Roots to research: a vertical development of mathematical problems
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2007]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xiii, 338 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821844038 0821844032 |
Internformat
MARC
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| 245 | 1 | 0 | |a Roots to research |b a vertical development of mathematical problems |c Judith D. Sally ; Paul J. Sally Jr. |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2007] | |
| 264 | 4 | |c © 2007 | |
| 300 | |a xiii, 338 Seiten |b Illustrationen, Diagramme | ||
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Datensatz im Suchindex
| _version_ | 1804137664419463168 |
|---|---|
| adam_text | Contents
Chapter
1.
The Four Numbers Problem
1
1.
Introduction
1
2.
The Four Numbers Game Rule
2
3.
Symmetry and the Four Numbers Game
5
4.
Does Every Four Numbers Game Have Finite Length?
11
5.
Games With Length Independent of the Size of the Numbers
15
6.
Long Games
17
6.1.
Some Formal Notation
18
6.2.
Constructing Long Games
19
7.
The Tribonacci Games
22
7.1.
Computation of
L
(Γη)
22
7.2.
Upper Bounds for Lengths of Games
23
8.
The Length of the Four Real Numbers Game
26
8.1.
Linear Algebra Comes Into Play
26
8.2.
Construction of a Four Numbers Game of Infinite Length
27
8.3.
Construction of All Four Numbers Games of Infinite Length
28
9.
The Probability that a Four Numbers Game Ends in
η
Steps
32
10.
The fc-Numbers Game
41
Bibliography
45
Chapter
2.
Rational Right Triangles and the Congruent Number
Problem
47
1.
Introduction
47
2.
Right Triangles
48
vii
viii Contents
3.
Pythagorean Triples
63
4.
Sums of Squares
69
4.1.
The Two Squares Theorem
69
4.2.
Characterization of the Length of the Hypotenuse of an Integer
Right Triangle
75
4.3.
The Number of Representations of
η
as a Sum of Two Squares
76
5.
Rational Right Triangles
84
6.
Congruent Numbers
90
7.
Equivalent Definitions of Congruent Number
94
8. 1,2,
and
3
Are Not Congruent Numbers
96
9.
Rational Right Triangles and Certain Cubic Curves
101
10.
Elliptic Curves
104
11.
The Abelian Group of Rational Points on an Elliptic Curve
109
12.
En(Q) and Congruent Numbers
112
Bibliography
121
Chapter
3.
Lattice Point Geometry
123
1.
Introduction
123
2.
Geometric Shapes as Lattice Polygons
126
2.1.
Properties of Lattice Polygons in the Plane
126
3.
Embedding Regular Polygons in a Lattice
130
3.1.
Regular Lattice n-gons
130
3.2.
Which Positive Integers Are Areas of Lattice Squares?
134
4.
Basic Algebraic and Geometric Tools
137
4.1.
Dissection of a Lattice Polygon Into Lattice Triangles
137
4.2.
The Algebraic Structure of the Lattice Z2
140
4.3.
The Isometry Group of a Lattice
143
5.
Pick s Theorem
145
5.1.
First Proof
145
5.2.
From
Euler
to Pick
149
5.3.
Visible Lattice Points
153
5.4.
Pick s Theorem for nP
156
6.
Applications of Pick s Theorem
157
6.1.
Lattice Triangles
T
with I(T)
= 0
and
1 157
6.2.
Farey Sequences
159
7.
Lattice Points In and On a Circle
162
8.
Integer Points in Bounded Convex Regions in K2
166
8.1.
Convex Plane Regions and Integer Points
167
Contents
8.2. An Application
of Pick s
Theorem
to Bounded Convex
Regions 168
9.
Minkowski s Theorem in R2
169
10.
Embedding Regular Plane Polygons as Lattice Polygons in Rk
172
11.
Lattice Hypercubes
176
12.
Minkowski s Theorem in
Шк
183
13.
Ehrhart s Theorem
185
13.1.
Convex Polytopes
186
13.2.
Ehrhart s Theorem for a /c-Simplex
188
13.3.
The Coefficients of the
Ehrhart
Polynomial
191
Bibliography
193
Chapter
4.
Rational Approximation
195
1.
Introduction
195
2.
Introduction to Approximation Theory
197
3.
Properties of Rational Numbers Close to a Real Number
201
4.
An Interesting Example, Part I
204
5.
Dirichlet s Theorem
205
6.
An Interesting Example, Part II
208
7.
Hurwitz s Theorem
210
8.
Liouville s Theorem
215
8.1.
Statement and Proofs of Liouville s Theorem
216
8.2.
Liouville s Theorem and Transcendental Numbers
219
9.
The Thue-Siegel-Roth Theorem
221
9.1.
Introduction
221
9.2.
Thue s Theorem
224
9.3.
