New approaches to circle packing in a square: with program codes
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
New York
Springer
2007
|
| Schriftenreihe: | Optimization and its applications
6 |
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of contents only Inhaltsverzeichnis |
| Beschreibung: | Includes bibliography (p. [219]-225) and index |
| Beschreibung: | xiv, 238 p. ill. |
| ISBN: | 9780387456737 0387456732 |
Internformat
MARC
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| 020 | |a 9780387456737 |9 978-0-387-45673-7 | ||
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| 245 | 1 | 0 | |a New approaches to circle packing in a square |b with program codes |c by P. G. Szábo ... [et al.] |
| 264 | 1 | |a New York |b Springer |c 2007 | |
| 300 | |a xiv, 238 p. |b ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Optimization and its applications |v 6 | |
| 500 | |a Includes bibliography (p. [219]-225) and index | ||
| 650 | 7 | |a Cirkels |2 gtt | |
| 650 | 7 | |a Numerieke wiskunde |2 gtt | |
| 650 | 7 | |a Pakking (natuurwetenschappen) |2 gtt | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Discrete geometry |x Data processing | |
| 650 | 4 | |a Circle packing |x Data processing | |
| 650 | 0 | 7 | |a Packung |g Mathematik |0 (DE-588)4209951-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kreis |0 (DE-588)4032962-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Kreis |0 (DE-588)4032962-8 |D s |
| 689 | 0 | 1 | |a Packung |g Mathematik |0 (DE-588)4209951-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Szabó, Péter Gábor |e Sonstige |4 oth | |
| 830 | 0 | |a Optimization and its applications |v 6 |w (DE-604)BV035278570 |9 6 | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0824/2006932708-d.html |3 Publisher description | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0824/2006932708-t.html |3 Table of contents only | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017055283&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017055283 | ||
Datensatz im Suchindex
| _version_ | 1804138527977373696 |
|---|---|
| adam_text | Contents
Preface
............................................................
IX
Glossary of Symbols
................................................ XIII
1
Introduction and Problem History
............................... 1
1.1
Problem description and motivation
........................... 1
1.2
The history of circle packing
................................. 2
1.2.1
The problem of
Malfatti
............................... 2
1.2.2
The circle packing studies of
Farkas Bolyai..............
2
1.2.3
Packing problems from ancient Japan
................... 5
1.2.4
A circle packing from Hieronymus Bosch
................ 6
1.2.5
The densest packing of circles in the plane
............... 6
1.2.6
Circle packings in bounded shapes
...................... 7
1.2.7
Generalizations and related problems
................... 8
1.2.8
Packing of equal circles in a unit square
................. 8
2
Problem Definitions and Formulations
............................ 13
2.1
Geometrical models
........................................ 13
2.2
Mathematical programming models
........................... 18
3
Bounds for the Optimum Values
................................. 23
3.1
Lower bounds for the maximal distance mn
.................... 23
3.2
Upper bounds for the optimal radius Fn
........................ 25
3.3
Asymptotic estimation and its error
............................ 28
4
Approximate Circle Packings Using Optimization Methods
......... 31
4.1
An earlier approach: energy function minimization
.............. 31
4.2
The TAMSASS-PECS algorithm
.............................. 32
4.3
Computational results
....................................... 37
VI
Contents
5
Other Methods for Finding Approximate Circle Packings
........... 43
5.1
A deterministic approach based on LP-relaxation
................ 43
5.2
A perturbation method
...................................... 44
5.3
Billiard simulation
.......................................... 44
5.4
Pulsating Disk Shaking
(PDS)
algorithm
....................... 45
5.5
Computational results
....................................... 46
6
Interval Methods for Validating Optimal Solutions
................. 51
6.1
Problem formulation for the interval algorithms
................. 51
6.2
Interval analysis
............................................ 53
6.3
A prototype interval global optimization algorithm
............... 54
6.4
An interval inclusion function for the objective function of the
point arrangement problem
.................................. 56
6.5
The subdivision step
........................................ 57
6.6
Accelerating devices
........................................ 57
6.6.1
The cutoff test
....................................... 58
6.6.2
The method of active regions
......................... 58
7
The First Fully Interval-based Optimization Method
............... 61
7.1
A monotonicity
test for the point arrangement problems
.......... 61
7.2
A special case of
monotonicity
—
handling the free points
......... 63
7.3
A rectangular approach for the method of active regions
......... 65
7.4
Numerical results
........................................... 68
7.4.1
Local verification
.................................... 68
7.4.2
Global solutions using tiles
............................ 70
8
Thelmproved Version of the Interval Optimization Method
........ 75
8.1
Method of active regions using polygons
....................... 76
8.2
A new technique for eliminating tile combinations
............... 86
8.2.1
Basic algorithms for the optimality proofs
............... 89
8.3
An improved method for handling free points
...............___ 93
8.4
Optimal packing of
η
= 28,29,
and
30
points
................... 95
8.4.1
Hardware and software environment
.................... 95
8.4.2
The global procedure
................................. 96
8.5
Summary of the results for the advanced algorithm
.............. 107
9
Interval Methods for Verifying Structural Optimality
.............. 109
9.1
Packing structures and previous numerical results
................ 109
9.2
Optimality and uniqueness properties
..........................
