Self-Similarities and Invariant Densities for Model Sets.- Model Sets and Self-Similarities.- Averaging Operators and Invariant Densities.- Further Remarks.- Outlook.- References.- Symmetry Operations in the Brain: Music and Reasoning.- Trion Model.- Music Enhances Spatial-Temporal Reasoning.- References.- Lie Modules of Bounded Multiplicities.- Simple L Modules with Finite-Dimensional Weight Spaces.- Completely Pointed Modules.- Completely Pointed Modules Tensored with Finite-Dimensional Modules.- References.- Moving Frames and Coframes.- References.- The Fibonacci-Deformed Harmonic Oscillator.- About Strictly Increasing Sequences of Positive Numbers.- Quantum Algebra Associated with the Spectrum ? = xn.- The ?-Natural Spectrum.- The Fibonacci Deformation of Weyl Algebra.- Coherent States and Some Special Functions.- References.- Continuous and Discrete Linearizable Systems: The Riccati Saga.- Brief Review of the Continuous Gambier Equation.- Discrete Analog of the Gambier Equation, Revisited.- Discrete Projective and Matrix Riccati Equations.- Discrete Conformai Riccati Equations.- Conclusions and Outlook.- References.- Superintegrability on Two-Dimensional Complex Euclidean Space.- Potential V5.- Potential V6.- Potential V7.- References.- Hydrodynamic Systems and the Higher-Dimensional Laplace Transformations of Cartan Submanifolds.- Hydrodynamic Systems Rich in Conservation Laws.- Applications of the Higher-Dimensional Laplace Transformation to Hydrodynamic Systems that are Rich in Conservation Laws.- References.- Branching Rules and Weight Multiplicities for Simple and Affine Lie Algebras.- Simple and Affine Lie Algebras.- Branching Rules for Simple Lie Algebras.- Young Diagrams and Branching Rules.- Weight Multiplicities of Simple Lie Algebras.- Young Tableaux and Weight Multiplicities.- Branching Rule Multiplicities for the Restriction from Affine to Simple Lie Algebras.- Branching Rules Derived from Characters.- Weight Multiplicities of Affine Lie Algebras.- References.- Conditions for the Existence of Higher Symmetries and Nonlinear Evolutionary Equations on the Lattice.- Construction of the Classifying Conditions.- The Toda Lattice Class.- References.- Complete Description of the Voronoï Cell of the Lie Algebra An Weight Lattice. On the Bounds for the Number of d-Faces of the n-Dimensional Voronoï Cells.- The Expression of the Bounds Nd(n) Obtained by Voronoï.- Detailed Description of the Voronoï Cells of the A(TM) Lattices.- The New Explicit Expression of Bounds Nd(n).- Expression of Nd(n) as Multiple of a Stirling Number of Second Kind.- Final Remarks.- References.- The Relativistic Oscillator and the Mass Spectra of Baryons.- The System of Three Relativistic Scalar Particles with Oscillator Interactions.- An Approach to the Spinorial Relativistic Three-Body System.- References.- Seiberg-Witten Theory Without Tears.- N = 2 Supersymmetry.- N = 2 Superaction.- Textbook Properties.- Spontaneous Symmetry-Breaking.- Holomorphy and Duality.- Perturbative and Nonperturbative F (A).- Preliminaries.- Fuchsian Maps.- The Schwarzian Derivatives.- SW Choice.- Correctness.- Uniqueness.- References.- Bargmann Representation for Some Deformed Harmonic Oscillators with Non-Fock Representation.- Representations.- Toward a Bargmann Representation.- The "q-Oscillator".- Generalization of the Previous Example.- Deformed Algebra Associated to a Given Weight function.- Bargmann Representations Corresponding to Different ?.- The Case of an Annulus.- Conclusion.- References.- The Vector-Coherent-State Inducing Construction for Clebsch-Gordan Coefficients.- Induced Representations of su(4).- SU(4) Clebsch-Gordan Coefficients.- Summary.- References.- Highest-Weight Representations of Borcherds Algebras.- Borcherds Algebras.- Cartan Subalgebra of an Affine Kac-Moody Algebra.- Adding Energy and Number Operators to the Cartan Subalgebra.- Conclusions.- References.- Graded Contractions of Lie Algebras of Physical Interest.- Notion of Graded
This book is devoted to a topic that has undergone rapid and fruitful development over the last few years: symmetries and integrability of difference equations and q-difference equations and the theory of special functions that occur as solutions of such equations. Techniques that have been traditionally applied to solve linear and nonlinear differential equations are now being successfully adapted and applied to discrete equations. This volume is based on contributions made by leading experts in the field during the workshop on Symmetries and Integrability of Difference Equations held Estérel, Québec, in May 1994. Giving an up-to-date review of the current status of the field, the book treats these specific topics: Lie group and quantum group symmetries of difference and q-difference equations, integrable and nonintegrable discretizations of continuous integrable systems, integrability of difference equations, discrete Painlevé property and singularity confinement, integrable mappings, applications in statistical mechanics and field theories, Yang-Baxter equations, q-special functions and discrete polynomials, and q-difference integrable systems.
