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Considers polynominal invariants & comitants of autonomous systems of differential equations with right-hand sides relative to various transformation groups of phase space. Contains an in-depth discussion of the two-dimensional system with quadratic right-hand sides. Features numerous applications to the qualitative theory of differential equations.
These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models.
The book is a self-contained introduction to the results and methods in classical invariant theory.
This volume of expository papers is the outgrowth of a conference in combinatorics and invariant theory. In recent years, newly developed techniques from algebraic geometry and combinatorics have been applied with great success to some of the outstanding problems of invariant theory, moving it back to the forefront of mathematical research once again. This collection of papers centers on constructive aspects of invariant theory and opens with an introduction to the subject by F. Grosshans. Its purpose is to make the current research more accesssible to mathematicians in related fields.
Classic monograph offers a brief account of the invariant theory connected with a single quadratic differential form. Includes historical overview; methods of Christoffel, Lie, Maschke; and geometrical, dynamical methods. 1960 edition.
A unique introduction to the subject, reflecting different approaches to the integration of differential equations.
Group Theory, with its emphasis on Lie Groups and their application to the study of symmetries of the fundamental constituents of matter is introduced at a level suitable for Seniors and advanced Juniors majoring in the Physical Sciences. The book has its origin in a one-semester course that Professor Firk taught at Yale University for more than ten years. It is not generally appreciated by Physicists that continuous transformation groups (Lie Groups) originated in the Theory of Differential Equations. The infinitesimal generators of Lie Groups therefore have forms that involve differential operators and their commutators, and these operators and their algebraic properties have found, and continue to find, a natural place in the development of Quantum Physics. Topics covered include:Galois Groups Algebraic Invariants Invariants of Physics Groups − Concrete and Abstract Lie's Differential Equation Lie's Continuous Transformation Groups Matrix Representations of Groups Lie Groups of Transformations Group Structure of Lorentz Transformations Groups and the Structure of Matter Lie Groups and the Conservation Laws of the Physical Universe
For most of the book the only prerequisites are the basic facts of algebraic geometry and number theory."--BOOK JACKET.
This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced – with big impact – in the 1990s. Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.
Differential algebra explores properties of solutions to systems of (ordinary or partial, linear or nonlinear) differential equations from an algebraic point of view. It includes as special cases algebraic systems as well as differential systems with algebraic constraints. This algebraic theory of Joseph F Ritt and Ellis R Kolchin is further enriched by its interactions with algebraic geometry, Diophantine geometry, differential geometry, model theory, control theory, automatic theorem proving, combinatorics, and difference equations. Differential algebra now plays an important role in computational methods such as symbolic integration, and symmetry analysis of differential equations. This volume includes tutorial and survey papers presented at workshop.