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• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
31

#### Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method

The rigid body has been one of the most noteworthy applications of Newtonian mechanics. Applying the principles of classical mechanics to the rigid body is by no means routine. The equations of motion, though discovered two hundred and fifty years ago by Euler, have remained quite elusive since their introduction. Understanding the rigid body has required the applications of concepts from integrable systems, algebraic geometry, Lie groups, representation theory, and symplectic geometry to name a few. Moreover, several important developments in these fields have in fact originated with the study of the rigid body and subsequently have grown into general theories with much wider applications.In this work, we study the stability of equilibria of non-degenerate orbits of the generalized rigid body. The energy-Casimir method introduced by V.I. Arnold in 1966 allows us to prove stability of certain non-degenerate equilibria of systems on Lie groups. Applied to the three dimensional rigid body, it recovers the classical Euler stability theorem [12]: rotations around the longest and shortest principal moments of inertia are stable equilibria. This method has not been applied to the analysis of rigid body dynamics beyond dimension n = 3. Furthermore, no conditions for the stability of equilibria are known at all beyond n = 4, in which case the conditions are not of the elegant longest/shortest type [10].Utilizing the rich geometric structures of the symmetry group G = SO(2n), we obtain stability results for generic equilibria of the even dimensional free rigid body. After obtaining a general expression for the generic equilibria, we apply the energy-Casimir method and find that indeed the classical longest/shortest conditions on the entries of the inertia matrix are suffcient to prove stability of generic equilibria for the generalized rigid body in even dimensions.
32

#### Resolving Multiplicities in the Tensor Product of Irreducible Representations of Semisimple Lie Algebras

Brooke, David John 20 January 2009 (has links)
When the tensor product of two irreducible representations contains multiple copies of some of its irreducible constituents, there is a problem of choosing specific copies: resolving the multiplicity. This is typically accomplished by some ad hoc method chosen primarily for convenience in labelling and calculations. This thesis addresses the possibility of making choices according to other criteria. One possible criterion is to choose copies for which the Clebsch-Gordan coefficients have a simple form. A method fulfilling this is introduced for the tensor product of three irreps of \$su(2)\$. This method is then extended to the tensor product of two irreps of \$su(3)\$. In both cases the method is shown to construct a full nested sequence of basis independent highest weight subspaces. Another possible criterion is to make choices which are intrinsic, independent of all choices of bases. This is investigated in the final part of the thesis with a basis independent method that applies to the tensor product of finite dimensional irreps of any semisimple Lie algebra over \$\mathbb{C}\$.
33

#### Scalar and Vector Coherent State Representations of Compact and Non-compact Symplectic Groups

Shorser, Lindsey 05 September 2012 (has links)
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation \$T\$ of a real algebraic Lie group \$G\$. This requires defining an inner product on the Hilbert space \$\mathbb{H}\$ that carries the representation \$T\$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of \$T\$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (\$\mathfrak{F}_{\mathbb{H}_D}\$, \$\mathbb{H}_D\$, \$\mathfrak{F}^{\mathfrak{H}_D}\$) of Bargmann spaces. Coherent state wave functions -- the elements of \$\mathfrak{F}_{\mathbb{H}_D}\$ and of \$\mathfrak{F}^{\mathbb{H}_D}\$ -- are used to define the inner product on \$\mathbb{H}_D\$ in a way that simplifies the calculation of the overlaps. This inner product and the group action \$\Gamma\$ of \$G\$ on \$\mathfrak{F}^{\mathbb{H}_D}\$ are used to formulate expressions for the matrix elements of \$T\$ with coefficients from the given subrepresentation. The process of finding an explicit expression for \$\Gamma\$ relies on matrix factorizations in the complexification of \$G\$ even though the representation \$T\$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and \$\Gamma\$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in \$\mathbb{H}\$. The scalar and vector coherent state methods will both be applied to Sp(\$n\$) and Sp(\$n,\mathbb{R}\$).
34

#### Resolving Multiplicities in the Tensor Product of Irreducible Representations of Semisimple Lie Algebras

Brooke, David John 20 January 2009 (has links)
When the tensor product of two irreducible representations contains multiple copies of some of its irreducible constituents, there is a problem of choosing specific copies: resolving the multiplicity. This is typically accomplished by some ad hoc method chosen primarily for convenience in labelling and calculations. This thesis addresses the possibility of making choices according to other criteria. One possible criterion is to choose copies for which the Clebsch-Gordan coefficients have a simple form. A method fulfilling this is introduced for the tensor product of three irreps of \$su(2)\$. This method is then extended to the tensor product of two irreps of \$su(3)\$. In both cases the method is shown to construct a full nested sequence of basis independent highest weight subspaces. Another possible criterion is to make choices which are intrinsic, independent of all choices of bases. This is investigated in the final part of the thesis with a basis independent method that applies to the tensor product of finite dimensional irreps of any semisimple Lie algebra over \$\mathbb{C}\$.
35

