Ici nous étudions les extensions de feuilletage

We study the Hamiltonian structure of an important class of nonlinear partial differential equations: the so-called systems of hydrodynamic type, which are first-order in tempo-spatial variables, and quasi-linear. Chapters 1 and 2 constitute a review of background material, while Chapters 3, 4, 5 contain new results, with additional review sections as necessary. In Chapter 3 we demonstrate, via the Nijenhuis tensor, the integrability of a system of hydrodynamic type derived from the classical Volterra system. In Chapter 4, families of Hamiltonian structures of hydrodynamic type are constructed, as well as a gauge transform acting on Hamiltonian structures of hydrodynamic type. In Chapter 5, we present necessary and sufficient criteria for a three-component system of hydrodynamic type to be Hamiltonian, and classify the Lie-algebraic structures induced by a Hamiltonian structure for four-component systems of hydrodynamic type. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2010-12-23 11:35:41.976

In this thesis a number of problems concerning proper curvature collineations, proper Weyl collineations and projective vector fields will be considered. The work on the above areas can be summarised as: (i) A study of proper curvature collineations in plane symmetric static, spherically symmetric static and Bianchi type <I>I</I> spacetimes will be presented by considering the rank of their 6 x 6 Riemann tensors and using a theorem which eliminates those space-times where proper curvature collineations can not exist; (ii) A study of proper Weyl collineations is given by using the algebraic classification and associated rank of the Weyl tensor and using a theorem similar to that used in (i); (iii) A technique is developed to study projective vector fields in the Friedmann Robertson-Walker models and plane symmetric static spacetimes; (iv) The situations when conformally flat spacetimes admit proper curvature collineations are fully explored.

This thesis proposes four novel geometric neurodynamical classifier models, namely GBAM, Lie-derivative, Lie-Poisson, and FAM, applied to breast cancer detection. All these models have been published in a paper and/or in a book form. All theoretical material of this thesis (Chapter 2) has been published in my monographs (see my publication list), as follows: 2.1 Tensorial Neurodynamics has been published in Natural Biodynamics (Chapters 3, 5 and 7), Geometrical Dynamics of Complex Systems; (Chapter 1 and Appendix), 2006) as well as Applied Differential Geometry:A Modern Introduction(Chapter 3) 2.2 GBAM Neurodynamical Classifier has been published in Natural Biodynamics (Chapter 7) and Neuro-Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling (Chapter 3), as well as in the KES–Conference paper with the same title; 2.3 Lie-Derivative Neurodynamical Classifier has been published in Geometrical Dynamics of Complex Systems; (Chapter 1) and Applied Differential Geometry: A Modern Introduction (Chapter 3); 2.4 Lie-Poisson Neurodynamical Classifier has been published in Geometrical Dynamics of Complex Systems; (Chapter 1) and Applied Differential Geometry: A Modern Introduction (Chapter 3); 2.5 Fuzzy Associative Dynamical Classifier has been published in Neuro-Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling (Chapter 4), as well as in the KES-Conference paper with the same title. Besides, Section 1.2 Artificial Neural Networks has been published in Natural Biodynamics (Chapter 7) and Neuro-Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling (Chapter 3). Also, Sections 4.1. and 4.5. have partially been published in Neuro-Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling (Chapters 3 and 4, respectively) and in the corresponding KES–Conference papers. A. The GBAM (generalized bidirectional associative memory) classifier is a neurodynamical, tensor-invariant classifier based on Riemannian geometry. The GBAM is a tensor-field system resembling a two-phase biological neural oscillator in which an excitatory neural field excites an inhibitory neural field, which reciprocally inhibits the excitatory one. This is a new generalization of Kosko’s BAM neural network, with a new biological (oscillatory, i.e., excitatory/inhibitory)interpretation. The model includes two nonlinearly-coupled (yet non-chaotic and Lyapunov stable) subsystems, activation dynamics and self-organized learning dynamics, including a symmetric synaptic 2-dimensional tensor-field, updated by differential Hebbian associative learning innovations. Biologically, the GBAM describes interacting excitatory and inhibitory populations of neurons found in the cerebellum, olfactory cortex, and neocortex, all representing the basic mechanisms for the generation of oscillating (EEG-monitored) activity in the brain. B. Lie-derivative neurodynamical classifier is an associative-memory, tensor-invariant neuro-classifier, based on the Lie-derivative operator from geometry of smooth manifolds. C. Lie-Poisson neurodynamical classifier is an associative-memory, tensor-invariant neuro-classifier based on the Lie-Poisson bracket from the generalized symplectic geometry. D. The FAM-matrix (fuzzy associative memory) dynamical classifier is a fuzzy-logic classifier based on a FAM-matrix (fuzzy phase-plane). All models are formulated and simulated in Mathematica computer algebra system. All models are applied to breast cancer detection, using the database from the University of Wisconsin and Mammography database. Classification results outperformed those obtained with standard MLP trained with backpropagation algorithm. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2008

