DEPRO - Departamento de Engenharia de Produção
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Item Matrix computations with the Omega calculus.(2021) Francisco Neto, AntônioIn this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [Matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264–280]. We obtain Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular. Our representation recovers the main results of Chen and Louck [The combinatorial power of the companion matrix. Linear Algebra Appl. 1996;232:261–278] on the powers of the companion matrix. Furthermore, we generalize previous work on the powers of tridiagonal matrices due to Gutiérrez–Gutiérrez in [Powers of tridiagonal matrices with constant diagonals. Appl Math Comput. 2008;206(2):885–891], Öteleş and Akbulak [Positive integer powers of certain complex tridiagonal matrices. Appl Math Comput. 2013;219(21):10448–10455], and triangular matrices following Shur [A simple closed form for triangular matrix powers. Electron J Linear Algebra. 2011;22:1000–1003].Item Extending putzer’s representation to all analytic matrix functions via omega matrix calculus.(2021) Francisco Neto, AntônioWe show that Putzer’s method to calculate the matrix exponential in [28] can be generalized to compute an arbitrary matrix function defined by a convergent power series. The main technical tool for adapting Putzer’s formulation to the general setting is the omega matrix calculus; that is, an extension of MacMahon’s partition analysis to the realm of matrix calculus and the method in [8]. Several results in the literature are shown to be special cases of our general formalism, including the computation of the fractional matrix exponentials introduced by Rodrigo [30]. Our formulation is a much more general, direct, and conceptually simple method for computing analytic matrix functions. In our approach the recursive system of equations the base for Putzer’s method is explicitly solved, and all we need to determine is the analytic matrix functions.Item Matrix analysis and omega calculus.(2020) Francisco Neto, AntônioIn this work we introduce a new operator based approach to matrix analysis. Our main technical tool comprises an extension of a tool introduced long ago by MacMahon to analyze the partitions of natural numbers: the Omega operator calculus. More precisely, we construct an operator acting linearly on absolutely convergent matrix valued expansions which selects appropriate terms of those expansions. In the context of our framework a new representation of matrix valued functions is available. Our representation is simple, requiring only the computation of matrix inverses and basic manipulations of the Taylor series of scalar functions. To show the usefulness of our approach we obtain fundamental results related to the basic theory of ODEs, perturbative calculations, multiple integrals involving the matrix exponential, the Sylvester equation, the multivariate Fa`a di Bruno formula, Hermite polynomials, queuing theory, and graph theory.Item An approach to isotropic tensor functions and their derivatives via omega matrix calculus.(2020) Francisco Neto, AntônioIn this work we show how to obtain a closed form expression of any isotropic tensor function F (A) and their associated derivatives with A a second order tensor in a finite dimensional space. Our approach is based on a recent work of the author (SIAM Rev. 62(1):264–280, 2020) extending the Omega operator calculus, originally devised by MacMahon to describe partitions of natural numbers, to the realm of matrix analysis, namely, the Omega Matrix Calculus (OMC). The OMC is conceptually simple and useful in practice. Indeed, we show that the Cayley-Hamilton theorem and an improvement for low-rank second order tensors due to Segercrantz (Am. Math. Mon. 99(1):42–44, 1992), the representation of isotropic tensor functions and their first derivative of Itskov (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 459(2034):1449–1457, 2003), and Theorem 1.1 of Norris (Q. Appl. Math. 66(4):725–741, 2008) are all special cases of a general Omega expression introduced in this work.Item An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players.