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Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed. / <p>Thesis for the degree of Doctor of Philosophy</p>

Orientador: Yuri Dimitrov Bozhkov / Tese (doutorado) ¿ Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T22:59:33Z (GMT). No. of bitstreams: 1 Silva_KenioAlexsomdeAlmeida_D.pdf: 5129062 bytes, checksum: 0bae8b75b0ea90b8799bc1dd7496d766 (MD5) Previous issue date: 2013 / Resumo: Nesta tese estudamos o conceito novo de equações diferenciais não - linearmente auto-adjuntas para duas classes gerais de equações evolutivas de segunda ordem quase lineares. Uma vez que essas equações não provêm de um problema variacional, não podemos obter leis de conservação via o Teorema de Noether. Por isto aplicamos tal conceito e o Novo Teorema sobre Leis de Conservação de Nail H. Ibragimov, o qual possibilita-nos a determinação de leis de conservação para qualquer equação diferencial. Obtivemos em ambas as classes, equações não - linearmente auto-adjuntos e leis de conservação para alguns casos particularmente importantes: a) as equações do fluxo de Ricci geométrico, do fluxo de Ricci 2D, do fluxo de Ricci modificada e a equação do calor não-linear, na primeira classe; b) as equações do fluxo geométrico hiperbólico e do fluxo geométrica hiperbólica modificada, na segunda classe de equações evolutivas / Abstract: In this thesis we study the new concept of nonlinear self-adjoint deferential equations for two general classes of quasilinear 2D second order evolution equations. Since these equations do not come from a variational problem, we cannot obtain conservation laws via the Noether's Theorem. Therefore we apply this concept and the New Conservation Theorem of Nail H. Ibragimov, which enables one to establish the conservation laws for any deferential equation. We obtain in classes, nonlinear self-adjoint equations and conservation laws for important particular cases: a) the Ricci flow geometric equation, Ricci flow 2D equation, the modified Ricci flow equation and the nonlinear heat equation in the first class; b) the hyperbolic geometric flow equation and the modified hyperbolic geometric flow equation in the second class of evolution equations / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

This thesis makes contributions to the solution of Hadamard's problem through an examination of the question of the validity of Huygens'principle for the non-self-adjoint scalar wave equation on a Petrov type D spacetime. The problem is split into five further sub-cases based on the alignment of the Maxwell and Weyl principal spinors of the underlying spacetime. Two of these sub-cases are considered, one of which is proved to be incompatible with Huygens' principle, while for the other, it is shown that Huygens' principle implies that the two principal null congruences of the Weyl tensor are geodesic and shear-free. Furthermore, an unpublished result of McLenaghan regarding symmetric spacetimes of Petrov type D is independently verified. This result suggests the possible existence of counter-examples of the Carminati-McLenaghan conjecture.

Nessa tese, mostramos que a Hamiltoniana H e o operador helicidade de uma partícula de Dirac que se movimenta em duas dimensões na presença de um tubo de fluxo magnético infinitamente fino na origem admitem, cada um, uma família de quatro parâmetros de extensões auto-adjuntas. Para cada extensão correspondem condições de contorno a serem satisfeitas pelas auto-fuções na origem. Apesar dos operadores H e formalmente comutarem antes da especificação das condições de contorno, para garantirmos a conservação da helicidade, não é suficiente obtermos as mesmas condições de contorno para ambos os operadores, ou seja, não é suficiente a determinação de um domínio comum a ambos. Mostramos que, para certas relações entre os parâmetros das extensões satisfeitas, é possível a determinação dos domínios mais gerais onde ambos os operadores H e são auto-adjuntos e onde a helicidade é conservada, simultaneamente com a preservação da simetria de Aharonov-Bohm ( + 1), onde é o fluxo magnético em unidades naturais. Nossos resultados implicam que, nem a conservação da helicidade nem a simetria de Aharonov-Bohn, resolvem o problema da escolha da condição de contorno fisicamente correta. / We show that both the Hamiltonian H and the helicity operator of a Dirac particle moving in two dimension in the presence of an infinitely thin magnetic flux tube admit each a four- parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BC\'s) to be satisfied by the eigenfunctions at the origin. Althou- gh the actions af these two operators commute before specification of boundary conditions, to ensure helicity conservation it is not sufficient to take the same BC\'s for both operators. We show that, given certain relations between the parameters of the extensions it is possible to write down the most general domain where both operators H and are self-adjoint with heli- city conservation and also Aharonov-Bohm symmetry ( + 1) preserved, where is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results im- ply that neither helicity conservation nor Aharonov-Bohm symmetry by themselves solves the problem of choosing the \"physical \"boundary conditions for this system.

We study several unbounded operators with view to extending von Neumann's theory of deficiency indices for single Hermitian operators with dense domain in Hilbert space. If the operators are non-commuting, the problems are difficult, but special cases may be understood with the use representation theory. We will further study the partial derivative operators in the coordinate directions on the L2 space on various covering surfaces of the punctured plane. The operators are defined on the common dense domain of C∞ functions with compact support, and they separately are essentially selfadjoint, but the unique selfadjoint extensions will be non-commuting. This problem is of a geometric flavor, and we study an index formulation for its solution. The applications include the study of vector fields, the theory of Dirichlet problems for second order partial differential operators (PDOs), Sturm-Liouville problems, H.Weyl's limit-point/limit-circle theory, Schrödinger equations, and more.

