Esther García

Tits-Kantor-Koecher Superalgebras of Jordan Superpairs Covered by Grids

Abstract In this paper we describe the Tits-Kantor-Koecher superalgebras associated to Jordan superpairs covered by grids,extending results from Neher (Neher,E. (1996). Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid. Amer. J. Math. 118; 439–491) to the supercase. These Lie superalgebras together with their central coverings are precisely the Lie superalgebras graded by a 3-graded root system.

A new method for comparing rankings through complex networks: Model and analysis of competitiveness of major European soccer leagues

In this paper, we show a new technique to analyze families of rankings. In particular, we focus on sports rankings and, more precisely, on soccer leagues. We consider that two teams compete when they change their relative positions in consecutive rankings. This allows to define a graph by linking teams that compete. We show how to use some structural properties of this competitivity graph to measure to what extend the teams in a league compete. These structural properties are the mean degree, the mean strength, and the clustering coefficient. We give a generalization of the Kendalls correlation coefficient to more than two rankings. We also show how to make a dynamic analysis of a league and how to compare different leagues. We apply this technique to analyze the four major European soccer leagues: Bundesliga, Italian Lega, Spanish Liga, and Premier League. We compare our results with the classical analysis of sport ranking based on measures of competitive balance.

A Perron–Frobenius theory for block matrices associated to a multiplex network

Abstract The uniqueness of the Perron vector of a nonnegative block matrix associated to a multiplex network is discussed. The conclusions come from the relationships between the irreducibility of some nonnegative block matrix associated to a multiplex network and the irreducibility of the corresponding matrices to each layer as well as the irreducibility of the adjacency matrix of the projection network. In addition the computation of that Perron vector in terms of the Perron vectors of the blocks is also addressed. Finally we present the precise relations that allow to express the Perron eigenvector of the multiplex network in terms of the Perron eigenvectors of its layers.

Inner ideal structure of nearly Artinian Lie algebras

In this paper we study the inner ideal structure of nondegenerate Lie algebras with essential socle, and characterize, in terms of the whole algebra, when the socle is Artinian.

SIMPLE, PRIMITIVE, AND STRONGLY PRIME JORDAN 3-GRADED LIE ALGEBRAS

Abstract In this paper we give full descriptions of Jordan 3-graded Lie algebras under conditions of simplicity, primitivity and strong primeness. For nondegenerate Jordan 3-graded Lie algebras, we also prove that their heart is either zero or simple.

Center, Centroid, Extended Centroid, and Quotients of Jordan Systems

In this article we prove that the extended centroid of a nondegenerate Jordan system is isomorphic to the centroid (and to the center in the case of Jordan algebras) of its maximal Martindale-like system of quotients with respect to the filter of all essential ideals.

Outer Inheritance of Simplicity in Jordan Systems

Abstract In this article, we show that outer ideals of simple Jordan systems have simple cube. In most cases we obtain that the outer ideals inherit simplicity and give precise descriptions in the cases in which simplicity is not obtained.

3-Graded Lie Algebras with Jordan Finiteness Conditions

Abstract A notion of socle is introduced for 3-graded Lie algebras (over a ring of scalars Φ containing ) whose associated Jordan pairs are non-degenerate. The socle turns out to be a 3-graded ideal and is the sum of minimal 3-graded inner ideals each of which is a central extension of the TKK-algebra of a division Jordan pair. Non-degenerate 3-graded Lie algebras having a large socle are essentially determined by TKK-algebras of simple Jordan pairs with minimal inner ideals and their derivation algebras, which are also 3-graded. Classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space provide a source of examples of infinite dimensional strongly prime 3-graded Lie algebras with non-zero socle. Other examples can be found within the class of finitary simple Lie algebras

On the α-nonbacktracking centrality for complex networks: Existence and limit cases

Abstract Nonbactracking centrality was introduced as an attempt to correct some deficiencies of eigenvector centrality. In this work the α -nonbacktracking centrality is introduced as an extension that interpolates between the nonbacktracking centrality of the edges of a directed network and the eigenvector centrality of the corresponding directed line graph. The existence of this new α -nonbacktracking centrality is proved in terms of the connectivity of the original network. We prove that the limit of the α -nonbacktracking centrality when α decreases to zero exists and is well defined. Moreover, it coincides with the nonbacktracking centrality when this measure is defined. With the same techniques we also prove the convergence of PageRank vectors to the eigenvector centrality vector when the damping factor tends to 1.

On the spectrum of two-layer approach and Multiplex PageRank

Abstract In this paper, we present some results about the spectrum of the matrix associated with the computation of the Multiplex PageRank defined by the authors in a previous paper. These results can be considered as a natural extension of the known results about the spectrum of the Google matrix. In particular, we show that the eigenvalues of the transition matrix associated with the multiplex network can be deduced from the eigenvalues of a block matrix containing the stochastic matrices defined for each layer. We also show that, as occurs in the classic PageRank, the spectrum is not affected by the personalization vectors defined on each layer but depends on the parameter α that controls the teleportation. We also give some analytical relations between the eigenvalues and we include some small examples illustrating the main results.