|
Polytopal Complex
In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements. Definition A polyhedral complex \mathcal is a set of polyhedra that satisfies the following conditions: :1. Every face of a polyhedron from \mathcal is also in \mathcal. :2. The intersection of any two polyhedra \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in \mathcal may be empty. Examples * Tropical varieties are polyhedral complexes satisfying a certain ''balancing condition''. * Simplicial complexes are polyhedral complexes in which every polyhedron is a simplex. * Voronoi diagrams. * Splines. Fans A fan is a polyhedral complex in which every polyhedron is a cone from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
