Combinatorial Matrix Theory

We introduce the characterization of a class of quantum PageRank algorithms in a scenario in which some kind of quantum network is realizable out of the current classical internet web, but no quantum computer is yet available. This class represents a quantization of the PageRank protocol currently employed to list web pages according to their importance. We have found an instance of this class of quantum protocols that outperforms its classical counterpart and may break the classical hierarchy of web pages depending on the topology of the web.

/pdf/google-in-a-quantum-network-li1vefz8u2.pdf

Google in a Quantum Network

The study of semigroup algebras has a long tradition in commutative algebra. Presentation ideals of semigroup algebras are called toric ideals, in reference to their prominent role in geometry. In this paper we consider the more general class of lattice ideals. Fix a polynomial ring S = k[x1, . . . , xn] over a field k and identify monomials x in S with vectors a ∈ N. Let L be any sublattice of Z. Then its associated lattice ideal in S is

Syzygies of codimension 2 lattice ideals

Electronic notes in discrete mathematics

We investigate the behaviour of the recently proposed Quantum PageRank algorithm, in large complex networks. We find that the algorithm is able to univocally reveal the underlying topology of the network and to identify and order the most relevant nodes. Furthermore, it is capable to clearly highlight the structure of secondary hubs and to resolve the degeneracy in importance of the low lying part of the list of rankings. The quantum algorithm displays an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance, as compared to the classical algorithm. We test the performance and confirm the listed features by applying it to real world examples from the WWW. Finally, we raise and partially address whether the increased sensitivity of the quantum algorithm persists under coordinated attacks in scale-free and random networks.

/pdf/quantum-google-in-a-complex-network-2d0i6f6w6a.pdf

Quantum Google in a Complex Network

Minimal systems of generators for ideals of semigroups

http://www.singacom.uva.es/Uploads/2007/54ed90bb728dc.pdf

The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs

© Université Bordeaux 1, 1991, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/higher-order-relations-for-a-numerical-semigroup-suf0v5ui8e.pdf

Higher order relations for a numerical semigroup

A map of sufficient conditions for the real nonnegative inverse eigenvalue problem

The nonnegative inverse eigenvalue problem (NIEP) asks which lists of $n$ complex numbers (counting multiplicity) occur as the eigenvalues of some $n$-by-$n$ entry-wise nonnegative matrix The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem Thus, there are many subproblems and relevant results in a variety of directions We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; c) low dimensional results; d) sufficient conditions; e) appending 0's to achieve realizability; f) the graph NIEP's; g) Perron similarities; and h) the relevance of Jordan structure

Carlos Marijuán

Papers

Minimal systems of generators for ideals of semigroups

The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs

Higher order relations for a numerical semigroup

A map of sufficient conditions for the real nonnegative inverse eigenvalue problem

The NIEP