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http://hdl.handle.net/2117/3227
Sat, 18 Apr 2015 20:50:19 GMT2015-04-18T20:50:19Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoNonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
http://hdl.handle.net/2117/22391
Title: Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
Authors: Cabré Vilagut, Xavier; Sire, Yannick
Abstract: This is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.http://hdl.handle.net/2117/22391Cabré Vilagut, Xavier; Sire, YannicknoThis is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.Estructuras A-infinito en la opérada de cactus
http://hdl.handle.net/2117/22097
Title: Estructuras A-infinito en la opérada de cactus
Authors: Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, Andrew
Abstract: Diversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterhttp://hdl.handle.net/2117/22097Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, AndrewnoDiversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterA note on symplectic and Poisson linearization of semisimple Lie algebra actions
http://hdl.handle.net/2117/26960
Title: A note on symplectic and Poisson linearization of semisimple Lie algebra actions
Authors: Miranda Galcerán, Eva
Abstract: In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson settingMon, 23 Mar 2015 16:46:48 GMThttp://hdl.handle.net/2117/269602015-03-23T16:46:48ZMiranda Galcerán, EvanoIn this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson settingGeneric bifurcations of low codimension of planar Filippov Systems
http://hdl.handle.net/2117/26671
Title: Generic bifurcations of low codimension of planar Filippov Systems
Authors: Martínez-Seara Alonso, M. Teresa; Guàrdia Munarriz, Marcel; Teixeira, Marco Antonio
Abstract: In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of
this article is to develop a systematic method for studying local(and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequencesThu, 12 Mar 2015 10:45:38 GMThttp://hdl.handle.net/2117/266712015-03-12T10:45:38ZMartínez-Seara Alonso, M. Teresa; Guàrdia Munarriz, Marcel; Teixeira, Marco AntonionoSingularity, Non-smooth vector field, Structural stability, BifurcationIn this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of
this article is to develop a systematic method for studying local(and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequencesA distributed attitude control law for formation flying based on the Cucker-smale model
http://hdl.handle.net/2117/26670
Title: A distributed attitude control law for formation flying based on the Cucker-smale model
Authors: Paita, Fabrizio; Gómez Muntané, Gerard; Masdemont Soler, Josep
Abstract: In this paper we consider the attitude synchronization problem for a swarm of spacecrafts flying in formation.
Starting from previous works on consensus dynamics, we construct a distributed attitude control law and derive analytically
sufficient conditions for the formation to converge asymptotically towards a synchronized, non–accelerating state (possibly defined a priori). Moreover, motivated by the results obtained on a particular consensus model, first introduced by F. Cucker and S. Smale to modellize the translational dynamics of flocks, we numerically explore
the dependence of the convergence process on the dimension of the formation and the relative initial conditions of the spacecrafts. Finally, we generalize the class of weights defined by the previous authors in order to dampen the aforementioned effects, thus making our control law suitable for very large formations.Thu, 12 Mar 2015 09:06:25 GMThttp://hdl.handle.net/2117/266702015-03-12T09:06:25ZPaita, Fabrizio; Gómez Muntané, Gerard; Masdemont Soler, JosepnoIn this paper we consider the attitude synchronization problem for a swarm of spacecrafts flying in formation.
