Articles de revista
http://hdl.handle.net/2117/79765
20210731T00:27:53Z

Of limit key polynomials
http://hdl.handle.net/2117/350273
Of limit key polynomials
Alberich Carramiñana, Maria; Fernández Boix, Albert.; Fernández González, Julio; Guàrdia Rubies, Jordi; Nart Vinyals, Enric; Roé Vellvé, Joaquim
Let ν be a valuation of arbitrary rank on the polynomial ring K[x] with coefficients in a field K. We prove comparison theorems between MacLaneVaquié key polynomials for valuations μ ≤ν and abstract key polynomials for ν. Also, some results on invariants associated to limit key polynomials are obtained. In particular, if char(K) = 0 we show that all the limit key polynomials of unbounded continuous families of augmentations have the numerical character equal to one.
20210729T11:23:04Z
Alberich Carramiñana, Maria
Fernández Boix, Albert.
Fernández González, Julio
Guàrdia Rubies, Jordi
Nart Vinyals, Enric
Roé Vellvé, Joaquim
Let ν be a valuation of arbitrary rank on the polynomial ring K[x] with coefficients in a field K. We prove comparison theorems between MacLaneVaquié key polynomials for valuations μ ≤ν and abstract key polynomials for ν. Also, some results on invariants associated to limit key polynomials are obtained. In particular, if char(K) = 0 we show that all the limit key polynomials of unbounded continuous families of augmentations have the numerical character equal to one.

The singular Weinstein conjecture
http://hdl.handle.net/2117/350252
The singular Weinstein conjecture
Miranda Galcerán, Eva; Oms, Cedric
In this article, we investigate Reeb dynamics on bmcontact manifolds, previously introduced in [37], which are contact away from a hypersurface Zbut satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the wellknown Weinstein conjecture. Contrary to the initial expectations, examples of compact bmcontact manifolds without periodic Reeb orbits outside Zare provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bmReeb flow exist in any dimension. This investigation goes handinhand with the Weinstein conjecture on noncompact manifolds having compact ends of convex
20210729T08:54:09Z
Miranda Galcerán, Eva
Oms, Cedric
In this article, we investigate Reeb dynamics on bmcontact manifolds, previously introduced in [37], which are contact away from a hypersurface Zbut satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the wellknown Weinstein conjecture. Contrary to the initial expectations, examples of compact bmcontact manifolds without periodic Reeb orbits outside Zare provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bmReeb flow exist in any dimension. This investigation goes handinhand with the Weinstein conjecture on noncompact manifolds having compact ends of convex

On the singular Weinstein conjecture and the existence of escape orbits for bBeltrami fields
http://hdl.handle.net/2117/349229
On the singular Weinstein conjecture and the existence of escape orbits for bBeltrami fields
Miranda Galcerán, Eva; Oms, Cedric; PeraltaSalas, Daniel
Motivated by Poincaré’s orbits going to infinity in the (restricted) threebody problem, we investigate the generic existence of heterocliniclike orbits in a neighbourhood of the critical set of a bcontact form. This is done by using the singular counterpart of Etnyre– Ghrist’s contact/Beltrami correspondence, and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck. Specifically, we analyze the bBeltrami vector fields on bmanifolds of dimension 3 and prove that for a generic asymptotically exact bmetric they exhibit escape orbits. We also show that a generic asymptotically symmetric bBeltrami vector field on an asymptotically flat bmanifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose α and ωlimit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjectur
20210714T08:11:57Z
Miranda Galcerán, Eva
Oms, Cedric
PeraltaSalas, Daniel
Motivated by Poincaré’s orbits going to infinity in the (restricted) threebody problem, we investigate the generic existence of heterocliniclike orbits in a neighbourhood of the critical set of a bcontact form. This is done by using the singular counterpart of Etnyre– Ghrist’s contact/Beltrami correspondence, and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck. Specifically, we analyze the bBeltrami vector fields on bmanifolds of dimension 3 and prove that for a generic asymptotically exact bmetric they exhibit escape orbits. We also show that a generic asymptotically symmetric bBeltrami vector field on an asymptotically flat bmanifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose α and ωlimit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjectur

Geometric quantization via cotangent models
http://hdl.handle.net/2117/348245
Geometric quantization via cotangent models
Mir Garcia, Pau; Miranda Galcerán, Eva
In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with nondegenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morsetype functions and include elliptic, hyperbolic and focusfocus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focusfocus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a localtoglobal principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.
20210701T12:48:49Z
Mir Garcia, Pau
Miranda Galcerán, Eva
In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with nondegenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morsetype functions and include elliptic, hyperbolic and focusfocus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focusfocus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a localtoglobal principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.

