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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 ...
In mathematics, the term "graded" has a number of meanings, mostly related: . In abstract algebra, it refers to a family of concepts: . An algebraic structure is said to be -graded for an index set if it has a gradation or grading, i.e. a decomposition into a direct sum = of structures; the elements of are said to be "homogeneous of degree i ".
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.
The 1-form λ does not descend to a genuine 1-form on M. However, it is homogeneous of degree 1, and so it defines a 1-form with values in the line bundle O(1), which is the dual of the fibrewise tautological line bundle of M. The kernel of this 1-form defines a contact distribution. Energy surfaces
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written (,) = (,), where f and g are homogeneous functions of the same degree of x and y. [1] In this case, the change of variable y = ux leads to an equation of the form
A nonzero element of is said to be homogeneous of degree . By definition of a direct sum, every nonzero element a {\displaystyle a} of R {\displaystyle R} can be uniquely written as a sum a = a 0 + a 1 + ⋯ + a n {\displaystyle a=a_{0}+a_{1}+\cdots +a_{n}} where each a i {\displaystyle a_{i}} is either 0 or homogeneous of degree i ...
In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1. [1]: 146 For example, in an economy with two goods ,, homothetic preferences can be represented by a utility function that has the following property: for every >:
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.