective randomised controlled trials are needed to compare the different techniques for clinical results, applicability, and cost-effectiveness

Chaos and Statistical Mechanics in the Hamiltonian Mean Field Model

We study the dynamical and statistical behavior of the Hamiltonian Mean Field (HMF) model in order to investigate the relation between microscopic chaos and phase transitions. HMF is a simple toy model of $N$ fully-coupled rotators which shows a second order phase transition. The canonical thermodynamical solution is briefly recalled and its predictions are tested numerically at finite $N$. The Vlasov stationary solution is shown to give the same consistency equation of the canonical solution and its predictions for rotator angle and momenta distribution functions agree very well with numerical simulations. A link is established between the behavior of the maximal Lyapunov exponent and that of thermodynamical fluctuations, expressed by kinetic energy fluctuations or specific heat. The extensivity of chaos in the $N \to \infty$ limit is tested through the scaling properties of Lyapunov spectra and of the Kolmogorov-Sinai entropy. Chaotic dynamics provides the mixing property in phase space necessary for obtaining equilibration; however, the relaxation time to equilibrium grows with $N$, at least near the critical point. Our results constitute an interesting bridge between Hamiltonian chaos in many degrees of freedom systems and equilibrium thermodynamics.Comment: 19 pages, 10 postscript figures included, Latex, Elsevier macros included. Invited talk at the conference ``Classical Chaos and its quantum manifestations'' in honour of Boris Chirikov, Sputnik conference of STATPHYS 20 - Toulouse, France - July 16-18, 1998. Revised version (added refs, changed part of the text and some figures) accepted for publication in Physica

Indium selenide: An insight into electronic band structure and surface excitations

We have investigated the electronic response of single crystals of indium selenide by means of angle-resolved photoemission spectroscopy, electron energy loss spectroscopy and density functional theory. The loss spectrum of indium selenide shows the direct free exciton at similar to 1.3 eV and several other peaks, which do not exhibit dispersion with the momentum. The joint analysis of the experimental band structure and the density of states indicates that spectral features in the loss function are strictly related to single-particle transitions. These excitations cannot be considered as fully coherent plasmons and they are damped even in the optical limit, i.e. for small momenta. The comparison of the calculated symmetry-projected density of states with electron energy loss spectra enables the assignment of the spectral features to transitions between specific electronic states. Furthermore, the effects of ambient gases on the band structure and on the loss function have been probed