When less is more: visualizing basic inequalities
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Washington, D.C.
Mathematical Association of America
2009
|
| Schriftenreihe: | The Dolciani mathematical expositions
36 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | The proofs in When Less is More are in the spirit of proofs without words, though most require at least a few words. The first inequalities presented in the book, such as the inequalities between the harmonic, geometric, and arithmetic mean, are familiar from analysis, but are given geometric proofs. The second and largest set of inequalities are geometric both in their statements and in their proofs. Toward the end of the book some inequalities are more analytical in their statements as well as their proofs. --from publisher description Includes bibliographical references (p. 171-177) and index |
| Beschreibung: | XIX, 181 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9780883853429 0883853426 |
Internformat
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| 245 | 1 | 0 | |a When less is more |b visualizing basic inequalities |c Claudi Alsina, Roger B. Nelsen |
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| 490 | 1 | |a The Dolciani mathematical expositions |v 36 | |
| 500 | |a The proofs in When Less is More are in the spirit of proofs without words, though most require at least a few words. The first inequalities presented in the book, such as the inequalities between the harmonic, geometric, and arithmetic mean, are familiar from analysis, but are given geometric proofs. The second and largest set of inequalities are geometric both in their statements and in their proofs. Toward the end of the book some inequalities are more analytical in their statements as well as their proofs. --from publisher description | ||
| 500 | |a Includes bibliographical references (p. 171-177) and index | ||
| 650 | 4 | |a Inequalities (Mathematics) | |
| 650 | 4 | |a Visualization | |
| 650 | 4 | |a Geometrical drawing | |
| 700 | 1 | |a Nelsen, Roger B. |d 1942- |e Sonstige |0 (DE-588)12076945X |4 oth | |
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Datensatz im Suchindex
| _version_ | 1804149367903354880 |
|---|---|
| adam_text | Titel: When less is more
Autor: Alsina Catalá, Claudi
Jahr: 2009
Contents
Preface xiii
Introduction xv
1 Representing positive numbers as lengths of Segments 1
1.1 Inequalities associated with triangles 2
1.2 Polygonal paths 4
1.3 n-gons inside m-gons 5
1.4 The arithmetic mean-geometric mean inequality 6
1.5 More inequalities for means 11
1.6 The Ravi Substitution 13
1.7 Comparing graphs of functions 15
1.8 Challenges 17
2 Representing positive numbers as areas or volumes 19
2.1 Three examples 20
2.2 Chebyshev s inequality 21
2.3 The AM-GM inequality for three numbers 24
2.4 Guha s inequality 29
2.5 The AM-GM inequality for n numbers 29
2.6 The HM-AM-GM-RMS inequality for n numbers 31
2.7 The mediant property and Simpson s paradox 32
2.8 Chebyshev s inequality revisited 34
2.9 Schur s inequality 37
2.10 Challenges 38
3 Inequalities and the existence of triangles 43
3.1 Inequalities and the altitudes of a triangle 44
3.2 Inequalities and the medians of a triangle 46
3.3 Inequalities and the angle-bisectors of a triangle 49
3.4 The Steiner-Lehmus theorem 52
3.5 Challenges 52
x Contents
4 Using incircles and circumcircles 55
4.1 Euler s triangle inequality 56
4.2 The isoperimetric inequality 60
4.3 Cyclic, tangential, and bicentric quadrilaterals 64
4.4 Some properties of n-gons 67
4.5 Areas of parallel domains 69
4.6 Challenges 70
5 Using reflections 73
5.1 An inscribed triangle with minimum perimeter 74
5.2 Altitudes and the orthic triangle 75
5.3 Steiner symmetrization 76
5.4 Another minimal path 78
5.5 An inscribed triangle with minimum area 79
5.6 Challenges 79
6 Using rotations 81
6.1 Ptolemy s inequality 82
6.2 Fermat s problem for Torricelli 83
6.3 The Weitzenböck and Hadwiger-Finsler inequalities .... 84
6.4 A maximal chord problem 87
6.5 The Pythagorean inequality and the law of cosines 88
6.6 Challenges 90
7 Employing non-isometric transformations 93
7.1 The Erdos-Mordell theorem 93
7.2 Another Erdos triangle inequality 99
7.3 The Cauchy-Schwarz inequality 101
7.4 Aczel s inequality 107
7.5 Challenges 108
8 Employing graphs of funetions 111
8.1 Boundedness and monotonicity 111
8.2 Continuity and uniform continuity 117
8.3 The Lipschitz condition 118
8.4 Subadditivity and superadditivity 119
8.5 Convexity and concavity 120
8.6 Tangent and secant lines 124
8.7 Using integrals 128
8.8 Bounded monotone sequences 133
8.9 Challenges 135
Contents xi
9 Additional topics 137
9.1 Combining inequalities 137
9.2 Majorization 139
9.3 Challenges 143
Solutions to the Challenges 145
Chapter 1 145
Chapter2 147
Chapter 3 152
Chapter 4 154
Chapter 5 156
Chapter 6 157
Chapter 7 159
Chapter 8 162
Chapter 9 166
Notation and Symbols 169
References 171
Index 179
About the Authors 183
|
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| id | DE-604.BV040336378 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:21:55Z |
| institution | BVB |
| isbn | 9780883853429 0883853426 |
| language | English |
| lccn | 2008942145 |
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| physical | XIX, 181 S. Ill., graph. Darst. 24 cm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Mathematical Association of America |
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| series | The Dolciani mathematical expositions |
| series2 | The Dolciani mathematical expositions |
| spelling | Alsina, Claudi 1952- Verfasser (DE-588)143990934 aut When less is more visualizing basic inequalities Claudi Alsina, Roger B. Nelsen Washington, D.C. Mathematical Association of America 2009 XIX, 181 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier The Dolciani mathematical expositions 36 The proofs in When Less is More are in the spirit of proofs without words, though most require at least a few words. The first inequalities presented in the book, such as the inequalities between the harmonic, geometric, and arithmetic mean, are familiar from analysis, but are given geometric proofs. The second and largest set of inequalities are geometric both in their statements and in their proofs. Toward the end of the book some inequalities are more analytical in their statements as well as their proofs. --from publisher description Includes bibliographical references (p. 171-177) and index Inequalities (Mathematics) Visualization Geometrical drawing Nelsen, Roger B. 1942- Sonstige (DE-588)12076945X oth The Dolciani mathematical expositions 36 (DE-604)BV001900740 36 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025190715&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Alsina, Claudi 1952- When less is more visualizing basic inequalities The Dolciani mathematical expositions Inequalities (Mathematics) Visualization Geometrical drawing |
| title | When less is more visualizing basic inequalities |
| title_auth | When less is more visualizing basic inequalities |
| title_exact_search | When less is more visualizing basic inequalities |
| title_full | When less is more visualizing basic inequalities Claudi Alsina, Roger B. Nelsen |
| title_fullStr | When less is more visualizing basic inequalities Claudi Alsina, Roger B. Nelsen |
| title_full_unstemmed | When less is more visualizing basic inequalities Claudi Alsina, Roger B. Nelsen |
| title_short | When less is more |
| title_sort | when less is more visualizing basic inequalities |
| title_sub | visualizing basic inequalities |
| topic | Inequalities (Mathematics) Visualization Geometrical drawing |
| topic_facet | Inequalities (Mathematics) Visualization Geometrical drawing |
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