TECHNICAL UNIVERSITY OF CIVIL ENGINEERING BUCHAREST THE 13TH WORKSHOP OF SCIENTIFIC COMMUNICATIONS, DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Bucharest, May 23, 2015 EDITORS: Pavel MATEI - Head of Department of Mathematics and Computer Science Ghiocel GROZA Daniel TUDOR Mariana ZAMFIR Organizing Committee: Romică TRANDAFIR Ion MIERLUŞ-MAZILU Gabriela-Roxana DOBRE CONTENT Sever Achimescu IHARA ZETA FUNCTIONS OF GRAPHS ASSOCIATED 1 Stelian Corneliu Andronescu TO NON-CONGRUENT NUMBERS Ileana Bucur CONTRACTIVE SEQUENCES IN METRIC SPACES 4 ON SOME EXAMPLES OF SPECTRAL GROUPS OF Dan Caragheorgheopol 9 AUTOMORPHISMS IN QUANTUM LOGICS CONSUMER MODELS AND THE COMMON INFLUENCE Daniel Ciuiu 15 OF INCREASING VAT AND DECREASING WEDGES SOLVING THE BILEVEL LINEAR PROGRAMMING Ştefania Constantinescu 20 PROBLEM BY THE MONTE CARLO METHOD SOME RESULTS IN THE EQUIVARIANT ALGEBRAIC Cristian Costinescu 26 TOPOLOGY Nicolae Dăneț CLOSURE SUBLINEAR OPERATORS 32 FROM VECTOR SPACES TO INTERVAL-SPACES via Rodica-Mihaela Dăneț THE TECHNIQUE OF THE AUXILIARY SUBLINEAR 38 FUNCTIONAL Gabriela-Roxana Dobre TABU SEARCH OPTIMIZATION WITH APPLICATIONS 44 IN GROUNDWATER MANAGEMENT Ștefania Donescu ON THE CIRCULAR EQUILIBRIUM FOR A CLASS OF 51 Ligia Munteanu NONLINEAR DYNAMICAL SYSTEMS Marinică Gavrilă EXTREMAL POINTS IN LINEAR SPACES 57 Corina Grosu MULTI INDEXED LAGUERRE POLYNOMIALS AND CHI 61 Marta Grosu SQUARE CORRELATION Ghiocel Groza ON BOUNDED ENTIRE REAL FUNCTIONS 67 Lucian Niță Ghiocel Groza ON A CLASS OF SYMMETRIC (0,1)-MATRICES 73 Alina Elisabeta Sandu PREDICTING THE EARTHQUAKE MAGNITUDE FROM Iuliana Iatan SEISMICITY INDICATORS USING A PROBABILISTIC 79 NEURAL NETWORK Ruxandra Diana Ilie A COMPUTATIONAL METHOD FOR DESCRIBING THE 84 Veturia Chiroiu BEHAVIOR OF THE NATURAL (MAIZE) COMPOSITES PARTIAL DIFFERENTIAL EQUATIONS NUMERICALLY Marilena Jianu 90 SOLVED USING TAYLOR EXPANSION Anca Nicoleta Marcoci APPLICATIONS OF SAWYER’S DUALITY PRINCIPLE 96 Liviu Gabriel Marcoci BOUNDEDNESS ON SOME WEIGHTED SPACES 102 George Daniel Mateescu DYNAMIC WEB PAGES 106 Pavel Matei A VARIATIONAL METHOD FOR THE p -LAPLACIAN 110 Maria Mihailova IGUAL PROJECT AS A SPRINGBOARD TO MODERN 116 Ion Mierluș-Mazilu EDUCATION IN LATIN AMERICA AFFINE SURFACES IN R4 AND R5. Adela Mihai LAPLACE OPERATOR AND CORRESPONDING 122 Daniel Tudor BELTRAMI FORMULAE AN INTEGRAL FOR VECTOR FUNCTIONS WITH Lucian Niță 127 RESPECT TO VECTOR MEASURES A DENSITY THEOREM IN THE SET OF CONTINUOUS Gavriil Păltineanu 130 FUNCTIONS WITH VALUES IN THE UNIT INTERVAL Viorel Petrehuș SOME CONSERVATIVE NUMERICAL 134 APPROXIMATIONS OF DYNAMICAL SYSTEMS Emil Popescu SYMMETRIES IN BUCKINGHAM-TYPE PROBLEMS 138 Iuliana Popescu BIFURCATION ANALYSIS OF NATURAL CONVECTION Simona Cristina Nartea 143 FLOWS - MATHEMATICAL MODELLING Adrian Gabriel Ghiauș Marian-Valentin Popescu A COLLECTIVELY COINCIDENCE RESULT IN Rodica-Mihaela Dăneț 149 ALGEBRAIC TOPOLOGY AND ITS APPLICATION Nicoleta Popescu Sever Angel Popescu SHIFT OPERATORS ON SEQUENCES 155 AN APPLICATION IN SPSS FOR CALCULATE THE Alina Elisabeta Sandu 