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List Of Things Named After Hermann Grassmann ...
{{Short description, none The following is a list of things named in the memory of the German scholar and polymath Hermann Grassmann: Color science * Grassmann's laws Mathematics *Grassmann algebra *Grassmann bundle *Grassmann dimensions *Grassmann graph *Grassmann integral *Grassmann number ** Grassmann variables *Grassmannian **Affine Grassmannian ** Affine Grassmannian (manifold) **Lagrangian Grassmannian * Grassmann–Cayley algebra * Grassmann–Plücker relations Linguistics * Grassmann's law Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. Biography Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics. Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann's Laws (color Science)
Grassmann's laws describe empirical results about how the perception of mixtures of colored lights (i.e., lights that co-stimulate the same area on the retina) composed of different spectral power distributions can be algebraically related to one another in a color matching context. Discovered by Hermann Grassmann these "laws" are actually principles used to predict color match responses to a good approximation under photopic and mesopic vision. A number of studies have examined how and why they provide poor predictions under specific conditions. Modern interpretation The four laws are described in modern texts with varying degrees of algebraic notation and are summarized as follows (the precise numbering and corollary definitions can vary across sources): ---- ---- These laws entail an algebraic representation of colored light. Assuming beam 1 and 2 each have a color, and the observer chooses (R_1,G_1,B_1) as the strengths of the primaries that match beam 1 and (R_2,G_2,B_2) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meanin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Bundle
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. Biography Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics. Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers. The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom. The word "superspace" was first used by John Wheeler in an unrelated sense to describe the configuration space of general relativity; for example, this usage may be seen in his 1973 textbook ''Gravitation''. Informal discussion There are several similar, but not equivalent, definitions of superspace that have been used, and continue to be used in the mathematical and physics literature. One such usage is as a synonym for super Minkowski space. In this case, one takes ordinary Minkowski space, and extends it with anti-commuting fermionic degrees of freedom, taken to be anti-co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Graph
In graph theory, Grassmann graphs are a special class of simple graphs defined from systems of subspaces. The vertices of the Grassmann graph are the -dimensional subspaces of an -dimensional vector space over a finite field of order ; two vertices are adjacent when their intersection is -dimensional. Many of the parameters of Grassmann graphs are -analogs of the parameters of Johnson graphs, and Grassmann graphs have several of the same graph properties as Johnson graphs. Graph-theoretic properties * is isomorphic to . * For all , the intersection of any pair of vertices at distance is -dimensional. * The clique number of is given by an expression in terms its least and greatest eigenvalues and : ::\omega \left ( J_q(n,k) \right ) = 1 - \frac Automorphism group There is a distance-transitive subgroup of \operatorname(J_q(n, k)) isomorphic to the projective linear group \operatorname(n, q). In fact, unless n = 2k or k \in \, \operatorname(J_q(n,k)) \operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Integral
In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions. Definition Let \Lambda^n be the exterior algebra of polynomials in anticommuting elements \theta_,\dots,\theta_ over the field of complex numbers. (The ordering of the generators \theta_1,\dots,\theta_n is fixed and defines the orientation of the exterior algebra.) One variable The ''Berezin integral'' over the sole Grassmann variable \theta = \theta_1 is defined to be a linear functional : \int f(\theta)+bg(\theta)\, d\theta = a\int f(\theta) \, d\theta + b\int g(\theta) \, d\theta, \qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed. Informal discussion Grassmann numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" that the forms were define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension k(n-k). The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space . Motivation By giving a collection of subspaces of some vector space a topological stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Grassmannian
In mathematics, the affine Grassmannian of an algebraic group ''G'' over a field ''k'' is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought of as a flag variety for the loop group ''G''(''k''((''t''))) and which describes the representation theory of the Langlands dual group ''L''''G'' through what is known as the geometric Satake correspondence. Definition of Gr via functor of points Let ''k'' be a field, and denote by k\text and \mathrm the category of commutative ''k''-algebras and the category of sets respectively. Through the Yoneda lemma, a scheme ''X'' over a field ''k'' is determined by its functor of points, which is the functor X:k\text \to \mathrm which takes ''A'' to the set ''X''(''A'') of ''A''-points of ''X''. We then say that this functor is representable by the scheme ''X''. The affine Grassmannian is a functor from ''k''-algebras to sets which is not itself representable, but which has a filtration by representable functors. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Grassmannian (manifold)
In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all ''k''-dimensional affine subspaces of R''n'' (described on this page), while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series. Formal definition Given a finite-dimensional vector space ''V'' and a non-negative integer ''k'', then Graff''k''(''V'') is the topological space of all affine ''k''-dimensional subspaces of ''V''. It has a natural projection ''p'':Graff''k''(''V'') → Gr''k''(''V''), the Grassmannian of all linear ''k''-dimensional subspaces of ''V'' by defining ''p''(''U'') to be the translation of ''U'' to a subspace through the origin. This projection is a fibration, and if ''V'' is given an inner product, the fibre containing ''U'' can be identified with p(U)^\perp, the orthogonal complement to ''p''(''U''). The fibres are therefore vector spaces, and the projection ''p'' is a vector bundle over the Grassm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
