The Resource An algebraic geometric approach to separation of variables, Konrad Schöbel

An algebraic geometric approach to separation of variables, Konrad Schöbel

Label
An algebraic geometric approach to separation of variables
Title
An algebraic geometric approach to separation of variables
Statement of responsibility
Konrad Schöbel
Creator
Author
Subject
Genre
Language
eng
Summary
Konrad Sch©œbel aims to lay the foundations for a consequent algebraic geometric treatment of variable separation, which is one of the oldest and most powerful methods to construct exact solutions for the fundamental equations in classical and quantum physics. The present work reveals a surprising algebraic geometric structure behind the famous list of separation coordinates, bringing together a great range of mathematics and mathematical physics, from the late 19th century theory of separation of variables to modern moduli space theory, Stasheff polytopes and operads. "I am particularly impressed by his mastery of a variety of techniques and his ability to show clearly how they interact to produce his results.ĺl℗l℗l (Jim Stasheff) ℗l Contents The Foundation: The Algebraic Integrability Conditions The Proof of Concept: A Complete Solution for the 3-Sphere The Generalisation: A Solution for Spheres of Arbitrary Dimension The Perspectives: Applications and Generalisations ℗l Target Groups Scientists in the fields of Mathematical Physics and Algebraic Geometry ℗l The Author Konrad Sch©œbel studied physics and mathematics at Friedrich-Schiller University Jena (Germany) and Universidad de Granada (Spain) and obtained his PhD at the Universit©♭ de Provence Aix-Marseille I (France). He now holds a postdoc position at Friedrich-Schiller University Jena and works as a research and development engineer for applications in clinical ultrasound diagnostics
Cataloging source
N$T
http://library.link/vocab/creatorName
Schöbel, Konrad
Dewey number
515.353
Index
no index present
LC call number
QA377
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
http://library.link/vocab/subjectName
  • Separation of variables
  • Differential equations
  • MATHEMATICS / Calculus
  • MATHEMATICS / Mathematical Analysis
  • Differential equations
  • Separation of variables
Label
An algebraic geometric approach to separation of variables, Konrad Schöbel
Link
https://ezproxy.lib.ou.edu/login?url=http://link.springer.com/10.1007/978-3-658-11408-4
Instantiates
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Preface; Contents; 0 Introduction; 0.1 Separation of variables; 0.2 History; 0.3 Aim; 0.4 Mathematical formulation; 0.4.1 Separation coordinates; 0.4.2 Killing tensors, Nijenhuis integrability and Stäckel systems ; 0.4.3 Observations; 0.4.4 The problems; 0.5 Method; 0.6 Scope; 0.7 Some examples; 0.7.1 Cofactor systems; 0.7.2 Elliptic and polyspherical coordinates; 0.7.3 A trivial and a simple example: Spheres of dimension one and two; 0.8 Overview over the main results; 1 The foundation: the algebraic integrability conditions; 1.1 Young tableaux; 1.2 The 1st integrability condition
  • 1.3 The 2nd integrability condition1.4 Redundancy of the 3rd integrability condition; 1.5 Commuting Killing tensors; 2 The proof of concept: a complete solution for the 3-dimensional sphere; 2.1 Properties of algebraic curvature tensors; 2.1.1 Decomposition; 2.1.2 The action of the isometry group; 2.1.3 Aligned algebraic curvature tensors; 2.1.4 Diagonal algebraic curvature tensors; 2.1.5 The residual action of the isometry group; 2.2 Solution of the algebraic integrability conditions; 2.2.1 Reformulation of the first integrability condition; 2.2.2 Integrability implies diagonalisability
  • 2.2.3 Solution of the second integrability condition2.3 The algebraic geometry of the Killing-Stäckel variety ; 2.4 Interpretation of the Killing-Stäckel variety ; 2.4.1 Stäckel systems and isokernel lines ; 2.4.2 Antisymmetric matrices and special conformal Killing tensors; 2.4.3 Isokernel planes and integrable Killing tensors from S ; 2.5 Separation coordinates; 2.6 The space of separation coordinates; 2.7 The variety of integrable Killing tensors; 3 The generalisation: a solution for spheres of arbitrary dimension; 3.1 An alternative definition of Stäckel systems