This book on integrable systems and symmetries presents new results on applications of symmetries and integrability techniques to the case of equations defined on the lattice. This relatively new field has many applications, for example, in describing the evolution of crystals and molecular systems defined on lattices, and in finding numerical approximations for differential equations preserving their symmetries. The book contains three chapters and five appendices. The first chapter is an introduction to the general ideas about symmetries, lattices, differential difference and partial difference equations and Lie point symmetries defined on them. Chapter 2 deals with integrable and linearizable systems in two dimensions. The authors start from the prototype of integrable and linearizable partial differential equations, the Korteweg de Vries and the Burgers equations. Then they consider the best known integrable differential difference and partial difference equations. Chapter 3 considers generalized symmetries and conserved densities as integrability criteria. The appendices provide details which may help the readers' understanding of the subjects presented in Chapters 2 and 3. This book is written for PhD students and early researchers, both in theoretical physics and in applied mathematics, who are interested in the study of symmetries and integrability of difference equations.
From Classical to Quantum : Proceedings of the 38th Session of the Seminaire de Mathématiques Supérieures, July 26-August 6, 1999 Montréal, Québec, Canada
From Classical to Quantum : Proceedings of the 38th Session of the Seminaire de Mathématiques Supérieures, July 26-August 6, 1999 Montréal, Québec, Canada
This volume presents the papers based upon lectures given at the 1999 Séminaire de Mathémathiques Supérieurs held in Montreal. It includes contributions from many of the most active researchers in the field. This subject has been in a remarkably active state of development throughout the past three decades, resulting in new motivation for study in r s3risingly different directions. Beyond the intrinsic interest in the study of integrable models of many-particle systems, spin chains, lattice and field theory models at both the classical and the quantum level, and completely solvable models in statistical mechanics, there have been new applications in relation to a number of other fields of current interest. These fields include theoretical physics and pure mathematics, for example the Seiberg-Witten approach to supersymmetric Yang-Mills theory, the spectral theory of random matrices, topological models of quantum gravity, conformal field theory, mirror symmetry, quantum cohomology, etc. This collection gives a nice cross-section of the current state of the work in the area of integrable systems which is presented by some of the leading active researchers in this field. The scope and quality of the articles in this volume make this a valuable resource for those interested in an up-to-date introduction and an overview of many of the main areas of study in the theory of integral systems.
This volume contains papers presented at a conference held in April 2007 at the CRM in Montreal honouring the remarkable contributions of John McKay over four decades of research. Papers by invitees who were unable to attend the conference are also included. The papers cover a wide range of topics, including group theory, symmetries, modular functions, and geometry, with particular focus on two areas in which John McKay has made pioneering contributions: ``Monstrous Moonshine'' and the ``McKay Correspondence''. This book will be a valuable reference for graduate students and researchers interested in these and related areas and serve as a stimulus for new ideas.
Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities à la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.
Papers in this volume are based on the Workshop on Symmetries in Physics held at the Centre de recherches mathématiques (University of Montreal) in memory of Robert T. Sharp. Contributed articles are on a variety of topics revolving around the theme of symmetry in physics. The preface presents a biographical and scientific retrospect of the life and work of Robert Sharp. Other articles in the volume represent his diverse range of interests, including representation theoretic methods for Lie algebras, quantization techniques and foundational considerations, modular group invariants and applicat.
