DSpace Collection:
http://hdl.handle.net/2117/3228
2014-04-21T02:59:05ZCenters of quasi-homogeneous polynomial differential equations of degree three
http://hdl.handle.net/2117/22462
Title: Centers of quasi-homogeneous polynomial differential equations of degree three
Authors: Aziz, Waleed; Llibre Saló, Jaume; Pantazi, Chara
Abstract: We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order2014-03-31T16:38:52ZMathematical Methods Applied to the Celestial Mechanics of Artificial Satellites 2013
http://hdl.handle.net/2117/22393
Title: Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites 2013
Authors: Prado, Antonio F. Bertachini A.; Masdemont Soler, Josep; Zanardi, Maria Cecilia; Winter, Silvia Maria Giuliatti; Yokoyama, Tadashi; Gomes, Vivian Martins2014-03-26T15:17:42ZComputation of limit cycles and their isochrons: Fast algorithms and their convergence
http://hdl.handle.net/2117/22392
Title: Computation of limit cycles and their isochrons: Fast algorithms and their convergence
Authors: Huguet Casades, Gemma; de la Llave Canosa, Rafael
Abstract: We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using N points, a Newton step requires O(N) storage and O(N log N) operations in Fourier discretization or O(N) operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem also shows that the isochrons are analytic and depend analytically on the base point. Moreover, it establishes smooth dependence of the solutions on parameters and provides efficient algorithms to compute perturbative expansions with respect to external parameters. We include a discussion on the numerical implementation of the algorithms as well as numerical results for representative examples.2014-03-26T15:06:49ZNonlinear 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.A Perturbation argument for a Monge–Ampère type equation arising in optimal transportation
http://hdl.handle.net/2117/22385
Title: A Perturbation argument for a Monge–Ampère type equation arising in optimal transportation
Authors: Caffarelli, Luis; González Nogueras, María del Mar; Nguyen, Truyen
Abstract: We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C1,1 estimates for the potential and, second, interior Hölder estimates for
second derivatives. In particular, we take a close look at the geometry of optimal
transportation when the cost function is close to quadratic in order to understand
how the equation degenerates near the boundary.2014-03-26T07:03:34ZGroupoids and Faà di Bruno formulae for Green functions in bialgebras of trees
http://hdl.handle.net/2117/22088
Title: Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees
Authors: Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew
Abstract: We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.2014-03-17T09:08:32ZSharp energy estimates for nonlinear fractional diffusion equations
http://hdl.handle.net/2117/21782
Title: Sharp energy estimates for nonlinear fractional diffusion equations
Authors: Cabré Vilagut, Xavier; Cinti, Eleonora
Abstract: We study the nonlinear fractional equation (−Δ)su=f(u) in Rn, for all fractions 0<s<1 and all nonlinearities f . For every fractional power s∈(0,1) , we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension n=3 whenever 1/2≤s<1 . This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation −Δu=f(u) in Rn . It remains open for n=3 and s<1/2 , and also for n≥4 and all s .2014-02-27T11:34:39ZEigenstructure of of singular systems. Perturbation analysis of simple eigenvalues
http://hdl.handle.net/2117/21372
Title: Eigenstructure of of singular systems. Perturbation analysis of simple eigenvalues
Authors: García Planas, María Isabel; Tarragona Romero, Sonia
Abstract: The problem to study small perturbations of
simple eigenvalues with a change of parameters is of general
interest in applied mathematics. After to introduce a systematic
way to know if an eigenvalue of a singular system is simple or
not, the aim of this work is to study the behavior of a simple
eigenvalue of singular linear system family2014-01-27T11:26:03ZMorphisms and inverse problems for Darboux integrating factors
http://hdl.handle.net/2117/21360
Title: Morphisms and inverse problems for Darboux integrating factors
Authors: Llibre Saló, Jaume; Pantazi, Chara; Walcher, Sebastian
Abstract: Polynomial vector fields which admit a prescribed Darboux integrat-
ing factor are quite well-understood when the geometry of the underlying curve is
nondegenerate. In the general setting morphisms of the affine plane may remove
degeneracies of the curve, and thus allow more structural insight. In the present
paper we establish some properties of integrating factors subjected to morphisms,
and we discuss in detail one particular class of morphisms related to finite reflection
groups. The results indicate that degeneracies for the underlying curve generally
impose restrictions on the nontrivial vector fields which admit a given integrating
factor.2014-01-24T12:27:35ZLocal cohomology supported on monomial ideals
http://hdl.handle.net/2117/21325
Title: Local cohomology supported on monomial ideals
Authors: Álvarez Montaner, Josep2014-01-22T11:02:19ZA Poincaré lemma in geometric quantisation
http://hdl.handle.net/2117/21165
