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University of Arizona
1.
Dinius, Joseph.
Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems
.
Degree: 2014, University of Arizona
URL: http://hdl.handle.net/10150/315858
► The theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the…
(more)
▼ The theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the Multiplicative Ergodic Theorem of Oseledec. The numerical challenges and algorithms to approximate Lyapunov exponents and vectors are described, with multiple illustrative examples. A novel generalized impulsive collision rule is derived for particle systems interacting pairwise. This collision rule is constructed to address the question of whether or not the quantitative measures of chaos (e.g. Lyapunov exponents and Kolmogorov-Sinai entropy) can be reduced in these systems. Major results from previous studies of hard-disk systems, which interact via elastic collisions, are summarized and used as a framework for the study of the generalized collision rule. Numerical comparisons between the elastic and new generalized rules reveal many qualitatively different features between the two rules. Chaos reduction in the new rule through appropriate parameter choice is demonstrated, but not without affecting the structural properties of the Lyapunov spectra (e.g. symmetry and conjugate-pairing) and of the tangent space decomposition (e.g. hyperbolicity and domination of the Oseledec splitting). A novel measure of the degree of domination of the Oseledec splitting is developed for assessing the impact of fluctuations in the local Lyapunov exponents on the observation of coherent structures in perturbation vectors corresponding to slowly growing (or contracting) modes. The qualitatively different features observed between the dynamics of generalized and elastic collisions are discussed in the context of numerical simulations. Source code and complete descriptions for the simulation models used are provided.
Advisors/Committee Members: Lega, Joceline (advisor), Lega, Joceline (committeemember), Flaschka, Hermann (committeemember), Sanfelice, Ricardo (committeemember).
Subjects/Keywords: Dynamical systems;
Lyapunov exponents;
Statistical mechanics;
Applied Mathematics;
chaos
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APA (6th Edition):
Dinius, J. (2014). Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/315858
Chicago Manual of Style (16th Edition):
Dinius, Joseph. “Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems
.” 2014. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/315858.
MLA Handbook (7th Edition):
Dinius, Joseph. “Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems
.” 2014. Web. 22 Jan 2021.
Vancouver:
Dinius J. Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems
. [Internet] [Doctoral dissertation]. University of Arizona; 2014. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/315858.
Council of Science Editors:
Dinius J. Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems
. [Doctoral Dissertation]. University of Arizona; 2014. Available from: http://hdl.handle.net/10150/315858
2.
Yang, Bole.
Algebraic Aspects of the Dispersionless Limit of the Discrete Nonlinear Schrödinger Equation
.
Degree: 2013, University of Arizona
URL: http://hdl.handle.net/10150/297064
► We study the DNLS and its dispersionless limit based on a family of matrices, named after Cantero, Moral, and Velazquez (CMV). The work is an…
(more)
▼ We study the DNLS and its dispersionless limit based on a family of matrices, named after Cantero, Moral, and Velazquez (CMV). The work is an analog to that of the Toda lattice and dispersionless Toda. We rigorously introduce the constants of motion and matrix symbols of the dispersionless limit of the DNLS. The thesis is an algebraic preparation for some potential geometry setup in the continuum sense as the next step.
Advisors/Committee Members: Flaschka, Hermann (advisor), McLaughlin, Kenneth (committeemember), Pickrell, Douglas (committeemember), Restrepo, Juan (committeemember), Flaschka, Hermann (committeemember).
Subjects/Keywords: Applied Mathematics
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APA (6th Edition):
Yang, B. (2013). Algebraic Aspects of the Dispersionless Limit of the Discrete Nonlinear Schrödinger Equation
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/297064
Chicago Manual of Style (16th Edition):
Yang, Bole. “Algebraic Aspects of the Dispersionless Limit of the Discrete Nonlinear Schrödinger Equation
.” 2013. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/297064.
MLA Handbook (7th Edition):
Yang, Bole. “Algebraic Aspects of the Dispersionless Limit of the Discrete Nonlinear Schrödinger Equation
.” 2013. Web. 22 Jan 2021.
Vancouver:
Yang B. Algebraic Aspects of the Dispersionless Limit of the Discrete Nonlinear Schrödinger Equation
. [Internet] [Doctoral dissertation]. University of Arizona; 2013. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/297064.
Council of Science Editors:
Yang B. Algebraic Aspects of the Dispersionless Limit of the Discrete Nonlinear Schrödinger Equation
. [Doctoral Dissertation]. University of Arizona; 2013. Available from: http://hdl.handle.net/10150/297064

University of Arizona
3.
Damianou, Pantelis Andrea.
Nonlinear Poisson brackets.
Degree: 1989, University of Arizona
URL: http://hdl.handle.net/10150/184704
► A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them…
(more)
▼ A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. Among their properties: new brackets are generated from old ones by using Lie-derivatives in the direction of certain vector fields; the infinite sequences obtained consist of compatible Poisson brackets in which the constants of motion for the Toda lattice are in involution. The vector fields in the construction are unique up to addition of a Hamiltonian vector field. Similarly the Poisson brackets are unique up to addition of a trivial Poisson bracket. These are Poisson tensors generated by wedge products of Hamiltonian vector fields. The non-trivial brackets may also be obtained by the use of r-matrices; we give formulas and prove this for the quadratic and cubic Toda brackets. We also indicate how these results can be generalized to other (semisimple) Toda flows and we give explicit formulas for the rank 2 Lie algebra of type B₂. The main tool in this calculation is Dirac's constraint bracket formula. Finally we study nonlinear Poisson brackets associated with orbits through nilpotent conjugacy classes in gl(n, R) and formulate some conjectures. We determine the degree of the transverse Poisson structure through such nilpotent elements in gl(n, R) for n ≤ 7. This is accomplished also by the use of Dirac's bracket formula.
Advisors/Committee Members: Flaschka, Hermann (advisor).
Subjects/Keywords: Poisson algebras.;
Poisson manifolds.
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APA ·
Chicago ·
MLA ·
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CSE |
Export
to Zotero / EndNote / Reference
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APA (6th Edition):
Damianou, P. A. (1989). Nonlinear Poisson brackets.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/184704
Chicago Manual of Style (16th Edition):
Damianou, Pantelis Andrea. “Nonlinear Poisson brackets.
” 1989. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/184704.
MLA Handbook (7th Edition):
Damianou, Pantelis Andrea. “Nonlinear Poisson brackets.
” 1989. Web. 22 Jan 2021.
Vancouver:
Damianou PA. Nonlinear Poisson brackets.
[Internet] [Doctoral dissertation]. University of Arizona; 1989. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/184704.
Council of Science Editors:
Damianou PA. Nonlinear Poisson brackets.
[Doctoral Dissertation]. University of Arizona; 1989. Available from: http://hdl.handle.net/10150/184704

