DOCSLIB.ORG

Goncalves De Assis Phd Thesis.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

City Research Online City, University of London Institutional Repository Citation: Goncalves de Assis, P. E (2009). Non-Hermitian Hamiltonians in Field Theory. (Doctoral thesis, City University London) This is the unspecified version of the paper. This version of the publication may differ from the final published version. Permanent repository link: https://openaccess.city.ac.uk/id/eprint/2118/ Link to published version: Copyright: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. Reuse: Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. City Research Online: http://openaccess.city.ac.uk/ [email protected] Non-Hermitian Hamiltonians in Field Theory Paulo Eduardo Gon»calves de Assis PhD Thesis City University London Centre for Mathematical Science October 2009 Supervisor: Prof. Dr. Andreas Fring A meus pais. Contents Acknowledgements VI Declaration VII Abstract VIII 1 Introduction 1 1.1 Outline . 7 2 Aspects of non-Hermitian Hamiltonians 10 2.1 Dissipative systems . 10 2.2 The role of PT -symmetry in Physics . 13 2.2.1 A family of non-Hermitian Hamiltonians with real spectra . 17 2.3 Non-Hermitian Hamiltonians as fundamental theories . 21 2.3.1 Anti-unitary symmetries in Quantum Mechanics . 21 2.3.2 Darboux transformations and superpartner potentials . 27 2.3.3 Dyson map and change of metric . 29 2.3.4 Quasi-Pseudo-Hermiticity . 31 2.3.5 Perturbative approach . 36 2.3.6 Uniqueness of the metric . 37 2.3.7 Non-Hermitian time evolution . 39 3 The quantum brachistochrone for non-Hermitian Hamiltonians 40 3.1 The classical brachistochrone problem . 40 3.2 Hermitian evolution in Quantum Physics . 41 3.3 Fast transitions with PT -symmetric non-Hermitian Hamiltonians . 44 3.4 Dissipative evolution . 47 III 3.5 Geometry of the state space for the quantum brachistochrone . 53 3.6 Interpretation of the physical setup . 54 4 Symmetries, Integrability and Solvability 56 4.1 The importance of symmetries in Physics . 56 4.1.1 Lie groups and Lie algebras . 58 4.1.2 Representation theory of semi-simple Lie algebras . 60 4.2 Integrability . 65 4.2.1 Solitons and Compactons . 72 4.3 Solvability . 75 5 Mapping non-Hermitian into Hermitian Hamiltonians 77 5.1 Lie algebras and similarity transformations with operators . 77 5.2 Hamiltonians of sl(2; R)-Lie algebraic type . 78 5.3 sl(2; R) Metrics and Hermitian counterparts . 80 5.4 Hamiltonians of su(1; 1)-Lie algebraic type and generalized Swanson models 81 5.5 su(1; 1) Metrics and Hermitian counterparts . 83 5.5.1 Non-Hermitian linear term and Hermitian bilinear combinations . 86 5.5.2 Hermitian linear term and non-Hermitian bilinear combinations . 86 5.5.3 Generic non-Hermitian reducible Hamiltonian . 87 5.5.4 Generic non-Hermitian non-reducible Hamiltonian . 90 5.5.5 Generalised Bogoliubov transformation in the construction of metrics 93 5.5.6 Some concrete realisations of the generalised Swanson Hamiltonian . 100 5.6 Hamiltonians of su(2)-Lie algebraic type and bosonic spin chains . 103 6 Moyal products and isospectral transformations 106 6.1 Moyal Products to construct metric operators . 106 6.2 Generic cubic PT -symmetric non-Hermitian Hamiltonians . 110 6.2.1 Non-vanishingp ^ x^2-term . 113 6.2.2 Non-vanishingp ^ x^2-term and vanishingx ^3-term . 115 6.2.3 Vanishingp ^ x^2-term and non-vanishingx ^2-term . 116 6.2.4 Vanishingp ^ x^2-term and non-vanishingp ^ -term or non-vanishing p^2-term . 117 6.3 The single site lattice Reggeon model . 118 6.3.1 Perturbative solution . 120 IV 6.4 Solvable examples . 122 6.5 Limitations of the method . 124 7 Integrable PT -symmetric deformations of classical models 126 7.1 PT -symmetry in classical theories . 126 7.2 Integrability and the Painlev¶etest . 128 7.3 PT -symmetrically deformed Burgers' equation . 131 7.4 PT -symmetrically deformed KdV equation . 138 7.5 PT -symmetrically generalized KdV Hamiltonian: Hl;m;p ........... 