Multigraded Hilbert Series of noncommutative modules

In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group. In fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.Comment: 28 pages, to appear in Journal of Algebr

Les Històries de Tobies d'Andrea Vaccaro: de Nàpols al Museu Nacional d'Art de Catalunya

[cat] Aquest article intenta reconstruir la història de quatre pintures de l'artista napolità Andrea Vaccaro, actualment al Museu Nacional d'Art de Catalunya, i originàriament part d'una sèrie de 12 peces dedicada a les històries bíbliques de Tobies, propietat del virrei de Nàpols Pedro Antonio de Aragón. L'anàlisi de les obres i del context de la seva comissió a la llum d'estudis recents permet corregir-ne la datació i comprendre la importància de les quatre peces de Vaccaro del MNAC, en el marc de la circulació d'obres d'art entre el Regne de Nàpols i la península Ibèrica al llarg del segle XVII.[spa] Este artículo intenta reconstruir la historia de cuatro pinturas del artista napolitano Andrea Vaccaro, actualmente en el Museu Nacional d'Art de Catalunya, y originariamente parte de una serie de 12 piezas dedicada a las historias bíblicas de Tobías, propiedad del virrey de Nápoles Pedro Antonio de Aragón. El análisis de las obras y del contexto de su comisión a la luz de recientes estudios permite corregir su datación y comprender la importancia de las cuatro piezas de Vaccaro del MNAC, en el marco de la circulación de obras de arte entre el Reino de Nápoles y la península Ibérica a lo largo del siglo XVII.[eng] This article reconstructs the history of four paintings by the Neapolitan painter Andrea Vaccaro (1604-1670), currently in the Museu Nacional d'Art de Catalunya. These masterpieces were originally part of a larger series of twelve works (in the collection of the Viceroy Pedro Antonio de Aragón), all depicting the Old Testament story of Tobias. The analysis of the four Tobias paintings by Vaccaro in the MNAC and the circumstances of their commission (in light of recent studies) enables us to establish their correct date, and to understand their importance in relation to other works of art which circulated between the kingdom of Naples and Spain in the seventeenth century

Les Històries de Tobies d’Andrea Vaccaro: de Nàpols al Museu Nacional d’Art de Catalunya

Aquest article intenta reconstruir la història de quatre pintures de l’artista napolità Andrea Vaccaro, actualment al Museu Nacional d’Art de Catalunya, i originàriament part d’una sèrie de 12 peces dedicada a les històries bíbliques de Tobies, propietat del virrei de Nàpols Pedro Antonio de Aragón. L’anàlisi de les obres i del context de la seva comissió –a la llum d’estudis recents– permet corregir-ne la datació i comprendre la importància de les quatre peces de Vaccaro del MNAC, en el marc de la circulació d’obres d’art entre el Regne de Nàpols i la península Ibèrica al llarg del segle XVII.This article reconstructs the history of four paintings by the Neapolitan painter Andrea Vaccaro (1604-1670), currently in the Museu Nacional d’Art de Catalunya. These masterpieces were originally part of a larger series of twelve works (in the collection of the Viceroy Pedro Antonio de Aragón), all depicting the Old Testament story of Tobias. The analysis of the four Tobias paintings by Vaccaro in the MNAC and the circumstances of their commission (in light of recent studies) enables us to establish their correct date, and to understand their importance in relation to other works of art which circulated between the kingdom of Naples and Spain in the seventeenth century.Este artículo intenta reconstruir la historia de cuatro pinturas del artista napolitano Andrea Vaccaro, actualmente en el Museu Nacional d’Art de Catalunya, y originariamente parte de una serie de 12 piezas dedicada a las historias bíblicas de Tobías, propiedad del virrey de Nápoles Pedro Antonio de Aragón. El análisis de las obras y del contexto de su comisión –a la luz de recientes estudios– permite corregir su datación y comprender la importancia de las cuatro piezas de Vaccaro del MNAC, en el marco de la circulación de obras de arte entre el Reino de Nápoles y la península Ibérica a lo largo del siglo XVII

