Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions:
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| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2019]
|
| Series: | Memoirs of the American Mathematical Society
volume 257, number 1232 (second of 6 numbers) |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | v, 106, 4 ungezählte Seiten |
| ISBN: | 9781470434557 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
Chapter 1. Introduction 1
1.1. Simultaneous dilations 2
1.2. Solution of the minimization problem (1.1) 4
1.3. Linear matrix inequalities (LMIs), spectrahedra
and general dilations 5
1.4. Interpretation in terms of completely positive maps 8
1.5. Matrix cube problem 8
1.6. Matrix balls 9
1.7. Adapting the Theory to Free Nonsymmetric Variables 10
1.8. Probabilistic theorems and interpretations 11
1.9. Reader’s guide 14
Chapter 2. Dilations and Free Spectrahedral Inclusions 17
Chapter 3. Lifting and Averaging 19
Chapter 4. A Simplified Form for d 23
Chapter 5. d is the Optimal Bound 27
5.1. Averages over 0(d) equal averages over S d_1 27
5.2. Dilating to commuting self-adjoint operators 29
5.3. Optimality of /c*(d) 31
Chapter 6. The Optimality Condition a = ¡3 inTerms of Beta Functions 35
Chapter 7. Rank versus Size for the Matrix Cube 41
7.1. Proof of Theorem 1.6 44
Chapter 8. Free Spectrahedral Inclusion Generalities 47
8.1. A general bound on the inclusion scale 47
8.2. The inclusion scale equals the commutability index 49
Chapter 9. Reformulation of the Optimization Problem 53
Chapter 10. Simmons’ Theorem for Half Integers 55
10.1. The upper boundary case 58
10.2. The lower boundary cases for d even 61
10.3. The lower boundary cases for d odd 62
iii
IV
CONTENTS
Chapter 11. Bounds on the Median and the Equipoint of the Beta Distribution 67
11.1. Lower bound for the equipoint es,t 67
11.2. New bounds on the median of the beta distribution 69
Chapter 12. Proof of Theorem 1.2 73
12.1. An auxiliary function 73
Chapter 13. Estimating #(d) for Odd d 79
13.1. Proof of Theorem 13.1 79
13.2. Explicit bounds on $(d) 83
Chapter 14. Dilations and Inclusions of Balls 85
14.1. The general dilation result 85
14.2. Four types of balls 86
14.3. Inclusions and dilations 91
Chapter 15. Probabilistic Theorems and Interpretations Continued 97
15.1. The nature of equipoints 97
Bibliography 101
Index 105
;
|
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| author | Helton, J. William 1944- Klep, Igor McCullough, Scott Schweighofer, Markus |
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| id | DE-604.BV045553279 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:21:19Z |
| institution | BVB |
| isbn | 9781470434557 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030937211 |
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| physical | v, 106, 4 ungezählte Seiten |
| publishDate | 2019 |
| publishDateSearch | 2019 |
| publishDateSort | 2019 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spelling | Helton, J. William 1944- (DE-588)132893320 aut Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions J. William Helton, Igor Klep, Scott McCullough, Markus Schweighofer Providence, Rhode Island American Mathematical Society [2019] © 2019 v, 106, 4 ungezählte Seiten txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society volume 257, number 1232 (second of 6 numbers) Klep, Igor (DE-588)115342004X aut McCullough, Scott (DE-588)1179943848 aut Schweighofer, Markus aut Memoirs of the American Mathematical Society volume 257, number 1232 (second of 6 numbers) (DE-604)BV008000141 1232 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030937211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Helton, J. William 1944- Klep, Igor McCullough, Scott Schweighofer, Markus Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions Memoirs of the American Mathematical Society |
| title | Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions |
| title_auth | Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions |
| title_exact_search | Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions |
| title_full | Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions J. William Helton, Igor Klep, Scott McCullough, Markus Schweighofer |
| title_fullStr | Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions J. William Helton, Igor Klep, Scott McCullough, Markus Schweighofer |
| title_full_unstemmed | Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions J. William Helton, Igor Klep, Scott McCullough, Markus Schweighofer |
| title_short | Dilations, linear Matrix inequalities, the Matrix cube problem and beta distributions |
| title_sort | dilations linear matrix inequalities the matrix cube problem and beta distributions |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030937211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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