Near polygons:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel u. a.
Birkhäuser
2006
|
| Series: | Frontiers in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Literaturverz. S. 253 - 259 |
| Physical Description: | XI, 263 S. 24 cm |
| ISBN: | 9783764375522 3764375523 |
Staff View
MARC
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| 264 | 1 | |a Basel u. a. |b Birkhäuser |c 2006 | |
| 300 | |a XI, 263 S. |c 24 cm | ||
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| 500 | |a Literaturverz. S. 253 - 259 | ||
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| 650 | 4 | |a Graph theory | |
| 650 | 4 | |a Steiner systems | |
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Record in the Search Index
| _version_ | 1804138328467963904 |
|---|---|
| adam_text | Contents
Preface
їх
1
Introduction
1
1.1 Definition
of near polygon
....................... 1
1.2
Genesis
................................. 2
1.3
Near polygons with an order
..................... 3
1.4
Parallel lines
.............................. 3
1.5
Substructures
.............................. 4
1.6
Product near polygons
......................... 7
1.7
Existence of quads
........................... 12
1.8
The point-quad and line-quad relations
................ 14
1.9
Some classes of near polygons
..................... 17
1.10
Generalized quadrangles of order
(2,
í)
................ 21
2
Dense near polygons
27
2.1
Main results
............................... 27
2.2
The existence of convex subpolygons
................. 28
2.3
Proof of Theorem
2.6.......................... 37
2.4
Upper bound for the diameter of Td{x)
................ 38
2.5
Upper bounds for
t
+ 1
in the case of slim dense near polygons
. . 40
2.6
Slim dense near polygons with a big convex subpolygon
...... 41
3
Regular near polygons
47
3.1
Introduction
............................... 47
3.2
Some restrictions on the parameters
................. 47
3.3
Eigenvalues of the collinearity matrix
................. 50
3.4
Upper bounds for
í
........................... 54
3.5
Slim dense regular near hexagons
................... 55
3.6
Slim dense regular near octagons
................... 56
vi
Contents
4
Glued near polygons
57
4.1
Characterizations of product near polygons
............. 57
4.2
Admissible
á-spreads
.......................... 62
4.3
Construction and elementary properties of glued near polygons
. . 63
4.4
Basic characterization result for glued near polygons
........ 68
4.5
Other characterizations of glued near polygons
........... 71
4.6
Subpolygons
............................... 75
4.7
Glued near polygons of type
б Є
{0,1}................ 77
5
Valuations
93
5.1
Nice near polygons
........................... 93
5.2
Valuations of nice near polygons
................... 94
5.3
Characterizations of classical and ovoidal valuations
........ 96
5.4
The partial linear space
G f
...................... 98
5.5
A property of valuations
........................ 98
5.6
Some classes of valuations
....................... 99
5.7
Valuations of dense near hexagons
.................. 109
5.8
Proof of Theorem
5.29.........................
Ill
5.9
Proof of Theorem
5.30......................... 115
5.10
Proof of Theorem
5.31......................... 115
5.11
Proof of Theorem
5.32......................... 116
6
The known slim dense near polygons
121
6.1
The classical near polygons DQ(2n,
2)
and DH(2n-
1,4)..... 121
6.2
The class
И„
.............................. 127
6.3
The class Gn
.............................. 129
6.4
The class
I„
............................... 140
6.5
The near hexagon Ei
.......................... 143
6.6
The near hexagon E2
.......................... 152
6.7
The near hexagon E3
.......................... 159
6.8
The known slim dense near polygons
................. 161
6.9
The elements of C3 and C4
...................... 162
7
Slim dense near hexagons
167
7.1
Introduction
............................... 167
7.2
Elementary properties of slim dense near hexagons
......... 168
7.3
Case
I: S
is a regular near hexagon
.................. 170
7.4
Case II:
S
contains grid- and W(2)-qaads but no Q(5,2)-quads
. . 171
7.5
Case III:
S
contains grid- and Q(5,2)-quads but no W(2)-quads
. . 175
7.6
Case IV:
S
contains
W
(2)-
and Q(5, 2)-quads but no grid-quads
.. 176
7.7
Case V: <S contains grid-quads, VF(2)-quads and Q(5,2)-quads
. 177
7.8
Appendix
................................ 181
Contents
vii
8
Slim dense near polygons with a nice chain of subpolygons
187
8.1
Overview
................................ 187
8.2
Proof of Theorem
8.1.......................... 189
8.3
Proof of Theorem
8.2.......................... 190
8.4
Proof of Theorem
8.3.......................... 193
8.5
Proof of Theorem
8.4.......................... 197
8.6
Proof of Theorem
8.5.......................... 197
8.7
Proof of Theorem
8.6.......................... 204
8.8
Proof of Theorem
8.7.......................... 204
8.9
Proof of Theorem
8.8.......................... 205
8.10
Proof of Theorem
8.9.......................... 208
9
Slim dense near octagons
211
9.1
Some properties of slim dense near octagons
............. 211
9.2
Existence of big hexes
......................... 212
9.3
Classification of the near octagons
.................. 219
10
Nondense
slim near hexagons
225
10.1
A few lemmas
.............................. 225
10.2
Slim near hexagons with special points
................ 226
10.3
Slim near hexagons without special points
.............. 228
10.4
Proof of Theorem
10.8......................... 230
10.5
Proof of Theorem
10.9......................... 233
10.6
Proof of Theorem
10.10........................ 235
10.7
Slim near hexagons with an order
................... 243
A Dense near polygons of order
(3,
t)
247
A.I Generalized quadrangles of order (3,i)
................ 247
A.