Title: A Poincaré lemma in geometric quantisation
Authors: Miranda Galcerán, Eva; Solha, Romero
Abstract: This article presents a Poincar e lemma for the Kostant complex,
used to compute geometric quantisation, when the polarisation is given by
a Lagrangian foliation de ned by an integrable system with nondegenerate
singularities.2014-01-08T13:10:07ZDescription of characteristic non-hyperinvariant subspaces over the field GF(2)
http://hdl.handle.net/2117/21028
Title: Description of characteristic non-hyperinvariant subspaces over the field GF(2)
Authors: Mingueza, David; Montoro López, María Eulalia; Pacha Andújar, Juan Ramón
Abstract: Given a square matrix A, an A-invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T) that commute with A. Shodaʼs Theorem gives a necessary and sufficient condition for the existence of characteristic non-hyperinvariant subspaces for a nilpotent matrix in GF(2). Here we present an explicit construction for all subspaces of this type.2013-12-17T12:27:24ZDeterminant of a matrix that commutes with a Jordan matrix
http://hdl.handle.net/2117/21026
Title: Determinant of a matrix that commutes with a Jordan matrix
Authors: Montoro López, María Eulalia; Ferrer Llop, Josep; Mingueza, David
Abstract: Let F be an arbitrary field, Mn(F) the set of all matrices n×n over F and J∈Mn(F) a Jordan matrix. In this paper, we obtain an explicit formula for the determinant of any matrix that commutes with J, i.e., the determinant of any element T∈Z(J), the centralizer of J. Our result can also be extended to any T′∈Z(A), where A∈Mn(F), can be reduced to J=S−1AS. This is because T=S−1T′S∈Z(J), and clearly View the MathML source. If F is algebraically closed, any matrix A can be reduced in this way to a suitable J.
In order to achieve our main result, we use an alternative canonical form W∈Mn(F) called the Weyr canonical form. This canonical form has the advantage that all matrices K∈Z(W) are upper block triangular. The permutation similarity of T∈Z(J) and K∈Z(W) is exploited to obtain a formula for the determinant of T.
The paper is organized as follows: Section 2 contains some definitions and notations that will be used through all the paper. In Section 3, matrices T∈Z(J) are described and the determinant of T is computed in a particular case. In Section 4, we recall the Weyr canonical form W of a matrix and the corresponding centralizer Z(W). A formula to compute the determinant of any K∈Z(W) is rewritten. Finally, in Section 5 an explicit formula for the determinant of any T∈Z(J) is obtained.2013-12-17T12:18:56ZA Cartan-Eilenberg approach to homotopical algebra
http://hdl.handle.net/2117/20911
Title: A Cartan-Eilenberg approach to homotopical algebra
Authors: Guillén Santos, Francisco; Navarro Aznar, Vicente; Pascual Gainza, Pere; Roig Martí, Agustín
Abstract: In this paper we propose an approach to homotopical algebra w
here the basic ingredient
is a category with two classes of distinguished morphisms: s
trong and weak equivalences. These data
determine the cofibrant objects by an extension property ana
logous to the classical lifting property
of projective modules. We define a Cartan-Eilenberg categor
y as a category with strong and weak
equivalences such that there is an equivalence of categorie
s between its localisation with respect to
weak equivalences and the relative localisation of the subc
ategory of cofibrant objets with respect to
strong equivalences. This equivalence of categories allow
s us to extend the classical theory of derived
additive functors to this non additive setting. The main exa
mples include Quillen model categories
and categories of functors defined on a category endowed with
a cotriple (comonad) and taking values
on a category of complexes of an abelian category. In the latt
er case there are examples in which the
class of strong equivalences is not determined by a homotopy
relation. Among other applications of
our theory, we establish a very general acyclic models theor
em2013-12-04T12:58:02ZA note on the dynamics around the Lagrange collinear points of the Earth-Moon system in a complete Solar System model
http://hdl.handle.net/2117/20896
Title: A note on the dynamics around the Lagrange collinear points of the Earth-Moon system in a complete Solar System model
Authors: Lian, Yijun; Gómez Muntané, Gerard; Masdemont Soler, Josep; Tang, Guojian
Abstract: In this paper we study the dynamics of a massless particle around the L 1,2 libration points of the Earth–Moon system in a full Solar System gravitational model. The study is based on the analysis of the quasi-periodic solutions around the two collinear equilibrium points. For the analysis and computation of the quasi-periodic orbits, a new iterative algorithm is introduced which is a combination of a multiple shooting method with a refined Fourier analysis of the orbits computed with the multiple shooting. Using as initial seeds for the algorithm the libration point orbits of Circular Restricted Three Body Problem, determined by Lindstedt-Poincaré methods, the procedure is able to refine them in the Solar System force-field model for large time-spans, that cover most of the relevant Sun–Earth–Moon periods.2013-12-03T12:00:25Z