Ill
10
Repeated Patterns in Circle Packings
............................. 115
10.1
Finite pattern classes
........................................ 115
10.1.1
The PAT(k2-l)(l =
0,1,2)
pattern classes
............. 115
10.1.2
The STR(fc2
- 0
(I
- 3,4,5)
structure classes
........... 123
10.1.3
The PAT(fc(fc
+ 1))
pattern class
....................... 128
10.1.4
The PAT(fc2
+
|_fe/2j> pattern class
..................... 128
Contents
VII
10.2
A conjectured
infinite
pattern class
............................ 130
10.2.1
Grid packings
....................................... 132
10.2.2
Conjectural infinite grid packing sequences
.............. 133
10.3
Improved theoretical lower bounds based on pattern classes
....... 138
10.3.1
Sharp, constant lower and upper bounds for the density of
circle packing problems
............................... 139
11
Minimal Polynomials of Point Arrangements
...................... 143
11.1
Determining the minimal polynomial for a point arrangement
..... 143
11.1.1
Optimal substructures
................................ 144
11.1.2
Examining the general case
............................ 149
11.1.3
Finding minimal polynomials using Maple
............... 153
11.1.4
Determining exact values
............................. 155
11.1.5
Solving the
Рц(т)
= 0
equation by radicals
............. 156
11.2
Classifying circle packings based on minimal polynomials
........ 156
11.2.1
The linear class
...................................... 158
11.2.2
The quadratic class
................................... 158
11.2.3
The quartic class
..................................... 159
11.2.4
Some classes with higher degree polynomials
............ 160
12
About the Codes Used
.......................................... 163
12.1
The threshold accepting approach
............................. 163
12.2
Pulsating Disk Shaking
...................................... 166
12.2.1
Installation
.......................................... 166
12.2.2
Why does the program use PostScript files?
.............. 168
12.2.3
Usage and command-line options
....................... 168
12.3
The interval branch-and-bound algorithm
...................... 169
12.3.1
Package components
................................. 169
12.3.2
Requirements
....................................... 171
12.3.3
Installing and using
сігсраск.і.З
....................... 172
12.4
A Java demo program
....................................... 173
13
Appendix A: Currently Best Known Results for Packing Congruent
Circles in a Square
............................................. 179
A.I. Numerical results for the packings
............................ 179
A.2. Figures of the packings
..................................... 179
Bibliography
....................................................... 219
Related websites
.................................................... 227
List of Figures
..................................................... 229
List of Tables
...................................................... 233
Index
............................................................. 237
|
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516/.11 |
| dewey-search | 516/.11 |
| dewey-sort | 3516 211 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035249617 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:29:37Z |
| institution | BVB |
| isbn | 9780387456737 0387456732 |
| language | English |
| lccn | 2006932708 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017055283 |
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| owner_facet | DE-703 |
| physical | xiv, 238 p. ill. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series | Optimization and its applications |
| series2 | Optimization and its applications |
| spelling | New approaches to circle packing in a square with program codes by P. G. Szábo ... [et al.] New York Springer 2007 xiv, 238 p. ill. txt rdacontent n rdamedia nc rdacarrier Optimization and its applications 6 Includes bibliography (p. [219]-225) and index Cirkels gtt Numerieke wiskunde gtt Pakking (natuurwetenschappen) gtt Datenverarbeitung Discrete geometry Data processing Circle packing Data processing Packung Mathematik (DE-588)4209951-1 gnd rswk-swf Kreis (DE-588)4032962-8 gnd rswk-swf Kreis (DE-588)4032962-8 s Packung Mathematik (DE-588)4209951-1 s DE-604 Szabó, Péter Gábor Sonstige oth Optimization and its applications 6 (DE-604)BV035278570 6 http://www.loc.gov/catdir/enhancements/fy0824/2006932708-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0824/2006932708-t.html Table of contents only Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017055283&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | New approaches to circle packing in a square with program codes Optimization and its applications Cirkels gtt Numerieke wiskunde gtt Pakking (natuurwetenschappen) gtt Datenverarbeitung Discrete geometry Data processing Circle packing Data processing Packung Mathematik (DE-588)4209951-1 gnd Kreis (DE-588)4032962-8 gnd |
| subject_GND | (DE-588)4209951-1 (DE-588)4032962-8 |
| title | New approaches to circle packing in a square with program codes |
| title_auth | New approaches to circle packing in a square with program codes |
| title_exact_search | New approaches to circle packing in a square with program codes |
| title_full | New approaches to circle packing in a square with program codes by P. G. Szábo ... [et al.] |
| title_fullStr | New approaches to circle packing in a square with program codes by P. G. Szábo ... [et al.] |
| title_full_unstemmed | New approaches to circle packing in a square with program codes by P. G. Szábo ... [et al.] |
| title_short | New approaches to circle packing in a square |
| title_sort | new approaches to circle packing in a square with program codes |
| title_sub | with program codes |
| topic | Cirkels gtt Numerieke wiskunde gtt Pakking (natuurwetenschappen) gtt Datenverarbeitung Discrete geometry Data processing Circle packing Data processing Packung Mathematik (DE-588)4209951-1 gnd Kreis (DE-588)4032962-8 gnd |
| topic_facet | Cirkels Numerieke wiskunde Pakking (natuurwetenschappen) Datenverarbeitung Discrete geometry Data processing Circle packing Data processing Packung Mathematik Kreis |
| url | http://www.loc.gov/catdir/enhancements/fy0824/2006932708-d.html http://www.loc.gov/catdir/enhancements/fy0824/2006932708-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017055283&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035278570 |
| work_keys_str_mv | AT szabopetergabor newapproachestocirclepackinginasquarewithprogramcodes |