University of Arizona
4.
El Hadrami, Mohamed Lemine Ould, 1962-.
Poisson algebras and convexity
.
Degree: 1996, University of Arizona
URL: http://hdl.handle.net/10150/290675
► In this dissertation, we identify a subgroup Tˢ of Dˢ(μ), the group of Sobolev symplectomorphisms of CP (n), n = 1,2 that has all the…
(more)
▼ In this dissertation, we identify a subgroup Tˢ of Dˢ(μ), the group of Sobolev symplectomorphisms of CP (n), n = 1,2 that has all the properties of a torus of a compact finite dimensional Lie group. We prove that Tˢ: (1) topologically is a submanifold of Dˢ(μ); (2) algebraically is a maximal abelian subgroup of Dˢ(μ); (3) geometrically is flat and totally geodesic. We also characterize the doubly stochastic operators on measurable spaces and use this result to extend the convexity Theorem of T. Bloch, H.
Flaschka and T. Ratiu.
Advisors/Committee Members: Flaschka, Hermann (advisor).
Subjects/Keywords: Mathematics.
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APA ·
Chicago ·
MLA ·
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Export
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APA (6th Edition):
El Hadrami, Mohamed Lemine Ould, 1. (1996). Poisson algebras and convexity
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/290675
Chicago Manual of Style (16th Edition):
El Hadrami, Mohamed Lemine Ould, 1962-. “Poisson algebras and convexity
.” 1996. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/290675.
MLA Handbook (7th Edition):
El Hadrami, Mohamed Lemine Ould, 1962-. “Poisson algebras and convexity
.” 1996. Web. 22 Jan 2021.
Vancouver:
El Hadrami, Mohamed Lemine Ould 1. Poisson algebras and convexity
. [Internet] [Doctoral dissertation]. University of Arizona; 1996. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/290675.
Council of Science Editors:
El Hadrami, Mohamed Lemine Ould 1. Poisson algebras and convexity
. [Doctoral Dissertation]. University of Arizona; 1996. Available from: http://hdl.handle.net/10150/290675
5.
Pittman-Polletta, Benjamin Rafael.
Factorization in unitary loop groups and reduced words in affine Weyl groups.
Degree: 2010, University of Arizona
URL: http://hdl.handle.net/10150/194348
► The purpose of this dissertation is to elaborate, with specific examples and calculations, on a new refinement of triangular factorization for the loop group of…
(more)
▼ The purpose of this dissertation is to elaborate, with specific examples and calculations, on a new refinement of triangular factorization for the loop group of a simple, compact Lie group K, first appearing in Pickrell & Pittman-Polletta 2010. This new factorization allows us to write a smooth map from the unit circle into K (having a triangular factorization) as a triply infinite product of loops, each of which depends on a single complex parameter. These parameters give a set of coordinates on the loop group of K.The order of the factors in this refinement is determined by an infinite sequence of simple generators in the affine Weyl group associated to K, having certain properties. The major results of this dissertation are examples of such sequences for all the classical Weyl groups.We also produce a variation of this refinement which allows us to write smooth maps from the unit circle into the special unitary group of n by n matrices as products of 2n+1 infinite products. By analogy with the semisimple analog of our factorization, we suggest that this variation of the refinement has simpler combinatorics than that appearing in Pickrell & Pittman-Polletta 2010.
Advisors/Committee Members: Pickrell, Doug (advisor), Pickrell, Doug (committeemember), Glickenstein, David (committeemember), Palmer, John (committeemember), Flaschka, Hermann (committeemember), Venkataramani, Shankar (committeemember).
Subjects/Keywords: affine weyl groups;
birkhoff factorization;
loop groups;
reduced words;
triangular factorization
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Pittman-Polletta, B. R. (2010). Factorization in unitary loop groups and reduced words in affine Weyl groups.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/194348
Chicago Manual of Style (16th Edition):
Pittman-Polletta, Benjamin Rafael. “Factorization in unitary loop groups and reduced words in affine Weyl groups.
” 2010. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/194348.
MLA Handbook (7th Edition):
Pittman-Polletta, Benjamin Rafael. “Factorization in unitary loop groups and reduced words in affine Weyl groups.
” 2010. Web. 22 Jan 2021.
Vancouver:
Pittman-Polletta BR. Factorization in unitary loop groups and reduced words in affine Weyl groups.
[Internet] [Doctoral dissertation]. University of Arizona; 2010. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/194348.
Council of Science Editors:
Pittman-Polletta BR. Factorization in unitary loop groups and reduced words in affine Weyl groups.
[Doctoral Dissertation]. University of Arizona; 2010. Available from: http://hdl.handle.net/10150/194348
6.
Pounder, Kyle.
Nearly Singular Jacobi Matrices and Applications to the Finite Toda Lattice
.
Degree: 2018, University of Arizona
URL: http://hdl.handle.net/10150/627692
► In this dissertation, we consider a singular limit of the inverse spectral map for Jacobi matrices. The main results of the analysis are quite general…
(more)
▼ In this dissertation, we consider a singular limit of the inverse spectral map for Jacobi matrices. The main results of the analysis are quite general estimates for the entries of a Jacobi matrix under certain assumptions about the relative sizes of the weights (norming constants). We apply these estimates to provide a detailed long time asymptotic analysis of the finite Toda lattice. The formulas we obtain improve upon the classical results of Moser by giving precise estimates of the associated error. Moreover, the Riemann-Hilbert techniques allow one, if they should so desire, to compute the complete asymptotic expansions for the various dynamical quantities. Finally, we apply our general estimates to study the time evolution of "nearly singular" Jacobi matrices.
Advisors/Committee Members: McLaughlin, Kenneth T-R (advisor), Ercolani, Nicholas (advisor), Sethuraman, Sunder (committeemember), Jenkins, Robert (committeemember), Flaschka, Hermann (committeemember).
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Pounder, K. (2018). Nearly Singular Jacobi Matrices and Applications to the Finite Toda Lattice
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/627692
Chicago Manual of Style (16th Edition):
Pounder, Kyle. “Nearly Singular Jacobi Matrices and Applications to the Finite Toda Lattice
.” 2018. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/627692.
MLA Handbook (7th Edition):
Pounder, Kyle. “Nearly Singular Jacobi Matrices and Applications to the Finite Toda Lattice
.” 2018. Web. 22 Jan 2021.
Vancouver:
Pounder K. Nearly Singular Jacobi Matrices and Applications to the Finite Toda Lattice
. [Internet] [Doctoral dissertation]. University of Arizona; 2018. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/627692.
Council of Science Editors:
Pounder K. Nearly Singular Jacobi Matrices and Applications to the Finite Toda Lattice
. [Doctoral Dissertation]. University of Arizona; 2018. Available from: http://hdl.handle.net/10150/627692
7.
Comeau, Darin.
Conceptual and Numerical Modeling of Ice in a Global Climate Framework
.
Degree: 2013, University of Arizona
URL: http://hdl.handle.net/10150/297044
► Ice is both an important indicator, and agent, of climate change. In this work we consider conceptual and numerical models of ice in the global…
(more)
▼ Ice is both an important indicator, and agent, of climate change. In this work we consider conceptual and numerical models of ice in the global climate system on two ends of the climate modeling spectrum. On the simple end of the spectrum, we introduce a low-dimensional global climate model to investigate the role of oceanic heat transport on ice cover, particularly in the initiation of global ice cover, known as Snowball Earth events. We find that oceanic heat transport is effective at keeping the ice margin at high latitudes, and neglecting to include oceanic heat transport can lead to drastically different climate states. On the complex end of the climate modeling spectrum, we implement an iceberg parameterization in the Los Alamos National Laboratory's sea ice model CICE. Novel to our approach is we model icebergs in two frameworks - as Lagrangian particles, and as an Eulerian field. We allow icebergs to interact dynamically with the surrounding sea ice, and the modeled iceberg thermodynamics allow them to melt as they drift, serving as vehicles of freshwater injection into the ocean from land ice sheets. We focus on Antarctic icebergs, which tend to be larger than those found in the Arctic and are more likely to encounter substantial sea ice pack.
Advisors/Committee Members: Restrepo, Juan M (advisor), Flaschka, Hermann (committeemember), Venkataramani, Shankar (committeemember), Lin, Kevin (committeemember), Restrepo, Juan M. (committeemember).
Subjects/Keywords: Applied Mathematics;
climate
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
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APA (6th Edition):
Comeau, D. (2013). Conceptual and Numerical Modeling of Ice in a Global Climate Framework
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/297044
Chicago Manual of Style (16th Edition):
Comeau, Darin. “Conceptual and Numerical Modeling of Ice in a Global Climate Framework
.” 2013. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/297044.
MLA Handbook (7th Edition):
Comeau, Darin. “Conceptual and Numerical Modeling of Ice in a Global Climate Framework
.” 2013. Web. 22 Jan 2021.
Vancouver:
Comeau D. Conceptual and Numerical Modeling of Ice in a Global Climate Framework
. [Internet] [Doctoral dissertation]. University of Arizona; 2013. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/297044.
Council of Science Editors:
Comeau D. Conceptual and Numerical Modeling of Ice in a Global Climate Framework
. [Doctoral Dissertation]. University of Arizona; 2013. Available from: http://hdl.handle.net/10150/297044
8.
Acosta Jaramillo, Enrique.
Leading Order Asymptotics of a Multi-Matrix Partition Function for Colored Triangulations
.
Degree: 2013, University of Arizona
URL: http://hdl.handle.net/10150/293410
► We study the leading order asymptotics of a Random Matrix theory partition function related to colored triangulations. This partition function comes from a three Hermitian…
(more)
▼ We study the leading order asymptotics of a Random Matrix theory partition function related to colored triangulations. This partition function comes from a three Hermitian matrix model that has been introduced in the physics literature. We provide a detailed and precise description of the combinatorial objects that the partition function counts that has not appeared previously in the literature. We also provide a general framework for studying the leading order asymptotics of an N dimensional integral that one encounters studying the partition function of colored triangulations. The results are obtained by generalizing well know results for integrals coming from Hermitian matrix models with only one matrix that give the leading order asymptiotics in terms of a finite dimensional variational problem. We apply these results to the partition function for colored triangulations to show that the minimizing density of the variational problem is unique, and agrees with the one proposed in the physics literature. This provides the first complete mathematically rigorous description of the leading order asymptotics of this matrix model for colored triangulations.
Advisors/Committee Members: McLaughlin, Kenneth D. T-R (advisor), Ercolani, Nicholas M. (committeemember), Flaschka, Hermann (committeemember), Kennedy, Thomas G. (committeemember), McLaughlin, Kenneth D. T-R. (committeemember).
Subjects/Keywords: Enumeration;
Maps;
Triangulations;
Mathematics;
Asymptotics
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
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APA (6th Edition):
Acosta Jaramillo, E. (2013). Leading Order Asymptotics of a Multi-Matrix Partition Function for Colored Triangulations
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/293410
Chicago Manual of Style (16th Edition):
Acosta Jaramillo, Enrique. “Leading Order Asymptotics of a Multi-Matrix Partition Function for Colored Triangulations
.” 2013. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/293410.
MLA Handbook (7th Edition):
Acosta Jaramillo, Enrique. “Leading Order Asymptotics of a Multi-Matrix Partition Function for Colored Triangulations
.” 2013. Web. 22 Jan 2021.
Vancouver:
Acosta Jaramillo E. Leading Order Asymptotics of a Multi-Matrix Partition Function for Colored Triangulations
. [Internet] [Doctoral dissertation]. University of Arizona; 2013. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/293410.
Council of Science Editors:
Acosta Jaramillo E. Leading Order Asymptotics of a Multi-Matrix Partition Function for Colored Triangulations
. [Doctoral Dissertation]. University of Arizona; 2013. Available from: http://hdl.handle.net/10150/293410

