Online Public Access Catalogue (OPAC)
Library,Documentation and Information Science Division

“A research journal serves that narrow

borderland which separates the known from the unknown”

-P.C.Mahalanobis


Normal view MARC view ISBD view

Transformation groups and Lie algebras / Nail H Ibragimov.

By: Ibragimov, Nail H.
Material type: TextTextPublisher: Singapore : World Scientific, c2013Description: x, 185 p. : illustrations ; 25 cm.ISBN: 9789814460842 (hardback).Subject(s): Transformation groups | Lie algebrasDDC classification: 512.55
Contents:
Machine generated contents note: pt. I Local Transformation Groups -- 1. Preliminaries -- 1.1. Changes of frames of reference and point transformations -- 1.1.1. Translations -- 1.1.2. Rotations -- 1.1.3. Galilean transformation -- 1.2. Introduction of transformation groups -- 1.2.1. Definitions and examples -- 1.2.2. Different types of groups -- 1.3. Some useful groups -- 1.3.1. Finite continuous groups on the straight line -- 1.3.2. Groups on the plane -- 1.3.3. Groups in IRn -- Exercises to Chapter 1 -- 2. One-parameter groups and their invariants -- 2.1. Local groups of transformations -- 2.1.1. Notation and definition -- 2.1.2. Groups written in a canonical parameter -- 2.1.3. Infinitesimal transformations and generators -- 2.1.4. Lie equations -- 2.1.5. Exponential map -- 2.1.6. Determination of a canonical parameter -- 2.2. Invariants -- 2.2.1. Definition and infinitesimal test -- 2.2.2. Canonical variables 2.2.3. Construction of groups using canonical variables -- 2.2.4. Frequently used groups in the plane -- 2.3. Invariant equations -- 2.3.1. Definition and infinitesimal test -- 2.3.2. Invariant representation of invariant manifolds -- 2.3.3. Proof of Theorem 2.9 -- 2.3.4. Examples on Theorem 2.9 -- Exercises to Chapter 2 -- 3. Groups admitted by differential equations -- 3.1. Preliminaries -- 3.1.1. Differential variables and functions -- 3.1.2. Point transformations -- 3.1.3. Frame of differential equations -- 3.2. Prolongation of group transformations -- 3.2.1. One-dimensional case -- 3.2.2. Prolongation with several differential variables -- 3.2.3. General case -- 3.3. Prolongation of group generators -- 3.3.1. One-dimensional case -- 3.3.2. Several differential variables -- 3.3.3. General case -- 3.4. First definition of symmetry groups -- 3.4.1. Definition -- 3.4.2. Examples -- 3.5. Second definition of symmetry groups -- 3.5.1. Definition and determining equations 3.5.2. Determining equation for second-order ODEs -- 3.5.3. Examples on solution of determining equations -- Exercises to Chapter 3 -- 4. Lie algebras of operators -- 4.1. Basic definitions -- 4.1.1. Commutator -- 4.1.2. Properties of the commutator -- 4.1.3. Properties of determining equations -- 4.1.4. Lie algebras -- 4.2. Basic properties -- 4.2.1. Notation -- 4.2.2. Subalgebra and ideal -- 4.2.3. Derived algebras -- 4.2.4. Solvable Lie algebras -- 4.3. Isomorphism and similarity -- 4.3.1. Isomorphic Lie algebras -- 