142 7.5.1 Generalized KdV-equation: m=2 . 145 7.5.2 PT -symmetric generalized KdV-equation . 146 7.5.3 Deformations of Burgers equation: m = 1; p = 1; l = 3 . 148 7.5.4 Coexistence of solitons and compactons . 148 8 PT -symmetric ¯elds and particles 150 8.1 PT -symmetric constraints on real ¯elds . 150 8.2 Calogero models . 151 8.3 Poles of nonlinear waves as interacting particles . 154 8.4 The motion of Boussinesq singularities . 157 8.5 Di®erent constraints in nonlinear wave equations . 162 8.6 Complex Calogero systems and PT -symmetric deformations . 164 9 Concluding remarks 170 Bibliography 176 V Acknowledgements Unfortunately - or should I say fortunately? - gratitude is not a measurable object. This makes this recognition demonstration imprecise. Nonetheless some people's contri- butions are colossal and cannot be left out. I do not intend to be long and must clarify that this is by no means an indication of lack of appreciation. First of all, I am enormously indebted with the person most directly involved in this work, Prof. Andreas Fring, supervisor and friend. Without him certainly none of this would have been accomplished. He has always demonstrated faith in me, even before we actually met. This support was crucial for my settlement in the United Kingdom and for the progress of this work. I owe so much to your help, Andreas. It has been a real pleasure working with you and I hope to collaborate with you in the future. Secondly, I give many thanks to the members of the panel who kindly accepted the arduous task of assessing this thesis, Prof. Carl Bender and Prof. Joe Chuang. I feel extremely honourned by their assistance. I am also very grateful for ¯nancial support provided by City University London. My stay in London is memorable, de¯nitely because of the amazing people I have met here. New friends who came into my life and will forever be remembered. Exactly like the friends I left in my home country and who have also contributed immensely in a very supportive way. I miss you, friends. I miss you, family. Wonderful family. I have no words to express my admiration and love for you. Thanks for everything. Thank you, life. You have been extraordinary, blessed and cursed and won. The writing of these words ¯lled my heart with joy. Few names but many faces. To everyone here briefly mentioned, one common sentiment. Simple but strong and sincere. Muito obrigado. VI Declaration The work presented in this thesis is based on investigations believed to be original and carried out at the Centre for Mathematical Science, City University London, in collabora- tion with Prof. Andreas Fring. It has not been presented elsewhere for a degree, diploma, or similar quali¯cation at any university or similar institution. I have clearly stated my contributions, in jointly-authored works, as well as referenced the contributions of other people working in the area. Powers of discretion are granted to the University Librarian to allow the thesis to be copied in whole or in part without further reference to the author. This permission covers only single copies made for study purposes, subject to normal conditions of acknowledge- ment. VII Abstract This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachis- tochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calcu- lated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of in¯nite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The ¯nding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commu- tation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of di®erential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our e®orts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear di®erential equations for which the presence of hidden symmetries allows one to assemble exact solu- tions. We use [4] the Painlev¶etest to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions.
Recommended publications
  • Contact Lax Pairs and Associated (3+ 1)-Dimensional Integrable Dispersionless Systems

