Dedication

pp ix-x

Contents

pp xi-xiii

Foreword

pp xiv-xvi

Preface

pp xvii-xxii

Part One - Preliminaries: Entrywise Powers Preserving Positivity in a Fixed Dimension

pp 1-2

1 - The Cone of Positive Semidefinite Matrices

pp 3-15

2 - The Schur Product Theorem and Nonzero Lower Bounds

pp 16-22

3 - Totally Positive (TP) and Totally Nonnegative (TN) Matrices

pp 23-31

4 - Totally Positive Matrices – Generalized Vandermonde and Hankel Moment Matrices

pp 32-39

5 - Entrywise Powers Preserving Positivity in a Fixed Dimension

pp 40-45

6 - Midconvex Implies Continuity, and 2 × 2 Preservers

pp 46-54

7 - Entrywise Preservers of Positivity on Matrices with Zero Patterns

pp 55-63

8 - Entrywise Powers Preserving Positivity, Monotonicity, and Superadditivity

pp 64-71

9 - Loewner Convexity and Single Matrix Encoders of Preservers

pp 72-83

10 - Exercises

pp 84-94

Part Two - Entrywise Functions Preserving Positivity in all Dimensions

pp 95-96

11 - History – Schoenberg, Rudin, Vasudeva, and Metric Geometry

pp 97-113

12 - Loewner’s Determinant Calculation in Horn’s Thesis

pp 114-119

13 - The Stronger Horn–Loewner Theorem via Mollifiers

pp 120-126

14 - Stronger Vasudeva and Schoenberg Theorems via Bernstein’s Theorem

pp 127-134

15 - Proof of the Stronger Schoenberg Theorem (Part I) – Positivity Certificates

pp 135-143

16 - Proof of the Stronger Schoenberg Theorem (Part II) – Real Analyticity

pp 144-150

17 - Proof of the Stronger Schoenberg Theorem (Part III) – Complex Analysis

pp 151-154

18 - Preservers of Loewner Positivity on Kernels

pp 155-158

19 - Preservers of Loewner Monotonicity and Convexity on Kernels

pp 159-167

20 - Functions Acting Outside Forbidden Diagonal Blocks

pp 168-175

21 - The Boas–Widder Theorem on Functions with Positive Differences

pp 176-189

22 - Menger’s Results and Euclidean Distance Geometry

pp 190-202

23 - Exercises

pp 203-210