#### Scalar and Vector Coherent State Representations of Compact and Non-compact Symplectic Groups

Shorser, Lindsey 05 September 2012 (has links)
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation \$T\$ of a real algebraic Lie group \$G\$. This requires defining an inner product on the Hilbert space \$\mathbb{H}\$ that carries the representation \$T\$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of \$T\$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (\$\mathfrak{F}_{\mathbb{H}_D}\$, \$\mathbb{H}_D\$, \$\mathfrak{F}^{\mathfrak{H}_D}\$) of Bargmann spaces. Coherent state wave functions -- the elements of \$\mathfrak{F}_{\mathbb{H}_D}\$ and of \$\mathfrak{F}^{\mathbb{H}_D}\$ -- are used to define the inner product on \$\mathbb{H}_D\$ in a way that simplifies the calculation of the overlaps. This inner product and the group action \$\Gamma\$ of \$G\$ on \$\mathfrak{F}^{\mathbb{H}_D}\$ are used to formulate expressions for the matrix elements of \$T\$ with coefficients from the given subrepresentation. The process of finding an explicit expression for \$\Gamma\$ relies on matrix factorizations in the complexification of \$G\$ even though the representation \$T\$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and \$\Gamma\$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in \$\mathbb{H}\$. The scalar and vector coherent state methods will both be applied to Sp(\$n\$) and Sp(\$n,\mathbb{R}\$).
36

#### Matrixordnung in der Lietheorie

Betz, Benedikt. Unknown Date (has links) (PDF)
Universiẗat, Diss., 2004--Saarbrücken.
37

#### Εισαγωγή στην θεωρία των συμμετρικών χώρων

Στουφής, Διονύσιος 27 June 2012 (has links)
Η θεωρία των συμμετρικών χώρων αποτελεί μια σπουδαία κλάση των ομογενών χώρων, με εφαρμογές σε πολλούς κλάδους των μαθηματικών όπως στην αλγεβρική και την διαφορική γεωμετρία. Σε αυτήν την εργασία θα δώσουμμε τον ορισμό των συμμετρικών χώρων, τα βασικά τους χαρακτηριστικά και την ταξινόμησή τους. Θα περιγράψουμε τους χώρους αυτούς κυρίως αλγεβρικά, οπότε δεν θεωρείται απαραίτητο από τον αναγνώστη να γνωρίζει εκτενώς την θεωρία της διαφορικής γεωμετρίας για να κατανοήσει πλήρως την εργασία. / The theory of symmetric spaces is an important class of homogeneous spaces, with applications in many branches of mathematics such as algebraic and differential geometry. In this work we will define the symmetric spaces, their key features and sort them. We will describe these spaces mainly algebraic, so it is not considered necessary by the reader to know in detail the theory of differential geometry to understand the work.
38

#### Monopolos magnéticos Z2 em teorias de Yang-Mills-Higgs com simetria de gauge SU(n)

Liebgott, Paulo Juliano January 2009 (has links)
Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-graduação em Física / Made available in DSpace on 2012-10-24T18:37:40Z (GMT). No. of bitstreams: 1 263132.pdf: 526017 bytes, checksum: 5840fcf1dc0e49a55cf108092da716f2 (MD5) / Monopolos magnéticos têm sido objetos de grande interesse nos últimos anos, principalmente por serem previstos em algumas teorias de grande unificação e por, possivelmente, serem relevantes no fenômeno do confinamento em QCD. Consideramos uma teoria de Yang-Mills-Higgs com simetria de gauge SU(n) quebrada espontaneamente em SO(n) que apresenta condições topológicas necessárias para a existência de monopolos Z2. Construímos as formas assintóticas desses monopolos, considerando duas quebras distintas do SU(n) em SO(n), e verificamos que os monopolos fundamentais estão associados aos pesos da representação definidora da álgebra so(n)v.
39

#### Álgebras de Lie e aplicações à sistemas alternantes /

Nascimento, Rildo Pinheiro do. January 2005 (has links)