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1
Fialowski, Alice, Michael Penkava, and fialowsk@cs elte hu. "Deformation Theory of Infinity Algebras." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi906.ps.
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Yaseen, Hogar M. "Generalized root graded Lie algebras." Thesis, University of Leicester, 2018. http://hdl.handle.net/2381/42765.
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Let g be a non-zero finite-dimensional split semisimple Lie algebra with root system Δ. Let Γ be a finite set of integral weights of g containing Δ and {0}. We say that a Lie algebra L over F is generalized root graded, or more exactly (Γ,g)-graded, if L contains a semisimple subalgebra isomorphic to g, the g-module L is the direct sum of its weight subspaces Lα (α ∈ Γ) and L is generated by all Lα with α ̸= 0 as a Lie algebra. If g is the split simple Lie algebra and Γ = Δ∪{0} then L is said to be root-graded. Let g∼= sln and Θn = {0,±εi±ε j,±εi,±2εi | 1 ≤ i ̸= j ≤ n} where {ε1, . . . , εn} is the set of weights of the natural sln-module. Then a Lie algebra L is (Θn,g)-graded if and only if L is generated by g as an ideal and the g-module L decomposes into copies of the adjoint module, the natural module V, its symmetric and exterior squares S2V and ∧2V, their duals and the one dimensional trivial g-module. In this thesis we study properties of generalized root graded Lie algebras and focus our attention on (Θn, sln)-graded Lie algebras. We describe the multiplicative structures and the coordinate algebras of (Θn, sln)-graded Lie algebras, classify these Lie algebras and determine their central extensions.
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at, Andreas Cap@esi ac. "Graded Lie Algebras and Dynamical Systems." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1086.ps.
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Yang, Qunfeng. "Some graded Lie algebra structures associated with Lie algebras and Lie algebroids." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0007/NQ41350.pdf.
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Bagnoli, Lucia. "Z-graded Lie superalgebras." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/14118/.
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This thesis investigates the role of filtrations and gradings in the study of Lie superalgebras. The main connections between the Z-grading of a Lie superalgebra and its structure are explained. As an example, the simplicity of the Lie superalgebras W(m,n) and S(m,n) is proved. Finally, the strongly symmetric gradings of length three and five of the Lie superalgebras W(m,n) and S(m,n) are classified.
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Pauksztello, David. "Homological properties of differential graded algebras." Thesis, University of Leeds, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.493288.
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In this thesis we consider various homological properties of differential graded algebras, and more generally, properties of arbitrary triangulated categories which have set indexed coproducts. A major example of such a triangulated category is the derived category of a differential graded algebra. We present the background material to the theory of triangulated categories, derived categories and differential graded algebras as well as a brief resume of classical homological algebra in Chapters 2 and 3.
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Shklyarov, Dmytro. "Hirzebruch-Riemann-Roch theorem for differential graded algebras." Diss., Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/1381.
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Maycock, Daniel. "Properties of triangular matrix and Gorenstein differential graded algebras." Thesis, University of Newcastle upon Tyne, 2011. http://hdl.handle.net/10443/1359.
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The main goal of this thesis is to investigate properties of two types of Differential Graded Algebras (or DGAs), namely upper triangular matrix DGAs and Gorenstein DGAs. In doing so we extend a number of corresponding ring theory results to the more general setting of DGAs and DG modules. Chapters 2 and 3 contain background material. In chapter 2 we give a brief summary of some important aspects of homological algebra. Starting with the definition of an abelian category we give the construction of the derived category and the definition of derived functors. In chapter 3 we present the basics about Differential Graded Algebras and Differential Graded Modules in particular extending the definitions of the derived category and derived functors to the Differential Graded case before providing some results on Recollement of DGAs, Dualising DG-modules and Gorenstein DGAs. Chapters 4 and 5 contain the bulk of the work for the Thesis. In chapter 4 we look at upper triangular matrix DGAs and in particular we generalise a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form [R M / 0 S] where R and S are DGAs and M is a DG R-Sop-bimodule. We show that under certain conditions on the DG-module M, and given the existence of a DG R-module X from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs [R M / 0 S] and [ S M’ / 0 R’], where the DG-bimodule M0 is obtained from M and X, and R0 is the endomorphism differential graded algebra of a K-projective resolution of X. In chapter 5 we turn our attention to Gorenstein DGAs and generalise some results from Gorenstein rings to Gorenstein DGAs. We present a number of Gorenstein Theorems which state, for certain types of DGAs, that being Gorenstein is equivalent to the bounded and finite versions of the Auslander and Bass classes being maximal. We also provide a new definition of a Gorenstein morphism for DGAs by considering a DG bimodule as a generalised morphism of DGAs. We then show that some existing results for Gorenstein morphism extend to these "Generalised Gorenstein Morphisms". We finally conclude with some examples of generalised Gorenstein morphisms for some well known DGAs.
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Chung, Myungsuk. "Lie derivations on rings of differential operators." Diss., Virginia Tech, 1995. http://hdl.handle.net/10919/37457.
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Martini, Alessio. "Algebras of differential operators on Lie groups and spectral multipliers." Doctoral thesis, Scuola Normale Superiore, 2010. http://hdl.handle.net/11384/85663.
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Let (X, μ) be a measure space, and let L1, . . . ,Ln be (possibly unbounded) selfadjoint
operators on L2(X, μ), which commute strongly pairwise, i.e., which
admit a joint spectral resolution E on Rn. A joint functional calculus is then
defined via spectral integration: for every Borel function m : Rn → C,
m(L) = m(L1, . . . ,Ln) =
∫ Rn m(λ) dE(λ) is a normal operator on L2(X, μ), which is bounded if and only if m - called
the joint spectral multiplier associated to m(L) - is (E-essentially) bounded.
However, the abstract theory of spectral integrals does not tackle the following
problem: to find conditions on the multiplier m ensuring the boundedness of
m(L) on Lp(X, μ) for some p ≠ 2.
