On two theorems for flat,affine group schemes over a discrete valuation ring

We include short and elementary proofs of two theorems that characterize reductive group schemes over a discrete valuation ring, in a slightly more general context.     

The Role of Information in Managing Interactions from a Multifractal Perspective

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”.     

Self-concordant barriers for hyperbolic means

The geometric mean and the function (det(·)) ^1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p ^1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t↦p(x+td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and concave (on a suitable domain): we present a self-concordant barrier for its hypograph, with barrier parameter O(m ²). Our approach is direct, and shows, for example, that the function −mlog(det(·)−1) is an m ²-self-concordant barrier on a natural domain. Such barriers suggest novel interior point approaches to convex programs involving hyperbolic means. Received: December 2, 1999 / Accepted: February 2001?Published online September 3, 2001     

Stable multi-sets

In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets.  For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined.  The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks. Received: February 14, 2001/Revised version: September 7, 2001     

Existence and data dependence of fixed points for multivalued operators on gauge spaces

The purpose of this note is to present some fixed point and data dependence theorems in complete gauge spaces and in hyperconvex metric spaces for the so-called Meir-Keeler multivalued operators and admissible multivalued aα-contractions. Our results extend and generalize several theorems of Espínola and Kirk [R. Espínola, W.A. Kirk, Set-valued contractions and fixed points, Nonlinear Anal. 54 (2003) 485-494] and Rus, Petru?el, and Sînt?m?rian [I.A. Rus, A. Petru?el, A. Sînt?m?rian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003) 1947-1959].     

Hard X‐ray phase‐contrast tomography of non‐homogeneous specimens: grating interferometry versus propagation‐based imaging

X‐ray phase‐contrast imaging is an effective approach to drastically increase the contrast and sensitivity of microtomographic techniques. Numerous approaches to depict the real part of the complex‐valued refractive index of a specimen are nowadays available. A comparative study using experimental data from grating‐based interferometry and propagation‐based phase contrast combined with single‐distance phase retrieval applied to a non‐homogeneous sample is presented (acquired at beamline ID19‐ESRF). It is shown that grating‐based interferometry can handle density gradients in a superior manner. The study underlines the complementarity of the two techniques for practical applications.     

Simultaneous optical coherence tomography--Indocyanine Green dye fluorescence imaging system for investigations of the eye's fundus

We have developed a dual-channel optical coherence tomography-Indocyanine Green dye (OCT-ICG) fluorescence system based on a previously reported ophthalmic OCT confocal imaging system. The confocal channel is tuned to the fluorescence wavelength range of the ICG, and light from the same optical source is used to generate the OCT image and to excite the ICG fluorescence. The system enables the clinician to visualize simultaneously en face OCT slices and corresponding ICG angiograms of the ocular fundus, displayed side by side. C-scan (constant depth) and B-scan (cross section) images are collected by a fast en face scan (T scan). The pixel-to-pixel correspondence between the OCT and angiography images allows the user to capture OCT B scans precisely at selected points on the ICG confocal images.     

Approximating the maximum 2- and 3-edge-colorable subgraph problems

For a fixed value of a parameter k≥2, the Maximum k-Edge-Colorable Subgraph Problem consists in finding k edge-disjoint matchings in a simple graph, with the goal of maximising the total number of edges used. The problem is known to be -hard for all k, but there exist polynomial time approximation algorithms with approximation ratios tending to 1 as k tends to infinity. Herein we propose improved approximation algorithms for the cases of k=2 and k=3, having approximation ratios of 5/6 and 4/5, respectively.