(2019) Francisco Neto, Antônio; Fonseca, Carolina RodriguesWe introduce a new generating function based method to compute the Banzhaf, Deegan– Packel, Public Good (a.k.a. the Holler power index) and Shapley–Shubik power indices in the presence of incompatibility among players. More precisely, given a graph G = (V, E) with V the set of players and E the edge set, our extension involves multiple weighted voting games (MWVG’s) and incompatible players, i.e., pairs of players belonging to E are not allowed to cooperate. The route to obtain the aforementioned generating functions comprises the use of a key lemma characterizing the set of minimal winning coalitions of the game with incompatibility due to Alonso-Meijide et al. (Appl Math Comput 252(1):377– 387, 2015), a tool from combinatorial analysis, namely, the Omega calculus in partition analysis, and basic tools borrowed from commutative algebra involving the computation of certain quotients of polynomial rings module polynomial ideals. Using partition analysis, we obtain new generating functions to compute the Deegan–Packel and Public Good power indices with incompatibility leading to lower time complexity than previous results of Chessa (TOP 22(2):658–673, 2014) and some results of Alonso-Meijide et al. (Appl Math Comput 219(8):3395–3402, 2012). Using a conjunction of partition analysis and commutative algebra, we extend to MWVG’s the generating function approach to compute the Banzhaf and Shapley–Shubik power indices in the presence of incompatibility. Finally, an example taken from the real-world, i.e., the European Union under the Lisbon Treaty, is used to illustrate the usefulness of the Omega package, a symbolic computational package that implements the Omega calculus in Mathematica, due to Andrews et al. (Eur J Comb 22(7):887–904, 2001) in the context of MWVG’s by computing the PG power index of the associated voting game.Item Generating functions of weighted voting games, MacMahon’s Partition Analysis, and Clifford algebras.(2019) Francisco Neto, AntônioMacMahon’s Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighted voting games. We introduce a new generalized generating function that gives, as special cases, extensions of the generating functions of the Banzhaf, Shapley-Shubik, Banzhaf-Owen, symmetric coalitional Banzhaf, and Owen power indices. Our extensions involve any coalition formation related to a linear diophantine system and multiple voting games. In addition, we show that a combination of ideas from MPA and Clifford algebras is useful in constructing generating functions for coalition configuration power indices. Finally, a brief account on how to design voting systems via MPA is advanced. More precisely, we obtain new generating functions that give, for fixed coalitions, all the distribution of weights of the players of the voting game such that a given player swings or not.Item The dual of Spivey’s bell number identity from Zeon algebra.(2017) Francisco Neto, AntônioIn this paper, we give a new short proof of the dual of Spivey’s Bell number identity due to Mezo. Our approach follows from basic manipulations involving a fundamental identity representing factorials in the Zeon algebra. This work, along with a previous one due to the author and dos Anjos, shows that Spivey’s and Mez˝o’s identities have at their root a common underlying algebraic origin.Item A note on a theorem of Schumacher.(2016) Francisco Neto, AntônioIn this paper, we give a short new proof of a recent result due to Schumacher con- cerning an extension of Faulhaber’s identity for the Bernoulli numbers. Our approach follows from basic manipulations involving the ordinary generating function for the Bernoulli polynomials in the context of the Zeon algebra.Item A note on a theorem of Guo, Mezo, and Qi.(2016) Francisco Neto, AntônioIn a recent paper, Guo, Mez˝o, and Qi proved an identity representing the Bernoulli polynomials at non-negative integer points m in terms of the m-Stirling numbers of the second kind. In this note, using a new representation of the Bernoulli polynomials in the context of the Zeon algebra, we give an alternative proof of the aforementioned identity.Item A bijection between rooted trees and fermionic Fock states : grafting and growth operators in Fock space and fermionic operators for rooted trees.(2013) Francisco Neto, AntônioWe showthat fermionic Fock states in the occupation number representation can be indexed uniquely by rooted trees. Our main ingredients in this construction are the Matula numbers, the fundamental theorem of arithmetic, and a relabeling of fermionic quantum states by natural numbers. As a byproduct of the correspondence mentioned above we realize the grafting and the growth operators, comprising important constructions in the context of Hopf algebras, in the fermionic Fock space. Also, we show how to construct fermionic creation and annihilation operators in the context of rooted trees. New representations of the solutions of combinatorial Dyson–Schwinger equations and of the antipode in the Connes–Kreimer Hopf algebra of rooted trees related to the occupation number picture are presented.