We are interested in this problem when the measure space is a connected Lie
group G with a right Haar measure, and L1, . . . ,Ln are left-invariant differential
operators on G. In fact, the question has been studied quite extensively in the
case of a single operator, namely, a sublaplacian or a higher-order analogue.
On the other hand, for multiple operators, only specific classes of groups and
specific choices of operators have been considered in the literature.
Suppose that L1, . . . ,Ln are formally self-adjoint, left-invariant differential
operators on a connected Lie group G, which commute pairwise (as operators
on smooth functions). Under the assumption that the algebra generated by
L1, . . . ,Ln contains a weighted subcoercive operator --- a notion due to [ER98],
including positive elliptic operators, sublaplacians and Rockland operators---we
prove that L1, . . . ,Ln are (essentially) self-adjoint and strongly commuting on
L2(G). Moreover, we perform an abstract study of such a system of operators,
in connection with the algebraic structure and the representation theory of G,
similarly as what is done in the literature for the algebras of differential operators
associated with Gelfand pairs.
Under the additional assumption that G has polynomial volume growth,
weighted L1 estimates are obtained for the convolution kernel of the operator
m(L) corresponding to a compactly supported multiplier m satisfying some
smoothness condition. The order of smoothness which we require on m is related
to the degree of polynomial growth of G. Some techniques are presented,
which allow, for some specific groups and operators, to lower the smoothness
requirement on the multiplier.
In the case G is a homogeneous Lie group and L1, . . . ,Ln are homogeneous
operators, a multiplier theorem of Mihlin-H\"ormander type is proved, extending
the result for a single operator of [Chr91] and [MM90]. Further, a product theory
is developed, by considering several homogeneous groups Gj , each of which with
its own system of operators; a non-conventional use of transference techniques
then yields a multiplier theorem of Marcinkiewicz type, not only on the direct
product of the Gj , but also on other (possibly non-homogeneous) groups, containing
homomorphic images of the Gj . Consequently, for certain non-nilpotent
groups of polynomial growth and for some distinguished sublaplacians, we are
able to improve the general result of [Ale94].
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Dolan, Peter. "A Z2-graded generalization of Kostant's version of the Bott-Borel-Weil theorem /." view abstract or download file of text, 2007. http://proquest.umi.com/pqdweb?did=1400959341&sid=2&Fmt=2&clientId=11238&RQT=309&VName=PQD.
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Thesis (Ph. D.)--University of Oregon, 2007.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 130-131). Also available for download via the World Wide Web; free to University of Oregon users.
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Gover, Ashwin Roderick. "A geometrical construction of conformally invariant differential operators." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329953.
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Hansen, Nils Bahne [Verfasser]. "Structure Analysis of the Pohlmeyer-Rehren Lie Algebra and Adaptations of the Hall Algorithm to Non-Free Graded Lie Algebras / Nils Bahne Hansen." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2021. http://d-nb.info/1236401646/34.
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Lavau, Sylvain. "Lie infini-algébroides et feuilletages singuliers." Thesis, Lyon, 2016. http://www.theses.fr/2016LYSE1215/document.
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On dit qu'une variété est feuilletée lorsqu'il existe une partition de celle-ci en sous-variétés immergées. La théorie des feuilletages a des applications très profondes dans divers champs des Mathématiques et de la Physique, et il semble d'autant plus intéressant de pouvoir analyser le feuilletage à partir de ce qui semble être une donnée plus fondamentale : sa distribution de champs de vecteurs associée. C'est ainsi que nous avons observé que si le feuilletage est résolu par un fibré gradué, on peut relever le crochet de Lie des champs de vecteurs en une structure de Lie infini-algébroide sur ce fibré. D'autre part, cette structure est universelle dans le sens où toute autre résolution du feuilletage sera isomorphe à celle-ci dans un sens L_infini, mais seulement à homotopie près. Lorsqu'on se limite à l'étude au dessus d'un point, on observe que la cohomologie associée à la résolution devient potentiellement non triviale. La structure de Lie infini-algébroide universelle se réduit alors à une algèbre de Lie graduée sur cette cohomologie. Cette structure algébrique peut être transportée (non canoniquement) tout le long de la feuille, transformant la cohomologie au dessus d'une feuille en algébroide de Lie gradué. Cela nous permet de retrouver des résultats déjà connus par ailleurs et de déduire des avancées prometteusesA smooth manifold is said to be foliated when it is partitioned into immersed submanifolds. Foliation Theory has profound applications in various fields of Mathematics and Physics, and it seems much more interesting to analyze the foliation from what seems to be a more fundamental point of view: its associated distribution of vector fields. Thus we have noticed that if the foliation is resolved by a graded fiber bundle, one can lift the Lie bracket of vector fields into a Lie infinity-algebroid structure on this fiber bundle. Moreover, this structure is universal in the sense that any other resolution of the foliation is isomorphic to it in the L_infinity setup, but only up to homotopy. When one restricts the analysis over a point, we observe that the cohomology associated to the resolution may become non trivial. The universal Lie infinity-algebroid structure hence reduces to a graded Lie algebra structure on this cohomology. This algebraic structure can be carried (non canonically) along the leaf, providing the cohomology over a leaf with a graded Lie algebroid structure. This enables us to retrieve former well-known results, as well as promising advances
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Souza, Manuela da Silva 1985. "Propriedade de Specht e crescimento das identidades polinomiais graduadas de sl_2." [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306362.
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Orientador: Plamen Emilov KochloukovTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Doutorado
Matematica
Doutora em Matemática
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Sánchez, Jesús. "About E-infinity-structures in L-algebras." Thesis, Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCC204/document.
