Building (mathematics)

In mathematics, a building (also Tits building, Bruhat–Tits building, named after François Bruhat and Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type, the theory has also been used to study the geometry and topology of homogeneous spaces of p-adic Lie groups and their discrete subgroups of symmetries, in the same way that trees have been used to study free groups.

Overview

The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field. Tits demonstrated how to every such group G one can associate a simplicial complex Δ = Δ(G) with an action of G, called the spherical building of G. The group G imposes very strong combinatorial regularity conditions on the complexes Δ that can arise in this fashion. By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at his first definition of a building. A part of the data defining a building Δ is a Coxeter group W, which determines a highly symmetrical simplicial complex Σ = Σ(W,S), called the Coxeter complex. A building Δ is glued together from multiple copies of Σ, called its apartments, in a certain regular fashion. When W is a finite Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of spherical type. When W is an affine Weyl group, the Coxeter complex is a subdivision of the affine plane and one speaks of affine, or Euclidean, buildings. An affine building of type {\scriptstyle {\tilde {A}}_{1}} is the same as an infinite tree without terminal vertices.

Building (disambiguation)

A building is a constructed object intended for occupancy by humans or animals.

Building may also refer to:

  • Building (mathematics), a type of geometric structure
  • Buildings (album), an album by Northern Irish band General Fiasco
  • In computer programming, building is the process by which source code is converted into executable object code; see compiler
  • Building or Online creation, the name for creating areas and objects in online games
  • Building (magazine), a British magazine
  • A song by Poi Dog Pondering on their album Volo Volo
  • Building, a classification used by the U.S. National Register of Historic Places
  • Building (Brian Larsen album)
  • Building (Sense Field album)
  • See also

  • Structure (disambiguation)
  • Structure & Nonbuilding structure
  • Architectural structure, a man-made structure used or intended for supporting or sheltering any use or continuous occupancy
  • Online creation

    Online creation, also referred to as OLC, online coding, online building, and online editing, is a software feature of MUDs that allows users to edit a virtual world from within the game itself. In the absence of online creation, content is created in a text editor or level editor, and the program generally requires a restart in order to implement the changes.

    Online creation as original content

    An aspect of online creation that separates it from "mere game play" is that online creation systems can generally be used to create new content new objects, new locations, new creatures rather than simply creating instances of predefined items in the game world. Some have observed that certain forms of online creation notably those associated with creating new commands can threaten the stability of the server.

    History

    Origins

    The first publicly available MUD that featured in-game creation of the game world was Skrenta's 1988 Monster.

  • Create and describe new objects and locations
  • Podcasts:

    PLAYLIST TIME:

    Latest News for: Leasing building

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    SANDAG moves into new, pricier downtown HQ

    San Diego Union-Tribune 05 Apr 2025
    ... building ... The agency leases 87,308 square feet of space spread across three floors in the mixed-use building ... “When I came in, there had already been realized a 15-year lease in the new building.
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    Clean Life partially leases residential building in Riyadh for SAR 525,000

    Eye Of Riyadh 05 Apr 2025
    to partially lease its owned residential building in Ar-Rimal district, Riyadh, for SAR 525,000.
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    After 14 years, Ponda council invites tender to lease out market complex

    The Times of India 05 Apr 2025
    ... shops and offices on auction, but no takers came forward at its latest auction in Sep 2024.The Ponda council’s decision comes after the Curchorem Municipal Council recently leased its entire building.
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    Fresh chaos looms for $1billion Sydney Fish Market: 'I'm getting out' | Daily Mail Online

    The Daily Mail 05 Apr 2025
    At least two key fish merchants have yet to sign their leases for the new market building ... Once completed, the market will lease the building under an agreement that the state government signed in 2019.
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    Fayetteville real estate: Have any of downtown's notable properties sold recently?

    The Fayetteville Observer 05 Apr 2025
    Downtown Fayetteville’s real estate market remains active when it comes to leasing opportunities, but property listings have lingered on the market, with several key buildings still up for sale.
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    The Buzz: Owners of Mister Taco spot south of Redding courting newcomers. Buzz on Walgreens

    Record Searchlight 05 Apr 2025
    Nothing anyone puts in that building will ever compare to the joy I got from eating at your restaurant ... They were working on the building and the surrounding property the day I met them ... The building ...
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    Victorian building in Sheffield to be turned into apartments

    BBC News 05 Apr 2025
    An historic Victorian building is to be converted into 18 new apartments ... "These proposals are sensitive to the historical significance of 14 East Parade whilst giving the building a new lease of life.".
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    GAO report on DoD-linked civilians' care points to GMH problems, new medical facility planned at ...

    Guam Pacific Daily News 05 Apr 2025
    The GAO report said the DoD has worked with Guam partners to.Attempt to lease land to build a new public hospitalHas provided funding to upgrade existing public medical facilities, andPlans to expand ...
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    Hundreds attend anti-Trump 'Hands Off' protests in downtown Wilmington, Brunswick County

    StarNews Online 05 Apr 2025
    Wilmington has been included in the Department of Government Efficiency’s cost-cutting initiatives since two local buildings were added to a list of lease terminations in early March.
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    Encompass Christian Counseling begins work on mental health addition at RiverTree church

    The Independent (inde) 05 Apr 2025
    They leased us that building for $1 a year for a long time, and in 2018, we purchased it from them ... "It was just time to build because we saw the need." ... Wagler-Mills said the new building will include a life skills room.
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    After fatal crash, Manesar civic body turns its lens on stray animals

    The Times of India 05 Apr 2025
    ... the cattle, which will be relocated to shelters in coordination with the municipal area,' said Yadav.Plans are on to lease a building within the municipal zone, to provide medical care to stray dogs.
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    Vintage Souls Antique Market offers old, new and handmade items

    The Herald-Times - Bloomington 05 Apr 2025
    Then one day, Ranee’s husband Rob was coming home from work and called to tell his wife that a building down the street from their home was for lease ... On the spot, Ranee rented part of the building for her dreamed-of antique store ... to 7.30 p.m ... What.
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    More than 10 companies show interest in new Simon High School project

    Guam Pacific Daily News 05 Apr 2025
    The Simon Sanchez High School project is a public-private partnership, utilizing a finance, demolition, design, build, lease/leaseback, and insure/capital maintenance (FDDBLM) delivery method.
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    State claims 50% of Waqf land is encroached, starts mapping properties

    Hindustan Times 05 Apr 2025
    “Barring four properties in Marathwada, none of the others has been developed for the welfare of the community, either by building educational or health institutions on them.
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    Glancing Back for April 5: No bills approved; Windsor park goes unfinished

    Journal Gazette 05 Apr 2025
    Fred Edgar, board administrator, said Friday talks with the hospital and the Area E-7 Hospital Association about leasing or purchasing the building have halted ... Teaming up with the right partner can be a game changer when building a business.
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