Applications Of Lie Algebras To Hyperbolic And Stochastic Differential Equations (mathematics And Its Applications)
by Constantin Vârsan /
1999 / English / PDF
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The main part of the book is based on a one semester graduate
course for students in mathematics. I have attempted to develop the
theory of hyperbolic systems of differen tial equations in a
systematic way, making as much use as possible ofgradient systems
and their algebraic representation. However, despite the strong
sim ilarities between the development of ideas here and that found
in a Lie alge bras course this is not a book on Lie algebras. The
order of presentation has been determined mainly by taking into
account that algebraic representation and homomorphism
correspondence with a full rank Lie algebra are the basic tools
which require a detailed presentation. I am aware that the
inclusion of the material on algebraic and homomorphism
correspondence with a full rank Lie algebra is not standard in
courses on the application of Lie algebras to hyperbolic equations.
I think it should be. Moreover, the Lie algebraic structure plays
an important role in integral representation for solutions of
nonlinear control systems and stochastic differential equations
yelding results that look quite different in their original
setting. Finite-dimensional nonlin ear filters for stochastic
differential equations and, say, decomposability of a nonlinear
control system receive a common understanding in this framework.
The main part of the book is based on a one semester graduate
course for students in mathematics. I have attempted to develop the
theory of hyperbolic systems of differen tial equations in a
systematic way, making as much use as possible ofgradient systems
and their algebraic representation. However, despite the strong
sim ilarities between the development of ideas here and that found
in a Lie alge bras course this is not a book on Lie algebras. The
order of presentation has been determined mainly by taking into
account that algebraic representation and homomorphism
correspondence with a full rank Lie algebra are the basic tools
which require a detailed presentation. I am aware that the
inclusion of the material on algebraic and homomorphism
correspondence with a full rank Lie algebra is not standard in
courses on the application of Lie algebras to hyperbolic equations.
I think it should be. Moreover, the Lie algebraic structure plays
an important role in integral representation for solutions of
nonlinear control systems and stochastic differential equations
yelding results that look quite different in their original
setting. Finite-dimensional nonlin ear filters for stochastic
differential equations and, say, decomposability of a nonlinear
control system receive a common understanding in this framework.