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Dans cette thèse nous rappelons la notion de L-algèbre, qui a pour objet d'être un modèle algébrique des types d'homotopie. L'objectif principal de cette thèse est la description d'une structure de E-infini-coalgèbre sur l'élément principal d'une L-algèbre. Ceci peut être vu comme une généralisation de la structure de E-infini-coalgèbre sur le complexe des chaînes d'un ensemble simplicial, telle que décrite par Smith dans Iterating the cobar construction, 1994. Nous construisons une E-infini-opérade, notée K, utilisée pour construire la E-infini-coalgèbre sur l'élément principal d'une L-algèbre. Cette structure de E-infini-coalgèbre montre que la L-algèbre canoniquement associée à un ensemble simplicial contient au moins autant d'information homotopique que la E-infini-coalgèbre couramment associée à un ensemble simplicialIn this thesis we recall the notion of L-algebra. L-algebras are intended as algebraic models for homotopy types. L-algebras were introduced by Alain Prouté in several talks since the eighties. The principal objective of this thesis is the description of an E-infinity-coalgebra structure on the main element of an L-algebra. This can be seen as a generalization of the E-infinity-coalgebra structure on the chain complex associated to a simplicial set given by Smith in Iterating the cobar construction, 1994. We construct an E-inifity-operad, denoted K, used to construct the E-inifity-coalgebra on the main element of a L-algebra. This E-inifity-coalgebra structure shows that the canonical L-algebra associated to a simplicial set contains at least as much homotopy information as the E-inifity-coalgebras usually associated to simplicial sets
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Sene, Renato Tolentino de. "Curvaturas de métricas invariantes em Grupos de Lie." Universidade Federal de Uberlândia, 2015. https://repositorio.ufu.br/handle/123456789/16821.
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In this work we study the geometric aspects of Lie groups from the view point of the
Riemannian geometry, by means of invariant geometric structures associated. We present
some properties on curvatures of metrics left invariants and bi-invariant one on Lie groups.
We also present a treatment of the Lie algebras unimodular, including the tridimensional
case. Most of the results studied are from the article of John Milnor: Curvatures of Left
Invariant Metrics on Lie Groups.Neste trabalho estudamos os aspectos geometricos de grupos de Lie, do ponto de vista da geometria Riemanniana, por meio de estruturas geometricas invariantes associadas. Nos apresentamos algumas propriedades de curvaturas com metricas invariante a esquerda e aquelas bi-invariantes em grupos de Lie. Apresentamos tambem um tratamento das algebras de Lie unimodulares, incluindo o caso tridimensional. A maioria dos resultados estudados foram retirados do artigo de John Milnor: Curvatures of Left Invariant Metrics on Lie Groups.
Mestre em Matemática
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Webster, Benjamin. "On Representations of the Jacobi Group and Differential Equations." UNF Digital Commons, 2018. https://digitalcommons.unf.edu/etd/858.
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In PDEs with nontrivial Lie symmetry algebras, the Lie symmetry naturally yield Fourier and Laplace transforms of fundamental solutions. Applying this fact we discuss the semidirect product of the metaplectic group and the Heisenberg group, then induce a representation our group and use it to investigate the invariant solutions of a general differential equation of the form .
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Kunz, Daniel. "Lieovy grupy a jejich fyzikální aplikace." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2020. http://www.nusl.cz/ntk/nusl-417088.
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In this thesis I describe construction of Lie group and Lie algebra and its following usage for physical problems. To be able to construct Lie groups and Lie algebras we need define basic terms such as topological manifold, tensor algebra and differential geometry. First part of my thesis is aimed on this topic. In second part I am dealing with construction of Lie groups and algebras. Furthermore, I am showing different properties of given structures. Next I am trying to show, that there exists some connection among Lie groups and Lie algebras. In last part of this thesis is used just for showing how this apparat can be used on physical problems. Best known usage is to find physical symmetries to establish conservation laws, all thanks to famous Noether theorem.
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Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.
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The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.
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Ouzina, Mostafa. "Théorème du support en théorie du filtrage non-linéaire." Rouen, 1998. http://www.theses.fr/1998ROUES029.
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La thèse comporte essentiellement quatre parties. Dans la première partie, la preuve simple de A. Millet et Marta-Sanz-Sole du théorème du support de Stroock-Varadhan dans le cas indépendant du temps est étendue au cas dépendant du temps. Dans la seconde partie, soit (x#t) la solution de l'équation différentielle stochastique x#t = x + #r#i# #=# #1#t#0#i(x#s)dw#i#s + #l#j# #=# #1#t#0$$#j(x#s)odw$$#j#s + #t#0b(x#s)ds nous considérons cette solution comme fonction de w a valeurs dans l'espace l#p des fonctions de w$$ a valeurs dans l'espace des fonctions continues. Le résultat concernant le théorème du support dans ce contexte est établi. Le reste de la thèse est une application de ces idées au filtrage non linéaire : le support de l’espérance conditionnelle déterminée soit par l'équation de Zakai soit par l'équation de Stratonovich est établi, puis nous donnons également le support de la loi conditionnelle du signal x sachant l'observation y, ou le couple (x,y) est donne par dx#t = #r#i# #=# #1#i(x#t)odw#i#t + #l#j# #=# #1$$#j(x#t)ody#j#t + b(x#t)dt dy#t = (x#t)dt + d$$#t, x#0 = x, y#0 = 0, t , 0. 1; dans deux situations différentes. Dans la première on suppose que l'algèbre de lie l($$#1,,$$#l) est commutative. Dans la seconde on suppose que les champs de vecteurs $$#j sont à support compact. Dans la dernière partie on étudie le principe de grandes déviations pour le filtre normalise perturbe obtenu en remplaçant y par y avec tend vers zero.
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Rocha, Eugénio Alexandre Miguel. "Uma Abordagem Algébrica à Teoria de Controlo Não Linear." Doctoral thesis, Universidade de Aveiro, 2003. http://hdl.handle.net/10773/21444.
