Programa de Pós-graduação em Ciência da Computação em Rede
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Navegando Programa de Pós-graduação em Ciência da Computação em Rede por Por Orientador "Castonguay, Diane"
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Item Definitividade de formas quadráticas – uma abordagem polinomial(Universidade Federal de Goiás, 2016-11-18) Alves, Jesmmer da Silveira; Brustle, Thomas; http://www2.ubishops.ca/algebra/brustleCv.pdf; Castonguay, Diane; http://lattes.cnpq.br/4005898623592261; Castonguay, Diane; http://lattes.cnpq.br/4005898623592261; Centeno, Carmen; Alvares, Edson Ribeiro; Martinez, Fabio Henrique Viduani; Longo, Humberto JoséQuadratic forms are algebraic expressions that have important role in different areas of computer science, mathematics, physics, statistics and others. We deal with rational quadratic forms and integral quadratic forms, with rational and integer coefficients respectively. Existing methods for recognition of rational quadratic forms have exponential time complexity or use approximation that weaken the result reliability. We develop a polinomial algorithm that improves the best-case of rational quadratic forms recognition in constant time. In addition, new strategies were used to guarantee the results reliability, by representing rational numbers as a fraction of integers, and to identify linear combinations that are linearly independent, using Gauss reduction. About the recognition of integral quadratic forms, we identified that the existing algorithms have exponential time complexity for weakly nonnegative type and are polynomial for weakly positive type, however the degree of the polynomial depends on the algebra dimension and can be very large. We have introduced a polynomial algorithm for the recognition of weakly nonnegative quadratic forms. The related algorithm identify hypercritical restrictions testing every subgraph of 9 vertices of the quadratic form associated graph. By adding Depth First Search approach, a similar strategy was used in the recognition of weakly positive type. We have also shown that the recognition of integral quadratic forms can be done by mutations in the related exchange matrix.Item Reconhecimento polinomial de álgebras cluster de tipo finito(Universidade Federal de Goiás, 2015-09-09) Dias, Elisângela SIlva; Castonguay, Diane; http://lattes.cnpq.br/4005898623592261; Castonguay, Diane; Schiffler, Ralf; Dourado, Mitre Costa; Carvalho, Marcelo Henrique de; Longo, Humberto JoséCluster algebras form a class of commutative algebra, introduced at the beginning of the millennium by Fomin and Zelevinsky. They are defined constructively from a set of generating variables (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. Since its inception, the theory of cluster algebras found applications in many areas of science, specially in mathematics. In this thesis, we study, with computational focus, the recognition of cluster algebras of finite type. In 2006, Barot, Geiss and Zelevinsky showed that a cluster algebra is of finite type whether the associated graph is cyclically oriented, i.e., all chordless cycles of the graph are cyclically oriented, and whether the skew-symmetrizable matrix associated has a positive quasi-Cartan companion. At first, we studied the two topics independently. Related to the first part of the criteria, we developed an algorithm that lists all chordless cycles (polynomial on the length of those cycles) and another that checks whether a graph is cyclically oriented and, if so, list all their chordless cycles (polynomial on the number of vertices). Related to the second part of the criteria, we developed some theoretical results and we also developed a polynomial algorithm that checks whether a quasi-Cartan companion matrix is positive. The latter algorithm is used to prove that the problem of deciding whether a skew-symmetrizable matrix has a positive quasi-Cartan companion for general graphs is in NP class. We conjecture that this problem is in NP-complete class.We show that the same problem belongs to the class of polynomial problems for cyclically oriented graphs and, finally, we show that deciding whether a cluster algebra is of finite type also belongs to this class.