Hurwitz's complex continued fractions: a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche
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| Main Author: | |
|---|---|
| Format: | Thesis Book |
| Language: | English |
| Published: |
Würzburg
2014
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| Subjects: | |
| Online Access: | Volltext Inhaltsverzeichnis |
| Physical Description: | 236 Seiten Illustrationen, Diagramme |
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Record in the Search Index
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|---|---|
| adam_text | HURWITZ S COMPLEX CONTINUED FRACTIONS
A HISTORICAL APPROACH AND MODERN PERSPECTIVES.
DISSERTATIONSSCHRIFT ZUR ERLANGUNG
DES NATURWISSENSCHAFTLICHEN DOKTORGRADES
DER JULIUS-MAXIMILIANS-UNIVERSITAT WURZBURG
VORGELEGT VON
NICOLA OSWALD
AUS KRONACH
WURZBURG 2014
HTTP://D-NB.INFO/1081452684
CONTENTS
INTRODUCTION 8
1 REAL CONTINUED FRACTIONS 11
1.1 A BRIEF MATHEMATICAL AND HISTORICAL INTRODUCTION 11
1.2 SOME CLASSICAL RESULTS 16
2 THE HURWITZ BROTHERS 19
2.1 THE YOUNGER: ADOLF HURWITZ 20
2.2 EXCURSION: THE PERSONAL HURWITZ ESTATE FROM THE ARCHIVE OF THE ETH
ZURICH 31
2.2.1 RECREATIONAL MATHEMATICS IN THE MATHEMATICAL DIARIES 34
2.2.2 ADOLF HURWITZ FOLDING AND CUTTING PAPER 44
2.2.3 RELATION TO HIS STUDENT DAVID HILBERT 53
2.3 THE ELDER BROTHER: JULIUS HURWITZ 76
2.4 EXCURSION: A LETTER EXCHANGE CONCERNING JULIUS HURWITZ S PHD THESIS
. . 85
3 HURWITZ S APPROACH TO COMPLEX CONTINUED FRACTIONS 95
3.1 CONTINUED FRACTIONS ACCORDING TO ADOLF 95
3.2 A COMPLEX CONTINUED FRACTION ACCORDING TO JULIUS 106
3.3 SOME HISTORICAL NOTES 113
4 MODERN DEVELOPMENTS OF COMPLEX CONTINUED FRACTIONS 117
4.1 J. HURWITZ S ALGORITHM = TANAKA S ALGORITHM 118
5
CONTENTS
4.1.1 BASICS 119
4.1.2 EXAMPLE 122
4.1.3 SOME CONSIDERATIONS AND CHARACTERISTICS 124
4.1.4 GEOMETRICAL APPROACH TO THE APPROXIMATION BEHAVIOUR 132
4.1.5 APPROXIMATION QUALITY 135
4.2 ERGODIC THEORY 139
4.2.1 TRANSFORMATIONS 139
4.2.2 CONTINUED FRACTION TRANSFORMATION 140
4.2.3 DUAL TRANSFORMATION AND NATURAL EXTENSION FOR TANAKA S ALGORITHM
142
4.2.4 VARIATION ON THE DOBLIN-LENSTRA CONJECTURE 146
4.3 TRANSCENDENTAL NUMBERS 151
4.3.1 ROTH S THEOREM 152
4.3.2 TRANSFER TO TANAKA S CONTINUED FRACTION 153
4.3.3 EXAMPLE OF A TRANSCENDENTAL NUMBER 155
5 REPRESENTATION OF REAL NUMBERS AS SUM OF CONTINUED FRACTIONS 159
5.1 INTRODUCTION AND MAIN RESULT 159
5.2 SOME PRELIMINARIES 161
5.3 A CUSICK AND LEE-TYPE THEOREM 165
5.4 TRANSFER TO THE COMPLEX CASE 170
6 A NEW TYPE OF CONTINUED FRACTIONS WITH PARTIAL QUOTIENTS FROM A
LATTICE 173
6.1 CYCLOTOMIC FIELDS: UNION OF LATTICES 173
6.2 GENERALIZATION: VORONO F
DIAGRAMS 181
7 APPROXIMATION QUALITY 185 J
7.1 PREVIOUS RESULTS 186 J
7.2 MINKOWSKI AND GINTNER 187
|
7.2.1 PRELIMINARIES 188
|
6
CONTENTS
7.2.2 APPLICATION OF LINEAR FORMS 189
7.2.3 IMPROVEMENT BY THE LATTICE POINT THEOREM 191
7.2.4 GEOMETRICAL INTERPRETATION 193
7.2.5 GENERALIZATION OF THE RESULT 194
7.3 MODULAR GROUP 198
7.4 ARITHMETICAL APPLICATION 201
7.5 LAW OF BEST APPROXIMATION 205
8 LAST WORDS 207
9 APPENDIX 209
9.1 APPENDIX I: RECREATIONAL MATHEMATICS IN THE MATHEMATICAL DIARIES 209
9.2 APPENDIX II: LINKS TO HILBERT IN THE ETH ESTATE OF HURWITZ 210
9.3 APPENDIX III: TABLE OF FIGURES 212
7
|
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| discipline | Mathematik |
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| spelling | Oswald, Nicola Verfasser (DE-588)1064704123 aut Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche vorgelegt von Nicola Oswald aus Kronach Hurwitz' komplexe Kettenbrüche Würzburg 2014 236 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Dissertation Julius-Maximilians-Universität Würzburg 2014 Kettenbruch (DE-588)4030401-2 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Kettenbruch (DE-588)4030401-2 s DE-604 Erscheint auch als Online-Ausgabe urn:nbn:de:bvb:20-opus-106040 https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-106040 Resolving-System kostenfrei Volltext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028740837&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Oswald, Nicola Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche Kettenbruch (DE-588)4030401-2 gnd |
| subject_GND | (DE-588)4030401-2 (DE-588)4113937-9 |
| title | Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche |
| title_alt | Hurwitz' komplexe Kettenbrüche |
| title_auth | Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche |
| title_exact_search | Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche |
| title_full | Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche vorgelegt von Nicola Oswald aus Kronach |
| title_fullStr | Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche vorgelegt von Nicola Oswald aus Kronach |
| title_full_unstemmed | Hurwitz's complex continued fractions a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche vorgelegt von Nicola Oswald aus Kronach |
| title_short | Hurwitz's complex continued fractions |
| title_sort | hurwitz s complex continued fractions a historical approach and modern perspectives hurwitz komplexe kettenbruche |
| title_sub | a historical approach and modern perspectives = Hurwitz' komplexe Kettenbrüche |
| topic | Kettenbruch (DE-588)4030401-2 gnd |
| topic_facet | Kettenbruch Hochschulschrift |
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