University of Arizona
9.
Shipman, Barbara Anne.
Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.
Degree: 1995, University of Arizona
URL: http://hdl.handle.net/10150/187199
► Our study describes the structure of the completely integrable system known as the full Kostant-Toda lattice in terms of the rich geometry of complex generalized…
(more)
▼ Our study describes the structure of the completely integrable system known as the full Kostant-Toda lattice in terms of the rich geometry of complex generalized flag manifolds and the information encoded in their momentum polytopes. The space in which the system evolves is a Poisson manifold which is essentially the dual of a Borel subalgebra of a Lie algebra, and the symplectic leaves are the coadjoint orbits. We extend the results of Ercolani,
Flaschka, and Singer in (4) in which an embedding of an isospectral submanifold of the phase space into the flag manifold is used to study the geometry of the "generic" compactified level sets of a particular family of constants of motion. In a detailed analysis of the full Sl(4,C) Kostant-Toda lattice, we consider all types of level sets, in particular those which do not satisfy the genericity conditions of (4). The breakdown of these conditions is reflected in the types of nongeneric strata to which the torus orbits in a "special" level set belong. This degeneration corresponds to certain decompositions of the momentum polytopes, which we explain in terms of representation theory. We discover a fundamental two-fold symmetry intrinsic to this geometry which appears in the phase space as an involution preserving the constants of motion, and we express it in terms of duality in the flag manifold and the pairing between a representation of a Lie algebra and its dual. Several chapters are devoted to the study of a double fibration of a generic symplectic leaf by the level sets of two distinct involutive families of integrals for the full Sl(4,C) Kostant-Toda lattice. We describe the symmetries of these two fibrations and determine the monodromy around their singular fibers. Finally, we show how the configuration of the lower-dimensional symplectic leaves of the Poisson structure in this example is revealed in the geometry of the flag manifold and its momentum polytope.
Advisors/Committee Members: Flaschka, Hermann (committeemember), Pickrell, Doug (committeemember).
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Shipman, B. A. (1995). Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/187199
Chicago Manual of Style (16th Edition):
Shipman, Barbara Anne. “Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.
” 1995. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/187199.
MLA Handbook (7th Edition):
Shipman, Barbara Anne. “Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.
” 1995. Web. 22 Jan 2021.
Vancouver:
Shipman BA. Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.
[Internet] [Doctoral dissertation]. University of Arizona; 1995. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/187199.
Council of Science Editors:
Shipman BA. Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.
[Doctoral Dissertation]. University of Arizona; 1995. Available from: http://hdl.handle.net/10150/187199