  • 3.2 Killing tensors with diagonal algebraic curvature tensor3.3 The residual action of the isometry group; 3.4 Gaudin subalgebras of the Kohno-Drinfeld Lie algebra and the moduli space M0̄,n+1 ; 3.5 The real version M0̄,n+1(R) and Stasheff polytopes ; 3.5.1 Topology; 3.5.2 Combinatorics; 3.5.3 Operad structure; 3.6 The correspondence; 3.7 Applications; 3.7.1 Enumerating separation coordinates; 3.7.2 Constructing separation coordinates via the mosaic operad; 3.7.3 Constructing Stäckel systems via the mosaic operad ; 4 The perspectives: applications and generalisations
  • 4.1 Other families of Riemannian manifolds4.2 Further notions of variable separation; 4.3 Applications; 4.3.1 Separable potentials.; 4.3.2 Special function theory; 4.3.3 Generalisations of M0̄,n+1(R) ; Acknowledgements; Bibliography
Dimensions
unknown
Extent
1 online resource
File format
unknown
Form of item
online
Isbn
9783658114084
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Note
SpringerLink
Other control number
10.1007/978-3-658-11408-4
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
  • (OCoLC)925478417
  • (OCoLC)ocn925478417
Label
An algebraic geometric approach to separation of variables, Konrad Schöbel
Link
https://ezproxy.lib.ou.edu/login?url=http://link.springer.com/10.1007/978-3-658-11408-4
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Preface; Contents; 0 Introduction; 0.1 Separation of variables; 0.2 History; 0.3 Aim; 0.4 Mathematical formulation; 0.4.1 Separation coordinates; 0.4.2 Killing tensors, Nijenhuis integrability and Stäckel systems ; 0.4.3 Observations; 0.4.4 The problems; 0.5 Method; 0.6 Scope; 0.7 Some examples; 0.7.1 Cofactor systems; 0.7.2 Elliptic and polyspherical coordinates; 0.7.3 A trivial and a simple example: Spheres of dimension one and two; 0.8 Overview over the main results; 1 The foundation: the algebraic integrability conditions; 1.1 Young tableaux; 1.2 The 1st integrability condition
  • 1.3 The 2nd integrability condition1.4 Redundancy of the 3rd integrability condition; 1.5 Commuting Killing tensors; 2 The proof of concept: a complete solution for the 3-dimensional sphere; 2.1 Properties of algebraic curvature tensors; 2.1.1 Decomposition; 2.1.2 The action of the isometry group; 2.1.3 Aligned algebraic curvature tensors; 2.1.4 Diagonal algebraic curvature tensors; 2.1.5 The residual action of the isometry group; 2.2 Solution of the algebraic integrability conditions; 2.2.1 Reformulation of the first integrability condition; 2.2.2 Integrability implies diagonalisability
  • 2.2.3 Solution of the second integrability condition2.3 The algebraic geometry of the Killing-Stäckel variety ; 2.4 Interpretation of the Killing-Stäckel variety ; 2.4.1 Stäckel systems and isokernel lines ; 2.4.2 Antisymmetric matrices and special conformal Killing tensors; 2.4.3 Isokernel planes and integrable Killing tensors from S ; 2.5 Separation coordinates; 2.6 The space of separation coordinates; 2.7 The variety of integrable Killing tensors; 3 The generalisation: a solution for spheres of arbitrary dimension; 3.1 An alternative definition of Stäckel systems
  • 3.2 Killing tensors with diagonal algebraic curvature tensor3.3 The residual action of the isometry group; 3.4 Gaudin subalgebras of the Kohno-Drinfeld Lie algebra and the moduli space M0̄,n+1 ; 3.5 The real version M0̄,n+1(R) and Stasheff polytopes ; 3.5.1 Topology; 3.5.2 Combinatorics; 3.5.3 Operad structure; 3.6 The correspondence; 3.7 Applications; 3.7.1 Enumerating separation coordinates; 3.7.2 Constructing separation coordinates via the mosaic operad; 3.7.3 Constructing Stäckel systems via the mosaic operad ; 4 The perspectives: applications and generalisations
  • 4.1 Other families of Riemannian manifolds4.2 Further notions of variable separation; 4.3 Applications; 4.3.1 Separable potentials.; 4.3.2 Special function theory; 4.3.3 Generalisations of M0̄,n+1(R) ; Acknowledgements; Bibliography
Dimensions
unknown
Extent
1 online resource
File format
unknown
Form of item
online
Isbn
9783658114084
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Note
SpringerLink
Other control number
10.1007/978-3-658-11408-4
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
  • (OCoLC)925478417
  • (OCoLC)ocn925478417

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