2
Dense near hexagons of order
(3,
t)
.................. 248
A.3 Dense near octagons of order
(3,
í)
.................. 249
A.
4
Some properties of dense near
2á-gons
of order
(3,
í)
........ 250
A.
5
Dense near polygons of order
(3.
ť)
with a nice chain of subpolygons
251
Bibliography
253
Index
261
Bart De Bruyn
Near
Polygons
Near polygons were introduced about
¿ъ
years ago ana stuaiea intensive^
the
1980s.
In recent years the subject has regained interest. This monograph gives
an extensive overview of the basic theory of general near polygons.
The first part of the book includes a discussion of the classes of dense near
polygons, regular near polygons, and glued near polygons. Also valuations, one
of the most important tools for classifying dense near polygons, are treated
in detail. The second part of the book discusses the classification of dense near
polygons with three points per line.
The book is self-contained and almost all theorems are accompanied with
proofs. Several new results are presented. Many known results occur in a more
general form and the proofs are often more streamlined than their original
versions. The volume is aimed at advanced graduate students and researchers in
the fields of combinatorics and finite geometry.
ISBN-1Q:3-7ó¿3-7552-3
ISBN-13:
978-3-7643-7552-2
www.birkhauser.ch
lathematics
|
| adam_txt |
Contents
Preface
їх
1
Introduction
1
1.1 Definition
of near polygon
. 1
1.2
Genesis
. 2
1.3
Near polygons with an order
. 3
1.4
Parallel lines
. 3
1.5
Substructures
. 4
1.6
Product near polygons
. 7
1.7
Existence of quads
. 12
1.8
The point-quad and line-quad relations
. 14
1.9
Some classes of near polygons
. 17
1.10
Generalized quadrangles of order
(2,
í)
. 21
2
Dense near polygons
27
2.1
Main results
. 27
2.2
The existence of convex subpolygons
. 28
2.3
Proof of Theorem
2.6. 37
2.4
Upper bound for the diameter of Td{x)
. 38
2.5
Upper bounds for
t
+ 1
in the case of slim dense near polygons
. . 40
2.6
Slim dense near polygons with a big convex subpolygon
. 41
3
Regular near polygons
47
3.1
Introduction
. 47
3.2
Some restrictions on the parameters
. 47
3.3
Eigenvalues of the collinearity matrix
. 50
3.4
Upper bounds for
í
. 54
3.5
Slim dense regular near hexagons
. 55
3.6
Slim dense regular near octagons
. 56
vi
Contents
4
Glued near polygons
57
4.1
Characterizations of product near polygons
. 57
4.2
Admissible
á-spreads
. 62
4.3
Construction and elementary properties of glued near polygons
. . 63
4.4
Basic characterization result for glued near polygons
. 68
4.5
Other characterizations of glued near polygons
. 71
4.6
Subpolygons
. 75
4.7
Glued near polygons of type
б Є
{0,1}. 77
5
Valuations
93
5.1
Nice near polygons
. 93
5.2
Valuations of nice near polygons
. 94
5.3
Characterizations of classical and ovoidal valuations
. 96
5.4
The partial linear space
G f
. 98
5.5
A property of valuations
. 98
5.6
Some classes of valuations
. 99
5.7
Valuations of dense near hexagons
. 109
5.8
Proof of Theorem
5.29.