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Doutoramento em MatemáticaNesta tese de Doutoramento desenvolve-se principalmente uma abordagem algébrica à teoria de sistemas de controlo não lineares. No entanto, outros tópicos são também estudados. Os tópicos tratados são os seguidamente enunciados: fórmulas para sistemas de controlo sobre álgebras de Lie livres, estabilidade de um sistema de corpos rolantes, algoritmos para aritmética digital, e equações integrais de Fredholm não lineares. No primeiro e principal tópico estudam-se representações para as soluções de sistemas de controlo lineares no controlo. As suas trajetórias são representadas pelas chamadas séries de Chen. Estuda-se a representação formal destas séries através da introdução de várias álgebras não associativas e técnicas específicas de álgebras de Lie livres. Sistemas de coordenadas para estes sistemas são estudados, nomeadamente, coordenadas de primeiro tipo e de segundo tipo. Apresenta-se uma demonstração alternativa para as coordenadas de segundo tipo e obtêm-se expressões explícitas para as coordenadas de primeiro tipo. Estas últimas estão intimamente ligadas ao logaritmo da série de Chen que, por sua vez, tem fortes relações com uma fórmula designada na literatura por “continuous Baker-Campbell- Hausdorff formula”. São ainda apresentadas aplicações à teoria de funções simétricas não comutativas. É, por fim, caracterizado o mapa de monodromia de um campo de vectores não linear e periódico no tempo em relação a uma truncatura do logaritmo de Chen. No segundo tópico é estudada a estabilizabilidade de um sistema de quaisquer dois corpos que rolem um sobre o outro sem deslizar ou torcer. Constroem-se controlos fechados e dependentes do tempo que tornam a origem do sistema de dois corpos num sistema localmente assimptoticamente estável. Vários exemplos e algumas implementações em Maple°c são discutidos. No terceiro tópico, em apêndice, constroem-se algoritmos para calcular o valor de várias funções fundamentais na aritmética digital, sendo possível a sua implementação em microprocessadores. São também obtidos os seus domínios de convergência. No último tópico, também em apêndice, demonstra-se a existência e unicidade de solução para uma classe de equações integrais não lineares com atraso. O atraso tem um carácter funcional, mostrando-se ainda a diferenciabilidade no sentido de Fréchet da solução em relação à função de atraso.
In this PhD thesis several subjects are studied regarding the following topics: formulas for nonlinear control systems on free Lie algebras, stabilizability of nonlinear control systems, digital arithmetic algorithms, and nonlinear Fredholm integral equations with delay. The first and principal topic is mainly related with a problem known as the continuous Baker-Campbell-Hausdorff exponents. We propose a calculus to deal with formal nonautonomous ordinary differential equations evolving on the algebra of formal series defined on an alphabet. We introduce and connect several (non)associative algebras as Lie, shuffle, zinbiel, pre-zinbiel, chronological (pre-Lie), pre-chronological, dendriform, D-I, and I-D. Most of those notions were also introduced into the universal enveloping algebra of a free Lie algebra. We study Chen series and iterated integrals by relating them with nonlinear control systems linear in control. At the heart of all the theory of Chen series resides a zinbiel and shuffle homomorphism that allows us to construct a purely formal representation of Chen series on algebras of words. It is also given a pre-zinbiel representation of the chronological exponential, introduced by A.Agrachev and R.Gamkrelidze on the context of a tool to deal with nonlinear nonautonomous ordinary differential equations over a manifold, the so-called chronological calculus. An extensive description of that calculus is made, collecting some fragmented results on several publications. It is a fundamental tool of study along the thesis. We also present an alternative demonstration of the result of H.Sussmann about coordinates of second kind using the mentioned tools. This simple and comprehensive proof shows that coordinates of second kind are exactly the image of elements of the dual basis of a Hall basis, under the above discussed homomorphism. We obtain explicit expressions for the logarithm of Chen series and the respective coordinates of first kind, by defining several operations on a forest of leaf-labelled trees. It is the same as saying that we have an explicit formula for the functional coefficients of the Lie brackets on a continuous Baker-Campbell-Hausdorff-Dynkin formula when a Hall basis is used. We apply those formulas to relate some noncommutative symmetric functions, and we also connect the monodromy map of a time-periodic nonlinear vector field with a truncation of the Chen logarithm. On the second topic, we study any system of two bodies rolling one over the other without twisting or slipping. By using the Chen logarithm expressions, the monodromy map of a flow and Lyapunov functions, we construct time-variant controls that turn the origin of a control system linear in control into a locally asymptotically stable equilibrium point. Stabilizers for control systems whose vector fields generate a nilpotent Lie algebra with degree of nilpotency · 3 are also given. Some examples are presented and Maple°c were implemented. The third topic, on appendix, concerns the construction of efficient algorithms for Digital Arithmetic, potentially for the implementation in microprocessors. The algorithms are intended for the computation of several functions as the division, square root, sines, cosines, exponential, logarithm, etc. By using redundant number representations and methods of Lyapunov stability for discrete dynamical systems, we obtain several algorithms (that can be glued together into an algorithm for parallel execution) having the same core and selection scheme in each iteration. We also prove their domains of convergence and discuss possible extensions. The last topic, also on appendix, studies the set of solutions of a class of nonlinear Fredholm integral equations with general delay. The delay is of functional character modelled by a continuous lag function. We ensure existence and uniqueness of a continuous (positive) solution of such equation. Moreover, under additional conditions, it is obtained the Fr´echet differentiability of the solution with respect to the lag function.
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Gartz, Kaj M. "A construction of a differential graded Lie algebra in the category of effective homological motives /." 2003. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:3088737.
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TORTORELLA, ALFONSO GIUSEPPE. "Deformations of coisotropic submanifolds in Jacobi manifolds." Doctoral thesis, 2017. http://hdl.handle.net/2158/1077777.
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In this thesis, we investigate deformation theory and moduli theory of coisotropic submanifolds in Jacobi manifolds. Originally introduced by Kirillov as local Lie algebras with one dimensional fibers, Jacobi manifolds encompass, unifying and generalizing, locally conformal symplectic manifolds, locally conformal Poisson manifolds, and non-necessarily coorientable contact manifolds.