In any research program we begin with a hypothesis and when our expected results do not concur with the observed results we must try and understand the dynamics behind the changed process. In this study we were trying to understand the flux between regional groundwater systems, surface waters and sedimentation processes in order to predict the fate of contaminants entering one of the larger bodies of water in the world- Lake Ontario. This lake has increased levels of tritium due to anthropogenic inputs. Our first approach to the problem was to look at tritium fluxes within the system . Hydrological balances were constructed and a series of sediment cores were taken longitudinally and laterally across the lake. The second approach was to quantify the sediment accumulation rate (SAR) within the depositional basins and zones of erosion in order to improve the linkage between erosion control (sedimentation) and the water quality program. In the last chapter the movement of tritium, by molecular diffusion, through the clayey-silts of Lake Ontario is quantified in terms of an effective diffusion coefficient. In these sediments effective diffusion equals molecular diffusion. In a laboratory experiment four cores of lake sediment were spiked with tritium . The resulting concentration gradient changes in the sediment porewaters after six weeks could be modeled by an analytical one- dimensional diffusive transport equation. Results calculated the average lab diffusion coefficient to be 2. 7 x 10 - 5cm 2. sec -1 which is twice that determined by Wang et al, 1952 but still reasonable. Short cores (50 cm) from lake Ontario had observed tritium concentrations with depth that reflected a variable diffusive profile. The increases and decreases in tritium with depth could be correlated between cores. Monthly tritium emission data was obtained and correlations between peaks in the tritium profile and emissions were observed. Monthly variations in release emissions corresponded to approximately a one centimeter slice of core. An average calculated diffusion coefficient of theses cores was 1. 0 x 10 -5 cm 2. sec -1 which compares to Wang's coefficient of 1. 39 x 10 -5 cm 2. sec -1. This implies that tritium is moving through the sediment column at a rate equal to diffusion. The results were obtained for smoothed values. It was not possible to model the perturbations of the data with a one dimensional model. The dynamics of the system imply that tritium could be used as a biomonitor for reactor emissions, mixing time and current direction scenarios and that a better understanding of this process could be gained by future coring studies and a new hypothesis.

This thesis makes contributions to the solution of Hadamard's problem through an examination of the question of the validity of Huygens'principle for the non-self-adjoint scalar wave equation on a Petrov type D spacetime. The problem is split into five further sub-cases based on the alignment of the Maxwell and Weyl principal spinors of the underlying spacetime. Two of these sub-cases are considered, one of which is proved to be incompatible with Huygens' principle, while for the other, it is shown that Huygens' principle implies that the two principal null congruences of the Weyl tensor are geodesic and shear-free. Furthermore, an unpublished result of McLenaghan regarding symmetric spacetimes of Petrov type D is independently verified. This result suggests the possible existence of counter-examples of the Carminati-McLenaghan conjecture.

In any research program we begin with a hypothesis and when our expected results do not concur with the observed results we must try and understand the dynamics behind the changed process. In this study we were trying to understand the flux between regional groundwater systems, surface waters and sedimentation processes in order to predict the fate of contaminants entering one of the larger bodies of water in the world- Lake Ontario. This lake has increased levels of tritium due to anthropogenic inputs. Our first approach to the problem was to look at tritium fluxes within the system . Hydrological balances were constructed and a series of sediment cores were taken longitudinally and laterally across the lake. The second approach was to quantify the sediment accumulation rate (SAR) within the depositional basins and zones of erosion in order to improve the linkage between erosion control (sedimentation) and the water quality program. In the last chapter the movement of tritium, by molecular diffusion, through the clayey-silts of Lake Ontario is quantified in terms of an effective diffusion coefficient. In these sediments effective diffusion equals molecular diffusion. In a laboratory experiment four cores of lake sediment were spiked with tritium . The resulting concentration gradient changes in the sediment porewaters after six weeks could be modeled by an analytical one- dimensional diffusive transport equation. Results calculated the average lab diffusion coefficient to be 2. 7 x 10 - 5cm 2. sec -1 which is twice that determined by Wang et al, 1952 but still reasonable. Short cores (50 cm) from lake Ontario had observed tritium concentrations with depth that reflected a variable diffusive profile. The increases and decreases in tritium with depth could be correlated between cores. Monthly tritium emission data was obtained and correlations between peaks in the tritium profile and emissions were observed. Monthly variations in release emissions corresponded to approximately a one centimeter slice of core. An average calculated diffusion coefficient of theses cores was 1. 0 x 10 -5 cm 2. sec -1 which compares to Wang's coefficient of 1. 39 x 10 -5 cm 2. sec -1. This implies that tritium is moving through the sediment column at a rate equal to diffusion. The results were obtained for smoothed values. It was not possible to model the perturbations of the data with a one dimensional model. The dynamics of the system imply that tritium could be used as a biomonitor for reactor emissions, mixing time and current direction scenarios and that a better understanding of this process could be gained by future coring studies and a new hypothesis.

Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions.