Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems

F.P. Gantmacher, M.G. Krein

318 pages, parution le 09/04/2003 (2^eme édition)

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Résumé

Fifty years after the original Russian Edition, this classic work is finally available in English for the general mathematical audience. This book lays the foundation of what later became "Krein's Theory of String". The original ideas stemming from mechanical considerations are developed with exceptional clarity. A unique feature is that it can be read profitably by both research mathematicians and engineers.

The authors study in depth small oscillations of one-dimensional continua with a finite or infinite number of degrees of freedom. They single out an algebraic property responsible for the qualitative behavior of eigenvalues and eigenfunctions of one-dimensional continua and introduce a subclass of totally positive matrices, which they call oscillatory matrices, as well as their infinite-dimensional generalization and oscillatory kernels. Totally positive matrices play an important role in several areas of modern mathematics, but this book is the only source that explains their simple and intuitively appealing relation to mechanics.

There are two supplements contained in the book, "A Method of Approximate Calculation of Eigenvalues and Eigenvectors of an Oscillatory Matrix", and Krein's famous paper which laid the groundwork for the broad research area of the inverse spectral problem: "On a Remarkable Problem for a String with Beads and Continued Fractions of Stieltjes".

The exposition is self-contained. The first chapter presents all necessary results (with proofs) on the theory of matrices which are not included in a standard linear algebra course. The only prerequisite in addition to standard linear algebra is the theory of linear integral equations used in Chapter 5. The book is suitable for graduate students, research mathematicians and engineers interested in ordinary differential equations, integral equations, and their applications.

Contents

Introduction
Review of matrices and quadratic forms
Oscillatory matrices
Small oscillations of mechanical systems with $n$ degrees of freedom
Small oscillations of mechanical systems with an infinite number of degrees of freedom
Sign-definite matrices
A method of approximate calculation of eigenvalues and eigenvectors of an oscillatory matrix
On a remarkable problem for a string with beads and continued fractions of Stieltjes

Remarks
References
Index

Caractéristiques techniques

	PAPIER
Éditeur(s)	AMS Chelsea
Auteur(s)	F.P. Gantmacher, M.G. Krein
Parution	09/04/2003
Édition	2^eme édition
Nb. de pages	318
Format	18,2 x 26
Couverture	Relié
Poids	758g
Intérieur	Noir et Blanc
EAN13	9780821831717
ISBN13	978-0-8218-3171-7