A geometrical analysis of the field equations in field theory

In this review paper we give a geometrical formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and non-uniqueness of solutions, as well as their integrability.Comment: 14 pages. LaTeX file. This is a review paper based on previous works by the same author

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

Very Rare Complementation between Mitochondria Carrying Different Mitochondrial DNA Mutations Points to Intrinsic Genetic Autonomy of the Organelles in Cultured Human Cells

In the present work, a large scale investigation was done regarding the capacity of cultured human cell lines (carrying in homoplasmic form either the mitochondrial tRNALys A8344G mutation associated with the myoclonic epilepsy and ragged red fiber (MERRF) encephalomyopathy or a frameshift mutation, isolated in vitro, in the gene for the ND4 subunit of NADH dehydrogenase) to undergo transcomplementation of their recessive mitochondrial DNA (mtDNA) mutations after cell fusion. The presence of appropriate nuclear drug resistance markers in the two cell lines allowed measurements of the frequency of cell fusion in glucose-containing medium, non-selective for respiratory capacity, whereas the frequency of transcomplementation of the two mtDNA mutations was determined by growing the same cell fusion mixture in galactose-containing medium, selective for respiratory competence. Transcomplementation of the two mutations was revealed by the re-establishment of normal mitochondrial protein synthesis and respiratory activity and by the relative rates synthesis of two isoforms of the ND3 subunit of NADH dehydrogenase. The results of several experiments showed a cell fusion frequency between 1.4 and 3.4% and an absolute transcomplementation frequency that varied between 1.2 × 10^-5 and 5.5 × 10^-4. Thus, only 0.3-1.6% of the fusion products exhibited transcomplementation of the two mutations. These rare transcomplementing clones were very sluggish in developing, grew very slowly thereafter, and showed a substantial rate of cell death (22-28%). The present results strongly support the conclusion that the capacity of mitochondria to fuse and mix their contents is not a general intrinsic property of these organelles in mammalian cells, although it may become activated in some developmental or physiological situations

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

Meiosis in a Bottle : New Approaches to Overcome Mammalian Meiocyte Study Limitations

The study of meiosis is limited because of the intrinsic nature of gametogenesis in mammals. One way to overcome these limitations would be the use of culture systems that would allow meiotic progression in vitro. There have been some attempts to culture mammalian meiocytes in recent years. In this review we will summarize all the efforts to-date in order to culture mammalian sperm and oocyte precursor cells