32 research outputs found
Symmetry breaking in the periodic Thomas--Fermi--Dirac--von Weizs{\"a}cker model
We consider the Thomas--Fermi--Dirac--von~Weizs{\"a}cker model for a system
composed of infinitely many nuclei placed on a periodic lattice and electrons
with a periodic density. We prove that if the Dirac constant is small enough,
the electrons have the same periodicity as the nuclei. On the other hand if the
Dirac constant is large enough, the 2-periodic electronic minimizer is not
1-periodic, hence symmetry breaking occurs. We analyze in detail the behavior
of the electrons when the Dirac constant tends to infinity and show that the
electrons all concentrate around exactly one of the 8 nuclei of the unit cell
of size 2, which is the explanation of the breaking of symmetry. Zooming at
this point, the electronic density solves an effective nonlinear Schr\"odinger
equation in the whole space with nonlinearity . Our results
rely on the analysis of this nonlinear equation, in particular on the
uniqueness and non-degeneracy of positive solutions
On uniqueness and non-degeneracy of anisotropic polarons
We study the anisotropic Choquard--Pekar equation which de-scribes a polaron
in an anisotropic medium. We prove the uniqueness and non-degeneracy of
minimizers in a weakly anisotropic medium. In addition, for a wide range of
anisotropic media, we derive the symmetry properties of minimizers and prove
that the kernel of the associated linearized operator is reduced, apart from
three functions coming from the translation invariance, to the kernel on the
subspace of functions that are even in each of the three principal directions
of the medium
Results on the spectral stability of standing wave solutions of the Soler model in 1-D
We study the spectral stability of the nonlinear Dirac operator in dimension
1+1, restricting our attention to nonlinearities of the form
. We obtain bounds
on eigenvalues for the linearized operator around standing wave solutions of
the form . For the case of power nonlinearities , , we obtain a range of frequencies such that the
linearized operator has no unstable eigenvalues on the axes of the complex
plane. As a crucial part of the proofs, we obtain a detailed description of the
spectra of the self-adjoint blocks in the linearized operator. In particular,
we show that the condition characterizes groundstates analogously to the Schr\"{o}dinger case.Comment: 46 pages, 3 figure
Sugarcane (Saccharum X officinarum): A Reference Study for the Regulation of Genetically Modified Cultivars in Brazil
Global interest in sugarcane has increased significantly in recent years due to its economic impact on sustainable energy production. Sugarcane breeding and better agronomic practices have contributed to a huge increase in sugarcane yield in the last 30Â years. Additional increases in sugarcane yield are expected to result from the use of biotechnology tools in the near future. Genetically modified (GM) sugarcane that incorporates genes to increase resistance to biotic and abiotic stresses could play a major role in achieving this goal. However, to bring GM sugarcane to the market, it is necessary to follow a regulatory process that will evaluate the environmental and health impacts of this crop. The regulatory review process is usually accomplished through a comparison of the biology and composition of the GM cultivar and a non-GM counterpart. This review intends to provide information on non-GM sugarcane biology, genetics, breeding, agronomic management, processing, products and byproducts, as well as the current technologies used to develop GM sugarcane, with the aim of assisting regulators in the decision-making process regarding the commercial release of GM sugarcane cultivars
SymeÌtrie et brisure de symeÌtrie pour certains probleÌmes non lineÌaires
This thesis is devoted to the mathematical study of two quantum systems described by nonlinear models: the anisotropic polaron and the electrons in a periodic crystal. We first prove the existence of minimizers, and then discuss the question of uniqueness for both problems. In the first part, we show the uniqueness and nondegeneracy of the minimizer for the polaron, described by the Choquard--Pekar anisotropic equation, assuming that the dielectric matrix of the medium is almost isotropic. In the strong anisotropic setting, we leave the question of uniqueness open but identify the symmetry that can possibly be degenerate. In the second part, we study the electrons of a crystal in the periodic Thomas--Fermi--Dirac--Von~WeizsĂ€cker model, varying the parameter in front of the Dirac term. We show uniqueness and nondegeneracy of the minimizer when this parameter is small enough et prove the occurrence of symmetry breaking when it is large.Cette thĂšse est consacrĂ©e Ă l'Ă©tude mathĂ©matique de deux systĂšmes quantiques dĂ©crits par des modĂšles non linĂ©aires : le polaron anisotrope et les Ă©lectrons d'un cristal