Roth s Theorem
227
10.
The Approximation Exponent
230
11.
An Interesting Example, Part III
232
12.
An Application to Diophantine Equations
233
13.
What About Transcendental Numbers?
236
Bibliography
241
Chapter
5.
Dissection
243
1.
Introduction
243
2.
Dissection and Area
245
3.
Basic Properties of Dissection
251
χ
Contents
4. Polygons
of Equal Area
257
5.
Dissection in Three Dimensions
259
6.
The Angles of a Polyhedron
266
7.
The
Dehn
Invariant
269
8.
A Solution of Hubert s Third Problem
277
9.
Congruence by Finite Decomposition and Equidecomposability
280
10.
Hausdorff s Paradox
283
11.
The Banach-Tarski Paradox
290
12.
Equidissectability and Equidecomposability
295
13.
Squaring the Circle
297
14.
Borsuk s Problem
298
14.1.
Borsuk s Conjecture in the Plane
301
14.2.
Borsuk s Conjecture in
Ш3
305
14.3.
Closed Convex Sets with Smooth Boundary
307
Bibliography
311
Appendix A. Volume
315
Appendix. Bibliography
323
Appendix B. Convexity
325
Appendix. Bibliography
333
Index
335
|
| adam_txt |
Contents
Chapter
1.
The Four Numbers Problem
1
1.
Introduction
1
2.
The Four Numbers Game Rule
2
3.
Symmetry and the Four Numbers Game
5
4.
Does Every Four Numbers Game Have Finite Length?
11
5.
Games With Length Independent of the Size of the Numbers
15
6.
Long Games
17
6.1.
Some Formal Notation
18
6.2.
Constructing Long Games
19
7.
The Tribonacci Games
22
7.1.
Computation of
L
(Γη)
22
7.2.
Upper Bounds for Lengths of Games
23
8.
The Length of the Four Real Numbers Game
26
8.1.
Linear Algebra Comes Into Play
26
8.2.
Construction of a Four Numbers Game of Infinite Length
27
8.3.
Construction of All Four Numbers Games of Infinite Length
28
9.
The Probability that a Four Numbers Game Ends in
η
Steps
32
10.
The fc-Numbers Game
41
Bibliography
45
Chapter
2.
Rational Right Triangles and the Congruent Number
Problem
47
1.
Introduction
47
2.
Right Triangles
48
vii
viii Contents
3.
Pythagorean Triples
63
4.
Sums of Squares
69
4.1.
The Two Squares Theorem
69
4.2.
Characterization of the Length of the Hypotenuse of an Integer
Right Triangle
75
4.3.
The Number of Representations of
η
as a Sum of Two Squares
76
5.
Rational Right Triangles
84
6.
Congruent Numbers
90
7.
Equivalent Definitions of Congruent Number
94
8. 1,2,
and
3
Are Not Congruent Numbers
96
9.
Rational Right Triangles and Certain Cubic Curves
101
10.
Elliptic Curves
104
11.
The Abelian Group of Rational Points on an Elliptic Curve
109
12.
En(Q) and Congruent Numbers
112
Bibliography
121
Chapter
3.
Lattice Point Geometry
123
1.
Introduction
123
2.
Geometric Shapes as Lattice Polygons
126
2.1.
Properties of Lattice Polygons in the Plane
126
3.
Embedding Regular Polygons in a Lattice
130
3.1.
Regular Lattice n-gons
130
3.2.
Which Positive Integers Are Areas of Lattice Squares?
134
4.
Basic Algebraic and Geometric Tools
137
4.1.
Dissection of a Lattice Polygon Into Lattice Triangles
137
4.2.
The Algebraic Structure of the Lattice Z2
140
4.3.
The Isometry Group of a Lattice
143
5.
Pick's Theorem
145
5.1.
First Proof
145
5.2.
From
Euler
to Pick
149
5.3.
Visible Lattice Points
153
5.4.
Pick's Theorem for nP
156
6.
Applications of Pick's Theorem
157
6.1.
Lattice Triangles
T
with I(T)
= 0
and
1 157
6.2.
Farey Sequences
159
7.
Lattice Points In and On a Circle
162
8.
Integer Points in Bounded Convex Regions in K2
166
8.1.
Convex Plane Regions and Integer Points
167
Contents
8.2. An Application
of Pick's
Theorem
to Bounded Convex
Regions 168
9.
Minkowski's Theorem in R2
169
10.
Embedding Regular Plane Polygons as Lattice Polygons in Rk
172
11.
Lattice Hypercubes
176
12.
Minkowski's Theorem in
Шк
183
13.
Ehrhart's Theorem
185
13.1.
Convex Polytopes
186
13.2.
Ehrhart's Theorem for a /c-Simplex
188
13.3.
