Institute of Mathematics of the Romanian Academy

Bucharest, Romania

Institute of Mathematics of the Romanian Academy

Bucharest, Romania
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Badea L.,Institute of Mathematics of the Romanian Academy
Lecture Notes in Computational Science and Engineering | Year: 2017

We introduce some multilevel and multigrid methods and derive their global convergence rate for variational inequalities and for variational inequalities containing a term introduced by a nonlinear operator. Also, we estimate the convergence rate of the one- and two-level methods for variational inequalities of the second kind and for quasi-variational inequalities. The methods are introduced as subspace correction algorithms in a reflexive Banach space, where general convergence results are derived. These algorithms become multilevel and multigrid methods by introducing the finite element spaces. In this case, the error estimates are written in function of the number of subdomains and the overlapping parameter for the one- and two-level methods, and in function of the number of levels for the multigrid methods. The obtained convergence rates for the multigrid methods are compared with those existing in the literature for the complementarity problems. © Springer International Publishing AG 2017.


Song T.,Huazhong University of Science and Technology | Pan L.,Huazhong University of Science and Technology | Paun G.,University of Seville | Paun G.,Institute of Mathematics of the Romanian Academy
Information Sciences | Year: 2013

Spiking neural P systems (SN P systems, for short) are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes. Asynchronous SN P systems are non-synchronized systems, where the use of spiking rules (even if they are enabled by the contents of neurons) is not obligatory. It remains open whether asynchronous SN P systems with standard spiking rules are equivalent with Turing machines. In this paper, with a biological inspiration (in order to achieve some specific biological functioning, neurons from the same functioning motif or community work synchronously to cooperate with each other), we introduce the notion of local synchronization into asynchronous SN P systems. The computation power of asynchronous SN P systems with local synchronization is investigated. Such systems consisting of general neurons (respectively, unbounded neurons) and using standard spiking rules are proved to be universal. Asynchronous SN P systems with local synchronization consisting of bounded neurons and using standard spiking rules characterize the semilinear sets of natural numbers. These results show that the local synchronization is useful, it provides some "programming capacity" useful for achieving a desired computation power. © 2012 Elsevier Inc. All rights reserved.


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.


Aberkane S.,University of Lorraine | Aberkane S.,Nancy Research Center for Automatic Control | Dragan V.,Institute of Mathematics of the Romanian Academy
Automatica | Year: 2012

In this note, a bounded real lemma (BRL) is established for discrete-time periodic Markov jump linear systems (MJLS) with nonhomogeneous finite state Markov chain. The BRL is formulated as a finite linear matrix inequality (LMI) feasibility problem. An application to H∞ filtering for systems subject to limited communication capacity is then presented. © 2012 Elsevier Ltd. All rights reserved.


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.


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.


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).


Mihailescu E.,Institute of Mathematics of the Romanian Academy
Journal of Statistical Physics | Year: 2010

We study new invariant probability measures, describing the distribution of multivalued inverse iterates (i. e. of different local inverse iterates) for a non-invertible smooth function f which is hyperbolic, but not necessarily expanding on a repellor Λ. The methods for the higher dimensional non-expanding and non-invertible case are different than the ones for diffeomorphisms, due to the lack of a nice unstable foliation (local unstable manifolds depend on prehistories and may intersect each other, both in Λ and outside Λ), and the fact that Markov partitions may not exist on Λ. We obtain that for Lebesgue almost all points z in a neighbourhood V of Λ, the normalized averages of Dirac measures on the consecutive preimage sets of z converge weakly to an equilibrium measure μ- on Λ; this implies that μ- is a physical measure for the local inverse iterates of f. It turns out that μ- is an inverse SRB measure in the sense that it is the only invariant measure satisfying a Pesin type formula for the negative Lyapunov exponents. Also we show that μ- has absolutely continuous conditional measures on local stable manifolds, by using the above convergence of measures. We prove then that f:(Λ,ℬ(Λ),μ-)→(Λ,ℬ(Λ),μ-) cannot be one-sided Bernoulli, although it is an exact endomorphism of Lebesgue spaces. Several classes of examples of hyperbolic non-invertible and non-expanding repellors, with their inverse SRB measures, are given in the end. © Springer Science+Business Media, LLC 2010.


Stoica O.C.,Institute of Mathematics of the Romanian Academy
European Physical Journal Plus | Year: 2012

It is shown that the Schwarzschild spacetime can be extended so that the metric becomes analytic at the singularity. The singularity continues to exist, but it is made degenerate and smooth, and the infinities are removed by an appropriate choice of coordinates. A family of analytic extensions is found, and one of these extensions is semi-regular. A degenerate singularity does not destroy the topology, and when it is semi-regular, it allows the field equations to be rewritten in a form which avoids the infinities, as it was shown elsewhere. In the new coordinates, the Schwarzschild solution extends beyond the singularity. This suggests a possibility that the information is not destroyed in the singularity, and can be restored after the evaporation. © 2012 Società Italiana di Fisica and Springer.


Ionescu-Kruse D.,Institute of Mathematics of the Romanian Academy
Nonlinear Analysis: Real World Applications | Year: 2015

We propose an exact implicit solution to the nonlinear geophysical edge-wave problem in the f-plane approximation. Adequate for this exact solution is the Lagrangian framework. We confirm that this solution is an example of a trapped wave. © 2015 Elsevier Ltd. All rights reserved.

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