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Quadratic Vector Equations on Complex Upper Half-Plane
 
Oskari Ajanki Institute of Science and Technology, Klosterneuberg, Austria
László Erdős Institute of Science and Technology, Klosterneuberg, Austria
Torben Krüger Institute of Science and Technology, Klosterneuberg, Austria
Quadratic Vector Equations on Complex Upper Half-Plane
Softcover ISBN:  978-1-4704-3683-4
Product Code:  MEMO/261/1261
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5414-2
Product Code:  MEMO/261/1261.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3683-4
eBook: ISBN:  978-1-4704-5414-2
Product Code:  MEMO/261/1261.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
Quadratic Vector Equations on Complex Upper Half-Plane
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Quadratic Vector Equations on Complex Upper Half-Plane
Oskari Ajanki Institute of Science and Technology, Klosterneuberg, Austria
László Erdős Institute of Science and Technology, Klosterneuberg, Austria
Torben Krüger Institute of Science and Technology, Klosterneuberg, Austria
Softcover ISBN:  978-1-4704-3683-4
Product Code:  MEMO/261/1261
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5414-2
Product Code:  MEMO/261/1261.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3683-4
eBook ISBN:  978-1-4704-5414-2
Product Code:  MEMO/261/1261.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2612019; 133 pp
    MSC: Primary 45; Secondary 46; 60; 15

    The authors consider the nonlinear equation \(-\frac 1m=z+Sm\) with a parameter \(z\) in the complex upper half plane \(\mathbb H \), where \(S\) is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \( \mathbb H\) is unique and its \(z\)-dependence is conveniently described as the Stieltjes transforms of a family of measures \(v\) on \(\mathbb R\). In a previous paper the authors qualitatively identified the possible singular behaviors of \(v\): under suitable conditions on \(S\) we showed that in the density of \(v\) only algebraic singularities of degree two or three may occur.

    In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any \(z\in \mathbb H\), including the vicinity of the singularities.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Set-up and main results
    • 3. Local laws for large random matrices
    • 4. Existence, uniqueness and $\mathrm {L}^{\!2}$-bound
    • 5. Properties of solution
    • 6. Uniform bounds
    • 7. Regularity of solution
    • 8. Perturbations when generating density is small
    • 9. Behavior of generating density where it is small
    • 10. Stability around small minima of generating density
    • 11. Examples
    • A. Appendix
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2612019; 133 pp
MSC: Primary 45; Secondary 46; 60; 15

The authors consider the nonlinear equation \(-\frac 1m=z+Sm\) with a parameter \(z\) in the complex upper half plane \(\mathbb H \), where \(S\) is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \( \mathbb H\) is unique and its \(z\)-dependence is conveniently described as the Stieltjes transforms of a family of measures \(v\) on \(\mathbb R\). In a previous paper the authors qualitatively identified the possible singular behaviors of \(v\): under suitable conditions on \(S\) we showed that in the density of \(v\) only algebraic singularities of degree two or three may occur.

In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any \(z\in \mathbb H\), including the vicinity of the singularities.

  • Chapters
  • 1. Introduction
  • 2. Set-up and main results
  • 3. Local laws for large random matrices
  • 4. Existence, uniqueness and $\mathrm {L}^{\!2}$-bound
  • 5. Properties of solution
  • 6. Uniform bounds
  • 7. Regularity of solution
  • 8. Perturbations when generating density is small
  • 9. Behavior of generating density where it is small
  • 10. Stability around small minima of generating density
  • 11. Examples
  • A. Appendix
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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