Starting from previous works on consensus dynamics, we construct a distributed attitude control law and derive analytically
sufficient conditions for the formation to converge asymptotically towards a synchronized, non–accelerating state (possibly defined a priori). Moreover, motivated by the results obtained on a particular consensus model, first introduced by F. Cucker and S. Smale to modellize the translational dynamics of flocks, we numerically explore
the dependence of the convergence process on the dimension of the formation and the relative initial conditions of the spacecrafts. Finally, we generalize the class of weights defined by the previous authors in order to dampen the aforementioned effects, thus making our control law suitable for very large formations.Matrices positivas y aplicaciones
http://hdl.handle.net/2117/26622
Title: Matrices positivas y aplicaciones
Authors: García Planas, María IsabelMon, 09 Mar 2015 12:14:12 GMThttp://hdl.handle.net/2117/266222015-03-09T12:14:12ZGarcía Planas, María IsabelnoCredit risk contributions under the Vasicek one-factor model: a fast wavelet expansion approximation
http://hdl.handle.net/2117/26597
Title: Credit risk contributions under the Vasicek one-factor model: a fast wavelet expansion approximation
Authors: Masdemont Soler, Josep; Ortiz-Gracia, Luis
Abstract: To measure the contribution of individual transactions inside the total risk of a credit portfolio is a major issue in financial institutions. VaR Contributions (VaRC) and Expected Shortfall Contributions (ESC) have become two popular ways of quantifying the risks. However, the usual Monte Carlo (MC) approach is known to be a very time consum-
ing method for computing these risk contributions. In this paper we consider the Wavelet Approximation (WA) method for Value at Risk (VaR) computation presented in [Mas10] in order to calculate the Expected Shortfall (ES) and the risk contributions under the Vasicek
one-factor model framework. We decompose the VaR and the ES as a sum of sensitivities representing the marginal impact on the total portfolio risk. Moreover, we present technical improvements in the Wavelet Approximation (WA) that considerably reduce the computa-
tional effort in the approximation while, at the same time, the accuracy increasesThu, 05 Mar 2015 12:53:03 GMThttp://hdl.handle.net/2117/265972015-03-05T12:53:03ZMasdemont Soler, Josep; Ortiz-Gracia, LuisnoTo measure the contribution of individual transactions inside the total risk of a credit portfolio is a major issue in financial institutions. VaR Contributions (VaRC) and Expected Shortfall Contributions (ESC) have become two popular ways of quantifying the risks. However, the usual Monte Carlo (MC) approach is known to be a very time consum-
ing method for computing these risk contributions. In this paper we consider the Wavelet Approximation (WA) method for Value at Risk (VaR) computation presented in [Mas10] in order to calculate the Expected Shortfall (ES) and the risk contributions under the Vasicek
one-factor model framework. We decompose the VaR and the ES as a sum of sensitivities representing the marginal impact on the total portfolio risk. Moreover, we present technical improvements in the Wavelet Approximation (WA) that considerably reduce the computa-
tional effort in the approximation while, at the same time, the accuracy increasesCom les Varietats Invariants formen Espirals i Anells en Galàxies barrades
http://hdl.handle.net/2117/26596
Title: Com les Varietats Invariants formen Espirals i Anells en Galàxies barrades
Authors: Romero Gómez, Mercè; Sánchez-Martín, Patricia; Masdemont Soler, Josep
Abstract: L'espectacularitat de les galàxies barrades consisteix no solament en la
presència de la barra, allargada en el centre de la galàxia, sinó també en els braços espirals o anells que es desenvolupen en les parts exteriors. No hi ha una teoria clara per a la formació d'anells i, fins fa poc, només n'hi havia una que explicava l'origen dels braços espirals en galàxies no barrades. En els darrers anys hem desenvolupat una teoria basada en els sistemes dinàmics que relaciona els braços espirals i els anells amb les varietats invariants hiperbòliques associades a òrbites periòdiques i quasiperiòdiques al voltant de punts d'equilibri colineals del sistemaThu, 05 Mar 2015 12:42:01 GMThttp://hdl.handle.net/2117/265962015-03-05T12:42:01ZRomero Gómez, Mercè; Sánchez-Martín, Patricia; Masdemont Soler, JosepnoL'espectacularitat de les galàxies barrades consisteix no solament en la
