In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups. The fundamental group of a topological space is denoted by .
An Internet Protocol address is a numerical label such as 192.0.2.1 that is connected to a computer network that uses the Internet Protocol for communication. An IP address serves two main functions: network interface identification and location addressing.
In mathematics, a Lie group is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be a group, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.
In the United States, a political action committee (PAC) is a 527 organization, that pools campaign contributions from members and donates those funds to campaigns for or against candidates, ballot initiatives, or legislation. The legal term PAC has been created in pursuit of campaign finance reform in the United States. This term is quite specific to all activities of campaign finance in the United States. Democracies of other countries use different terms for the units of campaign spending or spending on political competition. At the U.S. federal level, an organization becomes a PAC when it receives or spends more than $1,000 for the purpose of influencing a federal election, and registers with the Federal Election Commission (FEC), according to the Federal Election Campaign Act as amended by the Bipartisan Campaign Reform Act of 2002. At the state level, an organization becomes a PAC according to the state's election laws.
In mathematics, specifically algebraic topology, a covering map is a continuous function from a topological space to a topological space such that each point in has an open neighborhood evenly covered by . In this case, is called a covering space and the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.
In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.
In mathematics the spin group Spin(n) is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups
In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.
In mathematics, a compact (topological) group is a topological group whose topology is compact. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and semisimple algebraic groups are reductive.
The Diduni or Dunii were a Germanic tribe mentioned only by the 2nd century geographer Claudius Ptolemy. They apparently dwelt near the Asciburgius mountains which correspond to the north central parts of Sudetes in western-southern Poland. According to Ptolemy, they were part of the larger tribal group, the Lugii. The Diduni are may be connected to the town of Iugidunum, which Ptolemy places in the same area as he places the tribe.
A battery tower was a defensive tower built into the outermost defences of many castles, usually in the 16th century or later, after the advent of firearms. Its name is derived from the word battery, a group of several cannon.
Cultural framework is a term used in social science to explain traditions, value systems, myths and symbols that are common in a given society. A given society may have multiple cultural frameworks. Usually cultural frameworks are mixed; as certain individuals or entire groups can be familiar with many cultural frameworks.
A quarter is a section of an urban settlement.
500 Brickell is a residential complex in the Brickell neighborhood of Miami, Florida, United States. The complex consists of two condominium towers, 500 Brickell West Tower and 500 Brickell East Tower. The two buildings were designed as twin towers, and rise 426 feet, with 42 floors. Both skyscrapers were designed by the Arquitectonica architectural firm and the complex was developed by Thomas Kramer's Portofino Group in partnership with the Related Group of Florida. The towers' construction began in April 2005, was topped out in mid-2007 and was completed in 2008.
Serua is an almost extinct Austronesian language originally spoken on Serua Island in Maluku, Indonesia. Speakers were relocated to Seram due to volcanic activity on Serua. The language continues in communities in Waipia in Seram, where the islanders were resettled, along with those also from Nila and Teun. Here, the older generation retain the island language as a strong form of identity.
