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Stoica O.C.,Institute of Mathematics of the Romanian Academy
International Journal of Geometric Methods in Modern Physics | Year: 2014

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate. © World Scientific Publishing Company. Source

Song T.,Huazhong University of Science and Technology | Pan L.,Huazhong University of Science and Technology | Paun G.,Institute of Mathematics of the Romanian Academy
Theoretical Computer Science | Year: 2014

Spiking neural P systems (SN P systems, for short) are a class of membrane systems inspired from the way the neurons process information and communicate by means of spikes. In this paper, we introduce and investigate a new class of SN P systems, with spiking rules placed on synapses. The computational completeness is first proved, then two small universal SN P systems with rules on synapses for computing functions are constructed. Specifically, when using standard spiking rules, we obtain a universal system with 39 neurons, while when using extended spiking rules on synapses, a universal SN P system with 30 neurons is constructed. © 2014 Elsevier B.V. Source

Ionescu-Kruse D.,Institute of Mathematics of the Romanian Academy
Journal of Nonlinear Mathematical Physics | Year: 2012

We consider the two-dimensional irrotational water-wave problem with a free surface and a flat bottom. In the shallow-water regime and without smallness assumption on the wave amplitude we derive, by a variational approach in the Lagrangian formalism, the Green-Naghdi equations (1.1). The second equation in (1.1) is a transport equation, the free surface is advected by the fluid flow. We show that the first equation of the system (1.1) yields the critical points of an action functional in the space of paths with fixed endpoints, within the Lagrangian formalism. © 2012 2012 The Author(s). Source

Makhlouf A.,Upper Alsace University | Panaite F.,Institute of Mathematics of the Romanian Academy
Journal of Mathematical Physics | Year: 2014

The aim of this paper is to define and study Yetter-Drinfeld modules over Hombialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a Hombialgebra with bijective structure map provide solutions of the Hom-Yang-Baxter equation. The category H H U{cyrillic}D of Yetter-Drinfeld modules with bijective structure maps over a Hom-bialgebra H with bijective structure map can be organized, in two different ways, as a quasi-braided pre-tensor category. If H is quasitriangular (respectively, coquasitriangular) the first (respectively, second) quasi-braided pre-tensor category H H U{cyrillic}D contains, as a quasi-braided pre-tensor subcategory, the category of modules (respectively, comodules) with bijective structure maps over H. © 2014 AIP Publishing LLC. Source

Ion A.,Vienna University of Technology | Carreira J.,University of Coimbra | Sminchisescu C.,Lund University | Sminchisescu C.,Institute of Mathematics of the Romanian Academy
International Journal of Computer Vision | Year: 2014

We propose a layered statistical model for image segmentation and labeling obtained by combining independently extracted, possibly overlapping sets of figure-ground (FG) segmentations. The process of constructing consistent image segmentations, called tilings, is cast as optimization over sets of maximal cliques sampled from a graph connecting all non-overlapping figure-ground segment hypotheses. Potential functions over cliques combine unary, Gestalt-based figure qualities, and pairwise compatibilities among spatially neighboring segments, constrained by T-junctions and the boundary interface statistics of real scenes. Building on the segmentation layer, we further derive a joint image segmentation and labeling model (JSL) which, given a bag of FGs, constructs a joint probability distribution over both the compatible image interpretations (tilings) composed from those segments, and over their labeling into categories. The process of drawing samples from the joint distribution can be interpreted as first sampling tilings, followed by sampling labelings conditioned on the choice of a particular tiling. We learn the segmentation and labeling parameters jointly, based on maximum likelihood with a novel estimation procedure we refer to as incremental saddle-point approximation. The partition function over tilings and labelings is increasingly more accurately approximated by including incorrect configurations that are rated as probable by candidate models during learning. State of the art results are reported on the Berkeley, Stanford and Pascal VOC datasets, where an improvement of 28 % was achieved for the segmentation task only (tiling), and an accuracy of 47.8 % was obtained on the test set of VOC12 for semantic labeling (JSL). © 2013 Springer Science+Business Media New York. Source

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