161 PEARSON CORRELATION COEFFICIENT A NORMAL SCORE PARAMETERIZATION FOR Bogdan Sebacher CHANNELIZED RESERVOIRS ESTIMATION USING THE 167 ITERATIVE ADAPTIVE GAUSSIAN MIXTURE FILTER ABOUT METHODS OF SOLVING MULTI-OBJECTIVE Narcisa Teodorescu 173 FRACTIONAL PROGRAMMING PROBLEMS Romică Trandafir TRANSMUTED GENERALIZED PARETO 177 Vasile Preda DISTRIBUTION HOW TO TEACH ELEMENTARY GEOMETRIC Mariana Zamfir 183 PROPERTIES OF PLANE CURVES USING MATHCAD Mariana Zamfir ON RESTRICTIONS AND QUOTIENTS OF 189 Tania-Luminița Costache DECOMPOSABLE OPERATORS IN BANACH SPACES The 13th Workshop of Scientific Communications, Department of Mathematics and Computer Science, May 23, 2015, p. 1-3 IHARA ZETA FUNCTIONS OF GRAPHS ASSOCIATED TO NON-CONGRUENT NUMBERS Sever ACHIMESCU Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest Bd. Lacul Tei 124, sector 2, 38RO-020396 Bucharest, Romania E-mail: [email protected] Stelian Corneliu ANDRONESCU Department of Mathematics and Computer Science Technical University of Pitesti Str. Targu din Vale Nr.1 , Pitesti, Romania E-mail: [email protected] Abstract: For a number of the form n=p1p2…pt, t 3 we associate, following [1], a graph G(n). We compute the Ihara zeta function of G(n). Mathematics Subject Classification (2010): 68R10, 14G10, 11M99 Key words: congruent numbers, Ihara zeta function, finite digraphs. 1. Introduction Zeta Functions and L-function of an elliptic curve. We recall basic facts from [2]. For an elliptic curve E over Q we denote Δ its discriminant. For each prime p we denote Ep the reduction of E modulo p, defined over Fp. Let # Ep(Fp) be the number of projective solutions. Put app 1 # Ep(Fp) for all primes p. The local L-factor corresponding to the prime p is defined to be the formal power series 2 L p (u) 1 (1 a pu pu ) if p does not divide Δ 2 L p (u) 1 (1 a p u ) if p divides Δ The L-function of E is the product s L(s, E) L p ( p ) p The local zeta function of E is # Ep(F )un pn Z(u, E p ) exp( ) n1 n It is known that E(Q) is an abelian group The weak version of the Birch Swinnerton-Dyer conjecture is: Rank(E(Q))=ord s1 L(s,E(Q)) 1 Congruent Numbers. A positive integer n is said to be a congruent number if it is the area of a right triangle with sides positive integers. For example n=6 is a congruent number since it is the area of the right triangle with sides (3,4,5). Finding all the congruent numbers is perhaps one of the oldest problems in mathematics. Assuming the weak Birch Swinnerton-Dyer to be true, J. Tunnell gave in 1983 an algorithmic criterion to decide whether an arbitrarily fixed positive integer n is congruent or not. We send the reader to [4] for more details. There are other criteria for the congruence problem but they give an answer for particular positive integers n; they are not as general as Tunnell’s criterion. For example in [1] Feng proved the non-congruence for infinitely many positive numbers n using graphs. In [1] Feng proved also that the Birch Swinnerton-Dyer conjecture is true for infinitely many elliptic curves, given by the equation 2 En : y x(x n)(x