The Workshop on Group Theory and Numerical Analysis brought together scientists working in several different but related areas. The unifying theme was the application of group theory and geometrical methods to the solution of differential and difference equations. The emphasis was on the combination of analytical and numerical methods and also the use of symbolic computation. This meeting was organized under the auspices of the Centre de Recherches Mathématiques, Université de Montréal (Canada). This volume has the character of a monograph and should represent a useful reference book for scien.
from classical to quantum : proceedings of the 38th session of the seminaire de mathâematiques supâerieures, July 26-August 6, 1999 Montrâeal, Quâebec, Canada
from classical to quantum : proceedings of the 38th session of the seminaire de mathâematiques supâerieures, July 26-August 6, 1999 Montrâeal, Quâebec, Canada
This volume presents the papers based upon lectures given at the 1999 Séminaire de Mathémathiques Supérieurs held in Montréal. It includes contributions from many of the most active researchers in the field. This subject has been in a remarkably active state of development throughout the past three decades, resulting in new motivation for study in surprisingly different directions. Beyond the intrinsic interest in the study of integrable models of many-particle systems, spin chains, lattice and field theory models at both the classical and the quantum level, and completely solvable models in s.
This book on integrable systems and symmetries presents new results on applications of symmetries and integrability techniques to the case of equations defined on the lattice. This relatively new field has many applications, for example, in describing the evolution of crystals and molecular systems defined on lattices, and in finding numerical approximations for differential equations preserving their symmetries. The book contains three chapters and five appendices. The first chapter is an introduction to the general ideas about symmetries, lattices, differential difference and partial difference equations and Lie point symmetries defined on them. Chapter 2 deals with integrable and linearizable systems in two dimensions. The authors start from the prototype of integrable and linearizable partial differential equations, the Korteweg de Vries and the Burgers equations. Then they consider the best known integrable differential difference and partial difference equations. Chapter 3 considers generalized symmetries and conserved densities as integrability criteria. The appendices provide details which may help the readers' understanding of the subjects presented in Chapters 2 and 3. This book is written for PhD students and early researchers, both in theoretical physics and in applied mathematics, who are interested in the study of symmetries and integrability of difference equations.
Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities à la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.
This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Bäcklund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory. The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Bäcklund transformations of Painlevé equations, to the dressing method and Bäcklund and Darboux transformations, and to the classical geometry of Bäcklund transformations and their ap.
Self-Similarities and Invariant Densities for Model Sets.- Model Sets and Self-Similarities.- Averaging Operators and Invariant Densities.- Further Remarks.- Outlook.- References.- Symmetry Operations in the Brain: Music and Reasoning.- Trion Model.- Music Enhances Spatial-Temporal Reasoning.- References.- Lie Modules of Bounded Multiplicities.- Simple L Modules with Finite-Dimensional Weight Spaces.- Completely Pointed Modules.- Completely Pointed Modules Tensored with Finite-Dimensional Modules.- References.- Moving Frames and Coframes.- References.- The Fibonacci-Deformed Harmonic Oscillator.- About Strictly Increasing Sequences of Positive Numbers.- Quantum Algebra Associated with the Spectrum ? = xn.- The ?-Natural Spectrum.- The Fibonacci Deformation of Weyl Algebra.- Coherent States and Some Special Functions.- References.- Continuous and Discrete Linearizable Systems: The Riccati Saga.- Brief Review of the Continuous Gambier Equation.- Discrete Analog of the Gambier Equation, Revisited.- Discrete Projective and Matrix Riccati Equations.- Discrete Conformai Riccati Equations.- Conclusions and Outlook.- References.- Superintegrability on Two-Dimensional Complex Euclidean Space.- Potential V5.