pĂ©riodique. AprĂšs avoir prouvĂ© l'existence de minimiseurs, nous nous intĂ©ressons Ă la question de l'unicitĂ© pour chacun des deux modĂšles. Dans une premiĂšre partie, nous montrons l'unicitĂ© du minimiseur et sa non-dĂ©gĂ©nĂ©rescence pour le polaron dĂ©crit par l'Ă©quation de Choquard--Pekar anisotrope, sous la condition que la matrice diĂ©lectrique du milieu est presque isotrope. Dans le cas d'une forte anisotropie, nous laissons la question de l'unicitĂ© en suspens mais caractĂ©risons prĂ©cisĂ©ment les symĂ©tries pouvant ĂȘtre dĂ©gĂ©nĂ©rĂ©es. Dans une seconde partie, nous Ă©tudions les Ă©lectrons d'un cristal dans le modĂšle de Thomas--Fermi--Dirac--Von~WeizsĂ€cker pĂ©riodique, en faisant varier le paramĂštre devant le terme de Dirac. Nous montrons l'unicitĂ© et la non-dĂ©gĂ©nĂ©rescence du minimiseur lorsque ce paramĂštre est suffisamment petit et mettons en Ă©vidence une brisure de symĂ©trie lorsque celui-ci est grand
Symmetry and symmetry breaking for some nonlinear problems
Cette thĂšse est consacrĂ©e Ă l'Ă©tude mathĂ©matique de deux systĂšmes quantiques dĂ©crits par des modĂšles non linĂ©aires : le polaron anisotrope et les Ă©lectrons d'un cristal pĂ©riodique. AprĂšs avoir prouvĂ© l'existence de minimiseurs, nous nous intĂ©ressons Ă la question de l'unicitĂ© pour chacun des deux modĂšles. Dans une premiĂšre partie, nous montrons l'unicitĂ© du minimiseur et sa non-dĂ©gĂ©nĂ©rescence pour le polaron dĂ©crit par l'Ă©quation de Choquard--Pekar anisotrope, sous la condition que la matrice diĂ©lectrique du milieu est presque isotrope. Dans le cas d'une forte anisotropie, nous laissons la question de l'unicitĂ© en suspens mais caractĂ©risons prĂ©cisĂ©ment les symĂ©tries pouvant ĂȘtre dĂ©gĂ©nĂ©rĂ©es. Dans une seconde partie, nous Ă©tudions les Ă©lectrons d'un cristal dans le modĂšle de Thomas--Fermi--Dirac--Von~WeizsĂ€cker pĂ©riodique, en faisant varier le paramĂštre devant le terme de Dirac. Nous montrons l'unicitĂ© et la non-dĂ©gĂ©nĂ©rescence du minimiseur lorsque ce paramĂštre est suffisamment petit et mettons en Ă©vidence une brisure de symĂ©trie lorsque celui-ci est grand.This thesis is devoted to the mathematical study of two quantum systems described by nonlinear models: the anisotropic polaron and the electrons in a periodic crystal. We first prove the existence of minimizers, and then discuss the question of uniqueness for both problems. In the first part, we show the uniqueness and nondegeneracy of the minimizer for the polaron, described by the Choquard--Pekar anisotropic equation, assuming that the dielectric matrix of the medium is almost isotropic. In the strong anisotropic setting, we leave the question of uniqueness open but identify the symmetry that can possibly be degenerate. In the second part, we study the electrons of a crystal in the periodic Thomas--Fermi--Dirac--Von~WeizsĂ€cker model, varying the parameter in front of the Dirac term. We show uniqueness and nondegeneracy of the minimizer when this parameter is small enough et prove the occurrence of symmetry breaking when it is large
On uniqueness and non-degeneracy of anisotropic polarons
We study the anisotropic ChoquardâPekar equation which de-scribes a polaron in an anisotropic medium. We prove the uniqueness and non-degeneracy of minimizers in a weakly anisotropic medium. In addition, for a wide range of anisotropic media, we derive the symmetry properties of minimizers and prove that the kernel of the associated linearized operator is reduced, apart from three functions coming from the translation invariance, to the kernel on the subspace of functions that are even in each of the three principal directions of the medium.nonnonouirechercheInternationa
On one-dimensional Bose gases with two- and (critical) attractive three-body interactions
We consider a one-dimensional, trapped, focusing Bose gas where bosons interact with each other via both a two-body interaction potential of the form and an attractive three-body interaction potential of the form , where , , is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence is "faster" than the slow enough convergence
On one-dimensional Bose gases with two- and (critical) attractive three-body interactions
We consider a one-dimensional, trapped, focusing Bose gas where bosons interact with each other via both a two-body interaction potential of the form and an attractive three-body interaction potential of the form , where , , is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence is "faster" than the slow enough convergence
On one-dimensional Bose gases with two- and (critical) attractive three-body interactions
We consider a one-dimensional, trapped, focusing Bose gas where bosons interact with each other via both a two-body interaction potential of the form and an attractive three-body interaction potential of the form , where , , is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence is "faster" than the slow enough convergence