Bharucha-Reid / Sambandham / Brinbaum Random Polynomials

Probability and Mathematical Statistics: A Series of Monographs and Textbooks
1. Auflage 2014
ISBN: 978-1-4831-9146-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 222 Seiten, Web PDF

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Random Polynomials focuses on a comprehensive treatment of random algebraic, orthogonal, and trigonometric polynomials. The publication first offers information on the basic definitions and properties of random algebraic polynomials and random matrices. Discussions focus on Newton's formula for random algebraic polynomials, random characteristic polynomials, measurability of the zeros of a random algebraic polynomial, and random power series and random algebraic polynomials. The text then elaborates on the number and expected number of real zeros of random algebraic polynomials; number and expected number of real zeros of other random polynomials; and variance of the number of real zeros of random algebraic polynomials. Topics include the expected number of real zeros of random orthogonal polynomials and the number and expected number of real zeros of trigonometric polynomials. The book takes a look at convergence and limit theorems for random polynomials and distribution of the zeros of random algebraic polynomials, including limit theorems for random algebraic polynomials and random companion matrices and distribution of the zeros of random algebraic polynomials. The publication is a dependable reference for probabilists, statisticians, physicists, engineers, and economists.

Autoren/Hrsg.

Bharucha-Reid, A. T.

Sambandham, M.

Brinbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Random Polynomials;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;Acknowledgments;16
8;CHAPTER 1. Introduction;18
8.1;1.1. INTRODUCTION;18
8.2;1.2. ORIGINS OF SOME RANDOM ALGEBRAIC POLYNOMIALS;19
8.3;1.3. SOME HISTORICAL REMARKS;28
8.4;1.4. OTHER TYPES OF RANDOM POLYNOMIALS;31
8.5;REFERENCES;34
9;CHAPTER 2. Random Algebraic Polynomials: Basic Definitions and Properties;36
9.1;2.1. INTRODUCTION;36
9.2;2.2. RANDOM POWER SERIES AND RANDOM ALGEBRAIC POLYNOMIALS;36
9.3;2.3. OTHER DEFINITIONS OF RANDOM ALGEBRAIC POLYNOMIALS;39
9.4;2.4. MEASURABILITY OF THE ZEROS OF A RANDOM ALGEBRAIC POLYNOMIAL;40
9.5;2.5. MEASURABILITY OF THE NUMBER OF ZEROS OF A RANDOM ALGEBRAIC POLYNOMIAL;42
9.6;2.6. SOME PROPERTIES OF RANDOM ALGEBRAIC POLYNOMIALS;45
9.7;REFERENCES;50
10;CHAPTER 3. Random Matrices and Random Algebraic Polynomials;51
10.1;3.1. INTRODUCTION;51
10.2;3.2. SOME EXAMPLES OF RANDOM MATRICES;52
10.3;3.3. RANDOM CHARACTERISTIC POLYNOMIALS;58
10.4;3.4. NEWTON'S FORMULA FOR RANDOM ALGEBRAIC POLYNOMIALS;58
10.5;3.5. RANDOM COMPANION MATRICES;60
10.6;REFERENCES;64
11;CHAPTER 4. The Number and Expected Number of Real Zeros of Random Algebraic Polynomials;66
11.1;4.1. INTRODUCTION;66
11.2;4.2. ESTIMATES OF Nn(B, .);67
11.3;4.3. THE EXPECTED NUMBER OF REAL ZEROS OF RANDOM ALGEBRAIC POLYNOMIALS;97
11.4;4.4. THE AVERAGE NUMBER OF ZEROS OF RANDOM ALGEBRAIC POLYNOMIALS WITH COMPLEX COEFFICIENTS;115
11.5;REFERENCES;117
12;CHAPTER 5. The Number and Expected Number of Real Zeros of Other Random Polynomials;120
12.1;5.1. INTRODUCTION;120
12.2;5.2. THE NUMBER AND EXPECTED NUMBER OF REAL ZEROS OF TRIGONOMETRIC POLYNOMIALS;122
12.3;5.3. THE EXPECTED NUMBER OF REAL ZEROS OF RANDOM HYPERBOLIC POLYNOMIALS;126
12.4;5.4. THE EXPECTED NUMBER OF REAL ZEROS OF RANDOM ORTHOGONAL POLYNOMIALS;127
12.5;5.5. NUMERICAL RESULTS;128
12.6;REFERENCES;136
13;CHAPTER 6. The Variance of the Number of Real Zeros of Random Algebraic Polynomials;138
13.1;6.1. INTRODUCTION;138
13.2;6.2. THE MAIN THEOREM;139
13.3;6.3. FORMULA FOR THE VARIANCE;141
13.4;6.4. SOME LEMMAS;143
13.5;6.5. PROOF OF THEOREM 6.2(a);144
13.6;6.6. PROOF OF THEOREM 6.2(b);149
13.7;6.7. SOME COMPUTATIONAL RESULTS;152
13.8;REFERENCES;155
14;CHAPTER 7. Distribution of the Zeros of Random Algebraic Polynomials;156
14.1;7.1. INTRODUCTION;156
14.2;7.2. DISTRIBUTION OF THE REAL ZEROS OF RANDOM LINEAR AND QUADRATIC EQUATIONS;157
14.3;7.3. DISTRIBUTION OF THE ZEROS OF A RANDOM POLYNOMIAL WITH COMPLEX COEFFICIENTS;167
14.4;7.4. CONDENSED DISTRIBUTION OF THE ZEROS OF A RANDOM ALGEBRAIC POLYNOMIAL;169
14.5;7.5. DISTRIBUTION OF THE NUMBER OF REAL ZEROS;171
14.6;7.6. SOME NUMERICAL RESULTS;172
14.7;7.7. ON THE DISTRIBUTION OF THE ZEROS OF RANDOM ALGEBRAIC POLYNOMIALS;179
14.8;REFERENCES;189
15;CHAPTER 8. Convergence and Limit Theorems for Random Polynomials;190
15.1;8.1. INTRODUCTION;190
15.2;8.2. THE LIMITING BEHAVIOR OF n-1 Nn(B, .);191
15.3;8.3. THE LIMITING BEHAVIOR OF Fn,k(z, .) AND Nn,k(B, .);197
15.4;8.4. STABILITY OF THE ZEROS OF RANDOM ALGEBRAIC POLYNOMIALS;199
15.5;8.5. SOME LIMIT THEOREMS FOR RANDOM ALGEBRAIC POLYNOMIALS AND RANDOM COMPANION MATRICES;209
15.6;REFERENCES;212
16;APPENDIX: Fortran Programs;213
17;Index;220