    Contact Lax Pairs and Associated (3+ 1)-Dimensional Integrable Dispersionless Systems

    Contact Lax pairs and associated (3+1)-dimensional integrable dispersionless systems Maciej B laszak a and Artur Sergyeyev b a Faculty of Physics, Division of Mathematical Physics, A. Mickiewicz University Umultowska 85, 61-614 Pozna´n, Poland E-mail [email protected] b Mathematical Institute, Silesian University in Opava, Na Rybn´ıˇcku 1, 74601 Opava, Czech Republic E-mail [email protected] January 17, 2019 Abstract We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related R-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these examples to lower-dimensional integrable dispersionless systems. 1 Introduction Integrable systems play an important role in modern mathematics and theoretical and mathematical arXiv:1901.05181v1 [nlin.SI] 16 Jan 2019 physics, cf. e.g. [15, 34], and, since according to general relativity our spacetime is four-dimensional, integrable systems in four independent variables ((3+1)D for short; likewise (n+1)D is shorthand for n + 1 independent variables) are particularly interesting. For a long time it appeared that such systems were very difficult to find but in a recent paper by one of us [39] a novel systematic and effective construction for a large new class of integrable (3+1)D systems was introduced.
  • Arxiv:2012.03456V1 [Nlin.SI] 7 Dec 2020 Emphasis on the Kdv Equation

    Arxiv:2012.03456V1 [Nlin.SI] 7 Dec 2020 Emphasis on the Kdv Equation

    Integrable systems: From the inverse spectral transform to zero curvature condition Basir Ahamed Khan,1, ∗ Supriya Chatterjee,2 Sekh Golam Ali,3 and Benoy Talukdar4 1Department of Physics, Krishnath College, Berhampore, Murshidabad 742101, India 2Department of Physics, Bidhannagar College, EB-2, Sector-1, Salt Lake, Kolkata 700064, India 3Department of Physics, Kazi Nazrul University, Asansol 713303, India 4Department of Physics, Visva-Bharati University, Santiniketan 731235, India This `research-survey' is meant for beginners in the studies of integrable systems. Here we outline some analytical methods for dealing with a class of nonlinear partial differential equations. We pay special attention to `inverse spectral transform', `Lax pair representation', and `zero-curvature condition' as applied to these equations. We provide a number of interesting exmples to gain some physico-mathematical feeling for the methods presented. PACS numbers: Keywords: Nonlinear Partial Differential Equations, Integrable Systems, Inverse Spectral Method, Lax Pairs, Zero Curvature Condition 1. Introduction Integrable systems are represented by nonlinear partial differential equations (NLPDEs) which, in principle, can be solved by analytic methods. This necessarily implies that the solution of such equations can be constructed using a finite number of algebraic operations and integrations. The inverse scattering method as discovered by Gardner, Greene, Kruskal and Miura [1] represents a very useful tool to analytically solve a class of nonlinear differential equa- tions which support soliton solutions. Solitons are localized waves that propagate without change in their properties (shape, velocity etc.). These waves are stable against mutual collision and retain their identities except for some trivial phase change. Mechanistically, the linear and nonlinear terms in NLPDEs have opposite effects on the wave propagation.
  • The Idea of a Lax Pair–Part II∗ Continuum Wave Equations

    The Idea of a Lax Pair–Part II∗ Continuum Wave Equations

    GENERAL ARTICLE The Idea of a Lax Pair–Part II∗ Continuum Wave Equations Govind S. Krishnaswami and T R Vishnu In Part I [1], we introduced the idea of a Lax pair and ex- plained how it could be used to obtain conserved quantities for systems of particles. Here, we extend these ideas to con- tinuum mechanical systems of fields such as the linear wave equation for vibrations of a stretched string and the Korteweg- de Vries (KdV) equation for water waves. Unlike the Lax ma- trices for systems of particles, here Lax pairs are differential operators. A key idea is to view the Lax equation as a com- Govind Krishnaswami is on the faculty of the Chennai patibility condition between a pair of linear equations. This is Mathematical Institute. He used to obtain a geometric reformulation of the Lax equation works on various problems in as the condition for a certain curvature to vanish. This ‘zero theoretical and mathematical curvature representation’ then leads to a recipe for finding physics. (typically an infinite sequence of) conserved quantities. 1. Introduction In the first part of this article [1], we introduced the idea of a dy- namical system: one whose variables evolve in time, typically via T R Vishnu is a PhD student at the Chennai Mathematical differential equations. It was pointed out that conserved quanti- Institute. He has been ties, which are dynamical variables that are constant along trajec- working on integrable tories help in simplifying the dynamics and solving the equations systems. of motion (EOM) both in the classical and quantum settings.
  • On the Linear Stability of Plane Couette Flow for an Oldroyd-B Fluid and Its

