Symmetries and Integrability of Difference Equations.:
A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike.
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Weitere Verfasser: | , , |
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge :
Cambridge University Press,
2011.
|
| Schriftenreihe: | London Mathematical Society Lecture Note Series, 381.
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike. |
| Beschreibung: | 3.8 Soliton solutions. |
| Beschreibung: | 1 online resource (362 pages) |
| Bibliographie: | Includes bibliographical references. |
| ISBN: | 9781139117098 1139117092 9781139127752 1139127756 9780511997136 0511997132 1139114921 9781139114929 1280776099 9781280776090 1139122835 9781139122832 9786613686480 6613686484 1139112732 9781139112734 |
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| 100 | 1 | |a Levi, D. |q (Decio) |1 https://id.oclc.org/worldcat/entity/E39PCjvh9P8fhmgvbvMJb9fvf3 |0 http://id.loc.gov/authorities/names/n80151915 | |
| 245 | 1 | 0 | |a Symmetries and Integrability of Difference Equations. |
| 260 | |a Cambridge : |b Cambridge University Press, |c 2011. | ||
| 300 | |a 1 online resource (362 pages) | ||
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| 490 | 1 | |a London Mathematical Society Lecture Note Series, 381 ; |v v. 381 | |
| 505 | 0 | |a Cover; Title; Copyright; Contents; List of figures; List of contributors; Preface; Introduction; 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals V. orodnitsyn and R. Kozlov; Abstract; 1.1 Introduction; 1.2 Invariance of Euler-Lagrange equations; 1.3 Lagrangian formalism for second-order difference equations; 1.4 Hamiltonian formalism for differential equations; 1.4.1 Canonical Hamiltonian equations; 1.4.2 The Legendre transformation; 1.4.3 Invariance of canonical Hamiltonian equations; 1.5 Discrete Hamiltonian formalism. | |
| 505 | 8 | |a 1.5.1 Discrete Legendre transform1.5.2 Variational formulation of the discrete Hamiltonian equations; 1.5.3 Symplecticity of the discrete Hamiltonian equations; 1.5.4 Invariance of the Hamiltonian action; 1.5.5 Discrete Hamiltonian identity and discrete Noether theorem; 1.5.6 Invariance of the discrete Hamiltonian equations; 1.6 Examples; 1.6.1 Nonlinear motion; 1.6.2 A nonlinear ODE; 1.6.3 Discrete harmonic oscillator; 1.6.4 Modified discrete harmonic oscillator (exact scheme); 1.7 Conclusion; Acknowledgments; References. | |
| 505 | 8 | |a 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete B. Grammaticos and A. RamaniAbstract; 2.1 Introduction; 2.2 A rough sketch of the top-down description of the Painlevé equations; The Hamiltonian formulation of Painlevé equations; 2.3 A succinct presentation of the bottom-up description of the Painlevé equations; Derivation of continuous Painlevé equations; 2.4 Properties of the, continuous and discrete, Painlevé equations: a parallel presentation; 2.4.1 Degeneration cascade; 2.4.2 Lax pairs; 2.4.3 Miura and Bäcklund relations; 2.4.4 Particular solutions; 2.4.5 Contiguity relations. | |
| 505 | 8 | |a 2.5 The ultradiscrete Painlevé equations2.5.1 Degeneration cascade; 2.5.2 Lax pairs; 2.5.3 Miura and Bäcklund relations; 2.5.4 Particular solutions; 2.5.5 Contiguity relations; 2.6 Conclusion; References; 3 Definitions and Predictions of Integrability for Difference Equations J. Hietarinta; Abstract; 3.1 Preliminaries; 3.1.1 Points of view on integrability; 3.1.2 Preliminaries on discreteness and discrete integrability; 3.2 Conserved quantities; 3.2.1 Constants of motion for continuous ODE; 3.2.2 The standard discrete case; 3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization. | |
| 505 | 8 | |a 3.3 Singularity confinement and algebraic entropy3.3.1 Singularity analysis for difference equations; 3.3.2 Singularity confinement in projective space; 3.3.3 Singularity confinement is not sufficient; 3.4 Integrability in 2D; 3.4.1 Definitions and examples; 3.4.2 Quadrilateral lattices; 3.4.3 Continuum limit; 3.4.4 Conservation laws; 3.5 Singularity confinement in 2D; 3.6 Algebraic entropy for 2D lattices; 3.6.1 Default growth of degree and factorization; 3.6.2 Search based on factorization; 3.7 Consistency around a cube; 3.7.1 Definition; 3.7.2 Lax pair; 3.7.3 CAC as a search method. | |
| 500 | |a 3.8 Soliton solutions. | ||
| 520 | |a A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike. | ||
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references. | ||
| 546 | |a English. | ||
| 650 | 0 | |a Difference equations. |0 http://id.loc.gov/authorities/subjects/sh85037879 | |
| 650 | 0 | |a Symmetry (Mathematics) |0 http://id.loc.gov/authorities/subjects/sh2006001303 | |
| 650 | 0 | |a Integrals. |0 http://id.loc.gov/authorities/subjects/sh85067099 | |
| 650 | 6 | |a Équations aux différences. | |
| 650 | 6 | |a Symétrie (Mathématiques) | |
| 650 | 6 | |a Intégrales. | |
| 650 | 7 | |a MATHEMATICS |x Calculus. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Mathematical Analysis. |2 bisacsh | |
| 650 | 7 | |a Difference equations |2 fast | |
| 650 | 7 | |a Integrals |2 fast | |
| 650 | 7 | |a Symmetry (Mathematics) |2 fast | |
| 700 | 1 | |a Olver, Peter. | |
| 700 | 1 | |a Thomova, Zora. | |
| 700 | 1 | |a Winternitz, Pavel. | |
| 776 | 0 | 8 | |i Print version: |a Levi, Decio. |t Symmetries and Integrability of Difference Equations. |d Cambridge : Cambridge University Press, ©2011 |z 9780521136587 |
| 830 | 0 | |a London Mathematical Society Lecture Note Series, 381. | |
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| 880 | 0 | 0 | |6 505-00/(S |g Machine generated contents note: |g 1. |t Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals / |r R. Kozlov -- |g 1.1. |t Introduction -- |g 1.2. |t Invariance of Euler-Lagrange equations -- |g 1.3. |t Lagrangian formalism for second-order difference equations -- |g 1.4. |t Hamiltonian formalism for differential equations -- |g 1.4.1. |t Canonical Hamiltonian equations -- |g 1.4.2. |t Legendre transformation -- |g 1.4.3. |t Invariance of canonical Hamiltonian equations -- |g 1.5. |t Discrete Hamiltonian formalism -- |g 1.5.1. |t Discrete Legendre transform -- |g 1.5.2. |t Variational formulation of the discrete Hamiltonian equations -- |g 1.5.3. |t Symplecticity of the discrete Hamiltonian equations -- |g 1.5.4. |t Invariance of the Hamiltonian action -- |g 1.5.5. |t Discrete Hamiltonian identity and discrete Noether theorem -- |g 1.5.6. |t Invariance of the discrete Hamiltonian equations -- |g 1.6. |t Examples -- |g 1.6.1. |t Nonlinear motion -- |g 1.6.2. |t nonlinear ODE -- |g 1.6.3. |t Discrete harmonic oscillator -- |g 1.6.4. |t Modified discrete harmonic oscillator (exact scheme) -- |g 1.7. |t Conclusion -- |g 2. |t Painleve Equations: Continuous, Discrete and Ultradiscrete / |r A. Ramani -- |g 2.1. |t Introduction -- |g 2.2. |t rough sketch of the top-down description of the Painleve equations -- |g 2.3. |t succinct presentation of the bottom-up description of the Painleve equations -- |g 2.4. |t Properties of the, continuous and discrete, Painleve equations: a parallel presentation -- |g 2.4.1. |t Degeneration cascade -- |g 2.4.2. |t Lax pairs -- |g 2.4.3. |t Miura and Backlund relations -- |g 2.4.4. |t Particular solutions -- |g 2.4.5. |t Contiguity relations -- |g 2.5. |t ultradiscrete Painleve equations -- |g 2.5.1. |t Degeneration cascade -- |g 2.5.2. |t Lax pairs -- |g 2.5.3. |t Miura and Backlund relations -- |g 2.5.4. |t Particular solutions -- |g 2.5.5. |t Contiguity relations -- |g 2.6. |t Conclusion -- |g 3. |t Definitions and Predictions of Integrability for Difference Equations / |r J. Hietarinta -- |g 3.1. |t Preliminaries -- |g 3.1.1. |t Points of view on integrability -- |g 3.1.2. |t Preliminaries on discreteness and discrete integrability -- |g 3.2. |t Conserved quantities -- |g 3.2.1. |t Constants of motion for continuous ODE -- |g 3.2.2. |t standard discrete case -- |g 3.2.3. |t Hirota-Kimura-Yahagi (HKY) generalization -- |g 3.3. |t Singularity confinement and algebraic entropy -- |g 3.3.1. |t Singularity analysis for difference equations -- |g 3.3.2. |t Singularity confinement in projective space -- |g 3.3.3. |t Singularity confinement is not sufficient -- |g 3.4. |t Integrability in 2D -- |g 3.4.1. |t Definitions and examples -- |g 3.4.2. |t Quadrilateral lattices -- |g 3.4.3. |t Continuum limit -- |g 3.4.4. |t Conservation laws -- |g 3.5. |t Singularity confinement in 2D -- |g 3.6. |t Algebraic entropy for 2D lattices -- |g 3.6.1. |t Default growth of degree and factorization -- |g 3.6.2. |t Search based on factorization -- |g 3.7. |t Consistency around a cube -- |g 3.7.1. |t Definition -- |g 3.7.2. |t Lax pair -- |g 3.7.3. |t CAC as a search method -- |g 3.8. |t Soliton solutions -- |g 3.8.1. |t Background solutions -- |g 3.8.2. |t 1SS -- |g 3.8.3. |t NSS -- |g 3.9. |t Conclusions -- |g 4. |t Orthogonal Polynomials, their Recursions, and Functional Equations / |r M.E.H. Ismail -- |g 4.1. |t Introduction -- |g 4.2. |t Orthogonal polynomials -- |g 4.3. |t spectral theorem -- |g 4.4. |t Freud nonlinear recursions -- |g 4.5. |t Differential equations -- |g 4.6. |t q-difference equations -- |g 4.7. |t inverse problem -- |g 4.8. |t Applications -- |g 4.9. |t Askey-Wilson polynomials -- |g 5. |t Discrete Painleve Equations and Orthogonal Polynomials / |r A. Its -- |g 5.1. |t General setting -- |g 5.1.1. |t Orthogonal polynomials -- |g 5.1.2. |t Connections to integrable systems -- |g 5.1.3. |t Riemann-Hilbert representation of the orthogonal polynomials -- |g 5.1.4. |t Discrete Painleve equations -- |g 5.2. |t Examples -- |g 5.2.1. |t Gaussian weight -- |g 5.2.2. |t d-Painleve I -- |g 5.2.3. |t d-Painleve XXXIV -- |g 6. |t Generalized Lie Symmetries for Difference Equations / |r R.I. Yamilov -- |g 6.1. |t Introduction -- |g 6.1.1. |t Direct construction of generalized symmetries: an example -- |g 6.2. |t Generalized symmetries from the integrability properties -- |g 6.2.1. |t Toda Lattice -- |g 6.2.2. |t symmetry algebra for the Toda Lattice -- |g 6.2.3. |t continuous limit of the Toda symmetry algebras -- |g 6.2.4. |t Backlund transformations for the Toda equation -- |g 6.2.5. |t Backlund transformations vs. generalized symmetries -- |g 6.2.6. |t Generalized symmetries for PδE's -- |g 6.3. |t Formal symmetries and integrable lattice equations -- |g 6.3.1. |t Formal symmetries and further integrability conditions -- |g 6.3.2. |t Why integrable equations on the lattice must be symmetric -- |g 6.3.3. |t Example of classification problem -- |g 7. |t Four Lectures on Discrete Systems / |r S.P. Novikov -- |g 7.1. |t Introduction -- |g 7.2. |t Discrete symmetries and completely integrable systems -- |g 7.3. |t Discretization of linear operators -- |g 7.4. |t Discrete GLn connections and triangle equation -- |g 7.5. |t New discretization of complex analysis -- |g 8. |t Lectures on Moving Frames / |r P.J. Olver -- |g 8.1. |t Introduction -- |g 8.2. |t Equivariant moving frames -- |g 8.3. |t Moving frames on jet space and differential invariants -- |g 8.4. |t Equivalence and signatures -- |g 8.5. |t Joint invariants and joint differential invariants -- |g 8.6. |t Invariant numerical approximations -- |g 8.7. |t invariant bicomplex -- |g 8.8. |t Generating differential invariants -- |g 8.9. |t Invariant variational problems -- |g 8.10. |t Invariant curve flows -- |g 9. |t Lattices of Compact Semisimple Lie Groups / |r J. Patera -- |g 9.1. |t Introduction -- |g 9.2. |t Motivating example -- |g 9.3. |t Simple Lie groups and simple Lie algebras -- |g 9.3.1. |t Simple roots -- |g 9.3.2. |t Standard bases in Rn -- |g 9.3.3. |t Reflections and affine reflections in Rn -- |g 9.3.4. |t Weyl group and Affine Weyl group -- |g 9.4. |t Lattice grids FM [⊂] F [⊂] Rn -- |g 9.4.1. |t Examples of FM -- |g 9.5. |t W-invariant functions orthogonal on FM -- |g 9.6. |t Properties of elements of finite order -- |g 10. |t Lectures on Discrete Differential Geometry / |r Yu. B Suris -- |g 10.1. |t Basic notions -- |g 10.2. |t Backlund transformations -- |g 10.3. |t Q-nets -- |g 10.4. |t Circular nets -- |g 10.5. |t Q-nets in quadrics -- |g 10.6. |t T-nets -- |g 10.7. |t A-nets -- |g 10.8. |t T-nets in quadrics -- |g 10.9. |t K-nets -- |g 10.10. |t Hirota equation for K-nets -- |g 11. |t Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations / |r P. |
| 880 | 0 | 0 | |6 505-00/(S |t Winternitz -- |g 11.1. |t Symmetry preserving discretization of ODEs -- |g 11.1.1. |t Formulation of the problem -- |g 11.1.2. |t Lie point symmetries of ordinary difference schemes -- |g 11.1.3. |t continuous limit -- |g 11.2. |t Examples of symmetry preserving discretizations -- |g 11.2.1. |t Equations invariant under SL1(2, R) -- |g 11.2.2. |t Equations invariant under SL2(2, R) -- |g 11.2.3. |t Equations invariant under the similitude group of the Euclidean plane -- |g 11.3. |t Applications to numerical solutions of ODEs -- |g 11.3.1. |t General procedure for testing the numerical schemes -- |g 11.3.2. |t Numerical experiments for a third-order ODE invariant under SL1(2, R) -- |g 11.3.3. |t Numerical experiments for ODEs invariant under SL2(2, R) -- |g 11.3.4. |t Numerical experiments for third-order ODE invariant under Sim(2) -- |g 11.4. |t Exact solutions of invariant difference schemes -- |g 11.4.1. |t Lagrangian formulation for second-order ODEs -- |g 11.4.2. |t Lagrangian formulation for second order difference equations -- |g 11.4.3. |t Example: Second-order ODE with three-dimensional solvable symmetry algebra -- |g 11.5. |t Lie point symmetries of differential-difference equations -- |g 11.5.1. |t Formulation of the problem -- |g 11.5.2. |t evolutionary formalism and commuting flows for differential equations -- |g 11.5.3. |t evolutionary formalism and commuting flows for differential-difference equations -- |g 11.5.4. |t General algorithm for calculating Lie point symmetries of a differential-difference equation -- |g 11.5.5. |t Theorems simplifying the calculation of symmetries of DδE -- |g 11.5.6. |t Volterra type equations and their generalizations -- |g 11.5.7. |t Toda type equations -- |g 11.5.8. |t Toda field theory type equations -- |g 11.6. |t Examples of symmetries of DδE -- |g 11.6.1. |t YdKN equation -- |g 11.6.2. |t Toda lattice. |
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| author | Levi, D. (Decio) |
| author2 | Olver, Peter Thomova, Zora Winternitz, Pavel |
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| author_GND | http://id.loc.gov/authorities/names/n80151915 |
| author_facet | Levi, D. (Decio) Olver, Peter Thomova, Zora Winternitz, Pavel |
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| contents | Cover; Title; Copyright; Contents; List of figures; List of contributors; Preface; Introduction; 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals V. orodnitsyn and R. Kozlov; Abstract; 1.1 Introduction; 1.2 Invariance of Euler-Lagrange equations; 1.3 Lagrangian formalism for second-order difference equations; 1.4 Hamiltonian formalism for differential equations; 1.4.1 Canonical Hamiltonian equations; 1.4.2 The Legendre transformation; 1.4.3 Invariance of canonical Hamiltonian equations; 1.5 Discrete Hamiltonian formalism. 1.5.1 Discrete Legendre transform1.5.2 Variational formulation of the discrete Hamiltonian equations; 1.5.3 Symplecticity of the discrete Hamiltonian equations; 1.5.4 Invariance of the Hamiltonian action; 1.5.5 Discrete Hamiltonian identity and discrete Noether theorem; 1.5.6 Invariance of the discrete Hamiltonian equations; 1.6 Examples; 1.6.1 Nonlinear motion; 1.6.2 A nonlinear ODE; 1.6.3 Discrete harmonic oscillator; 1.6.4 Modified discrete harmonic oscillator (exact scheme); 1.7 Conclusion; Acknowledgments; References. 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete B. Grammaticos and A. RamaniAbstract; 2.1 Introduction; 2.2 A rough sketch of the top-down description of the Painlevé equations; The Hamiltonian formulation of Painlevé equations; 2.3 A succinct presentation of the bottom-up description of the Painlevé equations; Derivation of continuous Painlevé equations; 2.4 Properties of the, continuous and discrete, Painlevé equations: a parallel presentation; 2.4.1 Degeneration cascade; 2.4.2 Lax pairs; 2.4.3 Miura and Bäcklund relations; 2.4.4 Particular solutions; 2.4.5 Contiguity relations. 2.5 The ultradiscrete Painlevé equations2.5.1 Degeneration cascade; 2.5.2 Lax pairs; 2.5.3 Miura and Bäcklund relations; 2.5.4 Particular solutions; 2.5.5 Contiguity relations; 2.6 Conclusion; References; 3 Definitions and Predictions of Integrability for Difference Equations J. Hietarinta; Abstract; 3.1 Preliminaries; 3.1.1 Points of view on integrability; 3.1.2 Preliminaries on discreteness and discrete integrability; 3.2 Conserved quantities; 3.2.1 Constants of motion for continuous ODE; 3.2.2 The standard discrete case; 3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization. 3.3 Singularity confinement and algebraic entropy3.3.1 Singularity analysis for difference equations; 3.3.2 Singularity confinement in projective space; 3.3.3 Singularity confinement is not sufficient; 3.4 Integrability in 2D; 3.4.1 Definitions and examples; 3.4.2 Quadrilateral lattices; 3.4.3 Continuum limit; 3.4.4 Conservation laws; 3.5 Singularity confinement in 2D; 3.6 Algebraic entropy for 2D lattices; 3.6.1 Default growth of degree and factorization; 3.6.2 Search based on factorization; 3.7 Consistency around a cube; 3.7.1 Definition; 3.7.2 Lax pair; 3.7.3 CAC as a search method. |
| ctrlnum | (OCoLC)769341701 |
| dewey-full | 515.625 515/.625 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515.625 515/.625 |
| dewey-search | 515.625 515/.625 |
| dewey-sort | 3515.625 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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code="2">bicssc</subfield></datafield><datafield tag="072" ind1=" " ind2="7"><subfield code="a">MAT</subfield><subfield code="x">005000</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="072" ind1=" " ind2="7"><subfield code="a">MAT</subfield><subfield code="x">034000</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="082" ind1="7" ind2=" "><subfield code="a">515.625</subfield><subfield code="a">515/.625</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">MAIN</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Levi, D.</subfield><subfield code="q">(Decio)</subfield><subfield code="1">https://id.oclc.org/worldcat/entity/E39PCjvh9P8fhmgvbvMJb9fvf3</subfield><subfield code="0">http://id.loc.gov/authorities/names/n80151915</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Symmetries and Integrability of Difference Equations.</subfield></datafield><datafield tag="260" ind1=" " ind2=" "><subfield code="a">Cambridge :</subfield><subfield code="b">Cambridge University Press,</subfield><subfield code="c">2011.</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (362 pages)</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">computer</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">online resource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">London Mathematical Society Lecture Note Series, 381 ;</subfield><subfield code="v">v. 381</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">Cover; Title; Copyright; Contents; List of figures; List of contributors; Preface; Introduction; 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals V. orodnitsyn and R. Kozlov; Abstract; 1.1 Introduction; 1.2 Invariance of Euler-Lagrange equations; 1.3 Lagrangian formalism for second-order difference equations; 1.4 Hamiltonian formalism for differential equations; 1.4.1 Canonical Hamiltonian equations; 1.4.2 The Legendre transformation; 1.4.3 Invariance of canonical Hamiltonian equations; 1.5 Discrete Hamiltonian formalism.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">1.5.1 Discrete Legendre transform1.5.2 Variational formulation of the discrete Hamiltonian equations; 1.5.3 Symplecticity of the discrete Hamiltonian equations; 1.5.4 Invariance of the Hamiltonian action; 1.5.5 Discrete Hamiltonian identity and discrete Noether theorem; 1.5.6 Invariance of the discrete Hamiltonian equations; 1.6 Examples; 1.6.1 Nonlinear motion; 1.6.2 A nonlinear ODE; 1.6.3 Discrete harmonic oscillator; 1.6.4 Modified discrete harmonic oscillator (exact scheme); 1.7 Conclusion; Acknowledgments; References.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">2 Painlevé Equations: Continuous, Discrete and Ultradiscrete B. Grammaticos and A. RamaniAbstract; 2.1 Introduction; 2.2 A rough sketch of the top-down description of the Painlevé equations; The Hamiltonian formulation of Painlevé equations; 2.3 A succinct presentation of the bottom-up description of the Painlevé equations; Derivation of continuous Painlevé equations; 2.4 Properties of the, continuous and discrete, Painlevé equations: a parallel presentation; 2.4.1 Degeneration cascade; 2.4.2 Lax pairs; 2.4.3 Miura and Bäcklund relations; 2.4.4 Particular solutions; 2.4.5 Contiguity relations.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">2.5 The ultradiscrete Painlevé equations2.5.1 Degeneration cascade; 2.5.2 Lax pairs; 2.5.3 Miura and Bäcklund relations; 2.5.4 Particular solutions; 2.5.5 Contiguity relations; 2.6 Conclusion; References; 3 Definitions and Predictions of Integrability for Difference Equations J. Hietarinta; Abstract; 3.1 Preliminaries; 3.1.1 Points of view on integrability; 3.1.2 Preliminaries on discreteness and discrete integrability; 3.2 Conserved quantities; 3.2.1 Constants of motion for continuous ODE; 3.2.2 The standard discrete case; 3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">3.3 Singularity confinement and algebraic entropy3.3.1 Singularity analysis for difference equations; 3.3.2 Singularity confinement in projective space; 3.3.3 Singularity confinement is not sufficient; 3.4 Integrability in 2D; 3.4.1 Definitions and examples; 3.4.2 Quadrilateral lattices; 3.4.3 Continuum limit; 3.4.4 Conservation laws; 3.5 Singularity confinement in 2D; 3.6 Algebraic entropy for 2D lattices; 3.6.1 Default growth of degree and factorization; 3.6.2 Search based on factorization; 3.7 Consistency around a cube; 3.7.1 Definition; 3.7.2 Lax pair; 3.7.3 CAC as a search method.</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">3.8 Soliton solutions.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike.</subfield></datafield><datafield tag="588" ind1="0" ind2=" "><subfield code="a">Print version record.</subfield></datafield><datafield tag="504" ind1=" " ind2=" "><subfield code="a">Includes bibliographical references.</subfield></datafield><datafield tag="546" ind1=" " ind2=" "><subfield code="a">English.</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Difference equations.</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh85037879</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Symmetry (Mathematics)</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh2006001303</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Integrals.</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh85067099</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Équations aux différences.</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Symétrie (Mathématiques)</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Intégrales.</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS</subfield><subfield code="x">Calculus.</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS</subfield><subfield code="x">Mathematical Analysis.</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Difference equations</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Integrals</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Symmetry (Mathematics)</subfield><subfield code="2">fast</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Olver, Peter.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Thomova, Zora.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Winternitz, Pavel.</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Print version:</subfield><subfield code="a">Levi, Decio.</subfield><subfield code="t">Symmetries and Integrability of Difference Equations.</subfield><subfield code="d">Cambridge : Cambridge University Press, ©2011</subfield><subfield code="z">9780521136587</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">London Mathematical Society Lecture Note Series, 381.</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="l">FWS01</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=399282</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="880" ind1="0" ind2="0"><subfield code="6">505-00/(S</subfield><subfield code="g">Machine generated contents note:</subfield><subfield code="g">1.</subfield><subfield code="t">Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals /</subfield><subfield code="r">R. Kozlov --</subfield><subfield code="g">1.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">1.2.</subfield><subfield code="t">Invariance of Euler-Lagrange equations --</subfield><subfield code="g">1.3.</subfield><subfield code="t">Lagrangian formalism for second-order difference equations --</subfield><subfield code="g">1.4.</subfield><subfield code="t">Hamiltonian formalism for differential equations --</subfield><subfield code="g">1.4.1.</subfield><subfield code="t">Canonical Hamiltonian equations --</subfield><subfield code="g">1.4.2.</subfield><subfield code="t">Legendre transformation --</subfield><subfield code="g">1.4.3.</subfield><subfield code="t">Invariance of canonical Hamiltonian equations --</subfield><subfield code="g">1.5.</subfield><subfield code="t">Discrete Hamiltonian formalism --</subfield><subfield code="g">1.5.1.</subfield><subfield code="t">Discrete Legendre transform --</subfield><subfield code="g">1.5.2.</subfield><subfield code="t">Variational formulation of the discrete Hamiltonian equations --</subfield><subfield code="g">1.5.3.</subfield><subfield code="t">Symplecticity of the discrete Hamiltonian equations --</subfield><subfield code="g">1.5.4.</subfield><subfield code="t">Invariance of the Hamiltonian action --</subfield><subfield code="g">1.5.5.</subfield><subfield code="t">Discrete Hamiltonian identity and discrete Noether theorem --</subfield><subfield code="g">1.5.6.</subfield><subfield code="t">Invariance of the discrete Hamiltonian equations --</subfield><subfield code="g">1.6.</subfield><subfield code="t">Examples --</subfield><subfield code="g">1.6.1.</subfield><subfield code="t">Nonlinear motion --</subfield><subfield code="g">1.6.2.</subfield><subfield code="t">nonlinear ODE --</subfield><subfield code="g">1.6.3.</subfield><subfield code="t">Discrete harmonic oscillator --</subfield><subfield code="g">1.6.4.</subfield><subfield code="t">Modified discrete harmonic oscillator (exact scheme) --</subfield><subfield code="g">1.7.</subfield><subfield code="t">Conclusion --</subfield><subfield code="g">2.</subfield><subfield code="t">Painleve Equations: Continuous, Discrete and Ultradiscrete /</subfield><subfield code="r">A. Ramani --</subfield><subfield code="g">2.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">2.2.</subfield><subfield code="t">rough sketch of the top-down description of the Painleve equations --</subfield><subfield code="g">2.3.</subfield><subfield code="t">succinct presentation of the bottom-up description of the Painleve equations --</subfield><subfield code="g">2.4.</subfield><subfield code="t">Properties of the, continuous and discrete, Painleve equations: a parallel presentation --</subfield><subfield code="g">2.4.1.</subfield><subfield code="t">Degeneration cascade --</subfield><subfield code="g">2.4.2.</subfield><subfield code="t">Lax pairs --</subfield><subfield code="g">2.4.3.</subfield><subfield code="t">Miura and Backlund relations --</subfield><subfield code="g">2.4.4.</subfield><subfield code="t">Particular solutions --</subfield><subfield code="g">2.4.5.</subfield><subfield code="t">Contiguity relations --</subfield><subfield code="g">2.5.</subfield><subfield code="t">ultradiscrete Painleve equations --</subfield><subfield code="g">2.5.1.</subfield><subfield code="t">Degeneration cascade --</subfield><subfield code="g">2.5.2.</subfield><subfield code="t">Lax pairs --</subfield><subfield code="g">2.5.3.</subfield><subfield code="t">Miura and Backlund relations --</subfield><subfield code="g">2.5.4.</subfield><subfield code="t">Particular solutions --</subfield><subfield code="g">2.5.5.</subfield><subfield code="t">Contiguity relations --</subfield><subfield code="g">2.6.</subfield><subfield code="t">Conclusion --</subfield><subfield code="g">3.</subfield><subfield code="t">Definitions and Predictions of Integrability for Difference Equations /</subfield><subfield code="r">J. Hietarinta --</subfield><subfield code="g">3.1.</subfield><subfield code="t">Preliminaries --</subfield><subfield code="g">3.1.1.</subfield><subfield code="t">Points of view on integrability --</subfield><subfield code="g">3.1.2.</subfield><subfield code="t">Preliminaries on discreteness and discrete integrability --</subfield><subfield code="g">3.2.</subfield><subfield code="t">Conserved quantities --</subfield><subfield code="g">3.2.1.</subfield><subfield code="t">Constants of motion for continuous ODE --</subfield><subfield code="g">3.2.2.</subfield><subfield code="t">standard discrete case --</subfield><subfield code="g">3.2.3.</subfield><subfield code="t">Hirota-Kimura-Yahagi (HKY) generalization --</subfield><subfield code="g">3.3.</subfield><subfield code="t">Singularity confinement and algebraic entropy --</subfield><subfield code="g">3.3.1.</subfield><subfield code="t">Singularity analysis for difference equations --</subfield><subfield code="g">3.3.2.</subfield><subfield code="t">Singularity confinement in projective space --</subfield><subfield code="g">3.3.3.</subfield><subfield code="t">Singularity confinement is not sufficient --</subfield><subfield code="g">3.4.</subfield><subfield code="t">Integrability in 2D --</subfield><subfield code="g">3.4.1.</subfield><subfield code="t">Definitions and examples --</subfield><subfield code="g">3.4.2.</subfield><subfield code="t">Quadrilateral lattices --</subfield><subfield code="g">3.4.3.</subfield><subfield code="t">Continuum limit --</subfield><subfield code="g">3.4.4.</subfield><subfield code="t">Conservation laws --</subfield><subfield code="g">3.5.</subfield><subfield code="t">Singularity confinement in 2D --</subfield><subfield code="g">3.6.</subfield><subfield code="t">Algebraic entropy for 2D lattices --</subfield><subfield code="g">3.6.1.</subfield><subfield code="t">Default growth of degree and factorization --</subfield><subfield code="g">3.6.2.</subfield><subfield code="t">Search based on factorization --</subfield><subfield code="g">3.7.</subfield><subfield code="t">Consistency around a cube --</subfield><subfield code="g">3.7.1.</subfield><subfield code="t">Definition --</subfield><subfield code="g">3.7.2.</subfield><subfield code="t">Lax pair --</subfield><subfield code="g">3.7.3.</subfield><subfield code="t">CAC as a search method --</subfield><subfield code="g">3.8.</subfield><subfield code="t">Soliton solutions --</subfield><subfield code="g">3.8.1.</subfield><subfield code="t">Background solutions --</subfield><subfield code="g">3.8.2.</subfield><subfield code="t">1SS --</subfield><subfield code="g">3.8.3.</subfield><subfield code="t">NSS --</subfield><subfield code="g">3.9.</subfield><subfield code="t">Conclusions --</subfield><subfield code="g">4.</subfield><subfield code="t">Orthogonal Polynomials, their Recursions, and Functional Equations /</subfield><subfield code="r">M.E.H. Ismail --</subfield><subfield code="g">4.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">4.2.</subfield><subfield code="t">Orthogonal polynomials --</subfield><subfield code="g">4.3.</subfield><subfield code="t">spectral theorem --</subfield><subfield code="g">4.4.</subfield><subfield code="t">Freud nonlinear recursions --</subfield><subfield code="g">4.5.</subfield><subfield code="t">Differential equations --</subfield><subfield code="g">4.6.</subfield><subfield code="t">q-difference equations --</subfield><subfield code="g">4.7.</subfield><subfield code="t">inverse problem --</subfield><subfield code="g">4.8.</subfield><subfield code="t">Applications --</subfield><subfield code="g">4.9.</subfield><subfield code="t">Askey-Wilson polynomials --</subfield><subfield code="g">5.</subfield><subfield code="t">Discrete Painleve Equations and Orthogonal Polynomials /</subfield><subfield code="r">A. Its --</subfield><subfield code="g">5.1.</subfield><subfield code="t">General setting --</subfield><subfield code="g">5.1.1.</subfield><subfield code="t">Orthogonal polynomials --</subfield><subfield code="g">5.1.2.</subfield><subfield code="t">Connections to integrable systems --</subfield><subfield code="g">5.1.3.</subfield><subfield code="t">Riemann-Hilbert representation of the orthogonal polynomials --</subfield><subfield code="g">5.1.4.</subfield><subfield code="t">Discrete Painleve equations --</subfield><subfield code="g">5.2.</subfield><subfield code="t">Examples --</subfield><subfield code="g">5.2.1.</subfield><subfield code="t">Gaussian weight --</subfield><subfield code="g">5.2.2.</subfield><subfield code="t">d-Painleve I --</subfield><subfield code="g">5.2.3.</subfield><subfield code="t">d-Painleve XXXIV --</subfield><subfield code="g">6.</subfield><subfield code="t">Generalized Lie Symmetries for Difference Equations /</subfield><subfield code="r">R.I. Yamilov --</subfield><subfield code="g">6.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">6.1.1.</subfield><subfield code="t">Direct construction of generalized symmetries: an example --</subfield><subfield code="g">6.2.</subfield><subfield code="t">Generalized symmetries from the integrability properties --</subfield><subfield code="g">6.2.1.</subfield><subfield code="t">Toda Lattice --</subfield><subfield code="g">6.2.2.</subfield><subfield code="t">symmetry algebra for the Toda Lattice --</subfield><subfield code="g">6.2.3.</subfield><subfield code="t">continuous limit of the Toda symmetry algebras --</subfield><subfield code="g">6.2.4.</subfield><subfield code="t">Backlund transformations for the Toda equation --</subfield><subfield code="g">6.2.5.</subfield><subfield code="t">Backlund transformations vs. generalized symmetries --</subfield><subfield code="g">6.2.6.</subfield><subfield code="t">Generalized symmetries for PδE's --</subfield><subfield code="g">6.3.</subfield><subfield code="t">Formal symmetries and integrable lattice equations --</subfield><subfield code="g">6.3.1.</subfield><subfield code="t">Formal symmetries and further integrability conditions --</subfield><subfield code="g">6.3.2.</subfield><subfield code="t">Why integrable equations on the lattice must be symmetric --</subfield><subfield code="g">6.3.3.</subfield><subfield code="t">Example of classification problem --</subfield><subfield code="g">7.</subfield><subfield code="t">Four Lectures on Discrete Systems /</subfield><subfield code="r">S.P. Novikov --</subfield><subfield code="g">7.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">7.2.</subfield><subfield code="t">Discrete symmetries and completely integrable systems --</subfield><subfield code="g">7.3.</subfield><subfield code="t">Discretization of linear operators --</subfield><subfield code="g">7.4.</subfield><subfield code="t">Discrete GLn connections and triangle equation --</subfield><subfield code="g">7.5.</subfield><subfield code="t">New discretization of complex analysis --</subfield><subfield code="g">8.</subfield><subfield code="t">Lectures on Moving Frames /</subfield><subfield code="r">P.J. Olver --</subfield><subfield code="g">8.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">8.2.</subfield><subfield code="t">Equivariant moving frames --</subfield><subfield code="g">8.3.</subfield><subfield code="t">Moving frames on jet space and differential invariants --</subfield><subfield code="g">8.4.</subfield><subfield code="t">Equivalence and signatures --</subfield><subfield code="g">8.5.</subfield><subfield code="t">Joint invariants and joint differential invariants --</subfield><subfield code="g">8.6.</subfield><subfield code="t">Invariant numerical approximations --</subfield><subfield code="g">8.7.</subfield><subfield code="t">invariant bicomplex --</subfield><subfield code="g">8.8.</subfield><subfield code="t">Generating differential invariants --</subfield><subfield code="g">8.9.</subfield><subfield code="t">Invariant variational problems --</subfield><subfield code="g">8.10.</subfield><subfield code="t">Invariant curve flows --</subfield><subfield code="g">9.</subfield><subfield code="t">Lattices of Compact Semisimple Lie Groups /</subfield><subfield code="r">J. Patera --</subfield><subfield code="g">9.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">9.2.</subfield><subfield code="t">Motivating example --</subfield><subfield code="g">9.3.</subfield><subfield code="t">Simple Lie groups and simple Lie algebras --</subfield><subfield code="g">9.3.1.</subfield><subfield code="t">Simple roots --</subfield><subfield code="g">9.3.2.</subfield><subfield code="t">Standard bases in Rn --</subfield><subfield code="g">9.3.3.</subfield><subfield code="t">Reflections and affine reflections in Rn --</subfield><subfield code="g">9.3.4.</subfield><subfield code="t">Weyl group and Affine Weyl group --</subfield><subfield code="g">9.4.</subfield><subfield code="t">Lattice grids FM [⊂] F [⊂] Rn --</subfield><subfield code="g">9.4.1.</subfield><subfield code="t">Examples of FM --</subfield><subfield code="g">9.5.</subfield><subfield code="t">W-invariant functions orthogonal on FM --</subfield><subfield code="g">9.6.</subfield><subfield code="t">Properties of elements of finite order --</subfield><subfield code="g">10.</subfield><subfield code="t">Lectures on Discrete Differential Geometry /</subfield><subfield code="r">Yu. B Suris --</subfield><subfield code="g">10.1.</subfield><subfield code="t">Basic notions --</subfield><subfield code="g">10.2.</subfield><subfield code="t">Backlund transformations --</subfield><subfield code="g">10.3.</subfield><subfield code="t">Q-nets --</subfield><subfield code="g">10.4.</subfield><subfield code="t">Circular nets --</subfield><subfield code="g">10.5.</subfield><subfield code="t">Q-nets in quadrics --</subfield><subfield code="g">10.6.</subfield><subfield code="t">T-nets --</subfield><subfield code="g">10.7.</subfield><subfield code="t">A-nets --</subfield><subfield code="g">10.8.</subfield><subfield code="t">T-nets in quadrics --</subfield><subfield code="g">10.9.</subfield><subfield code="t">K-nets --</subfield><subfield code="g">10.10.</subfield><subfield code="t">Hirota equation for K-nets --</subfield><subfield code="g">11.</subfield><subfield code="t">Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations /</subfield><subfield code="r">P.</subfield></datafield><datafield tag="880" ind1="0" ind2="0"><subfield code="6">505-00/(S</subfield><subfield code="t">Winternitz --</subfield><subfield code="g">11.1.</subfield><subfield code="t">Symmetry preserving discretization of ODEs --</subfield><subfield code="g">11.1.1.</subfield><subfield code="t">Formulation of the problem --</subfield><subfield code="g">11.1.2.</subfield><subfield code="t">Lie point symmetries of ordinary difference schemes --</subfield><subfield code="g">11.1.3.</subfield><subfield code="t">continuous limit --</subfield><subfield code="g">11.2.</subfield><subfield code="t">Examples of symmetry preserving discretizations --</subfield><subfield code="g">11.2.1.</subfield><subfield code="t">Equations invariant under SL1(2, R) --</subfield><subfield code="g">11.2.2.</subfield><subfield code="t">Equations invariant under SL2(2, R) --</subfield><subfield code="g">11.2.3.</subfield><subfield code="t">Equations invariant under the similitude group of the Euclidean plane --</subfield><subfield code="g">11.3.</subfield><subfield code="t">Applications to numerical solutions of ODEs --</subfield><subfield code="g">11.3.1.</subfield><subfield code="t">General procedure for testing the numerical schemes --</subfield><subfield code="g">11.3.2.</subfield><subfield code="t">Numerical experiments for a third-order ODE invariant under SL1(2, R) --</subfield><subfield code="g">11.3.3.</subfield><subfield code="t">Numerical experiments for ODEs invariant under SL2(2, R) --</subfield><subfield code="g">11.3.4.</subfield><subfield code="t">Numerical experiments for third-order ODE invariant under Sim(2) --</subfield><subfield code="g">11.4.</subfield><subfield code="t">Exact solutions of invariant difference schemes --</subfield><subfield code="g">11.4.1.</subfield><subfield code="t">Lagrangian formulation for second-order ODEs --</subfield><subfield code="g">11.4.2.</subfield><subfield code="t">Lagrangian formulation for second order difference equations --</subfield><subfield code="g">11.4.3.</subfield><subfield code="t">Example: Second-order ODE with three-dimensional solvable symmetry algebra --</subfield><subfield code="g">11.5.</subfield><subfield code="t">Lie point symmetries of differential-difference equations --</subfield><subfield code="g">11.5.1.</subfield><subfield code="t">Formulation of the problem --</subfield><subfield code="g">11.5.2.</subfield><subfield code="t">evolutionary formalism and commuting flows for differential equations --</subfield><subfield code="g">11.5.3.</subfield><subfield code="t">evolutionary formalism and commuting flows for differential-difference equations --</subfield><subfield code="g">11.5.4.</subfield><subfield code="t">General algorithm for calculating Lie point symmetries of a differential-difference equation --</subfield><subfield code="g">11.5.5.</subfield><subfield code="t">Theorems simplifying the calculation of symmetries of DδE --</subfield><subfield code="g">11.5.6.</subfield><subfield code="t">Volterra type equations and their generalizations --</subfield><subfield code="g">11.5.7.</subfield><subfield code="t">Toda type equations --</subfield><subfield code="g">11.5.8.</subfield><subfield code="t">Toda field theory type equations --</subfield><subfield code="g">11.6.</subfield><subfield code="t">Examples of symmetries of DδE --</subfield><subfield code="g">11.6.1.</subfield><subfield code="t">YdKN equation --</subfield><subfield code="g">11.6.2.</subfield><subfield code="t">Toda lattice.</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">Askews and Holts Library Services</subfield><subfield code="b">ASKH</subfield><subfield code="n">AH13438962</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">Coutts Information Services</subfield><subfield code="b">COUT</subfield><subfield code="n">22772753</subfield></datafield><datafield 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| id | ZDB-4-EBA-ocn769341701 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:18:10Z |
| institution | BVB |
| isbn | 9781139117098 1139117092 9781139127752 1139127756 9780511997136 0511997132 1139114921 9781139114929 1280776099 9781280776090 1139122835 9781139122832 9786613686480 6613686484 1139112732 9781139112734 |
| language | English |
| oclc_num | 769341701 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (362 pages) |
| psigel | ZDB-4-EBA |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | London Mathematical Society Lecture Note Series, 381. |
| series2 | London Mathematical Society Lecture Note Series, 381 ; |
| spelling | Levi, D. (Decio) https://id.oclc.org/worldcat/entity/E39PCjvh9P8fhmgvbvMJb9fvf3 http://id.loc.gov/authorities/names/n80151915 Symmetries and Integrability of Difference Equations. Cambridge : Cambridge University Press, 2011. 1 online resource (362 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society Lecture Note Series, 381 ; v. 381 Cover; Title; Copyright; Contents; List of figures; List of contributors; Preface; Introduction; 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals V. orodnitsyn and R. Kozlov; Abstract; 1.1 Introduction; 1.2 Invariance of Euler-Lagrange equations; 1.3 Lagrangian formalism for second-order difference equations; 1.4 Hamiltonian formalism for differential equations; 1.4.1 Canonical Hamiltonian equations; 1.4.2 The Legendre transformation; 1.4.3 Invariance of canonical Hamiltonian equations; 1.5 Discrete Hamiltonian formalism. 1.5.1 Discrete Legendre transform1.5.2 Variational formulation of the discrete Hamiltonian equations; 1.5.3 Symplecticity of the discrete Hamiltonian equations; 1.5.4 Invariance of the Hamiltonian action; 1.5.5 Discrete Hamiltonian identity and discrete Noether theorem; 1.5.6 Invariance of the discrete Hamiltonian equations; 1.6 Examples; 1.6.1 Nonlinear motion; 1.6.2 A nonlinear ODE; 1.6.3 Discrete harmonic oscillator; 1.6.4 Modified discrete harmonic oscillator (exact scheme); 1.7 Conclusion; Acknowledgments; References. 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete B. Grammaticos and A. RamaniAbstract; 2.1 Introduction; 2.2 A rough sketch of the top-down description of the Painlevé equations; The Hamiltonian formulation of Painlevé equations; 2.3 A succinct presentation of the bottom-up description of the Painlevé equations; Derivation of continuous Painlevé equations; 2.4 Properties of the, continuous and discrete, Painlevé equations: a parallel presentation; 2.4.1 Degeneration cascade; 2.4.2 Lax pairs; 2.4.3 Miura and Bäcklund relations; 2.4.4 Particular solutions; 2.4.5 Contiguity relations. 2.5 The ultradiscrete Painlevé equations2.5.1 Degeneration cascade; 2.5.2 Lax pairs; 2.5.3 Miura and Bäcklund relations; 2.5.4 Particular solutions; 2.5.5 Contiguity relations; 2.6 Conclusion; References; 3 Definitions and Predictions of Integrability for Difference Equations J. Hietarinta; Abstract; 3.1 Preliminaries; 3.1.1 Points of view on integrability; 3.1.2 Preliminaries on discreteness and discrete integrability; 3.2 Conserved quantities; 3.2.1 Constants of motion for continuous ODE; 3.2.2 The standard discrete case; 3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization. 3.3 Singularity confinement and algebraic entropy3.3.1 Singularity analysis for difference equations; 3.3.2 Singularity confinement in projective space; 3.3.3 Singularity confinement is not sufficient; 3.4 Integrability in 2D; 3.4.1 Definitions and examples; 3.4.2 Quadrilateral lattices; 3.4.3 Continuum limit; 3.4.4 Conservation laws; 3.5 Singularity confinement in 2D; 3.6 Algebraic entropy for 2D lattices; 3.6.1 Default growth of degree and factorization; 3.6.2 Search based on factorization; 3.7 Consistency around a cube; 3.7.1 Definition; 3.7.2 Lax pair; 3.7.3 CAC as a search method. 3.8 Soliton solutions. A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike. Print version record. Includes bibliographical references. English. Difference equations. http://id.loc.gov/authorities/subjects/sh85037879 Symmetry (Mathematics) http://id.loc.gov/authorities/subjects/sh2006001303 Integrals. http://id.loc.gov/authorities/subjects/sh85067099 Équations aux différences. Symétrie (Mathématiques) Intégrales. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Difference equations fast Integrals fast Symmetry (Mathematics) fast Olver, Peter. Thomova, Zora. Winternitz, Pavel. Print version: Levi, Decio. Symmetries and Integrability of Difference Equations. Cambridge : Cambridge University Press, ©2011 9780521136587 London Mathematical Society Lecture Note Series, 381. FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=399282 Volltext 505-00/(S Machine generated contents note: 1. Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals / R. Kozlov -- 1.1. Introduction -- 1.2. Invariance of Euler-Lagrange equations -- 1.3. Lagrangian formalism for second-order difference equations -- 1.4. Hamiltonian formalism for differential equations -- 1.4.1. Canonical Hamiltonian equations -- 1.4.2. Legendre transformation -- 1.4.3. Invariance of canonical Hamiltonian equations -- 1.5. Discrete Hamiltonian formalism -- 1.5.1. Discrete Legendre transform -- 1.5.2. Variational formulation of the discrete Hamiltonian equations -- 1.5.3. Symplecticity of the discrete Hamiltonian equations -- 1.5.4. Invariance of the Hamiltonian action -- 1.5.5. Discrete Hamiltonian identity and discrete Noether theorem -- 1.5.6. Invariance of the discrete Hamiltonian equations -- 1.6. Examples -- 1.6.1. Nonlinear motion -- 1.6.2. nonlinear ODE -- 1.6.3. Discrete harmonic oscillator -- 1.6.4. Modified discrete harmonic oscillator (exact scheme) -- 1.7. Conclusion -- 2. Painleve Equations: Continuous, Discrete and Ultradiscrete / A. Ramani -- 2.1. Introduction -- 2.2. rough sketch of the top-down description of the Painleve equations -- 2.3. succinct presentation of the bottom-up description of the Painleve equations -- 2.4. Properties of the, continuous and discrete, Painleve equations: a parallel presentation -- 2.4.1. Degeneration cascade -- 2.4.2. Lax pairs -- 2.4.3. Miura and Backlund relations -- 2.4.4. Particular solutions -- 2.4.5. Contiguity relations -- 2.5. ultradiscrete Painleve equations -- 2.5.1. Degeneration cascade -- 2.5.2. Lax pairs -- 2.5.3. Miura and Backlund relations -- 2.5.4. Particular solutions -- 2.5.5. Contiguity relations -- 2.6. Conclusion -- 3. Definitions and Predictions of Integrability for Difference Equations / J. Hietarinta -- 3.1. Preliminaries -- 3.1.1. Points of view on integrability -- 3.1.2. Preliminaries on discreteness and discrete integrability -- 3.2. Conserved quantities -- 3.2.1. Constants of motion for continuous ODE -- 3.2.2. standard discrete case -- 3.2.3. Hirota-Kimura-Yahagi (HKY) generalization -- 3.3. Singularity confinement and algebraic entropy -- 3.3.1. Singularity analysis for difference equations -- 3.3.2. Singularity confinement in projective space -- 3.3.3. Singularity confinement is not sufficient -- 3.4. Integrability in 2D -- 3.4.1. Definitions and examples -- 3.4.2. Quadrilateral lattices -- 3.4.3. Continuum limit -- 3.4.4. Conservation laws -- 3.5. Singularity confinement in 2D -- 3.6. Algebraic entropy for 2D lattices -- 3.6.1. Default growth of degree and factorization -- 3.6.2. Search based on factorization -- 3.7. Consistency around a cube -- 3.7.1. Definition -- 3.7.2. Lax pair -- 3.7.3. CAC as a search method -- 3.8. Soliton solutions -- 3.8.1. Background solutions -- 3.8.2. 1SS -- 3.8.3. NSS -- 3.9. Conclusions -- 4. Orthogonal Polynomials, their Recursions, and Functional Equations / M.E.H. Ismail -- 4.1. Introduction -- 4.2. Orthogonal polynomials -- 4.3. spectral theorem -- 4.4. Freud nonlinear recursions -- 4.5. Differential equations -- 4.6. q-difference equations -- 4.7. inverse problem -- 4.8. Applications -- 4.9. Askey-Wilson polynomials -- 5. Discrete Painleve Equations and Orthogonal Polynomials / A. Its -- 5.1. General setting -- 5.1.1. Orthogonal polynomials -- 5.1.2. Connections to integrable systems -- 5.1.3. Riemann-Hilbert representation of the orthogonal polynomials -- 5.1.4. Discrete Painleve equations -- 5.2. Examples -- 5.2.1. Gaussian weight -- 5.2.2. d-Painleve I -- 5.2.3. d-Painleve XXXIV -- 6. Generalized Lie Symmetries for Difference Equations / R.I. Yamilov -- 6.1. Introduction -- 6.1.1. Direct construction of generalized symmetries: an example -- 6.2. Generalized symmetries from the integrability properties -- 6.2.1. Toda Lattice -- 6.2.2. symmetry algebra for the Toda Lattice -- 6.2.3. continuous limit of the Toda symmetry algebras -- 6.2.4. Backlund transformations for the Toda equation -- 6.2.5. Backlund transformations vs. generalized symmetries -- 6.2.6. Generalized symmetries for PδE's -- 6.3. Formal symmetries and integrable lattice equations -- 6.3.1. Formal symmetries and further integrability conditions -- 6.3.2. Why integrable equations on the lattice must be symmetric -- 6.3.3. Example of classification problem -- 7. Four Lectures on Discrete Systems / S.P. Novikov -- 7.1. Introduction -- 7.2. Discrete symmetries and completely integrable systems -- 7.3. Discretization of linear operators -- 7.4. Discrete GLn connections and triangle equation -- 7.5. New discretization of complex analysis -- 8. Lectures on Moving Frames / P.J. Olver -- 8.1. Introduction -- 8.2. Equivariant moving frames -- 8.3. Moving frames on jet space and differential invariants -- 8.4. Equivalence and signatures -- 8.5. Joint invariants and joint differential invariants -- 8.6. Invariant numerical approximations -- 8.7. invariant bicomplex -- 8.8. Generating differential invariants -- 8.9. Invariant variational problems -- 8.10. Invariant curve flows -- 9. Lattices of Compact Semisimple Lie Groups / J. Patera -- 9.1. Introduction -- 9.2. Motivating example -- 9.3. Simple Lie groups and simple Lie algebras -- 9.3.1. Simple roots -- 9.3.2. Standard bases in Rn -- 9.3.3. Reflections and affine reflections in Rn -- 9.3.4. Weyl group and Affine Weyl group -- 9.4. Lattice grids FM [⊂] F [⊂] Rn -- 9.4.1. Examples of FM -- 9.5. W-invariant functions orthogonal on FM -- 9.6. Properties of elements of finite order -- 10. Lectures on Discrete Differential Geometry / Yu. B Suris -- 10.1. Basic notions -- 10.2. Backlund transformations -- 10.3. Q-nets -- 10.4. Circular nets -- 10.5. Q-nets in quadrics -- 10.6. T-nets -- 10.7. A-nets -- 10.8. T-nets in quadrics -- 10.9. K-nets -- 10.10. Hirota equation for K-nets -- 11. Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations / P. 505-00/(S Winternitz -- 11.1. Symmetry preserving discretization of ODEs -- 11.1.1. Formulation of the problem -- 11.1.2. Lie point symmetries of ordinary difference schemes -- 11.1.3. continuous limit -- 11.2. Examples of symmetry preserving discretizations -- 11.2.1. Equations invariant under SL1(2, R) -- 11.2.2. Equations invariant under SL2(2, R) -- 11.2.3. Equations invariant under the similitude group of the Euclidean plane -- 11.3. Applications to numerical solutions of ODEs -- 11.3.1. General procedure for testing the numerical schemes -- 11.3.2. Numerical experiments for a third-order ODE invariant under SL1(2, R) -- 11.3.3. Numerical experiments for ODEs invariant under SL2(2, R) -- 11.3.4. Numerical experiments for third-order ODE invariant under Sim(2) -- 11.4. Exact solutions of invariant difference schemes -- 11.4.1. Lagrangian formulation for second-order ODEs -- 11.4.2. Lagrangian formulation for second order difference equations -- 11.4.3. Example: Second-order ODE with three-dimensional solvable symmetry algebra -- 11.5. Lie point symmetries of differential-difference equations -- 11.5.1. Formulation of the problem -- 11.5.2. evolutionary formalism and commuting flows for differential equations -- 11.5.3. evolutionary formalism and commuting flows for differential-difference equations -- 11.5.4. General algorithm for calculating Lie point symmetries of a differential-difference equation -- 11.5.5. Theorems simplifying the calculation of symmetries of DδE -- 11.5.6. Volterra type equations and their generalizations -- 11.5.7. Toda type equations -- 11.5.8. Toda field theory type equations -- 11.6. Examples of symmetries of DδE -- 11.6.1. YdKN equation -- 11.6.2. Toda lattice. |
| spellingShingle | Levi, D. (Decio) Symmetries and Integrability of Difference Equations. London Mathematical Society Lecture Note Series, 381. Cover; Title; Copyright; Contents; List of figures; List of contributors; Preface; Introduction; 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals V. orodnitsyn and R. Kozlov; Abstract; 1.1 Introduction; 1.2 Invariance of Euler-Lagrange equations; 1.3 Lagrangian formalism for second-order difference equations; 1.4 Hamiltonian formalism for differential equations; 1.4.1 Canonical Hamiltonian equations; 1.4.2 The Legendre transformation; 1.4.3 Invariance of canonical Hamiltonian equations; 1.5 Discrete Hamiltonian formalism. 1.5.1 Discrete Legendre transform1.5.2 Variational formulation of the discrete Hamiltonian equations; 1.5.3 Symplecticity of the discrete Hamiltonian equations; 1.5.4 Invariance of the Hamiltonian action; 1.5.5 Discrete Hamiltonian identity and discrete Noether theorem; 1.5.6 Invariance of the discrete Hamiltonian equations; 1.6 Examples; 1.6.1 Nonlinear motion; 1.6.2 A nonlinear ODE; 1.6.3 Discrete harmonic oscillator; 1.6.4 Modified discrete harmonic oscillator (exact scheme); 1.7 Conclusion; Acknowledgments; References. 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete B. Grammaticos and A. RamaniAbstract; 2.1 Introduction; 2.2 A rough sketch of the top-down description of the Painlevé equations; The Hamiltonian formulation of Painlevé equations; 2.3 A succinct presentation of the bottom-up description of the Painlevé equations; Derivation of continuous Painlevé equations; 2.4 Properties of the, continuous and discrete, Painlevé equations: a parallel presentation; 2.4.1 Degeneration cascade; 2.4.2 Lax pairs; 2.4.3 Miura and Bäcklund relations; 2.4.4 Particular solutions; 2.4.5 Contiguity relations. 2.5 The ultradiscrete Painlevé equations2.5.1 Degeneration cascade; 2.5.2 Lax pairs; 2.5.3 Miura and Bäcklund relations; 2.5.4 Particular solutions; 2.5.5 Contiguity relations; 2.6 Conclusion; References; 3 Definitions and Predictions of Integrability for Difference Equations J. Hietarinta; Abstract; 3.1 Preliminaries; 3.1.1 Points of view on integrability; 3.1.2 Preliminaries on discreteness and discrete integrability; 3.2 Conserved quantities; 3.2.1 Constants of motion for continuous ODE; 3.2.2 The standard discrete case; 3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization. 3.3 Singularity confinement and algebraic entropy3.3.1 Singularity analysis for difference equations; 3.3.2 Singularity confinement in projective space; 3.3.3 Singularity confinement is not sufficient; 3.4 Integrability in 2D; 3.4.1 Definitions and examples; 3.4.2 Quadrilateral lattices; 3.4.3 Continuum limit; 3.4.4 Conservation laws; 3.5 Singularity confinement in 2D; 3.6 Algebraic entropy for 2D lattices; 3.6.1 Default growth of degree and factorization; 3.6.2 Search based on factorization; 3.7 Consistency around a cube; 3.7.1 Definition; 3.7.2 Lax pair; 3.7.3 CAC as a search method. Difference equations. http://id.loc.gov/authorities/subjects/sh85037879 Symmetry (Mathematics) http://id.loc.gov/authorities/subjects/sh2006001303 Integrals. http://id.loc.gov/authorities/subjects/sh85067099 Équations aux différences. Symétrie (Mathématiques) Intégrales. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Difference equations fast Integrals fast Symmetry (Mathematics) fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85037879 http://id.loc.gov/authorities/subjects/sh2006001303 http://id.loc.gov/authorities/subjects/sh85067099 |
| title | Symmetries and Integrability of Difference Equations. |
| title_auth | Symmetries and Integrability of Difference Equations. |
| title_exact_search | Symmetries and Integrability of Difference Equations. |
| title_full | Symmetries and Integrability of Difference Equations. |
| title_fullStr | Symmetries and Integrability of Difference Equations. |
| title_full_unstemmed | Symmetries and Integrability of Difference Equations. |
| title_short | Symmetries and Integrability of Difference Equations. |
| title_sort | symmetries and integrability of difference equations |
| topic | Difference equations. http://id.loc.gov/authorities/subjects/sh85037879 Symmetry (Mathematics) http://id.loc.gov/authorities/subjects/sh2006001303 Integrals. http://id.loc.gov/authorities/subjects/sh85067099 Équations aux différences. Symétrie (Mathématiques) Intégrales. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Difference equations fast Integrals fast Symmetry (Mathematics) fast |
| topic_facet | Difference equations. Symmetry (Mathematics) Integrals. Équations aux différences. Symétrie (Mathématiques) Intégrales. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. Difference equations Integrals |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=399282 |
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