Ill
5.9
Proof of Theorem
5.30. 115
5.10
Proof of Theorem
5.31. 115
5.11
Proof of Theorem
5.32. 116
6
The known slim dense near polygons
121
6.1
The classical near polygons DQ(2n,
2)
and DH(2n-
1,4). 121
6.2
The class
И„
. 127
6.3
The class Gn
. 129
6.4
The class
I„
. 140
6.5
The near hexagon Ei
. 143
6.6
The near hexagon E2
. 152
6.7
The near hexagon E3
. 159
6.8
The known slim dense near polygons
. 161
6.9
The elements of C3 and C4
. 162
7
Slim dense near hexagons
167
7.1
Introduction
. 167
7.2
Elementary properties of slim dense near hexagons
. 168
7.3
Case
I: S
is a regular near hexagon
. 170
7.4
Case II:
S
contains grid- and W(2)-qaads but no Q(5,2)-quads
. . 171
7.5
Case III:
S
contains grid- and Q(5,2)-quads but no W(2)-quads
. . 175
7.6
Case IV:
S
contains
W
(2)-
and Q(5, 2)-quads but no grid-quads
. 176
7.7
Case V: <S contains grid-quads, VF(2)-quads and Q(5,2)-quads
. 177
7.8
Appendix
. 181
Contents
vii
8
Slim dense near polygons with a nice chain of subpolygons
187
8.1
Overview
. 187
8.2
Proof of Theorem
8.1. 189
8.3
Proof of Theorem
8.2. 190
8.4
Proof of Theorem
8.3. 193
8.5
Proof of Theorem
8.4. 197
8.6
Proof of Theorem
8.5. 197
8.7
Proof of Theorem
8.6. 204
8.8
Proof of Theorem
8.7. 204
8.9
Proof of Theorem
8.8. 205
8.10
Proof of Theorem
8.9. 208
9
Slim dense near octagons
211
9.1
Some properties of slim dense near octagons
. 211
9.2
Existence of big hexes
. 212
9.3
Classification of the near octagons
. 219
10
Nondense
slim near hexagons
225
10.1
A few lemmas
. 225
10.2
Slim near hexagons with special points
. 226
10.3
Slim near hexagons without special points
. 228
10.4
Proof of Theorem
10.8. 230
10.5
Proof of Theorem
10.9. 233
10.6
Proof of Theorem
10.10. 235
10.7
Slim near hexagons with an order
. 243
A Dense near polygons of order
(3,
t)
247
A.I Generalized quadrangles of order (3,i)
. 247
A.
2
Dense near hexagons of order
(3,
t)
. 248
A.3 Dense near octagons of order
(3,
í)
. 249
A.
4
Some properties of dense near
2á-gons
of order
(3,
í)
. 250
A.
5
Dense near polygons of order
(3.
ť)
with a nice chain of subpolygons
251
Bibliography
253
Index
261
Bart De Bruyn
Near
Polygons
Near polygons were introduced about
¿ъ
years ago ana stuaiea intensive^
the
1980s.
In recent years the subject has regained interest. This monograph gives
an extensive overview of the basic theory of general near polygons.
The first part of the book includes a discussion of the classes of dense near
polygons, regular near polygons, and glued near polygons. Also valuations, one
of the most important tools for classifying dense near polygons, are treated
in detail. The second part of the book discusses the classification of dense near
polygons with three points per line.
The book is self-contained and almost all theorems are accompanied with
proofs. Several new results are presented. Many known results occur in a more
general form and the proofs are often more streamlined than their original
versions. The volume is aimed at advanced graduate students and researchers in
the fields of combinatorics and finite geometry.
ISBN-1Q:3-7ó¿3-7552-3
ISBN-13:
978-3-7643-7552-2
www.birkhauser.ch
lathematics |
| any_adam_object | 1 |
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| author | De Bruyn, Bart 1974- |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 380 |
| ctrlnum | (OCoLC)64898223 (DE-599)BVBBV035163803 |
| dewey-full | 516/.154 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516/.154 |
| dewey-search | 516/.154 |
| dewey-sort | 3516 3154 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035163803 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:51:42Z |
| indexdate | 2024-07-09T21:26:27Z |
| institution | BVB |
| isbn | 9783764375522 3764375523 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016970877 |
| oclc_num | 64898223 |
| open_access_boolean | |
| owner | DE-384 DE-703 DE-188 |
| owner_facet | DE-384 DE-703 DE-188 |
| physical | XI, 263 S. 24 cm |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Frontiers in mathematics |
| spelling | De Bruyn, Bart 1974- Verfasser (DE-588)131586491 aut Near polygons Bart De Bruyn Basel u. a. Birkhäuser 2006 XI, 263 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Frontiers in mathematics Literaturverz. S. 253 - 259 Finite geometries Graph theory Steiner systems Polygon (DE-588)4175197-8 gnd rswk-swf Polygon (DE-588)4175197-8 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016970877&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016970877&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | De Bruyn, Bart 1974- Near polygons Finite geometries Graph theory Steiner systems Polygon (DE-588)4175197-8 gnd |
| subject_GND | (DE-588)4175197-8 |
| title | Near polygons |
| title_auth | Near polygons |
| title_exact_search | Near polygons |
| title_exact_search_txtP | Near polygons |
| title_full | Near polygons Bart De Bruyn |
| title_fullStr | Near polygons Bart De Bruyn |
| title_full_unstemmed | Near polygons Bart De Bruyn |
| title_short | Near polygons |
| title_sort | near polygons |
| topic | Finite geometries Graph theory Steiner systems Polygon (DE-588)4175197-8 gnd |
| topic_facet | Finite geometries Graph theory Steiner systems Polygon |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016970877&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016970877&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT debruynbart nearpolygons |