Discrepancies between observational data and theoretical forecast in single field slow roll inflation
http://hdl.handle.net/2117/347742
Discrepancies between observational data and theoretical forecast in single field slow roll inflation
Amorós Torrent, Jaume; Haro Cases, Jaume
The PLANCK collaboration has determined, or greatly constrained, values for the spectral parameters of the CMB radiation, namely the spectral index ns, its running as, the running of the running ßs, using a growing body of measurements of CMB anisotropies by the Planck satellite and other missions. These values do not follow the hierarchy of sizes predicted by single field, slow roll inflationary theory, and are thus difficult to fit for such inflation models. In this work we present first a study of 49 single field, slow roll inflationary potentials in which we assess the likelyhood of these models fitting the spectral parameters to their currently most accurate determination given by the PLANCK collaboration. We check numerically with a MATLAB program the spectral parameters that each model can yield for a very broad, comprehensive list of possible parameter and field values. The comparison of spectral parameter values supported by the models with their determinations by the PLANCK collaboration leads to the conclusion that the data provided by PLANCK2015 TT+lowP and PLANCK2015 TT,TE,EE+lowP taking into account the running of the running disfavours 40 of the 49 models with confidence level at least 92.8 %. Next, we discuss the reliability of the current computations of these spectral parameters. We identify a bias in the method of determination of the spectral parameters by least residue parameter fitting (using MCMC or any other scheme) currently used to reconstruct the power spectrum of scalar perturbations. This bias can explain the observed contradiction between theory and observations. Its removal is computationally costly, but necessary in order to compare the forecasts of single field, slow roll theories with observation
20210622T10:00:44Z
Amorós Torrent, Jaume
Haro Cases, Jaume
The PLANCK collaboration has determined, or greatly constrained, values for the spectral parameters of the CMB radiation, namely the spectral index ns, its running as, the running of the running ßs, using a growing body of measurements of CMB anisotropies by the Planck satellite and other missions. These values do not follow the hierarchy of sizes predicted by single field, slow roll inflationary theory, and are thus difficult to fit for such inflation models. In this work we present first a study of 49 single field, slow roll inflationary potentials in which we assess the likelyhood of these models fitting the spectral parameters to their currently most accurate determination given by the PLANCK collaboration. We check numerically with a MATLAB program the spectral parameters that each model can yield for a very broad, comprehensive list of possible parameter and field values. The comparison of spectral parameter values supported by the models with their determinations by the PLANCK collaboration leads to the conclusion that the data provided by PLANCK2015 TT+lowP and PLANCK2015 TT,TE,EE+lowP taking into account the running of the running disfavours 40 of the 49 models with confidence level at least 92.8 %. Next, we discuss the reliability of the current computations of these spectral parameters. We identify a bias in the method of determination of the spectral parameters by least residue parameter fitting (using MCMC or any other scheme) currently used to reconstruct the power spectrum of scalar perturbations. This bias can explain the observed contradiction between theory and observations. Its removal is computationally costly, but necessary in order to compare the forecasts of single field, slow roll theories with observation

Geometric quantization of almost toric manifolds
http://hdl.handle.net/2117/345182
Geometric quantization of almost toric manifolds
Miranda Galcerán, Eva; Presas, Francisco; Barbieri Solha, Romero
Kostant gave a model for the geometric quantization via the cohomology associated to the sheaf of flat sections of a prequantum line bundle. This model is welladapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed by counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one induced by an integrable system with focusfocus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme
20210505T14:36:51Z
Miranda Galcerán, Eva
Presas, Francisco
Barbieri Solha, Romero
Kostant gave a model for the geometric quantization via the cohomology associated to the sheaf of flat sections of a prequantum line bundle. This model is welladapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed by counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one induced by an integrable system with focusfocus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme

Algebraic statistics in practice: applications to networks
http://hdl.handle.net/2117/344004
Algebraic statistics in practice: applications to networks
Casanellas Rius, Marta; Petrovic, Sonja; Uhler, Caroline
Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra, and computational algebra), geometry, and combinatorics to provide insight into knotty problems in mathematical statistics. In this review, we illustrate this on three problems related to networks: network models for relational data, causal structure discovery, and phylogenetics. For each problem, we give an overview of recent results in algebraic statistics, with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines.
20210420T12:26:14Z
Casanellas Rius, Marta
Petrovic, Sonja
Uhler, Caroline
Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra, and computational algebra), geometry, and combinatorics to provide insight into knotty problems in mathematical statistics. In this review, we illustrate this on three problems related to networks: network models for relational data, causal structure discovery, and phylogenetics. For each problem, we give an overview of recent results in algebraic statistics, with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines.