University of Arizona
10.
Cruz-Pacheco, Gustavo.
The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation.
Degree: 1995, University of Arizona
URL: http://hdl.handle.net/10150/187238
► This work consists of a study of the complex Ginzburg-Landau equation (CGL) as a perturbation of the nonlinear Schrodinger equation (NLS) in one dimension under…
(more)
▼ This work consists of a study of the complex Ginzburg-Landau equation (CGL) as a perturbation of the nonlinear Schrodinger equation (NLS) in one dimension under periodic boundary conditions. Using an averaging technique which is similar to a Melnikov method for pde's, necessary conditions are derived for the persistence of NLS solutions under the CGL perturbation. For the traveling wave solutions, these conditions are derived for a general nonlinearity and written explicitly as two equations for the two continuous parameters which determine the NLS traveling wave. It is shown using a Melnikov argument that in this case these two conditions are sufficient provided they satisfy a transversality condition. As a concrete example, the equations for the parameters are solved numerically in the important case of the CGL equation with a cubic nonlinearity. For the case of the CGL equation with a general power nonlinearity, it is proved that the NLS homoclinic orbits to rotating waves are destroyed by the CGL perturbation. Special attention is dedicated to the cubic case. For this nonlinearity, the NLS equation is a completely integrable Hamiltonian system and a much larger family of its solutions can be written explicitly. The necessary conditions for the persistence of the NLS isospectral manifold are written explicitly as a system of equations for the simple periodic eigenvalues. As an example, the conditions for an even genus two solution are written down as a system of three equations with three unknowns.
Advisors/Committee Members: Ercolani, Nicholas M. (committeemember), Flaschka, Hermann (committeemember).
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Cruz-Pacheco, G. (1995). The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/187238
Chicago Manual of Style (16th Edition):
Cruz-Pacheco, Gustavo. “The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation.
” 1995. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/187238.
MLA Handbook (7th Edition):
Cruz-Pacheco, Gustavo. “The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation.
” 1995. Web. 22 Jan 2021.
Vancouver:
Cruz-Pacheco G. The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation.
[Internet] [Doctoral dissertation]. University of Arizona; 1995. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/187238.
Council of Science Editors:
Cruz-Pacheco G. The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation.
[Doctoral Dissertation]. University of Arizona; 1995. Available from: http://hdl.handle.net/10150/187238

University of Arizona
11.
Stark, Donald Richard.
Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.
Degree: 1995, University of Arizona
URL: http://hdl.handle.net/10150/187363
► Numerical and analytical studies are undertaken for the "inviscid" limit of the complex Ginzburg-Landau (CGL) equation with the objective of studying the applicability of paradigms…
(more)
▼ Numerical and analytical studies are undertaken for the "inviscid" limit of the complex Ginzburg-Landau (CGL) equation with the objective of studying the applicability of paradigms from finite dimensional dynamical systems and statistical mechanics to the case of an infinite dimensional dynamical system. The analytical results rely on exploiting the structure of this limit, which becomes the nonlinear Schrodinger (NLS) equation. In the NLS limit the CGL equation can exhibit strong spatio-temporal chaos. The initial growth of the bursts closely mimics the blowup solutions of the NLS equation. The study of this turbulent behavior focuses on the inertial range of the time-averaged wavenumber spectrum. Analytical estimates of the decay rate are constructed assuming both structure driven and homogeneous turbulence, and are compared with numerical simulations. The quintic case is observed to have a stronger decay rate than what is predicted by either theory. This reflects the dominance of dissipation in the dynamics. In the septic case, two distinct inertial ranges are observed. This combination suggests that the evolution of a single burst, on average, is predominantly due to the self-focusing mechanism of blowup NLS in the initial stage, and regularization effects of dissipation in the final stage. Because the initial stage is primarily influenced by the NLS structure, the rate of decay for this range is close to the decay predicted for the structure driven turbulence. In a numerical experiment it is observed that some NLS solutions survive the deformation due to a CGL perturbation. In some cases the question of persistence can be addressed analytically using an averaging technique similar to a Melnikov method for pde's.
Advisors/Committee Members: Flaschka, Hermann (committeemember), Hyman, James (committeemember).
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
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APA (6th Edition):
Stark, D. R. (1995). Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/187363
Chicago Manual of Style (16th Edition):
Stark, Donald Richard. “Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.
” 1995. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/187363.
MLA Handbook (7th Edition):
Stark, Donald Richard. “Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.
” 1995. Web. 22 Jan 2021.
Vancouver:
Stark DR. Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.
[Internet] [Doctoral dissertation]. University of Arizona; 1995. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/187363.
Council of Science Editors:
Stark DR. Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.
[Doctoral Dissertation]. University of Arizona; 1995. Available from: http://hdl.handle.net/10150/187363

