Search results
Results From The WOW.Com Content Network
For a given n the elements of are then called homogeneous elements of degree n. Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n.
Elements of R that lie inside for some are said to be homogeneous of grade i. The previously defined notion of "graded ring" now becomes the same thing as an N {\displaystyle \mathbb {N} } -graded ring, where N {\displaystyle \mathbb {N} } is the monoid of natural numbers under addition.
[3]: 273 However, in strongly-typed object-oriented programming languages it may be somewhat complicated for a developer to write reusable homogeneous containers. Because of differences in element types this results in a tedious process of writing and keeping a collection of containers for every elemental type. [3]: 274–276
In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. It corresponds to the origin in the affine space, which cannot be mapped to a point in the projective space.
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.
The Nijenhuis–Richardson bracket can be defined on the vector valued forms Ω * (M, T(M)) on a smooth manifold M in a similar way. Vector valued forms act as derivations on the supercommutative ring Ω * (M) of forms on M by taking K to the derivation i K, and the Nijenhuis–Richardson bracket then corresponds to the commutator of two derivations.
Let be a vector space over a field equipped with a linear mapping : from to the exterior product of with itself. It is possible to extend uniquely to a graded derivation (this means that, for any , which are homogeneous elements, () = + ()) of degree 1 on the exterior algebra of :
Let H be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of I. If I is homogeneous, then H=I. Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements of B.