Thesis (Ph.D.)--University of Adelaide, School of Mathematical Sciences, Discipline of Pure Mathematics, 2004. / "Submitted September 2002 ... Amended September 2004." Bibliography: p. 152-157.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisors: Willy Fischler and Yuval Neʼeman. Vita. Includes bibliographical references.

Differential geometry is deceptively simple. It is surprisingly easyto get the right answer with unclear and informal symbol manipulation.To address this problem we use computer programs to communicate aprecise understanding of the computations in differential geometry.Expressing the methods of differential geometry in a computer languageforces them to be unambiguous and computationally effective. The taskof formulating a method as a computer-executable program and debuggingthat program is a powerful exercise in the learning process. Also,once formalized procedurally, a mathematical idea becomes a tool thatcan be used directly to compute results.

In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constraint equations for near-constant-mean-curvature (near-CMC) data as well as for far-from-CMC data, a proof of the limit equation criterion in the near-CMC case, as well as a model problem on the relationship between the asymptotic constants of solutions and the ADM mass. We also prove a characterization of the Yamabe classes on asymptotically Euclidean manifolds and resolve the (conformally) prescribed scalar curvature problem on asymptotically Euclidean manifolds for the case of nonpositive scalar curvatures. This dissertation includes previously published coauthored material.

In this thesis we study near-horizon geometries of extreme black holes. We first consider stationary extreme black hole solutions to the Einstein-Yang-Mills theory with a compact semi-simple gauge group in four dimensions, allowing for a negative cosmological constant. We prove that any axisymmetric black hole of this kind possesses a near-horizon AdS2 symmetry and deduce its near-horizon geometry must be that of the abelian embedded extreme Kerr-Newman (AdS) black hole. We show that the near-horizon geometry of any static black hole is a direct product of AdS2 and a constant curvature space. We then consider near-horizon geometry in Einstein gravity coupled to a Maxwell field and a massive complex scalar field, with a cosmological constant. We prove that assuming non-zero coupling between the Maxwell and the scalar fields, there exists no solution with a compact horizon in any dimensions where the massive scalar is non-trivial. This result generalises to any scalar potential which is a monotonically increasing function of the modulus of the complex scalar. Next we determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS3 extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these as descendants of the extreme BTZ black hole. However descendants of the non-extreme BTZ black hole are absent from our general solution with a non-degenerate horizon. We then show that the first order deformation along transverse null geodesics about any near-horizon geometry with compact cross-sections always admits a finite-parameter family of solutions as the most general solution. As an application, we consider the first order expansion from the near-horizon geometry of the extreme Kerr black hole. We uncover a local uniqueness theorem by demonstrating that the only possible black hole solutions which admit a U(1) symmetry are gauge equivalent to the first order expansion of the extreme Kerr solution itself. We then investigate the first order expansion from the near-horizon geometry of the extreme self-dual Myers-Perry black hole in 5D. The only solutions which inherit the enhanced SU(2) X U(1) symmetry and are compatible with black holes correspond to the first order expansion of the extreme self-dual Myers-Perry black hole itself and the extreme J = 0 Kaluza-Klein black hole. These are the only known black holes to possess this near-horizon geometry. If only U(1) X U(1) symmetry is assumed in first order, we find that the most general solution is a three-parameter family which is more general than the two known black hole solutions. This hints the possibility of the existence of new black holes.

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