    On the Linear Stability of Plane Couette Flow for an Oldroyd-B Fluid and Its

    J. Non-Newtonian Fluid Mech. 127 (2005) 169–190 On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation Raz Kupferman Institute of Mathematics, The Hebrew University, Jerusalem, 91904, Israel Received 14 December 2004; received in revised form 10 February 2005; accepted 7 March 2005 Abstract It is well known that plane Couette flow for an Oldroyd-B fluid is linearly stable, yet, most numerical methods predict spurious instabilities at sufficiently high Weissenberg number. In this paper we examine the reasons which cause this qualitative discrepancy. We identify a family of distribution-valued eigenfunctions, which have been overlooked by previous analyses. These singular eigenfunctions span a family of non- modal stress perturbations which are divergence-free, and therefore do not couple back into the velocity field. Although these perturbations decay eventually, they exhibit transient amplification during which their “passive" transport by shearing streamlines generates large cross- stream gradients. This filamentation process produces numerical under-resolution, accompanied with a growth of truncation errors. We believe that the unphysical behavior has to be addressed by fine-scale modelling, such as artificial stress diffusivity, or other non-local couplings. © 2005 Elsevier B.V. All rights reserved. Keywords: Couette flow; Oldroyd-B model; Linear stability; Generalized functions; Non-normal operators; Stress diffusion 1. Introduction divided into two main groups: (i) numerical solutions of the boundary value problem defined by the linearized system The linear stability of Couette flow for viscoelastic fluids (e.g. [2,3]) and (ii) stability analysis of numerical methods is a classical problem whose study originated with the pi- for time-dependent flows, with the objective of understand- oneering work of Gorodtsov and Leonov (GL) back in the ing how spurious solutions emerge in computations.
  • A Short Introduction to the Quantum Formalism[S]

    A Short Introduction to the Quantum Formalism[S]

    A short introduction to the quantum formalism[s] François David Institut de Physique Théorique CNRS, URA 2306, F-91191 Gif-sur-Yvette, France CEA, IPhT, F-91191 Gif-sur-Yvette, France [email protected] These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May- June 2012; (ii) a previous letter [Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics. arXiv:1211.5627v1 [math-ph] 24 Nov 2012 Preprint IPhT t12/042 ii CONTENTS Contents 1 Introduction 1-1 1.1 Motivation . 1-1 1.2 Organization . 1-2 1.3 What this course is not! . 1-3 1.4 Acknowledgements . 1-3 2 Reminders 2-1 2.1 Classical mechanics .
  • A Highly Efficient Spectral-Galerkin Method Based on Tensor Product for Fourth-Order Steklov Equation with Boundary Eigenvalue

    A Highly Efficient Spectral-Galerkin Method Based on Tensor Product for Fourth-Order Steklov Equation with Boundary Eigenvalue

    An et al. Journal of Inequalities and Applications (2016)2016:211 DOI 10.1186/s13660-016-1158-1 R E S E A R C H Open Access A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue Jing An1,HaiBi1 and Zhendong Luo2* *Correspondence: [email protected] Abstract 2School of Mathematics and Physics, North China Electric Power In this study, a highly efficient spectral-Galerkin method is posed for the fourth-order University, No. 2, Bei Nong Road, Steklov equation with boundary eigenvalue. By making use of the spectral theory of Changping District, Beijing, 102206, compact operators and the error formulas of projective operators, we first obtain the China Full list of author information is error estimates of approximative eigenvalues and eigenfunctions. Then we build a 1 ∩ 2 available at the end of the article suitable set of basis functions included in H0() H () and establish the matrix model for the discrete spectral-Galerkin scheme by adopting the tensor product. Finally, we use some numerical experiments to verify the correctness of the theoretical results. MSC: 65N35; 65N30 Keywords: fourth-order Steklov equation with boundary eigenvalue; spectral-Galerkin method; error estimates; tensor product 1 Introduction Increasing attention has recently been paid to numerical approximations for Steklov equa- tions with boundary eigenvalue, arising in fluid mechanics, electromagnetism, etc. (see, e.g.,[–]). However, most existing work usually treated the second-order Steklov equa- tions with boundary eigenvalue and there are relatively few articles treating fourth-order ones. The fourth-order Steklov equations with boundary eigenvalue also have been used in both mathematics and physics, for example, the main eigenvalues play a very key role in the positivity-preserving properties for the biharmonic-operator under the border conditions w = w – χwν =on∂ (see []).
  • Chebyshev and Fourier Spectral Methods 2000

