Upper bound theorem: Difference between revisions

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{{DISPLAYTITLE:Near polygon}}
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In [[mathematics]], a '''near polygon''' is an [[incidence structure]] introduced by E.Shult and A. Yanushka in 1980.<ref>Shult, Ernest; Yanushka, Arthur. "Near n-gons and line systems".</ref> Shult and Yanushka showed the connection between the so called tetrahedrally closed line-systems in Euclidean spaces and a class of [[incidence geometry|point-line geometries]] which they called near polygons. These structures generalise the notion of [[generalized polygon]] as every generalized 2''n''-gon is a near 2''n''-gon of a particular kind. These structures were extensively studied and connection between them and dual polar space <ref>Cameron, Peter J. "Dual polar spaces".</ref> was shown in 1980s and early 1990s. Some sporadic [[simple groups]], for example the [[Hall-Janko group]], act as automorphism groups of some near polygons.  
 
== Definition ==
 
A near 2''d''-gon is an [[incidence structure]] (<math>P,L,I</math>), where <math>P</math> is the set of points, <math>L</math> is the set of lines and <math>I\subseteq P\times L</math> is the [[incidence relation]], such that:
* The maximum distance between two points (the so called diameter) is ''d''.
* For every point <math>x</math> and every line <math>L</math> there exists a unique point on <math>L</math> which is nearest to <math>x</math>.
 
A near quadrangle is same as a [[generalized quadrangle]] and it can be shown that every generalized 2''d''-gon is a near 2''d''-gon with the following two additional conditions:
* Every point is incident with at least two lines.  
*  For every two points ''x'',&nbsp;''y'' at distance ''i''&nbsp;<&nbsp;''d'', there exists a unique neighbour of ''y'' at distance ''i''&nbsp;−&nbsp;1 from&nbsp;''x''.
 
A near polygon is called dense if every line is incident with at least three points and if every two points at distance two have at least two common neighbours. It is said to have order (''s'',&nbsp;''t'') if every line is incident with precisely ''s''&nbsp;+&nbsp;1 points and every point is incident with precisely ''t''&nbsp;+&nbsp;1 lines.
 
== Examples ==
 
* All [[generalized polygon|generalized polygons]].  
* All dual polar spaces.
* The Hall–Janko near octagon<ref>http://cage.ugent.be/~hvm/artikels/187.pdf</ref> associated with the [[Hall–Janko group]].
 
== Regular near polygons ==
 
A finite near <math>2d</math>-gon S is called regular if it has an order <math>(s,t)</math> and if there exist constants <math>t_i, i \in \{0,1,\ldots,d\}</math>, such that for every two points <math>x</math> and <math>y</math> at distance <math>i</math>, there are precisely <math>t_i + 1</math> lines through <math>y</math> containing a (necessarily unique) point at distance <math>i - 1</math> from <math>x</math>. It turns out that regular near <math>2d</math>-gons are precisely those near <math>2d</math>-gons whose point graph is a [[distance-regular graph]]. A generalized <math>2d</math>-gon is a regular near <math>2d</math>-gon of order <math>(s, t; 0, 0, \ldots, 0)</math>.
 
== See also ==
 
* [[Finite geometry]]
* [[Polar space]]
* [[Partial linear space]]
* [[Association scheme]]
* [[Hall–Janko graph]]
 
== Notes ==
 
{{Reflist}}
 
== References ==
 
<!--- See [[Wikipedia:Footnotes]] on how to create references using<ref></ref> tags which will then appear here automatically -->
*{{Citation
| last1 = Shult | first1 = Ernest
| last2 = Yanushka | first2 = Arthur
| doi = 10.1007/BF00156473
| journal = Geom. Dedicata
| mr = 566437
| pages = 1––72
| title = Near n-gons and line systems
| volume = 9
| year = 1980}}.
 
*{{Citation
| last1 = Cameron | first1 = Peter J.  
| journal = Geom. Dedicata
| mr = 645040
| pages = 75–85
| title = Dual polar spaces
| volume = 12
| year = 1982}}.
 
*{{citation
| last = De Bruyn | first1 = Bart
| doi = 10.1007/978-3-7643-7553-9
| publisher = Birkhäuser Verlag
| mr = 2227553
| title = Near Polygons
| isbn = 3-7643-7552-3
| year = 2006
| series = Frontiers in Mathematics}}.
 