We attach an L-infinity-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), and Le-Oh (locally conformal symplectic case). As a completely new case we also associate an L-infinity-algebra with any coisotropic submanifold in a contact manifold. The L-infinity-algebra of a coisotropic submanifold S controls the formal coisotropic deformation problem of S, even under Hamiltonian equivalence, and provides criteria both for the obstructedness and for the unobstructedness at the formal level. Additionally we prove that if a certain condition ("fiberwise entireness") is satisfied then the L-infinity-algebra controls the non-formal coisotropic deformation problem, even under Hamiltonian equivalence.
We associate a BFV-complex with any coisotropic submanifold in a Jacobi manifold. Our construction extends an analogous construction by Schatz in the Poisson setting, and in particular it also applies in the locally conformal symplectic/Poisson setting and the contact setting. Unlike the L-infinity-algebra, we prove that, with no need of any restrictive hypothesis, the BFV-complex of a coisotropic submanifold S controls the non-formal coisotropic deformation problem of S, even under both Hamiltonian equivalence and Jacobi equivalence.
Notwithstanding the differences there is a close relation between the approaches to the coisotropic deformation problem via L-infinity-algebra and via BFV-complex. Indeed both the L-infinity-algebra and the BFV-complex of a coisotropic submanifold S provide a cohomological reduction of S. Moreover they are L-infinity quasi-isomorphic and so they encode equally well the moduli space of formal coisotropic deformations of S under Hamiltonian equivalence.
In addition we exhibit two examples of coisotropic submanifolds in the contact setting whose coisotropic deformation problem is obstructed at the formal level. Further we provide a conceptual explanation of this phenomenon both in terms of the L-infinity-algebra and in terms of the BFV-complex.
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Euler, Norbert. "Continuous symmetries, lie algebras and differential equations." Thesis, 2014. http://hdl.handle.net/10210/9131.
Full textAbstract:
D.Sc. (Mathematics)In this thesis aspects of continuous symmetries of differential equations are studied. In particular the following aspects are studied in detail: Lie algebras, the Lie derivative, the jet bundle formalism for differential equations, Lie point and Lie-Backlund symmetry vector fields, recursion operators, conservation laws, Lax pairs, the Painlcve test, Lie algebra valued differenmtial forms and Dose operators as a representation of differential operators. The purpose of the study is to gain a better understanding of complicated nonlinear dirrerential equations that describe nature and to construct solutions. The differential equations under consideration were derived [rom physics and engineering. They are the following: the Kortcweg-dc Vries equation, Burgers' cquation , the sine-Gordon equation, nonlinear diffusion equations, the Klein Gordon equation, the Schrodinger equation, nonlinear Dirac equations, Yang-Mills equations, the Lorentz model, the Lotka-Volterra model, damped unharrnonic oscillators, and others. The newly found results and insights are discussed in chapters 8 to 17. Details on the COli tents of each chapter and rcfernces to some of my articles arc given in chapter 1.
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Donin, Dmitry. "Lie Algebras of Differential Operators and D-modules." Thesis, 2008. http://hdl.handle.net/1807/16779.
Full textAbstract:
In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two
problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic
structure of differential and pseudodifferential operators. We define the Krichever-Novikov type Lie algebras of
differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and
central extensions. We show that the corresponding algebras of meromorphic differential operators and
pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic
vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols
holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the
similarities and differences with the case of smooth symbols on the circle.
The second problem is related to the geometry of differential operators and its connection with representations of
semi-simple Lie algebras. We show that the semiregular module, naturally associated with a graded semi-simple
complex Lie algebra, can be realized in geometric terms, using the Brion's construction of degeneration of
the diagonal in the square of the flag variety. Namely, we consider the Beilinson-Bernstein localization
of the semiregular module and show that it is isomorphic to the D-module obtained by applying the
Emerton-Nadler-Vilonen geometric Jacquet functor to the D-module of distributions on the square of the flag variety
with support on the diagonal.
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PASQUALI, MARCO. "On the structure of Borel stable abelian subalgebras in Z2-graded Lie algebras." Doctoral thesis, 2012. http://hdl.handle.net/11573/917818.
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Kara, A. H. "On lie and Noether symmetries of differential equations." Thesis, 1994. https://hdl.handle.net/10539/26093.
Full textAbstract:
A thesis submitted to the faculty of Science, University of the Witwatersrand, in
fulfilment of the requirements for the degree of Doctor of Philosophy,The inverse problem in the Calculus of Variations involves determining the Lagrangians, if any, associated with a given (system of) differential equation(s). One can classify Lagrangians according to the Lie algebra of symmetries of the Action integral (the Noether algebra). We give a complete classification of first-order Lagrangians defined on the line and produce results pertaining to the dimensionality of the algebra of Noether symmetries and compare and contrast these with similar results on the algebra of Lie symmetries of the corresponding Euler-Lagrange .equations. It is proved that the maximum dimension of the Noether point symmetry algebra of a particle Lagrangian. is five whereas it is known that the maximum dimension Qf the Lie algebra of the corresponding scalar second-order Euler-Lagrange equation is eight. Moreover, we show th'a.t a particle Lagrangian does not admit a maximal four-dimensional Noether point symmeiry algebra and consequently a particle Lagrangian admits the maximal r E {O, 1,2,3, 5}-dimensional Noether point symmetry algebra, It is well .known that an important means of analyzing differential equations lies in the knowledge of the first integrals of the equation. We deliver an algorithm for finding first integrals of partial differential equations and show how some of the symmetry properties of the first integrals help to 'further' reduce the order of the equations and sometimes completely solve the equations. Finally, we discuss some open questions. These include the inverse problem and classification of partial differential equations. ALo, there is the question of the extension of the results to 'higher' dimensions.