University of Arizona
12.
Calini, Annalisa Maria.
Integrable curve dynamics.
Degree: 1994, University of Arizona
URL: http://hdl.handle.net/10150/186987
► The Heisenberg Model of the integrable evolution of a continuous spin chain can be used to describe an integrable dynamics of curves in R ³.…
(more)
▼ The Heisenberg Model of the integrable evolution of a continuous spin chain can be used to describe an integrable dynamics of curves in R ³. The role of orthonormal frames of the curve is explored. In this framework a second Poisson structure for the Heisenberg Model is derived and the relation between the Heisenberg Model and the cubic Non-Linear Schrodinger Equation is explained. The Frenet frame of a curve is shown to be a Legendrian curve in the space of orthonormal frames with respect to a natural contact structure. As a consequence, generic singularities of the solution of the Heisenberg Model and topological invariants of the curve are computed. The family of multi-phase solutions of the Heisenberg Model and the corresponding curves are constructed with techniques of algebraic geometry. The relation with the Non-Linear Schrodinger Equation is explained also in this context. A formula for the Backlund transformation for the Heisenberg Model is derived and applied to construct orbits homoclinic to planar circles. As a result singular knots are obtained.
Advisors/Committee Members: Flaschka, Hermann (committeemember), Pickrell, Doug (committeemember).
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
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APA (6th Edition):
Calini, A. M. (1994). Integrable curve dynamics.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/186987
Chicago Manual of Style (16th Edition):
Calini, Annalisa Maria. “Integrable curve dynamics.
” 1994. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/186987.
MLA Handbook (7th Edition):
Calini, Annalisa Maria. “Integrable curve dynamics.
” 1994. Web. 22 Jan 2021.
Vancouver:
Calini AM. Integrable curve dynamics.
[Internet] [Doctoral dissertation]. University of Arizona; 1994. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/186987.
Council of Science Editors:
Calini AM. Integrable curve dynamics.
[Doctoral Dissertation]. University of Arizona; 1994. Available from: http://hdl.handle.net/10150/186987

University of Arizona
13.
McNicholas, Erin Mari.
Embedded Tree Structures and Eigenvalue Statistics of Genus Zero One-Face Maps
.
Degree: 2006, University of Arizona
URL: http://hdl.handle.net/10150/194030
► Using numerical simulations and combinatorics, this dissertation focuses on connections between random matrix theory and graph theory.We examine the adjacency matrices of three-regular graphs representing…
(more)
▼ Using numerical simulations and combinatorics, this dissertation focuses on connections between random matrix theory and graph theory.We examine the adjacency matrices of three-regular graphs representing one-face maps. Numerical studies have revealed that the limiting eigenvalue statistics of these matrices are the same as those of much larger, and more widely studied classes of random matrices. In particular, the eigenvalue density is described by the McKay density formula, and the distribution of scaled eigenvalue spacings appears to be that of the Gaussian Orthogonal Ensemble (GOE).A natural question is whether the eigenvalue statistics depend on the genus of the underlying map. We present an algorithm for generating random three-regular graphs representing genus zero one-face maps. Our numerical studies of these three-regular graphs have revealed that their eigenvalue statistics are strikingly different from those of three-regular graphs representing maps of higher genus. While our results indicate that there is a limiting eigenvalue density formula in the genus zero case, it is not described by any established density function. Furthermore, the scaled eigenvalue spacings appear to be described by the exponential distribution function, not the GOE spacing distribution.The embedded graph of a genus zero one-face map is a planar tree, and there is a correlation between its vertices and the primitive cycles of the associated three-regular graph. The second half of this dissertation examines the structure of these embedded planar trees. In particular, we show how the Dyck path representation can be used to recast questions about the probabilistic structure of random planar trees into straightforward counting problems. Using this Dyck path approach, we find:1. the expected number of degree k vertices adjacent to j degree d vertices in a random planar tree, 2. the structure of the planar tree's adjacency matrix under a natural labeling of the vertices, and 3. an explanation for the existence of eigenvalues with multiplicity greater than one in the tree's spectrum.
Advisors/Committee Members: Flaschka, Hermann (advisor), Kenneth, McLaughlin (committeemember), Grove, Larry (committeemember), Flaschka, Hermann (committeemember).
Subjects/Keywords: planar trees;
Dyck paths;
eigenvalue statistics
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
McNicholas, E. M. (2006). Embedded Tree Structures and Eigenvalue Statistics of Genus Zero One-Face Maps
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/194030
Chicago Manual of Style (16th Edition):
McNicholas, Erin Mari. “Embedded Tree Structures and Eigenvalue Statistics of Genus Zero One-Face Maps
.” 2006. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/194030.
MLA Handbook (7th Edition):
McNicholas, Erin Mari. “Embedded Tree Structures and Eigenvalue Statistics of Genus Zero One-Face Maps
.” 2006. Web. 22 Jan 2021.
Vancouver:
McNicholas EM. Embedded Tree Structures and Eigenvalue Statistics of Genus Zero One-Face Maps
. [Internet] [Doctoral dissertation]. University of Arizona; 2006. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/194030.
Council of Science Editors:
McNicholas EM. Embedded Tree Structures and Eigenvalue Statistics of Genus Zero One-Face Maps
. [Doctoral Dissertation]. University of Arizona; 2006. Available from: http://hdl.handle.net/10150/194030