*{{citation
| last1 = Brouwer | first1 = A.E. | authorlink=Andries Brouwer
| last2 = Cohen | first2 = A.M.  
| publisher = Berlin, New York: Springer-Verlag.
| mr = 1002568
| title = Distance Regular Graphs
| isbn = 3-540-50619-5
| year = 1989}}.
 
*{{Citation | last1=Cameron | first1=Peter J. | authorlink=Peter Cameron (mathematician) | title=Projective and polar spaces | url=http://www.maths.qmul.ac.uk/~pjc/pps/ | publisher=Queen Mary and Westfield College School of Mathematical Sciences | location=London | series=QMW Maths Notes | mr=1153019 | year=1991 | volume=13}}
 
[[Category:Incidence geometry]]
[[Category:Graph theory]]
[[Category:Finite geometry]]

Latest revision as of 13:47, 5 May 2014

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In the long run, New Hyde Park residents received their wish to protect the historic spot, and McDonald's had no other possibility however to restore the property to its former glory. Clara Kirk, who runs two women's shelters in Englewood, told DNAinfo the cost of growing a property into something more than a backyard or expanded yard may very well be a problem. The Mayor's officer instructed HuffPost that beneath Chicago's Massive Lots program, applicants would need to own a property for at least 5 years before promoting. Jade Residences is a brand new uncommon freehold residential property launching near Serangoon MRT and positioned at Lew Lian Vale. Enjoy the 50m Lap pool, Youngsters's Pool on this lovely development. You are not obliged to proceed to buy. Woodlands EC Forestville @ Woodlands

For more from ProPublica on fracking, learn concerning the latest health studies and our investigation on Native Americans being cheated out of $1 billion by schemes to buy drilling rights on their oil-wealthy reservation.

The Pancreas Transplant Programme in Singapore will provide precedence transplant to patients who qualify for simultaneous pancreas & kidney transplant. "Such a priority allocation is necessary as these patients face a considerably increased mortality danger in comparison with recipients of kidney alone transplant." minutes to Singapore Flyer, 9 minutes to The Esplanade, 10 minutes to Marina Bay Sands, eleven minutes to Central Business District, eleven minutes to Marina Bay Golf Course, 14 minutes to East Coast Park, 21 minutes to Resorts World Sentosa La Fiesta is the last condominium residential improvement that's withinwalking distance from Sengkang MRT station. Hottest Launch Bartley Ridge Hottest Launch DUO Residences The deliberate number of phases for the challenge launch

Freehold Condominium at Bukit Timah, near to imminent King Albert Park MRT. Whole of 536 models from 1BR to 4BR, with 162 luxurious brand new and improved apartments accessible on the market. Firms are expected to supply jobs in the north that complement labour-intensive industries across the Causeway. With the intense improvement of Iskandar simply throughout the border, more jobs are anticipated to be created inside the northern a part of Singapore. The new coming intergrated CIQ complicated can be located North of Woodlands MRT Station with the Northern terminus of the Thomson MRT Line Along with JB Sentral, they are going to act as the link of Fast Transit System (RTS), adding better connectivity between Malaysia and Singapore property agent. Non-public Rental @ EC Prices! From Only $8xxPSF!

Zest Carlton – Make it your house in Australia Zest Carlton is a condominium development at Melbourne. Strategically situated within the bustling heard of Carlton and approximately 3km away from Central Business District (CBD). It takes a couple of minutes to One Balmoral – Ultra High End Freehold Condominium To Match Your Status! One Balmoral is a ultra high end freehold condominium positioned at One Balmoral Road. One Balmoral A growth by Hong Leong Holdings Limited, consisting of 91 units in Sant Ritz near Potong Pasir MRT is now open for VVIP reserving! Sant Ritz has launched on 05 April 2013. Sant Ritz is an upcoming project situated just three minutes walk away from Potong Pasir MRT. Sant Ritz european impressed architecture is designed 14xxpsf view this venture

A major developer, Thor Equities, has snatched up stretches of beachside property with plans for brand spanking new outlets, restaurants, condos, lodges and different year-round leisure. Leveraging on the pure terrain of hills and valleys in the Braddell location. New launch apartment The Maisons are impressed by a baronial architectural style, the masonry conveying enduring strength and resilience. The terraced ranges, lavish facilities and modern interiors provide a balance of modernity and timeless tradition. Continuereading "The Midtown Hougang New Launch Combined deveopment nearHougang MRT." Apr 02, 2013 NeWest - Clementi/ West Coast New Launch Mixeddeveopment nearHougang MRT. Continuereading "NeWest West Coast New Launch Combined deveopment nearHougang MRT."

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