Andrew Chakane 2018
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Wills, Luis Alberto. "Finite group graded lie algebraic extensions and trefoil symmetric relativity, standard model, yang mills and gravity theories." Thesis, 2008. http://hdl.handle.net/10125/11725.
Full textAbstract:
Mode of access: World Wide Web.Thesis (Ph. D.)--University of Hawaii at Manoa, 2004.
Includes bibliographical references (leaves 159-164).
Electronic reproduction.
Also available by subscription via World Wide Web
x, 164 leaves, bound ill. 29 cm
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Schmidt, Karsten [Verfasser]. "Auslander-Reiten theory for simply connected differential graded algebras / vorgelegt von Karsten Schmidt." 2007. http://d-nb.info/987425641/34.
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Bhargava, Sandeep. "Realizations of BC(r)-graded intersection matrix algebras with grading subalgebras of type B(r), r greater than or equal to 3 /." 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR45986.
Full textAbstract:
Thesis (Ph.D.)--York University, 2008. Graduate Programme in Mathematics and Statistics.Typescript. Includes bibliographical references (leaves 275-278). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR45986
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"Applications of symmetry analysis to physically relevant differential equations." Thesis, 2005. http://hdl.handle.net/10413/2546.
Full textAbstract:
We investigate the role of Lie symmetries in generating solutions to differential equations that arise in particular physical systems. We first provide an overview of the Lie analysis and review the relevant symmetry analysis of differential equations in general. The Lie symmetries of some simple ordinary differential equations are found t. illustrate the general method. Then we study the properties of particular ordinary differential equations that arise in astrophysics and cosmology using the Lie analysis of differential equations. Firstly, a system of differential equations arising in the model of a relativistic star is generated and a governing nonlinear equation is obtained for a linear equation of state. A comprehensive symmetry analysis is performed on this equation. Secondly, a second order nonlinear ordinary differential equation arising in the model of the early universe is described and a detailed symmetry analysis of this equation is undertaken. Our objective in each case is to find explicit Lie symmetry generators that may help in analysing the model.Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2005.
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Kohler, Astri. "Conditional and approximate symmetries for nonlinear partial differential equations." Thesis, 2014. http://hdl.handle.net/10210/11449.
Full textAbstract:
M.Sc.In this work we concentrate on two generalizations of Lie symmetries namely conditional symmetries in the form of Q-symmetries and approximate symmetries. The theorems and definitions presented can be used to obtain exact and approximate solutions for nonlinear partial differential equations. These are then applied to various nonlinear heat and wave equations and many interesting solutions are given. Chapters 1 and 2 gives an introduction to the classical Lie approach. Chapters 3, 4 and 5 deals with conditional -, approximate -, and approximate conditional symmetries respectively. In chapter 6 we give a review of symbolic algebra computer packages available to aid in the search for symmetries, as well as useful REDUCE programs which were written to obtain the results given in chapters 2 to 5.
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Batistelli, Karina Haydeé. "Subálgebras de álgebra de Lie de operadores pseudo-diferenciales matriciales cuánticos y representaciones de módulos de peso máximo cuasifinitos de subálgebras de tipo ortogonal y simpléticos." Doctoral thesis, 2017. http://hdl.handle.net/11086/5845.
Full textAbstract:
En esta tesis caracterizamos los módulos irreducibles de peso máximo cuasifinitos de las sub\'algebras del álgebra de Lie de operadores pseudo-diferenciales matriciales cuánticos N x N.
En la primer parte, se presentan los resultados que obtenidos, dando una descripción completa de las anti-involuciones que preservan la graduación principal. Obtenemos, salvo conjugación, dos familias de anti-involuciones para un cierto parámetro n con resultados diferentes cuando n=N y n En la segunda parte, nos focalizamos en el estudio de las subálgebras de tipo "ortogonal" y "simpléctico" halladas para el caso n=N, puntualmente la clasificación y realización de los módulos irreducibles de peso máximo cuasifinitos.
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Mahomed, Komal Shahzadi. "Symmetry properties for first integrals." Thesis, 2015. http://hdl.handle.net/10539/16838.
Full textAbstract:
A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. July 2014.This is the study of Lie algebraic properties of first integrals of scalar second-, third and higher-order ordinary differential equations (ODEs). The Lie algebraic classification of such differential equations is now well-known from the works of Lie [10] as well as recently Mahomed and Leach [19]. However, the algebraic properties of first integrals are not known except in the maximal cases for the basic first integrals and some of their quotients. Here our intention is to investigate the complete problem for scalar second-order and maximal symmetry classes of higher-order ODEs using Lie algebras and Lie symmetry methods. We invoke the realizations of low-dimensional Lie algebras. Symmetries of the fundamental first integrals for scalar second-order ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the point symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classi cation of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0, 1, 2 or 3 point symmetry cases. It is proved that the maximal algebra case is unique. By use of Lie symmetry group methods we further analyze the relationship between the first integrals of the simplest linear third-order ODEs and their point symmetries. It is well-known that there are three classes of linear third-order ODEs for maximal and submaximal cases of point symmetries which are 4, 5 and 7. The simplest scalar linear third-order equation has seven point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y000 = 0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs of maximal type. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals. The relationship between rst integrals of sub-maximal linearizable third-order ODEs and their symmetries are investigated as well. All scalar linearizable third-order equations can be reduced to three classes by point transformations. We obtain the classifying relations between the symmetries and the first integral for sub-maximal cases of linear third-order ODEs. It is known, from the above, that the maximum Lie algebra of the first integral is achieved for the simplest equation. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the rst integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2 and 3 symmetries and for the 4 symmetry class of linear third-order ODEs they are 0, 1 and 2 symmetries respectively. In the case of sub-maximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals. For the n+2 symmetry class, the symmetries of the rst integral I2 and a two-dimensional subalgebra of I1 generate the symmetry algebra and for the n + 1 symmetry class, the full algebra is generated by the symmetries of I1 and a two-dimensional subalgebra of the quotient I3=I2. Finally, we completely classify the first integrals of scalar nonlinear second-order ODEs in terms of their Lie point symmetries. This is performed by first obtaining the classifying relations between point symmetries and first integrals of scalar nonlinear second order equations which admit 1, 2 and 3 point symmetries. We show that the maximum number of symmetries admitted by any first integral of a scalar second-order nonlinear (which is not linearizable by point transformation) ODE is one which in turn provides reduction to quadratures of the underlying dynamical equation. We provide physical examples of the generalized Emden-Fowler, Lane-Emden and modi ed Emden equations.
