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 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:

See also

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.

Partners (Scherrie & Susaye album)

Partners is a 1979 album recorded by Scherrie & Susaye for Motown Records. Following the demise of The Supremes in 1977, former group members Scherrie Payne and Susaye Greene recorded this album together on the Motown label. Both singers share songwriting duties as well as receiving associate producer credit for the album. Legendary performer Ray Charles makes a guest appearance on the album on the song "Love Bug". Joyce Vincent Wilson, a candidate to replace original Supreme Mary Wilson when Payne and Greene were considering continuing with the Supremes name, is heavily featured on background vocals.

The song "Leaving Me Was the Best Thing You've Ever Done" was released as the album's only single.

The album was only released in the US.

The album was finally released on compact disc in 2014.

Track listing

Side one

Side two

Partners (Paul Bley & Gary Peacock album)

Partners is an album by pianist Paul Bley and bassist Gary Peacock recorded in New York in 1989 and released on the French Owl label in 1991.

Reception

Allmusic awarded the album 4 stars stating "by the time this duo session was recorded, one could expect that a certain degree of musical empathy would be in play. And yes, here there is plenty of the give and take of two old friends who do not go along with the mainstream jazz program. Yet one could also call this an album of twin monologues, for ten of the 15 tracks here are solo improvisations for each player".

Track listing

All compositions by Paul Bley and Gary Peacock except as indicated

Partners (board game)

Partners is a cross and circle game for four players, working together in pairs. The gameplay is similar to Ludo, Parcheesi and many other cross and circle games, but moves are decided by cards played, not by dice throws. Among the cards available are cards that move pieces backwards, and cards that allow swapping any two pieces that are in circuit, including enemy pieces. The object is for each pair to get their total of 8 pieces lined up in the goal area. The obvious way is to move the pieces around the board, but e.g. using a card to move a piece backwards enables some shortcuts.

The game is designed by Thomas Bisgaard and marketed by Dan-Spil since 1998.

References

Partners at BoardGameGeek