Polar Actions on Berger Spheres

The object of this article is to study a torus action on a so-called Berger sphere. We also make some comments on polar actions on naturally reductive homogeneous spaces. Finally, we prove a rigidity-type theorem for Riemannian manifolds carrying a polar action with a fix point

Stream/block ciphers, difference equations and algebraic attacks

In this paper we introduce a general class of stream and block ciphers that are defined by means of systems of (ordinary) explicit difference equations over a finite field. We call this class "difference ciphers". Many important ciphers such as systems of LFSRs, Trivium/Bivium and Keeloq are difference ciphers. To the purpose of studying their underlying explicit difference systems, we introduce key notions as state transition endomorphisms and show conditions for their invertibility. Reducible and periodic systems are also considered. We then propose general algebraic attacks to difference ciphers which are experimented by means of Bivium and Keeloq.Comment: 22 page

Population trapping due to cavity losses

In population trapping the occupation of a decaying quantum level keeps a constant non-zero value. We show that an atom-cavity system interacting with an environment characterized by a non-flat spectrum, in the non-Markovian limit, exhibits such a behavior, effectively realizing the preservation of nonclassical states against dissipation. Our results allow to understand the role of cavity losses in hybrid solid state systems and pave the way to the proper description of leakage in the recently developed cavity quantum electrodynamic systems.Comment: 4 pages, 3 figures, version accepted for publication on Phys. Rev.

Noncommutative algebras, context-free grammars and algebraic Hilbert series

In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.Comment: 26 pages, to appear in Journal of Symbolic Computatio

Pathophysiological mechanisms in neurodevelopmental disorders caused by rac GTPases dysregulation: What’s behind neuro-RACopathies

Rho family guanosine triphosphatases (GTPases) regulate cellular signaling and cytoskele-tal dynamics, playing a pivotal role in cell adhesion, migration, and cell cycle progression. The Rac subfamily of Rho GTPases consists of three highly homologous proteins, Rac 1–3. The proper function of Rac1 and Rac3, and their correct interaction with guanine nucleotide-exchange factors (GEFs) and GTPase-activating proteins (GAPs) are crucial for neural development. Pathogenic variants affecting these delicate biological processes are implicated in different medical conditions in humans, primarily neurodevelopmental disorders (NDDs). In addition to a direct deleterious effect produced by genetic variants in the RAC genes, a dysregulated GTPase activity resulting from an abnormal function of GEFs and GAPs has been involved in the pathogenesis of distinctive emerging conditions. In this study, we reviewed the current pertinent literature on Rac-related disorders with a primary neurological involvement, providing an overview of the current knowledge on the pathophysiological mechanisms involved in the neuro-RACopathies

Asymptotic Entanglement Dynamics and Geometry of Quantum States

A given dynamics for a composite quantum system can exhibit several distinct properties for the asymptotic entanglement behavior, like entanglement sudden death, asymptotic death of entanglement, sudden birth of entanglement, etc. A classification of the possible situations was given in [M. O. Terra Cunha, {\emph{New J. Phys}} {\bf{9}}, 237 (2007)] but for some classes there were no known examples. In this work we give a better classification for the possibile relaxing dynamics at the light of the geometry of their set of asymptotic states and give explicit examples for all the classes. Although the classification is completely general, in the search of examples it is sufficient to use two qubits with dynamics given by differential equations in Lindblad form (some of them non-autonomous). We also investigate, in each case, the probabilities to find each possible behavior for random initial states.Comment: 9 pages, 2 figures; revised version accepted for publication in J. Phys. A: Math. Theo

Light interaction with extended quantum systems in dispersive media

We derive a light–matter interaction Hamiltonian to describe a quantum system embedded in a dispersive environment and coupled with the electromagnetic field. We include in this theory the spatial extension of the system, taken into account through its wavefunction. This enables us to overcome the divergence problem of the Green tensor propagator that arises from a point-like approximation of the quantum system. Thus the formalism can be applied to generalize the expressions for the spontaneous emission rate and the Lamb shift for a quantum system defined by a spatially extended dipole. In particular, these quantities can be modified by the asymmetry of the spatial structure of the atomic system as demonstrated in two test-bed examples