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.

Union

Union is the state of being united or joined.

Union may also refer to:

Labor

Education

History and politics

Union (Toni Childs album)

Union is the debut album of the American singer/songwriter Toni Childs. Released in 1988, the album peaked at #63 in the US (where it has since been certified Gold for sales of over 500,000 copies). It also peaked at #1 in New Zealand where it was certified 5x Platinum (75,000 copies sold).

The album was recorded in London, Paris, and Swaziland. Following its release, Childs was nominated for two Grammy Awards (for 'Best New Artist' and for 'Best Rock Vocal Performance (Female)' for the single "Don't Walk Away").

Track listing

All songs written by Toni Childs and David Ricketts except as indicated.

Personnel

Union (comics)

Union is a fictional character in the Wildstorm universe.

Fictional character biography

Ohmen was a member of a faction, called the Protectorate, on his homeworld of Aegena. The world was mired in a long civil war, as the Protectorate fought against another faction, the Directorate, for control of the planet. The members of both factions had special "stones" (in reality, a symbiotic organism) embedded in their chests. The stones imbued them with a variety of powers, including super strength, flight, energy manipulation and a staff composed of pure energy.

At some point during the war, Ohmen was propelled to Earth via a portal, crashing into a frozen lake in Maine. There, he was rescued by painter Jill Monroe. Left severely weakened by the transit through the portal, he spent some time recovering and learning about Earth from Jill. Much later, he journeyed to New York, after seeing photographs of two unidentified beings flying over a small town in that state. It turned out the two beings were Directorate soldiers, who belonged to a secret Directorate base under the town. Captured, he was brought before Darian, King of the Directorate forces. Knowing his fate, Union grabbed staff and stone of Darian, causing it to explode. Unbeknownst to Ohmen, Darian was a paranoid man, fearing a coup from his own men and had rigged their stones to explode if his own was destroyed. The result was a chain reaction and a massive explosion that destroyed the base and the town.