University of Arizona
14.
Jenkins, Robert M.
Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data
.
Degree: 2009, University of Arizona
URL: http://hdl.handle.net/10150/193553
► The small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non…
(more)
▼ The small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The asymptotics for reectionless WKB-like initial data was worked out in [KMM03] and for the family q(x, 0) = sech^(1+(i/∈)μ in [TVZ04]. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors. In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q(x; 0) = qx[-L,L] for real amplitudes q. Using Riemann- Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an O(1) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.
Advisors/Committee Members: McLaughlin, Ken (advisor), Ercolani, Nick (committeemember), Flaschka, Hermann (committeemember).
Subjects/Keywords: NLS;
nonlinear waves;
Schrodinger;
square barrier
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Jenkins, R. M. (2009). Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/193553
Chicago Manual of Style (16th Edition):
Jenkins, Robert M. “Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data
.” 2009. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/193553.
MLA Handbook (7th Edition):
Jenkins, Robert M. “Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data
.” 2009. Web. 22 Jan 2021.
Vancouver:
Jenkins RM. Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data
. [Internet] [Doctoral dissertation]. University of Arizona; 2009. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/193553.
Council of Science Editors:
Jenkins RM. Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data
. [Doctoral Dissertation]. University of Arizona; 2009. Available from: http://hdl.handle.net/10150/193553

University of Arizona
15.
Lu, Yixia.
Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations
.
Degree: 2005, University of Arizona
URL: http://hdl.handle.net/10150/193894
► The Painleve analysis plays an important role in investigating local structure of the solutions of differential equations, while Lie symmetries provide powerful tools in global…
(more)
▼ The Painleve analysis plays an important role in investigating local structure of the solutions of differential equations, while Lie symmetries provide powerful tools in global solvability of equations. In this research, the method of Painleve analysis is applied to discrete nonlinear Schrodinger equations and to a family of second order nonlinear ordinary differential equations. Lie symmetries are studied together with the Painleve property for second order nonlinear ordinary differential equations.In the study of the local singularity of discrete nonlinear Schrodinger equations, the Painleve method shows the existence of solution blow up at finite time. It also determines the rate of blow-up. For second order nonlinear ordinary differential equations, the Painleve test is introduced and demonstrated in detail using several examples. These examples are used throughout the research. The Painleve property is shown to be significant for the integrability of a differential equation.After introducing one-parameter groups, a family of differential equations is determined for discussing solvability and for drawing more meaningful conclusions. This is the most general family of differential equations invariant under a given one-parameter group. The first part of this research is the classification of the integrals in the general solutions of differential equations obtained by quadratures. The second part is the application of Riemann surfaces and algebraic curves in the projective complex space to the integrands. The theories of Riemann surfaces and algebraic curves lead us to an effective way to understand the nature of the integral defined on a curve. Our theoretical work then concentrates on the blowing-up of algebraic curves at singular points. The calculation of the genus, which essentially determines the shape of a curve, becomes possible after a sequence of blowing-ups.The research shows that when combining both the Painleve property and Lie symmetries possessed by the differential equations studied in the thesis, the general solutions can be represented by either elementary functions or elliptic integrals.
Advisors/Committee Members: Ercolani, Nicholas M. (committeemember), Flaschka, Hermann (committeemember), Tabor, Michael (committeemember).
Subjects/Keywords: Painleve analysis;
Lie symmetries;
Integrability;
ODE
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Lu, Y. (2005). Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/193894
Chicago Manual of Style (16th Edition):
Lu, Yixia. “Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations
.” 2005. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/193894.
MLA Handbook (7th Edition):
Lu, Yixia. “Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations
.” 2005. Web. 22 Jan 2021.
Vancouver:
Lu Y. Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations
. [Internet] [Doctoral dissertation]. University of Arizona; 2005. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/193894.
Council of Science Editors:
Lu Y. Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations
. [Doctoral Dissertation]. University of Arizona; 2005. Available from: http://hdl.handle.net/10150/193894

University of Arizona
16.
Caine, John Arlo.
Poisson Structures on U/K and Applications
.
Degree: 2007, University of Arizona
URL: http://hdl.handle.net/10150/195363
► Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group…
(more)
▼ Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let g denote the complexification of the Lie algebra of U, g=u^C. Each u-compatible triangular decomposition g= n_- + h + n_+ determines a Poisson Lie group structure pi_U on U. The Evens-Lu construction produces a (U, pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that these actions are Hamiltonian and the momentum maps are computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.
Advisors/Committee Members: Pickrell, Douglas M (advisor), Flaschka, Hermann (committeemember), Foth, Philip (committeemember).
Subjects/Keywords: Poisson Geometry;
Triangular Factorization;
Symmetric Spaces
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Caine, J. A. (2007). Poisson Structures on U/K and Applications
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/195363
Chicago Manual of Style (16th Edition):
Caine, John Arlo. “Poisson Structures on U/K and Applications
.” 2007. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/195363.
MLA Handbook (7th Edition):
Caine, John Arlo. “Poisson Structures on U/K and Applications
.” 2007. Web. 22 Jan 2021.
Vancouver:
Caine JA. Poisson Structures on U/K and Applications
. [Internet] [Doctoral dissertation]. University of Arizona; 2007. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/195363.
Council of Science Editors:
Caine JA. Poisson Structures on U/K and Applications
. [Doctoral Dissertation]. University of Arizona; 2007. Available from: http://hdl.handle.net/10150/195363