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Leach, Peter Gavin Lawrence. "Algebraic properties of ordinary differential equations." Thesis, 1995. http://hdl.handle.net/10413/4897.
Full textAbstract:
In Chapter One the theoretical basis for infinitesimal transformations is
presented with particular emphasis on the central theme of this thesis which
is the invariance of ordinary differential equations, and their first integrals,
under infinitesimal transformations. The differential operators associated with
these infinitesimal transformations constitute an algebra under the operation
of taking the Lie Bracket. Some of the major results of Lie's work are recalled.
The way to use the generators of symmetries to reduce the order of a differential
equation and/or to find its first integrals is explained. The chapter concludes
with a summary of the state of the art in the mid-seventies just before the
work described here was initiated.
Chapter Two describes the growing awareness of the algebraic properties of
the paradigms of differential equations. This essentially ad hoc period demonstrated
that there was value in studying the Lie method of extended groups
for finding first integrals and so solutions of equations and systems of equations.
This value was emphasised by the application of the method to a class of
nonautonomous anharmonic equations which did not belong to the then pantheon
of paradigms. The generalised Emden-Fowler equation provided a route
to major development in the area of the theory of the conditions for the linearisation
of second order equations. This was in addition to its own interest.
The stage was now set to establish broad theoretical results and retreat from
the particularism of the seventies.
Chapters Three and Four deal with the linearisation theorems for second
order equations and the classification of intrinsically nonlinear equations according
to their algebras. The rather meagre results for systems of second
order equations are recorded.
In the fifth chapter the investigation is extended to higher order equations
for which there are some major departures away from the pattern established
at the second order level and reinforced by the central role played by these
equations in a world still dominated by Newton. The classification of third
order equations by their algebras is presented, but it must be admitted that
the story of higher order equations is still very much incomplete.
In the sixth chapter the relationships between first integrals and their algebras
is explored for both first order integrals and those of higher orders. Again
the peculiar position of second order equations is revealed.
In the seventh chapter the generalised Emden-Fowler equation is given a
more modern and complete treatment.
The final chapter looks at one of the fundamental algebras associated with
ordinary differential equations, the three element 8£(2, R), which is found in all
higher order equations of maximal symmetry, is a fundamental feature of the
Pinney equation which has played so prominent a role in the study of nonautonomous
Hamiltonian systems in Physics and is the signature of Ermakov
systems and their generalisations.Thesis (Ph.D.)-University of Natal, 1995.
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Moyo, Sibusiso. "Noether's theorem and first integrals of ordinary differential equations." Thesis, 1997. http://hdl.handle.net/10413/5061.
Full textAbstract:
The Lie theory of extended groups is a practical tool in the analysis of differential equations, particularly in the construction of solutions. A formalism of the Lie theory is given and contrasted with Noether's theorem which plays a prominent role in the analysis of differential equations derivable from a Lagrangian. The relationship between the Lie and Noether approach to differential equations is investigated. The standard separation of Lie point symmetries into Noetherian and nonNoetherian symmetries is shown to be irrelevant within the context of nonlocality. This also emphasises the role played by nonlocal symmetries in such an approach.Thesis (M.Sc.)-University of Natal, Durban, 1997.
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Bashe, Mantombi Beryl. "Equivalence and symmetry groups of a nonlinear equation in plasma physics." Thesis, 2016. http://hdl.handle.net/10539/20598.
Full textAbstract:
Degree awarded with distinction on 6 December 1995.
A research report submitted to the Faculty of Science, University of the
Witwatersrand, in fulfilment of the requirements for the degree of Masters.
Johannesburg, 1995.In this work we give a brief overview of the existing group classification methods of partial differential equations by means of examples. On top of these methods we introduce another new method which classify according to low-dimensional Lie elgebras, One can ask: What is the aim of introducing a new method whilst there are existing methods? This question is answered in the following paragraph. Firstly we classify our system of non-linear partial differential equations using the preliminary group classification method (one of the existing methods). The results are not different from what; Euler, Steeb and Mulsor have obtained in 1991 and 1992. That is, this method does not yield new information. This new method which classifies according to low-dimensional Lie algebras is used to classify a general system of equations from plasma physics. Finally, using this method we completely classify our system for four-dimensionnl algebras. For a partial differential equation to be completely classified using this method, it must admit a low-dimensional Lie algebra.
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Govinder, Kesh S. "Ermakov systems : a group theoretic approach." Thesis, 1993. http://hdl.handle.net/10413/5951.
Full textAbstract:
The physical world is, for the most part, modelled using second order ordinary differential equations. The time-dependent simple harmonic oscillator and the Ermakov-Pinney equation (which together form an Ermakov system) are two examples that jointly and separately describe many physical situations. We study Ermakov systems from the point of view of the algebraic properties
of differential equations. The idea of generalised Ermakov systems is introduced and their relationship to the Lie algebra sl(2, R) is explained. We show that the 'compact' form of generalized Ermakov systems has an infinite dimensional Lie algebra. Such algebras are usually associated only with first order equations in the context of ordinary differential equations. Apart from the Ermakov invariant which shares the infinite-dimensional algebra of the 'compact' equation, the other three integrals force the dimension of the algebra to be reduced to the three of sl(2, R). Subsequently we establish a new class of Ermakov systems by considering equations invariant under sl(2, R) (in two dimensions) and sl(2, R) EB so(3) (in three dimensions). The former class contains the generalized Ermakov system as a special case in which the force is velocity-independent. The latter case is a generalization of the classical equation of motion of the magnetic monopole
which is well known to possess the conserved Poincare vector. We demonstrate that in fact there are three such vectors for all equations of this type.Thesis (M.Sc.)-University of Natal, 1993.
