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Isreie.Uav@Gmail.Com 22001144 IISSRREEIIEE Ministerul Educaţiei "Aurel Vlaicu" University of Arad, Romania 05~07 November 2014 PProceedings of the IInternational SSymposium RR e s e a r ch and EEducation in IInnovation EEra 5th Edition Mathematics & Computer Science Edited by Sorin Nădăban The International Symposium Research and Education in Innovation Era 5th Edition Arad, November 5th – 7th, 2014 IISSRREEIIEE 2014 Mathematics & Computer Science "Aurel Vlaicu" University of Arad, Romania 5th – 7th November 2014 Edited by Sorin Nădăban Co-editors: Adrian Palcu, Codruţa Stoica, Marius Tomescu The International Symposium Research and Education in Innovation Era Fourth Edition Mathematics & Computer Science Arad, 5th – 7th November 2014 Organized by "Aurel Vlaicu" University of Arad, Address: B-dul Revoluţiei nr. 77, 310130 Arad, Romania, P.O. BOX 2/158 AR Telephone:+40-257-283010 Fax:+40-257-280070 Email: [email protected] http://www.uav.ro/en/conferences/isreie-2014 Copyright © 2014 by Editura Universităţii “Aurel Vlaicu” Arad SCIENTIFIC COMMITTEE • Prof. Lizica Mihuţ, Ph.D., “Aurel Vlaicu” University of Arad • Prof. Michaela Dina Stănescu, Ph.D., ”Aurel Vlaicu”University of Arad • Prof. Adrian Nicolescu, Ph.D.,Politehnica University, Bucharest • Assoc. Prof. Alberto Folchini,Ph.D., Politecnico di Milano • Prof. Artur Cavaco-Paulo, Ph.D., Minho University • Prof. Bahrim Gabriela, Ph.D., "Dunarea de Jos"University, Galati • Prof. Burkhard Corves, Ph.D., RWTH Aachen • Doctor Craig Faulds, Food Research Institute, Norwich • Prof. Crisan Popescu, Ph.D., DWI at RWTH, Aachen • Prof. Gabriela Stanciulescu, Ph.D., Bucharest Academy of Economic Studies • Acad. Grozdanka Gojkov, Belgrade University • Prof. Hartwig Hocker, Ph.D.,RWTH Aachen • Prof. Iacob Hantiu, Ph.D., Romanian Council of Sport Science • Prof. Ioan I. Lador, Ph.D., Ministry of Education, Research, Youth and Sport, Bucharest • Prof. Ioan Oprean, Ph.D., University” Babes Bolyai”University, Cluj-Napoca • Prof. Jannick P. Rolland, Ph.D., University of Rochester, New York • Prof. Laura Muresan, Ph.D., Bucharest Academy of Economic Studies • Prof. Lieva van Langenhove, Ph.D., Ghent University • Prof. Luis Lezaun Martinez de Ubago, Ph.D., Zaragoza University • Prof. Miron Ionescu, Ph.D., Babes-Bolyai University, Cluj-Napoca • Prof. Nicanor Ursua Lezaun, Ph.D.del Pais Vasco/EHU, Bilbao University • Prof. Nicolae Mitrofan, Ph.D., Bucharest University. • Prof. Nicolae Pop, Ph.D. , Bucharest Academy of Economic Studies • Prof. Nicolae Ursu Fischer, Ph.D., Technical University of Cluj-Napoca • Prof. Nicolae Vaszilcsin, Ph.D., Politechnica University, Timisoara • Prof. Pierre de Hillerin, Ph.D., Romanian Institute of Sport Research • Prof. Paul Kickens,Ph.D, Ghent University • Prof. Paulo Brito,Ph.D., Polytechnic Institute, Portalegre • Prof. Renato Rizzo, Ph.D., Naples Federico II University • Prof. Ricardo Ferreira, Ph.D., Polytechnic Institute, Portalegre • Prof. Rodica Zafiu, Ph.D., Bucharest University • Prof. Vincent Nierstrasz, Ph.D., Ghent University • Prof. Viorel Lefter, Ph. D. , Bucharest Academy of Economic Studies • Prof. Werner Reichart, Ph.D., Zurich University ORGANIZING COMMITTEE • Prof. Ramona Lile, PhD • Prof. Florentina-Daniela Munteanu, Ph.D. • Prof. Alina Zamfir, Ph.D. • Prof. Anton Ilica, Ph.D. • Prof. Marian Spînu, Ph.D. • Prof. Dorin Herlo, Ph.D. • Prof. Florea Lucaci, Ph.D. • Prof. Ioan Tulcan, Ph.D. • Assoc. Prof. Păstorel Gaşpar, Ph.D. • Assoc. Prof. Cecilia Sîrghie, Ph.D. • Assoc. Prof. Virgil Ciutină, Ph.D. • Assoc. Prof. Dan Glăvan, Ph.D. • Assoc. Prof. Alina Roman, Ph.D. • Assoc. Prof. Gabriela Kelemen, Ph.D. • Assoc. Prof. Florin Isac, Ph.D. • Assoc. Prof. Ioan Galea, Ph.D. • Assoc. Prof. Sorin Nădăban, Ph.D. • Assoc. Prof. Lăcrămioara Ionescu Ph.D • Assist. Prof. Marius Tomescu Ph.D Reviewers committee Prof. Ph.D. Ioan Dzitac Assoc. Prof. Ph.D. Păstorel Gaşpar Assoc. Prof. Ph.D. Adrian Palcu Assoc. Prof. Ph.D. Codruţa Stoica Contents Codrut¸a Stoica, Diana Patras¸cu˘ Borlea Stability issues in infinite dimensional spaces ......................... 1 Octavian Cira, Florentin Smarandache Luhn prime numbers ................................................. 7 Mihai-Gabriel Babut¸ia, Traian Ceaus¸u Connections between some well-known concepts of uniform exponential dichotomy for discrete-time systems in Banach spaces . 15 Dominic Bucerzan, Crina Ratiu Secret communication using cryptography and steganography . 22 Violeta Chis Fuzzy logic applications in power systems ............................. 29 Octavian Cira Inverse narcissistic numbers ......................................... 36 Carmen T¸erei, Marius Tomescu A study on classification algorithms for imbalanced datasets . 