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Methods of intermediate problems for eigenvalues: theory and ramifications
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrang...
| Clasificación: | Libro Electrónico |
|---|---|
| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
New York,
Academic Press,
1972.
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| Colección: | Mathematics in science and engineering ;
v. 89. |
| Temas: | |
| Acceso en línea: | Texto completo Texto completo Texto completo |
Tabla de Contenidos:
- Front Cover; Methods of Intermediate Problems for Eigenvalues; Copyright Page; Contents; Preface; Acknowledgments; Introduction; CHAPTER ONE The Variational Characterization of Eigenvalues; 1. A Preliminary Survey of the Classical and Minimax Principles for Eigenvalues; 2. Operators of Class S; 3. Rayleigh's Principle and the Classical Characterization; CHAPTER TWO The Rayleigh-Ritz Method; 1. Poincar�e's Inequalities: The Theoretical Foundation of the Rayleigh-Rit Method; 2. The Minimum-Maximum Principle; 3. Upper Bounds for Eigenvalues.
- 4. A Necessary and Sufficient Criterion in the Minimum-Maximum Theory5. The Principle of Monotonicity; CHAPTER THREE The Classical Maximum-Minimum Theory and Its Extension to Unbounded Operators; 1. Weyl's First Fundamental Lemma; 2. The Maximum-Minimum Principle; 3. The Existence of Minima for Semibounded Operators; 4. Comparison of the Minimum-Maximum and Maximum-Minimum Principles; 5. Finite-Dimensional Spaces; 6. THe General Maxi-Mini-Max Principle; CHAPTER FOUR Intermediate Problems of The First Type; 1. Weinstein's General Scheme of Intermediate Problems.
- 2. The Basic Principles of Intermediate Problems of the First Type3. Nonpersistent Eigenvalues, Weinstein's Determinant; 4. The Distinguished Choice; 5. Lower Bounds Using a Distingushed Choice; 6. The Existence of Distinguished Choices; 7. The General Choice; 8. Aronszajn's Rule for the Theoretical Determination of Eigenvalues; 9. Comparison of the Various Rules; 10. Weinberger-Bazley-Fox Method of Truncation; 11. Convergence; 12. Computation of Lower Bounds for the Buckling of a Clamped plate; 13. On the Symmetries of the Eigenfunctions; 14. Weyl's Second Lemma.
- CHAPTER FIVE Intermediate Problems of The Second Type1. Formulation of Problems of the Second Type; 2. Finite Rank Perturbations and Intermediate Problems of the Second Type; 3. A Solution of the Second Type; 4. Bazley's Special Choice; 5. The General Choice and Truncation; 6. The Existence of a Base Problem for a Compact Operator; 7. The Spectrum of the Helium Atom; 8. Application of the Special Choice to the Helium Atom; 9. Application of Intermediate Problems to Temple's Formula; 10. Application of Truncation to Quantum Theory.
- CHAPTER SIX Various Other Methods and Their Connections with Intermediate Problems1. Quadratic Forms and Intermediate Problems; 2. Application of Quadratic Forms to Free and Cantilever Plates; 3. The Rhombical Membrane; 4. Bazley-Fox Method for Sums of Solvable Operators; 5. The Trefftz-Fichera Method for Integral Operators; 6. Fichera's Construction of Intermediate Green's Operators; CHAPTER SEVEN The New Maximum-Minimum Theory; 1. The Basis of Weinstein's New Maximum-Minimum Theory; 2. The Case of One Constraint; 3. The Case of Orthonormal Constraints; 4. The General Formulation.