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Adams, Conny Molatlhegi. "A Lie symmetry analysis of the heat equation through modified one-parameter local point transformation." Diss., 2014. http://hdl.handle.net/10500/18414.
Full textAbstract:
Using a Lie symmetry group generator and a generalized form of Manale's formula
for solving second order ordinary di erential equations, we determine new symmetries
for the one and two dimensional heat equations, leading to new solutions. As
an application, we test a formula resulting from this approach on thin plate heat
conduction.Applied Mathematics
M.Sc. (Applied Mathematics)
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Pantazi, Hara. "First integrals for the Bianchi universes : supplementation of the Noetherian integrals with first integrals obtained by using Lie symmetries." 1997. http://hdl.handle.net/10413/5103.
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Lemmer, Ryan Lee. "The paradigms of mechanics : a symmetry based approach." Thesis, 1996. http://hdl.handle.net/10413/4899.
Full textAbstract:
An overview of the historical developments of the paradigms of classical mechanics, the free particle, oscillator and the Kepler problem, is given ito (in terms of) their conserved quantities. Next, the orbits of the three paradigms are found from quadratic forms. The quadratic forms are constructed using first integrals found by the application of Poisson's theorem. The orbits are presented ito expanding surfaces defined by the quadratic forms. The Lie and Noether symmetries of the paradigms are investigated. The free particle is discussed in detail and an overview of the work done on the oscillator and Kepler problem is given. The Lie and Noether theories are compared from various aspects. A technical description of Lie groups and algebras is given. This provides a basis for a discussion of the historical development of the paradigms of mechanics ito their group properties. Lastly the paradigms are discussed ito of Quantum Mechanics.Thesis (M.Sc.)-University of Natal, 1996.
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Edelstein, R. M. "A classification of second order equations via nonlocal transformations." Thesis, 2000. http://hdl.handle.net/10413/3694.
Full textAbstract:
The study of second order ordinary differential equations is vital given their proliferation in
mechanics. The group theoretic approach devised by Lie is one of the most successful techniques
available for solving these equations. However, many second order equations cannot be reduced
to quadratures due to the lack of a sufficient number of point symmetries. We observe that
increasing the order will result in a third order differential equation which, when reduced via an
alternate symmetry, may result in a solvable second order equation. Thus the original second
order equation can be solved.
In this dissertation we will attempt to classify second order differential equations that can
be solved in this manner. We also provide the nonlocal transformations between the original
second order equations and the new solvable second order equations.
Our starting point is third order differential equations. Here we concentrate on those invariant
under two- and three-dimensional Lie algebras.Thesis (M.Sc.)-University of Natal, Durban, 2000.
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Ewert, Eske Ellen. "Index theory and groupoids for filtered manifolds." Doctoral thesis, 2020. http://hdl.handle.net/21.11130/00-1735-0000-0005-152D-2.
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Moas, Ruth Paola. "La energía de las secciones unitarias normales de la grassmanniana asociadas a productos cruz." Doctoral thesis, 2020. http://hdl.handle.net/11086/19767.
Full textAbstract:
Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2020.Sea G(k,n) la grassmanniana de subespacios orientados de Rn de dimensión k con su métrica riemanniana canónica. Estudiamos la energía de funciones que asignan a cada P en G(k,n) un vector unitario normal a P. Son secciones de un fibrado esférico E(k,n) sobre G(k,n). Los productos cruz doble y triple octoniónicos inducen de manera natural secciones de este tipo para k=2, n=7 y k=3, n=8, respectivamente. Probamos que son aplicaciones armónicas en E(k,n) munido de la métrica de Sasaki. Esto, junto con el resultado bien conocido de que los campos vectoriales de Hopf en esferas de dimensión impar son aplicaciones armónicas en su fibrado tangente unitario, nos permite concluir que todas las secciones normales unitarias de las grassmannianas asociadas a productos cruz son aplicaciones armónicas. También mostramos que estos fibrados esféricos no poseen secciones paralelas, que trivialmente habrían tenido energía mínima. En una segunda instancia analizamos la energía de aplicaciones que asignan a cada P en G(2,8) una estructura compleja ortogonal J(P) en el subespacio ortogonal a P. Estas asignaciones son secciones del subfibrado esférico unitario del fibrado vectorial sobre P en G(2,8) cuya fibra en cada P consiste esencialmente de las transformaciones antisimétricas del subespacio ortogonal a P. Probamos que la sección naturalmente inducida por el producto cruz triple octoniónico es una aplicación armónica. Comentamos la relación con la armonicidad de la estructura casi compleja canónica de la esfera de dimensión 6.
Let G(k,n) be the Grassmannian of oriented subspaces of Rn of dimension k with its canonical symmetric Riemannian metric. We study the energy of maps assigning a unit vector normal to P to each P in G(k,n) . They are sections of a sphere bundle E(k,n) over G(k,n). The octonionic double and triple cross products induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively. We prove that they are harmonic maps into E(k,n) endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. We also show that these sphere bundles do not have parallel sections, which trivially would have had minimum energy. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J(P) on P to each P in G(2,8). They are sections of the unit sphere bundle over G(2,8) whose fiber at each P consists essentially of the skewsymmetric transformations on P?. We prove that the section naturally induced by the octonionic triple product is a harmonic map. We comment on the relationship with the harmonicity of the canonical almost complex structure of the sphere of dimension 6.
publishedVersion
Fil: Moas, Ruth Paola. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.