n) with n noncongruent numbers of a particular form: n=p1p2…pt, t 1, p1 3(mod 8), pi 1(mod 8), i 2 or n=p1p2…pt, t 1, p1 5(mod 8), pi 1(mod 8), i 2 The following basic but very important result (proved in [4]) has been used in [1]: n is a noncongruent number iff rank( En (Q))=0 We find interesting the fact that by studying one graph one can conclude the non-congruence of infinitely many n’s. Ihara Zeta Functions of Finite Oriented (Directed) Connected Graphs. We follow [4],[5]. An oriented (directed) graph is denoted X=(V,E) with E a subset of the Cartesian product V V. Let e=(a,b) be an edge; we denote o(e)=a (the origin of e) and t(e)=b (the terminus of e) and e =(b,a). A prime cycle in X is an equivalent class of a sequence of edges such that each vertex of one of these edges is exactly once target for one edge and exactly once terminus for one edge. The length of a prime cycle p is defined to be the number of edges defining the cycle p and it is denoted by |p|. The Ihara Zeta Function of X is Z(u)= (1 u | p| ) 1 p where the product is taken on all prime cycles p. 2. Ihara Zeta Functions of G(n) For a number of the form n=p1p2…pt, t 3 we associate, following [1], a graph G(n). We compute the Ihara zeta function of G(n). We define a directed graph G(n) as follows: the vertices of G(n) are all prime factors of n and p j for pi p j there exists an edge ( pi , p j ) in G(n) iff the Legendre symbol ( ) 1 . We put pi m ( ) 1 for all odd m. 2 2 Examples. 1) n 105 357 We have: 3 5 3 7 7 5 ( ) 1 ( ) , ( ) =-1, ( ) =1, ( ) =-1= ( ) . 5 3 7 3 5 7 The edges are (3,5),(5,3),(3,7),(5,7),(7,5). The prime cycles are: [(3,5),(5,3)],[(5,7),(7,5)], [(3,7),(7,5),(5,3)]. The Ihara Zeta Function is Z(u)=1/((1-u2)(1-u2)(1-u3)) 2) n 1155 357 11 We have: 3 11 5 11 7 11 ( ) =1, ( ) =-1, ( ) =1= ( ) , ( ) -=1, ( ) =1. 11 3 11 5 11 7 The edges are (3,5),(5,3),(3,7),(5,7),(7,5), (11,3)(7,11). The prime cycles are: [(3,5),(5,3)],[(5,7),(7,5)], [(3,7),(7,5),(5,3)],[(7,11),(11,3),(3,7)] [(3,5),(5,7),(7,11),(11,3)]. The Ihara Zeta Function is Z(u)=1/((1-u2)(1-u2)(1-u3) (1-u3) (1-u4)) References [1] Feng.K.: Non-congruent numbers, odd graphs and the Birch-Swinnerton Dyer conjecture, Acta Arithmetica 75(1), 1996. [2] Knapp, A.W.:Elliptic Curves, Mathematical Notes 40, Princeton University Text, 1992. [3] Kotani,M. and Sunada,T.: Zeta funcions of Finite Graphs, J. Math. Sci. Univ. Tokyo 7, 7- 25, 2000. [4] Koblitz, N.: Introduction to Elliptic Curves and Modular Forms, GTM 97, Springer,1984. [5] Terras, A.:Zeta Functions of Graphs-A Stroll through the Garden, Cambridge Studies in Advanced Mathematics, 128 , 2010. 3 The 13th Workshop of Scientific Communications, Department of Mathematics and Computer Science, May 23, 2015, p. 4-8 CONTRACTIVE SEQUENCES IN METRIC SPACES Ileana BUCUR Technical University of Civil Engineering Bucharest, Bucharest, Romania E-mail: [email protected] Abstract: In this paper we define the notions of contractive and semi-contractive sequences in metric spaces and we show that every semi-contractive sequence is a Cauchy sequence.