The Coefficients of the
Ehrhart
Polynomial
191
Bibliography
193
Chapter
4.
Rational Approximation
195
1.
Introduction
195
2.
Introduction to Approximation Theory
197
3.
Properties of Rational Numbers Close to a Real Number
201
4.
An Interesting Example, Part I
204
5.
Dirichlet's Theorem
205
6.
An Interesting Example, Part II
208
7.
Hurwitz's Theorem
210
8.
Liouville's Theorem
215
8.1.
Statement and Proofs of Liouville's Theorem
216
8.2.
Liouville's Theorem and Transcendental Numbers
219
9.
The Thue-Siegel-Roth Theorem
221
9.1.
Introduction
221
9.2.
Thue's Theorem
224
9.3.
Roth's Theorem
227
10.
The Approximation Exponent
230
11.
An Interesting Example, Part III
232
12.
An Application to Diophantine Equations
233
13.
What About Transcendental Numbers?
236
Bibliography
241
Chapter
5.
Dissection
243
1.
Introduction
243
2.
Dissection and Area
245
3.
Basic Properties of Dissection
251
χ
Contents
4. Polygons
of Equal Area
257
5.
Dissection in Three Dimensions
259
6.
The Angles of a Polyhedron
266
7.
The
Dehn
Invariant
269
8.
A Solution of Hubert's Third Problem
277
9.
Congruence by Finite Decomposition and Equidecomposability
280
10.
Hausdorff's Paradox
283
11.
The Banach-Tarski Paradox
290
12.
Equidissectability and Equidecomposability
295
13.
Squaring the Circle
297
14.
Borsuk's Problem
298
14.1.
Borsuk's Conjecture in the Plane
301
14.2.
Borsuk's Conjecture in
Ш3
305
14.3.
Closed Convex Sets with Smooth Boundary
307
Bibliography
311
Appendix A. Volume
315
Appendix. Bibliography
323
Appendix B. Convexity
325
Appendix. Bibliography
333
Index
335 |
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| author | Sally, Judith D. 1937- Sally, Paul 1933- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023324209 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:54:54Z |
| indexdate | 2024-07-09T21:15:53Z |
| institution | BVB |
| isbn | 9780821844038 0821844032 |
| language | English |
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| publisher | American Mathematical Society |
| record_format | marc |
| spelling | Sally, Judith D. 1937- Verfasser (DE-588)136128548 aut Roots to research a vertical development of mathematical problems Judith D. Sally ; Paul J. Sally Jr. Providence, Rhode Island American Mathematical Society [2007] © 2007 xiii, 338 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematik Mathematics Research Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Problemlösen (DE-588)4076358-4 gnd rswk-swf Mathematik (DE-588)4037944-9 s Problemlösen (DE-588)4076358-4 s Geometrie (DE-588)4020236-7 s Zahlentheorie (DE-588)4067277-3 s DE-604 Sally, Paul 1933- Verfasser (DE-588)136128629 aut Erscheint auch als Online-Ausgabe 978-1-4704-1198-5 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016508252&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sally, Judith D. 1937- Sally, Paul 1933- Roots to research a vertical development of mathematical problems Mathematik Mathematics Research Zahlentheorie (DE-588)4067277-3 gnd Geometrie (DE-588)4020236-7 gnd Mathematik (DE-588)4037944-9 gnd Problemlösen (DE-588)4076358-4 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4020236-7 (DE-588)4037944-9 (DE-588)4076358-4 |
| title | Roots to research a vertical development of mathematical problems |
| title_auth | Roots to research a vertical development of mathematical problems |
| title_exact_search | Roots to research a vertical development of mathematical problems |
| title_exact_search_txtP | Roots to research a vertical development of mathematical problems |
| title_full | Roots to research a vertical development of mathematical problems Judith D. Sally ; Paul J. Sally Jr. |
| title_fullStr | Roots to research a vertical development of mathematical problems Judith D. Sally ; Paul J. Sally Jr. |
| title_full_unstemmed | Roots to research a vertical development of mathematical problems Judith D. Sally ; Paul J. Sally Jr. |
| title_short | Roots to research |
| title_sort | roots to research a vertical development of mathematical problems |
| title_sub | a vertical development of mathematical problems |
| topic | Mathematik Mathematics Research Zahlentheorie (DE-588)4067277-3 gnd Geometrie (DE-588)4020236-7 gnd Mathematik (DE-588)4037944-9 gnd Problemlösen (DE-588)4076358-4 gnd |
| topic_facet | Mathematik Mathematics Research Zahlentheorie Geometrie Problemlösen |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016508252&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT sallyjudithd rootstoresearchaverticaldevelopmentofmathematicalproblems AT sallypaul rootstoresearchaverticaldevelopmentofmathematicalproblems |