    Chebyshev and Fourier Spectral Methods 2000

    Chebyshev and Fourier Spectral Methods Second Edition John P. Boyd University of Michigan Ann Arbor, Michigan 48109-2143 email: [email protected] http://www-personal.engin.umich.edu/jpboyd/ 2000 DOVER Publications, Inc. 31 East 2nd Street Mineola, New York 11501 1 Dedication To Marilyn, Ian, and Emma “A computation is a temptation that should be resisted as long as possible.” — J. P. Boyd, paraphrasing T. S. Eliot i Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series .................................... 20 2.3 Orders of Convergence ............................... 25 2.4 Convergence Order ................................. 27 2.5 Assumption of Equal Errors ...........................
  • Functional Integration on Paracompact Manifods Pierre Grange, E

    Functional Integration on Paracompact Manifods Pierre Grange, E

    Functional Integration on Paracompact Manifods Pierre Grange, E. Werner To cite this version: Pierre Grange, E. Werner. Functional Integration on Paracompact Manifods: Functional Integra- tion on Manifold. Theoretical and Mathematical Physics, Consultants bureau, 2018, pp.1-29. hal- 01942764 HAL Id: hal-01942764 https://hal.archives-ouvertes.fr/hal-01942764 Submitted on 3 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Functional Integration on Paracompact Manifolds Pierre Grangé Laboratoire Univers et Particules, Université Montpellier II, CNRS/IN2P3, Place E. Bataillon F-34095 Montpellier Cedex 05, France E-mail: [email protected] Ernst Werner Institut fu¨r Theoretische Physik, Universita¨t Regensburg, Universita¨tstrasse 31, D-93053 Regensburg, Germany E-mail: [email protected] ........................................................................ Abstract. In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette (CDWM) focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. CDWM’s central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures.
  • Isospectral Families of High-Order Systems∗

    Isospectral Families of High-Order Systems∗

    Isospectral Families of High-order Systems∗ Peter Lancaster† Department of Mathematics and Statistics University of Calgary Calgary T2N 1N4, Alberta, Canada and Uwe Prells School of Mechanical, Materials, Manufacturing Engineering & Management University of Nottingham Nottingham NG7 2RD United Kingdom November 28, 2006 Keywords: Matrix Polynomials, Isospectral Systems, Structure Preserving Transformations, Palindromic Symmetries. Abstract Earlier work of the authors concerning the generation of isospectral families of second or- der (vibrating) systems is generalized to higher-order systems (with no spectrum at infinity). Results and techniques are developed first for systems without symmetries, then with Hermi- tian symmetry and, finally, with palindromic symmetry. The construction of linearizations which retain such symmetries is discussed. In both cases, the notion of strictly isospectral families of systems is introduced - implying that properties of both the spectrum and the sign-characteristic are preserved. Open questions remain in the case of strictly isospectral families of palindromic systems. Intimate connections between Hermitian and unitary sys- tems are discussed in an Appendix. ∗The authors are grateful to an anonymous reviewer for perceptive comments which led to significant improve- ments in exposition. †The research of Peter Lancaster was partly supported by a grant from the Natural Sciences and Engineering Research Council of Canada 1 1 Introduction ` By a high-order system we mean an n × n matrix polynomial L(λ) = A`λ + ··· + A1λ + A0 n×n with coefficients in C . Such a polynomial with A` 6= 0 is said to have degree `, (sometimes referred to as “order” `) and “high-order” implies that ` ≥ 2. It will be assumed throughout that the leading coefficient A` is nonsingular.
  • Well-Defined Lagrangian Flows for Absolutely Continuous Curves of Probabilities on the Line

    Well-Defined Lagrangian Flows for Absolutely Continuous Curves of Probabilities on the Line

