Search results
Results from the WOW.Com Content Network
Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g 1, ..., g n of degree 1, then the map which sends X i onto g i defines an homomorphism of graded rings from = [, …,] onto S.
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. [1] For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.
In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. [c] For example, x 3 y 2 + 7x 2 y 3 − 3x 5 is homogeneous of degree 5. For more details, see ...
The complete homogeneous symmetric polynomial of degree k in n variables X 1, ... Here each fraction in the final expression is the usual way to represent the formal ...
Since X is projective, each g i is a fraction of homogeneous elements of the same degree in the homogeneous coordinate ring k[X] of X. We can arrange the fractions so that they all have the same homogeneous denominator say f 0. Then we can write g i = f i /f 0 for some homogeneous elements f i 's in k[X]. Hence, going back to the homogeneous ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
By Euler's second theorem for homogeneous functions, ¯ is a homogeneous function of degree 0 (i.e., ¯ is an intensive ... mole fractions: ...