4.3.2. Similar Lie algebras -- 4.4. Low-dimensional Lie algebras -- 4.4.1. One-dimensional algebras -- 4.4.2. Two-dimensional algebras in the plane -- 4.4.3. Three-dimensional algebras in the plane -- 4.4.4. Three-dimensional algebras in IR3 -- 4.5. Lie algebras and multi-parameter groups -- 4.5.1. Definition of multi-parameter groups' -- 4.5.2. Construction of multi-parameter groups -- Exercises to Chapter 4 -- 5. Galois groups via symmetries 5.1. Preliminaries -- 5.2. Symmetries of algebraic equations -- 5.2.1. Determining equation -- 5.2.2. First example -- 5.2.3. Second example -- 5.2.4. Third example -- 5.3. Construction of Galois groups -- 5.3.1. First example -- 5.3.2. Second example -- 5.3.3. Third example -- 5.3.4. Concluding remarks -- Assignment to Part 1 -- pt. II Approximate Transformation Groups -- 6. Preliminaries -- 6.1. Motivation -- 6.2. A sketch on Lie transformation groups -- 6.2.1. One-parameter transformation groups -- 6.2.2. Canonical parameter -- 6.2.3. Group generator and Lie equations -- 6.2.4. Exponential map -- 6.3. Approximate Cauchy problem -- 6.3.1. Notation -- 6.3.2. Definition of the approximate Cauchy problem -- 7. Approximate transformations -- 7.1. Approximate transformations defined -- 7.2. Approximate one-parameter groups -- 7.2.1. Introductory remark -- 7.2.2. Definition of one-parameter approximate transformation groups -- 7.2.3. Generator of approximate transformation group 7.3. Infinitesimal description -- 7.3.1. Approximate Lie equations -- 7.3.2. Approximate exponential map -- Exercises to Chapter 7 -- 8. Approximate symmetries -- 8.1. Definition of approximate symmetries -- 8.2. Calculation of approximate symmetries -- 8.2.1. Determining equations -- 8.2.2. Stable symmetries -- 8.2.3. Algorithm for calculation -- 8.3. Examples -- 8.3.1. First example -- 8.3.2. Approximate commutator and Lie algebras -- 8.3.3. Second example -- 8.3.4. Third example -- Exercises to Chapter 8 -- 9. Applications -- 9.1. Integration of equations with a small parameter using approximate symmetries -- 9.1.1. Equation haying no exact point symmetries -- 9.1.2. Utilization of stable symmetries -- 9.2. Approximately invariant solutions -- 9.2.1. Nonlinear wave equation -- 9.2.2. Approximate travelling waves of KdV equation -- 9.3. Approximate conservation laws -- Exercises to Chapter 9 -- Assignment to Part II.
Summary: Part I of these book introduces the reader to the basic concepts of the classical theory of local transformation groups and their Lie algebras. It has been designed for the graduate course on Transformation groups and Lie algebras. Part II of these book provides an easy to follow introduction to the new topic. It is based on talks about various conferences, in particular on the plenary lecture at the International Workshop.
Tags from this library: No tags from this library for this title. Log in to add tags.
Item type Current location Call number Status Date due Barcode Item holds
Books Books ISI Library, Kolkata
 
