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Mnëv's Universality Theorem
In mathematics, Mnëv's universality theorem is a result in the intersection of combinatorics and algebraic geometry used to represent algebraic (or semialgebraic) varieties as realization spaces of oriented matroids. Informally it can also be understood as the statement that point configurations of a fixed combinatorics can show arbitrarily complicated behavior. The precise statement is as follows: : Let V be a semialgebraic variety in ^n defined over the integers. Then V is stably equivalent to the realization space of some oriented matroid. The theorem was discovered by Nikolai Mnëv in his 1986 Ph.D. thesis. Oriented matroids For the purposes of this article, an ''oriented matroid'' of a finite subset S\subset ^n is the list of partitions of S induced by hyperplanes in ^n (each oriented hyperplane partitions S into the points on the "positive side" of the hyperplane, the points on the "negative side" of the hyperplane, and the points that lie on the hyperplane). In part ... [...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] |
Kempe's Universality Theorem
In algebraic geometry, Kempe's universality theorem states that any bounded subset of an algebraic curve may be traced out by the motion of one of the joints in a suitably chosen linkage. It is named for British mathematician Alfred B. Kempe, who in 1876 published his article ''On a General Method of describing Plane Curves of the nth degree by Linkwork,'' which showed that for any arbitrary algebraic plane curve, a linkage can be constructed that draws the curve. However, Kempe's proof was flawed and the first complete proof was provided in 2002 based on his ideas. This theorem has been popularized by describing it as saying, "One can design a linkage which will sign your name!" Kempe recognized that his results demonstrate the existence of a drawing linkage but it would not be practical. He states It is hardly necessary to add, that this method would not be practically useful on account of the complexity of the linkwork employed, a necessary consequence of the perfect gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oriented Matroids
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, Surface (topology), surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loop (topology), loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
