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G. A. T. F. da Costa

Publications and source records attributed to G. A. T. F. da Costa.

9 recordsLinked to original sources
arXiv2016

Graphs and Generalized Witt identities

This paper is about the determinantal identities associated with the Ihara (Ih) zeta function of a non directed graph and the Bowen-Lanford (BL) zeta function of a directed graph. They will be called the Ih and the BL identities in this paper. We show that the Witt identity (WI) is a special case of the BL identity and inspired by the links the WI has with Lie algebras and combinatorics we investigate similar aspects of the Ih and BL identities. We show that they satisfy generalizations of the Strehl identity and Carlitz, Metropolis-Rota relations and each one of them can be interpreted as the denominator (or generalized Witt) identity of a free Lie superalgebra. Also, they can be associated to a coloring problem. New interpretations of the Ih and BL zeta functions are presented.

math.CO↗
arXiv2016

Feynman identity for planar graphs

The Feynman identity (FI) of a planar graph relates the Euler polynomial of the graph to an infinite product over the equivalence classes of closed nonperiodic signed cycles in the graph. The main objectives of this paper are to compute the number of equivalence classes of nonperiodic cycles of given length and sign in a planar graph and to interpret the data encoded by the FI in the context of free Lie superalgebras. This solves in the case of planar graphs a problem first raised by S. Sherman and sets the FI as the denominator identity of a free Lie superalgebra generated from a graph. Other results are obtained. For instance, in connection with zeta functions of graphs.

math-ph↗
arXiv2013

A Witt type formula

This paper investigates some combinatorial and algebraic properties of a Witt type formula for graphs.

math.CO↗
arXiv2013

An analogue to the Witt identity

In this paper we solve combinatorial and algebraic problems associated with a multivariate identity first considered by S. Sherman wich he called an analog to the Witt identity. We extend previous results obtained for the univariante case.

math.CO↗
arXiv2011

Generalized Symmetric Divergence Measures and Metric Spaces

Recently, Taneja studied two one parameter generalizations of J-divergence, Jensen-Shannon divergence and Arithmetic-Geometric divergence. These two generalizations in particular contain measures like: Hellinger discrimination, symmetric chi-square divergence, and triangular discrimination. These measures are well known in the literature of Statistics and Information theory. In this paper our aim is to prove metric space properties for square root of these two symmetric generalized divergence measures.

cs.IT↗
arXiv2003

Feynman Identity: a special case. II

In this paper, the results of part I regarding a special case of Feynman identity are extended. The sign rule for a path in terms of data encoded by its word and formulas for the numbers of distinct equivalence classes of nonperiodic paths of given length with positive or negative sign are obtained for this case. Also, a connection is found between these numbers and the generalized Witt formula for the dimension of certain graded Lie algebras. Convergence of the infinite product in the identity is proved.

math-ph↗
arXiv2003

Combinatorial formulation of Ising model revisited

In 1952, Kac and Ward developed a combinatorial formulation for the two dimensional Ising model which is another method of obtaining Onsager's formula for the free energy per site in the thermodynamic limit of the model. Feynman gave an important contribution to this formulation conjecturing a crucial mathematical relation which completed Kac and Ward ideas. In this paper, the method of Kac, Ward and Feynman for the free field Ising model in two dimensions is reviewed in a selfcontained way.

math-ph↗