A contact geometry framework for field theories with dissipation
http://hdl.handle.net/2117/343989
A contact geometry framework for field theories with dissipation
Gaset Rifà, Jordi; Gràcia Sabaté, Francesc Xavier; Muñoz Lecanda, Miguel Carlos; Rivas Guijarro, Xavier; Román Roy, Narciso
We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of kcontact structure and kcontact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the ksymplectic Hamiltonian systems in field theory. The concepts of symmetries and dissipation laws are introduced and developed. Two relevant examples are analyzed in detail: the damped vibrating string and Burgers’ equation.
© 2020. Elsevier
20210420T11:13:17Z
Gaset Rifà, Jordi
Gràcia Sabaté, Francesc Xavier
Muñoz Lecanda, Miguel Carlos
Rivas Guijarro, Xavier
Román Roy, Narciso
We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of kcontact structure and kcontact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the ksymplectic Hamiltonian systems in field theory. The concepts of symmetries and dissipation laws are introduced and developed. Two relevant examples are analyzed in detail: the damped vibrating string and Burgers’ equation.

Unified LagrangianHamiltonian formalism for contact systems
http://hdl.handle.net/2117/343824
Unified LagrangianHamiltonian formalism for contact systems
De León, Manuel; Gaset Rifà, Jordi; Lainz Valcázar, Manuel; Rivas Guijarro, Xavier; Román Roy, Narciso
We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of contact autonomous mechanical systems, which is based on the approach of the pioneering work of R. Skinner and R. Rusk. This framework permits to skip the second order differential equation problem, which is obtained as a part of the constraint algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method.
This is the peer reviewed version of the following article: De León, M. [et al.]. Unified LagrangianHamiltonian formalism for contact systems. "Fortschritte der physik. Progress of physics", 23 Juny 2020, vol. 68, núm. 8, p. 20000451200004512., which has been published in final form at 10.1002/prop.202000045. This article may be used for noncommercial purposes in accordance with Wiley Terms and Conditions for Use of SelfArchived Versions
20210416T11:44:09Z
De León, Manuel
Gaset Rifà, Jordi
Lainz Valcázar, Manuel
Rivas Guijarro, Xavier
Román Roy, Narciso
We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of contact autonomous mechanical systems, which is based on the approach of the pioneering work of R. Skinner and R. Rusk. This framework permits to skip the second order differential equation problem, which is obtained as a part of the constraint algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method.

La compacitat, una noció seminal en l'evolució de la topologia general (18951930)
http://hdl.handle.net/2117/342112
La compacitat, una noció seminal en l'evolució de la topologia general (18951930)
Moreno Montes, Júlia; Blanco Abellán, Mónica; Pascual Gainza, Pere
En aquest article es revisa l'evolució de la definició d'espai topològic compacte des de la primera definició de Fréchet el 1904, que va generalitzar la propietat de BolzanoWeierstrass, passant per la relació amb el teorema de Borel del 1895 per a recobriments numerables de l'interval [0, 1], fins a arribar a la definició general a partir de la propietat de HeineBorel dels recobriments oberts arbitraris l'any 1929. Centrem l'anàlisi en quatre publicacions clau: les tesis de Borel i de Fréchet, el llibre de Hausdorff en el qual s'estableixen les bases de la topologia general, i la memòria d'Alexandroff i Urysohn.
20210322T09:48:01Z
Moreno Montes, Júlia
Blanco Abellán, Mónica
Pascual Gainza, Pere
En aquest article es revisa l'evolució de la definició d'espai topològic compacte des de la primera definició de Fréchet el 1904, que va generalitzar la propietat de BolzanoWeierstrass, passant per la relació amb el teorema de Borel del 1895 per a recobriments numerables de l'interval [0, 1], fins a arribar a la definició general a partir de la propietat de HeineBorel dels recobriments oberts arbitraris l'any 1929. Centrem l'anàlisi en quatre publicacions clau: les tesis de Borel i de Fréchet, el llibre de Hausdorff en el qual s'estableixen les bases de la topologia general, i la memòria d'Alexandroff i Urysohn.