    Graduate Theses, Dissertations, and Problem Reports 2016 Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Amsaad, Mohamed H., "Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line" (2016). Graduate Theses, Dissertations, and Problem Reports. 5098. https://researchrepository.wvu.edu/etd/5098 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Adrian Tudorascu, Ph.D., Chair Harumi Hattori, Ph.D. Harry Gingold, Ph.D. Tudor Stanescu, Ph.D. Charis Tsikkou, Ph.D. Department Of Mathematics Morgantown, West Virginia 2016 Keywords: Continuity Equation, Lagrangian Flow, Optimal Transport, Wasserstein metric, Wasserstein space.
  • Introduction to Spectral Methods

    Introduction to Spectral Methods

    Introduction to spectral methods Eric Gourgoulhon Laboratoire de l’Univers et de ses Th´eories (LUTH) CNRS / Observatoire de Paris Meudon, France Based on a collaboration with Silvano Bonazzola, Philippe Grandcl´ement,Jean-Alain Marck & J´eromeNovak [email protected] http://www.luth.obspm.fr 4th EU Network Meeting, Palma de Mallorca, Sept. 2002 1 Plan 1. Basic principles 2. Legendre and Chebyshev expansions 3. An illustrative example 4. Spectral methods in numerical relativity 2 1 Basic principles 3 Solving a partial differential equation Consider the PDE with boundary condition Lu(x) = s(x); x 2 U ½ IRd (1) Bu(y) = 0; y 2 @U; (2) where L and B are linear differential operators. Question: What is a numerical solution of (1)-(2)? Answer: It is a function u¯ which satisfies (2) and makes the residual R := Lu¯ ¡ s small. 4 What do you mean by “small” ? Answer in the framework of Method of Weighted Residuals (MWR): Search for solutions u¯ in a finite-dimensional sub-space PN of some Hilbert space W (typically a L2 space). Expansion functions = trial functions : basis of PN : (Á0;:::;ÁN ) XN u¯ is expanded in terms of the trial functions: u¯(x) = u˜n Án(x) n=0 Test functions : family of functions (Â0;:::;ÂN ) to define the smallness of the residual R, by means of the Hilbert space scalar product: 8 n 2 f0;:::;Ng; (Ân;R) = 0 5 Various numerical methods Classification according to the trial functions Án: Finite difference: trial functions = overlapping local polynomials of low order Finite element: trial functions = local smooth functions (polynomial
  • Functional Derivative

    Functional Derivative

    Appendix Functional Derivative Appendix A: Calculus of Variation Consider a mapping from a vector space of functions to real number. Such mapping is called functional. One of the most representative examples of such a mapping is the action functional of analytical mechanics, A ,which is defined by Zt2 A½xtðÞ Lxt½ðÞ; vtðÞdt ðA:1Þ t1 where t is the independent variable, xtðÞis the trajectory of a particle, vtðÞ¼dx=dt is the velocity of the particle, and L is the Lagrangian which is the difference of kinetic energy and the potential: m L ¼ v2 À uxðÞ ðA:2Þ 2 Calculus of variation is the mathematical theory to find the extremal function xtðÞwhich maximizes or minimizes the functional such as Eq. (A.1). We are interested in the set of functions which satisfy the fixed boundary con- ditions such as xtðÞ¼1 x1 and xtðÞ¼2 x2. An arbitrary element of the function space may be defined from xtðÞby xtðÞ¼xtðÞþegðÞt ðA:3Þ where ε is a real number and gðÞt is any function satisfying gðÞ¼t1 gðÞ¼t2 0. Of course, it is obvious that xtðÞ¼1 x1 and xtðÞ¼2 x2. Varying ε and gðÞt generates any function of the vector space. Substitution of Eq. (A.3) to Eq. (A.1) gives Zt2 dx dg aðÞe A½¼xtðÞþegðÞt L xtðÞþegðÞt ; þ e dt ðA:4Þ dt dt t1 © Springer Science+Business Media Dordrecht 2016 601 K.S. Cho, Viscoelasticity of Polymers, Springer Series in Materials Science 241, DOI 10.1007/978-94-017-7564-9 602 Appendix: Functional Derivative From Eq.