University of Arizona
17.
Campini, Marco.
The fluid dynamical limits of the linearized Boltzmann equation.
Degree: 1991, University of Arizona
URL: http://hdl.handle.net/10150/185664
► The old question concerning the mathematical formulation of the fluid dynamic limits of kinetic theory is examined by studying the solution of the Cauchy problem…
(more)
▼ The old question concerning the mathematical formulation of the fluid dynamic limits of kinetic theory is examined by studying the solution of the Cauchy problem for two differently scaled linearized Boltzmann equations on periodic domain as the mean free path of the particles becomes small. Under minimal assumptions on the initial data, by using an a priori estimate, it is possible, in a Hilbert space functional frame, to prove the weak convergence of solutions toward a function that has the form of an infinitesimal maxwellian in the velocity variable. The velocity moments of this function are then proved to satisfy either the linearized Euler or the Stokes system of equations (depending on the chosen scaling), by passing to the limit in the conservation relations derived from the Boltzmann equation. A theorem injecting continuously the intersection of certain weak spaces into a normed one is proved. Together with properties of the Euler semigroup, this allows to show strong convergence of the first three moments of the distribution function toward the macroscopic quantities density, bulk velocity and temperature, solutions of the linearized Euler system. The Stokes case is treated somewhat differently, through the introduction of a result, proved by using the adjoint formulation for linear kinetic equations, that extends the averaging theory of Golse-Lions-Perthame-Sentis. The desired convergence for the divergence-free component of the second moment toward the macroscopic velocity is then shown.
Advisors/Committee Members: Levermore, David (advisor), Flaschka, Hermann (committeemember), Bayly, Bruce (committeemember).
Subjects/Keywords: Dissertations, Academic;
Mathematics;
Fluid dynamics;
Transport theory.
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Campini, M. (1991). The fluid dynamical limits of the linearized Boltzmann equation.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/185664
Chicago Manual of Style (16th Edition):
Campini, Marco. “The fluid dynamical limits of the linearized Boltzmann equation.
” 1991. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/185664.
MLA Handbook (7th Edition):
Campini, Marco. “The fluid dynamical limits of the linearized Boltzmann equation.
” 1991. Web. 22 Jan 2021.
Vancouver:
Campini M. The fluid dynamical limits of the linearized Boltzmann equation.
[Internet] [Doctoral dissertation]. University of Arizona; 1991. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/185664.
Council of Science Editors:
Campini M. The fluid dynamical limits of the linearized Boltzmann equation.
[Doctoral Dissertation]. University of Arizona; 1991. Available from: http://hdl.handle.net/10150/185664

University of Arizona
18.
Spiegler, Adam.
Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method
.
Degree: 2006, University of Arizona
URL: http://hdl.handle.net/10150/194821
► 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…
(more)
▼ 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.
Advisors/Committee Members: Flaschka, Hermann (advisor), Ercolani, Nicholas (committeemember), Pickrell, Doug (committeemember), Foth, Philip (committeemember), Zakharov, Vladimir (committeemember).
Subjects/Keywords: rigid body;
symplectic geometry;
Lie algebra;
energy-Casimir
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Spiegler, A. (2006). Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/194821
Chicago Manual of Style (16th Edition):
Spiegler, Adam. “Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method
.” 2006. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/194821.
MLA Handbook (7th Edition):
Spiegler, Adam. “Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method
.” 2006. Web. 22 Jan 2021.
Vancouver:
Spiegler A. Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method
. [Internet] [Doctoral dissertation]. University of Arizona; 2006. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/194821.
Council of Science Editors:
Spiegler A. Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method
. [Doctoral Dissertation]. University of Arizona; 2006. Available from: http://hdl.handle.net/10150/194821

University of Arizona
19.
Garcia-Naranjo, Luis Constantino.
Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
.
Degree: 2007, University of Arizona
URL: http://hdl.handle.net/10150/195845
► We present a geometric construction of almost Poisson brackets for nonholonomic mechanical systems whose configuration space is a Lie group G. We study the so-called…
(more)
▼ We present a geometric construction of almost Poisson brackets for nonholonomic mechanical systems whose configuration space is a Lie group G. We study the so-called LL and LR systems where the kinetic energy defines a left invariant metric on G and the constraints are invariant with respect to left (respectively right) translation on G.For LL systems, the equations on the momentum phase space, T*G, can be left translated onto g*, the dual space of the Lie algebra g. We show that the reduced equations on g* can be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the standard Lie-Poisson bracket onto the constraint space.For LR systems, we use ideas of semidirect product reduction to transfer the equations on T*G into the dual Lie algebra, s*, of a semidirect product. This provides a natural Lie algebraic setting for the equations of motion commonly found in the literature. We show that these equations can also be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the Lie-Poisson structure on s* onto a constraint submanifold.In both cases the constraint functions are Casimirs of the bracket and are satisfied automatically. Our construction is a natural generalization of the classical ideas of Lie-Poisson and semidirect product reduction to the nonholonomic case. It also sets a convenient stage for the study of Hamiltonization of certain nonholonomic systems.Our examples include the Suslov and the Veselova problems of constrained motion of a rigid body, and the Chaplygin sleigh.In addition we study the almost Poisson reduction of the Chaplygin sphere. We show that the bracket given byBorisov and Mamaev is obtained by reducing a nonstandard almost Poisson bracket that is obtained by projecting a non-canonical bivector onto the constraint submanifold using the Lagrange-D'Alembert principle.The examples that we treat show that it is possible to cast the reduced equations of motion of certain nonholonomic systems in Hamiltonian form (in the Poisson formulation) either by multiplication by a conformal factor, by the use of nonstandard brackets or simply by reduction methods.
Advisors/Committee Members: Flaschka, Hermann (advisor), Goriely, Alain (committeemember), Ercolani, Nicholas (committeemember), Newell, Alan (committeemember).
Subjects/Keywords: nonholonomic;
almost Poisson bracket;
Lie groups;
mechanics;
Hamiltonization;
reduction
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Garcia-Naranjo, L. C. (2007). Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/195845
Chicago Manual of Style (16th Edition):
Garcia-Naranjo, Luis Constantino. “Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
.” 2007. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/195845.
MLA Handbook (7th Edition):
Garcia-Naranjo, Luis Constantino. “Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
.” 2007. Web. 22 Jan 2021.
Vancouver:
Garcia-Naranjo LC. Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
. [Internet] [Doctoral dissertation]. University of Arizona; 2007. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/195845.
Council of Science Editors:
Garcia-Naranjo LC. Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
. [Doctoral Dissertation]. University of Arizona; 2007. Available from: http://hdl.handle.net/10150/195845

