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>

This thesis deals with non-linear non-quadratic optimal control problems in an autonomous system and a related iterative numerical method, the Kleinman-Newton method, for solving the problem. The thesis proves the local convergence of Kleinman-Newton method using the contraction mapping theorem and then describes how this Kleinman-Newton method may be used to numerically solve for the optimal control and the corresponding solution. In order to show the proof and the related numerical work, it is necessary to review some of earlier work in the beginning of Chapter 1 [Zhang], and to introduce the Kleinman-Newton method at the end of the chapter. In Chapter 2 we will demonstrate the proof. In Chapter 3 we will show the related numerical work and results. / Ph. D.

This work evaluates whether the branch of Reynolds Averaging in Computational Fluid Dynamics can be used to, based on real field measurements, find the source of the measured gas in question. The method to do this is via the adjoint to the Reynolds Averaged Scalar Transport equation, explained and derived herein. Since the Inverse is only as good as the main equation, forward runs are made to evaluate the turbulence model. Reynolds Averaged Navier Stokes is solved in a domain containing 4 cubes in a 2x2 grid, generating a velocity field for said domain. The turbulence model in question is a union of two modifications to the standard two equation k-ε model in order to capture blunt body turbulence but also to model the atmospheric boundary layer. This field is then inserted into the Reynolds Averaged Scalar Transport equation and the simulation is compared to data from the Environmental Flow wind tunnel in Surrey. Finally the adjoint scalar transport is solved, both for synthetic data that was generated in the forward run, but also for the data from EnFlo. It was discovered that the turbulent Schmidt number plays a major role in capturing the dispersed gas, three different Schmidt numbers were tested, the standard 0.7, the unconventional 0.3 and a height dependent Schmidt number. The widely accepted value of 0.7 did not capture the dispersion at all and gave a huge model error. As such the adjoint scalar transport was solved for 0.3 and a height dependent Schmidt number. The interaction between measurements, the real source strength (which is not used in the adjoint equation, but needed to find the source) and the location of the source is intricate indeed. Over estimation and under estimation of the forward model may cancel out in order to find the correct source, with the correct strength. It is found that Reynolds Averaged Computational fluid dynamics may prove useful in source term estimation. / Detta arbete utvärderar hurvida Reynolds medelvärdesmodellering inom flödessimuleringar kan användas till att finna källan till en viss gas baserat på verkliga mätningar ute i fält. Metoden går ut på att använda den adjungerade ekvationen till Reynolds tidsmedlade skalära transportekvationen, beskriven och härledd häri. Då bakåtmodellen bygger på framåtmodellen, måste såleds framåtmodellen utvärderas först. Navier-Stokes ekvationer med en turbulensmodell löses i en domän, innehållandes 4 kuber i en 2x2 orientering, för vilken en hastighetsprofil erhålles. Turbulensmodellen som användes är en union av två olika k-ε modeller, där den ena fångar turbulens runt tröga objekt och den andra som modellerar atmosfäriska gränsskiktet. Detta fält används sedan i framåtmodellen av skalära transportekvationen, som sedan jämförs med körningar från EnFlo windtunneln i Surrey. Slutligen testkörs även den adjungerade ekvationen, både för syntetiskt data genererat i framåtkörningen men även för data från EnFlo tunneln. Då det visade sig att det turbulenta Schmidttalet spelar stor roll inom spridning i det atmosfäriska gränsskiktet, gjordes testkörningar med tre olika Schmidttal, det normala 0.7, det väldigt låga talet 0.3 samt ett höjdberoende Schmidttal. Det visade sig att det vanligtvis använda talet 0.7 inte alls lyckas fånga spridningen tillfredställande och gav ett stort modellfel. Därför löstes den adjungerade ekvationen för 0.3 samt för ett höjdberoende Schmidttal. Interaktionen mellan mätningar, den riktiga källstyrkan (som är okänd i den adjungerade ekvationen) samt källpositionen är onekligen intrikat. Över- samt underestimationer av framåtmodellen kan ta ut varandra i bakåtmodellen för att finna rätt källa, med rätt källstyrka. Det ter sig som Reynolds turbulensmodellering mycket möjligt kan användas inom källtermsuppskattning.

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.

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此篇論文，主要是討論遞迴式神經網路。在文中，我們將架構一個單層的神經網路結構。並利用三種不同的學習法則來套用此架構。我們也做了圓軌跡和圖形８的模擬，以及討論了此架構的收斂性。 / Our task in this paper is to discuss the Recurrent Neural Network. We construct a singal layer neural network and apply three different learning rules to simulate circular trajectory and figure eight. Also, we present the proof of convergence.