512.55 Ib14 (Browse shelf) Available 135300
Total holds: 0

Includes bibliographical references (pages 181-182) and index.

Machine generated contents note: pt. I Local Transformation Groups --
1. Preliminaries --
1.1. Changes of frames of reference and point transformations --
1.1.1. Translations --
1.1.2. Rotations --
1.1.3. Galilean transformation --
1.2. Introduction of transformation groups --
1.2.1. Definitions and examples --
1.2.2. Different types of groups --
1.3. Some useful groups --
1.3.1. Finite continuous groups on the straight line --
1.3.2. Groups on the plane --
1.3.3. Groups in IRn --
Exercises to Chapter 1 --
2. One-parameter groups and their invariants --
2.1. Local groups of transformations --
2.1.1. Notation and definition --
2.1.2. Groups written in a canonical parameter --
2.1.3. Infinitesimal transformations and generators --
2.1.4. Lie equations --
2.1.5. Exponential map --
2.1.6. Determination of a canonical parameter --
2.2. Invariants --
2.2.1. Definition and infinitesimal test --
2.2.2. Canonical variables 2.2.3. Construction of groups using canonical variables --
2.2.4. Frequently used groups in the plane --
2.3. Invariant equations --
2.3.1. Definition and infinitesimal test --
2.3.2. Invariant representation of invariant manifolds --
2.3.3. Proof of Theorem 2.9 --
2.3.4. Examples on Theorem 2.9 --
Exercises to Chapter 2 --
3. Groups admitted by differential equations --
3.1. Preliminaries --
3.1.1. Differential variables and functions --
3.1.2. Point transformations --
3.1.3. Frame of differential equations --
3.2. Prolongation of group transformations --
3.2.1. One-dimensional case --
3.2.2. Prolongation with several differential variables --
3.2.3. General case --
3.3. Prolongation of group generators --
3.3.1. One-dimensional case --
3.3.2. Several differential variables --
3.3.3. General case --
3.4. First definition of symmetry groups --
3.4.1. Definition --
3.4.2. Examples --
3.5. Second definition of symmetry groups --
3.5.1. Definition and determining equations 3.5.2. Determining equation for second-order ODEs --
3.5.3. Examples on solution of determining equations --
Exercises to Chapter 3 --
4. Lie algebras of operators --
4.1. Basic definitions --
4.1.1. Commutator --
4.1.2. Properties of the commutator --
4.1.3. Properties of determining equations --
4.1.4. Lie algebras --
4.2. Basic properties --
4.2.1. Notation --
4.2.2. Subalgebra and ideal --
4.2.3. Derived algebras --
4.2.4. Solvable Lie algebras --
4.3. Isomorphism and similarity --
4.3.1. Isomorphic Lie algebras --
4.3.2. Similar Lie algebras --
4.4. Low-dimensional Lie algebras --
4.4.1. One-dimensional algebras --
4.4.2. Two-dimensional algebras in the plane --
4.4.3. Three-dimensional algebras in the plane --
4.4.4. Three-dimensional algebras in IR3 --
4.5. Lie algebras and multi-parameter groups --
4.5.1. Definition of multi-parameter groups' --
4.5.2. Construction of multi-parameter groups --
Exercises to Chapter 4 --
5. Galois groups via symmetries 5.1. Preliminaries --
5.2. Symmetries of algebraic equations --
5.2.1. Determining equation --
5.2.2. First example --
5.2.3. Second example --
5.2.4. Third example --
5.3. Construction of Galois groups --
5.3.1. First example --
5.3.2. Second example --
5.3.3. Third example --
5.3.4. Concluding remarks --
Assignment to Part 1 --
pt. II Approximate Transformation Groups --
6. Preliminaries --
6.1. Motivation --
6.2. A sketch on Lie transformation groups --
6.2.1. One-parameter transformation groups --
6.2.2. Canonical parameter --
6.2.3. Group generator and Lie equations --
6.2.4. Exponential map --
6.3. Approximate Cauchy problem --
6.3.1. Notation --
6.3.2. Definition of the approximate Cauchy problem --
7. Approximate transformations --
7.1. Approximate transformations defined --
7.2. Approximate one-parameter groups --
7.2.1. Introductory remark --
7.2.2. Definition of one-parameter approximate transformation groups --
7.2.3. Generator of approximate transformation group 7.3. Infinitesimal description --
7.3.1. Approximate Lie equations --
7.3.2. Approximate exponential map --
Exercises to Chapter 7 --
8. Approximate symmetries --
8.1. Definition of approximate symmetries --
8.2. Calculation of approximate symmetries --
8.2.1. Determining equations --
8.2.2. Stable symmetries --
8.2.3. Algorithm for calculation --
8.3. Examples --
8.3.1. First example --
8.3.2. Approximate commutator and Lie algebras --
8.3.3. Second example --
8.3.4. Third example --
Exercises to Chapter 8 --
9. Applications --
9.1. Integration of equations with a small parameter using approximate symmetries --
9.1.1. Equation haying no exact point symmetries --
9.1.2. Utilization of stable symmetries --
9.2. Approximately invariant solutions --
9.2.1. Nonlinear wave equation --
9.2.2. Approximate travelling waves of KdV equation --
9.3. Approximate conservation laws --
Exercises to Chapter 9 --
Assignment to Part II.

Part I of these book introduces the reader to the basic concepts of the classical theory of local transformation groups and their Lie algebras. It has been designed for the graduate course on Transformation groups and Lie algebras.
Part II of these book provides an easy to follow introduction to the new topic. It is based on talks about various conferences, in particular on the plenary lecture at the International Workshop.

There are no comments for this item.

Log in to your account to post a comment.
Library, Documentation and Information Science Division, Indian Statistical Institute, 203 B T Road, Kolkata 700108, INDIA
Phone no. 91-33-2575 2100, Fax no. 91-33-2578 1412, ksatpathy@isical.ac.in


Visitor Counter