Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and non-uniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentum bundle. Furthermore, the existence of first integrals of these Hamiltonian equations is considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl general symmetries} of the system is discussed. Noether's theorem is also stated in this context, both the ``classical'' version and its generalization to include higher-order Cartan-Noether symmetries. Finally, the equivalence between the Lagrangian and Hamiltonian formalisms is also discussed.Comment: Some minor mistakes are corrected. Bibliography is updated. To be published in J. Phys. A: Mathematical and Genera

Symplectic Cuts and Projection Quantization

The recently proposed projection quantization, which is a method to quantize particular subspaces of systems with known quantum theory, is shown to yield a genuine quantization in several cases. This may be inferred from exact results established within symplectic cutting.Comment: 12 pages, v2: additional examples and a new reference to related wor

On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories

The objective of this work is twofold: First, we analyze the relation between the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between k-symplectic field theories and the so-called autonomous k-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).Comment: 25 page

Invariant Forms and Automorphisms of Locally Homogeneous Multisymplectic Manifolds

It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree > 1, which is locally homogeneous of degree k with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms, and on the study of the local properties of Hamiltonian vector fields on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.Comment: 25 p.; LaTeX file. The paper has been partially rewritten. Some terminology has been changed. The proof of some theorems and lemmas have been revised. The title and the abstract are slightly modified. An appendix is added. The bibliography is update

Non-standard connections in classical mechanics

In the jet-bundle description of first-order classical field theories there are some elements, such as the lagrangian energy and the construction of the hamiltonian formalism, which require the prior choice of a connection. Bearing these facts in mind, we analyze the situation in the jet-bundle description of time-dependent classical mechanics. So we prove that this connection-dependence also occurs in this case, although it is usually hidden by the use of the ``natural'' connection given by the trivial bundle structure of the phase spaces in consideration. However, we also prove that this dependence is dynamically irrelevant, except where the dynamical variation of the energy is concerned. In addition, the relationship between first integrals and connections is shown for a large enough class of lagrangians.Comment: 17 pages, Latex fil

Agronomic Evaluation of Twenty Ecotypes of \u3cem\u3eLeucaena\u3c/em\u3e spp. for Acid Soil Conditions in México

Leucaena leucocephala Lam. (de Witt) has been shown to be a good forage producer and to posses good persistence under grazing conditions in México tolerating well the management of local cattlemen (Quero et al., 2004). The Leucaena genus is native to Central America and Mexico (Hughes, 1998), but L. leucocephala is a low producer under acid soil conditions. The natural diversity is a good source of resistance to acid soil conditions resistance and to other adverse factors. Several Leucaena accessions were evaluated for production under acid soil conditions in tropical Mexico

Gene expression parallels synaptic excitability and plasticity changes in Alzheimer's disease

Altres ajuts: CIBERNED CB06/05/0042 i BrightFocus Foundation (A2014417S)Alzheimer's disease (AD) is a neurodegenerative disorder characterized by abnormal accumulation of β-amyloid and tau and synapse dysfunction in memory-related neural circuits. Pathological and functional changes in the medial temporal lobe, a region essential for explicit memory encoding, contribute to cognitive decline in AD. Surprisingly, functional imaging studies show increased activity of the hippocampus and associated cortical regions during memory tasks in presymptomatic and early AD stages, whereas brain activity declines as the disease progresses. These findings suggest an emerging scenario where early pathogenic events might increase neuronal excitability leading to enhanced brain activity before clinical manifestations of the disease, a stage that is followed by decreased brain activity as neurodegeneration progresses. The mechanisms linking pathology with synaptic excitability and plasticity changes leading to memory loss in AD remain largely unclear. Recent studies suggest that increased brain activity parallels enhanced expression of genes involved in synaptic transmission and plasticity in preclinical stages, whereas expression of synaptic and activity-dependent genes are reduced by the onset of pathological and cognitive symptoms. Here, we review recent evidences indicating a relationship between transcriptional deregulation of synaptic genes and neuronal activity and memory loss in AD and mouse models. These findings provide the basis for potential clinical applications of memory-related transcriptional programs and their regulatory mechanisms as novel biomarkers and therapeutic targets to restore brain function in AD and other cognitive disorders

Properties of Multisymplectic Manifolds

This lecture is devoted to review some of the main properties of multisymplectic geometry. In particular, after reminding the standard definition of multisymplectic manifold, we introduce its characteristic submanifolds, the canonical models, and other relevant kinds of multisymplectic manifolds, such as those where the existence of Darboux-type coordinates is assured. The Hamiltonian structures that can be defined in these manifolds are also studied, as well as other important properties, such as their invariant forms and the characterization by automorphisms.Comment: 10 pp. Changes in Sections 5 and 7 (where brief guides to the proofs of theorems have been added). Lecture given at the workshop on {\sl Classical and Quantum Physics: Geometry, Dynamics and Control. (60 Years Alberto Ibort Fest), Instituto de Ciencias Matem\'aticas (ICMAT)}, Madrid (Spain), 5--9 March 201