University of Arizona
20.
McShane, Janet Marie.
Computation of polynomial invariants of finite groups.
Degree: 1992, University of Arizona
URL: http://hdl.handle.net/10150/186000
► If G is a finite subgroup of GL(n,K), K a field of characteristic 0, it is well known that the algebra I of polynomial invariants…
(more)
▼ If G is a finite subgroup of GL(n,K), K a field of characteristic 0, it is well known that the algebra I of polynomial invariants of G is Cohen-Macaulay. Consequently I has a subalgebra J of Krull dimension n so that I is a free J-module of finite rank. A sequence (f₁,...,f(n);g₁,...,g(m)) of homogeneous invariants is a Cohen-Macaulay (or CM) basis if J = K[f₁,...,f(n)] and {g₁,...,g(m)} is a basis for I as a J-module. We discuss an algorithm, and an implementation using the systems GAP and Maple, for the calculation of CM-bases.
Advisors/Committee Members: Fan, Paul (committeemember), Gay, David (committeemember), Flaschka, Hermann (committeemember), Laetsch, Theodore (committeemember).
Subjects/Keywords: Dissertations, Academic.;
Mathematics.
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APA ·
Chicago ·
MLA ·
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APA (6th Edition):
McShane, J. M. (1992). Computation of polynomial invariants of finite groups.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/186000
Chicago Manual of Style (16th Edition):
McShane, Janet Marie. “Computation of polynomial invariants of finite groups.
” 1992. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/186000.
MLA Handbook (7th Edition):
McShane, Janet Marie. “Computation of polynomial invariants of finite groups.
” 1992. Web. 22 Jan 2021.
Vancouver:
McShane JM. Computation of polynomial invariants of finite groups.
[Internet] [Doctoral dissertation]. University of Arizona; 1992. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/186000.
Council of Science Editors:
McShane JM. Computation of polynomial invariants of finite groups.
[Doctoral Dissertation]. University of Arizona; 1992. Available from: http://hdl.handle.net/10150/186000

University of Arizona
21.
Lamb, McKenzie Russell.
Ginzburg-Weinstein Isomorphisms for Pseudo-Unitary Groups
.
Degree: 2009, University of Arizona
URL: http://hdl.handle.net/10150/193755
► Ginzburg and Weinstein proved in [GW92] that for a compact, semisimple Lie group K endowed with the Lu-Weinstein Poisson structure, there exists a Poisson diffeomorphism…
(more)
▼ Ginzburg and Weinstein proved in [GW92] that for a compact, semisimple Lie group K endowed with the Lu-Weinstein Poisson structure, there exists a Poisson diffeomorphism from the dual Poisson Lie group K* to the dual k* of the Lie algebra of K endowed with the Lie-Poisson structure. We investigate the possibility of extending this result to the pseudo-unitary groups SU (p, q ), which are semisimple but not compact. The main results presented here are the following. (1) The Ginzburg-Weinstein proof hinges on the existence of a certain vector field X on k*. We prove that for any p, q, the analogous vector field for the SU (p, q ) case exists on an open subset of k*. (2) Each generic dressing orbit ψ(λ) in the Poisson dual AN can be embedded in the complex flag manifold K/T . We show that for SU (1, 1) and SU (1, 2), the induced Poisson structure π(λ) on ψ(λ) extends smoothly to the entire flag manifold. (3) Finally, we prove the Ginzburg-Weinstein theorem for the SU (1, 1) case in two different ways: first, by constructing the vector field X in coordinates and proving that it satisfies the necessary properties, and second, by adapting the approach of [FR96] to the SU (1, 1) case.
Advisors/Committee Members: Foth, Philip A (advisor), Foth, Philip A. (committeemember), Pickrell, Doug (committeemember), Flaschka, Hermann (committeemember), Glickenstein, David (committeemember).
Subjects/Keywords: geometry;
Lie groups;
Poisson
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Record Details
Similar Records
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Lamb, M. R. (2009). Ginzburg-Weinstein Isomorphisms for Pseudo-Unitary Groups
. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/193755
Chicago Manual of Style (16th Edition):
Lamb, McKenzie Russell. “Ginzburg-Weinstein Isomorphisms for Pseudo-Unitary Groups
.” 2009. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/193755.
MLA Handbook (7th Edition):
Lamb, McKenzie Russell. “Ginzburg-Weinstein Isomorphisms for Pseudo-Unitary Groups
.” 2009. Web. 22 Jan 2021.
Vancouver:
Lamb MR. Ginzburg-Weinstein Isomorphisms for Pseudo-Unitary Groups
. [Internet] [Doctoral dissertation]. University of Arizona; 2009. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/193755.
Council of Science Editors:
Lamb MR. Ginzburg-Weinstein Isomorphisms for Pseudo-Unitary Groups
. [Doctoral Dissertation]. University of Arizona; 2009. Available from: http://hdl.handle.net/10150/193755

University of Arizona
22.
Jin, Shan.
The semiclassical limit of the defocusing nonlinear Schroedinger flows.
Degree: 1991, University of Arizona
URL: http://hdl.handle.net/10150/185687
► The Lax-Levermore strategy for analyzing the zero-dispersion limit of the KdV equation through its inverse scattering transform can be adapted to study the semiclassical limits…
(more)
▼ The Lax-Levermore strategy for analyzing the zero-dispersion limit of the KdV equation through its inverse scattering transform can be adapted to study the semiclassical limits of the defocusing nonlinear Schrodinger (NLS) equation, which are in fact the limits of corresponding conservation laws. The weak limits of all conserved densities and their fluxes can be characterized in terms of the solution of a variational problem that in turn can be solved using function theory. These results rest on a new formula for the N-soliton solutions and a WKB analysis of the semiclassical limit for the direct and inverse Zakharov-Shabat scattering transform. Moreover, with Levermore's method, one can see that the limiting dynamics explored is hyperbolic and agrees with that obtained by classical nonlinear modulation theory. The result is extended to the whole NLS hierarchy.
Advisors/Committee Members: Levermore, C. David (advisor), McLaughlin, David W. (committeemember), Ercolani, Nick (committeemember), Faris, William G. (committeemember), Flaschka, Hermann (committeemember), Greenlee, Martin (committeemember).
Subjects/Keywords: Dissertations, Academic;
Mathematics.
Record Details
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Record Details
Similar Records
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Jin, S. (1991). The semiclassical limit of the defocusing nonlinear Schroedinger flows.
(Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/185687
Chicago Manual of Style (16th Edition):
Jin, Shan. “The semiclassical limit of the defocusing nonlinear Schroedinger flows.
” 1991. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021.
http://hdl.handle.net/10150/185687.
MLA Handbook (7th Edition):
Jin, Shan. “The semiclassical limit of the defocusing nonlinear Schroedinger flows.
” 1991. Web. 22 Jan 2021.
Vancouver:
Jin S. The semiclassical limit of the defocusing nonlinear Schroedinger flows.
[Internet] [Doctoral dissertation]. University of Arizona; 1991. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/10150/185687.
Council of Science Editors:
Jin S. The semiclassical limit of the defocusing nonlinear Schroedinger flows.
[Doctoral Dissertation]. University of Arizona; 1991. Available from: http://hdl.handle.net/10150/185687
.