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Free Regular Set
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Regular Set
Solid modeling (or solid modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes '' (solids)''. Solid modeling is distinguished from related areas of geometric modeling and computer graphics, such as ''3D modeling'', by its emphasis on physical fidelity. Together, the principles of geometric and solid modeling form the foundation of 3D-computer-aided design and in general support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects. Overview The use of solid modeling techniques allows for the automation process of several difficult engineering calculations that are carried out as a part of the design process. Simulation, planning, and verification of processes such as machining and assembly were one of the main catalysts for the development of solid modeling. More recently, the range of supported manufacturing applications has been greatly expanded to incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
μ-regular Set
In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Definition Let (''X'', ''T'') be a topological space and let Σ be a σ-algebra on ''X''. Let ''μ'' be a measure on (''X'', Σ). A measurable subset ''A'' of ''X'' is said to be inner regular if :\mu (A) = \sup \ and said to be outer regular if :\mu (A) = \inf \ *A measure is called inner regular if every measurable set is inner regular. Some authors use a different definition: a measure is called inner regular if every open measurable set is inner regular. *A measure is called outer regular if every measurable set is outer regular. *A measure is called regular if it is outer regular and inner regular. Examples Regular measures * Lebesgue measure on the real line is a regular measure: see the regularity theorem for Lebesgue measure. * Any Baire probability mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