presència de la barra, allargada en el centre de la galàxia, sinó també en els braços espirals o anells que es desenvolupen en les parts exteriors. No hi ha una teoria clara per a la formació d'anells i, fins fa poc, només n'hi havia una que explicava l'origen dels braços espirals en galàxies no barrades. En els darrers anys hem desenvolupat una teoria basada en els sistemes dinàmics que relaciona els braços espirals i els anells amb les varietats invariants hiperbòliques associades a òrbites periòdiques i quasiperiòdiques al voltant de punts d'equilibri colineals del sistemaAction-angle variables and a KAM theorem for b-Poisson manifolds
http://hdl.handle.net/2117/26390
Title: Action-angle variables and a KAM theorem for b-Poisson manifolds
Authors: Kiesenhofer, Anna; Miranda Galcerán, Eva; Scott, Geoffrey
Abstract: In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.Tue, 17 Feb 2015 12:12:30 GMThttp://hdl.handle.net/2117/263902015-02-17T12:12:30ZKiesenhofer, Anna; Miranda Galcerán, Eva; Scott, GeoffreynoIn this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.Structural stability of planar bimodal linear systems
http://hdl.handle.net/2117/26226
Title: Structural stability of planar bimodal linear systems
Authors: Ferrer Llop, Josep; Peña Carrera, Marta; Susín Sánchez, Antonio
Abstract: Structural stability ensures that the qualitative behavior of a system is preserved under small perturbations. We study it for planar bimodal linear dynamical systems, that is, systems consisting of two linear dynamics acting on each side of a given hyperplane and assuming continuity along the separating hyperplane. We describe which one of these systems is structurally stable when (real) spiral does not appear and when it does we give necessary and sufficient conditions concerning finite periodic orbits and saddle connections. In particular, we study the finite periodic orbits and the homoclinic orbits in the saddle/spiral case.Thu, 05 Feb 2015 10:53:33 GMThttp://hdl.handle.net/2117/262262015-02-05T10:53:33ZFerrer Llop, Josep; Peña Carrera, Marta; Susín Sánchez, AntonionoVECTOR-FIELDSStructural stability ensures that the qualitative behavior of a system is preserved under small perturbations. We study it for planar bimodal linear dynamical systems, that is, systems consisting of two linear dynamics acting on each side of a given hyperplane and assuming continuity along the separating hyperplane. We describe which one of these systems is structurally stable when (real) spiral does not appear and when it does we give necessary and sufficient conditions concerning finite periodic orbits and saddle connections. In particular, we study the finite periodic orbits and the homoclinic orbits in the saddle/spiral case.EMtree for phylogenetic topology reconstruction on nonhomogeneous data
http://hdl.handle.net/2117/26031
Title: EMtree for phylogenetic topology reconstruction on nonhomogeneous data
Authors: Ibáñez Marcelo, Esther; Casanellas Rius, MartaThu, 22 Jan 2015 12:04:00 GMThttp://hdl.handle.net/2117/260312015-01-22T12:04:00ZIbáñez Marcelo, Esther; Casanellas Rius, MartanoTree topology reconstruction, Expectation-maximization, Quartet-based method, Evolutionary Markov modelLow degree equations for phylogenetic group-based models
http://hdl.handle.net/2117/26029
Title: Low degree equations for phylogenetic group-based models
Authors: Casanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, Mateusz
Abstract: Motivated by phylogenetics, our aim is to obtain a system of low degree equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group G , we provide an explicit construction of codimX polynomial equations (phylogenetic invariants) of degree at most |G| that define the variety X on a Zariski open set U . The set U contains all biologically meaningful points when G is the group of the Kimura 3-parameter model. In particular, our main result confirms (Michalek, Toric varieties: phylogenetics and derived categories, PhD thesis, Conjecture 7.9, 2012) and, on the set U , Conjectures 29 and 30 of Sturmfels and Sullivant (J Comput Biol 12:204–228, 2005).Thu, 22 Jan 2015 12:01:03 GMThttp://hdl.handle.net/2117/260292015-01-22T12:01:03ZCasanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, MateusznoMotivated by phylogenetics, our aim is to obtain a system of low degree equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group G , we provide an explicit construction of codimX polynomial equations (phylogenetic invariants) of degree at most |G| that define the variety X on a Zariski open set U . The set U contains all biologically meaningful points when G is the group of the Kimura 3-parameter model. In particular, our main result confirms (Michalek, Toric varieties: phylogenetics and derived categories, PhD thesis, Conjecture 7.9, 2012) and, on the set U , Conjectures 29 and 30 of Sturmfels and Sullivant (J Comput Biol 12:204–228, 2005).On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory
http://hdl.handle.net/2117/26024
Title: On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory
Authors: Acosta-Humànez, Primitivo; Lázaro Ochoa, José Tomás; Morales Ruiz, Juan José; Pantazi, Chara
Abstract: We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Lienard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincare problem for some families is also approached.Thu, 22 Jan 2015 11:17:38 GMThttp://hdl.handle.net/2117/260242015-01-22T11:17:38ZAcosta-Humànez, Primitivo; Lázaro Ochoa, José Tomás; Morales Ruiz, Juan José; Pantazi, CharanoDifferential Galois theory, Darboux theory of integrability, Poincare problem, rational first integral, integrating factor, Riccati equation, Lienard equation, Liouvillian solution, INVARIANT ALGEBRAIC-CURVES, LINEAR-DIFFERENTIAL EQUATIONS, DARBOUX INTEGRATING FACTORS, INVERSE PROBLEMS, 1ST INTEGRALS, POINCARE PROBLEM, GALOIS THEORY, SYSTEMS, FOLIATIONS, MULTIPLICITYWe study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Lienard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincare problem for some families is also approached.Multi-agent linear dynamical systems, analyzing the consensus problem
http://hdl.handle.net/2117/25531
Title: Multi-agent linear dynamical systems, analyzing the consensus problem
Authors: García Planas, María Isabel
Abstract: In this paper the consensus problem is considered
for multi-agent systems having an independent agent
and fixed topology.Thu, 15 Jan 2015 09:18:49 GMThttp://hdl.handle.net/2117/255312015-01-15T09:18:49ZGarcía Planas, María IsabelnoMulti-agent systems, consensus, controlIn this paper the consensus problem is considered
for multi-agent systems having an independent agent
and fixed topology.Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties
http://hdl.handle.net/2117/25175
Title: Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties
Authors: González Nogueras, María del Mar; Saéz, Mariel; Sire, Yannick
Abstract: We investigate the equation; (-Delta(Hn))(gamma) w = f(w) in H-n,; where (-Delta(Hn))(gamma) corresponds to the fractional Laplacian on hyperbolic space for gamma is an element of(0, 1) and f is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connections in the following sense; a so-called layer solution is a smooth solution of the previous equation converging to +/- 1 at any point of the two hemispheres S-+/- subset of partial derivative H-infinity(n) and which is strictly increasing with respect to the signed distance to a totally geodesic hyperplane Pi. We prove that under additional conditions on the nonlinearity uniqueness holds up to isometry. Then we provide several symmetry results and qualitative properties of the layer solutions. Finally, we consider the multilayer case, at least when gamma is close to one.Thu, 08 Jan 2015 12:16:16 GMThttp://hdl.handle.net/2117/251752015-01-08T12:16:16ZGonzález Nogueras, María del Mar; Saéz, Mariel; Sire, YannicknoFractional Laplacian, Hyperbolic space, Layer solution, Symmetry, SEMILINEAR ELLIPTIC-EQUATIONS, PHASE-TRANSITIONS, SYMMETRY, CONJECTURE, REGULARITY, MANIFOLDS, GIORGIWe investigate the equation; (-Delta(Hn))(gamma) w = f(w) in H-n,; where (-Delta(Hn))(gamma) corresponds to the fractional Laplacian on hyperbolic space for gamma is an element of(0, 1) and f is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connections in the following sense; a so-called layer solution is a smooth solution of the previous equation converging to +/- 1 at any point of the two hemispheres S-+/- subset of partial derivative H-infinity(n) and which is strictly increasing with respect to the signed distance to a totally geodesic hyperplane Pi. We prove that under additional conditions on the nonlinearity uniqueness holds up to isometry. Then we provide several symmetry results and qualitative properties of the layer solutions. Finally, we consider the multilayer case, at least when gamma is close to one.