- Potential V6.- Potential V7.- References.- Hydrodynamic Systems and the Higher-Dimensional Laplace Transformations of Cartan Submanifolds.- Hydrodynamic Systems Rich in Conservation Laws.- Applications of the Higher-Dimensional Laplace Transformation to Hydrodynamic Systems that are Rich in Conservation Laws.- References.- Branching Rules and Weight Multiplicities for Simple and Affine Lie Algebras.- Simple and Affine Lie Algebras.- Branching Rules for Simple Lie Algebras.- Young Diagrams and Branching Rules.- Weight Multiplicities of Simple Lie Algebras.- Young Tableaux and Weight Multiplicities.- Branching Rule Multiplicities for the Restriction from Affine to Simple Lie Algebras.- Branching Rules Derived from Characters.- Weight Multiplicities of Affine Lie Algebras.- References.- Conditions for the Existence of Higher Symmetries and Nonlinear Evolutionary Equations on the Lattice.- Construction of the Classifying Conditions.- The Toda Lattice Class.- References.- Complete Description of the Voronoï Cell of the Lie Algebra An Weight Lattice. On the Bounds for the Number of d-Faces of the n-Dimensional Voronoï Cells.- The Expression of the Bounds Nd(n) Obtained by Voronoï.- Detailed Description of the Voronoï Cells of the A(TM) Lattices.- The New Explicit Expression of Bounds Nd(n).- Expression of Nd(n) as Multiple of a Stirling Number of Second Kind.- Final Remarks.- References.- The Relativistic Oscillator and the Mass Spectra of Baryons.- The System of Three Relativistic Scalar Particles with Oscillator Interactions.- An Approach to the Spinorial Relativistic Three-Body System.- References.- Seiberg-Witten Theory Without Tears.- N = 2 Supersymmetry.- N = 2 Superaction.- Textbook Properties.- Spontaneous Symmetry-Breaking.- Holomorphy and Duality.- Perturbative and Nonperturbative F (A).- Preliminaries.- Fuchsian Maps.- The Schwarzian Derivatives.- SW Choice.- Correctness.- Uniqueness.- References.- Bargmann Representation for Some Deformed Harmonic Oscillators with Non-Fock Representation.- Representations.- Toward a Bargmann Representation.- The "q-Oscillator".- Generalization of the Previous Example.- Deformed Algebra Associated to a Given Weight function.- Bargmann Representations Corresponding to Different ?.- The Case of an Annulus.- Conclusion.- References.- The Vector-Coherent-State Inducing Construction for Clebsch-Gordan Coefficients.- Induced Representations of su(4).- SU(4) Clebsch-Gordan Coefficients.- Summary.- References.- Highest-Weight Representations of Borcherds Algebras.- Borcherds Algebras.- Cartan Subalgebra of an Affine Kac-Moody Algebra.- Adding Energy and Number Operators to the Cartan Subalgebra.- Conclusions.- References.- Graded Contractions of Lie Algebras of Physical Interest.- Notion of Graded
This is a comprehensive rehabilitation textbook emphasizing functional approach based on the aspects of the Prague School of Rehabilitation (www.rehabps.com), which was developed by Professors Vojta, Janda, Lewit and others. The publication is divided into general and special sections. The diagnostic section describes in detail clinical evaluation approaches for the musculoskeletal system, testing and assessment of a motor and sensory involvement and limitations in the activities of daily living. The text also includes psychological assessment in rehabilitation of painful conditions, functional laboratory assessments and functional assessment using imaging methods. In the general section of the therapeutic approaches, the authors focus on treatment rehabilitaton of the motor system and also focus on disorders of other organ systems. The special section of the book includes rehabilitation of individual clinical specialties, in which treatment rehabilitation plays an important part. The individual chapters include rehabilitation in neurology, orthopedics, internal diseases, gynecology, oncology, psychiatry, pain and psychosomatic conditions. This book is unique in its presentation of human development and the options for its implementation into diagnostic and therapeutic approaches of the movement system. A chapter is also devoted to the original diagnostic-therapeutic approach of Dynamic Neuromuscular Stabilization according to Professor Pavel Kolar, the main editor of the book. - full-color resource - exceptional photographs of developmental sequences, radiographic images, diagrams and schematic drawings specific to the Prague School of Rehabilitation and Dynamic Neuromuscular Stabilization - 800 text pages - photographs of pediatric ontogenetic development
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