42 Carmen T¸erei, Marius Tomescu Techniques of evaluating classification algorithms for imbalanced data . 48 Mariana Nagy, Negrus¸a Adina, Burca Valentin Clusters describing IFRS adoption stage .............................. 54 Pastorel˘ Gas¸par, Lorena Popa Series representations for random distribution fields . 63 Mot¸Ghiocel, Popa Lorena On quaternions ...................................................... 70 Nicolae Marian Seimeanu, Mihail Megan Three concepts of uniform polynomial dichotomy for discrete-time linear systems in Banach spaces ............................................ 77 Lavinia Sida, Pastorel˘ Gas¸par The T -convolution product ........................................... 85 S¸andru Andrea, Palcu Adrian Applications of graph theory in chemistry ............................ 91 i Stability issues in infinite dimensional spaces Codrut¸a Stoica∗ Diana P˘atra¸scuBorleay Abstract The aim of this paper is to define various concepts of stability for co- cycles over non-autonomous dynamical systems, as generalizations of the skew-evolution semiflows, such as exponential stability, (α; β){exponential stability, and two more general concepts, the (h; k){stability and the (h; k){integral stability. Connections between these notions are also given. Mathematics Subject Classification: 34D05, 93D20 Keywords: Dynamical systems, skew-evolution cocycles, exponential stability, (α, β){exponential stability, (h; k){stability, (h; k){integral stability 1 Preliminaries In recent years, several concepts of the control theory, as stability, stabili- zability, controllability or observability were refined, based on the fact that the dynamical systems which describe processes from engineering, physics, biology or economics are extremely complex and the identification of the proper mathematical models is difficult. In the qualitative theory of evolution equations, the exponential sta- bility and instability are two of the most important asymptotic properties, approached lately from various perspectives. Recently, other asymptotic behaviors were studied, the dichotomy and the trichotomy, by generalizing the techniques used in the investigation of the stability. The stability theory puts into discussion the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Many parts of the qualitative theory of differential equations and dynamical systems deal with asymptotic proper- ties of solutions/ trajectories { what happens with the system after a long period of time. If an orbit is well understood, it is natural to ask whether a small change in the initial condition will lead to similar behavior. The study of the behaviors of the evolution equations by means of associated operator families has allowed to obtain answers to open prob- lems by involving techniques of functional analysis and operator theory. ∗[email protected] Department of Mathematics and Computer Science, "Aurel Vlaicu" University of Arad, Romania [email protected] Department of Mathematics, West University of Timi¸soara,Roma- nia 1 Stability issues in infinite dimensional spaces 2 Remarkable results of the stability theory are due to J.L. Daleckii and M.G. Krein (see [1]) and J.L. Massera and J.J. Sch¨affer(see [2]). The concept of evolution operators arises naturally from the theory of well-posed non-autonomous Cauchy problems, while the notion of lin- ear skew-product semiflows is involved when considering the linearization along an invariant manifold of a dynamical system generated by a nonli- near differential equation. The skew-evolution semiflows, defined in [3], are generalizations for both evolution operators and skew-product semi- flows. The applicability of this notion was emphasized by P. Viet Hai in [6] and [7], and by T. Yue, X.Q. Song, D.Q. Li in [8]. We consider in this paper the case of cocycles over multivalued dy- namical systems (see [5]). We define various concepts of stability, charac- terizations for these notions, as well as connections between them. 2 Definitions. Examples X = (X; dX ) denotes a complete metric space, P(X) the set of all non- empty subsets of X, V a Banach space with the dual V ∗, B(V ) the space of all bounded linear operators from V into itself. idX is the identity map on X, I the identity operator on V and Y = X × V . We consider αt the sets E = ff : R+ ! [1; 1) 9α 2 R+ such that f(t) = e g and 2 T = (t; s) 2 R+; t ≥ s . Definition 2.1 A multivalued map ue : T × X !P(X) with (s1) ue(t; t; ·) = idX ; 8(t; x) 2 R+ × X; (s2) ue(t; t0; x) ⊆ ue(t; s; ue(s; t0; x)); 8(t; s); (s; t0) 2 T; x 2 X is a gene- ralized multivalued non{autonomous dynamical system on X. Remark 2.2 In what follows we will consider the case of a mapping u : T × X ! X with the properties 0 (s1) u(t; t; x) = x; 8(t; x) 2 R+ × X; 0 (s2) u(t; t0; x) = u(t; s; u(s;
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