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.

Offshore

Offshore may refer to:

Finance and law

Technology

Arts

Offshore 2020

The Offshore 2020 is an annual market research conducted by Offshore Incorporations Limited Group (OIL). The research focused on the key trends and future developments of the offshore and related industries, and reviewed issues such as market drivers and constraints, impact of the increasing regulations and jurisdictional popularity.

One of the benefits of the market research is that it provides a window into an industry which is otherwise little understood from the outside, providing insight into common uses for offshore companies and countries where using offshore companies is most popular. It also enables trends to be observed across different jurisdictions and time periods. Recent reports reflect the reduction in the amount of offshore structuring which is tax driven (down to 15.9% in 2012, and 10.3% in 2013), possibly a result in the increase in offshore jurisdictions signing large numbers of tax information exchange agreements. They also appear to demonstrate the changes in popularity of certain offshore jurisdictions pursuant to different regulatory developments, and geopolitical events like the Cyprus financial crisis and the data leaks scandal.